LMFDB - Newform orbit 1122.2.l.g

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1122,2,Mod(463,1122)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1122, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 3]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1122.463");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1122 = 2 \cdot 3 \cdot 11 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1122.l (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$8.95921510679$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$10$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 44 x^{18} + 732 x^{16} + 6050 x^{14} + 27262 x^{12} + 69598 x^{10} + 100205 x^{8} + 77682 x^{6} + \cdots + 289 $ x^20 + 44x^18 + 732x^16 + 6050x^14 + 27262x^12 + 69598x^10 + 100205x^8 + 77682x^6 + 29931x^4 + 5170*x^2 + 289
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of a basis $1,\beta_1,\ldots,\beta_{19}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - \beta_{8} q^{2} - \beta_{2} q^{3} - q^{4} + \beta_{6} q^{5} + \beta_1 q^{6} + ( - \beta_{19} - \beta_{4}) q^{7} + \beta_{8} q^{8} - \beta_{8} q^{9}+O(q^{10}) $ q - b8 * q^2 - b2 * q^3 - q^4 + b6 * q^5 + b1 * q^6 + (-b19 - b4) * q^7 + b8 * q^8 - b8 * q^9	$ q - \beta_{8} q^{2} - \beta_{2} q^{3} - q^{4} + \beta_{6} q^{5} + \beta_1 q^{6} + ( - \beta_{19} - \beta_{4}) q^{7} + \beta_{8} q^{8} - \beta_{8} q^{9} - \beta_{7} q^{10} + \beta_1 q^{11} + \beta_{2} q^{12} - \beta_{4} q^{13} - \beta_{18} q^{14} + \beta_{9} q^{15} + q^{16} + (\beta_{16} + \beta_{10} - \beta_{5} - 1) q^{17} - q^{18} + ( - \beta_{14} + \beta_{12} + \cdots - \beta_1) q^{19}+ \cdots + \beta_{2} q^{99}+O(q^{100}) $ q - b8 * q^2 - b2 * q^3 - q^4 + b6 * q^5 + b1 * q^6 + (-b19 - b4) * q^7 + b8 * q^8 - b8 * q^9 - b7 * q^10 + b1 * q^11 + b2 * q^12 - b4 * q^13 - b18 * q^14 + b9 * q^15 + q^16 + (b16 + b10 - b5 - 1) * q^17 - q^18 + (-b14 + b12 + b11 - b10 - b9 + b2 - b1) * q^19 - b6 * q^20 + b5 * q^21 + b2 * q^22 + (b19 - b14 + b10 - b5 + b4) * q^23 - b1 * q^24 + (b12 + b11 - 3b8 + b7 - b6) q^25 + (b19 + b13 + b4) * q^26 + b1 * q^27 + (b19 + b4) * q^28 + (b18 - b16 + b15 + b12 - b8 + 1) * q^29 - b3 * q^30 + (-b18 + b15 - b9 - b3 - 2b2) q^31 - b8 * q^32 - q^33 + (-b15 - b11 + b8 + b5) * q^34 + (b19 + b18 - 2b15 - 2b14 + b12 - b11 + b4) * q^35 + b8 * q^36 + (b18 - b17 - b15 - b14 - b11 - b10 + b9 + b8 + b5 + b3 - 2b2 - 1) q^37 + (-b17 + b16 - b5 + b3 - b2 - b1) * q^38 + (-b15 - b14 - b10 + b5) * q^39 + b7 * q^40 + (-b19 - b14 - b13 - b10 + b8 - b5 - 2b4 + 1) q^41 + (-b14 - b10) * q^42 + (-b19 - b18 - b15 - 2b13 + b12 + b11 - b10 + b9 + b5 - b4 - b2 + b1) q^43 - b1 * q^44 - b7 * q^45 + (b18 - 2b15 - b14 - b10 + b5) q^46 + (-2b19 - 2b18 + b12 - b11 + b5 + b4 + b3 - b2 - b1) * q^47 - b2 * q^48 + (b15 - b14 + b12 + b11 + b8 - b5) * q^49 + (-b17 + b16 + b7 + b6 - 3) * q^50 + (b18 - b17 - b4 + b2) * q^51 + b4 * q^52 + (b19 + b15 - b14 + b13 - b5 + b4 + 4b2 - 4b1) * q^53 + b2 * q^54 - b3 * q^55 + b18 * q^56 + (b19 + b16 + b12 + b8 + b7 + b4 + 1) * q^57 + (-b19 - b17 - b14 + b11 - b8 - b5 - b4 - 1) * q^58 + (-b17 - b16 + b15 + b10 - b9 - 6b8 + b7 - b6 - b5 + b2 - b1) q^59 - b9 * q^60 + (b19 - b14 - b13 + b10 - 2b8 - b7 - b5 - 2) q^61 + (b19 - b14 - b9 - b5 + b4 + b3 + 2b1) q^62 - b18 * q^63 - q^64 + (b19 + 2b18 - b16 - 2b15 - 2b14 + b13 + b12 - 2b10 + 2b5) q^65 + b8 * q^66 + (-b19 - b18 + b7 + b6 + b4 - 2) * q^67 + (-b16 - b10 + b5 + 1) * q^68 + (b19 + b18 - b5 - b4) * q^69 + (-b19 + b18 - b17 - b16 - 2b15 - 2b10 + 2b5 - b4) q^70 + (-b19 - b18 - b13 + 2b8 - 2b6 - 2) * q^71 + q^72 + (-b19 - b18 - b17 - b16 - b14 - b13 + b12 - b11 - b10 + b9 - 2b8 + b6 + b5 + b3 + 2b2 + 2) * q^73 + (-b19 - b16 - b12 - b10 + b9 + b8 - b4 - b3 + 2b1 + 1) q^74 + (b16 + b12 - b9 + b3 + 3b1) q^75 + (b14 - b12 - b11 + b10 + b9 - b2 + b1) * q^76 + (-b14 - b10) * q^77 - b10 * q^78 + (-b14 + b13 + b10 - 2b8 - 2b7 - b5 + b4 - 2) * q^79 + b6 * q^80 - q^81 + (b19 + b14 + b13 + b10 - b8 - b5 + 1) * q^82 + (b19 + b18 - b17 - b16 - 2b15 + 2b13 - b12 - b11 - 2b10 + 2b8 - b7 + b6 + 2b5 + b4) q^83 - b5 * q^84 + (-4b19 - 3b18 - b16 - 2b14 - 3b13 + b11 - 2b10 + b8 - 2b6 - 3b4 - b3 + b2 + b1 - 1) q^85 + (b19 + b18 - b17 + b16 + b15 + b14 - b4 - b3 + b2 + b1) * q^86 + (-b18 + b17 + b16 + b14 - b13 + b10 - b2 + b1) * q^87 - b2 * q^88 + (b19 + b18 + b17 - b16 - b15 - b14 + b12 - b11 - b5 + b4) * q^89 - b6 * q^90 + (-b17 + b11 - b9 + 4b8 + b3 - 2b1 + 4) * q^91 + (-b19 + b14 - b10 + b5 - b4) * q^92 + (-b18 - b14 - b13 - b10 - 2b8 + b7 - b6) q^93 + (-b19 - 2b18 - b17 - b16 - b14 - 3b13 - b10 + b9 - b4 - b2 + b1) * q^94 + (b17 - b16 + 3b14 + 4b13 - b12 - b11 + b10 + b9 - 2b8 - 2b7 + 3b5 + 4b4 - b3 - 6b1 - 2) q^95 + b1 * q^96 + (b19 + b16 + 2b15 + 2b14 + b13 - b12 + 2b10 - b8 + 2b6 - 2b5 + 1) q^97 + (-b17 + b16 - b15 - b14 - b5 + 1) * q^98 + b2 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 20 q^{4} - 4 q^{5}+O(q^{10}) $ 20 * q - 20 * q^4 - 4 * q^5	$ 20 q - 20 q^{4} - 4 q^{5} + 4 q^{10} + 20 q^{16} - 12 q^{17} - 20 q^{18} + 4 q^{20} + 16 q^{23} + 4 q^{29} - 8 q^{31} - 20 q^{33} + 16 q^{35} - 20 q^{37} + 8 q^{39} - 4 q^{40} + 20 q^{41} + 4 q^{45} + 16 q^{46} - 16 q^{47} - 68 q^{50} + 8 q^{57} - 4 q^{58} - 20 q^{61} + 8 q^{62} - 20 q^{64} + 8 q^{65} - 48 q^{67} + 12 q^{68} - 32 q^{71} + 20 q^{72} + 20 q^{73} + 20 q^{74} - 8 q^{75} - 8 q^{78} - 16 q^{79} - 4 q^{80} - 20 q^{81} + 20 q^{82} - 4 q^{85} - 16 q^{86} + 4 q^{90} + 88 q^{91} - 16 q^{92} - 48 q^{95} + 4 q^{97} + 36 q^{98}+O(q^{100}) $ 20 * q - 20 * q^4 - 4 * q^5 + 4 * q^10 + 20 * q^16 - 12 * q^17 - 20 * q^18 + 4 * q^20 + 16 * q^23 + 4 * q^29 - 8 * q^31 - 20 * q^33 + 16 * q^35 - 20 * q^37 + 8 * q^39 - 4 * q^40 + 20 * q^41 + 4 * q^45 + 16 * q^46 - 16 * q^47 - 68 * q^50 + 8 * q^57 - 4 * q^58 - 20 * q^61 + 8 * q^62 - 20 * q^64 + 8 * q^65 - 48 * q^67 + 12 * q^68 - 32 * q^71 + 20 * q^72 + 20 * q^73 + 20 * q^74 - 8 * q^75 - 8 * q^78 - 16 * q^79 - 4 * q^80 - 20 * q^81 + 20 * q^82 - 4 * q^85 - 16 * q^86 + 4 * q^90 + 88 * q^91 - 16 * q^92 - 48 * q^95 + 4 * q^97 + 36 * q^98

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{20} + 44 x^{18} + 732 x^{16} + 6050 x^{14} + 27262 x^{12} + 69598 x^{10} + 100205 x^{8} + 77682 x^{6} + \cdots + 289 $ x^20 + 44*x^18 + 732*x^16 + 6050*x^14 + 27262*x^12 + 69598*x^10 + 100205*x^8 + 77682*x^6 + 29931*x^4 + 5170*x^2 + 289 :

$\beta_{1}$	$=$	$ ( - 478453232 \nu^{19} - 65607844609 \nu^{18} - 20790152935 \nu^{17} - 2865069509061 \nu^{16} + \cdots - 55604642864711 ) / 474432850376 $ (-478453232v^19 - 65607844609v^18 - 20790152935v^17 - 2865069509061v^16 - 338917545566v^15 - 47075635647853v^14 - 2712070921497v^13 - 381263160365035v^12 - 11607278696930v^11 - 1661009394596199v^10 - 27338720388297v^9 - 4006421429337605v^8 - 34709977991120v^7 - 5214676459361530v^6 - 22405456855914v^5 - 3316918140098058v^4 - 7637317749682v^3 - 828047778854719v^2 - 1629578068081*v - 55604642864711) / 474432850376
$\beta_{2}$	$=$	$ ( 478453232 \nu^{19} - 65607844609 \nu^{18} + 20790152935 \nu^{17} - 2865069509061 \nu^{16} + \cdots - 55604642864711 ) / 474432850376 $ (478453232v^19 - 65607844609v^18 + 20790152935v^17 - 2865069509061v^16 + 338917545566v^15 - 47075635647853v^14 + 2712070921497v^13 - 381263160365035v^12 + 11607278696930v^11 - 1661009394596199v^10 + 27338720388297v^9 - 4006421429337605v^8 + 34709977991120v^7 - 5214676459361530v^6 + 22405456855914v^5 - 3316918140098058v^4 + 7637317749682v^3 - 828047778854719v^2 + 1629578068081*v - 55604642864711) / 474432850376
$\beta_{3}$	$=$	$ ( 5058550639 \nu^{18} + 223694565153 \nu^{16} + 3744505387173 \nu^{14} + 31110636071745 \nu^{12} + \cdots + 5064152867485 ) / 27907814728 $ (5058550639v^18 + 223694565153v^16 + 3744505387173v^14 + 31110636071745v^12 + 139986142331567v^10 + 349648532161315v^8 + 470045438835854v^6 + 305955182420016v^4 + 76957686604497*v^2 + 5064152867485) / 27907814728
$\beta_{4}$	$=$	$ ( - 10148357483 \nu^{18} - 445041584131 \nu^{16} - 7358635881849 \nu^{14} + \cdots - 9081831922431 ) / 27907814728 $ (-10148357483v^18 - 445041584131v^16 - 7358635881849v^14 - 60122842198375v^12 - 264918808196019v^10 - 647067653590709v^8 - 852259999673914v^6 - 546632113102428v^4 - 136570802313397*v^2 - 9081831922431) / 27907814728
$\beta_{5}$	$=$	$ ( 20389568647 \nu^{18} + 890396850383 \nu^{16} + 14629854784637 \nu^{14} + 118485210525483 \nu^{12} + \cdots + 16926462334275 ) / 27907814728 $ (20389568647v^18 + 890396850383v^16 + 14629854784637v^14 + 118485210525483v^12 + 516186649960791v^10 + 1245019644242777v^8 + 1620162473611754v^6 + 1029558507628972v^4 + 255939812252649*v^2 + 16926462334275) / 27907814728
$\beta_{6}$	$=$	$ ( - 139301146690 \nu^{19} - 106669931977 \nu^{18} - 6058900080957 \nu^{17} + \cdots - 78163571880140 ) / 474432850376 $ (-139301146690v^19 - 106669931977v^18 - 6058900080957v^17 - 4617719070078v^16 - 98951904059896v^15 - 74873525236589v^14 - 794582992039069v^13 - 595084920757662v^12 - 3422960693279472v^11 - 2528760184100971v^10 - 8151299190023259v^9 - 5929193915290958v^8 - 10473397988916922v^7 - 7506889363788560v^6 - 6583682116882800v^5 - 4676748657443724v^4 - 1620046710347750v^3 - 1158702610325915v^2 - 104268340777307*v - 78163571880140) / 474432850376
$\beta_{7}$	$=$	$ ( 139301146690 \nu^{19} - 106669931977 \nu^{18} + 6058900080957 \nu^{17} + \cdots - 78163571880140 ) / 474432850376 $ (139301146690v^19 - 106669931977v^18 + 6058900080957v^17 - 4617719070078v^16 + 98951904059896v^15 - 74873525236589v^14 + 794582992039069v^13 - 595084920757662v^12 + 3422960693279472v^11 - 2528760184100971v^10 + 8151299190023259v^9 - 5929193915290958v^8 + 10473397988916922v^7 - 7506889363788560v^6 + 6583682116882800v^5 - 4676748657443724v^4 + 1620046710347750v^3 - 1158702610325915v^2 + 104268340777307*v - 78163571880140) / 474432850376
$\beta_{8}$	$=$	$ ( 1404907 \nu^{19} + 61326597 \nu^{17} + 1007052793 \nu^{15} + 8149764229 \nu^{13} + \cdots + 1196618885 \nu ) / 4194104 $ (1404907v^19 + 61326597v^17 + 1007052793v^15 + 8149764229v^13 + 35474164511v^11 + 85505262423v^9 + 111268672750v^7 + 70821737508v^5 + 17714444473v^3 + 1196618885v) / 4194104
$\beta_{9}$	$=$	$ ( 198021598279 \nu^{19} + 8680993741599 \nu^{17} + 143468768001205 \nu^{15} + \cdots + 186387049023435 \nu ) / 474432850376 $ (198021598279v^19 + 8680993741599v^17 + 143468768001205v^15 + 1171499492229951v^13 + 5159024662579791v^11 + 12599392653447497v^9 + 16613727939830606v^7 + 10702062017531120v^5 + 2708982607502525v^3 + 186387049023435v) / 474432850376
$\beta_{10}$	$=$	$ ( 233700662525 \nu^{19} + 48399845376 \nu^{18} + 10185869836434 \nu^{17} + \cdots + 48121187253053 ) / 474432850376 $ (233700662525v^19 + 48399845376v^18 + 10185869836434v^17 + 2140643049511v^16 + 166875104105745v^15 + 35838822387566v^14 + 1345979794093378v^13 + 297772407073891v^12 + 5832483487393567v^11 + 1339410340734166v^10 + 13982579149755490v^9 + 3341764982076613v^8 + 18085828026582976v^7 + 4482763069675646v^6 + 11433905888104068v^5 + 2907528387642272v^4 + 2832474662487223v^3 + 727089853605996v^2 + 187196882199640*v + 48121187253053) / 474432850376
$\beta_{11}$	$=$	$ ( 243072241811 \nu^{19} + 21820907889 \nu^{18} + 10624957886419 \nu^{17} + \cdots + 21177824929574 ) / 474432850376 $ (243072241811v^19 + 21820907889v^18 + 10624957886419v^17 + 960999407232v^16 + 174827238548909v^15 + 15990384178901v^14 + 1418757856390083v^13 + 131786781660972v^12 + 6197076267516619v^11 + 587160399443479v^10 + 14990771145384333v^9 + 1451892040944876v^8 + 19563396967649766v^7 + 1936329198013400v^6 + 12463245620673056v^5 + 1257797697422936v^4 + 3110069184579833v^3 + 319648245046351v^2 + 209945391510523*v + 21177824929574) / 474432850376
$\beta_{12}$	$=$	$ ( 243072241811 \nu^{19} - 21820907889 \nu^{18} + 10624957886419 \nu^{17} + \cdots - 21177824929574 ) / 474432850376 $ (243072241811v^19 - 21820907889v^18 + 10624957886419v^17 - 960999407232v^16 + 174827238548909v^15 - 15990384178901v^14 + 1418757856390083v^13 - 131786781660972v^12 + 6197076267516619v^11 - 587160399443479v^10 + 14990771145384333v^9 - 1451892040944876v^8 + 19563396967649766v^7 - 1936329198013400v^6 + 12463245620673056v^5 - 1257797697422936v^4 + 3110069184579833v^3 - 319648245046351v^2 + 209945391510523*v - 21177824929574) / 474432850376
$\beta_{13}$	$=$	$ ( 263502915318 \nu^{19} - 12271307302 \nu^{18} + 11525417122427 \nu^{17} + \cdots - 14368375102973 ) / 474432850376 $ (263502915318v^19 - 12271307302v^18 + 11525417122427v^17 - 536853252751v^16 + 189827094809752v^15 - 8847061459320v^14 + 1542568980992775v^13 - 71993516507291v^12 + 6749728887477256v^11 - 316095182926804v^10 + 16359775161018901v^9 - 772282550814425v^8 + 21390409747923306v^7 - 1027943994695914v^6 + 13644788022546956v^5 - 682252590539008v^4 + 3400856208910174v^3 - 186002166337706v^2 + 223932584435117*v - 14368375102973) / 474432850376
$\beta_{14}$	$=$	$ ( - 268820550276 \nu^{19} - 48399845376 \nu^{18} - 11721434280167 \nu^{17} + \cdots - 48121187253053 ) / 474432850376 $ (-268820550276v^19 - 48399845376v^18 - 11721434280167v^17 - 2140643049511v^16 - 192158923731954v^15 - 35838822387566v^14 - 1551490803933211v^13 - 297772407073891v^12 - 6733500920866650v^11 - 1339410340734166v^10 - 16180612474008077v^9 - 3341764982076613v^8 - 21007969325115622v^7 - 4482763069675646v^6 - 13375628622751672v^5 - 2907528387642272v^4 - 3369319232867232v^3 - 727089853605996v^2 - 231099634601005*v - 48121187253053) / 474432850376
$\beta_{15}$	$=$	$ ( 268820550276 \nu^{19} + 298222821623 \nu^{18} + 11721434280167 \nu^{17} + \cdots + 239628672429622 ) / 474432850376 $ (268820550276v^19 + 298222821623v^18 + 11721434280167v^17 + 12996103407000v^16 + 192158923731954v^15 + 212868708951263v^14 + 1551490803933211v^13 + 1716476171859320v^12 + 6733500920866650v^11 + 7435762708599281v^10 + 16180612474008077v^9 + 17823568970050596v^8 + 21007969325115622v^7 + 23059998981724172v^6 + 13375628622751672v^5 + 14594966242050252v^4 + 3369319232867232v^3 + 3623886954689037v^2 + 231099634601005*v + 239628672429622) / 474432850376
$\beta_{16}$	$=$	$ ( 307594400909 \nu^{19} + 243949611482 \nu^{18} + 13435191456985 \nu^{17} + \cdots + 196489229464517 ) / 474432850376 $ (307594400909v^19 + 243949611482v^18 + 13435191456985v^17 + 10625085031535v^16 + 220820843394427v^15 + 173887870494052v^14 + 1789254234156025v^13 + 1400511919577855v^12 + 7800355488722333v^11 + 6057732029310852v^10 + 18831760965679439v^9 + 14495357744130725v^8 + 24537567922790962v^7 + 18721462089406514v^6 + 15624305974619240v^5 + 11832168163819732v^4 + 3901481476781647v^3 + 2938673674388150v^2 + 261428316039753*v + 196489229464517) / 474432850376
$\beta_{17}$	$=$	$ ( 307594400909 \nu^{19} - 243949611482 \nu^{18} + 13435191456985 \nu^{17} + \cdots - 196489229464517 ) / 474432850376 $ (307594400909v^19 - 243949611482v^18 + 13435191456985v^17 - 10625085031535v^16 + 220820843394427v^15 - 173887870494052v^14 + 1789254234156025v^13 - 1400511919577855v^12 + 7800355488722333v^11 - 6057732029310852v^10 + 18831760965679439v^9 - 14495357744130725v^8 + 24537567922790962v^7 - 18721462089406514v^6 + 15624305974619240v^5 - 11832168163819732v^4 + 3901481476781647v^3 - 2938673674388150v^2 + 261428316039753*v - 196489229464517) / 474432850376
$\beta_{18}$	$=$	$ ( 419195646885 \nu^{19} + 12271307302 \nu^{18} + 18296108564424 \nu^{17} + \cdots + 14368375102973 ) / 474432850376 $ (419195646885v^19 + 12271307302v^18 + 18296108564424v^17 + 536853252751v^16 + 300375306097121v^15 + 8847061459320v^14 + 2429949457457728v^13 + 71993516507291v^12 + 10570592181351195v^11 + 316095182926804v^10 + 25452896162356456v^9 + 772282550814425v^8 + 33062531772210744v^7 + 1027943994695914v^6 + 20968476307573912v^5 + 682252590539008v^4 + 5197920075051431v^3 + 186002166337706v^2 + 341242752168506*v + 14368375102973) / 474432850376
$\beta_{19}$	$=$	$ ( - 419195646885 \nu^{19} + 184793384513 \nu^{18} - 18296108564424 \nu^{17} + \cdots + 168759517784300 ) / 474432850376 $ (-419195646885v^19 + 184793384513v^18 - 18296108564424v^17 + 8102560182978v^16 - 300375306097121v^15 + 133943871450753v^14 - 2429949457457728v^13 + 1094081833879666v^12 - 10570592181351195v^11 + 4819714922259127v^10 - 25452896162356456v^9 + 11772432661856478v^8 - 33062531772210744v^7 + 15516363989152452v^6 - 20968476307573912v^5 + 9974998513280284v^4 - 5197920075051431v^3 + 2507705805665455v^2 - 341242752168506*v + 168759517784300) / 474432850376

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$ ( - \beta_{17} - \beta_{16} - \beta_{15} - \beta_{14} + \beta_{12} + \beta_{11} - 2 \beta_{10} + \cdots + \beta_{5} ) / 4 $ (-b17 - b16 - b15 - b14 + b12 + b11 - 2b10 + 2b8 + b7 - b6 + b5) / 4
$\nu^{2}$	$=$	$ ( \beta_{12} - \beta_{11} - \beta_{5} - 2\beta_{4} - \beta_{3} - \beta_{2} - \beta _1 - 8 ) / 2 $ (b12 - b11 - b5 - 2*b4 - b3 - b2 - b1 - 8) / 2
$\nu^{3}$	$=$	$ ( - 6 \beta_{19} - 2 \beta_{18} + 15 \beta_{17} + 15 \beta_{16} + 3 \beta_{15} + 3 \beta_{14} + \cdots - 6 \beta_{4} ) / 4 $ (-6b19 - 2b18 + 15b17 + 15b16 + 3b15 + 3b14 - 8b13 - 7b12 - 7b11 + 6b10 - 4b9 - 22b8 - 9b7 + 9b6 - 3b5 - 6b4) / 4
$\nu^{4}$	$=$	$ ( 4 \beta_{19} + 4 \beta_{18} - 13 \beta_{17} + 13 \beta_{16} - 13 \beta_{15} - 13 \beta_{14} - 39 \beta_{12} + \cdots + 150 ) / 4 $ (4b19 + 4b18 - 13b17 + 13b16 - 13b15 - 13b14 - 39b12 + 39b11 + 7b7 + 7b6 + 31b5 + 56b4 + 10b3 + 46b2 + 46*b1 + 150) / 4
$\nu^{5}$	$=$	$ ( 120 \beta_{19} + 10 \beta_{18} - 231 \beta_{17} - 231 \beta_{16} + 23 \beta_{15} + 37 \beta_{14} + \cdots + 100 \beta_1 ) / 4 $ (120b19 + 10b18 - 231b17 - 231b16 + 23b15 + 37b14 + 130b13 + 95b12 + 95b11 + 60b10 + 76b9 + 346b8 + 99b7 - 99b6 - 23b5 + 120b4 - 100b2 + 100b1) / 4
$\nu^{6}$	$=$	$ ( - 38 \beta_{19} - 38 \beta_{18} + 149 \beta_{17} - 149 \beta_{16} + 148 \beta_{15} + 148 \beta_{14} + \cdots - 954 ) / 2 $ (-38b19 - 38b18 + 149b17 - 149b16 + 148b15 + 148b14 + 378b12 - 378b11 - 82b7 - 82b6 - 247b5 - 384b4 + 31b3 - 449b2 - 449*b1 - 954) / 2
$\nu^{7}$	$=$	$ ( - 1095 \beta_{19} + 31 \beta_{18} + 1893 \beta_{17} + 1893 \beta_{16} - 476 \beta_{15} + \cdots - 1448 \beta_1 ) / 2 $ (-1095b19 + 31b18 + 1893b17 + 1893b16 - 476b15 - 640b14 - 1064b13 - 773b12 - 773b11 - 1116b10 - 642b9 - 2908b8 - 631b7 + 631b6 + 476b5 - 1095b4 + 1448b2 - 1448b1) / 2
$\nu^{8}$	$=$	$ ( 1284 \beta_{19} + 1284 \beta_{18} - 5725 \beta_{17} + 5725 \beta_{16} - 5647 \beta_{15} - 5647 \beta_{14} + \cdots + 28266 ) / 4 $ (1284b19 + 1284b18 - 5725b17 + 5725b16 - 5647b15 - 5647b14 - 14053b12 + 14053b11 + 3177b7 + 3177b6 + 8171b5 + 11480b4 - 3100b3 + 16432b2 + 16432*b1 + 28266) / 4
$\nu^{9}$	$=$	$ ( 19637 \beta_{19} - 1550 \beta_{18} - 32182 \beta_{17} - 32182 \beta_{16} + 10566 \beta_{15} + \cdots + 30610 \beta_1 ) / 2 $ (19637b19 - 1550b18 - 32182b17 - 32182b16 + 10566b15 + 13743b14 + 18087b13 + 13220b12 + 13220b11 + 24309b10 + 10884b9 + 50312b8 + 9110b7 - 9110b6 - 10566b5 + 19637b4 - 30610b2 + 30610b1) / 2
$\nu^{10}$	$=$	$ ( - 21768 \beta_{19} - 21768 \beta_{18} + 104719 \beta_{17} - 104719 \beta_{16} + 102615 \beta_{15} + \cdots - 455854 ) / 4 $ (-21768b19 - 21768b18 + 104719b17 - 104719b16 + 102615b15 + 102615b14 + 254523b12 - 254523b11 - 58497b7 - 58497b6 - 139055b5 - 184716b4 + 71444b3 - 293888b2 - 293888*b1 - 455854) / 4
$\nu^{11}$	$=$	$ ( - 699534 \beta_{19} + 71444 \beta_{18} + 1115253 \beta_{17} + 1115253 \beta_{16} - 408309 \beta_{15} + \cdots - 1166208 \beta_1 ) / 4 $ (-699534b19 + 71444b18 + 1115253b17 + 1115253b16 - 408309b15 - 525303b14 - 628090b13 - 460073b12 - 460073b11 - 933612b10 - 374660b9 - 1762918b8 - 287651b7 + 287651b6 + 408309b5 - 699534b4 + 1166208b2 - 1166208b1) / 4
$\nu^{12}$	$=$	$ ( 374660 \beta_{19} + 374660 \beta_{18} - 1878963 \beta_{17} + 1878963 \beta_{16} - 1832237 \beta_{15} + \cdots + 7694554 ) / 4 $ (374660b19 + 374660b18 - 1878963b17 + 1878963b16 - 1832237b15 - 1832237b14 - 4549597b12 + 4549597b11 + 1054935b7 + 1054935b6 + 2407539b5 + 3109568b4 - 1400830b3 + 5212094b2 + 5212094*b1 + 7694554) / 4
$\nu^{13}$	$=$	$ ( 6206500 \beta_{19} - 700415 \beta_{18} - 9754708 \beta_{17} - 9754708 \beta_{16} + 3749086 \beta_{15} + \cdots + 10650490 \beta_1 ) / 2 $ (6206500b19 - 700415b18 - 9754708b17 - 9754708b16 + 3749086b15 + 4804021b14 + 5506085b13 + 4032828b12 + 4032828b11 + 8553107b10 + 3260034b9 + 15520604b8 + 2396695b7 - 2396695b6 - 3749086b5 + 6206500b4 - 10650490b2 + 10650490b1) / 2
$\nu^{14}$	$=$	$ ( - 3260034 \beta_{19} - 3260034 \beta_{18} + 16713065 \beta_{17} - 16713065 \beta_{16} + 16245625 \beta_{15} + \cdots - 66462252 ) / 2 $ (-3260034b19 - 3260034b18 + 16713065b17 - 16713065b16 + 16245625b15 + 16245625b14 + 40402329b12 - 40402329b11 - 9415771b7 - 9415771b6 - 21039015b5 - 26814078b4 + 12948750b3 - 46074650b2 - 46074650*b1 - 66462252) / 2
$\nu^{15}$	$=$	$ ( - 219774656 \beta_{19} + 25897500 \beta_{18} + 342889497 \beta_{17} + 342889497 \beta_{16} + \cdots - 381972568 \beta_1 ) / 4 $ (-219774656b19 + 25897500b18 + 342889497b17 + 342889497b16 - 134766031b15 - 172429115b14 - 193877156b13 - 141904653b12 - 141904653b11 - 307195146b10 - 114215248b9 - 547584562b8 - 82236901b7 + 82236901b6 + 134766031b5 - 219774656b4 + 381972568b2 - 381972568b1) / 4
$\nu^{16}$	$=$	$ ( 57107624 \beta_{19} + 57107624 \beta_{18} - 296132380 \beta_{17} + 296132380 \beta_{16} + \cdots + 1161250844 ) / 2 $ (57107624b19 + 57107624b18 - 296132380b17 + 296132380b16 - 287296190b15 - 287296190b14 - 715356113b12 + 715356113b11 + 167191944b7 + 167191944b6 + 369533091b5 + 468074396b4 - 233485653b3 + 813705239b2 + 813705239*b1 + 1161250844) / 2
$\nu^{17}$	$=$	$ ( 3886121434 \beta_{19} - 466971306 \beta_{18} - 6040320349 \beta_{17} - 6040320349 \beta_{16} + \cdots + 6792587328 \beta_1 ) / 4 $ (3886121434b19 - 466971306b18 - 6040320349b17 - 6040320349b16 + 2398960207b15 + 3067727983b14 + 3419150128b13 + 2500974985b12 + 2500974985b11 + 5466688190b10 + 2008093396b9 + 9665764714b8 + 1431819171b7 - 1431819171b6 - 2398960207b5 + 3886121434b4 - 6792587328b2 + 6792587328b1) / 4
$\nu^{18}$	$=$	$ ( - 2008093396 \beta_{19} - 2008093396 \beta_{18} + 10474196037 \beta_{17} - 10474196037 \beta_{16} + \cdots - 40801155470 ) / 4 $ (-2008093396b19 - 2008093396b18 + 10474196037b17 - 10474196037b16 + 10150382061b15 + 10150382061b14 + 25293859771b12 - 25293859771b11 - 5921036915b7 - 5921036915b6 - 13014020403b5 - 16438153508b4 + 8325906462b3 - 28731131782b2 - 28731131782*b1 - 40801155470) / 4
$\nu^{19}$	$=$	$ ( - 68665334220 \beta_{19} + 8325906462 \beta_{18} + 106524653389 \beta_{17} + 106524653389 \beta_{16} + \cdots - 120326053512 \beta_1 ) / 4 $ (-68665334220b19 + 8325906462b18 + 106524653389b17 + 106524653389b16 - 42515356177b15 - 54357430007b14 - 60339427758b13 - 44115831413b12 - 44115831413b11 - 96872786184b10 - 35375108520b9 - 170647052142b8 - 25109743089b7 + 25109743089b6 + 42515356177b5 - 68665334220b4 + 120326053512b2 - 120326053512b1) / 4

Display $\beta_i$ in terms of $\nu^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1122\mathbb{Z}\right)^\times$.

$n$	$409$	$749$	$1057$
$\chi(n)$	$1$	$1$	$-\beta_{8}$

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

463.1

	−	2.92702i
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		0.590127i
		2.83889i
	−	1.59091i
		2.92702i
	−	1.68287i
	−	4.20168i
	−	1.04301i
		0.586325i
		1.57275i
	−	0.320425i
	−	0.590127i
	−	2.83889i
		1.59091i

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1.00000i

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1.64641i

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1.00000i

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2.72487i

463.2

−

1.00000i

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0.707107i

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−2.39327

2.39327i

0.707107

0.707107i

2.34381

2.34381i

1.00000i

−

1.00000i

2.39327

2.39327i

463.3

−

1.00000i

−0.707107

0.707107i

−1.00000

0.653251

−

0.653251i

0.707107

0.707107i

−2.55471

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2.55471i

1.00000i

−

1.00000i

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0.653251i

463.4

−

1.00000i

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0.707107i

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1.68953

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1.68953i

0.707107

0.707107i

−0.936655

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0.936655i

1.00000i

−

1.00000i

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−

1.68953i

463.5

−

1.00000i

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0.707107i

−1.00000

1.77535

−

1.77535i

0.707107

0.707107i

2.79396

2.79396i

1.00000i

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1.00000i

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1.77535i

463.6

−

1.00000i

0.707107

−

0.707107i

−1.00000

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2.96448i

−0.707107

−

0.707107i

0.962955

0.962955i

1.00000i

−

1.00000i

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2.96448i

463.7

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1.00000i

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0.707107i

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3.02020i

1.00000i

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1.00000i

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2.11456i

463.8

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1.00000i

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0.707107i

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0.105899i

−0.707107

−

0.707107i

2.56507

2.56507i

1.00000i

−

1.00000i

0.105899

0.105899i

463.9

−

1.00000i

0.707107

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0.707107i

−1.00000

1.65274

−

1.65274i

−0.707107

−

0.707107i

−1.63746

−

1.63746i

1.00000i

−

1.00000i

−1.65274

−

1.65274i

463.10

−

1.00000i

0.707107

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0.707107i

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2.53219

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2.53219i

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0.707107i

1.12963

1.12963i

1.00000i

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1.00000i

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2.53219i

727.1

1.00000i

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1.00000i

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2.72487i

727.2

1.00000i

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1.00000i

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2.39327i

727.3

1.00000i

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1.68953i

0.707107

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0.707107i

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0.936655i

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1.00000i

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1.68953i

727.5

1.00000i

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0.707107i

−1.00000

1.77535

1.77535i

0.707107

−

0.707107i

2.79396

−

2.79396i

−

1.00000i

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1.77535i

727.6

1.00000i

0.707107

0.707107i

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−

2.96448i

−0.707107

0.707107i

0.962955

−

0.962955i

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1.00000i

2.96448

−

2.96448i

727.7

1.00000i

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0.707107i

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2.11456i

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3.02020i

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1.00000i

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−

2.11456i

727.8

1.00000i

0.707107

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0.707107i

2.56507

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1.00000i

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0.105899i

727.9

1.00000i

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0.707107i

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1.65274i

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0.707107i

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1.63746i

−

1.00000i

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1.65274i

727.10

1.00000i

0.707107

0.707107i

−1.00000

2.53219

2.53219i

−0.707107

0.707107i

1.12963

−

1.12963i

−

1.00000i

−2.53219

2.53219i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
17.c	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1122.2.l.g	✓	20
17.c	even	4	1	inner	1122.2.l.g	✓	20

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1122.2.l.g	✓	20	1.a	even	1	1	trivial
1122.2.l.g	✓	20	17.c	even	4	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(1122, [\chi])$:

$ T_{5}^{20} + 4 T_{5}^{19} + 8 T_{5}^{18} - 40 T_{5}^{17} + 312 T_{5}^{16} + 1056 T_{5}^{15} + \cdots + 1290496 $

$ T_{7}^{20} + 556 T_{7}^{16} - 48 T_{7}^{15} + 1408 T_{7}^{13} + 93188 T_{7}^{12} + 3168 T_{7}^{11} + \cdots + 129777664 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{2} + 1)^{10} $ (T^2 + 1)^10
$3$	$ (T^{4} + 1)^{5} $ (T^4 + 1)^5
$5$	$ T^{20} + 4 T^{19} + \cdots + 1290496 $ T^20 + 4T^19 + 8T^18 - 40T^17 + 312T^16 + 1056T^15 + 2528T^14 - 13280T^13 + 40196T^12 + 68016T^11 + 250528T^10 - 1383712T^9 + 3293616T^8 - 114816T^7 + 8764160T^6 - 47533824T^5 + 115503040T^4 - 94771456T^3 + 34877952T^2 + 9487872*T + 1290496
$7$	$ T^{20} + \cdots + 129777664 $ T^20 + 556T^16 - 48T^15 + 1408T^13 + 93188T^12 + 3168T^11 + 1152T^10 + 217984T^9 + 5000576T^8 + 1950208T^7 + 198656T^6 - 3645440T^5 + 55673856T^4 + 4988928T^3 + 524288T^2 - 11665408*T + 129777664
$11$	$ (T^{4} + 1)^{5} $ (T^4 + 1)^5
$13$	$ (T^{10} - 52 T^{8} + \cdots - 1024)^{2} $ (T^10 - 52T^8 - 8T^7 + 850T^6 + 160T^5 - 4960T^4 - 1792T^3 + 8576T^2 + 5120T - 1024)^2
$17$	$ T^{20} + \cdots + 2015993900449 $ T^20 + 12T^19 + 66T^18 + 316T^17 + 1489T^16 + 4560T^15 + 1056T^14 - 64976T^13 - 437106T^12 - 2702472T^11 - 13703492T^10 - 45942024T^9 - 126323634T^8 - 319227088T^7 + 88198176T^6 + 6474547920T^5 + 35940840241T^4 + 129667020668T^3 + 460399991106T^2 + 1423054517964*T + 2015993900449
$19$	$ T^{20} + \cdots + 10070523904 $ T^20 + 288T^18 + 35424T^16 + 2421632T^14 + 100037888T^12 + 2532349952T^10 + 37722148864T^8 + 293510578176T^6 + 889844662272T^4 + 829278912512*T^2 + 10070523904
$23$	$ T^{20} + \cdots + 20032570802176 $ T^20 - 16T^19 + 128T^18 - 176T^17 + 620T^16 - 20112T^15 + 257920T^14 - 123104T^13 - 715388T^12 - 1883936T^11 + 185714816T^10 + 226021376T^9 - 8781440T^8 + 1583945728T^7 + 50837835776T^6 + 116083238912T^5 + 128920994816T^4 + 256295968768T^3 + 4664484200448T^2 + 13670523764736*T + 20032570802176
$29$	$ T^{20} + \cdots + 269668335616 $ T^20 - 4T^19 + 8T^18 + 184T^17 + 8748T^16 - 25568T^15 + 49216T^14 + 1100992T^13 + 20125360T^12 - 26931264T^11 + 45624448T^10 + 972458880T^9 + 12091455040T^8 - 4266202112T^7 + 4323463168T^6 + 95369269248T^5 + 1254573344768T^4 - 317383966720T^3 + 51939934208T^2 + 167371177984*T + 269668335616
$31$	$ T^{20} + \cdots + 3619856318464 $ T^20 + 8T^19 + 32T^18 + 512T^17 + 16992T^16 + 159616T^15 + 864256T^14 + 5340672T^13 + 61852928T^12 + 552054784T^11 + 3224764416T^10 + 12986105856T^9 + 43349102592T^8 + 157416161280T^7 + 647248216064T^6 + 2304999686144T^5 + 6086740934656T^4 + 11145154920448T^3 + 13572633526272T^2 + 9912717410304*T + 3619856318464
$37$	$ T^{20} + \cdots + 294683951104 $ T^20 + 20T^19 + 200T^18 + 840T^17 + 10284T^16 + 170464T^15 + 1705280T^14 + 7290944T^13 + 32995760T^12 + 358204736T^11 + 3681223808T^10 + 15926942848T^9 + 31923502656T^8 - 12124132352T^7 + 266677882880T^6 + 928373964800T^5 + 1620295972864T^4 - 1019519533056T^3 + 618058514432T^2 + 603542749184*T + 294683951104
$41$	$ T^{20} + \cdots + 4960666624 $ T^20 - 20T^19 + 200T^18 - 936T^17 + 14036T^16 - 230464T^15 + 2240128T^14 - 11006080T^13 + 37636256T^12 - 191978368T^11 + 1728398080T^10 - 8531232512T^9 + 22351370880T^8 - 13014541312T^7 + 16425035776T^6 - 84059670528T^5 + 259829794048T^4 - 77140841472T^3 + 8216594432T^2 + 9028818944*T + 4960666624
$43$	$ T^{20} + \cdots + 326137016221696 $ T^20 + 608T^18 + 151072T^16 + 19897472T^14 + 1515004160T^12 + 68550494208T^10 + 1815089156096T^8 + 26275548102656T^6 + 177918606573568T^4 + 426208898580480*T^2 + 326137016221696
$47$	$ (T^{10} + 8 T^{9} + \cdots - 206235776)^{2} $ (T^10 + 8T^9 - 380T^8 - 2792T^7 + 53330T^6 + 360992T^5 - 3312864T^4 - 20490976T^3 + 79895136T^2 + 430758272*T - 206235776)^2
$53$	$ T^{20} + \cdots + 5079722622976 $ T^20 + 552T^18 + 117172T^16 + 12363984T^14 + 714900548T^12 + 23534412928T^10 + 442622434816T^8 + 4564036136960T^6 + 22956378374144T^4 + 41931622318080*T^2 + 5079722622976
$59$	$ T^{20} + \cdots + 66\!\cdots\!44 $ T^20 + 1056T^18 + 479968T^16 + 123013504T^14 + 19564158208T^12 + 2001243371520T^10 + 131889883828224T^8 + 5442996319944704T^6 + 131276730267074560T^4 + 1594458375286947840*T^2 + 6627532723814662144
$61$	$ T^{20} + \cdots + 41881192378624 $ T^20 + 20T^19 + 200T^18 + 760T^17 + 19448T^16 + 334272T^15 + 3084640T^14 + 14335840T^13 + 71596932T^12 + 658301488T^11 + 5620688032T^10 + 26841425632T^9 + 80021253552T^8 + 201842067840T^7 + 1080082205440T^6 + 5058448661760T^5 + 14251943856064T^4 + 16455075241728T^3 + 1876092973568T^2 - 12535789623808*T + 41881192378624
$67$	$ (T^{10} + 24 T^{9} + \cdots - 50864128)^{2} $ (T^10 + 24T^9 + 56T^8 - 2848T^7 - 23984T^6 + 30976T^5 + 1053440T^4 + 3457024T^3 - 4058112T^2 - 38060032*T - 50864128)^2
$71$	$ T^{20} + \cdots + 124716335104 $ T^20 + 32T^19 + 512T^18 + 3024T^17 + 9068T^16 + 70800T^15 + 2195072T^14 + 7859424T^13 + 17795972T^12 + 133392352T^11 + 1525322880T^10 + 6719745408T^9 + 16376083840T^8 + 21110205952T^7 + 45865379840T^6 + 168905138176T^5 + 436123296768T^4 + 474856316928T^3 + 166478348288T^2 - 203777179648*T + 124716335104
$73$	$ T^{20} + \cdots + 510936476041216 $ T^20 - 20T^19 + 200T^18 + 24T^17 + 40300T^16 - 777120T^15 + 7482688T^14 - 4941376T^13 + 280602800T^12 - 5539204416T^11 + 54950356096T^10 - 33595230848T^9 + 571057801792T^8 - 11665569628160T^7 + 120966125817856T^6 - 69937978687488T^5 + 105050903676928T^4 - 3519012552327168T^3 + 53432464032432128T^2 + 7389261787070464*T + 510936476041216
$79$	$ T^{20} + \cdots + 109990395904 $ T^20 + 16T^19 + 128T^18 - 1760T^17 + 11948T^16 + 68048T^15 + 1108224T^14 - 15900288T^13 + 94255876T^12 - 154131168T^11 + 2527860864T^10 - 32669843200T^9 + 213767936384T^8 - 648523388928T^7 + 1022009946112T^6 - 698819713024T^5 + 5896582259712T^4 - 17364023861248T^3 + 25267973685248T^2 + 2357640527872*T + 109990395904
$83$	$ T^{20} + \cdots + 11\!\cdots\!24 $ T^20 + 992T^18 + 408960T^16 + 92279808T^14 + 12600954880T^12 + 1082401161216T^10 + 58556684697600T^8 + 1923709291986944T^6 + 34793471411748864T^4 + 265858939478867968*T^2 + 113383030625665024
$89$	$ (T^{10} - 408 T^{8} + \cdots + 17440768)^{2} $ (T^10 - 408T^8 + 1056T^7 + 52432T^6 - 232064T^5 - 1990144T^4 + 10625024T^3 + 7420928T^2 - 45465600T + 17440768)^2
$97$	$ T^{20} + \cdots + 14479425273856 $ T^20 - 4T^19 + 8T^18 - 1928T^17 + 75340T^16 - 781984T^15 + 4383808T^14 - 18013248T^13 + 387892784T^12 - 4148565056T^11 + 23694053504T^10 - 41541933184T^9 + 52676949056T^8 - 205980903424T^7 + 1287923073024T^6 - 2102825451520T^5 + 1853695393792T^4 - 1759069077504T^3 + 11248821338112T^2 - 18048626982912*T + 14479425273856
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1122 = 2 \cdot 3 \cdot 11 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1122.l (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q - \beta_{8} q^{2} - \beta_{2} q^{3} - q^{4} + \beta_{6} q^{5} + \beta_1 q^{6} + ( - \beta_{19} - \beta_{4}) q^{7} + \beta_{8} q^{8} - \beta_{8} q^{9}+O(q^{10}) \) q - b8 * q^2 - b2 * q^3 - q^4 + b6 * q^5 + b1 * q^6 + (-b19 - b4) * q^7 + b8 * q^8 - b8 * q^9	\( q - \beta_{8} q^{2} - \beta_{2} q^{3} - q^{4} + \beta_{6} q^{5} + \beta_1 q^{6} + ( - \beta_{19} - \beta_{4}) q^{7} + \beta_{8} q^{8} - \beta_{8} q^{9} - \beta_{7} q^{10} + \beta_1 q^{11} + \beta_{2} q^{12} - \beta_{4} q^{13} - \beta_{18} q^{14} + \beta_{9} q^{15} + q^{16} + (\beta_{16} + \beta_{10} - \beta_{5} - 1) q^{17} - q^{18} + ( - \beta_{14} + \beta_{12} + \cdots - \beta_1) q^{19}+ \cdots + \beta_{2} q^{99}+O(q^{100}) \) q - b8 * q^2 - b2 * q^3 - q^4 + b6 * q^5 + b1 * q^6 + (-b19 - b4) * q^7 + b8 * q^8 - b8 * q^9 - b7 * q^10 + b1 * q^11 + b2 * q^12 - b4 * q^13 - b18 * q^14 + b9 * q^15 + q^16 + (b16 + b10 - b5 - 1) * q^17 - q^18 + (-b14 + b12 + b11 - b10 - b9 + b2 - b1) * q^19 - b6 * q^20 + b5 * q^21 + b2 * q^22 + (b19 - b14 + b10 - b5 + b4) * q^23 - b1 * q^24 + (b12 + b11 - 3b8 + b7 - b6) q^25 + (b19 + b13 + b4) * q^26 + b1 * q^27 + (b19 + b4) * q^28 + (b18 - b16 + b15 + b12 - b8 + 1) * q^29 - b3 * q^30 + (-b18 + b15 - b9 - b3 - 2b2) q^31 - b8 * q^32 - q^33 + (-b15 - b11 + b8 + b5) * q^34 + (b19 + b18 - 2b15 - 2b14 + b12 - b11 + b4) * q^35 + b8 * q^36 + (b18 - b17 - b15 - b14 - b11 - b10 + b9 + b8 + b5 + b3 - 2b2 - 1) q^37 + (-b17 + b16 - b5 + b3 - b2 - b1) * q^38 + (-b15 - b14 - b10 + b5) * q^39 + b7 * q^40 + (-b19 - b14 - b13 - b10 + b8 - b5 - 2b4 + 1) q^41 + (-b14 - b10) * q^42 + (-b19 - b18 - b15 - 2b13 + b12 + b11 - b10 + b9 + b5 - b4 - b2 + b1) q^43 - b1 * q^44 - b7 * q^45 + (b18 - 2b15 - b14 - b10 + b5) q^46 + (-2b19 - 2b18 + b12 - b11 + b5 + b4 + b3 - b2 - b1) * q^47 - b2 * q^48 + (b15 - b14 + b12 + b11 + b8 - b5) * q^49 + (-b17 + b16 + b7 + b6 - 3) * q^50 + (b18 - b17 - b4 + b2) * q^51 + b4 * q^52 + (b19 + b15 - b14 + b13 - b5 + b4 + 4b2 - 4b1) * q^53 + b2 * q^54 - b3 * q^55 + b18 * q^56 + (b19 + b16 + b12 + b8 + b7 + b4 + 1) * q^57 + (-b19 - b17 - b14 + b11 - b8 - b5 - b4 - 1) * q^58 + (-b17 - b16 + b15 + b10 - b9 - 6b8 + b7 - b6 - b5 + b2 - b1) q^59 - b9 * q^60 + (b19 - b14 - b13 + b10 - 2b8 - b7 - b5 - 2) q^61 + (b19 - b14 - b9 - b5 + b4 + b3 + 2b1) q^62 - b18 * q^63 - q^64 + (b19 + 2b18 - b16 - 2b15 - 2b14 + b13 + b12 - 2b10 + 2b5) q^65 + b8 * q^66 + (-b19 - b18 + b7 + b6 + b4 - 2) * q^67 + (-b16 - b10 + b5 + 1) * q^68 + (b19 + b18 - b5 - b4) * q^69 + (-b19 + b18 - b17 - b16 - 2b15 - 2b10 + 2b5 - b4) q^70 + (-b19 - b18 - b13 + 2b8 - 2b6 - 2) * q^71 + q^72 + (-b19 - b18 - b17 - b16 - b14 - b13 + b12 - b11 - b10 + b9 - 2b8 + b6 + b5 + b3 + 2b2 + 2) * q^73 + (-b19 - b16 - b12 - b10 + b9 + b8 - b4 - b3 + 2b1 + 1) q^74 + (b16 + b12 - b9 + b3 + 3b1) q^75 + (b14 - b12 - b11 + b10 + b9 - b2 + b1) * q^76 + (-b14 - b10) * q^77 - b10 * q^78 + (-b14 + b13 + b10 - 2b8 - 2b7 - b5 + b4 - 2) * q^79 + b6 * q^80 - q^81 + (b19 + b14 + b13 + b10 - b8 - b5 + 1) * q^82 + (b19 + b18 - b17 - b16 - 2b15 + 2b13 - b12 - b11 - 2b10 + 2b8 - b7 + b6 + 2b5 + b4) q^83 - b5 * q^84 + (-4b19 - 3b18 - b16 - 2b14 - 3b13 + b11 - 2b10 + b8 - 2b6 - 3b4 - b3 + b2 + b1 - 1) q^85 + (b19 + b18 - b17 + b16 + b15 + b14 - b4 - b3 + b2 + b1) * q^86 + (-b18 + b17 + b16 + b14 - b13 + b10 - b2 + b1) * q^87 - b2 * q^88 + (b19 + b18 + b17 - b16 - b15 - b14 + b12 - b11 - b5 + b4) * q^89 - b6 * q^90 + (-b17 + b11 - b9 + 4b8 + b3 - 2b1 + 4) * q^91 + (-b19 + b14 - b10 + b5 - b4) * q^92 + (-b18 - b14 - b13 - b10 - 2b8 + b7 - b6) q^93 + (-b19 - 2b18 - b17 - b16 - b14 - 3b13 - b10 + b9 - b4 - b2 + b1) * q^94 + (b17 - b16 + 3b14 + 4b13 - b12 - b11 + b10 + b9 - 2b8 - 2b7 + 3b5 + 4b4 - b3 - 6b1 - 2) q^95 + b1 * q^96 + (b19 + b16 + 2b15 + 2b14 + b13 - b12 + 2b10 - b8 + 2b6 - 2b5 + 1) q^97 + (-b17 + b16 - b15 - b14 - b5 + 1) * q^98 + b2 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 20 q^{4} - 4 q^{5}+O(q^{10}) \) 20 * q - 20 * q^4 - 4 * q^5	\( 20 q - 20 q^{4} - 4 q^{5} + 4 q^{10} + 20 q^{16} - 12 q^{17} - 20 q^{18} + 4 q^{20} + 16 q^{23} + 4 q^{29} - 8 q^{31} - 20 q^{33} + 16 q^{35} - 20 q^{37} + 8 q^{39} - 4 q^{40} + 20 q^{41} + 4 q^{45} + 16 q^{46} - 16 q^{47} - 68 q^{50} + 8 q^{57} - 4 q^{58} - 20 q^{61} + 8 q^{62} - 20 q^{64} + 8 q^{65} - 48 q^{67} + 12 q^{68} - 32 q^{71} + 20 q^{72} + 20 q^{73} + 20 q^{74} - 8 q^{75} - 8 q^{78} - 16 q^{79} - 4 q^{80} - 20 q^{81} + 20 q^{82} - 4 q^{85} - 16 q^{86} + 4 q^{90} + 88 q^{91} - 16 q^{92} - 48 q^{95} + 4 q^{97} + 36 q^{98}+O(q^{100}) \) 20 * q - 20 * q^4 - 4 * q^5 + 4 * q^10 + 20 * q^16 - 12 * q^17 - 20 * q^18 + 4 * q^20 + 16 * q^23 + 4 * q^29 - 8 * q^31 - 20 * q^33 + 16 * q^35 - 20 * q^37 + 8 * q^39 - 4 * q^40 + 20 * q^41 + 4 * q^45 + 16 * q^46 - 16 * q^47 - 68 * q^50 + 8 * q^57 - 4 * q^58 - 20 * q^61 + 8 * q^62 - 20 * q^64 + 8 * q^65 - 48 * q^67 + 12 * q^68 - 32 * q^71 + 20 * q^72 + 20 * q^73 + 20 * q^74 - 8 * q^75 - 8 * q^78 - 16 * q^79 - 4 * q^80 - 20 * q^81 + 20 * q^82 - 4 * q^85 - 16 * q^86 + 4 * q^90 + 88 * q^91 - 16 * q^92 - 48 * q^95 + 4 * q^97 + 36 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1122.2.l.g

Level	$1122$
Weight	$2$
Character orbit	1122.l
Analytic conductor	$8.959$
Analytic rank	$0$
Dimension	$20$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(8.95921510679\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Relative dimension:	\(10\) over \(\Q(i)\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{20} + 44 x^{18} + 732 x^{16} + 6050 x^{14} + 27262 x^{12} + 69598 x^{10} + 100205 x^{8} + 77682 x^{6} + \cdots + 289 \) x^20 + 44x^18 + 732x^16 + 6050x^14 + 27262x^12 + 69598x^10 + 100205x^8 + 77682x^6 + 29931x^4 + 5170*x^2 + 289
Coefficient ring:	\(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index:	\( 2^{12} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

\(\beta_{1}\)	\(=\)	\( ( - 478453232 \nu^{19} - 65607844609 \nu^{18} - 20790152935 \nu^{17} - 2865069509061 \nu^{16} + \cdots - 55604642864711 ) / 474432850376 \) (-478453232v^19 - 65607844609v^18 - 20790152935v^17 - 2865069509061v^16 - 338917545566v^15 - 47075635647853v^14 - 2712070921497v^13 - 381263160365035v^12 - 11607278696930v^11 - 1661009394596199v^10 - 27338720388297v^9 - 4006421429337605v^8 - 34709977991120v^7 - 5214676459361530v^6 - 22405456855914v^5 - 3316918140098058v^4 - 7637317749682v^3 - 828047778854719v^2 - 1629578068081*v - 55604642864711) / 474432850376
\(\beta_{2}\)	\(=\)	\( ( 478453232 \nu^{19} - 65607844609 \nu^{18} + 20790152935 \nu^{17} - 2865069509061 \nu^{16} + \cdots - 55604642864711 ) / 474432850376 \) (478453232v^19 - 65607844609v^18 + 20790152935v^17 - 2865069509061v^16 + 338917545566v^15 - 47075635647853v^14 + 2712070921497v^13 - 381263160365035v^12 + 11607278696930v^11 - 1661009394596199v^10 + 27338720388297v^9 - 4006421429337605v^8 + 34709977991120v^7 - 5214676459361530v^6 + 22405456855914v^5 - 3316918140098058v^4 + 7637317749682v^3 - 828047778854719v^2 + 1629578068081*v - 55604642864711) / 474432850376
\(\beta_{3}\)	\(=\)	\( ( 5058550639 \nu^{18} + 223694565153 \nu^{16} + 3744505387173 \nu^{14} + 31110636071745 \nu^{12} + \cdots + 5064152867485 ) / 27907814728 \) (5058550639v^18 + 223694565153v^16 + 3744505387173v^14 + 31110636071745v^12 + 139986142331567v^10 + 349648532161315v^8 + 470045438835854v^6 + 305955182420016v^4 + 76957686604497*v^2 + 5064152867485) / 27907814728
\(\beta_{4}\)	\(=\)	\( ( - 10148357483 \nu^{18} - 445041584131 \nu^{16} - 7358635881849 \nu^{14} + \cdots - 9081831922431 ) / 27907814728 \) (-10148357483v^18 - 445041584131v^16 - 7358635881849v^14 - 60122842198375v^12 - 264918808196019v^10 - 647067653590709v^8 - 852259999673914v^6 - 546632113102428v^4 - 136570802313397*v^2 - 9081831922431) / 27907814728
\(\beta_{5}\)	\(=\)	\( ( 20389568647 \nu^{18} + 890396850383 \nu^{16} + 14629854784637 \nu^{14} + 118485210525483 \nu^{12} + \cdots + 16926462334275 ) / 27907814728 \) (20389568647v^18 + 890396850383v^16 + 14629854784637v^14 + 118485210525483v^12 + 516186649960791v^10 + 1245019644242777v^8 + 1620162473611754v^6 + 1029558507628972v^4 + 255939812252649*v^2 + 16926462334275) / 27907814728
\(\beta_{6}\)	\(=\)	\( ( - 139301146690 \nu^{19} - 106669931977 \nu^{18} - 6058900080957 \nu^{17} + \cdots - 78163571880140 ) / 474432850376 \) (-139301146690v^19 - 106669931977v^18 - 6058900080957v^17 - 4617719070078v^16 - 98951904059896v^15 - 74873525236589v^14 - 794582992039069v^13 - 595084920757662v^12 - 3422960693279472v^11 - 2528760184100971v^10 - 8151299190023259v^9 - 5929193915290958v^8 - 10473397988916922v^7 - 7506889363788560v^6 - 6583682116882800v^5 - 4676748657443724v^4 - 1620046710347750v^3 - 1158702610325915v^2 - 104268340777307*v - 78163571880140) / 474432850376
\(\beta_{7}\)	\(=\)	\( ( 139301146690 \nu^{19} - 106669931977 \nu^{18} + 6058900080957 \nu^{17} + \cdots - 78163571880140 ) / 474432850376 \) (139301146690v^19 - 106669931977v^18 + 6058900080957v^17 - 4617719070078v^16 + 98951904059896v^15 - 74873525236589v^14 + 794582992039069v^13 - 595084920757662v^12 + 3422960693279472v^11 - 2528760184100971v^10 + 8151299190023259v^9 - 5929193915290958v^8 + 10473397988916922v^7 - 7506889363788560v^6 + 6583682116882800v^5 - 4676748657443724v^4 + 1620046710347750v^3 - 1158702610325915v^2 + 104268340777307*v - 78163571880140) / 474432850376
\(\beta_{8}\)	\(=\)	\( ( 1404907 \nu^{19} + 61326597 \nu^{17} + 1007052793 \nu^{15} + 8149764229 \nu^{13} + \cdots + 1196618885 \nu ) / 4194104 \) (1404907v^19 + 61326597v^17 + 1007052793v^15 + 8149764229v^13 + 35474164511v^11 + 85505262423v^9 + 111268672750v^7 + 70821737508v^5 + 17714444473v^3 + 1196618885v) / 4194104
\(\beta_{9}\)	\(=\)	\( ( 198021598279 \nu^{19} + 8680993741599 \nu^{17} + 143468768001205 \nu^{15} + \cdots + 186387049023435 \nu ) / 474432850376 \) (198021598279v^19 + 8680993741599v^17 + 143468768001205v^15 + 1171499492229951v^13 + 5159024662579791v^11 + 12599392653447497v^9 + 16613727939830606v^7 + 10702062017531120v^5 + 2708982607502525v^3 + 186387049023435v) / 474432850376
\(\beta_{10}\)	\(=\)	\( ( 233700662525 \nu^{19} + 48399845376 \nu^{18} + 10185869836434 \nu^{17} + \cdots + 48121187253053 ) / 474432850376 \) (233700662525v^19 + 48399845376v^18 + 10185869836434v^17 + 2140643049511v^16 + 166875104105745v^15 + 35838822387566v^14 + 1345979794093378v^13 + 297772407073891v^12 + 5832483487393567v^11 + 1339410340734166v^10 + 13982579149755490v^9 + 3341764982076613v^8 + 18085828026582976v^7 + 4482763069675646v^6 + 11433905888104068v^5 + 2907528387642272v^4 + 2832474662487223v^3 + 727089853605996v^2 + 187196882199640*v + 48121187253053) / 474432850376
\(\beta_{11}\)	\(=\)	\( ( 243072241811 \nu^{19} + 21820907889 \nu^{18} + 10624957886419 \nu^{17} + \cdots + 21177824929574 ) / 474432850376 \) (243072241811v^19 + 21820907889v^18 + 10624957886419v^17 + 960999407232v^16 + 174827238548909v^15 + 15990384178901v^14 + 1418757856390083v^13 + 131786781660972v^12 + 6197076267516619v^11 + 587160399443479v^10 + 14990771145384333v^9 + 1451892040944876v^8 + 19563396967649766v^7 + 1936329198013400v^6 + 12463245620673056v^5 + 1257797697422936v^4 + 3110069184579833v^3 + 319648245046351v^2 + 209945391510523*v + 21177824929574) / 474432850376
\(\beta_{12}\)	\(=\)	\( ( 243072241811 \nu^{19} - 21820907889 \nu^{18} + 10624957886419 \nu^{17} + \cdots - 21177824929574 ) / 474432850376 \) (243072241811v^19 - 21820907889v^18 + 10624957886419v^17 - 960999407232v^16 + 174827238548909v^15 - 15990384178901v^14 + 1418757856390083v^13 - 131786781660972v^12 + 6197076267516619v^11 - 587160399443479v^10 + 14990771145384333v^9 - 1451892040944876v^8 + 19563396967649766v^7 - 1936329198013400v^6 + 12463245620673056v^5 - 1257797697422936v^4 + 3110069184579833v^3 - 319648245046351v^2 + 209945391510523*v - 21177824929574) / 474432850376
\(\beta_{13}\)	\(=\)	\( ( 263502915318 \nu^{19} - 12271307302 \nu^{18} + 11525417122427 \nu^{17} + \cdots - 14368375102973 ) / 474432850376 \) (263502915318v^19 - 12271307302v^18 + 11525417122427v^17 - 536853252751v^16 + 189827094809752v^15 - 8847061459320v^14 + 1542568980992775v^13 - 71993516507291v^12 + 6749728887477256v^11 - 316095182926804v^10 + 16359775161018901v^9 - 772282550814425v^8 + 21390409747923306v^7 - 1027943994695914v^6 + 13644788022546956v^5 - 682252590539008v^4 + 3400856208910174v^3 - 186002166337706v^2 + 223932584435117*v - 14368375102973) / 474432850376
\(\beta_{14}\)	\(=\)	\( ( - 268820550276 \nu^{19} - 48399845376 \nu^{18} - 11721434280167 \nu^{17} + \cdots - 48121187253053 ) / 474432850376 \) (-268820550276v^19 - 48399845376v^18 - 11721434280167v^17 - 2140643049511v^16 - 192158923731954v^15 - 35838822387566v^14 - 1551490803933211v^13 - 297772407073891v^12 - 6733500920866650v^11 - 1339410340734166v^10 - 16180612474008077v^9 - 3341764982076613v^8 - 21007969325115622v^7 - 4482763069675646v^6 - 13375628622751672v^5 - 2907528387642272v^4 - 3369319232867232v^3 - 727089853605996v^2 - 231099634601005*v - 48121187253053) / 474432850376
\(\beta_{15}\)	\(=\)	\( ( 268820550276 \nu^{19} + 298222821623 \nu^{18} + 11721434280167 \nu^{17} + \cdots + 239628672429622 ) / 474432850376 \) (268820550276v^19 + 298222821623v^18 + 11721434280167v^17 + 12996103407000v^16 + 192158923731954v^15 + 212868708951263v^14 + 1551490803933211v^13 + 1716476171859320v^12 + 6733500920866650v^11 + 7435762708599281v^10 + 16180612474008077v^9 + 17823568970050596v^8 + 21007969325115622v^7 + 23059998981724172v^6 + 13375628622751672v^5 + 14594966242050252v^4 + 3369319232867232v^3 + 3623886954689037v^2 + 231099634601005*v + 239628672429622) / 474432850376
\(\beta_{16}\)	\(=\)	\( ( 307594400909 \nu^{19} + 243949611482 \nu^{18} + 13435191456985 \nu^{17} + \cdots + 196489229464517 ) / 474432850376 \) (307594400909v^19 + 243949611482v^18 + 13435191456985v^17 + 10625085031535v^16 + 220820843394427v^15 + 173887870494052v^14 + 1789254234156025v^13 + 1400511919577855v^12 + 7800355488722333v^11 + 6057732029310852v^10 + 18831760965679439v^9 + 14495357744130725v^8 + 24537567922790962v^7 + 18721462089406514v^6 + 15624305974619240v^5 + 11832168163819732v^4 + 3901481476781647v^3 + 2938673674388150v^2 + 261428316039753*v + 196489229464517) / 474432850376
\(\beta_{17}\)	\(=\)	\( ( 307594400909 \nu^{19} - 243949611482 \nu^{18} + 13435191456985 \nu^{17} + \cdots - 196489229464517 ) / 474432850376 \) (307594400909v^19 - 243949611482v^18 + 13435191456985v^17 - 10625085031535v^16 + 220820843394427v^15 - 173887870494052v^14 + 1789254234156025v^13 - 1400511919577855v^12 + 7800355488722333v^11 - 6057732029310852v^10 + 18831760965679439v^9 - 14495357744130725v^8 + 24537567922790962v^7 - 18721462089406514v^6 + 15624305974619240v^5 - 11832168163819732v^4 + 3901481476781647v^3 - 2938673674388150v^2 + 261428316039753*v - 196489229464517) / 474432850376
\(\beta_{18}\)	\(=\)	\( ( 419195646885 \nu^{19} + 12271307302 \nu^{18} + 18296108564424 \nu^{17} + \cdots + 14368375102973 ) / 474432850376 \) (419195646885v^19 + 12271307302v^18 + 18296108564424v^17 + 536853252751v^16 + 300375306097121v^15 + 8847061459320v^14 + 2429949457457728v^13 + 71993516507291v^12 + 10570592181351195v^11 + 316095182926804v^10 + 25452896162356456v^9 + 772282550814425v^8 + 33062531772210744v^7 + 1027943994695914v^6 + 20968476307573912v^5 + 682252590539008v^4 + 5197920075051431v^3 + 186002166337706v^2 + 341242752168506*v + 14368375102973) / 474432850376
\(\beta_{19}\)	\(=\)	\( ( - 419195646885 \nu^{19} + 184793384513 \nu^{18} - 18296108564424 \nu^{17} + \cdots + 168759517784300 ) / 474432850376 \) (-419195646885v^19 + 184793384513v^18 - 18296108564424v^17 + 8102560182978v^16 - 300375306097121v^15 + 133943871450753v^14 - 2429949457457728v^13 + 1094081833879666v^12 - 10570592181351195v^11 + 4819714922259127v^10 - 25452896162356456v^9 + 11772432661856478v^8 - 33062531772210744v^7 + 15516363989152452v^6 - 20968476307573912v^5 + 9974998513280284v^4 - 5197920075051431v^3 + 2507705805665455v^2 - 341242752168506*v + 168759517784300) / 474432850376

\(\nu\)	\(=\)	\( ( - \beta_{17} - \beta_{16} - \beta_{15} - \beta_{14} + \beta_{12} + \beta_{11} - 2 \beta_{10} + \cdots + \beta_{5} ) / 4 \) (-b17 - b16 - b15 - b14 + b12 + b11 - 2b10 + 2b8 + b7 - b6 + b5) / 4
\(\nu^{2}\)	\(=\)	\( ( \beta_{12} - \beta_{11} - \beta_{5} - 2\beta_{4} - \beta_{3} - \beta_{2} - \beta _1 - 8 ) / 2 \) (b12 - b11 - b5 - 2*b4 - b3 - b2 - b1 - 8) / 2
\(\nu^{3}\)	\(=\)	\( ( - 6 \beta_{19} - 2 \beta_{18} + 15 \beta_{17} + 15 \beta_{16} + 3 \beta_{15} + 3 \beta_{14} + \cdots - 6 \beta_{4} ) / 4 \) (-6b19 - 2b18 + 15b17 + 15b16 + 3b15 + 3b14 - 8b13 - 7b12 - 7b11 + 6b10 - 4b9 - 22b8 - 9b7 + 9b6 - 3b5 - 6b4) / 4
\(\nu^{4}\)	\(=\)	\( ( 4 \beta_{19} + 4 \beta_{18} - 13 \beta_{17} + 13 \beta_{16} - 13 \beta_{15} - 13 \beta_{14} - 39 \beta_{12} + \cdots + 150 ) / 4 \) (4b19 + 4b18 - 13b17 + 13b16 - 13b15 - 13b14 - 39b12 + 39b11 + 7b7 + 7b6 + 31b5 + 56b4 + 10b3 + 46b2 + 46*b1 + 150) / 4
\(\nu^{5}\)	\(=\)	\( ( 120 \beta_{19} + 10 \beta_{18} - 231 \beta_{17} - 231 \beta_{16} + 23 \beta_{15} + 37 \beta_{14} + \cdots + 100 \beta_1 ) / 4 \) (120b19 + 10b18 - 231b17 - 231b16 + 23b15 + 37b14 + 130b13 + 95b12 + 95b11 + 60b10 + 76b9 + 346b8 + 99b7 - 99b6 - 23b5 + 120b4 - 100b2 + 100b1) / 4
\(\nu^{6}\)	\(=\)	\( ( - 38 \beta_{19} - 38 \beta_{18} + 149 \beta_{17} - 149 \beta_{16} + 148 \beta_{15} + 148 \beta_{14} + \cdots - 954 ) / 2 \) (-38b19 - 38b18 + 149b17 - 149b16 + 148b15 + 148b14 + 378b12 - 378b11 - 82b7 - 82b6 - 247b5 - 384b4 + 31b3 - 449b2 - 449*b1 - 954) / 2
\(\nu^{7}\)	\(=\)	\( ( - 1095 \beta_{19} + 31 \beta_{18} + 1893 \beta_{17} + 1893 \beta_{16} - 476 \beta_{15} + \cdots - 1448 \beta_1 ) / 2 \) (-1095b19 + 31b18 + 1893b17 + 1893b16 - 476b15 - 640b14 - 1064b13 - 773b12 - 773b11 - 1116b10 - 642b9 - 2908b8 - 631b7 + 631b6 + 476b5 - 1095b4 + 1448b2 - 1448b1) / 2
\(\nu^{8}\)	\(=\)	\( ( 1284 \beta_{19} + 1284 \beta_{18} - 5725 \beta_{17} + 5725 \beta_{16} - 5647 \beta_{15} - 5647 \beta_{14} + \cdots + 28266 ) / 4 \) (1284b19 + 1284b18 - 5725b17 + 5725b16 - 5647b15 - 5647b14 - 14053b12 + 14053b11 + 3177b7 + 3177b6 + 8171b5 + 11480b4 - 3100b3 + 16432b2 + 16432*b1 + 28266) / 4
\(\nu^{9}\)	\(=\)	\( ( 19637 \beta_{19} - 1550 \beta_{18} - 32182 \beta_{17} - 32182 \beta_{16} + 10566 \beta_{15} + \cdots + 30610 \beta_1 ) / 2 \) (19637b19 - 1550b18 - 32182b17 - 32182b16 + 10566b15 + 13743b14 + 18087b13 + 13220b12 + 13220b11 + 24309b10 + 10884b9 + 50312b8 + 9110b7 - 9110b6 - 10566b5 + 19637b4 - 30610b2 + 30610b1) / 2
\(\nu^{10}\)	\(=\)	\( ( - 21768 \beta_{19} - 21768 \beta_{18} + 104719 \beta_{17} - 104719 \beta_{16} + 102615 \beta_{15} + \cdots - 455854 ) / 4 \) (-21768b19 - 21768b18 + 104719b17 - 104719b16 + 102615b15 + 102615b14 + 254523b12 - 254523b11 - 58497b7 - 58497b6 - 139055b5 - 184716b4 + 71444b3 - 293888b2 - 293888*b1 - 455854) / 4
\(\nu^{11}\)	\(=\)	\( ( - 699534 \beta_{19} + 71444 \beta_{18} + 1115253 \beta_{17} + 1115253 \beta_{16} - 408309 \beta_{15} + \cdots - 1166208 \beta_1 ) / 4 \) (-699534b19 + 71444b18 + 1115253b17 + 1115253b16 - 408309b15 - 525303b14 - 628090b13 - 460073b12 - 460073b11 - 933612b10 - 374660b9 - 1762918b8 - 287651b7 + 287651b6 + 408309b5 - 699534b4 + 1166208b2 - 1166208b1) / 4
\(\nu^{12}\)	\(=\)	\( ( 374660 \beta_{19} + 374660 \beta_{18} - 1878963 \beta_{17} + 1878963 \beta_{16} - 1832237 \beta_{15} + \cdots + 7694554 ) / 4 \) (374660b19 + 374660b18 - 1878963b17 + 1878963b16 - 1832237b15 - 1832237b14 - 4549597b12 + 4549597b11 + 1054935b7 + 1054935b6 + 2407539b5 + 3109568b4 - 1400830b3 + 5212094b2 + 5212094*b1 + 7694554) / 4
\(\nu^{13}\)	\(=\)	\( ( 6206500 \beta_{19} - 700415 \beta_{18} - 9754708 \beta_{17} - 9754708 \beta_{16} + 3749086 \beta_{15} + \cdots + 10650490 \beta_1 ) / 2 \) (6206500b19 - 700415b18 - 9754708b17 - 9754708b16 + 3749086b15 + 4804021b14 + 5506085b13 + 4032828b12 + 4032828b11 + 8553107b10 + 3260034b9 + 15520604b8 + 2396695b7 - 2396695b6 - 3749086b5 + 6206500b4 - 10650490b2 + 10650490b1) / 2
\(\nu^{14}\)	\(=\)	\( ( - 3260034 \beta_{19} - 3260034 \beta_{18} + 16713065 \beta_{17} - 16713065 \beta_{16} + 16245625 \beta_{15} + \cdots - 66462252 ) / 2 \) (-3260034b19 - 3260034b18 + 16713065b17 - 16713065b16 + 16245625b15 + 16245625b14 + 40402329b12 - 40402329b11 - 9415771b7 - 9415771b6 - 21039015b5 - 26814078b4 + 12948750b3 - 46074650b2 - 46074650*b1 - 66462252) / 2
\(\nu^{15}\)	\(=\)	\( ( - 219774656 \beta_{19} + 25897500 \beta_{18} + 342889497 \beta_{17} + 342889497 \beta_{16} + \cdots - 381972568 \beta_1 ) / 4 \) (-219774656b19 + 25897500b18 + 342889497b17 + 342889497b16 - 134766031b15 - 172429115b14 - 193877156b13 - 141904653b12 - 141904653b11 - 307195146b10 - 114215248b9 - 547584562b8 - 82236901b7 + 82236901b6 + 134766031b5 - 219774656b4 + 381972568b2 - 381972568b1) / 4
\(\nu^{16}\)	\(=\)	\( ( 57107624 \beta_{19} + 57107624 \beta_{18} - 296132380 \beta_{17} + 296132380 \beta_{16} + \cdots + 1161250844 ) / 2 \) (57107624b19 + 57107624b18 - 296132380b17 + 296132380b16 - 287296190b15 - 287296190b14 - 715356113b12 + 715356113b11 + 167191944b7 + 167191944b6 + 369533091b5 + 468074396b4 - 233485653b3 + 813705239b2 + 813705239*b1 + 1161250844) / 2
\(\nu^{17}\)	\(=\)	\( ( 3886121434 \beta_{19} - 466971306 \beta_{18} - 6040320349 \beta_{17} - 6040320349 \beta_{16} + \cdots + 6792587328 \beta_1 ) / 4 \) (3886121434b19 - 466971306b18 - 6040320349b17 - 6040320349b16 + 2398960207b15 + 3067727983b14 + 3419150128b13 + 2500974985b12 + 2500974985b11 + 5466688190b10 + 2008093396b9 + 9665764714b8 + 1431819171b7 - 1431819171b6 - 2398960207b5 + 3886121434b4 - 6792587328b2 + 6792587328b1) / 4
\(\nu^{18}\)	\(=\)	\( ( - 2008093396 \beta_{19} - 2008093396 \beta_{18} + 10474196037 \beta_{17} - 10474196037 \beta_{16} + \cdots - 40801155470 ) / 4 \) (-2008093396b19 - 2008093396b18 + 10474196037b17 - 10474196037b16 + 10150382061b15 + 10150382061b14 + 25293859771b12 - 25293859771b11 - 5921036915b7 - 5921036915b6 - 13014020403b5 - 16438153508b4 + 8325906462b3 - 28731131782b2 - 28731131782*b1 - 40801155470) / 4
\(\nu^{19}\)	\(=\)	\( ( - 68665334220 \beta_{19} + 8325906462 \beta_{18} + 106524653389 \beta_{17} + 106524653389 \beta_{16} + \cdots - 120326053512 \beta_1 ) / 4 \) (-68665334220b19 + 8325906462b18 + 106524653389b17 + 106524653389b16 - 42515356177b15 - 54357430007b14 - 60339427758b13 - 44115831413b12 - 44115831413b11 - 96872786184b10 - 35375108520b9 - 170647052142b8 - 25109743089b7 + 25109743089b6 + 42515356177b5 - 68665334220b4 + 120326053512b2 - 120326053512b1) / 4

\(n\)	\(409\)	\(749\)	\(1057\)
\(\chi(n)\)	\(1\)	\(1\)	\(-\beta_{8}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 463.10
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{2} + 1)^{10} \) (T^2 + 1)^10
$3$	\( (T^{4} + 1)^{5} \) (T^4 + 1)^5
$5$	\( T^{20} + 4 T^{19} + \cdots + 1290496 \) T^20 + 4T^19 + 8T^18 - 40T^17 + 312T^16 + 1056T^15 + 2528T^14 - 13280T^13 + 40196T^12 + 68016T^11 + 250528T^10 - 1383712T^9 + 3293616T^8 - 114816T^7 + 8764160T^6 - 47533824T^5 + 115503040T^4 - 94771456T^3 + 34877952T^2 + 9487872*T + 1290496
$7$	\( T^{20} + \cdots + 129777664 \) T^20 + 556T^16 - 48T^15 + 1408T^13 + 93188T^12 + 3168T^11 + 1152T^10 + 217984T^9 + 5000576T^8 + 1950208T^7 + 198656T^6 - 3645440T^5 + 55673856T^4 + 4988928T^3 + 524288T^2 - 11665408*T + 129777664
$11$	\( (T^{4} + 1)^{5} \) (T^4 + 1)^5
$13$	\( (T^{10} - 52 T^{8} + \cdots - 1024)^{2} \) (T^10 - 52T^8 - 8T^7 + 850T^6 + 160T^5 - 4960T^4 - 1792T^3 + 8576T^2 + 5120T - 1024)^2
$17$	\( T^{20} + \cdots + 2015993900449 \) T^20 + 12T^19 + 66T^18 + 316T^17 + 1489T^16 + 4560T^15 + 1056T^14 - 64976T^13 - 437106T^12 - 2702472T^11 - 13703492T^10 - 45942024T^9 - 126323634T^8 - 319227088T^7 + 88198176T^6 + 6474547920T^5 + 35940840241T^4 + 129667020668T^3 + 460399991106T^2 + 1423054517964*T + 2015993900449
$19$	\( T^{20} + \cdots + 10070523904 \) T^20 + 288T^18 + 35424T^16 + 2421632T^14 + 100037888T^12 + 2532349952T^10 + 37722148864T^8 + 293510578176T^6 + 889844662272T^4 + 829278912512*T^2 + 10070523904
$23$	\( T^{20} + \cdots + 20032570802176 \) T^20 - 16T^19 + 128T^18 - 176T^17 + 620T^16 - 20112T^15 + 257920T^14 - 123104T^13 - 715388T^12 - 1883936T^11 + 185714816T^10 + 226021376T^9 - 8781440T^8 + 1583945728T^7 + 50837835776T^6 + 116083238912T^5 + 128920994816T^4 + 256295968768T^3 + 4664484200448T^2 + 13670523764736*T + 20032570802176
$29$	\( T^{20} + \cdots + 269668335616 \) T^20 - 4T^19 + 8T^18 + 184T^17 + 8748T^16 - 25568T^15 + 49216T^14 + 1100992T^13 + 20125360T^12 - 26931264T^11 + 45624448T^10 + 972458880T^9 + 12091455040T^8 - 4266202112T^7 + 4323463168T^6 + 95369269248T^5 + 1254573344768T^4 - 317383966720T^3 + 51939934208T^2 + 167371177984*T + 269668335616
$31$	\( T^{20} + \cdots + 3619856318464 \) T^20 + 8T^19 + 32T^18 + 512T^17 + 16992T^16 + 159616T^15 + 864256T^14 + 5340672T^13 + 61852928T^12 + 552054784T^11 + 3224764416T^10 + 12986105856T^9 + 43349102592T^8 + 157416161280T^7 + 647248216064T^6 + 2304999686144T^5 + 6086740934656T^4 + 11145154920448T^3 + 13572633526272T^2 + 9912717410304*T + 3619856318464
$37$	\( T^{20} + \cdots + 294683951104 \) T^20 + 20T^19 + 200T^18 + 840T^17 + 10284T^16 + 170464T^15 + 1705280T^14 + 7290944T^13 + 32995760T^12 + 358204736T^11 + 3681223808T^10 + 15926942848T^9 + 31923502656T^8 - 12124132352T^7 + 266677882880T^6 + 928373964800T^5 + 1620295972864T^4 - 1019519533056T^3 + 618058514432T^2 + 603542749184*T + 294683951104
$41$	\( T^{20} + \cdots + 4960666624 \) T^20 - 20T^19 + 200T^18 - 936T^17 + 14036T^16 - 230464T^15 + 2240128T^14 - 11006080T^13 + 37636256T^12 - 191978368T^11 + 1728398080T^10 - 8531232512T^9 + 22351370880T^8 - 13014541312T^7 + 16425035776T^6 - 84059670528T^5 + 259829794048T^4 - 77140841472T^3 + 8216594432T^2 + 9028818944*T + 4960666624
$43$	\( T^{20} + \cdots + 326137016221696 \) T^20 + 608T^18 + 151072T^16 + 19897472T^14 + 1515004160T^12 + 68550494208T^10 + 1815089156096T^8 + 26275548102656T^6 + 177918606573568T^4 + 426208898580480*T^2 + 326137016221696
$47$	\( (T^{10} + 8 T^{9} + \cdots - 206235776)^{2} \) (T^10 + 8T^9 - 380T^8 - 2792T^7 + 53330T^6 + 360992T^5 - 3312864T^4 - 20490976T^3 + 79895136T^2 + 430758272*T - 206235776)^2
$53$	\( T^{20} + \cdots + 5079722622976 \) T^20 + 552T^18 + 117172T^16 + 12363984T^14 + 714900548T^12 + 23534412928T^10 + 442622434816T^8 + 4564036136960T^6 + 22956378374144T^4 + 41931622318080*T^2 + 5079722622976
$59$	\( T^{20} + \cdots + 66\!\cdots\!44 \) T^20 + 1056T^18 + 479968T^16 + 123013504T^14 + 19564158208T^12 + 2001243371520T^10 + 131889883828224T^8 + 5442996319944704T^6 + 131276730267074560T^4 + 1594458375286947840*T^2 + 6627532723814662144
$61$	\( T^{20} + \cdots + 41881192378624 \) T^20 + 20T^19 + 200T^18 + 760T^17 + 19448T^16 + 334272T^15 + 3084640T^14 + 14335840T^13 + 71596932T^12 + 658301488T^11 + 5620688032T^10 + 26841425632T^9 + 80021253552T^8 + 201842067840T^7 + 1080082205440T^6 + 5058448661760T^5 + 14251943856064T^4 + 16455075241728T^3 + 1876092973568T^2 - 12535789623808*T + 41881192378624
$67$	\( (T^{10} + 24 T^{9} + \cdots - 50864128)^{2} \) (T^10 + 24T^9 + 56T^8 - 2848T^7 - 23984T^6 + 30976T^5 + 1053440T^4 + 3457024T^3 - 4058112T^2 - 38060032*T - 50864128)^2
$71$	\( T^{20} + \cdots + 124716335104 \) T^20 + 32T^19 + 512T^18 + 3024T^17 + 9068T^16 + 70800T^15 + 2195072T^14 + 7859424T^13 + 17795972T^12 + 133392352T^11 + 1525322880T^10 + 6719745408T^9 + 16376083840T^8 + 21110205952T^7 + 45865379840T^6 + 168905138176T^5 + 436123296768T^4 + 474856316928T^3 + 166478348288T^2 - 203777179648*T + 124716335104
$73$	\( T^{20} + \cdots + 510936476041216 \) T^20 - 20T^19 + 200T^18 + 24T^17 + 40300T^16 - 777120T^15 + 7482688T^14 - 4941376T^13 + 280602800T^12 - 5539204416T^11 + 54950356096T^10 - 33595230848T^9 + 571057801792T^8 - 11665569628160T^7 + 120966125817856T^6 - 69937978687488T^5 + 105050903676928T^4 - 3519012552327168T^3 + 53432464032432128T^2 + 7389261787070464*T + 510936476041216
$79$	\( T^{20} + \cdots + 109990395904 \) T^20 + 16T^19 + 128T^18 - 1760T^17 + 11948T^16 + 68048T^15 + 1108224T^14 - 15900288T^13 + 94255876T^12 - 154131168T^11 + 2527860864T^10 - 32669843200T^9 + 213767936384T^8 - 648523388928T^7 + 1022009946112T^6 - 698819713024T^5 + 5896582259712T^4 - 17364023861248T^3 + 25267973685248T^2 + 2357640527872*T + 109990395904
$83$	\( T^{20} + \cdots + 11\!\cdots\!24 \) T^20 + 992T^18 + 408960T^16 + 92279808T^14 + 12600954880T^12 + 1082401161216T^10 + 58556684697600T^8 + 1923709291986944T^6 + 34793471411748864T^4 + 265858939478867968*T^2 + 113383030625665024
$89$	\( (T^{10} - 408 T^{8} + \cdots + 17440768)^{2} \) (T^10 - 408T^8 + 1056T^7 + 52432T^6 - 232064T^5 - 1990144T^4 + 10625024T^3 + 7420928T^2 - 45465600T + 17440768)^2
$97$	\( T^{20} + \cdots + 14479425273856 \) T^20 - 4T^19 + 8T^18 - 1928T^17 + 75340T^16 - 781984T^15 + 4383808T^14 - 18013248T^13 + 387892784T^12 - 4148565056T^11 + 23694053504T^10 - 41541933184T^9 + 52676949056T^8 - 205980903424T^7 + 1287923073024T^6 - 2102825451520T^5 + 1853695393792T^4 - 1759069077504T^3 + 11248821338112T^2 - 18048626982912*T + 14479425273856
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