LMFDB - Newform orbit 2010.2.i.g

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2010,2,Mod(841,2010)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2010, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

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

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

chi := DirichletCharacter("2010.841");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2010 = 2 \cdot 3 \cdot 5 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2010.i (of order $3$, 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:	$16.0499308063$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{11} + 14 x^{10} + 19 x^{9} + 128 x^{8} + 144 x^{7} + 522 x^{6} + 645 x^{5} + 1466 x^{4} + \cdots + 25 $ x^12 - x^11 + 14x^10 + 19x^9 + 128x^8 + 144x^7 + 522x^6 + 645x^5 + 1466x^4 + 1072x^3 + 1031x^2 - 145x + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-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 $1,\beta_1,\ldots,\beta_{11}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} - x^{11} + 14 x^{10} + 19 x^{9} + 128 x^{8} + 144 x^{7} + 522 x^{6} + 645 x^{5} + 1466 x^{4} + \cdots + 25 $ x^12 - x^11 + 14*x^10 + 19*x^9 + 128*x^8 + 144*x^7 + 522*x^6 + 645*x^5 + 1466*x^4 + 1072*x^3 + 1031*x^2 - 145*x + 25 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( 2819168 \nu^{11} + 265198160 \nu^{10} - 477773490 \nu^{9} + 3036341904 \nu^{8} + \cdots - 66262408515 ) / 35643585385 $ (2819168v^11 + 265198160v^10 - 477773490v^9 + 3036341904v^8 + 5959276720v^7 + 11385206295v^6 + 24081566144v^5 + 38035565760v^4 + 120293726640v^3 + 19484788432v^2 - 2703405440*v - 66262408515) / 35643585385
$\beta_{3}$	$=$	$ ( - 33921673 \nu^{11} + 63688670 \nu^{10} - 492954100 \nu^{9} - 243291424 \nu^{8} + \cdots + 5162245150 ) / 35643585385 $ (-33921673v^11 + 63688670v^10 - 492954100v^9 - 243291424v^8 - 3706572240v^7 - 841605200v^6 - 13078196499v^5 - 7186486390v^4 - 38185841100v^3 + 4751171423v^2 - 718061785*v + 5162245150) / 35643585385
$\beta_{4}$	$=$	$ ( - 39021658 \nu^{11} + 308911751 \nu^{10} + 455970818 \nu^{9} - 641732196 \nu^{8} + \cdots + 10904045450 ) / 35643585385 $ (-39021658v^11 + 308911751v^10 + 455970818v^9 - 641732196v^8 + 21125740804v^7 + 23663803342v^6 + 130545546630v^5 + 90576811580v^4 + 497658288514v^3 + 444758392548v^2 + 786214752751*v + 10904045450) / 35643585385
$\beta_{5}$	$=$	$ ( 94791175 \nu^{11} - 410626178 \nu^{10} + 1864902053 \nu^{9} - 1914357152 \nu^{8} + \cdots - 85788381500 ) / 35643585385 $ (94791175v^11 - 410626178v^10 + 1864902053v^9 - 1914357152v^8 + 5496983472v^7 - 5073079088v^6 + 16767819549v^5 - 12119261462v^4 - 2806839561v^3 - 30375533585v^2 + 4383131575*v - 85788381500) / 35643585385
$\beta_{6}$	$=$	$ ( - 206489806 \nu^{11} + 172568133 \nu^{10} - 2827168614 \nu^{9} - 4416260414 \nu^{8} + \cdots + 29222960085 ) / 35643585385 $ (-206489806v^11 + 172568133v^10 - 2827168614v^9 - 4416260414v^8 - 26673986592v^7 - 33441104304v^6 - 108629283932v^5 - 146264121369v^4 - 309900541986v^3 - 259542913132v^2 - 208139818563*v + 29222960085) / 35643585385
$\beta_{7}$	$=$	$ ( - 438214690 \nu^{11} + 564108370 \nu^{10} - 6874518668 \nu^{9} - 5490449467 \nu^{8} + \cdots + 71345728850 ) / 35643585385 $ (-438214690v^11 + 564108370v^10 - 6874518668v^9 - 5490449467v^8 - 60798887952v^7 - 59858636368v^6 - 244364748363v^5 - 276884025146v^4 - 685391076372v^3 - 554680872781v^2 - 506410838830*v + 71345728850) / 35643585385
$\beta_{8}$	$=$	$ ( 455587880 \nu^{11} - 253541502 \nu^{10} + 5420063988 \nu^{9} + 12661175037 \nu^{8} + \cdots - 55870601790 ) / 35643585385 $ (455587880v^11 - 253541502v^10 + 5420063988v^9 + 12661175037v^8 + 53310202896v^7 + 75588991248v^6 + 216308780363v^5 + 322545433110v^4 + 630582773772v^3 + 438344851516v^2 + 398871729762*v - 55870601790) / 35643585385
$\beta_{9}$	$=$	$ ( - 613353486 \nu^{11} + 1406905506 \nu^{10} - 10913994885 \nu^{9} + 1437365197 \nu^{8} + \cdots + 114285727520 ) / 35643585385 $ (-613353486v^11 + 1406905506v^10 - 10913994885v^9 + 1437365197v^8 - 82152276377v^7 - 19102441115v^6 - 340325290628v^5 - 160831627012v^4 - 846924705500v^3 - 317301195189v^2 - 805039653781*v + 114285727520) / 35643585385
$\beta_{10}$	$=$	$ ( 725477973 \nu^{11} + 748732528 \nu^{10} + 5519743294 \nu^{9} + 38781688065 \nu^{8} + \cdots - 54895060340 ) / 35643585385 $ (725477973v^11 + 748732528v^10 + 5519743294v^9 + 38781688065v^8 + 82593124990v^7 + 226503843354v^6 + 332054561353v^5 + 877409472252v^4 + 1117172383696v^3 + 1264790933665v^2 + 404117634802*v - 54895060340) / 35643585385
$\beta_{11}$	$=$	$ ( - 790732852 \nu^{11} + 2592244241 \nu^{10} - 13568854921 \nu^{9} + 6679372073 \nu^{8} + \cdots - 81596245665 ) / 35643585385 $ (-790732852v^11 + 2592244241v^10 - 13568854921v^9 + 6679372073v^8 - 63261184149v^7 + 31539977231v^6 - 210699805314v^5 - 14215939216v^4 - 219920015163v^3 + 192156998147v^2 - 28052877490*v - 81596245665) / 35643585385

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ -\beta_{8} + \beta_{7} - 4\beta_{6} - 2\beta_{3} + 2\beta_1 $ -b8 + b7 - 4b6 - 2b3 + 2*b1
$\nu^{3}$	$=$	$ \beta_{11} - \beta_{9} - 2\beta_{5} - \beta_{4} - 10\beta_{3} - 4\beta_{2} - 5 $ b11 - b9 - 2b5 - b4 - 10b3 - 4*b2 - 5
$\nu^{4}$	$=$	$ 4 \beta_{11} - 4 \beta_{10} + 18 \beta_{8} - 12 \beta_{7} + 31 \beta_{6} - 12 \beta_{5} - 5 \beta_{4} + \cdots - 31 $ 4b11 - 4b10 + 18b8 - 12b7 + 31b6 - 12b5 - 5b4 - 18b2 - 32*b1 - 31
$\nu^{5}$	$=$	$ -19\beta_{10} + 22\beta_{9} + 71\beta_{8} - 38\beta_{7} + 84\beta_{6} + 129\beta_{3} - 129\beta_1 $ -19b10 + 22b9 + 71b8 - 38b7 + 84b6 + 129b3 - 129*b1
$\nu^{6}$	$=$	$ -74\beta_{11} + 90\beta_{9} + 162\beta_{5} + 90\beta_{4} + 470\beta_{3} + 279\beta_{2} + 355 $ -74b11 + 90b9 + 162b5 + 90b4 + 470b3 + 279b2 + 355
$\nu^{7}$	$=$	$ - 295 \beta_{11} + 295 \beta_{10} - 1075 \beta_{8} + 587 \beta_{7} - 1234 \beta_{6} + 587 \beta_{5} + \cdots + 1234 $ -295b11 + 295b10 - 1075b8 + 587b7 - 1234b6 + 587b5 + 353b4 + 1075b2 + 1810*b1 + 1234
$\nu^{8}$	$=$	$ 1133\beta_{10} - 1370\beta_{9} - 4120\beta_{8} + 2298\beta_{7} - 4798\beta_{6} - 6811\beta_{3} + 6811\beta_1 $ 1133b10 - 1370b9 - 4120b8 + 2298b7 - 4798b6 - 6811b3 + 6811*b1
$\nu^{9}$	$=$	$ 4357\beta_{11} - 5253\beta_{9} - 8633\beta_{5} - 5253\beta_{4} - 25971\beta_{3} - 15732\beta_{2} - 17839 $ 4357b11 - 5253b9 - 8633b5 - 5253b4 - 25971b3 - 15732b2 - 17839
$\nu^{10}$	$=$	$ 16628 \beta_{11} - 16628 \beta_{10} + 59946 \beta_{8} - 33070 \beta_{7} + 68144 \beta_{6} - 33070 \beta_{5} + \cdots - 68144 $ 16628b11 - 16628b10 + 59946b8 - 33070b7 + 68144b6 - 33070b5 - 20089b4 - 59946b2 - 98503*b1 - 68144
$\nu^{11}$	$=$	$ - 63407 \beta_{10} + 76574 \beta_{9} + 228236 \beta_{8} - 125379 \beta_{7} + 257678 \beta_{6} + \cdots - 374794 \beta_1 $ -63407b10 + 76574b9 + 228236b8 - 125379b7 + 257678b6 + 374794b3 - 374794*b1

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2010\mathbb{Z}\right)^\times$.

$n$	$671$	$1141$	$1207$
$\chi(n)$	$1$	$-1 + \beta_{6}$	$1$

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} $

841.1

−0.690503	+	1.19599i
−0.853889	+	1.47898i
0.0726386	−	0.125814i
1.01533	−	1.75861i
−0.944909	+	1.63663i
1.90133	−	3.29320i
−0.690503	−	1.19599i
−0.853889	−	1.47898i
0.0726386	+	0.125814i
1.01533	+	1.75861i
−0.944909	−	1.63663i
1.90133	+	3.29320i

−0.500000

−

0.866025i

−1.00000

−0.500000

0.866025i

−1.00000

0.500000

0.866025i

−2.07324

3.59096i

1.00000

0.500000

0.866025i

841.2

−0.500000

−

0.866025i

−1.00000

−0.500000

0.866025i

−1.00000

0.500000

0.866025i

−1.49675

2.59245i

1.00000

0.500000

0.866025i

841.3

−0.500000

−

0.866025i

−1.00000

−0.500000

0.866025i

−1.00000

0.500000

0.866025i

−1.12165

1.94276i

1.00000

0.500000

0.866025i

841.4

−0.500000

−

0.866025i

−1.00000

−0.500000

0.866025i

−1.00000

0.500000

0.866025i

0.654717

−

1.13400i

1.00000

0.500000

0.866025i

841.5

−0.500000

−

0.866025i

−1.00000

−0.500000

0.866025i

−1.00000

0.500000

0.866025i

1.14811

−

1.98858i

1.00000

0.500000

0.866025i

841.6

−0.500000

−

0.866025i

−1.00000

−0.500000

0.866025i

−1.00000

0.500000

0.866025i

2.38882

−

4.13756i

1.00000

0.500000

0.866025i

1771.1

−0.500000

0.866025i

−1.00000

−0.500000

−

0.866025i

−1.00000

0.500000

−

0.866025i

−2.07324

−

3.59096i

1.00000

0.500000

−

0.866025i

1771.2

−0.500000

0.866025i

−1.00000

−0.500000

−

0.866025i

−1.00000

0.500000

−

0.866025i

−1.49675

−

2.59245i

1.00000

0.500000

−

0.866025i

1771.3

−0.500000

0.866025i

−1.00000

−0.500000

−

0.866025i

−1.00000

0.500000

−

0.866025i

−1.12165

−

1.94276i

1.00000

0.500000

−

0.866025i

1771.4

−0.500000

0.866025i

−1.00000

−0.500000

−

0.866025i

−1.00000

0.500000

−

0.866025i

0.654717

1.13400i

1.00000

0.500000

−

0.866025i

1771.5

−0.500000

0.866025i

−1.00000

−0.500000

−

0.866025i

−1.00000

0.500000

−

0.866025i

1.14811

1.98858i

1.00000

0.500000

−

0.866025i

1771.6

−0.500000

0.866025i

−1.00000

−0.500000

−

0.866025i

−1.00000

0.500000

−

0.866025i

2.38882

4.13756i

1.00000

0.500000

−

0.866025i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
67.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2010.2.i.g	✓	12
67.c	even	3	1	inner	2010.2.i.g	✓	12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2010.2.i.g	✓	12	1.a	even	1	1	trivial
2010.2.i.g	✓	12	67.c	even	3	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7}^{12} + T_{7}^{11} + 31 T_{7}^{10} + 40 T_{7}^{9} + 728 T_{7}^{8} + 791 T_{7}^{7} + 6480 T_{7}^{6} + \cdots + 160000 $ T7^12 + T7^11 + 31*T7^10 + 40*T7^9 + 728*T7^8 + 791*T7^7 + 6480*T7^6 + 1655*T7^5 + 36274*T7^4 + 4085*T7^3 + 106825*T7^2 - 62000*T7 + 160000 acting on $S_{2}^{\mathrm{new}}(2010, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{2} + T + 1)^{6} $ (T^2 + T + 1)^6
$3$	$ (T + 1)^{12} $ (T + 1)^12
$5$	$ (T + 1)^{12} $ (T + 1)^12
$7$	$ T^{12} + T^{11} + \cdots + 160000 $ T^12 + T^11 + 31T^10 + 40T^9 + 728T^8 + 791T^7 + 6480T^6 + 1655T^5 + 36274T^4 + 4085T^3 + 106825T^2 - 62000T + 160000
$11$	$ T^{12} + 2 T^{11} + \cdots + 5640625 $ T^12 + 2T^11 + 54T^10 + 6T^9 + 1861T^8 - 180T^7 + 35009T^6 - 29235T^5 + 444225T^4 - 140000T^3 + 1791875T^2 - 356250*T + 5640625
$13$	$ T^{12} - 6 T^{11} + \cdots + 1562500 $ T^12 - 6T^11 + 62T^10 - 210T^9 + 1752T^8 - 5637T^7 + 29703T^6 - 62910T^5 + 242153T^4 - 482646T^3 + 1278949T^2 - 1428750*T + 1562500
$17$	$ T^{12} - 5 T^{11} + \cdots + 49 $ T^12 - 5T^11 + 71T^10 - 256T^9 + 3388T^8 - 11881T^7 + 55776T^6 - 26122T^5 + 35890T^4 + 4179T^3 + 18088T^2 - 931*T + 49
$19$	$ T^{12} + 9 T^{11} + \cdots + 160000 $ T^12 + 9T^11 + 91T^10 + 234T^9 + 1501T^8 + 1259T^7 + 20029T^6 + 466T^5 + 106979T^4 - 91695T^3 + 465025T^2 - 266000*T + 160000
$23$	$ T^{12} + 8 T^{11} + \cdots + 77880625 $ T^12 + 8T^11 + 115T^10 + 790T^9 + 8389T^8 + 50005T^7 + 297495T^6 + 1026244T^5 + 3550571T^4 + 7066430T^3 + 21180100T^2 + 31064000*T + 77880625
$29$	$ T^{12} + 11 T^{11} + \cdots + 12496225 $ T^12 + 11T^11 + 124T^10 + 753T^9 + 5540T^8 + 28112T^7 + 153968T^6 + 515771T^5 + 1610170T^4 + 2347254T^3 + 4397729T^2 + 1261995*T + 12496225
$31$	$ T^{12} + \cdots + 222367744 $ T^12 + 2T^11 + 95T^10 + 130T^9 + 6287T^8 + 7329T^7 + 199644T^6 + 39414T^5 + 4328336T^4 + 782698T^3 + 39942521T^2 - 35147584*T + 222367744
$37$	$ T^{12} + 93 T^{10} + \cdots + 53728900 $ T^12 + 93T^10 - 130T^9 + 6448T^8 - 9044T^7 + 194258T^6 - 414749T^5 + 4357646T^4 - 5647899T^3 + 25127331T^2 + 21982670T + 53728900
$41$	$ T^{12} + 2 T^{11} + \cdots + 144400 $ T^12 + 2T^11 + 87T^10 - 10T^9 + 5813T^8 + 219T^7 + 111754T^6 - 315618T^5 + 1445858T^4 - 1575584T^3 + 1292769T^2 - 504260*T + 144400
$43$	$ (T^{6} - 8 T^{5} + 2 T^{4} + \cdots + 1)^{2} $ (T^6 - 8T^5 + 2T^4 + 108T^3 - 211T^2 + 7*T + 1)^2
$47$	$ T^{12} + 2 T^{11} + \cdots + 180625 $ T^12 + 2T^11 + 82T^10 + 432T^9 + 6539T^8 + 23045T^7 + 94670T^6 + 60668T^5 + 174169T^4 - 164115T^3 + 472550T^2 - 274125*T + 180625
$53$	$ (T^{6} + 2 T^{5} + \cdots - 96040)^{2} $ (T^6 + 2T^5 - 202T^4 - 245T^3 + 9310T^2 - 343*T - 96040)^2
$59$	$ (T^{6} - 24 T^{5} + \cdots - 6268)^{2} $ (T^6 - 24T^5 + 159T^4 + 195T^3 - 5119T^2 + 13431*T - 6268)^2
$61$	$ T^{12} - 5 T^{11} + \cdots + 702244 $ T^12 - 5T^11 + 186T^10 - 403T^9 + 27088T^8 - 82319T^7 + 646800T^6 - 45788T^5 + 5745947T^4 - 7692369T^3 + 11443211T^2 - 3020990*T + 702244
$67$	$ T^{12} + \cdots + 90458382169 $ T^12 - 3T^11 + 3T^10 + 945T^9 + 1437T^8 + 6042T^7 + 516805T^6 + 404814T^5 + 6450693T^4 + 284221035T^3 + 60453363T^2 - 4050375321*T + 90458382169
$71$	$ T^{12} + \cdots + 510724622500 $ T^12 + 308T^10 + 638T^9 + 66972T^8 + 129945T^7 + 7263197T^6 + 10625340T^5 + 567961531T^4 + 428034456T^3 + 20937464049T^2 - 22649402450T + 510724622500
$73$	$ T^{12} + \cdots + 111195571600 $ T^12 + 6T^11 + 375T^10 + 1016T^9 + 95608T^8 + 287998T^7 + 10632188T^6 + 30921727T^5 + 868782404T^4 + 2187283077T^3 + 22165303221T^2 - 37540593340*T + 111195571600
$79$	$ T^{12} + \cdots + 105673755625 $ T^12 - 18T^11 + 483T^10 - 2692T^9 + 63241T^8 - 103739T^7 + 7261337T^6 + 8995576T^5 + 357119315T^4 + 661409118T^3 + 12959449374T^2 + 30925034900*T + 105673755625
$83$	$ T^{12} + \cdots + 11607476644 $ T^12 + 18T^11 + 379T^10 + 2502T^9 + 36359T^8 + 206991T^7 + 2396952T^6 + 9925110T^5 + 83492796T^4 + 295511526T^3 + 1998447653T^2 + 4562165610*T + 11607476644
$89$	$ (T^{6} + T^{5} + \cdots + 12490)^{2} $ (T^6 + T^5 - 208T^4 + 118T^3 + 8869T^2 - 25271T + 12490)^2
$97$	$ T^{12} - 8 T^{11} + \cdots + 55696 $ T^12 - 8T^11 + 227T^10 - 1072T^9 + 38333T^8 - 228423T^7 + 1053540T^6 - 2232786T^5 + 3428504T^4 - 2560324T^3 + 1373801T^2 - 325916*T + 55696
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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:	\( N \)	\(=\)	\( 2010 = 2 \cdot 3 \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2010.i (of order \(3\), degree \(2\), minimal)

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

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	2010.2.i.g

Level	$2010$
Weight	$2$
Character orbit	2010.i
Analytic conductor	$16.050$
Analytic rank	$0$
Dimension	$12$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(16.0499308063\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(6\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} - x^{11} + 14 x^{10} + 19 x^{9} + 128 x^{8} + 144 x^{7} + 522 x^{6} + 645 x^{5} + 1466 x^{4} + \cdots + 25 \) x^12 - x^11 + 14x^10 + 19x^9 + 128x^8 + 144x^7 + 522x^6 + 645x^5 + 1466x^4 + 1072x^3 + 1031x^2 - 145x + 25
Coefficient ring:	\(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( ( 2819168 \nu^{11} + 265198160 \nu^{10} - 477773490 \nu^{9} + 3036341904 \nu^{8} + \cdots - 66262408515 ) / 35643585385 \) (2819168v^11 + 265198160v^10 - 477773490v^9 + 3036341904v^8 + 5959276720v^7 + 11385206295v^6 + 24081566144v^5 + 38035565760v^4 + 120293726640v^3 + 19484788432v^2 - 2703405440*v - 66262408515) / 35643585385
\(\beta_{3}\)	\(=\)	\( ( - 33921673 \nu^{11} + 63688670 \nu^{10} - 492954100 \nu^{9} - 243291424 \nu^{8} + \cdots + 5162245150 ) / 35643585385 \) (-33921673v^11 + 63688670v^10 - 492954100v^9 - 243291424v^8 - 3706572240v^7 - 841605200v^6 - 13078196499v^5 - 7186486390v^4 - 38185841100v^3 + 4751171423v^2 - 718061785*v + 5162245150) / 35643585385
\(\beta_{4}\)	\(=\)	\( ( - 39021658 \nu^{11} + 308911751 \nu^{10} + 455970818 \nu^{9} - 641732196 \nu^{8} + \cdots + 10904045450 ) / 35643585385 \) (-39021658v^11 + 308911751v^10 + 455970818v^9 - 641732196v^8 + 21125740804v^7 + 23663803342v^6 + 130545546630v^5 + 90576811580v^4 + 497658288514v^3 + 444758392548v^2 + 786214752751*v + 10904045450) / 35643585385
\(\beta_{5}\)	\(=\)	\( ( 94791175 \nu^{11} - 410626178 \nu^{10} + 1864902053 \nu^{9} - 1914357152 \nu^{8} + \cdots - 85788381500 ) / 35643585385 \) (94791175v^11 - 410626178v^10 + 1864902053v^9 - 1914357152v^8 + 5496983472v^7 - 5073079088v^6 + 16767819549v^5 - 12119261462v^4 - 2806839561v^3 - 30375533585v^2 + 4383131575*v - 85788381500) / 35643585385
\(\beta_{6}\)	\(=\)	\( ( - 206489806 \nu^{11} + 172568133 \nu^{10} - 2827168614 \nu^{9} - 4416260414 \nu^{8} + \cdots + 29222960085 ) / 35643585385 \) (-206489806v^11 + 172568133v^10 - 2827168614v^9 - 4416260414v^8 - 26673986592v^7 - 33441104304v^6 - 108629283932v^5 - 146264121369v^4 - 309900541986v^3 - 259542913132v^2 - 208139818563*v + 29222960085) / 35643585385
\(\beta_{7}\)	\(=\)	\( ( - 438214690 \nu^{11} + 564108370 \nu^{10} - 6874518668 \nu^{9} - 5490449467 \nu^{8} + \cdots + 71345728850 ) / 35643585385 \) (-438214690v^11 + 564108370v^10 - 6874518668v^9 - 5490449467v^8 - 60798887952v^7 - 59858636368v^6 - 244364748363v^5 - 276884025146v^4 - 685391076372v^3 - 554680872781v^2 - 506410838830*v + 71345728850) / 35643585385
\(\beta_{8}\)	\(=\)	\( ( 455587880 \nu^{11} - 253541502 \nu^{10} + 5420063988 \nu^{9} + 12661175037 \nu^{8} + \cdots - 55870601790 ) / 35643585385 \) (455587880v^11 - 253541502v^10 + 5420063988v^9 + 12661175037v^8 + 53310202896v^7 + 75588991248v^6 + 216308780363v^5 + 322545433110v^4 + 630582773772v^3 + 438344851516v^2 + 398871729762*v - 55870601790) / 35643585385
\(\beta_{9}\)	\(=\)	\( ( - 613353486 \nu^{11} + 1406905506 \nu^{10} - 10913994885 \nu^{9} + 1437365197 \nu^{8} + \cdots + 114285727520 ) / 35643585385 \) (-613353486v^11 + 1406905506v^10 - 10913994885v^9 + 1437365197v^8 - 82152276377v^7 - 19102441115v^6 - 340325290628v^5 - 160831627012v^4 - 846924705500v^3 - 317301195189v^2 - 805039653781*v + 114285727520) / 35643585385
\(\beta_{10}\)	\(=\)	\( ( 725477973 \nu^{11} + 748732528 \nu^{10} + 5519743294 \nu^{9} + 38781688065 \nu^{8} + \cdots - 54895060340 ) / 35643585385 \) (725477973v^11 + 748732528v^10 + 5519743294v^9 + 38781688065v^8 + 82593124990v^7 + 226503843354v^6 + 332054561353v^5 + 877409472252v^4 + 1117172383696v^3 + 1264790933665v^2 + 404117634802*v - 54895060340) / 35643585385
\(\beta_{11}\)	\(=\)	\( ( - 790732852 \nu^{11} + 2592244241 \nu^{10} - 13568854921 \nu^{9} + 6679372073 \nu^{8} + \cdots - 81596245665 ) / 35643585385 \) (-790732852v^11 + 2592244241v^10 - 13568854921v^9 + 6679372073v^8 - 63261184149v^7 + 31539977231v^6 - 210699805314v^5 - 14215939216v^4 - 219920015163v^3 + 192156998147v^2 - 28052877490*v - 81596245665) / 35643585385

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( -\beta_{8} + \beta_{7} - 4\beta_{6} - 2\beta_{3} + 2\beta_1 \) -b8 + b7 - 4b6 - 2b3 + 2*b1
\(\nu^{3}\)	\(=\)	\( \beta_{11} - \beta_{9} - 2\beta_{5} - \beta_{4} - 10\beta_{3} - 4\beta_{2} - 5 \) b11 - b9 - 2b5 - b4 - 10b3 - 4*b2 - 5
\(\nu^{4}\)	\(=\)	\( 4 \beta_{11} - 4 \beta_{10} + 18 \beta_{8} - 12 \beta_{7} + 31 \beta_{6} - 12 \beta_{5} - 5 \beta_{4} + \cdots - 31 \) 4b11 - 4b10 + 18b8 - 12b7 + 31b6 - 12b5 - 5b4 - 18b2 - 32*b1 - 31
\(\nu^{5}\)	\(=\)	\( -19\beta_{10} + 22\beta_{9} + 71\beta_{8} - 38\beta_{7} + 84\beta_{6} + 129\beta_{3} - 129\beta_1 \) -19b10 + 22b9 + 71b8 - 38b7 + 84b6 + 129b3 - 129*b1
\(\nu^{6}\)	\(=\)	\( -74\beta_{11} + 90\beta_{9} + 162\beta_{5} + 90\beta_{4} + 470\beta_{3} + 279\beta_{2} + 355 \) -74b11 + 90b9 + 162b5 + 90b4 + 470b3 + 279b2 + 355
\(\nu^{7}\)	\(=\)	\( - 295 \beta_{11} + 295 \beta_{10} - 1075 \beta_{8} + 587 \beta_{7} - 1234 \beta_{6} + 587 \beta_{5} + \cdots + 1234 \) -295b11 + 295b10 - 1075b8 + 587b7 - 1234b6 + 587b5 + 353b4 + 1075b2 + 1810*b1 + 1234
\(\nu^{8}\)	\(=\)	\( 1133\beta_{10} - 1370\beta_{9} - 4120\beta_{8} + 2298\beta_{7} - 4798\beta_{6} - 6811\beta_{3} + 6811\beta_1 \) 1133b10 - 1370b9 - 4120b8 + 2298b7 - 4798b6 - 6811b3 + 6811*b1
\(\nu^{9}\)	\(=\)	\( 4357\beta_{11} - 5253\beta_{9} - 8633\beta_{5} - 5253\beta_{4} - 25971\beta_{3} - 15732\beta_{2} - 17839 \) 4357b11 - 5253b9 - 8633b5 - 5253b4 - 25971b3 - 15732b2 - 17839
\(\nu^{10}\)	\(=\)	\( 16628 \beta_{11} - 16628 \beta_{10} + 59946 \beta_{8} - 33070 \beta_{7} + 68144 \beta_{6} - 33070 \beta_{5} + \cdots - 68144 \) 16628b11 - 16628b10 + 59946b8 - 33070b7 + 68144b6 - 33070b5 - 20089b4 - 59946b2 - 98503*b1 - 68144
\(\nu^{11}\)	\(=\)	\( - 63407 \beta_{10} + 76574 \beta_{9} + 228236 \beta_{8} - 125379 \beta_{7} + 257678 \beta_{6} + \cdots - 374794 \beta_1 \) -63407b10 + 76574b9 + 228236b8 - 125379b7 + 257678b6 + 374794b3 - 374794*b1

\(n\)	\(671\)	\(1141\)	\(1207\)
\(\chi(n)\)	\(1\)	\(-1 + \beta_{6}\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 841.6
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{2} + T + 1)^{6} \) (T^2 + T + 1)^6
$3$	\( (T + 1)^{12} \) (T + 1)^12
$5$	\( (T + 1)^{12} \) (T + 1)^12
$7$	\( T^{12} + T^{11} + \cdots + 160000 \) T^12 + T^11 + 31T^10 + 40T^9 + 728T^8 + 791T^7 + 6480T^6 + 1655T^5 + 36274T^4 + 4085T^3 + 106825T^2 - 62000T + 160000
$11$	\( T^{12} + 2 T^{11} + \cdots + 5640625 \) T^12 + 2T^11 + 54T^10 + 6T^9 + 1861T^8 - 180T^7 + 35009T^6 - 29235T^5 + 444225T^4 - 140000T^3 + 1791875T^2 - 356250*T + 5640625
$13$	\( T^{12} - 6 T^{11} + \cdots + 1562500 \) T^12 - 6T^11 + 62T^10 - 210T^9 + 1752T^8 - 5637T^7 + 29703T^6 - 62910T^5 + 242153T^4 - 482646T^3 + 1278949T^2 - 1428750*T + 1562500
$17$	\( T^{12} - 5 T^{11} + \cdots + 49 \) T^12 - 5T^11 + 71T^10 - 256T^9 + 3388T^8 - 11881T^7 + 55776T^6 - 26122T^5 + 35890T^4 + 4179T^3 + 18088T^2 - 931*T + 49
$19$	\( T^{12} + 9 T^{11} + \cdots + 160000 \) T^12 + 9T^11 + 91T^10 + 234T^9 + 1501T^8 + 1259T^7 + 20029T^6 + 466T^5 + 106979T^4 - 91695T^3 + 465025T^2 - 266000*T + 160000
$23$	\( T^{12} + 8 T^{11} + \cdots + 77880625 \) T^12 + 8T^11 + 115T^10 + 790T^9 + 8389T^8 + 50005T^7 + 297495T^6 + 1026244T^5 + 3550571T^4 + 7066430T^3 + 21180100T^2 + 31064000*T + 77880625
$29$	\( T^{12} + 11 T^{11} + \cdots + 12496225 \) T^12 + 11T^11 + 124T^10 + 753T^9 + 5540T^8 + 28112T^7 + 153968T^6 + 515771T^5 + 1610170T^4 + 2347254T^3 + 4397729T^2 + 1261995*T + 12496225
$31$	\( T^{12} + \cdots + 222367744 \) T^12 + 2T^11 + 95T^10 + 130T^9 + 6287T^8 + 7329T^7 + 199644T^6 + 39414T^5 + 4328336T^4 + 782698T^3 + 39942521T^2 - 35147584*T + 222367744
$37$	\( T^{12} + 93 T^{10} + \cdots + 53728900 \) T^12 + 93T^10 - 130T^9 + 6448T^8 - 9044T^7 + 194258T^6 - 414749T^5 + 4357646T^4 - 5647899T^3 + 25127331T^2 + 21982670T + 53728900
$41$	\( T^{12} + 2 T^{11} + \cdots + 144400 \) T^12 + 2T^11 + 87T^10 - 10T^9 + 5813T^8 + 219T^7 + 111754T^6 - 315618T^5 + 1445858T^4 - 1575584T^3 + 1292769T^2 - 504260*T + 144400
$43$	\( (T^{6} - 8 T^{5} + 2 T^{4} + \cdots + 1)^{2} \) (T^6 - 8T^5 + 2T^4 + 108T^3 - 211T^2 + 7*T + 1)^2
$47$	\( T^{12} + 2 T^{11} + \cdots + 180625 \) T^12 + 2T^11 + 82T^10 + 432T^9 + 6539T^8 + 23045T^7 + 94670T^6 + 60668T^5 + 174169T^4 - 164115T^3 + 472550T^2 - 274125*T + 180625
$53$	\( (T^{6} + 2 T^{5} + \cdots - 96040)^{2} \) (T^6 + 2T^5 - 202T^4 - 245T^3 + 9310T^2 - 343*T - 96040)^2
$59$	\( (T^{6} - 24 T^{5} + \cdots - 6268)^{2} \) (T^6 - 24T^5 + 159T^4 + 195T^3 - 5119T^2 + 13431*T - 6268)^2
$61$	\( T^{12} - 5 T^{11} + \cdots + 702244 \) T^12 - 5T^11 + 186T^10 - 403T^9 + 27088T^8 - 82319T^7 + 646800T^6 - 45788T^5 + 5745947T^4 - 7692369T^3 + 11443211T^2 - 3020990*T + 702244
$67$	\( T^{12} + \cdots + 90458382169 \) T^12 - 3T^11 + 3T^10 + 945T^9 + 1437T^8 + 6042T^7 + 516805T^6 + 404814T^5 + 6450693T^4 + 284221035T^3 + 60453363T^2 - 4050375321*T + 90458382169
$71$	\( T^{12} + \cdots + 510724622500 \) T^12 + 308T^10 + 638T^9 + 66972T^8 + 129945T^7 + 7263197T^6 + 10625340T^5 + 567961531T^4 + 428034456T^3 + 20937464049T^2 - 22649402450T + 510724622500
$73$	\( T^{12} + \cdots + 111195571600 \) T^12 + 6T^11 + 375T^10 + 1016T^9 + 95608T^8 + 287998T^7 + 10632188T^6 + 30921727T^5 + 868782404T^4 + 2187283077T^3 + 22165303221T^2 - 37540593340*T + 111195571600
$79$	\( T^{12} + \cdots + 105673755625 \) T^12 - 18T^11 + 483T^10 - 2692T^9 + 63241T^8 - 103739T^7 + 7261337T^6 + 8995576T^5 + 357119315T^4 + 661409118T^3 + 12959449374T^2 + 30925034900*T + 105673755625
$83$	\( T^{12} + \cdots + 11607476644 \) T^12 + 18T^11 + 379T^10 + 2502T^9 + 36359T^8 + 206991T^7 + 2396952T^6 + 9925110T^5 + 83492796T^4 + 295511526T^3 + 1998447653T^2 + 4562165610*T + 11607476644
$89$	\( (T^{6} + T^{5} + \cdots + 12490)^{2} \) (T^6 + T^5 - 208T^4 + 118T^3 + 8869T^2 - 25271T + 12490)^2
$97$	\( T^{12} - 8 T^{11} + \cdots + 55696 \) T^12 - 8T^11 + 227T^10 - 1072T^9 + 38333T^8 - 228423T^7 + 1053540T^6 - 2232786T^5 + 3428504T^4 - 2560324T^3 + 1373801T^2 - 325916*T + 55696
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