LMFDB - Newform orbit 4005.2.a.w

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4005,2,Mod(1,4005)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4005, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4005.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4005 = 3^{2} \cdot 5 \cdot 89 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4005.a (trivial)

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:	yes
Analytic conductor:	$31.9800860095$
Analytic rank:	$0$
Dimension:	$17$
Coefficient field:	$\mathbb{Q}[x]/(x^{17} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{17} - 5 x^{16} - 15 x^{15} + 105 x^{14} + 45 x^{13} - 849 x^{12} + 320 x^{11} + 3371 x^{10} + \cdots + 24 $ x^17 - 5x^16 - 15x^15 + 105x^14 + 45x^13 - 849x^12 + 320x^11 + 3371x^10 - 2456x^9 - 7002x^8 + 6279x^7 + 7299x^6 - 7119x^5 - 3066x^4 + 3184x^3 + 99x^2 - 231x + 24
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-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 $1,\beta_1,\ldots,\beta_{16}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{17} - 5 x^{16} - 15 x^{15} + 105 x^{14} + 45 x^{13} - 849 x^{12} + 320 x^{11} + 3371 x^{10} + \cdots + 24 $ x^17 - 5*x^16 - 15*x^15 + 105*x^14 + 45*x^13 - 849*x^12 + 320*x^11 + 3371*x^10 - 2456*x^9 - 7002*x^8 + 6279*x^7 + 7299*x^6 - 7119*x^5 - 3066*x^4 + 3184*x^3 + 99*x^2 - 231*x + 24 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ \nu^{2} - 3 $ v^2 - 3
$\beta_{3}$	$=$	$ \nu^{3} - 5\nu - 1 $ v^3 - 5*v - 1
$\beta_{4}$	$=$	$ ( 685 \nu^{16} - 1978 \nu^{15} - 23375 \nu^{14} + 80742 \nu^{13} + 247035 \nu^{12} - 1063752 \nu^{11} + \cdots - 368684 ) / 31022 $ (685v^16 - 1978v^15 - 23375v^14 + 80742v^13 + 247035v^12 - 1063752v^11 - 927300v^10 + 6222061v^9 + 79227v^8 - 17370991v^7 + 6231634v^6 + 22769593v^5 - 11572552v^4 - 11774626v^3 + 6374744v^2 + 982243v - 368684) / 31022
$\beta_{5}$	$=$	$ ( - 1070 \nu^{16} + 1165 \nu^{15} + 40249 \nu^{14} - 75174 \nu^{13} - 484380 \nu^{12} + 1137763 \nu^{11} + \cdots + 300552 ) / 31022 $ (-1070v^16 + 1165v^15 + 40249v^14 - 75174v^13 - 484380v^12 + 1137763v^11 + 2444697v^10 - 7082375v^9 - 4941450v^8 + 20342129v^7 + 1670236v^6 - 26893622v^5 + 5596139v^4 + 13977281v^3 - 4729743v^2 - 1384065v + 300552) / 31022
$\beta_{6}$	$=$	$ ( 6613 \nu^{16} - 17420 \nu^{15} - 180375 \nu^{14} + 520530 \nu^{13} + 1766251 \nu^{12} + \cdots - 1639268 ) / 62044 $ (6613v^16 - 17420v^15 - 180375v^14 + 520530v^13 + 1766251v^12 - 5814854v^11 - 7314342v^10 + 30806921v^9 + 9574065v^8 - 81028001v^7 + 12506146v^6 + 103349077v^5 - 39644890v^4 - 55867828v^3 + 25227292v^2 + 7528791v - 1639268) / 62044
$\beta_{7}$	$=$	$ ( - 7257 \nu^{16} + 30443 \nu^{15} + 137250 \nu^{14} - 693400 \nu^{13} - 837779 \nu^{12} + \cdots + 1063920 ) / 62044 $ (-7257v^16 + 30443v^15 + 137250v^14 - 693400v^13 - 837779v^12 + 6167935v^11 + 943861v^10 - 27194200v^9 + 8863387v^8 + 62567552v^7 - 34862246v^6 - 72621151v^5 + 46524231v^4 + 36780711v^3 - 21561069v^2 - 4918520v + 1063920) / 62044
$\beta_{8}$	$=$	$ ( 6850 \nu^{16} - 35291 \nu^{15} - 94151 \nu^{14} + 714354 \nu^{13} + 174722 \nu^{12} - 5503379 \nu^{11} + \cdots - 739750 ) / 31022 $ (6850v^16 - 35291v^15 - 94151v^14 + 714354v^13 + 174722v^12 - 5503379v^11 + 2779047v^10 + 20604597v^9 - 16766182v^8 - 40485931v^7 + 37622828v^6 + 41160644v^5 - 37720701v^4 - 18925679v^3 + 14593081v^2 + 2423683v - 739750) / 31022
$\beta_{9}$	$=$	$ ( - 14426 \nu^{16} + 61515 \nu^{15} + 278177 \nu^{14} - 1417322 \nu^{13} - 1771986 \nu^{12} + \cdots + 2544040 ) / 62044 $ (-14426v^16 + 61515v^15 + 278177v^14 - 1417322v^13 - 1771986v^12 + 12781181v^11 + 2666987v^10 - 57266037v^9 + 14936458v^8 + 134174293v^7 - 65290780v^6 - 158712488v^5 + 91148769v^4 + 81480431v^3 - 44547125v^2 - 10787091v + 2544040) / 62044
$\beta_{10}$	$=$	$ ( 7535 \nu^{16} - 37269 \nu^{15} - 117526 \nu^{14} + 795096 \nu^{13} + 421757 \nu^{12} + \cdots - 1263544 ) / 31022 $ (7535v^16 - 37269v^15 - 117526v^14 + 795096v^13 + 421757v^12 - 6567131v^11 + 1851747v^10 + 26826658v^9 - 16686955v^8 - 57856922v^7 + 43854462v^6 + 63961259v^5 - 49324275v^4 - 30979503v^3 + 21184979v^2 + 3902278v - 1263544) / 31022
$\beta_{11}$	$=$	$ ( 7535 \nu^{16} - 37269 \nu^{15} - 117526 \nu^{14} + 795096 \nu^{13} + 421757 \nu^{12} + \cdots - 1015368 ) / 31022 $ (7535v^16 - 37269v^15 - 117526v^14 + 795096v^13 + 421757v^12 - 6567131v^11 + 1851747v^10 + 26826658v^9 - 16686955v^8 - 57856922v^7 + 43854462v^6 + 63961259v^5 - 49293253v^4 - 30979503v^3 + 20936803v^2 + 3871256v - 1015368) / 31022
$\beta_{12}$	$=$	$ ( 15238 \nu^{16} - 70517 \nu^{15} - 270335 \nu^{14} + 1582686 \nu^{13} + 1432154 \nu^{12} + \cdots - 3029808 ) / 62044 $ (15238v^16 - 70517v^15 - 270335v^14 + 1582686v^13 + 1432154v^12 - 13900087v^11 + 185359v^10 + 60904047v^9 - 25130162v^8 - 141089899v^7 + 83217368v^6 + 167347268v^5 - 106903155v^4 - 87031309v^3 + 50859751v^2 + 11368909v - 3029808) / 62044
$\beta_{13}$	$=$	$ ( - 5003 \nu^{16} + 23640 \nu^{15} + 78789 \nu^{14} - 491641 \nu^{13} - 318258 \nu^{12} + \cdots + 580367 ) / 15511 $ (-5003v^16 + 23640v^15 + 78789v^14 - 491641v^13 - 318258v^12 + 3931692v^11 - 700684v^10 - 15463187v^9 + 8035982v^8 + 32135471v^7 - 21250233v^6 - 34474069v^5 + 23881305v^4 + 16421556v^3 - 10374066v^2 - 1994666v + 580367) / 15511
$\beta_{14}$	$=$	$ ( - 12523 \nu^{16} + 61545 \nu^{15} + 189010 \nu^{14} - 1274666 \nu^{13} - 638709 \nu^{12} + \cdots + 1508748 ) / 31022 $ (-12523v^16 + 61545v^15 + 189010v^14 - 1274666v^13 - 638709v^12 + 10147647v^11 - 2869597v^10 - 39770336v^9 + 23674113v^8 + 82744596v^7 - 58289788v^6 - 89735141v^5 + 62257615v^4 + 43991657v^3 - 25895951v^2 - 5969520v + 1508748) / 31022
$\beta_{15}$	$=$	$ ( - 16906 \nu^{16} + 80451 \nu^{15} + 276079 \nu^{14} - 1727532 \nu^{13} - 1200628 \nu^{12} + \cdots + 2614408 ) / 31022 $ (-16906v^16 + 80451v^15 + 276079v^14 - 1727532v^13 - 1200628v^12 + 14402921v^11 - 2192535v^10 - 59629455v^9 + 30378002v^8 + 130973989v^7 - 86449694v^6 - 148302258v^5 + 102577437v^4 + 74063549v^3 - 46121451v^2 - 9366361v + 2614408) / 31022
$\beta_{16}$	$=$	$ ( - 38615 \nu^{16} + 177511 \nu^{15} + 642124 \nu^{14} - 3780588 \nu^{13} - 3038785 \nu^{12} + \cdots + 4923384 ) / 62044 $ (-38615v^16 + 177511v^15 + 642124v^14 - 3780588v^13 - 3038785v^12 + 31183239v^11 - 2039975v^10 - 127378426v^9 + 54823177v^8 + 275564722v^7 - 161135054v^6 - 306983937v^5 + 190626927v^4 + 150613483v^3 - 84980589v^2 - 18623890v + 4923384) / 62044

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{2} + 3 $ b2 + 3
$\nu^{3}$	$=$	$ \beta_{3} + 5\beta _1 + 1 $ b3 + 5*b1 + 1
$\nu^{4}$	$=$	$ \beta_{11} - \beta_{10} + 8\beta_{2} + \beta _1 + 16 $ b11 - b10 + 8*b2 + b1 + 16
$\nu^{5}$	$=$	$ \beta_{11} - \beta_{8} - \beta_{4} + 9\beta_{3} + \beta_{2} + 30\beta _1 + 9 $ b11 - b8 - b4 + 9b3 + b2 + 30b1 + 9
$\nu^{6}$	$=$	$ 11 \beta_{11} - 9 \beta_{10} + \beta_{9} - \beta_{8} + \beta_{7} + \beta_{6} - \beta_{4} + 2 \beta_{3} + \cdots + 99 $ 11b11 - 9b10 + b9 - b8 + b7 + b6 - b4 + 2b3 + 57b2 + 11*b1 + 99
$\nu^{7}$	$=$	$ - \beta_{16} + \beta_{14} + \beta_{13} + 14 \beta_{11} - \beta_{10} + \beta_{9} - 11 \beta_{8} + \cdots + 71 $ -b16 + b14 + b13 + 14b11 - b10 + b9 - 11b8 + 2b7 + b6 - 12b4 + 69b3 + 15b2 + 195*b1 + 71
$\nu^{8}$	$=$	$ - 2 \beta_{16} - \beta_{15} + \beta_{14} + 4 \beta_{13} - 2 \beta_{12} + 95 \beta_{11} - 67 \beta_{10} + \cdots + 654 $ -2b16 - b15 + b14 + 4b13 - 2b12 + 95b11 - 67b10 + 12b9 - 13b8 + 16b7 + 14b6 + b5 - 15b4 + 33b3 + 396b2 + 100*b1 + 654
$\nu^{9}$	$=$	$ - 16 \beta_{16} - \beta_{15} + 15 \beta_{14} + 20 \beta_{13} - \beta_{12} + 141 \beta_{11} - 17 \beta_{10} + \cdots + 556 $ -16b16 - b15 + 15b14 + 20b13 - b12 + 141b11 - 17b10 + 16b9 - 89b8 + 35b7 + 19b6 - 110b4 + 510b3 + 160b2 + 1317*b1 + 556
$\nu^{10}$	$=$	$ - 34 \beta_{16} - 21 \beta_{15} + 18 \beta_{14} + 73 \beta_{13} - 38 \beta_{12} + 758 \beta_{11} + \cdots + 4461 $ -34b16 - 21b15 + 18b14 + 73b13 - 38b12 + 758b11 - 475b10 + 108b9 - 119b8 + 182b7 + 146b6 + 16b5 - 160b4 + 374b3 + 2735b2 + 863b1 + 4461
$\nu^{11}$	$=$	$ - 175 \beta_{16} - 25 \beta_{15} + 157 \beta_{14} + 258 \beta_{13} - 29 \beta_{12} + 1254 \beta_{11} + \cdots + 4378 $ -175b16 - 25b15 + 157b14 + 258b13 - 29b12 + 1254b11 - 192b10 + 177b9 - 643b8 + 419b7 + 240b6 - 3b5 - 920b4 + 3738b3 + 1489b2 + 9078b1 + 4378
$\nu^{12}$	$=$	$ - 387 \beta_{16} - 280 \beta_{15} + 209 \beta_{14} + 891 \beta_{13} - 477 \beta_{12} + 5849 \beta_{11} + \cdots + 30985 $ -387b16 - 280b15 + 209b14 + 891b13 - 477b12 + 5849b11 - 3319b10 + 891b9 - 946b8 + 1795b7 + 1366b6 + 166b5 - 1491b4 + 3637b3 + 18897b2 + 7255b1 + 30985
$\nu^{13}$	$=$	$ - 1639 \beta_{16} - 379 \beta_{15} + 1418 \beta_{14} + 2756 \beta_{13} - 465 \beta_{12} + 10502 \beta_{11} + \cdots + 34535 $ -1639b16 - 379b15 + 1418b14 + 2756b13 - 465b12 + 10502b11 - 1842b10 + 1700b9 - 4398b8 + 4260b7 + 2539b6 - 73b5 - 7381b4 + 27380b3 + 12932b2 + 63356b1 + 34535
$\nu^{14}$	$=$	$ - 3726 \beta_{16} - 3056 \beta_{15} + 2014 \beta_{14} + 9160 \beta_{13} - 4993 \beta_{12} + \cdots + 217789 $ -3726b16 - 3056b15 + 2014b14 + 9160b13 - 4993b12 + 44424b11 - 23120b10 + 7152b9 - 6988b8 + 16345b7 + 12081b6 + 1413b5 - 12968b4 + 32623b3 + 130973b2 + 59806b1 + 217789
$\nu^{15}$	$=$	$ - 14164 \beta_{16} - 4567 \beta_{15} + 11851 \beta_{14} + 26556 \beta_{13} - 5729 \beta_{12} + \cdots + 271845 $ -14164b16 - 4567b15 + 11851b14 + 26556b13 - 5729b12 + 85086b11 - 16278b10 + 15288b9 - 29152b8 + 39557b7 + 24381b6 - 1106b5 - 57918b4 + 200835b3 + 107939b2 + 446006b1 + 271845
$\nu^{16}$	$=$	$ - 32872 \beta_{16} - 29872 \beta_{15} + 17602 \beta_{14} + 85786 \beta_{13} - 47253 \beta_{12} + \cdots + 1544374 $ -32872b16 - 29872b15 + 17602b14 + 85786b13 - 47253b12 + 334746b11 - 161278b10 + 57104b9 - 49396b8 + 141417b7 + 103157b6 + 10648b5 - 108361b4 + 278635b3 + 911836b2 + 485008b1 + 1544374

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

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} $

1.1

2.74886
2.71108
2.50383
2.12059
1.89951
1.69575
1.08305
0.829580
0.184653
0.159847
−0.320785
−1.02928
−1.30924
−1.58492
−1.59605
−2.53341
−2.56306

−2.74886

5.55624

−1.00000

4.26511

−9.77563

2.74886

1.2

−2.71108

5.34994

−1.00000

−0.946340

−9.08194

2.71108

1.3

−2.50383

4.26915

−1.00000

−2.88177

−5.68155

2.50383

1.4

−2.12059

2.49688

−1.00000

4.22149

−1.05368

2.12059

1.5

−1.89951

1.60815

−1.00000

2.75204

0.744322

1.89951

1.6

−1.69575

0.875568

−1.00000

−1.30460

1.90676

1.69575

1.7

−1.08305

−0.826999

−1.00000

−2.85554

3.06179

1.08305

1.8

−0.829580

−1.31180

−1.00000

1.06171

2.74740

0.829580

1.9

−0.184653

−1.96590

−1.00000

2.68806

0.732314

0.184653

1.10

−0.159847

−1.97445

−1.00000

−1.46854

0.635302

0.159847

1.11

0.320785

−1.89710

−1.00000

4.94917

−1.25013

−0.320785

1.12

1.02928

−0.940578

−1.00000

−4.61830

−3.02668

−1.02928

1.13

1.30924

−0.285886

−1.00000

4.28141

−2.99278

−1.30924

1.14

1.58492

0.511971

−1.00000

−2.52227

−2.35841

−1.58492

1.15

1.59605

0.547372

−1.00000

1.65189

−2.31847

−1.59605

1.16

2.53341

4.41816

−1.00000

3.58669

6.12619

−2.53341

1.17

2.56306

4.56927

−1.00000

−0.860224

6.58519

−2.56306

Atkin-Lehner signs

$ p $	Sign
$3$	$1$
$5$	$1$
$89$	$-1$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	4005.2.a.w	✓	17
3.b	odd	2	1		4005.2.a.x	yes	17

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
4005.2.a.w	✓	17	1.a	even	1	1	trivial
4005.2.a.x	yes	17	3.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(4005))$:

$ T_{2}^{17} + 5 T_{2}^{16} - 15 T_{2}^{15} - 105 T_{2}^{14} + 45 T_{2}^{13} + 849 T_{2}^{12} + 320 T_{2}^{11} + \cdots - 24 $

$ T_{7}^{17} - 12 T_{7}^{16} - 8 T_{7}^{15} + 580 T_{7}^{14} - 1219 T_{7}^{13} - 10266 T_{7}^{12} + \cdots - 2654208 $

$ T_{11}^{17} + 2 T_{11}^{16} - 108 T_{11}^{15} - 178 T_{11}^{14} + 4744 T_{11}^{13} + 6384 T_{11}^{12} + \cdots - 18210816 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{17} + 5 T^{16} + \cdots - 24 $ T^17 + 5T^16 - 15T^15 - 105T^14 + 45T^13 + 849T^12 + 320T^11 - 3371T^10 - 2456T^9 + 7002T^8 + 6279T^7 - 7299T^6 - 7119T^5 + 3066T^4 + 3184T^3 - 99T^2 - 231T - 24
$3$	$ T^{17} $ T^17
$5$	$ (T + 1)^{17} $ (T + 1)^17
$7$	$ T^{17} - 12 T^{16} + \cdots - 2654208 $ T^17 - 12T^16 - 8T^15 + 580T^14 - 1219T^13 - 10266T^12 + 32862T^11 + 88986T^10 - 355798T^9 - 427474T^8 + 1955568T^7 + 1317270T^6 - 5627911T^5 - 3089632T^4 + 7875408T^3 + 4643712T^2 - 4003776T - 2654208
$11$	$ T^{17} + 2 T^{16} + \cdots - 18210816 $ T^17 + 2T^16 - 108T^15 - 178T^14 + 4744T^13 + 6384T^12 - 109444T^11 - 119712T^10 + 1434080T^9 + 1255896T^8 - 10775536T^7 - 7185504T^6 + 44484096T^5 + 19357824T^4 - 90317824T^3 - 12189696T^2 + 69894144T - 18210816
$13$	$ T^{17} - 8 T^{16} + \cdots - 71951312 $ T^17 - 8T^16 - 102T^15 + 966T^14 + 3199T^13 - 43152T^12 - 16674T^11 + 902622T^10 - 841538T^9 - 9147576T^8 + 14474786T^7 + 42807144T^6 - 75870801T^5 - 101030492T^4 + 151839996T^3 + 128556064T^2 - 97412164T - 71951312
$17$	$ T^{17} + 10 T^{16} + \cdots + 5355936 $ T^17 + 10T^16 - 100T^15 - 1288T^14 + 2043T^13 + 58152T^12 + 59672T^11 - 1180192T^10 - 2855258T^9 + 10667886T^8 + 39713014T^7 - 29589328T^6 - 224464341T^5 - 82109134T^4 + 452905096T^3 + 368094960T^2 - 94609296T + 5355936
$19$	$ T^{17} - 32 T^{16} + \cdots - 4195328 $ T^17 - 32T^16 + 326T^15 - 112T^14 - 19336T^13 + 103104T^12 + 178364T^11 - 2829704T^10 + 4435424T^9 + 23793096T^8 - 82470928T^7 - 9072480T^6 + 354767680T^5 - 448480992T^4 + 45886080T^3 + 122287232T^2 - 383744T - 4195328
$23$	$ T^{17} + 15 T^{16} + \cdots - 49152 $ T^17 + 15T^16 - 42T^15 - 1440T^14 - 2054T^13 + 46044T^12 + 124456T^11 - 569504T^10 - 1936616T^9 + 2074632T^8 + 10394616T^7 + 958304T^6 - 19323232T^5 - 10218688T^4 + 9459712T^3 + 5044224T^2 - 1449984T - 49152
$29$	$ T^{17} + \cdots + 5191403856 $ T^17 + T^16 - 216T^15 - 474T^14 + 18555T^13 + 60893T^12 - 781740T^11 - 3433646T^10 + 15734060T^9 + 94052930T^8 - 102388450T^7 - 1180958002T^6 - 655153031T^5 + 5274003397T^4 + 5255035520T^3 - 9248142648T^2 - 7964856144*T + 5191403856
$31$	$ T^{17} - 18 T^{16} + \cdots + 41579008 $ T^17 - 18T^16 - 28T^15 + 2246T^14 - 9052T^13 - 76980T^12 + 562904T^11 + 480696T^10 - 11455928T^9 + 15204504T^8 + 83999024T^7 - 214365568T^6 - 103278240T^5 + 579812000T^4 - 138542400T^3 - 273964928T^2 + 49672192T + 41579008
$37$	$ T^{17} + \cdots + 4930682024 $ T^17 - 12T^16 - 232T^15 + 3062T^14 + 17663T^13 - 275364T^12 - 572934T^11 + 11843924T^10 + 6310162T^9 - 261315512T^8 + 73525974T^7 + 2815862976T^6 - 2312282213T^5 - 11902932622T^4 + 15930529196T^3 + 8520195944T^2 - 18329650996T + 4930682024
$41$	$ T^{17} - 7 T^{16} + \cdots - 52072656 $ T^17 - 7T^16 - 248T^15 + 1840T^14 + 20335T^13 - 168029T^12 - 653852T^11 + 6966698T^10 + 4703988T^9 - 134163678T^8 + 139033842T^7 + 1042239484T^6 - 2209401889T^5 - 1864491877T^4 + 8189203360T^3 - 6215330520T^2 + 1069161552T - 52072656
$43$	$ T^{17} + \cdots + 34255008128 $ T^17 - 28T^16 + 110T^15 + 3360T^14 - 30299T^13 - 102146T^12 + 1774024T^11 - 241632T^10 - 47109682T^9 + 61808886T^8 + 658206106T^7 - 1126902380T^6 - 4944027251T^5 + 8428311974T^4 + 18635647440T^3 - 27537055392T^2 - 26194530496T + 34255008128
$47$	$ T^{17} + 26 T^{16} + \cdots - 192 $ T^17 + 26T^16 + 8T^15 - 3886T^14 - 12859T^13 + 201060T^12 + 535322T^11 - 4192016T^10 - 519394T^9 + 18044272T^8 - 664522T^7 - 26914298T^6 - 6149211T^5 + 11255676T^4 + 3843400T^3 - 1105632T^2 - 414384T - 192
$53$	$ T^{17} + \cdots - 97868430528 $ T^17 + 12T^16 - 202T^15 - 2458T^14 + 18179T^13 + 207368T^12 - 991632T^11 - 9238724T^10 + 36310366T^9 + 228113406T^8 - 880380834T^7 - 2886790108T^6 + 12729117507T^5 + 12324223452T^4 - 86949280456T^3 + 46323171264T^2 + 116954330928T - 97868430528
$59$	$ T^{17} + \cdots - 1190469383604 $ T^17 - 23T^16 - 270T^15 + 9208T^14 + 1597T^13 - 1210843T^12 + 3162618T^11 + 75309288T^10 - 264290570T^9 - 2461239654T^8 + 9036172008T^7 + 41546549564T^6 - 147125431325T^5 - 319274868069T^4 + 1037990637756T^3 + 808664016060T^2 - 1854573704244T - 1190469383604
$61$	$ T^{17} + \cdots + 290308096 $ T^17 - 26T^16 + 6T^15 + 5028T^14 - 34216T^13 - 249120T^12 + 3068644T^11 + 159088T^10 - 93316208T^9 + 205890376T^8 + 1064736432T^7 - 3824881408T^6 - 2837809280T^5 + 19286765696T^4 - 7633794304T^3 - 12917957632T^2 + 7050776576T + 290308096
$67$	$ T^{17} + \cdots - 469614592 $ T^17 - 31T^16 + 156T^15 + 3366T^14 - 27910T^13 - 155676T^12 + 1443964T^11 + 4595260T^10 - 32863320T^9 - 92761992T^8 + 305291880T^7 + 967782784T^6 - 514774432T^5 - 2569171776T^4 + 57172992T^3 + 2000680960T^2 + 65413120T - 469614592
$71$	$ T^{17} + \cdots - 2580234829824 $ T^17 - 2T^16 - 662T^15 + 226T^14 + 170388T^13 + 207444T^12 - 21480544T^11 - 55323296T^10 + 1362260056T^9 + 5137002952T^8 - 38751093632T^7 - 195615135072T^6 + 295302069952T^5 + 2541681297408T^4 + 1675031751680T^3 - 7231064782848T^2 - 9773465456640T - 2580234829824
$73$	$ T^{17} + \cdots - 41984050383872 $ T^17 - 33T^16 + 48T^15 + 9774T^14 - 99766T^13 - 743168T^12 + 16231716T^11 - 28332792T^10 - 939580960T^9 + 5803352264T^8 + 13211693784T^7 - 229956071360T^6 + 508653096928T^5 + 2310372980160T^4 - 13107194664320T^3 + 16740541011456T^2 + 17585525842432T - 41984050383872
$79$	$ T^{17} + \cdots + 10\!\cdots\!72 $ T^17 - 33T^16 - 278T^15 + 19256T^14 - 66253T^13 - 3962971T^12 + 30506894T^11 + 364405310T^10 - 3979839872T^9 - 14949975008T^8 + 243719181574T^7 + 139713425548T^6 - 7479221115125T^5 + 7801243826443T^4 + 105121159886960T^3 - 206257417829408T^2 - 459490391302656T + 1099147233686272
$83$	$ T^{17} + \cdots + 23\!\cdots\!96 $ T^17 + 13T^16 - 880T^15 - 12244T^14 + 297610T^13 + 4629468T^12 - 46957716T^11 - 889477856T^10 + 3034696408T^9 + 90566741720T^8 + 23229526024T^7 - 4635939126272T^6 - 10518115349344T^5 + 103976399881920T^4 + 324484586822656T^3 - 890695676525568T^2 - 2588669415512064T + 2390031586308096
$89$	$ (T - 1)^{17} $ (T - 1)^17
$97$	$ T^{17} + \cdots + 903930927182848 $ T^17 - 45T^16 - 56T^15 + 30660T^14 - 353614T^13 - 6031236T^12 + 133342984T^11 + 14835184T^10 - 17237774728T^9 + 106943873008T^8 + 624035113160T^7 - 9150212322528T^6 + 23190093587424T^5 + 133147624860480T^4 - 996552372477056T^3 + 2494434047964160T^2 - 2640499211331072T + 903930927182848
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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:	\( N \)	\(=\)	\( 4005 = 3^{2} \cdot 5 \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4005.a (trivial)

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$-expansion

Embeddings

Atkin-Lehner signs

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	4005.2.a.w

Level	$4005$
Weight	$2$
Character orbit	4005.a
Self dual	yes
Analytic conductor	$31.980$
Analytic rank	$0$
Dimension	$17$
CM	no
Inner twists	$1$

Self dual:	yes
Analytic conductor:	\(31.9800860095\)
Analytic rank:	\(0\)
Dimension:	\(17\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{17} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{17} - 5 x^{16} - 15 x^{15} + 105 x^{14} + 45 x^{13} - 849 x^{12} + 320 x^{11} + 3371 x^{10} + \cdots + 24 \) x^17 - 5x^16 - 15x^15 + 105x^14 + 45x^13 - 849x^12 + 320x^11 + 3371x^10 - 2456x^9 - 7002x^8 + 6279x^7 + 7299x^6 - 7119x^5 - 3066x^4 + 3184x^3 + 99x^2 - 231x + 24
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	yes
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( \nu^{2} - 3 \) v^2 - 3
\(\beta_{3}\)	\(=\)	\( \nu^{3} - 5\nu - 1 \) v^3 - 5*v - 1
\(\beta_{4}\)	\(=\)	\( ( 685 \nu^{16} - 1978 \nu^{15} - 23375 \nu^{14} + 80742 \nu^{13} + 247035 \nu^{12} - 1063752 \nu^{11} + \cdots - 368684 ) / 31022 \) (685v^16 - 1978v^15 - 23375v^14 + 80742v^13 + 247035v^12 - 1063752v^11 - 927300v^10 + 6222061v^9 + 79227v^8 - 17370991v^7 + 6231634v^6 + 22769593v^5 - 11572552v^4 - 11774626v^3 + 6374744v^2 + 982243v - 368684) / 31022
\(\beta_{5}\)	\(=\)	\( ( - 1070 \nu^{16} + 1165 \nu^{15} + 40249 \nu^{14} - 75174 \nu^{13} - 484380 \nu^{12} + 1137763 \nu^{11} + \cdots + 300552 ) / 31022 \) (-1070v^16 + 1165v^15 + 40249v^14 - 75174v^13 - 484380v^12 + 1137763v^11 + 2444697v^10 - 7082375v^9 - 4941450v^8 + 20342129v^7 + 1670236v^6 - 26893622v^5 + 5596139v^4 + 13977281v^3 - 4729743v^2 - 1384065v + 300552) / 31022
\(\beta_{6}\)	\(=\)	\( ( 6613 \nu^{16} - 17420 \nu^{15} - 180375 \nu^{14} + 520530 \nu^{13} + 1766251 \nu^{12} + \cdots - 1639268 ) / 62044 \) (6613v^16 - 17420v^15 - 180375v^14 + 520530v^13 + 1766251v^12 - 5814854v^11 - 7314342v^10 + 30806921v^9 + 9574065v^8 - 81028001v^7 + 12506146v^6 + 103349077v^5 - 39644890v^4 - 55867828v^3 + 25227292v^2 + 7528791v - 1639268) / 62044
\(\beta_{7}\)	\(=\)	\( ( - 7257 \nu^{16} + 30443 \nu^{15} + 137250 \nu^{14} - 693400 \nu^{13} - 837779 \nu^{12} + \cdots + 1063920 ) / 62044 \) (-7257v^16 + 30443v^15 + 137250v^14 - 693400v^13 - 837779v^12 + 6167935v^11 + 943861v^10 - 27194200v^9 + 8863387v^8 + 62567552v^7 - 34862246v^6 - 72621151v^5 + 46524231v^4 + 36780711v^3 - 21561069v^2 - 4918520v + 1063920) / 62044
\(\beta_{8}\)	\(=\)	\( ( 6850 \nu^{16} - 35291 \nu^{15} - 94151 \nu^{14} + 714354 \nu^{13} + 174722 \nu^{12} - 5503379 \nu^{11} + \cdots - 739750 ) / 31022 \) (6850v^16 - 35291v^15 - 94151v^14 + 714354v^13 + 174722v^12 - 5503379v^11 + 2779047v^10 + 20604597v^9 - 16766182v^8 - 40485931v^7 + 37622828v^6 + 41160644v^5 - 37720701v^4 - 18925679v^3 + 14593081v^2 + 2423683v - 739750) / 31022
\(\beta_{9}\)	\(=\)	\( ( - 14426 \nu^{16} + 61515 \nu^{15} + 278177 \nu^{14} - 1417322 \nu^{13} - 1771986 \nu^{12} + \cdots + 2544040 ) / 62044 \) (-14426v^16 + 61515v^15 + 278177v^14 - 1417322v^13 - 1771986v^12 + 12781181v^11 + 2666987v^10 - 57266037v^9 + 14936458v^8 + 134174293v^7 - 65290780v^6 - 158712488v^5 + 91148769v^4 + 81480431v^3 - 44547125v^2 - 10787091v + 2544040) / 62044
\(\beta_{10}\)	\(=\)	\( ( 7535 \nu^{16} - 37269 \nu^{15} - 117526 \nu^{14} + 795096 \nu^{13} + 421757 \nu^{12} + \cdots - 1263544 ) / 31022 \) (7535v^16 - 37269v^15 - 117526v^14 + 795096v^13 + 421757v^12 - 6567131v^11 + 1851747v^10 + 26826658v^9 - 16686955v^8 - 57856922v^7 + 43854462v^6 + 63961259v^5 - 49324275v^4 - 30979503v^3 + 21184979v^2 + 3902278v - 1263544) / 31022
\(\beta_{11}\)	\(=\)	\( ( 7535 \nu^{16} - 37269 \nu^{15} - 117526 \nu^{14} + 795096 \nu^{13} + 421757 \nu^{12} + \cdots - 1015368 ) / 31022 \) (7535v^16 - 37269v^15 - 117526v^14 + 795096v^13 + 421757v^12 - 6567131v^11 + 1851747v^10 + 26826658v^9 - 16686955v^8 - 57856922v^7 + 43854462v^6 + 63961259v^5 - 49293253v^4 - 30979503v^3 + 20936803v^2 + 3871256v - 1015368) / 31022
\(\beta_{12}\)	\(=\)	\( ( 15238 \nu^{16} - 70517 \nu^{15} - 270335 \nu^{14} + 1582686 \nu^{13} + 1432154 \nu^{12} + \cdots - 3029808 ) / 62044 \) (15238v^16 - 70517v^15 - 270335v^14 + 1582686v^13 + 1432154v^12 - 13900087v^11 + 185359v^10 + 60904047v^9 - 25130162v^8 - 141089899v^7 + 83217368v^6 + 167347268v^5 - 106903155v^4 - 87031309v^3 + 50859751v^2 + 11368909v - 3029808) / 62044
\(\beta_{13}\)	\(=\)	\( ( - 5003 \nu^{16} + 23640 \nu^{15} + 78789 \nu^{14} - 491641 \nu^{13} - 318258 \nu^{12} + \cdots + 580367 ) / 15511 \) (-5003v^16 + 23640v^15 + 78789v^14 - 491641v^13 - 318258v^12 + 3931692v^11 - 700684v^10 - 15463187v^9 + 8035982v^8 + 32135471v^7 - 21250233v^6 - 34474069v^5 + 23881305v^4 + 16421556v^3 - 10374066v^2 - 1994666v + 580367) / 15511
\(\beta_{14}\)	\(=\)	\( ( - 12523 \nu^{16} + 61545 \nu^{15} + 189010 \nu^{14} - 1274666 \nu^{13} - 638709 \nu^{12} + \cdots + 1508748 ) / 31022 \) (-12523v^16 + 61545v^15 + 189010v^14 - 1274666v^13 - 638709v^12 + 10147647v^11 - 2869597v^10 - 39770336v^9 + 23674113v^8 + 82744596v^7 - 58289788v^6 - 89735141v^5 + 62257615v^4 + 43991657v^3 - 25895951v^2 - 5969520v + 1508748) / 31022
\(\beta_{15}\)	\(=\)	\( ( - 16906 \nu^{16} + 80451 \nu^{15} + 276079 \nu^{14} - 1727532 \nu^{13} - 1200628 \nu^{12} + \cdots + 2614408 ) / 31022 \) (-16906v^16 + 80451v^15 + 276079v^14 - 1727532v^13 - 1200628v^12 + 14402921v^11 - 2192535v^10 - 59629455v^9 + 30378002v^8 + 130973989v^7 - 86449694v^6 - 148302258v^5 + 102577437v^4 + 74063549v^3 - 46121451v^2 - 9366361v + 2614408) / 31022
\(\beta_{16}\)	\(=\)	\( ( - 38615 \nu^{16} + 177511 \nu^{15} + 642124 \nu^{14} - 3780588 \nu^{13} - 3038785 \nu^{12} + \cdots + 4923384 ) / 62044 \) (-38615v^16 + 177511v^15 + 642124v^14 - 3780588v^13 - 3038785v^12 + 31183239v^11 - 2039975v^10 - 127378426v^9 + 54823177v^8 + 275564722v^7 - 161135054v^6 - 306983937v^5 + 190626927v^4 + 150613483v^3 - 84980589v^2 - 18623890v + 4923384) / 62044

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( \beta_{2} + 3 \) b2 + 3
\(\nu^{3}\)	\(=\)	\( \beta_{3} + 5\beta _1 + 1 \) b3 + 5*b1 + 1
\(\nu^{4}\)	\(=\)	\( \beta_{11} - \beta_{10} + 8\beta_{2} + \beta _1 + 16 \) b11 - b10 + 8*b2 + b1 + 16
\(\nu^{5}\)	\(=\)	\( \beta_{11} - \beta_{8} - \beta_{4} + 9\beta_{3} + \beta_{2} + 30\beta _1 + 9 \) b11 - b8 - b4 + 9b3 + b2 + 30b1 + 9
\(\nu^{6}\)	\(=\)	\( 11 \beta_{11} - 9 \beta_{10} + \beta_{9} - \beta_{8} + \beta_{7} + \beta_{6} - \beta_{4} + 2 \beta_{3} + \cdots + 99 \) 11b11 - 9b10 + b9 - b8 + b7 + b6 - b4 + 2b3 + 57b2 + 11*b1 + 99
\(\nu^{7}\)	\(=\)	\( - \beta_{16} + \beta_{14} + \beta_{13} + 14 \beta_{11} - \beta_{10} + \beta_{9} - 11 \beta_{8} + \cdots + 71 \) -b16 + b14 + b13 + 14b11 - b10 + b9 - 11b8 + 2b7 + b6 - 12b4 + 69b3 + 15b2 + 195*b1 + 71
\(\nu^{8}\)	\(=\)	\( - 2 \beta_{16} - \beta_{15} + \beta_{14} + 4 \beta_{13} - 2 \beta_{12} + 95 \beta_{11} - 67 \beta_{10} + \cdots + 654 \) -2b16 - b15 + b14 + 4b13 - 2b12 + 95b11 - 67b10 + 12b9 - 13b8 + 16b7 + 14b6 + b5 - 15b4 + 33b3 + 396b2 + 100*b1 + 654
\(\nu^{9}\)	\(=\)	\( - 16 \beta_{16} - \beta_{15} + 15 \beta_{14} + 20 \beta_{13} - \beta_{12} + 141 \beta_{11} - 17 \beta_{10} + \cdots + 556 \) -16b16 - b15 + 15b14 + 20b13 - b12 + 141b11 - 17b10 + 16b9 - 89b8 + 35b7 + 19b6 - 110b4 + 510b3 + 160b2 + 1317*b1 + 556
\(\nu^{10}\)	\(=\)	\( - 34 \beta_{16} - 21 \beta_{15} + 18 \beta_{14} + 73 \beta_{13} - 38 \beta_{12} + 758 \beta_{11} + \cdots + 4461 \) -34b16 - 21b15 + 18b14 + 73b13 - 38b12 + 758b11 - 475b10 + 108b9 - 119b8 + 182b7 + 146b6 + 16b5 - 160b4 + 374b3 + 2735b2 + 863b1 + 4461
\(\nu^{11}\)	\(=\)	\( - 175 \beta_{16} - 25 \beta_{15} + 157 \beta_{14} + 258 \beta_{13} - 29 \beta_{12} + 1254 \beta_{11} + \cdots + 4378 \) -175b16 - 25b15 + 157b14 + 258b13 - 29b12 + 1254b11 - 192b10 + 177b9 - 643b8 + 419b7 + 240b6 - 3b5 - 920b4 + 3738b3 + 1489b2 + 9078b1 + 4378
\(\nu^{12}\)	\(=\)	\( - 387 \beta_{16} - 280 \beta_{15} + 209 \beta_{14} + 891 \beta_{13} - 477 \beta_{12} + 5849 \beta_{11} + \cdots + 30985 \) -387b16 - 280b15 + 209b14 + 891b13 - 477b12 + 5849b11 - 3319b10 + 891b9 - 946b8 + 1795b7 + 1366b6 + 166b5 - 1491b4 + 3637b3 + 18897b2 + 7255b1 + 30985
\(\nu^{13}\)	\(=\)	\( - 1639 \beta_{16} - 379 \beta_{15} + 1418 \beta_{14} + 2756 \beta_{13} - 465 \beta_{12} + 10502 \beta_{11} + \cdots + 34535 \) -1639b16 - 379b15 + 1418b14 + 2756b13 - 465b12 + 10502b11 - 1842b10 + 1700b9 - 4398b8 + 4260b7 + 2539b6 - 73b5 - 7381b4 + 27380b3 + 12932b2 + 63356b1 + 34535
\(\nu^{14}\)	\(=\)	\( - 3726 \beta_{16} - 3056 \beta_{15} + 2014 \beta_{14} + 9160 \beta_{13} - 4993 \beta_{12} + \cdots + 217789 \) -3726b16 - 3056b15 + 2014b14 + 9160b13 - 4993b12 + 44424b11 - 23120b10 + 7152b9 - 6988b8 + 16345b7 + 12081b6 + 1413b5 - 12968b4 + 32623b3 + 130973b2 + 59806b1 + 217789
\(\nu^{15}\)	\(=\)	\( - 14164 \beta_{16} - 4567 \beta_{15} + 11851 \beta_{14} + 26556 \beta_{13} - 5729 \beta_{12} + \cdots + 271845 \) -14164b16 - 4567b15 + 11851b14 + 26556b13 - 5729b12 + 85086b11 - 16278b10 + 15288b9 - 29152b8 + 39557b7 + 24381b6 - 1106b5 - 57918b4 + 200835b3 + 107939b2 + 446006b1 + 271845
\(\nu^{16}\)	\(=\)	\( - 32872 \beta_{16} - 29872 \beta_{15} + 17602 \beta_{14} + 85786 \beta_{13} - 47253 \beta_{12} + \cdots + 1544374 \) -32872b16 - 29872b15 + 17602b14 + 85786b13 - 47253b12 + 334746b11 - 161278b10 + 57104b9 - 49396b8 + 141417b7 + 103157b6 + 10648b5 - 108361b4 + 278635b3 + 911836b2 + 485008b1 + 1544374

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1.17
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{17} + 5 T^{16} + \cdots - 24 \) T^17 + 5T^16 - 15T^15 - 105T^14 + 45T^13 + 849T^12 + 320T^11 - 3371T^10 - 2456T^9 + 7002T^8 + 6279T^7 - 7299T^6 - 7119T^5 + 3066T^4 + 3184T^3 - 99T^2 - 231T - 24
$3$	\( T^{17} \) T^17
$5$	\( (T + 1)^{17} \) (T + 1)^17
$7$	\( T^{17} - 12 T^{16} + \cdots - 2654208 \) T^17 - 12T^16 - 8T^15 + 580T^14 - 1219T^13 - 10266T^12 + 32862T^11 + 88986T^10 - 355798T^9 - 427474T^8 + 1955568T^7 + 1317270T^6 - 5627911T^5 - 3089632T^4 + 7875408T^3 + 4643712T^2 - 4003776T - 2654208
$11$	\( T^{17} + 2 T^{16} + \cdots - 18210816 \) T^17 + 2T^16 - 108T^15 - 178T^14 + 4744T^13 + 6384T^12 - 109444T^11 - 119712T^10 + 1434080T^9 + 1255896T^8 - 10775536T^7 - 7185504T^6 + 44484096T^5 + 19357824T^4 - 90317824T^3 - 12189696T^2 + 69894144T - 18210816
$13$	\( T^{17} - 8 T^{16} + \cdots - 71951312 \) T^17 - 8T^16 - 102T^15 + 966T^14 + 3199T^13 - 43152T^12 - 16674T^11 + 902622T^10 - 841538T^9 - 9147576T^8 + 14474786T^7 + 42807144T^6 - 75870801T^5 - 101030492T^4 + 151839996T^3 + 128556064T^2 - 97412164T - 71951312
$17$	\( T^{17} + 10 T^{16} + \cdots + 5355936 \) T^17 + 10T^16 - 100T^15 - 1288T^14 + 2043T^13 + 58152T^12 + 59672T^11 - 1180192T^10 - 2855258T^9 + 10667886T^8 + 39713014T^7 - 29589328T^6 - 224464341T^5 - 82109134T^4 + 452905096T^3 + 368094960T^2 - 94609296T + 5355936
$19$	\( T^{17} - 32 T^{16} + \cdots - 4195328 \) T^17 - 32T^16 + 326T^15 - 112T^14 - 19336T^13 + 103104T^12 + 178364T^11 - 2829704T^10 + 4435424T^9 + 23793096T^8 - 82470928T^7 - 9072480T^6 + 354767680T^5 - 448480992T^4 + 45886080T^3 + 122287232T^2 - 383744T - 4195328
$23$	\( T^{17} + 15 T^{16} + \cdots - 49152 \) T^17 + 15T^16 - 42T^15 - 1440T^14 - 2054T^13 + 46044T^12 + 124456T^11 - 569504T^10 - 1936616T^9 + 2074632T^8 + 10394616T^7 + 958304T^6 - 19323232T^5 - 10218688T^4 + 9459712T^3 + 5044224T^2 - 1449984T - 49152
$29$	\( T^{17} + \cdots + 5191403856 \) T^17 + T^16 - 216T^15 - 474T^14 + 18555T^13 + 60893T^12 - 781740T^11 - 3433646T^10 + 15734060T^9 + 94052930T^8 - 102388450T^7 - 1180958002T^6 - 655153031T^5 + 5274003397T^4 + 5255035520T^3 - 9248142648T^2 - 7964856144*T + 5191403856
$31$	\( T^{17} - 18 T^{16} + \cdots + 41579008 \) T^17 - 18T^16 - 28T^15 + 2246T^14 - 9052T^13 - 76980T^12 + 562904T^11 + 480696T^10 - 11455928T^9 + 15204504T^8 + 83999024T^7 - 214365568T^6 - 103278240T^5 + 579812000T^4 - 138542400T^3 - 273964928T^2 + 49672192T + 41579008
$37$	\( T^{17} + \cdots + 4930682024 \) T^17 - 12T^16 - 232T^15 + 3062T^14 + 17663T^13 - 275364T^12 - 572934T^11 + 11843924T^10 + 6310162T^9 - 261315512T^8 + 73525974T^7 + 2815862976T^6 - 2312282213T^5 - 11902932622T^4 + 15930529196T^3 + 8520195944T^2 - 18329650996T + 4930682024
$41$	\( T^{17} - 7 T^{16} + \cdots - 52072656 \) T^17 - 7T^16 - 248T^15 + 1840T^14 + 20335T^13 - 168029T^12 - 653852T^11 + 6966698T^10 + 4703988T^9 - 134163678T^8 + 139033842T^7 + 1042239484T^6 - 2209401889T^5 - 1864491877T^4 + 8189203360T^3 - 6215330520T^2 + 1069161552T - 52072656
$43$	\( T^{17} + \cdots + 34255008128 \) T^17 - 28T^16 + 110T^15 + 3360T^14 - 30299T^13 - 102146T^12 + 1774024T^11 - 241632T^10 - 47109682T^9 + 61808886T^8 + 658206106T^7 - 1126902380T^6 - 4944027251T^5 + 8428311974T^4 + 18635647440T^3 - 27537055392T^2 - 26194530496T + 34255008128
$47$	\( T^{17} + 26 T^{16} + \cdots - 192 \) T^17 + 26T^16 + 8T^15 - 3886T^14 - 12859T^13 + 201060T^12 + 535322T^11 - 4192016T^10 - 519394T^9 + 18044272T^8 - 664522T^7 - 26914298T^6 - 6149211T^5 + 11255676T^4 + 3843400T^3 - 1105632T^2 - 414384T - 192
$53$	\( T^{17} + \cdots - 97868430528 \) T^17 + 12T^16 - 202T^15 - 2458T^14 + 18179T^13 + 207368T^12 - 991632T^11 - 9238724T^10 + 36310366T^9 + 228113406T^8 - 880380834T^7 - 2886790108T^6 + 12729117507T^5 + 12324223452T^4 - 86949280456T^3 + 46323171264T^2 + 116954330928T - 97868430528
$59$	\( T^{17} + \cdots - 1190469383604 \) T^17 - 23T^16 - 270T^15 + 9208T^14 + 1597T^13 - 1210843T^12 + 3162618T^11 + 75309288T^10 - 264290570T^9 - 2461239654T^8 + 9036172008T^7 + 41546549564T^6 - 147125431325T^5 - 319274868069T^4 + 1037990637756T^3 + 808664016060T^2 - 1854573704244T - 1190469383604
$61$	\( T^{17} + \cdots + 290308096 \) T^17 - 26T^16 + 6T^15 + 5028T^14 - 34216T^13 - 249120T^12 + 3068644T^11 + 159088T^10 - 93316208T^9 + 205890376T^8 + 1064736432T^7 - 3824881408T^6 - 2837809280T^5 + 19286765696T^4 - 7633794304T^3 - 12917957632T^2 + 7050776576T + 290308096
$67$	\( T^{17} + \cdots - 469614592 \) T^17 - 31T^16 + 156T^15 + 3366T^14 - 27910T^13 - 155676T^12 + 1443964T^11 + 4595260T^10 - 32863320T^9 - 92761992T^8 + 305291880T^7 + 967782784T^6 - 514774432T^5 - 2569171776T^4 + 57172992T^3 + 2000680960T^2 + 65413120T - 469614592
$71$	\( T^{17} + \cdots - 2580234829824 \) T^17 - 2T^16 - 662T^15 + 226T^14 + 170388T^13 + 207444T^12 - 21480544T^11 - 55323296T^10 + 1362260056T^9 + 5137002952T^8 - 38751093632T^7 - 195615135072T^6 + 295302069952T^5 + 2541681297408T^4 + 1675031751680T^3 - 7231064782848T^2 - 9773465456640T - 2580234829824
$73$	\( T^{17} + \cdots - 41984050383872 \) T^17 - 33T^16 + 48T^15 + 9774T^14 - 99766T^13 - 743168T^12 + 16231716T^11 - 28332792T^10 - 939580960T^9 + 5803352264T^8 + 13211693784T^7 - 229956071360T^6 + 508653096928T^5 + 2310372980160T^4 - 13107194664320T^3 + 16740541011456T^2 + 17585525842432T - 41984050383872
$79$	\( T^{17} + \cdots + 10\!\cdots\!72 \) T^17 - 33T^16 - 278T^15 + 19256T^14 - 66253T^13 - 3962971T^12 + 30506894T^11 + 364405310T^10 - 3979839872T^9 - 14949975008T^8 + 243719181574T^7 + 139713425548T^6 - 7479221115125T^5 + 7801243826443T^4 + 105121159886960T^3 - 206257417829408T^2 - 459490391302656T + 1099147233686272
$83$	\( T^{17} + \cdots + 23\!\cdots\!96 \) T^17 + 13T^16 - 880T^15 - 12244T^14 + 297610T^13 + 4629468T^12 - 46957716T^11 - 889477856T^10 + 3034696408T^9 + 90566741720T^8 + 23229526024T^7 - 4635939126272T^6 - 10518115349344T^5 + 103976399881920T^4 + 324484586822656T^3 - 890695676525568T^2 - 2588669415512064T + 2390031586308096
$89$	\( (T - 1)^{17} \) (T - 1)^17
$97$	\( T^{17} + \cdots + 903930927182848 \) T^17 - 45T^16 - 56T^15 + 30660T^14 - 353614T^13 - 6031236T^12 + 133342984T^11 + 14835184T^10 - 17237774728T^9 + 106943873008T^8 + 624035113160T^7 - 9150212322528T^6 + 23190093587424T^5 + 133147624860480T^4 - 996552372477056T^3 + 2494434047964160T^2 - 2640499211331072T + 903930927182848
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\( p \)	Sign
\(3\)	\(1\)
\(5\)	\(1\)
\(89\)	\(-1\)