LMFDB - Newform orbit 833.2.j.d

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [833,2,Mod(67,833)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(833, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4, 3]))

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

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

chi := DirichletCharacter("833.67");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 833 = 7^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	833.j (of order $6$, degree $2$, not 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:	$6.65153848837$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$10$ over $\Q(\zeta_{6})$
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} - 12 x^{18} + 100 x^{16} - 416 x^{14} + 1248 x^{12} - 2081 x^{10} + 2420 x^{8} - 808 x^{6} + \cdots + 1 $ x^20 - 12x^18 + 100x^16 - 416x^14 + 1248x^12 - 2081x^10 + 2420x^8 - 808x^6 + 200x^4 - 16*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 119)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{20} - 12 x^{18} + 100 x^{16} - 416 x^{14} + 1248 x^{12} - 2081 x^{10} + 2420 x^{8} - 808 x^{6} + \cdots + 1 $ x^20 - 12*x^18 + 100*x^16 - 416*x^14 + 1248*x^12 - 2081*x^10 + 2420*x^8 - 808*x^6 + 200*x^4 - 16*x^2 + 1 :

$\beta_{1}$	$=$	$ ( 6039118451 \nu^{18} - 80813320476 \nu^{16} + 702193491548 \nu^{14} - 3322647400063 \nu^{12} + \cdots - 200301046652 ) / 272579072857 $ (6039118451v^18 - 80813320476v^16 + 702193491548v^14 - 3322647400063v^12 + 10803153835007v^10 - 22055779327087v^8 + 29144769089092v^6 - 19990788348982v^4 + 2497620480247*v^2 - 200301046652) / 272579072857
$\beta_{2}$	$=$	$ ( - 112298 \nu^{18} + 1378153 \nu^{16} - 11593224 \nu^{14} + 49712554 \nu^{12} - 152305402 \nu^{10} + \cdots + 2180136 ) / 4341883 $ (-112298v^18 + 1378153v^16 - 11593224v^14 + 49712554v^12 - 152305402v^10 + 268778066v^8 - 325490296v^6 + 145395424v^4 - 27228986*v^2 + 2180136) / 4341883
$\beta_{3}$	$=$	$ ( 8744563018 \nu^{18} - 102597652160 \nu^{16} + 845961433740 \nu^{14} - 3399224292638 \nu^{12} + \cdots + 81045928818 ) / 272579072857 $ (8744563018v^18 - 102597652160v^16 + 845961433740v^14 - 3399224292638v^12 + 9905052601520v^10 - 15168187038240v^8 + 16085794153571v^6 - 1346437785740v^4 + 108228011120*v^2 + 81045928818) / 272579072857
$\beta_{4}$	$=$	$ ( - 10697152006 \nu^{18} + 125411002048 \nu^{16} - 1034067240972 \nu^{14} + \cdots - 602508290000 ) / 272579072857 $ (-10697152006v^18 + 125411002048v^16 - 1034067240972v^14 + 4152521018519v^12 - 12107514606256v^10 + 18540946071072v^8 - 19753535979179v^6 + 1645828226572v^4 - 132293313136*v^2 - 602508290000) / 272579072857
$\beta_{5}$	$=$	$ ( 3385112076 \nu^{18} - 40342403628 \nu^{16} + 335252157616 \nu^{14} - 1381334365440 \nu^{12} + \cdots - 11784025377 ) / 38939867551 $ (3385112076v^18 - 40342403628v^16 + 335252157616v^14 - 1381334365440v^12 + 4117006052865v^10 - 6729780800908v^8 + 7710148504944v^6 - 2211207439464v^4 + 634252352224*v^2 - 11784025377) / 38939867551
$\beta_{6}$	$=$	$ ( 31106069601 \nu^{18} - 364761863456 \nu^{16} + 3007617255234 \nu^{14} - 12080364877729 \nu^{12} + \cdots + 68250470805 ) / 272579072857 $ (31106069601v^18 - 364761863456v^16 + 3007617255234v^14 - 12080364877729v^12 + 35215088927432v^10 - 53926927691184v^8 + 57244020028116v^6 - 4786943418434v^4 + 384779282792*v^2 + 68250470805) / 272579072857
$\beta_{7}$	$=$	$ ( 52488936218 \nu^{18} - 628499368091 \nu^{16} + 5233235408888 \nu^{14} - 21706288817845 \nu^{12} + \cdots - 823305474404 ) / 272579072857 $ (52488936218v^18 - 628499368091v^16 + 5233235408888v^14 - 21706288817845v^12 + 64997203567532v^10 - 107717790385294v^8 + 124708292536192v^6 - 39785293034960v^4 + 10292297068357*v^2 - 823305474404) / 272579072857
$\beta_{8}$	$=$	$ ( - 55820829126 \nu^{18} + 654055430352 \nu^{16} - 5392966193253 \nu^{14} + \cdots - 924503257198 ) / 272579072857 $ (-55820829126v^18 + 654055430352v^16 - 5392966193253v^14 + 21647692473180v^12 - 63144266029044v^10 + 96696511977528v^8 - 103044601241266v^6 + 8583480487653v^4 - 689948716164*v^2 - 924503257198) / 272579072857
$\beta_{9}$	$=$	$ ( 55820829126 \nu^{19} - 654055430352 \nu^{17} + 5392966193253 \nu^{15} + \cdots + 1197082330055 \nu ) / 272579072857 $ (55820829126v^19 - 654055430352v^17 + 5392966193253v^15 - 21647692473180v^13 + 63144266029044v^11 - 96696511977528v^9 + 103044601241266v^7 - 8583480487653v^5 + 689948716164v^3 + 1197082330055v) / 272579072857
$\beta_{10}$	$=$	$ ( - 62211352354 \nu^{19} + 769298398785 \nu^{17} - 6487615452408 \nu^{15} + \cdots + 1276485769020 \nu ) / 272579072857 $ (-62211352354v^19 + 769298398785v^17 - 6487615452408v^15 + 28077165710051v^13 - 86453798607362v^11 + 155188539345550v^9 - 189948297950192v^7 + 92399612089574v^5 - 15939417099187v^3 + 1276485769020v) / 272579072857
$\beta_{11}$	$=$	$ ( - 81045928818 \nu^{18} + 981295708834 \nu^{16} - 8207190533960 \nu^{14} + \cdots + 1404962872208 ) / 272579072857 $ (-81045928818v^18 + 981295708834v^16 - 8207190533960v^14 + 34561067822028v^12 - 104544543457502v^10 + 178561630471778v^8 - 211299334777800v^6 + 81570904638515v^4 - 17555623549340*v^2 + 1404962872208) / 272579072857
$\beta_{12}$	$=$	$ ( 71791025633 \nu^{19} - 840751345088 \nu^{17} + 6932353697532 \nu^{15} + \cdots + 2244868260487 \nu ) / 272579072857 $ (71791025633v^19 - 840751345088v^17 + 6932353697532v^15 - 27815795775993v^13 + 81168390529136v^11 - 124297909225632v^9 + 132759388300845v^7 - 11033579771132v^5 + 886890138416v^3 + 2244868260487v) / 272579072857
$\beta_{13}$	$=$	$ ( 11784025377 \nu^{19} - 138023192448 \nu^{17} + 1138060134072 \nu^{15} + \cdots + 406768078641 \nu ) / 38939867551 $ (11784025377v^19 - 138023192448v^17 + 1138060134072v^15 - 4566902399216v^13 + 13325129305056v^11 - 20405550756672v^9 + 21787560611432v^7 - 1811343999672v^5 + 145597635936v^3 + 406768078641v) / 38939867551
$\beta_{14}$	$=$	$ ( 100944506246 \nu^{19} - 1182699858656 \nu^{17} + 9751865145534 \nu^{15} + \cdots + 701340078682 \nu ) / 272579072857 $ (100944506246v^19 - 1182699858656v^17 + 9751865145534v^15 - 39142863927841v^13 + 114181017451832v^11 - 174852077883984v^9 + 186335666503353v^7 - 15521132748734v^5 + 1247604119192v^3 + 701340078682v) / 272579072857
$\beta_{15}$	$=$	$ ( 102897095234 \nu^{19} - 1205513208544 \nu^{17} + 9939970952766 \nu^{15} + \cdots + 2040539658435 \nu ) / 272579072857 $ (102897095234v^19 - 1205513208544v^17 + 9939970952766v^15 - 39896160653722v^13 + 116383479456568v^11 - 178224836916816v^9 + 190003408328961v^7 - 15820523189566v^5 + 1271669421208v^3 + 2040539658435v) / 272579072857
$\beta_{16}$	$=$	$ ( - 104741713350 \nu^{19} + 1263692534230 \nu^{17} - 10553955637272 \nu^{15} + \cdots + 1760030122704 \nu ) / 272579072857 $ (-104741713350v^19 + 1263692534230v^17 - 10553955637272v^15 + 44230408380108v^13 - 133363585827557v^11 + 225670096078134v^9 - 265270374312408v^7 + 97049356714763v^5 - 21995390014908v^3 + 1760030122704v) / 272579072857
$\beta_{17}$	$=$	$ ( - 178737686026 \nu^{19} + 2158469175877 \nu^{17} - 18033335161736 \nu^{15} + \cdots + 3028126236968 \nu ) / 272579072857 $ (-178737686026v^19 + 2158469175877v^17 - 18033335161736v^15 + 75670571774570v^13 - 228346548452901v^11 + 387358108344498v^9 - 456135753797624v^7 + 169492482029982v^5 - 37841605052154v^3 + 3028126236968v) / 272579072857
$\beta_{18}$	$=$	$ ( 196394352107 \nu^{19} - 2360052680797 \nu^{17} + 19677906773500 \nu^{15} + \cdots - 3182892440812 \nu ) / 272579072857 $ (196394352107v^19 - 2360052680797v^17 + 19677906773500v^15 - 82017264932587v^13 + 246362436987949v^11 - 412410794623767v^9 + 480962024243140v^7 - 164980145757248v^5 + 39783751037475v^3 - 3182892440812v) / 272579072857
$\beta_{19}$	$=$	$ ( 35554755475 \nu^{19} - 426936006984 \nu^{17} + 3558734597484 \nu^{15} + \cdots - 572313987888 \nu ) / 38939867551 $ (35554755475v^19 - 426936006984v^17 + 3558734597484v^15 - 14817650535776v^13 + 44479948650783v^11 - 74304083572723v^9 + 86524330968476v^7 - 29252205541744v^5 + 7153721157976v^3 - 572313987888v) / 38939867551

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

$\nu$	$=$	$ ( -\beta_{14} - \beta_{13} + \beta_{12} + 2\beta_{9} ) / 4 $ (-b14 - b13 + b12 + 2*b9) / 4
$\nu^{2}$	$=$	$ -\beta_{7} + 2\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} $ -b7 + 2*b5 + b4 + b3 - b2
$\nu^{3}$	$=$	$ ( - 5 \beta_{19} + 5 \beta_{18} + 4 \beta_{17} - 10 \beta_{16} - 4 \beta_{15} - \beta_{14} + \cdots + 10 \beta_{9} ) / 4 $ (-5b19 + 5b18 + 4b17 - 10b16 - 4b15 - b14 - 5b13 + 5b12 + b10 + 10b9) / 4
$\nu^{4}$	$=$	$ -\beta_{11} - 6\beta_{7} + 8\beta_{5} - 8\beta_{2} - 2\beta _1 - 8 $ -b11 - 6b7 + 8b5 - 8b2 - 2b1 - 8
$\nu^{5}$	$=$	$ ( -21\beta_{19} + 29\beta_{18} + 40\beta_{17} - 62\beta_{16} - 3\beta_{10} ) / 4 $ (-21b19 + 29b18 + 40b17 - 62b16 - 3*b10) / 4
$\nu^{6}$	$=$	$ 8\beta_{8} + 18\beta_{6} - 36\beta_{4} - 57\beta_{3} - 40 $ 8b8 + 18b6 - 36b4 - 57b3 - 40
$\nu^{7}$	$=$	$ ( 300\beta_{15} - 43\beta_{14} + 101\beta_{13} - 173\beta_{12} - 402\beta_{9} ) / 4 $ (300b15 - 43b14 + 101b13 - 173b12 - 402*b9) / 4
$\nu^{8}$	$=$	$ 54 \beta_{11} + 54 \beta_{8} + 223 \beta_{7} + 129 \beta_{6} - 227 \beta_{5} - 223 \beta_{4} + \cdots + 129 \beta_1 $ 54b11 + 54b8 + 223b7 + 129b6 - 227b5 - 223b4 - 387b3 + 387b2 + 129*b1
$\nu^{9}$	$=$	$ ( 549 \beta_{19} - 1065 \beta_{18} - 2064 \beta_{17} + 2626 \beta_{16} + 2064 \beta_{15} + \cdots - 2626 \beta_{9} ) / 4 $ (549b19 - 1065b18 - 2064b17 + 2626b16 + 2064b15 - 343b14 + 549b13 - 1065b12 + 343b10 - 2626b9) / 4
$\nu^{10}$	$=$	$ 352\beta_{11} + 1412\beta_{7} - 1381\beta_{5} + 2568\beta_{2} + 868\beta _1 + 1381 $ 352b11 + 1412b7 - 1381b5 + 2568b2 + 868*b1 + 1381
$\nu^{11}$	$=$	$ ( 3241\beta_{19} - 6713\beta_{18} - 13744\beta_{17} + 17138\beta_{16} + 2407\beta_{10} ) / 4 $ (3241b19 - 6713b18 - 13744b17 + 17138b16 + 2407*b10) / 4
$\nu^{12}$	$=$	$ -2280\beta_{8} - 5716\beta_{6} + 9053\beta_{4} + 16853\beta_{3} + 8698 $ -2280b8 - 5716b6 + 9053b4 + 16853b3 + 8698
$\nu^{13}$	$=$	$ ( -90276\beta_{15} + 16151\beta_{14} - 20061\beta_{13} + 42925\beta_{12} + 111642\beta_{9} ) / 4 $ (-90276b15 + 16151b14 - 20061b13 + 42925b12 + 111642*b9) / 4
$\nu^{14}$	$=$	$ - 14769 \beta_{11} - 14769 \beta_{8} - 58426 \beta_{7} - 37338 \beta_{6} + 55692 \beta_{5} + \cdots - 37338 \beta_1 $ -14769b11 - 14769b8 - 58426b7 - 37338b6 + 55692b5 + 58426b4 + 110012b3 - 110012b2 - 37338*b1
$\nu^{15}$	$=$	$ ( - 127277 \beta_{19} + 276629 \beta_{18} + 589400 \beta_{17} - 726318 \beta_{16} + \cdots + 726318 \beta_{9} ) / 4 $ (-127277b19 + 276629b18 + 589400b17 - 726318b16 - 589400b15 + 106427b14 - 127277b13 + 276629b12 - 106427b10 + 726318b9) / 4
$\nu^{16}$	$=$	$ -95764\beta_{11} - 378320\beta_{7} + 359336\beta_{5} - 716285\beta_{2} - 243114\beta _1 - 359336 $ -95764b11 - 378320b7 + 359336b5 - 716285b2 - 243114*b1 - 359336
$\nu^{17}$	$=$	$ ( -817453\beta_{19} + 1789909\beta_{18} + 3837596\beta_{17} - 4721682\beta_{16} - 695827\beta_{10} ) / 4 $ (-817453b19 + 1789909b18 + 3837596b17 - 4721682b16 - 695827*b10) / 4
$\nu^{18}$	$=$	$ 621434\beta_{8} + 1580833\beta_{6} - 2453695\beta_{4} - 4657959\beta_{3} - 2326803 $ 621434b8 + 1580833b6 - 2453695b4 - 4657959b3 - 2326803
$\nu^{19}$	$=$	$ ( 24955168\beta_{15} - 4533423\beta_{14} + 5281357\beta_{13} - 11604689\beta_{12} - 30682562\beta_{9} ) / 4 $ (24955168b15 - 4533423b14 + 5281357b13 - 11604689b12 - 30682562*b9) / 4

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/833\mathbb{Z}\right)^\times$.

$n$	$52$	$785$
$\chi(n)$	$-\beta_{5}$	$-1$

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} $

67.1

−1.51912	−	0.877065i
1.51912	+	0.877065i
0.455982	+	0.263261i
−0.455982	−	0.263261i
0.256168	+	0.147899i
−0.256168	−	0.147899i
−1.24379	−	0.718103i
1.24379	+	0.718103i
−2.20720	−	1.27433i
2.20720	+	1.27433i
−1.51912	+	0.877065i
1.51912	−	0.877065i
0.455982	−	0.263261i
−0.455982	+	0.263261i
0.256168	−	0.147899i
−0.256168	+	0.147899i
−1.24379	+	0.718103i
1.24379	−	0.718103i
−2.20720	+	1.27433i
2.20720	−	1.27433i

−0.877065

−

1.51912i

−1.02542

−

0.592024i

−0.538487

0.932687i

−0.469502

0.271067i

2.07697i

−1.61911

−0.799016

−

1.38394i

0.823568

0.475487i

67.2

−0.877065

−

1.51912i

1.02542

0.592024i

−0.538487

0.932687i

0.469502

−

0.271067i

−

2.07697i

−1.61911

−0.799016

−

1.38394i

−0.823568

−

0.475487i

67.3

−0.263261

−

0.455982i

−1.18882

−

0.686366i

0.861387

−

1.49197i

3.18948

−

1.84145i

0.722774i

−1.96012

−0.557805

−

0.966146i

−1.67934

−

0.969565i

67.4

−0.263261

−

0.455982i

1.18882

0.686366i

0.861387

−

1.49197i

−3.18948

1.84145i

−

0.722774i

−1.96012

−0.557805

−

0.966146i

1.67934

0.969565i

67.5

0.147899

0.256168i

−2.67160

−

1.54245i

0.956252

−

1.65628i

−0.654019

0.377598i

−

0.912504i

1.15731

3.25829

5.64353i

−0.193457

−

0.111692i

67.6

0.147899

0.256168i

2.67160

1.54245i

0.956252

−

1.65628i

0.654019

−

0.377598i

0.912504i

1.15731

3.25829

5.64353i

0.193457

0.111692i

67.7

0.718103

1.24379i

−0.640795

−

0.369963i

−0.0313433

0.0542882i

−2.46390

1.42253i

−

1.06269i

2.78238

−1.22625

−

2.12393i

−3.53867

−

2.04305i

67.8

0.718103

1.24379i

0.640795

0.369963i

−0.0313433

0.0542882i

2.46390

−

1.42253i

1.06269i

2.78238

−1.22625

−

2.12393i

3.53867

2.04305i

67.9

1.27433

2.20720i

−1.86740

−

1.07814i

−2.24781

3.89332i

1.61500

−

0.932419i

−

5.49562i

−6.36046

0.824784

1.42857i

4.11606

2.37641i

67.10

1.27433

2.20720i

1.86740

1.07814i

−2.24781

3.89332i

−1.61500

0.932419i

5.49562i

−6.36046

0.824784

1.42857i

−4.11606

−

2.37641i

373.1

−0.877065

1.51912i

−1.02542

0.592024i

−0.538487

−

0.932687i

−0.469502

−

0.271067i

−

2.07697i

−1.61911

−0.799016

1.38394i

0.823568

−

0.475487i

373.2

−0.877065

1.51912i

1.02542

−

0.592024i

−0.538487

−

0.932687i

0.469502

0.271067i

2.07697i

−1.61911

−0.799016

1.38394i

−0.823568

0.475487i

373.3

−0.263261

0.455982i

−1.18882

0.686366i

0.861387

1.49197i

3.18948

1.84145i

−

0.722774i

−1.96012

−0.557805

0.966146i

−1.67934

0.969565i

373.4

−0.263261

0.455982i

1.18882

−

0.686366i

0.861387

1.49197i

−3.18948

−

1.84145i

0.722774i

−1.96012

−0.557805

0.966146i

1.67934

−

0.969565i

373.5

0.147899

−

0.256168i

−2.67160

1.54245i

0.956252

1.65628i

−0.654019

−

0.377598i

0.912504i

1.15731

3.25829

−

5.64353i

−0.193457

0.111692i

373.6

0.147899

−

0.256168i

2.67160

−

1.54245i

0.956252

1.65628i

0.654019

0.377598i

−

0.912504i

1.15731

3.25829

−

5.64353i

0.193457

−

0.111692i

373.7

0.718103

−

1.24379i

−0.640795

0.369963i

−0.0313433

−

0.0542882i

−2.46390

−

1.42253i

1.06269i

2.78238

−1.22625

2.12393i

−3.53867

2.04305i

373.8

0.718103

−

1.24379i

0.640795

−

0.369963i

−0.0313433

−

0.0542882i

2.46390

1.42253i

−

1.06269i

2.78238

−1.22625

2.12393i

3.53867

−

2.04305i

373.9

1.27433

−

2.20720i

−1.86740

1.07814i

−2.24781

−

3.89332i

1.61500

0.932419i

5.49562i

−6.36046

0.824784

−

1.42857i

4.11606

−

2.37641i

373.10

1.27433

−

2.20720i

1.86740

−

1.07814i

−2.24781

−

3.89332i

−1.61500

−

0.932419i

−

5.49562i

−6.36046

0.824784

−

1.42857i

−4.11606

2.37641i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner
17.b	even	2	1	inner
119.j	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	833.2.j.d		20
7.b	odd	2	1		833.2.j.c		20
7.c	even	3	1		833.2.b.a		10
7.c	even	3	1	inner	833.2.j.d		20
7.d	odd	6	1		119.2.b.a	✓	10
7.d	odd	6	1		833.2.j.c		20
17.b	even	2	1	inner	833.2.j.d		20
21.g	even	6	1		1071.2.f.c		10
28.f	even	6	1		1904.2.c.i		10
119.d	odd	2	1		833.2.j.c		20
119.h	odd	6	1		119.2.b.a	✓	10
119.h	odd	6	1		833.2.j.c		20
119.j	even	6	1		833.2.b.a		10
119.j	even	6	1	inner	833.2.j.d		20
119.m	odd	12	1		2023.2.a.h		5
119.m	odd	12	1		2023.2.a.i		5
357.s	even	6	1		1071.2.f.c		10
476.q	even	6	1		1904.2.c.i		10

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
119.2.b.a	✓	10	7.d	odd	6	1
119.2.b.a	✓	10	119.h	odd	6	1
833.2.b.a		10	7.c	even	3	1
833.2.b.a		10	119.j	even	6	1
833.2.j.c		20	7.b	odd	2	1
833.2.j.c		20	7.d	odd	6	1
833.2.j.c		20	119.d	odd	2	1
833.2.j.c		20	119.h	odd	6	1
833.2.j.d		20	1.a	even	1	1	trivial
833.2.j.d		20	7.c	even	3	1	inner
833.2.j.d		20	17.b	even	2	1	inner
833.2.j.d		20	119.j	even	6	1	inner
1071.2.f.c		10	21.g	even	6	1
1071.2.f.c		10	357.s	even	6	1
1904.2.c.i		10	28.f	even	6	1
1904.2.c.i		10	476.q	even	6	1
2023.2.a.h		5	119.m	odd	12	1
2023.2.a.i		5	119.m	odd	12	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(833, [\chi])$:

$ T_{2}^{10} - 2T_{2}^{9} + 8T_{2}^{8} - 4T_{2}^{7} + 26T_{2}^{6} - 17T_{2}^{5} + 42T_{2}^{4} + 4T_{2}^{3} + 10T_{2}^{2} - 2T_{2} + 1 $

$ T_{13}^{5} - 4T_{13}^{4} - 36T_{13}^{3} + 176T_{13}^{2} - 16T_{13} - 416 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{10} - 2 T^{9} + 8 T^{8} + \cdots + 1)^{2} $ (T^10 - 2T^9 + 8T^8 - 4T^7 + 26T^6 - 17T^5 + 42T^4 + 4T^3 + 10T^2 - 2*T + 1)^2
$3$	$ T^{20} - 18 T^{18} + \cdots + 4096 $ T^20 - 18T^18 + 221T^16 - 1386T^14 + 6180T^12 - 16354T^10 + 31253T^8 - 37594T^6 + 32113T^4 - 13888*T^2 + 4096
$5$	$ T^{20} - 26 T^{18} + \cdots + 4096 $ T^20 - 26T^18 + 469T^16 - 4290T^14 + 28292T^12 - 94314T^10 + 221725T^8 - 170610T^6 + 95377T^4 - 23104*T^2 + 4096
$7$	$ T^{20} $ T^20
$11$	$ T^{20} + \cdots + 167772160000 $ T^20 - 80T^18 + 4064T^16 - 124800T^14 + 2787072T^12 - 43059200T^10 + 494759936T^8 - 3879157760T^6 + 22114533376T^4 - 76441190400*T^2 + 167772160000
$13$	$ (T^{5} - 4 T^{4} + \cdots - 416)^{4} $ (T^5 - 4T^4 - 36T^3 + 176T^2 - 16T - 416)^4
$17$	$ T^{20} + \cdots + 2015993900449 $ T^20 - 2T^19 + 15T^18 - 218T^17 + 191T^16 - 2060T^15 + 19210T^14 + 18060T^13 + 146413T^12 - 874310T^11 - 3201495T^10 - 14863270T^9 + 42313357T^8 + 88728780T^7 + 1604438410T^6 - 2924905420T^5 + 4610275679T^4 - 89453830714T^3 + 104636361615T^2 - 237175752994*T + 2015993900449
$19$	$ (T^{10} - 2 T^{9} + \cdots + 102400)^{2} $ (T^10 - 2T^9 + 48T^8 - 88T^7 + 1936T^6 - 3488T^5 + 14848T^4 - 12672T^3 + 59136T^2 - 56320*T + 102400)^2
$23$	$ T^{20} + \cdots + 268435456 $ T^20 - 84T^18 + 5104T^16 - 136832T^14 + 2641664T^12 - 21641216T^10 + 126246912T^8 - 328531968T^6 + 614531072T^4 - 473956352*T^2 + 268435456
$29$	$ (T^{10} + 208 T^{8} + \cdots + 1048576)^{2} $ (T^10 + 208T^8 + 15008T^6 + 454208T^4 + 4923648T^2 + 1048576)^2
$31$	$ T^{20} + \cdots + 65053903360000 $ T^20 - 146T^18 + 13645T^16 - 754506T^14 + 30174020T^12 - 828228258T^10 + 16934797189T^8 - 240190124730T^6 + 2492806784881T^4 - 16063779649600*T^2 + 65053903360000
$37$	$ T^{20} + \cdots + 23\!\cdots\!36 $ T^20 - 308T^18 + 58928T^16 - 7124864T^14 + 633559808T^12 - 40201967616T^10 + 1921830723584T^8 - 64831389827072T^6 + 1599503394209792T^4 - 24501311558909952*T^2 + 234938215784513536
$41$	$ (T^{10} + 146 T^{8} + \cdots + 1600)^{2} $ (T^10 + 146T^8 + 5111T^6 + 10922T^4 + 7449T^2 + 1600)^2
$43$	$ (T^{5} - 8 T^{4} - 59 T^{3} + \cdots + 4)^{4} $ (T^5 - 8T^4 - 59T^3 + 140T^2 + 49T + 4)^4
$47$	$ (T^{10} + 96 T^{8} + \cdots + 16384)^{2} $ (T^10 + 96T^8 - 400T^7 + 8912T^6 - 19328T^5 + 69184T^4 + 36224T^3 + 118016T^2 - 38912T + 16384)^2
$53$	$ (T^{10} + 4 T^{9} + \cdots + 2316484)^{2} $ (T^10 + 4T^9 + 97T^8 - 468T^7 + 5432T^6 - 14082T^5 + 79393T^4 - 186012T^3 + 816865T^2 - 1280002*T + 2316484)^2
$59$	$ (T^{10} - 28 T^{9} + \cdots + 17305600)^{2} $ (T^10 - 28T^9 + 516T^8 - 5648T^7 + 45856T^6 - 245440T^5 + 981952T^4 - 2214912T^3 + 3860736T^2 - 66560*T + 17305600)^2
$61$	$ T^{20} - 146 T^{18} + \cdots + 2560000 $ T^20 - 146T^18 + 16205T^16 - 724362T^14 + 24520260T^12 - 53648834T^10 + 80984645T^8 - 65002778T^6 + 38012401T^4 - 11918400*T^2 + 2560000
$67$	$ (T^{10} + 12 T^{9} + \cdots + 5779216)^{2} $ (T^10 + 12T^9 + 215T^8 + 1228T^7 + 20632T^6 + 146100T^5 + 889567T^4 + 2894072T^3 + 7178161T^2 + 7478844*T + 5779216)^2
$71$	$ (T^{10} + 172 T^{8} + \cdots + 409600)^{2} $ (T^10 + 172T^8 + 7632T^6 + 103360T^4 + 461056T^2 + 409600)^2
$73$	$ T^{20} + \cdots + 24\!\cdots\!76 $ T^20 - 426T^18 + 119925T^16 - 19583954T^14 + 2331001796T^12 - 167012484922T^10 + 8242839321853T^8 - 126203974286786T^6 + 1402297859777297T^4 - 6932844687240256*T^2 + 24949663242981376
$79$	$ T^{20} + \cdots + 42\!\cdots\!56 $ T^20 - 352T^18 + 78624T^16 - 10615680T^14 + 1042881280T^12 - 71477377024T^10 + 3659456786432T^8 - 129061068226560T^6 + 3253355863408640T^4 - 45787579584348160*T^2 + 420971005037510656
$83$	$ (T^{5} - 6 T^{4} - 28 T^{3} + \cdots + 64)^{4} $ (T^5 - 6T^4 - 28T^3 + 56T^2 + 208T + 64)^4
$89$	$ (T^{10} + 2 T^{9} + \cdots + 8522982400)^{2} $ (T^10 + 2T^9 + 316T^8 + 864T^7 + 77040T^6 + 237280T^5 + 7168000T^4 + 41394432T^3 + 543577344T^2 + 2011837440*T + 8522982400)^2
$97$	$ (T^{10} + 282 T^{8} + \cdots + 24049216)^{2} $ (T^10 + 282T^8 + 23055T^6 + 758658T^4 + 9600233T^2 + 24049216)^2
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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:	\( N \)	\(=\)	\( 833 = 7^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	833.j (of order \(6\), degree \(2\), not minimal)

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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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	833.2.j.d

Level	$833$
Weight	$2$
Character orbit	833.j
Analytic conductor	$6.652$
Analytic rank	$0$
Dimension	$20$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(6.65153848837\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Relative dimension:	\(10\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{20} - 12 x^{18} + 100 x^{16} - 416 x^{14} + 1248 x^{12} - 2081 x^{10} + 2420 x^{8} - 808 x^{6} + \cdots + 1 \) x^20 - 12x^18 + 100x^16 - 416x^14 + 1248x^12 - 2081x^10 + 2420x^8 - 808x^6 + 200x^4 - 16*x^2 + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index:	\( 2^{4} \)
Twist minimal:	no (minimal twist has level 119)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

\(\beta_{1}\)	\(=\)	\( ( 6039118451 \nu^{18} - 80813320476 \nu^{16} + 702193491548 \nu^{14} - 3322647400063 \nu^{12} + \cdots - 200301046652 ) / 272579072857 \) (6039118451v^18 - 80813320476v^16 + 702193491548v^14 - 3322647400063v^12 + 10803153835007v^10 - 22055779327087v^8 + 29144769089092v^6 - 19990788348982v^4 + 2497620480247*v^2 - 200301046652) / 272579072857
\(\beta_{2}\)	\(=\)	\( ( - 112298 \nu^{18} + 1378153 \nu^{16} - 11593224 \nu^{14} + 49712554 \nu^{12} - 152305402 \nu^{10} + \cdots + 2180136 ) / 4341883 \) (-112298v^18 + 1378153v^16 - 11593224v^14 + 49712554v^12 - 152305402v^10 + 268778066v^8 - 325490296v^6 + 145395424v^4 - 27228986*v^2 + 2180136) / 4341883
\(\beta_{3}\)	\(=\)	\( ( 8744563018 \nu^{18} - 102597652160 \nu^{16} + 845961433740 \nu^{14} - 3399224292638 \nu^{12} + \cdots + 81045928818 ) / 272579072857 \) (8744563018v^18 - 102597652160v^16 + 845961433740v^14 - 3399224292638v^12 + 9905052601520v^10 - 15168187038240v^8 + 16085794153571v^6 - 1346437785740v^4 + 108228011120*v^2 + 81045928818) / 272579072857
\(\beta_{4}\)	\(=\)	\( ( - 10697152006 \nu^{18} + 125411002048 \nu^{16} - 1034067240972 \nu^{14} + \cdots - 602508290000 ) / 272579072857 \) (-10697152006v^18 + 125411002048v^16 - 1034067240972v^14 + 4152521018519v^12 - 12107514606256v^10 + 18540946071072v^8 - 19753535979179v^6 + 1645828226572v^4 - 132293313136*v^2 - 602508290000) / 272579072857
\(\beta_{5}\)	\(=\)	\( ( 3385112076 \nu^{18} - 40342403628 \nu^{16} + 335252157616 \nu^{14} - 1381334365440 \nu^{12} + \cdots - 11784025377 ) / 38939867551 \) (3385112076v^18 - 40342403628v^16 + 335252157616v^14 - 1381334365440v^12 + 4117006052865v^10 - 6729780800908v^8 + 7710148504944v^6 - 2211207439464v^4 + 634252352224*v^2 - 11784025377) / 38939867551
\(\beta_{6}\)	\(=\)	\( ( 31106069601 \nu^{18} - 364761863456 \nu^{16} + 3007617255234 \nu^{14} - 12080364877729 \nu^{12} + \cdots + 68250470805 ) / 272579072857 \) (31106069601v^18 - 364761863456v^16 + 3007617255234v^14 - 12080364877729v^12 + 35215088927432v^10 - 53926927691184v^8 + 57244020028116v^6 - 4786943418434v^4 + 384779282792*v^2 + 68250470805) / 272579072857
\(\beta_{7}\)	\(=\)	\( ( 52488936218 \nu^{18} - 628499368091 \nu^{16} + 5233235408888 \nu^{14} - 21706288817845 \nu^{12} + \cdots - 823305474404 ) / 272579072857 \) (52488936218v^18 - 628499368091v^16 + 5233235408888v^14 - 21706288817845v^12 + 64997203567532v^10 - 107717790385294v^8 + 124708292536192v^6 - 39785293034960v^4 + 10292297068357*v^2 - 823305474404) / 272579072857
\(\beta_{8}\)	\(=\)	\( ( - 55820829126 \nu^{18} + 654055430352 \nu^{16} - 5392966193253 \nu^{14} + \cdots - 924503257198 ) / 272579072857 \) (-55820829126v^18 + 654055430352v^16 - 5392966193253v^14 + 21647692473180v^12 - 63144266029044v^10 + 96696511977528v^8 - 103044601241266v^6 + 8583480487653v^4 - 689948716164*v^2 - 924503257198) / 272579072857
\(\beta_{9}\)	\(=\)	\( ( 55820829126 \nu^{19} - 654055430352 \nu^{17} + 5392966193253 \nu^{15} + \cdots + 1197082330055 \nu ) / 272579072857 \) (55820829126v^19 - 654055430352v^17 + 5392966193253v^15 - 21647692473180v^13 + 63144266029044v^11 - 96696511977528v^9 + 103044601241266v^7 - 8583480487653v^5 + 689948716164v^3 + 1197082330055v) / 272579072857
\(\beta_{10}\)	\(=\)	\( ( - 62211352354 \nu^{19} + 769298398785 \nu^{17} - 6487615452408 \nu^{15} + \cdots + 1276485769020 \nu ) / 272579072857 \) (-62211352354v^19 + 769298398785v^17 - 6487615452408v^15 + 28077165710051v^13 - 86453798607362v^11 + 155188539345550v^9 - 189948297950192v^7 + 92399612089574v^5 - 15939417099187v^3 + 1276485769020v) / 272579072857
\(\beta_{11}\)	\(=\)	\( ( - 81045928818 \nu^{18} + 981295708834 \nu^{16} - 8207190533960 \nu^{14} + \cdots + 1404962872208 ) / 272579072857 \) (-81045928818v^18 + 981295708834v^16 - 8207190533960v^14 + 34561067822028v^12 - 104544543457502v^10 + 178561630471778v^8 - 211299334777800v^6 + 81570904638515v^4 - 17555623549340*v^2 + 1404962872208) / 272579072857
\(\beta_{12}\)	\(=\)	\( ( 71791025633 \nu^{19} - 840751345088 \nu^{17} + 6932353697532 \nu^{15} + \cdots + 2244868260487 \nu ) / 272579072857 \) (71791025633v^19 - 840751345088v^17 + 6932353697532v^15 - 27815795775993v^13 + 81168390529136v^11 - 124297909225632v^9 + 132759388300845v^7 - 11033579771132v^5 + 886890138416v^3 + 2244868260487v) / 272579072857
\(\beta_{13}\)	\(=\)	\( ( 11784025377 \nu^{19} - 138023192448 \nu^{17} + 1138060134072 \nu^{15} + \cdots + 406768078641 \nu ) / 38939867551 \) (11784025377v^19 - 138023192448v^17 + 1138060134072v^15 - 4566902399216v^13 + 13325129305056v^11 - 20405550756672v^9 + 21787560611432v^7 - 1811343999672v^5 + 145597635936v^3 + 406768078641v) / 38939867551
\(\beta_{14}\)	\(=\)	\( ( 100944506246 \nu^{19} - 1182699858656 \nu^{17} + 9751865145534 \nu^{15} + \cdots + 701340078682 \nu ) / 272579072857 \) (100944506246v^19 - 1182699858656v^17 + 9751865145534v^15 - 39142863927841v^13 + 114181017451832v^11 - 174852077883984v^9 + 186335666503353v^7 - 15521132748734v^5 + 1247604119192v^3 + 701340078682v) / 272579072857
\(\beta_{15}\)	\(=\)	\( ( 102897095234 \nu^{19} - 1205513208544 \nu^{17} + 9939970952766 \nu^{15} + \cdots + 2040539658435 \nu ) / 272579072857 \) (102897095234v^19 - 1205513208544v^17 + 9939970952766v^15 - 39896160653722v^13 + 116383479456568v^11 - 178224836916816v^9 + 190003408328961v^7 - 15820523189566v^5 + 1271669421208v^3 + 2040539658435v) / 272579072857
\(\beta_{16}\)	\(=\)	\( ( - 104741713350 \nu^{19} + 1263692534230 \nu^{17} - 10553955637272 \nu^{15} + \cdots + 1760030122704 \nu ) / 272579072857 \) (-104741713350v^19 + 1263692534230v^17 - 10553955637272v^15 + 44230408380108v^13 - 133363585827557v^11 + 225670096078134v^9 - 265270374312408v^7 + 97049356714763v^5 - 21995390014908v^3 + 1760030122704v) / 272579072857
\(\beta_{17}\)	\(=\)	\( ( - 178737686026 \nu^{19} + 2158469175877 \nu^{17} - 18033335161736 \nu^{15} + \cdots + 3028126236968 \nu ) / 272579072857 \) (-178737686026v^19 + 2158469175877v^17 - 18033335161736v^15 + 75670571774570v^13 - 228346548452901v^11 + 387358108344498v^9 - 456135753797624v^7 + 169492482029982v^5 - 37841605052154v^3 + 3028126236968v) / 272579072857
\(\beta_{18}\)	\(=\)	\( ( 196394352107 \nu^{19} - 2360052680797 \nu^{17} + 19677906773500 \nu^{15} + \cdots - 3182892440812 \nu ) / 272579072857 \) (196394352107v^19 - 2360052680797v^17 + 19677906773500v^15 - 82017264932587v^13 + 246362436987949v^11 - 412410794623767v^9 + 480962024243140v^7 - 164980145757248v^5 + 39783751037475v^3 - 3182892440812v) / 272579072857
\(\beta_{19}\)	\(=\)	\( ( 35554755475 \nu^{19} - 426936006984 \nu^{17} + 3558734597484 \nu^{15} + \cdots - 572313987888 \nu ) / 38939867551 \) (35554755475v^19 - 426936006984v^17 + 3558734597484v^15 - 14817650535776v^13 + 44479948650783v^11 - 74304083572723v^9 + 86524330968476v^7 - 29252205541744v^5 + 7153721157976v^3 - 572313987888v) / 38939867551

\(\nu\)	\(=\)	\( ( -\beta_{14} - \beta_{13} + \beta_{12} + 2\beta_{9} ) / 4 \) (-b14 - b13 + b12 + 2*b9) / 4
\(\nu^{2}\)	\(=\)	\( -\beta_{7} + 2\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} \) -b7 + 2*b5 + b4 + b3 - b2
\(\nu^{3}\)	\(=\)	\( ( - 5 \beta_{19} + 5 \beta_{18} + 4 \beta_{17} - 10 \beta_{16} - 4 \beta_{15} - \beta_{14} + \cdots + 10 \beta_{9} ) / 4 \) (-5b19 + 5b18 + 4b17 - 10b16 - 4b15 - b14 - 5b13 + 5b12 + b10 + 10b9) / 4
\(\nu^{4}\)	\(=\)	\( -\beta_{11} - 6\beta_{7} + 8\beta_{5} - 8\beta_{2} - 2\beta _1 - 8 \) -b11 - 6b7 + 8b5 - 8b2 - 2b1 - 8
\(\nu^{5}\)	\(=\)	\( ( -21\beta_{19} + 29\beta_{18} + 40\beta_{17} - 62\beta_{16} - 3\beta_{10} ) / 4 \) (-21b19 + 29b18 + 40b17 - 62b16 - 3*b10) / 4
\(\nu^{6}\)	\(=\)	\( 8\beta_{8} + 18\beta_{6} - 36\beta_{4} - 57\beta_{3} - 40 \) 8b8 + 18b6 - 36b4 - 57b3 - 40
\(\nu^{7}\)	\(=\)	\( ( 300\beta_{15} - 43\beta_{14} + 101\beta_{13} - 173\beta_{12} - 402\beta_{9} ) / 4 \) (300b15 - 43b14 + 101b13 - 173b12 - 402*b9) / 4
\(\nu^{8}\)	\(=\)	\( 54 \beta_{11} + 54 \beta_{8} + 223 \beta_{7} + 129 \beta_{6} - 227 \beta_{5} - 223 \beta_{4} + \cdots + 129 \beta_1 \) 54b11 + 54b8 + 223b7 + 129b6 - 227b5 - 223b4 - 387b3 + 387b2 + 129*b1
\(\nu^{9}\)	\(=\)	\( ( 549 \beta_{19} - 1065 \beta_{18} - 2064 \beta_{17} + 2626 \beta_{16} + 2064 \beta_{15} + \cdots - 2626 \beta_{9} ) / 4 \) (549b19 - 1065b18 - 2064b17 + 2626b16 + 2064b15 - 343b14 + 549b13 - 1065b12 + 343b10 - 2626b9) / 4
\(\nu^{10}\)	\(=\)	\( 352\beta_{11} + 1412\beta_{7} - 1381\beta_{5} + 2568\beta_{2} + 868\beta _1 + 1381 \) 352b11 + 1412b7 - 1381b5 + 2568b2 + 868*b1 + 1381
\(\nu^{11}\)	\(=\)	\( ( 3241\beta_{19} - 6713\beta_{18} - 13744\beta_{17} + 17138\beta_{16} + 2407\beta_{10} ) / 4 \) (3241b19 - 6713b18 - 13744b17 + 17138b16 + 2407*b10) / 4
\(\nu^{12}\)	\(=\)	\( -2280\beta_{8} - 5716\beta_{6} + 9053\beta_{4} + 16853\beta_{3} + 8698 \) -2280b8 - 5716b6 + 9053b4 + 16853b3 + 8698
\(\nu^{13}\)	\(=\)	\( ( -90276\beta_{15} + 16151\beta_{14} - 20061\beta_{13} + 42925\beta_{12} + 111642\beta_{9} ) / 4 \) (-90276b15 + 16151b14 - 20061b13 + 42925b12 + 111642*b9) / 4
\(\nu^{14}\)	\(=\)	\( - 14769 \beta_{11} - 14769 \beta_{8} - 58426 \beta_{7} - 37338 \beta_{6} + 55692 \beta_{5} + \cdots - 37338 \beta_1 \) -14769b11 - 14769b8 - 58426b7 - 37338b6 + 55692b5 + 58426b4 + 110012b3 - 110012b2 - 37338*b1
\(\nu^{15}\)	\(=\)	\( ( - 127277 \beta_{19} + 276629 \beta_{18} + 589400 \beta_{17} - 726318 \beta_{16} + \cdots + 726318 \beta_{9} ) / 4 \) (-127277b19 + 276629b18 + 589400b17 - 726318b16 - 589400b15 + 106427b14 - 127277b13 + 276629b12 - 106427b10 + 726318b9) / 4
\(\nu^{16}\)	\(=\)	\( -95764\beta_{11} - 378320\beta_{7} + 359336\beta_{5} - 716285\beta_{2} - 243114\beta _1 - 359336 \) -95764b11 - 378320b7 + 359336b5 - 716285b2 - 243114*b1 - 359336
\(\nu^{17}\)	\(=\)	\( ( -817453\beta_{19} + 1789909\beta_{18} + 3837596\beta_{17} - 4721682\beta_{16} - 695827\beta_{10} ) / 4 \) (-817453b19 + 1789909b18 + 3837596b17 - 4721682b16 - 695827*b10) / 4
\(\nu^{18}\)	\(=\)	\( 621434\beta_{8} + 1580833\beta_{6} - 2453695\beta_{4} - 4657959\beta_{3} - 2326803 \) 621434b8 + 1580833b6 - 2453695b4 - 4657959b3 - 2326803
\(\nu^{19}\)	\(=\)	\( ( 24955168\beta_{15} - 4533423\beta_{14} + 5281357\beta_{13} - 11604689\beta_{12} - 30682562\beta_{9} ) / 4 \) (24955168b15 - 4533423b14 + 5281357b13 - 11604689b12 - 30682562*b9) / 4

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 67.10
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{10} - 2 T^{9} + 8 T^{8} + \cdots + 1)^{2} \) (T^10 - 2T^9 + 8T^8 - 4T^7 + 26T^6 - 17T^5 + 42T^4 + 4T^3 + 10T^2 - 2*T + 1)^2
$3$	\( T^{20} - 18 T^{18} + \cdots + 4096 \) T^20 - 18T^18 + 221T^16 - 1386T^14 + 6180T^12 - 16354T^10 + 31253T^8 - 37594T^6 + 32113T^4 - 13888*T^2 + 4096
$5$	\( T^{20} - 26 T^{18} + \cdots + 4096 \) T^20 - 26T^18 + 469T^16 - 4290T^14 + 28292T^12 - 94314T^10 + 221725T^8 - 170610T^6 + 95377T^4 - 23104*T^2 + 4096
$7$	\( T^{20} \) T^20
$11$	\( T^{20} + \cdots + 167772160000 \) T^20 - 80T^18 + 4064T^16 - 124800T^14 + 2787072T^12 - 43059200T^10 + 494759936T^8 - 3879157760T^6 + 22114533376T^4 - 76441190400*T^2 + 167772160000
$13$	\( (T^{5} - 4 T^{4} + \cdots - 416)^{4} \) (T^5 - 4T^4 - 36T^3 + 176T^2 - 16T - 416)^4
$17$	\( T^{20} + \cdots + 2015993900449 \) T^20 - 2T^19 + 15T^18 - 218T^17 + 191T^16 - 2060T^15 + 19210T^14 + 18060T^13 + 146413T^12 - 874310T^11 - 3201495T^10 - 14863270T^9 + 42313357T^8 + 88728780T^7 + 1604438410T^6 - 2924905420T^5 + 4610275679T^4 - 89453830714T^3 + 104636361615T^2 - 237175752994*T + 2015993900449
$19$	\( (T^{10} - 2 T^{9} + \cdots + 102400)^{2} \) (T^10 - 2T^9 + 48T^8 - 88T^7 + 1936T^6 - 3488T^5 + 14848T^4 - 12672T^3 + 59136T^2 - 56320*T + 102400)^2
$23$	\( T^{20} + \cdots + 268435456 \) T^20 - 84T^18 + 5104T^16 - 136832T^14 + 2641664T^12 - 21641216T^10 + 126246912T^8 - 328531968T^6 + 614531072T^4 - 473956352*T^2 + 268435456
$29$	\( (T^{10} + 208 T^{8} + \cdots + 1048576)^{2} \) (T^10 + 208T^8 + 15008T^6 + 454208T^4 + 4923648T^2 + 1048576)^2
$31$	\( T^{20} + \cdots + 65053903360000 \) T^20 - 146T^18 + 13645T^16 - 754506T^14 + 30174020T^12 - 828228258T^10 + 16934797189T^8 - 240190124730T^6 + 2492806784881T^4 - 16063779649600*T^2 + 65053903360000
$37$	\( T^{20} + \cdots + 23\!\cdots\!36 \) T^20 - 308T^18 + 58928T^16 - 7124864T^14 + 633559808T^12 - 40201967616T^10 + 1921830723584T^8 - 64831389827072T^6 + 1599503394209792T^4 - 24501311558909952*T^2 + 234938215784513536
$41$	\( (T^{10} + 146 T^{8} + \cdots + 1600)^{2} \) (T^10 + 146T^8 + 5111T^6 + 10922T^4 + 7449T^2 + 1600)^2
$43$	\( (T^{5} - 8 T^{4} - 59 T^{3} + \cdots + 4)^{4} \) (T^5 - 8T^4 - 59T^3 + 140T^2 + 49T + 4)^4
$47$	\( (T^{10} + 96 T^{8} + \cdots + 16384)^{2} \) (T^10 + 96T^8 - 400T^7 + 8912T^6 - 19328T^5 + 69184T^4 + 36224T^3 + 118016T^2 - 38912T + 16384)^2
$53$	\( (T^{10} + 4 T^{9} + \cdots + 2316484)^{2} \) (T^10 + 4T^9 + 97T^8 - 468T^7 + 5432T^6 - 14082T^5 + 79393T^4 - 186012T^3 + 816865T^2 - 1280002*T + 2316484)^2
$59$	\( (T^{10} - 28 T^{9} + \cdots + 17305600)^{2} \) (T^10 - 28T^9 + 516T^8 - 5648T^7 + 45856T^6 - 245440T^5 + 981952T^4 - 2214912T^3 + 3860736T^2 - 66560*T + 17305600)^2
$61$	\( T^{20} - 146 T^{18} + \cdots + 2560000 \) T^20 - 146T^18 + 16205T^16 - 724362T^14 + 24520260T^12 - 53648834T^10 + 80984645T^8 - 65002778T^6 + 38012401T^4 - 11918400*T^2 + 2560000
$67$	\( (T^{10} + 12 T^{9} + \cdots + 5779216)^{2} \) (T^10 + 12T^9 + 215T^8 + 1228T^7 + 20632T^6 + 146100T^5 + 889567T^4 + 2894072T^3 + 7178161T^2 + 7478844*T + 5779216)^2
$71$	\( (T^{10} + 172 T^{8} + \cdots + 409600)^{2} \) (T^10 + 172T^8 + 7632T^6 + 103360T^4 + 461056T^2 + 409600)^2
$73$	\( T^{20} + \cdots + 24\!\cdots\!76 \) T^20 - 426T^18 + 119925T^16 - 19583954T^14 + 2331001796T^12 - 167012484922T^10 + 8242839321853T^8 - 126203974286786T^6 + 1402297859777297T^4 - 6932844687240256*T^2 + 24949663242981376
$79$	\( T^{20} + \cdots + 42\!\cdots\!56 \) T^20 - 352T^18 + 78624T^16 - 10615680T^14 + 1042881280T^12 - 71477377024T^10 + 3659456786432T^8 - 129061068226560T^6 + 3253355863408640T^4 - 45787579584348160*T^2 + 420971005037510656
$83$	\( (T^{5} - 6 T^{4} - 28 T^{3} + \cdots + 64)^{4} \) (T^5 - 6T^4 - 28T^3 + 56T^2 + 208T + 64)^4
$89$	\( (T^{10} + 2 T^{9} + \cdots + 8522982400)^{2} \) (T^10 + 2T^9 + 316T^8 + 864T^7 + 77040T^6 + 237280T^5 + 7168000T^4 + 41394432T^3 + 543577344T^2 + 2011837440*T + 8522982400)^2
$97$	\( (T^{10} + 282 T^{8} + \cdots + 24049216)^{2} \) (T^10 + 282T^8 + 23055T^6 + 758658T^4 + 9600233T^2 + 24049216)^2
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\(n\)	\(52\)	\(785\)
\(\chi(n)\)	\(-\beta_{5}\)	\(-1\)