LMFDB - Embedded newform 160.6.c.b.129.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [160,6,Mod(129,160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(160, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("160.129");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 160 = 2^{5} \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	160.c (of order $2$, degree $1$, 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:	$25.6614111701$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4}\cdot 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			129.1
Root			$-1.61803i$ of defining polynomial
Character	$\chi$	$=$	160.129
Dual form			160.6.c.b.129.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-30.1803i q^{3} -55.9017 q^{5} -194.180i q^{7} -667.853 q^{9} +O(q^{10})$	$q-30.1803i q^{3} -55.9017 q^{5} -194.180i q^{7} -667.853 q^{9} +1687.13i q^{15} -5860.43 q^{21} -5068.74i q^{23} +3125.00 q^{25} +12822.2i q^{27} +1686.00 q^{29} +10855.0i q^{35} +21041.4 q^{41} +10157.5i q^{43} +37334.1 q^{45} -3217.05i q^{47} -20899.0 q^{49} -52256.9 q^{61} +129684. i q^{63} +63577.5i q^{67} -152976. q^{69} -94313.6i q^{75} +224690. q^{81} -116480. i q^{83} -50884.1i q^{87} -149286. q^{89} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 972 q^{9}+O(q^{10}) $ 4 * q - 972 * q^9	$ 4 q - 972 q^{9} - 14408 q^{21} + 12500 q^{25} + 6744 q^{29} + 95000 q^{45} - 67228 q^{49} - 302344 q^{69} + 485804 q^{81} - 597144 q^{89}+O(q^{100}) $ 4 * q - 972 * q^9 - 14408 * q^21 + 12500 * q^25 + 6744 * q^29 + 95000 * q^45 - 67228 * q^49 - 302344 * q^69 + 485804 * q^81 - 597144 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/160\mathbb{Z}\right)^\times$.

$n$	$31$	$97$	$101$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	− 30.1803i	− 1.93607i	−0.250816	−	0.968035i	$-0.580699\pi$
$3$	− 30.1803i	− 1.93607i	0.250816	−	0.968035i	$-0.419301\pi$
$4$	0	0
$5$	−55.9017	−1.00000
$5$	−55.9017	−1.00000
$6$	0	0
$7$	− 194.180i	− 1.49782i	−0.662671	−	0.748911i	$-0.730577\pi$
$7$	− 194.180i	− 1.49782i	0.662671	−	0.748911i	$-0.269423\pi$
$8$	0	0
$9$	−667.853	−2.74837
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	1687.13i	1.93607i
$16$	0	0
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	−5860.43	−2.89989
$22$	0	0
$23$	− 5068.74i	− 1.99793i	−0.0454752	−	0.998965i	$-0.514480\pi$
$23$	− 5068.74i	− 1.99793i	0.0454752	−	0.998965i	$-0.485520\pi$
$24$	0	0
$25$	3125.00	1.00000
$26$	0	0
$27$	12822.2i	3.38496i
$28$	0	0
$29$	1686.00	0.372274	0.186137	−	0.982524i	$-0.440403\pi$
$29$	1686.00	0.372274	0.186137	+	0.982524i	$0.440403\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	10855.0i	1.49782i
$36$	0	0
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	21041.4	1.95486	0.977428	−	0.211267i	$-0.0677588\pi$
$41$	21041.4	1.95486	0.977428	+	0.211267i	$0.0677588\pi$
$42$	0	0
$43$	10157.5i	0.837753i	0.908043	+	0.418877i	$0.137576\pi$
$43$	10157.5i	0.837753i	−0.908043	+	0.418877i	$0.862424\pi$
$44$	0	0
$45$	37334.1	2.74837
$46$	0	0
$47$	− 3217.05i	− 0.212429i	−0.994343	−	0.106214i	$-0.966127\pi$
$47$	− 3217.05i	− 0.212429i	0.994343	−	0.106214i	$-0.0338730\pi$
$48$	0	0
$49$	−20899.0	−1.24347
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−52256.9	−1.79812	−0.899061	−	0.437824i	$-0.855749\pi$
$61$	−52256.9	−1.79812	−0.899061	+	0.437824i	$0.855749\pi$
$62$	0	0
$63$	129684.i	4.11656i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	63577.5i	1.73028i	0.501530	+	0.865140i	$0.332771\pi$
$67$	63577.5i	1.73028i	−0.501530	+	0.865140i	$0.667229\pi$
$68$	0	0
$69$	−152976.	−3.86813
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	− 94313.6i	− 1.93607i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	224690.	3.80515
$82$	0	0
$83$	− 116480.i	− 1.85591i	−0.372694	−	0.927954i	$-0.621566\pi$
$83$	− 116480.i	− 1.85591i	0.372694	−	0.927954i	$-0.378434\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 50884.1i	− 0.720748i
$88$	0	0
$89$	−149286.	−1.99776	−0.998882	−	0.0472789i	$-0.984945\pi$
$89$	−149286.	−1.99776	−0.998882	+	0.0472789i	$0.984945\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	137502.	1.34124	0.670619	−	0.741802i	$-0.266029\pi$
$101$	137502.	1.34124	0.670619	+	0.741802i	$0.266029\pi$
$102$	0	0
$103$	− 110367.i	− 1.02506i	−0.858670	−	0.512528i	$-0.828709\pi$
$103$	− 110367.i	− 1.02506i	0.858670	−	0.512528i	$-0.171291\pi$
$104$	0	0
$105$	327608.	2.89989
$106$	0	0
$107$	6085.43i	0.0513845i	0.999670	+	0.0256922i	$0.00817900\pi$
$107$	6085.43i	0.0513845i	−0.999670	+	0.0256922i	$0.991821\pi$
$108$	0	0
$109$	−84456.3	−0.680872	−0.340436	−	0.940268i	$-0.610575\pi$
$109$	−84456.3	−0.680872	−0.340436	+	0.940268i	$0.610575\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	283351.i	1.99793i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−161051.	−1.00000
$122$	0	0
$123$	− 635037.i	− 3.78474i
$124$	0	0
$125$	−174693.	−1.00000
$126$	0	0
$127$	58431.5i	0.321468i	0.986998	+	0.160734i	$0.0513861\pi$
$127$	58431.5i	0.321468i	−0.986998	+	0.160734i	$0.948614\pi$
$128$	0	0
$129$	306557.	1.62195
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	− 716783.i	− 3.38496i
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	−97091.7	−0.411277
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−94250.3	−0.372274
$146$	0	0
$147$	630739.i	2.40745i
$148$	0	0
$149$	−431583.	−1.59257	−0.796286	−	0.604920i	$-0.793205\pi$
$149$	−431583.	−1.59257	−0.796286	+	0.604920i	$0.793205\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−984250.	−2.99254
$162$	0	0
$163$	169773.i	0.500494i	0.968182	+	0.250247i	$0.0805118\pi$
$163$	169773.i	0.500494i	−0.968182	+	0.250247i	$0.919488\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 330923.i	− 0.918198i	−0.888385	−	0.459099i	$-0.848172\pi$
$167$	− 330923.i	− 0.918198i	0.888385	−	0.459099i	$-0.151828\pi$
$168$	0	0
$169$	371293.	1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	− 606814.i	− 1.49782i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	−320402.	−0.726940	−0.363470	−	0.931606i	$-0.618408\pi$
$181$	−320402.	−0.726940	−0.363470	+	0.931606i	$0.618408\pi$
$182$	0	0
$183$	1.57713e6i	3.48129i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	2.48982e6	5.07006
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	1.91879e6	3.34994
$202$	0	0
$203$	− 327388.i	− 0.557600i
$204$	0	0
$205$	−1.17625e6	−1.95486
$206$	0	0
$207$	3.38517e6i	5.49105i
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 567822.i	− 0.837753i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 374640.i	− 0.504490i	−0.967663	−	0.252245i	$-0.918831\pi$
$223$	− 374640.i	− 0.504490i	0.967663	−	0.252245i	$-0.0811689\pi$
$224$	0	0
$225$	−2.08704e6	−2.74837
$226$	0	0
$227$	979335.i	1.26144i	0.776011	+	0.630720i	$0.217240\pi$
$227$	979335.i	1.26144i	−0.776011	+	0.630720i	$0.782760\pi$
$228$	0	0
$229$	1.06681e6	1.34431	0.672156	−	0.740410i	$-0.265368\pi$
$229$	1.06681e6	1.34431	0.672156	+	0.740410i	$0.265368\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	179839.i	0.212429i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	−1.42086e6	−1.57583	−0.787916	−	0.615782i	$-0.788840\pi$
$241$	−1.42086e6	−1.57583	−0.787916	+	0.615782i	$0.788840\pi$
$242$	0	0
$243$	− 3.66543e6i	− 3.98208i
$244$	0	0
$245$	1.16829e6	1.24347
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−3.51541e6	−3.59317
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−1.12600e6	−1.02314
$262$	0	0
$263$	− 324007.i	− 0.288845i	−0.989516	−	0.144423i	$-0.953868\pi$
$263$	− 324007.i	− 0.288845i	0.989516	−	0.144423i	$-0.0461325\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	4.50550e6i	3.86781i
$268$	0	0
$269$	1.35718e6	1.14356	0.571778	−	0.820409i	$-0.306254\pi$
$269$	1.35718e6	1.14356	0.571778	+	0.820409i	$0.306254\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	679161.	0.513106	0.256553	−	0.966530i	$-0.417413\pi$
$281$	679161.	0.513106	0.256553	+	0.966530i	$0.417413\pi$
$282$	0	0
$283$	2.68372e6i	1.99192i	0.0898251	+	0.995958i	$0.471369\pi$
$283$	2.68372e6i	1.99192i	−0.0898251	+	0.995958i	$0.528631\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 4.08583e6i	− 2.92803i
$288$	0	0
$289$	1.41986e6	1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	1.97239e6	1.25480
$302$	0	0
$303$	− 4.14986e6i	− 2.59673i
$304$	0	0
$305$	2.92125e6	1.79812
$306$	0	0
$307$	− 586606.i	− 0.355223i	−0.984101	−	0.177611i	$-0.943163\pi$
$307$	− 586606.i	− 0.355223i	0.984101	−	0.177611i	$-0.0568370\pi$
$308$	0	0
$309$	−3.33093e6	−1.98458
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0	0
$315$	− 7.24955e6i	− 4.11656i
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	183660.	0.0994840
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	2.54892e6i	1.31822i
$328$	0	0
$329$	−624688.	−0.318180
$330$	0	0
$331$	0	0		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 3.55409e6i	− 1.73028i
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	794587.i	0.364675i
$344$	0	0
$345$	8.55164e6	3.86813
$346$	0	0
$347$	− 1.82853e6i	− 0.815225i	−0.913155	−	0.407613i	$-0.866361\pi$
$347$	− 1.82853e6i	− 0.815225i	0.913155	−	0.407613i	$-0.133639\pi$
$348$	0	0
$349$	2.95461e6	1.29849	0.649243	−	0.760581i	$-0.275086\pi$
$349$	2.95461e6	1.29849	0.649243	+	0.760581i	$0.275086\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−2.47610e6	−1.00000
$362$	0	0
$363$	4.86057e6i	1.93607i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	4.80709e6i	1.86302i	0.363716	+	0.931510i	$0.381508\pi$
$367$	4.80709e6i	1.86302i	−0.363716	+	0.931510i	$0.618492\pi$
$368$	0	0
$369$	−1.40526e7	−5.37266
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	5.27229e6i	1.93607i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	0	0		−	1.00000i	$-0.5\pi$
$379$	0	0			1.00000i	$0.5\pi$
$380$	0	0
$381$	1.76348e6	0.622384
$382$	0	0
$383$	− 5.68236e6i	− 1.97939i	−0.143188	−	0.989695i	$-0.545735\pi$
$383$	− 5.68236e6i	− 1.97939i	0.143188	−	0.989695i	$-0.454265\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 6.78372e6i	− 2.30245i
$388$	0	0
$389$	−5.91961e6	−1.98344	−0.991720	−	0.128419i	$-0.959010\pi$
$389$	−5.91961e6	−1.98344	−0.991720	+	0.128419i	$0.959010\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	4.66850e6	1.44983	0.724914	−	0.688840i	$-0.241880\pi$
$401$	4.66850e6	1.44983	0.724914	+	0.688840i	$0.241880\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−1.25606e7	−3.80515
$406$	0	0
$407$	0	0
$408$	0	0
$409$	5.95306e6	1.75967	0.879837	−	0.475276i	$-0.157652\pi$
$409$	5.95306e6	1.75967	0.879837	+	0.475276i	$0.157652\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	6.51144e6i	1.85591i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	−4.04525e6	−1.11235	−0.556173	−	0.831067i	$-0.687731\pi$
$421$	−4.04525e6	−1.11235	−0.556173	+	0.831067i	$0.687731\pi$
$422$	0	0
$423$	2.14852e6i	0.583832i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	1.01473e7i	2.69327i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	2.84451e6i	0.720748i
$436$	0	0
$437$	0	0
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	1.39575e7	3.41751
$442$	0	0
$443$	− 7.75688e6i	− 1.87792i	−0.344023	−	0.938961i	$-0.611790\pi$
$443$	− 7.75688e6i	− 1.87792i	0.344023	−	0.938961i	$-0.388210\pi$
$444$	0	0
$445$	8.34534e6	1.99776
$446$	0	0
$447$	1.30253e7i	3.08333i
$448$	0	0
$449$	3.02998e6	0.709291	0.354646	−	0.935001i	$-0.384602\pi$
$449$	3.02998e6	0.709291	0.354646	+	0.935001i	$0.384602\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−4.54780e6	−0.996665	−0.498333	−	0.866986i	$-0.666054\pi$
$461$	−4.54780e6	−0.996665	−0.498333	+	0.866986i	$0.666054\pi$
$462$	0	0
$463$	− 6.05420e6i	− 1.31252i	−0.754537	−	0.656258i	$-0.772138\pi$
$463$	− 6.05420e6i	− 1.31252i	0.754537	−	0.656258i	$-0.227862\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 8.90843e6i	− 1.89020i	−0.326774	−	0.945102i	$-0.605962\pi$
$467$	− 8.90843e6i	− 1.89020i	0.326774	−	0.945102i	$-0.394038\pi$
$468$	0	0
$469$	1.23455e7	2.59165
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	2.97050e7i	5.79377i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 1.00866e7i	− 1.92718i	−0.267388	−	0.963589i	$-0.586161\pi$
$487$	− 1.00866e7i	− 1.92718i	0.267388	−	0.963589i	$-0.413839\pi$
$488$	0	0
$489$	5.12380e6	0.968991
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	0	0		−	1.00000i	$-0.5\pi$
$499$	0	0			1.00000i	$0.5\pi$
$500$	0	0
$501$	−9.98738e6	−1.77770
$502$	0	0
$503$	− 3.13785e6i	− 0.552983i	−0.961016	−	0.276491i	$-0.910828\pi$
$503$	− 3.13785e6i	− 0.552983i	0.961016	−	0.276491i	$-0.0891717\pi$
$504$	0	0
$505$	−7.68660e6	−1.34124
$506$	0	0
$507$	− 1.12057e7i	− 1.93607i
$508$	0	0
$509$	7.23049e6	1.23701	0.618505	−	0.785781i	$-0.287739\pi$
$509$	7.23049e6	1.23701	0.618505	+	0.785781i	$0.287739\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	6.16972e6i	1.02506i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−5.30540e6	−0.856295	−0.428148	−	0.903709i	$-0.640834\pi$
$521$	−5.30540e6	−0.856295	−0.428148	+	0.903709i	$0.640834\pi$
$522$	0	0
$523$	1.15542e7i	1.84707i	0.383509	+	0.923537i	$0.374715\pi$
$523$	1.15542e7i	1.84707i	−0.383509	+	0.923537i	$0.625285\pi$
$524$	0	0
$525$	−1.83138e7	−2.89989
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−1.92558e7	−2.99173
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	− 340186.i	− 0.0513845i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−1.35842e7	−1.99545	−0.997725	−	0.0674113i	$-0.978526\pi$
$541$	−1.35842e7	−1.99545	−0.997725	+	0.0674113i	$0.978526\pi$
$542$	0	0
$543$	9.66984e6i	1.40741i
$544$	0	0
$545$	4.72125e6	0.680872
$546$	0	0
$547$	− 6.05134e6i	− 0.864736i	−0.901697	−	0.432368i	$-0.857678\pi$
$547$	− 6.05134e6i	− 0.864736i	0.901697	−	0.432368i	$-0.142322\pi$
$548$	0	0
$549$	3.48999e7	4.94190
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 1.20975e7i	− 1.60852i	−0.594281	−	0.804258i	$-0.702563\pi$
$563$	− 1.20975e7i	− 1.60852i	0.594281	−	0.804258i	$-0.297437\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 4.36304e7i	− 5.69944i
$568$	0	0
$569$	6.53945e6	0.846760	0.423380	−	0.905952i	$-0.360843\pi$
$569$	6.53945e6	0.846760	0.423380	+	0.905952i	$0.360843\pi$
$570$	0	0
$571$	0	0		−	1.00000i	$-0.5\pi$
$571$	0	0			1.00000i	$0.5\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 1.58398e7i	− 1.99793i
$576$	0	0
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−2.26181e7	−2.77982
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	4.02285e6i	0.481880i	0.970540	+	0.240940i	$0.0774557\pi$
$587$	4.02285e6i	0.481880i	−0.970540	+	0.240940i	$0.922544\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	−1.76169e7	−1.98950	−0.994748	−	0.102358i	$-0.967361\pi$
$601$	−1.76169e7	−1.98950	−0.994748	+	0.102358i	$0.967361\pi$
$602$	0	0
$603$	− 4.24604e7i	− 4.75544i
$604$	0	0
$605$	9.00302e6	1.00000
$606$	0	0
$607$	1.36145e7i	1.49979i	0.661555	+	0.749897i	$0.269897\pi$
$607$	1.36145e7i	1.49979i	−0.661555	+	0.749897i	$0.730103\pi$
$608$	0	0
$609$	−9.88068e6	−1.07955
$610$	0	0
$611$	0	0
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	3.54996e7i	3.78474i
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	0	0		−	1.00000i	$-0.5\pi$
$619$	0	0			1.00000i	$0.5\pi$
$620$	0	0
$621$	6.49924e7	6.76291
$622$	0	0
$623$	2.89884e7i	2.99229i
$624$	0	0
$625$	9.76562e6	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 3.26642e6i	− 0.321468i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−7.61048e6	−0.731589	−0.365794	−	0.930696i	$-0.619203\pi$
$641$	−7.61048e6	−0.731589	−0.365794	+	0.930696i	$0.619203\pi$
$642$	0	0
$643$	− 1.50256e7i	− 1.43319i	−0.697489	−	0.716596i	$-0.745699\pi$
$643$	− 1.50256e7i	− 1.43319i	0.697489	−	0.716596i	$-0.254301\pi$
$644$	0	0
$645$	−1.71371e7	−1.62195
$646$	0	0
$647$	− 1.03477e7i	− 0.971818i	−0.874009	−	0.485909i	$-0.838489\pi$
$647$	− 1.03477e7i	− 0.971818i	0.874009	−	0.485909i	$-0.161511\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	2.19397e7	1.95311	0.976554	−	0.215275i	$-0.0690647\pi$
$661$	2.19397e7	1.95311	0.976554	+	0.215275i	$0.0690647\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 8.54590e6i	− 0.743778i
$668$	0	0
$669$	−1.13068e7	−0.976728
$670$	0	0
$671$	0	0
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	4.00694e7i	3.38496i
$676$	0	0
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	2.95566e7	2.44223
$682$	0	0
$683$	− 1.21745e7i	− 0.998619i	−0.866424	−	0.499310i	$-0.833587\pi$
$683$	− 1.21745e7i	− 0.998619i	0.866424	−	0.499310i	$-0.166413\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 3.21968e7i	− 2.60268i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	0	0		−	1.00000i	$-0.5\pi$
$691$	0	0			1.00000i	$0.5\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−2.13434e7	−1.64047	−0.820235	−	0.572027i	$-0.806157\pi$
$701$	−2.13434e7	−1.64047	−0.820235	+	0.572027i	$0.806157\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	5.42759e6	0.411277
$706$	0	0
$707$	− 2.67002e7i	− 2.00893i
$708$	0	0
$709$	−2.34765e7	−1.75395	−0.876977	−	0.480533i	$-0.840443\pi$
$709$	−2.34765e7	−1.75395	−0.876977	+	0.480533i	$0.840443\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	−2.14312e7	−1.53535
$722$	0	0
$723$	4.28822e7i	3.05092i
$724$	0	0
$725$	5.26875e6	0.372274
$726$	0	0
$727$	2.63437e7i	1.84859i	0.381681	+	0.924294i	$0.375345\pi$
$727$	2.63437e7i	1.84859i	−0.381681	+	0.924294i	$0.624655\pi$
$728$	0	0
$729$	−5.60243e7	−3.90443
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	− 3.52594e7i	− 2.40745i
$736$	0	0
$737$	0	0
$738$	0	0
$739$	0	0		−	1.00000i	$-0.5\pi$
$739$	0	0			1.00000i	$0.5\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	2.93648e7i	1.95144i	0.219018	+	0.975721i	$0.429715\pi$
$743$	2.93648e7i	1.95144i	−0.219018	+	0.975721i	$0.570285\pi$
$744$	0	0
$745$	2.41262e7	1.59257
$746$	0	0
$747$	7.77916e7i	5.10072i
$748$	0	0
$749$	1.18167e6	0.0769648
$750$	0	0
$751$	0	0		−	1.00000i	$-0.5\pi$
$751$	0	0			1.00000i	$0.5\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−2.95982e7	−1.85269	−0.926347	−	0.376672i	$-0.877069\pi$
$761$	−2.95982e7	−1.85269	−0.926347	+	0.376672i	$0.877069\pi$
$762$	0	0
$763$	1.63998e7i	1.01983i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−3.13824e7	−1.91369	−0.956843	−	0.290607i	$-0.906143\pi$
$769$	−3.13824e7	−1.91369	−0.956843	+	0.290607i	$0.906143\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	2.16182e7i	1.26013i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 3.41617e7i	− 1.96609i	−0.183368	−	0.983044i	$-0.558700\pi$
$787$	− 3.41617e7i	− 1.96609i	0.183368	−	0.983044i	$-0.441300\pi$
$788$	0	0
$789$	−9.77864e6	−0.559224
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	9.97011e7	5.49058
$802$	0	0
$803$	0	0
$804$	0	0
$805$	5.50212e7	2.99254
$806$	0	0
$807$	− 4.09602e7i	− 2.21400i
$808$	0	0
$809$	1.06699e6	0.0573175	0.0286588	−	0.999589i	$-0.490876\pi$
$809$	1.06699e6	0.0573175	0.0286588	+	0.999589i	$0.490876\pi$
$810$	0	0
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 9.49058e6i	− 0.500494i
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	9.63269e6	0.498758	0.249379	−	0.968406i	$-0.419774\pi$
$821$	9.63269e6	0.498758	0.249379	+	0.968406i	$0.419774\pi$
$822$	0	0
$823$	3.18414e7i	1.63868i	0.573311	+	0.819338i	$0.305659\pi$
$823$	3.18414e7i	1.63868i	−0.573311	+	0.819338i	$0.694341\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	3.13336e7i	1.59311i	0.604565	+	0.796556i	$0.293347\pi$
$827$	3.13336e7i	1.59311i	−0.604565	+	0.796556i	$0.706653\pi$
$828$	0	0
$829$	3.65262e7	1.84594	0.922972	−	0.384867i	$-0.125753\pi$
$829$	3.65262e7	1.84594	0.922972	+	0.384867i	$0.125753\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	1.84992e7i	0.918198i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−1.76686e7	−0.861412
$842$	0	0
$843$	− 2.04973e7i	− 0.993409i
$844$	0	0
$845$	−2.07559e7	−1.00000
$846$	0	0
$847$	3.12729e7i	1.49782i
$848$	0	0
$849$	8.09955e7	3.85649
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	0	0		−	1.00000i	$-0.5\pi$
$859$	0	0			1.00000i	$0.5\pi$
$860$	0	0
$861$	−1.23312e8	−5.66887
$862$	0	0
$863$	− 4.37437e7i	− 1.99935i	−0.0254879	−	0.999675i	$-0.508114\pi$
$863$	− 4.37437e7i	− 1.99935i	0.0254879	−	0.999675i	$-0.491886\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 4.28518e7i	− 1.93607i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	3.39219e7i	1.49782i
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	2.42030e7	1.05058	0.525291	−	0.850922i	$-0.323956\pi$
$881$	2.42030e7	1.05058	0.525291	+	0.850922i	$0.323956\pi$
$882$	0	0
$883$	− 4.17673e7i	− 1.80275i	−0.433043	−	0.901373i	$-0.642560\pi$
$883$	− 4.17673e7i	− 1.80275i	0.433043	−	0.901373i	$-0.357440\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	4.64402e7i	1.98192i	0.134171	+	0.990958i	$0.457163\pi$
$887$	4.64402e7i	1.98192i	−0.134171	+	0.990958i	$0.542837\pi$
$888$	0	0
$889$	1.13462e7	0.481502
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	− 5.95274e7i	− 2.42939i
$904$	0	0
$905$	1.79110e7	0.726940
$906$	0	0
$907$	− 2.07876e7i	− 0.839046i	−0.907745	−	0.419523i	$-0.862197\pi$
$907$	− 2.07876e7i	− 0.839046i	0.907745	−	0.419523i	$-0.137803\pi$
$908$	0	0
$909$	−9.18311e7	−3.68621
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	− 8.81643e7i	− 3.48129i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	0	0		−	1.00000i	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	0	0
$921$	−1.77040e7	−0.687736
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	7.37092e7i	2.81723i
$928$	0	0
$929$	5.26024e7	1.99971	0.999853	−	0.0171747i	$-0.00546715\pi$
$929$	5.26024e7	1.99971	0.999853	+	0.0171747i	$0.00546715\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	3.19418e7	1.17594	0.587970	−	0.808883i	$-0.299927\pi$
$941$	3.19418e7	1.17594	0.587970	+	0.808883i	$0.299927\pi$
$942$	0	0
$943$	− 1.06653e8i	− 3.90567i
$944$	0	0
$945$	−1.39185e8	−5.07006
$946$	0	0
$947$	640893.i	0.0232226i	0.999933	+	0.0116113i	$0.00369607\pi$
$947$	640893.i	0.0232226i	−0.999933	+	0.0116113i	$0.996304\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−2.86292e7	−1.00000
$962$	0	0
$963$	− 4.06417e6i	− 0.141223i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	1.80869e7i	0.622012i	0.950408	+	0.311006i	$0.100666\pi$
$967$	1.80869e7i	0.622012i	−0.950408	+	0.311006i	$0.899334\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	5.64044e7	1.87129
$982$	0	0
$983$	5.66561e7i	1.87009i	0.354526	+	0.935046i	$0.384642\pi$
$983$	5.66561e7i	1.87009i	−0.354526	+	0.935046i	$0.615358\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	1.88533e7i	0.616020i
$988$	0	0
$989$	5.14858e7	1.67377
$990$	0	0
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	160.6.c.b.129.1	✓	4
4.3	odd	2	inner	160.6.c.b.129.4	yes	4
5.2	odd	4		800.6.a.f.1.1		2
5.3	odd	4		800.6.a.m.1.2		2
5.4	even	2	inner	160.6.c.b.129.4	yes	4
8.3	odd	2		320.6.c.h.129.1		4
8.5	even	2		320.6.c.h.129.4		4
20.3	even	4		800.6.a.f.1.1		2
20.7	even	4		800.6.a.m.1.2		2
20.19	odd	2	CM	160.6.c.b.129.1	✓	4
40.19	odd	2		320.6.c.h.129.4		4
40.29	even	2		320.6.c.h.129.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.6.c.b.129.1	✓	4	1.1	even	1	trivial
160.6.c.b.129.1	✓	4	20.19	odd	2	CM
160.6.c.b.129.4	yes	4	4.3	odd	2	inner
160.6.c.b.129.4	yes	4	5.4	even	2	inner
320.6.c.h.129.1		4	8.3	odd	2
320.6.c.h.129.1		4	40.29	even	2
320.6.c.h.129.4		4	8.5	even	2
320.6.c.h.129.4		4	40.19	odd	2
800.6.a.f.1.1		2	5.2	odd	4
800.6.a.f.1.1		2	20.3	even	4
800.6.a.m.1.2		2	5.3	odd	4
800.6.a.m.1.2		2	20.7	even	4

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:	\( N \)	\(=\)	\( 160 = 2^{5} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	160.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-30.1803i q^{3} -55.9017 q^{5} -194.180i q^{7} -667.853 q^{9} +O(q^{10})\)	\(q-30.1803i q^{3} -55.9017 q^{5} -194.180i q^{7} -667.853 q^{9} +1687.13i q^{15} -5860.43 q^{21} -5068.74i q^{23} +3125.00 q^{25} +12822.2i q^{27} +1686.00 q^{29} +10855.0i q^{35} +21041.4 q^{41} +10157.5i q^{43} +37334.1 q^{45} -3217.05i q^{47} -20899.0 q^{49} -52256.9 q^{61} +129684. i q^{63} +63577.5i q^{67} -152976. q^{69} -94313.6i q^{75} +224690. q^{81} -116480. i q^{83} -50884.1i q^{87} -149286. q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 972 q^{9}+O(q^{10}) \) 4 * q - 972 * q^9	\( 4 q - 972 q^{9} - 14408 q^{21} + 12500 q^{25} + 6744 q^{29} + 95000 q^{45} - 67228 q^{49} - 302344 q^{69} + 485804 q^{81} - 597144 q^{89}+O(q^{100}) \) 4 * q - 972 * q^9 - 14408 * q^21 + 12500 * q^25 + 6744 * q^29 + 95000 * q^45 - 67228 * q^49 - 302344 * q^69 + 485804 * q^81 - 597144 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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