LMFDB - Embedded newform 3200.2.d.i.1601.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3200,2,Mod(1601,3200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3200, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3200.1601");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3200 = 2^{7} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3200.d (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:	$25.5521286468$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 4x^{2} + 9 $ x^4 + 4*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 640)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			1601.3
Root			$-0.707107 + 1.58114i$ of defining polynomial
Character	$\chi$	$=$	3200.1601
Dual form			3200.2.d.i.1601.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.16228i q^{3} -4.24264 q^{7} -7.00000 q^{9} +O(q^{10})$	$q+3.16228i q^{3} -4.24264 q^{7} -7.00000 q^{9} -13.4164i q^{21} +1.41421 q^{23} -12.6491i q^{27} -8.94427i q^{29} -12.0000 q^{41} +3.16228i q^{43} +9.89949 q^{47} +11.0000 q^{49} -13.4164i q^{61} +29.6985 q^{63} +15.8114i q^{67} +4.47214i q^{69} +19.0000 q^{81} +9.48683i q^{83} +28.2843 q^{87} +6.00000 q^{89} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 28 q^{9}+O(q^{10}) $ 4 * q - 28 * q^9	$ 4 q - 28 q^{9} - 48 q^{41} + 44 q^{49} + 76 q^{81} + 24 q^{89}+O(q^{100}) $ 4 * q - 28 * q^9 - 48 * q^41 + 44 * q^49 + 76 * q^81 + 24 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3200\mathbb{Z}\right)^\times$.

$n$	$901$	$1151$	$2177$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	3.16228i	1.82574i	0.408248	+	0.912871i	$0.366140\pi$
$3$	3.16228i	1.82574i	−0.408248	+	0.912871i	$0.633860\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4.24264	−1.60357	−0.801784	−	0.597614i	$-0.796115\pi$
$7$	−4.24264	−1.60357	−0.801784	+	0.597614i	$0.796115\pi$
$8$	0	0
$9$	−7.00000	−2.33333
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	− 13.4164i	− 2.92770i
$22$	0	0
$23$	1.41421	0.294884	0.147442	−	0.989071i	$-0.452896\pi$
$23$	1.41421	0.294884	0.147442	+	0.989071i	$0.452896\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 12.6491i	− 2.43432i
$28$	0	0
$29$	− 8.94427i	− 1.66091i	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	− 8.94427i	− 1.66091i	0.557086	−	0.830455i	$-0.311919\pi$
$30$	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
$35$	0	0
$36$	0	0
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−12.0000	−1.87409	−0.937043	−	0.349215i	$-0.886448\pi$
$41$	−12.0000	−1.87409	−0.937043	+	0.349215i	$0.886448\pi$
$42$	0	0
$43$	3.16228i	0.482243i	0.970495	+	0.241121i	$0.0775152\pi$
$43$	3.16228i	0.482243i	−0.970495	+	0.241121i	$0.922485\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	9.89949	1.44399	0.721995	−	0.691898i	$-0.243225\pi$
$47$	9.89949	1.44399	0.721995	+	0.691898i	$0.243225\pi$
$48$	0	0
$49$	11.0000	1.57143
$50$	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	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	− 13.4164i	− 1.71780i	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	− 13.4164i	− 1.71780i	0.512148	−	0.858898i	$-0.328850\pi$
$62$	0	0
$63$	29.6985	3.74166
$64$	0	0
$65$	0	0
$66$	0	0
$67$	15.8114i	1.93167i	0.259161	+	0.965834i	$0.416554\pi$
$67$	15.8114i	1.93167i	−0.259161	+	0.965834i	$0.583446\pi$
$68$	0	0
$69$	4.47214i	0.538382i
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	0	0
$75$	0	0
$76$	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
$81$	19.0000	2.11111
$82$	0	0
$83$	9.48683i	1.04132i	0.853766	+	0.520658i	$0.174313\pi$
$83$	9.48683i	1.04132i	−0.853766	+	0.520658i	$0.825687\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	28.2843	3.03239
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 8.94427i	− 0.889988i	−0.895533	−	0.444994i	$-0.853206\pi$
$101$	− 8.94427i	− 0.889988i	0.895533	−	0.444994i	$-0.146794\pi$
$102$	0	0
$103$	12.7279	1.25412	0.627060	−	0.778971i	$-0.284258\pi$
$103$	12.7279	1.25412	0.627060	+	0.778971i	$0.284258\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 9.48683i	− 0.917127i	−0.888662	−	0.458563i	$-0.848364\pi$
$107$	− 9.48683i	− 0.917127i	0.888662	−	0.458563i	$-0.151636\pi$
$108$	0	0
$109$	− 13.4164i	− 1.28506i	−0.766261	−	0.642529i	$-0.777885\pi$
$109$	− 13.4164i	− 1.28506i	0.766261	−	0.642529i	$-0.222115\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	11.0000	1.00000
$122$	0	0
$123$	− 37.9473i	− 3.42160i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−4.24264	−0.376473	−0.188237	−	0.982124i	$-0.560277\pi$
$127$	−4.24264	−0.376473	−0.188237	+	0.982124i	$0.560277\pi$
$128$	0	0
$129$	−10.0000	−0.880451
$130$	0	0
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	31.3050i	2.63635i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	34.7851i	2.86902i
$148$	0	0
$149$	4.47214i	0.366372i	0.983078	+	0.183186i	$0.0586410\pi$
$149$	4.47214i	0.366372i	−0.983078	+	0.183186i	$0.941359\pi$
$150$	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
$161$	−6.00000	−0.472866
$162$	0	0
$163$	− 22.1359i	− 1.73382i	−0.498464	−	0.866910i	$-0.666102\pi$
$163$	− 22.1359i	− 1.73382i	0.498464	−	0.866910i	$-0.333898\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−24.0416	−1.86040	−0.930199	−	0.367057i	$-0.880366\pi$
$167$	−24.0416	−1.86040	−0.930199	+	0.367057i	$0.880366\pi$
$168$	0	0
$169$	13.0000	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
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	0	0
$181$	26.8328i	1.99447i	0.0743294	+	0.997234i	$0.476318\pi$
$181$	26.8328i	1.99447i	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	0	0
$183$	42.4264	3.13625
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	53.6656i	3.90360i
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	0	0		−	1.00000i	$-0.5\pi$
$193$	0	0			1.00000i	$0.5\pi$
$194$	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
$201$	−50.0000	−3.52673
$202$	0	0
$203$	37.9473i	2.66338i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−9.89949	−0.688062
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−29.6985	−1.98876	−0.994379	−	0.105881i	$-0.966234\pi$
$223$	−29.6985	−1.98876	−0.994379	+	0.105881i	$0.966234\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 28.4605i	− 1.88899i	−0.328526	−	0.944495i	$-0.606552\pi$
$227$	− 28.4605i	− 1.88899i	0.328526	−	0.944495i	$-0.393448\pi$
$228$	0	0
$229$	− 26.8328i	− 1.77316i	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	− 26.8328i	− 1.77316i	0.462573	−	0.886581i	$-0.346926\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	0	0
$236$	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
$241$	28.0000	1.80364	0.901819	−	0.432113i	$-0.142232\pi$
$241$	28.0000	1.80364	0.901819	+	0.432113i	$0.142232\pi$
$242$	0	0
$243$	22.1359i	1.42002i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−30.0000	−1.90117
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	62.6099i	3.87546i
$262$	0	0
$263$	15.5563	0.959246	0.479623	−	0.877475i	$-0.340774\pi$
$263$	15.5563	0.959246	0.479623	+	0.877475i	$0.340774\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	18.9737i	1.16117i
$268$	0	0
$269$	− 22.3607i	− 1.36335i	−0.731653	−	0.681677i	$-0.761251\pi$
$269$	− 22.3607i	− 1.36335i	0.731653	−	0.681677i	$-0.238749\pi$
$270$	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
$281$	−12.0000	−0.715860	−0.357930	−	0.933748i	$-0.616517\pi$
$281$	−12.0000	−0.715860	−0.357930	+	0.933748i	$0.616517\pi$
$282$	0	0
$283$	− 15.8114i	− 0.939889i	−0.882696	−	0.469945i	$-0.844274\pi$
$283$	− 15.8114i	− 0.939889i	0.882696	−	0.469945i	$-0.155726\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	50.9117	3.00522
$288$	0	0
$289$	−17.0000	−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
$301$	− 13.4164i	− 0.773309i
$302$	0	0
$303$	28.2843	1.62489
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 34.7851i	− 1.98529i	−0.121070	−	0.992644i	$-0.538633\pi$
$307$	− 34.7851i	− 1.98529i	0.121070	−	0.992644i	$-0.461367\pi$
$308$	0	0
$309$	40.2492i	2.28970i
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	0	0		−	1.00000i	$-0.5\pi$
$313$	0	0			1.00000i	$0.5\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	30.0000	1.67444
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	42.4264	2.34619
$328$	0	0
$329$	−42.0000	−2.31553
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	0	0		−	1.00000i	$-0.5\pi$
$337$	0	0			1.00000i	$0.5\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−16.9706	−0.916324
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 28.4605i	− 1.52784i	−0.645311	−	0.763920i	$-0.723272\pi$
$347$	− 28.4605i	− 1.52784i	0.645311	−	0.763920i	$-0.276728\pi$
$348$	0	0
$349$	26.8328i	1.43633i	0.695874	+	0.718164i	$0.255017\pi$
$349$	26.8328i	1.43633i	−0.695874	+	0.718164i	$0.744983\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	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
$361$	19.0000	1.00000
$362$	0	0
$363$	34.7851i	1.82574i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−38.1838	−1.99318	−0.996588	−	0.0825348i	$-0.973698\pi$
$367$	−38.1838	−1.99318	−0.996588	+	0.0825348i	$0.973698\pi$
$368$	0	0
$369$	84.0000	4.37287
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	− 13.4164i	− 0.687343i
$382$	0	0
$383$	26.8701	1.37300	0.686498	−	0.727132i	$-0.259147\pi$
$383$	26.8701	1.37300	0.686498	+	0.727132i	$0.259147\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 22.1359i	− 1.12523i
$388$	0	0
$389$	− 31.3050i	− 1.58722i	−0.608424	−	0.793612i	$-0.708198\pi$
$389$	− 31.3050i	− 1.58722i	0.608424	−	0.793612i	$-0.291802\pi$
$390$	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
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−4.00000	−0.197787	−0.0988936	−	0.995098i	$-0.531530\pi$
$409$	−4.00000	−0.197787	−0.0988936	+	0.995098i	$0.531530\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	− 40.2492i	− 1.96163i	−0.194948	−	0.980814i	$-0.562454\pi$
$421$	− 40.2492i	− 1.96163i	0.194948	−	0.980814i	$-0.437546\pi$
$422$	0	0
$423$	−69.2965	−3.36931
$424$	0	0
$425$	0	0
$426$	0	0
$427$	56.9210i	2.75460i
$428$	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		−	1.00000i	$-0.5\pi$
$433$	0	0			1.00000i	$0.5\pi$
$434$	0	0
$435$	0	0
$436$	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
$441$	−77.0000	−3.66667
$442$	0	0
$443$	9.48683i	0.450733i	0.974274	+	0.225367i	$0.0723580\pi$
$443$	9.48683i	0.450733i	−0.974274	+	0.225367i	$0.927642\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−14.1421	−0.668900
$448$	0	0
$449$	36.0000	1.69895	0.849473	−	0.527633i	$-0.176920\pi$
$449$	36.0000	1.69895	0.849473	+	0.527633i	$0.176920\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0	0		−	1.00000i	$-0.5\pi$
$457$	0	0			1.00000i	$0.5\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	8.94427i	0.416576i	0.978068	+	0.208288i	$0.0667892\pi$
$461$	8.94427i	0.416576i	−0.978068	+	0.208288i	$0.933211\pi$
$462$	0	0
$463$	12.7279	0.591517	0.295758	−	0.955263i	$-0.404428\pi$
$463$	12.7279	0.591517	0.295758	+	0.955263i	$0.404428\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 28.4605i	− 1.31699i	−0.752583	−	0.658497i	$-0.771192\pi$
$467$	− 28.4605i	− 1.31699i	0.752583	−	0.658497i	$-0.228808\pi$
$468$	0	0
$469$	− 67.0820i	− 3.09756i
$470$	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
$483$	− 18.9737i	− 0.863332i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−38.1838	−1.73027	−0.865136	−	0.501538i	$-0.832768\pi$
$487$	−38.1838	−1.73027	−0.865136	+	0.501538i	$0.832768\pi$
$488$	0	0
$489$	70.0000	3.16551
$490$	0	0
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	0	0
$501$	− 76.0263i	− 3.39661i
$502$	0	0
$503$	−43.8406	−1.95476	−0.977378	−	0.211498i	$-0.932166\pi$
$503$	−43.8406	−1.95476	−0.977378	+	0.211498i	$0.932166\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	41.1096i	1.82574i
$508$	0	0
$509$	− 44.7214i	− 1.98224i	−0.132973	−	0.991120i	$-0.542452\pi$
$509$	− 44.7214i	− 1.98224i	0.132973	−	0.991120i	$-0.457548\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	42.0000	1.84005	0.920027	−	0.391856i	$-0.128167\pi$
$521$	42.0000	1.84005	0.920027	+	0.391856i	$0.128167\pi$
$522$	0	0
$523$	34.7851i	1.52104i	0.649312	+	0.760522i	$0.275057\pi$
$523$	34.7851i	1.52104i	−0.649312	+	0.760522i	$0.724943\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−21.0000	−0.913043
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	− 26.8328i	− 1.15363i	−0.816874	−	0.576816i	$-0.804295\pi$
$541$	− 26.8328i	− 1.15363i	0.816874	−	0.576816i	$-0.195705\pi$
$542$	0	0
$543$	−84.8528	−3.64138
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 3.16228i	− 0.135209i	−0.997712	−	0.0676046i	$-0.978464\pi$
$547$	− 3.16228i	− 0.135209i	0.997712	−	0.0676046i	$-0.0215356\pi$
$548$	0	0
$549$	93.9149i	4.00819i
$550$	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
$563$	− 47.4342i	− 1.99911i	−0.0298010	−	0.999556i	$-0.509487\pi$
$563$	− 47.4342i	− 1.99911i	0.0298010	−	0.999556i	$-0.490513\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−80.6102	−3.38531
$568$	0	0
$569$	−36.0000	−1.50920	−0.754599	−	0.656186i	$-0.772169\pi$
$569$	−36.0000	−1.50920	−0.754599	+	0.656186i	$0.772169\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	0	0		−	1.00000i	$-0.5\pi$
$577$	0	0			1.00000i	$0.5\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 40.2492i	− 1.66982i
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	47.4342i	1.95782i	0.204298	+	0.978909i	$0.434509\pi$
$587$	47.4342i	1.95782i	−0.204298	+	0.978909i	$0.565491\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	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
$601$	−28.0000	−1.14214	−0.571072	−	0.820900i	$-0.693472\pi$
$601$	−28.0000	−1.14214	−0.571072	+	0.820900i	$0.693472\pi$
$602$	0	0
$603$	− 110.680i	− 4.50723i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−46.6690	−1.89424	−0.947119	−	0.320882i	$-0.896021\pi$
$607$	−46.6690	−1.89424	−0.947119	+	0.320882i	$0.896021\pi$
$608$	0	0
$609$	−120.000	−4.86265
$610$	0	0
$611$	0	0
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	− 17.8885i	− 0.717843i
$622$	0	0
$623$	−25.4558	−1.01987
$624$	0	0
$625$	0	0
$626$	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
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	12.0000	0.473972	0.236986	−	0.971513i	$-0.423841\pi$
$641$	12.0000	0.473972	0.236986	+	0.971513i	$0.423841\pi$
$642$	0	0
$643$	− 41.1096i	− 1.62120i	−0.585597	−	0.810602i	$-0.699140\pi$
$643$	− 41.1096i	− 1.62120i	0.585597	−	0.810602i	$-0.300860\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	18.3848	0.722780	0.361390	−	0.932415i	$-0.382302\pi$
$647$	18.3848	0.722780	0.361390	+	0.932415i	$0.382302\pi$
$648$	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	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	40.2492i	1.56551i	0.622328	+	0.782757i	$0.286187\pi$
$661$	40.2492i	1.56551i	−0.622328	+	0.782757i	$0.713813\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 12.6491i	− 0.489776i
$668$	0	0
$669$	− 93.9149i	− 3.63096i
$670$	0	0
$671$	0	0
$672$	0	0
$673$	0	0		−	1.00000i	$-0.5\pi$
$673$	0	0			1.00000i	$0.5\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	90.0000	3.44881
$682$	0	0
$683$	28.4605i	1.08901i	0.838757	+	0.544505i	$0.183283\pi$
$683$	28.4605i	1.08901i	−0.838757	+	0.544505i	$0.816717\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	84.8528	3.23734
$688$	0	0
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	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
$701$	− 22.3607i	− 0.844551i	−0.906467	−	0.422276i	$-0.861231\pi$
$701$	− 22.3607i	− 0.844551i	0.906467	−	0.422276i	$-0.138769\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	37.9473i	1.42716i
$708$	0	0
$709$	− 26.8328i	− 1.00773i	−0.863783	−	0.503864i	$-0.831911\pi$
$709$	− 26.8328i	− 1.00773i	0.863783	−	0.503864i	$-0.168089\pi$
$710$	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
$721$	−54.0000	−2.01107
$722$	0	0
$723$	88.5438i	3.29298i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−4.24264	−0.157351	−0.0786754	−	0.996900i	$-0.525069\pi$
$727$	−4.24264	−0.157351	−0.0786754	+	0.996900i	$0.525069\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	26.8701	0.985767	0.492883	−	0.870095i	$-0.335943\pi$
$743$	26.8701	0.985767	0.492883	+	0.870095i	$0.335943\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 66.4078i	− 2.42974i
$748$	0	0
$749$	40.2492i	1.47067i
$750$	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
$761$	−42.0000	−1.52250	−0.761249	−	0.648459i	$-0.775414\pi$
$761$	−42.0000	−1.52250	−0.761249	+	0.648459i	$0.775414\pi$
$762$	0	0
$763$	56.9210i	2.06068i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−14.0000	−0.504853	−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000	−0.504853	−0.252426	+	0.967616i	$0.581229\pi$
$770$	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
$783$	−113.137	−4.04319
$784$	0	0
$785$	0	0
$786$	0	0
$787$	41.1096i	1.46540i	0.680552	+	0.732700i	$0.261740\pi$
$787$	41.1096i	1.46540i	−0.680552	+	0.732700i	$0.738260\pi$
$788$	0	0
$789$	49.1935i	1.75133i
$790$	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
$801$	−42.0000	−1.48400
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	70.7107	2.48913
$808$	0	0
$809$	54.0000	1.89854	0.949269	−	0.314464i	$-0.101825\pi$
$809$	54.0000	1.89854	0.949269	+	0.314464i	$0.101825\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 31.3050i	− 1.09255i	−0.837606	−	0.546275i	$-0.816045\pi$
$821$	− 31.3050i	− 1.09255i	0.837606	−	0.546275i	$-0.183955\pi$
$822$	0	0
$823$	−55.1543	−1.92256	−0.961280	−	0.275575i	$-0.911132\pi$
$823$	−55.1543	−1.92256	−0.961280	+	0.275575i	$0.911132\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	47.4342i	1.64945i	0.565536	+	0.824724i	$0.308669\pi$
$827$	47.4342i	1.64945i	−0.565536	+	0.824724i	$0.691331\pi$
$828$	0	0
$829$	13.4164i	0.465971i	0.972480	+	0.232986i	$0.0748495\pi$
$829$	13.4164i	0.465971i	−0.972480	+	0.232986i	$0.925151\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	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
$841$	−51.0000	−1.75862
$842$	0	0
$843$	− 37.9473i	− 1.30698i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−46.6690	−1.60357
$848$	0	0
$849$	50.0000	1.71600
$850$	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		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	0	0
$861$	160.997i	5.48676i
$862$	0	0
$863$	57.9828	1.97376	0.986878	−	0.161468i	$-0.0516228\pi$
$863$	57.9828	1.97376	0.986878	+	0.161468i	$0.0516228\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 53.7587i	− 1.82574i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	12.0000	0.404290	0.202145	−	0.979356i	$-0.435209\pi$
$881$	12.0000	0.404290	0.202145	+	0.979356i	$0.435209\pi$
$882$	0	0
$883$	− 22.1359i	− 0.744934i	−0.928045	−	0.372467i	$-0.878512\pi$
$883$	− 22.1359i	− 0.744934i	0.928045	−	0.372467i	$-0.121488\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	52.3259	1.75693	0.878466	−	0.477805i	$-0.158567\pi$
$887$	52.3259	1.75693	0.878466	+	0.477805i	$0.158567\pi$
$888$	0	0
$889$	18.0000	0.603701
$890$	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
$903$	42.4264	1.41186
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 60.0833i	− 1.99503i	−0.0704373	−	0.997516i	$-0.522439\pi$
$907$	− 60.0833i	− 1.99503i	0.0704373	−	0.997516i	$-0.477561\pi$
$908$	0	0
$909$	62.6099i	2.07664i
$910$	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
$915$	0	0
$916$	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
$921$	110.000	3.62462
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−89.0955	−2.92628
$928$	0	0
$929$	36.0000	1.18112	0.590561	−	0.806993i	$-0.298907\pi$
$929$	36.0000	1.18112	0.590561	+	0.806993i	$0.298907\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0	0		−	1.00000i	$-0.5\pi$
$937$	0	0			1.00000i	$0.5\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	44.7214i	1.45787i	0.684580	+	0.728937i	$0.259985\pi$
$941$	44.7214i	1.45787i	−0.684580	+	0.728937i	$0.740015\pi$
$942$	0	0
$943$	−16.9706	−0.552638
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 9.48683i	− 0.308281i	−0.988049	−	0.154140i	$-0.950739\pi$
$947$	− 9.48683i	− 0.308281i	0.988049	−	0.154140i	$-0.0492608\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	66.4078i	2.13996i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−46.6690	−1.50078	−0.750388	−	0.660998i	$-0.770133\pi$
$967$	−46.6690	−1.50078	−0.750388	+	0.660998i	$0.770133\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0	0		−	1.00000i	$-0.5\pi$
$977$	0	0			1.00000i	$0.5\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	93.9149i	2.99847i
$982$	0	0
$983$	−41.0122	−1.30809	−0.654043	−	0.756457i	$-0.726928\pi$
$983$	−41.0122	−1.30809	−0.654043	+	0.756457i	$0.726928\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 132.816i	− 4.22757i
$988$	0	0
$989$	4.47214i	0.142206i
$990$	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	3200.2.d.i.1601.3		4
4.3	odd	2	inner	3200.2.d.i.1601.2		4
5.2	odd	4		640.2.f.f.449.4	yes	4
5.3	odd	4		640.2.f.f.449.2	yes	4
5.4	even	2	inner	3200.2.d.i.1601.2		4
8.3	odd	2	inner	3200.2.d.i.1601.4		4
8.5	even	2	inner	3200.2.d.i.1601.1		4
16.3	odd	4		6400.2.a.cv.1.3		4
16.5	even	4		6400.2.a.cv.1.4		4
16.11	odd	4		6400.2.a.cv.1.1		4
16.13	even	4		6400.2.a.cv.1.2		4
20.3	even	4		640.2.f.f.449.4	yes	4
20.7	even	4		640.2.f.f.449.2	yes	4
20.19	odd	2	CM	3200.2.d.i.1601.3		4
40.3	even	4		640.2.f.f.449.1	✓	4
40.13	odd	4		640.2.f.f.449.3	yes	4
40.19	odd	2	inner	3200.2.d.i.1601.1		4
40.27	even	4		640.2.f.f.449.3	yes	4
40.29	even	2	inner	3200.2.d.i.1601.4		4
40.37	odd	4		640.2.f.f.449.1	✓	4
80.3	even	4		1280.2.c.f.769.4		4
80.13	odd	4		1280.2.c.f.769.2		4
80.19	odd	4		6400.2.a.cv.1.2		4
80.27	even	4		1280.2.c.f.769.3		4
80.29	even	4		6400.2.a.cv.1.3		4
80.37	odd	4		1280.2.c.f.769.1		4
80.43	even	4		1280.2.c.f.769.1		4
80.53	odd	4		1280.2.c.f.769.3		4
80.59	odd	4		6400.2.a.cv.1.4		4
80.67	even	4		1280.2.c.f.769.2		4
80.69	even	4		6400.2.a.cv.1.1		4
80.77	odd	4		1280.2.c.f.769.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
640.2.f.f.449.1	✓	4	40.3	even	4
640.2.f.f.449.1	✓	4	40.37	odd	4
640.2.f.f.449.2	yes	4	5.3	odd	4
640.2.f.f.449.2	yes	4	20.7	even	4
640.2.f.f.449.3	yes	4	40.13	odd	4
640.2.f.f.449.3	yes	4	40.27	even	4
640.2.f.f.449.4	yes	4	5.2	odd	4
640.2.f.f.449.4	yes	4	20.3	even	4
1280.2.c.f.769.1		4	80.37	odd	4
1280.2.c.f.769.1		4	80.43	even	4
1280.2.c.f.769.2		4	80.13	odd	4
1280.2.c.f.769.2		4	80.67	even	4
1280.2.c.f.769.3		4	80.27	even	4
1280.2.c.f.769.3		4	80.53	odd	4
1280.2.c.f.769.4		4	80.3	even	4
1280.2.c.f.769.4		4	80.77	odd	4
3200.2.d.i.1601.1		4	8.5	even	2	inner
3200.2.d.i.1601.1		4	40.19	odd	2	inner
3200.2.d.i.1601.2		4	4.3	odd	2	inner
3200.2.d.i.1601.2		4	5.4	even	2	inner
3200.2.d.i.1601.3		4	1.1	even	1	trivial
3200.2.d.i.1601.3		4	20.19	odd	2	CM
3200.2.d.i.1601.4		4	8.3	odd	2	inner
3200.2.d.i.1601.4		4	40.29	even	2	inner
6400.2.a.cv.1.1		4	16.11	odd	4
6400.2.a.cv.1.1		4	80.69	even	4
6400.2.a.cv.1.2		4	16.13	even	4
6400.2.a.cv.1.2		4	80.19	odd	4
6400.2.a.cv.1.3		4	16.3	odd	4
6400.2.a.cv.1.3		4	80.29	even	4
6400.2.a.cv.1.4		4	16.5	even	4
6400.2.a.cv.1.4		4	80.59	odd	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:	\( N \)	\(=\)	\( 3200 = 2^{7} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3200.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.16228i q^{3} -4.24264 q^{7} -7.00000 q^{9} +O(q^{10})\)	\(q+3.16228i q^{3} -4.24264 q^{7} -7.00000 q^{9} -13.4164i q^{21} +1.41421 q^{23} -12.6491i q^{27} -8.94427i q^{29} -12.0000 q^{41} +3.16228i q^{43} +9.89949 q^{47} +11.0000 q^{49} -13.4164i q^{61} +29.6985 q^{63} +15.8114i q^{67} +4.47214i q^{69} +19.0000 q^{81} +9.48683i q^{83} +28.2843 q^{87} +6.00000 q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 28 q^{9}+O(q^{10}) \) 4 * q - 28 * q^9	\( 4 q - 28 q^{9} - 48 q^{41} + 44 q^{49} + 76 q^{81} + 24 q^{89}+O(q^{100}) \) 4 * q - 28 * q^9 - 48 * q^41 + 44 * q^49 + 76 * q^81 + 24 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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