LMFDB - Embedded newform 1600.2.f.j.1249.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1600,2,Mod(1249,1600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1600, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1600.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1600 = 2^{6} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1600.f (of order $2$, degree $1$, 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:	$12.7760643234$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			1249.1
Root			$1.22474 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	1600.1249
Dual form			1600.2.f.j.1249.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.44949 q^{3} -0.898979 q^{9} +O(q^{10})$	$q-1.44949 q^{3} -0.898979 q^{9} -0.550510i q^{11} +7.89898i q^{17} -8.34847i q^{19} +5.65153 q^{27} +0.797959i q^{33} +12.7980 q^{41} -10.0000 q^{43} +7.00000 q^{49} -11.4495i q^{51} +12.1010i q^{57} -6.00000i q^{59} +14.3485 q^{67} -13.6969i q^{73} -5.49490 q^{81} +11.4495 q^{83} +13.8990 q^{89} +10.0000i q^{97} +0.494897i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 16 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 + 16 * q^9	$ 4 q + 4 q^{3} + 16 q^{9} + 52 q^{27} + 12 q^{41} - 40 q^{43} + 28 q^{49} + 28 q^{67} + 76 q^{81} + 36 q^{83} + 36 q^{89}+O(q^{100}) $ 4 * q + 4 * q^3 + 16 * q^9 + 52 * q^27 + 12 * q^41 - 40 * q^43 + 28 * q^49 + 28 * q^67 + 76 * q^81 + 36 * q^83 + 36 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1151$
$\chi(n)$	$-1$	$-1$	$1$

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
$3$	−1.44949	−0.836863	−0.418432	−	0.908248i	$-0.637420\pi$
$3$	−1.44949	−0.836863	−0.418432	+	0.908248i	$0.637420\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	0	0
$9$	−0.898979	−0.299660
$10$	0	0
$11$	− 0.550510i	− 0.165985i	−0.996550	−	0.0829925i	$-0.973552\pi$
$11$	− 0.550510i	− 0.165985i	0.996550	−	0.0829925i	$-0.0264478\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	7.89898i	1.91578i	0.287129	+	0.957892i	$0.407299\pi$
$17$	7.89898i	1.91578i	−0.287129	+	0.957892i	$0.592701\pi$
$18$	0	0
$19$	− 8.34847i	− 1.91527i	−0.287984	−	0.957635i	$-0.592985\pi$
$19$	− 8.34847i	− 1.91527i	0.287984	−	0.957635i	$-0.407015\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.65153	1.08764
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	0.797959i	0.138907i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	12.7980	1.99871	0.999353	−	0.0359748i	$-0.0114536\pi$
$41$	12.7980	1.99871	0.999353	+	0.0359748i	$0.0114536\pi$
$42$	0	0
$43$	−10.0000	−1.52499	−0.762493	−	0.646997i	$-0.776025\pi$
$43$	−10.0000	−1.52499	−0.762493	+	0.646997i	$0.776025\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	7.00000	1.00000
$50$	0	0
$51$	− 11.4495i	− 1.60325i
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	12.1010i	1.60282i
$58$	0	0
$59$	− 6.00000i	− 0.781133i	−0.920575	−	0.390567i	$-0.872279\pi$
$59$	− 6.00000i	− 0.781133i	0.920575	−	0.390567i	$-0.127721\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	14.3485	1.75294	0.876472	−	0.481452i	$-0.159891\pi$
$67$	14.3485	1.75294	0.876472	+	0.481452i	$0.159891\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	− 13.6969i	− 1.60311i	−0.597924	−	0.801553i	$-0.704008\pi$
$73$	− 13.6969i	− 1.60311i	0.597924	−	0.801553i	$-0.295992\pi$
$74$	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
$81$	−5.49490	−0.610544
$82$	0	0
$83$	11.4495	1.25674	0.628372	−	0.777913i	$-0.283721\pi$
$83$	11.4495	1.25674	0.628372	+	0.777913i	$0.283721\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	13.8990	1.47329	0.736644	−	0.676280i	$-0.236409\pi$
$89$	13.8990	1.47329	0.736644	+	0.676280i	$0.236409\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	10.0000i	1.01535i	0.861550	+	0.507673i	$0.169494\pi$
$97$	10.0000i	1.01535i	−0.861550	+	0.507673i	$0.830506\pi$
$98$	0	0
$99$	0.494897i	0.0497391i
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	20.1464	1.94763	0.973814	−	0.227345i	$-0.0730044\pi$
$107$	20.1464	1.94763	0.973814	+	0.227345i	$0.0730044\pi$
$108$	0	0
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 0.797959i	− 0.0750657i	−0.999295	−	0.0375328i	$-0.988050\pi$
$113$	− 0.797959i	− 0.0750657i	0.999295	−	0.0375328i	$-0.0119499\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	10.6969	0.972449
$122$	0	0
$123$	−18.5505	−1.67264
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	0	0
$129$	14.4949	1.27620
$130$	0	0
$131$	− 18.0000i	− 1.57267i	−0.617802	−	0.786334i	$-0.711977\pi$
$131$	− 18.0000i	− 1.57267i	0.617802	−	0.786334i	$-0.288023\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 16.5959i	− 1.41788i	−0.705266	−	0.708942i	$-0.749173\pi$
$137$	− 16.5959i	− 1.41788i	0.705266	−	0.708942i	$-0.250827\pi$
$138$	0	0
$139$	18.3485i	1.55630i	0.628080	+	0.778148i	$0.283841\pi$
$139$	18.3485i	1.55630i	−0.628080	+	0.778148i	$0.716159\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−10.1464	−0.836863
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	− 7.10102i	− 0.574083i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	23.0454	1.80506	0.902528	−	0.430632i	$-0.141709\pi$
$163$	23.0454	1.80506	0.902528	+	0.430632i	$0.141709\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	7.50510i	0.573930i
$172$	0	0
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	8.69694i	0.653702i
$178$	0	0
$179$	− 26.1464i	− 1.95428i	−0.212607	−	0.977138i	$-0.568195\pi$
$179$	− 26.1464i	− 1.95428i	0.212607	−	0.977138i	$-0.431805\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	4.34847	0.317992
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	3.69694i	0.266111i	0.991109	+	0.133056i	$0.0424789\pi$
$193$	3.69694i	0.266111i	−0.991109	+	0.133056i	$0.957521\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	−20.7980	−1.46697
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−4.59592	−0.317906
$210$	0	0
$211$	15.0454i	1.03577i	0.855451	+	0.517884i	$0.173280\pi$
$211$	15.0454i	1.03577i	−0.855451	+	0.517884i	$0.826720\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	19.8536i	1.34158i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−30.0000	−1.99117	−0.995585	−	0.0938647i	$-0.970078\pi$
$227$	−30.0000	−1.99117	−0.995585	+	0.0938647i	$0.970078\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	30.0000i	1.96537i	0.185296	+	0.982683i	$0.440675\pi$
$233$	30.0000i	1.96537i	−0.185296	+	0.982683i	$0.559325\pi$
$234$	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
$241$	1.69694	0.109309	0.0546547	−	0.998505i	$-0.482594\pi$
$241$	1.69694	0.109309	0.0546547	+	0.998505i	$0.482594\pi$
$242$	0	0
$243$	−8.98979	−0.576696
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−16.5959	−1.05172
$250$	0	0
$251$	23.9444i	1.51136i	0.654943	+	0.755678i	$0.272693\pi$
$251$	23.9444i	1.51136i	−0.654943	+	0.755678i	$0.727307\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 30.0000i	− 1.87135i	−0.352865	−	0.935674i	$-0.614792\pi$
$257$	− 30.0000i	− 1.87135i	0.352865	−	0.935674i	$-0.385208\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−20.1464	−1.23294
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	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		−	1.00000i	$-0.5\pi$
$277$	0	0			1.00000i	$0.5\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	0	0
$283$	−33.0454	−1.96435	−0.982173	−	0.187980i	$-0.939806\pi$
$283$	−33.0454	−1.96435	−0.982173	+	0.187980i	$0.939806\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−45.3939	−2.67023
$290$	0	0
$291$	− 14.4949i	− 0.849706i
$292$	0	0
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 3.11123i	− 0.180532i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−24.3485	−1.38964	−0.694820	−	0.719183i	$-0.744516\pi$
$307$	−24.3485	−1.38964	−0.694820	+	0.719183i	$0.744516\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	− 10.0000i	− 0.565233i	−0.959233	−	0.282617i	$-0.908798\pi$
$313$	− 10.0000i	− 0.565233i	0.959233	−	0.282617i	$-0.0912024\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−29.2020	−1.62990
$322$	0	0
$323$	65.9444	3.66924
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	35.0454i	1.92627i	0.269019	+	0.963135i	$0.413301\pi$
$331$	35.0454i	1.92627i	−0.269019	+	0.963135i	$0.586699\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 22.3939i	− 1.21987i	−0.792451	−	0.609936i	$-0.791195\pi$
$337$	− 22.3939i	− 1.21987i	0.792451	−	0.609936i	$-0.208805\pi$
$338$	0	0
$339$	1.15663i	0.0628197i
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−28.8434	−1.54839	−0.774197	−	0.632945i	$-0.781846\pi$
$347$	−28.8434	−1.54839	−0.774197	+	0.632945i	$0.781846\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	30.0000i	1.59674i	0.602168	+	0.798369i	$0.294304\pi$
$353$	30.0000i	1.59674i	−0.602168	+	0.798369i	$0.705696\pi$
$354$	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
$361$	−50.6969	−2.66826
$362$	0	0
$363$	−15.5051	−0.813807
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	−11.5051	−0.598932
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0		−	1.00000i	$-0.5\pi$
$373$	0	0			1.00000i	$0.5\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	11.6515i	0.598499i	0.954175	+	0.299249i	$0.0967363\pi$
$379$	11.6515i	0.598499i	−0.954175	+	0.299249i	$0.903264\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	8.98979	0.456977
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	26.0908i	1.31611i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	37.2929	1.86232	0.931158	−	0.364615i	$-0.118800\pi$
$401$	37.2929	1.86232	0.931158	+	0.364615i	$0.118800\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	18.3939	0.909519	0.454759	−	0.890614i	$-0.349725\pi$
$409$	18.3939	0.909519	0.454759	+	0.890614i	$0.349725\pi$
$410$	0	0
$411$	24.0556i	1.18658i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 26.5959i	− 1.30241i
$418$	0	0
$419$	22.8434i	1.11597i	0.829851	+	0.557986i	$0.188426\pi$
$419$	22.8434i	1.11597i	−0.829851	+	0.557986i	$0.811574\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	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
$433$	− 33.6969i	− 1.61937i	−0.586864	−	0.809686i	$-0.699638\pi$
$433$	− 33.6969i	− 1.61937i	0.586864	−	0.809686i	$-0.300362\pi$
$434$	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
$441$	−6.29286	−0.299660
$442$	0	0
$443$	18.5505	0.881361	0.440681	−	0.897664i	$-0.354737\pi$
$443$	18.5505	0.881361	0.440681	+	0.897664i	$0.354737\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	16.1010	0.759854	0.379927	−	0.925016i	$-0.375949\pi$
$449$	16.1010	0.759854	0.379927	+	0.925016i	$0.375949\pi$
$450$	0	0
$451$	− 7.04541i	− 0.331755i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 42.3939i	− 1.98310i	−0.129718	−	0.991551i	$-0.541407\pi$
$457$	− 42.3939i	− 1.98310i	0.129718	−	0.991551i	$-0.458593\pi$
$458$	0	0
$459$	44.6413i	2.08368i
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−30.0000	−1.38823	−0.694117	−	0.719862i	$-0.744205\pi$
$467$	−30.0000	−1.38823	−0.694117	+	0.719862i	$0.744205\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	5.50510i	0.253125i
$474$	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
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	−33.4041	−1.51058
$490$	0	0
$491$	− 42.0000i	− 1.89543i	−0.319113	−	0.947717i	$-0.603385\pi$
$491$	− 42.0000i	− 1.89543i	0.319113	−	0.947717i	$-0.396615\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 14.0000i	− 0.626726i	−0.949633	−	0.313363i	$-0.898544\pi$
$499$	− 14.0000i	− 0.626726i	0.949633	−	0.313363i	$-0.101456\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	18.8434	0.836863
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	− 47.1816i	− 2.08312i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−36.1918	−1.58559	−0.792797	−	0.609486i	$-0.791376\pi$
$521$	−36.1918	−1.58559	−0.792797	+	0.609486i	$0.791376\pi$
$522$	0	0
$523$	3.04541	0.133166	0.0665832	−	0.997781i	$-0.478790\pi$
$523$	3.04541	0.133166	0.0665832	+	0.997781i	$0.478790\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	23.0000	1.00000
$530$	0	0
$531$	5.39388i	0.234074i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	37.8990i	1.63546i
$538$	0	0
$539$	− 3.85357i	− 0.165985i
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	15.6515	0.669211	0.334606	−	0.942358i	$-0.391397\pi$
$547$	15.6515	0.669211	0.334606	+	0.942358i	$0.391397\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−6.30306	−0.266115
$562$	0	0
$563$	30.0000	1.26435	0.632175	−	0.774826i	$-0.282163\pi$
$563$	30.0000	1.26435	0.632175	+	0.774826i	$0.282163\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	40.5959	1.70187	0.850935	−	0.525271i	$-0.176036\pi$
$569$	40.5959	1.70187	0.850935	+	0.525271i	$0.176036\pi$
$570$	0	0
$571$	22.0000i	0.920671i	0.887745	+	0.460336i	$0.152271\pi$
$571$	22.0000i	0.920671i	−0.887745	+	0.460336i	$0.847729\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	12.3939i	0.515964i	0.966150	+	0.257982i	$0.0830575\pi$
$577$	12.3939i	0.515964i	−0.966150	+	0.257982i	$0.916942\pi$
$578$	0	0
$579$	− 5.35867i	− 0.222699i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	44.6413	1.84254	0.921272	−	0.388918i	$-0.127151\pi$
$587$	44.6413	1.84254	0.921272	+	0.388918i	$0.127151\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	48.1918i	1.97900i	0.144528	+	0.989501i	$0.453834\pi$
$593$	48.1918i	1.97900i	−0.144528	+	0.989501i	$0.546166\pi$
$594$	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
$601$	8.30306	0.338689	0.169344	−	0.985557i	$-0.445835\pi$
$601$	8.30306	0.338689	0.169344	+	0.985557i	$0.445835\pi$
$602$	0	0
$603$	−12.8990	−0.525287
$604$	0	0
$605$	0	0
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	0	0		−	1.00000i	$-0.5\pi$
$613$	0	0			1.00000i	$0.5\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 30.0000i	− 1.20775i	−0.797077	−	0.603877i	$-0.793622\pi$
$617$	− 30.0000i	− 1.20775i	0.797077	−	0.603877i	$-0.206378\pi$
$618$	0	0
$619$	26.0000i	1.04503i	0.852631	+	0.522514i	$0.175006\pi$
$619$	26.0000i	1.04503i	−0.852631	+	0.522514i	$0.824994\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	6.66174	0.266044
$628$	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
$633$	− 21.8082i	− 0.866797i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	42.0000	1.65890	0.829450	−	0.558581i	$-0.188654\pi$
$641$	42.0000	1.65890	0.829450	+	0.558581i	$0.188654\pi$
$642$	0	0
$643$	−50.0000	−1.97181	−0.985904	−	0.167313i	$-0.946491\pi$
$643$	−50.0000	−1.97181	−0.985904	+	0.167313i	$0.946491\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	0	0
$649$	−3.30306	−0.129657
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0	0		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	12.3133i	0.480386i
$658$	0	0
$659$	− 50.6413i	− 1.97271i	−0.164644	−	0.986353i	$-0.552648\pi$
$659$	− 50.6413i	− 1.97271i	0.164644	−	0.986353i	$-0.447352\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	− 10.0000i	− 0.385472i	−0.981251	−	0.192736i	$-0.938264\pi$
$673$	− 10.0000i	− 0.385472i	0.981251	−	0.192736i	$-0.0617360\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	43.4847	1.66634
$682$	0	0
$683$	−5.94439	−0.227456	−0.113728	−	0.993512i	$-0.536279\pi$
$683$	−5.94439	−0.227456	−0.113728	+	0.993512i	$0.536279\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	− 45.0454i	− 1.71361i	−0.515642	−	0.856804i	$-0.672447\pi$
$691$	− 45.0454i	− 1.71361i	0.515642	−	0.856804i	$-0.327553\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	101.091i	3.82909i
$698$	0	0
$699$	− 43.4847i	− 1.64474i
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	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
$721$	0	0
$722$	0	0
$723$	−2.45969	−0.0914770
$724$	0	0
$725$	0	0
$726$	0	0
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	29.5153	1.09316
$730$	0	0
$731$	− 78.9898i	− 2.92154i
$732$	0	0
$733$	0	0		−	1.00000i	$-0.5\pi$
$733$	0	0			1.00000i	$0.5\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 7.89898i	− 0.290963i
$738$	0	0
$739$	34.0000i	1.25071i	0.780340	+	0.625355i	$0.215046\pi$
$739$	34.0000i	1.25071i	−0.780340	+	0.625355i	$0.784954\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−10.2929	−0.376596
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0		−	1.00000i	$-0.5\pi$
$751$	0	0			1.00000i	$0.5\pi$
$752$	0	0
$753$	− 34.7071i	− 1.26480i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0		−	1.00000i	$-0.5\pi$
$757$	0	0			1.00000i	$0.5\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	17.2020	0.623574	0.311787	−	0.950152i	$-0.399073\pi$
$761$	17.2020	0.623574	0.311787	+	0.950152i	$0.399073\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−55.0908	−1.98663	−0.993313	−	0.115454i	$-0.963168\pi$
$769$	−55.0908	−1.98663	−0.993313	+	0.115454i	$0.963168\pi$
$770$	0	0
$771$	43.4847i	1.56606i
$772$	0	0
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 106.843i	− 3.82806i
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	50.0000	1.78231	0.891154	−	0.453701i	$-0.149897\pi$
$787$	50.0000	1.78231	0.891154	+	0.453701i	$0.149897\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−12.4949	−0.441485
$802$	0	0
$803$	−7.54031	−0.266092
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	0	0
$811$	38.0000i	1.33436i	0.744896	+	0.667180i	$0.232499\pi$
$811$	38.0000i	1.33436i	−0.744896	+	0.667180i	$0.767501\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	83.4847i	2.92076i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	9.85357	0.342642	0.171321	−	0.985215i	$-0.445196\pi$
$827$	9.85357	0.342642	0.171321	+	0.985215i	$0.445196\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	55.2929i	1.91578i
$834$	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
$841$	29.0000	1.00000
$842$	0	0
$843$	−26.0908	−0.898615
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	47.8990	1.64389
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0	0		−	1.00000i	$-0.5\pi$
$853$	0	0			1.00000i	$0.5\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	46.5959i	1.59169i	0.605503	+	0.795843i	$0.292972\pi$
$857$	46.5959i	1.59169i	−0.605503	+	0.795843i	$0.707028\pi$
$858$	0	0
$859$	− 21.6515i	− 0.738741i	−0.929282	−	0.369370i	$-0.879573\pi$
$859$	− 21.6515i	− 0.738741i	0.929282	−	0.369370i	$-0.120427\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	65.7980	2.23462
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	− 8.98979i	− 0.304258i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	0	0		−	1.00000i	$-0.5\pi$
$877$	0	0			1.00000i	$0.5\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	0	0
$883$	−50.4393	−1.69742	−0.848709	−	0.528861i	$-0.822619\pi$
$883$	−50.4393	−1.69742	−0.848709	+	0.528861i	$0.822619\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	3.02500i	0.101341i
$892$	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
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	10.0000	0.332045	0.166022	−	0.986122i	$-0.446908\pi$
$907$	10.0000	0.332045	0.166022	+	0.986122i	$0.446908\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	− 6.30306i	− 0.208601i
$914$	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
$921$	35.2929	1.16294
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	54.0000	1.77168	0.885841	−	0.463988i	$-0.153582\pi$
$929$	54.0000	1.77168	0.885841	+	0.463988i	$0.153582\pi$
$930$	0	0
$931$	− 58.4393i	− 1.91527i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 61.0908i	− 1.99575i	−0.0651578	−	0.997875i	$-0.520755\pi$
$937$	− 61.0908i	− 1.99575i	0.0651578	−	0.997875i	$-0.479245\pi$
$938$	0	0
$939$	14.4949i	0.473023i
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−30.0000	−0.974869	−0.487435	−	0.873160i	$-0.662067\pi$
$947$	−30.0000	−0.974869	−0.487435	+	0.873160i	$0.662067\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 18.1918i	− 0.589291i	−0.955607	−	0.294646i	$-0.904798\pi$
$953$	− 18.1918i	− 0.589291i	0.955607	−	0.294646i	$-0.0952016\pi$
$954$	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
$963$	−18.1112	−0.583626
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	0	0
$969$	−95.5857	−3.07066
$970$	0	0
$971$	− 53.9444i	− 1.73116i	−0.500773	−	0.865579i	$-0.666951\pi$
$971$	− 53.9444i	− 1.73116i	0.500773	−	0.865579i	$-0.333049\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	56.8888i	1.82003i	0.414572	+	0.910017i	$0.363931\pi$
$977$	56.8888i	1.82003i	−0.414572	+	0.910017i	$0.636069\pi$
$978$	0	0
$979$	− 7.65153i	− 0.244544i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	0	0
$993$	− 50.7980i	− 1.61202i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	0	0		−	1.00000i	$-0.5\pi$
$997$	0	0			1.00000i	$0.5\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1600.2.f.j.1249.1		4
4.3	odd	2		1600.2.f.f.1249.4		4
5.2	odd	4		1600.2.d.c.801.3	yes	4
5.3	odd	4		1600.2.d.d.801.2	yes	4
5.4	even	2		1600.2.f.f.1249.3		4
8.3	odd	2	CM	1600.2.f.j.1249.1		4
8.5	even	2		1600.2.f.f.1249.4		4
20.3	even	4		1600.2.d.d.801.3	yes	4
20.7	even	4		1600.2.d.c.801.2	✓	4
20.19	odd	2	inner	1600.2.f.j.1249.2		4
40.3	even	4		1600.2.d.d.801.2	yes	4
40.13	odd	4		1600.2.d.d.801.3	yes	4
40.19	odd	2		1600.2.f.f.1249.3		4
40.27	even	4		1600.2.d.c.801.3	yes	4
40.29	even	2	inner	1600.2.f.j.1249.2		4
40.37	odd	4		1600.2.d.c.801.2	✓	4
80.3	even	4		6400.2.a.ci.1.1		2
80.13	odd	4		6400.2.a.bc.1.2		2
80.27	even	4		6400.2.a.ch.1.1		2
80.37	odd	4		6400.2.a.bd.1.2		2
80.43	even	4		6400.2.a.bc.1.2		2
80.53	odd	4		6400.2.a.ci.1.1		2
80.67	even	4		6400.2.a.bd.1.2		2
80.77	odd	4		6400.2.a.ch.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1600.2.d.c.801.2	✓	4	20.7	even	4
1600.2.d.c.801.2	✓	4	40.37	odd	4
1600.2.d.c.801.3	yes	4	5.2	odd	4
1600.2.d.c.801.3	yes	4	40.27	even	4
1600.2.d.d.801.2	yes	4	5.3	odd	4
1600.2.d.d.801.2	yes	4	40.3	even	4
1600.2.d.d.801.3	yes	4	20.3	even	4
1600.2.d.d.801.3	yes	4	40.13	odd	4
1600.2.f.f.1249.3		4	5.4	even	2
1600.2.f.f.1249.3		4	40.19	odd	2
1600.2.f.f.1249.4		4	4.3	odd	2
1600.2.f.f.1249.4		4	8.5	even	2
1600.2.f.j.1249.1		4	1.1	even	1	trivial
1600.2.f.j.1249.1		4	8.3	odd	2	CM
1600.2.f.j.1249.2		4	20.19	odd	2	inner
1600.2.f.j.1249.2		4	40.29	even	2	inner
6400.2.a.bc.1.2		2	80.13	odd	4
6400.2.a.bc.1.2		2	80.43	even	4
6400.2.a.bd.1.2		2	80.37	odd	4
6400.2.a.bd.1.2		2	80.67	even	4
6400.2.a.ch.1.1		2	80.27	even	4
6400.2.a.ch.1.1		2	80.77	odd	4
6400.2.a.ci.1.1		2	80.3	even	4
6400.2.a.ci.1.1		2	80.53	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:	\( N \)	\(=\)	\( 1600 = 2^{6} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1600.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.44949 q^{3} -0.898979 q^{9} +O(q^{10})\)	\(q-1.44949 q^{3} -0.898979 q^{9} -0.550510i q^{11} +7.89898i q^{17} -8.34847i q^{19} +5.65153 q^{27} +0.797959i q^{33} +12.7980 q^{41} -10.0000 q^{43} +7.00000 q^{49} -11.4495i q^{51} +12.1010i q^{57} -6.00000i q^{59} +14.3485 q^{67} -13.6969i q^{73} -5.49490 q^{81} +11.4495 q^{83} +13.8990 q^{89} +10.0000i q^{97} +0.494897i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 16 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + 16 * q^9	\( 4 q + 4 q^{3} + 16 q^{9} + 52 q^{27} + 12 q^{41} - 40 q^{43} + 28 q^{49} + 28 q^{67} + 76 q^{81} + 36 q^{83} + 36 q^{89}+O(q^{100}) \) 4 * q + 4 * q^3 + 16 * q^9 + 52 * q^27 + 12 * q^41 - 40 * q^43 + 28 * q^49 + 28 * q^67 + 76 * q^81 + 36 * q^83 + 36 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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