LMFDB - Embedded newform 3200.2.f.s.449.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3200,2,Mod(449,3200)]

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, 1]))

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

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

chi := DirichletCharacter("3200.449");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3200 = 2^{7} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3200.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:	$25.5521286468$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{10}\cdot 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			449.2
Root			$-0.258819 + 0.965926i$ of defining polynomial
Character	$\chi$	$=$	3200.449
Dual form			3200.2.f.s.449.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.14626 q^{3} +6.89898 q^{9} +O(q^{10})$	$q-3.14626 q^{3} +6.89898 q^{9} +6.61037i q^{11} +7.89898i q^{17} +2.51059i q^{19} -12.2672 q^{27} -20.7980i q^{33} +12.7980 q^{41} -8.48528 q^{43} +7.00000 q^{49} -24.8523i q^{51} -7.89898i q^{57} +14.1421i q^{59} +7.88171 q^{67} -13.6969i q^{73} +17.8990 q^{81} +14.1742 q^{83} -13.8990 q^{89} +10.0000i q^{97} +45.6048i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{9}+O(q^{10}) $ 8 * q + 16 * q^9	$ 8 q + 16 q^{9} + 24 q^{41} + 56 q^{49} + 104 q^{81} - 72 q^{89}+O(q^{100}) $ 8 * q + 16 * q^9 + 24 * q^41 + 56 * q^49 + 104 * q^81 - 72 * 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
$3$	−3.14626	−1.81650	−0.908248	−	0.418432i	$-0.862580\pi$
$3$	−3.14626	−1.81650	−0.908248	+	0.418432i	$0.862580\pi$
$4$	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
$9$	6.89898	2.29966
$10$	0	0
$11$	6.61037i	1.99310i	0.0829925	+	0.996550i	$0.473552\pi$
$11$	6.61037i	1.99310i	−0.0829925	+	0.996550i	$0.526448\pi$
$12$	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
$19$	2.51059i	0.575969i	0.957635	+	0.287984i	$0.0929851\pi$
$19$	2.51059i	0.575969i	−0.957635	+	0.287984i	$0.907015\pi$
$20$	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
$27$	−12.2672	−2.36083
$28$	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
$33$	− 20.7980i	− 3.62046i
$34$	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
$43$	−8.48528	−1.29399	−0.646997	−	0.762493i	$-0.723975\pi$
$43$	−8.48528	−1.29399	−0.646997	+	0.762493i	$0.723975\pi$
$44$	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
$51$	− 24.8523i	− 3.48001i
$52$	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
$57$	− 7.89898i	− 1.04625i
$58$	0	0
$59$	14.1421i	1.84115i	0.390567	+	0.920575i	$0.372279\pi$
$59$	14.1421i	1.84115i	−0.390567	+	0.920575i	$0.627721\pi$
$60$	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
$67$	7.88171	0.962905	0.481452	−	0.876472i	$-0.340109\pi$
$67$	7.88171	0.962905	0.481452	+	0.876472i	$0.340109\pi$
$68$	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
$81$	17.8990	1.98878
$82$	0	0
$83$	14.1742	1.55583	0.777913	−	0.628372i	$-0.216279\pi$
$83$	14.1742	1.55583	0.777913	+	0.628372i	$0.216279\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−13.8990	−1.47329	−0.736644	−	0.676280i	$-0.763591\pi$
$89$	−13.8990	−1.47329	−0.736644	+	0.676280i	$0.763591\pi$
$90$	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
$99$	45.6048i	4.58345i
$100$	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
$107$	−4.70334	−0.454689	−0.227345	−	0.973814i	$-0.573004\pi$
$107$	−4.70334	−0.454689	−0.227345	+	0.973814i	$0.573004\pi$
$108$	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	$0.0119499\pi$
$113$	0.797959i	0.0750657i	−0.999295	+	0.0375328i	$0.988050\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−32.6969	−2.97245
$122$	0	0
$123$	−40.2658	−3.63064
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	0	0
$129$	26.6969	2.35053
$130$	0	0
$131$	− 14.1421i	− 1.23560i	−0.786334	−	0.617802i	$-0.788023\pi$
$131$	− 14.1421i	− 1.23560i	0.786334	−	0.617802i	$-0.211977\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	16.5959i	1.41788i	0.705266	+	0.708942i	$0.250827\pi$
$137$	16.5959i	1.41788i	−0.705266	+	0.708942i	$0.749173\pi$
$138$	0	0
$139$	− 14.8099i	− 1.25616i	−0.778148	−	0.628080i	$-0.783841\pi$
$139$	− 14.8099i	− 1.25616i	0.778148	−	0.628080i	$-0.216159\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−22.0239	−1.81650
$148$	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
$153$	54.4949i	4.40565i
$154$	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
$163$	−10.9959	−0.861263	−0.430632	−	0.902528i	$-0.641709\pi$
$163$	−10.9959	−0.861263	−0.430632	+	0.902528i	$0.641709\pi$
$164$	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
$171$	17.3205i	1.32453i
$172$	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
$177$	− 44.4949i	− 3.34444i
$178$	0	0
$179$	− 5.68896i	− 0.425213i	−0.977138	−	0.212607i	$-0.931805\pi$
$179$	− 5.68896i	− 0.425213i	0.977138	−	0.212607i	$-0.0681952\pi$
$180$	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
$187$	−52.2151	−3.81835
$188$	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
$201$	−24.7980	−1.74911
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−16.5959	−1.14796
$210$	0	0
$211$	24.8523i	1.71090i	0.517884	+	0.855451i	$0.326720\pi$
$211$	24.8523i	1.71090i	−0.517884	+	0.855451i	$0.673280\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	43.0942i	2.91204i
$220$	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
$227$	−2.82843	−0.187729	−0.0938647	−	0.995585i	$-0.529922\pi$
$227$	−2.82843	−0.187729	−0.0938647	+	0.995585i	$0.529922\pi$
$228$	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	$-0.517406\pi$
$241$	−1.69694	−0.109309	−0.0546547	+	0.998505i	$0.517406\pi$
$242$	0	0
$243$	−19.5133	−1.25178
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−44.5959	−2.82615
$250$	0	0
$251$	− 20.7525i	− 1.30989i	−0.755678	−	0.654943i	$-0.772693\pi$
$251$	− 20.7525i	− 1.30989i	0.755678	−	0.654943i	$-0.227307\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	30.0000i	1.87135i	0.352865	+	0.935674i	$0.385208\pi$
$257$	30.0000i	1.87135i	−0.352865	+	0.935674i	$0.614792\pi$
$258$	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
$267$	43.7299	2.67622
$268$	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	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	0	0
$283$	6.32464	0.375961	0.187980	−	0.982173i	$-0.439806\pi$
$283$	6.32464	0.375961	0.187980	+	0.982173i	$0.439806\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−45.3939	−2.67023
$290$	0	0
$291$	− 31.4626i	− 1.84437i
$292$	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
$297$	− 81.0908i	− 4.70537i
$298$	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
$307$	25.2022	1.43837	0.719183	−	0.694820i	$-0.244516\pi$
$307$	25.2022	1.43837	0.719183	+	0.694820i	$0.244516\pi$
$308$	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	$0.0912024\pi$
$313$	10.0000i	0.565233i	−0.959233	+	0.282617i	$0.908798\pi$
$314$	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
$321$	14.7980	0.825942
$322$	0	0
$323$	−19.8311	−1.10343
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	− 9.78874i	− 0.538038i	−0.963135	−	0.269019i	$-0.913301\pi$
$331$	− 9.78874i	− 0.538038i	0.963135	−	0.269019i	$-0.0866994\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	22.3939i	1.21987i	0.792451	+	0.609936i	$0.208805\pi$
$337$	22.3939i	1.21987i	−0.792451	+	0.609936i	$0.791195\pi$
$338$	0	0
$339$	− 2.51059i	− 0.136357i
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	23.5809	1.26589	0.632945	−	0.774197i	$-0.281846\pi$
$347$	23.5809	1.26589	0.632945	+	0.774197i	$0.281846\pi$
$348$	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	$-0.705696\pi$
$353$	− 30.0000i	− 1.59674i	0.602168	−	0.798369i	$-0.294304\pi$
$354$	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
$361$	12.6969	0.668260
$362$	0	0
$363$	102.873	5.39944
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	88.2929	4.59634
$370$	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
$379$	37.1516i	1.90835i	0.299249	+	0.954175i	$0.403264\pi$
$379$	37.1516i	1.90835i	−0.299249	+	0.954175i	$0.596736\pi$
$380$	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
$387$	−58.5398	−2.97574
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	44.4949i	2.24447i
$394$	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	$-0.881200\pi$
$401$	−37.2929	−1.86232	−0.931158	+	0.364615i	$0.881200\pi$
$402$	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
$411$	− 52.2151i	− 2.57558i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	46.5959i	2.28181i
$418$	0	0
$419$	− 33.9732i	− 1.65970i	−0.557986	−	0.829851i	$-0.688426\pi$
$419$	− 33.9732i	− 1.65970i	0.557986	−	0.829851i	$-0.311574\pi$
$420$	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
$441$	48.2929	2.29966
$442$	0	0
$443$	−37.7873	−1.79533	−0.897664	−	0.440681i	$-0.854737\pi$
$443$	−37.7873	−1.79533	−0.897664	+	0.440681i	$0.854737\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−16.1010	−0.759854	−0.379927	−	0.925016i	$-0.624051\pi$
$449$	−16.1010	−0.759854	−0.379927	+	0.925016i	$0.624051\pi$
$450$	0	0
$451$	84.5992i	3.98362i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	42.3939i	1.98310i	0.129718	+	0.991551i	$0.458593\pi$
$457$	42.3939i	1.98310i	−0.129718	+	0.991551i	$0.541407\pi$
$458$	0	0
$459$	− 96.8985i	− 4.52284i
$460$	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
$467$	31.1127	1.43972	0.719862	−	0.694117i	$-0.244205\pi$
$467$	31.1127	1.43972	0.719862	+	0.694117i	$0.244205\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 56.0908i	− 2.57906i
$474$	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
$489$	34.5959	1.56448
$490$	0	0
$491$	− 14.1421i	− 0.638226i	−0.947717	−	0.319113i	$-0.896615\pi$
$491$	− 14.1421i	− 0.638226i	0.947717	−	0.319113i	$-0.103385\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 42.4264i	− 1.89927i	−0.313363	−	0.949633i	$-0.601456\pi$
$499$	− 42.4264i	− 1.89927i	0.313363	−	0.949633i	$-0.398544\pi$
$500$	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
$507$	40.9014	1.81650
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	− 30.7980i	− 1.35976i
$514$	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
$523$	−45.6369	−1.99556	−0.997781	−	0.0665832i	$-0.978790\pi$
$523$	−45.6369	−1.99556	−0.997781	+	0.0665832i	$0.978790\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	23.0000	1.00000
$530$	0	0
$531$	97.5663i	4.23402i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	17.8990i	0.772398i
$538$	0	0
$539$	46.2726i	1.99310i
$540$	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
$547$	−44.0798	−1.88472	−0.942358	−	0.334606i	$-0.891397\pi$
$547$	−44.0798	−1.88472	−0.942358	+	0.334606i	$0.891397\pi$
$548$	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
$561$	164.283	6.93602
$562$	0	0
$563$	36.7696	1.54965	0.774826	−	0.632175i	$-0.217837\pi$
$563$	36.7696	1.54965	0.774826	+	0.632175i	$0.217837\pi$
$564$	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
$571$	42.4264i	1.77549i	0.460336	+	0.887745i	$0.347729\pi$
$571$	42.4264i	1.77549i	−0.460336	+	0.887745i	$0.652271\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 12.3939i	− 0.515964i	−0.966150	−	0.257982i	$-0.916942\pi$
$577$	− 12.3939i	− 0.515964i	0.966150	−	0.257982i	$-0.0830575\pi$
$578$	0	0
$579$	− 11.6315i	− 0.483391i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	18.8455	0.777836	0.388918	−	0.921272i	$-0.372849\pi$
$587$	18.8455	0.777836	0.388918	+	0.921272i	$0.372849\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 48.1918i	− 1.97900i	−0.144528	−	0.989501i	$-0.546166\pi$
$593$	− 48.1918i	− 1.97900i	0.144528	−	0.989501i	$-0.453834\pi$
$594$	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	$-0.554165\pi$
$601$	−8.30306	−0.338689	−0.169344	+	0.985557i	$0.554165\pi$
$602$	0	0
$603$	54.3758	2.21435
$604$	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
$619$	− 42.4264i	− 1.70526i	−0.522514	−	0.852631i	$-0.675006\pi$
$619$	− 42.4264i	− 1.70526i	0.522514	−	0.852631i	$-0.324994\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	52.2151	2.08527
$628$	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
$633$	− 78.1918i	− 3.10785i
$634$	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	$-0.811346\pi$
$641$	−42.0000	−1.65890	−0.829450	+	0.558581i	$0.811346\pi$
$642$	0	0
$643$	−8.48528	−0.334627	−0.167313	−	0.985904i	$-0.553509\pi$
$643$	−8.48528	−0.334627	−0.167313	+	0.985904i	$0.553509\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	0	0
$649$	−93.4847	−3.66960
$650$	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
$657$	− 94.4949i	− 3.68660i
$658$	0	0
$659$	− 8.45317i	− 0.329289i	−0.986353	−	0.164644i	$-0.947352\pi$
$659$	− 8.45317i	− 0.329289i	0.986353	−	0.164644i	$-0.0526477\pi$
$660$	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
$681$	8.89898	0.341010
$682$	0	0
$683$	51.9294	1.98702	0.993512	−	0.113728i	$-0.0362792\pi$
$683$	51.9294	1.98702	0.993512	+	0.113728i	$0.0362792\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	− 27.1092i	− 1.03128i	−0.856804	−	0.515642i	$-0.827553\pi$
$691$	− 27.1092i	− 1.03128i	0.856804	−	0.515642i	$-0.172447\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	101.091i	3.82909i
$698$	0	0
$699$	− 94.3879i	− 3.57008i
$700$	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
$723$	5.33902	0.198560
$724$	0	0
$725$	0	0
$726$	0	0
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	7.69694	0.285072
$730$	0	0
$731$	− 67.0251i	− 2.47901i
$732$	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
$737$	52.1010i	1.91917i
$738$	0	0
$739$	− 42.4264i	− 1.56068i	−0.625355	−	0.780340i	$-0.715046\pi$
$739$	− 42.4264i	− 1.56068i	0.625355	−	0.780340i	$-0.284954\pi$
$740$	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
$747$	97.7878	3.57787
$748$	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
$753$	65.2929i	2.37940i
$754$	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	$-0.0368323\pi$
$769$	55.0908	1.98663	0.993313	+	0.115454i	$0.0368323\pi$
$770$	0	0
$771$	− 94.3879i	− 3.39930i
$772$	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
$779$	32.1304i	1.15119i
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−25.4558	−0.907403	−0.453701	−	0.891154i	$-0.649897\pi$
$787$	−25.4558	−0.907403	−0.453701	+	0.891154i	$0.649897\pi$
$788$	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
$801$	−95.8888	−3.38806
$802$	0	0
$803$	90.5418	3.19515
$804$	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
$811$	42.4264i	1.48979i	0.667180	+	0.744896i	$0.267501\pi$
$811$	42.4264i	1.48979i	−0.667180	+	0.744896i	$0.732499\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 21.3031i	− 0.745300i
$818$	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
$827$	−56.6649	−1.97043	−0.985215	−	0.171321i	$-0.945196\pi$
$827$	−56.6649	−1.97043	−0.985215	+	0.171321i	$0.945196\pi$
$828$	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
$843$	56.6328	1.95054
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−19.8990	−0.682931
$850$	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	$-0.707028\pi$
$857$	− 46.5959i	− 1.59169i	0.605503	−	0.795843i	$-0.292972\pi$
$858$	0	0
$859$	54.4721i	1.85856i	0.369370	+	0.929282i	$0.379573\pi$
$859$	54.4721i	1.85856i	−0.369370	+	0.929282i	$0.620427\pi$
$860$	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
$867$	142.821	4.85046
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	68.9898i	2.33495i
$874$	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
$883$	−31.4305	−1.05772	−0.528861	−	0.848709i	$-0.677381\pi$
$883$	−31.4305	−1.05772	−0.528861	+	0.848709i	$0.677381\pi$
$884$	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
$891$	118.319i	3.96383i
$892$	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
$907$	−59.3970	−1.97224	−0.986122	−	0.166022i	$-0.946908\pi$
$907$	−59.3970	−1.97224	−0.986122	+	0.166022i	$0.946908\pi$
$908$	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
$913$	93.6969i	3.10092i
$914$	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
$921$	−79.2929	−2.61279
$922$	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	$-0.846418\pi$
$929$	−54.0000	−1.77168	−0.885841	+	0.463988i	$0.846418\pi$
$930$	0	0
$931$	17.5741i	0.575969i
$932$	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
$939$	− 31.4626i	− 1.02674i
$940$	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
$947$	53.7401	1.74632	0.873160	−	0.487435i	$-0.162067\pi$
$947$	53.7401	1.74632	0.873160	+	0.487435i	$0.162067\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	18.1918i	0.589291i	0.955607	+	0.294646i	$0.0952016\pi$
$953$	18.1918i	0.589291i	−0.955607	+	0.294646i	$0.904798\pi$
$954$	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
$963$	−32.4483	−1.04563
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	0	0
$969$	62.3939	2.00438
$970$	0	0
$971$	31.2090i	1.00155i	0.865579	+	0.500773i	$0.166951\pi$
$971$	31.2090i	1.00155i	−0.865579	+	0.500773i	$0.833049\pi$
$972$	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
$979$	− 91.8773i	− 2.93641i
$980$	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
$993$	30.7980i	0.977344i
$994$	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	3200.2.f.s.449.2		8
4.3	odd	2	inner	3200.2.f.s.449.7		8
5.2	odd	4		3200.2.d.n.1601.4	yes	4
5.3	odd	4		3200.2.d.q.1601.1	yes	4
5.4	even	2	inner	3200.2.f.s.449.8		8
8.3	odd	2	CM	3200.2.f.s.449.2		8
8.5	even	2	inner	3200.2.f.s.449.7		8
20.3	even	4		3200.2.d.q.1601.4	yes	4
20.7	even	4		3200.2.d.n.1601.1	✓	4
20.19	odd	2	inner	3200.2.f.s.449.1		8
40.3	even	4		3200.2.d.q.1601.1	yes	4
40.13	odd	4		3200.2.d.q.1601.4	yes	4
40.19	odd	2	inner	3200.2.f.s.449.8		8
40.27	even	4		3200.2.d.n.1601.4	yes	4
40.29	even	2	inner	3200.2.f.s.449.1		8
40.37	odd	4		3200.2.d.n.1601.1	✓	4
80.3	even	4		6400.2.a.cr.1.1		4
80.13	odd	4		6400.2.a.cr.1.4		4
80.27	even	4		6400.2.a.cq.1.1		4
80.37	odd	4		6400.2.a.cq.1.4		4
80.43	even	4		6400.2.a.cr.1.4		4
80.53	odd	4		6400.2.a.cr.1.1		4
80.67	even	4		6400.2.a.cq.1.4		4
80.77	odd	4		6400.2.a.cq.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3200.2.d.n.1601.1	✓	4	20.7	even	4
3200.2.d.n.1601.1	✓	4	40.37	odd	4
3200.2.d.n.1601.4	yes	4	5.2	odd	4
3200.2.d.n.1601.4	yes	4	40.27	even	4
3200.2.d.q.1601.1	yes	4	5.3	odd	4
3200.2.d.q.1601.1	yes	4	40.3	even	4
3200.2.d.q.1601.4	yes	4	20.3	even	4
3200.2.d.q.1601.4	yes	4	40.13	odd	4
3200.2.f.s.449.1		8	20.19	odd	2	inner
3200.2.f.s.449.1		8	40.29	even	2	inner
3200.2.f.s.449.2		8	1.1	even	1	trivial
3200.2.f.s.449.2		8	8.3	odd	2	CM
3200.2.f.s.449.7		8	4.3	odd	2	inner
3200.2.f.s.449.7		8	8.5	even	2	inner
3200.2.f.s.449.8		8	5.4	even	2	inner
3200.2.f.s.449.8		8	40.19	odd	2	inner
6400.2.a.cq.1.1		4	80.27	even	4
6400.2.a.cq.1.1		4	80.77	odd	4
6400.2.a.cq.1.4		4	80.37	odd	4
6400.2.a.cq.1.4		4	80.67	even	4
6400.2.a.cr.1.1		4	80.3	even	4
6400.2.a.cr.1.1		4	80.53	odd	4
6400.2.a.cr.1.4		4	80.13	odd	4
6400.2.a.cr.1.4		4	80.43	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:	\( N \)	\(=\)	\( 3200 = 2^{7} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3200.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.14626 q^{3} +6.89898 q^{9} +O(q^{10})\)	\(q-3.14626 q^{3} +6.89898 q^{9} +6.61037i q^{11} +7.89898i q^{17} +2.51059i q^{19} -12.2672 q^{27} -20.7980i q^{33} +12.7980 q^{41} -8.48528 q^{43} +7.00000 q^{49} -24.8523i q^{51} -7.89898i q^{57} +14.1421i q^{59} +7.88171 q^{67} -13.6969i q^{73} +17.8990 q^{81} +14.1742 q^{83} -13.8990 q^{89} +10.0000i q^{97} +45.6048i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{9}+O(q^{10}) \) 8 * q + 16 * q^9	\( 8 q + 16 q^{9} + 24 q^{41} + 56 q^{49} + 104 q^{81} - 72 q^{89}+O(q^{100}) \) 8 * q + 16 * q^9 + 24 * q^41 + 56 * q^49 + 104 * q^81 - 72 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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