LMFDB - Embedded newform 3200.2.f.o.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:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 128)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			449.2
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	3200.449
Dual form			3200.2.f.o.449.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.82843 q^{3} +5.00000 q^{9} +O(q^{10})$	$q-2.82843 q^{3} +5.00000 q^{9} +2.82843i q^{11} -6.00000i q^{17} +8.48528i q^{19} -5.65685 q^{27} -8.00000i q^{33} -6.00000 q^{41} +8.48528 q^{43} +7.00000 q^{49} +16.9706i q^{51} -24.0000i q^{57} -14.1421i q^{59} -8.48528 q^{67} -2.00000i q^{73} +1.00000 q^{81} -2.82843 q^{83} +18.0000 q^{89} +10.0000i q^{97} +14.1421i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 20 q^{9}+O(q^{10}) $ 4 * q + 20 * q^9	$ 4 q + 20 q^{9} - 24 q^{41} + 28 q^{49} + 4 q^{81} + 72 q^{89}+O(q^{100}) $ 4 * q + 20 * q^9 - 24 * q^41 + 28 * q^49 + 4 * 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
$3$	−2.82843	−1.63299	−0.816497	−	0.577350i	$-0.804087\pi$
$3$	−2.82843	−1.63299	−0.816497	+	0.577350i	$0.804087\pi$
$4$	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
$9$	5.00000	1.66667
$10$	0	0
$11$	2.82843i	0.852803i	0.904534	+	0.426401i	$0.140219\pi$
$11$	2.82843i	0.852803i	−0.904534	+	0.426401i	$0.859781\pi$
$12$	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
$17$	− 6.00000i	− 1.45521i	−0.685994	−	0.727607i	$-0.740633\pi$
$17$	− 6.00000i	− 1.45521i	0.685994	−	0.727607i	$-0.259367\pi$
$18$	0	0
$19$	8.48528i	1.94666i	0.229416	+	0.973329i	$0.426318\pi$
$19$	8.48528i	1.94666i	−0.229416	+	0.973329i	$0.573682\pi$
$20$	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
$27$	−5.65685	−1.08866
$28$	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
$33$	− 8.00000i	− 1.39262i
$34$	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
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	8.48528	1.29399	0.646997	−	0.762493i	$-0.276025\pi$
$43$	8.48528	1.29399	0.646997	+	0.762493i	$0.276025\pi$
$44$	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
$51$	16.9706i	2.37635i
$52$	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
$57$	− 24.0000i	− 3.17888i
$58$	0	0
$59$	− 14.1421i	− 1.84115i	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	− 14.1421i	− 1.84115i	0.390567	−	0.920575i	$-0.372279\pi$
$60$	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
$67$	−8.48528	−1.03664	−0.518321	−	0.855186i	$-0.673443\pi$
$67$	−8.48528	−1.03664	−0.518321	+	0.855186i	$0.673443\pi$
$68$	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
$73$	− 2.00000i	− 0.234082i	−0.993127	−	0.117041i	$-0.962659\pi$
$73$	− 2.00000i	− 0.234082i	0.993127	−	0.117041i	$-0.0373409\pi$
$74$	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
$81$	1.00000	0.111111
$82$	0	0
$83$	−2.82843	−0.310460	−0.155230	−	0.987878i	$-0.549612\pi$
$83$	−2.82843	−0.310460	−0.155230	+	0.987878i	$0.549612\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	18.0000	1.90800	0.953998	−	0.299813i	$-0.0969242\pi$
$89$	18.0000	1.90800	0.953998	+	0.299813i	$0.0969242\pi$
$90$	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
$99$	14.1421i	1.42134i
$100$	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
$107$	−19.7990	−1.91404	−0.957020	−	0.290021i	$-0.906338\pi$
$107$	−19.7990	−1.91404	−0.957020	+	0.290021i	$0.906338\pi$
$108$	0	0
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	18.0000i	1.69330i	0.532152	+	0.846649i	$0.321383\pi$
$113$	18.0000i	1.69330i	−0.532152	+	0.846649i	$0.678617\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	3.00000	0.272727
$122$	0	0
$123$	16.9706	1.53018
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	0	0
$129$	−24.0000	−2.11308
$130$	0	0
$131$	14.1421i	1.23560i	0.786334	+	0.617802i	$0.211977\pi$
$131$	14.1421i	1.23560i	−0.786334	+	0.617802i	$0.788023\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	6.00000i	0.512615i	0.966595	+	0.256307i	$0.0825059\pi$
$137$	6.00000i	0.512615i	−0.966595	+	0.256307i	$0.917494\pi$
$138$	0	0
$139$	8.48528i	0.719712i	0.933008	+	0.359856i	$0.117174\pi$
$139$	8.48528i	0.719712i	−0.933008	+	0.359856i	$0.882826\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−19.7990	−1.63299
$148$	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
$153$	− 30.0000i	− 2.42536i
$154$	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
$163$	−25.4558	−1.99386	−0.996928	−	0.0783260i	$-0.975042\pi$
$163$	−25.4558	−1.99386	−0.996928	+	0.0783260i	$0.975042\pi$
$164$	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
$171$	42.4264i	3.24443i
$172$	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
$177$	40.0000i	3.00658i
$178$	0	0
$179$	19.7990i	1.47985i	0.672692	+	0.739923i	$0.265138\pi$
$179$	19.7990i	1.47985i	−0.672692	+	0.739923i	$0.734862\pi$
$180$	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
$187$	16.9706	1.24101
$188$	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
$193$	22.0000i	1.58359i	0.610784	+	0.791797i	$0.290854\pi$
$193$	22.0000i	1.58359i	−0.610784	+	0.791797i	$0.709146\pi$
$194$	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
$201$	24.0000	1.69283
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−24.0000	−1.66011
$210$	0	0
$211$	25.4558i	1.75245i	0.481900	+	0.876226i	$0.339947\pi$
$211$	25.4558i	1.75245i	−0.481900	+	0.876226i	$0.660053\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	5.65685i	0.382255i
$220$	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	$-0.470078\pi$
$227$	2.82843	0.187729	0.0938647	+	0.995585i	$0.470078\pi$
$228$	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
$241$	−26.0000	−1.67481	−0.837404	−	0.546585i	$-0.815928\pi$
$241$	−26.0000	−1.67481	−0.837404	+	0.546585i	$0.815928\pi$
$242$	0	0
$243$	14.1421	0.907218
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	8.00000	0.506979
$250$	0	0
$251$	− 31.1127i	− 1.96382i	−0.189358	−	0.981908i	$-0.560641\pi$
$251$	− 31.1127i	− 1.96382i	0.189358	−	0.981908i	$-0.439359\pi$
$252$	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
$267$	−50.9117	−3.11574
$268$	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
$283$	−25.4558	−1.51319	−0.756596	−	0.653882i	$-0.773139\pi$
$283$	−25.4558	−1.51319	−0.756596	+	0.653882i	$0.773139\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−19.0000	−1.11765
$290$	0	0
$291$	− 28.2843i	− 1.65805i
$292$	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
$297$	− 16.0000i	− 0.928414i
$298$	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
$307$	−8.48528	−0.484281	−0.242140	−	0.970241i	$-0.577849\pi$
$307$	−8.48528	−0.484281	−0.242140	+	0.970241i	$0.577849\pi$
$308$	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
$321$	56.0000	3.12562
$322$	0	0
$323$	50.9117	2.83280
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	25.4558i	1.39918i	0.714545	+	0.699590i	$0.246634\pi$
$331$	25.4558i	1.39918i	−0.714545	+	0.699590i	$0.753366\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	14.0000i	0.762629i	0.924445	+	0.381314i	$0.124528\pi$
$337$	14.0000i	0.762629i	−0.924445	+	0.381314i	$0.875472\pi$
$338$	0	0
$339$	− 50.9117i	− 2.76514i
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	36.7696	1.97389	0.986947	−	0.161048i	$-0.0514875\pi$
$347$	36.7696	1.97389	0.986947	+	0.161048i	$0.0514875\pi$
$348$	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
$361$	−53.0000	−2.78947
$362$	0	0
$363$	−8.48528	−0.445362
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	−30.0000	−1.56174
$370$	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
$379$	8.48528i	0.435860i	0.975964	+	0.217930i	$0.0699304\pi$
$379$	8.48528i	0.435860i	−0.975964	+	0.217930i	$0.930070\pi$
$380$	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
$387$	42.4264	2.15666
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	− 40.0000i	− 2.01773i
$394$	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
$401$	6.00000	0.299626	0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000	0.299626	0.149813	+	0.988714i	$0.452133\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	22.0000	1.08783	0.543915	−	0.839140i	$-0.316941\pi$
$409$	22.0000	1.08783	0.543915	+	0.839140i	$0.316941\pi$
$410$	0	0
$411$	− 16.9706i	− 0.837096i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 24.0000i	− 1.17529i
$418$	0	0
$419$	− 36.7696i	− 1.79631i	−0.439679	−	0.898155i	$-0.644908\pi$
$419$	− 36.7696i	− 1.79631i	0.439679	−	0.898155i	$-0.355092\pi$
$420$	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
$433$	38.0000i	1.82616i	0.407777	+	0.913082i	$0.366304\pi$
$433$	38.0000i	1.82616i	−0.407777	+	0.913082i	$0.633696\pi$
$434$	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
$441$	35.0000	1.66667
$442$	0	0
$443$	−2.82843	−0.134383	−0.0671913	−	0.997740i	$-0.521404\pi$
$443$	−2.82843	−0.134383	−0.0671913	+	0.997740i	$0.521404\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	42.0000	1.98210	0.991051	−	0.133482i	$-0.0426157\pi$
$449$	42.0000	1.98210	0.991051	+	0.133482i	$0.0426157\pi$
$450$	0	0
$451$	− 16.9706i	− 0.799113i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 26.0000i	− 1.21623i	−0.793849	−	0.608114i	$-0.791926\pi$
$457$	− 26.0000i	− 1.21623i	0.793849	−	0.608114i	$-0.208074\pi$
$458$	0	0
$459$	33.9411i	1.58424i
$460$	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	$-0.755795\pi$
$467$	−31.1127	−1.43972	−0.719862	+	0.694117i	$0.755795\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	24.0000i	1.10352i
$474$	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
$489$	72.0000	3.25595
$490$	0	0
$491$	14.1421i	0.638226i	0.947717	+	0.319113i	$0.103385\pi$
$491$	14.1421i	0.638226i	−0.947717	+	0.319113i	$0.896615\pi$
$492$	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	$0.398544\pi$
$499$	42.4264i	1.89927i	−0.313363	+	0.949633i	$0.601456\pi$
$500$	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
$507$	36.7696	1.63299
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	− 48.0000i	− 2.11925i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	0	0
$523$	−25.4558	−1.11311	−0.556553	−	0.830812i	$-0.687876\pi$
$523$	−25.4558	−1.11311	−0.556553	+	0.830812i	$0.687876\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	23.0000	1.00000
$530$	0	0
$531$	− 70.7107i	− 3.06858i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 56.0000i	− 2.41658i
$538$	0	0
$539$	19.7990i	0.852803i
$540$	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
$547$	−8.48528	−0.362804	−0.181402	−	0.983409i	$-0.558064\pi$
$547$	−8.48528	−0.362804	−0.181402	+	0.983409i	$0.558064\pi$
$548$	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
$561$	−48.0000	−2.02656
$562$	0	0
$563$	−36.7696	−1.54965	−0.774826	−	0.632175i	$-0.782163\pi$
$563$	−36.7696	−1.54965	−0.774826	+	0.632175i	$0.782163\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−42.0000	−1.76073	−0.880366	−	0.474295i	$-0.842703\pi$
$569$	−42.0000	−1.76073	−0.880366	+	0.474295i	$0.842703\pi$
$570$	0	0
$571$	− 42.4264i	− 1.77549i	−0.460336	−	0.887745i	$-0.652271\pi$
$571$	− 42.4264i	− 1.77549i	0.460336	−	0.887745i	$-0.347729\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 34.0000i	− 1.41544i	−0.706494	−	0.707719i	$-0.749724\pi$
$577$	− 34.0000i	− 1.41544i	0.706494	−	0.707719i	$-0.250276\pi$
$578$	0	0
$579$	− 62.2254i	− 2.58600i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	48.0833	1.98461	0.992304	−	0.123823i	$-0.0395156\pi$
$587$	48.0833	1.98461	0.992304	+	0.123823i	$0.0395156\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	18.0000i	0.739171i	0.929197	+	0.369586i	$0.120500\pi$
$593$	18.0000i	0.739171i	−0.929197	+	0.369586i	$0.879500\pi$
$594$	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
$601$	46.0000	1.87638	0.938190	−	0.346122i	$-0.112502\pi$
$601$	46.0000	1.87638	0.938190	+	0.346122i	$0.112502\pi$
$602$	0	0
$603$	−42.4264	−1.72774
$604$	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	$0.324994\pi$
$619$	42.4264i	1.70526i	−0.522514	+	0.852631i	$0.675006\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	67.8823	2.71096
$628$	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
$633$	− 72.0000i	− 2.86174i
$634$	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	$-0.446491\pi$
$643$	8.48528	0.334627	0.167313	+	0.985904i	$0.446491\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	0	0
$649$	40.0000	1.57014
$650$	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
$657$	− 10.0000i	− 0.390137i
$658$	0	0
$659$	− 48.0833i	− 1.87306i	−0.350590	−	0.936529i	$-0.614019\pi$
$659$	− 48.0833i	− 1.87306i	0.350590	−	0.936529i	$-0.385981\pi$
$660$	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
$681$	−8.00000	−0.306561
$682$	0	0
$683$	31.1127	1.19049	0.595247	−	0.803543i	$-0.297054\pi$
$683$	31.1127	1.19049	0.595247	+	0.803543i	$0.297054\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	25.4558i	0.968386i	0.874961	+	0.484193i	$0.160887\pi$
$691$	25.4558i	0.968386i	−0.874961	+	0.484193i	$0.839113\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	36.0000i	1.36360i
$698$	0	0
$699$	− 84.8528i	− 3.20943i
$700$	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
$723$	73.5391	2.73495
$724$	0	0
$725$	0	0
$726$	0	0
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	−43.0000	−1.59259
$730$	0	0
$731$	− 50.9117i	− 1.88304i
$732$	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
$737$	− 24.0000i	− 0.884051i
$738$	0	0
$739$	42.4264i	1.56068i	0.625355	+	0.780340i	$0.284954\pi$
$739$	42.4264i	1.56068i	−0.625355	+	0.780340i	$0.715046\pi$
$740$	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
$747$	−14.1421	−0.517434
$748$	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
$753$	88.0000i	3.20690i
$754$	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
$761$	−54.0000	−1.95750	−0.978749	−	0.205061i	$-0.934261\pi$
$761$	−54.0000	−1.95750	−0.978749	+	0.205061i	$0.934261\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−22.0000	−0.793340	−0.396670	−	0.917961i	$-0.629834\pi$
$769$	−22.0000	−0.793340	−0.396670	+	0.917961i	$0.629834\pi$
$770$	0	0
$771$	− 84.8528i	− 3.05590i
$772$	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
$779$	− 50.9117i	− 1.82410i
$780$	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	$-0.350103\pi$
$787$	25.4558	0.907403	0.453701	+	0.891154i	$0.350103\pi$
$788$	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
$801$	90.0000	3.17999
$802$	0	0
$803$	5.65685	0.199626
$804$	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	$-0.732499\pi$
$811$	− 42.4264i	− 1.48979i	0.667180	−	0.744896i	$-0.267501\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	72.0000i	2.51896i
$818$	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
$827$	−19.7990	−0.688478	−0.344239	−	0.938882i	$-0.611863\pi$
$827$	−19.7990	−0.688478	−0.344239	+	0.938882i	$0.611863\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 42.0000i	− 1.45521i
$834$	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
$843$	50.9117	1.75349
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	72.0000	2.47103
$850$	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
$857$	54.0000i	1.84460i	0.386469	+	0.922302i	$0.373695\pi$
$857$	54.0000i	1.84460i	−0.386469	+	0.922302i	$0.626305\pi$
$858$	0	0
$859$	8.48528i	0.289514i	0.989467	+	0.144757i	$0.0462401\pi$
$859$	8.48528i	0.289514i	−0.989467	+	0.144757i	$0.953760\pi$
$860$	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
$867$	53.7401	1.82511
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	50.0000i	1.69224i
$874$	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
$883$	−59.3970	−1.99887	−0.999434	−	0.0336527i	$-0.989286\pi$
$883$	−59.3970	−1.99887	−0.999434	+	0.0336527i	$0.989286\pi$
$884$	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
$891$	2.82843i	0.0947559i
$892$	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	$-0.0530924\pi$
$907$	59.3970	1.97224	0.986122	+	0.166022i	$0.0530924\pi$
$908$	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
$913$	− 8.00000i	− 0.264761i
$914$	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
$921$	24.0000	0.790827
$922$	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
$931$	59.3970i	1.94666i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	34.0000i	1.11073i	0.831606	+	0.555366i	$0.187422\pi$
$937$	34.0000i	1.11073i	−0.831606	+	0.555366i	$0.812578\pi$
$938$	0	0
$939$	− 28.2843i	− 0.923022i
$940$	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	$-0.837933\pi$
$947$	−53.7401	−1.74632	−0.873160	+	0.487435i	$0.837933\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	42.0000i	1.36051i	0.732974	+	0.680257i	$0.238132\pi$
$953$	42.0000i	1.36051i	−0.732974	+	0.680257i	$0.761868\pi$
$954$	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
$963$	−98.9949	−3.19007
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	0	0
$969$	−144.000	−4.62595
$970$	0	0
$971$	− 31.1127i	− 0.998454i	−0.866471	−	0.499227i	$-0.833617\pi$
$971$	− 31.1127i	− 0.998454i	0.866471	−	0.499227i	$-0.166383\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 6.00000i	− 0.191957i	−0.995383	−	0.0959785i	$-0.969402\pi$
$977$	− 6.00000i	− 0.191957i	0.995383	−	0.0959785i	$-0.0305980\pi$
$978$	0	0
$979$	50.9117i	1.62714i
$980$	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
$993$	− 72.0000i	− 2.28485i
$994$	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.o.449.2		4
4.3	odd	2	inner	3200.2.f.o.449.3		4
5.2	odd	4		128.2.b.a.65.2	yes	2
5.3	odd	4		3200.2.d.c.1601.1		2
5.4	even	2	inner	3200.2.f.o.449.4		4
8.3	odd	2	CM	3200.2.f.o.449.2		4
8.5	even	2	inner	3200.2.f.o.449.3		4
15.2	even	4		1152.2.d.c.577.1		2
20.3	even	4		3200.2.d.c.1601.2		2
20.7	even	4		128.2.b.a.65.1	✓	2
20.19	odd	2	inner	3200.2.f.o.449.1		4
40.3	even	4		3200.2.d.c.1601.1		2
40.13	odd	4		3200.2.d.c.1601.2		2
40.19	odd	2	inner	3200.2.f.o.449.4		4
40.27	even	4		128.2.b.a.65.2	yes	2
40.29	even	2	inner	3200.2.f.o.449.1		4
40.37	odd	4		128.2.b.a.65.1	✓	2
60.47	odd	4		1152.2.d.c.577.2		2
80.3	even	4		6400.2.a.by.1.1		2
80.13	odd	4		6400.2.a.by.1.2		2
80.27	even	4		256.2.a.e.1.1		2
80.37	odd	4		256.2.a.e.1.2		2
80.43	even	4		6400.2.a.by.1.2		2
80.53	odd	4		6400.2.a.by.1.1		2
80.67	even	4		256.2.a.e.1.2		2
80.77	odd	4		256.2.a.e.1.1		2
120.77	even	4		1152.2.d.c.577.2		2
120.107	odd	4		1152.2.d.c.577.1		2
160.27	even	8		1024.2.e.f.257.1		2
160.37	odd	8		1024.2.e.a.257.1		2
160.67	even	8		1024.2.e.f.769.1		2
160.77	odd	8		1024.2.e.f.769.1		2
160.107	even	8		1024.2.e.a.257.1		2
160.117	odd	8		1024.2.e.f.257.1		2
160.147	even	8		1024.2.e.a.769.1		2
160.157	odd	8		1024.2.e.a.769.1		2
240.77	even	4		2304.2.a.t.1.1		2
240.107	odd	4		2304.2.a.t.1.1		2
240.197	even	4		2304.2.a.t.1.2		2
240.227	odd	4		2304.2.a.t.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.2.b.a.65.1	✓	2	20.7	even	4
128.2.b.a.65.1	✓	2	40.37	odd	4
128.2.b.a.65.2	yes	2	5.2	odd	4
128.2.b.a.65.2	yes	2	40.27	even	4
256.2.a.e.1.1		2	80.27	even	4
256.2.a.e.1.1		2	80.77	odd	4
256.2.a.e.1.2		2	80.37	odd	4
256.2.a.e.1.2		2	80.67	even	4
1024.2.e.a.257.1		2	160.37	odd	8
1024.2.e.a.257.1		2	160.107	even	8
1024.2.e.a.769.1		2	160.147	even	8
1024.2.e.a.769.1		2	160.157	odd	8
1024.2.e.f.257.1		2	160.27	even	8
1024.2.e.f.257.1		2	160.117	odd	8
1024.2.e.f.769.1		2	160.67	even	8
1024.2.e.f.769.1		2	160.77	odd	8
1152.2.d.c.577.1		2	15.2	even	4
1152.2.d.c.577.1		2	120.107	odd	4
1152.2.d.c.577.2		2	60.47	odd	4
1152.2.d.c.577.2		2	120.77	even	4
2304.2.a.t.1.1		2	240.77	even	4
2304.2.a.t.1.1		2	240.107	odd	4
2304.2.a.t.1.2		2	240.197	even	4
2304.2.a.t.1.2		2	240.227	odd	4
3200.2.d.c.1601.1		2	5.3	odd	4
3200.2.d.c.1601.1		2	40.3	even	4
3200.2.d.c.1601.2		2	20.3	even	4
3200.2.d.c.1601.2		2	40.13	odd	4
3200.2.f.o.449.1		4	20.19	odd	2	inner
3200.2.f.o.449.1		4	40.29	even	2	inner
3200.2.f.o.449.2		4	1.1	even	1	trivial
3200.2.f.o.449.2		4	8.3	odd	2	CM
3200.2.f.o.449.3		4	4.3	odd	2	inner
3200.2.f.o.449.3		4	8.5	even	2	inner
3200.2.f.o.449.4		4	5.4	even	2	inner
3200.2.f.o.449.4		4	40.19	odd	2	inner
6400.2.a.by.1.1		2	80.3	even	4
6400.2.a.by.1.1		2	80.53	odd	4
6400.2.a.by.1.2		2	80.13	odd	4
6400.2.a.by.1.2		2	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)

\(f(q)\)	\(=\)	\(q-2.82843 q^{3} +5.00000 q^{9} +O(q^{10})\)	\(q-2.82843 q^{3} +5.00000 q^{9} +2.82843i q^{11} -6.00000i q^{17} +8.48528i q^{19} -5.65685 q^{27} -8.00000i q^{33} -6.00000 q^{41} +8.48528 q^{43} +7.00000 q^{49} +16.9706i q^{51} -24.0000i q^{57} -14.1421i q^{59} -8.48528 q^{67} -2.00000i q^{73} +1.00000 q^{81} -2.82843 q^{83} +18.0000 q^{89} +10.0000i q^{97} +14.1421i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 20 q^{9}+O(q^{10}) \) 4 * q + 20 * q^9	\( 4 q + 20 q^{9} - 24 q^{41} + 28 q^{49} + 4 q^{81} + 72 q^{89}+O(q^{100}) \) 4 * q + 20 * q^9 - 24 * q^41 + 28 * q^49 + 4 * q^81 + 72 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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