LMFDB - Embedded newform 800.3.e.a.399.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [800,3,Mod(399,800)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("800.399"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(800, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	800.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$21.7984211488$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 8)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			399.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	800.399
Dual form			800.3.e.a.399.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{3} +5.00000 q^{9} -14.0000 q^{11} -2.00000i q^{17} -34.0000 q^{19} +28.0000i q^{27} -28.0000i q^{33} -46.0000 q^{41} -14.0000i q^{43} -49.0000 q^{49} +4.00000 q^{51} -68.0000i q^{57} -82.0000 q^{59} +62.0000i q^{67} -142.000i q^{73} -11.0000 q^{81} -158.000i q^{83} -146.000 q^{89} +94.0000i q^{97} -70.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 10 q^{9} - 28 q^{11} - 68 q^{19} - 92 q^{41} - 98 q^{49} + 8 q^{51} - 164 q^{59} - 22 q^{81} - 292 q^{89} - 140 q^{99}+O(q^{100}) $ 2 * q + 10 * q^9 - 28 * q^11 - 68 * q^19 - 92 * q^41 - 98 * q^49 + 8 * q^51 - 164 * q^59 - 22 * q^81 - 292 * q^89 - 140 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/800\mathbb{Z}\right)^\times$.

$n$	$101$	$351$	$577$
$\chi(n)$	$-1$	$-1$	$-1$

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
$3$	2.00000i	0.666667i	0.942809	+	0.333333i	$0.108173\pi$
$3$	2.00000i	0.666667i	−0.942809	+	0.333333i	$0.891827\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	0	0
$9$	5.00000	0.555556
$10$	0	0
$11$	−14.0000	−1.27273	−0.636364	−	0.771389i	$-0.719562\pi$
$11$	−14.0000	−1.27273	−0.636364	+	0.771389i	$0.719562\pi$
$12$	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
$17$	− 2.00000i	− 0.117647i	−0.998268	−	0.0588235i	$-0.981265\pi$
$17$	− 2.00000i	− 0.117647i	0.998268	−	0.0588235i	$-0.0187349\pi$
$18$	0	0
$19$	−34.0000	−1.78947	−0.894737	−	0.446594i	$-0.852637\pi$
$19$	−34.0000	−1.78947	−0.894737	+	0.446594i	$0.852637\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	28.0000i	1.03704i
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	0	0
$33$	− 28.0000i	− 0.848485i
$34$	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
$41$	−46.0000	−1.12195	−0.560976	−	0.827832i	$-0.689574\pi$
$41$	−46.0000	−1.12195	−0.560976	+	0.827832i	$0.689574\pi$
$42$	0	0
$43$	− 14.0000i	− 0.325581i	−0.986661	−	0.162791i	$-0.947950\pi$
$43$	− 14.0000i	− 0.325581i	0.986661	−	0.162791i	$-0.0520495\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	−49.0000	−1.00000
$50$	0	0
$51$	4.00000	0.0784314
$52$	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
$57$	− 68.0000i	− 1.19298i
$58$	0	0
$59$	−82.0000	−1.38983	−0.694915	−	0.719092i	$-0.744558\pi$
$59$	−82.0000	−1.38983	−0.694915	+	0.719092i	$0.744558\pi$
$60$	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
$67$	62.0000i	0.925373i	0.886522	+	0.462687i	$0.153114\pi$
$67$	62.0000i	0.925373i	−0.886522	+	0.462687i	$0.846886\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0	0
$73$	− 142.000i	− 1.94521i	−0.232473	−	0.972603i	$-0.574682\pi$
$73$	− 142.000i	− 1.94521i	0.232473	−	0.972603i	$-0.425318\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	−11.0000	−0.135802
$82$	0	0
$83$	− 158.000i	− 1.90361i	−0.306697	−	0.951807i	$-0.599224\pi$
$83$	− 158.000i	− 1.90361i	0.306697	−	0.951807i	$-0.400776\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−146.000	−1.64045	−0.820225	−	0.572041i	$-0.806152\pi$
$89$	−146.000	−1.64045	−0.820225	+	0.572041i	$0.806152\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	94.0000i	0.969072i	0.874771	+	0.484536i	$0.161012\pi$
$97$	94.0000i	0.969072i	−0.874771	+	0.484536i	$0.838988\pi$
$98$	0	0
$99$	−70.0000	−0.707071
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 178.000i	− 1.66355i	−0.555112	−	0.831776i	$-0.687325\pi$
$107$	− 178.000i	− 1.66355i	0.555112	−	0.831776i	$-0.312675\pi$
$108$	0	0
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	98.0000i	0.867257i	0.901092	+	0.433628i	$0.142767\pi$
$113$	98.0000i	0.867257i	−0.901092	+	0.433628i	$0.857233\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	75.0000	0.619835
$122$	0	0
$123$	− 92.0000i	− 0.747967i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	0	0
$129$	28.0000	0.217054
$130$	0	0
$131$	−62.0000	−0.473282	−0.236641	−	0.971597i	$-0.576047\pi$
$131$	−62.0000	−0.473282	−0.236641	+	0.971597i	$0.576047\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	238.000i	1.73723i	0.495491	+	0.868613i	$0.334988\pi$
$137$	238.000i	1.73723i	−0.495491	+	0.868613i	$0.665012\pi$
$138$	0	0
$139$	206.000	1.48201	0.741007	−	0.671497i	$-0.234348\pi$
$139$	206.000	1.48201	0.741007	+	0.671497i	$0.234348\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 98.0000i	− 0.666667i
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	− 10.0000i	− 0.0653595i
$154$	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
$163$	322.000i	1.97546i	0.156171	+	0.987730i	$0.450085\pi$
$163$	322.000i	1.97546i	−0.156171	+	0.987730i	$0.549915\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	−169.000	−1.00000
$170$	0	0
$171$	−170.000	−0.994152
$172$	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
$177$	− 164.000i	− 0.926554i
$178$	0	0
$179$	−34.0000	−0.189944	−0.0949721	−	0.995480i	$-0.530276\pi$
$179$	−34.0000	−0.189944	−0.0949721	+	0.995480i	$0.530276\pi$
$180$	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
$187$	28.0000i	0.149733i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	0	0
$193$	98.0000i	0.507772i	0.967234	+	0.253886i	$0.0817088\pi$
$193$	98.0000i	0.507772i	−0.967234	+	0.253886i	$0.918291\pi$
$194$	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	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	0	0
$201$	−124.000	−0.616915
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	476.000	2.27751
$210$	0	0
$211$	226.000	1.07109	0.535545	−	0.844507i	$-0.320106\pi$
$211$	226.000	1.07109	0.535545	+	0.844507i	$0.320106\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	284.000	1.29680
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	446.000i	1.96476i	0.186900	+	0.982379i	$0.440156\pi$
$227$	446.000i	1.96476i	−0.186900	+	0.982379i	$0.559844\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	434.000i	1.86266i	0.364175	+	0.931330i	$0.381351\pi$
$233$	434.000i	1.86266i	−0.364175	+	0.931330i	$0.618649\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	194.000	0.804979	0.402490	−	0.915425i	$-0.368145\pi$
$241$	194.000	0.804979	0.402490	+	0.915425i	$0.368145\pi$
$242$	0	0
$243$	230.000i	0.946502i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	316.000	1.26908
$250$	0	0
$251$	466.000	1.85657	0.928287	−	0.371865i	$-0.121282\pi$
$251$	466.000	1.85657	0.928287	+	0.371865i	$0.121282\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 386.000i	− 1.50195i	−0.660333	−	0.750973i	$-0.729585\pi$
$257$	− 386.000i	− 1.50195i	0.660333	−	0.750973i	$-0.270415\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 292.000i	− 1.09363i
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	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
$281$	−238.000	−0.846975	−0.423488	−	0.905902i	$-0.639194\pi$
$281$	−238.000	−0.846975	−0.423488	+	0.905902i	$0.639194\pi$
$282$	0	0
$283$	82.0000i	0.289753i	0.989450	+	0.144876i	$0.0462784\pi$
$283$	82.0000i	0.289753i	−0.989450	+	0.144876i	$0.953722\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	285.000	0.986159
$290$	0	0
$291$	−188.000	−0.646048
$292$	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
$297$	− 392.000i	− 1.31987i
$298$	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
$307$	542.000i	1.76547i	0.469869	+	0.882736i	$0.344301\pi$
$307$	542.000i	1.76547i	−0.469869	+	0.882736i	$0.655699\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	− 526.000i	− 1.68051i	−0.542191	−	0.840256i	$-0.682405\pi$
$313$	− 526.000i	− 1.68051i	0.542191	−	0.840256i	$-0.317595\pi$
$314$	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
$321$	356.000	1.10903
$322$	0	0
$323$	68.0000i	0.210526i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−14.0000	−0.0422961	−0.0211480	−	0.999776i	$-0.506732\pi$
$331$	−14.0000	−0.0422961	−0.0211480	+	0.999776i	$0.506732\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	478.000i	1.41840i	0.705009	+	0.709199i	$0.250943\pi$
$337$	478.000i	1.41840i	−0.705009	+	0.709199i	$0.749057\pi$
$338$	0	0
$339$	−196.000	−0.578171
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 658.000i	− 1.89625i	−0.317892	−	0.948127i	$-0.602975\pi$
$347$	− 658.000i	− 1.89625i	0.317892	−	0.948127i	$-0.397025\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	194.000i	0.549575i	0.961505	+	0.274788i	$0.0886075\pi$
$353$	194.000i	0.549575i	−0.961505	+	0.274788i	$0.911392\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	795.000	2.20222
$362$	0	0
$363$	150.000i	0.413223i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	0	0
$369$	−230.000	−0.623306
$370$	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
$379$	686.000	1.81003	0.905013	−	0.425383i	$-0.139861\pi$
$379$	686.000	1.81003	0.905013	+	0.425383i	$0.139861\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 70.0000i	− 0.180879i
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	− 124.000i	− 0.315522i
$394$	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
$401$	−766.000	−1.91022	−0.955112	−	0.296244i	$-0.904266\pi$
$401$	−766.000	−1.91022	−0.955112	+	0.296244i	$0.904266\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	334.000	0.816626	0.408313	−	0.912842i	$-0.366117\pi$
$409$	334.000	0.816626	0.408313	+	0.912842i	$0.366117\pi$
$410$	0	0
$411$	−476.000	−1.15815
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	412.000i	0.988010i
$418$	0	0
$419$	−514.000	−1.22673	−0.613365	−	0.789799i	$-0.710185\pi$
$419$	−514.000	−1.22673	−0.613365	+	0.789799i	$0.710185\pi$
$420$	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	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0	0
$433$	578.000i	1.33487i	0.744667	+	0.667436i	$0.232608\pi$
$433$	578.000i	1.33487i	−0.744667	+	0.667436i	$0.767392\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	−245.000	−0.555556
$442$	0	0
$443$	− 878.000i	− 1.98194i	−0.134079	−	0.990971i	$-0.542808\pi$
$443$	− 878.000i	− 1.98194i	0.134079	−	0.990971i	$-0.457192\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−866.000	−1.92873	−0.964365	−	0.264574i	$-0.914769\pi$
$449$	−866.000	−1.92873	−0.964365	+	0.264574i	$0.914769\pi$
$450$	0	0
$451$	644.000	1.42794
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	238.000i	0.520788i	0.965502	+	0.260394i	$0.0838524\pi$
$457$	238.000i	0.520788i	−0.965502	+	0.260394i	$0.916148\pi$
$458$	0	0
$459$	56.0000	0.122004
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	0	0		−	1.00000i	$-0.5\pi$
$463$	0	0			1.00000i	$0.5\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 34.0000i	− 0.0728051i	−0.999337	−	0.0364026i	$-0.988410\pi$
$467$	− 34.0000i	− 0.0728051i	0.999337	−	0.0364026i	$-0.0115899\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	196.000i	0.414376i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	0	0		−	1.00000i	$-0.5\pi$
$487$	0	0			1.00000i	$0.5\pi$
$488$	0	0
$489$	−644.000	−1.31697
$490$	0	0
$491$	−782.000	−1.59267	−0.796334	−	0.604857i	$-0.793230\pi$
$491$	−782.000	−1.59267	−0.796334	+	0.604857i	$0.793230\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−802.000	−1.60721	−0.803607	−	0.595160i	$-0.797089\pi$
$499$	−802.000	−1.60721	−0.803607	+	0.595160i	$0.797089\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 338.000i	− 0.666667i
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	− 952.000i	− 1.85575i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−1006.00	−1.93090	−0.965451	−	0.260584i	$-0.916085\pi$
$521$	−1006.00	−1.93090	−0.965451	+	0.260584i	$0.916085\pi$
$522$	0	0
$523$	− 398.000i	− 0.760994i	−0.924782	−	0.380497i	$-0.875753\pi$
$523$	− 398.000i	− 0.760994i	0.924782	−	0.380497i	$-0.124247\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−529.000	−1.00000
$530$	0	0
$531$	−410.000	−0.772128
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 68.0000i	− 0.126629i
$538$	0	0
$539$	686.000	1.27273
$540$	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
$547$	1022.00i	1.86837i	0.356785	+	0.934186i	$0.383873\pi$
$547$	1022.00i	1.86837i	−0.356785	+	0.934186i	$0.616127\pi$
$548$	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
$561$	−56.0000	−0.0998217
$562$	0	0
$563$	226.000i	0.401421i	0.979651	+	0.200710i	$0.0643251\pi$
$563$	226.000i	0.401421i	−0.979651	+	0.200710i	$0.935675\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−626.000	−1.10018	−0.550088	−	0.835107i	$-0.685406\pi$
$569$	−626.000	−1.10018	−0.550088	+	0.835107i	$0.685406\pi$
$570$	0	0
$571$	658.000	1.15236	0.576182	−	0.817321i	$-0.304542\pi$
$571$	658.000	1.15236	0.576182	+	0.817321i	$0.304542\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 2.00000i	− 0.00346620i	−0.999998	−	0.00173310i	$-0.999448\pi$
$577$	− 2.00000i	− 0.00346620i	0.999998	−	0.00173310i	$-0.000551664\pi$
$578$	0	0
$579$	−196.000	−0.338515
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 1138.00i	− 1.93867i	−0.245741	−	0.969336i	$-0.579031\pi$
$587$	− 1138.00i	− 1.93867i	0.245741	−	0.969336i	$-0.420969\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 862.000i	− 1.45363i	−0.686836	−	0.726813i	$-0.741001\pi$
$593$	− 862.000i	− 1.45363i	0.686836	−	0.726813i	$-0.258999\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	914.000	1.52080	0.760399	−	0.649456i	$-0.225003\pi$
$601$	914.000	1.52080	0.760399	+	0.649456i	$0.225003\pi$
$602$	0	0
$603$	310.000i	0.514096i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	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
$617$	334.000i	0.541329i	0.962674	+	0.270665i	$0.0872434\pi$
$617$	334.000i	0.541329i	−0.962674	+	0.270665i	$0.912757\pi$
$618$	0	0
$619$	−562.000	−0.907916	−0.453958	−	0.891023i	$-0.649988\pi$
$619$	−562.000	−0.907916	−0.453958	+	0.891023i	$0.649988\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	952.000i	1.51834i
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	452.000i	0.714060i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	482.000	0.751950	0.375975	−	0.926630i	$-0.377308\pi$
$641$	482.000	0.751950	0.375975	+	0.926630i	$0.377308\pi$
$642$	0	0
$643$	− 1214.00i	− 1.88802i	−0.329910	−	0.944012i	$-0.607018\pi$
$643$	− 1214.00i	− 1.88802i	0.329910	−	0.944012i	$-0.392982\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	1148.00	1.76888
$650$	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
$657$	− 710.000i	− 1.08067i
$658$	0	0
$659$	−994.000	−1.50835	−0.754173	−	0.656676i	$-0.771962\pi$
$659$	−994.000	−1.50835	−0.754173	+	0.656676i	$0.771962\pi$
$660$	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
$673$	− 1246.00i	− 1.85141i	−0.378244	−	0.925706i	$-0.623472\pi$
$673$	− 1246.00i	− 1.85141i	0.378244	−	0.925706i	$-0.376528\pi$
$674$	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
$681$	−892.000	−1.30984
$682$	0	0
$683$	− 398.000i	− 0.582723i	−0.956613	−	0.291362i	$-0.905892\pi$
$683$	− 398.000i	− 0.582723i	0.956613	−	0.291362i	$-0.0941083\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−734.000	−1.06223	−0.531114	−	0.847300i	$-0.678227\pi$
$691$	−734.000	−1.06223	−0.531114	+	0.847300i	$0.678227\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	92.0000i	0.131994i
$698$	0	0
$699$	−868.000	−1.24177
$700$	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	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	388.000i	0.536653i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	0	0
$729$	−559.000	−0.766804
$730$	0	0
$731$	−28.0000	−0.0383037
$732$	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
$737$	− 868.000i	− 1.17775i
$738$	0	0
$739$	−322.000	−0.435724	−0.217862	−	0.975980i	$-0.569908\pi$
$739$	−322.000	−0.435724	−0.217862	+	0.975980i	$0.569908\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 790.000i	− 1.05756i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	932.000i	1.23772i
$754$	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
$761$	1394.00	1.83180	0.915900	−	0.401406i	$-0.131478\pi$
$761$	1394.00	1.83180	0.915900	+	0.401406i	$0.131478\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	1054.00	1.37061	0.685306	−	0.728256i	$-0.259669\pi$
$769$	1054.00	1.37061	0.685306	+	0.728256i	$0.259669\pi$
$770$	0	0
$771$	772.000	1.00130
$772$	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
$779$	1564.00	2.00770
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	926.000i	1.17662i	0.808635	+	0.588310i	$0.200207\pi$
$787$	926.000i	1.17662i	−0.808635	+	0.588310i	$0.799793\pi$
$788$	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
$801$	−730.000	−0.911361
$802$	0	0
$803$	1988.00i	2.47572i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1582.00	1.95550	0.977750	−	0.209772i	$-0.0672722\pi$
$809$	1582.00	1.95550	0.977750	+	0.209772i	$0.0672722\pi$
$810$	0	0
$811$	178.000	0.219482	0.109741	−	0.993960i	$-0.464998\pi$
$811$	178.000	0.219482	0.109741	+	0.993960i	$0.464998\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	476.000i	0.582619i
$818$	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		−	1.00000i	$-0.5\pi$
$823$	0	0			1.00000i	$0.5\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1262.00i	1.52600i	0.646400	+	0.762999i	$0.276274\pi$
$827$	1262.00i	1.52600i	−0.646400	+	0.762999i	$0.723726\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	98.0000i	0.117647i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	841.000	1.00000
$842$	0	0
$843$	− 476.000i	− 0.564650i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−164.000	−0.193168
$850$	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
$857$	− 1202.00i	− 1.40257i	−0.712882	−	0.701284i	$-0.752611\pi$
$857$	− 1202.00i	− 1.40257i	0.712882	−	0.701284i	$-0.247389\pi$
$858$	0	0
$859$	1646.00	1.91618	0.958091	−	0.286465i	$-0.0924801\pi$
$859$	1646.00	1.91618	0.958091	+	0.286465i	$0.0924801\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	570.000i	0.657439i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	470.000i	0.538373i
$874$	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
$881$	−1438.00	−1.63224	−0.816118	−	0.577885i	$-0.803878\pi$
$881$	−1438.00	−1.63224	−0.816118	+	0.577885i	$0.803878\pi$
$882$	0	0
$883$	1762.00i	1.99547i	0.0672672	+	0.997735i	$0.478572\pi$
$883$	1762.00i	1.99547i	−0.0672672	+	0.997735i	$0.521428\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	154.000	0.172840
$892$	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
$907$	− 1714.00i	− 1.88975i	−0.327436	−	0.944873i	$-0.606185\pi$
$907$	− 1714.00i	− 1.88975i	0.327436	−	0.944873i	$-0.393815\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	2212.00i	2.42278i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	−1084.00	−1.17698
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−1058.00	−1.13886	−0.569429	−	0.822040i	$-0.692836\pi$
$929$	−1058.00	−1.13886	−0.569429	+	0.822040i	$0.692836\pi$
$930$	0	0
$931$	1666.00	1.78947
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	718.000i	0.766275i	0.923691	+	0.383138i	$0.125157\pi$
$937$	718.000i	0.766275i	−0.923691	+	0.383138i	$0.874843\pi$
$938$	0	0
$939$	1052.00	1.12034
$940$	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
$947$	− 994.000i	− 1.04963i	−0.851216	−	0.524815i	$-0.824134\pi$
$947$	− 994.000i	− 1.04963i	0.851216	−	0.524815i	$-0.175866\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 142.000i	− 0.149003i	−0.997221	−	0.0745016i	$-0.976263\pi$
$953$	− 142.000i	− 0.149003i	0.997221	−	0.0745016i	$-0.0237366\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	961.000	1.00000
$962$	0	0
$963$	− 890.000i	− 0.924195i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0	0		−	1.00000i	$-0.5\pi$
$967$	0	0			1.00000i	$0.5\pi$
$968$	0	0
$969$	−136.000	−0.140351
$970$	0	0
$971$	−974.000	−1.00309	−0.501545	−	0.865132i	$-0.667235\pi$
$971$	−974.000	−1.00309	−0.501545	+	0.865132i	$0.667235\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1918.00i	1.96315i	0.191071	+	0.981576i	$0.438804\pi$
$977$	1918.00i	1.96315i	−0.191071	+	0.981576i	$0.561196\pi$
$978$	0	0
$979$	2044.00	2.08784
$980$	0	0
$981$	0	0
$982$	0	0
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	− 28.0000i	− 0.0281974i
$994$	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	800.3.e.a.399.2		2
4.3	odd	2		200.3.e.a.99.2		2
5.2	odd	4		32.3.d.a.15.1		1
5.3	odd	4		800.3.g.a.751.1		1
5.4	even	2	inner	800.3.e.a.399.1		2
8.3	odd	2	CM	800.3.e.a.399.2		2
8.5	even	2		200.3.e.a.99.2		2
15.2	even	4		288.3.b.a.271.1		1
20.3	even	4		200.3.g.a.51.1		1
20.7	even	4		8.3.d.a.3.1	✓	1
20.19	odd	2		200.3.e.a.99.1		2
35.27	even	4		1568.3.g.a.687.1		1
40.3	even	4		800.3.g.a.751.1		1
40.13	odd	4		200.3.g.a.51.1		1
40.19	odd	2	inner	800.3.e.a.399.1		2
40.27	even	4		32.3.d.a.15.1		1
40.29	even	2		200.3.e.a.99.1		2
40.37	odd	4		8.3.d.a.3.1	✓	1
60.47	odd	4		72.3.b.a.19.1		1
80.27	even	4		256.3.c.e.255.2		2
80.37	odd	4		256.3.c.e.255.1		2
80.67	even	4		256.3.c.e.255.1		2
80.77	odd	4		256.3.c.e.255.2		2
120.77	even	4		72.3.b.a.19.1		1
120.107	odd	4		288.3.b.a.271.1		1
140.27	odd	4		392.3.g.a.99.1		1
140.47	odd	12		392.3.k.b.67.1		2
140.67	even	12		392.3.k.d.275.1		2
140.87	odd	12		392.3.k.b.275.1		2
140.107	even	12		392.3.k.d.67.1		2
240.77	even	4		2304.3.g.j.1279.1		2
240.107	odd	4		2304.3.g.j.1279.1		2
240.197	even	4		2304.3.g.j.1279.2		2
240.227	odd	4		2304.3.g.j.1279.2		2
280.27	odd	4		1568.3.g.a.687.1		1
280.37	odd	12		392.3.k.d.67.1		2
280.117	even	12		392.3.k.b.67.1		2
280.157	even	12		392.3.k.b.275.1		2
280.237	even	4		392.3.g.a.99.1		1
280.277	odd	12		392.3.k.d.275.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.3.d.a.3.1	✓	1	20.7	even	4
8.3.d.a.3.1	✓	1	40.37	odd	4
32.3.d.a.15.1		1	5.2	odd	4
32.3.d.a.15.1		1	40.27	even	4
72.3.b.a.19.1		1	60.47	odd	4
72.3.b.a.19.1		1	120.77	even	4
200.3.e.a.99.1		2	20.19	odd	2
200.3.e.a.99.1		2	40.29	even	2
200.3.e.a.99.2		2	4.3	odd	2
200.3.e.a.99.2		2	8.5	even	2
200.3.g.a.51.1		1	20.3	even	4
200.3.g.a.51.1		1	40.13	odd	4
256.3.c.e.255.1		2	80.37	odd	4
256.3.c.e.255.1		2	80.67	even	4
256.3.c.e.255.2		2	80.27	even	4
256.3.c.e.255.2		2	80.77	odd	4
288.3.b.a.271.1		1	15.2	even	4
288.3.b.a.271.1		1	120.107	odd	4
392.3.g.a.99.1		1	140.27	odd	4
392.3.g.a.99.1		1	280.237	even	4
392.3.k.b.67.1		2	140.47	odd	12
392.3.k.b.67.1		2	280.117	even	12
392.3.k.b.275.1		2	140.87	odd	12
392.3.k.b.275.1		2	280.157	even	12
392.3.k.d.67.1		2	140.107	even	12
392.3.k.d.67.1		2	280.37	odd	12
392.3.k.d.275.1		2	140.67	even	12
392.3.k.d.275.1		2	280.277	odd	12
800.3.e.a.399.1		2	5.4	even	2	inner
800.3.e.a.399.1		2	40.19	odd	2	inner
800.3.e.a.399.2		2	1.1	even	1	trivial
800.3.e.a.399.2		2	8.3	odd	2	CM
800.3.g.a.751.1		1	5.3	odd	4
800.3.g.a.751.1		1	40.3	even	4
1568.3.g.a.687.1		1	35.27	even	4
1568.3.g.a.687.1		1	280.27	odd	4
2304.3.g.j.1279.1		2	240.77	even	4
2304.3.g.j.1279.1		2	240.107	odd	4
2304.3.g.j.1279.2		2	240.197	even	4
2304.3.g.j.1279.2		2	240.227	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	800.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{3} +5.00000 q^{9} -14.0000 q^{11} -2.00000i q^{17} -34.0000 q^{19} +28.0000i q^{27} -28.0000i q^{33} -46.0000 q^{41} -14.0000i q^{43} -49.0000 q^{49} +4.00000 q^{51} -68.0000i q^{57} -82.0000 q^{59} +62.0000i q^{67} -142.000i q^{73} -11.0000 q^{81} -158.000i q^{83} -146.000 q^{89} +94.0000i q^{97} -70.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 10 q^{9} - 28 q^{11} - 68 q^{19} - 92 q^{41} - 98 q^{49} + 8 q^{51} - 164 q^{59} - 22 q^{81} - 292 q^{89} - 140 q^{99}+O(q^{100}) \) 2 * q + 10 * q^9 - 28 * q^11 - 68 * q^19 - 92 * q^41 - 98 * q^49 + 8 * q^51 - 164 * q^59 - 22 * q^81 - 292 * q^89 - 140 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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