LMFDB - Embedded newform 363.4.d.a.362.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [363,4,Mod(362,363)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(363, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("363.362");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 363 = 3 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	363.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$21.4176933321$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 4x^{2} + 1 $ x^4 + 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2\cdot 3^{2}\cdot 11 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			362.4
Root			$0.517638i$ of defining polynomial
Character	$\chi$	$=$	363.362
Dual form			363.4.d.a.362.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.19615 q^{3} -8.00000 q^{4} +7.90327i q^{7} +27.0000 q^{9} +O(q^{10})$	$q+5.19615 q^{3} -8.00000 q^{4} +7.90327i q^{7} +27.0000 q^{9} -41.5692 q^{12} -93.5883i q^{13} +64.0000 q^{16} +70.6291i q^{19} +41.0666i q^{21} +125.000 q^{25} +140.296 q^{27} -63.2262i q^{28} +155.885 q^{31} -216.000 q^{36} +436.477 q^{37} -486.299i q^{39} -522.013i q^{43} +332.554 q^{48} +280.538 q^{49} +748.706i q^{52} +366.999i q^{57} -532.669i q^{61} +213.388i q^{63} -512.000 q^{64} -654.715 q^{67} +1106.00i q^{73} +649.519 q^{75} -565.032i q^{76} -1396.67i q^{79} +729.000 q^{81} -328.533i q^{84} +739.654 q^{91} +810.000 q^{93} +1330.00 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 32 q^{4} + 108 q^{9}+O(q^{10}) $ 4 * q - 32 * q^4 + 108 * q^9	$ 4 q - 32 q^{4} + 108 q^{9} + 256 q^{16} + 500 q^{25} - 864 q^{36} - 1372 q^{49} - 2048 q^{64} + 2916 q^{81} + 1088 q^{91} + 3240 q^{93} + 5320 q^{97}+O(q^{100}) $ 4 * q - 32 * q^4 + 108 * q^9 + 256 * q^16 + 500 * q^25 - 864 * q^36 - 1372 * q^49 - 2048 * q^64 + 2916 * q^81 + 1088 * q^91 + 3240 * q^93 + 5320 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/363\mathbb{Z}\right)^\times$.

$n$	$122$	$244$
$\chi(n)$	$-1$	$-1$

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		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	5.19615	1.00000
$3$	5.19615	1.00000
$4$	−8.00000	−1.00000
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	0	0
$7$	7.90327i	0.426737i	0.976972	+	0.213368i	$0.0684434\pi$
$7$	7.90327i	0.426737i	−0.976972	+	0.213368i	$0.931557\pi$
$8$	0	0
$9$	27.0000	1.00000
$10$	0	0
$11$	0	0
$11$	0	0
$12$	−41.5692	−1.00000
$13$	− 93.5883i	− 1.99667i	−0.0576745	−	0.998335i	$-0.518369\pi$
$13$	− 93.5883i	− 1.99667i	0.0576745	−	0.998335i	$-0.481631\pi$
$14$	0	0
$15$	0	0
$16$	64.0000	1.00000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	70.6291i	0.852811i	0.904532	+	0.426406i	$0.140220\pi$
$19$	70.6291i	0.852811i	−0.904532	+	0.426406i	$0.859780\pi$
$20$	0	0
$21$	41.0666i	0.426737i
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	125.000	1.00000
$26$	0	0
$27$	140.296	1.00000
$28$	− 63.2262i	− 0.426737i
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	155.885	0.903151	0.451576	−	0.892233i	$-0.350862\pi$
$31$	155.885	0.903151	0.451576	+	0.892233i	$0.350862\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	−216.000	−1.00000
$37$	436.477	1.93936	0.969680	−	0.244377i	$-0.0785834\pi$
$37$	436.477	1.93936	0.969680	+	0.244377i	$0.0785834\pi$
$38$	0	0
$39$	− 486.299i	− 1.99667i
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	− 522.013i	− 1.85131i	−0.378370	−	0.925655i	$-0.623515\pi$
$43$	− 522.013i	− 1.85131i	0.378370	−	0.925655i	$-0.376485\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	332.554	1.00000
$49$	280.538	0.817896
$50$	0	0
$51$	0	0
$52$	748.706i	1.99667i
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	366.999i	0.852811i
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	− 532.669i	− 1.11805i	−0.829150	−	0.559027i	$-0.811175\pi$
$61$	− 532.669i	− 1.11805i	0.829150	−	0.559027i	$-0.188825\pi$
$62$	0	0
$63$	213.388i	0.426737i
$64$	−512.000	−1.00000
$65$	0	0
$66$	0	0
$67$	−654.715	−1.19382	−0.596912	−	0.802307i	$-0.703606\pi$
$67$	−654.715	−1.19382	−0.596912	+	0.802307i	$0.703606\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0	0
$73$	1106.00i	1.77326i	0.462483	+	0.886628i	$0.346959\pi$
$73$	1106.00i	1.77326i	−0.462483	+	0.886628i	$0.653041\pi$
$74$	0	0
$75$	649.519	1.00000
$76$	− 565.032i	− 0.852811i
$77$	0	0
$78$	0	0
$79$	− 1396.67i	− 1.98909i	−0.104325	−	0.994543i	$-0.533268\pi$
$79$	− 1396.67i	− 1.98909i	0.104325	−	0.994543i	$-0.466732\pi$
$80$	0	0
$81$	729.000	1.00000
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	− 328.533i	− 0.426737i
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	739.654	0.852053
$92$	0	0
$93$	810.000	0.903151
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1330.00	1.39218	0.696088	−	0.717957i	$-0.254922\pi$
$97$	1330.00	1.39218	0.696088	+	0.717957i	$0.254922\pi$
$98$	0	0
$99$	0	0
$100$	−1000.00	−1.00000
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	−1820.00	−1.74107	−0.870534	−	0.492109i	$-0.836226\pi$
$103$	−1820.00	−1.74107	−0.870534	+	0.492109i	$0.836226\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	−1122.37	−1.00000
$109$	1999.97i	1.75745i	0.477326	+	0.878726i	$0.341606\pi$
$109$	1999.97i	1.75745i	−0.477326	+	0.878726i	$0.658394\pi$
$110$	0	0
$111$	2268.00	1.93936
$112$	505.809i	0.426737i
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 2526.88i	− 1.99667i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	0	0
$122$	0	0
$123$	0	0
$124$	−1247.08	−0.903151
$125$	0	0
$126$	0	0
$127$	2274.83i	1.58944i	0.606977	+	0.794720i	$0.292382\pi$
$127$	2274.83i	1.58944i	−0.606977	+	0.794720i	$0.707618\pi$
$128$	0	0
$129$	− 2712.46i	− 1.85131i
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	−558.201	−0.363926
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	388.556i	0.237100i	0.992948	+	0.118550i	$0.0378245\pi$
$139$	388.556i	0.237100i	−0.992948	+	0.118550i	$0.962175\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	1728.00	1.00000
$145$	0	0
$146$	0	0
$147$	1457.72	0.817896
$148$	−3491.81	−1.93936
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	− 1078.75i	− 0.581371i	−0.956819	−	0.290686i	$-0.906117\pi$
$151$	− 1078.75i	− 0.581371i	0.956819	−	0.290686i	$-0.0938834\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	3890.39i	1.99667i
$157$	810.600	0.412057	0.206028	−	0.978546i	$-0.433946\pi$
$157$	810.600	0.412057	0.206028	+	0.978546i	$0.433946\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−3400.00	−1.63379	−0.816897	−	0.576783i	$-0.804308\pi$
$163$	−3400.00	−1.63379	−0.816897	+	0.576783i	$0.804308\pi$
$164$	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
$169$	−6561.77	−2.98669
$170$	0	0
$171$	1906.98i	0.852811i
$172$	4176.11i	1.85131i
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	987.909i	0.426737i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	0	0
$181$	−3458.00	−1.42006	−0.710031	−	0.704171i	$-0.751319\pi$
$181$	−3458.00	−1.42006	−0.710031	+	0.704171i	$0.751319\pi$
$182$	0	0
$183$	− 2767.83i	− 1.11805i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	1108.80i	0.426737i
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	−2660.43	−1.00000
$193$	4516.80i	1.68459i	0.539014	+	0.842297i	$0.318797\pi$
$193$	4516.80i	1.68459i	−0.539014	+	0.842297i	$0.681203\pi$
$194$	0	0
$195$	0	0
$196$	−2244.31	−0.817896
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	−2026.50	−0.721883	−0.360942	−	0.932588i	$-0.617545\pi$
$199$	−2026.50	−0.721883	−0.360942	+	0.932588i	$0.617545\pi$
$200$	0	0
$201$	−3402.00	−1.19382
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	− 5989.65i	− 1.99667i
$209$	0	0
$210$	0	0
$211$	− 5036.86i	− 1.64337i	−0.569940	−	0.821686i	$-0.693034\pi$
$211$	− 5036.86i	− 1.64337i	0.569940	−	0.821686i	$-0.306966\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1232.00i	0.385408i
$218$	0	0
$219$	5746.95i	1.77326i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	3220.00	0.966938	0.483469	−	0.875362i	$-0.339377\pi$
$223$	3220.00	0.966938	0.483469	+	0.875362i	$0.339377\pi$
$224$	0	0
$225$	3375.00	1.00000
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	− 2935.99i	− 0.852811i
$229$	5300.08	1.52943	0.764714	−	0.644370i	$-0.222880\pi$
$229$	5300.08	1.52943	0.764714	+	0.644370i	$0.222880\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 7257.32i	− 1.98909i
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	4335.22i	1.15874i	0.815065	+	0.579369i	$0.196701\pi$
$241$	4335.22i	1.15874i	−0.815065	+	0.579369i	$0.803299\pi$
$242$	0	0
$243$	3788.00	1.00000
$244$	4261.35i	1.11805i
$245$	0	0
$246$	0	0
$247$	6610.05	1.70278
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	− 1707.11i	− 0.426737i
$253$	0	0
$254$	0	0
$255$	0	0
$256$	4096.00	1.00000
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	3449.59i	0.827596i
$260$	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
$267$	0	0
$268$	5237.72	1.19382
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	− 6857.11i	− 1.53705i	−0.639821	−	0.768524i	$-0.720992\pi$
$271$	− 6857.11i	− 1.53705i	0.639821	−	0.768524i	$-0.279008\pi$
$272$	0	0
$273$	3843.35	0.852053
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 8713.72i	− 1.89010i	−0.326931	−	0.945048i	$-0.606015\pi$
$277$	− 8713.72i	− 1.89010i	0.326931	−	0.945048i	$-0.393985\pi$
$278$	0	0
$279$	4208.88	0.903151
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	1485.42i	0.312010i	0.987756	+	0.156005i	$0.0498616\pi$
$283$	1485.42i	0.312010i	−0.987756	+	0.156005i	$0.950138\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−4913.00	−1.00000
$290$	0	0
$291$	6910.88	1.39218
$292$	− 8848.02i	− 1.77326i
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	−5196.15	−1.00000
$301$	4125.61	0.790021
$302$	0	0
$303$	0	0
$304$	4520.26i	0.852811i
$305$	0	0
$306$	0	0
$307$	− 8647.93i	− 1.60770i	−0.594833	−	0.803849i	$-0.702782\pi$
$307$	− 8647.93i	− 1.60770i	0.594833	−	0.803849i	$-0.297218\pi$
$308$	0	0
$309$	−9457.00	−1.74107
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	4738.89	0.855776	0.427888	−	0.903832i	$-0.359258\pi$
$313$	4738.89	0.855776	0.427888	+	0.903832i	$0.359258\pi$
$314$	0	0
$315$	0	0
$316$	11173.4i	1.98909i
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	−5832.00	−1.00000
$325$	− 11698.5i	− 1.99667i
$326$	0	0
$327$	10392.1i	1.75745i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	992.000	0.164729	0.0823644	−	0.996602i	$-0.473753\pi$
$331$	992.000	0.164729	0.0823644	+	0.996602i	$0.473753\pi$
$332$	0	0
$333$	11784.9	1.93936
$334$	0	0
$335$	0	0
$336$	2628.26i	0.426737i
$337$	11510.6i	1.86059i	0.366806	+	0.930297i	$0.380451\pi$
$337$	11510.6i	1.86059i	−0.366806	+	0.930297i	$0.619549\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	4927.99i	0.775763i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	− 4676.75i	− 0.717309i	−0.933470	−	0.358654i	$-0.883236\pi$
$349$	− 4676.75i	− 0.717309i	0.933470	−	0.358654i	$-0.116764\pi$
$350$	0	0
$351$	− 13130.1i	− 1.99667i
$352$	0	0
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	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
$361$	1870.54	0.272713
$362$	0	0
$363$	0	0
$364$	−5917.23	−0.852053
$365$	0	0
$366$	0	0
$367$	4340.00	0.617292	0.308646	−	0.951177i	$-0.400124\pi$
$367$	4340.00	0.617292	0.308646	+	0.951177i	$0.400124\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	−6480.00	−0.903151
$373$	3485.96i	0.483904i	0.970288	+	0.241952i	$0.0777877\pi$
$373$	3485.96i	0.483904i	−0.970288	+	0.241952i	$0.922212\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−8584.00	−1.16340	−0.581702	−	0.813402i	$-0.697613\pi$
$379$	−8584.00	−1.16340	−0.581702	+	0.813402i	$0.697613\pi$
$380$	0	0
$381$	11820.4i	1.58944i
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 14094.4i	− 1.85131i
$388$	−10640.0	−1.39218
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	1190.00	0.150439	0.0752196	−	0.997167i	$-0.476034\pi$
$397$	1190.00	0.150439	0.0752196	+	0.997167i	$0.476034\pi$
$398$	0	0
$399$	−2900.50	−0.363926
$400$	8000.00	1.00000
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	− 14589.0i	− 1.80330i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	15971.7i	1.93093i	0.260538	+	0.965464i	$0.416100\pi$
$409$	15971.7i	1.93093i	−0.260538	+	0.965464i	$0.583900\pi$
$410$	0	0
$411$	0	0
$412$	14560.0	1.74107
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	2018.99i	0.237100i
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	2182.38	0.252643	0.126322	−	0.991989i	$-0.459683\pi$
$421$	2182.38	0.252643	0.126322	+	0.991989i	$0.459683\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	4209.83	0.477114
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	8978.95	1.00000
$433$	−2590.00	−0.287454	−0.143727	−	0.989617i	$-0.545909\pi$
$433$	−2590.00	−0.287454	−0.143727	+	0.989617i	$0.545909\pi$
$434$	0	0
$435$	0	0
$436$	− 15999.8i	− 1.75745i
$437$	0	0
$438$	0	0
$439$	18158.5i	1.97417i	0.160207	+	0.987083i	$0.448784\pi$
$439$	18158.5i	1.97417i	−0.160207	+	0.987083i	$0.551216\pi$
$440$	0	0
$441$	7574.53	0.817896
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	−18144.0	−1.93936
$445$	0	0
$446$	0	0
$447$	0	0
$448$	− 4046.48i	− 0.426737i
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	− 5605.32i	− 0.581371i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	1506.29i	0.154182i	0.997024	+	0.0770909i	$0.0245632\pi$
$457$	1506.29i	0.154182i	−0.997024	+	0.0770909i	$0.975437\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	19780.0	1.98543	0.992716	−	0.120482i	$-0.0384440\pi$
$463$	19780.0	1.98543	0.992716	+	0.120482i	$0.0384440\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	20215.1i	1.99667i
$469$	− 5174.39i	− 0.509448i
$470$	0	0
$471$	4212.00	0.412057
$472$	0	0
$473$	0	0
$474$	0	0
$475$	8828.63i	0.852811i
$476$	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
$481$	− 40849.1i	− 3.87226i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−5019.48	−0.467052	−0.233526	−	0.972351i	$-0.575026\pi$
$487$	−5019.48	−0.467052	−0.233526	+	0.972351i	$0.575026\pi$
$488$	0	0
$489$	−17666.9	−1.63379
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	9976.61	0.903151
$497$	0	0
$498$	0	0
$499$	−16367.9	−1.46839	−0.734195	−	0.678938i	$-0.762440\pi$
$499$	−16367.9	−1.46839	−0.734195	+	0.678938i	$0.762440\pi$
$500$	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
$507$	−34095.9	−2.98669
$508$	− 18198.7i	− 1.58944i
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	−8741.03	−0.756713
$512$	0	0
$513$	9908.98i	0.852811i
$514$	0	0
$515$	0	0
$516$	21699.7i	1.85131i
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	23129.7i	1.93382i	0.255110	+	0.966912i	$0.417888\pi$
$523$	23129.7i	1.93382i	−0.255110	+	0.966912i	$0.582112\pi$
$524$	0	0
$525$	5133.33i	0.426737i
$526$	0	0
$527$	0	0
$528$	0	0
$529$	12167.0	1.00000
$530$	0	0
$531$	0	0
$532$	4465.61	0.363926
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	23751.7i	1.88755i	0.330591	+	0.943774i	$0.392752\pi$
$541$	23751.7i	1.88755i	−0.330591	+	0.943774i	$0.607248\pi$
$542$	0	0
$543$	−17968.3	−1.42006
$544$	0	0
$545$	0	0
$546$	0	0
$547$	16895.5i	1.32066i	0.750976	+	0.660330i	$0.229584\pi$
$547$	16895.5i	1.32066i	−0.750976	+	0.660330i	$0.770416\pi$
$548$	0	0
$549$	− 14382.1i	− 1.11805i
$550$	0	0
$551$	0	0
$552$	0	0
$553$	11038.3	0.848816
$554$	0	0
$555$	0	0
$556$	− 3108.44i	− 0.237100i
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	−48854.3	−3.69646
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	5761.49i	0.426737i
$568$	0	0
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	− 6453.41i	− 0.472972i	−0.971635	−	0.236486i	$-0.924004\pi$
$571$	− 6453.41i	− 0.472972i	0.971635	−	0.236486i	$-0.0759957\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	−13824.0	−1.00000
$577$	−21325.0	−1.53860	−0.769300	−	0.638888i	$-0.779395\pi$
$577$	−21325.0	−1.53860	−0.769300	+	0.638888i	$0.779395\pi$
$578$	0	0
$579$	23470.0i	1.68459i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	−11661.8	−0.817896
$589$	11010.0i	0.770218i
$590$	0	0
$591$	0	0
$592$	27934.5	1.93936
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−10530.0	−0.721883
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	− 18515.1i	− 1.25665i	−0.777951	−	0.628325i	$-0.783741\pi$
$601$	− 18515.1i	− 1.25665i	0.777951	−	0.628325i	$-0.216259\pi$
$602$	0	0
$603$	−17677.3	−1.19382
$604$	8629.96i	0.581371i
$605$	0	0
$606$	0	0
$607$	13504.4i	0.903009i	0.892269	+	0.451505i	$0.149113\pi$
$607$	13504.4i	0.903009i	−0.892269	+	0.451505i	$0.850887\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 5295.65i	− 0.348922i	−0.984664	−	0.174461i	$-0.944182\pi$
$613$	− 5295.65i	− 0.348922i	0.984664	−	0.174461i	$-0.0558183\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	−26656.0	−1.73085	−0.865424	−	0.501040i	$-0.832951\pi$
$619$	−26656.0	−1.73085	−0.865424	+	0.501040i	$0.832951\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	− 31123.1i	− 1.99667i
$625$	15625.0	1.00000
$626$	0	0
$627$	0	0
$628$	−6484.80	−0.412057
$629$	0	0
$630$	0	0
$631$	31644.6	1.99643	0.998217	−	0.0596825i	$-0.0190088\pi$
$631$	31644.6	1.99643	0.998217	+	0.0596825i	$0.0190088\pi$
$632$	0	0
$633$	− 26172.3i	− 1.64337i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 26255.1i	− 1.63307i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	−13160.0	−0.807122	−0.403561	−	0.914953i	$-0.632228\pi$
$643$	−13160.0	−0.807122	−0.403561	+	0.914953i	$0.632228\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	6401.65i	0.385408i
$652$	27200.0	1.63379
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	29862.1i	1.77326i
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	27123.9	1.59606	0.798032	−	0.602615i	$-0.205875\pi$
$661$	27123.9	1.59606	0.798032	+	0.602615i	$0.205875\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	16731.6	0.966938
$670$	0	0
$671$	0	0
$672$	0	0
$673$	894.953i	0.0512599i	0.999671	+	0.0256299i	$0.00815916\pi$
$673$	894.953i	0.0512599i	−0.999671	+	0.0256299i	$0.991841\pi$
$674$	0	0
$675$	17537.0	1.00000
$676$	52494.1	2.98669
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	10511.4i	0.594092i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	− 15255.9i	− 0.852811i
$685$	0	0
$686$	0	0
$687$	27540.0	1.52943
$688$	− 33408.9i	− 1.85131i
$689$	0	0
$690$	0	0
$691$	−16072.0	−0.884816	−0.442408	−	0.896814i	$-0.645876\pi$
$691$	−16072.0	−0.884816	−0.442408	+	0.896814i	$0.645876\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	− 7903.27i	− 0.426737i
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	30827.9i	1.65391i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	10911.9	0.578005	0.289003	−	0.957328i	$-0.406676\pi$
$709$	10911.9	0.578005	0.289003	+	0.957328i	$0.406676\pi$
$710$	0	0
$711$	− 37710.1i	− 1.98909i
$712$	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
$721$	− 14384.0i	− 0.742977i
$722$	0	0
$723$	22526.5i	1.15874i
$724$	27664.0	1.42006
$725$	0	0
$726$	0	0
$727$	−37692.9	−1.92290	−0.961452	−	0.274971i	$-0.911332\pi$
$727$	−37692.9	−1.92290	−0.961452	+	0.274971i	$0.911332\pi$
$728$	0	0
$729$	19683.0	1.00000
$730$	0	0
$731$	0	0
$732$	22142.6i	1.11805i
$733$	36611.4i	1.84485i	0.386177	+	0.922425i	$0.373795\pi$
$733$	36611.4i	1.84485i	−0.386177	+	0.922425i	$0.626205\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	− 39932.7i	− 1.98775i	−0.110497	−	0.993876i	$-0.535244\pi$
$739$	− 39932.7i	− 1.98775i	0.110497	−	0.993876i	$-0.464756\pi$
$740$	0	0
$741$	34346.8	1.70278
$742$	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
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−33827.0	−1.64363	−0.821813	−	0.569757i	$-0.807037\pi$
$751$	−33827.0	−1.64363	−0.821813	+	0.569757i	$0.807037\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	− 8870.39i	− 0.426737i
$757$	3928.29	0.188608	0.0943039	−	0.995543i	$-0.469937\pi$
$757$	3928.29	0.188608	0.0943039	+	0.995543i	$0.469937\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	0	0
$763$	−15806.3	−0.749969
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	21283.4	1.00000
$769$	− 26724.8i	− 1.25321i	−0.779335	−	0.626607i	$-0.784443\pi$
$769$	− 26724.8i	− 1.25321i	0.779335	−	0.626607i	$-0.215557\pi$
$770$	0	0
$771$	0	0
$772$	− 36134.4i	− 1.68459i
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	0	0
$775$	19485.6	0.903151
$776$	0	0
$777$	17924.6i	0.827596i
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	17954.5	0.817896
$785$	0	0
$786$	0	0
$787$	− 36442.3i	− 1.65061i	−0.564690	−	0.825303i	$-0.691004\pi$
$787$	− 36442.3i	− 1.65061i	0.564690	−	0.825303i	$-0.308996\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−49851.6	−2.23238
$794$	0	0
$795$	0	0
$796$	16212.0	0.721883
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	27216.0	1.19382
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	0	0		−	1.00000i	$-0.5\pi$
$809$	0	0			1.00000i	$0.5\pi$
$810$	0	0
$811$	44922.6i	1.94506i	0.232775	+	0.972531i	$0.425220\pi$
$811$	44922.6i	1.94506i	−0.232775	+	0.972531i	$0.574780\pi$
$812$	0	0
$813$	− 35630.6i	− 1.53705i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	36869.3	1.57882
$818$	0	0
$819$	19970.7	0.852053
$820$	0	0
$821$	0	0		−	1.00000i	$-0.5\pi$
$821$	0	0			1.00000i	$0.5\pi$
$822$	0	0
$823$	12220.0	0.517573	0.258786	−	0.965935i	$-0.416677\pi$
$823$	12220.0	0.517573	0.258786	+	0.965935i	$0.416677\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0		−	1.00000i	$-0.5\pi$
$827$	0	0			1.00000i	$0.5\pi$
$828$	0	0
$829$	−17066.0	−0.714990	−0.357495	−	0.933915i	$-0.616369\pi$
$829$	−17066.0	−0.714990	−0.357495	+	0.933915i	$0.616369\pi$
$830$	0	0
$831$	− 45277.8i	− 1.89010i
$832$	47917.2i	1.99667i
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	21870.0	0.903151
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	−24389.0	−1.00000
$842$	0	0
$843$	0	0
$844$	40294.9i	1.64337i
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	7718.46i	0.312010i
$850$	0	0
$851$	0	0
$852$	0	0
$853$	20713.5i	0.831437i	0.909493	+	0.415719i	$0.136470\pi$
$853$	20713.5i	0.831437i	−0.909493	+	0.415719i	$0.863530\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	−31304.0	−1.24340	−0.621699	−	0.783256i	$-0.713557\pi$
$859$	−31304.0	−1.24340	−0.621699	+	0.783256i	$0.713557\pi$
$860$	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
$867$	−25528.7	−1.00000
$868$	− 9855.99i	− 0.385408i
$869$	0	0
$870$	0	0
$871$	61273.7i	2.38367i
$872$	0	0
$873$	35910.0	1.39218
$874$	0	0
$875$	0	0
$876$	− 45975.6i	− 1.77326i
$877$	− 25893.7i	− 0.996999i	−0.866890	−	0.498499i	$-0.833885\pi$
$877$	− 25893.7i	− 0.996999i	0.866890	−	0.498499i	$-0.166115\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	0	0
$883$	−48230.7	−1.83816	−0.919078	−	0.394076i	$-0.871065\pi$
$883$	−48230.7	−1.83816	−0.919078	+	0.394076i	$0.871065\pi$
$884$	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
$889$	−17978.6	−0.678272
$890$	0	0
$891$	0	0
$892$	−25760.0	−0.966938
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	−27000.0	−1.00000
$901$	0	0
$902$	0	0
$903$	21437.3	0.790021
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−44840.0	−1.64155	−0.820776	−	0.571250i	$-0.806459\pi$
$907$	−44840.0	−1.64155	−0.820776	+	0.571250i	$0.806459\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	23488.0i	0.852811i
$913$	0	0
$914$	0	0
$915$	0	0
$916$	−42400.6	−1.52943
$917$	0	0
$918$	0	0
$919$	41299.8i	1.48243i	0.671266	+	0.741217i	$0.265751\pi$
$919$	41299.8i	1.48243i	−0.671266	+	0.741217i	$0.734249\pi$
$920$	0	0
$921$	− 44936.0i	− 1.60770i
$922$	0	0
$923$	0	0
$924$	0	0
$925$	54559.6	1.93936
$926$	0	0
$927$	−49140.0	−1.74107
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	19814.2i	0.697511i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 49480.6i	− 1.72514i	−0.505935	−	0.862572i	$-0.668852\pi$
$937$	− 49480.6i	− 1.72514i	0.505935	−	0.862572i	$-0.331148\pi$
$938$	0	0
$939$	24624.0	0.855776
$940$	0	0
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	58058.6i	1.98909i
$949$	103509.	3.54061
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−5491.00	−0.184317
$962$	0	0
$963$	0	0
$964$	− 34681.7i	− 1.15874i
$965$	0	0
$966$	0	0
$967$	− 11758.8i	− 0.391044i	−0.980699	−	0.195522i	$-0.937360\pi$
$967$	− 11758.8i	− 0.391044i	0.980699	−	0.195522i	$-0.0626400\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	−30304.0	−1.00000
$973$	−3070.86	−0.101179
$974$	0	0
$975$	− 60787.4i	− 1.99667i
$976$	− 34090.8i	− 1.11805i
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	53999.2i	1.75745i
$982$	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
$988$	−52880.4	−1.70278
$989$	0	0
$990$	0	0
$991$	42556.5	1.36413	0.682064	−	0.731292i	$-0.261082\pi$
$991$	42556.5	1.36413	0.682064	+	0.731292i	$0.261082\pi$
$992$	0	0
$993$	5154.58	0.164729
$994$	0	0
$995$	0	0
$996$	0	0
$997$	59991.9i	1.90568i	0.303473	+	0.952840i	$0.401854\pi$
$997$	59991.9i	1.90568i	−0.303473	+	0.952840i	$0.598146\pi$
$998$	0	0
$999$	61236.0	1.93936

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	363.4.d.a.362.4	yes	4
3.2	odd	2	CM	363.4.d.a.362.4	yes	4
11.10	odd	2	inner	363.4.d.a.362.3	✓	4
33.32	even	2	inner	363.4.d.a.362.3	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
363.4.d.a.362.3	✓	4	11.10	odd	2	inner
363.4.d.a.362.3	✓	4	33.32	even	2	inner
363.4.d.a.362.4	yes	4	1.1	even	1	trivial
363.4.d.a.362.4	yes	4	3.2	odd	2	CM

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:	\( N \)	\(=\)	\( 363 = 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	363.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+5.19615 q^{3} -8.00000 q^{4} +7.90327i q^{7} +27.0000 q^{9} +O(q^{10})\)	\(q+5.19615 q^{3} -8.00000 q^{4} +7.90327i q^{7} +27.0000 q^{9} -41.5692 q^{12} -93.5883i q^{13} +64.0000 q^{16} +70.6291i q^{19} +41.0666i q^{21} +125.000 q^{25} +140.296 q^{27} -63.2262i q^{28} +155.885 q^{31} -216.000 q^{36} +436.477 q^{37} -486.299i q^{39} -522.013i q^{43} +332.554 q^{48} +280.538 q^{49} +748.706i q^{52} +366.999i q^{57} -532.669i q^{61} +213.388i q^{63} -512.000 q^{64} -654.715 q^{67} +1106.00i q^{73} +649.519 q^{75} -565.032i q^{76} -1396.67i q^{79} +729.000 q^{81} -328.533i q^{84} +739.654 q^{91} +810.000 q^{93} +1330.00 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 32 q^{4} + 108 q^{9}+O(q^{10}) \) 4 * q - 32 * q^4 + 108 * q^9	\( 4 q - 32 q^{4} + 108 q^{9} + 256 q^{16} + 500 q^{25} - 864 q^{36} - 1372 q^{49} - 2048 q^{64} + 2916 q^{81} + 1088 q^{91} + 3240 q^{93} + 5320 q^{97}+O(q^{100}) \) 4 * q - 32 * q^4 + 108 * q^9 + 256 * q^16 + 500 * q^25 - 864 * q^36 - 1372 * q^49 - 2048 * q^64 + 2916 * q^81 + 1088 * q^91 + 3240 * q^93 + 5320 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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