LMFDB - Embedded newform 363.4.d.a.362.1

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			362.1
Root			$-1.93185i$ of defining polynomial
Character	$\chi$	$=$	363.362
Dual form			363.4.d.a.362.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.19615 q^{3} -8.00000 q^{4} -36.1875i q^{7} +27.0000 q^{9} +O(q^{10})$	$q-5.19615 q^{3} -8.00000 q^{4} -36.1875i q^{7} +27.0000 q^{9} +41.5692 q^{12} -5.40666i q^{13} +64.0000 q^{16} -149.825i q^{19} +188.036i q^{21} +125.000 q^{25} -140.296 q^{27} +289.500i q^{28} -155.885 q^{31} -216.000 q^{36} -436.477 q^{37} +28.0938i q^{39} -213.378i q^{43} -332.554 q^{48} -966.538 q^{49} +43.2533i q^{52} +778.514i q^{57} +790.056i q^{61} -977.064i q^{63} -512.000 q^{64} +654.715 q^{67} +576.912i q^{73} -649.519 q^{75} +1198.60i q^{76} +146.507i q^{79} +729.000 q^{81} -1504.29i q^{84} -195.654 q^{91} +810.000 q^{93} +1330.00 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	− 36.1875i	− 1.95394i	−0.213368	−	0.976972i	$-0.568443\pi$
$7$	− 36.1875i	− 1.95394i	0.213368	−	0.976972i	$-0.431557\pi$
$8$	0	0
$9$	27.0000	1.00000
$10$	0	0
$11$	0	0
$11$	0	0
$12$	41.5692	1.00000
$13$	− 5.40666i	− 0.115349i	−0.998335	−	0.0576745i	$-0.981631\pi$
$13$	− 5.40666i	− 0.115349i	0.998335	−	0.0576745i	$-0.0183686\pi$
$14$	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
$19$	− 149.825i	− 1.80906i	−0.426406	−	0.904532i	$-0.640220\pi$
$19$	− 149.825i	− 1.80906i	0.426406	−	0.904532i	$-0.359780\pi$
$20$	0	0
$21$	188.036i	1.95394i
$22$	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
$28$	289.500i	1.95394i
$29$	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	$-0.649138\pi$
$31$	−155.885	−0.903151	−0.451576	+	0.892233i	$0.649138\pi$
$32$	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	$-0.921417\pi$
$37$	−436.477	−1.93936	−0.969680	+	0.244377i	$0.921417\pi$
$38$	0	0
$39$	28.0938i	0.115349i
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	− 213.378i	− 0.756739i	−0.925655	−	0.378370i	$-0.876485\pi$
$43$	− 213.378i	− 0.756739i	0.925655	−	0.378370i	$-0.123515\pi$
$44$	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
$49$	−966.538	−2.81790
$50$	0	0
$51$	0	0
$52$	43.2533i	0.115349i
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	778.514i	1.80906i
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	790.056i	1.65830i	0.559027	+	0.829150i	$0.311175\pi$
$61$	790.056i	1.65830i	−0.559027	+	0.829150i	$0.688825\pi$
$62$	0	0
$63$	− 977.064i	− 1.95394i
$64$	−512.000	−1.00000
$65$	0	0
$66$	0	0
$67$	654.715	1.19382	0.596912	−	0.802307i	$-0.296394\pi$
$67$	654.715	1.19382	0.596912	+	0.802307i	$0.296394\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0	0
$73$	576.912i	0.924965i	0.886628	+	0.462483i	$0.153041\pi$
$73$	576.912i	0.924965i	−0.886628	+	0.462483i	$0.846959\pi$
$74$	0	0
$75$	−649.519	−1.00000
$76$	1198.60i	1.80906i
$77$	0	0
$78$	0	0
$79$	146.507i	0.208650i	0.994543	+	0.104325i	$0.0332681\pi$
$79$	146.507i	0.208650i	−0.994543	+	0.104325i	$0.966732\pi$
$80$	0	0
$81$	729.000	1.00000
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	− 1504.29i	− 1.95394i
$85$	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
$91$	−195.654	−0.225386
$92$	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
$109$	− 1086.39i	− 0.954652i	−0.878726	−	0.477326i	$-0.841606\pi$
$109$	− 1086.39i	− 0.954652i	0.878726	−	0.477326i	$-0.158394\pi$
$110$	0	0
$111$	2268.00	1.93936
$112$	− 2316.00i	− 1.95394i
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 145.980i	− 0.115349i
$118$	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
$127$	− 1737.43i	− 1.21395i	−0.794720	−	0.606977i	$-0.792382\pi$
$127$	− 1737.43i	− 1.21395i	0.794720	−	0.606977i	$-0.207618\pi$
$128$	0	0
$129$	1108.74i	0.756739i
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	−5421.80	−3.53481
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	3254.46i	1.98590i	0.118550	+	0.992948i	$0.462175\pi$
$139$	3254.46i	1.98590i	−0.118550	+	0.992948i	$0.537825\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	1728.00	1.00000
$145$	0	0
$146$	0	0
$147$	5022.28	2.81790
$148$	3491.81	1.93936
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	3550.79i	1.91364i	0.290686	+	0.956819i	$0.406117\pi$
$151$	3550.79i	1.91364i	−0.290686	+	0.956819i	$0.593883\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	− 224.751i	− 0.115349i
$157$	−810.600	−0.412057	−0.206028	−	0.978546i	$-0.566054\pi$
$157$	−810.600	−0.412057	−0.206028	+	0.978546i	$0.566054\pi$
$158$	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
$169$	2167.77	0.986695
$170$	0	0
$171$	− 4045.28i	− 1.80906i
$172$	1707.02i	0.756739i
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	− 4523.44i	− 1.95394i
$176$	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
$183$	− 4105.25i	− 1.65830i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	5076.97i	1.95394i
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	2660.43	1.00000
$193$	− 2890.46i	− 1.07803i	−0.842297	−	0.539014i	$-0.818797\pi$
$193$	− 2890.46i	− 1.07803i	0.842297	−	0.539014i	$-0.181203\pi$
$194$	0	0
$195$	0	0
$196$	7732.31	2.81790
$197$	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	$-0.382455\pi$
$199$	2026.50	0.721883	0.360942	+	0.932588i	$0.382455\pi$
$200$	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
$208$	− 346.026i	− 0.115349i
$209$	0	0
$210$	0	0
$211$	− 3493.68i	− 1.13988i	−0.821686	−	0.569940i	$-0.806966\pi$
$211$	− 3493.68i	− 1.13988i	0.821686	−	0.569940i	$-0.193034\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	5641.08i	1.76471i
$218$	0	0
$219$	− 2997.72i	− 0.924965i
$220$	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	$0.5\pi$
$228$	− 6228.11i	− 1.80906i
$229$	−5300.08	−1.52943	−0.764714	−	0.644370i	$-0.777120\pi$
$229$	−5300.08	−1.52943	−0.764714	+	0.644370i	$0.777120\pi$
$230$	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
$237$	− 761.272i	− 0.208650i
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	6098.85i	1.63013i	0.579369	+	0.815065i	$0.303299\pi$
$241$	6098.85i	1.63013i	−0.579369	+	0.815065i	$0.696701\pi$
$242$	0	0
$243$	−3788.00	−1.00000
$244$	− 6320.45i	− 1.65830i
$245$	0	0
$246$	0	0
$247$	−810.053	−0.208674
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	7816.51i	1.95394i
$253$	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
$259$	15795.0i	3.78940i
$260$	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
$271$	5708.77i	1.27964i	0.768524	+	0.639821i	$0.220992\pi$
$271$	5708.77i	1.27964i	−0.768524	+	0.639821i	$0.779008\pi$
$272$	0	0
$273$	1016.65	0.225386
$274$	0	0
$275$	0	0
$276$	0	0
$277$	3014.44i	0.653863i	0.945048	+	0.326931i	$0.106015\pi$
$277$	3014.44i	0.653863i	−0.945048	+	0.326931i	$0.893985\pi$
$278$	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
$283$	− 9405.01i	− 1.97551i	−0.156005	−	0.987756i	$-0.549862\pi$
$283$	− 9405.01i	− 1.97551i	0.156005	−	0.987756i	$-0.450138\pi$
$284$	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
$292$	− 4615.30i	− 0.924965i
$293$	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
$301$	−7721.61	−1.47863
$302$	0	0
$303$	0	0
$304$	− 9588.80i	− 1.80906i
$305$	0	0
$306$	0	0
$307$	− 6399.30i	− 1.18967i	−0.803849	−	0.594833i	$-0.797218\pi$
$307$	− 6399.30i	− 1.18967i	0.803849	−	0.594833i	$-0.202782\pi$
$308$	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	$-0.640742\pi$
$313$	−4738.89	−0.855776	−0.427888	+	0.903832i	$0.640742\pi$
$314$	0	0
$315$	0	0
$316$	− 1172.05i	− 0.208650i
$317$	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
$325$	− 675.832i	− 0.115349i
$326$	0	0
$327$	5645.04i	0.954652i
$328$	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
$336$	12034.3i	1.95394i
$337$	− 4538.49i	− 0.733612i	−0.930297	−	0.366806i	$-0.880451\pi$
$337$	− 4538.49i	− 0.733612i	0.930297	−	0.366806i	$-0.119549\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	22564.3i	3.55207i
$344$	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
$349$	− 12172.2i	− 1.86694i	−0.358654	−	0.933470i	$-0.616764\pi$
$349$	− 12172.2i	− 1.86694i	0.358654	−	0.933470i	$-0.383236\pi$
$350$	0	0
$351$	758.533i	0.115349i
$352$	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
$361$	−15588.5	−2.27271
$362$	0	0
$363$	0	0
$364$	1565.23	0.225386
$365$	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
$373$	13979.6i	1.94058i	0.241952	+	0.970288i	$0.422212\pi$
$373$	13979.6i	1.94058i	−0.241952	+	0.970288i	$0.577788\pi$
$374$	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
$381$	9027.96i	1.21395i
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 5761.20i	− 0.756739i
$388$	−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
$399$	28172.5	3.53481
$400$	8000.00	1.00000
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	842.815i	0.104178i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	− 4310.09i	− 0.521076i	−0.965464	−	0.260538i	$-0.916100\pi$
$409$	− 4310.09i	− 0.521076i	0.965464	−	0.260538i	$-0.0838999\pi$
$410$	0	0
$411$	0	0
$412$	14560.0	1.74107
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 16910.7i	− 1.98590i
$418$	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	$-0.540317\pi$
$421$	−2182.38	−0.252643	−0.126322	+	0.991989i	$0.540317\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	28590.2	3.24022
$428$	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
$436$	8691.10i	0.954652i
$437$	0	0
$438$	0	0
$439$	2947.20i	0.320415i	0.987083	+	0.160207i	$0.0512163\pi$
$439$	2947.20i	0.320415i	−0.987083	+	0.160207i	$0.948784\pi$
$440$	0	0
$441$	−26096.5	−2.81790
$442$	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
$448$	18528.0i	1.95394i
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	− 18450.4i	− 1.91364i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 19480.9i	− 1.99405i	−0.0770909	−	0.997024i	$-0.524563\pi$
$457$	− 19480.9i	− 1.99405i	0.0770909	−	0.997024i	$-0.475437\pi$
$458$	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			$\pi$
$468$	1167.84i	0.115349i
$469$	− 23692.5i	− 2.33266i
$470$	0	0
$471$	4212.00	0.412057
$472$	0	0
$473$	0	0
$474$	0	0
$475$	− 18728.1i	− 1.80906i
$476$	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
$481$	2359.88i	0.223703i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	5019.48	0.467052	0.233526	−	0.972351i	$-0.424974\pi$
$487$	5019.48	0.467052	0.233526	+	0.972351i	$0.424974\pi$
$488$	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	$-0.237560\pi$
$499$	16367.9	1.46839	0.734195	+	0.678938i	$0.237560\pi$
$500$	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
$507$	−11264.1	−0.986695
$508$	13899.5i	1.21395i
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	20877.0	1.80733
$512$	0	0
$513$	21019.9i	1.80906i
$514$	0	0
$515$	0	0
$516$	− 8869.94i	− 0.756739i
$517$	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
$523$	− 6102.54i	− 0.510221i	−0.966912	−	0.255110i	$-0.917888\pi$
$523$	− 6102.54i	− 0.510221i	0.966912	−	0.255110i	$-0.0821118\pi$
$524$	0	0
$525$	23504.5i	1.95394i
$526$	0	0
$527$	0	0
$528$	0	0
$529$	12167.0	1.00000
$530$	0	0
$531$	0	0
$532$	43374.4	3.53481
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	8319.87i	0.661182i	0.943774	+	0.330591i	$0.107248\pi$
$541$	8319.87i	0.661182i	−0.943774	+	0.330591i	$0.892752\pi$
$542$	0	0
$543$	17968.3	1.42006
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 19214.8i	− 1.50195i	−0.660330	−	0.750976i	$-0.729584\pi$
$547$	− 19214.8i	− 1.50195i	0.660330	−	0.750976i	$-0.270416\pi$
$548$	0	0
$549$	21331.5i	1.65830i
$550$	0	0
$551$	0	0
$552$	0	0
$553$	5301.72	0.407689
$554$	0	0
$555$	0	0
$556$	− 26035.7i	− 1.98590i
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	−1153.66	−0.0872892
$560$	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
$567$	− 26380.7i	− 1.95394i
$568$	0	0
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	− 26514.7i	− 1.94327i	−0.236486	−	0.971635i	$-0.575996\pi$
$571$	− 26514.7i	− 1.94327i	0.236486	−	0.971635i	$-0.424004\pi$
$572$	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	$-0.220605\pi$
$577$	21325.0	1.53860	0.769300	+	0.638888i	$0.220605\pi$
$578$	0	0
$579$	15019.2i	1.07803i
$580$	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			$\pi$
$588$	−40178.2	−2.81790
$589$	23355.4i	1.63386i
$590$	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
$601$	− 22924.2i	− 1.55590i	−0.628325	−	0.777951i	$-0.716259\pi$
$601$	− 22924.2i	− 1.55590i	0.628325	−	0.777951i	$-0.283741\pi$
$602$	0	0
$603$	17677.3	1.19382
$604$	− 28406.3i	− 1.91364i
$605$	0	0
$606$	0	0
$607$	26687.6i	1.78454i	0.451505	+	0.892269i	$0.350887\pi$
$607$	26687.6i	1.78454i	−0.451505	+	0.892269i	$0.649113\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	29888.8i	1.96933i	0.174461	+	0.984664i	$0.444182\pi$
$613$	29888.8i	1.96933i	−0.174461	+	0.984664i	$0.555818\pi$
$614$	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
$624$	1798.00i	0.115349i
$625$	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	$-0.980991\pi$
$631$	−31644.6	−1.99643	−0.998217	+	0.0596825i	$0.980991\pi$
$632$	0	0
$633$	18153.7i	1.13988i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	5225.74i	0.325042i
$638$	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
$651$	− 29311.9i	− 1.76471i
$652$	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
$657$	15576.6i	0.924965i
$658$	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	$-0.794125\pi$
$661$	−27123.9	−1.59606	−0.798032	+	0.602615i	$0.794125\pi$
$662$	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
$673$	− 34906.8i	− 1.99934i	−0.0256299	−	0.999671i	$-0.508159\pi$
$673$	− 34906.8i	− 1.99934i	0.0256299	−	0.999671i	$-0.491841\pi$
$674$	0	0
$675$	−17537.0	−1.00000
$676$	−17342.1	−0.986695
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	− 48129.4i	− 2.72023i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	32362.2i	1.80906i
$685$	0	0
$686$	0	0
$687$	27540.0	1.52943
$688$	− 13656.2i	− 0.756739i
$689$	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
$700$	36187.5i	1.95394i
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	65395.1i	3.50843i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−10911.9	−0.578005	−0.289003	−	0.957328i	$-0.593324\pi$
$709$	−10911.9	−0.578005	−0.289003	+	0.957328i	$0.593324\pi$
$710$	0	0
$711$	3955.69i	0.208650i
$712$	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
$721$	65861.3i	3.40195i
$722$	0	0
$723$	− 31690.6i	− 1.63013i
$724$	27664.0	1.42006
$725$	0	0
$726$	0	0
$727$	37692.9	1.92290	0.961452	−	0.274971i	$-0.0886683\pi$
$727$	37692.9	1.92290	0.961452	+	0.274971i	$0.0886683\pi$
$728$	0	0
$729$	19683.0	1.00000
$730$	0	0
$731$	0	0
$732$	32842.0i	1.65830i
$733$	− 15327.5i	− 0.772354i	−0.922425	−	0.386177i	$-0.873795\pi$
$733$	− 15327.5i	− 0.772354i	0.922425	−	0.386177i	$-0.126205\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	− 4439.63i	− 0.220994i	−0.993876	−	0.110497i	$-0.964756\pi$
$739$	− 4439.63i	− 0.220994i	0.993876	−	0.110497i	$-0.0352442\pi$
$740$	0	0
$741$	4209.16	0.208674
$742$	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	$-0.192963\pi$
$751$	33827.0	1.64363	0.821813	+	0.569757i	$0.192963\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	− 40615.8i	− 1.95394i
$757$	−3928.29	−0.188608	−0.0943039	−	0.995543i	$-0.530063\pi$
$757$	−3928.29	−0.188608	−0.0943039	+	0.995543i	$0.530063\pi$
$758$	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
$763$	−39313.7	−1.86534
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	−21283.4	−1.00000
$769$	33238.7i	1.55867i	0.626607	+	0.779335i	$0.284443\pi$
$769$	33238.7i	1.55867i	−0.626607	+	0.779335i	$0.715557\pi$
$770$	0	0
$771$	0	0
$772$	23123.6i	1.07803i
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	0	0
$775$	−19485.6	−0.903151
$776$	0	0
$777$	− 82073.3i	− 3.78940i
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−61858.5	−2.81790
$785$	0	0
$786$	0	0
$787$	− 24934.6i	− 1.12938i	−0.825303	−	0.564690i	$-0.808996\pi$
$787$	− 24934.6i	− 1.12938i	0.825303	−	0.564690i	$-0.191004\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	4271.56	0.191283
$794$	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
$811$	10752.2i	0.465549i	0.972531	+	0.232775i	$0.0747804\pi$
$811$	10752.2i	0.465549i	−0.972531	+	0.232775i	$0.925220\pi$
$812$	0	0
$813$	− 29663.6i	− 1.27964i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−31969.3	−1.36899
$818$	0	0
$819$	−5282.65	−0.225386
$820$	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
$831$	− 15663.5i	− 0.653863i
$832$	2768.21i	0.115349i
$833$	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
$844$	27949.4i	1.13988i
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	48869.9i	1.97551i
$850$	0	0
$851$	0	0
$852$	0	0
$853$	45316.2i	1.81899i	0.415719	+	0.909493i	$0.363530\pi$
$853$	45316.2i	1.81899i	−0.415719	+	0.909493i	$0.636470\pi$
$854$	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
$868$	− 45128.6i	− 1.76471i
$869$	0	0
$870$	0	0
$871$	− 3539.82i	− 0.137706i
$872$	0	0
$873$	35910.0	1.39218
$874$	0	0
$875$	0	0
$876$	23981.8i	0.924965i
$877$	− 45029.1i	− 1.73378i	−0.498499	−	0.866890i	$-0.666115\pi$
$877$	− 45029.1i	− 1.73378i	0.498499	−	0.866890i	$-0.333885\pi$
$878$	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	$-0.128935\pi$
$883$	48230.7	1.83816	0.919078	+	0.394076i	$0.128935\pi$
$884$	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
$889$	−62873.4	−2.37200
$890$	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
$903$	40122.7	1.47863
$904$	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			$\pi$
$912$	49824.9i	1.80906i
$913$	0	0
$914$	0	0
$915$	0	0
$916$	42400.6	1.52943
$917$	0	0
$918$	0	0
$919$	− 37402.3i	− 1.34253i	−0.741217	−	0.671266i	$-0.765751\pi$
$919$	− 37402.3i	− 1.34253i	0.741217	−	0.671266i	$-0.234249\pi$
$920$	0	0
$921$	33251.7i	1.18967i
$922$	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
$931$	144812.i	5.09775i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 29022.4i	− 1.01187i	−0.862572	−	0.505935i	$-0.831148\pi$
$937$	− 29022.4i	− 1.01187i	0.862572	−	0.505935i	$-0.168852\pi$
$938$	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			$\pi$
$948$	6090.18i	0.208650i
$949$	3119.17	0.106694
$950$	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
$964$	− 48790.8i	− 1.63013i
$965$	0	0
$966$	0	0
$967$	− 58980.1i	− 1.96140i	−0.195522	−	0.980699i	$-0.562640\pi$
$967$	− 58980.1i	− 1.96140i	0.195522	−	0.980699i	$-0.437360\pi$
$968$	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
$973$	117771.	3.88033
$974$	0	0
$975$	3511.73i	0.115349i
$976$	50563.6i	1.65830i
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	− 29332.5i	− 0.954652i
$982$	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
$988$	6480.42	0.208674
$989$	0	0
$990$	0	0
$991$	−42556.5	−1.36413	−0.682064	−	0.731292i	$-0.738918\pi$
$991$	−42556.5	−1.36413	−0.682064	+	0.731292i	$0.738918\pi$
$992$	0	0
$993$	−5154.58	−0.164729
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 19107.0i	− 0.606946i	−0.952840	−	0.303473i	$-0.901854\pi$
$997$	− 19107.0i	− 0.606946i	0.952840	−	0.303473i	$-0.0981462\pi$
$998$	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.362.1	✓	4
3.2	odd	2	CM	363.4.d.a.362.1	✓	4
11.10	odd	2	inner	363.4.d.a.362.2	yes	4
33.32	even	2	inner	363.4.d.a.362.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
363.4.d.a.362.1	✓	4	1.1	even	1	trivial
363.4.d.a.362.1	✓	4	3.2	odd	2	CM
363.4.d.a.362.2	yes	4	11.10	odd	2	inner
363.4.d.a.362.2	yes	4	33.32	even	2	inner

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)

\(f(q)\)	\(=\)	\(q-5.19615 q^{3} -8.00000 q^{4} -36.1875i q^{7} +27.0000 q^{9} +O(q^{10})\)	\(q-5.19615 q^{3} -8.00000 q^{4} -36.1875i q^{7} +27.0000 q^{9} +41.5692 q^{12} -5.40666i q^{13} +64.0000 q^{16} -149.825i q^{19} +188.036i q^{21} +125.000 q^{25} -140.296 q^{27} +289.500i q^{28} -155.885 q^{31} -216.000 q^{36} -436.477 q^{37} +28.0938i q^{39} -213.378i q^{43} -332.554 q^{48} -966.538 q^{49} +43.2533i q^{52} +778.514i q^{57} +790.056i q^{61} -977.064i q^{63} -512.000 q^{64} +654.715 q^{67} +576.912i q^{73} -649.519 q^{75} +1198.60i q^{76} +146.507i q^{79} +729.000 q^{81} -1504.29i q^{84} -195.654 q^{91} +810.000 q^{93} +1330.00 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 32 q^{4} + 108 q^{9}+O(q^{10}) \) 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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