LMFDB - Embedded newform 2400.4.a.bz.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2400,4,Mod(1,2400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2400, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

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

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

chi := DirichletCharacter("2400.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2400 = 2^{5} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2400.a (trivial)

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:	yes
Analytic conductor:	$141.604584014$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{21}, \sqrt{141})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 61x^{2} + 154x + 28 $ x^4 - x^3 - 61x^2 + 154x + 28
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 480)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-8.39882$ of defining polynomial
Character	$\chi$	$=$	2400.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} +33.1384 q^{7} +9.00000 q^{9} +63.9543 q^{11} -89.7781 q^{13} -71.4478 q^{17} -44.1541 q^{19} -99.4151 q^{21} +100.000 q^{23} -27.0000 q^{27} +33.4151 q^{29} +120.415 q^{31} -191.863 q^{33} -13.5172 q^{37} +269.334 q^{39} +130.616 q^{41} +410.767 q^{43} +216.490 q^{47} +755.151 q^{49} +214.343 q^{51} +279.768 q^{53} +132.462 q^{57} -133.529 q^{59} -297.321 q^{61} +298.245 q^{63} -501.044 q^{67} -300.000 q^{69} +289.538 q^{71} -756.211 q^{73} +2119.34 q^{77} -120.415 q^{79} +81.0000 q^{81} +65.6603 q^{83} -100.245 q^{87} +513.321 q^{89} -2975.10 q^{91} -361.245 q^{93} +828.207 q^{97} +575.589 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{3} + 60 q^{7} + 36 q^{9} - 180 q^{21} + 400 q^{23} - 108 q^{27} - 84 q^{29} + 1248 q^{41} + 192 q^{43} - 440 q^{47} + 844 q^{49} + 552 q^{61} + 540 q^{63} - 408 q^{67} - 1200 q^{69} + 324 q^{81}+ \cdots + 312 q^{89}+O(q^{100}) $ 4 * q - 12 * q^3 + 60 * q^7 + 36 * q^9 - 180 * q^21 + 400 * q^23 - 108 * q^27 - 84 * q^29 + 1248 * q^41 + 192 * q^43 - 440 * q^47 + 844 * q^49 + 552 * q^61 + 540 * q^63 - 408 * q^67 - 1200 * q^69 + 324 * q^81 - 608 * q^83 + 252 * q^87 + 312 * q^89

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	33.1384	1.78930	0.894652	−	0.446765i	$-0.147424\pi$
$7$	33.1384	1.78930	0.894652	+	0.446765i	$0.147424\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	63.9543	1.75299	0.876497	−	0.481407i	$-0.159874\pi$
$11$	63.9543	1.75299	0.876497	+	0.481407i	$0.159874\pi$
$12$	0	0
$13$	−89.7781	−1.91538	−0.957691	−	0.287798i	$-0.907077\pi$
$13$	−89.7781	−1.91538	−0.957691	+	0.287798i	$0.907077\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−71.4478	−1.01933	−0.509666	−	0.860372i	$-0.670231\pi$
$17$	−71.4478	−1.01933	−0.509666	+	0.860372i	$0.670231\pi$
$18$	0	0
$19$	−44.1541	−0.533140	−0.266570	−	0.963816i	$-0.585890\pi$
$19$	−44.1541	−0.533140	−0.266570	+	0.963816i	$0.585890\pi$
$20$	0	0
$21$	−99.4151	−1.03305
$22$	0	0
$23$	100.000	0.906584	0.453292	−	0.891362i	$-0.350249\pi$
$23$	100.000	0.906584	0.453292	+	0.891362i	$0.350249\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	33.4151	0.213966	0.106983	−	0.994261i	$-0.465881\pi$
$29$	33.4151	0.213966	0.106983	+	0.994261i	$0.465881\pi$
$30$	0	0
$31$	120.415	0.697651	0.348825	−	0.937188i	$-0.386581\pi$
$31$	120.415	0.697651	0.348825	+	0.937188i	$0.386581\pi$
$32$	0	0
$33$	−191.863	−1.01209
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−13.5172	−0.0600600	−0.0300300	−	0.999549i	$-0.509560\pi$
$37$	−13.5172	−0.0600600	−0.0300300	+	0.999549i	$0.509560\pi$
$38$	0	0
$39$	269.334	1.10585
$40$	0	0
$41$	130.616	0.497533	0.248767	−	0.968563i	$-0.419975\pi$
$41$	130.616	0.497533	0.248767	+	0.968563i	$0.419975\pi$
$42$	0	0
$43$	410.767	1.45678	0.728388	−	0.685164i	$-0.240270\pi$
$43$	410.767	1.45678	0.728388	+	0.685164i	$0.240270\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	216.490	0.671880	0.335940	−	0.941883i	$-0.390946\pi$
$47$	216.490	0.671880	0.335940	+	0.941883i	$0.390946\pi$
$48$	0	0
$49$	755.151	2.20161
$50$	0	0
$51$	214.343	0.588512
$52$	0	0
$53$	279.768	0.725076	0.362538	−	0.931969i	$-0.381910\pi$
$53$	279.768	0.725076	0.362538	+	0.931969i	$0.381910\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	132.462	0.307808
$58$	0	0
$59$	−133.529	−0.294643	−0.147322	−	0.989089i	$-0.547065\pi$
$59$	−133.529	−0.294643	−0.147322	+	0.989089i	$0.547065\pi$
$60$	0	0
$61$	−297.321	−0.624066	−0.312033	−	0.950071i	$-0.601010\pi$
$61$	−297.321	−0.624066	−0.312033	+	0.950071i	$0.601010\pi$
$62$	0	0
$63$	298.245	0.596434
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−501.044	−0.913616	−0.456808	−	0.889565i	$-0.651007\pi$
$67$	−501.044	−0.913616	−0.456808	+	0.889565i	$0.651007\pi$
$68$	0	0
$69$	−300.000	−0.523417
$70$	0	0
$71$	289.538	0.483970	0.241985	−	0.970280i	$-0.422202\pi$
$71$	289.538	0.483970	0.241985	+	0.970280i	$0.422202\pi$
$72$	0	0
$73$	−756.211	−1.21244	−0.606218	−	0.795299i	$-0.707314\pi$
$73$	−756.211	−1.21244	−0.606218	+	0.795299i	$0.707314\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2119.34	3.13664
$78$	0	0
$79$	−120.415	−0.171491	−0.0857453	−	0.996317i	$-0.527327\pi$
$79$	−120.415	−0.171491	−0.0857453	+	0.996317i	$0.527327\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	65.6603	0.0868332	0.0434166	−	0.999057i	$-0.486176\pi$
$83$	65.6603	0.0868332	0.0434166	+	0.999057i	$0.486176\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−100.245	−0.123534
$88$	0	0
$89$	513.321	0.611369	0.305685	−	0.952133i	$-0.401115\pi$
$89$	513.321	0.611369	0.305685	+	0.952133i	$0.401115\pi$
$90$	0	0
$91$	−2975.10	−3.42720
$92$	0	0
$93$	−361.245	−0.402789
$94$	0	0
$95$	0	0
$96$	0	0
$97$	828.207	0.866924	0.433462	−	0.901172i	$-0.357292\pi$
$97$	828.207	0.866924	0.433462	+	0.901172i	$0.357292\pi$
$98$	0	0
$99$	575.589	0.584332
$100$	0	0
$101$	−1469.72	−1.44794	−0.723972	−	0.689830i	$-0.757685\pi$
$101$	−1469.72	−1.44794	−0.723972	+	0.689830i	$0.757685\pi$
$102$	0	0
$103$	−100.522	−0.0961623	−0.0480812	−	0.998843i	$-0.515311\pi$
$103$	−100.522	−0.0961623	−0.0480812	+	0.998843i	$0.515311\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−710.340	−0.641786	−0.320893	−	0.947115i	$-0.603983\pi$
$107$	−710.340	−0.641786	−0.320893	+	0.947115i	$0.603983\pi$
$108$	0	0
$109$	1363.47	1.19814	0.599068	−	0.800698i	$-0.295538\pi$
$109$	1363.47	1.19814	0.599068	+	0.800698i	$0.295538\pi$
$110$	0	0
$111$	40.5517	0.0346757
$112$	0	0
$113$	727.303	0.605477	0.302739	−	0.953074i	$-0.402099\pi$
$113$	727.303	0.605477	0.302739	+	0.953074i	$0.402099\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−808.003	−0.638461
$118$	0	0
$119$	−2367.66	−1.82389
$120$	0	0
$121$	2759.15	2.07299
$122$	0	0
$123$	−391.849	−0.287251
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−369.327	−0.258051	−0.129026	−	0.991641i	$-0.541185\pi$
$127$	−369.327	−0.258051	−0.129026	+	0.991641i	$0.541185\pi$
$128$	0	0
$129$	−1232.30	−0.841070
$130$	0	0
$131$	2000.80	1.33443	0.667216	−	0.744864i	$-0.267486\pi$
$131$	2000.80	1.33443	0.667216	+	0.744864i	$0.267486\pi$
$132$	0	0
$133$	−1463.20	−0.953949
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2436.46	−1.51942	−0.759711	−	0.650261i	$-0.774660\pi$
$137$	−2436.46	−1.51942	−0.759711	+	0.650261i	$0.774660\pi$
$138$	0	0
$139$	1020.33	0.622613	0.311306	−	0.950310i	$-0.399233\pi$
$139$	1020.33	0.622613	0.311306	+	0.950310i	$0.399233\pi$
$140$	0	0
$141$	−649.471	−0.387910
$142$	0	0
$143$	−5741.70	−3.35766
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−2265.45	−1.27110
$148$	0	0
$149$	−2833.10	−1.55770	−0.778848	−	0.627213i	$-0.784195\pi$
$149$	−2833.10	−1.55770	−0.778848	+	0.627213i	$0.784195\pi$
$150$	0	0
$151$	1104.60	0.595306	0.297653	−	0.954674i	$-0.403796\pi$
$151$	1104.60	0.595306	0.297653	+	0.954674i	$0.403796\pi$
$152$	0	0
$153$	−643.030	−0.339777
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−3213.65	−1.63361	−0.816807	−	0.576911i	$-0.804258\pi$
$157$	−3213.65	−1.63361	−0.816807	+	0.576911i	$0.804258\pi$
$158$	0	0
$159$	−839.303	−0.418623
$160$	0	0
$161$	3313.84	1.62215
$162$	0	0
$163$	−1878.28	−0.902564	−0.451282	−	0.892381i	$-0.649033\pi$
$163$	−1878.28	−0.902564	−0.451282	+	0.892381i	$0.649033\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1220.60	0.565587	0.282794	−	0.959181i	$-0.408739\pi$
$167$	1220.60	0.565587	0.282794	+	0.959181i	$0.408739\pi$
$168$	0	0
$169$	5863.11	2.66869
$170$	0	0
$171$	−397.387	−0.177713
$172$	0	0
$173$	2090.84	0.918863	0.459431	−	0.888213i	$-0.348053\pi$
$173$	2090.84	0.918863	0.459431	+	0.888213i	$0.348053\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	400.586	0.170112
$178$	0	0
$179$	2447.70	1.02207	0.511033	−	0.859561i	$-0.329263\pi$
$179$	2447.70	1.02207	0.511033	+	0.859561i	$0.329263\pi$
$180$	0	0
$181$	4324.07	1.77572	0.887862	−	0.460111i	$-0.152190\pi$
$181$	4324.07	1.77572	0.887862	+	0.460111i	$0.152190\pi$
$182$	0	0
$183$	891.962	0.360304
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−4569.39	−1.78688
$188$	0	0
$189$	−894.736	−0.344352
$190$	0	0
$191$	3395.20	1.28622	0.643110	−	0.765774i	$-0.277644\pi$
$191$	3395.20	1.28622	0.643110	+	0.765774i	$0.277644\pi$
$192$	0	0
$193$	3824.17	1.42627	0.713135	−	0.701027i	$-0.247275\pi$
$193$	3824.17	1.42627	0.713135	+	0.701027i	$0.247275\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−2219.26	−0.802619	−0.401309	−	0.915943i	$-0.631445\pi$
$197$	−2219.26	−0.802619	−0.401309	+	0.915943i	$0.631445\pi$
$198$	0	0
$199$	1939.50	0.690890	0.345445	−	0.938439i	$-0.387728\pi$
$199$	1939.50	0.690890	0.345445	+	0.938439i	$0.387728\pi$
$200$	0	0
$201$	1503.13	0.527476
$202$	0	0
$203$	1107.32	0.382851
$204$	0	0
$205$	0	0
$206$	0	0
$207$	900.000	0.302195
$208$	0	0
$209$	−2823.85	−0.934591
$210$	0	0
$211$	2128.22	0.694372	0.347186	−	0.937796i	$-0.387137\pi$
$211$	2128.22	0.694372	0.347186	+	0.937796i	$0.387137\pi$
$212$	0	0
$213$	−868.614	−0.279420
$214$	0	0
$215$	0	0
$216$	0	0
$217$	3990.36	1.24831
$218$	0	0
$219$	2268.63	0.700000
$220$	0	0
$221$	6414.45	1.95241
$222$	0	0
$223$	4872.63	1.46321	0.731605	−	0.681729i	$-0.238772\pi$
$223$	4872.63	1.46321	0.731605	+	0.681729i	$0.238772\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2440.90	0.713694	0.356847	−	0.934163i	$-0.383852\pi$
$227$	2440.90	0.713694	0.356847	+	0.934163i	$0.383852\pi$
$228$	0	0
$229$	−2029.06	−0.585518	−0.292759	−	0.956186i	$-0.594573\pi$
$229$	−2029.06	−0.585518	−0.292759	+	0.956186i	$0.594573\pi$
$230$	0	0
$231$	−6358.02	−1.81094
$232$	0	0
$233$	−1626.41	−0.457295	−0.228647	−	0.973509i	$-0.573430\pi$
$233$	−1626.41	−0.457295	−0.228647	+	0.973509i	$0.573430\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	361.245	0.0990101
$238$	0	0
$239$	−3808.38	−1.03073	−0.515363	−	0.856972i	$-0.672343\pi$
$239$	−3808.38	−1.03073	−0.515363	+	0.856972i	$0.672343\pi$
$240$	0	0
$241$	1607.74	0.429724	0.214862	−	0.976644i	$-0.431070\pi$
$241$	1607.74	0.429724	0.214862	+	0.976644i	$0.431070\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	3964.08	1.02117
$248$	0	0
$249$	−196.981	−0.0501332
$250$	0	0
$251$	−1478.50	−0.371801	−0.185901	−	0.982569i	$-0.559520\pi$
$251$	−1478.50	−0.371801	−0.185901	+	0.982569i	$0.559520\pi$
$252$	0	0
$253$	6395.43	1.58924
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−427.043	−0.103651	−0.0518253	−	0.998656i	$-0.516504\pi$
$257$	−427.043	−0.103651	−0.0518253	+	0.998656i	$0.516504\pi$
$258$	0	0
$259$	−447.939	−0.107466
$260$	0	0
$261$	300.736	0.0713221
$262$	0	0
$263$	−2779.62	−0.651707	−0.325853	−	0.945420i	$-0.605652\pi$
$263$	−2779.62	−0.651707	−0.325853	+	0.945420i	$0.605652\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−1539.96	−0.352974
$268$	0	0
$269$	6832.04	1.54854	0.774270	−	0.632856i	$-0.218118\pi$
$269$	6832.04	1.54854	0.774270	+	0.632856i	$0.218118\pi$
$270$	0	0
$271$	6321.13	1.41691	0.708453	−	0.705758i	$-0.249394\pi$
$271$	6321.13	1.41691	0.708453	+	0.705758i	$0.249394\pi$
$272$	0	0
$273$	8925.30	1.97869
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−2445.91	−0.530544	−0.265272	−	0.964174i	$-0.585462\pi$
$277$	−2445.91	−0.530544	−0.265272	+	0.964174i	$0.585462\pi$
$278$	0	0
$279$	1083.74	0.232550
$280$	0	0
$281$	2387.91	0.506943	0.253471	−	0.967343i	$-0.418428\pi$
$281$	2387.91	0.506943	0.253471	+	0.967343i	$0.418428\pi$
$282$	0	0
$283$	7117.27	1.49497	0.747487	−	0.664276i	$-0.231260\pi$
$283$	7117.27	1.49497	0.747487	+	0.664276i	$0.231260\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4328.41	0.890238
$288$	0	0
$289$	191.792	0.0390376
$290$	0	0
$291$	−2484.62	−0.500519
$292$	0	0
$293$	−6952.40	−1.38622	−0.693112	−	0.720830i	$-0.743761\pi$
$293$	−6952.40	−1.38622	−0.693112	+	0.720830i	$0.743761\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−1726.77	−0.337364
$298$	0	0
$299$	−8977.81	−1.73646
$300$	0	0
$301$	13612.1	2.60662
$302$	0	0
$303$	4409.15	0.835970
$304$	0	0
$305$	0	0
$306$	0	0
$307$	2817.76	0.523837	0.261919	−	0.965090i	$-0.415645\pi$
$307$	2817.76	0.523837	0.261919	+	0.965090i	$0.415645\pi$
$308$	0	0
$309$	301.566	0.0555193
$310$	0	0
$311$	−7775.68	−1.41774	−0.708872	−	0.705337i	$-0.750795\pi$
$311$	−7775.68	−1.41774	−0.708872	+	0.705337i	$0.750795\pi$
$312$	0	0
$313$	−8222.87	−1.48493	−0.742466	−	0.669883i	$-0.766344\pi$
$313$	−8222.87	−1.48493	−0.742466	+	0.669883i	$0.766344\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	4264.76	0.755624	0.377812	−	0.925882i	$-0.376676\pi$
$317$	4264.76	0.755624	0.377812	+	0.925882i	$0.376676\pi$
$318$	0	0
$319$	2137.04	0.375082
$320$	0	0
$321$	2131.02	0.370535
$322$	0	0
$323$	3154.72	0.543446
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−4090.41	−0.691744
$328$	0	0
$329$	7174.14	1.20220
$330$	0	0
$331$	−7766.86	−1.28974	−0.644872	−	0.764291i	$-0.723089\pi$
$331$	−7766.86	−1.28974	−0.644872	+	0.764291i	$0.723089\pi$
$332$	0	0
$333$	−121.655	−0.0200200
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−2364.55	−0.382212	−0.191106	−	0.981569i	$-0.561207\pi$
$337$	−2364.55	−0.382212	−0.191106	+	0.981569i	$0.561207\pi$
$338$	0	0
$339$	−2181.91	−0.349572
$340$	0	0
$341$	7701.06	1.22298
$342$	0	0
$343$	13658.0	2.15004
$344$	0	0
$345$	0	0
$346$	0	0
$347$	4929.89	0.762680	0.381340	−	0.924435i	$-0.375463\pi$
$347$	4929.89	0.762680	0.381340	+	0.924435i	$0.375463\pi$
$348$	0	0
$349$	−772.263	−0.118448	−0.0592239	−	0.998245i	$-0.518863\pi$
$349$	−772.263	−0.118448	−0.0592239	+	0.998245i	$0.518863\pi$
$350$	0	0
$351$	2424.01	0.368616
$352$	0	0
$353$	−6732.15	−1.01506	−0.507530	−	0.861634i	$-0.669441\pi$
$353$	−6732.15	−1.01506	−0.507530	+	0.861634i	$0.669441\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	7102.99	1.05303
$358$	0	0
$359$	8062.68	1.18533	0.592663	−	0.805450i	$-0.298077\pi$
$359$	8062.68	1.18533	0.592663	+	0.805450i	$0.298077\pi$
$360$	0	0
$361$	−4909.41	−0.715762
$362$	0	0
$363$	−8277.45	−1.19684
$364$	0	0
$365$	0	0
$366$	0	0
$367$	11662.4	1.65878	0.829392	−	0.558668i	$-0.188687\pi$
$367$	11662.4	1.65878	0.829392	+	0.558668i	$0.188687\pi$
$368$	0	0
$369$	1175.55	0.165844
$370$	0	0
$371$	9271.04	1.29738
$372$	0	0
$373$	6728.52	0.934020	0.467010	−	0.884252i	$-0.345331\pi$
$373$	6728.52	0.934020	0.467010	+	0.884252i	$0.345331\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2999.94	−0.409827
$378$	0	0
$379$	281.815	0.0381949	0.0190974	−	0.999818i	$-0.493921\pi$
$379$	281.815	0.0381949	0.0190974	+	0.999818i	$0.493921\pi$
$380$	0	0
$381$	1107.98	0.148986
$382$	0	0
$383$	−2705.09	−0.360898	−0.180449	−	0.983584i	$-0.557755\pi$
$383$	−2705.09	−0.360898	−0.180449	+	0.983584i	$0.557755\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3696.90	0.485592
$388$	0	0
$389$	2182.49	0.284465	0.142232	−	0.989833i	$-0.454572\pi$
$389$	2182.49	0.284465	0.142232	+	0.989833i	$0.454572\pi$
$390$	0	0
$391$	−7144.78	−0.924110
$392$	0	0
$393$	−6002.39	−0.770435
$394$	0	0
$395$	0	0
$396$	0	0
$397$	4374.92	0.553075	0.276537	−	0.961003i	$-0.410813\pi$
$397$	4374.92	0.553075	0.276537	+	0.961003i	$0.410813\pi$
$398$	0	0
$399$	4389.59	0.550763
$400$	0	0
$401$	4012.53	0.499691	0.249846	−	0.968286i	$-0.419620\pi$
$401$	4012.53	0.499691	0.249846	+	0.968286i	$0.419620\pi$
$402$	0	0
$403$	−10810.6	−1.33627
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−864.486	−0.105285
$408$	0	0
$409$	5255.70	0.635397	0.317699	−	0.948192i	$-0.397090\pi$
$409$	5255.70	0.635397	0.317699	+	0.948192i	$0.397090\pi$
$410$	0	0
$411$	7309.38	0.877239
$412$	0	0
$413$	−4424.92	−0.527206
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−3060.99	−0.359466
$418$	0	0
$419$	−14338.0	−1.67173	−0.835865	−	0.548935i	$-0.815033\pi$
$419$	−14338.0	−1.67173	−0.835865	+	0.548935i	$0.815033\pi$
$420$	0	0
$421$	−4074.34	−0.471665	−0.235833	−	0.971794i	$-0.575782\pi$
$421$	−4074.34	−0.471665	−0.235833	+	0.971794i	$0.575782\pi$
$422$	0	0
$423$	1948.41	0.223960
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−9852.72	−1.11664
$428$	0	0
$429$	17225.1	1.93854
$430$	0	0
$431$	7266.75	0.812128	0.406064	−	0.913845i	$-0.366901\pi$
$431$	7266.75	0.812128	0.406064	+	0.913845i	$0.366901\pi$
$432$	0	0
$433$	−13061.5	−1.44964	−0.724821	−	0.688937i	$-0.758078\pi$
$433$	−13061.5	−1.44964	−0.724821	+	0.688937i	$0.758078\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−4415.41	−0.483336
$438$	0	0
$439$	7231.30	0.786176	0.393088	−	0.919501i	$-0.371407\pi$
$439$	7231.30	0.786176	0.393088	+	0.919501i	$0.371407\pi$
$440$	0	0
$441$	6796.36	0.733869
$442$	0	0
$443$	−4034.19	−0.432664	−0.216332	−	0.976320i	$-0.569409\pi$
$443$	−4034.19	−0.432664	−0.216332	+	0.976320i	$0.569409\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	8499.30	0.899336
$448$	0	0
$449$	1568.58	0.164868	0.0824341	−	0.996597i	$-0.473731\pi$
$449$	1568.58	0.164868	0.0824341	+	0.996597i	$0.473731\pi$
$450$	0	0
$451$	8353.48	0.872173
$452$	0	0
$453$	−3313.81	−0.343700
$454$	0	0
$455$	0	0
$456$	0	0
$457$	1116.77	0.114311	0.0571555	−	0.998365i	$-0.481797\pi$
$457$	1116.77	0.114311	0.0571555	+	0.998365i	$0.481797\pi$
$458$	0	0
$459$	1929.09	0.196171
$460$	0	0
$461$	−830.355	−0.0838905	−0.0419452	−	0.999120i	$-0.513355\pi$
$461$	−830.355	−0.0838905	−0.0419452	+	0.999120i	$0.513355\pi$
$462$	0	0
$463$	6056.60	0.607935	0.303968	−	0.952682i	$-0.401689\pi$
$463$	6056.60	0.607935	0.303968	+	0.952682i	$0.401689\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−11572.8	−1.14673	−0.573365	−	0.819300i	$-0.694362\pi$
$467$	−11572.8	−1.14673	−0.573365	+	0.819300i	$0.694362\pi$
$468$	0	0
$469$	−16603.8	−1.63474
$470$	0	0
$471$	9640.96	0.943168
$472$	0	0
$473$	26270.3	2.55372
$474$	0	0
$475$	0	0
$476$	0	0
$477$	2517.91	0.241692
$478$	0	0
$479$	4892.51	0.466690	0.233345	−	0.972394i	$-0.425033\pi$
$479$	4892.51	0.466690	0.233345	+	0.972394i	$0.425033\pi$
$480$	0	0
$481$	1213.55	0.115038
$482$	0	0
$483$	−9941.51	−0.936551
$484$	0	0
$485$	0	0
$486$	0	0
$487$	2124.68	0.197697	0.0988483	−	0.995103i	$-0.468484\pi$
$487$	2124.68	0.197697	0.0988483	+	0.995103i	$0.468484\pi$
$488$	0	0
$489$	5634.83	0.521096
$490$	0	0
$491$	−2438.59	−0.224139	−0.112069	−	0.993700i	$-0.535748\pi$
$491$	−2438.59	−0.224139	−0.112069	+	0.993700i	$0.535748\pi$
$492$	0	0
$493$	−2387.43	−0.218103
$494$	0	0
$495$	0	0
$496$	0	0
$497$	9594.82	0.865968
$498$	0	0
$499$	142.030	0.0127417	0.00637086	−	0.999980i	$-0.497972\pi$
$499$	142.030	0.0127417	0.00637086	+	0.999980i	$0.497972\pi$
$500$	0	0
$501$	−3661.81	−0.326542
$502$	0	0
$503$	−3249.40	−0.288039	−0.144019	−	0.989575i	$-0.546003\pi$
$503$	−3249.40	−0.288039	−0.144019	+	0.989575i	$0.546003\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−17589.3	−1.54077
$508$	0	0
$509$	22022.2	1.91771	0.958856	−	0.283893i	$-0.0916261\pi$
$509$	22022.2	1.91771	0.958856	+	0.283893i	$0.0916261\pi$
$510$	0	0
$511$	−25059.6	−2.16942
$512$	0	0
$513$	1192.16	0.102603
$514$	0	0
$515$	0	0
$516$	0	0
$517$	13845.5	1.17780
$518$	0	0
$519$	−6272.51	−0.530506
$520$	0	0
$521$	15560.5	1.30848	0.654241	−	0.756286i	$-0.272988\pi$
$521$	15560.5	1.30848	0.654241	+	0.756286i	$0.272988\pi$
$522$	0	0
$523$	−4931.76	−0.412334	−0.206167	−	0.978517i	$-0.566099\pi$
$523$	−4931.76	−0.412334	−0.206167	+	0.978517i	$0.566099\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−8603.39	−0.711138
$528$	0	0
$529$	−2167.00	−0.178105
$530$	0	0
$531$	−1201.76	−0.0982145
$532$	0	0
$533$	−11726.5	−0.952966
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−7343.10	−0.590090
$538$	0	0
$539$	48295.1	3.85940
$540$	0	0
$541$	6468.68	0.514067	0.257033	−	0.966403i	$-0.417255\pi$
$541$	6468.68	0.514067	0.257033	+	0.966403i	$0.417255\pi$
$542$	0	0
$543$	−12972.2	−1.02521
$544$	0	0
$545$	0	0
$546$	0	0
$547$	9309.53	0.727691	0.363845	−	0.931459i	$-0.381464\pi$
$547$	9309.53	0.727691	0.363845	+	0.931459i	$0.381464\pi$
$548$	0	0
$549$	−2675.89	−0.208022
$550$	0	0
$551$	−1475.41	−0.114074
$552$	0	0
$553$	−3990.36	−0.306848
$554$	0	0
$555$	0	0
$556$	0	0
$557$	11829.4	0.899867	0.449933	−	0.893062i	$-0.351448\pi$
$557$	11829.4	0.899867	0.449933	+	0.893062i	$0.351448\pi$
$558$	0	0
$559$	−36877.9	−2.79028
$560$	0	0
$561$	13708.2	1.03166
$562$	0	0
$563$	−26222.4	−1.96295	−0.981477	−	0.191581i	$-0.938639\pi$
$563$	−26222.4	−1.96295	−0.981477	+	0.191581i	$0.938639\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	2684.21	0.198811
$568$	0	0
$569$	4444.50	0.327457	0.163729	−	0.986505i	$-0.447648\pi$
$569$	4444.50	0.327457	0.163729	+	0.986505i	$0.447648\pi$
$570$	0	0
$571$	21409.4	1.56910	0.784551	−	0.620064i	$-0.212893\pi$
$571$	21409.4	1.56910	0.784551	+	0.620064i	$0.212893\pi$
$572$	0	0
$573$	−10185.6	−0.742599
$574$	0	0
$575$	0	0
$576$	0	0
$577$	17414.4	1.25645	0.628223	−	0.778033i	$-0.283783\pi$
$577$	17414.4	1.25645	0.628223	+	0.778033i	$0.283783\pi$
$578$	0	0
$579$	−11472.5	−0.823457
$580$	0	0
$581$	2175.87	0.155371
$582$	0	0
$583$	17892.3	1.27105
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−23489.4	−1.65163	−0.825817	−	0.563938i	$-0.809286\pi$
$587$	−23489.4	−1.65163	−0.825817	+	0.563938i	$0.809286\pi$
$588$	0	0
$589$	−5316.82	−0.371945
$590$	0	0
$591$	6657.79	0.463392
$592$	0	0
$593$	6536.04	0.452619	0.226310	−	0.974055i	$-0.427334\pi$
$593$	6536.04	0.452619	0.226310	+	0.974055i	$0.427334\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−5818.49	−0.398886
$598$	0	0
$599$	−4507.99	−0.307498	−0.153749	−	0.988110i	$-0.549135\pi$
$599$	−4507.99	−0.307498	−0.153749	+	0.988110i	$0.549135\pi$
$600$	0	0
$601$	19945.3	1.35372	0.676859	−	0.736113i	$-0.263341\pi$
$601$	19945.3	1.35372	0.676859	+	0.736113i	$0.263341\pi$
$602$	0	0
$603$	−4509.39	−0.304539
$604$	0	0
$605$	0	0
$606$	0	0
$607$	17948.0	1.20014	0.600071	−	0.799947i	$-0.295139\pi$
$607$	17948.0	1.20014	0.600071	+	0.799947i	$0.295139\pi$
$608$	0	0
$609$	−3321.96	−0.221039
$610$	0	0
$611$	−19436.1	−1.28691
$612$	0	0
$613$	16936.5	1.11592	0.557959	−	0.829868i	$-0.311585\pi$
$613$	16936.5	1.11592	0.557959	+	0.829868i	$0.311585\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	16338.2	1.06605	0.533024	−	0.846100i	$-0.321056\pi$
$617$	16338.2	1.06605	0.533024	+	0.846100i	$0.321056\pi$
$618$	0	0
$619$	−21081.7	−1.36890	−0.684448	−	0.729062i	$-0.739957\pi$
$619$	−21081.7	−1.36890	−0.684448	+	0.729062i	$0.739957\pi$
$620$	0	0
$621$	−2700.00	−0.174472
$622$	0	0
$623$	17010.6	1.09393
$624$	0	0
$625$	0	0
$626$	0	0
$627$	8471.54	0.539587
$628$	0	0
$629$	965.778	0.0612211
$630$	0	0
$631$	−4472.88	−0.282191	−0.141095	−	0.989996i	$-0.545062\pi$
$631$	−4472.88	−0.282191	−0.141095	+	0.989996i	$0.545062\pi$
$632$	0	0
$633$	−6384.65	−0.400896
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−67796.0	−4.21692
$638$	0	0
$639$	2605.84	0.161323
$640$	0	0
$641$	−4906.44	−0.302329	−0.151164	−	0.988509i	$-0.548302\pi$
$641$	−4906.44	−0.302329	−0.151164	+	0.988509i	$0.548302\pi$
$642$	0	0
$643$	18740.6	1.14939	0.574695	−	0.818368i	$-0.305121\pi$
$643$	18740.6	1.14939	0.574695	+	0.818368i	$0.305121\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	19178.1	1.16533	0.582665	−	0.812713i	$-0.302010\pi$
$647$	19178.1	1.16533	0.582665	+	0.812713i	$0.302010\pi$
$648$	0	0
$649$	−8539.73	−0.516508
$650$	0	0
$651$	−11971.1	−0.720712
$652$	0	0
$653$	25638.9	1.53649	0.768246	−	0.640155i	$-0.221130\pi$
$653$	25638.9	1.53649	0.768246	+	0.640155i	$0.221130\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−6805.90	−0.404145
$658$	0	0
$659$	−17516.7	−1.03544	−0.517719	−	0.855551i	$-0.673219\pi$
$659$	−17516.7	−1.03544	−0.517719	+	0.855551i	$0.673219\pi$
$660$	0	0
$661$	−15764.4	−0.927632	−0.463816	−	0.885932i	$-0.653520\pi$
$661$	−15764.4	−0.927632	−0.463816	+	0.885932i	$0.653520\pi$
$662$	0	0
$663$	−19243.4	−1.12722
$664$	0	0
$665$	0	0
$666$	0	0
$667$	3341.51	0.193979
$668$	0	0
$669$	−14617.9	−0.844784
$670$	0	0
$671$	−19014.9	−1.09398
$672$	0	0
$673$	2242.93	0.128467	0.0642337	−	0.997935i	$-0.479540\pi$
$673$	2242.93	0.128467	0.0642337	+	0.997935i	$0.479540\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−30022.8	−1.70439	−0.852194	−	0.523226i	$-0.824728\pi$
$677$	−30022.8	−1.70439	−0.852194	+	0.523226i	$0.824728\pi$
$678$	0	0
$679$	27445.4	1.55119
$680$	0	0
$681$	−7322.71	−0.412051
$682$	0	0
$683$	31713.5	1.77670	0.888348	−	0.459171i	$-0.151853\pi$
$683$	31713.5	1.77670	0.888348	+	0.459171i	$0.151853\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	6087.17	0.338049
$688$	0	0
$689$	−25117.0	−1.38880
$690$	0	0
$691$	11938.0	0.657224	0.328612	−	0.944465i	$-0.393419\pi$
$691$	11938.0	0.657224	0.328612	+	0.944465i	$0.393419\pi$
$692$	0	0
$693$	19074.1	1.04555
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−9332.26	−0.507151
$698$	0	0
$699$	4879.23	0.264019
$700$	0	0
$701$	24502.7	1.32019	0.660095	−	0.751182i	$-0.270516\pi$
$701$	24502.7	1.32019	0.660095	+	0.751182i	$0.270516\pi$
$702$	0	0
$703$	596.842	0.0320204
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−48704.0	−2.59081
$708$	0	0
$709$	−752.907	−0.0398815	−0.0199408	−	0.999801i	$-0.506348\pi$
$709$	−752.907	−0.0398815	−0.0199408	+	0.999801i	$0.506348\pi$
$710$	0	0
$711$	−1083.74	−0.0571635
$712$	0	0
$713$	12041.5	0.632479
$714$	0	0
$715$	0	0
$716$	0	0
$717$	11425.1	0.595090
$718$	0	0
$719$	26968.1	1.39881	0.699403	−	0.714727i	$-0.253449\pi$
$719$	26968.1	1.39881	0.699403	+	0.714727i	$0.253449\pi$
$720$	0	0
$721$	−3331.13	−0.172064
$722$	0	0
$723$	−4823.21	−0.248101
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−4430.65	−0.226030	−0.113015	−	0.993593i	$-0.536051\pi$
$727$	−4430.65	−0.226030	−0.113015	+	0.993593i	$0.536051\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−29348.4	−1.48494
$732$	0	0
$733$	−23967.1	−1.20770	−0.603850	−	0.797098i	$-0.706367\pi$
$733$	−23967.1	−1.20770	−0.603850	+	0.797098i	$0.706367\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−32043.9	−1.60156
$738$	0	0
$739$	14607.9	0.727143	0.363572	−	0.931566i	$-0.381557\pi$
$739$	14607.9	0.727143	0.363572	+	0.931566i	$0.381557\pi$
$740$	0	0
$741$	−11892.2	−0.589571
$742$	0	0
$743$	21349.8	1.05417	0.527084	−	0.849813i	$-0.323285\pi$
$743$	21349.8	1.05417	0.527084	+	0.849813i	$0.323285\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	590.943	0.0289444
$748$	0	0
$749$	−23539.5	−1.14835
$750$	0	0
$751$	34286.8	1.66597	0.832985	−	0.553295i	$-0.186630\pi$
$751$	34286.8	1.66597	0.832985	+	0.553295i	$0.186630\pi$
$752$	0	0
$753$	4435.50	0.214660
$754$	0	0
$755$	0	0
$756$	0	0
$757$	33290.8	1.59838	0.799190	−	0.601078i	$-0.205262\pi$
$757$	33290.8	1.59838	0.799190	+	0.601078i	$0.205262\pi$
$758$	0	0
$759$	−19186.3	−0.917547
$760$	0	0
$761$	−14994.4	−0.714254	−0.357127	−	0.934056i	$-0.616244\pi$
$761$	−14994.4	−0.714254	−0.357127	+	0.934056i	$0.616244\pi$
$762$	0	0
$763$	45183.2	2.14383
$764$	0	0
$765$	0	0
$766$	0	0
$767$	11988.0	0.564355
$768$	0	0
$769$	19734.3	0.925407	0.462704	−	0.886513i	$-0.346879\pi$
$769$	19734.3	0.925407	0.462704	+	0.886513i	$0.346879\pi$
$770$	0	0
$771$	1281.13	0.0598428
$772$	0	0
$773$	13919.1	0.647651	0.323825	−	0.946117i	$-0.395031\pi$
$773$	13919.1	0.647651	0.323825	+	0.946117i	$0.395031\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	1343.82	0.0620453
$778$	0	0
$779$	−5767.26	−0.265255
$780$	0	0
$781$	18517.2	0.848396
$782$	0	0
$783$	−902.207	−0.0411778
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−29089.1	−1.31755	−0.658776	−	0.752339i	$-0.728925\pi$
$787$	−29089.1	−1.31755	−0.658776	+	0.752339i	$0.728925\pi$
$788$	0	0
$789$	8338.87	0.376263
$790$	0	0
$791$	24101.6	1.08338
$792$	0	0
$793$	26692.9	1.19532
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−5048.75	−0.224386	−0.112193	−	0.993686i	$-0.535788\pi$
$797$	−5048.75	−0.224386	−0.112193	+	0.993686i	$0.535788\pi$
$798$	0	0
$799$	−15467.8	−0.684869
$800$	0	0
$801$	4619.89	0.203790
$802$	0	0
$803$	−48362.9	−2.12539
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−20496.1	−0.894049
$808$	0	0
$809$	−25168.3	−1.09378	−0.546891	−	0.837204i	$-0.684189\pi$
$809$	−25168.3	−1.09378	−0.546891	+	0.837204i	$0.684189\pi$
$810$	0	0
$811$	−4042.06	−0.175014	−0.0875068	−	0.996164i	$-0.527890\pi$
$811$	−4042.06	−0.175014	−0.0875068	+	0.996164i	$0.527890\pi$
$812$	0	0
$813$	−18963.4	−0.818051
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−18137.1	−0.776666
$818$	0	0
$819$	−26775.9	−1.14240
$820$	0	0
$821$	−29594.0	−1.25802	−0.629012	−	0.777396i	$-0.716540\pi$
$821$	−29594.0	−1.25802	−0.629012	+	0.777396i	$0.716540\pi$
$822$	0	0
$823$	20336.2	0.861331	0.430665	−	0.902512i	$-0.358279\pi$
$823$	20336.2	0.861331	0.430665	+	0.902512i	$0.358279\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−23156.1	−0.973661	−0.486831	−	0.873496i	$-0.661847\pi$
$827$	−23156.1	−0.973661	−0.486831	+	0.873496i	$0.661847\pi$
$828$	0	0
$829$	19876.3	0.832729	0.416365	−	0.909198i	$-0.363304\pi$
$829$	19876.3	0.832729	0.416365	+	0.909198i	$0.363304\pi$
$830$	0	0
$831$	7337.74	0.306310
$832$	0	0
$833$	−53953.9	−2.24417
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−3251.21	−0.134263
$838$	0	0
$839$	5649.88	0.232486	0.116243	−	0.993221i	$-0.462915\pi$
$839$	5649.88	0.232486	0.116243	+	0.993221i	$0.462915\pi$
$840$	0	0
$841$	−23272.4	−0.954218
$842$	0	0
$843$	−7163.74	−0.292684
$844$	0	0
$845$	0	0
$846$	0	0
$847$	91433.7	3.70921
$848$	0	0
$849$	−21351.8	−0.863124
$850$	0	0
$851$	−1351.72	−0.0544495
$852$	0	0
$853$	−12974.3	−0.520788	−0.260394	−	0.965502i	$-0.583852\pi$
$853$	−12974.3	−0.520788	−0.260394	+	0.965502i	$0.583852\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	5604.38	0.223386	0.111693	−	0.993743i	$-0.464373\pi$
$857$	5604.38	0.223386	0.111693	+	0.993743i	$0.464373\pi$
$858$	0	0
$859$	31863.4	1.26562	0.632809	−	0.774308i	$-0.281902\pi$
$859$	31863.4	1.26562	0.632809	+	0.774308i	$0.281902\pi$
$860$	0	0
$861$	−12985.2	−0.513979
$862$	0	0
$863$	−23695.6	−0.934656	−0.467328	−	0.884084i	$-0.654783\pi$
$863$	−23695.6	−0.934656	−0.467328	+	0.884084i	$0.654783\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−575.376	−0.0225384
$868$	0	0
$869$	−7701.06	−0.300622
$870$	0	0
$871$	44982.8	1.74992
$872$	0	0
$873$	7453.86	0.288975
$874$	0	0
$875$	0	0
$876$	0	0
$877$	49065.3	1.88919	0.944593	−	0.328243i	$-0.106456\pi$
$877$	49065.3	1.88919	0.944593	+	0.328243i	$0.106456\pi$
$878$	0	0
$879$	20857.2	0.800336
$880$	0	0
$881$	−14711.2	−0.562581	−0.281291	−	0.959623i	$-0.590763\pi$
$881$	−14711.2	−0.562581	−0.281291	+	0.959623i	$0.590763\pi$
$882$	0	0
$883$	21638.5	0.824681	0.412340	−	0.911030i	$-0.364711\pi$
$883$	21638.5	0.824681	0.412340	+	0.911030i	$0.364711\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	14331.7	0.542515	0.271258	−	0.962507i	$-0.412560\pi$
$887$	14331.7	0.542515	0.271258	+	0.962507i	$0.412560\pi$
$888$	0	0
$889$	−12238.9	−0.461732
$890$	0	0
$891$	5180.30	0.194777
$892$	0	0
$893$	−9558.95	−0.358206
$894$	0	0
$895$	0	0
$896$	0	0
$897$	26933.4	1.00254
$898$	0	0
$899$	4023.68	0.149274
$900$	0	0
$901$	−19988.8	−0.739093
$902$	0	0
$903$	−40836.4	−1.50493
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−26495.0	−0.969957	−0.484978	−	0.874526i	$-0.661173\pi$
$907$	−26495.0	−0.969957	−0.484978	+	0.874526i	$0.661173\pi$
$908$	0	0
$909$	−13227.4	−0.482648
$910$	0	0
$911$	−4927.97	−0.179222	−0.0896108	−	0.995977i	$-0.528562\pi$
$911$	−4927.97	−0.179222	−0.0896108	+	0.995977i	$0.528562\pi$
$912$	0	0
$913$	4199.26	0.152218
$914$	0	0
$915$	0	0
$916$	0	0
$917$	66303.2	2.38770
$918$	0	0
$919$	15904.9	0.570896	0.285448	−	0.958394i	$-0.407858\pi$
$919$	15904.9	0.570896	0.285448	+	0.958394i	$0.407858\pi$
$920$	0	0
$921$	−8453.28	−0.302438
$922$	0	0
$923$	−25994.2	−0.926987
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−904.697	−0.0320541
$928$	0	0
$929$	−8577.94	−0.302942	−0.151471	−	0.988462i	$-0.548401\pi$
$929$	−8577.94	−0.302942	−0.151471	+	0.988462i	$0.548401\pi$
$930$	0	0
$931$	−33343.0	−1.17376
$932$	0	0
$933$	23327.0	0.818535
$934$	0	0
$935$	0	0
$936$	0	0
$937$	12395.3	0.432164	0.216082	−	0.976375i	$-0.430672\pi$
$937$	12395.3	0.432164	0.216082	+	0.976375i	$0.430672\pi$
$938$	0	0
$939$	24668.6	0.857326
$940$	0	0
$941$	−266.126	−0.00921939	−0.00460969	−	0.999989i	$-0.501467\pi$
$941$	−266.126	−0.00921939	−0.00460969	+	0.999989i	$0.501467\pi$
$942$	0	0
$943$	13061.6	0.451056
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−25584.5	−0.877912	−0.438956	−	0.898508i	$-0.644652\pi$
$947$	−25584.5	−0.877912	−0.438956	+	0.898508i	$0.644652\pi$
$948$	0	0
$949$	67891.2	2.32228
$950$	0	0
$951$	−12794.3	−0.436260
$952$	0	0
$953$	−41960.3	−1.42626	−0.713130	−	0.701031i	$-0.752723\pi$
$953$	−41960.3	−1.42626	−0.713130	+	0.701031i	$0.752723\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−6411.11	−0.216554
$958$	0	0
$959$	−80740.3	−2.71871
$960$	0	0
$961$	−15291.2	−0.513283
$962$	0	0
$963$	−6393.06	−0.213929
$964$	0	0
$965$	0	0
$966$	0	0
$967$	9034.73	0.300452	0.150226	−	0.988652i	$-0.452000\pi$
$967$	9034.73	0.300452	0.150226	+	0.988652i	$0.452000\pi$
$968$	0	0
$969$	−9464.15	−0.313759
$970$	0	0
$971$	30649.8	1.01297	0.506487	−	0.862247i	$-0.330944\pi$
$971$	30649.8	1.01297	0.506487	+	0.862247i	$0.330944\pi$
$972$	0	0
$973$	33812.0	1.11404
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−26503.6	−0.867887	−0.433944	−	0.900940i	$-0.642878\pi$
$977$	−26503.6	−0.867887	−0.433944	+	0.900940i	$0.642878\pi$
$978$	0	0
$979$	32829.1	1.07173
$980$	0	0
$981$	12271.2	0.399379
$982$	0	0
$983$	14822.3	0.480934	0.240467	−	0.970657i	$-0.422699\pi$
$983$	14822.3	0.480934	0.240467	+	0.970657i	$0.422699\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−21522.4	−0.694089
$988$	0	0
$989$	41076.7	1.32069
$990$	0	0
$991$	16407.1	0.525921	0.262960	−	0.964807i	$-0.415301\pi$
$991$	16407.1	0.525921	0.262960	+	0.964807i	$0.415301\pi$
$992$	0	0
$993$	23300.6	0.744634
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−37538.5	−1.19243	−0.596216	−	0.802824i	$-0.703330\pi$
$997$	−37538.5	−1.19243	−0.596216	+	0.802824i	$0.703330\pi$
$998$	0	0
$999$	364.966	0.0115586

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2400.4.a.bz.1.4		4
4.3	odd	2		2400.4.a.ca.1.1		4
5.2	odd	4		480.4.f.f.289.8	yes	8
5.3	odd	4		480.4.f.f.289.3	✓	8
5.4	even	2		2400.4.a.ca.1.2		4
15.2	even	4		1440.4.f.m.289.2		8
15.8	even	4		1440.4.f.m.289.3		8
20.3	even	4		480.4.f.f.289.7	yes	8
20.7	even	4		480.4.f.f.289.4	yes	8
20.19	odd	2	inner	2400.4.a.bz.1.3		4
40.3	even	4		960.4.f.u.769.2		8
40.13	odd	4		960.4.f.u.769.6		8
40.27	even	4		960.4.f.u.769.5		8
40.37	odd	4		960.4.f.u.769.1		8
60.23	odd	4		1440.4.f.m.289.4		8
60.47	odd	4		1440.4.f.m.289.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
480.4.f.f.289.3	✓	8	5.3	odd	4
480.4.f.f.289.4	yes	8	20.7	even	4
480.4.f.f.289.7	yes	8	20.3	even	4
480.4.f.f.289.8	yes	8	5.2	odd	4
960.4.f.u.769.1		8	40.37	odd	4
960.4.f.u.769.2		8	40.3	even	4
960.4.f.u.769.5		8	40.27	even	4
960.4.f.u.769.6		8	40.13	odd	4
1440.4.f.m.289.1		8	60.47	odd	4
1440.4.f.m.289.2		8	15.2	even	4
1440.4.f.m.289.3		8	15.8	even	4
1440.4.f.m.289.4		8	60.23	odd	4
2400.4.a.bz.1.3		4	20.19	odd	2	inner
2400.4.a.bz.1.4		4	1.1	even	1	trivial
2400.4.a.ca.1.1		4	4.3	odd	2
2400.4.a.ca.1.2		4	5.4	even	2

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:	\( N \)	\(=\)	\( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2400.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +33.1384 q^{7} +9.00000 q^{9} +63.9543 q^{11} -89.7781 q^{13} -71.4478 q^{17} -44.1541 q^{19} -99.4151 q^{21} +100.000 q^{23} -27.0000 q^{27} +33.4151 q^{29} +120.415 q^{31} -191.863 q^{33} -13.5172 q^{37} +269.334 q^{39} +130.616 q^{41} +410.767 q^{43} +216.490 q^{47} +755.151 q^{49} +214.343 q^{51} +279.768 q^{53} +132.462 q^{57} -133.529 q^{59} -297.321 q^{61} +298.245 q^{63} -501.044 q^{67} -300.000 q^{69} +289.538 q^{71} -756.211 q^{73} +2119.34 q^{77} -120.415 q^{79} +81.0000 q^{81} +65.6603 q^{83} -100.245 q^{87} +513.321 q^{89} -2975.10 q^{91} -361.245 q^{93} +828.207 q^{97} +575.589 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{3} + 60 q^{7} + 36 q^{9} - 180 q^{21} + 400 q^{23} - 108 q^{27} - 84 q^{29} + 1248 q^{41} + 192 q^{43} - 440 q^{47} + 844 q^{49} + 552 q^{61} + 540 q^{63} - 408 q^{67} - 1200 q^{69} + 324 q^{81}+ \cdots + 312 q^{89}+O(q^{100}) \) 4 * q - 12 * q^3 + 60 * q^7 + 36 * q^9 - 180 * q^21 + 400 * q^23 - 108 * q^27 - 84 * q^29 + 1248 * q^41 + 192 * q^43 - 440 * q^47 + 844 * q^49 + 552 * q^61 + 540 * q^63 - 408 * q^67 - 1200 * q^69 + 324 * q^81 - 608 * q^83 + 252 * q^87 + 312 * q^89

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