LMFDB - Embedded newform 1216.4.a.x.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1216,4,Mod(1,1216)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1216.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1216, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1216 = 2^{6} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1216.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,5,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$71.7463225670$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.3221.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 9x + 2 $ x^3 - x^2 - 9*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 152)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$3.44437$ of defining polynomial
Character	$\chi$	$=$	1216.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+10.3081 q^{3} -14.1468 q^{5} -4.06119 q^{7} +79.2568 q^{9} +5.53059 q^{11} +45.6697 q^{13} -145.826 q^{15} -92.8486 q^{17} -19.0000 q^{19} -41.8631 q^{21} -57.0689 q^{23} +75.1314 q^{25} +538.668 q^{27} +290.147 q^{29} +163.494 q^{31} +57.0099 q^{33} +57.4527 q^{35} +81.3659 q^{37} +470.767 q^{39} +73.1997 q^{41} +346.600 q^{43} -1121.23 q^{45} +503.383 q^{47} -326.507 q^{49} -957.092 q^{51} +164.243 q^{53} -78.2401 q^{55} -195.854 q^{57} +763.585 q^{59} -545.832 q^{61} -321.877 q^{63} -646.079 q^{65} +510.800 q^{67} -588.272 q^{69} -294.139 q^{71} -172.903 q^{73} +774.461 q^{75} -22.4608 q^{77} -973.626 q^{79} +3412.70 q^{81} -510.932 q^{83} +1313.51 q^{85} +2990.87 q^{87} +845.158 q^{89} -185.473 q^{91} +1685.31 q^{93} +268.789 q^{95} -863.606 q^{97} +438.337 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 5 q^{3} - 2 q^{5} - 35 q^{7} + 48 q^{9} + 28 q^{11} + 109 q^{13} - 228 q^{15} - 123 q^{17} - 57 q^{19} - 25 q^{21} - 193 q^{23} + 187 q^{25} + 719 q^{27} + 297 q^{29} - 140 q^{31} + 30 q^{33} + 246 q^{35}+ \cdots - 86 q^{99}+O(q^{100}) $ 3 * q + 5 * q^3 - 2 * q^5 - 35 * q^7 + 48 * q^9 + 28 * q^11 + 109 * q^13 - 228 * q^15 - 123 * q^17 - 57 * q^19 - 25 * q^21 - 193 * q^23 + 187 * q^25 + 719 * q^27 + 297 * q^29 - 140 * q^31 + 30 * q^33 + 246 * q^35 - 38 * q^37 + 57 * q^39 + 736 * q^41 + 514 * q^43 - 1046 * q^45 + 134 * q^47 - 42 * q^49 - 929 * q^51 - 311 * q^53 - 130 * q^55 - 95 * q^57 - 199 * q^59 - 56 * q^61 + 508 * q^63 + 1156 * q^65 + 509 * q^67 - 621 * q^69 - 874 * q^71 + 203 * q^73 - 129 * q^75 - 616 * q^77 + 242 * q^79 + 3531 * q^81 + 62 * q^83 + 1430 * q^85 + 3237 * q^87 + 1764 * q^89 + 687 * q^91 + 2592 * q^93 + 38 * q^95 - 2178 * q^97 - 86 * q^99

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
$3$	10.3081	1.98379	0.991897	−	0.127047i	$-0.0405499\pi$
$3$	10.3081	1.98379	0.991897	+	0.127047i	$0.0405499\pi$
$4$	0	0
$5$	−14.1468	−1.26533	−0.632663	−	0.774427i	$-0.718038\pi$
$5$	−14.1468	−1.26533	−0.632663	+	0.774427i	$0.718038\pi$
$6$	0	0
$7$	−4.06119	−0.219283	−0.109642	−	0.993971i	$-0.534970\pi$
$7$	−4.06119	−0.219283	−0.109642	+	0.993971i	$0.534970\pi$
$8$	0	0
$9$	79.2568	2.93544
$10$	0	0
$11$	5.53059	0.151594	0.0757971	−	0.997123i	$-0.475850\pi$
$11$	5.53059	0.151594	0.0757971	+	0.997123i	$0.475850\pi$
$12$	0	0
$13$	45.6697	0.974345	0.487172	−	0.873306i	$-0.338028\pi$
$13$	45.6697	0.974345	0.487172	+	0.873306i	$0.338028\pi$
$14$	0	0
$15$	−145.826	−2.51015
$16$	0	0
$17$	−92.8486	−1.32465	−0.662326	−	0.749216i	$-0.730431\pi$
$17$	−92.8486	−1.32465	−0.662326	+	0.749216i	$0.730431\pi$
$18$	0	0
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	0	0
$21$	−41.8631	−0.435013
$22$	0	0
$23$	−57.0689	−0.517378	−0.258689	−	0.965961i	$-0.583290\pi$
$23$	−57.0689	−0.517378	−0.258689	+	0.965961i	$0.583290\pi$
$24$	0	0
$25$	75.1314	0.601051
$26$	0	0
$27$	538.668	3.83951
$28$	0	0
$29$	290.147	1.85790	0.928948	−	0.370209i	$-0.120714\pi$
$29$	290.147	1.85790	0.928948	+	0.370209i	$0.120714\pi$
$30$	0	0
$31$	163.494	0.947240	0.473620	−	0.880729i	$-0.342947\pi$
$31$	163.494	0.947240	0.473620	+	0.880729i	$0.342947\pi$
$32$	0	0
$33$	57.0099	0.300732
$34$	0	0
$35$	57.4527	0.277465
$36$	0	0
$37$	81.3659	0.361526	0.180763	−	0.983527i	$-0.442143\pi$
$37$	81.3659	0.361526	0.180763	+	0.983527i	$0.442143\pi$
$38$	0	0
$39$	470.767	1.93290
$40$	0	0
$41$	73.1997	0.278826	0.139413	−	0.990234i	$-0.455478\pi$
$41$	73.1997	0.278826	0.139413	+	0.990234i	$0.455478\pi$
$42$	0	0
$43$	346.600	1.22921	0.614605	−	0.788835i	$-0.289315\pi$
$43$	346.600	1.22921	0.614605	+	0.788835i	$0.289315\pi$
$44$	0	0
$45$	−1121.23	−3.71428
$46$	0	0
$47$	503.383	1.56225	0.781127	−	0.624372i	$-0.214645\pi$
$47$	503.383	1.56225	0.781127	+	0.624372i	$0.214645\pi$
$48$	0	0
$49$	−326.507	−0.951915
$50$	0	0
$51$	−957.092	−2.62784
$52$	0	0
$53$	164.243	0.425670	0.212835	−	0.977088i	$-0.431730\pi$
$53$	164.243	0.425670	0.212835	+	0.977088i	$0.431730\pi$
$54$	0	0
$55$	−78.2401	−0.191816
$56$	0	0
$57$	−195.854	−0.455113
$58$	0	0
$59$	763.585	1.68492	0.842460	−	0.538759i	$-0.181107\pi$
$59$	763.585	1.68492	0.842460	+	0.538759i	$0.181107\pi$
$60$	0	0
$61$	−545.832	−1.14568	−0.572841	−	0.819666i	$-0.694159\pi$
$61$	−545.832	−1.14568	−0.572841	+	0.819666i	$0.694159\pi$
$62$	0	0
$63$	−321.877	−0.643693
$64$	0	0
$65$	−646.079	−1.23286
$66$	0	0
$67$	510.800	0.931406	0.465703	−	0.884941i	$-0.345801\pi$
$67$	510.800	0.931406	0.465703	+	0.884941i	$0.345801\pi$
$68$	0	0
$69$	−588.272	−1.02637
$70$	0	0
$71$	−294.139	−0.491659	−0.245830	−	0.969313i	$-0.579060\pi$
$71$	−294.139	−0.491659	−0.245830	+	0.969313i	$0.579060\pi$
$72$	0	0
$73$	−172.903	−0.277217	−0.138608	−	0.990347i	$-0.544263\pi$
$73$	−172.903	−0.277217	−0.138608	+	0.990347i	$0.544263\pi$
$74$	0	0
$75$	774.461	1.19236
$76$	0	0
$77$	−22.4608	−0.0332421
$78$	0	0
$79$	−973.626	−1.38660	−0.693300	−	0.720649i	$-0.743844\pi$
$79$	−973.626	−1.38660	−0.693300	+	0.720649i	$0.743844\pi$
$80$	0	0
$81$	3412.70	4.68135
$82$	0	0
$83$	−510.932	−0.675688	−0.337844	−	0.941202i	$-0.609698\pi$
$83$	−510.932	−0.675688	−0.337844	+	0.941202i	$0.609698\pi$
$84$	0	0
$85$	1313.51	1.67612
$86$	0	0
$87$	2990.87	3.68568
$88$	0	0
$89$	845.158	1.00659	0.503295	−	0.864114i	$-0.332121\pi$
$89$	845.158	1.00659	0.503295	+	0.864114i	$0.332121\pi$
$90$	0	0
$91$	−185.473	−0.213658
$92$	0	0
$93$	1685.31	1.87913
$94$	0	0
$95$	268.789	0.290286
$96$	0	0
$97$	−863.606	−0.903979	−0.451989	−	0.892023i	$-0.649286\pi$
$97$	−863.606	−0.903979	−0.451989	+	0.892023i	$0.649286\pi$
$98$	0	0
$99$	438.337	0.444995
$100$	0	0
$101$	1016.53	1.00147	0.500735	−	0.865600i	$-0.333063\pi$
$101$	1016.53	1.00147	0.500735	+	0.865600i	$0.333063\pi$
$102$	0	0
$103$	−185.045	−0.177020	−0.0885100	−	0.996075i	$-0.528211\pi$
$103$	−185.045	−0.177020	−0.0885100	+	0.996075i	$0.528211\pi$
$104$	0	0
$105$	592.228	0.550434
$106$	0	0
$107$	396.796	0.358502	0.179251	−	0.983803i	$-0.442633\pi$
$107$	396.796	0.358502	0.179251	+	0.983803i	$0.442633\pi$
$108$	0	0
$109$	−1355.57	−1.19119	−0.595595	−	0.803285i	$-0.703084\pi$
$109$	−1355.57	−1.19119	−0.595595	+	0.803285i	$0.703084\pi$
$110$	0	0
$111$	838.727	0.717193
$112$	0	0
$113$	1915.38	1.59455	0.797275	−	0.603616i	$-0.206274\pi$
$113$	1915.38	1.59455	0.797275	+	0.603616i	$0.206274\pi$
$114$	0	0
$115$	807.341	0.654652
$116$	0	0
$117$	3619.63	2.86013
$118$	0	0
$119$	377.075	0.290474
$120$	0	0
$121$	−1300.41	−0.977019
$122$	0	0
$123$	754.549	0.553134
$124$	0	0
$125$	705.481	0.504801
$126$	0	0
$127$	985.870	0.688833	0.344417	−	0.938817i	$-0.388077\pi$
$127$	985.870	0.688833	0.344417	+	0.938817i	$0.388077\pi$
$128$	0	0
$129$	3572.79	2.43850
$130$	0	0
$131$	−2296.18	−1.53144	−0.765718	−	0.643176i	$-0.777616\pi$
$131$	−2296.18	−1.53144	−0.765718	+	0.643176i	$0.777616\pi$
$132$	0	0
$133$	77.1626	0.0503071
$134$	0	0
$135$	−7620.41	−4.85823
$136$	0	0
$137$	1344.91	0.838712	0.419356	−	0.907822i	$-0.362256\pi$
$137$	1344.91	0.838712	0.419356	+	0.907822i	$0.362256\pi$
$138$	0	0
$139$	−402.094	−0.245361	−0.122681	−	0.992446i	$-0.539149\pi$
$139$	−402.094	−0.245361	−0.122681	+	0.992446i	$0.539149\pi$
$140$	0	0
$141$	5188.92	3.09919
$142$	0	0
$143$	252.580	0.147705
$144$	0	0
$145$	−4104.65	−2.35085
$146$	0	0
$147$	−3365.66	−1.88840
$148$	0	0
$149$	738.796	0.406205	0.203102	−	0.979158i	$-0.434898\pi$
$149$	738.796	0.406205	0.203102	+	0.979158i	$0.434898\pi$
$150$	0	0
$151$	819.258	0.441525	0.220762	−	0.975328i	$-0.429145\pi$
$151$	819.258	0.441525	0.220762	+	0.975328i	$0.429145\pi$
$152$	0	0
$153$	−7358.88	−3.88843
$154$	0	0
$155$	−2312.92	−1.19857
$156$	0	0
$157$	1469.33	0.746913	0.373456	−	0.927648i	$-0.378173\pi$
$157$	1469.33	0.746913	0.373456	+	0.927648i	$0.378173\pi$
$158$	0	0
$159$	1693.03	0.844442
$160$	0	0
$161$	231.767	0.113452
$162$	0	0
$163$	582.247	0.279786	0.139893	−	0.990167i	$-0.455324\pi$
$163$	582.247	0.279786	0.139893	+	0.990167i	$0.455324\pi$
$164$	0	0
$165$	−806.506	−0.380524
$166$	0	0
$167$	2401.31	1.11269	0.556345	−	0.830952i	$-0.312203\pi$
$167$	2401.31	1.11269	0.556345	+	0.830952i	$0.312203\pi$
$168$	0	0
$169$	−111.282	−0.0506519
$170$	0	0
$171$	−1505.88	−0.673435
$172$	0	0
$173$	−1104.21	−0.485269	−0.242635	−	0.970118i	$-0.578012\pi$
$173$	−1104.21	−0.485269	−0.242635	+	0.970118i	$0.578012\pi$
$174$	0	0
$175$	−305.122	−0.131801
$176$	0	0
$177$	7871.10	3.34253
$178$	0	0
$179$	822.030	0.343248	0.171624	−	0.985163i	$-0.445099\pi$
$179$	822.030	0.343248	0.171624	+	0.985163i	$0.445099\pi$
$180$	0	0
$181$	−151.476	−0.0622052	−0.0311026	−	0.999516i	$-0.509902\pi$
$181$	−151.476	−0.0622052	−0.0311026	+	0.999516i	$0.509902\pi$
$182$	0	0
$183$	−5626.49	−2.27280
$184$	0	0
$185$	−1151.07	−0.457449
$186$	0	0
$187$	−513.508	−0.200810
$188$	0	0
$189$	−2187.63	−0.841940
$190$	0	0
$191$	477.884	0.181039	0.0905195	−	0.995895i	$-0.471147\pi$
$191$	477.884	0.181039	0.0905195	+	0.995895i	$0.471147\pi$
$192$	0	0
$193$	−2169.48	−0.809134	−0.404567	−	0.914508i	$-0.632578\pi$
$193$	−2169.48	−0.809134	−0.404567	+	0.914508i	$0.632578\pi$
$194$	0	0
$195$	−6659.84	−2.44575
$196$	0	0
$197$	−1444.93	−0.522574	−0.261287	−	0.965261i	$-0.584147\pi$
$197$	−1444.93	−0.522574	−0.261287	+	0.965261i	$0.584147\pi$
$198$	0	0
$199$	1490.17	0.530830	0.265415	−	0.964134i	$-0.414491\pi$
$199$	1490.17	0.530830	0.265415	+	0.964134i	$0.414491\pi$
$200$	0	0
$201$	5265.38	1.84772
$202$	0	0
$203$	−1178.34	−0.407406
$204$	0	0
$205$	−1035.54	−0.352806
$206$	0	0
$207$	−4523.10	−1.51873
$208$	0	0
$209$	−105.081	−0.0347781
$210$	0	0
$211$	2270.00	0.740630	0.370315	−	0.928906i	$-0.379250\pi$
$211$	2270.00	0.740630	0.370315	+	0.928906i	$0.379250\pi$
$212$	0	0
$213$	−3032.01	−0.975351
$214$	0	0
$215$	−4903.28	−1.55535
$216$	0	0
$217$	−663.981	−0.207714
$218$	0	0
$219$	−1782.30	−0.549941
$220$	0	0
$221$	−4240.36	−1.29067
$222$	0	0
$223$	2380.30	0.714783	0.357392	−	0.933955i	$-0.383666\pi$
$223$	2380.30	0.714783	0.357392	+	0.933955i	$0.383666\pi$
$224$	0	0
$225$	5954.67	1.76435
$226$	0	0
$227$	1972.94	0.576865	0.288432	−	0.957500i	$-0.406866\pi$
$227$	1972.94	0.576865	0.288432	+	0.957500i	$0.406866\pi$
$228$	0	0
$229$	−21.4526	−0.00619053	−0.00309526	−	0.999995i	$-0.500985\pi$
$229$	−21.4526	−0.00619053	−0.00309526	+	0.999995i	$0.500985\pi$
$230$	0	0
$231$	−231.528	−0.0659455
$232$	0	0
$233$	1711.23	0.481143	0.240571	−	0.970631i	$-0.422665\pi$
$233$	1711.23	0.481143	0.240571	+	0.970631i	$0.422665\pi$
$234$	0	0
$235$	−7121.24	−1.97676
$236$	0	0
$237$	−10036.2	−2.75073
$238$	0	0
$239$	−2193.99	−0.593796	−0.296898	−	0.954909i	$-0.595952\pi$
$239$	−2193.99	−0.593796	−0.296898	+	0.954909i	$0.595952\pi$
$240$	0	0
$241$	726.890	0.194287	0.0971434	−	0.995270i	$-0.469029\pi$
$241$	726.890	0.194287	0.0971434	+	0.995270i	$0.469029\pi$
$242$	0	0
$243$	20634.4	5.44732
$244$	0	0
$245$	4619.02	1.20448
$246$	0	0
$247$	−867.724	−0.223530
$248$	0	0
$249$	−5266.74	−1.34043
$250$	0	0
$251$	−3168.49	−0.796785	−0.398393	−	0.917215i	$-0.630432\pi$
$251$	−3168.49	−0.796785	−0.398393	+	0.917215i	$0.630432\pi$
$252$	0	0
$253$	−315.625	−0.0784315
$254$	0	0
$255$	13539.8	3.32507
$256$	0	0
$257$	4341.53	1.05376	0.526881	−	0.849939i	$-0.323361\pi$
$257$	4341.53	1.05376	0.526881	+	0.849939i	$0.323361\pi$
$258$	0	0
$259$	−330.442	−0.0792767
$260$	0	0
$261$	22996.1	5.45374
$262$	0	0
$263$	−6418.23	−1.50481	−0.752405	−	0.658701i	$-0.771106\pi$
$263$	−6418.23	−1.50481	−0.752405	+	0.658701i	$0.771106\pi$
$264$	0	0
$265$	−2323.51	−0.538612
$266$	0	0
$267$	8711.97	1.99687
$268$	0	0
$269$	−6476.36	−1.46792	−0.733961	−	0.679192i	$-0.762330\pi$
$269$	−6476.36	−1.46792	−0.733961	+	0.679192i	$0.762330\pi$
$270$	0	0
$271$	−6832.73	−1.53158	−0.765791	−	0.643090i	$-0.777652\pi$
$271$	−6832.73	−1.53158	−0.765791	+	0.643090i	$0.777652\pi$
$272$	0	0
$273$	−1911.87	−0.423853
$274$	0	0
$275$	415.521	0.0911158
$276$	0	0
$277$	3243.37	0.703521	0.351761	−	0.936090i	$-0.385583\pi$
$277$	3243.37	0.703521	0.351761	+	0.936090i	$0.385583\pi$
$278$	0	0
$279$	12958.0	2.78056
$280$	0	0
$281$	4524.91	0.960618	0.480309	−	0.877099i	$-0.340524\pi$
$281$	4524.91	0.960618	0.480309	+	0.877099i	$0.340524\pi$
$282$	0	0
$283$	363.865	0.0764293	0.0382147	−	0.999270i	$-0.487833\pi$
$283$	363.865	0.0764293	0.0382147	+	0.999270i	$0.487833\pi$
$284$	0	0
$285$	2770.70	0.575867
$286$	0	0
$287$	−297.278	−0.0611420
$288$	0	0
$289$	3707.85	0.754703
$290$	0	0
$291$	−8902.13	−1.79331
$292$	0	0
$293$	4565.73	0.910350	0.455175	−	0.890402i	$-0.349577\pi$
$293$	4565.73	0.910350	0.455175	+	0.890402i	$0.349577\pi$
$294$	0	0
$295$	−10802.3	−2.13197
$296$	0	0
$297$	2979.15	0.582047
$298$	0	0
$299$	−2606.32	−0.504104
$300$	0	0
$301$	−1407.61	−0.269546
$302$	0	0
$303$	10478.5	1.98671
$304$	0	0
$305$	7721.77	1.44966
$306$	0	0
$307$	−8554.23	−1.59028	−0.795139	−	0.606427i	$-0.792602\pi$
$307$	−8554.23	−1.59028	−0.795139	+	0.606427i	$0.792602\pi$
$308$	0	0
$309$	−1907.46	−0.351171
$310$	0	0
$311$	2620.19	0.477740	0.238870	−	0.971052i	$-0.423223\pi$
$311$	2620.19	0.477740	0.238870	+	0.971052i	$0.423223\pi$
$312$	0	0
$313$	−5233.04	−0.945013	−0.472507	−	0.881327i	$-0.656651\pi$
$313$	−5233.04	−0.945013	−0.472507	+	0.881327i	$0.656651\pi$
$314$	0	0
$315$	4553.52	0.814481
$316$	0	0
$317$	−8199.12	−1.45271	−0.726354	−	0.687320i	$-0.758787\pi$
$317$	−8199.12	−1.45271	−0.726354	+	0.687320i	$0.758787\pi$
$318$	0	0
$319$	1604.69	0.281646
$320$	0	0
$321$	4090.21	0.711194
$322$	0	0
$323$	1764.12	0.303896
$324$	0	0
$325$	3431.22	0.585631
$326$	0	0
$327$	−13973.3	−2.36307
$328$	0	0
$329$	−2044.33	−0.342576
$330$	0	0
$331$	−4942.10	−0.820672	−0.410336	−	0.911934i	$-0.634589\pi$
$331$	−4942.10	−0.820672	−0.410336	+	0.911934i	$0.634589\pi$
$332$	0	0
$333$	6448.80	1.06124
$334$	0	0
$335$	−7226.18	−1.17853
$336$	0	0
$337$	−11622.4	−1.87868	−0.939339	−	0.342991i	$-0.888560\pi$
$337$	−11622.4	−1.87868	−0.939339	+	0.342991i	$0.888560\pi$
$338$	0	0
$339$	19744.0	3.16326
$340$	0	0
$341$	904.221	0.143596
$342$	0	0
$343$	2718.99	0.428023
$344$	0	0
$345$	8322.15	1.29869
$346$	0	0
$347$	6401.72	0.990382	0.495191	−	0.868784i	$-0.335098\pi$
$347$	6401.72	0.990382	0.495191	+	0.868784i	$0.335098\pi$
$348$	0	0
$349$	9713.15	1.48978	0.744890	−	0.667188i	$-0.232502\pi$
$349$	9713.15	1.48978	0.744890	+	0.667188i	$0.232502\pi$
$350$	0	0
$351$	24600.8	3.74100
$352$	0	0
$353$	−10674.0	−1.60941	−0.804705	−	0.593675i	$-0.797677\pi$
$353$	−10674.0	−1.60941	−0.804705	+	0.593675i	$0.797677\pi$
$354$	0	0
$355$	4161.11	0.622110
$356$	0	0
$357$	3886.93	0.576241
$358$	0	0
$359$	−4991.30	−0.733791	−0.366895	−	0.930262i	$-0.619579\pi$
$359$	−4991.30	−0.733791	−0.366895	+	0.930262i	$0.619579\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	0	0
$363$	−13404.8	−1.93820
$364$	0	0
$365$	2446.03	0.350770
$366$	0	0
$367$	−7742.21	−1.10120	−0.550599	−	0.834770i	$-0.685601\pi$
$367$	−7742.21	−1.10120	−0.550599	+	0.834770i	$0.685601\pi$
$368$	0	0
$369$	5801.57	0.818477
$370$	0	0
$371$	−667.022	−0.0933425
$372$	0	0
$373$	1644.48	0.228279	0.114139	−	0.993465i	$-0.463589\pi$
$373$	1644.48	0.228279	0.114139	+	0.993465i	$0.463589\pi$
$374$	0	0
$375$	7272.16	1.00142
$376$	0	0
$377$	13250.9	1.81023
$378$	0	0
$379$	1385.48	0.187776	0.0938881	−	0.995583i	$-0.470070\pi$
$379$	1385.48	0.187776	0.0938881	+	0.995583i	$0.470070\pi$
$380$	0	0
$381$	10162.4	1.36650
$382$	0	0
$383$	−6651.19	−0.887363	−0.443682	−	0.896185i	$-0.646328\pi$
$383$	−6651.19	−0.887363	−0.443682	+	0.896185i	$0.646328\pi$
$384$	0	0
$385$	317.748	0.0420621
$386$	0	0
$387$	27470.4	3.60827
$388$	0	0
$389$	−9322.00	−1.21502	−0.607512	−	0.794311i	$-0.707832\pi$
$389$	−9322.00	−1.21502	−0.607512	+	0.794311i	$0.707832\pi$
$390$	0	0
$391$	5298.76	0.685345
$392$	0	0
$393$	−23669.2	−3.03805
$394$	0	0
$395$	13773.7	1.75450
$396$	0	0
$397$	−11403.0	−1.44157	−0.720783	−	0.693160i	$-0.756218\pi$
$397$	−11403.0	−1.44157	−0.720783	+	0.693160i	$0.756218\pi$
$398$	0	0
$399$	795.399	0.0997989
$400$	0	0
$401$	3794.17	0.472498	0.236249	−	0.971693i	$-0.424082\pi$
$401$	3794.17	0.472498	0.236249	+	0.971693i	$0.424082\pi$
$402$	0	0
$403$	7466.73	0.922939
$404$	0	0
$405$	−48278.8	−5.92343
$406$	0	0
$407$	450.002	0.0548053
$408$	0	0
$409$	3210.32	0.388118	0.194059	−	0.980990i	$-0.437835\pi$
$409$	3210.32	0.388118	0.194059	+	0.980990i	$0.437835\pi$
$410$	0	0
$411$	13863.5	1.66383
$412$	0	0
$413$	−3101.06	−0.369475
$414$	0	0
$415$	7228.05	0.854966
$416$	0	0
$417$	−4144.83	−0.486746
$418$	0	0
$419$	−5646.89	−0.658398	−0.329199	−	0.944261i	$-0.606779\pi$
$419$	−5646.89	−0.658398	−0.329199	+	0.944261i	$0.606779\pi$
$420$	0	0
$421$	1912.56	0.221407	0.110704	−	0.993853i	$-0.464690\pi$
$421$	1912.56	0.221407	0.110704	+	0.993853i	$0.464690\pi$
$422$	0	0
$423$	39896.5	4.58590
$424$	0	0
$425$	−6975.84	−0.796183
$426$	0	0
$427$	2216.73	0.251229
$428$	0	0
$429$	2603.62	0.293016
$430$	0	0
$431$	−4568.25	−0.510545	−0.255273	−	0.966869i	$-0.582165\pi$
$431$	−4568.25	−0.510545	−0.255273	+	0.966869i	$0.582165\pi$
$432$	0	0
$433$	7432.60	0.824915	0.412458	−	0.910977i	$-0.364671\pi$
$433$	7432.60	0.824915	0.412458	+	0.910977i	$0.364671\pi$
$434$	0	0
$435$	−42311.1	−4.66359
$436$	0	0
$437$	1084.31	0.118695
$438$	0	0
$439$	−5937.86	−0.645555	−0.322777	−	0.946475i	$-0.604617\pi$
$439$	−5937.86	−0.645555	−0.322777	+	0.946475i	$0.604617\pi$
$440$	0	0
$441$	−25877.9	−2.79428
$442$	0	0
$443$	−11714.5	−1.25638	−0.628188	−	0.778062i	$-0.716203\pi$
$443$	−11714.5	−1.25638	−0.628188	+	0.778062i	$0.716203\pi$
$444$	0	0
$445$	−11956.3	−1.27367
$446$	0	0
$447$	7615.57	0.805826
$448$	0	0
$449$	2007.87	0.211040	0.105520	−	0.994417i	$-0.466349\pi$
$449$	2007.87	0.211040	0.105520	+	0.994417i	$0.466349\pi$
$450$	0	0
$451$	404.838	0.0422684
$452$	0	0
$453$	8444.98	0.875894
$454$	0	0
$455$	2623.85	0.270347
$456$	0	0
$457$	1447.39	0.148153	0.0740764	−	0.997253i	$-0.476399\pi$
$457$	1447.39	0.148153	0.0740764	+	0.997253i	$0.476399\pi$
$458$	0	0
$459$	−50014.5	−5.08601
$460$	0	0
$461$	2893.39	0.292318	0.146159	−	0.989261i	$-0.453309\pi$
$461$	2893.39	0.292318	0.146159	+	0.989261i	$0.453309\pi$
$462$	0	0
$463$	−13259.1	−1.33089	−0.665445	−	0.746447i	$-0.731758\pi$
$463$	−13259.1	−1.33089	−0.665445	+	0.746447i	$0.731758\pi$
$464$	0	0
$465$	−23841.8	−2.37771
$466$	0	0
$467$	−4031.59	−0.399486	−0.199743	−	0.979848i	$-0.564011\pi$
$467$	−4031.59	−0.399486	−0.199743	+	0.979848i	$0.564011\pi$
$468$	0	0
$469$	−2074.46	−0.204242
$470$	0	0
$471$	15146.0	1.48172
$472$	0	0
$473$	1916.91	0.186341
$474$	0	0
$475$	−1427.50	−0.137891
$476$	0	0
$477$	13017.4	1.24953
$478$	0	0
$479$	−8616.81	−0.821946	−0.410973	−	0.911647i	$-0.634811\pi$
$479$	−8616.81	−0.821946	−0.410973	+	0.911647i	$0.634811\pi$
$480$	0	0
$481$	3715.95	0.352251
$482$	0	0
$483$	2389.08	0.225066
$484$	0	0
$485$	12217.2	1.14383
$486$	0	0
$487$	−10149.9	−0.944429	−0.472215	−	0.881484i	$-0.656545\pi$
$487$	−10149.9	−0.944429	−0.472215	+	0.881484i	$0.656545\pi$
$488$	0	0
$489$	6001.85	0.555037
$490$	0	0
$491$	5982.00	0.549825	0.274912	−	0.961469i	$-0.411351\pi$
$491$	5982.00	0.549825	0.274912	+	0.961469i	$0.411351\pi$
$492$	0	0
$493$	−26939.8	−2.46107
$494$	0	0
$495$	−6201.06	−0.563064
$496$	0	0
$497$	1194.55	0.107813
$498$	0	0
$499$	−2022.13	−0.181409	−0.0907044	−	0.995878i	$-0.528912\pi$
$499$	−2022.13	−0.181409	−0.0907044	+	0.995878i	$0.528912\pi$
$500$	0	0
$501$	24752.9	2.20735
$502$	0	0
$503$	−1621.54	−0.143739	−0.0718697	−	0.997414i	$-0.522897\pi$
$503$	−1621.54	−0.143739	−0.0718697	+	0.997414i	$0.522897\pi$
$504$	0	0
$505$	−14380.6	−1.26719
$506$	0	0
$507$	−1147.11	−0.100483
$508$	0	0
$509$	1108.58	0.0965358	0.0482679	−	0.998834i	$-0.484630\pi$
$509$	1108.58	0.0965358	0.0482679	+	0.998834i	$0.484630\pi$
$510$	0	0
$511$	702.193	0.0607890
$512$	0	0
$513$	−10234.7	−0.880843
$514$	0	0
$515$	2617.80	0.223988
$516$	0	0
$517$	2784.01	0.236829
$518$	0	0
$519$	−11382.3	−0.962674
$520$	0	0
$521$	5249.04	0.441391	0.220696	−	0.975343i	$-0.429167\pi$
$521$	5249.04	0.441391	0.220696	+	0.975343i	$0.429167\pi$
$522$	0	0
$523$	−6191.33	−0.517645	−0.258822	−	0.965925i	$-0.583334\pi$
$523$	−6191.33	−0.517645	−0.258822	+	0.965925i	$0.583334\pi$
$524$	0	0
$525$	−3145.23	−0.261465
$526$	0	0
$527$	−15180.2	−1.25476
$528$	0	0
$529$	−8910.14	−0.732320
$530$	0	0
$531$	60519.3	4.94597
$532$	0	0
$533$	3343.01	0.271673
$534$	0	0
$535$	−5613.38	−0.453622
$536$	0	0
$537$	8473.56	0.680933
$538$	0	0
$539$	−1805.78	−0.144305
$540$	0	0
$541$	16975.2	1.34903	0.674513	−	0.738263i	$-0.264354\pi$
$541$	16975.2	1.34903	0.674513	+	0.738263i	$0.264354\pi$
$542$	0	0
$543$	−1561.43	−0.123402
$544$	0	0
$545$	19176.9	1.50724
$546$	0	0
$547$	−16676.3	−1.30352	−0.651762	−	0.758424i	$-0.725970\pi$
$547$	−16676.3	−1.30352	−0.651762	+	0.758424i	$0.725970\pi$
$548$	0	0
$549$	−43260.9	−3.36308
$550$	0	0
$551$	−5512.80	−0.426231
$552$	0	0
$553$	3954.08	0.304059
$554$	0	0
$555$	−11865.3	−0.907484
$556$	0	0
$557$	12448.6	0.946973	0.473486	−	0.880801i	$-0.342995\pi$
$557$	12448.6	0.946973	0.473486	+	0.880801i	$0.342995\pi$
$558$	0	0
$559$	15829.1	1.19768
$560$	0	0
$561$	−5293.28	−0.398365
$562$	0	0
$563$	−12829.5	−0.960386	−0.480193	−	0.877163i	$-0.659433\pi$
$563$	−12829.5	−0.960386	−0.480193	+	0.877163i	$0.659433\pi$
$564$	0	0
$565$	−27096.5	−2.01763
$566$	0	0
$567$	−13859.6	−1.02654
$568$	0	0
$569$	−916.141	−0.0674985	−0.0337492	−	0.999430i	$-0.510745\pi$
$569$	−916.141	−0.0674985	−0.0337492	+	0.999430i	$0.510745\pi$
$570$	0	0
$571$	−1247.99	−0.0914654	−0.0457327	−	0.998954i	$-0.514562\pi$
$571$	−1247.99	−0.0914654	−0.0457327	+	0.998954i	$0.514562\pi$
$572$	0	0
$573$	4926.07	0.359144
$574$	0	0
$575$	−4287.66	−0.310970
$576$	0	0
$577$	12669.4	0.914095	0.457047	−	0.889442i	$-0.348907\pi$
$577$	12669.4	0.914095	0.457047	+	0.889442i	$0.348907\pi$
$578$	0	0
$579$	−22363.2	−1.60516
$580$	0	0
$581$	2074.99	0.148167
$582$	0	0
$583$	908.362	0.0645292
$584$	0	0
$585$	−51206.1	−3.61899
$586$	0	0
$587$	20350.0	1.43089	0.715446	−	0.698668i	$-0.246223\pi$
$587$	20350.0	1.43089	0.715446	+	0.698668i	$0.246223\pi$
$588$	0	0
$589$	−3106.39	−0.217312
$590$	0	0
$591$	−14894.5	−1.03668
$592$	0	0
$593$	2306.21	0.159705	0.0798523	−	0.996807i	$-0.474555\pi$
$593$	2306.21	0.159705	0.0798523	+	0.996807i	$0.474555\pi$
$594$	0	0
$595$	−5334.40	−0.367545
$596$	0	0
$597$	15360.8	1.05306
$598$	0	0
$599$	−21458.5	−1.46372	−0.731861	−	0.681454i	$-0.761348\pi$
$599$	−21458.5	−1.46372	−0.731861	+	0.681454i	$0.761348\pi$
$600$	0	0
$601$	8415.81	0.571195	0.285598	−	0.958350i	$-0.407808\pi$
$601$	8415.81	0.571195	0.285598	+	0.958350i	$0.407808\pi$
$602$	0	0
$603$	40484.4	2.73408
$604$	0	0
$605$	18396.6	1.23625
$606$	0	0
$607$	−3515.81	−0.235095	−0.117547	−	0.993067i	$-0.537503\pi$
$607$	−3515.81	−0.235095	−0.117547	+	0.993067i	$0.537503\pi$
$608$	0	0
$609$	−12146.5	−0.808209
$610$	0	0
$611$	22989.3	1.52217
$612$	0	0
$613$	−16618.5	−1.09497	−0.547484	−	0.836816i	$-0.684414\pi$
$613$	−16618.5	−1.09497	−0.547484	+	0.836816i	$0.684414\pi$
$614$	0	0
$615$	−10674.4	−0.699895
$616$	0	0
$617$	29469.3	1.92284	0.961419	−	0.275088i	$-0.0887069\pi$
$617$	29469.3	1.92284	0.961419	+	0.275088i	$0.0887069\pi$
$618$	0	0
$619$	−29408.3	−1.90957	−0.954783	−	0.297304i	$-0.903912\pi$
$619$	−29408.3	−1.90957	−0.954783	+	0.297304i	$0.903912\pi$
$620$	0	0
$621$	−30741.2	−1.98647
$622$	0	0
$623$	−3432.34	−0.220729
$624$	0	0
$625$	−19371.7	−1.23979
$626$	0	0
$627$	−1083.19	−0.0689926
$628$	0	0
$629$	−7554.70	−0.478896
$630$	0	0
$631$	1315.79	0.0830124	0.0415062	−	0.999138i	$-0.486784\pi$
$631$	1315.79	0.0830124	0.0415062	+	0.999138i	$0.486784\pi$
$632$	0	0
$633$	23399.3	1.46926
$634$	0	0
$635$	−13946.9	−0.871599
$636$	0	0
$637$	−14911.5	−0.927493
$638$	0	0
$639$	−23312.5	−1.44324
$640$	0	0
$641$	−28499.5	−1.75610	−0.878052	−	0.478565i	$-0.841157\pi$
$641$	−28499.5	−1.75610	−0.878052	+	0.478565i	$0.841157\pi$
$642$	0	0
$643$	−19337.7	−1.18601	−0.593006	−	0.805198i	$-0.702059\pi$
$643$	−19337.7	−1.18601	−0.593006	+	0.805198i	$0.702059\pi$
$644$	0	0
$645$	−50543.5	−3.08550
$646$	0	0
$647$	23409.8	1.42246	0.711232	−	0.702957i	$-0.248137\pi$
$647$	23409.8	1.42246	0.711232	+	0.702957i	$0.248137\pi$
$648$	0	0
$649$	4223.08	0.255424
$650$	0	0
$651$	−6844.38	−0.412062
$652$	0	0
$653$	−11324.7	−0.678665	−0.339332	−	0.940667i	$-0.610201\pi$
$653$	−11324.7	−0.678665	−0.339332	+	0.940667i	$0.610201\pi$
$654$	0	0
$655$	32483.5	1.93777
$656$	0	0
$657$	−13703.8	−0.813752
$658$	0	0
$659$	6171.58	0.364811	0.182406	−	0.983223i	$-0.441612\pi$
$659$	6171.58	0.364811	0.182406	+	0.983223i	$0.441612\pi$
$660$	0	0
$661$	10178.3	0.598926	0.299463	−	0.954108i	$-0.403193\pi$
$661$	10178.3	0.598926	0.299463	+	0.954108i	$0.403193\pi$
$662$	0	0
$663$	−43710.0	−2.56042
$664$	0	0
$665$	−1091.60	−0.0636549
$666$	0	0
$667$	−16558.4	−0.961234
$668$	0	0
$669$	24536.3	1.41798
$670$	0	0
$671$	−3018.78	−0.173679
$672$	0	0
$673$	−1615.15	−0.0925106	−0.0462553	−	0.998930i	$-0.514729\pi$
$673$	−1615.15	−0.0925106	−0.0462553	+	0.998930i	$0.514729\pi$
$674$	0	0
$675$	40470.8	2.30774
$676$	0	0
$677$	−33333.2	−1.89232	−0.946158	−	0.323706i	$-0.895071\pi$
$677$	−33333.2	−1.89232	−0.946158	+	0.323706i	$0.895071\pi$
$678$	0	0
$679$	3507.27	0.198228
$680$	0	0
$681$	20337.2	1.14438
$682$	0	0
$683$	−11254.4	−0.630507	−0.315253	−	0.949008i	$-0.602089\pi$
$683$	−11254.4	−0.630507	−0.315253	+	0.949008i	$0.602089\pi$
$684$	0	0
$685$	−19026.2	−1.06124
$686$	0	0
$687$	−221.136	−0.0122807
$688$	0	0
$689$	7500.93	0.414750
$690$	0	0
$691$	15873.6	0.873891	0.436945	−	0.899488i	$-0.356060\pi$
$691$	15873.6	0.873891	0.436945	+	0.899488i	$0.356060\pi$
$692$	0	0
$693$	−1780.17	−0.0975801
$694$	0	0
$695$	5688.34	0.310462
$696$	0	0
$697$	−6796.49	−0.369348
$698$	0	0
$699$	17639.5	0.954487
$700$	0	0
$701$	14151.8	0.762494	0.381247	−	0.924473i	$-0.375495\pi$
$701$	14151.8	0.762494	0.381247	+	0.924473i	$0.375495\pi$
$702$	0	0
$703$	−1545.95	−0.0829398
$704$	0	0
$705$	−73406.5	−3.92148
$706$	0	0
$707$	−4128.32	−0.219606
$708$	0	0
$709$	26805.8	1.41991	0.709953	−	0.704249i	$-0.248716\pi$
$709$	26805.8	1.41991	0.709953	+	0.704249i	$0.248716\pi$
$710$	0	0
$711$	−77166.4	−4.07028
$712$	0	0
$713$	−9330.44	−0.490081
$714$	0	0
$715$	−3573.20	−0.186895
$716$	0	0
$717$	−22615.8	−1.17797
$718$	0	0
$719$	−254.907	−0.0132217	−0.00661087	−	0.999978i	$-0.502104\pi$
$719$	−254.907	−0.0132217	−0.00661087	+	0.999978i	$0.502104\pi$
$720$	0	0
$721$	751.504	0.0388176
$722$	0	0
$723$	7492.85	0.385425
$724$	0	0
$725$	21799.2	1.11669
$726$	0	0
$727$	10874.1	0.554745	0.277373	−	0.960762i	$-0.410536\pi$
$727$	10874.1	0.554745	0.277373	+	0.960762i	$0.410536\pi$
$728$	0	0
$729$	120559.	6.12502
$730$	0	0
$731$	−32181.3	−1.62828
$732$	0	0
$733$	14048.2	0.707888	0.353944	−	0.935267i	$-0.384840\pi$
$733$	14048.2	0.707888	0.353944	+	0.935267i	$0.384840\pi$
$734$	0	0
$735$	47613.3	2.38945
$736$	0	0
$737$	2825.03	0.141196
$738$	0	0
$739$	334.036	0.0166275	0.00831374	−	0.999965i	$-0.497354\pi$
$739$	334.036	0.0166275	0.00831374	+	0.999965i	$0.497354\pi$
$740$	0	0
$741$	−8944.57	−0.443437
$742$	0	0
$743$	10340.4	0.510571	0.255286	−	0.966866i	$-0.417831\pi$
$743$	10340.4	0.510571	0.255286	+	0.966866i	$0.417831\pi$
$744$	0	0
$745$	−10451.6	−0.513981
$746$	0	0
$747$	−40494.9	−1.98344
$748$	0	0
$749$	−1611.46	−0.0786135
$750$	0	0
$751$	31144.7	1.51330	0.756648	−	0.653822i	$-0.226836\pi$
$751$	31144.7	1.51330	0.756648	+	0.653822i	$0.226836\pi$
$752$	0	0
$753$	−32661.0	−1.58066
$754$	0	0
$755$	−11589.9	−0.558673
$756$	0	0
$757$	1688.02	0.0810465	0.0405233	−	0.999179i	$-0.487098\pi$
$757$	1688.02	0.0810465	0.0405233	+	0.999179i	$0.487098\pi$
$758$	0	0
$759$	−3253.49	−0.155592
$760$	0	0
$761$	11804.3	0.562296	0.281148	−	0.959664i	$-0.409285\pi$
$761$	11804.3	0.562296	0.281148	+	0.959664i	$0.409285\pi$
$762$	0	0
$763$	5505.21	0.261208
$764$	0	0
$765$	104104.	4.92013
$766$	0	0
$767$	34872.7	1.64169
$768$	0	0
$769$	20685.2	0.969995	0.484998	−	0.874516i	$-0.338821\pi$
$769$	20685.2	0.969995	0.484998	+	0.874516i	$0.338821\pi$
$770$	0	0
$771$	44752.9	2.09045
$772$	0	0
$773$	24226.8	1.12727	0.563634	−	0.826024i	$-0.309403\pi$
$773$	24226.8	1.12727	0.563634	+	0.826024i	$0.309403\pi$
$774$	0	0
$775$	12283.6	0.569340
$776$	0	0
$777$	−3406.23	−0.157269
$778$	0	0
$779$	−1390.79	−0.0639671
$780$	0	0
$781$	−1626.76	−0.0745327
$782$	0	0
$783$	156293.	7.13340
$784$	0	0
$785$	−20786.3	−0.945089
$786$	0	0
$787$	2974.74	0.134737	0.0673685	−	0.997728i	$-0.478540\pi$
$787$	2974.74	0.134737	0.0673685	+	0.997728i	$0.478540\pi$
$788$	0	0
$789$	−66159.7	−2.98523
$790$	0	0
$791$	−7778.73	−0.349659
$792$	0	0
$793$	−24928.0	−1.11629
$794$	0	0
$795$	−23951.0	−1.06849
$796$	0	0
$797$	3979.23	0.176853	0.0884264	−	0.996083i	$-0.471816\pi$
$797$	3979.23	0.176853	0.0884264	+	0.996083i	$0.471816\pi$
$798$	0	0
$799$	−46738.4	−2.06944
$800$	0	0
$801$	66984.5	2.95478
$802$	0	0
$803$	−956.259	−0.0420245
$804$	0	0
$805$	−3278.76	−0.143554
$806$	0	0
$807$	−66759.0	−2.91205
$808$	0	0
$809$	27125.7	1.17885	0.589425	−	0.807823i	$-0.299354\pi$
$809$	27125.7	1.17885	0.589425	+	0.807823i	$0.299354\pi$
$810$	0	0
$811$	−8095.42	−0.350516	−0.175258	−	0.984523i	$-0.556076\pi$
$811$	−8095.42	−0.350516	−0.175258	+	0.984523i	$0.556076\pi$
$812$	0	0
$813$	−70432.4	−3.03834
$814$	0	0
$815$	−8236.91	−0.354020
$816$	0	0
$817$	−6585.41	−0.282000
$818$	0	0
$819$	−14700.0	−0.627179
$820$	0	0
$821$	19575.6	0.832148	0.416074	−	0.909331i	$-0.363406\pi$
$821$	19575.6	0.832148	0.416074	+	0.909331i	$0.363406\pi$
$822$	0	0
$823$	−857.808	−0.0363321	−0.0181660	−	0.999835i	$-0.505783\pi$
$823$	−857.808	−0.0363321	−0.0181660	+	0.999835i	$0.505783\pi$
$824$	0	0
$825$	4283.23	0.180755
$826$	0	0
$827$	33329.2	1.40142	0.700708	−	0.713448i	$-0.252867\pi$
$827$	33329.2	1.40142	0.700708	+	0.713448i	$0.252867\pi$
$828$	0	0
$829$	37545.6	1.57299	0.786497	−	0.617594i	$-0.211893\pi$
$829$	37545.6	1.57299	0.786497	+	0.617594i	$0.211893\pi$
$830$	0	0
$831$	33433.0	1.39564
$832$	0	0
$833$	30315.7	1.26096
$834$	0	0
$835$	−33970.8	−1.40791
$836$	0	0
$837$	88069.1	3.63693
$838$	0	0
$839$	−8699.05	−0.357955	−0.178978	−	0.983853i	$-0.557279\pi$
$839$	−8699.05	−0.357955	−0.178978	+	0.983853i	$0.557279\pi$
$840$	0	0
$841$	59796.5	2.45178
$842$	0	0
$843$	46643.2	1.90567
$844$	0	0
$845$	1574.28	0.0640911
$846$	0	0
$847$	5281.22	0.214244
$848$	0	0
$849$	3750.75	0.151620
$850$	0	0
$851$	−4643.46	−0.187046
$852$	0	0
$853$	39492.9	1.58524	0.792621	−	0.609715i	$-0.208716\pi$
$853$	39492.9	1.58524	0.792621	+	0.609715i	$0.208716\pi$
$854$	0	0
$855$	21303.3	0.852115
$856$	0	0
$857$	−49878.9	−1.98813	−0.994067	−	0.108774i	$-0.965308\pi$
$857$	−49878.9	−1.98813	−0.994067	+	0.108774i	$0.965308\pi$
$858$	0	0
$859$	21676.4	0.860988	0.430494	−	0.902593i	$-0.358339\pi$
$859$	21676.4	0.860988	0.430494	+	0.902593i	$0.358339\pi$
$860$	0	0
$861$	−3064.37	−0.121293
$862$	0	0
$863$	−19826.9	−0.782058	−0.391029	−	0.920378i	$-0.627881\pi$
$863$	−19826.9	−0.782058	−0.391029	+	0.920378i	$0.627881\pi$
$864$	0	0
$865$	15621.0	0.614024
$866$	0	0
$867$	38220.9	1.49717
$868$	0	0
$869$	−5384.73	−0.210201
$870$	0	0
$871$	23328.1	0.907511
$872$	0	0
$873$	−68446.7	−2.65357
$874$	0	0
$875$	−2865.09	−0.110694
$876$	0	0
$877$	28960.6	1.11509	0.557543	−	0.830148i	$-0.311744\pi$
$877$	28960.6	1.11509	0.557543	+	0.830148i	$0.311744\pi$
$878$	0	0
$879$	47063.9	1.80595
$880$	0	0
$881$	9612.36	0.367592	0.183796	−	0.982964i	$-0.441161\pi$
$881$	9612.36	0.367592	0.183796	+	0.982964i	$0.441161\pi$
$882$	0	0
$883$	37634.8	1.43433	0.717164	−	0.696905i	$-0.245440\pi$
$883$	37634.8	1.43433	0.717164	+	0.696905i	$0.245440\pi$
$884$	0	0
$885$	−111351.	−4.22939
$886$	0	0
$887$	11501.8	0.435390	0.217695	−	0.976017i	$-0.430146\pi$
$887$	11501.8	0.435390	0.217695	+	0.976017i	$0.430146\pi$
$888$	0	0
$889$	−4003.80	−0.151050
$890$	0	0
$891$	18874.3	0.709666
$892$	0	0
$893$	−9564.27	−0.358406
$894$	0	0
$895$	−11629.1	−0.434321
$896$	0	0
$897$	−26866.2	−1.00004
$898$	0	0
$899$	47437.4	1.75987
$900$	0	0
$901$	−15249.7	−0.563865
$902$	0	0
$903$	−14509.8	−0.534723
$904$	0	0
$905$	2142.90	0.0787099
$906$	0	0
$907$	18651.0	0.682796	0.341398	−	0.939919i	$-0.389100\pi$
$907$	18651.0	0.682796	0.341398	+	0.939919i	$0.389100\pi$
$908$	0	0
$909$	80566.9	2.93975
$910$	0	0
$911$	42680.1	1.55220	0.776100	−	0.630610i	$-0.217195\pi$
$911$	42680.1	1.55220	0.776100	+	0.630610i	$0.217195\pi$
$912$	0	0
$913$	−2825.76	−0.102430
$914$	0	0
$915$	79596.7	2.87583
$916$	0	0
$917$	9325.21	0.335819
$918$	0	0
$919$	−5918.21	−0.212431	−0.106215	−	0.994343i	$-0.533873\pi$
$919$	−5918.21	−0.212431	−0.106215	+	0.994343i	$0.533873\pi$
$920$	0	0
$921$	−88177.8	−3.15478
$922$	0	0
$923$	−13433.2	−0.479046
$924$	0	0
$925$	6113.13	0.217296
$926$	0	0
$927$	−14666.1	−0.519631
$928$	0	0
$929$	−21425.2	−0.756661	−0.378331	−	0.925671i	$-0.623502\pi$
$929$	−21425.2	−0.756661	−0.378331	+	0.925671i	$0.623502\pi$
$930$	0	0
$931$	6203.63	0.218384
$932$	0	0
$933$	27009.1	0.947738
$934$	0	0
$935$	7264.48	0.254090
$936$	0	0
$937$	−26280.8	−0.916283	−0.458142	−	0.888879i	$-0.651485\pi$
$937$	−26280.8	−0.916283	−0.458142	+	0.888879i	$0.651485\pi$
$938$	0	0
$939$	−53942.7	−1.87471
$940$	0	0
$941$	−17105.3	−0.592581	−0.296290	−	0.955098i	$-0.595750\pi$
$941$	−17105.3	−0.592581	−0.296290	+	0.955098i	$0.595750\pi$
$942$	0	0
$943$	−4177.43	−0.144258
$944$	0	0
$945$	30947.9	1.06533
$946$	0	0
$947$	−30353.4	−1.04155	−0.520777	−	0.853692i	$-0.674358\pi$
$947$	−30353.4	−1.04155	−0.520777	+	0.853692i	$0.674358\pi$
$948$	0	0
$949$	−7896.44	−0.270105
$950$	0	0
$951$	−84517.3	−2.88187
$952$	0	0
$953$	−16580.1	−0.563569	−0.281784	−	0.959478i	$-0.590926\pi$
$953$	−16580.1	−0.563569	−0.281784	+	0.959478i	$0.590926\pi$
$954$	0	0
$955$	−6760.52	−0.229074
$956$	0	0
$957$	16541.3	0.558728
$958$	0	0
$959$	−5461.94	−0.183916
$960$	0	0
$961$	−3060.60	−0.102736
$962$	0	0
$963$	31448.8	1.05236
$964$	0	0
$965$	30691.2	1.02382
$966$	0	0
$967$	1243.10	0.0413396	0.0206698	−	0.999786i	$-0.493420\pi$
$967$	1243.10	0.0413396	0.0206698	+	0.999786i	$0.493420\pi$
$968$	0	0
$969$	18184.7	0.602867
$970$	0	0
$971$	−19859.0	−0.656341	−0.328170	−	0.944619i	$-0.606432\pi$
$971$	−19859.0	−0.656341	−0.328170	+	0.944619i	$0.606432\pi$
$972$	0	0
$973$	1632.98	0.0538036
$974$	0	0
$975$	35369.4	1.16177
$976$	0	0
$977$	21378.0	0.700043	0.350021	−	0.936742i	$-0.386174\pi$
$977$	21378.0	0.700043	0.350021	+	0.936742i	$0.386174\pi$
$978$	0	0
$979$	4674.23	0.152593
$980$	0	0
$981$	−107438.	−3.49666
$982$	0	0
$983$	−37387.5	−1.21310	−0.606549	−	0.795046i	$-0.707447\pi$
$983$	−37387.5	−1.21310	−0.606549	+	0.795046i	$0.707447\pi$
$984$	0	0
$985$	20441.1	0.661227
$986$	0	0
$987$	−21073.2	−0.679601
$988$	0	0
$989$	−19780.1	−0.635966
$990$	0	0
$991$	−56543.4	−1.81247	−0.906236	−	0.422771i	$-0.861057\pi$
$991$	−56543.4	−1.81247	−0.906236	+	0.422771i	$0.861057\pi$
$992$	0	0
$993$	−50943.7	−1.62804
$994$	0	0
$995$	−21081.1	−0.671674
$996$	0	0
$997$	−17828.4	−0.566330	−0.283165	−	0.959071i	$-0.591384\pi$
$997$	−17828.4	−0.566330	−0.283165	+	0.959071i	$0.591384\pi$
$998$	0	0
$999$	43829.2	1.38808

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.4.a.x.1.3		3
4.3	odd	2		1216.4.a.q.1.1		3
8.3	odd	2		304.4.a.j.1.3		3
8.5	even	2		152.4.a.b.1.1	✓	3
24.5	odd	2		1368.4.a.e.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.4.a.b.1.1	✓	3	8.5	even	2
304.4.a.j.1.3		3	8.3	odd	2
1216.4.a.q.1.1		3	4.3	odd	2
1216.4.a.x.1.3		3	1.1	even	1	trivial
1368.4.a.e.1.1		3	24.5	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1216.a (trivial)

\(f(q)\)	\(=\)	\(q+10.3081 q^{3} -14.1468 q^{5} -4.06119 q^{7} +79.2568 q^{9} +5.53059 q^{11} +45.6697 q^{13} -145.826 q^{15} -92.8486 q^{17} -19.0000 q^{19} -41.8631 q^{21} -57.0689 q^{23} +75.1314 q^{25} +538.668 q^{27} +290.147 q^{29} +163.494 q^{31} +57.0099 q^{33} +57.4527 q^{35} +81.3659 q^{37} +470.767 q^{39} +73.1997 q^{41} +346.600 q^{43} -1121.23 q^{45} +503.383 q^{47} -326.507 q^{49} -957.092 q^{51} +164.243 q^{53} -78.2401 q^{55} -195.854 q^{57} +763.585 q^{59} -545.832 q^{61} -321.877 q^{63} -646.079 q^{65} +510.800 q^{67} -588.272 q^{69} -294.139 q^{71} -172.903 q^{73} +774.461 q^{75} -22.4608 q^{77} -973.626 q^{79} +3412.70 q^{81} -510.932 q^{83} +1313.51 q^{85} +2990.87 q^{87} +845.158 q^{89} -185.473 q^{91} +1685.31 q^{93} +268.789 q^{95} -863.606 q^{97} +438.337 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 5 q^{3} - 2 q^{5} - 35 q^{7} + 48 q^{9} + 28 q^{11} + 109 q^{13} - 228 q^{15} - 123 q^{17} - 57 q^{19} - 25 q^{21} - 193 q^{23} + 187 q^{25} + 719 q^{27} + 297 q^{29} - 140 q^{31} + 30 q^{33} + 246 q^{35}+ \cdots - 86 q^{99}+O(q^{100}) \) 3 * q + 5 * q^3 - 2 * q^5 - 35 * q^7 + 48 * q^9 + 28 * q^11 + 109 * q^13 - 228 * q^15 - 123 * q^17 - 57 * q^19 - 25 * q^21 - 193 * q^23 + 187 * q^25 + 719 * q^27 + 297 * q^29 - 140 * q^31 + 30 * q^33 + 246 * q^35 - 38 * q^37 + 57 * q^39 + 736 * q^41 + 514 * q^43 - 1046 * q^45 + 134 * q^47 - 42 * q^49 - 929 * q^51 - 311 * q^53 - 130 * q^55 - 95 * q^57 - 199 * q^59 - 56 * q^61 + 508 * q^63 + 1156 * q^65 + 509 * q^67 - 621 * q^69 - 874 * q^71 + 203 * q^73 - 129 * q^75 - 616 * q^77 + 242 * q^79 + 3531 * q^81 + 62 * q^83 + 1430 * q^85 + 3237 * q^87 + 1764 * q^89 + 687 * q^91 + 2592 * q^93 + 38 * q^95 - 2178 * q^97 - 86 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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