LMFDB - Embedded newform 5408.2.a.bh.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5408,2,Mod(1,5408)] 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("5408.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5408, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5408 = 2^{5} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5408.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,-2,0,0,0,2,0,0,0,0,0,0,0,8,0,0,0,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(21)] == traces)

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

Self dual:	yes
Analytic conductor:	$43.1830974131$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.9248.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 2 $ x^4 - 5*x^2 + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 416)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.13578$ of defining polynomial
Character	$\chi$	$=$	5408.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.19935 q^{3} +1.56155 q^{5} -3.07221 q^{7} -1.56155 q^{9} +5.47091 q^{11} -1.87285 q^{15} -2.12311 q^{17} +7.34376 q^{19} +3.68466 q^{21} -3.59806 q^{23} -2.56155 q^{25} +5.47091 q^{27} -5.00000 q^{29} -6.67026 q^{31} -6.56155 q^{33} -4.79741 q^{35} -6.12311 q^{37} +4.12311 q^{41} -0.673500 q^{43} -2.43845 q^{45} +0.525853 q^{47} +2.43845 q^{49} +2.54635 q^{51} +1.56155 q^{53} +8.54312 q^{55} -8.80776 q^{57} +10.2683 q^{59} +5.24621 q^{61} +4.79741 q^{63} +4.94506 q^{67} +4.31534 q^{69} -7.34376 q^{71} -16.6847 q^{73} +3.07221 q^{75} -16.8078 q^{77} +15.7392 q^{79} -1.87689 q^{81} +8.54312 q^{83} -3.31534 q^{85} +5.99676 q^{87} -1.68466 q^{89} +8.00000 q^{93} +11.4677 q^{95} +10.8078 q^{97} -8.54312 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{5} + 2 q^{9} + 8 q^{17} - 10 q^{21} - 2 q^{25} - 20 q^{29} - 18 q^{33} - 8 q^{37} - 18 q^{45} + 18 q^{49} - 2 q^{53} + 6 q^{57} - 12 q^{61} + 42 q^{69} - 42 q^{73} - 26 q^{77} - 24 q^{81} - 38 q^{85}+ \cdots + 2 q^{97}+O(q^{100}) $ 4 * q - 2 * q^5 + 2 * q^9 + 8 * q^17 - 10 * q^21 - 2 * q^25 - 20 * q^29 - 18 * q^33 - 8 * q^37 - 18 * q^45 + 18 * q^49 - 2 * q^53 + 6 * q^57 - 12 * q^61 + 42 * q^69 - 42 * q^73 - 26 * q^77 - 24 * q^81 - 38 * q^85 + 18 * q^89 + 32 * q^93 + 2 * q^97

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
$3$	−1.19935	−0.692447	−0.346223	−	0.938152i	$-0.612536\pi$
$3$	−1.19935	−0.692447	−0.346223	+	0.938152i	$0.612536\pi$
$4$	0	0
$5$	1.56155	0.698348	0.349174	−	0.937058i	$-0.386462\pi$
$5$	1.56155	0.698348	0.349174	+	0.937058i	$0.386462\pi$
$6$	0	0
$7$	−3.07221	−1.16118	−0.580592	−	0.814194i	$-0.697179\pi$
$7$	−3.07221	−1.16118	−0.580592	+	0.814194i	$0.697179\pi$
$8$	0	0
$9$	−1.56155	−0.520518
$10$	0	0
$11$	5.47091	1.64954	0.824771	−	0.565467i	$-0.191304\pi$
$11$	5.47091	1.64954	0.824771	+	0.565467i	$0.191304\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−1.87285	−0.483569
$16$	0	0
$17$	−2.12311	−0.514929	−0.257464	−	0.966288i	$-0.582887\pi$
$17$	−2.12311	−0.514929	−0.257464	+	0.966288i	$0.582887\pi$
$18$	0	0
$19$	7.34376	1.68478	0.842388	−	0.538872i	$-0.181149\pi$
$19$	7.34376	1.68478	0.842388	+	0.538872i	$0.181149\pi$
$20$	0	0
$21$	3.68466	0.804058
$22$	0	0
$23$	−3.59806	−0.750247	−0.375124	−	0.926975i	$-0.622400\pi$
$23$	−3.59806	−0.750247	−0.375124	+	0.926975i	$0.622400\pi$
$24$	0	0
$25$	−2.56155	−0.512311
$26$	0	0
$27$	5.47091	1.05288
$28$	0	0
$29$	−5.00000	−0.928477	−0.464238	−	0.885710i	$-0.653672\pi$
$29$	−5.00000	−0.928477	−0.464238	+	0.885710i	$0.653672\pi$
$30$	0	0
$31$	−6.67026	−1.19801	−0.599007	−	0.800743i	$-0.704438\pi$
$31$	−6.67026	−1.19801	−0.599007	+	0.800743i	$0.704438\pi$
$32$	0	0
$33$	−6.56155	−1.14222
$34$	0	0
$35$	−4.79741	−0.810911
$36$	0	0
$37$	−6.12311	−1.00663	−0.503316	−	0.864102i	$-0.667887\pi$
$37$	−6.12311	−1.00663	−0.503316	+	0.864102i	$0.667887\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	4.12311	0.643921	0.321960	−	0.946753i	$-0.395658\pi$
$41$	4.12311	0.643921	0.321960	+	0.946753i	$0.395658\pi$
$42$	0	0
$43$	−0.673500	−0.102708	−0.0513539	−	0.998681i	$-0.516354\pi$
$43$	−0.673500	−0.102708	−0.0513539	+	0.998681i	$0.516354\pi$
$44$	0	0
$45$	−2.43845	−0.363502
$46$	0	0
$47$	0.525853	0.0767035	0.0383518	−	0.999264i	$-0.487789\pi$
$47$	0.525853	0.0767035	0.0383518	+	0.999264i	$0.487789\pi$
$48$	0	0
$49$	2.43845	0.348350
$50$	0	0
$51$	2.54635	0.356561
$52$	0	0
$53$	1.56155	0.214496	0.107248	−	0.994232i	$-0.465796\pi$
$53$	1.56155	0.214496	0.107248	+	0.994232i	$0.465796\pi$
$54$	0	0
$55$	8.54312	1.15195
$56$	0	0
$57$	−8.80776	−1.16662
$58$	0	0
$59$	10.2683	1.33682	0.668411	−	0.743792i	$-0.266975\pi$
$59$	10.2683	1.33682	0.668411	+	0.743792i	$0.266975\pi$
$60$	0	0
$61$	5.24621	0.671709	0.335854	−	0.941914i	$-0.390975\pi$
$61$	5.24621	0.671709	0.335854	+	0.941914i	$0.390975\pi$
$62$	0	0
$63$	4.79741	0.604417
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.94506	0.604135	0.302068	−	0.953287i	$-0.402323\pi$
$67$	4.94506	0.604135	0.302068	+	0.953287i	$0.402323\pi$
$68$	0	0
$69$	4.31534	0.519506
$70$	0	0
$71$	−7.34376	−0.871544	−0.435772	−	0.900057i	$-0.643525\pi$
$71$	−7.34376	−0.871544	−0.435772	+	0.900057i	$0.643525\pi$
$72$	0	0
$73$	−16.6847	−1.95279	−0.976396	−	0.215989i	$-0.930702\pi$
$73$	−16.6847	−1.95279	−0.976396	+	0.215989i	$0.930702\pi$
$74$	0	0
$75$	3.07221	0.354748
$76$	0	0
$77$	−16.8078	−1.91542
$78$	0	0
$79$	15.7392	1.77080	0.885401	−	0.464828i	$-0.153884\pi$
$79$	15.7392	1.77080	0.885401	+	0.464828i	$0.153884\pi$
$80$	0	0
$81$	−1.87689	−0.208544
$82$	0	0
$83$	8.54312	0.937729	0.468864	−	0.883270i	$-0.344663\pi$
$83$	8.54312	0.937729	0.468864	+	0.883270i	$0.344663\pi$
$84$	0	0
$85$	−3.31534	−0.359599
$86$	0	0
$87$	5.99676	0.642921
$88$	0	0
$89$	−1.68466	−0.178573	−0.0892867	−	0.996006i	$-0.528459\pi$
$89$	−1.68466	−0.178573	−0.0892867	+	0.996006i	$0.528459\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	8.00000	0.829561
$94$	0	0
$95$	11.4677	1.17656
$96$	0	0
$97$	10.8078	1.09736	0.548681	−	0.836032i	$-0.315130\pi$
$97$	10.8078	1.09736	0.548681	+	0.836032i	$0.315130\pi$
$98$	0	0
$99$	−8.54312	−0.858616
$100$	0	0
$101$	8.12311	0.808279	0.404140	−	0.914697i	$-0.367571\pi$
$101$	8.12311	0.808279	0.404140	+	0.914697i	$0.367571\pi$
$102$	0	0
$103$	−10.4160	−1.02632	−0.513158	−	0.858294i	$-0.671525\pi$
$103$	−10.4160	−1.02632	−0.513158	+	0.858294i	$0.671525\pi$
$104$	0	0
$105$	5.75379	0.561512
$106$	0	0
$107$	−4.94506	−0.478057	−0.239028	−	0.971013i	$-0.576829\pi$
$107$	−4.94506	−0.478057	−0.239028	+	0.971013i	$0.576829\pi$
$108$	0	0
$109$	−7.12311	−0.682270	−0.341135	−	0.940014i	$-0.610811\pi$
$109$	−7.12311	−0.682270	−0.341135	+	0.940014i	$0.610811\pi$
$110$	0	0
$111$	7.34376	0.697039
$112$	0	0
$113$	−8.36932	−0.787319	−0.393660	−	0.919256i	$-0.628791\pi$
$113$	−8.36932	−0.787319	−0.393660	+	0.919256i	$0.628791\pi$
$114$	0	0
$115$	−5.61856	−0.523933
$116$	0	0
$117$	0	0
$118$	0	0
$119$	6.52262	0.597927
$120$	0	0
$121$	18.9309	1.72099
$122$	0	0
$123$	−4.94506	−0.445881
$124$	0	0
$125$	−11.8078	−1.05612
$126$	0	0
$127$	−9.74247	−0.864504	−0.432252	−	0.901753i	$-0.642281\pi$
$127$	−9.74247	−0.864504	−0.432252	+	0.901753i	$0.642281\pi$
$128$	0	0
$129$	0.807764	0.0711197
$130$	0	0
$131$	−18.9591	−1.65646	−0.828232	−	0.560386i	$-0.810653\pi$
$131$	−18.9591	−1.65646	−0.828232	+	0.560386i	$0.810653\pi$
$132$	0	0
$133$	−22.5616	−1.95633
$134$	0	0
$135$	8.54312	0.735274
$136$	0	0
$137$	−17.0000	−1.45241	−0.726204	−	0.687479i	$-0.758717\pi$
$137$	−17.0000	−1.45241	−0.726204	+	0.687479i	$0.758717\pi$
$138$	0	0
$139$	4.41921	0.374832	0.187416	−	0.982281i	$-0.439989\pi$
$139$	4.41921	0.374832	0.187416	+	0.982281i	$0.439989\pi$
$140$	0	0
$141$	−0.630683	−0.0531131
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−7.80776	−0.648400
$146$	0	0
$147$	−2.92456	−0.241214
$148$	0	0
$149$	−23.2462	−1.90440	−0.952202	−	0.305469i	$-0.901187\pi$
$149$	−23.2462	−1.90440	−0.952202	+	0.305469i	$0.901187\pi$
$150$	0	0
$151$	14.1617	1.15246	0.576230	−	0.817287i	$-0.304523\pi$
$151$	14.1617	1.15246	0.576230	+	0.817287i	$0.304523\pi$
$152$	0	0
$153$	3.31534	0.268029
$154$	0	0
$155$	−10.4160	−0.836631
$156$	0	0
$157$	−12.6847	−1.01235	−0.506173	−	0.862432i	$-0.668940\pi$
$157$	−12.6847	−1.01235	−0.506173	+	0.862432i	$0.668940\pi$
$158$	0	0
$159$	−1.87285	−0.148527
$160$	0	0
$161$	11.0540	0.871175
$162$	0	0
$163$	−4.41921	−0.346139	−0.173069	−	0.984910i	$-0.555369\pi$
$163$	−4.41921	−0.346139	−0.173069	+	0.984910i	$0.555369\pi$
$164$	0	0
$165$	−10.2462	−0.797666
$166$	0	0
$167$	−10.2683	−0.794587	−0.397293	−	0.917692i	$-0.630050\pi$
$167$	−10.2683	−0.794587	−0.397293	+	0.917692i	$0.630050\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−11.4677	−0.876955
$172$	0	0
$173$	12.5616	0.955037	0.477519	−	0.878622i	$-0.341536\pi$
$173$	12.5616	0.955037	0.477519	+	0.878622i	$0.341536\pi$
$174$	0	0
$175$	7.86962	0.594887
$176$	0	0
$177$	−12.3153	−0.925678
$178$	0	0
$179$	−7.86962	−0.588203	−0.294101	−	0.955774i	$-0.595020\pi$
$179$	−7.86962	−0.588203	−0.294101	+	0.955774i	$0.595020\pi$
$180$	0	0
$181$	−0.192236	−0.0142888	−0.00714439	−	0.999974i	$-0.502274\pi$
$181$	−0.192236	−0.0142888	−0.00714439	+	0.999974i	$0.502274\pi$
$182$	0	0
$183$	−6.29206	−0.465122
$184$	0	0
$185$	−9.56155	−0.702979
$186$	0	0
$187$	−11.6153	−0.849396
$188$	0	0
$189$	−16.8078	−1.22258
$190$	0	0
$191$	−13.4882	−0.975970	−0.487985	−	0.872852i	$-0.662268\pi$
$191$	−13.4882	−0.975970	−0.487985	+	0.872852i	$0.662268\pi$
$192$	0	0
$193$	−10.1231	−0.728677	−0.364339	−	0.931267i	$-0.618705\pi$
$193$	−10.1231	−0.728677	−0.364339	+	0.931267i	$0.618705\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	14.8078	1.05501	0.527505	−	0.849552i	$-0.323128\pi$
$197$	14.8078	1.05501	0.527505	+	0.849552i	$0.323128\pi$
$198$	0	0
$199$	−3.59806	−0.255060	−0.127530	−	0.991835i	$-0.540705\pi$
$199$	−3.59806	−0.255060	−0.127530	+	0.991835i	$0.540705\pi$
$200$	0	0
$201$	−5.93087	−0.418331
$202$	0	0
$203$	15.3610	1.07813
$204$	0	0
$205$	6.43845	0.449681
$206$	0	0
$207$	5.61856	0.390517
$208$	0	0
$209$	40.1771	2.77911
$210$	0	0
$211$	−1.19935	−0.0825669	−0.0412834	−	0.999147i	$-0.513145\pi$
$211$	−1.19935	−0.0825669	−0.0412834	+	0.999147i	$0.513145\pi$
$212$	0	0
$213$	8.80776	0.603498
$214$	0	0
$215$	−1.05171	−0.0717257
$216$	0	0
$217$	20.4924	1.39112
$218$	0	0
$219$	20.0108	1.35220
$220$	0	0
$221$	0	0
$222$	0	0
$223$	13.4882	0.903235	0.451618	−	0.892212i	$-0.350847\pi$
$223$	13.4882	0.903235	0.451618	+	0.892212i	$0.350847\pi$
$224$	0	0
$225$	4.00000	0.266667
$226$	0	0
$227$	−23.0830	−1.53207	−0.766036	−	0.642797i	$-0.777774\pi$
$227$	−23.0830	−1.53207	−0.766036	+	0.642797i	$0.777774\pi$
$228$	0	0
$229$	5.36932	0.354814	0.177407	−	0.984138i	$-0.443229\pi$
$229$	5.36932	0.354814	0.177407	+	0.984138i	$0.443229\pi$
$230$	0	0
$231$	20.1584	1.32633
$232$	0	0
$233$	9.36932	0.613804	0.306902	−	0.951741i	$-0.400708\pi$
$233$	9.36932	0.613804	0.306902	+	0.951741i	$0.400708\pi$
$234$	0	0
$235$	0.821147	0.0535657
$236$	0	0
$237$	−18.8769	−1.22619
$238$	0	0
$239$	−13.3405	−0.862927	−0.431464	−	0.902130i	$-0.642003\pi$
$239$	−13.3405	−0.862927	−0.431464	+	0.902130i	$0.642003\pi$
$240$	0	0
$241$	−26.1231	−1.68274	−0.841369	−	0.540462i	$-0.818250\pi$
$241$	−26.1231	−1.68274	−0.841369	+	0.540462i	$0.818250\pi$
$242$	0	0
$243$	−14.1617	−0.908472
$244$	0	0
$245$	3.80776	0.243269
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−10.2462	−0.649327
$250$	0	0
$251$	9.74247	0.614939	0.307470	−	0.951558i	$-0.400518\pi$
$251$	9.74247	0.614939	0.307470	+	0.951558i	$0.400518\pi$
$252$	0	0
$253$	−19.6847	−1.23756
$254$	0	0
$255$	3.97626	0.249003
$256$	0	0
$257$	4.12311	0.257192	0.128596	−	0.991697i	$-0.458953\pi$
$257$	4.12311	0.257192	0.128596	+	0.991697i	$0.458953\pi$
$258$	0	0
$259$	18.8114	1.16889
$260$	0	0
$261$	7.80776	0.483288
$262$	0	0
$263$	−31.1003	−1.91772	−0.958862	−	0.283872i	$-0.908381\pi$
$263$	−31.1003	−1.91772	−0.958862	+	0.283872i	$0.908381\pi$
$264$	0	0
$265$	2.43845	0.149793
$266$	0	0
$267$	2.02050	0.123653
$268$	0	0
$269$	−6.31534	−0.385053	−0.192527	−	0.981292i	$-0.561668\pi$
$269$	−6.31534	−0.385053	−0.192527	+	0.981292i	$0.561668\pi$
$270$	0	0
$271$	−10.2683	−0.623756	−0.311878	−	0.950122i	$-0.600958\pi$
$271$	−10.2683	−0.623756	−0.311878	+	0.950122i	$0.600958\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−14.0140	−0.845078
$276$	0	0
$277$	26.3693	1.58438	0.792189	−	0.610275i	$-0.208941\pi$
$277$	26.3693	1.58438	0.792189	+	0.610275i	$0.208941\pi$
$278$	0	0
$279$	10.4160	0.623588
$280$	0	0
$281$	−24.6847	−1.47256	−0.736282	−	0.676675i	$-0.763420\pi$
$281$	−24.6847	−1.47256	−0.736282	+	0.676675i	$0.763420\pi$
$282$	0	0
$283$	−2.02050	−0.120106	−0.0600531	−	0.998195i	$-0.519127\pi$
$283$	−2.02050	−0.120106	−0.0600531	+	0.998195i	$0.519127\pi$
$284$	0	0
$285$	−13.7538	−0.814704
$286$	0	0
$287$	−12.6670	−0.747711
$288$	0	0
$289$	−12.4924	−0.734848
$290$	0	0
$291$	−12.9623	−0.759865
$292$	0	0
$293$	−1.00000	−0.0584206	−0.0292103	−	0.999573i	$-0.509299\pi$
$293$	−1.00000	−0.0584206	−0.0292103	+	0.999573i	$0.509299\pi$
$294$	0	0
$295$	16.0345	0.933566
$296$	0	0
$297$	29.9309	1.73677
$298$	0	0
$299$	0	0
$300$	0	0
$301$	2.06913	0.119263
$302$	0	0
$303$	−9.74247	−0.559690
$304$	0	0
$305$	8.19224	0.469086
$306$	0	0
$307$	−24.8082	−1.41588	−0.707939	−	0.706273i	$-0.750375\pi$
$307$	−24.8082	−1.41588	−0.707939	+	0.706273i	$0.750375\pi$
$308$	0	0
$309$	12.4924	0.710669
$310$	0	0
$311$	20.0108	1.13471	0.567354	−	0.823474i	$-0.307967\pi$
$311$	20.0108	1.13471	0.567354	+	0.823474i	$0.307967\pi$
$312$	0	0
$313$	4.87689	0.275658	0.137829	−	0.990456i	$-0.455988\pi$
$313$	4.87689	0.275658	0.137829	+	0.990456i	$0.455988\pi$
$314$	0	0
$315$	7.49141	0.422093
$316$	0	0
$317$	−17.8078	−1.00018	−0.500092	−	0.865972i	$-0.666700\pi$
$317$	−17.8078	−1.00018	−0.500092	+	0.865972i	$0.666700\pi$
$318$	0	0
$319$	−27.3546	−1.53156
$320$	0	0
$321$	5.93087	0.331029
$322$	0	0
$323$	−15.5916	−0.867539
$324$	0	0
$325$	0	0
$326$	0	0
$327$	8.54312	0.472435
$328$	0	0
$329$	−1.61553	−0.0890669
$330$	0	0
$331$	12.6670	0.696243	0.348121	−	0.937449i	$-0.386820\pi$
$331$	12.6670	0.696243	0.348121	+	0.937449i	$0.386820\pi$
$332$	0	0
$333$	9.56155	0.523970
$334$	0	0
$335$	7.72197	0.421896
$336$	0	0
$337$	−27.4233	−1.49384	−0.746921	−	0.664913i	$-0.768469\pi$
$337$	−27.4233	−1.49384	−0.746921	+	0.664913i	$0.768469\pi$
$338$	0	0
$339$	10.0378	0.545176
$340$	0	0
$341$	−36.4924	−1.97618
$342$	0	0
$343$	14.0140	0.756686
$344$	0	0
$345$	6.73863	0.362796
$346$	0	0
$347$	−7.34376	−0.394234	−0.197117	−	0.980380i	$-0.563158\pi$
$347$	−7.34376	−0.394234	−0.197117	+	0.980380i	$0.563158\pi$
$348$	0	0
$349$	−32.5616	−1.74298	−0.871490	−	0.490413i	$-0.836846\pi$
$349$	−32.5616	−1.74298	−0.871490	+	0.490413i	$0.836846\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	19.4924	1.03748	0.518738	−	0.854933i	$-0.326402\pi$
$353$	19.4924	1.03748	0.518738	+	0.854933i	$0.326402\pi$
$354$	0	0
$355$	−11.4677	−0.608641
$356$	0	0
$357$	−7.82292	−0.414033
$358$	0	0
$359$	−30.4268	−1.60586	−0.802932	−	0.596071i	$-0.796728\pi$
$359$	−30.4268	−1.60586	−0.802932	+	0.596071i	$0.796728\pi$
$360$	0	0
$361$	34.9309	1.83847
$362$	0	0
$363$	−22.7048	−1.19169
$364$	0	0
$365$	−26.0540	−1.36373
$366$	0	0
$367$	23.0830	1.20492	0.602461	−	0.798148i	$-0.294187\pi$
$367$	23.0830	1.20492	0.602461	+	0.798148i	$0.294187\pi$
$368$	0	0
$369$	−6.43845	−0.335172
$370$	0	0
$371$	−4.79741	−0.249069
$372$	0	0
$373$	−0.369317	−0.0191225	−0.00956125	−	0.999954i	$-0.503043\pi$
$373$	−0.369317	−0.0191225	−0.00956125	+	0.999954i	$0.503043\pi$
$374$	0	0
$375$	14.1617	0.731306
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−1.19935	−0.0616066	−0.0308033	−	0.999525i	$-0.509807\pi$
$379$	−1.19935	−0.0616066	−0.0308033	+	0.999525i	$0.509807\pi$
$380$	0	0
$381$	11.6847	0.598623
$382$	0	0
$383$	17.7597	0.907480	0.453740	−	0.891134i	$-0.350089\pi$
$383$	17.7597	0.907480	0.453740	+	0.891134i	$0.350089\pi$
$384$	0	0
$385$	−26.2462	−1.33763
$386$	0	0
$387$	1.05171	0.0534612
$388$	0	0
$389$	−13.8078	−0.700081	−0.350041	−	0.936734i	$-0.613832\pi$
$389$	−13.8078	−0.700081	−0.350041	+	0.936734i	$0.613832\pi$
$390$	0	0
$391$	7.63906	0.386324
$392$	0	0
$393$	22.7386	1.14701
$394$	0	0
$395$	24.5776	1.23664
$396$	0	0
$397$	−6.31534	−0.316958	−0.158479	−	0.987362i	$-0.550659\pi$
$397$	−6.31534	−0.316958	−0.158479	+	0.987362i	$0.550659\pi$
$398$	0	0
$399$	27.0593	1.35466
$400$	0	0
$401$	10.3693	0.517819	0.258909	−	0.965902i	$-0.416637\pi$
$401$	10.3693	0.517819	0.258909	+	0.965902i	$0.416637\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−2.93087	−0.145636
$406$	0	0
$407$	−33.4990	−1.66048
$408$	0	0
$409$	4.12311	0.203874	0.101937	−	0.994791i	$-0.467496\pi$
$409$	4.12311	0.203874	0.101937	+	0.994791i	$0.467496\pi$
$410$	0	0
$411$	20.3890	1.00572
$412$	0	0
$413$	−31.5464	−1.55230
$414$	0	0
$415$	13.3405	0.654861
$416$	0	0
$417$	−5.30019	−0.259551
$418$	0	0
$419$	25.7770	1.25929	0.629644	−	0.776883i	$-0.283201\pi$
$419$	25.7770	1.25929	0.629644	+	0.776883i	$0.283201\pi$
$420$	0	0
$421$	3.31534	0.161580	0.0807899	−	0.996731i	$-0.474256\pi$
$421$	3.31534	0.161580	0.0807899	+	0.996731i	$0.474256\pi$
$422$	0	0
$423$	−0.821147	−0.0399255
$424$	0	0
$425$	5.43845	0.263803
$426$	0	0
$427$	−16.1174	−0.779978
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−26.0076	−1.25274	−0.626370	−	0.779526i	$-0.715460\pi$
$431$	−26.0076	−1.25274	−0.626370	+	0.779526i	$0.715460\pi$
$432$	0	0
$433$	−13.6307	−0.655049	−0.327524	−	0.944843i	$-0.606214\pi$
$433$	−13.6307	−0.655049	−0.327524	+	0.944843i	$0.606214\pi$
$434$	0	0
$435$	9.36426	0.448982
$436$	0	0
$437$	−26.4233	−1.26400
$438$	0	0
$439$	−12.6670	−0.604564	−0.302282	−	0.953219i	$-0.597748\pi$
$439$	−12.6670	−0.604564	−0.302282	+	0.953219i	$0.597748\pi$
$440$	0	0
$441$	−3.80776	−0.181322
$442$	0	0
$443$	−8.54312	−0.405896	−0.202948	−	0.979190i	$-0.565052\pi$
$443$	−8.54312	−0.405896	−0.202948	+	0.979190i	$0.565052\pi$
$444$	0	0
$445$	−2.63068	−0.124706
$446$	0	0
$447$	27.8804	1.31870
$448$	0	0
$449$	−12.5616	−0.592816	−0.296408	−	0.955061i	$-0.595789\pi$
$449$	−12.5616	−0.592816	−0.296408	+	0.955061i	$0.595789\pi$
$450$	0	0
$451$	22.5571	1.06217
$452$	0	0
$453$	−16.9848	−0.798018
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−4.50758	−0.210856	−0.105428	−	0.994427i	$-0.533621\pi$
$457$	−4.50758	−0.210856	−0.105428	+	0.994427i	$0.533621\pi$
$458$	0	0
$459$	−11.6153	−0.542157
$460$	0	0
$461$	−2.12311	−0.0988829	−0.0494414	−	0.998777i	$-0.515744\pi$
$461$	−2.12311	−0.0988829	−0.0494414	+	0.998777i	$0.515744\pi$
$462$	0	0
$463$	−13.3405	−0.619987	−0.309993	−	0.950739i	$-0.600327\pi$
$463$	−13.3405	−0.619987	−0.309993	+	0.950739i	$0.600327\pi$
$464$	0	0
$465$	12.4924	0.579322
$466$	0	0
$467$	−15.7392	−0.728325	−0.364162	−	0.931335i	$-0.618645\pi$
$467$	−15.7392	−0.728325	−0.364162	+	0.931335i	$0.618645\pi$
$468$	0	0
$469$	−15.1922	−0.701512
$470$	0	0
$471$	15.2134	0.700996
$472$	0	0
$473$	−3.68466	−0.169421
$474$	0	0
$475$	−18.8114	−0.863128
$476$	0	0
$477$	−2.43845	−0.111649
$478$	0	0
$479$	−5.99676	−0.273999	−0.137000	−	0.990571i	$-0.543746\pi$
$479$	−5.99676	−0.273999	−0.137000	+	0.990571i	$0.543746\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−13.2576	−0.603242
$484$	0	0
$485$	16.8769	0.766340
$486$	0	0
$487$	−33.2037	−1.50460	−0.752301	−	0.658820i	$-0.771056\pi$
$487$	−33.2037	−1.50460	−0.752301	+	0.658820i	$0.771056\pi$
$488$	0	0
$489$	5.30019	0.239683
$490$	0	0
$491$	42.0421	1.89733	0.948666	−	0.316279i	$-0.102433\pi$
$491$	42.0421	1.89733	0.948666	+	0.316279i	$0.102433\pi$
$492$	0	0
$493$	10.6155	0.478099
$494$	0	0
$495$	−13.3405	−0.599612
$496$	0	0
$497$	22.5616	1.01202
$498$	0	0
$499$	21.0625	0.942887	0.471443	−	0.881896i	$-0.343733\pi$
$499$	21.0625	0.942887	0.471443	+	0.881896i	$0.343733\pi$
$500$	0	0
$501$	12.3153	0.550209
$502$	0	0
$503$	8.16491	0.364055	0.182028	−	0.983293i	$-0.441734\pi$
$503$	8.16491	0.364055	0.182028	+	0.983293i	$0.441734\pi$
$504$	0	0
$505$	12.6847	0.564460
$506$	0	0
$507$	0	0
$508$	0	0
$509$	14.3693	0.636909	0.318454	−	0.947938i	$-0.396836\pi$
$509$	14.3693	0.636909	0.318454	+	0.947938i	$0.396836\pi$
$510$	0	0
$511$	51.2587	2.26755
$512$	0	0
$513$	40.1771	1.77386
$514$	0	0
$515$	−16.2651	−0.716725
$516$	0	0
$517$	2.87689	0.126526
$518$	0	0
$519$	−15.0657	−0.661312
$520$	0	0
$521$	37.5616	1.64560	0.822801	−	0.568330i	$-0.192410\pi$
$521$	37.5616	1.64560	0.822801	+	0.568330i	$0.192410\pi$
$522$	0	0
$523$	0.378206	0.0165378	0.00826889	−	0.999966i	$-0.497368\pi$
$523$	0.378206	0.0165378	0.00826889	+	0.999966i	$0.497368\pi$
$524$	0	0
$525$	−9.43845	−0.411928
$526$	0	0
$527$	14.1617	0.616892
$528$	0	0
$529$	−10.0540	−0.437129
$530$	0	0
$531$	−16.0345	−0.695839
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−7.72197	−0.333850
$536$	0	0
$537$	9.43845	0.407299
$538$	0	0
$539$	13.3405	0.574617
$540$	0	0
$541$	35.1771	1.51238	0.756190	−	0.654352i	$-0.227058\pi$
$541$	35.1771	1.51238	0.756190	+	0.654352i	$0.227058\pi$
$542$	0	0
$543$	0.230559	0.00989422
$544$	0	0
$545$	−11.1231	−0.476461
$546$	0	0
$547$	24.8082	1.06072	0.530361	−	0.847772i	$-0.322056\pi$
$547$	24.8082	1.06072	0.530361	+	0.847772i	$0.322056\pi$
$548$	0	0
$549$	−8.19224	−0.349636
$550$	0	0
$551$	−36.7188	−1.56427
$552$	0	0
$553$	−48.3542	−2.05623
$554$	0	0
$555$	11.4677	0.486776
$556$	0	0
$557$	−29.9848	−1.27050	−0.635249	−	0.772307i	$-0.719103\pi$
$557$	−29.9848	−1.27050	−0.635249	+	0.772307i	$0.719103\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	13.9309	0.588162
$562$	0	0
$563$	4.41921	0.186247	0.0931237	−	0.995655i	$-0.470315\pi$
$563$	4.41921	0.186247	0.0931237	+	0.995655i	$0.470315\pi$
$564$	0	0
$565$	−13.0691	−0.549822
$566$	0	0
$567$	5.76621	0.242158
$568$	0	0
$569$	−42.6695	−1.78880	−0.894399	−	0.447269i	$-0.852397\pi$
$569$	−42.6695	−1.78880	−0.894399	+	0.447269i	$0.852397\pi$
$570$	0	0
$571$	42.7156	1.78759	0.893796	−	0.448474i	$-0.148032\pi$
$571$	42.7156	1.78759	0.893796	+	0.448474i	$0.148032\pi$
$572$	0	0
$573$	16.1771	0.675807
$574$	0	0
$575$	9.21662	0.384359
$576$	0	0
$577$	−14.9309	−0.621580	−0.310790	−	0.950479i	$-0.600594\pi$
$577$	−14.9309	−0.621580	−0.310790	+	0.950479i	$0.600594\pi$
$578$	0	0
$579$	12.1412	0.504570
$580$	0	0
$581$	−26.2462	−1.08888
$582$	0	0
$583$	8.54312	0.353820
$584$	0	0
$585$	0	0
$586$	0	0
$587$	22.8524	0.943221	0.471611	−	0.881807i	$-0.343673\pi$
$587$	22.8524	0.943221	0.471611	+	0.881807i	$0.343673\pi$
$588$	0	0
$589$	−48.9848	−2.01839
$590$	0	0
$591$	−17.7597	−0.730538
$592$	0	0
$593$	26.0540	1.06991	0.534954	−	0.844881i	$-0.320329\pi$
$593$	26.0540	1.06991	0.534954	+	0.844881i	$0.320329\pi$
$594$	0	0
$595$	10.1854	0.417561
$596$	0	0
$597$	4.31534	0.176615
$598$	0	0
$599$	4.27156	0.174531	0.0872656	−	0.996185i	$-0.472187\pi$
$599$	4.27156	0.174531	0.0872656	+	0.996185i	$0.472187\pi$
$600$	0	0
$601$	−2.12311	−0.0866033	−0.0433016	−	0.999062i	$-0.513788\pi$
$601$	−2.12311	−0.0866033	−0.0433016	+	0.999062i	$0.513788\pi$
$602$	0	0
$603$	−7.72197	−0.314463
$604$	0	0
$605$	29.5616	1.20185
$606$	0	0
$607$	−43.6196	−1.77047	−0.885233	−	0.465147i	$-0.846001\pi$
$607$	−43.6196	−1.77047	−0.885233	+	0.465147i	$0.846001\pi$
$608$	0	0
$609$	−18.4233	−0.746549
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−1.63068	−0.0658627	−0.0329313	−	0.999458i	$-0.510484\pi$
$613$	−1.63068	−0.0658627	−0.0329313	+	0.999458i	$0.510484\pi$
$614$	0	0
$615$	−7.72197	−0.311380
$616$	0	0
$617$	−7.24621	−0.291721	−0.145861	−	0.989305i	$-0.546595\pi$
$617$	−7.24621	−0.291721	−0.145861	+	0.989305i	$0.546595\pi$
$618$	0	0
$619$	8.54312	0.343377	0.171688	−	0.985151i	$-0.445078\pi$
$619$	8.54312	0.343377	0.171688	+	0.985151i	$0.445078\pi$
$620$	0	0
$621$	−19.6847	−0.789918
$622$	0	0
$623$	5.17562	0.207357
$624$	0	0
$625$	−5.63068	−0.225227
$626$	0	0
$627$	−48.1865	−1.92438
$628$	0	0
$629$	13.0000	0.518344
$630$	0	0
$631$	23.9041	0.951609	0.475804	−	0.879551i	$-0.342157\pi$
$631$	23.9041	0.951609	0.475804	+	0.879551i	$0.342157\pi$
$632$	0	0
$633$	1.43845	0.0571731
$634$	0	0
$635$	−15.2134	−0.603725
$636$	0	0
$637$	0	0
$638$	0	0
$639$	11.4677	0.453654
$640$	0	0
$641$	−30.6155	−1.20924	−0.604620	−	0.796514i	$-0.706675\pi$
$641$	−30.6155	−1.20924	−0.604620	+	0.796514i	$0.706675\pi$
$642$	0	0
$643$	−8.16491	−0.321993	−0.160996	−	0.986955i	$-0.551471\pi$
$643$	−8.16491	−0.321993	−0.160996	+	0.986955i	$0.551471\pi$
$644$	0	0
$645$	1.26137	0.0496662
$646$	0	0
$647$	−25.1864	−0.990180	−0.495090	−	0.868842i	$-0.664865\pi$
$647$	−25.1864	−0.990180	−0.495090	+	0.868842i	$0.664865\pi$
$648$	0	0
$649$	56.1771	2.20514
$650$	0	0
$651$	−24.5776	−0.963274
$652$	0	0
$653$	−0.561553	−0.0219753	−0.0109876	−	0.999940i	$-0.503498\pi$
$653$	−0.561553	−0.0219753	−0.0109876	+	0.999940i	$0.503498\pi$
$654$	0	0
$655$	−29.6056	−1.15679
$656$	0	0
$657$	26.0540	1.01646
$658$	0	0
$659$	−25.2511	−0.983645	−0.491822	−	0.870696i	$-0.663669\pi$
$659$	−25.2511	−0.983645	−0.491822	+	0.870696i	$0.663669\pi$
$660$	0	0
$661$	−15.8769	−0.617540	−0.308770	−	0.951137i	$-0.599917\pi$
$661$	−15.8769	−0.617540	−0.308770	+	0.951137i	$0.599917\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−35.2311	−1.36620
$666$	0	0
$667$	17.9903	0.696587
$668$	0	0
$669$	−16.1771	−0.625442
$670$	0	0
$671$	28.7016	1.10801
$672$	0	0
$673$	25.7386	0.992151	0.496076	−	0.868279i	$-0.334774\pi$
$673$	25.7386	0.992151	0.496076	+	0.868279i	$0.334774\pi$
$674$	0	0
$675$	−14.0140	−0.539400
$676$	0	0
$677$	−23.1231	−0.888693	−0.444347	−	0.895855i	$-0.646564\pi$
$677$	−23.1231	−0.888693	−0.444347	+	0.895855i	$0.646564\pi$
$678$	0	0
$679$	−33.2037	−1.27424
$680$	0	0
$681$	27.6847	1.06088
$682$	0	0
$683$	−23.0830	−0.883246	−0.441623	−	0.897201i	$-0.645597\pi$
$683$	−23.0830	−0.883246	−0.441623	+	0.897201i	$0.645597\pi$
$684$	0	0
$685$	−26.5464	−1.01429
$686$	0	0
$687$	−6.43971	−0.245690
$688$	0	0
$689$	0	0
$690$	0	0
$691$	49.2382	1.87311	0.936555	−	0.350521i	$-0.113996\pi$
$691$	49.2382	1.87311	0.936555	+	0.350521i	$0.113996\pi$
$692$	0	0
$693$	26.2462	0.997011
$694$	0	0
$695$	6.90082	0.261763
$696$	0	0
$697$	−8.75379	−0.331573
$698$	0	0
$699$	−11.2371	−0.425027
$700$	0	0
$701$	−23.1231	−0.873348	−0.436674	−	0.899620i	$-0.643844\pi$
$701$	−23.1231	−0.873348	−0.436674	+	0.899620i	$0.643844\pi$
$702$	0	0
$703$	−44.9666	−1.69595
$704$	0	0
$705$	−0.984845	−0.0370914
$706$	0	0
$707$	−24.9559	−0.938561
$708$	0	0
$709$	−11.8769	−0.446046	−0.223023	−	0.974813i	$-0.571593\pi$
$709$	−11.8769	−0.446046	−0.223023	+	0.974813i	$0.571593\pi$
$710$	0	0
$711$	−24.5776	−0.921734
$712$	0	0
$713$	24.0000	0.898807
$714$	0	0
$715$	0	0
$716$	0	0
$717$	16.0000	0.597531
$718$	0	0
$719$	−37.2447	−1.38899	−0.694496	−	0.719497i	$-0.744373\pi$
$719$	−37.2447	−1.38899	−0.694496	+	0.719497i	$0.744373\pi$
$720$	0	0
$721$	32.0000	1.19174
$722$	0	0
$723$	31.3308	1.16521
$724$	0	0
$725$	12.8078	0.475668
$726$	0	0
$727$	−22.9354	−0.850625	−0.425313	−	0.905047i	$-0.639836\pi$
$727$	−22.9354	−0.850625	−0.425313	+	0.905047i	$0.639836\pi$
$728$	0	0
$729$	22.6155	0.837612
$730$	0	0
$731$	1.42991	0.0528872
$732$	0	0
$733$	16.9309	0.625356	0.312678	−	0.949859i	$-0.398774\pi$
$733$	16.9309	0.625356	0.312678	+	0.949859i	$0.398774\pi$
$734$	0	0
$735$	−4.56685	−0.168451
$736$	0	0
$737$	27.0540	0.996546
$738$	0	0
$739$	38.2964	1.40876	0.704378	−	0.709826i	$-0.251226\pi$
$739$	38.2964	1.40876	0.704378	+	0.709826i	$0.251226\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−10.5636	−0.387542	−0.193771	−	0.981047i	$-0.562072\pi$
$743$	−10.5636	−0.387542	−0.193771	+	0.981047i	$0.562072\pi$
$744$	0	0
$745$	−36.3002	−1.32994
$746$	0	0
$747$	−13.3405	−0.488104
$748$	0	0
$749$	15.1922	0.555112
$750$	0	0
$751$	26.0076	0.949029	0.474515	−	0.880248i	$-0.342624\pi$
$751$	26.0076	0.949029	0.474515	+	0.880248i	$0.342624\pi$
$752$	0	0
$753$	−11.6847	−0.425813
$754$	0	0
$755$	22.1142	0.804818
$756$	0	0
$757$	48.4233	1.75997	0.879987	−	0.474997i	$-0.157551\pi$
$757$	48.4233	1.75997	0.879987	+	0.474997i	$0.157551\pi$
$758$	0	0
$759$	23.6089	0.856947
$760$	0	0
$761$	31.9309	1.15749	0.578747	−	0.815507i	$-0.303542\pi$
$761$	31.9309	1.15749	0.578747	+	0.815507i	$0.303542\pi$
$762$	0	0
$763$	21.8836	0.792241
$764$	0	0
$765$	5.17708	0.187178
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−33.0540	−1.19196	−0.595978	−	0.803000i	$-0.703236\pi$
$769$	−33.0540	−1.19196	−0.595978	+	0.803000i	$0.703236\pi$
$770$	0	0
$771$	−4.94506	−0.178092
$772$	0	0
$773$	−6.31534	−0.227147	−0.113574	−	0.993530i	$-0.536230\pi$
$773$	−6.31534	−0.227147	−0.113574	+	0.993530i	$0.536230\pi$
$774$	0	0
$775$	17.0862	0.613756
$776$	0	0
$777$	−22.5616	−0.809391
$778$	0	0
$779$	30.2791	1.08486
$780$	0	0
$781$	−40.1771	−1.43765
$782$	0	0
$783$	−27.3546	−0.977572
$784$	0	0
$785$	−19.8078	−0.706969
$786$	0	0
$787$	30.5744	1.08986	0.544930	−	0.838482i	$-0.316556\pi$
$787$	30.5744	1.08986	0.544930	+	0.838482i	$0.316556\pi$
$788$	0	0
$789$	37.3002	1.32792
$790$	0	0
$791$	25.7123	0.914223
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−2.92456	−0.103723
$796$	0	0
$797$	−14.9460	−0.529415	−0.264708	−	0.964329i	$-0.585275\pi$
$797$	−14.9460	−0.529415	−0.264708	+	0.964329i	$0.585275\pi$
$798$	0	0
$799$	−1.11644	−0.0394968
$800$	0	0
$801$	2.63068	0.0929506
$802$	0	0
$803$	−91.2803	−3.22121
$804$	0	0
$805$	17.2614	0.608383
$806$	0	0
$807$	7.57432	0.266629
$808$	0	0
$809$	7.63068	0.268281	0.134140	−	0.990962i	$-0.457173\pi$
$809$	7.63068	0.268281	0.134140	+	0.990962i	$0.457173\pi$
$810$	0	0
$811$	−22.4095	−0.786904	−0.393452	−	0.919345i	$-0.628719\pi$
$811$	−22.4095	−0.786904	−0.393452	+	0.919345i	$0.628719\pi$
$812$	0	0
$813$	12.3153	0.431918
$814$	0	0
$815$	−6.90082	−0.241725
$816$	0	0
$817$	−4.94602	−0.173039
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1.19224	−0.0416093	−0.0208047	−	0.999784i	$-0.506623\pi$
$821$	−1.19224	−0.0416093	−0.0208047	+	0.999784i	$0.506623\pi$
$822$	0	0
$823$	16.4127	0.572112	0.286056	−	0.958213i	$-0.407656\pi$
$823$	16.4127	0.572112	0.286056	+	0.958213i	$0.407656\pi$
$824$	0	0
$825$	16.8078	0.585171
$826$	0	0
$827$	−19.7802	−0.687826	−0.343913	−	0.939001i	$-0.611753\pi$
$827$	−19.7802	−0.687826	−0.343913	+	0.939001i	$0.611753\pi$
$828$	0	0
$829$	19.9848	0.694102	0.347051	−	0.937846i	$-0.387183\pi$
$829$	19.9848	0.694102	0.347051	+	0.937846i	$0.387183\pi$
$830$	0	0
$831$	−31.6261	−1.09710
$832$	0	0
$833$	−5.17708	−0.179375
$834$	0	0
$835$	−16.0345	−0.554898
$836$	0	0
$837$	−36.4924	−1.26136
$838$	0	0
$839$	11.8459	0.408965	0.204483	−	0.978870i	$-0.434449\pi$
$839$	11.8459	0.408965	0.204483	+	0.978870i	$0.434449\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	0	0
$843$	29.6056	1.01967
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−58.1595	−1.99838
$848$	0	0
$849$	2.42329	0.0831672
$850$	0	0
$851$	22.0313	0.755223
$852$	0	0
$853$	14.0540	0.481199	0.240599	−	0.970624i	$-0.422656\pi$
$853$	14.0540	0.481199	0.240599	+	0.970624i	$0.422656\pi$
$854$	0	0
$855$	−17.9074	−0.612419
$856$	0	0
$857$	21.5616	0.736529	0.368264	−	0.929721i	$-0.379952\pi$
$857$	21.5616	0.736529	0.368264	+	0.929721i	$0.379952\pi$
$858$	0	0
$859$	−22.7048	−0.774678	−0.387339	−	0.921937i	$-0.626606\pi$
$859$	−22.7048	−0.774678	−0.387339	+	0.921937i	$0.626606\pi$
$860$	0	0
$861$	15.1922	0.517750
$862$	0	0
$863$	34.1725	1.16324	0.581622	−	0.813459i	$-0.302418\pi$
$863$	34.1725	1.16324	0.581622	+	0.813459i	$0.302418\pi$
$864$	0	0
$865$	19.6155	0.666948
$866$	0	0
$867$	14.9828	0.508843
$868$	0	0
$869$	86.1080	2.92101
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−16.8769	−0.571196
$874$	0	0
$875$	36.2759	1.22635
$876$	0	0
$877$	−7.24621	−0.244687	−0.122344	−	0.992488i	$-0.539041\pi$
$877$	−7.24621	−0.244687	−0.122344	+	0.992488i	$0.539041\pi$
$878$	0	0
$879$	1.19935	0.0404532
$880$	0	0
$881$	−28.8617	−0.972377	−0.486188	−	0.873854i	$-0.661613\pi$
$881$	−28.8617	−0.972377	−0.486188	+	0.873854i	$0.661613\pi$
$882$	0	0
$883$	−36.0453	−1.21302	−0.606511	−	0.795075i	$-0.707431\pi$
$883$	−36.0453	−1.21302	−0.606511	+	0.795075i	$0.707431\pi$
$884$	0	0
$885$	−19.2311	−0.646445
$886$	0	0
$887$	33.2037	1.11487	0.557435	−	0.830221i	$-0.311786\pi$
$887$	33.2037	1.11487	0.557435	+	0.830221i	$0.311786\pi$
$888$	0	0
$889$	29.9309	1.00385
$890$	0	0
$891$	−10.2683	−0.344002
$892$	0	0
$893$	3.86174	0.129228
$894$	0	0
$895$	−12.2888	−0.410770
$896$	0	0
$897$	0	0
$898$	0	0
$899$	33.3513	1.11233
$900$	0	0
$901$	−3.31534	−0.110450
$902$	0	0
$903$	−2.48162	−0.0825831
$904$	0	0
$905$	−0.300187	−0.00997854
$906$	0	0
$907$	−17.4644	−0.579897	−0.289949	−	0.957042i	$-0.593638\pi$
$907$	−17.4644	−0.579897	−0.289949	+	0.957042i	$0.593638\pi$
$908$	0	0
$909$	−12.6847	−0.420724
$910$	0	0
$911$	51.2587	1.69828	0.849138	−	0.528171i	$-0.177122\pi$
$911$	51.2587	1.69828	0.849138	+	0.528171i	$0.177122\pi$
$912$	0	0
$913$	46.7386	1.54682
$914$	0	0
$915$	−9.82538	−0.324817
$916$	0	0
$917$	58.2462	1.92346
$918$	0	0
$919$	−53.5098	−1.76512	−0.882562	−	0.470196i	$-0.844183\pi$
$919$	−53.5098	−1.76512	−0.882562	+	0.470196i	$0.844183\pi$
$920$	0	0
$921$	29.7538	0.980421
$922$	0	0
$923$	0	0
$924$	0	0
$925$	15.6847	0.515708
$926$	0	0
$927$	16.2651	0.534216
$928$	0	0
$929$	21.8769	0.717758	0.358879	−	0.933384i	$-0.383159\pi$
$929$	21.8769	0.717758	0.358879	+	0.933384i	$0.383159\pi$
$930$	0	0
$931$	17.9074	0.586891
$932$	0	0
$933$	−24.0000	−0.785725
$934$	0	0
$935$	−18.1379	−0.593174
$936$	0	0
$937$	−28.1922	−0.921000	−0.460500	−	0.887660i	$-0.652330\pi$
$937$	−28.1922	−0.921000	−0.460500	+	0.887660i	$0.652330\pi$
$938$	0	0
$939$	−5.84912	−0.190879
$940$	0	0
$941$	−2.00000	−0.0651981	−0.0325991	−	0.999469i	$-0.510378\pi$
$941$	−2.00000	−0.0651981	−0.0325991	+	0.999469i	$0.510378\pi$
$942$	0	0
$943$	−14.8352	−0.483100
$944$	0	0
$945$	−26.2462	−0.853789
$946$	0	0
$947$	23.3783	0.759692	0.379846	−	0.925050i	$-0.375977\pi$
$947$	23.3783	0.759692	0.379846	+	0.925050i	$0.375977\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	21.3578	0.692574
$952$	0	0
$953$	51.7926	1.67773	0.838864	−	0.544342i	$-0.183220\pi$
$953$	51.7926	1.67773	0.838864	+	0.544342i	$0.183220\pi$
$954$	0	0
$955$	−21.0625	−0.681566
$956$	0	0
$957$	32.8078	1.06052
$958$	0	0
$959$	52.2275	1.68651
$960$	0	0
$961$	13.4924	0.435239
$962$	0	0
$963$	7.72197	0.248837
$964$	0	0
$965$	−15.8078	−0.508870
$966$	0	0
$967$	10.4160	0.334955	0.167478	−	0.985876i	$-0.446438\pi$
$967$	10.4160	0.334955	0.167478	+	0.985876i	$0.446438\pi$
$968$	0	0
$969$	18.6998	0.600725
$970$	0	0
$971$	19.3373	0.620563	0.310282	−	0.950645i	$-0.399577\pi$
$971$	19.3373	0.620563	0.310282	+	0.950645i	$0.399577\pi$
$972$	0	0
$973$	−13.5767	−0.435249
$974$	0	0
$975$	0	0
$976$	0	0
$977$	33.7386	1.07939	0.539697	−	0.841859i	$-0.318539\pi$
$977$	33.7386	1.07939	0.539697	+	0.841859i	$0.318539\pi$
$978$	0	0
$979$	−9.21662	−0.294564
$980$	0	0
$981$	11.1231	0.355133
$982$	0	0
$983$	−60.0324	−1.91474	−0.957368	−	0.288872i	$-0.906720\pi$
$983$	−60.0324	−1.91474	−0.957368	+	0.288872i	$0.906720\pi$
$984$	0	0
$985$	23.1231	0.736763
$986$	0	0
$987$	1.93759	0.0616741
$988$	0	0
$989$	2.42329	0.0770562
$990$	0	0
$991$	11.0895	0.352269	0.176134	−	0.984366i	$-0.443641\pi$
$991$	11.0895	0.352269	0.176134	+	0.984366i	$0.443641\pi$
$992$	0	0
$993$	−15.1922	−0.482111
$994$	0	0
$995$	−5.61856	−0.178120
$996$	0	0
$997$	54.2311	1.71751	0.858757	−	0.512382i	$-0.171237\pi$
$997$	54.2311	1.71751	0.858757	+	0.512382i	$0.171237\pi$
$998$	0	0
$999$	−33.4990	−1.05986

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5408.2.a.bh.1.2		4
4.3	odd	2	inner	5408.2.a.bh.1.3		4
13.3	even	3		416.2.i.g.321.3	yes	8
13.9	even	3		416.2.i.g.289.3	yes	8
13.12	even	2		5408.2.a.bi.1.2		4
52.3	odd	6		416.2.i.g.321.2	yes	8
52.35	odd	6		416.2.i.g.289.2	✓	8
52.51	odd	2		5408.2.a.bi.1.3		4
104.3	odd	6		832.2.i.q.321.3		8
104.29	even	6		832.2.i.q.321.2		8
104.35	odd	6		832.2.i.q.705.3		8
104.61	even	6		832.2.i.q.705.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
416.2.i.g.289.2	✓	8	52.35	odd	6
416.2.i.g.289.3	yes	8	13.9	even	3
416.2.i.g.321.2	yes	8	52.3	odd	6
416.2.i.g.321.3	yes	8	13.3	even	3
832.2.i.q.321.2		8	104.29	even	6
832.2.i.q.321.3		8	104.3	odd	6
832.2.i.q.705.2		8	104.61	even	6
832.2.i.q.705.3		8	104.35	odd	6
5408.2.a.bh.1.2		4	1.1	even	1	trivial
5408.2.a.bh.1.3		4	4.3	odd	2	inner
5408.2.a.bi.1.2		4	13.12	even	2
5408.2.a.bi.1.3		4	52.51	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:	\( N \)	\(=\)	\( 5408 = 2^{5} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5408.a (trivial)

\(f(q)\)	\(=\)	\(q-1.19935 q^{3} +1.56155 q^{5} -3.07221 q^{7} -1.56155 q^{9} +5.47091 q^{11} -1.87285 q^{15} -2.12311 q^{17} +7.34376 q^{19} +3.68466 q^{21} -3.59806 q^{23} -2.56155 q^{25} +5.47091 q^{27} -5.00000 q^{29} -6.67026 q^{31} -6.56155 q^{33} -4.79741 q^{35} -6.12311 q^{37} +4.12311 q^{41} -0.673500 q^{43} -2.43845 q^{45} +0.525853 q^{47} +2.43845 q^{49} +2.54635 q^{51} +1.56155 q^{53} +8.54312 q^{55} -8.80776 q^{57} +10.2683 q^{59} +5.24621 q^{61} +4.79741 q^{63} +4.94506 q^{67} +4.31534 q^{69} -7.34376 q^{71} -16.6847 q^{73} +3.07221 q^{75} -16.8078 q^{77} +15.7392 q^{79} -1.87689 q^{81} +8.54312 q^{83} -3.31534 q^{85} +5.99676 q^{87} -1.68466 q^{89} +8.00000 q^{93} +11.4677 q^{95} +10.8078 q^{97} -8.54312 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{5} + 2 q^{9} + 8 q^{17} - 10 q^{21} - 2 q^{25} - 20 q^{29} - 18 q^{33} - 8 q^{37} - 18 q^{45} + 18 q^{49} - 2 q^{53} + 6 q^{57} - 12 q^{61} + 42 q^{69} - 42 q^{73} - 26 q^{77} - 24 q^{81} - 38 q^{85}+ \cdots + 2 q^{97}+O(q^{100}) \) 4 * q - 2 * q^5 + 2 * q^9 + 8 * q^17 - 10 * q^21 - 2 * q^25 - 20 * q^29 - 18 * q^33 - 8 * q^37 - 18 * q^45 + 18 * q^49 - 2 * q^53 + 6 * q^57 - 12 * q^61 + 42 * q^69 - 42 * q^73 - 26 * q^77 - 24 * q^81 - 38 * q^85 + 18 * q^89 + 32 * q^93 + 2 * q^97

Introduction

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