LMFDB - Embedded newform 2900.2.c.h.349.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2900,2,Mod(349,2900)] 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("2900.349"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2900, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2900 = 2^{2} \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2900.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [10,0,0,0,0,0,0,0,-14,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$23.1566165862$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	10.0.9689973693776896.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 22x^{8} + 179x^{6} + 639x^{4} + 847x^{2} + 9 $ x^10 + 22x^8 + 179x^6 + 639x^4 + 847x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			349.6
Root			$0.103499i$ of defining polynomial
Character	$\chi$	$=$	2900.349
Dual form			2900.2.c.h.349.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.103499i q^{3} -3.50567i q^{7} +2.98929 q^{9} -3.18972 q^{11} -2.79957i q^{13} -2.88579i q^{17} -1.66330 q^{19} +0.362834 q^{21} -3.41946i q^{23} +0.619885i q^{27} -1.00000 q^{29} -8.41531 q^{31} -0.330133i q^{33} +8.19629i q^{37} +0.289753 q^{39} -1.06416 q^{41} -0.444272i q^{43} +1.94241i q^{47} -5.28975 q^{49} +0.298676 q^{51} +0.393252i q^{53} -0.172150i q^{57} -8.68882 q^{59} +3.78886 q^{61} -10.4795i q^{63} +8.76087i q^{67} +0.353910 q^{69} -10.1790 q^{71} -12.9720i q^{73} +11.1821i q^{77} +0.626454 q^{79} +8.90371 q^{81} -7.45216i q^{83} -0.103499i q^{87} -1.02731 q^{89} -9.81438 q^{91} -0.870977i q^{93} +7.58304i q^{97} -9.53498 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 14 q^{9} - 4 q^{19} - 26 q^{21} - 10 q^{29} - 14 q^{31} - 24 q^{39} - 22 q^{41} - 26 q^{49} - 38 q^{51} - 26 q^{59} - 18 q^{61} - 12 q^{69} - 26 q^{71} - 8 q^{79} - 54 q^{81} - 20 q^{89} + 6 q^{91}+ \cdots - 60 q^{99}+O(q^{100}) $ 10 * q - 14 * q^9 - 4 * q^19 - 26 * q^21 - 10 * q^29 - 14 * q^31 - 24 * q^39 - 22 * q^41 - 26 * q^49 - 38 * q^51 - 26 * q^59 - 18 * q^61 - 12 * q^69 - 26 * q^71 - 8 * q^79 - 54 * q^81 - 20 * q^89 + 6 * q^91 - 60 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2900\mathbb{Z}\right)^\times$.

$n$	$901$	$1277$	$1451$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0.103499i	0.0597552i	0.999554	+	0.0298776i	$0.00951175\pi$
$3$	0.103499i	0.0597552i	−0.999554	+	0.0298776i	$0.990488\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 3.50567i	− 1.32502i	−0.749053	−	0.662510i	$-0.769491\pi$
$7$	− 3.50567i	− 1.32502i	0.749053	−	0.662510i	$-0.230509\pi$
$8$	0	0
$9$	2.98929	0.996429
$10$	0	0
$11$	−3.18972	−0.961736	−0.480868	−	0.876793i	$-0.659678\pi$
$11$	−3.18972	−0.961736	−0.480868	+	0.876793i	$0.659678\pi$
$12$	0	0
$13$	− 2.79957i	− 0.776461i	−0.921562	−	0.388231i	$-0.873086\pi$
$13$	− 2.79957i	− 0.776461i	0.921562	−	0.388231i	$-0.126914\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 2.88579i	− 0.699907i	−0.936767	−	0.349953i	$-0.886197\pi$
$17$	− 2.88579i	− 0.699907i	0.936767	−	0.349953i	$-0.113803\pi$
$18$	0	0
$19$	−1.66330	−0.381587	−0.190793	−	0.981630i	$-0.561106\pi$
$19$	−1.66330	−0.381587	−0.190793	+	0.981630i	$0.561106\pi$
$20$	0	0
$21$	0.362834	0.0791768
$22$	0	0
$23$	− 3.41946i	− 0.713006i	−0.934294	−	0.356503i	$-0.883969\pi$
$23$	− 3.41946i	− 0.713006i	0.934294	−	0.356503i	$-0.116031\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0.619885i	0.119297i
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−8.41531	−1.51143	−0.755717	−	0.654898i	$-0.772712\pi$
$31$	−8.41531	−1.51143	−0.755717	+	0.654898i	$0.772712\pi$
$32$	0	0
$33$	− 0.330133i	− 0.0574687i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8.19629i	1.34746i	0.738977	+	0.673731i	$0.235309\pi$
$37$	8.19629i	1.34746i	−0.738977	+	0.673731i	$0.764691\pi$
$38$	0	0
$39$	0.289753	0.0463976
$40$	0	0
$41$	−1.06416	−0.166193	−0.0830967	−	0.996541i	$-0.526481\pi$
$41$	−1.06416	−0.166193	−0.0830967	+	0.996541i	$0.526481\pi$
$42$	0	0
$43$	− 0.444272i	− 0.0677509i	−0.999426	−	0.0338754i	$-0.989215\pi$
$43$	− 0.444272i	− 0.0677509i	0.999426	−	0.0338754i	$-0.0107849\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.94241i	0.283330i	0.989915	+	0.141665i	$0.0452455\pi$
$47$	1.94241i	0.283330i	−0.989915	+	0.141665i	$0.954754\pi$
$48$	0	0
$49$	−5.28975	−0.755679
$50$	0	0
$51$	0.298676	0.0418231
$52$	0	0
$53$	0.393252i	0.0540173i	0.999635	+	0.0270086i	$0.00859816\pi$
$53$	0.393252i	0.0540173i	−0.999635	+	0.0270086i	$0.991402\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 0.172150i	− 0.0228018i
$58$	0	0
$59$	−8.68882	−1.13119	−0.565594	−	0.824684i	$-0.691353\pi$
$59$	−8.68882	−1.13119	−0.565594	+	0.824684i	$0.691353\pi$
$60$	0	0
$61$	3.78886	0.485114	0.242557	−	0.970137i	$-0.422014\pi$
$61$	3.78886	0.485114	0.242557	+	0.970137i	$0.422014\pi$
$62$	0	0
$63$	− 10.4795i	− 1.32029i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	8.76087i	1.07031i	0.844754	+	0.535155i	$0.179747\pi$
$67$	8.76087i	1.07031i	−0.844754	+	0.535155i	$0.820253\pi$
$68$	0	0
$69$	0.353910	0.0426058
$70$	0	0
$71$	−10.1790	−1.20803	−0.604013	−	0.796975i	$-0.706432\pi$
$71$	−10.1790	−1.20803	−0.604013	+	0.796975i	$0.706432\pi$
$72$	0	0
$73$	− 12.9720i	− 1.51826i	−0.650940	−	0.759129i	$-0.725625\pi$
$73$	− 12.9720i	− 1.51826i	0.650940	−	0.759129i	$-0.274375\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	11.1821i	1.27432i
$78$	0	0
$79$	0.626454	0.0704816	0.0352408	−	0.999379i	$-0.488780\pi$
$79$	0.626454	0.0704816	0.0352408	+	0.999379i	$0.488780\pi$
$80$	0	0
$81$	8.90371	0.989301
$82$	0	0
$83$	− 7.45216i	− 0.817981i	−0.912539	−	0.408990i	$-0.865881\pi$
$83$	− 7.45216i	− 0.817981i	0.912539	−	0.408990i	$-0.134119\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 0.103499i	− 0.0110963i
$88$	0	0
$89$	−1.02731	−0.108895	−0.0544475	−	0.998517i	$-0.517340\pi$
$89$	−1.02731	−0.108895	−0.0544475	+	0.998517i	$0.517340\pi$
$90$	0	0
$91$	−9.81438	−1.02883
$92$	0	0
$93$	− 0.870977i	− 0.0903161i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	7.58304i	0.769941i	0.922929	+	0.384971i	$0.125788\pi$
$97$	7.58304i	0.769941i	−0.922929	+	0.384971i	$0.874212\pi$
$98$	0	0
$99$	−9.53498	−0.958302
$100$	0	0
$101$	−1.37258	−0.136577	−0.0682884	−	0.997666i	$-0.521754\pi$
$101$	−1.37258	−0.136577	−0.0682884	+	0.997666i	$0.521754\pi$
$102$	0	0
$103$	6.06133i	0.597241i	0.954372	+	0.298620i	$0.0965264\pi$
$103$	6.06133i	0.597241i	−0.954372	+	0.298620i	$0.903474\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 2.81685i	− 0.272315i	−0.990687	−	0.136158i	$-0.956525\pi$
$107$	− 2.81685i	− 0.272315i	0.990687	−	0.136158i	$-0.0434754\pi$
$108$	0	0
$109$	−12.1507	−1.16383	−0.581914	−	0.813250i	$-0.697696\pi$
$109$	−12.1507	−1.16383	−0.581914	+	0.813250i	$0.697696\pi$
$110$	0	0
$111$	−0.848308	−0.0805178
$112$	0	0
$113$	− 5.61989i	− 0.528674i	−0.964430	−	0.264337i	$-0.914847\pi$
$113$	− 5.61989i	− 0.528674i	0.964430	−	0.264337i	$-0.0851532\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 8.36872i	− 0.773689i
$118$	0	0
$119$	−10.1166	−0.927391
$120$	0	0
$121$	−0.825703	−0.0750639
$122$	0	0
$123$	− 0.110139i	− 0.00993092i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	8.71156i	0.773026i	0.922284	+	0.386513i	$0.126321\pi$
$127$	8.71156i	0.773026i	−0.922284	+	0.386513i	$0.873679\pi$
$128$	0	0
$129$	0.0459817	0.00404847
$130$	0	0
$131$	17.2831	1.51004	0.755018	−	0.655705i	$-0.227628\pi$
$131$	17.2831	1.51004	0.755018	+	0.655705i	$0.227628\pi$
$132$	0	0
$133$	5.83098i	0.505610i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 11.8537i	− 1.01273i	−0.862320	−	0.506363i	$-0.830990\pi$
$137$	− 11.8537i	− 1.01273i	0.862320	−	0.506363i	$-0.169010\pi$
$138$	0	0
$139$	12.4939	1.05972	0.529860	−	0.848085i	$-0.322244\pi$
$139$	12.4939	1.05972	0.529860	+	0.848085i	$0.322244\pi$
$140$	0	0
$141$	−0.201038	−0.0169304
$142$	0	0
$143$	8.92984i	0.746751i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 0.547484i	− 0.0451557i
$148$	0	0
$149$	−9.19561	−0.753333	−0.376667	−	0.926349i	$-0.622930\pi$
$149$	−9.19561	−0.753333	−0.376667	+	0.926349i	$0.622930\pi$
$150$	0	0
$151$	−20.0661	−1.63296	−0.816478	−	0.577376i	$-0.804077\pi$
$151$	−20.0661	−1.63296	−0.816478	+	0.577376i	$0.804077\pi$
$152$	0	0
$153$	− 8.62645i	− 0.697407i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	10.9304i	0.872339i	0.899864	+	0.436170i	$0.143665\pi$
$157$	10.9304i	0.872339i	−0.899864	+	0.436170i	$0.856335\pi$
$158$	0	0
$159$	−0.0407012	−0.00322781
$160$	0	0
$161$	−11.9875	−0.944747
$162$	0	0
$163$	− 6.18557i	− 0.484492i	−0.970215	−	0.242246i	$-0.922116\pi$
$163$	− 6.18557i	− 0.484492i	0.970215	−	0.242246i	$-0.0778840\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 19.6437i	− 1.52007i	−0.649880	−	0.760037i	$-0.725181\pi$
$167$	− 19.6437i	− 1.52007i	0.649880	−	0.760037i	$-0.274819\pi$
$168$	0	0
$169$	5.16240	0.397108
$170$	0	0
$171$	−4.97208	−0.380224
$172$	0	0
$173$	− 19.4462i	− 1.47847i	−0.673448	−	0.739235i	$-0.735187\pi$
$173$	− 19.4462i	− 1.47847i	0.673448	−	0.739235i	$-0.264813\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 0.899285i	− 0.0675944i
$178$	0	0
$179$	−9.11367	−0.681187	−0.340594	−	0.940211i	$-0.610628\pi$
$179$	−9.11367	−0.681187	−0.340594	+	0.940211i	$0.610628\pi$
$180$	0	0
$181$	14.3939	1.06989	0.534945	−	0.844887i	$-0.320332\pi$
$181$	14.3939	1.06989	0.534945	+	0.844887i	$0.320332\pi$
$182$	0	0
$183$	0.392143i	0.0289881i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	9.20485i	0.673125i
$188$	0	0
$189$	2.17312	0.158071
$190$	0	0
$191$	−22.6038	−1.63556	−0.817779	−	0.575533i	$-0.804795\pi$
$191$	−22.6038	−1.63556	−0.817779	+	0.575533i	$0.804795\pi$
$192$	0	0
$193$	− 11.8055i	− 0.849776i	−0.905246	−	0.424888i	$-0.860314\pi$
$193$	− 11.8055i	− 0.849776i	0.905246	−	0.424888i	$-0.139686\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 11.1821i	− 0.796692i	−0.917235	−	0.398346i	$-0.869584\pi$
$197$	− 11.1821i	− 0.796692i	0.917235	−	0.398346i	$-0.130416\pi$
$198$	0	0
$199$	−5.50678	−0.390366	−0.195183	−	0.980767i	$-0.562530\pi$
$199$	−5.50678	−0.390366	−0.195183	+	0.980767i	$0.562530\pi$
$200$	0	0
$201$	−0.906741	−0.0639566
$202$	0	0
$203$	3.50567i	0.246050i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 10.2217i	− 0.710460i
$208$	0	0
$209$	5.30545	0.366986
$210$	0	0
$211$	−2.43495	−0.167629	−0.0838144	−	0.996481i	$-0.526710\pi$
$211$	−2.43495	−0.167629	−0.0838144	+	0.996481i	$0.526710\pi$
$212$	0	0
$213$	− 1.05352i	− 0.0721858i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	29.5013i	2.00268i
$218$	0	0
$219$	1.34259	0.0907238
$220$	0	0
$221$	−8.07897	−0.543450
$222$	0	0
$223$	− 8.60600i	− 0.576300i	−0.957585	−	0.288150i	$-0.906960\pi$
$223$	− 8.60600i	− 0.576300i	0.957585	−	0.288150i	$-0.0930402\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 11.8949i	− 0.789489i	−0.918791	−	0.394745i	$-0.870833\pi$
$227$	− 11.8949i	− 0.789489i	0.918791	−	0.394745i	$-0.129167\pi$
$228$	0	0
$229$	3.41835	0.225891	0.112945	−	0.993601i	$-0.463971\pi$
$229$	3.41835	0.225891	0.112945	+	0.993601i	$0.463971\pi$
$230$	0	0
$231$	−1.15734	−0.0761472
$232$	0	0
$233$	− 30.1333i	− 1.97410i	−0.160413	−	0.987050i	$-0.551282\pi$
$233$	− 30.1333i	− 1.97410i	0.160413	−	0.987050i	$-0.448718\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0.0648374i	0.00421164i
$238$	0	0
$239$	−13.1363	−0.849715	−0.424857	−	0.905260i	$-0.639676\pi$
$239$	−13.1363	−0.849715	−0.424857	+	0.905260i	$0.639676\pi$
$240$	0	0
$241$	−22.5284	−1.45118	−0.725590	−	0.688127i	$-0.758433\pi$
$241$	−22.5284	−1.45118	−0.725590	+	0.688127i	$0.758433\pi$
$242$	0	0
$243$	2.78118i	0.178413i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	4.65652i	0.296287i
$248$	0	0
$249$	0.771291	0.0488786
$250$	0	0
$251$	11.0138	0.695188	0.347594	−	0.937645i	$-0.386999\pi$
$251$	11.0138	0.695188	0.347594	+	0.937645i	$0.386999\pi$
$252$	0	0
$253$	10.9071i	0.685723i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2.52157i	0.157291i	0.996903	+	0.0786455i	$0.0250595\pi$
$257$	2.52157i	0.157291i	−0.996903	+	0.0786455i	$0.974940\pi$
$258$	0	0
$259$	28.7335	1.78541
$260$	0	0
$261$	−2.98929	−0.185032
$262$	0	0
$263$	− 15.4522i	− 0.952823i	−0.879223	−	0.476411i	$-0.841937\pi$
$263$	− 15.4522i	− 0.952823i	0.879223	−	0.476411i	$-0.158063\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 0.106326i	− 0.00650704i
$268$	0	0
$269$	28.5474	1.74056	0.870282	−	0.492554i	$-0.163937\pi$
$269$	28.5474	1.74056	0.870282	+	0.492554i	$0.163937\pi$
$270$	0	0
$271$	18.8841	1.14713	0.573563	−	0.819162i	$-0.305561\pi$
$271$	18.8841	1.14713	0.573563	+	0.819162i	$0.305561\pi$
$272$	0	0
$273$	− 1.01578i	− 0.0614777i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	12.3011i	0.739102i	0.929211	+	0.369551i	$0.120488\pi$
$277$	12.3011i	0.739102i	−0.929211	+	0.369551i	$0.879512\pi$
$278$	0	0
$279$	−25.1558	−1.50604
$280$	0	0
$281$	−9.27040	−0.553026	−0.276513	−	0.961010i	$-0.589179\pi$
$281$	−9.27040	−0.553026	−0.276513	+	0.961010i	$0.589179\pi$
$282$	0	0
$283$	− 4.83513i	− 0.287418i	−0.989620	−	0.143709i	$-0.954097\pi$
$283$	− 4.83513i	− 0.287418i	0.989620	−	0.143709i	$-0.0459030\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3.73059i	0.220210i
$288$	0	0
$289$	8.67222	0.510131
$290$	0	0
$291$	−0.784837	−0.0460080
$292$	0	0
$293$	17.6330i	1.03013i	0.857151	+	0.515065i	$0.172232\pi$
$293$	17.6330i	1.03013i	−0.857151	+	0.515065i	$0.827768\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 1.97726i	− 0.114732i
$298$	0	0
$299$	−9.57301	−0.553621
$300$	0	0
$301$	−1.55747	−0.0897713
$302$	0	0
$303$	− 0.142061i	− 0.00816117i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 25.7208i	− 1.46797i	−0.679168	−	0.733983i	$-0.737659\pi$
$307$	− 25.7208i	− 1.46797i	0.679168	−	0.733983i	$-0.262341\pi$
$308$	0	0
$309$	−0.627342	−0.0356882
$310$	0	0
$311$	13.9182	0.789226	0.394613	−	0.918847i	$-0.370879\pi$
$311$	13.9182	0.789226	0.394613	+	0.918847i	$0.370879\pi$
$312$	0	0
$313$	− 1.20886i	− 0.0683287i	−0.999416	−	0.0341644i	$-0.989123\pi$
$313$	− 1.20886i	− 0.0683287i	0.999416	−	0.0341644i	$-0.0108770\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	27.3432i	1.53575i	0.640601	+	0.767874i	$0.278685\pi$
$317$	27.3432i	1.53575i	−0.640601	+	0.767874i	$0.721315\pi$
$318$	0	0
$319$	3.18972	0.178590
$320$	0	0
$321$	0.291541	0.0162723
$322$	0	0
$323$	4.79993i	0.267075i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 1.25759i	− 0.0695448i
$328$	0	0
$329$	6.80946	0.375418
$330$	0	0
$331$	27.2276	1.49657	0.748283	−	0.663380i	$-0.230879\pi$
$331$	27.2276	1.49657	0.748283	+	0.663380i	$0.230879\pi$
$332$	0	0
$333$	24.5011i	1.34265i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 10.3552i	− 0.564085i	−0.959402	−	0.282043i	$-0.908988\pi$
$337$	− 10.3552i	− 0.564085i	0.959402	−	0.282043i	$-0.0910119\pi$
$338$	0	0
$339$	0.581653	0.0315910
$340$	0	0
$341$	26.8425	1.45360
$342$	0	0
$343$	− 5.99557i	− 0.323730i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	9.72991i	0.522329i	0.965294	+	0.261164i	$0.0841065\pi$
$347$	9.72991i	0.522329i	−0.965294	+	0.261164i	$0.915894\pi$
$348$	0	0
$349$	−28.3772	−1.51900	−0.759498	−	0.650510i	$-0.774555\pi$
$349$	−28.3772	−1.51900	−0.759498	+	0.650510i	$0.774555\pi$
$350$	0	0
$351$	1.73541	0.0926295
$352$	0	0
$353$	− 8.21420i	− 0.437198i	−0.975815	−	0.218599i	$-0.929851\pi$
$353$	− 8.21420i	− 0.437198i	0.975815	−	0.218599i	$-0.0701486\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 1.04706i	− 0.0554164i
$358$	0	0
$359$	−12.5044	−0.659958	−0.329979	−	0.943988i	$-0.607042\pi$
$359$	−12.5044	−0.659958	−0.329979	+	0.943988i	$0.607042\pi$
$360$	0	0
$361$	−16.2334	−0.854391
$362$	0	0
$363$	− 0.0854595i	− 0.00448546i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 1.47850i	− 0.0771773i	−0.999255	−	0.0385886i	$-0.987714\pi$
$367$	− 1.47850i	− 0.0771773i	0.999255	−	0.0385886i	$-0.0122862\pi$
$368$	0	0
$369$	−3.18107	−0.165600
$370$	0	0
$371$	1.37861	0.0715740
$372$	0	0
$373$	− 14.2554i	− 0.738117i	−0.929406	−	0.369058i	$-0.879680\pi$
$373$	− 14.2554i	− 0.738117i	0.929406	−	0.369058i	$-0.120320\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.79957i	0.144185i
$378$	0	0
$379$	0.732280	0.0376147	0.0188073	−	0.999823i	$-0.494013\pi$
$379$	0.732280	0.0376147	0.0188073	+	0.999823i	$0.494013\pi$
$380$	0	0
$381$	−0.901638	−0.0461923
$382$	0	0
$383$	30.7234i	1.56989i	0.619564	+	0.784947i	$0.287310\pi$
$383$	30.7234i	1.56989i	−0.619564	+	0.784947i	$0.712690\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 1.32806i	− 0.0675089i
$388$	0	0
$389$	−14.4218	−0.731215	−0.365607	−	0.930769i	$-0.619139\pi$
$389$	−14.4218	−0.731215	−0.365607	+	0.930769i	$0.619139\pi$
$390$	0	0
$391$	−9.86783	−0.499038
$392$	0	0
$393$	1.78879i	0.0902324i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	32.5133i	1.63180i	0.578196	+	0.815898i	$0.303757\pi$
$397$	32.5133i	1.63180i	−0.578196	+	0.815898i	$0.696243\pi$
$398$	0	0
$399$	−0.603501	−0.0302128
$400$	0	0
$401$	27.8651	1.39152	0.695758	−	0.718277i	$-0.255069\pi$
$401$	27.8651	1.39152	0.695758	+	0.718277i	$0.255069\pi$
$402$	0	0
$403$	23.5593i	1.17357i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 26.1438i	− 1.29590i
$408$	0	0
$409$	4.41896	0.218503	0.109252	−	0.994014i	$-0.465155\pi$
$409$	4.41896	0.218503	0.109252	+	0.994014i	$0.465155\pi$
$410$	0	0
$411$	1.22684	0.0605156
$412$	0	0
$413$	30.4602i	1.49885i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	1.29311i	0.0633238i
$418$	0	0
$419$	−9.48564	−0.463404	−0.231702	−	0.972787i	$-0.574429\pi$
$419$	−9.48564	−0.463404	−0.231702	+	0.972787i	$0.574429\pi$
$420$	0	0
$421$	31.9906	1.55913	0.779564	−	0.626323i	$-0.215441\pi$
$421$	31.9906	1.55913	0.779564	+	0.626323i	$0.215441\pi$
$422$	0	0
$423$	5.80643i	0.282318i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 13.2825i	− 0.642785i
$428$	0	0
$429$	−0.924230	−0.0446222
$430$	0	0
$431$	12.1321	0.584381	0.292190	−	0.956360i	$-0.405616\pi$
$431$	12.1321	0.584381	0.292190	+	0.956360i	$0.405616\pi$
$432$	0	0
$433$	39.0744i	1.87779i	0.344197	+	0.938897i	$0.388151\pi$
$433$	39.0744i	1.87779i	−0.344197	+	0.938897i	$0.611849\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	5.68758i	0.272074i
$438$	0	0
$439$	15.2458	0.727644	0.363822	−	0.931468i	$-0.381472\pi$
$439$	15.2458	0.727644	0.363822	+	0.931468i	$0.381472\pi$
$440$	0	0
$441$	−15.8126	−0.752981
$442$	0	0
$443$	21.9223i	1.04156i	0.853691	+	0.520780i	$0.174359\pi$
$443$	21.9223i	1.04156i	−0.853691	+	0.520780i	$0.825641\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 0.951736i	− 0.0450156i
$448$	0	0
$449$	24.4633	1.15450	0.577248	−	0.816569i	$-0.304127\pi$
$449$	24.4633	1.15450	0.577248	+	0.816569i	$0.304127\pi$
$450$	0	0
$451$	3.39436	0.159834
$452$	0	0
$453$	− 2.07682i	− 0.0975776i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 2.33595i	− 0.109271i	−0.998506	−	0.0546356i	$-0.982600\pi$
$457$	− 2.33595i	− 0.109271i	0.998506	−	0.0546356i	$-0.0173997\pi$
$458$	0	0
$459$	1.78886	0.0834968
$460$	0	0
$461$	−32.4124	−1.50960	−0.754799	−	0.655956i	$-0.772266\pi$
$461$	−32.4124	−1.50960	−0.754799	+	0.655956i	$0.772266\pi$
$462$	0	0
$463$	− 13.5518i	− 0.629806i	−0.949124	−	0.314903i	$-0.898028\pi$
$463$	− 13.5518i	− 0.629806i	0.949124	−	0.314903i	$-0.101972\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 29.3188i	− 1.35671i	−0.734733	−	0.678356i	$-0.762693\pi$
$467$	− 29.3188i	− 1.35671i	0.734733	−	0.678356i	$-0.237307\pi$
$468$	0	0
$469$	30.7127	1.41818
$470$	0	0
$471$	−1.13128	−0.0521268
$472$	0	0
$473$	1.41710i	0.0651584i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1.17554i	0.0538244i
$478$	0	0
$479$	9.09451	0.415539	0.207769	−	0.978178i	$-0.433380\pi$
$479$	9.09451	0.415539	0.207769	+	0.978178i	$0.433380\pi$
$480$	0	0
$481$	22.9461	1.04625
$482$	0	0
$483$	− 1.24069i	− 0.0564536i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 38.8247i	− 1.75932i	−0.475607	−	0.879658i	$-0.657771\pi$
$487$	− 38.8247i	− 1.75932i	0.475607	−	0.879658i	$-0.342229\pi$
$488$	0	0
$489$	0.640201	0.0289509
$490$	0	0
$491$	−18.4119	−0.830920	−0.415460	−	0.909612i	$-0.636379\pi$
$491$	−18.4119	−0.830920	−0.415460	+	0.909612i	$0.636379\pi$
$492$	0	0
$493$	2.88579i	0.129969i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	35.6843i	1.60066i
$498$	0	0
$499$	−5.77726	−0.258626	−0.129313	−	0.991604i	$-0.541277\pi$
$499$	−5.77726	−0.258626	−0.129313	+	0.991604i	$0.541277\pi$
$500$	0	0
$501$	2.03310	0.0908323
$502$	0	0
$503$	− 2.76044i	− 0.123082i	−0.998105	−	0.0615409i	$-0.980399\pi$
$503$	− 2.76044i	− 0.123082i	0.998105	−	0.0615409i	$-0.0196015\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0.534304i	0.0237293i
$508$	0	0
$509$	−13.6627	−0.605588	−0.302794	−	0.953056i	$-0.597919\pi$
$509$	−13.6627	−0.605588	−0.302794	+	0.953056i	$0.597919\pi$
$510$	0	0
$511$	−45.4756	−2.01172
$512$	0	0
$513$	− 1.03105i	− 0.0455222i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 6.19574i	− 0.272488i
$518$	0	0
$519$	2.01267	0.0883462
$520$	0	0
$521$	29.2870	1.28309	0.641543	−	0.767087i	$-0.278294\pi$
$521$	29.2870	1.28309	0.641543	+	0.767087i	$0.278294\pi$
$522$	0	0
$523$	28.3966i	1.24170i	0.783930	+	0.620849i	$0.213212\pi$
$523$	28.3966i	1.24170i	−0.783930	+	0.620849i	$0.786788\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	24.2848i	1.05786i
$528$	0	0
$529$	11.3073	0.491623
$530$	0	0
$531$	−25.9734	−1.12715
$532$	0	0
$533$	2.97918i	0.129043i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 0.943256i	− 0.0407045i
$538$	0	0
$539$	16.8728	0.726764
$540$	0	0
$541$	33.4272	1.43715	0.718573	−	0.695452i	$-0.244796\pi$
$541$	33.4272	1.43715	0.718573	+	0.695452i	$0.244796\pi$
$542$	0	0
$543$	1.48975i	0.0639314i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 16.1084i	− 0.688746i	−0.938833	−	0.344373i	$-0.888092\pi$
$547$	− 16.1084i	− 0.688746i	0.938833	−	0.344373i	$-0.111908\pi$
$548$	0	0
$549$	11.3260	0.483381
$550$	0	0
$551$	1.66330	0.0708589
$552$	0	0
$553$	− 2.19614i	− 0.0933895i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 26.3154i	− 1.11502i	−0.830170	−	0.557510i	$-0.811757\pi$
$557$	− 26.3154i	− 1.11502i	0.830170	−	0.557510i	$-0.188243\pi$
$558$	0	0
$559$	−1.24377	−0.0526059
$560$	0	0
$561$	−0.952693	−0.0402227
$562$	0	0
$563$	− 11.9921i	− 0.505405i	−0.967544	−	0.252703i	$-0.918681\pi$
$563$	− 11.9921i	− 0.505405i	0.967544	−	0.252703i	$-0.0813194\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 31.2135i	− 1.31084i
$568$	0	0
$569$	25.2445	1.05830	0.529152	−	0.848527i	$-0.322510\pi$
$569$	25.2445	1.05830	0.529152	+	0.848527i	$0.322510\pi$
$570$	0	0
$571$	−3.88718	−0.162674	−0.0813368	−	0.996687i	$-0.525919\pi$
$571$	−3.88718	−0.162674	−0.0813368	+	0.996687i	$0.525919\pi$
$572$	0	0
$573$	− 2.33948i	− 0.0977330i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	5.86380i	0.244113i	0.992523	+	0.122057i	$0.0389489\pi$
$577$	5.86380i	0.244113i	−0.992523	+	0.122057i	$0.961051\pi$
$578$	0	0
$579$	1.22185	0.0507785
$580$	0	0
$581$	−26.1248	−1.08384
$582$	0	0
$583$	− 1.25436i	− 0.0519504i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 35.8714i	− 1.48057i	−0.672292	−	0.740286i	$-0.734690\pi$
$587$	− 35.8714i	− 1.48057i	0.672292	−	0.740286i	$-0.265310\pi$
$588$	0	0
$589$	13.9972	0.576744
$590$	0	0
$591$	1.15734	0.0476065
$592$	0	0
$593$	11.0568i	0.454047i	0.973889	+	0.227023i	$0.0728994\pi$
$593$	11.0568i	0.454047i	−0.973889	+	0.227023i	$0.927101\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 0.569947i	− 0.0233264i
$598$	0	0
$599$	−46.5588	−1.90234	−0.951171	−	0.308665i	$-0.900118\pi$
$599$	−46.5588	−1.90234	−0.951171	+	0.308665i	$0.900118\pi$
$600$	0	0
$601$	29.0643	1.18556	0.592779	−	0.805365i	$-0.298031\pi$
$601$	29.0643	1.18556	0.592779	+	0.805365i	$0.298031\pi$
$602$	0	0
$603$	26.1888i	1.06649i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	27.6464i	1.12213i	0.827770	+	0.561067i	$0.189609\pi$
$607$	27.6464i	1.12213i	−0.827770	+	0.561067i	$0.810391\pi$
$608$	0	0
$609$	−0.362834	−0.0147028
$610$	0	0
$611$	5.43792	0.219995
$612$	0	0
$613$	− 27.9150i	− 1.12748i	−0.825954	−	0.563738i	$-0.809363\pi$
$613$	− 27.9150i	− 1.12748i	0.825954	−	0.563738i	$-0.190637\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	33.5412i	1.35032i	0.737672	+	0.675159i	$0.235925\pi$
$617$	33.5412i	1.35032i	−0.737672	+	0.675159i	$0.764075\pi$
$618$	0	0
$619$	1.91371	0.0769186	0.0384593	−	0.999260i	$-0.487755\pi$
$619$	1.91371	0.0769186	0.0384593	+	0.999260i	$0.487755\pi$
$620$	0	0
$621$	2.11967	0.0850595
$622$	0	0
$623$	3.60142i	0.144288i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0.549109i	0.0219293i
$628$	0	0
$629$	23.6528	0.943097
$630$	0	0
$631$	26.9261	1.07191	0.535956	−	0.844246i	$-0.319951\pi$
$631$	26.9261	1.07191	0.535956	+	0.844246i	$0.319951\pi$
$632$	0	0
$633$	− 0.252015i	− 0.0100167i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	14.8090i	0.586755i
$638$	0	0
$639$	−30.4280	−1.20371
$640$	0	0
$641$	4.31224	0.170323	0.0851617	−	0.996367i	$-0.472859\pi$
$641$	4.31224	0.170323	0.0851617	+	0.996367i	$0.472859\pi$
$642$	0	0
$643$	− 27.8513i	− 1.09835i	−0.835709	−	0.549173i	$-0.814943\pi$
$643$	− 27.8513i	− 1.09835i	0.835709	−	0.549173i	$-0.185057\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 25.0072i	− 0.983134i	−0.870840	−	0.491567i	$-0.836424\pi$
$647$	− 25.0072i	− 0.983134i	0.870840	−	0.491567i	$-0.163576\pi$
$648$	0	0
$649$	27.7149	1.08790
$650$	0	0
$651$	−3.05336	−0.119671
$652$	0	0
$653$	− 21.1675i	− 0.828347i	−0.910198	−	0.414174i	$-0.864071\pi$
$653$	− 21.1675i	− 0.828347i	0.910198	−	0.414174i	$-0.135929\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 38.7771i	− 1.51284i
$658$	0	0
$659$	43.6261	1.69943	0.849715	−	0.527242i	$-0.176774\pi$
$659$	43.6261	1.69943	0.849715	+	0.527242i	$0.176774\pi$
$660$	0	0
$661$	34.3854	1.33744	0.668719	−	0.743515i	$-0.266843\pi$
$661$	34.3854	1.33744	0.668719	+	0.743515i	$0.266843\pi$
$662$	0	0
$663$	− 0.836165i	− 0.0324740i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	3.41946i	0.132402i
$668$	0	0
$669$	0.890712	0.0344369
$670$	0	0
$671$	−12.0854	−0.466551
$672$	0	0
$673$	6.76708i	0.260852i	0.991458	+	0.130426i	$0.0416345\pi$
$673$	6.76708i	0.260852i	−0.991458	+	0.130426i	$0.958366\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 12.7515i	− 0.490082i	−0.969513	−	0.245041i	$-0.921199\pi$
$677$	− 12.7515i	− 0.490082i	0.969513	−	0.245041i	$-0.0788014\pi$
$678$	0	0
$679$	26.5837	1.02019
$680$	0	0
$681$	1.23111	0.0471761
$682$	0	0
$683$	44.7472i	1.71220i	0.516807	+	0.856102i	$0.327120\pi$
$683$	44.7472i	1.71220i	−0.516807	+	0.856102i	$0.672880\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0.353796i	0.0134981i
$688$	0	0
$689$	1.10094	0.0419423
$690$	0	0
$691$	30.7285	1.16897	0.584484	−	0.811405i	$-0.301297\pi$
$691$	30.7285	1.16897	0.584484	+	0.811405i	$0.301297\pi$
$692$	0	0
$693$	33.4265i	1.26977i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	3.07093i	0.116320i
$698$	0	0
$699$	3.11877	0.117963
$700$	0	0
$701$	24.5252	0.926305	0.463153	−	0.886278i	$-0.346718\pi$
$701$	24.5252	0.926305	0.463153	+	0.886278i	$0.346718\pi$
$702$	0	0
$703$	− 13.6329i	− 0.514174i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	4.81182i	0.180967i
$708$	0	0
$709$	28.6263	1.07508	0.537542	−	0.843237i	$-0.319353\pi$
$709$	28.6263	1.07508	0.537542	+	0.843237i	$0.319353\pi$
$710$	0	0
$711$	1.87265	0.0702299
$712$	0	0
$713$	28.7758i	1.07766i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 1.35959i	− 0.0507749i
$718$	0	0
$719$	4.81171	0.179446	0.0897232	−	0.995967i	$-0.471402\pi$
$719$	4.81171	0.179446	0.0897232	+	0.995967i	$0.471402\pi$
$720$	0	0
$721$	21.2491	0.791356
$722$	0	0
$723$	− 2.33166i	− 0.0867155i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	24.4967i	0.908532i	0.890866	+	0.454266i	$0.150098\pi$
$727$	24.4967i	0.908532i	−0.890866	+	0.454266i	$0.849902\pi$
$728$	0	0
$729$	26.4233	0.978640
$730$	0	0
$731$	−1.28208	−0.0474193
$732$	0	0
$733$	− 19.1809i	− 0.708464i	−0.935158	−	0.354232i	$-0.884742\pi$
$733$	− 19.1809i	− 0.708464i	0.935158	−	0.354232i	$-0.115258\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 27.9447i	− 1.02936i
$738$	0	0
$739$	16.3523	0.601530	0.300765	−	0.953698i	$-0.402758\pi$
$739$	16.3523	0.601530	0.300765	+	0.953698i	$0.402758\pi$
$740$	0	0
$741$	−0.481945	−0.0177047
$742$	0	0
$743$	37.5590i	1.37791i	0.724806	+	0.688953i	$0.241929\pi$
$743$	37.5590i	1.37791i	−0.724806	+	0.688953i	$0.758071\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 22.2766i	− 0.815060i
$748$	0	0
$749$	−9.87496	−0.360823
$750$	0	0
$751$	−13.9179	−0.507873	−0.253937	−	0.967221i	$-0.581725\pi$
$751$	−13.9179	−0.507873	−0.253937	+	0.967221i	$0.581725\pi$
$752$	0	0
$753$	1.13992i	0.0415411i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	3.95042i	0.143580i	0.997420	+	0.0717902i	$0.0228712\pi$
$757$	3.95042i	0.143580i	−0.997420	+	0.0717902i	$0.977129\pi$
$758$	0	0
$759$	−1.12887	−0.0409755
$760$	0	0
$761$	2.97493	0.107841	0.0539206	−	0.998545i	$-0.482828\pi$
$761$	2.97493	0.107841	0.0539206	+	0.998545i	$0.482828\pi$
$762$	0	0
$763$	42.5965i	1.54210i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	24.3250i	0.878324i
$768$	0	0
$769$	4.17676	0.150618	0.0753089	−	0.997160i	$-0.476006\pi$
$769$	4.17676	0.150618	0.0753089	+	0.997160i	$0.476006\pi$
$770$	0	0
$771$	−0.260980	−0.00939895
$772$	0	0
$773$	0.788509i	0.0283607i	0.999899	+	0.0141803i	$0.00451389\pi$
$773$	0.788509i	0.0283607i	−0.999899	+	0.0141803i	$0.995486\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	2.97389i	0.106688i
$778$	0	0
$779$	1.77001	0.0634173
$780$	0	0
$781$	32.4681	1.16180
$782$	0	0
$783$	− 0.619885i	− 0.0221529i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	17.4582i	0.622316i	0.950358	+	0.311158i	$0.100717\pi$
$787$	17.4582i	0.622316i	−0.950358	+	0.311158i	$0.899283\pi$
$788$	0	0
$789$	1.59929	0.0569361
$790$	0	0
$791$	−19.7015	−0.700504
$792$	0	0
$793$	− 10.6072i	− 0.376672i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	19.7775i	0.700556i	0.936646	+	0.350278i	$0.113913\pi$
$797$	19.7775i	0.700556i	−0.936646	+	0.350278i	$0.886087\pi$
$798$	0	0
$799$	5.60539	0.198304
$800$	0	0
$801$	−3.07093	−0.108506
$802$	0	0
$803$	41.3770i	1.46016i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	2.95462i	0.104008i
$808$	0	0
$809$	−35.9486	−1.26389	−0.631943	−	0.775015i	$-0.717742\pi$
$809$	−35.9486	−1.26389	−0.631943	+	0.775015i	$0.717742\pi$
$810$	0	0
$811$	−3.31810	−0.116514	−0.0582571	−	0.998302i	$-0.518554\pi$
$811$	−3.31810	−0.116514	−0.0582571	+	0.998302i	$0.518554\pi$
$812$	0	0
$813$	1.95448i	0.0685467i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0.738957i	0.0258528i
$818$	0	0
$819$	−29.3380	−1.02515
$820$	0	0
$821$	−26.8351	−0.936551	−0.468275	−	0.883583i	$-0.655125\pi$
$821$	−26.8351	−0.936551	−0.468275	+	0.883583i	$0.655125\pi$
$822$	0	0
$823$	− 35.5995i	− 1.24092i	−0.784238	−	0.620460i	$-0.786946\pi$
$823$	− 35.5995i	− 1.24092i	0.784238	−	0.620460i	$-0.213054\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	5.19953i	0.180806i	0.995905	+	0.0904028i	$0.0288154\pi$
$827$	5.19953i	0.180806i	−0.995905	+	0.0904028i	$0.971185\pi$
$828$	0	0
$829$	−25.9255	−0.900430	−0.450215	−	0.892920i	$-0.648653\pi$
$829$	−25.9255	−0.900430	−0.450215	+	0.892920i	$0.648653\pi$
$830$	0	0
$831$	−1.27315	−0.0441652
$832$	0	0
$833$	15.2651i	0.528905i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 5.21653i	− 0.180310i
$838$	0	0
$839$	23.1079	0.797775	0.398887	−	0.917000i	$-0.369396\pi$
$839$	23.1079	0.797775	0.398887	+	0.917000i	$0.369396\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	− 0.959477i	− 0.0330461i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	2.89465i	0.0994612i
$848$	0	0
$849$	0.500431	0.0171747
$850$	0	0
$851$	28.0268	0.960748
$852$	0	0
$853$	4.01364i	0.137424i	0.997637	+	0.0687122i	$0.0218890\pi$
$853$	4.01364i	0.137424i	−0.997637	+	0.0687122i	$0.978111\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	16.0821i	0.549355i	0.961536	+	0.274678i	$0.0885712\pi$
$857$	16.0821i	0.549355i	−0.961536	+	0.274678i	$0.911429\pi$
$858$	0	0
$859$	−31.2890	−1.06757	−0.533784	−	0.845621i	$-0.679230\pi$
$859$	−31.2890	−1.06757	−0.533784	+	0.845621i	$0.679230\pi$
$860$	0	0
$861$	−0.386112	−0.0131587
$862$	0	0
$863$	− 21.2705i	− 0.724058i	−0.932167	−	0.362029i	$-0.882084\pi$
$863$	− 21.2705i	− 0.724058i	0.932167	−	0.362029i	$-0.117916\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0.897567i	0.0304830i
$868$	0	0
$869$	−1.99821	−0.0677847
$870$	0	0
$871$	24.5267	0.831054
$872$	0	0
$873$	22.6679i	0.767192i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 5.88836i	− 0.198836i	−0.995046	−	0.0994179i	$-0.968302\pi$
$877$	− 5.88836i	− 0.198836i	0.995046	−	0.0994179i	$-0.0316980\pi$
$878$	0	0
$879$	−1.82500	−0.0615556
$880$	0	0
$881$	21.5053	0.724531	0.362266	−	0.932075i	$-0.382003\pi$
$881$	21.5053	0.724531	0.362266	+	0.932075i	$0.382003\pi$
$882$	0	0
$883$	26.2915i	0.884779i	0.896823	+	0.442390i	$0.145869\pi$
$883$	26.2915i	0.884779i	−0.896823	+	0.442390i	$0.854131\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 24.7353i	− 0.830531i	−0.909700	−	0.415266i	$-0.863689\pi$
$887$	− 24.7353i	− 0.830531i	0.909700	−	0.415266i	$-0.136311\pi$
$888$	0	0
$889$	30.5399	1.02428
$890$	0	0
$891$	−28.4003	−0.951446
$892$	0	0
$893$	− 3.23081i	− 0.108115i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 0.990797i	− 0.0330817i
$898$	0	0
$899$	8.41531	0.280666
$900$	0	0
$901$	1.13484	0.0378071
$902$	0	0
$903$	− 0.161197i	− 0.00536430i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	21.0838i	0.700076i	0.936736	+	0.350038i	$0.113831\pi$
$907$	21.0838i	0.700076i	−0.936736	+	0.350038i	$0.886169\pi$
$908$	0	0
$909$	−4.10303	−0.136089
$910$	0	0
$911$	−31.5086	−1.04393	−0.521964	−	0.852968i	$-0.674800\pi$
$911$	−31.5086	−1.04393	−0.521964	+	0.852968i	$0.674800\pi$
$912$	0	0
$913$	23.7703i	0.786681i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 60.5891i	− 2.00083i
$918$	0	0
$919$	−31.8202	−1.04965	−0.524825	−	0.851210i	$-0.675869\pi$
$919$	−31.8202	−1.04965	−0.524825	+	0.851210i	$0.675869\pi$
$920$	0	0
$921$	2.66208	0.0877186
$922$	0	0
$923$	28.4968i	0.937985i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	18.1191i	0.595108i
$928$	0	0
$929$	40.9628	1.34395	0.671974	−	0.740575i	$-0.265447\pi$
$929$	40.9628	1.34395	0.671974	+	0.740575i	$0.265447\pi$
$930$	0	0
$931$	8.79844	0.288357
$932$	0	0
$933$	1.44052i	0.0471604i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 45.9337i	− 1.50059i	−0.661104	−	0.750294i	$-0.729912\pi$
$937$	− 45.9337i	− 1.50059i	0.661104	−	0.750294i	$-0.270088\pi$
$938$	0	0
$939$	0.125116	0.00408299
$940$	0	0
$941$	−31.1076	−1.01408	−0.507039	−	0.861923i	$-0.669260\pi$
$941$	−31.1076	−1.01408	−0.507039	+	0.861923i	$0.669260\pi$
$942$	0	0
$943$	3.63884i	0.118497i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	7.28083i	0.236595i	0.992978	+	0.118298i	$0.0377437\pi$
$947$	7.28083i	0.236595i	−0.992978	+	0.118298i	$0.962256\pi$
$948$	0	0
$949$	−36.3161	−1.17887
$950$	0	0
$951$	−2.83000	−0.0917689
$952$	0	0
$953$	5.72786i	0.185543i	0.995687	+	0.0927717i	$0.0295727\pi$
$953$	5.72786i	0.185543i	−0.995687	+	0.0927717i	$0.970427\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0.330133i	0.0106717i
$958$	0	0
$959$	−41.5550	−1.34188
$960$	0	0
$961$	39.8175	1.28444
$962$	0	0
$963$	− 8.42038i	− 0.271343i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	22.3003i	0.717129i	0.933505	+	0.358564i	$0.116734\pi$
$967$	22.3003i	0.717129i	−0.933505	+	0.358564i	$0.883266\pi$
$968$	0	0
$969$	−0.496788	−0.0159591
$970$	0	0
$971$	−58.5388	−1.87860	−0.939299	−	0.343099i	$-0.888523\pi$
$971$	−58.5388	−1.87860	−0.939299	+	0.343099i	$0.888523\pi$
$972$	0	0
$973$	− 43.7996i	− 1.40415i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 44.2394i	− 1.41534i	−0.706542	−	0.707672i	$-0.749746\pi$
$977$	− 44.2394i	− 1.41534i	0.706542	−	0.707672i	$-0.250254\pi$
$978$	0	0
$979$	3.27684	0.104728
$980$	0	0
$981$	−36.3220	−1.15967
$982$	0	0
$983$	− 30.4106i	− 0.969947i	−0.874529	−	0.484974i	$-0.838829\pi$
$983$	− 30.4106i	− 0.969947i	0.874529	−	0.484974i	$-0.161171\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0.704773i	0.0224332i
$988$	0	0
$989$	−1.51917	−0.0483068
$990$	0	0
$991$	31.9150	1.01381	0.506906	−	0.862001i	$-0.330789\pi$
$991$	31.9150	1.01381	0.506906	+	0.862001i	$0.330789\pi$
$992$	0	0
$993$	2.81803i	0.0894276i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	58.0409i	1.83818i	0.394053	+	0.919088i	$0.371073\pi$
$997$	58.0409i	1.83818i	−0.394053	+	0.919088i	$0.628927\pi$
$998$	0	0
$999$	−5.08076	−0.160748

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2900.2.c.h.349.6		10
5.2	odd	4		2900.2.a.j.1.3	✓	5
5.3	odd	4		2900.2.a.l.1.3	yes	5
5.4	even	2	inner	2900.2.c.h.349.5		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2900.2.a.j.1.3	✓	5	5.2	odd	4
2900.2.a.l.1.3	yes	5	5.3	odd	4
2900.2.c.h.349.5		10	5.4	even	2	inner
2900.2.c.h.349.6		10	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2900 = 2^{2} \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2900.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.103499i q^{3} -3.50567i q^{7} +2.98929 q^{9} -3.18972 q^{11} -2.79957i q^{13} -2.88579i q^{17} -1.66330 q^{19} +0.362834 q^{21} -3.41946i q^{23} +0.619885i q^{27} -1.00000 q^{29} -8.41531 q^{31} -0.330133i q^{33} +8.19629i q^{37} +0.289753 q^{39} -1.06416 q^{41} -0.444272i q^{43} +1.94241i q^{47} -5.28975 q^{49} +0.298676 q^{51} +0.393252i q^{53} -0.172150i q^{57} -8.68882 q^{59} +3.78886 q^{61} -10.4795i q^{63} +8.76087i q^{67} +0.353910 q^{69} -10.1790 q^{71} -12.9720i q^{73} +11.1821i q^{77} +0.626454 q^{79} +8.90371 q^{81} -7.45216i q^{83} -0.103499i q^{87} -1.02731 q^{89} -9.81438 q^{91} -0.870977i q^{93} +7.58304i q^{97} -9.53498 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 14 q^{9} - 4 q^{19} - 26 q^{21} - 10 q^{29} - 14 q^{31} - 24 q^{39} - 22 q^{41} - 26 q^{49} - 38 q^{51} - 26 q^{59} - 18 q^{61} - 12 q^{69} - 26 q^{71} - 8 q^{79} - 54 q^{81} - 20 q^{89} + 6 q^{91}+ \cdots - 60 q^{99}+O(q^{100}) \) 10 * q - 14 * q^9 - 4 * q^19 - 26 * q^21 - 10 * q^29 - 14 * q^31 - 24 * q^39 - 22 * q^41 - 26 * q^49 - 38 * q^51 - 26 * q^59 - 18 * q^61 - 12 * q^69 - 26 * q^71 - 8 * q^79 - 54 * q^81 - 20 * q^89 + 6 * q^91 - 60 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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