LMFDB - Embedded newform 5328.2.e.f.2591.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5328,2,Mod(2591,5328)] 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("5328.2591"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5328, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5328 = 2^{4} \cdot 3^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5328.e (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [24,0,0,0,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$42.5442941969$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2591.6
Character	$\chi$	$=$	5328.2591
Dual form			5328.2.e.f.2591.19

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.80927i q^{5} +4.99414i q^{7} +2.88061 q^{11} +6.56921 q^{13} +4.32005i q^{17} +5.96095i q^{19} +4.50495 q^{23} -2.89201 q^{25} -8.08102i q^{29} +8.78197i q^{31} +14.0299 q^{35} -1.00000 q^{37} -2.06855i q^{41} +0.596608i q^{43} -11.3419 q^{47} -17.9415 q^{49} +9.48922i q^{53} -8.09243i q^{55} -6.93395 q^{59} +7.45537 q^{61} -18.4547i q^{65} -0.0130947i q^{67} -7.35239 q^{71} -9.27712 q^{73} +14.3862i q^{77} +11.9267i q^{79} +1.66304 q^{83} +12.1362 q^{85} -15.4386i q^{89} +32.8076i q^{91} +16.7459 q^{95} +5.35179 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 32 q^{25} - 24 q^{37} - 80 q^{49} - 48 q^{61} - 48 q^{73} - 40 q^{85} + 24 q^{97}+O(q^{100}) $ 24 * q - 32 * q^25 - 24 * q^37 - 80 * q^49 - 48 * q^61 - 48 * q^73 - 40 * q^85 + 24 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5328\mathbb{Z}\right)^\times$.

$n$	$1297$	$1333$	$1999$	$2369$
$\chi(n)$	$1$	$1$	$-1$	$-1$

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	0
$3$	0	0
$4$	0	0
$5$	− 2.80927i	− 1.25634i	−0.778074	−	0.628172i	$-0.783803\pi$
$5$	− 2.80927i	− 1.25634i	0.778074	−	0.628172i	$-0.216197\pi$
$6$	0	0
$7$	4.99414i	1.88761i	0.330505	+	0.943804i	$0.392781\pi$
$7$	4.99414i	1.88761i	−0.330505	+	0.943804i	$0.607219\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.88061	0.868538	0.434269	−	0.900783i	$-0.357007\pi$
$11$	2.88061	0.868538	0.434269	+	0.900783i	$0.357007\pi$
$12$	0	0
$13$	6.56921	1.82197	0.910985	−	0.412439i	$-0.135323\pi$
$13$	6.56921	1.82197	0.910985	+	0.412439i	$0.135323\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.32005i	1.04777i	0.851790	+	0.523883i	$0.175517\pi$
$17$	4.32005i	1.04777i	−0.851790	+	0.523883i	$0.824483\pi$
$18$	0	0
$19$	5.96095i	1.36753i	0.729700	+	0.683767i	$0.239660\pi$
$19$	5.96095i	1.36753i	−0.729700	+	0.683767i	$0.760340\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.50495	0.939348	0.469674	−	0.882840i	$-0.344371\pi$
$23$	4.50495	0.939348	0.469674	+	0.882840i	$0.344371\pi$
$24$	0	0
$25$	−2.89201	−0.578402
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 8.08102i	− 1.50061i	−0.661094	−	0.750303i	$-0.729907\pi$
$29$	− 8.08102i	− 1.50061i	0.661094	−	0.750303i	$-0.270093\pi$
$30$	0	0
$31$	8.78197i	1.57729i	0.614849	+	0.788644i	$0.289217\pi$
$31$	8.78197i	1.57729i	−0.614849	+	0.788644i	$0.710783\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	14.0299	2.37149
$36$	0	0
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 2.06855i	− 0.323053i	−0.986868	−	0.161526i	$-0.948358\pi$
$41$	− 2.06855i	− 0.323053i	0.986868	−	0.161526i	$-0.0516417\pi$
$42$	0	0
$43$	0.596608i	0.0909819i	0.998965	+	0.0454909i	$0.0144852\pi$
$43$	0.596608i	0.0909819i	−0.998965	+	0.0454909i	$0.985515\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−11.3419	−1.65439	−0.827195	−	0.561915i	$-0.810065\pi$
$47$	−11.3419	−1.65439	−0.827195	+	0.561915i	$0.810065\pi$
$48$	0	0
$49$	−17.9415	−2.56306
$50$	0	0
$51$	0	0
$52$	0	0
$53$	9.48922i	1.30345i	0.758457	+	0.651723i	$0.225953\pi$
$53$	9.48922i	1.30345i	−0.758457	+	0.651723i	$0.774047\pi$
$54$	0	0
$55$	− 8.09243i	− 1.09118i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−6.93395	−0.902723	−0.451361	−	0.892341i	$-0.649061\pi$
$59$	−6.93395	−0.902723	−0.451361	+	0.892341i	$0.649061\pi$
$60$	0	0
$61$	7.45537	0.954562	0.477281	−	0.878751i	$-0.341622\pi$
$61$	7.45537	0.954562	0.477281	+	0.878751i	$0.341622\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 18.4547i	− 2.28902i
$66$	0	0
$67$	− 0.0130947i	− 0.00159977i	−1.00000		0.000799886i	$-0.999745\pi$
$67$	− 0.0130947i	− 0.00159977i	1.00000		0.000799886i	$-0.000254612\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−7.35239	−0.872568	−0.436284	−	0.899809i	$-0.643706\pi$
$71$	−7.35239	−0.872568	−0.436284	+	0.899809i	$0.643706\pi$
$72$	0	0
$73$	−9.27712	−1.08580	−0.542902	−	0.839796i	$-0.682675\pi$
$73$	−9.27712	−1.08580	−0.542902	+	0.839796i	$0.682675\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	14.3862i	1.63946i
$78$	0	0
$79$	11.9267i	1.34186i	0.741520	+	0.670931i	$0.234105\pi$
$79$	11.9267i	1.34186i	−0.741520	+	0.670931i	$0.765895\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1.66304	0.182543	0.0912713	−	0.995826i	$-0.470907\pi$
$83$	1.66304	0.182543	0.0912713	+	0.995826i	$0.470907\pi$
$84$	0	0
$85$	12.1362	1.31635
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 15.4386i	− 1.63648i	−0.574874	−	0.818242i	$-0.694949\pi$
$89$	− 15.4386i	− 1.63648i	0.574874	−	0.818242i	$-0.305051\pi$
$90$	0	0
$91$	32.8076i	3.43917i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	16.7459	1.71810
$96$	0	0
$97$	5.35179	0.543392	0.271696	−	0.962383i	$-0.412415\pi$
$97$	5.35179	0.543392	0.271696	+	0.962383i	$0.412415\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	2.22126i	0.221024i	0.993875	+	0.110512i	$0.0352491\pi$
$101$	2.22126i	0.221024i	−0.993875	+	0.110512i	$0.964751\pi$
$102$	0	0
$103$	− 2.91996i	− 0.287712i	−0.989599	−	0.143856i	$-0.954050\pi$
$103$	− 2.91996i	− 0.287712i	0.989599	−	0.143856i	$-0.0459502\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−5.67767	−0.548881	−0.274441	−	0.961604i	$-0.588493\pi$
$107$	−5.67767	−0.548881	−0.274441	+	0.961604i	$0.588493\pi$
$108$	0	0
$109$	−4.78518	−0.458337	−0.229169	−	0.973387i	$-0.573601\pi$
$109$	−4.78518	−0.458337	−0.229169	+	0.973387i	$0.573601\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	14.9280i	1.40430i	0.712027	+	0.702152i	$0.247777\pi$
$113$	14.9280i	1.40430i	−0.712027	+	0.702152i	$0.752223\pi$
$114$	0	0
$115$	− 12.6556i	− 1.18014i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−21.5749	−1.97777
$120$	0	0
$121$	−2.70206	−0.245642
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 5.92191i	− 0.529672i
$126$	0	0
$127$	− 16.2265i	− 1.43987i	−0.694042	−	0.719934i	$-0.744172\pi$
$127$	− 16.2265i	− 1.43987i	0.694042	−	0.719934i	$-0.255828\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	12.7424	1.11331	0.556653	−	0.830745i	$-0.312085\pi$
$131$	12.7424	1.11331	0.556653	+	0.830745i	$0.312085\pi$
$132$	0	0
$133$	−29.7698	−2.58137
$134$	0	0
$135$	0	0
$136$	0	0
$137$	19.8647i	1.69716i	0.529067	+	0.848580i	$0.322542\pi$
$137$	19.8647i	1.69716i	−0.529067	+	0.848580i	$0.677458\pi$
$138$	0	0
$139$	− 11.4689i	− 0.972778i	−0.873742	−	0.486389i	$-0.838314\pi$
$139$	− 11.4689i	− 0.972778i	0.873742	−	0.486389i	$-0.161686\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	18.9234	1.58245
$144$	0	0
$145$	−22.7018	−1.88528
$146$	0	0
$147$	0	0
$148$	0	0
$149$	3.10383i	0.254275i	0.991885	+	0.127138i	$0.0405790\pi$
$149$	3.10383i	0.254275i	−0.991885	+	0.127138i	$0.959421\pi$
$150$	0	0
$151$	− 3.01200i	− 0.245113i	−0.992462	−	0.122557i	$-0.960891\pi$
$151$	− 3.01200i	− 0.245113i	0.992462	−	0.122557i	$-0.0391093\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	24.6710	1.98162
$156$	0	0
$157$	3.11574	0.248663	0.124331	−	0.992241i	$-0.460321\pi$
$157$	3.11574	0.248663	0.124331	+	0.992241i	$0.460321\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	22.4984i	1.77312i
$162$	0	0
$163$	− 2.86040i	− 0.224044i	−0.993706	−	0.112022i	$-0.964267\pi$
$163$	− 2.86040i	− 0.224044i	0.993706	−	0.112022i	$-0.0357327\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	19.8364	1.53499	0.767494	−	0.641056i	$-0.221503\pi$
$167$	19.8364	1.53499	0.767494	+	0.641056i	$0.221503\pi$
$168$	0	0
$169$	30.1545	2.31958
$170$	0	0
$171$	0	0
$172$	0	0
$173$	10.7176i	0.814846i	0.913240	+	0.407423i	$0.133573\pi$
$173$	10.7176i	0.814846i	−0.913240	+	0.407423i	$0.866427\pi$
$174$	0	0
$175$	− 14.4431i	− 1.09180i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−13.2555	−0.990763	−0.495381	−	0.868676i	$-0.664972\pi$
$179$	−13.2555	−0.990763	−0.495381	+	0.868676i	$0.664972\pi$
$180$	0	0
$181$	−8.85296	−0.658035	−0.329018	−	0.944324i	$-0.606718\pi$
$181$	−8.85296	−0.658035	−0.329018	+	0.944324i	$0.606718\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	2.80927i	0.206542i
$186$	0	0
$187$	12.4444i	0.910024i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−2.98799	−0.216203	−0.108102	−	0.994140i	$-0.534477\pi$
$191$	−2.98799	−0.216203	−0.108102	+	0.994140i	$0.534477\pi$
$192$	0	0
$193$	23.3101	1.67790	0.838948	−	0.544212i	$-0.183171\pi$
$193$	23.3101	1.67790	0.838948	+	0.544212i	$0.183171\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	12.9047i	0.919421i	0.888069	+	0.459711i	$0.152047\pi$
$197$	12.9047i	0.919421i	−0.888069	+	0.459711i	$0.847953\pi$
$198$	0	0
$199$	10.4496i	0.740753i	0.928882	+	0.370377i	$0.120771\pi$
$199$	10.4496i	0.740753i	−0.928882	+	0.370377i	$0.879229\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	40.3577	2.83256
$204$	0	0
$205$	−5.81111	−0.405866
$206$	0	0
$207$	0	0
$208$	0	0
$209$	17.1712i	1.18776i
$210$	0	0
$211$	− 10.7080i	− 0.737172i	−0.929594	−	0.368586i	$-0.879842\pi$
$211$	− 10.7080i	− 0.737172i	0.929594	−	0.368586i	$-0.120158\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1.67603	0.114305
$216$	0	0
$217$	−43.8584	−2.97730
$218$	0	0
$219$	0	0
$220$	0	0
$221$	28.3793i	1.90900i
$222$	0	0
$223$	− 7.09988i	− 0.475443i	−0.971333	−	0.237722i	$-0.923599\pi$
$223$	− 7.09988i	− 0.475443i	0.971333	−	0.237722i	$-0.0764006\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	26.9594	1.78936	0.894679	−	0.446710i	$-0.147405\pi$
$227$	26.9594	1.78936	0.894679	+	0.446710i	$0.147405\pi$
$228$	0	0
$229$	5.93993	0.392522	0.196261	−	0.980552i	$-0.437120\pi$
$229$	5.93993	0.392522	0.196261	+	0.980552i	$0.437120\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 27.4804i	− 1.80030i	−0.435577	−	0.900152i	$-0.643456\pi$
$233$	− 27.4804i	− 1.80030i	0.435577	−	0.900152i	$-0.356544\pi$
$234$	0	0
$235$	31.8626i	2.07848i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−11.6175	−0.751475	−0.375737	−	0.926726i	$-0.622611\pi$
$239$	−11.6175	−0.751475	−0.375737	+	0.926726i	$0.622611\pi$
$240$	0	0
$241$	−11.3986	−0.734248	−0.367124	−	0.930172i	$-0.619658\pi$
$241$	−11.3986	−0.734248	−0.367124	+	0.930172i	$0.619658\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	50.4024i	3.22009i
$246$	0	0
$247$	39.1587i	2.49161i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	17.2860	1.09108	0.545542	−	0.838083i	$-0.316324\pi$
$251$	17.2860	1.09108	0.545542	+	0.838083i	$0.316324\pi$
$252$	0	0
$253$	12.9770	0.815859
$254$	0	0
$255$	0	0
$256$	0	0
$257$	14.3791i	0.896941i	0.893798	+	0.448470i	$0.148031\pi$
$257$	14.3791i	0.896941i	−0.893798	+	0.448470i	$0.851969\pi$
$258$	0	0
$259$	− 4.99414i	− 0.310321i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−17.8206	−1.09887	−0.549434	−	0.835537i	$-0.685157\pi$
$263$	−17.8206	−1.09887	−0.549434	+	0.835537i	$0.685157\pi$
$264$	0	0
$265$	26.6578	1.63758
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 7.74849i	− 0.472434i	−0.971700	−	0.236217i	$-0.924092\pi$
$269$	− 7.74849i	− 0.472434i	0.971700	−	0.236217i	$-0.0759076\pi$
$270$	0	0
$271$	9.18463i	0.557927i	0.960302	+	0.278963i	$0.0899908\pi$
$271$	9.18463i	0.557927i	−0.960302	+	0.278963i	$0.910009\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−8.33077	−0.502365
$276$	0	0
$277$	−8.74135	−0.525217	−0.262608	−	0.964903i	$-0.584583\pi$
$277$	−8.74135	−0.525217	−0.262608	+	0.964903i	$0.584583\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 0.0344009i	− 0.00205218i	−0.999999	−	0.00102609i	$-0.999673\pi$
$281$	− 0.0344009i	− 0.00205218i	0.999999	−	0.00102609i	$-0.000326615\pi$
$282$	0	0
$283$	2.15402i	0.128043i	0.997949	+	0.0640216i	$0.0203927\pi$
$283$	2.15402i	0.128043i	−0.997949	+	0.0640216i	$0.979607\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	10.3306	0.609797
$288$	0	0
$289$	−1.66281	−0.0978125
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 19.6228i	− 1.14638i	−0.819424	−	0.573188i	$-0.805707\pi$
$293$	− 19.6228i	− 1.14638i	0.819424	−	0.573188i	$-0.194293\pi$
$294$	0	0
$295$	19.4793i	1.13413i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	29.5940	1.71146
$300$	0	0
$301$	−2.97955	−0.171738
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 20.9442i	− 1.19926i
$306$	0	0
$307$	− 4.15577i	− 0.237182i	−0.992943	−	0.118591i	$-0.962162\pi$
$307$	− 4.15577i	− 0.237182i	0.992943	−	0.118591i	$-0.0378378\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	1.66432	0.0943749	0.0471875	−	0.998886i	$-0.484974\pi$
$311$	1.66432	0.0943749	0.0471875	+	0.998886i	$0.484974\pi$
$312$	0	0
$313$	13.4910	0.762554	0.381277	−	0.924461i	$-0.375484\pi$
$313$	13.4910	0.762554	0.381277	+	0.924461i	$0.375484\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	15.5212i	0.871760i	0.900005	+	0.435880i	$0.143563\pi$
$317$	15.5212i	0.871760i	−0.900005	+	0.435880i	$0.856437\pi$
$318$	0	0
$319$	− 23.2783i	− 1.30333i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−25.7516	−1.43286
$324$	0	0
$325$	−18.9982	−1.05383
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 56.6432i	− 3.12284i
$330$	0	0
$331$	− 16.2371i	− 0.892470i	−0.894916	−	0.446235i	$-0.852765\pi$
$331$	− 16.2371i	− 0.892470i	0.894916	−	0.446235i	$-0.147235\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−0.0367866	−0.00200987
$336$	0	0
$337$	14.9415	0.813913	0.406956	−	0.913448i	$-0.366590\pi$
$337$	14.9415	0.813913	0.406956	+	0.913448i	$0.366590\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	25.2975i	1.36994i
$342$	0	0
$343$	− 54.6432i	− 2.95045i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	16.6889	0.895908	0.447954	−	0.894057i	$-0.352153\pi$
$347$	16.6889	0.895908	0.447954	+	0.894057i	$0.352153\pi$
$348$	0	0
$349$	−4.97254	−0.266174	−0.133087	−	0.991104i	$-0.542489\pi$
$349$	−4.97254	−0.266174	−0.133087	+	0.991104i	$0.542489\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 14.0170i	− 0.746050i	−0.927821	−	0.373025i	$-0.878321\pi$
$353$	− 14.0170i	− 0.746050i	0.927821	−	0.373025i	$-0.121679\pi$
$354$	0	0
$355$	20.6549i	1.09625i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	28.2867	1.49291	0.746456	−	0.665435i	$-0.231754\pi$
$359$	28.2867	1.49291	0.746456	+	0.665435i	$0.231754\pi$
$360$	0	0
$361$	−16.5329	−0.870152
$362$	0	0
$363$	0	0
$364$	0	0
$365$	26.0620i	1.36415i
$366$	0	0
$367$	10.0489i	0.524548i	0.964993	+	0.262274i	$0.0844724\pi$
$367$	10.0489i	0.524548i	−0.964993	+	0.262274i	$0.915528\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−47.3905	−2.46039
$372$	0	0
$373$	25.7348	1.33250	0.666248	−	0.745730i	$-0.267899\pi$
$373$	25.7348	1.33250	0.666248	+	0.745730i	$0.267899\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 53.0859i	− 2.73406i
$378$	0	0
$379$	− 24.8403i	− 1.27596i	−0.770054	−	0.637979i	$-0.779771\pi$
$379$	− 24.8403i	− 1.27596i	0.770054	−	0.637979i	$-0.220229\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	2.81256	0.143715	0.0718576	−	0.997415i	$-0.477107\pi$
$383$	2.81256	0.143715	0.0718576	+	0.997415i	$0.477107\pi$
$384$	0	0
$385$	40.4148	2.05973
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 17.3522i	− 0.879790i	−0.898049	−	0.439895i	$-0.855016\pi$
$389$	− 17.3522i	− 0.879790i	0.898049	−	0.439895i	$-0.144984\pi$
$390$	0	0
$391$	19.4616i	0.984216i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	33.5054	1.68584
$396$	0	0
$397$	−15.5886	−0.782369	−0.391185	−	0.920312i	$-0.627935\pi$
$397$	−15.5886	−0.782369	−0.391185	+	0.920312i	$0.627935\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	15.7383i	0.785931i	0.919553	+	0.392965i	$0.128551\pi$
$401$	15.7383i	0.785931i	−0.919553	+	0.392965i	$0.871449\pi$
$402$	0	0
$403$	57.6906i	2.87377i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−2.88061	−0.142787
$408$	0	0
$409$	−10.1860	−0.503663	−0.251832	−	0.967771i	$-0.581033\pi$
$409$	−10.1860	−0.503663	−0.251832	+	0.967771i	$0.581033\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 34.6291i	− 1.70399i
$414$	0	0
$415$	− 4.67194i	− 0.229336i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	2.23998	0.109430	0.0547150	−	0.998502i	$-0.482575\pi$
$419$	2.23998	0.109430	0.0547150	+	0.998502i	$0.482575\pi$
$420$	0	0
$421$	29.9767	1.46098	0.730488	−	0.682926i	$-0.239293\pi$
$421$	29.9767	1.46098	0.730488	+	0.682926i	$0.239293\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 12.4936i	− 0.606030i
$426$	0	0
$427$	37.2332i	1.80184i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−32.0400	−1.54331	−0.771655	−	0.636041i	$-0.780571\pi$
$431$	−32.0400	−1.54331	−0.771655	+	0.636041i	$0.780571\pi$
$432$	0	0
$433$	6.21916	0.298874	0.149437	−	0.988771i	$-0.452254\pi$
$433$	6.21916	0.298874	0.149437	+	0.988771i	$0.452254\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	26.8538i	1.28459i
$438$	0	0
$439$	23.4644i	1.11990i	0.828528	+	0.559948i	$0.189179\pi$
$439$	23.4644i	1.11990i	−0.828528	+	0.559948i	$0.810821\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−8.55338	−0.406383	−0.203192	−	0.979139i	$-0.565131\pi$
$443$	−8.55338	−0.406383	−0.203192	+	0.979139i	$0.565131\pi$
$444$	0	0
$445$	−43.3711	−2.05599
$446$	0	0
$447$	0	0
$448$	0	0
$449$	26.1931i	1.23613i	0.786128	+	0.618064i	$0.212083\pi$
$449$	26.1931i	1.23613i	−0.786128	+	0.618064i	$0.787917\pi$
$450$	0	0
$451$	− 5.95869i	− 0.280584i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	92.1654	4.32078
$456$	0	0
$457$	31.1055	1.45505	0.727527	−	0.686079i	$-0.240669\pi$
$457$	31.1055	1.45505	0.727527	+	0.686079i	$0.240669\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 17.8226i	− 0.830080i	−0.909803	−	0.415040i	$-0.863768\pi$
$461$	− 17.8226i	− 0.830080i	0.909803	−	0.415040i	$-0.136232\pi$
$462$	0	0
$463$	16.8725i	0.784134i	0.919937	+	0.392067i	$0.128240\pi$
$463$	16.8725i	0.784134i	−0.919937	+	0.392067i	$0.871760\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	7.73795	0.358070	0.179035	−	0.983843i	$-0.442703\pi$
$467$	7.73795	0.358070	0.179035	+	0.983843i	$0.442703\pi$
$468$	0	0
$469$	0.0653968	0.00301974
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1.71860i	0.0790212i
$474$	0	0
$475$	− 17.2391i	− 0.790986i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	13.8421	0.632464	0.316232	−	0.948682i	$-0.397582\pi$
$479$	13.8421	0.632464	0.316232	+	0.948682i	$0.397582\pi$
$480$	0	0
$481$	−6.56921	−0.299530
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 15.0346i	− 0.682688i
$486$	0	0
$487$	− 0.106116i	− 0.00480856i	−0.999997	−	0.00240428i	$-0.999235\pi$
$487$	− 0.106116i	− 0.00480856i	0.999997	−	0.00240428i	$-0.000765308\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	1.23946	0.0559359	0.0279680	−	0.999609i	$-0.491096\pi$
$491$	1.23946	0.0559359	0.0279680	+	0.999609i	$0.491096\pi$
$492$	0	0
$493$	34.9104	1.57228
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 36.7189i	− 1.64707i
$498$	0	0
$499$	10.3008i	0.461126i	0.973057	+	0.230563i	$0.0740568\pi$
$499$	10.3008i	0.461126i	−0.973057	+	0.230563i	$0.925943\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	1.50295	0.0670130	0.0335065	−	0.999438i	$-0.489333\pi$
$503$	1.50295	0.0670130	0.0335065	+	0.999438i	$0.489333\pi$
$504$	0	0
$505$	6.24014	0.277683
$506$	0	0
$507$	0	0
$508$	0	0
$509$	25.2718i	1.12015i	0.828441	+	0.560076i	$0.189228\pi$
$509$	25.2718i	1.12015i	−0.828441	+	0.560076i	$0.810772\pi$
$510$	0	0
$511$	− 46.3312i	− 2.04957i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−8.20296	−0.361466
$516$	0	0
$517$	−32.6717	−1.43690
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 29.2373i	− 1.28091i	−0.767995	−	0.640456i	$-0.778746\pi$
$521$	− 29.2373i	− 1.28091i	0.767995	−	0.640456i	$-0.221254\pi$
$522$	0	0
$523$	− 36.3425i	− 1.58915i	−0.607168	−	0.794574i	$-0.707694\pi$
$523$	− 36.3425i	− 1.58915i	0.607168	−	0.794574i	$-0.292306\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−37.9385	−1.65263
$528$	0	0
$529$	−2.70539	−0.117626
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 13.5887i	− 0.588593i
$534$	0	0
$535$	15.9501i	0.689584i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−51.6824	−2.22612
$540$	0	0
$541$	−43.7494	−1.88093	−0.940466	−	0.339887i	$-0.889611\pi$
$541$	−43.7494	−1.88093	−0.940466	+	0.339887i	$0.889611\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	13.4429i	0.575830i
$546$	0	0
$547$	− 37.5332i	− 1.60480i	−0.596785	−	0.802401i	$-0.703556\pi$
$547$	− 37.5332i	− 1.60480i	0.596785	−	0.802401i	$-0.296444\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	48.1705	2.05213
$552$	0	0
$553$	−59.5638	−2.53291
$554$	0	0
$555$	0	0
$556$	0	0
$557$	31.6108i	1.33939i	0.742634	+	0.669697i	$0.233576\pi$
$557$	31.6108i	1.33939i	−0.742634	+	0.669697i	$0.766424\pi$
$558$	0	0
$559$	3.91924i	0.165766i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−33.6405	−1.41778	−0.708888	−	0.705321i	$-0.750803\pi$
$563$	−33.6405	−1.41778	−0.708888	+	0.705321i	$0.750803\pi$
$564$	0	0
$565$	41.9367	1.76429
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 45.6502i	− 1.91376i	−0.290487	−	0.956879i	$-0.593817\pi$
$569$	− 45.6502i	− 1.91376i	0.290487	−	0.956879i	$-0.406183\pi$
$570$	0	0
$571$	29.5804i	1.23790i	0.785429	+	0.618951i	$0.212442\pi$
$571$	29.5804i	1.23790i	−0.785429	+	0.618951i	$0.787558\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−13.0284	−0.543321
$576$	0	0
$577$	0.645125	0.0268569	0.0134285	−	0.999910i	$-0.495725\pi$
$577$	0.645125	0.0268569	0.0134285	+	0.999910i	$0.495725\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	8.30547i	0.344569i
$582$	0	0
$583$	27.3348i	1.13209i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.10398	0.0455659	0.0227830	−	0.999740i	$-0.492747\pi$
$587$	1.10398	0.0455659	0.0227830	+	0.999740i	$0.492747\pi$
$588$	0	0
$589$	−52.3489	−2.15700
$590$	0	0
$591$	0	0
$592$	0	0
$593$	14.3175i	0.587951i	0.955813	+	0.293976i	$0.0949784\pi$
$593$	14.3175i	0.587951i	−0.955813	+	0.293976i	$0.905022\pi$
$594$	0	0
$595$	60.6099i	2.48476i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	11.2290	0.458805	0.229403	−	0.973332i	$-0.426323\pi$
$599$	11.2290	0.458805	0.229403	+	0.973332i	$0.426323\pi$
$600$	0	0
$601$	15.7720	0.643354	0.321677	−	0.946849i	$-0.395753\pi$
$601$	15.7720	0.643354	0.321677	+	0.946849i	$0.395753\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	7.59082i	0.308611i
$606$	0	0
$607$	24.5109i	0.994866i	0.867502	+	0.497433i	$0.165724\pi$
$607$	24.5109i	0.994866i	−0.867502	+	0.497433i	$0.834276\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−74.5075	−3.01425
$612$	0	0
$613$	−4.77579	−0.192892	−0.0964462	−	0.995338i	$-0.530748\pi$
$613$	−4.77579	−0.192892	−0.0964462	+	0.995338i	$0.530748\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 7.14769i	− 0.287755i	−0.989596	−	0.143878i	$-0.954043\pi$
$617$	− 7.14769i	− 0.287755i	0.989596	−	0.143878i	$-0.0459571\pi$
$618$	0	0
$619$	31.7803i	1.27736i	0.769473	+	0.638679i	$0.220519\pi$
$619$	31.7803i	1.27736i	−0.769473	+	0.638679i	$0.779481\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	77.1024	3.08904
$624$	0	0
$625$	−31.0963	−1.24385
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 4.32005i	− 0.172252i
$630$	0	0
$631$	48.1374i	1.91632i	0.286235	+	0.958159i	$0.407596\pi$
$631$	48.1374i	1.91632i	−0.286235	+	0.958159i	$0.592404\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−45.5846	−1.80897
$636$	0	0
$637$	−117.861	−4.66983
$638$	0	0
$639$	0	0
$640$	0	0
$641$	24.9698i	0.986249i	0.869959	+	0.493124i	$0.164145\pi$
$641$	24.9698i	0.986249i	−0.869959	+	0.493124i	$0.835855\pi$
$642$	0	0
$643$	− 6.97278i	− 0.274979i	−0.990503	−	0.137490i	$-0.956097\pi$
$643$	− 6.97278i	− 0.274979i	0.990503	−	0.137490i	$-0.0439034\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−3.72733	−0.146536	−0.0732681	−	0.997312i	$-0.523343\pi$
$647$	−3.72733	−0.146536	−0.0732681	+	0.997312i	$0.523343\pi$
$648$	0	0
$649$	−19.9740	−0.784049
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 21.8097i	− 0.853478i	−0.904375	−	0.426739i	$-0.859662\pi$
$653$	− 21.8097i	− 0.853478i	0.904375	−	0.426739i	$-0.140338\pi$
$654$	0	0
$655$	− 35.7968i	− 1.39870i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−20.8067	−0.810512	−0.405256	−	0.914203i	$-0.632818\pi$
$659$	−20.8067	−0.810512	−0.405256	+	0.914203i	$0.632818\pi$
$660$	0	0
$661$	−4.38317	−0.170485	−0.0852427	−	0.996360i	$-0.527167\pi$
$661$	−4.38317	−0.170485	−0.0852427	+	0.996360i	$0.527167\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	83.6315i	3.24309i
$666$	0	0
$667$	− 36.4046i	− 1.40959i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	21.4760	0.829074
$672$	0	0
$673$	41.9917	1.61866	0.809331	−	0.587353i	$-0.199830\pi$
$673$	41.9917	1.61866	0.809331	+	0.587353i	$0.199830\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	19.1585i	0.736320i	0.929762	+	0.368160i	$0.120012\pi$
$677$	19.1585i	0.736320i	−0.929762	+	0.368160i	$0.879988\pi$
$678$	0	0
$679$	26.7276i	1.02571i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−11.9290	−0.456449	−0.228224	−	0.973609i	$-0.573292\pi$
$683$	−11.9290	−0.456449	−0.228224	+	0.973609i	$0.573292\pi$
$684$	0	0
$685$	55.8055	2.13222
$686$	0	0
$687$	0	0
$688$	0	0
$689$	62.3367i	2.37484i
$690$	0	0
$691$	− 42.1880i	− 1.60491i	−0.596716	−	0.802453i	$-0.703528\pi$
$691$	− 42.1880i	− 1.60491i	0.596716	−	0.802453i	$-0.296472\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−32.2192	−1.22214
$696$	0	0
$697$	8.93622	0.338484
$698$	0	0
$699$	0	0
$700$	0	0
$701$	42.5643i	1.60763i	0.594879	+	0.803815i	$0.297200\pi$
$701$	42.5643i	1.60763i	−0.594879	+	0.803815i	$0.702800\pi$
$702$	0	0
$703$	− 5.96095i	− 0.224821i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−11.0933	−0.417207
$708$	0	0
$709$	45.0421	1.69159	0.845796	−	0.533507i	$-0.179126\pi$
$709$	45.0421	1.69159	0.845796	+	0.533507i	$0.179126\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	39.5624i	1.48162i
$714$	0	0
$715$	− 53.1609i	− 1.98810i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−9.08858	−0.338947	−0.169473	−	0.985535i	$-0.554207\pi$
$719$	−9.08858	−0.338947	−0.169473	+	0.985535i	$0.554207\pi$
$720$	0	0
$721$	14.5827	0.543088
$722$	0	0
$723$	0	0
$724$	0	0
$725$	23.3704i	0.867955i
$726$	0	0
$727$	8.38952i	0.311150i	0.987824	+	0.155575i	$0.0497230\pi$
$727$	8.38952i	0.311150i	−0.987824	+	0.155575i	$0.950277\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−2.57738	−0.0953277
$732$	0	0
$733$	21.3101	0.787106	0.393553	−	0.919302i	$-0.371246\pi$
$733$	21.3101	0.787106	0.393553	+	0.919302i	$0.371246\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 0.0377208i	− 0.00138946i
$738$	0	0
$739$	− 20.5320i	− 0.755282i	−0.925952	−	0.377641i	$-0.876735\pi$
$739$	− 20.5320i	− 0.755282i	0.925952	−	0.377641i	$-0.123265\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	10.1598	0.372726	0.186363	−	0.982481i	$-0.440330\pi$
$743$	10.1598	0.372726	0.186363	+	0.982481i	$0.440330\pi$
$744$	0	0
$745$	8.71950	0.319458
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 28.3551i	− 1.03607i
$750$	0	0
$751$	47.3726i	1.72865i	0.502933	+	0.864325i	$0.332254\pi$
$751$	47.3726i	1.72865i	−0.502933	+	0.864325i	$0.667746\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−8.46153	−0.307947
$756$	0	0
$757$	−10.0246	−0.364350	−0.182175	−	0.983266i	$-0.558314\pi$
$757$	−10.0246	−0.364350	−0.182175	+	0.983266i	$0.558314\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 31.9570i	− 1.15844i	−0.815171	−	0.579221i	$-0.803357\pi$
$761$	− 31.9570i	− 1.15844i	0.815171	−	0.579221i	$-0.196643\pi$
$762$	0	0
$763$	− 23.8979i	− 0.865162i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−45.5505	−1.64473
$768$	0	0
$769$	23.8880	0.861422	0.430711	−	0.902490i	$-0.358263\pi$
$769$	23.8880	0.861422	0.430711	+	0.902490i	$0.358263\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	16.2119i	0.583100i	0.956556	+	0.291550i	$0.0941709\pi$
$773$	16.2119i	0.583100i	−0.956556	+	0.291550i	$0.905829\pi$
$774$	0	0
$775$	− 25.3976i	− 0.912308i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	12.3305	0.441786
$780$	0	0
$781$	−21.1794	−0.757859
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 8.75295i	− 0.312406i
$786$	0	0
$787$	− 19.6053i	− 0.698853i	−0.936964	−	0.349426i	$-0.886376\pi$
$787$	− 19.6053i	− 0.698853i	0.936964	−	0.349426i	$-0.113624\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−74.5524	−2.65078
$792$	0	0
$793$	48.9759	1.73918
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 55.7424i	− 1.97450i	−0.159183	−	0.987249i	$-0.550886\pi$
$797$	− 55.7424i	− 1.97450i	0.159183	−	0.987249i	$-0.449114\pi$
$798$	0	0
$799$	− 48.9977i	− 1.73341i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−26.7238	−0.943063
$804$	0	0
$805$	63.2041	2.22765
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 29.6082i	− 1.04097i	−0.853871	−	0.520484i	$-0.825752\pi$
$809$	− 29.6082i	− 1.04097i	0.853871	−	0.520484i	$-0.174248\pi$
$810$	0	0
$811$	27.7499i	0.974432i	0.873281	+	0.487216i	$0.161988\pi$
$811$	27.7499i	0.974432i	−0.873281	+	0.487216i	$0.838012\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−8.03564	−0.281476
$816$	0	0
$817$	−3.55635	−0.124421
$818$	0	0
$819$	0	0
$820$	0	0
$821$	52.1379i	1.81962i	0.415020	+	0.909812i	$0.363774\pi$
$821$	52.1379i	1.81962i	−0.415020	+	0.909812i	$0.636226\pi$
$822$	0	0
$823$	40.0353i	1.39554i	0.716321	+	0.697771i	$0.245825\pi$
$823$	40.0353i	1.39554i	−0.716321	+	0.697771i	$0.754175\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	47.6451	1.65678	0.828391	−	0.560150i	$-0.189257\pi$
$827$	47.6451	1.65678	0.828391	+	0.560150i	$0.189257\pi$
$828$	0	0
$829$	−16.8437	−0.585005	−0.292503	−	0.956265i	$-0.594488\pi$
$829$	−16.8437	−0.585005	−0.292503	+	0.956265i	$0.594488\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 77.5079i	− 2.68549i
$834$	0	0
$835$	− 55.7259i	− 1.92847i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−16.3141	−0.563225	−0.281613	−	0.959528i	$-0.590869\pi$
$839$	−16.3141	−0.563225	−0.281613	+	0.959528i	$0.590869\pi$
$840$	0	0
$841$	−36.3028	−1.25182
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 84.7122i	− 2.91419i
$846$	0	0
$847$	− 13.4945i	− 0.463675i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−4.50495	−0.154428
$852$	0	0
$853$	−51.2845	−1.75595	−0.877974	−	0.478708i	$-0.841105\pi$
$853$	−51.2845	−1.75595	−0.877974	+	0.478708i	$0.841105\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 10.2021i	− 0.348496i	−0.984702	−	0.174248i	$-0.944251\pi$
$857$	− 10.2021i	− 0.348496i	0.984702	−	0.174248i	$-0.0557495\pi$
$858$	0	0
$859$	− 19.6655i	− 0.670977i	−0.942044	−	0.335489i	$-0.891099\pi$
$859$	− 19.6655i	− 0.670977i	0.942044	−	0.335489i	$-0.108901\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−25.6692	−0.873789	−0.436894	−	0.899513i	$-0.643922\pi$
$863$	−25.6692	−0.873789	−0.436894	+	0.899513i	$0.643922\pi$
$864$	0	0
$865$	30.1087	1.02373
$866$	0	0
$867$	0	0
$868$	0	0
$869$	34.3563i	1.16546i
$870$	0	0
$871$	− 0.0860218i	− 0.00291474i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	29.5749	0.999813
$876$	0	0
$877$	−11.4498	−0.386634	−0.193317	−	0.981136i	$-0.561925\pi$
$877$	−11.4498	−0.386634	−0.193317	+	0.981136i	$0.561925\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 30.6017i	− 1.03100i	−0.856890	−	0.515499i	$-0.827607\pi$
$881$	− 30.6017i	− 1.03100i	0.856890	−	0.515499i	$-0.172393\pi$
$882$	0	0
$883$	8.82961i	0.297140i	0.988902	+	0.148570i	$0.0474670\pi$
$883$	8.82961i	0.297140i	−0.988902	+	0.148570i	$0.952533\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−23.1925	−0.778726	−0.389363	−	0.921084i	$-0.627305\pi$
$887$	−23.1925	−0.778726	−0.389363	+	0.921084i	$0.627305\pi$
$888$	0	0
$889$	81.0374	2.71791
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 67.6086i	− 2.26244i
$894$	0	0
$895$	37.2383i	1.24474i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	70.9673	2.36689
$900$	0	0
$901$	−40.9939	−1.36570
$902$	0	0
$903$	0	0
$904$	0	0
$905$	24.8704i	0.826719i
$906$	0	0
$907$	17.9132i	0.594799i	0.954753	+	0.297400i	$0.0961193\pi$
$907$	17.9132i	0.594799i	−0.954753	+	0.297400i	$0.903881\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	22.2905	0.738517	0.369259	−	0.929327i	$-0.379612\pi$
$911$	22.2905	0.738517	0.369259	+	0.929327i	$0.379612\pi$
$912$	0	0
$913$	4.79058	0.158545
$914$	0	0
$915$	0	0
$916$	0	0
$917$	63.6372i	2.10149i
$918$	0	0
$919$	− 40.0238i	− 1.32026i	−0.751150	−	0.660131i	$-0.770501\pi$
$919$	− 40.0238i	− 1.32026i	0.751150	−	0.660131i	$-0.229499\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−48.2994	−1.58979
$924$	0	0
$925$	2.89201	0.0950888
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 42.3223i	− 1.38855i	−0.719710	−	0.694275i	$-0.755725\pi$
$929$	− 42.3223i	− 1.38855i	0.719710	−	0.694275i	$-0.244275\pi$
$930$	0	0
$931$	− 106.948i	− 3.50508i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	34.9597	1.14330
$936$	0	0
$937$	1.30170	0.0425247	0.0212624	−	0.999774i	$-0.493231\pi$
$937$	1.30170	0.0425247	0.0212624	+	0.999774i	$0.493231\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	20.1447i	0.656700i	0.944556	+	0.328350i	$0.106493\pi$
$941$	20.1447i	0.656700i	−0.944556	+	0.328350i	$0.893507\pi$
$942$	0	0
$943$	− 9.31871i	− 0.303459i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	14.8590	0.482853	0.241426	−	0.970419i	$-0.422385\pi$
$947$	14.8590	0.482853	0.241426	+	0.970419i	$0.422385\pi$
$948$	0	0
$949$	−60.9433	−1.97830
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 10.1548i	− 0.328945i	−0.986382	−	0.164473i	$-0.947408\pi$
$953$	− 10.1548i	− 0.328945i	0.986382	−	0.164473i	$-0.0525922\pi$
$954$	0	0
$955$	8.39408i	0.271626i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−99.2074	−3.20357
$960$	0	0
$961$	−46.1231	−1.48784
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 65.4844i	− 2.10802i
$966$	0	0
$967$	− 39.0327i	− 1.25521i	−0.778533	−	0.627603i	$-0.784036\pi$
$967$	− 39.0327i	− 1.25521i	0.778533	−	0.627603i	$-0.215964\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	38.0931	1.22247	0.611233	−	0.791451i	$-0.290674\pi$
$971$	38.0931	1.22247	0.611233	+	0.791451i	$0.290674\pi$
$972$	0	0
$973$	57.2772	1.83622
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 28.7066i	− 0.918406i	−0.888331	−	0.459203i	$-0.848135\pi$
$977$	− 28.7066i	− 0.918406i	0.888331	−	0.459203i	$-0.151865\pi$
$978$	0	0
$979$	− 44.4726i	− 1.42135i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	36.7424	1.17190	0.585951	−	0.810347i	$-0.300721\pi$
$983$	36.7424	1.17190	0.585951	+	0.810347i	$0.300721\pi$
$984$	0	0
$985$	36.2528	1.15511
$986$	0	0
$987$	0	0
$988$	0	0
$989$	2.68769i	0.0854636i
$990$	0	0
$991$	− 13.0344i	− 0.414050i	−0.978336	−	0.207025i	$-0.933622\pi$
$991$	− 13.0344i	− 0.414050i	0.978336	−	0.207025i	$-0.0663782\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	29.3558	0.930641
$996$	0	0
$997$	37.3268	1.18215	0.591075	−	0.806616i	$-0.298704\pi$
$997$	37.3268	1.18215	0.591075	+	0.806616i	$0.298704\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5328.2.e.f.2591.6	yes	24
3.2	odd	2	inner	5328.2.e.f.2591.20	yes	24
4.3	odd	2	inner	5328.2.e.f.2591.5	✓	24
12.11	even	2	inner	5328.2.e.f.2591.19	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5328.2.e.f.2591.5	✓	24	4.3	odd	2	inner
5328.2.e.f.2591.6	yes	24	1.1	even	1	trivial
5328.2.e.f.2591.19	yes	24	12.11	even	2	inner
5328.2.e.f.2591.20	yes	24	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 5328 = 2^{4} \cdot 3^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5328.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.80927i q^{5} +4.99414i q^{7} +2.88061 q^{11} +6.56921 q^{13} +4.32005i q^{17} +5.96095i q^{19} +4.50495 q^{23} -2.89201 q^{25} -8.08102i q^{29} +8.78197i q^{31} +14.0299 q^{35} -1.00000 q^{37} -2.06855i q^{41} +0.596608i q^{43} -11.3419 q^{47} -17.9415 q^{49} +9.48922i q^{53} -8.09243i q^{55} -6.93395 q^{59} +7.45537 q^{61} -18.4547i q^{65} -0.0130947i q^{67} -7.35239 q^{71} -9.27712 q^{73} +14.3862i q^{77} +11.9267i q^{79} +1.66304 q^{83} +12.1362 q^{85} -15.4386i q^{89} +32.8076i q^{91} +16.7459 q^{95} +5.35179 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 32 q^{25} - 24 q^{37} - 80 q^{49} - 48 q^{61} - 48 q^{73} - 40 q^{85} + 24 q^{97}+O(q^{100}) \) 24 * q - 32 * q^25 - 24 * q^37 - 80 * q^49 - 48 * q^61 - 48 * q^73 - 40 * q^85 + 24 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more