LMFDB - Embedded newform 5328.2.e.f.2591.17

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			2591.17
Character	$\chi$	$=$	5328.2591
Dual form			5328.2.e.f.2591.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.30074i q^{5} -4.07997i q^{7} -2.75773 q^{11} -4.51956 q^{13} +3.51408i q^{17} -5.45570i q^{19} +0.877830 q^{23} -0.293412 q^{25} -0.290393i q^{29} +1.57873i q^{31} +9.38696 q^{35} -1.00000 q^{37} +3.40784i q^{41} -2.37454i q^{43} +0.815705 q^{47} -9.64615 q^{49} +11.8309i q^{53} -6.34483i q^{55} -6.27292 q^{59} +3.66442 q^{61} -10.3983i q^{65} +4.98958i q^{67} +10.7639 q^{71} +11.0152 q^{73} +11.2515i q^{77} +10.8077i q^{79} +14.3417 q^{83} -8.08498 q^{85} +7.85597i q^{89} +18.4397i q^{91} +12.5521 q^{95} +14.8538 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	2.30074i	1.02892i	0.857513	+	0.514461i	$0.172008\pi$
$5$	2.30074i	1.02892i	−0.857513	+	0.514461i	$0.827992\pi$
$6$	0	0
$7$	− 4.07997i	− 1.54208i	−0.636785	−	0.771042i	$-0.719736\pi$
$7$	− 4.07997i	− 1.54208i	0.636785	−	0.771042i	$-0.280264\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−2.75773	−0.831488	−0.415744	−	0.909482i	$-0.636479\pi$
$11$	−2.75773	−0.831488	−0.415744	+	0.909482i	$0.636479\pi$
$12$	0	0
$13$	−4.51956	−1.25350	−0.626751	−	0.779220i	$-0.715615\pi$
$13$	−4.51956	−1.25350	−0.626751	+	0.779220i	$0.715615\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.51408i	0.852288i	0.904655	+	0.426144i	$0.140128\pi$
$17$	3.51408i	0.852288i	−0.904655	+	0.426144i	$0.859872\pi$
$18$	0	0
$19$	− 5.45570i	− 1.25162i	−0.779975	−	0.625811i	$-0.784768\pi$
$19$	− 5.45570i	− 1.25162i	0.779975	−	0.625811i	$-0.215232\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0.877830	0.183040	0.0915201	−	0.995803i	$-0.470827\pi$
$23$	0.877830	0.183040	0.0915201	+	0.995803i	$0.470827\pi$
$24$	0	0
$25$	−0.293412	−0.0586825
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 0.290393i	− 0.0539246i	−0.999636	−	0.0269623i	$-0.991417\pi$
$29$	− 0.290393i	− 0.0539246i	0.999636	−	0.0269623i	$-0.00858341\pi$
$30$	0	0
$31$	1.57873i	0.283548i	0.989899	+	0.141774i	$0.0452806\pi$
$31$	1.57873i	0.283548i	−0.989899	+	0.141774i	$0.954719\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	9.38696	1.58669
$36$	0	0
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	0	0
$39$	0	0
$40$	0	0
$41$	3.40784i	0.532216i	0.963943	+	0.266108i	$0.0857377\pi$
$41$	3.40784i	0.532216i	−0.963943	+	0.266108i	$0.914262\pi$
$42$	0	0
$43$	− 2.37454i	− 0.362114i	−0.983473	−	0.181057i	$-0.942048\pi$
$43$	− 2.37454i	− 0.362114i	0.983473	−	0.181057i	$-0.0579519\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0.815705	0.118983	0.0594914	−	0.998229i	$-0.481052\pi$
$47$	0.815705	0.118983	0.0594914	+	0.998229i	$0.481052\pi$
$48$	0	0
$49$	−9.64615	−1.37802
$50$	0	0
$51$	0	0
$52$	0	0
$53$	11.8309i	1.62510i	0.582894	+	0.812548i	$0.301920\pi$
$53$	11.8309i	1.62510i	−0.582894	+	0.812548i	$0.698080\pi$
$54$	0	0
$55$	− 6.34483i	− 0.855537i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−6.27292	−0.816665	−0.408333	−	0.912833i	$-0.633890\pi$
$59$	−6.27292	−0.816665	−0.408333	+	0.912833i	$0.633890\pi$
$60$	0	0
$61$	3.66442	0.469181	0.234590	−	0.972094i	$-0.424625\pi$
$61$	3.66442	0.469181	0.234590	+	0.972094i	$0.424625\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 10.3983i	− 1.28976i
$66$	0	0
$67$	4.98958i	0.609574i	0.952420	+	0.304787i	$0.0985853\pi$
$67$	4.98958i	0.609574i	−0.952420	+	0.304787i	$0.901415\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	10.7639	1.27744	0.638719	−	0.769440i	$-0.279465\pi$
$71$	10.7639	1.27744	0.638719	+	0.769440i	$0.279465\pi$
$72$	0	0
$73$	11.0152	1.28923	0.644617	−	0.764506i	$-0.277017\pi$
$73$	11.0152	1.28923	0.644617	+	0.764506i	$0.277017\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	11.2515i	1.28222i
$78$	0	0
$79$	10.8077i	1.21597i	0.793950	+	0.607983i	$0.208021\pi$
$79$	10.8077i	1.21597i	−0.793950	+	0.607983i	$0.791979\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	14.3417	1.57421	0.787105	−	0.616819i	$-0.211579\pi$
$83$	14.3417	1.57421	0.787105	+	0.616819i	$0.211579\pi$
$84$	0	0
$85$	−8.08498	−0.876939
$86$	0	0
$87$	0	0
$88$	0	0
$89$	7.85597i	0.832731i	0.909197	+	0.416365i	$0.136696\pi$
$89$	7.85597i	0.832731i	−0.909197	+	0.416365i	$0.863304\pi$
$90$	0	0
$91$	18.4397i	1.93300i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	12.5521	1.28782
$96$	0	0
$97$	14.8538	1.50817	0.754086	−	0.656775i	$-0.228080\pi$
$97$	14.8538	1.50817	0.754086	+	0.656775i	$0.228080\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0.419851i	0.0417767i	0.999782	+	0.0208884i	$0.00664945\pi$
$101$	0.419851i	0.0417767i	−0.999782	+	0.0208884i	$0.993351\pi$
$102$	0	0
$103$	7.21224i	0.710643i	0.934744	+	0.355322i	$0.115629\pi$
$103$	7.21224i	0.710643i	−0.934744	+	0.355322i	$0.884371\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−12.6662	−1.22449	−0.612245	−	0.790668i	$-0.709733\pi$
$107$	−12.6662	−1.22449	−0.612245	+	0.790668i	$0.709733\pi$
$108$	0	0
$109$	1.10639	0.105973	0.0529863	−	0.998595i	$-0.483126\pi$
$109$	1.10639	0.105973	0.0529863	+	0.998595i	$0.483126\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 5.86394i	− 0.551633i	−0.961210	−	0.275817i	$-0.911052\pi$
$113$	− 5.86394i	− 0.551633i	0.961210	−	0.275817i	$-0.0889482\pi$
$114$	0	0
$115$	2.01966i	0.188334i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	14.3373	1.31430
$120$	0	0
$121$	−3.39491	−0.308628
$122$	0	0
$123$	0	0
$124$	0	0
$125$	10.8286i	0.968543i
$126$	0	0
$127$	10.2253i	0.907345i	0.891168	+	0.453673i	$0.149886\pi$
$127$	10.2253i	0.907345i	−0.891168	+	0.453673i	$0.850114\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	6.12270	0.534943	0.267472	−	0.963566i	$-0.413812\pi$
$131$	6.12270	0.534943	0.267472	+	0.963566i	$0.413812\pi$
$132$	0	0
$133$	−22.2591	−1.93011
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3.21484i	0.274662i	0.990525	+	0.137331i	$0.0438524\pi$
$137$	3.21484i	0.274662i	−0.990525	+	0.137331i	$0.956148\pi$
$138$	0	0
$139$	− 4.63394i	− 0.393046i	−0.980499	−	0.196523i	$-0.937035\pi$
$139$	− 4.63394i	− 0.393046i	0.980499	−	0.196523i	$-0.0629650\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	12.4637	1.04227
$144$	0	0
$145$	0.668119	0.0554843
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 13.7619i	− 1.12742i	−0.825973	−	0.563709i	$-0.809374\pi$
$149$	− 13.7619i	− 1.12742i	0.825973	−	0.563709i	$-0.190626\pi$
$150$	0	0
$151$	− 5.92226i	− 0.481947i	−0.970532	−	0.240973i	$-0.922533\pi$
$151$	− 5.92226i	− 0.481947i	0.970532	−	0.240973i	$-0.0774666\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−3.63225	−0.291749
$156$	0	0
$157$	−10.3969	−0.829764	−0.414882	−	0.909875i	$-0.636177\pi$
$157$	−10.3969	−0.829764	−0.414882	+	0.909875i	$0.636177\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 3.58152i	− 0.282263i
$162$	0	0
$163$	1.02688i	0.0804316i	0.999191	+	0.0402158i	$0.0128045\pi$
$163$	1.02688i	0.0804316i	−0.999191	+	0.0402158i	$0.987195\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	10.0103	0.774621	0.387310	−	0.921949i	$-0.373404\pi$
$167$	10.0103	0.774621	0.387310	+	0.921949i	$0.373404\pi$
$168$	0	0
$169$	7.42644	0.571265
$170$	0	0
$171$	0	0
$172$	0	0
$173$	10.2780i	0.781424i	0.920513	+	0.390712i	$0.127771\pi$
$173$	10.2780i	0.781424i	−0.920513	+	0.390712i	$0.872229\pi$
$174$	0	0
$175$	1.19711i	0.0904933i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−13.6497	−1.02023	−0.510115	−	0.860106i	$-0.670397\pi$
$179$	−13.6497	−1.02023	−0.510115	+	0.860106i	$0.670397\pi$
$180$	0	0
$181$	8.59298	0.638711	0.319356	−	0.947635i	$-0.396534\pi$
$181$	8.59298	0.638711	0.319356	+	0.947635i	$0.396534\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 2.30074i	− 0.169154i
$186$	0	0
$187$	− 9.69088i	− 0.708667i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	12.3305	0.892203	0.446102	−	0.894982i	$-0.352812\pi$
$191$	12.3305	0.892203	0.446102	+	0.894982i	$0.352812\pi$
$192$	0	0
$193$	25.7483	1.85340	0.926701	−	0.375799i	$-0.122632\pi$
$193$	25.7483	1.85340	0.926701	+	0.375799i	$0.122632\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 22.1456i	− 1.57781i	−0.614518	−	0.788903i	$-0.710650\pi$
$197$	− 22.1456i	− 1.57781i	0.614518	−	0.788903i	$-0.289350\pi$
$198$	0	0
$199$	19.0606i	1.35117i	0.737283	+	0.675584i	$0.236108\pi$
$199$	19.0606i	1.35117i	−0.737283	+	0.675584i	$0.763892\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−1.18479	−0.0831563
$204$	0	0
$205$	−7.84056	−0.547609
$206$	0	0
$207$	0	0
$208$	0	0
$209$	15.0454i	1.04071i
$210$	0	0
$211$	21.7979i	1.50063i	0.661082	+	0.750314i	$0.270098\pi$
$211$	21.7979i	1.50063i	−0.661082	+	0.750314i	$0.729902\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	5.46321	0.372588
$216$	0	0
$217$	6.44116	0.437255
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 15.8821i	− 1.06834i
$222$	0	0
$223$	11.5860i	0.775854i	0.921690	+	0.387927i	$0.126809\pi$
$223$	11.5860i	0.775854i	−0.921690	+	0.387927i	$0.873191\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0.727076	0.0482577	0.0241289	−	0.999709i	$-0.492319\pi$
$227$	0.727076	0.0482577	0.0241289	+	0.999709i	$0.492319\pi$
$228$	0	0
$229$	−20.6665	−1.36568	−0.682840	−	0.730568i	$-0.739255\pi$
$229$	−20.6665	−1.36568	−0.682840	+	0.730568i	$0.739255\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	14.1922i	0.929762i	0.885373	+	0.464881i	$0.153903\pi$
$233$	14.1922i	0.929762i	−0.885373	+	0.464881i	$0.846097\pi$
$234$	0	0
$235$	1.87673i	0.122424i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	2.29090	0.148186	0.0740932	−	0.997251i	$-0.476394\pi$
$239$	2.29090	0.148186	0.0740932	+	0.997251i	$0.476394\pi$
$240$	0	0
$241$	−19.6367	−1.26491	−0.632454	−	0.774598i	$-0.717952\pi$
$241$	−19.6367	−1.26491	−0.632454	+	0.774598i	$0.717952\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 22.1933i	− 1.41788i
$246$	0	0
$247$	24.6574i	1.56891i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	17.7067	1.11764	0.558819	−	0.829290i	$-0.311255\pi$
$251$	17.7067	1.11764	0.558819	+	0.829290i	$0.311255\pi$
$252$	0	0
$253$	−2.42082	−0.152196
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 14.1303i	− 0.881421i	−0.897649	−	0.440710i	$-0.854727\pi$
$257$	− 14.1303i	− 0.881421i	0.897649	−	0.440710i	$-0.145273\pi$
$258$	0	0
$259$	4.07997i	0.253517i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−11.0143	−0.679170	−0.339585	−	0.940575i	$-0.610287\pi$
$263$	−11.0143	−0.679170	−0.339585	+	0.940575i	$0.610287\pi$
$264$	0	0
$265$	−27.2198	−1.67210
$266$	0	0
$267$	0	0
$268$	0	0
$269$	16.6306i	1.01399i	0.861950	+	0.506994i	$0.169243\pi$
$269$	16.6306i	1.01399i	−0.861950	+	0.506994i	$0.830757\pi$
$270$	0	0
$271$	19.4585i	1.18202i	0.806664	+	0.591011i	$0.201271\pi$
$271$	19.4585i	1.18202i	−0.806664	+	0.591011i	$0.798729\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0.809153	0.0487937
$276$	0	0
$277$	3.36314	0.202072	0.101036	−	0.994883i	$-0.467784\pi$
$277$	3.36314	0.202072	0.101036	+	0.994883i	$0.467784\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 15.0998i	− 0.900778i	−0.892832	−	0.450389i	$-0.851285\pi$
$281$	− 15.0998i	− 0.900778i	0.892832	−	0.450389i	$-0.148715\pi$
$282$	0	0
$283$	5.73030i	0.340631i	0.985390	+	0.170315i	$0.0544786\pi$
$283$	5.73030i	0.340631i	−0.985390	+	0.170315i	$0.945521\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	13.9039	0.820721
$288$	0	0
$289$	4.65127	0.273604
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 0.820684i	− 0.0479449i	−0.999713	−	0.0239724i	$-0.992369\pi$
$293$	− 0.820684i	− 0.0479449i	0.999713	−	0.0239724i	$-0.00763140\pi$
$294$	0	0
$295$	− 14.4324i	− 0.840286i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−3.96741	−0.229441
$300$	0	0
$301$	−9.68806	−0.558411
$302$	0	0
$303$	0	0
$304$	0	0
$305$	8.43088i	0.482751i
$306$	0	0
$307$	− 21.5111i	− 1.22770i	−0.789421	−	0.613852i	$-0.789619\pi$
$307$	− 21.5111i	− 1.22770i	0.789421	−	0.613852i	$-0.210381\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−5.89088	−0.334041	−0.167021	−	0.985953i	$-0.553415\pi$
$311$	−5.89088	−0.334041	−0.167021	+	0.985953i	$0.553415\pi$
$312$	0	0
$313$	2.59744	0.146816	0.0734081	−	0.997302i	$-0.476612\pi$
$313$	2.59744	0.146816	0.0734081	+	0.997302i	$0.476612\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	33.7173i	1.89375i	0.321603	+	0.946875i	$0.395778\pi$
$317$	33.7173i	1.89375i	−0.321603	+	0.946875i	$0.604222\pi$
$318$	0	0
$319$	0.800826i	0.0448377i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	19.1717	1.06674
$324$	0	0
$325$	1.32610	0.0735585
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 3.32805i	− 0.183481i
$330$	0	0
$331$	− 17.6870i	− 0.972163i	−0.873913	−	0.486082i	$-0.838426\pi$
$331$	− 17.6870i	− 0.972163i	0.873913	−	0.486082i	$-0.161574\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−11.4797	−0.627205
$336$	0	0
$337$	6.64615	0.362039	0.181020	−	0.983480i	$-0.442060\pi$
$337$	6.64615	0.362039	0.181020	+	0.983480i	$0.442060\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 4.35371i	− 0.235767i
$342$	0	0
$343$	10.7962i	0.582941i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−16.3655	−0.878545	−0.439273	−	0.898354i	$-0.644764\pi$
$347$	−16.3655	−0.878545	−0.439273	+	0.898354i	$0.644764\pi$
$348$	0	0
$349$	−22.2718	−1.19218	−0.596092	−	0.802916i	$-0.703281\pi$
$349$	−22.2718	−1.19218	−0.596092	+	0.802916i	$0.703281\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	16.9083i	0.899936i	0.893045	+	0.449968i	$0.148565\pi$
$353$	16.9083i	0.899936i	−0.893045	+	0.449968i	$0.851435\pi$
$354$	0	0
$355$	24.7649i	1.31439i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	20.0573	1.05858	0.529292	−	0.848440i	$-0.322458\pi$
$359$	20.0573	1.05858	0.529292	+	0.848440i	$0.322458\pi$
$360$	0	0
$361$	−10.7646	−0.566559
$362$	0	0
$363$	0	0
$364$	0	0
$365$	25.3432i	1.32652i
$366$	0	0
$367$	5.90420i	0.308197i	0.988056	+	0.154098i	$0.0492473\pi$
$367$	5.90420i	0.308197i	−0.988056	+	0.154098i	$0.950753\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	48.2696	2.50603
$372$	0	0
$373$	−29.0893	−1.50619	−0.753094	−	0.657913i	$-0.771439\pi$
$373$	−29.0893	−1.50619	−0.753094	+	0.657913i	$0.771439\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.31245i	0.0675946i
$378$	0	0
$379$	12.1310i	0.623127i	0.950225	+	0.311563i	$0.100853\pi$
$379$	12.1310i	0.623127i	−0.950225	+	0.311563i	$0.899147\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−35.3197	−1.80475	−0.902377	−	0.430947i	$-0.858180\pi$
$383$	−35.3197	−1.80475	−0.902377	+	0.430947i	$0.858180\pi$
$384$	0	0
$385$	−25.8867	−1.31931
$386$	0	0
$387$	0	0
$388$	0	0
$389$	33.9534i	1.72150i	0.509024	+	0.860752i	$0.330006\pi$
$389$	33.9534i	1.72150i	−0.509024	+	0.860752i	$0.669994\pi$
$390$	0	0
$391$	3.08476i	0.156003i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−24.8658	−1.25114
$396$	0	0
$397$	0.946692	0.0475131	0.0237566	−	0.999718i	$-0.492437\pi$
$397$	0.946692	0.0475131	0.0237566	+	0.999718i	$0.492437\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	23.3649i	1.16679i	0.812190	+	0.583393i	$0.198275\pi$
$401$	23.3649i	1.16679i	−0.812190	+	0.583393i	$0.801725\pi$
$402$	0	0
$403$	− 7.13516i	− 0.355428i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2.75773	0.136696
$408$	0	0
$409$	11.4734	0.567326	0.283663	−	0.958924i	$-0.408450\pi$
$409$	11.4734	0.567326	0.283663	+	0.958924i	$0.408450\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	25.5933i	1.25937i
$414$	0	0
$415$	32.9966i	1.61974i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	32.3760	1.58167	0.790835	−	0.612030i	$-0.209647\pi$
$419$	32.3760	1.58167	0.790835	+	0.612030i	$0.209647\pi$
$420$	0	0
$421$	10.3204	0.502985	0.251492	−	0.967859i	$-0.419079\pi$
$421$	10.3204	0.502985	0.251492	+	0.967859i	$0.419079\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 1.03107i	− 0.0500144i
$426$	0	0
$427$	− 14.9507i	− 0.723516i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	20.6789	0.996067	0.498034	−	0.867158i	$-0.334056\pi$
$431$	20.6789	0.996067	0.498034	+	0.867158i	$0.334056\pi$
$432$	0	0
$433$	9.30351	0.447098	0.223549	−	0.974693i	$-0.428236\pi$
$433$	9.30351	0.447098	0.223549	+	0.974693i	$0.428236\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 4.78918i	− 0.229097i
$438$	0	0
$439$	− 23.6321i	− 1.12790i	−0.825809	−	0.563950i	$-0.809281\pi$
$439$	− 23.6321i	− 1.12790i	0.825809	−	0.563950i	$-0.190719\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	19.8560	0.943388	0.471694	−	0.881762i	$-0.343643\pi$
$443$	19.8560	0.943388	0.471694	+	0.881762i	$0.343643\pi$
$444$	0	0
$445$	−18.0746	−0.856816
$446$	0	0
$447$	0	0
$448$	0	0
$449$	26.4791i	1.24962i	0.780775	+	0.624812i	$0.214825\pi$
$449$	26.4791i	1.24962i	−0.780775	+	0.624812i	$0.785175\pi$
$450$	0	0
$451$	− 9.39792i	− 0.442531i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−42.4249	−1.98891
$456$	0	0
$457$	33.6473	1.57395	0.786977	−	0.616983i	$-0.211645\pi$
$457$	33.6473	1.57395	0.786977	+	0.616983i	$0.211645\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	4.75677i	0.221545i	0.993846	+	0.110772i	$0.0353325\pi$
$461$	4.75677i	0.221545i	−0.993846	+	0.110772i	$0.964668\pi$
$462$	0	0
$463$	− 32.9378i	− 1.53075i	−0.643586	−	0.765374i	$-0.722554\pi$
$463$	− 32.9378i	− 1.53075i	0.643586	−	0.765374i	$-0.277446\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−13.1511	−0.608563	−0.304281	−	0.952582i	$-0.598416\pi$
$467$	−13.1511	−0.608563	−0.304281	+	0.952582i	$0.598416\pi$
$468$	0	0
$469$	20.3573	0.940015
$470$	0	0
$471$	0	0
$472$	0	0
$473$	6.54835i	0.301094i
$474$	0	0
$475$	1.60077i	0.0734483i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	2.98594	0.136431	0.0682155	−	0.997671i	$-0.478269\pi$
$479$	2.98594	0.136431	0.0682155	+	0.997671i	$0.478269\pi$
$480$	0	0
$481$	4.51956	0.206074
$482$	0	0
$483$	0	0
$484$	0	0
$485$	34.1747i	1.55179i
$486$	0	0
$487$	0.473703i	0.0214655i	0.999942	+	0.0107328i	$0.00341641\pi$
$487$	0.473703i	0.0214655i	−0.999942	+	0.0107328i	$0.996584\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	15.4571	0.697569	0.348784	−	0.937203i	$-0.386595\pi$
$491$	15.4571	0.697569	0.348784	+	0.937203i	$0.386595\pi$
$492$	0	0
$493$	1.02046	0.0459593
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 43.9163i	− 1.96992i
$498$	0	0
$499$	39.8856i	1.78552i	0.450530	+	0.892761i	$0.351235\pi$
$499$	39.8856i	1.78552i	−0.450530	+	0.892761i	$0.648765\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	16.7271	0.745823	0.372911	−	0.927867i	$-0.378360\pi$
$503$	16.7271	0.745823	0.372911	+	0.927867i	$0.378360\pi$
$504$	0	0
$505$	−0.965968	−0.0429850
$506$	0	0
$507$	0	0
$508$	0	0
$509$	35.3278i	1.56588i	0.622099	+	0.782939i	$0.286280\pi$
$509$	35.3278i	1.56588i	−0.622099	+	0.782939i	$0.713720\pi$
$510$	0	0
$511$	− 44.9418i	− 1.98811i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−16.5935	−0.731197
$516$	0	0
$517$	−2.24950	−0.0989327
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 2.61450i	− 0.114543i	−0.998359	−	0.0572716i	$-0.981760\pi$
$521$	− 2.61450i	− 0.114543i	0.998359	−	0.0572716i	$-0.0182401\pi$
$522$	0	0
$523$	− 39.3421i	− 1.72031i	−0.510034	−	0.860154i	$-0.670367\pi$
$523$	− 39.3421i	− 1.72031i	0.510034	−	0.860154i	$-0.329633\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−5.54777	−0.241665
$528$	0	0
$529$	−22.2294	−0.966496
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 15.4020i	− 0.667133i
$534$	0	0
$535$	− 29.1417i	− 1.25991i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	26.6015	1.14581
$540$	0	0
$541$	7.88682	0.339081	0.169540	−	0.985523i	$-0.445772\pi$
$541$	7.88682	0.339081	0.169540	+	0.985523i	$0.445772\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	2.54551i	0.109038i
$546$	0	0
$547$	1.97976i	0.0846486i	0.999104	+	0.0423243i	$0.0134763\pi$
$547$	1.97976i	0.0846486i	−0.999104	+	0.0423243i	$0.986524\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−1.58430	−0.0674933
$552$	0	0
$553$	44.0953	1.87512
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 40.3668i	− 1.71039i	−0.518303	−	0.855197i	$-0.673436\pi$
$557$	− 40.3668i	− 1.71039i	0.518303	−	0.855197i	$-0.326564\pi$
$558$	0	0
$559$	10.7319i	0.453911i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	25.6523	1.08111	0.540557	−	0.841307i	$-0.318214\pi$
$563$	25.6523	1.08111	0.540557	+	0.841307i	$0.318214\pi$
$564$	0	0
$565$	13.4914	0.567588
$566$	0	0
$567$	0	0
$568$	0	0
$569$	21.9227i	0.919046i	0.888166	+	0.459523i	$0.151980\pi$
$569$	21.9227i	0.919046i	−0.888166	+	0.459523i	$0.848020\pi$
$570$	0	0
$571$	39.5144i	1.65363i	0.562476	+	0.826814i	$0.309849\pi$
$571$	39.5144i	1.65363i	−0.562476	+	0.826814i	$0.690151\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−0.257566	−0.0107413
$576$	0	0
$577$	43.2569	1.80081	0.900405	−	0.435052i	$-0.143270\pi$
$577$	43.2569	1.80081	0.900405	+	0.435052i	$0.143270\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 58.5138i	− 2.42756i
$582$	0	0
$583$	− 32.6264i	− 1.35125i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−31.9324	−1.31799	−0.658996	−	0.752146i	$-0.729019\pi$
$587$	−31.9324	−1.31799	−0.658996	+	0.752146i	$0.729019\pi$
$588$	0	0
$589$	8.61306	0.354895
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 13.9688i	− 0.573629i	−0.957986	−	0.286815i	$-0.907404\pi$
$593$	− 13.9688i	− 0.573629i	0.957986	−	0.286815i	$-0.0925964\pi$
$594$	0	0
$595$	32.9865i	1.35231i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−4.48986	−0.183451	−0.0917254	−	0.995784i	$-0.529238\pi$
$599$	−4.48986	−0.183451	−0.0917254	+	0.995784i	$0.529238\pi$
$600$	0	0
$601$	−19.0954	−0.778918	−0.389459	−	0.921044i	$-0.627338\pi$
$601$	−19.0954	−0.778918	−0.389459	+	0.921044i	$0.627338\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 7.81081i	− 0.317555i
$606$	0	0
$607$	− 13.6112i	− 0.552462i	−0.961091	−	0.276231i	$-0.910914\pi$
$607$	− 13.6112i	− 0.552462i	0.961091	−	0.276231i	$-0.0890855\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−3.68663	−0.149145
$612$	0	0
$613$	−41.3213	−1.66895	−0.834476	−	0.551044i	$-0.814230\pi$
$613$	−41.3213	−1.66895	−0.834476	+	0.551044i	$0.814230\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 27.5877i	− 1.11064i	−0.831637	−	0.555320i	$-0.812596\pi$
$617$	− 27.5877i	− 1.11064i	0.831637	−	0.555320i	$-0.187404\pi$
$618$	0	0
$619$	− 1.51849i	− 0.0610331i	−0.999534	−	0.0305166i	$-0.990285\pi$
$619$	− 1.51849i	− 0.0610331i	0.999534	−	0.0305166i	$-0.00971523\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	32.0521	1.28414
$624$	0	0
$625$	−26.3810	−1.05524
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 3.51408i	− 0.140115i
$630$	0	0
$631$	31.3170i	1.24671i	0.781939	+	0.623355i	$0.214231\pi$
$631$	31.3170i	1.24671i	−0.781939	+	0.623355i	$0.785769\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−23.5257	−0.933588
$636$	0	0
$637$	43.5964	1.72735
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 45.9377i	− 1.81443i	−0.420667	−	0.907215i	$-0.638204\pi$
$641$	− 45.9377i	− 1.81443i	0.420667	−	0.907215i	$-0.361796\pi$
$642$	0	0
$643$	− 27.4382i	− 1.08206i	−0.841004	−	0.541029i	$-0.818035\pi$
$643$	− 27.4382i	− 1.08206i	0.841004	−	0.541029i	$-0.181965\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−47.5963	−1.87120	−0.935602	−	0.353057i	$-0.885142\pi$
$647$	−47.5963	−1.87120	−0.935602	+	0.353057i	$0.885142\pi$
$648$	0	0
$649$	17.2990	0.679047
$650$	0	0
$651$	0	0
$652$	0	0
$653$	33.0266i	1.29243i	0.763155	+	0.646216i	$0.223649\pi$
$653$	33.0266i	1.29243i	−0.763155	+	0.646216i	$0.776351\pi$
$654$	0	0
$655$	14.0868i	0.550415i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	28.2191	1.09926	0.549629	−	0.835409i	$-0.314769\pi$
$659$	28.2191	1.09926	0.549629	+	0.835409i	$0.314769\pi$
$660$	0	0
$661$	29.1453	1.13362	0.566810	−	0.823849i	$-0.308177\pi$
$661$	29.1453	1.13362	0.566810	+	0.823849i	$0.308177\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 51.2124i	− 1.98593i
$666$	0	0
$667$	− 0.254916i	− 0.00987038i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−10.1055	−0.390118
$672$	0	0
$673$	−44.2073	−1.70407	−0.852033	−	0.523489i	$-0.824630\pi$
$673$	−44.2073	−1.70407	−0.852033	+	0.523489i	$0.824630\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	48.3903i	1.85979i	0.367823	+	0.929896i	$0.380103\pi$
$677$	48.3903i	1.85979i	−0.367823	+	0.929896i	$0.619897\pi$
$678$	0	0
$679$	− 60.6030i	− 2.32573i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	25.5788	0.978744	0.489372	−	0.872075i	$-0.337226\pi$
$683$	25.5788	0.978744	0.489372	+	0.872075i	$0.337226\pi$
$684$	0	0
$685$	−7.39651	−0.282606
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 53.4704i	− 2.03706i
$690$	0	0
$691$	− 32.0759i	− 1.22023i	−0.792314	−	0.610113i	$-0.791124\pi$
$691$	− 32.0759i	− 1.22023i	0.792314	−	0.610113i	$-0.208876\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	10.6615	0.404414
$696$	0	0
$697$	−11.9754	−0.453601
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 18.9920i	− 0.717319i	−0.933468	−	0.358659i	$-0.883234\pi$
$701$	− 18.9920i	− 0.717319i	0.933468	−	0.358659i	$-0.116766\pi$
$702$	0	0
$703$	5.45570i	0.205765i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.71298	0.0644232
$708$	0	0
$709$	3.82391	0.143610	0.0718049	−	0.997419i	$-0.477124\pi$
$709$	3.82391	0.143610	0.0718049	+	0.997419i	$0.477124\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	1.38586i	0.0519007i
$714$	0	0
$715$	28.6759i	1.07242i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−1.21083	−0.0451565	−0.0225783	−	0.999745i	$-0.507187\pi$
$719$	−1.21083	−0.0451565	−0.0225783	+	0.999745i	$0.507187\pi$
$720$	0	0
$721$	29.4257	1.09587
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0.0852049i	0.00316443i
$726$	0	0
$727$	4.75896i	0.176500i	0.996098	+	0.0882501i	$0.0281275\pi$
$727$	4.75896i	0.176500i	−0.996098	+	0.0882501i	$0.971873\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	8.34432	0.308626
$732$	0	0
$733$	23.7483	0.877163	0.438582	−	0.898691i	$-0.355481\pi$
$733$	23.7483	0.877163	0.438582	+	0.898691i	$0.355481\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 13.7599i	− 0.506854i
$738$	0	0
$739$	− 17.9801i	− 0.661410i	−0.943734	−	0.330705i	$-0.892714\pi$
$739$	− 17.9801i	− 0.661410i	0.943734	−	0.330705i	$-0.107286\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−51.9803	−1.90697	−0.953486	−	0.301437i	$-0.902534\pi$
$743$	−51.9803	−1.90697	−0.953486	+	0.301437i	$0.902534\pi$
$744$	0	0
$745$	31.6626	1.16003
$746$	0	0
$747$	0	0
$748$	0	0
$749$	51.6778i	1.88827i
$750$	0	0
$751$	10.6853i	0.389912i	0.980812	+	0.194956i	$0.0624564\pi$
$751$	10.6853i	0.389912i	−0.980812	+	0.194956i	$0.937544\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	13.6256	0.495886
$756$	0	0
$757$	4.85514	0.176463	0.0882316	−	0.996100i	$-0.471878\pi$
$757$	4.85514	0.176463	0.0882316	+	0.996100i	$0.471878\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 18.6881i	− 0.677442i	−0.940887	−	0.338721i	$-0.890006\pi$
$761$	− 18.6881i	− 0.677442i	0.940887	−	0.338721i	$-0.109994\pi$
$762$	0	0
$763$	− 4.51403i	− 0.163419i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	28.3509	1.02369
$768$	0	0
$769$	24.4033	0.880006	0.440003	−	0.897996i	$-0.354977\pi$
$769$	24.4033	0.880006	0.440003	+	0.897996i	$0.354977\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	27.0749i	0.973817i	0.873453	+	0.486909i	$0.161875\pi$
$773$	27.0749i	0.973817i	−0.873453	+	0.486909i	$0.838125\pi$
$774$	0	0
$775$	− 0.463218i	− 0.0166393i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	18.5922	0.666133
$780$	0	0
$781$	−29.6839	−1.06217
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 23.9206i	− 0.853763i
$786$	0	0
$787$	19.8823i	0.708728i	0.935108	+	0.354364i	$0.115303\pi$
$787$	19.8823i	0.708728i	−0.935108	+	0.354364i	$0.884697\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−23.9247	−0.850664
$792$	0	0
$793$	−16.5616	−0.588118
$794$	0	0
$795$	0	0
$796$	0	0
$797$	7.44088i	0.263570i	0.991278	+	0.131785i	$0.0420708\pi$
$797$	7.44088i	0.263570i	−0.991278	+	0.131785i	$0.957929\pi$
$798$	0	0
$799$	2.86645i	0.101408i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−30.3770	−1.07198
$804$	0	0
$805$	8.24015	0.290427
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 29.7939i	− 1.04750i	−0.851873	−	0.523748i	$-0.824533\pi$
$809$	− 29.7939i	− 1.04750i	0.851873	−	0.523748i	$-0.175467\pi$
$810$	0	0
$811$	− 6.10131i	− 0.214246i	−0.994246	−	0.107123i	$-0.965836\pi$
$811$	− 6.10131i	− 0.214246i	0.994246	−	0.107123i	$-0.0341639\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−2.36259	−0.0827580
$816$	0	0
$817$	−12.9548	−0.453231
$818$	0	0
$819$	0	0
$820$	0	0
$821$	16.0683i	0.560787i	0.959885	+	0.280394i	$0.0904650\pi$
$821$	16.0683i	0.560787i	−0.959885	+	0.280394i	$0.909535\pi$
$822$	0	0
$823$	24.5792i	0.856777i	0.903595	+	0.428389i	$0.140919\pi$
$823$	24.5792i	0.856777i	−0.903595	+	0.428389i	$0.859081\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−12.2481	−0.425907	−0.212953	−	0.977062i	$-0.568308\pi$
$827$	−12.2481	−0.425907	−0.212953	+	0.977062i	$0.568308\pi$
$828$	0	0
$829$	10.1046	0.350949	0.175474	−	0.984484i	$-0.443854\pi$
$829$	10.1046	0.350949	0.175474	+	0.984484i	$0.443854\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 33.8973i	− 1.17447i
$834$	0	0
$835$	23.0311i	0.797025i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−52.5667	−1.81481	−0.907403	−	0.420262i	$-0.861938\pi$
$839$	−52.5667	−1.81481	−0.907403	+	0.420262i	$0.861938\pi$
$840$	0	0
$841$	28.9157	0.997092
$842$	0	0
$843$	0	0
$844$	0	0
$845$	17.0863i	0.587788i
$846$	0	0
$847$	13.8511i	0.475931i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−0.877830	−0.0300916
$852$	0	0
$853$	−13.9732	−0.478435	−0.239217	−	0.970966i	$-0.576891\pi$
$853$	−13.9732	−0.478435	−0.239217	+	0.970966i	$0.576891\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 24.9176i	− 0.851168i	−0.904919	−	0.425584i	$-0.860069\pi$
$857$	− 24.9176i	− 0.851168i	0.904919	−	0.425584i	$-0.139931\pi$
$858$	0	0
$859$	− 3.34275i	− 0.114053i	−0.998373	−	0.0570265i	$-0.981838\pi$
$859$	− 3.34275i	− 0.114053i	0.998373	−	0.0570265i	$-0.0181620\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−28.1595	−0.958559	−0.479280	−	0.877662i	$-0.659102\pi$
$863$	−28.1595	−0.958559	−0.479280	+	0.877662i	$0.659102\pi$
$864$	0	0
$865$	−23.6471	−0.804025
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 29.8049i	− 1.01106i
$870$	0	0
$871$	− 22.5507i	− 0.764102i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	44.1805	1.49357
$876$	0	0
$877$	−21.8138	−0.736599	−0.368299	−	0.929707i	$-0.620060\pi$
$877$	−21.8138	−0.736599	−0.368299	+	0.929707i	$0.620060\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 0.878391i	− 0.0295937i	−0.999891	−	0.0147969i	$-0.995290\pi$
$881$	− 0.878391i	− 0.0295937i	0.999891	−	0.0147969i	$-0.00471016\pi$
$882$	0	0
$883$	9.85401i	0.331614i	0.986158	+	0.165807i	$0.0530229\pi$
$883$	9.85401i	0.331614i	−0.986158	+	0.165807i	$0.946977\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−30.8827	−1.03694	−0.518470	−	0.855096i	$-0.673498\pi$
$887$	−30.8827	−1.03694	−0.518470	+	0.855096i	$0.673498\pi$
$888$	0	0
$889$	41.7188	1.39920
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 4.45024i	− 0.148922i
$894$	0	0
$895$	− 31.4045i	− 1.04974i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0.458452	0.0152902
$900$	0	0
$901$	−41.5746	−1.38505
$902$	0	0
$903$	0	0
$904$	0	0
$905$	19.7702i	0.657185i
$906$	0	0
$907$	− 50.1945i	− 1.66668i	−0.552759	−	0.833341i	$-0.686425\pi$
$907$	− 50.1945i	− 1.66668i	0.552759	−	0.833341i	$-0.313575\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	3.78170	0.125293	0.0626467	−	0.998036i	$-0.480046\pi$
$911$	3.78170	0.125293	0.0626467	+	0.998036i	$0.480046\pi$
$912$	0	0
$913$	−39.5507	−1.30894
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 24.9805i	− 0.824927i
$918$	0	0
$919$	− 31.7149i	− 1.04618i	−0.852279	−	0.523088i	$-0.824780\pi$
$919$	− 31.7149i	− 1.04618i	0.852279	−	0.523088i	$-0.175220\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−48.6480	−1.60127
$924$	0	0
$925$	0.293412	0.00964734
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 53.4909i	− 1.75498i	−0.479595	−	0.877490i	$-0.659216\pi$
$929$	− 53.4909i	− 1.75498i	0.479595	−	0.877490i	$-0.340784\pi$
$930$	0	0
$931$	52.6265i	1.72476i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	22.2962	0.729164
$936$	0	0
$937$	−26.5679	−0.867934	−0.433967	−	0.900929i	$-0.642887\pi$
$937$	−26.5679	−0.867934	−0.433967	+	0.900929i	$0.642887\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	10.0851i	0.328766i	0.986397	+	0.164383i	$0.0525633\pi$
$941$	10.0851i	0.328766i	−0.986397	+	0.164383i	$0.947437\pi$
$942$	0	0
$943$	2.99151i	0.0974169i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	3.23781	0.105215	0.0526074	−	0.998615i	$-0.483247\pi$
$947$	3.23781	0.105215	0.0526074	+	0.998615i	$0.483247\pi$
$948$	0	0
$949$	−49.7840	−1.61606
$950$	0	0
$951$	0	0
$952$	0	0
$953$	45.9043i	1.48699i	0.668744	+	0.743493i	$0.266832\pi$
$953$	45.9043i	1.48699i	−0.668744	+	0.743493i	$0.733168\pi$
$954$	0	0
$955$	28.3693i	0.918008i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	13.1164	0.423552
$960$	0	0
$961$	28.5076	0.919601
$962$	0	0
$963$	0	0
$964$	0	0
$965$	59.2402i	1.90701i
$966$	0	0
$967$	− 21.3639i	− 0.687018i	−0.939149	−	0.343509i	$-0.888384\pi$
$967$	− 21.3639i	− 0.687018i	0.939149	−	0.343509i	$-0.111616\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	13.0896	0.420065	0.210033	−	0.977694i	$-0.432643\pi$
$971$	13.0896	0.420065	0.210033	+	0.977694i	$0.432643\pi$
$972$	0	0
$973$	−18.9063	−0.606110
$974$	0	0
$975$	0	0
$976$	0	0
$977$	35.7900i	1.14502i	0.819896	+	0.572512i	$0.194031\pi$
$977$	35.7900i	1.14502i	−0.819896	+	0.572512i	$0.805969\pi$
$978$	0	0
$979$	− 21.6647i	− 0.692406i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	20.1771	0.643550	0.321775	−	0.946816i	$-0.395720\pi$
$983$	20.1771	0.643550	0.321775	+	0.946816i	$0.395720\pi$
$984$	0	0
$985$	50.9512	1.62344
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 2.08445i	− 0.0662815i
$990$	0	0
$991$	45.7778i	1.45418i	0.686543	+	0.727089i	$0.259127\pi$
$991$	45.7778i	1.45418i	−0.686543	+	0.727089i	$0.740873\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−43.8534	−1.39025
$996$	0	0
$997$	60.2332	1.90760	0.953802	−	0.300437i	$-0.0971324\pi$
$997$	60.2332	1.90760	0.953802	+	0.300437i	$0.0971324\pi$
$998$	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.2591.17	yes	24
3.2	odd	2	inner	5328.2.e.f.2591.7	✓	24
4.3	odd	2	inner	5328.2.e.f.2591.18	yes	24
12.11	even	2	inner	5328.2.e.f.2591.8	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5328.2.e.f.2591.7	✓	24	3.2	odd	2	inner
5328.2.e.f.2591.8	yes	24	12.11	even	2	inner
5328.2.e.f.2591.17	yes	24	1.1	even	1	trivial
5328.2.e.f.2591.18	yes	24	4.3	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)

\(f(q)\)	\(=\)	\(q+2.30074i q^{5} -4.07997i q^{7} -2.75773 q^{11} -4.51956 q^{13} +3.51408i q^{17} -5.45570i q^{19} +0.877830 q^{23} -0.293412 q^{25} -0.290393i q^{29} +1.57873i q^{31} +9.38696 q^{35} -1.00000 q^{37} +3.40784i q^{41} -2.37454i q^{43} +0.815705 q^{47} -9.64615 q^{49} +11.8309i q^{53} -6.34483i q^{55} -6.27292 q^{59} +3.66442 q^{61} -10.3983i q^{65} +4.98958i q^{67} +10.7639 q^{71} +11.0152 q^{73} +11.2515i q^{77} +10.8077i q^{79} +14.3417 q^{83} -8.08498 q^{85} +7.85597i q^{89} +18.4397i q^{91} +12.5521 q^{95} +14.8538 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 32 q^{25} - 24 q^{37} - 80 q^{49} - 48 q^{61} - 48 q^{73} - 40 q^{85} + 24 q^{97}+O(q^{100}) \) 24 * q - 32 * q^25 - 24 * q^37 - 80 * q^49 - 48 * q^61 - 48 * q^73 - 40 * q^85 + 24 * q^97

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