LMFDB - Embedded newform 4732.2.g.k.337.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4732,2,Mod(337,4732)] 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("4732.337"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4732, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4732 = 2^{2} \cdot 7 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4732.g (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	no
Analytic conductor:	$37.7852102365$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 38x^{14} + 587x^{12} + 4762x^{10} + 21849x^{8} + 56552x^{6} + 76456x^{4} + 42624x^{2} + 2704 $ x^16 + 38x^14 + 587x^12 + 4762x^10 + 21849x^8 + 56552x^6 + 76456x^4 + 42624*x^2 + 2704
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	no (minimal twist has level 364)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.7
Root			$-0.268953i$ of defining polynomial
Character	$\chi$	$=$	4732.337
Dual form			4732.2.g.k.337.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.268953 q^{3} -1.35585i q^{5} +1.00000i q^{7} -2.92766 q^{9} -2.80318i q^{11} +0.364660i q^{15} +5.81868 q^{17} +3.18458i q^{19} -0.268953i q^{21} -7.55347 q^{23} +3.16167 q^{25} +1.59426 q^{27} +7.21661 q^{29} +8.04407i q^{31} +0.753925i q^{33} +1.35585 q^{35} +6.95744i q^{37} -5.05949i q^{41} -6.26854 q^{43} +3.96948i q^{45} +2.64315i q^{47} -1.00000 q^{49} -1.56495 q^{51} -11.0887 q^{53} -3.80070 q^{55} -0.856503i q^{57} -6.29558i q^{59} +12.8081 q^{61} -2.92766i q^{63} -4.96294i q^{67} +2.03153 q^{69} -16.5114i q^{71} +15.4351i q^{73} -0.850340 q^{75} +2.80318 q^{77} +14.7661 q^{79} +8.35421 q^{81} -9.79310i q^{83} -7.88926i q^{85} -1.94093 q^{87} +8.36820i q^{89} -2.16348i q^{93} +4.31782 q^{95} -12.2034i q^{97} +8.20678i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 28 q^{9} - 4 q^{17} - 44 q^{25} - 12 q^{27} + 44 q^{29} + 12 q^{35} - 12 q^{43} - 16 q^{49} - 4 q^{51} + 8 q^{53} - 4 q^{55} - 8 q^{61} + 104 q^{69} + 20 q^{75} - 24 q^{77} + 8 q^{79} - 52 q^{87}+ \cdots + 60 q^{95}+O(q^{100}) $ 16 * q + 28 * q^9 - 4 * q^17 - 44 * q^25 - 12 * q^27 + 44 * q^29 + 12 * q^35 - 12 * q^43 - 16 * q^49 - 4 * q^51 + 8 * q^53 - 4 * q^55 - 8 * q^61 + 104 * q^69 + 20 * q^75 - 24 * q^77 + 8 * q^79 - 52 * q^87 + 60 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4732\mathbb{Z}\right)^\times$.

$n$	$2367$	$2705$	$4565$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	−0.268953	−0.155280	−0.0776400	−	0.996981i	$-0.524738\pi$
$3$	−0.268953	−0.155280	−0.0776400	+	0.996981i	$0.524738\pi$
$4$	0	0
$5$	− 1.35585i	− 0.606355i	−0.952934	−	0.303177i	$-0.901953\pi$
$5$	− 1.35585i	− 0.606355i	0.952934	−	0.303177i	$-0.0980475\pi$
$6$	0	0
$7$	1.00000i	0.377964i
$7$	1.00000i	0.377964i
$8$	0	0
$9$	−2.92766	−0.975888
$10$	0	0
$11$	− 2.80318i	− 0.845192i	−0.906318	−	0.422596i	$-0.861119\pi$
$11$	− 2.80318i	− 0.845192i	0.906318	−	0.422596i	$-0.138881\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	0.364660i	0.0941548i
$16$	0	0
$17$	5.81868	1.41124	0.705618	−	0.708592i	$-0.250669\pi$
$17$	5.81868	1.41124	0.705618	+	0.708592i	$0.250669\pi$
$18$	0	0
$19$	3.18458i	0.730594i	0.930891	+	0.365297i	$0.119032\pi$
$19$	3.18458i	0.730594i	−0.930891	+	0.365297i	$0.880968\pi$
$20$	0	0
$21$	− 0.268953i	− 0.0586903i
$22$	0	0
$23$	−7.55347	−1.57501	−0.787504	−	0.616310i	$-0.788627\pi$
$23$	−7.55347	−1.57501	−0.787504	+	0.616310i	$0.788627\pi$
$24$	0	0
$25$	3.16167	0.632334
$26$	0	0
$27$	1.59426	0.306816
$28$	0	0
$29$	7.21661	1.34009	0.670045	−	0.742320i	$-0.266275\pi$
$29$	7.21661	1.34009	0.670045	+	0.742320i	$0.266275\pi$
$30$	0	0
$31$	8.04407i	1.44476i	0.691498	+	0.722379i	$0.256951\pi$
$31$	8.04407i	1.44476i	−0.691498	+	0.722379i	$0.743049\pi$
$32$	0	0
$33$	0.753925i	0.131241i
$34$	0	0
$35$	1.35585	0.229181
$36$	0	0
$37$	6.95744i	1.14380i	0.820325	+	0.571898i	$0.193793\pi$
$37$	6.95744i	1.14380i	−0.820325	+	0.571898i	$0.806207\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 5.05949i	− 0.790159i	−0.918647	−	0.395079i	$-0.870717\pi$
$41$	− 5.05949i	− 0.790159i	0.918647	−	0.395079i	$-0.129283\pi$
$42$	0	0
$43$	−6.26854	−0.955944	−0.477972	−	0.878375i	$-0.658628\pi$
$43$	−6.26854	−0.955944	−0.477972	+	0.878375i	$0.658628\pi$
$44$	0	0
$45$	3.96948i	0.591734i
$46$	0	0
$47$	2.64315i	0.385542i	0.981244	+	0.192771i	$0.0617475\pi$
$47$	2.64315i	0.385542i	−0.981244	+	0.192771i	$0.938252\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	−1.56495	−0.219137
$52$	0	0
$53$	−11.0887	−1.52315	−0.761574	−	0.648079i	$-0.775573\pi$
$53$	−11.0887	−1.52315	−0.761574	+	0.648079i	$0.775573\pi$
$54$	0	0
$55$	−3.80070	−0.512486
$56$	0	0
$57$	− 0.856503i	− 0.113447i
$58$	0	0
$59$	− 6.29558i	− 0.819614i	−0.912172	−	0.409807i	$-0.865596\pi$
$59$	− 6.29558i	− 0.819614i	0.912172	−	0.409807i	$-0.134404\pi$
$60$	0	0
$61$	12.8081	1.63991	0.819953	−	0.572431i	$-0.193999\pi$
$61$	12.8081	1.63991	0.819953	+	0.572431i	$0.193999\pi$
$62$	0	0
$63$	− 2.92766i	− 0.368851i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 4.96294i	− 0.606319i	−0.952940	−	0.303160i	$-0.901958\pi$
$67$	− 4.96294i	− 0.606319i	0.952940	−	0.303160i	$-0.0980416\pi$
$68$	0	0
$69$	2.03153	0.244567
$70$	0	0
$71$	− 16.5114i	− 1.95955i	−0.200111	−	0.979773i	$-0.564130\pi$
$71$	− 16.5114i	− 1.95955i	0.200111	−	0.979773i	$-0.435870\pi$
$72$	0	0
$73$	15.4351i	1.80654i	0.429071	+	0.903271i	$0.358841\pi$
$73$	15.4351i	1.80654i	−0.429071	+	0.903271i	$0.641159\pi$
$74$	0	0
$75$	−0.850340	−0.0981888
$76$	0	0
$77$	2.80318	0.319452
$78$	0	0
$79$	14.7661	1.66131	0.830656	−	0.556786i	$-0.187966\pi$
$79$	14.7661	1.66131	0.830656	+	0.556786i	$0.187966\pi$
$80$	0	0
$81$	8.35421	0.928246
$82$	0	0
$83$	− 9.79310i	− 1.07493i	−0.843285	−	0.537466i	$-0.819382\pi$
$83$	− 9.79310i	− 1.07493i	0.843285	−	0.537466i	$-0.180618\pi$
$84$	0	0
$85$	− 7.88926i	− 0.855710i
$86$	0	0
$87$	−1.94093	−0.208089
$88$	0	0
$89$	8.36820i	0.887027i	0.896268	+	0.443513i	$0.146268\pi$
$89$	8.36820i	0.887027i	−0.896268	+	0.443513i	$0.853732\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	− 2.16348i	− 0.224342i
$94$	0	0
$95$	4.31782	0.442999
$96$	0	0
$97$	− 12.2034i	− 1.23907i	−0.784970	−	0.619534i	$-0.787322\pi$
$97$	− 12.2034i	− 1.23907i	0.784970	−	0.619534i	$-0.212678\pi$
$98$	0	0
$99$	8.20678i	0.824813i
$100$	0	0
$101$	2.42285	0.241083	0.120541	−	0.992708i	$-0.461537\pi$
$101$	2.42285	0.241083	0.120541	+	0.992708i	$0.461537\pi$
$102$	0	0
$103$	14.4909	1.42783	0.713917	−	0.700230i	$-0.246919\pi$
$103$	14.4909	1.42783	0.713917	+	0.700230i	$0.246919\pi$
$104$	0	0
$105$	−0.364660	−0.0355872
$106$	0	0
$107$	6.48741	0.627161	0.313581	−	0.949562i	$-0.398471\pi$
$107$	6.48741	0.627161	0.313581	+	0.949562i	$0.398471\pi$
$108$	0	0
$109$	− 2.93644i	− 0.281260i	−0.990062	−	0.140630i	$-0.955087\pi$
$109$	− 2.93644i	− 0.281260i	0.990062	−	0.140630i	$-0.0449129\pi$
$110$	0	0
$111$	− 1.87122i	− 0.177609i
$112$	0	0
$113$	−3.33818	−0.314029	−0.157015	−	0.987596i	$-0.550187\pi$
$113$	−3.33818	−0.314029	−0.157015	+	0.987596i	$0.550187\pi$
$114$	0	0
$115$	10.2414i	0.955013i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	5.81868i	0.533397i
$120$	0	0
$121$	3.14216	0.285651
$122$	0	0
$123$	1.36076i	0.122696i
$124$	0	0
$125$	− 11.0660i	− 0.989773i
$126$	0	0
$127$	12.5502	1.11365	0.556827	−	0.830629i	$-0.312019\pi$
$127$	12.5502	1.11365	0.556827	+	0.830629i	$0.312019\pi$
$128$	0	0
$129$	1.68594	0.148439
$130$	0	0
$131$	1.26001	0.110087	0.0550437	−	0.998484i	$-0.482470\pi$
$131$	1.26001	0.110087	0.0550437	+	0.998484i	$0.482470\pi$
$132$	0	0
$133$	−3.18458	−0.276138
$134$	0	0
$135$	− 2.16158i	− 0.186039i
$136$	0	0
$137$	− 8.32302i	− 0.711083i	−0.934660	−	0.355542i	$-0.884296\pi$
$137$	− 8.32302i	− 0.711083i	0.934660	−	0.355542i	$-0.115704\pi$
$138$	0	0
$139$	−5.78876	−0.490997	−0.245498	−	0.969397i	$-0.578952\pi$
$139$	−5.78876	−0.490997	−0.245498	+	0.969397i	$0.578952\pi$
$140$	0	0
$141$	− 0.710882i	− 0.0598671i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	− 9.78464i	− 0.812570i
$146$	0	0
$147$	0.268953	0.0221829
$148$	0	0
$149$	14.2039i	1.16362i	0.813323	+	0.581812i	$0.197656\pi$
$149$	14.2039i	1.16362i	−0.813323	+	0.581812i	$0.802344\pi$
$150$	0	0
$151$	11.1509i	0.907451i	0.891142	+	0.453725i	$0.149905\pi$
$151$	11.1509i	0.907451i	−0.891142	+	0.453725i	$0.850095\pi$
$152$	0	0
$153$	−17.0351	−1.37721
$154$	0	0
$155$	10.9066	0.876036
$156$	0	0
$157$	4.04749	0.323025	0.161513	−	0.986871i	$-0.448363\pi$
$157$	4.04749	0.323025	0.161513	+	0.986871i	$0.448363\pi$
$158$	0	0
$159$	2.98233	0.236514
$160$	0	0
$161$	− 7.55347i	− 0.595297i
$162$	0	0
$163$	− 5.27322i	− 0.413031i	−0.978443	−	0.206515i	$-0.933788\pi$
$163$	− 5.27322i	− 0.413031i	0.978443	−	0.206515i	$-0.0662124\pi$
$164$	0	0
$165$	1.02221	0.0795789
$166$	0	0
$167$	5.34981i	0.413981i	0.978343	+	0.206990i	$0.0663669\pi$
$167$	5.34981i	0.413981i	−0.978343	+	0.206990i	$0.933633\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	− 9.32339i	− 0.712978i
$172$	0	0
$173$	9.74309	0.740754	0.370377	−	0.928882i	$-0.379229\pi$
$173$	9.74309	0.740754	0.370377	+	0.928882i	$0.379229\pi$
$174$	0	0
$175$	3.16167i	0.239000i
$176$	0	0
$177$	1.69321i	0.127270i
$178$	0	0
$179$	−11.4053	−0.852469	−0.426234	−	0.904613i	$-0.640160\pi$
$179$	−11.4053	−0.852469	−0.426234	+	0.904613i	$0.640160\pi$
$180$	0	0
$181$	25.4727	1.89337	0.946685	−	0.322160i	$-0.104409\pi$
$181$	25.4727	1.89337	0.946685	+	0.322160i	$0.104409\pi$
$182$	0	0
$183$	−3.44477	−0.254645
$184$	0	0
$185$	9.43325	0.693546
$186$	0	0
$187$	− 16.3108i	− 1.19277i
$188$	0	0
$189$	1.59426i	0.115966i
$190$	0	0
$191$	16.3781	1.18508	0.592539	−	0.805542i	$-0.298126\pi$
$191$	16.3781	1.18508	0.592539	+	0.805542i	$0.298126\pi$
$192$	0	0
$193$	− 9.56245i	− 0.688320i	−0.938911	−	0.344160i	$-0.888164\pi$
$193$	− 9.56245i	− 0.688320i	0.938911	−	0.344160i	$-0.111836\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 3.01049i	− 0.214489i	−0.994233	−	0.107244i	$-0.965797\pi$
$197$	− 3.01049i	− 0.214489i	0.994233	−	0.107244i	$-0.0342027\pi$
$198$	0	0
$199$	4.61969	0.327481	0.163741	−	0.986503i	$-0.447644\pi$
$199$	4.61969	0.327481	0.163741	+	0.986503i	$0.447644\pi$
$200$	0	0
$201$	1.33480i	0.0941493i
$202$	0	0
$203$	7.21661i	0.506507i
$204$	0	0
$205$	−6.85991	−0.479117
$206$	0	0
$207$	22.1140	1.53703
$208$	0	0
$209$	8.92697	0.617492
$210$	0	0
$211$	2.33865	0.160999	0.0804995	−	0.996755i	$-0.474348\pi$
$211$	2.33865	0.160999	0.0804995	+	0.996755i	$0.474348\pi$
$212$	0	0
$213$	4.44080i	0.304279i
$214$	0	0
$215$	8.49921i	0.579641i
$216$	0	0
$217$	−8.04407	−0.546067
$218$	0	0
$219$	− 4.15132i	− 0.280520i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	4.86625i	0.325868i	0.986637	+	0.162934i	$0.0520958\pi$
$223$	4.86625i	0.325868i	−0.986637	+	0.162934i	$0.947904\pi$
$224$	0	0
$225$	−9.25631	−0.617087
$226$	0	0
$227$	− 28.0796i	− 1.86371i	−0.362832	−	0.931855i	$-0.618190\pi$
$227$	− 28.0796i	− 1.86371i	0.362832	−	0.931855i	$-0.381810\pi$
$228$	0	0
$229$	− 0.179280i	− 0.0118471i	−0.999982	−	0.00592357i	$-0.998114\pi$
$229$	− 0.179280i	− 0.0118471i	0.999982	−	0.00592357i	$-0.00188554\pi$
$230$	0	0
$231$	−0.753925	−0.0496046
$232$	0	0
$233$	15.1384	0.991747	0.495873	−	0.868395i	$-0.334848\pi$
$233$	15.1384	0.991747	0.495873	+	0.868395i	$0.334848\pi$
$234$	0	0
$235$	3.58371	0.233775
$236$	0	0
$237$	−3.97138	−0.257969
$238$	0	0
$239$	− 12.8703i	− 0.832512i	−0.909247	−	0.416256i	$-0.863342\pi$
$239$	− 12.8703i	− 0.832512i	0.909247	−	0.416256i	$-0.136658\pi$
$240$	0	0
$241$	9.83172i	0.633317i	0.948540	+	0.316659i	$0.102561\pi$
$241$	9.83172i	0.633317i	−0.948540	+	0.316659i	$0.897439\pi$
$242$	0	0
$243$	−7.02968	−0.450954
$244$	0	0
$245$	1.35585i	0.0866221i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	2.63388i	0.166916i
$250$	0	0
$251$	17.3220	1.09335	0.546677	−	0.837344i	$-0.315893\pi$
$251$	17.3220	1.09335	0.546677	+	0.837344i	$0.315893\pi$
$252$	0	0
$253$	21.1738i	1.33118i
$254$	0	0
$255$	2.12184i	0.132875i
$256$	0	0
$257$	10.7060	0.667824	0.333912	−	0.942604i	$-0.391631\pi$
$257$	10.7060	0.667824	0.333912	+	0.942604i	$0.391631\pi$
$258$	0	0
$259$	−6.95744	−0.432314
$260$	0	0
$261$	−21.1278	−1.30778
$262$	0	0
$263$	−14.0697	−0.867574	−0.433787	−	0.901015i	$-0.642823\pi$
$263$	−14.0697	−0.867574	−0.433787	+	0.901015i	$0.642823\pi$
$264$	0	0
$265$	15.0346i	0.923567i
$266$	0	0
$267$	− 2.25065i	− 0.137738i
$268$	0	0
$269$	−0.544023	−0.0331697	−0.0165848	−	0.999862i	$-0.505279\pi$
$269$	−0.544023	−0.0331697	−0.0165848	+	0.999862i	$0.505279\pi$
$270$	0	0
$271$	0.741214i	0.0450256i	0.999747	+	0.0225128i	$0.00716665\pi$
$271$	0.741214i	0.0450256i	−0.999747	+	0.0225128i	$0.992833\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 8.86274i	− 0.534443i
$276$	0	0
$277$	14.1335	0.849200	0.424600	−	0.905381i	$-0.360415\pi$
$277$	14.1335	0.849200	0.424600	+	0.905381i	$0.360415\pi$
$278$	0	0
$279$	− 23.5503i	− 1.40992i
$280$	0	0
$281$	25.8051i	1.53940i	0.638404	+	0.769702i	$0.279595\pi$
$281$	25.8051i	1.53940i	−0.638404	+	0.769702i	$0.720405\pi$
$282$	0	0
$283$	−27.2817	−1.62173	−0.810865	−	0.585233i	$-0.801003\pi$
$283$	−27.2817	−1.62173	−0.810865	+	0.585233i	$0.801003\pi$
$284$	0	0
$285$	−1.16129	−0.0687889
$286$	0	0
$287$	5.05949	0.298652
$288$	0	0
$289$	16.8570	0.991589
$290$	0	0
$291$	3.28214i	0.192403i
$292$	0	0
$293$	− 1.73217i	− 0.101194i	−0.998719	−	0.0505971i	$-0.983888\pi$
$293$	− 1.73217i	− 0.101194i	0.998719	−	0.0505971i	$-0.0161124\pi$
$294$	0	0
$295$	−8.53586	−0.496977
$296$	0	0
$297$	− 4.46901i	− 0.259318i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	− 6.26854i	− 0.361313i
$302$	0	0
$303$	−0.651633	−0.0374354
$304$	0	0
$305$	− 17.3658i	− 0.994365i
$306$	0	0
$307$	21.4365i	1.22345i	0.791072	+	0.611723i	$0.209523\pi$
$307$	21.4365i	1.22345i	−0.791072	+	0.611723i	$0.790477\pi$
$308$	0	0
$309$	−3.89738	−0.221714
$310$	0	0
$311$	−27.2724	−1.54647	−0.773237	−	0.634117i	$-0.781364\pi$
$311$	−27.2724	−1.54647	−0.773237	+	0.634117i	$0.781364\pi$
$312$	0	0
$313$	7.28221	0.411615	0.205807	−	0.978593i	$-0.434018\pi$
$313$	7.28221	0.411615	0.205807	+	0.978593i	$0.434018\pi$
$314$	0	0
$315$	−3.96948	−0.223655
$316$	0	0
$317$	− 2.54836i	− 0.143130i	−0.997436	−	0.0715651i	$-0.977201\pi$
$317$	− 2.54836i	− 0.143130i	0.997436	−	0.0715651i	$-0.0227994\pi$
$318$	0	0
$319$	− 20.2295i	− 1.13263i
$320$	0	0
$321$	−1.74481	−0.0973856
$322$	0	0
$323$	18.5301i	1.03104i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0.789766i	0.0436741i
$328$	0	0
$329$	−2.64315	−0.145721
$330$	0	0
$331$	0.195664i	0.0107547i	0.999986	+	0.00537733i	$0.00171167\pi$
$331$	0.195664i	0.0107547i	−0.999986	+	0.00537733i	$0.998288\pi$
$332$	0	0
$333$	− 20.3691i	− 1.11622i
$334$	0	0
$335$	−6.72900	−0.367645
$336$	0	0
$337$	2.00900	0.109437	0.0547185	−	0.998502i	$-0.482574\pi$
$337$	2.00900	0.109437	0.0547185	+	0.998502i	$0.482574\pi$
$338$	0	0
$339$	0.897813	0.0487625
$340$	0	0
$341$	22.5490	1.22110
$342$	0	0
$343$	− 1.00000i	− 0.0539949i
$344$	0	0
$345$	− 2.75445i	− 0.148295i
$346$	0	0
$347$	4.54901	0.244204	0.122102	−	0.992518i	$-0.461037\pi$
$347$	4.54901	0.244204	0.122102	+	0.992518i	$0.461037\pi$
$348$	0	0
$349$	− 5.37745i	− 0.287848i	−0.989589	−	0.143924i	$-0.954028\pi$
$349$	− 5.37745i	− 0.287848i	0.989589	−	0.143924i	$-0.0459721\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 8.20095i	− 0.436492i	−0.975894	−	0.218246i	$-0.929966\pi$
$353$	− 8.20095i	− 0.436492i	0.975894	−	0.218246i	$-0.0700336\pi$
$354$	0	0
$355$	−22.3870	−1.18818
$356$	0	0
$357$	− 1.56495i	− 0.0828260i
$358$	0	0
$359$	31.7664i	1.67657i	0.545233	+	0.838284i	$0.316441\pi$
$359$	31.7664i	1.67657i	−0.545233	+	0.838284i	$0.683559\pi$
$360$	0	0
$361$	8.85843	0.466233
$362$	0	0
$363$	−0.845093	−0.0443559
$364$	0	0
$365$	20.9277	1.09541
$366$	0	0
$367$	17.3374	0.905005	0.452502	−	0.891763i	$-0.350531\pi$
$367$	17.3374	0.905005	0.452502	+	0.891763i	$0.350531\pi$
$368$	0	0
$369$	14.8125i	0.771107i
$370$	0	0
$371$	− 11.0887i	− 0.575695i
$372$	0	0
$373$	16.0963	0.833434	0.416717	−	0.909036i	$-0.363181\pi$
$373$	16.0963	0.833434	0.416717	+	0.909036i	$0.363181\pi$
$374$	0	0
$375$	2.97623i	0.153692i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	17.6562i	0.906939i	0.891272	+	0.453470i	$0.149814\pi$
$379$	17.6562i	0.906939i	−0.891272	+	0.453470i	$0.850186\pi$
$380$	0	0
$381$	−3.37542	−0.172928
$382$	0	0
$383$	8.22450i	0.420252i	0.977674	+	0.210126i	$0.0673874\pi$
$383$	8.22450i	0.420252i	−0.977674	+	0.210126i	$0.932613\pi$
$384$	0	0
$385$	− 3.80070i	− 0.193702i
$386$	0	0
$387$	18.3522	0.932894
$388$	0	0
$389$	15.1688	0.769090	0.384545	−	0.923106i	$-0.374358\pi$
$389$	15.1688	0.769090	0.384545	+	0.923106i	$0.374358\pi$
$390$	0	0
$391$	−43.9512	−2.22271
$392$	0	0
$393$	−0.338883	−0.0170944
$394$	0	0
$395$	− 20.0206i	− 1.00734i
$396$	0	0
$397$	− 28.3063i	− 1.42065i	−0.703874	−	0.710325i	$-0.748548\pi$
$397$	− 28.3063i	− 1.42065i	0.703874	−	0.710325i	$-0.251452\pi$
$398$	0	0
$399$	0.856503	0.0428788
$400$	0	0
$401$	− 17.1192i	− 0.854894i	−0.904040	−	0.427447i	$-0.859413\pi$
$401$	− 17.1192i	− 0.854894i	0.904040	−	0.427447i	$-0.140587\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	− 11.3271i	− 0.562846i
$406$	0	0
$407$	19.5030	0.966727
$408$	0	0
$409$	11.3400i	0.560726i	0.959894	+	0.280363i	$0.0904549\pi$
$409$	11.3400i	0.560726i	−0.959894	+	0.280363i	$0.909545\pi$
$410$	0	0
$411$	2.23850i	0.110417i
$412$	0	0
$413$	6.29558	0.309785
$414$	0	0
$415$	−13.2780	−0.651790
$416$	0	0
$417$	1.55691	0.0762420
$418$	0	0
$419$	0.138855	0.00678352	0.00339176	−	0.999994i	$-0.498920\pi$
$419$	0.138855	0.00678352	0.00339176	+	0.999994i	$0.498920\pi$
$420$	0	0
$421$	− 24.3258i	− 1.18557i	−0.805362	−	0.592783i	$-0.798029\pi$
$421$	− 24.3258i	− 1.18557i	0.805362	−	0.592783i	$-0.201971\pi$
$422$	0	0
$423$	− 7.73824i	− 0.376246i
$424$	0	0
$425$	18.3967	0.892373
$426$	0	0
$427$	12.8081i	0.619826i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 28.7924i	− 1.38688i	−0.720514	−	0.693440i	$-0.756094\pi$
$431$	− 28.7924i	− 1.38688i	0.720514	−	0.693440i	$-0.243906\pi$
$432$	0	0
$433$	25.2733	1.21456	0.607279	−	0.794489i	$-0.292261\pi$
$433$	25.2733	1.21456	0.607279	+	0.794489i	$0.292261\pi$
$434$	0	0
$435$	2.63161i	0.126176i
$436$	0	0
$437$	− 24.0547i	− 1.15069i
$438$	0	0
$439$	−6.03023	−0.287807	−0.143904	−	0.989592i	$-0.545966\pi$
$439$	−6.03023	−0.287807	−0.143904	+	0.989592i	$0.545966\pi$
$440$	0	0
$441$	2.92766	0.139413
$442$	0	0
$443$	23.4988	1.11646	0.558230	−	0.829686i	$-0.311481\pi$
$443$	23.4988	1.11646	0.558230	+	0.829686i	$0.311481\pi$
$444$	0	0
$445$	11.3460	0.537853
$446$	0	0
$447$	− 3.82017i	− 0.180688i
$448$	0	0
$449$	23.3876i	1.10373i	0.833933	+	0.551865i	$0.186084\pi$
$449$	23.3876i	1.10373i	−0.833933	+	0.551865i	$0.813916\pi$
$450$	0	0
$451$	−14.1827	−0.667836
$452$	0	0
$453$	− 2.99908i	− 0.140909i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	26.1674i	1.22406i	0.790835	+	0.612030i	$0.209647\pi$
$457$	26.1674i	1.22406i	−0.790835	+	0.612030i	$0.790353\pi$
$458$	0	0
$459$	9.27650	0.432990
$460$	0	0
$461$	− 25.2330i	− 1.17522i	−0.809145	−	0.587609i	$-0.800069\pi$
$461$	− 25.2330i	− 1.17522i	0.809145	−	0.587609i	$-0.199931\pi$
$462$	0	0
$463$	− 18.1600i	− 0.843968i	−0.906603	−	0.421984i	$-0.861334\pi$
$463$	− 18.1600i	− 0.843968i	0.906603	−	0.421984i	$-0.138666\pi$
$464$	0	0
$465$	−2.93335	−0.136031
$466$	0	0
$467$	−12.6328	−0.584577	−0.292289	−	0.956330i	$-0.594417\pi$
$467$	−12.6328	−0.584577	−0.292289	+	0.956330i	$0.594417\pi$
$468$	0	0
$469$	4.96294	0.229167
$470$	0	0
$471$	−1.08859	−0.0501594
$472$	0	0
$473$	17.5719i	0.807956i
$474$	0	0
$475$	10.0686i	0.461979i
$476$	0	0
$477$	32.4639	1.48642
$478$	0	0
$479$	− 12.3074i	− 0.562338i	−0.959658	−	0.281169i	$-0.909278\pi$
$479$	− 12.3074i	− 0.562338i	0.959658	−	0.281169i	$-0.0907221\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	2.03153i	0.0924378i
$484$	0	0
$485$	−16.5460	−0.751315
$486$	0	0
$487$	15.6339i	0.708438i	0.935163	+	0.354219i	$0.115253\pi$
$487$	15.6339i	0.708438i	−0.935163	+	0.354219i	$0.884747\pi$
$488$	0	0
$489$	1.41825i	0.0641355i
$490$	0	0
$491$	−10.2109	−0.460809	−0.230405	−	0.973095i	$-0.574005\pi$
$491$	−10.2109	−0.460809	−0.230405	+	0.973095i	$0.574005\pi$
$492$	0	0
$493$	41.9911	1.89119
$494$	0	0
$495$	11.1272	0.500129
$496$	0	0
$497$	16.5114	0.740639
$498$	0	0
$499$	− 0.681003i	− 0.0304859i	−0.999884	−	0.0152430i	$-0.995148\pi$
$499$	− 0.681003i	− 0.0304859i	0.999884	−	0.0152430i	$-0.00485217\pi$
$500$	0	0
$501$	− 1.43885i	− 0.0642830i
$502$	0	0
$503$	−35.3403	−1.57575	−0.787873	−	0.615838i	$-0.788817\pi$
$503$	−35.3403	−1.57575	−0.787873	+	0.615838i	$0.788817\pi$
$504$	0	0
$505$	− 3.28503i	− 0.146182i
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 16.8260i	− 0.745800i	−0.927872	−	0.372900i	$-0.878364\pi$
$509$	− 16.8260i	− 0.745800i	0.927872	−	0.372900i	$-0.121636\pi$
$510$	0	0
$511$	−15.4351	−0.682809
$512$	0	0
$513$	5.07706i	0.224158i
$514$	0	0
$515$	− 19.6475i	− 0.865774i
$516$	0	0
$517$	7.40922	0.325857
$518$	0	0
$519$	−2.62043	−0.115024
$520$	0	0
$521$	4.76619	0.208811	0.104405	−	0.994535i	$-0.466706\pi$
$521$	4.76619	0.208811	0.104405	+	0.994535i	$0.466706\pi$
$522$	0	0
$523$	−0.638159	−0.0279048	−0.0139524	−	0.999903i	$-0.504441\pi$
$523$	−0.638159	−0.0279048	−0.0139524	+	0.999903i	$0.504441\pi$
$524$	0	0
$525$	− 0.850340i	− 0.0371119i
$526$	0	0
$527$	46.8058i	2.03889i
$528$	0	0
$529$	34.0549	1.48065
$530$	0	0
$531$	18.4313i	0.799852i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	− 8.79595i	− 0.380282i
$536$	0	0
$537$	3.06748	0.132371
$538$	0	0
$539$	2.80318i	0.120742i
$540$	0	0
$541$	22.2050i	0.954668i	0.878722	+	0.477334i	$0.158397\pi$
$541$	22.2050i	0.954668i	−0.878722	+	0.477334i	$0.841603\pi$
$542$	0	0
$543$	−6.85096	−0.294003
$544$	0	0
$545$	−3.98138	−0.170544
$546$	0	0
$547$	16.2608	0.695263	0.347631	−	0.937631i	$-0.386986\pi$
$547$	16.2608	0.695263	0.347631	+	0.937631i	$0.386986\pi$
$548$	0	0
$549$	−37.4977	−1.60036
$550$	0	0
$551$	22.9819i	0.979062i
$552$	0	0
$553$	14.7661i	0.627917i
$554$	0	0
$555$	−2.53710	−0.107694
$556$	0	0
$557$	21.4676i	0.909612i	0.890591	+	0.454806i	$0.150291\pi$
$557$	21.4676i	0.909612i	−0.890591	+	0.454806i	$0.849709\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	4.38685i	0.185213i
$562$	0	0
$563$	11.1821	0.471268	0.235634	−	0.971842i	$-0.424283\pi$
$563$	11.1821	0.471268	0.235634	+	0.971842i	$0.424283\pi$
$564$	0	0
$565$	4.52607i	0.190413i
$566$	0	0
$567$	8.35421i	0.350844i
$568$	0	0
$569$	13.1633	0.551835	0.275917	−	0.961181i	$-0.411018\pi$
$569$	13.1633	0.551835	0.275917	+	0.961181i	$0.411018\pi$
$570$	0	0
$571$	−17.4199	−0.729001	−0.364501	−	0.931203i	$-0.618760\pi$
$571$	−17.4199	−0.729001	−0.364501	+	0.931203i	$0.618760\pi$
$572$	0	0
$573$	−4.40494	−0.184019
$574$	0	0
$575$	−23.8816	−0.995931
$576$	0	0
$577$	− 8.47023i	− 0.352620i	−0.984335	−	0.176310i	$-0.943584\pi$
$577$	− 8.47023i	− 0.352620i	0.984335	−	0.176310i	$-0.0564161\pi$
$578$	0	0
$579$	2.57185i	0.106882i
$580$	0	0
$581$	9.79310	0.406286
$582$	0	0
$583$	31.0836i	1.28735i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 21.0685i	− 0.869590i	−0.900529	−	0.434795i	$-0.856821\pi$
$587$	− 21.0685i	− 0.869590i	0.900529	−	0.434795i	$-0.143179\pi$
$588$	0	0
$589$	−25.6170	−1.05553
$590$	0	0
$591$	0.809681i	0.0333058i
$592$	0	0
$593$	13.2761i	0.545185i	0.962130	+	0.272593i	$0.0878811\pi$
$593$	13.2761i	0.545185i	−0.962130	+	0.272593i	$0.912119\pi$
$594$	0	0
$595$	7.88926	0.323428
$596$	0	0
$597$	−1.24248	−0.0508513
$598$	0	0
$599$	5.39084	0.220264	0.110132	−	0.993917i	$-0.464873\pi$
$599$	5.39084	0.220264	0.110132	+	0.993917i	$0.464873\pi$
$600$	0	0
$601$	−24.4408	−0.996959	−0.498480	−	0.866901i	$-0.666108\pi$
$601$	−24.4408	−0.996959	−0.498480	+	0.866901i	$0.666108\pi$
$602$	0	0
$603$	14.5298i	0.591700i
$604$	0	0
$605$	− 4.26030i	− 0.173206i
$606$	0	0
$607$	−25.8582	−1.04955	−0.524776	−	0.851240i	$-0.675851\pi$
$607$	−25.8582	−1.04955	−0.524776	+	0.851240i	$0.675851\pi$
$608$	0	0
$609$	− 1.94093i	− 0.0786504i
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 14.1688i	− 0.572273i	−0.958189	−	0.286136i	$-0.907629\pi$
$613$	− 14.1688i	− 0.572273i	0.958189	−	0.286136i	$-0.0923711\pi$
$614$	0	0
$615$	1.84499	0.0743973
$616$	0	0
$617$	46.0452i	1.85371i	0.375419	+	0.926855i	$0.377499\pi$
$617$	46.0452i	1.85371i	−0.375419	+	0.926855i	$0.622501\pi$
$618$	0	0
$619$	− 12.0284i	− 0.483464i	−0.970343	−	0.241732i	$-0.922285\pi$
$619$	− 12.0284i	− 0.483464i	0.970343	−	0.241732i	$-0.0777155\pi$
$620$	0	0
$621$	−12.0422	−0.483238
$622$	0	0
$623$	−8.36820	−0.335265
$624$	0	0
$625$	0.804500	0.0321800
$626$	0	0
$627$	−2.40094	−0.0958841
$628$	0	0
$629$	40.4831i	1.61417i
$630$	0	0
$631$	24.3036i	0.967512i	0.875203	+	0.483756i	$0.160728\pi$
$631$	24.3036i	0.967512i	−0.875203	+	0.483756i	$0.839272\pi$
$632$	0	0
$633$	−0.628986	−0.0249999
$634$	0	0
$635$	− 17.0162i	− 0.675269i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	48.3399i	1.91230i
$640$	0	0
$641$	17.7655	0.701696	0.350848	−	0.936432i	$-0.385893\pi$
$641$	17.7655	0.701696	0.350848	+	0.936432i	$0.385893\pi$
$642$	0	0
$643$	− 43.1560i	− 1.70191i	−0.525242	−	0.850953i	$-0.676025\pi$
$643$	− 43.1560i	− 1.70191i	0.525242	−	0.850953i	$-0.323975\pi$
$644$	0	0
$645$	− 2.28589i	− 0.0900067i
$646$	0	0
$647$	10.7995	0.424571	0.212286	−	0.977208i	$-0.431909\pi$
$647$	10.7995	0.424571	0.212286	+	0.977208i	$0.431909\pi$
$648$	0	0
$649$	−17.6477	−0.692731
$650$	0	0
$651$	2.16348	0.0847933
$652$	0	0
$653$	−3.06560	−0.119966	−0.0599831	−	0.998199i	$-0.519105\pi$
$653$	−3.06560	−0.119966	−0.0599831	+	0.998199i	$0.519105\pi$
$654$	0	0
$655$	− 1.70838i	− 0.0667520i
$656$	0	0
$657$	− 45.1888i	− 1.76298i
$658$	0	0
$659$	−3.64872	−0.142134	−0.0710670	−	0.997472i	$-0.522640\pi$
$659$	−3.64872	−0.142134	−0.0710670	+	0.997472i	$0.522640\pi$
$660$	0	0
$661$	27.2252i	1.05894i	0.848329	+	0.529469i	$0.177609\pi$
$661$	27.2252i	1.05894i	−0.848329	+	0.529469i	$0.822391\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	4.31782i	0.167438i
$666$	0	0
$667$	−54.5105	−2.11065
$668$	0	0
$669$	− 1.30879i	− 0.0506008i
$670$	0	0
$671$	− 35.9034i	− 1.38604i
$672$	0	0
$673$	−28.2970	−1.09077	−0.545384	−	0.838186i	$-0.683616\pi$
$673$	−28.2970	−1.09077	−0.545384	+	0.838186i	$0.683616\pi$
$674$	0	0
$675$	5.04053	0.194010
$676$	0	0
$677$	−28.8361	−1.10826	−0.554131	−	0.832429i	$-0.686949\pi$
$677$	−28.8361	−1.10826	−0.554131	+	0.832429i	$0.686949\pi$
$678$	0	0
$679$	12.2034	0.468324
$680$	0	0
$681$	7.55209i	0.289397i
$682$	0	0
$683$	40.9045i	1.56517i	0.622546	+	0.782583i	$0.286098\pi$
$683$	40.9045i	1.56517i	−0.622546	+	0.782583i	$0.713902\pi$
$684$	0	0
$685$	−11.2848	−0.431169
$686$	0	0
$687$	0.0482178i	0.00183963i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	42.2544i	1.60743i	0.595011	+	0.803717i	$0.297148\pi$
$691$	42.2544i	1.60743i	−0.595011	+	0.803717i	$0.702852\pi$
$692$	0	0
$693$	−8.20678	−0.311750
$694$	0	0
$695$	7.84870i	0.297718i
$696$	0	0
$697$	− 29.4395i	− 1.11510i
$698$	0	0
$699$	−4.07151	−0.153998
$700$	0	0
$701$	−30.3734	−1.14719	−0.573594	−	0.819140i	$-0.694451\pi$
$701$	−30.3734	−1.14719	−0.573594	+	0.819140i	$0.694451\pi$
$702$	0	0
$703$	−22.1566	−0.835650
$704$	0	0
$705$	−0.963850	−0.0363007
$706$	0	0
$707$	2.42285i	0.0911208i
$708$	0	0
$709$	− 23.4297i	− 0.879920i	−0.898017	−	0.439960i	$-0.854993\pi$
$709$	− 23.4297i	− 0.879920i	0.898017	−	0.439960i	$-0.145007\pi$
$710$	0	0
$711$	−43.2301	−1.62125
$712$	0	0
$713$	− 60.7606i	− 2.27550i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	3.46151i	0.129273i
$718$	0	0
$719$	7.02982	0.262168	0.131084	−	0.991371i	$-0.458154\pi$
$719$	7.02982	0.262168	0.131084	+	0.991371i	$0.458154\pi$
$720$	0	0
$721$	14.4909i	0.539671i
$722$	0	0
$723$	− 2.64427i	− 0.0983415i
$724$	0	0
$725$	22.8165	0.847385
$726$	0	0
$727$	−20.6810	−0.767015	−0.383507	−	0.923538i	$-0.625284\pi$
$727$	−20.6810	−0.767015	−0.383507	+	0.923538i	$0.625284\pi$
$728$	0	0
$729$	−23.1720	−0.858222
$730$	0	0
$731$	−36.4746	−1.34906
$732$	0	0
$733$	− 39.0035i	− 1.44063i	−0.693648	−	0.720314i	$-0.743998\pi$
$733$	− 39.0035i	− 1.44063i	0.693648	−	0.720314i	$-0.256002\pi$
$734$	0	0
$735$	− 0.364660i	− 0.0134507i
$736$	0	0
$737$	−13.9120	−0.512456
$738$	0	0
$739$	− 28.7416i	− 1.05728i	−0.848847	−	0.528639i	$-0.822703\pi$
$739$	− 28.7416i	− 1.05728i	0.848847	−	0.528639i	$-0.177297\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	6.32843i	0.232168i	0.993239	+	0.116084i	$0.0370341\pi$
$743$	6.32843i	0.232168i	−0.993239	+	0.116084i	$0.962966\pi$
$744$	0	0
$745$	19.2583	0.705569
$746$	0	0
$747$	28.6709i	1.04901i
$748$	0	0
$749$	6.48741i	0.237045i
$750$	0	0
$751$	31.4405	1.14728	0.573641	−	0.819107i	$-0.305531\pi$
$751$	31.4405	1.14728	0.573641	+	0.819107i	$0.305531\pi$
$752$	0	0
$753$	−4.65880	−0.169776
$754$	0	0
$755$	15.1190	0.550237
$756$	0	0
$757$	25.4423	0.924716	0.462358	−	0.886693i	$-0.347003\pi$
$757$	25.4423	0.924716	0.462358	+	0.886693i	$0.347003\pi$
$758$	0	0
$759$	− 5.69475i	− 0.206706i
$760$	0	0
$761$	− 20.6105i	− 0.747130i	−0.927604	−	0.373565i	$-0.878135\pi$
$761$	− 20.6105i	− 0.747130i	0.927604	−	0.373565i	$-0.121865\pi$
$762$	0	0
$763$	2.93644	0.106306
$764$	0	0
$765$	23.0971i	0.835077i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	− 9.18186i	− 0.331106i	−0.986201	−	0.165553i	$-0.947059\pi$
$769$	− 9.18186i	− 0.331106i	0.986201	−	0.165553i	$-0.0529410\pi$
$770$	0	0
$771$	−2.87942	−0.103700
$772$	0	0
$773$	− 8.49555i	− 0.305564i	−0.988260	−	0.152782i	$-0.951177\pi$
$773$	− 8.49555i	− 0.305564i	0.988260	−	0.152782i	$-0.0488232\pi$
$774$	0	0
$775$	25.4327i	0.913569i
$776$	0	0
$777$	1.87122	0.0671298
$778$	0	0
$779$	16.1124	0.577285
$780$	0	0
$781$	−46.2846	−1.65619
$782$	0	0
$783$	11.5052	0.411161
$784$	0	0
$785$	− 5.48780i	− 0.195868i
$786$	0	0
$787$	1.96835i	0.0701640i	0.999384	+	0.0350820i	$0.0111692\pi$
$787$	1.96835i	0.0701640i	−0.999384	+	0.0350820i	$0.988831\pi$
$788$	0	0
$789$	3.78408	0.134717
$790$	0	0
$791$	− 3.33818i	− 0.118692i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	− 4.04360i	− 0.143412i
$796$	0	0
$797$	38.4898	1.36338	0.681690	−	0.731641i	$-0.261245\pi$
$797$	38.4898	1.36338	0.681690	+	0.731641i	$0.261245\pi$
$798$	0	0
$799$	15.3796i	0.544092i
$800$	0	0
$801$	− 24.4993i	− 0.865639i
$802$	0	0
$803$	43.2674	1.52687
$804$	0	0
$805$	−10.2414	−0.360961
$806$	0	0
$807$	0.146317	0.00515059
$808$	0	0
$809$	1.79976	0.0632760	0.0316380	−	0.999499i	$-0.489928\pi$
$809$	1.79976	0.0632760	0.0316380	+	0.999499i	$0.489928\pi$
$810$	0	0
$811$	49.9235i	1.75305i	0.481357	+	0.876525i	$0.340144\pi$
$811$	49.9235i	1.75305i	−0.481357	+	0.876525i	$0.659856\pi$
$812$	0	0
$813$	− 0.199352i	− 0.00699157i
$814$	0	0
$815$	−7.14970	−0.250443
$816$	0	0
$817$	− 19.9627i	− 0.698406i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 4.21824i	− 0.147217i	−0.997287	−	0.0736087i	$-0.976548\pi$
$821$	− 4.21824i	− 0.147217i	0.997287	−	0.0736087i	$-0.0234516\pi$
$822$	0	0
$823$	−14.4982	−0.505377	−0.252688	−	0.967548i	$-0.581315\pi$
$823$	−14.4982	−0.505377	−0.252688	+	0.967548i	$0.581315\pi$
$824$	0	0
$825$	2.38366i	0.0829884i
$826$	0	0
$827$	26.2618i	0.913213i	0.889669	+	0.456607i	$0.150935\pi$
$827$	26.2618i	0.913213i	−0.889669	+	0.456607i	$0.849065\pi$
$828$	0	0
$829$	55.1580	1.91572	0.957858	−	0.287242i	$-0.0927383\pi$
$829$	55.1580	1.91572	0.957858	+	0.287242i	$0.0927383\pi$
$830$	0	0
$831$	−3.80125	−0.131864
$832$	0	0
$833$	−5.81868	−0.201605
$834$	0	0
$835$	7.25355	0.251019
$836$	0	0
$837$	12.8244i	0.443275i
$838$	0	0
$839$	− 7.02028i	− 0.242367i	−0.992630	−	0.121184i	$-0.961331\pi$
$839$	− 7.02028i	− 0.242367i	0.992630	−	0.121184i	$-0.0386690\pi$
$840$	0	0
$841$	23.0795	0.795843
$842$	0	0
$843$	− 6.94036i	− 0.239039i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	3.14216i	0.107966i
$848$	0	0
$849$	7.33750	0.251822
$850$	0	0
$851$	− 52.5528i	− 1.80149i
$852$	0	0
$853$	− 57.6508i	− 1.97393i	−0.160945	−	0.986963i	$-0.551454\pi$
$853$	− 57.6508i	− 1.97393i	0.160945	−	0.986963i	$-0.448546\pi$
$854$	0	0
$855$	−12.6411	−0.432317
$856$	0	0
$857$	−2.50654	−0.0856217	−0.0428109	−	0.999083i	$-0.513631\pi$
$857$	−2.50654	−0.0856217	−0.0428109	+	0.999083i	$0.513631\pi$
$858$	0	0
$859$	17.1305	0.584486	0.292243	−	0.956344i	$-0.405598\pi$
$859$	17.1305	0.584486	0.292243	+	0.956344i	$0.405598\pi$
$860$	0	0
$861$	−1.36076	−0.0463747
$862$	0	0
$863$	− 55.6429i	− 1.89411i	−0.321079	−	0.947053i	$-0.604045\pi$
$863$	− 55.6429i	− 1.89411i	0.321079	−	0.947053i	$-0.395955\pi$
$864$	0	0
$865$	− 13.2102i	− 0.449159i
$866$	0	0
$867$	−4.53375	−0.153974
$868$	0	0
$869$	− 41.3920i	− 1.40413i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	35.7275i	1.20919i
$874$	0	0
$875$	11.0660	0.374099
$876$	0	0
$877$	40.1549i	1.35593i	0.735093	+	0.677967i	$0.237139\pi$
$877$	40.1549i	1.35593i	−0.735093	+	0.677967i	$0.762861\pi$
$878$	0	0
$879$	0.465871i	0.0157134i
$880$	0	0
$881$	−50.7673	−1.71039	−0.855197	−	0.518303i	$-0.826564\pi$
$881$	−50.7673	−1.71039	−0.855197	+	0.518303i	$0.826564\pi$
$882$	0	0
$883$	−46.3312	−1.55917	−0.779585	−	0.626297i	$-0.784570\pi$
$883$	−46.3312	−1.55917	−0.779585	+	0.626297i	$0.784570\pi$
$884$	0	0
$885$	2.29575	0.0771706
$886$	0	0
$887$	−10.2550	−0.344331	−0.172165	−	0.985068i	$-0.555076\pi$
$887$	−10.2550	−0.344331	−0.172165	+	0.985068i	$0.555076\pi$
$888$	0	0
$889$	12.5502i	0.420921i
$890$	0	0
$891$	− 23.4184i	− 0.784546i
$892$	0	0
$893$	−8.41732	−0.281675
$894$	0	0
$895$	15.4638i	0.516898i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	58.0509i	1.93611i
$900$	0	0
$901$	−64.5214	−2.14952
$902$	0	0
$903$	1.68594i	0.0561047i
$904$	0	0
$905$	− 34.5372i	− 1.14805i
$906$	0	0
$907$	−52.2945	−1.73641	−0.868205	−	0.496206i	$-0.834726\pi$
$907$	−52.2945	−1.73641	−0.868205	+	0.496206i	$0.834726\pi$
$908$	0	0
$909$	−7.09330	−0.235270
$910$	0	0
$911$	50.9172	1.68696	0.843481	−	0.537159i	$-0.180502\pi$
$911$	50.9172	1.68696	0.843481	+	0.537159i	$0.180502\pi$
$912$	0	0
$913$	−27.4519	−0.908524
$914$	0	0
$915$	4.67059i	0.154405i
$916$	0	0
$917$	1.26001i	0.0416091i
$918$	0	0
$919$	22.8646	0.754234	0.377117	−	0.926166i	$-0.376915\pi$
$919$	22.8646	0.754234	0.377117	+	0.926166i	$0.376915\pi$
$920$	0	0
$921$	− 5.76541i	− 0.189977i
$922$	0	0
$923$	0	0
$924$	0	0
$925$	21.9971i	0.723261i
$926$	0	0
$927$	−42.4246	−1.39341
$928$	0	0
$929$	33.2658i	1.09142i	0.837976	+	0.545708i	$0.183739\pi$
$929$	33.2658i	1.09142i	−0.837976	+	0.545708i	$0.816261\pi$
$930$	0	0
$931$	− 3.18458i	− 0.104371i
$932$	0	0
$933$	7.33498	0.240137
$934$	0	0
$935$	−22.1150	−0.723239
$936$	0	0
$937$	−51.2179	−1.67322	−0.836608	−	0.547802i	$-0.815465\pi$
$937$	−51.2179	−1.67322	−0.836608	+	0.547802i	$0.815465\pi$
$938$	0	0
$939$	−1.95857	−0.0639156
$940$	0	0
$941$	10.2918i	0.335502i	0.985829	+	0.167751i	$0.0536504\pi$
$941$	10.2918i	0.335502i	−0.985829	+	0.167751i	$0.946350\pi$
$942$	0	0
$943$	38.2167i	1.24451i
$944$	0	0
$945$	2.16158	0.0703163
$946$	0	0
$947$	− 35.4887i	− 1.15323i	−0.817017	−	0.576614i	$-0.804374\pi$
$947$	− 35.4887i	− 1.15323i	0.817017	−	0.576614i	$-0.195626\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0.685389i	0.0222253i
$952$	0	0
$953$	−33.3352	−1.07983	−0.539916	−	0.841719i	$-0.681544\pi$
$953$	−33.3352	−1.07983	−0.539916	+	0.841719i	$0.681544\pi$
$954$	0	0
$955$	− 22.2062i	− 0.718577i
$956$	0	0
$957$	5.44078i	0.175875i
$958$	0	0
$959$	8.32302	0.268764
$960$	0	0
$961$	−33.7070	−1.08732
$962$	0	0
$963$	−18.9929	−0.612039
$964$	0	0
$965$	−12.9653	−0.417366
$966$	0	0
$967$	29.7131i	0.955509i	0.878494	+	0.477754i	$0.158549\pi$
$967$	29.7131i	0.955509i	−0.878494	+	0.477754i	$0.841451\pi$
$968$	0	0
$969$	− 4.98372i	− 0.160100i
$970$	0	0
$971$	−8.83702	−0.283593	−0.141797	−	0.989896i	$-0.545288\pi$
$971$	−8.83702	−0.283593	−0.141797	+	0.989896i	$0.545288\pi$
$972$	0	0
$973$	− 5.78876i	− 0.185579i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	14.8307i	0.474475i	0.971452	+	0.237237i	$0.0762419\pi$
$977$	14.8307i	0.474475i	−0.971452	+	0.237237i	$0.923758\pi$
$978$	0	0
$979$	23.4576	0.749708
$980$	0	0
$981$	8.59692i	0.274479i
$982$	0	0
$983$	20.3390i	0.648714i	0.945935	+	0.324357i	$0.105148\pi$
$983$	20.3390i	0.648714i	−0.945935	+	0.324357i	$0.894852\pi$
$984$	0	0
$985$	−4.08178	−0.130056
$986$	0	0
$987$	0.710882	0.0226276
$988$	0	0
$989$	47.3493	1.50562
$990$	0	0
$991$	−50.8906	−1.61659	−0.808297	−	0.588775i	$-0.799611\pi$
$991$	−50.8906	−1.61659	−0.808297	+	0.588775i	$0.799611\pi$
$992$	0	0
$993$	− 0.0526244i	− 0.00166999i
$994$	0	0
$995$	− 6.26361i	− 0.198570i
$996$	0	0
$997$	−7.39394	−0.234168	−0.117084	−	0.993122i	$-0.537355\pi$
$997$	−7.39394	−0.234168	−0.117084	+	0.993122i	$0.537355\pi$
$998$	0	0
$999$	11.0920i	0.350935i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4732.2.g.k.337.7		16
13.3	even	3		364.2.u.a.225.5	✓	16
13.4	even	6		364.2.u.a.309.5	yes	16
13.5	odd	4		4732.2.a.s.1.4		8
13.8	odd	4		4732.2.a.t.1.4		8
13.12	even	2	inner	4732.2.g.k.337.8		16
39.17	odd	6		3276.2.cf.c.1765.4		16
39.29	odd	6		3276.2.cf.c.2773.5		16
52.3	odd	6		1456.2.cc.f.225.4		16
52.43	odd	6		1456.2.cc.f.673.4		16
91.3	odd	6		2548.2.bq.c.1941.5		16
91.4	even	6		2548.2.bb.d.569.5		16
91.16	even	3		2548.2.bb.d.1733.5		16
91.17	odd	6		2548.2.bb.c.569.4		16
91.30	even	6		2548.2.bq.e.361.4		16
91.55	odd	6		2548.2.u.c.589.4		16
91.68	odd	6		2548.2.bb.c.1733.4		16
91.69	odd	6		2548.2.u.c.1765.4		16
91.81	even	3		2548.2.bq.e.1941.4		16
91.82	odd	6		2548.2.bq.c.361.5		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
364.2.u.a.225.5	✓	16	13.3	even	3
364.2.u.a.309.5	yes	16	13.4	even	6
1456.2.cc.f.225.4		16	52.3	odd	6
1456.2.cc.f.673.4		16	52.43	odd	6
2548.2.u.c.589.4		16	91.55	odd	6
2548.2.u.c.1765.4		16	91.69	odd	6
2548.2.bb.c.569.4		16	91.17	odd	6
2548.2.bb.c.1733.4		16	91.68	odd	6
2548.2.bb.d.569.5		16	91.4	even	6
2548.2.bb.d.1733.5		16	91.16	even	3
2548.2.bq.c.361.5		16	91.82	odd	6
2548.2.bq.c.1941.5		16	91.3	odd	6
2548.2.bq.e.361.4		16	91.30	even	6
2548.2.bq.e.1941.4		16	91.81	even	3
3276.2.cf.c.1765.4		16	39.17	odd	6
3276.2.cf.c.2773.5		16	39.29	odd	6
4732.2.a.s.1.4		8	13.5	odd	4
4732.2.a.t.1.4		8	13.8	odd	4
4732.2.g.k.337.7		16	1.1	even	1	trivial
4732.2.g.k.337.8		16	13.12	even	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:	\( N \)	\(=\)	\( 4732 = 2^{2} \cdot 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4732.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.268953 q^{3} -1.35585i q^{5} +1.00000i q^{7} -2.92766 q^{9} -2.80318i q^{11} +0.364660i q^{15} +5.81868 q^{17} +3.18458i q^{19} -0.268953i q^{21} -7.55347 q^{23} +3.16167 q^{25} +1.59426 q^{27} +7.21661 q^{29} +8.04407i q^{31} +0.753925i q^{33} +1.35585 q^{35} +6.95744i q^{37} -5.05949i q^{41} -6.26854 q^{43} +3.96948i q^{45} +2.64315i q^{47} -1.00000 q^{49} -1.56495 q^{51} -11.0887 q^{53} -3.80070 q^{55} -0.856503i q^{57} -6.29558i q^{59} +12.8081 q^{61} -2.92766i q^{63} -4.96294i q^{67} +2.03153 q^{69} -16.5114i q^{71} +15.4351i q^{73} -0.850340 q^{75} +2.80318 q^{77} +14.7661 q^{79} +8.35421 q^{81} -9.79310i q^{83} -7.88926i q^{85} -1.94093 q^{87} +8.36820i q^{89} -2.16348i q^{93} +4.31782 q^{95} -12.2034i q^{97} +8.20678i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 28 q^{9} - 4 q^{17} - 44 q^{25} - 12 q^{27} + 44 q^{29} + 12 q^{35} - 12 q^{43} - 16 q^{49} - 4 q^{51} + 8 q^{53} - 4 q^{55} - 8 q^{61} + 104 q^{69} + 20 q^{75} - 24 q^{77} + 8 q^{79} - 52 q^{87}+ \cdots + 60 q^{95}+O(q^{100}) \) 16 * q + 28 * q^9 - 4 * q^17 - 44 * q^25 - 12 * q^27 + 44 * q^29 + 12 * q^35 - 12 * q^43 - 16 * q^49 - 4 * q^51 + 8 * q^53 - 4 * q^55 - 8 * q^61 + 104 * q^69 + 20 * q^75 - 24 * q^77 + 8 * q^79 - 52 * q^87 + 60 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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