LMFDB - Embedded newform 832.2.b.a.417.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [832,2,Mod(417,832)] 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("832.417"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(832, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 832 = 2^{6} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	832.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,-12,0,0,0,0,0,0,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)

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

Self dual:	no
Analytic conductor:	$6.64355344817$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			417.2
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	832.417
Dual form			832.2.b.a.417.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.414214i q^{3} -3.82843i q^{5} -1.58579 q^{7} +2.82843 q^{9} -4.82843i q^{11} +1.00000i q^{13} -1.58579 q^{15} +1.00000 q^{17} +5.65685i q^{19} +0.656854i q^{21} -3.17157 q^{23} -9.65685 q^{25} -2.41421i q^{27} -7.65685i q^{29} -7.65685 q^{31} -2.00000 q^{33} +6.07107i q^{35} -7.00000i q^{37} +0.414214 q^{39} +1.65685 q^{41} +5.58579i q^{43} -10.8284i q^{45} +9.24264 q^{47} -4.48528 q^{49} -0.414214i q^{51} +7.65685i q^{53} -18.4853 q^{55} +2.34315 q^{57} -8.00000i q^{59} -4.48528 q^{63} +3.82843 q^{65} +3.17157i q^{67} +1.31371i q^{69} -13.7279 q^{71} +9.65685 q^{73} +4.00000i q^{75} +7.65685i q^{77} +12.1421 q^{79} +7.48528 q^{81} -16.1421i q^{83} -3.82843i q^{85} -3.17157 q^{87} -2.34315 q^{89} -1.58579i q^{91} +3.17157i q^{93} +21.6569 q^{95} +3.65685 q^{97} -13.6569i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{7} - 12 q^{15} + 4 q^{17} - 24 q^{23} - 16 q^{25} - 8 q^{31} - 8 q^{33} - 4 q^{39} - 16 q^{41} + 20 q^{47} + 16 q^{49} - 40 q^{55} + 32 q^{57} + 16 q^{63} + 4 q^{65} - 4 q^{71} + 16 q^{73} - 8 q^{79}+ \cdots - 8 q^{97}+O(q^{100}) $ 4 * q - 12 * q^7 - 12 * q^15 + 4 * q^17 - 24 * q^23 - 16 * q^25 - 8 * q^31 - 8 * q^33 - 4 * q^39 - 16 * q^41 + 20 * q^47 + 16 * q^49 - 40 * q^55 + 32 * q^57 + 16 * q^63 + 4 * q^65 - 4 * q^71 + 16 * q^73 - 8 * q^79 - 4 * q^81 - 24 * q^87 - 32 * q^89 + 64 * q^95 - 8 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/832\mathbb{Z}\right)^\times$.

$n$	$261$	$703$	$769$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	− 0.414214i	− 0.239146i	−0.992825	−	0.119573i	$-0.961847\pi$
$3$	− 0.414214i	− 0.239146i	0.992825	−	0.119573i	$-0.0381526\pi$
$4$	0	0
$5$	− 3.82843i	− 1.71212i	−0.516873	−	0.856062i	$-0.672904\pi$
$5$	− 3.82843i	− 1.71212i	0.516873	−	0.856062i	$-0.327096\pi$
$6$	0	0
$7$	−1.58579	−0.599371	−0.299685	−	0.954038i	$-0.596882\pi$
$7$	−1.58579	−0.599371	−0.299685	+	0.954038i	$0.596882\pi$
$8$	0	0
$9$	2.82843	0.942809
$10$	0	0
$11$	− 4.82843i	− 1.45583i	−0.685670	−	0.727913i	$-0.740491\pi$
$11$	− 4.82843i	− 1.45583i	0.685670	−	0.727913i	$-0.259509\pi$
$12$	0	0
$13$	1.00000i	0.277350i
$13$	1.00000i	0.277350i
$14$	0	0
$15$	−1.58579	−0.409448
$16$	0	0
$17$	1.00000	0.242536	0.121268	−	0.992620i	$-0.461304\pi$
$17$	1.00000	0.242536	0.121268	+	0.992620i	$0.461304\pi$
$18$	0	0
$19$	5.65685i	1.29777i	0.760886	+	0.648886i	$0.224765\pi$
$19$	5.65685i	1.29777i	−0.760886	+	0.648886i	$0.775235\pi$
$20$	0	0
$21$	0.656854i	0.143337i
$22$	0	0
$23$	−3.17157	−0.661319	−0.330659	−	0.943750i	$-0.607271\pi$
$23$	−3.17157	−0.661319	−0.330659	+	0.943750i	$0.607271\pi$
$24$	0	0
$25$	−9.65685	−1.93137
$26$	0	0
$27$	− 2.41421i	− 0.464616i
$28$	0	0
$29$	− 7.65685i	− 1.42184i	−0.703272	−	0.710921i	$-0.748278\pi$
$29$	− 7.65685i	− 1.42184i	0.703272	−	0.710921i	$-0.251722\pi$
$30$	0	0
$31$	−7.65685	−1.37521	−0.687606	−	0.726084i	$-0.741338\pi$
$31$	−7.65685	−1.37521	−0.687606	+	0.726084i	$0.741338\pi$
$32$	0	0
$33$	−2.00000	−0.348155
$34$	0	0
$35$	6.07107i	1.02620i
$36$	0	0
$37$	− 7.00000i	− 1.15079i	−0.817875	−	0.575396i	$-0.804848\pi$
$37$	− 7.00000i	− 1.15079i	0.817875	−	0.575396i	$-0.195152\pi$
$38$	0	0
$39$	0.414214	0.0663273
$40$	0	0
$41$	1.65685	0.258757	0.129379	−	0.991595i	$-0.458702\pi$
$41$	1.65685	0.258757	0.129379	+	0.991595i	$0.458702\pi$
$42$	0	0
$43$	5.58579i	0.851824i	0.904764	+	0.425912i	$0.140047\pi$
$43$	5.58579i	0.851824i	−0.904764	+	0.425912i	$0.859953\pi$
$44$	0	0
$45$	− 10.8284i	− 1.61421i
$46$	0	0
$47$	9.24264	1.34818	0.674089	−	0.738650i	$-0.264536\pi$
$47$	9.24264	1.34818	0.674089	+	0.738650i	$0.264536\pi$
$48$	0	0
$49$	−4.48528	−0.640754
$50$	0	0
$51$	− 0.414214i	− 0.0580015i
$52$	0	0
$53$	7.65685i	1.05175i	0.850562	+	0.525875i	$0.176262\pi$
$53$	7.65685i	1.05175i	−0.850562	+	0.525875i	$0.823738\pi$
$54$	0	0
$55$	−18.4853	−2.49255
$56$	0	0
$57$	2.34315	0.310357
$58$	0	0
$59$	− 8.00000i	− 1.04151i	−0.853706	−	0.520756i	$-0.825650\pi$
$59$	− 8.00000i	− 1.04151i	0.853706	−	0.520756i	$-0.174350\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	−4.48528	−0.565092
$64$	0	0
$65$	3.82843	0.474858
$66$	0	0
$67$	3.17157i	0.387469i	0.981054	+	0.193735i	$0.0620601\pi$
$67$	3.17157i	0.387469i	−0.981054	+	0.193735i	$0.937940\pi$
$68$	0	0
$69$	1.31371i	0.158152i
$70$	0	0
$71$	−13.7279	−1.62920	−0.814602	−	0.580020i	$-0.803045\pi$
$71$	−13.7279	−1.62920	−0.814602	+	0.580020i	$0.803045\pi$
$72$	0	0
$73$	9.65685	1.13025	0.565125	−	0.825006i	$-0.308828\pi$
$73$	9.65685	1.13025	0.565125	+	0.825006i	$0.308828\pi$
$74$	0	0
$75$	4.00000i	0.461880i
$76$	0	0
$77$	7.65685i	0.872580i
$78$	0	0
$79$	12.1421	1.36610	0.683048	−	0.730373i	$-0.260654\pi$
$79$	12.1421	1.36610	0.683048	+	0.730373i	$0.260654\pi$
$80$	0	0
$81$	7.48528	0.831698
$82$	0	0
$83$	− 16.1421i	− 1.77183i	−0.463848	−	0.885915i	$-0.653532\pi$
$83$	− 16.1421i	− 1.77183i	0.463848	−	0.885915i	$-0.346468\pi$
$84$	0	0
$85$	− 3.82843i	− 0.415251i
$86$	0	0
$87$	−3.17157	−0.340028
$88$	0	0
$89$	−2.34315	−0.248373	−0.124186	−	0.992259i	$-0.539632\pi$
$89$	−2.34315	−0.248373	−0.124186	+	0.992259i	$0.539632\pi$
$90$	0	0
$91$	− 1.58579i	− 0.166236i
$92$	0	0
$93$	3.17157i	0.328877i
$94$	0	0
$95$	21.6569	2.22195
$96$	0	0
$97$	3.65685	0.371297	0.185649	−	0.982616i	$-0.440561\pi$
$97$	3.65685	0.371297	0.185649	+	0.982616i	$0.440561\pi$
$98$	0	0
$99$	− 13.6569i	− 1.37257i
$100$	0	0
$101$	6.34315i	0.631167i	0.948898	+	0.315583i	$0.102200\pi$
$101$	6.34315i	0.631167i	−0.948898	+	0.315583i	$0.897800\pi$
$102$	0	0
$103$	12.1421	1.19640	0.598200	−	0.801347i	$-0.295883\pi$
$103$	12.1421	1.19640	0.598200	+	0.801347i	$0.295883\pi$
$104$	0	0
$105$	2.51472	0.245411
$106$	0	0
$107$	6.00000i	0.580042i	0.957020	+	0.290021i	$0.0936623\pi$
$107$	6.00000i	0.580042i	−0.957020	+	0.290021i	$0.906338\pi$
$108$	0	0
$109$	− 14.6569i	− 1.40387i	−0.712240	−	0.701936i	$-0.752319\pi$
$109$	− 14.6569i	− 1.40387i	0.712240	−	0.701936i	$-0.247681\pi$
$110$	0	0
$111$	−2.89949	−0.275208
$112$	0	0
$113$	−10.0000	−0.940721	−0.470360	−	0.882474i	$-0.655876\pi$
$113$	−10.0000	−0.940721	−0.470360	+	0.882474i	$0.655876\pi$
$114$	0	0
$115$	12.1421i	1.13226i
$116$	0	0
$117$	2.82843i	0.261488i
$118$	0	0
$119$	−1.58579	−0.145369
$120$	0	0
$121$	−12.3137	−1.11943
$122$	0	0
$123$	− 0.686292i	− 0.0618808i
$124$	0	0
$125$	17.8284i	1.59462i
$126$	0	0
$127$	3.17157	0.281432	0.140716	−	0.990050i	$-0.455060\pi$
$127$	3.17157	0.281432	0.140716	+	0.990050i	$0.455060\pi$
$128$	0	0
$129$	2.31371	0.203711
$130$	0	0
$131$	− 5.58579i	− 0.488032i	−0.969771	−	0.244016i	$-0.921535\pi$
$131$	− 5.58579i	− 0.488032i	0.969771	−	0.244016i	$-0.0784650\pi$
$132$	0	0
$133$	− 8.97056i	− 0.777846i
$134$	0	0
$135$	−9.24264	−0.795480
$136$	0	0
$137$	15.6569	1.33766	0.668828	−	0.743417i	$-0.266796\pi$
$137$	15.6569	1.33766	0.668828	+	0.743417i	$0.266796\pi$
$138$	0	0
$139$	− 4.41421i	− 0.374409i	−0.982321	−	0.187204i	$-0.940057\pi$
$139$	− 4.41421i	− 0.374409i	0.982321	−	0.187204i	$-0.0599427\pi$
$140$	0	0
$141$	− 3.82843i	− 0.322412i
$142$	0	0
$143$	4.82843	0.403773
$144$	0	0
$145$	−29.3137	−2.43437
$146$	0	0
$147$	1.85786i	0.153234i
$148$	0	0
$149$	− 1.31371i	− 0.107623i	−0.998551	−	0.0538116i	$-0.982863\pi$
$149$	− 1.31371i	− 0.107623i	0.998551	−	0.0538116i	$-0.0171370\pi$
$150$	0	0
$151$	9.24264	0.752155	0.376078	−	0.926588i	$-0.377273\pi$
$151$	9.24264	0.752155	0.376078	+	0.926588i	$0.377273\pi$
$152$	0	0
$153$	2.82843	0.228665
$154$	0	0
$155$	29.3137i	2.35453i
$156$	0	0
$157$	21.6569i	1.72841i	0.503144	+	0.864203i	$0.332177\pi$
$157$	21.6569i	1.72841i	−0.503144	+	0.864203i	$0.667823\pi$
$158$	0	0
$159$	3.17157	0.251522
$160$	0	0
$161$	5.02944	0.396375
$162$	0	0
$163$	2.48528i	0.194662i	0.995252	+	0.0973311i	$0.0310306\pi$
$163$	2.48528i	0.194662i	−0.995252	+	0.0973311i	$0.968969\pi$
$164$	0	0
$165$	7.65685i	0.596085i
$166$	0	0
$167$	7.65685	0.592505	0.296253	−	0.955110i	$-0.404263\pi$
$167$	7.65685	0.592505	0.296253	+	0.955110i	$0.404263\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	16.0000i	1.22355i
$172$	0	0
$173$	14.0000i	1.06440i	0.846619	+	0.532200i	$0.178635\pi$
$173$	14.0000i	1.06440i	−0.846619	+	0.532200i	$0.821365\pi$
$174$	0	0
$175$	15.3137	1.15761
$176$	0	0
$177$	−3.31371	−0.249074
$178$	0	0
$179$	− 6.89949i	− 0.515692i	−0.966186	−	0.257846i	$-0.916987\pi$
$179$	− 6.89949i	− 0.515692i	0.966186	−	0.257846i	$-0.0830128\pi$
$180$	0	0
$181$	7.65685i	0.569129i	0.958657	+	0.284565i	$0.0918491\pi$
$181$	7.65685i	0.569129i	−0.958657	+	0.284565i	$0.908151\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−26.7990	−1.97030
$186$	0	0
$187$	− 4.82843i	− 0.353090i
$188$	0	0
$189$	3.82843i	0.278477i
$190$	0	0
$191$	24.8284	1.79652	0.898261	−	0.439462i	$-0.144831\pi$
$191$	24.8284	1.79652	0.898261	+	0.439462i	$0.144831\pi$
$192$	0	0
$193$	21.3137	1.53419	0.767097	−	0.641531i	$-0.221700\pi$
$193$	21.3137	1.53419	0.767097	+	0.641531i	$0.221700\pi$
$194$	0	0
$195$	− 1.58579i	− 0.113561i
$196$	0	0
$197$	7.00000i	0.498729i	0.968410	+	0.249365i	$0.0802218\pi$
$197$	7.00000i	0.498729i	−0.968410	+	0.249365i	$0.919778\pi$
$198$	0	0
$199$	−18.4853	−1.31039	−0.655193	−	0.755461i	$-0.727413\pi$
$199$	−18.4853	−1.31039	−0.655193	+	0.755461i	$0.727413\pi$
$200$	0	0
$201$	1.31371	0.0926619
$202$	0	0
$203$	12.1421i	0.852211i
$204$	0	0
$205$	− 6.34315i	− 0.443025i
$206$	0	0
$207$	−8.97056	−0.623497
$208$	0	0
$209$	27.3137	1.88933
$210$	0	0
$211$	− 1.58579i	− 0.109170i	−0.998509	−	0.0545850i	$-0.982616\pi$
$211$	− 1.58579i	− 0.109170i	0.998509	−	0.0545850i	$-0.0173836\pi$
$212$	0	0
$213$	5.68629i	0.389618i
$214$	0	0
$215$	21.3848	1.45843
$216$	0	0
$217$	12.1421	0.824262
$218$	0	0
$219$	− 4.00000i	− 0.270295i
$220$	0	0
$221$	1.00000i	0.0672673i
$222$	0	0
$223$	12.4142	0.831317	0.415659	−	0.909521i	$-0.363551\pi$
$223$	12.4142	0.831317	0.415659	+	0.909521i	$0.363551\pi$
$224$	0	0
$225$	−27.3137	−1.82091
$226$	0	0
$227$	− 20.9706i	− 1.39187i	−0.718107	−	0.695933i	$-0.754991\pi$
$227$	− 20.9706i	− 1.39187i	0.718107	−	0.695933i	$-0.245009\pi$
$228$	0	0
$229$	− 2.51472i	− 0.166177i	−0.996542	−	0.0830886i	$-0.973522\pi$
$229$	− 2.51472i	− 0.166177i	0.996542	−	0.0830886i	$-0.0264785\pi$
$230$	0	0
$231$	3.17157	0.208674
$232$	0	0
$233$	−12.1716	−0.797386	−0.398693	−	0.917084i	$-0.630536\pi$
$233$	−12.1716	−0.797386	−0.398693	+	0.917084i	$0.630536\pi$
$234$	0	0
$235$	− 35.3848i	− 2.30825i
$236$	0	0
$237$	− 5.02944i	− 0.326697i
$238$	0	0
$239$	7.92893	0.512880	0.256440	−	0.966560i	$-0.417450\pi$
$239$	7.92893	0.512880	0.256440	+	0.966560i	$0.417450\pi$
$240$	0	0
$241$	20.9706	1.35083	0.675416	−	0.737437i	$-0.263964\pi$
$241$	20.9706	1.35083	0.675416	+	0.737437i	$0.263964\pi$
$242$	0	0
$243$	− 10.3431i	− 0.663513i
$244$	0	0
$245$	17.1716i	1.09705i
$246$	0	0
$247$	−5.65685	−0.359937
$248$	0	0
$249$	−6.68629	−0.423727
$250$	0	0
$251$	− 9.31371i	− 0.587876i	−0.955824	−	0.293938i	$-0.905034\pi$
$251$	− 9.31371i	− 0.587876i	0.955824	−	0.293938i	$-0.0949659\pi$
$252$	0	0
$253$	15.3137i	0.962765i
$254$	0	0
$255$	−1.58579	−0.0993058
$256$	0	0
$257$	11.3431	0.707566	0.353783	−	0.935328i	$-0.384895\pi$
$257$	11.3431	0.707566	0.353783	+	0.935328i	$0.384895\pi$
$258$	0	0
$259$	11.1005i	0.689752i
$260$	0	0
$261$	− 21.6569i	− 1.34053i
$262$	0	0
$263$	6.34315	0.391135	0.195568	−	0.980690i	$-0.437345\pi$
$263$	6.34315	0.391135	0.195568	+	0.980690i	$0.437345\pi$
$264$	0	0
$265$	29.3137	1.80073
$266$	0	0
$267$	0.970563i	0.0593975i
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	−12.4142	−0.754110	−0.377055	−	0.926191i	$-0.623063\pi$
$271$	−12.4142	−0.754110	−0.377055	+	0.926191i	$0.623063\pi$
$272$	0	0
$273$	−0.656854	−0.0397546
$274$	0	0
$275$	46.6274i	2.81174i
$276$	0	0
$277$	− 8.97056i	− 0.538989i	−0.963002	−	0.269494i	$-0.913143\pi$
$277$	− 8.97056i	− 0.538989i	0.963002	−	0.269494i	$-0.0868566\pi$
$278$	0	0
$279$	−21.6569	−1.29656
$280$	0	0
$281$	4.00000	0.238620	0.119310	−	0.992857i	$-0.461932\pi$
$281$	4.00000	0.238620	0.119310	+	0.992857i	$0.461932\pi$
$282$	0	0
$283$	− 30.2843i	− 1.80021i	−0.435670	−	0.900107i	$-0.643488\pi$
$283$	− 30.2843i	− 1.80021i	0.435670	−	0.900107i	$-0.356512\pi$
$284$	0	0
$285$	− 8.97056i	− 0.531370i
$286$	0	0
$287$	−2.62742	−0.155092
$288$	0	0
$289$	−16.0000	−0.941176
$290$	0	0
$291$	− 1.51472i	− 0.0887944i
$292$	0	0
$293$	14.6569i	0.856263i	0.903717	+	0.428131i	$0.140828\pi$
$293$	14.6569i	0.856263i	−0.903717	+	0.428131i	$0.859172\pi$
$294$	0	0
$295$	−30.6274	−1.78320
$296$	0	0
$297$	−11.6569	−0.676399
$298$	0	0
$299$	− 3.17157i	− 0.183417i
$300$	0	0
$301$	− 8.85786i	− 0.510559i
$302$	0	0
$303$	2.62742	0.150941
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 20.0000i	− 1.14146i	−0.821138	−	0.570730i	$-0.806660\pi$
$307$	− 20.0000i	− 1.14146i	0.821138	−	0.570730i	$-0.193340\pi$
$308$	0	0
$309$	− 5.02944i	− 0.286115i
$310$	0	0
$311$	21.6569	1.22805	0.614024	−	0.789288i	$-0.289550\pi$
$311$	21.6569	1.22805	0.614024	+	0.789288i	$0.289550\pi$
$312$	0	0
$313$	4.85786	0.274583	0.137291	−	0.990531i	$-0.456160\pi$
$313$	4.85786	0.274583	0.137291	+	0.990531i	$0.456160\pi$
$314$	0	0
$315$	17.1716i	0.967509i
$316$	0	0
$317$	16.6274i	0.933889i	0.884287	+	0.466944i	$0.154645\pi$
$317$	16.6274i	0.933889i	−0.884287	+	0.466944i	$0.845355\pi$
$318$	0	0
$319$	−36.9706	−2.06995
$320$	0	0
$321$	2.48528	0.138715
$322$	0	0
$323$	5.65685i	0.314756i
$324$	0	0
$325$	− 9.65685i	− 0.535666i
$326$	0	0
$327$	−6.07107	−0.335731
$328$	0	0
$329$	−14.6569	−0.808059
$330$	0	0
$331$	− 8.82843i	− 0.485254i	−0.970120	−	0.242627i	$-0.921991\pi$
$331$	− 8.82843i	− 0.485254i	0.970120	−	0.242627i	$-0.0780092\pi$
$332$	0	0
$333$	− 19.7990i	− 1.08498i
$334$	0	0
$335$	12.1421	0.663396
$336$	0	0
$337$	−17.8284	−0.971176	−0.485588	−	0.874188i	$-0.661395\pi$
$337$	−17.8284	−0.971176	−0.485588	+	0.874188i	$0.661395\pi$
$338$	0	0
$339$	4.14214i	0.224970i
$340$	0	0
$341$	36.9706i	2.00207i
$342$	0	0
$343$	18.2132	0.983421
$344$	0	0
$345$	5.02944	0.270776
$346$	0	0
$347$	29.2426i	1.56983i	0.619605	+	0.784914i	$0.287293\pi$
$347$	29.2426i	1.56983i	−0.619605	+	0.784914i	$0.712707\pi$
$348$	0	0
$349$	3.82843i	0.204931i	0.994737	+	0.102466i	$0.0326731\pi$
$349$	3.82843i	0.204931i	−0.994737	+	0.102466i	$0.967327\pi$
$350$	0	0
$351$	2.41421	0.128861
$352$	0	0
$353$	−23.6569	−1.25913	−0.629564	−	0.776949i	$-0.716766\pi$
$353$	−23.6569	−1.25913	−0.629564	+	0.776949i	$0.716766\pi$
$354$	0	0
$355$	52.5563i	2.78940i
$356$	0	0
$357$	0.656854i	0.0347644i
$358$	0	0
$359$	−35.6569	−1.88190	−0.940948	−	0.338550i	$-0.890064\pi$
$359$	−35.6569	−1.88190	−0.940948	+	0.338550i	$0.890064\pi$
$360$	0	0
$361$	−13.0000	−0.684211
$362$	0	0
$363$	5.10051i	0.267707i
$364$	0	0
$365$	− 36.9706i	− 1.93513i
$366$	0	0
$367$	27.4558	1.43318	0.716592	−	0.697493i	$-0.245701\pi$
$367$	27.4558	1.43318	0.716592	+	0.697493i	$0.245701\pi$
$368$	0	0
$369$	4.68629	0.243959
$370$	0	0
$371$	− 12.1421i	− 0.630388i
$372$	0	0
$373$	− 30.6274i	− 1.58583i	−0.609334	−	0.792914i	$-0.708563\pi$
$373$	− 30.6274i	− 1.58583i	0.609334	−	0.792914i	$-0.291437\pi$
$374$	0	0
$375$	7.38478	0.381348
$376$	0	0
$377$	7.65685	0.394348
$378$	0	0
$379$	14.4853i	0.744059i	0.928221	+	0.372029i	$0.121338\pi$
$379$	14.4853i	0.744059i	−0.928221	+	0.372029i	$0.878662\pi$
$380$	0	0
$381$	− 1.31371i	− 0.0673033i
$382$	0	0
$383$	27.7279	1.41683	0.708415	−	0.705796i	$-0.249410\pi$
$383$	27.7279	1.41683	0.708415	+	0.705796i	$0.249410\pi$
$384$	0	0
$385$	29.3137	1.49396
$386$	0	0
$387$	15.7990i	0.803108i
$388$	0	0
$389$	− 28.0000i	− 1.41966i	−0.704375	−	0.709828i	$-0.748773\pi$
$389$	− 28.0000i	− 1.41966i	0.704375	−	0.709828i	$-0.251227\pi$
$390$	0	0
$391$	−3.17157	−0.160393
$392$	0	0
$393$	−2.31371	−0.116711
$394$	0	0
$395$	− 46.4853i	− 2.33893i
$396$	0	0
$397$	− 29.3137i	− 1.47121i	−0.677409	−	0.735606i	$-0.736897\pi$
$397$	− 29.3137i	− 1.47121i	0.677409	−	0.735606i	$-0.263103\pi$
$398$	0	0
$399$	−3.71573	−0.186019
$400$	0	0
$401$	−30.2843	−1.51232	−0.756162	−	0.654384i	$-0.772928\pi$
$401$	−30.2843	−1.51232	−0.756162	+	0.654384i	$0.772928\pi$
$402$	0	0
$403$	− 7.65685i	− 0.381415i
$404$	0	0
$405$	− 28.6569i	− 1.42397i
$406$	0	0
$407$	−33.7990	−1.67535
$408$	0	0
$409$	27.6569	1.36754	0.683772	−	0.729696i	$-0.260338\pi$
$409$	27.6569	1.36754	0.683772	+	0.729696i	$0.260338\pi$
$410$	0	0
$411$	− 6.48528i	− 0.319895i
$412$	0	0
$413$	12.6863i	0.624252i
$414$	0	0
$415$	−61.7990	−3.03359
$416$	0	0
$417$	−1.82843	−0.0895385
$418$	0	0
$419$	5.24264i	0.256120i	0.991766	+	0.128060i	$0.0408750\pi$
$419$	5.24264i	0.256120i	−0.991766	+	0.128060i	$0.959125\pi$
$420$	0	0
$421$	26.7990i	1.30610i	0.757314	+	0.653051i	$0.226511\pi$
$421$	26.7990i	1.30610i	−0.757314	+	0.653051i	$0.773489\pi$
$422$	0	0
$423$	26.1421	1.27107
$424$	0	0
$425$	−9.65685	−0.468426
$426$	0	0
$427$	0	0
$428$	0	0
$429$	− 2.00000i	− 0.0965609i
$430$	0	0
$431$	−7.38478	−0.355712	−0.177856	−	0.984057i	$-0.556916\pi$
$431$	−7.38478	−0.355712	−0.177856	+	0.984057i	$0.556916\pi$
$432$	0	0
$433$	10.4558	0.502476	0.251238	−	0.967925i	$-0.419162\pi$
$433$	10.4558	0.502476	0.251238	+	0.967925i	$0.419162\pi$
$434$	0	0
$435$	12.1421i	0.582171i
$436$	0	0
$437$	− 17.9411i	− 0.858240i
$438$	0	0
$439$	−24.2843	−1.15903	−0.579513	−	0.814963i	$-0.696757\pi$
$439$	−24.2843	−1.15903	−0.579513	+	0.814963i	$0.696757\pi$
$440$	0	0
$441$	−12.6863	−0.604109
$442$	0	0
$443$	− 7.92893i	− 0.376715i	−0.982101	−	0.188357i	$-0.939684\pi$
$443$	− 7.92893i	− 0.376715i	0.982101	−	0.188357i	$-0.0603164\pi$
$444$	0	0
$445$	8.97056i	0.425245i
$446$	0	0
$447$	−0.544156	−0.0257377
$448$	0	0
$449$	1.31371	0.0619977	0.0309989	−	0.999519i	$-0.490131\pi$
$449$	1.31371	0.0619977	0.0309989	+	0.999519i	$0.490131\pi$
$450$	0	0
$451$	− 8.00000i	− 0.376705i
$452$	0	0
$453$	− 3.82843i	− 0.179875i
$454$	0	0
$455$	−6.07107	−0.284616
$456$	0	0
$457$	−37.6569	−1.76151	−0.880757	−	0.473569i	$-0.842965\pi$
$457$	−37.6569	−1.76151	−0.880757	+	0.473569i	$0.842965\pi$
$458$	0	0
$459$	− 2.41421i	− 0.112686i
$460$	0	0
$461$	0.656854i	0.0305928i	0.999883	+	0.0152964i	$0.00486918\pi$
$461$	0.656854i	0.0305928i	−0.999883	+	0.0152964i	$0.995131\pi$
$462$	0	0
$463$	−22.9706	−1.06753	−0.533766	−	0.845632i	$-0.679224\pi$
$463$	−22.9706	−1.06753	−0.533766	+	0.845632i	$0.679224\pi$
$464$	0	0
$465$	12.1421	0.563078
$466$	0	0
$467$	2.00000i	0.0925490i	0.998929	+	0.0462745i	$0.0147349\pi$
$467$	2.00000i	0.0925490i	−0.998929	+	0.0462745i	$0.985265\pi$
$468$	0	0
$469$	− 5.02944i	− 0.232238i
$470$	0	0
$471$	8.97056	0.413342
$472$	0	0
$473$	26.9706	1.24011
$474$	0	0
$475$	− 54.6274i	− 2.50648i
$476$	0	0
$477$	21.6569i	0.991599i
$478$	0	0
$479$	−12.4142	−0.567220	−0.283610	−	0.958940i	$-0.591532\pi$
$479$	−12.4142	−0.567220	−0.283610	+	0.958940i	$0.591532\pi$
$480$	0	0
$481$	7.00000	0.319173
$482$	0	0
$483$	− 2.08326i	− 0.0947917i
$484$	0	0
$485$	− 14.0000i	− 0.635707i
$486$	0	0
$487$	22.9706	1.04090	0.520448	−	0.853894i	$-0.325765\pi$
$487$	22.9706	1.04090	0.520448	+	0.853894i	$0.325765\pi$
$488$	0	0
$489$	1.02944	0.0465528
$490$	0	0
$491$	25.8701i	1.16750i	0.811934	+	0.583750i	$0.198415\pi$
$491$	25.8701i	1.16750i	−0.811934	+	0.583750i	$0.801585\pi$
$492$	0	0
$493$	− 7.65685i	− 0.344847i
$494$	0	0
$495$	−52.2843	−2.35000
$496$	0	0
$497$	21.7696	0.976498
$498$	0	0
$499$	− 21.6569i	− 0.969494i	−0.874654	−	0.484747i	$-0.838912\pi$
$499$	− 21.6569i	− 0.969494i	0.874654	−	0.484747i	$-0.161088\pi$
$500$	0	0
$501$	− 3.17157i	− 0.141695i
$502$	0	0
$503$	6.34315	0.282827	0.141413	−	0.989951i	$-0.454835\pi$
$503$	6.34315	0.282827	0.141413	+	0.989951i	$0.454835\pi$
$504$	0	0
$505$	24.2843	1.08064
$506$	0	0
$507$	0.414214i	0.0183959i
$508$	0	0
$509$	− 1.31371i	− 0.0582291i	−0.999576	−	0.0291146i	$-0.990731\pi$
$509$	− 1.31371i	− 0.0582291i	0.999576	−	0.0291146i	$-0.00926876\pi$
$510$	0	0
$511$	−15.3137	−0.677439
$512$	0	0
$513$	13.6569	0.602965
$514$	0	0
$515$	− 46.4853i	− 2.04839i
$516$	0	0
$517$	− 44.6274i	− 1.96271i
$518$	0	0
$519$	5.79899	0.254547
$520$	0	0
$521$	41.1421	1.80247	0.901235	−	0.433331i	$-0.142662\pi$
$521$	41.1421	1.80247	0.901235	+	0.433331i	$0.142662\pi$
$522$	0	0
$523$	25.3137i	1.10689i	0.832885	+	0.553446i	$0.186687\pi$
$523$	25.3137i	1.10689i	−0.832885	+	0.553446i	$0.813313\pi$
$524$	0	0
$525$	− 6.34315i	− 0.276838i
$526$	0	0
$527$	−7.65685	−0.333538
$528$	0	0
$529$	−12.9411	−0.562658
$530$	0	0
$531$	− 22.6274i	− 0.981946i
$532$	0	0
$533$	1.65685i	0.0717663i
$534$	0	0
$535$	22.9706	0.993104
$536$	0	0
$537$	−2.85786	−0.123326
$538$	0	0
$539$	21.6569i	0.932827i
$540$	0	0
$541$	− 7.00000i	− 0.300954i	−0.988614	−	0.150477i	$-0.951919\pi$
$541$	− 7.00000i	− 0.300954i	0.988614	−	0.150477i	$-0.0480809\pi$
$542$	0	0
$543$	3.17157	0.136105
$544$	0	0
$545$	−56.1127	−2.40360
$546$	0	0
$547$	− 30.0711i	− 1.28575i	−0.765973	−	0.642873i	$-0.777742\pi$
$547$	− 30.0711i	− 1.28575i	0.765973	−	0.642873i	$-0.222258\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	43.3137	1.84523
$552$	0	0
$553$	−19.2548	−0.818799
$554$	0	0
$555$	11.1005i	0.471190i
$556$	0	0
$557$	1.97056i	0.0834954i	0.999128	+	0.0417477i	$0.0132926\pi$
$557$	1.97056i	0.0834954i	−0.999128	+	0.0417477i	$0.986707\pi$
$558$	0	0
$559$	−5.58579	−0.236254
$560$	0	0
$561$	−2.00000	−0.0844401
$562$	0	0
$563$	31.0416i	1.30825i	0.756387	+	0.654124i	$0.226963\pi$
$563$	31.0416i	1.30825i	−0.756387	+	0.654124i	$0.773037\pi$
$564$	0	0
$565$	38.2843i	1.61063i
$566$	0	0
$567$	−11.8701	−0.498496
$568$	0	0
$569$	14.6569	0.614447	0.307224	−	0.951637i	$-0.400600\pi$
$569$	14.6569	0.614447	0.307224	+	0.951637i	$0.400600\pi$
$570$	0	0
$571$	35.7279i	1.49517i	0.664168	+	0.747584i	$0.268786\pi$
$571$	35.7279i	1.49517i	−0.664168	+	0.747584i	$0.731214\pi$
$572$	0	0
$573$	− 10.2843i	− 0.429632i
$574$	0	0
$575$	30.6274	1.27725
$576$	0	0
$577$	−1.65685	−0.0689757	−0.0344879	−	0.999405i	$-0.510980\pi$
$577$	−1.65685	−0.0689757	−0.0344879	+	0.999405i	$0.510980\pi$
$578$	0	0
$579$	− 8.82843i	− 0.366897i
$580$	0	0
$581$	25.5980i	1.06198i
$582$	0	0
$583$	36.9706	1.53116
$584$	0	0
$585$	10.8284	0.447700
$586$	0	0
$587$	1.51472i	0.0625191i	0.999511	+	0.0312596i	$0.00995185\pi$
$587$	1.51472i	0.0625191i	−0.999511	+	0.0312596i	$0.990048\pi$
$588$	0	0
$589$	− 43.3137i	− 1.78471i
$590$	0	0
$591$	2.89949	0.119269
$592$	0	0
$593$	20.6274	0.847066	0.423533	−	0.905881i	$-0.360790\pi$
$593$	20.6274	0.847066	0.423533	+	0.905881i	$0.360790\pi$
$594$	0	0
$595$	6.07107i	0.248890i
$596$	0	0
$597$	7.65685i	0.313374i
$598$	0	0
$599$	−15.3137	−0.625701	−0.312851	−	0.949802i	$-0.601284\pi$
$599$	−15.3137	−0.625701	−0.312851	+	0.949802i	$0.601284\pi$
$600$	0	0
$601$	11.9706	0.488289	0.244145	−	0.969739i	$-0.421493\pi$
$601$	11.9706	0.488289	0.244145	+	0.969739i	$0.421493\pi$
$602$	0	0
$603$	8.97056i	0.365310i
$604$	0	0
$605$	47.1421i	1.91660i
$606$	0	0
$607$	−15.3137	−0.621564	−0.310782	−	0.950481i	$-0.600591\pi$
$607$	−15.3137	−0.621564	−0.310782	+	0.950481i	$0.600591\pi$
$608$	0	0
$609$	5.02944	0.203803
$610$	0	0
$611$	9.24264i	0.373917i
$612$	0	0
$613$	44.6274i	1.80248i	0.433316	+	0.901242i	$0.357344\pi$
$613$	44.6274i	1.80248i	−0.433316	+	0.901242i	$0.642656\pi$
$614$	0	0
$615$	−2.62742	−0.105948
$616$	0	0
$617$	−49.2548	−1.98292	−0.991462	−	0.130392i	$-0.958376\pi$
$617$	−49.2548	−1.98292	−0.991462	+	0.130392i	$0.958376\pi$
$618$	0	0
$619$	37.6569i	1.51356i	0.653672	+	0.756778i	$0.273228\pi$
$619$	37.6569i	1.51356i	−0.653672	+	0.756778i	$0.726772\pi$
$620$	0	0
$621$	7.65685i	0.307259i
$622$	0	0
$623$	3.71573	0.148868
$624$	0	0
$625$	19.9706	0.798823
$626$	0	0
$627$	− 11.3137i	− 0.451826i
$628$	0	0
$629$	− 7.00000i	− 0.279108i
$630$	0	0
$631$	6.61522	0.263348	0.131674	−	0.991293i	$-0.457965\pi$
$631$	6.61522	0.263348	0.131674	+	0.991293i	$0.457965\pi$
$632$	0	0
$633$	−0.656854	−0.0261076
$634$	0	0
$635$	− 12.1421i	− 0.481846i
$636$	0	0
$637$	− 4.48528i	− 0.177713i
$638$	0	0
$639$	−38.8284	−1.53603
$640$	0	0
$641$	−14.0000	−0.552967	−0.276483	−	0.961019i	$-0.589169\pi$
$641$	−14.0000	−0.552967	−0.276483	+	0.961019i	$0.589169\pi$
$642$	0	0
$643$	23.1716i	0.913798i	0.889519	+	0.456899i	$0.151040\pi$
$643$	23.1716i	0.913798i	−0.889519	+	0.456899i	$0.848960\pi$
$644$	0	0
$645$	− 8.85786i	− 0.348778i
$646$	0	0
$647$	8.97056	0.352669	0.176335	−	0.984330i	$-0.443576\pi$
$647$	8.97056	0.352669	0.176335	+	0.984330i	$0.443576\pi$
$648$	0	0
$649$	−38.6274	−1.51626
$650$	0	0
$651$	− 5.02944i	− 0.197119i
$652$	0	0
$653$	21.6569i	0.847498i	0.905780	+	0.423749i	$0.139286\pi$
$653$	21.6569i	0.847498i	−0.905780	+	0.423749i	$0.860714\pi$
$654$	0	0
$655$	−21.3848	−0.835572
$656$	0	0
$657$	27.3137	1.06561
$658$	0	0
$659$	− 35.9411i	− 1.40007i	−0.714110	−	0.700034i	$-0.753168\pi$
$659$	− 35.9411i	− 1.40007i	0.714110	−	0.700034i	$-0.246832\pi$
$660$	0	0
$661$	29.3137i	1.14017i	0.821585	+	0.570086i	$0.193090\pi$
$661$	29.3137i	1.14017i	−0.821585	+	0.570086i	$0.806910\pi$
$662$	0	0
$663$	0.414214	0.0160867
$664$	0	0
$665$	−34.3431	−1.33177
$666$	0	0
$667$	24.2843i	0.940291i
$668$	0	0
$669$	− 5.14214i	− 0.198806i
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−35.2843	−1.36011	−0.680054	−	0.733162i	$-0.738044\pi$
$673$	−35.2843	−1.36011	−0.680054	+	0.733162i	$0.738044\pi$
$674$	0	0
$675$	23.3137i	0.897345i
$676$	0	0
$677$	− 10.2843i	− 0.395257i	−0.980277	−	0.197628i	$-0.936676\pi$
$677$	− 10.2843i	− 0.395257i	0.980277	−	0.197628i	$-0.0633239\pi$
$678$	0	0
$679$	−5.79899	−0.222545
$680$	0	0
$681$	−8.68629	−0.332859
$682$	0	0
$683$	− 3.17157i	− 0.121357i	−0.998157	−	0.0606784i	$-0.980674\pi$
$683$	− 3.17157i	− 0.121357i	0.998157	−	0.0606784i	$-0.0193264\pi$
$684$	0	0
$685$	− 59.9411i	− 2.29023i
$686$	0	0
$687$	−1.04163	−0.0397407
$688$	0	0
$689$	−7.65685	−0.291703
$690$	0	0
$691$	34.6274i	1.31729i	0.752454	+	0.658645i	$0.228870\pi$
$691$	34.6274i	1.31729i	−0.752454	+	0.658645i	$0.771130\pi$
$692$	0	0
$693$	21.6569i	0.822676i
$694$	0	0
$695$	−16.8995	−0.641034
$696$	0	0
$697$	1.65685	0.0627578
$698$	0	0
$699$	5.04163i	0.190692i
$700$	0	0
$701$	14.0000i	0.528773i	0.964417	+	0.264386i	$0.0851694\pi$
$701$	14.0000i	0.528773i	−0.964417	+	0.264386i	$0.914831\pi$
$702$	0	0
$703$	39.5980	1.49347
$704$	0	0
$705$	−14.6569	−0.552009
$706$	0	0
$707$	− 10.0589i	− 0.378303i
$708$	0	0
$709$	− 29.3137i	− 1.10090i	−0.834868	−	0.550450i	$-0.814456\pi$
$709$	− 29.3137i	− 1.10090i	0.834868	−	0.550450i	$-0.185544\pi$
$710$	0	0
$711$	34.3431	1.28797
$712$	0	0
$713$	24.2843	0.909453
$714$	0	0
$715$	− 18.4853i	− 0.691310i
$716$	0	0
$717$	− 3.28427i	− 0.122653i
$718$	0	0
$719$	33.7990	1.26049	0.630245	−	0.776396i	$-0.282955\pi$
$719$	33.7990	1.26049	0.630245	+	0.776396i	$0.282955\pi$
$720$	0	0
$721$	−19.2548	−0.717087
$722$	0	0
$723$	− 8.68629i	− 0.323047i
$724$	0	0
$725$	73.9411i	2.74610i
$726$	0	0
$727$	31.1716	1.15609	0.578045	−	0.816005i	$-0.303816\pi$
$727$	31.1716	1.15609	0.578045	+	0.816005i	$0.303816\pi$
$728$	0	0
$729$	18.1716	0.673021
$730$	0	0
$731$	5.58579i	0.206598i
$732$	0	0
$733$	30.5147i	1.12709i	0.826086	+	0.563543i	$0.190562\pi$
$733$	30.5147i	1.12709i	−0.826086	+	0.563543i	$0.809438\pi$
$734$	0	0
$735$	7.11270	0.262356
$736$	0	0
$737$	15.3137	0.564088
$738$	0	0
$739$	− 7.31371i	− 0.269039i	−0.990911	−	0.134520i	$-0.957051\pi$
$739$	− 7.31371i	− 0.269039i	0.990911	−	0.134520i	$-0.0429491\pi$
$740$	0	0
$741$	2.34315i	0.0860776i
$742$	0	0
$743$	−0.272078	−0.00998157	−0.00499079	−	0.999988i	$-0.501589\pi$
$743$	−0.272078	−0.00998157	−0.00499079	+	0.999988i	$0.501589\pi$
$744$	0	0
$745$	−5.02944	−0.184264
$746$	0	0
$747$	− 45.6569i	− 1.67050i
$748$	0	0
$749$	− 9.51472i	− 0.347660i
$750$	0	0
$751$	−28.0000	−1.02173	−0.510867	−	0.859660i	$-0.670676\pi$
$751$	−28.0000	−1.02173	−0.510867	+	0.859660i	$0.670676\pi$
$752$	0	0
$753$	−3.85786	−0.140588
$754$	0	0
$755$	− 35.3848i	− 1.28778i
$756$	0	0
$757$	16.6274i	0.604334i	0.953255	+	0.302167i	$0.0977100\pi$
$757$	16.6274i	0.604334i	−0.953255	+	0.302167i	$0.902290\pi$
$758$	0	0
$759$	6.34315	0.230242
$760$	0	0
$761$	0.686292	0.0248780	0.0124390	−	0.999923i	$-0.496040\pi$
$761$	0.686292	0.0248780	0.0124390	+	0.999923i	$0.496040\pi$
$762$	0	0
$763$	23.2426i	0.841440i
$764$	0	0
$765$	− 10.8284i	− 0.391503i
$766$	0	0
$767$	8.00000	0.288863
$768$	0	0
$769$	22.6274	0.815966	0.407983	−	0.912990i	$-0.366232\pi$
$769$	22.6274	0.815966	0.407983	+	0.912990i	$0.366232\pi$
$770$	0	0
$771$	− 4.69848i	− 0.169212i
$772$	0	0
$773$	− 12.7990i	− 0.460348i	−0.973150	−	0.230174i	$-0.926071\pi$
$773$	− 12.7990i	− 0.460348i	0.973150	−	0.230174i	$-0.0739295\pi$
$774$	0	0
$775$	73.9411	2.65604
$776$	0	0
$777$	4.59798	0.164952
$778$	0	0
$779$	9.37258i	0.335808i
$780$	0	0
$781$	66.2843i	2.37184i
$782$	0	0
$783$	−18.4853	−0.660610
$784$	0	0
$785$	82.9117	2.95925
$786$	0	0
$787$	26.3431i	0.939032i	0.882924	+	0.469516i	$0.155572\pi$
$787$	26.3431i	0.939032i	−0.882924	+	0.469516i	$0.844428\pi$
$788$	0	0
$789$	− 2.62742i	− 0.0935385i
$790$	0	0
$791$	15.8579	0.563841
$792$	0	0
$793$	0	0
$794$	0	0
$795$	− 12.1421i	− 0.430637i
$796$	0	0
$797$	24.2843i	0.860193i	0.902783	+	0.430097i	$0.141520\pi$
$797$	24.2843i	0.860193i	−0.902783	+	0.430097i	$0.858480\pi$
$798$	0	0
$799$	9.24264	0.326981
$800$	0	0
$801$	−6.62742	−0.234168
$802$	0	0
$803$	− 46.6274i	− 1.64545i
$804$	0	0
$805$	− 19.2548i	− 0.678644i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	10.5147	0.369678	0.184839	−	0.982769i	$-0.440824\pi$
$809$	10.5147	0.369678	0.184839	+	0.982769i	$0.440824\pi$
$810$	0	0
$811$	4.97056i	0.174540i	0.996185	+	0.0872700i	$0.0278143\pi$
$811$	4.97056i	0.174540i	−0.996185	+	0.0872700i	$0.972186\pi$
$812$	0	0
$813$	5.14214i	0.180343i
$814$	0	0
$815$	9.51472	0.333286
$816$	0	0
$817$	−31.5980	−1.10547
$818$	0	0
$819$	− 4.48528i	− 0.156728i
$820$	0	0
$821$	− 28.1127i	− 0.981140i	−0.871402	−	0.490570i	$-0.836789\pi$
$821$	− 28.1127i	− 0.981140i	0.871402	−	0.490570i	$-0.163211\pi$
$822$	0	0
$823$	28.0000	0.976019	0.488009	−	0.872838i	$-0.337723\pi$
$823$	28.0000	0.976019	0.488009	+	0.872838i	$0.337723\pi$
$824$	0	0
$825$	19.3137	0.672417
$826$	0	0
$827$	− 33.9411i	− 1.18025i	−0.807312	−	0.590124i	$-0.799079\pi$
$827$	− 33.9411i	− 1.18025i	0.807312	−	0.590124i	$-0.200921\pi$
$828$	0	0
$829$	− 22.9706i	− 0.797801i	−0.916994	−	0.398900i	$-0.869392\pi$
$829$	− 22.9706i	− 0.797801i	0.916994	−	0.398900i	$-0.130608\pi$
$830$	0	0
$831$	−3.71573	−0.128897
$832$	0	0
$833$	−4.48528	−0.155406
$834$	0	0
$835$	− 29.3137i	− 1.01444i
$836$	0	0
$837$	18.4853i	0.638945i
$838$	0	0
$839$	−5.02944	−0.173635	−0.0868177	−	0.996224i	$-0.527670\pi$
$839$	−5.02944	−0.173635	−0.0868177	+	0.996224i	$0.527670\pi$
$840$	0	0
$841$	−29.6274	−1.02164
$842$	0	0
$843$	− 1.65685i	− 0.0570651i
$844$	0	0
$845$	3.82843i	0.131702i
$846$	0	0
$847$	19.5269	0.670953
$848$	0	0
$849$	−12.5442	−0.430514
$850$	0	0
$851$	22.2010i	0.761041i
$852$	0	0
$853$	34.4558i	1.17975i	0.807496	+	0.589873i	$0.200822\pi$
$853$	34.4558i	1.17975i	−0.807496	+	0.589873i	$0.799178\pi$
$854$	0	0
$855$	61.2548	2.09487
$856$	0	0
$857$	45.3137	1.54789	0.773943	−	0.633255i	$-0.218281\pi$
$857$	45.3137	1.54789	0.773943	+	0.633255i	$0.218281\pi$
$858$	0	0
$859$	28.6274i	0.976755i	0.872633	+	0.488377i	$0.162411\pi$
$859$	28.6274i	0.976755i	−0.872633	+	0.488377i	$0.837589\pi$
$860$	0	0
$861$	1.08831i	0.0370896i
$862$	0	0
$863$	−7.38478	−0.251381	−0.125690	−	0.992070i	$-0.540115\pi$
$863$	−7.38478	−0.251381	−0.125690	+	0.992070i	$0.540115\pi$
$864$	0	0
$865$	53.5980	1.82239
$866$	0	0
$867$	6.62742i	0.225079i
$868$	0	0
$869$	− 58.6274i	− 1.98880i
$870$	0	0
$871$	−3.17157	−0.107465
$872$	0	0
$873$	10.3431	0.350062
$874$	0	0
$875$	− 28.2721i	− 0.955771i
$876$	0	0
$877$	5.14214i	0.173638i	0.996224	+	0.0868188i	$0.0276701\pi$
$877$	5.14214i	0.173638i	−0.996224	+	0.0868188i	$0.972330\pi$
$878$	0	0
$879$	6.07107	0.204772
$880$	0	0
$881$	9.68629	0.326339	0.163170	−	0.986598i	$-0.447828\pi$
$881$	9.68629	0.326339	0.163170	+	0.986598i	$0.447828\pi$
$882$	0	0
$883$	16.2132i	0.545618i	0.962068	+	0.272809i	$0.0879527\pi$
$883$	16.2132i	0.545618i	−0.962068	+	0.272809i	$0.912047\pi$
$884$	0	0
$885$	12.6863i	0.426445i
$886$	0	0
$887$	−21.6569	−0.727166	−0.363583	−	0.931562i	$-0.618447\pi$
$887$	−21.6569	−0.727166	−0.363583	+	0.931562i	$0.618447\pi$
$888$	0	0
$889$	−5.02944	−0.168682
$890$	0	0
$891$	− 36.1421i	− 1.21081i
$892$	0	0
$893$	52.2843i	1.74963i
$894$	0	0
$895$	−26.4142	−0.882930
$896$	0	0
$897$	−1.31371	−0.0438635
$898$	0	0
$899$	58.6274i	1.95533i
$900$	0	0
$901$	7.65685i	0.255087i
$902$	0	0
$903$	−3.66905	−0.122098
$904$	0	0
$905$	29.3137	0.974421
$906$	0	0
$907$	42.5563i	1.41306i	0.707683	+	0.706530i	$0.249741\pi$
$907$	42.5563i	1.41306i	−0.707683	+	0.706530i	$0.750259\pi$
$908$	0	0
$909$	17.9411i	0.595070i
$910$	0	0
$911$	12.1421	0.402287	0.201143	−	0.979562i	$-0.435534\pi$
$911$	12.1421	0.402287	0.201143	+	0.979562i	$0.435534\pi$
$912$	0	0
$913$	−77.9411	−2.57947
$914$	0	0
$915$	0	0
$916$	0	0
$917$	8.85786i	0.292512i
$918$	0	0
$919$	−34.3431	−1.13288	−0.566438	−	0.824104i	$-0.691679\pi$
$919$	−34.3431	−1.13288	−0.566438	+	0.824104i	$0.691679\pi$
$920$	0	0
$921$	−8.28427	−0.272976
$922$	0	0
$923$	− 13.7279i	− 0.451860i
$924$	0	0
$925$	67.5980i	2.22261i
$926$	0	0
$927$	34.3431	1.12798
$928$	0	0
$929$	−3.02944	−0.0993926	−0.0496963	−	0.998764i	$-0.515825\pi$
$929$	−3.02944	−0.0993926	−0.0496963	+	0.998764i	$0.515825\pi$
$930$	0	0
$931$	− 25.3726i	− 0.831553i
$932$	0	0
$933$	− 8.97056i	− 0.293683i
$934$	0	0
$935$	−18.4853	−0.604533
$936$	0	0
$937$	−37.3137	−1.21899	−0.609493	−	0.792792i	$-0.708627\pi$
$937$	−37.3137	−1.21899	−0.609493	+	0.792792i	$0.708627\pi$
$938$	0	0
$939$	− 2.01219i	− 0.0656654i
$940$	0	0
$941$	− 40.7990i	− 1.33001i	−0.746839	−	0.665005i	$-0.768430\pi$
$941$	− 40.7990i	− 1.33001i	0.746839	−	0.665005i	$-0.231570\pi$
$942$	0	0
$943$	−5.25483	−0.171121
$944$	0	0
$945$	14.6569	0.476788
$946$	0	0
$947$	40.8284i	1.32675i	0.748289	+	0.663373i	$0.230876\pi$
$947$	40.8284i	1.32675i	−0.748289	+	0.663373i	$0.769124\pi$
$948$	0	0
$949$	9.65685i	0.313475i
$950$	0	0
$951$	6.88730	0.223336
$952$	0	0
$953$	−24.9411	−0.807922	−0.403961	−	0.914776i	$-0.632367\pi$
$953$	−24.9411	−0.807922	−0.403961	+	0.914776i	$0.632367\pi$
$954$	0	0
$955$	− 95.0538i	− 3.07587i
$956$	0	0
$957$	15.3137i	0.495022i
$958$	0	0
$959$	−24.8284	−0.801752
$960$	0	0
$961$	27.6274	0.891207
$962$	0	0
$963$	16.9706i	0.546869i
$964$	0	0
$965$	− 81.5980i	− 2.62673i
$966$	0	0
$967$	−26.4142	−0.849424	−0.424712	−	0.905329i	$-0.639625\pi$
$967$	−26.4142	−0.849424	−0.424712	+	0.905329i	$0.639625\pi$
$968$	0	0
$969$	2.34315	0.0752727
$970$	0	0
$971$	23.1005i	0.741330i	0.928767	+	0.370665i	$0.120870\pi$
$971$	23.1005i	0.741330i	−0.928767	+	0.370665i	$0.879130\pi$
$972$	0	0
$973$	7.00000i	0.224410i
$974$	0	0
$975$	−4.00000	−0.128103
$976$	0	0
$977$	−21.3137	−0.681886	−0.340943	−	0.940084i	$-0.610746\pi$
$977$	−21.3137	−0.681886	−0.340943	+	0.940084i	$0.610746\pi$
$978$	0	0
$979$	11.3137i	0.361588i
$980$	0	0
$981$	− 41.4558i	− 1.32358i
$982$	0	0
$983$	−1.58579	−0.0505787	−0.0252894	−	0.999680i	$-0.508051\pi$
$983$	−1.58579	−0.0505787	−0.0252894	+	0.999680i	$0.508051\pi$
$984$	0	0
$985$	26.7990	0.853887
$986$	0	0
$987$	6.07107i	0.193244i
$988$	0	0
$989$	− 17.7157i	− 0.563327i
$990$	0	0
$991$	−36.9706	−1.17441	−0.587204	−	0.809439i	$-0.699771\pi$
$991$	−36.9706	−1.17441	−0.587204	+	0.809439i	$0.699771\pi$
$992$	0	0
$993$	−3.65685	−0.116047
$994$	0	0
$995$	70.7696i	2.24355i
$996$	0	0
$997$	42.0000i	1.33015i	0.746775	+	0.665077i	$0.231601\pi$
$997$	42.0000i	1.33015i	−0.746775	+	0.665077i	$0.768399\pi$
$998$	0	0
$999$	−16.8995	−0.534676

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	832.2.b.a.417.2	✓	4
4.3	odd	2		832.2.b.b.417.3	yes	4
8.3	odd	2		832.2.b.b.417.2	yes	4
8.5	even	2	inner	832.2.b.a.417.3	yes	4
16.3	odd	4		3328.2.a.y.1.1		2
16.5	even	4		3328.2.a.bb.1.1		2
16.11	odd	4		3328.2.a.q.1.2		2
16.13	even	4		3328.2.a.p.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
832.2.b.a.417.2	✓	4	1.1	even	1	trivial
832.2.b.a.417.3	yes	4	8.5	even	2	inner
832.2.b.b.417.2	yes	4	8.3	odd	2
832.2.b.b.417.3	yes	4	4.3	odd	2
3328.2.a.p.1.2		2	16.13	even	4
3328.2.a.q.1.2		2	16.11	odd	4
3328.2.a.y.1.1		2	16.3	odd	4
3328.2.a.bb.1.1		2	16.5	even	4

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:	\( N \)	\(=\)	\( 832 = 2^{6} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	832.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.414214i q^{3} -3.82843i q^{5} -1.58579 q^{7} +2.82843 q^{9} -4.82843i q^{11} +1.00000i q^{13} -1.58579 q^{15} +1.00000 q^{17} +5.65685i q^{19} +0.656854i q^{21} -3.17157 q^{23} -9.65685 q^{25} -2.41421i q^{27} -7.65685i q^{29} -7.65685 q^{31} -2.00000 q^{33} +6.07107i q^{35} -7.00000i q^{37} +0.414214 q^{39} +1.65685 q^{41} +5.58579i q^{43} -10.8284i q^{45} +9.24264 q^{47} -4.48528 q^{49} -0.414214i q^{51} +7.65685i q^{53} -18.4853 q^{55} +2.34315 q^{57} -8.00000i q^{59} -4.48528 q^{63} +3.82843 q^{65} +3.17157i q^{67} +1.31371i q^{69} -13.7279 q^{71} +9.65685 q^{73} +4.00000i q^{75} +7.65685i q^{77} +12.1421 q^{79} +7.48528 q^{81} -16.1421i q^{83} -3.82843i q^{85} -3.17157 q^{87} -2.34315 q^{89} -1.58579i q^{91} +3.17157i q^{93} +21.6569 q^{95} +3.65685 q^{97} -13.6569i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{7} - 12 q^{15} + 4 q^{17} - 24 q^{23} - 16 q^{25} - 8 q^{31} - 8 q^{33} - 4 q^{39} - 16 q^{41} + 20 q^{47} + 16 q^{49} - 40 q^{55} + 32 q^{57} + 16 q^{63} + 4 q^{65} - 4 q^{71} + 16 q^{73} - 8 q^{79}+ \cdots - 8 q^{97}+O(q^{100}) \) 4 * q - 12 * q^7 - 12 * q^15 + 4 * q^17 - 24 * q^23 - 16 * q^25 - 8 * q^31 - 8 * q^33 - 4 * q^39 - 16 * q^41 + 20 * q^47 + 16 * q^49 - 40 * q^55 + 32 * q^57 + 16 * q^63 + 4 * q^65 - 4 * q^71 + 16 * q^73 - 8 * q^79 - 4 * q^81 - 24 * q^87 - 32 * q^89 + 64 * q^95 - 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more