LMFDB - Embedded newform 1664.2.b.b.833.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1664,2,Mod(833,1664)] 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("1664.833"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1664, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1664 = 2^{7} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1664.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,-2,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$13.2871068963$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			833.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1664.833
Dual form			1664.2.b.b.833.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +1.00000i q^{5} -1.00000 q^{7} +2.00000 q^{9} -2.00000i q^{11} +1.00000i q^{13} -1.00000 q^{15} +1.00000 q^{17} -8.00000i q^{19} -1.00000i q^{21} +6.00000 q^{23} +4.00000 q^{25} +5.00000i q^{27} +6.00000i q^{29} +8.00000 q^{31} +2.00000 q^{33} -1.00000i q^{35} +1.00000i q^{37} -1.00000 q^{39} +4.00000 q^{41} +5.00000i q^{43} +2.00000i q^{45} -11.0000 q^{47} -6.00000 q^{49} +1.00000i q^{51} +6.00000i q^{53} +2.00000 q^{55} +8.00000 q^{57} -8.00000i q^{59} +4.00000i q^{61} -2.00000 q^{63} -1.00000 q^{65} +2.00000i q^{67} +6.00000i q^{69} -3.00000 q^{71} +8.00000 q^{73} +4.00000i q^{75} +2.00000i q^{77} +10.0000 q^{79} +1.00000 q^{81} +6.00000i q^{83} +1.00000i q^{85} -6.00000 q^{87} +16.0000 q^{89} -1.00000i q^{91} +8.00000i q^{93} +8.00000 q^{95} -2.00000 q^{97} -4.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{7} + 4 q^{9} - 2 q^{15} + 2 q^{17} + 12 q^{23} + 8 q^{25} + 16 q^{31} + 4 q^{33} - 2 q^{39} + 8 q^{41} - 22 q^{47} - 12 q^{49} + 4 q^{55} + 16 q^{57} - 4 q^{63} - 2 q^{65} - 6 q^{71} + 16 q^{73}+ \cdots - 4 q^{97}+O(q^{100}) $ 2 * q - 2 * q^7 + 4 * q^9 - 2 * q^15 + 2 * q^17 + 12 * q^23 + 8 * q^25 + 16 * q^31 + 4 * q^33 - 2 * q^39 + 8 * q^41 - 22 * q^47 - 12 * q^49 + 4 * q^55 + 16 * q^57 - 4 * q^63 - 2 * q^65 - 6 * q^71 + 16 * q^73 + 20 * q^79 + 2 * q^81 - 12 * q^87 + 32 * q^89 + 16 * q^95 - 4 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1664\mathbb{Z}\right)^\times$.

$n$	$261$	$769$	$1535$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	1.00000i	0.577350i	0.957427	+	0.288675i	$0.0932147\pi$
$3$	1.00000i	0.577350i	−0.957427	+	0.288675i	$0.906785\pi$
$4$	0	0
$5$	1.00000i	0.447214i	0.974679	+	0.223607i	$0.0717831\pi$
$5$	1.00000i	0.447214i	−0.974679	+	0.223607i	$0.928217\pi$
$6$	0	0
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	0	0
$9$	2.00000	0.666667
$10$	0	0
$11$	− 2.00000i	− 0.603023i	−0.953463	−	0.301511i	$-0.902509\pi$
$11$	− 2.00000i	− 0.603023i	0.953463	−	0.301511i	$-0.0974911\pi$
$12$	0	0
$13$	1.00000i	0.277350i
$13$	1.00000i	0.277350i
$14$	0	0
$15$	−1.00000	−0.258199
$16$	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
$19$	− 8.00000i	− 1.83533i	−0.397360	−	0.917663i	$-0.630073\pi$
$19$	− 8.00000i	− 1.83533i	0.397360	−	0.917663i	$-0.369927\pi$
$20$	0	0
$21$	− 1.00000i	− 0.218218i
$22$	0	0
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	0	0
$27$	5.00000i	0.962250i
$28$	0	0
$29$	6.00000i	1.11417i	0.830455	+	0.557086i	$0.188081\pi$
$29$	6.00000i	1.11417i	−0.830455	+	0.557086i	$0.811919\pi$
$30$	0	0
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	0	0
$33$	2.00000	0.348155
$34$	0	0
$35$	− 1.00000i	− 0.169031i
$36$	0	0
$37$	1.00000i	0.164399i	0.996616	+	0.0821995i	$0.0261945\pi$
$37$	1.00000i	0.164399i	−0.996616	+	0.0821995i	$0.973806\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	4.00000	0.624695	0.312348	−	0.949968i	$-0.398885\pi$
$41$	4.00000	0.624695	0.312348	+	0.949968i	$0.398885\pi$
$42$	0	0
$43$	5.00000i	0.762493i	0.924473	+	0.381246i	$0.124505\pi$
$43$	5.00000i	0.762493i	−0.924473	+	0.381246i	$0.875495\pi$
$44$	0	0
$45$	2.00000i	0.298142i
$46$	0	0
$47$	−11.0000	−1.60451	−0.802257	−	0.596978i	$-0.796368\pi$
$47$	−11.0000	−1.60451	−0.802257	+	0.596978i	$0.796368\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	0	0
$51$	1.00000i	0.140028i
$52$	0	0
$53$	6.00000i	0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$	6.00000i	0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$	0	0
$55$	2.00000	0.269680
$56$	0	0
$57$	8.00000	1.05963
$58$	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
$61$	4.00000i	0.512148i	0.966657	+	0.256074i	$0.0824290\pi$
$61$	4.00000i	0.512148i	−0.966657	+	0.256074i	$0.917571\pi$
$62$	0	0
$63$	−2.00000	−0.251976
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	2.00000i	0.244339i	0.992509	+	0.122169i	$0.0389851\pi$
$67$	2.00000i	0.244339i	−0.992509	+	0.122169i	$0.961015\pi$
$68$	0	0
$69$	6.00000i	0.722315i
$70$	0	0
$71$	−3.00000	−0.356034	−0.178017	−	0.984027i	$-0.556968\pi$
$71$	−3.00000	−0.356034	−0.178017	+	0.984027i	$0.556968\pi$
$72$	0	0
$73$	8.00000	0.936329	0.468165	−	0.883641i	$-0.344915\pi$
$73$	8.00000	0.936329	0.468165	+	0.883641i	$0.344915\pi$
$74$	0	0
$75$	4.00000i	0.461880i
$76$	0	0
$77$	2.00000i	0.227921i
$78$	0	0
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	6.00000i	0.658586i	0.944228	+	0.329293i	$0.106810\pi$
$83$	6.00000i	0.658586i	−0.944228	+	0.329293i	$0.893190\pi$
$84$	0	0
$85$	1.00000i	0.108465i
$86$	0	0
$87$	−6.00000	−0.643268
$88$	0	0
$89$	16.0000	1.69600	0.847998	−	0.529999i	$-0.177808\pi$
$89$	16.0000	1.69600	0.847998	+	0.529999i	$0.177808\pi$
$90$	0	0
$91$	− 1.00000i	− 0.104828i
$92$	0	0
$93$	8.00000i	0.829561i
$94$	0	0
$95$	8.00000	0.820783
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	0	0
$99$	− 4.00000i	− 0.402015i
$100$	0	0
$101$	8.00000i	0.796030i	0.917379	+	0.398015i	$0.130301\pi$
$101$	8.00000i	0.796030i	−0.917379	+	0.398015i	$0.869699\pi$
$102$	0	0
$103$	−6.00000	−0.591198	−0.295599	−	0.955312i	$-0.595519\pi$
$103$	−6.00000	−0.591198	−0.295599	+	0.955312i	$0.595519\pi$
$104$	0	0
$105$	1.00000	0.0975900
$106$	0	0
$107$	− 4.00000i	− 0.386695i	−0.981130	−	0.193347i	$-0.938066\pi$
$107$	− 4.00000i	− 0.386695i	0.981130	−	0.193347i	$-0.0619344\pi$
$108$	0	0
$109$	− 1.00000i	− 0.0957826i	−0.998853	−	0.0478913i	$-0.984750\pi$
$109$	− 1.00000i	− 0.0957826i	0.998853	−	0.0478913i	$-0.0152501\pi$
$110$	0	0
$111$	−1.00000	−0.0949158
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	6.00000i	0.559503i
$116$	0	0
$117$	2.00000i	0.184900i
$118$	0	0
$119$	−1.00000	−0.0916698
$120$	0	0
$121$	7.00000	0.636364
$122$	0	0
$123$	4.00000i	0.360668i
$124$	0	0
$125$	9.00000i	0.804984i
$126$	0	0
$127$	−22.0000	−1.95218	−0.976092	−	0.217357i	$-0.930256\pi$
$127$	−22.0000	−1.95218	−0.976092	+	0.217357i	$0.930256\pi$
$128$	0	0
$129$	−5.00000	−0.440225
$130$	0	0
$131$	3.00000i	0.262111i	0.991375	+	0.131056i	$0.0418366\pi$
$131$	3.00000i	0.262111i	−0.991375	+	0.131056i	$0.958163\pi$
$132$	0	0
$133$	8.00000i	0.693688i
$134$	0	0
$135$	−5.00000	−0.430331
$136$	0	0
$137$	14.0000	1.19610	0.598050	−	0.801459i	$-0.295942\pi$
$137$	14.0000	1.19610	0.598050	+	0.801459i	$0.295942\pi$
$138$	0	0
$139$	13.0000i	1.10265i	0.834292	+	0.551323i	$0.185877\pi$
$139$	13.0000i	1.10265i	−0.834292	+	0.551323i	$0.814123\pi$
$140$	0	0
$141$	− 11.0000i	− 0.926367i
$142$	0	0
$143$	2.00000	0.167248
$144$	0	0
$145$	−6.00000	−0.498273
$146$	0	0
$147$	− 6.00000i	− 0.494872i
$148$	0	0
$149$	− 6.00000i	− 0.491539i	−0.969328	−	0.245770i	$-0.920959\pi$
$149$	− 6.00000i	− 0.491539i	0.969328	−	0.245770i	$-0.0790407\pi$
$150$	0	0
$151$	5.00000	0.406894	0.203447	−	0.979086i	$-0.434786\pi$
$151$	5.00000	0.406894	0.203447	+	0.979086i	$0.434786\pi$
$152$	0	0
$153$	2.00000	0.161690
$154$	0	0
$155$	8.00000i	0.642575i
$156$	0	0
$157$	− 4.00000i	− 0.319235i	−0.987179	−	0.159617i	$-0.948974\pi$
$157$	− 4.00000i	− 0.319235i	0.987179	−	0.159617i	$-0.0510260\pi$
$158$	0	0
$159$	−6.00000	−0.475831
$160$	0	0
$161$	−6.00000	−0.472866
$162$	0	0
$163$	− 6.00000i	− 0.469956i	−0.972001	−	0.234978i	$-0.924498\pi$
$163$	− 6.00000i	− 0.469956i	0.972001	−	0.234978i	$-0.0755019\pi$
$164$	0	0
$165$	2.00000i	0.155700i
$166$	0	0
$167$	16.0000	1.23812	0.619059	−	0.785345i	$-0.287514\pi$
$167$	16.0000	1.23812	0.619059	+	0.785345i	$0.287514\pi$
$168$	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	$-0.821365\pi$
$173$	− 14.0000i	− 1.06440i	0.846619	−	0.532200i	$-0.178635\pi$
$174$	0	0
$175$	−4.00000	−0.302372
$176$	0	0
$177$	8.00000	0.601317
$178$	0	0
$179$	− 13.0000i	− 0.971666i	−0.874052	−	0.485833i	$-0.838516\pi$
$179$	− 13.0000i	− 0.971666i	0.874052	−	0.485833i	$-0.161484\pi$
$180$	0	0
$181$	− 22.0000i	− 1.63525i	−0.575753	−	0.817624i	$-0.695291\pi$
$181$	− 22.0000i	− 1.63525i	0.575753	−	0.817624i	$-0.304709\pi$
$182$	0	0
$183$	−4.00000	−0.295689
$184$	0	0
$185$	−1.00000	−0.0735215
$186$	0	0
$187$	− 2.00000i	− 0.146254i
$188$	0	0
$189$	− 5.00000i	− 0.363696i
$190$	0	0
$191$	18.0000	1.30243	0.651217	−	0.758891i	$-0.274259\pi$
$191$	18.0000	1.30243	0.651217	+	0.758891i	$0.274259\pi$
$192$	0	0
$193$	6.00000	0.431889	0.215945	−	0.976406i	$-0.430717\pi$
$193$	6.00000	0.431889	0.215945	+	0.976406i	$0.430717\pi$
$194$	0	0
$195$	− 1.00000i	− 0.0716115i
$196$	0	0
$197$	23.0000i	1.63868i	0.573306	+	0.819341i	$0.305660\pi$
$197$	23.0000i	1.63868i	−0.573306	+	0.819341i	$0.694340\pi$
$198$	0	0
$199$	−2.00000	−0.141776	−0.0708881	−	0.997484i	$-0.522583\pi$
$199$	−2.00000	−0.141776	−0.0708881	+	0.997484i	$0.522583\pi$
$200$	0	0
$201$	−2.00000	−0.141069
$202$	0	0
$203$	− 6.00000i	− 0.421117i
$204$	0	0
$205$	4.00000i	0.279372i
$206$	0	0
$207$	12.0000	0.834058
$208$	0	0
$209$	−16.0000	−1.10674
$210$	0	0
$211$	15.0000i	1.03264i	0.856395	+	0.516321i	$0.172699\pi$
$211$	15.0000i	1.03264i	−0.856395	+	0.516321i	$0.827301\pi$
$212$	0	0
$213$	− 3.00000i	− 0.205557i
$214$	0	0
$215$	−5.00000	−0.340997
$216$	0	0
$217$	−8.00000	−0.543075
$218$	0	0
$219$	8.00000i	0.540590i
$220$	0	0
$221$	1.00000i	0.0672673i
$222$	0	0
$223$	3.00000	0.200895	0.100447	−	0.994942i	$-0.467973\pi$
$223$	3.00000	0.200895	0.100447	+	0.994942i	$0.467973\pi$
$224$	0	0
$225$	8.00000	0.533333
$226$	0	0
$227$	24.0000i	1.59294i	0.604681	+	0.796468i	$0.293301\pi$
$227$	24.0000i	1.59294i	−0.604681	+	0.796468i	$0.706699\pi$
$228$	0	0
$229$	− 21.0000i	− 1.38772i	−0.720110	−	0.693860i	$-0.755909\pi$
$229$	− 21.0000i	− 1.38772i	0.720110	−	0.693860i	$-0.244091\pi$
$230$	0	0
$231$	−2.00000	−0.131590
$232$	0	0
$233$	7.00000	0.458585	0.229293	−	0.973358i	$-0.426359\pi$
$233$	7.00000	0.458585	0.229293	+	0.973358i	$0.426359\pi$
$234$	0	0
$235$	− 11.0000i	− 0.717561i
$236$	0	0
$237$	10.0000i	0.649570i
$238$	0	0
$239$	−11.0000	−0.711531	−0.355765	−	0.934575i	$-0.615780\pi$
$239$	−11.0000	−0.711531	−0.355765	+	0.934575i	$0.615780\pi$
$240$	0	0
$241$	−20.0000	−1.28831	−0.644157	−	0.764894i	$-0.722792\pi$
$241$	−20.0000	−1.28831	−0.644157	+	0.764894i	$0.722792\pi$
$242$	0	0
$243$	16.0000i	1.02640i
$244$	0	0
$245$	− 6.00000i	− 0.383326i
$246$	0	0
$247$	8.00000	0.509028
$248$	0	0
$249$	−6.00000	−0.380235
$250$	0	0
$251$	− 12.0000i	− 0.757433i	−0.925513	−	0.378717i	$-0.876365\pi$
$251$	− 12.0000i	− 0.757433i	0.925513	−	0.378717i	$-0.123635\pi$
$252$	0	0
$253$	− 12.0000i	− 0.754434i
$254$	0	0
$255$	−1.00000	−0.0626224
$256$	0	0
$257$	−15.0000	−0.935674	−0.467837	−	0.883815i	$-0.654967\pi$
$257$	−15.0000	−0.935674	−0.467837	+	0.883815i	$0.654967\pi$
$258$	0	0
$259$	− 1.00000i	− 0.0621370i
$260$	0	0
$261$	12.0000i	0.742781i
$262$	0	0
$263$	−16.0000	−0.986602	−0.493301	−	0.869859i	$-0.664210\pi$
$263$	−16.0000	−0.986602	−0.493301	+	0.869859i	$0.664210\pi$
$264$	0	0
$265$	−6.00000	−0.368577
$266$	0	0
$267$	16.0000i	0.979184i
$268$	0	0
$269$	− 4.00000i	− 0.243884i	−0.992537	−	0.121942i	$-0.961088\pi$
$269$	− 4.00000i	− 0.243884i	0.992537	−	0.121942i	$-0.0389122\pi$
$270$	0	0
$271$	−3.00000	−0.182237	−0.0911185	−	0.995840i	$-0.529044\pi$
$271$	−3.00000	−0.182237	−0.0911185	+	0.995840i	$0.529044\pi$
$272$	0	0
$273$	1.00000	0.0605228
$274$	0	0
$275$	− 8.00000i	− 0.482418i
$276$	0	0
$277$	− 24.0000i	− 1.44202i	−0.692925	−	0.721010i	$-0.743678\pi$
$277$	− 24.0000i	− 1.44202i	0.692925	−	0.721010i	$-0.256322\pi$
$278$	0	0
$279$	16.0000	0.957895
$280$	0	0
$281$	8.00000	0.477240	0.238620	−	0.971113i	$-0.423305\pi$
$281$	8.00000	0.477240	0.238620	+	0.971113i	$0.423305\pi$
$282$	0	0
$283$	− 32.0000i	− 1.90220i	−0.308879	−	0.951101i	$-0.599954\pi$
$283$	− 32.0000i	− 1.90220i	0.308879	−	0.951101i	$-0.400046\pi$
$284$	0	0
$285$	8.00000i	0.473879i
$286$	0	0
$287$	−4.00000	−0.236113
$288$	0	0
$289$	−16.0000	−0.941176
$290$	0	0
$291$	− 2.00000i	− 0.117242i
$292$	0	0
$293$	9.00000i	0.525786i	0.964825	+	0.262893i	$0.0846766\pi$
$293$	9.00000i	0.525786i	−0.964825	+	0.262893i	$0.915323\pi$
$294$	0	0
$295$	8.00000	0.465778
$296$	0	0
$297$	10.0000	0.580259
$298$	0	0
$299$	6.00000i	0.346989i
$300$	0	0
$301$	− 5.00000i	− 0.288195i
$302$	0	0
$303$	−8.00000	−0.459588
$304$	0	0
$305$	−4.00000	−0.229039
$306$	0	0
$307$	− 8.00000i	− 0.456584i	−0.973593	−	0.228292i	$-0.926686\pi$
$307$	− 8.00000i	− 0.456584i	0.973593	−	0.228292i	$-0.0733141\pi$
$308$	0	0
$309$	− 6.00000i	− 0.341328i
$310$	0	0
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	0	0
$313$	−11.0000	−0.621757	−0.310878	−	0.950450i	$-0.600623\pi$
$313$	−11.0000	−0.621757	−0.310878	+	0.950450i	$0.600623\pi$
$314$	0	0
$315$	− 2.00000i	− 0.112687i
$316$	0	0
$317$	6.00000i	0.336994i	0.985702	+	0.168497i	$0.0538913\pi$
$317$	6.00000i	0.336994i	−0.985702	+	0.168497i	$0.946109\pi$
$318$	0	0
$319$	12.0000	0.671871
$320$	0	0
$321$	4.00000	0.223258
$322$	0	0
$323$	− 8.00000i	− 0.445132i
$324$	0	0
$325$	4.00000i	0.221880i
$326$	0	0
$327$	1.00000	0.0553001
$328$	0	0
$329$	11.0000	0.606450
$330$	0	0
$331$	− 26.0000i	− 1.42909i	−0.699590	−	0.714545i	$-0.746634\pi$
$331$	− 26.0000i	− 1.42909i	0.699590	−	0.714545i	$-0.253366\pi$
$332$	0	0
$333$	2.00000i	0.109599i
$334$	0	0
$335$	−2.00000	−0.109272
$336$	0	0
$337$	−25.0000	−1.36184	−0.680918	−	0.732359i	$-0.738419\pi$
$337$	−25.0000	−1.36184	−0.680918	+	0.732359i	$0.738419\pi$
$338$	0	0
$339$	− 6.00000i	− 0.325875i
$340$	0	0
$341$	− 16.0000i	− 0.866449i
$342$	0	0
$343$	13.0000	0.701934
$344$	0	0
$345$	−6.00000	−0.323029
$346$	0	0
$347$	− 15.0000i	− 0.805242i	−0.915367	−	0.402621i	$-0.868099\pi$
$347$	− 15.0000i	− 0.805242i	0.915367	−	0.402621i	$-0.131901\pi$
$348$	0	0
$349$	− 25.0000i	− 1.33822i	−0.743164	−	0.669110i	$-0.766676\pi$
$349$	− 25.0000i	− 1.33822i	0.743164	−	0.669110i	$-0.233324\pi$
$350$	0	0
$351$	−5.00000	−0.266880
$352$	0	0
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	0	0
$355$	− 3.00000i	− 0.159223i
$356$	0	0
$357$	− 1.00000i	− 0.0529256i
$358$	0	0
$359$	−32.0000	−1.68890	−0.844448	−	0.535638i	$-0.820071\pi$
$359$	−32.0000	−1.68890	−0.844448	+	0.535638i	$0.820071\pi$
$360$	0	0
$361$	−45.0000	−2.36842
$362$	0	0
$363$	7.00000i	0.367405i
$364$	0	0
$365$	8.00000i	0.418739i
$366$	0	0
$367$	22.0000	1.14839	0.574195	−	0.818718i	$-0.305315\pi$
$367$	22.0000	1.14839	0.574195	+	0.818718i	$0.305315\pi$
$368$	0	0
$369$	8.00000	0.416463
$370$	0	0
$371$	− 6.00000i	− 0.311504i
$372$	0	0
$373$	− 24.0000i	− 1.24267i	−0.783544	−	0.621336i	$-0.786590\pi$
$373$	− 24.0000i	− 1.24267i	0.783544	−	0.621336i	$-0.213410\pi$
$374$	0	0
$375$	−9.00000	−0.464758
$376$	0	0
$377$	−6.00000	−0.309016
$378$	0	0
$379$	34.0000i	1.74646i	0.487306	+	0.873231i	$0.337980\pi$
$379$	34.0000i	1.74646i	−0.487306	+	0.873231i	$0.662020\pi$
$380$	0	0
$381$	− 22.0000i	− 1.12709i
$382$	0	0
$383$	15.0000	0.766464	0.383232	−	0.923652i	$-0.374811\pi$
$383$	15.0000	0.766464	0.383232	+	0.923652i	$0.374811\pi$
$384$	0	0
$385$	−2.00000	−0.101929
$386$	0	0
$387$	10.0000i	0.508329i
$388$	0	0
$389$	20.0000i	1.01404i	0.861934	+	0.507020i	$0.169253\pi$
$389$	20.0000i	1.01404i	−0.861934	+	0.507020i	$0.830747\pi$
$390$	0	0
$391$	6.00000	0.303433
$392$	0	0
$393$	−3.00000	−0.151330
$394$	0	0
$395$	10.0000i	0.503155i
$396$	0	0
$397$	− 22.0000i	− 1.10415i	−0.833795	−	0.552074i	$-0.813837\pi$
$397$	− 22.0000i	− 1.10415i	0.833795	−	0.552074i	$-0.186163\pi$
$398$	0	0
$399$	−8.00000	−0.400501
$400$	0	0
$401$	10.0000	0.499376	0.249688	−	0.968326i	$-0.419672\pi$
$401$	10.0000	0.499376	0.249688	+	0.968326i	$0.419672\pi$
$402$	0	0
$403$	8.00000i	0.398508i
$404$	0	0
$405$	1.00000i	0.0496904i
$406$	0	0
$407$	2.00000	0.0991363
$408$	0	0
$409$	−6.00000	−0.296681	−0.148340	−	0.988936i	$-0.547393\pi$
$409$	−6.00000	−0.296681	−0.148340	+	0.988936i	$0.547393\pi$
$410$	0	0
$411$	14.0000i	0.690569i
$412$	0	0
$413$	8.00000i	0.393654i
$414$	0	0
$415$	−6.00000	−0.294528
$416$	0	0
$417$	−13.0000	−0.636613
$418$	0	0
$419$	− 15.0000i	− 0.732798i	−0.930458	−	0.366399i	$-0.880591\pi$
$419$	− 15.0000i	− 0.732798i	0.930458	−	0.366399i	$-0.119409\pi$
$420$	0	0
$421$	− 15.0000i	− 0.731055i	−0.930800	−	0.365528i	$-0.880889\pi$
$421$	− 15.0000i	− 0.731055i	0.930800	−	0.365528i	$-0.119111\pi$
$422$	0	0
$423$	−22.0000	−1.06968
$424$	0	0
$425$	4.00000	0.194029
$426$	0	0
$427$	− 4.00000i	− 0.193574i
$428$	0	0
$429$	2.00000i	0.0965609i
$430$	0	0
$431$	−15.0000	−0.722525	−0.361262	−	0.932464i	$-0.617654\pi$
$431$	−15.0000	−0.722525	−0.361262	+	0.932464i	$0.617654\pi$
$432$	0	0
$433$	31.0000	1.48976	0.744882	−	0.667196i	$-0.232506\pi$
$433$	31.0000	1.48976	0.744882	+	0.667196i	$0.232506\pi$
$434$	0	0
$435$	− 6.00000i	− 0.287678i
$436$	0	0
$437$	− 48.0000i	− 2.29615i
$438$	0	0
$439$	−20.0000	−0.954548	−0.477274	−	0.878755i	$-0.658375\pi$
$439$	−20.0000	−0.954548	−0.477274	+	0.878755i	$0.658375\pi$
$440$	0	0
$441$	−12.0000	−0.571429
$442$	0	0
$443$	35.0000i	1.66290i	0.555599	+	0.831450i	$0.312489\pi$
$443$	35.0000i	1.66290i	−0.555599	+	0.831450i	$0.687511\pi$
$444$	0	0
$445$	16.0000i	0.758473i
$446$	0	0
$447$	6.00000	0.283790
$448$	0	0
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	0	0
$451$	− 8.00000i	− 0.376705i
$452$	0	0
$453$	5.00000i	0.234920i
$454$	0	0
$455$	1.00000	0.0468807
$456$	0	0
$457$	−32.0000	−1.49690	−0.748448	−	0.663193i	$-0.769201\pi$
$457$	−32.0000	−1.49690	−0.748448	+	0.663193i	$0.769201\pi$
$458$	0	0
$459$	5.00000i	0.233380i
$460$	0	0
$461$	− 37.0000i	− 1.72326i	−0.507535	−	0.861631i	$-0.669443\pi$
$461$	− 37.0000i	− 1.72326i	0.507535	−	0.861631i	$-0.330557\pi$
$462$	0	0
$463$	16.0000	0.743583	0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000	0.743583	0.371792	+	0.928316i	$0.378744\pi$
$464$	0	0
$465$	−8.00000	−0.370991
$466$	0	0
$467$	12.0000i	0.555294i	0.960683	+	0.277647i	$0.0895545\pi$
$467$	12.0000i	0.555294i	−0.960683	+	0.277647i	$0.910445\pi$
$468$	0	0
$469$	− 2.00000i	− 0.0923514i
$470$	0	0
$471$	4.00000	0.184310
$472$	0	0
$473$	10.0000	0.459800
$474$	0	0
$475$	− 32.0000i	− 1.46826i
$476$	0	0
$477$	12.0000i	0.549442i
$478$	0	0
$479$	−35.0000	−1.59919	−0.799595	−	0.600539i	$-0.794953\pi$
$479$	−35.0000	−1.59919	−0.799595	+	0.600539i	$0.794953\pi$
$480$	0	0
$481$	−1.00000	−0.0455961
$482$	0	0
$483$	− 6.00000i	− 0.273009i
$484$	0	0
$485$	− 2.00000i	− 0.0908153i
$486$	0	0
$487$	−16.0000	−0.725029	−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000	−0.725029	−0.362515	+	0.931978i	$0.618082\pi$
$488$	0	0
$489$	6.00000	0.271329
$490$	0	0
$491$	13.0000i	0.586682i	0.956008	+	0.293341i	$0.0947671\pi$
$491$	13.0000i	0.586682i	−0.956008	+	0.293341i	$0.905233\pi$
$492$	0	0
$493$	6.00000i	0.270226i
$494$	0	0
$495$	4.00000	0.179787
$496$	0	0
$497$	3.00000	0.134568
$498$	0	0
$499$	24.0000i	1.07439i	0.843459	+	0.537194i	$0.180516\pi$
$499$	24.0000i	1.07439i	−0.843459	+	0.537194i	$0.819484\pi$
$500$	0	0
$501$	16.0000i	0.714827i
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	−8.00000	−0.355995
$506$	0	0
$507$	− 1.00000i	− 0.0444116i
$508$	0	0
$509$	14.0000i	0.620539i	0.950649	+	0.310270i	$0.100419\pi$
$509$	14.0000i	0.620539i	−0.950649	+	0.310270i	$0.899581\pi$
$510$	0	0
$511$	−8.00000	−0.353899
$512$	0	0
$513$	40.0000	1.76604
$514$	0	0
$515$	− 6.00000i	− 0.264392i
$516$	0	0
$517$	22.0000i	0.967559i
$518$	0	0
$519$	14.0000	0.614532
$520$	0	0
$521$	−11.0000	−0.481919	−0.240959	−	0.970535i	$-0.577462\pi$
$521$	−11.0000	−0.481919	−0.240959	+	0.970535i	$0.577462\pi$
$522$	0	0
$523$	4.00000i	0.174908i	0.996169	+	0.0874539i	$0.0278730\pi$
$523$	4.00000i	0.174908i	−0.996169	+	0.0874539i	$0.972127\pi$
$524$	0	0
$525$	− 4.00000i	− 0.174574i
$526$	0	0
$527$	8.00000	0.348485
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	− 16.0000i	− 0.694341i
$532$	0	0
$533$	4.00000i	0.173259i
$534$	0	0
$535$	4.00000	0.172935
$536$	0	0
$537$	13.0000	0.560991
$538$	0	0
$539$	12.0000i	0.516877i
$540$	0	0
$541$	1.00000i	0.0429934i	0.999769	+	0.0214967i	$0.00684313\pi$
$541$	1.00000i	0.0429934i	−0.999769	+	0.0214967i	$0.993157\pi$
$542$	0	0
$543$	22.0000	0.944110
$544$	0	0
$545$	1.00000	0.0428353
$546$	0	0
$547$	1.00000i	0.0427569i	0.999771	+	0.0213785i	$0.00680549\pi$
$547$	1.00000i	0.0427569i	−0.999771	+	0.0213785i	$0.993195\pi$
$548$	0	0
$549$	8.00000i	0.341432i
$550$	0	0
$551$	48.0000	2.04487
$552$	0	0
$553$	−10.0000	−0.425243
$554$	0	0
$555$	− 1.00000i	− 0.0424476i
$556$	0	0
$557$	− 15.0000i	− 0.635570i	−0.948163	−	0.317785i	$-0.897061\pi$
$557$	− 15.0000i	− 0.635570i	0.948163	−	0.317785i	$-0.102939\pi$
$558$	0	0
$559$	−5.00000	−0.211477
$560$	0	0
$561$	2.00000	0.0844401
$562$	0	0
$563$	31.0000i	1.30649i	0.757145	+	0.653247i	$0.226594\pi$
$563$	31.0000i	1.30649i	−0.757145	+	0.653247i	$0.773406\pi$
$564$	0	0
$565$	− 6.00000i	− 0.252422i
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	1.00000	0.0419222	0.0209611	−	0.999780i	$-0.493327\pi$
$569$	1.00000	0.0419222	0.0209611	+	0.999780i	$0.493327\pi$
$570$	0	0
$571$	7.00000i	0.292941i	0.989215	+	0.146470i	$0.0467913\pi$
$571$	7.00000i	0.292941i	−0.989215	+	0.146470i	$0.953209\pi$
$572$	0	0
$573$	18.0000i	0.751961i
$574$	0	0
$575$	24.0000	1.00087
$576$	0	0
$577$	−8.00000	−0.333044	−0.166522	−	0.986038i	$-0.553254\pi$
$577$	−8.00000	−0.333044	−0.166522	+	0.986038i	$0.553254\pi$
$578$	0	0
$579$	6.00000i	0.249351i
$580$	0	0
$581$	− 6.00000i	− 0.248922i
$582$	0	0
$583$	12.0000	0.496989
$584$	0	0
$585$	−2.00000	−0.0826898
$586$	0	0
$587$	− 18.0000i	− 0.742940i	−0.928445	−	0.371470i	$-0.878854\pi$
$587$	− 18.0000i	− 0.742940i	0.928445	−	0.371470i	$-0.121146\pi$
$588$	0	0
$589$	− 64.0000i	− 2.63707i
$590$	0	0
$591$	−23.0000	−0.946094
$592$	0	0
$593$	34.0000	1.39621	0.698106	−	0.715994i	$-0.254026\pi$
$593$	34.0000	1.39621	0.698106	+	0.715994i	$0.254026\pi$
$594$	0	0
$595$	− 1.00000i	− 0.0409960i
$596$	0	0
$597$	− 2.00000i	− 0.0818546i
$598$	0	0
$599$	−16.0000	−0.653742	−0.326871	−	0.945069i	$-0.605994\pi$
$599$	−16.0000	−0.653742	−0.326871	+	0.945069i	$0.605994\pi$
$600$	0	0
$601$	−45.0000	−1.83559	−0.917794	−	0.397057i	$-0.870032\pi$
$601$	−45.0000	−1.83559	−0.917794	+	0.397057i	$0.870032\pi$
$602$	0	0
$603$	4.00000i	0.162893i
$604$	0	0
$605$	7.00000i	0.284590i
$606$	0	0
$607$	−32.0000	−1.29884	−0.649420	−	0.760430i	$-0.724988\pi$
$607$	−32.0000	−1.29884	−0.649420	+	0.760430i	$0.724988\pi$
$608$	0	0
$609$	6.00000	0.243132
$610$	0	0
$611$	− 11.0000i	− 0.445012i
$612$	0	0
$613$	38.0000i	1.53481i	0.641165	+	0.767403i	$0.278451\pi$
$613$	38.0000i	1.53481i	−0.641165	+	0.767403i	$0.721549\pi$
$614$	0	0
$615$	−4.00000	−0.161296
$616$	0	0
$617$	20.0000	0.805170	0.402585	−	0.915383i	$-0.368112\pi$
$617$	20.0000	0.805170	0.402585	+	0.915383i	$0.368112\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	30.0000i	1.20386i
$622$	0	0
$623$	−16.0000	−0.641026
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	− 16.0000i	− 0.638978i
$628$	0	0
$629$	1.00000i	0.0398726i
$630$	0	0
$631$	−47.0000	−1.87104	−0.935520	−	0.353273i	$-0.885069\pi$
$631$	−47.0000	−1.87104	−0.935520	+	0.353273i	$0.885069\pi$
$632$	0	0
$633$	−15.0000	−0.596196
$634$	0	0
$635$	− 22.0000i	− 0.873043i
$636$	0	0
$637$	− 6.00000i	− 0.237729i
$638$	0	0
$639$	−6.00000	−0.237356
$640$	0	0
$641$	42.0000	1.65890	0.829450	−	0.558581i	$-0.188654\pi$
$641$	42.0000	1.65890	0.829450	+	0.558581i	$0.188654\pi$
$642$	0	0
$643$	− 14.0000i	− 0.552106i	−0.961142	−	0.276053i	$-0.910973\pi$
$643$	− 14.0000i	− 0.552106i	0.961142	−	0.276053i	$-0.0890266\pi$
$644$	0	0
$645$	− 5.00000i	− 0.196875i
$646$	0	0
$647$	4.00000	0.157256	0.0786281	−	0.996904i	$-0.474946\pi$
$647$	4.00000	0.157256	0.0786281	+	0.996904i	$0.474946\pi$
$648$	0	0
$649$	−16.0000	−0.628055
$650$	0	0
$651$	− 8.00000i	− 0.313545i
$652$	0	0
$653$	36.0000i	1.40879i	0.709809	+	0.704394i	$0.248781\pi$
$653$	36.0000i	1.40879i	−0.709809	+	0.704394i	$0.751219\pi$
$654$	0	0
$655$	−3.00000	−0.117220
$656$	0	0
$657$	16.0000	0.624219
$658$	0	0
$659$	− 12.0000i	− 0.467454i	−0.972302	−	0.233727i	$-0.924908\pi$
$659$	− 12.0000i	− 0.467454i	0.972302	−	0.233727i	$-0.0750921\pi$
$660$	0	0
$661$	− 42.0000i	− 1.63361i	−0.576913	−	0.816805i	$-0.695743\pi$
$661$	− 42.0000i	− 1.63361i	0.576913	−	0.816805i	$-0.304257\pi$
$662$	0	0
$663$	−1.00000	−0.0388368
$664$	0	0
$665$	−8.00000	−0.310227
$666$	0	0
$667$	36.0000i	1.39393i
$668$	0	0
$669$	3.00000i	0.115987i
$670$	0	0
$671$	8.00000	0.308837
$672$	0	0
$673$	1.00000	0.0385472	0.0192736	−	0.999814i	$-0.493865\pi$
$673$	1.00000	0.0385472	0.0192736	+	0.999814i	$0.493865\pi$
$674$	0	0
$675$	20.0000i	0.769800i
$676$	0	0
$677$	18.0000i	0.691796i	0.938272	+	0.345898i	$0.112426\pi$
$677$	18.0000i	0.691796i	−0.938272	+	0.345898i	$0.887574\pi$
$678$	0	0
$679$	2.00000	0.0767530
$680$	0	0
$681$	−24.0000	−0.919682
$682$	0	0
$683$	18.0000i	0.688751i	0.938832	+	0.344375i	$0.111909\pi$
$683$	18.0000i	0.688751i	−0.938832	+	0.344375i	$0.888091\pi$
$684$	0	0
$685$	14.0000i	0.534913i
$686$	0	0
$687$	21.0000	0.801200
$688$	0	0
$689$	−6.00000	−0.228582
$690$	0	0
$691$	− 16.0000i	− 0.608669i	−0.952565	−	0.304334i	$-0.901566\pi$
$691$	− 16.0000i	− 0.608669i	0.952565	−	0.304334i	$-0.0984340\pi$
$692$	0	0
$693$	4.00000i	0.151947i
$694$	0	0
$695$	−13.0000	−0.493118
$696$	0	0
$697$	4.00000	0.151511
$698$	0	0
$699$	7.00000i	0.264764i
$700$	0	0
$701$	22.0000i	0.830929i	0.909610	+	0.415464i	$0.136381\pi$
$701$	22.0000i	0.830929i	−0.909610	+	0.415464i	$0.863619\pi$
$702$	0	0
$703$	8.00000	0.301726
$704$	0	0
$705$	11.0000	0.414284
$706$	0	0
$707$	− 8.00000i	− 0.300871i
$708$	0	0
$709$	42.0000i	1.57734i	0.614815	+	0.788672i	$0.289231\pi$
$709$	42.0000i	1.57734i	−0.614815	+	0.788672i	$0.710769\pi$
$710$	0	0
$711$	20.0000	0.750059
$712$	0	0
$713$	48.0000	1.79761
$714$	0	0
$715$	2.00000i	0.0747958i
$716$	0	0
$717$	− 11.0000i	− 0.410803i
$718$	0	0
$719$	42.0000	1.56634	0.783168	−	0.621810i	$-0.213603\pi$
$719$	42.0000	1.56634	0.783168	+	0.621810i	$0.213603\pi$
$720$	0	0
$721$	6.00000	0.223452
$722$	0	0
$723$	− 20.0000i	− 0.743808i
$724$	0	0
$725$	24.0000i	0.891338i
$726$	0	0
$727$	−26.0000	−0.964287	−0.482143	−	0.876092i	$-0.660142\pi$
$727$	−26.0000	−0.964287	−0.482143	+	0.876092i	$0.660142\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	5.00000i	0.184932i
$732$	0	0
$733$	1.00000i	0.0369358i	0.999829	+	0.0184679i	$0.00587886\pi$
$733$	1.00000i	0.0369358i	−0.999829	+	0.0184679i	$0.994121\pi$
$734$	0	0
$735$	6.00000	0.221313
$736$	0	0
$737$	4.00000	0.147342
$738$	0	0
$739$	20.0000i	0.735712i	0.929883	+	0.367856i	$0.119908\pi$
$739$	20.0000i	0.735712i	−0.929883	+	0.367856i	$0.880092\pi$
$740$	0	0
$741$	8.00000i	0.293887i
$742$	0	0
$743$	3.00000	0.110059	0.0550297	−	0.998485i	$-0.482475\pi$
$743$	3.00000	0.110059	0.0550297	+	0.998485i	$0.482475\pi$
$744$	0	0
$745$	6.00000	0.219823
$746$	0	0
$747$	12.0000i	0.439057i
$748$	0	0
$749$	4.00000i	0.146157i
$750$	0	0
$751$	−20.0000	−0.729810	−0.364905	−	0.931045i	$-0.618899\pi$
$751$	−20.0000	−0.729810	−0.364905	+	0.931045i	$0.618899\pi$
$752$	0	0
$753$	12.0000	0.437304
$754$	0	0
$755$	5.00000i	0.181969i
$756$	0	0
$757$	− 22.0000i	− 0.799604i	−0.916602	−	0.399802i	$-0.869079\pi$
$757$	− 22.0000i	− 0.799604i	0.916602	−	0.399802i	$-0.130921\pi$
$758$	0	0
$759$	12.0000	0.435572
$760$	0	0
$761$	−12.0000	−0.435000	−0.217500	−	0.976060i	$-0.569790\pi$
$761$	−12.0000	−0.435000	−0.217500	+	0.976060i	$0.569790\pi$
$762$	0	0
$763$	1.00000i	0.0362024i
$764$	0	0
$765$	2.00000i	0.0723102i
$766$	0	0
$767$	8.00000	0.288863
$768$	0	0
$769$	−32.0000	−1.15395	−0.576975	−	0.816762i	$-0.695767\pi$
$769$	−32.0000	−1.15395	−0.576975	+	0.816762i	$0.695767\pi$
$770$	0	0
$771$	− 15.0000i	− 0.540212i
$772$	0	0
$773$	33.0000i	1.18693i	0.804861	+	0.593464i	$0.202240\pi$
$773$	33.0000i	1.18693i	−0.804861	+	0.593464i	$0.797760\pi$
$774$	0	0
$775$	32.0000	1.14947
$776$	0	0
$777$	1.00000	0.0358748
$778$	0	0
$779$	− 32.0000i	− 1.14652i
$780$	0	0
$781$	6.00000i	0.214697i
$782$	0	0
$783$	−30.0000	−1.07211
$784$	0	0
$785$	4.00000	0.142766
$786$	0	0
$787$	8.00000i	0.285169i	0.989783	+	0.142585i	$0.0455413\pi$
$787$	8.00000i	0.285169i	−0.989783	+	0.142585i	$0.954459\pi$
$788$	0	0
$789$	− 16.0000i	− 0.569615i
$790$	0	0
$791$	6.00000	0.213335
$792$	0	0
$793$	−4.00000	−0.142044
$794$	0	0
$795$	− 6.00000i	− 0.212798i
$796$	0	0
$797$	− 36.0000i	− 1.27519i	−0.770374	−	0.637593i	$-0.779930\pi$
$797$	− 36.0000i	− 1.27519i	0.770374	−	0.637593i	$-0.220070\pi$
$798$	0	0
$799$	−11.0000	−0.389152
$800$	0	0
$801$	32.0000	1.13066
$802$	0	0
$803$	− 16.0000i	− 0.564628i
$804$	0	0
$805$	− 6.00000i	− 0.211472i
$806$	0	0
$807$	4.00000	0.140807
$808$	0	0
$809$	−3.00000	−0.105474	−0.0527372	−	0.998608i	$-0.516795\pi$
$809$	−3.00000	−0.105474	−0.0527372	+	0.998608i	$0.516795\pi$
$810$	0	0
$811$	4.00000i	0.140459i	0.997531	+	0.0702295i	$0.0223732\pi$
$811$	4.00000i	0.140459i	−0.997531	+	0.0702295i	$0.977627\pi$
$812$	0	0
$813$	− 3.00000i	− 0.105215i
$814$	0	0
$815$	6.00000	0.210171
$816$	0	0
$817$	40.0000	1.39942
$818$	0	0
$819$	− 2.00000i	− 0.0698857i
$820$	0	0
$821$	45.0000i	1.57051i	0.619172	+	0.785255i	$0.287468\pi$
$821$	45.0000i	1.57051i	−0.619172	+	0.785255i	$0.712532\pi$
$822$	0	0
$823$	8.00000	0.278862	0.139431	−	0.990232i	$-0.455473\pi$
$823$	8.00000	0.278862	0.139431	+	0.990232i	$0.455473\pi$
$824$	0	0
$825$	8.00000	0.278524
$826$	0	0
$827$	− 52.0000i	− 1.80822i	−0.427303	−	0.904109i	$-0.640536\pi$
$827$	− 52.0000i	− 1.80822i	0.427303	−	0.904109i	$-0.359464\pi$
$828$	0	0
$829$	46.0000i	1.59765i	0.601566	+	0.798823i	$0.294544\pi$
$829$	46.0000i	1.59765i	−0.601566	+	0.798823i	$0.705456\pi$
$830$	0	0
$831$	24.0000	0.832551
$832$	0	0
$833$	−6.00000	−0.207888
$834$	0	0
$835$	16.0000i	0.553703i
$836$	0	0
$837$	40.0000i	1.38260i
$838$	0	0
$839$	24.0000	0.828572	0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000	0.828572	0.414286	+	0.910147i	$0.364031\pi$
$840$	0	0
$841$	−7.00000	−0.241379
$842$	0	0
$843$	8.00000i	0.275535i
$844$	0	0
$845$	− 1.00000i	− 0.0344010i
$846$	0	0
$847$	−7.00000	−0.240523
$848$	0	0
$849$	32.0000	1.09824
$850$	0	0
$851$	6.00000i	0.205677i
$852$	0	0
$853$	− 33.0000i	− 1.12990i	−0.825126	−	0.564949i	$-0.808896\pi$
$853$	− 33.0000i	− 1.12990i	0.825126	−	0.564949i	$-0.191104\pi$
$854$	0	0
$855$	16.0000	0.547188
$856$	0	0
$857$	18.0000	0.614868	0.307434	−	0.951569i	$-0.400530\pi$
$857$	18.0000	0.614868	0.307434	+	0.951569i	$0.400530\pi$
$858$	0	0
$859$	− 36.0000i	− 1.22830i	−0.789188	−	0.614152i	$-0.789498\pi$
$859$	− 36.0000i	− 1.22830i	0.789188	−	0.614152i	$-0.210502\pi$
$860$	0	0
$861$	− 4.00000i	− 0.136320i
$862$	0	0
$863$	−39.0000	−1.32758	−0.663788	−	0.747921i	$-0.731052\pi$
$863$	−39.0000	−1.32758	−0.663788	+	0.747921i	$0.731052\pi$
$864$	0	0
$865$	14.0000	0.476014
$866$	0	0
$867$	− 16.0000i	− 0.543388i
$868$	0	0
$869$	− 20.0000i	− 0.678454i
$870$	0	0
$871$	−2.00000	−0.0677674
$872$	0	0
$873$	−4.00000	−0.135379
$874$	0	0
$875$	− 9.00000i	− 0.304256i
$876$	0	0
$877$	− 7.00000i	− 0.236373i	−0.992991	−	0.118187i	$-0.962292\pi$
$877$	− 7.00000i	− 0.236373i	0.992991	−	0.118187i	$-0.0377081\pi$
$878$	0	0
$879$	−9.00000	−0.303562
$880$	0	0
$881$	−19.0000	−0.640126	−0.320063	−	0.947396i	$-0.603704\pi$
$881$	−19.0000	−0.640126	−0.320063	+	0.947396i	$0.603704\pi$
$882$	0	0
$883$	− 31.0000i	− 1.04323i	−0.853180	−	0.521617i	$-0.825329\pi$
$883$	− 31.0000i	− 1.04323i	0.853180	−	0.521617i	$-0.174671\pi$
$884$	0	0
$885$	8.00000i	0.268917i
$886$	0	0
$887$	−40.0000	−1.34307	−0.671534	−	0.740973i	$-0.734364\pi$
$887$	−40.0000	−1.34307	−0.671534	+	0.740973i	$0.734364\pi$
$888$	0	0
$889$	22.0000	0.737856
$890$	0	0
$891$	− 2.00000i	− 0.0670025i
$892$	0	0
$893$	88.0000i	2.94481i
$894$	0	0
$895$	13.0000	0.434542
$896$	0	0
$897$	−6.00000	−0.200334
$898$	0	0
$899$	48.0000i	1.60089i
$900$	0	0
$901$	6.00000i	0.199889i
$902$	0	0
$903$	5.00000	0.166390
$904$	0	0
$905$	22.0000	0.731305
$906$	0	0
$907$	17.0000i	0.564476i	0.959344	+	0.282238i	$0.0910767\pi$
$907$	17.0000i	0.564476i	−0.959344	+	0.282238i	$0.908923\pi$
$908$	0	0
$909$	16.0000i	0.530687i
$910$	0	0
$911$	18.0000	0.596367	0.298183	−	0.954509i	$-0.403619\pi$
$911$	18.0000	0.596367	0.298183	+	0.954509i	$0.403619\pi$
$912$	0	0
$913$	12.0000	0.397142
$914$	0	0
$915$	− 4.00000i	− 0.132236i
$916$	0	0
$917$	− 3.00000i	− 0.0990687i
$918$	0	0
$919$	36.0000	1.18753	0.593765	−	0.804638i	$-0.297641\pi$
$919$	36.0000	1.18753	0.593765	+	0.804638i	$0.297641\pi$
$920$	0	0
$921$	8.00000	0.263609
$922$	0	0
$923$	− 3.00000i	− 0.0987462i
$924$	0	0
$925$	4.00000i	0.131519i
$926$	0	0
$927$	−12.0000	−0.394132
$928$	0	0
$929$	48.0000	1.57483	0.787414	−	0.616424i	$-0.211419\pi$
$929$	48.0000	1.57483	0.787414	+	0.616424i	$0.211419\pi$
$930$	0	0
$931$	48.0000i	1.57314i
$932$	0	0
$933$	24.0000i	0.785725i
$934$	0	0
$935$	2.00000	0.0654070
$936$	0	0
$937$	−46.0000	−1.50275	−0.751377	−	0.659873i	$-0.770610\pi$
$937$	−46.0000	−1.50275	−0.751377	+	0.659873i	$0.770610\pi$
$938$	0	0
$939$	− 11.0000i	− 0.358971i
$940$	0	0
$941$	− 27.0000i	− 0.880175i	−0.897955	−	0.440087i	$-0.854947\pi$
$941$	− 27.0000i	− 0.880175i	0.897955	−	0.440087i	$-0.145053\pi$
$942$	0	0
$943$	24.0000	0.781548
$944$	0	0
$945$	5.00000	0.162650
$946$	0	0
$947$	42.0000i	1.36482i	0.730971	+	0.682408i	$0.239067\pi$
$947$	42.0000i	1.36482i	−0.730971	+	0.682408i	$0.760933\pi$
$948$	0	0
$949$	8.00000i	0.259691i
$950$	0	0
$951$	−6.00000	−0.194563
$952$	0	0
$953$	57.0000	1.84641	0.923206	−	0.384307i	$-0.125559\pi$
$953$	57.0000	1.84641	0.923206	+	0.384307i	$0.125559\pi$
$954$	0	0
$955$	18.0000i	0.582466i
$956$	0	0
$957$	12.0000i	0.387905i
$958$	0	0
$959$	−14.0000	−0.452084
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	− 8.00000i	− 0.257796i
$964$	0	0
$965$	6.00000i	0.193147i
$966$	0	0
$967$	−59.0000	−1.89731	−0.948656	−	0.316310i	$-0.897556\pi$
$967$	−59.0000	−1.89731	−0.948656	+	0.316310i	$0.897556\pi$
$968$	0	0
$969$	8.00000	0.256997
$970$	0	0
$971$	27.0000i	0.866471i	0.901281	+	0.433236i	$0.142628\pi$
$971$	27.0000i	0.866471i	−0.901281	+	0.433236i	$0.857372\pi$
$972$	0	0
$973$	− 13.0000i	− 0.416761i
$974$	0	0
$975$	−4.00000	−0.128103
$976$	0	0
$977$	−10.0000	−0.319928	−0.159964	−	0.987123i	$-0.551138\pi$
$977$	−10.0000	−0.319928	−0.159964	+	0.987123i	$0.551138\pi$
$978$	0	0
$979$	− 32.0000i	− 1.02272i
$980$	0	0
$981$	− 2.00000i	− 0.0638551i
$982$	0	0
$983$	23.0000	0.733586	0.366793	−	0.930303i	$-0.380456\pi$
$983$	23.0000	0.733586	0.366793	+	0.930303i	$0.380456\pi$
$984$	0	0
$985$	−23.0000	−0.732841
$986$	0	0
$987$	11.0000i	0.350134i
$988$	0	0
$989$	30.0000i	0.953945i
$990$	0	0
$991$	40.0000	1.27064	0.635321	−	0.772248i	$-0.280868\pi$
$991$	40.0000	1.27064	0.635321	+	0.772248i	$0.280868\pi$
$992$	0	0
$993$	26.0000	0.825085
$994$	0	0
$995$	− 2.00000i	− 0.0634043i
$996$	0	0
$997$	− 50.0000i	− 1.58352i	−0.610835	−	0.791758i	$-0.709166\pi$
$997$	− 50.0000i	− 1.58352i	0.610835	−	0.791758i	$-0.290834\pi$
$998$	0	0
$999$	−5.00000	−0.158193

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1664.2.b.b.833.2	yes	2
4.3	odd	2		1664.2.b.c.833.1	yes	2
8.3	odd	2		1664.2.b.c.833.2	yes	2
8.5	even	2	inner	1664.2.b.b.833.1	✓	2
16.3	odd	4		3328.2.a.i.1.1		1
16.5	even	4		3328.2.a.h.1.1		1
16.11	odd	4		3328.2.a.d.1.1		1
16.13	even	4		3328.2.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1664.2.b.b.833.1	✓	2	8.5	even	2	inner
1664.2.b.b.833.2	yes	2	1.1	even	1	trivial
1664.2.b.c.833.1	yes	2	4.3	odd	2
1664.2.b.c.833.2	yes	2	8.3	odd	2
3328.2.a.d.1.1		1	16.11	odd	4
3328.2.a.e.1.1		1	16.13	even	4
3328.2.a.h.1.1		1	16.5	even	4
3328.2.a.i.1.1		1	16.3	odd	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:	\( N \)	\(=\)	\( 1664 = 2^{7} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1664.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +1.00000i q^{5} -1.00000 q^{7} +2.00000 q^{9} -2.00000i q^{11} +1.00000i q^{13} -1.00000 q^{15} +1.00000 q^{17} -8.00000i q^{19} -1.00000i q^{21} +6.00000 q^{23} +4.00000 q^{25} +5.00000i q^{27} +6.00000i q^{29} +8.00000 q^{31} +2.00000 q^{33} -1.00000i q^{35} +1.00000i q^{37} -1.00000 q^{39} +4.00000 q^{41} +5.00000i q^{43} +2.00000i q^{45} -11.0000 q^{47} -6.00000 q^{49} +1.00000i q^{51} +6.00000i q^{53} +2.00000 q^{55} +8.00000 q^{57} -8.00000i q^{59} +4.00000i q^{61} -2.00000 q^{63} -1.00000 q^{65} +2.00000i q^{67} +6.00000i q^{69} -3.00000 q^{71} +8.00000 q^{73} +4.00000i q^{75} +2.00000i q^{77} +10.0000 q^{79} +1.00000 q^{81} +6.00000i q^{83} +1.00000i q^{85} -6.00000 q^{87} +16.0000 q^{89} -1.00000i q^{91} +8.00000i q^{93} +8.00000 q^{95} -2.00000 q^{97} -4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{7} + 4 q^{9} - 2 q^{15} + 2 q^{17} + 12 q^{23} + 8 q^{25} + 16 q^{31} + 4 q^{33} - 2 q^{39} + 8 q^{41} - 22 q^{47} - 12 q^{49} + 4 q^{55} + 16 q^{57} - 4 q^{63} - 2 q^{65} - 6 q^{71} + 16 q^{73}+ \cdots - 4 q^{97}+O(q^{100}) \) 2 * q - 2 * q^7 + 4 * q^9 - 2 * q^15 + 2 * q^17 + 12 * q^23 + 8 * q^25 + 16 * q^31 + 4 * q^33 - 2 * q^39 + 8 * q^41 - 22 * q^47 - 12 * q^49 + 4 * q^55 + 16 * q^57 - 4 * q^63 - 2 * q^65 - 6 * q^71 + 16 * q^73 + 20 * q^79 + 2 * q^81 - 12 * q^87 + 32 * q^89 + 16 * q^95 - 4 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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