LMFDB - Embedded newform 3328.2.a.bb.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3328,2,Mod(1,3328)] 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("3328.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3328, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3328 = 2^{8} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3328.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,2,0,2,0,6,0,0] 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:	yes
Analytic conductor:	$26.5742137927$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 832)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	3328.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.414214 q^{3} +3.82843 q^{5} +1.58579 q^{7} -2.82843 q^{9} +4.82843 q^{11} +1.00000 q^{13} -1.58579 q^{15} +1.00000 q^{17} +5.65685 q^{19} -0.656854 q^{21} +3.17157 q^{23} +9.65685 q^{25} +2.41421 q^{27} -7.65685 q^{29} -7.65685 q^{31} -2.00000 q^{33} +6.07107 q^{35} +7.00000 q^{37} -0.414214 q^{39} -1.65685 q^{41} -5.58579 q^{43} -10.8284 q^{45} +9.24264 q^{47} -4.48528 q^{49} -0.414214 q^{51} -7.65685 q^{53} +18.4853 q^{55} -2.34315 q^{57} +8.00000 q^{59} -4.48528 q^{63} +3.82843 q^{65} +3.17157 q^{67} -1.31371 q^{69} +13.7279 q^{71} -9.65685 q^{73} -4.00000 q^{75} +7.65685 q^{77} +12.1421 q^{79} +7.48528 q^{81} -16.1421 q^{83} +3.82843 q^{85} +3.17157 q^{87} +2.34315 q^{89} +1.58579 q^{91} +3.17157 q^{93} +21.6569 q^{95} +3.65685 q^{97} -13.6569 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{5} + 6 q^{7} + 4 q^{11} + 2 q^{13} - 6 q^{15} + 2 q^{17} + 10 q^{21} + 12 q^{23} + 8 q^{25} + 2 q^{27} - 4 q^{29} - 4 q^{31} - 4 q^{33} - 2 q^{35} + 14 q^{37} + 2 q^{39} + 8 q^{41}+ \cdots - 16 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^5 + 6 * q^7 + 4 * q^11 + 2 * q^13 - 6 * q^15 + 2 * q^17 + 10 * q^21 + 12 * q^23 + 8 * q^25 + 2 * q^27 - 4 * q^29 - 4 * q^31 - 4 * q^33 - 2 * q^35 + 14 * q^37 + 2 * q^39 + 8 * q^41 - 14 * q^43 - 16 * q^45 + 10 * q^47 + 8 * q^49 + 2 * q^51 - 4 * q^53 + 20 * q^55 - 16 * q^57 + 16 * q^59 + 8 * q^63 + 2 * q^65 + 12 * q^67 + 20 * q^69 + 2 * q^71 - 8 * q^73 - 8 * q^75 + 4 * q^77 - 4 * q^79 - 2 * q^81 - 4 * q^83 + 2 * q^85 + 12 * q^87 + 16 * q^89 + 6 * q^91 + 12 * q^93 + 32 * q^95 - 4 * q^97 - 16 * q^99

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
$3$	−0.414214	−0.239146	−0.119573	−	0.992825i	$-0.538153\pi$
$3$	−0.414214	−0.239146	−0.119573	+	0.992825i	$0.538153\pi$
$4$	0	0
$5$	3.82843	1.71212	0.856062	−	0.516873i	$-0.172904\pi$
$5$	3.82843	1.71212	0.856062	+	0.516873i	$0.172904\pi$
$6$	0	0
$7$	1.58579	0.599371	0.299685	−	0.954038i	$-0.403118\pi$
$7$	1.58579	0.599371	0.299685	+	0.954038i	$0.403118\pi$
$8$	0	0
$9$	−2.82843	−0.942809
$10$	0	0
$11$	4.82843	1.45583	0.727913	−	0.685670i	$-0.240491\pi$
$11$	4.82843	1.45583	0.727913	+	0.685670i	$0.240491\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	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
$19$	5.65685	1.29777	0.648886	−	0.760886i	$-0.275235\pi$
$19$	5.65685	1.29777	0.648886	+	0.760886i	$0.275235\pi$
$20$	0	0
$21$	−0.656854	−0.143337
$22$	0	0
$23$	3.17157	0.661319	0.330659	−	0.943750i	$-0.392729\pi$
$23$	3.17157	0.661319	0.330659	+	0.943750i	$0.392729\pi$
$24$	0	0
$25$	9.65685	1.93137
$26$	0	0
$27$	2.41421	0.464616
$28$	0	0
$29$	−7.65685	−1.42184	−0.710921	−	0.703272i	$-0.751722\pi$
$29$	−7.65685	−1.42184	−0.710921	+	0.703272i	$0.751722\pi$
$30$	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
$35$	6.07107	1.02620
$36$	0	0
$37$	7.00000	1.15079	0.575396	−	0.817875i	$-0.304848\pi$
$37$	7.00000	1.15079	0.575396	+	0.817875i	$0.304848\pi$
$38$	0	0
$39$	−0.414214	−0.0663273
$40$	0	0
$41$	−1.65685	−0.258757	−0.129379	−	0.991595i	$-0.541298\pi$
$41$	−1.65685	−0.258757	−0.129379	+	0.991595i	$0.541298\pi$
$42$	0	0
$43$	−5.58579	−0.851824	−0.425912	−	0.904764i	$-0.640047\pi$
$43$	−5.58579	−0.851824	−0.425912	+	0.904764i	$0.640047\pi$
$44$	0	0
$45$	−10.8284	−1.61421
$46$	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
$51$	−0.414214	−0.0580015
$52$	0	0
$53$	−7.65685	−1.05175	−0.525875	−	0.850562i	$-0.676262\pi$
$53$	−7.65685	−1.05175	−0.525875	+	0.850562i	$0.676262\pi$
$54$	0	0
$55$	18.4853	2.49255
$56$	0	0
$57$	−2.34315	−0.310357
$58$	0	0
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	0	0
$63$	−4.48528	−0.565092
$64$	0	0
$65$	3.82843	0.474858
$66$	0	0
$67$	3.17157	0.387469	0.193735	−	0.981054i	$-0.437940\pi$
$67$	3.17157	0.387469	0.193735	+	0.981054i	$0.437940\pi$
$68$	0	0
$69$	−1.31371	−0.158152
$70$	0	0
$71$	13.7279	1.62920	0.814602	−	0.580020i	$-0.196955\pi$
$71$	13.7279	1.62920	0.814602	+	0.580020i	$0.196955\pi$
$72$	0	0
$73$	−9.65685	−1.13025	−0.565125	−	0.825006i	$-0.691172\pi$
$73$	−9.65685	−1.13025	−0.565125	+	0.825006i	$0.691172\pi$
$74$	0	0
$75$	−4.00000	−0.461880
$76$	0	0
$77$	7.65685	0.872580
$78$	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
$83$	−16.1421	−1.77183	−0.885915	−	0.463848i	$-0.846468\pi$
$83$	−16.1421	−1.77183	−0.885915	+	0.463848i	$0.846468\pi$
$84$	0	0
$85$	3.82843	0.415251
$86$	0	0
$87$	3.17157	0.340028
$88$	0	0
$89$	2.34315	0.248373	0.124186	−	0.992259i	$-0.460368\pi$
$89$	2.34315	0.248373	0.124186	+	0.992259i	$0.460368\pi$
$90$	0	0
$91$	1.58579	0.166236
$92$	0	0
$93$	3.17157	0.328877
$94$	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
$99$	−13.6569	−1.37257
$100$	0	0
$101$	−6.34315	−0.631167	−0.315583	−	0.948898i	$-0.602200\pi$
$101$	−6.34315	−0.631167	−0.315583	+	0.948898i	$0.602200\pi$
$102$	0	0
$103$	−12.1421	−1.19640	−0.598200	−	0.801347i	$-0.704117\pi$
$103$	−12.1421	−1.19640	−0.598200	+	0.801347i	$0.704117\pi$
$104$	0	0
$105$	−2.51472	−0.245411
$106$	0	0
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	0	0
$109$	−14.6569	−1.40387	−0.701936	−	0.712240i	$-0.747681\pi$
$109$	−14.6569	−1.40387	−0.701936	+	0.712240i	$0.747681\pi$
$110$	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
$115$	12.1421	1.13226
$116$	0	0
$117$	−2.82843	−0.261488
$118$	0	0
$119$	1.58579	0.145369
$120$	0	0
$121$	12.3137	1.11943
$122$	0	0
$123$	0.686292	0.0618808
$124$	0	0
$125$	17.8284	1.59462
$126$	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
$131$	−5.58579	−0.488032	−0.244016	−	0.969771i	$-0.578465\pi$
$131$	−5.58579	−0.488032	−0.244016	+	0.969771i	$0.578465\pi$
$132$	0	0
$133$	8.97056	0.777846
$134$	0	0
$135$	9.24264	0.795480
$136$	0	0
$137$	−15.6569	−1.33766	−0.668828	−	0.743417i	$-0.733204\pi$
$137$	−15.6569	−1.33766	−0.668828	+	0.743417i	$0.733204\pi$
$138$	0	0
$139$	4.41421	0.374409	0.187204	−	0.982321i	$-0.440057\pi$
$139$	4.41421	0.374409	0.187204	+	0.982321i	$0.440057\pi$
$140$	0	0
$141$	−3.82843	−0.322412
$142$	0	0
$143$	4.82843	0.403773
$144$	0	0
$145$	−29.3137	−2.43437
$146$	0	0
$147$	1.85786	0.153234
$148$	0	0
$149$	1.31371	0.107623	0.0538116	−	0.998551i	$-0.482863\pi$
$149$	1.31371	0.107623	0.0538116	+	0.998551i	$0.482863\pi$
$150$	0	0
$151$	−9.24264	−0.752155	−0.376078	−	0.926588i	$-0.622727\pi$
$151$	−9.24264	−0.752155	−0.376078	+	0.926588i	$0.622727\pi$
$152$	0	0
$153$	−2.82843	−0.228665
$154$	0	0
$155$	−29.3137	−2.35453
$156$	0	0
$157$	21.6569	1.72841	0.864203	−	0.503144i	$-0.167823\pi$
$157$	21.6569	1.72841	0.864203	+	0.503144i	$0.167823\pi$
$158$	0	0
$159$	3.17157	0.251522
$160$	0	0
$161$	5.02944	0.396375
$162$	0	0
$163$	2.48528	0.194662	0.0973311	−	0.995252i	$-0.468969\pi$
$163$	2.48528	0.194662	0.0973311	+	0.995252i	$0.468969\pi$
$164$	0	0
$165$	−7.65685	−0.596085
$166$	0	0
$167$	−7.65685	−0.592505	−0.296253	−	0.955110i	$-0.595737\pi$
$167$	−7.65685	−0.592505	−0.296253	+	0.955110i	$0.595737\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−16.0000	−1.22355
$172$	0	0
$173$	14.0000	1.06440	0.532200	−	0.846619i	$-0.321365\pi$
$173$	14.0000	1.06440	0.532200	+	0.846619i	$0.321365\pi$
$174$	0	0
$175$	15.3137	1.15761
$176$	0	0
$177$	−3.31371	−0.249074
$178$	0	0
$179$	−6.89949	−0.515692	−0.257846	−	0.966186i	$-0.583013\pi$
$179$	−6.89949	−0.515692	−0.257846	+	0.966186i	$0.583013\pi$
$180$	0	0
$181$	−7.65685	−0.569129	−0.284565	−	0.958657i	$-0.591849\pi$
$181$	−7.65685	−0.569129	−0.284565	+	0.958657i	$0.591849\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	26.7990	1.97030
$186$	0	0
$187$	4.82843	0.353090
$188$	0	0
$189$	3.82843	0.278477
$190$	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
$195$	−1.58579	−0.113561
$196$	0	0
$197$	−7.00000	−0.498729	−0.249365	−	0.968410i	$-0.580222\pi$
$197$	−7.00000	−0.498729	−0.249365	+	0.968410i	$0.580222\pi$
$198$	0	0
$199$	18.4853	1.31039	0.655193	−	0.755461i	$-0.272587\pi$
$199$	18.4853	1.31039	0.655193	+	0.755461i	$0.272587\pi$
$200$	0	0
$201$	−1.31371	−0.0926619
$202$	0	0
$203$	−12.1421	−0.852211
$204$	0	0
$205$	−6.34315	−0.443025
$206$	0	0
$207$	−8.97056	−0.623497
$208$	0	0
$209$	27.3137	1.88933
$210$	0	0
$211$	−1.58579	−0.109170	−0.0545850	−	0.998509i	$-0.517384\pi$
$211$	−1.58579	−0.109170	−0.0545850	+	0.998509i	$0.517384\pi$
$212$	0	0
$213$	−5.68629	−0.389618
$214$	0	0
$215$	−21.3848	−1.45843
$216$	0	0
$217$	−12.1421	−0.824262
$218$	0	0
$219$	4.00000	0.270295
$220$	0	0
$221$	1.00000	0.0672673
$222$	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
$227$	−20.9706	−1.39187	−0.695933	−	0.718107i	$-0.745009\pi$
$227$	−20.9706	−1.39187	−0.695933	+	0.718107i	$0.745009\pi$
$228$	0	0
$229$	2.51472	0.166177	0.0830886	−	0.996542i	$-0.473522\pi$
$229$	2.51472	0.166177	0.0830886	+	0.996542i	$0.473522\pi$
$230$	0	0
$231$	−3.17157	−0.208674
$232$	0	0
$233$	12.1716	0.797386	0.398693	−	0.917084i	$-0.369464\pi$
$233$	12.1716	0.797386	0.398693	+	0.917084i	$0.369464\pi$
$234$	0	0
$235$	35.3848	2.30825
$236$	0	0
$237$	−5.02944	−0.326697
$238$	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
$243$	−10.3431	−0.663513
$244$	0	0
$245$	−17.1716	−1.09705
$246$	0	0
$247$	5.65685	0.359937
$248$	0	0
$249$	6.68629	0.423727
$250$	0	0
$251$	9.31371	0.587876	0.293938	−	0.955824i	$-0.405034\pi$
$251$	9.31371	0.587876	0.293938	+	0.955824i	$0.405034\pi$
$252$	0	0
$253$	15.3137	0.962765
$254$	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
$259$	11.1005	0.689752
$260$	0	0
$261$	21.6569	1.34053
$262$	0	0
$263$	−6.34315	−0.391135	−0.195568	−	0.980690i	$-0.562655\pi$
$263$	−6.34315	−0.391135	−0.195568	+	0.980690i	$0.562655\pi$
$264$	0	0
$265$	−29.3137	−1.80073
$266$	0	0
$267$	−0.970563	−0.0593975
$268$	0	0
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	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
$275$	46.6274	2.81174
$276$	0	0
$277$	8.97056	0.538989	0.269494	−	0.963002i	$-0.413143\pi$
$277$	8.97056	0.538989	0.269494	+	0.963002i	$0.413143\pi$
$278$	0	0
$279$	21.6569	1.29656
$280$	0	0
$281$	−4.00000	−0.238620	−0.119310	−	0.992857i	$-0.538068\pi$
$281$	−4.00000	−0.238620	−0.119310	+	0.992857i	$0.538068\pi$
$282$	0	0
$283$	30.2843	1.80021	0.900107	−	0.435670i	$-0.143488\pi$
$283$	30.2843	1.80021	0.900107	+	0.435670i	$0.143488\pi$
$284$	0	0
$285$	−8.97056	−0.531370
$286$	0	0
$287$	−2.62742	−0.155092
$288$	0	0
$289$	−16.0000	−0.941176
$290$	0	0
$291$	−1.51472	−0.0887944
$292$	0	0
$293$	−14.6569	−0.856263	−0.428131	−	0.903717i	$-0.640828\pi$
$293$	−14.6569	−0.856263	−0.428131	+	0.903717i	$0.640828\pi$
$294$	0	0
$295$	30.6274	1.78320
$296$	0	0
$297$	11.6569	0.676399
$298$	0	0
$299$	3.17157	0.183417
$300$	0	0
$301$	−8.85786	−0.510559
$302$	0	0
$303$	2.62742	0.150941
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−20.0000	−1.14146	−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000	−1.14146	−0.570730	+	0.821138i	$0.693340\pi$
$308$	0	0
$309$	5.02944	0.286115
$310$	0	0
$311$	−21.6569	−1.22805	−0.614024	−	0.789288i	$-0.710450\pi$
$311$	−21.6569	−1.22805	−0.614024	+	0.789288i	$0.710450\pi$
$312$	0	0
$313$	−4.85786	−0.274583	−0.137291	−	0.990531i	$-0.543840\pi$
$313$	−4.85786	−0.274583	−0.137291	+	0.990531i	$0.543840\pi$
$314$	0	0
$315$	−17.1716	−0.967509
$316$	0	0
$317$	16.6274	0.933889	0.466944	−	0.884287i	$-0.345355\pi$
$317$	16.6274	0.933889	0.466944	+	0.884287i	$0.345355\pi$
$318$	0	0
$319$	−36.9706	−2.06995
$320$	0	0
$321$	2.48528	0.138715
$322$	0	0
$323$	5.65685	0.314756
$324$	0	0
$325$	9.65685	0.535666
$326$	0	0
$327$	6.07107	0.335731
$328$	0	0
$329$	14.6569	0.808059
$330$	0	0
$331$	8.82843	0.485254	0.242627	−	0.970120i	$-0.421991\pi$
$331$	8.82843	0.485254	0.242627	+	0.970120i	$0.421991\pi$
$332$	0	0
$333$	−19.7990	−1.08498
$334$	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
$339$	4.14214	0.224970
$340$	0	0
$341$	−36.9706	−2.00207
$342$	0	0
$343$	−18.2132	−0.983421
$344$	0	0
$345$	−5.02944	−0.270776
$346$	0	0
$347$	−29.2426	−1.56983	−0.784914	−	0.619605i	$-0.787293\pi$
$347$	−29.2426	−1.56983	−0.784914	+	0.619605i	$0.787293\pi$
$348$	0	0
$349$	3.82843	0.204931	0.102466	−	0.994737i	$-0.467327\pi$
$349$	3.82843	0.204931	0.102466	+	0.994737i	$0.467327\pi$
$350$	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
$355$	52.5563	2.78940
$356$	0	0
$357$	−0.656854	−0.0347644
$358$	0	0
$359$	35.6569	1.88190	0.940948	−	0.338550i	$-0.109936\pi$
$359$	35.6569	1.88190	0.940948	+	0.338550i	$0.109936\pi$
$360$	0	0
$361$	13.0000	0.684211
$362$	0	0
$363$	−5.10051	−0.267707
$364$	0	0
$365$	−36.9706	−1.93513
$366$	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
$371$	−12.1421	−0.630388
$372$	0	0
$373$	30.6274	1.58583	0.792914	−	0.609334i	$-0.208563\pi$
$373$	30.6274	1.58583	0.792914	+	0.609334i	$0.208563\pi$
$374$	0	0
$375$	−7.38478	−0.381348
$376$	0	0
$377$	−7.65685	−0.394348
$378$	0	0
$379$	−14.4853	−0.744059	−0.372029	−	0.928221i	$-0.621338\pi$
$379$	−14.4853	−0.744059	−0.372029	+	0.928221i	$0.621338\pi$
$380$	0	0
$381$	−1.31371	−0.0673033
$382$	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
$387$	15.7990	0.803108
$388$	0	0
$389$	28.0000	1.41966	0.709828	−	0.704375i	$-0.248773\pi$
$389$	28.0000	1.41966	0.709828	+	0.704375i	$0.248773\pi$
$390$	0	0
$391$	3.17157	0.160393
$392$	0	0
$393$	2.31371	0.116711
$394$	0	0
$395$	46.4853	2.33893
$396$	0	0
$397$	−29.3137	−1.47121	−0.735606	−	0.677409i	$-0.763103\pi$
$397$	−29.3137	−1.47121	−0.735606	+	0.677409i	$0.763103\pi$
$398$	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
$403$	−7.65685	−0.381415
$404$	0	0
$405$	28.6569	1.42397
$406$	0	0
$407$	33.7990	1.67535
$408$	0	0
$409$	−27.6569	−1.36754	−0.683772	−	0.729696i	$-0.739662\pi$
$409$	−27.6569	−1.36754	−0.683772	+	0.729696i	$0.739662\pi$
$410$	0	0
$411$	6.48528	0.319895
$412$	0	0
$413$	12.6863	0.624252
$414$	0	0
$415$	−61.7990	−3.03359
$416$	0	0
$417$	−1.82843	−0.0895385
$418$	0	0
$419$	5.24264	0.256120	0.128060	−	0.991766i	$-0.459125\pi$
$419$	5.24264	0.256120	0.128060	+	0.991766i	$0.459125\pi$
$420$	0	0
$421$	−26.7990	−1.30610	−0.653051	−	0.757314i	$-0.726511\pi$
$421$	−26.7990	−1.30610	−0.653051	+	0.757314i	$0.726511\pi$
$422$	0	0
$423$	−26.1421	−1.27107
$424$	0	0
$425$	9.65685	0.468426
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−2.00000	−0.0965609
$430$	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
$435$	12.1421	0.582171
$436$	0	0
$437$	17.9411	0.858240
$438$	0	0
$439$	24.2843	1.15903	0.579513	−	0.814963i	$-0.303243\pi$
$439$	24.2843	1.15903	0.579513	+	0.814963i	$0.303243\pi$
$440$	0	0
$441$	12.6863	0.604109
$442$	0	0
$443$	7.92893	0.376715	0.188357	−	0.982101i	$-0.439684\pi$
$443$	7.92893	0.376715	0.188357	+	0.982101i	$0.439684\pi$
$444$	0	0
$445$	8.97056	0.425245
$446$	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
$451$	−8.00000	−0.376705
$452$	0	0
$453$	3.82843	0.179875
$454$	0	0
$455$	6.07107	0.284616
$456$	0	0
$457$	37.6569	1.76151	0.880757	−	0.473569i	$-0.157035\pi$
$457$	37.6569	1.76151	0.880757	+	0.473569i	$0.157035\pi$
$458$	0	0
$459$	2.41421	0.112686
$460$	0	0
$461$	0.656854	0.0305928	0.0152964	−	0.999883i	$-0.495131\pi$
$461$	0.656854	0.0305928	0.0152964	+	0.999883i	$0.495131\pi$
$462$	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
$467$	2.00000	0.0925490	0.0462745	−	0.998929i	$-0.485265\pi$
$467$	2.00000	0.0925490	0.0462745	+	0.998929i	$0.485265\pi$
$468$	0	0
$469$	5.02944	0.232238
$470$	0	0
$471$	−8.97056	−0.413342
$472$	0	0
$473$	−26.9706	−1.24011
$474$	0	0
$475$	54.6274	2.50648
$476$	0	0
$477$	21.6569	0.991599
$478$	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
$483$	−2.08326	−0.0947917
$484$	0	0
$485$	14.0000	0.635707
$486$	0	0
$487$	−22.9706	−1.04090	−0.520448	−	0.853894i	$-0.674235\pi$
$487$	−22.9706	−1.04090	−0.520448	+	0.853894i	$0.674235\pi$
$488$	0	0
$489$	−1.02944	−0.0465528
$490$	0	0
$491$	−25.8701	−1.16750	−0.583750	−	0.811934i	$-0.698415\pi$
$491$	−25.8701	−1.16750	−0.583750	+	0.811934i	$0.698415\pi$
$492$	0	0
$493$	−7.65685	−0.344847
$494$	0	0
$495$	−52.2843	−2.35000
$496$	0	0
$497$	21.7696	0.976498
$498$	0	0
$499$	−21.6569	−0.969494	−0.484747	−	0.874654i	$-0.661088\pi$
$499$	−21.6569	−0.969494	−0.484747	+	0.874654i	$0.661088\pi$
$500$	0	0
$501$	3.17157	0.141695
$502$	0	0
$503$	−6.34315	−0.282827	−0.141413	−	0.989951i	$-0.545165\pi$
$503$	−6.34315	−0.282827	−0.141413	+	0.989951i	$0.545165\pi$
$504$	0	0
$505$	−24.2843	−1.08064
$506$	0	0
$507$	−0.414214	−0.0183959
$508$	0	0
$509$	−1.31371	−0.0582291	−0.0291146	−	0.999576i	$-0.509269\pi$
$509$	−1.31371	−0.0582291	−0.0291146	+	0.999576i	$0.509269\pi$
$510$	0	0
$511$	−15.3137	−0.677439
$512$	0	0
$513$	13.6569	0.602965
$514$	0	0
$515$	−46.4853	−2.04839
$516$	0	0
$517$	44.6274	1.96271
$518$	0	0
$519$	−5.79899	−0.254547
$520$	0	0
$521$	−41.1421	−1.80247	−0.901235	−	0.433331i	$-0.857338\pi$
$521$	−41.1421	−1.80247	−0.901235	+	0.433331i	$0.857338\pi$
$522$	0	0
$523$	−25.3137	−1.10689	−0.553446	−	0.832885i	$-0.686687\pi$
$523$	−25.3137	−1.10689	−0.553446	+	0.832885i	$0.686687\pi$
$524$	0	0
$525$	−6.34315	−0.276838
$526$	0	0
$527$	−7.65685	−0.333538
$528$	0	0
$529$	−12.9411	−0.562658
$530$	0	0
$531$	−22.6274	−0.981946
$532$	0	0
$533$	−1.65685	−0.0717663
$534$	0	0
$535$	−22.9706	−0.993104
$536$	0	0
$537$	2.85786	0.123326
$538$	0	0
$539$	−21.6569	−0.932827
$540$	0	0
$541$	−7.00000	−0.300954	−0.150477	−	0.988614i	$-0.548081\pi$
$541$	−7.00000	−0.300954	−0.150477	+	0.988614i	$0.548081\pi$
$542$	0	0
$543$	3.17157	0.136105
$544$	0	0
$545$	−56.1127	−2.40360
$546$	0	0
$547$	−30.0711	−1.28575	−0.642873	−	0.765973i	$-0.722258\pi$
$547$	−30.0711	−1.28575	−0.642873	+	0.765973i	$0.722258\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−43.3137	−1.84523
$552$	0	0
$553$	19.2548	0.818799
$554$	0	0
$555$	−11.1005	−0.471190
$556$	0	0
$557$	1.97056	0.0834954	0.0417477	−	0.999128i	$-0.486707\pi$
$557$	1.97056	0.0834954	0.0417477	+	0.999128i	$0.486707\pi$
$558$	0	0
$559$	−5.58579	−0.236254
$560$	0	0
$561$	−2.00000	−0.0844401
$562$	0	0
$563$	31.0416	1.30825	0.654124	−	0.756387i	$-0.273037\pi$
$563$	31.0416	1.30825	0.654124	+	0.756387i	$0.273037\pi$
$564$	0	0
$565$	−38.2843	−1.61063
$566$	0	0
$567$	11.8701	0.498496
$568$	0	0
$569$	−14.6569	−0.614447	−0.307224	−	0.951637i	$-0.599400\pi$
$569$	−14.6569	−0.614447	−0.307224	+	0.951637i	$0.599400\pi$
$570$	0	0
$571$	−35.7279	−1.49517	−0.747584	−	0.664168i	$-0.768786\pi$
$571$	−35.7279	−1.49517	−0.747584	+	0.664168i	$0.768786\pi$
$572$	0	0
$573$	−10.2843	−0.429632
$574$	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
$579$	−8.82843	−0.366897
$580$	0	0
$581$	−25.5980	−1.06198
$582$	0	0
$583$	−36.9706	−1.53116
$584$	0	0
$585$	−10.8284	−0.447700
$586$	0	0
$587$	−1.51472	−0.0625191	−0.0312596	−	0.999511i	$-0.509952\pi$
$587$	−1.51472	−0.0625191	−0.0312596	+	0.999511i	$0.509952\pi$
$588$	0	0
$589$	−43.3137	−1.78471
$590$	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
$595$	6.07107	0.248890
$596$	0	0
$597$	−7.65685	−0.313374
$598$	0	0
$599$	15.3137	0.625701	0.312851	−	0.949802i	$-0.398716\pi$
$599$	15.3137	0.625701	0.312851	+	0.949802i	$0.398716\pi$
$600$	0	0
$601$	−11.9706	−0.488289	−0.244145	−	0.969739i	$-0.578507\pi$
$601$	−11.9706	−0.488289	−0.244145	+	0.969739i	$0.578507\pi$
$602$	0	0
$603$	−8.97056	−0.365310
$604$	0	0
$605$	47.1421	1.91660
$606$	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
$611$	9.24264	0.373917
$612$	0	0
$613$	−44.6274	−1.80248	−0.901242	−	0.433316i	$-0.857344\pi$
$613$	−44.6274	−1.80248	−0.901242	+	0.433316i	$0.857344\pi$
$614$	0	0
$615$	2.62742	0.105948
$616$	0	0
$617$	49.2548	1.98292	0.991462	−	0.130392i	$-0.0416237\pi$
$617$	49.2548	1.98292	0.991462	+	0.130392i	$0.0416237\pi$
$618$	0	0
$619$	−37.6569	−1.51356	−0.756778	−	0.653672i	$-0.773228\pi$
$619$	−37.6569	−1.51356	−0.756778	+	0.653672i	$0.773228\pi$
$620$	0	0
$621$	7.65685	0.307259
$622$	0	0
$623$	3.71573	0.148868
$624$	0	0
$625$	19.9706	0.798823
$626$	0	0
$627$	−11.3137	−0.451826
$628$	0	0
$629$	7.00000	0.279108
$630$	0	0
$631$	−6.61522	−0.263348	−0.131674	−	0.991293i	$-0.542035\pi$
$631$	−6.61522	−0.263348	−0.131674	+	0.991293i	$0.542035\pi$
$632$	0	0
$633$	0.656854	0.0261076
$634$	0	0
$635$	12.1421	0.481846
$636$	0	0
$637$	−4.48528	−0.177713
$638$	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
$643$	23.1716	0.913798	0.456899	−	0.889519i	$-0.348960\pi$
$643$	23.1716	0.913798	0.456899	+	0.889519i	$0.348960\pi$
$644$	0	0
$645$	8.85786	0.348778
$646$	0	0
$647$	−8.97056	−0.352669	−0.176335	−	0.984330i	$-0.556424\pi$
$647$	−8.97056	−0.352669	−0.176335	+	0.984330i	$0.556424\pi$
$648$	0	0
$649$	38.6274	1.51626
$650$	0	0
$651$	5.02944	0.197119
$652$	0	0
$653$	21.6569	0.847498	0.423749	−	0.905780i	$-0.360714\pi$
$653$	21.6569	0.847498	0.423749	+	0.905780i	$0.360714\pi$
$654$	0	0
$655$	−21.3848	−0.835572
$656$	0	0
$657$	27.3137	1.06561
$658$	0	0
$659$	−35.9411	−1.40007	−0.700034	−	0.714110i	$-0.746832\pi$
$659$	−35.9411	−1.40007	−0.700034	+	0.714110i	$0.746832\pi$
$660$	0	0
$661$	−29.3137	−1.14017	−0.570086	−	0.821585i	$-0.693090\pi$
$661$	−29.3137	−1.14017	−0.570086	+	0.821585i	$0.693090\pi$
$662$	0	0
$663$	−0.414214	−0.0160867
$664$	0	0
$665$	34.3431	1.33177
$666$	0	0
$667$	−24.2843	−0.940291
$668$	0	0
$669$	−5.14214	−0.198806
$670$	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
$675$	23.3137	0.897345
$676$	0	0
$677$	10.2843	0.395257	0.197628	−	0.980277i	$-0.436676\pi$
$677$	10.2843	0.395257	0.197628	+	0.980277i	$0.436676\pi$
$678$	0	0
$679$	5.79899	0.222545
$680$	0	0
$681$	8.68629	0.332859
$682$	0	0
$683$	3.17157	0.121357	0.0606784	−	0.998157i	$-0.480674\pi$
$683$	3.17157	0.121357	0.0606784	+	0.998157i	$0.480674\pi$
$684$	0	0
$685$	−59.9411	−2.29023
$686$	0	0
$687$	−1.04163	−0.0397407
$688$	0	0
$689$	−7.65685	−0.291703
$690$	0	0
$691$	34.6274	1.31729	0.658645	−	0.752454i	$-0.271130\pi$
$691$	34.6274	1.31729	0.658645	+	0.752454i	$0.271130\pi$
$692$	0	0
$693$	−21.6569	−0.822676
$694$	0	0
$695$	16.8995	0.641034
$696$	0	0
$697$	−1.65685	−0.0627578
$698$	0	0
$699$	−5.04163	−0.190692
$700$	0	0
$701$	14.0000	0.528773	0.264386	−	0.964417i	$-0.414831\pi$
$701$	14.0000	0.528773	0.264386	+	0.964417i	$0.414831\pi$
$702$	0	0
$703$	39.5980	1.49347
$704$	0	0
$705$	−14.6569	−0.552009
$706$	0	0
$707$	−10.0589	−0.378303
$708$	0	0
$709$	29.3137	1.10090	0.550450	−	0.834868i	$-0.314456\pi$
$709$	29.3137	1.10090	0.550450	+	0.834868i	$0.314456\pi$
$710$	0	0
$711$	−34.3431	−1.28797
$712$	0	0
$713$	−24.2843	−0.909453
$714$	0	0
$715$	18.4853	0.691310
$716$	0	0
$717$	−3.28427	−0.122653
$718$	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
$723$	−8.68629	−0.323047
$724$	0	0
$725$	−73.9411	−2.74610
$726$	0	0
$727$	−31.1716	−1.15609	−0.578045	−	0.816005i	$-0.696184\pi$
$727$	−31.1716	−1.15609	−0.578045	+	0.816005i	$0.696184\pi$
$728$	0	0
$729$	−18.1716	−0.673021
$730$	0	0
$731$	−5.58579	−0.206598
$732$	0	0
$733$	30.5147	1.12709	0.563543	−	0.826086i	$-0.309438\pi$
$733$	30.5147	1.12709	0.563543	+	0.826086i	$0.309438\pi$
$734$	0	0
$735$	7.11270	0.262356
$736$	0	0
$737$	15.3137	0.564088
$738$	0	0
$739$	−7.31371	−0.269039	−0.134520	−	0.990911i	$-0.542949\pi$
$739$	−7.31371	−0.269039	−0.134520	+	0.990911i	$0.542949\pi$
$740$	0	0
$741$	−2.34315	−0.0860776
$742$	0	0
$743$	0.272078	0.00998157	0.00499079	−	0.999988i	$-0.498411\pi$
$743$	0.272078	0.00998157	0.00499079	+	0.999988i	$0.498411\pi$
$744$	0	0
$745$	5.02944	0.184264
$746$	0	0
$747$	45.6569	1.67050
$748$	0	0
$749$	−9.51472	−0.347660
$750$	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
$755$	−35.3848	−1.28778
$756$	0	0
$757$	−16.6274	−0.604334	−0.302167	−	0.953255i	$-0.597710\pi$
$757$	−16.6274	−0.604334	−0.302167	+	0.953255i	$0.597710\pi$
$758$	0	0
$759$	−6.34315	−0.230242
$760$	0	0
$761$	−0.686292	−0.0248780	−0.0124390	−	0.999923i	$-0.503960\pi$
$761$	−0.686292	−0.0248780	−0.0124390	+	0.999923i	$0.503960\pi$
$762$	0	0
$763$	−23.2426	−0.841440
$764$	0	0
$765$	−10.8284	−0.391503
$766$	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
$771$	−4.69848	−0.169212
$772$	0	0
$773$	12.7990	0.460348	0.230174	−	0.973150i	$-0.426071\pi$
$773$	12.7990	0.460348	0.230174	+	0.973150i	$0.426071\pi$
$774$	0	0
$775$	−73.9411	−2.65604
$776$	0	0
$777$	−4.59798	−0.164952
$778$	0	0
$779$	−9.37258	−0.335808
$780$	0	0
$781$	66.2843	2.37184
$782$	0	0
$783$	−18.4853	−0.660610
$784$	0	0
$785$	82.9117	2.95925
$786$	0	0
$787$	26.3431	0.939032	0.469516	−	0.882924i	$-0.344428\pi$
$787$	26.3431	0.939032	0.469516	+	0.882924i	$0.344428\pi$
$788$	0	0
$789$	2.62742	0.0935385
$790$	0	0
$791$	−15.8579	−0.563841
$792$	0	0
$793$	0	0
$794$	0	0
$795$	12.1421	0.430637
$796$	0	0
$797$	24.2843	0.860193	0.430097	−	0.902783i	$-0.358480\pi$
$797$	24.2843	0.860193	0.430097	+	0.902783i	$0.358480\pi$
$798$	0	0
$799$	9.24264	0.326981
$800$	0	0
$801$	−6.62742	−0.234168
$802$	0	0
$803$	−46.6274	−1.64545
$804$	0	0
$805$	19.2548	0.678644
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−10.5147	−0.369678	−0.184839	−	0.982769i	$-0.559176\pi$
$809$	−10.5147	−0.369678	−0.184839	+	0.982769i	$0.559176\pi$
$810$	0	0
$811$	−4.97056	−0.174540	−0.0872700	−	0.996185i	$-0.527814\pi$
$811$	−4.97056	−0.174540	−0.0872700	+	0.996185i	$0.527814\pi$
$812$	0	0
$813$	5.14214	0.180343
$814$	0	0
$815$	9.51472	0.333286
$816$	0	0
$817$	−31.5980	−1.10547
$818$	0	0
$819$	−4.48528	−0.156728
$820$	0	0
$821$	28.1127	0.981140	0.490570	−	0.871402i	$-0.336789\pi$
$821$	28.1127	0.981140	0.490570	+	0.871402i	$0.336789\pi$
$822$	0	0
$823$	−28.0000	−0.976019	−0.488009	−	0.872838i	$-0.662277\pi$
$823$	−28.0000	−0.976019	−0.488009	+	0.872838i	$0.662277\pi$
$824$	0	0
$825$	−19.3137	−0.672417
$826$	0	0
$827$	33.9411	1.18025	0.590124	−	0.807312i	$-0.299079\pi$
$827$	33.9411	1.18025	0.590124	+	0.807312i	$0.299079\pi$
$828$	0	0
$829$	−22.9706	−0.797801	−0.398900	−	0.916994i	$-0.630608\pi$
$829$	−22.9706	−0.797801	−0.398900	+	0.916994i	$0.630608\pi$
$830$	0	0
$831$	−3.71573	−0.128897
$832$	0	0
$833$	−4.48528	−0.155406
$834$	0	0
$835$	−29.3137	−1.01444
$836$	0	0
$837$	−18.4853	−0.638945
$838$	0	0
$839$	5.02944	0.173635	0.0868177	−	0.996224i	$-0.472330\pi$
$839$	5.02944	0.173635	0.0868177	+	0.996224i	$0.472330\pi$
$840$	0	0
$841$	29.6274	1.02164
$842$	0	0
$843$	1.65685	0.0570651
$844$	0	0
$845$	3.82843	0.131702
$846$	0	0
$847$	19.5269	0.670953
$848$	0	0
$849$	−12.5442	−0.430514
$850$	0	0
$851$	22.2010	0.761041
$852$	0	0
$853$	−34.4558	−1.17975	−0.589873	−	0.807496i	$-0.700822\pi$
$853$	−34.4558	−1.17975	−0.589873	+	0.807496i	$0.700822\pi$
$854$	0	0
$855$	−61.2548	−2.09487
$856$	0	0
$857$	−45.3137	−1.54789	−0.773943	−	0.633255i	$-0.781719\pi$
$857$	−45.3137	−1.54789	−0.773943	+	0.633255i	$0.781719\pi$
$858$	0	0
$859$	−28.6274	−0.976755	−0.488377	−	0.872633i	$-0.662411\pi$
$859$	−28.6274	−0.976755	−0.488377	+	0.872633i	$0.662411\pi$
$860$	0	0
$861$	1.08831	0.0370896
$862$	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
$867$	6.62742	0.225079
$868$	0	0
$869$	58.6274	1.98880
$870$	0	0
$871$	3.17157	0.107465
$872$	0	0
$873$	−10.3431	−0.350062
$874$	0	0
$875$	28.2721	0.955771
$876$	0	0
$877$	5.14214	0.173638	0.0868188	−	0.996224i	$-0.472330\pi$
$877$	5.14214	0.173638	0.0868188	+	0.996224i	$0.472330\pi$
$878$	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
$883$	16.2132	0.545618	0.272809	−	0.962068i	$-0.412047\pi$
$883$	16.2132	0.545618	0.272809	+	0.962068i	$0.412047\pi$
$884$	0	0
$885$	−12.6863	−0.426445
$886$	0	0
$887$	21.6569	0.727166	0.363583	−	0.931562i	$-0.381553\pi$
$887$	21.6569	0.727166	0.363583	+	0.931562i	$0.381553\pi$
$888$	0	0
$889$	5.02944	0.168682
$890$	0	0
$891$	36.1421	1.21081
$892$	0	0
$893$	52.2843	1.74963
$894$	0	0
$895$	−26.4142	−0.882930
$896$	0	0
$897$	−1.31371	−0.0438635
$898$	0	0
$899$	58.6274	1.95533
$900$	0	0
$901$	−7.65685	−0.255087
$902$	0	0
$903$	3.66905	0.122098
$904$	0	0
$905$	−29.3137	−0.974421
$906$	0	0
$907$	−42.5563	−1.41306	−0.706530	−	0.707683i	$-0.749741\pi$
$907$	−42.5563	−1.41306	−0.706530	+	0.707683i	$0.749741\pi$
$908$	0	0
$909$	17.9411	0.595070
$910$	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
$917$	−8.85786	−0.292512
$918$	0	0
$919$	34.3431	1.13288	0.566438	−	0.824104i	$-0.308321\pi$
$919$	34.3431	1.13288	0.566438	+	0.824104i	$0.308321\pi$
$920$	0	0
$921$	8.28427	0.272976
$922$	0	0
$923$	13.7279	0.451860
$924$	0	0
$925$	67.5980	2.22261
$926$	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
$931$	−25.3726	−0.831553
$932$	0	0
$933$	8.97056	0.293683
$934$	0	0
$935$	18.4853	0.604533
$936$	0	0
$937$	37.3137	1.21899	0.609493	−	0.792792i	$-0.291373\pi$
$937$	37.3137	1.21899	0.609493	+	0.792792i	$0.291373\pi$
$938$	0	0
$939$	2.01219	0.0656654
$940$	0	0
$941$	−40.7990	−1.33001	−0.665005	−	0.746839i	$-0.731570\pi$
$941$	−40.7990	−1.33001	−0.665005	+	0.746839i	$0.731570\pi$
$942$	0	0
$943$	−5.25483	−0.171121
$944$	0	0
$945$	14.6569	0.476788
$946$	0	0
$947$	40.8284	1.32675	0.663373	−	0.748289i	$-0.269124\pi$
$947$	40.8284	1.32675	0.663373	+	0.748289i	$0.269124\pi$
$948$	0	0
$949$	−9.65685	−0.313475
$950$	0	0
$951$	−6.88730	−0.223336
$952$	0	0
$953$	24.9411	0.807922	0.403961	−	0.914776i	$-0.367633\pi$
$953$	24.9411	0.807922	0.403961	+	0.914776i	$0.367633\pi$
$954$	0	0
$955$	95.0538	3.07587
$956$	0	0
$957$	15.3137	0.495022
$958$	0	0
$959$	−24.8284	−0.801752
$960$	0	0
$961$	27.6274	0.891207
$962$	0	0
$963$	16.9706	0.546869
$964$	0	0
$965$	81.5980	2.62673
$966$	0	0
$967$	26.4142	0.849424	0.424712	−	0.905329i	$-0.360375\pi$
$967$	26.4142	0.849424	0.424712	+	0.905329i	$0.360375\pi$
$968$	0	0
$969$	−2.34315	−0.0752727
$970$	0	0
$971$	−23.1005	−0.741330	−0.370665	−	0.928767i	$-0.620870\pi$
$971$	−23.1005	−0.741330	−0.370665	+	0.928767i	$0.620870\pi$
$972$	0	0
$973$	7.00000	0.224410
$974$	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
$979$	11.3137	0.361588
$980$	0	0
$981$	41.4558	1.32358
$982$	0	0
$983$	1.58579	0.0505787	0.0252894	−	0.999680i	$-0.491949\pi$
$983$	1.58579	0.0505787	0.0252894	+	0.999680i	$0.491949\pi$
$984$	0	0
$985$	−26.7990	−0.853887
$986$	0	0
$987$	−6.07107	−0.193244
$988$	0	0
$989$	−17.7157	−0.563327
$990$	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
$995$	70.7696	2.24355
$996$	0	0
$997$	−42.0000	−1.33015	−0.665077	−	0.746775i	$-0.731601\pi$
$997$	−42.0000	−1.33015	−0.665077	+	0.746775i	$0.731601\pi$
$998$	0	0
$999$	16.8995	0.534676

Twists

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

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

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:	\( N \)	\(=\)	\( 3328 = 2^{8} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3328.a (trivial)

\(f(q)\)	\(=\)	\(q-0.414214 q^{3} +3.82843 q^{5} +1.58579 q^{7} -2.82843 q^{9} +4.82843 q^{11} +1.00000 q^{13} -1.58579 q^{15} +1.00000 q^{17} +5.65685 q^{19} -0.656854 q^{21} +3.17157 q^{23} +9.65685 q^{25} +2.41421 q^{27} -7.65685 q^{29} -7.65685 q^{31} -2.00000 q^{33} +6.07107 q^{35} +7.00000 q^{37} -0.414214 q^{39} -1.65685 q^{41} -5.58579 q^{43} -10.8284 q^{45} +9.24264 q^{47} -4.48528 q^{49} -0.414214 q^{51} -7.65685 q^{53} +18.4853 q^{55} -2.34315 q^{57} +8.00000 q^{59} -4.48528 q^{63} +3.82843 q^{65} +3.17157 q^{67} -1.31371 q^{69} +13.7279 q^{71} -9.65685 q^{73} -4.00000 q^{75} +7.65685 q^{77} +12.1421 q^{79} +7.48528 q^{81} -16.1421 q^{83} +3.82843 q^{85} +3.17157 q^{87} +2.34315 q^{89} +1.58579 q^{91} +3.17157 q^{93} +21.6569 q^{95} +3.65685 q^{97} -13.6569 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{5} + 6 q^{7} + 4 q^{11} + 2 q^{13} - 6 q^{15} + 2 q^{17} + 10 q^{21} + 12 q^{23} + 8 q^{25} + 2 q^{27} - 4 q^{29} - 4 q^{31} - 4 q^{33} - 2 q^{35} + 14 q^{37} + 2 q^{39} + 8 q^{41}+ \cdots - 16 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^5 + 6 * q^7 + 4 * q^11 + 2 * q^13 - 6 * q^15 + 2 * q^17 + 10 * q^21 + 12 * q^23 + 8 * q^25 + 2 * q^27 - 4 * q^29 - 4 * q^31 - 4 * q^33 - 2 * q^35 + 14 * q^37 + 2 * q^39 + 8 * q^41 - 14 * q^43 - 16 * q^45 + 10 * q^47 + 8 * q^49 + 2 * q^51 - 4 * q^53 + 20 * q^55 - 16 * q^57 + 16 * q^59 + 8 * q^63 + 2 * q^65 + 12 * q^67 + 20 * q^69 + 2 * q^71 - 8 * q^73 - 8 * q^75 + 4 * q^77 - 4 * q^79 - 2 * q^81 - 4 * q^83 + 2 * q^85 + 12 * q^87 + 16 * q^89 + 6 * q^91 + 12 * q^93 + 32 * q^95 - 4 * q^97 - 16 * q^99

Introduction

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