LMFDB - Embedded newform 6640.2.a.bb.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6640,2,Mod(1,6640)] 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("6640.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6640, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6640 = 2^{4} \cdot 5 \cdot 83 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6640.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,-3,0,-6,0,0,0,9,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$53.0206669421$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.7783241.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 5x^{4} + 9x^{3} + 5x^{2} - 6x - 1 $ x^6 - 2x^5 - 5x^4 + 9x^3 + 5x^2 - 6*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 415)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.92537$ of defining polynomial
Character	$\chi$	$=$	6640.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.431741 q^{3} -1.00000 q^{5} +3.01948 q^{7} -2.81360 q^{9} -0.386511 q^{11} +6.99372 q^{13} +0.431741 q^{15} +6.58773 q^{17} +5.57007 q^{19} -1.30363 q^{21} -2.83126 q^{23} +1.00000 q^{25} +2.50997 q^{27} +6.46422 q^{29} -3.65295 q^{31} +0.166873 q^{33} -3.01948 q^{35} -2.48887 q^{37} -3.01948 q^{39} +8.52322 q^{41} -6.95309 q^{43} +2.81360 q^{45} -1.08451 q^{47} +2.11723 q^{49} -2.84419 q^{51} +5.27625 q^{53} +0.386511 q^{55} -2.40483 q^{57} -4.70284 q^{59} +5.05383 q^{61} -8.49559 q^{63} -6.99372 q^{65} -4.19631 q^{67} +1.22237 q^{69} +10.3287 q^{71} -13.8414 q^{73} -0.431741 q^{75} -1.16706 q^{77} -7.75829 q^{79} +7.35714 q^{81} -1.00000 q^{83} -6.58773 q^{85} -2.79087 q^{87} +0.0634123 q^{89} +21.1174 q^{91} +1.57713 q^{93} -5.57007 q^{95} +6.29407 q^{97} +1.08749 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 3 q^{3} - 6 q^{5} + 9 q^{9} - 4 q^{11} + 7 q^{13} + 3 q^{15} + 21 q^{17} - 2 q^{19} - 9 q^{21} - 8 q^{23} + 6 q^{25} - 12 q^{27} + 10 q^{29} + 13 q^{31} + 6 q^{33} + q^{37} + 10 q^{41} + 3 q^{43}+ \cdots + 22 q^{99}+O(q^{100}) $ 6 * q - 3 * q^3 - 6 * q^5 + 9 * q^9 - 4 * q^11 + 7 * q^13 + 3 * q^15 + 21 * q^17 - 2 * q^19 - 9 * q^21 - 8 * q^23 + 6 * q^25 - 12 * q^27 + 10 * q^29 + 13 * q^31 + 6 * q^33 + q^37 + 10 * q^41 + 3 * q^43 - 9 * q^45 - 14 * q^47 - 12 * q^49 + 6 * q^51 + 9 * q^53 + 4 * q^55 + 19 * q^57 - 20 * q^59 - 13 * q^61 + 11 * q^63 - 7 * q^65 - 16 * q^67 - 11 * q^69 - 5 * q^71 + 3 * q^73 - 3 * q^75 + 30 * q^77 - 2 * q^79 + 2 * q^81 - 6 * q^83 - 21 * q^85 + 5 * q^87 + 45 * q^89 + 21 * q^91 - 27 * q^93 + 2 * q^95 + 22 * q^97 + 22 * 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
$3$	−0.431741	−0.249266	−0.124633	−	0.992203i	$-0.539775\pi$
$3$	−0.431741	−0.249266	−0.124633	+	0.992203i	$0.539775\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	3.01948	1.14125	0.570627	−	0.821209i	$-0.306700\pi$
$7$	3.01948	1.14125	0.570627	+	0.821209i	$0.306700\pi$
$8$	0	0
$9$	−2.81360	−0.937867
$10$	0	0
$11$	−0.386511	−0.116537	−0.0582687	−	0.998301i	$-0.518558\pi$
$11$	−0.386511	−0.116537	−0.0582687	+	0.998301i	$0.518558\pi$
$12$	0	0
$13$	6.99372	1.93971	0.969855	−	0.243685i	$-0.0783562\pi$
$13$	6.99372	1.93971	0.969855	+	0.243685i	$0.0783562\pi$
$14$	0	0
$15$	0.431741	0.111475
$16$	0	0
$17$	6.58773	1.59776	0.798880	−	0.601490i	$-0.205426\pi$
$17$	6.58773	1.59776	0.798880	+	0.601490i	$0.205426\pi$
$18$	0	0
$19$	5.57007	1.27786	0.638931	−	0.769264i	$-0.279377\pi$
$19$	5.57007	1.27786	0.638931	+	0.769264i	$0.279377\pi$
$20$	0	0
$21$	−1.30363	−0.284476
$22$	0	0
$23$	−2.83126	−0.590359	−0.295179	−	0.955442i	$-0.595379\pi$
$23$	−2.83126	−0.590359	−0.295179	+	0.955442i	$0.595379\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	2.50997	0.483044
$28$	0	0
$29$	6.46422	1.20038	0.600188	−	0.799859i	$-0.295092\pi$
$29$	6.46422	1.20038	0.600188	+	0.799859i	$0.295092\pi$
$30$	0	0
$31$	−3.65295	−0.656090	−0.328045	−	0.944662i	$-0.606390\pi$
$31$	−3.65295	−0.656090	−0.328045	+	0.944662i	$0.606390\pi$
$32$	0	0
$33$	0.166873	0.0290488
$34$	0	0
$35$	−3.01948	−0.510384
$36$	0	0
$37$	−2.48887	−0.409167	−0.204584	−	0.978849i	$-0.565584\pi$
$37$	−2.48887	−0.409167	−0.204584	+	0.978849i	$0.565584\pi$
$38$	0	0
$39$	−3.01948	−0.483503
$40$	0	0
$41$	8.52322	1.33110	0.665551	−	0.746352i	$-0.268196\pi$
$41$	8.52322	1.33110	0.665551	+	0.746352i	$0.268196\pi$
$42$	0	0
$43$	−6.95309	−1.06034	−0.530168	−	0.847893i	$-0.677871\pi$
$43$	−6.95309	−1.06034	−0.530168	+	0.847893i	$0.677871\pi$
$44$	0	0
$45$	2.81360	0.419427
$46$	0	0
$47$	−1.08451	−0.158192	−0.0790958	−	0.996867i	$-0.525203\pi$
$47$	−1.08451	−0.158192	−0.0790958	+	0.996867i	$0.525203\pi$
$48$	0	0
$49$	2.11723	0.302462
$50$	0	0
$51$	−2.84419	−0.398267
$52$	0	0
$53$	5.27625	0.724749	0.362374	−	0.932033i	$-0.381966\pi$
$53$	5.27625	0.724749	0.362374	+	0.932033i	$0.381966\pi$
$54$	0	0
$55$	0.386511	0.0521171
$56$	0	0
$57$	−2.40483	−0.318527
$58$	0	0
$59$	−4.70284	−0.612257	−0.306129	−	0.951990i	$-0.599034\pi$
$59$	−4.70284	−0.612257	−0.306129	+	0.951990i	$0.599034\pi$
$60$	0	0
$61$	5.05383	0.647076	0.323538	−	0.946215i	$-0.395128\pi$
$61$	5.05383	0.647076	0.323538	+	0.946215i	$0.395128\pi$
$62$	0	0
$63$	−8.49559	−1.07034
$64$	0	0
$65$	−6.99372	−0.867464
$66$	0	0
$67$	−4.19631	−0.512661	−0.256330	−	0.966589i	$-0.582514\pi$
$67$	−4.19631	−0.512661	−0.256330	+	0.966589i	$0.582514\pi$
$68$	0	0
$69$	1.22237	0.147156
$70$	0	0
$71$	10.3287	1.22579	0.612897	−	0.790163i	$-0.290004\pi$
$71$	10.3287	1.22579	0.612897	+	0.790163i	$0.290004\pi$
$72$	0	0
$73$	−13.8414	−1.62001	−0.810004	−	0.586424i	$-0.800535\pi$
$73$	−13.8414	−1.62001	−0.810004	+	0.586424i	$0.800535\pi$
$74$	0	0
$75$	−0.431741	−0.0498531
$76$	0	0
$77$	−1.16706	−0.132999
$78$	0	0
$79$	−7.75829	−0.872876	−0.436438	−	0.899734i	$-0.643760\pi$
$79$	−7.75829	−0.872876	−0.436438	+	0.899734i	$0.643760\pi$
$80$	0	0
$81$	7.35714	0.817460
$82$	0	0
$83$	−1.00000	−0.109764
$83$	−1.00000	−0.109764
$84$	0	0
$85$	−6.58773	−0.714540
$86$	0	0
$87$	−2.79087	−0.299213
$88$	0	0
$89$	0.0634123	0.00672169	0.00336084	−	0.999994i	$-0.498930\pi$
$89$	0.0634123	0.00672169	0.00336084	+	0.999994i	$0.498930\pi$
$90$	0	0
$91$	21.1174	2.21370
$92$	0	0
$93$	1.57713	0.163541
$94$	0	0
$95$	−5.57007	−0.571478
$96$	0	0
$97$	6.29407	0.639066	0.319533	−	0.947575i	$-0.396474\pi$
$97$	6.29407	0.639066	0.319533	+	0.947575i	$0.396474\pi$
$98$	0	0
$99$	1.08749	0.109297
$100$	0	0
$101$	−5.38701	−0.536028	−0.268014	−	0.963415i	$-0.586367\pi$
$101$	−5.38701	−0.536028	−0.268014	+	0.963415i	$0.586367\pi$
$102$	0	0
$103$	9.41240	0.927431	0.463716	−	0.885984i	$-0.346516\pi$
$103$	9.41240	0.927431	0.463716	+	0.885984i	$0.346516\pi$
$104$	0	0
$105$	1.30363	0.127221
$106$	0	0
$107$	−13.1270	−1.26903	−0.634517	−	0.772909i	$-0.718801\pi$
$107$	−13.1270	−1.26903	−0.634517	+	0.772909i	$0.718801\pi$
$108$	0	0
$109$	−16.5975	−1.58975	−0.794876	−	0.606773i	$-0.792464\pi$
$109$	−16.5975	−1.58975	−0.794876	+	0.606773i	$0.792464\pi$
$110$	0	0
$111$	1.07455	0.101991
$112$	0	0
$113$	−4.56004	−0.428972	−0.214486	−	0.976727i	$-0.568808\pi$
$113$	−4.56004	−0.428972	−0.214486	+	0.976727i	$0.568808\pi$
$114$	0	0
$115$	2.83126	0.264016
$116$	0	0
$117$	−19.6775	−1.81919
$118$	0	0
$119$	19.8915	1.82345
$120$	0	0
$121$	−10.8506	−0.986419
$122$	0	0
$123$	−3.67982	−0.331798
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	5.94008	0.527097	0.263549	−	0.964646i	$-0.415107\pi$
$127$	5.94008	0.527097	0.263549	+	0.964646i	$0.415107\pi$
$128$	0	0
$129$	3.00193	0.264306
$130$	0	0
$131$	13.6795	1.19518	0.597590	−	0.801802i	$-0.296125\pi$
$131$	13.6795	1.19518	0.597590	+	0.801802i	$0.296125\pi$
$132$	0	0
$133$	16.8187	1.45837
$134$	0	0
$135$	−2.50997	−0.216024
$136$	0	0
$137$	21.1218	1.80456	0.902278	−	0.431155i	$-0.141894\pi$
$137$	21.1218	1.80456	0.902278	+	0.431155i	$0.141894\pi$
$138$	0	0
$139$	−9.91452	−0.840939	−0.420469	−	0.907307i	$-0.638135\pi$
$139$	−9.91452	−0.840939	−0.420469	+	0.907307i	$0.638135\pi$
$140$	0	0
$141$	0.468226	0.0394318
$142$	0	0
$143$	−2.70315	−0.226049
$144$	0	0
$145$	−6.46422	−0.536825
$146$	0	0
$147$	−0.914095	−0.0753933
$148$	0	0
$149$	17.7934	1.45769	0.728847	−	0.684677i	$-0.240057\pi$
$149$	17.7934	1.45769	0.728847	+	0.684677i	$0.240057\pi$
$150$	0	0
$151$	16.2437	1.32189	0.660945	−	0.750434i	$-0.270155\pi$
$151$	16.2437	1.32189	0.660945	+	0.750434i	$0.270155\pi$
$152$	0	0
$153$	−18.5352	−1.49849
$154$	0	0
$155$	3.65295	0.293412
$156$	0	0
$157$	−10.1188	−0.807572	−0.403786	−	0.914854i	$-0.632306\pi$
$157$	−10.1188	−0.807572	−0.403786	+	0.914854i	$0.632306\pi$
$158$	0	0
$159$	−2.27797	−0.180655
$160$	0	0
$161$	−8.54892	−0.673749
$162$	0	0
$163$	−16.9814	−1.33008	−0.665041	−	0.746806i	$-0.731586\pi$
$163$	−16.9814	−1.33008	−0.665041	+	0.746806i	$0.731586\pi$
$164$	0	0
$165$	−0.166873	−0.0129910
$166$	0	0
$167$	4.22535	0.326967	0.163484	−	0.986546i	$-0.447727\pi$
$167$	4.22535	0.326967	0.163484	+	0.986546i	$0.447727\pi$
$168$	0	0
$169$	35.9121	2.76247
$170$	0	0
$171$	−15.6720	−1.19846
$172$	0	0
$173$	5.62999	0.428040	0.214020	−	0.976829i	$-0.431344\pi$
$173$	5.62999	0.428040	0.214020	+	0.976829i	$0.431344\pi$
$174$	0	0
$175$	3.01948	0.228251
$176$	0	0
$177$	2.03041	0.152615
$178$	0	0
$179$	19.6729	1.47042	0.735210	−	0.677839i	$-0.237084\pi$
$179$	19.6729	1.47042	0.735210	+	0.677839i	$0.237084\pi$
$180$	0	0
$181$	12.0848	0.898256	0.449128	−	0.893467i	$-0.351735\pi$
$181$	12.0848	0.898256	0.449128	+	0.893467i	$0.351735\pi$
$182$	0	0
$183$	−2.18194	−0.161294
$184$	0	0
$185$	2.48887	0.182985
$186$	0	0
$187$	−2.54623	−0.186199
$188$	0	0
$189$	7.57879	0.551276
$190$	0	0
$191$	−25.1044	−1.81649	−0.908246	−	0.418437i	$-0.862578\pi$
$191$	−25.1044	−1.81649	−0.908246	+	0.418437i	$0.862578\pi$
$192$	0	0
$193$	5.20785	0.374869	0.187435	−	0.982277i	$-0.439983\pi$
$193$	5.20785	0.374869	0.187435	+	0.982277i	$0.439983\pi$
$194$	0	0
$195$	3.01948	0.216229
$196$	0	0
$197$	14.6607	1.04453	0.522267	−	0.852782i	$-0.325087\pi$
$197$	14.6607	1.04453	0.522267	+	0.852782i	$0.325087\pi$
$198$	0	0
$199$	16.2836	1.15432	0.577159	−	0.816632i	$-0.304161\pi$
$199$	16.2836	1.15432	0.577159	+	0.816632i	$0.304161\pi$
$200$	0	0
$201$	1.81172	0.127789
$202$	0	0
$203$	19.5186	1.36993
$204$	0	0
$205$	−8.52322	−0.595287
$206$	0	0
$207$	7.96603	0.553678
$208$	0	0
$209$	−2.15289	−0.148919
$210$	0	0
$211$	−8.44791	−0.581578	−0.290789	−	0.956787i	$-0.593918\pi$
$211$	−8.44791	−0.581578	−0.290789	+	0.956787i	$0.593918\pi$
$212$	0	0
$213$	−4.45933	−0.305548
$214$	0	0
$215$	6.95309	0.474197
$216$	0	0
$217$	−11.0300	−0.748766
$218$	0	0
$219$	5.97588	0.403813
$220$	0	0
$221$	46.0728	3.09919
$222$	0	0
$223$	−12.9884	−0.869765	−0.434882	−	0.900487i	$-0.643210\pi$
$223$	−12.9884	−0.869765	−0.434882	+	0.900487i	$0.643210\pi$
$224$	0	0
$225$	−2.81360	−0.187573
$226$	0	0
$227$	22.1599	1.47080	0.735401	−	0.677632i	$-0.236994\pi$
$227$	22.1599	1.47080	0.735401	+	0.677632i	$0.236994\pi$
$228$	0	0
$229$	−9.23725	−0.610414	−0.305207	−	0.952286i	$-0.598726\pi$
$229$	−9.23725	−0.610414	−0.305207	+	0.952286i	$0.598726\pi$
$230$	0	0
$231$	0.503868	0.0331521
$232$	0	0
$233$	10.6482	0.697584	0.348792	−	0.937200i	$-0.386592\pi$
$233$	10.6482	0.697584	0.348792	+	0.937200i	$0.386592\pi$
$234$	0	0
$235$	1.08451	0.0707455
$236$	0	0
$237$	3.34957	0.217578
$238$	0	0
$239$	20.4925	1.32555	0.662775	−	0.748819i	$-0.269379\pi$
$239$	20.4925	1.32555	0.662775	+	0.748819i	$0.269379\pi$
$240$	0	0
$241$	−1.19273	−0.0768303	−0.0384151	−	0.999262i	$-0.512231\pi$
$241$	−1.19273	−0.0768303	−0.0384151	+	0.999262i	$0.512231\pi$
$242$	0	0
$243$	−10.7063	−0.686809
$244$	0	0
$245$	−2.11723	−0.135265
$246$	0	0
$247$	38.9555	2.47868
$248$	0	0
$249$	0.431741	0.0273605
$250$	0	0
$251$	−12.4779	−0.787598	−0.393799	−	0.919197i	$-0.628839\pi$
$251$	−12.4779	−0.787598	−0.393799	+	0.919197i	$0.628839\pi$
$252$	0	0
$253$	1.09431	0.0687989
$254$	0	0
$255$	2.84419	0.178110
$256$	0	0
$257$	−1.18701	−0.0740434	−0.0370217	−	0.999314i	$-0.511787\pi$
$257$	−1.18701	−0.0740434	−0.0370217	+	0.999314i	$0.511787\pi$
$258$	0	0
$259$	−7.51507	−0.466964
$260$	0	0
$261$	−18.1877	−1.12579
$262$	0	0
$263$	28.3723	1.74951	0.874757	−	0.484562i	$-0.161021\pi$
$263$	28.3723	1.74951	0.874757	+	0.484562i	$0.161021\pi$
$264$	0	0
$265$	−5.27625	−0.324117
$266$	0	0
$267$	−0.0273777	−0.00167549
$268$	0	0
$269$	0.759625	0.0463152	0.0231576	−	0.999732i	$-0.492628\pi$
$269$	0.759625	0.0463152	0.0231576	+	0.999732i	$0.492628\pi$
$270$	0	0
$271$	−10.0695	−0.611678	−0.305839	−	0.952083i	$-0.598937\pi$
$271$	−10.0695	−0.611678	−0.305839	+	0.952083i	$0.598937\pi$
$272$	0	0
$273$	−9.11723	−0.551800
$274$	0	0
$275$	−0.386511	−0.0233075
$276$	0	0
$277$	3.70252	0.222463	0.111232	−	0.993795i	$-0.464520\pi$
$277$	3.70252	0.222463	0.111232	+	0.993795i	$0.464520\pi$
$278$	0	0
$279$	10.2780	0.615325
$280$	0	0
$281$	−4.53568	−0.270576	−0.135288	−	0.990806i	$-0.543196\pi$
$281$	−4.53568	−0.270576	−0.135288	+	0.990806i	$0.543196\pi$
$282$	0	0
$283$	−27.3069	−1.62323	−0.811614	−	0.584194i	$-0.801411\pi$
$283$	−27.3069	−1.62323	−0.811614	+	0.584194i	$0.801411\pi$
$284$	0	0
$285$	2.40483	0.142450
$286$	0	0
$287$	25.7356	1.51913
$288$	0	0
$289$	26.3982	1.55284
$290$	0	0
$291$	−2.71741	−0.159297
$292$	0	0
$293$	−20.2487	−1.18294	−0.591471	−	0.806326i	$-0.701453\pi$
$293$	−20.2487	−1.18294	−0.591471	+	0.806326i	$0.701453\pi$
$294$	0	0
$295$	4.70284	0.273810
$296$	0	0
$297$	−0.970131	−0.0562927
$298$	0	0
$299$	−19.8010	−1.14512
$300$	0	0
$301$	−20.9947	−1.21011
$302$	0	0
$303$	2.32579	0.133613
$304$	0	0
$305$	−5.05383	−0.289381
$306$	0	0
$307$	−27.4709	−1.56785	−0.783925	−	0.620856i	$-0.786785\pi$
$307$	−27.4709	−1.56785	−0.783925	+	0.620856i	$0.786785\pi$
$308$	0	0
$309$	−4.06372	−0.231177
$310$	0	0
$311$	−4.51650	−0.256107	−0.128053	−	0.991767i	$-0.540873\pi$
$311$	−4.51650	−0.256107	−0.128053	+	0.991767i	$0.540873\pi$
$312$	0	0
$313$	3.91415	0.221241	0.110620	−	0.993863i	$-0.464716\pi$
$313$	3.91415	0.221241	0.110620	+	0.993863i	$0.464716\pi$
$314$	0	0
$315$	8.49559	0.478673
$316$	0	0
$317$	−15.5806	−0.875095	−0.437548	−	0.899195i	$-0.644153\pi$
$317$	−15.5806	−0.875095	−0.437548	+	0.899195i	$0.644153\pi$
$318$	0	0
$319$	−2.49849	−0.139889
$320$	0	0
$321$	5.66746	0.316327
$322$	0	0
$323$	36.6942	2.04172
$324$	0	0
$325$	6.99372	0.387942
$326$	0	0
$327$	7.16581	0.396270
$328$	0	0
$329$	−3.27464	−0.180537
$330$	0	0
$331$	−12.4747	−0.685672	−0.342836	−	0.939395i	$-0.611388\pi$
$331$	−12.4747	−0.685672	−0.342836	+	0.939395i	$0.611388\pi$
$332$	0	0
$333$	7.00267	0.383744
$334$	0	0
$335$	4.19631	0.229269
$336$	0	0
$337$	15.2315	0.829712	0.414856	−	0.909887i	$-0.363832\pi$
$337$	15.2315	0.829712	0.414856	+	0.909887i	$0.363832\pi$
$338$	0	0
$339$	1.96876	0.106928
$340$	0	0
$341$	1.41191	0.0764591
$342$	0	0
$343$	−14.7434	−0.796069
$344$	0	0
$345$	−1.22237	−0.0658102
$346$	0	0
$347$	10.9993	0.590472	0.295236	−	0.955424i	$-0.404602\pi$
$347$	10.9993	0.590472	0.295236	+	0.955424i	$0.404602\pi$
$348$	0	0
$349$	−17.1177	−0.916288	−0.458144	−	0.888878i	$-0.651486\pi$
$349$	−17.1177	−0.916288	−0.458144	+	0.888878i	$0.651486\pi$
$350$	0	0
$351$	17.5540	0.936964
$352$	0	0
$353$	13.7837	0.733631	0.366816	−	0.930294i	$-0.380448\pi$
$353$	13.7837	0.733631	0.366816	+	0.930294i	$0.380448\pi$
$354$	0	0
$355$	−10.3287	−0.548192
$356$	0	0
$357$	−8.58797	−0.454524
$358$	0	0
$359$	5.87777	0.310217	0.155109	−	0.987897i	$-0.450427\pi$
$359$	5.87777	0.310217	0.155109	+	0.987897i	$0.450427\pi$
$360$	0	0
$361$	12.0257	0.632933
$362$	0	0
$363$	4.68465	0.245880
$364$	0	0
$365$	13.8414	0.724490
$366$	0	0
$367$	−7.31349	−0.381761	−0.190880	−	0.981613i	$-0.561134\pi$
$367$	−7.31349	−0.381761	−0.190880	+	0.981613i	$0.561134\pi$
$368$	0	0
$369$	−23.9809	−1.24840
$370$	0	0
$371$	15.9315	0.827122
$372$	0	0
$373$	1.89691	0.0982185	0.0491092	−	0.998793i	$-0.484362\pi$
$373$	1.89691	0.0982185	0.0491092	+	0.998793i	$0.484362\pi$
$374$	0	0
$375$	0.431741	0.0222950
$376$	0	0
$377$	45.2090	2.32838
$378$	0	0
$379$	34.6799	1.78139	0.890693	−	0.454605i	$-0.150219\pi$
$379$	34.6799	1.78139	0.890693	+	0.454605i	$0.150219\pi$
$380$	0	0
$381$	−2.56458	−0.131387
$382$	0	0
$383$	1.13468	0.0579796	0.0289898	−	0.999580i	$-0.490771\pi$
$383$	1.13468	0.0579796	0.0289898	+	0.999580i	$0.490771\pi$
$384$	0	0
$385$	1.16706	0.0594789
$386$	0	0
$387$	19.5632	0.994454
$388$	0	0
$389$	13.4060	0.679709	0.339855	−	0.940478i	$-0.389622\pi$
$389$	13.4060	0.679709	0.339855	+	0.940478i	$0.389622\pi$
$390$	0	0
$391$	−18.6516	−0.943251
$392$	0	0
$393$	−5.90598	−0.297917
$394$	0	0
$395$	7.75829	0.390362
$396$	0	0
$397$	−24.1747	−1.21329	−0.606647	−	0.794972i	$-0.707486\pi$
$397$	−24.1747	−1.21329	−0.606647	+	0.794972i	$0.707486\pi$
$398$	0	0
$399$	−7.26132	−0.363521
$400$	0	0
$401$	16.0221	0.800105	0.400053	−	0.916492i	$-0.368992\pi$
$401$	16.0221	0.800105	0.400053	+	0.916492i	$0.368992\pi$
$402$	0	0
$403$	−25.5477	−1.27262
$404$	0	0
$405$	−7.35714	−0.365579
$406$	0	0
$407$	0.961974	0.0476833
$408$	0	0
$409$	15.4202	0.762480	0.381240	−	0.924476i	$-0.375497\pi$
$409$	15.4202	0.762480	0.381240	+	0.924476i	$0.375497\pi$
$410$	0	0
$411$	−9.11914	−0.449814
$412$	0	0
$413$	−14.2001	−0.698741
$414$	0	0
$415$	1.00000	0.0490881
$416$	0	0
$417$	4.28051	0.209617
$418$	0	0
$419$	−6.59479	−0.322177	−0.161088	−	0.986940i	$-0.551500\pi$
$419$	−6.59479	−0.322177	−0.161088	+	0.986940i	$0.551500\pi$
$420$	0	0
$421$	−36.1505	−1.76187	−0.880934	−	0.473240i	$-0.843084\pi$
$421$	−36.1505	−1.76187	−0.880934	+	0.473240i	$0.843084\pi$
$422$	0	0
$423$	3.05137	0.148363
$424$	0	0
$425$	6.58773	0.319552
$426$	0	0
$427$	15.2599	0.738478
$428$	0	0
$429$	1.16706	0.0563462
$430$	0	0
$431$	−25.4782	−1.22724	−0.613622	−	0.789600i	$-0.710288\pi$
$431$	−25.4782	−1.22724	−0.613622	+	0.789600i	$0.710288\pi$
$432$	0	0
$433$	14.7128	0.707053	0.353527	−	0.935424i	$-0.384982\pi$
$433$	14.7128	0.707053	0.353527	+	0.935424i	$0.384982\pi$
$434$	0	0
$435$	2.79087	0.133812
$436$	0	0
$437$	−15.7703	−0.754397
$438$	0	0
$439$	−6.66548	−0.318126	−0.159063	−	0.987268i	$-0.550847\pi$
$439$	−6.66548	−0.318126	−0.159063	+	0.987268i	$0.550847\pi$
$440$	0	0
$441$	−5.95704	−0.283669
$442$	0	0
$443$	−0.549488	−0.0261070	−0.0130535	−	0.999915i	$-0.504155\pi$
$443$	−0.549488	−0.0261070	−0.0130535	+	0.999915i	$0.504155\pi$
$444$	0	0
$445$	−0.0634123	−0.00300603
$446$	0	0
$447$	−7.68215	−0.363353
$448$	0	0
$449$	−4.52148	−0.213382	−0.106691	−	0.994292i	$-0.534026\pi$
$449$	−4.52148	−0.213382	−0.106691	+	0.994292i	$0.534026\pi$
$450$	0	0
$451$	−3.29432	−0.155123
$452$	0	0
$453$	−7.01306	−0.329502
$454$	0	0
$455$	−21.1174	−0.989997
$456$	0	0
$457$	32.9160	1.53974	0.769872	−	0.638199i	$-0.220320\pi$
$457$	32.9160	1.53974	0.769872	+	0.638199i	$0.220320\pi$
$458$	0	0
$459$	16.5350	0.771788
$460$	0	0
$461$	−0.628370	−0.0292661	−0.0146331	−	0.999893i	$-0.504658\pi$
$461$	−0.628370	−0.0292661	−0.0146331	+	0.999893i	$0.504658\pi$
$462$	0	0
$463$	−21.2255	−0.986432	−0.493216	−	0.869907i	$-0.664179\pi$
$463$	−21.2255	−0.986432	−0.493216	+	0.869907i	$0.664179\pi$
$464$	0	0
$465$	−1.57713	−0.0731376
$466$	0	0
$467$	−35.7171	−1.65279	−0.826396	−	0.563089i	$-0.809613\pi$
$467$	−35.7171	−1.65279	−0.826396	+	0.563089i	$0.809613\pi$
$468$	0	0
$469$	−12.6707	−0.585076
$470$	0	0
$471$	4.36872	0.201300
$472$	0	0
$473$	2.68745	0.123569
$474$	0	0
$475$	5.57007	0.255573
$476$	0	0
$477$	−14.8453	−0.679717
$478$	0	0
$479$	24.3814	1.11401	0.557007	−	0.830508i	$-0.311950\pi$
$479$	24.3814	1.11401	0.557007	+	0.830508i	$0.311950\pi$
$480$	0	0
$481$	−17.4064	−0.793665
$482$	0	0
$483$	3.69092	0.167943
$484$	0	0
$485$	−6.29407	−0.285799
$486$	0	0
$487$	23.1661	1.04976	0.524879	−	0.851177i	$-0.324111\pi$
$487$	23.1661	1.04976	0.524879	+	0.851177i	$0.324111\pi$
$488$	0	0
$489$	7.33155	0.331544
$490$	0	0
$491$	−0.895629	−0.0404192	−0.0202096	−	0.999796i	$-0.506433\pi$
$491$	−0.895629	−0.0404192	−0.0202096	+	0.999796i	$0.506433\pi$
$492$	0	0
$493$	42.5846	1.91791
$494$	0	0
$495$	−1.08749	−0.0488789
$496$	0	0
$497$	31.1873	1.39894
$498$	0	0
$499$	11.8061	0.528514	0.264257	−	0.964452i	$-0.414873\pi$
$499$	11.8061	0.528514	0.264257	+	0.964452i	$0.414873\pi$
$500$	0	0
$501$	−1.82426	−0.0815018
$502$	0	0
$503$	19.7931	0.882531	0.441265	−	0.897377i	$-0.354530\pi$
$503$	19.7931	0.882531	0.441265	+	0.897377i	$0.354530\pi$
$504$	0	0
$505$	5.38701	0.239719
$506$	0	0
$507$	−15.5047	−0.688589
$508$	0	0
$509$	3.69547	0.163799	0.0818995	−	0.996641i	$-0.473901\pi$
$509$	3.69547	0.163799	0.0818995	+	0.996641i	$0.473901\pi$
$510$	0	0
$511$	−41.7936	−1.84884
$512$	0	0
$513$	13.9807	0.617264
$514$	0	0
$515$	−9.41240	−0.414760
$516$	0	0
$517$	0.419174	0.0184353
$518$	0	0
$519$	−2.43070	−0.106696
$520$	0	0
$521$	17.7211	0.776376	0.388188	−	0.921580i	$-0.373101\pi$
$521$	17.7211	0.776376	0.388188	+	0.921580i	$0.373101\pi$
$522$	0	0
$523$	31.6186	1.38258	0.691292	−	0.722575i	$-0.257042\pi$
$523$	31.6186	1.38258	0.691292	+	0.722575i	$0.257042\pi$
$524$	0	0
$525$	−1.30363	−0.0568951
$526$	0	0
$527$	−24.0647	−1.04827
$528$	0	0
$529$	−14.9840	−0.651477
$530$	0	0
$531$	13.2319	0.574216
$532$	0	0
$533$	59.6090	2.58195
$534$	0	0
$535$	13.1270	0.567529
$536$	0	0
$537$	−8.49359	−0.366525
$538$	0	0
$539$	−0.818333	−0.0352481
$540$	0	0
$541$	4.35506	0.187239	0.0936193	−	0.995608i	$-0.470156\pi$
$541$	4.35506	0.187239	0.0936193	+	0.995608i	$0.470156\pi$
$542$	0	0
$543$	−5.21750	−0.223904
$544$	0	0
$545$	16.5975	0.710958
$546$	0	0
$547$	42.4490	1.81499	0.907493	−	0.420067i	$-0.137993\pi$
$547$	42.4490	1.81499	0.907493	+	0.420067i	$0.137993\pi$
$548$	0	0
$549$	−14.2194	−0.606871
$550$	0	0
$551$	36.0062	1.53392
$552$	0	0
$553$	−23.4260	−0.996173
$554$	0	0
$555$	−1.07455	−0.0456119
$556$	0	0
$557$	−6.63302	−0.281050	−0.140525	−	0.990077i	$-0.544879\pi$
$557$	−6.63302	−0.281050	−0.140525	+	0.990077i	$0.544879\pi$
$558$	0	0
$559$	−48.6280	−2.05674
$560$	0	0
$561$	1.09931	0.0464130
$562$	0	0
$563$	−35.7791	−1.50791	−0.753955	−	0.656927i	$-0.771856\pi$
$563$	−35.7791	−1.50791	−0.753955	+	0.656927i	$0.771856\pi$
$564$	0	0
$565$	4.56004	0.191842
$566$	0	0
$567$	22.2147	0.932930
$568$	0	0
$569$	−35.5696	−1.49116	−0.745578	−	0.666419i	$-0.767826\pi$
$569$	−35.5696	−1.49116	−0.745578	+	0.666419i	$0.767826\pi$
$570$	0	0
$571$	20.4274	0.854858	0.427429	−	0.904049i	$-0.359419\pi$
$571$	20.4274	0.854858	0.427429	+	0.904049i	$0.359419\pi$
$572$	0	0
$573$	10.8386	0.452789
$574$	0	0
$575$	−2.83126	−0.118072
$576$	0	0
$577$	30.7540	1.28030	0.640152	−	0.768248i	$-0.278871\pi$
$577$	30.7540	1.28030	0.640152	+	0.768248i	$0.278871\pi$
$578$	0	0
$579$	−2.24844	−0.0934420
$580$	0	0
$581$	−3.01948	−0.125269
$582$	0	0
$583$	−2.03933	−0.0844604
$584$	0	0
$585$	19.6775	0.813566
$586$	0	0
$587$	47.9182	1.97780	0.988898	−	0.148595i	$-0.0474751\pi$
$587$	47.9182	1.97780	0.988898	+	0.148595i	$0.0474751\pi$
$588$	0	0
$589$	−20.3472	−0.838393
$590$	0	0
$591$	−6.32963	−0.260366
$592$	0	0
$593$	−17.1170	−0.702910	−0.351455	−	0.936205i	$-0.614313\pi$
$593$	−17.1170	−0.702910	−0.351455	+	0.936205i	$0.614313\pi$
$594$	0	0
$595$	−19.8915	−0.815472
$596$	0	0
$597$	−7.03032	−0.287732
$598$	0	0
$599$	−1.61307	−0.0659081	−0.0329540	−	0.999457i	$-0.510491\pi$
$599$	−1.61307	−0.0659081	−0.0329540	+	0.999457i	$0.510491\pi$
$600$	0	0
$601$	−44.6538	−1.82147	−0.910734	−	0.412994i	$-0.864483\pi$
$601$	−44.6538	−1.82147	−0.910734	+	0.412994i	$0.864483\pi$
$602$	0	0
$603$	11.8067	0.480807
$604$	0	0
$605$	10.8506	0.441140
$606$	0	0
$607$	44.2902	1.79768	0.898841	−	0.438274i	$-0.144410\pi$
$607$	44.2902	1.79768	0.898841	+	0.438274i	$0.144410\pi$
$608$	0	0
$609$	−8.42696	−0.341478
$610$	0	0
$611$	−7.58474	−0.306846
$612$	0	0
$613$	−44.7545	−1.80762	−0.903808	−	0.427939i	$-0.859240\pi$
$613$	−44.7545	−1.80762	−0.903808	+	0.427939i	$0.859240\pi$
$614$	0	0
$615$	3.67982	0.148385
$616$	0	0
$617$	2.19344	0.0883047	0.0441523	−	0.999025i	$-0.485941\pi$
$617$	2.19344	0.0883047	0.0441523	+	0.999025i	$0.485941\pi$
$618$	0	0
$619$	20.1843	0.811277	0.405639	−	0.914033i	$-0.367049\pi$
$619$	20.1843	0.811277	0.405639	+	0.914033i	$0.367049\pi$
$620$	0	0
$621$	−7.10637	−0.285169
$622$	0	0
$623$	0.191472	0.00767116
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0.929493	0.0371204
$628$	0	0
$629$	−16.3960	−0.653751
$630$	0	0
$631$	6.65239	0.264827	0.132414	−	0.991195i	$-0.457727\pi$
$631$	6.65239	0.264827	0.132414	+	0.991195i	$0.457727\pi$
$632$	0	0
$633$	3.64731	0.144967
$634$	0	0
$635$	−5.94008	−0.235725
$636$	0	0
$637$	14.8073	0.586687
$638$	0	0
$639$	−29.0609	−1.14963
$640$	0	0
$641$	35.1221	1.38724	0.693620	−	0.720341i	$-0.256015\pi$
$641$	35.1221	1.38724	0.693620	+	0.720341i	$0.256015\pi$
$642$	0	0
$643$	33.0043	1.30156	0.650781	−	0.759266i	$-0.274442\pi$
$643$	33.0043	1.30156	0.650781	+	0.759266i	$0.274442\pi$
$644$	0	0
$645$	−3.00193	−0.118201
$646$	0	0
$647$	15.2774	0.600618	0.300309	−	0.953842i	$-0.402910\pi$
$647$	15.2774	0.600618	0.300309	+	0.953842i	$0.402910\pi$
$648$	0	0
$649$	1.81770	0.0713509
$650$	0	0
$651$	4.76210	0.186642
$652$	0	0
$653$	34.4208	1.34699	0.673495	−	0.739192i	$-0.264792\pi$
$653$	34.4208	1.34699	0.673495	+	0.739192i	$0.264792\pi$
$654$	0	0
$655$	−13.6795	−0.534500
$656$	0	0
$657$	38.9440	1.51935
$658$	0	0
$659$	24.7850	0.965488	0.482744	−	0.875762i	$-0.339640\pi$
$659$	24.7850	0.965488	0.482744	+	0.875762i	$0.339640\pi$
$660$	0	0
$661$	−3.33949	−0.129891	−0.0649456	−	0.997889i	$-0.520687\pi$
$661$	−3.33949	−0.129891	−0.0649456	+	0.997889i	$0.520687\pi$
$662$	0	0
$663$	−19.8915	−0.772522
$664$	0	0
$665$	−16.8187	−0.652201
$666$	0	0
$667$	−18.3019	−0.708652
$668$	0	0
$669$	5.60761	0.216803
$670$	0	0
$671$	−1.95336	−0.0754086
$672$	0	0
$673$	14.0825	0.542841	0.271421	−	0.962461i	$-0.412507\pi$
$673$	14.0825	0.542841	0.271421	+	0.962461i	$0.412507\pi$
$674$	0	0
$675$	2.50997	0.0966087
$676$	0	0
$677$	35.9538	1.38182	0.690909	−	0.722941i	$-0.257210\pi$
$677$	35.9538	1.38182	0.690909	+	0.722941i	$0.257210\pi$
$678$	0	0
$679$	19.0048	0.729336
$680$	0	0
$681$	−9.56732	−0.366620
$682$	0	0
$683$	3.40122	0.130144	0.0650720	−	0.997881i	$-0.479272\pi$
$683$	3.40122	0.130144	0.0650720	+	0.997881i	$0.479272\pi$
$684$	0	0
$685$	−21.1218	−0.807022
$686$	0	0
$687$	3.98810	0.152155
$688$	0	0
$689$	36.9006	1.40580
$690$	0	0
$691$	18.9887	0.722365	0.361182	−	0.932495i	$-0.382373\pi$
$691$	18.9887	0.722365	0.361182	+	0.932495i	$0.382373\pi$
$692$	0	0
$693$	3.28364	0.124735
$694$	0	0
$695$	9.91452	0.376079
$696$	0	0
$697$	56.1487	2.12678
$698$	0	0
$699$	−4.59725	−0.173884
$700$	0	0
$701$	−43.2740	−1.63444	−0.817219	−	0.576327i	$-0.804485\pi$
$701$	−43.2740	−1.63444	−0.817219	+	0.576327i	$0.804485\pi$
$702$	0	0
$703$	−13.8632	−0.522859
$704$	0	0
$705$	−0.468226	−0.0176344
$706$	0	0
$707$	−16.2660	−0.611744
$708$	0	0
$709$	−9.47103	−0.355692	−0.177846	−	0.984058i	$-0.556913\pi$
$709$	−9.47103	−0.355692	−0.177846	+	0.984058i	$0.556913\pi$
$710$	0	0
$711$	21.8287	0.818641
$712$	0	0
$713$	10.3425	0.387328
$714$	0	0
$715$	2.70315	0.101092
$716$	0	0
$717$	−8.84745	−0.330414
$718$	0	0
$719$	−24.6960	−0.921004	−0.460502	−	0.887659i	$-0.652331\pi$
$719$	−24.6960	−0.921004	−0.460502	+	0.887659i	$0.652331\pi$
$720$	0	0
$721$	28.4205	1.05844
$722$	0	0
$723$	0.514949	0.0191512
$724$	0	0
$725$	6.46422	0.240075
$726$	0	0
$727$	23.3386	0.865582	0.432791	−	0.901494i	$-0.357529\pi$
$727$	23.3386	0.865582	0.432791	+	0.901494i	$0.357529\pi$
$728$	0	0
$729$	−17.4491	−0.646263
$730$	0	0
$731$	−45.8051	−1.69416
$732$	0	0
$733$	40.6924	1.50301	0.751504	−	0.659728i	$-0.229329\pi$
$733$	40.6924	1.50301	0.751504	+	0.659728i	$0.229329\pi$
$734$	0	0
$735$	0.914095	0.0337169
$736$	0	0
$737$	1.62192	0.0597442
$738$	0	0
$739$	16.3218	0.600408	0.300204	−	0.953875i	$-0.402945\pi$
$739$	16.3218	0.600408	0.300204	+	0.953875i	$0.402945\pi$
$740$	0	0
$741$	−16.8187	−0.617850
$742$	0	0
$743$	1.50962	0.0553826	0.0276913	−	0.999617i	$-0.491184\pi$
$743$	1.50962	0.0553826	0.0276913	+	0.999617i	$0.491184\pi$
$744$	0	0
$745$	−17.7934	−0.651900
$746$	0	0
$747$	2.81360	0.102944
$748$	0	0
$749$	−39.6366	−1.44829
$750$	0	0
$751$	−19.7197	−0.719582	−0.359791	−	0.933033i	$-0.617152\pi$
$751$	−19.7197	−0.719582	−0.359791	+	0.933033i	$0.617152\pi$
$752$	0	0
$753$	5.38722	0.196321
$754$	0	0
$755$	−16.2437	−0.591168
$756$	0	0
$757$	14.9494	0.543346	0.271673	−	0.962390i	$-0.412423\pi$
$757$	14.9494	0.543346	0.271673	+	0.962390i	$0.412423\pi$
$758$	0	0
$759$	−0.472460	−0.0171492
$760$	0	0
$761$	9.48843	0.343955	0.171978	−	0.985101i	$-0.444984\pi$
$761$	9.48843	0.343955	0.171978	+	0.985101i	$0.444984\pi$
$762$	0	0
$763$	−50.1157	−1.81431
$764$	0	0
$765$	18.5352	0.670143
$766$	0	0
$767$	−32.8903	−1.18760
$768$	0	0
$769$	44.3031	1.59761	0.798806	−	0.601589i	$-0.205465\pi$
$769$	44.3031	1.59761	0.798806	+	0.601589i	$0.205465\pi$
$770$	0	0
$771$	0.512479	0.0184565
$772$	0	0
$773$	−53.9911	−1.94193	−0.970963	−	0.239231i	$-0.923105\pi$
$773$	−53.9911	−1.94193	−0.970963	+	0.239231i	$0.923105\pi$
$774$	0	0
$775$	−3.65295	−0.131218
$776$	0	0
$777$	3.24456	0.116398
$778$	0	0
$779$	47.4750	1.70097
$780$	0	0
$781$	−3.99216	−0.142851
$782$	0	0
$783$	16.2250	0.579834
$784$	0	0
$785$	10.1188	0.361157
$786$	0	0
$787$	−8.30642	−0.296092	−0.148046	−	0.988980i	$-0.547298\pi$
$787$	−8.30642	−0.296092	−0.148046	+	0.988980i	$0.547298\pi$
$788$	0	0
$789$	−12.2495	−0.436094
$790$	0	0
$791$	−13.7689	−0.489567
$792$	0	0
$793$	35.3450	1.25514
$794$	0	0
$795$	2.27797	0.0807914
$796$	0	0
$797$	10.8857	0.385590	0.192795	−	0.981239i	$-0.438245\pi$
$797$	10.8857	0.385590	0.192795	+	0.981239i	$0.438245\pi$
$798$	0	0
$799$	−7.14445	−0.252752
$800$	0	0
$801$	−0.178417	−0.00630405
$802$	0	0
$803$	5.34984	0.188792
$804$	0	0
$805$	8.54892	0.301310
$806$	0	0
$807$	−0.327961	−0.0115448
$808$	0	0
$809$	−49.2828	−1.73269	−0.866346	−	0.499444i	$-0.833537\pi$
$809$	−49.2828	−1.73269	−0.866346	+	0.499444i	$0.833537\pi$
$810$	0	0
$811$	7.57591	0.266026	0.133013	−	0.991114i	$-0.457535\pi$
$811$	7.57591	0.266026	0.133013	+	0.991114i	$0.457535\pi$
$812$	0	0
$813$	4.34741	0.152470
$814$	0	0
$815$	16.9814	0.594831
$816$	0	0
$817$	−38.7292	−1.35496
$818$	0	0
$819$	−59.4158	−2.07616
$820$	0	0
$821$	−13.1781	−0.459919	−0.229959	−	0.973200i	$-0.573859\pi$
$821$	−13.1781	−0.459919	−0.229959	+	0.973200i	$0.573859\pi$
$822$	0	0
$823$	−20.1232	−0.701450	−0.350725	−	0.936479i	$-0.614065\pi$
$823$	−20.1232	−0.701450	−0.350725	+	0.936479i	$0.614065\pi$
$824$	0	0
$825$	0.166873	0.00580976
$826$	0	0
$827$	0.667971	0.0232276	0.0116138	−	0.999933i	$-0.496303\pi$
$827$	0.667971	0.0232276	0.0116138	+	0.999933i	$0.496303\pi$
$828$	0	0
$829$	−52.7661	−1.83264	−0.916321	−	0.400445i	$-0.868855\pi$
$829$	−52.7661	−1.83264	−0.916321	+	0.400445i	$0.868855\pi$
$830$	0	0
$831$	−1.59853	−0.0554524
$832$	0	0
$833$	13.9478	0.483261
$834$	0	0
$835$	−4.22535	−0.146224
$836$	0	0
$837$	−9.16880	−0.316920
$838$	0	0
$839$	−46.4586	−1.60393	−0.801965	−	0.597372i	$-0.796212\pi$
$839$	−46.4586	−1.60393	−0.801965	+	0.597372i	$0.796212\pi$
$840$	0	0
$841$	12.7862	0.440903
$842$	0	0
$843$	1.95824	0.0674453
$844$	0	0
$845$	−35.9121	−1.23541
$846$	0	0
$847$	−32.7631	−1.12576
$848$	0	0
$849$	11.7895	0.404615
$850$	0	0
$851$	7.04663	0.241555
$852$	0	0
$853$	−34.0142	−1.16463	−0.582313	−	0.812965i	$-0.697852\pi$
$853$	−34.0142	−1.16463	−0.582313	+	0.812965i	$0.697852\pi$
$854$	0	0
$855$	15.6720	0.535970
$856$	0	0
$857$	−23.6364	−0.807404	−0.403702	−	0.914891i	$-0.632277\pi$
$857$	−23.6364	−0.807404	−0.403702	+	0.914891i	$0.632277\pi$
$858$	0	0
$859$	27.1844	0.927519	0.463759	−	0.885961i	$-0.346500\pi$
$859$	27.1844	0.927519	0.463759	+	0.885961i	$0.346500\pi$
$860$	0	0
$861$	−11.1111	−0.378666
$862$	0	0
$863$	55.4152	1.88636	0.943178	−	0.332289i	$-0.107821\pi$
$863$	55.4152	1.88636	0.943178	+	0.332289i	$0.107821\pi$
$864$	0	0
$865$	−5.62999	−0.191425
$866$	0	0
$867$	−11.3972	−0.387069
$868$	0	0
$869$	2.99866	0.101723
$870$	0	0
$871$	−29.3478	−0.994413
$872$	0	0
$873$	−17.7090	−0.599358
$874$	0	0
$875$	−3.01948	−0.102077
$876$	0	0
$877$	−31.3826	−1.05971	−0.529857	−	0.848087i	$-0.677754\pi$
$877$	−31.3826	−1.05971	−0.529857	+	0.848087i	$0.677754\pi$
$878$	0	0
$879$	8.74220	0.294867
$880$	0	0
$881$	−10.8571	−0.365787	−0.182893	−	0.983133i	$-0.558546\pi$
$881$	−10.8571	−0.365787	−0.182893	+	0.983133i	$0.558546\pi$
$882$	0	0
$883$	14.8728	0.500511	0.250255	−	0.968180i	$-0.419485\pi$
$883$	14.8728	0.500511	0.250255	+	0.968180i	$0.419485\pi$
$884$	0	0
$885$	−2.03041	−0.0682514
$886$	0	0
$887$	−34.0443	−1.14309	−0.571547	−	0.820569i	$-0.693657\pi$
$887$	−34.0443	−1.14309	−0.571547	+	0.820569i	$0.693657\pi$
$888$	0	0
$889$	17.9359	0.601552
$890$	0	0
$891$	−2.84362	−0.0952647
$892$	0	0
$893$	−6.04079	−0.202147
$894$	0	0
$895$	−19.6729	−0.657592
$896$	0	0
$897$	8.54892	0.285440
$898$	0	0
$899$	−23.6135	−0.787555
$900$	0	0
$901$	34.7585	1.15797
$902$	0	0
$903$	9.06426	0.301640
$904$	0	0
$905$	−12.0848	−0.401712
$906$	0	0
$907$	−2.39080	−0.0793851	−0.0396925	−	0.999212i	$-0.512638\pi$
$907$	−2.39080	−0.0793851	−0.0396925	+	0.999212i	$0.512638\pi$
$908$	0	0
$909$	15.1569	0.502723
$910$	0	0
$911$	−6.37553	−0.211231	−0.105615	−	0.994407i	$-0.533681\pi$
$911$	−6.37553	−0.211231	−0.105615	+	0.994407i	$0.533681\pi$
$912$	0	0
$913$	0.386511	0.0127916
$914$	0	0
$915$	2.18194	0.0721328
$916$	0	0
$917$	41.3048	1.36400
$918$	0	0
$919$	−59.7294	−1.97029	−0.985146	−	0.171719i	$-0.945068\pi$
$919$	−59.7294	−1.97029	−0.985146	+	0.171719i	$0.945068\pi$
$920$	0	0
$921$	11.8603	0.390811
$922$	0	0
$923$	72.2362	2.37768
$924$	0	0
$925$	−2.48887	−0.0818334
$926$	0	0
$927$	−26.4827	−0.869807
$928$	0	0
$929$	4.54835	0.149226	0.0746132	−	0.997213i	$-0.476228\pi$
$929$	4.54835	0.149226	0.0746132	+	0.997213i	$0.476228\pi$
$930$	0	0
$931$	11.7931	0.386504
$932$	0	0
$933$	1.94996	0.0638387
$934$	0	0
$935$	2.54623	0.0832707
$936$	0	0
$937$	−1.52795	−0.0499161	−0.0249580	−	0.999688i	$-0.507945\pi$
$937$	−1.52795	−0.0499161	−0.0249580	+	0.999688i	$0.507945\pi$
$938$	0	0
$939$	−1.68990	−0.0551477
$940$	0	0
$941$	−27.1537	−0.885184	−0.442592	−	0.896723i	$-0.645941\pi$
$941$	−27.1537	−0.885184	−0.442592	+	0.896723i	$0.645941\pi$
$942$	0	0
$943$	−24.1314	−0.785828
$944$	0	0
$945$	−7.57879	−0.246538
$946$	0	0
$947$	−58.1874	−1.89084	−0.945419	−	0.325858i	$-0.894347\pi$
$947$	−58.1874	−1.89084	−0.945419	+	0.325858i	$0.894347\pi$
$948$	0	0
$949$	−96.8026	−3.14234
$950$	0	0
$951$	6.72679	0.218131
$952$	0	0
$953$	−19.3719	−0.627518	−0.313759	−	0.949503i	$-0.601588\pi$
$953$	−19.3719	−0.627518	−0.313759	+	0.949503i	$0.601588\pi$
$954$	0	0
$955$	25.1044	0.812360
$956$	0	0
$957$	1.07870	0.0348695
$958$	0	0
$959$	63.7767	2.05946
$960$	0	0
$961$	−17.6559	−0.569546
$962$	0	0
$963$	36.9341	1.19018
$964$	0	0
$965$	−5.20785	−0.167647
$966$	0	0
$967$	19.4636	0.625908	0.312954	−	0.949768i	$-0.398681\pi$
$967$	19.4636	0.625908	0.312954	+	0.949768i	$0.398681\pi$
$968$	0	0
$969$	−15.8424	−0.508930
$970$	0	0
$971$	−44.6819	−1.43391	−0.716956	−	0.697119i	$-0.754465\pi$
$971$	−44.6819	−1.43391	−0.716956	+	0.697119i	$0.754465\pi$
$972$	0	0
$973$	−29.9367	−0.959725
$974$	0	0
$975$	−3.01948	−0.0967006
$976$	0	0
$977$	−17.7202	−0.566918	−0.283459	−	0.958984i	$-0.591482\pi$
$977$	−17.7202	−0.566918	−0.283459	+	0.958984i	$0.591482\pi$
$978$	0	0
$979$	−0.0245095	−0.000783328 0
$980$	0	0
$981$	46.6987	1.49097
$982$	0	0
$983$	−45.6673	−1.45656	−0.728280	−	0.685279i	$-0.759680\pi$
$983$	−45.6673	−1.45656	−0.728280	+	0.685279i	$0.759680\pi$
$984$	0	0
$985$	−14.6607	−0.467129
$986$	0	0
$987$	1.41380	0.0450017
$988$	0	0
$989$	19.6860	0.625979
$990$	0	0
$991$	−16.7135	−0.530922	−0.265461	−	0.964122i	$-0.585524\pi$
$991$	−16.7135	−0.530922	−0.265461	+	0.964122i	$0.585524\pi$
$992$	0	0
$993$	5.38584	0.170915
$994$	0	0
$995$	−16.2836	−0.516226
$996$	0	0
$997$	−14.3840	−0.455546	−0.227773	−	0.973714i	$-0.573144\pi$
$997$	−14.3840	−0.455546	−0.227773	+	0.973714i	$0.573144\pi$
$998$	0	0
$999$	−6.24698	−0.197646

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6640.2.a.bb.1.4		6
4.3	odd	2		415.2.a.c.1.1	✓	6
12.11	even	2		3735.2.a.l.1.6		6
20.19	odd	2		2075.2.a.f.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
415.2.a.c.1.1	✓	6	4.3	odd	2
2075.2.a.f.1.6		6	20.19	odd	2
3735.2.a.l.1.6		6	12.11	even	2
6640.2.a.bb.1.4		6	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:	\( N \)	\(=\)	\( 6640 = 2^{4} \cdot 5 \cdot 83 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6640.a (trivial)

\(f(q)\)	\(=\)	\(q-0.431741 q^{3} -1.00000 q^{5} +3.01948 q^{7} -2.81360 q^{9} -0.386511 q^{11} +6.99372 q^{13} +0.431741 q^{15} +6.58773 q^{17} +5.57007 q^{19} -1.30363 q^{21} -2.83126 q^{23} +1.00000 q^{25} +2.50997 q^{27} +6.46422 q^{29} -3.65295 q^{31} +0.166873 q^{33} -3.01948 q^{35} -2.48887 q^{37} -3.01948 q^{39} +8.52322 q^{41} -6.95309 q^{43} +2.81360 q^{45} -1.08451 q^{47} +2.11723 q^{49} -2.84419 q^{51} +5.27625 q^{53} +0.386511 q^{55} -2.40483 q^{57} -4.70284 q^{59} +5.05383 q^{61} -8.49559 q^{63} -6.99372 q^{65} -4.19631 q^{67} +1.22237 q^{69} +10.3287 q^{71} -13.8414 q^{73} -0.431741 q^{75} -1.16706 q^{77} -7.75829 q^{79} +7.35714 q^{81} -1.00000 q^{83} -6.58773 q^{85} -2.79087 q^{87} +0.0634123 q^{89} +21.1174 q^{91} +1.57713 q^{93} -5.57007 q^{95} +6.29407 q^{97} +1.08749 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3 q^{3} - 6 q^{5} + 9 q^{9} - 4 q^{11} + 7 q^{13} + 3 q^{15} + 21 q^{17} - 2 q^{19} - 9 q^{21} - 8 q^{23} + 6 q^{25} - 12 q^{27} + 10 q^{29} + 13 q^{31} + 6 q^{33} + q^{37} + 10 q^{41} + 3 q^{43}+ \cdots + 22 q^{99}+O(q^{100}) \) 6 * q - 3 * q^3 - 6 * q^5 + 9 * q^9 - 4 * q^11 + 7 * q^13 + 3 * q^15 + 21 * q^17 - 2 * q^19 - 9 * q^21 - 8 * q^23 + 6 * q^25 - 12 * q^27 + 10 * q^29 + 13 * q^31 + 6 * q^33 + q^37 + 10 * q^41 + 3 * q^43 - 9 * q^45 - 14 * q^47 - 12 * q^49 + 6 * q^51 + 9 * q^53 + 4 * q^55 + 19 * q^57 - 20 * q^59 - 13 * q^61 + 11 * q^63 - 7 * q^65 - 16 * q^67 - 11 * q^69 - 5 * q^71 + 3 * q^73 - 3 * q^75 + 30 * q^77 - 2 * q^79 + 2 * q^81 - 6 * q^83 - 21 * q^85 + 5 * q^87 + 45 * q^89 + 21 * q^91 - 27 * q^93 + 2 * q^95 + 22 * q^97 + 22 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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