LMFDB - Embedded newform 3328.2.a.x.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 = [2,0,0,0,4,0,0,0,10] 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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1664)
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)

$f(q)$	$=$	$q-2.82843 q^{3} +2.00000 q^{5} -4.24264 q^{7} +5.00000 q^{9} -1.41421 q^{11} -1.00000 q^{13} -5.65685 q^{15} -4.00000 q^{17} -1.41421 q^{19} +12.0000 q^{21} +5.65685 q^{23} -1.00000 q^{25} -5.65685 q^{27} -2.00000 q^{29} -9.89949 q^{31} +4.00000 q^{33} -8.48528 q^{35} +2.00000 q^{37} +2.82843 q^{39} -6.00000 q^{41} +11.3137 q^{43} +10.0000 q^{45} -7.07107 q^{47} +11.0000 q^{49} +11.3137 q^{51} +12.0000 q^{53} -2.82843 q^{55} +4.00000 q^{57} -9.89949 q^{59} -12.0000 q^{61} -21.2132 q^{63} -2.00000 q^{65} +9.89949 q^{67} -16.0000 q^{69} -4.24264 q^{71} -14.0000 q^{73} +2.82843 q^{75} +6.00000 q^{77} +8.48528 q^{79} +1.00000 q^{81} +7.07107 q^{83} -8.00000 q^{85} +5.65685 q^{87} +14.0000 q^{89} +4.24264 q^{91} +28.0000 q^{93} -2.82843 q^{95} -2.00000 q^{97} -7.07107 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{5} + 10 q^{9} - 2 q^{13} - 8 q^{17} + 24 q^{21} - 2 q^{25} - 4 q^{29} + 8 q^{33} + 4 q^{37} - 12 q^{41} + 20 q^{45} + 22 q^{49} + 24 q^{53} + 8 q^{57} - 24 q^{61} - 4 q^{65} - 32 q^{69} - 28 q^{73}+ \cdots - 4 q^{97}+O(q^{100}) $ 2 * q + 4 * q^5 + 10 * q^9 - 2 * q^13 - 8 * q^17 + 24 * q^21 - 2 * q^25 - 4 * q^29 + 8 * q^33 + 4 * q^37 - 12 * q^41 + 20 * q^45 + 22 * q^49 + 24 * q^53 + 8 * q^57 - 24 * q^61 - 4 * q^65 - 32 * q^69 - 28 * q^73 + 12 * q^77 + 2 * q^81 - 16 * q^85 + 28 * q^89 + 56 * q^93 - 4 * q^97

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
$3$	−2.82843	−1.63299	−0.816497	−	0.577350i	$-0.804087\pi$
$3$	−2.82843	−1.63299	−0.816497	+	0.577350i	$0.804087\pi$
$4$	0	0
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	0	0
$7$	−4.24264	−1.60357	−0.801784	−	0.597614i	$-0.796115\pi$
$7$	−4.24264	−1.60357	−0.801784	+	0.597614i	$0.796115\pi$
$8$	0	0
$9$	5.00000	1.66667
$10$	0	0
$11$	−1.41421	−0.426401	−0.213201	−	0.977008i	$-0.568389\pi$
$11$	−1.41421	−0.426401	−0.213201	+	0.977008i	$0.568389\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−5.65685	−1.46059
$16$	0	0
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	0	0
$19$	−1.41421	−0.324443	−0.162221	−	0.986754i	$-0.551866\pi$
$19$	−1.41421	−0.324443	−0.162221	+	0.986754i	$0.551866\pi$
$20$	0	0
$21$	12.0000	2.61861
$22$	0	0
$23$	5.65685	1.17954	0.589768	−	0.807573i	$-0.299219\pi$
$23$	5.65685	1.17954	0.589768	+	0.807573i	$0.299219\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−5.65685	−1.08866
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	−9.89949	−1.77800	−0.889001	−	0.457905i	$-0.848600\pi$
$31$	−9.89949	−1.77800	−0.889001	+	0.457905i	$0.848600\pi$
$32$	0	0
$33$	4.00000	0.696311
$34$	0	0
$35$	−8.48528	−1.43427
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	0	0
$39$	2.82843	0.452911
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	11.3137	1.72532	0.862662	−	0.505781i	$-0.168795\pi$
$43$	11.3137	1.72532	0.862662	+	0.505781i	$0.168795\pi$
$44$	0	0
$45$	10.0000	1.49071
$46$	0	0
$47$	−7.07107	−1.03142	−0.515711	−	0.856763i	$-0.672472\pi$
$47$	−7.07107	−1.03142	−0.515711	+	0.856763i	$0.672472\pi$
$48$	0	0
$49$	11.0000	1.57143
$50$	0	0
$51$	11.3137	1.58424
$52$	0	0
$53$	12.0000	1.64833	0.824163	−	0.566352i	$-0.191646\pi$
$53$	12.0000	1.64833	0.824163	+	0.566352i	$0.191646\pi$
$54$	0	0
$55$	−2.82843	−0.381385
$56$	0	0
$57$	4.00000	0.529813
$58$	0	0
$59$	−9.89949	−1.28880	−0.644402	−	0.764687i	$-0.722894\pi$
$59$	−9.89949	−1.28880	−0.644402	+	0.764687i	$0.722894\pi$
$60$	0	0
$61$	−12.0000	−1.53644	−0.768221	−	0.640184i	$-0.778858\pi$
$61$	−12.0000	−1.53644	−0.768221	+	0.640184i	$0.778858\pi$
$62$	0	0
$63$	−21.2132	−2.67261
$64$	0	0
$65$	−2.00000	−0.248069
$66$	0	0
$67$	9.89949	1.20942	0.604708	−	0.796447i	$-0.293290\pi$
$67$	9.89949	1.20942	0.604708	+	0.796447i	$0.293290\pi$
$68$	0	0
$69$	−16.0000	−1.92617
$70$	0	0
$71$	−4.24264	−0.503509	−0.251754	−	0.967791i	$-0.581008\pi$
$71$	−4.24264	−0.503509	−0.251754	+	0.967791i	$0.581008\pi$
$72$	0	0
$73$	−14.0000	−1.63858	−0.819288	−	0.573382i	$-0.805631\pi$
$73$	−14.0000	−1.63858	−0.819288	+	0.573382i	$0.805631\pi$
$74$	0	0
$75$	2.82843	0.326599
$76$	0	0
$77$	6.00000	0.683763
$78$	0	0
$79$	8.48528	0.954669	0.477334	−	0.878722i	$-0.341603\pi$
$79$	8.48528	0.954669	0.477334	+	0.878722i	$0.341603\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	7.07107	0.776151	0.388075	−	0.921628i	$-0.373140\pi$
$83$	7.07107	0.776151	0.388075	+	0.921628i	$0.373140\pi$
$84$	0	0
$85$	−8.00000	−0.867722
$86$	0	0
$87$	5.65685	0.606478
$88$	0	0
$89$	14.0000	1.48400	0.741999	−	0.670402i	$-0.233878\pi$
$89$	14.0000	1.48400	0.741999	+	0.670402i	$0.233878\pi$
$90$	0	0
$91$	4.24264	0.444750
$92$	0	0
$93$	28.0000	2.90346
$94$	0	0
$95$	−2.82843	−0.290191
$96$	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
$99$	−7.07107	−0.710669
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	24.0000	2.34216
$106$	0	0
$107$	−2.82843	−0.273434	−0.136717	−	0.990610i	$-0.543655\pi$
$107$	−2.82843	−0.273434	−0.136717	+	0.990610i	$0.543655\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	−5.65685	−0.536925
$112$	0	0
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	11.3137	1.05501
$116$	0	0
$117$	−5.00000	−0.462250
$118$	0	0
$119$	16.9706	1.55569
$120$	0	0
$121$	−9.00000	−0.818182
$122$	0	0
$123$	16.9706	1.53018
$124$	0	0
$125$	−12.0000	−1.07331
$126$	0	0
$127$	16.9706	1.50589	0.752947	−	0.658081i	$-0.228632\pi$
$127$	16.9706	1.50589	0.752947	+	0.658081i	$0.228632\pi$
$128$	0	0
$129$	−32.0000	−2.81744
$130$	0	0
$131$	−2.82843	−0.247121	−0.123560	−	0.992337i	$-0.539431\pi$
$131$	−2.82843	−0.247121	−0.123560	+	0.992337i	$0.539431\pi$
$132$	0	0
$133$	6.00000	0.520266
$134$	0	0
$135$	−11.3137	−0.973729
$136$	0	0
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	0	0
$139$	2.82843	0.239904	0.119952	−	0.992780i	$-0.461726\pi$
$139$	2.82843	0.239904	0.119952	+	0.992780i	$0.461726\pi$
$140$	0	0
$141$	20.0000	1.68430
$142$	0	0
$143$	1.41421	0.118262
$144$	0	0
$145$	−4.00000	−0.332182
$146$	0	0
$147$	−31.1127	−2.56613
$148$	0	0
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	12.7279	1.03578	0.517892	−	0.855446i	$-0.326717\pi$
$151$	12.7279	1.03578	0.517892	+	0.855446i	$0.326717\pi$
$152$	0	0
$153$	−20.0000	−1.61690
$154$	0	0
$155$	−19.7990	−1.59029
$156$	0	0
$157$	22.0000	1.75579	0.877896	−	0.478852i	$-0.158947\pi$
$157$	22.0000	1.75579	0.877896	+	0.478852i	$0.158947\pi$
$158$	0	0
$159$	−33.9411	−2.69171
$160$	0	0
$161$	−24.0000	−1.89146
$162$	0	0
$163$	−1.41421	−0.110770	−0.0553849	−	0.998465i	$-0.517639\pi$
$163$	−1.41421	−0.110770	−0.0553849	+	0.998465i	$0.517639\pi$
$164$	0	0
$165$	8.00000	0.622799
$166$	0	0
$167$	−18.3848	−1.42266	−0.711328	−	0.702860i	$-0.751906\pi$
$167$	−18.3848	−1.42266	−0.711328	+	0.702860i	$0.751906\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−7.07107	−0.540738
$172$	0	0
$173$	16.0000	1.21646	0.608229	−	0.793762i	$-0.291880\pi$
$173$	16.0000	1.21646	0.608229	+	0.793762i	$0.291880\pi$
$174$	0	0
$175$	4.24264	0.320713
$176$	0	0
$177$	28.0000	2.10461
$178$	0	0
$179$	22.6274	1.69125	0.845626	−	0.533775i	$-0.179227\pi$
$179$	22.6274	1.69125	0.845626	+	0.533775i	$0.179227\pi$
$180$	0	0
$181$	26.0000	1.93256	0.966282	−	0.257485i	$-0.0828937\pi$
$181$	26.0000	1.93256	0.966282	+	0.257485i	$0.0828937\pi$
$182$	0	0
$183$	33.9411	2.50900
$184$	0	0
$185$	4.00000	0.294086
$186$	0	0
$187$	5.65685	0.413670
$188$	0	0
$189$	24.0000	1.74574
$190$	0	0
$191$	14.1421	1.02329	0.511645	−	0.859197i	$-0.329036\pi$
$191$	14.1421	1.02329	0.511645	+	0.859197i	$0.329036\pi$
$192$	0	0
$193$	22.0000	1.58359	0.791797	−	0.610784i	$-0.209146\pi$
$193$	22.0000	1.58359	0.791797	+	0.610784i	$0.209146\pi$
$194$	0	0
$195$	5.65685	0.405096
$196$	0	0
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	0	0
$199$	5.65685	0.401004	0.200502	−	0.979693i	$-0.435743\pi$
$199$	5.65685	0.401004	0.200502	+	0.979693i	$0.435743\pi$
$200$	0	0
$201$	−28.0000	−1.97497
$202$	0	0
$203$	8.48528	0.595550
$204$	0	0
$205$	−12.0000	−0.838116
$206$	0	0
$207$	28.2843	1.96589
$208$	0	0
$209$	2.00000	0.138343
$210$	0	0
$211$	−14.1421	−0.973585	−0.486792	−	0.873518i	$-0.661833\pi$
$211$	−14.1421	−0.973585	−0.486792	+	0.873518i	$0.661833\pi$
$212$	0	0
$213$	12.0000	0.822226
$214$	0	0
$215$	22.6274	1.54318
$216$	0	0
$217$	42.0000	2.85115
$218$	0	0
$219$	39.5980	2.67578
$220$	0	0
$221$	4.00000	0.269069
$222$	0	0
$223$	15.5563	1.04173	0.520865	−	0.853639i	$-0.325609\pi$
$223$	15.5563	1.04173	0.520865	+	0.853639i	$0.325609\pi$
$224$	0	0
$225$	−5.00000	−0.333333
$226$	0	0
$227$	−18.3848	−1.22024	−0.610120	−	0.792309i	$-0.708879\pi$
$227$	−18.3848	−1.22024	−0.610120	+	0.792309i	$0.708879\pi$
$228$	0	0
$229$	2.00000	0.132164	0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000	0.132164	0.0660819	+	0.997814i	$0.478950\pi$
$230$	0	0
$231$	−16.9706	−1.11658
$232$	0	0
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	−14.1421	−0.922531
$236$	0	0
$237$	−24.0000	−1.55897
$238$	0	0
$239$	−1.41421	−0.0914779	−0.0457389	−	0.998953i	$-0.514564\pi$
$239$	−1.41421	−0.0914779	−0.0457389	+	0.998953i	$0.514564\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	0	0
$243$	14.1421	0.907218
$244$	0	0
$245$	22.0000	1.40553
$246$	0	0
$247$	1.41421	0.0899843
$248$	0	0
$249$	−20.0000	−1.26745
$250$	0	0
$251$	−5.65685	−0.357057	−0.178529	−	0.983935i	$-0.557134\pi$
$251$	−5.65685	−0.357057	−0.178529	+	0.983935i	$0.557134\pi$
$252$	0	0
$253$	−8.00000	−0.502956
$254$	0	0
$255$	22.6274	1.41698
$256$	0	0
$257$	30.0000	1.87135	0.935674	−	0.352865i	$-0.114792\pi$
$257$	30.0000	1.87135	0.935674	+	0.352865i	$0.114792\pi$
$258$	0	0
$259$	−8.48528	−0.527250
$260$	0	0
$261$	−10.0000	−0.618984
$262$	0	0
$263$	8.48528	0.523225	0.261612	−	0.965173i	$-0.415746\pi$
$263$	8.48528	0.523225	0.261612	+	0.965173i	$0.415746\pi$
$264$	0	0
$265$	24.0000	1.47431
$266$	0	0
$267$	−39.5980	−2.42336
$268$	0	0
$269$	−14.0000	−0.853595	−0.426798	−	0.904347i	$-0.640358\pi$
$269$	−14.0000	−0.853595	−0.426798	+	0.904347i	$0.640358\pi$
$270$	0	0
$271$	26.8701	1.63224	0.816120	−	0.577883i	$-0.196121\pi$
$271$	26.8701	1.63224	0.816120	+	0.577883i	$0.196121\pi$
$272$	0	0
$273$	−12.0000	−0.726273
$274$	0	0
$275$	1.41421	0.0852803
$276$	0	0
$277$	8.00000	0.480673	0.240337	−	0.970690i	$-0.422742\pi$
$277$	8.00000	0.480673	0.240337	+	0.970690i	$0.422742\pi$
$278$	0	0
$279$	−49.4975	−2.96334
$280$	0	0
$281$	−22.0000	−1.31241	−0.656205	−	0.754583i	$-0.727839\pi$
$281$	−22.0000	−1.31241	−0.656205	+	0.754583i	$0.727839\pi$
$282$	0	0
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	0	0
$285$	8.00000	0.473879
$286$	0	0
$287$	25.4558	1.50261
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	5.65685	0.331611
$292$	0	0
$293$	14.0000	0.817889	0.408944	−	0.912559i	$-0.365897\pi$
$293$	14.0000	0.817889	0.408944	+	0.912559i	$0.365897\pi$
$294$	0	0
$295$	−19.7990	−1.15274
$296$	0	0
$297$	8.00000	0.464207
$298$	0	0
$299$	−5.65685	−0.327144
$300$	0	0
$301$	−48.0000	−2.76667
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−24.0000	−1.37424
$306$	0	0
$307$	−7.07107	−0.403567	−0.201784	−	0.979430i	$-0.564674\pi$
$307$	−7.07107	−0.403567	−0.201784	+	0.979430i	$0.564674\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	22.6274	1.28308	0.641542	−	0.767088i	$-0.278295\pi$
$311$	22.6274	1.28308	0.641542	+	0.767088i	$0.278295\pi$
$312$	0	0
$313$	16.0000	0.904373	0.452187	−	0.891923i	$-0.350644\pi$
$313$	16.0000	0.904373	0.452187	+	0.891923i	$0.350644\pi$
$314$	0	0
$315$	−42.4264	−2.39046
$316$	0	0
$317$	22.0000	1.23564	0.617822	−	0.786318i	$-0.288015\pi$
$317$	22.0000	1.23564	0.617822	+	0.786318i	$0.288015\pi$
$318$	0	0
$319$	2.82843	0.158362
$320$	0	0
$321$	8.00000	0.446516
$322$	0	0
$323$	5.65685	0.314756
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	−5.65685	−0.312825
$328$	0	0
$329$	30.0000	1.65395
$330$	0	0
$331$	−9.89949	−0.544125	−0.272063	−	0.962280i	$-0.587706\pi$
$331$	−9.89949	−0.544125	−0.272063	+	0.962280i	$0.587706\pi$
$332$	0	0
$333$	10.0000	0.547997
$334$	0	0
$335$	19.7990	1.08173
$336$	0	0
$337$	−12.0000	−0.653682	−0.326841	−	0.945079i	$-0.605984\pi$
$337$	−12.0000	−0.653682	−0.326841	+	0.945079i	$0.605984\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	14.0000	0.758143
$342$	0	0
$343$	−16.9706	−0.916324
$344$	0	0
$345$	−32.0000	−1.72282
$346$	0	0
$347$	−2.82843	−0.151838	−0.0759190	−	0.997114i	$-0.524189\pi$
$347$	−2.82843	−0.151838	−0.0759190	+	0.997114i	$0.524189\pi$
$348$	0	0
$349$	−22.0000	−1.17763	−0.588817	−	0.808267i	$-0.700406\pi$
$349$	−22.0000	−1.17763	−0.588817	+	0.808267i	$0.700406\pi$
$350$	0	0
$351$	5.65685	0.301941
$352$	0	0
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	0	0
$355$	−8.48528	−0.450352
$356$	0	0
$357$	−48.0000	−2.54043
$358$	0	0
$359$	18.3848	0.970311	0.485156	−	0.874428i	$-0.338763\pi$
$359$	18.3848	0.970311	0.485156	+	0.874428i	$0.338763\pi$
$360$	0	0
$361$	−17.0000	−0.894737
$362$	0	0
$363$	25.4558	1.33609
$364$	0	0
$365$	−28.0000	−1.46559
$366$	0	0
$367$	8.48528	0.442928	0.221464	−	0.975169i	$-0.428916\pi$
$367$	8.48528	0.442928	0.221464	+	0.975169i	$0.428916\pi$
$368$	0	0
$369$	−30.0000	−1.56174
$370$	0	0
$371$	−50.9117	−2.64320
$372$	0	0
$373$	2.00000	0.103556	0.0517780	−	0.998659i	$-0.483511\pi$
$373$	2.00000	0.103556	0.0517780	+	0.998659i	$0.483511\pi$
$374$	0	0
$375$	33.9411	1.75271
$376$	0	0
$377$	2.00000	0.103005
$378$	0	0
$379$	−9.89949	−0.508503	−0.254251	−	0.967138i	$-0.581829\pi$
$379$	−9.89949	−0.508503	−0.254251	+	0.967138i	$0.581829\pi$
$380$	0	0
$381$	−48.0000	−2.45911
$382$	0	0
$383$	−24.0416	−1.22847	−0.614235	−	0.789123i	$-0.710535\pi$
$383$	−24.0416	−1.22847	−0.614235	+	0.789123i	$0.710535\pi$
$384$	0	0
$385$	12.0000	0.611577
$386$	0	0
$387$	56.5685	2.87554
$388$	0	0
$389$	10.0000	0.507020	0.253510	−	0.967333i	$-0.418415\pi$
$389$	10.0000	0.507020	0.253510	+	0.967333i	$0.418415\pi$
$390$	0	0
$391$	−22.6274	−1.14432
$392$	0	0
$393$	8.00000	0.403547
$394$	0	0
$395$	16.9706	0.853882
$396$	0	0
$397$	22.0000	1.10415	0.552074	−	0.833795i	$-0.313837\pi$
$397$	22.0000	1.10415	0.552074	+	0.833795i	$0.313837\pi$
$398$	0	0
$399$	−16.9706	−0.849591
$400$	0	0
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	0	0
$403$	9.89949	0.493129
$404$	0	0
$405$	2.00000	0.0993808
$406$	0	0
$407$	−2.82843	−0.140200
$408$	0	0
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	0	0
$411$	50.9117	2.51129
$412$	0	0
$413$	42.0000	2.06668
$414$	0	0
$415$	14.1421	0.694210
$416$	0	0
$417$	−8.00000	−0.391762
$418$	0	0
$419$	14.1421	0.690889	0.345444	−	0.938439i	$-0.387728\pi$
$419$	14.1421	0.690889	0.345444	+	0.938439i	$0.387728\pi$
$420$	0	0
$421$	−18.0000	−0.877266	−0.438633	−	0.898666i	$-0.644537\pi$
$421$	−18.0000	−0.877266	−0.438633	+	0.898666i	$0.644537\pi$
$422$	0	0
$423$	−35.3553	−1.71904
$424$	0	0
$425$	4.00000	0.194029
$426$	0	0
$427$	50.9117	2.46379
$428$	0	0
$429$	−4.00000	−0.193122
$430$	0	0
$431$	−15.5563	−0.749323	−0.374661	−	0.927162i	$-0.622241\pi$
$431$	−15.5563	−0.749323	−0.374661	+	0.927162i	$0.622241\pi$
$432$	0	0
$433$	−18.0000	−0.865025	−0.432512	−	0.901628i	$-0.642373\pi$
$433$	−18.0000	−0.865025	−0.432512	+	0.901628i	$0.642373\pi$
$434$	0	0
$435$	11.3137	0.542451
$436$	0	0
$437$	−8.00000	−0.382692
$438$	0	0
$439$	−16.9706	−0.809961	−0.404980	−	0.914325i	$-0.632722\pi$
$439$	−16.9706	−0.809961	−0.404980	+	0.914325i	$0.632722\pi$
$440$	0	0
$441$	55.0000	2.61905
$442$	0	0
$443$	−19.7990	−0.940678	−0.470339	−	0.882486i	$-0.655868\pi$
$443$	−19.7990	−0.940678	−0.470339	+	0.882486i	$0.655868\pi$
$444$	0	0
$445$	28.0000	1.32733
$446$	0	0
$447$	−16.9706	−0.802680
$448$	0	0
$449$	10.0000	0.471929	0.235965	−	0.971762i	$-0.424175\pi$
$449$	10.0000	0.471929	0.235965	+	0.971762i	$0.424175\pi$
$450$	0	0
$451$	8.48528	0.399556
$452$	0	0
$453$	−36.0000	−1.69143
$454$	0	0
$455$	8.48528	0.397796
$456$	0	0
$457$	−10.0000	−0.467780	−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000	−0.467780	−0.233890	+	0.972263i	$0.575146\pi$
$458$	0	0
$459$	22.6274	1.05616
$460$	0	0
$461$	−34.0000	−1.58354	−0.791769	−	0.610821i	$-0.790840\pi$
$461$	−34.0000	−1.58354	−0.791769	+	0.610821i	$0.790840\pi$
$462$	0	0
$463$	9.89949	0.460069	0.230034	−	0.973183i	$-0.426116\pi$
$463$	9.89949	0.460069	0.230034	+	0.973183i	$0.426116\pi$
$464$	0	0
$465$	56.0000	2.59694
$466$	0	0
$467$	−16.9706	−0.785304	−0.392652	−	0.919687i	$-0.628442\pi$
$467$	−16.9706	−0.785304	−0.392652	+	0.919687i	$0.628442\pi$
$468$	0	0
$469$	−42.0000	−1.93938
$470$	0	0
$471$	−62.2254	−2.86719
$472$	0	0
$473$	−16.0000	−0.735681
$474$	0	0
$475$	1.41421	0.0648886
$476$	0	0
$477$	60.0000	2.74721
$478$	0	0
$479$	−21.2132	−0.969256	−0.484628	−	0.874720i	$-0.661045\pi$
$479$	−21.2132	−0.969256	−0.484628	+	0.874720i	$0.661045\pi$
$480$	0	0
$481$	−2.00000	−0.0911922
$482$	0	0
$483$	67.8823	3.08875
$484$	0	0
$485$	−4.00000	−0.181631
$486$	0	0
$487$	15.5563	0.704925	0.352463	−	0.935826i	$-0.385344\pi$
$487$	15.5563	0.704925	0.352463	+	0.935826i	$0.385344\pi$
$488$	0	0
$489$	4.00000	0.180886
$490$	0	0
$491$	11.3137	0.510581	0.255290	−	0.966864i	$-0.417829\pi$
$491$	11.3137	0.510581	0.255290	+	0.966864i	$0.417829\pi$
$492$	0	0
$493$	8.00000	0.360302
$494$	0	0
$495$	−14.1421	−0.635642
$496$	0	0
$497$	18.0000	0.807410
$498$	0	0
$499$	4.24264	0.189927	0.0949633	−	0.995481i	$-0.469727\pi$
$499$	4.24264	0.189927	0.0949633	+	0.995481i	$0.469727\pi$
$500$	0	0
$501$	52.0000	2.32319
$502$	0	0
$503$	2.82843	0.126113	0.0630567	−	0.998010i	$-0.479915\pi$
$503$	2.82843	0.126113	0.0630567	+	0.998010i	$0.479915\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−2.82843	−0.125615
$508$	0	0
$509$	6.00000	0.265945	0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000	0.265945	0.132973	+	0.991120i	$0.457548\pi$
$510$	0	0
$511$	59.3970	2.62757
$512$	0	0
$513$	8.00000	0.353209
$514$	0	0
$515$	0	0
$516$	0	0
$517$	10.0000	0.439799
$518$	0	0
$519$	−45.2548	−1.98647
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	0	0
$523$	−42.4264	−1.85518	−0.927589	−	0.373603i	$-0.878122\pi$
$523$	−42.4264	−1.85518	−0.927589	+	0.373603i	$0.878122\pi$
$524$	0	0
$525$	−12.0000	−0.523723
$526$	0	0
$527$	39.5980	1.72492
$528$	0	0
$529$	9.00000	0.391304
$530$	0	0
$531$	−49.4975	−2.14801
$532$	0	0
$533$	6.00000	0.259889
$534$	0	0
$535$	−5.65685	−0.244567
$536$	0	0
$537$	−64.0000	−2.76180
$538$	0	0
$539$	−15.5563	−0.670059
$540$	0	0
$541$	6.00000	0.257960	0.128980	−	0.991647i	$-0.458830\pi$
$541$	6.00000	0.257960	0.128980	+	0.991647i	$0.458830\pi$
$542$	0	0
$543$	−73.5391	−3.15587
$544$	0	0
$545$	4.00000	0.171341
$546$	0	0
$547$	−25.4558	−1.08841	−0.544207	−	0.838951i	$-0.683169\pi$
$547$	−25.4558	−1.08841	−0.544207	+	0.838951i	$0.683169\pi$
$548$	0	0
$549$	−60.0000	−2.56074
$550$	0	0
$551$	2.82843	0.120495
$552$	0	0
$553$	−36.0000	−1.53088
$554$	0	0
$555$	−11.3137	−0.480240
$556$	0	0
$557$	38.0000	1.61011	0.805056	−	0.593199i	$-0.202135\pi$
$557$	38.0000	1.61011	0.805056	+	0.593199i	$0.202135\pi$
$558$	0	0
$559$	−11.3137	−0.478519
$560$	0	0
$561$	−16.0000	−0.675521
$562$	0	0
$563$	−16.9706	−0.715224	−0.357612	−	0.933870i	$-0.616409\pi$
$563$	−16.9706	−0.715224	−0.357612	+	0.933870i	$0.616409\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−4.24264	−0.178174
$568$	0	0
$569$	−26.0000	−1.08998	−0.544988	−	0.838444i	$-0.683466\pi$
$569$	−26.0000	−1.08998	−0.544988	+	0.838444i	$0.683466\pi$
$570$	0	0
$571$	5.65685	0.236732	0.118366	−	0.992970i	$-0.462234\pi$
$571$	5.65685	0.236732	0.118366	+	0.992970i	$0.462234\pi$
$572$	0	0
$573$	−40.0000	−1.67102
$574$	0	0
$575$	−5.65685	−0.235907
$576$	0	0
$577$	−18.0000	−0.749350	−0.374675	−	0.927156i	$-0.622246\pi$
$577$	−18.0000	−0.749350	−0.374675	+	0.927156i	$0.622246\pi$
$578$	0	0
$579$	−62.2254	−2.58600
$580$	0	0
$581$	−30.0000	−1.24461
$582$	0	0
$583$	−16.9706	−0.702849
$584$	0	0
$585$	−10.0000	−0.413449
$586$	0	0
$587$	35.3553	1.45927	0.729636	−	0.683836i	$-0.239690\pi$
$587$	35.3553	1.45927	0.729636	+	0.683836i	$0.239690\pi$
$588$	0	0
$589$	14.0000	0.576860
$590$	0	0
$591$	−5.65685	−0.232692
$592$	0	0
$593$	22.0000	0.903432	0.451716	−	0.892162i	$-0.350812\pi$
$593$	22.0000	0.903432	0.451716	+	0.892162i	$0.350812\pi$
$594$	0	0
$595$	33.9411	1.39145
$596$	0	0
$597$	−16.0000	−0.654836
$598$	0	0
$599$	−31.1127	−1.27123	−0.635615	−	0.772006i	$-0.719253\pi$
$599$	−31.1127	−1.27123	−0.635615	+	0.772006i	$0.719253\pi$
$600$	0	0
$601$	16.0000	0.652654	0.326327	−	0.945257i	$-0.394189\pi$
$601$	16.0000	0.652654	0.326327	+	0.945257i	$0.394189\pi$
$602$	0	0
$603$	49.4975	2.01569
$604$	0	0
$605$	−18.0000	−0.731804
$606$	0	0
$607$	14.1421	0.574012	0.287006	−	0.957929i	$-0.407340\pi$
$607$	14.1421	0.574012	0.287006	+	0.957929i	$0.407340\pi$
$608$	0	0
$609$	−24.0000	−0.972529
$610$	0	0
$611$	7.07107	0.286065
$612$	0	0
$613$	−14.0000	−0.565455	−0.282727	−	0.959200i	$-0.591239\pi$
$613$	−14.0000	−0.565455	−0.282727	+	0.959200i	$0.591239\pi$
$614$	0	0
$615$	33.9411	1.36864
$616$	0	0
$617$	−22.0000	−0.885687	−0.442843	−	0.896599i	$-0.646030\pi$
$617$	−22.0000	−0.885687	−0.442843	+	0.896599i	$0.646030\pi$
$618$	0	0
$619$	−24.0416	−0.966315	−0.483157	−	0.875534i	$-0.660510\pi$
$619$	−24.0416	−0.966315	−0.483157	+	0.875534i	$0.660510\pi$
$620$	0	0
$621$	−32.0000	−1.28412
$622$	0	0
$623$	−59.3970	−2.37969
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	−5.65685	−0.225913
$628$	0	0
$629$	−8.00000	−0.318981
$630$	0	0
$631$	−9.89949	−0.394093	−0.197046	−	0.980394i	$-0.563135\pi$
$631$	−9.89949	−0.394093	−0.197046	+	0.980394i	$0.563135\pi$
$632$	0	0
$633$	40.0000	1.58986
$634$	0	0
$635$	33.9411	1.34691
$636$	0	0
$637$	−11.0000	−0.435836
$638$	0	0
$639$	−21.2132	−0.839181
$640$	0	0
$641$	−12.0000	−0.473972	−0.236986	−	0.971513i	$-0.576159\pi$
$641$	−12.0000	−0.473972	−0.236986	+	0.971513i	$0.576159\pi$
$642$	0	0
$643$	12.7279	0.501940	0.250970	−	0.967995i	$-0.419250\pi$
$643$	12.7279	0.501940	0.250970	+	0.967995i	$0.419250\pi$
$644$	0	0
$645$	−64.0000	−2.52000
$646$	0	0
$647$	33.9411	1.33436	0.667182	−	0.744895i	$-0.267500\pi$
$647$	33.9411	1.33436	0.667182	+	0.744895i	$0.267500\pi$
$648$	0	0
$649$	14.0000	0.549548
$650$	0	0
$651$	−118.794	−4.65590
$652$	0	0
$653$	22.0000	0.860927	0.430463	−	0.902608i	$-0.358350\pi$
$653$	22.0000	0.860927	0.430463	+	0.902608i	$0.358350\pi$
$654$	0	0
$655$	−5.65685	−0.221032
$656$	0	0
$657$	−70.0000	−2.73096
$658$	0	0
$659$	−2.82843	−0.110180	−0.0550899	−	0.998481i	$-0.517545\pi$
$659$	−2.82843	−0.110180	−0.0550899	+	0.998481i	$0.517545\pi$
$660$	0	0
$661$	50.0000	1.94477	0.972387	−	0.233373i	$-0.0749763\pi$
$661$	50.0000	1.94477	0.972387	+	0.233373i	$0.0749763\pi$
$662$	0	0
$663$	−11.3137	−0.439388
$664$	0	0
$665$	12.0000	0.465340
$666$	0	0
$667$	−11.3137	−0.438069
$668$	0	0
$669$	−44.0000	−1.70114
$670$	0	0
$671$	16.9706	0.655141
$672$	0	0
$673$	−44.0000	−1.69608	−0.848038	−	0.529936i	$-0.822216\pi$
$673$	−44.0000	−1.69608	−0.848038	+	0.529936i	$0.822216\pi$
$674$	0	0
$675$	5.65685	0.217732
$676$	0	0
$677$	20.0000	0.768662	0.384331	−	0.923195i	$-0.374432\pi$
$677$	20.0000	0.768662	0.384331	+	0.923195i	$0.374432\pi$
$678$	0	0
$679$	8.48528	0.325635
$680$	0	0
$681$	52.0000	1.99264
$682$	0	0
$683$	−12.7279	−0.487020	−0.243510	−	0.969898i	$-0.578299\pi$
$683$	−12.7279	−0.487020	−0.243510	+	0.969898i	$0.578299\pi$
$684$	0	0
$685$	−36.0000	−1.37549
$686$	0	0
$687$	−5.65685	−0.215822
$688$	0	0
$689$	−12.0000	−0.457164
$690$	0	0
$691$	38.1838	1.45258	0.726289	−	0.687389i	$-0.241243\pi$
$691$	38.1838	1.45258	0.726289	+	0.687389i	$0.241243\pi$
$692$	0	0
$693$	30.0000	1.13961
$694$	0	0
$695$	5.65685	0.214577
$696$	0	0
$697$	24.0000	0.909065
$698$	0	0
$699$	−16.9706	−0.641886
$700$	0	0
$701$	18.0000	0.679851	0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000	0.679851	0.339925	+	0.940452i	$0.389598\pi$
$702$	0	0
$703$	−2.82843	−0.106676
$704$	0	0
$705$	40.0000	1.50649
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	0	0
$711$	42.4264	1.59111
$712$	0	0
$713$	−56.0000	−2.09722
$714$	0	0
$715$	2.82843	0.105777
$716$	0	0
$717$	4.00000	0.149383
$718$	0	0
$719$	−33.9411	−1.26579	−0.632895	−	0.774237i	$-0.718134\pi$
$719$	−33.9411	−1.26579	−0.632895	+	0.774237i	$0.718134\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−28.2843	−1.05190
$724$	0	0
$725$	2.00000	0.0742781
$726$	0	0
$727$	22.6274	0.839204	0.419602	−	0.907708i	$-0.362170\pi$
$727$	22.6274	0.839204	0.419602	+	0.907708i	$0.362170\pi$
$728$	0	0
$729$	−43.0000	−1.59259
$730$	0	0
$731$	−45.2548	−1.67381
$732$	0	0
$733$	−30.0000	−1.10808	−0.554038	−	0.832492i	$-0.686914\pi$
$733$	−30.0000	−1.10808	−0.554038	+	0.832492i	$0.686914\pi$
$734$	0	0
$735$	−62.2254	−2.29522
$736$	0	0
$737$	−14.0000	−0.515697
$738$	0	0
$739$	−4.24264	−0.156068	−0.0780340	−	0.996951i	$-0.524864\pi$
$739$	−4.24264	−0.156068	−0.0780340	+	0.996951i	$0.524864\pi$
$740$	0	0
$741$	−4.00000	−0.146944
$742$	0	0
$743$	−9.89949	−0.363177	−0.181589	−	0.983375i	$-0.558124\pi$
$743$	−9.89949	−0.363177	−0.181589	+	0.983375i	$0.558124\pi$
$744$	0	0
$745$	12.0000	0.439646
$746$	0	0
$747$	35.3553	1.29358
$748$	0	0
$749$	12.0000	0.438470
$750$	0	0
$751$	−22.6274	−0.825686	−0.412843	−	0.910802i	$-0.635464\pi$
$751$	−22.6274	−0.825686	−0.412843	+	0.910802i	$0.635464\pi$
$752$	0	0
$753$	16.0000	0.583072
$754$	0	0
$755$	25.4558	0.926433
$756$	0	0
$757$	−44.0000	−1.59921	−0.799604	−	0.600528i	$-0.794957\pi$
$757$	−44.0000	−1.59921	−0.799604	+	0.600528i	$0.794957\pi$
$758$	0	0
$759$	22.6274	0.821323
$760$	0	0
$761$	14.0000	0.507500	0.253750	−	0.967270i	$-0.418336\pi$
$761$	14.0000	0.507500	0.253750	+	0.967270i	$0.418336\pi$
$762$	0	0
$763$	−8.48528	−0.307188
$764$	0	0
$765$	−40.0000	−1.44620
$766$	0	0
$767$	9.89949	0.357450
$768$	0	0
$769$	18.0000	0.649097	0.324548	−	0.945869i	$-0.394788\pi$
$769$	18.0000	0.649097	0.324548	+	0.945869i	$0.394788\pi$
$770$	0	0
$771$	−84.8528	−3.05590
$772$	0	0
$773$	42.0000	1.51064	0.755318	−	0.655359i	$-0.227483\pi$
$773$	42.0000	1.51064	0.755318	+	0.655359i	$0.227483\pi$
$774$	0	0
$775$	9.89949	0.355600
$776$	0	0
$777$	24.0000	0.860995
$778$	0	0
$779$	8.48528	0.304017
$780$	0	0
$781$	6.00000	0.214697
$782$	0	0
$783$	11.3137	0.404319
$784$	0	0
$785$	44.0000	1.57043
$786$	0	0
$787$	21.2132	0.756169	0.378085	−	0.925771i	$-0.376583\pi$
$787$	21.2132	0.756169	0.378085	+	0.925771i	$0.376583\pi$
$788$	0	0
$789$	−24.0000	−0.854423
$790$	0	0
$791$	0	0
$792$	0	0
$793$	12.0000	0.426132
$794$	0	0
$795$	−67.8823	−2.40754
$796$	0	0
$797$	−18.0000	−0.637593	−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000	−0.637593	−0.318796	+	0.947823i	$0.603279\pi$
$798$	0	0
$799$	28.2843	1.00063
$800$	0	0
$801$	70.0000	2.47333
$802$	0	0
$803$	19.7990	0.698691
$804$	0	0
$805$	−48.0000	−1.69178
$806$	0	0
$807$	39.5980	1.39391
$808$	0	0
$809$	48.0000	1.68759	0.843795	−	0.536666i	$-0.180316\pi$
$809$	48.0000	1.68759	0.843795	+	0.536666i	$0.180316\pi$
$810$	0	0
$811$	−18.3848	−0.645577	−0.322788	−	0.946471i	$-0.604620\pi$
$811$	−18.3848	−0.645577	−0.322788	+	0.946471i	$0.604620\pi$
$812$	0	0
$813$	−76.0000	−2.66544
$814$	0	0
$815$	−2.82843	−0.0990755
$816$	0	0
$817$	−16.0000	−0.559769
$818$	0	0
$819$	21.2132	0.741249
$820$	0	0
$821$	−6.00000	−0.209401	−0.104701	−	0.994504i	$-0.533388\pi$
$821$	−6.00000	−0.209401	−0.104701	+	0.994504i	$0.533388\pi$
$822$	0	0
$823$	−5.65685	−0.197186	−0.0985928	−	0.995128i	$-0.531434\pi$
$823$	−5.65685	−0.197186	−0.0985928	+	0.995128i	$0.531434\pi$
$824$	0	0
$825$	−4.00000	−0.139262
$826$	0	0
$827$	35.3553	1.22943	0.614713	−	0.788751i	$-0.289272\pi$
$827$	35.3553	1.22943	0.614713	+	0.788751i	$0.289272\pi$
$828$	0	0
$829$	−40.0000	−1.38926	−0.694629	−	0.719368i	$-0.744431\pi$
$829$	−40.0000	−1.38926	−0.694629	+	0.719368i	$0.744431\pi$
$830$	0	0
$831$	−22.6274	−0.784936
$832$	0	0
$833$	−44.0000	−1.52451
$834$	0	0
$835$	−36.7696	−1.27246
$836$	0	0
$837$	56.0000	1.93564
$838$	0	0
$839$	12.7279	0.439417	0.219708	−	0.975566i	$-0.429489\pi$
$839$	12.7279	0.439417	0.219708	+	0.975566i	$0.429489\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	62.2254	2.14316
$844$	0	0
$845$	2.00000	0.0688021
$846$	0	0
$847$	38.1838	1.31201
$848$	0	0
$849$	0	0
$850$	0	0
$851$	11.3137	0.387829
$852$	0	0
$853$	−2.00000	−0.0684787	−0.0342393	−	0.999414i	$-0.510901\pi$
$853$	−2.00000	−0.0684787	−0.0342393	+	0.999414i	$0.510901\pi$
$854$	0	0
$855$	−14.1421	−0.483651
$856$	0	0
$857$	−22.0000	−0.751506	−0.375753	−	0.926720i	$-0.622616\pi$
$857$	−22.0000	−0.751506	−0.375753	+	0.926720i	$0.622616\pi$
$858$	0	0
$859$	36.7696	1.25456	0.627280	−	0.778793i	$-0.284168\pi$
$859$	36.7696	1.25456	0.627280	+	0.778793i	$0.284168\pi$
$860$	0	0
$861$	−72.0000	−2.45375
$862$	0	0
$863$	7.07107	0.240702	0.120351	−	0.992731i	$-0.461598\pi$
$863$	7.07107	0.240702	0.120351	+	0.992731i	$0.461598\pi$
$864$	0	0
$865$	32.0000	1.08803
$866$	0	0
$867$	2.82843	0.0960584
$868$	0	0
$869$	−12.0000	−0.407072
$870$	0	0
$871$	−9.89949	−0.335432
$872$	0	0
$873$	−10.0000	−0.338449
$874$	0	0
$875$	50.9117	1.72113
$876$	0	0
$877$	6.00000	0.202606	0.101303	−	0.994856i	$-0.467699\pi$
$877$	6.00000	0.202606	0.101303	+	0.994856i	$0.467699\pi$
$878$	0	0
$879$	−39.5980	−1.33561
$880$	0	0
$881$	−30.0000	−1.01073	−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000	−1.01073	−0.505363	+	0.862907i	$0.668641\pi$
$882$	0	0
$883$	−33.9411	−1.14221	−0.571105	−	0.820877i	$-0.693485\pi$
$883$	−33.9411	−1.14221	−0.571105	+	0.820877i	$0.693485\pi$
$884$	0	0
$885$	56.0000	1.88242
$886$	0	0
$887$	31.1127	1.04466	0.522331	−	0.852743i	$-0.325063\pi$
$887$	31.1127	1.04466	0.522331	+	0.852743i	$0.325063\pi$
$888$	0	0
$889$	−72.0000	−2.41480
$890$	0	0
$891$	−1.41421	−0.0473779
$892$	0	0
$893$	10.0000	0.334637
$894$	0	0
$895$	45.2548	1.51270
$896$	0	0
$897$	16.0000	0.534224
$898$	0	0
$899$	19.7990	0.660333
$900$	0	0
$901$	−48.0000	−1.59911
$902$	0	0
$903$	135.765	4.51796
$904$	0	0
$905$	52.0000	1.72854
$906$	0	0
$907$	−28.2843	−0.939164	−0.469582	−	0.882889i	$-0.655595\pi$
$907$	−28.2843	−0.939164	−0.469582	+	0.882889i	$0.655595\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−48.0833	−1.59307	−0.796535	−	0.604593i	$-0.793336\pi$
$911$	−48.0833	−1.59307	−0.796535	+	0.604593i	$0.793336\pi$
$912$	0	0
$913$	−10.0000	−0.330952
$914$	0	0
$915$	67.8823	2.24412
$916$	0	0
$917$	12.0000	0.396275
$918$	0	0
$919$	−36.7696	−1.21292	−0.606458	−	0.795116i	$-0.707410\pi$
$919$	−36.7696	−1.21292	−0.606458	+	0.795116i	$0.707410\pi$
$920$	0	0
$921$	20.0000	0.659022
$922$	0	0
$923$	4.24264	0.139648
$924$	0	0
$925$	−2.00000	−0.0657596
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−22.0000	−0.721797	−0.360898	−	0.932605i	$-0.617530\pi$
$929$	−22.0000	−0.721797	−0.360898	+	0.932605i	$0.617530\pi$
$930$	0	0
$931$	−15.5563	−0.509839
$932$	0	0
$933$	−64.0000	−2.09527
$934$	0	0
$935$	11.3137	0.369998
$936$	0	0
$937$	−26.0000	−0.849383	−0.424691	−	0.905338i	$-0.639617\pi$
$937$	−26.0000	−0.849383	−0.424691	+	0.905338i	$0.639617\pi$
$938$	0	0
$939$	−45.2548	−1.47684
$940$	0	0
$941$	−14.0000	−0.456387	−0.228193	−	0.973616i	$-0.573282\pi$
$941$	−14.0000	−0.456387	−0.228193	+	0.973616i	$0.573282\pi$
$942$	0	0
$943$	−33.9411	−1.10528
$944$	0	0
$945$	48.0000	1.56144
$946$	0	0
$947$	41.0122	1.33272	0.666359	−	0.745631i	$-0.267852\pi$
$947$	41.0122	1.33272	0.666359	+	0.745631i	$0.267852\pi$
$948$	0	0
$949$	14.0000	0.454459
$950$	0	0
$951$	−62.2254	−2.01780
$952$	0	0
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	0	0
$955$	28.2843	0.915258
$956$	0	0
$957$	−8.00000	−0.258603
$958$	0	0
$959$	76.3675	2.46604
$960$	0	0
$961$	67.0000	2.16129
$962$	0	0
$963$	−14.1421	−0.455724
$964$	0	0
$965$	44.0000	1.41641
$966$	0	0
$967$	−9.89949	−0.318346	−0.159173	−	0.987251i	$-0.550883\pi$
$967$	−9.89949	−0.318346	−0.159173	+	0.987251i	$0.550883\pi$
$968$	0	0
$969$	−16.0000	−0.513994
$970$	0	0
$971$	48.0833	1.54307	0.771533	−	0.636190i	$-0.219490\pi$
$971$	48.0833	1.54307	0.771533	+	0.636190i	$0.219490\pi$
$972$	0	0
$973$	−12.0000	−0.384702
$974$	0	0
$975$	−2.82843	−0.0905822
$976$	0	0
$977$	−58.0000	−1.85558	−0.927792	−	0.373097i	$-0.878296\pi$
$977$	−58.0000	−1.85558	−0.927792	+	0.373097i	$0.878296\pi$
$978$	0	0
$979$	−19.7990	−0.632778
$980$	0	0
$981$	10.0000	0.319275
$982$	0	0
$983$	−35.3553	−1.12766	−0.563830	−	0.825891i	$-0.690673\pi$
$983$	−35.3553	−1.12766	−0.563830	+	0.825891i	$0.690673\pi$
$984$	0	0
$985$	4.00000	0.127451
$986$	0	0
$987$	−84.8528	−2.70089
$988$	0	0
$989$	64.0000	2.03508
$990$	0	0
$991$	25.4558	0.808632	0.404316	−	0.914619i	$-0.367510\pi$
$991$	25.4558	0.808632	0.404316	+	0.914619i	$0.367510\pi$
$992$	0	0
$993$	28.0000	0.888553
$994$	0	0
$995$	11.3137	0.358669
$996$	0	0
$997$	−52.0000	−1.64686	−0.823428	−	0.567420i	$-0.807941\pi$
$997$	−52.0000	−1.64686	−0.823428	+	0.567420i	$0.807941\pi$
$998$	0	0
$999$	−11.3137	−0.357950

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3328.2.a.x.1.1		2
4.3	odd	2	inner	3328.2.a.x.1.2		2
8.3	odd	2		3328.2.a.t.1.1		2
8.5	even	2		3328.2.a.t.1.2		2
16.3	odd	4		1664.2.b.f.833.3	yes	4
16.5	even	4		1664.2.b.f.833.4	yes	4
16.11	odd	4		1664.2.b.f.833.2	yes	4
16.13	even	4		1664.2.b.f.833.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1664.2.b.f.833.1	✓	4	16.13	even	4
1664.2.b.f.833.2	yes	4	16.11	odd	4
1664.2.b.f.833.3	yes	4	16.3	odd	4
1664.2.b.f.833.4	yes	4	16.5	even	4
3328.2.a.t.1.1		2	8.3	odd	2
3328.2.a.t.1.2		2	8.5	even	2
3328.2.a.x.1.1		2	1.1	even	1	trivial
3328.2.a.x.1.2		2	4.3	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-2.82843 q^{3} +2.00000 q^{5} -4.24264 q^{7} +5.00000 q^{9} -1.41421 q^{11} -1.00000 q^{13} -5.65685 q^{15} -4.00000 q^{17} -1.41421 q^{19} +12.0000 q^{21} +5.65685 q^{23} -1.00000 q^{25} -5.65685 q^{27} -2.00000 q^{29} -9.89949 q^{31} +4.00000 q^{33} -8.48528 q^{35} +2.00000 q^{37} +2.82843 q^{39} -6.00000 q^{41} +11.3137 q^{43} +10.0000 q^{45} -7.07107 q^{47} +11.0000 q^{49} +11.3137 q^{51} +12.0000 q^{53} -2.82843 q^{55} +4.00000 q^{57} -9.89949 q^{59} -12.0000 q^{61} -21.2132 q^{63} -2.00000 q^{65} +9.89949 q^{67} -16.0000 q^{69} -4.24264 q^{71} -14.0000 q^{73} +2.82843 q^{75} +6.00000 q^{77} +8.48528 q^{79} +1.00000 q^{81} +7.07107 q^{83} -8.00000 q^{85} +5.65685 q^{87} +14.0000 q^{89} +4.24264 q^{91} +28.0000 q^{93} -2.82843 q^{95} -2.00000 q^{97} -7.07107 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{5} + 10 q^{9} - 2 q^{13} - 8 q^{17} + 24 q^{21} - 2 q^{25} - 4 q^{29} + 8 q^{33} + 4 q^{37} - 12 q^{41} + 20 q^{45} + 22 q^{49} + 24 q^{53} + 8 q^{57} - 24 q^{61} - 4 q^{65} - 32 q^{69} - 28 q^{73}+ \cdots - 4 q^{97}+O(q^{100}) \) 2 * q + 4 * q^5 + 10 * q^9 - 2 * q^13 - 8 * q^17 + 24 * q^21 - 2 * q^25 - 4 * q^29 + 8 * q^33 + 4 * q^37 - 12 * q^41 + 20 * q^45 + 22 * q^49 + 24 * q^53 + 8 * q^57 - 24 * q^61 - 4 * q^65 - 32 * q^69 - 28 * q^73 + 12 * q^77 + 2 * q^81 - 16 * q^85 + 28 * q^89 + 56 * q^93 - 4 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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