LMFDB - Embedded newform 3328.2.a.s.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,-6] 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:	$ 3 $
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)

$f(q)$	$=$	$q-2.00000 q^{5} -4.24264 q^{7} -3.00000 q^{9} -4.24264 q^{11} -1.00000 q^{13} +4.00000 q^{17} -4.24264 q^{19} -1.00000 q^{25} -2.00000 q^{29} -4.24264 q^{31} +8.48528 q^{35} -6.00000 q^{37} +2.00000 q^{41} +8.48528 q^{43} +6.00000 q^{45} -12.7279 q^{47} +11.0000 q^{49} -8.00000 q^{53} +8.48528 q^{55} +4.24264 q^{59} +12.7279 q^{63} +2.00000 q^{65} -4.24264 q^{67} -4.24264 q^{71} +6.00000 q^{73} +18.0000 q^{77} +8.48528 q^{79} +9.00000 q^{81} -12.7279 q^{83} -8.00000 q^{85} +10.0000 q^{89} +4.24264 q^{91} +8.48528 q^{95} +18.0000 q^{97} +12.7279 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{5} - 6 q^{9} - 2 q^{13} + 8 q^{17} - 2 q^{25} - 4 q^{29} - 12 q^{37} + 4 q^{41} + 12 q^{45} + 22 q^{49} - 16 q^{53} + 4 q^{65} + 12 q^{73} + 36 q^{77} + 18 q^{81} - 16 q^{85} + 20 q^{89} + 36 q^{97}+O(q^{100}) $ 2 * q - 4 * q^5 - 6 * q^9 - 2 * q^13 + 8 * q^17 - 2 * q^25 - 4 * q^29 - 12 * q^37 + 4 * q^41 + 12 * q^45 + 22 * q^49 - 16 * q^53 + 4 * q^65 + 12 * q^73 + 36 * q^77 + 18 * q^81 - 16 * q^85 + 20 * q^89 + 36 * 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
$3$	0	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	0	0
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	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
$9$	−3.00000	−1.00000
$10$	0	0
$11$	−4.24264	−1.27920	−0.639602	−	0.768706i	$-0.720901\pi$
$11$	−4.24264	−1.27920	−0.639602	+	0.768706i	$0.720901\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	0	0
$19$	−4.24264	−0.973329	−0.486664	−	0.873589i	$-0.661786\pi$
$19$	−4.24264	−0.973329	−0.486664	+	0.873589i	$0.661786\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	0	0
$28$	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
$31$	−4.24264	−0.762001	−0.381000	−	0.924575i	$-0.624420\pi$
$31$	−4.24264	−0.762001	−0.381000	+	0.924575i	$0.624420\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	8.48528	1.43427
$36$	0	0
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	8.48528	1.29399	0.646997	−	0.762493i	$-0.276025\pi$
$43$	8.48528	1.29399	0.646997	+	0.762493i	$0.276025\pi$
$44$	0	0
$45$	6.00000	0.894427
$46$	0	0
$47$	−12.7279	−1.85656	−0.928279	−	0.371884i	$-0.878712\pi$
$47$	−12.7279	−1.85656	−0.928279	+	0.371884i	$0.878712\pi$
$48$	0	0
$49$	11.0000	1.57143
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−8.00000	−1.09888	−0.549442	−	0.835532i	$-0.685160\pi$
$53$	−8.00000	−1.09888	−0.549442	+	0.835532i	$0.685160\pi$
$54$	0	0
$55$	8.48528	1.14416
$56$	0	0
$57$	0	0
$58$	0	0
$59$	4.24264	0.552345	0.276172	−	0.961108i	$-0.410934\pi$
$59$	4.24264	0.552345	0.276172	+	0.961108i	$0.410934\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	0	0
$63$	12.7279	1.60357
$64$	0	0
$65$	2.00000	0.248069
$66$	0	0
$67$	−4.24264	−0.518321	−0.259161	−	0.965834i	$-0.583446\pi$
$67$	−4.24264	−0.518321	−0.259161	+	0.965834i	$0.583446\pi$
$68$	0	0
$69$	0	0
$70$	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
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	18.0000	2.05129
$78$	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
$81$	9.00000	1.00000
$82$	0	0
$83$	−12.7279	−1.39707	−0.698535	−	0.715575i	$-0.746165\pi$
$83$	−12.7279	−1.39707	−0.698535	+	0.715575i	$0.746165\pi$
$84$	0	0
$85$	−8.00000	−0.867722
$86$	0	0
$87$	0	0
$88$	0	0
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	0	0
$91$	4.24264	0.444750
$92$	0	0
$93$	0	0
$94$	0	0
$95$	8.48528	0.870572
$96$	0	0
$97$	18.0000	1.82762	0.913812	−	0.406138i	$-0.133125\pi$
$97$	18.0000	1.82762	0.913812	+	0.406138i	$0.133125\pi$
$98$	0	0
$99$	12.7279	1.27920
$100$	0	0
$101$	−4.00000	−0.398015	−0.199007	−	0.979998i	$-0.563772\pi$
$101$	−4.00000	−0.398015	−0.199007	+	0.979998i	$0.563772\pi$
$102$	0	0
$103$	16.9706	1.67216	0.836080	−	0.548608i	$-0.184842\pi$
$103$	16.9706	1.67216	0.836080	+	0.548608i	$0.184842\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−16.9706	−1.64061	−0.820303	−	0.571929i	$-0.806195\pi$
$107$	−16.9706	−1.64061	−0.820303	+	0.571929i	$0.806195\pi$
$108$	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
$111$	0	0
$112$	0	0
$113$	16.0000	1.50515	0.752577	−	0.658505i	$-0.228811\pi$
$113$	16.0000	1.50515	0.752577	+	0.658505i	$0.228811\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	3.00000	0.277350
$118$	0	0
$119$	−16.9706	−1.55569
$120$	0	0
$121$	7.00000	0.636364
$122$	0	0
$123$	0	0
$124$	0	0
$125$	12.0000	1.07331
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	16.9706	1.48272	0.741362	−	0.671105i	$-0.234180\pi$
$131$	16.9706	1.48272	0.741362	+	0.671105i	$0.234180\pi$
$132$	0	0
$133$	18.0000	1.56080
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	0	0
$139$	−16.9706	−1.43942	−0.719712	−	0.694273i	$-0.755726\pi$
$139$	−16.9706	−1.43942	−0.719712	+	0.694273i	$0.755726\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	4.24264	0.354787
$144$	0	0
$145$	4.00000	0.332182
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	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
$153$	−12.0000	−0.970143
$154$	0	0
$155$	8.48528	0.681554
$156$	0	0
$157$	−18.0000	−1.43656	−0.718278	−	0.695756i	$-0.755069\pi$
$157$	−18.0000	−1.43656	−0.718278	+	0.695756i	$0.755069\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−21.2132	−1.66155	−0.830773	−	0.556611i	$-0.812101\pi$
$163$	−21.2132	−1.66155	−0.830773	+	0.556611i	$0.812101\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	4.24264	0.328305	0.164153	−	0.986435i	$-0.447511\pi$
$167$	4.24264	0.328305	0.164153	+	0.986435i	$0.447511\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	12.7279	0.973329
$172$	0	0
$173$	4.00000	0.304114	0.152057	−	0.988372i	$-0.451410\pi$
$173$	4.00000	0.304114	0.152057	+	0.988372i	$0.451410\pi$
$174$	0	0
$175$	4.24264	0.320713
$176$	0	0
$177$	0	0
$178$	0	0
$179$	8.48528	0.634220	0.317110	−	0.948389i	$-0.397288\pi$
$179$	8.48528	0.634220	0.317110	+	0.948389i	$0.397288\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	12.0000	0.882258
$186$	0	0
$187$	−16.9706	−1.24101
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−25.4558	−1.84192	−0.920960	−	0.389657i	$-0.872594\pi$
$191$	−25.4558	−1.84192	−0.920960	+	0.389657i	$0.872594\pi$
$192$	0	0
$193$	−22.0000	−1.58359	−0.791797	−	0.610784i	$-0.790854\pi$
$193$	−22.0000	−1.58359	−0.791797	+	0.610784i	$0.790854\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	26.0000	1.85242	0.926212	−	0.377004i	$-0.123046\pi$
$197$	26.0000	1.85242	0.926212	+	0.377004i	$0.123046\pi$
$198$	0	0
$199$	−16.9706	−1.20301	−0.601506	−	0.798869i	$-0.705432\pi$
$199$	−16.9706	−1.20301	−0.601506	+	0.798869i	$0.705432\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	8.48528	0.595550
$204$	0	0
$205$	−4.00000	−0.279372
$206$	0	0
$207$	0	0
$208$	0	0
$209$	18.0000	1.24509
$210$	0	0
$211$	−16.9706	−1.16830	−0.584151	−	0.811645i	$-0.698572\pi$
$211$	−16.9706	−1.16830	−0.584151	+	0.811645i	$0.698572\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−16.9706	−1.15738
$216$	0	0
$217$	18.0000	1.22192
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−4.00000	−0.269069
$222$	0	0
$223$	−12.7279	−0.852325	−0.426162	−	0.904647i	$-0.640135\pi$
$223$	−12.7279	−0.852325	−0.426162	+	0.904647i	$0.640135\pi$
$224$	0	0
$225$	3.00000	0.200000
$226$	0	0
$227$	−21.2132	−1.40797	−0.703985	−	0.710215i	$-0.748598\pi$
$227$	−21.2132	−1.40797	−0.703985	+	0.710215i	$0.748598\pi$
$228$	0	0
$229$	22.0000	1.45380	0.726900	−	0.686743i	$-0.240960\pi$
$229$	22.0000	1.45380	0.726900	+	0.686743i	$0.240960\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−10.0000	−0.655122	−0.327561	−	0.944830i	$-0.606227\pi$
$233$	−10.0000	−0.655122	−0.327561	+	0.944830i	$0.606227\pi$
$234$	0	0
$235$	25.4558	1.66056
$236$	0	0
$237$	0	0
$238$	0	0
$239$	4.24264	0.274434	0.137217	−	0.990541i	$-0.456184\pi$
$239$	4.24264	0.274434	0.137217	+	0.990541i	$0.456184\pi$
$240$	0	0
$241$	−18.0000	−1.15948	−0.579741	−	0.814801i	$-0.696846\pi$
$241$	−18.0000	−1.15948	−0.579741	+	0.814801i	$0.696846\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−22.0000	−1.40553
$246$	0	0
$247$	4.24264	0.269953
$248$	0	0
$249$	0	0
$250$	0	0
$251$	8.48528	0.535586	0.267793	−	0.963476i	$-0.413706\pi$
$251$	8.48528	0.535586	0.267793	+	0.963476i	$0.413706\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	14.0000	0.873296	0.436648	−	0.899632i	$-0.356166\pi$
$257$	14.0000	0.873296	0.436648	+	0.899632i	$0.356166\pi$
$258$	0	0
$259$	25.4558	1.58175
$260$	0	0
$261$	6.00000	0.371391
$262$	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
$265$	16.0000	0.982872
$266$	0	0
$267$	0	0
$268$	0	0
$269$	2.00000	0.121942	0.0609711	−	0.998140i	$-0.480580\pi$
$269$	2.00000	0.121942	0.0609711	+	0.998140i	$0.480580\pi$
$270$	0	0
$271$	−12.7279	−0.773166	−0.386583	−	0.922255i	$-0.626345\pi$
$271$	−12.7279	−0.773166	−0.386583	+	0.922255i	$0.626345\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4.24264	0.255841
$276$	0	0
$277$	−28.0000	−1.68236	−0.841178	−	0.540758i	$-0.818138\pi$
$277$	−28.0000	−1.68236	−0.841178	+	0.540758i	$0.818138\pi$
$278$	0	0
$279$	12.7279	0.762001
$280$	0	0
$281$	22.0000	1.31241	0.656205	−	0.754583i	$-0.272161\pi$
$281$	22.0000	1.31241	0.656205	+	0.754583i	$0.272161\pi$
$282$	0	0
$283$	−8.48528	−0.504398	−0.252199	−	0.967675i	$-0.581154\pi$
$283$	−8.48528	−0.504398	−0.252199	+	0.967675i	$0.581154\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−8.48528	−0.500870
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−2.00000	−0.116841	−0.0584206	−	0.998292i	$-0.518606\pi$
$293$	−2.00000	−0.116841	−0.0584206	+	0.998292i	$0.518606\pi$
$294$	0	0
$295$	−8.48528	−0.494032
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−36.0000	−2.07501
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	29.6985	1.69498	0.847491	−	0.530810i	$-0.178112\pi$
$307$	29.6985	1.69498	0.847491	+	0.530810i	$0.178112\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−16.9706	−0.962312	−0.481156	−	0.876635i	$-0.659783\pi$
$311$	−16.9706	−0.962312	−0.481156	+	0.876635i	$0.659783\pi$
$312$	0	0
$313$	−8.00000	−0.452187	−0.226093	−	0.974106i	$-0.572595\pi$
$313$	−8.00000	−0.452187	−0.226093	+	0.974106i	$0.572595\pi$
$314$	0	0
$315$	−25.4558	−1.43427
$316$	0	0
$317$	2.00000	0.112331	0.0561656	−	0.998421i	$-0.482113\pi$
$317$	2.00000	0.112331	0.0561656	+	0.998421i	$0.482113\pi$
$318$	0	0
$319$	8.48528	0.475085
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−16.9706	−0.944267
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	0	0
$328$	0	0
$329$	54.0000	2.97712
$330$	0	0
$331$	−12.7279	−0.699590	−0.349795	−	0.936826i	$-0.613749\pi$
$331$	−12.7279	−0.699590	−0.349795	+	0.936826i	$0.613749\pi$
$332$	0	0
$333$	18.0000	0.986394
$334$	0	0
$335$	8.48528	0.463600
$336$	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
$341$	18.0000	0.974755
$342$	0	0
$343$	−16.9706	−0.916324
$344$	0	0
$345$	0	0
$346$	0	0
$347$	33.9411	1.82206	0.911028	−	0.412346i	$-0.135290\pi$
$347$	33.9411	1.82206	0.911028	+	0.412346i	$0.135290\pi$
$348$	0	0
$349$	−18.0000	−0.963518	−0.481759	−	0.876304i	$-0.660002\pi$
$349$	−18.0000	−0.963518	−0.481759	+	0.876304i	$0.660002\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	10.0000	0.532246	0.266123	−	0.963939i	$-0.414257\pi$
$353$	10.0000	0.532246	0.266123	+	0.963939i	$0.414257\pi$
$354$	0	0
$355$	8.48528	0.450352
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−4.24264	−0.223918	−0.111959	−	0.993713i	$-0.535713\pi$
$359$	−4.24264	−0.223918	−0.111959	+	0.993713i	$0.535713\pi$
$360$	0	0
$361$	−1.00000	−0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−12.0000	−0.628109
$366$	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
$369$	−6.00000	−0.312348
$370$	0	0
$371$	33.9411	1.76214
$372$	0	0
$373$	−6.00000	−0.310668	−0.155334	−	0.987862i	$-0.549645\pi$
$373$	−6.00000	−0.310668	−0.155334	+	0.987862i	$0.549645\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.00000	0.103005
$378$	0	0
$379$	−12.7279	−0.653789	−0.326895	−	0.945061i	$-0.606002\pi$
$379$	−12.7279	−0.653789	−0.326895	+	0.945061i	$0.606002\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	4.24264	0.216789	0.108394	−	0.994108i	$-0.465429\pi$
$383$	4.24264	0.216789	0.108394	+	0.994108i	$0.465429\pi$
$384$	0	0
$385$	−36.0000	−1.83473
$386$	0	0
$387$	−25.4558	−1.29399
$388$	0	0
$389$	26.0000	1.31825	0.659126	−	0.752032i	$-0.270926\pi$
$389$	26.0000	1.31825	0.659126	+	0.752032i	$0.270926\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−16.9706	−0.853882
$396$	0	0
$397$	−30.0000	−1.50566	−0.752828	−	0.658217i	$-0.771311\pi$
$397$	−30.0000	−1.50566	−0.752828	+	0.658217i	$0.771311\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−26.0000	−1.29838	−0.649189	−	0.760627i	$-0.724892\pi$
$401$	−26.0000	−1.29838	−0.649189	+	0.760627i	$0.724892\pi$
$402$	0	0
$403$	4.24264	0.211341
$404$	0	0
$405$	−18.0000	−0.894427
$406$	0	0
$407$	25.4558	1.26180
$408$	0	0
$409$	30.0000	1.48340	0.741702	−	0.670729i	$-0.234019\pi$
$409$	30.0000	1.48340	0.741702	+	0.670729i	$0.234019\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−18.0000	−0.885722
$414$	0	0
$415$	25.4558	1.24958
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−16.9706	−0.829066	−0.414533	−	0.910034i	$-0.636055\pi$
$419$	−16.9706	−0.829066	−0.414533	+	0.910034i	$0.636055\pi$
$420$	0	0
$421$	26.0000	1.26716	0.633581	−	0.773676i	$-0.281584\pi$
$421$	26.0000	1.26716	0.633581	+	0.773676i	$0.281584\pi$
$422$	0	0
$423$	38.1838	1.85656
$424$	0	0
$425$	−4.00000	−0.194029
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	12.7279	0.613082	0.306541	−	0.951857i	$-0.400828\pi$
$431$	12.7279	0.613082	0.306541	+	0.951857i	$0.400828\pi$
$432$	0	0
$433$	−34.0000	−1.63394	−0.816968	−	0.576683i	$-0.804347\pi$
$433$	−34.0000	−1.63394	−0.816968	+	0.576683i	$0.804347\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	16.9706	0.809961	0.404980	−	0.914325i	$-0.367278\pi$
$439$	16.9706	0.809961	0.404980	+	0.914325i	$0.367278\pi$
$440$	0	0
$441$	−33.0000	−1.57143
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	−20.0000	−0.948091
$446$	0	0
$447$	0	0
$448$	0	0
$449$	14.0000	0.660701	0.330350	−	0.943858i	$-0.392833\pi$
$449$	14.0000	0.660701	0.330350	+	0.943858i	$0.392833\pi$
$450$	0	0
$451$	−8.48528	−0.399556
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−8.48528	−0.397796
$456$	0	0
$457$	−18.0000	−0.842004	−0.421002	−	0.907060i	$-0.638322\pi$
$457$	−18.0000	−0.842004	−0.421002	+	0.907060i	$0.638322\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	14.0000	0.652045	0.326023	−	0.945362i	$-0.394291\pi$
$461$	14.0000	0.652045	0.326023	+	0.945362i	$0.394291\pi$
$462$	0	0
$463$	38.1838	1.77455	0.887275	−	0.461241i	$-0.152596\pi$
$463$	38.1838	1.77455	0.887275	+	0.461241i	$0.152596\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−8.48528	−0.392652	−0.196326	−	0.980539i	$-0.562901\pi$
$467$	−8.48528	−0.392652	−0.196326	+	0.980539i	$0.562901\pi$
$468$	0	0
$469$	18.0000	0.831163
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−36.0000	−1.65528
$474$	0	0
$475$	4.24264	0.194666
$476$	0	0
$477$	24.0000	1.09888
$478$	0	0
$479$	−4.24264	−0.193851	−0.0969256	−	0.995292i	$-0.530901\pi$
$479$	−4.24264	−0.193851	−0.0969256	+	0.995292i	$0.530901\pi$
$480$	0	0
$481$	6.00000	0.273576
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−36.0000	−1.63468
$486$	0	0
$487$	−29.6985	−1.34577	−0.672883	−	0.739749i	$-0.734944\pi$
$487$	−29.6985	−1.34577	−0.672883	+	0.739749i	$0.734944\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	25.4558	1.14881	0.574403	−	0.818573i	$-0.305234\pi$
$491$	25.4558	1.14881	0.574403	+	0.818573i	$0.305234\pi$
$492$	0	0
$493$	−8.00000	−0.360302
$494$	0	0
$495$	−25.4558	−1.14416
$496$	0	0
$497$	18.0000	0.807410
$498$	0	0
$499$	−21.2132	−0.949633	−0.474817	−	0.880085i	$-0.657486\pi$
$499$	−21.2132	−0.949633	−0.474817	+	0.880085i	$0.657486\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	8.48528	0.378340	0.189170	−	0.981944i	$-0.439420\pi$
$503$	8.48528	0.378340	0.189170	+	0.981944i	$0.439420\pi$
$504$	0	0
$505$	8.00000	0.355995
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−38.0000	−1.68432	−0.842160	−	0.539227i	$-0.818716\pi$
$509$	−38.0000	−1.68432	−0.842160	+	0.539227i	$0.818716\pi$
$510$	0	0
$511$	−25.4558	−1.12610
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−33.9411	−1.49562
$516$	0	0
$517$	54.0000	2.37492
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−8.00000	−0.350486	−0.175243	−	0.984525i	$-0.556071\pi$
$521$	−8.00000	−0.350486	−0.175243	+	0.984525i	$0.556071\pi$
$522$	0	0
$523$	16.9706	0.742071	0.371035	−	0.928619i	$-0.379003\pi$
$523$	16.9706	0.742071	0.371035	+	0.928619i	$0.379003\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−16.9706	−0.739249
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	−12.7279	−0.552345
$532$	0	0
$533$	−2.00000	−0.0866296
$534$	0	0
$535$	33.9411	1.46740
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−46.6690	−2.01018
$540$	0	0
$541$	38.0000	1.63375	0.816874	−	0.576816i	$-0.195705\pi$
$541$	38.0000	1.63375	0.816874	+	0.576816i	$0.195705\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−4.00000	−0.171341
$546$	0	0
$547$	0	0		−	1.00000i	$-0.5\pi$
$547$	0	0			1.00000i	$0.5\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	8.48528	0.361485
$552$	0	0
$553$	−36.0000	−1.53088
$554$	0	0
$555$	0	0
$556$	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
$559$	−8.48528	−0.358889
$560$	0	0
$561$	0	0
$562$	0	0
$563$	25.4558	1.07284	0.536418	−	0.843952i	$-0.319777\pi$
$563$	25.4558	1.07284	0.536418	+	0.843952i	$0.319777\pi$
$564$	0	0
$565$	−32.0000	−1.34625
$566$	0	0
$567$	−38.1838	−1.60357
$568$	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
$571$	8.48528	0.355098	0.177549	−	0.984112i	$-0.443183\pi$
$571$	8.48528	0.355098	0.177549	+	0.984112i	$0.443183\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−34.0000	−1.41544	−0.707719	−	0.706494i	$-0.750276\pi$
$577$	−34.0000	−1.41544	−0.707719	+	0.706494i	$0.750276\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	54.0000	2.24030
$582$	0	0
$583$	33.9411	1.40570
$584$	0	0
$585$	−6.00000	−0.248069
$586$	0	0
$587$	−12.7279	−0.525338	−0.262669	−	0.964886i	$-0.584603\pi$
$587$	−12.7279	−0.525338	−0.262669	+	0.964886i	$0.584603\pi$
$588$	0	0
$589$	18.0000	0.741677
$590$	0	0
$591$	0	0
$592$	0	0
$593$	14.0000	0.574911	0.287456	−	0.957794i	$-0.407191\pi$
$593$	14.0000	0.574911	0.287456	+	0.957794i	$0.407191\pi$
$594$	0	0
$595$	33.9411	1.39145
$596$	0	0
$597$	0	0
$598$	0	0
$599$	8.48528	0.346699	0.173350	−	0.984860i	$-0.444541\pi$
$599$	8.48528	0.346699	0.173350	+	0.984860i	$0.444541\pi$
$600$	0	0
$601$	−8.00000	−0.326327	−0.163163	−	0.986599i	$-0.552170\pi$
$601$	−8.00000	−0.326327	−0.163163	+	0.986599i	$0.552170\pi$
$602$	0	0
$603$	12.7279	0.518321
$604$	0	0
$605$	−14.0000	−0.569181
$606$	0	0
$607$	42.4264	1.72203	0.861017	−	0.508576i	$-0.169828\pi$
$607$	42.4264	1.72203	0.861017	+	0.508576i	$0.169828\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	12.7279	0.514917
$612$	0	0
$613$	−6.00000	−0.242338	−0.121169	−	0.992632i	$-0.538664\pi$
$613$	−6.00000	−0.242338	−0.121169	+	0.992632i	$0.538664\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	22.0000	0.885687	0.442843	−	0.896599i	$-0.353970\pi$
$617$	22.0000	0.885687	0.442843	+	0.896599i	$0.353970\pi$
$618$	0	0
$619$	29.6985	1.19368	0.596841	−	0.802359i	$-0.296422\pi$
$619$	29.6985	1.19368	0.596841	+	0.802359i	$0.296422\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−42.4264	−1.69978
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−24.0000	−0.956943
$630$	0	0
$631$	−21.2132	−0.844484	−0.422242	−	0.906483i	$-0.638757\pi$
$631$	−21.2132	−0.844484	−0.422242	+	0.906483i	$0.638757\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−11.0000	−0.435836
$638$	0	0
$639$	12.7279	0.503509
$640$	0	0
$641$	−20.0000	−0.789953	−0.394976	−	0.918691i	$-0.629247\pi$
$641$	−20.0000	−0.789953	−0.394976	+	0.918691i	$0.629247\pi$
$642$	0	0
$643$	38.1838	1.50582	0.752910	−	0.658123i	$-0.228649\pi$
$643$	38.1838	1.50582	0.752910	+	0.658123i	$0.228649\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	−18.0000	−0.706562
$650$	0	0
$651$	0	0
$652$	0	0
$653$	46.0000	1.80012	0.900060	−	0.435767i	$-0.143523\pi$
$653$	46.0000	1.80012	0.900060	+	0.435767i	$0.143523\pi$
$654$	0	0
$655$	−33.9411	−1.32619
$656$	0	0
$657$	−18.0000	−0.702247
$658$	0	0
$659$	−16.9706	−0.661079	−0.330540	−	0.943792i	$-0.607231\pi$
$659$	−16.9706	−0.661079	−0.330540	+	0.943792i	$0.607231\pi$
$660$	0	0
$661$	−18.0000	−0.700119	−0.350059	−	0.936727i	$-0.613839\pi$
$661$	−18.0000	−0.700119	−0.350059	+	0.936727i	$0.613839\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−36.0000	−1.39602
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	28.0000	1.07932	0.539660	−	0.841883i	$-0.318553\pi$
$673$	28.0000	1.07932	0.539660	+	0.841883i	$0.318553\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	16.0000	0.614930	0.307465	−	0.951559i	$-0.400519\pi$
$677$	16.0000	0.614930	0.307465	+	0.951559i	$0.400519\pi$
$678$	0	0
$679$	−76.3675	−2.93072
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−38.1838	−1.46106	−0.730531	−	0.682880i	$-0.760727\pi$
$683$	−38.1838	−1.46106	−0.730531	+	0.682880i	$0.760727\pi$
$684$	0	0
$685$	4.00000	0.152832
$686$	0	0
$687$	0	0
$688$	0	0
$689$	8.00000	0.304776
$690$	0	0
$691$	12.7279	0.484193	0.242096	−	0.970252i	$-0.422165\pi$
$691$	12.7279	0.484193	0.242096	+	0.970252i	$0.422165\pi$
$692$	0	0
$693$	−54.0000	−2.05129
$694$	0	0
$695$	33.9411	1.28746
$696$	0	0
$697$	8.00000	0.303022
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−38.0000	−1.43524	−0.717620	−	0.696435i	$-0.754769\pi$
$701$	−38.0000	−1.43524	−0.717620	+	0.696435i	$0.754769\pi$
$702$	0	0
$703$	25.4558	0.960085
$704$	0	0
$705$	0	0
$706$	0	0
$707$	16.9706	0.638244
$708$	0	0
$709$	10.0000	0.375558	0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000	0.375558	0.187779	+	0.982211i	$0.439871\pi$
$710$	0	0
$711$	−25.4558	−0.954669
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−8.48528	−0.317332
$716$	0	0
$717$	0	0
$718$	0	0
$719$	16.9706	0.632895	0.316448	−	0.948610i	$-0.397510\pi$
$719$	16.9706	0.632895	0.316448	+	0.948610i	$0.397510\pi$
$720$	0	0
$721$	−72.0000	−2.68142
$722$	0	0
$723$	0	0
$724$	0	0
$725$	2.00000	0.0742781
$726$	0	0
$727$	−50.9117	−1.88821	−0.944105	−	0.329645i	$-0.893071\pi$
$727$	−50.9117	−1.88821	−0.944105	+	0.329645i	$0.893071\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	33.9411	1.25536
$732$	0	0
$733$	22.0000	0.812589	0.406294	−	0.913742i	$-0.366821\pi$
$733$	22.0000	0.812589	0.406294	+	0.913742i	$0.366821\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	18.0000	0.663039
$738$	0	0
$739$	38.1838	1.40461	0.702306	−	0.711875i	$-0.252154\pi$
$739$	38.1838	1.40461	0.702306	+	0.711875i	$0.252154\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	12.7279	0.466942	0.233471	−	0.972364i	$-0.424992\pi$
$743$	12.7279	0.466942	0.233471	+	0.972364i	$0.424992\pi$
$744$	0	0
$745$	20.0000	0.732743
$746$	0	0
$747$	38.1838	1.39707
$748$	0	0
$749$	72.0000	2.63082
$750$	0	0
$751$	−16.9706	−0.619265	−0.309632	−	0.950856i	$-0.600206\pi$
$751$	−16.9706	−0.619265	−0.309632	+	0.950856i	$0.600206\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−25.4558	−0.926433
$756$	0	0
$757$	16.0000	0.581530	0.290765	−	0.956795i	$-0.406090\pi$
$757$	16.0000	0.581530	0.290765	+	0.956795i	$0.406090\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−34.0000	−1.23250	−0.616250	−	0.787551i	$-0.711349\pi$
$761$	−34.0000	−1.23250	−0.616250	+	0.787551i	$0.711349\pi$
$762$	0	0
$763$	−8.48528	−0.307188
$764$	0	0
$765$	24.0000	0.867722
$766$	0	0
$767$	−4.24264	−0.153193
$768$	0	0
$769$	22.0000	0.793340	0.396670	−	0.917961i	$-0.370166\pi$
$769$	22.0000	0.793340	0.396670	+	0.917961i	$0.370166\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−26.0000	−0.935155	−0.467578	−	0.883952i	$-0.654873\pi$
$773$	−26.0000	−0.935155	−0.467578	+	0.883952i	$0.654873\pi$
$774$	0	0
$775$	4.24264	0.152400
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−8.48528	−0.304017
$780$	0	0
$781$	18.0000	0.644091
$782$	0	0
$783$	0	0
$784$	0	0
$785$	36.0000	1.28490
$786$	0	0
$787$	−21.2132	−0.756169	−0.378085	−	0.925771i	$-0.623417\pi$
$787$	−21.2132	−0.756169	−0.378085	+	0.925771i	$0.623417\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−67.8823	−2.41361
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−2.00000	−0.0708436	−0.0354218	−	0.999372i	$-0.511277\pi$
$797$	−2.00000	−0.0708436	−0.0354218	+	0.999372i	$0.511277\pi$
$798$	0	0
$799$	−50.9117	−1.80113
$800$	0	0
$801$	−30.0000	−1.06000
$802$	0	0
$803$	−25.4558	−0.898317
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−40.0000	−1.40633	−0.703163	−	0.711029i	$-0.748229\pi$
$809$	−40.0000	−1.40633	−0.703163	+	0.711029i	$0.748229\pi$
$810$	0	0
$811$	−4.24264	−0.148979	−0.0744896	−	0.997222i	$-0.523733\pi$
$811$	−4.24264	−0.148979	−0.0744896	+	0.997222i	$0.523733\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	42.4264	1.48613
$816$	0	0
$817$	−36.0000	−1.25948
$818$	0	0
$819$	−12.7279	−0.444750
$820$	0	0
$821$	−10.0000	−0.349002	−0.174501	−	0.984657i	$-0.555831\pi$
$821$	−10.0000	−0.349002	−0.174501	+	0.984657i	$0.555831\pi$
$822$	0	0
$823$	33.9411	1.18311	0.591557	−	0.806263i	$-0.298514\pi$
$823$	33.9411	1.18311	0.591557	+	0.806263i	$0.298514\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	4.24264	0.147531	0.0737655	−	0.997276i	$-0.476498\pi$
$827$	4.24264	0.147531	0.0737655	+	0.997276i	$0.476498\pi$
$828$	0	0
$829$	−52.0000	−1.80603	−0.903017	−	0.429604i	$-0.858653\pi$
$829$	−52.0000	−1.80603	−0.903017	+	0.429604i	$0.858653\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	44.0000	1.52451
$834$	0	0
$835$	−8.48528	−0.293645
$836$	0	0
$837$	0	0
$838$	0	0
$839$	46.6690	1.61119	0.805597	−	0.592464i	$-0.201845\pi$
$839$	46.6690	1.61119	0.805597	+	0.592464i	$0.201845\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−2.00000	−0.0688021
$846$	0	0
$847$	−29.6985	−1.02045
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−6.00000	−0.205436	−0.102718	−	0.994711i	$-0.532754\pi$
$853$	−6.00000	−0.205436	−0.102718	+	0.994711i	$0.532754\pi$
$854$	0	0
$855$	−25.4558	−0.870572
$856$	0	0
$857$	26.0000	0.888143	0.444072	−	0.895991i	$-0.353534\pi$
$857$	26.0000	0.888143	0.444072	+	0.895991i	$0.353534\pi$
$858$	0	0
$859$	0	0		−	1.00000i	$-0.5\pi$
$859$	0	0			1.00000i	$0.5\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	46.6690	1.58863	0.794316	−	0.607504i	$-0.207829\pi$
$863$	46.6690	1.58863	0.794316	+	0.607504i	$0.207829\pi$
$864$	0	0
$865$	−8.00000	−0.272008
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−36.0000	−1.22122
$870$	0	0
$871$	4.24264	0.143756
$872$	0	0
$873$	−54.0000	−1.82762
$874$	0	0
$875$	−50.9117	−1.72113
$876$	0	0
$877$	−54.0000	−1.82345	−0.911725	−	0.410801i	$-0.865249\pi$
$877$	−54.0000	−1.82345	−0.911725	+	0.410801i	$0.865249\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	2.00000	0.0673817	0.0336909	−	0.999432i	$-0.489274\pi$
$881$	2.00000	0.0673817	0.0336909	+	0.999432i	$0.489274\pi$
$882$	0	0
$883$	−42.4264	−1.42776	−0.713881	−	0.700267i	$-0.753064\pi$
$883$	−42.4264	−1.42776	−0.713881	+	0.700267i	$0.753064\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	42.4264	1.42454	0.712270	−	0.701906i	$-0.247667\pi$
$887$	42.4264	1.42454	0.712270	+	0.701906i	$0.247667\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−38.1838	−1.27920
$892$	0	0
$893$	54.0000	1.80704
$894$	0	0
$895$	−16.9706	−0.567263
$896$	0	0
$897$	0	0
$898$	0	0
$899$	8.48528	0.283000
$900$	0	0
$901$	−32.0000	−1.06607
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−4.00000	−0.132964
$906$	0	0
$907$	−42.4264	−1.40875	−0.704373	−	0.709830i	$-0.748772\pi$
$907$	−42.4264	−1.40875	−0.704373	+	0.709830i	$0.748772\pi$
$908$	0	0
$909$	12.0000	0.398015
$910$	0	0
$911$	8.48528	0.281130	0.140565	−	0.990071i	$-0.455108\pi$
$911$	8.48528	0.281130	0.140565	+	0.990071i	$0.455108\pi$
$912$	0	0
$913$	54.0000	1.78714
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−72.0000	−2.37765
$918$	0	0
$919$	42.4264	1.39952	0.699759	−	0.714379i	$-0.253291\pi$
$919$	42.4264	1.39952	0.699759	+	0.714379i	$0.253291\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	4.24264	0.139648
$924$	0	0
$925$	6.00000	0.197279
$926$	0	0
$927$	−50.9117	−1.67216
$928$	0	0
$929$	22.0000	0.721797	0.360898	−	0.932605i	$-0.382470\pi$
$929$	22.0000	0.721797	0.360898	+	0.932605i	$0.382470\pi$
$930$	0	0
$931$	−46.6690	−1.52952
$932$	0	0
$933$	0	0
$934$	0	0
$935$	33.9411	1.10999
$936$	0	0
$937$	−2.00000	−0.0653372	−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000	−0.0653372	−0.0326686	+	0.999466i	$0.510401\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−34.0000	−1.10837	−0.554184	−	0.832394i	$-0.686970\pi$
$941$	−34.0000	−1.10837	−0.554184	+	0.832394i	$0.686970\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	4.24264	0.137867	0.0689336	−	0.997621i	$-0.478040\pi$
$947$	4.24264	0.137867	0.0689336	+	0.997621i	$0.478040\pi$
$948$	0	0
$949$	−6.00000	−0.194768
$950$	0	0
$951$	0	0
$952$	0	0
$953$	10.0000	0.323932	0.161966	−	0.986796i	$-0.448217\pi$
$953$	10.0000	0.323932	0.161966	+	0.986796i	$0.448217\pi$
$954$	0	0
$955$	50.9117	1.64746
$956$	0	0
$957$	0	0
$958$	0	0
$959$	8.48528	0.274004
$960$	0	0
$961$	−13.0000	−0.419355
$962$	0	0
$963$	50.9117	1.64061
$964$	0	0
$965$	44.0000	1.41641
$966$	0	0
$967$	46.6690	1.50078	0.750388	−	0.660998i	$-0.229867\pi$
$967$	46.6690	1.50078	0.750388	+	0.660998i	$0.229867\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	33.9411	1.08922	0.544611	−	0.838689i	$-0.316677\pi$
$971$	33.9411	1.08922	0.544611	+	0.838689i	$0.316677\pi$
$972$	0	0
$973$	72.0000	2.30821
$974$	0	0
$975$	0	0
$976$	0	0
$977$	50.0000	1.59964	0.799821	−	0.600239i	$-0.204928\pi$
$977$	50.0000	1.59964	0.799821	+	0.600239i	$0.204928\pi$
$978$	0	0
$979$	−42.4264	−1.35595
$980$	0	0
$981$	−6.00000	−0.191565
$982$	0	0
$983$	21.2132	0.676596	0.338298	−	0.941039i	$-0.390149\pi$
$983$	21.2132	0.676596	0.338298	+	0.941039i	$0.390149\pi$
$984$	0	0
$985$	−52.0000	−1.65686
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−25.4558	−0.808632	−0.404316	−	0.914619i	$-0.632490\pi$
$991$	−25.4558	−0.808632	−0.404316	+	0.914619i	$0.632490\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	33.9411	1.07601
$996$	0	0
$997$	48.0000	1.52018	0.760088	−	0.649821i	$-0.225156\pi$
$997$	48.0000	1.52018	0.760088	+	0.649821i	$0.225156\pi$
$998$	0	0
$999$	0	0

Twists

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

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

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)

\(f(q)\)	\(=\)	\(q-2.00000 q^{5} -4.24264 q^{7} -3.00000 q^{9} -4.24264 q^{11} -1.00000 q^{13} +4.00000 q^{17} -4.24264 q^{19} -1.00000 q^{25} -2.00000 q^{29} -4.24264 q^{31} +8.48528 q^{35} -6.00000 q^{37} +2.00000 q^{41} +8.48528 q^{43} +6.00000 q^{45} -12.7279 q^{47} +11.0000 q^{49} -8.00000 q^{53} +8.48528 q^{55} +4.24264 q^{59} +12.7279 q^{63} +2.00000 q^{65} -4.24264 q^{67} -4.24264 q^{71} +6.00000 q^{73} +18.0000 q^{77} +8.48528 q^{79} +9.00000 q^{81} -12.7279 q^{83} -8.00000 q^{85} +10.0000 q^{89} +4.24264 q^{91} +8.48528 q^{95} +18.0000 q^{97} +12.7279 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{5} - 6 q^{9} - 2 q^{13} + 8 q^{17} - 2 q^{25} - 4 q^{29} - 12 q^{37} + 4 q^{41} + 12 q^{45} + 22 q^{49} - 16 q^{53} + 4 q^{65} + 12 q^{73} + 36 q^{77} + 18 q^{81} - 16 q^{85} + 20 q^{89} + 36 q^{97}+O(q^{100}) \) 2 * q - 4 * q^5 - 6 * q^9 - 2 * q^13 + 8 * q^17 - 2 * q^25 - 4 * q^29 - 12 * q^37 + 4 * q^41 + 12 * q^45 + 22 * q^49 - 16 * q^53 + 4 * q^65 + 12 * q^73 + 36 * q^77 + 18 * q^81 - 16 * q^85 + 20 * q^89 + 36 * q^97

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