LMFDB - Embedded newform 3328.2.b.bb.1665.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3328,2,Mod(1665,3328)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3328, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3328.1665");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3328 = 2^{8} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3328.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$26.5742137927$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.11574317056.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 45x^{4} + 4 $ x^8 + 45*x^4 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 416)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1665.8
Root			$1.83051 + 1.83051i$ of defining polynomial
Character	$\chi$	$=$	3328.1665
Dual form			3328.2.b.bb.1665.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.11473i q^{3} +3.70156i q^{5} -4.20732 q^{7} -6.70156 q^{9} +1.09259i q^{11} +1.00000i q^{13} -11.5294 q^{15} +0.298438 q^{17} +1.09259i q^{19} -13.1047i q^{21} -8.70156 q^{25} -11.5294i q^{27} -2.00000i q^{29} -5.13688 q^{31} -3.40312 q^{33} -15.5737i q^{35} +3.70156i q^{37} -3.11473 q^{39} +9.40312 q^{41} +5.29991i q^{43} -24.8062i q^{45} +4.20732 q^{47} +10.7016 q^{49} +0.929554i q^{51} -1.40312i q^{53} -4.04429 q^{55} -3.40312 q^{57} +13.5515i q^{59} +9.40312i q^{61} +28.1956 q^{63} -3.70156 q^{65} -11.3663i q^{67} -8.25161 q^{71} +6.00000 q^{73} -27.1030i q^{75} -4.59688i q^{77} -14.6441 q^{79} +15.8062 q^{81} +7.32206i q^{83} +1.10469i q^{85} +6.22947 q^{87} +6.00000 q^{89} -4.20732i q^{91} -16.0000i q^{93} -4.04429 q^{95} +8.80625 q^{97} -7.32206i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 28 q^{9} + 28 q^{17} - 44 q^{25} + 24 q^{33} + 24 q^{41} + 60 q^{49} + 24 q^{57} - 4 q^{65} + 48 q^{73} + 24 q^{81} + 48 q^{89} - 32 q^{97}+O(q^{100}) $ 8 * q - 28 * q^9 + 28 * q^17 - 44 * q^25 + 24 * q^33 + 24 * q^41 + 60 * q^49 + 24 * q^57 - 4 * q^65 + 48 * q^73 + 24 * q^81 + 48 * q^89 - 32 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3328\mathbb{Z}\right)^\times$.

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	3.11473i	1.79829i	0.437649	+	0.899146i	$0.355811\pi$
$3$	3.11473i	1.79829i	−0.437649	+	0.899146i	$0.644189\pi$
$4$	0	0
$5$	3.70156i	1.65539i	0.561179	+	0.827694i	$0.310348\pi$
$5$	3.70156i	1.65539i	−0.561179	+	0.827694i	$0.689652\pi$
$6$	0	0
$7$	−4.20732	−1.59022	−0.795109	−	0.606466i	$-0.792587\pi$
$7$	−4.20732	−1.59022	−0.795109	+	0.606466i	$0.792587\pi$
$8$	0	0
$9$	−6.70156	−2.23385
$10$	0	0
$11$	1.09259i	0.329428i	0.986341	+	0.164714i	$0.0526701\pi$
$11$	1.09259i	0.329428i	−0.986341	+	0.164714i	$0.947330\pi$
$12$	0	0
$13$	1.00000i	0.277350i
$13$	1.00000i	0.277350i
$14$	0	0
$15$	−11.5294	−2.97687
$16$	0	0
$17$	0.298438	0.0723818	0.0361909	−	0.999345i	$-0.488478\pi$
$17$	0.298438	0.0723818	0.0361909	+	0.999345i	$0.488478\pi$
$18$	0	0
$19$	1.09259i	0.250657i	0.992115	+	0.125329i	$0.0399985\pi$
$19$	1.09259i	0.250657i	−0.992115	+	0.125329i	$0.960001\pi$
$20$	0	0
$21$	− 13.1047i	− 2.85968i
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	−8.70156	−1.74031
$26$	0	0
$27$	− 11.5294i	− 2.21883i
$28$	0	0
$29$	− 2.00000i	− 0.371391i	−0.982607	−	0.185695i	$-0.940546\pi$
$29$	− 2.00000i	− 0.371391i	0.982607	−	0.185695i	$-0.0594537\pi$
$30$	0	0
$31$	−5.13688	−0.922610	−0.461305	−	0.887242i	$-0.652619\pi$
$31$	−5.13688	−0.922610	−0.461305	+	0.887242i	$0.652619\pi$
$32$	0	0
$33$	−3.40312	−0.592408
$34$	0	0
$35$	− 15.5737i	− 2.63243i
$36$	0	0
$37$	3.70156i	0.608533i	0.952587	+	0.304267i	$0.0984113\pi$
$37$	3.70156i	0.608533i	−0.952587	+	0.304267i	$0.901589\pi$
$38$	0	0
$39$	−3.11473	−0.498756
$40$	0	0
$41$	9.40312	1.46852	0.734261	−	0.678868i	$-0.237529\pi$
$41$	9.40312	1.46852	0.734261	+	0.678868i	$0.237529\pi$
$42$	0	0
$43$	5.29991i	0.808229i	0.914708	+	0.404114i	$0.132420\pi$
$43$	5.29991i	0.808229i	−0.914708	+	0.404114i	$0.867580\pi$
$44$	0	0
$45$	− 24.8062i	− 3.69790i
$46$	0	0
$47$	4.20732	0.613701	0.306851	−	0.951758i	$-0.400725\pi$
$47$	4.20732	0.613701	0.306851	+	0.951758i	$0.400725\pi$
$48$	0	0
$49$	10.7016	1.52879
$50$	0	0
$51$	0.929554i	0.130164i
$52$	0	0
$53$	− 1.40312i	− 0.192734i	−0.995346	−	0.0963670i	$-0.969278\pi$
$53$	− 1.40312i	− 0.192734i	0.995346	−	0.0963670i	$-0.0307222\pi$
$54$	0	0
$55$	−4.04429	−0.545332
$56$	0	0
$57$	−3.40312	−0.450755
$58$	0	0
$59$	13.5515i	1.76426i	0.471008	+	0.882129i	$0.343890\pi$
$59$	13.5515i	1.76426i	−0.471008	+	0.882129i	$0.656110\pi$
$60$	0	0
$61$	9.40312i	1.20395i	0.798516	+	0.601973i	$0.205619\pi$
$61$	9.40312i	1.20395i	−0.798516	+	0.601973i	$0.794381\pi$
$62$	0	0
$63$	28.1956	3.55232
$64$	0	0
$65$	−3.70156	−0.459122
$66$	0	0
$67$	− 11.3663i	− 1.38862i	−0.719676	−	0.694310i	$-0.755710\pi$
$67$	− 11.3663i	− 1.38862i	0.719676	−	0.694310i	$-0.244290\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−8.25161	−0.979286	−0.489643	−	0.871923i	$-0.662873\pi$
$71$	−8.25161	−0.979286	−0.489643	+	0.871923i	$0.662873\pi$
$72$	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
$75$	− 27.1030i	− 3.12959i
$76$	0	0
$77$	− 4.59688i	− 0.523863i
$78$	0	0
$79$	−14.6441	−1.64759	−0.823796	−	0.566887i	$-0.808148\pi$
$79$	−14.6441	−1.64759	−0.823796	+	0.566887i	$0.808148\pi$
$80$	0	0
$81$	15.8062	1.75625
$82$	0	0
$83$	7.32206i	0.803700i	0.915706	+	0.401850i	$0.131633\pi$
$83$	7.32206i	0.803700i	−0.915706	+	0.401850i	$0.868367\pi$
$84$	0	0
$85$	1.10469i	0.119820i
$86$	0	0
$87$	6.22947	0.667869
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	− 4.20732i	− 0.441047i
$92$	0	0
$93$	− 16.0000i	− 1.65912i
$94$	0	0
$95$	−4.04429	−0.414935
$96$	0	0
$97$	8.80625	0.894139	0.447070	−	0.894499i	$-0.352468\pi$
$97$	8.80625	0.894139	0.447070	+	0.894499i	$0.352468\pi$
$98$	0	0
$99$	− 7.32206i	− 0.735894i
$100$	0	0
$101$	5.40312i	0.537631i	0.963192	+	0.268815i	$0.0866322\pi$
$101$	5.40312i	0.537631i	−0.963192	+	0.268815i	$0.913368\pi$
$102$	0	0
$103$	−12.4589	−1.22762	−0.613808	−	0.789456i	$-0.710363\pi$
$103$	−12.4589	−1.22762	−0.613808	+	0.789456i	$0.710363\pi$
$104$	0	0
$105$	48.5078	4.73388
$106$	0	0
$107$	− 14.6441i	− 1.41570i	−0.706363	−	0.707850i	$-0.749665\pi$
$107$	− 14.6441i	− 1.41570i	0.706363	−	0.707850i	$-0.250335\pi$
$108$	0	0
$109$	4.29844i	0.411716i	0.978582	+	0.205858i	$0.0659984\pi$
$109$	4.29844i	0.411716i	−0.978582	+	0.205858i	$0.934002\pi$
$110$	0	0
$111$	−11.5294	−1.09432
$112$	0	0
$113$	10.0000	0.940721	0.470360	−	0.882474i	$-0.344124\pi$
$113$	10.0000	0.940721	0.470360	+	0.882474i	$0.344124\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 6.70156i	− 0.619560i
$118$	0	0
$119$	−1.25562	−0.115103
$120$	0	0
$121$	9.80625	0.891477
$122$	0	0
$123$	29.2882i	2.64083i
$124$	0	0
$125$	− 13.7016i	− 1.22550i
$126$	0	0
$127$	−16.8293	−1.49336	−0.746679	−	0.665185i	$-0.768353\pi$
$127$	−16.8293	−1.49336	−0.746679	+	0.665185i	$0.768353\pi$
$128$	0	0
$129$	−16.5078	−1.45343
$130$	0	0
$131$	− 7.48509i	− 0.653975i	−0.945029	−	0.326988i	$-0.893966\pi$
$131$	− 7.48509i	− 0.653975i	0.945029	−	0.326988i	$-0.106034\pi$
$132$	0	0
$133$	− 4.59688i	− 0.398600i
$134$	0	0
$135$	42.6767	3.67303
$136$	0	0
$137$	−20.2094	−1.72660	−0.863302	−	0.504688i	$-0.831607\pi$
$137$	−20.2094	−1.72660	−0.863302	+	0.504688i	$0.831607\pi$
$138$	0	0
$139$	15.5737i	1.32094i	0.750852	+	0.660471i	$0.229643\pi$
$139$	15.5737i	1.32094i	−0.750852	+	0.660471i	$0.770357\pi$
$140$	0	0
$141$	13.1047i	1.10361i
$142$	0	0
$143$	−1.09259	−0.0913669
$144$	0	0
$145$	7.40312	0.614796
$146$	0	0
$147$	33.3325i	2.74922i
$148$	0	0
$149$	− 12.8062i	− 1.04913i	−0.851371	−	0.524564i	$-0.824228\pi$
$149$	− 12.8062i	− 1.04913i	0.851371	−	0.524564i	$-0.175772\pi$
$150$	0	0
$151$	2.02214	0.164560	0.0822799	−	0.996609i	$-0.473780\pi$
$151$	2.02214	0.164560	0.0822799	+	0.996609i	$0.473780\pi$
$152$	0	0
$153$	−2.00000	−0.161690
$154$	0	0
$155$	− 19.0145i	− 1.52728i
$156$	0	0
$157$	− 2.00000i	− 0.159617i	−0.996810	−	0.0798087i	$-0.974569\pi$
$157$	− 2.00000i	− 0.159617i	0.996810	−	0.0798087i	$-0.0254309\pi$
$158$	0	0
$159$	4.37036	0.346592
$160$	0	0
$161$	0	0
$162$	0	0
$163$	9.50723i	0.744664i	0.928100	+	0.372332i	$0.121442\pi$
$163$	9.50723i	0.744664i	−0.928100	+	0.372332i	$0.878558\pi$
$164$	0	0
$165$	− 12.5969i	− 0.980665i
$166$	0	0
$167$	11.6924	0.904786	0.452393	−	0.891819i	$-0.350570\pi$
$167$	11.6924	0.904786	0.452393	+	0.891819i	$0.350570\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	− 7.32206i	− 0.559932i
$172$	0	0
$173$	9.40312i	0.714906i	0.933931	+	0.357453i	$0.116355\pi$
$173$	9.40312i	0.714906i	−0.933931	+	0.357453i	$0.883645\pi$
$174$	0	0
$175$	36.6103	2.76748
$176$	0	0
$177$	−42.2094	−3.17265
$178$	0	0
$179$	− 5.29991i	− 0.396134i	−0.980188	−	0.198067i	$-0.936534\pi$
$179$	− 5.29991i	− 0.396134i	0.980188	−	0.198067i	$-0.0634664\pi$
$180$	0	0
$181$	− 2.59688i	− 0.193024i	−0.995332	−	0.0965121i	$-0.969231\pi$
$181$	− 2.59688i	− 0.193024i	0.995332	−	0.0965121i	$-0.0307687\pi$
$182$	0	0
$183$	−29.2882	−2.16505
$184$	0	0
$185$	−13.7016	−1.00736
$186$	0	0
$187$	0.326070i	0.0238446i
$188$	0	0
$189$	48.5078i	3.52842i
$190$	0	0
$191$	4.04429	0.292634	0.146317	−	0.989238i	$-0.453258\pi$
$191$	4.04429	0.292634	0.146317	+	0.989238i	$0.453258\pi$
$192$	0	0
$193$	21.4031	1.54063	0.770315	−	0.637663i	$-0.220099\pi$
$193$	21.4031	1.54063	0.770315	+	0.637663i	$0.220099\pi$
$194$	0	0
$195$	− 11.5294i	− 0.825636i
$196$	0	0
$197$	16.2984i	1.16122i	0.814183	+	0.580608i	$0.197185\pi$
$197$	16.2984i	1.16122i	−0.814183	+	0.580608i	$0.802815\pi$
$198$	0	0
$199$	18.6884	1.32479	0.662393	−	0.749157i	$-0.269541\pi$
$199$	18.6884	1.32479	0.662393	+	0.749157i	$0.269541\pi$
$200$	0	0
$201$	35.4031	2.49714
$202$	0	0
$203$	8.41464i	0.590592i
$204$	0	0
$205$	34.8062i	2.43097i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−1.19375	−0.0825735
$210$	0	0
$211$	21.8031i	1.50099i	0.660876	+	0.750495i	$0.270185\pi$
$211$	21.8031i	1.50099i	−0.660876	+	0.750495i	$0.729815\pi$
$212$	0	0
$213$	− 25.7016i	− 1.76104i
$214$	0	0
$215$	−19.6180	−1.33793
$216$	0	0
$217$	21.6125	1.46715
$218$	0	0
$219$	18.6884i	1.26284i
$220$	0	0
$221$	0.298438i	0.0200751i
$222$	0	0
$223$	20.7105	1.38688	0.693440	−	0.720514i	$-0.256094\pi$
$223$	20.7105	1.38688	0.693440	+	0.720514i	$0.256094\pi$
$224$	0	0
$225$	58.3141	3.88760
$226$	0	0
$227$	− 5.13688i	− 0.340946i	−0.985362	−	0.170473i	$-0.945470\pi$
$227$	− 5.13688i	− 0.340946i	0.985362	−	0.170473i	$-0.0545296\pi$
$228$	0	0
$229$	− 8.89531i	− 0.587819i	−0.955833	−	0.293909i	$-0.905044\pi$
$229$	− 8.89531i	− 0.587819i	0.955833	−	0.293909i	$-0.0949564\pi$
$230$	0	0
$231$	14.3180	0.942058
$232$	0	0
$233$	−23.1047	−1.51364	−0.756819	−	0.653624i	$-0.773248\pi$
$233$	−23.1047	−1.51364	−0.756819	+	0.653624i	$0.773248\pi$
$234$	0	0
$235$	15.5737i	1.01591i
$236$	0	0
$237$	− 45.6125i	− 2.96285i
$238$	0	0
$239$	14.4811	0.936703	0.468351	−	0.883542i	$-0.344848\pi$
$239$	14.4811	0.936703	0.468351	+	0.883542i	$0.344848\pi$
$240$	0	0
$241$	−4.80625	−0.309598	−0.154799	−	0.987946i	$-0.549473\pi$
$241$	−4.80625	−0.309598	−0.154799	+	0.987946i	$0.549473\pi$
$242$	0	0
$243$	14.6441i	0.939420i
$244$	0	0
$245$	39.6125i	2.53075i
$246$	0	0
$247$	−1.09259	−0.0695198
$248$	0	0
$249$	−22.8062	−1.44529
$250$	0	0
$251$	− 16.8293i	− 1.06226i	−0.847292	−	0.531128i	$-0.821768\pi$
$251$	− 16.8293i	− 1.06226i	0.847292	−	0.531128i	$-0.178232\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−3.44080	−0.215471
$256$	0	0
$257$	−14.5078	−0.904972	−0.452486	−	0.891771i	$-0.649463\pi$
$257$	−14.5078	−0.904972	−0.452486	+	0.891771i	$0.649463\pi$
$258$	0	0
$259$	− 15.5737i	− 0.967700i
$260$	0	0
$261$	13.4031i	0.829633i
$262$	0	0
$263$	2.18518	0.134744	0.0673719	−	0.997728i	$-0.478539\pi$
$263$	2.18518	0.134744	0.0673719	+	0.997728i	$0.478539\pi$
$264$	0	0
$265$	5.19375	0.319050
$266$	0	0
$267$	18.6884i	1.14371i
$268$	0	0
$269$	− 12.2094i	− 0.744419i	−0.928149	−	0.372209i	$-0.878600\pi$
$269$	− 12.2094i	− 0.744419i	0.928149	−	0.372209i	$-0.121400\pi$
$270$	0	0
$271$	4.20732	0.255577	0.127788	−	0.991801i	$-0.459212\pi$
$271$	4.20732	0.255577	0.127788	+	0.991801i	$0.459212\pi$
$272$	0	0
$273$	13.1047	0.793132
$274$	0	0
$275$	− 9.50723i	− 0.573308i
$276$	0	0
$277$	− 24.2094i	− 1.45460i	−0.686320	−	0.727300i	$-0.740775\pi$
$277$	− 24.2094i	− 1.45460i	0.686320	−	0.727300i	$-0.259225\pi$
$278$	0	0
$279$	34.4251	2.06098
$280$	0	0
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0	0
$285$	− 12.5969i	− 0.746175i
$286$	0	0
$287$	−39.5620	−2.33527
$288$	0	0
$289$	−16.9109	−0.994761
$290$	0	0
$291$	27.4291i	1.60792i
$292$	0	0
$293$	− 19.1047i	− 1.11611i	−0.829805	−	0.558054i	$-0.811548\pi$
$293$	− 19.1047i	− 1.11611i	0.829805	−	0.558054i	$-0.188452\pi$
$294$	0	0
$295$	−50.1618	−2.92053
$296$	0	0
$297$	12.5969	0.730945
$298$	0	0
$299$	0	0
$300$	0	0
$301$	− 22.2984i	− 1.28526i
$302$	0	0
$303$	−16.8293	−0.966817
$304$	0	0
$305$	−34.8062	−1.99300
$306$	0	0
$307$	15.7367i	0.898141i	0.893496	+	0.449070i	$0.148245\pi$
$307$	15.7367i	0.898141i	−0.893496	+	0.449070i	$0.851755\pi$
$308$	0	0
$309$	− 38.8062i	− 2.20761i
$310$	0	0
$311$	18.6884	1.05972	0.529861	−	0.848085i	$-0.322244\pi$
$311$	18.6884	1.05972	0.529861	+	0.848085i	$0.322244\pi$
$312$	0	0
$313$	−2.50781	−0.141750	−0.0708749	−	0.997485i	$-0.522579\pi$
$313$	−2.50781	−0.141750	−0.0708749	+	0.997485i	$0.522579\pi$
$314$	0	0
$315$	104.368i	5.88046i
$316$	0	0
$317$	− 24.8062i	− 1.39326i	−0.717432	−	0.696629i	$-0.754682\pi$
$317$	− 24.8062i	− 1.39326i	0.717432	−	0.696629i	$-0.245318\pi$
$318$	0	0
$319$	2.18518	0.122347
$320$	0	0
$321$	45.6125	2.54584
$322$	0	0
$323$	0.326070i	0.0181430i
$324$	0	0
$325$	− 8.70156i	− 0.482676i
$326$	0	0
$327$	−13.3885	−0.740385
$328$	0	0
$329$	−17.7016	−0.975919
$330$	0	0
$331$	− 24.1513i	− 1.32748i	−0.747964	−	0.663739i	$-0.768969\pi$
$331$	− 24.1513i	− 1.32748i	0.747964	−	0.663739i	$-0.231031\pi$
$332$	0	0
$333$	− 24.8062i	− 1.35937i
$334$	0	0
$335$	42.0732	2.29871
$336$	0	0
$337$	−3.10469	−0.169123	−0.0845615	−	0.996418i	$-0.526949\pi$
$337$	−3.10469	−0.169123	−0.0845615	+	0.996418i	$0.526949\pi$
$338$	0	0
$339$	31.1473i	1.69169i
$340$	0	0
$341$	− 5.61250i	− 0.303934i
$342$	0	0
$343$	−15.5737	−0.840899
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 4.97384i	− 0.267010i	−0.991048	−	0.133505i	$-0.957377\pi$
$347$	− 4.97384i	− 0.267010i	0.991048	−	0.133505i	$-0.0426232\pi$
$348$	0	0
$349$	33.9109i	1.81521i	0.419824	+	0.907605i	$0.362092\pi$
$349$	33.9109i	1.81521i	−0.419824	+	0.907605i	$0.637908\pi$
$350$	0	0
$351$	11.5294	0.615393
$352$	0	0
$353$	14.5969	0.776913	0.388457	−	0.921467i	$-0.373008\pi$
$353$	14.5969	0.776913	0.388457	+	0.921467i	$0.373008\pi$
$354$	0	0
$355$	− 30.5438i	− 1.62110i
$356$	0	0
$357$	− 3.91093i	− 0.206989i
$358$	0	0
$359$	2.95170	0.155785	0.0778923	−	0.996962i	$-0.475181\pi$
$359$	2.95170	0.155785	0.0778923	+	0.996962i	$0.475181\pi$
$360$	0	0
$361$	17.8062	0.937171
$362$	0	0
$363$	30.5438i	1.60314i
$364$	0	0
$365$	22.2094i	1.16249i
$366$	0	0
$367$	−28.9622	−1.51181	−0.755906	−	0.654680i	$-0.772803\pi$
$367$	−28.9622	−1.51181	−0.755906	+	0.654680i	$0.772803\pi$
$368$	0	0
$369$	−63.0156	−3.28046
$370$	0	0
$371$	5.90340i	0.306489i
$372$	0	0
$373$	− 16.2094i	− 0.839290i	−0.907688	−	0.419645i	$-0.862155\pi$
$373$	− 16.2094i	− 0.839290i	0.907688	−	0.419645i	$-0.137845\pi$
$374$	0	0
$375$	42.6767	2.20382
$376$	0	0
$377$	2.00000	0.103005
$378$	0	0
$379$	− 27.8696i	− 1.43156i	−0.698324	−	0.715782i	$-0.746071\pi$
$379$	− 27.8696i	− 1.43156i	0.698324	−	0.715782i	$-0.253929\pi$
$380$	0	0
$381$	− 52.4187i	− 2.68549i
$382$	0	0
$383$	−8.57768	−0.438299	−0.219149	−	0.975691i	$-0.570328\pi$
$383$	−8.57768	−0.438299	−0.219149	+	0.975691i	$0.570328\pi$
$384$	0	0
$385$	17.0156	0.867196
$386$	0	0
$387$	− 35.5177i	− 1.80547i
$388$	0	0
$389$	32.8062i	1.66334i	0.555268	+	0.831671i	$0.312616\pi$
$389$	32.8062i	1.66334i	−0.555268	+	0.831671i	$0.687384\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	23.3141	1.17604
$394$	0	0
$395$	− 54.2061i	− 2.72740i
$396$	0	0
$397$	− 2.00000i	− 0.100377i	−0.998740	−	0.0501886i	$-0.984018\pi$
$397$	− 2.00000i	− 0.100377i	0.998740	−	0.0501886i	$-0.0159822\pi$
$398$	0	0
$399$	14.3180	0.716799
$400$	0	0
$401$	−24.2094	−1.20896	−0.604479	−	0.796621i	$-0.706619\pi$
$401$	−24.2094	−1.20896	−0.604479	+	0.796621i	$0.706619\pi$
$402$	0	0
$403$	− 5.13688i	− 0.255886i
$404$	0	0
$405$	58.5078i	2.90728i
$406$	0	0
$407$	−4.04429	−0.200468
$408$	0	0
$409$	9.40312	0.464955	0.232477	−	0.972602i	$-0.425317\pi$
$409$	9.40312	0.464955	0.232477	+	0.972602i	$0.425317\pi$
$410$	0	0
$411$	− 62.9468i	− 3.10494i
$412$	0	0
$413$	− 57.0156i	− 2.80556i
$414$	0	0
$415$	−27.1030	−1.33044
$416$	0	0
$417$	−48.5078	−2.37544
$418$	0	0
$419$	32.4030i	1.58299i	0.611177	+	0.791494i	$0.290696\pi$
$419$	32.4030i	1.58299i	−0.611177	+	0.791494i	$0.709304\pi$
$420$	0	0
$421$	− 19.1047i	− 0.931105i	−0.885020	−	0.465553i	$-0.845856\pi$
$421$	− 19.1047i	− 0.931105i	0.885020	−	0.465553i	$-0.154144\pi$
$422$	0	0
$423$	−28.1956	−1.37092
$424$	0	0
$425$	−2.59688	−0.125967
$426$	0	0
$427$	− 39.5620i	− 1.91454i
$428$	0	0
$429$	− 3.40312i	− 0.164304i
$430$	0	0
$431$	18.8514	0.908042	0.454021	−	0.890991i	$-0.349989\pi$
$431$	18.8514	0.908042	0.454021	+	0.890991i	$0.349989\pi$
$432$	0	0
$433$	4.89531	0.235254	0.117627	−	0.993058i	$-0.462471\pi$
$433$	4.89531	0.235254	0.117627	+	0.993058i	$0.462471\pi$
$434$	0	0
$435$	23.0588i	1.10558i
$436$	0	0
$437$	0	0
$438$	0	0
$439$	27.4291	1.30912	0.654560	−	0.756010i	$-0.272854\pi$
$439$	27.4291	1.30912	0.654560	+	0.756010i	$0.272854\pi$
$440$	0	0
$441$	−71.7172	−3.41510
$442$	0	0
$443$	− 11.8554i	− 0.563269i	−0.959522	−	0.281635i	$-0.909123\pi$
$443$	− 11.8554i	− 0.563269i	0.959522	−	0.281635i	$-0.0908766\pi$
$444$	0	0
$445$	22.2094i	1.05283i
$446$	0	0
$447$	39.8880	1.88664
$448$	0	0
$449$	24.8062	1.17068	0.585340	−	0.810788i	$-0.300961\pi$
$449$	24.8062	1.17068	0.585340	+	0.810788i	$0.300961\pi$
$450$	0	0
$451$	10.2738i	0.483772i
$452$	0	0
$453$	6.29844i	0.295926i
$454$	0	0
$455$	15.5737	0.730105
$456$	0	0
$457$	−0.806248	−0.0377147	−0.0188574	−	0.999822i	$-0.506003\pi$
$457$	−0.806248	−0.0377147	−0.0188574	+	0.999822i	$0.506003\pi$
$458$	0	0
$459$	− 3.44080i	− 0.160603i
$460$	0	0
$461$	− 11.7016i	− 0.544996i	−0.962156	−	0.272498i	$-0.912150\pi$
$461$	− 11.7016i	− 0.544996i	0.962156	−	0.272498i	$-0.0878498\pi$
$462$	0	0
$463$	−32.2399	−1.49832	−0.749158	−	0.662391i	$-0.769542\pi$
$463$	−32.2399	−1.49832	−0.749158	+	0.662391i	$0.769542\pi$
$464$	0	0
$465$	59.2250	2.74649
$466$	0	0
$467$	− 29.2882i	− 1.35530i	−0.735386	−	0.677649i	$-0.762999\pi$
$467$	− 29.2882i	− 1.35530i	0.735386	−	0.677649i	$-0.237001\pi$
$468$	0	0
$469$	47.8219i	2.20821i
$470$	0	0
$471$	6.22947	0.287039
$472$	0	0
$473$	−5.79063	−0.266253
$474$	0	0
$475$	− 9.50723i	− 0.436222i
$476$	0	0
$477$	9.40312i	0.430539i
$478$	0	0
$479$	−25.0809	−1.14598	−0.572988	−	0.819564i	$-0.694216\pi$
$479$	−25.0809	−1.14598	−0.572988	+	0.819564i	$0.694216\pi$
$480$	0	0
$481$	−3.70156	−0.168777
$482$	0	0
$483$	0	0
$484$	0	0
$485$	32.5969i	1.48015i
$486$	0	0
$487$	−2.95170	−0.133754	−0.0668771	−	0.997761i	$-0.521304\pi$
$487$	−2.95170	−0.133754	−0.0668771	+	0.997761i	$0.521304\pi$
$488$	0	0
$489$	−29.6125	−1.33912
$490$	0	0
$491$	34.5881i	1.56094i	0.625193	+	0.780470i	$0.285020\pi$
$491$	34.5881i	1.56094i	−0.625193	+	0.780470i	$0.714980\pi$
$492$	0	0
$493$	− 0.596876i	− 0.0268819i
$494$	0	0
$495$	27.1030	1.21819
$496$	0	0
$497$	34.7172	1.55728
$498$	0	0
$499$	− 38.7955i	− 1.73672i	−0.495932	−	0.868362i	$-0.665173\pi$
$499$	− 38.7955i	− 1.73672i	0.495932	−	0.868362i	$-0.334827\pi$
$500$	0	0
$501$	36.4187i	1.62707i
$502$	0	0
$503$	8.41464	0.375190	0.187595	−	0.982246i	$-0.439931\pi$
$503$	8.41464	0.375190	0.187595	+	0.982246i	$0.439931\pi$
$504$	0	0
$505$	−20.0000	−0.889988
$506$	0	0
$507$	− 3.11473i	− 0.138330i
$508$	0	0
$509$	14.0000i	0.620539i	0.950649	+	0.310270i	$0.100419\pi$
$509$	14.0000i	0.620539i	−0.950649	+	0.310270i	$0.899581\pi$
$510$	0	0
$511$	−25.2439	−1.11673
$512$	0	0
$513$	12.5969	0.556166
$514$	0	0
$515$	− 46.1175i	− 2.03218i
$516$	0	0
$517$	4.59688i	0.202170i
$518$	0	0
$519$	−29.2882	−1.28561
$520$	0	0
$521$	−18.5078	−0.810842	−0.405421	−	0.914130i	$-0.632875\pi$
$521$	−18.5078	−0.810842	−0.405421	+	0.914130i	$0.632875\pi$
$522$	0	0
$523$	− 19.0145i	− 0.831445i	−0.909492	−	0.415722i	$-0.863529\pi$
$523$	− 19.0145i	− 0.831445i	0.909492	−	0.415722i	$-0.136471\pi$
$524$	0	0
$525$	114.031i	4.97673i
$526$	0	0
$527$	−1.53304	−0.0667802
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	− 90.8164i	− 3.94109i
$532$	0	0
$533$	9.40312i	0.407295i
$534$	0	0
$535$	54.2061	2.34353
$536$	0	0
$537$	16.5078	0.712365
$538$	0	0
$539$	11.6924i	0.503628i
$540$	0	0
$541$	− 8.29844i	− 0.356778i	−0.983960	−	0.178389i	$-0.942912\pi$
$541$	− 8.29844i	− 0.356778i	0.983960	−	0.178389i	$-0.0570885\pi$
$542$	0	0
$543$	8.08857	0.347114
$544$	0	0
$545$	−15.9109	−0.681550
$546$	0	0
$547$	− 21.8031i	− 0.932235i	−0.884723	−	0.466117i	$-0.845652\pi$
$547$	− 21.8031i	− 0.932235i	0.884723	−	0.466117i	$-0.154348\pi$
$548$	0	0
$549$	− 63.0156i	− 2.68944i
$550$	0	0
$551$	2.18518	0.0930917
$552$	0	0
$553$	61.6125	2.62003
$554$	0	0
$555$	− 42.6767i	− 1.81153i
$556$	0	0
$557$	7.70156i	0.326326i	0.986599	+	0.163163i	$0.0521696\pi$
$557$	7.70156i	0.326326i	−0.986599	+	0.163163i	$0.947830\pi$
$558$	0	0
$559$	−5.29991	−0.224162
$560$	0	0
$561$	−1.01562	−0.0428796
$562$	0	0
$563$	− 9.01813i	− 0.380069i	−0.981777	−	0.190034i	$-0.939140\pi$
$563$	− 9.01813i	− 0.380069i	0.981777	−	0.190034i	$-0.0608600\pi$
$564$	0	0
$565$	37.0156i	1.55726i
$566$	0	0
$567$	−66.5020	−2.79282
$568$	0	0
$569$	−35.7016	−1.49669	−0.748344	−	0.663311i	$-0.769151\pi$
$569$	−35.7016	−1.49669	−0.748344	+	0.663311i	$0.769151\pi$
$570$	0	0
$571$	− 5.29991i	− 0.221794i	−0.993832	−	0.110897i	$-0.964628\pi$
$571$	− 5.29991i	− 0.221794i	0.993832	−	0.110897i	$-0.0353724\pi$
$572$	0	0
$573$	12.5969i	0.526242i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−7.19375	−0.299480	−0.149740	−	0.988725i	$-0.547844\pi$
$577$	−7.19375	−0.299480	−0.149740	+	0.988725i	$0.547844\pi$
$578$	0	0
$579$	66.6650i	2.77050i
$580$	0	0
$581$	− 30.8062i	− 1.27806i
$582$	0	0
$583$	1.53304	0.0634920
$584$	0	0
$585$	24.8062	1.02561
$586$	0	0
$587$	9.50723i	0.392406i	0.980563	+	0.196203i	$0.0628611\pi$
$587$	9.50723i	0.392406i	−0.980563	+	0.196203i	$0.937139\pi$
$588$	0	0
$589$	− 5.61250i	− 0.231259i
$590$	0	0
$591$	−50.7653	−2.08820
$592$	0	0
$593$	−30.0000	−1.23195	−0.615976	−	0.787765i	$-0.711238\pi$
$593$	−30.0000	−1.23195	−0.615976	+	0.787765i	$0.711238\pi$
$594$	0	0
$595$	− 4.64777i	− 0.190540i
$596$	0	0
$597$	58.2094i	2.38235i
$598$	0	0
$599$	−33.3325	−1.36193	−0.680965	−	0.732316i	$-0.738439\pi$
$599$	−33.3325	−1.36193	−0.680965	+	0.732316i	$0.738439\pi$
$600$	0	0
$601$	−24.2984	−0.991154	−0.495577	−	0.868564i	$-0.665043\pi$
$601$	−24.2984	−0.991154	−0.495577	+	0.868564i	$0.665043\pi$
$602$	0	0
$603$	76.1723i	3.10197i
$604$	0	0
$605$	36.2984i	1.47574i
$606$	0	0
$607$	−12.1329	−0.492458	−0.246229	−	0.969212i	$-0.579191\pi$
$607$	−12.1329	−0.492458	−0.246229	+	0.969212i	$0.579191\pi$
$608$	0	0
$609$	−26.2094	−1.06206
$610$	0	0
$611$	4.20732i	0.170210i
$612$	0	0
$613$	26.0000i	1.05013i	0.851062	+	0.525065i	$0.175959\pi$
$613$	26.0000i	1.05013i	−0.851062	+	0.525065i	$0.824041\pi$
$614$	0	0
$615$	−108.412	−4.37160
$616$	0	0
$617$	−6.59688	−0.265580	−0.132790	−	0.991144i	$-0.542394\pi$
$617$	−6.59688	−0.265580	−0.132790	+	0.991144i	$0.542394\pi$
$618$	0	0
$619$	36.6103i	1.47149i	0.677258	+	0.735746i	$0.263168\pi$
$619$	36.6103i	1.47149i	−0.677258	+	0.735746i	$0.736832\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−25.2439	−1.01138
$624$	0	0
$625$	7.20937	0.288375
$626$	0	0
$627$	− 3.71822i	− 0.148491i
$628$	0	0
$629$	1.10469i	0.0440467i
$630$	0	0
$631$	−1.69607	−0.0675196	−0.0337598	−	0.999430i	$-0.510748\pi$
$631$	−1.69607	−0.0675196	−0.0337598	+	0.999430i	$0.510748\pi$
$632$	0	0
$633$	−67.9109	−2.69922
$634$	0	0
$635$	− 62.2947i	− 2.47209i
$636$	0	0
$637$	10.7016i	0.424011i
$638$	0	0
$639$	55.2987	2.18758
$640$	0	0
$641$	32.8062	1.29577	0.647884	−	0.761739i	$-0.275654\pi$
$641$	32.8062	1.29577	0.647884	+	0.761739i	$0.275654\pi$
$642$	0	0
$643$	− 20.1071i	− 0.792945i	−0.918047	−	0.396472i	$-0.870234\pi$
$643$	− 20.1071i	− 0.792945i	0.918047	−	0.396472i	$-0.129766\pi$
$644$	0	0
$645$	− 61.1047i	− 2.40599i
$646$	0	0
$647$	18.6884	0.734717	0.367358	−	0.930079i	$-0.380262\pi$
$647$	18.6884	0.734717	0.367358	+	0.930079i	$0.380262\pi$
$648$	0	0
$649$	−14.8062	−0.581196
$650$	0	0
$651$	67.3172i	2.63837i
$652$	0	0
$653$	47.0156i	1.83986i	0.392079	+	0.919932i	$0.371756\pi$
$653$	47.0156i	1.83986i	−0.392079	+	0.919932i	$0.628244\pi$
$654$	0	0
$655$	27.7065	1.08258
$656$	0	0
$657$	−40.2094	−1.56872
$658$	0	0
$659$	19.0145i	0.740699i	0.928893	+	0.370349i	$0.120762\pi$
$659$	19.0145i	0.740699i	−0.928893	+	0.370349i	$0.879238\pi$
$660$	0	0
$661$	− 28.8062i	− 1.12043i	−0.828346	−	0.560217i	$-0.810718\pi$
$661$	− 28.8062i	− 1.12043i	0.828346	−	0.560217i	$-0.189282\pi$
$662$	0	0
$663$	−0.929554	−0.0361009
$664$	0	0
$665$	17.0156	0.659837
$666$	0	0
$667$	0	0
$668$	0	0
$669$	64.5078i	2.49402i
$670$	0	0
$671$	−10.2738	−0.396614
$672$	0	0
$673$	−29.3141	−1.12997	−0.564987	−	0.825100i	$-0.691119\pi$
$673$	−29.3141	−1.12997	−0.564987	+	0.825100i	$0.691119\pi$
$674$	0	0
$675$	100.324i	3.86146i
$676$	0	0
$677$	− 17.4031i	− 0.668856i	−0.942421	−	0.334428i	$-0.891457\pi$
$677$	− 17.4031i	− 0.668856i	0.942421	−	0.334428i	$-0.108543\pi$
$678$	0	0
$679$	−37.0507	−1.42188
$680$	0	0
$681$	16.0000	0.613121
$682$	0	0
$683$	49.0692i	1.87758i	0.344488	+	0.938791i	$0.388052\pi$
$683$	49.0692i	1.87758i	−0.344488	+	0.938791i	$0.611948\pi$
$684$	0	0
$685$	− 74.8062i	− 2.85820i
$686$	0	0
$687$	27.7065	1.05707
$688$	0	0
$689$	1.40312	0.0534548
$690$	0	0
$691$	28.5217i	1.08502i	0.840050	+	0.542508i	$0.182525\pi$
$691$	28.5217i	1.08502i	−0.840050	+	0.542508i	$0.817475\pi$
$692$	0	0
$693$	30.8062i	1.17023i
$694$	0	0
$695$	−57.6469	−2.18667
$696$	0	0
$697$	2.80625	0.106294
$698$	0	0
$699$	− 71.9649i	− 2.72196i
$700$	0	0
$701$	17.4031i	0.657307i	0.944451	+	0.328653i	$0.106595\pi$
$701$	17.4031i	0.657307i	−0.944451	+	0.328653i	$0.893405\pi$
$702$	0	0
$703$	−4.04429	−0.152533
$704$	0	0
$705$	−48.5078	−1.82691
$706$	0	0
$707$	− 22.7327i	− 0.854951i
$708$	0	0
$709$	3.19375i	0.119944i	0.998200	+	0.0599719i	$0.0191011\pi$
$709$	3.19375i	0.119944i	−0.998200	+	0.0599719i	$0.980899\pi$
$710$	0	0
$711$	98.1384	3.68048
$712$	0	0
$713$	0	0
$714$	0	0
$715$	− 4.04429i	− 0.151248i
$716$	0	0
$717$	45.1047i	1.68447i
$718$	0	0
$719$	−31.1473	−1.16160	−0.580800	−	0.814046i	$-0.697260\pi$
$719$	−31.1473	−1.16160	−0.580800	+	0.814046i	$0.697260\pi$
$720$	0	0
$721$	52.4187	1.95218
$722$	0	0
$723$	− 14.9702i	− 0.556747i
$724$	0	0
$725$	17.4031i	0.646336i
$726$	0	0
$727$	−2.51125	−0.0931371	−0.0465685	−	0.998915i	$-0.514829\pi$
$727$	−2.51125	−0.0931371	−0.0465685	+	0.998915i	$0.514829\pi$
$728$	0	0
$729$	1.80625	0.0668981
$730$	0	0
$731$	1.58169i	0.0585011i
$732$	0	0
$733$	− 37.9109i	− 1.40027i	−0.714009	−	0.700136i	$-0.753123\pi$
$733$	− 37.9109i	− 1.40027i	0.714009	−	0.700136i	$-0.246877\pi$
$734$	0	0
$735$	−123.382	−4.55103
$736$	0	0
$737$	12.4187	0.457450
$738$	0	0
$739$	17.5958i	0.647272i	0.946182	+	0.323636i	$0.104905\pi$
$739$	17.5958i	0.647272i	−0.946182	+	0.323636i	$0.895095\pi$
$740$	0	0
$741$	− 3.40312i	− 0.125017i
$742$	0	0
$743$	−2.02214	−0.0741853	−0.0370926	−	0.999312i	$-0.511810\pi$
$743$	−2.02214	−0.0741853	−0.0370926	+	0.999312i	$0.511810\pi$
$744$	0	0
$745$	47.4031	1.73672
$746$	0	0
$747$	− 49.0692i	− 1.79535i
$748$	0	0
$749$	61.6125i	2.25127i
$750$	0	0
$751$	49.8357	1.81853	0.909266	−	0.416216i	$-0.136644\pi$
$751$	49.8357	1.81853	0.909266	+	0.416216i	$0.136644\pi$
$752$	0	0
$753$	52.4187	1.91025
$754$	0	0
$755$	7.48509i	0.272410i
$756$	0	0
$757$	12.2094i	0.443757i	0.975074	+	0.221879i	$0.0712189\pi$
$757$	12.2094i	0.443757i	−0.975074	+	0.221879i	$0.928781\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−23.6125	−0.855952	−0.427976	−	0.903790i	$-0.640773\pi$
$761$	−23.6125	−0.855952	−0.427976	+	0.903790i	$0.640773\pi$
$762$	0	0
$763$	− 18.0849i	− 0.654718i
$764$	0	0
$765$	− 7.40312i	− 0.267661i
$766$	0	0
$767$	−13.5515	−0.489317
$768$	0	0
$769$	12.2094	0.440281	0.220141	−	0.975468i	$-0.429348\pi$
$769$	12.2094	0.440281	0.220141	+	0.975468i	$0.429348\pi$
$770$	0	0
$771$	− 45.1880i	− 1.62740i
$772$	0	0
$773$	− 3.10469i	− 0.111668i	−0.998440	−	0.0558339i	$-0.982218\pi$
$773$	− 3.10469i	− 0.111668i	0.998440	−	0.0558339i	$-0.0177817\pi$
$774$	0	0
$775$	44.6989	1.60563
$776$	0	0
$777$	48.5078	1.74021
$778$	0	0
$779$	10.2738i	0.368095i
$780$	0	0
$781$	− 9.01562i	− 0.322604i
$782$	0	0
$783$	−23.0588	−0.824053
$784$	0	0
$785$	7.40312	0.264229
$786$	0	0
$787$	0.766519i	0.0273235i	0.999907	+	0.0136617i	$0.00434880\pi$
$787$	0.766519i	0.0273235i	−0.999907	+	0.0136617i	$0.995651\pi$
$788$	0	0
$789$	6.80625i	0.242309i
$790$	0	0
$791$	−42.0732	−1.49595
$792$	0	0
$793$	−9.40312	−0.333915
$794$	0	0
$795$	16.1771i	0.573744i
$796$	0	0
$797$	− 2.00000i	− 0.0708436i	−0.999372	−	0.0354218i	$-0.988723\pi$
$797$	− 2.00000i	− 0.0708436i	0.999372	−	0.0354218i	$-0.0112775\pi$
$798$	0	0
$799$	1.25562	0.0444208
$800$	0	0
$801$	−40.2094	−1.42073
$802$	0	0
$803$	6.55554i	0.231340i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	38.0289	1.33868
$808$	0	0
$809$	27.1047	0.952950	0.476475	−	0.879188i	$-0.341914\pi$
$809$	27.1047	0.952950	0.476475	+	0.879188i	$0.341914\pi$
$810$	0	0
$811$	− 16.0628i	− 0.564040i	−0.959409	−	0.282020i	$-0.908996\pi$
$811$	− 16.0628i	− 0.564040i	0.959409	−	0.282020i	$-0.0910045\pi$
$812$	0	0
$813$	13.1047i	0.459601i
$814$	0	0
$815$	−35.1916	−1.23271
$816$	0	0
$817$	−5.79063	−0.202588
$818$	0	0
$819$	28.1956i	0.985235i
$820$	0	0
$821$	− 25.9109i	− 0.904298i	−0.891942	−	0.452149i	$-0.850658\pi$
$821$	− 25.9109i	− 0.904298i	0.891942	−	0.452149i	$-0.149342\pi$
$822$	0	0
$823$	−31.7995	−1.10846	−0.554230	−	0.832364i	$-0.686987\pi$
$823$	−31.7995	−1.10846	−0.554230	+	0.832364i	$0.686987\pi$
$824$	0	0
$825$	29.6125	1.03097
$826$	0	0
$827$	− 11.3663i	− 0.395246i	−0.980278	−	0.197623i	$-0.936678\pi$
$827$	− 11.3663i	− 0.395246i	0.980278	−	0.197623i	$-0.0633223\pi$
$828$	0	0
$829$	51.6125i	1.79258i	0.443472	+	0.896288i	$0.353746\pi$
$829$	51.6125i	1.79258i	−0.443472	+	0.896288i	$0.646254\pi$
$830$	0	0
$831$	75.4057	2.61580
$832$	0	0
$833$	3.19375	0.110657
$834$	0	0
$835$	43.2802i	1.49777i
$836$	0	0
$837$	59.2250i	2.04712i
$838$	0	0
$839$	32.2399	1.11305	0.556523	−	0.830832i	$-0.312135\pi$
$839$	32.2399	1.11305	0.556523	+	0.830832i	$0.312135\pi$
$840$	0	0
$841$	25.0000	0.862069
$842$	0	0
$843$	− 31.1473i	− 1.07277i
$844$	0	0
$845$	− 3.70156i	− 0.127338i
$846$	0	0
$847$	−41.2580	−1.41764
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	23.1047i	0.791089i	0.918447	+	0.395545i	$0.129444\pi$
$853$	23.1047i	0.791089i	−0.918447	+	0.395545i	$0.870556\pi$
$854$	0	0
$855$	27.1030	0.926905
$856$	0	0
$857$	12.8062	0.437453	0.218727	−	0.975786i	$-0.429810\pi$
$857$	12.8062	0.437453	0.218727	+	0.975786i	$0.429810\pi$
$858$	0	0
$859$	14.6441i	0.499651i	0.968291	+	0.249825i	$0.0803732\pi$
$859$	14.6441i	0.499651i	−0.968291	+	0.249825i	$0.919627\pi$
$860$	0	0
$861$	− 123.225i	− 4.19950i
$862$	0	0
$863$	−27.2661	−0.928148	−0.464074	−	0.885796i	$-0.653613\pi$
$863$	−27.2661	−0.928148	−0.464074	+	0.885796i	$0.653613\pi$
$864$	0	0
$865$	−34.8062	−1.18345
$866$	0	0
$867$	− 52.6730i	− 1.78887i
$868$	0	0
$869$	− 16.0000i	− 0.542763i
$870$	0	0
$871$	11.3663	0.385134
$872$	0	0
$873$	−59.0156	−1.99738
$874$	0	0
$875$	57.6469i	1.94882i
$876$	0	0
$877$	− 44.7172i	− 1.50999i	−0.655729	−	0.754996i	$-0.727639\pi$
$877$	− 44.7172i	− 1.50999i	0.655729	−	0.754996i	$-0.272361\pi$
$878$	0	0
$879$	59.5060	2.00709
$880$	0	0
$881$	−55.7016	−1.87663	−0.938317	−	0.345777i	$-0.887615\pi$
$881$	−55.7016	−1.87663	−0.938317	+	0.345777i	$0.887615\pi$
$882$	0	0
$883$	− 40.8176i	− 1.37362i	−0.726836	−	0.686811i	$-0.759010\pi$
$883$	− 40.8176i	− 1.37362i	0.726836	−	0.686811i	$-0.240990\pi$
$884$	0	0
$885$	− 156.241i	− 5.25197i
$886$	0	0
$887$	−19.0145	−0.638443	−0.319222	−	0.947680i	$-0.603421\pi$
$887$	−19.0145	−0.638443	−0.319222	+	0.947680i	$0.603421\pi$
$888$	0	0
$889$	70.8062	2.37477
$890$	0	0
$891$	17.2697i	0.578558i
$892$	0	0
$893$	4.59688i	0.153829i
$894$	0	0
$895$	19.6180	0.655756
$896$	0	0
$897$	0	0
$898$	0	0
$899$	10.2738i	0.342649i
$900$	0	0
$901$	− 0.418745i	− 0.0139504i
$902$	0	0
$903$	69.4537	2.31127
$904$	0	0
$905$	9.61250	0.319530
$906$	0	0
$907$	17.7588i	0.589673i	0.955548	+	0.294836i	$0.0952651\pi$
$907$	17.7588i	0.589673i	−0.955548	+	0.294836i	$0.904735\pi$
$908$	0	0
$909$	− 36.2094i	− 1.20099i
$910$	0	0
$911$	−25.2439	−0.836369	−0.418184	−	0.908362i	$-0.637333\pi$
$911$	−25.2439	−0.836369	−0.418184	+	0.908362i	$0.637333\pi$
$912$	0	0
$913$	−8.00000	−0.264761
$914$	0	0
$915$	− 108.412i	− 3.58400i
$916$	0	0
$917$	31.4922i	1.03996i
$918$	0	0
$919$	14.6441	0.483065	0.241532	−	0.970393i	$-0.422350\pi$
$919$	14.6441	0.483065	0.241532	+	0.970393i	$0.422350\pi$
$920$	0	0
$921$	−49.0156	−1.61512
$922$	0	0
$923$	− 8.25161i	− 0.271605i
$924$	0	0
$925$	− 32.2094i	− 1.05904i
$926$	0	0
$927$	83.4943	2.74231
$928$	0	0
$929$	5.40312	0.177271	0.0886354	−	0.996064i	$-0.471749\pi$
$929$	5.40312	0.177271	0.0886354	+	0.996064i	$0.471749\pi$
$930$	0	0
$931$	11.6924i	0.383203i
$932$	0	0
$933$	58.2094i	1.90569i
$934$	0	0
$935$	−1.20697	−0.0394721
$936$	0	0
$937$	−32.8062	−1.07173	−0.535867	−	0.844303i	$-0.680015\pi$
$937$	−32.8062	−1.07173	−0.535867	+	0.844303i	$0.680015\pi$
$938$	0	0
$939$	− 7.81116i	− 0.254908i
$940$	0	0
$941$	16.8953i	0.550771i	0.961334	+	0.275386i	$0.0888056\pi$
$941$	16.8953i	0.550771i	−0.961334	+	0.275386i	$0.911194\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−179.555	−5.84091
$946$	0	0
$947$	19.4549i	0.632200i	0.948726	+	0.316100i	$0.102373\pi$
$947$	19.4549i	0.632200i	−0.948726	+	0.316100i	$0.897627\pi$
$948$	0	0
$949$	6.00000i	0.194768i
$950$	0	0
$951$	77.2648	2.50548
$952$	0	0
$953$	−49.3141	−1.59744	−0.798720	−	0.601704i	$-0.794489\pi$
$953$	−49.3141	−1.59744	−0.798720	+	0.601704i	$0.794489\pi$
$954$	0	0
$955$	14.9702i	0.484424i
$956$	0	0
$957$	6.80625i	0.220015i
$958$	0	0
$959$	85.0273	2.74568
$960$	0	0
$961$	−4.61250	−0.148790
$962$	0	0
$963$	98.1384i	3.16247i
$964$	0	0
$965$	79.2250i	2.55034i
$966$	0	0
$967$	−0.163035	−0.00524285	−0.00262143	−	0.999997i	$-0.500834\pi$
$967$	−0.163035	−0.00524285	−0.00262143	+	0.999997i	$0.500834\pi$
$968$	0	0
$969$	−1.01562	−0.0326265
$970$	0	0
$971$	17.4328i	0.559444i	0.960081	+	0.279722i	$0.0902424\pi$
$971$	17.4328i	0.559444i	−0.960081	+	0.279722i	$0.909758\pi$
$972$	0	0
$973$	− 65.5234i	− 2.10058i
$974$	0	0
$975$	27.1030	0.867992
$976$	0	0
$977$	18.0000	0.575871	0.287936	−	0.957650i	$-0.407031\pi$
$977$	18.0000	0.575871	0.287936	+	0.957650i	$0.407031\pi$
$978$	0	0
$979$	6.55554i	0.209516i
$980$	0	0
$981$	− 28.8062i	− 0.919713i
$982$	0	0
$983$	49.9988	1.59471	0.797356	−	0.603509i	$-0.206231\pi$
$983$	49.9988	1.59471	0.797356	+	0.603509i	$0.206231\pi$
$984$	0	0
$985$	−60.3297	−1.92226
$986$	0	0
$987$	− 55.1356i	− 1.75499i
$988$	0	0
$989$	0	0
$990$	0	0
$991$	20.8736	0.663071	0.331536	−	0.943443i	$-0.392433\pi$
$991$	20.8736	0.663071	0.331536	+	0.943443i	$0.392433\pi$
$992$	0	0
$993$	75.2250	2.38719
$994$	0	0
$995$	69.1763i	2.19304i
$996$	0	0
$997$	− 4.80625i	− 0.152215i	−0.997100	−	0.0761077i	$-0.975751\pi$
$997$	− 4.80625i	− 0.152215i	0.997100	−	0.0761077i	$-0.0242493\pi$
$998$	0	0
$999$	42.6767	1.35023

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3328.2.b.bb.1665.8		8
4.3	odd	2	inner	3328.2.b.bb.1665.2		8
8.3	odd	2	inner	3328.2.b.bb.1665.7		8
8.5	even	2	inner	3328.2.b.bb.1665.1		8
16.3	odd	4		832.2.a.p.1.4		4
16.5	even	4		416.2.a.f.1.4	yes	4
16.11	odd	4		416.2.a.f.1.1	✓	4
16.13	even	4		832.2.a.p.1.1		4
48.5	odd	4		3744.2.a.be.1.4		4
48.11	even	4		3744.2.a.be.1.3		4
48.29	odd	4		7488.2.a.da.1.2		4
48.35	even	4		7488.2.a.da.1.1		4
208.155	odd	4		5408.2.a.bj.1.1		4
208.181	even	4		5408.2.a.bj.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
416.2.a.f.1.1	✓	4	16.11	odd	4
416.2.a.f.1.4	yes	4	16.5	even	4
832.2.a.p.1.1		4	16.13	even	4
832.2.a.p.1.4		4	16.3	odd	4
3328.2.b.bb.1665.1		8	8.5	even	2	inner
3328.2.b.bb.1665.2		8	4.3	odd	2	inner
3328.2.b.bb.1665.7		8	8.3	odd	2	inner
3328.2.b.bb.1665.8		8	1.1	even	1	trivial
3744.2.a.be.1.3		4	48.11	even	4
3744.2.a.be.1.4		4	48.5	odd	4
5408.2.a.bj.1.1		4	208.155	odd	4
5408.2.a.bj.1.4		4	208.181	even	4
7488.2.a.da.1.1		4	48.35	even	4
7488.2.a.da.1.2		4	48.29	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3328 = 2^{8} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3328.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.11473i q^{3} +3.70156i q^{5} -4.20732 q^{7} -6.70156 q^{9} +1.09259i q^{11} +1.00000i q^{13} -11.5294 q^{15} +0.298438 q^{17} +1.09259i q^{19} -13.1047i q^{21} -8.70156 q^{25} -11.5294i q^{27} -2.00000i q^{29} -5.13688 q^{31} -3.40312 q^{33} -15.5737i q^{35} +3.70156i q^{37} -3.11473 q^{39} +9.40312 q^{41} +5.29991i q^{43} -24.8062i q^{45} +4.20732 q^{47} +10.7016 q^{49} +0.929554i q^{51} -1.40312i q^{53} -4.04429 q^{55} -3.40312 q^{57} +13.5515i q^{59} +9.40312i q^{61} +28.1956 q^{63} -3.70156 q^{65} -11.3663i q^{67} -8.25161 q^{71} +6.00000 q^{73} -27.1030i q^{75} -4.59688i q^{77} -14.6441 q^{79} +15.8062 q^{81} +7.32206i q^{83} +1.10469i q^{85} +6.22947 q^{87} +6.00000 q^{89} -4.20732i q^{91} -16.0000i q^{93} -4.04429 q^{95} +8.80625 q^{97} -7.32206i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 28 q^{9} + 28 q^{17} - 44 q^{25} + 24 q^{33} + 24 q^{41} + 60 q^{49} + 24 q^{57} - 4 q^{65} + 48 q^{73} + 24 q^{81} + 48 q^{89} - 32 q^{97}+O(q^{100}) \) 8 * q - 28 * q^9 + 28 * q^17 - 44 * q^25 + 24 * q^33 + 24 * q^41 + 60 * q^49 + 24 * q^57 - 4 * q^65 + 48 * q^73 + 24 * q^81 + 48 * q^89 - 32 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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