LMFDB - Embedded newform 3332.2.a.u.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3332,2,Mod(1,3332)] 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("3332.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3332, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3332 = 2^{2} \cdot 7^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3332.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,0,4,0,4,0,0,0,8] 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:	$26.6061539535$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 8x^{6} + 36x^{5} + 17x^{4} - 76x^{3} - 20x^{2} + 44x + 17 $ x^8 - 4x^7 - 8x^6 + 36x^5 + 17x^4 - 76x^3 - 20x^2 + 44*x + 17
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.26400$ of defining polynomial
Character	$\chi$	$=$	3332.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.26400 q^{3} -1.77431 q^{5} -1.40230 q^{9} +0.318204 q^{11} +0.205935 q^{13} -2.24272 q^{15} -1.00000 q^{17} +5.45902 q^{19} +9.43694 q^{23} -1.85184 q^{25} -5.56451 q^{27} -8.61859 q^{29} +4.94814 q^{31} +0.402209 q^{33} +7.64938 q^{37} +0.260301 q^{39} +1.44153 q^{41} -1.98578 q^{43} +2.48812 q^{45} -1.56955 q^{47} -1.26400 q^{51} -13.5386 q^{53} -0.564590 q^{55} +6.90020 q^{57} +13.2075 q^{59} +3.37365 q^{61} -0.365391 q^{65} +13.3478 q^{67} +11.9283 q^{69} -2.62870 q^{71} +3.86935 q^{73} -2.34072 q^{75} +12.2324 q^{79} -2.82663 q^{81} +10.4915 q^{83} +1.77431 q^{85} -10.8939 q^{87} +0.00620724 q^{89} +6.25445 q^{93} -9.68597 q^{95} +4.93669 q^{97} -0.446218 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{3} + 4 q^{5} + 8 q^{9} - 4 q^{11} + 20 q^{13} + 12 q^{15} - 8 q^{17} + 8 q^{19} + 4 q^{23} + 8 q^{25} + 28 q^{27} - 16 q^{29} - 8 q^{31} + 16 q^{33} + 8 q^{37} + 20 q^{39} + 12 q^{41} - 4 q^{43}+ \cdots - 8 q^{99}+O(q^{100}) $ 8 * q + 4 * q^3 + 4 * q^5 + 8 * q^9 - 4 * q^11 + 20 * q^13 + 12 * q^15 - 8 * q^17 + 8 * q^19 + 4 * q^23 + 8 * q^25 + 28 * q^27 - 16 * q^29 - 8 * q^31 + 16 * q^33 + 8 * q^37 + 20 * q^39 + 12 * q^41 - 4 * q^43 + 20 * q^45 + 4 * q^47 - 4 * q^51 + 12 * q^55 - 16 * q^59 + 32 * q^61 + 44 * q^69 + 24 * q^73 + 24 * q^75 + 4 * q^79 + 36 * q^81 + 28 * q^83 - 4 * q^85 - 40 * q^87 + 20 * q^89 - 16 * q^93 - 20 * q^95 + 56 * q^97 - 8 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.26400	0.729771	0.364885	−	0.931052i	$-0.381108\pi$
$3$	1.26400	0.729771	0.364885	+	0.931052i	$0.381108\pi$
$4$	0	0
$5$	−1.77431	−0.793494	−0.396747	−	0.917928i	$-0.629861\pi$
$5$	−1.77431	−0.793494	−0.396747	+	0.917928i	$0.629861\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−1.40230	−0.467435
$10$	0	0
$11$	0.318204	0.0959420	0.0479710	−	0.998849i	$-0.484724\pi$
$11$	0.318204	0.0959420	0.0479710	+	0.998849i	$0.484724\pi$
$12$	0	0
$13$	0.205935	0.0571160	0.0285580	−	0.999592i	$-0.490908\pi$
$13$	0.205935	0.0571160	0.0285580	+	0.999592i	$0.490908\pi$
$14$	0	0
$15$	−2.24272	−0.579069
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	5.45902	1.25238	0.626192	−	0.779669i	$-0.284612\pi$
$19$	5.45902	1.25238	0.626192	+	0.779669i	$0.284612\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	9.43694	1.96774	0.983870	−	0.178888i	$-0.0572499\pi$
$23$	9.43694	1.96774	0.983870	+	0.178888i	$0.0572499\pi$
$24$	0	0
$25$	−1.85184	−0.370368
$26$	0	0
$27$	−5.56451	−1.07089
$28$	0	0
$29$	−8.61859	−1.60043	−0.800216	−	0.599712i	$-0.795282\pi$
$29$	−8.61859	−1.60043	−0.800216	+	0.599712i	$0.795282\pi$
$30$	0	0
$31$	4.94814	0.888712	0.444356	−	0.895850i	$-0.353432\pi$
$31$	4.94814	0.888712	0.444356	+	0.895850i	$0.353432\pi$
$32$	0	0
$33$	0.402209	0.0700157
$34$	0	0
$35$	0	0
$36$	0	0
$37$	7.64938	1.25755	0.628775	−	0.777587i	$-0.283557\pi$
$37$	7.64938	1.25755	0.628775	+	0.777587i	$0.283557\pi$
$38$	0	0
$39$	0.260301	0.0416816
$40$	0	0
$41$	1.44153	0.225129	0.112565	−	0.993644i	$-0.464093\pi$
$41$	1.44153	0.225129	0.112565	+	0.993644i	$0.464093\pi$
$42$	0	0
$43$	−1.98578	−0.302829	−0.151414	−	0.988470i	$-0.548383\pi$
$43$	−1.98578	−0.302829	−0.151414	+	0.988470i	$0.548383\pi$
$44$	0	0
$45$	2.48812	0.370906
$46$	0	0
$47$	−1.56955	−0.228942	−0.114471	−	0.993427i	$-0.536517\pi$
$47$	−1.56955	−0.228942	−0.114471	+	0.993427i	$0.536517\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−1.26400	−0.176995
$52$	0	0
$53$	−13.5386	−1.85968	−0.929838	−	0.367969i	$-0.880053\pi$
$53$	−13.5386	−1.85968	−0.929838	+	0.367969i	$0.880053\pi$
$54$	0	0
$55$	−0.564590	−0.0761293
$56$	0	0
$57$	6.90020	0.913954
$58$	0	0
$59$	13.2075	1.71947	0.859733	−	0.510743i	$-0.170630\pi$
$59$	13.2075	1.71947	0.859733	+	0.510743i	$0.170630\pi$
$60$	0	0
$61$	3.37365	0.431951	0.215976	−	0.976399i	$-0.430707\pi$
$61$	3.37365	0.431951	0.215976	+	0.976399i	$0.430707\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−0.365391	−0.0453212
$66$	0	0
$67$	13.3478	1.63070	0.815348	−	0.578971i	$-0.196545\pi$
$67$	13.3478	1.63070	0.815348	+	0.578971i	$0.196545\pi$
$68$	0	0
$69$	11.9283	1.43600
$70$	0	0
$71$	−2.62870	−0.311969	−0.155985	−	0.987759i	$-0.549855\pi$
$71$	−2.62870	−0.311969	−0.155985	+	0.987759i	$0.549855\pi$
$72$	0	0
$73$	3.86935	0.452873	0.226436	−	0.974026i	$-0.427292\pi$
$73$	3.86935	0.452873	0.226436	+	0.974026i	$0.427292\pi$
$74$	0	0
$75$	−2.34072	−0.270284
$76$	0	0
$77$	0	0
$78$	0	0
$79$	12.2324	1.37625	0.688124	−	0.725594i	$-0.258435\pi$
$79$	12.2324	1.37625	0.688124	+	0.725594i	$0.258435\pi$
$80$	0	0
$81$	−2.82663	−0.314070
$82$	0	0
$83$	10.4915	1.15159	0.575797	−	0.817592i	$-0.304692\pi$
$83$	10.4915	1.15159	0.575797	+	0.817592i	$0.304692\pi$
$84$	0	0
$85$	1.77431	0.192450
$86$	0	0
$87$	−10.8939	−1.16795
$88$	0	0
$89$	0.00620724	0.000657966 0	0.000328983	−	1.00000i	$-0.499895\pi$
$89$	0.00620724	0.000657966 0	0.000328983		1.00000i	$0.499895\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	6.25445	0.648556
$94$	0	0
$95$	−9.68597	−0.993759
$96$	0	0
$97$	4.93669	0.501245	0.250622	−	0.968085i	$-0.419365\pi$
$97$	4.93669	0.501245	0.250622	+	0.968085i	$0.419365\pi$
$98$	0	0
$99$	−0.446218	−0.0448466
$100$	0	0
$101$	8.69105	0.864792	0.432396	−	0.901684i	$-0.357668\pi$
$101$	8.69105	0.864792	0.432396	+	0.901684i	$0.357668\pi$
$102$	0	0
$103$	4.08166	0.402178	0.201089	−	0.979573i	$-0.435552\pi$
$103$	4.08166	0.402178	0.201089	+	0.979573i	$0.435552\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	14.4600	1.39790	0.698950	−	0.715171i	$-0.253651\pi$
$107$	14.4600	1.39790	0.698950	+	0.715171i	$0.253651\pi$
$108$	0	0
$109$	1.57368	0.150731	0.0753655	−	0.997156i	$-0.475988\pi$
$109$	1.57368	0.150731	0.0753655	+	0.997156i	$0.475988\pi$
$110$	0	0
$111$	9.66882	0.917723
$112$	0	0
$113$	−0.830486	−0.0781256	−0.0390628	−	0.999237i	$-0.512437\pi$
$113$	−0.830486	−0.0781256	−0.0390628	+	0.999237i	$0.512437\pi$
$114$	0	0
$115$	−16.7440	−1.56139
$116$	0	0
$117$	−0.288783	−0.0266980
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.8987	−0.990795
$122$	0	0
$123$	1.82210	0.164293
$124$	0	0
$125$	12.1573	1.08738
$126$	0	0
$127$	−4.93809	−0.438185	−0.219092	−	0.975704i	$-0.570310\pi$
$127$	−4.93809	−0.438185	−0.219092	+	0.975704i	$0.570310\pi$
$128$	0	0
$129$	−2.51003	−0.220995
$130$	0	0
$131$	18.2925	1.59823	0.799113	−	0.601181i	$-0.205303\pi$
$131$	18.2925	1.59823	0.799113	+	0.601181i	$0.205303\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	9.87315	0.849745
$136$	0	0
$137$	9.89826	0.845665	0.422833	−	0.906208i	$-0.361036\pi$
$137$	9.89826	0.845665	0.422833	+	0.906208i	$0.361036\pi$
$138$	0	0
$139$	−12.6946	−1.07675	−0.538373	−	0.842707i	$-0.680961\pi$
$139$	−12.6946	−1.07675	−0.538373	+	0.842707i	$0.680961\pi$
$140$	0	0
$141$	−1.98391	−0.167075
$142$	0	0
$143$	0.0655291	0.00547982
$144$	0	0
$145$	15.2920	1.26993
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1.30758	−0.107121	−0.0535607	−	0.998565i	$-0.517057\pi$
$149$	−1.30758	−0.107121	−0.0535607	+	0.998565i	$0.517057\pi$
$150$	0	0
$151$	3.68720	0.300060	0.150030	−	0.988681i	$-0.452063\pi$
$151$	3.68720	0.300060	0.150030	+	0.988681i	$0.452063\pi$
$152$	0	0
$153$	1.40230	0.113370
$154$	0	0
$155$	−8.77952	−0.705188
$156$	0	0
$157$	15.0157	1.19838	0.599190	−	0.800607i	$-0.295489\pi$
$157$	15.0157	1.19838	0.599190	+	0.800607i	$0.295489\pi$
$158$	0	0
$159$	−17.1128	−1.35714
$160$	0	0
$161$	0	0
$162$	0	0
$163$	9.11421	0.713880	0.356940	−	0.934127i	$-0.383820\pi$
$163$	9.11421	0.713880	0.356940	+	0.934127i	$0.383820\pi$
$164$	0	0
$165$	−0.713642	−0.0555570
$166$	0	0
$167$	9.18362	0.710650	0.355325	−	0.934743i	$-0.384370\pi$
$167$	9.18362	0.710650	0.355325	+	0.934743i	$0.384370\pi$
$168$	0	0
$169$	−12.9576	−0.996738
$170$	0	0
$171$	−7.65520	−0.585408
$172$	0	0
$173$	−15.5531	−1.18248	−0.591241	−	0.806495i	$-0.701362\pi$
$173$	−15.5531	−1.18248	−0.591241	+	0.806495i	$0.701362\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	16.6942	1.25482
$178$	0	0
$179$	−8.10572	−0.605850	−0.302925	−	0.953014i	$-0.597963\pi$
$179$	−8.10572	−0.605850	−0.302925	+	0.953014i	$0.597963\pi$
$180$	0	0
$181$	14.8172	1.10135	0.550677	−	0.834718i	$-0.314370\pi$
$181$	14.8172	1.10135	0.550677	+	0.834718i	$0.314370\pi$
$182$	0	0
$183$	4.26429	0.315226
$184$	0	0
$185$	−13.5723	−0.997858
$186$	0	0
$187$	−0.318204	−0.0232693
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−10.5824	−0.765719	−0.382860	−	0.923807i	$-0.625061\pi$
$191$	−10.5824	−0.765719	−0.382860	+	0.923807i	$0.625061\pi$
$192$	0	0
$193$	−20.4950	−1.47526	−0.737631	−	0.675204i	$-0.764055\pi$
$193$	−20.4950	−1.47526	−0.737631	+	0.675204i	$0.764055\pi$
$194$	0	0
$195$	−0.461854	−0.0330741
$196$	0	0
$197$	−12.3238	−0.878037	−0.439018	−	0.898478i	$-0.644674\pi$
$197$	−12.3238	−0.878037	−0.439018	+	0.898478i	$0.644674\pi$
$198$	0	0
$199$	−1.24045	−0.0879335	−0.0439668	−	0.999033i	$-0.514000\pi$
$199$	−1.24045	−0.0879335	−0.0439668	+	0.999033i	$0.514000\pi$
$200$	0	0
$201$	16.8717	1.19003
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−2.55772	−0.178639
$206$	0	0
$207$	−13.2335	−0.919789
$208$	0	0
$209$	1.73708	0.120156
$210$	0	0
$211$	6.88477	0.473967	0.236984	−	0.971514i	$-0.423841\pi$
$211$	6.88477	0.473967	0.236984	+	0.971514i	$0.423841\pi$
$212$	0	0
$213$	−3.32268	−0.227666
$214$	0	0
$215$	3.52338	0.240293
$216$	0	0
$217$	0	0
$218$	0	0
$219$	4.89086	0.330493
$220$	0	0
$221$	−0.205935	−0.0138527
$222$	0	0
$223$	13.5034	0.904258	0.452129	−	0.891953i	$-0.350665\pi$
$223$	13.5034	0.904258	0.452129	+	0.891953i	$0.350665\pi$
$224$	0	0
$225$	2.59684	0.173123
$226$	0	0
$227$	−0.645775	−0.0428616	−0.0214308	−	0.999770i	$-0.506822\pi$
$227$	−0.645775	−0.0428616	−0.0214308	+	0.999770i	$0.506822\pi$
$228$	0	0
$229$	17.8239	1.17784	0.588919	−	0.808192i	$-0.299554\pi$
$229$	17.8239	1.17784	0.588919	+	0.808192i	$0.299554\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−8.09800	−0.530518	−0.265259	−	0.964177i	$-0.585457\pi$
$233$	−8.09800	−0.530518	−0.265259	+	0.964177i	$0.585457\pi$
$234$	0	0
$235$	2.78485	0.181664
$236$	0	0
$237$	15.4617	1.00434
$238$	0	0
$239$	−3.12106	−0.201885	−0.100942	−	0.994892i	$-0.532186\pi$
$239$	−3.12106	−0.201885	−0.100942	+	0.994892i	$0.532186\pi$
$240$	0	0
$241$	11.3366	0.730252	0.365126	−	0.930958i	$-0.381026\pi$
$241$	11.3366	0.730252	0.365126	+	0.930958i	$0.381026\pi$
$242$	0	0
$243$	13.1207	0.841692
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.12420	0.0715312
$248$	0	0
$249$	13.2613	0.840400
$250$	0	0
$251$	3.57825	0.225857	0.112929	−	0.993603i	$-0.463977\pi$
$251$	3.57825	0.225857	0.112929	+	0.993603i	$0.463977\pi$
$252$	0	0
$253$	3.00287	0.188789
$254$	0	0
$255$	2.24272	0.140445
$256$	0	0
$257$	−9.06411	−0.565404	−0.282702	−	0.959208i	$-0.591231\pi$
$257$	−9.06411	−0.565404	−0.282702	+	0.959208i	$0.591231\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	12.0859	0.748097
$262$	0	0
$263$	4.93268	0.304162	0.152081	−	0.988368i	$-0.451403\pi$
$263$	4.93268	0.304162	0.152081	+	0.988368i	$0.451403\pi$
$264$	0	0
$265$	24.0217	1.47564
$266$	0	0
$267$	0.00784595	0.000480165 0
$268$	0	0
$269$	−17.2672	−1.05280	−0.526400	−	0.850237i	$-0.676458\pi$
$269$	−17.2672	−1.05280	−0.526400	+	0.850237i	$0.676458\pi$
$270$	0	0
$271$	29.4257	1.78748	0.893742	−	0.448581i	$-0.148070\pi$
$271$	29.4257	1.78748	0.893742	+	0.448581i	$0.148070\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−0.589262	−0.0355338
$276$	0	0
$277$	−29.4835	−1.77149	−0.885745	−	0.464172i	$-0.846352\pi$
$277$	−29.4835	−1.77149	−0.885745	+	0.464172i	$0.846352\pi$
$278$	0	0
$279$	−6.93880	−0.415415
$280$	0	0
$281$	5.51539	0.329021	0.164510	−	0.986375i	$-0.447396\pi$
$281$	5.51539	0.329021	0.164510	+	0.986375i	$0.447396\pi$
$282$	0	0
$283$	−7.46379	−0.443676	−0.221838	−	0.975084i	$-0.571206\pi$
$283$	−7.46379	−0.443676	−0.221838	+	0.975084i	$0.571206\pi$
$284$	0	0
$285$	−12.2431	−0.725217
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	6.23997	0.365794
$292$	0	0
$293$	−20.6687	−1.20748	−0.603740	−	0.797181i	$-0.706323\pi$
$293$	−20.6687	−1.20748	−0.603740	+	0.797181i	$0.706323\pi$
$294$	0	0
$295$	−23.4341	−1.36439
$296$	0	0
$297$	−1.77065	−0.102743
$298$	0	0
$299$	1.94339	0.112389
$300$	0	0
$301$	0	0
$302$	0	0
$303$	10.9855	0.631100
$304$	0	0
$305$	−5.98588	−0.342751
$306$	0	0
$307$	−14.0562	−0.802228	−0.401114	−	0.916028i	$-0.631377\pi$
$307$	−14.0562	−0.802228	−0.401114	+	0.916028i	$0.631377\pi$
$308$	0	0
$309$	5.15922	0.293498
$310$	0	0
$311$	−4.76105	−0.269975	−0.134987	−	0.990847i	$-0.543099\pi$
$311$	−4.76105	−0.269975	−0.134987	+	0.990847i	$0.543099\pi$
$312$	0	0
$313$	−26.9097	−1.52103	−0.760513	−	0.649323i	$-0.775052\pi$
$313$	−26.9097	−1.52103	−0.760513	+	0.649323i	$0.775052\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−20.3180	−1.14117	−0.570587	−	0.821237i	$-0.693284\pi$
$317$	−20.3180	−1.14117	−0.570587	+	0.821237i	$0.693284\pi$
$318$	0	0
$319$	−2.74246	−0.153549
$320$	0	0
$321$	18.2774	1.02015
$322$	0	0
$323$	−5.45902	−0.303748
$324$	0	0
$325$	−0.381358	−0.0211539
$326$	0	0
$327$	1.98913	0.109999
$328$	0	0
$329$	0	0
$330$	0	0
$331$	1.29216	0.0710238	0.0355119	−	0.999369i	$-0.488694\pi$
$331$	1.29216	0.0710238	0.0355119	+	0.999369i	$0.488694\pi$
$332$	0	0
$333$	−10.7268	−0.587822
$334$	0	0
$335$	−23.6831	−1.29395
$336$	0	0
$337$	4.35840	0.237417	0.118708	−	0.992929i	$-0.462125\pi$
$337$	4.35840	0.237417	0.118708	+	0.992929i	$0.462125\pi$
$338$	0	0
$339$	−1.04973	−0.0570138
$340$	0	0
$341$	1.57452	0.0852648
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−21.1645	−1.13946
$346$	0	0
$347$	5.83404	0.313188	0.156594	−	0.987663i	$-0.449949\pi$
$347$	5.83404	0.313188	0.156594	+	0.987663i	$0.449949\pi$
$348$	0	0
$349$	−35.2609	−1.88747	−0.943736	−	0.330700i	$-0.892715\pi$
$349$	−35.2609	−1.88747	−0.943736	+	0.330700i	$0.892715\pi$
$350$	0	0
$351$	−1.14593	−0.0611650
$352$	0	0
$353$	21.7586	1.15809	0.579046	−	0.815295i	$-0.303425\pi$
$353$	21.7586	1.15809	0.579046	+	0.815295i	$0.303425\pi$
$354$	0	0
$355$	4.66412	0.247546
$356$	0	0
$357$	0	0
$358$	0	0
$359$	3.68306	0.194385	0.0971923	−	0.995266i	$-0.469014\pi$
$359$	3.68306	0.194385	0.0971923	+	0.995266i	$0.469014\pi$
$360$	0	0
$361$	10.8009	0.568468
$362$	0	0
$363$	−13.7760	−0.723053
$364$	0	0
$365$	−6.86541	−0.359352
$366$	0	0
$367$	8.43369	0.440235	0.220117	−	0.975473i	$-0.429356\pi$
$367$	8.43369	0.440235	0.220117	+	0.975473i	$0.429356\pi$
$368$	0	0
$369$	−2.02146	−0.105233
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−31.3060	−1.62097	−0.810483	−	0.585762i	$-0.800795\pi$
$373$	−31.3060	−1.62097	−0.810483	+	0.585762i	$0.800795\pi$
$374$	0	0
$375$	15.3668	0.793537
$376$	0	0
$377$	−1.77486	−0.0914102
$378$	0	0
$379$	−27.3855	−1.40670	−0.703349	−	0.710844i	$-0.748313\pi$
$379$	−27.3855	−1.40670	−0.703349	+	0.710844i	$0.748313\pi$
$380$	0	0
$381$	−6.24175	−0.319774
$382$	0	0
$383$	−22.2937	−1.13916	−0.569579	−	0.821937i	$-0.692894\pi$
$383$	−22.2937	−1.13916	−0.569579	+	0.821937i	$0.692894\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2.78467	0.141553
$388$	0	0
$389$	−26.3008	−1.33351	−0.666753	−	0.745279i	$-0.732316\pi$
$389$	−26.3008	−1.33351	−0.666753	+	0.745279i	$0.732316\pi$
$390$	0	0
$391$	−9.43694	−0.477247
$392$	0	0
$393$	23.1218	1.16634
$394$	0	0
$395$	−21.7039	−1.09204
$396$	0	0
$397$	38.8034	1.94749	0.973745	−	0.227644i	$-0.0731021\pi$
$397$	38.8034	1.94749	0.973745	+	0.227644i	$0.0731021\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	14.9758	0.747855	0.373928	−	0.927458i	$-0.378011\pi$
$401$	14.9758	0.747855	0.373928	+	0.927458i	$0.378011\pi$
$402$	0	0
$403$	1.01899	0.0507597
$404$	0	0
$405$	5.01531	0.249213
$406$	0	0
$407$	2.43406	0.120652
$408$	0	0
$409$	38.3051	1.89407	0.947034	−	0.321134i	$-0.104064\pi$
$409$	38.3051	1.89407	0.947034	+	0.321134i	$0.104064\pi$
$410$	0	0
$411$	12.5114	0.617142
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−18.6152	−0.913783
$416$	0	0
$417$	−16.0460	−0.785778
$418$	0	0
$419$	23.3744	1.14192	0.570958	−	0.820979i	$-0.306572\pi$
$419$	23.3744	1.14192	0.570958	+	0.820979i	$0.306572\pi$
$420$	0	0
$421$	8.17711	0.398528	0.199264	−	0.979946i	$-0.436145\pi$
$421$	8.17711	0.398528	0.199264	+	0.979946i	$0.436145\pi$
$422$	0	0
$423$	2.20098	0.107015
$424$	0	0
$425$	1.85184	0.0898274
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0.0828288	0.00399901
$430$	0	0
$431$	−20.1422	−0.970217	−0.485108	−	0.874454i	$-0.661220\pi$
$431$	−20.1422	−0.970217	−0.485108	+	0.874454i	$0.661220\pi$
$432$	0	0
$433$	30.8004	1.48017	0.740085	−	0.672513i	$-0.234785\pi$
$433$	30.8004	1.48017	0.740085	+	0.672513i	$0.234785\pi$
$434$	0	0
$435$	19.3291	0.926759
$436$	0	0
$437$	51.5165	2.46437
$438$	0	0
$439$	−37.7595	−1.80216	−0.901082	−	0.433649i	$-0.857226\pi$
$439$	−37.7595	−1.80216	−0.901082	+	0.433649i	$0.857226\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	0.877302	0.0416819	0.0208409	−	0.999783i	$-0.493366\pi$
$443$	0.877302	0.0416819	0.0208409	+	0.999783i	$0.493366\pi$
$444$	0	0
$445$	−0.0110135	−0.000522092 0
$446$	0	0
$447$	−1.65279	−0.0781741
$448$	0	0
$449$	33.0313	1.55884	0.779421	−	0.626500i	$-0.215513\pi$
$449$	33.0313	1.55884	0.779421	+	0.626500i	$0.215513\pi$
$450$	0	0
$451$	0.458700	0.0215993
$452$	0	0
$453$	4.66062	0.218975
$454$	0	0
$455$	0	0
$456$	0	0
$457$	31.6690	1.48141	0.740707	−	0.671828i	$-0.234491\pi$
$457$	31.6690	1.48141	0.740707	+	0.671828i	$0.234491\pi$
$458$	0	0
$459$	5.56451	0.259729
$460$	0	0
$461$	−29.9124	−1.39316	−0.696580	−	0.717479i	$-0.745296\pi$
$461$	−29.9124	−1.39316	−0.696580	+	0.717479i	$0.745296\pi$
$462$	0	0
$463$	−15.8809	−0.738050	−0.369025	−	0.929419i	$-0.620308\pi$
$463$	−15.8809	−0.738050	−0.369025	+	0.929419i	$0.620308\pi$
$464$	0	0
$465$	−11.0973	−0.514625
$466$	0	0
$467$	−13.0111	−0.602081	−0.301041	−	0.953611i	$-0.597334\pi$
$467$	−13.0111	−0.602081	−0.301041	+	0.953611i	$0.597334\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	18.9798	0.874543
$472$	0	0
$473$	−0.631882	−0.0290540
$474$	0	0
$475$	−10.1092	−0.463843
$476$	0	0
$477$	18.9853	0.869277
$478$	0	0
$479$	−33.6377	−1.53695	−0.768473	−	0.639882i	$-0.778983\pi$
$479$	−33.6377	−1.53695	−0.768473	+	0.639882i	$0.778983\pi$
$480$	0	0
$481$	1.57527	0.0718262
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−8.75919	−0.397734
$486$	0	0
$487$	33.3793	1.51256	0.756281	−	0.654247i	$-0.227014\pi$
$487$	33.3793	1.51256	0.756281	+	0.654247i	$0.227014\pi$
$488$	0	0
$489$	11.5204	0.520969
$490$	0	0
$491$	1.61369	0.0728249	0.0364125	−	0.999337i	$-0.488407\pi$
$491$	1.61369	0.0728249	0.0364125	+	0.999337i	$0.488407\pi$
$492$	0	0
$493$	8.61859	0.388162
$494$	0	0
$495$	0.791727	0.0355855
$496$	0	0
$497$	0	0
$498$	0	0
$499$	7.08060	0.316971	0.158486	−	0.987361i	$-0.449339\pi$
$499$	7.08060	0.316971	0.158486	+	0.987361i	$0.449339\pi$
$500$	0	0
$501$	11.6081	0.518611
$502$	0	0
$503$	−17.2097	−0.767342	−0.383671	−	0.923470i	$-0.625340\pi$
$503$	−17.2097	−0.767342	−0.383671	+	0.923470i	$0.625340\pi$
$504$	0	0
$505$	−15.4206	−0.686207
$506$	0	0
$507$	−16.3784	−0.727390
$508$	0	0
$509$	0.569329	0.0252351	0.0126175	−	0.999920i	$-0.495984\pi$
$509$	0.569329	0.0252351	0.0126175	+	0.999920i	$0.495984\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−30.3768	−1.34117
$514$	0	0
$515$	−7.24211	−0.319125
$516$	0	0
$517$	−0.499435	−0.0219651
$518$	0	0
$519$	−19.6591	−0.862941
$520$	0	0
$521$	−4.96590	−0.217560	−0.108780	−	0.994066i	$-0.534694\pi$
$521$	−4.96590	−0.217560	−0.108780	+	0.994066i	$0.534694\pi$
$522$	0	0
$523$	10.5968	0.463364	0.231682	−	0.972792i	$-0.425577\pi$
$523$	10.5968	0.463364	0.231682	+	0.972792i	$0.425577\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−4.94814	−0.215544
$528$	0	0
$529$	66.0559	2.87200
$530$	0	0
$531$	−18.5209	−0.803738
$532$	0	0
$533$	0.296861	0.0128585
$534$	0	0
$535$	−25.6564	−1.10922
$536$	0	0
$537$	−10.2456	−0.442132
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−8.64750	−0.371785	−0.185892	−	0.982570i	$-0.559518\pi$
$541$	−8.64750	−0.371785	−0.185892	+	0.982570i	$0.559518\pi$
$542$	0	0
$543$	18.7289	0.803736
$544$	0	0
$545$	−2.79219	−0.119604
$546$	0	0
$547$	27.9601	1.19549	0.597744	−	0.801687i	$-0.296064\pi$
$547$	27.9601	1.19549	0.597744	+	0.801687i	$0.296064\pi$
$548$	0	0
$549$	−4.73088	−0.201909
$550$	0	0
$551$	−47.0490	−2.00436
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−17.1554	−0.728208
$556$	0	0
$557$	23.4737	0.994613	0.497306	−	0.867575i	$-0.334322\pi$
$557$	23.4737	0.994613	0.497306	+	0.867575i	$0.334322\pi$
$558$	0	0
$559$	−0.408941	−0.0172963
$560$	0	0
$561$	−0.402209	−0.0169813
$562$	0	0
$563$	26.6059	1.12130	0.560652	−	0.828052i	$-0.310551\pi$
$563$	26.6059	1.12130	0.560652	+	0.828052i	$0.310551\pi$
$564$	0	0
$565$	1.47354	0.0619921
$566$	0	0
$567$	0	0
$568$	0	0
$569$	28.0713	1.17681	0.588405	−	0.808566i	$-0.299756\pi$
$569$	28.0713	1.17681	0.588405	+	0.808566i	$0.299756\pi$
$570$	0	0
$571$	39.1974	1.64036	0.820180	−	0.572105i	$-0.193873\pi$
$571$	39.1974	1.64036	0.820180	+	0.572105i	$0.193873\pi$
$572$	0	0
$573$	−13.3762	−0.558799
$574$	0	0
$575$	−17.4757	−0.728787
$576$	0	0
$577$	6.80895	0.283460	0.141730	−	0.989905i	$-0.454733\pi$
$577$	6.80895	0.283460	0.141730	+	0.989905i	$0.454733\pi$
$578$	0	0
$579$	−25.9057	−1.07660
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−4.30804	−0.178421
$584$	0	0
$585$	0.512389	0.0211847
$586$	0	0
$587$	−23.8016	−0.982397	−0.491198	−	0.871048i	$-0.663441\pi$
$587$	−23.8016	−0.982397	−0.491198	+	0.871048i	$0.663441\pi$
$588$	0	0
$589$	27.0120	1.11301
$590$	0	0
$591$	−15.5773	−0.640766
$592$	0	0
$593$	−30.2650	−1.24284	−0.621418	−	0.783479i	$-0.713443\pi$
$593$	−30.2650	−1.24284	−0.621418	+	0.783479i	$0.713443\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−1.56793	−0.0641713
$598$	0	0
$599$	−20.3977	−0.833427	−0.416714	−	0.909038i	$-0.636818\pi$
$599$	−20.3977	−0.833427	−0.416714	+	0.909038i	$0.636818\pi$
$600$	0	0
$601$	−19.6591	−0.801912	−0.400956	−	0.916097i	$-0.631322\pi$
$601$	−19.6591	−0.801912	−0.400956	+	0.916097i	$0.631322\pi$
$602$	0	0
$603$	−18.7177	−0.762244
$604$	0	0
$605$	19.3377	0.786190
$606$	0	0
$607$	0.670435	0.0272121	0.0136061	−	0.999907i	$-0.495669\pi$
$607$	0.670435	0.0272121	0.0136061	+	0.999907i	$0.495669\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−0.323224	−0.0130762
$612$	0	0
$613$	20.1900	0.815466	0.407733	−	0.913101i	$-0.366319\pi$
$613$	20.1900	0.815466	0.407733	+	0.913101i	$0.366319\pi$
$614$	0	0
$615$	−3.23295	−0.130365
$616$	0	0
$617$	15.1299	0.609107	0.304554	−	0.952495i	$-0.401493\pi$
$617$	15.1299	0.609107	0.304554	+	0.952495i	$0.401493\pi$
$618$	0	0
$619$	−23.6956	−0.952407	−0.476204	−	0.879335i	$-0.657987\pi$
$619$	−23.6956	−0.952407	−0.476204	+	0.879335i	$0.657987\pi$
$620$	0	0
$621$	−52.5120	−2.10723
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−12.3115	−0.492460
$626$	0	0
$627$	2.19567	0.0876865
$628$	0	0
$629$	−7.64938	−0.305001
$630$	0	0
$631$	−0.879472	−0.0350112	−0.0175056	−	0.999847i	$-0.505572\pi$
$631$	−0.879472	−0.0350112	−0.0175056	+	0.999847i	$0.505572\pi$
$632$	0	0
$633$	8.70235	0.345887
$634$	0	0
$635$	8.76168	0.347697
$636$	0	0
$637$	0	0
$638$	0	0
$639$	3.68624	0.145825
$640$	0	0
$641$	−32.1852	−1.27124	−0.635620	−	0.772002i	$-0.719255\pi$
$641$	−32.1852	−1.27124	−0.635620	+	0.772002i	$0.719255\pi$
$642$	0	0
$643$	18.7957	0.741231	0.370615	−	0.928786i	$-0.379147\pi$
$643$	18.7957	0.741231	0.370615	+	0.928786i	$0.379147\pi$
$644$	0	0
$645$	4.45355	0.175358
$646$	0	0
$647$	−44.7477	−1.75921	−0.879607	−	0.475701i	$-0.842194\pi$
$647$	−44.7477	−1.75921	−0.879607	+	0.475701i	$0.842194\pi$
$648$	0	0
$649$	4.20266	0.164969
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−13.6090	−0.532560	−0.266280	−	0.963896i	$-0.585795\pi$
$653$	−13.6090	−0.532560	−0.266280	+	0.963896i	$0.585795\pi$
$654$	0	0
$655$	−32.4565	−1.26818
$656$	0	0
$657$	−5.42600	−0.211688
$658$	0	0
$659$	31.7616	1.23725	0.618627	−	0.785684i	$-0.287689\pi$
$659$	31.7616	1.23725	0.618627	+	0.785684i	$0.287689\pi$
$660$	0	0
$661$	20.4414	0.795078	0.397539	−	0.917585i	$-0.369864\pi$
$661$	20.4414	0.795078	0.397539	+	0.917585i	$0.369864\pi$
$662$	0	0
$663$	−0.260301	−0.0101093
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−81.3331	−3.14923
$668$	0	0
$669$	17.0684	0.659901
$670$	0	0
$671$	1.07351	0.0414423
$672$	0	0
$673$	−8.40756	−0.324087	−0.162044	−	0.986784i	$-0.551809\pi$
$673$	−8.40756	−0.324087	−0.162044	+	0.986784i	$0.551809\pi$
$674$	0	0
$675$	10.3046	0.396624
$676$	0	0
$677$	10.9126	0.419405	0.209703	−	0.977765i	$-0.432750\pi$
$677$	10.9126	0.419405	0.209703	+	0.977765i	$0.432750\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−0.816259	−0.0312791
$682$	0	0
$683$	0.402969	0.0154192	0.00770959	−	0.999970i	$-0.497546\pi$
$683$	0.402969	0.0154192	0.00770959	+	0.999970i	$0.497546\pi$
$684$	0	0
$685$	−17.5625	−0.671030
$686$	0	0
$687$	22.5294	0.859552
$688$	0	0
$689$	−2.78807	−0.106217
$690$	0	0
$691$	28.7597	1.09407	0.547036	−	0.837109i	$-0.315756\pi$
$691$	28.7597	1.09407	0.547036	+	0.837109i	$0.315756\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	22.5242	0.854391
$696$	0	0
$697$	−1.44153	−0.0546019
$698$	0	0
$699$	−10.2359	−0.387157
$700$	0	0
$701$	30.4661	1.15069	0.575345	−	0.817911i	$-0.304868\pi$
$701$	30.4661	1.15069	0.575345	+	0.817911i	$0.304868\pi$
$702$	0	0
$703$	41.7581	1.57494
$704$	0	0
$705$	3.52005	0.132573
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−13.1020	−0.492057	−0.246029	−	0.969263i	$-0.579126\pi$
$709$	−13.1020	−0.492057	−0.246029	+	0.969263i	$0.579126\pi$
$710$	0	0
$711$	−17.1535	−0.643305
$712$	0	0
$713$	46.6953	1.74875
$714$	0	0
$715$	−0.116269	−0.00434820
$716$	0	0
$717$	−3.94502	−0.147330
$718$	0	0
$719$	15.4948	0.577858	0.288929	−	0.957350i	$-0.406701\pi$
$719$	15.4948	0.577858	0.288929	+	0.957350i	$0.406701\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	14.3294	0.532917
$724$	0	0
$725$	15.9602	0.592748
$726$	0	0
$727$	−35.6680	−1.32285	−0.661427	−	0.750010i	$-0.730049\pi$
$727$	−35.6680	−1.32285	−0.661427	+	0.750010i	$0.730049\pi$
$728$	0	0
$729$	25.0644	0.928312
$730$	0	0
$731$	1.98578	0.0734467
$732$	0	0
$733$	−12.5978	−0.465311	−0.232656	−	0.972559i	$-0.574742\pi$
$733$	−12.5978	−0.465311	−0.232656	+	0.972559i	$0.574742\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4.24733	0.156452
$738$	0	0
$739$	32.5853	1.19867	0.599334	−	0.800499i	$-0.295432\pi$
$739$	32.5853	1.19867	0.599334	+	0.800499i	$0.295432\pi$
$740$	0	0
$741$	1.42099	0.0522014
$742$	0	0
$743$	−1.53311	−0.0562444	−0.0281222	−	0.999604i	$-0.508953\pi$
$743$	−1.53311	−0.0562444	−0.0281222	+	0.999604i	$0.508953\pi$
$744$	0	0
$745$	2.32005	0.0850001
$746$	0	0
$747$	−14.7123	−0.538295
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−24.5508	−0.895871	−0.447935	−	0.894066i	$-0.647841\pi$
$751$	−24.5508	−0.895871	−0.447935	+	0.894066i	$0.647841\pi$
$752$	0	0
$753$	4.52291	0.164824
$754$	0	0
$755$	−6.54221	−0.238095
$756$	0	0
$757$	30.8744	1.12215	0.561075	−	0.827765i	$-0.310388\pi$
$757$	30.8744	1.12215	0.561075	+	0.827765i	$0.310388\pi$
$758$	0	0
$759$	3.79563	0.137773
$760$	0	0
$761$	−26.8426	−0.973045	−0.486522	−	0.873668i	$-0.661735\pi$
$761$	−26.8426	−0.973045	−0.486522	+	0.873668i	$0.661735\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−2.48812	−0.0899580
$766$	0	0
$767$	2.71988	0.0982090
$768$	0	0
$769$	−52.2087	−1.88269	−0.941347	−	0.337440i	$-0.890439\pi$
$769$	−52.2087	−1.88269	−0.941347	+	0.337440i	$0.890439\pi$
$770$	0	0
$771$	−11.4570	−0.412615
$772$	0	0
$773$	15.3808	0.553207	0.276604	−	0.960984i	$-0.410791\pi$
$773$	15.3808	0.553207	0.276604	+	0.960984i	$0.410791\pi$
$774$	0	0
$775$	−9.16316	−0.329150
$776$	0	0
$777$	0	0
$778$	0	0
$779$	7.86934	0.281949
$780$	0	0
$781$	−0.836462	−0.0299309
$782$	0	0
$783$	47.9582	1.71389
$784$	0	0
$785$	−26.6424	−0.950907
$786$	0	0
$787$	25.6237	0.913387	0.456693	−	0.889624i	$-0.349034\pi$
$787$	25.6237	0.913387	0.456693	+	0.889624i	$0.349034\pi$
$788$	0	0
$789$	6.23491	0.221969
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0.694751	0.0246713
$794$	0	0
$795$	30.3634	1.07688
$796$	0	0
$797$	−15.4806	−0.548350	−0.274175	−	0.961680i	$-0.588405\pi$
$797$	−15.4806	−0.548350	−0.274175	+	0.961680i	$0.588405\pi$
$798$	0	0
$799$	1.56955	0.0555265
$800$	0	0
$801$	−0.00870444	−0.000307556 0
$802$	0	0
$803$	1.23124	0.0434495
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−21.8257	−0.768302
$808$	0	0
$809$	−38.5138	−1.35407	−0.677036	−	0.735950i	$-0.736736\pi$
$809$	−38.5138	−1.35407	−0.677036	+	0.735950i	$0.736736\pi$
$810$	0	0
$811$	−41.6065	−1.46100	−0.730502	−	0.682911i	$-0.760714\pi$
$811$	−41.6065	−1.46100	−0.730502	+	0.682911i	$0.760714\pi$
$812$	0	0
$813$	37.1941	1.30445
$814$	0	0
$815$	−16.1714	−0.566459
$816$	0	0
$817$	−10.8404	−0.379258
$818$	0	0
$819$	0	0
$820$	0	0
$821$	20.4853	0.714943	0.357472	−	0.933924i	$-0.383639\pi$
$821$	20.4853	0.714943	0.357472	+	0.933924i	$0.383639\pi$
$822$	0	0
$823$	17.1271	0.597014	0.298507	−	0.954407i	$-0.403511\pi$
$823$	17.1271	0.597014	0.298507	+	0.954407i	$0.403511\pi$
$824$	0	0
$825$	−0.744827	−0.0259315
$826$	0	0
$827$	−50.9089	−1.77028	−0.885138	−	0.465329i	$-0.845936\pi$
$827$	−50.9089	−1.77028	−0.885138	+	0.465329i	$0.845936\pi$
$828$	0	0
$829$	−37.5712	−1.30490	−0.652451	−	0.757831i	$-0.726259\pi$
$829$	−37.5712	−1.30490	−0.652451	+	0.757831i	$0.726259\pi$
$830$	0	0
$831$	−37.2671	−1.29278
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−16.2945	−0.563896
$836$	0	0
$837$	−27.5340	−0.951714
$838$	0	0
$839$	33.2696	1.14860	0.574298	−	0.818646i	$-0.305275\pi$
$839$	33.2696	1.14860	0.574298	+	0.818646i	$0.305275\pi$
$840$	0	0
$841$	45.2800	1.56138
$842$	0	0
$843$	6.97146	0.240110
$844$	0	0
$845$	22.9907	0.790905
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−9.43423	−0.323782
$850$	0	0
$851$	72.1868	2.47453
$852$	0	0
$853$	−30.3159	−1.03800	−0.518998	−	0.854776i	$-0.673695\pi$
$853$	−30.3159	−1.03800	−0.518998	+	0.854776i	$0.673695\pi$
$854$	0	0
$855$	13.5827	0.464517
$856$	0	0
$857$	−38.7778	−1.32463	−0.662313	−	0.749228i	$-0.730425\pi$
$857$	−38.7778	−1.32463	−0.662313	+	0.749228i	$0.730425\pi$
$858$	0	0
$859$	36.4346	1.24313	0.621566	−	0.783362i	$-0.286497\pi$
$859$	36.4346	1.24313	0.621566	+	0.783362i	$0.286497\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−41.0825	−1.39847	−0.699233	−	0.714894i	$-0.746475\pi$
$863$	−41.0825	−1.39847	−0.699233	+	0.714894i	$0.746475\pi$
$864$	0	0
$865$	27.5960	0.938292
$866$	0	0
$867$	1.26400	0.0429277
$868$	0	0
$869$	3.89238	0.132040
$870$	0	0
$871$	2.74878	0.0931388
$872$	0	0
$873$	−6.92273	−0.234299
$874$	0	0
$875$	0	0
$876$	0	0
$877$	6.12907	0.206964	0.103482	−	0.994631i	$-0.467002\pi$
$877$	6.12907	0.206964	0.103482	+	0.994631i	$0.467002\pi$
$878$	0	0
$879$	−26.1253	−0.881183
$880$	0	0
$881$	−32.4381	−1.09287	−0.546434	−	0.837502i	$-0.684015\pi$
$881$	−32.4381	−1.09287	−0.546434	+	0.837502i	$0.684015\pi$
$882$	0	0
$883$	−35.1411	−1.18259	−0.591296	−	0.806454i	$-0.701384\pi$
$883$	−35.1411	−1.18259	−0.591296	+	0.806454i	$0.701384\pi$
$884$	0	0
$885$	−29.6207	−0.995689
$886$	0	0
$887$	−26.9435	−0.904673	−0.452336	−	0.891847i	$-0.649409\pi$
$887$	−26.9435	−0.904673	−0.452336	+	0.891847i	$0.649409\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−0.899445	−0.0301325
$892$	0	0
$893$	−8.56818	−0.286723
$894$	0	0
$895$	14.3820	0.480738
$896$	0	0
$897$	2.45645	0.0820184
$898$	0	0
$899$	−42.6460	−1.42232
$900$	0	0
$901$	13.5386	0.451038
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−26.2903	−0.873918
$906$	0	0
$907$	−23.1647	−0.769170	−0.384585	−	0.923090i	$-0.625655\pi$
$907$	−23.1647	−0.769170	−0.384585	+	0.923090i	$0.625655\pi$
$908$	0	0
$909$	−12.1875	−0.404234
$910$	0	0
$911$	−28.5642	−0.946376	−0.473188	−	0.880962i	$-0.656897\pi$
$911$	−28.5642	−0.946376	−0.473188	+	0.880962i	$0.656897\pi$
$912$	0	0
$913$	3.33844	0.110486
$914$	0	0
$915$	−7.56616	−0.250129
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−29.4563	−0.971673	−0.485836	−	0.874050i	$-0.661485\pi$
$919$	−29.4563	−0.971673	−0.485836	+	0.874050i	$0.661485\pi$
$920$	0	0
$921$	−17.7670	−0.585442
$922$	0	0
$923$	−0.541340	−0.0178184
$924$	0	0
$925$	−14.1654	−0.465756
$926$	0	0
$927$	−5.72372	−0.187992
$928$	0	0
$929$	−6.91683	−0.226934	−0.113467	−	0.993542i	$-0.536196\pi$
$929$	−6.91683	−0.226934	−0.113467	+	0.993542i	$0.536196\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−6.01797	−0.197020
$934$	0	0
$935$	0.564590	0.0184641
$936$	0	0
$937$	39.1000	1.27734	0.638671	−	0.769480i	$-0.279484\pi$
$937$	39.1000	1.27734	0.638671	+	0.769480i	$0.279484\pi$
$938$	0	0
$939$	−34.0138	−1.11000
$940$	0	0
$941$	19.8018	0.645522	0.322761	−	0.946481i	$-0.395389\pi$
$941$	19.8018	0.645522	0.322761	+	0.946481i	$0.395389\pi$
$942$	0	0
$943$	13.6036	0.442996
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−6.95286	−0.225938	−0.112969	−	0.993599i	$-0.536036\pi$
$947$	−6.95286	−0.225938	−0.112969	+	0.993599i	$0.536036\pi$
$948$	0	0
$949$	0.796833	0.0258663
$950$	0	0
$951$	−25.6820	−0.832795
$952$	0	0
$953$	44.2489	1.43336	0.716681	−	0.697402i	$-0.245661\pi$
$953$	44.2489	1.43336	0.716681	+	0.697402i	$0.245661\pi$
$954$	0	0
$955$	18.7765	0.607593
$956$	0	0
$957$	−3.46648	−0.112055
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−6.51590	−0.210190
$962$	0	0
$963$	−20.2773	−0.653427
$964$	0	0
$965$	36.3644	1.17061
$966$	0	0
$967$	3.70560	0.119164	0.0595821	−	0.998223i	$-0.481023\pi$
$967$	3.70560	0.119164	0.0595821	+	0.998223i	$0.481023\pi$
$968$	0	0
$969$	−6.90020	−0.221666
$970$	0	0
$971$	7.59954	0.243881	0.121940	−	0.992537i	$-0.461088\pi$
$971$	7.59954	0.243881	0.121940	+	0.992537i	$0.461088\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−0.482036	−0.0154375
$976$	0	0
$977$	−37.4738	−1.19889	−0.599446	−	0.800415i	$-0.704612\pi$
$977$	−37.4738	−1.19889	−0.599446	+	0.800415i	$0.704612\pi$
$978$	0	0
$979$	0.00197517	6.31266e−5 0
$980$	0	0
$981$	−2.20678	−0.0704569
$982$	0	0
$983$	20.6504	0.658647	0.329323	−	0.944217i	$-0.393179\pi$
$983$	20.6504	0.658647	0.329323	+	0.944217i	$0.393179\pi$
$984$	0	0
$985$	21.8663	0.696717
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−18.7397	−0.595888
$990$	0	0
$991$	−17.2648	−0.548435	−0.274218	−	0.961668i	$-0.588419\pi$
$991$	−17.2648	−0.548435	−0.274218	+	0.961668i	$0.588419\pi$
$992$	0	0
$993$	1.63330	0.0518311
$994$	0	0
$995$	2.20095	0.0697747
$996$	0	0
$997$	13.6573	0.432530	0.216265	−	0.976335i	$-0.430613\pi$
$997$	13.6573	0.432530	0.216265	+	0.976335i	$0.430613\pi$
$998$	0	0
$999$	−42.5651	−1.34670

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3332.2.a.u.1.5	yes	8
7.6	odd	2		3332.2.a.t.1.4	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3332.2.a.t.1.4	✓	8	7.6	odd	2
3332.2.a.u.1.5	yes	8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3332 = 2^{2} \cdot 7^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3332.a (trivial)

\(f(q)\)	\(=\)	\(q+1.26400 q^{3} -1.77431 q^{5} -1.40230 q^{9} +0.318204 q^{11} +0.205935 q^{13} -2.24272 q^{15} -1.00000 q^{17} +5.45902 q^{19} +9.43694 q^{23} -1.85184 q^{25} -5.56451 q^{27} -8.61859 q^{29} +4.94814 q^{31} +0.402209 q^{33} +7.64938 q^{37} +0.260301 q^{39} +1.44153 q^{41} -1.98578 q^{43} +2.48812 q^{45} -1.56955 q^{47} -1.26400 q^{51} -13.5386 q^{53} -0.564590 q^{55} +6.90020 q^{57} +13.2075 q^{59} +3.37365 q^{61} -0.365391 q^{65} +13.3478 q^{67} +11.9283 q^{69} -2.62870 q^{71} +3.86935 q^{73} -2.34072 q^{75} +12.2324 q^{79} -2.82663 q^{81} +10.4915 q^{83} +1.77431 q^{85} -10.8939 q^{87} +0.00620724 q^{89} +6.25445 q^{93} -9.68597 q^{95} +4.93669 q^{97} -0.446218 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{3} + 4 q^{5} + 8 q^{9} - 4 q^{11} + 20 q^{13} + 12 q^{15} - 8 q^{17} + 8 q^{19} + 4 q^{23} + 8 q^{25} + 28 q^{27} - 16 q^{29} - 8 q^{31} + 16 q^{33} + 8 q^{37} + 20 q^{39} + 12 q^{41} - 4 q^{43}+ \cdots - 8 q^{99}+O(q^{100}) \) 8 * q + 4 * q^3 + 4 * q^5 + 8 * q^9 - 4 * q^11 + 20 * q^13 + 12 * q^15 - 8 * q^17 + 8 * q^19 + 4 * q^23 + 8 * q^25 + 28 * q^27 - 16 * q^29 - 8 * q^31 + 16 * q^33 + 8 * q^37 + 20 * q^39 + 12 * q^41 - 4 * q^43 + 20 * q^45 + 4 * q^47 - 4 * q^51 + 12 * q^55 - 16 * q^59 + 32 * q^61 + 44 * q^69 + 24 * q^73 + 24 * q^75 + 4 * q^79 + 36 * q^81 + 28 * q^83 - 4 * q^85 - 40 * q^87 + 20 * q^89 - 16 * q^93 - 20 * q^95 + 56 * q^97 - 8 * q^99

Introduction

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