LMFDB - Embedded newform 3332.2.a.u.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3332,2,Mod(1,3332)]

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

lf = mfeigenbasis(mf)

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")

//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);

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

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			1.2
Root			$-1.33282$ 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)

$f(q)$	$=$	$q-1.33282 q^{3} +2.63441 q^{5} -1.22360 q^{9} +3.39855 q^{11} +2.73620 q^{13} -3.51118 q^{15} -1.00000 q^{17} -0.404319 q^{19} +1.46779 q^{23} +1.94009 q^{25} +5.62928 q^{27} +5.14721 q^{29} -5.78955 q^{31} -4.52965 q^{33} +3.35268 q^{37} -3.64685 q^{39} +8.88513 q^{41} -7.39301 q^{43} -3.22346 q^{45} +10.8335 q^{47} +1.33282 q^{51} -5.85643 q^{53} +8.95317 q^{55} +0.538883 q^{57} -2.15676 q^{59} +1.64909 q^{61} +7.20825 q^{65} -12.4743 q^{67} -1.95630 q^{69} -5.10604 q^{71} +11.7695 q^{73} -2.58579 q^{75} +1.64849 q^{79} -3.83200 q^{81} +11.4159 q^{83} -2.63441 q^{85} -6.86029 q^{87} +5.01381 q^{89} +7.71641 q^{93} -1.06514 q^{95} -1.08733 q^{97} -4.15847 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.33282	−0.769502	−0.384751	−	0.923020i	$-0.625713\pi$
$3$	−1.33282	−0.769502	−0.384751	+	0.923020i	$0.625713\pi$
$4$	0	0
$5$	2.63441	1.17814	0.589071	−	0.808081i	$-0.299494\pi$
$5$	2.63441	1.17814	0.589071	+	0.808081i	$0.299494\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−1.22360	−0.407867
$10$	0	0
$11$	3.39855	1.02470	0.512351	−	0.858776i	$-0.328775\pi$
$11$	3.39855	1.02470	0.512351	+	0.858776i	$0.328775\pi$
$12$	0	0
$13$	2.73620	0.758885	0.379442	−	0.925215i	$-0.376116\pi$
$13$	2.73620	0.758885	0.379442	+	0.925215i	$0.376116\pi$
$14$	0	0
$15$	−3.51118	−0.906582
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−0.404319	−0.0927571	−0.0463785	−	0.998924i	$-0.514768\pi$
$19$	−0.404319	−0.0927571	−0.0463785	+	0.998924i	$0.514768\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.46779	0.306056	0.153028	−	0.988222i	$-0.451098\pi$
$23$	1.46779	0.306056	0.153028	+	0.988222i	$0.451098\pi$
$24$	0	0
$25$	1.94009	0.388019
$26$	0	0
$27$	5.62928	1.08336
$28$	0	0
$29$	5.14721	0.955813	0.477907	−	0.878411i	$-0.341396\pi$
$29$	5.14721	0.955813	0.477907	+	0.878411i	$0.341396\pi$
$30$	0	0
$31$	−5.78955	−1.03983	−0.519917	−	0.854217i	$-0.674037\pi$
$31$	−5.78955	−1.03983	−0.519917	+	0.854217i	$0.674037\pi$
$32$	0	0
$33$	−4.52965	−0.788511
$34$	0	0
$35$	0	0
$36$	0	0
$37$	3.35268	0.551177	0.275589	−	0.961276i	$-0.411127\pi$
$37$	3.35268	0.551177	0.275589	+	0.961276i	$0.411127\pi$
$38$	0	0
$39$	−3.64685	−0.583963
$40$	0	0
$41$	8.88513	1.38762	0.693812	−	0.720156i	$-0.255930\pi$
$41$	8.88513	1.38762	0.693812	+	0.720156i	$0.255930\pi$
$42$	0	0
$43$	−7.39301	−1.12742	−0.563712	−	0.825971i	$-0.690627\pi$
$43$	−7.39301	−1.12742	−0.563712	+	0.825971i	$0.690627\pi$
$44$	0	0
$45$	−3.22346	−0.480525
$46$	0	0
$47$	10.8335	1.58023	0.790113	−	0.612961i	$-0.210022\pi$
$47$	10.8335	1.58023	0.790113	+	0.612961i	$0.210022\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	1.33282	0.186632
$52$	0	0
$53$	−5.85643	−0.804443	−0.402222	−	0.915542i	$-0.631762\pi$
$53$	−5.85643	−0.804443	−0.402222	+	0.915542i	$0.631762\pi$
$54$	0	0
$55$	8.95317	1.20725
$56$	0	0
$57$	0.538883	0.0713767
$58$	0	0
$59$	−2.15676	−0.280786	−0.140393	−	0.990096i	$-0.544837\pi$
$59$	−2.15676	−0.280786	−0.140393	+	0.990096i	$0.544837\pi$
$60$	0	0
$61$	1.64909	0.211144	0.105572	−	0.994412i	$-0.466333\pi$
$61$	1.64909	0.211144	0.105572	+	0.994412i	$0.466333\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	7.20825	0.894074
$66$	0	0
$67$	−12.4743	−1.52398	−0.761989	−	0.647590i	$-0.775777\pi$
$67$	−12.4743	−1.52398	−0.761989	+	0.647590i	$0.775777\pi$
$68$	0	0
$69$	−1.95630	−0.235511
$70$	0	0
$71$	−5.10604	−0.605975	−0.302987	−	0.952995i	$-0.597984\pi$
$71$	−5.10604	−0.605975	−0.302987	+	0.952995i	$0.597984\pi$
$72$	0	0
$73$	11.7695	1.37752	0.688761	−	0.724989i	$-0.258155\pi$
$73$	11.7695	1.37752	0.688761	+	0.724989i	$0.258155\pi$
$74$	0	0
$75$	−2.58579	−0.298581
$76$	0	0
$77$	0	0
$78$	0	0
$79$	1.64849	0.185470	0.0927349	−	0.995691i	$-0.470439\pi$
$79$	1.64849	0.185470	0.0927349	+	0.995691i	$0.470439\pi$
$80$	0	0
$81$	−3.83200	−0.425778
$82$	0	0
$83$	11.4159	1.25306	0.626528	−	0.779399i	$-0.284475\pi$
$83$	11.4159	1.25306	0.626528	+	0.779399i	$0.284475\pi$
$84$	0	0
$85$	−2.63441	−0.285741
$86$	0	0
$87$	−6.86029	−0.735500
$88$	0	0
$89$	5.01381	0.531462	0.265731	−	0.964047i	$-0.414387\pi$
$89$	5.01381	0.531462	0.265731	+	0.964047i	$0.414387\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	7.71641	0.800154
$94$	0	0
$95$	−1.06514	−0.109281
$96$	0	0
$97$	−1.08733	−0.110401	−0.0552006	−	0.998475i	$-0.517580\pi$
$97$	−1.08733	−0.110401	−0.0552006	+	0.998475i	$0.517580\pi$
$98$	0	0
$99$	−4.15847	−0.417942
$100$	0	0
$101$	−3.77415	−0.375542	−0.187771	−	0.982213i	$-0.560126\pi$
$101$	−3.77415	−0.375542	−0.187771	+	0.982213i	$0.560126\pi$
$102$	0	0
$103$	6.99756	0.689490	0.344745	−	0.938696i	$-0.387965\pi$
$103$	6.99756	0.689490	0.344745	+	0.938696i	$0.387965\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−9.76321	−0.943846	−0.471923	−	0.881640i	$-0.656440\pi$
$107$	−9.76321	−0.943846	−0.471923	+	0.881640i	$0.656440\pi$
$108$	0	0
$109$	10.0135	0.959123	0.479562	−	0.877508i	$-0.340796\pi$
$109$	10.0135	0.959123	0.479562	+	0.877508i	$0.340796\pi$
$110$	0	0
$111$	−4.46851	−0.424132
$112$	0	0
$113$	13.8115	1.29927	0.649637	−	0.760245i	$-0.274921\pi$
$113$	13.8115	1.29927	0.649637	+	0.760245i	$0.274921\pi$
$114$	0	0
$115$	3.86676	0.360577
$116$	0	0
$117$	−3.34801	−0.309524
$118$	0	0
$119$	0	0
$120$	0	0
$121$	0.550175	0.0500159
$122$	0	0
$123$	−11.8423	−1.06778
$124$	0	0
$125$	−8.06103	−0.721001
$126$	0	0
$127$	−9.12622	−0.809821	−0.404911	−	0.914356i	$-0.632697\pi$
$127$	−9.12622	−0.809821	−0.404911	+	0.914356i	$0.632697\pi$
$128$	0	0
$129$	9.85353	0.867555
$130$	0	0
$131$	5.61422	0.490517	0.245258	−	0.969458i	$-0.421127\pi$
$131$	5.61422	0.490517	0.245258	+	0.969458i	$0.421127\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	14.8298	1.27635
$136$	0	0
$137$	−12.8360	−1.09666	−0.548329	−	0.836263i	$-0.684736\pi$
$137$	−12.8360	−1.09666	−0.548329	+	0.836263i	$0.684736\pi$
$138$	0	0
$139$	−4.04881	−0.343415	−0.171708	−	0.985148i	$-0.554928\pi$
$139$	−4.04881	−0.343415	−0.171708	+	0.985148i	$0.554928\pi$
$140$	0	0
$141$	−14.4390	−1.21599
$142$	0	0
$143$	9.29912	0.777631
$144$	0	0
$145$	13.5598	1.12608
$146$	0	0
$147$	0	0
$148$	0	0
$149$	8.50041	0.696380	0.348190	−	0.937424i	$-0.386796\pi$
$149$	8.50041	0.696380	0.348190	+	0.937424i	$0.386796\pi$
$150$	0	0
$151$	7.83954	0.637973	0.318987	−	0.947759i	$-0.396658\pi$
$151$	7.83954	0.637973	0.318987	+	0.947759i	$0.396658\pi$
$152$	0	0
$153$	1.22360	0.0989223
$154$	0	0
$155$	−15.2520	−1.22507
$156$	0	0
$157$	5.03137	0.401547	0.200774	−	0.979638i	$-0.435654\pi$
$157$	5.03137	0.401547	0.200774	+	0.979638i	$0.435654\pi$
$158$	0	0
$159$	7.80555	0.619020
$160$	0	0
$161$	0	0
$162$	0	0
$163$	3.29713	0.258251	0.129126	−	0.991628i	$-0.458783\pi$
$163$	3.29713	0.258251	0.129126	+	0.991628i	$0.458783\pi$
$164$	0	0
$165$	−11.9329	−0.928978
$166$	0	0
$167$	2.38360	0.184448	0.0922242	−	0.995738i	$-0.470602\pi$
$167$	2.38360	0.184448	0.0922242	+	0.995738i	$0.470602\pi$
$168$	0	0
$169$	−5.51323	−0.424094
$170$	0	0
$171$	0.494725	0.0378326
$172$	0	0
$173$	17.6282	1.34025	0.670124	−	0.742249i	$-0.266241\pi$
$173$	17.6282	1.34025	0.670124	+	0.742249i	$0.266241\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	2.87456	0.216065
$178$	0	0
$179$	−3.92787	−0.293583	−0.146792	−	0.989167i	$-0.546895\pi$
$179$	−3.92787	−0.293583	−0.146792	+	0.989167i	$0.546895\pi$
$180$	0	0
$181$	23.1186	1.71839	0.859197	−	0.511645i	$-0.170964\pi$
$181$	23.1186	1.71839	0.859197	+	0.511645i	$0.170964\pi$
$182$	0	0
$183$	−2.19793	−0.162476
$184$	0	0
$185$	8.83232	0.649365
$186$	0	0
$187$	−3.39855	−0.248527
$188$	0	0
$189$	0	0
$190$	0	0
$191$	22.5203	1.62951	0.814755	−	0.579805i	$-0.196871\pi$
$191$	22.5203	1.62951	0.814755	+	0.579805i	$0.196871\pi$
$192$	0	0
$193$	17.4315	1.25475	0.627373	−	0.778719i	$-0.284130\pi$
$193$	17.4315	1.25475	0.627373	+	0.778719i	$0.284130\pi$
$194$	0	0
$195$	−9.60728	−0.687991
$196$	0	0
$197$	9.55478	0.680750	0.340375	−	0.940290i	$-0.389446\pi$
$197$	9.55478	0.680750	0.340375	+	0.940290i	$0.389446\pi$
$198$	0	0
$199$	−3.19341	−0.226375	−0.113188	−	0.993574i	$-0.536106\pi$
$199$	−3.19341	−0.226375	−0.113188	+	0.993574i	$0.536106\pi$
$200$	0	0
$201$	16.6259	1.17270
$202$	0	0
$203$	0	0
$204$	0	0
$205$	23.4071	1.63482
$206$	0	0
$207$	−1.79599	−0.124830
$208$	0	0
$209$	−1.37410	−0.0950484
$210$	0	0
$211$	−23.5517	−1.62137	−0.810683	−	0.585485i	$-0.800904\pi$
$211$	−23.5517	−1.62137	−0.810683	+	0.585485i	$0.800904\pi$
$212$	0	0
$213$	6.80541	0.466299
$214$	0	0
$215$	−19.4762	−1.32827
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−15.6866	−1.06000
$220$	0	0
$221$	−2.73620	−0.184057
$222$	0	0
$223$	10.2413	0.685809	0.342904	−	0.939370i	$-0.388589\pi$
$223$	10.2413	0.685809	0.342904	+	0.939370i	$0.388589\pi$
$224$	0	0
$225$	−2.37390	−0.158260
$226$	0	0
$227$	−0.894724	−0.0593849	−0.0296925	−	0.999559i	$-0.509453\pi$
$227$	−0.894724	−0.0593849	−0.0296925	+	0.999559i	$0.509453\pi$
$228$	0	0
$229$	−15.6823	−1.03632	−0.518159	−	0.855284i	$-0.673383\pi$
$229$	−15.6823	−1.03632	−0.518159	+	0.855284i	$0.673383\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	19.6653	1.28832	0.644160	−	0.764891i	$-0.277207\pi$
$233$	19.6653	1.28832	0.644160	+	0.764891i	$0.277207\pi$
$234$	0	0
$235$	28.5398	1.86173
$236$	0	0
$237$	−2.19714	−0.142719
$238$	0	0
$239$	16.9468	1.09620	0.548100	−	0.836413i	$-0.315351\pi$
$239$	16.9468	1.09620	0.548100	+	0.836413i	$0.315351\pi$
$240$	0	0
$241$	23.8330	1.53522	0.767608	−	0.640920i	$-0.221447\pi$
$241$	23.8330	1.53522	0.767608	+	0.640920i	$0.221447\pi$
$242$	0	0
$243$	−11.7805	−0.755720
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1.10630	−0.0703919
$248$	0	0
$249$	−15.2153	−0.964230
$250$	0	0
$251$	15.3635	0.969737	0.484869	−	0.874587i	$-0.338867\pi$
$251$	15.3635	0.969737	0.484869	+	0.874587i	$0.338867\pi$
$252$	0	0
$253$	4.98838	0.313616
$254$	0	0
$255$	3.51118	0.219879
$256$	0	0
$257$	−0.186838	−0.0116546	−0.00582731	−	0.999983i	$-0.501855\pi$
$257$	−0.186838	−0.0116546	−0.00582731	+	0.999983i	$0.501855\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−6.29813	−0.389845
$262$	0	0
$263$	−21.3661	−1.31749	−0.658746	−	0.752365i	$-0.728913\pi$
$263$	−21.3661	−1.31749	−0.658746	+	0.752365i	$0.728913\pi$
$264$	0	0
$265$	−15.4282	−0.947748
$266$	0	0
$267$	−6.68248	−0.408961
$268$	0	0
$269$	−17.0143	−1.03738	−0.518689	−	0.854963i	$-0.673580\pi$
$269$	−17.0143	−1.03738	−0.518689	+	0.854963i	$0.673580\pi$
$270$	0	0
$271$	−23.7048	−1.43997	−0.719983	−	0.693992i	$-0.755850\pi$
$271$	−23.7048	−1.43997	−0.719983	+	0.693992i	$0.755850\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	6.59352	0.397604
$276$	0	0
$277$	−10.4722	−0.629211	−0.314606	−	0.949222i	$-0.601872\pi$
$277$	−10.4722	−0.629211	−0.314606	+	0.949222i	$0.601872\pi$
$278$	0	0
$279$	7.08410	0.424114
$280$	0	0
$281$	13.7779	0.821919	0.410960	−	0.911654i	$-0.365194\pi$
$281$	13.7779	0.821919	0.410960	+	0.911654i	$0.365194\pi$
$282$	0	0
$283$	−12.6862	−0.754118	−0.377059	−	0.926189i	$-0.623065\pi$
$283$	−12.6862	−0.754118	−0.377059	+	0.926189i	$0.623065\pi$
$284$	0	0
$285$	1.41964	0.0840919
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	1.44921	0.0849539
$292$	0	0
$293$	2.70276	0.157897	0.0789484	−	0.996879i	$-0.474844\pi$
$293$	2.70276	0.157897	0.0789484	+	0.996879i	$0.474844\pi$
$294$	0	0
$295$	−5.68177	−0.330805
$296$	0	0
$297$	19.1314	1.11012
$298$	0	0
$299$	4.01617	0.232261
$300$	0	0
$301$	0	0
$302$	0	0
$303$	5.03024	0.288980
$304$	0	0
$305$	4.34437	0.248758
$306$	0	0
$307$	14.6666	0.837065	0.418532	−	0.908202i	$-0.362545\pi$
$307$	14.6666	0.837065	0.418532	+	0.908202i	$0.362545\pi$
$308$	0	0
$309$	−9.32646	−0.530564
$310$	0	0
$311$	−12.1777	−0.690535	−0.345268	−	0.938504i	$-0.612212\pi$
$311$	−12.1777	−0.690535	−0.345268	+	0.938504i	$0.612212\pi$
$312$	0	0
$313$	25.0483	1.41582	0.707908	−	0.706305i	$-0.249639\pi$
$313$	25.0483	1.41582	0.707908	+	0.706305i	$0.249639\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	19.6043	1.10108	0.550542	−	0.834807i	$-0.314421\pi$
$317$	19.6043	1.10108	0.550542	+	0.834807i	$0.314421\pi$
$318$	0	0
$319$	17.4931	0.979425
$320$	0	0
$321$	13.0126	0.726291
$322$	0	0
$323$	0.404319	0.0224969
$324$	0	0
$325$	5.30848	0.294461
$326$	0	0
$327$	−13.3462	−0.738047
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−14.2160	−0.781384	−0.390692	−	0.920521i	$-0.627764\pi$
$331$	−14.2160	−0.781384	−0.390692	+	0.920521i	$0.627764\pi$
$332$	0	0
$333$	−4.10234	−0.224807
$334$	0	0
$335$	−32.8623	−1.79546
$336$	0	0
$337$	13.3893	0.729361	0.364680	−	0.931133i	$-0.381178\pi$
$337$	13.3893	0.729361	0.364680	+	0.931133i	$0.381178\pi$
$338$	0	0
$339$	−18.4081	−0.999793
$340$	0	0
$341$	−19.6761	−1.06552
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−5.15368	−0.277465
$346$	0	0
$347$	33.8546	1.81741	0.908706	−	0.417437i	$-0.137072\pi$
$347$	33.8546	1.81741	0.908706	+	0.417437i	$0.137072\pi$
$348$	0	0
$349$	27.2035	1.45617	0.728085	−	0.685487i	$-0.240411\pi$
$349$	27.2035	1.45617	0.728085	+	0.685487i	$0.240411\pi$
$350$	0	0
$351$	15.4028	0.822142
$352$	0	0
$353$	−14.6928	−0.782019	−0.391009	−	0.920387i	$-0.627874\pi$
$353$	−14.6928	−0.782019	−0.391009	+	0.920387i	$0.627874\pi$
$354$	0	0
$355$	−13.4514	−0.713925
$356$	0	0
$357$	0	0
$358$	0	0
$359$	9.33884	0.492885	0.246442	−	0.969157i	$-0.420738\pi$
$359$	9.33884	0.492885	0.246442	+	0.969157i	$0.420738\pi$
$360$	0	0
$361$	−18.8365	−0.991396
$362$	0	0
$363$	−0.733282	−0.0384873
$364$	0	0
$365$	31.0058	1.62292
$366$	0	0
$367$	0.578678	0.0302068	0.0151034	−	0.999886i	$-0.495192\pi$
$367$	0.578678	0.0302068	0.0151034	+	0.999886i	$0.495192\pi$
$368$	0	0
$369$	−10.8719	−0.565966
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−21.8602	−1.13188	−0.565940	−	0.824447i	$-0.691486\pi$
$373$	−21.8602	−1.13188	−0.565940	+	0.824447i	$0.691486\pi$
$374$	0	0
$375$	10.7439	0.554811
$376$	0	0
$377$	14.0838	0.725352
$378$	0	0
$379$	4.26768	0.219216	0.109608	−	0.993975i	$-0.465040\pi$
$379$	4.26768	0.219216	0.109608	+	0.993975i	$0.465040\pi$
$380$	0	0
$381$	12.1636	0.623159
$382$	0	0
$383$	28.5396	1.45830	0.729152	−	0.684351i	$-0.239915\pi$
$383$	28.5396	1.45830	0.729152	+	0.684351i	$0.239915\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	9.04610	0.459839
$388$	0	0
$389$	37.8618	1.91967	0.959836	−	0.280563i	$-0.0905211\pi$
$389$	37.8618	1.91967	0.959836	+	0.280563i	$0.0905211\pi$
$390$	0	0
$391$	−1.46779	−0.0742295
$392$	0	0
$393$	−7.48272	−0.377453
$394$	0	0
$395$	4.34280	0.218510
$396$	0	0
$397$	18.3336	0.920138	0.460069	−	0.887883i	$-0.347825\pi$
$397$	18.3336	0.920138	0.460069	+	0.887883i	$0.347825\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−30.2532	−1.51077	−0.755387	−	0.655279i	$-0.772551\pi$
$401$	−30.2532	−1.51077	−0.755387	+	0.655279i	$0.772551\pi$
$402$	0	0
$403$	−15.8414	−0.789114
$404$	0	0
$405$	−10.0950	−0.501626
$406$	0	0
$407$	11.3943	0.564793
$408$	0	0
$409$	−17.1292	−0.846986	−0.423493	−	0.905899i	$-0.639196\pi$
$409$	−17.1292	−0.846986	−0.423493	+	0.905899i	$0.639196\pi$
$410$	0	0
$411$	17.1081	0.843880
$412$	0	0
$413$	0	0
$414$	0	0
$415$	30.0741	1.47628
$416$	0	0
$417$	5.39632	0.264259
$418$	0	0
$419$	−12.7339	−0.622092	−0.311046	−	0.950395i	$-0.600679\pi$
$419$	−12.7339	−0.622092	−0.311046	+	0.950395i	$0.600679\pi$
$420$	0	0
$421$	27.3824	1.33454	0.667269	−	0.744817i	$-0.267463\pi$
$421$	27.3824	1.33454	0.667269	+	0.744817i	$0.267463\pi$
$422$	0	0
$423$	−13.2559	−0.644522
$424$	0	0
$425$	−1.94009	−0.0941084
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−12.3940	−0.598389
$430$	0	0
$431$	12.5372	0.603895	0.301947	−	0.953325i	$-0.402363\pi$
$431$	12.5372	0.603895	0.301947	+	0.953325i	$0.402363\pi$
$432$	0	0
$433$	−6.93267	−0.333163	−0.166581	−	0.986028i	$-0.553273\pi$
$433$	−6.93267	−0.333163	−0.166581	+	0.986028i	$0.553273\pi$
$434$	0	0
$435$	−18.0728	−0.866524
$436$	0	0
$437$	−0.593456	−0.0283889
$438$	0	0
$439$	1.59334	0.0760462	0.0380231	−	0.999277i	$-0.487894\pi$
$439$	1.59334	0.0760462	0.0380231	+	0.999277i	$0.487894\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	8.78641	0.417455	0.208728	−	0.977974i	$-0.433068\pi$
$443$	8.78641	0.417455	0.208728	+	0.977974i	$0.433068\pi$
$444$	0	0
$445$	13.2084	0.626138
$446$	0	0
$447$	−11.3295	−0.535866
$448$	0	0
$449$	−5.10893	−0.241105	−0.120553	−	0.992707i	$-0.538467\pi$
$449$	−5.10893	−0.241105	−0.120553	+	0.992707i	$0.538467\pi$
$450$	0	0
$451$	30.1966	1.42190
$452$	0	0
$453$	−10.4487	−0.490921
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−31.4086	−1.46923	−0.734616	−	0.678483i	$-0.762638\pi$
$457$	−31.4086	−1.46923	−0.734616	+	0.678483i	$0.762638\pi$
$458$	0	0
$459$	−5.62928	−0.262752
$460$	0	0
$461$	17.3719	0.809090	0.404545	−	0.914518i	$-0.367430\pi$
$461$	17.3719	0.809090	0.404545	+	0.914518i	$0.367430\pi$
$462$	0	0
$463$	−19.5143	−0.906905	−0.453453	−	0.891280i	$-0.649808\pi$
$463$	−19.5143	−0.906905	−0.453453	+	0.891280i	$0.649808\pi$
$464$	0	0
$465$	20.3282	0.942695
$466$	0	0
$467$	−11.5364	−0.533839	−0.266920	−	0.963719i	$-0.586006\pi$
$467$	−11.5364	−0.533839	−0.266920	+	0.963719i	$0.586006\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−6.70590	−0.308991
$472$	0	0
$473$	−25.1256	−1.15527
$474$	0	0
$475$	−0.784416	−0.0359915
$476$	0	0
$477$	7.16594	0.328106
$478$	0	0
$479$	−11.4533	−0.523315	−0.261657	−	0.965161i	$-0.584269\pi$
$479$	−11.4533	−0.523315	−0.261657	+	0.965161i	$0.584269\pi$
$480$	0	0
$481$	9.17359	0.418280
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−2.86446	−0.130068
$486$	0	0
$487$	−7.98161	−0.361681	−0.180841	−	0.983512i	$-0.557882\pi$
$487$	−7.98161	−0.361681	−0.180841	+	0.983512i	$0.557882\pi$
$488$	0	0
$489$	−4.39447	−0.198725
$490$	0	0
$491$	−21.2292	−0.958060	−0.479030	−	0.877798i	$-0.659012\pi$
$491$	−21.2292	−0.958060	−0.479030	+	0.877798i	$0.659012\pi$
$492$	0	0
$493$	−5.14721	−0.231819
$494$	0	0
$495$	−10.9551	−0.492396
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−36.1880	−1.62000	−0.809999	−	0.586432i	$-0.800532\pi$
$499$	−36.1880	−1.62000	−0.809999	+	0.586432i	$0.800532\pi$
$500$	0	0
$501$	−3.17690	−0.141933
$502$	0	0
$503$	−36.1481	−1.61176	−0.805882	−	0.592076i	$-0.798308\pi$
$503$	−36.1481	−1.61176	−0.805882	+	0.592076i	$0.798308\pi$
$504$	0	0
$505$	−9.94263	−0.442441
$506$	0	0
$507$	7.34812	0.326341
$508$	0	0
$509$	−31.4404	−1.39357	−0.696785	−	0.717280i	$-0.745387\pi$
$509$	−31.4404	−1.39357	−0.696785	+	0.717280i	$0.745387\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−2.27603	−0.100489
$514$	0	0
$515$	18.4344	0.812317
$516$	0	0
$517$	36.8182	1.61926
$518$	0	0
$519$	−23.4952	−1.03132
$520$	0	0
$521$	−22.0154	−0.964513	−0.482257	−	0.876030i	$-0.660183\pi$
$521$	−22.0154	−0.964513	−0.482257	+	0.876030i	$0.660183\pi$
$522$	0	0
$523$	14.6017	0.638487	0.319243	−	0.947673i	$-0.396571\pi$
$523$	14.6017	0.638487	0.319243	+	0.947673i	$0.396571\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5.78955	0.252197
$528$	0	0
$529$	−20.8456	−0.906330
$530$	0	0
$531$	2.63901	0.114523
$532$	0	0
$533$	24.3115	1.05305
$534$	0	0
$535$	−25.7203	−1.11198
$536$	0	0
$537$	5.23513	0.225913
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−38.4069	−1.65124	−0.825621	−	0.564225i	$-0.809175\pi$
$541$	−38.4069	−1.65124	−0.825621	+	0.564225i	$0.809175\pi$
$542$	0	0
$543$	−30.8129	−1.32231
$544$	0	0
$545$	26.3797	1.12998
$546$	0	0
$547$	−31.0157	−1.32614	−0.663069	−	0.748559i	$-0.730746\pi$
$547$	−31.0157	−1.32614	−0.663069	+	0.748559i	$0.730746\pi$
$548$	0	0
$549$	−2.01783	−0.0861188
$550$	0	0
$551$	−2.08111	−0.0886585
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−11.7719	−0.499688
$556$	0	0
$557$	6.94418	0.294235	0.147117	−	0.989119i	$-0.453001\pi$
$557$	6.94418	0.294235	0.147117	+	0.989119i	$0.453001\pi$
$558$	0	0
$559$	−20.2287	−0.855585
$560$	0	0
$561$	4.52965	0.191242
$562$	0	0
$563$	−4.13022	−0.174068	−0.0870340	−	0.996205i	$-0.527739\pi$
$563$	−4.13022	−0.174068	−0.0870340	+	0.996205i	$0.527739\pi$
$564$	0	0
$565$	36.3850	1.53073
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−20.7679	−0.870637	−0.435318	−	0.900277i	$-0.643364\pi$
$569$	−20.7679	−0.870637	−0.435318	+	0.900277i	$0.643364\pi$
$570$	0	0
$571$	34.9944	1.46447	0.732234	−	0.681053i	$-0.238478\pi$
$571$	34.9944	1.46447	0.732234	+	0.681053i	$0.238478\pi$
$572$	0	0
$573$	−30.0154	−1.25391
$574$	0	0
$575$	2.84766	0.118755
$576$	0	0
$577$	−32.3857	−1.34823	−0.674117	−	0.738624i	$-0.735476\pi$
$577$	−32.3857	−1.34823	−0.674117	+	0.738624i	$0.735476\pi$
$578$	0	0
$579$	−23.2330	−0.965529
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−19.9034	−0.824315
$584$	0	0
$585$	−8.82003	−0.364663
$586$	0	0
$587$	−41.0902	−1.69597	−0.847987	−	0.530017i	$-0.822186\pi$
$587$	−41.0902	−1.69597	−0.847987	+	0.530017i	$0.822186\pi$
$588$	0	0
$589$	2.34082	0.0964520
$590$	0	0
$591$	−12.7348	−0.523838
$592$	0	0
$593$	7.60694	0.312380	0.156190	−	0.987727i	$-0.450079\pi$
$593$	7.60694	0.312380	0.156190	+	0.987727i	$0.450079\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	4.25623	0.174196
$598$	0	0
$599$	−35.8096	−1.46314	−0.731570	−	0.681767i	$-0.761212\pi$
$599$	−35.8096	−1.46314	−0.731570	+	0.681767i	$0.761212\pi$
$600$	0	0
$601$	−36.6919	−1.49669	−0.748347	−	0.663307i	$-0.769152\pi$
$601$	−36.6919	−1.49669	−0.748347	+	0.663307i	$0.769152\pi$
$602$	0	0
$603$	15.2636	0.621580
$604$	0	0
$605$	1.44938	0.0589258
$606$	0	0
$607$	−2.83085	−0.114901	−0.0574503	−	0.998348i	$-0.518297\pi$
$607$	−2.83085	−0.114901	−0.0574503	+	0.998348i	$0.518297\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	29.6425	1.19921
$612$	0	0
$613$	26.3540	1.06443	0.532213	−	0.846610i	$-0.321361\pi$
$613$	26.3540	1.06443	0.532213	+	0.846610i	$0.321361\pi$
$614$	0	0
$615$	−31.1973	−1.25800
$616$	0	0
$617$	1.49863	0.0603326	0.0301663	−	0.999545i	$-0.490396\pi$
$617$	1.49863	0.0603326	0.0301663	+	0.999545i	$0.490396\pi$
$618$	0	0
$619$	4.29710	0.172715	0.0863576	−	0.996264i	$-0.472477\pi$
$619$	4.29710	0.172715	0.0863576	+	0.996264i	$0.472477\pi$
$620$	0	0
$621$	8.26262	0.331568
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−30.9365	−1.23746
$626$	0	0
$627$	1.83142	0.0731400
$628$	0	0
$629$	−3.35268	−0.133680
$630$	0	0
$631$	33.3092	1.32602	0.663010	−	0.748611i	$-0.269279\pi$
$631$	33.3092	1.32602	0.663010	+	0.748611i	$0.269279\pi$
$632$	0	0
$633$	31.3901	1.24764
$634$	0	0
$635$	−24.0422	−0.954084
$636$	0	0
$637$	0	0
$638$	0	0
$639$	6.24775	0.247157
$640$	0	0
$641$	−46.6140	−1.84114	−0.920572	−	0.390573i	$-0.872277\pi$
$641$	−46.6140	−1.84114	−0.920572	+	0.390573i	$0.872277\pi$
$642$	0	0
$643$	−1.83554	−0.0723867	−0.0361934	−	0.999345i	$-0.511523\pi$
$643$	−1.83554	−0.0723867	−0.0361934	+	0.999345i	$0.511523\pi$
$644$	0	0
$645$	25.9582	1.02210
$646$	0	0
$647$	−29.4260	−1.15686	−0.578428	−	0.815734i	$-0.696333\pi$
$647$	−29.4260	−1.15686	−0.578428	+	0.815734i	$0.696333\pi$
$648$	0	0
$649$	−7.32985	−0.287722
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−36.4919	−1.42804	−0.714018	−	0.700127i	$-0.753127\pi$
$653$	−36.4919	−1.42804	−0.714018	+	0.700127i	$0.753127\pi$
$654$	0	0
$655$	14.7901	0.577898
$656$	0	0
$657$	−14.4012	−0.561845
$658$	0	0
$659$	−39.2123	−1.52750	−0.763748	−	0.645515i	$-0.776643\pi$
$659$	−39.2123	−1.52750	−0.763748	+	0.645515i	$0.776643\pi$
$660$	0	0
$661$	23.6477	0.919790	0.459895	−	0.887973i	$-0.347887\pi$
$661$	23.6477	0.919790	0.459895	+	0.887973i	$0.347887\pi$
$662$	0	0
$663$	3.64685	0.141632
$664$	0	0
$665$	0	0
$666$	0	0
$667$	7.55504	0.292532
$668$	0	0
$669$	−13.6498	−0.527731
$670$	0	0
$671$	5.60452	0.216360
$672$	0	0
$673$	20.3812	0.785638	0.392819	−	0.919616i	$-0.371500\pi$
$673$	20.3812	0.785638	0.392819	+	0.919616i	$0.371500\pi$
$674$	0	0
$675$	10.9213	0.420363
$676$	0	0
$677$	−27.2439	−1.04707	−0.523533	−	0.852005i	$-0.675386\pi$
$677$	−27.2439	−1.04707	−0.523533	+	0.852005i	$0.675386\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	1.19250	0.0456968
$682$	0	0
$683$	−0.0918747	−0.00351549	−0.00175774	−	0.999998i	$-0.500560\pi$
$683$	−0.0918747	−0.00351549	−0.00175774	+	0.999998i	$0.500560\pi$
$684$	0	0
$685$	−33.8153	−1.29202
$686$	0	0
$687$	20.9017	0.797449
$688$	0	0
$689$	−16.0244	−0.610479
$690$	0	0
$691$	−2.31807	−0.0881834	−0.0440917	−	0.999027i	$-0.514039\pi$
$691$	−2.31807	−0.0881834	−0.0440917	+	0.999027i	$0.514039\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−10.6662	−0.404592
$696$	0	0
$697$	−8.88513	−0.336548
$698$	0	0
$699$	−26.2103	−0.991364
$700$	0	0
$701$	−16.9943	−0.641865	−0.320932	−	0.947102i	$-0.603996\pi$
$701$	−16.9943	−0.641865	−0.320932	+	0.947102i	$0.603996\pi$
$702$	0	0
$703$	−1.35555	−0.0511256
$704$	0	0
$705$	−38.0383	−1.43261
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−16.7430	−0.628798	−0.314399	−	0.949291i	$-0.601803\pi$
$709$	−16.7430	−0.628798	−0.314399	+	0.949291i	$0.601803\pi$
$710$	0	0
$711$	−2.01710	−0.0756470
$712$	0	0
$713$	−8.49786	−0.318247
$714$	0	0
$715$	24.4976	0.916160
$716$	0	0
$717$	−22.5870	−0.843528
$718$	0	0
$719$	17.4685	0.651465	0.325732	−	0.945462i	$-0.394389\pi$
$719$	17.4685	0.651465	0.325732	+	0.945462i	$0.394389\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−31.7649	−1.18135
$724$	0	0
$725$	9.98608	0.370874
$726$	0	0
$727$	29.0537	1.07754	0.538770	−	0.842453i	$-0.318889\pi$
$727$	29.0537	1.07754	0.538770	+	0.842453i	$0.318889\pi$
$728$	0	0
$729$	27.1972	1.00731
$730$	0	0
$731$	7.39301	0.273440
$732$	0	0
$733$	17.4825	0.645731	0.322866	−	0.946445i	$-0.395354\pi$
$733$	17.4825	0.645731	0.322866	+	0.946445i	$0.395354\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−42.3946	−1.56162
$738$	0	0
$739$	44.4652	1.63568	0.817839	−	0.575448i	$-0.195172\pi$
$739$	44.4652	1.63568	0.817839	+	0.575448i	$0.195172\pi$
$740$	0	0
$741$	1.47449	0.0541667
$742$	0	0
$743$	1.93478	0.0709802	0.0354901	−	0.999370i	$-0.488701\pi$
$743$	1.93478	0.0709802	0.0354901	+	0.999370i	$0.488701\pi$
$744$	0	0
$745$	22.3935	0.820435
$746$	0	0
$747$	−13.9685	−0.511081
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−10.3909	−0.379169	−0.189585	−	0.981864i	$-0.560714\pi$
$751$	−10.3909	−0.379169	−0.189585	+	0.981864i	$0.560714\pi$
$752$	0	0
$753$	−20.4768	−0.746214
$754$	0	0
$755$	20.6525	0.751623
$756$	0	0
$757$	−40.0024	−1.45391	−0.726956	−	0.686684i	$-0.759065\pi$
$757$	−40.0024	−1.45391	−0.726956	+	0.686684i	$0.759065\pi$
$758$	0	0
$759$	−6.64859	−0.241328
$760$	0	0
$761$	−35.3291	−1.28068	−0.640339	−	0.768092i	$-0.721206\pi$
$761$	−35.3291	−1.28068	−0.640339	+	0.768092i	$0.721206\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	3.22346	0.116544
$766$	0	0
$767$	−5.90131	−0.213084
$768$	0	0
$769$	−43.8275	−1.58046	−0.790230	−	0.612810i	$-0.790039\pi$
$769$	−43.8275	−1.58046	−0.790230	+	0.612810i	$0.790039\pi$
$770$	0	0
$771$	0.249020	0.00896825
$772$	0	0
$773$	−22.9608	−0.825844	−0.412922	−	0.910766i	$-0.635492\pi$
$773$	−22.9608	−0.825844	−0.412922	+	0.910766i	$0.635492\pi$
$774$	0	0
$775$	−11.2323	−0.403475
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−3.59243	−0.128712
$780$	0	0
$781$	−17.3531	−0.620944
$782$	0	0
$783$	28.9751	1.03549
$784$	0	0
$785$	13.2547	0.473080
$786$	0	0
$787$	27.7458	0.989032	0.494516	−	0.869168i	$-0.335345\pi$
$787$	27.7458	0.989032	0.494516	+	0.869168i	$0.335345\pi$
$788$	0	0
$789$	28.4771	1.01381
$790$	0	0
$791$	0	0
$792$	0	0
$793$	4.51223	0.160234
$794$	0	0
$795$	20.5630	0.729294
$796$	0	0
$797$	−22.4056	−0.793649	−0.396824	−	0.917895i	$-0.629888\pi$
$797$	−22.4056	−0.793649	−0.396824	+	0.917895i	$0.629888\pi$
$798$	0	0
$799$	−10.8335	−0.383261
$800$	0	0
$801$	−6.13490	−0.216766
$802$	0	0
$803$	39.9994	1.41155
$804$	0	0
$805$	0	0
$806$	0	0
$807$	22.6769	0.798265
$808$	0	0
$809$	9.66658	0.339859	0.169929	−	0.985456i	$-0.445646\pi$
$809$	9.66658	0.339859	0.169929	+	0.985456i	$0.445646\pi$
$810$	0	0
$811$	−15.8495	−0.556551	−0.278275	−	0.960501i	$-0.589763\pi$
$811$	−15.8495	−0.556551	−0.278275	+	0.960501i	$0.589763\pi$
$812$	0	0
$813$	31.5942	1.10806
$814$	0	0
$815$	8.68599	0.304257
$816$	0	0
$817$	2.98913	0.104577
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−32.7857	−1.14423	−0.572115	−	0.820173i	$-0.693877\pi$
$821$	−32.7857	−1.14423	−0.572115	+	0.820173i	$0.693877\pi$
$822$	0	0
$823$	51.2019	1.78479	0.892394	−	0.451257i	$-0.149024\pi$
$823$	51.2019	1.78479	0.892394	+	0.451257i	$0.149024\pi$
$824$	0	0
$825$	−8.78794	−0.305957
$826$	0	0
$827$	−2.48109	−0.0862758	−0.0431379	−	0.999069i	$-0.513735\pi$
$827$	−2.48109	−0.0862758	−0.0431379	+	0.999069i	$0.513735\pi$
$828$	0	0
$829$	25.4701	0.884612	0.442306	−	0.896864i	$-0.354161\pi$
$829$	25.4701	0.884612	0.442306	+	0.896864i	$0.354161\pi$
$830$	0	0
$831$	13.9575	0.484179
$832$	0	0
$833$	0	0
$834$	0	0
$835$	6.27936	0.217306
$836$	0	0
$837$	−32.5910	−1.12651
$838$	0	0
$839$	−47.3802	−1.63575	−0.817874	−	0.575398i	$-0.804847\pi$
$839$	−47.3802	−1.63575	−0.817874	+	0.575398i	$0.804847\pi$
$840$	0	0
$841$	−2.50621	−0.0864209
$842$	0	0
$843$	−18.3634	−0.632468
$844$	0	0
$845$	−14.5241	−0.499643
$846$	0	0
$847$	0	0
$848$	0	0
$849$	16.9084	0.580295
$850$	0	0
$851$	4.92104	0.168691
$852$	0	0
$853$	50.7362	1.73718	0.868588	−	0.495535i	$-0.165028\pi$
$853$	50.7362	1.73718	0.868588	+	0.495535i	$0.165028\pi$
$854$	0	0
$855$	1.30331	0.0445721
$856$	0	0
$857$	−6.95326	−0.237519	−0.118759	−	0.992923i	$-0.537892\pi$
$857$	−6.95326	−0.237519	−0.118759	+	0.992923i	$0.537892\pi$
$858$	0	0
$859$	−13.7889	−0.470471	−0.235236	−	0.971938i	$-0.575586\pi$
$859$	−13.7889	−0.470471	−0.235236	+	0.971938i	$0.575586\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	48.1645	1.63954	0.819770	−	0.572693i	$-0.194101\pi$
$863$	48.1645	1.63954	0.819770	+	0.572693i	$0.194101\pi$
$864$	0	0
$865$	46.4399	1.57900
$866$	0	0
$867$	−1.33282	−0.0452648
$868$	0	0
$869$	5.60249	0.190052
$870$	0	0
$871$	−34.1321	−1.15652
$872$	0	0
$873$	1.33045	0.0450290
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−32.1289	−1.08492	−0.542458	−	0.840083i	$-0.682506\pi$
$877$	−32.1289	−1.08492	−0.542458	+	0.840083i	$0.682506\pi$
$878$	0	0
$879$	−3.60228	−0.121502
$880$	0	0
$881$	23.8417	0.803246	0.401623	−	0.915805i	$-0.368446\pi$
$881$	23.8417	0.803246	0.401623	+	0.915805i	$0.368446\pi$
$882$	0	0
$883$	56.5522	1.90313	0.951566	−	0.307444i	$-0.0994736\pi$
$883$	56.5522	1.90313	0.951566	+	0.307444i	$0.0994736\pi$
$884$	0	0
$885$	7.57275	0.254555
$886$	0	0
$887$	3.79132	0.127300	0.0636500	−	0.997972i	$-0.479726\pi$
$887$	3.79132	0.127300	0.0636500	+	0.997972i	$0.479726\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−13.0233	−0.436295
$892$	0	0
$893$	−4.38018	−0.146577
$894$	0	0
$895$	−10.3476	−0.345883
$896$	0	0
$897$	−5.35282	−0.178725
$898$	0	0
$899$	−29.8001	−0.993887
$900$	0	0
$901$	5.85643	0.195106
$902$	0	0
$903$	0	0
$904$	0	0
$905$	60.9038	2.02451
$906$	0	0
$907$	17.0338	0.565598	0.282799	−	0.959179i	$-0.408737\pi$
$907$	17.0338	0.565598	0.282799	+	0.959179i	$0.408737\pi$
$908$	0	0
$909$	4.61805	0.153171
$910$	0	0
$911$	11.6353	0.385493	0.192747	−	0.981249i	$-0.438260\pi$
$911$	11.6353	0.385493	0.192747	+	0.981249i	$0.438260\pi$
$912$	0	0
$913$	38.7975	1.28401
$914$	0	0
$915$	−5.79025	−0.191420
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−4.79523	−0.158180	−0.0790899	−	0.996867i	$-0.525201\pi$
$919$	−4.79523	−0.158180	−0.0790899	+	0.996867i	$0.525201\pi$
$920$	0	0
$921$	−19.5478	−0.644123
$922$	0	0
$923$	−13.9711	−0.459865
$924$	0	0
$925$	6.50451	0.213867
$926$	0	0
$927$	−8.56222	−0.281220
$928$	0	0
$929$	−44.3224	−1.45417	−0.727086	−	0.686547i	$-0.759126\pi$
$929$	−44.3224	−1.45417	−0.727086	+	0.686547i	$0.759126\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	16.2307	0.531368
$934$	0	0
$935$	−8.95317	−0.292800
$936$	0	0
$937$	43.1735	1.41042	0.705208	−	0.709000i	$-0.250853\pi$
$937$	43.1735	1.41042	0.705208	+	0.709000i	$0.250853\pi$
$938$	0	0
$939$	−33.3848	−1.08947
$940$	0	0
$941$	28.1904	0.918982	0.459491	−	0.888183i	$-0.348032\pi$
$941$	28.1904	0.918982	0.459491	+	0.888183i	$0.348032\pi$
$942$	0	0
$943$	13.0415	0.424691
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−37.3180	−1.21267	−0.606337	−	0.795208i	$-0.707362\pi$
$947$	−37.3180	−1.21267	−0.606337	+	0.795208i	$0.707362\pi$
$948$	0	0
$949$	32.2038	1.04538
$950$	0	0
$951$	−26.1289	−0.847287
$952$	0	0
$953$	6.79062	0.219970	0.109985	−	0.993933i	$-0.464920\pi$
$953$	6.79062	0.219970	0.109985	+	0.993933i	$0.464920\pi$
$954$	0	0
$955$	59.3276	1.91980
$956$	0	0
$957$	−23.3151	−0.753669
$958$	0	0
$959$	0	0
$960$	0	0
$961$	2.51891	0.0812551
$962$	0	0
$963$	11.9463	0.384963
$964$	0	0
$965$	45.9216	1.47827
$966$	0	0
$967$	27.7347	0.891888	0.445944	−	0.895061i	$-0.352868\pi$
$967$	27.7347	0.891888	0.445944	+	0.895061i	$0.352868\pi$
$968$	0	0
$969$	−0.538883	−0.0173114
$970$	0	0
$971$	−32.6912	−1.04911	−0.524556	−	0.851376i	$-0.675769\pi$
$971$	−32.6912	−1.04911	−0.524556	+	0.851376i	$0.675769\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−7.07523	−0.226589
$976$	0	0
$977$	49.2549	1.57581	0.787903	−	0.615800i	$-0.211167\pi$
$977$	49.2549	1.57581	0.787903	+	0.615800i	$0.211167\pi$
$978$	0	0
$979$	17.0397	0.544591
$980$	0	0
$981$	−12.2526	−0.391195
$982$	0	0
$983$	26.0666	0.831396	0.415698	−	0.909503i	$-0.363537\pi$
$983$	26.0666	0.831396	0.415698	+	0.909503i	$0.363537\pi$
$984$	0	0
$985$	25.1712	0.802020
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−10.8514	−0.345055
$990$	0	0
$991$	42.6057	1.35341	0.676707	−	0.736252i	$-0.263406\pi$
$991$	42.6057	1.35341	0.676707	+	0.736252i	$0.263406\pi$
$992$	0	0
$993$	18.9474	0.601277
$994$	0	0
$995$	−8.41275	−0.266702
$996$	0	0
$997$	−22.0611	−0.698682	−0.349341	−	0.936996i	$-0.613594\pi$
$997$	−22.0611	−0.698682	−0.349341	+	0.936996i	$0.613594\pi$
$998$	0	0
$999$	18.8732	0.597121

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3332.2.a.u.1.2	yes	8
7.6	odd	2		3332.2.a.t.1.7	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3332.2.a.t.1.7	✓	8	7.6	odd	2
3332.2.a.u.1.2	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-1.33282 q^{3} +2.63441 q^{5} -1.22360 q^{9} +3.39855 q^{11} +2.73620 q^{13} -3.51118 q^{15} -1.00000 q^{17} -0.404319 q^{19} +1.46779 q^{23} +1.94009 q^{25} +5.62928 q^{27} +5.14721 q^{29} -5.78955 q^{31} -4.52965 q^{33} +3.35268 q^{37} -3.64685 q^{39} +8.88513 q^{41} -7.39301 q^{43} -3.22346 q^{45} +10.8335 q^{47} +1.33282 q^{51} -5.85643 q^{53} +8.95317 q^{55} +0.538883 q^{57} -2.15676 q^{59} +1.64909 q^{61} +7.20825 q^{65} -12.4743 q^{67} -1.95630 q^{69} -5.10604 q^{71} +11.7695 q^{73} -2.58579 q^{75} +1.64849 q^{79} -3.83200 q^{81} +11.4159 q^{83} -2.63441 q^{85} -6.86029 q^{87} +5.01381 q^{89} +7.71641 q^{93} -1.06514 q^{95} -1.08733 q^{97} -4.15847 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{3} + 4 q^{5} + 8 q^{9} - 4 q^{11} + 20 q^{13} + 12 q^{15} - 8 q^{17} + 8 q^{19} + 4 q^{23} + 8 q^{25} + 28 q^{27} - 16 q^{29} - 8 q^{31} + 16 q^{33} + 8 q^{37} + 20 q^{39} + 12 q^{41} - 4 q^{43}+ \cdots - 8 q^{99}+O(q^{100}) \) 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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Modular forms

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Database

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