LMFDB - Embedded newform 441.2.a.i.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [441,2,Mod(1,441)] 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("441.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(441, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	441.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,2,0,2,-4,0,0,6,0,0,4,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$3.52140272914$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 147)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	441.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.414214 q^{2} -1.82843 q^{4} -3.41421 q^{5} +1.58579 q^{8} +1.41421 q^{10} +2.00000 q^{11} +2.58579 q^{13} +3.00000 q^{16} +2.24264 q^{17} -2.82843 q^{19} +6.24264 q^{20} -0.828427 q^{22} +7.65685 q^{23} +6.65685 q^{25} -1.07107 q^{26} +6.82843 q^{29} -1.17157 q^{31} -4.41421 q^{32} -0.928932 q^{34} -4.00000 q^{37} +1.17157 q^{38} -5.41421 q^{40} -6.24264 q^{41} +5.65685 q^{43} -3.65685 q^{44} -3.17157 q^{46} +2.82843 q^{47} -2.75736 q^{50} -4.72792 q^{52} +2.00000 q^{53} -6.82843 q^{55} -2.82843 q^{58} +1.17157 q^{59} +12.2426 q^{61} +0.485281 q^{62} -4.17157 q^{64} -8.82843 q^{65} -5.65685 q^{67} -4.10051 q^{68} -9.31371 q^{71} +13.8995 q^{73} +1.65685 q^{74} +5.17157 q^{76} +13.6569 q^{79} -10.2426 q^{80} +2.58579 q^{82} -7.31371 q^{83} -7.65685 q^{85} -2.34315 q^{86} +3.17157 q^{88} +14.2426 q^{89} -14.0000 q^{92} -1.17157 q^{94} +9.65685 q^{95} +2.58579 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} - 4 q^{5} + 6 q^{8} + 4 q^{11} + 8 q^{13} + 6 q^{16} - 4 q^{17} + 4 q^{20} + 4 q^{22} + 4 q^{23} + 2 q^{25} + 12 q^{26} + 8 q^{29} - 8 q^{31} - 6 q^{32} - 16 q^{34} - 8 q^{37} + 8 q^{38}+ \cdots + 8 q^{97}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 - 4 * q^5 + 6 * q^8 + 4 * q^11 + 8 * q^13 + 6 * q^16 - 4 * q^17 + 4 * q^20 + 4 * q^22 + 4 * q^23 + 2 * q^25 + 12 * q^26 + 8 * q^29 - 8 * q^31 - 6 * q^32 - 16 * q^34 - 8 * q^37 + 8 * q^38 - 8 * q^40 - 4 * q^41 + 4 * q^44 - 12 * q^46 - 14 * q^50 + 16 * q^52 + 4 * q^53 - 8 * q^55 + 8 * q^59 + 16 * q^61 - 16 * q^62 - 14 * q^64 - 12 * q^65 - 28 * q^68 + 4 * q^71 + 8 * q^73 - 8 * q^74 + 16 * q^76 + 16 * q^79 - 12 * q^80 + 8 * q^82 + 8 * q^83 - 4 * q^85 - 16 * q^86 + 12 * q^88 + 20 * q^89 - 28 * q^92 - 8 * q^94 + 8 * q^95 + 8 * q^97

Coefficient data

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

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

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

$2$	−0.414214	−0.292893	−0.146447	−	0.989219i	$-0.546784\pi$
$2$	−0.414214	−0.292893	−0.146447	+	0.989219i	$0.546784\pi$
$3$	0	0
$3$	0	0
$4$	−1.82843	−0.914214
$5$	−3.41421	−1.52688	−0.763441	−	0.645877i	$-0.776492\pi$
$5$	−3.41421	−1.52688	−0.763441	+	0.645877i	$0.776492\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.58579	0.560660
$9$	0	0
$10$	1.41421	0.447214
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	2.58579	0.717168	0.358584	−	0.933497i	$-0.383260\pi$
$13$	2.58579	0.717168	0.358584	+	0.933497i	$0.383260\pi$
$14$	0	0
$15$	0	0
$16$	3.00000	0.750000
$17$	2.24264	0.543920	0.271960	−	0.962309i	$-0.412328\pi$
$17$	2.24264	0.543920	0.271960	+	0.962309i	$0.412328\pi$
$18$	0	0
$19$	−2.82843	−0.648886	−0.324443	−	0.945905i	$-0.605177\pi$
$19$	−2.82843	−0.648886	−0.324443	+	0.945905i	$0.605177\pi$
$20$	6.24264	1.39590
$21$	0	0
$22$	−0.828427	−0.176621
$23$	7.65685	1.59656	0.798282	−	0.602284i	$-0.205742\pi$
$23$	7.65685	1.59656	0.798282	+	0.602284i	$0.205742\pi$
$24$	0	0
$25$	6.65685	1.33137
$26$	−1.07107	−0.210054
$27$	0	0
$28$	0	0
$29$	6.82843	1.26801	0.634004	−	0.773330i	$-0.281410\pi$
$29$	6.82843	1.26801	0.634004	+	0.773330i	$0.281410\pi$
$30$	0	0
$31$	−1.17157	−0.210421	−0.105210	−	0.994450i	$-0.533552\pi$
$31$	−1.17157	−0.210421	−0.105210	+	0.994450i	$0.533552\pi$
$32$	−4.41421	−0.780330
$33$	0	0
$34$	−0.928932	−0.159311
$35$	0	0
$36$	0	0
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	1.17157	0.190054
$39$	0	0
$40$	−5.41421	−0.856062
$41$	−6.24264	−0.974937	−0.487468	−	0.873141i	$-0.662080\pi$
$41$	−6.24264	−0.974937	−0.487468	+	0.873141i	$0.662080\pi$
$42$	0	0
$43$	5.65685	0.862662	0.431331	−	0.902194i	$-0.358044\pi$
$43$	5.65685	0.862662	0.431331	+	0.902194i	$0.358044\pi$
$44$	−3.65685	−0.551292
$45$	0	0
$46$	−3.17157	−0.467623
$47$	2.82843	0.412568	0.206284	−	0.978492i	$-0.433863\pi$
$47$	2.82843	0.412568	0.206284	+	0.978492i	$0.433863\pi$
$48$	0	0
$49$	0	0
$50$	−2.75736	−0.389949
$51$	0	0
$52$	−4.72792	−0.655645
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	0	0
$55$	−6.82843	−0.920745
$56$	0	0
$57$	0	0
$58$	−2.82843	−0.371391
$59$	1.17157	0.152526	0.0762629	−	0.997088i	$-0.475701\pi$
$59$	1.17157	0.152526	0.0762629	+	0.997088i	$0.475701\pi$
$60$	0	0
$61$	12.2426	1.56751	0.783755	−	0.621070i	$-0.213302\pi$
$61$	12.2426	1.56751	0.783755	+	0.621070i	$0.213302\pi$
$62$	0.485281	0.0616308
$63$	0	0
$64$	−4.17157	−0.521447
$65$	−8.82843	−1.09503
$66$	0	0
$67$	−5.65685	−0.691095	−0.345547	−	0.938401i	$-0.612307\pi$
$67$	−5.65685	−0.691095	−0.345547	+	0.938401i	$0.612307\pi$
$68$	−4.10051	−0.497259
$69$	0	0
$70$	0	0
$71$	−9.31371	−1.10533	−0.552667	−	0.833402i	$-0.686390\pi$
$71$	−9.31371	−1.10533	−0.552667	+	0.833402i	$0.686390\pi$
$72$	0	0
$73$	13.8995	1.62681	0.813406	−	0.581696i	$-0.197611\pi$
$73$	13.8995	1.62681	0.813406	+	0.581696i	$0.197611\pi$
$74$	1.65685	0.192605
$75$	0	0
$76$	5.17157	0.593220
$77$	0	0
$78$	0	0
$79$	13.6569	1.53652	0.768258	−	0.640140i	$-0.221124\pi$
$79$	13.6569	1.53652	0.768258	+	0.640140i	$0.221124\pi$
$80$	−10.2426	−1.14516
$81$	0	0
$82$	2.58579	0.285552
$83$	−7.31371	−0.802784	−0.401392	−	0.915906i	$-0.631473\pi$
$83$	−7.31371	−0.802784	−0.401392	+	0.915906i	$0.631473\pi$
$84$	0	0
$85$	−7.65685	−0.830502
$86$	−2.34315	−0.252668
$87$	0	0
$88$	3.17157	0.338091
$89$	14.2426	1.50972	0.754858	−	0.655888i	$-0.227706\pi$
$89$	14.2426	1.50972	0.754858	+	0.655888i	$0.227706\pi$
$90$	0	0
$91$	0	0
$92$	−14.0000	−1.45960
$93$	0	0
$94$	−1.17157	−0.120839
$95$	9.65685	0.990772
$96$	0	0
$97$	2.58579	0.262547	0.131273	−	0.991346i	$-0.458093\pi$
$97$	2.58579	0.262547	0.131273	+	0.991346i	$0.458093\pi$
$98$	0	0
$99$	0	0
$100$	−12.1716	−1.21716
$101$	−2.92893	−0.291440	−0.145720	−	0.989326i	$-0.546550\pi$
$101$	−2.92893	−0.291440	−0.145720	+	0.989326i	$0.546550\pi$
$102$	0	0
$103$	4.48528	0.441948	0.220974	−	0.975280i	$-0.429076\pi$
$103$	4.48528	0.441948	0.220974	+	0.975280i	$0.429076\pi$
$104$	4.10051	0.402088
$105$	0	0
$106$	−0.828427	−0.0804640
$107$	0.343146	0.0331732	0.0165866	−	0.999862i	$-0.494720\pi$
$107$	0.343146	0.0331732	0.0165866	+	0.999862i	$0.494720\pi$
$108$	0	0
$109$	−5.65685	−0.541828	−0.270914	−	0.962604i	$-0.587326\pi$
$109$	−5.65685	−0.541828	−0.270914	+	0.962604i	$0.587326\pi$
$110$	2.82843	0.269680
$111$	0	0
$112$	0	0
$113$	5.31371	0.499872	0.249936	−	0.968262i	$-0.419590\pi$
$113$	5.31371	0.499872	0.249936	+	0.968262i	$0.419590\pi$
$114$	0	0
$115$	−26.1421	−2.43777
$116$	−12.4853	−1.15923
$117$	0	0
$118$	−0.485281	−0.0446738
$119$	0	0
$120$	0	0
$121$	−7.00000	−0.636364
$122$	−5.07107	−0.459113
$123$	0	0
$124$	2.14214	0.192369
$125$	−5.65685	−0.505964
$126$	0	0
$127$	−1.65685	−0.147022	−0.0735110	−	0.997294i	$-0.523420\pi$
$127$	−1.65685	−0.147022	−0.0735110	+	0.997294i	$0.523420\pi$
$128$	10.5563	0.933058
$129$	0	0
$130$	3.65685	0.320727
$131$	15.3137	1.33796	0.668982	−	0.743278i	$-0.266730\pi$
$131$	15.3137	1.33796	0.668982	+	0.743278i	$0.266730\pi$
$132$	0	0
$133$	0	0
$134$	2.34315	0.202417
$135$	0	0
$136$	3.55635	0.304954
$137$	−14.1421	−1.20824	−0.604122	−	0.796892i	$-0.706476\pi$
$137$	−14.1421	−1.20824	−0.604122	+	0.796892i	$0.706476\pi$
$138$	0	0
$139$	−17.6569	−1.49763	−0.748817	−	0.662776i	$-0.769378\pi$
$139$	−17.6569	−1.49763	−0.748817	+	0.662776i	$0.769378\pi$
$140$	0	0
$141$	0	0
$142$	3.85786	0.323745
$143$	5.17157	0.432469
$144$	0	0
$145$	−23.3137	−1.93610
$146$	−5.75736	−0.476482
$147$	0	0
$148$	7.31371	0.601183
$149$	−17.3137	−1.41839	−0.709197	−	0.705010i	$-0.750942\pi$
$149$	−17.3137	−1.41839	−0.709197	+	0.705010i	$0.750942\pi$
$150$	0	0
$151$	12.0000	0.976546	0.488273	−	0.872691i	$-0.337627\pi$
$151$	12.0000	0.976546	0.488273	+	0.872691i	$0.337627\pi$
$152$	−4.48528	−0.363804
$153$	0	0
$154$	0	0
$155$	4.00000	0.321288
$156$	0	0
$157$	11.7574	0.938339	0.469170	−	0.883108i	$-0.344553\pi$
$157$	11.7574	0.938339	0.469170	+	0.883108i	$0.344553\pi$
$158$	−5.65685	−0.450035
$159$	0	0
$160$	15.0711	1.19147
$161$	0	0
$162$	0	0
$163$	−11.3137	−0.886158	−0.443079	−	0.896483i	$-0.646114\pi$
$163$	−11.3137	−0.886158	−0.443079	+	0.896483i	$0.646114\pi$
$164$	11.4142	0.891300
$165$	0	0
$166$	3.02944	0.235130
$167$	19.7990	1.53209	0.766046	−	0.642786i	$-0.222221\pi$
$167$	19.7990	1.53209	0.766046	+	0.642786i	$0.222221\pi$
$168$	0	0
$169$	−6.31371	−0.485670
$170$	3.17157	0.243249
$171$	0	0
$172$	−10.3431	−0.788657
$173$	−21.0711	−1.60200	−0.801002	−	0.598662i	$-0.795699\pi$
$173$	−21.0711	−1.60200	−0.801002	+	0.598662i	$0.795699\pi$
$174$	0	0
$175$	0	0
$176$	6.00000	0.452267
$177$	0	0
$178$	−5.89949	−0.442186
$179$	19.6569	1.46922	0.734611	−	0.678488i	$-0.237365\pi$
$179$	19.6569	1.46922	0.734611	+	0.678488i	$0.237365\pi$
$180$	0	0
$181$	−2.58579	−0.192200	−0.0961000	−	0.995372i	$-0.530637\pi$
$181$	−2.58579	−0.192200	−0.0961000	+	0.995372i	$0.530637\pi$
$182$	0	0
$183$	0	0
$184$	12.1421	0.895130
$185$	13.6569	1.00407
$186$	0	0
$187$	4.48528	0.327996
$188$	−5.17157	−0.377176
$189$	0	0
$190$	−4.00000	−0.290191
$191$	18.0000	1.30243	0.651217	−	0.758891i	$-0.274259\pi$
$191$	18.0000	1.30243	0.651217	+	0.758891i	$0.274259\pi$
$192$	0	0
$193$	5.31371	0.382489	0.191245	−	0.981542i	$-0.438748\pi$
$193$	5.31371	0.382489	0.191245	+	0.981542i	$0.438748\pi$
$194$	−1.07107	−0.0768982
$195$	0	0
$196$	0	0
$197$	−2.00000	−0.142494	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000	−0.142494	−0.0712470	+	0.997459i	$0.522698\pi$
$198$	0	0
$199$	21.6569	1.53521	0.767607	−	0.640921i	$-0.221447\pi$
$199$	21.6569	1.53521	0.767607	+	0.640921i	$0.221447\pi$
$200$	10.5563	0.746447
$201$	0	0
$202$	1.21320	0.0853607
$203$	0	0
$204$	0	0
$205$	21.3137	1.48861
$206$	−1.85786	−0.129444
$207$	0	0
$208$	7.75736	0.537876
$209$	−5.65685	−0.391293
$210$	0	0
$211$	12.9706	0.892930	0.446465	−	0.894801i	$-0.352683\pi$
$211$	12.9706	0.892930	0.446465	+	0.894801i	$0.352683\pi$
$212$	−3.65685	−0.251154
$213$	0	0
$214$	−0.142136	−0.00971619
$215$	−19.3137	−1.31718
$216$	0	0
$217$	0	0
$218$	2.34315	0.158698
$219$	0	0
$220$	12.4853	0.841757
$221$	5.79899	0.390082
$222$	0	0
$223$	24.9706	1.67215	0.836076	−	0.548613i	$-0.184844\pi$
$223$	24.9706	1.67215	0.836076	+	0.548613i	$0.184844\pi$
$224$	0	0
$225$	0	0
$226$	−2.20101	−0.146409
$227$	−23.7990	−1.57959	−0.789797	−	0.613368i	$-0.789814\pi$
$227$	−23.7990	−1.57959	−0.789797	+	0.613368i	$0.789814\pi$
$228$	0	0
$229$	−0.242641	−0.0160341	−0.00801707	−	0.999968i	$-0.502552\pi$
$229$	−0.242641	−0.0160341	−0.00801707	+	0.999968i	$0.502552\pi$
$230$	10.8284	0.714005
$231$	0	0
$232$	10.8284	0.710921
$233$	6.14214	0.402385	0.201192	−	0.979552i	$-0.435518\pi$
$233$	6.14214	0.402385	0.201192	+	0.979552i	$0.435518\pi$
$234$	0	0
$235$	−9.65685	−0.629944
$236$	−2.14214	−0.139441
$237$	0	0
$238$	0	0
$239$	15.6569	1.01276	0.506379	−	0.862311i	$-0.330984\pi$
$239$	15.6569	1.01276	0.506379	+	0.862311i	$0.330984\pi$
$240$	0	0
$241$	−16.2426	−1.04628	−0.523140	−	0.852247i	$-0.675240\pi$
$241$	−16.2426	−1.04628	−0.523140	+	0.852247i	$0.675240\pi$
$242$	2.89949	0.186387
$243$	0	0
$244$	−22.3848	−1.43304
$245$	0	0
$246$	0	0
$247$	−7.31371	−0.465360
$248$	−1.85786	−0.117975
$249$	0	0
$250$	2.34315	0.148194
$251$	12.4853	0.788064	0.394032	−	0.919097i	$-0.371080\pi$
$251$	12.4853	0.788064	0.394032	+	0.919097i	$0.371080\pi$
$252$	0	0
$253$	15.3137	0.962765
$254$	0.686292	0.0430618
$255$	0	0
$256$	3.97056	0.248160
$257$	−23.2132	−1.44800	−0.724000	−	0.689800i	$-0.757698\pi$
$257$	−23.2132	−1.44800	−0.724000	+	0.689800i	$0.757698\pi$
$258$	0	0
$259$	0	0
$260$	16.1421	1.00109
$261$	0	0
$262$	−6.34315	−0.391881
$263$	−5.31371	−0.327657	−0.163829	−	0.986489i	$-0.552384\pi$
$263$	−5.31371	−0.327657	−0.163829	+	0.986489i	$0.552384\pi$
$264$	0	0
$265$	−6.82843	−0.419467
$266$	0	0
$267$	0	0
$268$	10.3431	0.631808
$269$	14.7279	0.897977	0.448989	−	0.893537i	$-0.351784\pi$
$269$	14.7279	0.897977	0.448989	+	0.893537i	$0.351784\pi$
$270$	0	0
$271$	−10.1421	−0.616091	−0.308045	−	0.951372i	$-0.599675\pi$
$271$	−10.1421	−0.616091	−0.308045	+	0.951372i	$0.599675\pi$
$272$	6.72792	0.407940
$273$	0	0
$274$	5.85786	0.353887
$275$	13.3137	0.802847
$276$	0	0
$277$	−9.31371	−0.559607	−0.279803	−	0.960057i	$-0.590269\pi$
$277$	−9.31371	−0.559607	−0.279803	+	0.960057i	$0.590269\pi$
$278$	7.31371	0.438647
$279$	0	0
$280$	0	0
$281$	−0.485281	−0.0289495	−0.0144747	−	0.999895i	$-0.504608\pi$
$281$	−0.485281	−0.0289495	−0.0144747	+	0.999895i	$0.504608\pi$
$282$	0	0
$283$	−8.48528	−0.504398	−0.252199	−	0.967675i	$-0.581154\pi$
$283$	−8.48528	−0.504398	−0.252199	+	0.967675i	$0.581154\pi$
$284$	17.0294	1.01051
$285$	0	0
$286$	−2.14214	−0.126667
$287$	0	0
$288$	0	0
$289$	−11.9706	−0.704151
$290$	9.65685	0.567070
$291$	0	0
$292$	−25.4142	−1.48725
$293$	−16.5858	−0.968952	−0.484476	−	0.874805i	$-0.660990\pi$
$293$	−16.5858	−0.968952	−0.484476	+	0.874805i	$0.660990\pi$
$294$	0	0
$295$	−4.00000	−0.232889
$296$	−6.34315	−0.368688
$297$	0	0
$298$	7.17157	0.415438
$299$	19.7990	1.14501
$300$	0	0
$301$	0	0
$302$	−4.97056	−0.286024
$303$	0	0
$304$	−8.48528	−0.486664
$305$	−41.7990	−2.39340
$306$	0	0
$307$	30.1421	1.72030	0.860151	−	0.510039i	$-0.170369\pi$
$307$	30.1421	1.72030	0.860151	+	0.510039i	$0.170369\pi$
$308$	0	0
$309$	0	0
$310$	−1.65685	−0.0941030
$311$	−6.14214	−0.348289	−0.174144	−	0.984720i	$-0.555716\pi$
$311$	−6.14214	−0.348289	−0.174144	+	0.984720i	$0.555716\pi$
$312$	0	0
$313$	1.89949	0.107366	0.0536829	−	0.998558i	$-0.482904\pi$
$313$	1.89949	0.107366	0.0536829	+	0.998558i	$0.482904\pi$
$314$	−4.87006	−0.274833
$315$	0	0
$316$	−24.9706	−1.40470
$317$	−10.0000	−0.561656	−0.280828	−	0.959758i	$-0.590609\pi$
$317$	−10.0000	−0.561656	−0.280828	+	0.959758i	$0.590609\pi$
$318$	0	0
$319$	13.6569	0.764637
$320$	14.2426	0.796188
$321$	0	0
$322$	0	0
$323$	−6.34315	−0.352942
$324$	0	0
$325$	17.2132	0.954817
$326$	4.68629	0.259550
$327$	0	0
$328$	−9.89949	−0.546608
$329$	0	0
$330$	0	0
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	13.3726	0.733916
$333$	0	0
$334$	−8.20101	−0.448739
$335$	19.3137	1.05522
$336$	0	0
$337$	−29.6569	−1.61551	−0.807756	−	0.589517i	$-0.799318\pi$
$337$	−29.6569	−1.61551	−0.807756	+	0.589517i	$0.799318\pi$
$338$	2.61522	0.142249
$339$	0	0
$340$	14.0000	0.759257
$341$	−2.34315	−0.126888
$342$	0	0
$343$	0	0
$344$	8.97056	0.483660
$345$	0	0
$346$	8.72792	0.469216
$347$	−33.3137	−1.78837	−0.894187	−	0.447694i	$-0.852245\pi$
$347$	−33.3137	−1.78837	−0.894187	+	0.447694i	$0.852245\pi$
$348$	0	0
$349$	−9.89949	−0.529908	−0.264954	−	0.964261i	$-0.585357\pi$
$349$	−9.89949	−0.529908	−0.264954	+	0.964261i	$0.585357\pi$
$350$	0	0
$351$	0	0
$352$	−8.82843	−0.470557
$353$	14.7279	0.783888	0.391944	−	0.919989i	$-0.371803\pi$
$353$	14.7279	0.783888	0.391944	+	0.919989i	$0.371803\pi$
$354$	0	0
$355$	31.7990	1.68772
$356$	−26.0416	−1.38020
$357$	0	0
$358$	−8.14214	−0.430325
$359$	0.343146	0.0181105	0.00905527	−	0.999959i	$-0.497118\pi$
$359$	0.343146	0.0181105	0.00905527	+	0.999959i	$0.497118\pi$
$360$	0	0
$361$	−11.0000	−0.578947
$362$	1.07107	0.0562941
$363$	0	0
$364$	0	0
$365$	−47.4558	−2.48395
$366$	0	0
$367$	−3.31371	−0.172974	−0.0864871	−	0.996253i	$-0.527564\pi$
$367$	−3.31371	−0.172974	−0.0864871	+	0.996253i	$0.527564\pi$
$368$	22.9706	1.19742
$369$	0	0
$370$	−5.65685	−0.294086
$371$	0	0
$372$	0	0
$373$	−10.6863	−0.553315	−0.276658	−	0.960969i	$-0.589227\pi$
$373$	−10.6863	−0.553315	−0.276658	+	0.960969i	$0.589227\pi$
$374$	−1.85786	−0.0960679
$375$	0	0
$376$	4.48528	0.231311
$377$	17.6569	0.909374
$378$	0	0
$379$	8.68629	0.446185	0.223092	−	0.974797i	$-0.428385\pi$
$379$	8.68629	0.446185	0.223092	+	0.974797i	$0.428385\pi$
$380$	−17.6569	−0.905778
$381$	0	0
$382$	−7.45584	−0.381474
$383$	18.3431	0.937291	0.468645	−	0.883386i	$-0.344742\pi$
$383$	18.3431	0.937291	0.468645	+	0.883386i	$0.344742\pi$
$384$	0	0
$385$	0	0
$386$	−2.20101	−0.112028
$387$	0	0
$388$	−4.72792	−0.240024
$389$	18.1421	0.919843	0.459921	−	0.887960i	$-0.347878\pi$
$389$	18.1421	0.919843	0.459921	+	0.887960i	$0.347878\pi$
$390$	0	0
$391$	17.1716	0.868404
$392$	0	0
$393$	0	0
$394$	0.828427	0.0417356
$395$	−46.6274	−2.34608
$396$	0	0
$397$	−2.38478	−0.119688	−0.0598442	−	0.998208i	$-0.519060\pi$
$397$	−2.38478	−0.119688	−0.0598442	+	0.998208i	$0.519060\pi$
$398$	−8.97056	−0.449654
$399$	0	0
$400$	19.9706	0.998528
$401$	6.14214	0.306724	0.153362	−	0.988170i	$-0.450990\pi$
$401$	6.14214	0.306724	0.153362	+	0.988170i	$0.450990\pi$
$402$	0	0
$403$	−3.02944	−0.150907
$404$	5.35534	0.266438
$405$	0	0
$406$	0	0
$407$	−8.00000	−0.396545
$408$	0	0
$409$	−21.4142	−1.05886	−0.529432	−	0.848352i	$-0.677595\pi$
$409$	−21.4142	−1.05886	−0.529432	+	0.848352i	$0.677595\pi$
$410$	−8.82843	−0.436005
$411$	0	0
$412$	−8.20101	−0.404035
$413$	0	0
$414$	0	0
$415$	24.9706	1.22576
$416$	−11.4142	−0.559628
$417$	0	0
$418$	2.34315	0.114607
$419$	−33.1716	−1.62054	−0.810269	−	0.586059i	$-0.800679\pi$
$419$	−33.1716	−1.62054	−0.810269	+	0.586059i	$0.800679\pi$
$420$	0	0
$421$	16.6274	0.810371	0.405185	−	0.914235i	$-0.367207\pi$
$421$	16.6274	0.810371	0.405185	+	0.914235i	$0.367207\pi$
$422$	−5.37258	−0.261533
$423$	0	0
$424$	3.17157	0.154025
$425$	14.9289	0.724160
$426$	0	0
$427$	0	0
$428$	−0.627417	−0.0303273
$429$	0	0
$430$	8.00000	0.385794
$431$	26.9706	1.29913	0.649563	−	0.760308i	$-0.274952\pi$
$431$	26.9706	1.29913	0.649563	+	0.760308i	$0.274952\pi$
$432$	0	0
$433$	−20.2426	−0.972799	−0.486400	−	0.873736i	$-0.661690\pi$
$433$	−20.2426	−0.972799	−0.486400	+	0.873736i	$0.661690\pi$
$434$	0	0
$435$	0	0
$436$	10.3431	0.495347
$437$	−21.6569	−1.03599
$438$	0	0
$439$	12.6863	0.605484	0.302742	−	0.953073i	$-0.402098\pi$
$439$	12.6863	0.605484	0.302742	+	0.953073i	$0.402098\pi$
$440$	−10.8284	−0.516225
$441$	0	0
$442$	−2.40202	−0.114252
$443$	−34.9706	−1.66150	−0.830751	−	0.556645i	$-0.812089\pi$
$443$	−34.9706	−1.66150	−0.830751	+	0.556645i	$0.812089\pi$
$444$	0	0
$445$	−48.6274	−2.30516
$446$	−10.3431	−0.489762
$447$	0	0
$448$	0	0
$449$	5.31371	0.250769	0.125385	−	0.992108i	$-0.459983\pi$
$449$	5.31371	0.250769	0.125385	+	0.992108i	$0.459983\pi$
$450$	0	0
$451$	−12.4853	−0.587909
$452$	−9.71573	−0.456989
$453$	0	0
$454$	9.85786	0.462652
$455$	0	0
$456$	0	0
$457$	−18.0000	−0.842004	−0.421002	−	0.907060i	$-0.638322\pi$
$457$	−18.0000	−0.842004	−0.421002	+	0.907060i	$0.638322\pi$
$458$	0.100505	0.00469629
$459$	0	0
$460$	47.7990	2.22864
$461$	16.5858	0.772477	0.386239	−	0.922399i	$-0.373774\pi$
$461$	16.5858	0.772477	0.386239	+	0.922399i	$0.373774\pi$
$462$	0	0
$463$	−26.6274	−1.23748	−0.618741	−	0.785595i	$-0.712357\pi$
$463$	−26.6274	−1.23748	−0.618741	+	0.785595i	$0.712357\pi$
$464$	20.4853	0.951005
$465$	0	0
$466$	−2.54416	−0.117856
$467$	−0.201010	−0.00930164	−0.00465082	−	0.999989i	$-0.501480\pi$
$467$	−0.201010	−0.00930164	−0.00465082	+	0.999989i	$0.501480\pi$
$468$	0	0
$469$	0	0
$470$	4.00000	0.184506
$471$	0	0
$472$	1.85786	0.0855151
$473$	11.3137	0.520205
$474$	0	0
$475$	−18.8284	−0.863907
$476$	0	0
$477$	0	0
$478$	−6.48528	−0.296630
$479$	−1.85786	−0.0848880	−0.0424440	−	0.999099i	$-0.513514\pi$
$479$	−1.85786	−0.0848880	−0.0424440	+	0.999099i	$0.513514\pi$
$480$	0	0
$481$	−10.3431	−0.471607
$482$	6.72792	0.306448
$483$	0	0
$484$	12.7990	0.581772
$485$	−8.82843	−0.400878
$486$	0	0
$487$	26.6274	1.20660	0.603302	−	0.797513i	$-0.293851\pi$
$487$	26.6274	1.20660	0.603302	+	0.797513i	$0.293851\pi$
$488$	19.4142	0.878840
$489$	0	0
$490$	0	0
$491$	−5.02944	−0.226975	−0.113488	−	0.993539i	$-0.536202\pi$
$491$	−5.02944	−0.226975	−0.113488	+	0.993539i	$0.536202\pi$
$492$	0	0
$493$	15.3137	0.689695
$494$	3.02944	0.136301
$495$	0	0
$496$	−3.51472	−0.157816
$497$	0	0
$498$	0	0
$499$	3.31371	0.148342	0.0741710	−	0.997246i	$-0.476369\pi$
$499$	3.31371	0.148342	0.0741710	+	0.997246i	$0.476369\pi$
$500$	10.3431	0.462560
$501$	0	0
$502$	−5.17157	−0.230819
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	10.0000	0.444994
$506$	−6.34315	−0.281987
$507$	0	0
$508$	3.02944	0.134410
$509$	−5.55635	−0.246281	−0.123140	−	0.992389i	$-0.539297\pi$
$509$	−5.55635	−0.246281	−0.123140	+	0.992389i	$0.539297\pi$
$510$	0	0
$511$	0	0
$512$	−22.7574	−1.00574
$513$	0	0
$514$	9.61522	0.424109
$515$	−15.3137	−0.674803
$516$	0	0
$517$	5.65685	0.248788
$518$	0	0
$519$	0	0
$520$	−14.0000	−0.613941
$521$	35.4142	1.55152	0.775762	−	0.631025i	$-0.217366\pi$
$521$	35.4142	1.55152	0.775762	+	0.631025i	$0.217366\pi$
$522$	0	0
$523$	−25.6569	−1.12190	−0.560948	−	0.827851i	$-0.689563\pi$
$523$	−25.6569	−1.12190	−0.560948	+	0.827851i	$0.689563\pi$
$524$	−28.0000	−1.22319
$525$	0	0
$526$	2.20101	0.0959686
$527$	−2.62742	−0.114452
$528$	0	0
$529$	35.6274	1.54902
$530$	2.82843	0.122859
$531$	0	0
$532$	0	0
$533$	−16.1421	−0.699194
$534$	0	0
$535$	−1.17157	−0.0506515
$536$	−8.97056	−0.387469
$537$	0	0
$538$	−6.10051	−0.263011
$539$	0	0
$540$	0	0
$541$	17.3137	0.744374	0.372187	−	0.928158i	$-0.378608\pi$
$541$	17.3137	0.744374	0.372187	+	0.928158i	$0.378608\pi$
$542$	4.20101	0.180449
$543$	0	0
$544$	−9.89949	−0.424437
$545$	19.3137	0.827308
$546$	0	0
$547$	−36.9706	−1.58075	−0.790374	−	0.612625i	$-0.790114\pi$
$547$	−36.9706	−1.58075	−0.790374	+	0.612625i	$0.790114\pi$
$548$	25.8579	1.10459
$549$	0	0
$550$	−5.51472	−0.235148
$551$	−19.3137	−0.822792
$552$	0	0
$553$	0	0
$554$	3.85786	0.163905
$555$	0	0
$556$	32.2843	1.36916
$557$	−26.0000	−1.10166	−0.550828	−	0.834619i	$-0.685688\pi$
$557$	−26.0000	−1.10166	−0.550828	+	0.834619i	$0.685688\pi$
$558$	0	0
$559$	14.6274	0.618674
$560$	0	0
$561$	0	0
$562$	0.201010	0.00847910
$563$	1.17157	0.0493759	0.0246880	−	0.999695i	$-0.492141\pi$
$563$	1.17157	0.0493759	0.0246880	+	0.999695i	$0.492141\pi$
$564$	0	0
$565$	−18.1421	−0.763245
$566$	3.51472	0.147735
$567$	0	0
$568$	−14.7696	−0.619717
$569$	16.4853	0.691099	0.345549	−	0.938401i	$-0.387693\pi$
$569$	16.4853	0.691099	0.345549	+	0.938401i	$0.387693\pi$
$570$	0	0
$571$	22.3431	0.935032	0.467516	−	0.883985i	$-0.345149\pi$
$571$	22.3431	0.935032	0.467516	+	0.883985i	$0.345149\pi$
$572$	−9.45584	−0.395369
$573$	0	0
$574$	0	0
$575$	50.9706	2.12562
$576$	0	0
$577$	−33.8995	−1.41125	−0.705627	−	0.708583i	$-0.749335\pi$
$577$	−33.8995	−1.41125	−0.705627	+	0.708583i	$0.749335\pi$
$578$	4.95837	0.206241
$579$	0	0
$580$	42.6274	1.77001
$581$	0	0
$582$	0	0
$583$	4.00000	0.165663
$584$	22.0416	0.912089
$585$	0	0
$586$	6.87006	0.283799
$587$	22.8284	0.942230	0.471115	−	0.882072i	$-0.343852\pi$
$587$	22.8284	0.942230	0.471115	+	0.882072i	$0.343852\pi$
$588$	0	0
$589$	3.31371	0.136539
$590$	1.65685	0.0682116
$591$	0	0
$592$	−12.0000	−0.493197
$593$	6.92893	0.284537	0.142269	−	0.989828i	$-0.454560\pi$
$593$	6.92893	0.284537	0.142269	+	0.989828i	$0.454560\pi$
$594$	0	0
$595$	0	0
$596$	31.6569	1.29672
$597$	0	0
$598$	−8.20101	−0.335364
$599$	2.00000	0.0817178	0.0408589	−	0.999165i	$-0.486991\pi$
$599$	2.00000	0.0817178	0.0408589	+	0.999165i	$0.486991\pi$
$600$	0	0
$601$	−15.0711	−0.614762	−0.307381	−	0.951587i	$-0.599453\pi$
$601$	−15.0711	−0.614762	−0.307381	+	0.951587i	$0.599453\pi$
$602$	0	0
$603$	0	0
$604$	−21.9411	−0.892772
$605$	23.8995	0.971653
$606$	0	0
$607$	−18.3431	−0.744525	−0.372263	−	0.928127i	$-0.621418\pi$
$607$	−18.3431	−0.744525	−0.372263	+	0.928127i	$0.621418\pi$
$608$	12.4853	0.506345
$609$	0	0
$610$	17.3137	0.701012
$611$	7.31371	0.295881
$612$	0	0
$613$	4.68629	0.189278	0.0946388	−	0.995512i	$-0.469830\pi$
$613$	4.68629	0.189278	0.0946388	+	0.995512i	$0.469830\pi$
$614$	−12.4853	−0.503865
$615$	0	0
$616$	0	0
$617$	24.4853	0.985740	0.492870	−	0.870103i	$-0.335948\pi$
$617$	24.4853	0.985740	0.492870	+	0.870103i	$0.335948\pi$
$618$	0	0
$619$	28.9706	1.16443	0.582213	−	0.813037i	$-0.302187\pi$
$619$	28.9706	1.16443	0.582213	+	0.813037i	$0.302187\pi$
$620$	−7.31371	−0.293726
$621$	0	0
$622$	2.54416	0.102011
$623$	0	0
$624$	0	0
$625$	−13.9706	−0.558823
$626$	−0.786797	−0.0314467
$627$	0	0
$628$	−21.4975	−0.857843
$629$	−8.97056	−0.357680
$630$	0	0
$631$	23.3137	0.928104	0.464052	−	0.885808i	$-0.346395\pi$
$631$	23.3137	0.928104	0.464052	+	0.885808i	$0.346395\pi$
$632$	21.6569	0.861463
$633$	0	0
$634$	4.14214	0.164505
$635$	5.65685	0.224485
$636$	0	0
$637$	0	0
$638$	−5.65685	−0.223957
$639$	0	0
$640$	−36.0416	−1.42467
$641$	10.8284	0.427697	0.213849	−	0.976867i	$-0.431400\pi$
$641$	10.8284	0.427697	0.213849	+	0.976867i	$0.431400\pi$
$642$	0	0
$643$	−34.4264	−1.35764	−0.678822	−	0.734302i	$-0.737509\pi$
$643$	−34.4264	−1.35764	−0.678822	+	0.734302i	$0.737509\pi$
$644$	0	0
$645$	0	0
$646$	2.62742	0.103374
$647$	−26.8284	−1.05473	−0.527367	−	0.849638i	$-0.676821\pi$
$647$	−26.8284	−1.05473	−0.527367	+	0.849638i	$0.676821\pi$
$648$	0	0
$649$	2.34315	0.0919765
$650$	−7.12994	−0.279659
$651$	0	0
$652$	20.6863	0.810138
$653$	−36.4853	−1.42778	−0.713890	−	0.700258i	$-0.753068\pi$
$653$	−36.4853	−1.42778	−0.713890	+	0.700258i	$0.753068\pi$
$654$	0	0
$655$	−52.2843	−2.04292
$656$	−18.7279	−0.731203
$657$	0	0
$658$	0	0
$659$	−9.31371	−0.362811	−0.181405	−	0.983408i	$-0.558065\pi$
$659$	−9.31371	−0.362811	−0.181405	+	0.983408i	$0.558065\pi$
$660$	0	0
$661$	−23.5563	−0.916236	−0.458118	−	0.888891i	$-0.651476\pi$
$661$	−23.5563	−0.916236	−0.458118	+	0.888891i	$0.651476\pi$
$662$	1.65685	0.0643955
$663$	0	0
$664$	−11.5980	−0.450089
$665$	0	0
$666$	0	0
$667$	52.2843	2.02446
$668$	−36.2010	−1.40066
$669$	0	0
$670$	−8.00000	−0.309067
$671$	24.4853	0.945244
$672$	0	0
$673$	23.3137	0.898677	0.449339	−	0.893361i	$-0.351660\pi$
$673$	23.3137	0.898677	0.449339	+	0.893361i	$0.351660\pi$
$674$	12.2843	0.473172
$675$	0	0
$676$	11.5442	0.444006
$677$	−31.4142	−1.20735	−0.603673	−	0.797232i	$-0.706297\pi$
$677$	−31.4142	−1.20735	−0.603673	+	0.797232i	$0.706297\pi$
$678$	0	0
$679$	0	0
$680$	−12.1421	−0.465630
$681$	0	0
$682$	0.970563	0.0371648
$683$	19.6569	0.752149	0.376074	−	0.926590i	$-0.377274\pi$
$683$	19.6569	0.752149	0.376074	+	0.926590i	$0.377274\pi$
$684$	0	0
$685$	48.2843	1.84485
$686$	0	0
$687$	0	0
$688$	16.9706	0.646997
$689$	5.17157	0.197021
$690$	0	0
$691$	−0.686292	−0.0261078	−0.0130539	−	0.999915i	$-0.504155\pi$
$691$	−0.686292	−0.0261078	−0.0130539	+	0.999915i	$0.504155\pi$
$692$	38.5269	1.46457
$693$	0	0
$694$	13.7990	0.523802
$695$	60.2843	2.28671
$696$	0	0
$697$	−14.0000	−0.530288
$698$	4.10051	0.155206
$699$	0	0
$700$	0	0
$701$	17.1716	0.648561	0.324281	−	0.945961i	$-0.394878\pi$
$701$	17.1716	0.648561	0.324281	+	0.945961i	$0.394878\pi$
$702$	0	0
$703$	11.3137	0.426705
$704$	−8.34315	−0.314444
$705$	0	0
$706$	−6.10051	−0.229596
$707$	0	0
$708$	0	0
$709$	−36.2843	−1.36268	−0.681342	−	0.731965i	$-0.738603\pi$
$709$	−36.2843	−1.36268	−0.681342	+	0.731965i	$0.738603\pi$
$710$	−13.1716	−0.494320
$711$	0	0
$712$	22.5858	0.846438
$713$	−8.97056	−0.335950
$714$	0	0
$715$	−17.6569	−0.660329
$716$	−35.9411	−1.34318
$717$	0	0
$718$	−0.142136	−0.00530445
$719$	−41.9411	−1.56414	−0.782070	−	0.623191i	$-0.785836\pi$
$719$	−41.9411	−1.56414	−0.782070	+	0.623191i	$0.785836\pi$
$720$	0	0
$721$	0	0
$722$	4.55635	0.169570
$723$	0	0
$724$	4.72792	0.175712
$725$	45.4558	1.68819
$726$	0	0
$727$	12.4853	0.463053	0.231527	−	0.972829i	$-0.425628\pi$
$727$	12.4853	0.463053	0.231527	+	0.972829i	$0.425628\pi$
$728$	0	0
$729$	0	0
$730$	19.6569	0.727533
$731$	12.6863	0.469219
$732$	0	0
$733$	49.6985	1.83566	0.917828	−	0.396979i	$-0.129941\pi$
$733$	49.6985	1.83566	0.917828	+	0.396979i	$0.129941\pi$
$734$	1.37258	0.0506630
$735$	0	0
$736$	−33.7990	−1.24585
$737$	−11.3137	−0.416746
$738$	0	0
$739$	4.68629	0.172388	0.0861940	−	0.996278i	$-0.472530\pi$
$739$	4.68629	0.172388	0.0861940	+	0.996278i	$0.472530\pi$
$740$	−24.9706	−0.917936
$741$	0	0
$742$	0	0
$743$	50.9706	1.86993	0.934964	−	0.354742i	$-0.115431\pi$
$743$	50.9706	1.86993	0.934964	+	0.354742i	$0.115431\pi$
$744$	0	0
$745$	59.1127	2.16572
$746$	4.42641	0.162062
$747$	0	0
$748$	−8.20101	−0.299859
$749$	0	0
$750$	0	0
$751$	13.6569	0.498346	0.249173	−	0.968459i	$-0.419841\pi$
$751$	13.6569	0.498346	0.249173	+	0.968459i	$0.419841\pi$
$752$	8.48528	0.309426
$753$	0	0
$754$	−7.31371	−0.266350
$755$	−40.9706	−1.49107
$756$	0	0
$757$	26.3431	0.957458	0.478729	−	0.877963i	$-0.341098\pi$
$757$	26.3431	0.957458	0.478729	+	0.877963i	$0.341098\pi$
$758$	−3.59798	−0.130685
$759$	0	0
$760$	15.3137	0.555487
$761$	18.5269	0.671600	0.335800	−	0.941933i	$-0.390993\pi$
$761$	18.5269	0.671600	0.335800	+	0.941933i	$0.390993\pi$
$762$	0	0
$763$	0	0
$764$	−32.9117	−1.19070
$765$	0	0
$766$	−7.59798	−0.274526
$767$	3.02944	0.109387
$768$	0	0
$769$	29.6985	1.07095	0.535477	−	0.844550i	$-0.320132\pi$
$769$	29.6985	1.07095	0.535477	+	0.844550i	$0.320132\pi$
$770$	0	0
$771$	0	0
$772$	−9.71573	−0.349677
$773$	9.55635	0.343718	0.171859	−	0.985122i	$-0.445023\pi$
$773$	9.55635	0.343718	0.171859	+	0.985122i	$0.445023\pi$
$774$	0	0
$775$	−7.79899	−0.280148
$776$	4.10051	0.147200
$777$	0	0
$778$	−7.51472	−0.269416
$779$	17.6569	0.632622
$780$	0	0
$781$	−18.6274	−0.666541
$782$	−7.11270	−0.254350
$783$	0	0
$784$	0	0
$785$	−40.1421	−1.43273
$786$	0	0
$787$	24.6863	0.879971	0.439986	−	0.898005i	$-0.354984\pi$
$787$	24.6863	0.879971	0.439986	+	0.898005i	$0.354984\pi$
$788$	3.65685	0.130270
$789$	0	0
$790$	19.3137	0.687151
$791$	0	0
$792$	0	0
$793$	31.6569	1.12417
$794$	0.987807	0.0350559
$795$	0	0
$796$	−39.5980	−1.40351
$797$	8.38478	0.297004	0.148502	−	0.988912i	$-0.452555\pi$
$797$	8.38478	0.297004	0.148502	+	0.988912i	$0.452555\pi$
$798$	0	0
$799$	6.34315	0.224404
$800$	−29.3848	−1.03891
$801$	0	0
$802$	−2.54416	−0.0898373
$803$	27.7990	0.981005
$804$	0	0
$805$	0	0
$806$	1.25483	0.0441996
$807$	0	0
$808$	−4.64466	−0.163399
$809$	−19.9411	−0.701093	−0.350546	−	0.936545i	$-0.614004\pi$
$809$	−19.9411	−0.701093	−0.350546	+	0.936545i	$0.614004\pi$
$810$	0	0
$811$	−17.6569	−0.620016	−0.310008	−	0.950734i	$-0.600332\pi$
$811$	−17.6569	−0.620016	−0.310008	+	0.950734i	$0.600332\pi$
$812$	0	0
$813$	0	0
$814$	3.31371	0.116145
$815$	38.6274	1.35306
$816$	0	0
$817$	−16.0000	−0.559769
$818$	8.87006	0.310134
$819$	0	0
$820$	−38.9706	−1.36091
$821$	10.6863	0.372954	0.186477	−	0.982459i	$-0.440293\pi$
$821$	10.6863	0.372954	0.186477	+	0.982459i	$0.440293\pi$
$822$	0	0
$823$	8.97056	0.312694	0.156347	−	0.987702i	$-0.450028\pi$
$823$	8.97056	0.312694	0.156347	+	0.987702i	$0.450028\pi$
$824$	7.11270	0.247783
$825$	0	0
$826$	0	0
$827$	−47.6569	−1.65719	−0.828596	−	0.559848i	$-0.810860\pi$
$827$	−47.6569	−1.65719	−0.828596	+	0.559848i	$0.810860\pi$
$828$	0	0
$829$	−0.727922	−0.0252818	−0.0126409	−	0.999920i	$-0.504024\pi$
$829$	−0.727922	−0.0252818	−0.0126409	+	0.999920i	$0.504024\pi$
$830$	−10.3431	−0.359016
$831$	0	0
$832$	−10.7868	−0.373965
$833$	0	0
$834$	0	0
$835$	−67.5980	−2.33932
$836$	10.3431	0.357725
$837$	0	0
$838$	13.7401	0.474644
$839$	−50.8284	−1.75479	−0.877396	−	0.479767i	$-0.840721\pi$
$839$	−50.8284	−1.75479	−0.877396	+	0.479767i	$0.840721\pi$
$840$	0	0
$841$	17.6274	0.607842
$842$	−6.88730	−0.237352
$843$	0	0
$844$	−23.7157	−0.816329
$845$	21.5563	0.741561
$846$	0	0
$847$	0	0
$848$	6.00000	0.206041
$849$	0	0
$850$	−6.18377	−0.212101
$851$	−30.6274	−1.04989
$852$	0	0
$853$	−49.4975	−1.69476	−0.847381	−	0.530986i	$-0.821822\pi$
$853$	−49.4975	−1.69476	−0.847381	+	0.530986i	$0.821822\pi$
$854$	0	0
$855$	0	0
$856$	0.544156	0.0185989
$857$	−15.4142	−0.526540	−0.263270	−	0.964722i	$-0.584801\pi$
$857$	−15.4142	−0.526540	−0.263270	+	0.964722i	$0.584801\pi$
$858$	0	0
$859$	57.4558	1.96037	0.980184	−	0.198089i	$-0.0634735\pi$
$859$	57.4558	1.96037	0.980184	+	0.198089i	$0.0634735\pi$
$860$	35.3137	1.20419
$861$	0	0
$862$	−11.1716	−0.380505
$863$	−17.3137	−0.589365	−0.294683	−	0.955595i	$-0.595214\pi$
$863$	−17.3137	−0.589365	−0.294683	+	0.955595i	$0.595214\pi$
$864$	0	0
$865$	71.9411	2.44607
$866$	8.38478	0.284926
$867$	0	0
$868$	0	0
$869$	27.3137	0.926554
$870$	0	0
$871$	−14.6274	−0.495631
$872$	−8.97056	−0.303782
$873$	0	0
$874$	8.97056	0.303434
$875$	0	0
$876$	0	0
$877$	−11.3137	−0.382037	−0.191018	−	0.981586i	$-0.561179\pi$
$877$	−11.3137	−0.382037	−0.191018	+	0.981586i	$0.561179\pi$
$878$	−5.25483	−0.177342
$879$	0	0
$880$	−20.4853	−0.690559
$881$	−21.7574	−0.733024	−0.366512	−	0.930413i	$-0.619448\pi$
$881$	−21.7574	−0.733024	−0.366512	+	0.930413i	$0.619448\pi$
$882$	0	0
$883$	−4.68629	−0.157706	−0.0788531	−	0.996886i	$-0.525126\pi$
$883$	−4.68629	−0.157706	−0.0788531	+	0.996886i	$0.525126\pi$
$884$	−10.6030	−0.356619
$885$	0	0
$886$	14.4853	0.486643
$887$	2.82843	0.0949693	0.0474846	−	0.998872i	$-0.484879\pi$
$887$	2.82843	0.0949693	0.0474846	+	0.998872i	$0.484879\pi$
$888$	0	0
$889$	0	0
$890$	20.1421	0.675166
$891$	0	0
$892$	−45.6569	−1.52870
$893$	−8.00000	−0.267710
$894$	0	0
$895$	−67.1127	−2.24333
$896$	0	0
$897$	0	0
$898$	−2.20101	−0.0734487
$899$	−8.00000	−0.266815
$900$	0	0
$901$	4.48528	0.149426
$902$	5.17157	0.172195
$903$	0	0
$904$	8.42641	0.280258
$905$	8.82843	0.293467
$906$	0	0
$907$	−16.0000	−0.531271	−0.265636	−	0.964073i	$-0.585582\pi$
$907$	−16.0000	−0.531271	−0.265636	+	0.964073i	$0.585582\pi$
$908$	43.5147	1.44409
$909$	0	0
$910$	0	0
$911$	1.02944	0.0341068	0.0170534	−	0.999855i	$-0.494571\pi$
$911$	1.02944	0.0341068	0.0170534	+	0.999855i	$0.494571\pi$
$912$	0	0
$913$	−14.6274	−0.484097
$914$	7.45584	0.246617
$915$	0	0
$916$	0.443651	0.0146586
$917$	0	0
$918$	0	0
$919$	−8.28427	−0.273273	−0.136636	−	0.990621i	$-0.543629\pi$
$919$	−8.28427	−0.273273	−0.136636	+	0.990621i	$0.543629\pi$
$920$	−41.4558	−1.36676
$921$	0	0
$922$	−6.87006	−0.226253
$923$	−24.0833	−0.792710
$924$	0	0
$925$	−26.6274	−0.875504
$926$	11.0294	0.362450
$927$	0	0
$928$	−30.1421	−0.989464
$929$	39.2132	1.28654	0.643272	−	0.765638i	$-0.277577\pi$
$929$	39.2132	1.28654	0.643272	+	0.765638i	$0.277577\pi$
$930$	0	0
$931$	0	0
$932$	−11.2304	−0.367866
$933$	0	0
$934$	0.0832611	0.00272439
$935$	−15.3137	−0.500812
$936$	0	0
$937$	30.5858	0.999194	0.499597	−	0.866258i	$-0.333481\pi$
$937$	30.5858	0.999194	0.499597	+	0.866258i	$0.333481\pi$
$938$	0	0
$939$	0	0
$940$	17.6569	0.575903
$941$	−35.2132	−1.14792	−0.573959	−	0.818884i	$-0.694593\pi$
$941$	−35.2132	−1.14792	−0.573959	+	0.818884i	$0.694593\pi$
$942$	0	0
$943$	−47.7990	−1.55655
$944$	3.51472	0.114394
$945$	0	0
$946$	−4.68629	−0.152364
$947$	−30.6863	−0.997170	−0.498585	−	0.866841i	$-0.666147\pi$
$947$	−30.6863	−0.997170	−0.498585	+	0.866841i	$0.666147\pi$
$948$	0	0
$949$	35.9411	1.16670
$950$	7.79899	0.253033
$951$	0	0
$952$	0	0
$953$	−2.00000	−0.0647864	−0.0323932	−	0.999475i	$-0.510313\pi$
$953$	−2.00000	−0.0647864	−0.0323932	+	0.999475i	$0.510313\pi$
$954$	0	0
$955$	−61.4558	−1.98866
$956$	−28.6274	−0.925877
$957$	0	0
$958$	0.769553	0.0248631
$959$	0	0
$960$	0	0
$961$	−29.6274	−0.955723
$962$	4.28427	0.138130
$963$	0	0
$964$	29.6985	0.956524
$965$	−18.1421	−0.584016
$966$	0	0
$967$	33.6569	1.08233	0.541166	−	0.840916i	$-0.317983\pi$
$967$	33.6569	1.08233	0.541166	+	0.840916i	$0.317983\pi$
$968$	−11.1005	−0.356784
$969$	0	0
$970$	3.65685	0.117415
$971$	50.6274	1.62471	0.812356	−	0.583162i	$-0.198185\pi$
$971$	50.6274	1.62471	0.812356	+	0.583162i	$0.198185\pi$
$972$	0	0
$973$	0	0
$974$	−11.0294	−0.353406
$975$	0	0
$976$	36.7279	1.17563
$977$	21.1716	0.677339	0.338669	−	0.940905i	$-0.390023\pi$
$977$	21.1716	0.677339	0.338669	+	0.940905i	$0.390023\pi$
$978$	0	0
$979$	28.4853	0.910394
$980$	0	0
$981$	0	0
$982$	2.08326	0.0664795
$983$	53.2548	1.69857	0.849283	−	0.527938i	$-0.177035\pi$
$983$	53.2548	1.69857	0.849283	+	0.527938i	$0.177035\pi$
$984$	0	0
$985$	6.82843	0.217572
$986$	−6.34315	−0.202007
$987$	0	0
$988$	13.3726	0.425439
$989$	43.3137	1.37730
$990$	0	0
$991$	−12.9706	−0.412024	−0.206012	−	0.978550i	$-0.566049\pi$
$991$	−12.9706	−0.412024	−0.206012	+	0.978550i	$0.566049\pi$
$992$	5.17157	0.164198
$993$	0	0
$994$	0	0
$995$	−73.9411	−2.34409
$996$	0	0
$997$	26.3848	0.835614	0.417807	−	0.908536i	$-0.362799\pi$
$997$	26.3848	0.835614	0.417807	+	0.908536i	$0.362799\pi$
$998$	−1.37258	−0.0434484
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.a.i.1.1		2
3.2	odd	2		147.2.a.e.1.2	yes	2
4.3	odd	2		7056.2.a.cf.1.1		2
7.2	even	3		441.2.e.g.361.2		4
7.3	odd	6		441.2.e.f.226.2		4
7.4	even	3		441.2.e.g.226.2		4
7.5	odd	6		441.2.e.f.361.2		4
7.6	odd	2		441.2.a.j.1.1		2
12.11	even	2		2352.2.a.bc.1.2		2
15.14	odd	2		3675.2.a.bd.1.1		2
21.2	odd	6		147.2.e.d.67.1		4
21.5	even	6		147.2.e.e.67.1		4
21.11	odd	6		147.2.e.d.79.1		4
21.17	even	6		147.2.e.e.79.1		4
21.20	even	2		147.2.a.d.1.2	✓	2
24.5	odd	2		9408.2.a.di.1.1		2
24.11	even	2		9408.2.a.dt.1.1		2
28.27	even	2		7056.2.a.cv.1.2		2
84.11	even	6		2352.2.q.bd.961.1		4
84.23	even	6		2352.2.q.bd.1537.1		4
84.47	odd	6		2352.2.q.bb.1537.2		4
84.59	odd	6		2352.2.q.bb.961.2		4
84.83	odd	2		2352.2.a.be.1.1		2
105.104	even	2		3675.2.a.bf.1.1		2
168.83	odd	2		9408.2.a.dq.1.2		2
168.125	even	2		9408.2.a.ef.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.2.a.d.1.2	✓	2	21.20	even	2
147.2.a.e.1.2	yes	2	3.2	odd	2
147.2.e.d.67.1		4	21.2	odd	6
147.2.e.d.79.1		4	21.11	odd	6
147.2.e.e.67.1		4	21.5	even	6
147.2.e.e.79.1		4	21.17	even	6
441.2.a.i.1.1		2	1.1	even	1	trivial
441.2.a.j.1.1		2	7.6	odd	2
441.2.e.f.226.2		4	7.3	odd	6
441.2.e.f.361.2		4	7.5	odd	6
441.2.e.g.226.2		4	7.4	even	3
441.2.e.g.361.2		4	7.2	even	3
2352.2.a.bc.1.2		2	12.11	even	2
2352.2.a.be.1.1		2	84.83	odd	2
2352.2.q.bb.961.2		4	84.59	odd	6
2352.2.q.bb.1537.2		4	84.47	odd	6
2352.2.q.bd.961.1		4	84.11	even	6
2352.2.q.bd.1537.1		4	84.23	even	6
3675.2.a.bd.1.1		2	15.14	odd	2
3675.2.a.bf.1.1		2	105.104	even	2
7056.2.a.cf.1.1		2	4.3	odd	2
7056.2.a.cv.1.2		2	28.27	even	2
9408.2.a.di.1.1		2	24.5	odd	2
9408.2.a.dq.1.2		2	168.83	odd	2
9408.2.a.dt.1.1		2	24.11	even	2
9408.2.a.ef.1.2		2	168.125	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	441.a (trivial)

\(f(q)\)	\(=\)	\(q-0.414214 q^{2} -1.82843 q^{4} -3.41421 q^{5} +1.58579 q^{8} +1.41421 q^{10} +2.00000 q^{11} +2.58579 q^{13} +3.00000 q^{16} +2.24264 q^{17} -2.82843 q^{19} +6.24264 q^{20} -0.828427 q^{22} +7.65685 q^{23} +6.65685 q^{25} -1.07107 q^{26} +6.82843 q^{29} -1.17157 q^{31} -4.41421 q^{32} -0.928932 q^{34} -4.00000 q^{37} +1.17157 q^{38} -5.41421 q^{40} -6.24264 q^{41} +5.65685 q^{43} -3.65685 q^{44} -3.17157 q^{46} +2.82843 q^{47} -2.75736 q^{50} -4.72792 q^{52} +2.00000 q^{53} -6.82843 q^{55} -2.82843 q^{58} +1.17157 q^{59} +12.2426 q^{61} +0.485281 q^{62} -4.17157 q^{64} -8.82843 q^{65} -5.65685 q^{67} -4.10051 q^{68} -9.31371 q^{71} +13.8995 q^{73} +1.65685 q^{74} +5.17157 q^{76} +13.6569 q^{79} -10.2426 q^{80} +2.58579 q^{82} -7.31371 q^{83} -7.65685 q^{85} -2.34315 q^{86} +3.17157 q^{88} +14.2426 q^{89} -14.0000 q^{92} -1.17157 q^{94} +9.65685 q^{95} +2.58579 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - 4 q^{5} + 6 q^{8} + 4 q^{11} + 8 q^{13} + 6 q^{16} - 4 q^{17} + 4 q^{20} + 4 q^{22} + 4 q^{23} + 2 q^{25} + 12 q^{26} + 8 q^{29} - 8 q^{31} - 6 q^{32} - 16 q^{34} - 8 q^{37} + 8 q^{38}+ \cdots + 8 q^{97}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 - 4 * q^5 + 6 * q^8 + 4 * q^11 + 8 * q^13 + 6 * q^16 - 4 * q^17 + 4 * q^20 + 4 * q^22 + 4 * q^23 + 2 * q^25 + 12 * q^26 + 8 * q^29 - 8 * q^31 - 6 * q^32 - 16 * q^34 - 8 * q^37 + 8 * q^38 - 8 * q^40 - 4 * q^41 + 4 * q^44 - 12 * q^46 - 14 * q^50 + 16 * q^52 + 4 * q^53 - 8 * q^55 + 8 * q^59 + 16 * q^61 - 16 * q^62 - 14 * q^64 - 12 * q^65 - 28 * q^68 + 4 * q^71 + 8 * q^73 - 8 * q^74 + 16 * q^76 + 16 * q^79 - 12 * q^80 + 8 * q^82 + 8 * q^83 - 4 * q^85 - 16 * q^86 + 12 * q^88 + 20 * q^89 - 28 * q^92 - 8 * q^94 + 8 * q^95 + 8 * q^97

Introduction

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