LMFDB - Embedded newform 9522.2.a.v.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9522,2,Mod(1,9522)] 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("9522.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9522, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9522 = 2 \cdot 3^{2} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9522.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,-2,0,2,0,0,0,-2,0,0,0,0,-4,0,0,2,0,0,0,0,0,0,0,0,2,4,0,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(29)] == traces)

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

Self dual:	yes
Analytic conductor:	$76.0335528047$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.44949$ of defining polynomial
Character	$\chi$	$=$	9522.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -2.44949 q^{5} -1.00000 q^{8} +2.44949 q^{10} +2.44949 q^{11} -2.00000 q^{13} +1.00000 q^{16} -4.89898 q^{17} -7.34847 q^{19} -2.44949 q^{20} -2.44949 q^{22} +1.00000 q^{25} +2.00000 q^{26} +6.00000 q^{29} -2.00000 q^{31} -1.00000 q^{32} +4.89898 q^{34} -7.34847 q^{37} +7.34847 q^{38} +2.44949 q^{40} +6.00000 q^{41} -7.34847 q^{43} +2.44949 q^{44} +6.00000 q^{47} -7.00000 q^{49} -1.00000 q^{50} -2.00000 q^{52} +2.44949 q^{53} -6.00000 q^{55} -6.00000 q^{58} -7.34847 q^{61} +2.00000 q^{62} +1.00000 q^{64} +4.89898 q^{65} +7.34847 q^{67} -4.89898 q^{68} -6.00000 q^{71} -8.00000 q^{73} +7.34847 q^{74} -7.34847 q^{76} -2.44949 q^{80} -6.00000 q^{82} -7.34847 q^{83} +12.0000 q^{85} +7.34847 q^{86} -2.44949 q^{88} +9.79796 q^{89} -6.00000 q^{94} +18.0000 q^{95} +14.6969 q^{97} +7.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 4 q^{13} + 2 q^{16} + 2 q^{25} + 4 q^{26} + 12 q^{29} - 4 q^{31} - 2 q^{32} + 12 q^{41} + 12 q^{47} - 14 q^{49} - 2 q^{50} - 4 q^{52} - 12 q^{55} - 12 q^{58} + 4 q^{62}+ \cdots + 14 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 - 4 * q^13 + 2 * q^16 + 2 * q^25 + 4 * q^26 + 12 * q^29 - 4 * q^31 - 2 * q^32 + 12 * q^41 + 12 * q^47 - 14 * q^49 - 2 * q^50 - 4 * q^52 - 12 * q^55 - 12 * q^58 + 4 * q^62 + 2 * q^64 - 12 * q^71 - 16 * q^73 - 12 * q^82 + 24 * q^85 - 12 * q^94 + 36 * q^95 + 14 * q^98

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−2.44949	−1.09545	−0.547723	−	0.836660i	$-0.684505\pi$
$5$	−2.44949	−1.09545	−0.547723	+	0.836660i	$0.684505\pi$
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	2.44949	0.774597
$11$	2.44949	0.738549	0.369274	−	0.929320i	$-0.379606\pi$
$11$	2.44949	0.738549	0.369274	+	0.929320i	$0.379606\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.89898	−1.18818	−0.594089	−	0.804400i	$-0.702487\pi$
$17$	−4.89898	−1.18818	−0.594089	+	0.804400i	$0.702487\pi$
$18$	0	0
$19$	−7.34847	−1.68585	−0.842927	−	0.538028i	$-0.819170\pi$
$19$	−7.34847	−1.68585	−0.842927	+	0.538028i	$0.819170\pi$
$20$	−2.44949	−0.547723
$21$	0	0
$22$	−2.44949	−0.522233
$23$	0	0
$23$	0	0
$24$	0	0
$25$	1.00000	0.200000
$26$	2.00000	0.392232
$27$	0	0
$28$	0	0
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	4.89898	0.840168
$35$	0	0
$36$	0	0
$37$	−7.34847	−1.20808	−0.604040	−	0.796954i	$-0.706443\pi$
$37$	−7.34847	−1.20808	−0.604040	+	0.796954i	$0.706443\pi$
$38$	7.34847	1.19208
$39$	0	0
$40$	2.44949	0.387298
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	−7.34847	−1.12063	−0.560316	−	0.828279i	$-0.689320\pi$
$43$	−7.34847	−1.12063	−0.560316	+	0.828279i	$0.689320\pi$
$44$	2.44949	0.369274
$45$	0	0
$46$	0	0
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	−1.00000	−0.141421
$51$	0	0
$52$	−2.00000	−0.277350
$53$	2.44949	0.336463	0.168232	−	0.985747i	$-0.446194\pi$
$53$	2.44949	0.336463	0.168232	+	0.985747i	$0.446194\pi$
$54$	0	0
$55$	−6.00000	−0.809040
$56$	0	0
$57$	0	0
$58$	−6.00000	−0.787839
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−7.34847	−0.940875	−0.470438	−	0.882433i	$-0.655904\pi$
$61$	−7.34847	−0.940875	−0.470438	+	0.882433i	$0.655904\pi$
$62$	2.00000	0.254000
$63$	0	0
$64$	1.00000	0.125000
$65$	4.89898	0.607644
$66$	0	0
$67$	7.34847	0.897758	0.448879	−	0.893592i	$-0.351823\pi$
$67$	7.34847	0.897758	0.448879	+	0.893592i	$0.351823\pi$
$68$	−4.89898	−0.594089
$69$	0	0
$70$	0	0
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	0	0
$73$	−8.00000	−0.936329	−0.468165	−	0.883641i	$-0.655085\pi$
$73$	−8.00000	−0.936329	−0.468165	+	0.883641i	$0.655085\pi$
$74$	7.34847	0.854242
$75$	0	0
$76$	−7.34847	−0.842927
$77$	0	0
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	−2.44949	−0.273861
$81$	0	0
$82$	−6.00000	−0.662589
$83$	−7.34847	−0.806599	−0.403300	−	0.915068i	$-0.632137\pi$
$83$	−7.34847	−0.806599	−0.403300	+	0.915068i	$0.632137\pi$
$84$	0	0
$85$	12.0000	1.30158
$86$	7.34847	0.792406
$87$	0	0
$88$	−2.44949	−0.261116
$89$	9.79796	1.03858	0.519291	−	0.854598i	$-0.326196\pi$
$89$	9.79796	1.03858	0.519291	+	0.854598i	$0.326196\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	−6.00000	−0.618853
$95$	18.0000	1.84676
$96$	0	0
$97$	14.6969	1.49225	0.746124	−	0.665807i	$-0.231913\pi$
$97$	14.6969	1.49225	0.746124	+	0.665807i	$0.231913\pi$
$98$	7.00000	0.707107
$99$	0	0
$100$	1.00000	0.100000
$101$	−18.0000	−1.79107	−0.895533	−	0.444994i	$-0.853206\pi$
$101$	−18.0000	−1.79107	−0.895533	+	0.444994i	$0.853206\pi$
$102$	0	0
$103$	14.6969	1.44813	0.724066	−	0.689730i	$-0.242271\pi$
$103$	14.6969	1.44813	0.724066	+	0.689730i	$0.242271\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	−2.44949	−0.237915
$107$	−17.1464	−1.65761	−0.828804	−	0.559539i	$-0.810978\pi$
$107$	−17.1464	−1.65761	−0.828804	+	0.559539i	$0.810978\pi$
$108$	0	0
$109$	−7.34847	−0.703856	−0.351928	−	0.936027i	$-0.614474\pi$
$109$	−7.34847	−0.703856	−0.351928	+	0.936027i	$0.614474\pi$
$110$	6.00000	0.572078
$111$	0	0
$112$	0	0
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	0	0
$116$	6.00000	0.557086
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−5.00000	−0.454545
$122$	7.34847	0.665299
$123$	0	0
$124$	−2.00000	−0.179605
$125$	9.79796	0.876356
$126$	0	0
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−4.89898	−0.429669
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	0	0
$133$	0	0
$134$	−7.34847	−0.634811
$135$	0	0
$136$	4.89898	0.420084
$137$	14.6969	1.25564	0.627822	−	0.778357i	$-0.283947\pi$
$137$	14.6969	1.25564	0.627822	+	0.778357i	$0.283947\pi$
$138$	0	0
$139$	20.0000	1.69638	0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000	1.69638	0.848189	+	0.529694i	$0.177693\pi$
$140$	0	0
$141$	0	0
$142$	6.00000	0.503509
$143$	−4.89898	−0.409673
$144$	0	0
$145$	−14.6969	−1.22051
$146$	8.00000	0.662085
$147$	0	0
$148$	−7.34847	−0.604040
$149$	−7.34847	−0.602010	−0.301005	−	0.953623i	$-0.597322\pi$
$149$	−7.34847	−0.602010	−0.301005	+	0.953623i	$0.597322\pi$
$150$	0	0
$151$	−10.0000	−0.813788	−0.406894	−	0.913475i	$-0.633388\pi$
$151$	−10.0000	−0.813788	−0.406894	+	0.913475i	$0.633388\pi$
$152$	7.34847	0.596040
$153$	0	0
$154$	0	0
$155$	4.89898	0.393496
$156$	0	0
$157$	7.34847	0.586472	0.293236	−	0.956040i	$-0.405268\pi$
$157$	7.34847	0.586472	0.293236	+	0.956040i	$0.405268\pi$
$158$	0	0
$159$	0	0
$160$	2.44949	0.193649
$161$	0	0
$162$	0	0
$163$	16.0000	1.25322	0.626608	−	0.779334i	$-0.284443\pi$
$163$	16.0000	1.25322	0.626608	+	0.779334i	$0.284443\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	7.34847	0.570352
$167$	18.0000	1.39288	0.696441	−	0.717614i	$-0.254766\pi$
$167$	18.0000	1.39288	0.696441	+	0.717614i	$0.254766\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	−12.0000	−0.920358
$171$	0	0
$172$	−7.34847	−0.560316
$173$	18.0000	1.36851	0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000	1.36851	0.684257	+	0.729241i	$0.260127\pi$
$174$	0	0
$175$	0	0
$176$	2.44949	0.184637
$177$	0	0
$178$	−9.79796	−0.734388
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	7.34847	0.546207	0.273104	−	0.961985i	$-0.411950\pi$
$181$	7.34847	0.546207	0.273104	+	0.961985i	$0.411950\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	18.0000	1.32339
$186$	0	0
$187$	−12.0000	−0.877527
$188$	6.00000	0.437595
$189$	0	0
$190$	−18.0000	−1.30586
$191$	−24.4949	−1.77239	−0.886194	−	0.463314i	$-0.846660\pi$
$191$	−24.4949	−1.77239	−0.886194	+	0.463314i	$0.846660\pi$
$192$	0	0
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	−14.6969	−1.05518
$195$	0	0
$196$	−7.00000	−0.500000
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	18.0000	1.26648
$203$	0	0
$204$	0	0
$205$	−14.6969	−1.02648
$206$	−14.6969	−1.02398
$207$	0	0
$208$	−2.00000	−0.138675
$209$	−18.0000	−1.24509
$210$	0	0
$211$	−20.0000	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$212$	2.44949	0.168232
$213$	0	0
$214$	17.1464	1.17211
$215$	18.0000	1.22759
$216$	0	0
$217$	0	0
$218$	7.34847	0.497701
$219$	0	0
$220$	−6.00000	−0.404520
$221$	9.79796	0.659082
$222$	0	0
$223$	14.0000	0.937509	0.468755	−	0.883328i	$-0.344703\pi$
$223$	14.0000	0.937509	0.468755	+	0.883328i	$0.344703\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−26.9444	−1.78836	−0.894181	−	0.447706i	$-0.852241\pi$
$227$	−26.9444	−1.78836	−0.894181	+	0.447706i	$0.852241\pi$
$228$	0	0
$229$	−22.0454	−1.45680	−0.728401	−	0.685151i	$-0.759736\pi$
$229$	−22.0454	−1.45680	−0.728401	+	0.685151i	$0.759736\pi$
$230$	0	0
$231$	0	0
$232$	−6.00000	−0.393919
$233$	24.0000	1.57229	0.786146	−	0.618041i	$-0.212073\pi$
$233$	24.0000	1.57229	0.786146	+	0.618041i	$0.212073\pi$
$234$	0	0
$235$	−14.6969	−0.958723
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	0	0
$241$	−14.6969	−0.946713	−0.473357	−	0.880871i	$-0.656958\pi$
$241$	−14.6969	−0.946713	−0.473357	+	0.880871i	$0.656958\pi$
$242$	5.00000	0.321412
$243$	0	0
$244$	−7.34847	−0.470438
$245$	17.1464	1.09545
$246$	0	0
$247$	14.6969	0.935144
$248$	2.00000	0.127000
$249$	0	0
$250$	−9.79796	−0.619677
$251$	−17.1464	−1.08227	−0.541136	−	0.840935i	$-0.682006\pi$
$251$	−17.1464	−1.08227	−0.541136	+	0.840935i	$0.682006\pi$
$252$	0	0
$253$	0	0
$254$	8.00000	0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	0	0
$259$	0	0
$260$	4.89898	0.303822
$261$	0	0
$262$	−12.0000	−0.741362
$263$	−9.79796	−0.604168	−0.302084	−	0.953281i	$-0.597682\pi$
$263$	−9.79796	−0.604168	−0.302084	+	0.953281i	$0.597682\pi$
$264$	0	0
$265$	−6.00000	−0.368577
$266$	0	0
$267$	0	0
$268$	7.34847	0.448879
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	0	0
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	−4.89898	−0.297044
$273$	0	0
$274$	−14.6969	−0.887875
$275$	2.44949	0.147710
$276$	0	0
$277$	22.0000	1.32185	0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000	1.32185	0.660926	+	0.750451i	$0.270164\pi$
$278$	−20.0000	−1.19952
$279$	0	0
$280$	0	0
$281$	−9.79796	−0.584497	−0.292249	−	0.956342i	$-0.594403\pi$
$281$	−9.79796	−0.584497	−0.292249	+	0.956342i	$0.594403\pi$
$282$	0	0
$283$	7.34847	0.436821	0.218411	−	0.975857i	$-0.429913\pi$
$283$	7.34847	0.436821	0.218411	+	0.975857i	$0.429913\pi$
$284$	−6.00000	−0.356034
$285$	0	0
$286$	4.89898	0.289683
$287$	0	0
$288$	0	0
$289$	7.00000	0.411765
$290$	14.6969	0.863034
$291$	0	0
$292$	−8.00000	−0.468165
$293$	−17.1464	−1.00171	−0.500853	−	0.865533i	$-0.666980\pi$
$293$	−17.1464	−1.00171	−0.500853	+	0.865533i	$0.666980\pi$
$294$	0	0
$295$	0	0
$296$	7.34847	0.427121
$297$	0	0
$298$	7.34847	0.425685
$299$	0	0
$300$	0	0
$301$	0	0
$302$	10.0000	0.575435
$303$	0	0
$304$	−7.34847	−0.421464
$305$	18.0000	1.03068
$306$	0	0
$307$	16.0000	0.913168	0.456584	−	0.889680i	$-0.349073\pi$
$307$	16.0000	0.913168	0.456584	+	0.889680i	$0.349073\pi$
$308$	0	0
$309$	0	0
$310$	−4.89898	−0.278243
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	0	0
$313$	14.6969	0.830720	0.415360	−	0.909657i	$-0.363656\pi$
$313$	14.6969	0.830720	0.415360	+	0.909657i	$0.363656\pi$
$314$	−7.34847	−0.414698
$315$	0	0
$316$	0	0
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	0	0
$319$	14.6969	0.822871
$320$	−2.44949	−0.136931
$321$	0	0
$322$	0	0
$323$	36.0000	2.00309
$324$	0	0
$325$	−2.00000	−0.110940
$326$	−16.0000	−0.886158
$327$	0	0
$328$	−6.00000	−0.331295
$329$	0	0
$330$	0	0
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	−7.34847	−0.403300
$333$	0	0
$334$	−18.0000	−0.984916
$335$	−18.0000	−0.983445
$336$	0	0
$337$	29.3939	1.60119	0.800593	−	0.599208i	$-0.204518\pi$
$337$	29.3939	1.60119	0.800593	+	0.599208i	$0.204518\pi$
$338$	9.00000	0.489535
$339$	0	0
$340$	12.0000	0.650791
$341$	−4.89898	−0.265295
$342$	0	0
$343$	0	0
$344$	7.34847	0.396203
$345$	0	0
$346$	−18.0000	−0.967686
$347$	−36.0000	−1.93258	−0.966291	−	0.257454i	$-0.917117\pi$
$347$	−36.0000	−1.93258	−0.966291	+	0.257454i	$0.917117\pi$
$348$	0	0
$349$	−22.0000	−1.17763	−0.588817	−	0.808267i	$-0.700406\pi$
$349$	−22.0000	−1.17763	−0.588817	+	0.808267i	$0.700406\pi$
$350$	0	0
$351$	0	0
$352$	−2.44949	−0.130558
$353$	24.0000	1.27739	0.638696	−	0.769460i	$-0.279474\pi$
$353$	24.0000	1.27739	0.638696	+	0.769460i	$0.279474\pi$
$354$	0	0
$355$	14.6969	0.780033
$356$	9.79796	0.519291
$357$	0	0
$358$	−12.0000	−0.634220
$359$	−24.4949	−1.29279	−0.646396	−	0.763002i	$-0.723724\pi$
$359$	−24.4949	−1.29279	−0.646396	+	0.763002i	$0.723724\pi$
$360$	0	0
$361$	35.0000	1.84211
$362$	−7.34847	−0.386227
$363$	0	0
$364$	0	0
$365$	19.5959	1.02570
$366$	0	0
$367$	29.3939	1.53435	0.767174	−	0.641439i	$-0.221662\pi$
$367$	29.3939	1.53435	0.767174	+	0.641439i	$0.221662\pi$
$368$	0	0
$369$	0	0
$370$	−18.0000	−0.935775
$371$	0	0
$372$	0	0
$373$	−22.0454	−1.14147	−0.570734	−	0.821135i	$-0.693341\pi$
$373$	−22.0454	−1.14147	−0.570734	+	0.821135i	$0.693341\pi$
$374$	12.0000	0.620505
$375$	0	0
$376$	−6.00000	−0.309426
$377$	−12.0000	−0.618031
$378$	0	0
$379$	−22.0454	−1.13240	−0.566198	−	0.824269i	$-0.691586\pi$
$379$	−22.0454	−1.13240	−0.566198	+	0.824269i	$0.691586\pi$
$380$	18.0000	0.923381
$381$	0	0
$382$	24.4949	1.25327
$383$	−9.79796	−0.500652	−0.250326	−	0.968162i	$-0.580538\pi$
$383$	−9.79796	−0.500652	−0.250326	+	0.968162i	$0.580538\pi$
$384$	0	0
$385$	0	0
$386$	−2.00000	−0.101797
$387$	0	0
$388$	14.6969	0.746124
$389$	12.2474	0.620970	0.310485	−	0.950578i	$-0.399508\pi$
$389$	12.2474	0.620970	0.310485	+	0.950578i	$0.399508\pi$
$390$	0	0
$391$	0	0
$392$	7.00000	0.353553
$393$	0	0
$394$	6.00000	0.302276
$395$	0	0
$396$	0	0
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	0	0
$399$	0	0
$400$	1.00000	0.0500000
$401$	34.2929	1.71250	0.856252	−	0.516559i	$-0.172787\pi$
$401$	34.2929	1.71250	0.856252	+	0.516559i	$0.172787\pi$
$402$	0	0
$403$	4.00000	0.199254
$404$	−18.0000	−0.895533
$405$	0	0
$406$	0	0
$407$	−18.0000	−0.892227
$408$	0	0
$409$	−22.0000	−1.08783	−0.543915	−	0.839140i	$-0.683059\pi$
$409$	−22.0000	−1.08783	−0.543915	+	0.839140i	$0.683059\pi$
$410$	14.6969	0.725830
$411$	0	0
$412$	14.6969	0.724066
$413$	0	0
$414$	0	0
$415$	18.0000	0.883585
$416$	2.00000	0.0980581
$417$	0	0
$418$	18.0000	0.880409
$419$	2.44949	0.119665	0.0598327	−	0.998208i	$-0.480943\pi$
$419$	2.44949	0.119665	0.0598327	+	0.998208i	$0.480943\pi$
$420$	0	0
$421$	36.7423	1.79071	0.895356	−	0.445351i	$-0.146921\pi$
$421$	36.7423	1.79071	0.895356	+	0.445351i	$0.146921\pi$
$422$	20.0000	0.973585
$423$	0	0
$424$	−2.44949	−0.118958
$425$	−4.89898	−0.237635
$426$	0	0
$427$	0	0
$428$	−17.1464	−0.828804
$429$	0	0
$430$	−18.0000	−0.868037
$431$	34.2929	1.65183	0.825914	−	0.563796i	$-0.190659\pi$
$431$	34.2929	1.65183	0.825914	+	0.563796i	$0.190659\pi$
$432$	0	0
$433$	14.6969	0.706290	0.353145	−	0.935569i	$-0.385112\pi$
$433$	14.6969	0.706290	0.353145	+	0.935569i	$0.385112\pi$
$434$	0	0
$435$	0	0
$436$	−7.34847	−0.351928
$437$	0	0
$438$	0	0
$439$	32.0000	1.52728	0.763638	−	0.645644i	$-0.223411\pi$
$439$	32.0000	1.52728	0.763638	+	0.645644i	$0.223411\pi$
$440$	6.00000	0.286039
$441$	0	0
$442$	−9.79796	−0.466041
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	0	0
$445$	−24.0000	−1.13771
$446$	−14.0000	−0.662919
$447$	0	0
$448$	0	0
$449$	18.0000	0.849473	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	0	0
$451$	14.6969	0.692052
$452$	0	0
$453$	0	0
$454$	26.9444	1.26456
$455$	0	0
$456$	0	0
$457$	−14.6969	−0.687494	−0.343747	−	0.939062i	$-0.611696\pi$
$457$	−14.6969	−0.687494	−0.343747	+	0.939062i	$0.611696\pi$
$458$	22.0454	1.03011
$459$	0	0
$460$	0	0
$461$	30.0000	1.39724	0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000	1.39724	0.698620	+	0.715493i	$0.253798\pi$
$462$	0	0
$463$	−16.0000	−0.743583	−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000	−0.743583	−0.371792	+	0.928316i	$0.621256\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	−24.0000	−1.11178
$467$	36.7423	1.70023	0.850117	−	0.526595i	$-0.176531\pi$
$467$	36.7423	1.70023	0.850117	+	0.526595i	$0.176531\pi$
$468$	0	0
$469$	0	0
$470$	14.6969	0.677919
$471$	0	0
$472$	0	0
$473$	−18.0000	−0.827641
$474$	0	0
$475$	−7.34847	−0.337171
$476$	0	0
$477$	0	0
$478$	24.0000	1.09773
$479$	9.79796	0.447680	0.223840	−	0.974626i	$-0.428141\pi$
$479$	9.79796	0.447680	0.223840	+	0.974626i	$0.428141\pi$
$480$	0	0
$481$	14.6969	0.670123
$482$	14.6969	0.669427
$483$	0	0
$484$	−5.00000	−0.227273
$485$	−36.0000	−1.63468
$486$	0	0
$487$	8.00000	0.362515	0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000	0.362515	0.181257	+	0.983436i	$0.441983\pi$
$488$	7.34847	0.332650
$489$	0	0
$490$	−17.1464	−0.774597
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	0	0
$493$	−29.3939	−1.32383
$494$	−14.6969	−0.661247
$495$	0	0
$496$	−2.00000	−0.0898027
$497$	0	0
$498$	0	0
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	9.79796	0.438178
$501$	0	0
$502$	17.1464	0.765283
$503$	4.89898	0.218435	0.109217	−	0.994018i	$-0.465166\pi$
$503$	4.89898	0.218435	0.109217	+	0.994018i	$0.465166\pi$
$504$	0	0
$505$	44.0908	1.96202
$506$	0	0
$507$	0	0
$508$	−8.00000	−0.354943
$509$	30.0000	1.32973	0.664863	−	0.746965i	$-0.268490\pi$
$509$	30.0000	1.32973	0.664863	+	0.746965i	$0.268490\pi$
$510$	0	0
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	6.00000	0.264649
$515$	−36.0000	−1.58635
$516$	0	0
$517$	14.6969	0.646371
$518$	0	0
$519$	0	0
$520$	−4.89898	−0.214834
$521$	14.6969	0.643885	0.321942	−	0.946759i	$-0.395664\pi$
$521$	14.6969	0.643885	0.321942	+	0.946759i	$0.395664\pi$
$522$	0	0
$523$	−7.34847	−0.321326	−0.160663	−	0.987009i	$-0.551363\pi$
$523$	−7.34847	−0.321326	−0.160663	+	0.987009i	$0.551363\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	9.79796	0.427211
$527$	9.79796	0.426806
$528$	0	0
$529$	0	0
$530$	6.00000	0.260623
$531$	0	0
$532$	0	0
$533$	−12.0000	−0.519778
$534$	0	0
$535$	42.0000	1.81582
$536$	−7.34847	−0.317406
$537$	0	0
$538$	−6.00000	−0.258678
$539$	−17.1464	−0.738549
$540$	0	0
$541$	38.0000	1.63375	0.816874	−	0.576816i	$-0.195705\pi$
$541$	38.0000	1.63375	0.816874	+	0.576816i	$0.195705\pi$
$542$	8.00000	0.343629
$543$	0	0
$544$	4.89898	0.210042
$545$	18.0000	0.771035
$546$	0	0
$547$	−8.00000	−0.342055	−0.171028	−	0.985266i	$-0.554709\pi$
$547$	−8.00000	−0.342055	−0.171028	+	0.985266i	$0.554709\pi$
$548$	14.6969	0.627822
$549$	0	0
$550$	−2.44949	−0.104447
$551$	−44.0908	−1.87833
$552$	0	0
$553$	0	0
$554$	−22.0000	−0.934690
$555$	0	0
$556$	20.0000	0.848189
$557$	36.7423	1.55682	0.778412	−	0.627754i	$-0.216026\pi$
$557$	36.7423	1.55682	0.778412	+	0.627754i	$0.216026\pi$
$558$	0	0
$559$	14.6969	0.621614
$560$	0	0
$561$	0	0
$562$	9.79796	0.413302
$563$	−2.44949	−0.103234	−0.0516168	−	0.998667i	$-0.516437\pi$
$563$	−2.44949	−0.103234	−0.0516168	+	0.998667i	$0.516437\pi$
$564$	0	0
$565$	0	0
$566$	−7.34847	−0.308879
$567$	0	0
$568$	6.00000	0.251754
$569$	−34.2929	−1.43763	−0.718816	−	0.695201i	$-0.755315\pi$
$569$	−34.2929	−1.43763	−0.718816	+	0.695201i	$0.755315\pi$
$570$	0	0
$571$	22.0454	0.922572	0.461286	−	0.887252i	$-0.347388\pi$
$571$	22.0454	0.922572	0.461286	+	0.887252i	$0.347388\pi$
$572$	−4.89898	−0.204837
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	8.00000	0.333044	0.166522	−	0.986038i	$-0.446746\pi$
$577$	8.00000	0.333044	0.166522	+	0.986038i	$0.446746\pi$
$578$	−7.00000	−0.291162
$579$	0	0
$580$	−14.6969	−0.610257
$581$	0	0
$582$	0	0
$583$	6.00000	0.248495
$584$	8.00000	0.331042
$585$	0	0
$586$	17.1464	0.708312
$587$	48.0000	1.98117	0.990586	−	0.136892i	$-0.0437113\pi$
$587$	48.0000	1.98117	0.990586	+	0.136892i	$0.0437113\pi$
$588$	0	0
$589$	14.6969	0.605577
$590$	0	0
$591$	0	0
$592$	−7.34847	−0.302020
$593$	−18.0000	−0.739171	−0.369586	−	0.929197i	$-0.620500\pi$
$593$	−18.0000	−0.739171	−0.369586	+	0.929197i	$0.620500\pi$
$594$	0	0
$595$	0	0
$596$	−7.34847	−0.301005
$597$	0	0
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	8.00000	0.326327	0.163163	−	0.986599i	$-0.447830\pi$
$601$	8.00000	0.326327	0.163163	+	0.986599i	$0.447830\pi$
$602$	0	0
$603$	0	0
$604$	−10.0000	−0.406894
$605$	12.2474	0.497930
$606$	0	0
$607$	2.00000	0.0811775	0.0405887	−	0.999176i	$-0.487077\pi$
$607$	2.00000	0.0811775	0.0405887	+	0.999176i	$0.487077\pi$
$608$	7.34847	0.298020
$609$	0	0
$610$	−18.0000	−0.728799
$611$	−12.0000	−0.485468
$612$	0	0
$613$	−22.0454	−0.890406	−0.445203	−	0.895430i	$-0.646868\pi$
$613$	−22.0454	−0.890406	−0.445203	+	0.895430i	$0.646868\pi$
$614$	−16.0000	−0.645707
$615$	0	0
$616$	0	0
$617$	−29.3939	−1.18335	−0.591676	−	0.806176i	$-0.701534\pi$
$617$	−29.3939	−1.18335	−0.591676	+	0.806176i	$0.701534\pi$
$618$	0	0
$619$	−7.34847	−0.295360	−0.147680	−	0.989035i	$-0.547181\pi$
$619$	−7.34847	−0.295360	−0.147680	+	0.989035i	$0.547181\pi$
$620$	4.89898	0.196748
$621$	0	0
$622$	−24.0000	−0.962312
$623$	0	0
$624$	0	0
$625$	−29.0000	−1.16000
$626$	−14.6969	−0.587408
$627$	0	0
$628$	7.34847	0.293236
$629$	36.0000	1.43541
$630$	0	0
$631$	−14.6969	−0.585076	−0.292538	−	0.956254i	$-0.594500\pi$
$631$	−14.6969	−0.585076	−0.292538	+	0.956254i	$0.594500\pi$
$632$	0	0
$633$	0	0
$634$	−18.0000	−0.714871
$635$	19.5959	0.777640
$636$	0	0
$637$	14.0000	0.554700
$638$	−14.6969	−0.581857
$639$	0	0
$640$	2.44949	0.0968246
$641$	14.6969	0.580494	0.290247	−	0.956952i	$-0.406263\pi$
$641$	14.6969	0.580494	0.290247	+	0.956952i	$0.406263\pi$
$642$	0	0
$643$	−36.7423	−1.44898	−0.724488	−	0.689287i	$-0.757924\pi$
$643$	−36.7423	−1.44898	−0.724488	+	0.689287i	$0.757924\pi$
$644$	0	0
$645$	0	0
$646$	−36.0000	−1.41640
$647$	24.0000	0.943537	0.471769	−	0.881722i	$-0.343616\pi$
$647$	24.0000	0.943537	0.471769	+	0.881722i	$0.343616\pi$
$648$	0	0
$649$	0	0
$650$	2.00000	0.0784465
$651$	0	0
$652$	16.0000	0.626608
$653$	−6.00000	−0.234798	−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000	−0.234798	−0.117399	+	0.993085i	$0.537456\pi$
$654$	0	0
$655$	−29.3939	−1.14851
$656$	6.00000	0.234261
$657$	0	0
$658$	0	0
$659$	−2.44949	−0.0954186	−0.0477093	−	0.998861i	$-0.515192\pi$
$659$	−2.44949	−0.0954186	−0.0477093	+	0.998861i	$0.515192\pi$
$660$	0	0
$661$	7.34847	0.285822	0.142911	−	0.989736i	$-0.454354\pi$
$661$	7.34847	0.285822	0.142911	+	0.989736i	$0.454354\pi$
$662$	−4.00000	−0.155464
$663$	0	0
$664$	7.34847	0.285176
$665$	0	0
$666$	0	0
$667$	0	0
$668$	18.0000	0.696441
$669$	0	0
$670$	18.0000	0.695401
$671$	−18.0000	−0.694882
$672$	0	0
$673$	28.0000	1.07932	0.539660	−	0.841883i	$-0.318553\pi$
$673$	28.0000	1.07932	0.539660	+	0.841883i	$0.318553\pi$
$674$	−29.3939	−1.13221
$675$	0	0
$676$	−9.00000	−0.346154
$677$	−26.9444	−1.03556	−0.517778	−	0.855515i	$-0.673241\pi$
$677$	−26.9444	−1.03556	−0.517778	+	0.855515i	$0.673241\pi$
$678$	0	0
$679$	0	0
$680$	−12.0000	−0.460179
$681$	0	0
$682$	4.89898	0.187592
$683$	24.0000	0.918334	0.459167	−	0.888350i	$-0.348148\pi$
$683$	24.0000	0.918334	0.459167	+	0.888350i	$0.348148\pi$
$684$	0	0
$685$	−36.0000	−1.37549
$686$	0	0
$687$	0	0
$688$	−7.34847	−0.280158
$689$	−4.89898	−0.186636
$690$	0	0
$691$	4.00000	0.152167	0.0760836	−	0.997101i	$-0.475758\pi$
$691$	4.00000	0.152167	0.0760836	+	0.997101i	$0.475758\pi$
$692$	18.0000	0.684257
$693$	0	0
$694$	36.0000	1.36654
$695$	−48.9898	−1.85829
$696$	0	0
$697$	−29.3939	−1.11337
$698$	22.0000	0.832712
$699$	0	0
$700$	0	0
$701$	31.8434	1.20271	0.601354	−	0.798983i	$-0.294628\pi$
$701$	31.8434	1.20271	0.601354	+	0.798983i	$0.294628\pi$
$702$	0	0
$703$	54.0000	2.03665
$704$	2.44949	0.0923186
$705$	0	0
$706$	−24.0000	−0.903252
$707$	0	0
$708$	0	0
$709$	−51.4393	−1.93184	−0.965921	−	0.258835i	$-0.916661\pi$
$709$	−51.4393	−1.93184	−0.965921	+	0.258835i	$0.916661\pi$
$710$	−14.6969	−0.551566
$711$	0	0
$712$	−9.79796	−0.367194
$713$	0	0
$714$	0	0
$715$	12.0000	0.448775
$716$	12.0000	0.448461
$717$	0	0
$718$	24.4949	0.914141
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	0	0
$722$	−35.0000	−1.30257
$723$	0	0
$724$	7.34847	0.273104
$725$	6.00000	0.222834
$726$	0	0
$727$	−29.3939	−1.09016	−0.545079	−	0.838385i	$-0.683500\pi$
$727$	−29.3939	−1.09016	−0.545079	+	0.838385i	$0.683500\pi$
$728$	0	0
$729$	0	0
$730$	−19.5959	−0.725277
$731$	36.0000	1.33151
$732$	0	0
$733$	22.0454	0.814266	0.407133	−	0.913369i	$-0.366529\pi$
$733$	22.0454	0.814266	0.407133	+	0.913369i	$0.366529\pi$
$734$	−29.3939	−1.08495
$735$	0	0
$736$	0	0
$737$	18.0000	0.663039
$738$	0	0
$739$	−28.0000	−1.03000	−0.514998	−	0.857191i	$-0.672207\pi$
$739$	−28.0000	−1.03000	−0.514998	+	0.857191i	$0.672207\pi$
$740$	18.0000	0.661693
$741$	0	0
$742$	0	0
$743$	−9.79796	−0.359452	−0.179726	−	0.983717i	$-0.557521\pi$
$743$	−9.79796	−0.359452	−0.179726	+	0.983717i	$0.557521\pi$
$744$	0	0
$745$	18.0000	0.659469
$746$	22.0454	0.807140
$747$	0	0
$748$	−12.0000	−0.438763
$749$	0	0
$750$	0	0
$751$	14.6969	0.536299	0.268149	−	0.963377i	$-0.413588\pi$
$751$	14.6969	0.536299	0.268149	+	0.963377i	$0.413588\pi$
$752$	6.00000	0.218797
$753$	0	0
$754$	12.0000	0.437014
$755$	24.4949	0.891461
$756$	0	0
$757$	−7.34847	−0.267085	−0.133542	−	0.991043i	$-0.542635\pi$
$757$	−7.34847	−0.267085	−0.133542	+	0.991043i	$0.542635\pi$
$758$	22.0454	0.800725
$759$	0	0
$760$	−18.0000	−0.652929
$761$	−12.0000	−0.435000	−0.217500	−	0.976060i	$-0.569790\pi$
$761$	−12.0000	−0.435000	−0.217500	+	0.976060i	$0.569790\pi$
$762$	0	0
$763$	0	0
$764$	−24.4949	−0.886194
$765$	0	0
$766$	9.79796	0.354015
$767$	0	0
$768$	0	0
$769$	44.0908	1.58996	0.794978	−	0.606639i	$-0.207482\pi$
$769$	44.0908	1.58996	0.794978	+	0.606639i	$0.207482\pi$
$770$	0	0
$771$	0	0
$772$	2.00000	0.0719816
$773$	31.8434	1.14533	0.572663	−	0.819791i	$-0.305910\pi$
$773$	31.8434	1.14533	0.572663	+	0.819791i	$0.305910\pi$
$774$	0	0
$775$	−2.00000	−0.0718421
$776$	−14.6969	−0.527589
$777$	0	0
$778$	−12.2474	−0.439092
$779$	−44.0908	−1.57972
$780$	0	0
$781$	−14.6969	−0.525898
$782$	0	0
$783$	0	0
$784$	−7.00000	−0.250000
$785$	−18.0000	−0.642448
$786$	0	0
$787$	36.7423	1.30972	0.654862	−	0.755749i	$-0.272727\pi$
$787$	36.7423	1.30972	0.654862	+	0.755749i	$0.272727\pi$
$788$	−6.00000	−0.213741
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	14.6969	0.521904
$794$	−2.00000	−0.0709773
$795$	0	0
$796$	0	0
$797$	7.34847	0.260296	0.130148	−	0.991495i	$-0.458455\pi$
$797$	7.34847	0.260296	0.130148	+	0.991495i	$0.458455\pi$
$798$	0	0
$799$	−29.3939	−1.03988
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	−34.2929	−1.21092
$803$	−19.5959	−0.691525
$804$	0	0
$805$	0	0
$806$	−4.00000	−0.140894
$807$	0	0
$808$	18.0000	0.633238
$809$	36.0000	1.26569	0.632846	−	0.774277i	$-0.281886\pi$
$809$	36.0000	1.26569	0.632846	+	0.774277i	$0.281886\pi$
$810$	0	0
$811$	16.0000	0.561836	0.280918	−	0.959732i	$-0.409361\pi$
$811$	16.0000	0.561836	0.280918	+	0.959732i	$0.409361\pi$
$812$	0	0
$813$	0	0
$814$	18.0000	0.630900
$815$	−39.1918	−1.37283
$816$	0	0
$817$	54.0000	1.88922
$818$	22.0000	0.769212
$819$	0	0
$820$	−14.6969	−0.513239
$821$	−18.0000	−0.628204	−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000	−0.628204	−0.314102	+	0.949389i	$0.601703\pi$
$822$	0	0
$823$	−14.0000	−0.488009	−0.244005	−	0.969774i	$-0.578461\pi$
$823$	−14.0000	−0.488009	−0.244005	+	0.969774i	$0.578461\pi$
$824$	−14.6969	−0.511992
$825$	0	0
$826$	0	0
$827$	31.8434	1.10730	0.553651	−	0.832749i	$-0.313234\pi$
$827$	31.8434	1.10730	0.553651	+	0.832749i	$0.313234\pi$
$828$	0	0
$829$	−46.0000	−1.59765	−0.798823	−	0.601566i	$-0.794544\pi$
$829$	−46.0000	−1.59765	−0.798823	+	0.601566i	$0.794544\pi$
$830$	−18.0000	−0.624789
$831$	0	0
$832$	−2.00000	−0.0693375
$833$	34.2929	1.18818
$834$	0	0
$835$	−44.0908	−1.52583
$836$	−18.0000	−0.622543
$837$	0	0
$838$	−2.44949	−0.0846162
$839$	9.79796	0.338263	0.169132	−	0.985593i	$-0.445904\pi$
$839$	9.79796	0.338263	0.169132	+	0.985593i	$0.445904\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	−36.7423	−1.26622
$843$	0	0
$844$	−20.0000	−0.688428
$845$	22.0454	0.758385
$846$	0	0
$847$	0	0
$848$	2.44949	0.0841158
$849$	0	0
$850$	4.89898	0.168034
$851$	0	0
$852$	0	0
$853$	14.0000	0.479351	0.239675	−	0.970853i	$-0.422959\pi$
$853$	14.0000	0.479351	0.239675	+	0.970853i	$0.422959\pi$
$854$	0	0
$855$	0	0
$856$	17.1464	0.586053
$857$	12.0000	0.409912	0.204956	−	0.978771i	$-0.434295\pi$
$857$	12.0000	0.409912	0.204956	+	0.978771i	$0.434295\pi$
$858$	0	0
$859$	−4.00000	−0.136478	−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000	−0.136478	−0.0682391	+	0.997669i	$0.521738\pi$
$860$	18.0000	0.613795
$861$	0	0
$862$	−34.2929	−1.16802
$863$	−24.0000	−0.816970	−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000	−0.816970	−0.408485	+	0.912765i	$0.633943\pi$
$864$	0	0
$865$	−44.0908	−1.49913
$866$	−14.6969	−0.499422
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−14.6969	−0.497987
$872$	7.34847	0.248851
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	34.0000	1.14810	0.574049	−	0.818821i	$-0.305372\pi$
$877$	34.0000	1.14810	0.574049	+	0.818821i	$0.305372\pi$
$878$	−32.0000	−1.07995
$879$	0	0
$880$	−6.00000	−0.202260
$881$	−39.1918	−1.32041	−0.660203	−	0.751087i	$-0.729530\pi$
$881$	−39.1918	−1.32041	−0.660203	+	0.751087i	$0.729530\pi$
$882$	0	0
$883$	−8.00000	−0.269221	−0.134611	−	0.990899i	$-0.542978\pi$
$883$	−8.00000	−0.269221	−0.134611	+	0.990899i	$0.542978\pi$
$884$	9.79796	0.329541
$885$	0	0
$886$	−12.0000	−0.403148
$887$	6.00000	0.201460	0.100730	−	0.994914i	$-0.467882\pi$
$887$	6.00000	0.201460	0.100730	+	0.994914i	$0.467882\pi$
$888$	0	0
$889$	0	0
$890$	24.0000	0.804482
$891$	0	0
$892$	14.0000	0.468755
$893$	−44.0908	−1.47544
$894$	0	0
$895$	−29.3939	−0.982529
$896$	0	0
$897$	0	0
$898$	−18.0000	−0.600668
$899$	−12.0000	−0.400222
$900$	0	0
$901$	−12.0000	−0.399778
$902$	−14.6969	−0.489355
$903$	0	0
$904$	0	0
$905$	−18.0000	−0.598340
$906$	0	0
$907$	−22.0454	−0.732006	−0.366003	−	0.930614i	$-0.619274\pi$
$907$	−22.0454	−0.732006	−0.366003	+	0.930614i	$0.619274\pi$
$908$	−26.9444	−0.894181
$909$	0	0
$910$	0	0
$911$	14.6969	0.486931	0.243466	−	0.969910i	$-0.421716\pi$
$911$	14.6969	0.486931	0.243466	+	0.969910i	$0.421716\pi$
$912$	0	0
$913$	−18.0000	−0.595713
$914$	14.6969	0.486132
$915$	0	0
$916$	−22.0454	−0.728401
$917$	0	0
$918$	0	0
$919$	−58.7878	−1.93923	−0.969615	−	0.244638i	$-0.921331\pi$
$919$	−58.7878	−1.93923	−0.969615	+	0.244638i	$0.921331\pi$
$920$	0	0
$921$	0	0
$922$	−30.0000	−0.987997
$923$	12.0000	0.394985
$924$	0	0
$925$	−7.34847	−0.241616
$926$	16.0000	0.525793
$927$	0	0
$928$	−6.00000	−0.196960
$929$	−30.0000	−0.984268	−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000	−0.984268	−0.492134	+	0.870519i	$0.663783\pi$
$930$	0	0
$931$	51.4393	1.68585
$932$	24.0000	0.786146
$933$	0	0
$934$	−36.7423	−1.20225
$935$	29.3939	0.961283
$936$	0	0
$937$	29.3939	0.960256	0.480128	−	0.877198i	$-0.340590\pi$
$937$	29.3939	0.960256	0.480128	+	0.877198i	$0.340590\pi$
$938$	0	0
$939$	0	0
$940$	−14.6969	−0.479361
$941$	31.8434	1.03806	0.519032	−	0.854755i	$-0.326293\pi$
$941$	31.8434	1.03806	0.519032	+	0.854755i	$0.326293\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	18.0000	0.585230
$947$	−12.0000	−0.389948	−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	0	0
$949$	16.0000	0.519382
$950$	7.34847	0.238416
$951$	0	0
$952$	0	0
$953$	4.89898	0.158694	0.0793468	−	0.996847i	$-0.474717\pi$
$953$	4.89898	0.158694	0.0793468	+	0.996847i	$0.474717\pi$
$954$	0	0
$955$	60.0000	1.94155
$956$	−24.0000	−0.776215
$957$	0	0
$958$	−9.79796	−0.316558
$959$	0	0
$960$	0	0
$961$	−27.0000	−0.870968
$962$	−14.6969	−0.473848
$963$	0	0
$964$	−14.6969	−0.473357
$965$	−4.89898	−0.157704
$966$	0	0
$967$	58.0000	1.86515	0.932577	−	0.360971i	$-0.117555\pi$
$967$	58.0000	1.86515	0.932577	+	0.360971i	$0.117555\pi$
$968$	5.00000	0.160706
$969$	0	0
$970$	36.0000	1.15589
$971$	41.6413	1.33633	0.668167	−	0.744011i	$-0.267079\pi$
$971$	41.6413	1.33633	0.668167	+	0.744011i	$0.267079\pi$
$972$	0	0
$973$	0	0
$974$	−8.00000	−0.256337
$975$	0	0
$976$	−7.34847	−0.235219
$977$	−14.6969	−0.470197	−0.235098	−	0.971972i	$-0.575541\pi$
$977$	−14.6969	−0.470197	−0.235098	+	0.971972i	$0.575541\pi$
$978$	0	0
$979$	24.0000	0.767043
$980$	17.1464	0.547723
$981$	0	0
$982$	−12.0000	−0.382935
$983$	53.8888	1.71878	0.859392	−	0.511316i	$-0.170842\pi$
$983$	53.8888	1.71878	0.859392	+	0.511316i	$0.170842\pi$
$984$	0	0
$985$	14.6969	0.468283
$986$	29.3939	0.936092
$987$	0	0
$988$	14.6969	0.467572
$989$	0	0
$990$	0	0
$991$	46.0000	1.46124	0.730619	−	0.682785i	$-0.239232\pi$
$991$	46.0000	1.46124	0.730619	+	0.682785i	$0.239232\pi$
$992$	2.00000	0.0635001
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−10.0000	−0.316703	−0.158352	−	0.987383i	$-0.550618\pi$
$997$	−10.0000	−0.316703	−0.158352	+	0.987383i	$0.550618\pi$
$998$	20.0000	0.633089
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9522.2.a.v.1.1	✓	2
3.2	odd	2		9522.2.a.bh.1.2	yes	2
23.22	odd	2	inner	9522.2.a.v.1.2	yes	2
69.68	even	2		9522.2.a.bh.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9522.2.a.v.1.1	✓	2	1.1	even	1	trivial
9522.2.a.v.1.2	yes	2	23.22	odd	2	inner
9522.2.a.bh.1.1	yes	2	69.68	even	2
9522.2.a.bh.1.2	yes	2	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 9522 = 2 \cdot 3^{2} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9522.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -2.44949 q^{5} -1.00000 q^{8} +2.44949 q^{10} +2.44949 q^{11} -2.00000 q^{13} +1.00000 q^{16} -4.89898 q^{17} -7.34847 q^{19} -2.44949 q^{20} -2.44949 q^{22} +1.00000 q^{25} +2.00000 q^{26} +6.00000 q^{29} -2.00000 q^{31} -1.00000 q^{32} +4.89898 q^{34} -7.34847 q^{37} +7.34847 q^{38} +2.44949 q^{40} +6.00000 q^{41} -7.34847 q^{43} +2.44949 q^{44} +6.00000 q^{47} -7.00000 q^{49} -1.00000 q^{50} -2.00000 q^{52} +2.44949 q^{53} -6.00000 q^{55} -6.00000 q^{58} -7.34847 q^{61} +2.00000 q^{62} +1.00000 q^{64} +4.89898 q^{65} +7.34847 q^{67} -4.89898 q^{68} -6.00000 q^{71} -8.00000 q^{73} +7.34847 q^{74} -7.34847 q^{76} -2.44949 q^{80} -6.00000 q^{82} -7.34847 q^{83} +12.0000 q^{85} +7.34847 q^{86} -2.44949 q^{88} +9.79796 q^{89} -6.00000 q^{94} +18.0000 q^{95} +14.6969 q^{97} +7.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 4 q^{13} + 2 q^{16} + 2 q^{25} + 4 q^{26} + 12 q^{29} - 4 q^{31} - 2 q^{32} + 12 q^{41} + 12 q^{47} - 14 q^{49} - 2 q^{50} - 4 q^{52} - 12 q^{55} - 12 q^{58} + 4 q^{62}+ \cdots + 14 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 - 4 * q^13 + 2 * q^16 + 2 * q^25 + 4 * q^26 + 12 * q^29 - 4 * q^31 - 2 * q^32 + 12 * q^41 + 12 * q^47 - 14 * q^49 - 2 * q^50 - 4 * q^52 - 12 * q^55 - 12 * q^58 + 4 * q^62 + 2 * q^64 - 12 * q^71 - 16 * q^73 - 12 * q^82 + 24 * q^85 - 12 * q^94 + 36 * q^95 + 14 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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