LMFDB - Embedded newform 9522.2.a.ci.1.3

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 = [10,10,0,10,-10,0,2,10,0,-10,-12,0,0,2,0,10,-24,0,-8,-10,0,-12, 0,0,8,0,0,2,4] 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:	$1$
Dimension:	$10$
Coefficient field:	10.10.52900342088704.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 18x^{8} + 123x^{6} - 390x^{4} + 548x^{2} - 241 $ x^10 - 18x^8 + 123x^6 - 390x^4 + 548x^2 - 241
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 414)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.70641$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -2.85684 q^{5} -4.48349 q^{7} +1.00000 q^{8} -2.85684 q^{10} -0.198365 q^{11} +3.70872 q^{13} -4.48349 q^{14} +1.00000 q^{16} +0.618594 q^{17} +0.259187 q^{19} -2.85684 q^{20} -0.198365 q^{22} +3.16155 q^{25} +3.70872 q^{26} -4.48349 q^{28} +7.49661 q^{29} -2.80927 q^{31} +1.00000 q^{32} +0.618594 q^{34} +12.8086 q^{35} -5.90747 q^{37} +0.259187 q^{38} -2.85684 q^{40} +8.45349 q^{41} +10.0715 q^{43} -0.198365 q^{44} -9.81259 q^{47} +13.1017 q^{49} +3.16155 q^{50} +3.70872 q^{52} +3.93580 q^{53} +0.566698 q^{55} -4.48349 q^{56} +7.49661 q^{58} -9.81277 q^{59} +5.11918 q^{61} -2.80927 q^{62} +1.00000 q^{64} -10.5952 q^{65} +7.35623 q^{67} +0.618594 q^{68} +12.8086 q^{70} +11.1005 q^{71} -15.6323 q^{73} -5.90747 q^{74} +0.259187 q^{76} +0.889368 q^{77} +14.9619 q^{79} -2.85684 q^{80} +8.45349 q^{82} -9.62975 q^{83} -1.76723 q^{85} +10.0715 q^{86} -0.198365 q^{88} -15.3355 q^{89} -16.6280 q^{91} -9.81259 q^{94} -0.740458 q^{95} -11.8464 q^{97} +13.1017 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 10 q^{2} + 10 q^{4} - 10 q^{5} + 2 q^{7} + 10 q^{8} - 10 q^{10} - 12 q^{11} + 2 q^{14} + 10 q^{16} - 24 q^{17} - 8 q^{19} - 10 q^{20} - 12 q^{22} + 8 q^{25} + 2 q^{28} + 4 q^{29} + 18 q^{31} + 10 q^{32}+ \cdots + 36 q^{98}+O(q^{100}) $ 10 * q + 10 * q^2 + 10 * q^4 - 10 * q^5 + 2 * q^7 + 10 * q^8 - 10 * q^10 - 12 * q^11 + 2 * q^14 + 10 * q^16 - 24 * q^17 - 8 * q^19 - 10 * q^20 - 12 * q^22 + 8 * q^25 + 2 * q^28 + 4 * q^29 + 18 * q^31 + 10 * q^32 - 24 * q^34 - 24 * q^35 - 12 * q^37 - 8 * q^38 - 10 * q^40 + 28 * q^41 - 8 * q^43 - 12 * q^44 - 16 * q^47 + 36 * q^49 + 8 * q^50 - 34 * q^53 + 30 * q^55 + 2 * q^56 + 4 * q^58 - 22 * q^59 - 30 * q^61 + 18 * q^62 + 10 * q^64 - 36 * q^65 - 18 * q^67 - 24 * q^68 - 24 * q^70 - 28 * q^71 - 20 * q^73 - 12 * q^74 - 8 * q^76 - 20 * q^77 - 2 * q^79 - 10 * q^80 + 28 * q^82 - 44 * q^83 + 16 * q^85 - 8 * q^86 - 12 * q^88 - 44 * q^89 - 22 * q^91 - 16 * q^94 - 10 * q^95 - 18 * q^97 + 36 * 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
$5$	−2.85684	−1.27762	−0.638810	−	0.769365i	$-0.720573\pi$
$5$	−2.85684	−1.27762	−0.638810	+	0.769365i	$0.720573\pi$
$6$	0	0
$7$	−4.48349	−1.69460	−0.847301	−	0.531113i	$-0.821774\pi$
$7$	−4.48349	−1.69460	−0.847301	+	0.531113i	$0.821774\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−2.85684	−0.903413
$11$	−0.198365	−0.0598093	−0.0299047	−	0.999553i	$-0.509520\pi$
$11$	−0.198365	−0.0598093	−0.0299047	+	0.999553i	$0.509520\pi$
$12$	0	0
$13$	3.70872	1.02861	0.514307	−	0.857606i	$-0.328049\pi$
$13$	3.70872	1.02861	0.514307	+	0.857606i	$0.328049\pi$
$14$	−4.48349	−1.19826
$15$	0	0
$16$	1.00000	0.250000
$17$	0.618594	0.150031	0.0750156	−	0.997182i	$-0.476099\pi$
$17$	0.618594	0.150031	0.0750156	+	0.997182i	$0.476099\pi$
$18$	0	0
$19$	0.259187	0.0594617	0.0297308	−	0.999558i	$-0.490535\pi$
$19$	0.259187	0.0594617	0.0297308	+	0.999558i	$0.490535\pi$
$20$	−2.85684	−0.638810
$21$	0	0
$22$	−0.198365	−0.0422916
$23$	0	0
$23$	0	0
$24$	0	0
$25$	3.16155	0.632310
$26$	3.70872	0.727340
$27$	0	0
$28$	−4.48349	−0.847301
$29$	7.49661	1.39209	0.696043	−	0.718000i	$-0.254942\pi$
$29$	7.49661	1.39209	0.696043	+	0.718000i	$0.254942\pi$
$30$	0	0
$31$	−2.80927	−0.504561	−0.252280	−	0.967654i	$-0.581181\pi$
$31$	−2.80927	−0.504561	−0.252280	+	0.967654i	$0.581181\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	0.618594	0.106088
$35$	12.8086	2.16506
$36$	0	0
$37$	−5.90747	−0.971182	−0.485591	−	0.874186i	$-0.661396\pi$
$37$	−5.90747	−0.971182	−0.485591	+	0.874186i	$0.661396\pi$
$38$	0.259187	0.0420457
$39$	0	0
$40$	−2.85684	−0.451707
$41$	8.45349	1.32021	0.660107	−	0.751172i	$-0.270511\pi$
$41$	8.45349	1.32021	0.660107	+	0.751172i	$0.270511\pi$
$42$	0	0
$43$	10.0715	1.53589	0.767943	−	0.640518i	$-0.221280\pi$
$43$	10.0715	1.53589	0.767943	+	0.640518i	$0.221280\pi$
$44$	−0.198365	−0.0299047
$45$	0	0
$46$	0	0
$47$	−9.81259	−1.43131	−0.715657	−	0.698452i	$-0.753872\pi$
$47$	−9.81259	−1.43131	−0.715657	+	0.698452i	$0.753872\pi$
$48$	0	0
$49$	13.1017	1.87167
$50$	3.16155	0.447111
$51$	0	0
$52$	3.70872	0.514307
$53$	3.93580	0.540624	0.270312	−	0.962773i	$-0.412873\pi$
$53$	3.93580	0.540624	0.270312	+	0.962773i	$0.412873\pi$
$54$	0	0
$55$	0.566698	0.0764135
$56$	−4.48349	−0.599132
$57$	0	0
$58$	7.49661	0.984353
$59$	−9.81277	−1.27751	−0.638757	−	0.769409i	$-0.720551\pi$
$59$	−9.81277	−1.27751	−0.638757	+	0.769409i	$0.720551\pi$
$60$	0	0
$61$	5.11918	0.655444	0.327722	−	0.944774i	$-0.393719\pi$
$61$	5.11918	0.655444	0.327722	+	0.944774i	$0.393719\pi$
$62$	−2.80927	−0.356778
$63$	0	0
$64$	1.00000	0.125000
$65$	−10.5952	−1.31418
$66$	0	0
$67$	7.35623	0.898707	0.449353	−	0.893354i	$-0.351654\pi$
$67$	7.35623	0.898707	0.449353	+	0.893354i	$0.351654\pi$
$68$	0.618594	0.0750156
$69$	0	0
$70$	12.8086	1.53093
$71$	11.1005	1.31739	0.658694	−	0.752411i	$-0.271109\pi$
$71$	11.1005	1.31739	0.658694	+	0.752411i	$0.271109\pi$
$72$	0	0
$73$	−15.6323	−1.82962	−0.914809	−	0.403888i	$-0.867659\pi$
$73$	−15.6323	−1.82962	−0.914809	+	0.403888i	$0.867659\pi$
$74$	−5.90747	−0.686729
$75$	0	0
$76$	0.259187	0.0297308
$77$	0.889368	0.101353
$78$	0	0
$79$	14.9619	1.68334	0.841670	−	0.539992i	$-0.181573\pi$
$79$	14.9619	1.68334	0.841670	+	0.539992i	$0.181573\pi$
$80$	−2.85684	−0.319405
$81$	0	0
$82$	8.45349	0.933532
$83$	−9.62975	−1.05700	−0.528501	−	0.848933i	$-0.677246\pi$
$83$	−9.62975	−1.05700	−0.528501	+	0.848933i	$0.677246\pi$
$84$	0	0
$85$	−1.76723	−0.191683
$86$	10.0715	1.08604
$87$	0	0
$88$	−0.198365	−0.0211458
$89$	−15.3355	−1.62556	−0.812781	−	0.582569i	$-0.802048\pi$
$89$	−15.3355	−1.62556	−0.812781	+	0.582569i	$0.802048\pi$
$90$	0	0
$91$	−16.6280	−1.74309
$92$	0	0
$93$	0	0
$94$	−9.81259	−1.01209
$95$	−0.740458	−0.0759693
$96$	0	0
$97$	−11.8464	−1.20282	−0.601410	−	0.798941i	$-0.705394\pi$
$97$	−11.8464	−1.20282	−0.601410	+	0.798941i	$0.705394\pi$
$98$	13.1017	1.32347
$99$	0	0
$100$	3.16155	0.316155
$101$	−18.8858	−1.87921	−0.939606	−	0.342258i	$-0.888808\pi$
$101$	−18.8858	−1.87921	−0.939606	+	0.342258i	$0.888808\pi$
$102$	0	0
$103$	3.33765	0.328868	0.164434	−	0.986388i	$-0.447420\pi$
$103$	3.33765	0.328868	0.164434	+	0.986388i	$0.447420\pi$
$104$	3.70872	0.363670
$105$	0	0
$106$	3.93580	0.382279
$107$	−4.94497	−0.478048	−0.239024	−	0.971014i	$-0.576827\pi$
$107$	−4.94497	−0.478048	−0.239024	+	0.971014i	$0.576827\pi$
$108$	0	0
$109$	−2.32856	−0.223036	−0.111518	−	0.993762i	$-0.535571\pi$
$109$	−2.32856	−0.223036	−0.111518	+	0.993762i	$0.535571\pi$
$110$	0.566698	0.0540325
$111$	0	0
$112$	−4.48349	−0.423650
$113$	−8.00742	−0.753275	−0.376637	−	0.926361i	$-0.622920\pi$
$113$	−8.00742	−0.753275	−0.376637	+	0.926361i	$0.622920\pi$
$114$	0	0
$115$	0	0
$116$	7.49661	0.696043
$117$	0	0
$118$	−9.81277	−0.903338
$119$	−2.77346	−0.254243
$120$	0	0
$121$	−10.9607	−0.996423
$122$	5.11918	0.463469
$123$	0	0
$124$	−2.80927	−0.252280
$125$	5.25216	0.469767
$126$	0	0
$127$	5.47728	0.486030	0.243015	−	0.970022i	$-0.421864\pi$
$127$	5.47728	0.486030	0.243015	+	0.970022i	$0.421864\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−10.5952	−0.929263
$131$	5.81959	0.508460	0.254230	−	0.967144i	$-0.418178\pi$
$131$	5.81959	0.508460	0.254230	+	0.967144i	$0.418178\pi$
$132$	0	0
$133$	−1.16206	−0.100764
$134$	7.35623	0.635482
$135$	0	0
$136$	0.618594	0.0530440
$137$	−6.21674	−0.531132	−0.265566	−	0.964093i	$-0.585559\pi$
$137$	−6.21674	−0.531132	−0.265566	+	0.964093i	$0.585559\pi$
$138$	0	0
$139$	3.51441	0.298089	0.149044	−	0.988831i	$-0.452380\pi$
$139$	3.51441	0.298089	0.149044	+	0.988831i	$0.452380\pi$
$140$	12.8086	1.08253
$141$	0	0
$142$	11.1005	0.931534
$143$	−0.735680	−0.0615207
$144$	0	0
$145$	−21.4166	−1.77856
$146$	−15.6323	−1.29373
$147$	0	0
$148$	−5.90747	−0.485591
$149$	−4.16095	−0.340879	−0.170439	−	0.985368i	$-0.554519\pi$
$149$	−4.16095	−0.340879	−0.170439	+	0.985368i	$0.554519\pi$
$150$	0	0
$151$	−7.91198	−0.643868	−0.321934	−	0.946762i	$-0.604333\pi$
$151$	−7.91198	−0.643868	−0.321934	+	0.946762i	$0.604333\pi$
$152$	0.259187	0.0210229
$153$	0	0
$154$	0.889368	0.0716673
$155$	8.02566	0.644636
$156$	0	0
$157$	−17.9469	−1.43232	−0.716158	−	0.697939i	$-0.754101\pi$
$157$	−17.9469	−1.43232	−0.716158	+	0.697939i	$0.754101\pi$
$158$	14.9619	1.19030
$159$	0	0
$160$	−2.85684	−0.225853
$161$	0	0
$162$	0	0
$163$	7.09440	0.555676	0.277838	−	0.960628i	$-0.410382\pi$
$163$	7.09440	0.555676	0.277838	+	0.960628i	$0.410382\pi$
$164$	8.45349	0.660107
$165$	0	0
$166$	−9.62975	−0.747413
$167$	−16.3201	−1.26289	−0.631443	−	0.775422i	$-0.717537\pi$
$167$	−16.3201	−1.26289	−0.631443	+	0.775422i	$0.717537\pi$
$168$	0	0
$169$	0.754609	0.0580469
$170$	−1.76723	−0.135540
$171$	0	0
$172$	10.0715	0.767943
$173$	15.4737	1.17645	0.588223	−	0.808699i	$-0.299828\pi$
$173$	15.4737	1.17645	0.588223	+	0.808699i	$0.299828\pi$
$174$	0	0
$175$	−14.1748	−1.07151
$176$	−0.198365	−0.0149523
$177$	0	0
$178$	−15.3355	−1.14945
$179$	−4.67890	−0.349717	−0.174859	−	0.984594i	$-0.555947\pi$
$179$	−4.67890	−0.349717	−0.174859	+	0.984594i	$0.555947\pi$
$180$	0	0
$181$	5.42873	0.403514	0.201757	−	0.979436i	$-0.435335\pi$
$181$	5.42873	0.403514	0.201757	+	0.979436i	$0.435335\pi$
$182$	−16.6280	−1.23255
$183$	0	0
$184$	0	0
$185$	16.8767	1.24080
$186$	0	0
$187$	−0.122707	−0.00897326
$188$	−9.81259	−0.715657
$189$	0	0
$190$	−0.740458	−0.0537184
$191$	5.76873	0.417411	0.208705	−	0.977979i	$-0.433075\pi$
$191$	5.76873	0.417411	0.208705	+	0.977979i	$0.433075\pi$
$192$	0	0
$193$	−2.78622	−0.200557	−0.100278	−	0.994959i	$-0.531973\pi$
$193$	−2.78622	−0.200557	−0.100278	+	0.994959i	$0.531973\pi$
$194$	−11.8464	−0.850522
$195$	0	0
$196$	13.1017	0.935837
$197$	−27.7303	−1.97570	−0.987852	−	0.155400i	$-0.950333\pi$
$197$	−27.7303	−1.97570	−0.987852	+	0.155400i	$0.950333\pi$
$198$	0	0
$199$	20.5232	1.45485	0.727426	−	0.686187i	$-0.240717\pi$
$199$	20.5232	1.45485	0.727426	+	0.686187i	$0.240717\pi$
$200$	3.16155	0.223555
$201$	0	0
$202$	−18.8858	−1.32880
$203$	−33.6110	−2.35903
$204$	0	0
$205$	−24.1503	−1.68673
$206$	3.33765	0.232545
$207$	0	0
$208$	3.70872	0.257154
$209$	−0.0514137	−0.00355636
$210$	0	0
$211$	−22.0376	−1.51713	−0.758566	−	0.651596i	$-0.774100\pi$
$211$	−22.0376	−1.51713	−0.758566	+	0.651596i	$0.774100\pi$
$212$	3.93580	0.270312
$213$	0	0
$214$	−4.94497	−0.338031
$215$	−28.7726	−1.96228
$216$	0	0
$217$	12.5954	0.855029
$218$	−2.32856	−0.157710
$219$	0	0
$220$	0.566698	0.0382068
$221$	2.29419	0.154324
$222$	0	0
$223$	16.0615	1.07556	0.537778	−	0.843087i	$-0.319264\pi$
$223$	16.0615	1.07556	0.537778	+	0.843087i	$0.319264\pi$
$224$	−4.48349	−0.299566
$225$	0	0
$226$	−8.00742	−0.532646
$227$	23.7222	1.57450	0.787248	−	0.616636i	$-0.211505\pi$
$227$	23.7222	1.57450	0.787248	+	0.616636i	$0.211505\pi$
$228$	0	0
$229$	19.2142	1.26971	0.634856	−	0.772630i	$-0.281059\pi$
$229$	19.2142	1.26971	0.634856	+	0.772630i	$0.281059\pi$
$230$	0	0
$231$	0	0
$232$	7.49661	0.492177
$233$	−21.4135	−1.40285	−0.701423	−	0.712745i	$-0.747452\pi$
$233$	−21.4135	−1.40285	−0.701423	+	0.712745i	$0.747452\pi$
$234$	0	0
$235$	28.0330	1.82867
$236$	−9.81277	−0.638757
$237$	0	0
$238$	−2.77346	−0.179777
$239$	−6.20486	−0.401359	−0.200679	−	0.979657i	$-0.564315\pi$
$239$	−6.20486	−0.401359	−0.200679	+	0.979657i	$0.564315\pi$
$240$	0	0
$241$	−7.77891	−0.501083	−0.250542	−	0.968106i	$-0.580609\pi$
$241$	−7.77891	−0.501083	−0.250542	+	0.968106i	$0.580609\pi$
$242$	−10.9607	−0.704577
$243$	0	0
$244$	5.11918	0.327722
$245$	−37.4296	−2.39129
$246$	0	0
$247$	0.961253	0.0611631
$248$	−2.80927	−0.178389
$249$	0	0
$250$	5.25216	0.332176
$251$	−13.7940	−0.870669	−0.435335	−	0.900269i	$-0.643370\pi$
$251$	−13.7940	−0.870669	−0.435335	+	0.900269i	$0.643370\pi$
$252$	0	0
$253$	0	0
$254$	5.47728	0.343675
$255$	0	0
$256$	1.00000	0.0625000
$257$	−12.2445	−0.763794	−0.381897	−	0.924205i	$-0.624729\pi$
$257$	−12.2445	−0.763794	−0.381897	+	0.924205i	$0.624729\pi$
$258$	0	0
$259$	26.4861	1.64577
$260$	−10.5952	−0.657088
$261$	0	0
$262$	5.81959	0.359535
$263$	13.4719	0.830712	0.415356	−	0.909659i	$-0.363657\pi$
$263$	13.4719	0.830712	0.415356	+	0.909659i	$0.363657\pi$
$264$	0	0
$265$	−11.2440	−0.690712
$266$	−1.16206	−0.0712508
$267$	0	0
$268$	7.35623	0.449353
$269$	10.5981	0.646176	0.323088	−	0.946369i	$-0.395279\pi$
$269$	10.5981	0.646176	0.323088	+	0.946369i	$0.395279\pi$
$270$	0	0
$271$	−6.37018	−0.386961	−0.193480	−	0.981104i	$-0.561978\pi$
$271$	−6.37018	−0.386961	−0.193480	+	0.981104i	$0.561978\pi$
$272$	0.618594	0.0375078
$273$	0	0
$274$	−6.21674	−0.375567
$275$	−0.627141	−0.0378180
$276$	0	0
$277$	−3.99485	−0.240027	−0.120014	−	0.992772i	$-0.538294\pi$
$277$	−3.99485	−0.240027	−0.120014	+	0.992772i	$0.538294\pi$
$278$	3.51441	0.210780
$279$	0	0
$280$	12.8086	0.765463
$281$	−3.63873	−0.217069	−0.108534	−	0.994093i	$-0.534616\pi$
$281$	−3.63873	−0.217069	−0.108534	+	0.994093i	$0.534616\pi$
$282$	0	0
$283$	12.1478	0.722113	0.361056	−	0.932544i	$-0.382416\pi$
$283$	12.1478	0.722113	0.361056	+	0.932544i	$0.382416\pi$
$284$	11.1005	0.658694
$285$	0	0
$286$	−0.735680	−0.0435017
$287$	−37.9012	−2.23724
$288$	0	0
$289$	−16.6173	−0.977491
$290$	−21.4166	−1.25763
$291$	0	0
$292$	−15.6323	−0.914809
$293$	−15.7068	−0.917603	−0.458802	−	0.888539i	$-0.651721\pi$
$293$	−15.7068	−0.917603	−0.458802	+	0.888539i	$0.651721\pi$
$294$	0	0
$295$	28.0335	1.63218
$296$	−5.90747	−0.343365
$297$	0	0
$298$	−4.16095	−0.241038
$299$	0	0
$300$	0	0
$301$	−45.1554	−2.60271
$302$	−7.91198	−0.455283
$303$	0	0
$304$	0.259187	0.0148654
$305$	−14.6247	−0.837408
$306$	0	0
$307$	1.74903	0.0998223	0.0499112	−	0.998754i	$-0.484106\pi$
$307$	1.74903	0.0998223	0.0499112	+	0.998754i	$0.484106\pi$
$308$	0.889368	0.0506765
$309$	0	0
$310$	8.02566	0.455827
$311$	−14.3459	−0.813484	−0.406742	−	0.913543i	$-0.633335\pi$
$311$	−14.3459	−0.813484	−0.406742	+	0.913543i	$0.633335\pi$
$312$	0	0
$313$	−3.21704	−0.181838	−0.0909190	−	0.995858i	$-0.528980\pi$
$313$	−3.21704	−0.181838	−0.0909190	+	0.995858i	$0.528980\pi$
$314$	−17.9469	−1.01280
$315$	0	0
$316$	14.9619	0.841670
$317$	−8.57906	−0.481848	−0.240924	−	0.970544i	$-0.577450\pi$
$317$	−8.57906	−0.481848	−0.240924	+	0.970544i	$0.577450\pi$
$318$	0	0
$319$	−1.48707	−0.0832597
$320$	−2.85684	−0.159702
$321$	0	0
$322$	0	0
$323$	0.160332	0.00892110
$324$	0	0
$325$	11.7253	0.650403
$326$	7.09440	0.392923
$327$	0	0
$328$	8.45349	0.466766
$329$	43.9947	2.42551
$330$	0	0
$331$	32.4766	1.78507	0.892537	−	0.450974i	$-0.148923\pi$
$331$	32.4766	1.78507	0.892537	+	0.450974i	$0.148923\pi$
$332$	−9.62975	−0.528501
$333$	0	0
$334$	−16.3201	−0.892995
$335$	−21.0156	−1.14820
$336$	0	0
$337$	19.8665	1.08220	0.541099	−	0.840959i	$-0.318008\pi$
$337$	19.8665	1.08220	0.541099	+	0.840959i	$0.318008\pi$
$338$	0.754609	0.0410453
$339$	0	0
$340$	−1.76723	−0.0958413
$341$	0.557262	0.0301774
$342$	0	0
$343$	−27.3570	−1.47714
$344$	10.0715	0.543018
$345$	0	0
$346$	15.4737	0.831873
$347$	18.9201	1.01569	0.507843	−	0.861449i	$-0.330443\pi$
$347$	18.9201	1.01569	0.507843	+	0.861449i	$0.330443\pi$
$348$	0	0
$349$	−0.333496	−0.0178516	−0.00892582	−	0.999960i	$-0.502841\pi$
$349$	−0.333496	−0.0178516	−0.00892582	+	0.999960i	$0.502841\pi$
$350$	−14.1748	−0.757675
$351$	0	0
$352$	−0.198365	−0.0105729
$353$	−7.64132	−0.406706	−0.203353	−	0.979105i	$-0.565184\pi$
$353$	−7.64132	−0.406706	−0.203353	+	0.979105i	$0.565184\pi$
$354$	0	0
$355$	−31.7124	−1.68312
$356$	−15.3355	−0.812781
$357$	0	0
$358$	−4.67890	−0.247287
$359$	−14.5582	−0.768350	−0.384175	−	0.923260i	$-0.625514\pi$
$359$	−14.5582	−0.768350	−0.384175	+	0.923260i	$0.625514\pi$
$360$	0	0
$361$	−18.9328	−0.996464
$362$	5.42873	0.285328
$363$	0	0
$364$	−16.6280	−0.871545
$365$	44.6589	2.33755
$366$	0	0
$367$	−24.3518	−1.27116	−0.635578	−	0.772037i	$-0.719238\pi$
$367$	−24.3518	−1.27116	−0.635578	+	0.772037i	$0.719238\pi$
$368$	0	0
$369$	0	0
$370$	16.8767	0.877378
$371$	−17.6462	−0.916143
$372$	0	0
$373$	−16.4397	−0.851214	−0.425607	−	0.904908i	$-0.639939\pi$
$373$	−16.4397	−0.851214	−0.425607	+	0.904908i	$0.639939\pi$
$374$	−0.122707	−0.00634505
$375$	0	0
$376$	−9.81259	−0.506046
$377$	27.8028	1.43192
$378$	0	0
$379$	−7.37024	−0.378584	−0.189292	−	0.981921i	$-0.560619\pi$
$379$	−7.37024	−0.378584	−0.189292	+	0.981921i	$0.560619\pi$
$380$	−0.740458	−0.0379847
$381$	0	0
$382$	5.76873	0.295154
$383$	−23.2014	−1.18554	−0.592768	−	0.805373i	$-0.701965\pi$
$383$	−23.2014	−1.18554	−0.592768	+	0.805373i	$0.701965\pi$
$384$	0	0
$385$	−2.54079	−0.129490
$386$	−2.78622	−0.141815
$387$	0	0
$388$	−11.8464	−0.601410
$389$	−20.2192	−1.02515	−0.512577	−	0.858641i	$-0.671309\pi$
$389$	−20.2192	−1.02515	−0.512577	+	0.858641i	$0.671309\pi$
$390$	0	0
$391$	0	0
$392$	13.1017	0.661737
$393$	0	0
$394$	−27.7303	−1.39703
$395$	−42.7437	−2.15067
$396$	0	0
$397$	18.2589	0.916389	0.458195	−	0.888852i	$-0.348496\pi$
$397$	18.2589	0.916389	0.458195	+	0.888852i	$0.348496\pi$
$398$	20.5232	1.02874
$399$	0	0
$400$	3.16155	0.158078
$401$	−35.9055	−1.79304	−0.896519	−	0.443006i	$-0.853912\pi$
$401$	−35.9055	−1.79304	−0.896519	+	0.443006i	$0.853912\pi$
$402$	0	0
$403$	−10.4188	−0.518998
$404$	−18.8858	−0.939606
$405$	0	0
$406$	−33.6110	−1.66809
$407$	1.17183	0.0580857
$408$	0	0
$409$	−26.2924	−1.30008	−0.650038	−	0.759901i	$-0.725247\pi$
$409$	−26.2924	−1.30008	−0.650038	+	0.759901i	$0.725247\pi$
$410$	−24.1503	−1.19270
$411$	0	0
$412$	3.33765	0.164434
$413$	43.9955	2.16488
$414$	0	0
$415$	27.5107	1.35045
$416$	3.70872	0.181835
$417$	0	0
$418$	−0.0514137	−0.00251473
$419$	−7.27755	−0.355532	−0.177766	−	0.984073i	$-0.556887\pi$
$419$	−7.27755	−0.355532	−0.177766	+	0.984073i	$0.556887\pi$
$420$	0	0
$421$	34.3333	1.67330	0.836652	−	0.547735i	$-0.184510\pi$
$421$	34.3333	1.67330	0.836652	+	0.547735i	$0.184510\pi$
$422$	−22.0376	−1.07277
$423$	0	0
$424$	3.93580	0.191140
$425$	1.95572	0.0948663
$426$	0	0
$427$	−22.9518	−1.11072
$428$	−4.94497	−0.239024
$429$	0	0
$430$	−28.7726	−1.38754
$431$	18.2265	0.877941	0.438971	−	0.898501i	$-0.355343\pi$
$431$	18.2265	0.877941	0.438971	+	0.898501i	$0.355343\pi$
$432$	0	0
$433$	−5.14943	−0.247466	−0.123733	−	0.992316i	$-0.539487\pi$
$433$	−5.14943	−0.247466	−0.123733	+	0.992316i	$0.539487\pi$
$434$	12.5954	0.604597
$435$	0	0
$436$	−2.32856	−0.111518
$437$	0	0
$438$	0	0
$439$	17.7487	0.847097	0.423548	−	0.905873i	$-0.360784\pi$
$439$	17.7487	0.847097	0.423548	+	0.905873i	$0.360784\pi$
$440$	0.566698	0.0270163
$441$	0	0
$442$	2.29419	0.109124
$443$	3.26754	0.155245	0.0776227	−	0.996983i	$-0.475267\pi$
$443$	3.26754	0.155245	0.0776227	+	0.996983i	$0.475267\pi$
$444$	0	0
$445$	43.8112	2.07685
$446$	16.0615	0.760533
$447$	0	0
$448$	−4.48349	−0.211825
$449$	40.2916	1.90148	0.950738	−	0.309995i	$-0.100327\pi$
$449$	40.2916	1.90148	0.950738	+	0.309995i	$0.100327\pi$
$450$	0	0
$451$	−1.67688	−0.0789610
$452$	−8.00742	−0.376637
$453$	0	0
$454$	23.7222	1.11334
$455$	47.5037	2.22701
$456$	0	0
$457$	−29.9182	−1.39952	−0.699758	−	0.714380i	$-0.746709\pi$
$457$	−29.9182	−1.39952	−0.699758	+	0.714380i	$0.746709\pi$
$458$	19.2142	0.897822
$459$	0	0
$460$	0	0
$461$	8.18649	0.381283	0.190642	−	0.981660i	$-0.438943\pi$
$461$	8.18649	0.381283	0.190642	+	0.981660i	$0.438943\pi$
$462$	0	0
$463$	−2.92694	−0.136026	−0.0680132	−	0.997684i	$-0.521666\pi$
$463$	−2.92694	−0.136026	−0.0680132	+	0.997684i	$0.521666\pi$
$464$	7.49661	0.348021
$465$	0	0
$466$	−21.4135	−0.991963
$467$	−33.2239	−1.53742	−0.768709	−	0.639598i	$-0.779101\pi$
$467$	−33.2239	−1.53742	−0.768709	+	0.639598i	$0.779101\pi$
$468$	0	0
$469$	−32.9816	−1.52295
$470$	28.0330	1.29307
$471$	0	0
$472$	−9.81277	−0.451669
$473$	−1.99783	−0.0918603
$474$	0	0
$475$	0.819434	0.0375982
$476$	−2.77346	−0.127122
$477$	0	0
$478$	−6.20486	−0.283803
$479$	−27.5307	−1.25791	−0.628955	−	0.777442i	$-0.716517\pi$
$479$	−27.5307	−1.25791	−0.628955	+	0.777442i	$0.716517\pi$
$480$	0	0
$481$	−21.9091	−0.998971
$482$	−7.77891	−0.354319
$483$	0	0
$484$	−10.9607	−0.498211
$485$	33.8433	1.53675
$486$	0	0
$487$	13.8291	0.626657	0.313329	−	0.949645i	$-0.398556\pi$
$487$	13.8291	0.626657	0.313329	+	0.949645i	$0.398556\pi$
$488$	5.11918	0.231734
$489$	0	0
$490$	−37.4296	−1.69089
$491$	6.40601	0.289099	0.144550	−	0.989498i	$-0.453827\pi$
$491$	6.40601	0.289099	0.144550	+	0.989498i	$0.453827\pi$
$492$	0	0
$493$	4.63736	0.208856
$494$	0.961253	0.0432488
$495$	0	0
$496$	−2.80927	−0.126140
$497$	−49.7690	−2.23245
$498$	0	0
$499$	4.42182	0.197948	0.0989738	−	0.995090i	$-0.468444\pi$
$499$	4.42182	0.197948	0.0989738	+	0.995090i	$0.468444\pi$
$500$	5.25216	0.234884
$501$	0	0
$502$	−13.7940	−0.615656
$503$	−24.5192	−1.09326	−0.546628	−	0.837376i	$-0.684089\pi$
$503$	−24.5192	−1.09326	−0.546628	+	0.837376i	$0.684089\pi$
$504$	0	0
$505$	53.9539	2.40092
$506$	0	0
$507$	0	0
$508$	5.47728	0.243015
$509$	−4.83296	−0.214217	−0.107109	−	0.994247i	$-0.534159\pi$
$509$	−4.83296	−0.214217	−0.107109	+	0.994247i	$0.534159\pi$
$510$	0	0
$511$	70.0871	3.10047
$512$	1.00000	0.0441942
$513$	0	0
$514$	−12.2445	−0.540084
$515$	−9.53513	−0.420168
$516$	0	0
$517$	1.94647	0.0856058
$518$	26.4861	1.16373
$519$	0	0
$520$	−10.5952	−0.464632
$521$	0.345015	0.0151154	0.00755769	−	0.999971i	$-0.497594\pi$
$521$	0.345015	0.0151154	0.00755769	+	0.999971i	$0.497594\pi$
$522$	0	0
$523$	−32.7245	−1.43094	−0.715471	−	0.698643i	$-0.753788\pi$
$523$	−32.7245	−1.43094	−0.715471	+	0.698643i	$0.753788\pi$
$524$	5.81959	0.254230
$525$	0	0
$526$	13.4719	0.587402
$527$	−1.73780	−0.0756998
$528$	0	0
$529$	0	0
$530$	−11.2440	−0.488407
$531$	0	0
$532$	−1.16206	−0.0503819
$533$	31.3516	1.35799
$534$	0	0
$535$	14.1270	0.610763
$536$	7.35623	0.317741
$537$	0	0
$538$	10.5981	0.456915
$539$	−2.59892	−0.111944
$540$	0	0
$541$	−5.90403	−0.253834	−0.126917	−	0.991913i	$-0.540508\pi$
$541$	−5.90403	−0.253834	−0.126917	+	0.991913i	$0.540508\pi$
$542$	−6.37018	−0.273623
$543$	0	0
$544$	0.618594	0.0265220
$545$	6.65233	0.284954
$546$	0	0
$547$	10.7026	0.457611	0.228806	−	0.973472i	$-0.426518\pi$
$547$	10.7026	0.457611	0.228806	+	0.973472i	$0.426518\pi$
$548$	−6.21674	−0.265566
$549$	0	0
$550$	−0.627141	−0.0267414
$551$	1.94303	0.0827757
$552$	0	0
$553$	−67.0814	−2.85259
$554$	−3.99485	−0.169725
$555$	0	0
$556$	3.51441	0.149044
$557$	−35.3336	−1.49713	−0.748566	−	0.663061i	$-0.769257\pi$
$557$	−35.3336	−1.49713	−0.748566	+	0.663061i	$0.769257\pi$
$558$	0	0
$559$	37.3523	1.57983
$560$	12.8086	0.541264
$561$	0	0
$562$	−3.63873	−0.153491
$563$	12.5021	0.526901	0.263450	−	0.964673i	$-0.415140\pi$
$563$	12.5021	0.526901	0.263450	+	0.964673i	$0.415140\pi$
$564$	0	0
$565$	22.8759	0.962398
$566$	12.1478	0.510611
$567$	0	0
$568$	11.1005	0.465767
$569$	−11.1853	−0.468913	−0.234457	−	0.972127i	$-0.575331\pi$
$569$	−11.1853	−0.468913	−0.234457	+	0.972127i	$0.575331\pi$
$570$	0	0
$571$	15.0602	0.630251	0.315126	−	0.949050i	$-0.397953\pi$
$571$	15.0602	0.630251	0.315126	+	0.949050i	$0.397953\pi$
$572$	−0.735680	−0.0307603
$573$	0	0
$574$	−37.9012	−1.58196
$575$	0	0
$576$	0	0
$577$	−29.2713	−1.21858	−0.609291	−	0.792947i	$-0.708546\pi$
$577$	−29.2713	−1.21858	−0.609291	+	0.792947i	$0.708546\pi$
$578$	−16.6173	−0.691190
$579$	0	0
$580$	−21.4166	−0.889278
$581$	43.1749	1.79120
$582$	0	0
$583$	−0.780726	−0.0323344
$584$	−15.6323	−0.646867
$585$	0	0
$586$	−15.7068	−0.648844
$587$	−22.6991	−0.936894	−0.468447	−	0.883492i	$-0.655186\pi$
$587$	−22.6991	−0.936894	−0.468447	+	0.883492i	$0.655186\pi$
$588$	0	0
$589$	−0.728128	−0.0300020
$590$	28.0335	1.15412
$591$	0	0
$592$	−5.90747	−0.242795
$593$	−8.22045	−0.337574	−0.168787	−	0.985653i	$-0.553985\pi$
$593$	−8.22045	−0.337574	−0.168787	+	0.985653i	$0.553985\pi$
$594$	0	0
$595$	7.92335	0.324826
$596$	−4.16095	−0.170439
$597$	0	0
$598$	0	0
$599$	28.8240	1.17771	0.588857	−	0.808237i	$-0.299578\pi$
$599$	28.8240	1.17771	0.588857	+	0.808237i	$0.299578\pi$
$600$	0	0
$601$	1.46517	0.0597656	0.0298828	−	0.999553i	$-0.490487\pi$
$601$	1.46517	0.0597656	0.0298828	+	0.999553i	$0.490487\pi$
$602$	−45.1554	−1.84040
$603$	0	0
$604$	−7.91198	−0.321934
$605$	31.3129	1.27305
$606$	0	0
$607$	−15.8819	−0.644626	−0.322313	−	0.946633i	$-0.604460\pi$
$607$	−15.8819	−0.644626	−0.322313	+	0.946633i	$0.604460\pi$
$608$	0.259187	0.0105114
$609$	0	0
$610$	−14.6247	−0.592137
$611$	−36.3922	−1.47227
$612$	0	0
$613$	26.7081	1.07873	0.539364	−	0.842073i	$-0.318665\pi$
$613$	26.7081	1.07873	0.539364	+	0.842073i	$0.318665\pi$
$614$	1.74903	0.0705851
$615$	0	0
$616$	0.889368	0.0358337
$617$	−40.7125	−1.63903	−0.819513	−	0.573061i	$-0.805756\pi$
$617$	−40.7125	−1.63903	−0.819513	+	0.573061i	$0.805756\pi$
$618$	0	0
$619$	20.7836	0.835362	0.417681	−	0.908594i	$-0.362843\pi$
$619$	20.7836	0.835362	0.417681	+	0.908594i	$0.362843\pi$
$620$	8.02566	0.322318
$621$	0	0
$622$	−14.3459	−0.575220
$623$	68.7567	2.75468
$624$	0	0
$625$	−30.8123	−1.23249
$626$	−3.21704	−0.128579
$627$	0	0
$628$	−17.9469	−0.716158
$629$	−3.65433	−0.145708
$630$	0	0
$631$	18.7972	0.748305	0.374153	−	0.927367i	$-0.377934\pi$
$631$	18.7972	0.748305	0.374153	+	0.927367i	$0.377934\pi$
$632$	14.9619	0.595151
$633$	0	0
$634$	−8.57906	−0.340718
$635$	−15.6477	−0.620962
$636$	0	0
$637$	48.5906	1.92523
$638$	−1.48707	−0.0588735
$639$	0	0
$640$	−2.85684	−0.112927
$641$	−22.2658	−0.879445	−0.439722	−	0.898134i	$-0.644923\pi$
$641$	−22.2658	−0.879445	−0.439722	+	0.898134i	$0.644923\pi$
$642$	0	0
$643$	25.8885	1.02094	0.510472	−	0.859894i	$-0.329471\pi$
$643$	25.8885	1.02094	0.510472	+	0.859894i	$0.329471\pi$
$644$	0	0
$645$	0	0
$646$	0.160332	0.00630817
$647$	14.5567	0.572283	0.286141	−	0.958187i	$-0.407627\pi$
$647$	14.5567	0.572283	0.286141	+	0.958187i	$0.407627\pi$
$648$	0	0
$649$	1.94651	0.0764072
$650$	11.7253	0.459905
$651$	0	0
$652$	7.09440	0.277838
$653$	−31.0445	−1.21486	−0.607432	−	0.794372i	$-0.707800\pi$
$653$	−31.0445	−1.21486	−0.607432	+	0.794372i	$0.707800\pi$
$654$	0	0
$655$	−16.6256	−0.649618
$656$	8.45349	0.330053
$657$	0	0
$658$	43.9947	1.71509
$659$	32.5712	1.26879	0.634397	−	0.773007i	$-0.281248\pi$
$659$	32.5712	1.26879	0.634397	+	0.773007i	$0.281248\pi$
$660$	0	0
$661$	−18.0721	−0.702924	−0.351462	−	0.936202i	$-0.614315\pi$
$661$	−18.0721	−0.702924	−0.351462	+	0.936202i	$0.614315\pi$
$662$	32.4766	1.26224
$663$	0	0
$664$	−9.62975	−0.373707
$665$	3.31984	0.128738
$666$	0	0
$667$	0	0
$668$	−16.3201	−0.631443
$669$	0	0
$670$	−21.0156	−0.811903
$671$	−1.01547	−0.0392016
$672$	0	0
$673$	12.5132	0.482350	0.241175	−	0.970482i	$-0.422467\pi$
$673$	12.5132	0.482350	0.241175	+	0.970482i	$0.422467\pi$
$674$	19.8665	0.765230
$675$	0	0
$676$	0.754609	0.0290234
$677$	−9.99506	−0.384141	−0.192071	−	0.981381i	$-0.561520\pi$
$677$	−9.99506	−0.384141	−0.192071	+	0.981381i	$0.561520\pi$
$678$	0	0
$679$	53.1133	2.03830
$680$	−1.76723	−0.0677701
$681$	0	0
$682$	0.557262	0.0213387
$683$	−42.2674	−1.61732	−0.808659	−	0.588278i	$-0.799806\pi$
$683$	−42.2674	−1.61732	−0.808659	+	0.588278i	$0.799806\pi$
$684$	0	0
$685$	17.7602	0.678584
$686$	−27.3570	−1.04450
$687$	0	0
$688$	10.0715	0.383971
$689$	14.5968	0.556094
$690$	0	0
$691$	−1.03010	−0.0391870	−0.0195935	−	0.999808i	$-0.506237\pi$
$691$	−1.03010	−0.0391870	−0.0195935	+	0.999808i	$0.506237\pi$
$692$	15.4737	0.588223
$693$	0	0
$694$	18.9201	0.718199
$695$	−10.0401	−0.380844
$696$	0	0
$697$	5.22928	0.198073
$698$	−0.333496	−0.0126230
$699$	0	0
$700$	−14.1748	−0.535757
$701$	−6.09647	−0.230260	−0.115130	−	0.993350i	$-0.536729\pi$
$701$	−6.09647	−0.230260	−0.115130	+	0.993350i	$0.536729\pi$
$702$	0	0
$703$	−1.53114	−0.0577481
$704$	−0.198365	−0.00747616
$705$	0	0
$706$	−7.64132	−0.287585
$707$	84.6746	3.18452
$708$	0	0
$709$	−18.8129	−0.706532	−0.353266	−	0.935523i	$-0.614929\pi$
$709$	−18.8129	−0.706532	−0.353266	+	0.935523i	$0.614929\pi$
$710$	−31.7124	−1.19015
$711$	0	0
$712$	−15.3355	−0.574723
$713$	0	0
$714$	0	0
$715$	2.10172	0.0786000
$716$	−4.67890	−0.174859
$717$	0	0
$718$	−14.5582	−0.543306
$719$	39.0013	1.45450	0.727252	−	0.686371i	$-0.240797\pi$
$719$	39.0013	1.45450	0.727252	+	0.686371i	$0.240797\pi$
$720$	0	0
$721$	−14.9643	−0.557300
$722$	−18.9328	−0.704607
$723$	0	0
$724$	5.42873	0.201757
$725$	23.7009	0.880230
$726$	0	0
$727$	15.5647	0.577263	0.288631	−	0.957440i	$-0.406800\pi$
$727$	15.5647	0.577263	0.288631	+	0.957440i	$0.406800\pi$
$728$	−16.6280	−0.616276
$729$	0	0
$730$	44.6589	1.65290
$731$	6.23016	0.230431
$732$	0	0
$733$	26.0598	0.962542	0.481271	−	0.876572i	$-0.340175\pi$
$733$	26.0598	0.962542	0.481271	+	0.876572i	$0.340175\pi$
$734$	−24.3518	−0.898842
$735$	0	0
$736$	0	0
$737$	−1.45922	−0.0537510
$738$	0	0
$739$	9.92255	0.365007	0.182503	−	0.983205i	$-0.441580\pi$
$739$	9.92255	0.365007	0.182503	+	0.983205i	$0.441580\pi$
$740$	16.8767	0.620400
$741$	0	0
$742$	−17.6462	−0.647811
$743$	28.1984	1.03450	0.517249	−	0.855835i	$-0.326956\pi$
$743$	28.1984	1.03450	0.517249	+	0.855835i	$0.326956\pi$
$744$	0	0
$745$	11.8872	0.435513
$746$	−16.4397	−0.601899
$747$	0	0
$748$	−0.122707	−0.00448663
$749$	22.1707	0.810101
$750$	0	0
$751$	−37.3162	−1.36169	−0.680843	−	0.732429i	$-0.738386\pi$
$751$	−37.3162	−1.36169	−0.680843	+	0.732429i	$0.738386\pi$
$752$	−9.81259	−0.357828
$753$	0	0
$754$	27.8028	1.01252
$755$	22.6033	0.822618
$756$	0	0
$757$	33.0571	1.20148	0.600740	−	0.799445i	$-0.294873\pi$
$757$	33.0571	1.20148	0.600740	+	0.799445i	$0.294873\pi$
$758$	−7.37024	−0.267699
$759$	0	0
$760$	−0.740458	−0.0268592
$761$	3.31453	0.120152	0.0600759	−	0.998194i	$-0.480866\pi$
$761$	3.31453	0.120152	0.0600759	+	0.998194i	$0.480866\pi$
$762$	0	0
$763$	10.4401	0.377956
$764$	5.76873	0.208705
$765$	0	0
$766$	−23.2014	−0.838301
$767$	−36.3928	−1.31407
$768$	0	0
$769$	4.88958	0.176323	0.0881614	−	0.996106i	$-0.471901\pi$
$769$	4.88958	0.176323	0.0881614	+	0.996106i	$0.471901\pi$
$770$	−2.54079	−0.0915636
$771$	0	0
$772$	−2.78622	−0.100278
$773$	19.5595	0.703506	0.351753	−	0.936093i	$-0.385586\pi$
$773$	19.5595	0.703506	0.351753	+	0.936093i	$0.385586\pi$
$774$	0	0
$775$	−8.88167	−0.319039
$776$	−11.8464	−0.425261
$777$	0	0
$778$	−20.2192	−0.724894
$779$	2.19104	0.0785021
$780$	0	0
$781$	−2.20195	−0.0787920
$782$	0	0
$783$	0	0
$784$	13.1017	0.467919
$785$	51.2714	1.82995
$786$	0	0
$787$	−14.3627	−0.511976	−0.255988	−	0.966680i	$-0.582401\pi$
$787$	−14.3627	−0.511976	−0.255988	+	0.966680i	$0.582401\pi$
$788$	−27.7303	−0.987852
$789$	0	0
$790$	−42.7437	−1.52075
$791$	35.9012	1.27650
$792$	0	0
$793$	18.9856	0.674199
$794$	18.2589	0.647985
$795$	0	0
$796$	20.5232	0.727426
$797$	34.6629	1.22782	0.613911	−	0.789375i	$-0.289595\pi$
$797$	34.6629	1.22782	0.613911	+	0.789375i	$0.289595\pi$
$798$	0	0
$799$	−6.07001	−0.214742
$800$	3.16155	0.111778
$801$	0	0
$802$	−35.9055	−1.26787
$803$	3.10089	0.109428
$804$	0	0
$805$	0	0
$806$	−10.4188	−0.366987
$807$	0	0
$808$	−18.8858	−0.664402
$809$	36.5839	1.28622	0.643111	−	0.765773i	$-0.277643\pi$
$809$	36.5839	1.28622	0.643111	+	0.765773i	$0.277643\pi$
$810$	0	0
$811$	−29.4096	−1.03271	−0.516356	−	0.856374i	$-0.672712\pi$
$811$	−29.4096	−1.03271	−0.516356	+	0.856374i	$0.672712\pi$
$812$	−33.6110	−1.17952
$813$	0	0
$814$	1.17183	0.0410728
$815$	−20.2676	−0.709943
$816$	0	0
$817$	2.61040	0.0913263
$818$	−26.2924	−0.919293
$819$	0	0
$820$	−24.1503	−0.843365
$821$	0.295199	0.0103025	0.00515126	−	0.999987i	$-0.498360\pi$
$821$	0.295199	0.0103025	0.00515126	+	0.999987i	$0.498360\pi$
$822$	0	0
$823$	44.2981	1.54414	0.772068	−	0.635540i	$-0.219223\pi$
$823$	44.2981	1.54414	0.772068	+	0.635540i	$0.219223\pi$
$824$	3.33765	0.116272
$825$	0	0
$826$	43.9955	1.53080
$827$	−28.7630	−1.00019	−0.500094	−	0.865971i	$-0.666701\pi$
$827$	−28.7630	−1.00019	−0.500094	+	0.865971i	$0.666701\pi$
$828$	0	0
$829$	22.7583	0.790428	0.395214	−	0.918589i	$-0.370670\pi$
$829$	22.7583	0.790428	0.395214	+	0.918589i	$0.370670\pi$
$830$	27.5107	0.954909
$831$	0	0
$832$	3.70872	0.128577
$833$	8.10465	0.280809
$834$	0	0
$835$	46.6239	1.61349
$836$	−0.0514137	−0.00177818
$837$	0	0
$838$	−7.27755	−0.251399
$839$	19.0736	0.658494	0.329247	−	0.944244i	$-0.393205\pi$
$839$	19.0736	0.658494	0.329247	+	0.944244i	$0.393205\pi$
$840$	0	0
$841$	27.1992	0.937903
$842$	34.3333	1.18320
$843$	0	0
$844$	−22.0376	−0.758566
$845$	−2.15580	−0.0741618
$846$	0	0
$847$	49.1420	1.68854
$848$	3.93580	0.135156
$849$	0	0
$850$	1.95572	0.0670806
$851$	0	0
$852$	0	0
$853$	25.3316	0.867336	0.433668	−	0.901073i	$-0.357219\pi$
$853$	25.3316	0.867336	0.433668	+	0.901073i	$0.357219\pi$
$854$	−22.9518	−0.785395
$855$	0	0
$856$	−4.94497	−0.169015
$857$	8.26978	0.282490	0.141245	−	0.989975i	$-0.454889\pi$
$857$	8.26978	0.282490	0.141245	+	0.989975i	$0.454889\pi$
$858$	0	0
$859$	1.89082	0.0645139	0.0322569	−	0.999480i	$-0.489731\pi$
$859$	1.89082	0.0645139	0.0322569	+	0.999480i	$0.489731\pi$
$860$	−28.7726	−0.981139
$861$	0	0
$862$	18.2265	0.620798
$863$	−8.19194	−0.278857	−0.139428	−	0.990232i	$-0.544527\pi$
$863$	−8.19194	−0.278857	−0.139428	+	0.990232i	$0.544527\pi$
$864$	0	0
$865$	−44.2060	−1.50305
$866$	−5.14943	−0.174985
$867$	0	0
$868$	12.5954	0.427515
$869$	−2.96791	−0.100679
$870$	0	0
$871$	27.2822	0.924422
$872$	−2.32856	−0.0788550
$873$	0	0
$874$	0	0
$875$	−23.5480	−0.796068
$876$	0	0
$877$	50.4191	1.70253	0.851266	−	0.524735i	$-0.175836\pi$
$877$	50.4191	1.70253	0.851266	+	0.524735i	$0.175836\pi$
$878$	17.7487	0.598988
$879$	0	0
$880$	0.566698	0.0191034
$881$	2.36032	0.0795212	0.0397606	−	0.999209i	$-0.487340\pi$
$881$	2.36032	0.0795212	0.0397606	+	0.999209i	$0.487340\pi$
$882$	0	0
$883$	−22.4264	−0.754708	−0.377354	−	0.926069i	$-0.623166\pi$
$883$	−22.4264	−0.754708	−0.377354	+	0.926069i	$0.623166\pi$
$884$	2.29419	0.0771621
$885$	0	0
$886$	3.26754	0.109775
$887$	−37.0984	−1.24564	−0.622821	−	0.782365i	$-0.714013\pi$
$887$	−37.0984	−1.24564	−0.622821	+	0.782365i	$0.714013\pi$
$888$	0	0
$889$	−24.5574	−0.823628
$890$	43.8112	1.46855
$891$	0	0
$892$	16.0615	0.537778
$893$	−2.54330	−0.0851082
$894$	0	0
$895$	13.3669	0.446805
$896$	−4.48349	−0.149783
$897$	0	0
$898$	40.2916	1.34455
$899$	−21.0600	−0.702392
$900$	0	0
$901$	2.43467	0.0811105
$902$	−1.67688	−0.0558339
$903$	0	0
$904$	−8.00742	−0.266323
$905$	−15.5090	−0.515537
$906$	0	0
$907$	12.3091	0.408717	0.204358	−	0.978896i	$-0.434489\pi$
$907$	12.3091	0.408717	0.204358	+	0.978896i	$0.434489\pi$
$908$	23.7222	0.787248
$909$	0	0
$910$	47.5037	1.57473
$911$	−15.7544	−0.521965	−0.260983	−	0.965343i	$-0.584047\pi$
$911$	−15.7544	−0.521965	−0.260983	+	0.965343i	$0.584047\pi$
$912$	0	0
$913$	1.91020	0.0632185
$914$	−29.9182	−0.989608
$915$	0	0
$916$	19.2142	0.634856
$917$	−26.0921	−0.861636
$918$	0	0
$919$	−5.16750	−0.170460	−0.0852300	−	0.996361i	$-0.527163\pi$
$919$	−5.16750	−0.170460	−0.0852300	+	0.996361i	$0.527163\pi$
$920$	0	0
$921$	0	0
$922$	8.18649	0.269608
$923$	41.1687	1.35508
$924$	0	0
$925$	−18.6768	−0.614088
$926$	−2.92694	−0.0961852
$927$	0	0
$928$	7.49661	0.246088
$929$	−19.1921	−0.629672	−0.314836	−	0.949146i	$-0.601949\pi$
$929$	−19.1921	−0.629672	−0.314836	+	0.949146i	$0.601949\pi$
$930$	0	0
$931$	3.39580	0.111293
$932$	−21.4135	−0.701423
$933$	0	0
$934$	−33.2239	−1.08712
$935$	0.350556	0.0114644
$936$	0	0
$937$	6.05012	0.197649	0.0988244	−	0.995105i	$-0.468492\pi$
$937$	6.05012	0.197649	0.0988244	+	0.995105i	$0.468492\pi$
$938$	−32.9816	−1.07689
$939$	0	0
$940$	28.0330	0.914336
$941$	−9.68489	−0.315718	−0.157859	−	0.987462i	$-0.550459\pi$
$941$	−9.68489	−0.315718	−0.157859	+	0.987462i	$0.550459\pi$
$942$	0	0
$943$	0	0
$944$	−9.81277	−0.319378
$945$	0	0
$946$	−1.99783	−0.0649550
$947$	−23.0242	−0.748185	−0.374093	−	0.927391i	$-0.622046\pi$
$947$	−23.0242	−0.748185	−0.374093	+	0.927391i	$0.622046\pi$
$948$	0	0
$949$	−57.9757	−1.88197
$950$	0.819434	0.0265860
$951$	0	0
$952$	−2.77346	−0.0898885
$953$	−29.5625	−0.957624	−0.478812	−	0.877917i	$-0.658932\pi$
$953$	−29.5625	−0.957624	−0.478812	+	0.877917i	$0.658932\pi$
$954$	0	0
$955$	−16.4804	−0.533292
$956$	−6.20486	−0.200679
$957$	0	0
$958$	−27.5307	−0.889476
$959$	27.8727	0.900057
$960$	0	0
$961$	−23.1080	−0.745419
$962$	−21.9091	−0.706379
$963$	0	0
$964$	−7.77891	−0.250542
$965$	7.95979	0.256235
$966$	0	0
$967$	−54.4705	−1.75165	−0.875826	−	0.482627i	$-0.839683\pi$
$967$	−54.4705	−1.75165	−0.875826	+	0.482627i	$0.839683\pi$
$968$	−10.9607	−0.352289
$969$	0	0
$970$	33.8433	1.08664
$971$	−52.6205	−1.68867	−0.844336	−	0.535813i	$-0.820005\pi$
$971$	−52.6205	−1.68867	−0.844336	+	0.535813i	$0.820005\pi$
$972$	0	0
$973$	−15.7569	−0.505141
$974$	13.8291	0.443114
$975$	0	0
$976$	5.11918	0.163861
$977$	−14.1829	−0.453752	−0.226876	−	0.973924i	$-0.572851\pi$
$977$	−14.1829	−0.453752	−0.226876	+	0.973924i	$0.572851\pi$
$978$	0	0
$979$	3.04203	0.0972238
$980$	−37.4296	−1.19564
$981$	0	0
$982$	6.40601	0.204424
$983$	−44.8753	−1.43130	−0.715650	−	0.698459i	$-0.753869\pi$
$983$	−44.8753	−1.43130	−0.715650	+	0.698459i	$0.753869\pi$
$984$	0	0
$985$	79.2212	2.52420
$986$	4.63736	0.147684
$987$	0	0
$988$	0.961253	0.0305815
$989$	0	0
$990$	0	0
$991$	6.31690	0.200663	0.100331	−	0.994954i	$-0.468010\pi$
$991$	6.31690	0.200663	0.100331	+	0.994954i	$0.468010\pi$
$992$	−2.80927	−0.0891946
$993$	0	0
$994$	−49.7690	−1.57858
$995$	−58.6316	−1.85875
$996$	0	0
$997$	31.8622	1.00909	0.504544	−	0.863386i	$-0.331661\pi$
$997$	31.8622	1.00909	0.504544	+	0.863386i	$0.331661\pi$
$998$	4.42182	0.139970
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9522.2.a.ci.1.3		10
3.2	odd	2		9522.2.a.ch.1.8		10
23.17	odd	22		414.2.i.g.289.2	✓	20
23.19	odd	22		414.2.i.g.361.2	yes	20
23.22	odd	2		9522.2.a.cj.1.8		10
69.17	even	22		414.2.i.h.289.1	yes	20
69.65	even	22		414.2.i.h.361.1	yes	20
69.68	even	2		9522.2.a.cg.1.3		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
414.2.i.g.289.2	✓	20	23.17	odd	22
414.2.i.g.361.2	yes	20	23.19	odd	22
414.2.i.h.289.1	yes	20	69.17	even	22
414.2.i.h.361.1	yes	20	69.65	even	22
9522.2.a.cg.1.3		10	69.68	even	2
9522.2.a.ch.1.8		10	3.2	odd	2
9522.2.a.ci.1.3		10	1.1	even	1	trivial
9522.2.a.cj.1.8		10	23.22	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -2.85684 q^{5} -4.48349 q^{7} +1.00000 q^{8} -2.85684 q^{10} -0.198365 q^{11} +3.70872 q^{13} -4.48349 q^{14} +1.00000 q^{16} +0.618594 q^{17} +0.259187 q^{19} -2.85684 q^{20} -0.198365 q^{22} +3.16155 q^{25} +3.70872 q^{26} -4.48349 q^{28} +7.49661 q^{29} -2.80927 q^{31} +1.00000 q^{32} +0.618594 q^{34} +12.8086 q^{35} -5.90747 q^{37} +0.259187 q^{38} -2.85684 q^{40} +8.45349 q^{41} +10.0715 q^{43} -0.198365 q^{44} -9.81259 q^{47} +13.1017 q^{49} +3.16155 q^{50} +3.70872 q^{52} +3.93580 q^{53} +0.566698 q^{55} -4.48349 q^{56} +7.49661 q^{58} -9.81277 q^{59} +5.11918 q^{61} -2.80927 q^{62} +1.00000 q^{64} -10.5952 q^{65} +7.35623 q^{67} +0.618594 q^{68} +12.8086 q^{70} +11.1005 q^{71} -15.6323 q^{73} -5.90747 q^{74} +0.259187 q^{76} +0.889368 q^{77} +14.9619 q^{79} -2.85684 q^{80} +8.45349 q^{82} -9.62975 q^{83} -1.76723 q^{85} +10.0715 q^{86} -0.198365 q^{88} -15.3355 q^{89} -16.6280 q^{91} -9.81259 q^{94} -0.740458 q^{95} -11.8464 q^{97} +13.1017 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 10 q^{2} + 10 q^{4} - 10 q^{5} + 2 q^{7} + 10 q^{8} - 10 q^{10} - 12 q^{11} + 2 q^{14} + 10 q^{16} - 24 q^{17} - 8 q^{19} - 10 q^{20} - 12 q^{22} + 8 q^{25} + 2 q^{28} + 4 q^{29} + 18 q^{31} + 10 q^{32}+ \cdots + 36 q^{98}+O(q^{100}) \) 10 * q + 10 * q^2 + 10 * q^4 - 10 * q^5 + 2 * q^7 + 10 * q^8 - 10 * q^10 - 12 * q^11 + 2 * q^14 + 10 * q^16 - 24 * q^17 - 8 * q^19 - 10 * q^20 - 12 * q^22 + 8 * q^25 + 2 * q^28 + 4 * q^29 + 18 * q^31 + 10 * q^32 - 24 * q^34 - 24 * q^35 - 12 * q^37 - 8 * q^38 - 10 * q^40 + 28 * q^41 - 8 * q^43 - 12 * q^44 - 16 * q^47 + 36 * q^49 + 8 * q^50 - 34 * q^53 + 30 * q^55 + 2 * q^56 + 4 * q^58 - 22 * q^59 - 30 * q^61 + 18 * q^62 + 10 * q^64 - 36 * q^65 - 18 * q^67 - 24 * q^68 - 24 * q^70 - 28 * q^71 - 20 * q^73 - 12 * q^74 - 8 * q^76 - 20 * q^77 - 2 * q^79 - 10 * q^80 + 28 * q^82 - 44 * q^83 + 16 * q^85 - 8 * q^86 - 12 * q^88 - 44 * q^89 - 22 * q^91 - 16 * q^94 - 10 * q^95 - 18 * q^97 + 36 * q^98

Introduction

L-functions

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