LMFDB - Embedded newform 9522.2.a.bw.1.5

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 = [5,5,0,5,-2,0,9,5,0,-2,-1,0,1,9,0,5,1,0,10,-2,0,-1,0,0,-11,1, 0,9,-10] 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:	$5$
Coefficient field:	$\Q(\zeta_{22})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 4x^{3} + 3x^{2} + 3x - 1 $ x^5 - x^4 - 4x^3 + 3x^2 + 3*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 46)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.68251$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.39788 q^{5} +4.29177 q^{7} +1.00000 q^{8} +1.39788 q^{10} +0.221566 q^{11} +1.98798 q^{13} +4.29177 q^{14} +1.00000 q^{16} +0.111215 q^{17} -3.23585 q^{19} +1.39788 q^{20} +0.221566 q^{22} -3.04594 q^{25} +1.98798 q^{26} +4.29177 q^{28} +3.56863 q^{29} +6.15787 q^{31} +1.00000 q^{32} +0.111215 q^{34} +5.99937 q^{35} -4.22808 q^{37} -3.23585 q^{38} +1.39788 q^{40} -3.41316 q^{41} +3.24797 q^{43} +0.221566 q^{44} -7.73852 q^{47} +11.4193 q^{49} -3.04594 q^{50} +1.98798 q^{52} +0.146111 q^{53} +0.309721 q^{55} +4.29177 q^{56} +3.56863 q^{58} +13.1693 q^{59} +5.68965 q^{61} +6.15787 q^{62} +1.00000 q^{64} +2.77896 q^{65} +14.3384 q^{67} +0.111215 q^{68} +5.99937 q^{70} +5.84511 q^{71} -3.79241 q^{73} -4.22808 q^{74} -3.23585 q^{76} +0.950909 q^{77} +8.72882 q^{79} +1.39788 q^{80} -3.41316 q^{82} -15.6494 q^{83} +0.155465 q^{85} +3.24797 q^{86} +0.221566 q^{88} +1.01934 q^{89} +8.53197 q^{91} -7.73852 q^{94} -4.52332 q^{95} +18.2132 q^{97} +11.4193 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{2} + 5 q^{4} - 2 q^{5} + 9 q^{7} + 5 q^{8} - 2 q^{10} - q^{11} + q^{13} + 9 q^{14} + 5 q^{16} + q^{17} + 10 q^{19} - 2 q^{20} - q^{22} - 11 q^{25} + q^{26} + 9 q^{28} - 10 q^{29} + 5 q^{32}+ \cdots + 12 q^{98}+O(q^{100}) $ 5 * q + 5 * q^2 + 5 * q^4 - 2 * q^5 + 9 * q^7 + 5 * q^8 - 2 * q^10 - q^11 + q^13 + 9 * q^14 + 5 * q^16 + q^17 + 10 * q^19 - 2 * q^20 - q^22 - 11 * q^25 + q^26 + 9 * q^28 - 10 * q^29 + 5 * q^32 + q^34 + 3 * q^35 + 20 * q^37 + 10 * q^38 - 2 * q^40 + 3 * q^41 + 27 * q^43 - q^44 + 11 * q^47 + 12 * q^49 - 11 * q^50 + q^52 + 2 * q^53 - 4 * q^55 + 9 * q^56 - 10 * q^58 + 17 * q^59 + 7 * q^61 + 5 * q^64 + 15 * q^65 + 28 * q^67 + q^68 + 3 * q^70 + 8 * q^73 + 20 * q^74 + 10 * q^76 - 26 * q^77 + 16 * q^79 - 2 * q^80 + 3 * q^82 - 18 * q^83 - 7 * q^85 + 27 * q^86 - q^88 + 14 * q^89 + 26 * q^91 + 11 * q^94 - 26 * q^95 + 47 * q^97 + 12 * 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
$5$	1.39788	0.625150	0.312575	−	0.949893i	$-0.398808\pi$
$5$	1.39788	0.625150	0.312575	+	0.949893i	$0.398808\pi$
$6$	0	0
$7$	4.29177	1.62214	0.811069	−	0.584951i	$-0.198886\pi$
$7$	4.29177	1.62214	0.811069	+	0.584951i	$0.198886\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	1.39788	0.442048
$11$	0.221566	0.0668045	0.0334023	−	0.999442i	$-0.489366\pi$
$11$	0.221566	0.0668045	0.0334023	+	0.999442i	$0.489366\pi$
$12$	0	0
$13$	1.98798	0.551367	0.275684	−	0.961248i	$-0.411096\pi$
$13$	1.98798	0.551367	0.275684	+	0.961248i	$0.411096\pi$
$14$	4.29177	1.14702
$15$	0	0
$16$	1.00000	0.250000
$17$	0.111215	0.0269736	0.0134868	−	0.999909i	$-0.495707\pi$
$17$	0.111215	0.0269736	0.0134868	+	0.999909i	$0.495707\pi$
$18$	0	0
$19$	−3.23585	−0.742355	−0.371177	−	0.928562i	$-0.621046\pi$
$19$	−3.23585	−0.742355	−0.371177	+	0.928562i	$0.621046\pi$
$20$	1.39788	0.312575
$21$	0	0
$22$	0.221566	0.0472379
$23$	0	0
$23$	0	0
$24$	0	0
$25$	−3.04594	−0.609188
$26$	1.98798	0.389876
$27$	0	0
$28$	4.29177	0.811069
$29$	3.56863	0.662678	0.331339	−	0.943512i	$-0.392500\pi$
$29$	3.56863	0.662678	0.331339	+	0.943512i	$0.392500\pi$
$30$	0	0
$31$	6.15787	1.10599	0.552993	−	0.833186i	$-0.313486\pi$
$31$	6.15787	1.10599	0.552993	+	0.833186i	$0.313486\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	0.111215	0.0190732
$35$	5.99937	1.01408
$36$	0	0
$37$	−4.22808	−0.695092	−0.347546	−	0.937663i	$-0.612985\pi$
$37$	−4.22808	−0.695092	−0.347546	+	0.937663i	$0.612985\pi$
$38$	−3.23585	−0.524924
$39$	0	0
$40$	1.39788	0.221024
$41$	−3.41316	−0.533047	−0.266523	−	0.963828i	$-0.585875\pi$
$41$	−3.41316	−0.533047	−0.266523	+	0.963828i	$0.585875\pi$
$42$	0	0
$43$	3.24797	0.495311	0.247656	−	0.968848i	$-0.420340\pi$
$43$	3.24797	0.495311	0.247656	+	0.968848i	$0.420340\pi$
$44$	0.221566	0.0334023
$45$	0	0
$46$	0	0
$47$	−7.73852	−1.12878	−0.564389	−	0.825509i	$-0.690888\pi$
$47$	−7.73852	−1.12878	−0.564389	+	0.825509i	$0.690888\pi$
$48$	0	0
$49$	11.4193	1.63133
$50$	−3.04594	−0.430761
$51$	0	0
$52$	1.98798	0.275684
$53$	0.146111	0.0200699	0.0100349	−	0.999950i	$-0.496806\pi$
$53$	0.146111	0.0200699	0.0100349	+	0.999950i	$0.496806\pi$
$54$	0	0
$55$	0.309721	0.0417628
$56$	4.29177	0.573512
$57$	0	0
$58$	3.56863	0.468584
$59$	13.1693	1.71449	0.857246	−	0.514908i	$-0.172174\pi$
$59$	13.1693	1.71449	0.857246	+	0.514908i	$0.172174\pi$
$60$	0	0
$61$	5.68965	0.728485	0.364242	−	0.931304i	$-0.381328\pi$
$61$	5.68965	0.728485	0.364242	+	0.931304i	$0.381328\pi$
$62$	6.15787	0.782051
$63$	0	0
$64$	1.00000	0.125000
$65$	2.77896	0.344687
$66$	0	0
$67$	14.3384	1.75172	0.875859	−	0.482567i	$-0.160296\pi$
$67$	14.3384	1.75172	0.875859	+	0.482567i	$0.160296\pi$
$68$	0.111215	0.0134868
$69$	0	0
$70$	5.99937	0.717062
$71$	5.84511	0.693687	0.346844	−	0.937923i	$-0.387253\pi$
$71$	5.84511	0.693687	0.346844	+	0.937923i	$0.387253\pi$
$72$	0	0
$73$	−3.79241	−0.443868	−0.221934	−	0.975062i	$-0.571237\pi$
$73$	−3.79241	−0.443868	−0.221934	+	0.975062i	$0.571237\pi$
$74$	−4.22808	−0.491504
$75$	0	0
$76$	−3.23585	−0.371177
$77$	0.950909	0.108366
$78$	0	0
$79$	8.72882	0.982069	0.491034	−	0.871140i	$-0.336619\pi$
$79$	8.72882	0.982069	0.491034	+	0.871140i	$0.336619\pi$
$80$	1.39788	0.156287
$81$	0	0
$82$	−3.41316	−0.376921
$83$	−15.6494	−1.71774	−0.858872	−	0.512191i	$-0.828834\pi$
$83$	−15.6494	−1.71774	−0.858872	+	0.512191i	$0.828834\pi$
$84$	0	0
$85$	0.155465	0.0168625
$86$	3.24797	0.350238
$87$	0	0
$88$	0.221566	0.0236190
$89$	1.01934	0.108050	0.0540248	−	0.998540i	$-0.482795\pi$
$89$	1.01934	0.108050	0.0540248	+	0.998540i	$0.482795\pi$
$90$	0	0
$91$	8.53197	0.894394
$92$	0	0
$93$	0	0
$94$	−7.73852	−0.798167
$95$	−4.52332	−0.464083
$96$	0	0
$97$	18.2132	1.84927	0.924633	−	0.380858i	$-0.124372\pi$
$97$	18.2132	1.84927	0.924633	+	0.380858i	$0.124372\pi$
$98$	11.4193	1.15352
$99$	0	0
$100$	−3.04594	−0.304594
$101$	−3.25891	−0.324273	−0.162137	−	0.986768i	$-0.551839\pi$
$101$	−3.25891	−0.324273	−0.162137	+	0.986768i	$0.551839\pi$
$102$	0	0
$103$	10.1126	0.996420	0.498210	−	0.867056i	$-0.333991\pi$
$103$	10.1126	0.996420	0.498210	+	0.867056i	$0.333991\pi$
$104$	1.98798	0.194938
$105$	0	0
$106$	0.146111	0.0141916
$107$	−0.475845	−0.0460017	−0.0230009	−	0.999735i	$-0.507322\pi$
$107$	−0.475845	−0.0460017	−0.0230009	+	0.999735i	$0.507322\pi$
$108$	0	0
$109$	−9.80872	−0.939505	−0.469753	−	0.882798i	$-0.655657\pi$
$109$	−9.80872	−0.939505	−0.469753	+	0.882798i	$0.655657\pi$
$110$	0.309721	0.0295308
$111$	0	0
$112$	4.29177	0.405534
$113$	8.86799	0.834231	0.417115	−	0.908854i	$-0.363041\pi$
$113$	8.86799	0.834231	0.417115	+	0.908854i	$0.363041\pi$
$114$	0	0
$115$	0	0
$116$	3.56863	0.331339
$117$	0	0
$118$	13.1693	1.21233
$119$	0.477309	0.0437548
$120$	0	0
$121$	−10.9509	−0.995537
$122$	5.68965	0.515117
$123$	0	0
$124$	6.15787	0.552993
$125$	−11.2472	−1.00598
$126$	0	0
$127$	2.61962	0.232454	0.116227	−	0.993223i	$-0.462920\pi$
$127$	2.61962	0.232454	0.116227	+	0.993223i	$0.462920\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	2.77896	0.243731
$131$	−15.3207	−1.33857	−0.669286	−	0.743005i	$-0.733400\pi$
$131$	−15.3207	−1.33857	−0.669286	+	0.743005i	$0.733400\pi$
$132$	0	0
$133$	−13.8875	−1.20420
$134$	14.3384	1.23865
$135$	0	0
$136$	0.111215	0.00953660
$137$	−10.3092	−0.880771	−0.440386	−	0.897809i	$-0.645158\pi$
$137$	−10.3092	−0.880771	−0.440386	+	0.897809i	$0.645158\pi$
$138$	0	0
$139$	−0.569454	−0.0483005	−0.0241502	−	0.999708i	$-0.507688\pi$
$139$	−0.569454	−0.0483005	−0.0241502	+	0.999708i	$0.507688\pi$
$140$	5.99937	0.507039
$141$	0	0
$142$	5.84511	0.490511
$143$	0.440469	0.0368338
$144$	0	0
$145$	4.98851	0.414273
$146$	−3.79241	−0.313862
$147$	0	0
$148$	−4.22808	−0.347546
$149$	13.1652	1.07853	0.539267	−	0.842135i	$-0.318701\pi$
$149$	13.1652	1.07853	0.539267	+	0.842135i	$0.318701\pi$
$150$	0	0
$151$	−9.08797	−0.739569	−0.369784	−	0.929118i	$-0.620568\pi$
$151$	−9.08797	−0.739569	−0.369784	+	0.929118i	$0.620568\pi$
$152$	−3.23585	−0.262462
$153$	0	0
$154$	0.950909	0.0766264
$155$	8.60795	0.691407
$156$	0	0
$157$	8.71215	0.695306	0.347653	−	0.937623i	$-0.386979\pi$
$157$	8.71215	0.695306	0.347653	+	0.937623i	$0.386979\pi$
$158$	8.72882	0.694428
$159$	0	0
$160$	1.39788	0.110512
$161$	0	0
$162$	0	0
$163$	10.6812	0.836616	0.418308	−	0.908305i	$-0.362623\pi$
$163$	10.6812	0.836616	0.418308	+	0.908305i	$0.362623\pi$
$164$	−3.41316	−0.266523
$165$	0	0
$166$	−15.6494	−1.21463
$167$	−7.93820	−0.614276	−0.307138	−	0.951665i	$-0.599371\pi$
$167$	−7.93820	−0.614276	−0.307138	+	0.951665i	$0.599371\pi$
$168$	0	0
$169$	−9.04792	−0.695994
$170$	0.155465	0.0119236
$171$	0	0
$172$	3.24797	0.247656
$173$	7.45918	0.567111	0.283555	−	0.958956i	$-0.408486\pi$
$173$	7.45918	0.567111	0.283555	+	0.958956i	$0.408486\pi$
$174$	0	0
$175$	−13.0725	−0.988186
$176$	0.221566	0.0167011
$177$	0	0
$178$	1.01934	0.0764026
$179$	−20.8381	−1.55751	−0.778757	−	0.627326i	$-0.784149\pi$
$179$	−20.8381	−1.55751	−0.778757	+	0.627326i	$0.784149\pi$
$180$	0	0
$181$	4.22222	0.313835	0.156917	−	0.987612i	$-0.449844\pi$
$181$	4.22222	0.313835	0.156917	+	0.987612i	$0.449844\pi$
$182$	8.53197	0.632432
$183$	0	0
$184$	0	0
$185$	−5.91033	−0.434536
$186$	0	0
$187$	0.0246414	0.00180196
$188$	−7.73852	−0.564389
$189$	0	0
$190$	−4.52332	−0.328156
$191$	−14.1891	−1.02669	−0.513344	−	0.858183i	$-0.671593\pi$
$191$	−14.1891	−1.02669	−0.513344	+	0.858183i	$0.671593\pi$
$192$	0	0
$193$	2.00602	0.144397	0.0721983	−	0.997390i	$-0.476999\pi$
$193$	2.00602	0.144397	0.0721983	+	0.997390i	$0.476999\pi$
$194$	18.2132	1.30763
$195$	0	0
$196$	11.4193	0.815664
$197$	−12.9688	−0.923985	−0.461993	−	0.886884i	$-0.652865\pi$
$197$	−12.9688	−0.923985	−0.461993	+	0.886884i	$0.652865\pi$
$198$	0	0
$199$	11.8687	0.841347	0.420673	−	0.907212i	$-0.361794\pi$
$199$	11.8687	0.841347	0.420673	+	0.907212i	$0.361794\pi$
$200$	−3.04594	−0.215380
$201$	0	0
$202$	−3.25891	−0.229296
$203$	15.3157	1.07495
$204$	0	0
$205$	−4.77119	−0.333234
$206$	10.1126	0.704576
$207$	0	0
$208$	1.98798	0.137842
$209$	−0.716953	−0.0495927
$210$	0	0
$211$	19.6017	1.34943	0.674717	−	0.738077i	$-0.264266\pi$
$211$	19.6017	1.34943	0.674717	+	0.738077i	$0.264266\pi$
$212$	0.146111	0.0100349
$213$	0	0
$214$	−0.475845	−0.0325281
$215$	4.54027	0.309644
$216$	0	0
$217$	26.4282	1.79406
$218$	−9.80872	−0.664331
$219$	0	0
$220$	0.309721	0.0208814
$221$	0.221093	0.0148724
$222$	0	0
$223$	−16.7226	−1.11983	−0.559913	−	0.828551i	$-0.689165\pi$
$223$	−16.7226	−1.11983	−0.559913	+	0.828551i	$0.689165\pi$
$224$	4.29177	0.286756
$225$	0	0
$226$	8.86799	0.589890
$227$	14.1479	0.939029	0.469514	−	0.882925i	$-0.344429\pi$
$227$	14.1479	0.939029	0.469514	+	0.882925i	$0.344429\pi$
$228$	0	0
$229$	23.3847	1.54531	0.772653	−	0.634828i	$-0.218929\pi$
$229$	23.3847	1.54531	0.772653	+	0.634828i	$0.218929\pi$
$230$	0	0
$231$	0	0
$232$	3.56863	0.234292
$233$	8.64601	0.566419	0.283209	−	0.959058i	$-0.408601\pi$
$233$	8.64601	0.566419	0.283209	+	0.959058i	$0.408601\pi$
$234$	0	0
$235$	−10.8175	−0.705656
$236$	13.1693	0.857246
$237$	0	0
$238$	0.477309	0.0309394
$239$	−8.84860	−0.572368	−0.286184	−	0.958175i	$-0.592387\pi$
$239$	−8.84860	−0.572368	−0.286184	+	0.958175i	$0.592387\pi$
$240$	0	0
$241$	−5.58741	−0.359917	−0.179958	−	0.983674i	$-0.557596\pi$
$241$	−5.58741	−0.359917	−0.179958	+	0.983674i	$0.557596\pi$
$242$	−10.9509	−0.703951
$243$	0	0
$244$	5.68965	0.364242
$245$	15.9628	1.01982
$246$	0	0
$247$	−6.43282	−0.409310
$248$	6.15787	0.391025
$249$	0	0
$250$	−11.2472	−0.711338
$251$	−19.1964	−1.21167	−0.605834	−	0.795591i	$-0.707160\pi$
$251$	−19.1964	−1.21167	−0.605834	+	0.795591i	$0.707160\pi$
$252$	0	0
$253$	0	0
$254$	2.61962	0.164370
$255$	0	0
$256$	1.00000	0.0625000
$257$	−8.43545	−0.526189	−0.263095	−	0.964770i	$-0.584743\pi$
$257$	−8.43545	−0.526189	−0.263095	+	0.964770i	$0.584743\pi$
$258$	0	0
$259$	−18.1459	−1.12753
$260$	2.77896	0.172344
$261$	0	0
$262$	−15.3207	−0.946513
$263$	21.4538	1.32290	0.661448	−	0.749991i	$-0.269942\pi$
$263$	21.4538	1.32290	0.661448	+	0.749991i	$0.269942\pi$
$264$	0	0
$265$	0.204245	0.0125467
$266$	−13.8875	−0.851499
$267$	0	0
$268$	14.3384	0.875859
$269$	21.5844	1.31602	0.658012	−	0.753008i	$-0.271398\pi$
$269$	21.5844	1.31602	0.658012	+	0.753008i	$0.271398\pi$
$270$	0	0
$271$	−12.8063	−0.777930	−0.388965	−	0.921252i	$-0.627167\pi$
$271$	−12.8063	−0.777930	−0.388965	+	0.921252i	$0.627167\pi$
$272$	0.111215	0.00674340
$273$	0	0
$274$	−10.3092	−0.622799
$275$	−0.674875	−0.0406965
$276$	0	0
$277$	−15.6781	−0.942007	−0.471004	−	0.882131i	$-0.656108\pi$
$277$	−15.6781	−0.942007	−0.471004	+	0.882131i	$0.656108\pi$
$278$	−0.569454	−0.0341536
$279$	0	0
$280$	5.99937	0.358531
$281$	−9.41093	−0.561409	−0.280705	−	0.959794i	$-0.590568\pi$
$281$	−9.41093	−0.561409	−0.280705	+	0.959794i	$0.590568\pi$
$282$	0	0
$283$	−7.08348	−0.421069	−0.210534	−	0.977586i	$-0.567520\pi$
$283$	−7.08348	−0.421069	−0.210534	+	0.977586i	$0.567520\pi$
$284$	5.84511	0.346844
$285$	0	0
$286$	0.440469	0.0260455
$287$	−14.6485	−0.864675
$288$	0	0
$289$	−16.9876	−0.999272
$290$	4.98851	0.292935
$291$	0	0
$292$	−3.79241	−0.221934
$293$	−26.4324	−1.54419	−0.772097	−	0.635504i	$-0.780792\pi$
$293$	−26.4324	−1.54419	−0.772097	+	0.635504i	$0.780792\pi$
$294$	0	0
$295$	18.4090	1.07181
$296$	−4.22808	−0.245752
$297$	0	0
$298$	13.1652	0.762639
$299$	0	0
$300$	0	0
$301$	13.9396	0.803463
$302$	−9.08797	−0.522954
$303$	0	0
$304$	−3.23585	−0.185589
$305$	7.95343	0.455412
$306$	0	0
$307$	11.2935	0.644554	0.322277	−	0.946645i	$-0.395552\pi$
$307$	11.2935	0.644554	0.322277	+	0.946645i	$0.395552\pi$
$308$	0.950909	0.0541831
$309$	0	0
$310$	8.60795	0.488899
$311$	8.16799	0.463164	0.231582	−	0.972815i	$-0.425610\pi$
$311$	8.16799	0.463164	0.231582	+	0.972815i	$0.425610\pi$
$312$	0	0
$313$	7.85449	0.443962	0.221981	−	0.975051i	$-0.428748\pi$
$313$	7.85449	0.443962	0.221981	+	0.975051i	$0.428748\pi$
$314$	8.71215	0.491655
$315$	0	0
$316$	8.72882	0.491034
$317$	−13.2797	−0.745861	−0.372931	−	0.927859i	$-0.621647\pi$
$317$	−13.2797	−0.745861	−0.372931	+	0.927859i	$0.621647\pi$
$318$	0	0
$319$	0.790685	0.0442699
$320$	1.39788	0.0781437
$321$	0	0
$322$	0	0
$323$	−0.359875	−0.0200240
$324$	0	0
$325$	−6.05528	−0.335886
$326$	10.6812	0.591577
$327$	0	0
$328$	−3.41316	−0.188461
$329$	−33.2120	−1.83103
$330$	0	0
$331$	1.31086	0.0720516	0.0360258	−	0.999351i	$-0.488530\pi$
$331$	1.31086	0.0720516	0.0360258	+	0.999351i	$0.488530\pi$
$332$	−15.6494	−0.858872
$333$	0	0
$334$	−7.93820	−0.434359
$335$	20.0434	1.09509
$336$	0	0
$337$	34.2683	1.86672	0.933358	−	0.358947i	$-0.116864\pi$
$337$	34.2683	1.86672	0.933358	+	0.358947i	$0.116864\pi$
$338$	−9.04792	−0.492142
$339$	0	0
$340$	0.155465	0.00843126
$341$	1.36437	0.0738849
$342$	0	0
$343$	18.9666	1.02410
$344$	3.24797	0.175119
$345$	0	0
$346$	7.45918	0.401008
$347$	−24.6201	−1.32168	−0.660838	−	0.750528i	$-0.729799\pi$
$347$	−24.6201	−1.32168	−0.660838	+	0.750528i	$0.729799\pi$
$348$	0	0
$349$	−20.5915	−1.10224	−0.551119	−	0.834427i	$-0.685799\pi$
$349$	−20.5915	−1.10224	−0.551119	+	0.834427i	$0.685799\pi$
$350$	−13.0725	−0.698753
$351$	0	0
$352$	0.221566	0.0118095
$353$	−27.3130	−1.45372	−0.726862	−	0.686784i	$-0.759022\pi$
$353$	−27.3130	−1.45372	−0.726862	+	0.686784i	$0.759022\pi$
$354$	0	0
$355$	8.17075	0.433659
$356$	1.01934	0.0540248
$357$	0	0
$358$	−20.8381	−1.10133
$359$	−37.5903	−1.98394	−0.991971	−	0.126468i	$-0.959636\pi$
$359$	−37.5903	−1.98394	−0.991971	+	0.126468i	$0.959636\pi$
$360$	0	0
$361$	−8.52928	−0.448909
$362$	4.22222	0.221915
$363$	0	0
$364$	8.53197	0.447197
$365$	−5.30132	−0.277484
$366$	0	0
$367$	22.3474	1.16653	0.583263	−	0.812283i	$-0.301776\pi$
$367$	22.3474	1.16653	0.583263	+	0.812283i	$0.301776\pi$
$368$	0	0
$369$	0	0
$370$	−5.91033	−0.307264
$371$	0.627075	0.0325561
$372$	0	0
$373$	−29.7918	−1.54256	−0.771279	−	0.636497i	$-0.780383\pi$
$373$	−29.7918	−1.54256	−0.771279	+	0.636497i	$0.780383\pi$
$374$	0.0246414	0.00127418
$375$	0	0
$376$	−7.73852	−0.399084
$377$	7.09438	0.365379
$378$	0	0
$379$	−21.5505	−1.10698	−0.553488	−	0.832857i	$-0.686703\pi$
$379$	−21.5505	−1.10698	−0.553488	+	0.832857i	$0.686703\pi$
$380$	−4.52332	−0.232041
$381$	0	0
$382$	−14.1891	−0.725978
$383$	12.8513	0.656668	0.328334	−	0.944562i	$-0.393513\pi$
$383$	12.8513	0.656668	0.328334	+	0.944562i	$0.393513\pi$
$384$	0	0
$385$	1.32925	0.0677450
$386$	2.00602	0.102104
$387$	0	0
$388$	18.2132	0.924633
$389$	−0.489944	−0.0248411	−0.0124206	−	0.999923i	$-0.503954\pi$
$389$	−0.489944	−0.0248411	−0.0124206	+	0.999923i	$0.503954\pi$
$390$	0	0
$391$	0	0
$392$	11.4193	0.576762
$393$	0	0
$394$	−12.9688	−0.653356
$395$	12.2018	0.613940
$396$	0	0
$397$	14.9274	0.749183	0.374592	−	0.927190i	$-0.377783\pi$
$397$	14.9274	0.749183	0.374592	+	0.927190i	$0.377783\pi$
$398$	11.8687	0.594922
$399$	0	0
$400$	−3.04594	−0.152297
$401$	36.8123	1.83832	0.919159	−	0.393886i	$-0.128870\pi$
$401$	36.8123	1.83832	0.919159	+	0.393886i	$0.128870\pi$
$402$	0	0
$403$	12.2417	0.609805
$404$	−3.25891	−0.162137
$405$	0	0
$406$	15.3157	0.760108
$407$	−0.936796	−0.0464353
$408$	0	0
$409$	−27.3001	−1.34990	−0.674951	−	0.737862i	$-0.735835\pi$
$409$	−27.3001	−1.34990	−0.674951	+	0.737862i	$0.735835\pi$
$410$	−4.77119	−0.235632
$411$	0	0
$412$	10.1126	0.498210
$413$	56.5194	2.78114
$414$	0	0
$415$	−21.8759	−1.07385
$416$	1.98798	0.0974689
$417$	0	0
$418$	−0.716953	−0.0350673
$419$	−16.9066	−0.825943	−0.412972	−	0.910744i	$-0.635509\pi$
$419$	−16.9066	−0.825943	−0.412972	+	0.910744i	$0.635509\pi$
$420$	0	0
$421$	−4.43267	−0.216035	−0.108017	−	0.994149i	$-0.534450\pi$
$421$	−4.43267	−0.216035	−0.108017	+	0.994149i	$0.534450\pi$
$422$	19.6017	0.954194
$423$	0	0
$424$	0.146111	0.00709578
$425$	−0.338754	−0.0164320
$426$	0	0
$427$	24.4187	1.18170
$428$	−0.475845	−0.0230009
$429$	0	0
$430$	4.54027	0.218951
$431$	27.9986	1.34865	0.674324	−	0.738436i	$-0.264435\pi$
$431$	27.9986	1.34865	0.674324	+	0.738436i	$0.264435\pi$
$432$	0	0
$433$	12.2280	0.587641	0.293820	−	0.955861i	$-0.405073\pi$
$433$	12.2280	0.587641	0.293820	+	0.955861i	$0.405073\pi$
$434$	26.4282	1.26859
$435$	0	0
$436$	−9.80872	−0.469753
$437$	0	0
$438$	0	0
$439$	15.3054	0.730487	0.365244	−	0.930912i	$-0.380986\pi$
$439$	15.3054	0.730487	0.365244	+	0.930912i	$0.380986\pi$
$440$	0.309721	0.0147654
$441$	0	0
$442$	0.221093	0.0105163
$443$	−7.54063	−0.358266	−0.179133	−	0.983825i	$-0.557329\pi$
$443$	−7.54063	−0.358266	−0.179133	+	0.983825i	$0.557329\pi$
$444$	0	0
$445$	1.42491	0.0675471
$446$	−16.7226	−0.791837
$447$	0	0
$448$	4.29177	0.202767
$449$	12.4935	0.589606	0.294803	−	0.955558i	$-0.404746\pi$
$449$	12.4935	0.589606	0.294803	+	0.955558i	$0.404746\pi$
$450$	0	0
$451$	−0.756240	−0.0356099
$452$	8.86799	0.417115
$453$	0	0
$454$	14.1479	0.663994
$455$	11.9267	0.559130
$456$	0	0
$457$	27.6630	1.29402	0.647011	−	0.762481i	$-0.276019\pi$
$457$	27.6630	1.29402	0.647011	+	0.762481i	$0.276019\pi$
$458$	23.3847	1.09270
$459$	0	0
$460$	0	0
$461$	−11.2174	−0.522445	−0.261223	−	0.965279i	$-0.584126\pi$
$461$	−11.2174	−0.522445	−0.261223	+	0.965279i	$0.584126\pi$
$462$	0	0
$463$	9.47539	0.440359	0.220179	−	0.975459i	$-0.429336\pi$
$463$	9.47539	0.440359	0.220179	+	0.975459i	$0.429336\pi$
$464$	3.56863	0.165669
$465$	0	0
$466$	8.64601	0.400519
$467$	9.33895	0.432155	0.216077	−	0.976376i	$-0.430674\pi$
$467$	9.33895	0.432155	0.216077	+	0.976376i	$0.430674\pi$
$468$	0	0
$469$	61.5373	2.84153
$470$	−10.8175	−0.498974
$471$	0	0
$472$	13.1693	0.606164
$473$	0.719639	0.0330890
$474$	0	0
$475$	9.85620	0.452233
$476$	0.477309	0.0218774
$477$	0	0
$478$	−8.84860	−0.404726
$479$	1.20307	0.0549699	0.0274849	−	0.999622i	$-0.491250\pi$
$479$	1.20307	0.0549699	0.0274849	+	0.999622i	$0.491250\pi$
$480$	0	0
$481$	−8.40535	−0.383251
$482$	−5.58741	−0.254500
$483$	0	0
$484$	−10.9509	−0.497769
$485$	25.4598	1.15607
$486$	0	0
$487$	−4.47248	−0.202667	−0.101334	−	0.994852i	$-0.532311\pi$
$487$	−4.47248	−0.202667	−0.101334	+	0.994852i	$0.532311\pi$
$488$	5.68965	0.257558
$489$	0	0
$490$	15.9628	0.721125
$491$	−12.0574	−0.544143	−0.272071	−	0.962277i	$-0.587709\pi$
$491$	−12.0574	−0.544143	−0.272071	+	0.962277i	$0.587709\pi$
$492$	0	0
$493$	0.396885	0.0178748
$494$	−6.43282	−0.289426
$495$	0	0
$496$	6.15787	0.276497
$497$	25.0859	1.12526
$498$	0	0
$499$	3.64574	0.163206	0.0816029	−	0.996665i	$-0.473996\pi$
$499$	3.64574	0.163206	0.0816029	+	0.996665i	$0.473996\pi$
$500$	−11.2472	−0.502992
$501$	0	0
$502$	−19.1964	−0.856779
$503$	−37.0763	−1.65315	−0.826575	−	0.562826i	$-0.809714\pi$
$503$	−37.0763	−1.65315	−0.826575	+	0.562826i	$0.809714\pi$
$504$	0	0
$505$	−4.55555	−0.202720
$506$	0	0
$507$	0	0
$508$	2.61962	0.116227
$509$	35.5360	1.57511	0.787554	−	0.616246i	$-0.211347\pi$
$509$	35.5360	1.57511	0.787554	+	0.616246i	$0.211347\pi$
$510$	0	0
$511$	−16.2761	−0.720014
$512$	1.00000	0.0441942
$513$	0	0
$514$	−8.43545	−0.372072
$515$	14.1361	0.622912
$516$	0	0
$517$	−1.71459	−0.0754075
$518$	−18.1459	−0.797287
$519$	0	0
$520$	2.77896	0.121865
$521$	−5.66152	−0.248036	−0.124018	−	0.992280i	$-0.539578\pi$
$521$	−5.66152	−0.248036	−0.124018	+	0.992280i	$0.539578\pi$
$522$	0	0
$523$	−3.79404	−0.165902	−0.0829508	−	0.996554i	$-0.526434\pi$
$523$	−3.79404	−0.165902	−0.0829508	+	0.996554i	$0.526434\pi$
$524$	−15.3207	−0.669286
$525$	0	0
$526$	21.4538	0.935429
$527$	0.684847	0.0298324
$528$	0	0
$529$	0	0
$530$	0.204245	0.00887185
$531$	0	0
$532$	−13.8875	−0.602101
$533$	−6.78532	−0.293905
$534$	0	0
$535$	−0.665174	−0.0287580
$536$	14.3384	0.619326
$537$	0	0
$538$	21.5844	0.930569
$539$	2.53012	0.108980
$540$	0	0
$541$	10.4793	0.450538	0.225269	−	0.974297i	$-0.427674\pi$
$541$	10.4793	0.450538	0.225269	+	0.974297i	$0.427674\pi$
$542$	−12.8063	−0.550080
$543$	0	0
$544$	0.111215	0.00476830
$545$	−13.7114	−0.587332
$546$	0	0
$547$	−28.6896	−1.22668	−0.613339	−	0.789820i	$-0.710174\pi$
$547$	−28.6896	−1.22668	−0.613339	+	0.789820i	$0.710174\pi$
$548$	−10.3092	−0.440386
$549$	0	0
$550$	−0.674875	−0.0287768
$551$	−11.5475	−0.491942
$552$	0	0
$553$	37.4621	1.59305
$554$	−15.6781	−0.666100
$555$	0	0
$556$	−0.569454	−0.0241502
$557$	40.9955	1.73704	0.868518	−	0.495658i	$-0.165073\pi$
$557$	40.9955	1.73704	0.868518	+	0.495658i	$0.165073\pi$
$558$	0	0
$559$	6.45692	0.273098
$560$	5.99937	0.253520
$561$	0	0
$562$	−9.41093	−0.396976
$563$	11.3662	0.479030	0.239515	−	0.970893i	$-0.423012\pi$
$563$	11.3662	0.479030	0.239515	+	0.970893i	$0.423012\pi$
$564$	0	0
$565$	12.3964	0.521519
$566$	−7.08348	−0.297741
$567$	0	0
$568$	5.84511	0.245256
$569$	−35.3055	−1.48008	−0.740042	−	0.672561i	$-0.765194\pi$
$569$	−35.3055	−1.48008	−0.740042	+	0.672561i	$0.765194\pi$
$570$	0	0
$571$	−14.3742	−0.601542	−0.300771	−	0.953696i	$-0.597244\pi$
$571$	−14.3742	−0.601542	−0.300771	+	0.953696i	$0.597244\pi$
$572$	0.440469	0.0184169
$573$	0	0
$574$	−14.6485	−0.611418
$575$	0	0
$576$	0	0
$577$	−19.1095	−0.795540	−0.397770	−	0.917485i	$-0.630216\pi$
$577$	−19.1095	−0.795540	−0.397770	+	0.917485i	$0.630216\pi$
$578$	−16.9876	−0.706592
$579$	0	0
$580$	4.98851	0.207136
$581$	−67.1636	−2.78642
$582$	0	0
$583$	0.0323732	0.00134076
$584$	−3.79241	−0.156931
$585$	0	0
$586$	−26.4324	−1.09191
$587$	42.3710	1.74884	0.874420	−	0.485170i	$-0.161242\pi$
$587$	42.3710	1.74884	0.874420	+	0.485170i	$0.161242\pi$
$588$	0	0
$589$	−19.9259	−0.821034
$590$	18.4090	0.757887
$591$	0	0
$592$	−4.22808	−0.173773
$593$	−32.5539	−1.33683	−0.668414	−	0.743789i	$-0.733027\pi$
$593$	−32.5539	−1.33683	−0.668414	+	0.743789i	$0.733027\pi$
$594$	0	0
$595$	0.667220	0.0273533
$596$	13.1652	0.539267
$597$	0	0
$598$	0	0
$599$	35.4803	1.44969	0.724843	−	0.688914i	$-0.241912\pi$
$599$	35.4803	1.44969	0.724843	+	0.688914i	$0.241912\pi$
$600$	0	0
$601$	13.9894	0.570641	0.285320	−	0.958432i	$-0.407900\pi$
$601$	13.9894	0.570641	0.285320	+	0.958432i	$0.407900\pi$
$602$	13.9396	0.568134
$603$	0	0
$604$	−9.08797	−0.369784
$605$	−15.3080	−0.622360
$606$	0	0
$607$	−30.3298	−1.23105	−0.615525	−	0.788118i	$-0.711056\pi$
$607$	−30.3298	−1.23105	−0.615525	+	0.788118i	$0.711056\pi$
$608$	−3.23585	−0.131231
$609$	0	0
$610$	7.95343	0.322025
$611$	−15.3840	−0.622372
$612$	0	0
$613$	−13.8504	−0.559413	−0.279706	−	0.960086i	$-0.590237\pi$
$613$	−13.8504	−0.559413	−0.279706	+	0.960086i	$0.590237\pi$
$614$	11.2935	0.455768
$615$	0	0
$616$	0.950909	0.0383132
$617$	−46.8999	−1.88812	−0.944060	−	0.329774i	$-0.893027\pi$
$617$	−46.8999	−1.88812	−0.944060	+	0.329774i	$0.893027\pi$
$618$	0	0
$619$	−15.3922	−0.618666	−0.309333	−	0.950954i	$-0.600106\pi$
$619$	−15.3922	−0.618666	−0.309333	+	0.950954i	$0.600106\pi$
$620$	8.60795	0.345704
$621$	0	0
$622$	8.16799	0.327507
$623$	4.37476	0.175271
$624$	0	0
$625$	−0.492563	−0.0197025
$626$	7.85449	0.313928
$627$	0	0
$628$	8.71215	0.347653
$629$	−0.470225	−0.0187491
$630$	0	0
$631$	26.2229	1.04392	0.521959	−	0.852971i	$-0.325201\pi$
$631$	26.2229	1.04392	0.521959	+	0.852971i	$0.325201\pi$
$632$	8.72882	0.347214
$633$	0	0
$634$	−13.2797	−0.527404
$635$	3.66191	0.145318
$636$	0	0
$637$	22.7014	0.899462
$638$	0.790685	0.0313035
$639$	0	0
$640$	1.39788	0.0552560
$641$	21.3508	0.843305	0.421653	−	0.906757i	$-0.361450\pi$
$641$	21.3508	0.843305	0.421653	+	0.906757i	$0.361450\pi$
$642$	0	0
$643$	−21.3545	−0.842139	−0.421069	−	0.907028i	$-0.638345\pi$
$643$	−21.3545	−0.842139	−0.421069	+	0.907028i	$0.638345\pi$
$644$	0	0
$645$	0	0
$646$	−0.359875	−0.0141591
$647$	10.7365	0.422095	0.211047	−	0.977476i	$-0.432313\pi$
$647$	10.7365	0.422095	0.211047	+	0.977476i	$0.432313\pi$
$648$	0	0
$649$	2.91785	0.114536
$650$	−6.05528	−0.237507
$651$	0	0
$652$	10.6812	0.418308
$653$	21.6445	0.847015	0.423507	−	0.905893i	$-0.360799\pi$
$653$	21.6445	0.847015	0.423507	+	0.905893i	$0.360799\pi$
$654$	0	0
$655$	−21.4164	−0.836808
$656$	−3.41316	−0.133262
$657$	0	0
$658$	−33.2120	−1.29474
$659$	−8.50264	−0.331216	−0.165608	−	0.986192i	$-0.552959\pi$
$659$	−8.50264	−0.331216	−0.165608	+	0.986192i	$0.552959\pi$
$660$	0	0
$661$	26.7131	1.03902	0.519510	−	0.854464i	$-0.326114\pi$
$661$	26.7131	1.03902	0.519510	+	0.854464i	$0.326114\pi$
$662$	1.31086	0.0509482
$663$	0	0
$664$	−15.6494	−0.607314
$665$	−19.4131	−0.752806
$666$	0	0
$667$	0	0
$668$	−7.93820	−0.307138
$669$	0	0
$670$	20.0434	0.774343
$671$	1.26063	0.0486661
$672$	0	0
$673$	−1.60196	−0.0617509	−0.0308754	−	0.999523i	$-0.509830\pi$
$673$	−1.60196	−0.0617509	−0.0308754	+	0.999523i	$0.509830\pi$
$674$	34.2683	1.31997
$675$	0	0
$676$	−9.04792	−0.347997
$677$	−17.2239	−0.661969	−0.330984	−	0.943636i	$-0.607381\pi$
$677$	−17.2239	−0.661969	−0.330984	+	0.943636i	$0.607381\pi$
$678$	0	0
$679$	78.1667	2.99976
$680$	0.155465	0.00596180
$681$	0	0
$682$	1.36437	0.0522445
$683$	9.47633	0.362602	0.181301	−	0.983428i	$-0.441969\pi$
$683$	9.47633	0.362602	0.181301	+	0.983428i	$0.441969\pi$
$684$	0	0
$685$	−14.4109	−0.550614
$686$	18.9666	0.724150
$687$	0	0
$688$	3.24797	0.123828
$689$	0.290466	0.0110659
$690$	0	0
$691$	−21.3629	−0.812685	−0.406342	−	0.913721i	$-0.633196\pi$
$691$	−21.3629	−0.812685	−0.406342	+	0.913721i	$0.633196\pi$
$692$	7.45918	0.283555
$693$	0	0
$694$	−24.6201	−0.934566
$695$	−0.796027	−0.0301950
$696$	0	0
$697$	−0.379595	−0.0143782
$698$	−20.5915	−0.779399
$699$	0	0
$700$	−13.0725	−0.494093
$701$	5.54455	0.209415	0.104707	−	0.994503i	$-0.466609\pi$
$701$	5.54455	0.209415	0.104707	+	0.994503i	$0.466609\pi$
$702$	0	0
$703$	13.6814	0.516005
$704$	0.221566	0.00835057
$705$	0	0
$706$	−27.3130	−1.02794
$707$	−13.9865	−0.526016
$708$	0	0
$709$	−4.00516	−0.150417	−0.0752086	−	0.997168i	$-0.523962\pi$
$709$	−4.00516	−0.150417	−0.0752086	+	0.997168i	$0.523962\pi$
$710$	8.17075	0.306643
$711$	0	0
$712$	1.01934	0.0382013
$713$	0	0
$714$	0	0
$715$	0.615721	0.0230267
$716$	−20.8381	−0.778757
$717$	0	0
$718$	−37.5903	−1.40286
$719$	22.6271	0.843849	0.421925	−	0.906631i	$-0.361355\pi$
$719$	22.6271	0.843849	0.421925	+	0.906631i	$0.361355\pi$
$720$	0	0
$721$	43.4008	1.61633
$722$	−8.52928	−0.317427
$723$	0	0
$724$	4.22222	0.156917
$725$	−10.8698	−0.403695
$726$	0	0
$727$	−37.3558	−1.38545	−0.692725	−	0.721201i	$-0.743590\pi$
$727$	−37.3558	−1.38545	−0.692725	+	0.721201i	$0.743590\pi$
$728$	8.53197	0.316216
$729$	0	0
$730$	−5.30132	−0.196211
$731$	0.361223	0.0133603
$732$	0	0
$733$	−2.09461	−0.0773661	−0.0386831	−	0.999252i	$-0.512316\pi$
$733$	−2.09461	−0.0773661	−0.0386831	+	0.999252i	$0.512316\pi$
$734$	22.3474	0.824859
$735$	0	0
$736$	0	0
$737$	3.17690	0.117023
$738$	0	0
$739$	18.7148	0.688436	0.344218	−	0.938890i	$-0.388144\pi$
$739$	18.7148	0.688436	0.344218	+	0.938890i	$0.388144\pi$
$740$	−5.91033	−0.217268
$741$	0	0
$742$	0.627075	0.0230206
$743$	31.2984	1.14823	0.574113	−	0.818776i	$-0.305347\pi$
$743$	31.2984	1.14823	0.574113	+	0.818776i	$0.305347\pi$
$744$	0	0
$745$	18.4033	0.674246
$746$	−29.7918	−1.09075
$747$	0	0
$748$	0.0246414	0.000900979 0
$749$	−2.04222	−0.0746211
$750$	0	0
$751$	−48.2216	−1.75963	−0.879816	−	0.475314i	$-0.842334\pi$
$751$	−48.2216	−1.75963	−0.879816	+	0.475314i	$0.842334\pi$
$752$	−7.73852	−0.282195
$753$	0	0
$754$	7.09438	0.258362
$755$	−12.7039	−0.462341
$756$	0	0
$757$	18.4533	0.670696	0.335348	−	0.942094i	$-0.391146\pi$
$757$	18.4533	0.670696	0.335348	+	0.942094i	$0.391146\pi$
$758$	−21.5505	−0.782750
$759$	0	0
$760$	−4.52332	−0.164078
$761$	−18.4322	−0.668166	−0.334083	−	0.942544i	$-0.608427\pi$
$761$	−18.4322	−0.668166	−0.334083	+	0.942544i	$0.608427\pi$
$762$	0	0
$763$	−42.0968	−1.52401
$764$	−14.1891	−0.513344
$765$	0	0
$766$	12.8513	0.464335
$767$	26.1803	0.945315
$768$	0	0
$769$	34.4332	1.24169	0.620846	−	0.783933i	$-0.286789\pi$
$769$	34.4332	1.24169	0.620846	+	0.783933i	$0.286789\pi$
$770$	1.32925	0.0479030
$771$	0	0
$772$	2.00602	0.0721983
$773$	5.84600	0.210266	0.105133	−	0.994458i	$-0.466473\pi$
$773$	5.84600	0.210266	0.105133	+	0.994458i	$0.466473\pi$
$774$	0	0
$775$	−18.7565	−0.673753
$776$	18.2132	0.653815
$777$	0	0
$778$	−0.489944	−0.0175653
$779$	11.0445	0.395710
$780$	0	0
$781$	1.29508	0.0463415
$782$	0	0
$783$	0	0
$784$	11.4193	0.407832
$785$	12.1785	0.434670
$786$	0	0
$787$	−5.47501	−0.195163	−0.0975816	−	0.995228i	$-0.531111\pi$
$787$	−5.47501	−0.195163	−0.0975816	+	0.995228i	$0.531111\pi$
$788$	−12.9688	−0.461993
$789$	0	0
$790$	12.2018	0.434121
$791$	38.0594	1.35324
$792$	0	0
$793$	11.3109	0.401663
$794$	14.9274	0.529753
$795$	0	0
$796$	11.8687	0.420673
$797$	2.24779	0.0796207	0.0398103	−	0.999207i	$-0.487325\pi$
$797$	2.24779	0.0796207	0.0398103	+	0.999207i	$0.487325\pi$
$798$	0	0
$799$	−0.860639	−0.0304472
$800$	−3.04594	−0.107690
$801$	0	0
$802$	36.8123	1.29989
$803$	−0.840267	−0.0296524
$804$	0	0
$805$	0	0
$806$	12.2417	0.431197
$807$	0	0
$808$	−3.25891	−0.114648
$809$	−12.4644	−0.438225	−0.219112	−	0.975700i	$-0.570316\pi$
$809$	−12.4644	−0.438225	−0.219112	+	0.975700i	$0.570316\pi$
$810$	0	0
$811$	23.9769	0.841944	0.420972	−	0.907074i	$-0.361689\pi$
$811$	23.9769	0.841944	0.420972	+	0.907074i	$0.361689\pi$
$812$	15.3157	0.537477
$813$	0	0
$814$	−0.936796	−0.0328347
$815$	14.9310	0.523010
$816$	0	0
$817$	−10.5099	−0.367697
$818$	−27.3001	−0.954525
$819$	0	0
$820$	−4.77119	−0.166617
$821$	29.8885	1.04311	0.521557	−	0.853216i	$-0.325351\pi$
$821$	29.8885	1.04311	0.521557	+	0.853216i	$0.325351\pi$
$822$	0	0
$823$	10.7506	0.374741	0.187370	−	0.982289i	$-0.440004\pi$
$823$	10.7506	0.374741	0.187370	+	0.982289i	$0.440004\pi$
$824$	10.1126	0.352288
$825$	0	0
$826$	56.5194	1.96656
$827$	4.42943	0.154026	0.0770132	−	0.997030i	$-0.475462\pi$
$827$	4.42943	0.154026	0.0770132	+	0.997030i	$0.475462\pi$
$828$	0	0
$829$	−7.87479	−0.273503	−0.136751	−	0.990605i	$-0.543666\pi$
$829$	−7.87479	−0.273503	−0.136751	+	0.990605i	$0.543666\pi$
$830$	−21.8759	−0.759324
$831$	0	0
$832$	1.98798	0.0689209
$833$	1.27000	0.0440028
$834$	0	0
$835$	−11.0966	−0.384015
$836$	−0.716953	−0.0247963
$837$	0	0
$838$	−16.9066	−0.584030
$839$	−11.8973	−0.410740	−0.205370	−	0.978684i	$-0.565840\pi$
$839$	−11.8973	−0.410740	−0.205370	+	0.978684i	$0.565840\pi$
$840$	0	0
$841$	−16.2649	−0.560858
$842$	−4.43267	−0.152760
$843$	0	0
$844$	19.6017	0.674717
$845$	−12.6479	−0.435100
$846$	0	0
$847$	−46.9988	−1.61490
$848$	0.146111	0.00501747
$849$	0	0
$850$	−0.338754	−0.0116192
$851$	0	0
$852$	0	0
$853$	−34.0150	−1.16465	−0.582326	−	0.812955i	$-0.697857\pi$
$853$	−34.0150	−1.16465	−0.582326	+	0.812955i	$0.697857\pi$
$854$	24.4187	0.835590
$855$	0	0
$856$	−0.475845	−0.0162641
$857$	−37.2645	−1.27293	−0.636466	−	0.771305i	$-0.719604\pi$
$857$	−37.2645	−1.27293	−0.636466	+	0.771305i	$0.719604\pi$
$858$	0	0
$859$	−37.7361	−1.28754	−0.643769	−	0.765220i	$-0.722630\pi$
$859$	−37.7361	−1.28754	−0.643769	+	0.765220i	$0.722630\pi$
$860$	4.54027	0.154822
$861$	0	0
$862$	27.9986	0.953637
$863$	−13.5482	−0.461188	−0.230594	−	0.973050i	$-0.574067\pi$
$863$	−13.5482	−0.461188	−0.230594	+	0.973050i	$0.574067\pi$
$864$	0	0
$865$	10.4270	0.354529
$866$	12.2280	0.415525
$867$	0	0
$868$	26.4282	0.897031
$869$	1.93401	0.0656067
$870$	0	0
$871$	28.5046	0.965840
$872$	−9.80872	−0.332165
$873$	0	0
$874$	0	0
$875$	−48.2706	−1.63184
$876$	0	0
$877$	7.07074	0.238762	0.119381	−	0.992849i	$-0.461909\pi$
$877$	7.07074	0.238762	0.119381	+	0.992849i	$0.461909\pi$
$878$	15.3054	0.516533
$879$	0	0
$880$	0.309721	0.0104407
$881$	8.19673	0.276155	0.138077	−	0.990421i	$-0.455908\pi$
$881$	8.19673	0.276155	0.138077	+	0.990421i	$0.455908\pi$
$882$	0	0
$883$	7.75231	0.260886	0.130443	−	0.991456i	$-0.458360\pi$
$883$	7.75231	0.260886	0.130443	+	0.991456i	$0.458360\pi$
$884$	0.221093	0.00743618
$885$	0	0
$886$	−7.54063	−0.253332
$887$	45.2584	1.51963	0.759815	−	0.650140i	$-0.225290\pi$
$887$	45.2584	1.51963	0.759815	+	0.650140i	$0.225290\pi$
$888$	0	0
$889$	11.2428	0.377072
$890$	1.42491	0.0477630
$891$	0	0
$892$	−16.7226	−0.559913
$893$	25.0407	0.837954
$894$	0	0
$895$	−29.1291	−0.973679
$896$	4.29177	0.143378
$897$	0	0
$898$	12.4935	0.416915
$899$	21.9752	0.732913
$900$	0	0
$901$	0.0162497	0.000541357 0
$902$	−0.756240	−0.0251800
$903$	0	0
$904$	8.86799	0.294945
$905$	5.90214	0.196194
$906$	0	0
$907$	−54.3258	−1.80386	−0.901929	−	0.431885i	$-0.857849\pi$
$907$	−54.3258	−1.80386	−0.901929	+	0.431885i	$0.857849\pi$
$908$	14.1479	0.469514
$909$	0	0
$910$	11.9267	0.395365
$911$	25.0359	0.829477	0.414739	−	0.909941i	$-0.363873\pi$
$911$	25.0359	0.829477	0.414739	+	0.909941i	$0.363873\pi$
$912$	0	0
$913$	−3.46737	−0.114753
$914$	27.6630	0.915011
$915$	0	0
$916$	23.3847	0.772653
$917$	−65.7528	−2.17135
$918$	0	0
$919$	23.8024	0.785170	0.392585	−	0.919716i	$-0.371581\pi$
$919$	23.8024	0.785170	0.392585	+	0.919716i	$0.371581\pi$
$920$	0	0
$921$	0	0
$922$	−11.2174	−0.369425
$923$	11.6200	0.382477
$924$	0	0
$925$	12.8785	0.423441
$926$	9.47539	0.311381
$927$	0	0
$928$	3.56863	0.117146
$929$	14.3204	0.469838	0.234919	−	0.972015i	$-0.424517\pi$
$929$	14.3204	0.469838	0.234919	+	0.972015i	$0.424517\pi$
$930$	0	0
$931$	−36.9511	−1.21102
$932$	8.64601	0.283209
$933$	0	0
$934$	9.33895	0.305580
$935$	0.0344456	0.00112649
$936$	0	0
$937$	−37.8951	−1.23798	−0.618989	−	0.785400i	$-0.712457\pi$
$937$	−37.8951	−1.23798	−0.618989	+	0.785400i	$0.712457\pi$
$938$	61.5373	2.00926
$939$	0	0
$940$	−10.8175	−0.352828
$941$	−26.3514	−0.859031	−0.429516	−	0.903059i	$-0.641316\pi$
$941$	−26.3514	−0.859031	−0.429516	+	0.903059i	$0.641316\pi$
$942$	0	0
$943$	0	0
$944$	13.1693	0.428623
$945$	0	0
$946$	0.719639	0.0233975
$947$	12.9255	0.420023	0.210011	−	0.977699i	$-0.432650\pi$
$947$	12.9255	0.420023	0.210011	+	0.977699i	$0.432650\pi$
$948$	0	0
$949$	−7.53924	−0.244734
$950$	9.85620	0.319777
$951$	0	0
$952$	0.477309	0.0154697
$953$	9.84574	0.318935	0.159467	−	0.987203i	$-0.449022\pi$
$953$	9.84574	0.318935	0.159467	+	0.987203i	$0.449022\pi$
$954$	0	0
$955$	−19.8346	−0.641833
$956$	−8.84860	−0.286184
$957$	0	0
$958$	1.20307	0.0388696
$959$	−44.2446	−1.42873
$960$	0	0
$961$	6.91939	0.223206
$962$	−8.40535	−0.270999
$963$	0	0
$964$	−5.58741	−0.179958
$965$	2.80417	0.0902695
$966$	0	0
$967$	−59.2387	−1.90499	−0.952495	−	0.304555i	$-0.901492\pi$
$967$	−59.2387	−1.90499	−0.952495	+	0.304555i	$0.901492\pi$
$968$	−10.9509	−0.351976
$969$	0	0
$970$	25.4598	0.817464
$971$	−6.72775	−0.215904	−0.107952	−	0.994156i	$-0.534429\pi$
$971$	−6.72775	−0.215904	−0.107952	+	0.994156i	$0.534429\pi$
$972$	0	0
$973$	−2.44397	−0.0783500
$974$	−4.47248	−0.143307
$975$	0	0
$976$	5.68965	0.182121
$977$	32.7644	1.04823	0.524113	−	0.851649i	$-0.324397\pi$
$977$	32.7644	1.04823	0.524113	+	0.851649i	$0.324397\pi$
$978$	0	0
$979$	0.225850	0.00721820
$980$	15.9628	0.509912
$981$	0	0
$982$	−12.0574	−0.384767
$983$	25.1809	0.803146	0.401573	−	0.915827i	$-0.368464\pi$
$983$	25.1809	0.803146	0.401573	+	0.915827i	$0.368464\pi$
$984$	0	0
$985$	−18.1287	−0.577629
$986$	0.396885	0.0126394
$987$	0	0
$988$	−6.43282	−0.204655
$989$	0	0
$990$	0	0
$991$	35.7334	1.13511	0.567554	−	0.823336i	$-0.307890\pi$
$991$	35.7334	1.13511	0.567554	+	0.823336i	$0.307890\pi$
$992$	6.15787	0.195513
$993$	0	0
$994$	25.0859	0.795676
$995$	16.5909	0.525968
$996$	0	0
$997$	5.86890	0.185870	0.0929350	−	0.995672i	$-0.470375\pi$
$997$	5.86890	0.185870	0.0929350	+	0.995672i	$0.470375\pi$
$998$	3.64574	0.115404
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9522.2.a.bw.1.5		5
3.2	odd	2		1058.2.a.k.1.1		5
12.11	even	2		8464.2.a.bv.1.5		5
23.5	odd	22		414.2.i.c.163.1		10
23.14	odd	22		414.2.i.c.127.1		10
23.22	odd	2		9522.2.a.bz.1.1		5
69.5	even	22		46.2.c.b.25.1	✓	10
69.14	even	22		46.2.c.b.35.1	yes	10
69.68	even	2		1058.2.a.j.1.1		5
276.83	odd	22		368.2.m.a.81.1		10
276.143	odd	22		368.2.m.a.209.1		10
276.275	odd	2		8464.2.a.bu.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
46.2.c.b.25.1	✓	10	69.5	even	22
46.2.c.b.35.1	yes	10	69.14	even	22
368.2.m.a.81.1		10	276.83	odd	22
368.2.m.a.209.1		10	276.143	odd	22
414.2.i.c.127.1		10	23.14	odd	22
414.2.i.c.163.1		10	23.5	odd	22
1058.2.a.j.1.1		5	69.68	even	2
1058.2.a.k.1.1		5	3.2	odd	2
8464.2.a.bu.1.5		5	276.275	odd	2
8464.2.a.bv.1.5		5	12.11	even	2
9522.2.a.bw.1.5		5	1.1	even	1	trivial
9522.2.a.bz.1.1		5	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.39788 q^{5} +4.29177 q^{7} +1.00000 q^{8} +1.39788 q^{10} +0.221566 q^{11} +1.98798 q^{13} +4.29177 q^{14} +1.00000 q^{16} +0.111215 q^{17} -3.23585 q^{19} +1.39788 q^{20} +0.221566 q^{22} -3.04594 q^{25} +1.98798 q^{26} +4.29177 q^{28} +3.56863 q^{29} +6.15787 q^{31} +1.00000 q^{32} +0.111215 q^{34} +5.99937 q^{35} -4.22808 q^{37} -3.23585 q^{38} +1.39788 q^{40} -3.41316 q^{41} +3.24797 q^{43} +0.221566 q^{44} -7.73852 q^{47} +11.4193 q^{49} -3.04594 q^{50} +1.98798 q^{52} +0.146111 q^{53} +0.309721 q^{55} +4.29177 q^{56} +3.56863 q^{58} +13.1693 q^{59} +5.68965 q^{61} +6.15787 q^{62} +1.00000 q^{64} +2.77896 q^{65} +14.3384 q^{67} +0.111215 q^{68} +5.99937 q^{70} +5.84511 q^{71} -3.79241 q^{73} -4.22808 q^{74} -3.23585 q^{76} +0.950909 q^{77} +8.72882 q^{79} +1.39788 q^{80} -3.41316 q^{82} -15.6494 q^{83} +0.155465 q^{85} +3.24797 q^{86} +0.221566 q^{88} +1.01934 q^{89} +8.53197 q^{91} -7.73852 q^{94} -4.52332 q^{95} +18.2132 q^{97} +11.4193 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{2} + 5 q^{4} - 2 q^{5} + 9 q^{7} + 5 q^{8} - 2 q^{10} - q^{11} + q^{13} + 9 q^{14} + 5 q^{16} + q^{17} + 10 q^{19} - 2 q^{20} - q^{22} - 11 q^{25} + q^{26} + 9 q^{28} - 10 q^{29} + 5 q^{32}+ \cdots + 12 q^{98}+O(q^{100}) \) 5 * q + 5 * q^2 + 5 * q^4 - 2 * q^5 + 9 * q^7 + 5 * q^8 - 2 * q^10 - q^11 + q^13 + 9 * q^14 + 5 * q^16 + q^17 + 10 * q^19 - 2 * q^20 - q^22 - 11 * q^25 + q^26 + 9 * q^28 - 10 * q^29 + 5 * q^32 + q^34 + 3 * q^35 + 20 * q^37 + 10 * q^38 - 2 * q^40 + 3 * q^41 + 27 * q^43 - q^44 + 11 * q^47 + 12 * q^49 - 11 * q^50 + q^52 + 2 * q^53 - 4 * q^55 + 9 * q^56 - 10 * q^58 + 17 * q^59 + 7 * q^61 + 5 * q^64 + 15 * q^65 + 28 * q^67 + q^68 + 3 * q^70 + 8 * q^73 + 20 * q^74 + 10 * q^76 - 26 * q^77 + 16 * q^79 - 2 * q^80 + 3 * q^82 - 18 * q^83 - 7 * q^85 + 27 * q^86 - q^88 + 14 * q^89 + 26 * q^91 + 11 * q^94 - 26 * q^95 + 47 * q^97 + 12 * q^98

Introduction

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