LMFDB - Embedded newform 9522.2.a.by.1.4

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 = [5,5,0,5,1,0,-1,5,0,1,3,0,-10,-1,0,5,11,0,11,1,0,3,0,0,6,-10, 0,-1,25] 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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 138)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.91899$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +2.77066 q^{5} -0.194262 q^{7} +1.00000 q^{8} +2.77066 q^{10} +2.90407 q^{11} +1.08260 q^{13} -0.194262 q^{14} +1.00000 q^{16} -4.64187 q^{17} +4.42518 q^{19} +2.77066 q^{20} +2.90407 q^{22} +2.67657 q^{25} +1.08260 q^{26} -0.194262 q^{28} +9.21076 q^{29} -6.87324 q^{31} +1.00000 q^{32} -4.64187 q^{34} -0.538234 q^{35} +4.84574 q^{37} +4.42518 q^{38} +2.77066 q^{40} +5.74494 q^{41} -7.56694 q^{43} +2.90407 q^{44} +5.61457 q^{47} -6.96226 q^{49} +2.67657 q^{50} +1.08260 q^{52} -4.14258 q^{53} +8.04621 q^{55} -0.194262 q^{56} +9.21076 q^{58} +10.7184 q^{59} +2.87961 q^{61} -6.87324 q^{62} +1.00000 q^{64} +2.99951 q^{65} +14.6373 q^{67} -4.64187 q^{68} -0.538234 q^{70} -12.7320 q^{71} +2.98812 q^{73} +4.84574 q^{74} +4.42518 q^{76} -0.564150 q^{77} -2.55897 q^{79} +2.77066 q^{80} +5.74494 q^{82} -0.811565 q^{83} -12.8611 q^{85} -7.56694 q^{86} +2.90407 q^{88} +6.77984 q^{89} -0.210307 q^{91} +5.61457 q^{94} +12.2607 q^{95} +12.3679 q^{97} -6.96226 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{2} + 5 q^{4} + q^{5} - q^{7} + 5 q^{8} + q^{10} + 3 q^{11} - 10 q^{13} - q^{14} + 5 q^{16} + 11 q^{17} + 11 q^{19} + q^{20} + 3 q^{22} + 6 q^{25} - 10 q^{26} - q^{28} + 25 q^{29} - 4 q^{31}+ \cdots - 4 q^{98}+O(q^{100}) $ 5 * q + 5 * q^2 + 5 * q^4 + q^5 - q^7 + 5 * q^8 + q^10 + 3 * q^11 - 10 * q^13 - q^14 + 5 * q^16 + 11 * q^17 + 11 * q^19 + q^20 + 3 * q^22 + 6 * q^25 - 10 * q^26 - q^28 + 25 * q^29 - 4 * q^31 + 5 * q^32 + 11 * q^34 - 9 * q^35 + 22 * q^37 + 11 * q^38 + q^40 - 6 * q^41 - 6 * q^43 + 3 * q^44 - 4 * q^47 - 4 * q^49 + 6 * q^50 - 10 * q^52 - q^53 - 6 * q^55 - q^56 + 25 * q^58 - 6 * q^59 + 35 * q^61 - 4 * q^62 + 5 * q^64 - 2 * q^65 + 5 * q^67 + 11 * q^68 - 9 * q^70 + 11 * q^71 + 16 * q^73 + 22 * q^74 + 11 * q^76 - 5 * q^77 + 8 * q^79 + q^80 - 6 * q^82 + 26 * q^83 - 33 * q^85 - 6 * q^86 + 3 * q^88 + 35 * q^91 - 4 * q^94 - 11 * q^95 + 19 * q^97 - 4 * 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
$5$	2.77066	1.23908	0.619539	−	0.784966i	$-0.287320\pi$
$5$	2.77066	1.23908	0.619539	+	0.784966i	$0.287320\pi$
$6$	0	0
$7$	−0.194262	−0.0734240	−0.0367120	−	0.999326i	$-0.511688\pi$
$7$	−0.194262	−0.0734240	−0.0367120	+	0.999326i	$0.511688\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	2.77066	0.876161
$11$	2.90407	0.875611	0.437805	−	0.899070i	$-0.355756\pi$
$11$	2.90407	0.875611	0.437805	+	0.899070i	$0.355756\pi$
$12$	0	0
$13$	1.08260	0.300258	0.150129	−	0.988666i	$-0.452031\pi$
$13$	1.08260	0.300258	0.150129	+	0.988666i	$0.452031\pi$
$14$	−0.194262	−0.0519186
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.64187	−1.12582	−0.562910	−	0.826518i	$-0.690318\pi$
$17$	−4.64187	−1.12582	−0.562910	+	0.826518i	$0.690318\pi$
$18$	0	0
$19$	4.42518	1.01521	0.507603	−	0.861591i	$-0.330532\pi$
$19$	4.42518	1.01521	0.507603	+	0.861591i	$0.330532\pi$
$20$	2.77066	0.619539
$21$	0	0
$22$	2.90407	0.619150
$23$	0	0
$23$	0	0
$24$	0	0
$25$	2.67657	0.535315
$26$	1.08260	0.212315
$27$	0	0
$28$	−0.194262	−0.0367120
$29$	9.21076	1.71039	0.855197	−	0.518302i	$-0.173436\pi$
$29$	9.21076	1.71039	0.855197	+	0.518302i	$0.173436\pi$
$30$	0	0
$31$	−6.87324	−1.23447	−0.617235	−	0.786779i	$-0.711747\pi$
$31$	−6.87324	−1.23447	−0.617235	+	0.786779i	$0.711747\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−4.64187	−0.796075
$35$	−0.538234	−0.0909781
$36$	0	0
$37$	4.84574	0.796635	0.398318	−	0.917248i	$-0.369594\pi$
$37$	4.84574	0.796635	0.398318	+	0.917248i	$0.369594\pi$
$38$	4.42518	0.717859
$39$	0	0
$40$	2.77066	0.438080
$41$	5.74494	0.897209	0.448605	−	0.893730i	$-0.351921\pi$
$41$	5.74494	0.897209	0.448605	+	0.893730i	$0.351921\pi$
$42$	0	0
$43$	−7.56694	−1.15395	−0.576974	−	0.816763i	$-0.695767\pi$
$43$	−7.56694	−1.15395	−0.576974	+	0.816763i	$0.695767\pi$
$44$	2.90407	0.437805
$45$	0	0
$46$	0	0
$47$	5.61457	0.818969	0.409484	−	0.912317i	$-0.365709\pi$
$47$	5.61457	0.818969	0.409484	+	0.912317i	$0.365709\pi$
$48$	0	0
$49$	−6.96226	−0.994609
$50$	2.67657	0.378525
$51$	0	0
$52$	1.08260	0.150129
$53$	−4.14258	−0.569028	−0.284514	−	0.958672i	$-0.591832\pi$
$53$	−4.14258	−0.569028	−0.284514	+	0.958672i	$0.591832\pi$
$54$	0	0
$55$	8.04621	1.08495
$56$	−0.194262	−0.0259593
$57$	0	0
$58$	9.21076	1.20943
$59$	10.7184	1.39541	0.697706	−	0.716385i	$-0.254204\pi$
$59$	10.7184	1.39541	0.697706	+	0.716385i	$0.254204\pi$
$60$	0	0
$61$	2.87961	0.368696	0.184348	−	0.982861i	$-0.440983\pi$
$61$	2.87961	0.368696	0.184348	+	0.982861i	$0.440983\pi$
$62$	−6.87324	−0.872903
$63$	0	0
$64$	1.00000	0.125000
$65$	2.99951	0.372043
$66$	0	0
$67$	14.6373	1.78824	0.894118	−	0.447832i	$-0.147804\pi$
$67$	14.6373	1.78824	0.894118	+	0.447832i	$0.147804\pi$
$68$	−4.64187	−0.562910
$69$	0	0
$70$	−0.538234	−0.0643313
$71$	−12.7320	−1.51101	−0.755503	−	0.655146i	$-0.772607\pi$
$71$	−12.7320	−1.51101	−0.755503	+	0.655146i	$0.772607\pi$
$72$	0	0
$73$	2.98812	0.349733	0.174867	−	0.984592i	$-0.444051\pi$
$73$	2.98812	0.349733	0.174867	+	0.984592i	$0.444051\pi$
$74$	4.84574	0.563306
$75$	0	0
$76$	4.42518	0.507603
$77$	−0.564150	−0.0642909
$78$	0	0
$79$	−2.55897	−0.287907	−0.143954	−	0.989584i	$-0.545982\pi$
$79$	−2.55897	−0.287907	−0.143954	+	0.989584i	$0.545982\pi$
$80$	2.77066	0.309770
$81$	0	0
$82$	5.74494	0.634423
$83$	−0.811565	−0.0890809	−0.0445404	−	0.999008i	$-0.514182\pi$
$83$	−0.811565	−0.0890809	−0.0445404	+	0.999008i	$0.514182\pi$
$84$	0	0
$85$	−12.8611	−1.39498
$86$	−7.56694	−0.815964
$87$	0	0
$88$	2.90407	0.309575
$89$	6.77984	0.718661	0.359331	−	0.933210i	$-0.383005\pi$
$89$	6.77984	0.718661	0.359331	+	0.933210i	$0.383005\pi$
$90$	0	0
$91$	−0.210307	−0.0220462
$92$	0	0
$93$	0	0
$94$	5.61457	0.579099
$95$	12.2607	1.25792
$96$	0	0
$97$	12.3679	1.25577	0.627887	−	0.778304i	$-0.283920\pi$
$97$	12.3679	1.25577	0.627887	+	0.778304i	$0.283920\pi$
$98$	−6.96226	−0.703295
$99$	0	0
$100$	2.67657	0.267657
$101$	−15.6967	−1.56188	−0.780939	−	0.624608i	$-0.785259\pi$
$101$	−15.6967	−1.56188	−0.780939	+	0.624608i	$0.785259\pi$
$102$	0	0
$103$	−9.44491	−0.930634	−0.465317	−	0.885144i	$-0.654060\pi$
$103$	−9.44491	−0.930634	−0.465317	+	0.885144i	$0.654060\pi$
$104$	1.08260	0.106157
$105$	0	0
$106$	−4.14258	−0.402363
$107$	9.51202	0.919562	0.459781	−	0.888032i	$-0.347928\pi$
$107$	9.51202	0.919562	0.459781	+	0.888032i	$0.347928\pi$
$108$	0	0
$109$	20.2217	1.93689	0.968443	−	0.249233i	$-0.0801786\pi$
$109$	20.2217	1.93689	0.968443	+	0.249233i	$0.0801786\pi$
$110$	8.04621	0.767176
$111$	0	0
$112$	−0.194262	−0.0183560
$113$	1.31667	0.123862	0.0619309	−	0.998080i	$-0.480274\pi$
$113$	1.31667	0.123862	0.0619309	+	0.998080i	$0.480274\pi$
$114$	0	0
$115$	0	0
$116$	9.21076	0.855197
$117$	0	0
$118$	10.7184	0.986705
$119$	0.901738	0.0826622
$120$	0	0
$121$	−2.56636	−0.233306
$122$	2.87961	0.260708
$123$	0	0
$124$	−6.87324	−0.617235
$125$	−6.43743	−0.575781
$126$	0	0
$127$	−4.81847	−0.427570	−0.213785	−	0.976881i	$-0.568579\pi$
$127$	−4.81847	−0.427570	−0.213785	+	0.976881i	$0.568579\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	2.99951	0.263074
$131$	−15.9778	−1.39598	−0.697991	−	0.716106i	$-0.745923\pi$
$131$	−15.9778	−1.39598	−0.697991	+	0.716106i	$0.745923\pi$
$132$	0	0
$133$	−0.859644	−0.0745405
$134$	14.6373	1.26447
$135$	0	0
$136$	−4.64187	−0.398037
$137$	18.7720	1.60380	0.801898	−	0.597460i	$-0.203824\pi$
$137$	18.7720	1.60380	0.801898	+	0.597460i	$0.203824\pi$
$138$	0	0
$139$	17.7540	1.50588	0.752938	−	0.658092i	$-0.228636\pi$
$139$	17.7540	1.50588	0.752938	+	0.658092i	$0.228636\pi$
$140$	−0.538234	−0.0454891
$141$	0	0
$142$	−12.7320	−1.06844
$143$	3.14394	0.262909
$144$	0	0
$145$	25.5199	2.11931
$146$	2.98812	0.247299
$147$	0	0
$148$	4.84574	0.398318
$149$	−9.51537	−0.779529	−0.389765	−	0.920914i	$-0.627444\pi$
$149$	−9.51537	−0.779529	−0.389765	+	0.920914i	$0.627444\pi$
$150$	0	0
$151$	−8.67872	−0.706265	−0.353132	−	0.935573i	$-0.614883\pi$
$151$	−8.67872	−0.706265	−0.353132	+	0.935573i	$0.614883\pi$
$152$	4.42518	0.358930
$153$	0	0
$154$	−0.564150	−0.0454605
$155$	−19.0434	−1.52961
$156$	0	0
$157$	−19.8163	−1.58152	−0.790758	−	0.612129i	$-0.790313\pi$
$157$	−19.8163	−1.58152	−0.790758	+	0.612129i	$0.790313\pi$
$158$	−2.55897	−0.203581
$159$	0	0
$160$	2.77066	0.219040
$161$	0	0
$162$	0	0
$163$	14.5014	1.13584	0.567919	−	0.823084i	$-0.307749\pi$
$163$	14.5014	1.13584	0.567919	+	0.823084i	$0.307749\pi$
$164$	5.74494	0.448605
$165$	0	0
$166$	−0.811565	−0.0629897
$167$	9.89119	0.765403	0.382702	−	0.923872i	$-0.374994\pi$
$167$	9.89119	0.765403	0.382702	+	0.923872i	$0.374994\pi$
$168$	0	0
$169$	−11.8280	−0.909845
$170$	−12.8611	−0.986399
$171$	0	0
$172$	−7.56694	−0.576974
$173$	−13.2434	−1.00688	−0.503440	−	0.864030i	$-0.667932\pi$
$173$	−13.2434	−1.00688	−0.503440	+	0.864030i	$0.667932\pi$
$174$	0	0
$175$	−0.519956	−0.0393050
$176$	2.90407	0.218903
$177$	0	0
$178$	6.77984	0.508170
$179$	−5.24531	−0.392053	−0.196026	−	0.980599i	$-0.562804\pi$
$179$	−5.24531	−0.392053	−0.196026	+	0.980599i	$0.562804\pi$
$180$	0	0
$181$	16.2553	1.20825	0.604123	−	0.796891i	$-0.293523\pi$
$181$	16.2553	1.20825	0.604123	+	0.796891i	$0.293523\pi$
$182$	−0.210307	−0.0155890
$183$	0	0
$184$	0	0
$185$	13.4259	0.987093
$186$	0	0
$187$	−13.4803	−0.985780
$188$	5.61457	0.409484
$189$	0	0
$190$	12.2607	0.889484
$191$	−7.69266	−0.556621	−0.278310	−	0.960491i	$-0.589774\pi$
$191$	−7.69266	−0.556621	−0.278310	+	0.960491i	$0.589774\pi$
$192$	0	0
$193$	−1.42745	−0.102750	−0.0513750	−	0.998679i	$-0.516360\pi$
$193$	−1.42745	−0.102750	−0.0513750	+	0.998679i	$0.516360\pi$
$194$	12.3679	0.887967
$195$	0	0
$196$	−6.96226	−0.497304
$197$	−6.25572	−0.445702	−0.222851	−	0.974853i	$-0.571536\pi$
$197$	−6.25572	−0.445702	−0.222851	+	0.974853i	$0.571536\pi$
$198$	0	0
$199$	−0.863136	−0.0611861	−0.0305931	−	0.999532i	$-0.509740\pi$
$199$	−0.863136	−0.0611861	−0.0305931	+	0.999532i	$0.509740\pi$
$200$	2.67657	0.189262
$201$	0	0
$202$	−15.6967	−1.10441
$203$	−1.78930	−0.125584
$204$	0	0
$205$	15.9173	1.11171
$206$	−9.44491	−0.658058
$207$	0	0
$208$	1.08260	0.0750646
$209$	12.8510	0.888926
$210$	0	0
$211$	19.0647	1.31247	0.656233	−	0.754559i	$-0.272149\pi$
$211$	19.0647	1.31247	0.656233	+	0.754559i	$0.272149\pi$
$212$	−4.14258	−0.284514
$213$	0	0
$214$	9.51202	0.650228
$215$	−20.9654	−1.42983
$216$	0	0
$217$	1.33521	0.0906398
$218$	20.2217	1.36959
$219$	0	0
$220$	8.04621	0.542475
$221$	−5.02527	−0.338037
$222$	0	0
$223$	6.41755	0.429751	0.214875	−	0.976641i	$-0.431065\pi$
$223$	6.41755	0.429751	0.214875	+	0.976641i	$0.431065\pi$
$224$	−0.194262	−0.0129797
$225$	0	0
$226$	1.31667	0.0875835
$227$	−1.19685	−0.0794376	−0.0397188	−	0.999211i	$-0.512646\pi$
$227$	−1.19685	−0.0794376	−0.0397188	+	0.999211i	$0.512646\pi$
$228$	0	0
$229$	28.0856	1.85595	0.927974	−	0.372646i	$-0.121549\pi$
$229$	28.0856	1.85595	0.927974	+	0.372646i	$0.121549\pi$
$230$	0	0
$231$	0	0
$232$	9.21076	0.604716
$233$	16.4128	1.07524	0.537618	−	0.843188i	$-0.319324\pi$
$233$	16.4128	1.07524	0.537618	+	0.843188i	$0.319324\pi$
$234$	0	0
$235$	15.5561	1.01477
$236$	10.7184	0.697706
$237$	0	0
$238$	0.901738	0.0584510
$239$	−10.3357	−0.668563	−0.334281	−	0.942473i	$-0.608494\pi$
$239$	−10.3357	−0.668563	−0.334281	+	0.942473i	$0.608494\pi$
$240$	0	0
$241$	21.9380	1.41315	0.706575	−	0.707638i	$-0.250239\pi$
$241$	21.9380	1.41315	0.706575	+	0.707638i	$0.250239\pi$
$242$	−2.56636	−0.164972
$243$	0	0
$244$	2.87961	0.184348
$245$	−19.2901	−1.23240
$246$	0	0
$247$	4.79069	0.304824
$248$	−6.87324	−0.436451
$249$	0	0
$250$	−6.43743	−0.407139
$251$	0.270958	0.0171027	0.00855134	−	0.999963i	$-0.497278\pi$
$251$	0.270958	0.0171027	0.00855134	+	0.999963i	$0.497278\pi$
$252$	0	0
$253$	0	0
$254$	−4.81847	−0.302338
$255$	0	0
$256$	1.00000	0.0625000
$257$	−14.6034	−0.910932	−0.455466	−	0.890253i	$-0.650527\pi$
$257$	−14.6034	−0.910932	−0.455466	+	0.890253i	$0.650527\pi$
$258$	0	0
$259$	−0.941343	−0.0584922
$260$	2.99951	0.186022
$261$	0	0
$262$	−15.9778	−0.987109
$263$	−17.0690	−1.05252	−0.526260	−	0.850324i	$-0.676406\pi$
$263$	−17.0690	−1.05252	−0.526260	+	0.850324i	$0.676406\pi$
$264$	0	0
$265$	−11.4777	−0.705070
$266$	−0.859644	−0.0527081
$267$	0	0
$268$	14.6373	0.894118
$269$	−32.2241	−1.96474	−0.982370	−	0.186949i	$-0.940140\pi$
$269$	−32.2241	−1.96474	−0.982370	+	0.186949i	$0.940140\pi$
$270$	0	0
$271$	3.06419	0.186136	0.0930680	−	0.995660i	$-0.470333\pi$
$271$	3.06419	0.186136	0.0930680	+	0.995660i	$0.470333\pi$
$272$	−4.64187	−0.281455
$273$	0	0
$274$	18.7720	1.13406
$275$	7.77296	0.468727
$276$	0	0
$277$	−5.05736	−0.303867	−0.151934	−	0.988391i	$-0.548550\pi$
$277$	−5.05736	−0.303867	−0.151934	+	0.988391i	$0.548550\pi$
$278$	17.7540	1.06481
$279$	0	0
$280$	−0.538234	−0.0321656
$281$	−8.39937	−0.501064	−0.250532	−	0.968108i	$-0.580606\pi$
$281$	−8.39937	−0.501064	−0.250532	+	0.968108i	$0.580606\pi$
$282$	0	0
$283$	−15.2299	−0.905321	−0.452661	−	0.891683i	$-0.649525\pi$
$283$	−15.2299	−0.905321	−0.452661	+	0.891683i	$0.649525\pi$
$284$	−12.7320	−0.755503
$285$	0	0
$286$	3.14394	0.185905
$287$	−1.11602	−0.0658767
$288$	0	0
$289$	4.54698	0.267469
$290$	25.5199	1.49858
$291$	0	0
$292$	2.98812	0.174867
$293$	28.9259	1.68987	0.844934	−	0.534870i	$-0.179639\pi$
$293$	28.9259	1.68987	0.844934	+	0.534870i	$0.179639\pi$
$294$	0	0
$295$	29.6970	1.72902
$296$	4.84574	0.281653
$297$	0	0
$298$	−9.51537	−0.551211
$299$	0	0
$300$	0	0
$301$	1.46997	0.0847275
$302$	−8.67872	−0.499405
$303$	0	0
$304$	4.42518	0.253802
$305$	7.97843	0.456844
$306$	0	0
$307$	−21.5835	−1.23184	−0.615918	−	0.787810i	$-0.711215\pi$
$307$	−21.5835	−1.23184	−0.615918	+	0.787810i	$0.711215\pi$
$308$	−0.564150	−0.0321454
$309$	0	0
$310$	−19.0434	−1.08159
$311$	−0.878874	−0.0498364	−0.0249182	−	0.999689i	$-0.507933\pi$
$311$	−0.878874	−0.0498364	−0.0249182	+	0.999689i	$0.507933\pi$
$312$	0	0
$313$	5.45556	0.308367	0.154183	−	0.988042i	$-0.450725\pi$
$313$	5.45556	0.308367	0.154183	+	0.988042i	$0.450725\pi$
$314$	−19.8163	−1.11830
$315$	0	0
$316$	−2.55897	−0.143954
$317$	5.08681	0.285704	0.142852	−	0.989744i	$-0.454373\pi$
$317$	5.08681	0.285704	0.142852	+	0.989744i	$0.454373\pi$
$318$	0	0
$319$	26.7487	1.49764
$320$	2.77066	0.154885
$321$	0	0
$322$	0	0
$323$	−20.5411	−1.14294
$324$	0	0
$325$	2.89765	0.160733
$326$	14.5014	0.803159
$327$	0	0
$328$	5.74494	0.317211
$329$	−1.09070	−0.0601320
$330$	0	0
$331$	−35.6131	−1.95747	−0.978737	−	0.205118i	$-0.934242\pi$
$331$	−35.6131	−1.95747	−0.978737	+	0.205118i	$0.934242\pi$
$332$	−0.811565	−0.0445404
$333$	0	0
$334$	9.89119	0.541222
$335$	40.5551	2.21576
$336$	0	0
$337$	−11.5044	−0.626686	−0.313343	−	0.949640i	$-0.601449\pi$
$337$	−11.5044	−0.626686	−0.313343	+	0.949640i	$0.601449\pi$
$338$	−11.8280	−0.643358
$339$	0	0
$340$	−12.8611	−0.697489
$341$	−19.9604	−1.08092
$342$	0	0
$343$	2.71233	0.146452
$344$	−7.56694	−0.407982
$345$	0	0
$346$	−13.2434	−0.711972
$347$	15.8035	0.848375	0.424188	−	0.905574i	$-0.360560\pi$
$347$	15.8035	0.848375	0.424188	+	0.905574i	$0.360560\pi$
$348$	0	0
$349$	12.2305	0.654686	0.327343	−	0.944906i	$-0.393847\pi$
$349$	12.2305	0.654686	0.327343	+	0.944906i	$0.393847\pi$
$350$	−0.519956	−0.0277928
$351$	0	0
$352$	2.90407	0.154788
$353$	−3.65673	−0.194628	−0.0973141	−	0.995254i	$-0.531025\pi$
$353$	−3.65673	−0.194628	−0.0973141	+	0.995254i	$0.531025\pi$
$354$	0	0
$355$	−35.2760	−1.87225
$356$	6.77984	0.359331
$357$	0	0
$358$	−5.24531	−0.277223
$359$	−13.1577	−0.694436	−0.347218	−	0.937784i	$-0.612874\pi$
$359$	−13.1577	−0.694436	−0.347218	+	0.937784i	$0.612874\pi$
$360$	0	0
$361$	0.582228	0.0306436
$362$	16.2553	0.854359
$363$	0	0
$364$	−0.210307	−0.0110231
$365$	8.27908	0.433347
$366$	0	0
$367$	12.3385	0.644064	0.322032	−	0.946729i	$-0.395634\pi$
$367$	12.3385	0.644064	0.322032	+	0.946729i	$0.395634\pi$
$368$	0	0
$369$	0	0
$370$	13.4259	0.697980
$371$	0.804746	0.0417803
$372$	0	0
$373$	18.5497	0.960468	0.480234	−	0.877140i	$-0.340552\pi$
$373$	18.5497	0.960468	0.480234	+	0.877140i	$0.340552\pi$
$374$	−13.4803	−0.697051
$375$	0	0
$376$	5.61457	0.289549
$377$	9.97153	0.513560
$378$	0	0
$379$	−2.85973	−0.146895	−0.0734473	−	0.997299i	$-0.523400\pi$
$379$	−2.85973	−0.146895	−0.0734473	+	0.997299i	$0.523400\pi$
$380$	12.2607	0.628960
$381$	0	0
$382$	−7.69266	−0.393590
$383$	−0.0804665	−0.00411165	−0.00205582	−	0.999998i	$-0.500654\pi$
$383$	−0.0804665	−0.00411165	−0.00205582	+	0.999998i	$0.500654\pi$
$384$	0	0
$385$	−1.56307	−0.0796614
$386$	−1.42745	−0.0726552
$387$	0	0
$388$	12.3679	0.627887
$389$	−13.7526	−0.697283	−0.348641	−	0.937256i	$-0.613357\pi$
$389$	−13.7526	−0.697283	−0.348641	+	0.937256i	$0.613357\pi$
$390$	0	0
$391$	0	0
$392$	−6.96226	−0.351647
$393$	0	0
$394$	−6.25572	−0.315159
$395$	−7.09006	−0.356739
$396$	0	0
$397$	−6.62898	−0.332699	−0.166349	−	0.986067i	$-0.553198\pi$
$397$	−6.62898	−0.332699	−0.166349	+	0.986067i	$0.553198\pi$
$398$	−0.863136	−0.0432651
$399$	0	0
$400$	2.67657	0.133829
$401$	−35.2896	−1.76228	−0.881139	−	0.472858i	$-0.843222\pi$
$401$	−35.2896	−1.76228	−0.881139	+	0.472858i	$0.843222\pi$
$402$	0	0
$403$	−7.44095	−0.370660
$404$	−15.6967	−0.780939
$405$	0	0
$406$	−1.78930	−0.0888014
$407$	14.0724	0.697543
$408$	0	0
$409$	−29.2533	−1.44648	−0.723241	−	0.690596i	$-0.757348\pi$
$409$	−29.2533	−1.44648	−0.723241	+	0.690596i	$0.757348\pi$
$410$	15.9173	0.786099
$411$	0	0
$412$	−9.44491	−0.465317
$413$	−2.08217	−0.102457
$414$	0	0
$415$	−2.24857	−0.110378
$416$	1.08260	0.0530787
$417$	0	0
$418$	12.8510	0.628565
$419$	0.728032	0.0355667	0.0177833	−	0.999842i	$-0.494339\pi$
$419$	0.728032	0.0355667	0.0177833	+	0.999842i	$0.494339\pi$
$420$	0	0
$421$	33.8531	1.64990	0.824950	−	0.565205i	$-0.191203\pi$
$421$	33.8531	1.64990	0.824950	+	0.565205i	$0.191203\pi$
$422$	19.0647	0.928053
$423$	0	0
$424$	−4.14258	−0.201182
$425$	−12.4243	−0.602668
$426$	0	0
$427$	−0.559398	−0.0270712
$428$	9.51202	0.459781
$429$	0	0
$430$	−20.9654	−1.01104
$431$	1.48395	0.0714793	0.0357397	−	0.999361i	$-0.488621\pi$
$431$	1.48395	0.0714793	0.0357397	+	0.999361i	$0.488621\pi$
$432$	0	0
$433$	6.97476	0.335186	0.167593	−	0.985856i	$-0.446401\pi$
$433$	6.97476	0.335186	0.167593	+	0.985856i	$0.446401\pi$
$434$	1.33521	0.0640920
$435$	0	0
$436$	20.2217	0.968443
$437$	0	0
$438$	0	0
$439$	9.32092	0.444863	0.222432	−	0.974948i	$-0.428601\pi$
$439$	9.32092	0.444863	0.222432	+	0.974948i	$0.428601\pi$
$440$	8.04621	0.383588
$441$	0	0
$442$	−5.02527	−0.239028
$443$	−31.6492	−1.50370	−0.751849	−	0.659335i	$-0.770838\pi$
$443$	−31.6492	−1.50370	−0.751849	+	0.659335i	$0.770838\pi$
$444$	0	0
$445$	18.7846	0.890478
$446$	6.41755	0.303880
$447$	0	0
$448$	−0.194262	−0.00917801
$449$	9.56061	0.451193	0.225596	−	0.974221i	$-0.427567\pi$
$449$	9.56061	0.451193	0.225596	+	0.974221i	$0.427567\pi$
$450$	0	0
$451$	16.6837	0.785606
$452$	1.31667	0.0619309
$453$	0	0
$454$	−1.19685	−0.0561709
$455$	−0.582690	−0.0273169
$456$	0	0
$457$	12.8604	0.601584	0.300792	−	0.953690i	$-0.402749\pi$
$457$	12.8604	0.601584	0.300792	+	0.953690i	$0.402749\pi$
$458$	28.0856	1.31235
$459$	0	0
$460$	0	0
$461$	−18.4038	−0.857151	−0.428575	−	0.903506i	$-0.640984\pi$
$461$	−18.4038	−0.857151	−0.428575	+	0.903506i	$0.640984\pi$
$462$	0	0
$463$	27.6569	1.28533	0.642663	−	0.766149i	$-0.277829\pi$
$463$	27.6569	1.28533	0.642663	+	0.766149i	$0.277829\pi$
$464$	9.21076	0.427599
$465$	0	0
$466$	16.4128	0.760307
$467$	24.3449	1.12655	0.563273	−	0.826271i	$-0.309542\pi$
$467$	24.3449	1.12655	0.563273	+	0.826271i	$0.309542\pi$
$468$	0	0
$469$	−2.84348	−0.131300
$470$	15.5561	0.717548
$471$	0	0
$472$	10.7184	0.493352
$473$	−21.9749	−1.01041
$474$	0	0
$475$	11.8443	0.543455
$476$	0.901738	0.0413311
$477$	0	0
$478$	−10.3357	−0.472745
$479$	−7.29209	−0.333184	−0.166592	−	0.986026i	$-0.553276\pi$
$479$	−7.29209	−0.333184	−0.166592	+	0.986026i	$0.553276\pi$
$480$	0	0
$481$	5.24598	0.239196
$482$	21.9380	0.999248
$483$	0	0
$484$	−2.56636	−0.116653
$485$	34.2674	1.55600
$486$	0	0
$487$	22.4092	1.01546	0.507728	−	0.861517i	$-0.330485\pi$
$487$	22.4092	1.01546	0.507728	+	0.861517i	$0.330485\pi$
$488$	2.87961	0.130354
$489$	0	0
$490$	−19.2901	−0.871437
$491$	−19.7002	−0.889057	−0.444528	−	0.895765i	$-0.646629\pi$
$491$	−19.7002	−0.889057	−0.444528	+	0.895765i	$0.646629\pi$
$492$	0	0
$493$	−42.7552	−1.92560
$494$	4.79069	0.215543
$495$	0	0
$496$	−6.87324	−0.308618
$497$	2.47333	0.110944
$498$	0	0
$499$	−17.2938	−0.774175	−0.387087	−	0.922043i	$-0.626519\pi$
$499$	−17.2938	−0.774175	−0.387087	+	0.922043i	$0.626519\pi$
$500$	−6.43743	−0.287891
$501$	0	0
$502$	0.270958	0.0120934
$503$	16.4299	0.732575	0.366287	−	0.930502i	$-0.380629\pi$
$503$	16.4299	0.732575	0.366287	+	0.930502i	$0.380629\pi$
$504$	0	0
$505$	−43.4902	−1.93529
$506$	0	0
$507$	0	0
$508$	−4.81847	−0.213785
$509$	−26.9951	−1.19654	−0.598268	−	0.801296i	$-0.704144\pi$
$509$	−26.9951	−1.19654	−0.598268	+	0.801296i	$0.704144\pi$
$510$	0	0
$511$	−0.580478	−0.0256788
$512$	1.00000	0.0441942
$513$	0	0
$514$	−14.6034	−0.644126
$515$	−26.1687	−1.15313
$516$	0	0
$517$	16.3051	0.717098
$518$	−0.941343	−0.0413602
$519$	0	0
$520$	2.99951	0.131537
$521$	23.4864	1.02896	0.514479	−	0.857503i	$-0.327985\pi$
$521$	23.4864	1.02896	0.514479	+	0.857503i	$0.327985\pi$
$522$	0	0
$523$	−29.3360	−1.28277	−0.641386	−	0.767218i	$-0.721640\pi$
$523$	−29.3360	−1.28277	−0.641386	+	0.767218i	$0.721640\pi$
$524$	−15.9778	−0.697991
$525$	0	0
$526$	−17.0690	−0.744244
$527$	31.9047	1.38979
$528$	0	0
$529$	0	0
$530$	−11.4777	−0.498560
$531$	0	0
$532$	−0.859644	−0.0372703
$533$	6.21945	0.269394
$534$	0	0
$535$	26.3546	1.13941
$536$	14.6373	0.632237
$537$	0	0
$538$	−32.2241	−1.38928
$539$	−20.2189	−0.870890
$540$	0	0
$541$	−4.79891	−0.206321	−0.103161	−	0.994665i	$-0.532896\pi$
$541$	−4.79891	−0.206321	−0.103161	+	0.994665i	$0.532896\pi$
$542$	3.06419	0.131618
$543$	0	0
$544$	−4.64187	−0.199019
$545$	56.0275	2.39995
$546$	0	0
$547$	27.4867	1.17525	0.587623	−	0.809135i	$-0.300064\pi$
$547$	27.4867	1.17525	0.587623	+	0.809135i	$0.300064\pi$
$548$	18.7720	0.801898
$549$	0	0
$550$	7.77296	0.331440
$551$	40.7593	1.73640
$552$	0	0
$553$	0.497111	0.0211393
$554$	−5.05736	−0.214866
$555$	0	0
$556$	17.7540	0.752938
$557$	−12.0051	−0.508672	−0.254336	−	0.967116i	$-0.581857\pi$
$557$	−12.0051	−0.508672	−0.254336	+	0.967116i	$0.581857\pi$
$558$	0	0
$559$	−8.19194	−0.346482
$560$	−0.538234	−0.0227445
$561$	0	0
$562$	−8.39937	−0.354306
$563$	−6.10797	−0.257420	−0.128710	−	0.991682i	$-0.541084\pi$
$563$	−6.10797	−0.257420	−0.128710	+	0.991682i	$0.541084\pi$
$564$	0	0
$565$	3.64804	0.153474
$566$	−15.2299	−0.640159
$567$	0	0
$568$	−12.7320	−0.534221
$569$	25.0080	1.04839	0.524194	−	0.851599i	$-0.324367\pi$
$569$	25.0080	1.04839	0.524194	+	0.851599i	$0.324367\pi$
$570$	0	0
$571$	23.3467	0.977027	0.488513	−	0.872556i	$-0.337539\pi$
$571$	23.3467	0.977027	0.488513	+	0.872556i	$0.337539\pi$
$572$	3.14394	0.131455
$573$	0	0
$574$	−1.11602	−0.0465819
$575$	0	0
$576$	0	0
$577$	36.7762	1.53101	0.765506	−	0.643429i	$-0.222489\pi$
$577$	36.7762	1.53101	0.765506	+	0.643429i	$0.222489\pi$
$578$	4.54698	0.189129
$579$	0	0
$580$	25.5199	1.05966
$581$	0.157656	0.00654068
$582$	0	0
$583$	−12.0304	−0.498247
$584$	2.98812	0.123649
$585$	0	0
$586$	28.9259	1.19492
$587$	−23.6864	−0.977641	−0.488821	−	0.872384i	$-0.662573\pi$
$587$	−23.6864	−0.977641	−0.488821	+	0.872384i	$0.662573\pi$
$588$	0	0
$589$	−30.4153	−1.25324
$590$	29.6970	1.22260
$591$	0	0
$592$	4.84574	0.199159
$593$	−26.0813	−1.07103	−0.535515	−	0.844525i	$-0.679883\pi$
$593$	−26.0813	−1.07103	−0.535515	+	0.844525i	$0.679883\pi$
$594$	0	0
$595$	2.49841	0.102425
$596$	−9.51537	−0.389765
$597$	0	0
$598$	0	0
$599$	−11.2311	−0.458891	−0.229445	−	0.973322i	$-0.573691\pi$
$599$	−11.2311	−0.458891	−0.229445	+	0.973322i	$0.573691\pi$
$600$	0	0
$601$	22.7180	0.926688	0.463344	−	0.886179i	$-0.346650\pi$
$601$	22.7180	0.926688	0.463344	+	0.886179i	$0.346650\pi$
$602$	1.46997	0.0599114
$603$	0	0
$604$	−8.67872	−0.353132
$605$	−7.11052	−0.289084
$606$	0	0
$607$	16.0618	0.651928	0.325964	−	0.945382i	$-0.394311\pi$
$607$	16.0618	0.651928	0.325964	+	0.945382i	$0.394311\pi$
$608$	4.42518	0.179465
$609$	0	0
$610$	7.97843	0.323037
$611$	6.07831	0.245902
$612$	0	0
$613$	−6.29694	−0.254331	−0.127165	−	0.991882i	$-0.540588\pi$
$613$	−6.29694	−0.254331	−0.127165	+	0.991882i	$0.540588\pi$
$614$	−21.5835	−0.871040
$615$	0	0
$616$	−0.564150	−0.0227303
$617$	−9.62759	−0.387592	−0.193796	−	0.981042i	$-0.562080\pi$
$617$	−9.62759	−0.387592	−0.193796	+	0.981042i	$0.562080\pi$
$618$	0	0
$619$	−24.7284	−0.993917	−0.496958	−	0.867774i	$-0.665550\pi$
$619$	−24.7284	−0.993917	−0.496958	+	0.867774i	$0.665550\pi$
$620$	−19.0434	−0.764803
$621$	0	0
$622$	−0.878874	−0.0352396
$623$	−1.31706	−0.0527670
$624$	0	0
$625$	−31.2188	−1.24875
$626$	5.45556	0.218048
$627$	0	0
$628$	−19.8163	−0.790758
$629$	−22.4933	−0.896867
$630$	0	0
$631$	−35.8294	−1.42635	−0.713173	−	0.700988i	$-0.752743\pi$
$631$	−35.8294	−1.42635	−0.713173	+	0.700988i	$0.752743\pi$
$632$	−2.55897	−0.101791
$633$	0	0
$634$	5.08681	0.202023
$635$	−13.3504	−0.529793
$636$	0	0
$637$	−7.53732	−0.298640
$638$	26.7487	1.05899
$639$	0	0
$640$	2.77066	0.109520
$641$	−12.6825	−0.500930	−0.250465	−	0.968126i	$-0.580583\pi$
$641$	−12.6825	−0.500930	−0.250465	+	0.968126i	$0.580583\pi$
$642$	0	0
$643$	27.3350	1.07799	0.538993	−	0.842310i	$-0.318805\pi$
$643$	27.3350	1.07799	0.538993	+	0.842310i	$0.318805\pi$
$644$	0	0
$645$	0	0
$646$	−20.5411	−0.808180
$647$	4.68503	0.184188	0.0920938	−	0.995750i	$-0.470644\pi$
$647$	4.68503	0.184188	0.0920938	+	0.995750i	$0.470644\pi$
$648$	0	0
$649$	31.1269	1.22184
$650$	2.89765	0.113655
$651$	0	0
$652$	14.5014	0.567919
$653$	28.7415	1.12474	0.562372	−	0.826885i	$-0.309889\pi$
$653$	28.7415	1.12474	0.562372	+	0.826885i	$0.309889\pi$
$654$	0	0
$655$	−44.2690	−1.72973
$656$	5.74494	0.224302
$657$	0	0
$658$	−1.09070	−0.0425198
$659$	3.61975	0.141006	0.0705028	−	0.997512i	$-0.477540\pi$
$659$	3.61975	0.141006	0.0705028	+	0.997512i	$0.477540\pi$
$660$	0	0
$661$	17.0756	0.664162	0.332081	−	0.943251i	$-0.392249\pi$
$661$	17.0756	0.664162	0.332081	+	0.943251i	$0.392249\pi$
$662$	−35.6131	−1.38414
$663$	0	0
$664$	−0.811565	−0.0314948
$665$	−2.38178	−0.0923616
$666$	0	0
$667$	0	0
$668$	9.89119	0.382702
$669$	0	0
$670$	40.5551	1.56678
$671$	8.36260	0.322835
$672$	0	0
$673$	44.6795	1.72227	0.861133	−	0.508379i	$-0.169755\pi$
$673$	44.6795	1.72227	0.861133	+	0.508379i	$0.169755\pi$
$674$	−11.5044	−0.443134
$675$	0	0
$676$	−11.8280	−0.454922
$677$	20.2343	0.777666	0.388833	−	0.921308i	$-0.372878\pi$
$677$	20.2343	0.777666	0.388833	+	0.921308i	$0.372878\pi$
$678$	0	0
$679$	−2.40262	−0.0922040
$680$	−12.8611	−0.493199
$681$	0	0
$682$	−19.9604	−0.764323
$683$	41.2263	1.57748	0.788740	−	0.614727i	$-0.210734\pi$
$683$	41.2263	1.57748	0.788740	+	0.614727i	$0.210734\pi$
$684$	0	0
$685$	52.0108	1.98723
$686$	2.71233	0.103557
$687$	0	0
$688$	−7.56694	−0.288487
$689$	−4.48475	−0.170855
$690$	0	0
$691$	−39.4196	−1.49959	−0.749795	−	0.661670i	$-0.769848\pi$
$691$	−39.4196	−1.49959	−0.749795	+	0.661670i	$0.769848\pi$
$692$	−13.2434	−0.503440
$693$	0	0
$694$	15.8035	0.599892
$695$	49.1904	1.86590
$696$	0	0
$697$	−26.6673	−1.01010
$698$	12.2305	0.462933
$699$	0	0
$700$	−0.519956	−0.0196525
$701$	−45.1856	−1.70663	−0.853317	−	0.521392i	$-0.825413\pi$
$701$	−45.1856	−1.70663	−0.853317	+	0.521392i	$0.825413\pi$
$702$	0	0
$703$	21.4433	0.808749
$704$	2.90407	0.109451
$705$	0	0
$706$	−3.65673	−0.137623
$707$	3.04926	0.114679
$708$	0	0
$709$	−31.8988	−1.19799	−0.598993	−	0.800755i	$-0.704432\pi$
$709$	−31.8988	−1.19799	−0.598993	+	0.800755i	$0.704432\pi$
$710$	−35.2760	−1.32388
$711$	0	0
$712$	6.77984	0.254085
$713$	0	0
$714$	0	0
$715$	8.71079	0.325765
$716$	−5.24531	−0.196026
$717$	0	0
$718$	−13.1577	−0.491041
$719$	25.1742	0.938839	0.469420	−	0.882975i	$-0.344463\pi$
$719$	25.1742	0.938839	0.469420	+	0.882975i	$0.344463\pi$
$720$	0	0
$721$	1.83478	0.0683309
$722$	0.582228	0.0216683
$723$	0	0
$724$	16.2553	0.604123
$725$	24.6533	0.915599
$726$	0	0
$727$	−2.36635	−0.0877629	−0.0438815	−	0.999037i	$-0.513972\pi$
$727$	−2.36635	−0.0877629	−0.0438815	+	0.999037i	$0.513972\pi$
$728$	−0.210307	−0.00779450
$729$	0	0
$730$	8.27908	0.306423
$731$	35.1248	1.29914
$732$	0	0
$733$	0.436382	0.0161181	0.00805907	−	0.999968i	$-0.497435\pi$
$733$	0.436382	0.0161181	0.00805907	+	0.999968i	$0.497435\pi$
$734$	12.3385	0.455422
$735$	0	0
$736$	0	0
$737$	42.5079	1.56580
$738$	0	0
$739$	−35.0846	−1.29061	−0.645304	−	0.763926i	$-0.723269\pi$
$739$	−35.0846	−1.29061	−0.645304	+	0.763926i	$0.723269\pi$
$740$	13.4259	0.493547
$741$	0	0
$742$	0.804746	0.0295431
$743$	33.6165	1.23327	0.616635	−	0.787249i	$-0.288495\pi$
$743$	33.6165	1.23327	0.616635	+	0.787249i	$0.288495\pi$
$744$	0	0
$745$	−26.3639	−0.965898
$746$	18.5497	0.679154
$747$	0	0
$748$	−13.4803	−0.492890
$749$	−1.84782	−0.0675180
$750$	0	0
$751$	−39.7647	−1.45104	−0.725518	−	0.688204i	$-0.758400\pi$
$751$	−39.7647	−1.45104	−0.725518	+	0.688204i	$0.758400\pi$
$752$	5.61457	0.204742
$753$	0	0
$754$	9.97153	0.363142
$755$	−24.0458	−0.875117
$756$	0	0
$757$	24.8420	0.902900	0.451450	−	0.892297i	$-0.350907\pi$
$757$	24.8420	0.902900	0.451450	+	0.892297i	$0.350907\pi$
$758$	−2.85973	−0.103870
$759$	0	0
$760$	12.2607	0.444742
$761$	24.3353	0.882153	0.441076	−	0.897470i	$-0.354597\pi$
$761$	24.3353	0.882153	0.441076	+	0.897470i	$0.354597\pi$
$762$	0	0
$763$	−3.92830	−0.142214
$764$	−7.69266	−0.278310
$765$	0	0
$766$	−0.0804665	−0.00290737
$767$	11.6037	0.418984
$768$	0	0
$769$	26.4421	0.953528	0.476764	−	0.879031i	$-0.341810\pi$
$769$	26.4421	0.953528	0.476764	+	0.879031i	$0.341810\pi$
$770$	−1.56307	−0.0563291
$771$	0	0
$772$	−1.42745	−0.0513750
$773$	22.1225	0.795691	0.397845	−	0.917452i	$-0.369758\pi$
$773$	22.1225	0.795691	0.397845	+	0.917452i	$0.369758\pi$
$774$	0	0
$775$	−18.3967	−0.660830
$776$	12.3679	0.443983
$777$	0	0
$778$	−13.7526	−0.493053
$779$	25.4224	0.910852
$780$	0	0
$781$	−36.9745	−1.32305
$782$	0	0
$783$	0	0
$784$	−6.96226	−0.248652
$785$	−54.9044	−1.95962
$786$	0	0
$787$	−4.51719	−0.161021	−0.0805103	−	0.996754i	$-0.525655\pi$
$787$	−4.51719	−0.161021	−0.0805103	+	0.996754i	$0.525655\pi$
$788$	−6.25572	−0.222851
$789$	0	0
$790$	−7.09006	−0.252253
$791$	−0.255778	−0.00909443
$792$	0	0
$793$	3.11746	0.110704
$794$	−6.62898	−0.235254
$795$	0	0
$796$	−0.863136	−0.0305931
$797$	−30.3071	−1.07353	−0.536765	−	0.843732i	$-0.680354\pi$
$797$	−30.3071	−1.07353	−0.536765	+	0.843732i	$0.680354\pi$
$798$	0	0
$799$	−26.0621	−0.922011
$800$	2.67657	0.0946312
$801$	0	0
$802$	−35.2896	−1.24612
$803$	8.67773	0.306230
$804$	0	0
$805$	0	0
$806$	−7.44095	−0.262096
$807$	0	0
$808$	−15.6967	−0.552207
$809$	−28.3620	−0.997155	−0.498578	−	0.866845i	$-0.666144\pi$
$809$	−28.3620	−0.997155	−0.498578	+	0.866845i	$0.666144\pi$
$810$	0	0
$811$	−10.6172	−0.372820	−0.186410	−	0.982472i	$-0.559685\pi$
$811$	−10.6172	−0.372820	−0.186410	+	0.982472i	$0.559685\pi$
$812$	−1.78930	−0.0627921
$813$	0	0
$814$	14.0724	0.493237
$815$	40.1785	1.40739
$816$	0	0
$817$	−33.4851	−1.17149
$818$	−29.2533	−1.02282
$819$	0	0
$820$	15.9173	0.555856
$821$	−15.5285	−0.541947	−0.270973	−	0.962587i	$-0.587346\pi$
$821$	−15.5285	−0.541947	−0.270973	+	0.962587i	$0.587346\pi$
$822$	0	0
$823$	−50.1664	−1.74869	−0.874346	−	0.485303i	$-0.838709\pi$
$823$	−50.1664	−1.74869	−0.874346	+	0.485303i	$0.838709\pi$
$824$	−9.44491	−0.329029
$825$	0	0
$826$	−2.08217	−0.0724478
$827$	−33.0620	−1.14968	−0.574839	−	0.818267i	$-0.694935\pi$
$827$	−33.0620	−1.14968	−0.574839	+	0.818267i	$0.694935\pi$
$828$	0	0
$829$	−11.7345	−0.407556	−0.203778	−	0.979017i	$-0.565322\pi$
$829$	−11.7345	−0.407556	−0.203778	+	0.979017i	$0.565322\pi$
$830$	−2.24857	−0.0780492
$831$	0	0
$832$	1.08260	0.0375323
$833$	32.3179	1.11975
$834$	0	0
$835$	27.4052	0.948395
$836$	12.8510	0.444463
$837$	0	0
$838$	0.728032	0.0251494
$839$	−7.70270	−0.265927	−0.132963	−	0.991121i	$-0.542449\pi$
$839$	−7.70270	−0.265927	−0.132963	+	0.991121i	$0.542449\pi$
$840$	0	0
$841$	55.8381	1.92545
$842$	33.8531	1.16666
$843$	0	0
$844$	19.0647	0.656233
$845$	−32.7714	−1.12737
$846$	0	0
$847$	0.498546	0.0171302
$848$	−4.14258	−0.142257
$849$	0	0
$850$	−12.4243	−0.426150
$851$	0	0
$852$	0	0
$853$	−42.2482	−1.44655	−0.723275	−	0.690560i	$-0.757364\pi$
$853$	−42.2482	−1.44655	−0.723275	+	0.690560i	$0.757364\pi$
$854$	−0.559398	−0.0191422
$855$	0	0
$856$	9.51202	0.325114
$857$	19.3318	0.660363	0.330181	−	0.943918i	$-0.392890\pi$
$857$	19.3318	0.660363	0.330181	+	0.943918i	$0.392890\pi$
$858$	0	0
$859$	−20.0821	−0.685192	−0.342596	−	0.939483i	$-0.611306\pi$
$859$	−20.0821	−0.685192	−0.342596	+	0.939483i	$0.611306\pi$
$860$	−20.9654	−0.714916
$861$	0	0
$862$	1.48395	0.0505435
$863$	−5.80470	−0.197594	−0.0987972	−	0.995108i	$-0.531500\pi$
$863$	−5.80470	−0.197594	−0.0987972	+	0.995108i	$0.531500\pi$
$864$	0	0
$865$	−36.6931	−1.24760
$866$	6.97476	0.237012
$867$	0	0
$868$	1.33521	0.0453199
$869$	−7.43145	−0.252095
$870$	0	0
$871$	15.8463	0.536933
$872$	20.2217	0.684793
$873$	0	0
$874$	0	0
$875$	1.25055	0.0422762
$876$	0	0
$877$	35.9926	1.21538	0.607692	−	0.794173i	$-0.292095\pi$
$877$	35.9926	1.21538	0.607692	+	0.794173i	$0.292095\pi$
$878$	9.32092	0.314566
$879$	0	0
$880$	8.04621	0.271238
$881$	37.4088	1.26033	0.630167	−	0.776460i	$-0.282987\pi$
$881$	37.4088	1.26033	0.630167	+	0.776460i	$0.282987\pi$
$882$	0	0
$883$	−50.3520	−1.69448	−0.847239	−	0.531211i	$-0.821737\pi$
$883$	−50.3520	−1.69448	−0.847239	+	0.531211i	$0.821737\pi$
$884$	−5.02527	−0.169018
$885$	0	0
$886$	−31.6492	−1.06328
$887$	−30.6942	−1.03061	−0.515305	−	0.857007i	$-0.672321\pi$
$887$	−30.6942	−1.03061	−0.515305	+	0.857007i	$0.672321\pi$
$888$	0	0
$889$	0.936045	0.0313939
$890$	18.7846	0.629663
$891$	0	0
$892$	6.41755	0.214875
$893$	24.8455	0.831422
$894$	0	0
$895$	−14.5330	−0.485784
$896$	−0.194262	−0.00648983
$897$	0	0
$898$	9.56061	0.319042
$899$	−63.3078	−2.11143
$900$	0	0
$901$	19.2293	0.640622
$902$	16.6837	0.555507
$903$	0	0
$904$	1.31667	0.0437917
$905$	45.0379	1.49711
$906$	0	0
$907$	5.40828	0.179579	0.0897895	−	0.995961i	$-0.471381\pi$
$907$	5.40828	0.179579	0.0897895	+	0.995961i	$0.471381\pi$
$908$	−1.19685	−0.0397188
$909$	0	0
$910$	−0.582690	−0.0193160
$911$	−45.2428	−1.49896	−0.749480	−	0.662027i	$-0.769696\pi$
$911$	−45.2428	−1.49896	−0.749480	+	0.662027i	$0.769696\pi$
$912$	0	0
$913$	−2.35684	−0.0780002
$914$	12.8604	0.425384
$915$	0	0
$916$	28.0856	0.927974
$917$	3.10387	0.102499
$918$	0	0
$919$	−10.7793	−0.355576	−0.177788	−	0.984069i	$-0.556894\pi$
$919$	−10.7793	−0.355576	−0.177788	+	0.984069i	$0.556894\pi$
$920$	0	0
$921$	0	0
$922$	−18.4038	−0.606097
$923$	−13.7836	−0.453692
$924$	0	0
$925$	12.9700	0.426451
$926$	27.6569	0.908863
$927$	0	0
$928$	9.21076	0.302358
$929$	31.3456	1.02842	0.514208	−	0.857665i	$-0.328086\pi$
$929$	31.3456	1.02842	0.514208	+	0.857665i	$0.328086\pi$
$930$	0	0
$931$	−30.8093	−1.00973
$932$	16.4128	0.537618
$933$	0	0
$934$	24.3449	0.796589
$935$	−37.3495	−1.22146
$936$	0	0
$937$	−16.1958	−0.529093	−0.264546	−	0.964373i	$-0.585222\pi$
$937$	−16.1958	−0.529093	−0.264546	+	0.964373i	$0.585222\pi$
$938$	−2.84348	−0.0928428
$939$	0	0
$940$	15.5561	0.507383
$941$	−24.0739	−0.784786	−0.392393	−	0.919798i	$-0.628353\pi$
$941$	−24.0739	−0.784786	−0.392393	+	0.919798i	$0.628353\pi$
$942$	0	0
$943$	0	0
$944$	10.7184	0.348853
$945$	0	0
$946$	−21.9749	−0.714467
$947$	−4.95630	−0.161058	−0.0805290	−	0.996752i	$-0.525661\pi$
$947$	−4.95630	−0.161058	−0.0805290	+	0.996752i	$0.525661\pi$
$948$	0	0
$949$	3.23493	0.105010
$950$	11.8443	0.384281
$951$	0	0
$952$	0.901738	0.0292255
$953$	−38.5775	−1.24965	−0.624824	−	0.780766i	$-0.714829\pi$
$953$	−38.5775	−1.24965	−0.624824	+	0.780766i	$0.714829\pi$
$954$	0	0
$955$	−21.3138	−0.689697
$956$	−10.3357	−0.334281
$957$	0	0
$958$	−7.29209	−0.235597
$959$	−3.64667	−0.117757
$960$	0	0
$961$	16.2415	0.523918
$962$	5.24598	0.169137
$963$	0	0
$964$	21.9380	0.706575
$965$	−3.95498	−0.127315
$966$	0	0
$967$	47.2484	1.51940	0.759702	−	0.650271i	$-0.225345\pi$
$967$	47.2484	1.51940	0.759702	+	0.650271i	$0.225345\pi$
$968$	−2.56636	−0.0824860
$969$	0	0
$970$	34.2674	1.10026
$971$	17.1237	0.549525	0.274763	−	0.961512i	$-0.411401\pi$
$971$	17.1237	0.549525	0.274763	+	0.961512i	$0.411401\pi$
$972$	0	0
$973$	−3.44893	−0.110567
$974$	22.4092	0.718036
$975$	0	0
$976$	2.87961	0.0921741
$977$	−23.5480	−0.753367	−0.376683	−	0.926342i	$-0.622936\pi$
$977$	−23.5480	−0.753367	−0.376683	+	0.926342i	$0.622936\pi$
$978$	0	0
$979$	19.6891	0.629268
$980$	−19.2901	−0.616199
$981$	0	0
$982$	−19.7002	−0.628658
$983$	−8.03154	−0.256166	−0.128083	−	0.991763i	$-0.540882\pi$
$983$	−8.03154	−0.256166	−0.128083	+	0.991763i	$0.540882\pi$
$984$	0	0
$985$	−17.3325	−0.552259
$986$	−42.7552	−1.36160
$987$	0	0
$988$	4.79069	0.152412
$989$	0	0
$990$	0	0
$991$	−53.1533	−1.68847	−0.844235	−	0.535973i	$-0.819945\pi$
$991$	−53.1533	−1.68847	−0.844235	+	0.535973i	$0.819945\pi$
$992$	−6.87324	−0.218226
$993$	0	0
$994$	2.47333	0.0784493
$995$	−2.39146	−0.0758144
$996$	0	0
$997$	−14.4036	−0.456167	−0.228083	−	0.973642i	$-0.573246\pi$
$997$	−14.4036	−0.456167	−0.228083	+	0.973642i	$0.573246\pi$
$998$	−17.2938	−0.547424
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9522.2.a.by.1.4		5
3.2	odd	2		3174.2.a.w.1.2		5
23.7	odd	22		414.2.i.a.325.1		10
23.10	odd	22		414.2.i.a.307.1		10
23.22	odd	2		9522.2.a.bx.1.2		5
69.53	even	22		138.2.e.d.49.1	yes	10
69.56	even	22		138.2.e.d.31.1	✓	10
69.68	even	2		3174.2.a.x.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.2.e.d.31.1	✓	10	69.56	even	22
138.2.e.d.49.1	yes	10	69.53	even	22
414.2.i.a.307.1		10	23.10	odd	22
414.2.i.a.325.1		10	23.7	odd	22
3174.2.a.w.1.2		5	3.2	odd	2
3174.2.a.x.1.4		5	69.68	even	2
9522.2.a.bx.1.2		5	23.22	odd	2
9522.2.a.by.1.4		5	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +2.77066 q^{5} -0.194262 q^{7} +1.00000 q^{8} +2.77066 q^{10} +2.90407 q^{11} +1.08260 q^{13} -0.194262 q^{14} +1.00000 q^{16} -4.64187 q^{17} +4.42518 q^{19} +2.77066 q^{20} +2.90407 q^{22} +2.67657 q^{25} +1.08260 q^{26} -0.194262 q^{28} +9.21076 q^{29} -6.87324 q^{31} +1.00000 q^{32} -4.64187 q^{34} -0.538234 q^{35} +4.84574 q^{37} +4.42518 q^{38} +2.77066 q^{40} +5.74494 q^{41} -7.56694 q^{43} +2.90407 q^{44} +5.61457 q^{47} -6.96226 q^{49} +2.67657 q^{50} +1.08260 q^{52} -4.14258 q^{53} +8.04621 q^{55} -0.194262 q^{56} +9.21076 q^{58} +10.7184 q^{59} +2.87961 q^{61} -6.87324 q^{62} +1.00000 q^{64} +2.99951 q^{65} +14.6373 q^{67} -4.64187 q^{68} -0.538234 q^{70} -12.7320 q^{71} +2.98812 q^{73} +4.84574 q^{74} +4.42518 q^{76} -0.564150 q^{77} -2.55897 q^{79} +2.77066 q^{80} +5.74494 q^{82} -0.811565 q^{83} -12.8611 q^{85} -7.56694 q^{86} +2.90407 q^{88} +6.77984 q^{89} -0.210307 q^{91} +5.61457 q^{94} +12.2607 q^{95} +12.3679 q^{97} -6.96226 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{2} + 5 q^{4} + q^{5} - q^{7} + 5 q^{8} + q^{10} + 3 q^{11} - 10 q^{13} - q^{14} + 5 q^{16} + 11 q^{17} + 11 q^{19} + q^{20} + 3 q^{22} + 6 q^{25} - 10 q^{26} - q^{28} + 25 q^{29} - 4 q^{31}+ \cdots - 4 q^{98}+O(q^{100}) \) 5 * q + 5 * q^2 + 5 * q^4 + q^5 - q^7 + 5 * q^8 + q^10 + 3 * q^11 - 10 * q^13 - q^14 + 5 * q^16 + 11 * q^17 + 11 * q^19 + q^20 + 3 * q^22 + 6 * q^25 - 10 * q^26 - q^28 + 25 * q^29 - 4 * q^31 + 5 * q^32 + 11 * q^34 - 9 * q^35 + 22 * q^37 + 11 * q^38 + q^40 - 6 * q^41 - 6 * q^43 + 3 * q^44 - 4 * q^47 - 4 * q^49 + 6 * q^50 - 10 * q^52 - q^53 - 6 * q^55 - q^56 + 25 * q^58 - 6 * q^59 + 35 * q^61 - 4 * q^62 + 5 * q^64 - 2 * q^65 + 5 * q^67 + 11 * q^68 - 9 * q^70 + 11 * q^71 + 16 * q^73 + 22 * q^74 + 11 * q^76 - 5 * q^77 + 8 * q^79 + q^80 - 6 * q^82 + 26 * q^83 - 33 * q^85 - 6 * q^86 + 3 * q^88 + 35 * q^91 - 4 * q^94 - 11 * q^95 + 19 * q^97 - 4 * q^98

Introduction

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