LMFDB - Embedded newform 9522.2.a.ca.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9522,2,Mod(1,9522)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9522.1"); S:= CuspForms(chi, 2); N := Newforms(S);

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

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

Newform invariants

comment:select newform

sage:traces = [5,5,0,5,11,0,1,5,0,11,11,0,10,1,0,5,11,0,1,11,0,11,0,0,30,10, 0,1,-3] 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			1.1
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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -2.43232 q^{5} -1.85592 q^{7} +1.00000 q^{8} -2.43232 q^{10} +5.55334 q^{11} +1.14832 q^{13} -1.85592 q^{14} +1.00000 q^{16} +7.50778 q^{17} +4.47536 q^{19} -2.43232 q^{20} +5.55334 q^{22} +0.916195 q^{25} +1.14832 q^{26} -1.85592 q^{28} -2.93814 q^{29} -2.07749 q^{31} +1.00000 q^{32} +7.50778 q^{34} +4.51420 q^{35} -7.73780 q^{37} +4.47536 q^{38} -2.43232 q^{40} -0.688441 q^{41} +8.56642 q^{43} +5.55334 q^{44} -7.09992 q^{47} -3.55555 q^{49} +0.916195 q^{50} +1.14832 q^{52} +5.38852 q^{53} -13.5075 q^{55} -1.85592 q^{56} -2.93814 q^{58} +4.03445 q^{59} +0.379242 q^{61} -2.07749 q^{62} +1.00000 q^{64} -2.79309 q^{65} -2.95291 q^{67} +7.50778 q^{68} +4.51420 q^{70} +15.0703 q^{71} -11.4896 q^{73} -7.73780 q^{74} +4.47536 q^{76} -10.3066 q^{77} -10.5366 q^{79} -2.43232 q^{80} -0.688441 q^{82} +10.4484 q^{83} -18.2613 q^{85} +8.56642 q^{86} +5.55334 q^{88} +10.7758 q^{89} -2.13120 q^{91} -7.09992 q^{94} -10.8855 q^{95} +9.94247 q^{97} -3.55555 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{2} + 5 q^{4} + 11 q^{5} + q^{7} + 5 q^{8} + 11 q^{10} + 11 q^{11} + 10 q^{13} + q^{14} + 5 q^{16} + 11 q^{17} + q^{19} + 11 q^{20} + 11 q^{22} + 30 q^{25} + 10 q^{26} + q^{28} - 3 q^{29}+ \cdots - 4 q^{98}+O(q^{100}) $ 5 * q + 5 * q^2 + 5 * q^4 + 11 * q^5 + q^7 + 5 * q^8 + 11 * q^10 + 11 * q^11 + 10 * q^13 + q^14 + 5 * q^16 + 11 * q^17 + q^19 + 11 * q^20 + 11 * q^22 + 30 * q^25 + 10 * q^26 + q^28 - 3 * q^29 + 2 * q^31 + 5 * q^32 + 11 * q^34 + 11 * q^35 - 16 * q^37 + q^38 + 11 * q^40 + 8 * q^41 + 12 * q^43 + 11 * q^44 - 4 * q^49 + 30 * q^50 + 10 * q^52 + 27 * q^53 + q^56 - 3 * q^58 + 6 * q^59 - 5 * q^61 + 2 * q^62 + 5 * q^64 + 22 * q^65 + 7 * q^67 + 11 * q^68 + 11 * q^70 + 11 * q^71 - 12 * q^73 - 16 * q^74 + q^76 - 11 * q^77 - 16 * q^79 + 11 * q^80 + 8 * q^82 + 22 * q^83 - 11 * q^85 + 12 * q^86 + 11 * q^88 + 38 * q^89 - 9 * q^91 - 11 * q^95 - 5 * 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
$5$	−2.43232	−1.08777	−0.543884	−	0.839160i	$-0.683047\pi$
$5$	−2.43232	−1.08777	−0.543884	+	0.839160i	$0.683047\pi$
$6$	0	0
$7$	−1.85592	−0.701473	−0.350736	−	0.936474i	$-0.614069\pi$
$7$	−1.85592	−0.701473	−0.350736	+	0.936474i	$0.614069\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−2.43232	−0.769168
$11$	5.55334	1.67440	0.837198	−	0.546900i	$-0.184192\pi$
$11$	5.55334	1.67440	0.837198	+	0.546900i	$0.184192\pi$
$12$	0	0
$13$	1.14832	0.318487	0.159244	−	0.987239i	$-0.449094\pi$
$13$	1.14832	0.318487	0.159244	+	0.987239i	$0.449094\pi$
$14$	−1.85592	−0.496016
$15$	0	0
$16$	1.00000	0.250000
$17$	7.50778	1.82090	0.910452	−	0.413615i	$-0.135734\pi$
$17$	7.50778	1.82090	0.910452	+	0.413615i	$0.135734\pi$
$18$	0	0
$19$	4.47536	1.02672	0.513360	−	0.858174i	$-0.328401\pi$
$19$	4.47536	1.02672	0.513360	+	0.858174i	$0.328401\pi$
$20$	−2.43232	−0.543884
$21$	0	0
$22$	5.55334	1.18398
$23$	0	0
$23$	0	0
$24$	0	0
$25$	0.916195	0.183239
$26$	1.14832	0.225205
$27$	0	0
$28$	−1.85592	−0.350736
$29$	−2.93814	−0.545600	−0.272800	−	0.962071i	$-0.587950\pi$
$29$	−2.93814	−0.545600	−0.272800	+	0.962071i	$0.587950\pi$
$30$	0	0
$31$	−2.07749	−0.373128	−0.186564	−	0.982443i	$-0.559735\pi$
$31$	−2.07749	−0.373128	−0.186564	+	0.982443i	$0.559735\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	7.50778	1.28757
$35$	4.51420	0.763039
$36$	0	0
$37$	−7.73780	−1.27209	−0.636043	−	0.771653i	$-0.719430\pi$
$37$	−7.73780	−1.27209	−0.636043	+	0.771653i	$0.719430\pi$
$38$	4.47536	0.726000
$39$	0	0
$40$	−2.43232	−0.384584
$41$	−0.688441	−0.107516	−0.0537582	−	0.998554i	$-0.517120\pi$
$41$	−0.688441	−0.107516	−0.0537582	+	0.998554i	$0.517120\pi$
$42$	0	0
$43$	8.56642	1.30637	0.653183	−	0.757200i	$-0.273433\pi$
$43$	8.56642	1.30637	0.653183	+	0.757200i	$0.273433\pi$
$44$	5.55334	0.837198
$45$	0	0
$46$	0	0
$47$	−7.09992	−1.03563	−0.517815	−	0.855493i	$-0.673254\pi$
$47$	−7.09992	−1.03563	−0.517815	+	0.855493i	$0.673254\pi$
$48$	0	0
$49$	−3.55555	−0.507936
$50$	0.916195	0.129570
$51$	0	0
$52$	1.14832	0.159244
$53$	5.38852	0.740171	0.370085	−	0.928998i	$-0.379328\pi$
$53$	5.38852	0.740171	0.370085	+	0.928998i	$0.379328\pi$
$54$	0	0
$55$	−13.5075	−1.82135
$56$	−1.85592	−0.248008
$57$	0	0
$58$	−2.93814	−0.385797
$59$	4.03445	0.525240	0.262620	−	0.964899i	$-0.415413\pi$
$59$	4.03445	0.525240	0.262620	+	0.964899i	$0.415413\pi$
$60$	0	0
$61$	0.379242	0.0485570	0.0242785	−	0.999705i	$-0.492271\pi$
$61$	0.379242	0.0485570	0.0242785	+	0.999705i	$0.492271\pi$
$62$	−2.07749	−0.263841
$63$	0	0
$64$	1.00000	0.125000
$65$	−2.79309	−0.346440
$66$	0	0
$67$	−2.95291	−0.360755	−0.180378	−	0.983597i	$-0.557732\pi$
$67$	−2.95291	−0.360755	−0.180378	+	0.983597i	$0.557732\pi$
$68$	7.50778	0.910452
$69$	0	0
$70$	4.51420	0.539550
$71$	15.0703	1.78851	0.894257	−	0.447553i	$-0.147704\pi$
$71$	15.0703	1.78851	0.894257	+	0.447553i	$0.147704\pi$
$72$	0	0
$73$	−11.4896	−1.34476	−0.672381	−	0.740205i	$-0.734728\pi$
$73$	−11.4896	−1.34476	−0.672381	+	0.740205i	$0.734728\pi$
$74$	−7.73780	−0.899501
$75$	0	0
$76$	4.47536	0.513360
$77$	−10.3066	−1.17454
$78$	0	0
$79$	−10.5366	−1.18546	−0.592730	−	0.805402i	$-0.701950\pi$
$79$	−10.5366	−1.18546	−0.592730	+	0.805402i	$0.701950\pi$
$80$	−2.43232	−0.271942
$81$	0	0
$82$	−0.688441	−0.0760256
$83$	10.4484	1.14687	0.573433	−	0.819253i	$-0.305611\pi$
$83$	10.4484	1.14687	0.573433	+	0.819253i	$0.305611\pi$
$84$	0	0
$85$	−18.2613	−1.98072
$86$	8.56642	0.923741
$87$	0	0
$88$	5.55334	0.591988
$89$	10.7758	1.14223	0.571115	−	0.820870i	$-0.306511\pi$
$89$	10.7758	1.14223	0.571115	+	0.820870i	$0.306511\pi$
$90$	0	0
$91$	−2.13120	−0.223410
$92$	0	0
$93$	0	0
$94$	−7.09992	−0.732301
$95$	−10.8855	−1.11683
$96$	0	0
$97$	9.94247	1.00951	0.504753	−	0.863264i	$-0.331584\pi$
$97$	9.94247	1.00951	0.504753	+	0.863264i	$0.331584\pi$
$98$	−3.55555	−0.359165
$99$	0	0
$100$	0.916195	0.0916195
$101$	4.91204	0.488766	0.244383	−	0.969679i	$-0.421415\pi$
$101$	4.91204	0.488766	0.244383	+	0.969679i	$0.421415\pi$
$102$	0	0
$103$	7.84290	0.772784	0.386392	−	0.922335i	$-0.373721\pi$
$103$	7.84290	0.772784	0.386392	+	0.922335i	$0.373721\pi$
$104$	1.14832	0.112602
$105$	0	0
$106$	5.38852	0.523380
$107$	−9.30388	−0.899440	−0.449720	−	0.893170i	$-0.648476\pi$
$107$	−9.30388	−0.899440	−0.449720	+	0.893170i	$0.648476\pi$
$108$	0	0
$109$	−14.4885	−1.38775	−0.693874	−	0.720097i	$-0.744098\pi$
$109$	−14.4885	−1.38775	−0.693874	+	0.720097i	$0.744098\pi$
$110$	−13.5075	−1.28789
$111$	0	0
$112$	−1.85592	−0.175368
$113$	14.7045	1.38328	0.691639	−	0.722243i	$-0.256889\pi$
$113$	14.7045	1.38328	0.691639	+	0.722243i	$0.256889\pi$
$114$	0	0
$115$	0	0
$116$	−2.93814	−0.272800
$117$	0	0
$118$	4.03445	0.371401
$119$	−13.9338	−1.27731
$120$	0	0
$121$	19.8396	1.80360
$122$	0.379242	0.0343350
$123$	0	0
$124$	−2.07749	−0.186564
$125$	9.93313	0.888446
$126$	0	0
$127$	6.93650	0.615515	0.307757	−	0.951465i	$-0.400421\pi$
$127$	6.93650	0.615515	0.307757	+	0.951465i	$0.400421\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−2.79309	−0.244970
$131$	−3.39284	−0.296434	−0.148217	−	0.988955i	$-0.547353\pi$
$131$	−3.39284	−0.296434	−0.148217	+	0.988955i	$0.547353\pi$
$132$	0	0
$133$	−8.30593	−0.720215
$134$	−2.95291	−0.255092
$135$	0	0
$136$	7.50778	0.643787
$137$	9.21973	0.787695	0.393847	−	0.919176i	$-0.371144\pi$
$137$	9.21973	0.787695	0.393847	+	0.919176i	$0.371144\pi$
$138$	0	0
$139$	−13.4691	−1.14244	−0.571218	−	0.820798i	$-0.693529\pi$
$139$	−13.4691	−1.14244	−0.571218	+	0.820798i	$0.693529\pi$
$140$	4.51420	0.381520
$141$	0	0
$142$	15.0703	1.26467
$143$	6.37703	0.533274
$144$	0	0
$145$	7.14652	0.593486
$146$	−11.4896	−0.950890
$147$	0	0
$148$	−7.73780	−0.636043
$149$	−11.1562	−0.913948	−0.456974	−	0.889480i	$-0.651067\pi$
$149$	−11.1562	−0.913948	−0.456974	+	0.889480i	$0.651067\pi$
$150$	0	0
$151$	15.3006	1.24514	0.622572	−	0.782563i	$-0.286088\pi$
$151$	15.3006	1.24514	0.622572	+	0.782563i	$0.286088\pi$
$152$	4.47536	0.363000
$153$	0	0
$154$	−10.3066	−0.830527
$155$	5.05312	0.405876
$156$	0	0
$157$	−2.83359	−0.226145	−0.113072	−	0.993587i	$-0.536069\pi$
$157$	−2.83359	−0.226145	−0.113072	+	0.993587i	$0.536069\pi$
$158$	−10.5366	−0.838246
$159$	0	0
$160$	−2.43232	−0.192292
$161$	0	0
$162$	0	0
$163$	24.9646	1.95538	0.977690	−	0.210053i	$-0.0673638\pi$
$163$	24.9646	1.95538	0.977690	+	0.210053i	$0.0673638\pi$
$164$	−0.688441	−0.0537582
$165$	0	0
$166$	10.4484	0.810957
$167$	−8.67950	−0.671640	−0.335820	−	0.941926i	$-0.609013\pi$
$167$	−8.67950	−0.671640	−0.335820	+	0.941926i	$0.609013\pi$
$168$	0	0
$169$	−11.6814	−0.898566
$170$	−18.2613	−1.40058
$171$	0	0
$172$	8.56642	0.653183
$173$	−15.4272	−1.17291	−0.586454	−	0.809983i	$-0.699476\pi$
$173$	−15.4272	−1.17291	−0.586454	+	0.809983i	$0.699476\pi$
$174$	0	0
$175$	−1.70039	−0.128537
$176$	5.55334	0.418599
$177$	0	0
$178$	10.7758	0.807678
$179$	6.04991	0.452192	0.226096	−	0.974105i	$-0.427404\pi$
$179$	6.04991	0.452192	0.226096	+	0.974105i	$0.427404\pi$
$180$	0	0
$181$	6.26785	0.465885	0.232943	−	0.972490i	$-0.425165\pi$
$181$	6.26785	0.465885	0.232943	+	0.972490i	$0.425165\pi$
$182$	−2.13120	−0.157975
$183$	0	0
$184$	0	0
$185$	18.8208	1.38373
$186$	0	0
$187$	41.6933	3.04891
$188$	−7.09992	−0.517815
$189$	0	0
$190$	−10.8855	−0.789720
$191$	19.2911	1.39585	0.697926	−	0.716170i	$-0.254106\pi$
$191$	19.2911	1.39585	0.697926	+	0.716170i	$0.254106\pi$
$192$	0	0
$193$	−25.8804	−1.86291	−0.931457	−	0.363850i	$-0.881462\pi$
$193$	−25.8804	−1.86291	−0.931457	+	0.363850i	$0.881462\pi$
$194$	9.94247	0.713828
$195$	0	0
$196$	−3.55555	−0.253968
$197$	−18.3833	−1.30975	−0.654877	−	0.755735i	$-0.727280\pi$
$197$	−18.3833	−1.30975	−0.654877	+	0.755735i	$0.727280\pi$
$198$	0	0
$199$	−8.65762	−0.613722	−0.306861	−	0.951754i	$-0.599279\pi$
$199$	−8.65762	−0.613722	−0.306861	+	0.951754i	$0.599279\pi$
$200$	0.916195	0.0647848
$201$	0	0
$202$	4.91204	0.345610
$203$	5.45297	0.382723
$204$	0	0
$205$	1.67451	0.116953
$206$	7.84290	0.546441
$207$	0	0
$208$	1.14832	0.0796219
$209$	24.8532	1.71913
$210$	0	0
$211$	13.8857	0.955931	0.477966	−	0.878379i	$-0.341374\pi$
$211$	13.8857	0.955931	0.477966	+	0.878379i	$0.341374\pi$
$212$	5.38852	0.370085
$213$	0	0
$214$	−9.30388	−0.636000
$215$	−20.8363	−1.42102
$216$	0	0
$217$	3.85565	0.261739
$218$	−14.4885	−0.981286
$219$	0	0
$220$	−13.5075	−0.910677
$221$	8.62135	0.579935
$222$	0	0
$223$	3.78498	0.253461	0.126730	−	0.991937i	$-0.459552\pi$
$223$	3.78498	0.253461	0.126730	+	0.991937i	$0.459552\pi$
$224$	−1.85592	−0.124004
$225$	0	0
$226$	14.7045	0.978126
$227$	9.83222	0.652588	0.326294	−	0.945268i	$-0.394200\pi$
$227$	9.83222	0.652588	0.326294	+	0.945268i	$0.394200\pi$
$228$	0	0
$229$	19.7726	1.30661	0.653307	−	0.757093i	$-0.273381\pi$
$229$	19.7726	1.30661	0.653307	+	0.757093i	$0.273381\pi$
$230$	0	0
$231$	0	0
$232$	−2.93814	−0.192899
$233$	16.7160	1.09510	0.547550	−	0.836773i	$-0.315561\pi$
$233$	16.7160	1.09510	0.547550	+	0.836773i	$0.315561\pi$
$234$	0	0
$235$	17.2693	1.12652
$236$	4.03445	0.262620
$237$	0	0
$238$	−13.9338	−0.903197
$239$	8.55341	0.553274	0.276637	−	0.960974i	$-0.410780\pi$
$239$	8.55341	0.553274	0.276637	+	0.960974i	$0.410780\pi$
$240$	0	0
$241$	−21.1421	−1.36188	−0.680942	−	0.732337i	$-0.738430\pi$
$241$	−21.1421	−1.36188	−0.680942	+	0.732337i	$0.738430\pi$
$242$	19.8396	1.27534
$243$	0	0
$244$	0.379242	0.0242785
$245$	8.64826	0.552517
$246$	0	0
$247$	5.13916	0.326997
$248$	−2.07749	−0.131921
$249$	0	0
$250$	9.93313	0.628226
$251$	−11.5974	−0.732020	−0.366010	−	0.930611i	$-0.619276\pi$
$251$	−11.5974	−0.732020	−0.366010	+	0.930611i	$0.619276\pi$
$252$	0	0
$253$	0	0
$254$	6.93650	0.435235
$255$	0	0
$256$	1.00000	0.0625000
$257$	8.44761	0.526947	0.263474	−	0.964667i	$-0.415132\pi$
$257$	8.44761	0.526947	0.263474	+	0.964667i	$0.415132\pi$
$258$	0	0
$259$	14.3608	0.892334
$260$	−2.79309	−0.173220
$261$	0	0
$262$	−3.39284	−0.209610
$263$	−31.2911	−1.92949	−0.964745	−	0.263185i	$-0.915227\pi$
$263$	−31.2911	−1.92949	−0.964745	+	0.263185i	$0.915227\pi$
$264$	0	0
$265$	−13.1066	−0.805134
$266$	−8.30593	−0.509269
$267$	0	0
$268$	−2.95291	−0.180378
$269$	9.25889	0.564525	0.282262	−	0.959337i	$-0.408915\pi$
$269$	9.25889	0.564525	0.282262	+	0.959337i	$0.408915\pi$
$270$	0	0
$271$	16.4713	1.00056	0.500280	−	0.865864i	$-0.333230\pi$
$271$	16.4713	1.00056	0.500280	+	0.865864i	$0.333230\pi$
$272$	7.50778	0.455226
$273$	0	0
$274$	9.21973	0.556984
$275$	5.08795	0.306815
$276$	0	0
$277$	29.2151	1.75537	0.877684	−	0.479240i	$-0.159088\pi$
$277$	29.2151	1.75537	0.877684	+	0.479240i	$0.159088\pi$
$278$	−13.4691	−0.807824
$279$	0	0
$280$	4.51420	0.269775
$281$	12.0319	0.717763	0.358881	−	0.933383i	$-0.383158\pi$
$281$	12.0319	0.717763	0.358881	+	0.933383i	$0.383158\pi$
$282$	0	0
$283$	10.2784	0.610986	0.305493	−	0.952194i	$-0.401179\pi$
$283$	10.2784	0.610986	0.305493	+	0.952194i	$0.401179\pi$
$284$	15.0703	0.894257
$285$	0	0
$286$	6.37703	0.377082
$287$	1.27769	0.0754198
$288$	0	0
$289$	39.3667	2.31569
$290$	7.14652	0.419658
$291$	0	0
$292$	−11.4896	−0.672381
$293$	−0.252336	−0.0147416	−0.00737080	−	0.999973i	$-0.502346\pi$
$293$	−0.252336	−0.0147416	−0.00737080	+	0.999973i	$0.502346\pi$
$294$	0	0
$295$	−9.81308	−0.571339
$296$	−7.73780	−0.449750
$297$	0	0
$298$	−11.1562	−0.646259
$299$	0	0
$300$	0	0
$301$	−15.8986	−0.916380
$302$	15.3006	0.880450
$303$	0	0
$304$	4.47536	0.256680
$305$	−0.922440	−0.0528188
$306$	0	0
$307$	−0.292252	−0.0166797	−0.00833986	−	0.999965i	$-0.502655\pi$
$307$	−0.292252	−0.0166797	−0.00833986	+	0.999965i	$0.502655\pi$
$308$	−10.3066	−0.587271
$309$	0	0
$310$	5.05312	0.286998
$311$	9.05821	0.513644	0.256822	−	0.966459i	$-0.417325\pi$
$311$	9.05821	0.513644	0.256822	+	0.966459i	$0.417325\pi$
$312$	0	0
$313$	−5.71933	−0.323275	−0.161638	−	0.986850i	$-0.551678\pi$
$313$	−5.71933	−0.323275	−0.161638	+	0.986850i	$0.551678\pi$
$314$	−2.83359	−0.159909
$315$	0	0
$316$	−10.5366	−0.592730
$317$	18.3109	1.02844	0.514221	−	0.857658i	$-0.328081\pi$
$317$	18.3109	1.02844	0.514221	+	0.857658i	$0.328081\pi$
$318$	0	0
$319$	−16.3165	−0.913550
$320$	−2.43232	−0.135971
$321$	0	0
$322$	0	0
$323$	33.6000	1.86956
$324$	0	0
$325$	1.05209	0.0583594
$326$	24.9646	1.38266
$327$	0	0
$328$	−0.688441	−0.0380128
$329$	13.1769	0.726465
$330$	0	0
$331$	−1.15479	−0.0634728	−0.0317364	−	0.999496i	$-0.510104\pi$
$331$	−1.15479	−0.0634728	−0.0317364	+	0.999496i	$0.510104\pi$
$332$	10.4484	0.573433
$333$	0	0
$334$	−8.67950	−0.474921
$335$	7.18243	0.392418
$336$	0	0
$337$	16.6139	0.905015	0.452507	−	0.891761i	$-0.350530\pi$
$337$	16.6139	0.905015	0.452507	+	0.891761i	$0.350530\pi$
$338$	−11.6814	−0.635382
$339$	0	0
$340$	−18.2613	−0.990360
$341$	−11.5370	−0.624763
$342$	0	0
$343$	19.5903	1.05778
$344$	8.56642	0.461870
$345$	0	0
$346$	−15.4272	−0.829371
$347$	0.130856	0.00702474	0.00351237	−	0.999994i	$-0.498882\pi$
$347$	0.130856	0.00702474	0.00351237	+	0.999994i	$0.498882\pi$
$348$	0	0
$349$	18.6878	1.00034	0.500168	−	0.865929i	$-0.333272\pi$
$349$	18.6878	1.00034	0.500168	+	0.865929i	$0.333272\pi$
$350$	−1.70039	−0.0908895
$351$	0	0
$352$	5.55334	0.295994
$353$	32.8820	1.75013	0.875066	−	0.484004i	$-0.160818\pi$
$353$	32.8820	1.75013	0.875066	+	0.484004i	$0.160818\pi$
$354$	0	0
$355$	−36.6558	−1.94549
$356$	10.7758	0.571115
$357$	0	0
$358$	6.04991	0.319748
$359$	6.56339	0.346402	0.173201	−	0.984886i	$-0.444589\pi$
$359$	6.56339	0.346402	0.173201	+	0.984886i	$0.444589\pi$
$360$	0	0
$361$	1.02889	0.0541521
$362$	6.26785	0.329431
$363$	0	0
$364$	−2.13120	−0.111705
$365$	27.9465	1.46279
$366$	0	0
$367$	15.7482	0.822052	0.411026	−	0.911624i	$-0.365171\pi$
$367$	15.7482	0.822052	0.411026	+	0.911624i	$0.365171\pi$
$368$	0	0
$369$	0	0
$370$	18.8208	0.978448
$371$	−10.0007	−0.519209
$372$	0	0
$373$	−16.1393	−0.835663	−0.417831	−	0.908525i	$-0.637210\pi$
$373$	−16.1393	−0.835663	−0.417831	+	0.908525i	$0.637210\pi$
$374$	41.6933	2.15591
$375$	0	0
$376$	−7.09992	−0.366150
$377$	−3.37394	−0.173767
$378$	0	0
$379$	−21.2866	−1.09342	−0.546709	−	0.837323i	$-0.684120\pi$
$379$	−21.2866	−1.09342	−0.546709	+	0.837323i	$0.684120\pi$
$380$	−10.8855	−0.558416
$381$	0	0
$382$	19.2911	0.987017
$383$	26.8982	1.37444	0.687218	−	0.726452i	$-0.258832\pi$
$383$	26.8982	1.37444	0.687218	+	0.726452i	$0.258832\pi$
$384$	0	0
$385$	25.0689	1.27763
$386$	−25.8804	−1.31728
$387$	0	0
$388$	9.94247	0.504753
$389$	29.1942	1.48021	0.740103	−	0.672494i	$-0.234777\pi$
$389$	29.1942	1.48021	0.740103	+	0.672494i	$0.234777\pi$
$390$	0	0
$391$	0	0
$392$	−3.55555	−0.179583
$393$	0	0
$394$	−18.3833	−0.926137
$395$	25.6284	1.28950
$396$	0	0
$397$	−2.16397	−0.108607	−0.0543033	−	0.998524i	$-0.517294\pi$
$397$	−2.16397	−0.108607	−0.0543033	+	0.998524i	$0.517294\pi$
$398$	−8.65762	−0.433967
$399$	0	0
$400$	0.916195	0.0458098
$401$	−6.81856	−0.340503	−0.170251	−	0.985401i	$-0.554458\pi$
$401$	−6.81856	−0.340503	−0.170251	+	0.985401i	$0.554458\pi$
$402$	0	0
$403$	−2.38563	−0.118837
$404$	4.91204	0.244383
$405$	0	0
$406$	5.45297	0.270626
$407$	−42.9706	−2.12998
$408$	0	0
$409$	−12.8837	−0.637057	−0.318529	−	0.947913i	$-0.603189\pi$
$409$	−12.8837	−0.637057	−0.318529	+	0.947913i	$0.603189\pi$
$410$	1.67451	0.0826982
$411$	0	0
$412$	7.84290	0.386392
$413$	−7.48762	−0.368441
$414$	0	0
$415$	−25.4140	−1.24752
$416$	1.14832	0.0563012
$417$	0	0
$418$	24.8532	1.21561
$419$	7.45842	0.364368	0.182184	−	0.983264i	$-0.441683\pi$
$419$	7.45842	0.364368	0.182184	+	0.983264i	$0.441683\pi$
$420$	0	0
$421$	−0.856168	−0.0417271	−0.0208635	−	0.999782i	$-0.506642\pi$
$421$	−0.856168	−0.0417271	−0.0208635	+	0.999782i	$0.506642\pi$
$422$	13.8857	0.675946
$423$	0	0
$424$	5.38852	0.261690
$425$	6.87859	0.333661
$426$	0	0
$427$	−0.703844	−0.0340614
$428$	−9.30388	−0.449720
$429$	0	0
$430$	−20.8363	−1.00482
$431$	−27.0358	−1.30227	−0.651134	−	0.758963i	$-0.725706\pi$
$431$	−27.0358	−1.30227	−0.651134	+	0.758963i	$0.725706\pi$
$432$	0	0
$433$	21.7849	1.04691	0.523457	−	0.852052i	$-0.324642\pi$
$433$	21.7849	1.04691	0.523457	+	0.852052i	$0.324642\pi$
$434$	3.85565	0.185077
$435$	0	0
$436$	−14.4885	−0.693874
$437$	0	0
$438$	0	0
$439$	15.1364	0.722420	0.361210	−	0.932485i	$-0.382364\pi$
$439$	15.1364	0.722420	0.361210	+	0.932485i	$0.382364\pi$
$440$	−13.5075	−0.643946
$441$	0	0
$442$	8.62135	0.410076
$443$	15.2638	0.725205	0.362603	−	0.931944i	$-0.381888\pi$
$443$	15.2638	0.725205	0.362603	+	0.931944i	$0.381888\pi$
$444$	0	0
$445$	−26.2102	−1.24248
$446$	3.78498	0.179224
$447$	0	0
$448$	−1.85592	−0.0876841
$449$	−28.3538	−1.33810	−0.669050	−	0.743217i	$-0.733299\pi$
$449$	−28.3538	−1.33810	−0.669050	+	0.743217i	$0.733299\pi$
$450$	0	0
$451$	−3.82315	−0.180025
$452$	14.7045	0.691639
$453$	0	0
$454$	9.83222	0.461449
$455$	5.18376	0.243018
$456$	0	0
$457$	19.7830	0.925411	0.462706	−	0.886512i	$-0.346879\pi$
$457$	19.7830	0.925411	0.462706	+	0.886512i	$0.346879\pi$
$458$	19.7726	0.923915
$459$	0	0
$460$	0	0
$461$	−10.9505	−0.510014	−0.255007	−	0.966939i	$-0.582078\pi$
$461$	−10.9505	−0.510014	−0.255007	+	0.966939i	$0.582078\pi$
$462$	0	0
$463$	−9.61368	−0.446786	−0.223393	−	0.974729i	$-0.571713\pi$
$463$	−9.61368	−0.446786	−0.223393	+	0.974729i	$0.571713\pi$
$464$	−2.93814	−0.136400
$465$	0	0
$466$	16.7160	0.774352
$467$	−15.8605	−0.733938	−0.366969	−	0.930233i	$-0.619605\pi$
$467$	−15.8605	−0.733938	−0.366969	+	0.930233i	$0.619605\pi$
$468$	0	0
$469$	5.48037	0.253060
$470$	17.2693	0.796573
$471$	0	0
$472$	4.03445	0.185700
$473$	47.5722	2.18737
$474$	0	0
$475$	4.10031	0.188135
$476$	−13.9338	−0.638657
$477$	0	0
$478$	8.55341	0.391224
$479$	28.2506	1.29080	0.645402	−	0.763843i	$-0.276690\pi$
$479$	28.2506	1.29080	0.645402	+	0.763843i	$0.276690\pi$
$480$	0	0
$481$	−8.88549	−0.405144
$482$	−21.1421	−0.962998
$483$	0	0
$484$	19.8396	0.901800
$485$	−24.1833	−1.09811
$486$	0	0
$487$	−2.02212	−0.0916309	−0.0458155	−	0.998950i	$-0.514589\pi$
$487$	−2.02212	−0.0916309	−0.0458155	+	0.998950i	$0.514589\pi$
$488$	0.379242	0.0171675
$489$	0	0
$490$	8.64826	0.390688
$491$	−18.4236	−0.831444	−0.415722	−	0.909492i	$-0.636471\pi$
$491$	−18.4236	−0.831444	−0.415722	+	0.909492i	$0.636471\pi$
$492$	0	0
$493$	−22.0589	−0.993484
$494$	5.13916	0.231222
$495$	0	0
$496$	−2.07749	−0.0932819
$497$	−27.9693	−1.25459
$498$	0	0
$499$	−12.6295	−0.565372	−0.282686	−	0.959213i	$-0.591225\pi$
$499$	−12.6295	−0.565372	−0.282686	+	0.959213i	$0.591225\pi$
$500$	9.93313	0.444223
$501$	0	0
$502$	−11.5974	−0.517616
$503$	−0.388069	−0.0173031	−0.00865156	−	0.999963i	$-0.502754\pi$
$503$	−0.388069	−0.0173031	−0.00865156	+	0.999963i	$0.502754\pi$
$504$	0	0
$505$	−11.9477	−0.531664
$506$	0	0
$507$	0	0
$508$	6.93650	0.307757
$509$	−33.5838	−1.48857	−0.744287	−	0.667860i	$-0.767211\pi$
$509$	−33.5838	−1.48857	−0.744287	+	0.667860i	$0.767211\pi$
$510$	0	0
$511$	21.3239	0.943313
$512$	1.00000	0.0441942
$513$	0	0
$514$	8.44761	0.372608
$515$	−19.0765	−0.840610
$516$	0	0
$517$	−39.4283	−1.73405
$518$	14.3608	0.630975
$519$	0	0
$520$	−2.79309	−0.122485
$521$	0.873591	0.0382727	0.0191364	−	0.999817i	$-0.493908\pi$
$521$	0.873591	0.0382727	0.0191364	+	0.999817i	$0.493908\pi$
$522$	0	0
$523$	20.3134	0.888245	0.444123	−	0.895966i	$-0.353515\pi$
$523$	20.3134	0.888245	0.444123	+	0.895966i	$0.353515\pi$
$524$	−3.39284	−0.148217
$525$	0	0
$526$	−31.2911	−1.36436
$527$	−15.5973	−0.679430
$528$	0	0
$529$	0	0
$530$	−13.1066	−0.569316
$531$	0	0
$532$	−8.30593	−0.360108
$533$	−0.790552	−0.0342426
$534$	0	0
$535$	22.6300	0.978382
$536$	−2.95291	−0.127546
$537$	0	0
$538$	9.25889	0.399179
$539$	−19.7452	−0.850486
$540$	0	0
$541$	−6.66716	−0.286644	−0.143322	−	0.989676i	$-0.545778\pi$
$541$	−6.66716	−0.286644	−0.143322	+	0.989676i	$0.545778\pi$
$542$	16.4713	0.707503
$543$	0	0
$544$	7.50778	0.321893
$545$	35.2407	1.50955
$546$	0	0
$547$	−13.5792	−0.580606	−0.290303	−	0.956935i	$-0.593756\pi$
$547$	−13.5792	−0.580606	−0.290303	+	0.956935i	$0.593756\pi$
$548$	9.21973	0.393847
$549$	0	0
$550$	5.08795	0.216951
$551$	−13.1493	−0.560178
$552$	0	0
$553$	19.5551	0.831567
$554$	29.2151	1.24123
$555$	0	0
$556$	−13.4691	−0.571218
$557$	16.1593	0.684690	0.342345	−	0.939574i	$-0.388779\pi$
$557$	16.1593	0.684690	0.342345	+	0.939574i	$0.388779\pi$
$558$	0	0
$559$	9.83701	0.416061
$560$	4.51420	0.190760
$561$	0	0
$562$	12.0319	0.507535
$563$	−3.28795	−0.138570	−0.0692852	−	0.997597i	$-0.522072\pi$
$563$	−3.28795	−0.138570	−0.0692852	+	0.997597i	$0.522072\pi$
$564$	0	0
$565$	−35.7660	−1.50469
$566$	10.2784	0.432032
$567$	0	0
$568$	15.0703	0.632336
$569$	−26.1825	−1.09763	−0.548814	−	0.835944i	$-0.684921\pi$
$569$	−26.1825	−1.09763	−0.548814	+	0.835944i	$0.684921\pi$
$570$	0	0
$571$	−13.1072	−0.548518	−0.274259	−	0.961656i	$-0.588433\pi$
$571$	−13.1072	−0.548518	−0.274259	+	0.961656i	$0.588433\pi$
$572$	6.37703	0.266637
$573$	0	0
$574$	1.27769	0.0533298
$575$	0	0
$576$	0	0
$577$	37.9892	1.58151	0.790756	−	0.612132i	$-0.209688\pi$
$577$	37.9892	1.58151	0.790756	+	0.612132i	$0.209688\pi$
$578$	39.3667	1.63744
$579$	0	0
$580$	7.14652	0.296743
$581$	−19.3915	−0.804495
$582$	0	0
$583$	29.9243	1.23934
$584$	−11.4896	−0.475445
$585$	0	0
$586$	−0.252336	−0.0104239
$587$	−26.2669	−1.08415	−0.542075	−	0.840330i	$-0.682361\pi$
$587$	−26.2669	−1.08415	−0.542075	+	0.840330i	$0.682361\pi$
$588$	0	0
$589$	−9.29751	−0.383097
$590$	−9.81308	−0.403998
$591$	0	0
$592$	−7.73780	−0.318022
$593$	1.67757	0.0688895	0.0344448	−	0.999407i	$-0.489034\pi$
$593$	1.67757	0.0688895	0.0344448	+	0.999407i	$0.489034\pi$
$594$	0	0
$595$	33.8916	1.38942
$596$	−11.1562	−0.456974
$597$	0	0
$598$	0	0
$599$	−10.8835	−0.444689	−0.222344	−	0.974968i	$-0.571371\pi$
$599$	−10.8835	−0.444689	−0.222344	+	0.974968i	$0.571371\pi$
$600$	0	0
$601$	−27.5866	−1.12528	−0.562640	−	0.826702i	$-0.690214\pi$
$601$	−27.5866	−1.12528	−0.562640	+	0.826702i	$0.690214\pi$
$602$	−15.8986	−0.647979
$603$	0	0
$604$	15.3006	0.622572
$605$	−48.2563	−1.96190
$606$	0	0
$607$	−37.0048	−1.50198	−0.750989	−	0.660315i	$-0.770423\pi$
$607$	−37.0048	−1.50198	−0.750989	+	0.660315i	$0.770423\pi$
$608$	4.47536	0.181500
$609$	0	0
$610$	−0.922440	−0.0373485
$611$	−8.15300	−0.329835
$612$	0	0
$613$	−10.2632	−0.414528	−0.207264	−	0.978285i	$-0.566456\pi$
$613$	−10.2632	−0.414528	−0.207264	+	0.978285i	$0.566456\pi$
$614$	−0.292252	−0.0117943
$615$	0	0
$616$	−10.3066	−0.415263
$617$	−7.03247	−0.283117	−0.141558	−	0.989930i	$-0.545211\pi$
$617$	−7.03247	−0.283117	−0.141558	+	0.989930i	$0.545211\pi$
$618$	0	0
$619$	36.6789	1.47425	0.737124	−	0.675757i	$-0.236183\pi$
$619$	36.6789	1.47425	0.737124	+	0.675757i	$0.236183\pi$
$620$	5.05312	0.202938
$621$	0	0
$622$	9.05821	0.363201
$623$	−19.9990	−0.801243
$624$	0	0
$625$	−28.7416	−1.14966
$626$	−5.71933	−0.228590
$627$	0	0
$628$	−2.83359	−0.113072
$629$	−58.0937	−2.31635
$630$	0	0
$631$	18.2039	0.724688	0.362344	−	0.932044i	$-0.381977\pi$
$631$	18.2039	0.724688	0.362344	+	0.932044i	$0.381977\pi$
$632$	−10.5366	−0.419123
$633$	0	0
$634$	18.3109	0.727219
$635$	−16.8718	−0.669537
$636$	0	0
$637$	−4.08292	−0.161771
$638$	−16.3165	−0.645977
$639$	0	0
$640$	−2.43232	−0.0961460
$641$	5.09612	0.201284	0.100642	−	0.994923i	$-0.467910\pi$
$641$	5.09612	0.201284	0.100642	+	0.994923i	$0.467910\pi$
$642$	0	0
$643$	5.24644	0.206899	0.103450	−	0.994635i	$-0.467012\pi$
$643$	5.24644	0.206899	0.103450	+	0.994635i	$0.467012\pi$
$644$	0	0
$645$	0	0
$646$	33.6000	1.32198
$647$	40.4161	1.58892	0.794461	−	0.607316i	$-0.207754\pi$
$647$	40.4161	1.58892	0.794461	+	0.607316i	$0.207754\pi$
$648$	0	0
$649$	22.4047	0.879460
$650$	1.05209	0.0412663
$651$	0	0
$652$	24.9646	0.977690
$653$	−14.8666	−0.581776	−0.290888	−	0.956757i	$-0.593951\pi$
$653$	−14.8666	−0.581776	−0.290888	+	0.956757i	$0.593951\pi$
$654$	0	0
$655$	8.25249	0.322451
$656$	−0.688441	−0.0268791
$657$	0	0
$658$	13.1769	0.513689
$659$	−25.9501	−1.01087	−0.505437	−	0.862864i	$-0.668669\pi$
$659$	−25.9501	−1.01087	−0.505437	+	0.862864i	$0.668669\pi$
$660$	0	0
$661$	48.1966	1.87463	0.937315	−	0.348482i	$-0.113303\pi$
$661$	48.1966	1.87463	0.937315	+	0.348482i	$0.113303\pi$
$662$	−1.15479	−0.0448820
$663$	0	0
$664$	10.4484	0.405478
$665$	20.2027	0.783427
$666$	0	0
$667$	0	0
$668$	−8.67950	−0.335820
$669$	0	0
$670$	7.18243	0.277481
$671$	2.10606	0.0813036
$672$	0	0
$673$	−15.5480	−0.599333	−0.299666	−	0.954044i	$-0.596875\pi$
$673$	−15.5480	−0.599333	−0.299666	+	0.954044i	$0.596875\pi$
$674$	16.6139	0.639942
$675$	0	0
$676$	−11.6814	−0.449283
$677$	−2.09267	−0.0804277	−0.0402138	−	0.999191i	$-0.512804\pi$
$677$	−2.09267	−0.0804277	−0.0402138	+	0.999191i	$0.512804\pi$
$678$	0	0
$679$	−18.4525	−0.708140
$680$	−18.2613	−0.700290
$681$	0	0
$682$	−11.5370	−0.441774
$683$	13.6536	0.522440	0.261220	−	0.965279i	$-0.415875\pi$
$683$	13.6536	0.522440	0.261220	+	0.965279i	$0.415875\pi$
$684$	0	0
$685$	−22.4254	−0.856829
$686$	19.5903	0.747960
$687$	0	0
$688$	8.56642	0.326592
$689$	6.18777	0.235735
$690$	0	0
$691$	−19.2615	−0.732744	−0.366372	−	0.930469i	$-0.619400\pi$
$691$	−19.2615	−0.732744	−0.366372	+	0.930469i	$0.619400\pi$
$692$	−15.4272	−0.586454
$693$	0	0
$694$	0.130856	0.00496724
$695$	32.7613	1.24271
$696$	0	0
$697$	−5.16866	−0.195777
$698$	18.6878	0.707344
$699$	0	0
$700$	−1.70039	−0.0642686
$701$	−5.92552	−0.223804	−0.111902	−	0.993719i	$-0.535694\pi$
$701$	−5.92552	−0.223804	−0.111902	+	0.993719i	$0.535694\pi$
$702$	0	0
$703$	−34.6295	−1.30608
$704$	5.55334	0.209299
$705$	0	0
$706$	32.8820	1.23753
$707$	−9.11636	−0.342856
$708$	0	0
$709$	6.46376	0.242751	0.121376	−	0.992607i	$-0.461269\pi$
$709$	6.46376	0.242751	0.121376	+	0.992607i	$0.461269\pi$
$710$	−36.6558	−1.37567
$711$	0	0
$712$	10.7758	0.403839
$713$	0	0
$714$	0	0
$715$	−15.5110	−0.580078
$716$	6.04991	0.226096
$717$	0	0
$718$	6.56339	0.244944
$719$	49.9658	1.86341	0.931704	−	0.363218i	$-0.118322\pi$
$719$	49.9658	1.86341	0.931704	+	0.363218i	$0.118322\pi$
$720$	0	0
$721$	−14.5558	−0.542087
$722$	1.02889	0.0382913
$723$	0	0
$724$	6.26785	0.232943
$725$	−2.69191	−0.0999752
$726$	0	0
$727$	−22.2104	−0.823738	−0.411869	−	0.911243i	$-0.635124\pi$
$727$	−22.2104	−0.823738	−0.411869	+	0.911243i	$0.635124\pi$
$728$	−2.13120	−0.0789874
$729$	0	0
$730$	27.9465	1.03435
$731$	64.3148	2.37877
$732$	0	0
$733$	−38.7367	−1.43077	−0.715386	−	0.698730i	$-0.753749\pi$
$733$	−38.7367	−1.43077	−0.715386	+	0.698730i	$0.753749\pi$
$734$	15.7482	0.581278
$735$	0	0
$736$	0	0
$737$	−16.3985	−0.604047
$738$	0	0
$739$	−8.17813	−0.300837	−0.150419	−	0.988622i	$-0.548062\pi$
$739$	−8.17813	−0.300837	−0.150419	+	0.988622i	$0.548062\pi$
$740$	18.8208	0.691867
$741$	0	0
$742$	−10.0007	−0.367136
$743$	6.88895	0.252731	0.126366	−	0.991984i	$-0.459669\pi$
$743$	6.88895	0.252731	0.126366	+	0.991984i	$0.459669\pi$
$744$	0	0
$745$	27.1354	0.994164
$746$	−16.1393	−0.590903
$747$	0	0
$748$	41.6933	1.52446
$749$	17.2673	0.630932
$750$	0	0
$751$	15.2937	0.558076	0.279038	−	0.960280i	$-0.409985\pi$
$751$	15.2937	0.558076	0.279038	+	0.960280i	$0.409985\pi$
$752$	−7.09992	−0.258907
$753$	0	0
$754$	−3.37394	−0.122872
$755$	−37.2160	−1.35443
$756$	0	0
$757$	10.6999	0.388895	0.194448	−	0.980913i	$-0.437709\pi$
$757$	10.6999	0.388895	0.194448	+	0.980913i	$0.437709\pi$
$758$	−21.2866	−0.773163
$759$	0	0
$760$	−10.8855	−0.394860
$761$	−5.31801	−0.192778	−0.0963889	−	0.995344i	$-0.530729\pi$
$761$	−5.31801	−0.192778	−0.0963889	+	0.995344i	$0.530729\pi$
$762$	0	0
$763$	26.8895	0.973467
$764$	19.2911	0.697926
$765$	0	0
$766$	26.8982	0.971872
$767$	4.63285	0.167282
$768$	0	0
$769$	−20.6987	−0.746415	−0.373207	−	0.927748i	$-0.621742\pi$
$769$	−20.6987	−0.746415	−0.373207	+	0.927748i	$0.621742\pi$
$770$	25.0689	0.903421
$771$	0	0
$772$	−25.8804	−0.931457
$773$	−26.7365	−0.961644	−0.480822	−	0.876818i	$-0.659662\pi$
$773$	−26.7365	−0.961644	−0.480822	+	0.876818i	$0.659662\pi$
$774$	0	0
$775$	−1.90338	−0.0683716
$776$	9.94247	0.356914
$777$	0	0
$778$	29.1942	1.04666
$779$	−3.08102	−0.110389
$780$	0	0
$781$	83.6905	2.99468
$782$	0	0
$783$	0	0
$784$	−3.55555	−0.126984
$785$	6.89220	0.245993
$786$	0	0
$787$	43.3493	1.54524	0.772618	−	0.634871i	$-0.218947\pi$
$787$	43.3493	1.54524	0.772618	+	0.634871i	$0.218947\pi$
$788$	−18.3833	−0.654877
$789$	0	0
$790$	25.6284	0.911817
$791$	−27.2903	−0.970332
$792$	0	0
$793$	0.435493	0.0154648
$794$	−2.16397	−0.0767965
$795$	0	0
$796$	−8.65762	−0.306861
$797$	−38.3534	−1.35855	−0.679273	−	0.733885i	$-0.737705\pi$
$797$	−38.3534	−1.35855	−0.679273	+	0.733885i	$0.737705\pi$
$798$	0	0
$799$	−53.3046	−1.88578
$800$	0.916195	0.0323924
$801$	0	0
$802$	−6.81856	−0.240772
$803$	−63.8060	−2.25166
$804$	0	0
$805$	0	0
$806$	−2.38563	−0.0840301
$807$	0	0
$808$	4.91204	0.172805
$809$	−6.85847	−0.241131	−0.120565	−	0.992705i	$-0.538471\pi$
$809$	−6.85847	−0.241131	−0.120565	+	0.992705i	$0.538471\pi$
$810$	0	0
$811$	−2.35596	−0.0827290	−0.0413645	−	0.999144i	$-0.513170\pi$
$811$	−2.35596	−0.0827290	−0.0413645	+	0.999144i	$0.513170\pi$
$812$	5.45297	0.191362
$813$	0	0
$814$	−42.9706	−1.50612
$815$	−60.7220	−2.12700
$816$	0	0
$817$	38.3378	1.34127
$818$	−12.8837	−0.450468
$819$	0	0
$820$	1.67451	0.0584764
$821$	−15.8274	−0.552378	−0.276189	−	0.961103i	$-0.589072\pi$
$821$	−15.8274	−0.552378	−0.276189	+	0.961103i	$0.589072\pi$
$822$	0	0
$823$	4.52293	0.157660	0.0788298	−	0.996888i	$-0.474882\pi$
$823$	4.52293	0.157660	0.0788298	+	0.996888i	$0.474882\pi$
$824$	7.84290	0.273220
$825$	0	0
$826$	−7.48762	−0.260527
$827$	−6.29678	−0.218961	−0.109480	−	0.993989i	$-0.534919\pi$
$827$	−6.29678	−0.218961	−0.109480	+	0.993989i	$0.534919\pi$
$828$	0	0
$829$	6.30959	0.219141	0.109570	−	0.993979i	$-0.465052\pi$
$829$	6.30959	0.219141	0.109570	+	0.993979i	$0.465052\pi$
$830$	−25.4140	−0.882133
$831$	0	0
$832$	1.14832	0.0398109
$833$	−26.6943	−0.924903
$834$	0	0
$835$	21.1114	0.730589
$836$	24.8532	0.859567
$837$	0	0
$838$	7.45842	0.257647
$839$	−4.39889	−0.151867	−0.0759333	−	0.997113i	$-0.524194\pi$
$839$	−4.39889	−0.151867	−0.0759333	+	0.997113i	$0.524194\pi$
$840$	0	0
$841$	−20.3673	−0.702321
$842$	−0.856168	−0.0295055
$843$	0	0
$844$	13.8857	0.477966
$845$	28.4128	0.977431
$846$	0	0
$847$	−36.8208	−1.26518
$848$	5.38852	0.185043
$849$	0	0
$850$	6.87859	0.235934
$851$	0	0
$852$	0	0
$853$	3.54886	0.121511	0.0607553	−	0.998153i	$-0.480649\pi$
$853$	3.54886	0.121511	0.0607553	+	0.998153i	$0.480649\pi$
$854$	−0.703844	−0.0240851
$855$	0	0
$856$	−9.30388	−0.318000
$857$	−47.7154	−1.62993	−0.814963	−	0.579513i	$-0.803243\pi$
$857$	−47.7154	−1.62993	−0.814963	+	0.579513i	$0.803243\pi$
$858$	0	0
$859$	−33.8479	−1.15487	−0.577437	−	0.816435i	$-0.695947\pi$
$859$	−33.8479	−1.15487	−0.577437	+	0.816435i	$0.695947\pi$
$860$	−20.8363	−0.710512
$861$	0	0
$862$	−27.0358	−0.920842
$863$	12.8090	0.436023	0.218011	−	0.975946i	$-0.430043\pi$
$863$	12.8090	0.436023	0.218011	+	0.975946i	$0.430043\pi$
$864$	0	0
$865$	37.5239	1.27585
$866$	21.7849	0.740280
$867$	0	0
$868$	3.85565	0.130869
$869$	−58.5133	−1.98493
$870$	0	0
$871$	−3.39089	−0.114896
$872$	−14.4885	−0.490643
$873$	0	0
$874$	0	0
$875$	−18.4351	−0.623221
$876$	0	0
$877$	−42.2324	−1.42609	−0.713043	−	0.701121i	$-0.752683\pi$
$877$	−42.2324	−1.42609	−0.713043	+	0.701121i	$0.752683\pi$
$878$	15.1364	0.510828
$879$	0	0
$880$	−13.5075	−0.455338
$881$	40.9954	1.38117	0.690586	−	0.723251i	$-0.257353\pi$
$881$	40.9954	1.38117	0.690586	+	0.723251i	$0.257353\pi$
$882$	0	0
$883$	55.3897	1.86401	0.932006	−	0.362443i	$-0.118057\pi$
$883$	55.3897	1.86401	0.932006	+	0.362443i	$0.118057\pi$
$884$	8.62135	0.289967
$885$	0	0
$886$	15.2638	0.512798
$887$	−28.9699	−0.972714	−0.486357	−	0.873760i	$-0.661675\pi$
$887$	−28.9699	−0.972714	−0.486357	+	0.873760i	$0.661675\pi$
$888$	0	0
$889$	−12.8736	−0.431767
$890$	−26.2102	−0.878567
$891$	0	0
$892$	3.78498	0.126730
$893$	−31.7747	−1.06330
$894$	0	0
$895$	−14.7153	−0.491880
$896$	−1.85592	−0.0620020
$897$	0	0
$898$	−28.3538	−0.946180
$899$	6.10396	0.203578
$900$	0	0
$901$	40.4558	1.34778
$902$	−3.82315	−0.127297
$903$	0	0
$904$	14.7045	0.489063
$905$	−15.2454	−0.506775
$906$	0	0
$907$	22.3513	0.742164	0.371082	−	0.928600i	$-0.378987\pi$
$907$	22.3513	0.742164	0.371082	+	0.928600i	$0.378987\pi$
$908$	9.83222	0.326294
$909$	0	0
$910$	5.18376	0.171840
$911$	−33.7128	−1.11696	−0.558478	−	0.829519i	$-0.688614\pi$
$911$	−33.7128	−1.11696	−0.558478	+	0.829519i	$0.688614\pi$
$912$	0	0
$913$	58.0238	1.92031
$914$	19.7830	0.654365
$915$	0	0
$916$	19.7726	0.653307
$917$	6.29685	0.207940
$918$	0	0
$919$	−41.0403	−1.35380	−0.676898	−	0.736077i	$-0.736676\pi$
$919$	−41.0403	−1.35380	−0.676898	+	0.736077i	$0.736676\pi$
$920$	0	0
$921$	0	0
$922$	−10.9505	−0.360635
$923$	17.3056	0.569620
$924$	0	0
$925$	−7.08934	−0.233096
$926$	−9.61368	−0.315925
$927$	0	0
$928$	−2.93814	−0.0964493
$929$	34.2650	1.12420	0.562100	−	0.827070i	$-0.309994\pi$
$929$	34.2650	1.12420	0.562100	+	0.827070i	$0.309994\pi$
$930$	0	0
$931$	−15.9124	−0.521508
$932$	16.7160	0.547550
$933$	0	0
$934$	−15.8605	−0.518973
$935$	−101.411	−3.31651
$936$	0	0
$937$	−9.92813	−0.324338	−0.162169	−	0.986763i	$-0.551849\pi$
$937$	−9.92813	−0.324338	−0.162169	+	0.986763i	$0.551849\pi$
$938$	5.48037	0.178940
$939$	0	0
$940$	17.2693	0.563262
$941$	17.9116	0.583901	0.291950	−	0.956433i	$-0.405696\pi$
$941$	17.9116	0.583901	0.291950	+	0.956433i	$0.405696\pi$
$942$	0	0
$943$	0	0
$944$	4.03445	0.131310
$945$	0	0
$946$	47.5722	1.54671
$947$	−36.4199	−1.18349	−0.591744	−	0.806126i	$-0.701560\pi$
$947$	−36.4199	−1.18349	−0.591744	+	0.806126i	$0.701560\pi$
$948$	0	0
$949$	−13.1938	−0.428290
$950$	4.10031	0.133032
$951$	0	0
$952$	−13.9338	−0.451599
$953$	−44.8270	−1.45209	−0.726044	−	0.687648i	$-0.758643\pi$
$953$	−44.8270	−1.45209	−0.726044	+	0.687648i	$0.758643\pi$
$954$	0	0
$955$	−46.9221	−1.51836
$956$	8.55341	0.276637
$957$	0	0
$958$	28.2506	0.912736
$959$	−17.1111	−0.552546
$960$	0	0
$961$	−26.6840	−0.860776
$962$	−8.88549	−0.286480
$963$	0	0
$964$	−21.1421	−0.680942
$965$	62.9496	2.02642
$966$	0	0
$967$	−15.8234	−0.508846	−0.254423	−	0.967093i	$-0.581885\pi$
$967$	−15.8234	−0.508846	−0.254423	+	0.967093i	$0.581885\pi$
$968$	19.8396	0.637669
$969$	0	0
$970$	−24.1833	−0.776479
$971$	−24.3680	−0.782006	−0.391003	−	0.920389i	$-0.627872\pi$
$971$	−24.3680	−0.782006	−0.391003	+	0.920389i	$0.627872\pi$
$972$	0	0
$973$	24.9976	0.801387
$974$	−2.02212	−0.0647929
$975$	0	0
$976$	0.379242	0.0121393
$977$	−2.11250	−0.0675847	−0.0337924	−	0.999429i	$-0.510758\pi$
$977$	−2.11250	−0.0675847	−0.0337924	+	0.999429i	$0.510758\pi$
$978$	0	0
$979$	59.8415	1.91254
$980$	8.64826	0.276258
$981$	0	0
$982$	−18.4236	−0.587920
$983$	−5.87030	−0.187233	−0.0936167	−	0.995608i	$-0.529843\pi$
$983$	−5.87030	−0.187233	−0.0936167	+	0.995608i	$0.529843\pi$
$984$	0	0
$985$	44.7141	1.42471
$986$	−22.0589	−0.702500
$987$	0	0
$988$	5.13916	0.163499
$989$	0	0
$990$	0	0
$991$	−49.5458	−1.57387	−0.786936	−	0.617034i	$-0.788334\pi$
$991$	−49.5458	−1.57387	−0.786936	+	0.617034i	$0.788334\pi$
$992$	−2.07749	−0.0659603
$993$	0	0
$994$	−27.9693	−0.887132
$995$	21.0581	0.667588
$996$	0	0
$997$	−1.76220	−0.0558095	−0.0279047	−	0.999611i	$-0.508884\pi$
$997$	−1.76220	−0.0558095	−0.0279047	+	0.999611i	$0.508884\pi$
$998$	−12.6295	−0.399778
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9522.2.a.ca.1.1		5
3.2	odd	2		3174.2.a.y.1.5		5
23.7	odd	22		414.2.i.b.325.1		10
23.10	odd	22		414.2.i.b.307.1		10
23.22	odd	2		9522.2.a.bv.1.5		5
69.53	even	22		138.2.e.c.49.1	yes	10
69.56	even	22		138.2.e.c.31.1	✓	10
69.68	even	2		3174.2.a.z.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.2.e.c.31.1	✓	10	69.56	even	22
138.2.e.c.49.1	yes	10	69.53	even	22
414.2.i.b.307.1		10	23.10	odd	22
414.2.i.b.325.1		10	23.7	odd	22
3174.2.a.y.1.5		5	3.2	odd	2
3174.2.a.z.1.1		5	69.68	even	2
9522.2.a.bv.1.5		5	23.22	odd	2
9522.2.a.ca.1.1		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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -2.43232 q^{5} -1.85592 q^{7} +1.00000 q^{8} -2.43232 q^{10} +5.55334 q^{11} +1.14832 q^{13} -1.85592 q^{14} +1.00000 q^{16} +7.50778 q^{17} +4.47536 q^{19} -2.43232 q^{20} +5.55334 q^{22} +0.916195 q^{25} +1.14832 q^{26} -1.85592 q^{28} -2.93814 q^{29} -2.07749 q^{31} +1.00000 q^{32} +7.50778 q^{34} +4.51420 q^{35} -7.73780 q^{37} +4.47536 q^{38} -2.43232 q^{40} -0.688441 q^{41} +8.56642 q^{43} +5.55334 q^{44} -7.09992 q^{47} -3.55555 q^{49} +0.916195 q^{50} +1.14832 q^{52} +5.38852 q^{53} -13.5075 q^{55} -1.85592 q^{56} -2.93814 q^{58} +4.03445 q^{59} +0.379242 q^{61} -2.07749 q^{62} +1.00000 q^{64} -2.79309 q^{65} -2.95291 q^{67} +7.50778 q^{68} +4.51420 q^{70} +15.0703 q^{71} -11.4896 q^{73} -7.73780 q^{74} +4.47536 q^{76} -10.3066 q^{77} -10.5366 q^{79} -2.43232 q^{80} -0.688441 q^{82} +10.4484 q^{83} -18.2613 q^{85} +8.56642 q^{86} +5.55334 q^{88} +10.7758 q^{89} -2.13120 q^{91} -7.09992 q^{94} -10.8855 q^{95} +9.94247 q^{97} -3.55555 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{2} + 5 q^{4} + 11 q^{5} + q^{7} + 5 q^{8} + 11 q^{10} + 11 q^{11} + 10 q^{13} + q^{14} + 5 q^{16} + 11 q^{17} + q^{19} + 11 q^{20} + 11 q^{22} + 30 q^{25} + 10 q^{26} + q^{28} - 3 q^{29}+ \cdots - 4 q^{98}+O(q^{100}) \) 5 * q + 5 * q^2 + 5 * q^4 + 11 * q^5 + q^7 + 5 * q^8 + 11 * q^10 + 11 * q^11 + 10 * q^13 + q^14 + 5 * q^16 + 11 * q^17 + q^19 + 11 * q^20 + 11 * q^22 + 30 * q^25 + 10 * q^26 + q^28 - 3 * q^29 + 2 * q^31 + 5 * q^32 + 11 * q^34 + 11 * q^35 - 16 * q^37 + q^38 + 11 * q^40 + 8 * q^41 + 12 * q^43 + 11 * q^44 - 4 * q^49 + 30 * q^50 + 10 * q^52 + 27 * q^53 + q^56 - 3 * q^58 + 6 * q^59 - 5 * q^61 + 2 * q^62 + 5 * q^64 + 22 * q^65 + 7 * q^67 + 11 * q^68 + 11 * q^70 + 11 * q^71 - 12 * q^73 - 16 * q^74 + q^76 - 11 * q^77 - 16 * q^79 + 11 * q^80 + 8 * q^82 + 22 * q^83 - 11 * q^85 + 12 * q^86 + 11 * q^88 + 38 * q^89 - 9 * q^91 - 11 * q^95 - 5 * q^97 - 4 * q^98

Introduction

L-functions

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Representations

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Database

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