LMFDB - Embedded newform 9522.2.a.cj.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

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

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

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

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

Newform invariants

comment:select newform

sage:traces = [10,10,0,10,10,0,-2,10,0,10,12,0,0,-2,0,10,24,0,8,10,0,12,0,0, 8,0,0,-2,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(29)] == traces)

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

Self dual:	yes
Analytic conductor:	$76.0335528047$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	10.10.52900342088704.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 18x^{8} + 123x^{6} - 390x^{4} + 548x^{2} - 241 $ x^10 - 18x^8 + 123x^6 - 390x^4 + 548x^2 - 241
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 414)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$2.09112$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.77954 q^{5} +0.833869 q^{7} +1.00000 q^{8} +1.77954 q^{10} +3.92552 q^{11} +4.58435 q^{13} +0.833869 q^{14} +1.00000 q^{16} +1.16645 q^{17} -3.74764 q^{19} +1.77954 q^{20} +3.92552 q^{22} -1.83324 q^{25} +4.58435 q^{26} +0.833869 q^{28} +1.08844 q^{29} +7.05081 q^{31} +1.00000 q^{32} +1.16645 q^{34} +1.48390 q^{35} +5.44758 q^{37} -3.74764 q^{38} +1.77954 q^{40} -1.45280 q^{41} +3.91817 q^{43} +3.92552 q^{44} -5.02025 q^{47} -6.30466 q^{49} -1.83324 q^{50} +4.58435 q^{52} +11.2107 q^{53} +6.98562 q^{55} +0.833869 q^{56} +1.08844 q^{58} -3.91737 q^{59} -4.19099 q^{61} +7.05081 q^{62} +1.00000 q^{64} +8.15804 q^{65} +2.05678 q^{67} +1.16645 q^{68} +1.48390 q^{70} -7.83463 q^{71} +13.8102 q^{73} +5.44758 q^{74} -3.74764 q^{76} +3.27337 q^{77} -9.56891 q^{79} +1.77954 q^{80} -1.45280 q^{82} -12.8937 q^{83} +2.07574 q^{85} +3.91817 q^{86} +3.92552 q^{88} +2.07244 q^{89} +3.82275 q^{91} -5.02025 q^{94} -6.66907 q^{95} -12.5515 q^{97} -6.30466 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 10 q^{2} + 10 q^{4} + 10 q^{5} - 2 q^{7} + 10 q^{8} + 10 q^{10} + 12 q^{11} - 2 q^{14} + 10 q^{16} + 24 q^{17} + 8 q^{19} + 10 q^{20} + 12 q^{22} + 8 q^{25} - 2 q^{28} + 4 q^{29} + 18 q^{31} + 10 q^{32}+ \cdots + 36 q^{98}+O(q^{100}) $ 10 * q + 10 * q^2 + 10 * q^4 + 10 * q^5 - 2 * q^7 + 10 * q^8 + 10 * q^10 + 12 * q^11 - 2 * q^14 + 10 * q^16 + 24 * q^17 + 8 * q^19 + 10 * q^20 + 12 * q^22 + 8 * q^25 - 2 * q^28 + 4 * q^29 + 18 * q^31 + 10 * q^32 + 24 * q^34 - 24 * q^35 + 12 * q^37 + 8 * q^38 + 10 * q^40 + 28 * q^41 + 8 * q^43 + 12 * q^44 - 16 * q^47 + 36 * q^49 + 8 * q^50 + 34 * q^53 + 30 * q^55 - 2 * q^56 + 4 * q^58 - 22 * q^59 + 30 * q^61 + 18 * q^62 + 10 * q^64 + 36 * q^65 + 18 * q^67 + 24 * q^68 - 24 * q^70 - 28 * q^71 - 20 * q^73 + 12 * q^74 + 8 * q^76 - 20 * q^77 + 2 * q^79 + 10 * q^80 + 28 * q^82 + 44 * q^83 + 16 * q^85 + 8 * q^86 + 12 * q^88 + 44 * q^89 + 22 * q^91 - 16 * q^94 - 10 * q^95 + 18 * q^97 + 36 * q^98

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	1.77954	0.795834	0.397917	−	0.917421i	$-0.369733\pi$
$5$	1.77954	0.795834	0.397917	+	0.917421i	$0.369733\pi$
$6$	0	0
$7$	0.833869	0.315173	0.157586	−	0.987505i	$-0.449629\pi$
$7$	0.833869	0.315173	0.157586	+	0.987505i	$0.449629\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	1.77954	0.562740
$11$	3.92552	1.18359	0.591794	−	0.806089i	$-0.298420\pi$
$11$	3.92552	1.18359	0.591794	+	0.806089i	$0.298420\pi$
$12$	0	0
$13$	4.58435	1.27147	0.635736	−	0.771907i	$-0.280697\pi$
$13$	4.58435	1.27147	0.635736	+	0.771907i	$0.280697\pi$
$14$	0.833869	0.222861
$15$	0	0
$16$	1.00000	0.250000
$17$	1.16645	0.282905	0.141452	−	0.989945i	$-0.454823\pi$
$17$	1.16645	0.282905	0.141452	+	0.989945i	$0.454823\pi$
$18$	0	0
$19$	−3.74764	−0.859767	−0.429883	−	0.902884i	$-0.641445\pi$
$19$	−3.74764	−0.859767	−0.429883	+	0.902884i	$0.641445\pi$
$20$	1.77954	0.397917
$21$	0	0
$22$	3.92552	0.836923
$23$	0	0
$23$	0	0
$24$	0	0
$25$	−1.83324	−0.366648
$26$	4.58435	0.899066
$27$	0	0
$28$	0.833869	0.157586
$29$	1.08844	0.202118	0.101059	−	0.994880i	$-0.467777\pi$
$29$	1.08844	0.202118	0.101059	+	0.994880i	$0.467777\pi$
$30$	0	0
$31$	7.05081	1.26636	0.633182	−	0.774003i	$-0.281749\pi$
$31$	7.05081	1.26636	0.633182	+	0.774003i	$0.281749\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	1.16645	0.200044
$35$	1.48390	0.250825
$36$	0	0
$37$	5.44758	0.895577	0.447788	−	0.894140i	$-0.352212\pi$
$37$	5.44758	0.895577	0.447788	+	0.894140i	$0.352212\pi$
$38$	−3.74764	−0.607947
$39$	0	0
$40$	1.77954	0.281370
$41$	−1.45280	−0.226889	−0.113445	−	0.993544i	$-0.536188\pi$
$41$	−1.45280	−0.226889	−0.113445	+	0.993544i	$0.536188\pi$
$42$	0	0
$43$	3.91817	0.597515	0.298757	−	0.954329i	$-0.403428\pi$
$43$	3.91817	0.597515	0.298757	+	0.954329i	$0.403428\pi$
$44$	3.92552	0.591794
$45$	0	0
$46$	0	0
$47$	−5.02025	−0.732279	−0.366139	−	0.930560i	$-0.619321\pi$
$47$	−5.02025	−0.732279	−0.366139	+	0.930560i	$0.619321\pi$
$48$	0	0
$49$	−6.30466	−0.900666
$50$	−1.83324	−0.259259
$51$	0	0
$52$	4.58435	0.635736
$53$	11.2107	1.53990	0.769952	−	0.638102i	$-0.220280\pi$
$53$	11.2107	1.53990	0.769952	+	0.638102i	$0.220280\pi$
$54$	0	0
$55$	6.98562	0.941940
$56$	0.833869	0.111430
$57$	0	0
$58$	1.08844	0.142919
$59$	−3.91737	−0.509999	−0.254999	−	0.966941i	$-0.582075\pi$
$59$	−3.91737	−0.509999	−0.254999	+	0.966941i	$0.582075\pi$
$60$	0	0
$61$	−4.19099	−0.536601	−0.268301	−	0.963335i	$-0.586462\pi$
$61$	−4.19099	−0.536601	−0.268301	+	0.963335i	$0.586462\pi$
$62$	7.05081	0.895454
$63$	0	0
$64$	1.00000	0.125000
$65$	8.15804	1.01188
$66$	0	0
$67$	2.05678	0.251276	0.125638	−	0.992076i	$-0.459902\pi$
$67$	2.05678	0.251276	0.125638	+	0.992076i	$0.459902\pi$
$68$	1.16645	0.141452
$69$	0	0
$70$	1.48390	0.177360
$71$	−7.83463	−0.929800	−0.464900	−	0.885363i	$-0.653910\pi$
$71$	−7.83463	−0.929800	−0.464900	+	0.885363i	$0.653910\pi$
$72$	0	0
$73$	13.8102	1.61636	0.808178	−	0.588938i	$-0.200454\pi$
$73$	13.8102	1.61636	0.808178	+	0.588938i	$0.200454\pi$
$74$	5.44758	0.633268
$75$	0	0
$76$	−3.74764	−0.429883
$77$	3.27337	0.373035
$78$	0	0
$79$	−9.56891	−1.07659	−0.538293	−	0.842758i	$-0.680931\pi$
$79$	−9.56891	−1.07659	−0.538293	+	0.842758i	$0.680931\pi$
$80$	1.77954	0.198959
$81$	0	0
$82$	−1.45280	−0.160435
$83$	−12.8937	−1.41527	−0.707634	−	0.706579i	$-0.750238\pi$
$83$	−12.8937	−1.41527	−0.707634	+	0.706579i	$0.750238\pi$
$84$	0	0
$85$	2.07574	0.225145
$86$	3.91817	0.422507
$87$	0	0
$88$	3.92552	0.418462
$89$	2.07244	0.219678	0.109839	−	0.993949i	$-0.464967\pi$
$89$	2.07244	0.219678	0.109839	+	0.993949i	$0.464967\pi$
$90$	0	0
$91$	3.82275	0.400733
$92$	0	0
$93$	0	0
$94$	−5.02025	−0.517799
$95$	−6.66907	−0.684232
$96$	0	0
$97$	−12.5515	−1.27441	−0.637203	−	0.770696i	$-0.719909\pi$
$97$	−12.5515	−1.27441	−0.637203	+	0.770696i	$0.719909\pi$
$98$	−6.30466	−0.636867
$99$	0	0
$100$	−1.83324	−0.183324
$101$	12.0480	1.19882	0.599409	−	0.800443i	$-0.295402\pi$
$101$	12.0480	1.19882	0.599409	+	0.800443i	$0.295402\pi$
$102$	0	0
$103$	13.0230	1.28320	0.641598	−	0.767041i	$-0.278272\pi$
$103$	13.0230	1.28320	0.641598	+	0.767041i	$0.278272\pi$
$104$	4.58435	0.449533
$105$	0	0
$106$	11.2107	1.08888
$107$	4.83960	0.467862	0.233931	−	0.972253i	$-0.424841\pi$
$107$	4.83960	0.467862	0.233931	+	0.972253i	$0.424841\pi$
$108$	0	0
$109$	1.00790	0.0965392	0.0482696	−	0.998834i	$-0.484629\pi$
$109$	1.00790	0.0965392	0.0482696	+	0.998834i	$0.484629\pi$
$110$	6.98562	0.666052
$111$	0	0
$112$	0.833869	0.0787932
$113$	10.5034	0.988072	0.494036	−	0.869441i	$-0.335521\pi$
$113$	10.5034	0.988072	0.494036	+	0.869441i	$0.335521\pi$
$114$	0	0
$115$	0	0
$116$	1.08844	0.101059
$117$	0	0
$118$	−3.91737	−0.360624
$119$	0.972663	0.0891639
$120$	0	0
$121$	4.40970	0.400882
$122$	−4.19099	−0.379434
$123$	0	0
$124$	7.05081	0.633182
$125$	−12.1600	−1.08763
$126$	0	0
$127$	−5.34381	−0.474187	−0.237093	−	0.971487i	$-0.576195\pi$
$127$	−5.34381	−0.474187	−0.237093	+	0.971487i	$0.576195\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	8.15804	0.715507
$131$	10.6159	0.927517	0.463759	−	0.885962i	$-0.346501\pi$
$131$	10.6159	0.927517	0.463759	+	0.885962i	$0.346501\pi$
$132$	0	0
$133$	−3.12504	−0.270975
$134$	2.05678	0.177679
$135$	0	0
$136$	1.16645	0.100022
$137$	11.2545	0.961534	0.480767	−	0.876848i	$-0.340358\pi$
$137$	11.2545	0.961534	0.480767	+	0.876848i	$0.340358\pi$
$138$	0	0
$139$	−5.20054	−0.441104	−0.220552	−	0.975375i	$-0.570786\pi$
$139$	−5.20054	−0.441104	−0.220552	+	0.975375i	$0.570786\pi$
$140$	1.48390	0.125413
$141$	0	0
$142$	−7.83463	−0.657468
$143$	17.9960	1.50490
$144$	0	0
$145$	1.93692	0.160852
$146$	13.8102	1.14294
$147$	0	0
$148$	5.44758	0.447788
$149$	−19.1329	−1.56743	−0.783715	−	0.621121i	$-0.786678\pi$
$149$	−19.1329	−1.56743	−0.783715	+	0.621121i	$0.786678\pi$
$150$	0	0
$151$	−11.6172	−0.945393	−0.472697	−	0.881225i	$-0.656719\pi$
$151$	−11.6172	−0.945393	−0.472697	+	0.881225i	$0.656719\pi$
$152$	−3.74764	−0.303973
$153$	0	0
$154$	3.27337	0.263776
$155$	12.5472	1.00782
$156$	0	0
$157$	−2.80726	−0.224044	−0.112022	−	0.993706i	$-0.535733\pi$
$157$	−2.80726	−0.224044	−0.112022	+	0.993706i	$0.535733\pi$
$158$	−9.56891	−0.761262
$159$	0	0
$160$	1.77954	0.140685
$161$	0	0
$162$	0	0
$163$	−22.2284	−1.74107	−0.870533	−	0.492110i	$-0.836226\pi$
$163$	−22.2284	−1.74107	−0.870533	+	0.492110i	$0.836226\pi$
$164$	−1.45280	−0.113445
$165$	0	0
$166$	−12.8937	−1.00075
$167$	−19.2351	−1.48846	−0.744228	−	0.667925i	$-0.767183\pi$
$167$	−19.2351	−1.48846	−0.744228	+	0.667925i	$0.767183\pi$
$168$	0	0
$169$	8.01630	0.616639
$170$	2.07574	0.159202
$171$	0	0
$172$	3.91817	0.298757
$173$	−22.1624	−1.68498	−0.842488	−	0.538715i	$-0.818910\pi$
$173$	−22.1624	−1.68498	−0.842488	+	0.538715i	$0.818910\pi$
$174$	0	0
$175$	−1.52868	−0.115557
$176$	3.92552	0.295897
$177$	0	0
$178$	2.07244	0.155336
$179$	8.71096	0.651088	0.325544	−	0.945527i	$-0.394453\pi$
$179$	8.71096	0.651088	0.325544	+	0.945527i	$0.394453\pi$
$180$	0	0
$181$	20.2676	1.50648	0.753240	−	0.657746i	$-0.228490\pi$
$181$	20.2676	1.50648	0.753240	+	0.657746i	$0.228490\pi$
$182$	3.82275	0.283361
$183$	0	0
$184$	0	0
$185$	9.69419	0.712731
$186$	0	0
$187$	4.57890	0.334843
$188$	−5.02025	−0.366139
$189$	0	0
$190$	−6.66907	−0.483825
$191$	22.2062	1.60678	0.803391	−	0.595452i	$-0.203027\pi$
$191$	22.2062	1.60678	0.803391	+	0.595452i	$0.203027\pi$
$192$	0	0
$193$	−19.1010	−1.37492	−0.687460	−	0.726222i	$-0.741274\pi$
$193$	−19.1010	−1.37492	−0.687460	+	0.726222i	$0.741274\pi$
$194$	−12.5515	−0.901142
$195$	0	0
$196$	−6.30466	−0.450333
$197$	−23.1223	−1.64740	−0.823699	−	0.567028i	$-0.808093\pi$
$197$	−23.1223	−1.64740	−0.823699	+	0.567028i	$0.808093\pi$
$198$	0	0
$199$	15.2797	1.08315	0.541576	−	0.840652i	$-0.317828\pi$
$199$	15.2797	1.08315	0.541576	+	0.840652i	$0.317828\pi$
$200$	−1.83324	−0.129630
$201$	0	0
$202$	12.0480	0.847692
$203$	0.907615	0.0637021
$204$	0	0
$205$	−2.58532	−0.180566
$206$	13.0230	0.907357
$207$	0	0
$208$	4.58435	0.317868
$209$	−14.7114	−1.01761
$210$	0	0
$211$	−6.20798	−0.427375	−0.213688	−	0.976902i	$-0.568547\pi$
$211$	−6.20798	−0.427375	−0.213688	+	0.976902i	$0.568547\pi$
$212$	11.2107	0.769952
$213$	0	0
$214$	4.83960	0.330828
$215$	6.97253	0.475523
$216$	0	0
$217$	5.87946	0.399123
$218$	1.00790	0.0682635
$219$	0	0
$220$	6.98562	0.470970
$221$	5.34740	0.359705
$222$	0	0
$223$	15.4778	1.03647	0.518234	−	0.855239i	$-0.326590\pi$
$223$	15.4778	1.03647	0.518234	+	0.855239i	$0.326590\pi$
$224$	0.833869	0.0557152
$225$	0	0
$226$	10.5034	0.698673
$227$	13.0742	0.867762	0.433881	−	0.900970i	$-0.357144\pi$
$227$	13.0742	0.867762	0.433881	+	0.900970i	$0.357144\pi$
$228$	0	0
$229$	24.8420	1.64160	0.820801	−	0.571214i	$-0.193527\pi$
$229$	24.8420	1.64160	0.820801	+	0.571214i	$0.193527\pi$
$230$	0	0
$231$	0	0
$232$	1.08844	0.0714594
$233$	−5.11474	−0.335078	−0.167539	−	0.985865i	$-0.553582\pi$
$233$	−5.11474	−0.335078	−0.167539	+	0.985865i	$0.553582\pi$
$234$	0	0
$235$	−8.93374	−0.582773
$236$	−3.91737	−0.254999
$237$	0	0
$238$	0.972663	0.0630484
$239$	24.3989	1.57824	0.789118	−	0.614241i	$-0.210538\pi$
$239$	24.3989	1.57824	0.789118	+	0.614241i	$0.210538\pi$
$240$	0	0
$241$	−9.08856	−0.585445	−0.292723	−	0.956197i	$-0.594561\pi$
$241$	−9.08856	−0.585445	−0.292723	+	0.956197i	$0.594561\pi$
$242$	4.40970	0.283466
$243$	0	0
$244$	−4.19099	−0.268301
$245$	−11.2194	−0.716781
$246$	0	0
$247$	−17.1805	−1.09317
$248$	7.05081	0.447727
$249$	0	0
$250$	−12.1600	−0.769067
$251$	2.53733	0.160155	0.0800776	−	0.996789i	$-0.474483\pi$
$251$	2.53733	0.160155	0.0800776	+	0.996789i	$0.474483\pi$
$252$	0	0
$253$	0	0
$254$	−5.34381	−0.335300
$255$	0	0
$256$	1.00000	0.0625000
$257$	−29.1594	−1.81891	−0.909457	−	0.415797i	$-0.863503\pi$
$257$	−29.1594	−1.81891	−0.909457	+	0.415797i	$0.863503\pi$
$258$	0	0
$259$	4.54257	0.282262
$260$	8.15804	0.505940
$261$	0	0
$262$	10.6159	0.655854
$263$	−3.88223	−0.239389	−0.119694	−	0.992811i	$-0.538191\pi$
$263$	−3.88223	−0.239389	−0.119694	+	0.992811i	$0.538191\pi$
$264$	0	0
$265$	19.9498	1.22551
$266$	−3.12504	−0.191608
$267$	0	0
$268$	2.05678	0.125638
$269$	3.55327	0.216647	0.108323	−	0.994116i	$-0.465452\pi$
$269$	3.55327	0.216647	0.108323	+	0.994116i	$0.465452\pi$
$270$	0	0
$271$	−22.1590	−1.34606	−0.673031	−	0.739615i	$-0.735008\pi$
$271$	−22.1590	−1.34606	−0.673031	+	0.739615i	$0.735008\pi$
$272$	1.16645	0.0707261
$273$	0	0
$274$	11.2545	0.679907
$275$	−7.19641	−0.433960
$276$	0	0
$277$	−13.4979	−0.811008	−0.405504	−	0.914093i	$-0.632904\pi$
$277$	−13.4979	−0.811008	−0.405504	+	0.914093i	$0.632904\pi$
$278$	−5.20054	−0.311908
$279$	0	0
$280$	1.48390	0.0886802
$281$	30.7477	1.83425	0.917126	−	0.398597i	$-0.130503\pi$
$281$	30.7477	1.83425	0.917126	+	0.398597i	$0.130503\pi$
$282$	0	0
$283$	−20.5356	−1.22072	−0.610358	−	0.792125i	$-0.708975\pi$
$283$	−20.5356	−1.22072	−0.610358	+	0.792125i	$0.708975\pi$
$284$	−7.83463	−0.464900
$285$	0	0
$286$	17.9960	1.06412
$287$	−1.21145	−0.0715094
$288$	0	0
$289$	−15.6394	−0.919965
$290$	1.93692	0.113740
$291$	0	0
$292$	13.8102	0.808178
$293$	−7.66535	−0.447815	−0.223907	−	0.974610i	$-0.571881\pi$
$293$	−7.66535	−0.447815	−0.223907	+	0.974610i	$0.571881\pi$
$294$	0	0
$295$	−6.97112	−0.405875
$296$	5.44758	0.316634
$297$	0	0
$298$	−19.1329	−1.10834
$299$	0	0
$300$	0	0
$301$	3.26724	0.188320
$302$	−11.6172	−0.668494
$303$	0	0
$304$	−3.74764	−0.214942
$305$	−7.45803	−0.427046
$306$	0	0
$307$	0.803500	0.0458582	0.0229291	−	0.999737i	$-0.492701\pi$
$307$	0.803500	0.0458582	0.0229291	+	0.999737i	$0.492701\pi$
$308$	3.27337	0.186518
$309$	0	0
$310$	12.5472	0.712633
$311$	26.2869	1.49059	0.745297	−	0.666732i	$-0.232307\pi$
$311$	26.2869	1.49059	0.745297	+	0.666732i	$0.232307\pi$
$312$	0	0
$313$	16.3715	0.925371	0.462685	−	0.886523i	$-0.346886\pi$
$313$	16.3715	0.925371	0.462685	+	0.886523i	$0.346886\pi$
$314$	−2.80726	−0.158423
$315$	0	0
$316$	−9.56891	−0.538293
$317$	12.8264	0.720400	0.360200	−	0.932875i	$-0.382708\pi$
$317$	12.8264	0.720400	0.360200	+	0.932875i	$0.382708\pi$
$318$	0	0
$319$	4.27268	0.239224
$320$	1.77954	0.0994793
$321$	0	0
$322$	0	0
$323$	−4.37141	−0.243232
$324$	0	0
$325$	−8.40421	−0.466182
$326$	−22.2284	−1.23112
$327$	0	0
$328$	−1.45280	−0.0802175
$329$	−4.18623	−0.230794
$330$	0	0
$331$	13.6654	0.751118	0.375559	−	0.926798i	$-0.377451\pi$
$331$	13.6654	0.751118	0.375559	+	0.926798i	$0.377451\pi$
$332$	−12.8937	−0.707634
$333$	0	0
$334$	−19.2351	−1.05250
$335$	3.66012	0.199974
$336$	0	0
$337$	−10.3043	−0.561312	−0.280656	−	0.959808i	$-0.590552\pi$
$337$	−10.3043	−0.561312	−0.280656	+	0.959808i	$0.590552\pi$
$338$	8.01630	0.436029
$339$	0	0
$340$	2.07574	0.112573
$341$	27.6781	1.49885
$342$	0	0
$343$	−11.0943	−0.599038
$344$	3.91817	0.211253
$345$	0	0
$346$	−22.1624	−1.19146
$347$	−28.8990	−1.55138	−0.775689	−	0.631116i	$-0.782597\pi$
$347$	−28.8990	−1.55138	−0.775689	+	0.631116i	$0.782597\pi$
$348$	0	0
$349$	12.4738	0.667710	0.333855	−	0.942624i	$-0.391650\pi$
$349$	12.4738	0.667710	0.333855	+	0.942624i	$0.391650\pi$
$350$	−1.52868	−0.0817114
$351$	0	0
$352$	3.92552	0.209231
$353$	36.8722	1.96251	0.981255	−	0.192712i	$-0.0617284\pi$
$353$	36.8722	1.96251	0.981255	+	0.192712i	$0.0617284\pi$
$354$	0	0
$355$	−13.9420	−0.739966
$356$	2.07244	0.109839
$357$	0	0
$358$	8.71096	0.460389
$359$	6.19788	0.327112	0.163556	−	0.986534i	$-0.447704\pi$
$359$	6.19788	0.327112	0.163556	+	0.986534i	$0.447704\pi$
$360$	0	0
$361$	−4.95522	−0.260801
$362$	20.2676	1.06524
$363$	0	0
$364$	3.82275	0.200367
$365$	24.5757	1.28635
$366$	0	0
$367$	−9.47250	−0.494460	−0.247230	−	0.968957i	$-0.579520\pi$
$367$	−9.47250	−0.494460	−0.247230	+	0.968957i	$0.579520\pi$
$368$	0	0
$369$	0	0
$370$	9.69419	0.503977
$371$	9.34823	0.485336
$372$	0	0
$373$	30.7862	1.59405	0.797025	−	0.603946i	$-0.206406\pi$
$373$	30.7862	1.59405	0.797025	+	0.603946i	$0.206406\pi$
$374$	4.57890	0.236769
$375$	0	0
$376$	−5.02025	−0.258900
$377$	4.98978	0.256987
$378$	0	0
$379$	−27.9229	−1.43430	−0.717152	−	0.696917i	$-0.754555\pi$
$379$	−27.9229	−1.43430	−0.717152	+	0.696917i	$0.754555\pi$
$380$	−6.66907	−0.342116
$381$	0	0
$382$	22.2062	1.13617
$383$	−3.92038	−0.200322	−0.100161	−	0.994971i	$-0.531936\pi$
$383$	−3.92038	−0.200322	−0.100161	+	0.994971i	$0.531936\pi$
$384$	0	0
$385$	5.82509	0.296874
$386$	−19.1010	−0.972215
$387$	0	0
$388$	−12.5515	−0.637203
$389$	34.8692	1.76794	0.883969	−	0.467545i	$-0.154861\pi$
$389$	34.8692	1.76794	0.883969	+	0.467545i	$0.154861\pi$
$390$	0	0
$391$	0	0
$392$	−6.30466	−0.318434
$393$	0	0
$394$	−23.1223	−1.16489
$395$	−17.0283	−0.856784
$396$	0	0
$397$	−2.01697	−0.101229	−0.0506145	−	0.998718i	$-0.516118\pi$
$397$	−2.01697	−0.101229	−0.0506145	+	0.998718i	$0.516118\pi$
$398$	15.2797	0.765905
$399$	0	0
$400$	−1.83324	−0.0916619
$401$	−21.4584	−1.07158	−0.535791	−	0.844351i	$-0.679986\pi$
$401$	−21.4584	−1.07158	−0.535791	+	0.844351i	$0.679986\pi$
$402$	0	0
$403$	32.3234	1.61014
$404$	12.0480	0.599409
$405$	0	0
$406$	0.907615	0.0450442
$407$	21.3846	1.05999
$408$	0	0
$409$	−15.8889	−0.785656	−0.392828	−	0.919612i	$-0.628503\pi$
$409$	−15.8889	−0.785656	−0.392828	+	0.919612i	$0.628503\pi$
$410$	−2.58532	−0.127680
$411$	0	0
$412$	13.0230	0.641598
$413$	−3.26658	−0.160738
$414$	0	0
$415$	−22.9449	−1.12632
$416$	4.58435	0.224766
$417$	0	0
$418$	−14.7114	−0.719559
$419$	27.8283	1.35950	0.679751	−	0.733443i	$-0.262088\pi$
$419$	27.8283	1.35950	0.679751	+	0.733443i	$0.262088\pi$
$420$	0	0
$421$	12.7105	0.619471	0.309735	−	0.950823i	$-0.399760\pi$
$421$	12.7105	0.619471	0.309735	+	0.950823i	$0.399760\pi$
$422$	−6.20798	−0.302200
$423$	0	0
$424$	11.2107	0.544438
$425$	−2.13837	−0.103726
$426$	0	0
$427$	−3.49474	−0.169122
$428$	4.83960	0.233931
$429$	0	0
$430$	6.97253	0.336245
$431$	36.6618	1.76593	0.882967	−	0.469435i	$-0.155542\pi$
$431$	36.6618	1.76593	0.882967	+	0.469435i	$0.155542\pi$
$432$	0	0
$433$	−6.94566	−0.333787	−0.166893	−	0.985975i	$-0.553374\pi$
$433$	−6.94566	−0.333787	−0.166893	+	0.985975i	$0.553374\pi$
$434$	5.87946	0.282223
$435$	0	0
$436$	1.00790	0.0482696
$437$	0	0
$438$	0	0
$439$	22.3866	1.06845	0.534227	−	0.845341i	$-0.320603\pi$
$439$	22.3866	1.06845	0.534227	+	0.845341i	$0.320603\pi$
$440$	6.98562	0.333026
$441$	0	0
$442$	5.34740	0.254350
$443$	−29.0072	−1.37817	−0.689087	−	0.724679i	$-0.741988\pi$
$443$	−29.0072	−1.37817	−0.689087	+	0.724679i	$0.741988\pi$
$444$	0	0
$445$	3.68798	0.174827
$446$	15.4778	0.732893
$447$	0	0
$448$	0.833869	0.0393966
$449$	37.0126	1.74673	0.873367	−	0.487063i	$-0.161932\pi$
$449$	37.0126	1.74673	0.873367	+	0.487063i	$0.161932\pi$
$450$	0	0
$451$	−5.70300	−0.268544
$452$	10.5034	0.494036
$453$	0	0
$454$	13.0742	0.613600
$455$	6.80274	0.318917
$456$	0	0
$457$	−6.38589	−0.298719	−0.149360	−	0.988783i	$-0.547721\pi$
$457$	−6.38589	−0.298719	−0.149360	+	0.988783i	$0.547721\pi$
$458$	24.8420	1.16079
$459$	0	0
$460$	0	0
$461$	6.03086	0.280885	0.140443	−	0.990089i	$-0.455147\pi$
$461$	6.03086	0.280885	0.140443	+	0.990089i	$0.455147\pi$
$462$	0	0
$463$	1.06935	0.0496970	0.0248485	−	0.999691i	$-0.492090\pi$
$463$	1.06935	0.0496970	0.0248485	+	0.999691i	$0.492090\pi$
$464$	1.08844	0.0505295
$465$	0	0
$466$	−5.11474	−0.236936
$467$	−0.947753	−0.0438568	−0.0219284	−	0.999760i	$-0.506981\pi$
$467$	−0.947753	−0.0438568	−0.0219284	+	0.999760i	$0.506981\pi$
$468$	0	0
$469$	1.71509	0.0791953
$470$	−8.93374	−0.412083
$471$	0	0
$472$	−3.91737	−0.180312
$473$	15.3808	0.707212
$474$	0	0
$475$	6.87031	0.315232
$476$	0.972663	0.0445819
$477$	0	0
$478$	24.3989	1.11598
$479$	−17.5466	−0.801726	−0.400863	−	0.916138i	$-0.631290\pi$
$479$	−17.5466	−0.801726	−0.400863	+	0.916138i	$0.631290\pi$
$480$	0	0
$481$	24.9736	1.13870
$482$	−9.08856	−0.413972
$483$	0	0
$484$	4.40970	0.200441
$485$	−22.3358	−1.01422
$486$	0	0
$487$	32.5801	1.47635	0.738174	−	0.674611i	$-0.235688\pi$
$487$	32.5801	1.47635	0.738174	+	0.674611i	$0.235688\pi$
$488$	−4.19099	−0.189717
$489$	0	0
$490$	−11.2194	−0.506841
$491$	−20.2982	−0.916047	−0.458023	−	0.888940i	$-0.651442\pi$
$491$	−20.2982	−0.916047	−0.458023	+	0.888940i	$0.651442\pi$
$492$	0	0
$493$	1.26960	0.0571801
$494$	−17.1805	−0.772987
$495$	0	0
$496$	7.05081	0.316591
$497$	−6.53306	−0.293048
$498$	0	0
$499$	−7.83390	−0.350694	−0.175347	−	0.984507i	$-0.556105\pi$
$499$	−7.83390	−0.350694	−0.175347	+	0.984507i	$0.556105\pi$
$500$	−12.1600	−0.543813
$501$	0	0
$502$	2.53733	0.113247
$503$	12.8212	0.571667	0.285833	−	0.958279i	$-0.407730\pi$
$503$	12.8212	0.571667	0.285833	+	0.958279i	$0.407730\pi$
$504$	0	0
$505$	21.4398	0.954060
$506$	0	0
$507$	0	0
$508$	−5.34381	−0.237093
$509$	−7.07163	−0.313444	−0.156722	−	0.987643i	$-0.550093\pi$
$509$	−7.07163	−0.313444	−0.156722	+	0.987643i	$0.550093\pi$
$510$	0	0
$511$	11.5159	0.509432
$512$	1.00000	0.0441942
$513$	0	0
$514$	−29.1594	−1.28617
$515$	23.1750	1.02121
$516$	0	0
$517$	−19.7071	−0.866717
$518$	4.54257	0.199589
$519$	0	0
$520$	8.15804	0.357754
$521$	26.4413	1.15841	0.579207	−	0.815180i	$-0.303362\pi$
$521$	26.4413	1.15841	0.579207	+	0.815180i	$0.303362\pi$
$522$	0	0
$523$	−29.7344	−1.30019	−0.650097	−	0.759851i	$-0.725272\pi$
$523$	−29.7344	−1.30019	−0.650097	+	0.759851i	$0.725272\pi$
$524$	10.6159	0.463759
$525$	0	0
$526$	−3.88223	−0.169273
$527$	8.22439	0.358260
$528$	0	0
$529$	0	0
$530$	19.9498	0.866565
$531$	0	0
$532$	−3.12504	−0.135488
$533$	−6.66016	−0.288483
$534$	0	0
$535$	8.61226	0.372340
$536$	2.05678	0.0888394
$537$	0	0
$538$	3.55327	0.153192
$539$	−24.7491	−1.06602
$540$	0	0
$541$	−6.72627	−0.289185	−0.144593	−	0.989491i	$-0.546187\pi$
$541$	−6.72627	−0.289185	−0.144593	+	0.989491i	$0.546187\pi$
$542$	−22.1590	−0.951809
$543$	0	0
$544$	1.16645	0.0500109
$545$	1.79360	0.0768292
$546$	0	0
$547$	−43.1456	−1.84477	−0.922386	−	0.386269i	$-0.873764\pi$
$547$	−43.1456	−1.84477	−0.922386	+	0.386269i	$0.873764\pi$
$548$	11.2545	0.480767
$549$	0	0
$550$	−7.19641	−0.306856
$551$	−4.07907	−0.173774
$552$	0	0
$553$	−7.97922	−0.339311
$554$	−13.4979	−0.573469
$555$	0	0
$556$	−5.20054	−0.220552
$557$	20.2095	0.856302	0.428151	−	0.903707i	$-0.359165\pi$
$557$	20.2095	0.856302	0.428151	+	0.903707i	$0.359165\pi$
$558$	0	0
$559$	17.9623	0.759723
$560$	1.48390	0.0627064
$561$	0	0
$562$	30.7477	1.29701
$563$	−3.33199	−0.140427	−0.0702133	−	0.997532i	$-0.522368\pi$
$563$	−3.33199	−0.140427	−0.0702133	+	0.997532i	$0.522368\pi$
$564$	0	0
$565$	18.6911	0.786342
$566$	−20.5356	−0.863177
$567$	0	0
$568$	−7.83463	−0.328734
$569$	−10.9839	−0.460469	−0.230234	−	0.973135i	$-0.573949\pi$
$569$	−10.9839	−0.460469	−0.230234	+	0.973135i	$0.573949\pi$
$570$	0	0
$571$	−17.7979	−0.744818	−0.372409	−	0.928069i	$-0.621468\pi$
$571$	−17.7979	−0.744818	−0.372409	+	0.928069i	$0.621468\pi$
$572$	17.9960	0.752449
$573$	0	0
$574$	−1.21145	−0.0505648
$575$	0	0
$576$	0	0
$577$	−6.39513	−0.266233	−0.133116	−	0.991100i	$-0.542498\pi$
$577$	−6.39513	−0.266233	−0.133116	+	0.991100i	$0.542498\pi$
$578$	−15.6394	−0.650513
$579$	0	0
$580$	1.93692	0.0804261
$581$	−10.7517	−0.446054
$582$	0	0
$583$	44.0077	1.82261
$584$	13.8102	0.571468
$585$	0	0
$586$	−7.66535	−0.316653
$587$	−9.11251	−0.376114	−0.188057	−	0.982158i	$-0.560219\pi$
$587$	−9.11251	−0.376114	−0.188057	+	0.982158i	$0.560219\pi$
$588$	0	0
$589$	−26.4239	−1.08878
$590$	−6.97112	−0.286997
$591$	0	0
$592$	5.44758	0.223894
$593$	37.1777	1.52670	0.763352	−	0.645983i	$-0.223552\pi$
$593$	37.1777	1.52670	0.763352	+	0.645983i	$0.223552\pi$
$594$	0	0
$595$	1.73089	0.0709597
$596$	−19.1329	−0.783715
$597$	0	0
$598$	0	0
$599$	7.27761	0.297355	0.148677	−	0.988886i	$-0.452498\pi$
$599$	7.27761	0.297355	0.148677	+	0.988886i	$0.452498\pi$
$600$	0	0
$601$	34.8840	1.42295	0.711475	−	0.702712i	$-0.248028\pi$
$601$	34.8840	1.42295	0.711475	+	0.702712i	$0.248028\pi$
$602$	3.26724	0.133163
$603$	0	0
$604$	−11.6172	−0.472697
$605$	7.84724	0.319036
$606$	0	0
$607$	−30.8471	−1.25205	−0.626024	−	0.779804i	$-0.715319\pi$
$607$	−30.8471	−1.25205	−0.626024	+	0.779804i	$0.715319\pi$
$608$	−3.74764	−0.151987
$609$	0	0
$610$	−7.45803	−0.301967
$611$	−23.0146	−0.931071
$612$	0	0
$613$	12.1703	0.491553	0.245777	−	0.969327i	$-0.420957\pi$
$613$	12.1703	0.491553	0.245777	+	0.969327i	$0.420957\pi$
$614$	0.803500	0.0324266
$615$	0	0
$616$	3.27337	0.131888
$617$	4.11400	0.165623	0.0828116	−	0.996565i	$-0.473610\pi$
$617$	4.11400	0.165623	0.0828116	+	0.996565i	$0.473610\pi$
$618$	0	0
$619$	−28.2972	−1.13736	−0.568680	−	0.822559i	$-0.692546\pi$
$619$	−28.2972	−1.13736	−0.568680	+	0.822559i	$0.692546\pi$
$620$	12.5472	0.503908
$621$	0	0
$622$	26.2869	1.05401
$623$	1.72814	0.0692365
$624$	0	0
$625$	−12.4730	−0.498922
$626$	16.3715	0.654336
$627$	0	0
$628$	−2.80726	−0.112022
$629$	6.35431	0.253363
$630$	0	0
$631$	−17.8555	−0.710816	−0.355408	−	0.934711i	$-0.615658\pi$
$631$	−17.8555	−0.710816	−0.355408	+	0.934711i	$0.615658\pi$
$632$	−9.56891	−0.380631
$633$	0	0
$634$	12.8264	0.509400
$635$	−9.50952	−0.377374
$636$	0	0
$637$	−28.9028	−1.14517
$638$	4.27268	0.169157
$639$	0	0
$640$	1.77954	0.0703425
$641$	−40.9767	−1.61848	−0.809241	−	0.587476i	$-0.800122\pi$
$641$	−40.9767	−1.61848	−0.809241	+	0.587476i	$0.800122\pi$
$642$	0	0
$643$	−22.3907	−0.883003	−0.441501	−	0.897261i	$-0.645554\pi$
$643$	−22.3907	−0.883003	−0.441501	+	0.897261i	$0.645554\pi$
$644$	0	0
$645$	0	0
$646$	−4.37141	−0.171991
$647$	6.61041	0.259882	0.129941	−	0.991522i	$-0.458521\pi$
$647$	6.61041	0.259882	0.129941	+	0.991522i	$0.458521\pi$
$648$	0	0
$649$	−15.3777	−0.603629
$650$	−8.40421	−0.329640
$651$	0	0
$652$	−22.2284	−0.870533
$653$	40.7556	1.59489	0.797444	−	0.603393i	$-0.206185\pi$
$653$	40.7556	1.59489	0.797444	+	0.603393i	$0.206185\pi$
$654$	0	0
$655$	18.8914	0.738150
$656$	−1.45280	−0.0567224
$657$	0	0
$658$	−4.18623	−0.163196
$659$	−27.3344	−1.06480	−0.532399	−	0.846494i	$-0.678709\pi$
$659$	−27.3344	−1.06480	−0.532399	+	0.846494i	$0.678709\pi$
$660$	0	0
$661$	10.0559	0.391129	0.195564	−	0.980691i	$-0.437346\pi$
$661$	10.0559	0.391129	0.195564	+	0.980691i	$0.437346\pi$
$662$	13.6654	0.531121
$663$	0	0
$664$	−12.8937	−0.500373
$665$	−5.56113	−0.215651
$666$	0	0
$667$	0	0
$668$	−19.2351	−0.744228
$669$	0	0
$670$	3.66012	0.141403
$671$	−16.4518	−0.635115
$672$	0	0
$673$	−17.5255	−0.675557	−0.337779	−	0.941226i	$-0.609676\pi$
$673$	−17.5255	−0.675557	−0.337779	+	0.941226i	$0.609676\pi$
$674$	−10.3043	−0.396908
$675$	0	0
$676$	8.01630	0.308319
$677$	−9.23238	−0.354829	−0.177415	−	0.984136i	$-0.556773\pi$
$677$	−9.23238	−0.354829	−0.177415	+	0.984136i	$0.556773\pi$
$678$	0	0
$679$	−10.4663	−0.401659
$680$	2.07574	0.0796008
$681$	0	0
$682$	27.6781	1.05985
$683$	30.7576	1.17691	0.588453	−	0.808531i	$-0.299737\pi$
$683$	30.7576	1.17691	0.588453	+	0.808531i	$0.299737\pi$
$684$	0	0
$685$	20.0278	0.765222
$686$	−11.0943	−0.423584
$687$	0	0
$688$	3.91817	0.149379
$689$	51.3937	1.95794
$690$	0	0
$691$	39.1989	1.49119	0.745597	−	0.666397i	$-0.232164\pi$
$691$	39.1989	1.49119	0.745597	+	0.666397i	$0.232164\pi$
$692$	−22.1624	−0.842488
$693$	0	0
$694$	−28.8990	−1.09699
$695$	−9.25457	−0.351046
$696$	0	0
$697$	−1.69461	−0.0641881
$698$	12.4738	0.472142
$699$	0	0
$700$	−1.52868	−0.0577787
$701$	−33.3259	−1.25870	−0.629351	−	0.777121i	$-0.716679\pi$
$701$	−33.3259	−1.25870	−0.629351	+	0.777121i	$0.716679\pi$
$702$	0	0
$703$	−20.4156	−0.769987
$704$	3.92552	0.147949
$705$	0	0
$706$	36.8722	1.38770
$707$	10.0464	0.377835
$708$	0	0
$709$	−2.15309	−0.0808608	−0.0404304	−	0.999182i	$-0.512873\pi$
$709$	−2.15309	−0.0808608	−0.0404304	+	0.999182i	$0.512873\pi$
$710$	−13.9420	−0.523235
$711$	0	0
$712$	2.07244	0.0776678
$713$	0	0
$714$	0	0
$715$	32.0245	1.19765
$716$	8.71096	0.325544
$717$	0	0
$718$	6.19788	0.231303
$719$	−4.28248	−0.159710	−0.0798548	−	0.996807i	$-0.525446\pi$
$719$	−4.28248	−0.159710	−0.0798548	+	0.996807i	$0.525446\pi$
$720$	0	0
$721$	10.8595	0.404429
$722$	−4.95522	−0.184414
$723$	0	0
$724$	20.2676	0.753240
$725$	−1.99537	−0.0741060
$726$	0	0
$727$	−37.7807	−1.40121	−0.700603	−	0.713551i	$-0.747086\pi$
$727$	−37.7807	−1.40121	−0.700603	+	0.713551i	$0.747086\pi$
$728$	3.82275	0.141681
$729$	0	0
$730$	24.5757	0.909588
$731$	4.57033	0.169040
$732$	0	0
$733$	44.6316	1.64851	0.824254	−	0.566221i	$-0.191595\pi$
$733$	44.6316	1.64851	0.824254	+	0.566221i	$0.191595\pi$
$734$	−9.47250	−0.349636
$735$	0	0
$736$	0	0
$737$	8.07393	0.297407
$738$	0	0
$739$	−22.7631	−0.837356	−0.418678	−	0.908135i	$-0.637506\pi$
$739$	−22.7631	−0.837356	−0.418678	+	0.908135i	$0.637506\pi$
$740$	9.69419	0.356365
$741$	0	0
$742$	9.34823	0.343184
$743$	3.30466	0.121236	0.0606181	−	0.998161i	$-0.480693\pi$
$743$	3.30466	0.121236	0.0606181	+	0.998161i	$0.480693\pi$
$744$	0	0
$745$	−34.0478	−1.24741
$746$	30.7862	1.12716
$747$	0	0
$748$	4.57890	0.167421
$749$	4.03559	0.147457
$750$	0	0
$751$	−52.8035	−1.92683	−0.963413	−	0.268020i	$-0.913631\pi$
$751$	−52.8035	−1.92683	−0.963413	+	0.268020i	$0.913631\pi$
$752$	−5.02025	−0.183070
$753$	0	0
$754$	4.98978	0.181717
$755$	−20.6732	−0.752376
$756$	0	0
$757$	−14.6559	−0.532677	−0.266338	−	0.963880i	$-0.585814\pi$
$757$	−14.6559	−0.532677	−0.266338	+	0.963880i	$0.585814\pi$
$758$	−27.9229	−1.01421
$759$	0	0
$760$	−6.66907	−0.241913
$761$	25.9170	0.939492	0.469746	−	0.882802i	$-0.344346\pi$
$761$	25.9170	0.939492	0.469746	+	0.882802i	$0.344346\pi$
$762$	0	0
$763$	0.840456	0.0304266
$764$	22.2062	0.803391
$765$	0	0
$766$	−3.92038	−0.141649
$767$	−17.9586	−0.648449
$768$	0	0
$769$	−52.5864	−1.89631	−0.948157	−	0.317801i	$-0.897055\pi$
$769$	−52.5864	−1.89631	−0.948157	+	0.317801i	$0.897055\pi$
$770$	5.82509	0.209922
$771$	0	0
$772$	−19.1010	−0.687460
$773$	4.07182	0.146453	0.0732266	−	0.997315i	$-0.476670\pi$
$773$	4.07182	0.146453	0.0732266	+	0.997315i	$0.476670\pi$
$774$	0	0
$775$	−12.9258	−0.464309
$776$	−12.5515	−0.450571
$777$	0	0
$778$	34.8692	1.25012
$779$	5.44457	0.195072
$780$	0	0
$781$	−30.7550	−1.10050
$782$	0	0
$783$	0	0
$784$	−6.30466	−0.225167
$785$	−4.99563	−0.178302
$786$	0	0
$787$	−0.0689694	−0.00245849	−0.00122925	−	0.999999i	$-0.500391\pi$
$787$	−0.0689694	−0.00245849	−0.00122925	+	0.999999i	$0.500391\pi$
$788$	−23.1223	−0.823699
$789$	0	0
$790$	−17.0283	−0.605838
$791$	8.75842	0.311414
$792$	0	0
$793$	−19.2130	−0.682273
$794$	−2.01697	−0.0715797
$795$	0	0
$796$	15.2797	0.541576
$797$	2.14410	0.0759481	0.0379740	−	0.999279i	$-0.487910\pi$
$797$	2.14410	0.0759481	0.0379740	+	0.999279i	$0.487910\pi$
$798$	0	0
$799$	−5.85585	−0.207165
$800$	−1.83324	−0.0648148
$801$	0	0
$802$	−21.4584	−0.757723
$803$	54.2120	1.91310
$804$	0	0
$805$	0	0
$806$	32.3234	1.13854
$807$	0	0
$808$	12.0480	0.423846
$809$	−25.7785	−0.906325	−0.453162	−	0.891428i	$-0.649704\pi$
$809$	−25.7785	−0.906325	−0.453162	+	0.891428i	$0.649704\pi$
$810$	0	0
$811$	−21.1268	−0.741861	−0.370930	−	0.928661i	$-0.620961\pi$
$811$	−21.1268	−0.741861	−0.370930	+	0.928661i	$0.620961\pi$
$812$	0.907615	0.0318510
$813$	0	0
$814$	21.3846	0.749529
$815$	−39.5564	−1.38560
$816$	0	0
$817$	−14.6839	−0.513723
$818$	−15.8889	−0.555543
$819$	0	0
$820$	−2.58532	−0.0902832
$821$	−30.0621	−1.04917	−0.524587	−	0.851357i	$-0.675780\pi$
$821$	−30.0621	−1.04917	−0.524587	+	0.851357i	$0.675780\pi$
$822$	0	0
$823$	18.7965	0.655204	0.327602	−	0.944816i	$-0.393760\pi$
$823$	18.7965	0.655204	0.327602	+	0.944816i	$0.393760\pi$
$824$	13.0230	0.453678
$825$	0	0
$826$	−3.26658	−0.113659
$827$	49.2602	1.71295	0.856473	−	0.516192i	$-0.172651\pi$
$827$	49.2602	1.71295	0.856473	+	0.516192i	$0.172651\pi$
$828$	0	0
$829$	10.0455	0.348896	0.174448	−	0.984666i	$-0.444186\pi$
$829$	10.0455	0.348896	0.174448	+	0.984666i	$0.444186\pi$
$830$	−22.9449	−0.796428
$831$	0	0
$832$	4.58435	0.158934
$833$	−7.35404	−0.254803
$834$	0	0
$835$	−34.2296	−1.18457
$836$	−14.7114	−0.508805
$837$	0	0
$838$	27.8283	0.961312
$839$	1.36255	0.0470406	0.0235203	−	0.999723i	$-0.492513\pi$
$839$	1.36255	0.0470406	0.0235203	+	0.999723i	$0.492513\pi$
$840$	0	0
$841$	−27.8153	−0.959148
$842$	12.7105	0.438032
$843$	0	0
$844$	−6.20798	−0.213688
$845$	14.2653	0.490742
$846$	0	0
$847$	3.67711	0.126347
$848$	11.2107	0.384976
$849$	0	0
$850$	−2.13837	−0.0733456
$851$	0	0
$852$	0	0
$853$	−30.5275	−1.04524	−0.522620	−	0.852565i	$-0.675045\pi$
$853$	−30.5275	−1.04524	−0.522620	+	0.852565i	$0.675045\pi$
$854$	−3.49474	−0.119587
$855$	0	0
$856$	4.83960	0.165414
$857$	−5.62391	−0.192109	−0.0960545	−	0.995376i	$-0.530622\pi$
$857$	−5.62391	−0.192109	−0.0960545	+	0.995376i	$0.530622\pi$
$858$	0	0
$859$	−30.4852	−1.04014	−0.520070	−	0.854123i	$-0.674094\pi$
$859$	−30.4852	−1.04014	−0.520070	+	0.854123i	$0.674094\pi$
$860$	6.97253	0.237761
$861$	0	0
$862$	36.6618	1.24870
$863$	−30.7008	−1.04507	−0.522535	−	0.852618i	$-0.675013\pi$
$863$	−30.7008	−1.04507	−0.522535	+	0.852618i	$0.675013\pi$
$864$	0	0
$865$	−39.4389	−1.34096
$866$	−6.94566	−0.236023
$867$	0	0
$868$	5.87946	0.199562
$869$	−37.5629	−1.27424
$870$	0	0
$871$	9.42901	0.319490
$872$	1.00790	0.0341318
$873$	0	0
$874$	0	0
$875$	−10.1399	−0.342790
$876$	0	0
$877$	52.4974	1.77271	0.886356	−	0.463005i	$-0.153229\pi$
$877$	52.4974	1.77271	0.886356	+	0.463005i	$0.153229\pi$
$878$	22.3866	0.755510
$879$	0	0
$880$	6.98562	0.235485
$881$	−5.56489	−0.187486	−0.0937429	−	0.995596i	$-0.529883\pi$
$881$	−5.56489	−0.187486	−0.0937429	+	0.995596i	$0.529883\pi$
$882$	0	0
$883$	27.8536	0.937350	0.468675	−	0.883371i	$-0.344732\pi$
$883$	27.8536	0.937350	0.468675	+	0.883371i	$0.344732\pi$
$884$	5.34740	0.179852
$885$	0	0
$886$	−29.0072	−0.974516
$887$	8.02769	0.269544	0.134772	−	0.990877i	$-0.456970\pi$
$887$	8.02769	0.269544	0.134772	+	0.990877i	$0.456970\pi$
$888$	0	0
$889$	−4.45604	−0.149451
$890$	3.68798	0.123621
$891$	0	0
$892$	15.4778	0.518234
$893$	18.8141	0.629589
$894$	0	0
$895$	15.5015	0.518158
$896$	0.833869	0.0278576
$897$	0	0
$898$	37.0126	1.23513
$899$	7.67437	0.255955
$900$	0	0
$901$	13.0766	0.435646
$902$	−5.70300	−0.189889
$903$	0	0
$904$	10.5034	0.349336
$905$	36.0670	1.19891
$906$	0	0
$907$	−52.8322	−1.75427	−0.877133	−	0.480248i	$-0.840547\pi$
$907$	−52.8322	−1.75427	−0.877133	+	0.480248i	$0.840547\pi$
$908$	13.0742	0.433881
$909$	0	0
$910$	6.80274	0.225509
$911$	9.32564	0.308972	0.154486	−	0.987995i	$-0.450628\pi$
$911$	9.32564	0.308972	0.154486	+	0.987995i	$0.450628\pi$
$912$	0	0
$913$	−50.6145	−1.67510
$914$	−6.38589	−0.211227
$915$	0	0
$916$	24.8420	0.820801
$917$	8.85229	0.292328
$918$	0	0
$919$	13.1566	0.433997	0.216998	−	0.976172i	$-0.430373\pi$
$919$	13.1566	0.433997	0.216998	+	0.976172i	$0.430373\pi$
$920$	0	0
$921$	0	0
$922$	6.03086	0.198616
$923$	−35.9167	−1.18221
$924$	0	0
$925$	−9.98672	−0.328361
$926$	1.06935	0.0351411
$927$	0	0
$928$	1.08844	0.0357297
$929$	−29.9879	−0.983872	−0.491936	−	0.870631i	$-0.663711\pi$
$929$	−29.9879	−0.983872	−0.491936	+	0.870631i	$0.663711\pi$
$930$	0	0
$931$	23.6276	0.774363
$932$	−5.11474	−0.167539
$933$	0	0
$934$	−0.947753	−0.0310114
$935$	8.14834	0.266479
$936$	0	0
$937$	21.3742	0.698265	0.349132	−	0.937073i	$-0.386476\pi$
$937$	21.3742	0.698265	0.349132	+	0.937073i	$0.386476\pi$
$938$	1.71509	0.0559995
$939$	0	0
$940$	−8.93374	−0.291386
$941$	−5.27352	−0.171912	−0.0859559	−	0.996299i	$-0.527394\pi$
$941$	−5.27352	−0.171912	−0.0859559	+	0.996299i	$0.527394\pi$
$942$	0	0
$943$	0	0
$944$	−3.91737	−0.127500
$945$	0	0
$946$	15.3808	0.500074
$947$	6.75876	0.219630	0.109815	−	0.993952i	$-0.464974\pi$
$947$	6.75876	0.219630	0.109815	+	0.993952i	$0.464974\pi$
$948$	0	0
$949$	63.3106	2.05515
$950$	6.87031	0.222902
$951$	0	0
$952$	0.972663	0.0315242
$953$	8.65623	0.280403	0.140201	−	0.990123i	$-0.455225\pi$
$953$	8.65623	0.280403	0.140201	+	0.990123i	$0.455225\pi$
$954$	0	0
$955$	39.5168	1.27873
$956$	24.3989	0.789118
$957$	0	0
$958$	−17.5466	−0.566906
$959$	9.38475	0.303049
$960$	0	0
$961$	18.7140	0.603676
$962$	24.9736	0.805182
$963$	0	0
$964$	−9.08856	−0.292723
$965$	−33.9910	−1.09421
$966$	0	0
$967$	9.42139	0.302971	0.151486	−	0.988459i	$-0.451594\pi$
$967$	9.42139	0.302971	0.151486	+	0.988459i	$0.451594\pi$
$968$	4.40970	0.141733
$969$	0	0
$970$	−22.3358	−0.717160
$971$	36.0574	1.15714	0.578569	−	0.815633i	$-0.303611\pi$
$971$	36.0574	1.15714	0.578569	+	0.815633i	$0.303611\pi$
$972$	0	0
$973$	−4.33657	−0.139024
$974$	32.5801	1.04394
$975$	0	0
$976$	−4.19099	−0.134150
$977$	43.1184	1.37948	0.689740	−	0.724057i	$-0.257725\pi$
$977$	43.1184	1.37948	0.689740	+	0.724057i	$0.257725\pi$
$978$	0	0
$979$	8.13539	0.260008
$980$	−11.2194	−0.358390
$981$	0	0
$982$	−20.2982	−0.647743
$983$	−7.15082	−0.228076	−0.114038	−	0.993476i	$-0.536379\pi$
$983$	−7.15082	−0.228076	−0.114038	+	0.993476i	$0.536379\pi$
$984$	0	0
$985$	−41.1471	−1.31106
$986$	1.26960	0.0404324
$987$	0	0
$988$	−17.1805	−0.546584
$989$	0	0
$990$	0	0
$991$	27.9221	0.886974	0.443487	−	0.896281i	$-0.353741\pi$
$991$	27.9221	0.886974	0.443487	+	0.896281i	$0.353741\pi$
$992$	7.05081	0.223864
$993$	0	0
$994$	−6.53306	−0.207216
$995$	27.1909	0.862010
$996$	0	0
$997$	7.05093	0.223305	0.111653	−	0.993747i	$-0.464386\pi$
$997$	7.05093	0.223305	0.111653	+	0.993747i	$0.464386\pi$
$998$	−7.83390	−0.247978
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9522.2.a.cj.1.6		10
3.2	odd	2		9522.2.a.cg.1.5		10
23.13	even	11		414.2.i.g.307.1	✓	20
23.16	even	11		414.2.i.g.325.1	yes	20
23.22	odd	2		9522.2.a.ci.1.5		10
69.59	odd	22		414.2.i.h.307.2	yes	20
69.62	odd	22		414.2.i.h.325.2	yes	20
69.68	even	2		9522.2.a.ch.1.6		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
414.2.i.g.307.1	✓	20	23.13	even	11
414.2.i.g.325.1	yes	20	23.16	even	11
414.2.i.h.307.2	yes	20	69.59	odd	22
414.2.i.h.325.2	yes	20	69.62	odd	22
9522.2.a.cg.1.5		10	3.2	odd	2
9522.2.a.ch.1.6		10	69.68	even	2
9522.2.a.ci.1.5		10	23.22	odd	2
9522.2.a.cj.1.6		10	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.77954 q^{5} +0.833869 q^{7} +1.00000 q^{8} +1.77954 q^{10} +3.92552 q^{11} +4.58435 q^{13} +0.833869 q^{14} +1.00000 q^{16} +1.16645 q^{17} -3.74764 q^{19} +1.77954 q^{20} +3.92552 q^{22} -1.83324 q^{25} +4.58435 q^{26} +0.833869 q^{28} +1.08844 q^{29} +7.05081 q^{31} +1.00000 q^{32} +1.16645 q^{34} +1.48390 q^{35} +5.44758 q^{37} -3.74764 q^{38} +1.77954 q^{40} -1.45280 q^{41} +3.91817 q^{43} +3.92552 q^{44} -5.02025 q^{47} -6.30466 q^{49} -1.83324 q^{50} +4.58435 q^{52} +11.2107 q^{53} +6.98562 q^{55} +0.833869 q^{56} +1.08844 q^{58} -3.91737 q^{59} -4.19099 q^{61} +7.05081 q^{62} +1.00000 q^{64} +8.15804 q^{65} +2.05678 q^{67} +1.16645 q^{68} +1.48390 q^{70} -7.83463 q^{71} +13.8102 q^{73} +5.44758 q^{74} -3.74764 q^{76} +3.27337 q^{77} -9.56891 q^{79} +1.77954 q^{80} -1.45280 q^{82} -12.8937 q^{83} +2.07574 q^{85} +3.91817 q^{86} +3.92552 q^{88} +2.07244 q^{89} +3.82275 q^{91} -5.02025 q^{94} -6.66907 q^{95} -12.5515 q^{97} -6.30466 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 10 q^{2} + 10 q^{4} + 10 q^{5} - 2 q^{7} + 10 q^{8} + 10 q^{10} + 12 q^{11} - 2 q^{14} + 10 q^{16} + 24 q^{17} + 8 q^{19} + 10 q^{20} + 12 q^{22} + 8 q^{25} - 2 q^{28} + 4 q^{29} + 18 q^{31} + 10 q^{32}+ \cdots + 36 q^{98}+O(q^{100}) \) 10 * q + 10 * q^2 + 10 * q^4 + 10 * q^5 - 2 * q^7 + 10 * q^8 + 10 * q^10 + 12 * q^11 - 2 * q^14 + 10 * q^16 + 24 * q^17 + 8 * q^19 + 10 * q^20 + 12 * q^22 + 8 * q^25 - 2 * q^28 + 4 * q^29 + 18 * q^31 + 10 * q^32 + 24 * q^34 - 24 * q^35 + 12 * q^37 + 8 * q^38 + 10 * q^40 + 28 * q^41 + 8 * q^43 + 12 * q^44 - 16 * q^47 + 36 * q^49 + 8 * q^50 + 34 * q^53 + 30 * q^55 - 2 * q^56 + 4 * q^58 - 22 * q^59 + 30 * q^61 + 18 * q^62 + 10 * q^64 + 36 * q^65 + 18 * q^67 + 24 * q^68 - 24 * q^70 - 28 * q^71 - 20 * q^73 + 12 * q^74 + 8 * q^76 - 20 * q^77 + 2 * q^79 + 10 * q^80 + 28 * q^82 + 44 * q^83 + 16 * q^85 + 8 * q^86 + 12 * q^88 + 44 * q^89 + 22 * q^91 - 16 * q^94 - 10 * q^95 + 18 * q^97 + 36 * q^98

Introduction

L-functions

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