LMFDB - Embedded newform 9522.2.a.cb.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 = [6,-6,0,6,0,0,0,-6,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,18,0,0,0,-24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(29)] == traces)

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

Self dual:	yes
Analytic conductor:	$76.0335528047$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.197448192.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 18x^{4} - 6x^{3} + 48x^{2} - 23 $ x^6 - 18x^4 - 6x^3 + 48*x^2 - 23
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.25487$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -3.77465 q^{5} -0.643332 q^{7} -1.00000 q^{8} +3.77465 q^{10} +5.83219 q^{11} +5.33816 q^{13} +0.643332 q^{14} +1.00000 q^{16} +2.05755 q^{17} -5.18886 q^{19} -3.77465 q^{20} -5.83219 q^{22} +9.24797 q^{25} -5.33816 q^{26} -0.643332 q^{28} -8.42835 q^{29} -8.24797 q^{31} -1.00000 q^{32} -2.05755 q^{34} +2.42835 q^{35} +0.302889 q^{37} +5.18886 q^{38} +3.77465 q^{40} -7.58612 q^{41} +9.90973 q^{43} +5.83219 q^{44} -3.09019 q^{47} -6.58612 q^{49} -9.24797 q^{50} +5.33816 q^{52} -0.340442 q^{53} -22.0145 q^{55} +0.643332 q^{56} +8.42835 q^{58} -5.81962 q^{59} +10.6806 q^{61} +8.24797 q^{62} +1.00000 q^{64} -20.1497 q^{65} +6.12485 q^{67} +2.05755 q^{68} -2.42835 q^{70} -3.09019 q^{71} +11.3382 q^{73} -0.302889 q^{74} -5.18886 q^{76} -3.75203 q^{77} +11.0211 q^{79} -3.77465 q^{80} +7.58612 q^{82} -10.2024 q^{83} -7.76651 q^{85} -9.90973 q^{86} -5.83219 q^{88} +9.60684 q^{89} -3.43421 q^{91} +3.09019 q^{94} +19.5861 q^{95} +2.70088 q^{97} +6.58612 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{2} + 6 q^{4} - 6 q^{8} + 6 q^{16} + 18 q^{25} - 24 q^{29} - 12 q^{31} - 6 q^{32} - 12 q^{35} + 24 q^{41} - 24 q^{47} + 30 q^{49} - 18 q^{50} - 36 q^{55} + 24 q^{58} - 24 q^{59} + 12 q^{62} + 6 q^{64}+ \cdots - 30 q^{98}+O(q^{100}) $ 6 * q - 6 * q^2 + 6 * q^4 - 6 * q^8 + 6 * q^16 + 18 * q^25 - 24 * q^29 - 12 * q^31 - 6 * q^32 - 12 * q^35 + 24 * q^41 - 24 * q^47 + 30 * q^49 - 18 * q^50 - 36 * q^55 + 24 * q^58 - 24 * q^59 + 12 * q^62 + 6 * q^64 + 12 * q^70 - 24 * q^71 + 36 * q^73 - 60 * q^77 - 24 * q^82 + 12 * q^85 + 24 * q^94 + 48 * q^95 - 30 * 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
$5$	−3.77465	−1.68807	−0.844037	−	0.536285i	$-0.819827\pi$
$5$	−3.77465	−1.68807	−0.844037	+	0.536285i	$0.819827\pi$
$6$	0	0
$7$	−0.643332	−0.243156	−0.121578	−	0.992582i	$-0.538796\pi$
$7$	−0.643332	−0.243156	−0.121578	+	0.992582i	$0.538796\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	3.77465	1.19365
$11$	5.83219	1.75847	0.879236	−	0.476386i	$-0.158054\pi$
$11$	5.83219	1.75847	0.879236	+	0.476386i	$0.158054\pi$
$12$	0	0
$13$	5.33816	1.48054	0.740269	−	0.672310i	$-0.234698\pi$
$13$	5.33816	1.48054	0.740269	+	0.672310i	$0.234698\pi$
$14$	0.643332	0.171938
$15$	0	0
$16$	1.00000	0.250000
$17$	2.05755	0.499028	0.249514	−	0.968371i	$-0.419729\pi$
$17$	2.05755	0.499028	0.249514	+	0.968371i	$0.419729\pi$
$18$	0	0
$19$	−5.18886	−1.19041	−0.595203	−	0.803575i	$-0.702928\pi$
$19$	−5.18886	−1.19041	−0.595203	+	0.803575i	$0.702928\pi$
$20$	−3.77465	−0.844037
$21$	0	0
$22$	−5.83219	−1.24343
$23$	0	0
$23$	0	0
$24$	0	0
$25$	9.24797	1.84959
$26$	−5.33816	−1.04690
$27$	0	0
$28$	−0.643332	−0.121578
$29$	−8.42835	−1.56511	−0.782553	−	0.622584i	$-0.786083\pi$
$29$	−8.42835	−1.56511	−0.782553	+	0.622584i	$0.786083\pi$
$30$	0	0
$31$	−8.24797	−1.48138	−0.740689	−	0.671848i	$-0.765501\pi$
$31$	−8.24797	−1.48138	−0.740689	+	0.671848i	$0.765501\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−2.05755	−0.352866
$35$	2.42835	0.410466
$36$	0	0
$37$	0.302889	0.0497947	0.0248973	−	0.999690i	$-0.492074\pi$
$37$	0.302889	0.0497947	0.0248973	+	0.999690i	$0.492074\pi$
$38$	5.18886	0.841744
$39$	0	0
$40$	3.77465	0.596824
$41$	−7.58612	−1.18475	−0.592377	−	0.805661i	$-0.701810\pi$
$41$	−7.58612	−1.18475	−0.592377	+	0.805661i	$0.701810\pi$
$42$	0	0
$43$	9.90973	1.51122	0.755610	−	0.655022i	$-0.227341\pi$
$43$	9.90973	1.51122	0.755610	+	0.655022i	$0.227341\pi$
$44$	5.83219	0.879236
$45$	0	0
$46$	0	0
$47$	−3.09019	−0.450751	−0.225375	−	0.974272i	$-0.572361\pi$
$47$	−3.09019	−0.450751	−0.225375	+	0.974272i	$0.572361\pi$
$48$	0	0
$49$	−6.58612	−0.940875
$50$	−9.24797	−1.30786
$51$	0	0
$52$	5.33816	0.740269
$53$	−0.340442	−0.0467634	−0.0233817	−	0.999727i	$-0.507443\pi$
$53$	−0.340442	−0.0467634	−0.0233817	+	0.999727i	$0.507443\pi$
$54$	0	0
$55$	−22.0145	−2.96843
$56$	0.643332	0.0859688
$57$	0	0
$58$	8.42835	1.10670
$59$	−5.81962	−0.757650	−0.378825	−	0.925468i	$-0.623672\pi$
$59$	−5.81962	−0.757650	−0.378825	+	0.925468i	$0.623672\pi$
$60$	0	0
$61$	10.6806	1.36751	0.683756	−	0.729711i	$-0.260345\pi$
$61$	10.6806	1.36751	0.683756	+	0.729711i	$0.260345\pi$
$62$	8.24797	1.04749
$63$	0	0
$64$	1.00000	0.125000
$65$	−20.1497	−2.49926
$66$	0	0
$67$	6.12485	0.748269	0.374135	−	0.927374i	$-0.377940\pi$
$67$	6.12485	0.748269	0.374135	+	0.927374i	$0.377940\pi$
$68$	2.05755	0.249514
$69$	0	0
$70$	−2.42835	−0.290243
$71$	−3.09019	−0.366738	−0.183369	−	0.983044i	$-0.558700\pi$
$71$	−3.09019	−0.366738	−0.183369	+	0.983044i	$0.558700\pi$
$72$	0	0
$73$	11.3382	1.32703	0.663516	−	0.748163i	$-0.269064\pi$
$73$	11.3382	1.32703	0.663516	+	0.748163i	$0.269064\pi$
$74$	−0.302889	−0.0352102
$75$	0	0
$76$	−5.18886	−0.595203
$77$	−3.75203	−0.427584
$78$	0	0
$79$	11.0211	1.23997	0.619983	−	0.784615i	$-0.287140\pi$
$79$	11.0211	1.23997	0.619983	+	0.784615i	$0.287140\pi$
$80$	−3.77465	−0.422018
$81$	0	0
$82$	7.58612	0.837747
$83$	−10.2024	−1.11986	−0.559929	−	0.828541i	$-0.689171\pi$
$83$	−10.2024	−1.11986	−0.559929	+	0.828541i	$0.689171\pi$
$84$	0	0
$85$	−7.76651	−0.842396
$86$	−9.90973	−1.06859
$87$	0	0
$88$	−5.83219	−0.621714
$89$	9.60684	1.01832	0.509162	−	0.860671i	$-0.329956\pi$
$89$	9.60684	1.01832	0.509162	+	0.860671i	$0.329956\pi$
$90$	0	0
$91$	−3.43421	−0.360003
$92$	0	0
$93$	0	0
$94$	3.09019	0.318729
$95$	19.5861	2.00949
$96$	0	0
$97$	2.70088	0.274232	0.137116	−	0.990555i	$-0.456217\pi$
$97$	2.70088	0.274232	0.137116	+	0.990555i	$0.456217\pi$
$98$	6.58612	0.665299
$99$	0	0
$100$	9.24797	0.924797
$101$	−11.3382	−1.12819	−0.564094	−	0.825710i	$-0.690775\pi$
$101$	−11.3382	−1.12819	−0.564094	+	0.825710i	$0.690775\pi$
$102$	0	0
$103$	−0.292654	−0.0288361	−0.0144180	−	0.999896i	$-0.504590\pi$
$103$	−0.292654	−0.0288361	−0.0144180	+	0.999896i	$0.504590\pi$
$104$	−5.33816	−0.523449
$105$	0	0
$106$	0.340442	0.0330667
$107$	−4.89621	−0.473334	−0.236667	−	0.971591i	$-0.576055\pi$
$107$	−4.89621	−0.473334	−0.236667	+	0.971591i	$0.576055\pi$
$108$	0	0
$109$	−4.06730	−0.389577	−0.194788	−	0.980845i	$-0.562402\pi$
$109$	−4.06730	−0.389577	−0.194788	+	0.980845i	$0.562402\pi$
$110$	22.0145	2.09900
$111$	0	0
$112$	−0.643332	−0.0607891
$113$	−5.49175	−0.516620	−0.258310	−	0.966062i	$-0.583166\pi$
$113$	−5.49175	−0.516620	−0.258310	+	0.966062i	$0.583166\pi$
$114$	0	0
$115$	0	0
$116$	−8.42835	−0.782553
$117$	0	0
$118$	5.81962	0.535739
$119$	−1.32368	−0.121342
$120$	0	0
$121$	23.0145	2.09222
$122$	−10.6806	−0.966977
$123$	0	0
$124$	−8.24797	−0.740689
$125$	−16.0346	−1.43418
$126$	0	0
$127$	2.72942	0.242197	0.121099	−	0.992640i	$-0.461358\pi$
$127$	2.72942	0.242197	0.121099	+	0.992640i	$0.461358\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	20.1497	1.76724
$131$	−8.24797	−0.720628	−0.360314	−	0.932831i	$-0.617330\pi$
$131$	−8.24797	−0.720628	−0.360314	+	0.932831i	$0.617330\pi$
$132$	0	0
$133$	3.33816	0.289455
$134$	−6.12485	−0.529106
$135$	0	0
$136$	−2.05755	−0.176433
$137$	6.42774	0.549159	0.274579	−	0.961564i	$-0.411461\pi$
$137$	6.42774	0.549159	0.274579	+	0.961564i	$0.411461\pi$
$138$	0	0
$139$	−12.4959	−1.05989	−0.529946	−	0.848032i	$-0.677788\pi$
$139$	−12.4959	−1.05989	−0.529946	+	0.848032i	$0.677788\pi$
$140$	2.42835	0.205233
$141$	0	0
$142$	3.09019	0.259323
$143$	31.1332	2.60349
$144$	0	0
$145$	31.8141	2.64201
$146$	−11.3382	−0.938353
$147$	0	0
$148$	0.302889	0.0248973
$149$	8.82572	0.723031	0.361516	−	0.932366i	$-0.382259\pi$
$149$	8.82572	0.723031	0.361516	+	0.932366i	$0.382259\pi$
$150$	0	0
$151$	−2.48146	−0.201938	−0.100969	−	0.994890i	$-0.532194\pi$
$151$	−2.48146	−0.201938	−0.100969	+	0.994890i	$0.532194\pi$
$152$	5.18886	0.420872
$153$	0	0
$154$	3.75203	0.302347
$155$	31.1332	2.50068
$156$	0	0
$157$	−20.7077	−1.65265	−0.826325	−	0.563193i	$-0.809573\pi$
$157$	−20.7077	−1.65265	−0.826325	+	0.563193i	$0.809573\pi$
$158$	−11.0211	−0.876788
$159$	0	0
$160$	3.77465	0.298412
$161$	0	0
$162$	0	0
$163$	−5.15777	−0.403988	−0.201994	−	0.979387i	$-0.564742\pi$
$163$	−5.15777	−0.403988	−0.201994	+	0.979387i	$0.564742\pi$
$164$	−7.58612	−0.592377
$165$	0	0
$166$	10.2024	0.791859
$167$	−20.9098	−1.61805	−0.809025	−	0.587775i	$-0.800004\pi$
$167$	−20.9098	−1.61805	−0.809025	+	0.587775i	$0.800004\pi$
$168$	0	0
$169$	15.4959	1.19199
$170$	7.76651	0.595664
$171$	0	0
$172$	9.90973	0.755610
$173$	−8.42835	−0.640796	−0.320398	−	0.947283i	$-0.603817\pi$
$173$	−8.42835	−0.640796	−0.320398	+	0.947283i	$0.603817\pi$
$174$	0	0
$175$	−5.94951	−0.449741
$176$	5.83219	0.439618
$177$	0	0
$178$	−9.60684	−0.720063
$179$	20.2480	1.51340	0.756702	−	0.653760i	$-0.226809\pi$
$179$	20.2480	1.51340	0.756702	+	0.653760i	$0.226809\pi$
$180$	0	0
$181$	2.87622	0.213787	0.106894	−	0.994270i	$-0.465910\pi$
$181$	2.87622	0.213787	0.106894	+	0.994270i	$0.465910\pi$
$182$	3.43421	0.254560
$183$	0	0
$184$	0	0
$185$	−1.14330	−0.0840571
$186$	0	0
$187$	12.0000	0.877527
$188$	−3.09019	−0.225375
$189$	0	0
$190$	−19.5861	−1.42093
$191$	−5.30618	−0.383941	−0.191971	−	0.981401i	$-0.561488\pi$
$191$	−5.30618	−0.383941	−0.191971	+	0.981401i	$0.561488\pi$
$192$	0	0
$193$	22.2624	1.60249	0.801243	−	0.598339i	$-0.204173\pi$
$193$	22.2624	1.60249	0.801243	+	0.598339i	$0.204173\pi$
$194$	−2.70088	−0.193912
$195$	0	0
$196$	−6.58612	−0.470437
$197$	16.3155	1.16243	0.581217	−	0.813748i	$-0.302577\pi$
$197$	16.3155	1.16243	0.581217	+	0.813748i	$0.302577\pi$
$198$	0	0
$199$	5.01352	0.355399	0.177700	−	0.984085i	$-0.443134\pi$
$199$	5.01352	0.355399	0.177700	+	0.984085i	$0.443134\pi$
$200$	−9.24797	−0.653930
$201$	0	0
$202$	11.3382	0.797750
$203$	5.42222	0.380565
$204$	0	0
$205$	28.6349	1.99995
$206$	0.292654	0.0203902
$207$	0	0
$208$	5.33816	0.370135
$209$	−30.2624	−2.09330
$210$	0	0
$211$	17.1578	1.18119	0.590595	−	0.806968i	$-0.298893\pi$
$211$	17.1578	1.18119	0.590595	+	0.806968i	$0.298893\pi$
$212$	−0.340442	−0.0233817
$213$	0	0
$214$	4.89621	0.334698
$215$	−37.4057	−2.55105
$216$	0	0
$217$	5.30618	0.360207
$218$	4.06730	0.275472
$219$	0	0
$220$	−22.0145	−1.48422
$221$	10.9835	0.738830
$222$	0	0
$223$	−2.48146	−0.166171	−0.0830854	−	0.996542i	$-0.526477\pi$
$223$	−2.48146	−0.166171	−0.0830854	+	0.996542i	$0.526477\pi$
$224$	0.643332	0.0429844
$225$	0	0
$226$	5.49175	0.365306
$227$	8.33041	0.552909	0.276454	−	0.961027i	$-0.410841\pi$
$227$	8.33041	0.552909	0.276454	+	0.961027i	$0.410841\pi$
$228$	0	0
$229$	15.7317	1.03958	0.519790	−	0.854294i	$-0.326010\pi$
$229$	15.7317	1.03958	0.519790	+	0.854294i	$0.326010\pi$
$230$	0	0
$231$	0	0
$232$	8.42835	0.553348
$233$	27.5330	1.80375	0.901874	−	0.431999i	$-0.142192\pi$
$233$	27.5330	1.80375	0.901874	+	0.431999i	$0.142192\pi$
$234$	0	0
$235$	11.6644	0.760901
$236$	−5.81962	−0.378825
$237$	0	0
$238$	1.32368	0.0858017
$239$	−6.18038	−0.399776	−0.199888	−	0.979819i	$-0.564058\pi$
$239$	−6.18038	−0.399776	−0.199888	+	0.979819i	$0.564058\pi$
$240$	0	0
$241$	−7.32617	−0.471920	−0.235960	−	0.971763i	$-0.575823\pi$
$241$	−7.32617	−0.471920	−0.235960	+	0.971763i	$0.575823\pi$
$242$	−23.0145	−1.47943
$243$	0	0
$244$	10.6806	0.683756
$245$	24.8603	1.58827
$246$	0	0
$247$	−27.6990	−1.76244
$248$	8.24797	0.523746
$249$	0	0
$250$	16.0346	1.01412
$251$	−17.4966	−1.10437	−0.552187	−	0.833720i	$-0.686207\pi$
$251$	−17.4966	−1.10437	−0.552187	+	0.833720i	$0.686207\pi$
$252$	0	0
$253$	0	0
$254$	−2.72942	−0.171259
$255$	0	0
$256$	1.00000	0.0625000
$257$	−15.0902	−0.941300	−0.470650	−	0.882320i	$-0.655981\pi$
$257$	−15.0902	−0.941300	−0.470650	+	0.882320i	$0.655981\pi$
$258$	0	0
$259$	−0.194858	−0.0121079
$260$	−20.1497	−1.24963
$261$	0	0
$262$	8.24797	0.509561
$263$	30.1972	1.86204	0.931019	−	0.364971i	$-0.118921\pi$
$263$	30.1972	1.86204	0.931019	+	0.364971i	$0.118921\pi$
$264$	0	0
$265$	1.28505	0.0789400
$266$	−3.33816	−0.204676
$267$	0	0
$268$	6.12485	0.374135
$269$	−0.180384	−0.0109982	−0.00549909	−	0.999985i	$-0.501750\pi$
$269$	−0.180384	−0.0109982	−0.00549909	+	0.999985i	$0.501750\pi$
$270$	0	0
$271$	−31.5861	−1.91872	−0.959360	−	0.282184i	$-0.908941\pi$
$271$	−31.5861	−1.91872	−0.959360	+	0.282184i	$0.908941\pi$
$272$	2.05755	0.124757
$273$	0	0
$274$	−6.42774	−0.388314
$275$	53.9359	3.25246
$276$	0	0
$277$	−3.75203	−0.225438	−0.112719	−	0.993627i	$-0.535956\pi$
$277$	−3.75203	−0.225438	−0.112719	+	0.993627i	$0.535956\pi$
$278$	12.4959	0.749456
$279$	0	0
$280$	−2.42835	−0.145122
$281$	−7.73487	−0.461424	−0.230712	−	0.973022i	$-0.574105\pi$
$281$	−7.73487	−0.461424	−0.230712	+	0.973022i	$0.574105\pi$
$282$	0	0
$283$	−27.5816	−1.63956	−0.819779	−	0.572679i	$-0.805904\pi$
$283$	−27.5816	−1.63956	−0.819779	+	0.572679i	$0.805904\pi$
$284$	−3.09019	−0.183369
$285$	0	0
$286$	−31.1332	−1.84094
$287$	4.88039	0.288080
$288$	0	0
$289$	−12.7665	−0.750971
$290$	−31.8141	−1.86819
$291$	0	0
$292$	11.3382	0.663516
$293$	−16.1199	−0.941736	−0.470868	−	0.882204i	$-0.656059\pi$
$293$	−16.1199	−0.941736	−0.470868	+	0.882204i	$0.656059\pi$
$294$	0	0
$295$	21.9670	1.27897
$296$	−0.302889	−0.0176051
$297$	0	0
$298$	−8.82572	−0.511260
$299$	0	0
$300$	0	0
$301$	−6.37524	−0.367463
$302$	2.48146	0.142792
$303$	0	0
$304$	−5.18886	−0.297602
$305$	−40.3155	−2.30846
$306$	0	0
$307$	−6.48146	−0.369916	−0.184958	−	0.982746i	$-0.559215\pi$
$307$	−6.48146	−0.369916	−0.184958	+	0.982746i	$0.559215\pi$
$308$	−3.75203	−0.213792
$309$	0	0
$310$	−31.1332	−1.76825
$311$	15.1722	0.860339	0.430170	−	0.902748i	$-0.358454\pi$
$311$	15.1722	0.860339	0.430170	+	0.902748i	$0.358454\pi$
$312$	0	0
$313$	−32.2923	−1.82527	−0.912634	−	0.408778i	$-0.865955\pi$
$313$	−32.2923	−1.82527	−0.912634	+	0.408778i	$0.865955\pi$
$314$	20.7077	1.16860
$315$	0	0
$316$	11.0211	0.619983
$317$	3.57165	0.200604	0.100302	−	0.994957i	$-0.468019\pi$
$317$	3.57165	0.200604	0.100302	+	0.994957i	$0.468019\pi$
$318$	0	0
$319$	−49.1558	−2.75219
$320$	−3.77465	−0.211009
$321$	0	0
$322$	0	0
$323$	−10.6763	−0.594046
$324$	0	0
$325$	49.3671	2.73839
$326$	5.15777	0.285663
$327$	0	0
$328$	7.58612	0.418874
$329$	1.98802	0.109603
$330$	0	0
$331$	−6.84223	−0.376083	−0.188041	−	0.982161i	$-0.560214\pi$
$331$	−6.84223	−0.376083	−0.188041	+	0.982161i	$0.560214\pi$
$332$	−10.2024	−0.559929
$333$	0	0
$334$	20.9098	1.14413
$335$	−23.1191	−1.26313
$336$	0	0
$337$	−32.8025	−1.78687	−0.893433	−	0.449197i	$-0.851710\pi$
$337$	−32.8025	−1.78687	−0.893433	+	0.449197i	$0.851710\pi$
$338$	−15.4959	−0.842868
$339$	0	0
$340$	−7.76651	−0.421198
$341$	−48.1037	−2.60496
$342$	0	0
$343$	8.74038	0.471936
$344$	−9.90973	−0.534297
$345$	0	0
$346$	8.42835	0.453111
$347$	29.9614	1.60841	0.804205	−	0.594352i	$-0.202591\pi$
$347$	29.9614	1.60841	0.804205	+	0.594352i	$0.202591\pi$
$348$	0	0
$349$	−30.7584	−1.64646	−0.823229	−	0.567709i	$-0.807830\pi$
$349$	−30.7584	−1.64646	−0.823229	+	0.567709i	$0.807830\pi$
$350$	5.94951	0.318015
$351$	0	0
$352$	−5.83219	−0.310857
$353$	12.2624	0.652664	0.326332	−	0.945255i	$-0.394187\pi$
$353$	12.2624	0.652664	0.326332	+	0.945255i	$0.394187\pi$
$354$	0	0
$355$	11.6644	0.619081
$356$	9.60684	0.509162
$357$	0	0
$358$	−20.2480	−1.07014
$359$	−7.29419	−0.384973	−0.192486	−	0.981300i	$-0.561655\pi$
$359$	−7.29419	−0.384973	−0.192486	+	0.981300i	$0.561655\pi$
$360$	0	0
$361$	7.92428	0.417068
$362$	−2.87622	−0.151171
$363$	0	0
$364$	−3.43421	−0.180001
$365$	−42.7976	−2.24013
$366$	0	0
$367$	12.8179	0.669090	0.334545	−	0.942380i	$-0.391417\pi$
$367$	12.8179	0.669090	0.334545	+	0.942380i	$0.391417\pi$
$368$	0	0
$369$	0	0
$370$	1.14330	0.0594373
$371$	0.219017	0.0113708
$372$	0	0
$373$	−20.1975	−1.04578	−0.522892	−	0.852399i	$-0.675147\pi$
$373$	−20.1975	−1.04578	−0.522892	+	0.852399i	$0.675147\pi$
$374$	−12.0000	−0.620505
$375$	0	0
$376$	3.09019	0.159364
$377$	−44.9919	−2.31720
$378$	0	0
$379$	−29.8043	−1.53094	−0.765472	−	0.643469i	$-0.777494\pi$
$379$	−29.8043	−1.53094	−0.765472	+	0.643469i	$0.777494\pi$
$380$	19.5861	1.00475
$381$	0	0
$382$	5.30618	0.271488
$383$	12.6004	0.643849	0.321924	−	0.946765i	$-0.395670\pi$
$383$	12.6004	0.643849	0.321924	+	0.946765i	$0.395670\pi$
$384$	0	0
$385$	14.1626	0.721793
$386$	−22.2624	−1.13313
$387$	0	0
$388$	2.70088	0.137116
$389$	14.5030	0.735334	0.367667	−	0.929958i	$-0.380157\pi$
$389$	14.5030	0.735334	0.367667	+	0.929958i	$0.380157\pi$
$390$	0	0
$391$	0	0
$392$	6.58612	0.332650
$393$	0	0
$394$	−16.3155	−0.821965
$395$	−41.6006	−2.09315
$396$	0	0
$397$	−30.9919	−1.55544	−0.777719	−	0.628613i	$-0.783623\pi$
$397$	−30.9919	−1.55544	−0.777719	+	0.628613i	$0.783623\pi$
$398$	−5.01352	−0.251305
$399$	0	0
$400$	9.24797	0.462398
$401$	−21.2712	−1.06223	−0.531117	−	0.847298i	$-0.678228\pi$
$401$	−21.2712	−1.06223	−0.531117	+	0.847298i	$0.678228\pi$
$402$	0	0
$403$	−44.0289	−2.19324
$404$	−11.3382	−0.564094
$405$	0	0
$406$	−5.42222	−0.269100
$407$	1.76651	0.0875626
$408$	0	0
$409$	10.7294	0.530536	0.265268	−	0.964175i	$-0.414540\pi$
$409$	10.7294	0.530536	0.265268	+	0.964175i	$0.414540\pi$
$410$	−28.6349	−1.41418
$411$	0	0
$412$	−0.292654	−0.0144180
$413$	3.74394	0.184227
$414$	0	0
$415$	38.5104	1.89040
$416$	−5.33816	−0.261725
$417$	0	0
$418$	30.2624	1.48018
$419$	30.0970	1.47033	0.735166	−	0.677887i	$-0.237104\pi$
$419$	30.0970	1.47033	0.735166	+	0.677887i	$0.237104\pi$
$420$	0	0
$421$	−11.7122	−0.570816	−0.285408	−	0.958406i	$-0.592129\pi$
$421$	−11.7122	−0.570816	−0.285408	+	0.958406i	$0.592129\pi$
$422$	−17.1578	−0.835227
$423$	0	0
$424$	0.340442	0.0165333
$425$	19.0281	0.922999
$426$	0	0
$427$	−6.87117	−0.332519
$428$	−4.89621	−0.236667
$429$	0	0
$430$	37.4057	1.80387
$431$	−35.5034	−1.71014	−0.855068	−	0.518515i	$-0.826485\pi$
$431$	−35.5034	−1.71014	−0.855068	+	0.518515i	$0.826485\pi$
$432$	0	0
$433$	−3.89196	−0.187036	−0.0935179	−	0.995618i	$-0.529811\pi$
$433$	−3.89196	−0.187036	−0.0935179	+	0.995618i	$0.529811\pi$
$434$	−5.30618	−0.254705
$435$	0	0
$436$	−4.06730	−0.194788
$437$	0	0
$438$	0	0
$439$	11.5330	0.550441	0.275220	−	0.961381i	$-0.411249\pi$
$439$	11.5330	0.550441	0.275220	+	0.961381i	$0.411249\pi$
$440$	22.0145	1.04950
$441$	0	0
$442$	−10.9835	−0.522432
$443$	−6.18038	−0.293639	−0.146819	−	0.989163i	$-0.546904\pi$
$443$	−6.18038	−0.293639	−0.146819	+	0.989163i	$0.546904\pi$
$444$	0	0
$445$	−36.2624	−1.71900
$446$	2.48146	0.117500
$447$	0	0
$448$	−0.643332	−0.0303946
$449$	2.72942	0.128810	0.0644048	−	0.997924i	$-0.479485\pi$
$449$	2.72942	0.128810	0.0644048	+	0.997924i	$0.479485\pi$
$450$	0	0
$451$	−44.2437	−2.08336
$452$	−5.49175	−0.258310
$453$	0	0
$454$	−8.33041	−0.390966
$455$	12.9629	0.607711
$456$	0	0
$457$	−9.21861	−0.431228	−0.215614	−	0.976479i	$-0.569175\pi$
$457$	−9.21861	−0.431228	−0.215614	+	0.976479i	$0.569175\pi$
$458$	−15.7317	−0.735093
$459$	0	0
$460$	0	0
$461$	4.67632	0.217798	0.108899	−	0.994053i	$-0.465268\pi$
$461$	4.67632	0.217798	0.108899	+	0.994053i	$0.465268\pi$
$462$	0	0
$463$	−8.00000	−0.371792	−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000	−0.371792	−0.185896	+	0.982569i	$0.559519\pi$
$464$	−8.42835	−0.391276
$465$	0	0
$466$	−27.5330	−1.27544
$467$	−3.33397	−0.154278	−0.0771389	−	0.997020i	$-0.524579\pi$
$467$	−3.33397	−0.154278	−0.0771389	+	0.997020i	$0.524579\pi$
$468$	0	0
$469$	−3.94031	−0.181946
$470$	−11.6644	−0.538038
$471$	0	0
$472$	5.81962	0.267870
$473$	57.7955	2.65744
$474$	0	0
$475$	−47.9864	−2.20177
$476$	−1.32368	−0.0606709
$477$	0	0
$478$	6.18038	0.282684
$479$	−4.37019	−0.199679	−0.0998396	−	0.995004i	$-0.531833\pi$
$479$	−4.37019	−0.199679	−0.0998396	+	0.995004i	$0.531833\pi$
$480$	0	0
$481$	1.61687	0.0737229
$482$	7.32617	0.333698
$483$	0	0
$484$	23.0145	1.04611
$485$	−10.1949	−0.462925
$486$	0	0
$487$	−16.0000	−0.725029	−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000	−0.725029	−0.362515	+	0.931978i	$0.618082\pi$
$488$	−10.6806	−0.483489
$489$	0	0
$490$	−24.8603	−1.12307
$491$	−31.2850	−1.41187	−0.705937	−	0.708274i	$-0.749474\pi$
$491$	−31.2850	−1.41187	−0.705937	+	0.708274i	$0.749474\pi$
$492$	0	0
$493$	−17.3417	−0.781031
$494$	27.6990	1.24624
$495$	0	0
$496$	−8.24797	−0.370345
$497$	1.98802	0.0891748
$498$	0	0
$499$	26.5104	1.18677	0.593384	−	0.804919i	$-0.297791\pi$
$499$	26.5104	1.18677	0.593384	+	0.804919i	$0.297791\pi$
$500$	−16.0346	−0.717088
$501$	0	0
$502$	17.4966	0.780911
$503$	−16.0346	−0.714946	−0.357473	−	0.933923i	$-0.616362\pi$
$503$	−16.0346	−0.714946	−0.357473	+	0.933923i	$0.616362\pi$
$504$	0	0
$505$	42.7976	1.90447
$506$	0	0
$507$	0	0
$508$	2.72942	0.121099
$509$	3.93242	0.174301	0.0871507	−	0.996195i	$-0.472224\pi$
$509$	3.93242	0.174301	0.0871507	+	0.996195i	$0.472224\pi$
$510$	0	0
$511$	−7.29419	−0.322676
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	15.0902	0.665600
$515$	1.10467	0.0486774
$516$	0	0
$517$	−18.0226	−0.792633
$518$	0.194858	0.00856158
$519$	0	0
$520$	20.1497	0.883621
$521$	15.1681	0.664527	0.332264	−	0.943187i	$-0.392188\pi$
$521$	15.1681	0.664527	0.332264	+	0.943187i	$0.392188\pi$
$522$	0	0
$523$	17.7892	0.777869	0.388934	−	0.921265i	$-0.372843\pi$
$523$	17.7892	0.777869	0.388934	+	0.921265i	$0.372843\pi$
$524$	−8.24797	−0.360314
$525$	0	0
$526$	−30.1972	−1.31666
$527$	−16.9706	−0.739249
$528$	0	0
$529$	0	0
$530$	−1.28505	−0.0558190
$531$	0	0
$532$	3.33816	0.144728
$533$	−40.4959	−1.75407
$534$	0	0
$535$	18.4815	0.799023
$536$	−6.12485	−0.264553
$537$	0	0
$538$	0.180384	0.00777689
$539$	−38.4115	−1.65450
$540$	0	0
$541$	−10.7294	−0.461294	−0.230647	−	0.973037i	$-0.574084\pi$
$541$	−10.7294	−0.461294	−0.230647	+	0.973037i	$0.574084\pi$
$542$	31.5861	1.35674
$543$	0	0
$544$	−2.05755	−0.0882165
$545$	15.3526	0.657635
$546$	0	0
$547$	4.19486	0.179359	0.0896796	−	0.995971i	$-0.471416\pi$
$547$	4.19486	0.179359	0.0896796	+	0.995971i	$0.471416\pi$
$548$	6.42774	0.274579
$549$	0	0
$550$	−53.9359	−2.29984
$551$	43.7335	1.86311
$552$	0	0
$553$	−7.09019	−0.301506
$554$	3.75203	0.159409
$555$	0	0
$556$	−12.4959	−0.529946
$557$	−25.9123	−1.09794	−0.548970	−	0.835842i	$-0.684980\pi$
$557$	−25.9123	−1.09794	−0.548970	+	0.835842i	$0.684980\pi$
$558$	0	0
$559$	52.8997	2.23742
$560$	2.42835	0.102617
$561$	0	0
$562$	7.73487	0.326276
$563$	37.3911	1.57585	0.787924	−	0.615772i	$-0.211156\pi$
$563$	37.3911	1.57585	0.787924	+	0.615772i	$0.211156\pi$
$564$	0	0
$565$	20.7294	0.872093
$566$	27.5816	1.15934
$567$	0	0
$568$	3.09019	0.129662
$569$	18.7730	0.787005	0.393503	−	0.919323i	$-0.371263\pi$
$569$	18.7730	0.787005	0.393503	+	0.919323i	$0.371263\pi$
$570$	0	0
$571$	−7.17688	−0.300343	−0.150172	−	0.988660i	$-0.547983\pi$
$571$	−7.17688	−0.300343	−0.150172	+	0.988660i	$0.547983\pi$
$572$	31.1332	1.30174
$573$	0	0
$574$	−4.88039	−0.203704
$575$	0	0
$576$	0	0
$577$	23.2850	0.969369	0.484685	−	0.874689i	$-0.338934\pi$
$577$	23.2850	0.969369	0.484685	+	0.874689i	$0.338934\pi$
$578$	12.7665	0.531017
$579$	0	0
$580$	31.8141	1.32101
$581$	6.56352	0.272301
$582$	0	0
$583$	−1.98553	−0.0822321
$584$	−11.3382	−0.469176
$585$	0	0
$586$	16.1199	0.665908
$587$	−13.7068	−0.565741	−0.282870	−	0.959158i	$-0.591287\pi$
$587$	−13.7068	−0.565741	−0.282870	+	0.959158i	$0.591287\pi$
$588$	0	0
$589$	42.7976	1.76344
$590$	−21.9670	−0.904367
$591$	0	0
$592$	0.302889	0.0124487
$593$	2.12728	0.0873567	0.0436784	−	0.999046i	$-0.486092\pi$
$593$	2.12728	0.0873567	0.0436784	+	0.999046i	$0.486092\pi$
$594$	0	0
$595$	4.99644	0.204834
$596$	8.82572	0.361516
$597$	0	0
$598$	0	0
$599$	−37.8486	−1.54645	−0.773225	−	0.634131i	$-0.781358\pi$
$599$	−37.8486	−1.54645	−0.773225	+	0.634131i	$0.781358\pi$
$600$	0	0
$601$	1.21088	0.0493929	0.0246965	−	0.999695i	$-0.492138\pi$
$601$	1.21088	0.0493929	0.0246965	+	0.999695i	$0.492138\pi$
$602$	6.37524	0.259835
$603$	0	0
$604$	−2.48146	−0.100969
$605$	−86.8715	−3.53183
$606$	0	0
$607$	22.5104	0.913669	0.456835	−	0.889552i	$-0.348983\pi$
$607$	22.5104	0.913669	0.456835	+	0.889552i	$0.348983\pi$
$608$	5.18886	0.210436
$609$	0	0
$610$	40.3155	1.63233
$611$	−16.4959	−0.667354
$612$	0	0
$613$	−30.8098	−1.24440	−0.622198	−	0.782860i	$-0.713760\pi$
$613$	−30.8098	−1.24440	−0.622198	+	0.782860i	$0.713760\pi$
$614$	6.48146	0.261570
$615$	0	0
$616$	3.75203	0.151174
$617$	−26.3223	−1.05970	−0.529848	−	0.848093i	$-0.677751\pi$
$617$	−26.3223	−1.05970	−0.529848	+	0.848093i	$0.677751\pi$
$618$	0	0
$619$	38.1189	1.53213	0.766064	−	0.642764i	$-0.222213\pi$
$619$	38.1189	1.53213	0.766064	+	0.642764i	$0.222213\pi$
$620$	31.1332	1.25034
$621$	0	0
$622$	−15.1722	−0.608352
$623$	−6.18038	−0.247612
$624$	0	0
$625$	14.2850	0.571402
$626$	32.2923	1.29066
$627$	0	0
$628$	−20.7077	−0.826325
$629$	0.623208	0.0248489
$630$	0	0
$631$	−7.16107	−0.285078	−0.142539	−	0.989789i	$-0.545527\pi$
$631$	−7.16107	−0.285078	−0.142539	+	0.989789i	$0.545527\pi$
$632$	−11.0211	−0.438394
$633$	0	0
$634$	−3.57165	−0.141848
$635$	−10.3026	−0.408847
$636$	0	0
$637$	−35.1578	−1.39300
$638$	49.1558	1.94610
$639$	0	0
$640$	3.77465	0.149206
$641$	−23.3983	−0.924177	−0.462089	−	0.886834i	$-0.652900\pi$
$641$	−23.3983	−0.924177	−0.462089	+	0.886834i	$0.652900\pi$
$642$	0	0
$643$	−33.2385	−1.31080	−0.655399	−	0.755283i	$-0.727500\pi$
$643$	−33.2385	−1.31080	−0.655399	+	0.755283i	$0.727500\pi$
$644$	0	0
$645$	0	0
$646$	10.6763	0.420054
$647$	−48.8856	−1.92189	−0.960947	−	0.276734i	$-0.910748\pi$
$647$	−48.8856	−1.92189	−0.960947	+	0.276734i	$0.910748\pi$
$648$	0	0
$649$	−33.9411	−1.33231
$650$	−49.3671	−1.93634
$651$	0	0
$652$	−5.15777	−0.201994
$653$	21.6537	0.847375	0.423688	−	0.905808i	$-0.360735\pi$
$653$	21.6537	0.847375	0.423688	+	0.905808i	$0.360735\pi$
$654$	0	0
$655$	31.1332	1.21647
$656$	−7.58612	−0.296188
$657$	0	0
$658$	−1.98802	−0.0775010
$659$	27.1729	1.05851	0.529254	−	0.848464i	$-0.322472\pi$
$659$	27.1729	1.05851	0.529254	+	0.848464i	$0.322472\pi$
$660$	0	0
$661$	−35.8062	−1.39270	−0.696351	−	0.717702i	$-0.745194\pi$
$661$	−35.8062	−1.39270	−0.696351	+	0.717702i	$0.745194\pi$
$662$	6.84223	0.265931
$663$	0	0
$664$	10.2024	0.395929
$665$	−12.6004	−0.488621
$666$	0	0
$667$	0	0
$668$	−20.9098	−0.809025
$669$	0	0
$670$	23.1191	0.893170
$671$	62.2914	2.40473
$672$	0	0
$673$	−8.06758	−0.310982	−0.155491	−	0.987837i	$-0.549696\pi$
$673$	−8.06758	−0.310982	−0.155491	+	0.987837i	$0.549696\pi$
$674$	32.8025	1.26350
$675$	0	0
$676$	15.4959	0.595997
$677$	−6.95375	−0.267254	−0.133627	−	0.991032i	$-0.542662\pi$
$677$	−6.95375	−0.267254	−0.133627	+	0.991032i	$0.542662\pi$
$678$	0	0
$679$	−1.73756	−0.0666814
$680$	7.76651	0.297832
$681$	0	0
$682$	48.1037	1.84199
$683$	−20.6087	−0.788571	−0.394286	−	0.918988i	$-0.629008\pi$
$683$	−20.6087	−0.788571	−0.394286	+	0.918988i	$0.629008\pi$
$684$	0	0
$685$	−24.2624	−0.927020
$686$	−8.74038	−0.333709
$687$	0	0
$688$	9.90973	0.377805
$689$	−1.81734	−0.0692350
$690$	0	0
$691$	−5.51854	−0.209935	−0.104968	−	0.994476i	$-0.533474\pi$
$691$	−5.51854	−0.209935	−0.104968	+	0.994476i	$0.533474\pi$
$692$	−8.42835	−0.320398
$693$	0	0
$694$	−29.9614	−1.13732
$695$	47.1677	1.78917
$696$	0	0
$697$	−15.6088	−0.591225
$698$	30.7584	1.16422
$699$	0	0
$700$	−5.94951	−0.224870
$701$	−41.5211	−1.56823	−0.784116	−	0.620615i	$-0.786883\pi$
$701$	−41.5211	−1.56823	−0.784116	+	0.620615i	$0.786883\pi$
$702$	0	0
$703$	−1.57165	−0.0592759
$704$	5.83219	0.219809
$705$	0	0
$706$	−12.2624	−0.461503
$707$	7.29419	0.274326
$708$	0	0
$709$	39.6662	1.48970	0.744848	−	0.667234i	$-0.232522\pi$
$709$	39.6662	1.48970	0.744848	+	0.667234i	$0.232522\pi$
$710$	−11.6644	−0.437757
$711$	0	0
$712$	−9.60684	−0.360032
$713$	0	0
$714$	0	0
$715$	−117.517	−4.39488
$716$	20.2480	0.756702
$717$	0	0
$718$	7.29419	0.272217
$719$	13.8486	0.516464	0.258232	−	0.966083i	$-0.416860\pi$
$719$	13.8486	0.516464	0.258232	+	0.966083i	$0.416860\pi$
$720$	0	0
$721$	0.188274	0.00701168
$722$	−7.92428	−0.294911
$723$	0	0
$724$	2.87622	0.106894
$725$	−77.9451	−2.89481
$726$	0	0
$727$	−1.57932	−0.0585736	−0.0292868	−	0.999571i	$-0.509324\pi$
$727$	−1.57932	−0.0585736	−0.0292868	+	0.999571i	$0.509324\pi$
$728$	3.43421	0.127280
$729$	0	0
$730$	42.7976	1.58401
$731$	20.3897	0.754141
$732$	0	0
$733$	16.4126	0.606212	0.303106	−	0.952957i	$-0.401976\pi$
$733$	16.4126	0.606212	0.303106	+	0.952957i	$0.401976\pi$
$734$	−12.8179	−0.473118
$735$	0	0
$736$	0	0
$737$	35.7213	1.31581
$738$	0	0
$739$	−43.5330	−1.60139	−0.800694	−	0.599074i	$-0.795536\pi$
$739$	−43.5330	−1.60139	−0.800694	+	0.599074i	$0.795536\pi$
$740$	−1.14330	−0.0420285
$741$	0	0
$742$	−0.219017	−0.00804038
$743$	−45.2958	−1.66174	−0.830870	−	0.556466i	$-0.812157\pi$
$743$	−45.2958	−1.66174	−0.830870	+	0.556466i	$0.812157\pi$
$744$	0	0
$745$	−33.3140	−1.22053
$746$	20.1975	0.739481
$747$	0	0
$748$	12.0000	0.438763
$749$	3.14988	0.115094
$750$	0	0
$751$	25.5343	0.931761	0.465881	−	0.884848i	$-0.345738\pi$
$751$	25.5343	0.931761	0.465881	+	0.884848i	$0.345738\pi$
$752$	−3.09019	−0.112688
$753$	0	0
$754$	44.9919	1.63851
$755$	9.36663	0.340887
$756$	0	0
$757$	15.1464	0.550505	0.275252	−	0.961372i	$-0.411239\pi$
$757$	15.1464	0.550505	0.275252	+	0.961372i	$0.411239\pi$
$758$	29.8043	1.08254
$759$	0	0
$760$	−19.5861	−0.710463
$761$	−1.58612	−0.0574970	−0.0287485	−	0.999587i	$-0.509152\pi$
$761$	−1.58612	−0.0574970	−0.0287485	+	0.999587i	$0.509152\pi$
$762$	0	0
$763$	2.61662	0.0947281
$764$	−5.30618	−0.191971
$765$	0	0
$766$	−12.6004	−0.455270
$767$	−31.0660	−1.12173
$768$	0	0
$769$	21.7439	0.784104	0.392052	−	0.919943i	$-0.371765\pi$
$769$	21.7439	0.784104	0.392052	+	0.919943i	$0.371765\pi$
$770$	−14.1626	−0.510385
$771$	0	0
$772$	22.2624	0.801243
$773$	−22.9883	−0.826833	−0.413416	−	0.910542i	$-0.635665\pi$
$773$	−22.9883	−0.826833	−0.413416	+	0.910542i	$0.635665\pi$
$774$	0	0
$775$	−76.2769	−2.73995
$776$	−2.70088	−0.0969558
$777$	0	0
$778$	−14.5030	−0.519959
$779$	39.3633	1.41034
$780$	0	0
$781$	−18.0226	−0.644899
$782$	0	0
$783$	0	0
$784$	−6.58612	−0.235219
$785$	78.1641	2.78980
$786$	0	0
$787$	33.8238	1.20569	0.602844	−	0.797859i	$-0.294034\pi$
$787$	33.8238	1.20569	0.602844	+	0.797859i	$0.294034\pi$
$788$	16.3155	0.581217
$789$	0	0
$790$	41.6006	1.48008
$791$	3.53302	0.125620
$792$	0	0
$793$	57.0148	2.02465
$794$	30.9919	1.09986
$795$	0	0
$796$	5.01352	0.177700
$797$	40.3300	1.42856	0.714282	−	0.699858i	$-0.246754\pi$
$797$	40.3300	1.42856	0.714282	+	0.699858i	$0.246754\pi$
$798$	0	0
$799$	−6.35821	−0.224937
$800$	−9.24797	−0.326965
$801$	0	0
$802$	21.2712	0.751113
$803$	66.1263	2.33355
$804$	0	0
$805$	0	0
$806$	44.0289	1.55085
$807$	0	0
$808$	11.3382	0.398875
$809$	4.23349	0.148842	0.0744208	−	0.997227i	$-0.476289\pi$
$809$	4.23349	0.148842	0.0744208	+	0.997227i	$0.476289\pi$
$810$	0	0
$811$	55.0063	1.93153	0.965767	−	0.259411i	$-0.0835285\pi$
$811$	55.0063	1.93153	0.965767	+	0.259411i	$0.0835285\pi$
$812$	5.42222	0.190283
$813$	0	0
$814$	−1.76651	−0.0619161
$815$	19.4688	0.681962
$816$	0	0
$817$	−51.4202	−1.79897
$818$	−10.7294	−0.375146
$819$	0	0
$820$	28.6349	0.999976
$821$	−31.3671	−1.09472	−0.547360	−	0.836897i	$-0.684367\pi$
$821$	−31.3671	−1.09472	−0.547360	+	0.836897i	$0.684367\pi$
$822$	0	0
$823$	−30.9243	−1.07795	−0.538976	−	0.842321i	$-0.681189\pi$
$823$	−30.9243	−1.07795	−0.538976	+	0.842321i	$0.681189\pi$
$824$	0.292654	0.0101951
$825$	0	0
$826$	−3.74394	−0.130268
$827$	−5.32199	−0.185064	−0.0925319	−	0.995710i	$-0.529496\pi$
$827$	−5.32199	−0.185064	−0.0925319	+	0.995710i	$0.529496\pi$
$828$	0	0
$829$	−38.2624	−1.32891	−0.664455	−	0.747328i	$-0.731336\pi$
$829$	−38.2624	−1.32891	−0.664455	+	0.747328i	$0.731336\pi$
$830$	−38.5104	−1.33672
$831$	0	0
$832$	5.33816	0.185067
$833$	−13.5512	−0.469523
$834$	0	0
$835$	78.9272	2.73139
$836$	−30.2624	−1.04665
$837$	0	0
$838$	−30.0970	−1.03968
$839$	31.5590	1.08954	0.544768	−	0.838587i	$-0.316618\pi$
$839$	31.5590	1.08954	0.544768	+	0.838587i	$0.316618\pi$
$840$	0	0
$841$	42.0371	1.44955
$842$	11.7122	0.403628
$843$	0	0
$844$	17.1578	0.590595
$845$	−58.4917	−2.01218
$846$	0	0
$847$	−14.8059	−0.508738
$848$	−0.340442	−0.0116908
$849$	0	0
$850$	−19.0281	−0.652659
$851$	0	0
$852$	0	0
$853$	35.4878	1.21508	0.607540	−	0.794289i	$-0.292157\pi$
$853$	35.4878	1.21508	0.607540	+	0.794289i	$0.292157\pi$
$854$	6.87117	0.235127
$855$	0	0
$856$	4.89621	0.167349
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	−19.5330	−0.666458	−0.333229	−	0.942846i	$-0.608138\pi$
$859$	−19.5330	−0.666458	−0.333229	+	0.942846i	$0.608138\pi$
$860$	−37.4057	−1.27553
$861$	0	0
$862$	35.5034	1.20925
$863$	30.3445	1.03294	0.516469	−	0.856306i	$-0.327246\pi$
$863$	30.3445	1.03294	0.516469	+	0.856306i	$0.327246\pi$
$864$	0	0
$865$	31.8141	1.08171
$866$	3.89196	0.132254
$867$	0	0
$868$	5.30618	0.180103
$869$	64.2769	2.18045
$870$	0	0
$871$	32.6954	1.10784
$872$	4.06730	0.137736
$873$	0	0
$874$	0	0
$875$	10.3155	0.348729
$876$	0	0
$877$	−38.9532	−1.31536	−0.657679	−	0.753299i	$-0.728461\pi$
$877$	−38.9532	−1.31536	−0.657679	+	0.753299i	$0.728461\pi$
$878$	−11.5330	−0.389220
$879$	0	0
$880$	−22.0145	−0.742108
$881$	26.8325	0.904010	0.452005	−	0.892015i	$-0.350709\pi$
$881$	26.8325	0.904010	0.452005	+	0.892015i	$0.350709\pi$
$882$	0	0
$883$	12.3011	0.413964	0.206982	−	0.978345i	$-0.433636\pi$
$883$	12.3011	0.413964	0.206982	+	0.978345i	$0.433636\pi$
$884$	10.9835	0.369415
$885$	0	0
$886$	6.18038	0.207634
$887$	−29.2995	−0.983782	−0.491891	−	0.870657i	$-0.663694\pi$
$887$	−29.2995	−0.983782	−0.491891	+	0.870657i	$0.663694\pi$
$888$	0	0
$889$	−1.75592	−0.0588918
$890$	36.2624	1.21552
$891$	0	0
$892$	−2.48146	−0.0830854
$893$	16.0346	0.536577
$894$	0	0
$895$	−76.4289	−2.55474
$896$	0.643332	0.0214922
$897$	0	0
$898$	−2.72942	−0.0910821
$899$	69.5167	2.31851
$900$	0	0
$901$	−0.700475	−0.0233362
$902$	44.2437	1.47316
$903$	0	0
$904$	5.49175	0.182653
$905$	−10.8567	−0.360889
$906$	0	0
$907$	−34.1745	−1.13475	−0.567373	−	0.823461i	$-0.692040\pi$
$907$	−34.1745	−1.13475	−0.567373	+	0.823461i	$0.692040\pi$
$908$	8.33041	0.276454
$909$	0	0
$910$	−12.9629	−0.429716
$911$	−30.5069	−1.01074	−0.505370	−	0.862903i	$-0.668644\pi$
$911$	−30.5069	−1.01074	−0.505370	+	0.862903i	$0.668644\pi$
$912$	0	0
$913$	−59.5023	−1.96924
$914$	9.21861	0.304925
$915$	0	0
$916$	15.7317	0.519790
$917$	5.30618	0.175225
$918$	0	0
$919$	33.4138	1.10222	0.551110	−	0.834432i	$-0.314204\pi$
$919$	33.4138	1.10222	0.551110	+	0.834432i	$0.314204\pi$
$920$	0	0
$921$	0	0
$922$	−4.67632	−0.154006
$923$	−16.4959	−0.542970
$924$	0	0
$925$	2.80111	0.0920999
$926$	8.00000	0.262896
$927$	0	0
$928$	8.42835	0.276674
$929$	25.4057	0.833535	0.416768	−	0.909013i	$-0.363163\pi$
$929$	25.4057	0.833535	0.416768	+	0.909013i	$0.363163\pi$
$930$	0	0
$931$	34.1745	1.12002
$932$	27.5330	0.901874
$933$	0	0
$934$	3.33397	0.109091
$935$	−45.2958	−1.48133
$936$	0	0
$937$	−30.1447	−0.984786	−0.492393	−	0.870373i	$-0.663878\pi$
$937$	−30.1447	−0.984786	−0.492393	+	0.870373i	$0.663878\pi$
$938$	3.94031	0.128656
$939$	0	0
$940$	11.6644	0.380450
$941$	−6.27287	−0.204490	−0.102245	−	0.994759i	$-0.532603\pi$
$941$	−6.27287	−0.204490	−0.102245	+	0.994759i	$0.532603\pi$
$942$	0	0
$943$	0	0
$944$	−5.81962	−0.189412
$945$	0	0
$946$	−57.7955	−1.87909
$947$	−42.9243	−1.39485	−0.697426	−	0.716657i	$-0.745671\pi$
$947$	−42.9243	−1.39485	−0.697426	+	0.716657i	$0.745671\pi$
$948$	0	0
$949$	60.5249	1.96472
$950$	47.9864	1.55688
$951$	0	0
$952$	1.32368	0.0429008
$953$	15.2842	0.495103	0.247551	−	0.968875i	$-0.420374\pi$
$953$	15.2842	0.495103	0.247551	+	0.968875i	$0.420374\pi$
$954$	0	0
$955$	20.0289	0.648122
$956$	−6.18038	−0.199888
$957$	0	0
$958$	4.37019	0.141194
$959$	−4.13517	−0.133531
$960$	0	0
$961$	37.0289	1.19448
$962$	−1.61687	−0.0521300
$963$	0	0
$964$	−7.32617	−0.235960
$965$	−84.0329	−2.70511
$966$	0	0
$967$	−6.92428	−0.222670	−0.111335	−	0.993783i	$-0.535513\pi$
$967$	−6.92428	−0.222670	−0.111335	+	0.993783i	$0.535513\pi$
$968$	−23.0145	−0.739713
$969$	0	0
$970$	10.1949	0.327337
$971$	−3.33397	−0.106992	−0.0534961	−	0.998568i	$-0.517036\pi$
$971$	−3.33397	−0.106992	−0.0534961	+	0.998568i	$0.517036\pi$
$972$	0	0
$973$	8.03903	0.257719
$974$	16.0000	0.512673
$975$	0	0
$976$	10.6806	0.341878
$977$	−12.7859	−0.409059	−0.204529	−	0.978860i	$-0.565566\pi$
$977$	−12.7859	−0.409059	−0.204529	+	0.978860i	$0.565566\pi$
$978$	0	0
$979$	56.0289	1.79069
$980$	24.8603	0.794133
$981$	0	0
$982$	31.2850	0.998346
$983$	9.67637	0.308628	0.154314	−	0.988022i	$-0.450683\pi$
$983$	9.67637	0.308628	0.154314	+	0.988022i	$0.450683\pi$
$984$	0	0
$985$	−61.5855	−1.96228
$986$	17.3417	0.552273
$987$	0	0
$988$	−27.6990	−0.881221
$989$	0	0
$990$	0	0
$991$	26.9532	0.856198	0.428099	−	0.903732i	$-0.359184\pi$
$991$	26.9532	0.856198	0.428099	+	0.903732i	$0.359184\pi$
$992$	8.24797	0.261873
$993$	0	0
$994$	−1.98802	−0.0630561
$995$	−18.9243	−0.599940
$996$	0	0
$997$	−7.69234	−0.243619	−0.121809	−	0.992554i	$-0.538870\pi$
$997$	−7.69234	−0.243619	−0.121809	+	0.992554i	$0.538870\pi$
$998$	−26.5104	−0.839172
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9522.2.a.cb.1.1	✓	6
3.2	odd	2		9522.2.a.cc.1.6	yes	6
23.22	odd	2	inner	9522.2.a.cb.1.6	yes	6
69.68	even	2		9522.2.a.cc.1.1	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9522.2.a.cb.1.1	✓	6	1.1	even	1	trivial
9522.2.a.cb.1.6	yes	6	23.22	odd	2	inner
9522.2.a.cc.1.1	yes	6	69.68	even	2
9522.2.a.cc.1.6	yes	6	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -3.77465 q^{5} -0.643332 q^{7} -1.00000 q^{8} +3.77465 q^{10} +5.83219 q^{11} +5.33816 q^{13} +0.643332 q^{14} +1.00000 q^{16} +2.05755 q^{17} -5.18886 q^{19} -3.77465 q^{20} -5.83219 q^{22} +9.24797 q^{25} -5.33816 q^{26} -0.643332 q^{28} -8.42835 q^{29} -8.24797 q^{31} -1.00000 q^{32} -2.05755 q^{34} +2.42835 q^{35} +0.302889 q^{37} +5.18886 q^{38} +3.77465 q^{40} -7.58612 q^{41} +9.90973 q^{43} +5.83219 q^{44} -3.09019 q^{47} -6.58612 q^{49} -9.24797 q^{50} +5.33816 q^{52} -0.340442 q^{53} -22.0145 q^{55} +0.643332 q^{56} +8.42835 q^{58} -5.81962 q^{59} +10.6806 q^{61} +8.24797 q^{62} +1.00000 q^{64} -20.1497 q^{65} +6.12485 q^{67} +2.05755 q^{68} -2.42835 q^{70} -3.09019 q^{71} +11.3382 q^{73} -0.302889 q^{74} -5.18886 q^{76} -3.75203 q^{77} +11.0211 q^{79} -3.77465 q^{80} +7.58612 q^{82} -10.2024 q^{83} -7.76651 q^{85} -9.90973 q^{86} -5.83219 q^{88} +9.60684 q^{89} -3.43421 q^{91} +3.09019 q^{94} +19.5861 q^{95} +2.70088 q^{97} +6.58612 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{2} + 6 q^{4} - 6 q^{8} + 6 q^{16} + 18 q^{25} - 24 q^{29} - 12 q^{31} - 6 q^{32} - 12 q^{35} + 24 q^{41} - 24 q^{47} + 30 q^{49} - 18 q^{50} - 36 q^{55} + 24 q^{58} - 24 q^{59} + 12 q^{62} + 6 q^{64}+ \cdots - 30 q^{98}+O(q^{100}) \) 6 * q - 6 * q^2 + 6 * q^4 - 6 * q^8 + 6 * q^16 + 18 * q^25 - 24 * q^29 - 12 * q^31 - 6 * q^32 - 12 * q^35 + 24 * q^41 - 24 * q^47 + 30 * q^49 - 18 * q^50 - 36 * q^55 + 24 * q^58 - 24 * q^59 + 12 * q^62 + 6 * q^64 + 12 * q^70 - 24 * q^71 + 36 * q^73 - 60 * q^77 - 24 * q^82 + 12 * q^85 + 24 * q^94 + 48 * q^95 - 30 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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