LMFDB - Embedded newform 7623.2.a.db.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7623,2,Mod(1,7623)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7623, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("7623.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7623 = 3^{2} \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7623.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$60.8699614608$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 26x^{14} + 268x^{12} - 1395x^{10} + 3876x^{8} - 5635x^{6} + 4042x^{4} - 1272x^{2} + 121 $ x^16 - 26x^14 + 268x^12 - 1395x^10 + 3876x^8 - 5635x^6 + 4042x^4 - 1272*x^2 + 121
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 693)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.10399$ of defining polynomial
Character	$\chi$	$=$	7623.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.10399 q^{2} +2.42677 q^{4} +3.75046 q^{5} -1.00000 q^{7} -0.897910 q^{8} +O(q^{10})$	$q-2.10399 q^{2} +2.42677 q^{4} +3.75046 q^{5} -1.00000 q^{7} -0.897910 q^{8} -7.89093 q^{10} -3.46575 q^{13} +2.10399 q^{14} -2.96434 q^{16} +1.52683 q^{17} +5.16008 q^{19} +9.10149 q^{20} +4.87487 q^{23} +9.06597 q^{25} +7.29190 q^{26} -2.42677 q^{28} +10.7274 q^{29} +8.12735 q^{31} +8.03275 q^{32} -3.21244 q^{34} -3.75046 q^{35} +9.25848 q^{37} -10.8568 q^{38} -3.36758 q^{40} -10.8973 q^{41} +0.137677 q^{43} -10.2567 q^{46} +2.73030 q^{47} +1.00000 q^{49} -19.0747 q^{50} -8.41057 q^{52} +2.79394 q^{53} +0.897910 q^{56} -22.5702 q^{58} +4.84485 q^{59} +0.297827 q^{61} -17.0998 q^{62} -10.9721 q^{64} -12.9982 q^{65} +0.816899 q^{67} +3.70527 q^{68} +7.89093 q^{70} -5.72995 q^{71} +2.47516 q^{73} -19.4797 q^{74} +12.5223 q^{76} -11.4243 q^{79} -11.1176 q^{80} +22.9278 q^{82} -10.6460 q^{83} +5.72633 q^{85} -0.289670 q^{86} -8.91589 q^{89} +3.46575 q^{91} +11.8302 q^{92} -5.74451 q^{94} +19.3527 q^{95} +8.08471 q^{97} -2.10399 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 20 q^{4} - 16 q^{7}+O(q^{10}) $ 16 * q + 20 * q^4 - 16 * q^7	$ 16 q + 20 q^{4} - 16 q^{7} + 6 q^{10} + 32 q^{16} - 10 q^{19} + 44 q^{25} - 20 q^{28} + 30 q^{31} + 12 q^{34} + 38 q^{37} + 68 q^{40} - 16 q^{43} - 80 q^{46} + 16 q^{49} - 2 q^{52} + 18 q^{58} + 28 q^{61} + 34 q^{64} + 52 q^{67} - 6 q^{70} + 14 q^{73} + 14 q^{76} - 54 q^{79} + 64 q^{82} - 30 q^{85} + 60 q^{94} + 108 q^{97}+O(q^{100}) $ 16 * q + 20 * q^4 - 16 * q^7 + 6 * q^10 + 32 * q^16 - 10 * q^19 + 44 * q^25 - 20 * q^28 + 30 * q^31 + 12 * q^34 + 38 * q^37 + 68 * q^40 - 16 * q^43 - 80 * q^46 + 16 * q^49 - 2 * q^52 + 18 * q^58 + 28 * q^61 + 34 * q^64 + 52 * q^67 - 6 * q^70 + 14 * q^73 + 14 * q^76 - 54 * q^79 + 64 * q^82 - 30 * q^85 + 60 * q^94 + 108 * q^97

Display 100 coefficients 10 coefficients

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 $

$2$	−2.10399	−1.48774	−0.743872	−	0.668322i	$-0.767013\pi$
$2$	−2.10399	−1.48774	−0.743872	+	0.668322i	$0.767013\pi$
$3$	0	0
$3$	0	0
$4$	2.42677	1.21338
$5$	3.75046	1.67726	0.838629	−	0.544703i	$-0.183358\pi$
$5$	3.75046	1.67726	0.838629	+	0.544703i	$0.183358\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−0.897910	−0.317459
$9$	0	0
$10$	−7.89093	−2.49533
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−3.46575	−0.961227	−0.480613	−	0.876933i	$-0.659586\pi$
$13$	−3.46575	−0.961227	−0.480613	+	0.876933i	$0.659586\pi$
$14$	2.10399	0.562314
$15$	0	0
$16$	−2.96434	−0.741085
$17$	1.52683	0.370311	0.185156	−	0.982709i	$-0.440721\pi$
$17$	1.52683	0.370311	0.185156	+	0.982709i	$0.440721\pi$
$18$	0	0
$19$	5.16008	1.18380	0.591902	−	0.806010i	$-0.298377\pi$
$19$	5.16008	1.18380	0.591902	+	0.806010i	$0.298377\pi$
$20$	9.10149	2.03516
$21$	0	0
$22$	0	0
$23$	4.87487	1.01648	0.508240	−	0.861215i	$-0.330296\pi$
$23$	4.87487	1.01648	0.508240	+	0.861215i	$0.330296\pi$
$24$	0	0
$25$	9.06597	1.81319
$26$	7.29190	1.43006
$27$	0	0
$28$	−2.42677	−0.458616
$29$	10.7274	1.99202	0.996010	−	0.0892450i	$-0.0284454\pi$
$29$	10.7274	1.99202	0.996010	+	0.0892450i	$0.0284454\pi$
$30$	0	0
$31$	8.12735	1.45971	0.729857	−	0.683599i	$-0.239586\pi$
$31$	8.12735	1.45971	0.729857	+	0.683599i	$0.239586\pi$
$32$	8.03275	1.42000
$33$	0	0
$34$	−3.21244	−0.550929
$35$	−3.75046	−0.633944
$36$	0	0
$37$	9.25848	1.52208	0.761042	−	0.648703i	$-0.224688\pi$
$37$	9.25848	1.52208	0.761042	+	0.648703i	$0.224688\pi$
$38$	−10.8568	−1.76120
$39$	0	0
$40$	−3.36758	−0.532461
$41$	−10.8973	−1.70187	−0.850937	−	0.525267i	$-0.823965\pi$
$41$	−10.8973	−1.70187	−0.850937	+	0.525267i	$0.823965\pi$
$42$	0	0
$43$	0.137677	0.0209955	0.0104978	−	0.999945i	$-0.496658\pi$
$43$	0.137677	0.0209955	0.0104978	+	0.999945i	$0.496658\pi$
$44$	0	0
$45$	0	0
$46$	−10.2567	−1.51226
$47$	2.73030	0.398255	0.199127	−	0.979974i	$-0.436189\pi$
$47$	2.73030	0.398255	0.199127	+	0.979974i	$0.436189\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−19.0747	−2.69757
$51$	0	0
$52$	−8.41057	−1.16634
$53$	2.79394	0.383777	0.191889	−	0.981417i	$-0.438539\pi$
$53$	2.79394	0.383777	0.191889	+	0.981417i	$0.438539\pi$
$54$	0	0
$55$	0	0
$56$	0.897910	0.119988
$57$	0	0
$58$	−22.5702	−2.96362
$59$	4.84485	0.630746	0.315373	−	0.948968i	$-0.397870\pi$
$59$	4.84485	0.630746	0.315373	+	0.948968i	$0.397870\pi$
$60$	0	0
$61$	0.297827	0.0381328	0.0190664	−	0.999818i	$-0.493931\pi$
$61$	0.297827	0.0381328	0.0190664	+	0.999818i	$0.493931\pi$
$62$	−17.0998	−2.17168
$63$	0	0
$64$	−10.9721	−1.37152
$65$	−12.9982	−1.61223
$66$	0	0
$67$	0.816899	0.0998001	0.0499001	−	0.998754i	$-0.484110\pi$
$67$	0.816899	0.0998001	0.0499001	+	0.998754i	$0.484110\pi$
$68$	3.70527	0.449329
$69$	0	0
$70$	7.89093	0.943146
$71$	−5.72995	−0.680020	−0.340010	−	0.940422i	$-0.610430\pi$
$71$	−5.72995	−0.680020	−0.340010	+	0.940422i	$0.610430\pi$
$72$	0	0
$73$	2.47516	0.289695	0.144848	−	0.989454i	$-0.453731\pi$
$73$	2.47516	0.289695	0.144848	+	0.989454i	$0.453731\pi$
$74$	−19.4797	−2.26447
$75$	0	0
$76$	12.5223	1.43641
$77$	0	0
$78$	0	0
$79$	−11.4243	−1.28533	−0.642664	−	0.766148i	$-0.722171\pi$
$79$	−11.4243	−1.28533	−0.642664	+	0.766148i	$0.722171\pi$
$80$	−11.1176	−1.24299
$81$	0	0
$82$	22.9278	2.53195
$83$	−10.6460	−1.16855	−0.584273	−	0.811557i	$-0.698620\pi$
$83$	−10.6460	−1.16855	−0.584273	+	0.811557i	$0.698620\pi$
$84$	0	0
$85$	5.72633	0.621108
$86$	−0.289670	−0.0312359
$87$	0	0
$88$	0	0
$89$	−8.91589	−0.945082	−0.472541	−	0.881309i	$-0.656663\pi$
$89$	−8.91589	−0.945082	−0.472541	+	0.881309i	$0.656663\pi$
$90$	0	0
$91$	3.46575	0.363310
$92$	11.8302	1.23338
$93$	0	0
$94$	−5.74451	−0.592501
$95$	19.3527	1.98555
$96$	0	0
$97$	8.08471	0.820878	0.410439	−	0.911888i	$-0.365375\pi$
$97$	8.08471	0.820878	0.410439	+	0.911888i	$0.365375\pi$
$98$	−2.10399	−0.212535
$99$	0	0
$100$	22.0010	2.20010
$101$	−1.72198	−0.171343	−0.0856715	−	0.996323i	$-0.527304\pi$
$101$	−1.72198	−0.171343	−0.0856715	+	0.996323i	$0.527304\pi$
$102$	0	0
$103$	−2.69398	−0.265446	−0.132723	−	0.991153i	$-0.542372\pi$
$103$	−2.69398	−0.265446	−0.132723	+	0.991153i	$0.542372\pi$
$104$	3.11194	0.305150
$105$	0	0
$106$	−5.87842	−0.570962
$107$	−4.97994	−0.481429	−0.240715	−	0.970596i	$-0.577382\pi$
$107$	−4.97994	−0.481429	−0.240715	+	0.970596i	$0.577382\pi$
$108$	0	0
$109$	−11.2061	−1.07335	−0.536676	−	0.843788i	$-0.680320\pi$
$109$	−11.2061	−1.07335	−0.536676	+	0.843788i	$0.680320\pi$
$110$	0	0
$111$	0	0
$112$	2.96434	0.280104
$113$	−16.4261	−1.54524	−0.772620	−	0.634869i	$-0.781054\pi$
$113$	−16.4261	−1.54524	−0.772620	+	0.634869i	$0.781054\pi$
$114$	0	0
$115$	18.2830	1.70490
$116$	26.0328	2.41708
$117$	0	0
$118$	−10.1935	−0.938388
$119$	−1.52683	−0.139965
$120$	0	0
$121$	0	0
$122$	−0.626624	−0.0567318
$123$	0	0
$124$	19.7232	1.77119
$125$	15.2493	1.36394
$126$	0	0
$127$	19.6058	1.73973	0.869865	−	0.493291i	$-0.164206\pi$
$127$	19.6058	1.73973	0.869865	+	0.493291i	$0.164206\pi$
$128$	7.01975	0.620464
$129$	0	0
$130$	27.3480	2.39858
$131$	−13.3659	−1.16778	−0.583891	−	0.811832i	$-0.698470\pi$
$131$	−13.3659	−1.16778	−0.583891	+	0.811832i	$0.698470\pi$
$132$	0	0
$133$	−5.16008	−0.447436
$134$	−1.71875	−0.148477
$135$	0	0
$136$	−1.37096	−0.117559
$137$	−5.27980	−0.451084	−0.225542	−	0.974233i	$-0.572415\pi$
$137$	−5.27980	−0.451084	−0.225542	+	0.974233i	$0.572415\pi$
$138$	0	0
$139$	21.3757	1.81306	0.906531	−	0.422139i	$-0.138720\pi$
$139$	21.3757	1.81306	0.906531	+	0.422139i	$0.138720\pi$
$140$	−9.10149	−0.769217
$141$	0	0
$142$	12.0557	1.01170
$143$	0	0
$144$	0	0
$145$	40.2325	3.34113
$146$	−5.20770	−0.430993
$147$	0	0
$148$	22.4682	1.84687
$149$	10.9966	0.900873	0.450437	−	0.892808i	$-0.351268\pi$
$149$	10.9966	0.900873	0.450437	+	0.892808i	$0.351268\pi$
$150$	0	0
$151$	3.54170	0.288219	0.144110	−	0.989562i	$-0.453968\pi$
$151$	3.54170	0.288219	0.144110	+	0.989562i	$0.453968\pi$
$152$	−4.63329	−0.375810
$153$	0	0
$154$	0	0
$155$	30.4813	2.44832
$156$	0	0
$157$	12.7591	1.01828	0.509142	−	0.860682i	$-0.329963\pi$
$157$	12.7591	1.01828	0.509142	+	0.860682i	$0.329963\pi$
$158$	24.0365	1.91224
$159$	0	0
$160$	30.1265	2.38171
$161$	−4.87487	−0.384194
$162$	0	0
$163$	−16.2850	−1.27554	−0.637769	−	0.770228i	$-0.720142\pi$
$163$	−16.2850	−1.27554	−0.637769	+	0.770228i	$0.720142\pi$
$164$	−26.4452	−2.06503
$165$	0	0
$166$	22.3990	1.73850
$167$	−5.66281	−0.438201	−0.219101	−	0.975702i	$-0.570312\pi$
$167$	−5.66281	−0.438201	−0.219101	+	0.975702i	$0.570312\pi$
$168$	0	0
$169$	−0.988556	−0.0760427
$170$	−12.0481	−0.924049
$171$	0	0
$172$	0.334109	0.0254756
$173$	13.4214	1.02041	0.510205	−	0.860053i	$-0.329570\pi$
$173$	13.4214	1.02041	0.510205	+	0.860053i	$0.329570\pi$
$174$	0	0
$175$	−9.06597	−0.685323
$176$	0	0
$177$	0	0
$178$	18.7589	1.40604
$179$	−3.80842	−0.284654	−0.142327	−	0.989820i	$-0.545459\pi$
$179$	−3.80842	−0.284654	−0.142327	+	0.989820i	$0.545459\pi$
$180$	0	0
$181$	−9.78859	−0.727580	−0.363790	−	0.931481i	$-0.618517\pi$
$181$	−9.78859	−0.727580	−0.363790	+	0.931481i	$0.618517\pi$
$182$	−7.29190	−0.540512
$183$	0	0
$184$	−4.37720	−0.322691
$185$	34.7236	2.55293
$186$	0	0
$187$	0	0
$188$	6.62579	0.483236
$189$	0	0
$190$	−40.7179	−2.95398
$191$	−8.98965	−0.650469	−0.325234	−	0.945633i	$-0.605443\pi$
$191$	−8.98965	−0.650469	−0.325234	+	0.945633i	$0.605443\pi$
$192$	0	0
$193$	5.84941	0.421049	0.210525	−	0.977589i	$-0.432483\pi$
$193$	5.84941	0.421049	0.210525	+	0.977589i	$0.432483\pi$
$194$	−17.0101	−1.22126
$195$	0	0
$196$	2.42677	0.173340
$197$	6.19555	0.441414	0.220707	−	0.975340i	$-0.429163\pi$
$197$	6.19555	0.441414	0.220707	+	0.975340i	$0.429163\pi$
$198$	0	0
$199$	−18.8634	−1.33719	−0.668596	−	0.743626i	$-0.733105\pi$
$199$	−18.8634	−1.33719	−0.668596	+	0.743626i	$0.733105\pi$
$200$	−8.14043	−0.575615
$201$	0	0
$202$	3.62302	0.254914
$203$	−10.7274	−0.752913
$204$	0	0
$205$	−40.8700	−2.85448
$206$	5.66810	0.394915
$207$	0	0
$208$	10.2737	0.712351
$209$	0	0
$210$	0	0
$211$	−0.640180	−0.0440718	−0.0220359	−	0.999757i	$-0.507015\pi$
$211$	−0.640180	−0.0440718	−0.0220359	+	0.999757i	$0.507015\pi$
$212$	6.78024	0.465669
$213$	0	0
$214$	10.4777	0.716243
$215$	0.516352	0.0352149
$216$	0	0
$217$	−8.12735	−0.551720
$218$	23.5775	1.59687
$219$	0	0
$220$	0	0
$221$	−5.29163	−0.355953
$222$	0	0
$223$	−16.7123	−1.11914	−0.559568	−	0.828785i	$-0.689033\pi$
$223$	−16.7123	−1.11914	−0.559568	+	0.828785i	$0.689033\pi$
$224$	−8.03275	−0.536711
$225$	0	0
$226$	34.5604	2.29892
$227$	9.32634	0.619011	0.309506	−	0.950898i	$-0.399836\pi$
$227$	9.32634	0.619011	0.309506	+	0.950898i	$0.399836\pi$
$228$	0	0
$229$	21.3716	1.41228	0.706138	−	0.708074i	$-0.250436\pi$
$229$	21.3716	1.41228	0.706138	+	0.708074i	$0.250436\pi$
$230$	−38.4672	−2.53646
$231$	0	0
$232$	−9.63220	−0.632385
$233$	−17.1164	−1.12133	−0.560666	−	0.828042i	$-0.689455\pi$
$233$	−17.1164	−1.12133	−0.560666	+	0.828042i	$0.689455\pi$
$234$	0	0
$235$	10.2399	0.667976
$236$	11.7573	0.765336
$237$	0	0
$238$	3.21244	0.208231
$239$	3.37044	0.218016	0.109008	−	0.994041i	$-0.465233\pi$
$239$	3.37044	0.218016	0.109008	+	0.994041i	$0.465233\pi$
$240$	0	0
$241$	−5.77583	−0.372054	−0.186027	−	0.982545i	$-0.559561\pi$
$241$	−5.77583	−0.372054	−0.186027	+	0.982545i	$0.559561\pi$
$242$	0	0
$243$	0	0
$244$	0.722755	0.0462697
$245$	3.75046	0.239608
$246$	0	0
$247$	−17.8836	−1.13790
$248$	−7.29763	−0.463400
$249$	0	0
$250$	−32.0843	−2.02919
$251$	−11.7616	−0.742383	−0.371191	−	0.928556i	$-0.621051\pi$
$251$	−11.7616	−0.742383	−0.371191	+	0.928556i	$0.621051\pi$
$252$	0	0
$253$	0	0
$254$	−41.2503	−2.58827
$255$	0	0
$256$	7.17482	0.448426
$257$	14.2660	0.889891	0.444945	−	0.895558i	$-0.353223\pi$
$257$	14.2660	0.889891	0.444945	+	0.895558i	$0.353223\pi$
$258$	0	0
$259$	−9.25848	−0.575294
$260$	−31.5435	−1.95625
$261$	0	0
$262$	28.1216	1.73736
$263$	−4.85285	−0.299239	−0.149620	−	0.988744i	$-0.547805\pi$
$263$	−4.85285	−0.299239	−0.149620	+	0.988744i	$0.547805\pi$
$264$	0	0
$265$	10.4786	0.643693
$266$	10.8568	0.665670
$267$	0	0
$268$	1.98242	0.121096
$269$	6.35465	0.387450	0.193725	−	0.981056i	$-0.437943\pi$
$269$	6.35465	0.387450	0.193725	+	0.981056i	$0.437943\pi$
$270$	0	0
$271$	15.2155	0.924273	0.462137	−	0.886809i	$-0.347083\pi$
$271$	15.2155	0.924273	0.462137	+	0.886809i	$0.347083\pi$
$272$	−4.52605	−0.274432
$273$	0	0
$274$	11.1086	0.671098
$275$	0	0
$276$	0	0
$277$	0.379296	0.0227897	0.0113948	−	0.999935i	$-0.496373\pi$
$277$	0.379296	0.0227897	0.0113948	+	0.999935i	$0.496373\pi$
$278$	−44.9742	−2.69737
$279$	0	0
$280$	3.36758	0.201251
$281$	5.04347	0.300868	0.150434	−	0.988620i	$-0.451933\pi$
$281$	5.04347	0.300868	0.150434	+	0.988620i	$0.451933\pi$
$282$	0	0
$283$	32.1351	1.91023	0.955116	−	0.296233i	$-0.0957305\pi$
$283$	32.1351	1.91023	0.955116	+	0.296233i	$0.0957305\pi$
$284$	−13.9052	−0.825125
$285$	0	0
$286$	0	0
$287$	10.8973	0.643248
$288$	0	0
$289$	−14.6688	−0.862870
$290$	−84.6488	−4.97075
$291$	0	0
$292$	6.00663	0.351511
$293$	26.8246	1.56711	0.783554	−	0.621324i	$-0.213405\pi$
$293$	26.8246	1.56711	0.783554	+	0.621324i	$0.213405\pi$
$294$	0	0
$295$	18.1704	1.05792
$296$	−8.31328	−0.483200
$297$	0	0
$298$	−23.1366	−1.34027
$299$	−16.8951	−0.977069
$300$	0	0
$301$	−0.137677	−0.00793556
$302$	−7.45169	−0.428797
$303$	0	0
$304$	−15.2962	−0.877300
$305$	1.11699	0.0639585
$306$	0	0
$307$	5.75200	0.328284	0.164142	−	0.986437i	$-0.447514\pi$
$307$	5.75200	0.328284	0.164142	+	0.986437i	$0.447514\pi$
$308$	0	0
$309$	0	0
$310$	−64.1323	−3.64247
$311$	−17.8807	−1.01392	−0.506960	−	0.861970i	$-0.669231\pi$
$311$	−17.8807	−1.01392	−0.506960	+	0.861970i	$0.669231\pi$
$312$	0	0
$313$	19.5005	1.10223	0.551117	−	0.834428i	$-0.314202\pi$
$313$	19.5005	1.10223	0.551117	+	0.834428i	$0.314202\pi$
$314$	−26.8449	−1.51495
$315$	0	0
$316$	−27.7240	−1.55960
$317$	17.5109	0.983508	0.491754	−	0.870734i	$-0.336356\pi$
$317$	17.5109	0.983508	0.491754	+	0.870734i	$0.336356\pi$
$318$	0	0
$319$	0	0
$320$	−41.1506	−2.30039
$321$	0	0
$322$	10.2567	0.571582
$323$	7.87859	0.438376
$324$	0	0
$325$	−31.4204	−1.74289
$326$	34.2634	1.89767
$327$	0	0
$328$	9.78481	0.540276
$329$	−2.73030	−0.150526
$330$	0	0
$331$	−17.6231	−0.968652	−0.484326	−	0.874888i	$-0.660935\pi$
$331$	−17.6231	−0.968652	−0.484326	+	0.874888i	$0.660935\pi$
$332$	−25.8352	−1.41789
$333$	0	0
$334$	11.9145	0.651931
$335$	3.06375	0.167391
$336$	0	0
$337$	−1.60109	−0.0872167	−0.0436084	−	0.999049i	$-0.513885\pi$
$337$	−1.60109	−0.0872167	−0.0436084	+	0.999049i	$0.513885\pi$
$338$	2.07991	0.113132
$339$	0	0
$340$	13.8965	0.753641
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−0.123621	−0.00666522
$345$	0	0
$346$	−28.2384	−1.51811
$347$	3.44462	0.184917	0.0924585	−	0.995717i	$-0.470527\pi$
$347$	3.44462	0.184917	0.0924585	+	0.995717i	$0.470527\pi$
$348$	0	0
$349$	−3.39745	−0.181862	−0.0909308	−	0.995857i	$-0.528984\pi$
$349$	−3.39745	−0.181862	−0.0909308	+	0.995857i	$0.528984\pi$
$350$	19.0747	1.01959
$351$	0	0
$352$	0	0
$353$	−10.4083	−0.553977	−0.276989	−	0.960873i	$-0.589336\pi$
$353$	−10.4083	−0.553977	−0.276989	+	0.960873i	$0.589336\pi$
$354$	0	0
$355$	−21.4900	−1.14057
$356$	−21.6368	−1.14675
$357$	0	0
$358$	8.01286	0.423493
$359$	22.9948	1.21362	0.606811	−	0.794846i	$-0.292449\pi$
$359$	22.9948	1.21362	0.606811	+	0.794846i	$0.292449\pi$
$360$	0	0
$361$	7.62647	0.401393
$362$	20.5951	1.08245
$363$	0	0
$364$	8.41057	0.440834
$365$	9.28299	0.485894
$366$	0	0
$367$	19.6357	1.02498	0.512488	−	0.858695i	$-0.328724\pi$
$367$	19.6357	1.02498	0.512488	+	0.858695i	$0.328724\pi$
$368$	−14.4508	−0.753298
$369$	0	0
$370$	−73.0580	−3.79810
$371$	−2.79394	−0.145054
$372$	0	0
$373$	31.9047	1.65196	0.825981	−	0.563698i	$-0.190622\pi$
$373$	31.9047	1.65196	0.825981	+	0.563698i	$0.190622\pi$
$374$	0	0
$375$	0	0
$376$	−2.45156	−0.126430
$377$	−37.1784	−1.91478
$378$	0	0
$379$	−1.71958	−0.0883287	−0.0441644	−	0.999024i	$-0.514063\pi$
$379$	−1.71958	−0.0883287	−0.0441644	+	0.999024i	$0.514063\pi$
$380$	46.9645	2.40923
$381$	0	0
$382$	18.9141	0.967731
$383$	−8.68039	−0.443547	−0.221774	−	0.975098i	$-0.571185\pi$
$383$	−8.68039	−0.443547	−0.221774	+	0.975098i	$0.571185\pi$
$384$	0	0
$385$	0	0
$386$	−12.3071	−0.626414
$387$	0	0
$388$	19.6197	0.996039
$389$	26.8018	1.35890	0.679452	−	0.733720i	$-0.262218\pi$
$389$	26.8018	1.35890	0.679452	+	0.733720i	$0.262218\pi$
$390$	0	0
$391$	7.44311	0.376414
$392$	−0.897910	−0.0453513
$393$	0	0
$394$	−13.0354	−0.656712
$395$	−42.8462	−2.15583
$396$	0	0
$397$	−17.8714	−0.896938	−0.448469	−	0.893798i	$-0.648031\pi$
$397$	−17.8714	−0.896938	−0.448469	+	0.893798i	$0.648031\pi$
$398$	39.6884	1.98940
$399$	0	0
$400$	−26.8746	−1.34373
$401$	28.9136	1.44388	0.721938	−	0.691958i	$-0.243252\pi$
$401$	28.9136	1.44388	0.721938	+	0.691958i	$0.243252\pi$
$402$	0	0
$403$	−28.1674	−1.40312
$404$	−4.17883	−0.207905
$405$	0	0
$406$	22.5702	1.12014
$407$	0	0
$408$	0	0
$409$	−27.6583	−1.36761	−0.683807	−	0.729663i	$-0.739677\pi$
$409$	−27.6583	−1.36761	−0.683807	+	0.729663i	$0.739677\pi$
$410$	85.9899	4.24674
$411$	0	0
$412$	−6.53766	−0.322087
$413$	−4.84485	−0.238399
$414$	0	0
$415$	−39.9273	−1.95995
$416$	−27.8395	−1.36495
$417$	0	0
$418$	0	0
$419$	4.51003	0.220330	0.110165	−	0.993913i	$-0.464862\pi$
$419$	4.51003	0.220330	0.110165	+	0.993913i	$0.464862\pi$
$420$	0	0
$421$	35.9445	1.75183	0.875914	−	0.482467i	$-0.160259\pi$
$421$	35.9445	1.75183	0.875914	+	0.482467i	$0.160259\pi$
$422$	1.34693	0.0655676
$423$	0	0
$424$	−2.50871	−0.121834
$425$	13.8422	0.671446
$426$	0	0
$427$	−0.297827	−0.0144128
$428$	−12.0851	−0.584158
$429$	0	0
$430$	−1.08640	−0.0523907
$431$	−22.2209	−1.07034	−0.535172	−	0.844743i	$-0.679753\pi$
$431$	−22.2209	−1.07034	−0.535172	+	0.844743i	$0.679753\pi$
$432$	0	0
$433$	25.4892	1.22493	0.612467	−	0.790496i	$-0.290177\pi$
$433$	25.4892	1.22493	0.612467	+	0.790496i	$0.290177\pi$
$434$	17.0998	0.820819
$435$	0	0
$436$	−27.1946	−1.30239
$437$	25.1547	1.20331
$438$	0	0
$439$	−11.1407	−0.531715	−0.265857	−	0.964012i	$-0.585655\pi$
$439$	−11.1407	−0.531715	−0.265857	+	0.964012i	$0.585655\pi$
$440$	0	0
$441$	0	0
$442$	11.1335	0.529567
$443$	−13.8512	−0.658092	−0.329046	−	0.944314i	$-0.606727\pi$
$443$	−13.8512	−0.658092	−0.329046	+	0.944314i	$0.606727\pi$
$444$	0	0
$445$	−33.4387	−1.58515
$446$	35.1624	1.66499
$447$	0	0
$448$	10.9721	0.518385
$449$	−10.6878	−0.504390	−0.252195	−	0.967676i	$-0.581152\pi$
$449$	−10.6878	−0.504390	−0.252195	+	0.967676i	$0.581152\pi$
$450$	0	0
$451$	0	0
$452$	−39.8624	−1.87497
$453$	0	0
$454$	−19.6225	−0.920930
$455$	12.9982	0.609364
$456$	0	0
$457$	−31.3778	−1.46779	−0.733895	−	0.679263i	$-0.762300\pi$
$457$	−31.3778	−1.46779	−0.733895	+	0.679263i	$0.762300\pi$
$458$	−44.9656	−2.10111
$459$	0	0
$460$	44.3686	2.06870
$461$	41.1844	1.91815	0.959075	−	0.283151i	$-0.0913797\pi$
$461$	41.1844	1.91815	0.959075	+	0.283151i	$0.0913797\pi$
$462$	0	0
$463$	35.6274	1.65575	0.827874	−	0.560915i	$-0.189550\pi$
$463$	35.6274	1.65575	0.827874	+	0.560915i	$0.189550\pi$
$464$	−31.7995	−1.47626
$465$	0	0
$466$	36.0127	1.66826
$467$	−6.45945	−0.298908	−0.149454	−	0.988769i	$-0.547752\pi$
$467$	−6.45945	−0.298908	−0.149454	+	0.988769i	$0.547752\pi$
$468$	0	0
$469$	−0.816899	−0.0377209
$470$	−21.5446	−0.993777
$471$	0	0
$472$	−4.35024	−0.200236
$473$	0	0
$474$	0	0
$475$	46.7812	2.14647
$476$	−3.70527	−0.169831
$477$	0	0
$478$	−7.09137	−0.324352
$479$	−6.80306	−0.310840	−0.155420	−	0.987849i	$-0.549673\pi$
$479$	−6.80306	−0.310840	−0.155420	+	0.987849i	$0.549673\pi$
$480$	0	0
$481$	−32.0876	−1.46307
$482$	12.1523	0.553521
$483$	0	0
$484$	0	0
$485$	30.3214	1.37682
$486$	0	0
$487$	26.7244	1.21100	0.605499	−	0.795846i	$-0.292974\pi$
$487$	26.7244	1.21100	0.605499	+	0.795846i	$0.292974\pi$
$488$	−0.267422	−0.0121056
$489$	0	0
$490$	−7.89093	−0.356476
$491$	−6.09257	−0.274954	−0.137477	−	0.990505i	$-0.543899\pi$
$491$	−6.09257	−0.274954	−0.137477	+	0.990505i	$0.543899\pi$
$492$	0	0
$493$	16.3789	0.737667
$494$	37.6268	1.69291
$495$	0	0
$496$	−24.0922	−1.08177
$497$	5.72995	0.257023
$498$	0	0
$499$	1.49552	0.0669488	0.0334744	−	0.999440i	$-0.489343\pi$
$499$	1.49552	0.0669488	0.0334744	+	0.999440i	$0.489343\pi$
$500$	37.0064	1.65498
$501$	0	0
$502$	24.7462	1.10448
$503$	14.3833	0.641321	0.320660	−	0.947194i	$-0.396095\pi$
$503$	14.3833	0.641321	0.320660	+	0.947194i	$0.396095\pi$
$504$	0	0
$505$	−6.45820	−0.287386
$506$	0	0
$507$	0	0
$508$	47.5786	2.11096
$509$	2.78397	0.123397	0.0616986	−	0.998095i	$-0.480348\pi$
$509$	2.78397	0.123397	0.0616986	+	0.998095i	$0.480348\pi$
$510$	0	0
$511$	−2.47516	−0.109495
$512$	−29.1352	−1.28761
$513$	0	0
$514$	−30.0156	−1.32393
$515$	−10.1037	−0.445221
$516$	0	0
$517$	0	0
$518$	19.4797	0.855890
$519$	0	0
$520$	11.6712	0.511816
$521$	−40.5289	−1.77560	−0.887801	−	0.460227i	$-0.847768\pi$
$521$	−40.5289	−1.77560	−0.887801	+	0.460227i	$0.847768\pi$
$522$	0	0
$523$	−0.895277	−0.0391477	−0.0195739	−	0.999808i	$-0.506231\pi$
$523$	−0.895277	−0.0391477	−0.0195739	+	0.999808i	$0.506231\pi$
$524$	−32.4358	−1.41697
$525$	0	0
$526$	10.2103	0.445192
$527$	12.4091	0.540549
$528$	0	0
$529$	0.764354	0.0332328
$530$	−22.0468	−0.957651
$531$	0	0
$532$	−12.5223	−0.542911
$533$	37.7674	1.63589
$534$	0	0
$535$	−18.6771	−0.807481
$536$	−0.733502	−0.0316825
$537$	0	0
$538$	−13.3701	−0.576426
$539$	0	0
$540$	0	0
$541$	2.86106	0.123007	0.0615033	−	0.998107i	$-0.480411\pi$
$541$	2.86106	0.123007	0.0615033	+	0.998107i	$0.480411\pi$
$542$	−32.0131	−1.37508
$543$	0	0
$544$	12.2647	0.525844
$545$	−42.0281	−1.80029
$546$	0	0
$547$	−26.7191	−1.14243	−0.571214	−	0.820801i	$-0.693527\pi$
$547$	−26.7191	−1.14243	−0.571214	+	0.820801i	$0.693527\pi$
$548$	−12.8128	−0.547338
$549$	0	0
$550$	0	0
$551$	55.3540	2.35816
$552$	0	0
$553$	11.4243	0.485809
$554$	−0.798034	−0.0339052
$555$	0	0
$556$	51.8738	2.19994
$557$	−11.4407	−0.484756	−0.242378	−	0.970182i	$-0.577927\pi$
$557$	−11.4407	−0.484756	−0.242378	+	0.970182i	$0.577927\pi$
$558$	0	0
$559$	−0.477154	−0.0201814
$560$	11.1176	0.469806
$561$	0	0
$562$	−10.6114	−0.447615
$563$	12.2142	0.514769	0.257385	−	0.966309i	$-0.417139\pi$
$563$	12.2142	0.514769	0.257385	+	0.966309i	$0.417139\pi$
$564$	0	0
$565$	−61.6056	−2.59177
$566$	−67.6118	−2.84194
$567$	0	0
$568$	5.14498	0.215879
$569$	20.3769	0.854243	0.427121	−	0.904194i	$-0.359528\pi$
$569$	20.3769	0.854243	0.427121	+	0.904194i	$0.359528\pi$
$570$	0	0
$571$	20.3621	0.852129	0.426065	−	0.904693i	$-0.359900\pi$
$571$	20.3621	0.852129	0.426065	+	0.904693i	$0.359900\pi$
$572$	0	0
$573$	0	0
$574$	−22.9278	−0.956989
$575$	44.1954	1.84308
$576$	0	0
$577$	6.04479	0.251648	0.125824	−	0.992053i	$-0.459843\pi$
$577$	6.04479	0.251648	0.125824	+	0.992053i	$0.459843\pi$
$578$	30.8629	1.28373
$579$	0	0
$580$	97.6349	4.05407
$581$	10.6460	0.441669
$582$	0	0
$583$	0	0
$584$	−2.22247	−0.0919665
$585$	0	0
$586$	−56.4386	−2.33146
$587$	−36.5418	−1.50824	−0.754120	−	0.656736i	$-0.771936\pi$
$587$	−36.5418	−1.50824	−0.754120	+	0.656736i	$0.771936\pi$
$588$	0	0
$589$	41.9378	1.72802
$590$	−38.2304	−1.57392
$591$	0	0
$592$	−27.4453	−1.12799
$593$	15.6666	0.643349	0.321675	−	0.946850i	$-0.395754\pi$
$593$	15.6666	0.643349	0.321675	+	0.946850i	$0.395754\pi$
$594$	0	0
$595$	−5.72633	−0.234757
$596$	26.6861	1.09310
$597$	0	0
$598$	35.5471	1.45363
$599$	−28.4274	−1.16151	−0.580756	−	0.814078i	$-0.697243\pi$
$599$	−28.4274	−1.16151	−0.580756	+	0.814078i	$0.697243\pi$
$600$	0	0
$601$	12.3125	0.502239	0.251120	−	0.967956i	$-0.419201\pi$
$601$	12.3125	0.502239	0.251120	+	0.967956i	$0.419201\pi$
$602$	0.289670	0.0118061
$603$	0	0
$604$	8.59487	0.349720
$605$	0	0
$606$	0	0
$607$	33.5574	1.36205	0.681026	−	0.732259i	$-0.261534\pi$
$607$	33.5574	1.36205	0.681026	+	0.732259i	$0.261534\pi$
$608$	41.4497	1.68101
$609$	0	0
$610$	−2.35013	−0.0951539
$611$	−9.46254	−0.382813
$612$	0	0
$613$	5.38046	0.217315	0.108657	−	0.994079i	$-0.465345\pi$
$613$	5.38046	0.217315	0.108657	+	0.994079i	$0.465345\pi$
$614$	−12.1021	−0.488402
$615$	0	0
$616$	0	0
$617$	29.3058	1.17981	0.589904	−	0.807473i	$-0.299166\pi$
$617$	29.3058	1.17981	0.589904	+	0.807473i	$0.299166\pi$
$618$	0	0
$619$	3.35835	0.134984	0.0674918	−	0.997720i	$-0.478500\pi$
$619$	3.35835	0.134984	0.0674918	+	0.997720i	$0.478500\pi$
$620$	73.9710	2.97075
$621$	0	0
$622$	37.6207	1.50845
$623$	8.91589	0.357207
$624$	0	0
$625$	11.8620	0.474478
$626$	−41.0289	−1.63984
$627$	0	0
$628$	30.9633	1.23557
$629$	14.1361	0.563645
$630$	0	0
$631$	−31.1413	−1.23972	−0.619858	−	0.784714i	$-0.712810\pi$
$631$	−31.1413	−1.23972	−0.619858	+	0.784714i	$0.712810\pi$
$632$	10.2580	0.408040
$633$	0	0
$634$	−36.8426	−1.46321
$635$	73.5306	2.91797
$636$	0	0
$637$	−3.46575	−0.137318
$638$	0	0
$639$	0	0
$640$	26.3273	1.04068
$641$	−21.2928	−0.841017	−0.420508	−	0.907289i	$-0.638148\pi$
$641$	−21.2928	−0.841017	−0.420508	+	0.907289i	$0.638148\pi$
$642$	0	0
$643$	22.7558	0.897399	0.448700	−	0.893683i	$-0.351887\pi$
$643$	22.7558	0.897399	0.448700	+	0.893683i	$0.351887\pi$
$644$	−11.8302	−0.466174
$645$	0	0
$646$	−16.5765	−0.652192
$647$	−43.0823	−1.69374	−0.846870	−	0.531799i	$-0.821516\pi$
$647$	−43.0823	−1.69374	−0.846870	+	0.531799i	$0.821516\pi$
$648$	0	0
$649$	0	0
$650$	66.1082	2.59298
$651$	0	0
$652$	−39.5198	−1.54772
$653$	−35.7629	−1.39951	−0.699756	−	0.714382i	$-0.746708\pi$
$653$	−35.7629	−1.39951	−0.699756	+	0.714382i	$0.746708\pi$
$654$	0	0
$655$	−50.1282	−1.95867
$656$	32.3033	1.26123
$657$	0	0
$658$	5.74451	0.223944
$659$	16.6988	0.650495	0.325247	−	0.945629i	$-0.394552\pi$
$659$	16.6988	0.650495	0.325247	+	0.945629i	$0.394552\pi$
$660$	0	0
$661$	36.5422	1.42133	0.710664	−	0.703532i	$-0.248395\pi$
$661$	36.5422	1.42133	0.710664	+	0.703532i	$0.248395\pi$
$662$	37.0787	1.44111
$663$	0	0
$664$	9.55912	0.370966
$665$	−19.3527	−0.750466
$666$	0	0
$667$	52.2944	2.02485
$668$	−13.7423	−0.531706
$669$	0	0
$670$	−6.44609	−0.249034
$671$	0	0
$672$	0	0
$673$	−28.6689	−1.10510	−0.552552	−	0.833478i	$-0.686346\pi$
$673$	−28.6689	−1.10510	−0.552552	+	0.833478i	$0.686346\pi$
$674$	3.36867	0.129756
$675$	0	0
$676$	−2.39899	−0.0922690
$677$	10.9806	0.422019	0.211010	−	0.977484i	$-0.432325\pi$
$677$	10.9806	0.422019	0.211010	+	0.977484i	$0.432325\pi$
$678$	0	0
$679$	−8.08471	−0.310263
$680$	−5.14173	−0.197176
$681$	0	0
$682$	0	0
$683$	−25.3941	−0.971679	−0.485840	−	0.874048i	$-0.661486\pi$
$683$	−25.3941	−0.971679	−0.485840	+	0.874048i	$0.661486\pi$
$684$	0	0
$685$	−19.8017	−0.756584
$686$	2.10399	0.0803306
$687$	0	0
$688$	−0.408121	−0.0155595
$689$	−9.68310	−0.368897
$690$	0	0
$691$	−26.5965	−1.01178	−0.505889	−	0.862599i	$-0.668835\pi$
$691$	−26.5965	−1.01178	−0.505889	+	0.862599i	$0.668835\pi$
$692$	32.5706	1.23815
$693$	0	0
$694$	−7.24744	−0.275109
$695$	80.1687	3.04097
$696$	0	0
$697$	−16.6384	−0.630224
$698$	7.14820	0.270563
$699$	0	0
$700$	−22.0010	−0.831559
$701$	−45.1579	−1.70559	−0.852796	−	0.522245i	$-0.825095\pi$
$701$	−45.1579	−1.70559	−0.852796	+	0.522245i	$0.825095\pi$
$702$	0	0
$703$	47.7745	1.80185
$704$	0	0
$705$	0	0
$706$	21.8989	0.824177
$707$	1.72198	0.0647615
$708$	0	0
$709$	42.7235	1.60452	0.802258	−	0.596978i	$-0.203632\pi$
$709$	42.7235	1.60452	0.802258	+	0.596978i	$0.203632\pi$
$710$	45.2146	1.69687
$711$	0	0
$712$	8.00567	0.300025
$713$	39.6198	1.48377
$714$	0	0
$715$	0	0
$716$	−9.24214	−0.345395
$717$	0	0
$718$	−48.3809	−1.80556
$719$	15.4401	0.575819	0.287909	−	0.957658i	$-0.407040\pi$
$719$	15.4401	0.575819	0.287909	+	0.957658i	$0.407040\pi$
$720$	0	0
$721$	2.69398	0.100329
$722$	−16.0460	−0.597171
$723$	0	0
$724$	−23.7546	−0.882834
$725$	97.2539	3.61192
$726$	0	0
$727$	14.8932	0.552357	0.276178	−	0.961106i	$-0.410932\pi$
$727$	14.8932	0.552357	0.276178	+	0.961106i	$0.410932\pi$
$728$	−3.11194	−0.115336
$729$	0	0
$730$	−19.5313	−0.722886
$731$	0.210209	0.00777488
$732$	0	0
$733$	7.38664	0.272832	0.136416	−	0.990652i	$-0.456442\pi$
$733$	7.38664	0.272832	0.136416	+	0.990652i	$0.456442\pi$
$734$	−41.3133	−1.52490
$735$	0	0
$736$	39.1586	1.44341
$737$	0	0
$738$	0	0
$739$	−34.8552	−1.28217	−0.641084	−	0.767471i	$-0.721515\pi$
$739$	−34.8552	−1.28217	−0.641084	+	0.767471i	$0.721515\pi$
$740$	84.2660	3.09768
$741$	0	0
$742$	5.87842	0.215803
$743$	−3.33330	−0.122287	−0.0611434	−	0.998129i	$-0.519475\pi$
$743$	−3.33330	−0.122287	−0.0611434	+	0.998129i	$0.519475\pi$
$744$	0	0
$745$	41.2422	1.51100
$746$	−67.1271	−2.45770
$747$	0	0
$748$	0	0
$749$	4.97994	0.181963
$750$	0	0
$751$	51.2589	1.87046	0.935232	−	0.354036i	$-0.115191\pi$
$751$	51.2589	1.87046	0.935232	+	0.354036i	$0.115191\pi$
$752$	−8.09353	−0.295141
$753$	0	0
$754$	78.2228	2.84871
$755$	13.2830	0.483418
$756$	0	0
$757$	4.07426	0.148081	0.0740407	−	0.997255i	$-0.476411\pi$
$757$	4.07426	0.148081	0.0740407	+	0.997255i	$0.476411\pi$
$758$	3.61797	0.131411
$759$	0	0
$760$	−17.3770	−0.630330
$761$	6.01136	0.217912	0.108956	−	0.994047i	$-0.465249\pi$
$761$	6.01136	0.217912	0.108956	+	0.994047i	$0.465249\pi$
$762$	0	0
$763$	11.2061	0.405689
$764$	−21.8158	−0.789267
$765$	0	0
$766$	18.2634	0.659885
$767$	−16.7910	−0.606290
$768$	0	0
$769$	−43.2310	−1.55895	−0.779474	−	0.626434i	$-0.784514\pi$
$769$	−43.2310	−1.55895	−0.779474	+	0.626434i	$0.784514\pi$
$770$	0	0
$771$	0	0
$772$	14.1951	0.510894
$773$	9.64530	0.346917	0.173459	−	0.984841i	$-0.444506\pi$
$773$	9.64530	0.346917	0.173459	+	0.984841i	$0.444506\pi$
$774$	0	0
$775$	73.6823	2.64675
$776$	−7.25935	−0.260595
$777$	0	0
$778$	−56.3906	−2.02170
$779$	−56.2311	−2.01469
$780$	0	0
$781$	0	0
$782$	−15.6602	−0.560008
$783$	0	0
$784$	−2.96434	−0.105869
$785$	47.8524	1.70793
$786$	0	0
$787$	24.1192	0.859757	0.429879	−	0.902887i	$-0.358556\pi$
$787$	24.1192	0.859757	0.429879	+	0.902887i	$0.358556\pi$
$788$	15.0351	0.535605
$789$	0	0
$790$	90.1480	3.20732
$791$	16.4261	0.584046
$792$	0	0
$793$	−1.03219	−0.0366543
$794$	37.6011	1.33441
$795$	0	0
$796$	−45.7771	−1.62253
$797$	42.4069	1.50213	0.751066	−	0.660228i	$-0.229540\pi$
$797$	42.4069	1.50213	0.751066	+	0.660228i	$0.229540\pi$
$798$	0	0
$799$	4.16871	0.147478
$800$	72.8247	2.57474
$801$	0	0
$802$	−60.8338	−2.14812
$803$	0	0
$804$	0	0
$805$	−18.2830	−0.644392
$806$	59.2638	2.08748
$807$	0	0
$808$	1.54618	0.0543944
$809$	−3.46232	−0.121729	−0.0608643	−	0.998146i	$-0.519386\pi$
$809$	−3.46232	−0.121729	−0.0608643	+	0.998146i	$0.519386\pi$
$810$	0	0
$811$	−15.7547	−0.553224	−0.276612	−	0.960982i	$-0.589212\pi$
$811$	−15.7547	−0.553224	−0.276612	+	0.960982i	$0.589212\pi$
$812$	−26.0328	−0.913571
$813$	0	0
$814$	0	0
$815$	−61.0762	−2.13941
$816$	0	0
$817$	0.710424	0.0248546
$818$	58.1927	2.03466
$819$	0	0
$820$	−99.1819	−3.46358
$821$	7.93147	0.276810	0.138405	−	0.990376i	$-0.455802\pi$
$821$	7.93147	0.276810	0.138405	+	0.990376i	$0.455802\pi$
$822$	0	0
$823$	32.9402	1.14822	0.574111	−	0.818777i	$-0.305348\pi$
$823$	32.9402	1.14822	0.574111	+	0.818777i	$0.305348\pi$
$824$	2.41895	0.0842682
$825$	0	0
$826$	10.1935	0.354677
$827$	30.1775	1.04937	0.524687	−	0.851295i	$-0.324182\pi$
$827$	30.1775	1.04937	0.524687	+	0.851295i	$0.324182\pi$
$828$	0	0
$829$	−21.7848	−0.756619	−0.378309	−	0.925679i	$-0.623494\pi$
$829$	−21.7848	−0.756619	−0.378309	+	0.925679i	$0.623494\pi$
$830$	84.0065	2.91591
$831$	0	0
$832$	38.0267	1.31834
$833$	1.52683	0.0529016
$834$	0	0
$835$	−21.2381	−0.734976
$836$	0	0
$837$	0	0
$838$	−9.48906	−0.327794
$839$	13.4974	0.465982	0.232991	−	0.972479i	$-0.425149\pi$
$839$	13.4974	0.465982	0.232991	+	0.972479i	$0.425149\pi$
$840$	0	0
$841$	86.0761	2.96814
$842$	−75.6268	−2.60627
$843$	0	0
$844$	−1.55357	−0.0534760
$845$	−3.70754	−0.127543
$846$	0	0
$847$	0	0
$848$	−8.28218	−0.284411
$849$	0	0
$850$	−29.1239	−0.998940
$851$	45.1339	1.54717
$852$	0	0
$853$	−22.5171	−0.770971	−0.385485	−	0.922714i	$-0.625966\pi$
$853$	−22.5171	−0.770971	−0.385485	+	0.922714i	$0.625966\pi$
$854$	0.626624	0.0214426
$855$	0	0
$856$	4.47154	0.152834
$857$	12.2772	0.419383	0.209691	−	0.977768i	$-0.432754\pi$
$857$	12.2772	0.419383	0.209691	+	0.977768i	$0.432754\pi$
$858$	0	0
$859$	−46.4901	−1.58622	−0.793110	−	0.609078i	$-0.791540\pi$
$859$	−46.4901	−1.58622	−0.793110	+	0.609078i	$0.791540\pi$
$860$	1.25306	0.0427291
$861$	0	0
$862$	46.7526	1.59240
$863$	−54.4619	−1.85391	−0.926953	−	0.375178i	$-0.877581\pi$
$863$	−54.4619	−1.85391	−0.926953	+	0.375178i	$0.877581\pi$
$864$	0	0
$865$	50.3364	1.71149
$866$	−53.6290	−1.82239
$867$	0	0
$868$	−19.7232	−0.669448
$869$	0	0
$870$	0	0
$871$	−2.83117	−0.0959306
$872$	10.0621	0.340746
$873$	0	0
$874$	−52.9253	−1.79022
$875$	−15.2493	−0.515519
$876$	0	0
$877$	29.4571	0.994697	0.497348	−	0.867551i	$-0.334307\pi$
$877$	29.4571	0.994697	0.497348	+	0.867551i	$0.334307\pi$
$878$	23.4398	0.791056
$879$	0	0
$880$	0	0
$881$	29.9763	1.00993	0.504964	−	0.863141i	$-0.331506\pi$
$881$	29.9763	1.00993	0.504964	+	0.863141i	$0.331506\pi$
$882$	0	0
$883$	30.2544	1.01814	0.509071	−	0.860725i	$-0.329989\pi$
$883$	30.2544	1.01814	0.509071	+	0.860725i	$0.329989\pi$
$884$	−12.8415	−0.431908
$885$	0	0
$886$	29.1428	0.979073
$887$	47.3156	1.58870	0.794352	−	0.607458i	$-0.207811\pi$
$887$	47.3156	1.58870	0.794352	+	0.607458i	$0.207811\pi$
$888$	0	0
$889$	−19.6058	−0.657556
$890$	70.3546	2.35829
$891$	0	0
$892$	−40.5567	−1.35794
$893$	14.0886	0.471456
$894$	0	0
$895$	−14.2833	−0.477439
$896$	−7.01975	−0.234513
$897$	0	0
$898$	22.4871	0.750404
$899$	87.1849	2.90778
$900$	0	0
$901$	4.26588	0.142117
$902$	0	0
$903$	0	0
$904$	14.7492	0.490551
$905$	−36.7118	−1.22034
$906$	0	0
$907$	−49.6479	−1.64853	−0.824265	−	0.566204i	$-0.808411\pi$
$907$	−49.6479	−1.64853	−0.824265	+	0.566204i	$0.808411\pi$
$908$	22.6328	0.751097
$909$	0	0
$910$	−27.3480	−0.906578
$911$	21.5478	0.713909	0.356954	−	0.934122i	$-0.383815\pi$
$911$	21.5478	0.713909	0.356954	+	0.934122i	$0.383815\pi$
$912$	0	0
$913$	0	0
$914$	66.0185	2.18370
$915$	0	0
$916$	51.8639	1.71363
$917$	13.3659	0.441380
$918$	0	0
$919$	−44.2085	−1.45830	−0.729151	−	0.684353i	$-0.760085\pi$
$919$	−44.2085	−1.45830	−0.729151	+	0.684353i	$0.760085\pi$
$920$	−16.4165	−0.541236
$921$	0	0
$922$	−86.6516	−2.85372
$923$	19.8586	0.653654
$924$	0	0
$925$	83.9371	2.75983
$926$	−74.9597	−2.46333
$927$	0	0
$928$	86.1702	2.82868
$929$	−1.70847	−0.0560532	−0.0280266	−	0.999607i	$-0.508922\pi$
$929$	−1.70847	−0.0560532	−0.0280266	+	0.999607i	$0.508922\pi$
$930$	0	0
$931$	5.16008	0.169115
$932$	−41.5375	−1.36061
$933$	0	0
$934$	13.5906	0.444698
$935$	0	0
$936$	0	0
$937$	33.3716	1.09020	0.545101	−	0.838371i	$-0.316491\pi$
$937$	33.3716	1.09020	0.545101	+	0.838371i	$0.316491\pi$
$938$	1.71875	0.0561191
$939$	0	0
$940$	24.8498	0.810511
$941$	−22.2132	−0.724128	−0.362064	−	0.932153i	$-0.617928\pi$
$941$	−22.2132	−0.724128	−0.362064	+	0.932153i	$0.617928\pi$
$942$	0	0
$943$	−53.1230	−1.72992
$944$	−14.3618	−0.467436
$945$	0	0
$946$	0	0
$947$	−16.6969	−0.542578	−0.271289	−	0.962498i	$-0.587450\pi$
$947$	−16.6969	−0.542578	−0.271289	+	0.962498i	$0.587450\pi$
$948$	0	0
$949$	−8.57829	−0.278463
$950$	−98.4270	−3.19339
$951$	0	0
$952$	1.37096	0.0444330
$953$	−19.0864	−0.618269	−0.309135	−	0.951018i	$-0.600039\pi$
$953$	−19.0864	−0.618269	−0.309135	+	0.951018i	$0.600039\pi$
$954$	0	0
$955$	−33.7154	−1.09100
$956$	8.17927	0.264537
$957$	0	0
$958$	14.3136	0.462450
$959$	5.27980	0.170494
$960$	0	0
$961$	35.0538	1.13077
$962$	67.5119	2.17667
$963$	0	0
$964$	−14.0166	−0.451444
$965$	21.9380	0.706208
$966$	0	0
$967$	−36.9443	−1.18805	−0.594024	−	0.804447i	$-0.702462\pi$
$967$	−36.9443	−1.18805	−0.594024	+	0.804447i	$0.702462\pi$
$968$	0	0
$969$	0	0
$970$	−63.7959	−2.04836
$971$	31.3049	1.00462	0.502310	−	0.864687i	$-0.332484\pi$
$971$	31.3049	1.00462	0.502310	+	0.864687i	$0.332484\pi$
$972$	0	0
$973$	−21.3757	−0.685273
$974$	−56.2278	−1.80165
$975$	0	0
$976$	−0.882859	−0.0282596
$977$	44.7409	1.43139	0.715695	−	0.698413i	$-0.246110\pi$
$977$	44.7409	1.43139	0.715695	+	0.698413i	$0.246110\pi$
$978$	0	0
$979$	0	0
$980$	9.10149	0.290737
$981$	0	0
$982$	12.8187	0.409061
$983$	−57.2232	−1.82514	−0.912568	−	0.408926i	$-0.865903\pi$
$983$	−57.2232	−1.82514	−0.912568	+	0.408926i	$0.865903\pi$
$984$	0	0
$985$	23.2362	0.740366
$986$	−34.4610	−1.09746
$987$	0	0
$988$	−43.3993	−1.38071
$989$	0.671156	0.0213415
$990$	0	0
$991$	−26.9663	−0.856614	−0.428307	−	0.903633i	$-0.640890\pi$
$991$	−26.9663	−0.856614	−0.428307	+	0.903633i	$0.640890\pi$
$992$	65.2850	2.07280
$993$	0	0
$994$	−12.0557	−0.382385
$995$	−70.7466	−2.24282
$996$	0	0
$997$	−25.3305	−0.802224	−0.401112	−	0.916029i	$-0.631376\pi$
$997$	−25.3305	−0.802224	−0.401112	+	0.916029i	$0.631376\pi$
$998$	−3.14656	−0.0996027
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.db.1.3		16
3.2	odd	2	inner	7623.2.a.db.1.14		16
11.2	odd	10		693.2.m.k.631.2	yes	32
11.6	odd	10		693.2.m.k.190.2	✓	32
11.10	odd	2		7623.2.a.dc.1.14		16
33.2	even	10		693.2.m.k.631.7	yes	32
33.17	even	10		693.2.m.k.190.7	yes	32
33.32	even	2		7623.2.a.dc.1.3		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
693.2.m.k.190.2	✓	32	11.6	odd	10
693.2.m.k.190.7	yes	32	33.17	even	10
693.2.m.k.631.2	yes	32	11.2	odd	10
693.2.m.k.631.7	yes	32	33.2	even	10
7623.2.a.db.1.3		16	1.1	even	1	trivial
7623.2.a.db.1.14		16	3.2	odd	2	inner
7623.2.a.dc.1.3		16	33.32	even	2
7623.2.a.dc.1.14		16	11.10	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7623.a (trivial)

\(f(q)\)	\(=\)	\(q-2.10399 q^{2} +2.42677 q^{4} +3.75046 q^{5} -1.00000 q^{7} -0.897910 q^{8} +O(q^{10})\)	\(q-2.10399 q^{2} +2.42677 q^{4} +3.75046 q^{5} -1.00000 q^{7} -0.897910 q^{8} -7.89093 q^{10} -3.46575 q^{13} +2.10399 q^{14} -2.96434 q^{16} +1.52683 q^{17} +5.16008 q^{19} +9.10149 q^{20} +4.87487 q^{23} +9.06597 q^{25} +7.29190 q^{26} -2.42677 q^{28} +10.7274 q^{29} +8.12735 q^{31} +8.03275 q^{32} -3.21244 q^{34} -3.75046 q^{35} +9.25848 q^{37} -10.8568 q^{38} -3.36758 q^{40} -10.8973 q^{41} +0.137677 q^{43} -10.2567 q^{46} +2.73030 q^{47} +1.00000 q^{49} -19.0747 q^{50} -8.41057 q^{52} +2.79394 q^{53} +0.897910 q^{56} -22.5702 q^{58} +4.84485 q^{59} +0.297827 q^{61} -17.0998 q^{62} -10.9721 q^{64} -12.9982 q^{65} +0.816899 q^{67} +3.70527 q^{68} +7.89093 q^{70} -5.72995 q^{71} +2.47516 q^{73} -19.4797 q^{74} +12.5223 q^{76} -11.4243 q^{79} -11.1176 q^{80} +22.9278 q^{82} -10.6460 q^{83} +5.72633 q^{85} -0.289670 q^{86} -8.91589 q^{89} +3.46575 q^{91} +11.8302 q^{92} -5.74451 q^{94} +19.3527 q^{95} +8.08471 q^{97} -2.10399 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 20 q^{4} - 16 q^{7}+O(q^{10}) \) 16 * q + 20 * q^4 - 16 * q^7	\( 16 q + 20 q^{4} - 16 q^{7} + 6 q^{10} + 32 q^{16} - 10 q^{19} + 44 q^{25} - 20 q^{28} + 30 q^{31} + 12 q^{34} + 38 q^{37} + 68 q^{40} - 16 q^{43} - 80 q^{46} + 16 q^{49} - 2 q^{52} + 18 q^{58} + 28 q^{61} + 34 q^{64} + 52 q^{67} - 6 q^{70} + 14 q^{73} + 14 q^{76} - 54 q^{79} + 64 q^{82} - 30 q^{85} + 60 q^{94} + 108 q^{97}+O(q^{100}) \) 16 * q + 20 * q^4 - 16 * q^7 + 6 * q^10 + 32 * q^16 - 10 * q^19 + 44 * q^25 - 20 * q^28 + 30 * q^31 + 12 * q^34 + 38 * q^37 + 68 * q^40 - 16 * q^43 - 80 * q^46 + 16 * q^49 - 2 * q^52 + 18 * q^58 + 28 * q^61 + 34 * q^64 + 52 * q^67 - 6 * q^70 + 14 * q^73 + 14 * q^76 - 54 * q^79 + 64 * q^82 - 30 * q^85 + 60 * q^94 + 108 * q^97

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