LMFDB - Embedded newform 7623.2.a.dc.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	$-0.816642\pi$
$5$	−3.75046	−1.67726	−0.838629	+	0.544703i	$0.816642\pi$
$6$	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	$-0.340414\pi$
$13$	3.46575	0.961227	0.480613	+	0.876933i	$0.340414\pi$
$14$	−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	$-0.701623\pi$
$19$	−5.16008	−1.18380	−0.591902	+	0.806010i	$0.701623\pi$
$20$	−9.10149	−2.03516
$21$	0	0
$22$	0	0
$23$	−4.87487	−1.01648	−0.508240	−	0.861215i	$-0.669704\pi$
$23$	−4.87487	−1.01648	−0.508240	+	0.861215i	$0.669704\pi$
$24$	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	$-0.503342\pi$
$43$	−0.137677	−0.0209955	−0.0104978	+	0.999945i	$0.503342\pi$
$44$	0	0
$45$	0	0
$46$	10.2567	1.51226
$47$	−2.73030	−0.398255	−0.199127	−	0.979974i	$-0.563811\pi$
$47$	−2.73030	−0.398255	−0.199127	+	0.979974i	$0.563811\pi$
$48$	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	$-0.561461\pi$
$53$	−2.79394	−0.383777	−0.191889	+	0.981417i	$0.561461\pi$
$54$	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	$-0.602130\pi$
$59$	−4.84485	−0.630746	−0.315373	+	0.948968i	$0.602130\pi$
$60$	0	0
$61$	−0.297827	−0.0381328	−0.0190664	−	0.999818i	$-0.506069\pi$
$61$	−0.297827	−0.0381328	−0.0190664	+	0.999818i	$0.506069\pi$
$62$	−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	$-0.389570\pi$
$71$	5.72995	0.680020	0.340010	+	0.940422i	$0.389570\pi$
$72$	0	0
$73$	−2.47516	−0.289695	−0.144848	−	0.989454i	$-0.546269\pi$
$73$	−2.47516	−0.289695	−0.144848	+	0.989454i	$0.546269\pi$
$74$	−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	$-0.277829\pi$
$79$	11.4243	1.28533	0.642664	+	0.766148i	$0.277829\pi$
$80$	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	$-0.343337\pi$
$89$	8.91589	0.945082	0.472541	+	0.881309i	$0.343337\pi$
$90$	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	$-0.319680\pi$
$109$	11.2061	1.07335	0.536676	+	0.843788i	$0.319680\pi$
$110$	0	0
$111$	0	0
$112$	−2.96434	−0.280104
$113$	16.4261	1.54524	0.772620	−	0.634869i	$-0.218946\pi$
$113$	16.4261	1.54524	0.772620	+	0.634869i	$0.218946\pi$
$114$	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	$-0.835794\pi$
$127$	−19.6058	−1.73973	−0.869865	+	0.493291i	$0.835794\pi$
$128$	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	$-0.427585\pi$
$137$	5.27980	0.451084	0.225542	+	0.974233i	$0.427585\pi$
$138$	0	0
$139$	−21.3757	−1.81306	−0.906531	−	0.422139i	$-0.861280\pi$
$139$	−21.3757	−1.81306	−0.906531	+	0.422139i	$0.861280\pi$
$140$	−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	$-0.546032\pi$
$151$	−3.54170	−0.288219	−0.144110	+	0.989562i	$0.546032\pi$
$152$	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	$-0.454541\pi$
$179$	3.80842	0.284654	0.142327	+	0.989820i	$0.454541\pi$
$180$	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	$-0.394557\pi$
$191$	8.98965	0.650469	0.325234	+	0.945633i	$0.394557\pi$
$192$	0	0
$193$	−5.84941	−0.421049	−0.210525	−	0.977589i	$-0.567517\pi$
$193$	−5.84941	−0.421049	−0.210525	+	0.977589i	$0.567517\pi$
$194$	−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	$-0.492985\pi$
$211$	0.640180	0.0440718	0.0220359	+	0.999757i	$0.492985\pi$
$212$	−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	$-0.440439\pi$
$241$	5.77583	0.372054	0.186027	+	0.982545i	$0.440439\pi$
$242$	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	$-0.378949\pi$
$251$	11.7616	0.742383	0.371191	+	0.928556i	$0.378949\pi$
$252$	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	$-0.646777\pi$
$257$	−14.2660	−0.889891	−0.444945	+	0.895558i	$0.646777\pi$
$258$	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	$-0.562057\pi$
$269$	−6.35465	−0.387450	−0.193725	+	0.981056i	$0.562057\pi$
$270$	0	0
$271$	−15.2155	−0.924273	−0.462137	−	0.886809i	$-0.652917\pi$
$271$	−15.2155	−0.924273	−0.462137	+	0.886809i	$0.652917\pi$
$272$	−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	$-0.503627\pi$
$277$	−0.379296	−0.0227897	−0.0113948	+	0.999935i	$0.503627\pi$
$278$	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	$-0.904270\pi$
$283$	−32.1351	−1.91023	−0.955116	+	0.296233i	$0.904270\pi$
$284$	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	$-0.552486\pi$
$307$	−5.75200	−0.328284	−0.164142	+	0.986437i	$0.552486\pi$
$308$	0	0
$309$	0	0
$310$	64.1323	3.64247
$311$	17.8807	1.01392	0.506960	−	0.861970i	$-0.330769\pi$
$311$	17.8807	1.01392	0.506960	+	0.861970i	$0.330769\pi$
$312$	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	$-0.663644\pi$
$317$	−17.5109	−0.983508	−0.491754	+	0.870734i	$0.663644\pi$
$318$	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	$-0.486115\pi$
$337$	1.60109	0.0872167	0.0436084	+	0.999049i	$0.486115\pi$
$338$	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	$-0.471016\pi$
$349$	3.39745	0.181862	0.0909308	+	0.995857i	$0.471016\pi$
$350$	−19.0747	−1.01959
$351$	0	0
$352$	0	0
$353$	10.4083	0.553977	0.276989	−	0.960873i	$-0.410664\pi$
$353$	10.4083	0.553977	0.276989	+	0.960873i	$0.410664\pi$
$354$	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	$-0.809378\pi$
$373$	−31.9047	−1.65196	−0.825981	+	0.563698i	$0.809378\pi$
$374$	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	$-0.428815\pi$
$383$	8.68039	0.443547	0.221774	+	0.975098i	$0.428815\pi$
$384$	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	$-0.737782\pi$
$389$	−26.8018	−1.35890	−0.679452	+	0.733720i	$0.737782\pi$
$390$	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	$-0.756748\pi$
$401$	−28.9136	−1.44388	−0.721938	+	0.691958i	$0.756748\pi$
$402$	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	$-0.260323\pi$
$409$	27.6583	1.36761	0.683807	+	0.729663i	$0.260323\pi$
$410$	−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	$-0.535138\pi$
$419$	−4.51003	−0.220330	−0.110165	+	0.993913i	$0.535138\pi$
$420$	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	$-0.414345\pi$
$439$	11.1407	0.531715	0.265857	+	0.964012i	$0.414345\pi$
$440$	0	0
$441$	0	0
$442$	−11.1335	−0.529567
$443$	13.8512	0.658092	0.329046	−	0.944314i	$-0.393273\pi$
$443$	13.8512	0.658092	0.329046	+	0.944314i	$0.393273\pi$
$444$	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	$-0.418848\pi$
$449$	10.6878	0.504390	0.252195	+	0.967676i	$0.418848\pi$
$450$	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	$-0.237700\pi$
$457$	31.3778	1.46779	0.733895	+	0.679263i	$0.237700\pi$
$458$	−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	$-0.452248\pi$
$467$	6.45945	0.298908	0.149454	+	0.988769i	$0.452248\pi$
$468$	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	$-0.519652\pi$
$509$	−2.78397	−0.123397	−0.0616986	+	0.998095i	$0.519652\pi$
$510$	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	$-0.152232\pi$
$521$	40.5289	1.77560	0.887801	+	0.460227i	$0.152232\pi$
$522$	0	0
$523$	0.895277	0.0391477	0.0195739	−	0.999808i	$-0.493769\pi$
$523$	0.895277	0.0391477	0.0195739	+	0.999808i	$0.493769\pi$
$524$	−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	$-0.519589\pi$
$541$	−2.86106	−0.123007	−0.0615033	+	0.998107i	$0.519589\pi$
$542$	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	$-0.306473\pi$
$547$	26.7191	1.14243	0.571214	+	0.820801i	$0.306473\pi$
$548$	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	$-0.640100\pi$
$571$	−20.3621	−0.852129	−0.426065	+	0.904693i	$0.640100\pi$
$572$	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	$-0.228064\pi$
$587$	36.5418	1.50824	0.754120	+	0.656736i	$0.228064\pi$
$588$	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	$-0.302757\pi$
$599$	28.4274	1.16151	0.580756	+	0.814078i	$0.302757\pi$
$600$	0	0
$601$	−12.3125	−0.502239	−0.251120	−	0.967956i	$-0.580799\pi$
$601$	−12.3125	−0.502239	−0.251120	+	0.967956i	$0.580799\pi$
$602$	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	$-0.738466\pi$
$607$	−33.5574	−1.36205	−0.681026	+	0.732259i	$0.738466\pi$
$608$	−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	$-0.534655\pi$
$613$	−5.38046	−0.217315	−0.108657	+	0.994079i	$0.534655\pi$
$614$	12.1021	0.488402
$615$	0	0
$616$	0	0
$617$	−29.3058	−1.17981	−0.589904	−	0.807473i	$-0.700834\pi$
$617$	−29.3058	−1.17981	−0.589904	+	0.807473i	$0.700834\pi$
$618$	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	$-0.361852\pi$
$641$	21.2928	0.841017	0.420508	+	0.907289i	$0.361852\pi$
$642$	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	$-0.178484\pi$
$647$	43.0823	1.69374	0.846870	+	0.531799i	$0.178484\pi$
$648$	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	$-0.253292\pi$
$653$	35.7629	1.39951	0.699756	+	0.714382i	$0.253292\pi$
$654$	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	$-0.313654\pi$
$673$	28.6689	1.10510	0.552552	+	0.833478i	$0.313654\pi$
$674$	−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	$-0.338514\pi$
$683$	25.3941	0.971679	0.485840	+	0.874048i	$0.338514\pi$
$684$	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	$-0.592960\pi$
$719$	−15.4401	−0.575819	−0.287909	+	0.957658i	$0.592960\pi$
$720$	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	$-0.543558\pi$
$733$	−7.38664	−0.272832	−0.136416	+	0.990652i	$0.543558\pi$
$734$	−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	$-0.278485\pi$
$739$	34.8552	1.28217	0.641084	+	0.767471i	$0.278485\pi$
$740$	−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	$-0.215486\pi$
$769$	43.2310	1.55895	0.779474	+	0.626434i	$0.215486\pi$
$770$	0	0
$771$	0	0
$772$	−14.1951	−0.510894
$773$	−9.64530	−0.346917	−0.173459	−	0.984841i	$-0.555494\pi$
$773$	−9.64530	−0.346917	−0.173459	+	0.984841i	$0.555494\pi$
$774$	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	$-0.641444\pi$
$787$	−24.1192	−0.859757	−0.429879	+	0.902887i	$0.641444\pi$
$788$	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	$-0.770460\pi$
$797$	−42.4069	−1.50213	−0.751066	+	0.660228i	$0.770460\pi$
$798$	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	$-0.410788\pi$
$811$	15.7547	0.553224	0.276612	+	0.960982i	$0.410788\pi$
$812$	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	$-0.574851\pi$
$839$	−13.4974	−0.465982	−0.232991	+	0.972479i	$0.574851\pi$
$840$	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	$-0.374034\pi$
$853$	22.5171	0.770971	0.385485	+	0.922714i	$0.374034\pi$
$854$	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	$-0.122419\pi$
$863$	54.4619	1.85391	0.926953	+	0.375178i	$0.122419\pi$
$864$	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	$-0.665693\pi$
$877$	−29.4571	−0.994697	−0.497348	+	0.867551i	$0.665693\pi$
$878$	−23.4398	−0.791056
$879$	0	0
$880$	0	0
$881$	−29.9763	−1.00993	−0.504964	−	0.863141i	$-0.668494\pi$
$881$	−29.9763	−1.00993	−0.504964	+	0.863141i	$0.668494\pi$
$882$	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	$-0.616185\pi$
$911$	−21.5478	−0.713909	−0.356954	+	0.934122i	$0.616185\pi$
$912$	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	$-0.239915\pi$
$919$	44.2085	1.45830	0.729151	+	0.684353i	$0.239915\pi$
$920$	−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	$-0.491078\pi$
$929$	1.70847	0.0560532	0.0280266	+	0.999607i	$0.491078\pi$
$930$	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	$-0.683509\pi$
$937$	−33.3716	−1.09020	−0.545101	+	0.838371i	$0.683509\pi$
$938$	−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	$-0.412550\pi$
$947$	16.6969	0.542578	0.271289	+	0.962498i	$0.412550\pi$
$948$	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	$-0.297538\pi$
$967$	36.9443	1.18805	0.594024	+	0.804447i	$0.297538\pi$
$968$	0	0
$969$	0	0
$970$	63.7959	2.04836
$971$	−31.3049	−1.00462	−0.502310	−	0.864687i	$-0.667516\pi$
$971$	−31.3049	−1.00462	−0.502310	+	0.864687i	$0.667516\pi$
$972$	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	$-0.753890\pi$
$977$	−44.7409	−1.43139	−0.715695	+	0.698413i	$0.753890\pi$
$978$	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	$-0.134097\pi$
$983$	57.2232	1.82514	0.912568	+	0.408926i	$0.134097\pi$
$984$	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	$-0.368624\pi$
$997$	25.3305	0.802224	0.401112	+	0.916029i	$0.368624\pi$
$998$	−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.dc.1.3		16
3.2	odd	2	inner	7623.2.a.dc.1.14		16
11.5	even	5		693.2.m.k.190.7	yes	32
11.9	even	5		693.2.m.k.631.7	yes	32
11.10	odd	2		7623.2.a.db.1.14		16
33.5	odd	10		693.2.m.k.190.2	✓	32
33.20	odd	10		693.2.m.k.631.2	yes	32
33.32	even	2		7623.2.a.db.1.3		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
693.2.m.k.190.2	✓	32	33.5	odd	10
693.2.m.k.190.7	yes	32	11.5	even	5
693.2.m.k.631.2	yes	32	33.20	odd	10
693.2.m.k.631.7	yes	32	11.9	even	5
7623.2.a.db.1.3		16	33.32	even	2
7623.2.a.db.1.14		16	11.10	odd	2
7623.2.a.dc.1.3		16	1.1	even	1	trivial
7623.2.a.dc.1.14		16	3.2	odd	2	inner

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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