LMFDB - Embedded newform 7623.2.a.cz.1.6

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:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 22x^{10} + 181x^{8} - 692x^{6} + 1240x^{4} - 936x^{2} + 244 $ x^12 - 22x^10 + 181x^8 - 692x^6 + 1240x^4 - 936*x^2 + 244
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-0.790737$ 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)

$f(q)$	$=$	$q-0.790737 q^{2} -1.37474 q^{4} +4.15216 q^{5} -1.00000 q^{7} +2.66853 q^{8} +O(q^{10})$	$q-0.790737 q^{2} -1.37474 q^{4} +4.15216 q^{5} -1.00000 q^{7} +2.66853 q^{8} -3.28327 q^{10} -2.46926 q^{13} +0.790737 q^{14} +0.639367 q^{16} -1.99183 q^{17} -3.11195 q^{19} -5.70812 q^{20} -3.45926 q^{23} +12.2404 q^{25} +1.95254 q^{26} +1.37474 q^{28} -3.32309 q^{29} -0.288427 q^{31} -5.84263 q^{32} +1.57501 q^{34} -4.15216 q^{35} +7.59863 q^{37} +2.46073 q^{38} +11.0802 q^{40} +2.32456 q^{41} +8.35116 q^{43} +2.73537 q^{46} -2.94175 q^{47} +1.00000 q^{49} -9.67896 q^{50} +3.39458 q^{52} -13.1332 q^{53} -2.66853 q^{56} +2.62769 q^{58} +7.82330 q^{59} -15.3255 q^{61} +0.228070 q^{62} +3.34125 q^{64} -10.2528 q^{65} -8.39127 q^{67} +2.73823 q^{68} +3.28327 q^{70} +13.6523 q^{71} -8.86448 q^{73} -6.00852 q^{74} +4.27810 q^{76} +15.5540 q^{79} +2.65475 q^{80} -1.83812 q^{82} -4.58873 q^{83} -8.27038 q^{85} -6.60357 q^{86} -11.8128 q^{89} +2.46926 q^{91} +4.75557 q^{92} +2.32615 q^{94} -12.9213 q^{95} -7.28658 q^{97} -0.790737 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 20 q^{4} - 12 q^{7}+O(q^{10}) $ 12 * q + 20 * q^4 - 12 * q^7	$ 12 q + 20 q^{4} - 12 q^{7} - 20 q^{10} - 20 q^{13} + 28 q^{16} - 12 q^{19} + 32 q^{25} - 20 q^{28} - 16 q^{31} - 24 q^{34} + 4 q^{37} - 48 q^{40} - 16 q^{43} - 24 q^{46} + 12 q^{49} - 96 q^{52} + 20 q^{58} - 44 q^{61} + 76 q^{64} + 20 q^{70} - 52 q^{73} + 8 q^{79} - 68 q^{82} - 72 q^{85} + 20 q^{91} - 20 q^{94} - 32 q^{97}+O(q^{100}) $ 12 * q + 20 * q^4 - 12 * q^7 - 20 * q^10 - 20 * q^13 + 28 * q^16 - 12 * q^19 + 32 * q^25 - 20 * q^28 - 16 * q^31 - 24 * q^34 + 4 * q^37 - 48 * q^40 - 16 * q^43 - 24 * q^46 + 12 * q^49 - 96 * q^52 + 20 * q^58 - 44 * q^61 + 76 * q^64 + 20 * q^70 - 52 * q^73 + 8 * q^79 - 68 * q^82 - 72 * q^85 + 20 * q^91 - 20 * q^94 - 32 * 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 $

$2$	−0.790737	−0.559135	−0.279568	−	0.960126i	$-0.590191\pi$
$2$	−0.790737	−0.559135	−0.279568	+	0.960126i	$0.590191\pi$
$3$	0	0
$3$	0	0
$4$	−1.37474	−0.687368
$5$	4.15216	1.85690	0.928451	−	0.371455i	$-0.121141\pi$
$5$	4.15216	1.85690	0.928451	+	0.371455i	$0.121141\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	2.66853	0.943467
$9$	0	0
$10$	−3.28327	−1.03826
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−2.46926	−0.684850	−0.342425	−	0.939545i	$-0.611248\pi$
$13$	−2.46926	−0.684850	−0.342425	+	0.939545i	$0.611248\pi$
$14$	0.790737	0.211333
$15$	0	0
$16$	0.639367	0.159842
$17$	−1.99183	−0.483089	−0.241544	−	0.970390i	$-0.577654\pi$
$17$	−1.99183	−0.483089	−0.241544	+	0.970390i	$0.577654\pi$
$18$	0	0
$19$	−3.11195	−0.713930	−0.356965	−	0.934118i	$-0.616188\pi$
$19$	−3.11195	−0.713930	−0.356965	+	0.934118i	$0.616188\pi$
$20$	−5.70812	−1.27637
$21$	0	0
$22$	0	0
$23$	−3.45926	−0.721307	−0.360653	−	0.932700i	$-0.617446\pi$
$23$	−3.45926	−0.721307	−0.360653	+	0.932700i	$0.617446\pi$
$24$	0	0
$25$	12.2404	2.44809
$26$	1.95254	0.382924
$27$	0	0
$28$	1.37474	0.259801
$29$	−3.32309	−0.617083	−0.308542	−	0.951211i	$-0.599841\pi$
$29$	−3.32309	−0.617083	−0.308542	+	0.951211i	$0.599841\pi$
$30$	0	0
$31$	−0.288427	−0.0518031	−0.0259015	−	0.999664i	$-0.508246\pi$
$31$	−0.288427	−0.0518031	−0.0259015	+	0.999664i	$0.508246\pi$
$32$	−5.84263	−1.03284
$33$	0	0
$34$	1.57501	0.270112
$35$	−4.15216	−0.701843
$36$	0	0
$37$	7.59863	1.24921	0.624604	−	0.780942i	$-0.285261\pi$
$37$	7.59863	1.24921	0.624604	+	0.780942i	$0.285261\pi$
$38$	2.46073	0.399183
$39$	0	0
$40$	11.0802	1.75193
$41$	2.32456	0.363036	0.181518	−	0.983388i	$-0.441899\pi$
$41$	2.32456	0.363036	0.181518	+	0.983388i	$0.441899\pi$
$42$	0	0
$43$	8.35116	1.27354	0.636770	−	0.771054i	$-0.280270\pi$
$43$	8.35116	1.27354	0.636770	+	0.771054i	$0.280270\pi$
$44$	0	0
$45$	0	0
$46$	2.73537	0.403308
$47$	−2.94175	−0.429098	−0.214549	−	0.976713i	$-0.568828\pi$
$47$	−2.94175	−0.429098	−0.214549	+	0.976713i	$0.568828\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−9.67896	−1.36881
$51$	0	0
$52$	3.39458	0.470744
$53$	−13.1332	−1.80398	−0.901990	−	0.431756i	$-0.857894\pi$
$53$	−13.1332	−1.80398	−0.901990	+	0.431756i	$0.857894\pi$
$54$	0	0
$55$	0	0
$56$	−2.66853	−0.356597
$57$	0	0
$58$	2.62769	0.345033
$59$	7.82330	1.01851	0.509254	−	0.860617i	$-0.329922\pi$
$59$	7.82330	1.01851	0.509254	+	0.860617i	$0.329922\pi$
$60$	0	0
$61$	−15.3255	−1.96223	−0.981116	−	0.193421i	$-0.938042\pi$
$61$	−15.3255	−1.96223	−0.981116	+	0.193421i	$0.938042\pi$
$62$	0.228070	0.0289649
$63$	0	0
$64$	3.34125	0.417656
$65$	−10.2528	−1.27170
$66$	0	0
$67$	−8.39127	−1.02516	−0.512578	−	0.858641i	$-0.671310\pi$
$67$	−8.39127	−1.02516	−0.512578	+	0.858641i	$0.671310\pi$
$68$	2.73823	0.332060
$69$	0	0
$70$	3.28327	0.392425
$71$	13.6523	1.62023	0.810114	−	0.586273i	$-0.199405\pi$
$71$	13.6523	1.62023	0.810114	+	0.586273i	$0.199405\pi$
$72$	0	0
$73$	−8.86448	−1.03751	−0.518754	−	0.854923i	$-0.673604\pi$
$73$	−8.86448	−1.03751	−0.518754	+	0.854923i	$0.673604\pi$
$74$	−6.00852	−0.698476
$75$	0	0
$76$	4.27810	0.490732
$77$	0	0
$78$	0	0
$79$	15.5540	1.74996	0.874981	−	0.484158i	$-0.160874\pi$
$79$	15.5540	1.74996	0.874981	+	0.484158i	$0.160874\pi$
$80$	2.65475	0.296810
$81$	0	0
$82$	−1.83812	−0.202986
$83$	−4.58873	−0.503678	−0.251839	−	0.967769i	$-0.581035\pi$
$83$	−4.58873	−0.503678	−0.251839	+	0.967769i	$0.581035\pi$
$84$	0	0
$85$	−8.27038	−0.897048
$86$	−6.60357	−0.712081
$87$	0	0
$88$	0	0
$89$	−11.8128	−1.25216	−0.626078	−	0.779760i	$-0.715341\pi$
$89$	−11.8128	−1.25216	−0.626078	+	0.779760i	$0.715341\pi$
$90$	0	0
$91$	2.46926	0.258849
$92$	4.75557	0.495803
$93$	0	0
$94$	2.32615	0.239924
$95$	−12.9213	−1.32570
$96$	0	0
$97$	−7.28658	−0.739840	−0.369920	−	0.929064i	$-0.620615\pi$
$97$	−7.28658	−0.739840	−0.369920	+	0.929064i	$0.620615\pi$
$98$	−0.790737	−0.0798765
$99$	0	0
$100$	−16.8273	−1.68273
$101$	−7.13224	−0.709684	−0.354842	−	0.934926i	$-0.615465\pi$
$101$	−7.13224	−0.709684	−0.354842	+	0.934926i	$0.615465\pi$
$102$	0	0
$103$	−18.0166	−1.77523	−0.887613	−	0.460590i	$-0.847638\pi$
$103$	−18.0166	−1.77523	−0.887613	+	0.460590i	$0.847638\pi$
$104$	−6.58930	−0.646134
$105$	0	0
$106$	10.3849	1.00867
$107$	−0.654567	−0.0632793	−0.0316397	−	0.999499i	$-0.510073\pi$
$107$	−0.654567	−0.0632793	−0.0316397	+	0.999499i	$0.510073\pi$
$108$	0	0
$109$	−13.7937	−1.32120	−0.660601	−	0.750738i	$-0.729698\pi$
$109$	−13.7937	−1.32120	−0.660601	+	0.750738i	$0.729698\pi$
$110$	0	0
$111$	0	0
$112$	−0.639367	−0.0604145
$113$	5.25262	0.494125	0.247062	−	0.969000i	$-0.420535\pi$
$113$	5.25262	0.494125	0.247062	+	0.969000i	$0.420535\pi$
$114$	0	0
$115$	−14.3634	−1.33940
$116$	4.56837	0.424163
$117$	0	0
$118$	−6.18617	−0.569483
$119$	1.99183	0.182590
$120$	0	0
$121$	0	0
$122$	12.1185	1.09715
$123$	0	0
$124$	0.396511	0.0356077
$125$	30.0634	2.68895
$126$	0	0
$127$	−7.93578	−0.704187	−0.352093	−	0.935965i	$-0.614530\pi$
$127$	−7.93578	−0.704187	−0.352093	+	0.935965i	$0.614530\pi$
$128$	9.04321	0.799314
$129$	0	0
$130$	8.10725	0.711053
$131$	−9.93190	−0.867754	−0.433877	−	0.900972i	$-0.642855\pi$
$131$	−9.93190	−0.867754	−0.433877	+	0.900972i	$0.642855\pi$
$132$	0	0
$133$	3.11195	0.269840
$134$	6.63528	0.573201
$135$	0	0
$136$	−5.31524	−0.455778
$137$	8.80747	0.752473	0.376237	−	0.926524i	$-0.377218\pi$
$137$	8.80747	0.752473	0.376237	+	0.926524i	$0.377218\pi$
$138$	0	0
$139$	1.21132	0.102743	0.0513713	−	0.998680i	$-0.483641\pi$
$139$	1.21132	0.102743	0.0513713	+	0.998680i	$0.483641\pi$
$140$	5.70812	0.482424
$141$	0	0
$142$	−10.7954	−0.905927
$143$	0	0
$144$	0	0
$145$	−13.7980	−1.14586
$146$	7.00947	0.580108
$147$	0	0
$148$	−10.4461	−0.858664
$149$	13.1877	1.08037	0.540187	−	0.841545i	$-0.318353\pi$
$149$	13.1877	1.08037	0.540187	+	0.841545i	$0.318353\pi$
$150$	0	0
$151$	5.82558	0.474079	0.237040	−	0.971500i	$-0.423823\pi$
$151$	5.82558	0.474079	0.237040	+	0.971500i	$0.423823\pi$
$152$	−8.30432	−0.673569
$153$	0	0
$154$	0	0
$155$	−1.19760	−0.0961932
$156$	0	0
$157$	16.3405	1.30412	0.652058	−	0.758169i	$-0.273906\pi$
$157$	16.3405	1.30412	0.652058	+	0.758169i	$0.273906\pi$
$158$	−12.2991	−0.978466
$159$	0	0
$160$	−24.2595	−1.91788
$161$	3.45926	0.272628
$162$	0	0
$163$	13.8875	1.08775	0.543875	−	0.839166i	$-0.316956\pi$
$163$	13.8875	1.08775	0.543875	+	0.839166i	$0.316956\pi$
$164$	−3.19566	−0.249539
$165$	0	0
$166$	3.62848	0.281624
$167$	−23.8599	−1.84633	−0.923167	−	0.384398i	$-0.874409\pi$
$167$	−23.8599	−1.84633	−0.923167	+	0.384398i	$0.874409\pi$
$168$	0	0
$169$	−6.90274	−0.530980
$170$	6.53969	0.501572
$171$	0	0
$172$	−11.4806	−0.875390
$173$	−15.0814	−1.14662	−0.573310	−	0.819338i	$-0.694341\pi$
$173$	−15.0814	−1.14662	−0.573310	+	0.819338i	$0.694341\pi$
$174$	0	0
$175$	−12.2404	−0.925289
$176$	0	0
$177$	0	0
$178$	9.34083	0.700125
$179$	−17.9434	−1.34115	−0.670576	−	0.741841i	$-0.733953\pi$
$179$	−17.9434	−1.34115	−0.670576	+	0.741841i	$0.733953\pi$
$180$	0	0
$181$	−2.36612	−0.175872	−0.0879362	−	0.996126i	$-0.528027\pi$
$181$	−2.36612	−0.175872	−0.0879362	+	0.996126i	$0.528027\pi$
$182$	−1.95254	−0.144732
$183$	0	0
$184$	−9.23114	−0.680529
$185$	31.5507	2.31966
$186$	0	0
$187$	0	0
$188$	4.04413	0.294948
$189$	0	0
$190$	10.2174	0.741245
$191$	−13.6654	−0.988790	−0.494395	−	0.869237i	$-0.664610\pi$
$191$	−13.6654	−0.988790	−0.494395	+	0.869237i	$0.664610\pi$
$192$	0	0
$193$	0.705416	0.0507770	0.0253885	−	0.999678i	$-0.491918\pi$
$193$	0.705416	0.0507770	0.0253885	+	0.999678i	$0.491918\pi$
$194$	5.76177	0.413671
$195$	0	0
$196$	−1.37474	−0.0981954
$197$	−1.84782	−0.131652	−0.0658260	−	0.997831i	$-0.520968\pi$
$197$	−1.84782	−0.131652	−0.0658260	+	0.997831i	$0.520968\pi$
$198$	0	0
$199$	−2.13069	−0.151040	−0.0755202	−	0.997144i	$-0.524062\pi$
$199$	−2.13069	−0.151040	−0.0755202	+	0.997144i	$0.524062\pi$
$200$	32.6639	2.30969
$201$	0	0
$202$	5.63972	0.396809
$203$	3.32309	0.233235
$204$	0	0
$205$	9.65195	0.674121
$206$	14.2464	0.992592
$207$	0	0
$208$	−1.57877	−0.109468
$209$	0	0
$210$	0	0
$211$	10.1028	0.695508	0.347754	−	0.937586i	$-0.386944\pi$
$211$	10.1028	0.695508	0.347754	+	0.937586i	$0.386944\pi$
$212$	18.0546	1.24000
$213$	0	0
$214$	0.517590	0.0353817
$215$	34.6753	2.36484
$216$	0	0
$217$	0.288427	0.0195797
$218$	10.9072	0.738730
$219$	0	0
$220$	0	0
$221$	4.91834	0.330844
$222$	0	0
$223$	10.6422	0.712655	0.356327	−	0.934361i	$-0.384029\pi$
$223$	10.6422	0.712655	0.356327	+	0.934361i	$0.384029\pi$
$224$	5.84263	0.390377
$225$	0	0
$226$	−4.15344	−0.276283
$227$	−12.2922	−0.815865	−0.407933	−	0.913012i	$-0.633750\pi$
$227$	−12.2922	−0.815865	−0.407933	+	0.913012i	$0.633750\pi$
$228$	0	0
$229$	−1.77421	−0.117243	−0.0586214	−	0.998280i	$-0.518670\pi$
$229$	−1.77421	−0.117243	−0.0586214	+	0.998280i	$0.518670\pi$
$230$	11.3577	0.748904
$231$	0	0
$232$	−8.86777	−0.582198
$233$	22.3208	1.46229	0.731143	−	0.682224i	$-0.238987\pi$
$233$	22.3208	1.46229	0.731143	+	0.682224i	$0.238987\pi$
$234$	0	0
$235$	−12.2146	−0.796793
$236$	−10.7550	−0.700089
$237$	0	0
$238$	−1.57501	−0.102093
$239$	27.8180	1.79940	0.899700	−	0.436509i	$-0.143785\pi$
$239$	27.8180	1.79940	0.899700	+	0.436509i	$0.143785\pi$
$240$	0	0
$241$	−15.4997	−0.998426	−0.499213	−	0.866479i	$-0.666378\pi$
$241$	−15.4997	−0.998426	−0.499213	+	0.866479i	$0.666378\pi$
$242$	0	0
$243$	0	0
$244$	21.0685	1.34877
$245$	4.15216	0.265272
$246$	0	0
$247$	7.68422	0.488935
$248$	−0.769676	−0.0488745
$249$	0	0
$250$	−23.7722	−1.50349
$251$	24.3459	1.53670	0.768351	−	0.640028i	$-0.221077\pi$
$251$	24.3459	1.53670	0.768351	+	0.640028i	$0.221077\pi$
$252$	0	0
$253$	0	0
$254$	6.27511	0.393736
$255$	0	0
$256$	−13.8333	−0.864581
$257$	−25.8654	−1.61344	−0.806720	−	0.590933i	$-0.798760\pi$
$257$	−25.8654	−1.61344	−0.806720	+	0.590933i	$0.798760\pi$
$258$	0	0
$259$	−7.59863	−0.472156
$260$	14.0948	0.874125
$261$	0	0
$262$	7.85352	0.485192
$263$	−11.0433	−0.680957	−0.340478	−	0.940252i	$-0.610589\pi$
$263$	−11.0433	−0.680957	−0.340478	+	0.940252i	$0.610589\pi$
$264$	0	0
$265$	−54.5311	−3.34982
$266$	−2.46073	−0.150877
$267$	0	0
$268$	11.5358	0.704659
$269$	−16.9534	−1.03367	−0.516835	−	0.856085i	$-0.672890\pi$
$269$	−16.9534	−1.03367	−0.516835	+	0.856085i	$0.672890\pi$
$270$	0	0
$271$	−5.05998	−0.307372	−0.153686	−	0.988120i	$-0.549114\pi$
$271$	−5.05998	−0.307372	−0.153686	+	0.988120i	$0.549114\pi$
$272$	−1.27351	−0.0772177
$273$	0	0
$274$	−6.96439	−0.420734
$275$	0	0
$276$	0	0
$277$	−2.79487	−0.167928	−0.0839639	−	0.996469i	$-0.526758\pi$
$277$	−2.79487	−0.167928	−0.0839639	+	0.996469i	$0.526758\pi$
$278$	−0.957834	−0.0574471
$279$	0	0
$280$	−11.0802	−0.662166
$281$	8.83145	0.526840	0.263420	−	0.964681i	$-0.415150\pi$
$281$	8.83145	0.526840	0.263420	+	0.964681i	$0.415150\pi$
$282$	0	0
$283$	−27.2151	−1.61777	−0.808886	−	0.587965i	$-0.799929\pi$
$283$	−27.2151	−1.61777	−0.808886	+	0.587965i	$0.799929\pi$
$284$	−18.7683	−1.11369
$285$	0	0
$286$	0	0
$287$	−2.32456	−0.137215
$288$	0	0
$289$	−13.0326	−0.766625
$290$	10.9106	0.640693
$291$	0	0
$292$	12.1863	0.713150
$293$	−25.2909	−1.47751	−0.738756	−	0.673973i	$-0.764586\pi$
$293$	−25.2909	−1.47751	−0.738756	+	0.673973i	$0.764586\pi$
$294$	0	0
$295$	32.4836	1.89127
$296$	20.2772	1.17859
$297$	0	0
$298$	−10.4280	−0.604076
$299$	8.54184	0.493987
$300$	0	0
$301$	−8.35116	−0.481353
$302$	−4.60650	−0.265075
$303$	0	0
$304$	−1.98968	−0.114116
$305$	−63.6340	−3.64367
$306$	0	0
$307$	15.0569	0.859343	0.429671	−	0.902985i	$-0.358629\pi$
$307$	15.0569	0.859343	0.429671	+	0.902985i	$0.358629\pi$
$308$	0	0
$309$	0	0
$310$	0.946983	0.0537850
$311$	19.1453	1.08563	0.542814	−	0.839853i	$-0.317359\pi$
$311$	19.1453	1.08563	0.542814	+	0.839853i	$0.317359\pi$
$312$	0	0
$313$	27.4656	1.55245	0.776223	−	0.630458i	$-0.217133\pi$
$313$	27.4656	1.55245	0.776223	+	0.630458i	$0.217133\pi$
$314$	−12.9211	−0.729177
$315$	0	0
$316$	−21.3826	−1.20287
$317$	16.9268	0.950702	0.475351	−	0.879796i	$-0.342321\pi$
$317$	16.9268	0.950702	0.475351	+	0.879796i	$0.342321\pi$
$318$	0	0
$319$	0	0
$320$	13.8734	0.775546
$321$	0	0
$322$	−2.73537	−0.152436
$323$	6.19846	0.344891
$324$	0	0
$325$	−30.2248	−1.67657
$326$	−10.9813	−0.608200
$327$	0	0
$328$	6.20316	0.342512
$329$	2.94175	0.162184
$330$	0	0
$331$	−3.22515	−0.177271	−0.0886353	−	0.996064i	$-0.528251\pi$
$331$	−3.22515	−0.177271	−0.0886353	+	0.996064i	$0.528251\pi$
$332$	6.30829	0.346212
$333$	0	0
$334$	18.8669	1.03235
$335$	−34.8419	−1.90361
$336$	0	0
$337$	−14.7110	−0.801359	−0.400679	−	0.916218i	$-0.631226\pi$
$337$	−14.7110	−0.801359	−0.400679	+	0.916218i	$0.631226\pi$
$338$	5.45825	0.296890
$339$	0	0
$340$	11.3696	0.616602
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	22.2853	1.20154
$345$	0	0
$346$	11.9254	0.641116
$347$	32.2505	1.73130	0.865648	−	0.500652i	$-0.166906\pi$
$347$	32.2505	1.73130	0.865648	+	0.500652i	$0.166906\pi$
$348$	0	0
$349$	−10.1495	−0.543291	−0.271646	−	0.962397i	$-0.587568\pi$
$349$	−10.1495	−0.543291	−0.271646	+	0.962397i	$0.587568\pi$
$350$	9.67896	0.517362
$351$	0	0
$352$	0	0
$353$	−12.1692	−0.647702	−0.323851	−	0.946108i	$-0.604978\pi$
$353$	−12.1692	−0.647702	−0.323851	+	0.946108i	$0.604978\pi$
$354$	0	0
$355$	56.6864	3.00860
$356$	16.2395	0.860691
$357$	0	0
$358$	14.1885	0.749886
$359$	−1.04853	−0.0553392	−0.0276696	−	0.999617i	$-0.508809\pi$
$359$	−1.04853	−0.0553392	−0.0276696	+	0.999617i	$0.508809\pi$
$360$	0	0
$361$	−9.31578	−0.490304
$362$	1.87098	0.0983364
$363$	0	0
$364$	−3.39458	−0.177924
$365$	−36.8067	−1.92655
$366$	0	0
$367$	−27.7306	−1.44752	−0.723762	−	0.690050i	$-0.757589\pi$
$367$	−27.7306	−1.44752	−0.723762	+	0.690050i	$0.757589\pi$
$368$	−2.21174	−0.115295
$369$	0	0
$370$	−24.9483	−1.29700
$371$	13.1332	0.681841
$372$	0	0
$373$	3.96398	0.205247	0.102624	−	0.994720i	$-0.467276\pi$
$373$	3.96398	0.205247	0.102624	+	0.994720i	$0.467276\pi$
$374$	0	0
$375$	0	0
$376$	−7.85014	−0.404840
$377$	8.20559	0.422610
$378$	0	0
$379$	8.00517	0.411198	0.205599	−	0.978636i	$-0.434086\pi$
$379$	8.00517	0.411198	0.205599	+	0.978636i	$0.434086\pi$
$380$	17.7634	0.911242
$381$	0	0
$382$	10.8057	0.552868
$383$	−2.10740	−0.107683	−0.0538415	−	0.998549i	$-0.517147\pi$
$383$	−2.10740	−0.107683	−0.0538415	+	0.998549i	$0.517147\pi$
$384$	0	0
$385$	0	0
$386$	−0.557799	−0.0283912
$387$	0	0
$388$	10.0171	0.508542
$389$	−35.7563	−1.81292	−0.906459	−	0.422294i	$-0.861225\pi$
$389$	−35.7563	−1.81292	−0.906459	+	0.422294i	$0.861225\pi$
$390$	0	0
$391$	6.89025	0.348455
$392$	2.66853	0.134781
$393$	0	0
$394$	1.46114	0.0736113
$395$	64.5827	3.24951
$396$	0	0
$397$	13.2280	0.663894	0.331947	−	0.943298i	$-0.392295\pi$
$397$	13.2280	0.663894	0.331947	+	0.943298i	$0.392295\pi$
$398$	1.68481	0.0844520
$399$	0	0
$400$	7.82612	0.391306
$401$	−11.5815	−0.578352	−0.289176	−	0.957276i	$-0.593381\pi$
$401$	−11.5815	−0.578352	−0.289176	+	0.957276i	$0.593381\pi$
$402$	0	0
$403$	0.712203	0.0354774
$404$	9.80494	0.487814
$405$	0	0
$406$	−2.62769	−0.130410
$407$	0	0
$408$	0	0
$409$	−37.4266	−1.85063	−0.925313	−	0.379203i	$-0.876198\pi$
$409$	−37.4266	−1.85063	−0.925313	+	0.379203i	$0.876198\pi$
$410$	−7.63215	−0.376925
$411$	0	0
$412$	24.7680	1.22023
$413$	−7.82330	−0.384959
$414$	0	0
$415$	−19.0531	−0.935281
$416$	14.4270	0.707341
$417$	0	0
$418$	0	0
$419$	10.4672	0.511356	0.255678	−	0.966762i	$-0.417701\pi$
$419$	10.4672	0.511356	0.255678	+	0.966762i	$0.417701\pi$
$420$	0	0
$421$	25.6365	1.24945	0.624724	−	0.780846i	$-0.285212\pi$
$421$	25.6365	1.24945	0.624724	+	0.780846i	$0.285212\pi$
$422$	−7.98869	−0.388883
$423$	0	0
$424$	−35.0463	−1.70200
$425$	−24.3808	−1.18264
$426$	0	0
$427$	15.3255	0.741654
$428$	0.899856	0.0434962
$429$	0	0
$430$	−27.4191	−1.32227
$431$	27.2723	1.31366	0.656829	−	0.754039i	$-0.271897\pi$
$431$	27.2723	1.31366	0.656829	+	0.754039i	$0.271897\pi$
$432$	0	0
$433$	−16.2430	−0.780588	−0.390294	−	0.920690i	$-0.627627\pi$
$433$	−16.2430	−0.780588	−0.390294	+	0.920690i	$0.627627\pi$
$434$	−0.228070	−0.0109477
$435$	0	0
$436$	18.9627	0.908151
$437$	10.7651	0.514962
$438$	0	0
$439$	−26.1777	−1.24940	−0.624698	−	0.780867i	$-0.714778\pi$
$439$	−26.1777	−1.24940	−0.624698	+	0.780867i	$0.714778\pi$
$440$	0	0
$441$	0	0
$442$	−3.88911	−0.184986
$443$	−36.5333	−1.73575	−0.867874	−	0.496784i	$-0.834514\pi$
$443$	−36.5333	−1.73575	−0.867874	+	0.496784i	$0.834514\pi$
$444$	0	0
$445$	−49.0487	−2.32513
$446$	−8.41518	−0.398471
$447$	0	0
$448$	−3.34125	−0.157859
$449$	15.5547	0.734070	0.367035	−	0.930207i	$-0.380373\pi$
$449$	15.5547	0.734070	0.367035	+	0.930207i	$0.380373\pi$
$450$	0	0
$451$	0	0
$452$	−7.22096	−0.339645
$453$	0	0
$454$	9.71994	0.456179
$455$	10.2528	0.480657
$456$	0	0
$457$	−30.6951	−1.43586	−0.717928	−	0.696117i	$-0.754909\pi$
$457$	−30.6951	−1.43586	−0.717928	+	0.696117i	$0.754909\pi$
$458$	1.40293	0.0655547
$459$	0	0
$460$	19.7459	0.920657
$461$	−7.29317	−0.339677	−0.169838	−	0.985472i	$-0.554325\pi$
$461$	−7.29317	−0.339677	−0.169838	+	0.985472i	$0.554325\pi$
$462$	0	0
$463$	11.5106	0.534944	0.267472	−	0.963566i	$-0.413812\pi$
$463$	11.5106	0.534944	0.267472	+	0.963566i	$0.413812\pi$
$464$	−2.12468	−0.0986356
$465$	0	0
$466$	−17.6499	−0.817616
$467$	32.4827	1.50312	0.751559	−	0.659665i	$-0.229302\pi$
$467$	32.4827	1.50312	0.751559	+	0.659665i	$0.229302\pi$
$468$	0	0
$469$	8.39127	0.387473
$470$	9.65854	0.445515
$471$	0	0
$472$	20.8767	0.960928
$473$	0	0
$474$	0	0
$475$	−38.0916	−1.74776
$476$	−2.73823	−0.125507
$477$	0	0
$478$	−21.9968	−1.00611
$479$	17.6561	0.806728	0.403364	−	0.915040i	$-0.367841\pi$
$479$	17.6561	0.806728	0.403364	+	0.915040i	$0.367841\pi$
$480$	0	0
$481$	−18.7630	−0.855520
$482$	12.2562	0.558256
$483$	0	0
$484$	0	0
$485$	−30.2551	−1.37381
$486$	0	0
$487$	−0.132320	−0.00599599	−0.00299800	−	0.999996i	$-0.500954\pi$
$487$	−0.132320	−0.00599599	−0.00299800	+	0.999996i	$0.500954\pi$
$488$	−40.8966	−1.85130
$489$	0	0
$490$	−3.28327	−0.148323
$491$	10.9347	0.493477	0.246738	−	0.969082i	$-0.420641\pi$
$491$	10.9347	0.493477	0.246738	+	0.969082i	$0.420641\pi$
$492$	0	0
$493$	6.61902	0.298106
$494$	−6.07620	−0.273381
$495$	0	0
$496$	−0.184411	−0.00828029
$497$	−13.6523	−0.612388
$498$	0	0
$499$	−19.8867	−0.890252	−0.445126	−	0.895468i	$-0.646841\pi$
$499$	−19.8867	−0.890252	−0.445126	+	0.895468i	$0.646841\pi$
$500$	−41.3292	−1.84830
$501$	0	0
$502$	−19.2512	−0.859225
$503$	24.2650	1.08192	0.540962	−	0.841047i	$-0.318060\pi$
$503$	24.2650	1.08192	0.540962	+	0.841047i	$0.318060\pi$
$504$	0	0
$505$	−29.6142	−1.31781
$506$	0	0
$507$	0	0
$508$	10.9096	0.484035
$509$	12.5909	0.558082	0.279041	−	0.960279i	$-0.409983\pi$
$509$	12.5909	0.558082	0.279041	+	0.960279i	$0.409983\pi$
$510$	0	0
$511$	8.86448	0.392141
$512$	−7.14792	−0.315896
$513$	0	0
$514$	20.4527	0.902132
$515$	−74.8077	−3.29642
$516$	0	0
$517$	0	0
$518$	6.00852	0.263999
$519$	0	0
$520$	−27.3598	−1.19981
$521$	8.30507	0.363852	0.181926	−	0.983312i	$-0.441767\pi$
$521$	8.30507	0.363852	0.181926	+	0.983312i	$0.441767\pi$
$522$	0	0
$523$	17.8685	0.781334	0.390667	−	0.920532i	$-0.372244\pi$
$523$	17.8685	0.781334	0.390667	+	0.920532i	$0.372244\pi$
$524$	13.6537	0.596466
$525$	0	0
$526$	8.73232	0.380747
$527$	0.574497	0.0250255
$528$	0	0
$529$	−11.0335	−0.479717
$530$	43.1197	1.87300
$531$	0	0
$532$	−4.27810	−0.185479
$533$	−5.73995	−0.248625
$534$	0	0
$535$	−2.71787	−0.117504
$536$	−22.3923	−0.967201
$537$	0	0
$538$	13.4057	0.577961
$539$	0	0
$540$	0	0
$541$	19.7117	0.847472	0.423736	−	0.905786i	$-0.360718\pi$
$541$	19.7117	0.847472	0.423736	+	0.905786i	$0.360718\pi$
$542$	4.00112	0.171863
$543$	0	0
$544$	11.6375	0.498953
$545$	−57.2738	−2.45334
$546$	0	0
$547$	−28.3120	−1.21053	−0.605267	−	0.796022i	$-0.706934\pi$
$547$	−28.3120	−1.21053	−0.605267	+	0.796022i	$0.706934\pi$
$548$	−12.1079	−0.517226
$549$	0	0
$550$	0	0
$551$	10.3413	0.440554
$552$	0	0
$553$	−15.5540	−0.661423
$554$	2.21001	0.0938943
$555$	0	0
$556$	−1.66524	−0.0706220
$557$	16.6409	0.705099	0.352549	−	0.935793i	$-0.385315\pi$
$557$	16.6409	0.705099	0.352549	+	0.935793i	$0.385315\pi$
$558$	0	0
$559$	−20.6212	−0.872184
$560$	−2.65475	−0.112184
$561$	0	0
$562$	−6.98335	−0.294575
$563$	−40.0765	−1.68902	−0.844512	−	0.535536i	$-0.820110\pi$
$563$	−40.0765	−1.68902	−0.844512	+	0.535536i	$0.820110\pi$
$564$	0	0
$565$	21.8097	0.917541
$566$	21.5200	0.904554
$567$	0	0
$568$	36.4315	1.52863
$569$	8.81813	0.369675	0.184838	−	0.982769i	$-0.440824\pi$
$569$	8.81813	0.369675	0.184838	+	0.982769i	$0.440824\pi$
$570$	0	0
$571$	−13.2461	−0.554333	−0.277166	−	0.960822i	$-0.589395\pi$
$571$	−13.2461	−0.554333	−0.277166	+	0.960822i	$0.589395\pi$
$572$	0	0
$573$	0	0
$574$	1.83812	0.0767215
$575$	−42.3429	−1.76582
$576$	0	0
$577$	−20.6017	−0.857661	−0.428831	−	0.903385i	$-0.641074\pi$
$577$	−20.6017	−0.857661	−0.428831	+	0.903385i	$0.641074\pi$
$578$	10.3054	0.428647
$579$	0	0
$580$	18.9686	0.787629
$581$	4.58873	0.190373
$582$	0	0
$583$	0	0
$584$	−23.6551	−0.978855
$585$	0	0
$586$	19.9985	0.826129
$587$	24.2499	1.00090	0.500450	−	0.865765i	$-0.333168\pi$
$587$	24.2499	1.00090	0.500450	+	0.865765i	$0.333168\pi$
$588$	0	0
$589$	0.897571	0.0369838
$590$	−25.6860	−1.05747
$591$	0	0
$592$	4.85831	0.199675
$593$	23.0301	0.945732	0.472866	−	0.881134i	$-0.343219\pi$
$593$	23.0301	0.945732	0.472866	+	0.881134i	$0.343219\pi$
$594$	0	0
$595$	8.27038	0.339052
$596$	−18.1295	−0.742614
$597$	0	0
$598$	−6.75434	−0.276206
$599$	16.0896	0.657403	0.328701	−	0.944434i	$-0.393389\pi$
$599$	16.0896	0.657403	0.328701	+	0.944434i	$0.393389\pi$
$600$	0	0
$601$	−32.1691	−1.31220	−0.656102	−	0.754673i	$-0.727796\pi$
$601$	−32.1691	−1.31220	−0.656102	+	0.754673i	$0.727796\pi$
$602$	6.60357	0.269141
$603$	0	0
$604$	−8.00864	−0.325867
$605$	0	0
$606$	0	0
$607$	17.3198	0.702990	0.351495	−	0.936190i	$-0.385673\pi$
$607$	17.3198	0.702990	0.351495	+	0.936190i	$0.385673\pi$
$608$	18.1819	0.737375
$609$	0	0
$610$	50.3178	2.03731
$611$	7.26395	0.293868
$612$	0	0
$613$	−24.6052	−0.993794	−0.496897	−	0.867810i	$-0.665527\pi$
$613$	−24.6052	−0.993794	−0.496897	+	0.867810i	$0.665527\pi$
$614$	−11.9060	−0.480489
$615$	0	0
$616$	0	0
$617$	−22.8127	−0.918405	−0.459203	−	0.888332i	$-0.651865\pi$
$617$	−22.8127	−0.918405	−0.459203	+	0.888332i	$0.651865\pi$
$618$	0	0
$619$	−45.7647	−1.83944	−0.919720	−	0.392575i	$-0.871584\pi$
$619$	−45.7647	−1.83944	−0.919720	+	0.392575i	$0.871584\pi$
$620$	1.64638	0.0661201
$621$	0	0
$622$	−15.1389	−0.607014
$623$	11.8128	0.473270
$624$	0	0
$625$	63.6259	2.54504
$626$	−21.7180	−0.868028
$627$	0	0
$628$	−22.4639	−0.896407
$629$	−15.1351	−0.603478
$630$	0	0
$631$	35.6172	1.41790	0.708948	−	0.705260i	$-0.249170\pi$
$631$	35.6172	1.41790	0.708948	+	0.705260i	$0.249170\pi$
$632$	41.5063	1.65103
$633$	0	0
$634$	−13.3846	−0.531571
$635$	−32.9506	−1.30761
$636$	0	0
$637$	−2.46926	−0.0978358
$638$	0	0
$639$	0	0
$640$	37.5488	1.48425
$641$	−15.4422	−0.609931	−0.304965	−	0.952363i	$-0.598645\pi$
$641$	−15.4422	−0.609931	−0.304965	+	0.952363i	$0.598645\pi$
$642$	0	0
$643$	−16.7543	−0.660726	−0.330363	−	0.943854i	$-0.607171\pi$
$643$	−16.7543	−0.660726	−0.330363	+	0.943854i	$0.607171\pi$
$644$	−4.75557	−0.187396
$645$	0	0
$646$	−4.90135	−0.192841
$647$	6.35881	0.249991	0.124995	−	0.992157i	$-0.460108\pi$
$647$	6.35881	0.249991	0.124995	+	0.992157i	$0.460108\pi$
$648$	0	0
$649$	0	0
$650$	23.8999	0.937431
$651$	0	0
$652$	−19.0916	−0.747684
$653$	−42.3077	−1.65563	−0.827815	−	0.561001i	$-0.810416\pi$
$653$	−42.3077	−1.65563	−0.827815	+	0.561001i	$0.810416\pi$
$654$	0	0
$655$	−41.2388	−1.61134
$656$	1.48625	0.0580282
$657$	0	0
$658$	−2.32615	−0.0906828
$659$	−38.7796	−1.51064	−0.755319	−	0.655358i	$-0.772518\pi$
$659$	−38.7796	−1.51064	−0.755319	+	0.655358i	$0.772518\pi$
$660$	0	0
$661$	3.77231	0.146726	0.0733629	−	0.997305i	$-0.476627\pi$
$661$	3.77231	0.146726	0.0733629	+	0.997305i	$0.476627\pi$
$662$	2.55025	0.0991182
$663$	0	0
$664$	−12.2451	−0.475204
$665$	12.9213	0.501067
$666$	0	0
$667$	11.4955	0.445106
$668$	32.8010	1.26911
$669$	0	0
$670$	27.5508	1.06438
$671$	0	0
$672$	0	0
$673$	−9.15255	−0.352805	−0.176402	−	0.984318i	$-0.556446\pi$
$673$	−9.15255	−0.352805	−0.176402	+	0.984318i	$0.556446\pi$
$674$	11.6325	0.448068
$675$	0	0
$676$	9.48944	0.364978
$677$	−18.1182	−0.696337	−0.348169	−	0.937432i	$-0.613196\pi$
$677$	−18.1182	−0.696337	−0.348169	+	0.937432i	$0.613196\pi$
$678$	0	0
$679$	7.28658	0.279633
$680$	−22.0697	−0.846336
$681$	0	0
$682$	0	0
$683$	−6.63504	−0.253883	−0.126941	−	0.991910i	$-0.540516\pi$
$683$	−6.63504	−0.253883	−0.126941	+	0.991910i	$0.540516\pi$
$684$	0	0
$685$	36.5700	1.39727
$686$	0.790737	0.0301905
$687$	0	0
$688$	5.33946	0.203565
$689$	32.4293	1.23546
$690$	0	0
$691$	11.1452	0.423982	0.211991	−	0.977272i	$-0.432005\pi$
$691$	11.1452	0.423982	0.211991	+	0.977272i	$0.432005\pi$
$692$	20.7330	0.788150
$693$	0	0
$694$	−25.5017	−0.968029
$695$	5.02959	0.190783
$696$	0	0
$697$	−4.63012	−0.175378
$698$	8.02560	0.303773
$699$	0	0
$700$	16.8273	0.636014
$701$	−13.0455	−0.492721	−0.246361	−	0.969178i	$-0.579235\pi$
$701$	−13.0455	−0.492721	−0.246361	+	0.969178i	$0.579235\pi$
$702$	0	0
$703$	−23.6465	−0.891846
$704$	0	0
$705$	0	0
$706$	9.62266	0.362153
$707$	7.13224	0.268235
$708$	0	0
$709$	−36.8687	−1.38463	−0.692317	−	0.721594i	$-0.743410\pi$
$709$	−36.8687	−1.38463	−0.692317	+	0.721594i	$0.743410\pi$
$710$	−44.8241	−1.68222
$711$	0	0
$712$	−31.5228	−1.18137
$713$	0.997746	0.0373659
$714$	0	0
$715$	0	0
$716$	24.6674	0.921865
$717$	0	0
$718$	0.829110	0.0309421
$719$	11.7771	0.439211	0.219605	−	0.975589i	$-0.429523\pi$
$719$	11.7771	0.439211	0.219605	+	0.975589i	$0.429523\pi$
$720$	0	0
$721$	18.0166	0.670972
$722$	7.36633	0.274147
$723$	0	0
$724$	3.25279	0.120889
$725$	−40.6761	−1.51067
$726$	0	0
$727$	−14.7544	−0.547209	−0.273604	−	0.961842i	$-0.588216\pi$
$727$	−14.7544	−0.547209	−0.273604	+	0.961842i	$0.588216\pi$
$728$	6.58930	0.244216
$729$	0	0
$730$	29.1044	1.07720
$731$	−16.6341	−0.615233
$732$	0	0
$733$	35.0202	1.29350	0.646751	−	0.762702i	$-0.276127\pi$
$733$	35.0202	1.29350	0.646751	+	0.762702i	$0.276127\pi$
$734$	21.9276	0.809362
$735$	0	0
$736$	20.2112	0.744994
$737$	0	0
$738$	0	0
$739$	−19.6512	−0.722880	−0.361440	−	0.932395i	$-0.617715\pi$
$739$	−19.6512	−0.722880	−0.361440	+	0.932395i	$0.617715\pi$
$740$	−43.3739	−1.59446
$741$	0	0
$742$	−10.3849	−0.381241
$743$	−36.8616	−1.35232	−0.676161	−	0.736754i	$-0.736358\pi$
$743$	−36.8616	−1.35232	−0.676161	+	0.736754i	$0.736358\pi$
$744$	0	0
$745$	54.7572	2.00615
$746$	−3.13447	−0.114761
$747$	0	0
$748$	0	0
$749$	0.654567	0.0239173
$750$	0	0
$751$	25.6185	0.934834	0.467417	−	0.884037i	$-0.345185\pi$
$751$	25.6185	0.934834	0.467417	+	0.884037i	$0.345185\pi$
$752$	−1.88086	−0.0685878
$753$	0	0
$754$	−6.48847	−0.236296
$755$	24.1888	0.880319
$756$	0	0
$757$	−27.8087	−1.01073	−0.505363	−	0.862907i	$-0.668641\pi$
$757$	−27.8087	−1.01073	−0.505363	+	0.862907i	$0.668641\pi$
$758$	−6.32998	−0.229915
$759$	0	0
$760$	−34.4809	−1.25075
$761$	1.55442	0.0563478	0.0281739	−	0.999603i	$-0.491031\pi$
$761$	1.55442	0.0563478	0.0281739	+	0.999603i	$0.491031\pi$
$762$	0	0
$763$	13.7937	0.499367
$764$	18.7862	0.679662
$765$	0	0
$766$	1.66640	0.0602094
$767$	−19.3178	−0.697525
$768$	0	0
$769$	17.2867	0.623374	0.311687	−	0.950185i	$-0.399106\pi$
$769$	17.2867	0.623374	0.311687	+	0.950185i	$0.399106\pi$
$770$	0	0
$771$	0	0
$772$	−0.969761	−0.0349025
$773$	28.0198	1.00780	0.503900	−	0.863762i	$-0.331898\pi$
$773$	28.0198	1.00780	0.503900	+	0.863762i	$0.331898\pi$
$774$	0	0
$775$	−3.53047	−0.126818
$776$	−19.4444	−0.698015
$777$	0	0
$778$	28.2738	1.01367
$779$	−7.23391	−0.259182
$780$	0	0
$781$	0	0
$782$	−5.44838	−0.194834
$783$	0	0
$784$	0.639367	0.0228345
$785$	67.8485	2.42162
$786$	0	0
$787$	8.85062	0.315490	0.157745	−	0.987480i	$-0.449578\pi$
$787$	8.85062	0.315490	0.157745	+	0.987480i	$0.449578\pi$
$788$	2.54027	0.0904933
$789$	0	0
$790$	−51.0679	−1.81691
$791$	−5.25262	−0.186762
$792$	0	0
$793$	37.8427	1.34384
$794$	−10.4599	−0.371206
$795$	0	0
$796$	2.92913	0.103820
$797$	20.7761	0.735927	0.367964	−	0.929840i	$-0.380055\pi$
$797$	20.7761	0.735927	0.367964	+	0.929840i	$0.380055\pi$
$798$	0	0
$799$	5.85945	0.207293
$800$	−71.5162	−2.52848
$801$	0	0
$802$	9.15792	0.323377
$803$	0	0
$804$	0	0
$805$	14.3634	0.506244
$806$	−0.563165	−0.0198366
$807$	0	0
$808$	−19.0326	−0.669563
$809$	−31.1956	−1.09678	−0.548389	−	0.836224i	$-0.684759\pi$
$809$	−31.1956	−1.09678	−0.548389	+	0.836224i	$0.684759\pi$
$810$	0	0
$811$	54.0417	1.89766	0.948831	−	0.315784i	$-0.102268\pi$
$811$	54.0417	1.89766	0.948831	+	0.315784i	$0.102268\pi$
$812$	−4.56837	−0.160319
$813$	0	0
$814$	0	0
$815$	57.6630	2.01985
$816$	0	0
$817$	−25.9884	−0.909218
$818$	29.5946	1.03475
$819$	0	0
$820$	−13.2689	−0.463369
$821$	10.0826	0.351884	0.175942	−	0.984401i	$-0.443703\pi$
$821$	10.0826	0.351884	0.175942	+	0.984401i	$0.443703\pi$
$822$	0	0
$823$	−21.5840	−0.752370	−0.376185	−	0.926545i	$-0.622764\pi$
$823$	−21.5840	−0.752370	−0.376185	+	0.926545i	$0.622764\pi$
$824$	−48.0777	−1.67487
$825$	0	0
$826$	6.18617	0.215244
$827$	−28.0544	−0.975548	−0.487774	−	0.872970i	$-0.662191\pi$
$827$	−28.0544	−0.975548	−0.487774	+	0.872970i	$0.662191\pi$
$828$	0	0
$829$	21.6288	0.751200	0.375600	−	0.926782i	$-0.377437\pi$
$829$	21.6288	0.751200	0.375600	+	0.926782i	$0.377437\pi$
$830$	15.0660	0.522949
$831$	0	0
$832$	−8.25042	−0.286032
$833$	−1.99183	−0.0690127
$834$	0	0
$835$	−99.0701	−3.42846
$836$	0	0
$837$	0	0
$838$	−8.27680	−0.285917
$839$	−56.3305	−1.94475	−0.972373	−	0.233432i	$-0.925004\pi$
$839$	−56.3305	−1.94475	−0.972373	+	0.233432i	$0.925004\pi$
$840$	0	0
$841$	−17.9570	−0.619208
$842$	−20.2717	−0.698610
$843$	0	0
$844$	−13.8887	−0.478070
$845$	−28.6613	−0.985978
$846$	0	0
$847$	0	0
$848$	−8.39692	−0.288351
$849$	0	0
$850$	19.2788	0.661257
$851$	−26.2857	−0.901061
$852$	0	0
$853$	2.96838	0.101635	0.0508177	−	0.998708i	$-0.483817\pi$
$853$	2.96838	0.101635	0.0508177	+	0.998708i	$0.483817\pi$
$854$	−12.1185	−0.414685
$855$	0	0
$856$	−1.74673	−0.0597020
$857$	22.2592	0.760360	0.380180	−	0.924913i	$-0.375862\pi$
$857$	22.2592	0.760360	0.380180	+	0.924913i	$0.375862\pi$
$858$	0	0
$859$	30.5186	1.04128	0.520641	−	0.853776i	$-0.325693\pi$
$859$	30.5186	1.04128	0.520641	+	0.853776i	$0.325693\pi$
$860$	−47.6694	−1.62551
$861$	0	0
$862$	−21.5652	−0.734513
$863$	−38.7492	−1.31904	−0.659519	−	0.751688i	$-0.729240\pi$
$863$	−38.7492	−1.31904	−0.659519	+	0.751688i	$0.729240\pi$
$864$	0	0
$865$	−62.6205	−2.12916
$866$	12.8439	0.436455
$867$	0	0
$868$	−0.396511	−0.0134585
$869$	0	0
$870$	0	0
$871$	20.7202	0.702079
$872$	−36.8090	−1.24651
$873$	0	0
$874$	−8.51232	−0.287934
$875$	−30.0634	−1.01633
$876$	0	0
$877$	12.6839	0.428303	0.214152	−	0.976800i	$-0.431301\pi$
$877$	12.6839	0.428303	0.214152	+	0.976800i	$0.431301\pi$
$878$	20.6997	0.698581
$879$	0	0
$880$	0	0
$881$	−42.9950	−1.44854	−0.724269	−	0.689518i	$-0.757823\pi$
$881$	−42.9950	−1.44854	−0.724269	+	0.689518i	$0.757823\pi$
$882$	0	0
$883$	15.2586	0.513494	0.256747	−	0.966479i	$-0.417349\pi$
$883$	15.2586	0.513494	0.256747	+	0.966479i	$0.417349\pi$
$884$	−6.76142	−0.227411
$885$	0	0
$886$	28.8882	0.970519
$887$	37.4888	1.25875	0.629375	−	0.777102i	$-0.283311\pi$
$887$	37.4888	1.25875	0.629375	+	0.777102i	$0.283311\pi$
$888$	0	0
$889$	7.93578	0.266157
$890$	38.7846	1.30006
$891$	0	0
$892$	−14.6302	−0.489856
$893$	9.15457	0.306346
$894$	0	0
$895$	−74.5038	−2.49039
$896$	−9.04321	−0.302112
$897$	0	0
$898$	−12.2997	−0.410445
$899$	0.958471	0.0319668
$900$	0	0
$901$	26.1590	0.871483
$902$	0	0
$903$	0	0
$904$	14.0168	0.466190
$905$	−9.82451	−0.326578
$906$	0	0
$907$	19.2679	0.639779	0.319890	−	0.947455i	$-0.396354\pi$
$907$	19.2679	0.639779	0.319890	+	0.947455i	$0.396354\pi$
$908$	16.8986	0.560799
$909$	0	0
$910$	−8.10725	−0.268753
$911$	−33.5705	−1.11224	−0.556121	−	0.831102i	$-0.687711\pi$
$911$	−33.5705	−1.11224	−0.556121	+	0.831102i	$0.687711\pi$
$912$	0	0
$913$	0	0
$914$	24.2717	0.802838
$915$	0	0
$916$	2.43906	0.0805890
$917$	9.93190	0.327980
$918$	0	0
$919$	46.3335	1.52840	0.764200	−	0.644979i	$-0.223134\pi$
$919$	46.3335	1.52840	0.764200	+	0.644979i	$0.223134\pi$
$920$	−38.3292	−1.26368
$921$	0	0
$922$	5.76698	0.189925
$923$	−33.7111	−1.10961
$924$	0	0
$925$	93.0105	3.05817
$926$	−9.10187	−0.299106
$927$	0	0
$928$	19.4156	0.637348
$929$	−35.4715	−1.16378	−0.581892	−	0.813266i	$-0.697687\pi$
$929$	−35.4715	−1.16378	−0.581892	+	0.813266i	$0.697687\pi$
$930$	0	0
$931$	−3.11195	−0.101990
$932$	−30.6852	−1.00513
$933$	0	0
$934$	−25.6852	−0.840447
$935$	0	0
$936$	0	0
$937$	−9.72792	−0.317797	−0.158899	−	0.987295i	$-0.550794\pi$
$937$	−9.72792	−0.317797	−0.158899	+	0.987295i	$0.550794\pi$
$938$	−6.63528	−0.216650
$939$	0	0
$940$	16.7919	0.547690
$941$	30.3039	0.987880	0.493940	−	0.869496i	$-0.335556\pi$
$941$	30.3039	0.987880	0.493940	+	0.869496i	$0.335556\pi$
$942$	0	0
$943$	−8.04127	−0.261860
$944$	5.00196	0.162800
$945$	0	0
$946$	0	0
$947$	32.6334	1.06044	0.530222	−	0.847859i	$-0.322108\pi$
$947$	32.6334	1.06044	0.530222	+	0.847859i	$0.322108\pi$
$948$	0	0
$949$	21.8887	0.710538
$950$	30.1204	0.977235
$951$	0	0
$952$	5.31524	0.172268
$953$	45.5467	1.47540	0.737701	−	0.675128i	$-0.235911\pi$
$953$	45.5467	1.47540	0.737701	+	0.675128i	$0.235911\pi$
$954$	0	0
$955$	−56.7407	−1.83609
$956$	−38.2424	−1.23685
$957$	0	0
$958$	−13.9613	−0.451070
$959$	−8.80747	−0.284408
$960$	0	0
$961$	−30.9168	−0.997316
$962$	14.8366	0.478352
$963$	0	0
$964$	21.3080	0.686286
$965$	2.92900	0.0942879
$966$	0	0
$967$	−35.0814	−1.12814	−0.564070	−	0.825727i	$-0.690765\pi$
$967$	−35.0814	−1.12814	−0.564070	+	0.825727i	$0.690765\pi$
$968$	0	0
$969$	0	0
$970$	23.9238	0.768146
$971$	−4.88397	−0.156734	−0.0783670	−	0.996925i	$-0.524971\pi$
$971$	−4.88397	−0.156734	−0.0783670	+	0.996925i	$0.524971\pi$
$972$	0	0
$973$	−1.21132	−0.0388331
$974$	0.104630	0.00335257
$975$	0	0
$976$	−9.79863	−0.313647
$977$	−26.0425	−0.833173	−0.416587	−	0.909096i	$-0.636774\pi$
$977$	−26.0425	−0.833173	−0.416587	+	0.909096i	$0.636774\pi$
$978$	0	0
$979$	0	0
$980$	−5.70812	−0.182339
$981$	0	0
$982$	−8.64648	−0.275920
$983$	−20.5479	−0.655377	−0.327688	−	0.944786i	$-0.606270\pi$
$983$	−20.5479	−0.655377	−0.327688	+	0.944786i	$0.606270\pi$
$984$	0	0
$985$	−7.67246	−0.244465
$986$	−5.23391	−0.166682
$987$	0	0
$988$	−10.5638	−0.336078
$989$	−28.8889	−0.918613
$990$	0	0
$991$	−14.3028	−0.454344	−0.227172	−	0.973855i	$-0.572948\pi$
$991$	−14.3028	−0.454344	−0.227172	+	0.973855i	$0.572948\pi$
$992$	1.68517	0.0535043
$993$	0	0
$994$	10.7954	0.342408
$995$	−8.84695	−0.280467
$996$	0	0
$997$	−10.6687	−0.337881	−0.168940	−	0.985626i	$-0.554035\pi$
$997$	−10.6687	−0.337881	−0.168940	+	0.985626i	$0.554035\pi$
$998$	15.7252	0.497772
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.cz.1.6	✓	12
3.2	odd	2	inner	7623.2.a.cz.1.7	yes	12
11.10	odd	2		7623.2.a.da.1.7	yes	12
33.32	even	2		7623.2.a.da.1.6	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7623.2.a.cz.1.6	✓	12	1.1	even	1	trivial
7623.2.a.cz.1.7	yes	12	3.2	odd	2	inner
7623.2.a.da.1.6	yes	12	33.32	even	2
7623.2.a.da.1.7	yes	12	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)

\(f(q)\)	\(=\)	\(q-0.790737 q^{2} -1.37474 q^{4} +4.15216 q^{5} -1.00000 q^{7} +2.66853 q^{8} +O(q^{10})\)	\(q-0.790737 q^{2} -1.37474 q^{4} +4.15216 q^{5} -1.00000 q^{7} +2.66853 q^{8} -3.28327 q^{10} -2.46926 q^{13} +0.790737 q^{14} +0.639367 q^{16} -1.99183 q^{17} -3.11195 q^{19} -5.70812 q^{20} -3.45926 q^{23} +12.2404 q^{25} +1.95254 q^{26} +1.37474 q^{28} -3.32309 q^{29} -0.288427 q^{31} -5.84263 q^{32} +1.57501 q^{34} -4.15216 q^{35} +7.59863 q^{37} +2.46073 q^{38} +11.0802 q^{40} +2.32456 q^{41} +8.35116 q^{43} +2.73537 q^{46} -2.94175 q^{47} +1.00000 q^{49} -9.67896 q^{50} +3.39458 q^{52} -13.1332 q^{53} -2.66853 q^{56} +2.62769 q^{58} +7.82330 q^{59} -15.3255 q^{61} +0.228070 q^{62} +3.34125 q^{64} -10.2528 q^{65} -8.39127 q^{67} +2.73823 q^{68} +3.28327 q^{70} +13.6523 q^{71} -8.86448 q^{73} -6.00852 q^{74} +4.27810 q^{76} +15.5540 q^{79} +2.65475 q^{80} -1.83812 q^{82} -4.58873 q^{83} -8.27038 q^{85} -6.60357 q^{86} -11.8128 q^{89} +2.46926 q^{91} +4.75557 q^{92} +2.32615 q^{94} -12.9213 q^{95} -7.28658 q^{97} -0.790737 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 20 q^{4} - 12 q^{7}+O(q^{10}) \) 12 * q + 20 * q^4 - 12 * q^7	\( 12 q + 20 q^{4} - 12 q^{7} - 20 q^{10} - 20 q^{13} + 28 q^{16} - 12 q^{19} + 32 q^{25} - 20 q^{28} - 16 q^{31} - 24 q^{34} + 4 q^{37} - 48 q^{40} - 16 q^{43} - 24 q^{46} + 12 q^{49} - 96 q^{52} + 20 q^{58} - 44 q^{61} + 76 q^{64} + 20 q^{70} - 52 q^{73} + 8 q^{79} - 68 q^{82} - 72 q^{85} + 20 q^{91} - 20 q^{94} - 32 q^{97}+O(q^{100}) \) 12 * q + 20 * q^4 - 12 * q^7 - 20 * q^10 - 20 * q^13 + 28 * q^16 - 12 * q^19 + 32 * q^25 - 20 * q^28 - 16 * q^31 - 24 * q^34 + 4 * q^37 - 48 * q^40 - 16 * q^43 - 24 * q^46 + 12 * q^49 - 96 * q^52 + 20 * q^58 - 44 * q^61 + 76 * q^64 + 20 * q^70 - 52 * q^73 + 8 * q^79 - 68 * q^82 - 72 * q^85 + 20 * q^91 - 20 * q^94 - 32 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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