LMFDB - Embedded newform 7623.2.a.o.1.1

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:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2541)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
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)

$f(q)$

$=$

$q+1.00000 q^{2} -1.00000 q^{4} +3.00000 q^{5} -1.00000 q^{7} -3.00000 q^{8} +O(q^{10})$

$q+1.00000 q^{2} -1.00000 q^{4} +3.00000 q^{5} -1.00000 q^{7} -3.00000 q^{8} +3.00000 q^{10} -7.00000 q^{13} -1.00000 q^{14} -1.00000 q^{16} +3.00000 q^{17} +2.00000 q^{19} -3.00000 q^{20} +4.00000 q^{23} +4.00000 q^{25} -7.00000 q^{26} +1.00000 q^{28} +7.00000 q^{29} -10.0000 q^{31} +5.00000 q^{32} +3.00000 q^{34} -3.00000 q^{35} +1.00000 q^{37} +2.00000 q^{38} -9.00000 q^{40} -5.00000 q^{41} -6.00000 q^{43} +4.00000 q^{46} +6.00000 q^{47} +1.00000 q^{49} +4.00000 q^{50} +7.00000 q^{52} +5.00000 q^{53} +3.00000 q^{56} +7.00000 q^{58} +6.00000 q^{59} -10.0000 q^{61} -10.0000 q^{62} +7.00000 q^{64} -21.0000 q^{65} -8.00000 q^{67} -3.00000 q^{68} -3.00000 q^{70} +10.0000 q^{71} -10.0000 q^{73} +1.00000 q^{74} -2.00000 q^{76} +2.00000 q^{79} -3.00000 q^{80} -5.00000 q^{82} -16.0000 q^{83} +9.00000 q^{85} -6.00000 q^{86} -3.00000 q^{89} +7.00000 q^{91} -4.00000 q^{92} +6.00000 q^{94} +6.00000 q^{95} -19.0000 q^{97} +1.00000 q^{98} +O(q^{100})$

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 $

$2$	1.00000	0.707107	0.353553	−	0.935414i	$-0.384973\pi$
$2$	1.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	3.00000	1.34164	0.670820	−	0.741620i	$-0.265942\pi$
$5$	3.00000	1.34164	0.670820	+	0.741620i	$0.265942\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−3.00000	−1.06066
$9$	0	0
$10$	3.00000	0.948683
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−7.00000	−1.94145	−0.970725	−	0.240192i	$-0.922790\pi$
$13$	−7.00000	−1.94145	−0.970725	+	0.240192i	$0.922790\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	−1.00000	−0.250000
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	−3.00000	−0.670820
$21$	0	0
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	−7.00000	−1.37281
$27$	0	0
$28$	1.00000	0.188982
$29$	7.00000	1.29987	0.649934	−	0.759991i	$-0.274797\pi$
$29$	7.00000	1.29987	0.649934	+	0.759991i	$0.274797\pi$
$30$	0	0
$31$	−10.0000	−1.79605	−0.898027	−	0.439941i	$-0.854999\pi$
$31$	−10.0000	−1.79605	−0.898027	+	0.439941i	$0.854999\pi$
$32$	5.00000	0.883883
$33$	0	0
$34$	3.00000	0.514496
$35$	−3.00000	−0.507093
$36$	0	0
$37$	1.00000	0.164399	0.0821995	−	0.996616i	$-0.473806\pi$
$37$	1.00000	0.164399	0.0821995	+	0.996616i	$0.473806\pi$
$38$	2.00000	0.324443
$39$	0	0
$40$	−9.00000	−1.42302
$41$	−5.00000	−0.780869	−0.390434	−	0.920631i	$-0.627675\pi$
$41$	−5.00000	−0.780869	−0.390434	+	0.920631i	$0.627675\pi$
$42$	0	0
$43$	−6.00000	−0.914991	−0.457496	−	0.889212i	$-0.651253\pi$
$43$	−6.00000	−0.914991	−0.457496	+	0.889212i	$0.651253\pi$
$44$	0	0
$45$	0	0
$46$	4.00000	0.589768
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	4.00000	0.565685
$51$	0	0
$52$	7.00000	0.970725
$53$	5.00000	0.686803	0.343401	−	0.939189i	$-0.388421\pi$
$53$	5.00000	0.686803	0.343401	+	0.939189i	$0.388421\pi$
$54$	0	0
$55$	0	0
$56$	3.00000	0.400892
$57$	0	0
$58$	7.00000	0.919145
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	−10.0000	−1.27000
$63$	0	0
$64$	7.00000	0.875000
$65$	−21.0000	−2.60473
$66$	0	0
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	−3.00000	−0.363803
$69$	0	0
$70$	−3.00000	−0.358569
$71$	10.0000	1.18678	0.593391	−	0.804914i	$-0.297789\pi$
$71$	10.0000	1.18678	0.593391	+	0.804914i	$0.297789\pi$
$72$	0	0
$73$	−10.0000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	1.00000	0.116248
$75$	0	0
$76$	−2.00000	−0.229416
$77$	0	0
$78$	0	0
$79$	2.00000	0.225018	0.112509	−	0.993651i	$-0.464111\pi$
$79$	2.00000	0.225018	0.112509	+	0.993651i	$0.464111\pi$
$80$	−3.00000	−0.335410
$81$	0	0
$82$	−5.00000	−0.552158
$83$	−16.0000	−1.75623	−0.878114	−	0.478451i	$-0.841198\pi$
$83$	−16.0000	−1.75623	−0.878114	+	0.478451i	$0.841198\pi$
$84$	0	0
$85$	9.00000	0.976187
$86$	−6.00000	−0.646997
$87$	0	0
$88$	0	0
$89$	−3.00000	−0.317999	−0.159000	−	0.987279i	$-0.550827\pi$
$89$	−3.00000	−0.317999	−0.159000	+	0.987279i	$0.550827\pi$
$90$	0	0
$91$	7.00000	0.733799
$92$	−4.00000	−0.417029
$93$	0	0
$94$	6.00000	0.618853
$95$	6.00000	0.615587
$96$	0	0
$97$	−19.0000	−1.92916	−0.964579	−	0.263795i	$-0.915026\pi$
$97$	−19.0000	−1.92916	−0.964579	+	0.263795i	$0.915026\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	−4.00000	−0.400000
$101$	−2.00000	−0.199007	−0.0995037	−	0.995037i	$-0.531726\pi$
$101$	−2.00000	−0.199007	−0.0995037	+	0.995037i	$0.531726\pi$
$102$	0	0
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	21.0000	2.05922
$105$	0	0
$106$	5.00000	0.485643
$107$	−14.0000	−1.35343	−0.676716	−	0.736245i	$-0.736597\pi$
$107$	−14.0000	−1.35343	−0.676716	+	0.736245i	$0.736597\pi$
$108$	0	0
$109$	7.00000	0.670478	0.335239	−	0.942133i	$-0.391183\pi$
$109$	7.00000	0.670478	0.335239	+	0.942133i	$0.391183\pi$
$110$	0	0
$111$	0	0
$112$	1.00000	0.0944911
$113$	−9.00000	−0.846649	−0.423324	−	0.905978i	$-0.639137\pi$
$113$	−9.00000	−0.846649	−0.423324	+	0.905978i	$0.639137\pi$
$114$	0	0
$115$	12.0000	1.11901
$116$	−7.00000	−0.649934
$117$	0	0
$118$	6.00000	0.552345
$119$	−3.00000	−0.275010
$120$	0	0
$121$	0	0
$122$	−10.0000	−0.905357
$123$	0	0
$124$	10.0000	0.898027
$125$	−3.00000	−0.268328
$126$	0	0
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	−3.00000	−0.265165
$129$	0	0
$130$	−21.0000	−1.84182
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	0	0
$133$	−2.00000	−0.173422
$134$	−8.00000	−0.691095
$135$	0	0
$136$	−9.00000	−0.771744
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	0	0
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	3.00000	0.253546
$141$	0	0
$142$	10.0000	0.839181
$143$	0	0
$144$	0	0
$145$	21.0000	1.74396
$146$	−10.0000	−0.827606
$147$	0	0
$148$	−1.00000	−0.0821995
$149$	−21.0000	−1.72039	−0.860194	−	0.509968i	$-0.829657\pi$
$149$	−21.0000	−1.72039	−0.860194	+	0.509968i	$0.829657\pi$
$150$	0	0
$151$	−2.00000	−0.162758	−0.0813788	−	0.996683i	$-0.525932\pi$
$151$	−2.00000	−0.162758	−0.0813788	+	0.996683i	$0.525932\pi$
$152$	−6.00000	−0.486664
$153$	0	0
$154$	0	0
$155$	−30.0000	−2.40966
$156$	0	0
$157$	−14.0000	−1.11732	−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000	−1.11732	−0.558661	+	0.829396i	$0.688685\pi$
$158$	2.00000	0.159111
$159$	0	0
$160$	15.0000	1.18585
$161$	−4.00000	−0.315244
$162$	0	0
$163$	10.0000	0.783260	0.391630	−	0.920123i	$-0.371911\pi$
$163$	10.0000	0.783260	0.391630	+	0.920123i	$0.371911\pi$
$164$	5.00000	0.390434
$165$	0	0
$166$	−16.0000	−1.24184
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	0	0
$169$	36.0000	2.76923
$170$	9.00000	0.690268
$171$	0	0
$172$	6.00000	0.457496
$173$	−10.0000	−0.760286	−0.380143	−	0.924928i	$-0.624125\pi$
$173$	−10.0000	−0.760286	−0.380143	+	0.924928i	$0.624125\pi$
$174$	0	0
$175$	−4.00000	−0.302372
$176$	0	0
$177$	0	0
$178$	−3.00000	−0.224860
$179$	10.0000	0.747435	0.373718	−	0.927543i	$-0.378083\pi$
$179$	10.0000	0.747435	0.373718	+	0.927543i	$0.378083\pi$
$180$	0	0
$181$	23.0000	1.70958	0.854788	−	0.518977i	$-0.173687\pi$
$181$	23.0000	1.70958	0.854788	+	0.518977i	$0.173687\pi$
$182$	7.00000	0.518875
$183$	0	0
$184$	−12.0000	−0.884652
$185$	3.00000	0.220564
$186$	0	0
$187$	0	0
$188$	−6.00000	−0.437595
$189$	0	0
$190$	6.00000	0.435286
$191$	−22.0000	−1.59186	−0.795932	−	0.605386i	$-0.793019\pi$
$191$	−22.0000	−1.59186	−0.795932	+	0.605386i	$0.793019\pi$
$192$	0	0
$193$	−11.0000	−0.791797	−0.395899	−	0.918294i	$-0.629567\pi$
$193$	−11.0000	−0.791797	−0.395899	+	0.918294i	$0.629567\pi$
$194$	−19.0000	−1.36412
$195$	0	0
$196$	−1.00000	−0.0714286
$197$	−1.00000	−0.0712470	−0.0356235	−	0.999365i	$-0.511342\pi$
$197$	−1.00000	−0.0712470	−0.0356235	+	0.999365i	$0.511342\pi$
$198$	0	0
$199$	−2.00000	−0.141776	−0.0708881	−	0.997484i	$-0.522583\pi$
$199$	−2.00000	−0.141776	−0.0708881	+	0.997484i	$0.522583\pi$
$200$	−12.0000	−0.848528
$201$	0	0
$202$	−2.00000	−0.140720
$203$	−7.00000	−0.491304
$204$	0	0
$205$	−15.0000	−1.04765
$206$	−16.0000	−1.11477
$207$	0	0
$208$	7.00000	0.485363
$209$	0	0
$210$	0	0
$211$	−8.00000	−0.550743	−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000	−0.550743	−0.275371	+	0.961338i	$0.588801\pi$
$212$	−5.00000	−0.343401
$213$	0	0
$214$	−14.0000	−0.957020
$215$	−18.0000	−1.22759
$216$	0	0
$217$	10.0000	0.678844
$218$	7.00000	0.474100
$219$	0	0
$220$	0	0
$221$	−21.0000	−1.41261
$222$	0	0
$223$	−24.0000	−1.60716	−0.803579	−	0.595198i	$-0.797074\pi$
$223$	−24.0000	−1.60716	−0.803579	+	0.595198i	$0.797074\pi$
$224$	−5.00000	−0.334077
$225$	0	0
$226$	−9.00000	−0.598671
$227$	2.00000	0.132745	0.0663723	−	0.997795i	$-0.478857\pi$
$227$	2.00000	0.132745	0.0663723	+	0.997795i	$0.478857\pi$
$228$	0	0
$229$	−13.0000	−0.859064	−0.429532	−	0.903052i	$-0.641321\pi$
$229$	−13.0000	−0.859064	−0.429532	+	0.903052i	$0.641321\pi$
$230$	12.0000	0.791257
$231$	0	0
$232$	−21.0000	−1.37872
$233$	−23.0000	−1.50678	−0.753390	−	0.657574i	$-0.771583\pi$
$233$	−23.0000	−1.50678	−0.753390	+	0.657574i	$0.771583\pi$
$234$	0	0
$235$	18.0000	1.17419
$236$	−6.00000	−0.390567
$237$	0	0
$238$	−3.00000	−0.194461
$239$	12.0000	0.776215	0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000	0.776215	0.388108	+	0.921614i	$0.373129\pi$
$240$	0	0
$241$	−14.0000	−0.901819	−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000	−0.901819	−0.450910	+	0.892570i	$0.648900\pi$
$242$	0	0
$243$	0	0
$244$	10.0000	0.640184
$245$	3.00000	0.191663
$246$	0	0
$247$	−14.0000	−0.890799
$248$	30.0000	1.90500
$249$	0	0
$250$	−3.00000	−0.189737
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	0	0
$254$	−8.00000	−0.501965
$255$	0	0
$256$	−17.0000	−1.06250
$257$	29.0000	1.80897	0.904485	−	0.426505i	$-0.140255\pi$
$257$	29.0000	1.80897	0.904485	+	0.426505i	$0.140255\pi$
$258$	0	0
$259$	−1.00000	−0.0621370
$260$	21.0000	1.30236
$261$	0	0
$262$	−4.00000	−0.247121
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	0	0
$265$	15.0000	0.921443
$266$	−2.00000	−0.122628
$267$	0	0
$268$	8.00000	0.488678
$269$	23.0000	1.40233	0.701167	−	0.712997i	$-0.252663\pi$
$269$	23.0000	1.40233	0.701167	+	0.712997i	$0.252663\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	−3.00000	−0.181902
$273$	0	0
$274$	−2.00000	−0.120824
$275$	0	0
$276$	0	0
$277$	−5.00000	−0.300421	−0.150210	−	0.988654i	$-0.547995\pi$
$277$	−5.00000	−0.300421	−0.150210	+	0.988654i	$0.547995\pi$
$278$	4.00000	0.239904
$279$	0	0
$280$	9.00000	0.537853
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	0	0
$283$	22.0000	1.30776	0.653882	−	0.756596i	$-0.273139\pi$
$283$	22.0000	1.30776	0.653882	+	0.756596i	$0.273139\pi$
$284$	−10.0000	−0.593391
$285$	0	0
$286$	0	0
$287$	5.00000	0.295141
$288$	0	0
$289$	−8.00000	−0.470588
$290$	21.0000	1.23316
$291$	0	0
$292$	10.0000	0.585206
$293$	−27.0000	−1.57736	−0.788678	−	0.614806i	$-0.789234\pi$
$293$	−27.0000	−1.57736	−0.788678	+	0.614806i	$0.789234\pi$
$294$	0	0
$295$	18.0000	1.04800
$296$	−3.00000	−0.174371
$297$	0	0
$298$	−21.0000	−1.21650
$299$	−28.0000	−1.61928
$300$	0	0
$301$	6.00000	0.345834
$302$	−2.00000	−0.115087
$303$	0	0
$304$	−2.00000	−0.114708
$305$	−30.0000	−1.71780
$306$	0	0
$307$	12.0000	0.684876	0.342438	−	0.939540i	$-0.388747\pi$
$307$	12.0000	0.684876	0.342438	+	0.939540i	$0.388747\pi$
$308$	0	0
$309$	0	0
$310$	−30.0000	−1.70389
$311$	10.0000	0.567048	0.283524	−	0.958965i	$-0.408496\pi$
$311$	10.0000	0.567048	0.283524	+	0.958965i	$0.408496\pi$
$312$	0	0
$313$	9.00000	0.508710	0.254355	−	0.967111i	$-0.418137\pi$
$313$	9.00000	0.508710	0.254355	+	0.967111i	$0.418137\pi$
$314$	−14.0000	−0.790066
$315$	0	0
$316$	−2.00000	−0.112509
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	0	0
$319$	0	0
$320$	21.0000	1.17394
$321$	0	0
$322$	−4.00000	−0.222911
$323$	6.00000	0.333849
$324$	0	0
$325$	−28.0000	−1.55316
$326$	10.0000	0.553849
$327$	0	0
$328$	15.0000	0.828236
$329$	−6.00000	−0.330791
$330$	0	0
$331$	18.0000	0.989369	0.494685	−	0.869072i	$-0.335284\pi$
$331$	18.0000	0.989369	0.494685	+	0.869072i	$0.335284\pi$
$332$	16.0000	0.878114
$333$	0	0
$334$	12.0000	0.656611
$335$	−24.0000	−1.31126
$336$	0	0
$337$	5.00000	0.272367	0.136184	−	0.990684i	$-0.456516\pi$
$337$	5.00000	0.272367	0.136184	+	0.990684i	$0.456516\pi$
$338$	36.0000	1.95814
$339$	0	0
$340$	−9.00000	−0.488094
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	18.0000	0.970495
$345$	0	0
$346$	−10.0000	−0.537603
$347$	26.0000	1.39575	0.697877	−	0.716218i	$-0.254128\pi$
$347$	26.0000	1.39575	0.697877	+	0.716218i	$0.254128\pi$
$348$	0	0
$349$	−3.00000	−0.160586	−0.0802932	−	0.996771i	$-0.525586\pi$
$349$	−3.00000	−0.160586	−0.0802932	+	0.996771i	$0.525586\pi$
$350$	−4.00000	−0.213809
$351$	0	0
$352$	0	0
$353$	21.0000	1.11772	0.558859	−	0.829263i	$-0.311239\pi$
$353$	21.0000	1.11772	0.558859	+	0.829263i	$0.311239\pi$
$354$	0	0
$355$	30.0000	1.59223
$356$	3.00000	0.159000
$357$	0	0
$358$	10.0000	0.528516
$359$	12.0000	0.633336	0.316668	−	0.948536i	$-0.397436\pi$
$359$	12.0000	0.633336	0.316668	+	0.948536i	$0.397436\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	23.0000	1.20885
$363$	0	0
$364$	−7.00000	−0.366900
$365$	−30.0000	−1.57027
$366$	0	0
$367$	10.0000	0.521996	0.260998	−	0.965339i	$-0.415948\pi$
$367$	10.0000	0.521996	0.260998	+	0.965339i	$0.415948\pi$
$368$	−4.00000	−0.208514
$369$	0	0
$370$	3.00000	0.155963
$371$	−5.00000	−0.259587
$372$	0	0
$373$	−6.00000	−0.310668	−0.155334	−	0.987862i	$-0.549645\pi$
$373$	−6.00000	−0.310668	−0.155334	+	0.987862i	$0.549645\pi$
$374$	0	0
$375$	0	0
$376$	−18.0000	−0.928279
$377$	−49.0000	−2.52363
$378$	0	0
$379$	16.0000	0.821865	0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000	0.821865	0.410932	+	0.911666i	$0.365203\pi$
$380$	−6.00000	−0.307794
$381$	0	0
$382$	−22.0000	−1.12562
$383$	−16.0000	−0.817562	−0.408781	−	0.912633i	$-0.634046\pi$
$383$	−16.0000	−0.817562	−0.408781	+	0.912633i	$0.634046\pi$
$384$	0	0
$385$	0	0
$386$	−11.0000	−0.559885
$387$	0	0
$388$	19.0000	0.964579
$389$	−7.00000	−0.354914	−0.177457	−	0.984129i	$-0.556787\pi$
$389$	−7.00000	−0.354914	−0.177457	+	0.984129i	$0.556787\pi$
$390$	0	0
$391$	12.0000	0.606866
$392$	−3.00000	−0.151523
$393$	0	0
$394$	−1.00000	−0.0503793
$395$	6.00000	0.301893
$396$	0	0
$397$	−33.0000	−1.65622	−0.828111	−	0.560564i	$-0.810584\pi$
$397$	−33.0000	−1.65622	−0.828111	+	0.560564i	$0.810584\pi$
$398$	−2.00000	−0.100251
$399$	0	0
$400$	−4.00000	−0.200000
$401$	15.0000	0.749064	0.374532	−	0.927214i	$-0.377803\pi$
$401$	15.0000	0.749064	0.374532	+	0.927214i	$0.377803\pi$
$402$	0	0
$403$	70.0000	3.48695
$404$	2.00000	0.0995037
$405$	0	0
$406$	−7.00000	−0.347404
$407$	0	0
$408$	0	0
$409$	−5.00000	−0.247234	−0.123617	−	0.992330i	$-0.539449\pi$
$409$	−5.00000	−0.247234	−0.123617	+	0.992330i	$0.539449\pi$
$410$	−15.0000	−0.740797
$411$	0	0
$412$	16.0000	0.788263
$413$	−6.00000	−0.295241
$414$	0	0
$415$	−48.0000	−2.35623
$416$	−35.0000	−1.71602
$417$	0	0
$418$	0	0
$419$	−16.0000	−0.781651	−0.390826	−	0.920465i	$-0.627810\pi$
$419$	−16.0000	−0.781651	−0.390826	+	0.920465i	$0.627810\pi$
$420$	0	0
$421$	−15.0000	−0.731055	−0.365528	−	0.930800i	$-0.619111\pi$
$421$	−15.0000	−0.731055	−0.365528	+	0.930800i	$0.619111\pi$
$422$	−8.00000	−0.389434
$423$	0	0
$424$	−15.0000	−0.728464
$425$	12.0000	0.582086
$426$	0	0
$427$	10.0000	0.483934
$428$	14.0000	0.676716
$429$	0	0
$430$	−18.0000	−0.868037
$431$	−14.0000	−0.674356	−0.337178	−	0.941441i	$-0.609472\pi$
$431$	−14.0000	−0.674356	−0.337178	+	0.941441i	$0.609472\pi$
$432$	0	0
$433$	1.00000	0.0480569	0.0240285	−	0.999711i	$-0.492351\pi$
$433$	1.00000	0.0480569	0.0240285	+	0.999711i	$0.492351\pi$
$434$	10.0000	0.480015
$435$	0	0
$436$	−7.00000	−0.335239
$437$	8.00000	0.382692
$438$	0	0
$439$	−4.00000	−0.190910	−0.0954548	−	0.995434i	$-0.530431\pi$
$439$	−4.00000	−0.190910	−0.0954548	+	0.995434i	$0.530431\pi$
$440$	0	0
$441$	0	0
$442$	−21.0000	−0.998868
$443$	−4.00000	−0.190046	−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000	−0.190046	−0.0950229	+	0.995475i	$0.530292\pi$
$444$	0	0
$445$	−9.00000	−0.426641
$446$	−24.0000	−1.13643
$447$	0	0
$448$	−7.00000	−0.330719
$449$	27.0000	1.27421	0.637104	−	0.770778i	$-0.280132\pi$
$449$	27.0000	1.27421	0.637104	+	0.770778i	$0.280132\pi$
$450$	0	0
$451$	0	0
$452$	9.00000	0.423324
$453$	0	0
$454$	2.00000	0.0938647
$455$	21.0000	0.984495
$456$	0	0
$457$	33.0000	1.54367	0.771837	−	0.635820i	$-0.219338\pi$
$457$	33.0000	1.54367	0.771837	+	0.635820i	$0.219338\pi$
$458$	−13.0000	−0.607450
$459$	0	0
$460$	−12.0000	−0.559503
$461$	−15.0000	−0.698620	−0.349310	−	0.937007i	$-0.613584\pi$
$461$	−15.0000	−0.698620	−0.349310	+	0.937007i	$0.613584\pi$
$462$	0	0
$463$	22.0000	1.02243	0.511213	−	0.859454i	$-0.329196\pi$
$463$	22.0000	1.02243	0.511213	+	0.859454i	$0.329196\pi$
$464$	−7.00000	−0.324967
$465$	0	0
$466$	−23.0000	−1.06545
$467$	−8.00000	−0.370196	−0.185098	−	0.982720i	$-0.559260\pi$
$467$	−8.00000	−0.370196	−0.185098	+	0.982720i	$0.559260\pi$
$468$	0	0
$469$	8.00000	0.369406
$470$	18.0000	0.830278
$471$	0	0
$472$	−18.0000	−0.828517
$473$	0	0
$474$	0	0
$475$	8.00000	0.367065
$476$	3.00000	0.137505
$477$	0	0
$478$	12.0000	0.548867
$479$	−26.0000	−1.18797	−0.593985	−	0.804476i	$-0.702446\pi$
$479$	−26.0000	−1.18797	−0.593985	+	0.804476i	$0.702446\pi$
$480$	0	0
$481$	−7.00000	−0.319173
$482$	−14.0000	−0.637683
$483$	0	0
$484$	0	0
$485$	−57.0000	−2.58824
$486$	0	0
$487$	−38.0000	−1.72194	−0.860972	−	0.508652i	$-0.830144\pi$
$487$	−38.0000	−1.72194	−0.860972	+	0.508652i	$0.830144\pi$
$488$	30.0000	1.35804
$489$	0	0
$490$	3.00000	0.135526
$491$	−36.0000	−1.62466	−0.812329	−	0.583200i	$-0.801800\pi$
$491$	−36.0000	−1.62466	−0.812329	+	0.583200i	$0.801800\pi$
$492$	0	0
$493$	21.0000	0.945792
$494$	−14.0000	−0.629890
$495$	0	0
$496$	10.0000	0.449013
$497$	−10.0000	−0.448561
$498$	0	0
$499$	−32.0000	−1.43252	−0.716258	−	0.697835i	$-0.754147\pi$
$499$	−32.0000	−1.43252	−0.716258	+	0.697835i	$0.754147\pi$
$500$	3.00000	0.134164
$501$	0	0
$502$	12.0000	0.535586
$503$	16.0000	0.713405	0.356702	−	0.934218i	$-0.383901\pi$
$503$	16.0000	0.713405	0.356702	+	0.934218i	$0.383901\pi$
$504$	0	0
$505$	−6.00000	−0.266996
$506$	0	0
$507$	0	0
$508$	8.00000	0.354943
$509$	42.0000	1.86162	0.930809	−	0.365507i	$-0.119104\pi$
$509$	42.0000	1.86162	0.930809	+	0.365507i	$0.119104\pi$
$510$	0	0
$511$	10.0000	0.442374
$512$	−11.0000	−0.486136
$513$	0	0
$514$	29.0000	1.27914
$515$	−48.0000	−2.11513
$516$	0	0
$517$	0	0
$518$	−1.00000	−0.0439375
$519$	0	0
$520$	63.0000	2.76273
$521$	2.00000	0.0876216	0.0438108	−	0.999040i	$-0.486050\pi$
$521$	2.00000	0.0876216	0.0438108	+	0.999040i	$0.486050\pi$
$522$	0	0
$523$	−20.0000	−0.874539	−0.437269	−	0.899331i	$-0.644054\pi$
$523$	−20.0000	−0.874539	−0.437269	+	0.899331i	$0.644054\pi$
$524$	4.00000	0.174741
$525$	0	0
$526$	24.0000	1.04645
$527$	−30.0000	−1.30682
$528$	0	0
$529$	−7.00000	−0.304348
$530$	15.0000	0.651558
$531$	0	0
$532$	2.00000	0.0867110
$533$	35.0000	1.51602
$534$	0	0
$535$	−42.0000	−1.81582
$536$	24.0000	1.03664
$537$	0	0
$538$	23.0000	0.991600
$539$	0	0
$540$	0	0
$541$	−30.0000	−1.28980	−0.644900	−	0.764267i	$-0.723101\pi$
$541$	−30.0000	−1.28980	−0.644900	+	0.764267i	$0.723101\pi$
$542$	8.00000	0.343629
$543$	0	0
$544$	15.0000	0.643120
$545$	21.0000	0.899541
$546$	0	0
$547$	−2.00000	−0.0855138	−0.0427569	−	0.999086i	$-0.513614\pi$
$547$	−2.00000	−0.0855138	−0.0427569	+	0.999086i	$0.513614\pi$
$548$	2.00000	0.0854358
$549$	0	0
$550$	0	0
$551$	14.0000	0.596420
$552$	0	0
$553$	−2.00000	−0.0850487
$554$	−5.00000	−0.212430
$555$	0	0
$556$	−4.00000	−0.169638
$557$	−6.00000	−0.254228	−0.127114	−	0.991888i	$-0.540571\pi$
$557$	−6.00000	−0.254228	−0.127114	+	0.991888i	$0.540571\pi$
$558$	0	0
$559$	42.0000	1.77641
$560$	3.00000	0.126773
$561$	0	0
$562$	18.0000	0.759284
$563$	−18.0000	−0.758610	−0.379305	−	0.925272i	$-0.623837\pi$
$563$	−18.0000	−0.758610	−0.379305	+	0.925272i	$0.623837\pi$
$564$	0	0
$565$	−27.0000	−1.13590
$566$	22.0000	0.924729
$567$	0	0
$568$	−30.0000	−1.25877
$569$	10.0000	0.419222	0.209611	−	0.977785i	$-0.432780\pi$
$569$	10.0000	0.419222	0.209611	+	0.977785i	$0.432780\pi$
$570$	0	0
$571$	16.0000	0.669579	0.334790	−	0.942293i	$-0.391335\pi$
$571$	16.0000	0.669579	0.334790	+	0.942293i	$0.391335\pi$
$572$	0	0
$573$	0	0
$574$	5.00000	0.208696
$575$	16.0000	0.667246
$576$	0	0
$577$	37.0000	1.54033	0.770165	−	0.637845i	$-0.220174\pi$
$577$	37.0000	1.54033	0.770165	+	0.637845i	$0.220174\pi$
$578$	−8.00000	−0.332756
$579$	0	0
$580$	−21.0000	−0.871978
$581$	16.0000	0.663792
$582$	0	0
$583$	0	0
$584$	30.0000	1.24141
$585$	0	0
$586$	−27.0000	−1.11536
$587$	42.0000	1.73353	0.866763	−	0.498721i	$-0.166197\pi$
$587$	42.0000	1.73353	0.866763	+	0.498721i	$0.166197\pi$
$588$	0	0
$589$	−20.0000	−0.824086
$590$	18.0000	0.741048
$591$	0	0
$592$	−1.00000	−0.0410997
$593$	7.00000	0.287456	0.143728	−	0.989617i	$-0.454091\pi$
$593$	7.00000	0.287456	0.143728	+	0.989617i	$0.454091\pi$
$594$	0	0
$595$	−9.00000	−0.368964
$596$	21.0000	0.860194
$597$	0	0
$598$	−28.0000	−1.14501
$599$	−10.0000	−0.408589	−0.204294	−	0.978909i	$-0.565490\pi$
$599$	−10.0000	−0.408589	−0.204294	+	0.978909i	$0.565490\pi$
$600$	0	0
$601$	−33.0000	−1.34610	−0.673049	−	0.739598i	$-0.735016\pi$
$601$	−33.0000	−1.34610	−0.673049	+	0.739598i	$0.735016\pi$
$602$	6.00000	0.244542
$603$	0	0
$604$	2.00000	0.0813788
$605$	0	0
$606$	0	0
$607$	−4.00000	−0.162355	−0.0811775	−	0.996700i	$-0.525868\pi$
$607$	−4.00000	−0.162355	−0.0811775	+	0.996700i	$0.525868\pi$
$608$	10.0000	0.405554
$609$	0	0
$610$	−30.0000	−1.21466
$611$	−42.0000	−1.69914
$612$	0	0
$613$	7.00000	0.282727	0.141364	−	0.989958i	$-0.454851\pi$
$613$	7.00000	0.282727	0.141364	+	0.989958i	$0.454851\pi$
$614$	12.0000	0.484281
$615$	0	0
$616$	0	0
$617$	3.00000	0.120775	0.0603877	−	0.998175i	$-0.480766\pi$
$617$	3.00000	0.120775	0.0603877	+	0.998175i	$0.480766\pi$
$618$	0	0
$619$	−42.0000	−1.68812	−0.844061	−	0.536247i	$-0.819842\pi$
$619$	−42.0000	−1.68812	−0.844061	+	0.536247i	$0.819842\pi$
$620$	30.0000	1.20483
$621$	0	0
$622$	10.0000	0.400963
$623$	3.00000	0.120192
$624$	0	0
$625$	−29.0000	−1.16000
$626$	9.00000	0.359712
$627$	0	0
$628$	14.0000	0.558661
$629$	3.00000	0.119618
$630$	0	0
$631$	14.0000	0.557331	0.278666	−	0.960388i	$-0.410108\pi$
$631$	14.0000	0.557331	0.278666	+	0.960388i	$0.410108\pi$
$632$	−6.00000	−0.238667
$633$	0	0
$634$	−18.0000	−0.714871
$635$	−24.0000	−0.952411
$636$	0	0
$637$	−7.00000	−0.277350
$638$	0	0
$639$	0	0
$640$	−9.00000	−0.355756
$641$	−9.00000	−0.355479	−0.177739	−	0.984078i	$-0.556878\pi$
$641$	−9.00000	−0.355479	−0.177739	+	0.984078i	$0.556878\pi$
$642$	0	0
$643$	12.0000	0.473234	0.236617	−	0.971603i	$-0.423961\pi$
$643$	12.0000	0.473234	0.236617	+	0.971603i	$0.423961\pi$
$644$	4.00000	0.157622
$645$	0	0
$646$	6.00000	0.236067
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	0	0
$650$	−28.0000	−1.09825
$651$	0	0
$652$	−10.0000	−0.391630
$653$	18.0000	0.704394	0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000	0.704394	0.352197	+	0.935926i	$0.385435\pi$
$654$	0	0
$655$	−12.0000	−0.468879
$656$	5.00000	0.195217
$657$	0	0
$658$	−6.00000	−0.233904
$659$	6.00000	0.233727	0.116863	−	0.993148i	$-0.462716\pi$
$659$	6.00000	0.233727	0.116863	+	0.993148i	$0.462716\pi$
$660$	0	0
$661$	35.0000	1.36134	0.680671	−	0.732589i	$-0.261688\pi$
$661$	35.0000	1.36134	0.680671	+	0.732589i	$0.261688\pi$
$662$	18.0000	0.699590
$663$	0	0
$664$	48.0000	1.86276
$665$	−6.00000	−0.232670
$666$	0	0
$667$	28.0000	1.08416
$668$	−12.0000	−0.464294
$669$	0	0
$670$	−24.0000	−0.927201
$671$	0	0
$672$	0	0
$673$	−46.0000	−1.77317	−0.886585	−	0.462566i	$-0.846929\pi$
$673$	−46.0000	−1.77317	−0.886585	+	0.462566i	$0.846929\pi$
$674$	5.00000	0.192593
$675$	0	0
$676$	−36.0000	−1.38462
$677$	33.0000	1.26829	0.634147	−	0.773213i	$-0.281352\pi$
$677$	33.0000	1.26829	0.634147	+	0.773213i	$0.281352\pi$
$678$	0	0
$679$	19.0000	0.729153
$680$	−27.0000	−1.03540
$681$	0	0
$682$	0	0
$683$	4.00000	0.153056	0.0765279	−	0.997067i	$-0.475617\pi$
$683$	4.00000	0.153056	0.0765279	+	0.997067i	$0.475617\pi$
$684$	0	0
$685$	−6.00000	−0.229248
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	6.00000	0.228748
$689$	−35.0000	−1.33339
$690$	0	0
$691$	22.0000	0.836919	0.418460	−	0.908235i	$-0.362570\pi$
$691$	22.0000	0.836919	0.418460	+	0.908235i	$0.362570\pi$
$692$	10.0000	0.380143
$693$	0	0
$694$	26.0000	0.986947
$695$	12.0000	0.455186
$696$	0	0
$697$	−15.0000	−0.568166
$698$	−3.00000	−0.113552
$699$	0	0
$700$	4.00000	0.151186
$701$	−29.0000	−1.09531	−0.547657	−	0.836703i	$-0.684480\pi$
$701$	−29.0000	−1.09531	−0.547657	+	0.836703i	$0.684480\pi$
$702$	0	0
$703$	2.00000	0.0754314
$704$	0	0
$705$	0	0
$706$	21.0000	0.790345
$707$	2.00000	0.0752177
$708$	0	0
$709$	−6.00000	−0.225335	−0.112667	−	0.993633i	$-0.535939\pi$
$709$	−6.00000	−0.225335	−0.112667	+	0.993633i	$0.535939\pi$
$710$	30.0000	1.12588
$711$	0	0
$712$	9.00000	0.337289
$713$	−40.0000	−1.49801
$714$	0	0
$715$	0	0
$716$	−10.0000	−0.373718
$717$	0	0
$718$	12.0000	0.447836
$719$	−30.0000	−1.11881	−0.559406	−	0.828894i	$-0.688971\pi$
$719$	−30.0000	−1.11881	−0.559406	+	0.828894i	$0.688971\pi$
$720$	0	0
$721$	16.0000	0.595871
$722$	−15.0000	−0.558242
$723$	0	0
$724$	−23.0000	−0.854788
$725$	28.0000	1.03989
$726$	0	0
$727$	−14.0000	−0.519231	−0.259616	−	0.965712i	$-0.583596\pi$
$727$	−14.0000	−0.519231	−0.259616	+	0.965712i	$0.583596\pi$
$728$	−21.0000	−0.778312
$729$	0	0
$730$	−30.0000	−1.11035
$731$	−18.0000	−0.665754
$732$	0	0
$733$	1.00000	0.0369358	0.0184679	−	0.999829i	$-0.494121\pi$
$733$	1.00000	0.0369358	0.0184679	+	0.999829i	$0.494121\pi$
$734$	10.0000	0.369107
$735$	0	0
$736$	20.0000	0.737210
$737$	0	0
$738$	0	0
$739$	30.0000	1.10357	0.551784	−	0.833987i	$-0.313947\pi$
$739$	30.0000	1.10357	0.551784	+	0.833987i	$0.313947\pi$
$740$	−3.00000	−0.110282
$741$	0	0
$742$	−5.00000	−0.183556
$743$	−6.00000	−0.220119	−0.110059	−	0.993925i	$-0.535104\pi$
$743$	−6.00000	−0.220119	−0.110059	+	0.993925i	$0.535104\pi$
$744$	0	0
$745$	−63.0000	−2.30814
$746$	−6.00000	−0.219676
$747$	0	0
$748$	0	0
$749$	14.0000	0.511549
$750$	0	0
$751$	14.0000	0.510867	0.255434	−	0.966827i	$-0.417782\pi$
$751$	14.0000	0.510867	0.255434	+	0.966827i	$0.417782\pi$
$752$	−6.00000	−0.218797
$753$	0	0
$754$	−49.0000	−1.78447
$755$	−6.00000	−0.218362
$756$	0	0
$757$	−39.0000	−1.41748	−0.708740	−	0.705470i	$-0.750736\pi$
$757$	−39.0000	−1.41748	−0.708740	+	0.705470i	$0.750736\pi$
$758$	16.0000	0.581146
$759$	0	0
$760$	−18.0000	−0.652929
$761$	15.0000	0.543750	0.271875	−	0.962333i	$-0.412356\pi$
$761$	15.0000	0.543750	0.271875	+	0.962333i	$0.412356\pi$
$762$	0	0
$763$	−7.00000	−0.253417
$764$	22.0000	0.795932
$765$	0	0
$766$	−16.0000	−0.578103
$767$	−42.0000	−1.51653
$768$	0	0
$769$	15.0000	0.540914	0.270457	−	0.962732i	$-0.412825\pi$
$769$	15.0000	0.540914	0.270457	+	0.962732i	$0.412825\pi$
$770$	0	0
$771$	0	0
$772$	11.0000	0.395899
$773$	18.0000	0.647415	0.323708	−	0.946157i	$-0.395071\pi$
$773$	18.0000	0.647415	0.323708	+	0.946157i	$0.395071\pi$
$774$	0	0
$775$	−40.0000	−1.43684
$776$	57.0000	2.04618
$777$	0	0
$778$	−7.00000	−0.250962
$779$	−10.0000	−0.358287
$780$	0	0
$781$	0	0
$782$	12.0000	0.429119
$783$	0	0
$784$	−1.00000	−0.0357143
$785$	−42.0000	−1.49904
$786$	0	0
$787$	−40.0000	−1.42585	−0.712923	−	0.701242i	$-0.752629\pi$
$787$	−40.0000	−1.42585	−0.712923	+	0.701242i	$0.752629\pi$
$788$	1.00000	0.0356235
$789$	0	0
$790$	6.00000	0.213470
$791$	9.00000	0.320003
$792$	0	0
$793$	70.0000	2.48577
$794$	−33.0000	−1.17113
$795$	0	0
$796$	2.00000	0.0708881
$797$	−2.00000	−0.0708436	−0.0354218	−	0.999372i	$-0.511277\pi$
$797$	−2.00000	−0.0708436	−0.0354218	+	0.999372i	$0.511277\pi$
$798$	0	0
$799$	18.0000	0.636794
$800$	20.0000	0.707107
$801$	0	0
$802$	15.0000	0.529668
$803$	0	0
$804$	0	0
$805$	−12.0000	−0.422944
$806$	70.0000	2.46564
$807$	0	0
$808$	6.00000	0.211079
$809$	18.0000	0.632846	0.316423	−	0.948618i	$-0.397518\pi$
$809$	18.0000	0.632846	0.316423	+	0.948618i	$0.397518\pi$
$810$	0	0
$811$	−38.0000	−1.33436	−0.667180	−	0.744896i	$-0.732499\pi$
$811$	−38.0000	−1.33436	−0.667180	+	0.744896i	$0.732499\pi$
$812$	7.00000	0.245652
$813$	0	0
$814$	0	0
$815$	30.0000	1.05085
$816$	0	0
$817$	−12.0000	−0.419827
$818$	−5.00000	−0.174821
$819$	0	0
$820$	15.0000	0.523823
$821$	10.0000	0.349002	0.174501	−	0.984657i	$-0.444169\pi$
$821$	10.0000	0.349002	0.174501	+	0.984657i	$0.444169\pi$
$822$	0	0
$823$	28.0000	0.976019	0.488009	−	0.872838i	$-0.337723\pi$
$823$	28.0000	0.976019	0.488009	+	0.872838i	$0.337723\pi$
$824$	48.0000	1.67216
$825$	0	0
$826$	−6.00000	−0.208767
$827$	46.0000	1.59958	0.799788	−	0.600282i	$-0.204945\pi$
$827$	46.0000	1.59958	0.799788	+	0.600282i	$0.204945\pi$
$828$	0	0
$829$	−25.0000	−0.868286	−0.434143	−	0.900844i	$-0.642949\pi$
$829$	−25.0000	−0.868286	−0.434143	+	0.900844i	$0.642949\pi$
$830$	−48.0000	−1.66610
$831$	0	0
$832$	−49.0000	−1.69877
$833$	3.00000	0.103944
$834$	0	0
$835$	36.0000	1.24583
$836$	0	0
$837$	0	0
$838$	−16.0000	−0.552711
$839$	−30.0000	−1.03572	−0.517858	−	0.855467i	$-0.673270\pi$
$839$	−30.0000	−1.03572	−0.517858	+	0.855467i	$0.673270\pi$
$840$	0	0
$841$	20.0000	0.689655
$842$	−15.0000	−0.516934
$843$	0	0
$844$	8.00000	0.275371
$845$	108.000	3.71531
$846$	0	0
$847$	0	0
$848$	−5.00000	−0.171701
$849$	0	0
$850$	12.0000	0.411597
$851$	4.00000	0.137118
$852$	0	0
$853$	37.0000	1.26686	0.633428	−	0.773802i	$-0.281647\pi$
$853$	37.0000	1.26686	0.633428	+	0.773802i	$0.281647\pi$
$854$	10.0000	0.342193
$855$	0	0
$856$	42.0000	1.43553
$857$	−26.0000	−0.888143	−0.444072	−	0.895991i	$-0.646466\pi$
$857$	−26.0000	−0.888143	−0.444072	+	0.895991i	$0.646466\pi$
$858$	0	0
$859$	0	0		−	1.00000i	$-0.5\pi$
$859$	0	0			1.00000i	$0.5\pi$
$860$	18.0000	0.613795
$861$	0	0
$862$	−14.0000	−0.476842
$863$	14.0000	0.476566	0.238283	−	0.971196i	$-0.423415\pi$
$863$	14.0000	0.476566	0.238283	+	0.971196i	$0.423415\pi$
$864$	0	0
$865$	−30.0000	−1.02003
$866$	1.00000	0.0339814
$867$	0	0
$868$	−10.0000	−0.339422
$869$	0	0
$870$	0	0
$871$	56.0000	1.89749
$872$	−21.0000	−0.711150
$873$	0	0
$874$	8.00000	0.270604
$875$	3.00000	0.101419
$876$	0	0
$877$	−45.0000	−1.51954	−0.759771	−	0.650191i	$-0.774689\pi$
$877$	−45.0000	−1.51954	−0.759771	+	0.650191i	$0.774689\pi$
$878$	−4.00000	−0.134993
$879$	0	0
$880$	0	0
$881$	−35.0000	−1.17918	−0.589590	−	0.807703i	$-0.700711\pi$
$881$	−35.0000	−1.17918	−0.589590	+	0.807703i	$0.700711\pi$
$882$	0	0
$883$	6.00000	0.201916	0.100958	−	0.994891i	$-0.467809\pi$
$883$	6.00000	0.201916	0.100958	+	0.994891i	$0.467809\pi$
$884$	21.0000	0.706306
$885$	0	0
$886$	−4.00000	−0.134383
$887$	−8.00000	−0.268614	−0.134307	−	0.990940i	$-0.542881\pi$
$887$	−8.00000	−0.268614	−0.134307	+	0.990940i	$0.542881\pi$
$888$	0	0
$889$	8.00000	0.268311
$890$	−9.00000	−0.301681
$891$	0	0
$892$	24.0000	0.803579
$893$	12.0000	0.401565
$894$	0	0
$895$	30.0000	1.00279
$896$	3.00000	0.100223
$897$	0	0
$898$	27.0000	0.901002
$899$	−70.0000	−2.33463
$900$	0	0
$901$	15.0000	0.499722
$902$	0	0
$903$	0	0
$904$	27.0000	0.898007
$905$	69.0000	2.29364
$906$	0	0
$907$	18.0000	0.597680	0.298840	−	0.954303i	$-0.403400\pi$
$907$	18.0000	0.597680	0.298840	+	0.954303i	$0.403400\pi$
$908$	−2.00000	−0.0663723
$909$	0	0
$910$	21.0000	0.696143
$911$	−12.0000	−0.397578	−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000	−0.397578	−0.198789	+	0.980042i	$0.563701\pi$
$912$	0	0
$913$	0	0
$914$	33.0000	1.09154
$915$	0	0
$916$	13.0000	0.429532
$917$	4.00000	0.132092
$918$	0	0
$919$	46.0000	1.51740	0.758700	−	0.651440i	$-0.225835\pi$
$919$	46.0000	1.51740	0.758700	+	0.651440i	$0.225835\pi$
$920$	−36.0000	−1.18688
$921$	0	0
$922$	−15.0000	−0.493999
$923$	−70.0000	−2.30408
$924$	0	0
$925$	4.00000	0.131519
$926$	22.0000	0.722965
$927$	0	0
$928$	35.0000	1.14893
$929$	21.0000	0.688988	0.344494	−	0.938789i	$-0.388051\pi$
$929$	21.0000	0.688988	0.344494	+	0.938789i	$0.388051\pi$
$930$	0	0
$931$	2.00000	0.0655474
$932$	23.0000	0.753390
$933$	0	0
$934$	−8.00000	−0.261768
$935$	0	0
$936$	0	0
$937$	−25.0000	−0.816714	−0.408357	−	0.912822i	$-0.633898\pi$
$937$	−25.0000	−0.816714	−0.408357	+	0.912822i	$0.633898\pi$
$938$	8.00000	0.261209
$939$	0	0
$940$	−18.0000	−0.587095
$941$	17.0000	0.554184	0.277092	−	0.960843i	$-0.410629\pi$
$941$	17.0000	0.554184	0.277092	+	0.960843i	$0.410629\pi$
$942$	0	0
$943$	−20.0000	−0.651290
$944$	−6.00000	−0.195283
$945$	0	0
$946$	0	0
$947$	54.0000	1.75476	0.877382	−	0.479792i	$-0.159288\pi$
$947$	54.0000	1.75476	0.877382	+	0.479792i	$0.159288\pi$
$948$	0	0
$949$	70.0000	2.27230
$950$	8.00000	0.259554
$951$	0	0
$952$	9.00000	0.291692
$953$	29.0000	0.939402	0.469701	−	0.882826i	$-0.344362\pi$
$953$	29.0000	0.939402	0.469701	+	0.882826i	$0.344362\pi$
$954$	0	0
$955$	−66.0000	−2.13571
$956$	−12.0000	−0.388108
$957$	0	0
$958$	−26.0000	−0.840022
$959$	2.00000	0.0645834
$960$	0	0
$961$	69.0000	2.22581
$962$	−7.00000	−0.225689
$963$	0	0
$964$	14.0000	0.450910
$965$	−33.0000	−1.06231
$966$	0	0
$967$	−14.0000	−0.450210	−0.225105	−	0.974335i	$-0.572272\pi$
$967$	−14.0000	−0.450210	−0.225105	+	0.974335i	$0.572272\pi$
$968$	0	0
$969$	0	0
$970$	−57.0000	−1.83016
$971$	−4.00000	−0.128366	−0.0641831	−	0.997938i	$-0.520444\pi$
$971$	−4.00000	−0.128366	−0.0641831	+	0.997938i	$0.520444\pi$
$972$	0	0
$973$	−4.00000	−0.128234
$974$	−38.0000	−1.21760
$975$	0	0
$976$	10.0000	0.320092
$977$	23.0000	0.735835	0.367918	−	0.929858i	$-0.380071\pi$
$977$	23.0000	0.735835	0.367918	+	0.929858i	$0.380071\pi$
$978$	0	0
$979$	0	0
$980$	−3.00000	−0.0958315
$981$	0	0
$982$	−36.0000	−1.14881
$983$	−58.0000	−1.84991	−0.924956	−	0.380073i	$-0.875899\pi$
$983$	−58.0000	−1.84991	−0.924956	+	0.380073i	$0.875899\pi$
$984$	0	0
$985$	−3.00000	−0.0955879
$986$	21.0000	0.668776
$987$	0	0
$988$	14.0000	0.445399
$989$	−24.0000	−0.763156
$990$	0	0
$991$	2.00000	0.0635321	0.0317660	−	0.999495i	$-0.489887\pi$
$991$	2.00000	0.0635321	0.0317660	+	0.999495i	$0.489887\pi$
$992$	−50.0000	−1.58750
$993$	0	0
$994$	−10.0000	−0.317181
$995$	−6.00000	−0.190213
$996$	0	0
$997$	−43.0000	−1.36182	−0.680912	−	0.732365i	$-0.738416\pi$
$997$	−43.0000	−1.36182	−0.680912	+	0.732365i	$0.738416\pi$
$998$	−32.0000	−1.01294
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.o.1.1		1
3.2	odd	2		2541.2.a.c.1.1	✓	1
11.10	odd	2		7623.2.a.h.1.1		1
33.32	even	2		2541.2.a.g.1.1	yes	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2541.2.a.c.1.1	✓	1	3.2	odd	2
2541.2.a.g.1.1	yes	1	33.32	even	2
7623.2.a.h.1.1		1	11.10	odd	2
7623.2.a.o.1.1		1	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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