LMFDB - Embedded newform 3762.2.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3762,2,Mod(1,3762)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3762, 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("3762.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3762 = 2 \cdot 3^{2} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3762.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:	$30.0397212404$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 418)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	3762.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$

$q-1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} -2.00000 q^{10} -1.00000 q^{11} -7.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +3.00000 q^{17} +1.00000 q^{19} +2.00000 q^{20} +1.00000 q^{22} -3.00000 q^{23} -1.00000 q^{25} +7.00000 q^{26} +1.00000 q^{28} -1.00000 q^{29} +2.00000 q^{31} -1.00000 q^{32} -3.00000 q^{34} +2.00000 q^{35} -6.00000 q^{37} -1.00000 q^{38} -2.00000 q^{40} +2.00000 q^{41} +4.00000 q^{43} -1.00000 q^{44} +3.00000 q^{46} -6.00000 q^{49} +1.00000 q^{50} -7.00000 q^{52} -3.00000 q^{53} -2.00000 q^{55} -1.00000 q^{56} +1.00000 q^{58} -7.00000 q^{59} -12.0000 q^{61} -2.00000 q^{62} +1.00000 q^{64} -14.0000 q^{65} +15.0000 q^{67} +3.00000 q^{68} -2.00000 q^{70} -6.00000 q^{71} -9.00000 q^{73} +6.00000 q^{74} +1.00000 q^{76} -1.00000 q^{77} -8.00000 q^{79} +2.00000 q^{80} -2.00000 q^{82} -16.0000 q^{83} +6.00000 q^{85} -4.00000 q^{86} +1.00000 q^{88} +16.0000 q^{89} -7.00000 q^{91} -3.00000 q^{92} +2.00000 q^{95} +8.00000 q^{97} +6.00000 q^{98} +O(q^{100})$

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
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	0	0
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−2.00000	−0.632456
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	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
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	2.00000	0.447214
$21$	0	0
$22$	1.00000	0.213201
$23$	−3.00000	−0.625543	−0.312772	−	0.949828i	$-0.601257\pi$
$23$	−3.00000	−0.625543	−0.312772	+	0.949828i	$0.601257\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	7.00000	1.37281
$27$	0	0
$28$	1.00000	0.188982
$29$	−1.00000	−0.185695	−0.0928477	−	0.995680i	$-0.529597\pi$
$29$	−1.00000	−0.185695	−0.0928477	+	0.995680i	$0.529597\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−3.00000	−0.514496
$35$	2.00000	0.338062
$36$	0	0
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	−2.00000	−0.316228
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	3.00000	0.442326
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	1.00000	0.141421
$51$	0	0
$52$	−7.00000	−0.970725
$53$	−3.00000	−0.412082	−0.206041	−	0.978543i	$-0.566058\pi$
$53$	−3.00000	−0.412082	−0.206041	+	0.978543i	$0.566058\pi$
$54$	0	0
$55$	−2.00000	−0.269680
$56$	−1.00000	−0.133631
$57$	0	0
$58$	1.00000	0.131306
$59$	−7.00000	−0.911322	−0.455661	−	0.890153i	$-0.650597\pi$
$59$	−7.00000	−0.911322	−0.455661	+	0.890153i	$0.650597\pi$
$60$	0	0
$61$	−12.0000	−1.53644	−0.768221	−	0.640184i	$-0.778858\pi$
$61$	−12.0000	−1.53644	−0.768221	+	0.640184i	$0.778858\pi$
$62$	−2.00000	−0.254000
$63$	0	0
$64$	1.00000	0.125000
$65$	−14.0000	−1.73649
$66$	0	0
$67$	15.0000	1.83254	0.916271	−	0.400559i	$-0.131184\pi$
$67$	15.0000	1.83254	0.916271	+	0.400559i	$0.131184\pi$
$68$	3.00000	0.363803
$69$	0	0
$70$	−2.00000	−0.239046
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	0	0
$73$	−9.00000	−1.05337	−0.526685	−	0.850060i	$-0.676565\pi$
$73$	−9.00000	−1.05337	−0.526685	+	0.850060i	$0.676565\pi$
$74$	6.00000	0.697486
$75$	0	0
$76$	1.00000	0.114708
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	2.00000	0.223607
$81$	0	0
$82$	−2.00000	−0.220863
$83$	−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
$85$	6.00000	0.650791
$86$	−4.00000	−0.431331
$87$	0	0
$88$	1.00000	0.106600
$89$	16.0000	1.69600	0.847998	−	0.529999i	$-0.177808\pi$
$89$	16.0000	1.69600	0.847998	+	0.529999i	$0.177808\pi$
$90$	0	0
$91$	−7.00000	−0.733799
$92$	−3.00000	−0.312772
$93$	0	0
$94$	0	0
$95$	2.00000	0.205196
$96$	0	0
$97$	8.00000	0.812277	0.406138	−	0.913812i	$-0.366875\pi$
$97$	8.00000	0.812277	0.406138	+	0.913812i	$0.366875\pi$
$98$	6.00000	0.606092
$99$	0	0
$100$	−1.00000	−0.100000
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	0	0
$103$	−12.0000	−1.18240	−0.591198	−	0.806527i	$-0.701345\pi$
$103$	−12.0000	−1.18240	−0.591198	+	0.806527i	$0.701345\pi$
$104$	7.00000	0.686406
$105$	0	0
$106$	3.00000	0.291386
$107$	−11.0000	−1.06341	−0.531705	−	0.846930i	$-0.678449\pi$
$107$	−11.0000	−1.06341	−0.531705	+	0.846930i	$0.678449\pi$
$108$	0	0
$109$	11.0000	1.05361	0.526804	−	0.849987i	$-0.323390\pi$
$109$	11.0000	1.05361	0.526804	+	0.849987i	$0.323390\pi$
$110$	2.00000	0.190693
$111$	0	0
$112$	1.00000	0.0944911
$113$	−20.0000	−1.88144	−0.940721	−	0.339182i	$-0.889850\pi$
$113$	−20.0000	−1.88144	−0.940721	+	0.339182i	$0.889850\pi$
$114$	0	0
$115$	−6.00000	−0.559503
$116$	−1.00000	−0.0928477
$117$	0	0
$118$	7.00000	0.644402
$119$	3.00000	0.275010
$120$	0	0
$121$	1.00000	0.0909091
$122$	12.0000	1.08643
$123$	0	0
$124$	2.00000	0.179605
$125$	−12.0000	−1.07331
$126$	0	0
$127$	−16.0000	−1.41977	−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000	−1.41977	−0.709885	+	0.704317i	$0.751253\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	14.0000	1.22788
$131$	14.0000	1.22319	0.611593	−	0.791173i	$-0.290529\pi$
$131$	14.0000	1.22319	0.611593	+	0.791173i	$0.290529\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	−15.0000	−1.29580
$135$	0	0
$136$	−3.00000	−0.257248
$137$	17.0000	1.45241	0.726204	−	0.687479i	$-0.241283\pi$
$137$	17.0000	1.45241	0.726204	+	0.687479i	$0.241283\pi$
$138$	0	0
$139$	10.0000	0.848189	0.424094	−	0.905618i	$-0.360592\pi$
$139$	10.0000	0.848189	0.424094	+	0.905618i	$0.360592\pi$
$140$	2.00000	0.169031
$141$	0	0
$142$	6.00000	0.503509
$143$	7.00000	0.585369
$144$	0	0
$145$	−2.00000	−0.166091
$146$	9.00000	0.744845
$147$	0	0
$148$	−6.00000	−0.493197
$149$	−14.0000	−1.14692	−0.573462	−	0.819232i	$-0.694400\pi$
$149$	−14.0000	−1.14692	−0.573462	+	0.819232i	$0.694400\pi$
$150$	0	0
$151$	−12.0000	−0.976546	−0.488273	−	0.872691i	$-0.662373\pi$
$151$	−12.0000	−0.976546	−0.488273	+	0.872691i	$0.662373\pi$
$152$	−1.00000	−0.0811107
$153$	0	0
$154$	1.00000	0.0805823
$155$	4.00000	0.321288
$156$	0	0
$157$	18.0000	1.43656	0.718278	−	0.695756i	$-0.244931\pi$
$157$	18.0000	1.43656	0.718278	+	0.695756i	$0.244931\pi$
$158$	8.00000	0.636446
$159$	0	0
$160$	−2.00000	−0.158114
$161$	−3.00000	−0.236433
$162$	0	0
$163$	14.0000	1.09656	0.548282	−	0.836293i	$-0.315282\pi$
$163$	14.0000	1.09656	0.548282	+	0.836293i	$0.315282\pi$
$164$	2.00000	0.156174
$165$	0	0
$166$	16.0000	1.24184
$167$	−18.0000	−1.39288	−0.696441	−	0.717614i	$-0.745234\pi$
$167$	−18.0000	−1.39288	−0.696441	+	0.717614i	$0.745234\pi$
$168$	0	0
$169$	36.0000	2.76923
$170$	−6.00000	−0.460179
$171$	0	0
$172$	4.00000	0.304997
$173$	2.00000	0.152057	0.0760286	−	0.997106i	$-0.475776\pi$
$173$	2.00000	0.152057	0.0760286	+	0.997106i	$0.475776\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	−16.0000	−1.19925
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	0	0
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	7.00000	0.518875
$183$	0	0
$184$	3.00000	0.221163
$185$	−12.0000	−0.882258
$186$	0	0
$187$	−3.00000	−0.219382
$188$	0	0
$189$	0	0
$190$	−2.00000	−0.145095
$191$	−3.00000	−0.217072	−0.108536	−	0.994092i	$-0.534616\pi$
$191$	−3.00000	−0.217072	−0.108536	+	0.994092i	$0.534616\pi$
$192$	0	0
$193$	4.00000	0.287926	0.143963	−	0.989583i	$-0.454015\pi$
$193$	4.00000	0.287926	0.143963	+	0.989583i	$0.454015\pi$
$194$	−8.00000	−0.574367
$195$	0	0
$196$	−6.00000	−0.428571
$197$	28.0000	1.99492	0.997459	−	0.0712470i	$-0.0226979\pi$
$197$	28.0000	1.99492	0.997459	+	0.0712470i	$0.0226979\pi$
$198$	0	0
$199$	7.00000	0.496217	0.248108	−	0.968732i	$-0.420191\pi$
$199$	7.00000	0.496217	0.248108	+	0.968732i	$0.420191\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	−2.00000	−0.140720
$203$	−1.00000	−0.0701862
$204$	0	0
$205$	4.00000	0.279372
$206$	12.0000	0.836080
$207$	0	0
$208$	−7.00000	−0.485363
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	−23.0000	−1.58339	−0.791693	−	0.610920i	$-0.790800\pi$
$211$	−23.0000	−1.58339	−0.791693	+	0.610920i	$0.790800\pi$
$212$	−3.00000	−0.206041
$213$	0	0
$214$	11.0000	0.751945
$215$	8.00000	0.545595
$216$	0	0
$217$	2.00000	0.135769
$218$	−11.0000	−0.745014
$219$	0	0
$220$	−2.00000	−0.134840
$221$	−21.0000	−1.41261
$222$	0	0
$223$	2.00000	0.133930	0.0669650	−	0.997755i	$-0.478668\pi$
$223$	2.00000	0.133930	0.0669650	+	0.997755i	$0.478668\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	20.0000	1.33038
$227$	−25.0000	−1.65931	−0.829654	−	0.558278i	$-0.811462\pi$
$227$	−25.0000	−1.65931	−0.829654	+	0.558278i	$0.811462\pi$
$228$	0	0
$229$	20.0000	1.32164	0.660819	−	0.750546i	$-0.270209\pi$
$229$	20.0000	1.32164	0.660819	+	0.750546i	$0.270209\pi$
$230$	6.00000	0.395628
$231$	0	0
$232$	1.00000	0.0656532
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	0	0
$235$	0	0
$236$	−7.00000	−0.455661
$237$	0	0
$238$	−3.00000	−0.194461
$239$	−13.0000	−0.840900	−0.420450	−	0.907316i	$-0.638128\pi$
$239$	−13.0000	−0.840900	−0.420450	+	0.907316i	$0.638128\pi$
$240$	0	0
$241$	−20.0000	−1.28831	−0.644157	−	0.764894i	$-0.722792\pi$
$241$	−20.0000	−1.28831	−0.644157	+	0.764894i	$0.722792\pi$
$242$	−1.00000	−0.0642824
$243$	0	0
$244$	−12.0000	−0.768221
$245$	−12.0000	−0.766652
$246$	0	0
$247$	−7.00000	−0.445399
$248$	−2.00000	−0.127000
$249$	0	0
$250$	12.0000	0.758947
$251$	30.0000	1.89358	0.946792	−	0.321847i	$-0.104304\pi$
$251$	30.0000	1.89358	0.946792	+	0.321847i	$0.104304\pi$
$252$	0	0
$253$	3.00000	0.188608
$254$	16.0000	1.00393
$255$	0	0
$256$	1.00000	0.0625000
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	0	0
$259$	−6.00000	−0.372822
$260$	−14.0000	−0.868243
$261$	0	0
$262$	−14.0000	−0.864923
$263$	−12.0000	−0.739952	−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000	−0.739952	−0.369976	+	0.929041i	$0.620634\pi$
$264$	0	0
$265$	−6.00000	−0.368577
$266$	−1.00000	−0.0613139
$267$	0	0
$268$	15.0000	0.916271
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	0	0
$271$	−3.00000	−0.182237	−0.0911185	−	0.995840i	$-0.529044\pi$
$271$	−3.00000	−0.182237	−0.0911185	+	0.995840i	$0.529044\pi$
$272$	3.00000	0.181902
$273$	0	0
$274$	−17.0000	−1.02701
$275$	1.00000	0.0603023
$276$	0	0
$277$	−8.00000	−0.480673	−0.240337	−	0.970690i	$-0.577258\pi$
$277$	−8.00000	−0.480673	−0.240337	+	0.970690i	$0.577258\pi$
$278$	−10.0000	−0.599760
$279$	0	0
$280$	−2.00000	−0.119523
$281$	16.0000	0.954480	0.477240	−	0.878773i	$-0.341637\pi$
$281$	16.0000	0.954480	0.477240	+	0.878773i	$0.341637\pi$
$282$	0	0
$283$	−22.0000	−1.30776	−0.653882	−	0.756596i	$-0.726861\pi$
$283$	−22.0000	−1.30776	−0.653882	+	0.756596i	$0.726861\pi$
$284$	−6.00000	−0.356034
$285$	0	0
$286$	−7.00000	−0.413919
$287$	2.00000	0.118056
$288$	0	0
$289$	−8.00000	−0.470588
$290$	2.00000	0.117444
$291$	0	0
$292$	−9.00000	−0.526685
$293$	−15.0000	−0.876309	−0.438155	−	0.898900i	$-0.644368\pi$
$293$	−15.0000	−0.876309	−0.438155	+	0.898900i	$0.644368\pi$
$294$	0	0
$295$	−14.0000	−0.815112
$296$	6.00000	0.348743
$297$	0	0
$298$	14.0000	0.810998
$299$	21.0000	1.21446
$300$	0	0
$301$	4.00000	0.230556
$302$	12.0000	0.690522
$303$	0	0
$304$	1.00000	0.0573539
$305$	−24.0000	−1.37424
$306$	0	0
$307$	16.0000	0.913168	0.456584	−	0.889680i	$-0.349073\pi$
$307$	16.0000	0.913168	0.456584	+	0.889680i	$0.349073\pi$
$308$	−1.00000	−0.0569803
$309$	0	0
$310$	−4.00000	−0.227185
$311$	−11.0000	−0.623753	−0.311876	−	0.950123i	$-0.600957\pi$
$311$	−11.0000	−0.623753	−0.311876	+	0.950123i	$0.600957\pi$
$312$	0	0
$313$	−27.0000	−1.52613	−0.763065	−	0.646322i	$-0.776306\pi$
$313$	−27.0000	−1.52613	−0.763065	+	0.646322i	$0.776306\pi$
$314$	−18.0000	−1.01580
$315$	0	0
$316$	−8.00000	−0.450035
$317$	−1.00000	−0.0561656	−0.0280828	−	0.999606i	$-0.508940\pi$
$317$	−1.00000	−0.0561656	−0.0280828	+	0.999606i	$0.508940\pi$
$318$	0	0
$319$	1.00000	0.0559893
$320$	2.00000	0.111803
$321$	0	0
$322$	3.00000	0.167183
$323$	3.00000	0.166924
$324$	0	0
$325$	7.00000	0.388290
$326$	−14.0000	−0.775388
$327$	0	0
$328$	−2.00000	−0.110432
$329$	0	0
$330$	0	0
$331$	17.0000	0.934405	0.467202	−	0.884150i	$-0.345262\pi$
$331$	17.0000	0.934405	0.467202	+	0.884150i	$0.345262\pi$
$332$	−16.0000	−0.878114
$333$	0	0
$334$	18.0000	0.984916
$335$	30.0000	1.63908
$336$	0	0
$337$	4.00000	0.217894	0.108947	−	0.994048i	$-0.465252\pi$
$337$	4.00000	0.217894	0.108947	+	0.994048i	$0.465252\pi$
$338$	−36.0000	−1.95814
$339$	0	0
$340$	6.00000	0.325396
$341$	−2.00000	−0.108306
$342$	0	0
$343$	−13.0000	−0.701934
$344$	−4.00000	−0.215666
$345$	0	0
$346$	−2.00000	−0.107521
$347$	−6.00000	−0.322097	−0.161048	−	0.986947i	$-0.551488\pi$
$347$	−6.00000	−0.322097	−0.161048	+	0.986947i	$0.551488\pi$
$348$	0	0
$349$	10.0000	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$350$	1.00000	0.0534522
$351$	0	0
$352$	1.00000	0.0533002
$353$	31.0000	1.64996	0.824982	−	0.565159i	$-0.191185\pi$
$353$	31.0000	1.64996	0.824982	+	0.565159i	$0.191185\pi$
$354$	0	0
$355$	−12.0000	−0.636894
$356$	16.0000	0.847998
$357$	0	0
$358$	−4.00000	−0.211407
$359$	−27.0000	−1.42501	−0.712503	−	0.701669i	$-0.752438\pi$
$359$	−27.0000	−1.42501	−0.712503	+	0.701669i	$0.752438\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	2.00000	0.105118
$363$	0	0
$364$	−7.00000	−0.366900
$365$	−18.0000	−0.942163
$366$	0	0
$367$	16.0000	0.835193	0.417597	−	0.908633i	$-0.362873\pi$
$367$	16.0000	0.835193	0.417597	+	0.908633i	$0.362873\pi$
$368$	−3.00000	−0.156386
$369$	0	0
$370$	12.0000	0.623850
$371$	−3.00000	−0.155752
$372$	0	0
$373$	−13.0000	−0.673114	−0.336557	−	0.941663i	$-0.609263\pi$
$373$	−13.0000	−0.673114	−0.336557	+	0.941663i	$0.609263\pi$
$374$	3.00000	0.155126
$375$	0	0
$376$	0	0
$377$	7.00000	0.360518
$378$	0	0
$379$	−1.00000	−0.0513665	−0.0256833	−	0.999670i	$-0.508176\pi$
$379$	−1.00000	−0.0513665	−0.0256833	+	0.999670i	$0.508176\pi$
$380$	2.00000	0.102598
$381$	0	0
$382$	3.00000	0.153493
$383$	−26.0000	−1.32854	−0.664269	−	0.747494i	$-0.731257\pi$
$383$	−26.0000	−1.32854	−0.664269	+	0.747494i	$0.731257\pi$
$384$	0	0
$385$	−2.00000	−0.101929
$386$	−4.00000	−0.203595
$387$	0	0
$388$	8.00000	0.406138
$389$	14.0000	0.709828	0.354914	−	0.934899i	$-0.384510\pi$
$389$	14.0000	0.709828	0.354914	+	0.934899i	$0.384510\pi$
$390$	0	0
$391$	−9.00000	−0.455150
$392$	6.00000	0.303046
$393$	0	0
$394$	−28.0000	−1.41062
$395$	−16.0000	−0.805047
$396$	0	0
$397$	−30.0000	−1.50566	−0.752828	−	0.658217i	$-0.771311\pi$
$397$	−30.0000	−1.50566	−0.752828	+	0.658217i	$0.771311\pi$
$398$	−7.00000	−0.350878
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	2.00000	0.0998752	0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000	0.0998752	0.0499376	+	0.998752i	$0.484098\pi$
$402$	0	0
$403$	−14.0000	−0.697390
$404$	2.00000	0.0995037
$405$	0	0
$406$	1.00000	0.0496292
$407$	6.00000	0.297409
$408$	0	0
$409$	−14.0000	−0.692255	−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$410$	−4.00000	−0.197546
$411$	0	0
$412$	−12.0000	−0.591198
$413$	−7.00000	−0.344447
$414$	0	0
$415$	−32.0000	−1.57082
$416$	7.00000	0.343203
$417$	0	0
$418$	1.00000	0.0489116
$419$	6.00000	0.293119	0.146560	−	0.989202i	$-0.453180\pi$
$419$	6.00000	0.293119	0.146560	+	0.989202i	$0.453180\pi$
$420$	0	0
$421$	3.00000	0.146211	0.0731055	−	0.997324i	$-0.476709\pi$
$421$	3.00000	0.146211	0.0731055	+	0.997324i	$0.476709\pi$
$422$	23.0000	1.11962
$423$	0	0
$424$	3.00000	0.145693
$425$	−3.00000	−0.145521
$426$	0	0
$427$	−12.0000	−0.580721
$428$	−11.0000	−0.531705
$429$	0	0
$430$	−8.00000	−0.385794
$431$	22.0000	1.05970	0.529851	−	0.848091i	$-0.322248\pi$
$431$	22.0000	1.05970	0.529851	+	0.848091i	$0.322248\pi$
$432$	0	0
$433$	0	0		−	1.00000i	$-0.5\pi$
$433$	0	0			1.00000i	$0.5\pi$
$434$	−2.00000	−0.0960031
$435$	0	0
$436$	11.0000	0.526804
$437$	−3.00000	−0.143509
$438$	0	0
$439$	−22.0000	−1.05000	−0.525001	−	0.851101i	$-0.675935\pi$
$439$	−22.0000	−1.05000	−0.525001	+	0.851101i	$0.675935\pi$
$440$	2.00000	0.0953463
$441$	0	0
$442$	21.0000	0.998868
$443$	14.0000	0.665160	0.332580	−	0.943075i	$-0.392081\pi$
$443$	14.0000	0.665160	0.332580	+	0.943075i	$0.392081\pi$
$444$	0	0
$445$	32.0000	1.51695
$446$	−2.00000	−0.0947027
$447$	0	0
$448$	1.00000	0.0472456
$449$	32.0000	1.51017	0.755087	−	0.655625i	$-0.227595\pi$
$449$	32.0000	1.51017	0.755087	+	0.655625i	$0.227595\pi$
$450$	0	0
$451$	−2.00000	−0.0941763
$452$	−20.0000	−0.940721
$453$	0	0
$454$	25.0000	1.17331
$455$	−14.0000	−0.656330
$456$	0	0
$457$	23.0000	1.07589	0.537947	−	0.842978i	$-0.319200\pi$
$457$	23.0000	1.07589	0.537947	+	0.842978i	$0.319200\pi$
$458$	−20.0000	−0.934539
$459$	0	0
$460$	−6.00000	−0.279751
$461$	40.0000	1.86299	0.931493	−	0.363760i	$-0.118507\pi$
$461$	40.0000	1.86299	0.931493	+	0.363760i	$0.118507\pi$
$462$	0	0
$463$	−32.0000	−1.48717	−0.743583	−	0.668644i	$-0.766875\pi$
$463$	−32.0000	−1.48717	−0.743583	+	0.668644i	$0.766875\pi$
$464$	−1.00000	−0.0464238
$465$	0	0
$466$	6.00000	0.277945
$467$	12.0000	0.555294	0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000	0.555294	0.277647	+	0.960683i	$0.410445\pi$
$468$	0	0
$469$	15.0000	0.692636
$470$	0	0
$471$	0	0
$472$	7.00000	0.322201
$473$	−4.00000	−0.183920
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	3.00000	0.137505
$477$	0	0
$478$	13.0000	0.594606
$479$	−28.0000	−1.27935	−0.639676	−	0.768644i	$-0.720932\pi$
$479$	−28.0000	−1.27935	−0.639676	+	0.768644i	$0.720932\pi$
$480$	0	0
$481$	42.0000	1.91504
$482$	20.0000	0.910975
$483$	0	0
$484$	1.00000	0.0454545
$485$	16.0000	0.726523
$486$	0	0
$487$	−20.0000	−0.906287	−0.453143	−	0.891438i	$-0.649697\pi$
$487$	−20.0000	−0.906287	−0.453143	+	0.891438i	$0.649697\pi$
$488$	12.0000	0.543214
$489$	0	0
$490$	12.0000	0.542105
$491$	6.00000	0.270776	0.135388	−	0.990793i	$-0.456772\pi$
$491$	6.00000	0.270776	0.135388	+	0.990793i	$0.456772\pi$
$492$	0	0
$493$	−3.00000	−0.135113
$494$	7.00000	0.314945
$495$	0	0
$496$	2.00000	0.0898027
$497$	−6.00000	−0.269137
$498$	0	0
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	−12.0000	−0.536656
$501$	0	0
$502$	−30.0000	−1.33897
$503$	23.0000	1.02552	0.512760	−	0.858532i	$-0.328623\pi$
$503$	23.0000	1.02552	0.512760	+	0.858532i	$0.328623\pi$
$504$	0	0
$505$	4.00000	0.177998
$506$	−3.00000	−0.133366
$507$	0	0
$508$	−16.0000	−0.709885
$509$	18.0000	0.797836	0.398918	−	0.916987i	$-0.369386\pi$
$509$	18.0000	0.797836	0.398918	+	0.916987i	$0.369386\pi$
$510$	0	0
$511$	−9.00000	−0.398137
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	18.0000	0.793946
$515$	−24.0000	−1.05757
$516$	0	0
$517$	0	0
$518$	6.00000	0.263625
$519$	0	0
$520$	14.0000	0.613941
$521$	4.00000	0.175243	0.0876216	−	0.996154i	$-0.472073\pi$
$521$	4.00000	0.175243	0.0876216	+	0.996154i	$0.472073\pi$
$522$	0	0
$523$	11.0000	0.480996	0.240498	−	0.970650i	$-0.422689\pi$
$523$	11.0000	0.480996	0.240498	+	0.970650i	$0.422689\pi$
$524$	14.0000	0.611593
$525$	0	0
$526$	12.0000	0.523225
$527$	6.00000	0.261364
$528$	0	0
$529$	−14.0000	−0.608696
$530$	6.00000	0.260623
$531$	0	0
$532$	1.00000	0.0433555
$533$	−14.0000	−0.606407
$534$	0	0
$535$	−22.0000	−0.951143
$536$	−15.0000	−0.647901
$537$	0	0
$538$	18.0000	0.776035
$539$	6.00000	0.258438
$540$	0	0
$541$	8.00000	0.343947	0.171973	−	0.985102i	$-0.444986\pi$
$541$	8.00000	0.343947	0.171973	+	0.985102i	$0.444986\pi$
$542$	3.00000	0.128861
$543$	0	0
$544$	−3.00000	−0.128624
$545$	22.0000	0.942376
$546$	0	0
$547$	8.00000	0.342055	0.171028	−	0.985266i	$-0.445291\pi$
$547$	8.00000	0.342055	0.171028	+	0.985266i	$0.445291\pi$
$548$	17.0000	0.726204
$549$	0	0
$550$	−1.00000	−0.0426401
$551$	−1.00000	−0.0426014
$552$	0	0
$553$	−8.00000	−0.340195
$554$	8.00000	0.339887
$555$	0	0
$556$	10.0000	0.424094
$557$	30.0000	1.27114	0.635570	−	0.772043i	$-0.280765\pi$
$557$	30.0000	1.27114	0.635570	+	0.772043i	$0.280765\pi$
$558$	0	0
$559$	−28.0000	−1.18427
$560$	2.00000	0.0845154
$561$	0	0
$562$	−16.0000	−0.674919
$563$	8.00000	0.337160	0.168580	−	0.985688i	$-0.446082\pi$
$563$	8.00000	0.337160	0.168580	+	0.985688i	$0.446082\pi$
$564$	0	0
$565$	−40.0000	−1.68281
$566$	22.0000	0.924729
$567$	0	0
$568$	6.00000	0.251754
$569$	−4.00000	−0.167689	−0.0838444	−	0.996479i	$-0.526720\pi$
$569$	−4.00000	−0.167689	−0.0838444	+	0.996479i	$0.526720\pi$
$570$	0	0
$571$	2.00000	0.0836974	0.0418487	−	0.999124i	$-0.486675\pi$
$571$	2.00000	0.0836974	0.0418487	+	0.999124i	$0.486675\pi$
$572$	7.00000	0.292685
$573$	0	0
$574$	−2.00000	−0.0834784
$575$	3.00000	0.125109
$576$	0	0
$577$	−17.0000	−0.707719	−0.353860	−	0.935299i	$-0.615131\pi$
$577$	−17.0000	−0.707719	−0.353860	+	0.935299i	$0.615131\pi$
$578$	8.00000	0.332756
$579$	0	0
$580$	−2.00000	−0.0830455
$581$	−16.0000	−0.663792
$582$	0	0
$583$	3.00000	0.124247
$584$	9.00000	0.372423
$585$	0	0
$586$	15.0000	0.619644
$587$	−20.0000	−0.825488	−0.412744	−	0.910847i	$-0.635430\pi$
$587$	−20.0000	−0.825488	−0.412744	+	0.910847i	$0.635430\pi$
$588$	0	0
$589$	2.00000	0.0824086
$590$	14.0000	0.576371
$591$	0	0
$592$	−6.00000	−0.246598
$593$	−26.0000	−1.06769	−0.533846	−	0.845582i	$-0.679254\pi$
$593$	−26.0000	−1.06769	−0.533846	+	0.845582i	$0.679254\pi$
$594$	0	0
$595$	6.00000	0.245976
$596$	−14.0000	−0.573462
$597$	0	0
$598$	−21.0000	−0.858754
$599$	4.00000	0.163436	0.0817178	−	0.996656i	$-0.473959\pi$
$599$	4.00000	0.163436	0.0817178	+	0.996656i	$0.473959\pi$
$600$	0	0
$601$	0	0		−	1.00000i	$-0.5\pi$
$601$	0	0			1.00000i	$0.5\pi$
$602$	−4.00000	−0.163028
$603$	0	0
$604$	−12.0000	−0.488273
$605$	2.00000	0.0813116
$606$	0	0
$607$	−14.0000	−0.568242	−0.284121	−	0.958788i	$-0.591702\pi$
$607$	−14.0000	−0.568242	−0.284121	+	0.958788i	$0.591702\pi$
$608$	−1.00000	−0.0405554
$609$	0	0
$610$	24.0000	0.971732
$611$	0	0
$612$	0	0
$613$	28.0000	1.13091	0.565455	−	0.824779i	$-0.308701\pi$
$613$	28.0000	1.13091	0.565455	+	0.824779i	$0.308701\pi$
$614$	−16.0000	−0.645707
$615$	0	0
$616$	1.00000	0.0402911
$617$	−18.0000	−0.724653	−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000	−0.724653	−0.362326	+	0.932051i	$0.618017\pi$
$618$	0	0
$619$	−20.0000	−0.803868	−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000	−0.803868	−0.401934	+	0.915669i	$0.631662\pi$
$620$	4.00000	0.160644
$621$	0	0
$622$	11.0000	0.441060
$623$	16.0000	0.641026
$624$	0	0
$625$	−19.0000	−0.760000
$626$	27.0000	1.07914
$627$	0	0
$628$	18.0000	0.718278
$629$	−18.0000	−0.717707
$630$	0	0
$631$	8.00000	0.318475	0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000	0.318475	0.159237	+	0.987240i	$0.449096\pi$
$632$	8.00000	0.318223
$633$	0	0
$634$	1.00000	0.0397151
$635$	−32.0000	−1.26988
$636$	0	0
$637$	42.0000	1.66410
$638$	−1.00000	−0.0395904
$639$	0	0
$640$	−2.00000	−0.0790569
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	0	0
$643$	−34.0000	−1.34083	−0.670415	−	0.741987i	$-0.733884\pi$
$643$	−34.0000	−1.34083	−0.670415	+	0.741987i	$0.733884\pi$
$644$	−3.00000	−0.118217
$645$	0	0
$646$	−3.00000	−0.118033
$647$	−3.00000	−0.117942	−0.0589711	−	0.998260i	$-0.518782\pi$
$647$	−3.00000	−0.117942	−0.0589711	+	0.998260i	$0.518782\pi$
$648$	0	0
$649$	7.00000	0.274774
$650$	−7.00000	−0.274563
$651$	0	0
$652$	14.0000	0.548282
$653$	−12.0000	−0.469596	−0.234798	−	0.972044i	$-0.575443\pi$
$653$	−12.0000	−0.469596	−0.234798	+	0.972044i	$0.575443\pi$
$654$	0	0
$655$	28.0000	1.09405
$656$	2.00000	0.0780869
$657$	0	0
$658$	0	0
$659$	−27.0000	−1.05177	−0.525885	−	0.850555i	$-0.676266\pi$
$659$	−27.0000	−1.05177	−0.525885	+	0.850555i	$0.676266\pi$
$660$	0	0
$661$	25.0000	0.972387	0.486194	−	0.873851i	$-0.338385\pi$
$661$	25.0000	0.972387	0.486194	+	0.873851i	$0.338385\pi$
$662$	−17.0000	−0.660724
$663$	0	0
$664$	16.0000	0.620920
$665$	2.00000	0.0775567
$666$	0	0
$667$	3.00000	0.116160
$668$	−18.0000	−0.696441
$669$	0	0
$670$	−30.0000	−1.15900
$671$	12.0000	0.463255
$672$	0	0
$673$	8.00000	0.308377	0.154189	−	0.988041i	$-0.450724\pi$
$673$	8.00000	0.308377	0.154189	+	0.988041i	$0.450724\pi$
$674$	−4.00000	−0.154074
$675$	0	0
$676$	36.0000	1.38462
$677$	−27.0000	−1.03769	−0.518847	−	0.854867i	$-0.673639\pi$
$677$	−27.0000	−1.03769	−0.518847	+	0.854867i	$0.673639\pi$
$678$	0	0
$679$	8.00000	0.307012
$680$	−6.00000	−0.230089
$681$	0	0
$682$	2.00000	0.0765840
$683$	16.0000	0.612223	0.306111	−	0.951996i	$-0.400972\pi$
$683$	16.0000	0.612223	0.306111	+	0.951996i	$0.400972\pi$
$684$	0	0
$685$	34.0000	1.29907
$686$	13.0000	0.496342
$687$	0	0
$688$	4.00000	0.152499
$689$	21.0000	0.800036
$690$	0	0
$691$	10.0000	0.380418	0.190209	−	0.981744i	$-0.439083\pi$
$691$	10.0000	0.380418	0.190209	+	0.981744i	$0.439083\pi$
$692$	2.00000	0.0760286
$693$	0	0
$694$	6.00000	0.227757
$695$	20.0000	0.758643
$696$	0	0
$697$	6.00000	0.227266
$698$	−10.0000	−0.378506
$699$	0	0
$700$	−1.00000	−0.0377964
$701$	24.0000	0.906467	0.453234	−	0.891392i	$-0.350270\pi$
$701$	24.0000	0.906467	0.453234	+	0.891392i	$0.350270\pi$
$702$	0	0
$703$	−6.00000	−0.226294
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	−31.0000	−1.16670
$707$	2.00000	0.0752177
$708$	0	0
$709$	−50.0000	−1.87779	−0.938895	−	0.344204i	$-0.888149\pi$
$709$	−50.0000	−1.87779	−0.938895	+	0.344204i	$0.888149\pi$
$710$	12.0000	0.450352
$711$	0	0
$712$	−16.0000	−0.599625
$713$	−6.00000	−0.224702
$714$	0	0
$715$	14.0000	0.523570
$716$	4.00000	0.149487
$717$	0	0
$718$	27.0000	1.00763
$719$	25.0000	0.932343	0.466171	−	0.884694i	$-0.345633\pi$
$719$	25.0000	0.932343	0.466171	+	0.884694i	$0.345633\pi$
$720$	0	0
$721$	−12.0000	−0.446903
$722$	−1.00000	−0.0372161
$723$	0	0
$724$	−2.00000	−0.0743294
$725$	1.00000	0.0371391
$726$	0	0
$727$	23.0000	0.853023	0.426511	−	0.904482i	$-0.359742\pi$
$727$	23.0000	0.853023	0.426511	+	0.904482i	$0.359742\pi$
$728$	7.00000	0.259437
$729$	0	0
$730$	18.0000	0.666210
$731$	12.0000	0.443836
$732$	0	0
$733$	−6.00000	−0.221615	−0.110808	−	0.993842i	$-0.535344\pi$
$733$	−6.00000	−0.221615	−0.110808	+	0.993842i	$0.535344\pi$
$734$	−16.0000	−0.590571
$735$	0	0
$736$	3.00000	0.110581
$737$	−15.0000	−0.552532
$738$	0	0
$739$	52.0000	1.91285	0.956425	−	0.291977i	$-0.0943129\pi$
$739$	52.0000	1.91285	0.956425	+	0.291977i	$0.0943129\pi$
$740$	−12.0000	−0.441129
$741$	0	0
$742$	3.00000	0.110133
$743$	−8.00000	−0.293492	−0.146746	−	0.989174i	$-0.546880\pi$
$743$	−8.00000	−0.293492	−0.146746	+	0.989174i	$0.546880\pi$
$744$	0	0
$745$	−28.0000	−1.02584
$746$	13.0000	0.475964
$747$	0	0
$748$	−3.00000	−0.109691
$749$	−11.0000	−0.401931
$750$	0	0
$751$	−34.0000	−1.24068	−0.620339	−	0.784334i	$-0.713005\pi$
$751$	−34.0000	−1.24068	−0.620339	+	0.784334i	$0.713005\pi$
$752$	0	0
$753$	0	0
$754$	−7.00000	−0.254925
$755$	−24.0000	−0.873449
$756$	0	0
$757$	−2.00000	−0.0726912	−0.0363456	−	0.999339i	$-0.511572\pi$
$757$	−2.00000	−0.0726912	−0.0363456	+	0.999339i	$0.511572\pi$
$758$	1.00000	0.0363216
$759$	0	0
$760$	−2.00000	−0.0725476
$761$	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
$763$	11.0000	0.398227
$764$	−3.00000	−0.108536
$765$	0	0
$766$	26.0000	0.939418
$767$	49.0000	1.76929
$768$	0	0
$769$	−37.0000	−1.33425	−0.667127	−	0.744944i	$-0.732476\pi$
$769$	−37.0000	−1.33425	−0.667127	+	0.744944i	$0.732476\pi$
$770$	2.00000	0.0720750
$771$	0	0
$772$	4.00000	0.143963
$773$	19.0000	0.683383	0.341691	−	0.939812i	$-0.389000\pi$
$773$	19.0000	0.683383	0.341691	+	0.939812i	$0.389000\pi$
$774$	0	0
$775$	−2.00000	−0.0718421
$776$	−8.00000	−0.287183
$777$	0	0
$778$	−14.0000	−0.501924
$779$	2.00000	0.0716574
$780$	0	0
$781$	6.00000	0.214697
$782$	9.00000	0.321839
$783$	0	0
$784$	−6.00000	−0.214286
$785$	36.0000	1.28490
$786$	0	0
$787$	41.0000	1.46149	0.730746	−	0.682649i	$-0.239172\pi$
$787$	41.0000	1.46149	0.730746	+	0.682649i	$0.239172\pi$
$788$	28.0000	0.997459
$789$	0	0
$790$	16.0000	0.569254
$791$	−20.0000	−0.711118
$792$	0	0
$793$	84.0000	2.98293
$794$	30.0000	1.06466
$795$	0	0
$796$	7.00000	0.248108
$797$	33.0000	1.16892	0.584460	−	0.811423i	$-0.301306\pi$
$797$	33.0000	1.16892	0.584460	+	0.811423i	$0.301306\pi$
$798$	0	0
$799$	0	0
$800$	1.00000	0.0353553
$801$	0	0
$802$	−2.00000	−0.0706225
$803$	9.00000	0.317603
$804$	0	0
$805$	−6.00000	−0.211472
$806$	14.0000	0.493129
$807$	0	0
$808$	−2.00000	−0.0703598
$809$	17.0000	0.597688	0.298844	−	0.954302i	$-0.403399\pi$
$809$	17.0000	0.597688	0.298844	+	0.954302i	$0.403399\pi$
$810$	0	0
$811$	−25.0000	−0.877869	−0.438934	−	0.898519i	$-0.644644\pi$
$811$	−25.0000	−0.877869	−0.438934	+	0.898519i	$0.644644\pi$
$812$	−1.00000	−0.0350931
$813$	0	0
$814$	−6.00000	−0.210300
$815$	28.0000	0.980797
$816$	0	0
$817$	4.00000	0.139942
$818$	14.0000	0.489499
$819$	0	0
$820$	4.00000	0.139686
$821$	8.00000	0.279202	0.139601	−	0.990208i	$-0.455418\pi$
$821$	8.00000	0.279202	0.139601	+	0.990208i	$0.455418\pi$
$822$	0	0
$823$	53.0000	1.84746	0.923732	−	0.383040i	$-0.125123\pi$
$823$	53.0000	1.84746	0.923732	+	0.383040i	$0.125123\pi$
$824$	12.0000	0.418040
$825$	0	0
$826$	7.00000	0.243561
$827$	−13.0000	−0.452054	−0.226027	−	0.974121i	$-0.572574\pi$
$827$	−13.0000	−0.452054	−0.226027	+	0.974121i	$0.572574\pi$
$828$	0	0
$829$	53.0000	1.84077	0.920383	−	0.391018i	$-0.127877\pi$
$829$	53.0000	1.84077	0.920383	+	0.391018i	$0.127877\pi$
$830$	32.0000	1.11074
$831$	0	0
$832$	−7.00000	−0.242681
$833$	−18.0000	−0.623663
$834$	0	0
$835$	−36.0000	−1.24583
$836$	−1.00000	−0.0345857
$837$	0	0
$838$	−6.00000	−0.207267
$839$	−32.0000	−1.10476	−0.552381	−	0.833592i	$-0.686281\pi$
$839$	−32.0000	−1.10476	−0.552381	+	0.833592i	$0.686281\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	−3.00000	−0.103387
$843$	0	0
$844$	−23.0000	−0.791693
$845$	72.0000	2.47688
$846$	0	0
$847$	1.00000	0.0343604
$848$	−3.00000	−0.103020
$849$	0	0
$850$	3.00000	0.102899
$851$	18.0000	0.617032
$852$	0	0
$853$	30.0000	1.02718	0.513590	−	0.858036i	$-0.328315\pi$
$853$	30.0000	1.02718	0.513590	+	0.858036i	$0.328315\pi$
$854$	12.0000	0.410632
$855$	0	0
$856$	11.0000	0.375972
$857$	6.00000	0.204956	0.102478	−	0.994735i	$-0.467323\pi$
$857$	6.00000	0.204956	0.102478	+	0.994735i	$0.467323\pi$
$858$	0	0
$859$	−14.0000	−0.477674	−0.238837	−	0.971060i	$-0.576766\pi$
$859$	−14.0000	−0.477674	−0.238837	+	0.971060i	$0.576766\pi$
$860$	8.00000	0.272798
$861$	0	0
$862$	−22.0000	−0.749323
$863$	38.0000	1.29354	0.646768	−	0.762687i	$-0.276120\pi$
$863$	38.0000	1.29354	0.646768	+	0.762687i	$0.276120\pi$
$864$	0	0
$865$	4.00000	0.136004
$866$	0	0
$867$	0	0
$868$	2.00000	0.0678844
$869$	8.00000	0.271381
$870$	0	0
$871$	−105.000	−3.55779
$872$	−11.0000	−0.372507
$873$	0	0
$874$	3.00000	0.101477
$875$	−12.0000	−0.405674
$876$	0	0
$877$	27.0000	0.911725	0.455863	−	0.890050i	$-0.349331\pi$
$877$	27.0000	0.911725	0.455863	+	0.890050i	$0.349331\pi$
$878$	22.0000	0.742464
$879$	0	0
$880$	−2.00000	−0.0674200
$881$	38.0000	1.28025	0.640126	−	0.768270i	$-0.278882\pi$
$881$	38.0000	1.28025	0.640126	+	0.768270i	$0.278882\pi$
$882$	0	0
$883$	20.0000	0.673054	0.336527	−	0.941674i	$-0.390748\pi$
$883$	20.0000	0.673054	0.336527	+	0.941674i	$0.390748\pi$
$884$	−21.0000	−0.706306
$885$	0	0
$886$	−14.0000	−0.470339
$887$	20.0000	0.671534	0.335767	−	0.941945i	$-0.391004\pi$
$887$	20.0000	0.671534	0.335767	+	0.941945i	$0.391004\pi$
$888$	0	0
$889$	−16.0000	−0.536623
$890$	−32.0000	−1.07264
$891$	0	0
$892$	2.00000	0.0669650
$893$	0	0
$894$	0	0
$895$	8.00000	0.267411
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	−32.0000	−1.06785
$899$	−2.00000	−0.0667037
$900$	0	0
$901$	−9.00000	−0.299833
$902$	2.00000	0.0665927
$903$	0	0
$904$	20.0000	0.665190
$905$	−4.00000	−0.132964
$906$	0	0
$907$	1.00000	0.0332045	0.0166022	−	0.999862i	$-0.494715\pi$
$907$	1.00000	0.0332045	0.0166022	+	0.999862i	$0.494715\pi$
$908$	−25.0000	−0.829654
$909$	0	0
$910$	14.0000	0.464095
$911$	30.0000	0.993944	0.496972	−	0.867766i	$-0.334445\pi$
$911$	30.0000	0.993944	0.496972	+	0.867766i	$0.334445\pi$
$912$	0	0
$913$	16.0000	0.529523
$914$	−23.0000	−0.760772
$915$	0	0
$916$	20.0000	0.660819
$917$	14.0000	0.462321
$918$	0	0
$919$	39.0000	1.28649	0.643246	−	0.765660i	$-0.277587\pi$
$919$	39.0000	1.28649	0.643246	+	0.765660i	$0.277587\pi$
$920$	6.00000	0.197814
$921$	0	0
$922$	−40.0000	−1.31733
$923$	42.0000	1.38245
$924$	0	0
$925$	6.00000	0.197279
$926$	32.0000	1.05159
$927$	0	0
$928$	1.00000	0.0328266
$929$	51.0000	1.67326	0.836628	−	0.547772i	$-0.184524\pi$
$929$	51.0000	1.67326	0.836628	+	0.547772i	$0.184524\pi$
$930$	0	0
$931$	−6.00000	−0.196642
$932$	−6.00000	−0.196537
$933$	0	0
$934$	−12.0000	−0.392652
$935$	−6.00000	−0.196221
$936$	0	0
$937$	−41.0000	−1.33941	−0.669706	−	0.742627i	$-0.733580\pi$
$937$	−41.0000	−1.33941	−0.669706	+	0.742627i	$0.733580\pi$
$938$	−15.0000	−0.489767
$939$	0	0
$940$	0	0
$941$	15.0000	0.488986	0.244493	−	0.969651i	$-0.421378\pi$
$941$	15.0000	0.488986	0.244493	+	0.969651i	$0.421378\pi$
$942$	0	0
$943$	−6.00000	−0.195387
$944$	−7.00000	−0.227831
$945$	0	0
$946$	4.00000	0.130051
$947$	8.00000	0.259965	0.129983	−	0.991516i	$-0.458508\pi$
$947$	8.00000	0.259965	0.129983	+	0.991516i	$0.458508\pi$
$948$	0	0
$949$	63.0000	2.04507
$950$	1.00000	0.0324443
$951$	0	0
$952$	−3.00000	−0.0972306
$953$	−4.00000	−0.129573	−0.0647864	−	0.997899i	$-0.520637\pi$
$953$	−4.00000	−0.129573	−0.0647864	+	0.997899i	$0.520637\pi$
$954$	0	0
$955$	−6.00000	−0.194155
$956$	−13.0000	−0.420450
$957$	0	0
$958$	28.0000	0.904639
$959$	17.0000	0.548959
$960$	0	0
$961$	−27.0000	−0.870968
$962$	−42.0000	−1.35413
$963$	0	0
$964$	−20.0000	−0.644157
$965$	8.00000	0.257529
$966$	0	0
$967$	56.0000	1.80084	0.900419	−	0.435023i	$-0.143260\pi$
$967$	56.0000	1.80084	0.900419	+	0.435023i	$0.143260\pi$
$968$	−1.00000	−0.0321412
$969$	0	0
$970$	−16.0000	−0.513729
$971$	−28.0000	−0.898563	−0.449281	−	0.893390i	$-0.648320\pi$
$971$	−28.0000	−0.898563	−0.449281	+	0.893390i	$0.648320\pi$
$972$	0	0
$973$	10.0000	0.320585
$974$	20.0000	0.640841
$975$	0	0
$976$	−12.0000	−0.384111
$977$	−12.0000	−0.383914	−0.191957	−	0.981403i	$-0.561483\pi$
$977$	−12.0000	−0.383914	−0.191957	+	0.981403i	$0.561483\pi$
$978$	0	0
$979$	−16.0000	−0.511362
$980$	−12.0000	−0.383326
$981$	0	0
$982$	−6.00000	−0.191468
$983$	−50.0000	−1.59475	−0.797376	−	0.603483i	$-0.793779\pi$
$983$	−50.0000	−1.59475	−0.797376	+	0.603483i	$0.793779\pi$
$984$	0	0
$985$	56.0000	1.78431
$986$	3.00000	0.0955395
$987$	0	0
$988$	−7.00000	−0.222700
$989$	−12.0000	−0.381578
$990$	0	0
$991$	−44.0000	−1.39771	−0.698853	−	0.715265i	$-0.746306\pi$
$991$	−44.0000	−1.39771	−0.698853	+	0.715265i	$0.746306\pi$
$992$	−2.00000	−0.0635001
$993$	0	0
$994$	6.00000	0.190308
$995$	14.0000	0.443830
$996$	0	0
$997$	−2.00000	−0.0633406	−0.0316703	−	0.999498i	$-0.510083\pi$
$997$	−2.00000	−0.0633406	−0.0316703	+	0.999498i	$0.510083\pi$
$998$	20.0000	0.633089
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3762.2.a.j.1.1		1
3.2	odd	2		418.2.a.c.1.1	✓	1
12.11	even	2		3344.2.a.a.1.1		1
33.32	even	2		4598.2.a.j.1.1		1
57.56	even	2		7942.2.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
418.2.a.c.1.1	✓	1	3.2	odd	2
3344.2.a.a.1.1		1	12.11	even	2
3762.2.a.j.1.1		1	1.1	even	1	trivial
4598.2.a.j.1.1		1	33.32	even	2
7942.2.a.a.1.1		1	57.56	even	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:	\( N \)	\(=\)	\( 3762 = 2 \cdot 3^{2} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3762.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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