LMFDB - Embedded newform 342.2.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [342,2,Mod(1,342)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(342, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("342.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 342 = 2 \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	342.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:	$2.73088374913$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 38)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	342.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} +1.00000 q^{4} +4.00000 q^{5} +3.00000 q^{7} -1.00000 q^{8} +O(q^{10})$

\(q-1.00000 q^{2} +1.00000 q^{4} +4.00000 q^{5} +3.00000 q^{7} -1.00000 q^{8} -4.00000 q^{10} -2.00000 q^{11} -1.00000 q^{13} -3.00000 q^{14} +1.00000 q^{16} -3.00000 q^{17} -1.00000 q^{19} +4.00000 q^{20} +2.00000 q^{22} +1.00000 q^{23} +11.0000 q^{25} +1.00000 q^{26} +3.00000 q^{28} +5.00000 q^{29} -8.00000 q^{31} -1.00000 q^{32} +3.00000 q^{34} +12.0000 q^{35} -2.00000 q^{37} +1.00000 q^{38} -4.00000 q^{40} +8.00000 q^{41} +4.00000 q^{43} -2.00000 q^{44} -1.00000 q^{46} -8.00000 q^{47} +2.00000 q^{49} -11.0000 q^{50} -1.00000 q^{52} +1.00000 q^{53} -8.00000 q^{55} -3.00000 q^{56} -5.00000 q^{58} -15.0000 q^{59} +2.00000 q^{61} +8.00000 q^{62} +1.00000 q^{64} -4.00000 q^{65} +3.00000 q^{67} -3.00000 q^{68} -12.0000 q^{70} -2.00000 q^{71} +9.00000 q^{73} +2.00000 q^{74} -1.00000 q^{76} -6.00000 q^{77} -10.0000 q^{79} +4.00000 q^{80} -8.00000 q^{82} +6.00000 q^{83} -12.0000 q^{85} -4.00000 q^{86} +2.00000 q^{88} -3.00000 q^{91} +1.00000 q^{92} +8.00000 q^{94} -4.00000 q^{95} -2.00000 q^{97} -2.00000 q^{98} +O(q^{100})\)

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
$5$	4.00000	1.78885	0.894427	−	0.447214i	$-0.147584\pi$
$5$	4.00000	1.78885	0.894427	+	0.447214i	$0.147584\pi$
$6$	0	0
$7$	3.00000	1.13389	0.566947	−	0.823754i	$-0.308125\pi$
$7$	3.00000	1.13389	0.566947	+	0.823754i	$0.308125\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−4.00000	−1.26491
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	−3.00000	−0.801784
$15$	0	0
$16$	1.00000	0.250000
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	4.00000	0.894427
$21$	0	0
$22$	2.00000	0.426401
$23$	1.00000	0.208514	0.104257	−	0.994550i	$-0.466753\pi$
$23$	1.00000	0.208514	0.104257	+	0.994550i	$0.466753\pi$
$24$	0	0
$25$	11.0000	2.20000
$26$	1.00000	0.196116
$27$	0	0
$28$	3.00000	0.566947
$29$	5.00000	0.928477	0.464238	−	0.885710i	$-0.346328\pi$
$29$	5.00000	0.928477	0.464238	+	0.885710i	$0.346328\pi$
$30$	0	0
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	3.00000	0.514496
$35$	12.0000	2.02837
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	−4.00000	−0.632456
$41$	8.00000	1.24939	0.624695	−	0.780869i	$-0.285223\pi$
$41$	8.00000	1.24939	0.624695	+	0.780869i	$0.285223\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	−2.00000	−0.301511
$45$	0	0
$46$	−1.00000	−0.147442
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	−11.0000	−1.55563
$51$	0	0
$52$	−1.00000	−0.138675
$53$	1.00000	0.137361	0.0686803	−	0.997639i	$-0.478121\pi$
$53$	1.00000	0.137361	0.0686803	+	0.997639i	$0.478121\pi$
$54$	0	0
$55$	−8.00000	−1.07872
$56$	−3.00000	−0.400892
$57$	0	0
$58$	−5.00000	−0.656532
$59$	−15.0000	−1.95283	−0.976417	−	0.215894i	$-0.930733\pi$
$59$	−15.0000	−1.95283	−0.976417	+	0.215894i	$0.930733\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	8.00000	1.01600
$63$	0	0
$64$	1.00000	0.125000
$65$	−4.00000	−0.496139
$66$	0	0
$67$	3.00000	0.366508	0.183254	−	0.983066i	$-0.441337\pi$
$67$	3.00000	0.366508	0.183254	+	0.983066i	$0.441337\pi$
$68$	−3.00000	−0.363803
$69$	0	0
$70$	−12.0000	−1.43427
$71$	−2.00000	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	0	0
$73$	9.00000	1.05337	0.526685	−	0.850060i	$-0.323435\pi$
$73$	9.00000	1.05337	0.526685	+	0.850060i	$0.323435\pi$
$74$	2.00000	0.232495
$75$	0	0
$76$	−1.00000	−0.114708
$77$	−6.00000	−0.683763
$78$	0	0
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	4.00000	0.447214
$81$	0	0
$82$	−8.00000	−0.883452
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	0	0
$85$	−12.0000	−1.30158
$86$	−4.00000	−0.431331
$87$	0	0
$88$	2.00000	0.213201
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	−3.00000	−0.314485
$92$	1.00000	0.104257
$93$	0	0
$94$	8.00000	0.825137
$95$	−4.00000	−0.410391
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	−2.00000	−0.202031
$99$	0	0
$100$	11.0000	1.10000
$101$	−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
$103$	−6.00000	−0.591198	−0.295599	−	0.955312i	$-0.595519\pi$
$103$	−6.00000	−0.591198	−0.295599	+	0.955312i	$0.595519\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	−1.00000	−0.0971286
$107$	7.00000	0.676716	0.338358	−	0.941018i	$-0.390129\pi$
$107$	7.00000	0.676716	0.338358	+	0.941018i	$0.390129\pi$
$108$	0	0
$109$	−15.0000	−1.43674	−0.718370	−	0.695662i	$-0.755111\pi$
$109$	−15.0000	−1.43674	−0.718370	+	0.695662i	$0.755111\pi$
$110$	8.00000	0.762770
$111$	0	0
$112$	3.00000	0.283473
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	0	0
$115$	4.00000	0.373002
$116$	5.00000	0.464238
$117$	0	0
$118$	15.0000	1.38086
$119$	−9.00000	−0.825029
$120$	0	0
$121$	−7.00000	−0.636364
$122$	−2.00000	−0.181071
$123$	0	0
$124$	−8.00000	−0.718421
$125$	24.0000	2.14663
$126$	0	0
$127$	18.0000	1.59724	0.798621	−	0.601834i	$-0.205563\pi$
$127$	18.0000	1.59724	0.798621	+	0.601834i	$0.205563\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	4.00000	0.350823
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	0	0
$133$	−3.00000	−0.260133
$134$	−3.00000	−0.259161
$135$	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
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	12.0000	1.01419
$141$	0	0
$142$	2.00000	0.167836
$143$	2.00000	0.167248
$144$	0	0
$145$	20.0000	1.66091
$146$	−9.00000	−0.744845
$147$	0	0
$148$	−2.00000	−0.164399
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	2.00000	0.162758	0.0813788	−	0.996683i	$-0.474068\pi$
$151$	2.00000	0.162758	0.0813788	+	0.996683i	$0.474068\pi$
$152$	1.00000	0.0811107
$153$	0	0
$154$	6.00000	0.483494
$155$	−32.0000	−2.57030
$156$	0	0
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	10.0000	0.795557
$159$	0	0
$160$	−4.00000	−0.316228
$161$	3.00000	0.236433
$162$	0	0
$163$	−16.0000	−1.25322	−0.626608	−	0.779334i	$-0.715557\pi$
$163$	−16.0000	−1.25322	−0.626608	+	0.779334i	$0.715557\pi$
$164$	8.00000	0.624695
$165$	0	0
$166$	−6.00000	−0.465690
$167$	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
$169$	−12.0000	−0.923077
$170$	12.0000	0.920358
$171$	0	0
$172$	4.00000	0.304997
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	0	0
$175$	33.0000	2.49457
$176$	−2.00000	−0.150756
$177$	0	0
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	22.0000	1.63525	0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000	1.63525	0.817624	+	0.575753i	$0.195291\pi$
$182$	3.00000	0.222375
$183$	0	0
$184$	−1.00000	−0.0737210
$185$	−8.00000	−0.588172
$186$	0	0
$187$	6.00000	0.438763
$188$	−8.00000	−0.583460
$189$	0	0
$190$	4.00000	0.290191
$191$	−7.00000	−0.506502	−0.253251	−	0.967401i	$-0.581500\pi$
$191$	−7.00000	−0.506502	−0.253251	+	0.967401i	$0.581500\pi$
$192$	0	0
$193$	−6.00000	−0.431889	−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000	−0.431889	−0.215945	+	0.976406i	$0.569283\pi$
$194$	2.00000	0.143592
$195$	0	0
$196$	2.00000	0.142857
$197$	−8.00000	−0.569976	−0.284988	−	0.958531i	$-0.591990\pi$
$197$	−8.00000	−0.569976	−0.284988	+	0.958531i	$0.591990\pi$
$198$	0	0
$199$	−25.0000	−1.77220	−0.886102	−	0.463491i	$-0.846597\pi$
$199$	−25.0000	−1.77220	−0.886102	+	0.463491i	$0.846597\pi$
$200$	−11.0000	−0.777817
$201$	0	0
$202$	2.00000	0.140720
$203$	15.0000	1.05279
$204$	0	0
$205$	32.0000	2.23498
$206$	6.00000	0.418040
$207$	0	0
$208$	−1.00000	−0.0693375
$209$	2.00000	0.138343
$210$	0	0
$211$	27.0000	1.85876	0.929378	−	0.369129i	$-0.120344\pi$
$211$	27.0000	1.85876	0.929378	+	0.369129i	$0.120344\pi$
$212$	1.00000	0.0686803
$213$	0	0
$214$	−7.00000	−0.478510
$215$	16.0000	1.09119
$216$	0	0
$217$	−24.0000	−1.62923
$218$	15.0000	1.01593
$219$	0	0
$220$	−8.00000	−0.539360
$221$	3.00000	0.201802
$222$	0	0
$223$	14.0000	0.937509	0.468755	−	0.883328i	$-0.344703\pi$
$223$	14.0000	0.937509	0.468755	+	0.883328i	$0.344703\pi$
$224$	−3.00000	−0.200446
$225$	0	0
$226$	14.0000	0.931266
$227$	17.0000	1.12833	0.564165	−	0.825662i	$-0.309198\pi$
$227$	17.0000	1.12833	0.564165	+	0.825662i	$0.309198\pi$
$228$	0	0
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	−4.00000	−0.263752
$231$	0	0
$232$	−5.00000	−0.328266
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	−32.0000	−2.08745
$236$	−15.0000	−0.976417
$237$	0	0
$238$	9.00000	0.583383
$239$	−15.0000	−0.970269	−0.485135	−	0.874439i	$-0.661229\pi$
$239$	−15.0000	−0.970269	−0.485135	+	0.874439i	$0.661229\pi$
$240$	0	0
$241$	−8.00000	−0.515325	−0.257663	−	0.966235i	$-0.582952\pi$
$241$	−8.00000	−0.515325	−0.257663	+	0.966235i	$0.582952\pi$
$242$	7.00000	0.449977
$243$	0	0
$244$	2.00000	0.128037
$245$	8.00000	0.511101
$246$	0	0
$247$	1.00000	0.0636285
$248$	8.00000	0.508001
$249$	0	0
$250$	−24.0000	−1.51789
$251$	−2.00000	−0.126239	−0.0631194	−	0.998006i	$-0.520105\pi$
$251$	−2.00000	−0.126239	−0.0631194	+	0.998006i	$0.520105\pi$
$252$	0	0
$253$	−2.00000	−0.125739
$254$	−18.0000	−1.12942
$255$	0	0
$256$	1.00000	0.0625000
$257$	−8.00000	−0.499026	−0.249513	−	0.968371i	$-0.580271\pi$
$257$	−8.00000	−0.499026	−0.249513	+	0.968371i	$0.580271\pi$
$258$	0	0
$259$	−6.00000	−0.372822
$260$	−4.00000	−0.248069
$261$	0	0
$262$	12.0000	0.741362
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	0	0
$265$	4.00000	0.245718
$266$	3.00000	0.183942
$267$	0	0
$268$	3.00000	0.183254
$269$	−30.0000	−1.82913	−0.914566	−	0.404436i	$-0.867468\pi$
$269$	−30.0000	−1.82913	−0.914566	+	0.404436i	$0.867468\pi$
$270$	0	0
$271$	7.00000	0.425220	0.212610	−	0.977137i	$-0.431804\pi$
$271$	7.00000	0.425220	0.212610	+	0.977137i	$0.431804\pi$
$272$	−3.00000	−0.181902
$273$	0	0
$274$	−17.0000	−1.02701
$275$	−22.0000	−1.32665
$276$	0	0
$277$	28.0000	1.68236	0.841178	−	0.540758i	$-0.181862\pi$
$277$	28.0000	1.68236	0.841178	+	0.540758i	$0.181862\pi$
$278$	0	0
$279$	0	0
$280$	−12.0000	−0.717137
$281$	8.00000	0.477240	0.238620	−	0.971113i	$-0.423305\pi$
$281$	8.00000	0.477240	0.238620	+	0.971113i	$0.423305\pi$
$282$	0	0
$283$	−6.00000	−0.356663	−0.178331	−	0.983970i	$-0.557070\pi$
$283$	−6.00000	−0.356663	−0.178331	+	0.983970i	$0.557070\pi$
$284$	−2.00000	−0.118678
$285$	0	0
$286$	−2.00000	−0.118262
$287$	24.0000	1.41668
$288$	0	0
$289$	−8.00000	−0.470588
$290$	−20.0000	−1.17444
$291$	0	0
$292$	9.00000	0.526685
$293$	−9.00000	−0.525786	−0.262893	−	0.964825i	$-0.584677\pi$
$293$	−9.00000	−0.525786	−0.262893	+	0.964825i	$0.584677\pi$
$294$	0	0
$295$	−60.0000	−3.49334
$296$	2.00000	0.116248
$297$	0	0
$298$	0	0
$299$	−1.00000	−0.0578315
$300$	0	0
$301$	12.0000	0.691669
$302$	−2.00000	−0.115087
$303$	0	0
$304$	−1.00000	−0.0573539
$305$	8.00000	0.458079
$306$	0	0
$307$	−12.0000	−0.684876	−0.342438	−	0.939540i	$-0.611253\pi$
$307$	−12.0000	−0.684876	−0.342438	+	0.939540i	$0.611253\pi$
$308$	−6.00000	−0.341882
$309$	0	0
$310$	32.0000	1.81748
$311$	−7.00000	−0.396934	−0.198467	−	0.980108i	$-0.563596\pi$
$311$	−7.00000	−0.396934	−0.198467	+	0.980108i	$0.563596\pi$
$312$	0	0
$313$	29.0000	1.63918	0.819588	−	0.572953i	$-0.194202\pi$
$313$	29.0000	1.63918	0.819588	+	0.572953i	$0.194202\pi$
$314$	2.00000	0.112867
$315$	0	0
$316$	−10.0000	−0.562544
$317$	27.0000	1.51647	0.758236	−	0.651981i	$-0.226062\pi$
$317$	27.0000	1.51647	0.758236	+	0.651981i	$0.226062\pi$
$318$	0	0
$319$	−10.0000	−0.559893
$320$	4.00000	0.223607
$321$	0	0
$322$	−3.00000	−0.167183
$323$	3.00000	0.166924
$324$	0	0
$325$	−11.0000	−0.610170
$326$	16.0000	0.886158
$327$	0	0
$328$	−8.00000	−0.441726
$329$	−24.0000	−1.32316
$330$	0	0
$331$	17.0000	0.934405	0.467202	−	0.884150i	$-0.345262\pi$
$331$	17.0000	0.934405	0.467202	+	0.884150i	$0.345262\pi$
$332$	6.00000	0.329293
$333$	0	0
$334$	−12.0000	−0.656611
$335$	12.0000	0.655630
$336$	0	0
$337$	−32.0000	−1.74315	−0.871576	−	0.490261i	$-0.836901\pi$
$337$	−32.0000	−1.74315	−0.871576	+	0.490261i	$0.836901\pi$
$338$	12.0000	0.652714
$339$	0	0
$340$	−12.0000	−0.650791
$341$	16.0000	0.866449
$342$	0	0
$343$	−15.0000	−0.809924
$344$	−4.00000	−0.215666
$345$	0	0
$346$	−6.00000	−0.322562
$347$	2.00000	0.107366	0.0536828	−	0.998558i	$-0.482904\pi$
$347$	2.00000	0.107366	0.0536828	+	0.998558i	$0.482904\pi$
$348$	0	0
$349$	10.0000	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$350$	−33.0000	−1.76392
$351$	0	0
$352$	2.00000	0.106600
$353$	−9.00000	−0.479022	−0.239511	−	0.970894i	$-0.576987\pi$
$353$	−9.00000	−0.479022	−0.239511	+	0.970894i	$0.576987\pi$
$354$	0	0
$355$	−8.00000	−0.424596
$356$	0	0
$357$	0	0
$358$	0	0
$359$	15.0000	0.791670	0.395835	−	0.918322i	$-0.370455\pi$
$359$	15.0000	0.791670	0.395835	+	0.918322i	$0.370455\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−22.0000	−1.15629
$363$	0	0
$364$	−3.00000	−0.157243
$365$	36.0000	1.88433
$366$	0	0
$367$	28.0000	1.46159	0.730794	−	0.682598i	$-0.239150\pi$
$367$	28.0000	1.46159	0.730794	+	0.682598i	$0.239150\pi$
$368$	1.00000	0.0521286
$369$	0	0
$370$	8.00000	0.415900
$371$	3.00000	0.155752
$372$	0	0
$373$	29.0000	1.50156	0.750782	−	0.660551i	$-0.229677\pi$
$373$	29.0000	1.50156	0.750782	+	0.660551i	$0.229677\pi$
$374$	−6.00000	−0.310253
$375$	0	0
$376$	8.00000	0.412568
$377$	−5.00000	−0.257513
$378$	0	0
$379$	15.0000	0.770498	0.385249	−	0.922813i	$-0.374116\pi$
$379$	15.0000	0.770498	0.385249	+	0.922813i	$0.374116\pi$
$380$	−4.00000	−0.205196
$381$	0	0
$382$	7.00000	0.358151
$383$	26.0000	1.32854	0.664269	−	0.747494i	$-0.268743\pi$
$383$	26.0000	1.32854	0.664269	+	0.747494i	$0.268743\pi$
$384$	0	0
$385$	−24.0000	−1.22315
$386$	6.00000	0.305392
$387$	0	0
$388$	−2.00000	−0.101535
$389$	30.0000	1.52106	0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000	1.52106	0.760530	+	0.649303i	$0.224939\pi$
$390$	0	0
$391$	−3.00000	−0.151717
$392$	−2.00000	−0.101015
$393$	0	0
$394$	8.00000	0.403034
$395$	−40.0000	−2.01262
$396$	0	0
$397$	8.00000	0.401508	0.200754	−	0.979642i	$-0.435661\pi$
$397$	8.00000	0.401508	0.200754	+	0.979642i	$0.435661\pi$
$398$	25.0000	1.25314
$399$	0	0
$400$	11.0000	0.550000
$401$	8.00000	0.399501	0.199750	−	0.979847i	$-0.435987\pi$
$401$	8.00000	0.399501	0.199750	+	0.979847i	$0.435987\pi$
$402$	0	0
$403$	8.00000	0.398508
$404$	−2.00000	−0.0995037
$405$	0	0
$406$	−15.0000	−0.744438
$407$	4.00000	0.198273
$408$	0	0
$409$	−20.0000	−0.988936	−0.494468	−	0.869196i	$-0.664637\pi$
$409$	−20.0000	−0.988936	−0.494468	+	0.869196i	$0.664637\pi$
$410$	−32.0000	−1.58037
$411$	0	0
$412$	−6.00000	−0.295599
$413$	−45.0000	−2.21431
$414$	0	0
$415$	24.0000	1.17811
$416$	1.00000	0.0490290
$417$	0	0
$418$	−2.00000	−0.0978232
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	−13.0000	−0.633581	−0.316791	−	0.948495i	$-0.602605\pi$
$421$	−13.0000	−0.633581	−0.316791	+	0.948495i	$0.602605\pi$
$422$	−27.0000	−1.31434
$423$	0	0
$424$	−1.00000	−0.0485643
$425$	−33.0000	−1.60074
$426$	0	0
$427$	6.00000	0.290360
$428$	7.00000	0.338358
$429$	0	0
$430$	−16.0000	−0.771589
$431$	18.0000	0.867029	0.433515	−	0.901146i	$-0.357273\pi$
$431$	18.0000	0.867029	0.433515	+	0.901146i	$0.357273\pi$
$432$	0	0
$433$	14.0000	0.672797	0.336399	−	0.941720i	$-0.390791\pi$
$433$	14.0000	0.672797	0.336399	+	0.941720i	$0.390791\pi$
$434$	24.0000	1.15204
$435$	0	0
$436$	−15.0000	−0.718370
$437$	−1.00000	−0.0478365
$438$	0	0
$439$	20.0000	0.954548	0.477274	−	0.878755i	$-0.341625\pi$
$439$	20.0000	0.954548	0.477274	+	0.878755i	$0.341625\pi$
$440$	8.00000	0.381385
$441$	0	0
$442$	−3.00000	−0.142695
$443$	26.0000	1.23530	0.617649	−	0.786454i	$-0.288085\pi$
$443$	26.0000	1.23530	0.617649	+	0.786454i	$0.288085\pi$
$444$	0	0
$445$	0	0
$446$	−14.0000	−0.662919
$447$	0	0
$448$	3.00000	0.141737
$449$	−10.0000	−0.471929	−0.235965	−	0.971762i	$-0.575825\pi$
$449$	−10.0000	−0.471929	−0.235965	+	0.971762i	$0.575825\pi$
$450$	0	0
$451$	−16.0000	−0.753411
$452$	−14.0000	−0.658505
$453$	0	0
$454$	−17.0000	−0.797850
$455$	−12.0000	−0.562569
$456$	0	0
$457$	−7.00000	−0.327446	−0.163723	−	0.986506i	$-0.552350\pi$
$457$	−7.00000	−0.327446	−0.163723	+	0.986506i	$0.552350\pi$
$458$	10.0000	0.467269
$459$	0	0
$460$	4.00000	0.186501
$461$	28.0000	1.30409	0.652045	−	0.758180i	$-0.273911\pi$
$461$	28.0000	1.30409	0.652045	+	0.758180i	$0.273911\pi$
$462$	0	0
$463$	4.00000	0.185896	0.0929479	−	0.995671i	$-0.470371\pi$
$463$	4.00000	0.185896	0.0929479	+	0.995671i	$0.470371\pi$
$464$	5.00000	0.232119
$465$	0	0
$466$	−6.00000	−0.277945
$467$	2.00000	0.0925490	0.0462745	−	0.998929i	$-0.485265\pi$
$467$	2.00000	0.0925490	0.0462745	+	0.998929i	$0.485265\pi$
$468$	0	0
$469$	9.00000	0.415581
$470$	32.0000	1.47605
$471$	0	0
$472$	15.0000	0.690431
$473$	−8.00000	−0.367840
$474$	0	0
$475$	−11.0000	−0.504715
$476$	−9.00000	−0.412514
$477$	0	0
$478$	15.0000	0.686084
$479$	20.0000	0.913823	0.456912	−	0.889512i	$-0.348956\pi$
$479$	20.0000	0.913823	0.456912	+	0.889512i	$0.348956\pi$
$480$	0	0
$481$	2.00000	0.0911922
$482$	8.00000	0.364390
$483$	0	0
$484$	−7.00000	−0.318182
$485$	−8.00000	−0.363261
$486$	0	0
$487$	−2.00000	−0.0906287	−0.0453143	−	0.998973i	$-0.514429\pi$
$487$	−2.00000	−0.0906287	−0.0453143	+	0.998973i	$0.514429\pi$
$488$	−2.00000	−0.0905357
$489$	0	0
$490$	−8.00000	−0.361403
$491$	28.0000	1.26362	0.631811	−	0.775122i	$-0.282312\pi$
$491$	28.0000	1.26362	0.631811	+	0.775122i	$0.282312\pi$
$492$	0	0
$493$	−15.0000	−0.675566
$494$	−1.00000	−0.0449921
$495$	0	0
$496$	−8.00000	−0.359211
$497$	−6.00000	−0.269137
$498$	0	0
$499$	40.0000	1.79065	0.895323	−	0.445418i	$-0.146945\pi$
$499$	40.0000	1.79065	0.895323	+	0.445418i	$0.146945\pi$
$500$	24.0000	1.07331
$501$	0	0
$502$	2.00000	0.0892644
$503$	−39.0000	−1.73892	−0.869462	−	0.494000i	$-0.835534\pi$
$503$	−39.0000	−1.73892	−0.869462	+	0.494000i	$0.835534\pi$
$504$	0	0
$505$	−8.00000	−0.355995
$506$	2.00000	0.0889108
$507$	0	0
$508$	18.0000	0.798621
$509$	30.0000	1.32973	0.664863	−	0.746965i	$-0.268490\pi$
$509$	30.0000	1.32973	0.664863	+	0.746965i	$0.268490\pi$
$510$	0	0
$511$	27.0000	1.19441
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	8.00000	0.352865
$515$	−24.0000	−1.05757
$516$	0	0
$517$	16.0000	0.703679
$518$	6.00000	0.263625
$519$	0	0
$520$	4.00000	0.175412
$521$	28.0000	1.22670	0.613351	−	0.789810i	$-0.289821\pi$
$521$	28.0000	1.22670	0.613351	+	0.789810i	$0.289821\pi$
$522$	0	0
$523$	29.0000	1.26808	0.634041	−	0.773300i	$-0.281395\pi$
$523$	29.0000	1.26808	0.634041	+	0.773300i	$0.281395\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	24.0000	1.04645
$527$	24.0000	1.04546
$528$	0	0
$529$	−22.0000	−0.956522
$530$	−4.00000	−0.173749
$531$	0	0
$532$	−3.00000	−0.130066
$533$	−8.00000	−0.346518
$534$	0	0
$535$	28.0000	1.21055
$536$	−3.00000	−0.129580
$537$	0	0
$538$	30.0000	1.29339
$539$	−4.00000	−0.172292
$540$	0	0
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	−7.00000	−0.300676
$543$	0	0
$544$	3.00000	0.128624
$545$	−60.0000	−2.57012
$546$	0	0
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	17.0000	0.726204
$549$	0	0
$550$	22.0000	0.938083
$551$	−5.00000	−0.213007
$552$	0	0
$553$	−30.0000	−1.27573
$554$	−28.0000	−1.18961
$555$	0	0
$556$	0	0
$557$	−28.0000	−1.18640	−0.593199	−	0.805056i	$-0.702135\pi$
$557$	−28.0000	−1.18640	−0.593199	+	0.805056i	$0.702135\pi$
$558$	0	0
$559$	−4.00000	−0.169182
$560$	12.0000	0.507093
$561$	0	0
$562$	−8.00000	−0.337460
$563$	36.0000	1.51722	0.758610	−	0.651546i	$-0.225879\pi$
$563$	36.0000	1.51722	0.758610	+	0.651546i	$0.225879\pi$
$564$	0	0
$565$	−56.0000	−2.35594
$566$	6.00000	0.252199
$567$	0	0
$568$	2.00000	0.0839181
$569$	−40.0000	−1.67689	−0.838444	−	0.544988i	$-0.816534\pi$
$569$	−40.0000	−1.67689	−0.838444	+	0.544988i	$0.816534\pi$
$570$	0	0
$571$	−28.0000	−1.17176	−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000	−1.17176	−0.585882	+	0.810397i	$0.699252\pi$
$572$	2.00000	0.0836242
$573$	0	0
$574$	−24.0000	−1.00174
$575$	11.0000	0.458732
$576$	0	0
$577$	−37.0000	−1.54033	−0.770165	−	0.637845i	$-0.779826\pi$
$577$	−37.0000	−1.54033	−0.770165	+	0.637845i	$0.779826\pi$
$578$	8.00000	0.332756
$579$	0	0
$580$	20.0000	0.830455
$581$	18.0000	0.746766
$582$	0	0
$583$	−2.00000	−0.0828315
$584$	−9.00000	−0.372423
$585$	0	0
$586$	9.00000	0.371787
$587$	12.0000	0.495293	0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000	0.495293	0.247647	+	0.968850i	$0.420343\pi$
$588$	0	0
$589$	8.00000	0.329634
$590$	60.0000	2.47016
$591$	0	0
$592$	−2.00000	−0.0821995
$593$	−34.0000	−1.39621	−0.698106	−	0.715994i	$-0.745974\pi$
$593$	−34.0000	−1.39621	−0.698106	+	0.715994i	$0.745974\pi$
$594$	0	0
$595$	−36.0000	−1.47586
$596$	0	0
$597$	0	0
$598$	1.00000	0.0408930
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	−8.00000	−0.326327	−0.163163	−	0.986599i	$-0.552170\pi$
$601$	−8.00000	−0.326327	−0.163163	+	0.986599i	$0.552170\pi$
$602$	−12.0000	−0.489083
$603$	0	0
$604$	2.00000	0.0813788
$605$	−28.0000	−1.13836
$606$	0	0
$607$	−22.0000	−0.892952	−0.446476	−	0.894795i	$-0.647321\pi$
$607$	−22.0000	−0.892952	−0.446476	+	0.894795i	$0.647321\pi$
$608$	1.00000	0.0405554
$609$	0	0
$610$	−8.00000	−0.323911
$611$	8.00000	0.323645
$612$	0	0
$613$	34.0000	1.37325	0.686624	−	0.727013i	$-0.259092\pi$
$613$	34.0000	1.37325	0.686624	+	0.727013i	$0.259092\pi$
$614$	12.0000	0.484281
$615$	0	0
$616$	6.00000	0.241747
$617$	−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
$619$	10.0000	0.401934	0.200967	−	0.979598i	$-0.435592\pi$
$619$	10.0000	0.401934	0.200967	+	0.979598i	$0.435592\pi$
$620$	−32.0000	−1.28515
$621$	0	0
$622$	7.00000	0.280674
$623$	0	0
$624$	0	0
$625$	41.0000	1.64000
$626$	−29.0000	−1.15907
$627$	0	0
$628$	−2.00000	−0.0798087
$629$	6.00000	0.239236
$630$	0	0
$631$	32.0000	1.27390	0.636950	−	0.770905i	$-0.280196\pi$
$631$	32.0000	1.27390	0.636950	+	0.770905i	$0.280196\pi$
$632$	10.0000	0.397779
$633$	0	0
$634$	−27.0000	−1.07231
$635$	72.0000	2.85723
$636$	0	0
$637$	−2.00000	−0.0792429
$638$	10.0000	0.395904
$639$	0	0
$640$	−4.00000	−0.158114
$641$	−42.0000	−1.65890	−0.829450	−	0.558581i	$-0.811346\pi$
$641$	−42.0000	−1.65890	−0.829450	+	0.558581i	$0.811346\pi$
$642$	0	0
$643$	−26.0000	−1.02534	−0.512670	−	0.858586i	$-0.671344\pi$
$643$	−26.0000	−1.02534	−0.512670	+	0.858586i	$0.671344\pi$
$644$	3.00000	0.118217
$645$	0	0
$646$	−3.00000	−0.118033
$647$	−23.0000	−0.904223	−0.452112	−	0.891961i	$-0.649329\pi$
$647$	−23.0000	−0.904223	−0.452112	+	0.891961i	$0.649329\pi$
$648$	0	0
$649$	30.0000	1.17760
$650$	11.0000	0.431455
$651$	0	0
$652$	−16.0000	−0.626608
$653$	36.0000	1.40879	0.704394	−	0.709809i	$-0.251219\pi$
$653$	36.0000	1.40879	0.704394	+	0.709809i	$0.251219\pi$
$654$	0	0
$655$	−48.0000	−1.87552
$656$	8.00000	0.312348
$657$	0	0
$658$	24.0000	0.935617
$659$	−5.00000	−0.194772	−0.0973862	−	0.995247i	$-0.531048\pi$
$659$	−5.00000	−0.194772	−0.0973862	+	0.995247i	$0.531048\pi$
$660$	0	0
$661$	−23.0000	−0.894596	−0.447298	−	0.894385i	$-0.647614\pi$
$661$	−23.0000	−0.894596	−0.447298	+	0.894385i	$0.647614\pi$
$662$	−17.0000	−0.660724
$663$	0	0
$664$	−6.00000	−0.232845
$665$	−12.0000	−0.465340
$666$	0	0
$667$	5.00000	0.193601
$668$	12.0000	0.464294
$669$	0	0
$670$	−12.0000	−0.463600
$671$	−4.00000	−0.154418
$672$	0	0
$673$	44.0000	1.69608	0.848038	−	0.529936i	$-0.177784\pi$
$673$	44.0000	1.69608	0.848038	+	0.529936i	$0.177784\pi$
$674$	32.0000	1.23259
$675$	0	0
$676$	−12.0000	−0.461538
$677$	−13.0000	−0.499631	−0.249815	−	0.968294i	$-0.580370\pi$
$677$	−13.0000	−0.499631	−0.249815	+	0.968294i	$0.580370\pi$
$678$	0	0
$679$	−6.00000	−0.230259
$680$	12.0000	0.460179
$681$	0	0
$682$	−16.0000	−0.612672
$683$	−4.00000	−0.153056	−0.0765279	−	0.997067i	$-0.524383\pi$
$683$	−4.00000	−0.153056	−0.0765279	+	0.997067i	$0.524383\pi$
$684$	0	0
$685$	68.0000	2.59815
$686$	15.0000	0.572703
$687$	0	0
$688$	4.00000	0.152499
$689$	−1.00000	−0.0380970
$690$	0	0
$691$	42.0000	1.59776	0.798878	−	0.601494i	$-0.205427\pi$
$691$	42.0000	1.59776	0.798878	+	0.601494i	$0.205427\pi$
$692$	6.00000	0.228086
$693$	0	0
$694$	−2.00000	−0.0759190
$695$	0	0
$696$	0	0
$697$	−24.0000	−0.909065
$698$	−10.0000	−0.378506
$699$	0	0
$700$	33.0000	1.24728
$701$	28.0000	1.05755	0.528773	−	0.848763i	$-0.322652\pi$
$701$	28.0000	1.05755	0.528773	+	0.848763i	$0.322652\pi$
$702$	0	0
$703$	2.00000	0.0754314
$704$	−2.00000	−0.0753778
$705$	0	0
$706$	9.00000	0.338719
$707$	−6.00000	−0.225653
$708$	0	0
$709$	−30.0000	−1.12667	−0.563337	−	0.826227i	$-0.690483\pi$
$709$	−30.0000	−1.12667	−0.563337	+	0.826227i	$0.690483\pi$
$710$	8.00000	0.300235
$711$	0	0
$712$	0	0
$713$	−8.00000	−0.299602
$714$	0	0
$715$	8.00000	0.299183
$716$	0	0
$717$	0	0
$718$	−15.0000	−0.559795
$719$	5.00000	0.186469	0.0932343	−	0.995644i	$-0.470279\pi$
$719$	5.00000	0.186469	0.0932343	+	0.995644i	$0.470279\pi$
$720$	0	0
$721$	−18.0000	−0.670355
$722$	−1.00000	−0.0372161
$723$	0	0
$724$	22.0000	0.817624
$725$	55.0000	2.04265
$726$	0	0
$727$	−17.0000	−0.630495	−0.315248	−	0.949009i	$-0.602088\pi$
$727$	−17.0000	−0.630495	−0.315248	+	0.949009i	$0.602088\pi$
$728$	3.00000	0.111187
$729$	0	0
$730$	−36.0000	−1.33242
$731$	−12.0000	−0.443836
$732$	0	0
$733$	−36.0000	−1.32969	−0.664845	−	0.746981i	$-0.731502\pi$
$733$	−36.0000	−1.32969	−0.664845	+	0.746981i	$0.731502\pi$
$734$	−28.0000	−1.03350
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	−6.00000	−0.221013
$738$	0	0
$739$	−40.0000	−1.47142	−0.735712	−	0.677295i	$-0.763152\pi$
$739$	−40.0000	−1.47142	−0.735712	+	0.677295i	$0.763152\pi$
$740$	−8.00000	−0.294086
$741$	0	0
$742$	−3.00000	−0.110133
$743$	16.0000	0.586983	0.293492	−	0.955962i	$-0.405183\pi$
$743$	16.0000	0.586983	0.293492	+	0.955962i	$0.405183\pi$
$744$	0	0
$745$	0	0
$746$	−29.0000	−1.06177
$747$	0	0
$748$	6.00000	0.219382
$749$	21.0000	0.767323
$750$	0	0
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	−8.00000	−0.291730
$753$	0	0
$754$	5.00000	0.182089
$755$	8.00000	0.291150
$756$	0	0
$757$	−2.00000	−0.0726912	−0.0363456	−	0.999339i	$-0.511572\pi$
$757$	−2.00000	−0.0726912	−0.0363456	+	0.999339i	$0.511572\pi$
$758$	−15.0000	−0.544825
$759$	0	0
$760$	4.00000	0.145095
$761$	−27.0000	−0.978749	−0.489375	−	0.872074i	$-0.662775\pi$
$761$	−27.0000	−0.978749	−0.489375	+	0.872074i	$0.662775\pi$
$762$	0	0
$763$	−45.0000	−1.62911
$764$	−7.00000	−0.253251
$765$	0	0
$766$	−26.0000	−0.939418
$767$	15.0000	0.541619
$768$	0	0
$769$	−35.0000	−1.26213	−0.631066	−	0.775729i	$-0.717382\pi$
$769$	−35.0000	−1.26213	−0.631066	+	0.775729i	$0.717382\pi$
$770$	24.0000	0.864900
$771$	0	0
$772$	−6.00000	−0.215945
$773$	−9.00000	−0.323708	−0.161854	−	0.986815i	$-0.551747\pi$
$773$	−9.00000	−0.323708	−0.161854	+	0.986815i	$0.551747\pi$
$774$	0	0
$775$	−88.0000	−3.16105
$776$	2.00000	0.0717958
$777$	0	0
$778$	−30.0000	−1.07555
$779$	−8.00000	−0.286630
$780$	0	0
$781$	4.00000	0.143131
$782$	3.00000	0.107280
$783$	0	0
$784$	2.00000	0.0714286
$785$	−8.00000	−0.285532
$786$	0	0
$787$	−17.0000	−0.605985	−0.302992	−	0.952993i	$-0.597986\pi$
$787$	−17.0000	−0.605985	−0.302992	+	0.952993i	$0.597986\pi$
$788$	−8.00000	−0.284988
$789$	0	0
$790$	40.0000	1.42314
$791$	−42.0000	−1.49335
$792$	0	0
$793$	−2.00000	−0.0710221
$794$	−8.00000	−0.283909
$795$	0	0
$796$	−25.0000	−0.886102
$797$	−3.00000	−0.106265	−0.0531327	−	0.998587i	$-0.516921\pi$
$797$	−3.00000	−0.106265	−0.0531327	+	0.998587i	$0.516921\pi$
$798$	0	0
$799$	24.0000	0.849059
$800$	−11.0000	−0.388909
$801$	0	0
$802$	−8.00000	−0.282490
$803$	−18.0000	−0.635206
$804$	0	0
$805$	12.0000	0.422944
$806$	−8.00000	−0.281788
$807$	0	0
$808$	2.00000	0.0703598
$809$	15.0000	0.527372	0.263686	−	0.964609i	$-0.415062\pi$
$809$	15.0000	0.527372	0.263686	+	0.964609i	$0.415062\pi$
$810$	0	0
$811$	−3.00000	−0.105344	−0.0526721	−	0.998612i	$-0.516774\pi$
$811$	−3.00000	−0.105344	−0.0526721	+	0.998612i	$0.516774\pi$
$812$	15.0000	0.526397
$813$	0	0
$814$	−4.00000	−0.140200
$815$	−64.0000	−2.24182
$816$	0	0
$817$	−4.00000	−0.139942
$818$	20.0000	0.699284
$819$	0	0
$820$	32.0000	1.11749
$821$	−12.0000	−0.418803	−0.209401	−	0.977830i	$-0.567152\pi$
$821$	−12.0000	−0.418803	−0.209401	+	0.977830i	$0.567152\pi$
$822$	0	0
$823$	29.0000	1.01088	0.505438	−	0.862863i	$-0.331331\pi$
$823$	29.0000	1.01088	0.505438	+	0.862863i	$0.331331\pi$
$824$	6.00000	0.209020
$825$	0	0
$826$	45.0000	1.56575
$827$	−23.0000	−0.799788	−0.399894	−	0.916561i	$-0.630953\pi$
$827$	−23.0000	−0.799788	−0.399894	+	0.916561i	$0.630953\pi$
$828$	0	0
$829$	−15.0000	−0.520972	−0.260486	−	0.965478i	$-0.583883\pi$
$829$	−15.0000	−0.520972	−0.260486	+	0.965478i	$0.583883\pi$
$830$	−24.0000	−0.833052
$831$	0	0
$832$	−1.00000	−0.0346688
$833$	−6.00000	−0.207888
$834$	0	0
$835$	48.0000	1.66111
$836$	2.00000	0.0691714
$837$	0	0
$838$	0	0
$839$	−20.0000	−0.690477	−0.345238	−	0.938515i	$-0.612202\pi$
$839$	−20.0000	−0.690477	−0.345238	+	0.938515i	$0.612202\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	13.0000	0.448010
$843$	0	0
$844$	27.0000	0.929378
$845$	−48.0000	−1.65125
$846$	0	0
$847$	−21.0000	−0.721569
$848$	1.00000	0.0343401
$849$	0	0
$850$	33.0000	1.13189
$851$	−2.00000	−0.0685591
$852$	0	0
$853$	−6.00000	−0.205436	−0.102718	−	0.994711i	$-0.532754\pi$
$853$	−6.00000	−0.205436	−0.102718	+	0.994711i	$0.532754\pi$
$854$	−6.00000	−0.205316
$855$	0	0
$856$	−7.00000	−0.239255
$857$	12.0000	0.409912	0.204956	−	0.978771i	$-0.434295\pi$
$857$	12.0000	0.409912	0.204956	+	0.978771i	$0.434295\pi$
$858$	0	0
$859$	−50.0000	−1.70598	−0.852989	−	0.521929i	$-0.825213\pi$
$859$	−50.0000	−1.70598	−0.852989	+	0.521929i	$0.825213\pi$
$860$	16.0000	0.545595
$861$	0	0
$862$	−18.0000	−0.613082
$863$	−54.0000	−1.83818	−0.919091	−	0.394046i	$-0.871075\pi$
$863$	−54.0000	−1.83818	−0.919091	+	0.394046i	$0.871075\pi$
$864$	0	0
$865$	24.0000	0.816024
$866$	−14.0000	−0.475739
$867$	0	0
$868$	−24.0000	−0.814613
$869$	20.0000	0.678454
$870$	0	0
$871$	−3.00000	−0.101651
$872$	15.0000	0.507964
$873$	0	0
$874$	1.00000	0.0338255
$875$	72.0000	2.43404
$876$	0	0
$877$	13.0000	0.438979	0.219489	−	0.975615i	$-0.429561\pi$
$877$	13.0000	0.438979	0.219489	+	0.975615i	$0.429561\pi$
$878$	−20.0000	−0.674967
$879$	0	0
$880$	−8.00000	−0.269680
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	0	0
$883$	34.0000	1.14419	0.572096	−	0.820187i	$-0.306131\pi$
$883$	34.0000	1.14419	0.572096	+	0.820187i	$0.306131\pi$
$884$	3.00000	0.100901
$885$	0	0
$886$	−26.0000	−0.873487
$887$	2.00000	0.0671534	0.0335767	−	0.999436i	$-0.489310\pi$
$887$	2.00000	0.0671534	0.0335767	+	0.999436i	$0.489310\pi$
$888$	0	0
$889$	54.0000	1.81110
$890$	0	0
$891$	0	0
$892$	14.0000	0.468755
$893$	8.00000	0.267710
$894$	0	0
$895$	0	0
$896$	−3.00000	−0.100223
$897$	0	0
$898$	10.0000	0.333704
$899$	−40.0000	−1.33407
$900$	0	0
$901$	−3.00000	−0.0999445
$902$	16.0000	0.532742
$903$	0	0
$904$	14.0000	0.465633
$905$	88.0000	2.92522
$906$	0	0
$907$	53.0000	1.75984	0.879918	−	0.475125i	$-0.157597\pi$
$907$	53.0000	1.75984	0.879918	+	0.475125i	$0.157597\pi$
$908$	17.0000	0.564165
$909$	0	0
$910$	12.0000	0.397796
$911$	−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
$913$	−12.0000	−0.397142
$914$	7.00000	0.231539
$915$	0	0
$916$	−10.0000	−0.330409
$917$	−36.0000	−1.18882
$918$	0	0
$919$	5.00000	0.164935	0.0824674	−	0.996594i	$-0.473720\pi$
$919$	5.00000	0.164935	0.0824674	+	0.996594i	$0.473720\pi$
$920$	−4.00000	−0.131876
$921$	0	0
$922$	−28.0000	−0.922131
$923$	2.00000	0.0658308
$924$	0	0
$925$	−22.0000	−0.723356
$926$	−4.00000	−0.131448
$927$	0	0
$928$	−5.00000	−0.164133
$929$	55.0000	1.80449	0.902246	−	0.431222i	$-0.141918\pi$
$929$	55.0000	1.80449	0.902246	+	0.431222i	$0.141918\pi$
$930$	0	0
$931$	−2.00000	−0.0655474
$932$	6.00000	0.196537
$933$	0	0
$934$	−2.00000	−0.0654420
$935$	24.0000	0.784884
$936$	0	0
$937$	−7.00000	−0.228680	−0.114340	−	0.993442i	$-0.536475\pi$
$937$	−7.00000	−0.228680	−0.114340	+	0.993442i	$0.536475\pi$
$938$	−9.00000	−0.293860
$939$	0	0
$940$	−32.0000	−1.04372
$941$	−7.00000	−0.228193	−0.114097	−	0.993470i	$-0.536397\pi$
$941$	−7.00000	−0.228193	−0.114097	+	0.993470i	$0.536397\pi$
$942$	0	0
$943$	8.00000	0.260516
$944$	−15.0000	−0.488208
$945$	0	0
$946$	8.00000	0.260102
$947$	12.0000	0.389948	0.194974	−	0.980808i	$-0.437538\pi$
$947$	12.0000	0.389948	0.194974	+	0.980808i	$0.437538\pi$
$948$	0	0
$949$	−9.00000	−0.292152
$950$	11.0000	0.356887
$951$	0	0
$952$	9.00000	0.291692
$953$	46.0000	1.49009	0.745043	−	0.667016i	$-0.232429\pi$
$953$	46.0000	1.49009	0.745043	+	0.667016i	$0.232429\pi$
$954$	0	0
$955$	−28.0000	−0.906059
$956$	−15.0000	−0.485135
$957$	0	0
$958$	−20.0000	−0.646171
$959$	51.0000	1.64688
$960$	0	0
$961$	33.0000	1.06452
$962$	−2.00000	−0.0644826
$963$	0	0
$964$	−8.00000	−0.257663
$965$	−24.0000	−0.772587
$966$	0	0
$967$	48.0000	1.54358	0.771788	−	0.635880i	$-0.219363\pi$
$967$	48.0000	1.54358	0.771788	+	0.635880i	$0.219363\pi$
$968$	7.00000	0.224989
$969$	0	0
$970$	8.00000	0.256865
$971$	28.0000	0.898563	0.449281	−	0.893390i	$-0.351680\pi$
$971$	28.0000	0.898563	0.449281	+	0.893390i	$0.351680\pi$
$972$	0	0
$973$	0	0
$974$	2.00000	0.0640841
$975$	0	0
$976$	2.00000	0.0640184
$977$	−8.00000	−0.255943	−0.127971	−	0.991778i	$-0.540847\pi$
$977$	−8.00000	−0.255943	−0.127971	+	0.991778i	$0.540847\pi$
$978$	0	0
$979$	0	0
$980$	8.00000	0.255551
$981$	0	0
$982$	−28.0000	−0.893516
$983$	6.00000	0.191370	0.0956851	−	0.995412i	$-0.469496\pi$
$983$	6.00000	0.191370	0.0956851	+	0.995412i	$0.469496\pi$
$984$	0	0
$985$	−32.0000	−1.01960
$986$	15.0000	0.477697
$987$	0	0
$988$	1.00000	0.0318142
$989$	4.00000	0.127193
$990$	0	0
$991$	−8.00000	−0.254128	−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000	−0.254128	−0.127064	+	0.991894i	$0.540555\pi$
$992$	8.00000	0.254000
$993$	0	0
$994$	6.00000	0.190308
$995$	−100.000	−3.17021
$996$	0	0
$997$	28.0000	0.886769	0.443384	−	0.896332i	$-0.353778\pi$
$997$	28.0000	0.886769	0.443384	+	0.896332i	$0.353778\pi$
$998$	−40.0000	−1.26618
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	342.2.a.d.1.1		1
3.2	odd	2		38.2.a.b.1.1	✓	1
4.3	odd	2		2736.2.a.w.1.1		1
5.4	even	2		8550.2.a.u.1.1		1
12.11	even	2		304.2.a.d.1.1		1
15.2	even	4		950.2.b.c.799.2		2
15.8	even	4		950.2.b.c.799.1		2
15.14	odd	2		950.2.a.b.1.1		1
19.18	odd	2		6498.2.a.y.1.1		1
21.20	even	2		1862.2.a.f.1.1		1
24.5	odd	2		1216.2.a.n.1.1		1
24.11	even	2		1216.2.a.g.1.1		1
33.32	even	2		4598.2.a.a.1.1		1
39.38	odd	2		6422.2.a.b.1.1		1
57.2	even	18		722.2.e.d.99.1		6
57.5	odd	18		722.2.e.c.595.1		6
57.8	even	6		722.2.c.f.653.1		2
57.11	odd	6		722.2.c.d.653.1		2
57.14	even	18		722.2.e.d.595.1		6
57.17	odd	18		722.2.e.c.99.1		6
57.23	odd	18		722.2.e.c.415.1		6
57.26	odd	6		722.2.c.d.429.1		2
57.29	even	18		722.2.e.d.423.1		6
57.32	even	18		722.2.e.d.245.1		6
57.35	odd	18		722.2.e.c.389.1		6
57.41	even	18		722.2.e.d.389.1		6
57.44	odd	18		722.2.e.c.245.1		6
57.47	odd	18		722.2.e.c.423.1		6
57.50	even	6		722.2.c.f.429.1		2
57.53	even	18		722.2.e.d.415.1		6
57.56	even	2		722.2.a.b.1.1		1
60.59	even	2		7600.2.a.h.1.1		1
228.227	odd	2		5776.2.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.2.a.b.1.1	✓	1	3.2	odd	2
304.2.a.d.1.1		1	12.11	even	2
342.2.a.d.1.1		1	1.1	even	1	trivial
722.2.a.b.1.1		1	57.56	even	2
722.2.c.d.429.1		2	57.26	odd	6
722.2.c.d.653.1		2	57.11	odd	6
722.2.c.f.429.1		2	57.50	even	6
722.2.c.f.653.1		2	57.8	even	6
722.2.e.c.99.1		6	57.17	odd	18
722.2.e.c.245.1		6	57.44	odd	18
722.2.e.c.389.1		6	57.35	odd	18
722.2.e.c.415.1		6	57.23	odd	18
722.2.e.c.423.1		6	57.47	odd	18
722.2.e.c.595.1		6	57.5	odd	18
722.2.e.d.99.1		6	57.2	even	18
722.2.e.d.245.1		6	57.32	even	18
722.2.e.d.389.1		6	57.41	even	18
722.2.e.d.415.1		6	57.53	even	18
722.2.e.d.423.1		6	57.29	even	18
722.2.e.d.595.1		6	57.14	even	18
950.2.a.b.1.1		1	15.14	odd	2
950.2.b.c.799.1		2	15.8	even	4
950.2.b.c.799.2		2	15.2	even	4
1216.2.a.g.1.1		1	24.11	even	2
1216.2.a.n.1.1		1	24.5	odd	2
1862.2.a.f.1.1		1	21.20	even	2
2736.2.a.w.1.1		1	4.3	odd	2
4598.2.a.a.1.1		1	33.32	even	2
5776.2.a.d.1.1		1	228.227	odd	2
6422.2.a.b.1.1		1	39.38	odd	2
6498.2.a.y.1.1		1	19.18	odd	2
7600.2.a.h.1.1		1	60.59	even	2
8550.2.a.u.1.1		1	5.4	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:	\( N \)	\(=\)	\( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	342.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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