LMFDB - Embedded newform 1422.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1422,2,Mod(1,1422)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1422, 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("1422.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1422 = 2 \cdot 3^{2} \cdot 79 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1422.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:	$11.3547271674$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 158)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1422.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{10} -2.00000 q^{11} -1.00000 q^{13} -3.00000 q^{14} +1.00000 q^{16} +2.00000 q^{17} -1.00000 q^{20} +2.00000 q^{22} +6.00000 q^{23} -4.00000 q^{25} +1.00000 q^{26} +3.00000 q^{28} +10.0000 q^{29} +2.00000 q^{31} -1.00000 q^{32} -2.00000 q^{34} -3.00000 q^{35} -2.00000 q^{37} +1.00000 q^{40} -2.00000 q^{41} +4.00000 q^{43} -2.00000 q^{44} -6.00000 q^{46} -3.00000 q^{47} +2.00000 q^{49} +4.00000 q^{50} -1.00000 q^{52} -4.00000 q^{53} +2.00000 q^{55} -3.00000 q^{56} -10.0000 q^{58} -5.00000 q^{59} +12.0000 q^{61} -2.00000 q^{62} +1.00000 q^{64} +1.00000 q^{65} +8.00000 q^{67} +2.00000 q^{68} +3.00000 q^{70} +13.0000 q^{71} -6.00000 q^{73} +2.00000 q^{74} -6.00000 q^{77} -1.00000 q^{79} -1.00000 q^{80} +2.00000 q^{82} +6.00000 q^{83} -2.00000 q^{85} -4.00000 q^{86} +2.00000 q^{88} +15.0000 q^{89} -3.00000 q^{91} +6.00000 q^{92} +3.00000 q^{94} +13.0000 q^{97} -2.00000 q^{98} +O(q^{100})$

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
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	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
$10$	1.00000	0.316228
$11$	−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
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	2.00000	0.426401
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	1.00000	0.196116
$27$	0	0
$28$	3.00000	0.566947
$29$	10.0000	1.85695	0.928477	−	0.371391i	$-0.121119\pi$
$29$	10.0000	1.85695	0.928477	+	0.371391i	$0.121119\pi$
$30$	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
$34$	−2.00000	−0.342997
$35$	−3.00000	−0.507093
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	0	0
$39$	0	0
$40$	1.00000	0.158114
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	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
$46$	−6.00000	−0.884652
$47$	−3.00000	−0.437595	−0.218797	−	0.975770i	$-0.570213\pi$
$47$	−3.00000	−0.437595	−0.218797	+	0.975770i	$0.570213\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	4.00000	0.565685
$51$	0	0
$52$	−1.00000	−0.138675
$53$	−4.00000	−0.549442	−0.274721	−	0.961524i	$-0.588586\pi$
$53$	−4.00000	−0.549442	−0.274721	+	0.961524i	$0.588586\pi$
$54$	0	0
$55$	2.00000	0.269680
$56$	−3.00000	−0.400892
$57$	0	0
$58$	−10.0000	−1.31306
$59$	−5.00000	−0.650945	−0.325472	−	0.945552i	$-0.605523\pi$
$59$	−5.00000	−0.650945	−0.325472	+	0.945552i	$0.605523\pi$
$60$	0	0
$61$	12.0000	1.53644	0.768221	−	0.640184i	$-0.221142\pi$
$61$	12.0000	1.53644	0.768221	+	0.640184i	$0.221142\pi$
$62$	−2.00000	−0.254000
$63$	0	0
$64$	1.00000	0.125000
$65$	1.00000	0.124035
$66$	0	0
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	2.00000	0.242536
$69$	0	0
$70$	3.00000	0.358569
$71$	13.0000	1.54282	0.771408	−	0.636341i	$-0.219553\pi$
$71$	13.0000	1.54282	0.771408	+	0.636341i	$0.219553\pi$
$72$	0	0
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	2.00000	0.232495
$75$	0	0
$76$	0	0
$77$	−6.00000	−0.683763
$78$	0	0
$79$	−1.00000	−0.112509
$79$	−1.00000	−0.112509
$80$	−1.00000	−0.111803
$81$	0	0
$82$	2.00000	0.220863
$83$	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
$85$	−2.00000	−0.216930
$86$	−4.00000	−0.431331
$87$	0	0
$88$	2.00000	0.213201
$89$	15.0000	1.59000	0.794998	−	0.606612i	$-0.207472\pi$
$89$	15.0000	1.59000	0.794998	+	0.606612i	$0.207472\pi$
$90$	0	0
$91$	−3.00000	−0.314485
$92$	6.00000	0.625543
$93$	0	0
$94$	3.00000	0.309426
$95$	0	0
$96$	0	0
$97$	13.0000	1.31995	0.659975	−	0.751288i	$-0.270567\pi$
$97$	13.0000	1.31995	0.659975	+	0.751288i	$0.270567\pi$
$98$	−2.00000	−0.202031
$99$	0	0
$100$	−4.00000	−0.400000
$101$	−7.00000	−0.696526	−0.348263	−	0.937397i	$-0.613228\pi$
$101$	−7.00000	−0.696526	−0.348263	+	0.937397i	$0.613228\pi$
$102$	0	0
$103$	19.0000	1.87213	0.936063	−	0.351833i	$-0.114441\pi$
$103$	19.0000	1.87213	0.936063	+	0.351833i	$0.114441\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	4.00000	0.388514
$107$	−13.0000	−1.25676	−0.628379	−	0.777908i	$-0.716281\pi$
$107$	−13.0000	−1.25676	−0.628379	+	0.777908i	$0.716281\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	−2.00000	−0.190693
$111$	0	0
$112$	3.00000	0.283473
$113$	−4.00000	−0.376288	−0.188144	−	0.982141i	$-0.560247\pi$
$113$	−4.00000	−0.376288	−0.188144	+	0.982141i	$0.560247\pi$
$114$	0	0
$115$	−6.00000	−0.559503
$116$	10.0000	0.928477
$117$	0	0
$118$	5.00000	0.460287
$119$	6.00000	0.550019
$120$	0	0
$121$	−7.00000	−0.636364
$122$	−12.0000	−1.08643
$123$	0	0
$124$	2.00000	0.179605
$125$	9.00000	0.804984
$126$	0	0
$127$	−7.00000	−0.621150	−0.310575	−	0.950549i	$-0.600522\pi$
$127$	−7.00000	−0.621150	−0.310575	+	0.950549i	$0.600522\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−1.00000	−0.0877058
$131$	18.0000	1.57267	0.786334	−	0.617802i	$-0.211977\pi$
$131$	18.0000	1.57267	0.786334	+	0.617802i	$0.211977\pi$
$132$	0	0
$133$	0	0
$134$	−8.00000	−0.691095
$135$	0	0
$136$	−2.00000	−0.171499
$137$	12.0000	1.02523	0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000	1.02523	0.512615	+	0.858619i	$0.328677\pi$
$138$	0	0
$139$	−5.00000	−0.424094	−0.212047	−	0.977259i	$-0.568013\pi$
$139$	−5.00000	−0.424094	−0.212047	+	0.977259i	$0.568013\pi$
$140$	−3.00000	−0.253546
$141$	0	0
$142$	−13.0000	−1.09094
$143$	2.00000	0.167248
$144$	0	0
$145$	−10.0000	−0.830455
$146$	6.00000	0.496564
$147$	0	0
$148$	−2.00000	−0.164399
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	0	0
$151$	2.00000	0.162758	0.0813788	−	0.996683i	$-0.474068\pi$
$151$	2.00000	0.162758	0.0813788	+	0.996683i	$0.474068\pi$
$152$	0	0
$153$	0	0
$154$	6.00000	0.483494
$155$	−2.00000	−0.160644
$156$	0	0
$157$	18.0000	1.43656	0.718278	−	0.695756i	$-0.244931\pi$
$157$	18.0000	1.43656	0.718278	+	0.695756i	$0.244931\pi$
$158$	1.00000	0.0795557
$159$	0	0
$160$	1.00000	0.0790569
$161$	18.0000	1.41860
$162$	0	0
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	−2.00000	−0.156174
$165$	0	0
$166$	−6.00000	−0.465690
$167$	2.00000	0.154765	0.0773823	−	0.997001i	$-0.475344\pi$
$167$	2.00000	0.154765	0.0773823	+	0.997001i	$0.475344\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	2.00000	0.153393
$171$	0	0
$172$	4.00000	0.304997
$173$	−4.00000	−0.304114	−0.152057	−	0.988372i	$-0.548590\pi$
$173$	−4.00000	−0.304114	−0.152057	+	0.988372i	$0.548590\pi$
$174$	0	0
$175$	−12.0000	−0.907115
$176$	−2.00000	−0.150756
$177$	0	0
$178$	−15.0000	−1.12430
$179$	−20.0000	−1.49487	−0.747435	−	0.664335i	$-0.768715\pi$
$179$	−20.0000	−1.49487	−0.747435	+	0.664335i	$0.768715\pi$
$180$	0	0
$181$	−18.0000	−1.33793	−0.668965	−	0.743294i	$-0.733262\pi$
$181$	−18.0000	−1.33793	−0.668965	+	0.743294i	$0.733262\pi$
$182$	3.00000	0.222375
$183$	0	0
$184$	−6.00000	−0.442326
$185$	2.00000	0.147043
$186$	0	0
$187$	−4.00000	−0.292509
$188$	−3.00000	−0.218797
$189$	0	0
$190$	0	0
$191$	3.00000	0.217072	0.108536	−	0.994092i	$-0.465384\pi$
$191$	3.00000	0.217072	0.108536	+	0.994092i	$0.465384\pi$
$192$	0	0
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	−13.0000	−0.933346
$195$	0	0
$196$	2.00000	0.142857
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	0	0
$199$	15.0000	1.06332	0.531661	−	0.846957i	$-0.321568\pi$
$199$	15.0000	1.06332	0.531661	+	0.846957i	$0.321568\pi$
$200$	4.00000	0.282843
$201$	0	0
$202$	7.00000	0.492518
$203$	30.0000	2.10559
$204$	0	0
$205$	2.00000	0.139686
$206$	−19.0000	−1.32379
$207$	0	0
$208$	−1.00000	−0.0693375
$209$	0	0
$210$	0	0
$211$	−28.0000	−1.92760	−0.963800	−	0.266627i	$-0.914091\pi$
$211$	−28.0000	−1.92760	−0.963800	+	0.266627i	$0.914091\pi$
$212$	−4.00000	−0.274721
$213$	0	0
$214$	13.0000	0.888662
$215$	−4.00000	−0.272798
$216$	0	0
$217$	6.00000	0.407307
$218$	−10.0000	−0.677285
$219$	0	0
$220$	2.00000	0.134840
$221$	−2.00000	−0.134535
$222$	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
$226$	4.00000	0.266076
$227$	−8.00000	−0.530979	−0.265489	−	0.964114i	$-0.585534\pi$
$227$	−8.00000	−0.530979	−0.265489	+	0.964114i	$0.585534\pi$
$228$	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
$232$	−10.0000	−0.656532
$233$	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
$235$	3.00000	0.195698
$236$	−5.00000	−0.325472
$237$	0	0
$238$	−6.00000	−0.388922
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	17.0000	1.09507	0.547533	−	0.836784i	$-0.315567\pi$
$241$	17.0000	1.09507	0.547533	+	0.836784i	$0.315567\pi$
$242$	7.00000	0.449977
$243$	0	0
$244$	12.0000	0.768221
$245$	−2.00000	−0.127775
$246$	0	0
$247$	0	0
$248$	−2.00000	−0.127000
$249$	0	0
$250$	−9.00000	−0.569210
$251$	−27.0000	−1.70422	−0.852112	−	0.523359i	$-0.824679\pi$
$251$	−27.0000	−1.70422	−0.852112	+	0.523359i	$0.824679\pi$
$252$	0	0
$253$	−12.0000	−0.754434
$254$	7.00000	0.439219
$255$	0	0
$256$	1.00000	0.0625000
$257$	2.00000	0.124757	0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000	0.124757	0.0623783	+	0.998053i	$0.480131\pi$
$258$	0	0
$259$	−6.00000	−0.372822
$260$	1.00000	0.0620174
$261$	0	0
$262$	−18.0000	−1.11204
$263$	6.00000	0.369976	0.184988	−	0.982741i	$-0.440775\pi$
$263$	6.00000	0.369976	0.184988	+	0.982741i	$0.440775\pi$
$264$	0	0
$265$	4.00000	0.245718
$266$	0	0
$267$	0	0
$268$	8.00000	0.488678
$269$	−25.0000	−1.52428	−0.762138	−	0.647414i	$-0.775850\pi$
$269$	−25.0000	−1.52428	−0.762138	+	0.647414i	$0.775850\pi$
$270$	0	0
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	2.00000	0.121268
$273$	0	0
$274$	−12.0000	−0.724947
$275$	8.00000	0.482418
$276$	0	0
$277$	−17.0000	−1.02143	−0.510716	−	0.859750i	$-0.670619\pi$
$277$	−17.0000	−1.02143	−0.510716	+	0.859750i	$0.670619\pi$
$278$	5.00000	0.299880
$279$	0	0
$280$	3.00000	0.179284
$281$	33.0000	1.96861	0.984307	−	0.176462i	$-0.0564652\pi$
$281$	33.0000	1.96861	0.984307	+	0.176462i	$0.0564652\pi$
$282$	0	0
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	13.0000	0.771408
$285$	0	0
$286$	−2.00000	−0.118262
$287$	−6.00000	−0.354169
$288$	0	0
$289$	−13.0000	−0.764706
$290$	10.0000	0.587220
$291$	0	0
$292$	−6.00000	−0.351123
$293$	−24.0000	−1.40209	−0.701047	−	0.713115i	$-0.747284\pi$
$293$	−24.0000	−1.40209	−0.701047	+	0.713115i	$0.747284\pi$
$294$	0	0
$295$	5.00000	0.291111
$296$	2.00000	0.116248
$297$	0	0
$298$	−10.0000	−0.579284
$299$	−6.00000	−0.346989
$300$	0	0
$301$	12.0000	0.691669
$302$	−2.00000	−0.115087
$303$	0	0
$304$	0	0
$305$	−12.0000	−0.687118
$306$	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
$310$	2.00000	0.113592
$311$	−12.0000	−0.680458	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000	−0.680458	−0.340229	+	0.940343i	$0.610505\pi$
$312$	0	0
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	−18.0000	−1.01580
$315$	0	0
$316$	−1.00000	−0.0562544
$317$	−3.00000	−0.168497	−0.0842484	−	0.996445i	$-0.526849\pi$
$317$	−3.00000	−0.168497	−0.0842484	+	0.996445i	$0.526849\pi$
$318$	0	0
$319$	−20.0000	−1.11979
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	−18.0000	−1.00310
$323$	0	0
$324$	0	0
$325$	4.00000	0.221880
$326$	−4.00000	−0.221540
$327$	0	0
$328$	2.00000	0.110432
$329$	−9.00000	−0.496186
$330$	0	0
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	6.00000	0.329293
$333$	0	0
$334$	−2.00000	−0.109435
$335$	−8.00000	−0.437087
$336$	0	0
$337$	23.0000	1.25289	0.626445	−	0.779466i	$-0.284509\pi$
$337$	23.0000	1.25289	0.626445	+	0.779466i	$0.284509\pi$
$338$	12.0000	0.652714
$339$	0	0
$340$	−2.00000	−0.108465
$341$	−4.00000	−0.216612
$342$	0	0
$343$	−15.0000	−0.809924
$344$	−4.00000	−0.215666
$345$	0	0
$346$	4.00000	0.215041
$347$	−18.0000	−0.966291	−0.483145	−	0.875540i	$-0.660506\pi$
$347$	−18.0000	−0.966291	−0.483145	+	0.875540i	$0.660506\pi$
$348$	0	0
$349$	−20.0000	−1.07058	−0.535288	−	0.844670i	$-0.679797\pi$
$349$	−20.0000	−1.07058	−0.535288	+	0.844670i	$0.679797\pi$
$350$	12.0000	0.641427
$351$	0	0
$352$	2.00000	0.106600
$353$	36.0000	1.91609	0.958043	−	0.286623i	$-0.0925328\pi$
$353$	36.0000	1.91609	0.958043	+	0.286623i	$0.0925328\pi$
$354$	0	0
$355$	−13.0000	−0.689968
$356$	15.0000	0.794998
$357$	0	0
$358$	20.0000	1.05703
$359$	−25.0000	−1.31945	−0.659725	−	0.751507i	$-0.729327\pi$
$359$	−25.0000	−1.31945	−0.659725	+	0.751507i	$0.729327\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	18.0000	0.946059
$363$	0	0
$364$	−3.00000	−0.157243
$365$	6.00000	0.314054
$366$	0	0
$367$	−22.0000	−1.14839	−0.574195	−	0.818718i	$-0.694685\pi$
$367$	−22.0000	−1.14839	−0.574195	+	0.818718i	$0.694685\pi$
$368$	6.00000	0.312772
$369$	0	0
$370$	−2.00000	−0.103975
$371$	−12.0000	−0.623009
$372$	0	0
$373$	14.0000	0.724893	0.362446	−	0.932005i	$-0.381942\pi$
$373$	14.0000	0.724893	0.362446	+	0.932005i	$0.381942\pi$
$374$	4.00000	0.206835
$375$	0	0
$376$	3.00000	0.154713
$377$	−10.0000	−0.515026
$378$	0	0
$379$	−15.0000	−0.770498	−0.385249	−	0.922813i	$-0.625884\pi$
$379$	−15.0000	−0.770498	−0.385249	+	0.922813i	$0.625884\pi$
$380$	0	0
$381$	0	0
$382$	−3.00000	−0.153493
$383$	6.00000	0.306586	0.153293	−	0.988181i	$-0.451012\pi$
$383$	6.00000	0.306586	0.153293	+	0.988181i	$0.451012\pi$
$384$	0	0
$385$	6.00000	0.305788
$386$	−14.0000	−0.712581
$387$	0	0
$388$	13.0000	0.659975
$389$	5.00000	0.253510	0.126755	−	0.991934i	$-0.459544\pi$
$389$	5.00000	0.253510	0.126755	+	0.991934i	$0.459544\pi$
$390$	0	0
$391$	12.0000	0.606866
$392$	−2.00000	−0.101015
$393$	0	0
$394$	18.0000	0.906827
$395$	1.00000	0.0503155
$396$	0	0
$397$	−37.0000	−1.85698	−0.928488	−	0.371361i	$-0.878891\pi$
$397$	−37.0000	−1.85698	−0.928488	+	0.371361i	$0.878891\pi$
$398$	−15.0000	−0.751882
$399$	0	0
$400$	−4.00000	−0.200000
$401$	−2.00000	−0.0998752	−0.0499376	−	0.998752i	$-0.515902\pi$
$401$	−2.00000	−0.0998752	−0.0499376	+	0.998752i	$0.515902\pi$
$402$	0	0
$403$	−2.00000	−0.0996271
$404$	−7.00000	−0.348263
$405$	0	0
$406$	−30.0000	−1.48888
$407$	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	$0.664637\pi$
$410$	−2.00000	−0.0987730
$411$	0	0
$412$	19.0000	0.936063
$413$	−15.0000	−0.738102
$414$	0	0
$415$	−6.00000	−0.294528
$416$	1.00000	0.0490290
$417$	0	0
$418$	0	0
$419$	−25.0000	−1.22133	−0.610665	−	0.791889i	$-0.709098\pi$
$419$	−25.0000	−1.22133	−0.610665	+	0.791889i	$0.709098\pi$
$420$	0	0
$421$	7.00000	0.341159	0.170580	−	0.985344i	$-0.445436\pi$
$421$	7.00000	0.341159	0.170580	+	0.985344i	$0.445436\pi$
$422$	28.0000	1.36302
$423$	0	0
$424$	4.00000	0.194257
$425$	−8.00000	−0.388057
$426$	0	0
$427$	36.0000	1.74216
$428$	−13.0000	−0.628379
$429$	0	0
$430$	4.00000	0.192897
$431$	−32.0000	−1.54139	−0.770693	−	0.637207i	$-0.780090\pi$
$431$	−32.0000	−1.54139	−0.770693	+	0.637207i	$0.780090\pi$
$432$	0	0
$433$	−11.0000	−0.528626	−0.264313	−	0.964437i	$-0.585145\pi$
$433$	−11.0000	−0.528626	−0.264313	+	0.964437i	$0.585145\pi$
$434$	−6.00000	−0.288009
$435$	0	0
$436$	10.0000	0.478913
$437$	0	0
$438$	0	0
$439$	−40.0000	−1.90910	−0.954548	−	0.298057i	$-0.903661\pi$
$439$	−40.0000	−1.90910	−0.954548	+	0.298057i	$0.903661\pi$
$440$	−2.00000	−0.0953463
$441$	0	0
$442$	2.00000	0.0951303
$443$	−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
$445$	−15.0000	−0.711068
$446$	−14.0000	−0.662919
$447$	0	0
$448$	3.00000	0.141737
$449$	10.0000	0.471929	0.235965	−	0.971762i	$-0.424175\pi$
$449$	10.0000	0.471929	0.235965	+	0.971762i	$0.424175\pi$
$450$	0	0
$451$	4.00000	0.188353
$452$	−4.00000	−0.188144
$453$	0	0
$454$	8.00000	0.375459
$455$	3.00000	0.140642
$456$	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
$461$	18.0000	0.838344	0.419172	−	0.907907i	$-0.362320\pi$
$461$	18.0000	0.838344	0.419172	+	0.907907i	$0.362320\pi$
$462$	0	0
$463$	−31.0000	−1.44069	−0.720346	−	0.693615i	$-0.756017\pi$
$463$	−31.0000	−1.44069	−0.720346	+	0.693615i	$0.756017\pi$
$464$	10.0000	0.464238
$465$	0	0
$466$	−6.00000	−0.277945
$467$	42.0000	1.94353	0.971764	−	0.235954i	$-0.0758216\pi$
$467$	42.0000	1.94353	0.971764	+	0.235954i	$0.0758216\pi$
$468$	0	0
$469$	24.0000	1.10822
$470$	−3.00000	−0.138380
$471$	0	0
$472$	5.00000	0.230144
$473$	−8.00000	−0.367840
$474$	0	0
$475$	0	0
$476$	6.00000	0.275010
$477$	0	0
$478$	0	0
$479$	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
$482$	−17.0000	−0.774329
$483$	0	0
$484$	−7.00000	−0.318182
$485$	−13.0000	−0.590300
$486$	0	0
$487$	38.0000	1.72194	0.860972	−	0.508652i	$-0.169856\pi$
$487$	38.0000	1.72194	0.860972	+	0.508652i	$0.169856\pi$
$488$	−12.0000	−0.543214
$489$	0	0
$490$	2.00000	0.0903508
$491$	3.00000	0.135388	0.0676941	−	0.997706i	$-0.478436\pi$
$491$	3.00000	0.135388	0.0676941	+	0.997706i	$0.478436\pi$
$492$	0	0
$493$	20.0000	0.900755
$494$	0	0
$495$	0	0
$496$	2.00000	0.0898027
$497$	39.0000	1.74939
$498$	0	0
$499$	−10.0000	−0.447661	−0.223831	−	0.974628i	$-0.571856\pi$
$499$	−10.0000	−0.447661	−0.223831	+	0.974628i	$0.571856\pi$
$500$	9.00000	0.402492
$501$	0	0
$502$	27.0000	1.20507
$503$	−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
$505$	7.00000	0.311496
$506$	12.0000	0.533465
$507$	0	0
$508$	−7.00000	−0.310575
$509$	20.0000	0.886484	0.443242	−	0.896402i	$-0.353828\pi$
$509$	20.0000	0.886484	0.443242	+	0.896402i	$0.353828\pi$
$510$	0	0
$511$	−18.0000	−0.796273
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−2.00000	−0.0882162
$515$	−19.0000	−0.837240
$516$	0	0
$517$	6.00000	0.263880
$518$	6.00000	0.263625
$519$	0	0
$520$	−1.00000	−0.0438529
$521$	−12.0000	−0.525730	−0.262865	−	0.964833i	$-0.584667\pi$
$521$	−12.0000	−0.525730	−0.262865	+	0.964833i	$0.584667\pi$
$522$	0	0
$523$	−26.0000	−1.13690	−0.568450	−	0.822718i	$-0.692457\pi$
$523$	−26.0000	−1.13690	−0.568450	+	0.822718i	$0.692457\pi$
$524$	18.0000	0.786334
$525$	0	0
$526$	−6.00000	−0.261612
$527$	4.00000	0.174243
$528$	0	0
$529$	13.0000	0.565217
$530$	−4.00000	−0.173749
$531$	0	0
$532$	0	0
$533$	2.00000	0.0866296
$534$	0	0
$535$	13.0000	0.562039
$536$	−8.00000	−0.345547
$537$	0	0
$538$	25.0000	1.07783
$539$	−4.00000	−0.172292
$540$	0	0
$541$	7.00000	0.300954	0.150477	−	0.988614i	$-0.451919\pi$
$541$	7.00000	0.300954	0.150477	+	0.988614i	$0.451919\pi$
$542$	8.00000	0.343629
$543$	0	0
$544$	−2.00000	−0.0857493
$545$	−10.0000	−0.428353
$546$	0	0
$547$	−2.00000	−0.0855138	−0.0427569	−	0.999086i	$-0.513614\pi$
$547$	−2.00000	−0.0855138	−0.0427569	+	0.999086i	$0.513614\pi$
$548$	12.0000	0.512615
$549$	0	0
$550$	−8.00000	−0.341121
$551$	0	0
$552$	0	0
$553$	−3.00000	−0.127573
$554$	17.0000	0.722261
$555$	0	0
$556$	−5.00000	−0.212047
$557$	7.00000	0.296600	0.148300	−	0.988942i	$-0.452620\pi$
$557$	7.00000	0.296600	0.148300	+	0.988942i	$0.452620\pi$
$558$	0	0
$559$	−4.00000	−0.169182
$560$	−3.00000	−0.126773
$561$	0	0
$562$	−33.0000	−1.39202
$563$	−14.0000	−0.590030	−0.295015	−	0.955493i	$-0.595325\pi$
$563$	−14.0000	−0.590030	−0.295015	+	0.955493i	$0.595325\pi$
$564$	0	0
$565$	4.00000	0.168281
$566$	−4.00000	−0.168133
$567$	0	0
$568$	−13.0000	−0.545468
$569$	30.0000	1.25767	0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000	1.25767	0.628833	+	0.777541i	$0.283533\pi$
$570$	0	0
$571$	22.0000	0.920671	0.460336	−	0.887745i	$-0.347729\pi$
$571$	22.0000	0.920671	0.460336	+	0.887745i	$0.347729\pi$
$572$	2.00000	0.0836242
$573$	0	0
$574$	6.00000	0.250435
$575$	−24.0000	−1.00087
$576$	0	0
$577$	−12.0000	−0.499567	−0.249783	−	0.968302i	$-0.580359\pi$
$577$	−12.0000	−0.499567	−0.249783	+	0.968302i	$0.580359\pi$
$578$	13.0000	0.540729
$579$	0	0
$580$	−10.0000	−0.415227
$581$	18.0000	0.746766
$582$	0	0
$583$	8.00000	0.331326
$584$	6.00000	0.248282
$585$	0	0
$586$	24.0000	0.991431
$587$	7.00000	0.288921	0.144460	−	0.989511i	$-0.453855\pi$
$587$	7.00000	0.288921	0.144460	+	0.989511i	$0.453855\pi$
$588$	0	0
$589$	0	0
$590$	−5.00000	−0.205847
$591$	0	0
$592$	−2.00000	−0.0821995
$593$	21.0000	0.862367	0.431183	−	0.902264i	$-0.358096\pi$
$593$	21.0000	0.862367	0.431183	+	0.902264i	$0.358096\pi$
$594$	0	0
$595$	−6.00000	−0.245976
$596$	10.0000	0.409616
$597$	0	0
$598$	6.00000	0.245358
$599$	−40.0000	−1.63436	−0.817178	−	0.576386i	$-0.804463\pi$
$599$	−40.0000	−1.63436	−0.817178	+	0.576386i	$0.804463\pi$
$600$	0	0
$601$	12.0000	0.489490	0.244745	−	0.969587i	$-0.421296\pi$
$601$	12.0000	0.489490	0.244745	+	0.969587i	$0.421296\pi$
$602$	−12.0000	−0.489083
$603$	0	0
$604$	2.00000	0.0813788
$605$	7.00000	0.284590
$606$	0	0
$607$	33.0000	1.33943	0.669714	−	0.742619i	$-0.266417\pi$
$607$	33.0000	1.33943	0.669714	+	0.742619i	$0.266417\pi$
$608$	0	0
$609$	0	0
$610$	12.0000	0.485866
$611$	3.00000	0.121367
$612$	0	0
$613$	14.0000	0.565455	0.282727	−	0.959200i	$-0.408761\pi$
$613$	14.0000	0.565455	0.282727	+	0.959200i	$0.408761\pi$
$614$	12.0000	0.484281
$615$	0	0
$616$	6.00000	0.241747
$617$	27.0000	1.08698	0.543490	−	0.839416i	$-0.317103\pi$
$617$	27.0000	1.08698	0.543490	+	0.839416i	$0.317103\pi$
$618$	0	0
$619$	−35.0000	−1.40677	−0.703384	−	0.710810i	$-0.748329\pi$
$619$	−35.0000	−1.40677	−0.703384	+	0.710810i	$0.748329\pi$
$620$	−2.00000	−0.0803219
$621$	0	0
$622$	12.0000	0.481156
$623$	45.0000	1.80289
$624$	0	0
$625$	11.0000	0.440000
$626$	−14.0000	−0.559553
$627$	0	0
$628$	18.0000	0.718278
$629$	−4.00000	−0.159490
$630$	0	0
$631$	37.0000	1.47295	0.736473	−	0.676467i	$-0.236490\pi$
$631$	37.0000	1.47295	0.736473	+	0.676467i	$0.236490\pi$
$632$	1.00000	0.0397779
$633$	0	0
$634$	3.00000	0.119145
$635$	7.00000	0.277787
$636$	0	0
$637$	−2.00000	−0.0792429
$638$	20.0000	0.791808
$639$	0	0
$640$	1.00000	0.0395285
$641$	33.0000	1.30342	0.651711	−	0.758468i	$-0.274052\pi$
$641$	33.0000	1.30342	0.651711	+	0.758468i	$0.274052\pi$
$642$	0	0
$643$	4.00000	0.157745	0.0788723	−	0.996885i	$-0.474868\pi$
$643$	4.00000	0.157745	0.0788723	+	0.996885i	$0.474868\pi$
$644$	18.0000	0.709299
$645$	0	0
$646$	0	0
$647$	32.0000	1.25805	0.629025	−	0.777385i	$-0.283454\pi$
$647$	32.0000	1.25805	0.629025	+	0.777385i	$0.283454\pi$
$648$	0	0
$649$	10.0000	0.392534
$650$	−4.00000	−0.156893
$651$	0	0
$652$	4.00000	0.156652
$653$	11.0000	0.430463	0.215232	−	0.976563i	$-0.430949\pi$
$653$	11.0000	0.430463	0.215232	+	0.976563i	$0.430949\pi$
$654$	0	0
$655$	−18.0000	−0.703318
$656$	−2.00000	−0.0780869
$657$	0	0
$658$	9.00000	0.350857
$659$	−15.0000	−0.584317	−0.292159	−	0.956370i	$-0.594373\pi$
$659$	−15.0000	−0.584317	−0.292159	+	0.956370i	$0.594373\pi$
$660$	0	0
$661$	2.00000	0.0777910	0.0388955	−	0.999243i	$-0.487616\pi$
$661$	2.00000	0.0777910	0.0388955	+	0.999243i	$0.487616\pi$
$662$	28.0000	1.08825
$663$	0	0
$664$	−6.00000	−0.232845
$665$	0	0
$666$	0	0
$667$	60.0000	2.32321
$668$	2.00000	0.0773823
$669$	0	0
$670$	8.00000	0.309067
$671$	−24.0000	−0.926510
$672$	0	0
$673$	−26.0000	−1.00223	−0.501113	−	0.865382i	$-0.667076\pi$
$673$	−26.0000	−1.00223	−0.501113	+	0.865382i	$0.667076\pi$
$674$	−23.0000	−0.885927
$675$	0	0
$676$	−12.0000	−0.461538
$677$	27.0000	1.03769	0.518847	−	0.854867i	$-0.326361\pi$
$677$	27.0000	1.03769	0.518847	+	0.854867i	$0.326361\pi$
$678$	0	0
$679$	39.0000	1.49668
$680$	2.00000	0.0766965
$681$	0	0
$682$	4.00000	0.153168
$683$	−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
$685$	−12.0000	−0.458496
$686$	15.0000	0.572703
$687$	0	0
$688$	4.00000	0.152499
$689$	4.00000	0.152388
$690$	0	0
$691$	17.0000	0.646710	0.323355	−	0.946278i	$-0.395189\pi$
$691$	17.0000	0.646710	0.323355	+	0.946278i	$0.395189\pi$
$692$	−4.00000	−0.152057
$693$	0	0
$694$	18.0000	0.683271
$695$	5.00000	0.189661
$696$	0	0
$697$	−4.00000	−0.151511
$698$	20.0000	0.757011
$699$	0	0
$700$	−12.0000	−0.453557
$701$	18.0000	0.679851	0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000	0.679851	0.339925	+	0.940452i	$0.389598\pi$
$702$	0	0
$703$	0	0
$704$	−2.00000	−0.0753778
$705$	0	0
$706$	−36.0000	−1.35488
$707$	−21.0000	−0.789786
$708$	0	0
$709$	−50.0000	−1.87779	−0.938895	−	0.344204i	$-0.888149\pi$
$709$	−50.0000	−1.87779	−0.938895	+	0.344204i	$0.888149\pi$
$710$	13.0000	0.487881
$711$	0	0
$712$	−15.0000	−0.562149
$713$	12.0000	0.449404
$714$	0	0
$715$	−2.00000	−0.0747958
$716$	−20.0000	−0.747435
$717$	0	0
$718$	25.0000	0.932992
$719$	−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
$721$	57.0000	2.12279
$722$	19.0000	0.707107
$723$	0	0
$724$	−18.0000	−0.668965
$725$	−40.0000	−1.48556
$726$	0	0
$727$	18.0000	0.667583	0.333792	−	0.942647i	$-0.391672\pi$
$727$	18.0000	0.667583	0.333792	+	0.942647i	$0.391672\pi$
$728$	3.00000	0.111187
$729$	0	0
$730$	−6.00000	−0.222070
$731$	8.00000	0.295891
$732$	0	0
$733$	34.0000	1.25582	0.627909	−	0.778287i	$-0.283911\pi$
$733$	34.0000	1.25582	0.627909	+	0.778287i	$0.283911\pi$
$734$	22.0000	0.812035
$735$	0	0
$736$	−6.00000	−0.221163
$737$	−16.0000	−0.589368
$738$	0	0
$739$	−5.00000	−0.183928	−0.0919640	−	0.995762i	$-0.529314\pi$
$739$	−5.00000	−0.183928	−0.0919640	+	0.995762i	$0.529314\pi$
$740$	2.00000	0.0735215
$741$	0	0
$742$	12.0000	0.440534
$743$	−24.0000	−0.880475	−0.440237	−	0.897881i	$-0.645106\pi$
$743$	−24.0000	−0.880475	−0.440237	+	0.897881i	$0.645106\pi$
$744$	0	0
$745$	−10.0000	−0.366372
$746$	−14.0000	−0.512576
$747$	0	0
$748$	−4.00000	−0.146254
$749$	−39.0000	−1.42503
$750$	0	0
$751$	42.0000	1.53260	0.766301	−	0.642482i	$-0.222095\pi$
$751$	42.0000	1.53260	0.766301	+	0.642482i	$0.222095\pi$
$752$	−3.00000	−0.109399
$753$	0	0
$754$	10.0000	0.364179
$755$	−2.00000	−0.0727875
$756$	0	0
$757$	−37.0000	−1.34479	−0.672394	−	0.740193i	$-0.734734\pi$
$757$	−37.0000	−1.34479	−0.672394	+	0.740193i	$0.734734\pi$
$758$	15.0000	0.544825
$759$	0	0
$760$	0	0
$761$	−22.0000	−0.797499	−0.398750	−	0.917060i	$-0.630556\pi$
$761$	−22.0000	−0.797499	−0.398750	+	0.917060i	$0.630556\pi$
$762$	0	0
$763$	30.0000	1.08607
$764$	3.00000	0.108536
$765$	0	0
$766$	−6.00000	−0.216789
$767$	5.00000	0.180540
$768$	0	0
$769$	−40.0000	−1.44244	−0.721218	−	0.692708i	$-0.756418\pi$
$769$	−40.0000	−1.44244	−0.721218	+	0.692708i	$0.756418\pi$
$770$	−6.00000	−0.216225
$771$	0	0
$772$	14.0000	0.503871
$773$	−34.0000	−1.22290	−0.611448	−	0.791285i	$-0.709412\pi$
$773$	−34.0000	−1.22290	−0.611448	+	0.791285i	$0.709412\pi$
$774$	0	0
$775$	−8.00000	−0.287368
$776$	−13.0000	−0.466673
$777$	0	0
$778$	−5.00000	−0.179259
$779$	0	0
$780$	0	0
$781$	−26.0000	−0.930353
$782$	−12.0000	−0.429119
$783$	0	0
$784$	2.00000	0.0714286
$785$	−18.0000	−0.642448
$786$	0	0
$787$	18.0000	0.641631	0.320815	−	0.947142i	$-0.396043\pi$
$787$	18.0000	0.641631	0.320815	+	0.947142i	$0.396043\pi$
$788$	−18.0000	−0.641223
$789$	0	0
$790$	−1.00000	−0.0355784
$791$	−12.0000	−0.426671
$792$	0	0
$793$	−12.0000	−0.426132
$794$	37.0000	1.31308
$795$	0	0
$796$	15.0000	0.531661
$797$	42.0000	1.48772	0.743858	−	0.668338i	$-0.232994\pi$
$797$	42.0000	1.48772	0.743858	+	0.668338i	$0.232994\pi$
$798$	0	0
$799$	−6.00000	−0.212265
$800$	4.00000	0.141421
$801$	0	0
$802$	2.00000	0.0706225
$803$	12.0000	0.423471
$804$	0	0
$805$	−18.0000	−0.634417
$806$	2.00000	0.0704470
$807$	0	0
$808$	7.00000	0.246259
$809$	−15.0000	−0.527372	−0.263686	−	0.964609i	$-0.584938\pi$
$809$	−15.0000	−0.527372	−0.263686	+	0.964609i	$0.584938\pi$
$810$	0	0
$811$	22.0000	0.772524	0.386262	−	0.922389i	$-0.373766\pi$
$811$	22.0000	0.772524	0.386262	+	0.922389i	$0.373766\pi$
$812$	30.0000	1.05279
$813$	0	0
$814$	−4.00000	−0.140200
$815$	−4.00000	−0.140114
$816$	0	0
$817$	0	0
$818$	20.0000	0.699284
$819$	0	0
$820$	2.00000	0.0698430
$821$	−42.0000	−1.46581	−0.732905	−	0.680331i	$-0.761836\pi$
$821$	−42.0000	−1.46581	−0.732905	+	0.680331i	$0.761836\pi$
$822$	0	0
$823$	−16.0000	−0.557725	−0.278862	−	0.960331i	$-0.589957\pi$
$823$	−16.0000	−0.557725	−0.278862	+	0.960331i	$0.589957\pi$
$824$	−19.0000	−0.661896
$825$	0	0
$826$	15.0000	0.521917
$827$	−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
$829$	−20.0000	−0.694629	−0.347314	−	0.937749i	$-0.612906\pi$
$829$	−20.0000	−0.694629	−0.347314	+	0.937749i	$0.612906\pi$
$830$	6.00000	0.208263
$831$	0	0
$832$	−1.00000	−0.0346688
$833$	4.00000	0.138592
$834$	0	0
$835$	−2.00000	−0.0692129
$836$	0	0
$837$	0	0
$838$	25.0000	0.863611
$839$	−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
$841$	71.0000	2.44828
$842$	−7.00000	−0.241236
$843$	0	0
$844$	−28.0000	−0.963800
$845$	12.0000	0.412813
$846$	0	0
$847$	−21.0000	−0.721569
$848$	−4.00000	−0.137361
$849$	0	0
$850$	8.00000	0.274398
$851$	−12.0000	−0.411355
$852$	0	0
$853$	−16.0000	−0.547830	−0.273915	−	0.961754i	$-0.588319\pi$
$853$	−16.0000	−0.547830	−0.273915	+	0.961754i	$0.588319\pi$
$854$	−36.0000	−1.23189
$855$	0	0
$856$	13.0000	0.444331
$857$	−33.0000	−1.12726	−0.563629	−	0.826028i	$-0.690595\pi$
$857$	−33.0000	−1.12726	−0.563629	+	0.826028i	$0.690595\pi$
$858$	0	0
$859$	−35.0000	−1.19418	−0.597092	−	0.802173i	$-0.703677\pi$
$859$	−35.0000	−1.19418	−0.597092	+	0.802173i	$0.703677\pi$
$860$	−4.00000	−0.136399
$861$	0	0
$862$	32.0000	1.08992
$863$	46.0000	1.56586	0.782929	−	0.622111i	$-0.213725\pi$
$863$	46.0000	1.56586	0.782929	+	0.622111i	$0.213725\pi$
$864$	0	0
$865$	4.00000	0.136004
$866$	11.0000	0.373795
$867$	0	0
$868$	6.00000	0.203653
$869$	2.00000	0.0678454
$870$	0	0
$871$	−8.00000	−0.271070
$872$	−10.0000	−0.338643
$873$	0	0
$874$	0	0
$875$	27.0000	0.912767
$876$	0	0
$877$	58.0000	1.95852	0.979260	−	0.202606i	$-0.0649409\pi$
$877$	58.0000	1.95852	0.979260	+	0.202606i	$0.0649409\pi$
$878$	40.0000	1.34993
$879$	0	0
$880$	2.00000	0.0674200
$881$	−12.0000	−0.404290	−0.202145	−	0.979356i	$-0.564791\pi$
$881$	−12.0000	−0.404290	−0.202145	+	0.979356i	$0.564791\pi$
$882$	0	0
$883$	−1.00000	−0.0336527	−0.0168263	−	0.999858i	$-0.505356\pi$
$883$	−1.00000	−0.0336527	−0.0168263	+	0.999858i	$0.505356\pi$
$884$	−2.00000	−0.0672673
$885$	0	0
$886$	4.00000	0.134383
$887$	22.0000	0.738688	0.369344	−	0.929293i	$-0.379582\pi$
$887$	22.0000	0.738688	0.369344	+	0.929293i	$0.379582\pi$
$888$	0	0
$889$	−21.0000	−0.704317
$890$	15.0000	0.502801
$891$	0	0
$892$	14.0000	0.468755
$893$	0	0
$894$	0	0
$895$	20.0000	0.668526
$896$	−3.00000	−0.100223
$897$	0	0
$898$	−10.0000	−0.333704
$899$	20.0000	0.667037
$900$	0	0
$901$	−8.00000	−0.266519
$902$	−4.00000	−0.133185
$903$	0	0
$904$	4.00000	0.133038
$905$	18.0000	0.598340
$906$	0	0
$907$	−12.0000	−0.398453	−0.199227	−	0.979953i	$-0.563843\pi$
$907$	−12.0000	−0.398453	−0.199227	+	0.979953i	$0.563843\pi$
$908$	−8.00000	−0.265489
$909$	0	0
$910$	−3.00000	−0.0994490
$911$	−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
$914$	−23.0000	−0.760772
$915$	0	0
$916$	20.0000	0.660819
$917$	54.0000	1.78324
$918$	0	0
$919$	0	0		−	1.00000i	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	6.00000	0.197814
$921$	0	0
$922$	−18.0000	−0.592798
$923$	−13.0000	−0.427900
$924$	0	0
$925$	8.00000	0.263038
$926$	31.0000	1.01872
$927$	0	0
$928$	−10.0000	−0.328266
$929$	0	0		−	1.00000i	$-0.5\pi$
$929$	0	0			1.00000i	$0.5\pi$
$930$	0	0
$931$	0	0
$932$	6.00000	0.196537
$933$	0	0
$934$	−42.0000	−1.37428
$935$	4.00000	0.130814
$936$	0	0
$937$	−12.0000	−0.392023	−0.196011	−	0.980602i	$-0.562799\pi$
$937$	−12.0000	−0.392023	−0.196011	+	0.980602i	$0.562799\pi$
$938$	−24.0000	−0.783628
$939$	0	0
$940$	3.00000	0.0978492
$941$	38.0000	1.23876	0.619382	−	0.785090i	$-0.287383\pi$
$941$	38.0000	1.23876	0.619382	+	0.785090i	$0.287383\pi$
$942$	0	0
$943$	−12.0000	−0.390774
$944$	−5.00000	−0.162736
$945$	0	0
$946$	8.00000	0.260102
$947$	32.0000	1.03986	0.519930	−	0.854209i	$-0.325958\pi$
$947$	32.0000	1.03986	0.519930	+	0.854209i	$0.325958\pi$
$948$	0	0
$949$	6.00000	0.194768
$950$	0	0
$951$	0	0
$952$	−6.00000	−0.194461
$953$	31.0000	1.00419	0.502094	−	0.864813i	$-0.332563\pi$
$953$	31.0000	1.00419	0.502094	+	0.864813i	$0.332563\pi$
$954$	0	0
$955$	−3.00000	−0.0970777
$956$	0	0
$957$	0	0
$958$	−20.0000	−0.646171
$959$	36.0000	1.16250
$960$	0	0
$961$	−27.0000	−0.870968
$962$	−2.00000	−0.0644826
$963$	0	0
$964$	17.0000	0.547533
$965$	−14.0000	−0.450676
$966$	0	0
$967$	−52.0000	−1.67221	−0.836104	−	0.548572i	$-0.815172\pi$
$967$	−52.0000	−1.67221	−0.836104	+	0.548572i	$0.815172\pi$
$968$	7.00000	0.224989
$969$	0	0
$970$	13.0000	0.417405
$971$	−22.0000	−0.706014	−0.353007	−	0.935621i	$-0.614841\pi$
$971$	−22.0000	−0.706014	−0.353007	+	0.935621i	$0.614841\pi$
$972$	0	0
$973$	−15.0000	−0.480878
$974$	−38.0000	−1.21760
$975$	0	0
$976$	12.0000	0.384111
$977$	−58.0000	−1.85558	−0.927792	−	0.373097i	$-0.878296\pi$
$977$	−58.0000	−1.85558	−0.927792	+	0.373097i	$0.878296\pi$
$978$	0	0
$979$	−30.0000	−0.958804
$980$	−2.00000	−0.0638877
$981$	0	0
$982$	−3.00000	−0.0957338
$983$	−24.0000	−0.765481	−0.382741	−	0.923856i	$-0.625020\pi$
$983$	−24.0000	−0.765481	−0.382741	+	0.923856i	$0.625020\pi$
$984$	0	0
$985$	18.0000	0.573528
$986$	−20.0000	−0.636930
$987$	0	0
$988$	0	0
$989$	24.0000	0.763156
$990$	0	0
$991$	−13.0000	−0.412959	−0.206479	−	0.978451i	$-0.566201\pi$
$991$	−13.0000	−0.412959	−0.206479	+	0.978451i	$0.566201\pi$
$992$	−2.00000	−0.0635001
$993$	0	0
$994$	−39.0000	−1.23700
$995$	−15.0000	−0.475532
$996$	0	0
$997$	−22.0000	−0.696747	−0.348373	−	0.937356i	$-0.613266\pi$
$997$	−22.0000	−0.696747	−0.348373	+	0.937356i	$0.613266\pi$
$998$	10.0000	0.316544
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1422.2.a.a.1.1		1
3.2	odd	2		158.2.a.d.1.1	✓	1
12.11	even	2		1264.2.a.f.1.1		1
15.14	odd	2		3950.2.a.d.1.1		1
21.20	even	2		7742.2.a.m.1.1		1
24.5	odd	2		5056.2.a.m.1.1		1
24.11	even	2		5056.2.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
158.2.a.d.1.1	✓	1	3.2	odd	2
1264.2.a.f.1.1		1	12.11	even	2
1422.2.a.a.1.1		1	1.1	even	1	trivial
3950.2.a.d.1.1		1	15.14	odd	2
5056.2.a.e.1.1		1	24.11	even	2
5056.2.a.m.1.1		1	24.5	odd	2
7742.2.a.m.1.1		1	21.20	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:	\( N \)	\(=\)	\( 1422 = 2 \cdot 3^{2} \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1422.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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