LMFDB - Embedded newform 1170.2.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1170,2,Mod(1,1170)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1170, 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("1170.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1170.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:	$9.34249703649$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1170.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +2.00000 q^{7} +1.00000 q^{8} -1.00000 q^{10} +4.00000 q^{11} -1.00000 q^{13} +2.00000 q^{14} +1.00000 q^{16} -4.00000 q^{17} +6.00000 q^{19} -1.00000 q^{20} +4.00000 q^{22} +1.00000 q^{25} -1.00000 q^{26} +2.00000 q^{28} -4.00000 q^{29} +6.00000 q^{31} +1.00000 q^{32} -4.00000 q^{34} -2.00000 q^{35} -2.00000 q^{37} +6.00000 q^{38} -1.00000 q^{40} +10.0000 q^{41} +8.00000 q^{43} +4.00000 q^{44} -3.00000 q^{49} +1.00000 q^{50} -1.00000 q^{52} -4.00000 q^{53} -4.00000 q^{55} +2.00000 q^{56} -4.00000 q^{58} +4.00000 q^{59} +2.00000 q^{61} +6.00000 q^{62} +1.00000 q^{64} +1.00000 q^{65} +6.00000 q^{67} -4.00000 q^{68} -2.00000 q^{70} -8.00000 q^{71} +10.0000 q^{73} -2.00000 q^{74} +6.00000 q^{76} +8.00000 q^{77} -4.00000 q^{79} -1.00000 q^{80} +10.0000 q^{82} -12.0000 q^{83} +4.00000 q^{85} +8.00000 q^{86} +4.00000 q^{88} -2.00000 q^{89} -2.00000 q^{91} -6.00000 q^{95} +2.00000 q^{97} -3.00000 q^{98} +O(q^{100})$

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
$5$	−1.00000	−0.447214
$6$	0	0
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−1.00000	−0.316228
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	2.00000	0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	0	0
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	4.00000	0.852803
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−1.00000	−0.196116
$27$	0	0
$28$	2.00000	0.377964
$29$	−4.00000	−0.742781	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000	−0.742781	−0.371391	+	0.928477i	$0.621119\pi$
$30$	0	0
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−4.00000	−0.685994
$35$	−2.00000	−0.338062
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	6.00000	0.973329
$39$	0	0
$40$	−1.00000	−0.158114
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	4.00000	0.603023
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	1.00000	0.141421
$51$	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
$55$	−4.00000	−0.539360
$56$	2.00000	0.267261
$57$	0	0
$58$	−4.00000	−0.525226
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	6.00000	0.762001
$63$	0	0
$64$	1.00000	0.125000
$65$	1.00000	0.124035
$66$	0	0
$67$	6.00000	0.733017	0.366508	−	0.930415i	$-0.380553\pi$
$67$	6.00000	0.733017	0.366508	+	0.930415i	$0.380553\pi$
$68$	−4.00000	−0.485071
$69$	0	0
$70$	−2.00000	−0.239046
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	0	0
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	−2.00000	−0.232495
$75$	0	0
$76$	6.00000	0.688247
$77$	8.00000	0.911685
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	10.0000	1.10432
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	0	0
$85$	4.00000	0.433861
$86$	8.00000	0.862662
$87$	0	0
$88$	4.00000	0.426401
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−6.00000	−0.615587
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	−3.00000	−0.303046
$99$	0	0
$100$	1.00000	0.100000
$101$	−4.00000	−0.398015	−0.199007	−	0.979998i	$-0.563772\pi$
$101$	−4.00000	−0.398015	−0.199007	+	0.979998i	$0.563772\pi$
$102$	0	0
$103$	12.0000	1.18240	0.591198	−	0.806527i	$-0.298655\pi$
$103$	12.0000	1.18240	0.591198	+	0.806527i	$0.298655\pi$
$104$	−1.00000	−0.0980581
$105$	0	0
$106$	−4.00000	−0.388514
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	0	0
$109$	−18.0000	−1.72409	−0.862044	−	0.506834i	$-0.830816\pi$
$109$	−18.0000	−1.72409	−0.862044	+	0.506834i	$0.830816\pi$
$110$	−4.00000	−0.381385
$111$	0	0
$112$	2.00000	0.188982
$113$	−16.0000	−1.50515	−0.752577	−	0.658505i	$-0.771189\pi$
$113$	−16.0000	−1.50515	−0.752577	+	0.658505i	$0.771189\pi$
$114$	0	0
$115$	0	0
$116$	−4.00000	−0.371391
$117$	0	0
$118$	4.00000	0.368230
$119$	−8.00000	−0.733359
$120$	0	0
$121$	5.00000	0.454545
$122$	2.00000	0.181071
$123$	0	0
$124$	6.00000	0.538816
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	1.00000	0.0877058
$131$	16.0000	1.39793	0.698963	−	0.715158i	$-0.253645\pi$
$131$	16.0000	1.39793	0.698963	+	0.715158i	$0.253645\pi$
$132$	0	0
$133$	12.0000	1.04053
$134$	6.00000	0.518321
$135$	0	0
$136$	−4.00000	−0.342997
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	−20.0000	−1.69638	−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000	−1.69638	−0.848189	+	0.529694i	$0.822307\pi$
$140$	−2.00000	−0.169031
$141$	0	0
$142$	−8.00000	−0.671345
$143$	−4.00000	−0.334497
$144$	0	0
$145$	4.00000	0.332182
$146$	10.0000	0.827606
$147$	0	0
$148$	−2.00000	−0.164399
$149$	14.0000	1.14692	0.573462	−	0.819232i	$-0.305600\pi$
$149$	14.0000	1.14692	0.573462	+	0.819232i	$0.305600\pi$
$150$	0	0
$151$	−6.00000	−0.488273	−0.244137	−	0.969741i	$-0.578505\pi$
$151$	−6.00000	−0.488273	−0.244137	+	0.969741i	$0.578505\pi$
$152$	6.00000	0.486664
$153$	0	0
$154$	8.00000	0.644658
$155$	−6.00000	−0.481932
$156$	0	0
$157$	−14.0000	−1.11732	−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000	−1.11732	−0.558661	+	0.829396i	$0.688685\pi$
$158$	−4.00000	−0.318223
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	0	0
$162$	0	0
$163$	−10.0000	−0.783260	−0.391630	−	0.920123i	$-0.628089\pi$
$163$	−10.0000	−0.783260	−0.391630	+	0.920123i	$0.628089\pi$
$164$	10.0000	0.780869
$165$	0	0
$166$	−12.0000	−0.931381
$167$	−16.0000	−1.23812	−0.619059	−	0.785345i	$-0.712486\pi$
$167$	−16.0000	−1.23812	−0.619059	+	0.785345i	$0.712486\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	4.00000	0.306786
$171$	0	0
$172$	8.00000	0.609994
$173$	−8.00000	−0.608229	−0.304114	−	0.952636i	$-0.598361\pi$
$173$	−8.00000	−0.608229	−0.304114	+	0.952636i	$0.598361\pi$
$174$	0	0
$175$	2.00000	0.151186
$176$	4.00000	0.301511
$177$	0	0
$178$	−2.00000	−0.149906
$179$	−16.0000	−1.19590	−0.597948	−	0.801535i	$-0.704017\pi$
$179$	−16.0000	−1.19590	−0.597948	+	0.801535i	$0.704017\pi$
$180$	0	0
$181$	22.0000	1.63525	0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000	1.63525	0.817624	+	0.575753i	$0.195291\pi$
$182$	−2.00000	−0.148250
$183$	0	0
$184$	0	0
$185$	2.00000	0.147043
$186$	0	0
$187$	−16.0000	−1.17004
$188$	0	0
$189$	0	0
$190$	−6.00000	−0.435286
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	0	0
$193$	−10.0000	−0.719816	−0.359908	−	0.932988i	$-0.617192\pi$
$193$	−10.0000	−0.719816	−0.359908	+	0.932988i	$0.617192\pi$
$194$	2.00000	0.143592
$195$	0	0
$196$	−3.00000	−0.214286
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	0	0
$199$	24.0000	1.70131	0.850657	−	0.525720i	$-0.176204\pi$
$199$	24.0000	1.70131	0.850657	+	0.525720i	$0.176204\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	−4.00000	−0.281439
$203$	−8.00000	−0.561490
$204$	0	0
$205$	−10.0000	−0.698430
$206$	12.0000	0.836080
$207$	0	0
$208$	−1.00000	−0.0693375
$209$	24.0000	1.66011
$210$	0	0
$211$	16.0000	1.10149	0.550743	−	0.834675i	$-0.314345\pi$
$211$	16.0000	1.10149	0.550743	+	0.834675i	$0.314345\pi$
$212$	−4.00000	−0.274721
$213$	0	0
$214$	−12.0000	−0.820303
$215$	−8.00000	−0.545595
$216$	0	0
$217$	12.0000	0.814613
$218$	−18.0000	−1.21911
$219$	0	0
$220$	−4.00000	−0.269680
$221$	4.00000	0.269069
$222$	0	0
$223$	−14.0000	−0.937509	−0.468755	−	0.883328i	$-0.655297\pi$
$223$	−14.0000	−0.937509	−0.468755	+	0.883328i	$0.655297\pi$
$224$	2.00000	0.133631
$225$	0	0
$226$	−16.0000	−1.06430
$227$	20.0000	1.32745	0.663723	−	0.747978i	$-0.268975\pi$
$227$	20.0000	1.32745	0.663723	+	0.747978i	$0.268975\pi$
$228$	0	0
$229$	−26.0000	−1.71813	−0.859064	−	0.511868i	$-0.828954\pi$
$229$	−26.0000	−1.71813	−0.859064	+	0.511868i	$0.828954\pi$
$230$	0	0
$231$	0	0
$232$	−4.00000	−0.262613
$233$	−24.0000	−1.57229	−0.786146	−	0.618041i	$-0.787927\pi$
$233$	−24.0000	−1.57229	−0.786146	+	0.618041i	$0.787927\pi$
$234$	0	0
$235$	0	0
$236$	4.00000	0.260378
$237$	0	0
$238$	−8.00000	−0.518563
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	5.00000	0.321412
$243$	0	0
$244$	2.00000	0.128037
$245$	3.00000	0.191663
$246$	0	0
$247$	−6.00000	−0.381771
$248$	6.00000	0.381000
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	0	0
$254$	−8.00000	−0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	24.0000	1.49708	0.748539	−	0.663090i	$-0.230755\pi$
$257$	24.0000	1.49708	0.748539	+	0.663090i	$0.230755\pi$
$258$	0	0
$259$	−4.00000	−0.248548
$260$	1.00000	0.0620174
$261$	0	0
$262$	16.0000	0.988483
$263$	−4.00000	−0.246651	−0.123325	−	0.992366i	$-0.539356\pi$
$263$	−4.00000	−0.246651	−0.123325	+	0.992366i	$0.539356\pi$
$264$	0	0
$265$	4.00000	0.245718
$266$	12.0000	0.735767
$267$	0	0
$268$	6.00000	0.366508
$269$	12.0000	0.731653	0.365826	−	0.930683i	$-0.380786\pi$
$269$	12.0000	0.731653	0.365826	+	0.930683i	$0.380786\pi$
$270$	0	0
$271$	−2.00000	−0.121491	−0.0607457	−	0.998153i	$-0.519348\pi$
$271$	−2.00000	−0.121491	−0.0607457	+	0.998153i	$0.519348\pi$
$272$	−4.00000	−0.242536
$273$	0	0
$274$	−6.00000	−0.362473
$275$	4.00000	0.241209
$276$	0	0
$277$	−22.0000	−1.32185	−0.660926	−	0.750451i	$-0.729836\pi$
$277$	−22.0000	−1.32185	−0.660926	+	0.750451i	$0.729836\pi$
$278$	−20.0000	−1.19952
$279$	0	0
$280$	−2.00000	−0.119523
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	0	0
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	−8.00000	−0.474713
$285$	0	0
$286$	−4.00000	−0.236525
$287$	20.0000	1.18056
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	4.00000	0.234888
$291$	0	0
$292$	10.0000	0.585206
$293$	−2.00000	−0.116841	−0.0584206	−	0.998292i	$-0.518606\pi$
$293$	−2.00000	−0.116841	−0.0584206	+	0.998292i	$0.518606\pi$
$294$	0	0
$295$	−4.00000	−0.232889
$296$	−2.00000	−0.116248
$297$	0	0
$298$	14.0000	0.810998
$299$	0	0
$300$	0	0
$301$	16.0000	0.922225
$302$	−6.00000	−0.345261
$303$	0	0
$304$	6.00000	0.344124
$305$	−2.00000	−0.114520
$306$	0	0
$307$	−6.00000	−0.342438	−0.171219	−	0.985233i	$-0.554771\pi$
$307$	−6.00000	−0.342438	−0.171219	+	0.985233i	$0.554771\pi$
$308$	8.00000	0.455842
$309$	0	0
$310$	−6.00000	−0.340777
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	0	0
$313$	−30.0000	−1.69570	−0.847850	−	0.530236i	$-0.822103\pi$
$313$	−30.0000	−1.69570	−0.847850	+	0.530236i	$0.822103\pi$
$314$	−14.0000	−0.790066
$315$	0	0
$316$	−4.00000	−0.225018
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	0	0
$319$	−16.0000	−0.895828
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	0	0
$323$	−24.0000	−1.33540
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	−10.0000	−0.553849
$327$	0	0
$328$	10.0000	0.552158
$329$	0	0
$330$	0	0
$331$	−14.0000	−0.769510	−0.384755	−	0.923019i	$-0.625714\pi$
$331$	−14.0000	−0.769510	−0.384755	+	0.923019i	$0.625714\pi$
$332$	−12.0000	−0.658586
$333$	0	0
$334$	−16.0000	−0.875481
$335$	−6.00000	−0.327815
$336$	0	0
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	1.00000	0.0543928
$339$	0	0
$340$	4.00000	0.216930
$341$	24.0000	1.29967
$342$	0	0
$343$	−20.0000	−1.07990
$344$	8.00000	0.431331
$345$	0	0
$346$	−8.00000	−0.430083
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	−22.0000	−1.17763	−0.588817	−	0.808267i	$-0.700406\pi$
$349$	−22.0000	−1.17763	−0.588817	+	0.808267i	$0.700406\pi$
$350$	2.00000	0.106904
$351$	0	0
$352$	4.00000	0.213201
$353$	10.0000	0.532246	0.266123	−	0.963939i	$-0.414257\pi$
$353$	10.0000	0.532246	0.266123	+	0.963939i	$0.414257\pi$
$354$	0	0
$355$	8.00000	0.424596
$356$	−2.00000	−0.106000
$357$	0	0
$358$	−16.0000	−0.845626
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	22.0000	1.15629
$363$	0	0
$364$	−2.00000	−0.104828
$365$	−10.0000	−0.523424
$366$	0	0
$367$	−24.0000	−1.25279	−0.626395	−	0.779506i	$-0.715470\pi$
$367$	−24.0000	−1.25279	−0.626395	+	0.779506i	$0.715470\pi$
$368$	0	0
$369$	0	0
$370$	2.00000	0.103975
$371$	−8.00000	−0.415339
$372$	0	0
$373$	22.0000	1.13912	0.569558	−	0.821951i	$-0.307114\pi$
$373$	22.0000	1.13912	0.569558	+	0.821951i	$0.307114\pi$
$374$	−16.0000	−0.827340
$375$	0	0
$376$	0	0
$377$	4.00000	0.206010
$378$	0	0
$379$	30.0000	1.54100	0.770498	−	0.637442i	$-0.220007\pi$
$379$	30.0000	1.54100	0.770498	+	0.637442i	$0.220007\pi$
$380$	−6.00000	−0.307794
$381$	0	0
$382$	−12.0000	−0.613973
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	−8.00000	−0.407718
$386$	−10.0000	−0.508987
$387$	0	0
$388$	2.00000	0.101535
$389$	8.00000	0.405616	0.202808	−	0.979219i	$-0.434993\pi$
$389$	8.00000	0.405616	0.202808	+	0.979219i	$0.434993\pi$
$390$	0	0
$391$	0	0
$392$	−3.00000	−0.151523
$393$	0	0
$394$	18.0000	0.906827
$395$	4.00000	0.201262
$396$	0	0
$397$	14.0000	0.702640	0.351320	−	0.936255i	$-0.385733\pi$
$397$	14.0000	0.702640	0.351320	+	0.936255i	$0.385733\pi$
$398$	24.0000	1.20301
$399$	0	0
$400$	1.00000	0.0500000
$401$	6.00000	0.299626	0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000	0.299626	0.149813	+	0.988714i	$0.452133\pi$
$402$	0	0
$403$	−6.00000	−0.298881
$404$	−4.00000	−0.199007
$405$	0	0
$406$	−8.00000	−0.397033
$407$	−8.00000	−0.396545
$408$	0	0
$409$	−2.00000	−0.0988936	−0.0494468	−	0.998777i	$-0.515746\pi$
$409$	−2.00000	−0.0988936	−0.0494468	+	0.998777i	$0.515746\pi$
$410$	−10.0000	−0.493865
$411$	0	0
$412$	12.0000	0.591198
$413$	8.00000	0.393654
$414$	0	0
$415$	12.0000	0.589057
$416$	−1.00000	−0.0490290
$417$	0	0
$418$	24.0000	1.17388
$419$	16.0000	0.781651	0.390826	−	0.920465i	$-0.372190\pi$
$419$	16.0000	0.781651	0.390826	+	0.920465i	$0.372190\pi$
$420$	0	0
$421$	−26.0000	−1.26716	−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000	−1.26716	−0.633581	+	0.773676i	$0.718416\pi$
$422$	16.0000	0.778868
$423$	0	0
$424$	−4.00000	−0.194257
$425$	−4.00000	−0.194029
$426$	0	0
$427$	4.00000	0.193574
$428$	−12.0000	−0.580042
$429$	0	0
$430$	−8.00000	−0.385794
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	−2.00000	−0.0961139	−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000	−0.0961139	−0.0480569	+	0.998845i	$0.515303\pi$
$434$	12.0000	0.576018
$435$	0	0
$436$	−18.0000	−0.862044
$437$	0	0
$438$	0	0
$439$	28.0000	1.33637	0.668184	−	0.743996i	$-0.267072\pi$
$439$	28.0000	1.33637	0.668184	+	0.743996i	$0.267072\pi$
$440$	−4.00000	−0.190693
$441$	0	0
$442$	4.00000	0.190261
$443$	4.00000	0.190046	0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000	0.190046	0.0950229	+	0.995475i	$0.469708\pi$
$444$	0	0
$445$	2.00000	0.0948091
$446$	−14.0000	−0.662919
$447$	0	0
$448$	2.00000	0.0944911
$449$	−18.0000	−0.849473	−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000	−0.849473	−0.424736	+	0.905317i	$0.639633\pi$
$450$	0	0
$451$	40.0000	1.88353
$452$	−16.0000	−0.752577
$453$	0	0
$454$	20.0000	0.938647
$455$	2.00000	0.0937614
$456$	0	0
$457$	26.0000	1.21623	0.608114	−	0.793849i	$-0.291926\pi$
$457$	26.0000	1.21623	0.608114	+	0.793849i	$0.291926\pi$
$458$	−26.0000	−1.21490
$459$	0	0
$460$	0	0
$461$	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
$463$	−14.0000	−0.650635	−0.325318	−	0.945605i	$-0.605471\pi$
$463$	−14.0000	−0.650635	−0.325318	+	0.945605i	$0.605471\pi$
$464$	−4.00000	−0.185695
$465$	0	0
$466$	−24.0000	−1.11178
$467$	−8.00000	−0.370196	−0.185098	−	0.982720i	$-0.559260\pi$
$467$	−8.00000	−0.370196	−0.185098	+	0.982720i	$0.559260\pi$
$468$	0	0
$469$	12.0000	0.554109
$470$	0	0
$471$	0	0
$472$	4.00000	0.184115
$473$	32.0000	1.47136
$474$	0	0
$475$	6.00000	0.275299
$476$	−8.00000	−0.366679
$477$	0	0
$478$	−8.00000	−0.365911
$479$	−16.0000	−0.731059	−0.365529	−	0.930800i	$-0.619112\pi$
$479$	−16.0000	−0.731059	−0.365529	+	0.930800i	$0.619112\pi$
$480$	0	0
$481$	2.00000	0.0911922
$482$	−10.0000	−0.455488
$483$	0	0
$484$	5.00000	0.227273
$485$	−2.00000	−0.0908153
$486$	0	0
$487$	−22.0000	−0.996915	−0.498458	−	0.866914i	$-0.666100\pi$
$487$	−22.0000	−0.996915	−0.498458	+	0.866914i	$0.666100\pi$
$488$	2.00000	0.0905357
$489$	0	0
$490$	3.00000	0.135526
$491$	36.0000	1.62466	0.812329	−	0.583200i	$-0.198200\pi$
$491$	36.0000	1.62466	0.812329	+	0.583200i	$0.198200\pi$
$492$	0	0
$493$	16.0000	0.720604
$494$	−6.00000	−0.269953
$495$	0	0
$496$	6.00000	0.269408
$497$	−16.0000	−0.717698
$498$	0	0
$499$	26.0000	1.16392	0.581960	−	0.813217i	$-0.302286\pi$
$499$	26.0000	1.16392	0.581960	+	0.813217i	$0.302286\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	12.0000	0.535586
$503$	−20.0000	−0.891756	−0.445878	−	0.895094i	$-0.647108\pi$
$503$	−20.0000	−0.891756	−0.445878	+	0.895094i	$0.647108\pi$
$504$	0	0
$505$	4.00000	0.177998
$506$	0	0
$507$	0	0
$508$	−8.00000	−0.354943
$509$	42.0000	1.86162	0.930809	−	0.365507i	$-0.119104\pi$
$509$	42.0000	1.86162	0.930809	+	0.365507i	$0.119104\pi$
$510$	0	0
$511$	20.0000	0.884748
$512$	1.00000	0.0441942
$513$	0	0
$514$	24.0000	1.05859
$515$	−12.0000	−0.528783
$516$	0	0
$517$	0	0
$518$	−4.00000	−0.175750
$519$	0	0
$520$	1.00000	0.0438529
$521$	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
$523$	36.0000	1.57417	0.787085	−	0.616844i	$-0.211589\pi$
$523$	36.0000	1.57417	0.787085	+	0.616844i	$0.211589\pi$
$524$	16.0000	0.698963
$525$	0	0
$526$	−4.00000	−0.174408
$527$	−24.0000	−1.04546
$528$	0	0
$529$	−23.0000	−1.00000
$530$	4.00000	0.173749
$531$	0	0
$532$	12.0000	0.520266
$533$	−10.0000	−0.433148
$534$	0	0
$535$	12.0000	0.518805
$536$	6.00000	0.259161
$537$	0	0
$538$	12.0000	0.517357
$539$	−12.0000	−0.516877
$540$	0	0
$541$	−2.00000	−0.0859867	−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000	−0.0859867	−0.0429934	+	0.999075i	$0.513689\pi$
$542$	−2.00000	−0.0859074
$543$	0	0
$544$	−4.00000	−0.171499
$545$	18.0000	0.771035
$546$	0	0
$547$	−20.0000	−0.855138	−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000	−0.855138	−0.427569	+	0.903983i	$0.640630\pi$
$548$	−6.00000	−0.256307
$549$	0	0
$550$	4.00000	0.170561
$551$	−24.0000	−1.02243
$552$	0	0
$553$	−8.00000	−0.340195
$554$	−22.0000	−0.934690
$555$	0	0
$556$	−20.0000	−0.848189
$557$	38.0000	1.61011	0.805056	−	0.593199i	$-0.202135\pi$
$557$	38.0000	1.61011	0.805056	+	0.593199i	$0.202135\pi$
$558$	0	0
$559$	−8.00000	−0.338364
$560$	−2.00000	−0.0845154
$561$	0	0
$562$	18.0000	0.759284
$563$	−36.0000	−1.51722	−0.758610	−	0.651546i	$-0.774121\pi$
$563$	−36.0000	−1.51722	−0.758610	+	0.651546i	$0.774121\pi$
$564$	0	0
$565$	16.0000	0.673125
$566$	−4.00000	−0.168133
$567$	0	0
$568$	−8.00000	−0.335673
$569$	−36.0000	−1.50920	−0.754599	−	0.656186i	$-0.772169\pi$
$569$	−36.0000	−1.50920	−0.754599	+	0.656186i	$0.772169\pi$
$570$	0	0
$571$	4.00000	0.167395	0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000	0.167395	0.0836974	+	0.996491i	$0.473327\pi$
$572$	−4.00000	−0.167248
$573$	0	0
$574$	20.0000	0.834784
$575$	0	0
$576$	0	0
$577$	−42.0000	−1.74848	−0.874241	−	0.485491i	$-0.838641\pi$
$577$	−42.0000	−1.74848	−0.874241	+	0.485491i	$0.838641\pi$
$578$	−1.00000	−0.0415945
$579$	0	0
$580$	4.00000	0.166091
$581$	−24.0000	−0.995688
$582$	0	0
$583$	−16.0000	−0.662652
$584$	10.0000	0.413803
$585$	0	0
$586$	−2.00000	−0.0826192
$587$	36.0000	1.48588	0.742940	−	0.669359i	$-0.233431\pi$
$587$	36.0000	1.48588	0.742940	+	0.669359i	$0.233431\pi$
$588$	0	0
$589$	36.0000	1.48335
$590$	−4.00000	−0.164677
$591$	0	0
$592$	−2.00000	−0.0821995
$593$	−2.00000	−0.0821302	−0.0410651	−	0.999156i	$-0.513075\pi$
$593$	−2.00000	−0.0821302	−0.0410651	+	0.999156i	$0.513075\pi$
$594$	0	0
$595$	8.00000	0.327968
$596$	14.0000	0.573462
$597$	0	0
$598$	0	0
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	0	0
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	16.0000	0.652111
$603$	0	0
$604$	−6.00000	−0.244137
$605$	−5.00000	−0.203279
$606$	0	0
$607$	8.00000	0.324710	0.162355	−	0.986732i	$-0.448091\pi$
$607$	8.00000	0.324710	0.162355	+	0.986732i	$0.448091\pi$
$608$	6.00000	0.243332
$609$	0	0
$610$	−2.00000	−0.0809776
$611$	0	0
$612$	0	0
$613$	38.0000	1.53481	0.767403	−	0.641165i	$-0.221549\pi$
$613$	38.0000	1.53481	0.767403	+	0.641165i	$0.221549\pi$
$614$	−6.00000	−0.242140
$615$	0	0
$616$	8.00000	0.322329
$617$	−30.0000	−1.20775	−0.603877	−	0.797077i	$-0.706378\pi$
$617$	−30.0000	−1.20775	−0.603877	+	0.797077i	$0.706378\pi$
$618$	0	0
$619$	−10.0000	−0.401934	−0.200967	−	0.979598i	$-0.564408\pi$
$619$	−10.0000	−0.401934	−0.200967	+	0.979598i	$0.564408\pi$
$620$	−6.00000	−0.240966
$621$	0	0
$622$	12.0000	0.481156
$623$	−4.00000	−0.160257
$624$	0	0
$625$	1.00000	0.0400000
$626$	−30.0000	−1.19904
$627$	0	0
$628$	−14.0000	−0.558661
$629$	8.00000	0.318981
$630$	0	0
$631$	−30.0000	−1.19428	−0.597141	−	0.802137i	$-0.703697\pi$
$631$	−30.0000	−1.19428	−0.597141	+	0.802137i	$0.703697\pi$
$632$	−4.00000	−0.159111
$633$	0	0
$634$	18.0000	0.714871
$635$	8.00000	0.317470
$636$	0	0
$637$	3.00000	0.118864
$638$	−16.0000	−0.633446
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	−24.0000	−0.947943	−0.473972	−	0.880540i	$-0.657180\pi$
$641$	−24.0000	−0.947943	−0.473972	+	0.880540i	$0.657180\pi$
$642$	0	0
$643$	−34.0000	−1.34083	−0.670415	−	0.741987i	$-0.733884\pi$
$643$	−34.0000	−1.34083	−0.670415	+	0.741987i	$0.733884\pi$
$644$	0	0
$645$	0	0
$646$	−24.0000	−0.944267
$647$	−12.0000	−0.471769	−0.235884	−	0.971781i	$-0.575799\pi$
$647$	−12.0000	−0.471769	−0.235884	+	0.971781i	$0.575799\pi$
$648$	0	0
$649$	16.0000	0.628055
$650$	−1.00000	−0.0392232
$651$	0	0
$652$	−10.0000	−0.391630
$653$	48.0000	1.87839	0.939193	−	0.343391i	$-0.111576\pi$
$653$	48.0000	1.87839	0.939193	+	0.343391i	$0.111576\pi$
$654$	0	0
$655$	−16.0000	−0.625172
$656$	10.0000	0.390434
$657$	0	0
$658$	0	0
$659$	20.0000	0.779089	0.389545	−	0.921008i	$-0.372632\pi$
$659$	20.0000	0.779089	0.389545	+	0.921008i	$0.372632\pi$
$660$	0	0
$661$	42.0000	1.63361	0.816805	−	0.576913i	$-0.195743\pi$
$661$	42.0000	1.63361	0.816805	+	0.576913i	$0.195743\pi$
$662$	−14.0000	−0.544125
$663$	0	0
$664$	−12.0000	−0.465690
$665$	−12.0000	−0.465340
$666$	0	0
$667$	0	0
$668$	−16.0000	−0.619059
$669$	0	0
$670$	−6.00000	−0.231800
$671$	8.00000	0.308837
$672$	0	0
$673$	14.0000	0.539660	0.269830	−	0.962908i	$-0.413032\pi$
$673$	14.0000	0.539660	0.269830	+	0.962908i	$0.413032\pi$
$674$	14.0000	0.539260
$675$	0	0
$676$	1.00000	0.0384615
$677$	−36.0000	−1.38359	−0.691796	−	0.722093i	$-0.743180\pi$
$677$	−36.0000	−1.38359	−0.691796	+	0.722093i	$0.743180\pi$
$678$	0	0
$679$	4.00000	0.153506
$680$	4.00000	0.153393
$681$	0	0
$682$	24.0000	0.919007
$683$	28.0000	1.07139	0.535695	−	0.844411i	$-0.320050\pi$
$683$	28.0000	1.07139	0.535695	+	0.844411i	$0.320050\pi$
$684$	0	0
$685$	6.00000	0.229248
$686$	−20.0000	−0.763604
$687$	0	0
$688$	8.00000	0.304997
$689$	4.00000	0.152388
$690$	0	0
$691$	34.0000	1.29342	0.646710	−	0.762736i	$-0.276144\pi$
$691$	34.0000	1.29342	0.646710	+	0.762736i	$0.276144\pi$
$692$	−8.00000	−0.304114
$693$	0	0
$694$	0	0
$695$	20.0000	0.758643
$696$	0	0
$697$	−40.0000	−1.51511
$698$	−22.0000	−0.832712
$699$	0	0
$700$	2.00000	0.0755929
$701$	8.00000	0.302156	0.151078	−	0.988522i	$-0.451726\pi$
$701$	8.00000	0.302156	0.151078	+	0.988522i	$0.451726\pi$
$702$	0	0
$703$	−12.0000	−0.452589
$704$	4.00000	0.150756
$705$	0	0
$706$	10.0000	0.376355
$707$	−8.00000	−0.300871
$708$	0	0
$709$	−34.0000	−1.27690	−0.638448	−	0.769665i	$-0.720423\pi$
$709$	−34.0000	−1.27690	−0.638448	+	0.769665i	$0.720423\pi$
$710$	8.00000	0.300235
$711$	0	0
$712$	−2.00000	−0.0749532
$713$	0	0
$714$	0	0
$715$	4.00000	0.149592
$716$	−16.0000	−0.597948
$717$	0	0
$718$	−24.0000	−0.895672
$719$	−36.0000	−1.34257	−0.671287	−	0.741198i	$-0.734258\pi$
$719$	−36.0000	−1.34257	−0.671287	+	0.741198i	$0.734258\pi$
$720$	0	0
$721$	24.0000	0.893807
$722$	17.0000	0.632674
$723$	0	0
$724$	22.0000	0.817624
$725$	−4.00000	−0.148556
$726$	0	0
$727$	40.0000	1.48352	0.741759	−	0.670667i	$-0.233992\pi$
$727$	40.0000	1.48352	0.741759	+	0.670667i	$0.233992\pi$
$728$	−2.00000	−0.0741249
$729$	0	0
$730$	−10.0000	−0.370117
$731$	−32.0000	−1.18356
$732$	0	0
$733$	2.00000	0.0738717	0.0369358	−	0.999318i	$-0.488240\pi$
$733$	2.00000	0.0738717	0.0369358	+	0.999318i	$0.488240\pi$
$734$	−24.0000	−0.885856
$735$	0	0
$736$	0	0
$737$	24.0000	0.884051
$738$	0	0
$739$	30.0000	1.10357	0.551784	−	0.833987i	$-0.313947\pi$
$739$	30.0000	1.10357	0.551784	+	0.833987i	$0.313947\pi$
$740$	2.00000	0.0735215
$741$	0	0
$742$	−8.00000	−0.293689
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	−14.0000	−0.512920
$746$	22.0000	0.805477
$747$	0	0
$748$	−16.0000	−0.585018
$749$	−24.0000	−0.876941
$750$	0	0
$751$	20.0000	0.729810	0.364905	−	0.931045i	$-0.381101\pi$
$751$	20.0000	0.729810	0.364905	+	0.931045i	$0.381101\pi$
$752$	0	0
$753$	0	0
$754$	4.00000	0.145671
$755$	6.00000	0.218362
$756$	0	0
$757$	30.0000	1.09037	0.545184	−	0.838316i	$-0.316460\pi$
$757$	30.0000	1.09037	0.545184	+	0.838316i	$0.316460\pi$
$758$	30.0000	1.08965
$759$	0	0
$760$	−6.00000	−0.217643
$761$	34.0000	1.23250	0.616250	−	0.787551i	$-0.288651\pi$
$761$	34.0000	1.23250	0.616250	+	0.787551i	$0.288651\pi$
$762$	0	0
$763$	−36.0000	−1.30329
$764$	−12.0000	−0.434145
$765$	0	0
$766$	0	0
$767$	−4.00000	−0.144432
$768$	0	0
$769$	34.0000	1.22607	0.613036	−	0.790055i	$-0.289948\pi$
$769$	34.0000	1.22607	0.613036	+	0.790055i	$0.289948\pi$
$770$	−8.00000	−0.288300
$771$	0	0
$772$	−10.0000	−0.359908
$773$	14.0000	0.503545	0.251773	−	0.967786i	$-0.418987\pi$
$773$	14.0000	0.503545	0.251773	+	0.967786i	$0.418987\pi$
$774$	0	0
$775$	6.00000	0.215526
$776$	2.00000	0.0717958
$777$	0	0
$778$	8.00000	0.286814
$779$	60.0000	2.14972
$780$	0	0
$781$	−32.0000	−1.14505
$782$	0	0
$783$	0	0
$784$	−3.00000	−0.107143
$785$	14.0000	0.499681
$786$	0	0
$787$	−38.0000	−1.35455	−0.677277	−	0.735728i	$-0.736840\pi$
$787$	−38.0000	−1.35455	−0.677277	+	0.735728i	$0.736840\pi$
$788$	18.0000	0.641223
$789$	0	0
$790$	4.00000	0.142314
$791$	−32.0000	−1.13779
$792$	0	0
$793$	−2.00000	−0.0710221
$794$	14.0000	0.496841
$795$	0	0
$796$	24.0000	0.850657
$797$	−28.0000	−0.991811	−0.495905	−	0.868377i	$-0.665164\pi$
$797$	−28.0000	−0.991811	−0.495905	+	0.868377i	$0.665164\pi$
$798$	0	0
$799$	0	0
$800$	1.00000	0.0353553
$801$	0	0
$802$	6.00000	0.211867
$803$	40.0000	1.41157
$804$	0	0
$805$	0	0
$806$	−6.00000	−0.211341
$807$	0	0
$808$	−4.00000	−0.140720
$809$	44.0000	1.54696	0.773479	−	0.633822i	$-0.218515\pi$
$809$	44.0000	1.54696	0.773479	+	0.633822i	$0.218515\pi$
$810$	0	0
$811$	−2.00000	−0.0702295	−0.0351147	−	0.999383i	$-0.511180\pi$
$811$	−2.00000	−0.0702295	−0.0351147	+	0.999383i	$0.511180\pi$
$812$	−8.00000	−0.280745
$813$	0	0
$814$	−8.00000	−0.280400
$815$	10.0000	0.350285
$816$	0	0
$817$	48.0000	1.67931
$818$	−2.00000	−0.0699284
$819$	0	0
$820$	−10.0000	−0.349215
$821$	−10.0000	−0.349002	−0.174501	−	0.984657i	$-0.555831\pi$
$821$	−10.0000	−0.349002	−0.174501	+	0.984657i	$0.555831\pi$
$822$	0	0
$823$	44.0000	1.53374	0.766872	−	0.641800i	$-0.221812\pi$
$823$	44.0000	1.53374	0.766872	+	0.641800i	$0.221812\pi$
$824$	12.0000	0.418040
$825$	0	0
$826$	8.00000	0.278356
$827$	36.0000	1.25184	0.625921	−	0.779886i	$-0.284723\pi$
$827$	36.0000	1.25184	0.625921	+	0.779886i	$0.284723\pi$
$828$	0	0
$829$	−34.0000	−1.18087	−0.590434	−	0.807086i	$-0.701044\pi$
$829$	−34.0000	−1.18087	−0.590434	+	0.807086i	$0.701044\pi$
$830$	12.0000	0.416526
$831$	0	0
$832$	−1.00000	−0.0346688
$833$	12.0000	0.415775
$834$	0	0
$835$	16.0000	0.553703
$836$	24.0000	0.830057
$837$	0	0
$838$	16.0000	0.552711
$839$	32.0000	1.10476	0.552381	−	0.833592i	$-0.313719\pi$
$839$	32.0000	1.10476	0.552381	+	0.833592i	$0.313719\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	−26.0000	−0.896019
$843$	0	0
$844$	16.0000	0.550743
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	10.0000	0.343604
$848$	−4.00000	−0.137361
$849$	0	0
$850$	−4.00000	−0.137199
$851$	0	0
$852$	0	0
$853$	−2.00000	−0.0684787	−0.0342393	−	0.999414i	$-0.510901\pi$
$853$	−2.00000	−0.0684787	−0.0342393	+	0.999414i	$0.510901\pi$
$854$	4.00000	0.136877
$855$	0	0
$856$	−12.0000	−0.410152
$857$	−32.0000	−1.09310	−0.546550	−	0.837427i	$-0.684059\pi$
$857$	−32.0000	−1.09310	−0.546550	+	0.837427i	$0.684059\pi$
$858$	0	0
$859$	0	0		−	1.00000i	$-0.5\pi$
$859$	0	0			1.00000i	$0.5\pi$
$860$	−8.00000	−0.272798
$861$	0	0
$862$	0	0
$863$	−24.0000	−0.816970	−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000	−0.816970	−0.408485	+	0.912765i	$0.633943\pi$
$864$	0	0
$865$	8.00000	0.272008
$866$	−2.00000	−0.0679628
$867$	0	0
$868$	12.0000	0.407307
$869$	−16.0000	−0.542763
$870$	0	0
$871$	−6.00000	−0.203302
$872$	−18.0000	−0.609557
$873$	0	0
$874$	0	0
$875$	−2.00000	−0.0676123
$876$	0	0
$877$	22.0000	0.742887	0.371444	−	0.928456i	$-0.378863\pi$
$877$	22.0000	0.742887	0.371444	+	0.928456i	$0.378863\pi$
$878$	28.0000	0.944954
$879$	0	0
$880$	−4.00000	−0.134840
$881$	36.0000	1.21287	0.606435	−	0.795133i	$-0.292599\pi$
$881$	36.0000	1.21287	0.606435	+	0.795133i	$0.292599\pi$
$882$	0	0
$883$	−4.00000	−0.134611	−0.0673054	−	0.997732i	$-0.521440\pi$
$883$	−4.00000	−0.134611	−0.0673054	+	0.997732i	$0.521440\pi$
$884$	4.00000	0.134535
$885$	0	0
$886$	4.00000	0.134383
$887$	−8.00000	−0.268614	−0.134307	−	0.990940i	$-0.542881\pi$
$887$	−8.00000	−0.268614	−0.134307	+	0.990940i	$0.542881\pi$
$888$	0	0
$889$	−16.0000	−0.536623
$890$	2.00000	0.0670402
$891$	0	0
$892$	−14.0000	−0.468755
$893$	0	0
$894$	0	0
$895$	16.0000	0.534821
$896$	2.00000	0.0668153
$897$	0	0
$898$	−18.0000	−0.600668
$899$	−24.0000	−0.800445
$900$	0	0
$901$	16.0000	0.533037
$902$	40.0000	1.33185
$903$	0	0
$904$	−16.0000	−0.532152
$905$	−22.0000	−0.731305
$906$	0	0
$907$	−20.0000	−0.664089	−0.332045	−	0.943264i	$-0.607738\pi$
$907$	−20.0000	−0.664089	−0.332045	+	0.943264i	$0.607738\pi$
$908$	20.0000	0.663723
$909$	0	0
$910$	2.00000	0.0662994
$911$	−20.0000	−0.662630	−0.331315	−	0.943520i	$-0.607492\pi$
$911$	−20.0000	−0.662630	−0.331315	+	0.943520i	$0.607492\pi$
$912$	0	0
$913$	−48.0000	−1.58857
$914$	26.0000	0.860004
$915$	0	0
$916$	−26.0000	−0.859064
$917$	32.0000	1.05673
$918$	0	0
$919$	28.0000	0.923635	0.461817	−	0.886975i	$-0.347198\pi$
$919$	28.0000	0.923635	0.461817	+	0.886975i	$0.347198\pi$
$920$	0	0
$921$	0	0
$922$	18.0000	0.592798
$923$	8.00000	0.263323
$924$	0	0
$925$	−2.00000	−0.0657596
$926$	−14.0000	−0.460069
$927$	0	0
$928$	−4.00000	−0.131306
$929$	−6.00000	−0.196854	−0.0984268	−	0.995144i	$-0.531381\pi$
$929$	−6.00000	−0.196854	−0.0984268	+	0.995144i	$0.531381\pi$
$930$	0	0
$931$	−18.0000	−0.589926
$932$	−24.0000	−0.786146
$933$	0	0
$934$	−8.00000	−0.261768
$935$	16.0000	0.523256
$936$	0	0
$937$	−42.0000	−1.37208	−0.686040	−	0.727564i	$-0.740653\pi$
$937$	−42.0000	−1.37208	−0.686040	+	0.727564i	$0.740653\pi$
$938$	12.0000	0.391814
$939$	0	0
$940$	0	0
$941$	−58.0000	−1.89075	−0.945373	−	0.325991i	$-0.894302\pi$
$941$	−58.0000	−1.89075	−0.945373	+	0.325991i	$0.894302\pi$
$942$	0	0
$943$	0	0
$944$	4.00000	0.130189
$945$	0	0
$946$	32.0000	1.04041
$947$	−44.0000	−1.42981	−0.714904	−	0.699223i	$-0.753530\pi$
$947$	−44.0000	−1.42981	−0.714904	+	0.699223i	$0.753530\pi$
$948$	0	0
$949$	−10.0000	−0.324614
$950$	6.00000	0.194666
$951$	0	0
$952$	−8.00000	−0.259281
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	12.0000	0.388311
$956$	−8.00000	−0.258738
$957$	0	0
$958$	−16.0000	−0.516937
$959$	−12.0000	−0.387500
$960$	0	0
$961$	5.00000	0.161290
$962$	2.00000	0.0644826
$963$	0	0
$964$	−10.0000	−0.322078
$965$	10.0000	0.321911
$966$	0	0
$967$	−38.0000	−1.22200	−0.610999	−	0.791632i	$-0.709232\pi$
$967$	−38.0000	−1.22200	−0.610999	+	0.791632i	$0.709232\pi$
$968$	5.00000	0.160706
$969$	0	0
$970$	−2.00000	−0.0642161
$971$	−4.00000	−0.128366	−0.0641831	−	0.997938i	$-0.520444\pi$
$971$	−4.00000	−0.128366	−0.0641831	+	0.997938i	$0.520444\pi$
$972$	0	0
$973$	−40.0000	−1.28234
$974$	−22.0000	−0.704925
$975$	0	0
$976$	2.00000	0.0640184
$977$	6.00000	0.191957	0.0959785	−	0.995383i	$-0.469402\pi$
$977$	6.00000	0.191957	0.0959785	+	0.995383i	$0.469402\pi$
$978$	0	0
$979$	−8.00000	−0.255681
$980$	3.00000	0.0958315
$981$	0	0
$982$	36.0000	1.14881
$983$	8.00000	0.255160	0.127580	−	0.991828i	$-0.459279\pi$
$983$	8.00000	0.255160	0.127580	+	0.991828i	$0.459279\pi$
$984$	0	0
$985$	−18.0000	−0.573528
$986$	16.0000	0.509544
$987$	0	0
$988$	−6.00000	−0.190885
$989$	0	0
$990$	0	0
$991$	−48.0000	−1.52477	−0.762385	−	0.647124i	$-0.775972\pi$
$991$	−48.0000	−1.52477	−0.762385	+	0.647124i	$0.775972\pi$
$992$	6.00000	0.190500
$993$	0	0
$994$	−16.0000	−0.507489
$995$	−24.0000	−0.760851
$996$	0	0
$997$	−26.0000	−0.823428	−0.411714	−	0.911313i	$-0.635070\pi$
$997$	−26.0000	−0.823428	−0.411714	+	0.911313i	$0.635070\pi$
$998$	26.0000	0.823016
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1170.2.a.l.1.1	yes	1
3.2	odd	2		1170.2.a.f.1.1	✓	1
4.3	odd	2		9360.2.a.f.1.1		1
5.2	odd	4		5850.2.e.bc.5149.2		2
5.3	odd	4		5850.2.e.bc.5149.1		2
5.4	even	2		5850.2.a.h.1.1		1
12.11	even	2		9360.2.a.bj.1.1		1
15.2	even	4		5850.2.e.g.5149.1		2
15.8	even	4		5850.2.e.g.5149.2		2
15.14	odd	2		5850.2.a.bh.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1170.2.a.f.1.1	✓	1	3.2	odd	2
1170.2.a.l.1.1	yes	1	1.1	even	1	trivial
5850.2.a.h.1.1		1	5.4	even	2
5850.2.a.bh.1.1		1	15.14	odd	2
5850.2.e.g.5149.1		2	15.2	even	4
5850.2.e.g.5149.2		2	15.8	even	4
5850.2.e.bc.5149.1		2	5.3	odd	4
5850.2.e.bc.5149.2		2	5.2	odd	4
9360.2.a.f.1.1		1	4.3	odd	2
9360.2.a.bj.1.1		1	12.11	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:	\( N \)	\(=\)	\( 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1170.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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