LMFDB - Embedded newform 6050.2.a.bl.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6050,2,Mod(1,6050)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6050, 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("6050.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6050 = 2 \cdot 5^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6050.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:	$48.3094932229$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 242)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	6050.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} +2.00000 q^{6} -2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} +2.00000 q^{6} -2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{12} -5.00000 q^{13} -2.00000 q^{14} +1.00000 q^{16} -3.00000 q^{17} +1.00000 q^{18} +2.00000 q^{19} -4.00000 q^{21} -6.00000 q^{23} +2.00000 q^{24} -5.00000 q^{26} -4.00000 q^{27} -2.00000 q^{28} -3.00000 q^{29} +2.00000 q^{31} +1.00000 q^{32} -3.00000 q^{34} +1.00000 q^{36} +7.00000 q^{37} +2.00000 q^{38} -10.0000 q^{39} +3.00000 q^{41} -4.00000 q^{42} -8.00000 q^{43} -6.00000 q^{46} -6.00000 q^{47} +2.00000 q^{48} -3.00000 q^{49} -6.00000 q^{51} -5.00000 q^{52} +3.00000 q^{53} -4.00000 q^{54} -2.00000 q^{56} +4.00000 q^{57} -3.00000 q^{58} -10.0000 q^{61} +2.00000 q^{62} -2.00000 q^{63} +1.00000 q^{64} +10.0000 q^{67} -3.00000 q^{68} -12.0000 q^{69} +12.0000 q^{71} +1.00000 q^{72} -14.0000 q^{73} +7.00000 q^{74} +2.00000 q^{76} -10.0000 q^{78} +2.00000 q^{79} -11.0000 q^{81} +3.00000 q^{82} -18.0000 q^{83} -4.00000 q^{84} -8.00000 q^{86} -6.00000 q^{87} -9.00000 q^{89} +10.0000 q^{91} -6.00000 q^{92} +4.00000 q^{93} -6.00000 q^{94} +2.00000 q^{96} -11.0000 q^{97} -3.00000 q^{98} +O(q^{100})\)

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
$3$	2.00000	1.15470	0.577350	−	0.816497i	$-0.304087\pi$
$3$	2.00000	1.15470	0.577350	+	0.816497i	$0.304087\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	2.00000	0.816497
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0
$11$	0	0
$12$	2.00000	0.577350
$13$	−5.00000	−1.38675	−0.693375	−	0.720577i	$-0.743877\pi$
$13$	−5.00000	−1.38675	−0.693375	+	0.720577i	$0.743877\pi$
$14$	−2.00000	−0.534522
$15$	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	$0.618522\pi$
$18$	1.00000	0.235702
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	−4.00000	−0.872872
$22$	0	0
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	2.00000	0.408248
$25$	0	0
$26$	−5.00000	−0.980581
$27$	−4.00000	−0.769800
$28$	−2.00000	−0.377964
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−3.00000	−0.514496
$35$	0	0
$36$	1.00000	0.166667
$37$	7.00000	1.15079	0.575396	−	0.817875i	$-0.304848\pi$
$37$	7.00000	1.15079	0.575396	+	0.817875i	$0.304848\pi$
$38$	2.00000	0.324443
$39$	−10.0000	−1.60128
$40$	0	0
$41$	3.00000	0.468521	0.234261	−	0.972174i	$-0.424733\pi$
$41$	3.00000	0.468521	0.234261	+	0.972174i	$0.424733\pi$
$42$	−4.00000	−0.617213
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	0	0
$45$	0	0
$46$	−6.00000	−0.884652
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	2.00000	0.288675
$49$	−3.00000	−0.428571
$50$	0	0
$51$	−6.00000	−0.840168
$52$	−5.00000	−0.693375
$53$	3.00000	0.412082	0.206041	−	0.978543i	$-0.433942\pi$
$53$	3.00000	0.412082	0.206041	+	0.978543i	$0.433942\pi$
$54$	−4.00000	−0.544331
$55$	0	0
$56$	−2.00000	−0.267261
$57$	4.00000	0.529813
$58$	−3.00000	−0.393919
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	2.00000	0.254000
$63$	−2.00000	−0.251976
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	10.0000	1.22169	0.610847	−	0.791748i	$-0.290829\pi$
$67$	10.0000	1.22169	0.610847	+	0.791748i	$0.290829\pi$
$68$	−3.00000	−0.363803
$69$	−12.0000	−1.44463
$70$	0	0
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	1.00000	0.117851
$73$	−14.0000	−1.63858	−0.819288	−	0.573382i	$-0.805631\pi$
$73$	−14.0000	−1.63858	−0.819288	+	0.573382i	$0.805631\pi$
$74$	7.00000	0.813733
$75$	0	0
$76$	2.00000	0.229416
$77$	0	0
$78$	−10.0000	−1.13228
$79$	2.00000	0.225018	0.112509	−	0.993651i	$-0.464111\pi$
$79$	2.00000	0.225018	0.112509	+	0.993651i	$0.464111\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	3.00000	0.331295
$83$	−18.0000	−1.97576	−0.987878	−	0.155230i	$-0.950388\pi$
$83$	−18.0000	−1.97576	−0.987878	+	0.155230i	$0.950388\pi$
$84$	−4.00000	−0.436436
$85$	0	0
$86$	−8.00000	−0.862662
$87$	−6.00000	−0.643268
$88$	0	0
$89$	−9.00000	−0.953998	−0.476999	−	0.878904i	$-0.658275\pi$
$89$	−9.00000	−0.953998	−0.476999	+	0.878904i	$0.658275\pi$
$90$	0	0
$91$	10.0000	1.04828
$92$	−6.00000	−0.625543
$93$	4.00000	0.414781
$94$	−6.00000	−0.618853
$95$	0	0
$96$	2.00000	0.204124
$97$	−11.0000	−1.11688	−0.558440	−	0.829545i	$-0.688600\pi$
$97$	−11.0000	−1.11688	−0.558440	+	0.829545i	$0.688600\pi$
$98$	−3.00000	−0.303046
$99$	0	0
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	−6.00000	−0.594089
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	−5.00000	−0.490290
$105$	0	0
$106$	3.00000	0.291386
$107$	6.00000	0.580042	0.290021	−	0.957020i	$-0.406338\pi$
$107$	6.00000	0.580042	0.290021	+	0.957020i	$0.406338\pi$
$108$	−4.00000	−0.384900
$109$	17.0000	1.62830	0.814152	−	0.580651i	$-0.197202\pi$
$109$	17.0000	1.62830	0.814152	+	0.580651i	$0.197202\pi$
$110$	0	0
$111$	14.0000	1.32882
$112$	−2.00000	−0.188982
$113$	9.00000	0.846649	0.423324	−	0.905978i	$-0.360863\pi$
$113$	9.00000	0.846649	0.423324	+	0.905978i	$0.360863\pi$
$114$	4.00000	0.374634
$115$	0	0
$116$	−3.00000	−0.278543
$117$	−5.00000	−0.462250
$118$	0	0
$119$	6.00000	0.550019
$120$	0	0
$121$	0	0
$122$	−10.0000	−0.905357
$123$	6.00000	0.541002
$124$	2.00000	0.179605
$125$	0	0
$126$	−2.00000	−0.178174
$127$	−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
$129$	−16.0000	−1.40872
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	−4.00000	−0.346844
$134$	10.0000	0.863868
$135$	0	0
$136$	−3.00000	−0.257248
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	−12.0000	−1.02151
$139$	−22.0000	−1.86602	−0.933008	−	0.359856i	$-0.882826\pi$
$139$	−22.0000	−1.86602	−0.933008	+	0.359856i	$0.882826\pi$
$140$	0	0
$141$	−12.0000	−1.01058
$142$	12.0000	1.00702
$143$	0	0
$144$	1.00000	0.0833333
$145$	0	0
$146$	−14.0000	−1.15865
$147$	−6.00000	−0.494872
$148$	7.00000	0.575396
$149$	−3.00000	−0.245770	−0.122885	−	0.992421i	$-0.539215\pi$
$149$	−3.00000	−0.245770	−0.122885	+	0.992421i	$0.539215\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	2.00000	0.162221
$153$	−3.00000	−0.242536
$154$	0	0
$155$	0	0
$156$	−10.0000	−0.800641
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	2.00000	0.159111
$159$	6.00000	0.475831
$160$	0	0
$161$	12.0000	0.945732
$162$	−11.0000	−0.864242
$163$	22.0000	1.72317	0.861586	−	0.507611i	$-0.169471\pi$
$163$	22.0000	1.72317	0.861586	+	0.507611i	$0.169471\pi$
$164$	3.00000	0.234261
$165$	0	0
$166$	−18.0000	−1.39707
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	−4.00000	−0.308607
$169$	12.0000	0.923077
$170$	0	0
$171$	2.00000	0.152944
$172$	−8.00000	−0.609994
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	−6.00000	−0.454859
$175$	0	0
$176$	0	0
$177$	0	0
$178$	−9.00000	−0.674579
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	5.00000	0.371647	0.185824	−	0.982583i	$-0.440505\pi$
$181$	5.00000	0.371647	0.185824	+	0.982583i	$0.440505\pi$
$182$	10.0000	0.741249
$183$	−20.0000	−1.47844
$184$	−6.00000	−0.442326
$185$	0	0
$186$	4.00000	0.293294
$187$	0	0
$188$	−6.00000	−0.437595
$189$	8.00000	0.581914
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	2.00000	0.144338
$193$	13.0000	0.935760	0.467880	−	0.883792i	$-0.345018\pi$
$193$	13.0000	0.935760	0.467880	+	0.883792i	$0.345018\pi$
$194$	−11.0000	−0.789754
$195$	0	0
$196$	−3.00000	−0.214286
$197$	15.0000	1.06871	0.534353	−	0.845262i	$-0.320555\pi$
$197$	15.0000	1.06871	0.534353	+	0.845262i	$0.320555\pi$
$198$	0	0
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	0	0
$201$	20.0000	1.41069
$202$	6.00000	0.422159
$203$	6.00000	0.421117
$204$	−6.00000	−0.420084
$205$	0	0
$206$	−8.00000	−0.557386
$207$	−6.00000	−0.417029
$208$	−5.00000	−0.346688
$209$	0	0
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	3.00000	0.206041
$213$	24.0000	1.64445
$214$	6.00000	0.410152
$215$	0	0
$216$	−4.00000	−0.272166
$217$	−4.00000	−0.271538
$218$	17.0000	1.15139
$219$	−28.0000	−1.89206
$220$	0	0
$221$	15.0000	1.00901
$222$	14.0000	0.939618
$223$	4.00000	0.267860	0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000	0.267860	0.133930	+	0.990991i	$0.457240\pi$
$224$	−2.00000	−0.133631
$225$	0	0
$226$	9.00000	0.598671
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	4.00000	0.264906
$229$	5.00000	0.330409	0.165205	−	0.986259i	$-0.447172\pi$
$229$	5.00000	0.330409	0.165205	+	0.986259i	$0.447172\pi$
$230$	0	0
$231$	0	0
$232$	−3.00000	−0.196960
$233$	21.0000	1.37576	0.687878	−	0.725826i	$-0.258542\pi$
$233$	21.0000	1.37576	0.687878	+	0.725826i	$0.258542\pi$
$234$	−5.00000	−0.326860
$235$	0	0
$236$	0	0
$237$	4.00000	0.259828
$238$	6.00000	0.388922
$239$	−30.0000	−1.94054	−0.970269	−	0.242028i	$-0.922188\pi$
$239$	−30.0000	−1.94054	−0.970269	+	0.242028i	$0.922188\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	0	0
$243$	−10.0000	−0.641500
$244$	−10.0000	−0.640184
$245$	0	0
$246$	6.00000	0.382546
$247$	−10.0000	−0.636285
$248$	2.00000	0.127000
$249$	−36.0000	−2.28141
$250$	0	0
$251$	18.0000	1.13615	0.568075	−	0.822977i	$-0.307688\pi$
$251$	18.0000	1.13615	0.568075	+	0.822977i	$0.307688\pi$
$252$	−2.00000	−0.125988
$253$	0	0
$254$	−8.00000	−0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	−3.00000	−0.187135	−0.0935674	−	0.995613i	$-0.529827\pi$
$257$	−3.00000	−0.187135	−0.0935674	+	0.995613i	$0.529827\pi$
$258$	−16.0000	−0.996116
$259$	−14.0000	−0.869918
$260$	0	0
$261$	−3.00000	−0.185695
$262$	0	0
$263$	−6.00000	−0.369976	−0.184988	−	0.982741i	$-0.559225\pi$
$263$	−6.00000	−0.369976	−0.184988	+	0.982741i	$0.559225\pi$
$264$	0	0
$265$	0	0
$266$	−4.00000	−0.245256
$267$	−18.0000	−1.10158
$268$	10.0000	0.610847
$269$	−3.00000	−0.182913	−0.0914566	−	0.995809i	$-0.529152\pi$
$269$	−3.00000	−0.182913	−0.0914566	+	0.995809i	$0.529152\pi$
$270$	0	0
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	−3.00000	−0.181902
$273$	20.0000	1.21046
$274$	−6.00000	−0.362473
$275$	0	0
$276$	−12.0000	−0.722315
$277$	−5.00000	−0.300421	−0.150210	−	0.988654i	$-0.547995\pi$
$277$	−5.00000	−0.300421	−0.150210	+	0.988654i	$0.547995\pi$
$278$	−22.0000	−1.31947
$279$	2.00000	0.119737
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	−12.0000	−0.714590
$283$	−20.0000	−1.18888	−0.594438	−	0.804141i	$-0.702626\pi$
$283$	−20.0000	−1.18888	−0.594438	+	0.804141i	$0.702626\pi$
$284$	12.0000	0.712069
$285$	0	0
$286$	0	0
$287$	−6.00000	−0.354169
$288$	1.00000	0.0589256
$289$	−8.00000	−0.470588
$290$	0	0
$291$	−22.0000	−1.28966
$292$	−14.0000	−0.819288
$293$	−21.0000	−1.22683	−0.613417	−	0.789760i	$-0.710205\pi$
$293$	−21.0000	−1.22683	−0.613417	+	0.789760i	$0.710205\pi$
$294$	−6.00000	−0.349927
$295$	0	0
$296$	7.00000	0.406867
$297$	0	0
$298$	−3.00000	−0.173785
$299$	30.0000	1.73494
$300$	0	0
$301$	16.0000	0.922225
$302$	8.00000	0.460348
$303$	12.0000	0.689382
$304$	2.00000	0.114708
$305$	0	0
$306$	−3.00000	−0.171499
$307$	10.0000	0.570730	0.285365	−	0.958419i	$-0.407885\pi$
$307$	10.0000	0.570730	0.285365	+	0.958419i	$0.407885\pi$
$308$	0	0
$309$	−16.0000	−0.910208
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	−10.0000	−0.566139
$313$	1.00000	0.0565233	0.0282617	−	0.999601i	$-0.491003\pi$
$313$	1.00000	0.0565233	0.0282617	+	0.999601i	$0.491003\pi$
$314$	−2.00000	−0.112867
$315$	0	0
$316$	2.00000	0.112509
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	6.00000	0.336463
$319$	0	0
$320$	0	0
$321$	12.0000	0.669775
$322$	12.0000	0.668734
$323$	−6.00000	−0.333849
$324$	−11.0000	−0.611111
$325$	0	0
$326$	22.0000	1.21847
$327$	34.0000	1.88020
$328$	3.00000	0.165647
$329$	12.0000	0.661581
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	−18.0000	−0.987878
$333$	7.00000	0.383598
$334$	12.0000	0.656611
$335$	0	0
$336$	−4.00000	−0.218218
$337$	13.0000	0.708155	0.354078	−	0.935216i	$-0.384795\pi$
$337$	13.0000	0.708155	0.354078	+	0.935216i	$0.384795\pi$
$338$	12.0000	0.652714
$339$	18.0000	0.977626
$340$	0	0
$341$	0	0
$342$	2.00000	0.108148
$343$	20.0000	1.07990
$344$	−8.00000	−0.431331
$345$	0	0
$346$	−6.00000	−0.322562
$347$	−36.0000	−1.93258	−0.966291	−	0.257454i	$-0.917117\pi$
$347$	−36.0000	−1.93258	−0.966291	+	0.257454i	$0.917117\pi$
$348$	−6.00000	−0.321634
$349$	17.0000	0.909989	0.454995	−	0.890494i	$-0.349641\pi$
$349$	17.0000	0.909989	0.454995	+	0.890494i	$0.349641\pi$
$350$	0	0
$351$	20.0000	1.06752
$352$	0	0
$353$	9.00000	0.479022	0.239511	−	0.970894i	$-0.423013\pi$
$353$	9.00000	0.479022	0.239511	+	0.970894i	$0.423013\pi$
$354$	0	0
$355$	0	0
$356$	−9.00000	−0.476999
$357$	12.0000	0.635107
$358$	0	0
$359$	18.0000	0.950004	0.475002	−	0.879985i	$-0.342447\pi$
$359$	18.0000	0.950004	0.475002	+	0.879985i	$0.342447\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	5.00000	0.262794
$363$	0	0
$364$	10.0000	0.524142
$365$	0	0
$366$	−20.0000	−1.04542
$367$	10.0000	0.521996	0.260998	−	0.965339i	$-0.415948\pi$
$367$	10.0000	0.521996	0.260998	+	0.965339i	$0.415948\pi$
$368$	−6.00000	−0.312772
$369$	3.00000	0.156174
$370$	0	0
$371$	−6.00000	−0.311504
$372$	4.00000	0.207390
$373$	10.0000	0.517780	0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000	0.517780	0.258890	+	0.965907i	$0.416643\pi$
$374$	0	0
$375$	0	0
$376$	−6.00000	−0.309426
$377$	15.0000	0.772539
$378$	8.00000	0.411476
$379$	−16.0000	−0.821865	−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000	−0.821865	−0.410932	+	0.911666i	$0.634797\pi$
$380$	0	0
$381$	−16.0000	−0.819705
$382$	0	0
$383$	−12.0000	−0.613171	−0.306586	−	0.951843i	$-0.599187\pi$
$383$	−12.0000	−0.613171	−0.306586	+	0.951843i	$0.599187\pi$
$384$	2.00000	0.102062
$385$	0	0
$386$	13.0000	0.661683
$387$	−8.00000	−0.406663
$388$	−11.0000	−0.558440
$389$	33.0000	1.67317	0.836583	−	0.547840i	$-0.184550\pi$
$389$	33.0000	1.67317	0.836583	+	0.547840i	$0.184550\pi$
$390$	0	0
$391$	18.0000	0.910299
$392$	−3.00000	−0.151523
$393$	0	0
$394$	15.0000	0.755689
$395$	0	0
$396$	0	0
$397$	−17.0000	−0.853206	−0.426603	−	0.904439i	$-0.640290\pi$
$397$	−17.0000	−0.853206	−0.426603	+	0.904439i	$0.640290\pi$
$398$	−16.0000	−0.802008
$399$	−8.00000	−0.400501
$400$	0	0
$401$	−33.0000	−1.64794	−0.823971	−	0.566632i	$-0.808246\pi$
$401$	−33.0000	−1.64794	−0.823971	+	0.566632i	$0.808246\pi$
$402$	20.0000	0.997509
$403$	−10.0000	−0.498135
$404$	6.00000	0.298511
$405$	0	0
$406$	6.00000	0.297775
$407$	0	0
$408$	−6.00000	−0.297044
$409$	−13.0000	−0.642809	−0.321404	−	0.946942i	$-0.604155\pi$
$409$	−13.0000	−0.642809	−0.321404	+	0.946942i	$0.604155\pi$
$410$	0	0
$411$	−12.0000	−0.591916
$412$	−8.00000	−0.394132
$413$	0	0
$414$	−6.00000	−0.294884
$415$	0	0
$416$	−5.00000	−0.245145
$417$	−44.0000	−2.15469
$418$	0	0
$419$	−18.0000	−0.879358	−0.439679	−	0.898155i	$-0.644908\pi$
$419$	−18.0000	−0.879358	−0.439679	+	0.898155i	$0.644908\pi$
$420$	0	0
$421$	17.0000	0.828529	0.414265	−	0.910156i	$-0.364039\pi$
$421$	17.0000	0.828529	0.414265	+	0.910156i	$0.364039\pi$
$422$	−4.00000	−0.194717
$423$	−6.00000	−0.291730
$424$	3.00000	0.145693
$425$	0	0
$426$	24.0000	1.16280
$427$	20.0000	0.967868
$428$	6.00000	0.290021
$429$	0	0
$430$	0	0
$431$	36.0000	1.73406	0.867029	−	0.498257i	$-0.166026\pi$
$431$	36.0000	1.73406	0.867029	+	0.498257i	$0.166026\pi$
$432$	−4.00000	−0.192450
$433$	37.0000	1.77811	0.889053	−	0.457804i	$-0.151364\pi$
$433$	37.0000	1.77811	0.889053	+	0.457804i	$0.151364\pi$
$434$	−4.00000	−0.192006
$435$	0	0
$436$	17.0000	0.814152
$437$	−12.0000	−0.574038
$438$	−28.0000	−1.33789
$439$	−22.0000	−1.05000	−0.525001	−	0.851101i	$-0.675935\pi$
$439$	−22.0000	−1.05000	−0.525001	+	0.851101i	$0.675935\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	15.0000	0.713477
$443$	−36.0000	−1.71041	−0.855206	−	0.518289i	$-0.826569\pi$
$443$	−36.0000	−1.71041	−0.855206	+	0.518289i	$0.826569\pi$
$444$	14.0000	0.664411
$445$	0	0
$446$	4.00000	0.189405
$447$	−6.00000	−0.283790
$448$	−2.00000	−0.0944911
$449$	−21.0000	−0.991051	−0.495526	−	0.868593i	$-0.665025\pi$
$449$	−21.0000	−0.991051	−0.495526	+	0.868593i	$0.665025\pi$
$450$	0	0
$451$	0	0
$452$	9.00000	0.423324
$453$	16.0000	0.751746
$454$	0	0
$455$	0	0
$456$	4.00000	0.187317
$457$	1.00000	0.0467780	0.0233890	−	0.999726i	$-0.492554\pi$
$457$	1.00000	0.0467780	0.0233890	+	0.999726i	$0.492554\pi$
$458$	5.00000	0.233635
$459$	12.0000	0.560112
$460$	0	0
$461$	21.0000	0.978068	0.489034	−	0.872265i	$-0.337349\pi$
$461$	21.0000	0.978068	0.489034	+	0.872265i	$0.337349\pi$
$462$	0	0
$463$	4.00000	0.185896	0.0929479	−	0.995671i	$-0.470371\pi$
$463$	4.00000	0.185896	0.0929479	+	0.995671i	$0.470371\pi$
$464$	−3.00000	−0.139272
$465$	0	0
$466$	21.0000	0.972806
$467$	12.0000	0.555294	0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000	0.555294	0.277647	+	0.960683i	$0.410445\pi$
$468$	−5.00000	−0.231125
$469$	−20.0000	−0.923514
$470$	0	0
$471$	−4.00000	−0.184310
$472$	0	0
$473$	0	0
$474$	4.00000	0.183726
$475$	0	0
$476$	6.00000	0.275010
$477$	3.00000	0.137361
$478$	−30.0000	−1.37217
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	−35.0000	−1.59586
$482$	−10.0000	−0.455488
$483$	24.0000	1.09204
$484$	0	0
$485$	0	0
$486$	−10.0000	−0.453609
$487$	34.0000	1.54069	0.770344	−	0.637629i	$-0.220085\pi$
$487$	34.0000	1.54069	0.770344	+	0.637629i	$0.220085\pi$
$488$	−10.0000	−0.452679
$489$	44.0000	1.98975
$490$	0	0
$491$	−30.0000	−1.35388	−0.676941	−	0.736038i	$-0.736695\pi$
$491$	−30.0000	−1.35388	−0.676941	+	0.736038i	$0.736695\pi$
$492$	6.00000	0.270501
$493$	9.00000	0.405340
$494$	−10.0000	−0.449921
$495$	0	0
$496$	2.00000	0.0898027
$497$	−24.0000	−1.07655
$498$	−36.0000	−1.61320
$499$	−16.0000	−0.716258	−0.358129	−	0.933672i	$-0.616585\pi$
$499$	−16.0000	−0.716258	−0.358129	+	0.933672i	$0.616585\pi$
$500$	0	0
$501$	24.0000	1.07224
$502$	18.0000	0.803379
$503$	−30.0000	−1.33763	−0.668817	−	0.743427i	$-0.733199\pi$
$503$	−30.0000	−1.33763	−0.668817	+	0.743427i	$0.733199\pi$
$504$	−2.00000	−0.0890871
$505$	0	0
$506$	0	0
$507$	24.0000	1.06588
$508$	−8.00000	−0.354943
$509$	6.00000	0.265945	0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000	0.265945	0.132973	+	0.991120i	$0.457548\pi$
$510$	0	0
$511$	28.0000	1.23865
$512$	1.00000	0.0441942
$513$	−8.00000	−0.353209
$514$	−3.00000	−0.132324
$515$	0	0
$516$	−16.0000	−0.704361
$517$	0	0
$518$	−14.0000	−0.615125
$519$	−12.0000	−0.526742
$520$	0	0
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	−3.00000	−0.131306
$523$	16.0000	0.699631	0.349816	−	0.936819i	$-0.386244\pi$
$523$	16.0000	0.699631	0.349816	+	0.936819i	$0.386244\pi$
$524$	0	0
$525$	0	0
$526$	−6.00000	−0.261612
$527$	−6.00000	−0.261364
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	0	0
$532$	−4.00000	−0.173422
$533$	−15.0000	−0.649722
$534$	−18.0000	−0.778936
$535$	0	0
$536$	10.0000	0.431934
$537$	0	0
$538$	−3.00000	−0.129339
$539$	0	0
$540$	0	0
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	20.0000	0.859074
$543$	10.0000	0.429141
$544$	−3.00000	−0.128624
$545$	0	0
$546$	20.0000	0.855921
$547$	−8.00000	−0.342055	−0.171028	−	0.985266i	$-0.554709\pi$
$547$	−8.00000	−0.342055	−0.171028	+	0.985266i	$0.554709\pi$
$548$	−6.00000	−0.256307
$549$	−10.0000	−0.426790
$550$	0	0
$551$	−6.00000	−0.255609
$552$	−12.0000	−0.510754
$553$	−4.00000	−0.170097
$554$	−5.00000	−0.212430
$555$	0	0
$556$	−22.0000	−0.933008
$557$	18.0000	0.762684	0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000	0.762684	0.381342	+	0.924434i	$0.375462\pi$
$558$	2.00000	0.0846668
$559$	40.0000	1.69182
$560$	0	0
$561$	0	0
$562$	6.00000	0.253095
$563$	42.0000	1.77009	0.885044	−	0.465506i	$-0.154128\pi$
$563$	42.0000	1.77009	0.885044	+	0.465506i	$0.154128\pi$
$564$	−12.0000	−0.505291
$565$	0	0
$566$	−20.0000	−0.840663
$567$	22.0000	0.923913
$568$	12.0000	0.503509
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	−10.0000	−0.418487	−0.209243	−	0.977864i	$-0.567100\pi$
$571$	−10.0000	−0.418487	−0.209243	+	0.977864i	$0.567100\pi$
$572$	0	0
$573$	0	0
$574$	−6.00000	−0.250435
$575$	0	0
$576$	1.00000	0.0416667
$577$	−11.0000	−0.457936	−0.228968	−	0.973434i	$-0.573535\pi$
$577$	−11.0000	−0.457936	−0.228968	+	0.973434i	$0.573535\pi$
$578$	−8.00000	−0.332756
$579$	26.0000	1.08052
$580$	0	0
$581$	36.0000	1.49353
$582$	−22.0000	−0.911929
$583$	0	0
$584$	−14.0000	−0.579324
$585$	0	0
$586$	−21.0000	−0.867502
$587$	18.0000	0.742940	0.371470	−	0.928445i	$-0.378854\pi$
$587$	18.0000	0.742940	0.371470	+	0.928445i	$0.378854\pi$
$588$	−6.00000	−0.247436
$589$	4.00000	0.164817
$590$	0	0
$591$	30.0000	1.23404
$592$	7.00000	0.287698
$593$	−27.0000	−1.10876	−0.554379	−	0.832265i	$-0.687044\pi$
$593$	−27.0000	−1.10876	−0.554379	+	0.832265i	$0.687044\pi$
$594$	0	0
$595$	0	0
$596$	−3.00000	−0.122885
$597$	−32.0000	−1.30967
$598$	30.0000	1.22679
$599$	−42.0000	−1.71607	−0.858037	−	0.513588i	$-0.828316\pi$
$599$	−42.0000	−1.71607	−0.858037	+	0.513588i	$0.828316\pi$
$600$	0	0
$601$	−13.0000	−0.530281	−0.265141	−	0.964210i	$-0.585418\pi$
$601$	−13.0000	−0.530281	−0.265141	+	0.964210i	$0.585418\pi$
$602$	16.0000	0.652111
$603$	10.0000	0.407231
$604$	8.00000	0.325515
$605$	0	0
$606$	12.0000	0.487467
$607$	22.0000	0.892952	0.446476	−	0.894795i	$-0.352679\pi$
$607$	22.0000	0.892952	0.446476	+	0.894795i	$0.352679\pi$
$608$	2.00000	0.0811107
$609$	12.0000	0.486265
$610$	0	0
$611$	30.0000	1.21367
$612$	−3.00000	−0.121268
$613$	−5.00000	−0.201948	−0.100974	−	0.994889i	$-0.532196\pi$
$613$	−5.00000	−0.201948	−0.100974	+	0.994889i	$0.532196\pi$
$614$	10.0000	0.403567
$615$	0	0
$616$	0	0
$617$	−39.0000	−1.57008	−0.785040	−	0.619445i	$-0.787358\pi$
$617$	−39.0000	−1.57008	−0.785040	+	0.619445i	$0.787358\pi$
$618$	−16.0000	−0.643614
$619$	−10.0000	−0.401934	−0.200967	−	0.979598i	$-0.564408\pi$
$619$	−10.0000	−0.401934	−0.200967	+	0.979598i	$0.564408\pi$
$620$	0	0
$621$	24.0000	0.963087
$622$	0	0
$623$	18.0000	0.721155
$624$	−10.0000	−0.400320
$625$	0	0
$626$	1.00000	0.0399680
$627$	0	0
$628$	−2.00000	−0.0798087
$629$	−21.0000	−0.837325
$630$	0	0
$631$	38.0000	1.51276	0.756378	−	0.654135i	$-0.226967\pi$
$631$	38.0000	1.51276	0.756378	+	0.654135i	$0.226967\pi$
$632$	2.00000	0.0795557
$633$	−8.00000	−0.317971
$634$	18.0000	0.714871
$635$	0	0
$636$	6.00000	0.237915
$637$	15.0000	0.594322
$638$	0	0
$639$	12.0000	0.474713
$640$	0	0
$641$	−33.0000	−1.30342	−0.651711	−	0.758468i	$-0.725948\pi$
$641$	−33.0000	−1.30342	−0.651711	+	0.758468i	$0.725948\pi$
$642$	12.0000	0.473602
$643$	−2.00000	−0.0788723	−0.0394362	−	0.999222i	$-0.512556\pi$
$643$	−2.00000	−0.0788723	−0.0394362	+	0.999222i	$0.512556\pi$
$644$	12.0000	0.472866
$645$	0	0
$646$	−6.00000	−0.236067
$647$	12.0000	0.471769	0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000	0.471769	0.235884	+	0.971781i	$0.424201\pi$
$648$	−11.0000	−0.432121
$649$	0	0
$650$	0	0
$651$	−8.00000	−0.313545
$652$	22.0000	0.861586
$653$	−18.0000	−0.704394	−0.352197	−	0.935926i	$-0.614565\pi$
$653$	−18.0000	−0.704394	−0.352197	+	0.935926i	$0.614565\pi$
$654$	34.0000	1.32951
$655$	0	0
$656$	3.00000	0.117130
$657$	−14.0000	−0.546192
$658$	12.0000	0.467809
$659$	−6.00000	−0.233727	−0.116863	−	0.993148i	$-0.537284\pi$
$659$	−6.00000	−0.233727	−0.116863	+	0.993148i	$0.537284\pi$
$660$	0	0
$661$	17.0000	0.661223	0.330612	−	0.943767i	$-0.392745\pi$
$661$	17.0000	0.661223	0.330612	+	0.943767i	$0.392745\pi$
$662$	20.0000	0.777322
$663$	30.0000	1.16510
$664$	−18.0000	−0.698535
$665$	0	0
$666$	7.00000	0.271244
$667$	18.0000	0.696963
$668$	12.0000	0.464294
$669$	8.00000	0.309298
$670$	0	0
$671$	0	0
$672$	−4.00000	−0.154303
$673$	−38.0000	−1.46479	−0.732396	−	0.680879i	$-0.761598\pi$
$673$	−38.0000	−1.46479	−0.732396	+	0.680879i	$0.761598\pi$
$674$	13.0000	0.500741
$675$	0	0
$676$	12.0000	0.461538
$677$	15.0000	0.576497	0.288248	−	0.957556i	$-0.406927\pi$
$677$	15.0000	0.576497	0.288248	+	0.957556i	$0.406927\pi$
$678$	18.0000	0.691286
$679$	22.0000	0.844283
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−6.00000	−0.229584	−0.114792	−	0.993390i	$-0.536620\pi$
$683$	−6.00000	−0.229584	−0.114792	+	0.993390i	$0.536620\pi$
$684$	2.00000	0.0764719
$685$	0	0
$686$	20.0000	0.763604
$687$	10.0000	0.381524
$688$	−8.00000	−0.304997
$689$	−15.0000	−0.571454
$690$	0	0
$691$	−4.00000	−0.152167	−0.0760836	−	0.997101i	$-0.524242\pi$
$691$	−4.00000	−0.152167	−0.0760836	+	0.997101i	$0.524242\pi$
$692$	−6.00000	−0.228086
$693$	0	0
$694$	−36.0000	−1.36654
$695$	0	0
$696$	−6.00000	−0.227429
$697$	−9.00000	−0.340899
$698$	17.0000	0.643459
$699$	42.0000	1.58859
$700$	0	0
$701$	−27.0000	−1.01978	−0.509888	−	0.860241i	$-0.670313\pi$
$701$	−27.0000	−1.01978	−0.509888	+	0.860241i	$0.670313\pi$
$702$	20.0000	0.754851
$703$	14.0000	0.528020
$704$	0	0
$705$	0	0
$706$	9.00000	0.338719
$707$	−12.0000	−0.451306
$708$	0	0
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	0	0
$711$	2.00000	0.0750059
$712$	−9.00000	−0.337289
$713$	−12.0000	−0.449404
$714$	12.0000	0.449089
$715$	0	0
$716$	0	0
$717$	−60.0000	−2.24074
$718$	18.0000	0.671754
$719$	18.0000	0.671287	0.335643	−	0.941989i	$-0.391046\pi$
$719$	18.0000	0.671287	0.335643	+	0.941989i	$0.391046\pi$
$720$	0	0
$721$	16.0000	0.595871
$722$	−15.0000	−0.558242
$723$	−20.0000	−0.743808
$724$	5.00000	0.185824
$725$	0	0
$726$	0	0
$727$	−26.0000	−0.964287	−0.482143	−	0.876092i	$-0.660142\pi$
$727$	−26.0000	−0.964287	−0.482143	+	0.876092i	$0.660142\pi$
$728$	10.0000	0.370625
$729$	13.0000	0.481481
$730$	0	0
$731$	24.0000	0.887672
$732$	−20.0000	−0.739221
$733$	−5.00000	−0.184679	−0.0923396	−	0.995728i	$-0.529435\pi$
$733$	−5.00000	−0.184679	−0.0923396	+	0.995728i	$0.529435\pi$
$734$	10.0000	0.369107
$735$	0	0
$736$	−6.00000	−0.221163
$737$	0	0
$738$	3.00000	0.110432
$739$	26.0000	0.956425	0.478213	−	0.878244i	$-0.341285\pi$
$739$	26.0000	0.956425	0.478213	+	0.878244i	$0.341285\pi$
$740$	0	0
$741$	−20.0000	−0.734718
$742$	−6.00000	−0.220267
$743$	18.0000	0.660356	0.330178	−	0.943919i	$-0.392891\pi$
$743$	18.0000	0.660356	0.330178	+	0.943919i	$0.392891\pi$
$744$	4.00000	0.146647
$745$	0	0
$746$	10.0000	0.366126
$747$	−18.0000	−0.658586
$748$	0	0
$749$	−12.0000	−0.438470
$750$	0	0
$751$	44.0000	1.60558	0.802791	−	0.596260i	$-0.203347\pi$
$751$	44.0000	1.60558	0.802791	+	0.596260i	$0.203347\pi$
$752$	−6.00000	−0.218797
$753$	36.0000	1.31191
$754$	15.0000	0.546268
$755$	0	0
$756$	8.00000	0.290957
$757$	−41.0000	−1.49017	−0.745085	−	0.666969i	$-0.767591\pi$
$757$	−41.0000	−1.49017	−0.745085	+	0.666969i	$0.767591\pi$
$758$	−16.0000	−0.581146
$759$	0	0
$760$	0	0
$761$	3.00000	0.108750	0.0543750	−	0.998521i	$-0.482683\pi$
$761$	3.00000	0.108750	0.0543750	+	0.998521i	$0.482683\pi$
$762$	−16.0000	−0.579619
$763$	−34.0000	−1.23088
$764$	0	0
$765$	0	0
$766$	−12.0000	−0.433578
$767$	0	0
$768$	2.00000	0.0721688
$769$	35.0000	1.26213	0.631066	−	0.775729i	$-0.282618\pi$
$769$	35.0000	1.26213	0.631066	+	0.775729i	$0.282618\pi$
$770$	0	0
$771$	−6.00000	−0.216085
$772$	13.0000	0.467880
$773$	42.0000	1.51064	0.755318	−	0.655359i	$-0.227483\pi$
$773$	42.0000	1.51064	0.755318	+	0.655359i	$0.227483\pi$
$774$	−8.00000	−0.287554
$775$	0	0
$776$	−11.0000	−0.394877
$777$	−28.0000	−1.00449
$778$	33.0000	1.18311
$779$	6.00000	0.214972
$780$	0	0
$781$	0	0
$782$	18.0000	0.643679
$783$	12.0000	0.428845
$784$	−3.00000	−0.107143
$785$	0	0
$786$	0	0
$787$	4.00000	0.142585	0.0712923	−	0.997455i	$-0.477288\pi$
$787$	4.00000	0.142585	0.0712923	+	0.997455i	$0.477288\pi$
$788$	15.0000	0.534353
$789$	−12.0000	−0.427211
$790$	0	0
$791$	−18.0000	−0.640006
$792$	0	0
$793$	50.0000	1.77555
$794$	−17.0000	−0.603307
$795$	0	0
$796$	−16.0000	−0.567105
$797$	42.0000	1.48772	0.743858	−	0.668338i	$-0.232994\pi$
$797$	42.0000	1.48772	0.743858	+	0.668338i	$0.232994\pi$
$798$	−8.00000	−0.283197
$799$	18.0000	0.636794
$800$	0	0
$801$	−9.00000	−0.317999
$802$	−33.0000	−1.16527
$803$	0	0
$804$	20.0000	0.705346
$805$	0	0
$806$	−10.0000	−0.352235
$807$	−6.00000	−0.211210
$808$	6.00000	0.211079
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	0	0
$811$	−28.0000	−0.983213	−0.491606	−	0.870817i	$-0.663590\pi$
$811$	−28.0000	−0.983213	−0.491606	+	0.870817i	$0.663590\pi$
$812$	6.00000	0.210559
$813$	40.0000	1.40286
$814$	0	0
$815$	0	0
$816$	−6.00000	−0.210042
$817$	−16.0000	−0.559769
$818$	−13.0000	−0.454534
$819$	10.0000	0.349428
$820$	0	0
$821$	30.0000	1.04701	0.523504	−	0.852023i	$-0.324625\pi$
$821$	30.0000	1.04701	0.523504	+	0.852023i	$0.324625\pi$
$822$	−12.0000	−0.418548
$823$	−32.0000	−1.11545	−0.557725	−	0.830026i	$-0.688326\pi$
$823$	−32.0000	−1.11545	−0.557725	+	0.830026i	$0.688326\pi$
$824$	−8.00000	−0.278693
$825$	0	0
$826$	0	0
$827$	−18.0000	−0.625921	−0.312961	−	0.949766i	$-0.601321\pi$
$827$	−18.0000	−0.625921	−0.312961	+	0.949766i	$0.601321\pi$
$828$	−6.00000	−0.208514
$829$	−43.0000	−1.49345	−0.746726	−	0.665132i	$-0.768375\pi$
$829$	−43.0000	−1.49345	−0.746726	+	0.665132i	$0.768375\pi$
$830$	0	0
$831$	−10.0000	−0.346896
$832$	−5.00000	−0.173344
$833$	9.00000	0.311832
$834$	−44.0000	−1.52360
$835$	0	0
$836$	0	0
$837$	−8.00000	−0.276520
$838$	−18.0000	−0.621800
$839$	18.0000	0.621429	0.310715	−	0.950503i	$-0.399432\pi$
$839$	18.0000	0.621429	0.310715	+	0.950503i	$0.399432\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	17.0000	0.585859
$843$	12.0000	0.413302
$844$	−4.00000	−0.137686
$845$	0	0
$846$	−6.00000	−0.206284
$847$	0	0
$848$	3.00000	0.103020
$849$	−40.0000	−1.37280
$850$	0	0
$851$	−42.0000	−1.43974
$852$	24.0000	0.822226
$853$	−29.0000	−0.992941	−0.496471	−	0.868054i	$-0.665371\pi$
$853$	−29.0000	−0.992941	−0.496471	+	0.868054i	$0.665371\pi$
$854$	20.0000	0.684386
$855$	0	0
$856$	6.00000	0.205076
$857$	54.0000	1.84460	0.922302	−	0.386469i	$-0.126305\pi$
$857$	54.0000	1.84460	0.922302	+	0.386469i	$0.126305\pi$
$858$	0	0
$859$	32.0000	1.09183	0.545913	−	0.837842i	$-0.316183\pi$
$859$	32.0000	1.09183	0.545913	+	0.837842i	$0.316183\pi$
$860$	0	0
$861$	−12.0000	−0.408959
$862$	36.0000	1.22616
$863$	−6.00000	−0.204242	−0.102121	−	0.994772i	$-0.532563\pi$
$863$	−6.00000	−0.204242	−0.102121	+	0.994772i	$0.532563\pi$
$864$	−4.00000	−0.136083
$865$	0	0
$866$	37.0000	1.25731
$867$	−16.0000	−0.543388
$868$	−4.00000	−0.135769
$869$	0	0
$870$	0	0
$871$	−50.0000	−1.69419
$872$	17.0000	0.575693
$873$	−11.0000	−0.372294
$874$	−12.0000	−0.405906
$875$	0	0
$876$	−28.0000	−0.946032
$877$	−41.0000	−1.38447	−0.692236	−	0.721671i	$-0.743374\pi$
$877$	−41.0000	−1.38447	−0.692236	+	0.721671i	$0.743374\pi$
$878$	−22.0000	−0.742464
$879$	−42.0000	−1.41662
$880$	0	0
$881$	−45.0000	−1.51609	−0.758044	−	0.652203i	$-0.773845\pi$
$881$	−45.0000	−1.51609	−0.758044	+	0.652203i	$0.773845\pi$
$882$	−3.00000	−0.101015
$883$	4.00000	0.134611	0.0673054	−	0.997732i	$-0.478560\pi$
$883$	4.00000	0.134611	0.0673054	+	0.997732i	$0.478560\pi$
$884$	15.0000	0.504505
$885$	0	0
$886$	−36.0000	−1.20944
$887$	18.0000	0.604381	0.302190	−	0.953248i	$-0.402282\pi$
$887$	18.0000	0.604381	0.302190	+	0.953248i	$0.402282\pi$
$888$	14.0000	0.469809
$889$	16.0000	0.536623
$890$	0	0
$891$	0	0
$892$	4.00000	0.133930
$893$	−12.0000	−0.401565
$894$	−6.00000	−0.200670
$895$	0	0
$896$	−2.00000	−0.0668153
$897$	60.0000	2.00334
$898$	−21.0000	−0.700779
$899$	−6.00000	−0.200111
$900$	0	0
$901$	−9.00000	−0.299833
$902$	0	0
$903$	32.0000	1.06489
$904$	9.00000	0.299336
$905$	0	0
$906$	16.0000	0.531564
$907$	−44.0000	−1.46100	−0.730498	−	0.682915i	$-0.760712\pi$
$907$	−44.0000	−1.46100	−0.730498	+	0.682915i	$0.760712\pi$
$908$	0	0
$909$	6.00000	0.199007
$910$	0	0
$911$	−24.0000	−0.795155	−0.397578	−	0.917568i	$-0.630149\pi$
$911$	−24.0000	−0.795155	−0.397578	+	0.917568i	$0.630149\pi$
$912$	4.00000	0.132453
$913$	0	0
$914$	1.00000	0.0330771
$915$	0	0
$916$	5.00000	0.165205
$917$	0	0
$918$	12.0000	0.396059
$919$	−28.0000	−0.923635	−0.461817	−	0.886975i	$-0.652802\pi$
$919$	−28.0000	−0.923635	−0.461817	+	0.886975i	$0.652802\pi$
$920$	0	0
$921$	20.0000	0.659022
$922$	21.0000	0.691598
$923$	−60.0000	−1.97492
$924$	0	0
$925$	0	0
$926$	4.00000	0.131448
$927$	−8.00000	−0.262754
$928$	−3.00000	−0.0984798
$929$	51.0000	1.67326	0.836628	−	0.547772i	$-0.184524\pi$
$929$	51.0000	1.67326	0.836628	+	0.547772i	$0.184524\pi$
$930$	0	0
$931$	−6.00000	−0.196642
$932$	21.0000	0.687878
$933$	0	0
$934$	12.0000	0.392652
$935$	0	0
$936$	−5.00000	−0.163430
$937$	−23.0000	−0.751377	−0.375689	−	0.926746i	$-0.622594\pi$
$937$	−23.0000	−0.751377	−0.375689	+	0.926746i	$0.622594\pi$
$938$	−20.0000	−0.653023
$939$	2.00000	0.0652675
$940$	0	0
$941$	−39.0000	−1.27136	−0.635682	−	0.771951i	$-0.719281\pi$
$941$	−39.0000	−1.27136	−0.635682	+	0.771951i	$0.719281\pi$
$942$	−4.00000	−0.130327
$943$	−18.0000	−0.586161
$944$	0	0
$945$	0	0
$946$	0	0
$947$	30.0000	0.974869	0.487435	−	0.873160i	$-0.337933\pi$
$947$	30.0000	0.974869	0.487435	+	0.873160i	$0.337933\pi$
$948$	4.00000	0.129914
$949$	70.0000	2.27230
$950$	0	0
$951$	36.0000	1.16738
$952$	6.00000	0.194461
$953$	−39.0000	−1.26333	−0.631667	−	0.775240i	$-0.717629\pi$
$953$	−39.0000	−1.26333	−0.631667	+	0.775240i	$0.717629\pi$
$954$	3.00000	0.0971286
$955$	0	0
$956$	−30.0000	−0.970269
$957$	0	0
$958$	0	0
$959$	12.0000	0.387500
$960$	0	0
$961$	−27.0000	−0.870968
$962$	−35.0000	−1.12845
$963$	6.00000	0.193347
$964$	−10.0000	−0.322078
$965$	0	0
$966$	24.0000	0.772187
$967$	−50.0000	−1.60789	−0.803946	−	0.594703i	$-0.797270\pi$
$967$	−50.0000	−1.60789	−0.803946	+	0.594703i	$0.797270\pi$
$968$	0	0
$969$	−12.0000	−0.385496
$970$	0	0
$971$	6.00000	0.192549	0.0962746	−	0.995355i	$-0.469307\pi$
$971$	6.00000	0.192549	0.0962746	+	0.995355i	$0.469307\pi$
$972$	−10.0000	−0.320750
$973$	44.0000	1.41058
$974$	34.0000	1.08943
$975$	0	0
$976$	−10.0000	−0.320092
$977$	33.0000	1.05576	0.527882	−	0.849318i	$-0.322986\pi$
$977$	33.0000	1.05576	0.527882	+	0.849318i	$0.322986\pi$
$978$	44.0000	1.40696
$979$	0	0
$980$	0	0
$981$	17.0000	0.542768
$982$	−30.0000	−0.957338
$983$	−60.0000	−1.91370	−0.956851	−	0.290578i	$-0.906153\pi$
$983$	−60.0000	−1.91370	−0.956851	+	0.290578i	$0.906153\pi$
$984$	6.00000	0.191273
$985$	0	0
$986$	9.00000	0.286618
$987$	24.0000	0.763928
$988$	−10.0000	−0.318142
$989$	48.0000	1.52631
$990$	0	0
$991$	−52.0000	−1.65183	−0.825917	−	0.563791i	$-0.809342\pi$
$991$	−52.0000	−1.65183	−0.825917	+	0.563791i	$0.809342\pi$
$992$	2.00000	0.0635001
$993$	40.0000	1.26936
$994$	−24.0000	−0.761234
$995$	0	0
$996$	−36.0000	−1.14070
$997$	31.0000	0.981780	0.490890	−	0.871222i	$-0.336672\pi$
$997$	31.0000	0.981780	0.490890	+	0.871222i	$0.336672\pi$
$998$	−16.0000	−0.506471
$999$	−28.0000	−0.885881

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6050.2.a.bl.1.1		1
5.4	even	2		242.2.a.a.1.1	✓	1
11.10	odd	2		6050.2.a.s.1.1		1
15.14	odd	2		2178.2.a.l.1.1		1
20.19	odd	2		1936.2.a.j.1.1		1
40.19	odd	2		7744.2.a.g.1.1		1
40.29	even	2		7744.2.a.bi.1.1		1
55.4	even	10		242.2.c.e.27.1		4
55.9	even	10		242.2.c.e.81.1		4
55.14	even	10		242.2.c.e.9.1		4
55.19	odd	10		242.2.c.b.9.1		4
55.24	odd	10		242.2.c.b.81.1		4
55.29	odd	10		242.2.c.b.27.1		4
55.39	odd	10		242.2.c.b.3.1		4
55.49	even	10		242.2.c.e.3.1		4
55.54	odd	2		242.2.a.b.1.1	yes	1
165.164	even	2		2178.2.a.f.1.1		1
220.219	even	2		1936.2.a.k.1.1		1
440.109	odd	2		7744.2.a.bh.1.1		1
440.219	even	2		7744.2.a.h.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
242.2.a.a.1.1	✓	1	5.4	even	2
242.2.a.b.1.1	yes	1	55.54	odd	2
242.2.c.b.3.1		4	55.39	odd	10
242.2.c.b.9.1		4	55.19	odd	10
242.2.c.b.27.1		4	55.29	odd	10
242.2.c.b.81.1		4	55.24	odd	10
242.2.c.e.3.1		4	55.49	even	10
242.2.c.e.9.1		4	55.14	even	10
242.2.c.e.27.1		4	55.4	even	10
242.2.c.e.81.1		4	55.9	even	10
1936.2.a.j.1.1		1	20.19	odd	2
1936.2.a.k.1.1		1	220.219	even	2
2178.2.a.f.1.1		1	165.164	even	2
2178.2.a.l.1.1		1	15.14	odd	2
6050.2.a.s.1.1		1	11.10	odd	2
6050.2.a.bl.1.1		1	1.1	even	1	trivial
7744.2.a.g.1.1		1	40.19	odd	2
7744.2.a.h.1.1		1	440.219	even	2
7744.2.a.bh.1.1		1	440.109	odd	2
7744.2.a.bi.1.1		1	40.29	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:	\( N \)	\(=\)	\( 6050 = 2 \cdot 5^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6050.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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