LMFDB - Embedded newform 1050.2.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1050,2,Mod(1,1050)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1050, 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("1050.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1050.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:	$8.38429221223$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 210)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1050.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{11} -1.00000 q^{12} +2.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +8.00000 q^{17} +1.00000 q^{18} -2.00000 q^{19} -1.00000 q^{21} -2.00000 q^{22} -1.00000 q^{24} +2.00000 q^{26} -1.00000 q^{27} +1.00000 q^{28} -6.00000 q^{29} +6.00000 q^{31} +1.00000 q^{32} +2.00000 q^{33} +8.00000 q^{34} +1.00000 q^{36} +8.00000 q^{37} -2.00000 q^{38} -2.00000 q^{39} +6.00000 q^{41} -1.00000 q^{42} +8.00000 q^{43} -2.00000 q^{44} +4.00000 q^{47} -1.00000 q^{48} +1.00000 q^{49} -8.00000 q^{51} +2.00000 q^{52} -2.00000 q^{53} -1.00000 q^{54} +1.00000 q^{56} +2.00000 q^{57} -6.00000 q^{58} -8.00000 q^{59} +10.0000 q^{61} +6.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} +2.00000 q^{66} -12.0000 q^{67} +8.00000 q^{68} -14.0000 q^{71} +1.00000 q^{72} -10.0000 q^{73} +8.00000 q^{74} -2.00000 q^{76} -2.00000 q^{77} -2.00000 q^{78} +4.00000 q^{79} +1.00000 q^{81} +6.00000 q^{82} +16.0000 q^{83} -1.00000 q^{84} +8.00000 q^{86} +6.00000 q^{87} -2.00000 q^{88} +10.0000 q^{89} +2.00000 q^{91} -6.00000 q^{93} +4.00000 q^{94} -1.00000 q^{96} +10.0000 q^{97} +1.00000 q^{98} -2.00000 q^{99} +O(q^{100})\)

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	−1.00000	−0.288675
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	8.00000	1.94029	0.970143	−	0.242536i	$-0.0779791\pi$
$17$	8.00000	1.94029	0.970143	+	0.242536i	$0.0779791\pi$
$18$	1.00000	0.235702
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	−2.00000	−0.426401
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	2.00000	0.392232
$27$	−1.00000	−0.192450
$28$	1.00000	0.188982
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	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
$33$	2.00000	0.348155
$34$	8.00000	1.37199
$35$	0	0
$36$	1.00000	0.166667
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	−2.00000	−0.324443
$39$	−2.00000	−0.320256
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	−1.00000	−0.154303
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	−2.00000	−0.301511
$45$	0	0
$46$	0	0
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	−1.00000	−0.144338
$49$	1.00000	0.142857
$50$	0	0
$51$	−8.00000	−1.12022
$52$	2.00000	0.277350
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	1.00000	0.133631
$57$	2.00000	0.264906
$58$	−6.00000	−0.787839
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	0	0
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	6.00000	0.762001
$63$	1.00000	0.125988
$64$	1.00000	0.125000
$65$	0	0
$66$	2.00000	0.246183
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	8.00000	0.970143
$69$	0	0
$70$	0	0
$71$	−14.0000	−1.66149	−0.830747	−	0.556650i	$-0.812086\pi$
$71$	−14.0000	−1.66149	−0.830747	+	0.556650i	$0.812086\pi$
$72$	1.00000	0.117851
$73$	−10.0000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	8.00000	0.929981
$75$	0	0
$76$	−2.00000	−0.229416
$77$	−2.00000	−0.227921
$78$	−2.00000	−0.226455
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	6.00000	0.662589
$83$	16.0000	1.75623	0.878114	−	0.478451i	$-0.158802\pi$
$83$	16.0000	1.75623	0.878114	+	0.478451i	$0.158802\pi$
$84$	−1.00000	−0.109109
$85$	0	0
$86$	8.00000	0.862662
$87$	6.00000	0.643268
$88$	−2.00000	−0.213201
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	0	0
$93$	−6.00000	−0.622171
$94$	4.00000	0.412568
$95$	0	0
$96$	−1.00000	−0.102062
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	1.00000	0.101015
$99$	−2.00000	−0.201008
$100$	0	0
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	−8.00000	−0.792118
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	−2.00000	−0.194257
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	−1.00000	−0.0962250
$109$	−6.00000	−0.574696	−0.287348	−	0.957826i	$-0.592774\pi$
$109$	−6.00000	−0.574696	−0.287348	+	0.957826i	$0.592774\pi$
$110$	0	0
$111$	−8.00000	−0.759326
$112$	1.00000	0.0944911
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	2.00000	0.187317
$115$	0	0
$116$	−6.00000	−0.557086
$117$	2.00000	0.184900
$118$	−8.00000	−0.736460
$119$	8.00000	0.733359
$120$	0	0
$121$	−7.00000	−0.636364
$122$	10.0000	0.905357
$123$	−6.00000	−0.541002
$124$	6.00000	0.538816
$125$	0	0
$126$	1.00000	0.0890871
$127$	12.0000	1.06483	0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000	1.06483	0.532414	+	0.846484i	$0.321285\pi$
$128$	1.00000	0.0883883
$129$	−8.00000	−0.704361
$130$	0	0
$131$	−20.0000	−1.74741	−0.873704	−	0.486458i	$-0.838289\pi$
$131$	−20.0000	−1.74741	−0.873704	+	0.486458i	$0.838289\pi$
$132$	2.00000	0.174078
$133$	−2.00000	−0.173422
$134$	−12.0000	−1.03664
$135$	0	0
$136$	8.00000	0.685994
$137$	−14.0000	−1.19610	−0.598050	−	0.801459i	$-0.704058\pi$
$137$	−14.0000	−1.19610	−0.598050	+	0.801459i	$0.704058\pi$
$138$	0	0
$139$	−2.00000	−0.169638	−0.0848189	−	0.996396i	$-0.527031\pi$
$139$	−2.00000	−0.169638	−0.0848189	+	0.996396i	$0.527031\pi$
$140$	0	0
$141$	−4.00000	−0.336861
$142$	−14.0000	−1.17485
$143$	−4.00000	−0.334497
$144$	1.00000	0.0833333
$145$	0	0
$146$	−10.0000	−0.827606
$147$	−1.00000	−0.0824786
$148$	8.00000	0.657596
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	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
$153$	8.00000	0.646762
$154$	−2.00000	−0.161165
$155$	0	0
$156$	−2.00000	−0.160128
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	4.00000	0.318223
$159$	2.00000	0.158610
$160$	0	0
$161$	0	0
$162$	1.00000	0.0785674
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	16.0000	1.24184
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	−1.00000	−0.0771517
$169$	−9.00000	−0.692308
$170$	0	0
$171$	−2.00000	−0.152944
$172$	8.00000	0.609994
$173$	−16.0000	−1.21646	−0.608229	−	0.793762i	$-0.708120\pi$
$173$	−16.0000	−1.21646	−0.608229	+	0.793762i	$0.708120\pi$
$174$	6.00000	0.454859
$175$	0	0
$176$	−2.00000	−0.150756
$177$	8.00000	0.601317
$178$	10.0000	0.749532
$179$	−2.00000	−0.149487	−0.0747435	−	0.997203i	$-0.523814\pi$
$179$	−2.00000	−0.149487	−0.0747435	+	0.997203i	$0.523814\pi$
$180$	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
$183$	−10.0000	−0.739221
$184$	0	0
$185$	0	0
$186$	−6.00000	−0.439941
$187$	−16.0000	−1.17004
$188$	4.00000	0.291730
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	14.0000	1.01300	0.506502	−	0.862239i	$-0.330938\pi$
$191$	14.0000	1.01300	0.506502	+	0.862239i	$0.330938\pi$
$192$	−1.00000	−0.0721688
$193$	0	0		−	1.00000i	$-0.5\pi$
$193$	0	0			1.00000i	$0.5\pi$
$194$	10.0000	0.717958
$195$	0	0
$196$	1.00000	0.0714286
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	−2.00000	−0.142134
$199$	−26.0000	−1.84309	−0.921546	−	0.388270i	$-0.873073\pi$
$199$	−26.0000	−1.84309	−0.921546	+	0.388270i	$0.873073\pi$
$200$	0	0
$201$	12.0000	0.846415
$202$	−10.0000	−0.703598
$203$	−6.00000	−0.421117
$204$	−8.00000	−0.560112
$205$	0	0
$206$	0	0
$207$	0	0
$208$	2.00000	0.138675
$209$	4.00000	0.276686
$210$	0	0
$211$	−8.00000	−0.550743	−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000	−0.550743	−0.275371	+	0.961338i	$0.588801\pi$
$212$	−2.00000	−0.137361
$213$	14.0000	0.959264
$214$	−12.0000	−0.820303
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	6.00000	0.407307
$218$	−6.00000	−0.406371
$219$	10.0000	0.675737
$220$	0	0
$221$	16.0000	1.07628
$222$	−8.00000	−0.536925
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	6.00000	0.399114
$227$	−8.00000	−0.530979	−0.265489	−	0.964114i	$-0.585534\pi$
$227$	−8.00000	−0.530979	−0.265489	+	0.964114i	$0.585534\pi$
$228$	2.00000	0.132453
$229$	26.0000	1.71813	0.859064	−	0.511868i	$-0.171046\pi$
$229$	26.0000	1.71813	0.859064	+	0.511868i	$0.171046\pi$
$230$	0	0
$231$	2.00000	0.131590
$232$	−6.00000	−0.393919
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	2.00000	0.130744
$235$	0	0
$236$	−8.00000	−0.520756
$237$	−4.00000	−0.259828
$238$	8.00000	0.518563
$239$	−22.0000	−1.42306	−0.711531	−	0.702655i	$-0.751998\pi$
$239$	−22.0000	−1.42306	−0.711531	+	0.702655i	$0.751998\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	−7.00000	−0.449977
$243$	−1.00000	−0.0641500
$244$	10.0000	0.640184
$245$	0	0
$246$	−6.00000	−0.382546
$247$	−4.00000	−0.254514
$248$	6.00000	0.381000
$249$	−16.0000	−1.01396
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	1.00000	0.0629941
$253$	0	0
$254$	12.0000	0.752947
$255$	0	0
$256$	1.00000	0.0625000
$257$	−20.0000	−1.24757	−0.623783	−	0.781598i	$-0.714405\pi$
$257$	−20.0000	−1.24757	−0.623783	+	0.781598i	$0.714405\pi$
$258$	−8.00000	−0.498058
$259$	8.00000	0.497096
$260$	0	0
$261$	−6.00000	−0.371391
$262$	−20.0000	−1.23560
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	2.00000	0.123091
$265$	0	0
$266$	−2.00000	−0.122628
$267$	−10.0000	−0.611990
$268$	−12.0000	−0.733017
$269$	14.0000	0.853595	0.426798	−	0.904347i	$-0.359642\pi$
$269$	14.0000	0.853595	0.426798	+	0.904347i	$0.359642\pi$
$270$	0	0
$271$	2.00000	0.121491	0.0607457	−	0.998153i	$-0.480652\pi$
$271$	2.00000	0.121491	0.0607457	+	0.998153i	$0.480652\pi$
$272$	8.00000	0.485071
$273$	−2.00000	−0.121046
$274$	−14.0000	−0.845771
$275$	0	0
$276$	0	0
$277$	−8.00000	−0.480673	−0.240337	−	0.970690i	$-0.577258\pi$
$277$	−8.00000	−0.480673	−0.240337	+	0.970690i	$0.577258\pi$
$278$	−2.00000	−0.119952
$279$	6.00000	0.359211
$280$	0	0
$281$	22.0000	1.31241	0.656205	−	0.754583i	$-0.272161\pi$
$281$	22.0000	1.31241	0.656205	+	0.754583i	$0.272161\pi$
$282$	−4.00000	−0.238197
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	−14.0000	−0.830747
$285$	0	0
$286$	−4.00000	−0.236525
$287$	6.00000	0.354169
$288$	1.00000	0.0589256
$289$	47.0000	2.76471
$290$	0	0
$291$	−10.0000	−0.586210
$292$	−10.0000	−0.585206
$293$	−16.0000	−0.934730	−0.467365	−	0.884064i	$-0.654797\pi$
$293$	−16.0000	−0.934730	−0.467365	+	0.884064i	$0.654797\pi$
$294$	−1.00000	−0.0583212
$295$	0	0
$296$	8.00000	0.464991
$297$	2.00000	0.116052
$298$	−18.0000	−1.04271
$299$	0	0
$300$	0	0
$301$	8.00000	0.461112
$302$	8.00000	0.460348
$303$	10.0000	0.574485
$304$	−2.00000	−0.114708
$305$	0	0
$306$	8.00000	0.457330
$307$	28.0000	1.59804	0.799022	−	0.601302i	$-0.205351\pi$
$307$	28.0000	1.59804	0.799022	+	0.601302i	$0.205351\pi$
$308$	−2.00000	−0.113961
$309$	0	0
$310$	0	0
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	−2.00000	−0.113228
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	−10.0000	−0.564333
$315$	0	0
$316$	4.00000	0.225018
$317$	−2.00000	−0.112331	−0.0561656	−	0.998421i	$-0.517887\pi$
$317$	−2.00000	−0.112331	−0.0561656	+	0.998421i	$0.517887\pi$
$318$	2.00000	0.112154
$319$	12.0000	0.671871
$320$	0	0
$321$	12.0000	0.669775
$322$	0	0
$323$	−16.0000	−0.890264
$324$	1.00000	0.0555556
$325$	0	0
$326$	0	0
$327$	6.00000	0.331801
$328$	6.00000	0.331295
$329$	4.00000	0.220527
$330$	0	0
$331$	0	0		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	16.0000	0.878114
$333$	8.00000	0.438397
$334$	−8.00000	−0.437741
$335$	0	0
$336$	−1.00000	−0.0545545
$337$	32.0000	1.74315	0.871576	−	0.490261i	$-0.163099\pi$
$337$	32.0000	1.74315	0.871576	+	0.490261i	$0.163099\pi$
$338$	−9.00000	−0.489535
$339$	−6.00000	−0.325875
$340$	0	0
$341$	−12.0000	−0.649836
$342$	−2.00000	−0.108148
$343$	1.00000	0.0539949
$344$	8.00000	0.431331
$345$	0	0
$346$	−16.0000	−0.860165
$347$	4.00000	0.214731	0.107366	−	0.994220i	$-0.465758\pi$
$347$	4.00000	0.214731	0.107366	+	0.994220i	$0.465758\pi$
$348$	6.00000	0.321634
$349$	18.0000	0.963518	0.481759	−	0.876304i	$-0.339998\pi$
$349$	18.0000	0.963518	0.481759	+	0.876304i	$0.339998\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	−2.00000	−0.106600
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	8.00000	0.425195
$355$	0	0
$356$	10.0000	0.529999
$357$	−8.00000	−0.423405
$358$	−2.00000	−0.105703
$359$	14.0000	0.738892	0.369446	−	0.929252i	$-0.379548\pi$
$359$	14.0000	0.738892	0.369446	+	0.929252i	$0.379548\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	22.0000	1.15629
$363$	7.00000	0.367405
$364$	2.00000	0.104828
$365$	0	0
$366$	−10.0000	−0.522708
$367$	−32.0000	−1.67039	−0.835193	−	0.549957i	$-0.814644\pi$
$367$	−32.0000	−1.67039	−0.835193	+	0.549957i	$0.814644\pi$
$368$	0	0
$369$	6.00000	0.312348
$370$	0	0
$371$	−2.00000	−0.103835
$372$	−6.00000	−0.311086
$373$	36.0000	1.86401	0.932005	−	0.362446i	$-0.118058\pi$
$373$	36.0000	1.86401	0.932005	+	0.362446i	$0.118058\pi$
$374$	−16.0000	−0.827340
$375$	0	0
$376$	4.00000	0.206284
$377$	−12.0000	−0.618031
$378$	−1.00000	−0.0514344
$379$	−28.0000	−1.43826	−0.719132	−	0.694874i	$-0.755460\pi$
$379$	−28.0000	−1.43826	−0.719132	+	0.694874i	$0.755460\pi$
$380$	0	0
$381$	−12.0000	−0.614779
$382$	14.0000	0.716302
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	0	0
$387$	8.00000	0.406663
$388$	10.0000	0.507673
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	0	0
$392$	1.00000	0.0505076
$393$	20.0000	1.00887
$394$	−6.00000	−0.302276
$395$	0	0
$396$	−2.00000	−0.100504
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	−26.0000	−1.30326
$399$	2.00000	0.100125
$400$	0	0
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	12.0000	0.598506
$403$	12.0000	0.597763
$404$	−10.0000	−0.497519
$405$	0	0
$406$	−6.00000	−0.297775
$407$	−16.0000	−0.793091
$408$	−8.00000	−0.396059
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	0	0
$411$	14.0000	0.690569
$412$	0	0
$413$	−8.00000	−0.393654
$414$	0	0
$415$	0	0
$416$	2.00000	0.0980581
$417$	2.00000	0.0979404
$418$	4.00000	0.195646
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	10.0000	0.487370	0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000	0.487370	0.243685	+	0.969854i	$0.421644\pi$
$422$	−8.00000	−0.389434
$423$	4.00000	0.194487
$424$	−2.00000	−0.0971286
$425$	0	0
$426$	14.0000	0.678302
$427$	10.0000	0.483934
$428$	−12.0000	−0.580042
$429$	4.00000	0.193122
$430$	0	0
$431$	−6.00000	−0.289010	−0.144505	−	0.989504i	$-0.546159\pi$
$431$	−6.00000	−0.289010	−0.144505	+	0.989504i	$0.546159\pi$
$432$	−1.00000	−0.0481125
$433$	26.0000	1.24948	0.624740	−	0.780833i	$-0.285205\pi$
$433$	26.0000	1.24948	0.624740	+	0.780833i	$0.285205\pi$
$434$	6.00000	0.288009
$435$	0	0
$436$	−6.00000	−0.287348
$437$	0	0
$438$	10.0000	0.477818
$439$	6.00000	0.286364	0.143182	−	0.989696i	$-0.454267\pi$
$439$	6.00000	0.286364	0.143182	+	0.989696i	$0.454267\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	16.0000	0.761042
$443$	36.0000	1.71041	0.855206	−	0.518289i	$-0.173431\pi$
$443$	36.0000	1.71041	0.855206	+	0.518289i	$0.173431\pi$
$444$	−8.00000	−0.379663
$445$	0	0
$446$	−16.0000	−0.757622
$447$	18.0000	0.851371
$448$	1.00000	0.0472456
$449$	−30.0000	−1.41579	−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000	−1.41579	−0.707894	+	0.706319i	$0.750354\pi$
$450$	0	0
$451$	−12.0000	−0.565058
$452$	6.00000	0.282216
$453$	−8.00000	−0.375873
$454$	−8.00000	−0.375459
$455$	0	0
$456$	2.00000	0.0936586
$457$	−28.0000	−1.30978	−0.654892	−	0.755722i	$-0.727286\pi$
$457$	−28.0000	−1.30978	−0.654892	+	0.755722i	$0.727286\pi$
$458$	26.0000	1.21490
$459$	−8.00000	−0.373408
$460$	0	0
$461$	18.0000	0.838344	0.419172	−	0.907907i	$-0.362320\pi$
$461$	18.0000	0.838344	0.419172	+	0.907907i	$0.362320\pi$
$462$	2.00000	0.0930484
$463$	−12.0000	−0.557687	−0.278844	−	0.960337i	$-0.589951\pi$
$463$	−12.0000	−0.557687	−0.278844	+	0.960337i	$0.589951\pi$
$464$	−6.00000	−0.278543
$465$	0	0
$466$	6.00000	0.277945
$467$	16.0000	0.740392	0.370196	−	0.928954i	$-0.379291\pi$
$467$	16.0000	0.740392	0.370196	+	0.928954i	$0.379291\pi$
$468$	2.00000	0.0924500
$469$	−12.0000	−0.554109
$470$	0	0
$471$	10.0000	0.460776
$472$	−8.00000	−0.368230
$473$	−16.0000	−0.735681
$474$	−4.00000	−0.183726
$475$	0	0
$476$	8.00000	0.366679
$477$	−2.00000	−0.0915737
$478$	−22.0000	−1.00626
$479$	8.00000	0.365529	0.182765	−	0.983157i	$-0.441495\pi$
$479$	8.00000	0.365529	0.182765	+	0.983157i	$0.441495\pi$
$480$	0	0
$481$	16.0000	0.729537
$482$	−10.0000	−0.455488
$483$	0	0
$484$	−7.00000	−0.318182
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	36.0000	1.63132	0.815658	−	0.578535i	$-0.196375\pi$
$487$	36.0000	1.63132	0.815658	+	0.578535i	$0.196375\pi$
$488$	10.0000	0.452679
$489$	0	0
$490$	0	0
$491$	−42.0000	−1.89543	−0.947717	−	0.319113i	$-0.896615\pi$
$491$	−42.0000	−1.89543	−0.947717	+	0.319113i	$0.896615\pi$
$492$	−6.00000	−0.270501
$493$	−48.0000	−2.16181
$494$	−4.00000	−0.179969
$495$	0	0
$496$	6.00000	0.269408
$497$	−14.0000	−0.627986
$498$	−16.0000	−0.716977
$499$	−28.0000	−1.25345	−0.626726	−	0.779240i	$-0.715605\pi$
$499$	−28.0000	−1.25345	−0.626726	+	0.779240i	$0.715605\pi$
$500$	0	0
$501$	8.00000	0.357414
$502$	0	0
$503$	24.0000	1.07011	0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000	1.07011	0.535054	+	0.844818i	$0.320291\pi$
$504$	1.00000	0.0445435
$505$	0	0
$506$	0	0
$507$	9.00000	0.399704
$508$	12.0000	0.532414
$509$	2.00000	0.0886484	0.0443242	−	0.999017i	$-0.485887\pi$
$509$	2.00000	0.0886484	0.0443242	+	0.999017i	$0.485887\pi$
$510$	0	0
$511$	−10.0000	−0.442374
$512$	1.00000	0.0441942
$513$	2.00000	0.0883022
$514$	−20.0000	−0.882162
$515$	0	0
$516$	−8.00000	−0.352180
$517$	−8.00000	−0.351840
$518$	8.00000	0.351500
$519$	16.0000	0.702322
$520$	0	0
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	−6.00000	−0.262613
$523$	−12.0000	−0.524723	−0.262362	−	0.964970i	$-0.584501\pi$
$523$	−12.0000	−0.524723	−0.262362	+	0.964970i	$0.584501\pi$
$524$	−20.0000	−0.873704
$525$	0	0
$526$	−24.0000	−1.04645
$527$	48.0000	2.09091
$528$	2.00000	0.0870388
$529$	−23.0000	−1.00000
$530$	0	0
$531$	−8.00000	−0.347170
$532$	−2.00000	−0.0867110
$533$	12.0000	0.519778
$534$	−10.0000	−0.432742
$535$	0	0
$536$	−12.0000	−0.518321
$537$	2.00000	0.0863064
$538$	14.0000	0.603583
$539$	−2.00000	−0.0861461
$540$	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
$543$	−22.0000	−0.944110
$544$	8.00000	0.342997
$545$	0	0
$546$	−2.00000	−0.0855921
$547$	20.0000	0.855138	0.427569	−	0.903983i	$-0.359370\pi$
$547$	20.0000	0.855138	0.427569	+	0.903983i	$0.359370\pi$
$548$	−14.0000	−0.598050
$549$	10.0000	0.426790
$550$	0	0
$551$	12.0000	0.511217
$552$	0	0
$553$	4.00000	0.170097
$554$	−8.00000	−0.339887
$555$	0	0
$556$	−2.00000	−0.0848189
$557$	6.00000	0.254228	0.127114	−	0.991888i	$-0.459429\pi$
$557$	6.00000	0.254228	0.127114	+	0.991888i	$0.459429\pi$
$558$	6.00000	0.254000
$559$	16.0000	0.676728
$560$	0	0
$561$	16.0000	0.675521
$562$	22.0000	0.928014
$563$	12.0000	0.505740	0.252870	−	0.967500i	$-0.418626\pi$
$563$	12.0000	0.505740	0.252870	+	0.967500i	$0.418626\pi$
$564$	−4.00000	−0.168430
$565$	0	0
$566$	4.00000	0.168133
$567$	1.00000	0.0419961
$568$	−14.0000	−0.587427
$569$	18.0000	0.754599	0.377300	−	0.926091i	$-0.376853\pi$
$569$	18.0000	0.754599	0.377300	+	0.926091i	$0.376853\pi$
$570$	0	0
$571$	−44.0000	−1.84134	−0.920671	−	0.390339i	$-0.872358\pi$
$571$	−44.0000	−1.84134	−0.920671	+	0.390339i	$0.872358\pi$
$572$	−4.00000	−0.167248
$573$	−14.0000	−0.584858
$574$	6.00000	0.250435
$575$	0	0
$576$	1.00000	0.0416667
$577$	18.0000	0.749350	0.374675	−	0.927156i	$-0.377754\pi$
$577$	18.0000	0.749350	0.374675	+	0.927156i	$0.377754\pi$
$578$	47.0000	1.95494
$579$	0	0
$580$	0	0
$581$	16.0000	0.663792
$582$	−10.0000	−0.414513
$583$	4.00000	0.165663
$584$	−10.0000	−0.413803
$585$	0	0
$586$	−16.0000	−0.660954
$587$	4.00000	0.165098	0.0825488	−	0.996587i	$-0.473694\pi$
$587$	4.00000	0.165098	0.0825488	+	0.996587i	$0.473694\pi$
$588$	−1.00000	−0.0412393
$589$	−12.0000	−0.494451
$590$	0	0
$591$	6.00000	0.246807
$592$	8.00000	0.328798
$593$	−32.0000	−1.31408	−0.657041	−	0.753855i	$-0.728192\pi$
$593$	−32.0000	−1.31408	−0.657041	+	0.753855i	$0.728192\pi$
$594$	2.00000	0.0820610
$595$	0	0
$596$	−18.0000	−0.737309
$597$	26.0000	1.06411
$598$	0	0
$599$	14.0000	0.572024	0.286012	−	0.958226i	$-0.407670\pi$
$599$	14.0000	0.572024	0.286012	+	0.958226i	$0.407670\pi$
$600$	0	0
$601$	−42.0000	−1.71322	−0.856608	−	0.515968i	$-0.827432\pi$
$601$	−42.0000	−1.71322	−0.856608	+	0.515968i	$0.827432\pi$
$602$	8.00000	0.326056
$603$	−12.0000	−0.488678
$604$	8.00000	0.325515
$605$	0	0
$606$	10.0000	0.406222
$607$	−16.0000	−0.649420	−0.324710	−	0.945814i	$-0.605267\pi$
$607$	−16.0000	−0.649420	−0.324710	+	0.945814i	$0.605267\pi$
$608$	−2.00000	−0.0811107
$609$	6.00000	0.243132
$610$	0	0
$611$	8.00000	0.323645
$612$	8.00000	0.323381
$613$	32.0000	1.29247	0.646234	−	0.763139i	$-0.276343\pi$
$613$	32.0000	1.29247	0.646234	+	0.763139i	$0.276343\pi$
$614$	28.0000	1.12999
$615$	0	0
$616$	−2.00000	−0.0805823
$617$	−18.0000	−0.724653	−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000	−0.724653	−0.362326	+	0.932051i	$0.618017\pi$
$618$	0	0
$619$	−18.0000	−0.723481	−0.361741	−	0.932279i	$-0.617817\pi$
$619$	−18.0000	−0.723481	−0.361741	+	0.932279i	$0.617817\pi$
$620$	0	0
$621$	0	0
$622$	−8.00000	−0.320771
$623$	10.0000	0.400642
$624$	−2.00000	−0.0800641
$625$	0	0
$626$	10.0000	0.399680
$627$	−4.00000	−0.159745
$628$	−10.0000	−0.399043
$629$	64.0000	2.55185
$630$	0	0
$631$	12.0000	0.477712	0.238856	−	0.971055i	$-0.423228\pi$
$631$	12.0000	0.477712	0.238856	+	0.971055i	$0.423228\pi$
$632$	4.00000	0.159111
$633$	8.00000	0.317971
$634$	−2.00000	−0.0794301
$635$	0	0
$636$	2.00000	0.0793052
$637$	2.00000	0.0792429
$638$	12.0000	0.475085
$639$	−14.0000	−0.553831
$640$	0	0
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	12.0000	0.473602
$643$	−20.0000	−0.788723	−0.394362	−	0.918955i	$-0.629034\pi$
$643$	−20.0000	−0.788723	−0.394362	+	0.918955i	$0.629034\pi$
$644$	0	0
$645$	0	0
$646$	−16.0000	−0.629512
$647$	16.0000	0.629025	0.314512	−	0.949253i	$-0.398159\pi$
$647$	16.0000	0.629025	0.314512	+	0.949253i	$0.398159\pi$
$648$	1.00000	0.0392837
$649$	16.0000	0.628055
$650$	0	0
$651$	−6.00000	−0.235159
$652$	0	0
$653$	−14.0000	−0.547862	−0.273931	−	0.961749i	$-0.588324\pi$
$653$	−14.0000	−0.547862	−0.273931	+	0.961749i	$0.588324\pi$
$654$	6.00000	0.234619
$655$	0	0
$656$	6.00000	0.234261
$657$	−10.0000	−0.390137
$658$	4.00000	0.155936
$659$	−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
$661$	10.0000	0.388955	0.194477	−	0.980907i	$-0.437699\pi$
$661$	10.0000	0.388955	0.194477	+	0.980907i	$0.437699\pi$
$662$	0	0
$663$	−16.0000	−0.621389
$664$	16.0000	0.620920
$665$	0	0
$666$	8.00000	0.309994
$667$	0	0
$668$	−8.00000	−0.309529
$669$	16.0000	0.618596
$670$	0	0
$671$	−20.0000	−0.772091
$672$	−1.00000	−0.0385758
$673$	12.0000	0.462566	0.231283	−	0.972887i	$-0.425708\pi$
$673$	12.0000	0.462566	0.231283	+	0.972887i	$0.425708\pi$
$674$	32.0000	1.23259
$675$	0	0
$676$	−9.00000	−0.346154
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	−6.00000	−0.230429
$679$	10.0000	0.383765
$680$	0	0
$681$	8.00000	0.306561
$682$	−12.0000	−0.459504
$683$	28.0000	1.07139	0.535695	−	0.844411i	$-0.320050\pi$
$683$	28.0000	1.07139	0.535695	+	0.844411i	$0.320050\pi$
$684$	−2.00000	−0.0764719
$685$	0	0
$686$	1.00000	0.0381802
$687$	−26.0000	−0.991962
$688$	8.00000	0.304997
$689$	−4.00000	−0.152388
$690$	0	0
$691$	−14.0000	−0.532585	−0.266293	−	0.963892i	$-0.585799\pi$
$691$	−14.0000	−0.532585	−0.266293	+	0.963892i	$0.585799\pi$
$692$	−16.0000	−0.608229
$693$	−2.00000	−0.0759737
$694$	4.00000	0.151838
$695$	0	0
$696$	6.00000	0.227429
$697$	48.0000	1.81813
$698$	18.0000	0.681310
$699$	−6.00000	−0.226941
$700$	0	0
$701$	−30.0000	−1.13308	−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000	−1.13308	−0.566542	+	0.824033i	$0.691719\pi$
$702$	−2.00000	−0.0754851
$703$	−16.0000	−0.603451
$704$	−2.00000	−0.0753778
$705$	0	0
$706$	0	0
$707$	−10.0000	−0.376089
$708$	8.00000	0.300658
$709$	−38.0000	−1.42712	−0.713560	−	0.700594i	$-0.752918\pi$
$709$	−38.0000	−1.42712	−0.713560	+	0.700594i	$0.752918\pi$
$710$	0	0
$711$	4.00000	0.150012
$712$	10.0000	0.374766
$713$	0	0
$714$	−8.00000	−0.299392
$715$	0	0
$716$	−2.00000	−0.0747435
$717$	22.0000	0.821605
$718$	14.0000	0.522475
$719$	−20.0000	−0.745874	−0.372937	−	0.927857i	$-0.621649\pi$
$719$	−20.0000	−0.745874	−0.372937	+	0.927857i	$0.621649\pi$
$720$	0	0
$721$	0	0
$722$	−15.0000	−0.558242
$723$	10.0000	0.371904
$724$	22.0000	0.817624
$725$	0	0
$726$	7.00000	0.259794
$727$	16.0000	0.593407	0.296704	−	0.954970i	$-0.404113\pi$
$727$	16.0000	0.593407	0.296704	+	0.954970i	$0.404113\pi$
$728$	2.00000	0.0741249
$729$	1.00000	0.0370370
$730$	0	0
$731$	64.0000	2.36713
$732$	−10.0000	−0.369611
$733$	−30.0000	−1.10808	−0.554038	−	0.832492i	$-0.686914\pi$
$733$	−30.0000	−1.10808	−0.554038	+	0.832492i	$0.686914\pi$
$734$	−32.0000	−1.18114
$735$	0	0
$736$	0	0
$737$	24.0000	0.884051
$738$	6.00000	0.220863
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	0	0
$741$	4.00000	0.146944
$742$	−2.00000	−0.0734223
$743$	16.0000	0.586983	0.293492	−	0.955962i	$-0.405183\pi$
$743$	16.0000	0.586983	0.293492	+	0.955962i	$0.405183\pi$
$744$	−6.00000	−0.219971
$745$	0	0
$746$	36.0000	1.31805
$747$	16.0000	0.585409
$748$	−16.0000	−0.585018
$749$	−12.0000	−0.438470
$750$	0	0
$751$	4.00000	0.145962	0.0729810	−	0.997333i	$-0.476749\pi$
$751$	4.00000	0.145962	0.0729810	+	0.997333i	$0.476749\pi$
$752$	4.00000	0.145865
$753$	0	0
$754$	−12.0000	−0.437014
$755$	0	0
$756$	−1.00000	−0.0363696
$757$	−4.00000	−0.145382	−0.0726912	−	0.997354i	$-0.523159\pi$
$757$	−4.00000	−0.145382	−0.0726912	+	0.997354i	$0.523159\pi$
$758$	−28.0000	−1.01701
$759$	0	0
$760$	0	0
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	−12.0000	−0.434714
$763$	−6.00000	−0.217215
$764$	14.0000	0.506502
$765$	0	0
$766$	0	0
$767$	−16.0000	−0.577727
$768$	−1.00000	−0.0360844
$769$	−42.0000	−1.51456	−0.757279	−	0.653091i	$-0.773472\pi$
$769$	−42.0000	−1.51456	−0.757279	+	0.653091i	$0.773472\pi$
$770$	0	0
$771$	20.0000	0.720282
$772$	0	0
$773$	−32.0000	−1.15096	−0.575480	−	0.817816i	$-0.695185\pi$
$773$	−32.0000	−1.15096	−0.575480	+	0.817816i	$0.695185\pi$
$774$	8.00000	0.287554
$775$	0	0
$776$	10.0000	0.358979
$777$	−8.00000	−0.286998
$778$	6.00000	0.215110
$779$	−12.0000	−0.429945
$780$	0	0
$781$	28.0000	1.00192
$782$	0	0
$783$	6.00000	0.214423
$784$	1.00000	0.0357143
$785$	0	0
$786$	20.0000	0.713376
$787$	−44.0000	−1.56843	−0.784215	−	0.620489i	$-0.786934\pi$
$787$	−44.0000	−1.56843	−0.784215	+	0.620489i	$0.786934\pi$
$788$	−6.00000	−0.213741
$789$	24.0000	0.854423
$790$	0	0
$791$	6.00000	0.213335
$792$	−2.00000	−0.0710669
$793$	20.0000	0.710221
$794$	2.00000	0.0709773
$795$	0	0
$796$	−26.0000	−0.921546
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	2.00000	0.0707992
$799$	32.0000	1.13208
$800$	0	0
$801$	10.0000	0.353333
$802$	18.0000	0.635602
$803$	20.0000	0.705785
$804$	12.0000	0.423207
$805$	0	0
$806$	12.0000	0.422682
$807$	−14.0000	−0.492823
$808$	−10.0000	−0.351799
$809$	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
$811$	14.0000	0.491606	0.245803	−	0.969320i	$-0.420948\pi$
$811$	14.0000	0.491606	0.245803	+	0.969320i	$0.420948\pi$
$812$	−6.00000	−0.210559
$813$	−2.00000	−0.0701431
$814$	−16.0000	−0.560800
$815$	0	0
$816$	−8.00000	−0.280056
$817$	−16.0000	−0.559769
$818$	10.0000	0.349642
$819$	2.00000	0.0698857
$820$	0	0
$821$	−6.00000	−0.209401	−0.104701	−	0.994504i	$-0.533388\pi$
$821$	−6.00000	−0.209401	−0.104701	+	0.994504i	$0.533388\pi$
$822$	14.0000	0.488306
$823$	−4.00000	−0.139431	−0.0697156	−	0.997567i	$-0.522209\pi$
$823$	−4.00000	−0.139431	−0.0697156	+	0.997567i	$0.522209\pi$
$824$	0	0
$825$	0	0
$826$	−8.00000	−0.278356
$827$	−36.0000	−1.25184	−0.625921	−	0.779886i	$-0.715277\pi$
$827$	−36.0000	−1.25184	−0.625921	+	0.779886i	$0.715277\pi$
$828$	0	0
$829$	−34.0000	−1.18087	−0.590434	−	0.807086i	$-0.701044\pi$
$829$	−34.0000	−1.18087	−0.590434	+	0.807086i	$0.701044\pi$
$830$	0	0
$831$	8.00000	0.277517
$832$	2.00000	0.0693375
$833$	8.00000	0.277184
$834$	2.00000	0.0692543
$835$	0	0
$836$	4.00000	0.138343
$837$	−6.00000	−0.207390
$838$	12.0000	0.414533
$839$	44.0000	1.51905	0.759524	−	0.650479i	$-0.225432\pi$
$839$	44.0000	1.51905	0.759524	+	0.650479i	$0.225432\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	10.0000	0.344623
$843$	−22.0000	−0.757720
$844$	−8.00000	−0.275371
$845$	0	0
$846$	4.00000	0.137523
$847$	−7.00000	−0.240523
$848$	−2.00000	−0.0686803
$849$	−4.00000	−0.137280
$850$	0	0
$851$	0	0
$852$	14.0000	0.479632
$853$	10.0000	0.342393	0.171197	−	0.985237i	$-0.445237\pi$
$853$	10.0000	0.342393	0.171197	+	0.985237i	$0.445237\pi$
$854$	10.0000	0.342193
$855$	0	0
$856$	−12.0000	−0.410152
$857$	20.0000	0.683187	0.341593	−	0.939848i	$-0.389033\pi$
$857$	20.0000	0.683187	0.341593	+	0.939848i	$0.389033\pi$
$858$	4.00000	0.136558
$859$	−6.00000	−0.204717	−0.102359	−	0.994748i	$-0.532639\pi$
$859$	−6.00000	−0.204717	−0.102359	+	0.994748i	$0.532639\pi$
$860$	0	0
$861$	−6.00000	−0.204479
$862$	−6.00000	−0.204361
$863$	48.0000	1.63394	0.816970	−	0.576681i	$-0.195652\pi$
$863$	48.0000	1.63394	0.816970	+	0.576681i	$0.195652\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	26.0000	0.883516
$867$	−47.0000	−1.59620
$868$	6.00000	0.203653
$869$	−8.00000	−0.271381
$870$	0	0
$871$	−24.0000	−0.813209
$872$	−6.00000	−0.203186
$873$	10.0000	0.338449
$874$	0	0
$875$	0	0
$876$	10.0000	0.337869
$877$	−48.0000	−1.62084	−0.810422	−	0.585846i	$-0.800762\pi$
$877$	−48.0000	−1.62084	−0.810422	+	0.585846i	$0.800762\pi$
$878$	6.00000	0.202490
$879$	16.0000	0.539667
$880$	0	0
$881$	−6.00000	−0.202145	−0.101073	−	0.994879i	$-0.532227\pi$
$881$	−6.00000	−0.202145	−0.101073	+	0.994879i	$0.532227\pi$
$882$	1.00000	0.0336718
$883$	52.0000	1.74994	0.874970	−	0.484178i	$-0.160881\pi$
$883$	52.0000	1.74994	0.874970	+	0.484178i	$0.160881\pi$
$884$	16.0000	0.538138
$885$	0	0
$886$	36.0000	1.20944
$887$	36.0000	1.20876	0.604381	−	0.796696i	$-0.293421\pi$
$887$	36.0000	1.20876	0.604381	+	0.796696i	$0.293421\pi$
$888$	−8.00000	−0.268462
$889$	12.0000	0.402467
$890$	0	0
$891$	−2.00000	−0.0670025
$892$	−16.0000	−0.535720
$893$	−8.00000	−0.267710
$894$	18.0000	0.602010
$895$	0	0
$896$	1.00000	0.0334077
$897$	0	0
$898$	−30.0000	−1.00111
$899$	−36.0000	−1.20067
$900$	0	0
$901$	−16.0000	−0.533037
$902$	−12.0000	−0.399556
$903$	−8.00000	−0.266223
$904$	6.00000	0.199557
$905$	0	0
$906$	−8.00000	−0.265782
$907$	−44.0000	−1.46100	−0.730498	−	0.682915i	$-0.760712\pi$
$907$	−44.0000	−1.46100	−0.730498	+	0.682915i	$0.760712\pi$
$908$	−8.00000	−0.265489
$909$	−10.0000	−0.331679
$910$	0	0
$911$	6.00000	0.198789	0.0993944	−	0.995048i	$-0.468309\pi$
$911$	6.00000	0.198789	0.0993944	+	0.995048i	$0.468309\pi$
$912$	2.00000	0.0662266
$913$	−32.0000	−1.05905
$914$	−28.0000	−0.926158
$915$	0	0
$916$	26.0000	0.859064
$917$	−20.0000	−0.660458
$918$	−8.00000	−0.264039
$919$	−4.00000	−0.131948	−0.0659739	−	0.997821i	$-0.521015\pi$
$919$	−4.00000	−0.131948	−0.0659739	+	0.997821i	$0.521015\pi$
$920$	0	0
$921$	−28.0000	−0.922631
$922$	18.0000	0.592798
$923$	−28.0000	−0.921631
$924$	2.00000	0.0657952
$925$	0	0
$926$	−12.0000	−0.394344
$927$	0	0
$928$	−6.00000	−0.196960
$929$	−18.0000	−0.590561	−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000	−0.590561	−0.295280	+	0.955411i	$0.595413\pi$
$930$	0	0
$931$	−2.00000	−0.0655474
$932$	6.00000	0.196537
$933$	8.00000	0.261908
$934$	16.0000	0.523536
$935$	0	0
$936$	2.00000	0.0653720
$937$	−26.0000	−0.849383	−0.424691	−	0.905338i	$-0.639617\pi$
$937$	−26.0000	−0.849383	−0.424691	+	0.905338i	$0.639617\pi$
$938$	−12.0000	−0.391814
$939$	−10.0000	−0.326338
$940$	0	0
$941$	−6.00000	−0.195594	−0.0977972	−	0.995206i	$-0.531180\pi$
$941$	−6.00000	−0.195594	−0.0977972	+	0.995206i	$0.531180\pi$
$942$	10.0000	0.325818
$943$	0	0
$944$	−8.00000	−0.260378
$945$	0	0
$946$	−16.0000	−0.520205
$947$	12.0000	0.389948	0.194974	−	0.980808i	$-0.437538\pi$
$947$	12.0000	0.389948	0.194974	+	0.980808i	$0.437538\pi$
$948$	−4.00000	−0.129914
$949$	−20.0000	−0.649227
$950$	0	0
$951$	2.00000	0.0648544
$952$	8.00000	0.259281
$953$	−18.0000	−0.583077	−0.291539	−	0.956559i	$-0.594167\pi$
$953$	−18.0000	−0.583077	−0.291539	+	0.956559i	$0.594167\pi$
$954$	−2.00000	−0.0647524
$955$	0	0
$956$	−22.0000	−0.711531
$957$	−12.0000	−0.387905
$958$	8.00000	0.258468
$959$	−14.0000	−0.452084
$960$	0	0
$961$	5.00000	0.161290
$962$	16.0000	0.515861
$963$	−12.0000	−0.386695
$964$	−10.0000	−0.322078
$965$	0	0
$966$	0	0
$967$	−24.0000	−0.771788	−0.385894	−	0.922543i	$-0.626107\pi$
$967$	−24.0000	−0.771788	−0.385894	+	0.922543i	$0.626107\pi$
$968$	−7.00000	−0.224989
$969$	16.0000	0.513994
$970$	0	0
$971$	24.0000	0.770197	0.385098	−	0.922876i	$-0.374168\pi$
$971$	24.0000	0.770197	0.385098	+	0.922876i	$0.374168\pi$
$972$	−1.00000	−0.0320750
$973$	−2.00000	−0.0641171
$974$	36.0000	1.15351
$975$	0	0
$976$	10.0000	0.320092
$977$	−18.0000	−0.575871	−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000	−0.575871	−0.287936	+	0.957650i	$0.592969\pi$
$978$	0	0
$979$	−20.0000	−0.639203
$980$	0	0
$981$	−6.00000	−0.191565
$982$	−42.0000	−1.34027
$983$	12.0000	0.382741	0.191370	−	0.981518i	$-0.438707\pi$
$983$	12.0000	0.382741	0.191370	+	0.981518i	$0.438707\pi$
$984$	−6.00000	−0.191273
$985$	0	0
$986$	−48.0000	−1.52863
$987$	−4.00000	−0.127321
$988$	−4.00000	−0.127257
$989$	0	0
$990$	0	0
$991$	−60.0000	−1.90596	−0.952981	−	0.303029i	$-0.902002\pi$
$991$	−60.0000	−1.90596	−0.952981	+	0.303029i	$0.902002\pi$
$992$	6.00000	0.190500
$993$	0	0
$994$	−14.0000	−0.444053
$995$	0	0
$996$	−16.0000	−0.506979
$997$	−38.0000	−1.20347	−0.601736	−	0.798695i	$-0.705524\pi$
$997$	−38.0000	−1.20347	−0.601736	+	0.798695i	$0.705524\pi$
$998$	−28.0000	−0.886325
$999$	−8.00000	−0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1050.2.a.m.1.1		1
3.2	odd	2		3150.2.a.q.1.1		1
4.3	odd	2		8400.2.a.ca.1.1		1
5.2	odd	4		210.2.g.a.169.2	yes	2
5.3	odd	4		210.2.g.a.169.1	✓	2
5.4	even	2		1050.2.a.g.1.1		1
7.6	odd	2		7350.2.a.co.1.1		1
15.2	even	4		630.2.g.d.379.1		2
15.8	even	4		630.2.g.d.379.2		2
15.14	odd	2		3150.2.a.be.1.1		1
20.3	even	4		1680.2.t.d.1009.2		2
20.7	even	4		1680.2.t.d.1009.1		2
20.19	odd	2		8400.2.a.bd.1.1		1
35.2	odd	12		1470.2.n.g.949.2		4
35.3	even	12		1470.2.n.c.79.2		4
35.12	even	12		1470.2.n.c.949.2		4
35.13	even	4		1470.2.g.e.589.1		2
35.17	even	12		1470.2.n.c.79.1		4
35.18	odd	12		1470.2.n.g.79.2		4
35.23	odd	12		1470.2.n.g.949.1		4
35.27	even	4		1470.2.g.e.589.2		2
35.32	odd	12		1470.2.n.g.79.1		4
35.33	even	12		1470.2.n.c.949.1		4
35.34	odd	2		7350.2.a.g.1.1		1
60.23	odd	4		5040.2.t.k.1009.2		2
60.47	odd	4		5040.2.t.k.1009.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.2.g.a.169.1	✓	2	5.3	odd	4
210.2.g.a.169.2	yes	2	5.2	odd	4
630.2.g.d.379.1		2	15.2	even	4
630.2.g.d.379.2		2	15.8	even	4
1050.2.a.g.1.1		1	5.4	even	2
1050.2.a.m.1.1		1	1.1	even	1	trivial
1470.2.g.e.589.1		2	35.13	even	4
1470.2.g.e.589.2		2	35.27	even	4
1470.2.n.c.79.1		4	35.17	even	12
1470.2.n.c.79.2		4	35.3	even	12
1470.2.n.c.949.1		4	35.33	even	12
1470.2.n.c.949.2		4	35.12	even	12
1470.2.n.g.79.1		4	35.32	odd	12
1470.2.n.g.79.2		4	35.18	odd	12
1470.2.n.g.949.1		4	35.23	odd	12
1470.2.n.g.949.2		4	35.2	odd	12
1680.2.t.d.1009.1		2	20.7	even	4
1680.2.t.d.1009.2		2	20.3	even	4
3150.2.a.q.1.1		1	3.2	odd	2
3150.2.a.be.1.1		1	15.14	odd	2
5040.2.t.k.1009.1		2	60.47	odd	4
5040.2.t.k.1009.2		2	60.23	odd	4
7350.2.a.g.1.1		1	35.34	odd	2
7350.2.a.co.1.1		1	7.6	odd	2
8400.2.a.bd.1.1		1	20.19	odd	2
8400.2.a.ca.1.1		1	4.3	odd	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:	\( N \)	\(=\)	\( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1050.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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