LMFDB - Embedded newform 1050.2.a.h.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)

$f(q)$

$=$

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

\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +4.00000 q^{11} +1.00000 q^{12} +2.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} -2.00000 q^{17} -1.00000 q^{18} -4.00000 q^{19} -1.00000 q^{21} -4.00000 q^{22} +8.00000 q^{23} -1.00000 q^{24} -2.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} +6.00000 q^{29} -8.00000 q^{31} -1.00000 q^{32} +4.00000 q^{33} +2.00000 q^{34} +1.00000 q^{36} +2.00000 q^{37} +4.00000 q^{38} +2.00000 q^{39} +2.00000 q^{41} +1.00000 q^{42} +12.0000 q^{43} +4.00000 q^{44} -8.00000 q^{46} +8.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} -2.00000 q^{51} +2.00000 q^{52} -6.00000 q^{53} -1.00000 q^{54} +1.00000 q^{56} -4.00000 q^{57} -6.00000 q^{58} +4.00000 q^{59} -2.00000 q^{61} +8.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} -4.00000 q^{66} -12.0000 q^{67} -2.00000 q^{68} +8.00000 q^{69} +8.00000 q^{71} -1.00000 q^{72} +14.0000 q^{73} -2.00000 q^{74} -4.00000 q^{76} -4.00000 q^{77} -2.00000 q^{78} +1.00000 q^{81} -2.00000 q^{82} -12.0000 q^{83} -1.00000 q^{84} -12.0000 q^{86} +6.00000 q^{87} -4.00000 q^{88} +2.00000 q^{89} -2.00000 q^{91} +8.00000 q^{92} -8.00000 q^{93} -8.00000 q^{94} -1.00000 q^{96} -10.0000 q^{97} -1.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

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
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	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
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	−1.00000	−0.235702
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	−4.00000	−0.852803
$23$	8.00000	1.66812	0.834058	−	0.551677i	$-0.186012\pi$
$23$	8.00000	1.66812	0.834058	+	0.551677i	$0.186012\pi$
$24$	−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	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	−1.00000	−0.176777
$33$	4.00000	0.696311
$34$	2.00000	0.342997
$35$	0	0
$36$	1.00000	0.166667
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	4.00000	0.648886
$39$	2.00000	0.320256
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	1.00000	0.154303
$43$	12.0000	1.82998	0.914991	−	0.403473i	$-0.132197\pi$
$43$	12.0000	1.82998	0.914991	+	0.403473i	$0.132197\pi$
$44$	4.00000	0.603023
$45$	0	0
$46$	−8.00000	−1.17954
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	1.00000	0.144338
$49$	1.00000	0.142857
$50$	0	0
$51$	−2.00000	−0.280056
$52$	2.00000	0.277350
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	1.00000	0.133631
$57$	−4.00000	−0.529813
$58$	−6.00000	−0.787839
$59$	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	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	8.00000	1.01600
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	0	0
$66$	−4.00000	−0.492366
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	−2.00000	−0.242536
$69$	8.00000	0.963087
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	−1.00000	−0.117851
$73$	14.0000	1.63858	0.819288	−	0.573382i	$-0.194369\pi$
$73$	14.0000	1.63858	0.819288	+	0.573382i	$0.194369\pi$
$74$	−2.00000	−0.232495
$75$	0	0
$76$	−4.00000	−0.458831
$77$	−4.00000	−0.455842
$78$	−2.00000	−0.226455
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−2.00000	−0.220863
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	−1.00000	−0.109109
$85$	0	0
$86$	−12.0000	−1.29399
$87$	6.00000	0.643268
$88$	−4.00000	−0.426401
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	8.00000	0.834058
$93$	−8.00000	−0.829561
$94$	−8.00000	−0.825137
$95$	0	0
$96$	−1.00000	−0.102062
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	−1.00000	−0.101015
$99$	4.00000	0.402015
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	2.00000	0.198030
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	6.00000	0.582772
$107$	20.0000	1.93347	0.966736	−	0.255774i	$-0.0823304\pi$
$107$	20.0000	1.93347	0.966736	+	0.255774i	$0.0823304\pi$
$108$	1.00000	0.0962250
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	−1.00000	−0.0944911
$113$	14.0000	1.31701	0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000	1.31701	0.658505	+	0.752577i	$0.271189\pi$
$114$	4.00000	0.374634
$115$	0	0
$116$	6.00000	0.557086
$117$	2.00000	0.184900
$118$	−4.00000	−0.368230
$119$	2.00000	0.183340
$120$	0	0
$121$	5.00000	0.454545
$122$	2.00000	0.181071
$123$	2.00000	0.180334
$124$	−8.00000	−0.718421
$125$	0	0
$126$	1.00000	0.0890871
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	−1.00000	−0.0883883
$129$	12.0000	1.05654
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	4.00000	0.348155
$133$	4.00000	0.346844
$134$	12.0000	1.03664
$135$	0	0
$136$	2.00000	0.171499
$137$	−10.0000	−0.854358	−0.427179	−	0.904167i	$-0.640493\pi$
$137$	−10.0000	−0.854358	−0.427179	+	0.904167i	$0.640493\pi$
$138$	−8.00000	−0.681005
$139$	20.0000	1.69638	0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000	1.69638	0.848189	+	0.529694i	$0.177693\pi$
$140$	0	0
$141$	8.00000	0.673722
$142$	−8.00000	−0.671345
$143$	8.00000	0.668994
$144$	1.00000	0.0833333
$145$	0	0
$146$	−14.0000	−1.15865
$147$	1.00000	0.0824786
$148$	2.00000	0.164399
$149$	−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	$0.394462\pi$
$152$	4.00000	0.324443
$153$	−2.00000	−0.161690
$154$	4.00000	0.322329
$155$	0	0
$156$	2.00000	0.160128
$157$	−14.0000	−1.11732	−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000	−1.11732	−0.558661	+	0.829396i	$0.688685\pi$
$158$	0	0
$159$	−6.00000	−0.475831
$160$	0	0
$161$	−8.00000	−0.630488
$162$	−1.00000	−0.0785674
$163$	−12.0000	−0.939913	−0.469956	−	0.882690i	$-0.655730\pi$
$163$	−12.0000	−0.939913	−0.469956	+	0.882690i	$0.655730\pi$
$164$	2.00000	0.156174
$165$	0	0
$166$	12.0000	0.931381
$167$	16.0000	1.23812	0.619059	−	0.785345i	$-0.287514\pi$
$167$	16.0000	1.23812	0.619059	+	0.785345i	$0.287514\pi$
$168$	1.00000	0.0771517
$169$	−9.00000	−0.692308
$170$	0	0
$171$	−4.00000	−0.305888
$172$	12.0000	0.914991
$173$	−14.0000	−1.06440	−0.532200	−	0.846619i	$-0.678635\pi$
$173$	−14.0000	−1.06440	−0.532200	+	0.846619i	$0.678635\pi$
$174$	−6.00000	−0.454859
$175$	0	0
$176$	4.00000	0.301511
$177$	4.00000	0.300658
$178$	−2.00000	−0.149906
$179$	−20.0000	−1.49487	−0.747435	−	0.664335i	$-0.768715\pi$
$179$	−20.0000	−1.49487	−0.747435	+	0.664335i	$0.768715\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	2.00000	0.148250
$183$	−2.00000	−0.147844
$184$	−8.00000	−0.589768
$185$	0	0
$186$	8.00000	0.586588
$187$	−8.00000	−0.585018
$188$	8.00000	0.583460
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−16.0000	−1.15772	−0.578860	−	0.815427i	$-0.696502\pi$
$191$	−16.0000	−1.15772	−0.578860	+	0.815427i	$0.696502\pi$
$192$	1.00000	0.0721688
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	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	$0.568565\pi$
$198$	−4.00000	−0.284268
$199$	−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
$201$	−12.0000	−0.846415
$202$	−6.00000	−0.422159
$203$	−6.00000	−0.421117
$204$	−2.00000	−0.140028
$205$	0	0
$206$	8.00000	0.557386
$207$	8.00000	0.556038
$208$	2.00000	0.138675
$209$	−16.0000	−1.10674
$210$	0	0
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	−6.00000	−0.412082
$213$	8.00000	0.548151
$214$	−20.0000	−1.36717
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	8.00000	0.543075
$218$	2.00000	0.135457
$219$	14.0000	0.946032
$220$	0	0
$221$	−4.00000	−0.269069
$222$	−2.00000	−0.134231
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	−14.0000	−0.931266
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	−4.00000	−0.264906
$229$	−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
$231$	−4.00000	−0.263181
$232$	−6.00000	−0.393919
$233$	−10.0000	−0.655122	−0.327561	−	0.944830i	$-0.606227\pi$
$233$	−10.0000	−0.655122	−0.327561	+	0.944830i	$0.606227\pi$
$234$	−2.00000	−0.130744
$235$	0	0
$236$	4.00000	0.260378
$237$	0	0
$238$	−2.00000	−0.129641
$239$	−16.0000	−1.03495	−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000	−1.03495	−0.517477	+	0.855697i	$0.673129\pi$
$240$	0	0
$241$	18.0000	1.15948	0.579741	−	0.814801i	$-0.303154\pi$
$241$	18.0000	1.15948	0.579741	+	0.814801i	$0.303154\pi$
$242$	−5.00000	−0.321412
$243$	1.00000	0.0641500
$244$	−2.00000	−0.128037
$245$	0	0
$246$	−2.00000	−0.127515
$247$	−8.00000	−0.509028
$248$	8.00000	0.508001
$249$	−12.0000	−0.760469
$250$	0	0
$251$	4.00000	0.252478	0.126239	−	0.992000i	$-0.459709\pi$
$251$	4.00000	0.252478	0.126239	+	0.992000i	$0.459709\pi$
$252$	−1.00000	−0.0629941
$253$	32.0000	2.01182
$254$	0	0
$255$	0	0
$256$	1.00000	0.0625000
$257$	−2.00000	−0.124757	−0.0623783	−	0.998053i	$-0.519869\pi$
$257$	−2.00000	−0.124757	−0.0623783	+	0.998053i	$0.519869\pi$
$258$	−12.0000	−0.747087
$259$	−2.00000	−0.124274
$260$	0	0
$261$	6.00000	0.371391
$262$	−12.0000	−0.741362
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	−4.00000	−0.246183
$265$	0	0
$266$	−4.00000	−0.245256
$267$	2.00000	0.122398
$268$	−12.0000	−0.733017
$269$	30.0000	1.82913	0.914566	−	0.404436i	$-0.132532\pi$
$269$	30.0000	1.82913	0.914566	+	0.404436i	$0.132532\pi$
$270$	0	0
$271$	−24.0000	−1.45790	−0.728948	−	0.684569i	$-0.759990\pi$
$271$	−24.0000	−1.45790	−0.728948	+	0.684569i	$0.759990\pi$
$272$	−2.00000	−0.121268
$273$	−2.00000	−0.121046
$274$	10.0000	0.604122
$275$	0	0
$276$	8.00000	0.481543
$277$	−14.0000	−0.841178	−0.420589	−	0.907251i	$-0.638177\pi$
$277$	−14.0000	−0.841178	−0.420589	+	0.907251i	$0.638177\pi$
$278$	−20.0000	−1.19952
$279$	−8.00000	−0.478947
$280$	0	0
$281$	10.0000	0.596550	0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000	0.596550	0.298275	+	0.954480i	$0.403589\pi$
$282$	−8.00000	−0.476393
$283$	−20.0000	−1.18888	−0.594438	−	0.804141i	$-0.702626\pi$
$283$	−20.0000	−1.18888	−0.594438	+	0.804141i	$0.702626\pi$
$284$	8.00000	0.474713
$285$	0	0
$286$	−8.00000	−0.473050
$287$	−2.00000	−0.118056
$288$	−1.00000	−0.0589256
$289$	−13.0000	−0.764706
$290$	0	0
$291$	−10.0000	−0.586210
$292$	14.0000	0.819288
$293$	−6.00000	−0.350524	−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000	−0.350524	−0.175262	+	0.984522i	$0.556077\pi$
$294$	−1.00000	−0.0583212
$295$	0	0
$296$	−2.00000	−0.116248
$297$	4.00000	0.232104
$298$	18.0000	1.04271
$299$	16.0000	0.925304
$300$	0	0
$301$	−12.0000	−0.691669
$302$	−8.00000	−0.460348
$303$	6.00000	0.344691
$304$	−4.00000	−0.229416
$305$	0	0
$306$	2.00000	0.114332
$307$	4.00000	0.228292	0.114146	−	0.993464i	$-0.463587\pi$
$307$	4.00000	0.228292	0.114146	+	0.993464i	$0.463587\pi$
$308$	−4.00000	−0.227921
$309$	−8.00000	−0.455104
$310$	0	0
$311$	8.00000	0.453638	0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000	0.453638	0.226819	+	0.973937i	$0.427167\pi$
$312$	−2.00000	−0.113228
$313$	−34.0000	−1.92179	−0.960897	−	0.276907i	$-0.910691\pi$
$313$	−34.0000	−1.92179	−0.960897	+	0.276907i	$0.910691\pi$
$314$	14.0000	0.790066
$315$	0	0
$316$	0	0
$317$	−14.0000	−0.786318	−0.393159	−	0.919470i	$-0.628618\pi$
$317$	−14.0000	−0.786318	−0.393159	+	0.919470i	$0.628618\pi$
$318$	6.00000	0.336463
$319$	24.0000	1.34374
$320$	0	0
$321$	20.0000	1.11629
$322$	8.00000	0.445823
$323$	8.00000	0.445132
$324$	1.00000	0.0555556
$325$	0	0
$326$	12.0000	0.664619
$327$	−2.00000	−0.110600
$328$	−2.00000	−0.110432
$329$	−8.00000	−0.441054
$330$	0	0
$331$	28.0000	1.53902	0.769510	−	0.638635i	$-0.220501\pi$
$331$	28.0000	1.53902	0.769510	+	0.638635i	$0.220501\pi$
$332$	−12.0000	−0.658586
$333$	2.00000	0.109599
$334$	−16.0000	−0.875481
$335$	0	0
$336$	−1.00000	−0.0545545
$337$	−2.00000	−0.108947	−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000	−0.108947	−0.0544735	+	0.998515i	$0.517348\pi$
$338$	9.00000	0.489535
$339$	14.0000	0.760376
$340$	0	0
$341$	−32.0000	−1.73290
$342$	4.00000	0.216295
$343$	−1.00000	−0.0539949
$344$	−12.0000	−0.646997
$345$	0	0
$346$	14.0000	0.752645
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	6.00000	0.321634
$349$	−34.0000	−1.81998	−0.909989	−	0.414632i	$-0.863910\pi$
$349$	−34.0000	−1.81998	−0.909989	+	0.414632i	$0.863910\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	−4.00000	−0.213201
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	−4.00000	−0.212598
$355$	0	0
$356$	2.00000	0.106000
$357$	2.00000	0.105851
$358$	20.0000	1.05703
$359$	−8.00000	−0.422224	−0.211112	−	0.977462i	$-0.567708\pi$
$359$	−8.00000	−0.422224	−0.211112	+	0.977462i	$0.567708\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	10.0000	0.525588
$363$	5.00000	0.262432
$364$	−2.00000	−0.104828
$365$	0	0
$366$	2.00000	0.104542
$367$	32.0000	1.67039	0.835193	−	0.549957i	$-0.185356\pi$
$367$	32.0000	1.67039	0.835193	+	0.549957i	$0.185356\pi$
$368$	8.00000	0.417029
$369$	2.00000	0.104116
$370$	0	0
$371$	6.00000	0.311504
$372$	−8.00000	−0.414781
$373$	−14.0000	−0.724893	−0.362446	−	0.932005i	$-0.618058\pi$
$373$	−14.0000	−0.724893	−0.362446	+	0.932005i	$0.618058\pi$
$374$	8.00000	0.413670
$375$	0	0
$376$	−8.00000	−0.412568
$377$	12.0000	0.618031
$378$	1.00000	0.0514344
$379$	28.0000	1.43826	0.719132	−	0.694874i	$-0.244540\pi$
$379$	28.0000	1.43826	0.719132	+	0.694874i	$0.244540\pi$
$380$	0	0
$381$	0	0
$382$	16.0000	0.818631
$383$	−24.0000	−1.22634	−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000	−1.22634	−0.613171	+	0.789950i	$0.710106\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−14.0000	−0.712581
$387$	12.0000	0.609994
$388$	−10.0000	−0.507673
$389$	−18.0000	−0.912636	−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000	−0.912636	−0.456318	+	0.889817i	$0.650832\pi$
$390$	0	0
$391$	−16.0000	−0.809155
$392$	−1.00000	−0.0505076
$393$	12.0000	0.605320
$394$	6.00000	0.302276
$395$	0	0
$396$	4.00000	0.201008
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	16.0000	0.802008
$399$	4.00000	0.200250
$400$	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
$403$	−16.0000	−0.797017
$404$	6.00000	0.298511
$405$	0	0
$406$	6.00000	0.297775
$407$	8.00000	0.396545
$408$	2.00000	0.0990148
$409$	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
$411$	−10.0000	−0.493264
$412$	−8.00000	−0.394132
$413$	−4.00000	−0.196827
$414$	−8.00000	−0.393179
$415$	0	0
$416$	−2.00000	−0.0980581
$417$	20.0000	0.979404
$418$	16.0000	0.782586
$419$	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
$421$	−26.0000	−1.26716	−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000	−1.26716	−0.633581	+	0.773676i	$0.718416\pi$
$422$	−20.0000	−0.973585
$423$	8.00000	0.388973
$424$	6.00000	0.291386
$425$	0	0
$426$	−8.00000	−0.387601
$427$	2.00000	0.0967868
$428$	20.0000	0.966736
$429$	8.00000	0.386244
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	1.00000	0.0481125
$433$	6.00000	0.288342	0.144171	−	0.989553i	$-0.453949\pi$
$433$	6.00000	0.288342	0.144171	+	0.989553i	$0.453949\pi$
$434$	−8.00000	−0.384012
$435$	0	0
$436$	−2.00000	−0.0957826
$437$	−32.0000	−1.53077
$438$	−14.0000	−0.668946
$439$	32.0000	1.52728	0.763638	−	0.645644i	$-0.223411\pi$
$439$	32.0000	1.52728	0.763638	+	0.645644i	$0.223411\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	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	$0.469708\pi$
$444$	2.00000	0.0949158
$445$	0	0
$446$	0	0
$447$	−18.0000	−0.851371
$448$	−1.00000	−0.0472456
$449$	−14.0000	−0.660701	−0.330350	−	0.943858i	$-0.607167\pi$
$449$	−14.0000	−0.660701	−0.330350	+	0.943858i	$0.607167\pi$
$450$	0	0
$451$	8.00000	0.376705
$452$	14.0000	0.658505
$453$	8.00000	0.375873
$454$	12.0000	0.563188
$455$	0	0
$456$	4.00000	0.187317
$457$	38.0000	1.77757	0.888783	−	0.458329i	$-0.151552\pi$
$457$	38.0000	1.77757	0.888783	+	0.458329i	$0.151552\pi$
$458$	26.0000	1.21490
$459$	−2.00000	−0.0933520
$460$	0	0
$461$	−34.0000	−1.58354	−0.791769	−	0.610821i	$-0.790840\pi$
$461$	−34.0000	−1.58354	−0.791769	+	0.610821i	$0.790840\pi$
$462$	4.00000	0.186097
$463$	16.0000	0.743583	0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000	0.743583	0.371792	+	0.928316i	$0.378744\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	10.0000	0.463241
$467$	20.0000	0.925490	0.462745	−	0.886492i	$-0.346865\pi$
$467$	20.0000	0.925490	0.462745	+	0.886492i	$0.346865\pi$
$468$	2.00000	0.0924500
$469$	12.0000	0.554109
$470$	0	0
$471$	−14.0000	−0.645086
$472$	−4.00000	−0.184115
$473$	48.0000	2.20704
$474$	0	0
$475$	0	0
$476$	2.00000	0.0916698
$477$	−6.00000	−0.274721
$478$	16.0000	0.731823
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	−18.0000	−0.819878
$483$	−8.00000	−0.364013
$484$	5.00000	0.227273
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	24.0000	1.08754	0.543772	−	0.839233i	$-0.316996\pi$
$487$	24.0000	1.08754	0.543772	+	0.839233i	$0.316996\pi$
$488$	2.00000	0.0905357
$489$	−12.0000	−0.542659
$490$	0	0
$491$	36.0000	1.62466	0.812329	−	0.583200i	$-0.198200\pi$
$491$	36.0000	1.62466	0.812329	+	0.583200i	$0.198200\pi$
$492$	2.00000	0.0901670
$493$	−12.0000	−0.540453
$494$	8.00000	0.359937
$495$	0	0
$496$	−8.00000	−0.359211
$497$	−8.00000	−0.358849
$498$	12.0000	0.537733
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	0	0
$501$	16.0000	0.714827
$502$	−4.00000	−0.178529
$503$	−16.0000	−0.713405	−0.356702	−	0.934218i	$-0.616099\pi$
$503$	−16.0000	−0.713405	−0.356702	+	0.934218i	$0.616099\pi$
$504$	1.00000	0.0445435
$505$	0	0
$506$	−32.0000	−1.42257
$507$	−9.00000	−0.399704
$508$	0	0
$509$	14.0000	0.620539	0.310270	−	0.950649i	$-0.399581\pi$
$509$	14.0000	0.620539	0.310270	+	0.950649i	$0.399581\pi$
$510$	0	0
$511$	−14.0000	−0.619324
$512$	−1.00000	−0.0441942
$513$	−4.00000	−0.176604
$514$	2.00000	0.0882162
$515$	0	0
$516$	12.0000	0.528271
$517$	32.0000	1.40736
$518$	2.00000	0.0878750
$519$	−14.0000	−0.614532
$520$	0	0
$521$	−30.0000	−1.31432	−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000	−1.31432	−0.657162	+	0.753749i	$0.728243\pi$
$522$	−6.00000	−0.262613
$523$	28.0000	1.22435	0.612177	−	0.790721i	$-0.290294\pi$
$523$	28.0000	1.22435	0.612177	+	0.790721i	$0.290294\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	24.0000	1.04645
$527$	16.0000	0.696971
$528$	4.00000	0.174078
$529$	41.0000	1.78261
$530$	0	0
$531$	4.00000	0.173585
$532$	4.00000	0.173422
$533$	4.00000	0.173259
$534$	−2.00000	−0.0865485
$535$	0	0
$536$	12.0000	0.518321
$537$	−20.0000	−0.863064
$538$	−30.0000	−1.29339
$539$	4.00000	0.172292
$540$	0	0
$541$	14.0000	0.601907	0.300954	−	0.953639i	$-0.402695\pi$
$541$	14.0000	0.601907	0.300954	+	0.953639i	$0.402695\pi$
$542$	24.0000	1.03089
$543$	−10.0000	−0.429141
$544$	2.00000	0.0857493
$545$	0	0
$546$	2.00000	0.0855921
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	−10.0000	−0.427179
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	−24.0000	−1.02243
$552$	−8.00000	−0.340503
$553$	0	0
$554$	14.0000	0.594803
$555$	0	0
$556$	20.0000	0.848189
$557$	2.00000	0.0847427	0.0423714	−	0.999102i	$-0.486509\pi$
$557$	2.00000	0.0847427	0.0423714	+	0.999102i	$0.486509\pi$
$558$	8.00000	0.338667
$559$	24.0000	1.01509
$560$	0	0
$561$	−8.00000	−0.337760
$562$	−10.0000	−0.421825
$563$	36.0000	1.51722	0.758610	−	0.651546i	$-0.225879\pi$
$563$	36.0000	1.51722	0.758610	+	0.651546i	$0.225879\pi$
$564$	8.00000	0.336861
$565$	0	0
$566$	20.0000	0.840663
$567$	−1.00000	−0.0419961
$568$	−8.00000	−0.335673
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	−20.0000	−0.836974	−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000	−0.836974	−0.418487	+	0.908223i	$0.637439\pi$
$572$	8.00000	0.334497
$573$	−16.0000	−0.668410
$574$	2.00000	0.0834784
$575$	0	0
$576$	1.00000	0.0416667
$577$	−26.0000	−1.08239	−0.541197	−	0.840896i	$-0.682029\pi$
$577$	−26.0000	−1.08239	−0.541197	+	0.840896i	$0.682029\pi$
$578$	13.0000	0.540729
$579$	14.0000	0.581820
$580$	0	0
$581$	12.0000	0.497844
$582$	10.0000	0.414513
$583$	−24.0000	−0.993978
$584$	−14.0000	−0.579324
$585$	0	0
$586$	6.00000	0.247858
$587$	12.0000	0.495293	0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000	0.495293	0.247647	+	0.968850i	$0.420343\pi$
$588$	1.00000	0.0412393
$589$	32.0000	1.31854
$590$	0	0
$591$	−6.00000	−0.246807
$592$	2.00000	0.0821995
$593$	−18.0000	−0.739171	−0.369586	−	0.929197i	$-0.620500\pi$
$593$	−18.0000	−0.739171	−0.369586	+	0.929197i	$0.620500\pi$
$594$	−4.00000	−0.164122
$595$	0	0
$596$	−18.0000	−0.737309
$597$	−16.0000	−0.654836
$598$	−16.0000	−0.654289
$599$	8.00000	0.326871	0.163436	−	0.986554i	$-0.447742\pi$
$599$	8.00000	0.326871	0.163436	+	0.986554i	$0.447742\pi$
$600$	0	0
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	12.0000	0.489083
$603$	−12.0000	−0.488678
$604$	8.00000	0.325515
$605$	0	0
$606$	−6.00000	−0.243733
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	4.00000	0.162221
$609$	−6.00000	−0.243132
$610$	0	0
$611$	16.0000	0.647291
$612$	−2.00000	−0.0808452
$613$	34.0000	1.37325	0.686624	−	0.727013i	$-0.259092\pi$
$613$	34.0000	1.37325	0.686624	+	0.727013i	$0.259092\pi$
$614$	−4.00000	−0.161427
$615$	0	0
$616$	4.00000	0.161165
$617$	−10.0000	−0.402585	−0.201292	−	0.979531i	$-0.564514\pi$
$617$	−10.0000	−0.402585	−0.201292	+	0.979531i	$0.564514\pi$
$618$	8.00000	0.321807
$619$	−44.0000	−1.76851	−0.884255	−	0.467005i	$-0.845333\pi$
$619$	−44.0000	−1.76851	−0.884255	+	0.467005i	$0.845333\pi$
$620$	0	0
$621$	8.00000	0.321029
$622$	−8.00000	−0.320771
$623$	−2.00000	−0.0801283
$624$	2.00000	0.0800641
$625$	0	0
$626$	34.0000	1.35891
$627$	−16.0000	−0.638978
$628$	−14.0000	−0.558661
$629$	−4.00000	−0.159490
$630$	0	0
$631$	24.0000	0.955425	0.477712	−	0.878516i	$-0.341466\pi$
$631$	24.0000	0.955425	0.477712	+	0.878516i	$0.341466\pi$
$632$	0	0
$633$	20.0000	0.794929
$634$	14.0000	0.556011
$635$	0	0
$636$	−6.00000	−0.237915
$637$	2.00000	0.0792429
$638$	−24.0000	−0.950169
$639$	8.00000	0.316475
$640$	0	0
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	−20.0000	−0.789337
$643$	20.0000	0.788723	0.394362	−	0.918955i	$-0.370966\pi$
$643$	20.0000	0.788723	0.394362	+	0.918955i	$0.370966\pi$
$644$	−8.00000	−0.315244
$645$	0	0
$646$	−8.00000	−0.314756
$647$	32.0000	1.25805	0.629025	−	0.777385i	$-0.283454\pi$
$647$	32.0000	1.25805	0.629025	+	0.777385i	$0.283454\pi$
$648$	−1.00000	−0.0392837
$649$	16.0000	0.628055
$650$	0	0
$651$	8.00000	0.313545
$652$	−12.0000	−0.469956
$653$	34.0000	1.33052	0.665261	−	0.746611i	$-0.268320\pi$
$653$	34.0000	1.33052	0.665261	+	0.746611i	$0.268320\pi$
$654$	2.00000	0.0782062
$655$	0	0
$656$	2.00000	0.0780869
$657$	14.0000	0.546192
$658$	8.00000	0.311872
$659$	28.0000	1.09073	0.545363	−	0.838200i	$-0.316392\pi$
$659$	28.0000	1.09073	0.545363	+	0.838200i	$0.316392\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	−28.0000	−1.08825
$663$	−4.00000	−0.155347
$664$	12.0000	0.465690
$665$	0	0
$666$	−2.00000	−0.0774984
$667$	48.0000	1.85857
$668$	16.0000	0.619059
$669$	0	0
$670$	0	0
$671$	−8.00000	−0.308837
$672$	1.00000	0.0385758
$673$	−2.00000	−0.0770943	−0.0385472	−	0.999257i	$-0.512273\pi$
$673$	−2.00000	−0.0770943	−0.0385472	+	0.999257i	$0.512273\pi$
$674$	2.00000	0.0770371
$675$	0	0
$676$	−9.00000	−0.346154
$677$	−6.00000	−0.230599	−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000	−0.230599	−0.115299	+	0.993331i	$0.536783\pi$
$678$	−14.0000	−0.537667
$679$	10.0000	0.383765
$680$	0	0
$681$	−12.0000	−0.459841
$682$	32.0000	1.22534
$683$	4.00000	0.153056	0.0765279	−	0.997067i	$-0.475617\pi$
$683$	4.00000	0.153056	0.0765279	+	0.997067i	$0.475617\pi$
$684$	−4.00000	−0.152944
$685$	0	0
$686$	1.00000	0.0381802
$687$	−26.0000	−0.991962
$688$	12.0000	0.457496
$689$	−12.0000	−0.457164
$690$	0	0
$691$	−20.0000	−0.760836	−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000	−0.760836	−0.380418	+	0.924815i	$0.624220\pi$
$692$	−14.0000	−0.532200
$693$	−4.00000	−0.151947
$694$	12.0000	0.455514
$695$	0	0
$696$	−6.00000	−0.227429
$697$	−4.00000	−0.151511
$698$	34.0000	1.28692
$699$	−10.0000	−0.378235
$700$	0	0
$701$	6.00000	0.226617	0.113308	−	0.993560i	$-0.463855\pi$
$701$	6.00000	0.226617	0.113308	+	0.993560i	$0.463855\pi$
$702$	−2.00000	−0.0754851
$703$	−8.00000	−0.301726
$704$	4.00000	0.150756
$705$	0	0
$706$	18.0000	0.677439
$707$	−6.00000	−0.225653
$708$	4.00000	0.150329
$709$	−26.0000	−0.976450	−0.488225	−	0.872718i	$-0.662356\pi$
$709$	−26.0000	−0.976450	−0.488225	+	0.872718i	$0.662356\pi$
$710$	0	0
$711$	0	0
$712$	−2.00000	−0.0749532
$713$	−64.0000	−2.39682
$714$	−2.00000	−0.0748481
$715$	0	0
$716$	−20.0000	−0.747435
$717$	−16.0000	−0.597531
$718$	8.00000	0.298557
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	8.00000	0.297936
$722$	3.00000	0.111648
$723$	18.0000	0.669427
$724$	−10.0000	−0.371647
$725$	0	0
$726$	−5.00000	−0.185567
$727$	24.0000	0.890111	0.445055	−	0.895503i	$-0.353184\pi$
$727$	24.0000	0.890111	0.445055	+	0.895503i	$0.353184\pi$
$728$	2.00000	0.0741249
$729$	1.00000	0.0370370
$730$	0	0
$731$	−24.0000	−0.887672
$732$	−2.00000	−0.0739221
$733$	−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
$736$	−8.00000	−0.294884
$737$	−48.0000	−1.76810
$738$	−2.00000	−0.0736210
$739$	36.0000	1.32428	0.662141	−	0.749380i	$-0.269648\pi$
$739$	36.0000	1.32428	0.662141	+	0.749380i	$0.269648\pi$
$740$	0	0
$741$	−8.00000	−0.293887
$742$	−6.00000	−0.220267
$743$	−24.0000	−0.880475	−0.440237	−	0.897881i	$-0.645106\pi$
$743$	−24.0000	−0.880475	−0.440237	+	0.897881i	$0.645106\pi$
$744$	8.00000	0.293294
$745$	0	0
$746$	14.0000	0.512576
$747$	−12.0000	−0.439057
$748$	−8.00000	−0.292509
$749$	−20.0000	−0.730784
$750$	0	0
$751$	−32.0000	−1.16770	−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000	−1.16770	−0.583848	+	0.811863i	$0.698454\pi$
$752$	8.00000	0.291730
$753$	4.00000	0.145768
$754$	−12.0000	−0.437014
$755$	0	0
$756$	−1.00000	−0.0363696
$757$	34.0000	1.23575	0.617876	−	0.786276i	$-0.287994\pi$
$757$	34.0000	1.23575	0.617876	+	0.786276i	$0.287994\pi$
$758$	−28.0000	−1.01701
$759$	32.0000	1.16153
$760$	0	0
$761$	2.00000	0.0724999	0.0362500	−	0.999343i	$-0.488459\pi$
$761$	2.00000	0.0724999	0.0362500	+	0.999343i	$0.488459\pi$
$762$	0	0
$763$	2.00000	0.0724049
$764$	−16.0000	−0.578860
$765$	0	0
$766$	24.0000	0.867155
$767$	8.00000	0.288863
$768$	1.00000	0.0360844
$769$	−14.0000	−0.504853	−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000	−0.504853	−0.252426	+	0.967616i	$0.581229\pi$
$770$	0	0
$771$	−2.00000	−0.0720282
$772$	14.0000	0.503871
$773$	26.0000	0.935155	0.467578	−	0.883952i	$-0.345127\pi$
$773$	26.0000	0.935155	0.467578	+	0.883952i	$0.345127\pi$
$774$	−12.0000	−0.431331
$775$	0	0
$776$	10.0000	0.358979
$777$	−2.00000	−0.0717496
$778$	18.0000	0.645331
$779$	−8.00000	−0.286630
$780$	0	0
$781$	32.0000	1.14505
$782$	16.0000	0.572159
$783$	6.00000	0.214423
$784$	1.00000	0.0357143
$785$	0	0
$786$	−12.0000	−0.428026
$787$	4.00000	0.142585	0.0712923	−	0.997455i	$-0.477288\pi$
$787$	4.00000	0.142585	0.0712923	+	0.997455i	$0.477288\pi$
$788$	−6.00000	−0.213741
$789$	−24.0000	−0.854423
$790$	0	0
$791$	−14.0000	−0.497783
$792$	−4.00000	−0.142134
$793$	−4.00000	−0.142044
$794$	−2.00000	−0.0709773
$795$	0	0
$796$	−16.0000	−0.567105
$797$	−30.0000	−1.06265	−0.531327	−	0.847167i	$-0.678307\pi$
$797$	−30.0000	−1.06265	−0.531327	+	0.847167i	$0.678307\pi$
$798$	−4.00000	−0.141598
$799$	−16.0000	−0.566039
$800$	0	0
$801$	2.00000	0.0706665
$802$	−18.0000	−0.635602
$803$	56.0000	1.97620
$804$	−12.0000	−0.423207
$805$	0	0
$806$	16.0000	0.563576
$807$	30.0000	1.05605
$808$	−6.00000	−0.211079
$809$	−38.0000	−1.33601	−0.668004	−	0.744157i	$-0.732851\pi$
$809$	−38.0000	−1.33601	−0.668004	+	0.744157i	$0.732851\pi$
$810$	0	0
$811$	36.0000	1.26413	0.632065	−	0.774915i	$-0.282207\pi$
$811$	36.0000	1.26413	0.632065	+	0.774915i	$0.282207\pi$
$812$	−6.00000	−0.210559
$813$	−24.0000	−0.841717
$814$	−8.00000	−0.280400
$815$	0	0
$816$	−2.00000	−0.0700140
$817$	−48.0000	−1.67931
$818$	−10.0000	−0.349642
$819$	−2.00000	−0.0698857
$820$	0	0
$821$	46.0000	1.60541	0.802706	−	0.596376i	$-0.203393\pi$
$821$	46.0000	1.60541	0.802706	+	0.596376i	$0.203393\pi$
$822$	10.0000	0.348790
$823$	8.00000	0.278862	0.139431	−	0.990232i	$-0.455473\pi$
$823$	8.00000	0.278862	0.139431	+	0.990232i	$0.455473\pi$
$824$	8.00000	0.278693
$825$	0	0
$826$	4.00000	0.139178
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	8.00000	0.278019
$829$	−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
$831$	−14.0000	−0.485655
$832$	2.00000	0.0693375
$833$	−2.00000	−0.0692959
$834$	−20.0000	−0.692543
$835$	0	0
$836$	−16.0000	−0.553372
$837$	−8.00000	−0.276520
$838$	−12.0000	−0.414533
$839$	−24.0000	−0.828572	−0.414286	−	0.910147i	$-0.635969\pi$
$839$	−24.0000	−0.828572	−0.414286	+	0.910147i	$0.635969\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	26.0000	0.896019
$843$	10.0000	0.344418
$844$	20.0000	0.688428
$845$	0	0
$846$	−8.00000	−0.275046
$847$	−5.00000	−0.171802
$848$	−6.00000	−0.206041
$849$	−20.0000	−0.686398
$850$	0	0
$851$	16.0000	0.548473
$852$	8.00000	0.274075
$853$	26.0000	0.890223	0.445112	−	0.895475i	$-0.353164\pi$
$853$	26.0000	0.890223	0.445112	+	0.895475i	$0.353164\pi$
$854$	−2.00000	−0.0684386
$855$	0	0
$856$	−20.0000	−0.683586
$857$	22.0000	0.751506	0.375753	−	0.926720i	$-0.377384\pi$
$857$	22.0000	0.751506	0.375753	+	0.926720i	$0.377384\pi$
$858$	−8.00000	−0.273115
$859$	−28.0000	−0.955348	−0.477674	−	0.878537i	$-0.658520\pi$
$859$	−28.0000	−0.955348	−0.477674	+	0.878537i	$0.658520\pi$
$860$	0	0
$861$	−2.00000	−0.0681598
$862$	0	0
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	−6.00000	−0.203888
$867$	−13.0000	−0.441503
$868$	8.00000	0.271538
$869$	0	0
$870$	0	0
$871$	−24.0000	−0.813209
$872$	2.00000	0.0677285
$873$	−10.0000	−0.338449
$874$	32.0000	1.08242
$875$	0	0
$876$	14.0000	0.473016
$877$	10.0000	0.337676	0.168838	−	0.985644i	$-0.445999\pi$
$877$	10.0000	0.337676	0.168838	+	0.985644i	$0.445999\pi$
$878$	−32.0000	−1.07995
$879$	−6.00000	−0.202375
$880$	0	0
$881$	42.0000	1.41502	0.707508	−	0.706705i	$-0.249819\pi$
$881$	42.0000	1.41502	0.707508	+	0.706705i	$0.249819\pi$
$882$	−1.00000	−0.0336718
$883$	−28.0000	−0.942275	−0.471138	−	0.882060i	$-0.656156\pi$
$883$	−28.0000	−0.942275	−0.471138	+	0.882060i	$0.656156\pi$
$884$	−4.00000	−0.134535
$885$	0	0
$886$	−4.00000	−0.134383
$887$	−16.0000	−0.537227	−0.268614	−	0.963248i	$-0.586566\pi$
$887$	−16.0000	−0.537227	−0.268614	+	0.963248i	$0.586566\pi$
$888$	−2.00000	−0.0671156
$889$	0	0
$890$	0	0
$891$	4.00000	0.134005
$892$	0	0
$893$	−32.0000	−1.07084
$894$	18.0000	0.602010
$895$	0	0
$896$	1.00000	0.0334077
$897$	16.0000	0.534224
$898$	14.0000	0.467186
$899$	−48.0000	−1.60089
$900$	0	0
$901$	12.0000	0.399778
$902$	−8.00000	−0.266371
$903$	−12.0000	−0.399335
$904$	−14.0000	−0.465633
$905$	0	0
$906$	−8.00000	−0.265782
$907$	12.0000	0.398453	0.199227	−	0.979953i	$-0.436157\pi$
$907$	12.0000	0.398453	0.199227	+	0.979953i	$0.436157\pi$
$908$	−12.0000	−0.398234
$909$	6.00000	0.199007
$910$	0	0
$911$	48.0000	1.59031	0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000	1.59031	0.795155	+	0.606406i	$0.207389\pi$
$912$	−4.00000	−0.132453
$913$	−48.0000	−1.58857
$914$	−38.0000	−1.25693
$915$	0	0
$916$	−26.0000	−0.859064
$917$	−12.0000	−0.396275
$918$	2.00000	0.0660098
$919$	−8.00000	−0.263896	−0.131948	−	0.991257i	$-0.542123\pi$
$919$	−8.00000	−0.263896	−0.131948	+	0.991257i	$0.542123\pi$
$920$	0	0
$921$	4.00000	0.131804
$922$	34.0000	1.11973
$923$	16.0000	0.526646
$924$	−4.00000	−0.131590
$925$	0	0
$926$	−16.0000	−0.525793
$927$	−8.00000	−0.262754
$928$	−6.00000	−0.196960
$929$	10.0000	0.328089	0.164045	−	0.986453i	$-0.447546\pi$
$929$	10.0000	0.328089	0.164045	+	0.986453i	$0.447546\pi$
$930$	0	0
$931$	−4.00000	−0.131095
$932$	−10.0000	−0.327561
$933$	8.00000	0.261908
$934$	−20.0000	−0.654420
$935$	0	0
$936$	−2.00000	−0.0653720
$937$	−18.0000	−0.588034	−0.294017	−	0.955800i	$-0.594992\pi$
$937$	−18.0000	−0.588034	−0.294017	+	0.955800i	$0.594992\pi$
$938$	−12.0000	−0.391814
$939$	−34.0000	−1.10955
$940$	0	0
$941$	−18.0000	−0.586783	−0.293392	−	0.955992i	$-0.594784\pi$
$941$	−18.0000	−0.586783	−0.293392	+	0.955992i	$0.594784\pi$
$942$	14.0000	0.456145
$943$	16.0000	0.521032
$944$	4.00000	0.130189
$945$	0	0
$946$	−48.0000	−1.56061
$947$	12.0000	0.389948	0.194974	−	0.980808i	$-0.437538\pi$
$947$	12.0000	0.389948	0.194974	+	0.980808i	$0.437538\pi$
$948$	0	0
$949$	28.0000	0.908918
$950$	0	0
$951$	−14.0000	−0.453981
$952$	−2.00000	−0.0648204
$953$	6.00000	0.194359	0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000	0.194359	0.0971795	+	0.995267i	$0.469018\pi$
$954$	6.00000	0.194257
$955$	0	0
$956$	−16.0000	−0.517477
$957$	24.0000	0.775810
$958$	0	0
$959$	10.0000	0.322917
$960$	0	0
$961$	33.0000	1.06452
$962$	−4.00000	−0.128965
$963$	20.0000	0.644491
$964$	18.0000	0.579741
$965$	0	0
$966$	8.00000	0.257396
$967$	8.00000	0.257263	0.128631	−	0.991692i	$-0.458942\pi$
$967$	8.00000	0.257263	0.128631	+	0.991692i	$0.458942\pi$
$968$	−5.00000	−0.160706
$969$	8.00000	0.256997
$970$	0	0
$971$	36.0000	1.15529	0.577647	−	0.816286i	$-0.303971\pi$
$971$	36.0000	1.15529	0.577647	+	0.816286i	$0.303971\pi$
$972$	1.00000	0.0320750
$973$	−20.0000	−0.641171
$974$	−24.0000	−0.769010
$975$	0	0
$976$	−2.00000	−0.0640184
$977$	−18.0000	−0.575871	−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000	−0.575871	−0.287936	+	0.957650i	$0.592969\pi$
$978$	12.0000	0.383718
$979$	8.00000	0.255681
$980$	0	0
$981$	−2.00000	−0.0638551
$982$	−36.0000	−1.14881
$983$	−32.0000	−1.02064	−0.510321	−	0.859984i	$-0.670473\pi$
$983$	−32.0000	−1.02064	−0.510321	+	0.859984i	$0.670473\pi$
$984$	−2.00000	−0.0637577
$985$	0	0
$986$	12.0000	0.382158
$987$	−8.00000	−0.254643
$988$	−8.00000	−0.254514
$989$	96.0000	3.05262
$990$	0	0
$991$	−32.0000	−1.01651	−0.508257	−	0.861206i	$-0.669710\pi$
$991$	−32.0000	−1.01651	−0.508257	+	0.861206i	$0.669710\pi$
$992$	8.00000	0.254000
$993$	28.0000	0.888553
$994$	8.00000	0.253745
$995$	0	0
$996$	−12.0000	−0.380235
$997$	−38.0000	−1.20347	−0.601736	−	0.798695i	$-0.705524\pi$
$997$	−38.0000	−1.20347	−0.601736	+	0.798695i	$0.705524\pi$
$998$	−4.00000	−0.126618
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1050.2.a.h.1.1		1
3.2	odd	2		3150.2.a.w.1.1		1
4.3	odd	2		8400.2.a.p.1.1		1
5.2	odd	4		1050.2.g.d.799.1		2
5.3	odd	4		1050.2.g.d.799.2		2
5.4	even	2		210.2.a.c.1.1	✓	1
7.6	odd	2		7350.2.a.p.1.1		1
15.2	even	4		3150.2.g.e.2899.2		2
15.8	even	4		3150.2.g.e.2899.1		2
15.14	odd	2		630.2.a.b.1.1		1
20.19	odd	2		1680.2.a.q.1.1		1
35.4	even	6		1470.2.i.f.961.1		2
35.9	even	6		1470.2.i.f.361.1		2
35.19	odd	6		1470.2.i.b.361.1		2
35.24	odd	6		1470.2.i.b.961.1		2
35.34	odd	2		1470.2.a.q.1.1		1
40.19	odd	2		6720.2.a.k.1.1		1
40.29	even	2		6720.2.a.bp.1.1		1
60.59	even	2		5040.2.a.i.1.1		1
105.104	even	2		4410.2.a.l.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.2.a.c.1.1	✓	1	5.4	even	2
630.2.a.b.1.1		1	15.14	odd	2
1050.2.a.h.1.1		1	1.1	even	1	trivial
1050.2.g.d.799.1		2	5.2	odd	4
1050.2.g.d.799.2		2	5.3	odd	4
1470.2.a.q.1.1		1	35.34	odd	2
1470.2.i.b.361.1		2	35.19	odd	6
1470.2.i.b.961.1		2	35.24	odd	6
1470.2.i.f.361.1		2	35.9	even	6
1470.2.i.f.961.1		2	35.4	even	6
1680.2.a.q.1.1		1	20.19	odd	2
3150.2.a.w.1.1		1	3.2	odd	2
3150.2.g.e.2899.1		2	15.8	even	4
3150.2.g.e.2899.2		2	15.2	even	4
4410.2.a.l.1.1		1	105.104	even	2
5040.2.a.i.1.1		1	60.59	even	2
6720.2.a.k.1.1		1	40.19	odd	2
6720.2.a.bp.1.1		1	40.29	even	2
7350.2.a.p.1.1		1	7.6	odd	2
8400.2.a.p.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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