LMFDB - Embedded newform 210.2.a.e.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [210,2,Mod(1,210)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("210.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(210, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 210 = 2 \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	210.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,1,1,1,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$1.67685844245$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	210.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -4.00000 q^{11} +1.00000 q^{12} -2.00000 q^{13} -1.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} +4.00000 q^{19} +1.00000 q^{20} -1.00000 q^{21} -4.00000 q^{22} -8.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} -2.00000 q^{29} +1.00000 q^{30} +1.00000 q^{32} -4.00000 q^{33} +2.00000 q^{34} -1.00000 q^{35} +1.00000 q^{36} +6.00000 q^{37} +4.00000 q^{38} -2.00000 q^{39} +1.00000 q^{40} -6.00000 q^{41} -1.00000 q^{42} -4.00000 q^{43} -4.00000 q^{44} +1.00000 q^{45} -8.00000 q^{46} +1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} +2.00000 q^{51} -2.00000 q^{52} -10.0000 q^{53} +1.00000 q^{54} -4.00000 q^{55} -1.00000 q^{56} +4.00000 q^{57} -2.00000 q^{58} +12.0000 q^{59} +1.00000 q^{60} +14.0000 q^{61} -1.00000 q^{63} +1.00000 q^{64} -2.00000 q^{65} -4.00000 q^{66} -12.0000 q^{67} +2.00000 q^{68} -8.00000 q^{69} -1.00000 q^{70} -8.00000 q^{71} +1.00000 q^{72} +10.0000 q^{73} +6.00000 q^{74} +1.00000 q^{75} +4.00000 q^{76} +4.00000 q^{77} -2.00000 q^{78} +16.0000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} -12.0000 q^{83} -1.00000 q^{84} +2.00000 q^{85} -4.00000 q^{86} -2.00000 q^{87} -4.00000 q^{88} +10.0000 q^{89} +1.00000 q^{90} +2.00000 q^{91} -8.00000 q^{92} +4.00000 q^{95} +1.00000 q^{96} +2.00000 q^{97} +1.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

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
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	1.00000	0.408248
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	1.00000	0.288675
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	−1.00000	−0.267261
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	1.00000	0.235702
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	1.00000	0.223607
$21$	−1.00000	−0.218218
$22$	−4.00000	−0.852803
$23$	−8.00000	−1.66812	−0.834058	−	0.551677i	$-0.813988\pi$
$23$	−8.00000	−1.66812	−0.834058	+	0.551677i	$0.813988\pi$
$24$	1.00000	0.204124
$25$	1.00000	0.200000
$26$	−2.00000	−0.392232
$27$	1.00000	0.192450
$28$	−1.00000	−0.188982
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	1.00000	0.182574
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	1.00000	0.176777
$33$	−4.00000	−0.696311
$34$	2.00000	0.342997
$35$	−1.00000	−0.169031
$36$	1.00000	0.166667
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	4.00000	0.648886
$39$	−2.00000	−0.320256
$40$	1.00000	0.158114
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	−1.00000	−0.154303
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	−4.00000	−0.603023
$45$	1.00000	0.149071
$46$	−8.00000	−1.17954
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	1.00000	0.144338
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	2.00000	0.280056
$52$	−2.00000	−0.277350
$53$	−10.0000	−1.37361	−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000	−1.37361	−0.686803	+	0.726844i	$0.740986\pi$
$54$	1.00000	0.136083
$55$	−4.00000	−0.539360
$56$	−1.00000	−0.133631
$57$	4.00000	0.529813
$58$	−2.00000	−0.262613
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	1.00000	0.129099
$61$	14.0000	1.79252	0.896258	−	0.443533i	$-0.146275\pi$
$61$	14.0000	1.79252	0.896258	+	0.443533i	$0.146275\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	−2.00000	−0.248069
$66$	−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
$70$	−1.00000	−0.119523
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	1.00000	0.117851
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	6.00000	0.697486
$75$	1.00000	0.115470
$76$	4.00000	0.458831
$77$	4.00000	0.455842
$78$	−2.00000	−0.226455
$79$	16.0000	1.80014	0.900070	−	0.435745i	$-0.143515\pi$
$79$	16.0000	1.80014	0.900070	+	0.435745i	$0.143515\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	−6.00000	−0.662589
$83$	−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
$85$	2.00000	0.216930
$86$	−4.00000	−0.431331
$87$	−2.00000	−0.214423
$88$	−4.00000	−0.426401
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	1.00000	0.105409
$91$	2.00000	0.209657
$92$	−8.00000	−0.834058
$93$	0	0
$94$	0	0
$95$	4.00000	0.410391
$96$	1.00000	0.102062
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	1.00000	0.101015
$99$	−4.00000	−0.402015
$100$	1.00000	0.100000
$101$	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
$105$	−1.00000	−0.0975900
$106$	−10.0000	−0.971286
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	1.00000	0.0962250
$109$	14.0000	1.34096	0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000	1.34096	0.670478	+	0.741929i	$0.266089\pi$
$110$	−4.00000	−0.381385
$111$	6.00000	0.569495
$112$	−1.00000	−0.0944911
$113$	18.0000	1.69330	0.846649	−	0.532152i	$-0.178617\pi$
$113$	18.0000	1.69330	0.846649	+	0.532152i	$0.178617\pi$
$114$	4.00000	0.374634
$115$	−8.00000	−0.746004
$116$	−2.00000	−0.185695
$117$	−2.00000	−0.184900
$118$	12.0000	1.10469
$119$	−2.00000	−0.183340
$120$	1.00000	0.0912871
$121$	5.00000	0.454545
$122$	14.0000	1.26750
$123$	−6.00000	−0.541002
$124$	0	0
$125$	1.00000	0.0894427
$126$	−1.00000	−0.0890871
$127$	−16.0000	−1.41977	−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000	−1.41977	−0.709885	+	0.704317i	$0.751253\pi$
$128$	1.00000	0.0883883
$129$	−4.00000	−0.352180
$130$	−2.00000	−0.175412
$131$	20.0000	1.74741	0.873704	−	0.486458i	$-0.161711\pi$
$131$	20.0000	1.74741	0.873704	+	0.486458i	$0.161711\pi$
$132$	−4.00000	−0.348155
$133$	−4.00000	−0.346844
$134$	−12.0000	−1.03664
$135$	1.00000	0.0860663
$136$	2.00000	0.171499
$137$	10.0000	0.854358	0.427179	−	0.904167i	$-0.359507\pi$
$137$	10.0000	0.854358	0.427179	+	0.904167i	$0.359507\pi$
$138$	−8.00000	−0.681005
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	−1.00000	−0.0845154
$141$	0	0
$142$	−8.00000	−0.671345
$143$	8.00000	0.668994
$144$	1.00000	0.0833333
$145$	−2.00000	−0.166091
$146$	10.0000	0.827606
$147$	1.00000	0.0824786
$148$	6.00000	0.493197
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	1.00000	0.0816497
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	4.00000	0.324443
$153$	2.00000	0.161690
$154$	4.00000	0.322329
$155$	0	0
$156$	−2.00000	−0.160128
$157$	−18.0000	−1.43656	−0.718278	−	0.695756i	$-0.755069\pi$
$157$	−18.0000	−1.43656	−0.718278	+	0.695756i	$0.755069\pi$
$158$	16.0000	1.27289
$159$	−10.0000	−0.793052
$160$	1.00000	0.0790569
$161$	8.00000	0.630488
$162$	1.00000	0.0785674
$163$	20.0000	1.56652	0.783260	−	0.621694i	$-0.213555\pi$
$163$	20.0000	1.56652	0.783260	+	0.621694i	$0.213555\pi$
$164$	−6.00000	−0.468521
$165$	−4.00000	−0.311400
$166$	−12.0000	−0.931381
$167$	−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
$170$	2.00000	0.153393
$171$	4.00000	0.305888
$172$	−4.00000	−0.304997
$173$	−18.0000	−1.36851	−0.684257	−	0.729241i	$-0.739873\pi$
$173$	−18.0000	−1.36851	−0.684257	+	0.729241i	$0.739873\pi$
$174$	−2.00000	−0.151620
$175$	−1.00000	−0.0755929
$176$	−4.00000	−0.301511
$177$	12.0000	0.901975
$178$	10.0000	0.749532
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	1.00000	0.0745356
$181$	6.00000	0.445976	0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000	0.445976	0.222988	+	0.974821i	$0.428419\pi$
$182$	2.00000	0.148250
$183$	14.0000	1.03491
$184$	−8.00000	−0.589768
$185$	6.00000	0.441129
$186$	0	0
$187$	−8.00000	−0.585018
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	4.00000	0.290191
$191$	−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
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	2.00000	0.143592
$195$	−2.00000	−0.143223
$196$	1.00000	0.0714286
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	−4.00000	−0.284268
$199$	−24.0000	−1.70131	−0.850657	−	0.525720i	$-0.823796\pi$
$199$	−24.0000	−1.70131	−0.850657	+	0.525720i	$0.823796\pi$
$200$	1.00000	0.0707107
$201$	−12.0000	−0.846415
$202$	6.00000	0.422159
$203$	2.00000	0.140372
$204$	2.00000	0.140028
$205$	−6.00000	−0.419058
$206$	−8.00000	−0.557386
$207$	−8.00000	−0.556038
$208$	−2.00000	−0.138675
$209$	−16.0000	−1.10674
$210$	−1.00000	−0.0690066
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	−10.0000	−0.686803
$213$	−8.00000	−0.548151
$214$	12.0000	0.820303
$215$	−4.00000	−0.272798
$216$	1.00000	0.0680414
$217$	0	0
$218$	14.0000	0.948200
$219$	10.0000	0.675737
$220$	−4.00000	−0.269680
$221$	−4.00000	−0.269069
$222$	6.00000	0.402694
$223$	−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
$225$	1.00000	0.0666667
$226$	18.0000	1.19734
$227$	4.00000	0.265489	0.132745	−	0.991150i	$-0.457621\pi$
$227$	4.00000	0.265489	0.132745	+	0.991150i	$0.457621\pi$
$228$	4.00000	0.264906
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	−8.00000	−0.527504
$231$	4.00000	0.263181
$232$	−2.00000	−0.131306
$233$	−22.0000	−1.44127	−0.720634	−	0.693316i	$-0.756149\pi$
$233$	−22.0000	−1.44127	−0.720634	+	0.693316i	$0.756149\pi$
$234$	−2.00000	−0.130744
$235$	0	0
$236$	12.0000	0.781133
$237$	16.0000	1.03931
$238$	−2.00000	−0.129641
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	1.00000	0.0645497
$241$	−14.0000	−0.901819	−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000	−0.901819	−0.450910	+	0.892570i	$0.648900\pi$
$242$	5.00000	0.321412
$243$	1.00000	0.0641500
$244$	14.0000	0.896258
$245$	1.00000	0.0638877
$246$	−6.00000	−0.382546
$247$	−8.00000	−0.509028
$248$	0	0
$249$	−12.0000	−0.760469
$250$	1.00000	0.0632456
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	−1.00000	−0.0629941
$253$	32.0000	2.01182
$254$	−16.0000	−1.00393
$255$	2.00000	0.125245
$256$	1.00000	0.0625000
$257$	−14.0000	−0.873296	−0.436648	−	0.899632i	$-0.643834\pi$
$257$	−14.0000	−0.873296	−0.436648	+	0.899632i	$0.643834\pi$
$258$	−4.00000	−0.249029
$259$	−6.00000	−0.372822
$260$	−2.00000	−0.124035
$261$	−2.00000	−0.123797
$262$	20.0000	1.23560
$263$	8.00000	0.493301	0.246651	−	0.969104i	$-0.420670\pi$
$263$	8.00000	0.493301	0.246651	+	0.969104i	$0.420670\pi$
$264$	−4.00000	−0.246183
$265$	−10.0000	−0.614295
$266$	−4.00000	−0.245256
$267$	10.0000	0.611990
$268$	−12.0000	−0.733017
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	1.00000	0.0608581
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	2.00000	0.121268
$273$	2.00000	0.121046
$274$	10.0000	0.604122
$275$	−4.00000	−0.241209
$276$	−8.00000	−0.481543
$277$	22.0000	1.32185	0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000	1.32185	0.660926	+	0.750451i	$0.270164\pi$
$278$	−4.00000	−0.239904
$279$	0	0
$280$	−1.00000	−0.0597614
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	0	0
$283$	28.0000	1.66443	0.832214	−	0.554455i	$-0.187073\pi$
$283$	28.0000	1.66443	0.832214	+	0.554455i	$0.187073\pi$
$284$	−8.00000	−0.474713
$285$	4.00000	0.236940
$286$	8.00000	0.473050
$287$	6.00000	0.354169
$288$	1.00000	0.0589256
$289$	−13.0000	−0.764706
$290$	−2.00000	−0.117444
$291$	2.00000	0.117242
$292$	10.0000	0.585206
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	1.00000	0.0583212
$295$	12.0000	0.698667
$296$	6.00000	0.348743
$297$	−4.00000	−0.232104
$298$	−10.0000	−0.579284
$299$	16.0000	0.925304
$300$	1.00000	0.0577350
$301$	4.00000	0.230556
$302$	−8.00000	−0.460348
$303$	6.00000	0.344691
$304$	4.00000	0.229416
$305$	14.0000	0.801638
$306$	2.00000	0.114332
$307$	20.0000	1.14146	0.570730	−	0.821138i	$-0.306660\pi$
$307$	20.0000	1.14146	0.570730	+	0.821138i	$0.306660\pi$
$308$	4.00000	0.227921
$309$	−8.00000	−0.455104
$310$	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
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	−18.0000	−1.01580
$315$	−1.00000	−0.0563436
$316$	16.0000	0.900070
$317$	−2.00000	−0.112331	−0.0561656	−	0.998421i	$-0.517887\pi$
$317$	−2.00000	−0.112331	−0.0561656	+	0.998421i	$0.517887\pi$
$318$	−10.0000	−0.560772
$319$	8.00000	0.447914
$320$	1.00000	0.0559017
$321$	12.0000	0.669775
$322$	8.00000	0.445823
$323$	8.00000	0.445132
$324$	1.00000	0.0555556
$325$	−2.00000	−0.110940
$326$	20.0000	1.10770
$327$	14.0000	0.774202
$328$	−6.00000	−0.331295
$329$	0	0
$330$	−4.00000	−0.220193
$331$	12.0000	0.659580	0.329790	−	0.944054i	$-0.393022\pi$
$331$	12.0000	0.659580	0.329790	+	0.944054i	$0.393022\pi$
$332$	−12.0000	−0.658586
$333$	6.00000	0.328798
$334$	−8.00000	−0.437741
$335$	−12.0000	−0.655630
$336$	−1.00000	−0.0545545
$337$	18.0000	0.980522	0.490261	−	0.871576i	$-0.336901\pi$
$337$	18.0000	0.980522	0.490261	+	0.871576i	$0.336901\pi$
$338$	−9.00000	−0.489535
$339$	18.0000	0.977626
$340$	2.00000	0.108465
$341$	0	0
$342$	4.00000	0.216295
$343$	−1.00000	−0.0539949
$344$	−4.00000	−0.215666
$345$	−8.00000	−0.430706
$346$	−18.0000	−0.967686
$347$	−4.00000	−0.214731	−0.107366	−	0.994220i	$-0.534242\pi$
$347$	−4.00000	−0.214731	−0.107366	+	0.994220i	$0.534242\pi$
$348$	−2.00000	−0.107211
$349$	14.0000	0.749403	0.374701	−	0.927146i	$-0.377745\pi$
$349$	14.0000	0.749403	0.374701	+	0.927146i	$0.377745\pi$
$350$	−1.00000	−0.0534522
$351$	−2.00000	−0.106752
$352$	−4.00000	−0.213201
$353$	−14.0000	−0.745145	−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000	−0.745145	−0.372572	+	0.928003i	$0.621524\pi$
$354$	12.0000	0.637793
$355$	−8.00000	−0.424596
$356$	10.0000	0.529999
$357$	−2.00000	−0.105851
$358$	4.00000	0.211407
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	1.00000	0.0527046
$361$	−3.00000	−0.157895
$362$	6.00000	0.315353
$363$	5.00000	0.262432
$364$	2.00000	0.104828
$365$	10.0000	0.523424
$366$	14.0000	0.731792
$367$	−32.0000	−1.67039	−0.835193	−	0.549957i	$-0.814644\pi$
$367$	−32.0000	−1.67039	−0.835193	+	0.549957i	$0.814644\pi$
$368$	−8.00000	−0.417029
$369$	−6.00000	−0.312348
$370$	6.00000	0.311925
$371$	10.0000	0.519174
$372$	0	0
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	−8.00000	−0.413670
$375$	1.00000	0.0516398
$376$	0	0
$377$	4.00000	0.206010
$378$	−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	$0.244540\pi$
$380$	4.00000	0.205196
$381$	−16.0000	−0.819705
$382$	−16.0000	−0.818631
$383$	16.0000	0.817562	0.408781	−	0.912633i	$-0.365954\pi$
$383$	16.0000	0.817562	0.408781	+	0.912633i	$0.365954\pi$
$384$	1.00000	0.0510310
$385$	4.00000	0.203859
$386$	2.00000	0.101797
$387$	−4.00000	−0.203331
$388$	2.00000	0.101535
$389$	−26.0000	−1.31825	−0.659126	−	0.752032i	$-0.729074\pi$
$389$	−26.0000	−1.31825	−0.659126	+	0.752032i	$0.729074\pi$
$390$	−2.00000	−0.101274
$391$	−16.0000	−0.809155
$392$	1.00000	0.0505076
$393$	20.0000	1.00887
$394$	6.00000	0.302276
$395$	16.0000	0.805047
$396$	−4.00000	−0.201008
$397$	30.0000	1.50566	0.752828	−	0.658217i	$-0.228689\pi$
$397$	30.0000	1.50566	0.752828	+	0.658217i	$0.228689\pi$
$398$	−24.0000	−1.20301
$399$	−4.00000	−0.200250
$400$	1.00000	0.0500000
$401$	−14.0000	−0.699127	−0.349563	−	0.936913i	$-0.613670\pi$
$401$	−14.0000	−0.699127	−0.349563	+	0.936913i	$0.613670\pi$
$402$	−12.0000	−0.598506
$403$	0	0
$404$	6.00000	0.298511
$405$	1.00000	0.0496904
$406$	2.00000	0.0992583
$407$	−24.0000	−1.18964
$408$	2.00000	0.0990148
$409$	−38.0000	−1.87898	−0.939490	−	0.342578i	$-0.888700\pi$
$409$	−38.0000	−1.87898	−0.939490	+	0.342578i	$0.888700\pi$
$410$	−6.00000	−0.296319
$411$	10.0000	0.493264
$412$	−8.00000	−0.394132
$413$	−12.0000	−0.590481
$414$	−8.00000	−0.393179
$415$	−12.0000	−0.589057
$416$	−2.00000	−0.0980581
$417$	−4.00000	−0.195881
$418$	−16.0000	−0.782586
$419$	20.0000	0.977064	0.488532	−	0.872546i	$-0.337533\pi$
$419$	20.0000	0.977064	0.488532	+	0.872546i	$0.337533\pi$
$420$	−1.00000	−0.0487950
$421$	−26.0000	−1.26716	−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000	−1.26716	−0.633581	+	0.773676i	$0.718416\pi$
$422$	−12.0000	−0.584151
$423$	0	0
$424$	−10.0000	−0.485643
$425$	2.00000	0.0970143
$426$	−8.00000	−0.387601
$427$	−14.0000	−0.677507
$428$	12.0000	0.580042
$429$	8.00000	0.386244
$430$	−4.00000	−0.192897
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	1.00000	0.0481125
$433$	−14.0000	−0.672797	−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000	−0.672797	−0.336399	+	0.941720i	$0.609209\pi$
$434$	0	0
$435$	−2.00000	−0.0958927
$436$	14.0000	0.670478
$437$	−32.0000	−1.53077
$438$	10.0000	0.477818
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	−4.00000	−0.190693
$441$	1.00000	0.0476190
$442$	−4.00000	−0.190261
$443$	−36.0000	−1.71041	−0.855206	−	0.518289i	$-0.826569\pi$
$443$	−36.0000	−1.71041	−0.855206	+	0.518289i	$0.826569\pi$
$444$	6.00000	0.284747
$445$	10.0000	0.474045
$446$	−16.0000	−0.757622
$447$	−10.0000	−0.472984
$448$	−1.00000	−0.0472456
$449$	2.00000	0.0943858	0.0471929	−	0.998886i	$-0.484972\pi$
$449$	2.00000	0.0943858	0.0471929	+	0.998886i	$0.484972\pi$
$450$	1.00000	0.0471405
$451$	24.0000	1.13012
$452$	18.0000	0.846649
$453$	−8.00000	−0.375873
$454$	4.00000	0.187729
$455$	2.00000	0.0937614
$456$	4.00000	0.187317
$457$	10.0000	0.467780	0.233890	−	0.972263i	$-0.424854\pi$
$457$	10.0000	0.467780	0.233890	+	0.972263i	$0.424854\pi$
$458$	−10.0000	−0.467269
$459$	2.00000	0.0933520
$460$	−8.00000	−0.373002
$461$	14.0000	0.652045	0.326023	−	0.945362i	$-0.394291\pi$
$461$	14.0000	0.652045	0.326023	+	0.945362i	$0.394291\pi$
$462$	4.00000	0.186097
$463$	0	0		−	1.00000i	$-0.5\pi$
$463$	0	0			1.00000i	$0.5\pi$
$464$	−2.00000	−0.0928477
$465$	0	0
$466$	−22.0000	−1.01913
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	−2.00000	−0.0924500
$469$	12.0000	0.554109
$470$	0	0
$471$	−18.0000	−0.829396
$472$	12.0000	0.552345
$473$	16.0000	0.735681
$474$	16.0000	0.734904
$475$	4.00000	0.183533
$476$	−2.00000	−0.0916698
$477$	−10.0000	−0.457869
$478$	0	0
$479$	32.0000	1.46212	0.731059	−	0.682315i	$-0.239027\pi$
$479$	32.0000	1.46212	0.731059	+	0.682315i	$0.239027\pi$
$480$	1.00000	0.0456435
$481$	−12.0000	−0.547153
$482$	−14.0000	−0.637683
$483$	8.00000	0.364013
$484$	5.00000	0.227273
$485$	2.00000	0.0908153
$486$	1.00000	0.0453609
$487$	−8.00000	−0.362515	−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000	−0.362515	−0.181257	+	0.983436i	$0.558017\pi$
$488$	14.0000	0.633750
$489$	20.0000	0.904431
$490$	1.00000	0.0451754
$491$	−36.0000	−1.62466	−0.812329	−	0.583200i	$-0.801800\pi$
$491$	−36.0000	−1.62466	−0.812329	+	0.583200i	$0.801800\pi$
$492$	−6.00000	−0.270501
$493$	−4.00000	−0.180151
$494$	−8.00000	−0.359937
$495$	−4.00000	−0.179787
$496$	0	0
$497$	8.00000	0.358849
$498$	−12.0000	−0.537733
$499$	−12.0000	−0.537194	−0.268597	−	0.963253i	$-0.586560\pi$
$499$	−12.0000	−0.537194	−0.268597	+	0.963253i	$0.586560\pi$
$500$	1.00000	0.0447214
$501$	−8.00000	−0.357414
$502$	12.0000	0.535586
$503$	40.0000	1.78351	0.891756	−	0.452517i	$-0.149474\pi$
$503$	40.0000	1.78351	0.891756	+	0.452517i	$0.149474\pi$
$504$	−1.00000	−0.0445435
$505$	6.00000	0.266996
$506$	32.0000	1.42257
$507$	−9.00000	−0.399704
$508$	−16.0000	−0.709885
$509$	30.0000	1.32973	0.664863	−	0.746965i	$-0.268490\pi$
$509$	30.0000	1.32973	0.664863	+	0.746965i	$0.268490\pi$
$510$	2.00000	0.0885615
$511$	−10.0000	−0.442374
$512$	1.00000	0.0441942
$513$	4.00000	0.176604
$514$	−14.0000	−0.617514
$515$	−8.00000	−0.352522
$516$	−4.00000	−0.176090
$517$	0	0
$518$	−6.00000	−0.263625
$519$	−18.0000	−0.790112
$520$	−2.00000	−0.0877058
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	−2.00000	−0.0875376
$523$	−20.0000	−0.874539	−0.437269	−	0.899331i	$-0.644054\pi$
$523$	−20.0000	−0.874539	−0.437269	+	0.899331i	$0.644054\pi$
$524$	20.0000	0.873704
$525$	−1.00000	−0.0436436
$526$	8.00000	0.348817
$527$	0	0
$528$	−4.00000	−0.174078
$529$	41.0000	1.78261
$530$	−10.0000	−0.434372
$531$	12.0000	0.520756
$532$	−4.00000	−0.173422
$533$	12.0000	0.519778
$534$	10.0000	0.432742
$535$	12.0000	0.518805
$536$	−12.0000	−0.518321
$537$	4.00000	0.172613
$538$	−18.0000	−0.776035
$539$	−4.00000	−0.172292
$540$	1.00000	0.0430331
$541$	−34.0000	−1.46177	−0.730887	−	0.682498i	$-0.760893\pi$
$541$	−34.0000	−1.46177	−0.730887	+	0.682498i	$0.760893\pi$
$542$	16.0000	0.687259
$543$	6.00000	0.257485
$544$	2.00000	0.0857493
$545$	14.0000	0.599694
$546$	2.00000	0.0855921
$547$	−12.0000	−0.513083	−0.256541	−	0.966533i	$-0.582583\pi$
$547$	−12.0000	−0.513083	−0.256541	+	0.966533i	$0.582583\pi$
$548$	10.0000	0.427179
$549$	14.0000	0.597505
$550$	−4.00000	−0.170561
$551$	−8.00000	−0.340811
$552$	−8.00000	−0.340503
$553$	−16.0000	−0.680389
$554$	22.0000	0.934690
$555$	6.00000	0.254686
$556$	−4.00000	−0.169638
$557$	−18.0000	−0.762684	−0.381342	−	0.924434i	$-0.624538\pi$
$557$	−18.0000	−0.762684	−0.381342	+	0.924434i	$0.624538\pi$
$558$	0	0
$559$	8.00000	0.338364
$560$	−1.00000	−0.0422577
$561$	−8.00000	−0.337760
$562$	−6.00000	−0.253095
$563$	20.0000	0.842900	0.421450	−	0.906852i	$-0.361521\pi$
$563$	20.0000	0.842900	0.421450	+	0.906852i	$0.361521\pi$
$564$	0	0
$565$	18.0000	0.757266
$566$	28.0000	1.17693
$567$	−1.00000	−0.0419961
$568$	−8.00000	−0.335673
$569$	26.0000	1.08998	0.544988	−	0.838444i	$-0.316534\pi$
$569$	26.0000	1.08998	0.544988	+	0.838444i	$0.316534\pi$
$570$	4.00000	0.167542
$571$	28.0000	1.17176	0.585882	−	0.810397i	$-0.300748\pi$
$571$	28.0000	1.17176	0.585882	+	0.810397i	$0.300748\pi$
$572$	8.00000	0.334497
$573$	−16.0000	−0.668410
$574$	6.00000	0.250435
$575$	−8.00000	−0.333623
$576$	1.00000	0.0416667
$577$	34.0000	1.41544	0.707719	−	0.706494i	$-0.249724\pi$
$577$	34.0000	1.41544	0.707719	+	0.706494i	$0.249724\pi$
$578$	−13.0000	−0.540729
$579$	2.00000	0.0831172
$580$	−2.00000	−0.0830455
$581$	12.0000	0.497844
$582$	2.00000	0.0829027
$583$	40.0000	1.65663
$584$	10.0000	0.413803
$585$	−2.00000	−0.0826898
$586$	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
$589$	0	0
$590$	12.0000	0.494032
$591$	6.00000	0.246807
$592$	6.00000	0.246598
$593$	−30.0000	−1.23195	−0.615976	−	0.787765i	$-0.711238\pi$
$593$	−30.0000	−1.23195	−0.615976	+	0.787765i	$0.711238\pi$
$594$	−4.00000	−0.164122
$595$	−2.00000	−0.0819920
$596$	−10.0000	−0.409616
$597$	−24.0000	−0.982255
$598$	16.0000	0.654289
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	1.00000	0.0408248
$601$	−38.0000	−1.55005	−0.775026	−	0.631929i	$-0.782263\pi$
$601$	−38.0000	−1.55005	−0.775026	+	0.631929i	$0.782263\pi$
$602$	4.00000	0.163028
$603$	−12.0000	−0.488678
$604$	−8.00000	−0.325515
$605$	5.00000	0.203279
$606$	6.00000	0.243733
$607$	16.0000	0.649420	0.324710	−	0.945814i	$-0.394733\pi$
$607$	16.0000	0.649420	0.324710	+	0.945814i	$0.394733\pi$
$608$	4.00000	0.162221
$609$	2.00000	0.0810441
$610$	14.0000	0.566843
$611$	0	0
$612$	2.00000	0.0808452
$613$	6.00000	0.242338	0.121169	−	0.992632i	$-0.461336\pi$
$613$	6.00000	0.242338	0.121169	+	0.992632i	$0.461336\pi$
$614$	20.0000	0.807134
$615$	−6.00000	−0.241943
$616$	4.00000	0.161165
$617$	−22.0000	−0.885687	−0.442843	−	0.896599i	$-0.646030\pi$
$617$	−22.0000	−0.885687	−0.442843	+	0.896599i	$0.646030\pi$
$618$	−8.00000	−0.321807
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	0	0
$621$	−8.00000	−0.321029
$622$	−8.00000	−0.320771
$623$	−10.0000	−0.400642
$624$	−2.00000	−0.0800641
$625$	1.00000	0.0400000
$626$	−6.00000	−0.239808
$627$	−16.0000	−0.638978
$628$	−18.0000	−0.718278
$629$	12.0000	0.478471
$630$	−1.00000	−0.0398410
$631$	−40.0000	−1.59237	−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000	−1.59237	−0.796187	+	0.605050i	$0.793153\pi$
$632$	16.0000	0.636446
$633$	−12.0000	−0.476957
$634$	−2.00000	−0.0794301
$635$	−16.0000	−0.634941
$636$	−10.0000	−0.396526
$637$	−2.00000	−0.0792429
$638$	8.00000	0.316723
$639$	−8.00000	−0.316475
$640$	1.00000	0.0395285
$641$	34.0000	1.34292	0.671460	−	0.741041i	$-0.265668\pi$
$641$	34.0000	1.34292	0.671460	+	0.741041i	$0.265668\pi$
$642$	12.0000	0.473602
$643$	−28.0000	−1.10421	−0.552106	−	0.833774i	$-0.686176\pi$
$643$	−28.0000	−1.10421	−0.552106	+	0.833774i	$0.686176\pi$
$644$	8.00000	0.315244
$645$	−4.00000	−0.157500
$646$	8.00000	0.314756
$647$	24.0000	0.943537	0.471769	−	0.881722i	$-0.343616\pi$
$647$	24.0000	0.943537	0.471769	+	0.881722i	$0.343616\pi$
$648$	1.00000	0.0392837
$649$	−48.0000	−1.88416
$650$	−2.00000	−0.0784465
$651$	0	0
$652$	20.0000	0.783260
$653$	14.0000	0.547862	0.273931	−	0.961749i	$-0.411676\pi$
$653$	14.0000	0.547862	0.273931	+	0.961749i	$0.411676\pi$
$654$	14.0000	0.547443
$655$	20.0000	0.781465
$656$	−6.00000	−0.234261
$657$	10.0000	0.390137
$658$	0	0
$659$	36.0000	1.40236	0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000	1.40236	0.701180	+	0.712984i	$0.252657\pi$
$660$	−4.00000	−0.155700
$661$	38.0000	1.47803	0.739014	−	0.673690i	$-0.235292\pi$
$661$	38.0000	1.47803	0.739014	+	0.673690i	$0.235292\pi$
$662$	12.0000	0.466393
$663$	−4.00000	−0.155347
$664$	−12.0000	−0.465690
$665$	−4.00000	−0.155113
$666$	6.00000	0.232495
$667$	16.0000	0.619522
$668$	−8.00000	−0.309529
$669$	−16.0000	−0.618596
$670$	−12.0000	−0.463600
$671$	−56.0000	−2.16186
$672$	−1.00000	−0.0385758
$673$	−30.0000	−1.15642	−0.578208	−	0.815890i	$-0.696248\pi$
$673$	−30.0000	−1.15642	−0.578208	+	0.815890i	$0.696248\pi$
$674$	18.0000	0.693334
$675$	1.00000	0.0384900
$676$	−9.00000	−0.346154
$677$	−26.0000	−0.999261	−0.499631	−	0.866239i	$-0.666531\pi$
$677$	−26.0000	−0.999261	−0.499631	+	0.866239i	$0.666531\pi$
$678$	18.0000	0.691286
$679$	−2.00000	−0.0767530
$680$	2.00000	0.0766965
$681$	4.00000	0.153280
$682$	0	0
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	4.00000	0.152944
$685$	10.0000	0.382080
$686$	−1.00000	−0.0381802
$687$	−10.0000	−0.381524
$688$	−4.00000	−0.152499
$689$	20.0000	0.761939
$690$	−8.00000	−0.304555
$691$	36.0000	1.36950	0.684752	−	0.728776i	$-0.259910\pi$
$691$	36.0000	1.36950	0.684752	+	0.728776i	$0.259910\pi$
$692$	−18.0000	−0.684257
$693$	4.00000	0.151947
$694$	−4.00000	−0.151838
$695$	−4.00000	−0.151729
$696$	−2.00000	−0.0758098
$697$	−12.0000	−0.454532
$698$	14.0000	0.529908
$699$	−22.0000	−0.832116
$700$	−1.00000	−0.0377964
$701$	30.0000	1.13308	0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000	1.13308	0.566542	+	0.824033i	$0.308281\pi$
$702$	−2.00000	−0.0754851
$703$	24.0000	0.905177
$704$	−4.00000	−0.150756
$705$	0	0
$706$	−14.0000	−0.526897
$707$	−6.00000	−0.225653
$708$	12.0000	0.450988
$709$	−26.0000	−0.976450	−0.488225	−	0.872718i	$-0.662356\pi$
$709$	−26.0000	−0.976450	−0.488225	+	0.872718i	$0.662356\pi$
$710$	−8.00000	−0.300235
$711$	16.0000	0.600047
$712$	10.0000	0.374766
$713$	0	0
$714$	−2.00000	−0.0748481
$715$	8.00000	0.299183
$716$	4.00000	0.149487
$717$	0	0
$718$	24.0000	0.895672
$719$	−16.0000	−0.596699	−0.298350	−	0.954457i	$-0.596436\pi$
$719$	−16.0000	−0.596699	−0.298350	+	0.954457i	$0.596436\pi$
$720$	1.00000	0.0372678
$721$	8.00000	0.297936
$722$	−3.00000	−0.111648
$723$	−14.0000	−0.520666
$724$	6.00000	0.222988
$725$	−2.00000	−0.0742781
$726$	5.00000	0.185567
$727$	8.00000	0.296704	0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000	0.296704	0.148352	+	0.988935i	$0.452603\pi$
$728$	2.00000	0.0741249
$729$	1.00000	0.0370370
$730$	10.0000	0.370117
$731$	−8.00000	−0.295891
$732$	14.0000	0.517455
$733$	46.0000	1.69905	0.849524	−	0.527549i	$-0.176889\pi$
$733$	46.0000	1.69905	0.849524	+	0.527549i	$0.176889\pi$
$734$	−32.0000	−1.18114
$735$	1.00000	0.0368856
$736$	−8.00000	−0.294884
$737$	48.0000	1.76810
$738$	−6.00000	−0.220863
$739$	−28.0000	−1.03000	−0.514998	−	0.857191i	$-0.672207\pi$
$739$	−28.0000	−1.03000	−0.514998	+	0.857191i	$0.672207\pi$
$740$	6.00000	0.220564
$741$	−8.00000	−0.293887
$742$	10.0000	0.367112
$743$	−24.0000	−0.880475	−0.440237	−	0.897881i	$-0.645106\pi$
$743$	−24.0000	−0.880475	−0.440237	+	0.897881i	$0.645106\pi$
$744$	0	0
$745$	−10.0000	−0.366372
$746$	−10.0000	−0.366126
$747$	−12.0000	−0.439057
$748$	−8.00000	−0.292509
$749$	−12.0000	−0.438470
$750$	1.00000	0.0365148
$751$	16.0000	0.583848	0.291924	−	0.956441i	$-0.405705\pi$
$751$	16.0000	0.583848	0.291924	+	0.956441i	$0.405705\pi$
$752$	0	0
$753$	12.0000	0.437304
$754$	4.00000	0.145671
$755$	−8.00000	−0.291150
$756$	−1.00000	−0.0363696
$757$	22.0000	0.799604	0.399802	−	0.916602i	$-0.369079\pi$
$757$	22.0000	0.799604	0.399802	+	0.916602i	$0.369079\pi$
$758$	28.0000	1.01701
$759$	32.0000	1.16153
$760$	4.00000	0.145095
$761$	10.0000	0.362500	0.181250	−	0.983437i	$-0.441986\pi$
$761$	10.0000	0.362500	0.181250	+	0.983437i	$0.441986\pi$
$762$	−16.0000	−0.579619
$763$	−14.0000	−0.506834
$764$	−16.0000	−0.578860
$765$	2.00000	0.0723102
$766$	16.0000	0.578103
$767$	−24.0000	−0.866590
$768$	1.00000	0.0360844
$769$	34.0000	1.22607	0.613036	−	0.790055i	$-0.289948\pi$
$769$	34.0000	1.22607	0.613036	+	0.790055i	$0.289948\pi$
$770$	4.00000	0.144150
$771$	−14.0000	−0.504198
$772$	2.00000	0.0719816
$773$	−26.0000	−0.935155	−0.467578	−	0.883952i	$-0.654873\pi$
$773$	−26.0000	−0.935155	−0.467578	+	0.883952i	$0.654873\pi$
$774$	−4.00000	−0.143777
$775$	0	0
$776$	2.00000	0.0717958
$777$	−6.00000	−0.215249
$778$	−26.0000	−0.932145
$779$	−24.0000	−0.859889
$780$	−2.00000	−0.0716115
$781$	32.0000	1.14505
$782$	−16.0000	−0.572159
$783$	−2.00000	−0.0714742
$784$	1.00000	0.0357143
$785$	−18.0000	−0.642448
$786$	20.0000	0.713376
$787$	20.0000	0.712923	0.356462	−	0.934310i	$-0.383983\pi$
$787$	20.0000	0.712923	0.356462	+	0.934310i	$0.383983\pi$
$788$	6.00000	0.213741
$789$	8.00000	0.284808
$790$	16.0000	0.569254
$791$	−18.0000	−0.640006
$792$	−4.00000	−0.142134
$793$	−28.0000	−0.994309
$794$	30.0000	1.06466
$795$	−10.0000	−0.354663
$796$	−24.0000	−0.850657
$797$	30.0000	1.06265	0.531327	−	0.847167i	$-0.321693\pi$
$797$	30.0000	1.06265	0.531327	+	0.847167i	$0.321693\pi$
$798$	−4.00000	−0.141598
$799$	0	0
$800$	1.00000	0.0353553
$801$	10.0000	0.353333
$802$	−14.0000	−0.494357
$803$	−40.0000	−1.41157
$804$	−12.0000	−0.423207
$805$	8.00000	0.281963
$806$	0	0
$807$	−18.0000	−0.633630
$808$	6.00000	0.211079
$809$	−22.0000	−0.773479	−0.386739	−	0.922189i	$-0.626399\pi$
$809$	−22.0000	−0.773479	−0.386739	+	0.922189i	$0.626399\pi$
$810$	1.00000	0.0351364
$811$	28.0000	0.983213	0.491606	−	0.870817i	$-0.336410\pi$
$811$	28.0000	0.983213	0.491606	+	0.870817i	$0.336410\pi$
$812$	2.00000	0.0701862
$813$	16.0000	0.561144
$814$	−24.0000	−0.841200
$815$	20.0000	0.700569
$816$	2.00000	0.0700140
$817$	−16.0000	−0.559769
$818$	−38.0000	−1.32864
$819$	2.00000	0.0698857
$820$	−6.00000	−0.209529
$821$	22.0000	0.767805	0.383903	−	0.923374i	$-0.374580\pi$
$821$	22.0000	0.767805	0.383903	+	0.923374i	$0.374580\pi$
$822$	10.0000	0.348790
$823$	−24.0000	−0.836587	−0.418294	−	0.908312i	$-0.637372\pi$
$823$	−24.0000	−0.836587	−0.418294	+	0.908312i	$0.637372\pi$
$824$	−8.00000	−0.278693
$825$	−4.00000	−0.139262
$826$	−12.0000	−0.417533
$827$	28.0000	0.973655	0.486828	−	0.873498i	$-0.338154\pi$
$827$	28.0000	0.973655	0.486828	+	0.873498i	$0.338154\pi$
$828$	−8.00000	−0.278019
$829$	−18.0000	−0.625166	−0.312583	−	0.949890i	$-0.601194\pi$
$829$	−18.0000	−0.625166	−0.312583	+	0.949890i	$0.601194\pi$
$830$	−12.0000	−0.416526
$831$	22.0000	0.763172
$832$	−2.00000	−0.0693375
$833$	2.00000	0.0692959
$834$	−4.00000	−0.138509
$835$	−8.00000	−0.276851
$836$	−16.0000	−0.553372
$837$	0	0
$838$	20.0000	0.690889
$839$	8.00000	0.276191	0.138095	−	0.990419i	$-0.455902\pi$
$839$	8.00000	0.276191	0.138095	+	0.990419i	$0.455902\pi$
$840$	−1.00000	−0.0345033
$841$	−25.0000	−0.862069
$842$	−26.0000	−0.896019
$843$	−6.00000	−0.206651
$844$	−12.0000	−0.413057
$845$	−9.00000	−0.309609
$846$	0	0
$847$	−5.00000	−0.171802
$848$	−10.0000	−0.343401
$849$	28.0000	0.960958
$850$	2.00000	0.0685994
$851$	−48.0000	−1.64542
$852$	−8.00000	−0.274075
$853$	−26.0000	−0.890223	−0.445112	−	0.895475i	$-0.646836\pi$
$853$	−26.0000	−0.890223	−0.445112	+	0.895475i	$0.646836\pi$
$854$	−14.0000	−0.479070
$855$	4.00000	0.136797
$856$	12.0000	0.410152
$857$	−22.0000	−0.751506	−0.375753	−	0.926720i	$-0.622616\pi$
$857$	−22.0000	−0.751506	−0.375753	+	0.926720i	$0.622616\pi$
$858$	8.00000	0.273115
$859$	−20.0000	−0.682391	−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000	−0.682391	−0.341196	+	0.939992i	$0.610832\pi$
$860$	−4.00000	−0.136399
$861$	6.00000	0.204479
$862$	0	0
$863$	32.0000	1.08929	0.544646	−	0.838666i	$-0.316664\pi$
$863$	32.0000	1.08929	0.544646	+	0.838666i	$0.316664\pi$
$864$	1.00000	0.0340207
$865$	−18.0000	−0.612018
$866$	−14.0000	−0.475739
$867$	−13.0000	−0.441503
$868$	0	0
$869$	−64.0000	−2.17105
$870$	−2.00000	−0.0678064
$871$	24.0000	0.813209
$872$	14.0000	0.474100
$873$	2.00000	0.0676897
$874$	−32.0000	−1.08242
$875$	−1.00000	−0.0338062
$876$	10.0000	0.337869
$877$	14.0000	0.472746	0.236373	−	0.971662i	$-0.424041\pi$
$877$	14.0000	0.472746	0.236373	+	0.971662i	$0.424041\pi$
$878$	−8.00000	−0.269987
$879$	6.00000	0.202375
$880$	−4.00000	−0.134840
$881$	2.00000	0.0673817	0.0336909	−	0.999432i	$-0.489274\pi$
$881$	2.00000	0.0673817	0.0336909	+	0.999432i	$0.489274\pi$
$882$	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
$885$	12.0000	0.403376
$886$	−36.0000	−1.20944
$887$	8.00000	0.268614	0.134307	−	0.990940i	$-0.457119\pi$
$887$	8.00000	0.268614	0.134307	+	0.990940i	$0.457119\pi$
$888$	6.00000	0.201347
$889$	16.0000	0.536623
$890$	10.0000	0.335201
$891$	−4.00000	−0.134005
$892$	−16.0000	−0.535720
$893$	0	0
$894$	−10.0000	−0.334450
$895$	4.00000	0.133705
$896$	−1.00000	−0.0334077
$897$	16.0000	0.534224
$898$	2.00000	0.0667409
$899$	0	0
$900$	1.00000	0.0333333
$901$	−20.0000	−0.666297
$902$	24.0000	0.799113
$903$	4.00000	0.133112
$904$	18.0000	0.598671
$905$	6.00000	0.199447
$906$	−8.00000	−0.265782
$907$	28.0000	0.929725	0.464862	−	0.885383i	$-0.346104\pi$
$907$	28.0000	0.929725	0.464862	+	0.885383i	$0.346104\pi$
$908$	4.00000	0.132745
$909$	6.00000	0.199007
$910$	2.00000	0.0662994
$911$	32.0000	1.06021	0.530104	−	0.847933i	$-0.322153\pi$
$911$	32.0000	1.06021	0.530104	+	0.847933i	$0.322153\pi$
$912$	4.00000	0.132453
$913$	48.0000	1.58857
$914$	10.0000	0.330771
$915$	14.0000	0.462826
$916$	−10.0000	−0.330409
$917$	−20.0000	−0.660458
$918$	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	$0.542123\pi$
$920$	−8.00000	−0.263752
$921$	20.0000	0.659022
$922$	14.0000	0.461065
$923$	16.0000	0.526646
$924$	4.00000	0.131590
$925$	6.00000	0.197279
$926$	0	0
$927$	−8.00000	−0.262754
$928$	−2.00000	−0.0656532
$929$	−14.0000	−0.459325	−0.229663	−	0.973270i	$-0.573762\pi$
$929$	−14.0000	−0.459325	−0.229663	+	0.973270i	$0.573762\pi$
$930$	0	0
$931$	4.00000	0.131095
$932$	−22.0000	−0.720634
$933$	−8.00000	−0.261908
$934$	−12.0000	−0.392652
$935$	−8.00000	−0.261628
$936$	−2.00000	−0.0653720
$937$	10.0000	0.326686	0.163343	−	0.986569i	$-0.447772\pi$
$937$	10.0000	0.326686	0.163343	+	0.986569i	$0.447772\pi$
$938$	12.0000	0.391814
$939$	−6.00000	−0.195803
$940$	0	0
$941$	−18.0000	−0.586783	−0.293392	−	0.955992i	$-0.594784\pi$
$941$	−18.0000	−0.586783	−0.293392	+	0.955992i	$0.594784\pi$
$942$	−18.0000	−0.586472
$943$	48.0000	1.56310
$944$	12.0000	0.390567
$945$	−1.00000	−0.0325300
$946$	16.0000	0.520205
$947$	52.0000	1.68977	0.844886	−	0.534946i	$-0.179668\pi$
$947$	52.0000	1.68977	0.844886	+	0.534946i	$0.179668\pi$
$948$	16.0000	0.519656
$949$	−20.0000	−0.649227
$950$	4.00000	0.129777
$951$	−2.00000	−0.0648544
$952$	−2.00000	−0.0648204
$953$	26.0000	0.842223	0.421111	−	0.907009i	$-0.361640\pi$
$953$	26.0000	0.842223	0.421111	+	0.907009i	$0.361640\pi$
$954$	−10.0000	−0.323762
$955$	−16.0000	−0.517748
$956$	0	0
$957$	8.00000	0.258603
$958$	32.0000	1.03387
$959$	−10.0000	−0.322917
$960$	1.00000	0.0322749
$961$	−31.0000	−1.00000
$962$	−12.0000	−0.386896
$963$	12.0000	0.386695
$964$	−14.0000	−0.450910
$965$	2.00000	0.0643823
$966$	8.00000	0.257396
$967$	−8.00000	−0.257263	−0.128631	−	0.991692i	$-0.541058\pi$
$967$	−8.00000	−0.257263	−0.128631	+	0.991692i	$0.541058\pi$
$968$	5.00000	0.160706
$969$	8.00000	0.256997
$970$	2.00000	0.0642161
$971$	28.0000	0.898563	0.449281	−	0.893390i	$-0.351680\pi$
$971$	28.0000	0.898563	0.449281	+	0.893390i	$0.351680\pi$
$972$	1.00000	0.0320750
$973$	4.00000	0.128234
$974$	−8.00000	−0.256337
$975$	−2.00000	−0.0640513
$976$	14.0000	0.448129
$977$	18.0000	0.575871	0.287936	−	0.957650i	$-0.407031\pi$
$977$	18.0000	0.575871	0.287936	+	0.957650i	$0.407031\pi$
$978$	20.0000	0.639529
$979$	−40.0000	−1.27841
$980$	1.00000	0.0319438
$981$	14.0000	0.446986
$982$	−36.0000	−1.14881
$983$	−24.0000	−0.765481	−0.382741	−	0.923856i	$-0.625020\pi$
$983$	−24.0000	−0.765481	−0.382741	+	0.923856i	$0.625020\pi$
$984$	−6.00000	−0.191273
$985$	6.00000	0.191176
$986$	−4.00000	−0.127386
$987$	0	0
$988$	−8.00000	−0.254514
$989$	32.0000	1.01754
$990$	−4.00000	−0.127128
$991$	−32.0000	−1.01651	−0.508257	−	0.861206i	$-0.669710\pi$
$991$	−32.0000	−1.01651	−0.508257	+	0.861206i	$0.669710\pi$
$992$	0	0
$993$	12.0000	0.380808
$994$	8.00000	0.253745
$995$	−24.0000	−0.760851
$996$	−12.0000	−0.380235
$997$	22.0000	0.696747	0.348373	−	0.937356i	$-0.386734\pi$
$997$	22.0000	0.696747	0.348373	+	0.937356i	$0.386734\pi$
$998$	−12.0000	−0.379853
$999$	6.00000	0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	210.2.a.e.1.1	✓	1
3.2	odd	2		630.2.a.a.1.1		1
4.3	odd	2		1680.2.a.j.1.1		1
5.2	odd	4		1050.2.g.g.799.2		2
5.3	odd	4		1050.2.g.g.799.1		2
5.4	even	2		1050.2.a.c.1.1		1
7.2	even	3		1470.2.i.a.361.1		2
7.3	odd	6		1470.2.i.j.961.1		2
7.4	even	3		1470.2.i.a.961.1		2
7.5	odd	6		1470.2.i.j.361.1		2
7.6	odd	2		1470.2.a.j.1.1		1
8.3	odd	2		6720.2.a.bq.1.1		1
8.5	even	2		6720.2.a.j.1.1		1
12.11	even	2		5040.2.a.k.1.1		1
15.2	even	4		3150.2.g.q.2899.1		2
15.8	even	4		3150.2.g.q.2899.2		2
15.14	odd	2		3150.2.a.bp.1.1		1
20.19	odd	2		8400.2.a.ce.1.1		1
21.20	even	2		4410.2.a.t.1.1		1
35.34	odd	2		7350.2.a.w.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.2.a.e.1.1	✓	1	1.1	even	1	trivial
630.2.a.a.1.1		1	3.2	odd	2
1050.2.a.c.1.1		1	5.4	even	2
1050.2.g.g.799.1		2	5.3	odd	4
1050.2.g.g.799.2		2	5.2	odd	4
1470.2.a.j.1.1		1	7.6	odd	2
1470.2.i.a.361.1		2	7.2	even	3
1470.2.i.a.961.1		2	7.4	even	3
1470.2.i.j.361.1		2	7.5	odd	6
1470.2.i.j.961.1		2	7.3	odd	6
1680.2.a.j.1.1		1	4.3	odd	2
3150.2.a.bp.1.1		1	15.14	odd	2
3150.2.g.q.2899.1		2	15.2	even	4
3150.2.g.q.2899.2		2	15.8	even	4
4410.2.a.t.1.1		1	21.20	even	2
5040.2.a.k.1.1		1	12.11	even	2
6720.2.a.j.1.1		1	8.5	even	2
6720.2.a.bq.1.1		1	8.3	odd	2
7350.2.a.w.1.1		1	35.34	odd	2
8400.2.a.ce.1.1		1	20.19	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	210.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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