LMFDB - Embedded newform 210.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [210,2,Mod(1,210)]

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

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$1.67685844245$
Analytic rank:	$1$
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)

$f(q)$

$=$

$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -4.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} +1.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} -6.00000 q^{17} -1.00000 q^{18} -1.00000 q^{20} +1.00000 q^{21} +4.00000 q^{22} -8.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} +2.00000 q^{26} -1.00000 q^{27} -1.00000 q^{28} +10.0000 q^{29} -1.00000 q^{30} -8.00000 q^{31} -1.00000 q^{32} +4.00000 q^{33} +6.00000 q^{34} +1.00000 q^{35} +1.00000 q^{36} +2.00000 q^{37} +2.00000 q^{39} +1.00000 q^{40} -2.00000 q^{41} -1.00000 q^{42} +8.00000 q^{43} -4.00000 q^{44} -1.00000 q^{45} +8.00000 q^{46} +4.00000 q^{47} -1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} +6.00000 q^{51} -2.00000 q^{52} +10.0000 q^{53} +1.00000 q^{54} +4.00000 q^{55} +1.00000 q^{56} -10.0000 q^{58} +4.00000 q^{59} +1.00000 q^{60} -6.00000 q^{61} +8.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} +2.00000 q^{65} -4.00000 q^{66} -6.00000 q^{68} +8.00000 q^{69} -1.00000 q^{70} -12.0000 q^{71} -1.00000 q^{72} -6.00000 q^{73} -2.00000 q^{74} -1.00000 q^{75} +4.00000 q^{77} -2.00000 q^{78} -8.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} -4.00000 q^{83} +1.00000 q^{84} +6.00000 q^{85} -8.00000 q^{86} -10.0000 q^{87} +4.00000 q^{88} +14.0000 q^{89} +1.00000 q^{90} +2.00000 q^{91} -8.00000 q^{92} +8.00000 q^{93} -4.00000 q^{94} +1.00000 q^{96} +2.00000 q^{97} -1.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

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
$5$	−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
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	−1.00000	−0.235702
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	−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
$29$	10.0000	1.85695	0.928477	−	0.371391i	$-0.121119\pi$
$29$	10.0000	1.85695	0.928477	+	0.371391i	$0.121119\pi$
$30$	−1.00000	−0.182574
$31$	−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
$34$	6.00000	1.02899
$35$	1.00000	0.169031
$36$	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	$0.447432\pi$
$38$	0	0
$39$	2.00000	0.320256
$40$	1.00000	0.158114
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	−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	$0.291172\pi$
$44$	−4.00000	−0.603023
$45$	−1.00000	−0.149071
$46$	8.00000	1.17954
$47$	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
$50$	−1.00000	−0.141421
$51$	6.00000	0.840168
$52$	−2.00000	−0.277350
$53$	10.0000	1.37361	0.686803	−	0.726844i	$-0.259014\pi$
$53$	10.0000	1.37361	0.686803	+	0.726844i	$0.259014\pi$
$54$	1.00000	0.136083
$55$	4.00000	0.539360
$56$	1.00000	0.133631
$57$	0	0
$58$	−10.0000	−1.31306
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	1.00000	0.129099
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	8.00000	1.01600
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	2.00000	0.248069
$66$	−4.00000	−0.492366
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	−6.00000	−0.727607
$69$	8.00000	0.963087
$70$	−1.00000	−0.119523
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	−1.00000	−0.117851
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	−2.00000	−0.232495
$75$	−1.00000	−0.115470
$76$	0	0
$77$	4.00000	0.455842
$78$	−2.00000	−0.226455
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	2.00000	0.220863
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	1.00000	0.109109
$85$	6.00000	0.650791
$86$	−8.00000	−0.862662
$87$	−10.0000	−1.07211
$88$	4.00000	0.426401
$89$	14.0000	1.48400	0.741999	−	0.670402i	$-0.233878\pi$
$89$	14.0000	1.48400	0.741999	+	0.670402i	$0.233878\pi$
$90$	1.00000	0.105409
$91$	2.00000	0.209657
$92$	−8.00000	−0.834058
$93$	8.00000	0.829561
$94$	−4.00000	−0.412568
$95$	0	0
$96$	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	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	−6.00000	−0.594089
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	2.00000	0.196116
$105$	−1.00000	−0.0975900
$106$	−10.0000	−0.971286
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	−1.00000	−0.0962250
$109$	−18.0000	−1.72409	−0.862044	−	0.506834i	$-0.830816\pi$
$109$	−18.0000	−1.72409	−0.862044	+	0.506834i	$0.830816\pi$
$110$	−4.00000	−0.381385
$111$	−2.00000	−0.189832
$112$	−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	$0.408930\pi$
$114$	0	0
$115$	8.00000	0.746004
$116$	10.0000	0.928477
$117$	−2.00000	−0.184900
$118$	−4.00000	−0.368230
$119$	6.00000	0.550019
$120$	−1.00000	−0.0912871
$121$	5.00000	0.454545
$122$	6.00000	0.543214
$123$	2.00000	0.180334
$124$	−8.00000	−0.718421
$125$	−1.00000	−0.0894427
$126$	1.00000	0.0890871
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	−1.00000	−0.0883883
$129$	−8.00000	−0.704361
$130$	−2.00000	−0.175412
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	4.00000	0.348155
$133$	0	0
$134$	0	0
$135$	1.00000	0.0860663
$136$	6.00000	0.514496
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	−8.00000	−0.681005
$139$	−16.0000	−1.35710	−0.678551	−	0.734553i	$-0.737392\pi$
$139$	−16.0000	−1.35710	−0.678551	+	0.734553i	$0.737392\pi$
$140$	1.00000	0.0845154
$141$	−4.00000	−0.336861
$142$	12.0000	1.00702
$143$	8.00000	0.668994
$144$	1.00000	0.0833333
$145$	−10.0000	−0.830455
$146$	6.00000	0.496564
$147$	−1.00000	−0.0824786
$148$	2.00000	0.164399
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	1.00000	0.0816497
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	−6.00000	−0.485071
$154$	−4.00000	−0.322329
$155$	8.00000	0.642575
$156$	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	$0.630658\pi$
$158$	8.00000	0.636446
$159$	−10.0000	−0.793052
$160$	1.00000	0.0790569
$161$	8.00000	0.630488
$162$	−1.00000	−0.0785674
$163$	16.0000	1.25322	0.626608	−	0.779334i	$-0.284443\pi$
$163$	16.0000	1.25322	0.626608	+	0.779334i	$0.284443\pi$
$164$	−2.00000	−0.156174
$165$	−4.00000	−0.311400
$166$	4.00000	0.310460
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	−1.00000	−0.0771517
$169$	−9.00000	−0.692308
$170$	−6.00000	−0.460179
$171$	0	0
$172$	8.00000	0.609994
$173$	18.0000	1.36851	0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000	1.36851	0.684257	+	0.729241i	$0.260127\pi$
$174$	10.0000	0.758098
$175$	−1.00000	−0.0755929
$176$	−4.00000	−0.301511
$177$	−4.00000	−0.300658
$178$	−14.0000	−1.04934
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	−1.00000	−0.0745356
$181$	18.0000	1.33793	0.668965	−	0.743294i	$-0.266738\pi$
$181$	18.0000	1.33793	0.668965	+	0.743294i	$0.266738\pi$
$182$	−2.00000	−0.148250
$183$	6.00000	0.443533
$184$	8.00000	0.589768
$185$	−2.00000	−0.147043
$186$	−8.00000	−0.586588
$187$	24.0000	1.75505
$188$	4.00000	0.291730
$189$	1.00000	0.0727393
$190$	0	0
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	−1.00000	−0.0721688
$193$	10.0000	0.719816	0.359908	−	0.932988i	$-0.382808\pi$
$193$	10.0000	0.719816	0.359908	+	0.932988i	$0.382808\pi$
$194$	−2.00000	−0.143592
$195$	−2.00000	−0.143223
$196$	1.00000	0.0714286
$197$	−22.0000	−1.56744	−0.783718	−	0.621117i	$-0.786679\pi$
$197$	−22.0000	−1.56744	−0.783718	+	0.621117i	$0.786679\pi$
$198$	4.00000	0.284268
$199$	24.0000	1.70131	0.850657	−	0.525720i	$-0.176204\pi$
$199$	24.0000	1.70131	0.850657	+	0.525720i	$0.176204\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	6.00000	0.422159
$203$	−10.0000	−0.701862
$204$	6.00000	0.420084
$205$	2.00000	0.139686
$206$	16.0000	1.11477
$207$	−8.00000	−0.556038
$208$	−2.00000	−0.138675
$209$	0	0
$210$	1.00000	0.0690066
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	10.0000	0.686803
$213$	12.0000	0.822226
$214$	4.00000	0.273434
$215$	−8.00000	−0.545595
$216$	1.00000	0.0680414
$217$	8.00000	0.543075
$218$	18.0000	1.21911
$219$	6.00000	0.405442
$220$	4.00000	0.269680
$221$	12.0000	0.807207
$222$	2.00000	0.134231
$223$	−24.0000	−1.60716	−0.803579	−	0.595198i	$-0.797074\pi$
$223$	−24.0000	−1.60716	−0.803579	+	0.595198i	$0.797074\pi$
$224$	1.00000	0.0668153
$225$	1.00000	0.0666667
$226$	−6.00000	−0.399114
$227$	20.0000	1.32745	0.663723	−	0.747978i	$-0.268975\pi$
$227$	20.0000	1.32745	0.663723	+	0.747978i	$0.268975\pi$
$228$	0	0
$229$	−22.0000	−1.45380	−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−22.0000	−1.45380	−0.726900	+	0.686743i	$0.759040\pi$
$230$	−8.00000	−0.527504
$231$	−4.00000	−0.263181
$232$	−10.0000	−0.656532
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	2.00000	0.130744
$235$	−4.00000	−0.260931
$236$	4.00000	0.260378
$237$	8.00000	0.519656
$238$	−6.00000	−0.388922
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	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
$244$	−6.00000	−0.384111
$245$	−1.00000	−0.0638877
$246$	−2.00000	−0.127515
$247$	0	0
$248$	8.00000	0.508001
$249$	4.00000	0.253490
$250$	1.00000	0.0632456
$251$	−20.0000	−1.26239	−0.631194	−	0.775625i	$-0.717435\pi$
$251$	−20.0000	−1.26239	−0.631194	+	0.775625i	$0.717435\pi$
$252$	−1.00000	−0.0629941
$253$	32.0000	2.01182
$254$	−16.0000	−1.00393
$255$	−6.00000	−0.375735
$256$	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	$0.643834\pi$
$258$	8.00000	0.498058
$259$	−2.00000	−0.124274
$260$	2.00000	0.124035
$261$	10.0000	0.618984
$262$	−4.00000	−0.247121
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	−4.00000	−0.246183
$265$	−10.0000	−0.614295
$266$	0	0
$267$	−14.0000	−0.856786
$268$	0	0
$269$	2.00000	0.121942	0.0609711	−	0.998140i	$-0.480580\pi$
$269$	2.00000	0.121942	0.0609711	+	0.998140i	$0.480580\pi$
$270$	−1.00000	−0.0608581
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	−6.00000	−0.363803
$273$	−2.00000	−0.121046
$274$	−6.00000	−0.362473
$275$	−4.00000	−0.241209
$276$	8.00000	0.481543
$277$	−22.0000	−1.32185	−0.660926	−	0.750451i	$-0.729836\pi$
$277$	−22.0000	−1.32185	−0.660926	+	0.750451i	$0.729836\pi$
$278$	16.0000	0.959616
$279$	−8.00000	−0.478947
$280$	−1.00000	−0.0597614
$281$	−22.0000	−1.31241	−0.656205	−	0.754583i	$-0.727839\pi$
$281$	−22.0000	−1.31241	−0.656205	+	0.754583i	$0.727839\pi$
$282$	4.00000	0.238197
$283$	28.0000	1.66443	0.832214	−	0.554455i	$-0.187073\pi$
$283$	28.0000	1.66443	0.832214	+	0.554455i	$0.187073\pi$
$284$	−12.0000	−0.712069
$285$	0	0
$286$	−8.00000	−0.473050
$287$	2.00000	0.118056
$288$	−1.00000	−0.0589256
$289$	19.0000	1.11765
$290$	10.0000	0.587220
$291$	−2.00000	−0.117242
$292$	−6.00000	−0.351123
$293$	−30.0000	−1.75262	−0.876309	−	0.481749i	$-0.840002\pi$
$293$	−30.0000	−1.75262	−0.876309	+	0.481749i	$0.840002\pi$
$294$	1.00000	0.0583212
$295$	−4.00000	−0.232889
$296$	−2.00000	−0.116248
$297$	4.00000	0.232104
$298$	6.00000	0.347571
$299$	16.0000	0.925304
$300$	−1.00000	−0.0577350
$301$	−8.00000	−0.461112
$302$	0	0
$303$	6.00000	0.344691
$304$	0	0
$305$	6.00000	0.343559
$306$	6.00000	0.342997
$307$	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
$309$	16.0000	0.910208
$310$	−8.00000	−0.454369
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	−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	$0.554238\pi$
$314$	10.0000	0.564333
$315$	1.00000	0.0563436
$316$	−8.00000	−0.450035
$317$	−30.0000	−1.68497	−0.842484	−	0.538721i	$-0.818908\pi$
$317$	−30.0000	−1.68497	−0.842484	+	0.538721i	$0.818908\pi$
$318$	10.0000	0.560772
$319$	−40.0000	−2.23957
$320$	−1.00000	−0.0559017
$321$	4.00000	0.223258
$322$	−8.00000	−0.445823
$323$	0	0
$324$	1.00000	0.0555556
$325$	−2.00000	−0.110940
$326$	−16.0000	−0.886158
$327$	18.0000	0.995402
$328$	2.00000	0.110432
$329$	−4.00000	−0.220527
$330$	4.00000	0.220193
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	−4.00000	−0.219529
$333$	2.00000	0.109599
$334$	12.0000	0.656611
$335$	0	0
$336$	1.00000	0.0545545
$337$	−6.00000	−0.326841	−0.163420	−	0.986557i	$-0.552253\pi$
$337$	−6.00000	−0.326841	−0.163420	+	0.986557i	$0.552253\pi$
$338$	9.00000	0.489535
$339$	−6.00000	−0.325875
$340$	6.00000	0.325396
$341$	32.0000	1.73290
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−8.00000	−0.431331
$345$	−8.00000	−0.430706
$346$	−18.0000	−0.967686
$347$	20.0000	1.07366	0.536828	−	0.843692i	$-0.319622\pi$
$347$	20.0000	1.07366	0.536828	+	0.843692i	$0.319622\pi$
$348$	−10.0000	−0.536056
$349$	18.0000	0.963518	0.481759	−	0.876304i	$-0.339998\pi$
$349$	18.0000	0.963518	0.481759	+	0.876304i	$0.339998\pi$
$350$	1.00000	0.0534522
$351$	2.00000	0.106752
$352$	4.00000	0.213201
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	4.00000	0.212598
$355$	12.0000	0.636894
$356$	14.0000	0.741999
$357$	−6.00000	−0.317554
$358$	4.00000	0.211407
$359$	−20.0000	−1.05556	−0.527780	−	0.849381i	$-0.676975\pi$
$359$	−20.0000	−1.05556	−0.527780	+	0.849381i	$0.676975\pi$
$360$	1.00000	0.0527046
$361$	−19.0000	−1.00000
$362$	−18.0000	−0.946059
$363$	−5.00000	−0.262432
$364$	2.00000	0.104828
$365$	6.00000	0.314054
$366$	−6.00000	−0.313625
$367$	16.0000	0.835193	0.417597	−	0.908633i	$-0.362873\pi$
$367$	16.0000	0.835193	0.417597	+	0.908633i	$0.362873\pi$
$368$	−8.00000	−0.417029
$369$	−2.00000	−0.104116
$370$	2.00000	0.103975
$371$	−10.0000	−0.519174
$372$	8.00000	0.414781
$373$	−30.0000	−1.55334	−0.776671	−	0.629907i	$-0.783093\pi$
$373$	−30.0000	−1.55334	−0.776671	+	0.629907i	$0.783093\pi$
$374$	−24.0000	−1.24101
$375$	1.00000	0.0516398
$376$	−4.00000	−0.206284
$377$	−20.0000	−1.03005
$378$	−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
$381$	−16.0000	−0.819705
$382$	−12.0000	−0.613973
$383$	−28.0000	−1.43073	−0.715367	−	0.698749i	$-0.753740\pi$
$383$	−28.0000	−1.43073	−0.715367	+	0.698749i	$0.753740\pi$
$384$	1.00000	0.0510310
$385$	−4.00000	−0.203859
$386$	−10.0000	−0.508987
$387$	8.00000	0.406663
$388$	2.00000	0.101535
$389$	10.0000	0.507020	0.253510	−	0.967333i	$-0.418415\pi$
$389$	10.0000	0.507020	0.253510	+	0.967333i	$0.418415\pi$
$390$	2.00000	0.101274
$391$	48.0000	2.42746
$392$	−1.00000	−0.0505076
$393$	−4.00000	−0.201773
$394$	22.0000	1.10834
$395$	8.00000	0.402524
$396$	−4.00000	−0.201008
$397$	14.0000	0.702640	0.351320	−	0.936255i	$-0.385733\pi$
$397$	14.0000	0.702640	0.351320	+	0.936255i	$0.385733\pi$
$398$	−24.0000	−1.20301
$399$	0	0
$400$	1.00000	0.0500000
$401$	34.0000	1.69788	0.848939	−	0.528490i	$-0.177242\pi$
$401$	34.0000	1.69788	0.848939	+	0.528490i	$0.177242\pi$
$402$	0	0
$403$	16.0000	0.797017
$404$	−6.00000	−0.298511
$405$	−1.00000	−0.0496904
$406$	10.0000	0.496292
$407$	−8.00000	−0.396545
$408$	−6.00000	−0.297044
$409$	26.0000	1.28562	0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000	1.28562	0.642809	+	0.766027i	$0.277769\pi$
$410$	−2.00000	−0.0987730
$411$	−6.00000	−0.295958
$412$	−16.0000	−0.788263
$413$	−4.00000	−0.196827
$414$	8.00000	0.393179
$415$	4.00000	0.196352
$416$	2.00000	0.0980581
$417$	16.0000	0.783523
$418$	0	0
$419$	28.0000	1.36789	0.683945	−	0.729534i	$-0.260263\pi$
$419$	28.0000	1.36789	0.683945	+	0.729534i	$0.260263\pi$
$420$	−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	$0.718416\pi$
$422$	4.00000	0.194717
$423$	4.00000	0.194487
$424$	−10.0000	−0.485643
$425$	−6.00000	−0.291043
$426$	−12.0000	−0.581402
$427$	6.00000	0.290360
$428$	−4.00000	−0.193347
$429$	−8.00000	−0.386244
$430$	8.00000	0.385794
$431$	−36.0000	−1.73406	−0.867029	−	0.498257i	$-0.833974\pi$
$431$	−36.0000	−1.73406	−0.867029	+	0.498257i	$0.833974\pi$
$432$	−1.00000	−0.0481125
$433$	34.0000	1.63394	0.816968	−	0.576683i	$-0.195653\pi$
$433$	34.0000	1.63394	0.816968	+	0.576683i	$0.195653\pi$
$434$	−8.00000	−0.384012
$435$	10.0000	0.479463
$436$	−18.0000	−0.862044
$437$	0	0
$438$	−6.00000	−0.286691
$439$	32.0000	1.52728	0.763638	−	0.645644i	$-0.223411\pi$
$439$	32.0000	1.52728	0.763638	+	0.645644i	$0.223411\pi$
$440$	−4.00000	−0.190693
$441$	1.00000	0.0476190
$442$	−12.0000	−0.570782
$443$	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
$445$	−14.0000	−0.663664
$446$	24.0000	1.13643
$447$	6.00000	0.283790
$448$	−1.00000	−0.0472456
$449$	26.0000	1.22702	0.613508	−	0.789689i	$-0.289758\pi$
$449$	26.0000	1.22702	0.613508	+	0.789689i	$0.289758\pi$
$450$	−1.00000	−0.0471405
$451$	8.00000	0.376705
$452$	6.00000	0.282216
$453$	0	0
$454$	−20.0000	−0.938647
$455$	−2.00000	−0.0937614
$456$	0	0
$457$	−6.00000	−0.280668	−0.140334	−	0.990104i	$-0.544818\pi$
$457$	−6.00000	−0.280668	−0.140334	+	0.990104i	$0.544818\pi$
$458$	22.0000	1.02799
$459$	6.00000	0.280056
$460$	8.00000	0.373002
$461$	18.0000	0.838344	0.419172	−	0.907907i	$-0.362320\pi$
$461$	18.0000	0.838344	0.419172	+	0.907907i	$0.362320\pi$
$462$	4.00000	0.186097
$463$	8.00000	0.371792	0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000	0.371792	0.185896	+	0.982569i	$0.440481\pi$
$464$	10.0000	0.464238
$465$	−8.00000	−0.370991
$466$	18.0000	0.833834
$467$	12.0000	0.555294	0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000	0.555294	0.277647	+	0.960683i	$0.410445\pi$
$468$	−2.00000	−0.0924500
$469$	0	0
$470$	4.00000	0.184506
$471$	10.0000	0.460776
$472$	−4.00000	−0.184115
$473$	−32.0000	−1.47136
$474$	−8.00000	−0.367452
$475$	0	0
$476$	6.00000	0.275010
$477$	10.0000	0.457869
$478$	12.0000	0.548867
$479$	−8.00000	−0.365529	−0.182765	−	0.983157i	$-0.558505\pi$
$479$	−8.00000	−0.365529	−0.182765	+	0.983157i	$0.558505\pi$
$480$	−1.00000	−0.0456435
$481$	−4.00000	−0.182384
$482$	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	$-0.441983\pi$
$487$	8.00000	0.362515	0.181257	+	0.983436i	$0.441983\pi$
$488$	6.00000	0.271607
$489$	−16.0000	−0.723545
$490$	1.00000	0.0451754
$491$	20.0000	0.902587	0.451294	−	0.892375i	$-0.350963\pi$
$491$	20.0000	0.902587	0.451294	+	0.892375i	$0.350963\pi$
$492$	2.00000	0.0901670
$493$	−60.0000	−2.70226
$494$	0	0
$495$	4.00000	0.179787
$496$	−8.00000	−0.359211
$497$	12.0000	0.538274
$498$	−4.00000	−0.179244
$499$	28.0000	1.25345	0.626726	−	0.779240i	$-0.284395\pi$
$499$	28.0000	1.25345	0.626726	+	0.779240i	$0.284395\pi$
$500$	−1.00000	−0.0447214
$501$	12.0000	0.536120
$502$	20.0000	0.892644
$503$	−12.0000	−0.535054	−0.267527	−	0.963550i	$-0.586206\pi$
$503$	−12.0000	−0.535054	−0.267527	+	0.963550i	$0.586206\pi$
$504$	1.00000	0.0445435
$505$	6.00000	0.266996
$506$	−32.0000	−1.42257
$507$	9.00000	0.399704
$508$	16.0000	0.709885
$509$	2.00000	0.0886484	0.0443242	−	0.999017i	$-0.485887\pi$
$509$	2.00000	0.0886484	0.0443242	+	0.999017i	$0.485887\pi$
$510$	6.00000	0.265684
$511$	6.00000	0.265424
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	14.0000	0.617514
$515$	16.0000	0.705044
$516$	−8.00000	−0.352180
$517$	−16.0000	−0.703679
$518$	2.00000	0.0878750
$519$	−18.0000	−0.790112
$520$	−2.00000	−0.0877058
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	−10.0000	−0.437688
$523$	−28.0000	−1.22435	−0.612177	−	0.790721i	$-0.709706\pi$
$523$	−28.0000	−1.22435	−0.612177	+	0.790721i	$0.709706\pi$
$524$	4.00000	0.174741
$525$	1.00000	0.0436436
$526$	0	0
$527$	48.0000	2.09091
$528$	4.00000	0.174078
$529$	41.0000	1.78261
$530$	10.0000	0.434372
$531$	4.00000	0.173585
$532$	0	0
$533$	4.00000	0.173259
$534$	14.0000	0.605839
$535$	4.00000	0.172935
$536$	0	0
$537$	4.00000	0.172613
$538$	−2.00000	−0.0862261
$539$	−4.00000	−0.172292
$540$	1.00000	0.0430331
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	−8.00000	−0.343629
$543$	−18.0000	−0.772454
$544$	6.00000	0.257248
$545$	18.0000	0.771035
$546$	2.00000	0.0855921
$547$	−32.0000	−1.36822	−0.684111	−	0.729378i	$-0.739809\pi$
$547$	−32.0000	−1.36822	−0.684111	+	0.729378i	$0.739809\pi$
$548$	6.00000	0.256307
$549$	−6.00000	−0.256074
$550$	4.00000	0.170561
$551$	0	0
$552$	−8.00000	−0.340503
$553$	8.00000	0.340195
$554$	22.0000	0.934690
$555$	2.00000	0.0848953
$556$	−16.0000	−0.678551
$557$	18.0000	0.762684	0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000	0.762684	0.381342	+	0.924434i	$0.375462\pi$
$558$	8.00000	0.338667
$559$	−16.0000	−0.676728
$560$	1.00000	0.0422577
$561$	−24.0000	−1.01328
$562$	22.0000	0.928014
$563$	4.00000	0.168580	0.0842900	−	0.996441i	$-0.473138\pi$
$563$	4.00000	0.168580	0.0842900	+	0.996441i	$0.473138\pi$
$564$	−4.00000	−0.168430
$565$	−6.00000	−0.252422
$566$	−28.0000	−1.17693
$567$	−1.00000	−0.0419961
$568$	12.0000	0.503509
$569$	−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
$573$	−12.0000	−0.501307
$574$	−2.00000	−0.0834784
$575$	−8.00000	−0.333623
$576$	1.00000	0.0416667
$577$	−14.0000	−0.582828	−0.291414	−	0.956597i	$-0.594126\pi$
$577$	−14.0000	−0.582828	−0.291414	+	0.956597i	$0.594126\pi$
$578$	−19.0000	−0.790296
$579$	−10.0000	−0.415586
$580$	−10.0000	−0.415227
$581$	4.00000	0.165948
$582$	2.00000	0.0829027
$583$	−40.0000	−1.65663
$584$	6.00000	0.248282
$585$	2.00000	0.0826898
$586$	30.0000	1.23929
$587$	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
$590$	4.00000	0.164677
$591$	22.0000	0.904959
$592$	2.00000	0.0821995
$593$	−38.0000	−1.56047	−0.780236	−	0.625485i	$-0.784901\pi$
$593$	−38.0000	−1.56047	−0.780236	+	0.625485i	$0.784901\pi$
$594$	−4.00000	−0.164122
$595$	−6.00000	−0.245976
$596$	−6.00000	−0.245770
$597$	−24.0000	−0.982255
$598$	−16.0000	−0.654289
$599$	20.0000	0.817178	0.408589	−	0.912719i	$-0.366021\pi$
$599$	20.0000	0.817178	0.408589	+	0.912719i	$0.366021\pi$
$600$	1.00000	0.0408248
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	8.00000	0.326056
$603$	0	0
$604$	0	0
$605$	−5.00000	−0.203279
$606$	−6.00000	−0.243733
$607$	−40.0000	−1.62355	−0.811775	−	0.583970i	$-0.801498\pi$
$607$	−40.0000	−1.62355	−0.811775	+	0.583970i	$0.801498\pi$
$608$	0	0
$609$	10.0000	0.405220
$610$	−6.00000	−0.242933
$611$	−8.00000	−0.323645
$612$	−6.00000	−0.242536
$613$	−14.0000	−0.565455	−0.282727	−	0.959200i	$-0.591239\pi$
$613$	−14.0000	−0.565455	−0.282727	+	0.959200i	$0.591239\pi$
$614$	−20.0000	−0.807134
$615$	−2.00000	−0.0806478
$616$	−4.00000	−0.161165
$617$	−2.00000	−0.0805170	−0.0402585	−	0.999189i	$-0.512818\pi$
$617$	−2.00000	−0.0805170	−0.0402585	+	0.999189i	$0.512818\pi$
$618$	−16.0000	−0.643614
$619$	8.00000	0.321547	0.160774	−	0.986991i	$-0.448601\pi$
$619$	8.00000	0.321547	0.160774	+	0.986991i	$0.448601\pi$
$620$	8.00000	0.321288
$621$	8.00000	0.321029
$622$	0	0
$623$	−14.0000	−0.560898
$624$	2.00000	0.0800641
$625$	1.00000	0.0400000
$626$	6.00000	0.239808
$627$	0	0
$628$	−10.0000	−0.399043
$629$	−12.0000	−0.478471
$630$	−1.00000	−0.0398410
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	8.00000	0.318223
$633$	4.00000	0.158986
$634$	30.0000	1.19145
$635$	−16.0000	−0.634941
$636$	−10.0000	−0.396526
$637$	−2.00000	−0.0792429
$638$	40.0000	1.58362
$639$	−12.0000	−0.474713
$640$	1.00000	0.0395285
$641$	−6.00000	−0.236986	−0.118493	−	0.992955i	$-0.537806\pi$
$641$	−6.00000	−0.236986	−0.118493	+	0.992955i	$0.537806\pi$
$642$	−4.00000	−0.157867
$643$	−4.00000	−0.157745	−0.0788723	−	0.996885i	$-0.525132\pi$
$643$	−4.00000	−0.157745	−0.0788723	+	0.996885i	$0.525132\pi$
$644$	8.00000	0.315244
$645$	8.00000	0.315000
$646$	0	0
$647$	28.0000	1.10079	0.550397	−	0.834903i	$-0.314476\pi$
$647$	28.0000	1.10079	0.550397	+	0.834903i	$0.314476\pi$
$648$	−1.00000	−0.0392837
$649$	−16.0000	−0.628055
$650$	2.00000	0.0784465
$651$	−8.00000	−0.313545
$652$	16.0000	0.626608
$653$	−14.0000	−0.547862	−0.273931	−	0.961749i	$-0.588324\pi$
$653$	−14.0000	−0.547862	−0.273931	+	0.961749i	$0.588324\pi$
$654$	−18.0000	−0.703856
$655$	−4.00000	−0.156293
$656$	−2.00000	−0.0780869
$657$	−6.00000	−0.234082
$658$	4.00000	0.155936
$659$	−12.0000	−0.467454	−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000	−0.467454	−0.233727	+	0.972302i	$0.575092\pi$
$660$	−4.00000	−0.155700
$661$	−30.0000	−1.16686	−0.583432	−	0.812162i	$-0.698291\pi$
$661$	−30.0000	−1.16686	−0.583432	+	0.812162i	$0.698291\pi$
$662$	−20.0000	−0.777322
$663$	−12.0000	−0.466041
$664$	4.00000	0.155230
$665$	0	0
$666$	−2.00000	−0.0774984
$667$	−80.0000	−3.09761
$668$	−12.0000	−0.464294
$669$	24.0000	0.927894
$670$	0	0
$671$	24.0000	0.926510
$672$	−1.00000	−0.0385758
$673$	10.0000	0.385472	0.192736	−	0.981251i	$-0.438264\pi$
$673$	10.0000	0.385472	0.192736	+	0.981251i	$0.438264\pi$
$674$	6.00000	0.231111
$675$	−1.00000	−0.0384900
$676$	−9.00000	−0.346154
$677$	−22.0000	−0.845529	−0.422764	−	0.906240i	$-0.638940\pi$
$677$	−22.0000	−0.845529	−0.422764	+	0.906240i	$0.638940\pi$
$678$	6.00000	0.230429
$679$	−2.00000	−0.0767530
$680$	−6.00000	−0.230089
$681$	−20.0000	−0.766402
$682$	−32.0000	−1.22534
$683$	28.0000	1.07139	0.535695	−	0.844411i	$-0.320050\pi$
$683$	28.0000	1.07139	0.535695	+	0.844411i	$0.320050\pi$
$684$	0	0
$685$	−6.00000	−0.229248
$686$	1.00000	0.0381802
$687$	22.0000	0.839352
$688$	8.00000	0.304997
$689$	−20.0000	−0.761939
$690$	8.00000	0.304555
$691$	0	0		−	1.00000i	$-0.5\pi$
$691$	0	0			1.00000i	$0.5\pi$
$692$	18.0000	0.684257
$693$	4.00000	0.151947
$694$	−20.0000	−0.759190
$695$	16.0000	0.606915
$696$	10.0000	0.379049
$697$	12.0000	0.454532
$698$	−18.0000	−0.681310
$699$	18.0000	0.680823
$700$	−1.00000	−0.0377964
$701$	42.0000	1.58632	0.793159	−	0.609015i	$-0.208435\pi$
$701$	42.0000	1.58632	0.793159	+	0.609015i	$0.208435\pi$
$702$	−2.00000	−0.0754851
$703$	0	0
$704$	−4.00000	−0.150756
$705$	4.00000	0.150649
$706$	−18.0000	−0.677439
$707$	6.00000	0.225653
$708$	−4.00000	−0.150329
$709$	14.0000	0.525781	0.262891	−	0.964826i	$-0.415324\pi$
$709$	14.0000	0.525781	0.262891	+	0.964826i	$0.415324\pi$
$710$	−12.0000	−0.450352
$711$	−8.00000	−0.300023
$712$	−14.0000	−0.524672
$713$	64.0000	2.39682
$714$	6.00000	0.224544
$715$	−8.00000	−0.299183
$716$	−4.00000	−0.149487
$717$	12.0000	0.448148
$718$	20.0000	0.746393
$719$	8.00000	0.298350	0.149175	−	0.988811i	$-0.452338\pi$
$719$	8.00000	0.298350	0.149175	+	0.988811i	$0.452338\pi$
$720$	−1.00000	−0.0372678
$721$	16.0000	0.595871
$722$	19.0000	0.707107
$723$	14.0000	0.520666
$724$	18.0000	0.668965
$725$	10.0000	0.371391
$726$	5.00000	0.185567
$727$	−16.0000	−0.593407	−0.296704	−	0.954970i	$-0.595887\pi$
$727$	−16.0000	−0.593407	−0.296704	+	0.954970i	$0.595887\pi$
$728$	−2.00000	−0.0741249
$729$	1.00000	0.0370370
$730$	−6.00000	−0.222070
$731$	−48.0000	−1.77534
$732$	6.00000	0.221766
$733$	14.0000	0.517102	0.258551	−	0.965998i	$-0.416755\pi$
$733$	14.0000	0.517102	0.258551	+	0.965998i	$0.416755\pi$
$734$	−16.0000	−0.590571
$735$	1.00000	0.0368856
$736$	8.00000	0.294884
$737$	0	0
$738$	2.00000	0.0736210
$739$	−4.00000	−0.147142	−0.0735712	−	0.997290i	$-0.523440\pi$
$739$	−4.00000	−0.147142	−0.0735712	+	0.997290i	$0.523440\pi$
$740$	−2.00000	−0.0735215
$741$	0	0
$742$	10.0000	0.367112
$743$	−8.00000	−0.293492	−0.146746	−	0.989174i	$-0.546880\pi$
$743$	−8.00000	−0.293492	−0.146746	+	0.989174i	$0.546880\pi$
$744$	−8.00000	−0.293294
$745$	6.00000	0.219823
$746$	30.0000	1.09838
$747$	−4.00000	−0.146352
$748$	24.0000	0.877527
$749$	4.00000	0.146157
$750$	−1.00000	−0.0365148
$751$	−16.0000	−0.583848	−0.291924	−	0.956441i	$-0.594295\pi$
$751$	−16.0000	−0.583848	−0.291924	+	0.956441i	$0.594295\pi$
$752$	4.00000	0.145865
$753$	20.0000	0.728841
$754$	20.0000	0.728357
$755$	0	0
$756$	1.00000	0.0363696
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	28.0000	1.01701
$759$	−32.0000	−1.16153
$760$	0	0
$761$	−42.0000	−1.52250	−0.761249	−	0.648459i	$-0.775414\pi$
$761$	−42.0000	−1.52250	−0.761249	+	0.648459i	$0.775414\pi$
$762$	16.0000	0.579619
$763$	18.0000	0.651644
$764$	12.0000	0.434145
$765$	6.00000	0.216930
$766$	28.0000	1.01168
$767$	−8.00000	−0.288863
$768$	−1.00000	−0.0360844
$769$	10.0000	0.360609	0.180305	−	0.983611i	$-0.442292\pi$
$769$	10.0000	0.360609	0.180305	+	0.983611i	$0.442292\pi$
$770$	4.00000	0.144150
$771$	14.0000	0.504198
$772$	10.0000	0.359908
$773$	18.0000	0.647415	0.323708	−	0.946157i	$-0.395071\pi$
$773$	18.0000	0.647415	0.323708	+	0.946157i	$0.395071\pi$
$774$	−8.00000	−0.287554
$775$	−8.00000	−0.287368
$776$	−2.00000	−0.0717958
$777$	2.00000	0.0717496
$778$	−10.0000	−0.358517
$779$	0	0
$780$	−2.00000	−0.0716115
$781$	48.0000	1.71758
$782$	−48.0000	−1.71648
$783$	−10.0000	−0.357371
$784$	1.00000	0.0357143
$785$	10.0000	0.356915
$786$	4.00000	0.142675
$787$	4.00000	0.142585	0.0712923	−	0.997455i	$-0.477288\pi$
$787$	4.00000	0.142585	0.0712923	+	0.997455i	$0.477288\pi$
$788$	−22.0000	−0.783718
$789$	0	0
$790$	−8.00000	−0.284627
$791$	−6.00000	−0.213335
$792$	4.00000	0.142134
$793$	12.0000	0.426132
$794$	−14.0000	−0.496841
$795$	10.0000	0.354663
$796$	24.0000	0.850657
$797$	18.0000	0.637593	0.318796	−	0.947823i	$-0.396721\pi$
$797$	18.0000	0.637593	0.318796	+	0.947823i	$0.396721\pi$
$798$	0	0
$799$	−24.0000	−0.849059
$800$	−1.00000	−0.0353553
$801$	14.0000	0.494666
$802$	−34.0000	−1.20058
$803$	24.0000	0.846942
$804$	0	0
$805$	−8.00000	−0.281963
$806$	−16.0000	−0.563576
$807$	−2.00000	−0.0704033
$808$	6.00000	0.211079
$809$	42.0000	1.47664	0.738321	−	0.674450i	$-0.235619\pi$
$809$	42.0000	1.47664	0.738321	+	0.674450i	$0.235619\pi$
$810$	1.00000	0.0351364
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	−10.0000	−0.350931
$813$	−8.00000	−0.280572
$814$	8.00000	0.280400
$815$	−16.0000	−0.560456
$816$	6.00000	0.210042
$817$	0	0
$818$	−26.0000	−0.909069
$819$	2.00000	0.0698857
$820$	2.00000	0.0698430
$821$	18.0000	0.628204	0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000	0.628204	0.314102	+	0.949389i	$0.398297\pi$
$822$	6.00000	0.209274
$823$	0	0		−	1.00000i	$-0.5\pi$
$823$	0	0			1.00000i	$0.5\pi$
$824$	16.0000	0.557386
$825$	4.00000	0.139262
$826$	4.00000	0.139178
$827$	−28.0000	−0.973655	−0.486828	−	0.873498i	$-0.661846\pi$
$827$	−28.0000	−0.973655	−0.486828	+	0.873498i	$0.661846\pi$
$828$	−8.00000	−0.278019
$829$	34.0000	1.18087	0.590434	−	0.807086i	$-0.298956\pi$
$829$	34.0000	1.18087	0.590434	+	0.807086i	$0.298956\pi$
$830$	−4.00000	−0.138842
$831$	22.0000	0.763172
$832$	−2.00000	−0.0693375
$833$	−6.00000	−0.207888
$834$	−16.0000	−0.554035
$835$	12.0000	0.415277
$836$	0	0
$837$	8.00000	0.276520
$838$	−28.0000	−0.967244
$839$	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
$841$	71.0000	2.44828
$842$	26.0000	0.896019
$843$	22.0000	0.757720
$844$	−4.00000	−0.137686
$845$	9.00000	0.309609
$846$	−4.00000	−0.137523
$847$	−5.00000	−0.171802
$848$	10.0000	0.343401
$849$	−28.0000	−0.960958
$850$	6.00000	0.205798
$851$	−16.0000	−0.548473
$852$	12.0000	0.411113
$853$	22.0000	0.753266	0.376633	−	0.926363i	$-0.377082\pi$
$853$	22.0000	0.753266	0.376633	+	0.926363i	$0.377082\pi$
$854$	−6.00000	−0.205316
$855$	0	0
$856$	4.00000	0.136717
$857$	−38.0000	−1.29806	−0.649028	−	0.760765i	$-0.724824\pi$
$857$	−38.0000	−1.29806	−0.649028	+	0.760765i	$0.724824\pi$
$858$	8.00000	0.273115
$859$	24.0000	0.818869	0.409435	−	0.912339i	$-0.365726\pi$
$859$	24.0000	0.818869	0.409435	+	0.912339i	$0.365726\pi$
$860$	−8.00000	−0.272798
$861$	−2.00000	−0.0681598
$862$	36.0000	1.22616
$863$	−8.00000	−0.272323	−0.136162	−	0.990687i	$-0.543477\pi$
$863$	−8.00000	−0.272323	−0.136162	+	0.990687i	$0.543477\pi$
$864$	1.00000	0.0340207
$865$	−18.0000	−0.612018
$866$	−34.0000	−1.15537
$867$	−19.0000	−0.645274
$868$	8.00000	0.271538
$869$	32.0000	1.08553
$870$	−10.0000	−0.339032
$871$	0	0
$872$	18.0000	0.609557
$873$	2.00000	0.0676897
$874$	0	0
$875$	1.00000	0.0338062
$876$	6.00000	0.202721
$877$	2.00000	0.0675352	0.0337676	−	0.999430i	$-0.489249\pi$
$877$	2.00000	0.0675352	0.0337676	+	0.999430i	$0.489249\pi$
$878$	−32.0000	−1.07995
$879$	30.0000	1.01187
$880$	4.00000	0.134840
$881$	−2.00000	−0.0673817	−0.0336909	−	0.999432i	$-0.510726\pi$
$881$	−2.00000	−0.0673817	−0.0336909	+	0.999432i	$0.510726\pi$
$882$	−1.00000	−0.0336718
$883$	48.0000	1.61533	0.807664	−	0.589643i	$-0.200731\pi$
$883$	48.0000	1.61533	0.807664	+	0.589643i	$0.200731\pi$
$884$	12.0000	0.403604
$885$	4.00000	0.134459
$886$	−4.00000	−0.134383
$887$	20.0000	0.671534	0.335767	−	0.941945i	$-0.391004\pi$
$887$	20.0000	0.671534	0.335767	+	0.941945i	$0.391004\pi$
$888$	2.00000	0.0671156
$889$	−16.0000	−0.536623
$890$	14.0000	0.469281
$891$	−4.00000	−0.134005
$892$	−24.0000	−0.803579
$893$	0	0
$894$	−6.00000	−0.200670
$895$	4.00000	0.133705
$896$	1.00000	0.0334077
$897$	−16.0000	−0.534224
$898$	−26.0000	−0.867631
$899$	−80.0000	−2.66815
$900$	1.00000	0.0333333
$901$	−60.0000	−1.99889
$902$	−8.00000	−0.266371
$903$	8.00000	0.266223
$904$	−6.00000	−0.199557
$905$	−18.0000	−0.598340
$906$	0	0
$907$	8.00000	0.265636	0.132818	−	0.991140i	$-0.457597\pi$
$907$	8.00000	0.265636	0.132818	+	0.991140i	$0.457597\pi$
$908$	20.0000	0.663723
$909$	−6.00000	−0.199007
$910$	2.00000	0.0662994
$911$	−20.0000	−0.662630	−0.331315	−	0.943520i	$-0.607492\pi$
$911$	−20.0000	−0.662630	−0.331315	+	0.943520i	$0.607492\pi$
$912$	0	0
$913$	16.0000	0.529523
$914$	6.00000	0.198462
$915$	−6.00000	−0.198354
$916$	−22.0000	−0.726900
$917$	−4.00000	−0.132092
$918$	−6.00000	−0.198030
$919$	−24.0000	−0.791687	−0.395843	−	0.918318i	$-0.629548\pi$
$919$	−24.0000	−0.791687	−0.395843	+	0.918318i	$0.629548\pi$
$920$	−8.00000	−0.263752
$921$	−20.0000	−0.659022
$922$	−18.0000	−0.592798
$923$	24.0000	0.789970
$924$	−4.00000	−0.131590
$925$	2.00000	0.0657596
$926$	−8.00000	−0.262896
$927$	−16.0000	−0.525509
$928$	−10.0000	−0.328266
$929$	−18.0000	−0.590561	−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000	−0.590561	−0.295280	+	0.955411i	$0.595413\pi$
$930$	8.00000	0.262330
$931$	0	0
$932$	−18.0000	−0.589610
$933$	0	0
$934$	−12.0000	−0.392652
$935$	−24.0000	−0.784884
$936$	2.00000	0.0653720
$937$	42.0000	1.37208	0.686040	−	0.727564i	$-0.259347\pi$
$937$	42.0000	1.37208	0.686040	+	0.727564i	$0.259347\pi$
$938$	0	0
$939$	6.00000	0.195803
$940$	−4.00000	−0.130466
$941$	18.0000	0.586783	0.293392	−	0.955992i	$-0.405216\pi$
$941$	18.0000	0.586783	0.293392	+	0.955992i	$0.405216\pi$
$942$	−10.0000	−0.325818
$943$	16.0000	0.521032
$944$	4.00000	0.130189
$945$	−1.00000	−0.0325300
$946$	32.0000	1.04041
$947$	52.0000	1.68977	0.844886	−	0.534946i	$-0.179668\pi$
$947$	52.0000	1.68977	0.844886	+	0.534946i	$0.179668\pi$
$948$	8.00000	0.259828
$949$	12.0000	0.389536
$950$	0	0
$951$	30.0000	0.972817
$952$	−6.00000	−0.194461
$953$	−50.0000	−1.61966	−0.809829	−	0.586665i	$-0.800440\pi$
$953$	−50.0000	−1.61966	−0.809829	+	0.586665i	$0.800440\pi$
$954$	−10.0000	−0.323762
$955$	−12.0000	−0.388311
$956$	−12.0000	−0.388108
$957$	40.0000	1.29302
$958$	8.00000	0.258468
$959$	−6.00000	−0.193750
$960$	1.00000	0.0322749
$961$	33.0000	1.06452
$962$	4.00000	0.128965
$963$	−4.00000	−0.128898
$964$	−14.0000	−0.450910
$965$	−10.0000	−0.321911
$966$	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
$969$	0	0
$970$	2.00000	0.0642161
$971$	−36.0000	−1.15529	−0.577647	−	0.816286i	$-0.696029\pi$
$971$	−36.0000	−1.15529	−0.577647	+	0.816286i	$0.696029\pi$
$972$	−1.00000	−0.0320750
$973$	16.0000	0.512936
$974$	−8.00000	−0.256337
$975$	2.00000	0.0640513
$976$	−6.00000	−0.192055
$977$	6.00000	0.191957	0.0959785	−	0.995383i	$-0.469402\pi$
$977$	6.00000	0.191957	0.0959785	+	0.995383i	$0.469402\pi$
$978$	16.0000	0.511624
$979$	−56.0000	−1.78977
$980$	−1.00000	−0.0319438
$981$	−18.0000	−0.574696
$982$	−20.0000	−0.638226
$983$	28.0000	0.893061	0.446531	−	0.894768i	$-0.352659\pi$
$983$	28.0000	0.893061	0.446531	+	0.894768i	$0.352659\pi$
$984$	−2.00000	−0.0637577
$985$	22.0000	0.700978
$986$	60.0000	1.91079
$987$	4.00000	0.127321
$988$	0	0
$989$	−64.0000	−2.03508
$990$	−4.00000	−0.127128
$991$	16.0000	0.508257	0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000	0.508257	0.254128	+	0.967170i	$0.418211\pi$
$992$	8.00000	0.254000
$993$	−20.0000	−0.634681
$994$	−12.0000	−0.380617
$995$	−24.0000	−0.760851
$996$	4.00000	0.126745
$997$	−10.0000	−0.316703	−0.158352	−	0.987383i	$-0.550618\pi$
$997$	−10.0000	−0.316703	−0.158352	+	0.987383i	$0.550618\pi$
$998$	−28.0000	−0.886325
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	210.2.a.a.1.1	✓	1
3.2	odd	2		630.2.a.i.1.1		1
4.3	odd	2		1680.2.a.o.1.1		1
5.2	odd	4		1050.2.g.f.799.1		2
5.3	odd	4		1050.2.g.f.799.2		2
5.4	even	2		1050.2.a.q.1.1		1
7.2	even	3		1470.2.i.t.361.1		2
7.3	odd	6		1470.2.i.n.961.1		2
7.4	even	3		1470.2.i.t.961.1		2
7.5	odd	6		1470.2.i.n.361.1		2
7.6	odd	2		1470.2.a.g.1.1		1
8.3	odd	2		6720.2.a.z.1.1		1
8.5	even	2		6720.2.a.cg.1.1		1
12.11	even	2		5040.2.a.bg.1.1		1
15.2	even	4		3150.2.g.t.2899.2		2
15.8	even	4		3150.2.g.t.2899.1		2
15.14	odd	2		3150.2.a.t.1.1		1
20.19	odd	2		8400.2.a.m.1.1		1
21.20	even	2		4410.2.a.bc.1.1		1
35.34	odd	2		7350.2.a.bo.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.2.a.a.1.1	✓	1	1.1	even	1	trivial
630.2.a.i.1.1		1	3.2	odd	2
1050.2.a.q.1.1		1	5.4	even	2
1050.2.g.f.799.1		2	5.2	odd	4
1050.2.g.f.799.2		2	5.3	odd	4
1470.2.a.g.1.1		1	7.6	odd	2
1470.2.i.n.361.1		2	7.5	odd	6
1470.2.i.n.961.1		2	7.3	odd	6
1470.2.i.t.361.1		2	7.2	even	3
1470.2.i.t.961.1		2	7.4	even	3
1680.2.a.o.1.1		1	4.3	odd	2
3150.2.a.t.1.1		1	15.14	odd	2
3150.2.g.t.2899.1		2	15.8	even	4
3150.2.g.t.2899.2		2	15.2	even	4
4410.2.a.bc.1.1		1	21.20	even	2
5040.2.a.bg.1.1		1	12.11	even	2
6720.2.a.z.1.1		1	8.3	odd	2
6720.2.a.cg.1.1		1	8.5	even	2
7350.2.a.bo.1.1		1	35.34	odd	2
8400.2.a.m.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 $\F_{q}$

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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