LMFDB - Embedded newform 210.2.a.b.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:	$0$
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)

$f(q)$

$=$

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

\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.00000 q^{12} +2.00000 q^{13} -1.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} -6.00000 q^{17} -1.00000 q^{18} +8.00000 q^{19} +1.00000 q^{20} +1.00000 q^{21} -1.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} +1.00000 q^{28} +6.00000 q^{29} -1.00000 q^{30} -4.00000 q^{31} -1.00000 q^{32} +6.00000 q^{34} +1.00000 q^{35} +1.00000 q^{36} -10.0000 q^{37} -8.00000 q^{38} +2.00000 q^{39} -1.00000 q^{40} -6.00000 q^{41} -1.00000 q^{42} -4.00000 q^{43} +1.00000 q^{45} +1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} -6.00000 q^{51} +2.00000 q^{52} -6.00000 q^{53} -1.00000 q^{54} -1.00000 q^{56} +8.00000 q^{57} -6.00000 q^{58} -12.0000 q^{59} +1.00000 q^{60} -10.0000 q^{61} +4.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} +2.00000 q^{65} -4.00000 q^{67} -6.00000 q^{68} -1.00000 q^{70} +12.0000 q^{71} -1.00000 q^{72} -10.0000 q^{73} +10.0000 q^{74} +1.00000 q^{75} +8.00000 q^{76} -2.00000 q^{78} +8.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} +12.0000 q^{83} +1.00000 q^{84} -6.00000 q^{85} +4.00000 q^{86} +6.00000 q^{87} -6.00000 q^{89} -1.00000 q^{90} +2.00000 q^{91} -4.00000 q^{93} +8.00000 q^{95} -1.00000 q^{96} -10.0000 q^{97} -1.00000 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	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
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	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
$15$	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
$19$	8.00000	1.83533	0.917663	−	0.397360i	$-0.130073\pi$
$19$	8.00000	1.83533	0.917663	+	0.397360i	$0.130073\pi$
$20$	1.00000	0.223607
$21$	1.00000	0.218218
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	−2.00000	−0.392232
$27$	1.00000	0.192450
$28$	1.00000	0.188982
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	−1.00000	−0.182574
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	6.00000	1.02899
$35$	1.00000	0.169031
$36$	1.00000	0.166667
$37$	−10.0000	−1.64399	−0.821995	−	0.569495i	$-0.807139\pi$
$37$	−10.0000	−1.64399	−0.821995	+	0.569495i	$0.807139\pi$
$38$	−8.00000	−1.29777
$39$	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	$0.598656\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	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
$51$	−6.00000	−0.840168
$52$	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
$57$	8.00000	1.05963
$58$	−6.00000	−0.787839
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	1.00000	0.129099
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	4.00000	0.508001
$63$	1.00000	0.125988
$64$	1.00000	0.125000
$65$	2.00000	0.248069
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	−6.00000	−0.727607
$69$	0	0
$70$	−1.00000	−0.119523
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	−1.00000	−0.117851
$73$	−10.0000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	10.0000	1.16248
$75$	1.00000	0.115470
$76$	8.00000	0.917663
$77$	0	0
$78$	−2.00000	−0.226455
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	6.00000	0.662589
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	1.00000	0.109109
$85$	−6.00000	−0.650791
$86$	4.00000	0.431331
$87$	6.00000	0.643268
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	−1.00000	−0.105409
$91$	2.00000	0.209657
$92$	0	0
$93$	−4.00000	−0.414781
$94$	0	0
$95$	8.00000	0.820783
$96$	−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
$99$	0	0
$100$	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	$0.403510\pi$
$102$	6.00000	0.594089
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	−2.00000	−0.196116
$105$	1.00000	0.0975900
$106$	6.00000	0.582772
$107$	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	$0.266089\pi$
$110$	0	0
$111$	−10.0000	−0.949158
$112$	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	$0.408930\pi$
$114$	−8.00000	−0.749269
$115$	0	0
$116$	6.00000	0.557086
$117$	2.00000	0.184900
$118$	12.0000	1.10469
$119$	−6.00000	−0.550019
$120$	−1.00000	−0.0912871
$121$	−11.0000	−1.00000
$122$	10.0000	0.905357
$123$	−6.00000	−0.541002
$124$	−4.00000	−0.359211
$125$	1.00000	0.0894427
$126$	−1.00000	−0.0890871
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	−1.00000	−0.0883883
$129$	−4.00000	−0.352180
$130$	−2.00000	−0.175412
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	0	0
$133$	8.00000	0.693688
$134$	4.00000	0.345547
$135$	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	$0.417494\pi$
$138$	0	0
$139$	−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
$141$	0	0
$142$	−12.0000	−1.00702
$143$	0	0
$144$	1.00000	0.0833333
$145$	6.00000	0.498273
$146$	10.0000	0.827606
$147$	1.00000	0.0824786
$148$	−10.0000	−0.821995
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	−1.00000	−0.0816497
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	−8.00000	−0.648886
$153$	−6.00000	−0.485071
$154$	0	0
$155$	−4.00000	−0.321288
$156$	2.00000	0.160128
$157$	−22.0000	−1.75579	−0.877896	−	0.478852i	$-0.841053\pi$
$157$	−22.0000	−1.75579	−0.877896	+	0.478852i	$0.841053\pi$
$158$	−8.00000	−0.636446
$159$	−6.00000	−0.475831
$160$	−1.00000	−0.0790569
$161$	0	0
$162$	−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
$165$	0	0
$166$	−12.0000	−0.931381
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	−1.00000	−0.0771517
$169$	−9.00000	−0.692308
$170$	6.00000	0.460179
$171$	8.00000	0.611775
$172$	−4.00000	−0.304997
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	−6.00000	−0.454859
$175$	1.00000	0.0755929
$176$	0	0
$177$	−12.0000	−0.901975
$178$	6.00000	0.449719
$179$	−24.0000	−1.79384	−0.896922	−	0.442189i	$-0.854202\pi$
$179$	−24.0000	−1.79384	−0.896922	+	0.442189i	$0.854202\pi$
$180$	1.00000	0.0745356
$181$	−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
$183$	−10.0000	−0.739221
$184$	0	0
$185$	−10.0000	−0.735215
$186$	4.00000	0.293294
$187$	0	0
$188$	0	0
$189$	1.00000	0.0727393
$190$	−8.00000	−0.580381
$191$	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
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	10.0000	0.717958
$195$	2.00000	0.143223
$196$	1.00000	0.0714286
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	0	0
$199$	20.0000	1.41776	0.708881	−	0.705328i	$-0.249200\pi$
$199$	20.0000	1.41776	0.708881	+	0.705328i	$0.249200\pi$
$200$	−1.00000	−0.0707107
$201$	−4.00000	−0.282138
$202$	−6.00000	−0.422159
$203$	6.00000	0.421117
$204$	−6.00000	−0.420084
$205$	−6.00000	−0.419058
$206$	−8.00000	−0.557386
$207$	0	0
$208$	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	$0.543966\pi$
$212$	−6.00000	−0.412082
$213$	12.0000	0.822226
$214$	−12.0000	−0.820303
$215$	−4.00000	−0.272798
$216$	−1.00000	−0.0680414
$217$	−4.00000	−0.271538
$218$	−14.0000	−0.948200
$219$	−10.0000	−0.675737
$220$	0	0
$221$	−12.0000	−0.807207
$222$	10.0000	0.671156
$223$	8.00000	0.535720	0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000	0.535720	0.267860	+	0.963458i	$0.413684\pi$
$224$	−1.00000	−0.0668153
$225$	1.00000	0.0666667
$226$	−6.00000	−0.399114
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	8.00000	0.529813
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	0	0
$231$	0	0
$232$	−6.00000	−0.393919
$233$	30.0000	1.96537	0.982683	−	0.185296i	$-0.0593245\pi$
$233$	30.0000	1.96537	0.982683	+	0.185296i	$0.0593245\pi$
$234$	−2.00000	−0.130744
$235$	0	0
$236$	−12.0000	−0.781133
$237$	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
$241$	26.0000	1.67481	0.837404	−	0.546585i	$-0.184072\pi$
$241$	26.0000	1.67481	0.837404	+	0.546585i	$0.184072\pi$
$242$	11.0000	0.707107
$243$	1.00000	0.0641500
$244$	−10.0000	−0.640184
$245$	1.00000	0.0638877
$246$	6.00000	0.382546
$247$	16.0000	1.01806
$248$	4.00000	0.254000
$249$	12.0000	0.760469
$250$	−1.00000	−0.0632456
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	1.00000	0.0629941
$253$	0	0
$254$	−8.00000	−0.501965
$255$	−6.00000	−0.375735
$256$	1.00000	0.0625000
$257$	−30.0000	−1.87135	−0.935674	−	0.352865i	$-0.885208\pi$
$257$	−30.0000	−1.87135	−0.935674	+	0.352865i	$0.885208\pi$
$258$	4.00000	0.249029
$259$	−10.0000	−0.621370
$260$	2.00000	0.124035
$261$	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	$0.765152\pi$
$264$	0	0
$265$	−6.00000	−0.368577
$266$	−8.00000	−0.490511
$267$	−6.00000	−0.367194
$268$	−4.00000	−0.244339
$269$	−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
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	−6.00000	−0.363803
$273$	2.00000	0.121046
$274$	−6.00000	−0.362473
$275$	0	0
$276$	0	0
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	16.0000	0.959616
$279$	−4.00000	−0.239474
$280$	−1.00000	−0.0597614
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	0	0
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	12.0000	0.712069
$285$	8.00000	0.473879
$286$	0	0
$287$	−6.00000	−0.354169
$288$	−1.00000	−0.0589256
$289$	19.0000	1.11765
$290$	−6.00000	−0.352332
$291$	−10.0000	−0.586210
$292$	−10.0000	−0.585206
$293$	30.0000	1.75262	0.876309	−	0.481749i	$-0.159998\pi$
$293$	30.0000	1.75262	0.876309	+	0.481749i	$0.159998\pi$
$294$	−1.00000	−0.0583212
$295$	−12.0000	−0.698667
$296$	10.0000	0.581238
$297$	0	0
$298$	−6.00000	−0.347571
$299$	0	0
$300$	1.00000	0.0577350
$301$	−4.00000	−0.230556
$302$	−8.00000	−0.460348
$303$	6.00000	0.344691
$304$	8.00000	0.458831
$305$	−10.0000	−0.572598
$306$	6.00000	0.342997
$307$	−4.00000	−0.228292	−0.114146	−	0.993464i	$-0.536413\pi$
$307$	−4.00000	−0.228292	−0.114146	+	0.993464i	$0.536413\pi$
$308$	0	0
$309$	8.00000	0.455104
$310$	4.00000	0.227185
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	−2.00000	−0.113228
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	22.0000	1.24153
$315$	1.00000	0.0563436
$316$	8.00000	0.450035
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	6.00000	0.336463
$319$	0	0
$320$	1.00000	0.0559017
$321$	12.0000	0.669775
$322$	0	0
$323$	−48.0000	−2.67079
$324$	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
$330$	0	0
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	12.0000	0.658586
$333$	−10.0000	−0.547997
$334$	0	0
$335$	−4.00000	−0.218543
$336$	1.00000	0.0545545
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	9.00000	0.489535
$339$	6.00000	0.325875
$340$	−6.00000	−0.325396
$341$	0	0
$342$	−8.00000	−0.432590
$343$	1.00000	0.0539949
$344$	4.00000	0.215666
$345$	0	0
$346$	−6.00000	−0.322562
$347$	−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
$349$	−10.0000	−0.535288	−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000	−0.535288	−0.267644	+	0.963518i	$0.586245\pi$
$350$	−1.00000	−0.0534522
$351$	2.00000	0.106752
$352$	0	0
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	12.0000	0.637793
$355$	12.0000	0.636894
$356$	−6.00000	−0.317999
$357$	−6.00000	−0.317554
$358$	24.0000	1.26844
$359$	−12.0000	−0.633336	−0.316668	−	0.948536i	$-0.602564\pi$
$359$	−12.0000	−0.633336	−0.316668	+	0.948536i	$0.602564\pi$
$360$	−1.00000	−0.0527046
$361$	45.0000	2.36842
$362$	10.0000	0.525588
$363$	−11.0000	−0.577350
$364$	2.00000	0.104828
$365$	−10.0000	−0.523424
$366$	10.0000	0.522708
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	10.0000	0.519875
$371$	−6.00000	−0.311504
$372$	−4.00000	−0.207390
$373$	−34.0000	−1.76045	−0.880227	−	0.474554i	$-0.842610\pi$
$373$	−34.0000	−1.76045	−0.880227	+	0.474554i	$0.842610\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	12.0000	0.618031
$378$	−1.00000	−0.0514344
$379$	−4.00000	−0.205466	−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000	−0.205466	−0.102733	+	0.994709i	$0.532759\pi$
$380$	8.00000	0.410391
$381$	8.00000	0.409852
$382$	−12.0000	−0.613973
$383$	24.0000	1.22634	0.613171	−	0.789950i	$-0.289894\pi$
$383$	24.0000	1.22634	0.613171	+	0.789950i	$0.289894\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−2.00000	−0.101797
$387$	−4.00000	−0.203331
$388$	−10.0000	−0.507673
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	−2.00000	−0.101274
$391$	0	0
$392$	−1.00000	−0.0505076
$393$	−12.0000	−0.605320
$394$	−18.0000	−0.906827
$395$	8.00000	0.402524
$396$	0	0
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	−20.0000	−1.00251
$399$	8.00000	0.400501
$400$	1.00000	0.0500000
$401$	−30.0000	−1.49813	−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000	−1.49813	−0.749064	+	0.662497i	$0.769497\pi$
$402$	4.00000	0.199502
$403$	−8.00000	−0.398508
$404$	6.00000	0.298511
$405$	1.00000	0.0496904
$406$	−6.00000	−0.297775
$407$	0	0
$408$	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	$0.277769\pi$
$410$	6.00000	0.296319
$411$	6.00000	0.295958
$412$	8.00000	0.394132
$413$	−12.0000	−0.590481
$414$	0	0
$415$	12.0000	0.589057
$416$	−2.00000	−0.0980581
$417$	−16.0000	−0.783523
$418$	0	0
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	1.00000	0.0487950
$421$	38.0000	1.85201	0.926003	−	0.377515i	$-0.123221\pi$
$421$	38.0000	1.85201	0.926003	+	0.377515i	$0.123221\pi$
$422$	4.00000	0.194717
$423$	0	0
$424$	6.00000	0.291386
$425$	−6.00000	−0.291043
$426$	−12.0000	−0.581402
$427$	−10.0000	−0.483934
$428$	12.0000	0.580042
$429$	0	0
$430$	4.00000	0.192897
$431$	−12.0000	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$432$	1.00000	0.0481125
$433$	−10.0000	−0.480569	−0.240285	−	0.970702i	$-0.577241\pi$
$433$	−10.0000	−0.480569	−0.240285	+	0.970702i	$0.577241\pi$
$434$	4.00000	0.192006
$435$	6.00000	0.287678
$436$	14.0000	0.670478
$437$	0	0
$438$	10.0000	0.477818
$439$	−28.0000	−1.33637	−0.668184	−	0.743996i	$-0.732928\pi$
$439$	−28.0000	−1.33637	−0.668184	+	0.743996i	$0.732928\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	12.0000	0.570782
$443$	−12.0000	−0.570137	−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000	−0.570137	−0.285069	+	0.958507i	$0.592016\pi$
$444$	−10.0000	−0.474579
$445$	−6.00000	−0.284427
$446$	−8.00000	−0.378811
$447$	6.00000	0.283790
$448$	1.00000	0.0472456
$449$	18.0000	0.849473	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	−1.00000	−0.0471405
$451$	0	0
$452$	6.00000	0.282216
$453$	8.00000	0.375873
$454$	−12.0000	−0.563188
$455$	2.00000	0.0937614
$456$	−8.00000	−0.374634
$457$	2.00000	0.0935561	0.0467780	−	0.998905i	$-0.485105\pi$
$457$	2.00000	0.0935561	0.0467780	+	0.998905i	$0.485105\pi$
$458$	−14.0000	−0.654177
$459$	−6.00000	−0.280056
$460$	0	0
$461$	−18.0000	−0.838344	−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000	−0.838344	−0.419172	+	0.907907i	$0.637680\pi$
$462$	0	0
$463$	−40.0000	−1.85896	−0.929479	−	0.368875i	$-0.879743\pi$
$463$	−40.0000	−1.85896	−0.929479	+	0.368875i	$0.879743\pi$
$464$	6.00000	0.278543
$465$	−4.00000	−0.185496
$466$	−30.0000	−1.38972
$467$	−36.0000	−1.66588	−0.832941	−	0.553362i	$-0.813345\pi$
$467$	−36.0000	−1.66588	−0.832941	+	0.553362i	$0.813345\pi$
$468$	2.00000	0.0924500
$469$	−4.00000	−0.184703
$470$	0	0
$471$	−22.0000	−1.01371
$472$	12.0000	0.552345
$473$	0	0
$474$	−8.00000	−0.367452
$475$	8.00000	0.367065
$476$	−6.00000	−0.275010
$477$	−6.00000	−0.274721
$478$	12.0000	0.548867
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	−1.00000	−0.0456435
$481$	−20.0000	−0.911922
$482$	−26.0000	−1.18427
$483$	0	0
$484$	−11.0000	−0.500000
$485$	−10.0000	−0.454077
$486$	−1.00000	−0.0453609
$487$	32.0000	1.45006	0.725029	−	0.688718i	$-0.241826\pi$
$487$	32.0000	1.45006	0.725029	+	0.688718i	$0.241826\pi$
$488$	10.0000	0.452679
$489$	20.0000	0.904431
$490$	−1.00000	−0.0451754
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	−6.00000	−0.270501
$493$	−36.0000	−1.62136
$494$	−16.0000	−0.719874
$495$	0	0
$496$	−4.00000	−0.179605
$497$	12.0000	0.538274
$498$	−12.0000	−0.537733
$499$	20.0000	0.895323	0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000	0.895323	0.447661	+	0.894203i	$0.352257\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	12.0000	0.535586
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	−1.00000	−0.0445435
$505$	6.00000	0.266996
$506$	0	0
$507$	−9.00000	−0.399704
$508$	8.00000	0.354943
$509$	30.0000	1.32973	0.664863	−	0.746965i	$-0.268490\pi$
$509$	30.0000	1.32973	0.664863	+	0.746965i	$0.268490\pi$
$510$	6.00000	0.265684
$511$	−10.0000	−0.442374
$512$	−1.00000	−0.0441942
$513$	8.00000	0.353209
$514$	30.0000	1.32324
$515$	8.00000	0.352522
$516$	−4.00000	−0.176090
$517$	0	0
$518$	10.0000	0.439375
$519$	6.00000	0.263371
$520$	−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	$0.541958\pi$
$522$	−6.00000	−0.262613
$523$	−28.0000	−1.22435	−0.612177	−	0.790721i	$-0.709706\pi$
$523$	−28.0000	−1.22435	−0.612177	+	0.790721i	$0.709706\pi$
$524$	−12.0000	−0.524222
$525$	1.00000	0.0436436
$526$	24.0000	1.04645
$527$	24.0000	1.04546
$528$	0	0
$529$	−23.0000	−1.00000
$530$	6.00000	0.260623
$531$	−12.0000	−0.520756
$532$	8.00000	0.346844
$533$	−12.0000	−0.519778
$534$	6.00000	0.259645
$535$	12.0000	0.518805
$536$	4.00000	0.172774
$537$	−24.0000	−1.03568
$538$	18.0000	0.776035
$539$	0	0
$540$	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	$0.760893\pi$
$542$	−20.0000	−0.859074
$543$	−10.0000	−0.429141
$544$	6.00000	0.257248
$545$	14.0000	0.599694
$546$	−2.00000	−0.0855921
$547$	44.0000	1.88130	0.940652	−	0.339372i	$-0.110215\pi$
$547$	44.0000	1.88130	0.940652	+	0.339372i	$0.110215\pi$
$548$	6.00000	0.256307
$549$	−10.0000	−0.426790
$550$	0	0
$551$	48.0000	2.04487
$552$	0	0
$553$	8.00000	0.340195
$554$	10.0000	0.424859
$555$	−10.0000	−0.424476
$556$	−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	$0.375462\pi$
$558$	4.00000	0.169334
$559$	−8.00000	−0.338364
$560$	1.00000	0.0422577
$561$	0	0
$562$	−18.0000	−0.759284
$563$	36.0000	1.51722	0.758610	−	0.651546i	$-0.225879\pi$
$563$	36.0000	1.51722	0.758610	+	0.651546i	$0.225879\pi$
$564$	0	0
$565$	6.00000	0.252422
$566$	4.00000	0.168133
$567$	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	$0.540139\pi$
$570$	−8.00000	−0.335083
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	0	0
$573$	12.0000	0.501307
$574$	6.00000	0.250435
$575$	0	0
$576$	1.00000	0.0416667
$577$	−10.0000	−0.416305	−0.208153	−	0.978096i	$-0.566745\pi$
$577$	−10.0000	−0.416305	−0.208153	+	0.978096i	$0.566745\pi$
$578$	−19.0000	−0.790296
$579$	2.00000	0.0831172
$580$	6.00000	0.249136
$581$	12.0000	0.497844
$582$	10.0000	0.414513
$583$	0	0
$584$	10.0000	0.413803
$585$	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
$589$	−32.0000	−1.31854
$590$	12.0000	0.494032
$591$	18.0000	0.740421
$592$	−10.0000	−0.410997
$593$	−6.00000	−0.246390	−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000	−0.246390	−0.123195	+	0.992382i	$0.539314\pi$
$594$	0	0
$595$	−6.00000	−0.245976
$596$	6.00000	0.245770
$597$	20.0000	0.818546
$598$	0	0
$599$	36.0000	1.47092	0.735460	−	0.677568i	$-0.236966\pi$
$599$	36.0000	1.47092	0.735460	+	0.677568i	$0.236966\pi$
$600$	−1.00000	−0.0408248
$601$	2.00000	0.0815817	0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000	0.0815817	0.0407909	+	0.999168i	$0.487012\pi$
$602$	4.00000	0.163028
$603$	−4.00000	−0.162893
$604$	8.00000	0.325515
$605$	−11.0000	−0.447214
$606$	−6.00000	−0.243733
$607$	8.00000	0.324710	0.162355	−	0.986732i	$-0.448091\pi$
$607$	8.00000	0.324710	0.162355	+	0.986732i	$0.448091\pi$
$608$	−8.00000	−0.324443
$609$	6.00000	0.243132
$610$	10.0000	0.404888
$611$	0	0
$612$	−6.00000	−0.242536
$613$	38.0000	1.53481	0.767403	−	0.641165i	$-0.221549\pi$
$613$	38.0000	1.53481	0.767403	+	0.641165i	$0.221549\pi$
$614$	4.00000	0.161427
$615$	−6.00000	−0.241943
$616$	0	0
$617$	−18.0000	−0.724653	−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000	−0.724653	−0.362326	+	0.932051i	$0.618017\pi$
$618$	−8.00000	−0.321807
$619$	32.0000	1.28619	0.643094	−	0.765787i	$-0.277650\pi$
$619$	32.0000	1.28619	0.643094	+	0.765787i	$0.277650\pi$
$620$	−4.00000	−0.160644
$621$	0	0
$622$	−24.0000	−0.962312
$623$	−6.00000	−0.240385
$624$	2.00000	0.0800641
$625$	1.00000	0.0400000
$626$	−14.0000	−0.559553
$627$	0	0
$628$	−22.0000	−0.877896
$629$	60.0000	2.39236
$630$	−1.00000	−0.0398410
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	−8.00000	−0.318223
$633$	−4.00000	−0.158986
$634$	−18.0000	−0.714871
$635$	8.00000	0.317470
$636$	−6.00000	−0.237915
$637$	2.00000	0.0792429
$638$	0	0
$639$	12.0000	0.474713
$640$	−1.00000	−0.0395285
$641$	42.0000	1.65890	0.829450	−	0.558581i	$-0.188654\pi$
$641$	42.0000	1.65890	0.829450	+	0.558581i	$0.188654\pi$
$642$	−12.0000	−0.473602
$643$	−4.00000	−0.157745	−0.0788723	−	0.996885i	$-0.525132\pi$
$643$	−4.00000	−0.157745	−0.0788723	+	0.996885i	$0.525132\pi$
$644$	0	0
$645$	−4.00000	−0.157500
$646$	48.0000	1.88853
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	−1.00000	−0.0392837
$649$	0	0
$650$	−2.00000	−0.0784465
$651$	−4.00000	−0.156772
$652$	20.0000	0.783260
$653$	−6.00000	−0.234798	−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000	−0.234798	−0.117399	+	0.993085i	$0.537456\pi$
$654$	−14.0000	−0.547443
$655$	−12.0000	−0.468879
$656$	−6.00000	−0.234261
$657$	−10.0000	−0.390137
$658$	0	0
$659$	48.0000	1.86981	0.934907	−	0.354892i	$-0.115482\pi$
$659$	48.0000	1.86981	0.934907	+	0.354892i	$0.115482\pi$
$660$	0	0
$661$	−34.0000	−1.32245	−0.661223	−	0.750189i	$-0.729962\pi$
$661$	−34.0000	−1.32245	−0.661223	+	0.750189i	$0.729962\pi$
$662$	28.0000	1.08825
$663$	−12.0000	−0.466041
$664$	−12.0000	−0.465690
$665$	8.00000	0.310227
$666$	10.0000	0.387492
$667$	0	0
$668$	0	0
$669$	8.00000	0.309298
$670$	4.00000	0.154533
$671$	0	0
$672$	−1.00000	−0.0385758
$673$	26.0000	1.00223	0.501113	−	0.865382i	$-0.332924\pi$
$673$	26.0000	1.00223	0.501113	+	0.865382i	$0.332924\pi$
$674$	−2.00000	−0.0770371
$675$	1.00000	0.0384900
$676$	−9.00000	−0.346154
$677$	−18.0000	−0.691796	−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000	−0.691796	−0.345898	+	0.938272i	$0.612426\pi$
$678$	−6.00000	−0.230429
$679$	−10.0000	−0.383765
$680$	6.00000	0.230089
$681$	12.0000	0.459841
$682$	0	0
$683$	36.0000	1.37750	0.688751	−	0.724998i	$-0.258159\pi$
$683$	36.0000	1.37750	0.688751	+	0.724998i	$0.258159\pi$
$684$	8.00000	0.305888
$685$	6.00000	0.229248
$686$	−1.00000	−0.0381802
$687$	14.0000	0.534133
$688$	−4.00000	−0.152499
$689$	−12.0000	−0.457164
$690$	0	0
$691$	−16.0000	−0.608669	−0.304334	−	0.952565i	$-0.598434\pi$
$691$	−16.0000	−0.608669	−0.304334	+	0.952565i	$0.598434\pi$
$692$	6.00000	0.228086
$693$	0	0
$694$	12.0000	0.455514
$695$	−16.0000	−0.606915
$696$	−6.00000	−0.227429
$697$	36.0000	1.36360
$698$	10.0000	0.378506
$699$	30.0000	1.13470
$700$	1.00000	0.0377964
$701$	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
$703$	−80.0000	−3.01726
$704$	0	0
$705$	0	0
$706$	6.00000	0.225813
$707$	6.00000	0.225653
$708$	−12.0000	−0.450988
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	−12.0000	−0.450352
$711$	8.00000	0.300023
$712$	6.00000	0.224860
$713$	0	0
$714$	6.00000	0.224544
$715$	0	0
$716$	−24.0000	−0.896922
$717$	−12.0000	−0.448148
$718$	12.0000	0.447836
$719$	−24.0000	−0.895049	−0.447524	−	0.894272i	$-0.647694\pi$
$719$	−24.0000	−0.895049	−0.447524	+	0.894272i	$0.647694\pi$
$720$	1.00000	0.0372678
$721$	8.00000	0.297936
$722$	−45.0000	−1.67473
$723$	26.0000	0.966950
$724$	−10.0000	−0.371647
$725$	6.00000	0.222834
$726$	11.0000	0.408248
$727$	32.0000	1.18681	0.593407	−	0.804902i	$-0.297782\pi$
$727$	32.0000	1.18681	0.593407	+	0.804902i	$0.297782\pi$
$728$	−2.00000	−0.0741249
$729$	1.00000	0.0370370
$730$	10.0000	0.370117
$731$	24.0000	0.887672
$732$	−10.0000	−0.369611
$733$	2.00000	0.0738717	0.0369358	−	0.999318i	$-0.488240\pi$
$733$	2.00000	0.0738717	0.0369358	+	0.999318i	$0.488240\pi$
$734$	−8.00000	−0.295285
$735$	1.00000	0.0368856
$736$	0	0
$737$	0	0
$738$	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	$0.672207\pi$
$740$	−10.0000	−0.367607
$741$	16.0000	0.587775
$742$	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	$0.645106\pi$
$744$	4.00000	0.146647
$745$	6.00000	0.219823
$746$	34.0000	1.24483
$747$	12.0000	0.439057
$748$	0	0
$749$	12.0000	0.438470
$750$	−1.00000	−0.0365148
$751$	8.00000	0.291924	0.145962	−	0.989290i	$-0.453372\pi$
$751$	8.00000	0.291924	0.145962	+	0.989290i	$0.453372\pi$
$752$	0	0
$753$	−12.0000	−0.437304
$754$	−12.0000	−0.437014
$755$	8.00000	0.291150
$756$	1.00000	0.0363696
$757$	−34.0000	−1.23575	−0.617876	−	0.786276i	$-0.712006\pi$
$757$	−34.0000	−1.23575	−0.617876	+	0.786276i	$0.712006\pi$
$758$	4.00000	0.145287
$759$	0	0
$760$	−8.00000	−0.290191
$761$	−6.00000	−0.217500	−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000	−0.217500	−0.108750	+	0.994069i	$0.534685\pi$
$762$	−8.00000	−0.289809
$763$	14.0000	0.506834
$764$	12.0000	0.434145
$765$	−6.00000	−0.216930
$766$	−24.0000	−0.867155
$767$	−24.0000	−0.866590
$768$	1.00000	0.0360844
$769$	26.0000	0.937584	0.468792	−	0.883309i	$-0.344689\pi$
$769$	26.0000	0.937584	0.468792	+	0.883309i	$0.344689\pi$
$770$	0	0
$771$	−30.0000	−1.08042
$772$	2.00000	0.0719816
$773$	−18.0000	−0.647415	−0.323708	−	0.946157i	$-0.604929\pi$
$773$	−18.0000	−0.647415	−0.323708	+	0.946157i	$0.604929\pi$
$774$	4.00000	0.143777
$775$	−4.00000	−0.143684
$776$	10.0000	0.358979
$777$	−10.0000	−0.358748
$778$	−6.00000	−0.215110
$779$	−48.0000	−1.71978
$780$	2.00000	0.0716115
$781$	0	0
$782$	0	0
$783$	6.00000	0.214423
$784$	1.00000	0.0357143
$785$	−22.0000	−0.785214
$786$	12.0000	0.428026
$787$	−4.00000	−0.142585	−0.0712923	−	0.997455i	$-0.522712\pi$
$787$	−4.00000	−0.142585	−0.0712923	+	0.997455i	$0.522712\pi$
$788$	18.0000	0.641223
$789$	−24.0000	−0.854423
$790$	−8.00000	−0.284627
$791$	6.00000	0.213335
$792$	0	0
$793$	−20.0000	−0.710221
$794$	−2.00000	−0.0709773
$795$	−6.00000	−0.212798
$796$	20.0000	0.708881
$797$	−42.0000	−1.48772	−0.743858	−	0.668338i	$-0.767006\pi$
$797$	−42.0000	−1.48772	−0.743858	+	0.668338i	$0.767006\pi$
$798$	−8.00000	−0.283197
$799$	0	0
$800$	−1.00000	−0.0353553
$801$	−6.00000	−0.212000
$802$	30.0000	1.05934
$803$	0	0
$804$	−4.00000	−0.141069
$805$	0	0
$806$	8.00000	0.281788
$807$	−18.0000	−0.633630
$808$	−6.00000	−0.211079
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	−1.00000	−0.0351364
$811$	−16.0000	−0.561836	−0.280918	−	0.959732i	$-0.590639\pi$
$811$	−16.0000	−0.561836	−0.280918	+	0.959732i	$0.590639\pi$
$812$	6.00000	0.210559
$813$	20.0000	0.701431
$814$	0	0
$815$	20.0000	0.700569
$816$	−6.00000	−0.210042
$817$	−32.0000	−1.11954
$818$	−26.0000	−0.909069
$819$	2.00000	0.0698857
$820$	−6.00000	−0.209529
$821$	−18.0000	−0.628204	−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000	−0.628204	−0.314102	+	0.949389i	$0.601703\pi$
$822$	−6.00000	−0.209274
$823$	32.0000	1.11545	0.557725	−	0.830026i	$-0.311674\pi$
$823$	32.0000	1.11545	0.557725	+	0.830026i	$0.311674\pi$
$824$	−8.00000	−0.278693
$825$	0	0
$826$	12.0000	0.417533
$827$	−36.0000	−1.25184	−0.625921	−	0.779886i	$-0.715277\pi$
$827$	−36.0000	−1.25184	−0.625921	+	0.779886i	$0.715277\pi$
$828$	0	0
$829$	−34.0000	−1.18087	−0.590434	−	0.807086i	$-0.701044\pi$
$829$	−34.0000	−1.18087	−0.590434	+	0.807086i	$0.701044\pi$
$830$	−12.0000	−0.416526
$831$	−10.0000	−0.346896
$832$	2.00000	0.0693375
$833$	−6.00000	−0.207888
$834$	16.0000	0.554035
$835$	0	0
$836$	0	0
$837$	−4.00000	−0.138260
$838$	12.0000	0.414533
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	−1.00000	−0.0345033
$841$	7.00000	0.241379
$842$	−38.0000	−1.30957
$843$	18.0000	0.619953
$844$	−4.00000	−0.137686
$845$	−9.00000	−0.309609
$846$	0	0
$847$	−11.0000	−0.377964
$848$	−6.00000	−0.206041
$849$	−4.00000	−0.137280
$850$	6.00000	0.205798
$851$	0	0
$852$	12.0000	0.411113
$853$	26.0000	0.890223	0.445112	−	0.895475i	$-0.353164\pi$
$853$	26.0000	0.890223	0.445112	+	0.895475i	$0.353164\pi$
$854$	10.0000	0.342193
$855$	8.00000	0.273594
$856$	−12.0000	−0.410152
$857$	−6.00000	−0.204956	−0.102478	−	0.994735i	$-0.532677\pi$
$857$	−6.00000	−0.204956	−0.102478	+	0.994735i	$0.532677\pi$
$858$	0	0
$859$	−40.0000	−1.36478	−0.682391	−	0.730987i	$-0.739060\pi$
$859$	−40.0000	−1.36478	−0.682391	+	0.730987i	$0.739060\pi$
$860$	−4.00000	−0.136399
$861$	−6.00000	−0.204479
$862$	12.0000	0.408722
$863$	−24.0000	−0.816970	−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000	−0.816970	−0.408485	+	0.912765i	$0.633943\pi$
$864$	−1.00000	−0.0340207
$865$	6.00000	0.204006
$866$	10.0000	0.339814
$867$	19.0000	0.645274
$868$	−4.00000	−0.135769
$869$	0	0
$870$	−6.00000	−0.203419
$871$	−8.00000	−0.271070
$872$	−14.0000	−0.474100
$873$	−10.0000	−0.338449
$874$	0	0
$875$	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	$0.424041\pi$
$878$	28.0000	0.944954
$879$	30.0000	1.01187
$880$	0	0
$881$	−30.0000	−1.01073	−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000	−1.01073	−0.505363	+	0.862907i	$0.668641\pi$
$882$	−1.00000	−0.0336718
$883$	44.0000	1.48072	0.740359	−	0.672212i	$-0.234656\pi$
$883$	44.0000	1.48072	0.740359	+	0.672212i	$0.234656\pi$
$884$	−12.0000	−0.403604
$885$	−12.0000	−0.403376
$886$	12.0000	0.403148
$887$	24.0000	0.805841	0.402921	−	0.915235i	$-0.367995\pi$
$887$	24.0000	0.805841	0.402921	+	0.915235i	$0.367995\pi$
$888$	10.0000	0.335578
$889$	8.00000	0.268311
$890$	6.00000	0.201120
$891$	0	0
$892$	8.00000	0.267860
$893$	0	0
$894$	−6.00000	−0.200670
$895$	−24.0000	−0.802232
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	−18.0000	−0.600668
$899$	−24.0000	−0.800445
$900$	1.00000	0.0333333
$901$	36.0000	1.19933
$902$	0	0
$903$	−4.00000	−0.133112
$904$	−6.00000	−0.199557
$905$	−10.0000	−0.332411
$906$	−8.00000	−0.265782
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	12.0000	0.398234
$909$	6.00000	0.199007
$910$	−2.00000	−0.0662994
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	8.00000	0.264906
$913$	0	0
$914$	−2.00000	−0.0661541
$915$	−10.0000	−0.330590
$916$	14.0000	0.462573
$917$	−12.0000	−0.396275
$918$	6.00000	0.198030
$919$	32.0000	1.05558	0.527791	−	0.849374i	$-0.323020\pi$
$919$	32.0000	1.05558	0.527791	+	0.849374i	$0.323020\pi$
$920$	0	0
$921$	−4.00000	−0.131804
$922$	18.0000	0.592798
$923$	24.0000	0.789970
$924$	0	0
$925$	−10.0000	−0.328798
$926$	40.0000	1.31448
$927$	8.00000	0.262754
$928$	−6.00000	−0.196960
$929$	−30.0000	−0.984268	−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000	−0.984268	−0.492134	+	0.870519i	$0.663783\pi$
$930$	4.00000	0.131165
$931$	8.00000	0.262189
$932$	30.0000	0.982683
$933$	24.0000	0.785725
$934$	36.0000	1.17796
$935$	0	0
$936$	−2.00000	−0.0653720
$937$	38.0000	1.24141	0.620703	−	0.784046i	$-0.286847\pi$
$937$	38.0000	1.24141	0.620703	+	0.784046i	$0.286847\pi$
$938$	4.00000	0.130605
$939$	14.0000	0.456873
$940$	0	0
$941$	−18.0000	−0.586783	−0.293392	−	0.955992i	$-0.594784\pi$
$941$	−18.0000	−0.586783	−0.293392	+	0.955992i	$0.594784\pi$
$942$	22.0000	0.716799
$943$	0	0
$944$	−12.0000	−0.390567
$945$	1.00000	0.0325300
$946$	0	0
$947$	−36.0000	−1.16984	−0.584921	−	0.811090i	$-0.698875\pi$
$947$	−36.0000	−1.16984	−0.584921	+	0.811090i	$0.698875\pi$
$948$	8.00000	0.259828
$949$	−20.0000	−0.649227
$950$	−8.00000	−0.259554
$951$	18.0000	0.583690
$952$	6.00000	0.194461
$953$	30.0000	0.971795	0.485898	−	0.874016i	$-0.338493\pi$
$953$	30.0000	0.971795	0.485898	+	0.874016i	$0.338493\pi$
$954$	6.00000	0.194257
$955$	12.0000	0.388311
$956$	−12.0000	−0.388108
$957$	0	0
$958$	0	0
$959$	6.00000	0.193750
$960$	1.00000	0.0322749
$961$	−15.0000	−0.483871
$962$	20.0000	0.644826
$963$	12.0000	0.386695
$964$	26.0000	0.837404
$965$	2.00000	0.0643823
$966$	0	0
$967$	8.00000	0.257263	0.128631	−	0.991692i	$-0.458942\pi$
$967$	8.00000	0.257263	0.128631	+	0.991692i	$0.458942\pi$
$968$	11.0000	0.353553
$969$	−48.0000	−1.54198
$970$	10.0000	0.321081
$971$	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
$973$	−16.0000	−0.512936
$974$	−32.0000	−1.02535
$975$	2.00000	0.0640513
$976$	−10.0000	−0.320092
$977$	6.00000	0.191957	0.0959785	−	0.995383i	$-0.469402\pi$
$977$	6.00000	0.191957	0.0959785	+	0.995383i	$0.469402\pi$
$978$	−20.0000	−0.639529
$979$	0	0
$980$	1.00000	0.0319438
$981$	14.0000	0.446986
$982$	0	0
$983$	24.0000	0.765481	0.382741	−	0.923856i	$-0.374980\pi$
$983$	24.0000	0.765481	0.382741	+	0.923856i	$0.374980\pi$
$984$	6.00000	0.191273
$985$	18.0000	0.573528
$986$	36.0000	1.14647
$987$	0	0
$988$	16.0000	0.509028
$989$	0	0
$990$	0	0
$991$	8.00000	0.254128	0.127064	−	0.991894i	$-0.459445\pi$
$991$	8.00000	0.254128	0.127064	+	0.991894i	$0.459445\pi$
$992$	4.00000	0.127000
$993$	−28.0000	−0.888553
$994$	−12.0000	−0.380617
$995$	20.0000	0.634043
$996$	12.0000	0.380235
$997$	26.0000	0.823428	0.411714	−	0.911313i	$-0.364930\pi$
$997$	26.0000	0.823428	0.411714	+	0.911313i	$0.364930\pi$
$998$	−20.0000	−0.633089
$999$	−10.0000	−0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	210.2.a.b.1.1	✓	1
3.2	odd	2		630.2.a.h.1.1		1
4.3	odd	2		1680.2.a.g.1.1		1
5.2	odd	4		1050.2.g.c.799.1		2
5.3	odd	4		1050.2.g.c.799.2		2
5.4	even	2		1050.2.a.k.1.1		1
7.2	even	3		1470.2.i.l.361.1		2
7.3	odd	6		1470.2.i.s.961.1		2
7.4	even	3		1470.2.i.l.961.1		2
7.5	odd	6		1470.2.i.s.361.1		2
7.6	odd	2		1470.2.a.b.1.1		1
8.3	odd	2		6720.2.a.bi.1.1		1
8.5	even	2		6720.2.a.n.1.1		1
12.11	even	2		5040.2.a.g.1.1		1
15.2	even	4		3150.2.g.i.2899.2		2
15.8	even	4		3150.2.g.i.2899.1		2
15.14	odd	2		3150.2.a.f.1.1		1
20.19	odd	2		8400.2.a.cm.1.1		1
21.20	even	2		4410.2.a.bi.1.1		1
35.34	odd	2		7350.2.a.cs.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.2.a.b.1.1	✓	1	1.1	even	1	trivial
630.2.a.h.1.1		1	3.2	odd	2
1050.2.a.k.1.1		1	5.4	even	2
1050.2.g.c.799.1		2	5.2	odd	4
1050.2.g.c.799.2		2	5.3	odd	4
1470.2.a.b.1.1		1	7.6	odd	2
1470.2.i.l.361.1		2	7.2	even	3
1470.2.i.l.961.1		2	7.4	even	3
1470.2.i.s.361.1		2	7.5	odd	6
1470.2.i.s.961.1		2	7.3	odd	6
1680.2.a.g.1.1		1	4.3	odd	2
3150.2.a.f.1.1		1	15.14	odd	2
3150.2.g.i.2899.1		2	15.8	even	4
3150.2.g.i.2899.2		2	15.2	even	4
4410.2.a.bi.1.1		1	21.20	even	2
5040.2.a.g.1.1		1	12.11	even	2
6720.2.a.n.1.1		1	8.5	even	2
6720.2.a.bi.1.1		1	8.3	odd	2
7350.2.a.cs.1.1		1	35.34	odd	2
8400.2.a.cm.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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