LMFDB - Embedded newform 1050.2.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1050, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1050.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1050.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$8.38429221223$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 210)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

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

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
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	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	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	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	$0.130073\pi$
$20$	0	0
$21$	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
$25$	0	0
$26$	−2.00000	−0.392232
$27$	−1.00000	−0.192450
$28$	−1.00000	−0.188982
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	−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
$35$	0	0
$36$	1.00000	0.166667
$37$	10.0000	1.64399	0.821995	−	0.569495i	$-0.192861\pi$
$37$	10.0000	1.64399	0.821995	+	0.569495i	$0.192861\pi$
$38$	8.00000	1.29777
$39$	2.00000	0.320256
$40$	0	0
$41$	−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	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	0	0
$45$	0	0
$46$	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
$50$	0	0
$51$	−6.00000	−0.840168
$52$	−2.00000	−0.277350
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	−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	$0.785358\pi$
$60$	0	0
$61$	−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
$65$	0	0
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	6.00000	0.727607
$69$	0	0
$70$	0	0
$71$	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	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	10.0000	1.16248
$75$	0	0
$76$	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	$0.351411\pi$
$80$	0	0
$81$	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
$85$	0	0
$86$	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	$0.603011\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	0	0
$93$	4.00000	0.414781
$94$	0	0
$95$	0	0
$96$	−1.00000	−0.102062
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	−6.00000	−0.594089
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	6.00000	0.582772
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	−1.00000	−0.0962250
$109$	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	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	−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
$120$	0	0
$121$	−11.0000	−1.00000
$122$	−10.0000	−0.905357
$123$	6.00000	0.541002
$124$	−4.00000	−0.359211
$125$	0	0
$126$	−1.00000	−0.0890871
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	1.00000	0.0883883
$129$	−4.00000	−0.352180
$130$	0	0
$131$	−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
$135$	0	0
$136$	6.00000	0.514496
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	−16.0000	−1.35710	−0.678551	−	0.734553i	$-0.737392\pi$
$139$	−16.0000	−1.35710	−0.678551	+	0.734553i	$0.737392\pi$
$140$	0	0
$141$	0	0
$142$	12.0000	1.00702
$143$	0	0
$144$	1.00000	0.0833333
$145$	0	0
$146$	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	$0.420959\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	8.00000	0.648886
$153$	6.00000	0.485071
$154$	0	0
$155$	0	0
$156$	2.00000	0.160128
$157$	22.0000	1.75579	0.877896	−	0.478852i	$-0.158947\pi$
$157$	22.0000	1.75579	0.877896	+	0.478852i	$0.158947\pi$
$158$	8.00000	0.636446
$159$	−6.00000	−0.475831
$160$	0	0
$161$	0	0
$162$	1.00000	0.0785674
$163$	−20.0000	−1.56652	−0.783260	−	0.621694i	$-0.786445\pi$
$163$	−20.0000	−1.56652	−0.783260	+	0.621694i	$0.786445\pi$
$164$	−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
$170$	0	0
$171$	8.00000	0.611775
$172$	4.00000	0.304997
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	−6.00000	−0.454859
$175$	0	0
$176$	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	$0.854202\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	2.00000	0.148250
$183$	10.0000	0.739221
$184$	0	0
$185$	0	0
$186$	4.00000	0.293294
$187$	0	0
$188$	0	0
$189$	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
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	10.0000	0.717958
$195$	0	0
$196$	1.00000	0.0714286
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	0	0
$199$	20.0000	1.41776	0.708881	−	0.705328i	$-0.249200\pi$
$199$	20.0000	1.41776	0.708881	+	0.705328i	$0.249200\pi$
$200$	0	0
$201$	−4.00000	−0.282138
$202$	6.00000	0.422159
$203$	−6.00000	−0.421117
$204$	−6.00000	−0.420084
$205$	0	0
$206$	−8.00000	−0.557386
$207$	0	0
$208$	−2.00000	−0.138675
$209$	0	0
$210$	0	0
$211$	−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
$215$	0	0
$216$	−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	$-0.586316\pi$
$223$	−8.00000	−0.535720	−0.267860	+	0.963458i	$0.586316\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	−6.00000	−0.399114
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	−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	$-0.940675\pi$
$233$	−30.0000	−1.96537	−0.982683	+	0.185296i	$0.940675\pi$
$234$	−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	$0.626871\pi$
$240$	0	0
$241$	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
$245$	0	0
$246$	6.00000	0.382546
$247$	−16.0000	−1.01806
$248$	−4.00000	−0.254000
$249$	12.0000	0.760469
$250$	0	0
$251$	−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
$255$	0	0
$256$	1.00000	0.0625000
$257$	30.0000	1.87135	0.935674	−	0.352865i	$-0.114792\pi$
$257$	30.0000	1.87135	0.935674	+	0.352865i	$0.114792\pi$
$258$	−4.00000	−0.249029
$259$	−10.0000	−0.621370
$260$	0	0
$261$	6.00000	0.371391
$262$	−12.0000	−0.741362
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	0	0
$265$	0	0
$266$	−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	$0.684892\pi$
$270$	0	0
$271$	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	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	−16.0000	−0.959616
$279$	−4.00000	−0.239474
$280$	0	0
$281$	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	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	12.0000	0.712069
$285$	0	0
$286$	0	0
$287$	6.00000	0.354169
$288$	1.00000	0.0589256
$289$	19.0000	1.11765
$290$	0	0
$291$	−10.0000	−0.586210
$292$	10.0000	0.585206
$293$	−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
$295$	0	0
$296$	10.0000	0.581238
$297$	0	0
$298$	6.00000	0.347571
$299$	0	0
$300$	0	0
$301$	−4.00000	−0.230556
$302$	8.00000	0.460348
$303$	−6.00000	−0.344691
$304$	8.00000	0.458831
$305$	0	0
$306$	6.00000	0.342997
$307$	4.00000	0.228292	0.114146	−	0.993464i	$-0.463587\pi$
$307$	4.00000	0.228292	0.114146	+	0.993464i	$0.463587\pi$
$308$	0	0
$309$	8.00000	0.455104
$310$	0	0
$311$	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	$-0.629485\pi$
$313$	−14.0000	−0.791327	−0.395663	+	0.918396i	$0.629485\pi$
$314$	22.0000	1.24153
$315$	0	0
$316$	8.00000	0.450035
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	−6.00000	−0.336463
$319$	0	0
$320$	0	0
$321$	12.0000	0.669775
$322$	0	0
$323$	48.0000	2.67079
$324$	1.00000	0.0555556
$325$	0	0
$326$	−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
$335$	0	0
$336$	1.00000	0.0545545
$337$	−2.00000	−0.108947	−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000	−0.108947	−0.0544735	+	0.998515i	$0.517348\pi$
$338$	−9.00000	−0.489535
$339$	6.00000	0.325875
$340$	0	0
$341$	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	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	−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	$0.586245\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	0	0
$353$	6.00000	0.319348	0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000	0.319348	0.159674	+	0.987170i	$0.448956\pi$
$354$	12.0000	0.637793
$355$	0	0
$356$	−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	$0.602564\pi$
$360$	0	0
$361$	45.0000	2.36842
$362$	−10.0000	−0.525588
$363$	11.0000	0.577350
$364$	2.00000	0.104828
$365$	0	0
$366$	10.0000	0.522708
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	0	0
$371$	−6.00000	−0.311504
$372$	4.00000	0.207390
$373$	34.0000	1.76045	0.880227	−	0.474554i	$-0.157390\pi$
$373$	34.0000	1.76045	0.880227	+	0.474554i	$0.157390\pi$
$374$	0	0
$375$	0	0
$376$	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	$0.532759\pi$
$380$	0	0
$381$	8.00000	0.409852
$382$	12.0000	0.613973
$383$	−24.0000	−1.22634	−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000	−1.22634	−0.613171	+	0.789950i	$0.710106\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−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	$0.451394\pi$
$390$	0	0
$391$	0	0
$392$	1.00000	0.0505076
$393$	12.0000	0.605320
$394$	−18.0000	−0.906827
$395$	0	0
$396$	0	0
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	20.0000	1.00251
$399$	8.00000	0.400501
$400$	0	0
$401$	−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
$405$	0	0
$406$	−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	$0.277769\pi$
$410$	0	0
$411$	6.00000	0.295958
$412$	−8.00000	−0.394132
$413$	12.0000	0.590481
$414$	0	0
$415$	0	0
$416$	−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	$0.594693\pi$
$420$	0	0
$421$	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
$425$	0	0
$426$	−12.0000	−0.581402
$427$	10.0000	0.483934
$428$	−12.0000	−0.580042
$429$	0	0
$430$	0	0
$431$	−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	$-0.422759\pi$
$433$	10.0000	0.480569	0.240285	+	0.970702i	$0.422759\pi$
$434$	4.00000	0.192006
$435$	0	0
$436$	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	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	−10.0000	−0.474579
$445$	0	0
$446$	−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	$0.360367\pi$
$450$	0	0
$451$	0	0
$452$	−6.00000	−0.282216
$453$	−8.00000	−0.375873
$454$	−12.0000	−0.563188
$455$	0	0
$456$	−8.00000	−0.374634
$457$	−2.00000	−0.0935561	−0.0467780	−	0.998905i	$-0.514895\pi$
$457$	−2.00000	−0.0935561	−0.0467780	+	0.998905i	$0.514895\pi$
$458$	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	$-0.120257\pi$
$463$	40.0000	1.85896	0.929479	+	0.368875i	$0.120257\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	−30.0000	−1.38972
$467$	36.0000	1.66588	0.832941	−	0.553362i	$-0.186655\pi$
$467$	36.0000	1.66588	0.832941	+	0.553362i	$0.186655\pi$
$468$	−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
$475$	0	0
$476$	−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	$0.5\pi$
$480$	0	0
$481$	−20.0000	−0.911922
$482$	26.0000	1.18427
$483$	0	0
$484$	−11.0000	−0.500000
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	−32.0000	−1.45006	−0.725029	−	0.688718i	$-0.758174\pi$
$487$	−32.0000	−1.45006	−0.725029	+	0.688718i	$0.758174\pi$
$488$	−10.0000	−0.452679
$489$	20.0000	0.904431
$490$	0	0
$491$	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	$0.352257\pi$
$500$	0	0
$501$	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
$505$	0	0
$506$	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	$0.268490\pi$
$510$	0	0
$511$	−10.0000	−0.442374
$512$	1.00000	0.0441942
$513$	−8.00000	−0.353209
$514$	30.0000	1.32324
$515$	0	0
$516$	−4.00000	−0.176090
$517$	0	0
$518$	−10.0000	−0.439375
$519$	6.00000	0.263371
$520$	0	0
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	6.00000	0.262613
$523$	28.0000	1.22435	0.612177	−	0.790721i	$-0.290294\pi$
$523$	28.0000	1.22435	0.612177	+	0.790721i	$0.290294\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	24.0000	1.04645
$527$	−24.0000	−1.04546
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	−12.0000	−0.520756
$532$	−8.00000	−0.346844
$533$	12.0000	0.519778
$534$	6.00000	0.259645
$535$	0	0
$536$	4.00000	0.172774
$537$	24.0000	1.03568
$538$	−18.0000	−0.776035
$539$	0	0
$540$	0	0
$541$	−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
$545$	0	0
$546$	−2.00000	−0.0855921
$547$	−44.0000	−1.88130	−0.940652	−	0.339372i	$-0.889785\pi$
$547$	−44.0000	−1.88130	−0.940652	+	0.339372i	$0.889785\pi$
$548$	−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
$555$	0	0
$556$	−16.0000	−0.678551
$557$	−18.0000	−0.762684	−0.381342	−	0.924434i	$-0.624538\pi$
$557$	−18.0000	−0.762684	−0.381342	+	0.924434i	$0.624538\pi$
$558$	−4.00000	−0.169334
$559$	−8.00000	−0.338364
$560$	0	0
$561$	0	0
$562$	18.0000	0.759284
$563$	−36.0000	−1.51722	−0.758610	−	0.651546i	$-0.774121\pi$
$563$	−36.0000	−1.51722	−0.758610	+	0.651546i	$0.774121\pi$
$564$	0	0
$565$	0	0
$566$	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	$0.540139\pi$
$570$	0	0
$571$	−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	$-0.433255\pi$
$577$	10.0000	0.416305	0.208153	+	0.978096i	$0.433255\pi$
$578$	19.0000	0.790296
$579$	2.00000	0.0831172
$580$	0	0
$581$	12.0000	0.497844
$582$	−10.0000	−0.414513
$583$	0	0
$584$	10.0000	0.413803
$585$	0	0
$586$	−30.0000	−1.23929
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	−1.00000	−0.0412393
$589$	−32.0000	−1.31854
$590$	0	0
$591$	18.0000	0.740421
$592$	10.0000	0.410997
$593$	6.00000	0.246390	0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000	0.246390	0.123195	+	0.992382i	$0.460686\pi$
$594$	0	0
$595$	0	0
$596$	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	$0.236966\pi$
$600$	0	0
$601$	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
$605$	0	0
$606$	−6.00000	−0.243733
$607$	−8.00000	−0.324710	−0.162355	−	0.986732i	$-0.551909\pi$
$607$	−8.00000	−0.324710	−0.162355	+	0.986732i	$0.551909\pi$
$608$	8.00000	0.324443
$609$	6.00000	0.243132
$610$	0	0
$611$	0	0
$612$	6.00000	0.242536
$613$	−38.0000	−1.53481	−0.767403	−	0.641165i	$-0.778451\pi$
$613$	−38.0000	−1.53481	−0.767403	+	0.641165i	$0.778451\pi$
$614$	4.00000	0.161427
$615$	0	0
$616$	0	0
$617$	18.0000	0.724653	0.362326	−	0.932051i	$-0.381983\pi$
$617$	18.0000	0.724653	0.362326	+	0.932051i	$0.381983\pi$
$618$	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	$0.277650\pi$
$620$	0	0
$621$	0	0
$622$	24.0000	0.962312
$623$	6.00000	0.240385
$624$	2.00000	0.0800641
$625$	0	0
$626$	−14.0000	−0.559553
$627$	0	0
$628$	22.0000	0.877896
$629$	60.0000	2.39236
$630$	0	0
$631$	−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
$635$	0	0
$636$	−6.00000	−0.237915
$637$	−2.00000	−0.0792429
$638$	0	0
$639$	12.0000	0.474713
$640$	0	0
$641$	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	$-0.474868\pi$
$643$	4.00000	0.157745	0.0788723	+	0.996885i	$0.474868\pi$
$644$	0	0
$645$	0	0
$646$	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
$650$	0	0
$651$	−4.00000	−0.156772
$652$	−20.0000	−0.783260
$653$	6.00000	0.234798	0.117399	−	0.993085i	$-0.462544\pi$
$653$	6.00000	0.234798	0.117399	+	0.993085i	$0.462544\pi$
$654$	−14.0000	−0.547443
$655$	0	0
$656$	−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
$665$	0	0
$666$	10.0000	0.387492
$667$	0	0
$668$	0	0
$669$	8.00000	0.309298
$670$	0	0
$671$	0	0
$672$	1.00000	0.0385758
$673$	−26.0000	−1.00223	−0.501113	−	0.865382i	$-0.667076\pi$
$673$	−26.0000	−1.00223	−0.501113	+	0.865382i	$0.667076\pi$
$674$	−2.00000	−0.0770371
$675$	0	0
$676$	−9.00000	−0.346154
$677$	18.0000	0.691796	0.345898	−	0.938272i	$-0.387574\pi$
$677$	18.0000	0.691796	0.345898	+	0.938272i	$0.387574\pi$
$678$	6.00000	0.230429
$679$	−10.0000	−0.383765
$680$	0	0
$681$	12.0000	0.459841
$682$	0	0
$683$	−36.0000	−1.37750	−0.688751	−	0.724998i	$-0.741841\pi$
$683$	−36.0000	−1.37750	−0.688751	+	0.724998i	$0.741841\pi$
$684$	8.00000	0.305888
$685$	0	0
$686$	−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
$695$	0	0
$696$	−6.00000	−0.227429
$697$	−36.0000	−1.36360
$698$	−10.0000	−0.378506
$699$	30.0000	1.13470
$700$	0	0
$701$	6.00000	0.226617	0.113308	−	0.993560i	$-0.463855\pi$
$701$	6.00000	0.226617	0.113308	+	0.993560i	$0.463855\pi$
$702$	2.00000	0.0754851
$703$	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	$0.560129\pi$
$710$	0	0
$711$	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	$0.647694\pi$
$720$	0	0
$721$	8.00000	0.297936
$722$	45.0000	1.67473
$723$	−26.0000	−0.966950
$724$	−10.0000	−0.371647
$725$	0	0
$726$	11.0000	0.408248
$727$	−32.0000	−1.18681	−0.593407	−	0.804902i	$-0.702218\pi$
$727$	−32.0000	−1.18681	−0.593407	+	0.804902i	$0.702218\pi$
$728$	2.00000	0.0741249
$729$	1.00000	0.0370370
$730$	0	0
$731$	24.0000	0.887672
$732$	10.0000	0.369611
$733$	−2.00000	−0.0738717	−0.0369358	−	0.999318i	$-0.511760\pi$
$733$	−2.00000	−0.0738717	−0.0369358	+	0.999318i	$0.511760\pi$
$734$	−8.00000	−0.295285
$735$	0	0
$736$	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	$0.672207\pi$
$740$	0	0
$741$	16.0000	0.587775
$742$	−6.00000	−0.220267
$743$	24.0000	0.880475	0.440237	−	0.897881i	$-0.354894\pi$
$743$	24.0000	0.880475	0.440237	+	0.897881i	$0.354894\pi$
$744$	4.00000	0.146647
$745$	0	0
$746$	34.0000	1.24483
$747$	−12.0000	−0.439057
$748$	0	0
$749$	12.0000	0.438470
$750$	0	0
$751$	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
$755$	0	0
$756$	1.00000	0.0363696
$757$	34.0000	1.23575	0.617876	−	0.786276i	$-0.287994\pi$
$757$	34.0000	1.23575	0.617876	+	0.786276i	$0.287994\pi$
$758$	−4.00000	−0.145287
$759$	0	0
$760$	0	0
$761$	−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
$765$	0	0
$766$	−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	$-0.395071\pi$
$773$	18.0000	0.647415	0.323708	+	0.946157i	$0.395071\pi$
$774$	4.00000	0.143777
$775$	0	0
$776$	10.0000	0.358979
$777$	10.0000	0.358748
$778$	6.00000	0.215110
$779$	−48.0000	−1.71978
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−6.00000	−0.214423
$784$	1.00000	0.0357143
$785$	0	0
$786$	12.0000	0.428026
$787$	4.00000	0.142585	0.0712923	−	0.997455i	$-0.477288\pi$
$787$	4.00000	0.142585	0.0712923	+	0.997455i	$0.477288\pi$
$788$	−18.0000	−0.641223
$789$	−24.0000	−0.854423
$790$	0	0
$791$	6.00000	0.213335
$792$	0	0
$793$	20.0000	0.710221
$794$	−2.00000	−0.0709773
$795$	0	0
$796$	20.0000	0.708881
$797$	42.0000	1.48772	0.743858	−	0.668338i	$-0.232994\pi$
$797$	42.0000	1.48772	0.743858	+	0.668338i	$0.232994\pi$
$798$	8.00000	0.283197
$799$	0	0
$800$	0	0
$801$	−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	$0.676823\pi$
$810$	0	0
$811$	−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
$815$	0	0
$816$	−6.00000	−0.210042
$817$	32.0000	1.11954
$818$	26.0000	0.909069
$819$	2.00000	0.0698857
$820$	0	0
$821$	−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	$-0.688326\pi$
$823$	−32.0000	−1.11545	−0.557725	+	0.830026i	$0.688326\pi$
$824$	−8.00000	−0.278693
$825$	0	0
$826$	12.0000	0.417533
$827$	36.0000	1.25184	0.625921	−	0.779886i	$-0.284723\pi$
$827$	36.0000	1.25184	0.625921	+	0.779886i	$0.284723\pi$
$828$	0	0
$829$	−34.0000	−1.18087	−0.590434	−	0.807086i	$-0.701044\pi$
$829$	−34.0000	−1.18087	−0.590434	+	0.807086i	$0.701044\pi$
$830$	0	0
$831$	−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	$0.5\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	38.0000	1.30957
$843$	−18.0000	−0.619953
$844$	−4.00000	−0.137686
$845$	0	0
$846$	0	0
$847$	11.0000	0.377964
$848$	6.00000	0.206041
$849$	−4.00000	−0.137280
$850$	0	0
$851$	0	0
$852$	−12.0000	−0.411113
$853$	−26.0000	−0.890223	−0.445112	−	0.895475i	$-0.646836\pi$
$853$	−26.0000	−0.890223	−0.445112	+	0.895475i	$0.646836\pi$
$854$	10.0000	0.342193
$855$	0	0
$856$	−12.0000	−0.410152
$857$	6.00000	0.204956	0.102478	−	0.994735i	$-0.467323\pi$
$857$	6.00000	0.204956	0.102478	+	0.994735i	$0.467323\pi$
$858$	0	0
$859$	−40.0000	−1.36478	−0.682391	−	0.730987i	$-0.739060\pi$
$859$	−40.0000	−1.36478	−0.682391	+	0.730987i	$0.739060\pi$
$860$	0	0
$861$	−6.00000	−0.204479
$862$	−12.0000	−0.408722
$863$	24.0000	0.816970	0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000	0.816970	0.408485	+	0.912765i	$0.366057\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	10.0000	0.339814
$867$	−19.0000	−0.645274
$868$	4.00000	0.135769
$869$	0	0
$870$	0	0
$871$	−8.00000	−0.271070
$872$	14.0000	0.474100
$873$	10.0000	0.338449
$874$	0	0
$875$	0	0
$876$	−10.0000	−0.337869
$877$	−14.0000	−0.472746	−0.236373	−	0.971662i	$-0.575959\pi$
$877$	−14.0000	−0.472746	−0.236373	+	0.971662i	$0.575959\pi$
$878$	−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	$-0.765344\pi$
$883$	−44.0000	−1.48072	−0.740359	+	0.672212i	$0.765344\pi$
$884$	−12.0000	−0.403604
$885$	0	0
$886$	12.0000	0.403148
$887$	−24.0000	−0.805841	−0.402921	−	0.915235i	$-0.632005\pi$
$887$	−24.0000	−0.805841	−0.402921	+	0.915235i	$0.632005\pi$
$888$	−10.0000	−0.335578
$889$	8.00000	0.268311
$890$	0	0
$891$	0	0
$892$	−8.00000	−0.267860
$893$	0	0
$894$	−6.00000	−0.200670
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	18.0000	0.600668
$899$	−24.0000	−0.800445
$900$	0	0
$901$	36.0000	1.19933
$902$	0	0
$903$	4.00000	0.133112
$904$	−6.00000	−0.199557
$905$	0	0
$906$	−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	$0.346104\pi$
$908$	−12.0000	−0.398234
$909$	6.00000	0.199007
$910$	0	0
$911$	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
$915$	0	0
$916$	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
$925$	0	0
$926$	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	$0.663783\pi$
$930$	0	0
$931$	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	$-0.713153\pi$
$937$	−38.0000	−1.24141	−0.620703	+	0.784046i	$0.713153\pi$
$938$	−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
$945$	0	0
$946$	0	0
$947$	36.0000	1.16984	0.584921	−	0.811090i	$-0.301125\pi$
$947$	36.0000	1.16984	0.584921	+	0.811090i	$0.301125\pi$
$948$	−8.00000	−0.259828
$949$	−20.0000	−0.649227
$950$	0	0
$951$	18.0000	0.583690
$952$	−6.00000	−0.194461
$953$	−30.0000	−0.971795	−0.485898	−	0.874016i	$-0.661507\pi$
$953$	−30.0000	−0.971795	−0.485898	+	0.874016i	$0.661507\pi$
$954$	6.00000	0.194257
$955$	0	0
$956$	−12.0000	−0.388108
$957$	0	0
$958$	0	0
$959$	6.00000	0.193750
$960$	0	0
$961$	−15.0000	−0.483871
$962$	−20.0000	−0.644826
$963$	−12.0000	−0.386695
$964$	26.0000	0.837404
$965$	0	0
$966$	0	0
$967$	−8.00000	−0.257263	−0.128631	−	0.991692i	$-0.541058\pi$
$967$	−8.00000	−0.257263	−0.128631	+	0.991692i	$0.541058\pi$
$968$	−11.0000	−0.353553
$969$	−48.0000	−1.54198
$970$	0	0
$971$	36.0000	1.15529	0.577647	−	0.816286i	$-0.303971\pi$
$971$	36.0000	1.15529	0.577647	+	0.816286i	$0.303971\pi$
$972$	−1.00000	−0.0320750
$973$	16.0000	0.512936
$974$	−32.0000	−1.02535
$975$	0	0
$976$	−10.0000	−0.320092
$977$	−6.00000	−0.191957	−0.0959785	−	0.995383i	$-0.530598\pi$
$977$	−6.00000	−0.191957	−0.0959785	+	0.995383i	$0.530598\pi$
$978$	20.0000	0.639529
$979$	0	0
$980$	0	0
$981$	14.0000	0.446986
$982$	0	0
$983$	−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
$985$	0	0
$986$	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
$995$	0	0
$996$	12.0000	0.380235
$997$	−26.0000	−0.823428	−0.411714	−	0.911313i	$-0.635070\pi$
$997$	−26.0000	−0.823428	−0.411714	+	0.911313i	$0.635070\pi$
$998$	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	1050.2.a.k.1.1		1
3.2	odd	2		3150.2.a.f.1.1		1
4.3	odd	2		8400.2.a.cm.1.1		1
5.2	odd	4		1050.2.g.c.799.2		2
5.3	odd	4		1050.2.g.c.799.1		2
5.4	even	2		210.2.a.b.1.1	✓	1
7.6	odd	2		7350.2.a.cs.1.1		1
15.2	even	4		3150.2.g.i.2899.1		2
15.8	even	4		3150.2.g.i.2899.2		2
15.14	odd	2		630.2.a.h.1.1		1
20.19	odd	2		1680.2.a.g.1.1		1
35.4	even	6		1470.2.i.l.961.1		2
35.9	even	6		1470.2.i.l.361.1		2
35.19	odd	6		1470.2.i.s.361.1		2
35.24	odd	6		1470.2.i.s.961.1		2
35.34	odd	2		1470.2.a.b.1.1		1
40.19	odd	2		6720.2.a.bi.1.1		1
40.29	even	2		6720.2.a.n.1.1		1
60.59	even	2		5040.2.a.g.1.1		1
105.104	even	2		4410.2.a.bi.1.1		1

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1050.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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