LMFDB - Embedded newform 930.2.a.k.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [930,2,Mod(1,930)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("930.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(930, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 930 = 2 \cdot 3 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	930.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,1,-1,1,-1,-1,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$7.42608738798$
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	$\chi$	$=$	930.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.00000 q^{12} +4.00000 q^{13} -2.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} +6.00000 q^{17} +1.00000 q^{18} -1.00000 q^{20} +2.00000 q^{21} -1.00000 q^{24} +1.00000 q^{25} +4.00000 q^{26} -1.00000 q^{27} -2.00000 q^{28} +8.00000 q^{29} +1.00000 q^{30} -1.00000 q^{31} +1.00000 q^{32} +6.00000 q^{34} +2.00000 q^{35} +1.00000 q^{36} +4.00000 q^{37} -4.00000 q^{39} -1.00000 q^{40} +10.0000 q^{41} +2.00000 q^{42} +8.00000 q^{43} -1.00000 q^{45} -4.00000 q^{47} -1.00000 q^{48} -3.00000 q^{49} +1.00000 q^{50} -6.00000 q^{51} +4.00000 q^{52} -14.0000 q^{53} -1.00000 q^{54} -2.00000 q^{56} +8.00000 q^{58} +14.0000 q^{59} +1.00000 q^{60} -6.00000 q^{61} -1.00000 q^{62} -2.00000 q^{63} +1.00000 q^{64} -4.00000 q^{65} -10.0000 q^{67} +6.00000 q^{68} +2.00000 q^{70} +6.00000 q^{71} +1.00000 q^{72} -8.00000 q^{73} +4.00000 q^{74} -1.00000 q^{75} -4.00000 q^{78} +8.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +10.0000 q^{82} -12.0000 q^{83} +2.00000 q^{84} -6.00000 q^{85} +8.00000 q^{86} -8.00000 q^{87} +16.0000 q^{89} -1.00000 q^{90} -8.00000 q^{91} +1.00000 q^{93} -4.00000 q^{94} -1.00000 q^{96} -10.0000 q^{97} -3.00000 q^{98} +O(q^{100})\)

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−1.00000	−0.408248
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	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
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	−2.00000	−0.534522
$15$	1.00000	0.258199
$16$	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
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	−1.00000	−0.223607
$21$	2.00000	0.436436
$22$	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
$26$	4.00000	0.784465
$27$	−1.00000	−0.192450
$28$	−2.00000	−0.377964
$29$	8.00000	1.48556	0.742781	−	0.669534i	$-0.233506\pi$
$29$	8.00000	1.48556	0.742781	+	0.669534i	$0.233506\pi$
$30$	1.00000	0.182574
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	1.00000	0.176777
$33$	0	0
$34$	6.00000	1.02899
$35$	2.00000	0.338062
$36$	1.00000	0.166667
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	0	0
$39$	−4.00000	−0.640513
$40$	−1.00000	−0.158114
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	2.00000	0.308607
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−4.00000	−0.583460	−0.291730	−	0.956501i	$-0.594231\pi$
$47$	−4.00000	−0.583460	−0.291730	+	0.956501i	$0.594231\pi$
$48$	−1.00000	−0.144338
$49$	−3.00000	−0.428571
$50$	1.00000	0.141421
$51$	−6.00000	−0.840168
$52$	4.00000	0.554700
$53$	−14.0000	−1.92305	−0.961524	−	0.274721i	$-0.911414\pi$
$53$	−14.0000	−1.92305	−0.961524	+	0.274721i	$0.911414\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	−2.00000	−0.267261
$57$	0	0
$58$	8.00000	1.05045
$59$	14.0000	1.82264	0.911322	−	0.411693i	$-0.135063\pi$
$59$	14.0000	1.82264	0.911322	+	0.411693i	$0.135063\pi$
$60$	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	$0.625492\pi$
$62$	−1.00000	−0.127000
$63$	−2.00000	−0.251976
$64$	1.00000	0.125000
$65$	−4.00000	−0.496139
$66$	0	0
$67$	−10.0000	−1.22169	−0.610847	−	0.791748i	$-0.709171\pi$
$67$	−10.0000	−1.22169	−0.610847	+	0.791748i	$0.709171\pi$
$68$	6.00000	0.727607
$69$	0	0
$70$	2.00000	0.239046
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	1.00000	0.117851
$73$	−8.00000	−0.936329	−0.468165	−	0.883641i	$-0.655085\pi$
$73$	−8.00000	−0.936329	−0.468165	+	0.883641i	$0.655085\pi$
$74$	4.00000	0.464991
$75$	−1.00000	−0.115470
$76$	0	0
$77$	0	0
$78$	−4.00000	−0.452911
$79$	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
$82$	10.0000	1.10432
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	2.00000	0.218218
$85$	−6.00000	−0.650791
$86$	8.00000	0.862662
$87$	−8.00000	−0.857690
$88$	0	0
$89$	16.0000	1.69600	0.847998	−	0.529999i	$-0.177808\pi$
$89$	16.0000	1.69600	0.847998	+	0.529999i	$0.177808\pi$
$90$	−1.00000	−0.105409
$91$	−8.00000	−0.838628
$92$	0	0
$93$	1.00000	0.103695
$94$	−4.00000	−0.412568
$95$	0	0
$96$	−1.00000	−0.102062
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	−3.00000	−0.303046
$99$	0	0
$100$	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
$103$	−6.00000	−0.591198	−0.295599	−	0.955312i	$-0.595519\pi$
$103$	−6.00000	−0.591198	−0.295599	+	0.955312i	$0.595519\pi$
$104$	4.00000	0.392232
$105$	−2.00000	−0.195180
$106$	−14.0000	−1.35980
$107$	8.00000	0.773389	0.386695	−	0.922208i	$-0.373617\pi$
$107$	8.00000	0.773389	0.386695	+	0.922208i	$0.373617\pi$
$108$	−1.00000	−0.0962250
$109$	18.0000	1.72409	0.862044	−	0.506834i	$-0.169184\pi$
$109$	18.0000	1.72409	0.862044	+	0.506834i	$0.169184\pi$
$110$	0	0
$111$	−4.00000	−0.379663
$112$	−2.00000	−0.188982
$113$	14.0000	1.31701	0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000	1.31701	0.658505	+	0.752577i	$0.271189\pi$
$114$	0	0
$115$	0	0
$116$	8.00000	0.742781
$117$	4.00000	0.369800
$118$	14.0000	1.28880
$119$	−12.0000	−1.10004
$120$	1.00000	0.0912871
$121$	−11.0000	−1.00000
$122$	−6.00000	−0.543214
$123$	−10.0000	−0.901670
$124$	−1.00000	−0.0898027
$125$	−1.00000	−0.0894427
$126$	−2.00000	−0.178174
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	1.00000	0.0883883
$129$	−8.00000	−0.704361
$130$	−4.00000	−0.350823
$131$	−18.0000	−1.57267	−0.786334	−	0.617802i	$-0.788023\pi$
$131$	−18.0000	−1.57267	−0.786334	+	0.617802i	$0.788023\pi$
$132$	0	0
$133$	0	0
$134$	−10.0000	−0.863868
$135$	1.00000	0.0860663
$136$	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
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	2.00000	0.169031
$141$	4.00000	0.336861
$142$	6.00000	0.503509
$143$	0	0
$144$	1.00000	0.0833333
$145$	−8.00000	−0.664364
$146$	−8.00000	−0.662085
$147$	3.00000	0.247436
$148$	4.00000	0.328798
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	−1.00000	−0.0816497
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	0	0
$153$	6.00000	0.485071
$154$	0	0
$155$	1.00000	0.0803219
$156$	−4.00000	−0.320256
$157$	−18.0000	−1.43656	−0.718278	−	0.695756i	$-0.755069\pi$
$157$	−18.0000	−1.43656	−0.718278	+	0.695756i	$0.755069\pi$
$158$	8.00000	0.636446
$159$	14.0000	1.11027
$160$	−1.00000	−0.0790569
$161$	0	0
$162$	1.00000	0.0785674
$163$	6.00000	0.469956	0.234978	−	0.972001i	$-0.424498\pi$
$163$	6.00000	0.469956	0.234978	+	0.972001i	$0.424498\pi$
$164$	10.0000	0.780869
$165$	0	0
$166$	−12.0000	−0.931381
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	2.00000	0.154303
$169$	3.00000	0.230769
$170$	−6.00000	−0.460179
$171$	0	0
$172$	8.00000	0.609994
$173$	10.0000	0.760286	0.380143	−	0.924928i	$-0.375875\pi$
$173$	10.0000	0.760286	0.380143	+	0.924928i	$0.375875\pi$
$174$	−8.00000	−0.606478
$175$	−2.00000	−0.151186
$176$	0	0
$177$	−14.0000	−1.05230
$178$	16.0000	1.19925
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	−1.00000	−0.0745356
$181$	−6.00000	−0.445976	−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000	−0.445976	−0.222988	+	0.974821i	$0.571581\pi$
$182$	−8.00000	−0.592999
$183$	6.00000	0.443533
$184$	0	0
$185$	−4.00000	−0.294086
$186$	1.00000	0.0733236
$187$	0	0
$188$	−4.00000	−0.291730
$189$	2.00000	0.145479
$190$	0	0
$191$	−14.0000	−1.01300	−0.506502	−	0.862239i	$-0.669062\pi$
$191$	−14.0000	−1.01300	−0.506502	+	0.862239i	$0.669062\pi$
$192$	−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
$195$	4.00000	0.286446
$196$	−3.00000	−0.214286
$197$	−2.00000	−0.142494	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000	−0.142494	−0.0712470	+	0.997459i	$0.522698\pi$
$198$	0	0
$199$	−24.0000	−1.70131	−0.850657	−	0.525720i	$-0.823796\pi$
$199$	−24.0000	−1.70131	−0.850657	+	0.525720i	$0.823796\pi$
$200$	1.00000	0.0707107
$201$	10.0000	0.705346
$202$	−6.00000	−0.422159
$203$	−16.0000	−1.12298
$204$	−6.00000	−0.420084
$205$	−10.0000	−0.698430
$206$	−6.00000	−0.418040
$207$	0	0
$208$	4.00000	0.277350
$209$	0	0
$210$	−2.00000	−0.138013
$211$	16.0000	1.10149	0.550743	−	0.834675i	$-0.314345\pi$
$211$	16.0000	1.10149	0.550743	+	0.834675i	$0.314345\pi$
$212$	−14.0000	−0.961524
$213$	−6.00000	−0.411113
$214$	8.00000	0.546869
$215$	−8.00000	−0.545595
$216$	−1.00000	−0.0680414
$217$	2.00000	0.135769
$218$	18.0000	1.21911
$219$	8.00000	0.540590
$220$	0	0
$221$	24.0000	1.61441
$222$	−4.00000	−0.268462
$223$	−20.0000	−1.33930	−0.669650	−	0.742677i	$-0.733556\pi$
$223$	−20.0000	−1.33930	−0.669650	+	0.742677i	$0.733556\pi$
$224$	−2.00000	−0.133631
$225$	1.00000	0.0666667
$226$	14.0000	0.931266
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	0	0
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	0	0
$231$	0	0
$232$	8.00000	0.525226
$233$	−14.0000	−0.917170	−0.458585	−	0.888650i	$-0.651644\pi$
$233$	−14.0000	−0.917170	−0.458585	+	0.888650i	$0.651644\pi$
$234$	4.00000	0.261488
$235$	4.00000	0.260931
$236$	14.0000	0.911322
$237$	−8.00000	−0.519656
$238$	−12.0000	−0.777844
$239$	−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	$0.648900\pi$
$242$	−11.0000	−0.707107
$243$	−1.00000	−0.0641500
$244$	−6.00000	−0.384111
$245$	3.00000	0.191663
$246$	−10.0000	−0.637577
$247$	0	0
$248$	−1.00000	−0.0635001
$249$	12.0000	0.760469
$250$	−1.00000	−0.0632456
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	−2.00000	−0.125988
$253$	0	0
$254$	0	0
$255$	6.00000	0.375735
$256$	1.00000	0.0625000
$257$	2.00000	0.124757	0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000	0.124757	0.0623783	+	0.998053i	$0.480131\pi$
$258$	−8.00000	−0.498058
$259$	−8.00000	−0.497096
$260$	−4.00000	−0.248069
$261$	8.00000	0.495188
$262$	−18.0000	−1.11204
$263$	16.0000	0.986602	0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000	0.986602	0.493301	+	0.869859i	$0.335790\pi$
$264$	0	0
$265$	14.0000	0.860013
$266$	0	0
$267$	−16.0000	−0.979184
$268$	−10.0000	−0.610847
$269$	24.0000	1.46331	0.731653	−	0.681677i	$-0.238749\pi$
$269$	24.0000	1.46331	0.731653	+	0.681677i	$0.238749\pi$
$270$	1.00000	0.0608581
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	6.00000	0.363803
$273$	8.00000	0.484182
$274$	−6.00000	−0.362473
$275$	0	0
$276$	0	0
$277$	16.0000	0.961347	0.480673	−	0.876900i	$-0.340392\pi$
$277$	16.0000	0.961347	0.480673	+	0.876900i	$0.340392\pi$
$278$	4.00000	0.239904
$279$	−1.00000	−0.0598684
$280$	2.00000	0.119523
$281$	30.0000	1.78965	0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000	1.78965	0.894825	+	0.446417i	$0.147300\pi$
$282$	4.00000	0.238197
$283$	14.0000	0.832214	0.416107	−	0.909316i	$-0.363394\pi$
$283$	14.0000	0.832214	0.416107	+	0.909316i	$0.363394\pi$
$284$	6.00000	0.356034
$285$	0	0
$286$	0	0
$287$	−20.0000	−1.18056
$288$	1.00000	0.0589256
$289$	19.0000	1.11765
$290$	−8.00000	−0.469776
$291$	10.0000	0.586210
$292$	−8.00000	−0.468165
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	3.00000	0.174964
$295$	−14.0000	−0.815112
$296$	4.00000	0.232495
$297$	0	0
$298$	18.0000	1.04271
$299$	0	0
$300$	−1.00000	−0.0577350
$301$	−16.0000	−0.922225
$302$	−8.00000	−0.460348
$303$	6.00000	0.344691
$304$	0	0
$305$	6.00000	0.343559
$306$	6.00000	0.342997
$307$	10.0000	0.570730	0.285365	−	0.958419i	$-0.407885\pi$
$307$	10.0000	0.570730	0.285365	+	0.958419i	$0.407885\pi$
$308$	0	0
$309$	6.00000	0.341328
$310$	1.00000	0.0567962
$311$	18.0000	1.02069	0.510343	−	0.859971i	$-0.329518\pi$
$311$	18.0000	1.02069	0.510343	+	0.859971i	$0.329518\pi$
$312$	−4.00000	−0.226455
$313$	−16.0000	−0.904373	−0.452187	−	0.891923i	$-0.649356\pi$
$313$	−16.0000	−0.904373	−0.452187	+	0.891923i	$0.649356\pi$
$314$	−18.0000	−1.01580
$315$	2.00000	0.112687
$316$	8.00000	0.450035
$317$	−10.0000	−0.561656	−0.280828	−	0.959758i	$-0.590609\pi$
$317$	−10.0000	−0.561656	−0.280828	+	0.959758i	$0.590609\pi$
$318$	14.0000	0.785081
$319$	0	0
$320$	−1.00000	−0.0559017
$321$	−8.00000	−0.446516
$322$	0	0
$323$	0	0
$324$	1.00000	0.0555556
$325$	4.00000	0.221880
$326$	6.00000	0.332309
$327$	−18.0000	−0.995402
$328$	10.0000	0.552158
$329$	8.00000	0.441054
$330$	0	0
$331$	−12.0000	−0.659580	−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000	−0.659580	−0.329790	+	0.944054i	$0.606978\pi$
$332$	−12.0000	−0.658586
$333$	4.00000	0.219199
$334$	−8.00000	−0.437741
$335$	10.0000	0.546358
$336$	2.00000	0.109109
$337$	4.00000	0.217894	0.108947	−	0.994048i	$-0.465252\pi$
$337$	4.00000	0.217894	0.108947	+	0.994048i	$0.465252\pi$
$338$	3.00000	0.163178
$339$	−14.0000	−0.760376
$340$	−6.00000	−0.325396
$341$	0	0
$342$	0	0
$343$	20.0000	1.07990
$344$	8.00000	0.431331
$345$	0	0
$346$	10.0000	0.537603
$347$	20.0000	1.07366	0.536828	−	0.843692i	$-0.319622\pi$
$347$	20.0000	1.07366	0.536828	+	0.843692i	$0.319622\pi$
$348$	−8.00000	−0.428845
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	−2.00000	−0.106904
$351$	−4.00000	−0.213504
$352$	0	0
$353$	14.0000	0.745145	0.372572	−	0.928003i	$-0.378476\pi$
$353$	14.0000	0.745145	0.372572	+	0.928003i	$0.378476\pi$
$354$	−14.0000	−0.744092
$355$	−6.00000	−0.318447
$356$	16.0000	0.847998
$357$	12.0000	0.635107
$358$	0	0
$359$	18.0000	0.950004	0.475002	−	0.879985i	$-0.342447\pi$
$359$	18.0000	0.950004	0.475002	+	0.879985i	$0.342447\pi$
$360$	−1.00000	−0.0527046
$361$	−19.0000	−1.00000
$362$	−6.00000	−0.315353
$363$	11.0000	0.577350
$364$	−8.00000	−0.419314
$365$	8.00000	0.418739
$366$	6.00000	0.313625
$367$	−36.0000	−1.87918	−0.939592	−	0.342296i	$-0.888796\pi$
$367$	−36.0000	−1.87918	−0.939592	+	0.342296i	$0.888796\pi$
$368$	0	0
$369$	10.0000	0.520579
$370$	−4.00000	−0.207950
$371$	28.0000	1.45369
$372$	1.00000	0.0518476
$373$	6.00000	0.310668	0.155334	−	0.987862i	$-0.450355\pi$
$373$	6.00000	0.310668	0.155334	+	0.987862i	$0.450355\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	−4.00000	−0.206284
$377$	32.0000	1.64808
$378$	2.00000	0.102869
$379$	8.00000	0.410932	0.205466	−	0.978664i	$-0.434129\pi$
$379$	8.00000	0.410932	0.205466	+	0.978664i	$0.434129\pi$
$380$	0	0
$381$	0	0
$382$	−14.0000	−0.716302
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−2.00000	−0.101797
$387$	8.00000	0.406663
$388$	−10.0000	−0.507673
$389$	−32.0000	−1.62246	−0.811232	−	0.584724i	$-0.801203\pi$
$389$	−32.0000	−1.62246	−0.811232	+	0.584724i	$0.801203\pi$
$390$	4.00000	0.202548
$391$	0	0
$392$	−3.00000	−0.151523
$393$	18.0000	0.907980
$394$	−2.00000	−0.100759
$395$	−8.00000	−0.402524
$396$	0	0
$397$	18.0000	0.903394	0.451697	−	0.892171i	$-0.350819\pi$
$397$	18.0000	0.903394	0.451697	+	0.892171i	$0.350819\pi$
$398$	−24.0000	−1.20301
$399$	0	0
$400$	1.00000	0.0500000
$401$	4.00000	0.199750	0.0998752	−	0.995000i	$-0.468156\pi$
$401$	4.00000	0.199750	0.0998752	+	0.995000i	$0.468156\pi$
$402$	10.0000	0.498755
$403$	−4.00000	−0.199254
$404$	−6.00000	−0.298511
$405$	−1.00000	−0.0496904
$406$	−16.0000	−0.794067
$407$	0	0
$408$	−6.00000	−0.297044
$409$	30.0000	1.48340	0.741702	−	0.670729i	$-0.234019\pi$
$409$	30.0000	1.48340	0.741702	+	0.670729i	$0.234019\pi$
$410$	−10.0000	−0.493865
$411$	6.00000	0.295958
$412$	−6.00000	−0.295599
$413$	−28.0000	−1.37779
$414$	0	0
$415$	12.0000	0.589057
$416$	4.00000	0.196116
$417$	−4.00000	−0.195881
$418$	0	0
$419$	−6.00000	−0.293119	−0.146560	−	0.989202i	$-0.546820\pi$
$419$	−6.00000	−0.293119	−0.146560	+	0.989202i	$0.546820\pi$
$420$	−2.00000	−0.0975900
$421$	−6.00000	−0.292422	−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000	−0.292422	−0.146211	+	0.989253i	$0.546708\pi$
$422$	16.0000	0.778868
$423$	−4.00000	−0.194487
$424$	−14.0000	−0.679900
$425$	6.00000	0.291043
$426$	−6.00000	−0.290701
$427$	12.0000	0.580721
$428$	8.00000	0.386695
$429$	0	0
$430$	−8.00000	−0.385794
$431$	−6.00000	−0.289010	−0.144505	−	0.989504i	$-0.546159\pi$
$431$	−6.00000	−0.289010	−0.144505	+	0.989504i	$0.546159\pi$
$432$	−1.00000	−0.0481125
$433$	−16.0000	−0.768911	−0.384455	−	0.923144i	$-0.625611\pi$
$433$	−16.0000	−0.768911	−0.384455	+	0.923144i	$0.625611\pi$
$434$	2.00000	0.0960031
$435$	8.00000	0.383571
$436$	18.0000	0.862044
$437$	0	0
$438$	8.00000	0.382255
$439$	24.0000	1.14546	0.572729	−	0.819745i	$-0.305885\pi$
$439$	24.0000	1.14546	0.572729	+	0.819745i	$0.305885\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	24.0000	1.14156
$443$	−36.0000	−1.71041	−0.855206	−	0.518289i	$-0.826569\pi$
$443$	−36.0000	−1.71041	−0.855206	+	0.518289i	$0.826569\pi$
$444$	−4.00000	−0.189832
$445$	−16.0000	−0.758473
$446$	−20.0000	−0.947027
$447$	−18.0000	−0.851371
$448$	−2.00000	−0.0944911
$449$	−20.0000	−0.943858	−0.471929	−	0.881636i	$-0.656442\pi$
$449$	−20.0000	−0.943858	−0.471929	+	0.881636i	$0.656442\pi$
$450$	1.00000	0.0471405
$451$	0	0
$452$	14.0000	0.658505
$453$	8.00000	0.375873
$454$	12.0000	0.563188
$455$	8.00000	0.375046
$456$	0	0
$457$	8.00000	0.374224	0.187112	−	0.982339i	$-0.440087\pi$
$457$	8.00000	0.374224	0.187112	+	0.982339i	$0.440087\pi$
$458$	−10.0000	−0.467269
$459$	−6.00000	−0.280056
$460$	0	0
$461$	−12.0000	−0.558896	−0.279448	−	0.960161i	$-0.590151\pi$
$461$	−12.0000	−0.558896	−0.279448	+	0.960161i	$0.590151\pi$
$462$	0	0
$463$	0	0		−	1.00000i	$-0.5\pi$
$463$	0	0			1.00000i	$0.5\pi$
$464$	8.00000	0.371391
$465$	−1.00000	−0.0463739
$466$	−14.0000	−0.648537
$467$	−32.0000	−1.48078	−0.740392	−	0.672176i	$-0.765360\pi$
$467$	−32.0000	−1.48078	−0.740392	+	0.672176i	$0.765360\pi$
$468$	4.00000	0.184900
$469$	20.0000	0.923514
$470$	4.00000	0.184506
$471$	18.0000	0.829396
$472$	14.0000	0.644402
$473$	0	0
$474$	−8.00000	−0.367452
$475$	0	0
$476$	−12.0000	−0.550019
$477$	−14.0000	−0.641016
$478$	−12.0000	−0.548867
$479$	−42.0000	−1.91903	−0.959514	−	0.281659i	$-0.909115\pi$
$479$	−42.0000	−1.91903	−0.959514	+	0.281659i	$0.909115\pi$
$480$	1.00000	0.0456435
$481$	16.0000	0.729537
$482$	−14.0000	−0.637683
$483$	0	0
$484$	−11.0000	−0.500000
$485$	10.0000	0.454077
$486$	−1.00000	−0.0453609
$487$	−8.00000	−0.362515	−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000	−0.362515	−0.181257	+	0.983436i	$0.558017\pi$
$488$	−6.00000	−0.271607
$489$	−6.00000	−0.271329
$490$	3.00000	0.135526
$491$	36.0000	1.62466	0.812329	−	0.583200i	$-0.198200\pi$
$491$	36.0000	1.62466	0.812329	+	0.583200i	$0.198200\pi$
$492$	−10.0000	−0.450835
$493$	48.0000	2.16181
$494$	0	0
$495$	0	0
$496$	−1.00000	−0.0449013
$497$	−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
$501$	8.00000	0.357414
$502$	0	0
$503$	−20.0000	−0.891756	−0.445878	−	0.895094i	$-0.647108\pi$
$503$	−20.0000	−0.891756	−0.445878	+	0.895094i	$0.647108\pi$
$504$	−2.00000	−0.0890871
$505$	6.00000	0.266996
$506$	0	0
$507$	−3.00000	−0.133235
$508$	0	0
$509$	4.00000	0.177297	0.0886484	−	0.996063i	$-0.471745\pi$
$509$	4.00000	0.177297	0.0886484	+	0.996063i	$0.471745\pi$
$510$	6.00000	0.265684
$511$	16.0000	0.707798
$512$	1.00000	0.0441942
$513$	0	0
$514$	2.00000	0.0882162
$515$	6.00000	0.264392
$516$	−8.00000	−0.352180
$517$	0	0
$518$	−8.00000	−0.351500
$519$	−10.0000	−0.438951
$520$	−4.00000	−0.175412
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	8.00000	0.350150
$523$	24.0000	1.04945	0.524723	−	0.851273i	$-0.324169\pi$
$523$	24.0000	1.04945	0.524723	+	0.851273i	$0.324169\pi$
$524$	−18.0000	−0.786334
$525$	2.00000	0.0872872
$526$	16.0000	0.697633
$527$	−6.00000	−0.261364
$528$	0	0
$529$	−23.0000	−1.00000
$530$	14.0000	0.608121
$531$	14.0000	0.607548
$532$	0	0
$533$	40.0000	1.73259
$534$	−16.0000	−0.692388
$535$	−8.00000	−0.345870
$536$	−10.0000	−0.431934
$537$	0	0
$538$	24.0000	1.03471
$539$	0	0
$540$	1.00000	0.0430331
$541$	18.0000	0.773880	0.386940	−	0.922105i	$-0.373532\pi$
$541$	18.0000	0.773880	0.386940	+	0.922105i	$0.373532\pi$
$542$	−16.0000	−0.687259
$543$	6.00000	0.257485
$544$	6.00000	0.257248
$545$	−18.0000	−0.771035
$546$	8.00000	0.342368
$547$	6.00000	0.256541	0.128271	−	0.991739i	$-0.459057\pi$
$547$	6.00000	0.256541	0.128271	+	0.991739i	$0.459057\pi$
$548$	−6.00000	−0.256307
$549$	−6.00000	−0.256074
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−16.0000	−0.680389
$554$	16.0000	0.679775
$555$	4.00000	0.169791
$556$	4.00000	0.169638
$557$	−30.0000	−1.27114	−0.635570	−	0.772043i	$-0.719235\pi$
$557$	−30.0000	−1.27114	−0.635570	+	0.772043i	$0.719235\pi$
$558$	−1.00000	−0.0423334
$559$	32.0000	1.35346
$560$	2.00000	0.0845154
$561$	0	0
$562$	30.0000	1.26547
$563$	−32.0000	−1.34864	−0.674320	−	0.738440i	$-0.735563\pi$
$563$	−32.0000	−1.34864	−0.674320	+	0.738440i	$0.735563\pi$
$564$	4.00000	0.168430
$565$	−14.0000	−0.588984
$566$	14.0000	0.588464
$567$	−2.00000	−0.0839921
$568$	6.00000	0.251754
$569$	4.00000	0.167689	0.0838444	−	0.996479i	$-0.473280\pi$
$569$	4.00000	0.167689	0.0838444	+	0.996479i	$0.473280\pi$
$570$	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
$573$	14.0000	0.584858
$574$	−20.0000	−0.834784
$575$	0	0
$576$	1.00000	0.0416667
$577$	−6.00000	−0.249783	−0.124892	−	0.992170i	$-0.539858\pi$
$577$	−6.00000	−0.249783	−0.124892	+	0.992170i	$0.539858\pi$
$578$	19.0000	0.790296
$579$	2.00000	0.0831172
$580$	−8.00000	−0.332182
$581$	24.0000	0.995688
$582$	10.0000	0.414513
$583$	0	0
$584$	−8.00000	−0.331042
$585$	−4.00000	−0.165380
$586$	6.00000	0.247858
$587$	12.0000	0.495293	0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000	0.495293	0.247647	+	0.968850i	$0.420343\pi$
$588$	3.00000	0.123718
$589$	0	0
$590$	−14.0000	−0.576371
$591$	2.00000	0.0822690
$592$	4.00000	0.164399
$593$	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
$595$	12.0000	0.491952
$596$	18.0000	0.737309
$597$	24.0000	0.982255
$598$	0	0
$599$	42.0000	1.71607	0.858037	−	0.513588i	$-0.171684\pi$
$599$	42.0000	1.71607	0.858037	+	0.513588i	$0.171684\pi$
$600$	−1.00000	−0.0408248
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	−16.0000	−0.652111
$603$	−10.0000	−0.407231
$604$	−8.00000	−0.325515
$605$	11.0000	0.447214
$606$	6.00000	0.243733
$607$	2.00000	0.0811775	0.0405887	−	0.999176i	$-0.487077\pi$
$607$	2.00000	0.0811775	0.0405887	+	0.999176i	$0.487077\pi$
$608$	0	0
$609$	16.0000	0.648353
$610$	6.00000	0.242933
$611$	−16.0000	−0.647291
$612$	6.00000	0.242536
$613$	16.0000	0.646234	0.323117	−	0.946359i	$-0.395269\pi$
$613$	16.0000	0.646234	0.323117	+	0.946359i	$0.395269\pi$
$614$	10.0000	0.403567
$615$	10.0000	0.403239
$616$	0	0
$617$	38.0000	1.52982	0.764911	−	0.644136i	$-0.222783\pi$
$617$	38.0000	1.52982	0.764911	+	0.644136i	$0.222783\pi$
$618$	6.00000	0.241355
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	1.00000	0.0401610
$621$	0	0
$622$	18.0000	0.721734
$623$	−32.0000	−1.28205
$624$	−4.00000	−0.160128
$625$	1.00000	0.0400000
$626$	−16.0000	−0.639489
$627$	0	0
$628$	−18.0000	−0.718278
$629$	24.0000	0.956943
$630$	2.00000	0.0796819
$631$	−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
$633$	−16.0000	−0.635943
$634$	−10.0000	−0.397151
$635$	0	0
$636$	14.0000	0.555136
$637$	−12.0000	−0.475457
$638$	0	0
$639$	6.00000	0.237356
$640$	−1.00000	−0.0395285
$641$	−44.0000	−1.73790	−0.868948	−	0.494904i	$-0.835203\pi$
$641$	−44.0000	−1.73790	−0.868948	+	0.494904i	$0.835203\pi$
$642$	−8.00000	−0.315735
$643$	−16.0000	−0.630978	−0.315489	−	0.948929i	$-0.602169\pi$
$643$	−16.0000	−0.630978	−0.315489	+	0.948929i	$0.602169\pi$
$644$	0	0
$645$	8.00000	0.315000
$646$	0	0
$647$	−16.0000	−0.629025	−0.314512	−	0.949253i	$-0.601841\pi$
$647$	−16.0000	−0.629025	−0.314512	+	0.949253i	$0.601841\pi$
$648$	1.00000	0.0392837
$649$	0	0
$650$	4.00000	0.156893
$651$	−2.00000	−0.0783862
$652$	6.00000	0.234978
$653$	−10.0000	−0.391330	−0.195665	−	0.980671i	$-0.562687\pi$
$653$	−10.0000	−0.391330	−0.195665	+	0.980671i	$0.562687\pi$
$654$	−18.0000	−0.703856
$655$	18.0000	0.703318
$656$	10.0000	0.390434
$657$	−8.00000	−0.312110
$658$	8.00000	0.311872
$659$	−30.0000	−1.16863	−0.584317	−	0.811525i	$-0.698638\pi$
$659$	−30.0000	−1.16863	−0.584317	+	0.811525i	$0.698638\pi$
$660$	0	0
$661$	−42.0000	−1.63361	−0.816805	−	0.576913i	$-0.804257\pi$
$661$	−42.0000	−1.63361	−0.816805	+	0.576913i	$0.804257\pi$
$662$	−12.0000	−0.466393
$663$	−24.0000	−0.932083
$664$	−12.0000	−0.465690
$665$	0	0
$666$	4.00000	0.154997
$667$	0	0
$668$	−8.00000	−0.309529
$669$	20.0000	0.773245
$670$	10.0000	0.386334
$671$	0	0
$672$	2.00000	0.0771517
$673$	−20.0000	−0.770943	−0.385472	−	0.922720i	$-0.625961\pi$
$673$	−20.0000	−0.770943	−0.385472	+	0.922720i	$0.625961\pi$
$674$	4.00000	0.154074
$675$	−1.00000	−0.0384900
$676$	3.00000	0.115385
$677$	−22.0000	−0.845529	−0.422764	−	0.906240i	$-0.638940\pi$
$677$	−22.0000	−0.845529	−0.422764	+	0.906240i	$0.638940\pi$
$678$	−14.0000	−0.537667
$679$	20.0000	0.767530
$680$	−6.00000	−0.230089
$681$	−12.0000	−0.459841
$682$	0	0
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	0	0
$685$	6.00000	0.229248
$686$	20.0000	0.763604
$687$	10.0000	0.381524
$688$	8.00000	0.304997
$689$	−56.0000	−2.13343
$690$	0	0
$691$	−4.00000	−0.152167	−0.0760836	−	0.997101i	$-0.524242\pi$
$691$	−4.00000	−0.152167	−0.0760836	+	0.997101i	$0.524242\pi$
$692$	10.0000	0.380143
$693$	0	0
$694$	20.0000	0.759190
$695$	−4.00000	−0.151729
$696$	−8.00000	−0.303239
$697$	60.0000	2.27266
$698$	2.00000	0.0757011
$699$	14.0000	0.529529
$700$	−2.00000	−0.0755929
$701$	30.0000	1.13308	0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000	1.13308	0.566542	+	0.824033i	$0.308281\pi$
$702$	−4.00000	−0.150970
$703$	0	0
$704$	0	0
$705$	−4.00000	−0.150649
$706$	14.0000	0.526897
$707$	12.0000	0.451306
$708$	−14.0000	−0.526152
$709$	46.0000	1.72757	0.863783	−	0.503864i	$-0.168089\pi$
$709$	46.0000	1.72757	0.863783	+	0.503864i	$0.168089\pi$
$710$	−6.00000	−0.225176
$711$	8.00000	0.300023
$712$	16.0000	0.599625
$713$	0	0
$714$	12.0000	0.449089
$715$	0	0
$716$	0	0
$717$	12.0000	0.448148
$718$	18.0000	0.671754
$719$	−40.0000	−1.49175	−0.745874	−	0.666087i	$-0.767968\pi$
$719$	−40.0000	−1.49175	−0.745874	+	0.666087i	$0.767968\pi$
$720$	−1.00000	−0.0372678
$721$	12.0000	0.446903
$722$	−19.0000	−0.707107
$723$	14.0000	0.520666
$724$	−6.00000	−0.222988
$725$	8.00000	0.297113
$726$	11.0000	0.408248
$727$	−10.0000	−0.370879	−0.185440	−	0.982656i	$-0.559371\pi$
$727$	−10.0000	−0.370879	−0.185440	+	0.982656i	$0.559371\pi$
$728$	−8.00000	−0.296500
$729$	1.00000	0.0370370
$730$	8.00000	0.296093
$731$	48.0000	1.77534
$732$	6.00000	0.221766
$733$	−22.0000	−0.812589	−0.406294	−	0.913742i	$-0.633179\pi$
$733$	−22.0000	−0.812589	−0.406294	+	0.913742i	$0.633179\pi$
$734$	−36.0000	−1.32878
$735$	−3.00000	−0.110657
$736$	0	0
$737$	0	0
$738$	10.0000	0.368105
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	−4.00000	−0.147043
$741$	0	0
$742$	28.0000	1.02791
$743$	24.0000	0.880475	0.440237	−	0.897881i	$-0.354894\pi$
$743$	24.0000	0.880475	0.440237	+	0.897881i	$0.354894\pi$
$744$	1.00000	0.0366618
$745$	−18.0000	−0.659469
$746$	6.00000	0.219676
$747$	−12.0000	−0.439057
$748$	0	0
$749$	−16.0000	−0.584627
$750$	1.00000	0.0365148
$751$	−32.0000	−1.16770	−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000	−1.16770	−0.583848	+	0.811863i	$0.698454\pi$
$752$	−4.00000	−0.145865
$753$	0	0
$754$	32.0000	1.16537
$755$	8.00000	0.291150
$756$	2.00000	0.0727393
$757$	−12.0000	−0.436147	−0.218074	−	0.975932i	$-0.569977\pi$
$757$	−12.0000	−0.436147	−0.218074	+	0.975932i	$0.569977\pi$
$758$	8.00000	0.290573
$759$	0	0
$760$	0	0
$761$	−20.0000	−0.724999	−0.362500	−	0.931984i	$-0.618077\pi$
$761$	−20.0000	−0.724999	−0.362500	+	0.931984i	$0.618077\pi$
$762$	0	0
$763$	−36.0000	−1.30329
$764$	−14.0000	−0.506502
$765$	−6.00000	−0.216930
$766$	0	0
$767$	56.0000	2.02204
$768$	−1.00000	−0.0360844
$769$	30.0000	1.08183	0.540914	−	0.841078i	$-0.318079\pi$
$769$	30.0000	1.08183	0.540914	+	0.841078i	$0.318079\pi$
$770$	0	0
$771$	−2.00000	−0.0720282
$772$	−2.00000	−0.0719816
$773$	−46.0000	−1.65451	−0.827253	−	0.561830i	$-0.810097\pi$
$773$	−46.0000	−1.65451	−0.827253	+	0.561830i	$0.810097\pi$
$774$	8.00000	0.287554
$775$	−1.00000	−0.0359211
$776$	−10.0000	−0.358979
$777$	8.00000	0.286998
$778$	−32.0000	−1.14726
$779$	0	0
$780$	4.00000	0.143223
$781$	0	0
$782$	0	0
$783$	−8.00000	−0.285897
$784$	−3.00000	−0.107143
$785$	18.0000	0.642448
$786$	18.0000	0.642039
$787$	44.0000	1.56843	0.784215	−	0.620489i	$-0.213066\pi$
$787$	44.0000	1.56843	0.784215	+	0.620489i	$0.213066\pi$
$788$	−2.00000	−0.0712470
$789$	−16.0000	−0.569615
$790$	−8.00000	−0.284627
$791$	−28.0000	−0.995565
$792$	0	0
$793$	−24.0000	−0.852265
$794$	18.0000	0.638796
$795$	−14.0000	−0.496529
$796$	−24.0000	−0.850657
$797$	−2.00000	−0.0708436	−0.0354218	−	0.999372i	$-0.511277\pi$
$797$	−2.00000	−0.0708436	−0.0354218	+	0.999372i	$0.511277\pi$
$798$	0	0
$799$	−24.0000	−0.849059
$800$	1.00000	0.0353553
$801$	16.0000	0.565332
$802$	4.00000	0.141245
$803$	0	0
$804$	10.0000	0.352673
$805$	0	0
$806$	−4.00000	−0.140894
$807$	−24.0000	−0.844840
$808$	−6.00000	−0.211079
$809$	4.00000	0.140633	0.0703163	−	0.997525i	$-0.477599\pi$
$809$	4.00000	0.140633	0.0703163	+	0.997525i	$0.477599\pi$
$810$	−1.00000	−0.0351364
$811$	56.0000	1.96643	0.983213	−	0.182462i	$-0.0584065\pi$
$811$	56.0000	1.96643	0.983213	+	0.182462i	$0.0584065\pi$
$812$	−16.0000	−0.561490
$813$	16.0000	0.561144
$814$	0	0
$815$	−6.00000	−0.210171
$816$	−6.00000	−0.210042
$817$	0	0
$818$	30.0000	1.04893
$819$	−8.00000	−0.279543
$820$	−10.0000	−0.349215
$821$	−48.0000	−1.67521	−0.837606	−	0.546275i	$-0.816045\pi$
$821$	−48.0000	−1.67521	−0.837606	+	0.546275i	$0.816045\pi$
$822$	6.00000	0.209274
$823$	0	0		−	1.00000i	$-0.5\pi$
$823$	0	0			1.00000i	$0.5\pi$
$824$	−6.00000	−0.209020
$825$	0	0
$826$	−28.0000	−0.974245
$827$	12.0000	0.417281	0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000	0.417281	0.208640	+	0.977992i	$0.433096\pi$
$828$	0	0
$829$	2.00000	0.0694629	0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000	0.0694629	0.0347314	+	0.999397i	$0.488942\pi$
$830$	12.0000	0.416526
$831$	−16.0000	−0.555034
$832$	4.00000	0.138675
$833$	−18.0000	−0.623663
$834$	−4.00000	−0.138509
$835$	8.00000	0.276851
$836$	0	0
$837$	1.00000	0.0345651
$838$	−6.00000	−0.207267
$839$	−30.0000	−1.03572	−0.517858	−	0.855467i	$-0.673270\pi$
$839$	−30.0000	−1.03572	−0.517858	+	0.855467i	$0.673270\pi$
$840$	−2.00000	−0.0690066
$841$	35.0000	1.20690
$842$	−6.00000	−0.206774
$843$	−30.0000	−1.03325
$844$	16.0000	0.550743
$845$	−3.00000	−0.103203
$846$	−4.00000	−0.137523
$847$	22.0000	0.755929
$848$	−14.0000	−0.480762
$849$	−14.0000	−0.480479
$850$	6.00000	0.205798
$851$	0	0
$852$	−6.00000	−0.205557
$853$	18.0000	0.616308	0.308154	−	0.951336i	$-0.400289\pi$
$853$	18.0000	0.616308	0.308154	+	0.951336i	$0.400289\pi$
$854$	12.0000	0.410632
$855$	0	0
$856$	8.00000	0.273434
$857$	−10.0000	−0.341593	−0.170797	−	0.985306i	$-0.554634\pi$
$857$	−10.0000	−0.341593	−0.170797	+	0.985306i	$0.554634\pi$
$858$	0	0
$859$	−20.0000	−0.682391	−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000	−0.682391	−0.341196	+	0.939992i	$0.610832\pi$
$860$	−8.00000	−0.272798
$861$	20.0000	0.681598
$862$	−6.00000	−0.204361
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	−1.00000	−0.0340207
$865$	−10.0000	−0.340010
$866$	−16.0000	−0.543702
$867$	−19.0000	−0.645274
$868$	2.00000	0.0678844
$869$	0	0
$870$	8.00000	0.271225
$871$	−40.0000	−1.35535
$872$	18.0000	0.609557
$873$	−10.0000	−0.338449
$874$	0	0
$875$	2.00000	0.0676123
$876$	8.00000	0.270295
$877$	−6.00000	−0.202606	−0.101303	−	0.994856i	$-0.532301\pi$
$877$	−6.00000	−0.202606	−0.101303	+	0.994856i	$0.532301\pi$
$878$	24.0000	0.809961
$879$	−6.00000	−0.202375
$880$	0	0
$881$	−48.0000	−1.61716	−0.808581	−	0.588386i	$-0.799764\pi$
$881$	−48.0000	−1.61716	−0.808581	+	0.588386i	$0.799764\pi$
$882$	−3.00000	−0.101015
$883$	−12.0000	−0.403832	−0.201916	−	0.979403i	$-0.564717\pi$
$883$	−12.0000	−0.403832	−0.201916	+	0.979403i	$0.564717\pi$
$884$	24.0000	0.807207
$885$	14.0000	0.470605
$886$	−36.0000	−1.20944
$887$	−56.0000	−1.88030	−0.940148	−	0.340766i	$-0.889313\pi$
$887$	−56.0000	−1.88030	−0.940148	+	0.340766i	$0.889313\pi$
$888$	−4.00000	−0.134231
$889$	0	0
$890$	−16.0000	−0.536321
$891$	0	0
$892$	−20.0000	−0.669650
$893$	0	0
$894$	−18.0000	−0.602010
$895$	0	0
$896$	−2.00000	−0.0668153
$897$	0	0
$898$	−20.0000	−0.667409
$899$	−8.00000	−0.266815
$900$	1.00000	0.0333333
$901$	−84.0000	−2.79845
$902$	0	0
$903$	16.0000	0.532447
$904$	14.0000	0.465633
$905$	6.00000	0.199447
$906$	8.00000	0.265782
$907$	2.00000	0.0664089	0.0332045	−	0.999449i	$-0.489429\pi$
$907$	2.00000	0.0664089	0.0332045	+	0.999449i	$0.489429\pi$
$908$	12.0000	0.398234
$909$	−6.00000	−0.199007
$910$	8.00000	0.265197
$911$	−24.0000	−0.795155	−0.397578	−	0.917568i	$-0.630149\pi$
$911$	−24.0000	−0.795155	−0.397578	+	0.917568i	$0.630149\pi$
$912$	0	0
$913$	0	0
$914$	8.00000	0.264616
$915$	−6.00000	−0.198354
$916$	−10.0000	−0.330409
$917$	36.0000	1.18882
$918$	−6.00000	−0.198030
$919$	−60.0000	−1.97922	−0.989609	−	0.143787i	$-0.954072\pi$
$919$	−60.0000	−1.97922	−0.989609	+	0.143787i	$0.954072\pi$
$920$	0	0
$921$	−10.0000	−0.329511
$922$	−12.0000	−0.395199
$923$	24.0000	0.789970
$924$	0	0
$925$	4.00000	0.131519
$926$	0	0
$927$	−6.00000	−0.197066
$928$	8.00000	0.262613
$929$	0	0		−	1.00000i	$-0.5\pi$
$929$	0	0			1.00000i	$0.5\pi$
$930$	−1.00000	−0.0327913
$931$	0	0
$932$	−14.0000	−0.458585
$933$	−18.0000	−0.589294
$934$	−32.0000	−1.04707
$935$	0	0
$936$	4.00000	0.130744
$937$	−42.0000	−1.37208	−0.686040	−	0.727564i	$-0.740653\pi$
$937$	−42.0000	−1.37208	−0.686040	+	0.727564i	$0.740653\pi$
$938$	20.0000	0.653023
$939$	16.0000	0.522140
$940$	4.00000	0.130466
$941$	−8.00000	−0.260793	−0.130396	−	0.991462i	$-0.541625\pi$
$941$	−8.00000	−0.260793	−0.130396	+	0.991462i	$0.541625\pi$
$942$	18.0000	0.586472
$943$	0	0
$944$	14.0000	0.455661
$945$	−2.00000	−0.0650600
$946$	0	0
$947$	−28.0000	−0.909878	−0.454939	−	0.890523i	$-0.650339\pi$
$947$	−28.0000	−0.909878	−0.454939	+	0.890523i	$0.650339\pi$
$948$	−8.00000	−0.259828
$949$	−32.0000	−1.03876
$950$	0	0
$951$	10.0000	0.324272
$952$	−12.0000	−0.388922
$953$	−46.0000	−1.49009	−0.745043	−	0.667016i	$-0.767571\pi$
$953$	−46.0000	−1.49009	−0.745043	+	0.667016i	$0.767571\pi$
$954$	−14.0000	−0.453267
$955$	14.0000	0.453029
$956$	−12.0000	−0.388108
$957$	0	0
$958$	−42.0000	−1.35696
$959$	12.0000	0.387500
$960$	1.00000	0.0322749
$961$	1.00000	0.0322581
$962$	16.0000	0.515861
$963$	8.00000	0.257796
$964$	−14.0000	−0.450910
$965$	2.00000	0.0643823
$966$	0	0
$967$	52.0000	1.67221	0.836104	−	0.548572i	$-0.184828\pi$
$967$	52.0000	1.67221	0.836104	+	0.548572i	$0.184828\pi$
$968$	−11.0000	−0.353553
$969$	0	0
$970$	10.0000	0.321081
$971$	18.0000	0.577647	0.288824	−	0.957382i	$-0.406736\pi$
$971$	18.0000	0.577647	0.288824	+	0.957382i	$0.406736\pi$
$972$	−1.00000	−0.0320750
$973$	−8.00000	−0.256468
$974$	−8.00000	−0.256337
$975$	−4.00000	−0.128103
$976$	−6.00000	−0.192055
$977$	−18.0000	−0.575871	−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000	−0.575871	−0.287936	+	0.957650i	$0.592969\pi$
$978$	−6.00000	−0.191859
$979$	0	0
$980$	3.00000	0.0958315
$981$	18.0000	0.574696
$982$	36.0000	1.14881
$983$	−40.0000	−1.27580	−0.637901	−	0.770118i	$-0.720197\pi$
$983$	−40.0000	−1.27580	−0.637901	+	0.770118i	$0.720197\pi$
$984$	−10.0000	−0.318788
$985$	2.00000	0.0637253
$986$	48.0000	1.52863
$987$	−8.00000	−0.254643
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	−1.00000	−0.0317500
$993$	12.0000	0.380808
$994$	−12.0000	−0.380617
$995$	24.0000	0.760851
$996$	12.0000	0.380235
$997$	6.00000	0.190022	0.0950110	−	0.995476i	$-0.469711\pi$
$997$	6.00000	0.190022	0.0950110	+	0.995476i	$0.469711\pi$
$998$	20.0000	0.633089
$999$	−4.00000	−0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	930.2.a.k.1.1	✓	1
3.2	odd	2		2790.2.a.f.1.1		1
4.3	odd	2		7440.2.a.u.1.1		1
5.2	odd	4		4650.2.d.g.3349.2		2
5.3	odd	4		4650.2.d.g.3349.1		2
5.4	even	2		4650.2.a.t.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.a.k.1.1	✓	1	1.1	even	1	trivial
2790.2.a.f.1.1		1	3.2	odd	2
4650.2.a.t.1.1		1	5.4	even	2
4650.2.d.g.3349.1		2	5.3	odd	4
4650.2.d.g.3349.2		2	5.2	odd	4
7440.2.a.u.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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	930.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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