LMFDB - Embedded newform 1785.2.a.g.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1785,2,Mod(1,1785)] 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("1785.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1785, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1785 = 3 \cdot 5 \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1785.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,-1,-2,1,0,-1] 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:	$14.2532967608$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1785.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q-1.00000 q^{3} -2.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} -2.00000 q^{11} +2.00000 q^{12} +3.00000 q^{13} -1.00000 q^{15} +4.00000 q^{16} +1.00000 q^{17} -2.00000 q^{19} -2.00000 q^{20} +1.00000 q^{21} -3.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +2.00000 q^{28} +4.00000 q^{29} -5.00000 q^{31} +2.00000 q^{33} -1.00000 q^{35} -2.00000 q^{36} -1.00000 q^{37} -3.00000 q^{39} +11.0000 q^{41} +4.00000 q^{43} +4.00000 q^{44} +1.00000 q^{45} -9.00000 q^{47} -4.00000 q^{48} +1.00000 q^{49} -1.00000 q^{51} -6.00000 q^{52} -2.00000 q^{55} +2.00000 q^{57} -12.0000 q^{59} +2.00000 q^{60} -5.00000 q^{61} -1.00000 q^{63} -8.00000 q^{64} +3.00000 q^{65} -12.0000 q^{67} -2.00000 q^{68} +3.00000 q^{69} -2.00000 q^{71} -4.00000 q^{73} -1.00000 q^{75} +4.00000 q^{76} +2.00000 q^{77} -10.0000 q^{79} +4.00000 q^{80} +1.00000 q^{81} -5.00000 q^{83} -2.00000 q^{84} +1.00000 q^{85} -4.00000 q^{87} -14.0000 q^{89} -3.00000 q^{91} +6.00000 q^{92} +5.00000 q^{93} -2.00000 q^{95} -2.00000 q^{97} -2.00000 q^{99} +O(q^{100})\)

Coefficient data

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

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

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

$2$	0	0		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−2.00000	−1.00000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	2.00000	0.577350
$13$	3.00000	0.832050	0.416025	−	0.909353i	$-0.363423\pi$
$13$	3.00000	0.832050	0.416025	+	0.909353i	$0.363423\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	4.00000	1.00000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	−2.00000	−0.447214
$21$	1.00000	0.218218
$22$	0	0
$23$	−3.00000	−0.625543	−0.312772	−	0.949828i	$-0.601257\pi$
$23$	−3.00000	−0.625543	−0.312772	+	0.949828i	$0.601257\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	2.00000	0.377964
$29$	4.00000	0.742781	0.371391	−	0.928477i	$-0.378881\pi$
$29$	4.00000	0.742781	0.371391	+	0.928477i	$0.378881\pi$
$30$	0	0
$31$	−5.00000	−0.898027	−0.449013	−	0.893525i	$-0.648224\pi$
$31$	−5.00000	−0.898027	−0.449013	+	0.893525i	$0.648224\pi$
$32$	0	0
$33$	2.00000	0.348155
$34$	0	0
$35$	−1.00000	−0.169031
$36$	−2.00000	−0.333333
$37$	−1.00000	−0.164399	−0.0821995	−	0.996616i	$-0.526194\pi$
$37$	−1.00000	−0.164399	−0.0821995	+	0.996616i	$0.526194\pi$
$38$	0	0
$39$	−3.00000	−0.480384
$40$	0	0
$41$	11.0000	1.71791	0.858956	−	0.512050i	$-0.171114\pi$
$41$	11.0000	1.71791	0.858956	+	0.512050i	$0.171114\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	4.00000	0.603023
$45$	1.00000	0.149071
$46$	0	0
$47$	−9.00000	−1.31278	−0.656392	−	0.754420i	$-0.727918\pi$
$47$	−9.00000	−1.31278	−0.656392	+	0.754420i	$0.727918\pi$
$48$	−4.00000	−0.577350
$49$	1.00000	0.142857
$50$	0	0
$51$	−1.00000	−0.140028
$52$	−6.00000	−0.832050
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	−2.00000	−0.269680
$56$	0	0
$57$	2.00000	0.264906
$58$	0	0
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	2.00000	0.258199
$61$	−5.00000	−0.640184	−0.320092	−	0.947386i	$-0.603714\pi$
$61$	−5.00000	−0.640184	−0.320092	+	0.947386i	$0.603714\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	−8.00000	−1.00000
$65$	3.00000	0.372104
$66$	0	0
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	−2.00000	−0.242536
$69$	3.00000	0.361158
$70$	0	0
$71$	−2.00000	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	0	0
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	4.00000	0.458831
$77$	2.00000	0.227921
$78$	0	0
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	4.00000	0.447214
$81$	1.00000	0.111111
$82$	0	0
$83$	−5.00000	−0.548821	−0.274411	−	0.961613i	$-0.588483\pi$
$83$	−5.00000	−0.548821	−0.274411	+	0.961613i	$0.588483\pi$
$84$	−2.00000	−0.218218
$85$	1.00000	0.108465
$86$	0	0
$87$	−4.00000	−0.428845
$88$	0	0
$89$	−14.0000	−1.48400	−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000	−1.48400	−0.741999	+	0.670402i	$0.766122\pi$
$90$	0	0
$91$	−3.00000	−0.314485
$92$	6.00000	0.625543
$93$	5.00000	0.518476
$94$	0	0
$95$	−2.00000	−0.205196
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	0	0
$99$	−2.00000	−0.201008
$100$	−2.00000	−0.200000
$101$	−8.00000	−0.796030	−0.398015	−	0.917379i	$-0.630301\pi$
$101$	−8.00000	−0.796030	−0.398015	+	0.917379i	$0.630301\pi$
$102$	0	0
$103$	16.0000	1.57653	0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000	1.57653	0.788263	+	0.615338i	$0.210980\pi$
$104$	0	0
$105$	1.00000	0.0975900
$106$	0	0
$107$	13.0000	1.25676	0.628379	−	0.777908i	$-0.283719\pi$
$107$	13.0000	1.25676	0.628379	+	0.777908i	$0.283719\pi$
$108$	2.00000	0.192450
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	1.00000	0.0949158
$112$	−4.00000	−0.377964
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	−3.00000	−0.279751
$116$	−8.00000	−0.742781
$117$	3.00000	0.277350
$118$	0	0
$119$	−1.00000	−0.0916698
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	−11.0000	−0.991837
$124$	10.0000	0.898027
$125$	1.00000	0.0894427
$126$	0	0
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	0	0
$129$	−4.00000	−0.352180
$130$	0	0
$131$	−3.00000	−0.262111	−0.131056	−	0.991375i	$-0.541837\pi$
$131$	−3.00000	−0.262111	−0.131056	+	0.991375i	$0.541837\pi$
$132$	−4.00000	−0.348155
$133$	2.00000	0.173422
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−8.00000	−0.683486	−0.341743	−	0.939793i	$-0.611017\pi$
$137$	−8.00000	−0.683486	−0.341743	+	0.939793i	$0.611017\pi$
$138$	0	0
$139$	12.0000	1.01783	0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	2.00000	0.169031
$141$	9.00000	0.757937
$142$	0	0
$143$	−6.00000	−0.501745
$144$	4.00000	0.333333
$145$	4.00000	0.332182
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	2.00000	0.164399
$149$	−11.0000	−0.901155	−0.450578	−	0.892737i	$-0.648782\pi$
$149$	−11.0000	−0.901155	−0.450578	+	0.892737i	$0.648782\pi$
$150$	0	0
$151$	19.0000	1.54620	0.773099	−	0.634285i	$-0.218706\pi$
$151$	19.0000	1.54620	0.773099	+	0.634285i	$0.218706\pi$
$152$	0	0
$153$	1.00000	0.0808452
$154$	0	0
$155$	−5.00000	−0.401610
$156$	6.00000	0.480384
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	3.00000	0.236433
$162$	0	0
$163$	19.0000	1.48819	0.744097	−	0.668071i	$-0.232880\pi$
$163$	19.0000	1.48819	0.744097	+	0.668071i	$0.232880\pi$
$164$	−22.0000	−1.71791
$165$	2.00000	0.155700
$166$	0	0
$167$	2.00000	0.154765	0.0773823	−	0.997001i	$-0.475344\pi$
$167$	2.00000	0.154765	0.0773823	+	0.997001i	$0.475344\pi$
$168$	0	0
$169$	−4.00000	−0.307692
$170$	0	0
$171$	−2.00000	−0.152944
$172$	−8.00000	−0.609994
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	−8.00000	−0.603023
$177$	12.0000	0.901975
$178$	0	0
$179$	−19.0000	−1.42013	−0.710063	−	0.704138i	$-0.751334\pi$
$179$	−19.0000	−1.42013	−0.710063	+	0.704138i	$0.751334\pi$
$180$	−2.00000	−0.149071
$181$	−1.00000	−0.0743294	−0.0371647	−	0.999309i	$-0.511833\pi$
$181$	−1.00000	−0.0743294	−0.0371647	+	0.999309i	$0.511833\pi$
$182$	0	0
$183$	5.00000	0.369611
$184$	0	0
$185$	−1.00000	−0.0735215
$186$	0	0
$187$	−2.00000	−0.146254
$188$	18.0000	1.31278
$189$	1.00000	0.0727393
$190$	0	0
$191$	−27.0000	−1.95365	−0.976826	−	0.214036i	$-0.931339\pi$
$191$	−27.0000	−1.95365	−0.976826	+	0.214036i	$0.931339\pi$
$192$	8.00000	0.577350
$193$	23.0000	1.65558	0.827788	−	0.561041i	$-0.189599\pi$
$193$	23.0000	1.65558	0.827788	+	0.561041i	$0.189599\pi$
$194$	0	0
$195$	−3.00000	−0.214834
$196$	−2.00000	−0.142857
$197$	−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
$199$	−12.0000	−0.850657	−0.425329	−	0.905039i	$-0.639842\pi$
$199$	−12.0000	−0.850657	−0.425329	+	0.905039i	$0.639842\pi$
$200$	0	0
$201$	12.0000	0.846415
$202$	0	0
$203$	−4.00000	−0.280745
$204$	2.00000	0.140028
$205$	11.0000	0.768273
$206$	0	0
$207$	−3.00000	−0.208514
$208$	12.0000	0.832050
$209$	4.00000	0.276686
$210$	0	0
$211$	−14.0000	−0.963800	−0.481900	−	0.876226i	$-0.660053\pi$
$211$	−14.0000	−0.963800	−0.481900	+	0.876226i	$0.660053\pi$
$212$	0	0
$213$	2.00000	0.137038
$214$	0	0
$215$	4.00000	0.272798
$216$	0	0
$217$	5.00000	0.339422
$218$	0	0
$219$	4.00000	0.270295
$220$	4.00000	0.269680
$221$	3.00000	0.201802
$222$	0	0
$223$	−15.0000	−1.00447	−0.502237	−	0.864730i	$-0.667490\pi$
$223$	−15.0000	−1.00447	−0.502237	+	0.864730i	$0.667490\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	2.00000	0.132745	0.0663723	−	0.997795i	$-0.478857\pi$
$227$	2.00000	0.132745	0.0663723	+	0.997795i	$0.478857\pi$
$228$	−4.00000	−0.264906
$229$	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
$231$	−2.00000	−0.131590
$232$	0	0
$233$	3.00000	0.196537	0.0982683	−	0.995160i	$-0.468670\pi$
$233$	3.00000	0.196537	0.0982683	+	0.995160i	$0.468670\pi$
$234$	0	0
$235$	−9.00000	−0.587095
$236$	24.0000	1.56227
$237$	10.0000	0.649570
$238$	0	0
$239$	−15.0000	−0.970269	−0.485135	−	0.874439i	$-0.661229\pi$
$239$	−15.0000	−0.970269	−0.485135	+	0.874439i	$0.661229\pi$
$240$	−4.00000	−0.258199
$241$	17.0000	1.09507	0.547533	−	0.836784i	$-0.315567\pi$
$241$	17.0000	1.09507	0.547533	+	0.836784i	$0.315567\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	10.0000	0.640184
$245$	1.00000	0.0638877
$246$	0	0
$247$	−6.00000	−0.381771
$248$	0	0
$249$	5.00000	0.316862
$250$	0	0
$251$	16.0000	1.00991	0.504956	−	0.863145i	$-0.331509\pi$
$251$	16.0000	1.00991	0.504956	+	0.863145i	$0.331509\pi$
$252$	2.00000	0.125988
$253$	6.00000	0.377217
$254$	0	0
$255$	−1.00000	−0.0626224
$256$	16.0000	1.00000
$257$	−7.00000	−0.436648	−0.218324	−	0.975876i	$-0.570059\pi$
$257$	−7.00000	−0.436648	−0.218324	+	0.975876i	$0.570059\pi$
$258$	0	0
$259$	1.00000	0.0621370
$260$	−6.00000	−0.372104
$261$	4.00000	0.247594
$262$	0	0
$263$	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
$266$	0	0
$267$	14.0000	0.856786
$268$	24.0000	1.46603
$269$	−21.0000	−1.28039	−0.640196	−	0.768211i	$-0.721147\pi$
$269$	−21.0000	−1.28039	−0.640196	+	0.768211i	$0.721147\pi$
$270$	0	0
$271$	26.0000	1.57939	0.789694	−	0.613501i	$-0.210239\pi$
$271$	26.0000	1.57939	0.789694	+	0.613501i	$0.210239\pi$
$272$	4.00000	0.242536
$273$	3.00000	0.181568
$274$	0	0
$275$	−2.00000	−0.120605
$276$	−6.00000	−0.361158
$277$	7.00000	0.420589	0.210295	−	0.977638i	$-0.432558\pi$
$277$	7.00000	0.420589	0.210295	+	0.977638i	$0.432558\pi$
$278$	0	0
$279$	−5.00000	−0.299342
$280$	0	0
$281$	−1.00000	−0.0596550	−0.0298275	−	0.999555i	$-0.509496\pi$
$281$	−1.00000	−0.0596550	−0.0298275	+	0.999555i	$0.509496\pi$
$282$	0	0
$283$	28.0000	1.66443	0.832214	−	0.554455i	$-0.187073\pi$
$283$	28.0000	1.66443	0.832214	+	0.554455i	$0.187073\pi$
$284$	4.00000	0.237356
$285$	2.00000	0.118470
$286$	0	0
$287$	−11.0000	−0.649309
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	2.00000	0.117242
$292$	8.00000	0.468165
$293$	14.0000	0.817889	0.408944	−	0.912559i	$-0.365897\pi$
$293$	14.0000	0.817889	0.408944	+	0.912559i	$0.365897\pi$
$294$	0	0
$295$	−12.0000	−0.698667
$296$	0	0
$297$	2.00000	0.116052
$298$	0	0
$299$	−9.00000	−0.520483
$300$	2.00000	0.115470
$301$	−4.00000	−0.230556
$302$	0	0
$303$	8.00000	0.459588
$304$	−8.00000	−0.458831
$305$	−5.00000	−0.286299
$306$	0	0
$307$	20.0000	1.14146	0.570730	−	0.821138i	$-0.306660\pi$
$307$	20.0000	1.14146	0.570730	+	0.821138i	$0.306660\pi$
$308$	−4.00000	−0.227921
$309$	−16.0000	−0.910208
$310$	0	0
$311$	−5.00000	−0.283524	−0.141762	−	0.989901i	$-0.545277\pi$
$311$	−5.00000	−0.283524	−0.141762	+	0.989901i	$0.545277\pi$
$312$	0	0
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	0	0
$315$	−1.00000	−0.0563436
$316$	20.0000	1.12509
$317$	−5.00000	−0.280828	−0.140414	−	0.990093i	$-0.544843\pi$
$317$	−5.00000	−0.280828	−0.140414	+	0.990093i	$0.544843\pi$
$318$	0	0
$319$	−8.00000	−0.447914
$320$	−8.00000	−0.447214
$321$	−13.0000	−0.725589
$322$	0	0
$323$	−2.00000	−0.111283
$324$	−2.00000	−0.111111
$325$	3.00000	0.166410
$326$	0	0
$327$	10.0000	0.553001
$328$	0	0
$329$	9.00000	0.496186
$330$	0	0
$331$	−17.0000	−0.934405	−0.467202	−	0.884150i	$-0.654738\pi$
$331$	−17.0000	−0.934405	−0.467202	+	0.884150i	$0.654738\pi$
$332$	10.0000	0.548821
$333$	−1.00000	−0.0547997
$334$	0	0
$335$	−12.0000	−0.655630
$336$	4.00000	0.218218
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	0	0
$339$	6.00000	0.325875
$340$	−2.00000	−0.108465
$341$	10.0000	0.541530
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	3.00000	0.161515
$346$	0	0
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	8.00000	0.428845
$349$	12.0000	0.642345	0.321173	−	0.947021i	$-0.395923\pi$
$349$	12.0000	0.642345	0.321173	+	0.947021i	$0.395923\pi$
$350$	0	0
$351$	−3.00000	−0.160128
$352$	0	0
$353$	−3.00000	−0.159674	−0.0798369	−	0.996808i	$-0.525440\pi$
$353$	−3.00000	−0.159674	−0.0798369	+	0.996808i	$0.525440\pi$
$354$	0	0
$355$	−2.00000	−0.106149
$356$	28.0000	1.48400
$357$	1.00000	0.0529256
$358$	0	0
$359$	3.00000	0.158334	0.0791670	−	0.996861i	$-0.474774\pi$
$359$	3.00000	0.158334	0.0791670	+	0.996861i	$0.474774\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	7.00000	0.367405
$364$	6.00000	0.314485
$365$	−4.00000	−0.209370
$366$	0	0
$367$	26.0000	1.35719	0.678594	−	0.734513i	$-0.262589\pi$
$367$	26.0000	1.35719	0.678594	+	0.734513i	$0.262589\pi$
$368$	−12.0000	−0.625543
$369$	11.0000	0.572637
$370$	0	0
$371$	0	0
$372$	−10.0000	−0.518476
$373$	20.0000	1.03556	0.517780	−	0.855514i	$-0.326758\pi$
$373$	20.0000	1.03556	0.517780	+	0.855514i	$0.326758\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	12.0000	0.618031
$378$	0	0
$379$	−16.0000	−0.821865	−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000	−0.821865	−0.410932	+	0.911666i	$0.634797\pi$
$380$	4.00000	0.205196
$381$	8.00000	0.409852
$382$	0	0
$383$	4.00000	0.204390	0.102195	−	0.994764i	$-0.467413\pi$
$383$	4.00000	0.204390	0.102195	+	0.994764i	$0.467413\pi$
$384$	0	0
$385$	2.00000	0.101929
$386$	0	0
$387$	4.00000	0.203331
$388$	4.00000	0.203069
$389$	11.0000	0.557722	0.278861	−	0.960331i	$-0.410043\pi$
$389$	11.0000	0.557722	0.278861	+	0.960331i	$0.410043\pi$
$390$	0	0
$391$	−3.00000	−0.151717
$392$	0	0
$393$	3.00000	0.151330
$394$	0	0
$395$	−10.0000	−0.503155
$396$	4.00000	0.201008
$397$	10.0000	0.501886	0.250943	−	0.968002i	$-0.419259\pi$
$397$	10.0000	0.501886	0.250943	+	0.968002i	$0.419259\pi$
$398$	0	0
$399$	−2.00000	−0.100125
$400$	4.00000	0.200000
$401$	−10.0000	−0.499376	−0.249688	−	0.968326i	$-0.580328\pi$
$401$	−10.0000	−0.499376	−0.249688	+	0.968326i	$0.580328\pi$
$402$	0	0
$403$	−15.0000	−0.747203
$404$	16.0000	0.796030
$405$	1.00000	0.0496904
$406$	0	0
$407$	2.00000	0.0991363
$408$	0	0
$409$	32.0000	1.58230	0.791149	−	0.611623i	$-0.209483\pi$
$409$	32.0000	1.58230	0.791149	+	0.611623i	$0.209483\pi$
$410$	0	0
$411$	8.00000	0.394611
$412$	−32.0000	−1.57653
$413$	12.0000	0.590481
$414$	0	0
$415$	−5.00000	−0.245440
$416$	0	0
$417$	−12.0000	−0.587643
$418$	0	0
$419$	7.00000	0.341972	0.170986	−	0.985273i	$-0.445305\pi$
$419$	7.00000	0.341972	0.170986	+	0.985273i	$0.445305\pi$
$420$	−2.00000	−0.0975900
$421$	−21.0000	−1.02348	−0.511739	−	0.859141i	$-0.670998\pi$
$421$	−21.0000	−1.02348	−0.511739	+	0.859141i	$0.670998\pi$
$422$	0	0
$423$	−9.00000	−0.437595
$424$	0	0
$425$	1.00000	0.0485071
$426$	0	0
$427$	5.00000	0.241967
$428$	−26.0000	−1.25676
$429$	6.00000	0.289683
$430$	0	0
$431$	28.0000	1.34871	0.674356	−	0.738406i	$-0.264421\pi$
$431$	28.0000	1.34871	0.674356	+	0.738406i	$0.264421\pi$
$432$	−4.00000	−0.192450
$433$	1.00000	0.0480569	0.0240285	−	0.999711i	$-0.492351\pi$
$433$	1.00000	0.0480569	0.0240285	+	0.999711i	$0.492351\pi$
$434$	0	0
$435$	−4.00000	−0.191785
$436$	20.0000	0.957826
$437$	6.00000	0.287019
$438$	0	0
$439$	−24.0000	−1.14546	−0.572729	−	0.819745i	$-0.694115\pi$
$439$	−24.0000	−1.14546	−0.572729	+	0.819745i	$0.694115\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−24.0000	−1.14027	−0.570137	−	0.821549i	$-0.693110\pi$
$443$	−24.0000	−1.14027	−0.570137	+	0.821549i	$0.693110\pi$
$444$	−2.00000	−0.0949158
$445$	−14.0000	−0.663664
$446$	0	0
$447$	11.0000	0.520282
$448$	8.00000	0.377964
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	0	0
$451$	−22.0000	−1.03594
$452$	12.0000	0.564433
$453$	−19.0000	−0.892698
$454$	0	0
$455$	−3.00000	−0.140642
$456$	0	0
$457$	8.00000	0.374224	0.187112	−	0.982339i	$-0.440087\pi$
$457$	8.00000	0.374224	0.187112	+	0.982339i	$0.440087\pi$
$458$	0	0
$459$	−1.00000	−0.0466760
$460$	6.00000	0.279751
$461$	−34.0000	−1.58354	−0.791769	−	0.610821i	$-0.790840\pi$
$461$	−34.0000	−1.58354	−0.791769	+	0.610821i	$0.790840\pi$
$462$	0	0
$463$	−2.00000	−0.0929479	−0.0464739	−	0.998920i	$-0.514798\pi$
$463$	−2.00000	−0.0929479	−0.0464739	+	0.998920i	$0.514798\pi$
$464$	16.0000	0.742781
$465$	5.00000	0.231869
$466$	0	0
$467$	33.0000	1.52706	0.763529	−	0.645774i	$-0.223465\pi$
$467$	33.0000	1.52706	0.763529	+	0.645774i	$0.223465\pi$
$468$	−6.00000	−0.277350
$469$	12.0000	0.554109
$470$	0	0
$471$	10.0000	0.460776
$472$	0	0
$473$	−8.00000	−0.367840
$474$	0	0
$475$	−2.00000	−0.0917663
$476$	2.00000	0.0916698
$477$	0	0
$478$	0	0
$479$	−11.0000	−0.502603	−0.251301	−	0.967909i	$-0.580859\pi$
$479$	−11.0000	−0.502603	−0.251301	+	0.967909i	$0.580859\pi$
$480$	0	0
$481$	−3.00000	−0.136788
$482$	0	0
$483$	−3.00000	−0.136505
$484$	14.0000	0.636364
$485$	−2.00000	−0.0908153
$486$	0	0
$487$	−23.0000	−1.04223	−0.521115	−	0.853487i	$-0.674484\pi$
$487$	−23.0000	−1.04223	−0.521115	+	0.853487i	$0.674484\pi$
$488$	0	0
$489$	−19.0000	−0.859210
$490$	0	0
$491$	9.00000	0.406164	0.203082	−	0.979162i	$-0.434904\pi$
$491$	9.00000	0.406164	0.203082	+	0.979162i	$0.434904\pi$
$492$	22.0000	0.991837
$493$	4.00000	0.180151
$494$	0	0
$495$	−2.00000	−0.0898933
$496$	−20.0000	−0.898027
$497$	2.00000	0.0897123
$498$	0	0
$499$	16.0000	0.716258	0.358129	−	0.933672i	$-0.383415\pi$
$499$	16.0000	0.716258	0.358129	+	0.933672i	$0.383415\pi$
$500$	−2.00000	−0.0894427
$501$	−2.00000	−0.0893534
$502$	0	0
$503$	16.0000	0.713405	0.356702	−	0.934218i	$-0.383901\pi$
$503$	16.0000	0.713405	0.356702	+	0.934218i	$0.383901\pi$
$504$	0	0
$505$	−8.00000	−0.355995
$506$	0	0
$507$	4.00000	0.177646
$508$	16.0000	0.709885
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	4.00000	0.176950
$512$	0	0
$513$	2.00000	0.0883022
$514$	0	0
$515$	16.0000	0.705044
$516$	8.00000	0.352180
$517$	18.0000	0.791639
$518$	0	0
$519$	6.00000	0.263371
$520$	0	0
$521$	−42.0000	−1.84005	−0.920027	−	0.391856i	$-0.871833\pi$
$521$	−42.0000	−1.84005	−0.920027	+	0.391856i	$0.871833\pi$
$522$	0	0
$523$	−35.0000	−1.53044	−0.765222	−	0.643767i	$-0.777371\pi$
$523$	−35.0000	−1.53044	−0.765222	+	0.643767i	$0.777371\pi$
$524$	6.00000	0.262111
$525$	1.00000	0.0436436
$526$	0	0
$527$	−5.00000	−0.217803
$528$	8.00000	0.348155
$529$	−14.0000	−0.608696
$530$	0	0
$531$	−12.0000	−0.520756
$532$	−4.00000	−0.173422
$533$	33.0000	1.42939
$534$	0	0
$535$	13.0000	0.562039
$536$	0	0
$537$	19.0000	0.819911
$538$	0	0
$539$	−2.00000	−0.0861461
$540$	2.00000	0.0860663
$541$	16.0000	0.687894	0.343947	−	0.938989i	$-0.388236\pi$
$541$	16.0000	0.687894	0.343947	+	0.938989i	$0.388236\pi$
$542$	0	0
$543$	1.00000	0.0429141
$544$	0	0
$545$	−10.0000	−0.428353
$546$	0	0
$547$	−19.0000	−0.812381	−0.406191	−	0.913788i	$-0.633143\pi$
$547$	−19.0000	−0.812381	−0.406191	+	0.913788i	$0.633143\pi$
$548$	16.0000	0.683486
$549$	−5.00000	−0.213395
$550$	0	0
$551$	−8.00000	−0.340811
$552$	0	0
$553$	10.0000	0.425243
$554$	0	0
$555$	1.00000	0.0424476
$556$	−24.0000	−1.01783
$557$	−2.00000	−0.0847427	−0.0423714	−	0.999102i	$-0.513491\pi$
$557$	−2.00000	−0.0847427	−0.0423714	+	0.999102i	$0.513491\pi$
$558$	0	0
$559$	12.0000	0.507546
$560$	−4.00000	−0.169031
$561$	2.00000	0.0844401
$562$	0	0
$563$	−17.0000	−0.716465	−0.358232	−	0.933632i	$-0.616620\pi$
$563$	−17.0000	−0.716465	−0.358232	+	0.933632i	$0.616620\pi$
$564$	−18.0000	−0.757937
$565$	−6.00000	−0.252422
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−39.0000	−1.63497	−0.817483	−	0.575953i	$-0.804631\pi$
$569$	−39.0000	−1.63497	−0.817483	+	0.575953i	$0.804631\pi$
$570$	0	0
$571$	−38.0000	−1.59025	−0.795125	−	0.606445i	$-0.792595\pi$
$571$	−38.0000	−1.59025	−0.795125	+	0.606445i	$0.792595\pi$
$572$	12.0000	0.501745
$573$	27.0000	1.12794
$574$	0	0
$575$	−3.00000	−0.125109
$576$	−8.00000	−0.333333
$577$	−14.0000	−0.582828	−0.291414	−	0.956597i	$-0.594126\pi$
$577$	−14.0000	−0.582828	−0.291414	+	0.956597i	$0.594126\pi$
$578$	0	0
$579$	−23.0000	−0.955847
$580$	−8.00000	−0.332182
$581$	5.00000	0.207435
$582$	0	0
$583$	0	0
$584$	0	0
$585$	3.00000	0.124035
$586$	0	0
$587$	20.0000	0.825488	0.412744	−	0.910847i	$-0.364570\pi$
$587$	20.0000	0.825488	0.412744	+	0.910847i	$0.364570\pi$
$588$	2.00000	0.0824786
$589$	10.0000	0.412043
$590$	0	0
$591$	18.0000	0.740421
$592$	−4.00000	−0.164399
$593$	−11.0000	−0.451716	−0.225858	−	0.974160i	$-0.572519\pi$
$593$	−11.0000	−0.451716	−0.225858	+	0.974160i	$0.572519\pi$
$594$	0	0
$595$	−1.00000	−0.0409960
$596$	22.0000	0.901155
$597$	12.0000	0.491127
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	14.0000	0.571072	0.285536	−	0.958368i	$-0.407828\pi$
$601$	14.0000	0.571072	0.285536	+	0.958368i	$0.407828\pi$
$602$	0	0
$603$	−12.0000	−0.488678
$604$	−38.0000	−1.54620
$605$	−7.00000	−0.284590
$606$	0	0
$607$	40.0000	1.62355	0.811775	−	0.583970i	$-0.198502\pi$
$607$	40.0000	1.62355	0.811775	+	0.583970i	$0.198502\pi$
$608$	0	0
$609$	4.00000	0.162088
$610$	0	0
$611$	−27.0000	−1.09230
$612$	−2.00000	−0.0808452
$613$	6.00000	0.242338	0.121169	−	0.992632i	$-0.461336\pi$
$613$	6.00000	0.242338	0.121169	+	0.992632i	$0.461336\pi$
$614$	0	0
$615$	−11.0000	−0.443563
$616$	0	0
$617$	7.00000	0.281809	0.140905	−	0.990023i	$-0.454999\pi$
$617$	7.00000	0.281809	0.140905	+	0.990023i	$0.454999\pi$
$618$	0	0
$619$	−20.0000	−0.803868	−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000	−0.803868	−0.401934	+	0.915669i	$0.631662\pi$
$620$	10.0000	0.401610
$621$	3.00000	0.120386
$622$	0	0
$623$	14.0000	0.560898
$624$	−12.0000	−0.480384
$625$	1.00000	0.0400000
$626$	0	0
$627$	−4.00000	−0.159745
$628$	20.0000	0.798087
$629$	−1.00000	−0.0398726
$630$	0	0
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	0	0
$633$	14.0000	0.556450
$634$	0	0
$635$	−8.00000	−0.317470
$636$	0	0
$637$	3.00000	0.118864
$638$	0	0
$639$	−2.00000	−0.0791188
$640$	0	0
$641$	10.0000	0.394976	0.197488	−	0.980305i	$-0.436722\pi$
$641$	10.0000	0.394976	0.197488	+	0.980305i	$0.436722\pi$
$642$	0	0
$643$	6.00000	0.236617	0.118308	−	0.992977i	$-0.462253\pi$
$643$	6.00000	0.236617	0.118308	+	0.992977i	$0.462253\pi$
$644$	−6.00000	−0.236433
$645$	−4.00000	−0.157500
$646$	0	0
$647$	12.0000	0.471769	0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000	0.471769	0.235884	+	0.971781i	$0.424201\pi$
$648$	0	0
$649$	24.0000	0.942082
$650$	0	0
$651$	−5.00000	−0.195965
$652$	−38.0000	−1.48819
$653$	−27.0000	−1.05659	−0.528296	−	0.849060i	$-0.677169\pi$
$653$	−27.0000	−1.05659	−0.528296	+	0.849060i	$0.677169\pi$
$654$	0	0
$655$	−3.00000	−0.117220
$656$	44.0000	1.71791
$657$	−4.00000	−0.156055
$658$	0	0
$659$	−44.0000	−1.71400	−0.856998	−	0.515319i	$-0.827673\pi$
$659$	−44.0000	−1.71400	−0.856998	+	0.515319i	$0.827673\pi$
$660$	−4.00000	−0.155700
$661$	14.0000	0.544537	0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000	0.544537	0.272268	+	0.962221i	$0.412226\pi$
$662$	0	0
$663$	−3.00000	−0.116510
$664$	0	0
$665$	2.00000	0.0775567
$666$	0	0
$667$	−12.0000	−0.464642
$668$	−4.00000	−0.154765
$669$	15.0000	0.579934
$670$	0	0
$671$	10.0000	0.386046
$672$	0	0
$673$	5.00000	0.192736	0.0963679	−	0.995346i	$-0.469277\pi$
$673$	5.00000	0.192736	0.0963679	+	0.995346i	$0.469277\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	8.00000	0.307692
$677$	34.0000	1.30673	0.653363	−	0.757045i	$-0.273358\pi$
$677$	34.0000	1.30673	0.653363	+	0.757045i	$0.273358\pi$
$678$	0	0
$679$	2.00000	0.0767530
$680$	0	0
$681$	−2.00000	−0.0766402
$682$	0	0
$683$	28.0000	1.07139	0.535695	−	0.844411i	$-0.320050\pi$
$683$	28.0000	1.07139	0.535695	+	0.844411i	$0.320050\pi$
$684$	4.00000	0.152944
$685$	−8.00000	−0.305664
$686$	0	0
$687$	−14.0000	−0.534133
$688$	16.0000	0.609994
$689$	0	0
$690$	0	0
$691$	15.0000	0.570627	0.285313	−	0.958434i	$-0.407902\pi$
$691$	15.0000	0.570627	0.285313	+	0.958434i	$0.407902\pi$
$692$	12.0000	0.456172
$693$	2.00000	0.0759737
$694$	0	0
$695$	12.0000	0.455186
$696$	0	0
$697$	11.0000	0.416655
$698$	0	0
$699$	−3.00000	−0.113470
$700$	2.00000	0.0755929
$701$	−41.0000	−1.54855	−0.774274	−	0.632850i	$-0.781885\pi$
$701$	−41.0000	−1.54855	−0.774274	+	0.632850i	$0.781885\pi$
$702$	0	0
$703$	2.00000	0.0754314
$704$	16.0000	0.603023
$705$	9.00000	0.338960
$706$	0	0
$707$	8.00000	0.300871
$708$	−24.0000	−0.901975
$709$	−12.0000	−0.450669	−0.225335	−	0.974281i	$-0.572348\pi$
$709$	−12.0000	−0.450669	−0.225335	+	0.974281i	$0.572348\pi$
$710$	0	0
$711$	−10.0000	−0.375029
$712$	0	0
$713$	15.0000	0.561754
$714$	0	0
$715$	−6.00000	−0.224387
$716$	38.0000	1.42013
$717$	15.0000	0.560185
$718$	0	0
$719$	−45.0000	−1.67822	−0.839108	−	0.543964i	$-0.816923\pi$
$719$	−45.0000	−1.67822	−0.839108	+	0.543964i	$0.816923\pi$
$720$	4.00000	0.149071
$721$	−16.0000	−0.595871
$722$	0	0
$723$	−17.0000	−0.632237
$724$	2.00000	0.0743294
$725$	4.00000	0.148556
$726$	0	0
$727$	−37.0000	−1.37225	−0.686127	−	0.727482i	$-0.740691\pi$
$727$	−37.0000	−1.37225	−0.686127	+	0.727482i	$0.740691\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	4.00000	0.147945
$732$	−10.0000	−0.369611
$733$	11.0000	0.406294	0.203147	−	0.979148i	$-0.434883\pi$
$733$	11.0000	0.406294	0.203147	+	0.979148i	$0.434883\pi$
$734$	0	0
$735$	−1.00000	−0.0368856
$736$	0	0
$737$	24.0000	0.884051
$738$	0	0
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	2.00000	0.0735215
$741$	6.00000	0.220416
$742$	0	0
$743$	−51.0000	−1.87101	−0.935504	−	0.353315i	$-0.885054\pi$
$743$	−51.0000	−1.87101	−0.935504	+	0.353315i	$0.885054\pi$
$744$	0	0
$745$	−11.0000	−0.403009
$746$	0	0
$747$	−5.00000	−0.182940
$748$	4.00000	0.146254
$749$	−13.0000	−0.475010
$750$	0	0
$751$	50.0000	1.82453	0.912263	−	0.409605i	$-0.134333\pi$
$751$	50.0000	1.82453	0.912263	+	0.409605i	$0.134333\pi$
$752$	−36.0000	−1.31278
$753$	−16.0000	−0.583072
$754$	0	0
$755$	19.0000	0.691481
$756$	−2.00000	−0.0727393
$757$	−34.0000	−1.23575	−0.617876	−	0.786276i	$-0.712006\pi$
$757$	−34.0000	−1.23575	−0.617876	+	0.786276i	$0.712006\pi$
$758$	0	0
$759$	−6.00000	−0.217786
$760$	0	0
$761$	−6.00000	−0.217500	−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000	−0.217500	−0.108750	+	0.994069i	$0.534685\pi$
$762$	0	0
$763$	10.0000	0.362024
$764$	54.0000	1.95365
$765$	1.00000	0.0361551
$766$	0	0
$767$	−36.0000	−1.29988
$768$	−16.0000	−0.577350
$769$	−16.0000	−0.576975	−0.288487	−	0.957484i	$-0.593152\pi$
$769$	−16.0000	−0.576975	−0.288487	+	0.957484i	$0.593152\pi$
$770$	0	0
$771$	7.00000	0.252099
$772$	−46.0000	−1.65558
$773$	39.0000	1.40273	0.701366	−	0.712801i	$-0.252574\pi$
$773$	39.0000	1.40273	0.701366	+	0.712801i	$0.252574\pi$
$774$	0	0
$775$	−5.00000	−0.179605
$776$	0	0
$777$	−1.00000	−0.0358748
$778$	0	0
$779$	−22.0000	−0.788232
$780$	6.00000	0.214834
$781$	4.00000	0.143131
$782$	0	0
$783$	−4.00000	−0.142948
$784$	4.00000	0.142857
$785$	−10.0000	−0.356915
$786$	0	0
$787$	44.0000	1.56843	0.784215	−	0.620489i	$-0.213066\pi$
$787$	44.0000	1.56843	0.784215	+	0.620489i	$0.213066\pi$
$788$	36.0000	1.28245
$789$	−24.0000	−0.854423
$790$	0	0
$791$	6.00000	0.213335
$792$	0	0
$793$	−15.0000	−0.532666
$794$	0	0
$795$	0	0
$796$	24.0000	0.850657
$797$	27.0000	0.956389	0.478195	−	0.878254i	$-0.341291\pi$
$797$	27.0000	0.956389	0.478195	+	0.878254i	$0.341291\pi$
$798$	0	0
$799$	−9.00000	−0.318397
$800$	0	0
$801$	−14.0000	−0.494666
$802$	0	0
$803$	8.00000	0.282314
$804$	−24.0000	−0.846415
$805$	3.00000	0.105736
$806$	0	0
$807$	21.0000	0.739235
$808$	0	0
$809$	2.00000	0.0703163	0.0351581	−	0.999382i	$-0.488807\pi$
$809$	2.00000	0.0703163	0.0351581	+	0.999382i	$0.488807\pi$
$810$	0	0
$811$	31.0000	1.08856	0.544279	−	0.838905i	$-0.316803\pi$
$811$	31.0000	1.08856	0.544279	+	0.838905i	$0.316803\pi$
$812$	8.00000	0.280745
$813$	−26.0000	−0.911860
$814$	0	0
$815$	19.0000	0.665541
$816$	−4.00000	−0.140028
$817$	−8.00000	−0.279885
$818$	0	0
$819$	−3.00000	−0.104828
$820$	−22.0000	−0.768273
$821$	22.0000	0.767805	0.383903	−	0.923374i	$-0.374580\pi$
$821$	22.0000	0.767805	0.383903	+	0.923374i	$0.374580\pi$
$822$	0	0
$823$	23.0000	0.801730	0.400865	−	0.916137i	$-0.368710\pi$
$823$	23.0000	0.801730	0.400865	+	0.916137i	$0.368710\pi$
$824$	0	0
$825$	2.00000	0.0696311
$826$	0	0
$827$	13.0000	0.452054	0.226027	−	0.974121i	$-0.427426\pi$
$827$	13.0000	0.452054	0.226027	+	0.974121i	$0.427426\pi$
$828$	6.00000	0.208514
$829$	−26.0000	−0.903017	−0.451509	−	0.892267i	$-0.649114\pi$
$829$	−26.0000	−0.903017	−0.451509	+	0.892267i	$0.649114\pi$
$830$	0	0
$831$	−7.00000	−0.242827
$832$	−24.0000	−0.832050
$833$	1.00000	0.0346479
$834$	0	0
$835$	2.00000	0.0692129
$836$	−8.00000	−0.276686
$837$	5.00000	0.172825
$838$	0	0
$839$	40.0000	1.38095	0.690477	−	0.723355i	$-0.257401\pi$
$839$	40.0000	1.38095	0.690477	+	0.723355i	$0.257401\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	0	0
$843$	1.00000	0.0344418
$844$	28.0000	0.963800
$845$	−4.00000	−0.137604
$846$	0	0
$847$	7.00000	0.240523
$848$	0	0
$849$	−28.0000	−0.960958
$850$	0	0
$851$	3.00000	0.102839
$852$	−4.00000	−0.137038
$853$	−26.0000	−0.890223	−0.445112	−	0.895475i	$-0.646836\pi$
$853$	−26.0000	−0.890223	−0.445112	+	0.895475i	$0.646836\pi$
$854$	0	0
$855$	−2.00000	−0.0683986
$856$	0	0
$857$	18.0000	0.614868	0.307434	−	0.951569i	$-0.400530\pi$
$857$	18.0000	0.614868	0.307434	+	0.951569i	$0.400530\pi$
$858$	0	0
$859$	44.0000	1.50126	0.750630	−	0.660722i	$-0.229750\pi$
$859$	44.0000	1.50126	0.750630	+	0.660722i	$0.229750\pi$
$860$	−8.00000	−0.272798
$861$	11.0000	0.374879
$862$	0	0
$863$	−14.0000	−0.476566	−0.238283	−	0.971196i	$-0.576585\pi$
$863$	−14.0000	−0.476566	−0.238283	+	0.971196i	$0.576585\pi$
$864$	0	0
$865$	−6.00000	−0.204006
$866$	0	0
$867$	−1.00000	−0.0339618
$868$	−10.0000	−0.339422
$869$	20.0000	0.678454
$870$	0	0
$871$	−36.0000	−1.21981
$872$	0	0
$873$	−2.00000	−0.0676897
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	−8.00000	−0.270295
$877$	−2.00000	−0.0675352	−0.0337676	−	0.999430i	$-0.510751\pi$
$877$	−2.00000	−0.0675352	−0.0337676	+	0.999430i	$0.510751\pi$
$878$	0	0
$879$	−14.0000	−0.472208
$880$	−8.00000	−0.269680
$881$	−14.0000	−0.471672	−0.235836	−	0.971793i	$-0.575783\pi$
$881$	−14.0000	−0.471672	−0.235836	+	0.971793i	$0.575783\pi$
$882$	0	0
$883$	−44.0000	−1.48072	−0.740359	−	0.672212i	$-0.765344\pi$
$883$	−44.0000	−1.48072	−0.740359	+	0.672212i	$0.765344\pi$
$884$	−6.00000	−0.201802
$885$	12.0000	0.403376
$886$	0	0
$887$	−42.0000	−1.41022	−0.705111	−	0.709097i	$-0.749103\pi$
$887$	−42.0000	−1.41022	−0.705111	+	0.709097i	$0.749103\pi$
$888$	0	0
$889$	8.00000	0.268311
$890$	0	0
$891$	−2.00000	−0.0670025
$892$	30.0000	1.00447
$893$	18.0000	0.602347
$894$	0	0
$895$	−19.0000	−0.635100
$896$	0	0
$897$	9.00000	0.300501
$898$	0	0
$899$	−20.0000	−0.667037
$900$	−2.00000	−0.0666667
$901$	0	0
$902$	0	0
$903$	4.00000	0.133112
$904$	0	0
$905$	−1.00000	−0.0332411
$906$	0	0
$907$	−40.0000	−1.32818	−0.664089	−	0.747653i	$-0.731180\pi$
$907$	−40.0000	−1.32818	−0.664089	+	0.747653i	$0.731180\pi$
$908$	−4.00000	−0.132745
$909$	−8.00000	−0.265343
$910$	0	0
$911$	4.00000	0.132526	0.0662630	−	0.997802i	$-0.478892\pi$
$911$	4.00000	0.132526	0.0662630	+	0.997802i	$0.478892\pi$
$912$	8.00000	0.264906
$913$	10.0000	0.330952
$914$	0	0
$915$	5.00000	0.165295
$916$	−28.0000	−0.925146
$917$	3.00000	0.0990687
$918$	0	0
$919$	8.00000	0.263896	0.131948	−	0.991257i	$-0.457877\pi$
$919$	8.00000	0.263896	0.131948	+	0.991257i	$0.457877\pi$
$920$	0	0
$921$	−20.0000	−0.659022
$922$	0	0
$923$	−6.00000	−0.197492
$924$	4.00000	0.131590
$925$	−1.00000	−0.0328798
$926$	0	0
$927$	16.0000	0.525509
$928$	0	0
$929$	−1.00000	−0.0328089	−0.0164045	−	0.999865i	$-0.505222\pi$
$929$	−1.00000	−0.0328089	−0.0164045	+	0.999865i	$0.505222\pi$
$930$	0	0
$931$	−2.00000	−0.0655474
$932$	−6.00000	−0.196537
$933$	5.00000	0.163693
$934$	0	0
$935$	−2.00000	−0.0654070
$936$	0	0
$937$	−47.0000	−1.53542	−0.767712	−	0.640796i	$-0.778605\pi$
$937$	−47.0000	−1.53542	−0.767712	+	0.640796i	$0.778605\pi$
$938$	0	0
$939$	10.0000	0.326338
$940$	18.0000	0.587095
$941$	27.0000	0.880175	0.440087	−	0.897955i	$-0.354947\pi$
$941$	27.0000	0.880175	0.440087	+	0.897955i	$0.354947\pi$
$942$	0	0
$943$	−33.0000	−1.07463
$944$	−48.0000	−1.56227
$945$	1.00000	0.0325300
$946$	0	0
$947$	57.0000	1.85225	0.926126	−	0.377215i	$-0.123118\pi$
$947$	57.0000	1.85225	0.926126	+	0.377215i	$0.123118\pi$
$948$	−20.0000	−0.649570
$949$	−12.0000	−0.389536
$950$	0	0
$951$	5.00000	0.162136
$952$	0	0
$953$	−50.0000	−1.61966	−0.809829	−	0.586665i	$-0.800440\pi$
$953$	−50.0000	−1.61966	−0.809829	+	0.586665i	$0.800440\pi$
$954$	0	0
$955$	−27.0000	−0.873699
$956$	30.0000	0.970269
$957$	8.00000	0.258603
$958$	0	0
$959$	8.00000	0.258333
$960$	8.00000	0.258199
$961$	−6.00000	−0.193548
$962$	0	0
$963$	13.0000	0.418919
$964$	−34.0000	−1.09507
$965$	23.0000	0.740396
$966$	0	0
$967$	22.0000	0.707472	0.353736	−	0.935345i	$-0.384911\pi$
$967$	22.0000	0.707472	0.353736	+	0.935345i	$0.384911\pi$
$968$	0	0
$969$	2.00000	0.0642493
$970$	0	0
$971$	−16.0000	−0.513464	−0.256732	−	0.966483i	$-0.582646\pi$
$971$	−16.0000	−0.513464	−0.256732	+	0.966483i	$0.582646\pi$
$972$	2.00000	0.0641500
$973$	−12.0000	−0.384702
$974$	0	0
$975$	−3.00000	−0.0960769
$976$	−20.0000	−0.640184
$977$	−18.0000	−0.575871	−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000	−0.575871	−0.287936	+	0.957650i	$0.592969\pi$
$978$	0	0
$979$	28.0000	0.894884
$980$	−2.00000	−0.0638877
$981$	−10.0000	−0.319275
$982$	0	0
$983$	−34.0000	−1.08443	−0.542216	−	0.840239i	$-0.682414\pi$
$983$	−34.0000	−1.08443	−0.542216	+	0.840239i	$0.682414\pi$
$984$	0	0
$985$	−18.0000	−0.573528
$986$	0	0
$987$	−9.00000	−0.286473
$988$	12.0000	0.381771
$989$	−12.0000	−0.381578
$990$	0	0
$991$	−58.0000	−1.84243	−0.921215	−	0.389053i	$-0.872802\pi$
$991$	−58.0000	−1.84243	−0.921215	+	0.389053i	$0.872802\pi$
$992$	0	0
$993$	17.0000	0.539479
$994$	0	0
$995$	−12.0000	−0.380426
$996$	−10.0000	−0.316862
$997$	42.0000	1.33015	0.665077	−	0.746775i	$-0.268399\pi$
$997$	42.0000	1.33015	0.665077	+	0.746775i	$0.268399\pi$
$998$	0	0
$999$	1.00000	0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1785.2.a.g.1.1	✓	1
3.2	odd	2		5355.2.a.k.1.1		1
5.4	even	2		8925.2.a.r.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1785.2.a.g.1.1	✓	1	1.1	even	1	trivial
5355.2.a.k.1.1		1	3.2	odd	2
8925.2.a.r.1.1		1	5.4	even	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:	\( N \)	\(=\)	\( 1785 = 3 \cdot 5 \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1785.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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