LMFDB - Embedded newform 786.2.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [786,2,Mod(1,786)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(786, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("786.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 786 = 2 \cdot 3 \cdot 131 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	786.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:	$6.27624159887$
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	$\chi$	$=$	786.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} +4.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} -2.00000 q^{15} +1.00000 q^{16} -2.00000 q^{17} -1.00000 q^{18} +8.00000 q^{19} +2.00000 q^{20} -4.00000 q^{22} +1.00000 q^{24} -1.00000 q^{25} +2.00000 q^{26} -1.00000 q^{27} +2.00000 q^{29} +2.00000 q^{30} -8.00000 q^{31} -1.00000 q^{32} -4.00000 q^{33} +2.00000 q^{34} +1.00000 q^{36} +2.00000 q^{37} -8.00000 q^{38} +2.00000 q^{39} -2.00000 q^{40} +10.0000 q^{41} -4.00000 q^{43} +4.00000 q^{44} +2.00000 q^{45} +8.00000 q^{47} -1.00000 q^{48} -7.00000 q^{49} +1.00000 q^{50} +2.00000 q^{51} -2.00000 q^{52} +10.0000 q^{53} +1.00000 q^{54} +8.00000 q^{55} -8.00000 q^{57} -2.00000 q^{58} +12.0000 q^{59} -2.00000 q^{60} +6.00000 q^{61} +8.00000 q^{62} +1.00000 q^{64} -4.00000 q^{65} +4.00000 q^{66} -2.00000 q^{68} +8.00000 q^{71} -1.00000 q^{72} -14.0000 q^{73} -2.00000 q^{74} +1.00000 q^{75} +8.00000 q^{76} -2.00000 q^{78} +16.0000 q^{79} +2.00000 q^{80} +1.00000 q^{81} -10.0000 q^{82} +12.0000 q^{83} -4.00000 q^{85} +4.00000 q^{86} -2.00000 q^{87} -4.00000 q^{88} -14.0000 q^{89} -2.00000 q^{90} +8.00000 q^{93} -8.00000 q^{94} +16.0000 q^{95} +1.00000 q^{96} +2.00000 q^{97} +7.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	1.00000	0.408248
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−2.00000	−0.632456
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	−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	$0.589456\pi$
$14$	0	0
$15$	−2.00000	−0.516398
$16$	1.00000	0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	−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	$0.130073\pi$
$20$	2.00000	0.447214
$21$	0	0
$22$	−4.00000	−0.852803
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	1.00000	0.204124
$25$	−1.00000	−0.200000
$26$	2.00000	0.392232
$27$	−1.00000	−0.192450
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	2.00000	0.365148
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	−1.00000	−0.176777
$33$	−4.00000	−0.696311
$34$	2.00000	0.342997
$35$	0	0
$36$	1.00000	0.166667
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	−8.00000	−1.29777
$39$	2.00000	0.320256
$40$	−2.00000	−0.316228
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	4.00000	0.603023
$45$	2.00000	0.298142
$46$	0	0
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	−1.00000	−0.144338
$49$	−7.00000	−1.00000
$50$	1.00000	0.141421
$51$	2.00000	0.280056
$52$	−2.00000	−0.277350
$53$	10.0000	1.37361	0.686803	−	0.726844i	$-0.259014\pi$
$53$	10.0000	1.37361	0.686803	+	0.726844i	$0.259014\pi$
$54$	1.00000	0.136083
$55$	8.00000	1.07872
$56$	0	0
$57$	−8.00000	−1.05963
$58$	−2.00000	−0.262613
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	−2.00000	−0.258199
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	8.00000	1.01600
$63$	0	0
$64$	1.00000	0.125000
$65$	−4.00000	−0.496139
$66$	4.00000	0.492366
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	−2.00000	−0.242536
$69$	0	0
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	−1.00000	−0.117851
$73$	−14.0000	−1.63858	−0.819288	−	0.573382i	$-0.805631\pi$
$73$	−14.0000	−1.63858	−0.819288	+	0.573382i	$0.805631\pi$
$74$	−2.00000	−0.232495
$75$	1.00000	0.115470
$76$	8.00000	0.917663
$77$	0	0
$78$	−2.00000	−0.226455
$79$	16.0000	1.80014	0.900070	−	0.435745i	$-0.143515\pi$
$79$	16.0000	1.80014	0.900070	+	0.435745i	$0.143515\pi$
$80$	2.00000	0.223607
$81$	1.00000	0.111111
$82$	−10.0000	−1.10432
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	0	0
$85$	−4.00000	−0.433861
$86$	4.00000	0.431331
$87$	−2.00000	−0.214423
$88$	−4.00000	−0.426401
$89$	−14.0000	−1.48400	−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000	−1.48400	−0.741999	+	0.670402i	$0.766122\pi$
$90$	−2.00000	−0.210819
$91$	0	0
$92$	0	0
$93$	8.00000	0.829561
$94$	−8.00000	−0.825137
$95$	16.0000	1.64157
$96$	1.00000	0.102062
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	7.00000	0.707107
$99$	4.00000	0.402015
$100$	−1.00000	−0.100000
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	−2.00000	−0.198030
$103$	16.0000	1.57653	0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000	1.57653	0.788263	+	0.615338i	$0.210980\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	−10.0000	−0.971286
$107$	−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
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	−8.00000	−0.762770
$111$	−2.00000	−0.189832
$112$	0	0
$113$	10.0000	0.940721	0.470360	−	0.882474i	$-0.344124\pi$
$113$	10.0000	0.940721	0.470360	+	0.882474i	$0.344124\pi$
$114$	8.00000	0.749269
$115$	0	0
$116$	2.00000	0.185695
$117$	−2.00000	−0.184900
$118$	−12.0000	−1.10469
$119$	0	0
$120$	2.00000	0.182574
$121$	5.00000	0.454545
$122$	−6.00000	−0.543214
$123$	−10.0000	−0.901670
$124$	−8.00000	−0.718421
$125$	−12.0000	−1.07331
$126$	0	0
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	−1.00000	−0.0883883
$129$	4.00000	0.352180
$130$	4.00000	0.350823
$131$	1.00000	0.0873704
$131$	1.00000	0.0873704
$132$	−4.00000	−0.348155
$133$	0	0
$134$	0	0
$135$	−2.00000	−0.172133
$136$	2.00000	0.171499
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	0	0
$139$	8.00000	0.678551	0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000	0.678551	0.339276	+	0.940687i	$0.389818\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	−8.00000	−0.671345
$143$	−8.00000	−0.668994
$144$	1.00000	0.0833333
$145$	4.00000	0.332182
$146$	14.0000	1.15865
$147$	7.00000	0.577350
$148$	2.00000	0.164399
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	−1.00000	−0.0816497
$151$	−24.0000	−1.95309	−0.976546	−	0.215308i	$-0.930924\pi$
$151$	−24.0000	−1.95309	−0.976546	+	0.215308i	$0.930924\pi$
$152$	−8.00000	−0.648886
$153$	−2.00000	−0.161690
$154$	0	0
$155$	−16.0000	−1.28515
$156$	2.00000	0.160128
$157$	−6.00000	−0.478852	−0.239426	−	0.970915i	$-0.576959\pi$
$157$	−6.00000	−0.478852	−0.239426	+	0.970915i	$0.576959\pi$
$158$	−16.0000	−1.27289
$159$	−10.0000	−0.793052
$160$	−2.00000	−0.158114
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	−16.0000	−1.25322	−0.626608	−	0.779334i	$-0.715557\pi$
$163$	−16.0000	−1.25322	−0.626608	+	0.779334i	$0.715557\pi$
$164$	10.0000	0.780869
$165$	−8.00000	−0.622799
$166$	−12.0000	−0.931381
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	4.00000	0.306786
$171$	8.00000	0.611775
$172$	−4.00000	−0.304997
$173$	2.00000	0.152057	0.0760286	−	0.997106i	$-0.475776\pi$
$173$	2.00000	0.152057	0.0760286	+	0.997106i	$0.475776\pi$
$174$	2.00000	0.151620
$175$	0	0
$176$	4.00000	0.301511
$177$	−12.0000	−0.901975
$178$	14.0000	1.04934
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	2.00000	0.149071
$181$	−6.00000	−0.445976	−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000	−0.445976	−0.222988	+	0.974821i	$0.571581\pi$
$182$	0	0
$183$	−6.00000	−0.443533
$184$	0	0
$185$	4.00000	0.294086
$186$	−8.00000	−0.586588
$187$	−8.00000	−0.585018
$188$	8.00000	0.583460
$189$	0	0
$190$	−16.0000	−1.16076
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	−1.00000	−0.0721688
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	−2.00000	−0.143592
$195$	4.00000	0.286446
$196$	−7.00000	−0.500000
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	−4.00000	−0.284268
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	6.00000	0.422159
$203$	0	0
$204$	2.00000	0.140028
$205$	20.0000	1.39686
$206$	−16.0000	−1.11477
$207$	0	0
$208$	−2.00000	−0.138675
$209$	32.0000	2.21349
$210$	0	0
$211$	−20.0000	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$212$	10.0000	0.686803
$213$	−8.00000	−0.548151
$214$	12.0000	0.820303
$215$	−8.00000	−0.545595
$216$	1.00000	0.0680414
$217$	0	0
$218$	2.00000	0.135457
$219$	14.0000	0.946032
$220$	8.00000	0.539360
$221$	4.00000	0.269069
$222$	2.00000	0.134231
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	−10.0000	−0.665190
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	−8.00000	−0.529813
$229$	2.00000	0.132164	0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000	0.132164	0.0660819	+	0.997814i	$0.478950\pi$
$230$	0	0
$231$	0	0
$232$	−2.00000	−0.131306
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	2.00000	0.130744
$235$	16.0000	1.04372
$236$	12.0000	0.781133
$237$	−16.0000	−1.03931
$238$	0	0
$239$	−28.0000	−1.81117	−0.905585	−	0.424165i	$-0.860568\pi$
$239$	−28.0000	−1.81117	−0.905585	+	0.424165i	$0.860568\pi$
$240$	−2.00000	−0.129099
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	−5.00000	−0.321412
$243$	−1.00000	−0.0641500
$244$	6.00000	0.384111
$245$	−14.0000	−0.894427
$246$	10.0000	0.637577
$247$	−16.0000	−1.01806
$248$	8.00000	0.508001
$249$	−12.0000	−0.760469
$250$	12.0000	0.758947
$251$	−20.0000	−1.26239	−0.631194	−	0.775625i	$-0.717435\pi$
$251$	−20.0000	−1.26239	−0.631194	+	0.775625i	$0.717435\pi$
$252$	0	0
$253$	0	0
$254$	−8.00000	−0.501965
$255$	4.00000	0.250490
$256$	1.00000	0.0625000
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	−4.00000	−0.249029
$259$	0	0
$260$	−4.00000	−0.248069
$261$	2.00000	0.123797
$262$	−1.00000	−0.0617802
$263$	12.0000	0.739952	0.369976	−	0.929041i	$-0.379366\pi$
$263$	12.0000	0.739952	0.369976	+	0.929041i	$0.379366\pi$
$264$	4.00000	0.246183
$265$	20.0000	1.22859
$266$	0	0
$267$	14.0000	0.856786
$268$	0	0
$269$	−30.0000	−1.82913	−0.914566	−	0.404436i	$-0.867468\pi$
$269$	−30.0000	−1.82913	−0.914566	+	0.404436i	$0.867468\pi$
$270$	2.00000	0.121716
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	−2.00000	−0.121268
$273$	0	0
$274$	18.0000	1.08742
$275$	−4.00000	−0.241209
$276$	0	0
$277$	22.0000	1.32185	0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000	1.32185	0.660926	+	0.750451i	$0.270164\pi$
$278$	−8.00000	−0.479808
$279$	−8.00000	−0.478947
$280$	0	0
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	8.00000	0.476393
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	8.00000	0.474713
$285$	−16.0000	−0.947758
$286$	8.00000	0.473050
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	−13.0000	−0.764706
$290$	−4.00000	−0.234888
$291$	−2.00000	−0.117242
$292$	−14.0000	−0.819288
$293$	−6.00000	−0.350524	−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000	−0.350524	−0.175262	+	0.984522i	$0.556077\pi$
$294$	−7.00000	−0.408248
$295$	24.0000	1.39733
$296$	−2.00000	−0.116248
$297$	−4.00000	−0.232104
$298$	6.00000	0.347571
$299$	0	0
$300$	1.00000	0.0577350
$301$	0	0
$302$	24.0000	1.38104
$303$	6.00000	0.344691
$304$	8.00000	0.458831
$305$	12.0000	0.687118
$306$	2.00000	0.114332
$307$	−4.00000	−0.228292	−0.114146	−	0.993464i	$-0.536413\pi$
$307$	−4.00000	−0.228292	−0.114146	+	0.993464i	$0.536413\pi$
$308$	0	0
$309$	−16.0000	−0.910208
$310$	16.0000	0.908739
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	−2.00000	−0.113228
$313$	2.00000	0.113047	0.0565233	−	0.998401i	$-0.481998\pi$
$313$	2.00000	0.113047	0.0565233	+	0.998401i	$0.481998\pi$
$314$	6.00000	0.338600
$315$	0	0
$316$	16.0000	0.900070
$317$	2.00000	0.112331	0.0561656	−	0.998421i	$-0.482113\pi$
$317$	2.00000	0.112331	0.0561656	+	0.998421i	$0.482113\pi$
$318$	10.0000	0.560772
$319$	8.00000	0.447914
$320$	2.00000	0.111803
$321$	12.0000	0.669775
$322$	0	0
$323$	−16.0000	−0.890264
$324$	1.00000	0.0555556
$325$	2.00000	0.110940
$326$	16.0000	0.886158
$327$	2.00000	0.110600
$328$	−10.0000	−0.552158
$329$	0	0
$330$	8.00000	0.440386
$331$	−16.0000	−0.879440	−0.439720	−	0.898135i	$-0.644922\pi$
$331$	−16.0000	−0.879440	−0.439720	+	0.898135i	$0.644922\pi$
$332$	12.0000	0.658586
$333$	2.00000	0.109599
$334$	12.0000	0.656611
$335$	0	0
$336$	0	0
$337$	−14.0000	−0.762629	−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000	−0.762629	−0.381314	+	0.924445i	$0.624528\pi$
$338$	9.00000	0.489535
$339$	−10.0000	−0.543125
$340$	−4.00000	−0.216930
$341$	−32.0000	−1.73290
$342$	−8.00000	−0.432590
$343$	0	0
$344$	4.00000	0.215666
$345$	0	0
$346$	−2.00000	−0.107521
$347$	−20.0000	−1.07366	−0.536828	−	0.843692i	$-0.680378\pi$
$347$	−20.0000	−1.07366	−0.536828	+	0.843692i	$0.680378\pi$
$348$	−2.00000	−0.107211
$349$	10.0000	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	−4.00000	−0.213201
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	12.0000	0.637793
$355$	16.0000	0.849192
$356$	−14.0000	−0.741999
$357$	0	0
$358$	−12.0000	−0.634220
$359$	−8.00000	−0.422224	−0.211112	−	0.977462i	$-0.567708\pi$
$359$	−8.00000	−0.422224	−0.211112	+	0.977462i	$0.567708\pi$
$360$	−2.00000	−0.105409
$361$	45.0000	2.36842
$362$	6.00000	0.315353
$363$	−5.00000	−0.262432
$364$	0	0
$365$	−28.0000	−1.46559
$366$	6.00000	0.313625
$367$	−16.0000	−0.835193	−0.417597	−	0.908633i	$-0.637127\pi$
$367$	−16.0000	−0.835193	−0.417597	+	0.908633i	$0.637127\pi$
$368$	0	0
$369$	10.0000	0.520579
$370$	−4.00000	−0.207950
$371$	0	0
$372$	8.00000	0.414781
$373$	18.0000	0.932005	0.466002	−	0.884783i	$-0.345694\pi$
$373$	18.0000	0.932005	0.466002	+	0.884783i	$0.345694\pi$
$374$	8.00000	0.413670
$375$	12.0000	0.619677
$376$	−8.00000	−0.412568
$377$	−4.00000	−0.206010
$378$	0	0
$379$	−12.0000	−0.616399	−0.308199	−	0.951322i	$-0.599726\pi$
$379$	−12.0000	−0.616399	−0.308199	+	0.951322i	$0.599726\pi$
$380$	16.0000	0.820783
$381$	−8.00000	−0.409852
$382$	−12.0000	−0.613973
$383$	4.00000	0.204390	0.102195	−	0.994764i	$-0.467413\pi$
$383$	4.00000	0.204390	0.102195	+	0.994764i	$0.467413\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−2.00000	−0.101797
$387$	−4.00000	−0.203331
$388$	2.00000	0.101535
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	−4.00000	−0.202548
$391$	0	0
$392$	7.00000	0.353553
$393$	−1.00000	−0.0504433
$394$	6.00000	0.302276
$395$	32.0000	1.61009
$396$	4.00000	0.201008
$397$	−18.0000	−0.903394	−0.451697	−	0.892171i	$-0.649181\pi$
$397$	−18.0000	−0.903394	−0.451697	+	0.892171i	$0.649181\pi$
$398$	0	0
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	0	0
$403$	16.0000	0.797017
$404$	−6.00000	−0.298511
$405$	2.00000	0.0993808
$406$	0	0
$407$	8.00000	0.396545
$408$	−2.00000	−0.0990148
$409$	26.0000	1.28562	0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000	1.28562	0.642809	+	0.766027i	$0.277769\pi$
$410$	−20.0000	−0.987730
$411$	18.0000	0.887875
$412$	16.0000	0.788263
$413$	0	0
$414$	0	0
$415$	24.0000	1.17811
$416$	2.00000	0.0980581
$417$	−8.00000	−0.391762
$418$	−32.0000	−1.56517
$419$	4.00000	0.195413	0.0977064	−	0.995215i	$-0.468849\pi$
$419$	4.00000	0.195413	0.0977064	+	0.995215i	$0.468849\pi$
$420$	0	0
$421$	14.0000	0.682318	0.341159	−	0.940006i	$-0.389181\pi$
$421$	14.0000	0.682318	0.341159	+	0.940006i	$0.389181\pi$
$422$	20.0000	0.973585
$423$	8.00000	0.388973
$424$	−10.0000	−0.485643
$425$	2.00000	0.0970143
$426$	8.00000	0.387601
$427$	0	0
$428$	−12.0000	−0.580042
$429$	8.00000	0.386244
$430$	8.00000	0.385794
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	−1.00000	−0.0481125
$433$	−38.0000	−1.82616	−0.913082	−	0.407777i	$-0.866304\pi$
$433$	−38.0000	−1.82616	−0.913082	+	0.407777i	$0.866304\pi$
$434$	0	0
$435$	−4.00000	−0.191785
$436$	−2.00000	−0.0957826
$437$	0	0
$438$	−14.0000	−0.668946
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	−8.00000	−0.381385
$441$	−7.00000	−0.333333
$442$	−4.00000	−0.190261
$443$	−4.00000	−0.190046	−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000	−0.190046	−0.0950229	+	0.995475i	$0.530292\pi$
$444$	−2.00000	−0.0949158
$445$	−28.0000	−1.32733
$446$	−16.0000	−0.757622
$447$	6.00000	0.283790
$448$	0	0
$449$	14.0000	0.660701	0.330350	−	0.943858i	$-0.392833\pi$
$449$	14.0000	0.660701	0.330350	+	0.943858i	$0.392833\pi$
$450$	1.00000	0.0471405
$451$	40.0000	1.88353
$452$	10.0000	0.470360
$453$	24.0000	1.12762
$454$	−12.0000	−0.563188
$455$	0	0
$456$	8.00000	0.374634
$457$	−6.00000	−0.280668	−0.140334	−	0.990104i	$-0.544818\pi$
$457$	−6.00000	−0.280668	−0.140334	+	0.990104i	$0.544818\pi$
$458$	−2.00000	−0.0934539
$459$	2.00000	0.0933520
$460$	0	0
$461$	18.0000	0.838344	0.419172	−	0.907907i	$-0.362320\pi$
$461$	18.0000	0.838344	0.419172	+	0.907907i	$0.362320\pi$
$462$	0	0
$463$	−32.0000	−1.48717	−0.743583	−	0.668644i	$-0.766875\pi$
$463$	−32.0000	−1.48717	−0.743583	+	0.668644i	$0.766875\pi$
$464$	2.00000	0.0928477
$465$	16.0000	0.741982
$466$	6.00000	0.277945
$467$	−28.0000	−1.29569	−0.647843	−	0.761774i	$-0.724329\pi$
$467$	−28.0000	−1.29569	−0.647843	+	0.761774i	$0.724329\pi$
$468$	−2.00000	−0.0924500
$469$	0	0
$470$	−16.0000	−0.738025
$471$	6.00000	0.276465
$472$	−12.0000	−0.552345
$473$	−16.0000	−0.735681
$474$	16.0000	0.734904
$475$	−8.00000	−0.367065
$476$	0	0
$477$	10.0000	0.457869
$478$	28.0000	1.28069
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	2.00000	0.0912871
$481$	−4.00000	−0.182384
$482$	−2.00000	−0.0910975
$483$	0	0
$484$	5.00000	0.227273
$485$	4.00000	0.181631
$486$	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	$0.758174\pi$
$488$	−6.00000	−0.271607
$489$	16.0000	0.723545
$490$	14.0000	0.632456
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	−10.0000	−0.450835
$493$	−4.00000	−0.180151
$494$	16.0000	0.719874
$495$	8.00000	0.359573
$496$	−8.00000	−0.359211
$497$	0	0
$498$	12.0000	0.537733
$499$	−40.0000	−1.79065	−0.895323	−	0.445418i	$-0.853055\pi$
$499$	−40.0000	−1.79065	−0.895323	+	0.445418i	$0.853055\pi$
$500$	−12.0000	−0.536656
$501$	12.0000	0.536120
$502$	20.0000	0.892644
$503$	24.0000	1.07011	0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000	1.07011	0.535054	+	0.844818i	$0.320291\pi$
$504$	0	0
$505$	−12.0000	−0.533993
$506$	0	0
$507$	9.00000	0.399704
$508$	8.00000	0.354943
$509$	2.00000	0.0886484	0.0443242	−	0.999017i	$-0.485887\pi$
$509$	2.00000	0.0886484	0.0443242	+	0.999017i	$0.485887\pi$
$510$	−4.00000	−0.177123
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	−8.00000	−0.353209
$514$	−6.00000	−0.264649
$515$	32.0000	1.41009
$516$	4.00000	0.176090
$517$	32.0000	1.40736
$518$	0	0
$519$	−2.00000	−0.0877903
$520$	4.00000	0.175412
$521$	−26.0000	−1.13908	−0.569540	−	0.821963i	$-0.692879\pi$
$521$	−26.0000	−1.13908	−0.569540	+	0.821963i	$0.692879\pi$
$522$	−2.00000	−0.0875376
$523$	40.0000	1.74908	0.874539	−	0.484955i	$-0.161164\pi$
$523$	40.0000	1.74908	0.874539	+	0.484955i	$0.161164\pi$
$524$	1.00000	0.0436852
$525$	0	0
$526$	−12.0000	−0.523225
$527$	16.0000	0.696971
$528$	−4.00000	−0.174078
$529$	−23.0000	−1.00000
$530$	−20.0000	−0.868744
$531$	12.0000	0.520756
$532$	0	0
$533$	−20.0000	−0.866296
$534$	−14.0000	−0.605839
$535$	−24.0000	−1.03761
$536$	0	0
$537$	−12.0000	−0.517838
$538$	30.0000	1.29339
$539$	−28.0000	−1.20605
$540$	−2.00000	−0.0860663
$541$	18.0000	0.773880	0.386940	−	0.922105i	$-0.373532\pi$
$541$	18.0000	0.773880	0.386940	+	0.922105i	$0.373532\pi$
$542$	−8.00000	−0.343629
$543$	6.00000	0.257485
$544$	2.00000	0.0857493
$545$	−4.00000	−0.171341
$546$	0	0
$547$	0	0		−	1.00000i	$-0.5\pi$
$547$	0	0			1.00000i	$0.5\pi$
$548$	−18.0000	−0.768922
$549$	6.00000	0.256074
$550$	4.00000	0.170561
$551$	16.0000	0.681623
$552$	0	0
$553$	0	0
$554$	−22.0000	−0.934690
$555$	−4.00000	−0.169791
$556$	8.00000	0.339276
$557$	−22.0000	−0.932170	−0.466085	−	0.884740i	$-0.654336\pi$
$557$	−22.0000	−0.932170	−0.466085	+	0.884740i	$0.654336\pi$
$558$	8.00000	0.338667
$559$	8.00000	0.338364
$560$	0	0
$561$	8.00000	0.337760
$562$	10.0000	0.421825
$563$	−4.00000	−0.168580	−0.0842900	−	0.996441i	$-0.526862\pi$
$563$	−4.00000	−0.168580	−0.0842900	+	0.996441i	$0.526862\pi$
$564$	−8.00000	−0.336861
$565$	20.0000	0.841406
$566$	4.00000	0.168133
$567$	0	0
$568$	−8.00000	−0.335673
$569$	26.0000	1.08998	0.544988	−	0.838444i	$-0.316534\pi$
$569$	26.0000	1.08998	0.544988	+	0.838444i	$0.316534\pi$
$570$	16.0000	0.670166
$571$	32.0000	1.33916	0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000	1.33916	0.669579	+	0.742741i	$0.266474\pi$
$572$	−8.00000	−0.334497
$573$	−12.0000	−0.501307
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	−14.0000	−0.582828	−0.291414	−	0.956597i	$-0.594126\pi$
$577$	−14.0000	−0.582828	−0.291414	+	0.956597i	$0.594126\pi$
$578$	13.0000	0.540729
$579$	−2.00000	−0.0831172
$580$	4.00000	0.166091
$581$	0	0
$582$	2.00000	0.0829027
$583$	40.0000	1.65663
$584$	14.0000	0.579324
$585$	−4.00000	−0.165380
$586$	6.00000	0.247858
$587$	−36.0000	−1.48588	−0.742940	−	0.669359i	$-0.766569\pi$
$587$	−36.0000	−1.48588	−0.742940	+	0.669359i	$0.766569\pi$
$588$	7.00000	0.288675
$589$	−64.0000	−2.63707
$590$	−24.0000	−0.988064
$591$	6.00000	0.246807
$592$	2.00000	0.0821995
$593$	14.0000	0.574911	0.287456	−	0.957794i	$-0.407191\pi$
$593$	14.0000	0.574911	0.287456	+	0.957794i	$0.407191\pi$
$594$	4.00000	0.164122
$595$	0	0
$596$	−6.00000	−0.245770
$597$	0	0
$598$	0	0
$599$	−12.0000	−0.490307	−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000	−0.490307	−0.245153	+	0.969484i	$0.578838\pi$
$600$	−1.00000	−0.0408248
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	0	0
$603$	0	0
$604$	−24.0000	−0.976546
$605$	10.0000	0.406558
$606$	−6.00000	−0.243733
$607$	−32.0000	−1.29884	−0.649420	−	0.760430i	$-0.724988\pi$
$607$	−32.0000	−1.29884	−0.649420	+	0.760430i	$0.724988\pi$
$608$	−8.00000	−0.324443
$609$	0	0
$610$	−12.0000	−0.485866
$611$	−16.0000	−0.647291
$612$	−2.00000	−0.0808452
$613$	30.0000	1.21169	0.605844	−	0.795583i	$-0.292835\pi$
$613$	30.0000	1.21169	0.605844	+	0.795583i	$0.292835\pi$
$614$	4.00000	0.161427
$615$	−20.0000	−0.806478
$616$	0	0
$617$	6.00000	0.241551	0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000	0.241551	0.120775	+	0.992680i	$0.461462\pi$
$618$	16.0000	0.643614
$619$	32.0000	1.28619	0.643094	−	0.765787i	$-0.277650\pi$
$619$	32.0000	1.28619	0.643094	+	0.765787i	$0.277650\pi$
$620$	−16.0000	−0.642575
$621$	0	0
$622$	−12.0000	−0.481156
$623$	0	0
$624$	2.00000	0.0800641
$625$	−19.0000	−0.760000
$626$	−2.00000	−0.0799361
$627$	−32.0000	−1.27796
$628$	−6.00000	−0.239426
$629$	−4.00000	−0.159490
$630$	0	0
$631$	−40.0000	−1.59237	−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000	−1.59237	−0.796187	+	0.605050i	$0.793153\pi$
$632$	−16.0000	−0.636446
$633$	20.0000	0.794929
$634$	−2.00000	−0.0794301
$635$	16.0000	0.634941
$636$	−10.0000	−0.396526
$637$	14.0000	0.554700
$638$	−8.00000	−0.316723
$639$	8.00000	0.316475
$640$	−2.00000	−0.0790569
$641$	−46.0000	−1.81689	−0.908445	−	0.418004i	$-0.862730\pi$
$641$	−46.0000	−1.81689	−0.908445	+	0.418004i	$0.862730\pi$
$642$	−12.0000	−0.473602
$643$	−48.0000	−1.89294	−0.946468	−	0.322799i	$-0.895376\pi$
$643$	−48.0000	−1.89294	−0.946468	+	0.322799i	$0.895376\pi$
$644$	0	0
$645$	8.00000	0.315000
$646$	16.0000	0.629512
$647$	12.0000	0.471769	0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000	0.471769	0.235884	+	0.971781i	$0.424201\pi$
$648$	−1.00000	−0.0392837
$649$	48.0000	1.88416
$650$	−2.00000	−0.0784465
$651$	0	0
$652$	−16.0000	−0.626608
$653$	−38.0000	−1.48705	−0.743527	−	0.668705i	$-0.766849\pi$
$653$	−38.0000	−1.48705	−0.743527	+	0.668705i	$0.766849\pi$
$654$	−2.00000	−0.0782062
$655$	2.00000	0.0781465
$656$	10.0000	0.390434
$657$	−14.0000	−0.546192
$658$	0	0
$659$	20.0000	0.779089	0.389545	−	0.921008i	$-0.372632\pi$
$659$	20.0000	0.779089	0.389545	+	0.921008i	$0.372632\pi$
$660$	−8.00000	−0.311400
$661$	−22.0000	−0.855701	−0.427850	−	0.903850i	$-0.640729\pi$
$661$	−22.0000	−0.855701	−0.427850	+	0.903850i	$0.640729\pi$
$662$	16.0000	0.621858
$663$	−4.00000	−0.155347
$664$	−12.0000	−0.465690
$665$	0	0
$666$	−2.00000	−0.0774984
$667$	0	0
$668$	−12.0000	−0.464294
$669$	−16.0000	−0.618596
$670$	0	0
$671$	24.0000	0.926510
$672$	0	0
$673$	10.0000	0.385472	0.192736	−	0.981251i	$-0.438264\pi$
$673$	10.0000	0.385472	0.192736	+	0.981251i	$0.438264\pi$
$674$	14.0000	0.539260
$675$	1.00000	0.0384900
$676$	−9.00000	−0.346154
$677$	−22.0000	−0.845529	−0.422764	−	0.906240i	$-0.638940\pi$
$677$	−22.0000	−0.845529	−0.422764	+	0.906240i	$0.638940\pi$
$678$	10.0000	0.384048
$679$	0	0
$680$	4.00000	0.153393
$681$	−12.0000	−0.459841
$682$	32.0000	1.22534
$683$	−4.00000	−0.153056	−0.0765279	−	0.997067i	$-0.524383\pi$
$683$	−4.00000	−0.153056	−0.0765279	+	0.997067i	$0.524383\pi$
$684$	8.00000	0.305888
$685$	−36.0000	−1.37549
$686$	0	0
$687$	−2.00000	−0.0763048
$688$	−4.00000	−0.152499
$689$	−20.0000	−0.761939
$690$	0	0
$691$	−12.0000	−0.456502	−0.228251	−	0.973602i	$-0.573301\pi$
$691$	−12.0000	−0.456502	−0.228251	+	0.973602i	$0.573301\pi$
$692$	2.00000	0.0760286
$693$	0	0
$694$	20.0000	0.759190
$695$	16.0000	0.606915
$696$	2.00000	0.0758098
$697$	−20.0000	−0.757554
$698$	−10.0000	−0.378506
$699$	6.00000	0.226941
$700$	0	0
$701$	2.00000	0.0755390	0.0377695	−	0.999286i	$-0.487975\pi$
$701$	2.00000	0.0755390	0.0377695	+	0.999286i	$0.487975\pi$
$702$	−2.00000	−0.0754851
$703$	16.0000	0.603451
$704$	4.00000	0.150756
$705$	−16.0000	−0.602595
$706$	−18.0000	−0.677439
$707$	0	0
$708$	−12.0000	−0.450988
$709$	2.00000	0.0751116	0.0375558	−	0.999295i	$-0.488043\pi$
$709$	2.00000	0.0751116	0.0375558	+	0.999295i	$0.488043\pi$
$710$	−16.0000	−0.600469
$711$	16.0000	0.600047
$712$	14.0000	0.524672
$713$	0	0
$714$	0	0
$715$	−16.0000	−0.598366
$716$	12.0000	0.448461
$717$	28.0000	1.04568
$718$	8.00000	0.298557
$719$	20.0000	0.745874	0.372937	−	0.927857i	$-0.378351\pi$
$719$	20.0000	0.745874	0.372937	+	0.927857i	$0.378351\pi$
$720$	2.00000	0.0745356
$721$	0	0
$722$	−45.0000	−1.67473
$723$	−2.00000	−0.0743808
$724$	−6.00000	−0.222988
$725$	−2.00000	−0.0742781
$726$	5.00000	0.185567
$727$	−32.0000	−1.18681	−0.593407	−	0.804902i	$-0.702218\pi$
$727$	−32.0000	−1.18681	−0.593407	+	0.804902i	$0.702218\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	28.0000	1.03633
$731$	8.00000	0.295891
$732$	−6.00000	−0.221766
$733$	26.0000	0.960332	0.480166	−	0.877178i	$-0.340576\pi$
$733$	26.0000	0.960332	0.480166	+	0.877178i	$0.340576\pi$
$734$	16.0000	0.590571
$735$	14.0000	0.516398
$736$	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
$741$	16.0000	0.587775
$742$	0	0
$743$	16.0000	0.586983	0.293492	−	0.955962i	$-0.405183\pi$
$743$	16.0000	0.586983	0.293492	+	0.955962i	$0.405183\pi$
$744$	−8.00000	−0.293294
$745$	−12.0000	−0.439646
$746$	−18.0000	−0.659027
$747$	12.0000	0.439057
$748$	−8.00000	−0.292509
$749$	0	0
$750$	−12.0000	−0.438178
$751$	40.0000	1.45962	0.729810	−	0.683650i	$-0.239608\pi$
$751$	40.0000	1.45962	0.729810	+	0.683650i	$0.239608\pi$
$752$	8.00000	0.291730
$753$	20.0000	0.728841
$754$	4.00000	0.145671
$755$	−48.0000	−1.74690
$756$	0	0
$757$	−10.0000	−0.363456	−0.181728	−	0.983349i	$-0.558169\pi$
$757$	−10.0000	−0.363456	−0.181728	+	0.983349i	$0.558169\pi$
$758$	12.0000	0.435860
$759$	0	0
$760$	−16.0000	−0.580381
$761$	−42.0000	−1.52250	−0.761249	−	0.648459i	$-0.775414\pi$
$761$	−42.0000	−1.52250	−0.761249	+	0.648459i	$0.775414\pi$
$762$	8.00000	0.289809
$763$	0	0
$764$	12.0000	0.434145
$765$	−4.00000	−0.144620
$766$	−4.00000	−0.144526
$767$	−24.0000	−0.866590
$768$	−1.00000	−0.0360844
$769$	50.0000	1.80305	0.901523	−	0.432731i	$-0.142450\pi$
$769$	50.0000	1.80305	0.901523	+	0.432731i	$0.142450\pi$
$770$	0	0
$771$	−6.00000	−0.216085
$772$	2.00000	0.0719816
$773$	−6.00000	−0.215805	−0.107903	−	0.994161i	$-0.534413\pi$
$773$	−6.00000	−0.215805	−0.107903	+	0.994161i	$0.534413\pi$
$774$	4.00000	0.143777
$775$	8.00000	0.287368
$776$	−2.00000	−0.0717958
$777$	0	0
$778$	6.00000	0.215110
$779$	80.0000	2.86630
$780$	4.00000	0.143223
$781$	32.0000	1.14505
$782$	0	0
$783$	−2.00000	−0.0714742
$784$	−7.00000	−0.250000
$785$	−12.0000	−0.428298
$786$	1.00000	0.0356688
$787$	−28.0000	−0.998092	−0.499046	−	0.866575i	$-0.666316\pi$
$787$	−28.0000	−0.998092	−0.499046	+	0.866575i	$0.666316\pi$
$788$	−6.00000	−0.213741
$789$	−12.0000	−0.427211
$790$	−32.0000	−1.13851
$791$	0	0
$792$	−4.00000	−0.142134
$793$	−12.0000	−0.426132
$794$	18.0000	0.638796
$795$	−20.0000	−0.709327
$796$	0	0
$797$	−54.0000	−1.91278	−0.956389	−	0.292096i	$-0.905647\pi$
$797$	−54.0000	−1.91278	−0.956389	+	0.292096i	$0.905647\pi$
$798$	0	0
$799$	−16.0000	−0.566039
$800$	1.00000	0.0353553
$801$	−14.0000	−0.494666
$802$	−30.0000	−1.05934
$803$	−56.0000	−1.97620
$804$	0	0
$805$	0	0
$806$	−16.0000	−0.563576
$807$	30.0000	1.05605
$808$	6.00000	0.211079
$809$	46.0000	1.61727	0.808637	−	0.588308i	$-0.200206\pi$
$809$	46.0000	1.61727	0.808637	+	0.588308i	$0.200206\pi$
$810$	−2.00000	−0.0702728
$811$	−44.0000	−1.54505	−0.772524	−	0.634985i	$-0.781006\pi$
$811$	−44.0000	−1.54505	−0.772524	+	0.634985i	$0.781006\pi$
$812$	0	0
$813$	−8.00000	−0.280572
$814$	−8.00000	−0.280400
$815$	−32.0000	−1.12091
$816$	2.00000	0.0700140
$817$	−32.0000	−1.11954
$818$	−26.0000	−0.909069
$819$	0	0
$820$	20.0000	0.698430
$821$	−6.00000	−0.209401	−0.104701	−	0.994504i	$-0.533388\pi$
$821$	−6.00000	−0.209401	−0.104701	+	0.994504i	$0.533388\pi$
$822$	−18.0000	−0.627822
$823$	−32.0000	−1.11545	−0.557725	−	0.830026i	$-0.688326\pi$
$823$	−32.0000	−1.11545	−0.557725	+	0.830026i	$0.688326\pi$
$824$	−16.0000	−0.557386
$825$	4.00000	0.139262
$826$	0	0
$827$	−4.00000	−0.139094	−0.0695468	−	0.997579i	$-0.522155\pi$
$827$	−4.00000	−0.139094	−0.0695468	+	0.997579i	$0.522155\pi$
$828$	0	0
$829$	22.0000	0.764092	0.382046	−	0.924143i	$-0.375220\pi$
$829$	22.0000	0.764092	0.382046	+	0.924143i	$0.375220\pi$
$830$	−24.0000	−0.833052
$831$	−22.0000	−0.763172
$832$	−2.00000	−0.0693375
$833$	14.0000	0.485071
$834$	8.00000	0.277017
$835$	−24.0000	−0.830554
$836$	32.0000	1.10674
$837$	8.00000	0.276520
$838$	−4.00000	−0.138178
$839$	−12.0000	−0.414286	−0.207143	−	0.978311i	$-0.566417\pi$
$839$	−12.0000	−0.414286	−0.207143	+	0.978311i	$0.566417\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	−14.0000	−0.482472
$843$	10.0000	0.344418
$844$	−20.0000	−0.688428
$845$	−18.0000	−0.619219
$846$	−8.00000	−0.275046
$847$	0	0
$848$	10.0000	0.343401
$849$	4.00000	0.137280
$850$	−2.00000	−0.0685994
$851$	0	0
$852$	−8.00000	−0.274075
$853$	42.0000	1.43805	0.719026	−	0.694983i	$-0.244588\pi$
$853$	42.0000	1.43805	0.719026	+	0.694983i	$0.244588\pi$
$854$	0	0
$855$	16.0000	0.547188
$856$	12.0000	0.410152
$857$	38.0000	1.29806	0.649028	−	0.760765i	$-0.275176\pi$
$857$	38.0000	1.29806	0.649028	+	0.760765i	$0.275176\pi$
$858$	−8.00000	−0.273115
$859$	0	0		−	1.00000i	$-0.5\pi$
$859$	0	0			1.00000i	$0.5\pi$
$860$	−8.00000	−0.272798
$861$	0	0
$862$	−12.0000	−0.408722
$863$	44.0000	1.49778	0.748889	−	0.662696i	$-0.230588\pi$
$863$	44.0000	1.49778	0.748889	+	0.662696i	$0.230588\pi$
$864$	1.00000	0.0340207
$865$	4.00000	0.136004
$866$	38.0000	1.29129
$867$	13.0000	0.441503
$868$	0	0
$869$	64.0000	2.17105
$870$	4.00000	0.135613
$871$	0	0
$872$	2.00000	0.0677285
$873$	2.00000	0.0676897
$874$	0	0
$875$	0	0
$876$	14.0000	0.473016
$877$	38.0000	1.28317	0.641584	−	0.767052i	$-0.278277\pi$
$877$	38.0000	1.28317	0.641584	+	0.767052i	$0.278277\pi$
$878$	−8.00000	−0.269987
$879$	6.00000	0.202375
$880$	8.00000	0.269680
$881$	6.00000	0.202145	0.101073	−	0.994879i	$-0.467773\pi$
$881$	6.00000	0.202145	0.101073	+	0.994879i	$0.467773\pi$
$882$	7.00000	0.235702
$883$	−8.00000	−0.269221	−0.134611	−	0.990899i	$-0.542978\pi$
$883$	−8.00000	−0.269221	−0.134611	+	0.990899i	$0.542978\pi$
$884$	4.00000	0.134535
$885$	−24.0000	−0.806751
$886$	4.00000	0.134383
$887$	28.0000	0.940148	0.470074	−	0.882627i	$-0.344227\pi$
$887$	28.0000	0.940148	0.470074	+	0.882627i	$0.344227\pi$
$888$	2.00000	0.0671156
$889$	0	0
$890$	28.0000	0.938562
$891$	4.00000	0.134005
$892$	16.0000	0.535720
$893$	64.0000	2.14168
$894$	−6.00000	−0.200670
$895$	24.0000	0.802232
$896$	0	0
$897$	0	0
$898$	−14.0000	−0.467186
$899$	−16.0000	−0.533630
$900$	−1.00000	−0.0333333
$901$	−20.0000	−0.666297
$902$	−40.0000	−1.33185
$903$	0	0
$904$	−10.0000	−0.332595
$905$	−12.0000	−0.398893
$906$	−24.0000	−0.797347
$907$	−52.0000	−1.72663	−0.863316	−	0.504664i	$-0.831616\pi$
$907$	−52.0000	−1.72663	−0.863316	+	0.504664i	$0.831616\pi$
$908$	12.0000	0.398234
$909$	−6.00000	−0.199007
$910$	0	0
$911$	52.0000	1.72284	0.861418	−	0.507896i	$-0.169577\pi$
$911$	52.0000	1.72284	0.861418	+	0.507896i	$0.169577\pi$
$912$	−8.00000	−0.264906
$913$	48.0000	1.58857
$914$	6.00000	0.198462
$915$	−12.0000	−0.396708
$916$	2.00000	0.0660819
$917$	0	0
$918$	−2.00000	−0.0660098
$919$	−40.0000	−1.31948	−0.659739	−	0.751495i	$-0.729333\pi$
$919$	−40.0000	−1.31948	−0.659739	+	0.751495i	$0.729333\pi$
$920$	0	0
$921$	4.00000	0.131804
$922$	−18.0000	−0.592798
$923$	−16.0000	−0.526646
$924$	0	0
$925$	−2.00000	−0.0657596
$926$	32.0000	1.05159
$927$	16.0000	0.525509
$928$	−2.00000	−0.0656532
$929$	−6.00000	−0.196854	−0.0984268	−	0.995144i	$-0.531381\pi$
$929$	−6.00000	−0.196854	−0.0984268	+	0.995144i	$0.531381\pi$
$930$	−16.0000	−0.524661
$931$	−56.0000	−1.83533
$932$	−6.00000	−0.196537
$933$	−12.0000	−0.392862
$934$	28.0000	0.916188
$935$	−16.0000	−0.523256
$936$	2.00000	0.0653720
$937$	42.0000	1.37208	0.686040	−	0.727564i	$-0.259347\pi$
$937$	42.0000	1.37208	0.686040	+	0.727564i	$0.259347\pi$
$938$	0	0
$939$	−2.00000	−0.0652675
$940$	16.0000	0.521862
$941$	−14.0000	−0.456387	−0.228193	−	0.973616i	$-0.573282\pi$
$941$	−14.0000	−0.456387	−0.228193	+	0.973616i	$0.573282\pi$
$942$	−6.00000	−0.195491
$943$	0	0
$944$	12.0000	0.390567
$945$	0	0
$946$	16.0000	0.520205
$947$	60.0000	1.94974	0.974869	−	0.222779i	$-0.0715128\pi$
$947$	60.0000	1.94974	0.974869	+	0.222779i	$0.0715128\pi$
$948$	−16.0000	−0.519656
$949$	28.0000	0.908918
$950$	8.00000	0.259554
$951$	−2.00000	−0.0648544
$952$	0	0
$953$	−22.0000	−0.712650	−0.356325	−	0.934362i	$-0.615970\pi$
$953$	−22.0000	−0.712650	−0.356325	+	0.934362i	$0.615970\pi$
$954$	−10.0000	−0.323762
$955$	24.0000	0.776622
$956$	−28.0000	−0.905585
$957$	−8.00000	−0.258603
$958$	0	0
$959$	0	0
$960$	−2.00000	−0.0645497
$961$	33.0000	1.06452
$962$	4.00000	0.128965
$963$	−12.0000	−0.386695
$964$	2.00000	0.0644157
$965$	4.00000	0.128765
$966$	0	0
$967$	0	0		−	1.00000i	$-0.5\pi$
$967$	0	0			1.00000i	$0.5\pi$
$968$	−5.00000	−0.160706
$969$	16.0000	0.513994
$970$	−4.00000	−0.128432
$971$	28.0000	0.898563	0.449281	−	0.893390i	$-0.351680\pi$
$971$	28.0000	0.898563	0.449281	+	0.893390i	$0.351680\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	32.0000	1.02535
$975$	−2.00000	−0.0640513
$976$	6.00000	0.192055
$977$	−6.00000	−0.191957	−0.0959785	−	0.995383i	$-0.530598\pi$
$977$	−6.00000	−0.191957	−0.0959785	+	0.995383i	$0.530598\pi$
$978$	−16.0000	−0.511624
$979$	−56.0000	−1.78977
$980$	−14.0000	−0.447214
$981$	−2.00000	−0.0638551
$982$	12.0000	0.382935
$983$	56.0000	1.78612	0.893061	−	0.449935i	$-0.148553\pi$
$983$	56.0000	1.78612	0.893061	+	0.449935i	$0.148553\pi$
$984$	10.0000	0.318788
$985$	−12.0000	−0.382352
$986$	4.00000	0.127386
$987$	0	0
$988$	−16.0000	−0.509028
$989$	0	0
$990$	−8.00000	−0.254257
$991$	8.00000	0.254128	0.127064	−	0.991894i	$-0.459445\pi$
$991$	8.00000	0.254128	0.127064	+	0.991894i	$0.459445\pi$
$992$	8.00000	0.254000
$993$	16.0000	0.507745
$994$	0	0
$995$	0	0
$996$	−12.0000	−0.380235
$997$	22.0000	0.696747	0.348373	−	0.937356i	$-0.386734\pi$
$997$	22.0000	0.696747	0.348373	+	0.937356i	$0.386734\pi$
$998$	40.0000	1.26618
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	786.2.a.d.1.1	✓	1
3.2	odd	2		2358.2.a.m.1.1		1
4.3	odd	2		6288.2.a.n.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
786.2.a.d.1.1	✓	1	1.1	even	1	trivial
2358.2.a.m.1.1		1	3.2	odd	2
6288.2.a.n.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:	\( N \)	\(=\)	\( 786 = 2 \cdot 3 \cdot 131 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	786.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more