LMFDB - Embedded newform 783.2.a.j.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [783,2,Mod(1,783)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(783, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("783.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 783 = 3^{3} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	783.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,1,0,11] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	yes
Analytic conductor:	$6.25228647827$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 13x^{6} + 11x^{5} + 49x^{4} - 37x^{3} - 43x^{2} + 35x - 3 $ x^8 - x^7 - 13x^6 + 11x^5 + 49x^4 - 37x^3 - 43x^2 + 35x - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.19689$ of defining polynomial
Character	$\chi$	$=$	783.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.19689 q^{2} -0.567458 q^{4} -0.563409 q^{5} -0.343748 q^{7} +3.07296 q^{8} +0.674338 q^{10} -3.12248 q^{11} +4.44814 q^{13} +0.411428 q^{14} -2.54308 q^{16} -7.44614 q^{17} +3.15603 q^{19} +0.319711 q^{20} +3.73726 q^{22} +3.69833 q^{23} -4.68257 q^{25} -5.32393 q^{26} +0.195063 q^{28} -1.00000 q^{29} +9.66146 q^{31} -3.10214 q^{32} +8.91219 q^{34} +0.193671 q^{35} +9.34175 q^{37} -3.77742 q^{38} -1.73133 q^{40} +8.00721 q^{41} -0.957186 q^{43} +1.77187 q^{44} -4.42648 q^{46} +2.49817 q^{47} -6.88184 q^{49} +5.60451 q^{50} -2.52413 q^{52} +10.0552 q^{53} +1.75923 q^{55} -1.05632 q^{56} +1.19689 q^{58} +6.40694 q^{59} +13.9651 q^{61} -11.5637 q^{62} +8.79907 q^{64} -2.50612 q^{65} -11.0168 q^{67} +4.22537 q^{68} -0.231803 q^{70} +9.26894 q^{71} +8.09644 q^{73} -11.1810 q^{74} -1.79091 q^{76} +1.07335 q^{77} -8.37007 q^{79} +1.43279 q^{80} -9.58373 q^{82} +3.27202 q^{83} +4.19522 q^{85} +1.14565 q^{86} -9.59525 q^{88} -5.67667 q^{89} -1.52904 q^{91} -2.09864 q^{92} -2.99003 q^{94} -1.77814 q^{95} +6.01316 q^{97} +8.23679 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + q^{2} + 11 q^{4} - 2 q^{5} + 5 q^{7} + 3 q^{8} + 7 q^{10} - 2 q^{11} + 11 q^{13} + 8 q^{14} + 21 q^{16} + 6 q^{17} + 5 q^{19} - 27 q^{20} + 17 q^{22} - 4 q^{23} + 30 q^{25} + 16 q^{26} + 2 q^{28}+ \cdots - 11 q^{98}+O(q^{100}) $ 8 * q + q^2 + 11 * q^4 - 2 * q^5 + 5 * q^7 + 3 * q^8 + 7 * q^10 - 2 * q^11 + 11 * q^13 + 8 * q^14 + 21 * q^16 + 6 * q^17 + 5 * q^19 - 27 * q^20 + 17 * q^22 - 4 * q^23 + 30 * q^25 + 16 * q^26 + 2 * q^28 - 8 * q^29 - q^31 + 37 * q^32 - 9 * q^34 + q^35 + 26 * q^37 - 14 * q^38 + 2 * q^40 + 19 * q^41 + 16 * q^43 - q^44 - 15 * q^46 - 18 * q^47 + 27 * q^49 + 4 * q^50 + 7 * q^52 - 19 * q^53 + 6 * q^55 - 27 * q^56 - q^58 + 13 * q^59 + 2 * q^61 + 7 * q^62 + 13 * q^64 - 19 * q^65 + 25 * q^67 + 22 * q^68 - 22 * q^70 - 7 * q^71 + 11 * q^73 - 9 * q^74 - 31 * q^76 + 26 * q^77 + 16 * q^79 - 19 * q^80 + 7 * q^82 + 19 * q^83 + 10 * q^85 + 63 * q^86 + 32 * q^88 - 16 * q^89 + 6 * q^91 - 101 * q^92 - 34 * q^94 + 43 * q^95 + 31 * q^97 - 11 * q^98

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 $

$2$	−1.19689	−0.846328	−0.423164	−	0.906053i	$-0.639081\pi$
$2$	−1.19689	−0.846328	−0.423164	+	0.906053i	$0.639081\pi$
$3$	0	0
$3$	0	0
$4$	−0.567458	−0.283729
$5$	−0.563409	−0.251964	−0.125982	−	0.992033i	$-0.540208\pi$
$5$	−0.563409	−0.251964	−0.125982	+	0.992033i	$0.540208\pi$
$6$	0	0
$7$	−0.343748	−0.129925	−0.0649623	−	0.997888i	$-0.520693\pi$
$7$	−0.343748	−0.129925	−0.0649623	+	0.997888i	$0.520693\pi$
$8$	3.07296	1.08646
$9$	0	0
$10$	0.674338	0.213244
$11$	−3.12248	−0.941462	−0.470731	−	0.882277i	$-0.656010\pi$
$11$	−3.12248	−0.941462	−0.470731	+	0.882277i	$0.656010\pi$
$12$	0	0
$13$	4.44814	1.23369	0.616846	−	0.787084i	$-0.288410\pi$
$13$	4.44814	1.23369	0.616846	+	0.787084i	$0.288410\pi$
$14$	0.411428	0.109959
$15$	0	0
$16$	−2.54308	−0.635769
$17$	−7.44614	−1.80595	−0.902977	−	0.429690i	$-0.858623\pi$
$17$	−7.44614	−1.80595	−0.902977	+	0.429690i	$0.858623\pi$
$18$	0	0
$19$	3.15603	0.724043	0.362021	−	0.932170i	$-0.382087\pi$
$19$	3.15603	0.724043	0.362021	+	0.932170i	$0.382087\pi$
$20$	0.319711	0.0714896
$21$	0	0
$22$	3.73726	0.796786
$23$	3.69833	0.771154	0.385577	−	0.922676i	$-0.374002\pi$
$23$	3.69833	0.771154	0.385577	+	0.922676i	$0.374002\pi$
$24$	0	0
$25$	−4.68257	−0.936514
$26$	−5.32393	−1.04411
$27$	0	0
$28$	0.195063	0.0368634
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	9.66146	1.73525	0.867624	−	0.497220i	$-0.165646\pi$
$31$	9.66146	1.73525	0.867624	+	0.497220i	$0.165646\pi$
$32$	−3.10214	−0.548387
$33$	0	0
$34$	8.91219	1.52843
$35$	0.193671	0.0327364
$36$	0	0
$37$	9.34175	1.53577	0.767887	−	0.640586i	$-0.221308\pi$
$37$	9.34175	1.53577	0.767887	+	0.640586i	$0.221308\pi$
$38$	−3.77742	−0.612778
$39$	0	0
$40$	−1.73133	−0.273748
$41$	8.00721	1.25052	0.625258	−	0.780418i	$-0.284994\pi$
$41$	8.00721	1.25052	0.625258	+	0.780418i	$0.284994\pi$
$42$	0	0
$43$	−0.957186	−0.145970	−0.0729848	−	0.997333i	$-0.523252\pi$
$43$	−0.957186	−0.145970	−0.0729848	+	0.997333i	$0.523252\pi$
$44$	1.77187	0.267120
$45$	0	0
$46$	−4.42648	−0.652649
$47$	2.49817	0.364395	0.182198	−	0.983262i	$-0.441679\pi$
$47$	2.49817	0.364395	0.182198	+	0.983262i	$0.441679\pi$
$48$	0	0
$49$	−6.88184	−0.983120
$50$	5.60451	0.792598
$51$	0	0
$52$	−2.52413	−0.350034
$53$	10.0552	1.38119	0.690596	−	0.723240i	$-0.257348\pi$
$53$	10.0552	1.38119	0.690596	+	0.723240i	$0.257348\pi$
$54$	0	0
$55$	1.75923	0.237215
$56$	−1.05632	−0.141157
$57$	0	0
$58$	1.19689	0.157159
$59$	6.40694	0.834112	0.417056	−	0.908881i	$-0.363062\pi$
$59$	6.40694	0.834112	0.417056	+	0.908881i	$0.363062\pi$
$60$	0	0
$61$	13.9651	1.78805	0.894025	−	0.448017i	$-0.147870\pi$
$61$	13.9651	1.78805	0.894025	+	0.448017i	$0.147870\pi$
$62$	−11.5637	−1.46859
$63$	0	0
$64$	8.79907	1.09988
$65$	−2.50612	−0.310846
$66$	0	0
$67$	−11.0168	−1.34591	−0.672955	−	0.739683i	$-0.734975\pi$
$67$	−11.0168	−1.34591	−0.672955	+	0.739683i	$0.734975\pi$
$68$	4.22537	0.512401
$69$	0	0
$70$	−0.231803	−0.0277057
$71$	9.26894	1.10002	0.550011	−	0.835158i	$-0.314624\pi$
$71$	9.26894	1.10002	0.550011	+	0.835158i	$0.314624\pi$
$72$	0	0
$73$	8.09644	0.947616	0.473808	−	0.880628i	$-0.342879\pi$
$73$	8.09644	0.947616	0.473808	+	0.880628i	$0.342879\pi$
$74$	−11.1810	−1.29977
$75$	0	0
$76$	−1.79091	−0.205432
$77$	1.07335	0.122319
$78$	0	0
$79$	−8.37007	−0.941706	−0.470853	−	0.882212i	$-0.656054\pi$
$79$	−8.37007	−0.941706	−0.470853	+	0.882212i	$0.656054\pi$
$80$	1.43279	0.160191
$81$	0	0
$82$	−9.58373	−1.05835
$83$	3.27202	0.359151	0.179575	−	0.983744i	$-0.442528\pi$
$83$	3.27202	0.359151	0.179575	+	0.983744i	$0.442528\pi$
$84$	0	0
$85$	4.19522	0.455036
$86$	1.14565	0.123538
$87$	0	0
$88$	−9.59525	−1.02286
$89$	−5.67667	−0.601726	−0.300863	−	0.953667i	$-0.597275\pi$
$89$	−5.67667	−0.601726	−0.300863	+	0.953667i	$0.597275\pi$
$90$	0	0
$91$	−1.52904	−0.160287
$92$	−2.09864	−0.218799
$93$	0	0
$94$	−2.99003	−0.308398
$95$	−1.77814	−0.182433
$96$	0	0
$97$	6.01316	0.610544	0.305272	−	0.952265i	$-0.401253\pi$
$97$	6.01316	0.610544	0.305272	+	0.952265i	$0.401253\pi$
$98$	8.23679	0.832042
$99$	0	0
$100$	2.65716	0.265716
$101$	−1.66941	−0.166113	−0.0830563	−	0.996545i	$-0.526468\pi$
$101$	−1.66941	−0.166113	−0.0830563	+	0.996545i	$0.526468\pi$
$102$	0	0
$103$	6.49460	0.639932	0.319966	−	0.947429i	$-0.396329\pi$
$103$	6.49460	0.639932	0.319966	+	0.947429i	$0.396329\pi$
$104$	13.6690	1.34035
$105$	0	0
$106$	−12.0350	−1.16894
$107$	−7.97526	−0.770998	−0.385499	−	0.922708i	$-0.625971\pi$
$107$	−7.97526	−0.770998	−0.385499	+	0.922708i	$0.625971\pi$
$108$	0	0
$109$	14.8682	1.42412	0.712059	−	0.702119i	$-0.247763\pi$
$109$	14.8682	1.42412	0.712059	+	0.702119i	$0.247763\pi$
$110$	−2.10561	−0.200762
$111$	0	0
$112$	0.874178	0.0826020
$113$	11.1400	1.04796	0.523981	−	0.851730i	$-0.324446\pi$
$113$	11.1400	1.04796	0.523981	+	0.851730i	$0.324446\pi$
$114$	0	0
$115$	−2.08367	−0.194303
$116$	0.567458	0.0526871
$117$	0	0
$118$	−7.66839	−0.705932
$119$	2.55960	0.234638
$120$	0	0
$121$	−1.25013	−0.113649
$122$	−16.7147	−1.51328
$123$	0	0
$124$	−5.48247	−0.492340
$125$	5.45525	0.487932
$126$	0	0
$127$	−6.75722	−0.599607	−0.299803	−	0.954001i	$-0.596921\pi$
$127$	−6.75722	−0.599607	−0.299803	+	0.954001i	$0.596921\pi$
$128$	−4.32722	−0.382476
$129$	0	0
$130$	2.99955	0.263078
$131$	−9.69793	−0.847312	−0.423656	−	0.905823i	$-0.639253\pi$
$131$	−9.69793	−0.847312	−0.423656	+	0.905823i	$0.639253\pi$
$132$	0	0
$133$	−1.08488	−0.0940710
$134$	13.1858	1.13908
$135$	0	0
$136$	−22.8817	−1.96209
$137$	2.85279	0.243731	0.121865	−	0.992547i	$-0.461112\pi$
$137$	2.85279	0.243731	0.121865	+	0.992547i	$0.461112\pi$
$138$	0	0
$139$	3.87677	0.328824	0.164412	−	0.986392i	$-0.447427\pi$
$139$	3.87677	0.328824	0.164412	+	0.986392i	$0.447427\pi$
$140$	−0.109900	−0.00928825
$141$	0	0
$142$	−11.0939	−0.930979
$143$	−13.8892	−1.16147
$144$	0	0
$145$	0.563409	0.0467886
$146$	−9.69053	−0.801994
$147$	0	0
$148$	−5.30105	−0.435743
$149$	20.6157	1.68891	0.844454	−	0.535629i	$-0.179925\pi$
$149$	20.6157	1.68891	0.844454	+	0.535629i	$0.179925\pi$
$150$	0	0
$151$	−17.3083	−1.40853	−0.704264	−	0.709938i	$-0.748723\pi$
$151$	−17.3083	−1.40853	−0.704264	+	0.709938i	$0.748723\pi$
$152$	9.69836	0.786641
$153$	0	0
$154$	−1.28468	−0.103522
$155$	−5.44335	−0.437221
$156$	0	0
$157$	−13.5590	−1.08213	−0.541065	−	0.840981i	$-0.681979\pi$
$157$	−13.5590	−1.08213	−0.541065	+	0.840981i	$0.681979\pi$
$158$	10.0180	0.796992
$159$	0	0
$160$	1.74778	0.138174
$161$	−1.27129	−0.100192
$162$	0	0
$163$	5.62357	0.440472	0.220236	−	0.975447i	$-0.429317\pi$
$163$	5.62357	0.440472	0.220236	+	0.975447i	$0.429317\pi$
$164$	−4.54375	−0.354808
$165$	0	0
$166$	−3.91624	−0.303959
$167$	14.3289	1.10880	0.554400	−	0.832250i	$-0.312948\pi$
$167$	14.3289	1.10880	0.554400	+	0.832250i	$0.312948\pi$
$168$	0	0
$169$	6.78594	0.521996
$170$	−5.02121	−0.385109
$171$	0	0
$172$	0.543163	0.0414158
$173$	−1.38627	−0.105396	−0.0526979	−	0.998610i	$-0.516782\pi$
$173$	−1.38627	−0.105396	−0.0526979	+	0.998610i	$0.516782\pi$
$174$	0	0
$175$	1.60963	0.121676
$176$	7.94070	0.598552
$177$	0	0
$178$	6.79434	0.509257
$179$	−19.4439	−1.45331	−0.726654	−	0.687004i	$-0.758926\pi$
$179$	−19.4439	−1.45331	−0.726654	+	0.687004i	$0.758926\pi$
$180$	0	0
$181$	−6.89110	−0.512211	−0.256106	−	0.966649i	$-0.582440\pi$
$181$	−6.89110	−0.512211	−0.256106	+	0.966649i	$0.582440\pi$
$182$	1.83009	0.135655
$183$	0	0
$184$	11.3648	0.837825
$185$	−5.26323	−0.386960
$186$	0	0
$187$	23.2504	1.70024
$188$	−1.41761	−0.103389
$189$	0	0
$190$	2.12823	0.154398
$191$	13.7672	0.996157	0.498078	−	0.867132i	$-0.334039\pi$
$191$	13.7672	0.996157	0.498078	+	0.867132i	$0.334039\pi$
$192$	0	0
$193$	−7.15801	−0.515245	−0.257623	−	0.966246i	$-0.582939\pi$
$193$	−7.15801	−0.515245	−0.257623	+	0.966246i	$0.582939\pi$
$194$	−7.19708	−0.516720
$195$	0	0
$196$	3.90515	0.278940
$197$	−7.61071	−0.542241	−0.271120	−	0.962545i	$-0.587394\pi$
$197$	−7.61071	−0.542241	−0.271120	+	0.962545i	$0.587394\pi$
$198$	0	0
$199$	−20.6993	−1.46734	−0.733668	−	0.679508i	$-0.762193\pi$
$199$	−20.6993	−1.46734	−0.733668	+	0.679508i	$0.762193\pi$
$200$	−14.3894	−1.01748
$201$	0	0
$202$	1.99810	0.140586
$203$	0.343748	0.0241264
$204$	0	0
$205$	−4.51133	−0.315085
$206$	−7.77331	−0.541592
$207$	0	0
$208$	−11.3120	−0.784343
$209$	−9.85463	−0.681659
$210$	0	0
$211$	−16.0184	−1.10275	−0.551375	−	0.834257i	$-0.685897\pi$
$211$	−16.0184	−1.10275	−0.551375	+	0.834257i	$0.685897\pi$
$212$	−5.70592	−0.391884
$213$	0	0
$214$	9.54550	0.652517
$215$	0.539288	0.0367791
$216$	0	0
$217$	−3.32111	−0.225452
$218$	−17.7956	−1.20527
$219$	0	0
$220$	−0.998291	−0.0673047
$221$	−33.1215	−2.22799
$222$	0	0
$223$	0.779025	0.0521674	0.0260837	−	0.999660i	$-0.491696\pi$
$223$	0.779025	0.0521674	0.0260837	+	0.999660i	$0.491696\pi$
$224$	1.06636	0.0712489
$225$	0	0
$226$	−13.3333	−0.886919
$227$	−7.29804	−0.484388	−0.242194	−	0.970228i	$-0.577867\pi$
$227$	−7.29804	−0.484388	−0.242194	+	0.970228i	$0.577867\pi$
$228$	0	0
$229$	−6.91007	−0.456631	−0.228315	−	0.973587i	$-0.573322\pi$
$229$	−6.91007	−0.456631	−0.228315	+	0.973587i	$0.573322\pi$
$230$	2.49392	0.164444
$231$	0	0
$232$	−3.07296	−0.201750
$233$	−8.83891	−0.579056	−0.289528	−	0.957170i	$-0.593498\pi$
$233$	−8.83891	−0.579056	−0.289528	+	0.957170i	$0.593498\pi$
$234$	0	0
$235$	−1.40749	−0.0918146
$236$	−3.63567	−0.236662
$237$	0	0
$238$	−3.06355	−0.198581
$239$	−29.8678	−1.93199	−0.965994	−	0.258566i	$-0.916750\pi$
$239$	−29.8678	−1.93199	−0.965994	+	0.258566i	$0.916750\pi$
$240$	0	0
$241$	14.0103	0.902481	0.451240	−	0.892402i	$-0.350982\pi$
$241$	14.0103	0.902481	0.451240	+	0.892402i	$0.350982\pi$
$242$	1.49627	0.0961840
$243$	0	0
$244$	−7.92462	−0.507321
$245$	3.87729	0.247711
$246$	0	0
$247$	14.0385	0.893246
$248$	29.6893	1.88527
$249$	0	0
$250$	−6.52933	−0.412951
$251$	17.3119	1.09272	0.546358	−	0.837552i	$-0.316014\pi$
$251$	17.3119	1.09272	0.546358	+	0.837552i	$0.316014\pi$
$252$	0	0
$253$	−11.5479	−0.726013
$254$	8.08764	0.507464
$255$	0	0
$256$	−12.4189	−0.776184
$257$	−2.08186	−0.129863	−0.0649314	−	0.997890i	$-0.520683\pi$
$257$	−2.08186	−0.129863	−0.0649314	+	0.997890i	$0.520683\pi$
$258$	0	0
$259$	−3.21121	−0.199535
$260$	1.42212	0.0881961
$261$	0	0
$262$	11.6073	0.717104
$263$	25.1625	1.55159	0.775794	−	0.630986i	$-0.217349\pi$
$263$	25.1625	1.55159	0.775794	+	0.630986i	$0.217349\pi$
$264$	0	0
$265$	−5.66521	−0.348011
$266$	1.29848	0.0796149
$267$	0	0
$268$	6.25154	0.381874
$269$	17.1141	1.04347	0.521733	−	0.853109i	$-0.325286\pi$
$269$	17.1141	1.04347	0.521733	+	0.853109i	$0.325286\pi$
$270$	0	0
$271$	1.37081	0.0832710	0.0416355	−	0.999133i	$-0.486743\pi$
$271$	1.37081	0.0832710	0.0416355	+	0.999133i	$0.486743\pi$
$272$	18.9361	1.14817
$273$	0	0
$274$	−3.41448	−0.206276
$275$	14.6212	0.881693
$276$	0	0
$277$	28.4761	1.71097	0.855483	−	0.517831i	$-0.173260\pi$
$277$	28.4761	1.71097	0.855483	+	0.517831i	$0.173260\pi$
$278$	−4.64007	−0.278293
$279$	0	0
$280$	0.595143	0.0355666
$281$	−20.2714	−1.20929	−0.604645	−	0.796495i	$-0.706685\pi$
$281$	−20.2714	−1.20929	−0.604645	+	0.796495i	$0.706685\pi$
$282$	0	0
$283$	−6.84558	−0.406928	−0.203464	−	0.979082i	$-0.565220\pi$
$283$	−6.84558	−0.406928	−0.203464	+	0.979082i	$0.565220\pi$
$284$	−5.25973	−0.312108
$285$	0	0
$286$	16.6238	0.982988
$287$	−2.75246	−0.162473
$288$	0	0
$289$	38.4449	2.26147
$290$	−0.674338	−0.0395985
$291$	0	0
$292$	−4.59439	−0.268866
$293$	−14.5425	−0.849583	−0.424792	−	0.905291i	$-0.639653\pi$
$293$	−14.5425	−0.849583	−0.424792	+	0.905291i	$0.639653\pi$
$294$	0	0
$295$	−3.60973	−0.210166
$296$	28.7068	1.66855
$297$	0	0
$298$	−24.6747	−1.42937
$299$	16.4507	0.951367
$300$	0	0
$301$	0.329031	0.0189650
$302$	20.7161	1.19208
$303$	0	0
$304$	−8.02602	−0.460324
$305$	−7.86807	−0.450525
$306$	0	0
$307$	34.2876	1.95690	0.978448	−	0.206493i	$-0.0662051\pi$
$307$	34.2876	1.95690	0.978448	+	0.206493i	$0.0662051\pi$
$308$	−0.609079	−0.0347055
$309$	0	0
$310$	6.51509	0.370032
$311$	−5.01658	−0.284464	−0.142232	−	0.989833i	$-0.545428\pi$
$311$	−5.01658	−0.284464	−0.142232	+	0.989833i	$0.545428\pi$
$312$	0	0
$313$	−14.5131	−0.820327	−0.410163	−	0.912012i	$-0.634528\pi$
$313$	−14.5131	−0.820327	−0.410163	+	0.912012i	$0.634528\pi$
$314$	16.2287	0.915837
$315$	0	0
$316$	4.74966	0.267189
$317$	12.3715	0.694851	0.347426	−	0.937708i	$-0.387056\pi$
$317$	12.3715	0.694851	0.347426	+	0.937708i	$0.387056\pi$
$318$	0	0
$319$	3.12248	0.174825
$320$	−4.95748	−0.277131
$321$	0	0
$322$	1.52160	0.0847952
$323$	−23.5002	−1.30759
$324$	0	0
$325$	−20.8287	−1.15537
$326$	−6.73078	−0.372784
$327$	0	0
$328$	24.6058	1.35863
$329$	−0.858741	−0.0473439
$330$	0	0
$331$	−26.8869	−1.47784	−0.738919	−	0.673794i	$-0.764663\pi$
$331$	−26.8869	−1.47784	−0.738919	+	0.673794i	$0.764663\pi$
$332$	−1.85673	−0.101901
$333$	0	0
$334$	−17.1501	−0.938409
$335$	6.20694	0.339121
$336$	0	0
$337$	34.9709	1.90499	0.952493	−	0.304559i	$-0.0985091\pi$
$337$	34.9709	1.90499	0.952493	+	0.304559i	$0.0985091\pi$
$338$	−8.12202	−0.441780
$339$	0	0
$340$	−2.38061	−0.129107
$341$	−30.1677	−1.63367
$342$	0	0
$343$	4.77186	0.257656
$344$	−2.94140	−0.158589
$345$	0	0
$346$	1.65921	0.0891994
$347$	15.2222	0.817173	0.408587	−	0.912720i	$-0.366022\pi$
$347$	15.2222	0.817173	0.408587	+	0.912720i	$0.366022\pi$
$348$	0	0
$349$	2.15850	0.115542	0.0577709	−	0.998330i	$-0.481601\pi$
$349$	2.15850	0.115542	0.0577709	+	0.998330i	$0.481601\pi$
$350$	−1.92654	−0.102978
$351$	0	0
$352$	9.68637	0.516285
$353$	19.6993	1.04849	0.524243	−	0.851569i	$-0.324348\pi$
$353$	19.6993	1.04849	0.524243	+	0.851569i	$0.324348\pi$
$354$	0	0
$355$	−5.22221	−0.277166
$356$	3.22127	0.170727
$357$	0	0
$358$	23.2722	1.22998
$359$	17.7305	0.935782	0.467891	−	0.883786i	$-0.345014\pi$
$359$	17.7305	0.935782	0.467891	+	0.883786i	$0.345014\pi$
$360$	0	0
$361$	−9.03948	−0.475762
$362$	8.24788	0.433499
$363$	0	0
$364$	0.867666	0.0454780
$365$	−4.56161	−0.238765
$366$	0	0
$367$	8.11400	0.423547	0.211774	−	0.977319i	$-0.432076\pi$
$367$	8.11400	0.423547	0.211774	+	0.977319i	$0.432076\pi$
$368$	−9.40512	−0.490276
$369$	0	0
$370$	6.29949	0.327495
$371$	−3.45647	−0.179451
$372$	0	0
$373$	−2.18932	−0.113359	−0.0566793	−	0.998392i	$-0.518051\pi$
$373$	−2.18932	−0.113359	−0.0566793	+	0.998392i	$0.518051\pi$
$374$	−27.8281	−1.43896
$375$	0	0
$376$	7.67677	0.395899
$377$	−4.44814	−0.229091
$378$	0	0
$379$	0.629514	0.0323359	0.0161680	−	0.999869i	$-0.494853\pi$
$379$	0.629514	0.0323359	0.0161680	+	0.999869i	$0.494853\pi$
$380$	1.00902	0.0517615
$381$	0	0
$382$	−16.4778	−0.843075
$383$	9.97756	0.509830	0.254915	−	0.966964i	$-0.417953\pi$
$383$	9.97756	0.509830	0.254915	+	0.966964i	$0.417953\pi$
$384$	0	0
$385$	−0.604733	−0.0308201
$386$	8.56734	0.436066
$387$	0	0
$388$	−3.41221	−0.173229
$389$	27.6562	1.40222	0.701112	−	0.713051i	$-0.252687\pi$
$389$	27.6562	1.40222	0.701112	+	0.713051i	$0.252687\pi$
$390$	0	0
$391$	−27.5382	−1.39267
$392$	−21.1476	−1.06812
$393$	0	0
$394$	9.10918	0.458914
$395$	4.71577	0.237276
$396$	0	0
$397$	15.4661	0.776222	0.388111	−	0.921613i	$-0.373128\pi$
$397$	15.4661	0.776222	0.388111	+	0.921613i	$0.373128\pi$
$398$	24.7748	1.24185
$399$	0	0
$400$	11.9081	0.595406
$401$	10.9366	0.546145	0.273073	−	0.961993i	$-0.411960\pi$
$401$	10.9366	0.546145	0.273073	+	0.961993i	$0.411960\pi$
$402$	0	0
$403$	42.9755	2.14076
$404$	0.947320	0.0471309
$405$	0	0
$406$	−0.411428	−0.0204188
$407$	−29.1694	−1.44587
$408$	0	0
$409$	36.5030	1.80496	0.902479	−	0.430734i	$-0.141745\pi$
$409$	36.5030	1.80496	0.902479	+	0.430734i	$0.141745\pi$
$410$	5.39956	0.266665
$411$	0	0
$412$	−3.68541	−0.181567
$413$	−2.20237	−0.108372
$414$	0	0
$415$	−1.84349	−0.0904932
$416$	−13.7988	−0.676540
$417$	0	0
$418$	11.7949	0.576907
$419$	−15.2539	−0.745202	−0.372601	−	0.927992i	$-0.621534\pi$
$419$	−15.2539	−0.745202	−0.372601	+	0.927992i	$0.621534\pi$
$420$	0	0
$421$	0.529285	0.0257958	0.0128979	−	0.999917i	$-0.495894\pi$
$421$	0.529285	0.0257958	0.0128979	+	0.999917i	$0.495894\pi$
$422$	19.1722	0.933288
$423$	0	0
$424$	30.8993	1.50060
$425$	34.8671	1.69130
$426$	0	0
$427$	−4.80048	−0.232312
$428$	4.52563	0.218754
$429$	0	0
$430$	−0.645467	−0.0311272
$431$	17.8969	0.862062	0.431031	−	0.902337i	$-0.358150\pi$
$431$	17.8969	0.862062	0.431031	+	0.902337i	$0.358150\pi$
$432$	0	0
$433$	−1.20711	−0.0580098	−0.0290049	−	0.999579i	$-0.509234\pi$
$433$	−1.20711	−0.0580098	−0.0290049	+	0.999579i	$0.509234\pi$
$434$	3.97500	0.190806
$435$	0	0
$436$	−8.43710	−0.404064
$437$	11.6720	0.558349
$438$	0	0
$439$	−31.5608	−1.50631	−0.753157	−	0.657841i	$-0.771470\pi$
$439$	−31.5608	−1.50631	−0.753157	+	0.657841i	$0.771470\pi$
$440$	5.40605	0.257723
$441$	0	0
$442$	39.6427	1.88561
$443$	3.11586	0.148039	0.0740196	−	0.997257i	$-0.476417\pi$
$443$	3.11586	0.148039	0.0740196	+	0.997257i	$0.476417\pi$
$444$	0	0
$445$	3.19829	0.151613
$446$	−0.932406	−0.0441507
$447$	0	0
$448$	−3.02467	−0.142902
$449$	−28.6953	−1.35421	−0.677107	−	0.735884i	$-0.736767\pi$
$449$	−28.6953	−1.35421	−0.677107	+	0.735884i	$0.736767\pi$
$450$	0	0
$451$	−25.0023	−1.17731
$452$	−6.32147	−0.297337
$453$	0	0
$454$	8.73494	0.409951
$455$	0.861475	0.0403866
$456$	0	0
$457$	19.6815	0.920660	0.460330	−	0.887748i	$-0.347731\pi$
$457$	19.6815	0.920660	0.460330	+	0.887748i	$0.347731\pi$
$458$	8.27059	0.386459
$459$	0	0
$460$	1.18240	0.0551295
$461$	−27.5053	−1.28105	−0.640526	−	0.767937i	$-0.721283\pi$
$461$	−27.5053	−1.28105	−0.640526	+	0.767937i	$0.721283\pi$
$462$	0	0
$463$	−11.0266	−0.512450	−0.256225	−	0.966617i	$-0.582479\pi$
$463$	−11.0266	−0.512450	−0.256225	+	0.966617i	$0.582479\pi$
$464$	2.54308	0.118059
$465$	0	0
$466$	10.5792	0.490071
$467$	11.9937	0.555001	0.277501	−	0.960725i	$-0.410494\pi$
$467$	11.9937	0.555001	0.277501	+	0.960725i	$0.410494\pi$
$468$	0	0
$469$	3.78699	0.174867
$470$	1.68461	0.0777053
$471$	0	0
$472$	19.6883	0.906226
$473$	2.98879	0.137425
$474$	0	0
$475$	−14.7783	−0.678076
$476$	−1.45246	−0.0665735
$477$	0	0
$478$	35.7484	1.63509
$479$	−18.5329	−0.846792	−0.423396	−	0.905945i	$-0.639162\pi$
$479$	−18.5329	−0.846792	−0.423396	+	0.905945i	$0.639162\pi$
$480$	0	0
$481$	41.5534	1.89467
$482$	−16.7687	−0.763795
$483$	0	0
$484$	0.709399	0.0322454
$485$	−3.38787	−0.153835
$486$	0	0
$487$	2.63814	0.119545	0.0597727	−	0.998212i	$-0.480962\pi$
$487$	2.63814	0.119545	0.0597727	+	0.998212i	$0.480962\pi$
$488$	42.9143	1.94264
$489$	0	0
$490$	−4.64068	−0.209645
$491$	−35.7876	−1.61507	−0.807535	−	0.589819i	$-0.799199\pi$
$491$	−35.7876	−1.61507	−0.807535	+	0.589819i	$0.799199\pi$
$492$	0	0
$493$	7.44614	0.335357
$494$	−16.8025	−0.755979
$495$	0	0
$496$	−24.5698	−1.10322
$497$	−3.18618	−0.142920
$498$	0	0
$499$	−26.7020	−1.19535	−0.597673	−	0.801740i	$-0.703908\pi$
$499$	−26.7020	−1.19535	−0.597673	+	0.801740i	$0.703908\pi$
$500$	−3.09562	−0.138441
$501$	0	0
$502$	−20.7204	−0.924796
$503$	23.8994	1.06562	0.532810	−	0.846235i	$-0.321136\pi$
$503$	23.8994	1.06562	0.532810	+	0.846235i	$0.321136\pi$
$504$	0	0
$505$	0.940561	0.0418544
$506$	13.8216	0.614445
$507$	0	0
$508$	3.83444	0.170126
$509$	5.67013	0.251324	0.125662	−	0.992073i	$-0.459894\pi$
$509$	5.67013	0.251324	0.125662	+	0.992073i	$0.459894\pi$
$510$	0	0
$511$	−2.78314	−0.123119
$512$	23.5185	1.03938
$513$	0	0
$514$	2.49175	0.109907
$515$	−3.65912	−0.161240
$516$	0	0
$517$	−7.80047	−0.343064
$518$	3.84346	0.168872
$519$	0	0
$520$	−7.70122	−0.337721
$521$	−27.0996	−1.18726	−0.593628	−	0.804739i	$-0.702305\pi$
$521$	−27.0996	−1.18726	−0.593628	+	0.804739i	$0.702305\pi$
$522$	0	0
$523$	−31.7417	−1.38797	−0.693985	−	0.719990i	$-0.744146\pi$
$523$	−31.7417	−1.38797	−0.693985	+	0.719990i	$0.744146\pi$
$524$	5.50317	0.240407
$525$	0	0
$526$	−30.1168	−1.31315
$527$	−71.9405	−3.13378
$528$	0	0
$529$	−9.32239	−0.405321
$530$	6.78063	0.294532
$531$	0	0
$532$	0.615624	0.0266907
$533$	35.6172	1.54275
$534$	0	0
$535$	4.49334	0.194264
$536$	−33.8541	−1.46227
$537$	0	0
$538$	−20.4837	−0.883115
$539$	21.4884	0.925570
$540$	0	0
$541$	10.9163	0.469329	0.234665	−	0.972076i	$-0.424601\pi$
$541$	10.9163	0.469329	0.234665	+	0.972076i	$0.424601\pi$
$542$	−1.64071	−0.0704746
$543$	0	0
$544$	23.0990	0.990361
$545$	−8.37690	−0.358827
$546$	0	0
$547$	−8.70334	−0.372128	−0.186064	−	0.982538i	$-0.559573\pi$
$547$	−8.70334	−0.372128	−0.186064	+	0.982538i	$0.559573\pi$
$548$	−1.61884	−0.0691535
$549$	0	0
$550$	−17.5000	−0.746201
$551$	−3.15603	−0.134451
$552$	0	0
$553$	2.87720	0.122351
$554$	−34.0828	−1.44804
$555$	0	0
$556$	−2.19991	−0.0932968
$557$	−28.7237	−1.21706	−0.608530	−	0.793531i	$-0.708241\pi$
$557$	−28.7237	−1.21706	−0.608530	+	0.793531i	$0.708241\pi$
$558$	0	0
$559$	−4.25770	−0.180081
$560$	−0.492520	−0.0208128
$561$	0	0
$562$	24.2626	1.02346
$563$	−21.9401	−0.924664	−0.462332	−	0.886707i	$-0.652987\pi$
$563$	−21.9401	−0.924664	−0.462332	+	0.886707i	$0.652987\pi$
$564$	0	0
$565$	−6.27637	−0.264049
$566$	8.19340	0.344394
$567$	0	0
$568$	28.4831	1.19512
$569$	32.0917	1.34535	0.672677	−	0.739937i	$-0.265145\pi$
$569$	32.0917	1.34535	0.672677	+	0.739937i	$0.265145\pi$
$570$	0	0
$571$	−7.53172	−0.315193	−0.157596	−	0.987504i	$-0.550374\pi$
$571$	−7.53172	−0.315193	−0.157596	+	0.987504i	$0.550374\pi$
$572$	7.88155	0.329544
$573$	0	0
$574$	3.29439	0.137505
$575$	−17.3177	−0.722197
$576$	0	0
$577$	5.37930	0.223943	0.111972	−	0.993711i	$-0.464283\pi$
$577$	5.37930	0.223943	0.111972	+	0.993711i	$0.464283\pi$
$578$	−46.0143	−1.91394
$579$	0	0
$580$	−0.319711	−0.0132753
$581$	−1.12475	−0.0466625
$582$	0	0
$583$	−31.3972	−1.30034
$584$	24.8800	1.02954
$585$	0	0
$586$	17.4058	0.719026
$587$	25.4186	1.04914	0.524568	−	0.851368i	$-0.324227\pi$
$587$	25.4186	1.04914	0.524568	+	0.851368i	$0.324227\pi$
$588$	0	0
$589$	30.4918	1.25639
$590$	4.32044	0.177870
$591$	0	0
$592$	−23.7568	−0.976397
$593$	45.4128	1.86488	0.932440	−	0.361325i	$-0.117676\pi$
$593$	45.4128	1.86488	0.932440	+	0.361325i	$0.117676\pi$
$594$	0	0
$595$	−1.44210	−0.0591203
$596$	−11.6986	−0.479192
$597$	0	0
$598$	−19.6896	−0.805168
$599$	7.66157	0.313043	0.156522	−	0.987675i	$-0.449972\pi$
$599$	7.66157	0.313043	0.156522	+	0.987675i	$0.449972\pi$
$600$	0	0
$601$	1.51529	0.0618099	0.0309049	−	0.999522i	$-0.490161\pi$
$601$	1.51529	0.0618099	0.0309049	+	0.999522i	$0.490161\pi$
$602$	−0.393813	−0.0160506
$603$	0	0
$604$	9.82173	0.399640
$605$	0.704338	0.0286354
$606$	0	0
$607$	−34.7165	−1.40910	−0.704550	−	0.709654i	$-0.748851\pi$
$607$	−34.7165	−1.40910	−0.704550	+	0.709654i	$0.748851\pi$
$608$	−9.79046	−0.397055
$609$	0	0
$610$	9.41721	0.381292
$611$	11.1122	0.449551
$612$	0	0
$613$	−3.30485	−0.133481	−0.0667407	−	0.997770i	$-0.521260\pi$
$613$	−3.30485	−0.133481	−0.0667407	+	0.997770i	$0.521260\pi$
$614$	−41.0384	−1.65618
$615$	0	0
$616$	3.29835	0.132894
$617$	16.3926	0.659941	0.329970	−	0.943991i	$-0.392961\pi$
$617$	16.3926	0.659941	0.329970	+	0.943991i	$0.392961\pi$
$618$	0	0
$619$	−7.70263	−0.309595	−0.154797	−	0.987946i	$-0.549472\pi$
$619$	−7.70263	−0.309595	−0.154797	+	0.987946i	$0.549472\pi$
$620$	3.08887	0.124052
$621$	0	0
$622$	6.00429	0.240750
$623$	1.95134	0.0781790
$624$	0	0
$625$	20.3393	0.813572
$626$	17.3705	0.694265
$627$	0	0
$628$	7.69419	0.307032
$629$	−69.5599	−2.77354
$630$	0	0
$631$	13.4016	0.533508	0.266754	−	0.963765i	$-0.414049\pi$
$631$	13.4016	0.533508	0.266754	+	0.963765i	$0.414049\pi$
$632$	−25.7209	−1.02312
$633$	0	0
$634$	−14.8073	−0.588072
$635$	3.80708	0.151079
$636$	0	0
$637$	−30.6114	−1.21287
$638$	−3.73726	−0.147959
$639$	0	0
$640$	2.43800	0.0963702
$641$	17.4225	0.688148	0.344074	−	0.938942i	$-0.388193\pi$
$641$	17.4225	0.688148	0.344074	+	0.938942i	$0.388193\pi$
$642$	0	0
$643$	−24.7401	−0.975656	−0.487828	−	0.872940i	$-0.662211\pi$
$643$	−24.7401	−0.975656	−0.487828	+	0.872940i	$0.662211\pi$
$644$	0.721405	0.0284273
$645$	0	0
$646$	28.1272	1.10665
$647$	−6.81224	−0.267817	−0.133908	−	0.990994i	$-0.542753\pi$
$647$	−6.81224	−0.267817	−0.133908	+	0.990994i	$0.542753\pi$
$648$	0	0
$649$	−20.0055	−0.785285
$650$	24.9297	0.977822
$651$	0	0
$652$	−3.19114	−0.124975
$653$	−10.2028	−0.399265	−0.199633	−	0.979871i	$-0.563975\pi$
$653$	−10.2028	−0.399265	−0.199633	+	0.979871i	$0.563975\pi$
$654$	0	0
$655$	5.46390	0.213492
$656$	−20.3629	−0.795039
$657$	0	0
$658$	1.02782	0.0400685
$659$	−9.67722	−0.376971	−0.188485	−	0.982076i	$-0.560358\pi$
$659$	−9.67722	−0.376971	−0.188485	+	0.982076i	$0.560358\pi$
$660$	0	0
$661$	−33.3035	−1.29536	−0.647678	−	0.761914i	$-0.724260\pi$
$661$	−33.3035	−1.29536	−0.647678	+	0.761914i	$0.724260\pi$
$662$	32.1806	1.25074
$663$	0	0
$664$	10.0548	0.390201
$665$	0.611231	0.0237025
$666$	0	0
$667$	−3.69833	−0.143200
$668$	−8.13103	−0.314599
$669$	0	0
$670$	−7.42902	−0.287008
$671$	−43.6058	−1.68338
$672$	0	0
$673$	−8.90841	−0.343394	−0.171697	−	0.985150i	$-0.554925\pi$
$673$	−8.90841	−0.343394	−0.171697	+	0.985150i	$0.554925\pi$
$674$	−41.8563	−1.61224
$675$	0	0
$676$	−3.85074	−0.148105
$677$	33.4097	1.28404	0.642020	−	0.766688i	$-0.278097\pi$
$677$	33.4097	1.28404	0.642020	+	0.766688i	$0.278097\pi$
$678$	0	0
$679$	−2.06701	−0.0793247
$680$	12.8918	0.494376
$681$	0	0
$682$	36.1073	1.38262
$683$	−13.2057	−0.505304	−0.252652	−	0.967557i	$-0.581303\pi$
$683$	−13.2057	−0.505304	−0.252652	+	0.967557i	$0.581303\pi$
$684$	0	0
$685$	−1.60729	−0.0614114
$686$	−5.71138	−0.218062
$687$	0	0
$688$	2.43420	0.0928029
$689$	44.7271	1.70397
$690$	0	0
$691$	−44.1972	−1.68134	−0.840670	−	0.541547i	$-0.817839\pi$
$691$	−44.1972	−1.68134	−0.840670	+	0.541547i	$0.817839\pi$
$692$	0.786648	0.0299039
$693$	0	0
$694$	−18.2193	−0.691596
$695$	−2.18421	−0.0828518
$696$	0	0
$697$	−59.6228	−2.25837
$698$	−2.58348	−0.0977863
$699$	0	0
$700$	−0.913395	−0.0345231
$701$	−1.53167	−0.0578505	−0.0289253	−	0.999582i	$-0.509208\pi$
$701$	−1.53167	−0.0578505	−0.0289253	+	0.999582i	$0.509208\pi$
$702$	0	0
$703$	29.4828	1.11197
$704$	−27.4749	−1.03550
$705$	0	0
$706$	−23.5778	−0.887363
$707$	0.573857	0.0215821
$708$	0	0
$709$	−12.9659	−0.486946	−0.243473	−	0.969908i	$-0.578287\pi$
$709$	−12.9659	−0.486946	−0.243473	+	0.969908i	$0.578287\pi$
$710$	6.25040	0.234573
$711$	0	0
$712$	−17.4442	−0.653748
$713$	35.7312	1.33814
$714$	0	0
$715$	7.82531	0.292650
$716$	11.0336	0.412346
$717$	0	0
$718$	−21.2215	−0.791979
$719$	47.3995	1.76770	0.883852	−	0.467767i	$-0.154941\pi$
$719$	47.3995	1.76770	0.883852	+	0.467767i	$0.154941\pi$
$720$	0	0
$721$	−2.23251	−0.0831429
$722$	10.8192	0.402651
$723$	0	0
$724$	3.91041	0.145329
$725$	4.68257	0.173906
$726$	0	0
$727$	32.5408	1.20687	0.603436	−	0.797411i	$-0.293798\pi$
$727$	32.5408	1.20687	0.603436	+	0.797411i	$0.293798\pi$
$728$	−4.69868	−0.174145
$729$	0	0
$730$	5.45974	0.202074
$731$	7.12734	0.263614
$732$	0	0
$733$	−21.2440	−0.784664	−0.392332	−	0.919824i	$-0.628332\pi$
$733$	−21.2440	−0.784664	−0.392332	+	0.919824i	$0.628332\pi$
$734$	−9.71155	−0.358460
$735$	0	0
$736$	−11.4727	−0.422891
$737$	34.3996	1.26712
$738$	0	0
$739$	−12.9289	−0.475597	−0.237799	−	0.971314i	$-0.576426\pi$
$739$	−12.9289	−0.475597	−0.237799	+	0.971314i	$0.576426\pi$
$740$	2.98666	0.109792
$741$	0	0
$742$	4.13701	0.151874
$743$	1.92982	0.0707984	0.0353992	−	0.999373i	$-0.488730\pi$
$743$	1.92982	0.0707984	0.0353992	+	0.999373i	$0.488730\pi$
$744$	0	0
$745$	−11.6151	−0.425544
$746$	2.62037	0.0959385
$747$	0	0
$748$	−13.1936	−0.482407
$749$	2.74148	0.100172
$750$	0	0
$751$	42.2181	1.54056	0.770281	−	0.637705i	$-0.220116\pi$
$751$	42.2181	1.54056	0.770281	+	0.637705i	$0.220116\pi$
$752$	−6.35303	−0.231671
$753$	0	0
$754$	5.32393	0.193886
$755$	9.75165	0.354899
$756$	0	0
$757$	−15.6503	−0.568819	−0.284410	−	0.958703i	$-0.591798\pi$
$757$	−15.6503	−0.568819	−0.284410	+	0.958703i	$0.591798\pi$
$758$	−0.753458	−0.0273668
$759$	0	0
$760$	−5.46414	−0.198205
$761$	−36.0119	−1.30543	−0.652715	−	0.757603i	$-0.726370\pi$
$761$	−36.0119	−1.30543	−0.652715	+	0.757603i	$0.726370\pi$
$762$	0	0
$763$	−5.11093	−0.185028
$764$	−7.81228	−0.282639
$765$	0	0
$766$	−11.9420	−0.431483
$767$	28.4989	1.02904
$768$	0	0
$769$	45.9039	1.65534	0.827668	−	0.561217i	$-0.189667\pi$
$769$	45.9039	1.65534	0.827668	+	0.561217i	$0.189667\pi$
$770$	0.723798	0.0260839
$771$	0	0
$772$	4.06187	0.146190
$773$	4.98199	0.179190	0.0895949	−	0.995978i	$-0.471443\pi$
$773$	4.98199	0.179190	0.0895949	+	0.995978i	$0.471443\pi$
$774$	0	0
$775$	−45.2404	−1.62508
$776$	18.4782	0.663329
$777$	0	0
$778$	−33.1014	−1.18674
$779$	25.2710	0.905427
$780$	0	0
$781$	−28.9421	−1.03563
$782$	32.9602	1.17865
$783$	0	0
$784$	17.5010	0.625037
$785$	7.63929	0.272658
$786$	0	0
$787$	23.0001	0.819866	0.409933	−	0.912116i	$-0.365552\pi$
$787$	23.0001	0.819866	0.409933	+	0.912116i	$0.365552\pi$
$788$	4.31876	0.153849
$789$	0	0
$790$	−5.64425	−0.200814
$791$	−3.82935	−0.136156
$792$	0	0
$793$	62.1188	2.20590
$794$	−18.5112	−0.656938
$795$	0	0
$796$	11.7460	0.416326
$797$	−29.0865	−1.03030	−0.515149	−	0.857101i	$-0.672263\pi$
$797$	−29.0865	−1.03030	−0.515149	+	0.857101i	$0.672263\pi$
$798$	0	0
$799$	−18.6017	−0.658081
$800$	14.5260	0.513572
$801$	0	0
$802$	−13.0898	−0.462218
$803$	−25.2809	−0.892145
$804$	0	0
$805$	0.716258	0.0252448
$806$	−51.4369	−1.81179
$807$	0	0
$808$	−5.13003	−0.180474
$809$	3.35592	0.117988	0.0589940	−	0.998258i	$-0.481211\pi$
$809$	3.35592	0.117988	0.0589940	+	0.998258i	$0.481211\pi$
$810$	0	0
$811$	−6.08002	−0.213498	−0.106749	−	0.994286i	$-0.534044\pi$
$811$	−6.08002	−0.213498	−0.106749	+	0.994286i	$0.534044\pi$
$812$	−0.195063	−0.00684536
$813$	0	0
$814$	34.9125	1.22368
$815$	−3.16837	−0.110983
$816$	0	0
$817$	−3.02091	−0.105688
$818$	−43.6900	−1.52759
$819$	0	0
$820$	2.55999	0.0893988
$821$	−36.1402	−1.26130	−0.630650	−	0.776067i	$-0.717212\pi$
$821$	−36.1402	−1.26130	−0.630650	+	0.776067i	$0.717212\pi$
$822$	0	0
$823$	−13.0576	−0.455158	−0.227579	−	0.973760i	$-0.573081\pi$
$823$	−13.0576	−0.455158	−0.227579	+	0.973760i	$0.573081\pi$
$824$	19.9576	0.695258
$825$	0	0
$826$	2.63599	0.0917180
$827$	−37.9302	−1.31896	−0.659481	−	0.751721i	$-0.729224\pi$
$827$	−37.9302	−1.31896	−0.659481	+	0.751721i	$0.729224\pi$
$828$	0	0
$829$	−31.1409	−1.08157	−0.540785	−	0.841161i	$-0.681873\pi$
$829$	−31.1409	−1.08157	−0.540785	+	0.841161i	$0.681873\pi$
$830$	2.20645	0.0765869
$831$	0	0
$832$	39.1395	1.35692
$833$	51.2431	1.77547
$834$	0	0
$835$	−8.07301	−0.279378
$836$	5.59209	0.193406
$837$	0	0
$838$	18.2572	0.630685
$839$	6.64645	0.229461	0.114730	−	0.993397i	$-0.463400\pi$
$839$	6.64645	0.229461	0.114730	+	0.993397i	$0.463400\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−0.633495	−0.0218317
$843$	0	0
$844$	9.08975	0.312882
$845$	−3.82326	−0.131524
$846$	0	0
$847$	0.429732	0.0147658
$848$	−25.5712	−0.878119
$849$	0	0
$850$	−41.7320	−1.43139
$851$	34.5488	1.18432
$852$	0	0
$853$	45.5778	1.56055	0.780277	−	0.625434i	$-0.215078\pi$
$853$	45.5778	1.56055	0.780277	+	0.625434i	$0.215078\pi$
$854$	5.74564	0.196612
$855$	0	0
$856$	−24.5077	−0.837655
$857$	−4.74936	−0.162235	−0.0811175	−	0.996705i	$-0.525849\pi$
$857$	−4.74936	−0.162235	−0.0811175	+	0.996705i	$0.525849\pi$
$858$	0	0
$859$	−26.5881	−0.907174	−0.453587	−	0.891212i	$-0.649856\pi$
$859$	−26.5881	−0.907174	−0.453587	+	0.891212i	$0.649856\pi$
$860$	−0.306023	−0.0104353
$861$	0	0
$862$	−21.4206	−0.729587
$863$	16.0940	0.547848	0.273924	−	0.961751i	$-0.411678\pi$
$863$	16.0940	0.547848	0.273924	+	0.961751i	$0.411678\pi$
$864$	0	0
$865$	0.781035	0.0265560
$866$	1.44477	0.0490953
$867$	0	0
$868$	1.88459	0.0639671
$869$	26.1353	0.886581
$870$	0	0
$871$	−49.0041	−1.66044
$872$	45.6895	1.54724
$873$	0	0
$874$	−13.9701	−0.472546
$875$	−1.87523	−0.0633944
$876$	0	0
$877$	−8.36466	−0.282455	−0.141227	−	0.989977i	$-0.545105\pi$
$877$	−8.36466	−0.282455	−0.141227	+	0.989977i	$0.545105\pi$
$878$	37.7747	1.27484
$879$	0	0
$880$	−4.47386	−0.150814
$881$	−15.7376	−0.530215	−0.265107	−	0.964219i	$-0.585407\pi$
$881$	−15.7376	−0.530215	−0.265107	+	0.964219i	$0.585407\pi$
$882$	0	0
$883$	33.6912	1.13380	0.566900	−	0.823787i	$-0.308143\pi$
$883$	33.6912	1.13380	0.566900	+	0.823787i	$0.308143\pi$
$884$	18.7950	0.632145
$885$	0	0
$886$	−3.72934	−0.125290
$887$	−46.5853	−1.56418	−0.782091	−	0.623164i	$-0.785847\pi$
$887$	−46.5853	−1.56418	−0.782091	+	0.623164i	$0.785847\pi$
$888$	0	0
$889$	2.32278	0.0779036
$890$	−3.82799	−0.128315
$891$	0	0
$892$	−0.442064	−0.0148014
$893$	7.88429	0.263838
$894$	0	0
$895$	10.9549	0.366182
$896$	1.48747	0.0496930
$897$	0	0
$898$	34.3451	1.14611
$899$	−9.66146	−0.322228
$900$	0	0
$901$	−74.8726	−2.49437
$902$	29.9250	0.996393
$903$	0	0
$904$	34.2327	1.13856
$905$	3.88251	0.129059
$906$	0	0
$907$	6.83215	0.226858	0.113429	−	0.993546i	$-0.463817\pi$
$907$	6.83215	0.226858	0.113429	+	0.993546i	$0.463817\pi$
$908$	4.14133	0.137435
$909$	0	0
$910$	−1.03109	−0.0341803
$911$	57.7016	1.91174	0.955869	−	0.293792i	$-0.0949173\pi$
$911$	57.7016	1.91174	0.955869	+	0.293792i	$0.0949173\pi$
$912$	0	0
$913$	−10.2168	−0.338127
$914$	−23.5565	−0.779180
$915$	0	0
$916$	3.92118	0.129559
$917$	3.33365	0.110087
$918$	0	0
$919$	−44.5911	−1.47092	−0.735462	−	0.677566i	$-0.763035\pi$
$919$	−44.5911	−1.47092	−0.735462	+	0.677566i	$0.763035\pi$
$920$	−6.40304	−0.211102
$921$	0	0
$922$	32.9208	1.08419
$923$	41.2295	1.35709
$924$	0	0
$925$	−43.7434	−1.43827
$926$	13.1976	0.433700
$927$	0	0
$928$	3.10214	0.101833
$929$	−50.3234	−1.65106	−0.825529	−	0.564359i	$-0.809123\pi$
$929$	−50.3234	−1.65106	−0.825529	+	0.564359i	$0.809123\pi$
$930$	0	0
$931$	−21.7193	−0.711821
$932$	5.01571	0.164295
$933$	0	0
$934$	−14.3551	−0.469713
$935$	−13.0995	−0.428399
$936$	0	0
$937$	40.0577	1.30863	0.654314	−	0.756223i	$-0.272957\pi$
$937$	40.0577	1.30863	0.654314	+	0.756223i	$0.272957\pi$
$938$	−4.53260	−0.147995
$939$	0	0
$940$	0.798692	0.0260505
$941$	51.3528	1.67405	0.837026	−	0.547163i	$-0.184292\pi$
$941$	51.3528	1.67405	0.837026	+	0.547163i	$0.184292\pi$
$942$	0	0
$943$	29.6133	0.964340
$944$	−16.2933	−0.530302
$945$	0	0
$946$	−3.57725	−0.116306
$947$	−10.2326	−0.332516	−0.166258	−	0.986082i	$-0.553168\pi$
$947$	−10.2326	−0.332516	−0.166258	+	0.986082i	$0.553168\pi$
$948$	0	0
$949$	36.0141	1.16907
$950$	17.6880	0.573875
$951$	0	0
$952$	7.86554	0.254924
$953$	−19.1875	−0.621544	−0.310772	−	0.950484i	$-0.600588\pi$
$953$	−19.1875	−0.621544	−0.310772	+	0.950484i	$0.600588\pi$
$954$	0	0
$955$	−7.75655	−0.250996
$956$	16.9487	0.548161
$957$	0	0
$958$	22.1819	0.716664
$959$	−0.980643	−0.0316666
$960$	0	0
$961$	62.3437	2.01109
$962$	−49.7348	−1.60351
$963$	0	0
$964$	−7.95024	−0.256060
$965$	4.03289	0.129823
$966$	0	0
$967$	−27.3710	−0.880190	−0.440095	−	0.897951i	$-0.645055\pi$
$967$	−27.3710	−0.880190	−0.440095	+	0.897951i	$0.645055\pi$
$968$	−3.84162	−0.123474
$969$	0	0
$970$	4.05490	0.130195
$971$	56.6753	1.81880	0.909398	−	0.415926i	$-0.136543\pi$
$971$	56.6753	1.81880	0.909398	+	0.415926i	$0.136543\pi$
$972$	0	0
$973$	−1.33263	−0.0427223
$974$	−3.15756	−0.101175
$975$	0	0
$976$	−35.5143	−1.13679
$977$	−14.2809	−0.456885	−0.228443	−	0.973557i	$-0.573363\pi$
$977$	−14.2809	−0.456885	−0.228443	+	0.973557i	$0.573363\pi$
$978$	0	0
$979$	17.7253	0.566502
$980$	−2.20020	−0.0702828
$981$	0	0
$982$	42.8337	1.36688
$983$	−56.8075	−1.81188	−0.905940	−	0.423407i	$-0.860834\pi$
$983$	−56.8075	−1.81188	−0.905940	+	0.423407i	$0.860834\pi$
$984$	0	0
$985$	4.28795	0.136625
$986$	−8.91219	−0.283822
$987$	0	0
$988$	−7.96624	−0.253440
$989$	−3.53999	−0.112565
$990$	0	0
$991$	15.0319	0.477503	0.238752	−	0.971081i	$-0.423262\pi$
$991$	15.0319	0.477503	0.238752	+	0.971081i	$0.423262\pi$
$992$	−29.9712	−0.951587
$993$	0	0
$994$	3.81350	0.120957
$995$	11.6622	0.369716
$996$	0	0
$997$	34.1487	1.08150	0.540750	−	0.841184i	$-0.318141\pi$
$997$	34.1487	1.08150	0.540750	+	0.841184i	$0.318141\pi$
$998$	31.9593	1.01165
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	783.2.a.j.1.3	yes	8
3.2	odd	2		783.2.a.i.1.6	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
783.2.a.i.1.6	✓	8	3.2	odd	2
783.2.a.j.1.3	yes	8	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 783 = 3^{3} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	783.a (trivial)

\(f(q)\)	\(=\)	\(q-1.19689 q^{2} -0.567458 q^{4} -0.563409 q^{5} -0.343748 q^{7} +3.07296 q^{8} +0.674338 q^{10} -3.12248 q^{11} +4.44814 q^{13} +0.411428 q^{14} -2.54308 q^{16} -7.44614 q^{17} +3.15603 q^{19} +0.319711 q^{20} +3.73726 q^{22} +3.69833 q^{23} -4.68257 q^{25} -5.32393 q^{26} +0.195063 q^{28} -1.00000 q^{29} +9.66146 q^{31} -3.10214 q^{32} +8.91219 q^{34} +0.193671 q^{35} +9.34175 q^{37} -3.77742 q^{38} -1.73133 q^{40} +8.00721 q^{41} -0.957186 q^{43} +1.77187 q^{44} -4.42648 q^{46} +2.49817 q^{47} -6.88184 q^{49} +5.60451 q^{50} -2.52413 q^{52} +10.0552 q^{53} +1.75923 q^{55} -1.05632 q^{56} +1.19689 q^{58} +6.40694 q^{59} +13.9651 q^{61} -11.5637 q^{62} +8.79907 q^{64} -2.50612 q^{65} -11.0168 q^{67} +4.22537 q^{68} -0.231803 q^{70} +9.26894 q^{71} +8.09644 q^{73} -11.1810 q^{74} -1.79091 q^{76} +1.07335 q^{77} -8.37007 q^{79} +1.43279 q^{80} -9.58373 q^{82} +3.27202 q^{83} +4.19522 q^{85} +1.14565 q^{86} -9.59525 q^{88} -5.67667 q^{89} -1.52904 q^{91} -2.09864 q^{92} -2.99003 q^{94} -1.77814 q^{95} +6.01316 q^{97} +8.23679 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + q^{2} + 11 q^{4} - 2 q^{5} + 5 q^{7} + 3 q^{8} + 7 q^{10} - 2 q^{11} + 11 q^{13} + 8 q^{14} + 21 q^{16} + 6 q^{17} + 5 q^{19} - 27 q^{20} + 17 q^{22} - 4 q^{23} + 30 q^{25} + 16 q^{26} + 2 q^{28}+ \cdots - 11 q^{98}+O(q^{100}) \) 8 * q + q^2 + 11 * q^4 - 2 * q^5 + 5 * q^7 + 3 * q^8 + 7 * q^10 - 2 * q^11 + 11 * q^13 + 8 * q^14 + 21 * q^16 + 6 * q^17 + 5 * q^19 - 27 * q^20 + 17 * q^22 - 4 * q^23 + 30 * q^25 + 16 * q^26 + 2 * q^28 - 8 * q^29 - q^31 + 37 * q^32 - 9 * q^34 + q^35 + 26 * q^37 - 14 * q^38 + 2 * q^40 + 19 * q^41 + 16 * q^43 - q^44 - 15 * q^46 - 18 * q^47 + 27 * q^49 + 4 * q^50 + 7 * q^52 - 19 * q^53 + 6 * q^55 - 27 * q^56 - q^58 + 13 * q^59 + 2 * q^61 + 7 * q^62 + 13 * q^64 - 19 * q^65 + 25 * q^67 + 22 * q^68 - 22 * q^70 - 7 * q^71 + 11 * q^73 - 9 * q^74 - 31 * q^76 + 26 * q^77 + 16 * q^79 - 19 * q^80 + 7 * q^82 + 19 * q^83 + 10 * q^85 + 63 * q^86 + 32 * q^88 - 16 * q^89 + 6 * q^91 - 101 * q^92 - 34 * q^94 + 43 * q^95 + 31 * q^97 - 11 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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