LMFDB - Embedded newform 8512.2.a.t.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8512,2,Mod(1,8512)] 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("8512.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8512, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8512 = 2^{6} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8512.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,1,0,-6,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$67.9686622005$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{21}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 5 $ x^2 - x - 5
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 532)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.79129$ of defining polynomial
Character	$\chi$	$=$	8512.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.79129 q^{3} -3.00000 q^{5} +1.00000 q^{7} +0.208712 q^{9} -0.791288 q^{11} +1.00000 q^{13} +5.37386 q^{15} -0.791288 q^{17} -1.00000 q^{19} -1.79129 q^{21} -4.58258 q^{23} +4.00000 q^{25} +5.00000 q^{27} +0.791288 q^{29} -6.37386 q^{31} +1.41742 q^{33} -3.00000 q^{35} -5.00000 q^{37} -1.79129 q^{39} +0.791288 q^{41} -2.00000 q^{43} -0.626136 q^{45} +1.41742 q^{47} +1.00000 q^{49} +1.41742 q^{51} -5.37386 q^{53} +2.37386 q^{55} +1.79129 q^{57} +6.16515 q^{59} +1.00000 q^{61} +0.208712 q^{63} -3.00000 q^{65} -4.37386 q^{67} +8.20871 q^{69} +6.16515 q^{71} +2.62614 q^{73} -7.16515 q^{75} -0.791288 q^{77} -10.0000 q^{79} -9.58258 q^{81} -0.626136 q^{83} +2.37386 q^{85} -1.41742 q^{87} -1.58258 q^{89} +1.00000 q^{91} +11.4174 q^{93} +3.00000 q^{95} -7.00000 q^{97} -0.165151 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} - 6 q^{5} + 2 q^{7} + 5 q^{9} + 3 q^{11} + 2 q^{13} - 3 q^{15} + 3 q^{17} - 2 q^{19} + q^{21} + 8 q^{25} + 10 q^{27} - 3 q^{29} + q^{31} + 12 q^{33} - 6 q^{35} - 10 q^{37} + q^{39} - 3 q^{41}+ \cdots + 18 q^{99}+O(q^{100}) $ 2 * q + q^3 - 6 * q^5 + 2 * q^7 + 5 * q^9 + 3 * q^11 + 2 * q^13 - 3 * q^15 + 3 * q^17 - 2 * q^19 + q^21 + 8 * q^25 + 10 * q^27 - 3 * q^29 + q^31 + 12 * q^33 - 6 * q^35 - 10 * q^37 + q^39 - 3 * q^41 - 4 * q^43 - 15 * q^45 + 12 * q^47 + 2 * q^49 + 12 * q^51 + 3 * q^53 - 9 * q^55 - q^57 - 6 * q^59 + 2 * q^61 + 5 * q^63 - 6 * q^65 + 5 * q^67 + 21 * q^69 - 6 * q^71 + 19 * q^73 + 4 * q^75 + 3 * q^77 - 20 * q^79 - 10 * q^81 - 15 * q^83 - 9 * q^85 - 12 * q^87 + 6 * q^89 + 2 * q^91 + 32 * q^93 + 6 * q^95 - 14 * q^97 + 18 * q^99

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
$2$	0	0
$3$	−1.79129	−1.03420	−0.517100	−	0.855925i	$-0.672989\pi$
$3$	−1.79129	−1.03420	−0.517100	+	0.855925i	$0.672989\pi$
$4$	0	0
$5$	−3.00000	−1.34164	−0.670820	−	0.741620i	$-0.734058\pi$
$5$	−3.00000	−1.34164	−0.670820	+	0.741620i	$0.734058\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0.208712	0.0695707
$10$	0	0
$11$	−0.791288	−0.238582	−0.119291	−	0.992859i	$-0.538062\pi$
$11$	−0.791288	−0.238582	−0.119291	+	0.992859i	$0.538062\pi$
$12$	0	0
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	0	0
$15$	5.37386	1.38753
$16$	0	0
$17$	−0.791288	−0.191915	−0.0959577	−	0.995385i	$-0.530591\pi$
$17$	−0.791288	−0.191915	−0.0959577	+	0.995385i	$0.530591\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−1.79129	−0.390891
$22$	0	0
$23$	−4.58258	−0.955533	−0.477767	−	0.878487i	$-0.658554\pi$
$23$	−4.58258	−0.955533	−0.477767	+	0.878487i	$0.658554\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	0	0
$27$	5.00000	0.962250
$28$	0	0
$29$	0.791288	0.146938	0.0734692	−	0.997297i	$-0.476593\pi$
$29$	0.791288	0.146938	0.0734692	+	0.997297i	$0.476593\pi$
$30$	0	0
$31$	−6.37386	−1.14478	−0.572390	−	0.819982i	$-0.693984\pi$
$31$	−6.37386	−1.14478	−0.572390	+	0.819982i	$0.693984\pi$
$32$	0	0
$33$	1.41742	0.246742
$34$	0	0
$35$	−3.00000	−0.507093
$36$	0	0
$37$	−5.00000	−0.821995	−0.410997	−	0.911636i	$-0.634819\pi$
$37$	−5.00000	−0.821995	−0.410997	+	0.911636i	$0.634819\pi$
$38$	0	0
$39$	−1.79129	−0.286836
$40$	0	0
$41$	0.791288	0.123578	0.0617892	−	0.998089i	$-0.480319\pi$
$41$	0.791288	0.123578	0.0617892	+	0.998089i	$0.480319\pi$
$42$	0	0
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	0	0
$45$	−0.626136	−0.0933389
$46$	0	0
$47$	1.41742	0.206753	0.103376	−	0.994642i	$-0.467035\pi$
$47$	1.41742	0.206753	0.103376	+	0.994642i	$0.467035\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	1.41742	0.198479
$52$	0	0
$53$	−5.37386	−0.738157	−0.369078	−	0.929398i	$-0.620327\pi$
$53$	−5.37386	−0.738157	−0.369078	+	0.929398i	$0.620327\pi$
$54$	0	0
$55$	2.37386	0.320092
$56$	0	0
$57$	1.79129	0.237262
$58$	0	0
$59$	6.16515	0.802634	0.401317	−	0.915939i	$-0.368553\pi$
$59$	6.16515	0.802634	0.401317	+	0.915939i	$0.368553\pi$
$60$	0	0
$61$	1.00000	0.128037	0.0640184	−	0.997949i	$-0.479608\pi$
$61$	1.00000	0.128037	0.0640184	+	0.997949i	$0.479608\pi$
$62$	0	0
$63$	0.208712	0.0262953
$64$	0	0
$65$	−3.00000	−0.372104
$66$	0	0
$67$	−4.37386	−0.534352	−0.267176	−	0.963648i	$-0.586091\pi$
$67$	−4.37386	−0.534352	−0.267176	+	0.963648i	$0.586091\pi$
$68$	0	0
$69$	8.20871	0.988213
$70$	0	0
$71$	6.16515	0.731669	0.365834	−	0.930680i	$-0.380784\pi$
$71$	6.16515	0.731669	0.365834	+	0.930680i	$0.380784\pi$
$72$	0	0
$73$	2.62614	0.307366	0.153683	−	0.988120i	$-0.450887\pi$
$73$	2.62614	0.307366	0.153683	+	0.988120i	$0.450887\pi$
$74$	0	0
$75$	−7.16515	−0.827360
$76$	0	0
$77$	−0.791288	−0.0901756
$78$	0	0
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	0	0
$81$	−9.58258	−1.06473
$82$	0	0
$83$	−0.626136	−0.0687274	−0.0343637	−	0.999409i	$-0.510940\pi$
$83$	−0.626136	−0.0687274	−0.0343637	+	0.999409i	$0.510940\pi$
$84$	0	0
$85$	2.37386	0.257482
$86$	0	0
$87$	−1.41742	−0.151964
$88$	0	0
$89$	−1.58258	−0.167753	−0.0838763	−	0.996476i	$-0.526730\pi$
$89$	−1.58258	−0.167753	−0.0838763	+	0.996476i	$0.526730\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	11.4174	1.18393
$94$	0	0
$95$	3.00000	0.307794
$96$	0	0
$97$	−7.00000	−0.710742	−0.355371	−	0.934725i	$-0.615646\pi$
$97$	−7.00000	−0.710742	−0.355371	+	0.934725i	$0.615646\pi$
$98$	0	0
$99$	−0.165151	−0.0165983
$100$	0	0
$101$	−19.7477	−1.96497	−0.982486	−	0.186336i	$-0.940339\pi$
$101$	−19.7477	−1.96497	−0.982486	+	0.186336i	$0.940339\pi$
$102$	0	0
$103$	−13.0000	−1.28093	−0.640464	−	0.767988i	$-0.721258\pi$
$103$	−13.0000	−1.28093	−0.640464	+	0.767988i	$0.721258\pi$
$104$	0	0
$105$	5.37386	0.524435
$106$	0	0
$107$	18.1652	1.75609	0.878046	−	0.478577i	$-0.158847\pi$
$107$	18.1652	1.75609	0.878046	+	0.478577i	$0.158847\pi$
$108$	0	0
$109$	−15.7477	−1.50836	−0.754179	−	0.656668i	$-0.771965\pi$
$109$	−15.7477	−1.50836	−0.754179	+	0.656668i	$0.771965\pi$
$110$	0	0
$111$	8.95644	0.850108
$112$	0	0
$113$	−5.37386	−0.505531	−0.252765	−	0.967528i	$-0.581340\pi$
$113$	−5.37386	−0.505531	−0.252765	+	0.967528i	$0.581340\pi$
$114$	0	0
$115$	13.7477	1.28198
$116$	0	0
$117$	0.208712	0.0192954
$118$	0	0
$119$	−0.791288	−0.0725372
$120$	0	0
$121$	−10.3739	−0.943079
$122$	0	0
$123$	−1.41742	−0.127805
$124$	0	0
$125$	3.00000	0.268328
$126$	0	0
$127$	−20.7477	−1.84106	−0.920532	−	0.390668i	$-0.872244\pi$
$127$	−20.7477	−1.84106	−0.920532	+	0.390668i	$0.872244\pi$
$128$	0	0
$129$	3.58258	0.315428
$130$	0	0
$131$	−19.1216	−1.67066	−0.835331	−	0.549748i	$-0.814724\pi$
$131$	−19.1216	−1.67066	−0.835331	+	0.549748i	$0.814724\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	0	0
$135$	−15.0000	−1.29099
$136$	0	0
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	0	0
$139$	17.7477	1.50534	0.752671	−	0.658396i	$-0.228765\pi$
$139$	17.7477	1.50534	0.752671	+	0.658396i	$0.228765\pi$
$140$	0	0
$141$	−2.53901	−0.213824
$142$	0	0
$143$	−0.791288	−0.0661708
$144$	0	0
$145$	−2.37386	−0.197139
$146$	0	0
$147$	−1.79129	−0.147743
$148$	0	0
$149$	−3.00000	−0.245770	−0.122885	−	0.992421i	$-0.539215\pi$
$149$	−3.00000	−0.245770	−0.122885	+	0.992421i	$0.539215\pi$
$150$	0	0
$151$	−0.373864	−0.0304246	−0.0152123	−	0.999884i	$-0.504842\pi$
$151$	−0.373864	−0.0304246	−0.0152123	+	0.999884i	$0.504842\pi$
$152$	0	0
$153$	−0.165151	−0.0133517
$154$	0	0
$155$	19.1216	1.53588
$156$	0	0
$157$	−21.1216	−1.68569	−0.842843	−	0.538159i	$-0.819120\pi$
$157$	−21.1216	−1.68569	−0.842843	+	0.538159i	$0.819120\pi$
$158$	0	0
$159$	9.62614	0.763402
$160$	0	0
$161$	−4.58258	−0.361158
$162$	0	0
$163$	9.37386	0.734218	0.367109	−	0.930178i	$-0.380348\pi$
$163$	9.37386	0.734218	0.367109	+	0.930178i	$0.380348\pi$
$164$	0	0
$165$	−4.25227	−0.331039
$166$	0	0
$167$	22.5826	1.74749	0.873746	−	0.486382i	$-0.161684\pi$
$167$	22.5826	1.74749	0.873746	+	0.486382i	$0.161684\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	−0.208712	−0.0159606
$172$	0	0
$173$	4.74773	0.360963	0.180482	−	0.983578i	$-0.442234\pi$
$173$	4.74773	0.360963	0.180482	+	0.983578i	$0.442234\pi$
$174$	0	0
$175$	4.00000	0.302372
$176$	0	0
$177$	−11.0436	−0.830085
$178$	0	0
$179$	11.2087	0.837778	0.418889	−	0.908037i	$-0.362420\pi$
$179$	11.2087	0.837778	0.418889	+	0.908037i	$0.362420\pi$
$180$	0	0
$181$	−4.37386	−0.325107	−0.162553	−	0.986700i	$-0.551973\pi$
$181$	−4.37386	−0.325107	−0.162553	+	0.986700i	$0.551973\pi$
$182$	0	0
$183$	−1.79129	−0.132416
$184$	0	0
$185$	15.0000	1.10282
$186$	0	0
$187$	0.626136	0.0457876
$188$	0	0
$189$	5.00000	0.363696
$190$	0	0
$191$	−8.53901	−0.617861	−0.308931	−	0.951085i	$-0.599971\pi$
$191$	−8.53901	−0.617861	−0.308931	+	0.951085i	$0.599971\pi$
$192$	0	0
$193$	−6.37386	−0.458801	−0.229400	−	0.973332i	$-0.573677\pi$
$193$	−6.37386	−0.458801	−0.229400	+	0.973332i	$0.573677\pi$
$194$	0	0
$195$	5.37386	0.384830
$196$	0	0
$197$	−11.3739	−0.810354	−0.405177	−	0.914238i	$-0.632790\pi$
$197$	−11.3739	−0.810354	−0.405177	+	0.914238i	$0.632790\pi$
$198$	0	0
$199$	3.74773	0.265669	0.132835	−	0.991138i	$-0.457592\pi$
$199$	3.74773	0.265669	0.132835	+	0.991138i	$0.457592\pi$
$200$	0	0
$201$	7.83485	0.552628
$202$	0	0
$203$	0.791288	0.0555375
$204$	0	0
$205$	−2.37386	−0.165798
$206$	0	0
$207$	−0.956439	−0.0664771
$208$	0	0
$209$	0.791288	0.0547345
$210$	0	0
$211$	10.6261	0.731533	0.365767	−	0.930707i	$-0.380807\pi$
$211$	10.6261	0.731533	0.365767	+	0.930707i	$0.380807\pi$
$212$	0	0
$213$	−11.0436	−0.756692
$214$	0	0
$215$	6.00000	0.409197
$216$	0	0
$217$	−6.37386	−0.432686
$218$	0	0
$219$	−4.70417	−0.317878
$220$	0	0
$221$	−0.791288	−0.0532278
$222$	0	0
$223$	3.74773	0.250966	0.125483	−	0.992096i	$-0.459952\pi$
$223$	3.74773	0.250966	0.125483	+	0.992096i	$0.459952\pi$
$224$	0	0
$225$	0.834849	0.0556566
$226$	0	0
$227$	−0.791288	−0.0525196	−0.0262598	−	0.999655i	$-0.508360\pi$
$227$	−0.791288	−0.0525196	−0.0262598	+	0.999655i	$0.508360\pi$
$228$	0	0
$229$	22.0000	1.45380	0.726900	−	0.686743i	$-0.240960\pi$
$229$	22.0000	1.45380	0.726900	+	0.686743i	$0.240960\pi$
$230$	0	0
$231$	1.41742	0.0932597
$232$	0	0
$233$	13.1216	0.859624	0.429812	−	0.902918i	$-0.358580\pi$
$233$	13.1216	0.859624	0.429812	+	0.902918i	$0.358580\pi$
$234$	0	0
$235$	−4.25227	−0.277388
$236$	0	0
$237$	17.9129	1.16357
$238$	0	0
$239$	19.7477	1.27737	0.638687	−	0.769467i	$-0.279478\pi$
$239$	19.7477	1.27737	0.638687	+	0.769467i	$0.279478\pi$
$240$	0	0
$241$	18.7477	1.20765	0.603824	−	0.797118i	$-0.293643\pi$
$241$	18.7477	1.20765	0.603824	+	0.797118i	$0.293643\pi$
$242$	0	0
$243$	2.16515	0.138895
$244$	0	0
$245$	−3.00000	−0.191663
$246$	0	0
$247$	−1.00000	−0.0636285
$248$	0	0
$249$	1.12159	0.0710779
$250$	0	0
$251$	−5.20871	−0.328771	−0.164385	−	0.986396i	$-0.552564\pi$
$251$	−5.20871	−0.328771	−0.164385	+	0.986396i	$0.552564\pi$
$252$	0	0
$253$	3.62614	0.227973
$254$	0	0
$255$	−4.25227	−0.266288
$256$	0	0
$257$	15.7913	0.985033	0.492517	−	0.870303i	$-0.336077\pi$
$257$	15.7913	0.985033	0.492517	+	0.870303i	$0.336077\pi$
$258$	0	0
$259$	−5.00000	−0.310685
$260$	0	0
$261$	0.165151	0.0102226
$262$	0	0
$263$	2.20871	0.136195	0.0680975	−	0.997679i	$-0.478307\pi$
$263$	2.20871	0.136195	0.0680975	+	0.997679i	$0.478307\pi$
$264$	0	0
$265$	16.1216	0.990341
$266$	0	0
$267$	2.83485	0.173490
$268$	0	0
$269$	−12.9564	−0.789968	−0.394984	−	0.918688i	$-0.629250\pi$
$269$	−12.9564	−0.789968	−0.394984	+	0.918688i	$0.629250\pi$
$270$	0	0
$271$	27.1216	1.64752	0.823760	−	0.566939i	$-0.191873\pi$
$271$	27.1216	1.64752	0.823760	+	0.566939i	$0.191873\pi$
$272$	0	0
$273$	−1.79129	−0.108414
$274$	0	0
$275$	−3.16515	−0.190866
$276$	0	0
$277$	−30.7477	−1.84745	−0.923726	−	0.383054i	$-0.874872\pi$
$277$	−30.7477	−1.84745	−0.923726	+	0.383054i	$0.874872\pi$
$278$	0	0
$279$	−1.33030	−0.0796431
$280$	0	0
$281$	28.5826	1.70509	0.852547	−	0.522651i	$-0.175057\pi$
$281$	28.5826	1.70509	0.852547	+	0.522651i	$0.175057\pi$
$282$	0	0
$283$	−7.37386	−0.438331	−0.219165	−	0.975688i	$-0.570333\pi$
$283$	−7.37386	−0.438331	−0.219165	+	0.975688i	$0.570333\pi$
$284$	0	0
$285$	−5.37386	−0.318320
$286$	0	0
$287$	0.791288	0.0467082
$288$	0	0
$289$	−16.3739	−0.963168
$290$	0	0
$291$	12.5390	0.735050
$292$	0	0
$293$	7.74773	0.452627	0.226314	−	0.974055i	$-0.427333\pi$
$293$	7.74773	0.452627	0.226314	+	0.974055i	$0.427333\pi$
$294$	0	0
$295$	−18.4955	−1.07685
$296$	0	0
$297$	−3.95644	−0.229576
$298$	0	0
$299$	−4.58258	−0.265017
$300$	0	0
$301$	−2.00000	−0.115278
$302$	0	0
$303$	35.3739	2.03218
$304$	0	0
$305$	−3.00000	−0.171780
$306$	0	0
$307$	13.6261	0.777685	0.388842	−	0.921304i	$-0.372875\pi$
$307$	13.6261	0.777685	0.388842	+	0.921304i	$0.372875\pi$
$308$	0	0
$309$	23.2867	1.32474
$310$	0	0
$311$	30.7913	1.74601	0.873007	−	0.487708i	$-0.162167\pi$
$311$	30.7913	1.74601	0.873007	+	0.487708i	$0.162167\pi$
$312$	0	0
$313$	−14.7477	−0.833591	−0.416795	−	0.909000i	$-0.636847\pi$
$313$	−14.7477	−0.833591	−0.416795	+	0.909000i	$0.636847\pi$
$314$	0	0
$315$	−0.626136	−0.0352788
$316$	0	0
$317$	−33.1652	−1.86274	−0.931370	−	0.364073i	$-0.881386\pi$
$317$	−33.1652	−1.86274	−0.931370	+	0.364073i	$0.881386\pi$
$318$	0	0
$319$	−0.626136	−0.0350569
$320$	0	0
$321$	−32.5390	−1.81615
$322$	0	0
$323$	0.791288	0.0440284
$324$	0	0
$325$	4.00000	0.221880
$326$	0	0
$327$	28.2087	1.55995
$328$	0	0
$329$	1.41742	0.0781451
$330$	0	0
$331$	−14.6261	−0.803925	−0.401963	−	0.915656i	$-0.631672\pi$
$331$	−14.6261	−0.803925	−0.401963	+	0.915656i	$0.631672\pi$
$332$	0	0
$333$	−1.04356	−0.0571868
$334$	0	0
$335$	13.1216	0.716909
$336$	0	0
$337$	−12.3739	−0.674047	−0.337024	−	0.941496i	$-0.609420\pi$
$337$	−12.3739	−0.674047	−0.337024	+	0.941496i	$0.609420\pi$
$338$	0	0
$339$	9.62614	0.522820
$340$	0	0
$341$	5.04356	0.273124
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−24.6261	−1.32583
$346$	0	0
$347$	3.79129	0.203527	0.101763	−	0.994809i	$-0.467552\pi$
$347$	3.79129	0.203527	0.101763	+	0.994809i	$0.467552\pi$
$348$	0	0
$349$	15.3739	0.822944	0.411472	−	0.911422i	$-0.365015\pi$
$349$	15.3739	0.822944	0.411472	+	0.911422i	$0.365015\pi$
$350$	0	0
$351$	5.00000	0.266880
$352$	0	0
$353$	23.5390	1.25286	0.626428	−	0.779480i	$-0.284516\pi$
$353$	23.5390	1.25286	0.626428	+	0.779480i	$0.284516\pi$
$354$	0	0
$355$	−18.4955	−0.981637
$356$	0	0
$357$	1.41742	0.0750180
$358$	0	0
$359$	−14.3739	−0.758624	−0.379312	−	0.925269i	$-0.623839\pi$
$359$	−14.3739	−0.758624	−0.379312	+	0.925269i	$0.623839\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	18.5826	0.975332
$364$	0	0
$365$	−7.87841	−0.412375
$366$	0	0
$367$	32.4955	1.69625	0.848124	−	0.529797i	$-0.177732\pi$
$367$	32.4955	1.69625	0.848124	+	0.529797i	$0.177732\pi$
$368$	0	0
$369$	0.165151	0.00859744
$370$	0	0
$371$	−5.37386	−0.278997
$372$	0	0
$373$	−2.62614	−0.135976	−0.0679881	−	0.997686i	$-0.521658\pi$
$373$	−2.62614	−0.135976	−0.0679881	+	0.997686i	$0.521658\pi$
$374$	0	0
$375$	−5.37386	−0.277505
$376$	0	0
$377$	0.791288	0.0407534
$378$	0	0
$379$	5.74773	0.295241	0.147620	−	0.989044i	$-0.452839\pi$
$379$	5.74773	0.295241	0.147620	+	0.989044i	$0.452839\pi$
$380$	0	0
$381$	37.1652	1.90403
$382$	0	0
$383$	21.0000	1.07305	0.536525	−	0.843884i	$-0.319737\pi$
$383$	21.0000	1.07305	0.536525	+	0.843884i	$0.319737\pi$
$384$	0	0
$385$	2.37386	0.120983
$386$	0	0
$387$	−0.417424	−0.0212189
$388$	0	0
$389$	32.3739	1.64142	0.820710	−	0.571345i	$-0.193578\pi$
$389$	32.3739	1.64142	0.820710	+	0.571345i	$0.193578\pi$
$390$	0	0
$391$	3.62614	0.183382
$392$	0	0
$393$	34.2523	1.72780
$394$	0	0
$395$	30.0000	1.50946
$396$	0	0
$397$	32.7477	1.64356	0.821781	−	0.569804i	$-0.192981\pi$
$397$	32.7477	1.64356	0.821781	+	0.569804i	$0.192981\pi$
$398$	0	0
$399$	1.79129	0.0896766
$400$	0	0
$401$	−26.3739	−1.31705	−0.658524	−	0.752560i	$-0.728819\pi$
$401$	−26.3739	−1.31705	−0.658524	+	0.752560i	$0.728819\pi$
$402$	0	0
$403$	−6.37386	−0.317505
$404$	0	0
$405$	28.7477	1.42849
$406$	0	0
$407$	3.95644	0.196113
$408$	0	0
$409$	−14.1216	−0.698268	−0.349134	−	0.937073i	$-0.613524\pi$
$409$	−14.1216	−0.698268	−0.349134	+	0.937073i	$0.613524\pi$
$410$	0	0
$411$	−10.7477	−0.530146
$412$	0	0
$413$	6.16515	0.303367
$414$	0	0
$415$	1.87841	0.0922075
$416$	0	0
$417$	−31.7913	−1.55683
$418$	0	0
$419$	−1.74773	−0.0853821	−0.0426910	−	0.999088i	$-0.513593\pi$
$419$	−1.74773	−0.0853821	−0.0426910	+	0.999088i	$0.513593\pi$
$420$	0	0
$421$	19.0000	0.926003	0.463002	−	0.886357i	$-0.346772\pi$
$421$	19.0000	0.926003	0.463002	+	0.886357i	$0.346772\pi$
$422$	0	0
$423$	0.295834	0.0143839
$424$	0	0
$425$	−3.16515	−0.153532
$426$	0	0
$427$	1.00000	0.0483934
$428$	0	0
$429$	1.41742	0.0684339
$430$	0	0
$431$	−6.00000	−0.289010	−0.144505	−	0.989504i	$-0.546159\pi$
$431$	−6.00000	−0.289010	−0.144505	+	0.989504i	$0.546159\pi$
$432$	0	0
$433$	−17.7477	−0.852901	−0.426451	−	0.904511i	$-0.640236\pi$
$433$	−17.7477	−0.852901	−0.426451	+	0.904511i	$0.640236\pi$
$434$	0	0
$435$	4.25227	0.203881
$436$	0	0
$437$	4.58258	0.219214
$438$	0	0
$439$	−13.4955	−0.644103	−0.322051	−	0.946722i	$-0.604372\pi$
$439$	−13.4955	−0.644103	−0.322051	+	0.946722i	$0.604372\pi$
$440$	0	0
$441$	0.208712	0.00993867
$442$	0	0
$443$	36.9564	1.75585	0.877927	−	0.478795i	$-0.158926\pi$
$443$	36.9564	1.75585	0.877927	+	0.478795i	$0.158926\pi$
$444$	0	0
$445$	4.74773	0.225064
$446$	0	0
$447$	5.37386	0.254175
$448$	0	0
$449$	−16.1216	−0.760825	−0.380412	−	0.924817i	$-0.624218\pi$
$449$	−16.1216	−0.760825	−0.380412	+	0.924817i	$0.624218\pi$
$450$	0	0
$451$	−0.626136	−0.0294836
$452$	0	0
$453$	0.669697	0.0314651
$454$	0	0
$455$	−3.00000	−0.140642
$456$	0	0
$457$	28.8693	1.35045	0.675225	−	0.737612i	$-0.264047\pi$
$457$	28.8693	1.35045	0.675225	+	0.737612i	$0.264047\pi$
$458$	0	0
$459$	−3.95644	−0.184671
$460$	0	0
$461$	−20.5390	−0.956597	−0.478299	−	0.878197i	$-0.658746\pi$
$461$	−20.5390	−0.956597	−0.478299	+	0.878197i	$0.658746\pi$
$462$	0	0
$463$	20.4955	0.952505	0.476252	−	0.879309i	$-0.341995\pi$
$463$	20.4955	0.952505	0.476252	+	0.879309i	$0.341995\pi$
$464$	0	0
$465$	−34.2523	−1.58841
$466$	0	0
$467$	25.2867	1.17013	0.585065	−	0.810986i	$-0.301069\pi$
$467$	25.2867	1.17013	0.585065	+	0.810986i	$0.301069\pi$
$468$	0	0
$469$	−4.37386	−0.201966
$470$	0	0
$471$	37.8348	1.74334
$472$	0	0
$473$	1.58258	0.0727669
$474$	0	0
$475$	−4.00000	−0.183533
$476$	0	0
$477$	−1.12159	−0.0513541
$478$	0	0
$479$	26.3739	1.20505	0.602526	−	0.798099i	$-0.294161\pi$
$479$	26.3739	1.20505	0.602526	+	0.798099i	$0.294161\pi$
$480$	0	0
$481$	−5.00000	−0.227980
$482$	0	0
$483$	8.20871	0.373509
$484$	0	0
$485$	21.0000	0.953561
$486$	0	0
$487$	11.0000	0.498458	0.249229	−	0.968445i	$-0.419823\pi$
$487$	11.0000	0.498458	0.249229	+	0.968445i	$0.419823\pi$
$488$	0	0
$489$	−16.7913	−0.759328
$490$	0	0
$491$	−3.16515	−0.142841	−0.0714206	−	0.997446i	$-0.522753\pi$
$491$	−3.16515	−0.142841	−0.0714206	+	0.997446i	$0.522753\pi$
$492$	0	0
$493$	−0.626136	−0.0281998
$494$	0	0
$495$	0.495454	0.0222690
$496$	0	0
$497$	6.16515	0.276545
$498$	0	0
$499$	−31.3739	−1.40449	−0.702244	−	0.711937i	$-0.747818\pi$
$499$	−31.3739	−1.40449	−0.702244	+	0.711937i	$0.747818\pi$
$500$	0	0
$501$	−40.4519	−1.80726
$502$	0	0
$503$	30.3303	1.35236	0.676181	−	0.736736i	$-0.263634\pi$
$503$	30.3303	1.35236	0.676181	+	0.736736i	$0.263634\pi$
$504$	0	0
$505$	59.2432	2.63629
$506$	0	0
$507$	21.4955	0.954647
$508$	0	0
$509$	36.3303	1.61031	0.805156	−	0.593063i	$-0.202081\pi$
$509$	36.3303	1.61031	0.805156	+	0.593063i	$0.202081\pi$
$510$	0	0
$511$	2.62614	0.116173
$512$	0	0
$513$	−5.00000	−0.220755
$514$	0	0
$515$	39.0000	1.71855
$516$	0	0
$517$	−1.12159	−0.0493275
$518$	0	0
$519$	−8.50455	−0.373308
$520$	0	0
$521$	−22.5826	−0.989361	−0.494680	−	0.869075i	$-0.664715\pi$
$521$	−22.5826	−0.989361	−0.494680	+	0.869075i	$0.664715\pi$
$522$	0	0
$523$	−27.7477	−1.21332	−0.606662	−	0.794960i	$-0.707492\pi$
$523$	−27.7477	−1.21332	−0.606662	+	0.794960i	$0.707492\pi$
$524$	0	0
$525$	−7.16515	−0.312713
$526$	0	0
$527$	5.04356	0.219701
$528$	0	0
$529$	−2.00000	−0.0869565
$530$	0	0
$531$	1.28674	0.0558398
$532$	0	0
$533$	0.791288	0.0342745
$534$	0	0
$535$	−54.4955	−2.35604
$536$	0	0
$537$	−20.0780	−0.866431
$538$	0	0
$539$	−0.791288	−0.0340832
$540$	0	0
$541$	−2.00000	−0.0859867	−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000	−0.0859867	−0.0429934	+	0.999075i	$0.513689\pi$
$542$	0	0
$543$	7.83485	0.336226
$544$	0	0
$545$	47.2432	2.02368
$546$	0	0
$547$	−1.37386	−0.0587422	−0.0293711	−	0.999569i	$-0.509350\pi$
$547$	−1.37386	−0.0587422	−0.0293711	+	0.999569i	$0.509350\pi$
$548$	0	0
$549$	0.208712	0.00890762
$550$	0	0
$551$	−0.791288	−0.0337100
$552$	0	0
$553$	−10.0000	−0.425243
$554$	0	0
$555$	−26.8693	−1.14054
$556$	0	0
$557$	−14.3739	−0.609040	−0.304520	−	0.952506i	$-0.598496\pi$
$557$	−14.3739	−0.609040	−0.304520	+	0.952506i	$0.598496\pi$
$558$	0	0
$559$	−2.00000	−0.0845910
$560$	0	0
$561$	−1.12159	−0.0473536
$562$	0	0
$563$	−6.16515	−0.259830	−0.129915	−	0.991525i	$-0.541470\pi$
$563$	−6.16515	−0.259830	−0.129915	+	0.991525i	$0.541470\pi$
$564$	0	0
$565$	16.1216	0.678240
$566$	0	0
$567$	−9.58258	−0.402430
$568$	0	0
$569$	11.8348	0.496143	0.248071	−	0.968742i	$-0.420203\pi$
$569$	11.8348	0.496143	0.248071	+	0.968742i	$0.420203\pi$
$570$	0	0
$571$	22.4955	0.941405	0.470703	−	0.882292i	$-0.344000\pi$
$571$	22.4955	0.941405	0.470703	+	0.882292i	$0.344000\pi$
$572$	0	0
$573$	15.2958	0.638993
$574$	0	0
$575$	−18.3303	−0.764426
$576$	0	0
$577$	18.1216	0.754412	0.377206	−	0.926129i	$-0.376885\pi$
$577$	18.1216	0.754412	0.377206	+	0.926129i	$0.376885\pi$
$578$	0	0
$579$	11.4174	0.474492
$580$	0	0
$581$	−0.626136	−0.0259765
$582$	0	0
$583$	4.25227	0.176111
$584$	0	0
$585$	−0.626136	−0.0258876
$586$	0	0
$587$	−9.16515	−0.378286	−0.189143	−	0.981950i	$-0.560571\pi$
$587$	−9.16515	−0.378286	−0.189143	+	0.981950i	$0.560571\pi$
$588$	0	0
$589$	6.37386	0.262630
$590$	0	0
$591$	20.3739	0.838069
$592$	0	0
$593$	16.9129	0.694529	0.347264	−	0.937767i	$-0.387111\pi$
$593$	16.9129	0.694529	0.347264	+	0.937767i	$0.387111\pi$
$594$	0	0
$595$	2.37386	0.0973189
$596$	0	0
$597$	−6.71326	−0.274755
$598$	0	0
$599$	−6.62614	−0.270737	−0.135368	−	0.990795i	$-0.543222\pi$
$599$	−6.62614	−0.270737	−0.135368	+	0.990795i	$0.543222\pi$
$600$	0	0
$601$	25.3739	1.03502	0.517511	−	0.855677i	$-0.326859\pi$
$601$	25.3739	1.03502	0.517511	+	0.855677i	$0.326859\pi$
$602$	0	0
$603$	−0.912878	−0.0371753
$604$	0	0
$605$	31.1216	1.26527
$606$	0	0
$607$	−28.4955	−1.15659	−0.578297	−	0.815826i	$-0.696283\pi$
$607$	−28.4955	−1.15659	−0.578297	+	0.815826i	$0.696283\pi$
$608$	0	0
$609$	−1.41742	−0.0574369
$610$	0	0
$611$	1.41742	0.0573428
$612$	0	0
$613$	−20.6261	−0.833082	−0.416541	−	0.909117i	$-0.636758\pi$
$613$	−20.6261	−0.833082	−0.416541	+	0.909117i	$0.636758\pi$
$614$	0	0
$615$	4.25227	0.171468
$616$	0	0
$617$	15.9564	0.642382	0.321191	−	0.947014i	$-0.395917\pi$
$617$	15.9564	0.642382	0.321191	+	0.947014i	$0.395917\pi$
$618$	0	0
$619$	9.37386	0.376767	0.188384	−	0.982096i	$-0.439675\pi$
$619$	9.37386	0.376767	0.188384	+	0.982096i	$0.439675\pi$
$620$	0	0
$621$	−22.9129	−0.919462
$622$	0	0
$623$	−1.58258	−0.0634046
$624$	0	0
$625$	−29.0000	−1.16000
$626$	0	0
$627$	−1.41742	−0.0566065
$628$	0	0
$629$	3.95644	0.157754
$630$	0	0
$631$	−25.0000	−0.995234	−0.497617	−	0.867397i	$-0.665792\pi$
$631$	−25.0000	−0.995234	−0.497617	+	0.867397i	$0.665792\pi$
$632$	0	0
$633$	−19.0345	−0.756552
$634$	0	0
$635$	62.2432	2.47005
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	1.28674	0.0509027
$640$	0	0
$641$	−24.7913	−0.979197	−0.489598	−	0.871948i	$-0.662857\pi$
$641$	−24.7913	−0.979197	−0.489598	+	0.871948i	$0.662857\pi$
$642$	0	0
$643$	−23.0000	−0.907031	−0.453516	−	0.891248i	$-0.649830\pi$
$643$	−23.0000	−0.907031	−0.453516	+	0.891248i	$0.649830\pi$
$644$	0	0
$645$	−10.7477	−0.423191
$646$	0	0
$647$	−22.5826	−0.887813	−0.443906	−	0.896073i	$-0.646408\pi$
$647$	−22.5826	−0.887813	−0.443906	+	0.896073i	$0.646408\pi$
$648$	0	0
$649$	−4.87841	−0.191494
$650$	0	0
$651$	11.4174	0.447484
$652$	0	0
$653$	−28.5826	−1.11852	−0.559261	−	0.828991i	$-0.688915\pi$
$653$	−28.5826	−1.11852	−0.559261	+	0.828991i	$0.688915\pi$
$654$	0	0
$655$	57.3648	2.24143
$656$	0	0
$657$	0.548107	0.0213837
$658$	0	0
$659$	14.0436	0.547059	0.273530	−	0.961864i	$-0.411809\pi$
$659$	14.0436	0.547059	0.273530	+	0.961864i	$0.411809\pi$
$660$	0	0
$661$	34.0000	1.32245	0.661223	−	0.750189i	$-0.270038\pi$
$661$	34.0000	1.32245	0.661223	+	0.750189i	$0.270038\pi$
$662$	0	0
$663$	1.41742	0.0550482
$664$	0	0
$665$	3.00000	0.116335
$666$	0	0
$667$	−3.62614	−0.140405
$668$	0	0
$669$	−6.71326	−0.259550
$670$	0	0
$671$	−0.791288	−0.0305473
$672$	0	0
$673$	24.1216	0.929819	0.464909	−	0.885358i	$-0.346087\pi$
$673$	24.1216	0.929819	0.464909	+	0.885358i	$0.346087\pi$
$674$	0	0
$675$	20.0000	0.769800
$676$	0	0
$677$	−0.956439	−0.0367589	−0.0183795	−	0.999831i	$-0.505851\pi$
$677$	−0.956439	−0.0367589	−0.0183795	+	0.999831i	$0.505851\pi$
$678$	0	0
$679$	−7.00000	−0.268635
$680$	0	0
$681$	1.41742	0.0543158
$682$	0	0
$683$	17.8348	0.682432	0.341216	−	0.939985i	$-0.389161\pi$
$683$	17.8348	0.682432	0.341216	+	0.939985i	$0.389161\pi$
$684$	0	0
$685$	−18.0000	−0.687745
$686$	0	0
$687$	−39.4083	−1.50352
$688$	0	0
$689$	−5.37386	−0.204728
$690$	0	0
$691$	−21.2523	−0.808475	−0.404237	−	0.914654i	$-0.632463\pi$
$691$	−21.2523	−0.808475	−0.404237	+	0.914654i	$0.632463\pi$
$692$	0	0
$693$	−0.165151	−0.00627358
$694$	0	0
$695$	−53.2432	−2.01963
$696$	0	0
$697$	−0.626136	−0.0237166
$698$	0	0
$699$	−23.5045	−0.889024
$700$	0	0
$701$	−5.66970	−0.214142	−0.107071	−	0.994251i	$-0.534147\pi$
$701$	−5.66970	−0.214142	−0.107071	+	0.994251i	$0.534147\pi$
$702$	0	0
$703$	5.00000	0.188579
$704$	0	0
$705$	7.61704	0.286875
$706$	0	0
$707$	−19.7477	−0.742690
$708$	0	0
$709$	51.2432	1.92448	0.962239	−	0.272206i	$-0.0877532\pi$
$709$	51.2432	1.92448	0.962239	+	0.272206i	$0.0877532\pi$
$710$	0	0
$711$	−2.08712	−0.0782732
$712$	0	0
$713$	29.2087	1.09387
$714$	0	0
$715$	2.37386	0.0887775
$716$	0	0
$717$	−35.3739	−1.32106
$718$	0	0
$719$	−12.0000	−0.447524	−0.223762	−	0.974644i	$-0.571834\pi$
$719$	−12.0000	−0.447524	−0.223762	+	0.974644i	$0.571834\pi$
$720$	0	0
$721$	−13.0000	−0.484145
$722$	0	0
$723$	−33.5826	−1.24895
$724$	0	0
$725$	3.16515	0.117551
$726$	0	0
$727$	11.0000	0.407967	0.203984	−	0.978974i	$-0.434611\pi$
$727$	11.0000	0.407967	0.203984	+	0.978974i	$0.434611\pi$
$728$	0	0
$729$	24.8693	0.921086
$730$	0	0
$731$	1.58258	0.0585337
$732$	0	0
$733$	13.4955	0.498466	0.249233	−	0.968444i	$-0.419822\pi$
$733$	13.4955	0.498466	0.249233	+	0.968444i	$0.419822\pi$
$734$	0	0
$735$	5.37386	0.198218
$736$	0	0
$737$	3.46099	0.127487
$738$	0	0
$739$	8.25227	0.303565	0.151782	−	0.988414i	$-0.451499\pi$
$739$	8.25227	0.303565	0.151782	+	0.988414i	$0.451499\pi$
$740$	0	0
$741$	1.79129	0.0658046
$742$	0	0
$743$	−15.3303	−0.562414	−0.281207	−	0.959647i	$-0.590735\pi$
$743$	−15.3303	−0.562414	−0.281207	+	0.959647i	$0.590735\pi$
$744$	0	0
$745$	9.00000	0.329734
$746$	0	0
$747$	−0.130682	−0.00478141
$748$	0	0
$749$	18.1652	0.663740
$750$	0	0
$751$	25.3739	0.925905	0.462953	−	0.886383i	$-0.346790\pi$
$751$	25.3739	0.925905	0.462953	+	0.886383i	$0.346790\pi$
$752$	0	0
$753$	9.33030	0.340015
$754$	0	0
$755$	1.12159	0.0408189
$756$	0	0
$757$	−2.00000	−0.0726912	−0.0363456	−	0.999339i	$-0.511572\pi$
$757$	−2.00000	−0.0726912	−0.0363456	+	0.999339i	$0.511572\pi$
$758$	0	0
$759$	−6.49545	−0.235770
$760$	0	0
$761$	−2.66970	−0.0967764	−0.0483882	−	0.998829i	$-0.515408\pi$
$761$	−2.66970	−0.0967764	−0.0483882	+	0.998829i	$0.515408\pi$
$762$	0	0
$763$	−15.7477	−0.570106
$764$	0	0
$765$	0.495454	0.0179132
$766$	0	0
$767$	6.16515	0.222611
$768$	0	0
$769$	41.4955	1.49636	0.748182	−	0.663493i	$-0.230927\pi$
$769$	41.4955	1.49636	0.748182	+	0.663493i	$0.230927\pi$
$770$	0	0
$771$	−28.2867	−1.01872
$772$	0	0
$773$	10.7477	0.386569	0.193284	−	0.981143i	$-0.438086\pi$
$773$	10.7477	0.386569	0.193284	+	0.981143i	$0.438086\pi$
$774$	0	0
$775$	−25.4955	−0.915824
$776$	0	0
$777$	8.95644	0.321310
$778$	0	0
$779$	−0.791288	−0.0283508
$780$	0	0
$781$	−4.87841	−0.174563
$782$	0	0
$783$	3.95644	0.141392
$784$	0	0
$785$	63.3648	2.26159
$786$	0	0
$787$	4.00000	0.142585	0.0712923	−	0.997455i	$-0.477288\pi$
$787$	4.00000	0.142585	0.0712923	+	0.997455i	$0.477288\pi$
$788$	0	0
$789$	−3.95644	−0.140853
$790$	0	0
$791$	−5.37386	−0.191073
$792$	0	0
$793$	1.00000	0.0355110
$794$	0	0
$795$	−28.8784	−1.02421
$796$	0	0
$797$	−35.8693	−1.27056	−0.635278	−	0.772283i	$-0.719115\pi$
$797$	−35.8693	−1.27056	−0.635278	+	0.772283i	$0.719115\pi$
$798$	0	0
$799$	−1.12159	−0.0396790
$800$	0	0
$801$	−0.330303	−0.0116707
$802$	0	0
$803$	−2.07803	−0.0733321
$804$	0	0
$805$	13.7477	0.484544
$806$	0	0
$807$	23.2087	0.816985
$808$	0	0
$809$	22.9129	0.805574	0.402787	−	0.915294i	$-0.368042\pi$
$809$	22.9129	0.805574	0.402787	+	0.915294i	$0.368042\pi$
$810$	0	0
$811$	−23.4955	−0.825037	−0.412518	−	0.910949i	$-0.635351\pi$
$811$	−23.4955	−0.825037	−0.412518	+	0.910949i	$0.635351\pi$
$812$	0	0
$813$	−48.5826	−1.70387
$814$	0	0
$815$	−28.1216	−0.985056
$816$	0	0
$817$	2.00000	0.0699711
$818$	0	0
$819$	0.208712	0.00729299
$820$	0	0
$821$	−5.66970	−0.197874	−0.0989369	−	0.995094i	$-0.531544\pi$
$821$	−5.66970	−0.197874	−0.0989369	+	0.995094i	$0.531544\pi$
$822$	0	0
$823$	28.2432	0.984495	0.492248	−	0.870455i	$-0.336175\pi$
$823$	28.2432	0.984495	0.492248	+	0.870455i	$0.336175\pi$
$824$	0	0
$825$	5.66970	0.197394
$826$	0	0
$827$	33.6606	1.17049	0.585247	−	0.810855i	$-0.300998\pi$
$827$	33.6606	1.17049	0.585247	+	0.810855i	$0.300998\pi$
$828$	0	0
$829$	4.00000	0.138926	0.0694629	−	0.997585i	$-0.477871\pi$
$829$	4.00000	0.138926	0.0694629	+	0.997585i	$0.477871\pi$
$830$	0	0
$831$	55.0780	1.91064
$832$	0	0
$833$	−0.791288	−0.0274165
$834$	0	0
$835$	−67.7477	−2.34451
$836$	0	0
$837$	−31.8693	−1.10156
$838$	0	0
$839$	27.1652	0.937845	0.468923	−	0.883239i	$-0.344642\pi$
$839$	27.1652	0.937845	0.468923	+	0.883239i	$0.344642\pi$
$840$	0	0
$841$	−28.3739	−0.978409
$842$	0	0
$843$	−51.1996	−1.76341
$844$	0	0
$845$	36.0000	1.23844
$846$	0	0
$847$	−10.3739	−0.356450
$848$	0	0
$849$	13.2087	0.453322
$850$	0	0
$851$	22.9129	0.785443
$852$	0	0
$853$	32.1216	1.09982	0.549911	−	0.835223i	$-0.314662\pi$
$853$	32.1216	1.09982	0.549911	+	0.835223i	$0.314662\pi$
$854$	0	0
$855$	0.626136	0.0214134
$856$	0	0
$857$	38.7042	1.32211	0.661055	−	0.750338i	$-0.270109\pi$
$857$	38.7042	1.32211	0.661055	+	0.750338i	$0.270109\pi$
$858$	0	0
$859$	6.37386	0.217473	0.108737	−	0.994071i	$-0.465319\pi$
$859$	6.37386	0.217473	0.108737	+	0.994071i	$0.465319\pi$
$860$	0	0
$861$	−1.41742	−0.0483057
$862$	0	0
$863$	11.7042	0.398414	0.199207	−	0.979957i	$-0.436163\pi$
$863$	11.7042	0.398414	0.199207	+	0.979957i	$0.436163\pi$
$864$	0	0
$865$	−14.2432	−0.484283
$866$	0	0
$867$	29.3303	0.996109
$868$	0	0
$869$	7.91288	0.268426
$870$	0	0
$871$	−4.37386	−0.148203
$872$	0	0
$873$	−1.46099	−0.0494469
$874$	0	0
$875$	3.00000	0.101419
$876$	0	0
$877$	11.2523	0.379962	0.189981	−	0.981788i	$-0.439157\pi$
$877$	11.2523	0.379962	0.189981	+	0.981788i	$0.439157\pi$
$878$	0	0
$879$	−13.8784	−0.468107
$880$	0	0
$881$	−11.3739	−0.383195	−0.191598	−	0.981474i	$-0.561367\pi$
$881$	−11.3739	−0.383195	−0.191598	+	0.981474i	$0.561367\pi$
$882$	0	0
$883$	−51.7477	−1.74145	−0.870725	−	0.491771i	$-0.836350\pi$
$883$	−51.7477	−1.74145	−0.870725	+	0.491771i	$0.836350\pi$
$884$	0	0
$885$	33.1307	1.11368
$886$	0	0
$887$	−4.58258	−0.153868	−0.0769339	−	0.997036i	$-0.524513\pi$
$887$	−4.58258	−0.153868	−0.0769339	+	0.997036i	$0.524513\pi$
$888$	0	0
$889$	−20.7477	−0.695856
$890$	0	0
$891$	7.58258	0.254026
$892$	0	0
$893$	−1.41742	−0.0474323
$894$	0	0
$895$	−33.6261	−1.12400
$896$	0	0
$897$	8.20871	0.274081
$898$	0	0
$899$	−5.04356	−0.168212
$900$	0	0
$901$	4.25227	0.141664
$902$	0	0
$903$	3.58258	0.119221
$904$	0	0
$905$	13.1216	0.436176
$906$	0	0
$907$	51.2432	1.70150	0.850751	−	0.525569i	$-0.176148\pi$
$907$	51.2432	1.70150	0.850751	+	0.525569i	$0.176148\pi$
$908$	0	0
$909$	−4.12159	−0.136705
$910$	0	0
$911$	−51.4955	−1.70612	−0.853060	−	0.521812i	$-0.825256\pi$
$911$	−51.4955	−1.70612	−0.853060	+	0.521812i	$0.825256\pi$
$912$	0	0
$913$	0.495454	0.0163971
$914$	0	0
$915$	5.37386	0.177654
$916$	0	0
$917$	−19.1216	−0.631451
$918$	0	0
$919$	20.4955	0.676083	0.338041	−	0.941131i	$-0.390236\pi$
$919$	20.4955	0.676083	0.338041	+	0.941131i	$0.390236\pi$
$920$	0	0
$921$	−24.4083	−0.804282
$922$	0	0
$923$	6.16515	0.202928
$924$	0	0
$925$	−20.0000	−0.657596
$926$	0	0
$927$	−2.71326	−0.0891151
$928$	0	0
$929$	−20.8693	−0.684700	−0.342350	−	0.939572i	$-0.611223\pi$
$929$	−20.8693	−0.684700	−0.342350	+	0.939572i	$0.611223\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	−55.1561	−1.80573
$934$	0	0
$935$	−1.87841	−0.0614306
$936$	0	0
$937$	−1.62614	−0.0531236	−0.0265618	−	0.999647i	$-0.508456\pi$
$937$	−1.62614	−0.0531236	−0.0265618	+	0.999647i	$0.508456\pi$
$938$	0	0
$939$	26.4174	0.862100
$940$	0	0
$941$	27.4955	0.896326	0.448163	−	0.893952i	$-0.352078\pi$
$941$	27.4955	0.896326	0.448163	+	0.893952i	$0.352078\pi$
$942$	0	0
$943$	−3.62614	−0.118083
$944$	0	0
$945$	−15.0000	−0.487950
$946$	0	0
$947$	−43.6170	−1.41736	−0.708682	−	0.705528i	$-0.750710\pi$
$947$	−43.6170	−1.41736	−0.708682	+	0.705528i	$0.750710\pi$
$948$	0	0
$949$	2.62614	0.0852480
$950$	0	0
$951$	59.4083	1.92645
$952$	0	0
$953$	53.5390	1.73430	0.867149	−	0.498048i	$-0.165950\pi$
$953$	53.5390	1.73430	0.867149	+	0.498048i	$0.165950\pi$
$954$	0	0
$955$	25.6170	0.828948
$956$	0	0
$957$	1.12159	0.0362559
$958$	0	0
$959$	6.00000	0.193750
$960$	0	0
$961$	9.62614	0.310521
$962$	0	0
$963$	3.79129	0.122173
$964$	0	0
$965$	19.1216	0.615546
$966$	0	0
$967$	−29.1216	−0.936487	−0.468244	−	0.883599i	$-0.655113\pi$
$967$	−29.1216	−0.936487	−0.468244	+	0.883599i	$0.655113\pi$
$968$	0	0
$969$	−1.41742	−0.0455342
$970$	0	0
$971$	−15.0000	−0.481373	−0.240686	−	0.970603i	$-0.577373\pi$
$971$	−15.0000	−0.481373	−0.240686	+	0.970603i	$0.577373\pi$
$972$	0	0
$973$	17.7477	0.568966
$974$	0	0
$975$	−7.16515	−0.229468
$976$	0	0
$977$	−43.5826	−1.39433	−0.697165	−	0.716911i	$-0.745556\pi$
$977$	−43.5826	−1.39433	−0.697165	+	0.716911i	$0.745556\pi$
$978$	0	0
$979$	1.25227	0.0400228
$980$	0	0
$981$	−3.28674	−0.104938
$982$	0	0
$983$	45.3303	1.44581	0.722906	−	0.690946i	$-0.242806\pi$
$983$	45.3303	1.44581	0.722906	+	0.690946i	$0.242806\pi$
$984$	0	0
$985$	34.1216	1.08720
$986$	0	0
$987$	−2.53901	−0.0808177
$988$	0	0
$989$	9.16515	0.291435
$990$	0	0
$991$	18.7477	0.595541	0.297771	−	0.954637i	$-0.403757\pi$
$991$	18.7477	0.595541	0.297771	+	0.954637i	$0.403757\pi$
$992$	0	0
$993$	26.1996	0.831420
$994$	0	0
$995$	−11.2432	−0.356433
$996$	0	0
$997$	44.1216	1.39734	0.698672	−	0.715442i	$-0.253775\pi$
$997$	44.1216	1.39734	0.698672	+	0.715442i	$0.253775\pi$
$998$	0	0
$999$	−25.0000	−0.790965

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8512.2.a.t.1.1		2
4.3	odd	2		8512.2.a.m.1.2		2
8.3	odd	2		2128.2.a.k.1.1		2
8.5	even	2		532.2.a.c.1.2	✓	2
24.5	odd	2		4788.2.a.g.1.1		2
56.13	odd	2		3724.2.a.e.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
532.2.a.c.1.2	✓	2	8.5	even	2
2128.2.a.k.1.1		2	8.3	odd	2
3724.2.a.e.1.1		2	56.13	odd	2
4788.2.a.g.1.1		2	24.5	odd	2
8512.2.a.m.1.2		2	4.3	odd	2
8512.2.a.t.1.1		2	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:	\( N \)	\(=\)	\( 8512 = 2^{6} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8512.a (trivial)

\(f(q)\)	\(=\)	\(q-1.79129 q^{3} -3.00000 q^{5} +1.00000 q^{7} +0.208712 q^{9} -0.791288 q^{11} +1.00000 q^{13} +5.37386 q^{15} -0.791288 q^{17} -1.00000 q^{19} -1.79129 q^{21} -4.58258 q^{23} +4.00000 q^{25} +5.00000 q^{27} +0.791288 q^{29} -6.37386 q^{31} +1.41742 q^{33} -3.00000 q^{35} -5.00000 q^{37} -1.79129 q^{39} +0.791288 q^{41} -2.00000 q^{43} -0.626136 q^{45} +1.41742 q^{47} +1.00000 q^{49} +1.41742 q^{51} -5.37386 q^{53} +2.37386 q^{55} +1.79129 q^{57} +6.16515 q^{59} +1.00000 q^{61} +0.208712 q^{63} -3.00000 q^{65} -4.37386 q^{67} +8.20871 q^{69} +6.16515 q^{71} +2.62614 q^{73} -7.16515 q^{75} -0.791288 q^{77} -10.0000 q^{79} -9.58258 q^{81} -0.626136 q^{83} +2.37386 q^{85} -1.41742 q^{87} -1.58258 q^{89} +1.00000 q^{91} +11.4174 q^{93} +3.00000 q^{95} -7.00000 q^{97} -0.165151 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} - 6 q^{5} + 2 q^{7} + 5 q^{9} + 3 q^{11} + 2 q^{13} - 3 q^{15} + 3 q^{17} - 2 q^{19} + q^{21} + 8 q^{25} + 10 q^{27} - 3 q^{29} + q^{31} + 12 q^{33} - 6 q^{35} - 10 q^{37} + q^{39} - 3 q^{41}+ \cdots + 18 q^{99}+O(q^{100}) \) 2 * q + q^3 - 6 * q^5 + 2 * q^7 + 5 * q^9 + 3 * q^11 + 2 * q^13 - 3 * q^15 + 3 * q^17 - 2 * q^19 + q^21 + 8 * q^25 + 10 * q^27 - 3 * q^29 + q^31 + 12 * q^33 - 6 * q^35 - 10 * q^37 + q^39 - 3 * q^41 - 4 * q^43 - 15 * q^45 + 12 * q^47 + 2 * q^49 + 12 * q^51 + 3 * q^53 - 9 * q^55 - q^57 - 6 * q^59 + 2 * q^61 + 5 * q^63 - 6 * q^65 + 5 * q^67 + 21 * q^69 - 6 * q^71 + 19 * q^73 + 4 * q^75 + 3 * q^77 - 20 * q^79 - 10 * q^81 - 15 * q^83 - 9 * q^85 - 12 * q^87 + 6 * q^89 + 2 * q^91 + 32 * q^93 + 6 * q^95 - 14 * q^97 + 18 * q^99

Introduction

L-functions

Modular forms

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Database

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