LMFDB - Embedded newform 6028.2.a.f.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6028,2,Mod(1,6028)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6028, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6028.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6028 = 2^{2} \cdot 11 \cdot 137 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6028.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:	$48.1338223384$
Analytic rank:	$0$
Dimension:	$29$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Character	$\chi$	$=$	6028.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.84910 q^{3} -0.463178 q^{5} -1.96260 q^{7} +0.419175 q^{9} +O(q^{10})$	$q-1.84910 q^{3} -0.463178 q^{5} -1.96260 q^{7} +0.419175 q^{9} +1.00000 q^{11} +5.77846 q^{13} +0.856463 q^{15} -0.0856389 q^{17} +4.69155 q^{19} +3.62905 q^{21} -3.17781 q^{23} -4.78547 q^{25} +4.77221 q^{27} -1.91934 q^{29} -0.614404 q^{31} -1.84910 q^{33} +0.909033 q^{35} +1.90430 q^{37} -10.6850 q^{39} +0.505352 q^{41} +0.806582 q^{43} -0.194153 q^{45} +2.69618 q^{47} -3.14820 q^{49} +0.158355 q^{51} -2.68878 q^{53} -0.463178 q^{55} -8.67516 q^{57} +5.35396 q^{59} -3.44350 q^{61} -0.822673 q^{63} -2.67646 q^{65} +11.0371 q^{67} +5.87609 q^{69} +13.8404 q^{71} -9.79702 q^{73} +8.84881 q^{75} -1.96260 q^{77} -14.3205 q^{79} -10.0818 q^{81} -14.3064 q^{83} +0.0396661 q^{85} +3.54905 q^{87} -12.4691 q^{89} -11.3408 q^{91} +1.13610 q^{93} -2.17302 q^{95} -18.2623 q^{97} +0.419175 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 29 q + 14 q^{3} + 9 q^{5} + 14 q^{7} + 43 q^{9}+O(q^{10}) $ 29 * q + 14 * q^3 + 9 * q^5 + 14 * q^7 + 43 * q^9	$ 29 q + 14 q^{3} + 9 q^{5} + 14 q^{7} + 43 q^{9} + 29 q^{11} + 10 q^{15} + 29 q^{17} + 7 q^{19} + 2 q^{21} + 36 q^{23} + 36 q^{25} + 50 q^{27} + 9 q^{29} + 28 q^{31} + 14 q^{33} + 15 q^{35} + 25 q^{37} + 9 q^{39} + 19 q^{41} + 23 q^{43} + 5 q^{45} + 27 q^{47} + 27 q^{49} + 13 q^{51} + 4 q^{53} + 9 q^{55} + 14 q^{57} + 40 q^{59} + 20 q^{61} - 17 q^{63} + 9 q^{65} + 59 q^{67} + 30 q^{69} + 29 q^{71} - 5 q^{73} + 46 q^{75} + 14 q^{77} + 29 q^{79} + 61 q^{81} + 35 q^{83} - 57 q^{85} + 45 q^{87} + 39 q^{89} + 45 q^{91} - 8 q^{93} + q^{95} + 55 q^{97} + 43 q^{99}+O(q^{100}) $ 29 * q + 14 * q^3 + 9 * q^5 + 14 * q^7 + 43 * q^9 + 29 * q^11 + 10 * q^15 + 29 * q^17 + 7 * q^19 + 2 * q^21 + 36 * q^23 + 36 * q^25 + 50 * q^27 + 9 * q^29 + 28 * q^31 + 14 * q^33 + 15 * q^35 + 25 * q^37 + 9 * q^39 + 19 * q^41 + 23 * q^43 + 5 * q^45 + 27 * q^47 + 27 * q^49 + 13 * q^51 + 4 * q^53 + 9 * q^55 + 14 * q^57 + 40 * q^59 + 20 * q^61 - 17 * q^63 + 9 * q^65 + 59 * q^67 + 30 * q^69 + 29 * q^71 - 5 * q^73 + 46 * q^75 + 14 * q^77 + 29 * q^79 + 61 * q^81 + 35 * q^83 - 57 * q^85 + 45 * q^87 + 39 * q^89 + 45 * q^91 - 8 * q^93 + q^95 + 55 * q^97 + 43 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	−1.84910	−1.06758	−0.533790	−	0.845617i	$-0.679233\pi$
$3$	−1.84910	−1.06758	−0.533790	+	0.845617i	$0.679233\pi$
$4$	0	0
$5$	−0.463178	−0.207140	−0.103570	−	0.994622i	$-0.533027\pi$
$5$	−0.463178	−0.207140	−0.103570	+	0.994622i	$0.533027\pi$
$6$	0	0
$7$	−1.96260	−0.741793	−0.370897	−	0.928674i	$-0.620950\pi$
$7$	−1.96260	−0.741793	−0.370897	+	0.928674i	$0.620950\pi$
$8$	0	0
$9$	0.419175	0.139725
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	5.77846	1.60266	0.801328	−	0.598225i	$-0.204127\pi$
$13$	5.77846	1.60266	0.801328	+	0.598225i	$0.204127\pi$
$14$	0	0
$15$	0.856463	0.221138
$16$	0	0
$17$	−0.0856389	−0.0207705	−0.0103852	−	0.999946i	$-0.503306\pi$
$17$	−0.0856389	−0.0207705	−0.0103852	+	0.999946i	$0.503306\pi$
$18$	0	0
$19$	4.69155	1.07632	0.538158	−	0.842844i	$-0.319120\pi$
$19$	4.69155	1.07632	0.538158	+	0.842844i	$0.319120\pi$
$20$	0	0
$21$	3.62905	0.791923
$22$	0	0
$23$	−3.17781	−0.662619	−0.331309	−	0.943522i	$-0.607490\pi$
$23$	−3.17781	−0.662619	−0.331309	+	0.943522i	$0.607490\pi$
$24$	0	0
$25$	−4.78547	−0.957093
$26$	0	0
$27$	4.77221	0.918412
$28$	0	0
$29$	−1.91934	−0.356412	−0.178206	−	0.983993i	$-0.557029\pi$
$29$	−1.91934	−0.356412	−0.178206	+	0.983993i	$0.557029\pi$
$30$	0	0
$31$	−0.614404	−0.110350	−0.0551751	−	0.998477i	$-0.517572\pi$
$31$	−0.614404	−0.110350	−0.0551751	+	0.998477i	$0.517572\pi$
$32$	0	0
$33$	−1.84910	−0.321887
$34$	0	0
$35$	0.909033	0.153655
$36$	0	0
$37$	1.90430	0.313064	0.156532	−	0.987673i	$-0.449969\pi$
$37$	1.90430	0.313064	0.156532	+	0.987673i	$0.449969\pi$
$38$	0	0
$39$	−10.6850	−1.71096
$40$	0	0
$41$	0.505352	0.0789227	0.0394614	−	0.999221i	$-0.487436\pi$
$41$	0.505352	0.0789227	0.0394614	+	0.999221i	$0.487436\pi$
$42$	0	0
$43$	0.806582	0.123003	0.0615013	−	0.998107i	$-0.480411\pi$
$43$	0.806582	0.123003	0.0615013	+	0.998107i	$0.480411\pi$
$44$	0	0
$45$	−0.194153	−0.0289426
$46$	0	0
$47$	2.69618	0.393278	0.196639	−	0.980476i	$-0.436997\pi$
$47$	2.69618	0.393278	0.196639	+	0.980476i	$0.436997\pi$
$48$	0	0
$49$	−3.14820	−0.449743
$50$	0	0
$51$	0.158355	0.0221741
$52$	0	0
$53$	−2.68878	−0.369332	−0.184666	−	0.982801i	$-0.559120\pi$
$53$	−2.68878	−0.369332	−0.184666	+	0.982801i	$0.559120\pi$
$54$	0	0
$55$	−0.463178	−0.0624549
$56$	0	0
$57$	−8.67516	−1.14905
$58$	0	0
$59$	5.35396	0.697027	0.348513	−	0.937304i	$-0.386687\pi$
$59$	5.35396	0.697027	0.348513	+	0.937304i	$0.386687\pi$
$60$	0	0
$61$	−3.44350	−0.440895	−0.220448	−	0.975399i	$-0.570752\pi$
$61$	−3.44350	−0.440895	−0.220448	+	0.975399i	$0.570752\pi$
$62$	0	0
$63$	−0.822673	−0.103647
$64$	0	0
$65$	−2.67646	−0.331974
$66$	0	0
$67$	11.0371	1.34839	0.674195	−	0.738553i	$-0.264491\pi$
$67$	11.0371	1.34839	0.674195	+	0.738553i	$0.264491\pi$
$68$	0	0
$69$	5.87609	0.707398
$70$	0	0
$71$	13.8404	1.64255	0.821277	−	0.570530i	$-0.193262\pi$
$71$	13.8404	1.64255	0.821277	+	0.570530i	$0.193262\pi$
$72$	0	0
$73$	−9.79702	−1.14665	−0.573327	−	0.819326i	$-0.694348\pi$
$73$	−9.79702	−1.14665	−0.573327	+	0.819326i	$0.694348\pi$
$74$	0	0
$75$	8.84881	1.02177
$76$	0	0
$77$	−1.96260	−0.223659
$78$	0	0
$79$	−14.3205	−1.61118	−0.805591	−	0.592472i	$-0.798152\pi$
$79$	−14.3205	−1.61118	−0.805591	+	0.592472i	$0.798152\pi$
$80$	0	0
$81$	−10.0818	−1.12020
$82$	0	0
$83$	−14.3064	−1.57033	−0.785164	−	0.619289i	$-0.787421\pi$
$83$	−14.3064	−1.57033	−0.785164	+	0.619289i	$0.787421\pi$
$84$	0	0
$85$	0.0396661	0.00430239
$86$	0	0
$87$	3.54905	0.380498
$88$	0	0
$89$	−12.4691	−1.32173	−0.660863	−	0.750506i	$-0.729810\pi$
$89$	−12.4691	−1.32173	−0.660863	+	0.750506i	$0.729810\pi$
$90$	0	0
$91$	−11.3408	−1.18884
$92$	0	0
$93$	1.13610	0.117808
$94$	0	0
$95$	−2.17302	−0.222948
$96$	0	0
$97$	−18.2623	−1.85426	−0.927130	−	0.374741i	$-0.877732\pi$
$97$	−18.2623	−1.85426	−0.927130	+	0.374741i	$0.877732\pi$
$98$	0	0
$99$	0.419175	0.0421287
$100$	0	0
$101$	−4.78956	−0.476579	−0.238290	−	0.971194i	$-0.576587\pi$
$101$	−4.78956	−0.476579	−0.238290	+	0.971194i	$0.576587\pi$
$102$	0	0
$103$	−3.33661	−0.328766	−0.164383	−	0.986397i	$-0.552563\pi$
$103$	−3.33661	−0.328766	−0.164383	+	0.986397i	$0.552563\pi$
$104$	0	0
$105$	−1.68089	−0.164039
$106$	0	0
$107$	10.2261	0.988599	0.494300	−	0.869292i	$-0.335425\pi$
$107$	10.2261	0.988599	0.494300	+	0.869292i	$0.335425\pi$
$108$	0	0
$109$	6.91368	0.662210	0.331105	−	0.943594i	$-0.392578\pi$
$109$	6.91368	0.662210	0.331105	+	0.943594i	$0.392578\pi$
$110$	0	0
$111$	−3.52124	−0.334221
$112$	0	0
$113$	10.1240	0.952390	0.476195	−	0.879340i	$-0.342016\pi$
$113$	10.1240	0.952390	0.476195	+	0.879340i	$0.342016\pi$
$114$	0	0
$115$	1.47189	0.137255
$116$	0	0
$117$	2.42219	0.223931
$118$	0	0
$119$	0.168075	0.0154074
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−0.934447	−0.0842562
$124$	0	0
$125$	4.53241	0.405391
$126$	0	0
$127$	13.9046	1.23384	0.616918	−	0.787027i	$-0.288381\pi$
$127$	13.9046	1.23384	0.616918	+	0.787027i	$0.288381\pi$
$128$	0	0
$129$	−1.49145	−0.131315
$130$	0	0
$131$	16.7403	1.46261	0.731303	−	0.682052i	$-0.238912\pi$
$131$	16.7403	1.46261	0.731303	+	0.682052i	$0.238912\pi$
$132$	0	0
$133$	−9.20764	−0.798404
$134$	0	0
$135$	−2.21038	−0.190239
$136$	0	0
$137$	1.00000	0.0854358
$137$	1.00000	0.0854358
$138$	0	0
$139$	9.67390	0.820530	0.410265	−	0.911966i	$-0.365436\pi$
$139$	9.67390	0.820530	0.410265	+	0.911966i	$0.365436\pi$
$140$	0	0
$141$	−4.98551	−0.419856
$142$	0	0
$143$	5.77846	0.483219
$144$	0	0
$145$	0.888996	0.0738271
$146$	0	0
$147$	5.82134	0.480136
$148$	0	0
$149$	16.8638	1.38154	0.690769	−	0.723075i	$-0.257272\pi$
$149$	16.8638	1.38154	0.690769	+	0.723075i	$0.257272\pi$
$150$	0	0
$151$	−13.6673	−1.11223	−0.556113	−	0.831107i	$-0.687708\pi$
$151$	−13.6673	−1.11223	−0.556113	+	0.831107i	$0.687708\pi$
$152$	0	0
$153$	−0.0358977	−0.00290216
$154$	0	0
$155$	0.284579	0.0228579
$156$	0	0
$157$	−11.3267	−0.903970	−0.451985	−	0.892025i	$-0.649284\pi$
$157$	−11.3267	−0.903970	−0.451985	+	0.892025i	$0.649284\pi$
$158$	0	0
$159$	4.97182	0.394291
$160$	0	0
$161$	6.23677	0.491526
$162$	0	0
$163$	4.31928	0.338312	0.169156	−	0.985589i	$-0.445896\pi$
$163$	4.31928	0.338312	0.169156	+	0.985589i	$0.445896\pi$
$164$	0	0
$165$	0.856463	0.0666756
$166$	0	0
$167$	10.5505	0.816426	0.408213	−	0.912887i	$-0.366152\pi$
$167$	10.5505	0.816426	0.408213	+	0.912887i	$0.366152\pi$
$168$	0	0
$169$	20.3906	1.56851
$170$	0	0
$171$	1.96658	0.150388
$172$	0	0
$173$	18.9554	1.44116	0.720578	−	0.693374i	$-0.243877\pi$
$173$	18.9554	1.44116	0.720578	+	0.693374i	$0.243877\pi$
$174$	0	0
$175$	9.39195	0.709965
$176$	0	0
$177$	−9.90002	−0.744131
$178$	0	0
$179$	3.76938	0.281737	0.140868	−	0.990028i	$-0.455011\pi$
$179$	3.76938	0.281737	0.140868	+	0.990028i	$0.455011\pi$
$180$	0	0
$181$	10.1319	0.753096	0.376548	−	0.926397i	$-0.377111\pi$
$181$	10.1319	0.753096	0.376548	+	0.926397i	$0.377111\pi$
$182$	0	0
$183$	6.36738	0.470690
$184$	0	0
$185$	−0.882028	−0.0648480
$186$	0	0
$187$	−0.0856389	−0.00626254
$188$	0	0
$189$	−9.36593	−0.681271
$190$	0	0
$191$	−18.1453	−1.31294	−0.656472	−	0.754350i	$-0.727952\pi$
$191$	−18.1453	−1.31294	−0.656472	+	0.754350i	$0.727952\pi$
$192$	0	0
$193$	17.1682	1.23579	0.617896	−	0.786260i	$-0.287985\pi$
$193$	17.1682	1.23579	0.617896	+	0.786260i	$0.287985\pi$
$194$	0	0
$195$	4.94904	0.354408
$196$	0	0
$197$	−3.90061	−0.277907	−0.138954	−	0.990299i	$-0.544374\pi$
$197$	−3.90061	−0.277907	−0.138954	+	0.990299i	$0.544374\pi$
$198$	0	0
$199$	−5.24735	−0.371975	−0.185988	−	0.982552i	$-0.559548\pi$
$199$	−5.24735	−0.371975	−0.185988	+	0.982552i	$0.559548\pi$
$200$	0	0
$201$	−20.4086	−1.43951
$202$	0	0
$203$	3.76690	0.264384
$204$	0	0
$205$	−0.234068	−0.0163480
$206$	0	0
$207$	−1.33206	−0.0925844
$208$	0	0
$209$	4.69155	0.324521
$210$	0	0
$211$	−1.63338	−0.112446	−0.0562231	−	0.998418i	$-0.517906\pi$
$211$	−1.63338	−0.112446	−0.0562231	+	0.998418i	$0.517906\pi$
$212$	0	0
$213$	−25.5923	−1.75356
$214$	0	0
$215$	−0.373591	−0.0254787
$216$	0	0
$217$	1.20583	0.0818571
$218$	0	0
$219$	18.1157	1.22414
$220$	0	0
$221$	−0.494861	−0.0332880
$222$	0	0
$223$	−5.63370	−0.377260	−0.188630	−	0.982048i	$-0.560405\pi$
$223$	−5.63370	−0.377260	−0.188630	+	0.982048i	$0.560405\pi$
$224$	0	0
$225$	−2.00595	−0.133730
$226$	0	0
$227$	15.3837	1.02105	0.510527	−	0.859861i	$-0.329450\pi$
$227$	15.3837	1.02105	0.510527	+	0.859861i	$0.329450\pi$
$228$	0	0
$229$	−0.325251	−0.0214932	−0.0107466	−	0.999942i	$-0.503421\pi$
$229$	−0.325251	−0.0214932	−0.0107466	+	0.999942i	$0.503421\pi$
$230$	0	0
$231$	3.62905	0.238774
$232$	0	0
$233$	−1.29223	−0.0846565	−0.0423283	−	0.999104i	$-0.513478\pi$
$233$	−1.29223	−0.0846565	−0.0423283	+	0.999104i	$0.513478\pi$
$234$	0	0
$235$	−1.24881	−0.0814635
$236$	0	0
$237$	26.4801	1.72006
$238$	0	0
$239$	30.3215	1.96134	0.980668	−	0.195680i	$-0.0626914\pi$
$239$	30.3215	1.96134	0.980668	+	0.195680i	$0.0626914\pi$
$240$	0	0
$241$	18.6808	1.20334	0.601669	−	0.798746i	$-0.294503\pi$
$241$	18.6808	1.20334	0.601669	+	0.798746i	$0.294503\pi$
$242$	0	0
$243$	4.32568	0.277493
$244$	0	0
$245$	1.45818	0.0931596
$246$	0	0
$247$	27.1100	1.72497
$248$	0	0
$249$	26.4539	1.67645
$250$	0	0
$251$	14.9660	0.944644	0.472322	−	0.881426i	$-0.343416\pi$
$251$	14.9660	0.944644	0.472322	+	0.881426i	$0.343416\pi$
$252$	0	0
$253$	−3.17781	−0.199787
$254$	0	0
$255$	−0.0733466	−0.00459314
$256$	0	0
$257$	−10.1557	−0.633496	−0.316748	−	0.948510i	$-0.602591\pi$
$257$	−10.1557	−0.633496	−0.316748	+	0.948510i	$0.602591\pi$
$258$	0	0
$259$	−3.73737	−0.232229
$260$	0	0
$261$	−0.804539	−0.0497997
$262$	0	0
$263$	−14.7233	−0.907880	−0.453940	−	0.891032i	$-0.649982\pi$
$263$	−14.7233	−0.907880	−0.453940	+	0.891032i	$0.649982\pi$
$264$	0	0
$265$	1.24538	0.0765033
$266$	0	0
$267$	23.0567	1.41105
$268$	0	0
$269$	−4.95511	−0.302119	−0.151059	−	0.988525i	$-0.548268\pi$
$269$	−4.95511	−0.302119	−0.151059	+	0.988525i	$0.548268\pi$
$270$	0	0
$271$	−0.0825849	−0.00501667	−0.00250834	−	0.999997i	$-0.500798\pi$
$271$	−0.0825849	−0.00501667	−0.00250834	+	0.999997i	$0.500798\pi$
$272$	0	0
$273$	20.9703	1.26918
$274$	0	0
$275$	−4.78547	−0.288574
$276$	0	0
$277$	11.5895	0.696343	0.348172	−	0.937431i	$-0.386803\pi$
$277$	11.5895	0.696343	0.348172	+	0.937431i	$0.386803\pi$
$278$	0	0
$279$	−0.257543	−0.0154187
$280$	0	0
$281$	7.47453	0.445893	0.222946	−	0.974831i	$-0.428432\pi$
$281$	7.47453	0.445893	0.222946	+	0.974831i	$0.428432\pi$
$282$	0	0
$283$	12.8497	0.763838	0.381919	−	0.924196i	$-0.375263\pi$
$283$	12.8497	0.763838	0.381919	+	0.924196i	$0.375263\pi$
$284$	0	0
$285$	4.01814	0.238014
$286$	0	0
$287$	−0.991804	−0.0585443
$288$	0	0
$289$	−16.9927	−0.999569
$290$	0	0
$291$	33.7689	1.97957
$292$	0	0
$293$	−9.07779	−0.530330	−0.265165	−	0.964203i	$-0.585426\pi$
$293$	−9.07779	−0.530330	−0.265165	+	0.964203i	$0.585426\pi$
$294$	0	0
$295$	−2.47984	−0.144382
$296$	0	0
$297$	4.77221	0.276911
$298$	0	0
$299$	−18.3628	−1.06195
$300$	0	0
$301$	−1.58300	−0.0912425
$302$	0	0
$303$	8.85639	0.508786
$304$	0	0
$305$	1.59495	0.0913268
$306$	0	0
$307$	0.849319	0.0484732	0.0242366	−	0.999706i	$-0.492284\pi$
$307$	0.849319	0.0484732	0.0242366	+	0.999706i	$0.492284\pi$
$308$	0	0
$309$	6.16972	0.350983
$310$	0	0
$311$	2.22658	0.126258	0.0631288	−	0.998005i	$-0.479892\pi$
$311$	2.22658	0.126258	0.0631288	+	0.998005i	$0.479892\pi$
$312$	0	0
$313$	−24.4324	−1.38100	−0.690502	−	0.723331i	$-0.742610\pi$
$313$	−24.4324	−1.38100	−0.690502	+	0.723331i	$0.742610\pi$
$314$	0	0
$315$	0.381044	0.0214694
$316$	0	0
$317$	−8.62261	−0.484294	−0.242147	−	0.970240i	$-0.577852\pi$
$317$	−8.62261	−0.484294	−0.242147	+	0.970240i	$0.577852\pi$
$318$	0	0
$319$	−1.91934	−0.107462
$320$	0	0
$321$	−18.9092	−1.05541
$322$	0	0
$323$	−0.401779	−0.0223556
$324$	0	0
$325$	−27.6526	−1.53389
$326$	0	0
$327$	−12.7841	−0.706962
$328$	0	0
$329$	−5.29152	−0.291731
$330$	0	0
$331$	−20.5866	−1.13154	−0.565770	−	0.824563i	$-0.691421\pi$
$331$	−20.5866	−1.13154	−0.565770	+	0.824563i	$0.691421\pi$
$332$	0	0
$333$	0.798233	0.0437429
$334$	0	0
$335$	−5.11212	−0.279305
$336$	0	0
$337$	2.22469	0.121186	0.0605932	−	0.998163i	$-0.480701\pi$
$337$	2.22469	0.121186	0.0605932	+	0.998163i	$0.480701\pi$
$338$	0	0
$339$	−18.7204	−1.01675
$340$	0	0
$341$	−0.614404	−0.0332719
$342$	0	0
$343$	19.9169	1.07541
$344$	0	0
$345$	−2.72168	−0.146530
$346$	0	0
$347$	30.6124	1.64336	0.821680	−	0.569949i	$-0.193037\pi$
$347$	30.6124	1.64336	0.821680	+	0.569949i	$0.193037\pi$
$348$	0	0
$349$	−8.51228	−0.455652	−0.227826	−	0.973702i	$-0.573162\pi$
$349$	−8.51228	−0.455652	−0.227826	+	0.973702i	$0.573162\pi$
$350$	0	0
$351$	27.5760	1.47190
$352$	0	0
$353$	29.8628	1.58944	0.794719	−	0.606978i	$-0.207618\pi$
$353$	29.8628	1.58944	0.794719	+	0.606978i	$0.207618\pi$
$354$	0	0
$355$	−6.41057	−0.340238
$356$	0	0
$357$	−0.310787	−0.0164486
$358$	0	0
$359$	13.8521	0.731084	0.365542	−	0.930795i	$-0.380884\pi$
$359$	13.8521	0.731084	0.365542	+	0.930795i	$0.380884\pi$
$360$	0	0
$361$	3.01067	0.158456
$362$	0	0
$363$	−1.84910	−0.0970526
$364$	0	0
$365$	4.53777	0.237518
$366$	0	0
$367$	12.1927	0.636456	0.318228	−	0.948014i	$-0.396912\pi$
$367$	12.1927	0.636456	0.318228	+	0.948014i	$0.396912\pi$
$368$	0	0
$369$	0.211831	0.0110275
$370$	0	0
$371$	5.27700	0.273968
$372$	0	0
$373$	35.7700	1.85210	0.926049	−	0.377402i	$-0.123183\pi$
$373$	35.7700	1.85210	0.926049	+	0.377402i	$0.123183\pi$
$374$	0	0
$375$	−8.38089	−0.432787
$376$	0	0
$377$	−11.0908	−0.571207
$378$	0	0
$379$	−6.74056	−0.346239	−0.173120	−	0.984901i	$-0.555385\pi$
$379$	−6.74056	−0.346239	−0.173120	+	0.984901i	$0.555385\pi$
$380$	0	0
$381$	−25.7111	−1.31722
$382$	0	0
$383$	7.01349	0.358372	0.179186	−	0.983815i	$-0.442654\pi$
$383$	7.01349	0.358372	0.179186	+	0.983815i	$0.442654\pi$
$384$	0	0
$385$	0.909033	0.0463286
$386$	0	0
$387$	0.338099	0.0171865
$388$	0	0
$389$	11.8723	0.601950	0.300975	−	0.953632i	$-0.402688\pi$
$389$	11.8723	0.601950	0.300975	+	0.953632i	$0.402688\pi$
$390$	0	0
$391$	0.272144	0.0137629
$392$	0	0
$393$	−30.9545	−1.56145
$394$	0	0
$395$	6.63294	0.333740
$396$	0	0
$397$	−9.47863	−0.475719	−0.237859	−	0.971300i	$-0.576446\pi$
$397$	−9.47863	−0.475719	−0.237859	+	0.971300i	$0.576446\pi$
$398$	0	0
$399$	17.0259	0.852359
$400$	0	0
$401$	−5.74802	−0.287042	−0.143521	−	0.989647i	$-0.545842\pi$
$401$	−5.74802	−0.287042	−0.143521	+	0.989647i	$0.545842\pi$
$402$	0	0
$403$	−3.55031	−0.176854
$404$	0	0
$405$	4.66968	0.232038
$406$	0	0
$407$	1.90430	0.0943924
$408$	0	0
$409$	12.5921	0.622640	0.311320	−	0.950305i	$-0.399229\pi$
$409$	12.5921	0.622640	0.311320	+	0.950305i	$0.399229\pi$
$410$	0	0
$411$	−1.84910	−0.0912094
$412$	0	0
$413$	−10.5077	−0.517050
$414$	0	0
$415$	6.62639	0.325277
$416$	0	0
$417$	−17.8880	−0.875980
$418$	0	0
$419$	30.7024	1.49991	0.749956	−	0.661487i	$-0.230074\pi$
$419$	30.7024	1.49991	0.749956	+	0.661487i	$0.230074\pi$
$420$	0	0
$421$	−0.211716	−0.0103184	−0.00515920	−	0.999987i	$-0.501642\pi$
$421$	−0.211716	−0.0103184	−0.00515920	+	0.999987i	$0.501642\pi$
$422$	0	0
$423$	1.13017	0.0549508
$424$	0	0
$425$	0.409822	0.0198793
$426$	0	0
$427$	6.75821	0.327053
$428$	0	0
$429$	−10.6850	−0.515875
$430$	0	0
$431$	−31.7875	−1.53115	−0.765574	−	0.643348i	$-0.777545\pi$
$431$	−31.7875	−1.53115	−0.765574	+	0.643348i	$0.777545\pi$
$432$	0	0
$433$	36.5203	1.75506	0.877528	−	0.479526i	$-0.159191\pi$
$433$	36.5203	1.75506	0.877528	+	0.479526i	$0.159191\pi$
$434$	0	0
$435$	−1.64384	−0.0788163
$436$	0	0
$437$	−14.9089	−0.713187
$438$	0	0
$439$	15.9881	0.763069	0.381534	−	0.924355i	$-0.375396\pi$
$439$	15.9881	0.763069	0.381534	+	0.924355i	$0.375396\pi$
$440$	0	0
$441$	−1.31965	−0.0628404
$442$	0	0
$443$	13.1570	0.625108	0.312554	−	0.949900i	$-0.398815\pi$
$443$	13.1570	0.625108	0.312554	+	0.949900i	$0.398815\pi$
$444$	0	0
$445$	5.77543	0.273782
$446$	0	0
$447$	−31.1829	−1.47490
$448$	0	0
$449$	−1.12835	−0.0532502	−0.0266251	−	0.999645i	$-0.508476\pi$
$449$	−1.12835	−0.0532502	−0.0266251	+	0.999645i	$0.508476\pi$
$450$	0	0
$451$	0.505352	0.0237961
$452$	0	0
$453$	25.2721	1.18739
$454$	0	0
$455$	5.25281	0.246256
$456$	0	0
$457$	17.1206	0.800867	0.400433	−	0.916326i	$-0.368860\pi$
$457$	17.1206	0.800867	0.400433	+	0.916326i	$0.368860\pi$
$458$	0	0
$459$	−0.408687	−0.0190759
$460$	0	0
$461$	30.8011	1.43455	0.717276	−	0.696789i	$-0.245389\pi$
$461$	30.8011	1.43455	0.717276	+	0.696789i	$0.245389\pi$
$462$	0	0
$463$	29.4367	1.36804	0.684019	−	0.729464i	$-0.260230\pi$
$463$	29.4367	1.36804	0.684019	+	0.729464i	$0.260230\pi$
$464$	0	0
$465$	−0.526215	−0.0244026
$466$	0	0
$467$	12.7277	0.588968	0.294484	−	0.955656i	$-0.404852\pi$
$467$	12.7277	0.588968	0.294484	+	0.955656i	$0.404852\pi$
$468$	0	0
$469$	−21.6613	−1.00023
$470$	0	0
$471$	20.9442	0.965060
$472$	0	0
$473$	0.806582	0.0370867
$474$	0	0
$475$	−22.4513	−1.03013
$476$	0	0
$477$	−1.12707	−0.0516050
$478$	0	0
$479$	15.4946	0.707968	0.353984	−	0.935251i	$-0.384827\pi$
$479$	15.4946	0.707968	0.353984	+	0.935251i	$0.384827\pi$
$480$	0	0
$481$	11.0039	0.501735
$482$	0	0
$483$	−11.5324	−0.524743
$484$	0	0
$485$	8.45871	0.384090
$486$	0	0
$487$	0.859847	0.0389634	0.0194817	−	0.999810i	$-0.493798\pi$
$487$	0.859847	0.0389634	0.0194817	+	0.999810i	$0.493798\pi$
$488$	0	0
$489$	−7.98678	−0.361175
$490$	0	0
$491$	−21.3620	−0.964051	−0.482026	−	0.876157i	$-0.660099\pi$
$491$	−21.3620	−0.964051	−0.482026	+	0.876157i	$0.660099\pi$
$492$	0	0
$493$	0.164370	0.00740286
$494$	0	0
$495$	−0.194153	−0.00872652
$496$	0	0
$497$	−27.1632	−1.21843
$498$	0	0
$499$	−24.0824	−1.07807	−0.539037	−	0.842282i	$-0.681212\pi$
$499$	−24.0824	−1.07807	−0.539037	+	0.842282i	$0.681212\pi$
$500$	0	0
$501$	−19.5090	−0.871599
$502$	0	0
$503$	−6.47308	−0.288621	−0.144310	−	0.989532i	$-0.546096\pi$
$503$	−6.47308	−0.288621	−0.144310	+	0.989532i	$0.546096\pi$
$504$	0	0
$505$	2.21842	0.0987184
$506$	0	0
$507$	−37.7043	−1.67451
$508$	0	0
$509$	12.1844	0.540063	0.270031	−	0.962852i	$-0.412966\pi$
$509$	12.1844	0.540063	0.270031	+	0.962852i	$0.412966\pi$
$510$	0	0
$511$	19.2276	0.850581
$512$	0	0
$513$	22.3891	0.988501
$514$	0	0
$515$	1.54544	0.0681004
$516$	0	0
$517$	2.69618	0.118578
$518$	0	0
$519$	−35.0505	−1.53855
$520$	0	0
$521$	−26.1033	−1.14361	−0.571803	−	0.820391i	$-0.693756\pi$
$521$	−26.1033	−1.14361	−0.571803	+	0.820391i	$0.693756\pi$
$522$	0	0
$523$	20.9224	0.914874	0.457437	−	0.889242i	$-0.348767\pi$
$523$	20.9224	0.914874	0.457437	+	0.889242i	$0.348767\pi$
$524$	0	0
$525$	−17.3667	−0.757944
$526$	0	0
$527$	0.0526169	0.00229203
$528$	0	0
$529$	−12.9015	−0.560936
$530$	0	0
$531$	2.24425	0.0973921
$532$	0	0
$533$	2.92016	0.126486
$534$	0	0
$535$	−4.73653	−0.204778
$536$	0	0
$537$	−6.96997	−0.300776
$538$	0	0
$539$	−3.14820	−0.135603
$540$	0	0
$541$	32.7101	1.40632	0.703159	−	0.711032i	$-0.251772\pi$
$541$	32.7101	1.40632	0.703159	+	0.711032i	$0.251772\pi$
$542$	0	0
$543$	−18.7349	−0.803990
$544$	0	0
$545$	−3.20226	−0.137170
$546$	0	0
$547$	−3.25436	−0.139146	−0.0695732	−	0.997577i	$-0.522164\pi$
$547$	−3.25436	−0.139146	−0.0695732	+	0.997577i	$0.522164\pi$
$548$	0	0
$549$	−1.44343	−0.0616041
$550$	0	0
$551$	−9.00468	−0.383612
$552$	0	0
$553$	28.1054	1.19516
$554$	0	0
$555$	1.63096	0.0692304
$556$	0	0
$557$	17.5830	0.745015	0.372508	−	0.928029i	$-0.378498\pi$
$557$	17.5830	0.745015	0.372508	+	0.928029i	$0.378498\pi$
$558$	0	0
$559$	4.66080	0.197131
$560$	0	0
$561$	0.158355	0.00668575
$562$	0	0
$563$	−23.9393	−1.00892	−0.504460	−	0.863435i	$-0.668309\pi$
$563$	−23.9393	−1.00892	−0.504460	+	0.863435i	$0.668309\pi$
$564$	0	0
$565$	−4.68924	−0.197278
$566$	0	0
$567$	19.7866	0.830958
$568$	0	0
$569$	24.1826	1.01379	0.506894	−	0.862008i	$-0.330793\pi$
$569$	24.1826	1.01379	0.506894	+	0.862008i	$0.330793\pi$
$570$	0	0
$571$	−2.38787	−0.0999291	−0.0499645	−	0.998751i	$-0.515911\pi$
$571$	−2.38787	−0.0999291	−0.0499645	+	0.998751i	$0.515911\pi$
$572$	0	0
$573$	33.5524	1.40167
$574$	0	0
$575$	15.2073	0.634188
$576$	0	0
$577$	−0.0623330	−0.00259495	−0.00129748	−	0.999999i	$-0.500413\pi$
$577$	−0.0623330	−0.00259495	−0.00129748	+	0.999999i	$0.500413\pi$
$578$	0	0
$579$	−31.7457	−1.31931
$580$	0	0
$581$	28.0777	1.16486
$582$	0	0
$583$	−2.68878	−0.111358
$584$	0	0
$585$	−1.12190	−0.0463850
$586$	0	0
$587$	28.0626	1.15827	0.579133	−	0.815233i	$-0.303391\pi$
$587$	28.0626	1.15827	0.579133	+	0.815233i	$0.303391\pi$
$588$	0	0
$589$	−2.88251	−0.118772
$590$	0	0
$591$	7.21263	0.296688
$592$	0	0
$593$	3.05688	0.125531	0.0627654	−	0.998028i	$-0.480008\pi$
$593$	3.05688	0.125531	0.0627654	+	0.998028i	$0.480008\pi$
$594$	0	0
$595$	−0.0778486	−0.00319148
$596$	0	0
$597$	9.70289	0.397113
$598$	0	0
$599$	−18.7915	−0.767801	−0.383900	−	0.923375i	$-0.625419\pi$
$599$	−18.7915	−0.767801	−0.383900	+	0.923375i	$0.625419\pi$
$600$	0	0
$601$	−25.8047	−1.05260	−0.526298	−	0.850300i	$-0.676420\pi$
$601$	−25.8047	−1.05260	−0.526298	+	0.850300i	$0.676420\pi$
$602$	0	0
$603$	4.62646	0.188404
$604$	0	0
$605$	−0.463178	−0.0188309
$606$	0	0
$607$	−15.8301	−0.642522	−0.321261	−	0.946991i	$-0.604107\pi$
$607$	−15.8301	−0.642522	−0.321261	+	0.946991i	$0.604107\pi$
$608$	0	0
$609$	−6.96537	−0.282251
$610$	0	0
$611$	15.5798	0.630290
$612$	0	0
$613$	−23.4924	−0.948848	−0.474424	−	0.880296i	$-0.657344\pi$
$613$	−23.4924	−0.948848	−0.474424	+	0.880296i	$0.657344\pi$
$614$	0	0
$615$	0.432815	0.0174528
$616$	0	0
$617$	−13.5855	−0.546930	−0.273465	−	0.961882i	$-0.588170\pi$
$617$	−13.5855	−0.546930	−0.273465	+	0.961882i	$0.588170\pi$
$618$	0	0
$619$	18.5507	0.745617	0.372808	−	0.927908i	$-0.378395\pi$
$619$	18.5507	0.745617	0.372808	+	0.927908i	$0.378395\pi$
$620$	0	0
$621$	−15.1652	−0.608557
$622$	0	0
$623$	24.4719	0.980447
$624$	0	0
$625$	21.8280	0.873121
$626$	0	0
$627$	−8.67516	−0.346452
$628$	0	0
$629$	−0.163082	−0.00650250
$630$	0	0
$631$	12.1849	0.485073	0.242536	−	0.970142i	$-0.422021\pi$
$631$	12.1849	0.485073	0.242536	+	0.970142i	$0.422021\pi$
$632$	0	0
$633$	3.02028	0.120045
$634$	0	0
$635$	−6.44032	−0.255576
$636$	0	0
$637$	−18.1918	−0.720784
$638$	0	0
$639$	5.80155	0.229506
$640$	0	0
$641$	4.94764	0.195420	0.0977099	−	0.995215i	$-0.468848\pi$
$641$	4.94764	0.195420	0.0977099	+	0.995215i	$0.468848\pi$
$642$	0	0
$643$	34.0534	1.34293	0.671467	−	0.741034i	$-0.265665\pi$
$643$	34.0534	1.34293	0.671467	+	0.741034i	$0.265665\pi$
$644$	0	0
$645$	0.690808	0.0272005
$646$	0	0
$647$	29.8194	1.17232	0.586161	−	0.810195i	$-0.300639\pi$
$647$	29.8194	1.17232	0.586161	+	0.810195i	$0.300639\pi$
$648$	0	0
$649$	5.35396	0.210161
$650$	0	0
$651$	−2.22970	−0.0873889
$652$	0	0
$653$	2.01589	0.0788879	0.0394439	−	0.999222i	$-0.487441\pi$
$653$	2.01589	0.0788879	0.0394439	+	0.999222i	$0.487441\pi$
$654$	0	0
$655$	−7.75374	−0.302964
$656$	0	0
$657$	−4.10667	−0.160216
$658$	0	0
$659$	−4.02091	−0.156632	−0.0783161	−	0.996929i	$-0.524954\pi$
$659$	−4.02091	−0.156632	−0.0783161	+	0.996929i	$0.524954\pi$
$660$	0	0
$661$	16.2124	0.630590	0.315295	−	0.948994i	$-0.397897\pi$
$661$	16.2124	0.630590	0.315295	+	0.948994i	$0.397897\pi$
$662$	0	0
$663$	0.915048	0.0355375
$664$	0	0
$665$	4.26478	0.165381
$666$	0	0
$667$	6.09929	0.236166
$668$	0	0
$669$	10.4173	0.402755
$670$	0	0
$671$	−3.44350	−0.132935
$672$	0	0
$673$	44.8013	1.72696	0.863481	−	0.504381i	$-0.168279\pi$
$673$	44.8013	1.72696	0.863481	+	0.504381i	$0.168279\pi$
$674$	0	0
$675$	−22.8372	−0.879005
$676$	0	0
$677$	−3.46282	−0.133087	−0.0665434	−	0.997784i	$-0.521197\pi$
$677$	−3.46282	−0.133087	−0.0665434	+	0.997784i	$0.521197\pi$
$678$	0	0
$679$	35.8417	1.37548
$680$	0	0
$681$	−28.4461	−1.09006
$682$	0	0
$683$	2.45186	0.0938177	0.0469088	−	0.998899i	$-0.485063\pi$
$683$	2.45186	0.0938177	0.0469088	+	0.998899i	$0.485063\pi$
$684$	0	0
$685$	−0.463178	−0.0176971
$686$	0	0
$687$	0.601421	0.0229457
$688$	0	0
$689$	−15.5370	−0.591913
$690$	0	0
$691$	−7.98379	−0.303718	−0.151859	−	0.988402i	$-0.548526\pi$
$691$	−7.98379	−0.303718	−0.151859	+	0.988402i	$0.548526\pi$
$692$	0	0
$693$	−0.822673	−0.0312508
$694$	0	0
$695$	−4.48074	−0.169964
$696$	0	0
$697$	−0.0432778	−0.00163926
$698$	0	0
$699$	2.38946	0.0903775
$700$	0	0
$701$	−35.1116	−1.32615	−0.663074	−	0.748554i	$-0.730748\pi$
$701$	−35.1116	−1.32615	−0.663074	+	0.748554i	$0.730748\pi$
$702$	0	0
$703$	8.93410	0.336956
$704$	0	0
$705$	2.30918	0.0869687
$706$	0	0
$707$	9.40000	0.353523
$708$	0	0
$709$	−1.78654	−0.0670948	−0.0335474	−	0.999437i	$-0.510680\pi$
$709$	−1.78654	−0.0670948	−0.0335474	+	0.999437i	$0.510680\pi$
$710$	0	0
$711$	−6.00280	−0.225122
$712$	0	0
$713$	1.95246	0.0731202
$714$	0	0
$715$	−2.67646	−0.100094
$716$	0	0
$717$	−56.0675	−2.09388
$718$	0	0
$719$	23.8951	0.891137	0.445569	−	0.895248i	$-0.353002\pi$
$719$	23.8951	0.891137	0.445569	+	0.895248i	$0.353002\pi$
$720$	0	0
$721$	6.54842	0.243876
$722$	0	0
$723$	−34.5427	−1.28466
$724$	0	0
$725$	9.18493	0.341120
$726$	0	0
$727$	−20.2460	−0.750882	−0.375441	−	0.926846i	$-0.622509\pi$
$727$	−20.2460	−0.750882	−0.375441	+	0.926846i	$0.622509\pi$
$728$	0	0
$729$	22.2468	0.823957
$730$	0	0
$731$	−0.0690748	−0.00255482
$732$	0	0
$733$	−48.0846	−1.77604	−0.888022	−	0.459801i	$-0.847921\pi$
$733$	−48.0846	−1.77604	−0.888022	+	0.459801i	$0.847921\pi$
$734$	0	0
$735$	−2.69632	−0.0994552
$736$	0	0
$737$	11.0371	0.406555
$738$	0	0
$739$	−11.7935	−0.433830	−0.216915	−	0.976190i	$-0.569600\pi$
$739$	−11.7935	−0.433830	−0.216915	+	0.976190i	$0.569600\pi$
$740$	0	0
$741$	−50.1290	−1.84154
$742$	0	0
$743$	14.5828	0.534990	0.267495	−	0.963559i	$-0.413804\pi$
$743$	14.5828	0.534990	0.267495	+	0.963559i	$0.413804\pi$
$744$	0	0
$745$	−7.81095	−0.286171
$746$	0	0
$747$	−5.99687	−0.219414
$748$	0	0
$749$	−20.0698	−0.733336
$750$	0	0
$751$	38.4165	1.40184	0.700919	−	0.713241i	$-0.252774\pi$
$751$	38.4165	1.40184	0.700919	+	0.713241i	$0.252774\pi$
$752$	0	0
$753$	−27.6736	−1.00848
$754$	0	0
$755$	6.33037	0.230386
$756$	0	0
$757$	24.5383	0.891861	0.445931	−	0.895068i	$-0.352873\pi$
$757$	24.5383	0.891861	0.445931	+	0.895068i	$0.352873\pi$
$758$	0	0
$759$	5.87609	0.213289
$760$	0	0
$761$	−46.6265	−1.69021	−0.845104	−	0.534602i	$-0.820462\pi$
$761$	−46.6265	−1.69021	−0.845104	+	0.534602i	$0.820462\pi$
$762$	0	0
$763$	−13.5688	−0.491223
$764$	0	0
$765$	0.0166270	0.000601151 0
$766$	0	0
$767$	30.9377	1.11709
$768$	0	0
$769$	4.71019	0.169854	0.0849269	−	0.996387i	$-0.472934\pi$
$769$	4.71019	0.169854	0.0849269	+	0.996387i	$0.472934\pi$
$770$	0	0
$771$	18.7789	0.676307
$772$	0	0
$773$	−16.4025	−0.589958	−0.294979	−	0.955504i	$-0.595313\pi$
$773$	−16.4025	−0.589958	−0.294979	+	0.955504i	$0.595313\pi$
$774$	0	0
$775$	2.94021	0.105616
$776$	0	0
$777$	6.91078	0.247923
$778$	0	0
$779$	2.37089	0.0849458
$780$	0	0
$781$	13.8404	0.495248
$782$	0	0
$783$	−9.15948	−0.327333
$784$	0	0
$785$	5.24629	0.187248
$786$	0	0
$787$	3.33552	0.118898	0.0594492	−	0.998231i	$-0.481066\pi$
$787$	3.33552	0.118898	0.0594492	+	0.998231i	$0.481066\pi$
$788$	0	0
$789$	27.2249	0.969233
$790$	0	0
$791$	−19.8695	−0.706476
$792$	0	0
$793$	−19.8981	−0.706603
$794$	0	0
$795$	−2.30284	−0.0816733
$796$	0	0
$797$	18.3784	0.650997	0.325499	−	0.945543i	$-0.394468\pi$
$797$	18.3784	0.650997	0.325499	+	0.945543i	$0.394468\pi$
$798$	0	0
$799$	−0.230898	−0.00816858
$800$	0	0
$801$	−5.22675	−0.184678
$802$	0	0
$803$	−9.79702	−0.345729
$804$	0	0
$805$	−2.88873	−0.101814
$806$	0	0
$807$	9.16250	0.322535
$808$	0	0
$809$	−8.01745	−0.281878	−0.140939	−	0.990018i	$-0.545012\pi$
$809$	−8.01745	−0.281878	−0.140939	+	0.990018i	$0.545012\pi$
$810$	0	0
$811$	−7.50505	−0.263538	−0.131769	−	0.991280i	$-0.542066\pi$
$811$	−7.50505	−0.263538	−0.131769	+	0.991280i	$0.542066\pi$
$812$	0	0
$813$	0.152708	0.00535570
$814$	0	0
$815$	−2.00059	−0.0700778
$816$	0	0
$817$	3.78412	0.132390
$818$	0	0
$819$	−4.75378	−0.166111
$820$	0	0
$821$	7.56760	0.264111	0.132056	−	0.991242i	$-0.457842\pi$
$821$	7.56760	0.264111	0.132056	+	0.991242i	$0.457842\pi$
$822$	0	0
$823$	−30.1616	−1.05137	−0.525683	−	0.850680i	$-0.676190\pi$
$823$	−30.1616	−1.05137	−0.525683	+	0.850680i	$0.676190\pi$
$824$	0	0
$825$	8.84881	0.308076
$826$	0	0
$827$	35.5177	1.23507	0.617535	−	0.786543i	$-0.288131\pi$
$827$	35.5177	1.23507	0.617535	+	0.786543i	$0.288131\pi$
$828$	0	0
$829$	−43.9313	−1.52580	−0.762898	−	0.646519i	$-0.776224\pi$
$829$	−43.9313	−1.52580	−0.762898	+	0.646519i	$0.776224\pi$
$830$	0	0
$831$	−21.4301	−0.743401
$832$	0	0
$833$	0.269609	0.00934138
$834$	0	0
$835$	−4.88678	−0.169114
$836$	0	0
$837$	−2.93206	−0.101347
$838$	0	0
$839$	13.1526	0.454077	0.227038	−	0.973886i	$-0.427096\pi$
$839$	13.1526	0.454077	0.227038	+	0.973886i	$0.427096\pi$
$840$	0	0
$841$	−25.3161	−0.872970
$842$	0	0
$843$	−13.8212	−0.476026
$844$	0	0
$845$	−9.44448	−0.324900
$846$	0	0
$847$	−1.96260	−0.0674357
$848$	0	0
$849$	−23.7605	−0.815457
$850$	0	0
$851$	−6.05149	−0.207442
$852$	0	0
$853$	11.4581	0.392317	0.196158	−	0.980572i	$-0.437153\pi$
$853$	11.4581	0.392317	0.196158	+	0.980572i	$0.437153\pi$
$854$	0	0
$855$	−0.910878	−0.0311514
$856$	0	0
$857$	−2.57746	−0.0880442	−0.0440221	−	0.999031i	$-0.514017\pi$
$857$	−2.57746	−0.0880442	−0.0440221	+	0.999031i	$0.514017\pi$
$858$	0	0
$859$	−31.8008	−1.08503	−0.542514	−	0.840047i	$-0.682528\pi$
$859$	−31.8008	−1.08503	−0.542514	+	0.840047i	$0.682528\pi$
$860$	0	0
$861$	1.83395	0.0625007
$862$	0	0
$863$	−7.51927	−0.255959	−0.127979	−	0.991777i	$-0.540849\pi$
$863$	−7.51927	−0.255959	−0.127979	+	0.991777i	$0.540849\pi$
$864$	0	0
$865$	−8.77974	−0.298520
$866$	0	0
$867$	31.4212	1.06712
$868$	0	0
$869$	−14.3205	−0.485790
$870$	0	0
$871$	63.7772	2.16101
$872$	0	0
$873$	−7.65511	−0.259086
$874$	0	0
$875$	−8.89531	−0.300717
$876$	0	0
$877$	26.0070	0.878194	0.439097	−	0.898440i	$-0.355299\pi$
$877$	26.0070	0.878194	0.439097	+	0.898440i	$0.355299\pi$
$878$	0	0
$879$	16.7858	0.566169
$880$	0	0
$881$	34.7838	1.17190	0.585948	−	0.810349i	$-0.300722\pi$
$881$	34.7838	1.17190	0.585948	+	0.810349i	$0.300722\pi$
$882$	0	0
$883$	9.08067	0.305589	0.152794	−	0.988258i	$-0.451173\pi$
$883$	9.08067	0.305589	0.152794	+	0.988258i	$0.451173\pi$
$884$	0	0
$885$	4.58547	0.154139
$886$	0	0
$887$	−19.9262	−0.669057	−0.334528	−	0.942386i	$-0.608577\pi$
$887$	−19.9262	−0.669057	−0.334528	+	0.942386i	$0.608577\pi$
$888$	0	0
$889$	−27.2892	−0.915251
$890$	0	0
$891$	−10.0818	−0.337754
$892$	0	0
$893$	12.6493	0.423292
$894$	0	0
$895$	−1.74590	−0.0583589
$896$	0	0
$897$	33.9548	1.13372
$898$	0	0
$899$	1.17925	0.0393302
$900$	0	0
$901$	0.230264	0.00767121
$902$	0	0
$903$	2.92712	0.0974086
$904$	0	0
$905$	−4.69286	−0.155996
$906$	0	0
$907$	−46.3807	−1.54005	−0.770023	−	0.638016i	$-0.779755\pi$
$907$	−46.3807	−1.54005	−0.770023	+	0.638016i	$0.779755\pi$
$908$	0	0
$909$	−2.00767	−0.0665901
$910$	0	0
$911$	−38.7233	−1.28296	−0.641479	−	0.767140i	$-0.721679\pi$
$911$	−38.7233	−1.28296	−0.641479	+	0.767140i	$0.721679\pi$
$912$	0	0
$913$	−14.3064	−0.473471
$914$	0	0
$915$	−2.94923	−0.0974986
$916$	0	0
$917$	−32.8545	−1.08495
$918$	0	0
$919$	39.1339	1.29091	0.645454	−	0.763799i	$-0.276668\pi$
$919$	39.1339	1.29091	0.645454	+	0.763799i	$0.276668\pi$
$920$	0	0
$921$	−1.57048	−0.0517490
$922$	0	0
$923$	79.9762	2.63245
$924$	0	0
$925$	−9.11294	−0.299632
$926$	0	0
$927$	−1.39862	−0.0459368
$928$	0	0
$929$	−3.54438	−0.116287	−0.0581437	−	0.998308i	$-0.518518\pi$
$929$	−3.54438	−0.116287	−0.0581437	+	0.998308i	$0.518518\pi$
$930$	0	0
$931$	−14.7700	−0.484066
$932$	0	0
$933$	−4.11717	−0.134790
$934$	0	0
$935$	0.0396661	0.00129722
$936$	0	0
$937$	35.2620	1.15196	0.575979	−	0.817464i	$-0.304621\pi$
$937$	35.2620	1.15196	0.575979	+	0.817464i	$0.304621\pi$
$938$	0	0
$939$	45.1781	1.47433
$940$	0	0
$941$	−42.1429	−1.37382	−0.686910	−	0.726743i	$-0.741033\pi$
$941$	−42.1429	−1.37382	−0.686910	+	0.726743i	$0.741033\pi$
$942$	0	0
$943$	−1.60591	−0.0522957
$944$	0	0
$945$	4.33809	0.141118
$946$	0	0
$947$	15.7020	0.510247	0.255124	−	0.966908i	$-0.417884\pi$
$947$	15.7020	0.510247	0.255124	+	0.966908i	$0.417884\pi$
$948$	0	0
$949$	−56.6117	−1.83769
$950$	0	0
$951$	15.9441	0.517022
$952$	0	0
$953$	−19.3375	−0.626403	−0.313202	−	0.949687i	$-0.601402\pi$
$953$	−19.3375	−0.626403	−0.313202	+	0.949687i	$0.601402\pi$
$954$	0	0
$955$	8.40449	0.271963
$956$	0	0
$957$	3.54905	0.114725
$958$	0	0
$959$	−1.96260	−0.0633757
$960$	0	0
$961$	−30.6225	−0.987823
$962$	0	0
$963$	4.28655	0.138132
$964$	0	0
$965$	−7.95192	−0.255981
$966$	0	0
$967$	5.02544	0.161607	0.0808036	−	0.996730i	$-0.474251\pi$
$967$	5.02544	0.161607	0.0808036	+	0.996730i	$0.474251\pi$
$968$	0	0
$969$	0.742931	0.0238664
$970$	0	0
$971$	−8.71099	−0.279549	−0.139774	−	0.990183i	$-0.544638\pi$
$971$	−8.71099	−0.279549	−0.139774	+	0.990183i	$0.544638\pi$
$972$	0	0
$973$	−18.9860	−0.608663
$974$	0	0
$975$	51.1325	1.63755
$976$	0	0
$977$	44.9653	1.43857	0.719283	−	0.694717i	$-0.244470\pi$
$977$	44.9653	1.43857	0.719283	+	0.694717i	$0.244470\pi$
$978$	0	0
$979$	−12.4691	−0.398515
$980$	0	0
$981$	2.89804	0.0925274
$982$	0	0
$983$	34.4501	1.09879	0.549393	−	0.835564i	$-0.314859\pi$
$983$	34.4501	1.09879	0.549393	+	0.835564i	$0.314859\pi$
$984$	0	0
$985$	1.80668	0.0575655
$986$	0	0
$987$	9.78456	0.311446
$988$	0	0
$989$	−2.56316	−0.0815039
$990$	0	0
$991$	−29.6159	−0.940780	−0.470390	−	0.882459i	$-0.655887\pi$
$991$	−29.6159	−0.940780	−0.470390	+	0.882459i	$0.655887\pi$
$992$	0	0
$993$	38.0667	1.20801
$994$	0	0
$995$	2.43046	0.0770507
$996$	0	0
$997$	−28.4901	−0.902292	−0.451146	−	0.892450i	$-0.648985\pi$
$997$	−28.4901	−0.902292	−0.451146	+	0.892450i	$0.648985\pi$
$998$	0	0
$999$	9.08769	0.287522

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6028.2.a.f.1.6	✓	29

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6028.2.a.f.1.6	✓	29	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 6028 = 2^{2} \cdot 11 \cdot 137 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6028.a (trivial)

\(f(q)\)	\(=\)	\(q-1.84910 q^{3} -0.463178 q^{5} -1.96260 q^{7} +0.419175 q^{9} +O(q^{10})\)	\(q-1.84910 q^{3} -0.463178 q^{5} -1.96260 q^{7} +0.419175 q^{9} +1.00000 q^{11} +5.77846 q^{13} +0.856463 q^{15} -0.0856389 q^{17} +4.69155 q^{19} +3.62905 q^{21} -3.17781 q^{23} -4.78547 q^{25} +4.77221 q^{27} -1.91934 q^{29} -0.614404 q^{31} -1.84910 q^{33} +0.909033 q^{35} +1.90430 q^{37} -10.6850 q^{39} +0.505352 q^{41} +0.806582 q^{43} -0.194153 q^{45} +2.69618 q^{47} -3.14820 q^{49} +0.158355 q^{51} -2.68878 q^{53} -0.463178 q^{55} -8.67516 q^{57} +5.35396 q^{59} -3.44350 q^{61} -0.822673 q^{63} -2.67646 q^{65} +11.0371 q^{67} +5.87609 q^{69} +13.8404 q^{71} -9.79702 q^{73} +8.84881 q^{75} -1.96260 q^{77} -14.3205 q^{79} -10.0818 q^{81} -14.3064 q^{83} +0.0396661 q^{85} +3.54905 q^{87} -12.4691 q^{89} -11.3408 q^{91} +1.13610 q^{93} -2.17302 q^{95} -18.2623 q^{97} +0.419175 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 29 q + 14 q^{3} + 9 q^{5} + 14 q^{7} + 43 q^{9}+O(q^{10}) \) 29 * q + 14 * q^3 + 9 * q^5 + 14 * q^7 + 43 * q^9	\( 29 q + 14 q^{3} + 9 q^{5} + 14 q^{7} + 43 q^{9} + 29 q^{11} + 10 q^{15} + 29 q^{17} + 7 q^{19} + 2 q^{21} + 36 q^{23} + 36 q^{25} + 50 q^{27} + 9 q^{29} + 28 q^{31} + 14 q^{33} + 15 q^{35} + 25 q^{37} + 9 q^{39} + 19 q^{41} + 23 q^{43} + 5 q^{45} + 27 q^{47} + 27 q^{49} + 13 q^{51} + 4 q^{53} + 9 q^{55} + 14 q^{57} + 40 q^{59} + 20 q^{61} - 17 q^{63} + 9 q^{65} + 59 q^{67} + 30 q^{69} + 29 q^{71} - 5 q^{73} + 46 q^{75} + 14 q^{77} + 29 q^{79} + 61 q^{81} + 35 q^{83} - 57 q^{85} + 45 q^{87} + 39 q^{89} + 45 q^{91} - 8 q^{93} + q^{95} + 55 q^{97} + 43 q^{99}+O(q^{100}) \) 29 * q + 14 * q^3 + 9 * q^5 + 14 * q^7 + 43 * q^9 + 29 * q^11 + 10 * q^15 + 29 * q^17 + 7 * q^19 + 2 * q^21 + 36 * q^23 + 36 * q^25 + 50 * q^27 + 9 * q^29 + 28 * q^31 + 14 * q^33 + 15 * q^35 + 25 * q^37 + 9 * q^39 + 19 * q^41 + 23 * q^43 + 5 * q^45 + 27 * q^47 + 27 * q^49 + 13 * q^51 + 4 * q^53 + 9 * q^55 + 14 * q^57 + 40 * q^59 + 20 * q^61 - 17 * q^63 + 9 * q^65 + 59 * q^67 + 30 * q^69 + 29 * q^71 - 5 * q^73 + 46 * q^75 + 14 * q^77 + 29 * q^79 + 61 * q^81 + 35 * q^83 - 57 * q^85 + 45 * q^87 + 39 * q^89 + 45 * q^91 - 8 * q^93 + q^95 + 55 * q^97 + 43 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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