LMFDB - Embedded newform 6036.2.a.i.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6036,2,Mod(1,6036)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6036, 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("6036.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6036 = 2^{2} \cdot 3 \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6036.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:	$48.1977026600$
Analytic rank:	$0$
Dimension:	$26$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	6036.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +2.53458 q^{5} +3.60576 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +2.53458 q^{5} +3.60576 q^{7} +1.00000 q^{9} +4.66613 q^{11} -2.60384 q^{13} -2.53458 q^{15} +2.74474 q^{17} +0.716395 q^{19} -3.60576 q^{21} +7.76678 q^{23} +1.42411 q^{25} -1.00000 q^{27} +8.07933 q^{29} -4.58770 q^{31} -4.66613 q^{33} +9.13909 q^{35} -10.4330 q^{37} +2.60384 q^{39} +10.0989 q^{41} -1.83235 q^{43} +2.53458 q^{45} +2.79183 q^{47} +6.00147 q^{49} -2.74474 q^{51} -10.2851 q^{53} +11.8267 q^{55} -0.716395 q^{57} +5.44275 q^{59} +8.24320 q^{61} +3.60576 q^{63} -6.59965 q^{65} -1.77928 q^{67} -7.76678 q^{69} +0.897417 q^{71} +0.515081 q^{73} -1.42411 q^{75} +16.8249 q^{77} +6.23609 q^{79} +1.00000 q^{81} -7.92540 q^{83} +6.95678 q^{85} -8.07933 q^{87} +3.24000 q^{89} -9.38882 q^{91} +4.58770 q^{93} +1.81576 q^{95} -0.898636 q^{97} +4.66613 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 26 q - 26 q^{3} + 6 q^{5} + 5 q^{7} + 26 q^{9}+O(q^{10}) $ 26 * q - 26 * q^3 + 6 * q^5 + 5 * q^7 + 26 * q^9	$ 26 q - 26 q^{3} + 6 q^{5} + 5 q^{7} + 26 q^{9} - 11 q^{11} + 13 q^{13} - 6 q^{15} + 12 q^{17} - q^{19} - 5 q^{21} - 22 q^{23} + 48 q^{25} - 26 q^{27} + 6 q^{29} + 19 q^{31} + 11 q^{33} - 21 q^{35} + 20 q^{37} - 13 q^{39} + 25 q^{41} + 4 q^{43} + 6 q^{45} + 8 q^{47} + 67 q^{49} - 12 q^{51} - 5 q^{53} + 20 q^{55} + q^{57} - 18 q^{59} + 43 q^{61} + 5 q^{63} + 41 q^{65} + 5 q^{67} + 22 q^{69} - q^{71} + 22 q^{73} - 48 q^{75} + 23 q^{77} + 16 q^{79} + 26 q^{81} - 19 q^{83} + 29 q^{85} - 6 q^{87} + 49 q^{89} - 13 q^{91} - 19 q^{93} - 26 q^{95} + 25 q^{97} - 11 q^{99}+O(q^{100}) $ 26 * q - 26 * q^3 + 6 * q^5 + 5 * q^7 + 26 * q^9 - 11 * q^11 + 13 * q^13 - 6 * q^15 + 12 * q^17 - q^19 - 5 * q^21 - 22 * q^23 + 48 * q^25 - 26 * q^27 + 6 * q^29 + 19 * q^31 + 11 * q^33 - 21 * q^35 + 20 * q^37 - 13 * q^39 + 25 * q^41 + 4 * q^43 + 6 * q^45 + 8 * q^47 + 67 * q^49 - 12 * q^51 - 5 * q^53 + 20 * q^55 + q^57 - 18 * q^59 + 43 * q^61 + 5 * q^63 + 41 * q^65 + 5 * q^67 + 22 * q^69 - q^71 + 22 * q^73 - 48 * q^75 + 23 * q^77 + 16 * q^79 + 26 * q^81 - 19 * q^83 + 29 * q^85 - 6 * q^87 + 49 * q^89 - 13 * q^91 - 19 * q^93 - 26 * q^95 + 25 * q^97 - 11 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	2.53458	1.13350	0.566750	−	0.823890i	$-0.308201\pi$
$5$	2.53458	1.13350	0.566750	+	0.823890i	$0.308201\pi$
$6$	0	0
$7$	3.60576	1.36285	0.681424	−	0.731889i	$-0.261361\pi$
$7$	3.60576	1.36285	0.681424	+	0.731889i	$0.261361\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.66613	1.40689	0.703446	−	0.710749i	$-0.251644\pi$
$11$	4.66613	1.40689	0.703446	+	0.710749i	$0.251644\pi$
$12$	0	0
$13$	−2.60384	−0.722176	−0.361088	−	0.932532i	$-0.617595\pi$
$13$	−2.60384	−0.722176	−0.361088	+	0.932532i	$0.617595\pi$
$14$	0	0
$15$	−2.53458	−0.654426
$16$	0	0
$17$	2.74474	0.665698	0.332849	−	0.942980i	$-0.391990\pi$
$17$	2.74474	0.665698	0.332849	+	0.942980i	$0.391990\pi$
$18$	0	0
$19$	0.716395	0.164352	0.0821762	−	0.996618i	$-0.473813\pi$
$19$	0.716395	0.164352	0.0821762	+	0.996618i	$0.473813\pi$
$20$	0	0
$21$	−3.60576	−0.786840
$22$	0	0
$23$	7.76678	1.61948	0.809742	−	0.586786i	$-0.199607\pi$
$23$	7.76678	1.61948	0.809742	+	0.586786i	$0.199607\pi$
$24$	0	0
$25$	1.42411	0.284822
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	8.07933	1.50029	0.750147	−	0.661271i	$-0.229983\pi$
$29$	8.07933	1.50029	0.750147	+	0.661271i	$0.229983\pi$
$30$	0	0
$31$	−4.58770	−0.823976	−0.411988	−	0.911189i	$-0.635165\pi$
$31$	−4.58770	−0.823976	−0.411988	+	0.911189i	$0.635165\pi$
$32$	0	0
$33$	−4.66613	−0.812269
$34$	0	0
$35$	9.13909	1.54479
$36$	0	0
$37$	−10.4330	−1.71517	−0.857586	−	0.514341i	$-0.828037\pi$
$37$	−10.4330	−1.71517	−0.857586	+	0.514341i	$0.828037\pi$
$38$	0	0
$39$	2.60384	0.416948
$40$	0	0
$41$	10.0989	1.57718	0.788591	−	0.614918i	$-0.210811\pi$
$41$	10.0989	1.57718	0.788591	+	0.614918i	$0.210811\pi$
$42$	0	0
$43$	−1.83235	−0.279431	−0.139715	−	0.990192i	$-0.544619\pi$
$43$	−1.83235	−0.279431	−0.139715	+	0.990192i	$0.544619\pi$
$44$	0	0
$45$	2.53458	0.377833
$46$	0	0
$47$	2.79183	0.407230	0.203615	−	0.979051i	$-0.434731\pi$
$47$	2.79183	0.407230	0.203615	+	0.979051i	$0.434731\pi$
$48$	0	0
$49$	6.00147	0.857353
$50$	0	0
$51$	−2.74474	−0.384341
$52$	0	0
$53$	−10.2851	−1.41277	−0.706386	−	0.707827i	$-0.749676\pi$
$53$	−10.2851	−1.41277	−0.706386	+	0.707827i	$0.749676\pi$
$54$	0	0
$55$	11.8267	1.59471
$56$	0	0
$57$	−0.716395	−0.0948889
$58$	0	0
$59$	5.44275	0.708586	0.354293	−	0.935134i	$-0.384722\pi$
$59$	5.44275	0.708586	0.354293	+	0.935134i	$0.384722\pi$
$60$	0	0
$61$	8.24320	1.05543	0.527717	−	0.849421i	$-0.323048\pi$
$61$	8.24320	1.05543	0.527717	+	0.849421i	$0.323048\pi$
$62$	0	0
$63$	3.60576	0.454282
$64$	0	0
$65$	−6.59965	−0.818586
$66$	0	0
$67$	−1.77928	−0.217373	−0.108687	−	0.994076i	$-0.534664\pi$
$67$	−1.77928	−0.217373	−0.108687	+	0.994076i	$0.534664\pi$
$68$	0	0
$69$	−7.76678	−0.935010
$70$	0	0
$71$	0.897417	0.106504	0.0532519	−	0.998581i	$-0.483041\pi$
$71$	0.897417	0.106504	0.0532519	+	0.998581i	$0.483041\pi$
$72$	0	0
$73$	0.515081	0.0602857	0.0301428	−	0.999546i	$-0.490404\pi$
$73$	0.515081	0.0602857	0.0301428	+	0.999546i	$0.490404\pi$
$74$	0	0
$75$	−1.42411	−0.164442
$76$	0	0
$77$	16.8249	1.91738
$78$	0	0
$79$	6.23609	0.701615	0.350807	−	0.936448i	$-0.385907\pi$
$79$	6.23609	0.701615	0.350807	+	0.936448i	$0.385907\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−7.92540	−0.869925	−0.434963	−	0.900448i	$-0.643238\pi$
$83$	−7.92540	−0.869925	−0.434963	+	0.900448i	$0.643238\pi$
$84$	0	0
$85$	6.95678	0.754568
$86$	0	0
$87$	−8.07933	−0.866195
$88$	0	0
$89$	3.24000	0.343439	0.171720	−	0.985146i	$-0.445068\pi$
$89$	3.24000	0.343439	0.171720	+	0.985146i	$0.445068\pi$
$90$	0	0
$91$	−9.38882	−0.984215
$92$	0	0
$93$	4.58770	0.475723
$94$	0	0
$95$	1.81576	0.186293
$96$	0	0
$97$	−0.898636	−0.0912426	−0.0456213	−	0.998959i	$-0.514527\pi$
$97$	−0.898636	−0.0912426	−0.0456213	+	0.998959i	$0.514527\pi$
$98$	0	0
$99$	4.66613	0.468964
$100$	0	0
$101$	−1.73787	−0.172924	−0.0864621	−	0.996255i	$-0.527556\pi$
$101$	−1.73787	−0.172924	−0.0864621	+	0.996255i	$0.527556\pi$
$102$	0	0
$103$	−8.02139	−0.790371	−0.395185	−	0.918601i	$-0.629320\pi$
$103$	−8.02139	−0.790371	−0.395185	+	0.918601i	$0.629320\pi$
$104$	0	0
$105$	−9.13909	−0.891883
$106$	0	0
$107$	−8.60577	−0.831951	−0.415976	−	0.909376i	$-0.636560\pi$
$107$	−8.60577	−0.831951	−0.415976	+	0.909376i	$0.636560\pi$
$108$	0	0
$109$	2.78090	0.266362	0.133181	−	0.991092i	$-0.457481\pi$
$109$	2.78090	0.266362	0.133181	+	0.991092i	$0.457481\pi$
$110$	0	0
$111$	10.4330	0.990255
$112$	0	0
$113$	−10.0371	−0.944212	−0.472106	−	0.881542i	$-0.656506\pi$
$113$	−10.0371	−0.944212	−0.472106	+	0.881542i	$0.656506\pi$
$114$	0	0
$115$	19.6855	1.83569
$116$	0	0
$117$	−2.60384	−0.240725
$118$	0	0
$119$	9.89687	0.907245
$120$	0	0
$121$	10.7728	0.979345
$122$	0	0
$123$	−10.0989	−0.910586
$124$	0	0
$125$	−9.06339	−0.810654
$126$	0	0
$127$	−5.09845	−0.452415	−0.226207	−	0.974079i	$-0.572633\pi$
$127$	−5.09845	−0.452415	−0.226207	+	0.974079i	$0.572633\pi$
$128$	0	0
$129$	1.83235	0.161329
$130$	0	0
$131$	−12.0250	−1.05063	−0.525315	−	0.850908i	$-0.676052\pi$
$131$	−12.0250	−1.05063	−0.525315	+	0.850908i	$0.676052\pi$
$132$	0	0
$133$	2.58315	0.223987
$134$	0	0
$135$	−2.53458	−0.218142
$136$	0	0
$137$	11.7745	1.00596	0.502980	−	0.864298i	$-0.332237\pi$
$137$	11.7745	1.00596	0.502980	+	0.864298i	$0.332237\pi$
$138$	0	0
$139$	1.29925	0.110201	0.0551005	−	0.998481i	$-0.482452\pi$
$139$	1.29925	0.110201	0.0551005	+	0.998481i	$0.482452\pi$
$140$	0	0
$141$	−2.79183	−0.235114
$142$	0	0
$143$	−12.1499	−1.01602
$144$	0	0
$145$	20.4777	1.70058
$146$	0	0
$147$	−6.00147	−0.494993
$148$	0	0
$149$	2.16907	0.177697	0.0888487	−	0.996045i	$-0.471681\pi$
$149$	2.16907	0.177697	0.0888487	+	0.996045i	$0.471681\pi$
$150$	0	0
$151$	2.72560	0.221806	0.110903	−	0.993831i	$-0.464626\pi$
$151$	2.72560	0.221806	0.110903	+	0.993831i	$0.464626\pi$
$152$	0	0
$153$	2.74474	0.221899
$154$	0	0
$155$	−11.6279	−0.933976
$156$	0	0
$157$	−6.40827	−0.511436	−0.255718	−	0.966751i	$-0.582312\pi$
$157$	−6.40827	−0.511436	−0.255718	+	0.966751i	$0.582312\pi$
$158$	0	0
$159$	10.2851	0.815664
$160$	0	0
$161$	28.0051	2.20711
$162$	0	0
$163$	8.87797	0.695376	0.347688	−	0.937610i	$-0.386967\pi$
$163$	8.87797	0.695376	0.347688	+	0.937610i	$0.386967\pi$
$164$	0	0
$165$	−11.8267	−0.920707
$166$	0	0
$167$	−20.3375	−1.57376	−0.786881	−	0.617105i	$-0.788305\pi$
$167$	−20.3375	−1.57376	−0.786881	+	0.617105i	$0.788305\pi$
$168$	0	0
$169$	−6.22001	−0.478462
$170$	0	0
$171$	0.716395	0.0547841
$172$	0	0
$173$	−22.7571	−1.73019	−0.865094	−	0.501610i	$-0.832741\pi$
$173$	−22.7571	−1.73019	−0.865094	+	0.501610i	$0.832741\pi$
$174$	0	0
$175$	5.13499	0.388169
$176$	0	0
$177$	−5.44275	−0.409102
$178$	0	0
$179$	−22.2626	−1.66398	−0.831992	−	0.554787i	$-0.812800\pi$
$179$	−22.2626	−1.66398	−0.831992	+	0.554787i	$0.812800\pi$
$180$	0	0
$181$	14.1664	1.05298	0.526491	−	0.850181i	$-0.323508\pi$
$181$	14.1664	1.05298	0.526491	+	0.850181i	$0.323508\pi$
$182$	0	0
$183$	−8.24320	−0.609355
$184$	0	0
$185$	−26.4433	−1.94415
$186$	0	0
$187$	12.8073	0.936565
$188$	0	0
$189$	−3.60576	−0.262280
$190$	0	0
$191$	8.82642	0.638657	0.319329	−	0.947644i	$-0.396543\pi$
$191$	8.82642	0.638657	0.319329	+	0.947644i	$0.396543\pi$
$192$	0	0
$193$	−11.3649	−0.818065	−0.409033	−	0.912520i	$-0.634134\pi$
$193$	−11.3649	−0.818065	−0.409033	+	0.912520i	$0.634134\pi$
$194$	0	0
$195$	6.59965	0.472611
$196$	0	0
$197$	11.8414	0.843664	0.421832	−	0.906674i	$-0.361387\pi$
$197$	11.8414	0.843664	0.421832	+	0.906674i	$0.361387\pi$
$198$	0	0
$199$	0.591504	0.0419306	0.0209653	−	0.999780i	$-0.493326\pi$
$199$	0.591504	0.0419306	0.0209653	+	0.999780i	$0.493326\pi$
$200$	0	0
$201$	1.77928	0.125500
$202$	0	0
$203$	29.1321	2.04467
$204$	0	0
$205$	25.5965	1.78774
$206$	0	0
$207$	7.76678	0.539828
$208$	0	0
$209$	3.34280	0.231226
$210$	0	0
$211$	−27.5368	−1.89571	−0.947857	−	0.318695i	$-0.896755\pi$
$211$	−27.5368	−1.89571	−0.947857	+	0.318695i	$0.896755\pi$
$212$	0	0
$213$	−0.897417	−0.0614900
$214$	0	0
$215$	−4.64424	−0.316734
$216$	0	0
$217$	−16.5421	−1.12295
$218$	0	0
$219$	−0.515081	−0.0348059
$220$	0	0
$221$	−7.14687	−0.480751
$222$	0	0
$223$	−7.63357	−0.511182	−0.255591	−	0.966785i	$-0.582270\pi$
$223$	−7.63357	−0.511182	−0.255591	+	0.966785i	$0.582270\pi$
$224$	0	0
$225$	1.42411	0.0949407
$226$	0	0
$227$	14.4554	0.959442	0.479721	−	0.877421i	$-0.340738\pi$
$227$	14.4554	0.959442	0.479721	+	0.877421i	$0.340738\pi$
$228$	0	0
$229$	0.542149	0.0358262	0.0179131	−	0.999840i	$-0.494298\pi$
$229$	0.542149	0.0358262	0.0179131	+	0.999840i	$0.494298\pi$
$230$	0	0
$231$	−16.8249	−1.10700
$232$	0	0
$233$	−26.2667	−1.72079	−0.860393	−	0.509631i	$-0.829782\pi$
$233$	−26.2667	−1.72079	−0.860393	+	0.509631i	$0.829782\pi$
$234$	0	0
$235$	7.07611	0.461595
$236$	0	0
$237$	−6.23609	−0.405077
$238$	0	0
$239$	7.55778	0.488872	0.244436	−	0.969665i	$-0.421397\pi$
$239$	7.55778	0.488872	0.244436	+	0.969665i	$0.421397\pi$
$240$	0	0
$241$	15.2058	0.979495	0.489747	−	0.871864i	$-0.337089\pi$
$241$	15.2058	0.979495	0.489747	+	0.871864i	$0.337089\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	15.2112	0.971810
$246$	0	0
$247$	−1.86538	−0.118691
$248$	0	0
$249$	7.92540	0.502252
$250$	0	0
$251$	−13.4480	−0.848831	−0.424415	−	0.905468i	$-0.639520\pi$
$251$	−13.4480	−0.848831	−0.424415	+	0.905468i	$0.639520\pi$
$252$	0	0
$253$	36.2408	2.27844
$254$	0	0
$255$	−6.95678	−0.435650
$256$	0	0
$257$	15.3341	0.956513	0.478256	−	0.878220i	$-0.341269\pi$
$257$	15.3341	0.956513	0.478256	+	0.878220i	$0.341269\pi$
$258$	0	0
$259$	−37.6188	−2.33752
$260$	0	0
$261$	8.07933	0.500098
$262$	0	0
$263$	6.84467	0.422060	0.211030	−	0.977480i	$-0.432318\pi$
$263$	6.84467	0.422060	0.211030	+	0.977480i	$0.432318\pi$
$264$	0	0
$265$	−26.0685	−1.60138
$266$	0	0
$267$	−3.24000	−0.198285
$268$	0	0
$269$	−25.6870	−1.56617	−0.783083	−	0.621918i	$-0.786354\pi$
$269$	−25.6870	−1.56617	−0.783083	+	0.621918i	$0.786354\pi$
$270$	0	0
$271$	−22.0183	−1.33752	−0.668758	−	0.743480i	$-0.733174\pi$
$271$	−22.0183	−1.33752	−0.668758	+	0.743480i	$0.733174\pi$
$272$	0	0
$273$	9.38882	0.568237
$274$	0	0
$275$	6.64508	0.400714
$276$	0	0
$277$	25.7727	1.54853	0.774264	−	0.632862i	$-0.218120\pi$
$277$	25.7727	1.54853	0.774264	+	0.632862i	$0.218120\pi$
$278$	0	0
$279$	−4.58770	−0.274659
$280$	0	0
$281$	5.79858	0.345914	0.172957	−	0.984929i	$-0.444668\pi$
$281$	5.79858	0.345914	0.172957	+	0.984929i	$0.444668\pi$
$282$	0	0
$283$	−0.919794	−0.0546761	−0.0273380	−	0.999626i	$-0.508703\pi$
$283$	−0.919794	−0.0546761	−0.0273380	+	0.999626i	$0.508703\pi$
$284$	0	0
$285$	−1.81576	−0.107557
$286$	0	0
$287$	36.4141	2.14946
$288$	0	0
$289$	−9.46639	−0.556846
$290$	0	0
$291$	0.898636	0.0526789
$292$	0	0
$293$	13.7089	0.800883	0.400442	−	0.916322i	$-0.368857\pi$
$293$	13.7089	0.800883	0.400442	+	0.916322i	$0.368857\pi$
$294$	0	0
$295$	13.7951	0.803182
$296$	0	0
$297$	−4.66613	−0.270756
$298$	0	0
$299$	−20.2235	−1.16955
$300$	0	0
$301$	−6.60700	−0.380821
$302$	0	0
$303$	1.73787	0.0998379
$304$	0	0
$305$	20.8931	1.19633
$306$	0	0
$307$	−11.9361	−0.681232	−0.340616	−	0.940203i	$-0.610636\pi$
$307$	−11.9361	−0.681232	−0.340616	+	0.940203i	$0.610636\pi$
$308$	0	0
$309$	8.02139	0.456321
$310$	0	0
$311$	−8.91744	−0.505662	−0.252831	−	0.967510i	$-0.581362\pi$
$311$	−8.91744	−0.505662	−0.252831	+	0.967510i	$0.581362\pi$
$312$	0	0
$313$	−13.0506	−0.737661	−0.368831	−	0.929497i	$-0.620242\pi$
$313$	−13.0506	−0.737661	−0.368831	+	0.929497i	$0.620242\pi$
$314$	0	0
$315$	9.13909	0.514929
$316$	0	0
$317$	22.5539	1.26675	0.633377	−	0.773844i	$-0.281668\pi$
$317$	22.5539	1.26675	0.633377	+	0.773844i	$0.281668\pi$
$318$	0	0
$319$	37.6992	2.11075
$320$	0	0
$321$	8.60577	0.480327
$322$	0	0
$323$	1.96632	0.109409
$324$	0	0
$325$	−3.70816	−0.205692
$326$	0	0
$327$	−2.78090	−0.153784
$328$	0	0
$329$	10.0666	0.554992
$330$	0	0
$331$	4.18703	0.230140	0.115070	−	0.993357i	$-0.463291\pi$
$331$	4.18703	0.230140	0.115070	+	0.993357i	$0.463291\pi$
$332$	0	0
$333$	−10.4330	−0.571724
$334$	0	0
$335$	−4.50972	−0.246392
$336$	0	0
$337$	21.6046	1.17688	0.588439	−	0.808542i	$-0.299743\pi$
$337$	21.6046	1.17688	0.588439	+	0.808542i	$0.299743\pi$
$338$	0	0
$339$	10.0371	0.545141
$340$	0	0
$341$	−21.4068	−1.15924
$342$	0	0
$343$	−3.60044	−0.194406
$344$	0	0
$345$	−19.6855	−1.05983
$346$	0	0
$347$	19.5054	1.04711	0.523553	−	0.851993i	$-0.324606\pi$
$347$	19.5054	1.04711	0.523553	+	0.851993i	$0.324606\pi$
$348$	0	0
$349$	34.9855	1.87273	0.936365	−	0.351027i	$-0.114168\pi$
$349$	34.9855	1.87273	0.936365	+	0.351027i	$0.114168\pi$
$350$	0	0
$351$	2.60384	0.138983
$352$	0	0
$353$	27.9902	1.48977	0.744883	−	0.667195i	$-0.232505\pi$
$353$	27.9902	1.48977	0.744883	+	0.667195i	$0.232505\pi$
$354$	0	0
$355$	2.27458	0.120722
$356$	0	0
$357$	−9.89687	−0.523798
$358$	0	0
$359$	34.0080	1.79487	0.897436	−	0.441145i	$-0.145428\pi$
$359$	34.0080	1.79487	0.897436	+	0.441145i	$0.145428\pi$
$360$	0	0
$361$	−18.4868	−0.972988
$362$	0	0
$363$	−10.7728	−0.565425
$364$	0	0
$365$	1.30552	0.0683338
$366$	0	0
$367$	0.261922	0.0136722	0.00683612	−	0.999977i	$-0.497824\pi$
$367$	0.261922	0.0136722	0.00683612	+	0.999977i	$0.497824\pi$
$368$	0	0
$369$	10.0989	0.525727
$370$	0	0
$371$	−37.0857	−1.92539
$372$	0	0
$373$	4.57946	0.237115	0.118558	−	0.992947i	$-0.462173\pi$
$373$	4.57946	0.237115	0.118558	+	0.992947i	$0.462173\pi$
$374$	0	0
$375$	9.06339	0.468031
$376$	0	0
$377$	−21.0373	−1.08348
$378$	0	0
$379$	−10.7251	−0.550913	−0.275456	−	0.961314i	$-0.588829\pi$
$379$	−10.7251	−0.550913	−0.275456	+	0.961314i	$0.588829\pi$
$380$	0	0
$381$	5.09845	0.261202
$382$	0	0
$383$	32.9032	1.68128	0.840638	−	0.541598i	$-0.182180\pi$
$383$	32.9032	1.68128	0.840638	+	0.541598i	$0.182180\pi$
$384$	0	0
$385$	42.6442	2.17335
$386$	0	0
$387$	−1.83235	−0.0931435
$388$	0	0
$389$	−11.6460	−0.590477	−0.295239	−	0.955424i	$-0.595399\pi$
$389$	−11.6460	−0.590477	−0.295239	+	0.955424i	$0.595399\pi$
$390$	0	0
$391$	21.3178	1.07809
$392$	0	0
$393$	12.0250	0.606581
$394$	0	0
$395$	15.8059	0.795280
$396$	0	0
$397$	27.5160	1.38099	0.690493	−	0.723339i	$-0.257394\pi$
$397$	27.5160	1.38099	0.690493	+	0.723339i	$0.257394\pi$
$398$	0	0
$399$	−2.58315	−0.129319
$400$	0	0
$401$	34.8087	1.73826	0.869132	−	0.494580i	$-0.164678\pi$
$401$	34.8087	1.73826	0.869132	+	0.494580i	$0.164678\pi$
$402$	0	0
$403$	11.9457	0.595055
$404$	0	0
$405$	2.53458	0.125944
$406$	0	0
$407$	−48.6817	−2.41306
$408$	0	0
$409$	36.7083	1.81511	0.907554	−	0.419935i	$-0.137947\pi$
$409$	36.7083	1.81511	0.907554	+	0.419935i	$0.137947\pi$
$410$	0	0
$411$	−11.7745	−0.580792
$412$	0	0
$413$	19.6252	0.965695
$414$	0	0
$415$	−20.0876	−0.986060
$416$	0	0
$417$	−1.29925	−0.0636246
$418$	0	0
$419$	32.0237	1.56446	0.782230	−	0.622989i	$-0.214082\pi$
$419$	32.0237	1.56446	0.782230	+	0.622989i	$0.214082\pi$
$420$	0	0
$421$	−9.64139	−0.469893	−0.234946	−	0.972008i	$-0.575491\pi$
$421$	−9.64139	−0.469893	−0.234946	+	0.972008i	$0.575491\pi$
$422$	0	0
$423$	2.79183	0.135743
$424$	0	0
$425$	3.90881	0.189605
$426$	0	0
$427$	29.7230	1.43839
$428$	0	0
$429$	12.1499	0.586601
$430$	0	0
$431$	9.65654	0.465139	0.232570	−	0.972580i	$-0.425287\pi$
$431$	9.65654	0.465139	0.232570	+	0.972580i	$0.425287\pi$
$432$	0	0
$433$	24.5946	1.18194	0.590970	−	0.806694i	$-0.298745\pi$
$433$	24.5946	1.18194	0.590970	+	0.806694i	$0.298745\pi$
$434$	0	0
$435$	−20.4777	−0.981832
$436$	0	0
$437$	5.56408	0.266166
$438$	0	0
$439$	−26.7881	−1.27853	−0.639264	−	0.768988i	$-0.720761\pi$
$439$	−26.7881	−1.27853	−0.639264	+	0.768988i	$0.720761\pi$
$440$	0	0
$441$	6.00147	0.285784
$442$	0	0
$443$	−39.4079	−1.87232	−0.936162	−	0.351568i	$-0.885649\pi$
$443$	−39.4079	−1.87232	−0.936162	+	0.351568i	$0.885649\pi$
$444$	0	0
$445$	8.21205	0.389288
$446$	0	0
$447$	−2.16907	−0.102594
$448$	0	0
$449$	−2.09208	−0.0987312	−0.0493656	−	0.998781i	$-0.515720\pi$
$449$	−2.09208	−0.0987312	−0.0493656	+	0.998781i	$0.515720\pi$
$450$	0	0
$451$	47.1228	2.21892
$452$	0	0
$453$	−2.72560	−0.128060
$454$	0	0
$455$	−23.7967	−1.11561
$456$	0	0
$457$	−18.4503	−0.863069	−0.431535	−	0.902096i	$-0.642028\pi$
$457$	−18.4503	−0.863069	−0.431535	+	0.902096i	$0.642028\pi$
$458$	0	0
$459$	−2.74474	−0.128114
$460$	0	0
$461$	22.0069	1.02496	0.512482	−	0.858698i	$-0.328726\pi$
$461$	22.0069	1.02496	0.512482	+	0.858698i	$0.328726\pi$
$462$	0	0
$463$	21.7884	1.01259	0.506297	−	0.862359i	$-0.331014\pi$
$463$	21.7884	1.01259	0.506297	+	0.862359i	$0.331014\pi$
$464$	0	0
$465$	11.6279	0.539231
$466$	0	0
$467$	−39.9361	−1.84802	−0.924011	−	0.382365i	$-0.875110\pi$
$467$	−39.9361	−1.84802	−0.924011	+	0.382365i	$0.875110\pi$
$468$	0	0
$469$	−6.41564	−0.296247
$470$	0	0
$471$	6.40827	0.295278
$472$	0	0
$473$	−8.54998	−0.393129
$474$	0	0
$475$	1.02023	0.0468112
$476$	0	0
$477$	−10.2851	−0.470924
$478$	0	0
$479$	−23.9302	−1.09340	−0.546701	−	0.837328i	$-0.684116\pi$
$479$	−23.9302	−1.09340	−0.546701	+	0.837328i	$0.684116\pi$
$480$	0	0
$481$	27.1658	1.23866
$482$	0	0
$483$	−28.0051	−1.27428
$484$	0	0
$485$	−2.27767	−0.103423
$486$	0	0
$487$	−33.2562	−1.50698	−0.753491	−	0.657458i	$-0.771632\pi$
$487$	−33.2562	−1.50698	−0.753491	+	0.657458i	$0.771632\pi$
$488$	0	0
$489$	−8.87797	−0.401475
$490$	0	0
$491$	−36.3830	−1.64194	−0.820970	−	0.570971i	$-0.806567\pi$
$491$	−36.3830	−1.64194	−0.820970	+	0.570971i	$0.806567\pi$
$492$	0	0
$493$	22.1757	0.998742
$494$	0	0
$495$	11.8267	0.531571
$496$	0	0
$497$	3.23586	0.145148
$498$	0	0
$499$	−6.52720	−0.292198	−0.146099	−	0.989270i	$-0.546672\pi$
$499$	−6.52720	−0.292198	−0.146099	+	0.989270i	$0.546672\pi$
$500$	0	0
$501$	20.3375	0.908611
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	−4.40477	−0.196010
$506$	0	0
$507$	6.22001	0.276240
$508$	0	0
$509$	21.7241	0.962905	0.481452	−	0.876472i	$-0.340109\pi$
$509$	21.7241	0.962905	0.481452	+	0.876472i	$0.340109\pi$
$510$	0	0
$511$	1.85726	0.0821601
$512$	0	0
$513$	−0.716395	−0.0316296
$514$	0	0
$515$	−20.3309	−0.895885
$516$	0	0
$517$	13.0270	0.572928
$518$	0	0
$519$	22.7571	0.998924
$520$	0	0
$521$	−4.03880	−0.176943	−0.0884716	−	0.996079i	$-0.528198\pi$
$521$	−4.03880	−0.176943	−0.0884716	+	0.996079i	$0.528198\pi$
$522$	0	0
$523$	35.9547	1.57219	0.786094	−	0.618107i	$-0.212100\pi$
$523$	35.9547	1.57219	0.786094	+	0.618107i	$0.212100\pi$
$524$	0	0
$525$	−5.13499	−0.224109
$526$	0	0
$527$	−12.5921	−0.548519
$528$	0	0
$529$	37.3228	1.62273
$530$	0	0
$531$	5.44275	0.236195
$532$	0	0
$533$	−26.2959	−1.13900
$534$	0	0
$535$	−21.8120	−0.943017
$536$	0	0
$537$	22.2626	0.960702
$538$	0	0
$539$	28.0037	1.20620
$540$	0	0
$541$	19.5129	0.838924	0.419462	−	0.907773i	$-0.362219\pi$
$541$	19.5129	0.838924	0.419462	+	0.907773i	$0.362219\pi$
$542$	0	0
$543$	−14.1664	−0.607939
$544$	0	0
$545$	7.04843	0.301922
$546$	0	0
$547$	6.92271	0.295994	0.147997	−	0.988988i	$-0.452717\pi$
$547$	6.92271	0.295994	0.147997	+	0.988988i	$0.452717\pi$
$548$	0	0
$549$	8.24320	0.351811
$550$	0	0
$551$	5.78799	0.246577
$552$	0	0
$553$	22.4858	0.956194
$554$	0	0
$555$	26.4433	1.12245
$556$	0	0
$557$	−31.5605	−1.33726	−0.668631	−	0.743594i	$-0.733120\pi$
$557$	−31.5605	−1.33726	−0.668631	+	0.743594i	$0.733120\pi$
$558$	0	0
$559$	4.77115	0.201798
$560$	0	0
$561$	−12.8073	−0.540726
$562$	0	0
$563$	10.2354	0.431371	0.215686	−	0.976463i	$-0.430801\pi$
$563$	10.2354	0.431371	0.215686	+	0.976463i	$0.430801\pi$
$564$	0	0
$565$	−25.4399	−1.07026
$566$	0	0
$567$	3.60576	0.151427
$568$	0	0
$569$	4.06701	0.170498	0.0852489	−	0.996360i	$-0.472831\pi$
$569$	4.06701	0.170498	0.0852489	+	0.996360i	$0.472831\pi$
$570$	0	0
$571$	22.5816	0.945008	0.472504	−	0.881328i	$-0.343350\pi$
$571$	22.5816	0.945008	0.472504	+	0.881328i	$0.343350\pi$
$572$	0	0
$573$	−8.82642	−0.368729
$574$	0	0
$575$	11.0607	0.461265
$576$	0	0
$577$	−28.4583	−1.18474	−0.592368	−	0.805667i	$-0.701807\pi$
$577$	−28.4583	−1.18474	−0.592368	+	0.805667i	$0.701807\pi$
$578$	0	0
$579$	11.3649	0.472310
$580$	0	0
$581$	−28.5770	−1.18558
$582$	0	0
$583$	−47.9918	−1.98762
$584$	0	0
$585$	−6.59965	−0.272862
$586$	0	0
$587$	36.0094	1.48627	0.743133	−	0.669143i	$-0.233339\pi$
$587$	36.0094	1.48627	0.743133	+	0.669143i	$0.233339\pi$
$588$	0	0
$589$	−3.28661	−0.135422
$590$	0	0
$591$	−11.8414	−0.487090
$592$	0	0
$593$	−6.77699	−0.278297	−0.139149	−	0.990271i	$-0.544437\pi$
$593$	−6.77699	−0.278297	−0.139149	+	0.990271i	$0.544437\pi$
$594$	0	0
$595$	25.0844	1.02836
$596$	0	0
$597$	−0.591504	−0.0242086
$598$	0	0
$599$	−19.4825	−0.796035	−0.398017	−	0.917378i	$-0.630302\pi$
$599$	−19.4825	−0.796035	−0.398017	+	0.917378i	$0.630302\pi$
$600$	0	0
$601$	8.85452	0.361183	0.180592	−	0.983558i	$-0.442199\pi$
$601$	8.85452	0.361183	0.180592	+	0.983558i	$0.442199\pi$
$602$	0	0
$603$	−1.77928	−0.0724577
$604$	0	0
$605$	27.3045	1.11009
$606$	0	0
$607$	0.631788	0.0256435	0.0128217	−	0.999918i	$-0.495919\pi$
$607$	0.631788	0.0256435	0.0128217	+	0.999918i	$0.495919\pi$
$608$	0	0
$609$	−29.1321	−1.18049
$610$	0	0
$611$	−7.26947	−0.294091
$612$	0	0
$613$	−26.5371	−1.07182	−0.535911	−	0.844275i	$-0.680032\pi$
$613$	−26.5371	−1.07182	−0.535911	+	0.844275i	$0.680032\pi$
$614$	0	0
$615$	−25.5965	−1.03215
$616$	0	0
$617$	13.5304	0.544713	0.272356	−	0.962196i	$-0.412197\pi$
$617$	13.5304	0.544713	0.272356	+	0.962196i	$0.412197\pi$
$618$	0	0
$619$	2.03673	0.0818632	0.0409316	−	0.999162i	$-0.486967\pi$
$619$	2.03673	0.0818632	0.0409316	+	0.999162i	$0.486967\pi$
$620$	0	0
$621$	−7.76678	−0.311670
$622$	0	0
$623$	11.6827	0.468056
$624$	0	0
$625$	−30.0925	−1.20370
$626$	0	0
$627$	−3.34280	−0.133498
$628$	0	0
$629$	−28.6358	−1.14179
$630$	0	0
$631$	−47.9419	−1.90854	−0.954269	−	0.298948i	$-0.903364\pi$
$631$	−47.9419	−1.90854	−0.954269	+	0.298948i	$0.903364\pi$
$632$	0	0
$633$	27.5368	1.09449
$634$	0	0
$635$	−12.9225	−0.512812
$636$	0	0
$637$	−15.6269	−0.619160
$638$	0	0
$639$	0.897417	0.0355013
$640$	0	0
$641$	−13.6629	−0.539654	−0.269827	−	0.962909i	$-0.586966\pi$
$641$	−13.6629	−0.539654	−0.269827	+	0.962909i	$0.586966\pi$
$642$	0	0
$643$	−33.0345	−1.30275	−0.651377	−	0.758754i	$-0.725809\pi$
$643$	−33.0345	−1.30275	−0.651377	+	0.758754i	$0.725809\pi$
$644$	0	0
$645$	4.64424	0.182867
$646$	0	0
$647$	−2.85739	−0.112335	−0.0561677	−	0.998421i	$-0.517888\pi$
$647$	−2.85739	−0.112335	−0.0561677	+	0.998421i	$0.517888\pi$
$648$	0	0
$649$	25.3966	0.996904
$650$	0	0
$651$	16.5421	0.648337
$652$	0	0
$653$	2.37650	0.0929997	0.0464998	−	0.998918i	$-0.485193\pi$
$653$	2.37650	0.0929997	0.0464998	+	0.998918i	$0.485193\pi$
$654$	0	0
$655$	−30.4784	−1.19089
$656$	0	0
$657$	0.515081	0.0200952
$658$	0	0
$659$	31.9662	1.24523	0.622613	−	0.782530i	$-0.286071\pi$
$659$	31.9662	1.24523	0.622613	+	0.782530i	$0.286071\pi$
$660$	0	0
$661$	−27.7011	−1.07745	−0.538725	−	0.842482i	$-0.681094\pi$
$661$	−27.7011	−1.07745	−0.538725	+	0.842482i	$0.681094\pi$
$662$	0	0
$663$	7.14687	0.277562
$664$	0	0
$665$	6.54720	0.253889
$666$	0	0
$667$	62.7503	2.42970
$668$	0	0
$669$	7.63357	0.295131
$670$	0	0
$671$	38.4638	1.48488
$672$	0	0
$673$	−4.83451	−0.186357	−0.0931784	−	0.995649i	$-0.529703\pi$
$673$	−4.83451	−0.186357	−0.0931784	+	0.995649i	$0.529703\pi$
$674$	0	0
$675$	−1.42411	−0.0548140
$676$	0	0
$677$	25.4646	0.978683	0.489341	−	0.872092i	$-0.337237\pi$
$677$	25.4646	0.978683	0.489341	+	0.872092i	$0.337237\pi$
$678$	0	0
$679$	−3.24026	−0.124350
$680$	0	0
$681$	−14.4554	−0.553934
$682$	0	0
$683$	19.3731	0.741291	0.370646	−	0.928774i	$-0.379136\pi$
$683$	19.3731	0.741291	0.370646	+	0.928774i	$0.379136\pi$
$684$	0	0
$685$	29.8434	1.14026
$686$	0	0
$687$	−0.542149	−0.0206843
$688$	0	0
$689$	26.7809	1.02027
$690$	0	0
$691$	−28.3147	−1.07714	−0.538571	−	0.842580i	$-0.681036\pi$
$691$	−28.3147	−1.07714	−0.538571	+	0.842580i	$0.681036\pi$
$692$	0	0
$693$	16.8249	0.639126
$694$	0	0
$695$	3.29306	0.124913
$696$	0	0
$697$	27.7189	1.04993
$698$	0	0
$699$	26.2667	0.993496
$700$	0	0
$701$	−2.49563	−0.0942588	−0.0471294	−	0.998889i	$-0.515007\pi$
$701$	−2.49563	−0.0942588	−0.0471294	+	0.998889i	$0.515007\pi$
$702$	0	0
$703$	−7.47414	−0.281893
$704$	0	0
$705$	−7.07611	−0.266502
$706$	0	0
$707$	−6.26632	−0.235669
$708$	0	0
$709$	23.1400	0.869041	0.434520	−	0.900662i	$-0.356918\pi$
$709$	23.1400	0.869041	0.434520	+	0.900662i	$0.356918\pi$
$710$	0	0
$711$	6.23609	0.233872
$712$	0	0
$713$	−35.6317	−1.33442
$714$	0	0
$715$	−30.7949	−1.15166
$716$	0	0
$717$	−7.55778	−0.282251
$718$	0	0
$719$	−18.3342	−0.683752	−0.341876	−	0.939745i	$-0.611062\pi$
$719$	−18.3342	−0.683752	−0.341876	+	0.939745i	$0.611062\pi$
$720$	0	0
$721$	−28.9232	−1.07715
$722$	0	0
$723$	−15.2058	−0.565512
$724$	0	0
$725$	11.5058	0.427317
$726$	0	0
$727$	41.8186	1.55096	0.775482	−	0.631369i	$-0.217507\pi$
$727$	41.8186	1.55096	0.775482	+	0.631369i	$0.217507\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−5.02933	−0.186016
$732$	0	0
$733$	10.1949	0.376558	0.188279	−	0.982116i	$-0.439709\pi$
$733$	10.1949	0.376558	0.188279	+	0.982116i	$0.439709\pi$
$734$	0	0
$735$	−15.2112	−0.561075
$736$	0	0
$737$	−8.30234	−0.305821
$738$	0	0
$739$	−42.8021	−1.57450	−0.787251	−	0.616633i	$-0.788496\pi$
$739$	−42.8021	−1.57450	−0.787251	+	0.616633i	$0.788496\pi$
$740$	0	0
$741$	1.86538	0.0685264
$742$	0	0
$743$	8.80981	0.323200	0.161600	−	0.986856i	$-0.448334\pi$
$743$	8.80981	0.323200	0.161600	+	0.986856i	$0.448334\pi$
$744$	0	0
$745$	5.49770	0.201420
$746$	0	0
$747$	−7.92540	−0.289975
$748$	0	0
$749$	−31.0303	−1.13382
$750$	0	0
$751$	1.00494	0.0366708	0.0183354	−	0.999832i	$-0.494163\pi$
$751$	1.00494	0.0366708	0.0183354	+	0.999832i	$0.494163\pi$
$752$	0	0
$753$	13.4480	0.490073
$754$	0	0
$755$	6.90826	0.251417
$756$	0	0
$757$	−14.7193	−0.534982	−0.267491	−	0.963560i	$-0.586195\pi$
$757$	−14.7193	−0.534982	−0.267491	+	0.963560i	$0.586195\pi$
$758$	0	0
$759$	−36.2408	−1.31546
$760$	0	0
$761$	12.0817	0.437961	0.218980	−	0.975729i	$-0.429727\pi$
$761$	12.0817	0.437961	0.218980	+	0.975729i	$0.429727\pi$
$762$	0	0
$763$	10.0273	0.363011
$764$	0	0
$765$	6.95678	0.251523
$766$	0	0
$767$	−14.1721	−0.511724
$768$	0	0
$769$	45.9933	1.65856	0.829281	−	0.558832i	$-0.188750\pi$
$769$	45.9933	1.65856	0.829281	+	0.558832i	$0.188750\pi$
$770$	0	0
$771$	−15.3341	−0.552243
$772$	0	0
$773$	4.51290	0.162318	0.0811589	−	0.996701i	$-0.474138\pi$
$773$	4.51290	0.162318	0.0811589	+	0.996701i	$0.474138\pi$
$774$	0	0
$775$	−6.53339	−0.234686
$776$	0	0
$777$	37.6188	1.34957
$778$	0	0
$779$	7.23480	0.259214
$780$	0	0
$781$	4.18746	0.149839
$782$	0	0
$783$	−8.07933	−0.288732
$784$	0	0
$785$	−16.2423	−0.579712
$786$	0	0
$787$	−18.8514	−0.671980	−0.335990	−	0.941865i	$-0.609071\pi$
$787$	−18.8514	−0.671980	−0.335990	+	0.941865i	$0.609071\pi$
$788$	0	0
$789$	−6.84467	−0.243677
$790$	0	0
$791$	−36.1914	−1.28682
$792$	0	0
$793$	−21.4640	−0.762208
$794$	0	0
$795$	26.0685	0.924555
$796$	0	0
$797$	−27.0011	−0.956430	−0.478215	−	0.878243i	$-0.658716\pi$
$797$	−27.0011	−0.956430	−0.478215	+	0.878243i	$0.658716\pi$
$798$	0	0
$799$	7.66284	0.271092
$800$	0	0
$801$	3.24000	0.114480
$802$	0	0
$803$	2.40344	0.0848154
$804$	0	0
$805$	70.9812	2.50176
$806$	0	0
$807$	25.6870	0.904226
$808$	0	0
$809$	18.5046	0.650588	0.325294	−	0.945613i	$-0.394537\pi$
$809$	18.5046	0.650588	0.325294	+	0.945613i	$0.394537\pi$
$810$	0	0
$811$	1.32514	0.0465320	0.0232660	−	0.999729i	$-0.492594\pi$
$811$	1.32514	0.0465320	0.0232660	+	0.999729i	$0.492594\pi$
$812$	0	0
$813$	22.0183	0.772215
$814$	0	0
$815$	22.5019	0.788208
$816$	0	0
$817$	−1.31269	−0.0459251
$818$	0	0
$819$	−9.38882	−0.328072
$820$	0	0
$821$	1.13965	0.0397739	0.0198870	−	0.999802i	$-0.493669\pi$
$821$	1.13965	0.0397739	0.0198870	+	0.999802i	$0.493669\pi$
$822$	0	0
$823$	−29.2285	−1.01884	−0.509421	−	0.860518i	$-0.670140\pi$
$823$	−29.2285	−1.01884	−0.509421	+	0.860518i	$0.670140\pi$
$824$	0	0
$825$	−6.64508	−0.231352
$826$	0	0
$827$	−16.8206	−0.584908	−0.292454	−	0.956280i	$-0.594472\pi$
$827$	−16.8206	−0.584908	−0.292454	+	0.956280i	$0.594472\pi$
$828$	0	0
$829$	−44.4361	−1.54333	−0.771665	−	0.636029i	$-0.780576\pi$
$829$	−44.4361	−1.54333	−0.771665	+	0.636029i	$0.780576\pi$
$830$	0	0
$831$	−25.7727	−0.894044
$832$	0	0
$833$	16.4725	0.570738
$834$	0	0
$835$	−51.5470	−1.78386
$836$	0	0
$837$	4.58770	0.158574
$838$	0	0
$839$	48.4989	1.67437	0.837184	−	0.546922i	$-0.184201\pi$
$839$	48.4989	1.67437	0.837184	+	0.546922i	$0.184201\pi$
$840$	0	0
$841$	36.2755	1.25088
$842$	0	0
$843$	−5.79858	−0.199714
$844$	0	0
$845$	−15.7651	−0.542337
$846$	0	0
$847$	38.8441	1.33470
$848$	0	0
$849$	0.919794	0.0315672
$850$	0	0
$851$	−81.0306	−2.77769
$852$	0	0
$853$	25.3564	0.868185	0.434093	−	0.900868i	$-0.357069\pi$
$853$	25.3564	0.868185	0.434093	+	0.900868i	$0.357069\pi$
$854$	0	0
$855$	1.81576	0.0620978
$856$	0	0
$857$	8.99949	0.307417	0.153708	−	0.988116i	$-0.450878\pi$
$857$	8.99949	0.307417	0.153708	+	0.988116i	$0.450878\pi$
$858$	0	0
$859$	−22.1970	−0.757352	−0.378676	−	0.925529i	$-0.623620\pi$
$859$	−22.1970	−0.757352	−0.378676	+	0.925529i	$0.623620\pi$
$860$	0	0
$861$	−36.4141	−1.24099
$862$	0	0
$863$	−35.0196	−1.19208	−0.596040	−	0.802955i	$-0.703260\pi$
$863$	−35.0196	−1.19208	−0.596040	+	0.802955i	$0.703260\pi$
$864$	0	0
$865$	−57.6797	−1.96117
$866$	0	0
$867$	9.46639	0.321495
$868$	0	0
$869$	29.0984	0.987096
$870$	0	0
$871$	4.63295	0.156982
$872$	0	0
$873$	−0.898636	−0.0304142
$874$	0	0
$875$	−32.6804	−1.10480
$876$	0	0
$877$	4.60081	0.155358	0.0776791	−	0.996978i	$-0.475249\pi$
$877$	4.60081	0.155358	0.0776791	+	0.996978i	$0.475249\pi$
$878$	0	0
$879$	−13.7089	−0.462390
$880$	0	0
$881$	31.5850	1.06412	0.532062	−	0.846705i	$-0.321417\pi$
$881$	31.5850	1.06412	0.532062	+	0.846705i	$0.321417\pi$
$882$	0	0
$883$	0.793183	0.0266927	0.0133464	−	0.999911i	$-0.495752\pi$
$883$	0.793183	0.0266927	0.0133464	+	0.999911i	$0.495752\pi$
$884$	0	0
$885$	−13.7951	−0.463717
$886$	0	0
$887$	−12.4746	−0.418858	−0.209429	−	0.977824i	$-0.567160\pi$
$887$	−12.4746	−0.418858	−0.209429	+	0.977824i	$0.567160\pi$
$888$	0	0
$889$	−18.3838	−0.616572
$890$	0	0
$891$	4.66613	0.156321
$892$	0	0
$893$	2.00005	0.0669291
$894$	0	0
$895$	−56.4264	−1.88613
$896$	0	0
$897$	20.2235	0.675242
$898$	0	0
$899$	−37.0656	−1.23621
$900$	0	0
$901$	−28.2300	−0.940479
$902$	0	0
$903$	6.60700	0.219867
$904$	0	0
$905$	35.9059	1.19355
$906$	0	0
$907$	−12.4391	−0.413035	−0.206518	−	0.978443i	$-0.566213\pi$
$907$	−12.4391	−0.413035	−0.206518	+	0.978443i	$0.566213\pi$
$908$	0	0
$909$	−1.73787	−0.0576414
$910$	0	0
$911$	−60.0276	−1.98880	−0.994402	−	0.105660i	$-0.966305\pi$
$911$	−60.0276	−1.98880	−0.994402	+	0.105660i	$0.966305\pi$
$912$	0	0
$913$	−36.9810	−1.22389
$914$	0	0
$915$	−20.8931	−0.690703
$916$	0	0
$917$	−43.3592	−1.43185
$918$	0	0
$919$	23.5259	0.776046	0.388023	−	0.921650i	$-0.373158\pi$
$919$	23.5259	0.776046	0.388023	+	0.921650i	$0.373158\pi$
$920$	0	0
$921$	11.9361	0.393309
$922$	0	0
$923$	−2.33673	−0.0769144
$924$	0	0
$925$	−14.8577	−0.488519
$926$	0	0
$927$	−8.02139	−0.263457
$928$	0	0
$929$	57.0540	1.87188	0.935940	−	0.352159i	$-0.114552\pi$
$929$	57.0540	1.87188	0.935940	+	0.352159i	$0.114552\pi$
$930$	0	0
$931$	4.29943	0.140908
$932$	0	0
$933$	8.91744	0.291944
$934$	0	0
$935$	32.4612	1.06160
$936$	0	0
$937$	−23.1246	−0.755449	−0.377725	−	0.925918i	$-0.623293\pi$
$937$	−23.1246	−0.755449	−0.377725	+	0.925918i	$0.623293\pi$
$938$	0	0
$939$	13.0506	0.425889
$940$	0	0
$941$	−48.1987	−1.57123	−0.785617	−	0.618713i	$-0.787654\pi$
$941$	−48.1987	−1.57123	−0.785617	+	0.618713i	$0.787654\pi$
$942$	0	0
$943$	78.4358	2.55422
$944$	0	0
$945$	−9.13909	−0.297294
$946$	0	0
$947$	−2.36134	−0.0767332	−0.0383666	−	0.999264i	$-0.512215\pi$
$947$	−2.36134	−0.0767332	−0.0383666	+	0.999264i	$0.512215\pi$
$948$	0	0
$949$	−1.34119	−0.0435368
$950$	0	0
$951$	−22.5539	−0.731360
$952$	0	0
$953$	12.2459	0.396683	0.198341	−	0.980133i	$-0.436445\pi$
$953$	12.2459	0.396683	0.198341	+	0.980133i	$0.436445\pi$
$954$	0	0
$955$	22.3713	0.723918
$956$	0	0
$957$	−37.6992	−1.21864
$958$	0	0
$959$	42.4558	1.37097
$960$	0	0
$961$	−9.95299	−0.321064
$962$	0	0
$963$	−8.60577	−0.277317
$964$	0	0
$965$	−28.8053	−0.927277
$966$	0	0
$967$	12.5200	0.402618	0.201309	−	0.979528i	$-0.435481\pi$
$967$	12.5200	0.402618	0.201309	+	0.979528i	$0.435481\pi$
$968$	0	0
$969$	−1.96632	−0.0631673
$970$	0	0
$971$	−23.8316	−0.764793	−0.382396	−	0.923998i	$-0.624901\pi$
$971$	−23.8316	−0.764793	−0.382396	+	0.923998i	$0.624901\pi$
$972$	0	0
$973$	4.68478	0.150187
$974$	0	0
$975$	3.70816	0.118756
$976$	0	0
$977$	−30.7902	−0.985064	−0.492532	−	0.870294i	$-0.663929\pi$
$977$	−30.7902	−0.985064	−0.492532	+	0.870294i	$0.663929\pi$
$978$	0	0
$979$	15.1183	0.483182
$980$	0	0
$981$	2.78090	0.0887874
$982$	0	0
$983$	−35.3799	−1.12844	−0.564221	−	0.825624i	$-0.690823\pi$
$983$	−35.3799	−1.12844	−0.564221	+	0.825624i	$0.690823\pi$
$984$	0	0
$985$	30.0130	0.956293
$986$	0	0
$987$	−10.0666	−0.320425
$988$	0	0
$989$	−14.2314	−0.452534
$990$	0	0
$991$	27.5363	0.874719	0.437360	−	0.899287i	$-0.355914\pi$
$991$	27.5363	0.874719	0.437360	+	0.899287i	$0.355914\pi$
$992$	0	0
$993$	−4.18703	−0.132871
$994$	0	0
$995$	1.49921	0.0475283
$996$	0	0
$997$	−34.2241	−1.08389	−0.541944	−	0.840415i	$-0.682312\pi$
$997$	−34.2241	−1.08389	−0.541944	+	0.840415i	$0.682312\pi$
$998$	0	0
$999$	10.4330	0.330085

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6036.2.a.i.1.20	✓	26

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6036.2.a.i.1.20	✓	26	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:	\( N \)	\(=\)	\( 6036 = 2^{2} \cdot 3 \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6036.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +2.53458 q^{5} +3.60576 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +2.53458 q^{5} +3.60576 q^{7} +1.00000 q^{9} +4.66613 q^{11} -2.60384 q^{13} -2.53458 q^{15} +2.74474 q^{17} +0.716395 q^{19} -3.60576 q^{21} +7.76678 q^{23} +1.42411 q^{25} -1.00000 q^{27} +8.07933 q^{29} -4.58770 q^{31} -4.66613 q^{33} +9.13909 q^{35} -10.4330 q^{37} +2.60384 q^{39} +10.0989 q^{41} -1.83235 q^{43} +2.53458 q^{45} +2.79183 q^{47} +6.00147 q^{49} -2.74474 q^{51} -10.2851 q^{53} +11.8267 q^{55} -0.716395 q^{57} +5.44275 q^{59} +8.24320 q^{61} +3.60576 q^{63} -6.59965 q^{65} -1.77928 q^{67} -7.76678 q^{69} +0.897417 q^{71} +0.515081 q^{73} -1.42411 q^{75} +16.8249 q^{77} +6.23609 q^{79} +1.00000 q^{81} -7.92540 q^{83} +6.95678 q^{85} -8.07933 q^{87} +3.24000 q^{89} -9.38882 q^{91} +4.58770 q^{93} +1.81576 q^{95} -0.898636 q^{97} +4.66613 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 26 q - 26 q^{3} + 6 q^{5} + 5 q^{7} + 26 q^{9}+O(q^{10}) \) 26 * q - 26 * q^3 + 6 * q^5 + 5 * q^7 + 26 * q^9	\( 26 q - 26 q^{3} + 6 q^{5} + 5 q^{7} + 26 q^{9} - 11 q^{11} + 13 q^{13} - 6 q^{15} + 12 q^{17} - q^{19} - 5 q^{21} - 22 q^{23} + 48 q^{25} - 26 q^{27} + 6 q^{29} + 19 q^{31} + 11 q^{33} - 21 q^{35} + 20 q^{37} - 13 q^{39} + 25 q^{41} + 4 q^{43} + 6 q^{45} + 8 q^{47} + 67 q^{49} - 12 q^{51} - 5 q^{53} + 20 q^{55} + q^{57} - 18 q^{59} + 43 q^{61} + 5 q^{63} + 41 q^{65} + 5 q^{67} + 22 q^{69} - q^{71} + 22 q^{73} - 48 q^{75} + 23 q^{77} + 16 q^{79} + 26 q^{81} - 19 q^{83} + 29 q^{85} - 6 q^{87} + 49 q^{89} - 13 q^{91} - 19 q^{93} - 26 q^{95} + 25 q^{97} - 11 q^{99}+O(q^{100}) \) 26 * q - 26 * q^3 + 6 * q^5 + 5 * q^7 + 26 * q^9 - 11 * q^11 + 13 * q^13 - 6 * q^15 + 12 * q^17 - q^19 - 5 * q^21 - 22 * q^23 + 48 * q^25 - 26 * q^27 + 6 * q^29 + 19 * q^31 + 11 * q^33 - 21 * q^35 + 20 * q^37 - 13 * q^39 + 25 * q^41 + 4 * q^43 + 6 * q^45 + 8 * q^47 + 67 * q^49 - 12 * q^51 - 5 * q^53 + 20 * q^55 + q^57 - 18 * q^59 + 43 * q^61 + 5 * q^63 + 41 * q^65 + 5 * q^67 + 22 * q^69 - q^71 + 22 * q^73 - 48 * q^75 + 23 * q^77 + 16 * q^79 + 26 * q^81 - 19 * q^83 + 29 * q^85 - 6 * q^87 + 49 * q^89 - 13 * q^91 - 19 * q^93 - 26 * q^95 + 25 * q^97 - 11 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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