LMFDB - Embedded newform 6080.2.a.cl.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6080,2,Mod(1,6080)] 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("6080.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6080, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6080 = 2^{6} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6080.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,0,4,0,5,0,4,0,7,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$48.5490444289$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.387268.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 7x^{3} + 4x^{2} + 12x - 2 $ x^5 - x^4 - 7x^3 + 4x^2 + 12*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 3040)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.88930$ of defining polynomial
Character	$\chi$	$=$	6080.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.45876 q^{3} +1.00000 q^{5} +4.18444 q^{7} +8.96304 q^{9} -1.91967 q^{11} +5.31770 q^{13} +3.45876 q^{15} -6.96519 q^{17} -1.00000 q^{19} +14.4730 q^{21} -2.55761 q^{23} +1.00000 q^{25} +20.6248 q^{27} -2.90445 q^{29} +4.26651 q^{31} -6.63968 q^{33} +4.18444 q^{35} -6.07710 q^{37} +18.3927 q^{39} -6.91753 q^{41} -3.64887 q^{43} +8.96304 q^{45} +6.29283 q^{47} +10.5095 q^{49} -24.0909 q^{51} +12.7150 q^{53} -1.91967 q^{55} -3.45876 q^{57} -5.24303 q^{59} +3.37102 q^{61} +37.5053 q^{63} +5.31770 q^{65} -8.20081 q^{67} -8.84616 q^{69} -16.3533 q^{71} -3.30133 q^{73} +3.45876 q^{75} -8.03274 q^{77} -1.02418 q^{79} +44.4470 q^{81} -4.34256 q^{83} -6.96519 q^{85} -10.0458 q^{87} -1.45564 q^{89} +22.2516 q^{91} +14.7569 q^{93} -1.00000 q^{95} -8.51561 q^{97} -17.2061 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 4 q^{3} + 5 q^{5} + 4 q^{7} + 7 q^{9} + 2 q^{11} + 4 q^{13} + 4 q^{15} - 12 q^{17} - 5 q^{19} + 10 q^{21} + 8 q^{23} + 5 q^{25} + 16 q^{27} + 6 q^{29} + 10 q^{31} - 18 q^{33} + 4 q^{35} + 6 q^{37}+ \cdots - 14 q^{99}+O(q^{100}) $ 5 * q + 4 * q^3 + 5 * q^5 + 4 * q^7 + 7 * q^9 + 2 * q^11 + 4 * q^13 + 4 * q^15 - 12 * q^17 - 5 * q^19 + 10 * q^21 + 8 * q^23 + 5 * q^25 + 16 * q^27 + 6 * q^29 + 10 * q^31 - 18 * q^33 + 4 * q^35 + 6 * q^37 + 18 * q^39 - 8 * q^41 + 12 * q^43 + 7 * q^45 + 16 * q^47 + 7 * q^49 - 14 * q^51 + 18 * q^53 + 2 * q^55 - 4 * q^57 + 8 * q^59 - 2 * q^61 + 36 * q^63 + 4 * q^65 + 10 * q^67 + 22 * q^69 - 18 * q^71 - 28 * q^73 + 4 * q^75 + 28 * q^77 + 14 * q^79 + 25 * q^81 + 8 * q^83 - 12 * q^85 + 24 * q^87 - 30 * q^89 + 28 * q^91 + 24 * q^93 - 5 * q^95 - 18 * q^97 - 14 * 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
$3$	3.45876	1.99692	0.998459	−	0.0554941i	$-0.0176734\pi$
$3$	3.45876	1.99692	0.998459	+	0.0554941i	$0.0176734\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	4.18444	1.58157	0.790785	−	0.612094i	$-0.209673\pi$
$7$	4.18444	1.58157	0.790785	+	0.612094i	$0.209673\pi$
$8$	0	0
$9$	8.96304	2.98768
$10$	0	0
$11$	−1.91967	−0.578802	−0.289401	−	0.957208i	$-0.593456\pi$
$11$	−1.91967	−0.578802	−0.289401	+	0.957208i	$0.593456\pi$
$12$	0	0
$13$	5.31770	1.47486	0.737432	−	0.675421i	$-0.236038\pi$
$13$	5.31770	1.47486	0.737432	+	0.675421i	$0.236038\pi$
$14$	0	0
$15$	3.45876	0.893049
$16$	0	0
$17$	−6.96519	−1.68931	−0.844653	−	0.535314i	$-0.820193\pi$
$17$	−6.96519	−1.68931	−0.844653	+	0.535314i	$0.820193\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	14.4730	3.15827
$22$	0	0
$23$	−2.55761	−0.533298	−0.266649	−	0.963794i	$-0.585916\pi$
$23$	−2.55761	−0.533298	−0.266649	+	0.963794i	$0.585916\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	20.6248	3.96924
$28$	0	0
$29$	−2.90445	−0.539343	−0.269672	−	0.962952i	$-0.586915\pi$
$29$	−2.90445	−0.539343	−0.269672	+	0.962952i	$0.586915\pi$
$30$	0	0
$31$	4.26651	0.766289	0.383144	−	0.923688i	$-0.374841\pi$
$31$	4.26651	0.766289	0.383144	+	0.923688i	$0.374841\pi$
$32$	0	0
$33$	−6.63968	−1.15582
$34$	0	0
$35$	4.18444	0.707300
$36$	0	0
$37$	−6.07710	−0.999070	−0.499535	−	0.866294i	$-0.666496\pi$
$37$	−6.07710	−0.999070	−0.499535	+	0.866294i	$0.666496\pi$
$38$	0	0
$39$	18.3927	2.94518
$40$	0	0
$41$	−6.91753	−1.08034	−0.540168	−	0.841557i	$-0.681639\pi$
$41$	−6.91753	−1.08034	−0.540168	+	0.841557i	$0.681639\pi$
$42$	0	0
$43$	−3.64887	−0.556448	−0.278224	−	0.960516i	$-0.589746\pi$
$43$	−3.64887	−0.556448	−0.278224	+	0.960516i	$0.589746\pi$
$44$	0	0
$45$	8.96304	1.33613
$46$	0	0
$47$	6.29283	0.917904	0.458952	−	0.888461i	$-0.348225\pi$
$47$	6.29283	0.917904	0.458952	+	0.888461i	$0.348225\pi$
$48$	0	0
$49$	10.5095	1.50136
$50$	0	0
$51$	−24.0909	−3.37341
$52$	0	0
$53$	12.7150	1.74655	0.873273	−	0.487232i	$-0.161993\pi$
$53$	12.7150	1.74655	0.873273	+	0.487232i	$0.161993\pi$
$54$	0	0
$55$	−1.91967	−0.258848
$56$	0	0
$57$	−3.45876	−0.458124
$58$	0	0
$59$	−5.24303	−0.682585	−0.341292	−	0.939957i	$-0.610865\pi$
$59$	−5.24303	−0.682585	−0.341292	+	0.939957i	$0.610865\pi$
$60$	0	0
$61$	3.37102	0.431615	0.215808	−	0.976436i	$-0.430762\pi$
$61$	3.37102	0.431615	0.215808	+	0.976436i	$0.430762\pi$
$62$	0	0
$63$	37.5053	4.72523
$64$	0	0
$65$	5.31770	0.659579
$66$	0	0
$67$	−8.20081	−1.00189	−0.500944	−	0.865480i	$-0.667014\pi$
$67$	−8.20081	−1.00189	−0.500944	+	0.865480i	$0.667014\pi$
$68$	0	0
$69$	−8.84616	−1.06495
$70$	0	0
$71$	−16.3533	−1.94078	−0.970388	−	0.241552i	$-0.922344\pi$
$71$	−16.3533	−1.94078	−0.970388	+	0.241552i	$0.922344\pi$
$72$	0	0
$73$	−3.30133	−0.386391	−0.193196	−	0.981160i	$-0.561885\pi$
$73$	−3.30133	−0.386391	−0.193196	+	0.981160i	$0.561885\pi$
$74$	0	0
$75$	3.45876	0.399384
$76$	0	0
$77$	−8.03274	−0.915416
$78$	0	0
$79$	−1.02418	−0.115229	−0.0576145	−	0.998339i	$-0.518349\pi$
$79$	−1.02418	−0.115229	−0.0576145	+	0.998339i	$0.518349\pi$
$80$	0	0
$81$	44.4470	4.93856
$82$	0	0
$83$	−4.34256	−0.476658	−0.238329	−	0.971184i	$-0.576600\pi$
$83$	−4.34256	−0.476658	−0.238329	+	0.971184i	$0.576600\pi$
$84$	0	0
$85$	−6.96519	−0.755481
$86$	0	0
$87$	−10.0458	−1.07702
$88$	0	0
$89$	−1.45564	−0.154297	−0.0771487	−	0.997020i	$-0.524582\pi$
$89$	−1.45564	−0.154297	−0.0771487	+	0.997020i	$0.524582\pi$
$90$	0	0
$91$	22.2516	2.33260
$92$	0	0
$93$	14.7569	1.53022
$94$	0	0
$95$	−1.00000	−0.102598
$96$	0	0
$97$	−8.51561	−0.864630	−0.432315	−	0.901723i	$-0.642303\pi$
$97$	−8.51561	−0.864630	−0.432315	+	0.901723i	$0.642303\pi$
$98$	0	0
$99$	−17.2061	−1.72928
$100$	0	0
$101$	2.08033	0.207001	0.103500	−	0.994629i	$-0.466996\pi$
$101$	2.08033	0.207001	0.103500	+	0.994629i	$0.466996\pi$
$102$	0	0
$103$	14.2598	1.40506	0.702530	−	0.711654i	$-0.252054\pi$
$103$	14.2598	1.40506	0.702530	+	0.711654i	$0.252054\pi$
$104$	0	0
$105$	14.4730	1.41242
$106$	0	0
$107$	8.01874	0.775201	0.387600	−	0.921827i	$-0.373304\pi$
$107$	8.01874	0.775201	0.387600	+	0.921827i	$0.373304\pi$
$108$	0	0
$109$	8.22558	0.787867	0.393934	−	0.919139i	$-0.371114\pi$
$109$	8.22558	0.787867	0.393934	+	0.919139i	$0.371114\pi$
$110$	0	0
$111$	−21.0193	−1.99506
$112$	0	0
$113$	16.3280	1.53601	0.768005	−	0.640444i	$-0.221250\pi$
$113$	16.3280	1.53601	0.768005	+	0.640444i	$0.221250\pi$
$114$	0	0
$115$	−2.55761	−0.238498
$116$	0	0
$117$	47.6628	4.40642
$118$	0	0
$119$	−29.1454	−2.67176
$120$	0	0
$121$	−7.31487	−0.664988
$122$	0	0
$123$	−23.9261	−2.15734
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−9.89012	−0.877606	−0.438803	−	0.898583i	$-0.644597\pi$
$127$	−9.89012	−0.877606	−0.438803	+	0.898583i	$0.644597\pi$
$128$	0	0
$129$	−12.6206	−1.11118
$130$	0	0
$131$	−5.98438	−0.522858	−0.261429	−	0.965223i	$-0.584194\pi$
$131$	−5.98438	−0.522858	−0.261429	+	0.965223i	$0.584194\pi$
$132$	0	0
$133$	−4.18444	−0.362837
$134$	0	0
$135$	20.6248	1.77510
$136$	0	0
$137$	−11.1258	−0.950545	−0.475273	−	0.879839i	$-0.657651\pi$
$137$	−11.1258	−0.950545	−0.475273	+	0.879839i	$0.657651\pi$
$138$	0	0
$139$	16.5011	1.39960	0.699801	−	0.714338i	$-0.253272\pi$
$139$	16.5011	1.39960	0.699801	+	0.714338i	$0.253272\pi$
$140$	0	0
$141$	21.7654	1.83298
$142$	0	0
$143$	−10.2082	−0.853654
$144$	0	0
$145$	−2.90445	−0.241202
$146$	0	0
$147$	36.3500	2.99810
$148$	0	0
$149$	−17.7917	−1.45756	−0.728778	−	0.684750i	$-0.759912\pi$
$149$	−17.7917	−1.45756	−0.728778	+	0.684750i	$0.759912\pi$
$150$	0	0
$151$	−0.913243	−0.0743187	−0.0371593	−	0.999309i	$-0.511831\pi$
$151$	−0.913243	−0.0743187	−0.0371593	+	0.999309i	$0.511831\pi$
$152$	0	0
$153$	−62.4293	−5.04711
$154$	0	0
$155$	4.26651	0.342695
$156$	0	0
$157$	2.28213	0.182134	0.0910669	−	0.995845i	$-0.470972\pi$
$157$	2.28213	0.182134	0.0910669	+	0.995845i	$0.470972\pi$
$158$	0	0
$159$	43.9783	3.48771
$160$	0	0
$161$	−10.7022	−0.843448
$162$	0	0
$163$	2.34685	0.183819	0.0919096	−	0.995767i	$-0.470703\pi$
$163$	2.34685	0.183819	0.0919096	+	0.995767i	$0.470703\pi$
$164$	0	0
$165$	−6.63968	−0.516898
$166$	0	0
$167$	−14.3130	−1.10757	−0.553787	−	0.832658i	$-0.686818\pi$
$167$	−14.3130	−1.10757	−0.553787	+	0.832658i	$0.686818\pi$
$168$	0	0
$169$	15.2779	1.17522
$170$	0	0
$171$	−8.96304	−0.685421
$172$	0	0
$173$	−11.4702	−0.872061	−0.436030	−	0.899932i	$-0.643616\pi$
$173$	−11.4702	−0.872061	−0.436030	+	0.899932i	$0.643616\pi$
$174$	0	0
$175$	4.18444	0.316314
$176$	0	0
$177$	−18.1344	−1.36307
$178$	0	0
$179$	−0.494634	−0.0369707	−0.0184853	−	0.999829i	$-0.505884\pi$
$179$	−0.494634	−0.0369707	−0.0184853	+	0.999829i	$0.505884\pi$
$180$	0	0
$181$	−5.02418	−0.373444	−0.186722	−	0.982413i	$-0.559786\pi$
$181$	−5.02418	−0.373444	−0.186722	+	0.982413i	$0.559786\pi$
$182$	0	0
$183$	11.6596	0.861901
$184$	0	0
$185$	−6.07710	−0.446798
$186$	0	0
$187$	13.3709	0.977774
$188$	0	0
$189$	86.3031	6.27763
$190$	0	0
$191$	−4.01736	−0.290686	−0.145343	−	0.989381i	$-0.546429\pi$
$191$	−4.01736	−0.290686	−0.145343	+	0.989381i	$0.546429\pi$
$192$	0	0
$193$	−15.3265	−1.10322	−0.551612	−	0.834101i	$-0.685987\pi$
$193$	−15.3265	−1.10322	−0.551612	+	0.834101i	$0.685987\pi$
$194$	0	0
$195$	18.3927	1.31713
$196$	0	0
$197$	2.20197	0.156884	0.0784420	−	0.996919i	$-0.475005\pi$
$197$	2.20197	0.156884	0.0784420	+	0.996919i	$0.475005\pi$
$198$	0	0
$199$	−11.4972	−0.815013	−0.407506	−	0.913202i	$-0.633602\pi$
$199$	−11.4972	−0.815013	−0.407506	+	0.913202i	$0.633602\pi$
$200$	0	0
$201$	−28.3647	−2.00069
$202$	0	0
$203$	−12.1535	−0.853009
$204$	0	0
$205$	−6.91753	−0.483141
$206$	0	0
$207$	−22.9239	−1.59332
$208$	0	0
$209$	1.91967	0.132786
$210$	0	0
$211$	2.98080	0.205207	0.102603	−	0.994722i	$-0.467283\pi$
$211$	2.98080	0.205207	0.102603	+	0.994722i	$0.467283\pi$
$212$	0	0
$213$	−56.5621	−3.87557
$214$	0	0
$215$	−3.64887	−0.248851
$216$	0	0
$217$	17.8530	1.21194
$218$	0	0
$219$	−11.4185	−0.771592
$220$	0	0
$221$	−37.0388	−2.49150
$222$	0	0
$223$	−16.1105	−1.07884	−0.539419	−	0.842038i	$-0.681356\pi$
$223$	−16.1105	−1.07884	−0.539419	+	0.842038i	$0.681356\pi$
$224$	0	0
$225$	8.96304	0.597536
$226$	0	0
$227$	16.7992	1.11500	0.557500	−	0.830177i	$-0.311760\pi$
$227$	16.7992	1.11500	0.557500	+	0.830177i	$0.311760\pi$
$228$	0	0
$229$	17.1521	1.13344	0.566720	−	0.823910i	$-0.308212\pi$
$229$	17.1521	1.13344	0.566720	+	0.823910i	$0.308212\pi$
$230$	0	0
$231$	−27.7834	−1.82801
$232$	0	0
$233$	−12.4713	−0.817019	−0.408509	−	0.912754i	$-0.633951\pi$
$233$	−12.4713	−0.817019	−0.408509	+	0.912754i	$0.633951\pi$
$234$	0	0
$235$	6.29283	0.410499
$236$	0	0
$237$	−3.54239	−0.230103
$238$	0	0
$239$	−0.600913	−0.0388698	−0.0194349	−	0.999811i	$-0.506187\pi$
$239$	−0.600913	−0.0388698	−0.0194349	+	0.999811i	$0.506187\pi$
$240$	0	0
$241$	21.5643	1.38908	0.694538	−	0.719456i	$-0.255609\pi$
$241$	21.5643	1.38908	0.694538	+	0.719456i	$0.255609\pi$
$242$	0	0
$243$	91.8575	5.89266
$244$	0	0
$245$	10.5095	0.671430
$246$	0	0
$247$	−5.31770	−0.338357
$248$	0	0
$249$	−15.0199	−0.951847
$250$	0	0
$251$	6.34667	0.400598	0.200299	−	0.979735i	$-0.435809\pi$
$251$	6.34667	0.400598	0.200299	+	0.979735i	$0.435809\pi$
$252$	0	0
$253$	4.90976	0.308674
$254$	0	0
$255$	−24.0909	−1.50863
$256$	0	0
$257$	−24.1830	−1.50849	−0.754246	−	0.656592i	$-0.771997\pi$
$257$	−24.1830	−1.50849	−0.754246	+	0.656592i	$0.771997\pi$
$258$	0	0
$259$	−25.4293	−1.58010
$260$	0	0
$261$	−26.0327	−1.61139
$262$	0	0
$263$	4.80017	0.295991	0.147996	−	0.988988i	$-0.452718\pi$
$263$	4.80017	0.295991	0.147996	+	0.988988i	$0.452718\pi$
$264$	0	0
$265$	12.7150	0.781079
$266$	0	0
$267$	−5.03471	−0.308119
$268$	0	0
$269$	4.88050	0.297569	0.148785	−	0.988870i	$-0.452464\pi$
$269$	4.88050	0.297569	0.148785	+	0.988870i	$0.452464\pi$
$270$	0	0
$271$	−1.60226	−0.0973301	−0.0486651	−	0.998815i	$-0.515497\pi$
$271$	−1.60226	−0.0973301	−0.0486651	+	0.998815i	$0.515497\pi$
$272$	0	0
$273$	76.9630	4.65801
$274$	0	0
$275$	−1.91967	−0.115760
$276$	0	0
$277$	−17.3012	−1.03953	−0.519765	−	0.854309i	$-0.673980\pi$
$277$	−17.3012	−1.03953	−0.519765	+	0.854309i	$0.673980\pi$
$278$	0	0
$279$	38.2410	2.28943
$280$	0	0
$281$	−24.7440	−1.47610	−0.738052	−	0.674744i	$-0.764254\pi$
$281$	−24.7440	−1.47610	−0.738052	+	0.674744i	$0.764254\pi$
$282$	0	0
$283$	3.78687	0.225106	0.112553	−	0.993646i	$-0.464097\pi$
$283$	3.78687	0.225106	0.112553	+	0.993646i	$0.464097\pi$
$284$	0	0
$285$	−3.45876	−0.204879
$286$	0	0
$287$	−28.9460	−1.70863
$288$	0	0
$289$	31.5138	1.85375
$290$	0	0
$291$	−29.4535	−1.72659
$292$	0	0
$293$	14.1812	0.828475	0.414238	−	0.910169i	$-0.364048\pi$
$293$	14.1812	0.828475	0.414238	+	0.910169i	$0.364048\pi$
$294$	0	0
$295$	−5.24303	−0.305261
$296$	0	0
$297$	−39.5927	−2.29740
$298$	0	0
$299$	−13.6006	−0.786542
$300$	0	0
$301$	−15.2685	−0.880061
$302$	0	0
$303$	7.19537	0.413363
$304$	0	0
$305$	3.37102	0.193024
$306$	0	0
$307$	−12.9379	−0.738404	−0.369202	−	0.929349i	$-0.620369\pi$
$307$	−12.9379	−0.738404	−0.369202	+	0.929349i	$0.620369\pi$
$308$	0	0
$309$	49.3213	2.80579
$310$	0	0
$311$	9.62875	0.545996	0.272998	−	0.962015i	$-0.411985\pi$
$311$	9.62875	0.545996	0.272998	+	0.962015i	$0.411985\pi$
$312$	0	0
$313$	−17.5614	−0.992628	−0.496314	−	0.868143i	$-0.665314\pi$
$313$	−17.5614	−0.992628	−0.496314	+	0.868143i	$0.665314\pi$
$314$	0	0
$315$	37.5053	2.11319
$316$	0	0
$317$	7.85153	0.440986	0.220493	−	0.975389i	$-0.429233\pi$
$317$	7.85153	0.440986	0.220493	+	0.975389i	$0.429233\pi$
$318$	0	0
$319$	5.57559	0.312173
$320$	0	0
$321$	27.7349	1.54801
$322$	0	0
$323$	6.96519	0.387553
$324$	0	0
$325$	5.31770	0.294973
$326$	0	0
$327$	28.4503	1.57331
$328$	0	0
$329$	26.3320	1.45173
$330$	0	0
$331$	22.3426	1.22806	0.614031	−	0.789282i	$-0.289547\pi$
$331$	22.3426	1.22806	0.614031	+	0.789282i	$0.289547\pi$
$332$	0	0
$333$	−54.4694	−2.98490
$334$	0	0
$335$	−8.20081	−0.448058
$336$	0	0
$337$	32.9016	1.79227	0.896133	−	0.443786i	$-0.146365\pi$
$337$	32.9016	1.79227	0.896133	+	0.443786i	$0.146365\pi$
$338$	0	0
$339$	56.4747	3.06728
$340$	0	0
$341$	−8.19030	−0.443529
$342$	0	0
$343$	14.6855	0.792942
$344$	0	0
$345$	−8.84616	−0.476261
$346$	0	0
$347$	−8.38387	−0.450070	−0.225035	−	0.974351i	$-0.572250\pi$
$347$	−8.38387	−0.450070	−0.225035	+	0.974351i	$0.572250\pi$
$348$	0	0
$349$	−4.05969	−0.217310	−0.108655	−	0.994080i	$-0.534654\pi$
$349$	−4.05969	−0.217310	−0.108655	+	0.994080i	$0.534654\pi$
$350$	0	0
$351$	109.676	5.85409
$352$	0	0
$353$	−9.56784	−0.509245	−0.254622	−	0.967041i	$-0.581951\pi$
$353$	−9.56784	−0.509245	−0.254622	+	0.967041i	$0.581951\pi$
$354$	0	0
$355$	−16.3533	−0.867941
$356$	0	0
$357$	−100.807	−5.33528
$358$	0	0
$359$	25.1347	1.32656	0.663280	−	0.748372i	$-0.269164\pi$
$359$	25.1347	1.32656	0.663280	+	0.748372i	$0.269164\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−25.3004	−1.32793
$364$	0	0
$365$	−3.30133	−0.172799
$366$	0	0
$367$	13.1236	0.685047	0.342523	−	0.939509i	$-0.388718\pi$
$367$	13.1236	0.685047	0.342523	+	0.939509i	$0.388718\pi$
$368$	0	0
$369$	−62.0021	−3.22770
$370$	0	0
$371$	53.2053	2.76228
$372$	0	0
$373$	19.6739	1.01867	0.509337	−	0.860567i	$-0.329891\pi$
$373$	19.6739	1.01867	0.509337	+	0.860567i	$0.329891\pi$
$374$	0	0
$375$	3.45876	0.178610
$376$	0	0
$377$	−15.4450	−0.795458
$378$	0	0
$379$	33.9979	1.74636	0.873178	−	0.487401i	$-0.162055\pi$
$379$	33.9979	1.74636	0.873178	+	0.487401i	$0.162055\pi$
$380$	0	0
$381$	−34.2076	−1.75251
$382$	0	0
$383$	0.534885	0.0273314	0.0136657	−	0.999907i	$-0.495650\pi$
$383$	0.534885	0.0273314	0.0136657	+	0.999907i	$0.495650\pi$
$384$	0	0
$385$	−8.03274	−0.409386
$386$	0	0
$387$	−32.7050	−1.66249
$388$	0	0
$389$	13.2296	0.670769	0.335385	−	0.942081i	$-0.391134\pi$
$389$	13.2296	0.670769	0.335385	+	0.942081i	$0.391134\pi$
$390$	0	0
$391$	17.8142	0.900903
$392$	0	0
$393$	−20.6986	−1.04410
$394$	0	0
$395$	−1.02418	−0.0515320
$396$	0	0
$397$	−11.5381	−0.579081	−0.289541	−	0.957166i	$-0.593502\pi$
$397$	−11.5381	−0.579081	−0.289541	+	0.957166i	$0.593502\pi$
$398$	0	0
$399$	−14.4730	−0.724556
$400$	0	0
$401$	−23.6658	−1.18181	−0.590907	−	0.806739i	$-0.701230\pi$
$401$	−23.6658	−1.18181	−0.590907	+	0.806739i	$0.701230\pi$
$402$	0	0
$403$	22.6880	1.13017
$404$	0	0
$405$	44.4470	2.20859
$406$	0	0
$407$	11.6660	0.578264
$408$	0	0
$409$	−18.0206	−0.891062	−0.445531	−	0.895267i	$-0.646985\pi$
$409$	−18.0206	−0.891062	−0.445531	+	0.895267i	$0.646985\pi$
$410$	0	0
$411$	−38.4817	−1.89816
$412$	0	0
$413$	−21.9392	−1.07956
$414$	0	0
$415$	−4.34256	−0.213168
$416$	0	0
$417$	57.0732	2.79489
$418$	0	0
$419$	18.6880	0.912970	0.456485	−	0.889731i	$-0.349108\pi$
$419$	18.6880	0.912970	0.456485	+	0.889731i	$0.349108\pi$
$420$	0	0
$421$	2.27530	0.110892	0.0554458	−	0.998462i	$-0.482342\pi$
$421$	2.27530	0.110892	0.0554458	+	0.998462i	$0.482342\pi$
$422$	0	0
$423$	56.4030	2.74241
$424$	0	0
$425$	−6.96519	−0.337861
$426$	0	0
$427$	14.1059	0.682630
$428$	0	0
$429$	−35.3078	−1.70468
$430$	0	0
$431$	−3.25578	−0.156826	−0.0784128	−	0.996921i	$-0.524985\pi$
$431$	−3.25578	−0.156826	−0.0784128	+	0.996921i	$0.524985\pi$
$432$	0	0
$433$	−0.690637	−0.0331899	−0.0165949	−	0.999862i	$-0.505283\pi$
$433$	−0.690637	−0.0331899	−0.0165949	+	0.999862i	$0.505283\pi$
$434$	0	0
$435$	−10.0458	−0.481660
$436$	0	0
$437$	2.55761	0.122347
$438$	0	0
$439$	11.5529	0.551391	0.275695	−	0.961245i	$-0.411092\pi$
$439$	11.5529	0.551391	0.275695	+	0.961245i	$0.411092\pi$
$440$	0	0
$441$	94.1975	4.48560
$442$	0	0
$443$	35.0042	1.66310	0.831551	−	0.555448i	$-0.187453\pi$
$443$	35.0042	1.66310	0.831551	+	0.555448i	$0.187453\pi$
$444$	0	0
$445$	−1.45564	−0.0690039
$446$	0	0
$447$	−61.5374	−2.91062
$448$	0	0
$449$	−7.59989	−0.358661	−0.179330	−	0.983789i	$-0.557393\pi$
$449$	−7.59989	−0.358661	−0.179330	+	0.983789i	$0.557393\pi$
$450$	0	0
$451$	13.2794	0.625301
$452$	0	0
$453$	−3.15869	−0.148408
$454$	0	0
$455$	22.2516	1.04317
$456$	0	0
$457$	−4.49322	−0.210184	−0.105092	−	0.994463i	$-0.533514\pi$
$457$	−4.49322	−0.210184	−0.105092	+	0.994463i	$0.533514\pi$
$458$	0	0
$459$	−143.655	−6.70526
$460$	0	0
$461$	28.3207	1.31902	0.659512	−	0.751694i	$-0.270763\pi$
$461$	28.3207	1.31902	0.659512	+	0.751694i	$0.270763\pi$
$462$	0	0
$463$	−11.3519	−0.527569	−0.263784	−	0.964582i	$-0.584971\pi$
$463$	−11.3519	−0.527569	−0.263784	+	0.964582i	$0.584971\pi$
$464$	0	0
$465$	14.7569	0.684333
$466$	0	0
$467$	−4.27998	−0.198054	−0.0990270	−	0.995085i	$-0.531573\pi$
$467$	−4.27998	−0.198054	−0.0990270	+	0.995085i	$0.531573\pi$
$468$	0	0
$469$	−34.3158	−1.58456
$470$	0	0
$471$	7.89335	0.363706
$472$	0	0
$473$	7.00462	0.322073
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	113.966	5.21812
$478$	0	0
$479$	18.9310	0.864979	0.432490	−	0.901639i	$-0.357635\pi$
$479$	18.9310	0.864979	0.432490	+	0.901639i	$0.357635\pi$
$480$	0	0
$481$	−32.3162	−1.47349
$482$	0	0
$483$	−37.0162	−1.68430
$484$	0	0
$485$	−8.51561	−0.386674
$486$	0	0
$487$	29.3181	1.32853	0.664265	−	0.747497i	$-0.268745\pi$
$487$	29.3181	1.32853	0.664265	+	0.747497i	$0.268745\pi$
$488$	0	0
$489$	8.11718	0.367072
$490$	0	0
$491$	−3.21099	−0.144910	−0.0724550	−	0.997372i	$-0.523083\pi$
$491$	−3.21099	−0.144910	−0.0724550	+	0.997372i	$0.523083\pi$
$492$	0	0
$493$	20.2301	0.911116
$494$	0	0
$495$	−17.2061	−0.773356
$496$	0	0
$497$	−68.4293	−3.06947
$498$	0	0
$499$	−3.85861	−0.172735	−0.0863675	−	0.996263i	$-0.527526\pi$
$499$	−3.85861	−0.172735	−0.0863675	+	0.996263i	$0.527526\pi$
$500$	0	0
$501$	−49.5053	−2.21174
$502$	0	0
$503$	−21.4587	−0.956797	−0.478399	−	0.878143i	$-0.658783\pi$
$503$	−21.4587	−0.956797	−0.478399	+	0.878143i	$0.658783\pi$
$504$	0	0
$505$	2.08033	0.0925735
$506$	0	0
$507$	52.8427	2.34683
$508$	0	0
$509$	13.6077	0.603152	0.301576	−	0.953442i	$-0.402487\pi$
$509$	13.6077	0.603152	0.301576	+	0.953442i	$0.402487\pi$
$510$	0	0
$511$	−13.8142	−0.611105
$512$	0	0
$513$	−20.6248	−0.910605
$514$	0	0
$515$	14.2598	0.628362
$516$	0	0
$517$	−12.0802	−0.531285
$518$	0	0
$519$	−39.6726	−1.74143
$520$	0	0
$521$	−36.3168	−1.59107	−0.795535	−	0.605908i	$-0.792810\pi$
$521$	−36.3168	−1.59107	−0.795535	+	0.605908i	$0.792810\pi$
$522$	0	0
$523$	−38.9982	−1.70527	−0.852636	−	0.522506i	$-0.824997\pi$
$523$	−38.9982	−1.70527	−0.852636	+	0.522506i	$0.824997\pi$
$524$	0	0
$525$	14.4730	0.631653
$526$	0	0
$527$	−29.7171	−1.29450
$528$	0	0
$529$	−16.4586	−0.715593
$530$	0	0
$531$	−46.9935	−2.03935
$532$	0	0
$533$	−36.7853	−1.59335
$534$	0	0
$535$	8.01874	0.346680
$536$	0	0
$537$	−1.71082	−0.0738274
$538$	0	0
$539$	−20.1749	−0.868992
$540$	0	0
$541$	−18.8267	−0.809421	−0.404711	−	0.914445i	$-0.632628\pi$
$541$	−18.8267	−0.809421	−0.404711	+	0.914445i	$0.632628\pi$
$542$	0	0
$543$	−17.3774	−0.745738
$544$	0	0
$545$	8.22558	0.352345
$546$	0	0
$547$	19.3493	0.827317	0.413659	−	0.910432i	$-0.364251\pi$
$547$	19.3493	0.827317	0.413659	+	0.910432i	$0.364251\pi$
$548$	0	0
$549$	30.2146	1.28953
$550$	0	0
$551$	2.90445	0.123734
$552$	0	0
$553$	−4.28561	−0.182243
$554$	0	0
$555$	−21.0193	−0.892218
$556$	0	0
$557$	29.3012	1.24153	0.620766	−	0.783996i	$-0.286822\pi$
$557$	29.3012	1.24153	0.620766	+	0.783996i	$0.286822\pi$
$558$	0	0
$559$	−19.4036	−0.820685
$560$	0	0
$561$	46.2466	1.95253
$562$	0	0
$563$	−20.2988	−0.855492	−0.427746	−	0.903899i	$-0.640692\pi$
$563$	−20.2988	−0.855492	−0.427746	+	0.903899i	$0.640692\pi$
$564$	0	0
$565$	16.3280	0.686924
$566$	0	0
$567$	185.986	7.81068
$568$	0	0
$569$	−1.49266	−0.0625757	−0.0312879	−	0.999510i	$-0.509961\pi$
$569$	−1.49266	−0.0625757	−0.0312879	+	0.999510i	$0.509961\pi$
$570$	0	0
$571$	23.4085	0.979615	0.489808	−	0.871830i	$-0.337067\pi$
$571$	23.4085	0.979615	0.489808	+	0.871830i	$0.337067\pi$
$572$	0	0
$573$	−13.8951	−0.580476
$574$	0	0
$575$	−2.55761	−0.106660
$576$	0	0
$577$	−29.4279	−1.22510	−0.612549	−	0.790433i	$-0.709856\pi$
$577$	−29.4279	−1.22510	−0.612549	+	0.790433i	$0.709856\pi$
$578$	0	0
$579$	−53.0107	−2.20305
$580$	0	0
$581$	−18.1712	−0.753868
$582$	0	0
$583$	−24.4087	−1.01090
$584$	0	0
$585$	47.6628	1.97061
$586$	0	0
$587$	38.3731	1.58383	0.791914	−	0.610632i	$-0.209085\pi$
$587$	38.3731	1.58383	0.791914	+	0.610632i	$0.209085\pi$
$588$	0	0
$589$	−4.26651	−0.175799
$590$	0	0
$591$	7.61610	0.313284
$592$	0	0
$593$	−12.1564	−0.499203	−0.249601	−	0.968349i	$-0.580300\pi$
$593$	−12.1564	−0.499203	−0.249601	+	0.968349i	$0.580300\pi$
$594$	0	0
$595$	−29.1454	−1.19485
$596$	0	0
$597$	−39.7660	−1.62751
$598$	0	0
$599$	4.87208	0.199068	0.0995340	−	0.995034i	$-0.468265\pi$
$599$	4.87208	0.199068	0.0995340	+	0.995034i	$0.468265\pi$
$600$	0	0
$601$	−4.82141	−0.196669	−0.0983347	−	0.995153i	$-0.531352\pi$
$601$	−4.82141	−0.196669	−0.0983347	+	0.995153i	$0.531352\pi$
$602$	0	0
$603$	−73.5042	−2.99332
$604$	0	0
$605$	−7.31487	−0.297392
$606$	0	0
$607$	−14.9897	−0.608414	−0.304207	−	0.952606i	$-0.598391\pi$
$607$	−14.9897	−0.608414	−0.304207	+	0.952606i	$0.598391\pi$
$608$	0	0
$609$	−42.0361	−1.70339
$610$	0	0
$611$	33.4634	1.35378
$612$	0	0
$613$	−11.0133	−0.444823	−0.222412	−	0.974953i	$-0.571393\pi$
$613$	−11.0133	−0.444823	−0.222412	+	0.974953i	$0.571393\pi$
$614$	0	0
$615$	−23.9261	−0.964793
$616$	0	0
$617$	11.3041	0.455088	0.227544	−	0.973768i	$-0.426931\pi$
$617$	11.3041	0.455088	0.227544	+	0.973768i	$0.426931\pi$
$618$	0	0
$619$	15.8466	0.636927	0.318463	−	0.947935i	$-0.396833\pi$
$619$	15.8466	0.636927	0.318463	+	0.947935i	$0.396833\pi$
$620$	0	0
$621$	−52.7500	−2.11679
$622$	0	0
$623$	−6.09104	−0.244032
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	6.63968	0.265163
$628$	0	0
$629$	42.3282	1.68773
$630$	0	0
$631$	22.3835	0.891074	0.445537	−	0.895263i	$-0.353013\pi$
$631$	22.3835	0.891074	0.445537	+	0.895263i	$0.353013\pi$
$632$	0	0
$633$	10.3099	0.409781
$634$	0	0
$635$	−9.89012	−0.392478
$636$	0	0
$637$	55.8866	2.21431
$638$	0	0
$639$	−146.575	−5.79842
$640$	0	0
$641$	31.2552	1.23451	0.617253	−	0.786765i	$-0.288246\pi$
$641$	31.2552	1.23451	0.617253	+	0.786765i	$0.288246\pi$
$642$	0	0
$643$	−28.6100	−1.12827	−0.564134	−	0.825683i	$-0.690790\pi$
$643$	−28.6100	−1.12827	−0.564134	+	0.825683i	$0.690790\pi$
$644$	0	0
$645$	−12.6206	−0.496935
$646$	0	0
$647$	17.0323	0.669610	0.334805	−	0.942287i	$-0.391329\pi$
$647$	17.0323	0.669610	0.334805	+	0.942287i	$0.391329\pi$
$648$	0	0
$649$	10.0649	0.395081
$650$	0	0
$651$	61.7492	2.42014
$652$	0	0
$653$	27.2694	1.06714	0.533568	−	0.845757i	$-0.320851\pi$
$653$	27.2694	1.06714	0.533568	+	0.845757i	$0.320851\pi$
$654$	0	0
$655$	−5.98438	−0.233829
$656$	0	0
$657$	−29.5899	−1.15441
$658$	0	0
$659$	−37.0311	−1.44253	−0.721264	−	0.692660i	$-0.756438\pi$
$659$	−37.0311	−1.44253	−0.721264	+	0.692660i	$0.756438\pi$
$660$	0	0
$661$	−6.03237	−0.234632	−0.117316	−	0.993095i	$-0.537429\pi$
$661$	−6.03237	−0.234632	−0.117316	+	0.993095i	$0.537429\pi$
$662$	0	0
$663$	−128.108	−4.97531
$664$	0	0
$665$	−4.18444	−0.162266
$666$	0	0
$667$	7.42845	0.287631
$668$	0	0
$669$	−55.7223	−2.15435
$670$	0	0
$671$	−6.47125	−0.249820
$672$	0	0
$673$	−37.8118	−1.45754	−0.728768	−	0.684761i	$-0.759907\pi$
$673$	−37.8118	−1.45754	−0.728768	+	0.684761i	$0.759907\pi$
$674$	0	0
$675$	20.6248	0.793847
$676$	0	0
$677$	31.0356	1.19279	0.596397	−	0.802690i	$-0.296598\pi$
$677$	31.0356	1.19279	0.596397	+	0.802690i	$0.296598\pi$
$678$	0	0
$679$	−35.6331	−1.36747
$680$	0	0
$681$	58.1044	2.22657
$682$	0	0
$683$	−6.68835	−0.255923	−0.127961	−	0.991779i	$-0.540843\pi$
$683$	−6.68835	−0.255923	−0.127961	+	0.991779i	$0.540843\pi$
$684$	0	0
$685$	−11.1258	−0.425097
$686$	0	0
$687$	59.3249	2.26339
$688$	0	0
$689$	67.6148	2.57592
$690$	0	0
$691$	27.6759	1.05284	0.526421	−	0.850224i	$-0.323534\pi$
$691$	27.6759	1.05284	0.526421	+	0.850224i	$0.323534\pi$
$692$	0	0
$693$	−71.9978	−2.73497
$694$	0	0
$695$	16.5011	0.625921
$696$	0	0
$697$	48.1819	1.82502
$698$	0	0
$699$	−43.1351	−1.63152
$700$	0	0
$701$	10.6234	0.401242	0.200621	−	0.979669i	$-0.435704\pi$
$701$	10.6234	0.401242	0.200621	+	0.979669i	$0.435704\pi$
$702$	0	0
$703$	6.07710	0.229202
$704$	0	0
$705$	21.7654	0.819733
$706$	0	0
$707$	8.70502	0.327386
$708$	0	0
$709$	−34.0768	−1.27978	−0.639891	−	0.768466i	$-0.721020\pi$
$709$	−34.0768	−1.27978	−0.639891	+	0.768466i	$0.721020\pi$
$710$	0	0
$711$	−9.17976	−0.344268
$712$	0	0
$713$	−10.9121	−0.408660
$714$	0	0
$715$	−10.2082	−0.381766
$716$	0	0
$717$	−2.07842	−0.0776199
$718$	0	0
$719$	25.8319	0.963367	0.481683	−	0.876345i	$-0.340026\pi$
$719$	25.8319	0.963367	0.481683	+	0.876345i	$0.340026\pi$
$720$	0	0
$721$	59.6693	2.22220
$722$	0	0
$723$	74.5857	2.77387
$724$	0	0
$725$	−2.90445	−0.107869
$726$	0	0
$727$	41.7046	1.54674	0.773369	−	0.633956i	$-0.218570\pi$
$727$	41.7046	1.54674	0.773369	+	0.633956i	$0.218570\pi$
$728$	0	0
$729$	184.372	6.82860
$730$	0	0
$731$	25.4151	0.940010
$732$	0	0
$733$	−13.3033	−0.491370	−0.245685	−	0.969350i	$-0.579013\pi$
$733$	−13.3033	−0.491370	−0.245685	+	0.969350i	$0.579013\pi$
$734$	0	0
$735$	36.3500	1.34079
$736$	0	0
$737$	15.7428	0.579895
$738$	0	0
$739$	37.7826	1.38985	0.694927	−	0.719080i	$-0.255437\pi$
$739$	37.7826	1.38985	0.694927	+	0.719080i	$0.255437\pi$
$740$	0	0
$741$	−18.3927	−0.675671
$742$	0	0
$743$	−9.99052	−0.366517	−0.183258	−	0.983065i	$-0.558665\pi$
$743$	−9.99052	−0.366517	−0.183258	+	0.983065i	$0.558665\pi$
$744$	0	0
$745$	−17.7917	−0.651839
$746$	0	0
$747$	−38.9226	−1.42410
$748$	0	0
$749$	33.5539	1.22603
$750$	0	0
$751$	44.6739	1.63017	0.815087	−	0.579338i	$-0.196689\pi$
$751$	44.6739	1.63017	0.815087	+	0.579338i	$0.196689\pi$
$752$	0	0
$753$	21.9516	0.799962
$754$	0	0
$755$	−0.913243	−0.0332363
$756$	0	0
$757$	−42.9608	−1.56144	−0.780718	−	0.624883i	$-0.785147\pi$
$757$	−42.9608	−1.56144	−0.780718	+	0.624883i	$0.785147\pi$
$758$	0	0
$759$	16.9817	0.616396
$760$	0	0
$761$	−45.0314	−1.63239	−0.816194	−	0.577778i	$-0.803920\pi$
$761$	−45.0314	−1.63239	−0.816194	+	0.577778i	$0.803920\pi$
$762$	0	0
$763$	34.4194	1.24607
$764$	0	0
$765$	−62.4293	−2.25714
$766$	0	0
$767$	−27.8809	−1.00672
$768$	0	0
$769$	41.6483	1.50188	0.750938	−	0.660372i	$-0.229602\pi$
$769$	41.6483	1.50188	0.750938	+	0.660372i	$0.229602\pi$
$770$	0	0
$771$	−83.6431	−3.01233
$772$	0	0
$773$	−42.6831	−1.53520	−0.767602	−	0.640927i	$-0.778550\pi$
$773$	−42.6831	−1.53520	−0.767602	+	0.640927i	$0.778550\pi$
$774$	0	0
$775$	4.26651	0.153258
$776$	0	0
$777$	−87.9539	−3.15533
$778$	0	0
$779$	6.91753	0.247846
$780$	0	0
$781$	31.3929	1.12332
$782$	0	0
$783$	−59.9036	−2.14078
$784$	0	0
$785$	2.28213	0.0814527
$786$	0	0
$787$	−16.4161	−0.585170	−0.292585	−	0.956240i	$-0.594515\pi$
$787$	−16.4161	−0.585170	−0.292585	+	0.956240i	$0.594515\pi$
$788$	0	0
$789$	16.6027	0.591070
$790$	0	0
$791$	68.3236	2.42931
$792$	0	0
$793$	17.9261	0.636574
$794$	0	0
$795$	43.9783	1.55975
$796$	0	0
$797$	−6.04841	−0.214246	−0.107123	−	0.994246i	$-0.534164\pi$
$797$	−6.04841	−0.214246	−0.107123	+	0.994246i	$0.534164\pi$
$798$	0	0
$799$	−43.8308	−1.55062
$800$	0	0
$801$	−13.0470	−0.460992
$802$	0	0
$803$	6.33746	0.223644
$804$	0	0
$805$	−10.7022	−0.377201
$806$	0	0
$807$	16.8805	0.594222
$808$	0	0
$809$	−40.1166	−1.41043	−0.705213	−	0.708996i	$-0.749149\pi$
$809$	−40.1166	−1.41043	−0.705213	+	0.708996i	$0.749149\pi$
$810$	0	0
$811$	−50.6027	−1.77690	−0.888450	−	0.458973i	$-0.848218\pi$
$811$	−50.6027	−1.77690	−0.888450	+	0.458973i	$0.848218\pi$
$812$	0	0
$813$	−5.54182	−0.194360
$814$	0	0
$815$	2.34685	0.0822064
$816$	0	0
$817$	3.64887	0.127658
$818$	0	0
$819$	199.442	6.96907
$820$	0	0
$821$	7.91047	0.276077	0.138039	−	0.990427i	$-0.455920\pi$
$821$	7.91047	0.276077	0.138039	+	0.990427i	$0.455920\pi$
$822$	0	0
$823$	−18.5164	−0.645442	−0.322721	−	0.946494i	$-0.604598\pi$
$823$	−18.5164	−0.645442	−0.322721	+	0.946494i	$0.604598\pi$
$824$	0	0
$825$	−6.63968	−0.231164
$826$	0	0
$827$	−16.9938	−0.590931	−0.295466	−	0.955353i	$-0.595475\pi$
$827$	−16.9938	−0.590931	−0.295466	+	0.955353i	$0.595475\pi$
$828$	0	0
$829$	−17.7915	−0.617926	−0.308963	−	0.951074i	$-0.599982\pi$
$829$	−17.7915	−0.617926	−0.308963	+	0.951074i	$0.599982\pi$
$830$	0	0
$831$	−59.8408	−2.07586
$832$	0	0
$833$	−73.2010	−2.53626
$834$	0	0
$835$	−14.3130	−0.495322
$836$	0	0
$837$	87.9958	3.04158
$838$	0	0
$839$	42.6250	1.47158	0.735789	−	0.677211i	$-0.236811\pi$
$839$	42.6250	1.47158	0.735789	+	0.677211i	$0.236811\pi$
$840$	0	0
$841$	−20.5642	−0.709109
$842$	0	0
$843$	−85.5837	−2.94766
$844$	0	0
$845$	15.2779	0.525576
$846$	0	0
$847$	−30.6086	−1.05173
$848$	0	0
$849$	13.0979	0.449518
$850$	0	0
$851$	15.5428	0.532802
$852$	0	0
$853$	9.19246	0.314744	0.157372	−	0.987539i	$-0.449698\pi$
$853$	9.19246	0.314744	0.157372	+	0.987539i	$0.449698\pi$
$854$	0	0
$855$	−8.96304	−0.306530
$856$	0	0
$857$	4.17610	0.142653	0.0713265	−	0.997453i	$-0.477277\pi$
$857$	4.17610	0.142653	0.0713265	+	0.997453i	$0.477277\pi$
$858$	0	0
$859$	26.0354	0.888317	0.444159	−	0.895948i	$-0.353503\pi$
$859$	26.0354	0.888317	0.444159	+	0.895948i	$0.353503\pi$
$860$	0	0
$861$	−100.117	−3.41199
$862$	0	0
$863$	34.0880	1.16037	0.580185	−	0.814485i	$-0.302980\pi$
$863$	34.0880	1.16037	0.580185	+	0.814485i	$0.302980\pi$
$864$	0	0
$865$	−11.4702	−0.389997
$866$	0	0
$867$	108.999	3.70180
$868$	0	0
$869$	1.96608	0.0666948
$870$	0	0
$871$	−43.6094	−1.47765
$872$	0	0
$873$	−76.3258	−2.58324
$874$	0	0
$875$	4.18444	0.141460
$876$	0	0
$877$	17.2442	0.582297	0.291148	−	0.956678i	$-0.405963\pi$
$877$	17.2442	0.582297	0.291148	+	0.956678i	$0.405963\pi$
$878$	0	0
$879$	49.0495	1.65440
$880$	0	0
$881$	1.50151	0.0505873	0.0252937	−	0.999680i	$-0.491948\pi$
$881$	1.50151	0.0505873	0.0252937	+	0.999680i	$0.491948\pi$
$882$	0	0
$883$	9.28230	0.312374	0.156187	−	0.987727i	$-0.450080\pi$
$883$	9.28230	0.312374	0.156187	+	0.987727i	$0.450080\pi$
$884$	0	0
$885$	−18.1344	−0.609582
$886$	0	0
$887$	−26.3714	−0.885467	−0.442733	−	0.896653i	$-0.645991\pi$
$887$	−26.3714	−0.885467	−0.442733	+	0.896653i	$0.645991\pi$
$888$	0	0
$889$	−41.3846	−1.38800
$890$	0	0
$891$	−85.3236	−2.85845
$892$	0	0
$893$	−6.29283	−0.210582
$894$	0	0
$895$	−0.494634	−0.0165338
$896$	0	0
$897$	−47.0412	−1.57066
$898$	0	0
$899$	−12.3919	−0.413293
$900$	0	0
$901$	−88.5626	−2.95045
$902$	0	0
$903$	−52.8101	−1.75741
$904$	0	0
$905$	−5.02418	−0.167009
$906$	0	0
$907$	15.6537	0.519772	0.259886	−	0.965639i	$-0.416315\pi$
$907$	15.6537	0.519772	0.259886	+	0.965639i	$0.416315\pi$
$908$	0	0
$909$	18.6461	0.618452
$910$	0	0
$911$	55.5024	1.83888	0.919438	−	0.393235i	$-0.128644\pi$
$911$	55.5024	1.83888	0.919438	+	0.393235i	$0.128644\pi$
$912$	0	0
$913$	8.33628	0.275891
$914$	0	0
$915$	11.6596	0.385454
$916$	0	0
$917$	−25.0413	−0.826937
$918$	0	0
$919$	−49.9719	−1.64842	−0.824211	−	0.566284i	$-0.808381\pi$
$919$	−49.9719	−1.64842	−0.824211	+	0.566284i	$0.808381\pi$
$920$	0	0
$921$	−44.7491	−1.47453
$922$	0	0
$923$	−86.9617	−2.86238
$924$	0	0
$925$	−6.07710	−0.199814
$926$	0	0
$927$	127.811	4.19787
$928$	0	0
$929$	4.21840	0.138401	0.0692006	−	0.997603i	$-0.477955\pi$
$929$	4.21840	0.138401	0.0692006	+	0.997603i	$0.477955\pi$
$930$	0	0
$931$	−10.5095	−0.344437
$932$	0	0
$933$	33.3036	1.09031
$934$	0	0
$935$	13.3709	0.437274
$936$	0	0
$937$	7.31403	0.238939	0.119469	−	0.992838i	$-0.461881\pi$
$937$	7.31403	0.238939	0.119469	+	0.992838i	$0.461881\pi$
$938$	0	0
$939$	−60.7407	−1.98220
$940$	0	0
$941$	47.3976	1.54512	0.772558	−	0.634944i	$-0.218977\pi$
$941$	47.3976	1.54512	0.772558	+	0.634944i	$0.218977\pi$
$942$	0	0
$943$	17.6923	0.576141
$944$	0	0
$945$	86.3031	2.80744
$946$	0	0
$947$	−34.0009	−1.10488	−0.552440	−	0.833553i	$-0.686303\pi$
$947$	−34.0009	−1.10488	−0.552440	+	0.833553i	$0.686303\pi$
$948$	0	0
$949$	−17.5555	−0.569875
$950$	0	0
$951$	27.1566	0.880612
$952$	0	0
$953$	−56.6039	−1.83358	−0.916790	−	0.399370i	$-0.869229\pi$
$953$	−56.6039	−1.83358	−0.916790	+	0.399370i	$0.869229\pi$
$954$	0	0
$955$	−4.01736	−0.129999
$956$	0	0
$957$	19.2846	0.623384
$958$	0	0
$959$	−46.5555	−1.50335
$960$	0	0
$961$	−12.7969	−0.412802
$962$	0	0
$963$	71.8723	2.31605
$964$	0	0
$965$	−15.3265	−0.493377
$966$	0	0
$967$	33.7007	1.08374	0.541871	−	0.840462i	$-0.317716\pi$
$967$	33.7007	1.08374	0.541871	+	0.840462i	$0.317716\pi$
$968$	0	0
$969$	24.0909	0.773912
$970$	0	0
$971$	14.4952	0.465173	0.232587	−	0.972576i	$-0.425281\pi$
$971$	14.4952	0.465173	0.232587	+	0.972576i	$0.425281\pi$
$972$	0	0
$973$	69.0477	2.21357
$974$	0	0
$975$	18.3927	0.589037
$976$	0	0
$977$	8.67759	0.277621	0.138810	−	0.990319i	$-0.455672\pi$
$977$	8.67759	0.277621	0.138810	+	0.990319i	$0.455672\pi$
$978$	0	0
$979$	2.79434	0.0893077
$980$	0	0
$981$	73.7262	2.35390
$982$	0	0
$983$	15.5754	0.496778	0.248389	−	0.968660i	$-0.420099\pi$
$983$	15.5754	0.496778	0.248389	+	0.968660i	$0.420099\pi$
$984$	0	0
$985$	2.20197	0.0701606
$986$	0	0
$987$	91.0762	2.89899
$988$	0	0
$989$	9.33238	0.296752
$990$	0	0
$991$	−35.5941	−1.13068	−0.565342	−	0.824857i	$-0.691256\pi$
$991$	−35.5941	−1.13068	−0.565342	+	0.824857i	$0.691256\pi$
$992$	0	0
$993$	77.2779	2.45234
$994$	0	0
$995$	−11.4972	−0.364485
$996$	0	0
$997$	−32.2203	−1.02043	−0.510213	−	0.860048i	$-0.670433\pi$
$997$	−32.2203	−1.02043	−0.510213	+	0.860048i	$0.670433\pi$
$998$	0	0
$999$	−125.339	−3.96555

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6080.2.a.cl.1.5		5
4.3	odd	2		6080.2.a.ci.1.1		5
8.3	odd	2		3040.2.a.x.1.5	yes	5
8.5	even	2		3040.2.a.u.1.1	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3040.2.a.u.1.1	✓	5	8.5	even	2
3040.2.a.x.1.5	yes	5	8.3	odd	2
6080.2.a.ci.1.1		5	4.3	odd	2
6080.2.a.cl.1.5		5	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:	\( N \)	\(=\)	\( 6080 = 2^{6} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6080.a (trivial)

\(f(q)\)	\(=\)	\(q+3.45876 q^{3} +1.00000 q^{5} +4.18444 q^{7} +8.96304 q^{9} -1.91967 q^{11} +5.31770 q^{13} +3.45876 q^{15} -6.96519 q^{17} -1.00000 q^{19} +14.4730 q^{21} -2.55761 q^{23} +1.00000 q^{25} +20.6248 q^{27} -2.90445 q^{29} +4.26651 q^{31} -6.63968 q^{33} +4.18444 q^{35} -6.07710 q^{37} +18.3927 q^{39} -6.91753 q^{41} -3.64887 q^{43} +8.96304 q^{45} +6.29283 q^{47} +10.5095 q^{49} -24.0909 q^{51} +12.7150 q^{53} -1.91967 q^{55} -3.45876 q^{57} -5.24303 q^{59} +3.37102 q^{61} +37.5053 q^{63} +5.31770 q^{65} -8.20081 q^{67} -8.84616 q^{69} -16.3533 q^{71} -3.30133 q^{73} +3.45876 q^{75} -8.03274 q^{77} -1.02418 q^{79} +44.4470 q^{81} -4.34256 q^{83} -6.96519 q^{85} -10.0458 q^{87} -1.45564 q^{89} +22.2516 q^{91} +14.7569 q^{93} -1.00000 q^{95} -8.51561 q^{97} -17.2061 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 4 q^{3} + 5 q^{5} + 4 q^{7} + 7 q^{9} + 2 q^{11} + 4 q^{13} + 4 q^{15} - 12 q^{17} - 5 q^{19} + 10 q^{21} + 8 q^{23} + 5 q^{25} + 16 q^{27} + 6 q^{29} + 10 q^{31} - 18 q^{33} + 4 q^{35} + 6 q^{37}+ \cdots - 14 q^{99}+O(q^{100}) \) 5 * q + 4 * q^3 + 5 * q^5 + 4 * q^7 + 7 * q^9 + 2 * q^11 + 4 * q^13 + 4 * q^15 - 12 * q^17 - 5 * q^19 + 10 * q^21 + 8 * q^23 + 5 * q^25 + 16 * q^27 + 6 * q^29 + 10 * q^31 - 18 * q^33 + 4 * q^35 + 6 * q^37 + 18 * q^39 - 8 * q^41 + 12 * q^43 + 7 * q^45 + 16 * q^47 + 7 * q^49 - 14 * q^51 + 18 * q^53 + 2 * q^55 - 4 * q^57 + 8 * q^59 - 2 * q^61 + 36 * q^63 + 4 * q^65 + 10 * q^67 + 22 * q^69 - 18 * q^71 - 28 * q^73 + 4 * q^75 + 28 * q^77 + 14 * q^79 + 25 * q^81 + 8 * q^83 - 12 * q^85 + 24 * q^87 - 30 * q^89 + 28 * q^91 + 24 * q^93 - 5 * q^95 - 18 * q^97 - 14 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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