LMFDB - Embedded newform 6020.2.a.f.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6020,2,Mod(1,6020)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6020, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6020.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6020 = 2^{2} \cdot 5 \cdot 7 \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6020.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:	$48.0699420168$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 12x^{6} + 26x^{5} + 55x^{4} - 52x^{3} - 82x^{2} + 22x + 27 $ x^8 - 3x^7 - 12x^6 + 26x^5 + 55x^4 - 52x^3 - 82x^2 + 22*x + 27
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.16692$ of defining polynomial
Character	$\chi$	$=$	6020.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.16692 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.69555 q^{9} +O(q^{10})$	$q-2.16692 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.69555 q^{9} +4.88853 q^{11} -0.835652 q^{13} +2.16692 q^{15} -1.74600 q^{17} -0.528632 q^{19} -2.16692 q^{21} -5.66214 q^{23} +1.00000 q^{25} +2.82663 q^{27} +5.57521 q^{29} -10.1489 q^{31} -10.5931 q^{33} -1.00000 q^{35} +3.13845 q^{37} +1.81079 q^{39} -9.30569 q^{41} -1.00000 q^{43} -1.69555 q^{45} +12.3512 q^{47} +1.00000 q^{49} +3.78345 q^{51} +7.15237 q^{53} -4.88853 q^{55} +1.14550 q^{57} -13.6307 q^{59} -5.07522 q^{61} +1.69555 q^{63} +0.835652 q^{65} +7.35963 q^{67} +12.2694 q^{69} -7.04426 q^{71} +8.08723 q^{73} -2.16692 q^{75} +4.88853 q^{77} -2.86464 q^{79} -11.2118 q^{81} -4.08703 q^{83} +1.74600 q^{85} -12.0811 q^{87} +2.90613 q^{89} -0.835652 q^{91} +21.9920 q^{93} +0.528632 q^{95} +9.57243 q^{97} +8.28877 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 5 q^{3} - 8 q^{5} + 8 q^{7} + 11 q^{9}+O(q^{10}) $ 8 * q - 5 * q^3 - 8 * q^5 + 8 * q^7 + 11 * q^9	$ 8 q - 5 q^{3} - 8 q^{5} + 8 q^{7} + 11 q^{9} + 6 q^{11} - 13 q^{13} + 5 q^{15} - 2 q^{17} - 14 q^{19} - 5 q^{21} + 6 q^{23} + 8 q^{25} - 11 q^{27} - 7 q^{29} - 18 q^{31} - q^{33} - 8 q^{35} + 2 q^{37} + 9 q^{39} - 18 q^{41} - 8 q^{43} - 11 q^{45} - q^{47} + 8 q^{49} - 19 q^{51} + 5 q^{53} - 6 q^{55} - 4 q^{57} - 12 q^{59} - 23 q^{61} + 11 q^{63} + 13 q^{65} + 8 q^{67} - 18 q^{69} + 20 q^{71} - 4 q^{73} - 5 q^{75} + 6 q^{77} + 24 q^{79} - 8 q^{81} - 14 q^{83} + 2 q^{85} + 10 q^{87} - 21 q^{89} - 13 q^{91} + q^{93} + 14 q^{95} + 7 q^{97} + 4 q^{99}+O(q^{100}) $ 8 * q - 5 * q^3 - 8 * q^5 + 8 * q^7 + 11 * q^9 + 6 * q^11 - 13 * q^13 + 5 * q^15 - 2 * q^17 - 14 * q^19 - 5 * q^21 + 6 * q^23 + 8 * q^25 - 11 * q^27 - 7 * q^29 - 18 * q^31 - q^33 - 8 * q^35 + 2 * q^37 + 9 * q^39 - 18 * q^41 - 8 * q^43 - 11 * q^45 - q^47 + 8 * q^49 - 19 * q^51 + 5 * q^53 - 6 * q^55 - 4 * q^57 - 12 * q^59 - 23 * q^61 + 11 * q^63 + 13 * q^65 + 8 * q^67 - 18 * q^69 + 20 * q^71 - 4 * q^73 - 5 * q^75 + 6 * q^77 + 24 * q^79 - 8 * q^81 - 14 * q^83 + 2 * q^85 + 10 * q^87 - 21 * q^89 - 13 * q^91 + q^93 + 14 * q^95 + 7 * q^97 + 4 * 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
$3$	−2.16692	−1.25107	−0.625537	−	0.780195i	$-0.715120\pi$
$3$	−2.16692	−1.25107	−0.625537	+	0.780195i	$0.715120\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.69555	0.565185
$10$	0	0
$11$	4.88853	1.47395	0.736973	−	0.675922i	$-0.236254\pi$
$11$	4.88853	1.47395	0.736973	+	0.675922i	$0.236254\pi$
$12$	0	0
$13$	−0.835652	−0.231768	−0.115884	−	0.993263i	$-0.536970\pi$
$13$	−0.835652	−0.231768	−0.115884	+	0.993263i	$0.536970\pi$
$14$	0	0
$15$	2.16692	0.559497
$16$	0	0
$17$	−1.74600	−0.423468	−0.211734	−	0.977327i	$-0.567911\pi$
$17$	−1.74600	−0.423468	−0.211734	+	0.977327i	$0.567911\pi$
$18$	0	0
$19$	−0.528632	−0.121276	−0.0606382	−	0.998160i	$-0.519314\pi$
$19$	−0.528632	−0.121276	−0.0606382	+	0.998160i	$0.519314\pi$
$20$	0	0
$21$	−2.16692	−0.472861
$22$	0	0
$23$	−5.66214	−1.18064	−0.590319	−	0.807170i	$-0.700998\pi$
$23$	−5.66214	−1.18064	−0.590319	+	0.807170i	$0.700998\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	2.82663	0.543986
$28$	0	0
$29$	5.57521	1.03529	0.517645	−	0.855595i	$-0.326809\pi$
$29$	5.57521	1.03529	0.517645	+	0.855595i	$0.326809\pi$
$30$	0	0
$31$	−10.1489	−1.82280	−0.911401	−	0.411519i	$-0.864998\pi$
$31$	−10.1489	−1.82280	−0.911401	+	0.411519i	$0.864998\pi$
$32$	0	0
$33$	−10.5931	−1.84402
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	3.13845	0.515958	0.257979	−	0.966150i	$-0.416943\pi$
$37$	3.13845	0.515958	0.257979	+	0.966150i	$0.416943\pi$
$38$	0	0
$39$	1.81079	0.289959
$40$	0	0
$41$	−9.30569	−1.45330	−0.726652	−	0.687005i	$-0.758925\pi$
$41$	−9.30569	−1.45330	−0.726652	+	0.687005i	$0.758925\pi$
$42$	0	0
$43$	−1.00000	−0.152499
$43$	−1.00000	−0.152499
$44$	0	0
$45$	−1.69555	−0.252758
$46$	0	0
$47$	12.3512	1.80161	0.900807	−	0.434219i	$-0.142976\pi$
$47$	12.3512	1.80161	0.900807	+	0.434219i	$0.142976\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	3.78345	0.529790
$52$	0	0
$53$	7.15237	0.982454	0.491227	−	0.871032i	$-0.336549\pi$
$53$	7.15237	0.982454	0.491227	+	0.871032i	$0.336549\pi$
$54$	0	0
$55$	−4.88853	−0.659169
$56$	0	0
$57$	1.14550	0.151726
$58$	0	0
$59$	−13.6307	−1.77456	−0.887281	−	0.461230i	$-0.847408\pi$
$59$	−13.6307	−1.77456	−0.887281	+	0.461230i	$0.847408\pi$
$60$	0	0
$61$	−5.07522	−0.649815	−0.324907	−	0.945746i	$-0.605333\pi$
$61$	−5.07522	−0.649815	−0.324907	+	0.945746i	$0.605333\pi$
$62$	0	0
$63$	1.69555	0.213620
$64$	0	0
$65$	0.835652	0.103650
$66$	0	0
$67$	7.35963	0.899122	0.449561	−	0.893250i	$-0.351580\pi$
$67$	7.35963	0.899122	0.449561	+	0.893250i	$0.351580\pi$
$68$	0	0
$69$	12.2694	1.47706
$70$	0	0
$71$	−7.04426	−0.835999	−0.418000	−	0.908447i	$-0.637269\pi$
$71$	−7.04426	−0.835999	−0.418000	+	0.908447i	$0.637269\pi$
$72$	0	0
$73$	8.08723	0.946539	0.473270	−	0.880918i	$-0.343074\pi$
$73$	8.08723	0.946539	0.473270	+	0.880918i	$0.343074\pi$
$74$	0	0
$75$	−2.16692	−0.250215
$76$	0	0
$77$	4.88853	0.557100
$78$	0	0
$79$	−2.86464	−0.322297	−0.161149	−	0.986930i	$-0.551520\pi$
$79$	−2.86464	−0.322297	−0.161149	+	0.986930i	$0.551520\pi$
$80$	0	0
$81$	−11.2118	−1.24575
$82$	0	0
$83$	−4.08703	−0.448610	−0.224305	−	0.974519i	$-0.572011\pi$
$83$	−4.08703	−0.448610	−0.224305	+	0.974519i	$0.572011\pi$
$84$	0	0
$85$	1.74600	0.189381
$86$	0	0
$87$	−12.0811	−1.29522
$88$	0	0
$89$	2.90613	0.308049	0.154025	−	0.988067i	$-0.450776\pi$
$89$	2.90613	0.308049	0.154025	+	0.988067i	$0.450776\pi$
$90$	0	0
$91$	−0.835652	−0.0876001
$92$	0	0
$93$	21.9920	2.28046
$94$	0	0
$95$	0.528632	0.0542365
$96$	0	0
$97$	9.57243	0.971933	0.485966	−	0.873978i	$-0.338468\pi$
$97$	9.57243	0.971933	0.485966	+	0.873978i	$0.338468\pi$
$98$	0	0
$99$	8.28877	0.833052
$100$	0	0
$101$	−8.08584	−0.804571	−0.402286	−	0.915514i	$-0.631784\pi$
$101$	−8.08584	−0.804571	−0.402286	+	0.915514i	$0.631784\pi$
$102$	0	0
$103$	6.79146	0.669183	0.334591	−	0.942363i	$-0.391402\pi$
$103$	6.79146	0.669183	0.334591	+	0.942363i	$0.391402\pi$
$104$	0	0
$105$	2.16692	0.211470
$106$	0	0
$107$	7.56565	0.731399	0.365700	−	0.930733i	$-0.380830\pi$
$107$	7.56565	0.731399	0.365700	+	0.930733i	$0.380830\pi$
$108$	0	0
$109$	−2.66301	−0.255070	−0.127535	−	0.991834i	$-0.540707\pi$
$109$	−2.66301	−0.255070	−0.127535	+	0.991834i	$0.540707\pi$
$110$	0	0
$111$	−6.80078	−0.645502
$112$	0	0
$113$	7.84552	0.738044	0.369022	−	0.929421i	$-0.379693\pi$
$113$	7.84552	0.738044	0.369022	+	0.929421i	$0.379693\pi$
$114$	0	0
$115$	5.66214	0.527997
$116$	0	0
$117$	−1.41689	−0.130992
$118$	0	0
$119$	−1.74600	−0.160056
$120$	0	0
$121$	12.8977	1.17252
$122$	0	0
$123$	20.1647	1.81819
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	11.6403	1.03291	0.516455	−	0.856314i	$-0.327251\pi$
$127$	11.6403	1.03291	0.516455	+	0.856314i	$0.327251\pi$
$128$	0	0
$129$	2.16692	0.190787
$130$	0	0
$131$	14.5283	1.26935	0.634673	−	0.772780i	$-0.281135\pi$
$131$	14.5283	1.26935	0.634673	+	0.772780i	$0.281135\pi$
$132$	0	0
$133$	−0.528632	−0.0458382
$134$	0	0
$135$	−2.82663	−0.243278
$136$	0	0
$137$	−11.7168	−1.00103	−0.500517	−	0.865727i	$-0.666857\pi$
$137$	−11.7168	−1.00103	−0.500517	+	0.865727i	$0.666857\pi$
$138$	0	0
$139$	5.19070	0.440269	0.220135	−	0.975470i	$-0.429350\pi$
$139$	5.19070	0.440269	0.220135	+	0.975470i	$0.429350\pi$
$140$	0	0
$141$	−26.7642	−2.25395
$142$	0	0
$143$	−4.08511	−0.341614
$144$	0	0
$145$	−5.57521	−0.462996
$146$	0	0
$147$	−2.16692	−0.178725
$148$	0	0
$149$	−13.4644	−1.10304	−0.551521	−	0.834161i	$-0.685952\pi$
$149$	−13.4644	−1.10304	−0.551521	+	0.834161i	$0.685952\pi$
$150$	0	0
$151$	17.6645	1.43751	0.718757	−	0.695261i	$-0.244711\pi$
$151$	17.6645	1.43751	0.718757	+	0.695261i	$0.244711\pi$
$152$	0	0
$153$	−2.96044	−0.239338
$154$	0	0
$155$	10.1489	0.815182
$156$	0	0
$157$	−22.7585	−1.81633	−0.908163	−	0.418618i	$-0.862515\pi$
$157$	−22.7585	−1.81633	−0.908163	+	0.418618i	$0.862515\pi$
$158$	0	0
$159$	−15.4986	−1.22912
$160$	0	0
$161$	−5.66214	−0.446239
$162$	0	0
$163$	−12.6057	−0.987353	−0.493676	−	0.869646i	$-0.664347\pi$
$163$	−12.6057	−0.987353	−0.493676	+	0.869646i	$0.664347\pi$
$164$	0	0
$165$	10.5931	0.824669
$166$	0	0
$167$	−1.88431	−0.145812	−0.0729060	−	0.997339i	$-0.523227\pi$
$167$	−1.88431	−0.145812	−0.0729060	+	0.997339i	$0.523227\pi$
$168$	0	0
$169$	−12.3017	−0.946284
$170$	0	0
$171$	−0.896323	−0.0685436
$172$	0	0
$173$	17.6714	1.34353	0.671767	−	0.740762i	$-0.265536\pi$
$173$	17.6714	1.34353	0.671767	+	0.740762i	$0.265536\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	29.5366	2.22011
$178$	0	0
$179$	−4.60914	−0.344503	−0.172252	−	0.985053i	$-0.555104\pi$
$179$	−4.60914	−0.344503	−0.172252	+	0.985053i	$0.555104\pi$
$180$	0	0
$181$	−18.8383	−1.40024	−0.700121	−	0.714024i	$-0.746871\pi$
$181$	−18.8383	−1.40024	−0.700121	+	0.714024i	$0.746871\pi$
$182$	0	0
$183$	10.9976	0.812966
$184$	0	0
$185$	−3.13845	−0.230744
$186$	0	0
$187$	−8.53539	−0.624169
$188$	0	0
$189$	2.82663	0.205607
$190$	0	0
$191$	−11.7805	−0.852407	−0.426204	−	0.904627i	$-0.640149\pi$
$191$	−11.7805	−0.852407	−0.426204	+	0.904627i	$0.640149\pi$
$192$	0	0
$193$	−9.13052	−0.657229	−0.328615	−	0.944464i	$-0.606582\pi$
$193$	−9.13052	−0.657229	−0.328615	+	0.944464i	$0.606582\pi$
$194$	0	0
$195$	−1.81079	−0.129674
$196$	0	0
$197$	−5.51316	−0.392797	−0.196398	−	0.980524i	$-0.562925\pi$
$197$	−5.51316	−0.392797	−0.196398	+	0.980524i	$0.562925\pi$
$198$	0	0
$199$	9.29102	0.658623	0.329312	−	0.944221i	$-0.393183\pi$
$199$	9.29102	0.658623	0.329312	+	0.944221i	$0.393183\pi$
$200$	0	0
$201$	−15.9478	−1.12487
$202$	0	0
$203$	5.57521	0.391303
$204$	0	0
$205$	9.30569	0.649938
$206$	0	0
$207$	−9.60046	−0.667278
$208$	0	0
$209$	−2.58423	−0.178755
$210$	0	0
$211$	14.5229	0.999795	0.499897	−	0.866085i	$-0.333371\pi$
$211$	14.5229	0.999795	0.499897	+	0.866085i	$0.333371\pi$
$212$	0	0
$213$	15.2644	1.04590
$214$	0	0
$215$	1.00000	0.0681994
$216$	0	0
$217$	−10.1489	−0.688954
$218$	0	0
$219$	−17.5244	−1.18419
$220$	0	0
$221$	1.45905	0.0981464
$222$	0	0
$223$	12.5896	0.843062	0.421531	−	0.906814i	$-0.361493\pi$
$223$	12.5896	0.843062	0.421531	+	0.906814i	$0.361493\pi$
$224$	0	0
$225$	1.69555	0.113037
$226$	0	0
$227$	−24.8839	−1.65160	−0.825800	−	0.563963i	$-0.809276\pi$
$227$	−24.8839	−1.65160	−0.825800	+	0.563963i	$0.809276\pi$
$228$	0	0
$229$	−10.7860	−0.712758	−0.356379	−	0.934341i	$-0.615989\pi$
$229$	−10.7860	−0.712758	−0.356379	+	0.934341i	$0.615989\pi$
$230$	0	0
$231$	−10.5931	−0.696972
$232$	0	0
$233$	−25.8045	−1.69051	−0.845253	−	0.534366i	$-0.820550\pi$
$233$	−25.8045	−1.69051	−0.845253	+	0.534366i	$0.820550\pi$
$234$	0	0
$235$	−12.3512	−0.805707
$236$	0	0
$237$	6.20745	0.403217
$238$	0	0
$239$	23.4853	1.51914	0.759570	−	0.650425i	$-0.225409\pi$
$239$	23.4853	1.51914	0.759570	+	0.650425i	$0.225409\pi$
$240$	0	0
$241$	−14.5251	−0.935647	−0.467823	−	0.883822i	$-0.654962\pi$
$241$	−14.5251	−0.935647	−0.467823	+	0.883822i	$0.654962\pi$
$242$	0	0
$243$	15.8151	1.01454
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	0.441752	0.0281080
$248$	0	0
$249$	8.85628	0.561244
$250$	0	0
$251$	1.69897	0.107238	0.0536189	−	0.998561i	$-0.482924\pi$
$251$	1.69897	0.107238	0.0536189	+	0.998561i	$0.482924\pi$
$252$	0	0
$253$	−27.6795	−1.74020
$254$	0	0
$255$	−3.78345	−0.236929
$256$	0	0
$257$	−10.3112	−0.643198	−0.321599	−	0.946876i	$-0.604220\pi$
$257$	−10.3112	−0.643198	−0.321599	+	0.946876i	$0.604220\pi$
$258$	0	0
$259$	3.13845	0.195014
$260$	0	0
$261$	9.45307	0.585131
$262$	0	0
$263$	−27.9130	−1.72119	−0.860596	−	0.509289i	$-0.829909\pi$
$263$	−27.9130	−1.72119	−0.860596	+	0.509289i	$0.829909\pi$
$264$	0	0
$265$	−7.15237	−0.439367
$266$	0	0
$267$	−6.29736	−0.385392
$268$	0	0
$269$	11.7594	0.716985	0.358493	−	0.933533i	$-0.383291\pi$
$269$	11.7594	0.716985	0.358493	+	0.933533i	$0.383291\pi$
$270$	0	0
$271$	−25.9903	−1.57880	−0.789398	−	0.613882i	$-0.789607\pi$
$271$	−25.9903	−1.57880	−0.789398	+	0.613882i	$0.789607\pi$
$272$	0	0
$273$	1.81079	0.109594
$274$	0	0
$275$	4.88853	0.294789
$276$	0	0
$277$	−13.3042	−0.799369	−0.399684	−	0.916653i	$-0.630880\pi$
$277$	−13.3042	−0.799369	−0.399684	+	0.916653i	$0.630880\pi$
$278$	0	0
$279$	−17.2081	−1.03022
$280$	0	0
$281$	29.1750	1.74043	0.870217	−	0.492669i	$-0.163979\pi$
$281$	29.1750	1.74043	0.870217	+	0.492669i	$0.163979\pi$
$282$	0	0
$283$	−19.9366	−1.18511	−0.592554	−	0.805531i	$-0.701880\pi$
$283$	−19.9366	−1.18511	−0.592554	+	0.805531i	$0.701880\pi$
$284$	0	0
$285$	−1.14550	−0.0678538
$286$	0	0
$287$	−9.30569	−0.549297
$288$	0	0
$289$	−13.9515	−0.820675
$290$	0	0
$291$	−20.7427	−1.21596
$292$	0	0
$293$	17.1922	1.00438	0.502189	−	0.864758i	$-0.332528\pi$
$293$	17.1922	1.00438	0.502189	+	0.864758i	$0.332528\pi$
$294$	0	0
$295$	13.6307	0.793608
$296$	0	0
$297$	13.8181	0.801806
$298$	0	0
$299$	4.73158	0.273634
$300$	0	0
$301$	−1.00000	−0.0576390
$302$	0	0
$303$	17.5214	1.00658
$304$	0	0
$305$	5.07522	0.290606
$306$	0	0
$307$	−6.65105	−0.379596	−0.189798	−	0.981823i	$-0.560783\pi$
$307$	−6.65105	−0.379596	−0.189798	+	0.981823i	$0.560783\pi$
$308$	0	0
$309$	−14.7166	−0.837197
$310$	0	0
$311$	−24.7389	−1.40281	−0.701406	−	0.712762i	$-0.747444\pi$
$311$	−24.7389	−1.40281	−0.701406	+	0.712762i	$0.747444\pi$
$312$	0	0
$313$	17.1245	0.967932	0.483966	−	0.875087i	$-0.339196\pi$
$313$	17.1245	0.967932	0.483966	+	0.875087i	$0.339196\pi$
$314$	0	0
$315$	−1.69555	−0.0955337
$316$	0	0
$317$	−27.3890	−1.53832	−0.769160	−	0.639056i	$-0.779325\pi$
$317$	−27.3890	−1.53832	−0.769160	+	0.639056i	$0.779325\pi$
$318$	0	0
$319$	27.2546	1.52596
$320$	0	0
$321$	−16.3942	−0.915034
$322$	0	0
$323$	0.922993	0.0513567
$324$	0	0
$325$	−0.835652	−0.0463536
$326$	0	0
$327$	5.77054	0.319112
$328$	0	0
$329$	12.3512	0.680946
$330$	0	0
$331$	−31.4452	−1.72838	−0.864191	−	0.503164i	$-0.832169\pi$
$331$	−31.4452	−1.72838	−0.864191	+	0.503164i	$0.832169\pi$
$332$	0	0
$333$	5.32142	0.291612
$334$	0	0
$335$	−7.35963	−0.402100
$336$	0	0
$337$	−26.1498	−1.42447	−0.712235	−	0.701941i	$-0.752317\pi$
$337$	−26.1498	−1.42447	−0.712235	+	0.701941i	$0.752317\pi$
$338$	0	0
$339$	−17.0006	−0.923347
$340$	0	0
$341$	−49.6133	−2.68671
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−12.2694	−0.660563
$346$	0	0
$347$	16.2419	0.871913	0.435956	−	0.899968i	$-0.356410\pi$
$347$	16.2419	0.871913	0.435956	+	0.899968i	$0.356410\pi$
$348$	0	0
$349$	−17.7410	−0.949655	−0.474828	−	0.880079i	$-0.657490\pi$
$349$	−17.7410	−0.949655	−0.474828	+	0.880079i	$0.657490\pi$
$350$	0	0
$351$	−2.36208	−0.126079
$352$	0	0
$353$	−2.35511	−0.125350	−0.0626748	−	0.998034i	$-0.519963\pi$
$353$	−2.35511	−0.125350	−0.0626748	+	0.998034i	$0.519963\pi$
$354$	0	0
$355$	7.04426	0.373870
$356$	0	0
$357$	3.78345	0.200242
$358$	0	0
$359$	27.1484	1.43284	0.716419	−	0.697670i	$-0.245780\pi$
$359$	27.1484	1.43284	0.716419	+	0.697670i	$0.245780\pi$
$360$	0	0
$361$	−18.7205	−0.985292
$362$	0	0
$363$	−27.9483	−1.46691
$364$	0	0
$365$	−8.08723	−0.423305
$366$	0	0
$367$	24.8618	1.29777	0.648887	−	0.760885i	$-0.275235\pi$
$367$	24.8618	1.29777	0.648887	+	0.760885i	$0.275235\pi$
$368$	0	0
$369$	−15.7783	−0.821386
$370$	0	0
$371$	7.15237	0.371333
$372$	0	0
$373$	−1.93214	−0.100043	−0.0500213	−	0.998748i	$-0.515929\pi$
$373$	−1.93214	−0.100043	−0.0500213	+	0.998748i	$0.515929\pi$
$374$	0	0
$375$	2.16692	0.111899
$376$	0	0
$377$	−4.65894	−0.239947
$378$	0	0
$379$	−26.3910	−1.35562	−0.677808	−	0.735239i	$-0.737070\pi$
$379$	−26.3910	−1.35562	−0.677808	+	0.735239i	$0.737070\pi$
$380$	0	0
$381$	−25.2236	−1.29225
$382$	0	0
$383$	−8.21560	−0.419797	−0.209899	−	0.977723i	$-0.567313\pi$
$383$	−8.21560	−0.419797	−0.209899	+	0.977723i	$0.567313\pi$
$384$	0	0
$385$	−4.88853	−0.249142
$386$	0	0
$387$	−1.69555	−0.0861899
$388$	0	0
$389$	18.5811	0.942098	0.471049	−	0.882107i	$-0.343876\pi$
$389$	18.5811	0.942098	0.471049	+	0.882107i	$0.343876\pi$
$390$	0	0
$391$	9.88611	0.499962
$392$	0	0
$393$	−31.4818	−1.58805
$394$	0	0
$395$	2.86464	0.144136
$396$	0	0
$397$	−5.67323	−0.284731	−0.142366	−	0.989814i	$-0.545471\pi$
$397$	−5.67323	−0.284731	−0.142366	+	0.989814i	$0.545471\pi$
$398$	0	0
$399$	1.14550	0.0573469
$400$	0	0
$401$	−28.0202	−1.39926	−0.699632	−	0.714503i	$-0.746653\pi$
$401$	−28.0202	−1.39926	−0.699632	+	0.714503i	$0.746653\pi$
$402$	0	0
$403$	8.48098	0.422467
$404$	0	0
$405$	11.2118	0.557117
$406$	0	0
$407$	15.3424	0.760495
$408$	0	0
$409$	−22.8276	−1.12875	−0.564376	−	0.825518i	$-0.690883\pi$
$409$	−22.8276	−1.12875	−0.564376	+	0.825518i	$0.690883\pi$
$410$	0	0
$411$	25.3894	1.25237
$412$	0	0
$413$	−13.6307	−0.670721
$414$	0	0
$415$	4.08703	0.200624
$416$	0	0
$417$	−11.2478	−0.550809
$418$	0	0
$419$	−3.27211	−0.159853	−0.0799264	−	0.996801i	$-0.525469\pi$
$419$	−3.27211	−0.159853	−0.0799264	+	0.996801i	$0.525469\pi$
$420$	0	0
$421$	14.0965	0.687023	0.343511	−	0.939148i	$-0.388384\pi$
$421$	14.0965	0.687023	0.343511	+	0.939148i	$0.388384\pi$
$422$	0	0
$423$	20.9422	1.01825
$424$	0	0
$425$	−1.74600	−0.0846936
$426$	0	0
$427$	−5.07522	−0.245607
$428$	0	0
$429$	8.85212	0.427384
$430$	0	0
$431$	36.8238	1.77374	0.886870	−	0.462019i	$-0.152875\pi$
$431$	36.8238	1.77374	0.886870	+	0.462019i	$0.152875\pi$
$432$	0	0
$433$	−25.6781	−1.23401	−0.617005	−	0.786959i	$-0.711654\pi$
$433$	−25.6781	−1.23401	−0.617005	+	0.786959i	$0.711654\pi$
$434$	0	0
$435$	12.0811	0.579242
$436$	0	0
$437$	2.99318	0.143183
$438$	0	0
$439$	1.00908	0.0481607	0.0240803	−	0.999710i	$-0.492334\pi$
$439$	1.00908	0.0481607	0.0240803	+	0.999710i	$0.492334\pi$
$440$	0	0
$441$	1.69555	0.0807407
$442$	0	0
$443$	−27.7021	−1.31617	−0.658084	−	0.752944i	$-0.728633\pi$
$443$	−27.7021	−1.31617	−0.658084	+	0.752944i	$0.728633\pi$
$444$	0	0
$445$	−2.90613	−0.137764
$446$	0	0
$447$	29.1762	1.37999
$448$	0	0
$449$	9.69430	0.457502	0.228751	−	0.973485i	$-0.426536\pi$
$449$	9.69430	0.457502	0.228751	+	0.973485i	$0.426536\pi$
$450$	0	0
$451$	−45.4911	−2.14209
$452$	0	0
$453$	−38.2775	−1.79844
$454$	0	0
$455$	0.835652	0.0391760
$456$	0	0
$457$	−15.2623	−0.713941	−0.356971	−	0.934116i	$-0.616190\pi$
$457$	−15.2623	−0.713941	−0.356971	+	0.934116i	$0.616190\pi$
$458$	0	0
$459$	−4.93531	−0.230361
$460$	0	0
$461$	−18.0408	−0.840244	−0.420122	−	0.907468i	$-0.638013\pi$
$461$	−18.0408	−0.840244	−0.420122	+	0.907468i	$0.638013\pi$
$462$	0	0
$463$	26.8086	1.24590	0.622950	−	0.782262i	$-0.285934\pi$
$463$	26.8086	1.24590	0.622950	+	0.782262i	$0.285934\pi$
$464$	0	0
$465$	−21.9920	−1.01985
$466$	0	0
$467$	1.37855	0.0637916	0.0318958	−	0.999491i	$-0.489846\pi$
$467$	1.37855	0.0637916	0.0318958	+	0.999491i	$0.489846\pi$
$468$	0	0
$469$	7.35963	0.339836
$470$	0	0
$471$	49.3159	2.27236
$472$	0	0
$473$	−4.88853	−0.224775
$474$	0	0
$475$	−0.528632	−0.0242553
$476$	0	0
$477$	12.1272	0.555268
$478$	0	0
$479$	−8.35855	−0.381912	−0.190956	−	0.981599i	$-0.561159\pi$
$479$	−8.35855	−0.381912	−0.190956	+	0.981599i	$0.561159\pi$
$480$	0	0
$481$	−2.62265	−0.119583
$482$	0	0
$483$	12.2694	0.558278
$484$	0	0
$485$	−9.57243	−0.434661
$486$	0	0
$487$	7.28119	0.329942	0.164971	−	0.986298i	$-0.447247\pi$
$487$	7.28119	0.329942	0.164971	+	0.986298i	$0.447247\pi$
$488$	0	0
$489$	27.3155	1.23525
$490$	0	0
$491$	1.87773	0.0847410	0.0423705	−	0.999102i	$-0.486509\pi$
$491$	1.87773	0.0847410	0.0423705	+	0.999102i	$0.486509\pi$
$492$	0	0
$493$	−9.73434	−0.438413
$494$	0	0
$495$	−8.28877	−0.372552
$496$	0	0
$497$	−7.04426	−0.315978
$498$	0	0
$499$	21.6961	0.971253	0.485626	−	0.874166i	$-0.338592\pi$
$499$	21.6961	0.971253	0.485626	+	0.874166i	$0.338592\pi$
$500$	0	0
$501$	4.08315	0.182422
$502$	0	0
$503$	3.18111	0.141839	0.0709193	−	0.997482i	$-0.477407\pi$
$503$	3.18111	0.141839	0.0709193	+	0.997482i	$0.477407\pi$
$504$	0	0
$505$	8.08584	0.359815
$506$	0	0
$507$	26.6568	1.18387
$508$	0	0
$509$	−3.30080	−0.146305	−0.0731526	−	0.997321i	$-0.523306\pi$
$509$	−3.30080	−0.146305	−0.0731526	+	0.997321i	$0.523306\pi$
$510$	0	0
$511$	8.08723	0.357758
$512$	0	0
$513$	−1.49425	−0.0659726
$514$	0	0
$515$	−6.79146	−0.299268
$516$	0	0
$517$	60.3794	2.65548
$518$	0	0
$519$	−38.2926	−1.68086
$520$	0	0
$521$	9.04412	0.396230	0.198115	−	0.980179i	$-0.436518\pi$
$521$	9.04412	0.396230	0.198115	+	0.980179i	$0.436518\pi$
$522$	0	0
$523$	−27.9215	−1.22092	−0.610461	−	0.792047i	$-0.709016\pi$
$523$	−27.9215	−1.22092	−0.610461	+	0.792047i	$0.709016\pi$
$524$	0	0
$525$	−2.16692	−0.0945723
$526$	0	0
$527$	17.7201	0.771898
$528$	0	0
$529$	9.05980	0.393905
$530$	0	0
$531$	−23.1115	−1.00295
$532$	0	0
$533$	7.77632	0.336830
$534$	0	0
$535$	−7.56565	−0.327092
$536$	0	0
$537$	9.98765	0.430999
$538$	0	0
$539$	4.88853	0.210564
$540$	0	0
$541$	−21.2753	−0.914694	−0.457347	−	0.889288i	$-0.651200\pi$
$541$	−21.2753	−0.914694	−0.457347	+	0.889288i	$0.651200\pi$
$542$	0	0
$543$	40.8212	1.75181
$544$	0	0
$545$	2.66301	0.114071
$546$	0	0
$547$	22.3758	0.956720	0.478360	−	0.878164i	$-0.341231\pi$
$547$	22.3758	0.956720	0.478360	+	0.878164i	$0.341231\pi$
$548$	0	0
$549$	−8.60531	−0.367265
$550$	0	0
$551$	−2.94723	−0.125556
$552$	0	0
$553$	−2.86464	−0.121817
$554$	0	0
$555$	6.80078	0.288677
$556$	0	0
$557$	−15.0948	−0.639587	−0.319794	−	0.947487i	$-0.603614\pi$
$557$	−15.0948	−0.639587	−0.319794	+	0.947487i	$0.603614\pi$
$558$	0	0
$559$	0.835652	0.0353443
$560$	0	0
$561$	18.4955	0.780882
$562$	0	0
$563$	26.2244	1.10523	0.552613	−	0.833438i	$-0.313631\pi$
$563$	26.2244	1.10523	0.552613	+	0.833438i	$0.313631\pi$
$564$	0	0
$565$	−7.84552	−0.330063
$566$	0	0
$567$	−11.2118	−0.470850
$568$	0	0
$569$	21.3194	0.893757	0.446879	−	0.894595i	$-0.352536\pi$
$569$	21.3194	0.893757	0.446879	+	0.894595i	$0.352536\pi$
$570$	0	0
$571$	−34.6305	−1.44924	−0.724621	−	0.689147i	$-0.757985\pi$
$571$	−34.6305	−1.44924	−0.724621	+	0.689147i	$0.757985\pi$
$572$	0	0
$573$	25.5274	1.06642
$574$	0	0
$575$	−5.66214	−0.236127
$576$	0	0
$577$	7.87990	0.328045	0.164022	−	0.986457i	$-0.447553\pi$
$577$	7.87990	0.328045	0.164022	+	0.986457i	$0.447553\pi$
$578$	0	0
$579$	19.7851	0.822242
$580$	0	0
$581$	−4.08703	−0.169559
$582$	0	0
$583$	34.9646	1.44808
$584$	0	0
$585$	1.41689	0.0585813
$586$	0	0
$587$	0.618451	0.0255262	0.0127631	−	0.999919i	$-0.495937\pi$
$587$	0.618451	0.0255262	0.0127631	+	0.999919i	$0.495937\pi$
$588$	0	0
$589$	5.36505	0.221063
$590$	0	0
$591$	11.9466	0.491418
$592$	0	0
$593$	−32.0424	−1.31583	−0.657913	−	0.753094i	$-0.728560\pi$
$593$	−32.0424	−1.31583	−0.657913	+	0.753094i	$0.728560\pi$
$594$	0	0
$595$	1.74600	0.0715792
$596$	0	0
$597$	−20.1329	−0.823986
$598$	0	0
$599$	28.5593	1.16690	0.583451	−	0.812148i	$-0.301702\pi$
$599$	28.5593	1.16690	0.583451	+	0.812148i	$0.301702\pi$
$600$	0	0
$601$	−41.6854	−1.70038	−0.850192	−	0.526474i	$-0.823514\pi$
$601$	−41.6854	−1.70038	−0.850192	+	0.526474i	$0.823514\pi$
$602$	0	0
$603$	12.4787	0.508170
$604$	0	0
$605$	−12.8977	−0.524367
$606$	0	0
$607$	17.1790	0.697273	0.348636	−	0.937258i	$-0.386645\pi$
$607$	17.1790	0.697273	0.348636	+	0.937258i	$0.386645\pi$
$608$	0	0
$609$	−12.0811	−0.489549
$610$	0	0
$611$	−10.3213	−0.417557
$612$	0	0
$613$	−39.7920	−1.60718	−0.803592	−	0.595180i	$-0.797081\pi$
$613$	−39.7920	−1.60718	−0.803592	+	0.595180i	$0.797081\pi$
$614$	0	0
$615$	−20.1647	−0.813120
$616$	0	0
$617$	45.2971	1.82359	0.911796	−	0.410643i	$-0.134696\pi$
$617$	45.2971	1.82359	0.911796	+	0.410643i	$0.134696\pi$
$618$	0	0
$619$	−2.53738	−0.101986	−0.0509930	−	0.998699i	$-0.516239\pi$
$619$	−2.53738	−0.101986	−0.0509930	+	0.998699i	$0.516239\pi$
$620$	0	0
$621$	−16.0048	−0.642250
$622$	0	0
$623$	2.90613	0.116432
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	5.59983	0.223636
$628$	0	0
$629$	−5.47975	−0.218492
$630$	0	0
$631$	0.527923	0.0210163	0.0105081	−	0.999945i	$-0.496655\pi$
$631$	0.527923	0.0210163	0.0105081	+	0.999945i	$0.496655\pi$
$632$	0	0
$633$	−31.4699	−1.25082
$634$	0	0
$635$	−11.6403	−0.461932
$636$	0	0
$637$	−0.835652	−0.0331097
$638$	0	0
$639$	−11.9439	−0.472494
$640$	0	0
$641$	−48.8551	−1.92966	−0.964829	−	0.262877i	$-0.915329\pi$
$641$	−48.8551	−1.92966	−0.964829	+	0.262877i	$0.915329\pi$
$642$	0	0
$643$	1.49998	0.0591536	0.0295768	−	0.999563i	$-0.490584\pi$
$643$	1.49998	0.0591536	0.0295768	+	0.999563i	$0.490584\pi$
$644$	0	0
$645$	−2.16692	−0.0853225
$646$	0	0
$647$	−35.7872	−1.40694	−0.703470	−	0.710725i	$-0.748367\pi$
$647$	−35.7872	−1.40694	−0.703470	+	0.710725i	$0.748367\pi$
$648$	0	0
$649$	−66.6339	−2.61561
$650$	0	0
$651$	21.9920	0.861933
$652$	0	0
$653$	−6.98648	−0.273402	−0.136701	−	0.990612i	$-0.543650\pi$
$653$	−6.98648	−0.273402	−0.136701	+	0.990612i	$0.543650\pi$
$654$	0	0
$655$	−14.5283	−0.567669
$656$	0	0
$657$	13.7123	0.534969
$658$	0	0
$659$	−11.1650	−0.434927	−0.217463	−	0.976068i	$-0.569778\pi$
$659$	−11.1650	−0.434927	−0.217463	+	0.976068i	$0.569778\pi$
$660$	0	0
$661$	−42.7855	−1.66416	−0.832082	−	0.554653i	$-0.812851\pi$
$661$	−42.7855	−1.66416	−0.832082	+	0.554653i	$0.812851\pi$
$662$	0	0
$663$	−3.16165	−0.122788
$664$	0	0
$665$	0.528632	0.0204995
$666$	0	0
$667$	−31.5676	−1.22230
$668$	0	0
$669$	−27.2807	−1.05473
$670$	0	0
$671$	−24.8103	−0.957793
$672$	0	0
$673$	20.5499	0.792141	0.396070	−	0.918220i	$-0.370374\pi$
$673$	20.5499	0.792141	0.396070	+	0.918220i	$0.370374\pi$
$674$	0	0
$675$	2.82663	0.108797
$676$	0	0
$677$	−36.5462	−1.40458	−0.702292	−	0.711889i	$-0.747840\pi$
$677$	−36.5462	−1.40458	−0.702292	+	0.711889i	$0.747840\pi$
$678$	0	0
$679$	9.57243	0.367356
$680$	0	0
$681$	53.9214	2.06627
$682$	0	0
$683$	14.4754	0.553885	0.276942	−	0.960887i	$-0.410679\pi$
$683$	14.4754	0.553885	0.276942	+	0.960887i	$0.410679\pi$
$684$	0	0
$685$	11.7168	0.447676
$686$	0	0
$687$	23.3724	0.891713
$688$	0	0
$689$	−5.97689	−0.227702
$690$	0	0
$691$	31.9488	1.21539	0.607694	−	0.794171i	$-0.292095\pi$
$691$	31.9488	1.21539	0.607694	+	0.794171i	$0.292095\pi$
$692$	0	0
$693$	8.28877	0.314864
$694$	0	0
$695$	−5.19070	−0.196894
$696$	0	0
$697$	16.2478	0.615428
$698$	0	0
$699$	55.9163	2.11495
$700$	0	0
$701$	32.8569	1.24099	0.620495	−	0.784211i	$-0.286932\pi$
$701$	32.8569	1.24099	0.620495	+	0.784211i	$0.286932\pi$
$702$	0	0
$703$	−1.65909	−0.0625736
$704$	0	0
$705$	26.7642	1.00800
$706$	0	0
$707$	−8.08584	−0.304099
$708$	0	0
$709$	25.9750	0.975513	0.487756	−	0.872980i	$-0.337815\pi$
$709$	25.9750	0.975513	0.487756	+	0.872980i	$0.337815\pi$
$710$	0	0
$711$	−4.85715	−0.182157
$712$	0	0
$713$	57.4646	2.15207
$714$	0	0
$715$	4.08511	0.152774
$716$	0	0
$717$	−50.8909	−1.90056
$718$	0	0
$719$	−49.9084	−1.86127	−0.930634	−	0.365950i	$-0.880744\pi$
$719$	−49.9084	−1.86127	−0.930634	+	0.365950i	$0.880744\pi$
$720$	0	0
$721$	6.79146	0.252927
$722$	0	0
$723$	31.4749	1.17056
$724$	0	0
$725$	5.57521	0.207058
$726$	0	0
$727$	22.2488	0.825162	0.412581	−	0.910921i	$-0.364627\pi$
$727$	22.2488	0.825162	0.412581	+	0.910921i	$0.364627\pi$
$728$	0	0
$729$	−0.634859	−0.0235133
$730$	0	0
$731$	1.74600	0.0645783
$732$	0	0
$733$	−10.8610	−0.401158	−0.200579	−	0.979677i	$-0.564282\pi$
$733$	−10.8610	−0.401158	−0.200579	+	0.979677i	$0.564282\pi$
$734$	0	0
$735$	2.16692	0.0799282
$736$	0	0
$737$	35.9778	1.32526
$738$	0	0
$739$	−36.8706	−1.35631	−0.678153	−	0.734920i	$-0.737220\pi$
$739$	−36.8706	−1.35631	−0.678153	+	0.734920i	$0.737220\pi$
$740$	0	0
$741$	−0.957243	−0.0351652
$742$	0	0
$743$	10.9710	0.402486	0.201243	−	0.979541i	$-0.435502\pi$
$743$	10.9710	0.402486	0.201243	+	0.979541i	$0.435502\pi$
$744$	0	0
$745$	13.4644	0.493296
$746$	0	0
$747$	−6.92978	−0.253547
$748$	0	0
$749$	7.56565	0.276443
$750$	0	0
$751$	−20.4785	−0.747269	−0.373635	−	0.927576i	$-0.621889\pi$
$751$	−20.4785	−0.747269	−0.373635	+	0.927576i	$0.621889\pi$
$752$	0	0
$753$	−3.68153	−0.134162
$754$	0	0
$755$	−17.6645	−0.642876
$756$	0	0
$757$	−9.86984	−0.358725	−0.179363	−	0.983783i	$-0.557404\pi$
$757$	−9.86984	−0.358725	−0.179363	+	0.983783i	$0.557404\pi$
$758$	0	0
$759$	59.9794	2.17711
$760$	0	0
$761$	51.1737	1.85504	0.927522	−	0.373768i	$-0.121934\pi$
$761$	51.1737	1.85504	0.927522	+	0.373768i	$0.121934\pi$
$762$	0	0
$763$	−2.66301	−0.0964075
$764$	0	0
$765$	2.96044	0.107035
$766$	0	0
$767$	11.3905	0.411287
$768$	0	0
$769$	33.0401	1.19146	0.595728	−	0.803186i	$-0.296864\pi$
$769$	33.0401	1.19146	0.595728	+	0.803186i	$0.296864\pi$
$770$	0	0
$771$	22.3437	0.804687
$772$	0	0
$773$	33.8199	1.21642	0.608208	−	0.793777i	$-0.291888\pi$
$773$	33.8199	1.21642	0.608208	+	0.793777i	$0.291888\pi$
$774$	0	0
$775$	−10.1489	−0.364560
$776$	0	0
$777$	−6.80078	−0.243977
$778$	0	0
$779$	4.91928	0.176252
$780$	0	0
$781$	−34.4360	−1.23222
$782$	0	0
$783$	15.7591	0.563183
$784$	0	0
$785$	22.7585	0.812285
$786$	0	0
$787$	−22.1409	−0.789238	−0.394619	−	0.918845i	$-0.629123\pi$
$787$	−22.1409	−0.789238	−0.394619	+	0.918845i	$0.629123\pi$
$788$	0	0
$789$	60.4854	2.15334
$790$	0	0
$791$	7.84552	0.278954
$792$	0	0
$793$	4.24112	0.150606
$794$	0	0
$795$	15.4986	0.549680
$796$	0	0
$797$	−47.0716	−1.66736	−0.833681	−	0.552246i	$-0.813771\pi$
$797$	−47.0716	−1.66736	−0.833681	+	0.552246i	$0.813771\pi$
$798$	0	0
$799$	−21.5653	−0.762926
$800$	0	0
$801$	4.92750	0.174105
$802$	0	0
$803$	39.5347	1.39515
$804$	0	0
$805$	5.66214	0.199564
$806$	0	0
$807$	−25.4818	−0.897001
$808$	0	0
$809$	32.8038	1.15332	0.576660	−	0.816984i	$-0.304356\pi$
$809$	32.8038	1.15332	0.576660	+	0.816984i	$0.304356\pi$
$810$	0	0
$811$	−29.0726	−1.02088	−0.510439	−	0.859914i	$-0.670517\pi$
$811$	−29.0726	−1.02088	−0.510439	+	0.859914i	$0.670517\pi$
$812$	0	0
$813$	56.3189	1.97519
$814$	0	0
$815$	12.6057	0.441558
$816$	0	0
$817$	0.528632	0.0184945
$818$	0	0
$819$	−1.41689	−0.0495103
$820$	0	0
$821$	−40.7292	−1.42146	−0.710730	−	0.703465i	$-0.751635\pi$
$821$	−40.7292	−1.42146	−0.710730	+	0.703465i	$0.751635\pi$
$822$	0	0
$823$	−13.4056	−0.467288	−0.233644	−	0.972322i	$-0.575065\pi$
$823$	−13.4056	−0.467288	−0.233644	+	0.972322i	$0.575065\pi$
$824$	0	0
$825$	−10.5931	−0.368803
$826$	0	0
$827$	−5.71450	−0.198713	−0.0993563	−	0.995052i	$-0.531678\pi$
$827$	−5.71450	−0.198713	−0.0993563	+	0.995052i	$0.531678\pi$
$828$	0	0
$829$	−47.2559	−1.64127	−0.820633	−	0.571456i	$-0.806379\pi$
$829$	−47.2559	−1.64127	−0.820633	+	0.571456i	$0.806379\pi$
$830$	0	0
$831$	28.8291	1.00007
$832$	0	0
$833$	−1.74600	−0.0604954
$834$	0	0
$835$	1.88431	0.0652091
$836$	0	0
$837$	−28.6873	−0.991578
$838$	0	0
$839$	48.9419	1.68966	0.844831	−	0.535033i	$-0.179701\pi$
$839$	48.9419	1.68966	0.844831	+	0.535033i	$0.179701\pi$
$840$	0	0
$841$	2.08298	0.0718270
$842$	0	0
$843$	−63.2199	−2.17741
$844$	0	0
$845$	12.3017	0.423191
$846$	0	0
$847$	12.8977	0.443171
$848$	0	0
$849$	43.2010	1.48266
$850$	0	0
$851$	−17.7704	−0.609160
$852$	0	0
$853$	31.6464	1.08355	0.541775	−	0.840523i	$-0.317752\pi$
$853$	31.6464	1.08355	0.541775	+	0.840523i	$0.317752\pi$
$854$	0	0
$855$	0.896323	0.0306536
$856$	0	0
$857$	4.02838	0.137607	0.0688033	−	0.997630i	$-0.478082\pi$
$857$	4.02838	0.137607	0.0688033	+	0.997630i	$0.478082\pi$
$858$	0	0
$859$	20.5000	0.699452	0.349726	−	0.936852i	$-0.386275\pi$
$859$	20.5000	0.699452	0.349726	+	0.936852i	$0.386275\pi$
$860$	0	0
$861$	20.1647	0.687211
$862$	0	0
$863$	−15.7958	−0.537695	−0.268848	−	0.963183i	$-0.586643\pi$
$863$	−15.7958	−0.537695	−0.268848	+	0.963183i	$0.586643\pi$
$864$	0	0
$865$	−17.6714	−0.600847
$866$	0	0
$867$	30.2318	1.02672
$868$	0	0
$869$	−14.0039	−0.475049
$870$	0	0
$871$	−6.15009	−0.208388
$872$	0	0
$873$	16.2306	0.549321
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	−0.771222	−0.0260423	−0.0130212	−	0.999915i	$-0.504145\pi$
$877$	−0.771222	−0.0260423	−0.0130212	+	0.999915i	$0.504145\pi$
$878$	0	0
$879$	−37.2541	−1.25655
$880$	0	0
$881$	−4.06319	−0.136892	−0.0684461	−	0.997655i	$-0.521804\pi$
$881$	−4.06319	−0.136892	−0.0684461	+	0.997655i	$0.521804\pi$
$882$	0	0
$883$	17.7639	0.597801	0.298901	−	0.954284i	$-0.403380\pi$
$883$	17.7639	0.597801	0.298901	+	0.954284i	$0.403380\pi$
$884$	0	0
$885$	−29.5366	−0.992862
$886$	0	0
$887$	20.5021	0.688393	0.344196	−	0.938898i	$-0.388151\pi$
$887$	20.5021	0.688393	0.344196	+	0.938898i	$0.388151\pi$
$888$	0	0
$889$	11.6403	0.390403
$890$	0	0
$891$	−54.8090	−1.83617
$892$	0	0
$893$	−6.52926	−0.218493
$894$	0	0
$895$	4.60914	0.154067
$896$	0	0
$897$	−10.2530	−0.342336
$898$	0	0
$899$	−56.5824	−1.88713
$900$	0	0
$901$	−12.4881	−0.416038
$902$	0	0
$903$	2.16692	0.0721107
$904$	0	0
$905$	18.8383	0.626207
$906$	0	0
$907$	0.335218	0.0111307	0.00556536	−	0.999985i	$-0.498228\pi$
$907$	0.335218	0.0111307	0.00556536	+	0.999985i	$0.498228\pi$
$908$	0	0
$909$	−13.7100	−0.454732
$910$	0	0
$911$	−13.9570	−0.462416	−0.231208	−	0.972904i	$-0.574268\pi$
$911$	−13.9570	−0.462416	−0.231208	+	0.972904i	$0.574268\pi$
$912$	0	0
$913$	−19.9796	−0.661227
$914$	0	0
$915$	−10.9976	−0.363570
$916$	0	0
$917$	14.5283	0.479768
$918$	0	0
$919$	25.9059	0.854556	0.427278	−	0.904120i	$-0.359473\pi$
$919$	25.9059	0.854556	0.427278	+	0.904120i	$0.359473\pi$
$920$	0	0
$921$	14.4123	0.474902
$922$	0	0
$923$	5.88655	0.193758
$924$	0	0
$925$	3.13845	0.103192
$926$	0	0
$927$	11.5153	0.378212
$928$	0	0
$929$	47.2127	1.54900	0.774500	−	0.632574i	$-0.218002\pi$
$929$	47.2127	1.54900	0.774500	+	0.632574i	$0.218002\pi$
$930$	0	0
$931$	−0.528632	−0.0173252
$932$	0	0
$933$	53.6072	1.75502
$934$	0	0
$935$	8.53539	0.279137
$936$	0	0
$937$	−31.7924	−1.03861	−0.519307	−	0.854588i	$-0.673810\pi$
$937$	−31.7924	−1.03861	−0.519307	+	0.854588i	$0.673810\pi$
$938$	0	0
$939$	−37.1074	−1.21095
$940$	0	0
$941$	−18.0952	−0.589888	−0.294944	−	0.955515i	$-0.595301\pi$
$941$	−18.0952	−0.589888	−0.294944	+	0.955515i	$0.595301\pi$
$942$	0	0
$943$	52.6901	1.71583
$944$	0	0
$945$	−2.82663	−0.0919504
$946$	0	0
$947$	−14.5054	−0.471362	−0.235681	−	0.971831i	$-0.575732\pi$
$947$	−14.5054	−0.471362	−0.235681	+	0.971831i	$0.575732\pi$
$948$	0	0
$949$	−6.75811	−0.219378
$950$	0	0
$951$	59.3499	1.92455
$952$	0	0
$953$	−56.9166	−1.84371	−0.921854	−	0.387536i	$-0.873326\pi$
$953$	−56.9166	−1.84371	−0.921854	+	0.387536i	$0.873326\pi$
$954$	0	0
$955$	11.7805	0.381208
$956$	0	0
$957$	−59.0586	−1.90909
$958$	0	0
$959$	−11.7168	−0.378355
$960$	0	0
$961$	72.0008	2.32261
$962$	0	0
$963$	12.8280	0.413376
$964$	0	0
$965$	9.13052	0.293922
$966$	0	0
$967$	6.33031	0.203569	0.101785	−	0.994806i	$-0.467545\pi$
$967$	6.33031	0.203569	0.101785	+	0.994806i	$0.467545\pi$
$968$	0	0
$969$	−2.00005	−0.0642510
$970$	0	0
$971$	0.134604	0.00431966	0.00215983	−	0.999998i	$-0.499313\pi$
$971$	0.134604	0.00431966	0.00215983	+	0.999998i	$0.499313\pi$
$972$	0	0
$973$	5.19070	0.166406
$974$	0	0
$975$	1.81079	0.0579918
$976$	0	0
$977$	25.4122	0.813009	0.406505	−	0.913649i	$-0.366748\pi$
$977$	25.4122	0.813009	0.406505	+	0.913649i	$0.366748\pi$
$978$	0	0
$979$	14.2067	0.454048
$980$	0	0
$981$	−4.51528	−0.144162
$982$	0	0
$983$	20.1381	0.642304	0.321152	−	0.947028i	$-0.395930\pi$
$983$	20.1381	0.642304	0.321152	+	0.947028i	$0.395930\pi$
$984$	0	0
$985$	5.51316	0.175664
$986$	0	0
$987$	−26.7642	−0.851914
$988$	0	0
$989$	5.66214	0.180046
$990$	0	0
$991$	−23.7703	−0.755088	−0.377544	−	0.925992i	$-0.623231\pi$
$991$	−23.7703	−0.755088	−0.377544	+	0.925992i	$0.623231\pi$
$992$	0	0
$993$	68.1392	2.16233
$994$	0	0
$995$	−9.29102	−0.294545
$996$	0	0
$997$	−0.0544599	−0.00172476	−0.000862381	−	1.00000i	$-0.500275\pi$
$997$	−0.0544599	−0.00172476	−0.000862381		1.00000i	$0.500275\pi$
$998$	0	0
$999$	8.87125	0.280674

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6020.2.a.f.1.3	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6020.2.a.f.1.3	✓	8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6020 = 2^{2} \cdot 5 \cdot 7 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6020.a (trivial)

\(f(q)\)	\(=\)	\(q-2.16692 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.69555 q^{9} +O(q^{10})\)	\(q-2.16692 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.69555 q^{9} +4.88853 q^{11} -0.835652 q^{13} +2.16692 q^{15} -1.74600 q^{17} -0.528632 q^{19} -2.16692 q^{21} -5.66214 q^{23} +1.00000 q^{25} +2.82663 q^{27} +5.57521 q^{29} -10.1489 q^{31} -10.5931 q^{33} -1.00000 q^{35} +3.13845 q^{37} +1.81079 q^{39} -9.30569 q^{41} -1.00000 q^{43} -1.69555 q^{45} +12.3512 q^{47} +1.00000 q^{49} +3.78345 q^{51} +7.15237 q^{53} -4.88853 q^{55} +1.14550 q^{57} -13.6307 q^{59} -5.07522 q^{61} +1.69555 q^{63} +0.835652 q^{65} +7.35963 q^{67} +12.2694 q^{69} -7.04426 q^{71} +8.08723 q^{73} -2.16692 q^{75} +4.88853 q^{77} -2.86464 q^{79} -11.2118 q^{81} -4.08703 q^{83} +1.74600 q^{85} -12.0811 q^{87} +2.90613 q^{89} -0.835652 q^{91} +21.9920 q^{93} +0.528632 q^{95} +9.57243 q^{97} +8.28877 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 5 q^{3} - 8 q^{5} + 8 q^{7} + 11 q^{9}+O(q^{10}) \) 8 * q - 5 * q^3 - 8 * q^5 + 8 * q^7 + 11 * q^9	\( 8 q - 5 q^{3} - 8 q^{5} + 8 q^{7} + 11 q^{9} + 6 q^{11} - 13 q^{13} + 5 q^{15} - 2 q^{17} - 14 q^{19} - 5 q^{21} + 6 q^{23} + 8 q^{25} - 11 q^{27} - 7 q^{29} - 18 q^{31} - q^{33} - 8 q^{35} + 2 q^{37} + 9 q^{39} - 18 q^{41} - 8 q^{43} - 11 q^{45} - q^{47} + 8 q^{49} - 19 q^{51} + 5 q^{53} - 6 q^{55} - 4 q^{57} - 12 q^{59} - 23 q^{61} + 11 q^{63} + 13 q^{65} + 8 q^{67} - 18 q^{69} + 20 q^{71} - 4 q^{73} - 5 q^{75} + 6 q^{77} + 24 q^{79} - 8 q^{81} - 14 q^{83} + 2 q^{85} + 10 q^{87} - 21 q^{89} - 13 q^{91} + q^{93} + 14 q^{95} + 7 q^{97} + 4 q^{99}+O(q^{100}) \) 8 * q - 5 * q^3 - 8 * q^5 + 8 * q^7 + 11 * q^9 + 6 * q^11 - 13 * q^13 + 5 * q^15 - 2 * q^17 - 14 * q^19 - 5 * q^21 + 6 * q^23 + 8 * q^25 - 11 * q^27 - 7 * q^29 - 18 * q^31 - q^33 - 8 * q^35 + 2 * q^37 + 9 * q^39 - 18 * q^41 - 8 * q^43 - 11 * q^45 - q^47 + 8 * q^49 - 19 * q^51 + 5 * q^53 - 6 * q^55 - 4 * q^57 - 12 * q^59 - 23 * q^61 + 11 * q^63 + 13 * q^65 + 8 * q^67 - 18 * q^69 + 20 * q^71 - 4 * q^73 - 5 * q^75 + 6 * q^77 + 24 * q^79 - 8 * q^81 - 14 * q^83 + 2 * q^85 + 10 * q^87 - 21 * q^89 - 13 * q^91 + q^93 + 14 * q^95 + 7 * q^97 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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