LMFDB - Embedded newform 6080.2.a.cl.1.2

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			1.2
Root			$1.81079$ 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)

$f(q)$	$=$	$q-0.531822 q^{3} +1.00000 q^{5} -0.0955795 q^{7} -2.71717 q^{9} -0.337570 q^{11} -4.49098 q^{13} -0.531822 q^{15} -1.68405 q^{17} -1.00000 q^{19} +0.0508313 q^{21} -2.51569 q^{23} +1.00000 q^{25} +3.04051 q^{27} +6.61268 q^{29} -6.79080 q^{31} +0.179527 q^{33} -0.0955795 q^{35} -6.28059 q^{37} +2.38840 q^{39} +1.06364 q^{41} -0.325938 q^{43} -2.71717 q^{45} +8.94884 q^{47} -6.99086 q^{49} +0.895614 q^{51} +14.4587 q^{53} -0.337570 q^{55} +0.531822 q^{57} +1.20007 q^{59} +1.21006 q^{61} +0.259705 q^{63} -4.49098 q^{65} +0.111712 q^{67} +1.33790 q^{69} +14.9003 q^{71} +2.47485 q^{73} -0.531822 q^{75} +0.0322648 q^{77} -8.33842 q^{79} +6.53448 q^{81} +17.9308 q^{83} -1.68405 q^{85} -3.51677 q^{87} -13.6749 q^{89} +0.429246 q^{91} +3.61150 q^{93} -1.00000 q^{95} -10.7213 q^{97} +0.917233 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.531822	−0.307048	−0.153524	−	0.988145i	$-0.549062\pi$
$3$	−0.531822	−0.307048	−0.153524	+	0.988145i	$0.549062\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−0.0955795	−0.0361257	−0.0180628	−	0.999837i	$-0.505750\pi$
$7$	−0.0955795	−0.0361257	−0.0180628	+	0.999837i	$0.505750\pi$
$8$	0	0
$9$	−2.71717	−0.905722
$10$	0	0
$11$	−0.337570	−0.101781	−0.0508906	−	0.998704i	$-0.516206\pi$
$11$	−0.337570	−0.101781	−0.0508906	+	0.998704i	$0.516206\pi$
$12$	0	0
$13$	−4.49098	−1.24557	−0.622787	−	0.782392i	$-0.714000\pi$
$13$	−4.49098	−1.24557	−0.622787	+	0.782392i	$0.714000\pi$
$14$	0	0
$15$	−0.531822	−0.137316
$16$	0	0
$17$	−1.68405	−0.408442	−0.204221	−	0.978925i	$-0.565466\pi$
$17$	−1.68405	−0.408442	−0.204221	+	0.978925i	$0.565466\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0.0508313	0.0110923
$22$	0	0
$23$	−2.51569	−0.524558	−0.262279	−	0.964992i	$-0.584474\pi$
$23$	−2.51569	−0.524558	−0.262279	+	0.964992i	$0.584474\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	3.04051	0.585147
$28$	0	0
$29$	6.61268	1.22794	0.613972	−	0.789328i	$-0.289571\pi$
$29$	6.61268	1.22794	0.613972	+	0.789328i	$0.289571\pi$
$30$	0	0
$31$	−6.79080	−1.21966	−0.609832	−	0.792531i	$-0.708763\pi$
$31$	−6.79080	−1.21966	−0.609832	+	0.792531i	$0.708763\pi$
$32$	0	0
$33$	0.179527	0.0312517
$34$	0	0
$35$	−0.0955795	−0.0161559
$36$	0	0
$37$	−6.28059	−1.03252	−0.516262	−	0.856431i	$-0.672677\pi$
$37$	−6.28059	−1.03252	−0.516262	+	0.856431i	$0.672677\pi$
$38$	0	0
$39$	2.38840	0.382450
$40$	0	0
$41$	1.06364	0.166113	0.0830567	−	0.996545i	$-0.473532\pi$
$41$	1.06364	0.166113	0.0830567	+	0.996545i	$0.473532\pi$
$42$	0	0
$43$	−0.325938	−0.0497051	−0.0248525	−	0.999691i	$-0.507912\pi$
$43$	−0.325938	−0.0497051	−0.0248525	+	0.999691i	$0.507912\pi$
$44$	0	0
$45$	−2.71717	−0.405051
$46$	0	0
$47$	8.94884	1.30532	0.652661	−	0.757650i	$-0.273653\pi$
$47$	8.94884	1.30532	0.652661	+	0.757650i	$0.273653\pi$
$48$	0	0
$49$	−6.99086	−0.998695
$50$	0	0
$51$	0.895614	0.125411
$52$	0	0
$53$	14.4587	1.98606	0.993028	−	0.117875i	$-0.0376081\pi$
$53$	14.4587	1.98606	0.993028	+	0.117875i	$0.0376081\pi$
$54$	0	0
$55$	−0.337570	−0.0455179
$56$	0	0
$57$	0.531822	0.0704416
$58$	0	0
$59$	1.20007	0.156236	0.0781178	−	0.996944i	$-0.475109\pi$
$59$	1.20007	0.156236	0.0781178	+	0.996944i	$0.475109\pi$
$60$	0	0
$61$	1.21006	0.154932	0.0774658	−	0.996995i	$-0.475317\pi$
$61$	1.21006	0.154932	0.0774658	+	0.996995i	$0.475317\pi$
$62$	0	0
$63$	0.259705	0.0327198
$64$	0	0
$65$	−4.49098	−0.557037
$66$	0	0
$67$	0.111712	0.0136478	0.00682389	−	0.999977i	$-0.497828\pi$
$67$	0.111712	0.0136478	0.00682389	+	0.999977i	$0.497828\pi$
$68$	0	0
$69$	1.33790	0.161064
$70$	0	0
$71$	14.9003	1.76834	0.884168	−	0.467169i	$-0.154726\pi$
$71$	14.9003	1.76834	0.884168	+	0.467169i	$0.154726\pi$
$72$	0	0
$73$	2.47485	0.289659	0.144829	−	0.989457i	$-0.453737\pi$
$73$	2.47485	0.289659	0.144829	+	0.989457i	$0.453737\pi$
$74$	0	0
$75$	−0.531822	−0.0614095
$76$	0	0
$77$	0.0322648	0.00367691
$78$	0	0
$79$	−8.33842	−0.938146	−0.469073	−	0.883159i	$-0.655412\pi$
$79$	−8.33842	−0.938146	−0.469073	+	0.883159i	$0.655412\pi$
$80$	0	0
$81$	6.53448	0.726054
$82$	0	0
$83$	17.9308	1.96816	0.984080	−	0.177725i	$-0.0568737\pi$
$83$	17.9308	1.96816	0.984080	+	0.177725i	$0.0568737\pi$
$84$	0	0
$85$	−1.68405	−0.182661
$86$	0	0
$87$	−3.51677	−0.377037
$88$	0	0
$89$	−13.6749	−1.44954	−0.724769	−	0.688992i	$-0.758054\pi$
$89$	−13.6749	−1.44954	−0.724769	+	0.688992i	$0.758054\pi$
$90$	0	0
$91$	0.429246	0.0449972
$92$	0	0
$93$	3.61150	0.374495
$94$	0	0
$95$	−1.00000	−0.102598
$96$	0	0
$97$	−10.7213	−1.08859	−0.544293	−	0.838895i	$-0.683202\pi$
$97$	−10.7213	−1.08859	−0.544293	+	0.838895i	$0.683202\pi$
$98$	0	0
$99$	0.917233	0.0921854
$100$	0	0
$101$	3.66243	0.364425	0.182213	−	0.983259i	$-0.441674\pi$
$101$	3.66243	0.364425	0.182213	+	0.983259i	$0.441674\pi$
$102$	0	0
$103$	18.5427	1.82706	0.913531	−	0.406768i	$-0.133344\pi$
$103$	18.5427	1.82706	0.913531	+	0.406768i	$0.133344\pi$
$104$	0	0
$105$	0.0508313	0.00496063
$106$	0	0
$107$	14.5024	1.40200	0.700999	−	0.713163i	$-0.252738\pi$
$107$	14.5024	1.40200	0.700999	+	0.713163i	$0.252738\pi$
$108$	0	0
$109$	10.8354	1.03785	0.518924	−	0.854821i	$-0.326333\pi$
$109$	10.8354	1.03785	0.518924	+	0.854821i	$0.326333\pi$
$110$	0	0
$111$	3.34016	0.317034
$112$	0	0
$113$	−17.2193	−1.61986	−0.809928	−	0.586529i	$-0.800494\pi$
$113$	−17.2193	−1.61986	−0.809928	+	0.586529i	$0.800494\pi$
$114$	0	0
$115$	−2.51569	−0.234589
$116$	0	0
$117$	12.2027	1.12814
$118$	0	0
$119$	0.160961	0.0147552
$120$	0	0
$121$	−10.8860	−0.989641
$122$	0	0
$123$	−0.565670	−0.0510047
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	6.78390	0.601974	0.300987	−	0.953628i	$-0.402684\pi$
$127$	6.78390	0.601974	0.300987	+	0.953628i	$0.402684\pi$
$128$	0	0
$129$	0.173341	0.0152618
$130$	0	0
$131$	16.7091	1.45988	0.729941	−	0.683510i	$-0.239548\pi$
$131$	16.7091	1.45988	0.729941	+	0.683510i	$0.239548\pi$
$132$	0	0
$133$	0.0955795	0.00828780
$134$	0	0
$135$	3.04051	0.261686
$136$	0	0
$137$	−9.00891	−0.769683	−0.384842	−	0.922983i	$-0.625744\pi$
$137$	−9.00891	−0.769683	−0.384842	+	0.922983i	$0.625744\pi$
$138$	0	0
$139$	7.43282	0.630444	0.315222	−	0.949018i	$-0.397921\pi$
$139$	7.43282	0.630444	0.315222	+	0.949018i	$0.397921\pi$
$140$	0	0
$141$	−4.75919	−0.400796
$142$	0	0
$143$	1.51602	0.126776
$144$	0	0
$145$	6.61268	0.549153
$146$	0	0
$147$	3.71790	0.306647
$148$	0	0
$149$	−4.98045	−0.408014	−0.204007	−	0.978969i	$-0.565397\pi$
$149$	−4.98045	−0.408014	−0.204007	+	0.978969i	$0.565397\pi$
$150$	0	0
$151$	19.8661	1.61668	0.808339	−	0.588717i	$-0.200367\pi$
$151$	19.8661	1.61668	0.808339	+	0.588717i	$0.200367\pi$
$152$	0	0
$153$	4.57584	0.369935
$154$	0	0
$155$	−6.79080	−0.545450
$156$	0	0
$157$	13.9183	1.11080	0.555401	−	0.831583i	$-0.312565\pi$
$157$	13.9183	1.11080	0.555401	+	0.831583i	$0.312565\pi$
$158$	0	0
$159$	−7.68946	−0.609814
$160$	0	0
$161$	0.240448	0.0189500
$162$	0	0
$163$	−7.12837	−0.558337	−0.279168	−	0.960242i	$-0.590059\pi$
$163$	−7.12837	−0.558337	−0.279168	+	0.960242i	$0.590059\pi$
$164$	0	0
$165$	0.179527	0.0139762
$166$	0	0
$167$	23.0523	1.78384	0.891919	−	0.452195i	$-0.149359\pi$
$167$	23.0523	1.78384	0.891919	+	0.452195i	$0.149359\pi$
$168$	0	0
$169$	7.16888	0.551452
$170$	0	0
$171$	2.71717	0.207787
$172$	0	0
$173$	−10.4279	−0.792815	−0.396408	−	0.918075i	$-0.629743\pi$
$173$	−10.4279	−0.792815	−0.396408	+	0.918075i	$0.629743\pi$
$174$	0	0
$175$	−0.0955795	−0.00722513
$176$	0	0
$177$	−0.638223	−0.0479718
$178$	0	0
$179$	−13.2047	−0.986967	−0.493484	−	0.869755i	$-0.664277\pi$
$179$	−13.2047	−0.986967	−0.493484	+	0.869755i	$0.664277\pi$
$180$	0	0
$181$	−12.3384	−0.917108	−0.458554	−	0.888667i	$-0.651632\pi$
$181$	−12.3384	−0.917108	−0.458554	+	0.888667i	$0.651632\pi$
$182$	0	0
$183$	−0.643534	−0.0475714
$184$	0	0
$185$	−6.28059	−0.461758
$186$	0	0
$187$	0.568484	0.0415717
$188$	0	0
$189$	−0.290611	−0.0211388
$190$	0	0
$191$	−18.3515	−1.32787	−0.663933	−	0.747792i	$-0.731114\pi$
$191$	−18.3515	−1.32787	−0.663933	+	0.747792i	$0.731114\pi$
$192$	0	0
$193$	5.74437	0.413489	0.206745	−	0.978395i	$-0.433713\pi$
$193$	5.74437	0.413489	0.206745	+	0.978395i	$0.433713\pi$
$194$	0	0
$195$	2.38840	0.171037
$196$	0	0
$197$	22.8975	1.63138	0.815688	−	0.578492i	$-0.196359\pi$
$197$	22.8975	1.63138	0.815688	+	0.578492i	$0.196359\pi$
$198$	0	0
$199$	−4.38925	−0.311146	−0.155573	−	0.987824i	$-0.549722\pi$
$199$	−4.38925	−0.311146	−0.155573	+	0.987824i	$0.549722\pi$
$200$	0	0
$201$	−0.0594109	−0.00419052
$202$	0	0
$203$	−0.632037	−0.0443603
$204$	0	0
$205$	1.06364	0.0742881
$206$	0	0
$207$	6.83554	0.475103
$208$	0	0
$209$	0.337570	0.0233502
$210$	0	0
$211$	20.3932	1.40392	0.701961	−	0.712215i	$-0.252308\pi$
$211$	20.3932	1.40392	0.701961	+	0.712215i	$0.252308\pi$
$212$	0	0
$213$	−7.92429	−0.542963
$214$	0	0
$215$	−0.325938	−0.0222288
$216$	0	0
$217$	0.649061	0.0440611
$218$	0	0
$219$	−1.31618	−0.0889390
$220$	0	0
$221$	7.56303	0.508744
$222$	0	0
$223$	−27.1245	−1.81639	−0.908195	−	0.418548i	$-0.862539\pi$
$223$	−27.1245	−1.81639	−0.908195	+	0.418548i	$0.862539\pi$
$224$	0	0
$225$	−2.71717	−0.181144
$226$	0	0
$227$	−15.8638	−1.05292	−0.526460	−	0.850200i	$-0.676481\pi$
$227$	−15.8638	−1.05292	−0.526460	+	0.850200i	$0.676481\pi$
$228$	0	0
$229$	11.1600	0.737472	0.368736	−	0.929534i	$-0.379791\pi$
$229$	11.1600	0.737472	0.368736	+	0.929534i	$0.379791\pi$
$230$	0	0
$231$	−0.0171591	−0.00112899
$232$	0	0
$233$	−6.40848	−0.419833	−0.209917	−	0.977719i	$-0.567319\pi$
$233$	−6.40848	−0.419833	−0.209917	+	0.977719i	$0.567319\pi$
$234$	0	0
$235$	8.94884	0.583758
$236$	0	0
$237$	4.43456	0.288055
$238$	0	0
$239$	2.59204	0.167665	0.0838327	−	0.996480i	$-0.473284\pi$
$239$	2.59204	0.167665	0.0838327	+	0.996480i	$0.473284\pi$
$240$	0	0
$241$	3.86108	0.248714	0.124357	−	0.992238i	$-0.460313\pi$
$241$	3.86108	0.248714	0.124357	+	0.992238i	$0.460313\pi$
$242$	0	0
$243$	−12.5967	−0.808080
$244$	0	0
$245$	−6.99086	−0.446630
$246$	0	0
$247$	4.49098	0.285754
$248$	0	0
$249$	−9.53599	−0.604319
$250$	0	0
$251$	−13.7699	−0.869151	−0.434575	−	0.900635i	$-0.643102\pi$
$251$	−13.7699	−0.869151	−0.434575	+	0.900635i	$0.643102\pi$
$252$	0	0
$253$	0.849221	0.0533901
$254$	0	0
$255$	0.895614	0.0560856
$256$	0	0
$257$	−10.1649	−0.634071	−0.317036	−	0.948414i	$-0.602687\pi$
$257$	−10.1649	−0.634071	−0.317036	+	0.948414i	$0.602687\pi$
$258$	0	0
$259$	0.600296	0.0373006
$260$	0	0
$261$	−17.9677	−1.11217
$262$	0	0
$263$	−9.49624	−0.585563	−0.292782	−	0.956179i	$-0.594581\pi$
$263$	−9.49624	−0.585563	−0.292782	+	0.956179i	$0.594581\pi$
$264$	0	0
$265$	14.4587	0.888192
$266$	0	0
$267$	7.27262	0.445077
$268$	0	0
$269$	−7.83381	−0.477636	−0.238818	−	0.971064i	$-0.576760\pi$
$269$	−7.83381	−0.477636	−0.238818	+	0.971064i	$0.576760\pi$
$270$	0	0
$271$	24.7086	1.50094	0.750469	−	0.660906i	$-0.229828\pi$
$271$	24.7086	1.50094	0.750469	+	0.660906i	$0.229828\pi$
$272$	0	0
$273$	−0.228282	−0.0138163
$274$	0	0
$275$	−0.337570	−0.0203562
$276$	0	0
$277$	6.06342	0.364315	0.182158	−	0.983269i	$-0.441692\pi$
$277$	6.06342	0.364315	0.182158	+	0.983269i	$0.441692\pi$
$278$	0	0
$279$	18.4517	1.10468
$280$	0	0
$281$	24.7958	1.47919	0.739596	−	0.673051i	$-0.235016\pi$
$281$	24.7958	1.47919	0.739596	+	0.673051i	$0.235016\pi$
$282$	0	0
$283$	−16.1626	−0.960765	−0.480382	−	0.877059i	$-0.659502\pi$
$283$	−16.1626	−0.960765	−0.480382	+	0.877059i	$0.659502\pi$
$284$	0	0
$285$	0.531822	0.0315024
$286$	0	0
$287$	−0.101663	−0.00600096
$288$	0	0
$289$	−14.1640	−0.833175
$290$	0	0
$291$	5.70184	0.334248
$292$	0	0
$293$	8.52258	0.497895	0.248947	−	0.968517i	$-0.419915\pi$
$293$	8.52258	0.497895	0.248947	+	0.968517i	$0.419915\pi$
$294$	0	0
$295$	1.20007	0.0698707
$296$	0	0
$297$	−1.02639	−0.0595570
$298$	0	0
$299$	11.2979	0.653375
$300$	0	0
$301$	0.0311530	0.00179563
$302$	0	0
$303$	−1.94776	−0.111896
$304$	0	0
$305$	1.21006	0.0692876
$306$	0	0
$307$	24.4232	1.39390	0.696952	−	0.717118i	$-0.254539\pi$
$307$	24.4232	1.39390	0.696952	+	0.717118i	$0.254539\pi$
$308$	0	0
$309$	−9.86140	−0.560995
$310$	0	0
$311$	7.67265	0.435076	0.217538	−	0.976052i	$-0.430197\pi$
$311$	7.67265	0.435076	0.217538	+	0.976052i	$0.430197\pi$
$312$	0	0
$313$	2.02932	0.114704	0.0573518	−	0.998354i	$-0.481734\pi$
$313$	2.02932	0.114704	0.0573518	+	0.998354i	$0.481734\pi$
$314$	0	0
$315$	0.259705	0.0146327
$316$	0	0
$317$	5.44515	0.305830	0.152915	−	0.988239i	$-0.451134\pi$
$317$	5.44515	0.305830	0.152915	+	0.988239i	$0.451134\pi$
$318$	0	0
$319$	−2.23224	−0.124982
$320$	0	0
$321$	−7.71268	−0.430480
$322$	0	0
$323$	1.68405	0.0937030
$324$	0	0
$325$	−4.49098	−0.249115
$326$	0	0
$327$	−5.76253	−0.318668
$328$	0	0
$329$	−0.855326	−0.0471556
$330$	0	0
$331$	−6.87780	−0.378038	−0.189019	−	0.981973i	$-0.560531\pi$
$331$	−6.87780	−0.378038	−0.189019	+	0.981973i	$0.560531\pi$
$332$	0	0
$333$	17.0654	0.935179
$334$	0	0
$335$	0.111712	0.00610348
$336$	0	0
$337$	−4.21120	−0.229398	−0.114699	−	0.993400i	$-0.536590\pi$
$337$	−4.21120	−0.229398	−0.114699	+	0.993400i	$0.536590\pi$
$338$	0	0
$339$	9.15761	0.497373
$340$	0	0
$341$	2.29237	0.124139
$342$	0	0
$343$	1.33724	0.0722042
$344$	0	0
$345$	1.33790	0.0720301
$346$	0	0
$347$	−3.64180	−0.195502	−0.0977510	−	0.995211i	$-0.531165\pi$
$347$	−3.64180	−0.195502	−0.0977510	+	0.995211i	$0.531165\pi$
$348$	0	0
$349$	−28.5835	−1.53004	−0.765020	−	0.644006i	$-0.777271\pi$
$349$	−28.5835	−1.53004	−0.765020	+	0.644006i	$0.777271\pi$
$350$	0	0
$351$	−13.6549	−0.728844
$352$	0	0
$353$	7.26564	0.386711	0.193356	−	0.981129i	$-0.438063\pi$
$353$	7.26564	0.386711	0.193356	+	0.981129i	$0.438063\pi$
$354$	0	0
$355$	14.9003	0.790824
$356$	0	0
$357$	−0.0856024	−0.00453056
$358$	0	0
$359$	4.80851	0.253784	0.126892	−	0.991917i	$-0.459500\pi$
$359$	4.80851	0.253784	0.126892	+	0.991917i	$0.459500\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	5.78944	0.303867
$364$	0	0
$365$	2.47485	0.129539
$366$	0	0
$367$	27.9947	1.46131	0.730655	−	0.682747i	$-0.239215\pi$
$367$	27.9947	1.46131	0.730655	+	0.682747i	$0.239215\pi$
$368$	0	0
$369$	−2.89010	−0.150452
$370$	0	0
$371$	−1.38196	−0.0717476
$372$	0	0
$373$	30.9677	1.60344	0.801722	−	0.597697i	$-0.203917\pi$
$373$	30.9677	1.60344	0.801722	+	0.597697i	$0.203917\pi$
$374$	0	0
$375$	−0.531822	−0.0274632
$376$	0	0
$377$	−29.6974	−1.52949
$378$	0	0
$379$	−20.3238	−1.04396	−0.521981	−	0.852957i	$-0.674807\pi$
$379$	−20.3238	−1.04396	−0.521981	+	0.852957i	$0.674807\pi$
$380$	0	0
$381$	−3.60783	−0.184835
$382$	0	0
$383$	19.3567	0.989082	0.494541	−	0.869154i	$-0.335336\pi$
$383$	19.3567	0.989082	0.494541	+	0.869154i	$0.335336\pi$
$384$	0	0
$385$	0.0322648	0.00164437
$386$	0	0
$387$	0.885627	0.0450190
$388$	0	0
$389$	24.5206	1.24324	0.621621	−	0.783318i	$-0.286474\pi$
$389$	24.5206	1.24324	0.621621	+	0.783318i	$0.286474\pi$
$390$	0	0
$391$	4.23654	0.214251
$392$	0	0
$393$	−8.88627	−0.448253
$394$	0	0
$395$	−8.33842	−0.419552
$396$	0	0
$397$	9.23699	0.463591	0.231796	−	0.972765i	$-0.425540\pi$
$397$	9.23699	0.463591	0.231796	+	0.972765i	$0.425540\pi$
$398$	0	0
$399$	−0.0508313	−0.00254475
$400$	0	0
$401$	21.0570	1.05154	0.525768	−	0.850628i	$-0.323778\pi$
$401$	21.0570	1.05154	0.525768	+	0.850628i	$0.323778\pi$
$402$	0	0
$403$	30.4973	1.51918
$404$	0	0
$405$	6.53448	0.324701
$406$	0	0
$407$	2.12014	0.105091
$408$	0	0
$409$	29.4567	1.45654	0.728269	−	0.685291i	$-0.240325\pi$
$409$	29.4567	1.45654	0.728269	+	0.685291i	$0.240325\pi$
$410$	0	0
$411$	4.79114	0.236329
$412$	0	0
$413$	−0.114702	−0.00564411
$414$	0	0
$415$	17.9308	0.880188
$416$	0	0
$417$	−3.95294	−0.193576
$418$	0	0
$419$	26.4973	1.29448	0.647239	−	0.762287i	$-0.275923\pi$
$419$	26.4973	1.29448	0.647239	+	0.762287i	$0.275923\pi$
$420$	0	0
$421$	−20.0442	−0.976894	−0.488447	−	0.872594i	$-0.662436\pi$
$421$	−20.0442	−0.976894	−0.488447	+	0.872594i	$0.662436\pi$
$422$	0	0
$423$	−24.3155	−1.18226
$424$	0	0
$425$	−1.68405	−0.0816884
$426$	0	0
$427$	−0.115657	−0.00559701
$428$	0	0
$429$	−0.806253	−0.0389262
$430$	0	0
$431$	−17.6186	−0.848660	−0.424330	−	0.905508i	$-0.639490\pi$
$431$	−17.6186	−0.848660	−0.424330	+	0.905508i	$0.639490\pi$
$432$	0	0
$433$	8.99700	0.432368	0.216184	−	0.976353i	$-0.430639\pi$
$433$	8.99700	0.432368	0.216184	+	0.976353i	$0.430639\pi$
$434$	0	0
$435$	−3.51677	−0.168616
$436$	0	0
$437$	2.51569	0.120342
$438$	0	0
$439$	−16.0456	−0.765815	−0.382907	−	0.923787i	$-0.625077\pi$
$439$	−16.0456	−0.765815	−0.382907	+	0.923787i	$0.625077\pi$
$440$	0	0
$441$	18.9953	0.904540
$442$	0	0
$443$	−17.7087	−0.841365	−0.420683	−	0.907208i	$-0.638209\pi$
$443$	−17.7087	−0.841365	−0.420683	+	0.907208i	$0.638209\pi$
$444$	0	0
$445$	−13.6749	−0.648253
$446$	0	0
$447$	2.64871	0.125280
$448$	0	0
$449$	29.2271	1.37931	0.689655	−	0.724138i	$-0.257762\pi$
$449$	29.2271	1.37931	0.689655	+	0.724138i	$0.257762\pi$
$450$	0	0
$451$	−0.359054	−0.0169072
$452$	0	0
$453$	−10.5652	−0.496397
$454$	0	0
$455$	0.429246	0.0201233
$456$	0	0
$457$	−6.29269	−0.294359	−0.147180	−	0.989110i	$-0.547020\pi$
$457$	−6.29269	−0.294359	−0.147180	+	0.989110i	$0.547020\pi$
$458$	0	0
$459$	−5.12038	−0.238999
$460$	0	0
$461$	−8.76201	−0.408087	−0.204044	−	0.978962i	$-0.565408\pi$
$461$	−8.76201	−0.408087	−0.204044	+	0.978962i	$0.565408\pi$
$462$	0	0
$463$	−3.21624	−0.149471	−0.0747357	−	0.997203i	$-0.523811\pi$
$463$	−3.21624	−0.149471	−0.0747357	+	0.997203i	$0.523811\pi$
$464$	0	0
$465$	3.61150	0.167479
$466$	0	0
$467$	31.4584	1.45572	0.727862	−	0.685724i	$-0.240514\pi$
$467$	31.4584	1.45572	0.727862	+	0.685724i	$0.240514\pi$
$468$	0	0
$469$	−0.0106774	−0.000493035 0
$470$	0	0
$471$	−7.40207	−0.341069
$472$	0	0
$473$	0.110027	0.00505904
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	−39.2867	−1.79881
$478$	0	0
$479$	−13.7313	−0.627399	−0.313699	−	0.949522i	$-0.601568\pi$
$479$	−13.7313	−0.627399	−0.313699	+	0.949522i	$0.601568\pi$
$480$	0	0
$481$	28.2060	1.28608
$482$	0	0
$483$	−0.127876	−0.00581855
$484$	0	0
$485$	−10.7213	−0.486831
$486$	0	0
$487$	−6.70766	−0.303953	−0.151977	−	0.988384i	$-0.548564\pi$
$487$	−6.70766	−0.303953	−0.151977	+	0.988384i	$0.548564\pi$
$488$	0	0
$489$	3.79102	0.171436
$490$	0	0
$491$	−16.7613	−0.756429	−0.378214	−	0.925718i	$-0.623462\pi$
$491$	−16.7613	−0.756429	−0.378214	+	0.925718i	$0.623462\pi$
$492$	0	0
$493$	−11.1361	−0.501543
$494$	0	0
$495$	0.917233	0.0412266
$496$	0	0
$497$	−1.42416	−0.0638823
$498$	0	0
$499$	−17.3105	−0.774926	−0.387463	−	0.921885i	$-0.626648\pi$
$499$	−17.3105	−0.774926	−0.387463	+	0.921885i	$0.626648\pi$
$500$	0	0
$501$	−12.2597	−0.547723
$502$	0	0
$503$	38.7748	1.72888	0.864441	−	0.502734i	$-0.167673\pi$
$503$	38.7748	1.72888	0.864441	+	0.502734i	$0.167673\pi$
$504$	0	0
$505$	3.66243	0.162976
$506$	0	0
$507$	−3.81257	−0.169322
$508$	0	0
$509$	3.39492	0.150477	0.0752385	−	0.997166i	$-0.476028\pi$
$509$	3.39492	0.150477	0.0752385	+	0.997166i	$0.476028\pi$
$510$	0	0
$511$	−0.236545	−0.0104641
$512$	0	0
$513$	−3.04051	−0.134242
$514$	0	0
$515$	18.5427	0.817087
$516$	0	0
$517$	−3.02086	−0.132857
$518$	0	0
$519$	5.54576	0.243432
$520$	0	0
$521$	5.32985	0.233505	0.116753	−	0.993161i	$-0.462752\pi$
$521$	5.32985	0.233505	0.116753	+	0.993161i	$0.462752\pi$
$522$	0	0
$523$	−7.55465	−0.330342	−0.165171	−	0.986265i	$-0.552818\pi$
$523$	−7.55465	−0.330342	−0.165171	+	0.986265i	$0.552818\pi$
$524$	0	0
$525$	0.0508313	0.00221846
$526$	0	0
$527$	11.4360	0.498161
$528$	0	0
$529$	−16.6713	−0.724839
$530$	0	0
$531$	−3.26078	−0.141506
$532$	0	0
$533$	−4.77680	−0.206906
$534$	0	0
$535$	14.5024	0.626992
$536$	0	0
$537$	7.02256	0.303046
$538$	0	0
$539$	2.35991	0.101648
$540$	0	0
$541$	−28.8850	−1.24186	−0.620931	−	0.783865i	$-0.713245\pi$
$541$	−28.8850	−1.24186	−0.620931	+	0.783865i	$0.713245\pi$
$542$	0	0
$543$	6.56185	0.281596
$544$	0	0
$545$	10.8354	0.464139
$546$	0	0
$547$	28.7106	1.22757	0.613787	−	0.789471i	$-0.289645\pi$
$547$	28.7106	1.22757	0.613787	+	0.789471i	$0.289645\pi$
$548$	0	0
$549$	−3.28792	−0.140325
$550$	0	0
$551$	−6.61268	−0.281709
$552$	0	0
$553$	0.796983	0.0338911
$554$	0	0
$555$	3.34016	0.141782
$556$	0	0
$557$	5.93658	0.251541	0.125771	−	0.992059i	$-0.459860\pi$
$557$	5.93658	0.251541	0.125771	+	0.992059i	$0.459860\pi$
$558$	0	0
$559$	1.46378	0.0619113
$560$	0	0
$561$	−0.302333	−0.0127645
$562$	0	0
$563$	15.9031	0.670235	0.335117	−	0.942176i	$-0.391224\pi$
$563$	15.9031	0.670235	0.335117	+	0.942176i	$0.391224\pi$
$564$	0	0
$565$	−17.2193	−0.724422
$566$	0	0
$567$	−0.624563	−0.0262292
$568$	0	0
$569$	−18.4451	−0.773258	−0.386629	−	0.922235i	$-0.626361\pi$
$569$	−18.4451	−0.773258	−0.386629	+	0.922235i	$0.626361\pi$
$570$	0	0
$571$	34.2148	1.43184	0.715922	−	0.698180i	$-0.246006\pi$
$571$	34.2148	1.43184	0.715922	+	0.698180i	$0.246006\pi$
$572$	0	0
$573$	9.75971	0.407718
$574$	0	0
$575$	−2.51569	−0.104912
$576$	0	0
$577$	7.51233	0.312742	0.156371	−	0.987698i	$-0.450020\pi$
$577$	7.51233	0.312742	0.156371	+	0.987698i	$0.450020\pi$
$578$	0	0
$579$	−3.05498	−0.126961
$580$	0	0
$581$	−1.71382	−0.0711011
$582$	0	0
$583$	−4.88083	−0.202143
$584$	0	0
$585$	12.2027	0.504521
$586$	0	0
$587$	18.0757	0.746064	0.373032	−	0.927819i	$-0.378318\pi$
$587$	18.0757	0.746064	0.373032	+	0.927819i	$0.378318\pi$
$588$	0	0
$589$	6.79080	0.279810
$590$	0	0
$591$	−12.1774	−0.500910
$592$	0	0
$593$	−2.52243	−0.103584	−0.0517919	−	0.998658i	$-0.516493\pi$
$593$	−2.52243	−0.103584	−0.0517919	+	0.998658i	$0.516493\pi$
$594$	0	0
$595$	0.160961	0.00659874
$596$	0	0
$597$	2.33430	0.0955366
$598$	0	0
$599$	34.6184	1.41447	0.707235	−	0.706978i	$-0.249942\pi$
$599$	34.6184	1.41447	0.707235	+	0.706978i	$0.249942\pi$
$600$	0	0
$601$	38.0768	1.55318	0.776592	−	0.630004i	$-0.216947\pi$
$601$	38.0768	1.55318	0.776592	+	0.630004i	$0.216947\pi$
$602$	0	0
$603$	−0.303540	−0.0123611
$604$	0	0
$605$	−10.8860	−0.442581
$606$	0	0
$607$	24.4616	0.992867	0.496433	−	0.868075i	$-0.334643\pi$
$607$	24.4616	0.992867	0.496433	+	0.868075i	$0.334643\pi$
$608$	0	0
$609$	0.336131	0.0136207
$610$	0	0
$611$	−40.1890	−1.62587
$612$	0	0
$613$	−16.6663	−0.673146	−0.336573	−	0.941657i	$-0.609268\pi$
$613$	−16.6663	−0.673146	−0.336573	+	0.941657i	$0.609268\pi$
$614$	0	0
$615$	−0.565670	−0.0228100
$616$	0	0
$617$	40.2955	1.62224	0.811118	−	0.584883i	$-0.198860\pi$
$617$	40.2955	1.62224	0.811118	+	0.584883i	$0.198860\pi$
$618$	0	0
$619$	20.4210	0.820788	0.410394	−	0.911908i	$-0.365391\pi$
$619$	20.4210	0.820788	0.410394	+	0.911908i	$0.365391\pi$
$620$	0	0
$621$	−7.64899	−0.306943
$622$	0	0
$623$	1.30704	0.0523655
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−0.179527	−0.00716962
$628$	0	0
$629$	10.5768	0.421726
$630$	0	0
$631$	−3.64134	−0.144960	−0.0724798	−	0.997370i	$-0.523091\pi$
$631$	−3.64134	−0.144960	−0.0724798	+	0.997370i	$0.523091\pi$
$632$	0	0
$633$	−10.8455	−0.431071
$634$	0	0
$635$	6.78390	0.269211
$636$	0	0
$637$	31.3958	1.24395
$638$	0	0
$639$	−40.4865	−1.60162
$640$	0	0
$641$	−30.6730	−1.21151	−0.605756	−	0.795651i	$-0.707129\pi$
$641$	−30.6730	−1.21151	−0.605756	+	0.795651i	$0.707129\pi$
$642$	0	0
$643$	−10.9021	−0.429938	−0.214969	−	0.976621i	$-0.568965\pi$
$643$	−10.9021	−0.429938	−0.214969	+	0.976621i	$0.568965\pi$
$644$	0	0
$645$	0.173341	0.00682530
$646$	0	0
$647$	−5.79113	−0.227673	−0.113836	−	0.993500i	$-0.536314\pi$
$647$	−5.79113	−0.227673	−0.113836	+	0.993500i	$0.536314\pi$
$648$	0	0
$649$	−0.405107	−0.0159018
$650$	0	0
$651$	−0.345185	−0.0135289
$652$	0	0
$653$	27.5926	1.07978	0.539890	−	0.841735i	$-0.318466\pi$
$653$	27.5926	1.07978	0.539890	+	0.841735i	$0.318466\pi$
$654$	0	0
$655$	16.7091	0.652879
$656$	0	0
$657$	−6.72456	−0.262350
$658$	0	0
$659$	−23.8541	−0.929224	−0.464612	−	0.885514i	$-0.653806\pi$
$659$	−23.8541	−0.929224	−0.464612	+	0.885514i	$0.653806\pi$
$660$	0	0
$661$	33.2311	1.29254	0.646270	−	0.763109i	$-0.276328\pi$
$661$	33.2311	1.29254	0.646270	+	0.763109i	$0.276328\pi$
$662$	0	0
$663$	−4.02218	−0.156209
$664$	0	0
$665$	0.0955795	0.00370642
$666$	0	0
$667$	−16.6354	−0.644127
$668$	0	0
$669$	14.4254	0.557718
$670$	0	0
$671$	−0.408478	−0.0157691
$672$	0	0
$673$	25.6622	0.989206	0.494603	−	0.869119i	$-0.335313\pi$
$673$	25.6622	0.989206	0.494603	+	0.869119i	$0.335313\pi$
$674$	0	0
$675$	3.04051	0.117029
$676$	0	0
$677$	9.59071	0.368601	0.184300	−	0.982870i	$-0.440998\pi$
$677$	9.59071	0.368601	0.184300	+	0.982870i	$0.440998\pi$
$678$	0	0
$679$	1.02474	0.0393259
$680$	0	0
$681$	8.43674	0.323296
$682$	0	0
$683$	36.4798	1.39586	0.697929	−	0.716166i	$-0.254105\pi$
$683$	36.4798	1.39586	0.697929	+	0.716166i	$0.254105\pi$
$684$	0	0
$685$	−9.00891	−0.344213
$686$	0	0
$687$	−5.93512	−0.226439
$688$	0	0
$689$	−64.9338	−2.47378
$690$	0	0
$691$	−20.3671	−0.774799	−0.387400	−	0.921912i	$-0.626627\pi$
$691$	−20.3671	−0.774799	−0.387400	+	0.921912i	$0.626627\pi$
$692$	0	0
$693$	−0.0876687	−0.00333026
$694$	0	0
$695$	7.43282	0.281943
$696$	0	0
$697$	−1.79123	−0.0678476
$698$	0	0
$699$	3.40817	0.128909
$700$	0	0
$701$	−37.7648	−1.42636	−0.713178	−	0.700983i	$-0.752745\pi$
$701$	−37.7648	−1.42636	−0.713178	+	0.700983i	$0.752745\pi$
$702$	0	0
$703$	6.28059	0.236877
$704$	0	0
$705$	−4.75919	−0.179241
$706$	0	0
$707$	−0.350053	−0.0131651
$708$	0	0
$709$	−27.8421	−1.04563	−0.522817	−	0.852445i	$-0.675119\pi$
$709$	−27.8421	−1.04563	−0.522817	+	0.852445i	$0.675119\pi$
$710$	0	0
$711$	22.6569	0.849699
$712$	0	0
$713$	17.0835	0.639783
$714$	0	0
$715$	1.51602	0.0566959
$716$	0	0
$717$	−1.37851	−0.0514813
$718$	0	0
$719$	−30.1635	−1.12491	−0.562455	−	0.826828i	$-0.690143\pi$
$719$	−30.1635	−1.12491	−0.562455	+	0.826828i	$0.690143\pi$
$720$	0	0
$721$	−1.77230	−0.0660039
$722$	0	0
$723$	−2.05341	−0.0763670
$724$	0	0
$725$	6.61268	0.245589
$726$	0	0
$727$	17.8911	0.663544	0.331772	−	0.943360i	$-0.392354\pi$
$727$	17.8911	0.663544	0.331772	+	0.943360i	$0.392354\pi$
$728$	0	0
$729$	−12.9042	−0.477934
$730$	0	0
$731$	0.548896	0.0203016
$732$	0	0
$733$	−12.7778	−0.471958	−0.235979	−	0.971758i	$-0.575830\pi$
$733$	−12.7778	−0.471958	−0.235979	+	0.971758i	$0.575830\pi$
$734$	0	0
$735$	3.71790	0.137137
$736$	0	0
$737$	−0.0377106	−0.00138909
$738$	0	0
$739$	−19.5006	−0.717340	−0.358670	−	0.933464i	$-0.616770\pi$
$739$	−19.5006	−0.717340	−0.358670	+	0.933464i	$0.616770\pi$
$740$	0	0
$741$	−2.38840	−0.0877401
$742$	0	0
$743$	−41.0316	−1.50530	−0.752652	−	0.658418i	$-0.771226\pi$
$743$	−41.0316	−1.50530	−0.752652	+	0.658418i	$0.771226\pi$
$744$	0	0
$745$	−4.98045	−0.182469
$746$	0	0
$747$	−48.7209	−1.78261
$748$	0	0
$749$	−1.38613	−0.0506481
$750$	0	0
$751$	−23.6623	−0.863449	−0.431724	−	0.902006i	$-0.642095\pi$
$751$	−23.6623	−0.863449	−0.431724	+	0.902006i	$0.642095\pi$
$752$	0	0
$753$	7.32316	0.266871
$754$	0	0
$755$	19.8661	0.723000
$756$	0	0
$757$	−7.29305	−0.265070	−0.132535	−	0.991178i	$-0.542312\pi$
$757$	−7.29305	−0.265070	−0.132535	+	0.991178i	$0.542312\pi$
$758$	0	0
$759$	−0.451635	−0.0163933
$760$	0	0
$761$	34.8131	1.26197	0.630987	−	0.775793i	$-0.282650\pi$
$761$	34.8131	1.26197	0.630987	+	0.775793i	$0.282650\pi$
$762$	0	0
$763$	−1.03565	−0.0374929
$764$	0	0
$765$	4.57584	0.165440
$766$	0	0
$767$	−5.38948	−0.194603
$768$	0	0
$769$	12.5028	0.450862	0.225431	−	0.974259i	$-0.427621\pi$
$769$	12.5028	0.450862	0.225431	+	0.974259i	$0.427621\pi$
$770$	0	0
$771$	5.40594	0.194690
$772$	0	0
$773$	−0.0576140	−0.00207223	−0.00103612	−	0.999999i	$-0.500330\pi$
$773$	−0.0576140	−0.00207223	−0.00103612	+	0.999999i	$0.500330\pi$
$774$	0	0
$775$	−6.79080	−0.243933
$776$	0	0
$777$	−0.319251	−0.0114531
$778$	0	0
$779$	−1.06364	−0.0381090
$780$	0	0
$781$	−5.02988	−0.179983
$782$	0	0
$783$	20.1059	0.718528
$784$	0	0
$785$	13.9183	0.496766
$786$	0	0
$787$	6.52348	0.232537	0.116268	−	0.993218i	$-0.462907\pi$
$787$	6.52348	0.232537	0.116268	+	0.993218i	$0.462907\pi$
$788$	0	0
$789$	5.05031	0.179796
$790$	0	0
$791$	1.64581	0.0585184
$792$	0	0
$793$	−5.43433	−0.192979
$794$	0	0
$795$	−7.68946	−0.272717
$796$	0	0
$797$	17.9775	0.636797	0.318399	−	0.947957i	$-0.396855\pi$
$797$	17.9775	0.636797	0.318399	+	0.947957i	$0.396855\pi$
$798$	0	0
$799$	−15.0703	−0.533148
$800$	0	0
$801$	37.1570	1.31288
$802$	0	0
$803$	−0.835433	−0.0294818
$804$	0	0
$805$	0.240448	0.00847470
$806$	0	0
$807$	4.16619	0.146657
$808$	0	0
$809$	−9.96787	−0.350452	−0.175226	−	0.984528i	$-0.556066\pi$
$809$	−9.96787	−0.350452	−0.175226	+	0.984528i	$0.556066\pi$
$810$	0	0
$811$	−7.56773	−0.265739	−0.132870	−	0.991134i	$-0.542419\pi$
$811$	−7.56773	−0.265739	−0.132870	+	0.991134i	$0.542419\pi$
$812$	0	0
$813$	−13.1406	−0.460859
$814$	0	0
$815$	−7.12837	−0.249696
$816$	0	0
$817$	0.325938	0.0114031
$818$	0	0
$819$	−1.16633	−0.0407549
$820$	0	0
$821$	−38.1434	−1.33122	−0.665608	−	0.746302i	$-0.731828\pi$
$821$	−38.1434	−1.33122	−0.665608	+	0.746302i	$0.731828\pi$
$822$	0	0
$823$	12.9509	0.451440	0.225720	−	0.974192i	$-0.427527\pi$
$823$	12.9509	0.451440	0.225720	+	0.974192i	$0.427527\pi$
$824$	0	0
$825$	0.179527	0.00625033
$826$	0	0
$827$	13.3538	0.464356	0.232178	−	0.972673i	$-0.425415\pi$
$827$	13.3538	0.464356	0.232178	+	0.972673i	$0.425415\pi$
$828$	0	0
$829$	15.5768	0.541005	0.270503	−	0.962719i	$-0.412810\pi$
$829$	15.5768	0.541005	0.270503	+	0.962719i	$0.412810\pi$
$830$	0	0
$831$	−3.22466	−0.111862
$832$	0	0
$833$	11.7730	0.407909
$834$	0	0
$835$	23.0523	0.797757
$836$	0	0
$837$	−20.6475	−0.713683
$838$	0	0
$839$	28.7351	0.992045	0.496023	−	0.868310i	$-0.334793\pi$
$839$	28.7351	0.992045	0.496023	+	0.868310i	$0.334793\pi$
$840$	0	0
$841$	14.7275	0.507845
$842$	0	0
$843$	−13.1869	−0.454183
$844$	0	0
$845$	7.16888	0.246617
$846$	0	0
$847$	1.04048	0.0357514
$848$	0	0
$849$	8.59561	0.295000
$850$	0	0
$851$	15.8000	0.541618
$852$	0	0
$853$	−52.3067	−1.79095	−0.895473	−	0.445116i	$-0.853163\pi$
$853$	−52.3067	−1.79095	−0.895473	+	0.445116i	$0.853163\pi$
$854$	0	0
$855$	2.71717	0.0929251
$856$	0	0
$857$	−53.7133	−1.83481	−0.917405	−	0.397955i	$-0.869720\pi$
$857$	−53.7133	−1.83481	−0.917405	+	0.397955i	$0.869720\pi$
$858$	0	0
$859$	−28.2653	−0.964399	−0.482199	−	0.876061i	$-0.660162\pi$
$859$	−28.2653	−0.964399	−0.482199	+	0.876061i	$0.660162\pi$
$860$	0	0
$861$	0.0540664	0.00184258
$862$	0	0
$863$	−21.4628	−0.730603	−0.365301	−	0.930889i	$-0.619034\pi$
$863$	−21.4628	−0.730603	−0.365301	+	0.930889i	$0.619034\pi$
$864$	0	0
$865$	−10.4279	−0.354558
$866$	0	0
$867$	7.53272	0.255824
$868$	0	0
$869$	2.81480	0.0954856
$870$	0	0
$871$	−0.501696	−0.0169993
$872$	0	0
$873$	29.1316	0.985957
$874$	0	0
$875$	−0.0955795	−0.00323118
$876$	0	0
$877$	−7.69073	−0.259697	−0.129849	−	0.991534i	$-0.541449\pi$
$877$	−7.69073	−0.259697	−0.129849	+	0.991534i	$0.541449\pi$
$878$	0	0
$879$	−4.53250	−0.152877
$880$	0	0
$881$	41.6429	1.40299	0.701493	−	0.712676i	$-0.252517\pi$
$881$	41.6429	1.40299	0.701493	+	0.712676i	$0.252517\pi$
$882$	0	0
$883$	−9.41566	−0.316862	−0.158431	−	0.987370i	$-0.550644\pi$
$883$	−9.41566	−0.316862	−0.158431	+	0.987370i	$0.550644\pi$
$884$	0	0
$885$	−0.638223	−0.0214536
$886$	0	0
$887$	24.2210	0.813263	0.406632	−	0.913592i	$-0.366703\pi$
$887$	24.2210	0.813263	0.406632	+	0.913592i	$0.366703\pi$
$888$	0	0
$889$	−0.648402	−0.0217467
$890$	0	0
$891$	−2.20585	−0.0738986
$892$	0	0
$893$	−8.94884	−0.299461
$894$	0	0
$895$	−13.2047	−0.441385
$896$	0	0
$897$	−6.00848	−0.200617
$898$	0	0
$899$	−44.9053	−1.49768
$900$	0	0
$901$	−24.3492	−0.811189
$902$	0	0
$903$	−0.0165679	−0.000551344 0
$904$	0	0
$905$	−12.3384	−0.410143
$906$	0	0
$907$	35.2614	1.17083	0.585417	−	0.810732i	$-0.300931\pi$
$907$	35.2614	1.17083	0.585417	+	0.810732i	$0.300931\pi$
$908$	0	0
$909$	−9.95143	−0.330068
$910$	0	0
$911$	−58.6348	−1.94266	−0.971328	−	0.237742i	$-0.923593\pi$
$911$	−58.6348	−1.94266	−0.971328	+	0.237742i	$0.923593\pi$
$912$	0	0
$913$	−6.05290	−0.200322
$914$	0	0
$915$	−0.643534	−0.0212746
$916$	0	0
$917$	−1.59705	−0.0527392
$918$	0	0
$919$	−20.0824	−0.662458	−0.331229	−	0.943550i	$-0.607463\pi$
$919$	−20.0824	−0.662458	−0.331229	+	0.943550i	$0.607463\pi$
$920$	0	0
$921$	−12.9888	−0.427995
$922$	0	0
$923$	−66.9168	−2.20259
$924$	0	0
$925$	−6.28059	−0.206505
$926$	0	0
$927$	−50.3835	−1.65481
$928$	0	0
$929$	−17.7731	−0.583116	−0.291558	−	0.956553i	$-0.594174\pi$
$929$	−17.7731	−0.583116	−0.291558	+	0.956553i	$0.594174\pi$
$930$	0	0
$931$	6.99086	0.229116
$932$	0	0
$933$	−4.08049	−0.133589
$934$	0	0
$935$	0.568484	0.0185914
$936$	0	0
$937$	−28.1246	−0.918792	−0.459396	−	0.888232i	$-0.651934\pi$
$937$	−28.1246	−0.918792	−0.459396	+	0.888232i	$0.651934\pi$
$938$	0	0
$939$	−1.07923	−0.0352195
$940$	0	0
$941$	22.0914	0.720160	0.360080	−	0.932921i	$-0.382749\pi$
$941$	22.0914	0.720160	0.360080	+	0.932921i	$0.382749\pi$
$942$	0	0
$943$	−2.67580	−0.0871360
$944$	0	0
$945$	−0.290611	−0.00945358
$946$	0	0
$947$	38.3266	1.24545	0.622724	−	0.782442i	$-0.286026\pi$
$947$	38.3266	1.24545	0.622724	+	0.782442i	$0.286026\pi$
$948$	0	0
$949$	−11.1145	−0.360791
$950$	0	0
$951$	−2.89585	−0.0939044
$952$	0	0
$953$	14.8386	0.480670	0.240335	−	0.970690i	$-0.422743\pi$
$953$	14.8386	0.480670	0.240335	+	0.970690i	$0.422743\pi$
$954$	0	0
$955$	−18.3515	−0.593839
$956$	0	0
$957$	1.18716	0.0383753
$958$	0	0
$959$	0.861067	0.0278053
$960$	0	0
$961$	15.1149	0.487578
$962$	0	0
$963$	−39.4053	−1.26982
$964$	0	0
$965$	5.74437	0.184918
$966$	0	0
$967$	28.2875	0.909664	0.454832	−	0.890577i	$-0.349699\pi$
$967$	28.2875	0.909664	0.454832	+	0.890577i	$0.349699\pi$
$968$	0	0
$969$	−0.895614	−0.0287713
$970$	0	0
$971$	−41.8695	−1.34366	−0.671828	−	0.740707i	$-0.734490\pi$
$971$	−41.8695	−1.34366	−0.671828	+	0.740707i	$0.734490\pi$
$972$	0	0
$973$	−0.710426	−0.0227752
$974$	0	0
$975$	2.38840	0.0764901
$976$	0	0
$977$	7.36966	0.235776	0.117888	−	0.993027i	$-0.462388\pi$
$977$	7.36966	0.235776	0.117888	+	0.993027i	$0.462388\pi$
$978$	0	0
$979$	4.61624	0.147536
$980$	0	0
$981$	−29.4417	−0.940001
$982$	0	0
$983$	−18.5640	−0.592098	−0.296049	−	0.955173i	$-0.595669\pi$
$983$	−18.5640	−0.592098	−0.296049	+	0.955173i	$0.595669\pi$
$984$	0	0
$985$	22.8975	0.729573
$986$	0	0
$987$	0.454881	0.0144790
$988$	0	0
$989$	0.819959	0.0260732
$990$	0	0
$991$	42.5301	1.35101	0.675506	−	0.737354i	$-0.263925\pi$
$991$	42.5301	1.35101	0.675506	+	0.737354i	$0.263925\pi$
$992$	0	0
$993$	3.65776	0.116076
$994$	0	0
$995$	−4.38925	−0.139149
$996$	0	0
$997$	52.5775	1.66515	0.832574	−	0.553914i	$-0.186866\pi$
$997$	52.5775	1.66515	0.832574	+	0.553914i	$0.186866\pi$
$998$	0	0
$999$	−19.0962	−0.604178

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6080.2.a.cl.1.2		5
4.3	odd	2		6080.2.a.ci.1.4		5
8.3	odd	2		3040.2.a.x.1.2	yes	5
8.5	even	2		3040.2.a.u.1.4	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3040.2.a.u.1.4	✓	5	8.5	even	2
3040.2.a.x.1.2	yes	5	8.3	odd	2
6080.2.a.ci.1.4		5	4.3	odd	2
6080.2.a.cl.1.2		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)

\(f(q)\)	\(=\)	\(q-0.531822 q^{3} +1.00000 q^{5} -0.0955795 q^{7} -2.71717 q^{9} -0.337570 q^{11} -4.49098 q^{13} -0.531822 q^{15} -1.68405 q^{17} -1.00000 q^{19} +0.0508313 q^{21} -2.51569 q^{23} +1.00000 q^{25} +3.04051 q^{27} +6.61268 q^{29} -6.79080 q^{31} +0.179527 q^{33} -0.0955795 q^{35} -6.28059 q^{37} +2.38840 q^{39} +1.06364 q^{41} -0.325938 q^{43} -2.71717 q^{45} +8.94884 q^{47} -6.99086 q^{49} +0.895614 q^{51} +14.4587 q^{53} -0.337570 q^{55} +0.531822 q^{57} +1.20007 q^{59} +1.21006 q^{61} +0.259705 q^{63} -4.49098 q^{65} +0.111712 q^{67} +1.33790 q^{69} +14.9003 q^{71} +2.47485 q^{73} -0.531822 q^{75} +0.0322648 q^{77} -8.33842 q^{79} +6.53448 q^{81} +17.9308 q^{83} -1.68405 q^{85} -3.51677 q^{87} -13.6749 q^{89} +0.429246 q^{91} +3.61150 q^{93} -1.00000 q^{95} -10.7213 q^{97} +0.917233 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 4 q^{3} + 5 q^{5} + 4 q^{7} + 7 q^{9} + 2 q^{11} + 4 q^{13} + 4 q^{15} - 12 q^{17} - 5 q^{19} + 10 q^{21} + 8 q^{23} + 5 q^{25} + 16 q^{27} + 6 q^{29} + 10 q^{31} - 18 q^{33} + 4 q^{35} + 6 q^{37}+ \cdots - 14 q^{99}+O(q^{100}) \) 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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