LMFDB - Embedded newform 8464.2.a.by.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8464,2,Mod(1,8464)] 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("8464.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8464, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8464 = 2^{4} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8464.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,-4,0,0,0,0,0,10,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$67.5853802708$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.26849792.2
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 14x^{4} - 2x^{3} + 28x^{2} - 4x - 7 $ x^6 - 14x^4 - 2x^3 + 28x^2 - 4x - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 4232)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-1.66832$ of defining polynomial
Character	$\chi$	$=$	8464.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.68133 q^{3} +1.77357 q^{5} -4.15133 q^{7} -0.173127 q^{9} +5.56554 q^{11} +1.68133 q^{13} +2.98195 q^{15} +0.359354 q^{17} -6.88844 q^{19} -6.97976 q^{21} -1.85446 q^{25} -5.33508 q^{27} -10.0604 q^{29} +1.17313 q^{31} +9.35752 q^{33} -7.36266 q^{35} +6.52909 q^{37} +2.82687 q^{39} -0.697737 q^{41} -5.90169 q^{43} -0.307053 q^{45} -1.81047 q^{47} +10.2335 q^{49} +0.604193 q^{51} -0.450667 q^{53} +9.87086 q^{55} -11.5818 q^{57} -6.50820 q^{59} +12.7901 q^{61} +0.718708 q^{63} +2.98195 q^{65} -11.0398 q^{67} -7.68133 q^{71} -9.42306 q^{73} -3.11796 q^{75} -23.1044 q^{77} -11.5818 q^{79} -8.45065 q^{81} +6.50589 q^{83} +0.637339 q^{85} -16.9149 q^{87} +11.3759 q^{89} -6.97976 q^{91} +1.97241 q^{93} -12.2171 q^{95} -12.2091 q^{97} -0.963547 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{3} + 10 q^{9} - 4 q^{13} + 14 q^{25} - 40 q^{27} - 12 q^{29} - 4 q^{31} - 16 q^{35} + 28 q^{39} + 16 q^{41} - 28 q^{47} + 2 q^{49} + 28 q^{55} - 36 q^{59} - 32 q^{71} + 20 q^{73} - 80 q^{75}+ \cdots - 20 q^{95}+O(q^{100}) $ 6 * q - 4 * q^3 + 10 * q^9 - 4 * q^13 + 14 * q^25 - 40 * q^27 - 12 * q^29 - 4 * q^31 - 16 * q^35 + 28 * q^39 + 16 * q^41 - 28 * q^47 + 2 * q^49 + 28 * q^55 - 36 * q^59 - 32 * q^71 + 20 * q^73 - 80 * q^75 - 48 * q^77 + 62 * q^81 + 32 * q^85 - 28 * q^87 + 48 * q^93 - 20 * q^95

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.68133	0.970717	0.485358	−	0.874315i	$-0.338689\pi$
$3$	1.68133	0.970717	0.485358	+	0.874315i	$0.338689\pi$
$4$	0	0
$5$	1.77357	0.793164	0.396582	−	0.917999i	$-0.370196\pi$
$5$	1.77357	0.793164	0.396582	+	0.917999i	$0.370196\pi$
$6$	0	0
$7$	−4.15133	−1.56905	−0.784527	−	0.620094i	$-0.787094\pi$
$7$	−4.15133	−1.56905	−0.784527	+	0.620094i	$0.787094\pi$
$8$	0	0
$9$	−0.173127	−0.0577091
$10$	0	0
$11$	5.56554	1.67807	0.839037	−	0.544074i	$-0.183119\pi$
$11$	5.56554	1.67807	0.839037	+	0.544074i	$0.183119\pi$
$12$	0	0
$13$	1.68133	0.466317	0.233159	−	0.972439i	$-0.425094\pi$
$13$	1.68133	0.466317	0.233159	+	0.972439i	$0.425094\pi$
$14$	0	0
$15$	2.98195	0.769937
$16$	0	0
$17$	0.359354	0.0871562	0.0435781	−	0.999050i	$-0.486124\pi$
$17$	0.359354	0.0871562	0.0435781	+	0.999050i	$0.486124\pi$
$18$	0	0
$19$	−6.88844	−1.58032	−0.790159	−	0.612902i	$-0.790002\pi$
$19$	−6.88844	−1.58032	−0.790159	+	0.612902i	$0.790002\pi$
$20$	0	0
$21$	−6.97976	−1.52311
$22$	0	0
$23$	0	0
$23$	0	0
$24$	0	0
$25$	−1.85446	−0.370892
$26$	0	0
$27$	−5.33508	−1.02674
$28$	0	0
$29$	−10.0604	−1.86817	−0.934085	−	0.357052i	$-0.883782\pi$
$29$	−10.0604	−1.86817	−0.934085	+	0.357052i	$0.883782\pi$
$30$	0	0
$31$	1.17313	0.210700	0.105350	−	0.994435i	$-0.466404\pi$
$31$	1.17313	0.210700	0.105350	+	0.994435i	$0.466404\pi$
$32$	0	0
$33$	9.35752	1.62893
$34$	0	0
$35$	−7.36266	−1.24452
$36$	0	0
$37$	6.52909	1.07338	0.536688	−	0.843781i	$-0.319675\pi$
$37$	6.52909	1.07338	0.536688	+	0.843781i	$0.319675\pi$
$38$	0	0
$39$	2.82687	0.452662
$40$	0	0
$41$	−0.697737	−0.108968	−0.0544841	−	0.998515i	$-0.517351\pi$
$41$	−0.697737	−0.108968	−0.0544841	+	0.998515i	$0.517351\pi$
$42$	0	0
$43$	−5.90169	−0.900000	−0.450000	−	0.893029i	$-0.648576\pi$
$43$	−5.90169	−0.900000	−0.450000	+	0.893029i	$0.648576\pi$
$44$	0	0
$45$	−0.307053	−0.0457727
$46$	0	0
$47$	−1.81047	−0.264084	−0.132042	−	0.991244i	$-0.542153\pi$
$47$	−1.81047	−0.264084	−0.132042	+	0.991244i	$0.542153\pi$
$48$	0	0
$49$	10.2335	1.46193
$50$	0	0
$51$	0.604193	0.0846039
$52$	0	0
$53$	−0.450667	−0.0619038	−0.0309519	−	0.999521i	$-0.509854\pi$
$53$	−0.450667	−0.0619038	−0.0309519	+	0.999521i	$0.509854\pi$
$54$	0	0
$55$	9.87086	1.33099
$56$	0	0
$57$	−11.5818	−1.53404
$58$	0	0
$59$	−6.50820	−0.847296	−0.423648	−	0.905827i	$-0.639251\pi$
$59$	−6.50820	−0.847296	−0.423648	+	0.905827i	$0.639251\pi$
$60$	0	0
$61$	12.7901	1.63761	0.818805	−	0.574072i	$-0.194637\pi$
$61$	12.7901	1.63761	0.818805	+	0.574072i	$0.194637\pi$
$62$	0	0
$63$	0.718708	0.0905487
$64$	0	0
$65$	2.98195	0.369866
$66$	0	0
$67$	−11.0398	−1.34872	−0.674361	−	0.738401i	$-0.735581\pi$
$67$	−11.0398	−1.34872	−0.674361	+	0.738401i	$0.735581\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−7.68133	−0.911606	−0.455803	−	0.890081i	$-0.650648\pi$
$71$	−7.68133	−0.911606	−0.455803	+	0.890081i	$0.650648\pi$
$72$	0	0
$73$	−9.42306	−1.10289	−0.551443	−	0.834213i	$-0.685923\pi$
$73$	−9.42306	−1.10289	−0.551443	+	0.834213i	$0.685923\pi$
$74$	0	0
$75$	−3.11796	−0.360031
$76$	0	0
$77$	−23.1044	−2.63299
$78$	0	0
$79$	−11.5818	−1.30305	−0.651524	−	0.758628i	$-0.725870\pi$
$79$	−11.5818	−1.30305	−0.651524	+	0.758628i	$0.725870\pi$
$80$	0	0
$81$	−8.45065	−0.938961
$82$	0	0
$83$	6.50589	0.714114	0.357057	−	0.934083i	$-0.383780\pi$
$83$	6.50589	0.714114	0.357057	+	0.934083i	$0.383780\pi$
$84$	0	0
$85$	0.637339	0.0691291
$86$	0	0
$87$	−16.9149	−1.81346
$88$	0	0
$89$	11.3759	1.20585	0.602923	−	0.797800i	$-0.294003\pi$
$89$	11.3759	1.20585	0.602923	+	0.797800i	$0.294003\pi$
$90$	0	0
$91$	−6.97976	−0.731677
$92$	0	0
$93$	1.97241	0.204530
$94$	0	0
$95$	−12.2171	−1.25345
$96$	0	0
$97$	−12.2091	−1.23965	−0.619825	−	0.784740i	$-0.712797\pi$
$97$	−12.2091	−1.23965	−0.619825	+	0.784740i	$0.712797\pi$
$98$	0	0
$99$	−0.963547	−0.0968401
$100$	0	0
$101$	−0.129135	−0.0128494	−0.00642472	−	0.999979i	$-0.502045\pi$
$101$	−0.129135	−0.0128494	−0.00642472	+	0.999979i	$0.502045\pi$
$102$	0	0
$103$	11.1992	1.10349	0.551745	−	0.834013i	$-0.313962\pi$
$103$	11.1992	1.10349	0.551745	+	0.834013i	$0.313962\pi$
$104$	0	0
$105$	−12.3791	−1.20807
$106$	0	0
$107$	9.53425	0.921710	0.460855	−	0.887475i	$-0.347543\pi$
$107$	9.53425	0.921710	0.460855	+	0.887475i	$0.347543\pi$
$108$	0	0
$109$	−3.72976	−0.357246	−0.178623	−	0.983918i	$-0.557164\pi$
$109$	−3.72976	−0.357246	−0.178623	+	0.983918i	$0.557164\pi$
$110$	0	0
$111$	10.9776	1.04194
$112$	0	0
$113$	−19.2570	−1.81155	−0.905774	−	0.423762i	$-0.860709\pi$
$113$	−19.2570	−1.81155	−0.905774	+	0.423762i	$0.860709\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−0.291084	−0.0269107
$118$	0	0
$119$	−1.49180	−0.136753
$120$	0	0
$121$	19.9753	1.81593
$122$	0	0
$123$	−1.17313	−0.105777
$124$	0	0
$125$	−12.1568	−1.08734
$126$	0	0
$127$	−6.02759	−0.534862	−0.267431	−	0.963577i	$-0.586175\pi$
$127$	−6.02759	−0.534862	−0.267431	+	0.963577i	$0.586175\pi$
$128$	0	0
$129$	−9.92270	−0.873645
$130$	0	0
$131$	7.68133	0.671121	0.335560	−	0.942019i	$-0.391074\pi$
$131$	7.68133	0.671121	0.335560	+	0.942019i	$0.391074\pi$
$132$	0	0
$133$	28.5962	2.47960
$134$	0	0
$135$	−9.46212	−0.814369
$136$	0	0
$137$	14.3870	1.22916	0.614581	−	0.788854i	$-0.289325\pi$
$137$	14.3870	1.22916	0.614581	+	0.788854i	$0.289325\pi$
$138$	0	0
$139$	−16.3187	−1.38413	−0.692066	−	0.721834i	$-0.743299\pi$
$139$	−16.3187	−1.38413	−0.692066	+	0.721834i	$0.743299\pi$
$140$	0	0
$141$	−3.04399	−0.256350
$142$	0	0
$143$	9.35752	0.782515
$144$	0	0
$145$	−17.8428	−1.48176
$146$	0	0
$147$	17.2059	1.41912
$148$	0	0
$149$	8.08102	0.662023	0.331011	−	0.943627i	$-0.392610\pi$
$149$	8.08102	0.662023	0.331011	+	0.943627i	$0.392610\pi$
$150$	0	0
$151$	−2.28586	−0.186020	−0.0930102	−	0.995665i	$-0.529649\pi$
$151$	−2.28586	−0.186020	−0.0930102	+	0.995665i	$0.529649\pi$
$152$	0	0
$153$	−0.0622140	−0.00502970
$154$	0	0
$155$	2.08062	0.167119
$156$	0	0
$157$	2.10972	0.168374	0.0841870	−	0.996450i	$-0.473171\pi$
$157$	2.10972	0.168374	0.0841870	+	0.996450i	$0.473171\pi$
$158$	0	0
$159$	−0.757719	−0.0600911
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−17.0440	−1.33499	−0.667494	−	0.744615i	$-0.732633\pi$
$163$	−17.0440	−1.33499	−0.667494	+	0.744615i	$0.732633\pi$
$164$	0	0
$165$	16.5962	1.29201
$166$	0	0
$167$	15.9037	1.23066	0.615332	−	0.788268i	$-0.289022\pi$
$167$	15.9037	1.23066	0.615332	+	0.788268i	$0.289022\pi$
$168$	0	0
$169$	−10.1731	−0.782548
$170$	0	0
$171$	1.19258	0.0911987
$172$	0	0
$173$	21.7417	1.65299	0.826497	−	0.562942i	$-0.190330\pi$
$173$	21.7417	1.65299	0.826497	+	0.562942i	$0.190330\pi$
$174$	0	0
$175$	7.69846	0.581949
$176$	0	0
$177$	−10.9424	−0.822484
$178$	0	0
$179$	−13.2059	−0.987058	−0.493529	−	0.869729i	$-0.664293\pi$
$179$	−13.2059	−0.987058	−0.493529	+	0.869729i	$0.664293\pi$
$180$	0	0
$181$	6.71171	0.498878	0.249439	−	0.968391i	$-0.419754\pi$
$181$	6.71171	0.498878	0.249439	+	0.968391i	$0.419754\pi$
$182$	0	0
$183$	21.5044	1.58965
$184$	0	0
$185$	11.5798	0.851362
$186$	0	0
$187$	2.00000	0.146254
$188$	0	0
$189$	22.1477	1.61100
$190$	0	0
$191$	−4.37296	−0.316417	−0.158208	−	0.987406i	$-0.550572\pi$
$191$	−4.37296	−0.316417	−0.158208	+	0.987406i	$0.550572\pi$
$192$	0	0
$193$	−9.55220	−0.687582	−0.343791	−	0.939046i	$-0.611711\pi$
$193$	−9.55220	−0.687582	−0.343791	+	0.939046i	$0.611711\pi$
$194$	0	0
$195$	5.01365	0.359035
$196$	0	0
$197$	21.3267	1.51947	0.759734	−	0.650234i	$-0.225329\pi$
$197$	21.3267	1.51947	0.759734	+	0.650234i	$0.225329\pi$
$198$	0	0
$199$	−10.7949	−0.765232	−0.382616	−	0.923907i	$-0.624977\pi$
$199$	−10.7949	−0.765232	−0.382616	+	0.923907i	$0.624977\pi$
$200$	0	0
$201$	−18.5615	−1.30923
$202$	0	0
$203$	41.7640	2.93126
$204$	0	0
$205$	−1.23748	−0.0864297
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−38.3379	−2.65189
$210$	0	0
$211$	−6.82164	−0.469621	−0.234811	−	0.972041i	$-0.575447\pi$
$211$	−6.82164	−0.469621	−0.234811	+	0.972041i	$0.575447\pi$
$212$	0	0
$213$	−12.9149	−0.884911
$214$	0	0
$215$	−10.4671	−0.713847
$216$	0	0
$217$	−4.87004	−0.330600
$218$	0	0
$219$	−15.8433	−1.07059
$220$	0	0
$221$	0.604193	0.0406424
$222$	0	0
$223$	−22.8461	−1.52989	−0.764945	−	0.644096i	$-0.777234\pi$
$223$	−22.8461	−1.52989	−0.764945	+	0.644096i	$0.777234\pi$
$224$	0	0
$225$	0.321057	0.0214038
$226$	0	0
$227$	−5.06847	−0.336406	−0.168203	−	0.985752i	$-0.553796\pi$
$227$	−5.06847	−0.336406	−0.168203	+	0.985752i	$0.553796\pi$
$228$	0	0
$229$	22.2158	1.46806	0.734029	−	0.679118i	$-0.237637\pi$
$229$	22.2158	1.46806	0.734029	+	0.679118i	$0.237637\pi$
$230$	0	0
$231$	−38.8461	−2.55589
$232$	0	0
$233$	−13.4231	−0.879374	−0.439687	−	0.898151i	$-0.644911\pi$
$233$	−13.4231	−0.879374	−0.439687	+	0.898151i	$0.644911\pi$
$234$	0	0
$235$	−3.21098	−0.209461
$236$	0	0
$237$	−19.4728	−1.26489
$238$	0	0
$239$	−17.6730	−1.14317	−0.571585	−	0.820543i	$-0.693671\pi$
$239$	−17.6730	−1.14317	−0.571585	+	0.820543i	$0.693671\pi$
$240$	0	0
$241$	−21.1451	−1.36208	−0.681038	−	0.732248i	$-0.738471\pi$
$241$	−21.1451	−1.36208	−0.681038	+	0.732248i	$0.738471\pi$
$242$	0	0
$243$	1.79690	0.115271
$244$	0	0
$245$	18.1499	1.15955
$246$	0	0
$247$	−11.5818	−0.736929
$248$	0	0
$249$	10.9385	0.693202
$250$	0	0
$251$	28.5854	1.80430	0.902148	−	0.431426i	$-0.141990\pi$
$251$	28.5854	1.80430	0.902148	+	0.431426i	$0.141990\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	1.07158	0.0671048
$256$	0	0
$257$	9.71414	0.605952	0.302976	−	0.952998i	$-0.402020\pi$
$257$	9.71414	0.605952	0.302976	+	0.952998i	$0.402020\pi$
$258$	0	0
$259$	−27.1044	−1.68418
$260$	0	0
$261$	1.74173	0.107810
$262$	0	0
$263$	−19.1657	−1.18181	−0.590904	−	0.806742i	$-0.701229\pi$
$263$	−19.1657	−1.18181	−0.590904	+	0.806742i	$0.701229\pi$
$264$	0	0
$265$	−0.799288	−0.0490998
$266$	0	0
$267$	19.1267	1.17053
$268$	0	0
$269$	9.04399	0.551422	0.275711	−	0.961241i	$-0.411087\pi$
$269$	9.04399	0.551422	0.275711	+	0.961241i	$0.411087\pi$
$270$	0	0
$271$	24.4999	1.48826	0.744130	−	0.668034i	$-0.232864\pi$
$271$	24.4999	1.48826	0.744130	+	0.668034i	$0.232864\pi$
$272$	0	0
$273$	−11.7353	−0.710251
$274$	0	0
$275$	−10.3211	−0.622384
$276$	0	0
$277$	−24.6894	−1.48344	−0.741721	−	0.670708i	$-0.765990\pi$
$277$	−24.6894	−1.48344	−0.741721	+	0.670708i	$0.765990\pi$
$278$	0	0
$279$	−0.203100	−0.0121593
$280$	0	0
$281$	10.2821	0.613376	0.306688	−	0.951810i	$-0.400779\pi$
$281$	10.2821	0.613376	0.306688	+	0.951810i	$0.400779\pi$
$282$	0	0
$283$	6.46687	0.384416	0.192208	−	0.981354i	$-0.438435\pi$
$283$	6.46687	0.384416	0.192208	+	0.981354i	$0.438435\pi$
$284$	0	0
$285$	−20.5410	−1.21674
$286$	0	0
$287$	2.89654	0.170977
$288$	0	0
$289$	−16.8709	−0.992404
$290$	0	0
$291$	−20.5276	−1.20335
$292$	0	0
$293$	8.83874	0.516365	0.258182	−	0.966096i	$-0.416876\pi$
$293$	8.83874	0.516365	0.258182	+	0.966096i	$0.416876\pi$
$294$	0	0
$295$	−11.5427	−0.672044
$296$	0	0
$297$	−29.6926	−1.72294
$298$	0	0
$299$	0	0
$300$	0	0
$301$	24.4999	1.41215
$302$	0	0
$303$	−0.217119	−0.0124732
$304$	0	0
$305$	22.6842	1.29889
$306$	0	0
$307$	−21.5798	−1.23162	−0.615812	−	0.787893i	$-0.711172\pi$
$307$	−21.5798	−1.23162	−0.615812	+	0.787893i	$0.711172\pi$
$308$	0	0
$309$	18.8295	1.07118
$310$	0	0
$311$	−5.97241	−0.338665	−0.169332	−	0.985559i	$-0.554161\pi$
$311$	−5.97241	−0.338665	−0.169332	+	0.985559i	$0.554161\pi$
$312$	0	0
$313$	14.2043	0.802877	0.401439	−	0.915886i	$-0.368510\pi$
$313$	14.2043	0.802877	0.401439	+	0.915886i	$0.368510\pi$
$314$	0	0
$315$	1.27468	0.0718199
$316$	0	0
$317$	8.50820	0.477868	0.238934	−	0.971036i	$-0.423202\pi$
$317$	8.50820	0.477868	0.238934	+	0.971036i	$0.423202\pi$
$318$	0	0
$319$	−55.9916	−3.13493
$320$	0	0
$321$	16.0302	0.894720
$322$	0	0
$323$	−2.47539	−0.137734
$324$	0	0
$325$	−3.11796	−0.172953
$326$	0	0
$327$	−6.27096	−0.346785
$328$	0	0
$329$	7.51584	0.414362
$330$	0	0
$331$	−19.4231	−1.06759	−0.533794	−	0.845615i	$-0.679234\pi$
$331$	−19.4231	−1.06759	−0.533794	+	0.845615i	$0.679234\pi$
$332$	0	0
$333$	−1.13036	−0.0619435
$334$	0	0
$335$	−19.5798	−1.06976
$336$	0	0
$337$	−0.0622140	−0.00338901	−0.00169451	−	0.999999i	$-0.500539\pi$
$337$	−0.0622140	−0.00338901	−0.00169451	+	0.999999i	$0.500539\pi$
$338$	0	0
$339$	−32.3774	−1.75850
$340$	0	0
$341$	6.52909	0.353570
$342$	0	0
$343$	−13.4234	−0.724797
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−19.7089	−1.05803	−0.529015	−	0.848612i	$-0.677438\pi$
$347$	−19.7089	−1.05803	−0.529015	+	0.848612i	$0.677438\pi$
$348$	0	0
$349$	0.826873	0.0442615	0.0221307	−	0.999755i	$-0.492955\pi$
$349$	0.826873	0.0442615	0.0221307	+	0.999755i	$0.492955\pi$
$350$	0	0
$351$	−8.97003	−0.478785
$352$	0	0
$353$	14.1484	0.753042	0.376521	−	0.926408i	$-0.377120\pi$
$353$	14.1484	0.753042	0.376521	+	0.926408i	$0.377120\pi$
$354$	0	0
$355$	−13.6234	−0.723053
$356$	0	0
$357$	−2.50820	−0.132748
$358$	0	0
$359$	−10.0081	−0.528208	−0.264104	−	0.964494i	$-0.585076\pi$
$359$	−10.0081	−0.528208	−0.264104	+	0.964494i	$0.585076\pi$
$360$	0	0
$361$	28.4506	1.49740
$362$	0	0
$363$	33.5850	1.76276
$364$	0	0
$365$	−16.7124	−0.874769
$366$	0	0
$367$	−15.6359	−0.816186	−0.408093	−	0.912940i	$-0.633806\pi$
$367$	−15.6359	−0.816186	−0.408093	+	0.912940i	$0.633806\pi$
$368$	0	0
$369$	0.120797	0.00628846
$370$	0	0
$371$	1.87086	0.0971305
$372$	0	0
$373$	2.08062	0.107730	0.0538652	−	0.998548i	$-0.482846\pi$
$373$	2.08062	0.107730	0.0538652	+	0.998548i	$0.482846\pi$
$374$	0	0
$375$	−20.4397	−1.05550
$376$	0	0
$377$	−16.9149	−0.871159
$378$	0	0
$379$	−3.79197	−0.194781	−0.0973903	−	0.995246i	$-0.531050\pi$
$379$	−3.79197	−0.194781	−0.0973903	+	0.995246i	$0.531050\pi$
$380$	0	0
$381$	−10.1344	−0.519199
$382$	0	0
$383$	−6.19294	−0.316444	−0.158222	−	0.987404i	$-0.550576\pi$
$383$	−6.19294	−0.316444	−0.158222	+	0.987404i	$0.550576\pi$
$384$	0	0
$385$	−40.9772	−2.08839
$386$	0	0
$387$	1.02174	0.0519382
$388$	0	0
$389$	8.34906	0.423314	0.211657	−	0.977344i	$-0.432114\pi$
$389$	8.34906	0.423314	0.211657	+	0.977344i	$0.432114\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	12.9149	0.651468
$394$	0	0
$395$	−20.5410	−1.03353
$396$	0	0
$397$	6.22235	0.312291	0.156145	−	0.987734i	$-0.450093\pi$
$397$	6.22235	0.312291	0.156145	+	0.987734i	$0.450093\pi$
$398$	0	0
$399$	48.0796	2.40699
$400$	0	0
$401$	12.8424	0.641321	0.320660	−	0.947194i	$-0.396095\pi$
$401$	12.8424	0.641321	0.320660	+	0.947194i	$0.396095\pi$
$402$	0	0
$403$	1.97241	0.0982530
$404$	0	0
$405$	−14.9878	−0.744749
$406$	0	0
$407$	36.3379	1.80120
$408$	0	0
$409$	−3.74696	−0.185275	−0.0926376	−	0.995700i	$-0.529530\pi$
$409$	−3.74696	−0.185275	−0.0926376	+	0.995700i	$0.529530\pi$
$410$	0	0
$411$	24.1893	1.19317
$412$	0	0
$413$	27.0177	1.32945
$414$	0	0
$415$	11.5386	0.566409
$416$	0	0
$417$	−27.4371	−1.34360
$418$	0	0
$419$	−3.74557	−0.182983	−0.0914915	−	0.995806i	$-0.529163\pi$
$419$	−3.74557	−0.182983	−0.0914915	+	0.995806i	$0.529163\pi$
$420$	0	0
$421$	−32.2530	−1.57191	−0.785957	−	0.618281i	$-0.787829\pi$
$421$	−32.2530	−1.57191	−0.785957	+	0.618281i	$0.787829\pi$
$422$	0	0
$423$	0.313441	0.0152400
$424$	0	0
$425$	−0.666407	−0.0323255
$426$	0	0
$427$	−53.0961	−2.56950
$428$	0	0
$429$	15.7331	0.759600
$430$	0	0
$431$	−18.5906	−0.895478	−0.447739	−	0.894164i	$-0.647771\pi$
$431$	−18.5906	−0.895478	−0.447739	+	0.894164i	$0.647771\pi$
$432$	0	0
$433$	−17.5398	−0.842907	−0.421454	−	0.906850i	$-0.638480\pi$
$433$	−17.5398	−0.842907	−0.421454	+	0.906850i	$0.638480\pi$
$434$	0	0
$435$	−29.9996	−1.43837
$436$	0	0
$437$	0	0
$438$	0	0
$439$	13.8761	0.662270	0.331135	−	0.943583i	$-0.392568\pi$
$439$	13.8761	0.662270	0.331135	+	0.943583i	$0.392568\pi$
$440$	0	0
$441$	−1.77170	−0.0843668
$442$	0	0
$443$	7.29392	0.346545	0.173272	−	0.984874i	$-0.444566\pi$
$443$	7.29392	0.346545	0.173272	+	0.984874i	$0.444566\pi$
$444$	0	0
$445$	20.1760	0.956433
$446$	0	0
$447$	13.5869	0.642637
$448$	0	0
$449$	18.4671	0.871514	0.435757	−	0.900064i	$-0.356481\pi$
$449$	18.4671	0.871514	0.435757	+	0.900064i	$0.356481\pi$
$450$	0	0
$451$	−3.88329	−0.182857
$452$	0	0
$453$	−3.84328	−0.180573
$454$	0	0
$455$	−12.3791	−0.580340
$456$	0	0
$457$	−22.6215	−1.05819	−0.529095	−	0.848563i	$-0.677469\pi$
$457$	−22.6215	−1.05819	−0.529095	+	0.848563i	$0.677469\pi$
$458$	0	0
$459$	−1.91718	−0.0894864
$460$	0	0
$461$	−20.8820	−0.972574	−0.486287	−	0.873799i	$-0.661649\pi$
$461$	−20.8820	−0.972574	−0.486287	+	0.873799i	$0.661649\pi$
$462$	0	0
$463$	−36.8789	−1.71391	−0.856955	−	0.515392i	$-0.827646\pi$
$463$	−36.8789	−1.71391	−0.856955	+	0.515392i	$0.827646\pi$
$464$	0	0
$465$	3.49821	0.162226
$466$	0	0
$467$	21.1550	0.978937	0.489468	−	0.872021i	$-0.337191\pi$
$467$	21.1550	0.978937	0.489468	+	0.872021i	$0.337191\pi$
$468$	0	0
$469$	45.8297	2.11622
$470$	0	0
$471$	3.54714	0.163443
$472$	0	0
$473$	−32.8461	−1.51027
$474$	0	0
$475$	12.7743	0.586126
$476$	0	0
$477$	0.0780227	0.00357241
$478$	0	0
$479$	7.74487	0.353872	0.176936	−	0.984222i	$-0.443381\pi$
$479$	7.74487	0.353872	0.176936	+	0.984222i	$0.443381\pi$
$480$	0	0
$481$	10.9776	0.500534
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−21.6537	−0.983246
$486$	0	0
$487$	−0.859686	−0.0389561	−0.0194781	−	0.999810i	$-0.506200\pi$
$487$	−0.859686	−0.0389561	−0.0194781	+	0.999810i	$0.506200\pi$
$488$	0	0
$489$	−28.6566	−1.29590
$490$	0	0
$491$	−19.0112	−0.857962	−0.428981	−	0.903313i	$-0.641127\pi$
$491$	−19.0112	−0.857962	−0.428981	+	0.903313i	$0.641127\pi$
$492$	0	0
$493$	−3.61525	−0.162822
$494$	0	0
$495$	−1.70892	−0.0768101
$496$	0	0
$497$	31.8877	1.43036
$498$	0	0
$499$	14.6566	0.656119	0.328059	−	0.944657i	$-0.393605\pi$
$499$	14.6566	0.656119	0.328059	+	0.944657i	$0.393605\pi$
$500$	0	0
$501$	26.7393	1.19463
$502$	0	0
$503$	15.6650	0.698466	0.349233	−	0.937036i	$-0.386442\pi$
$503$	15.6650	0.698466	0.349233	+	0.937036i	$0.386442\pi$
$504$	0	0
$505$	−0.229030	−0.0101917
$506$	0	0
$507$	−17.1044	−0.759633
$508$	0	0
$509$	40.4863	1.79452	0.897262	−	0.441499i	$-0.145553\pi$
$509$	40.4863	1.79452	0.897262	+	0.441499i	$0.145553\pi$
$510$	0	0
$511$	39.1182	1.73049
$512$	0	0
$513$	36.7504	1.62257
$514$	0	0
$515$	19.8625	0.875248
$516$	0	0
$517$	−10.0762	−0.443152
$518$	0	0
$519$	36.5550	1.60459
$520$	0	0
$521$	−23.2257	−1.01754	−0.508769	−	0.860903i	$-0.669899\pi$
$521$	−23.2257	−1.01754	−0.508769	+	0.860903i	$0.669899\pi$
$522$	0	0
$523$	21.5376	0.941772	0.470886	−	0.882194i	$-0.343934\pi$
$523$	21.5376	0.941772	0.470886	+	0.882194i	$0.343934\pi$
$524$	0	0
$525$	12.9437	0.564908
$526$	0	0
$527$	0.421568	0.0183638
$528$	0	0
$529$	0	0
$530$	0	0
$531$	1.12675	0.0488967
$532$	0	0
$533$	−1.17313	−0.0508138
$534$	0	0
$535$	16.9096	0.731067
$536$	0	0
$537$	−22.2036	−0.958154
$538$	0	0
$539$	56.9551	2.45323
$540$	0	0
$541$	−42.7118	−1.83632	−0.918161	−	0.396208i	$-0.870326\pi$
$541$	−42.7118	−1.83632	−0.918161	+	0.396208i	$0.870326\pi$
$542$	0	0
$543$	11.2846	0.484269
$544$	0	0
$545$	−6.61498	−0.283355
$546$	0	0
$547$	21.0112	0.898373	0.449187	−	0.893438i	$-0.351714\pi$
$547$	21.0112	0.898373	0.449187	+	0.893438i	$0.351714\pi$
$548$	0	0
$549$	−2.21432	−0.0945049
$550$	0	0
$551$	69.3005	2.95230
$552$	0	0
$553$	48.0796	2.04455
$554$	0	0
$555$	19.4694	0.826432
$556$	0	0
$557$	14.4492	0.612232	0.306116	−	0.951994i	$-0.400970\pi$
$557$	14.4492	0.612232	0.306116	+	0.951994i	$0.400970\pi$
$558$	0	0
$559$	−9.92270	−0.419685
$560$	0	0
$561$	3.36266	0.141972
$562$	0	0
$563$	−9.14178	−0.385280	−0.192640	−	0.981270i	$-0.561705\pi$
$563$	−9.14178	−0.385280	−0.192640	+	0.981270i	$0.561705\pi$
$564$	0	0
$565$	−34.1536	−1.43685
$566$	0	0
$567$	35.0814	1.47328
$568$	0	0
$569$	−22.1244	−0.927505	−0.463753	−	0.885965i	$-0.653497\pi$
$569$	−22.1244	−0.927505	−0.463753	+	0.885965i	$0.653497\pi$
$570$	0	0
$571$	2.24004	0.0937429	0.0468714	−	0.998901i	$-0.485075\pi$
$571$	2.24004	0.0937429	0.0468714	+	0.998901i	$0.485075\pi$
$572$	0	0
$573$	−7.35240	−0.307151
$574$	0	0
$575$	0	0
$576$	0	0
$577$	3.49702	0.145583	0.0727915	−	0.997347i	$-0.476809\pi$
$577$	3.49702	0.145583	0.0727915	+	0.997347i	$0.476809\pi$
$578$	0	0
$579$	−16.0604	−0.667447
$580$	0	0
$581$	−27.0081	−1.12048
$582$	0	0
$583$	−2.50820	−0.103879
$584$	0	0
$585$	−0.516257	−0.0213446
$586$	0	0
$587$	15.0028	0.619233	0.309617	−	0.950861i	$-0.399799\pi$
$587$	15.0028	0.619233	0.309617	+	0.950861i	$0.399799\pi$
$588$	0	0
$589$	−8.08102	−0.332973
$590$	0	0
$591$	35.8573	1.47497
$592$	0	0
$593$	7.86253	0.322875	0.161438	−	0.986883i	$-0.448387\pi$
$593$	7.86253	0.322875	0.161438	+	0.986883i	$0.448387\pi$
$594$	0	0
$595$	−2.64580	−0.108467
$596$	0	0
$597$	−18.1499	−0.742824
$598$	0	0
$599$	21.2747	0.869260	0.434630	−	0.900609i	$-0.356879\pi$
$599$	21.2747	0.869260	0.434630	+	0.900609i	$0.356879\pi$
$600$	0	0
$601$	−1.08514	−0.0442639	−0.0221320	−	0.999755i	$-0.507045\pi$
$601$	−1.08514	−0.0442639	−0.0221320	+	0.999755i	$0.507045\pi$
$602$	0	0
$603$	1.91129	0.0778336
$604$	0	0
$605$	35.4275	1.44033
$606$	0	0
$607$	7.95885	0.323040	0.161520	−	0.986869i	$-0.448360\pi$
$607$	7.95885	0.323040	0.161520	+	0.986869i	$0.448360\pi$
$608$	0	0
$609$	70.2191	2.84542
$610$	0	0
$611$	−3.04399	−0.123147
$612$	0	0
$613$	−26.1845	−1.05758	−0.528790	−	0.848753i	$-0.677354\pi$
$613$	−26.1845	−1.05758	−0.528790	+	0.848753i	$0.677354\pi$
$614$	0	0
$615$	−2.08062	−0.0838987
$616$	0	0
$617$	−4.54969	−0.183164	−0.0915819	−	0.995798i	$-0.529192\pi$
$617$	−4.54969	−0.183164	−0.0915819	+	0.995798i	$0.529192\pi$
$618$	0	0
$619$	27.6160	1.10998	0.554990	−	0.831857i	$-0.312722\pi$
$619$	27.6160	1.10998	0.554990	+	0.831857i	$0.312722\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−47.2252	−1.89204
$624$	0	0
$625$	−12.2887	−0.491548
$626$	0	0
$627$	−64.4587	−2.57423
$628$	0	0
$629$	2.34625	0.0935513
$630$	0	0
$631$	27.2467	1.08467	0.542337	−	0.840161i	$-0.317539\pi$
$631$	27.2467	1.08467	0.542337	+	0.840161i	$0.317539\pi$
$632$	0	0
$633$	−11.4694	−0.455869
$634$	0	0
$635$	−10.6903	−0.424233
$636$	0	0
$637$	17.2059	0.681724
$638$	0	0
$639$	1.32985	0.0526080
$640$	0	0
$641$	36.1477	1.42775	0.713874	−	0.700275i	$-0.246939\pi$
$641$	36.1477	1.42775	0.713874	+	0.700275i	$0.246939\pi$
$642$	0	0
$643$	10.8107	0.426334	0.213167	−	0.977016i	$-0.431622\pi$
$643$	10.8107	0.426334	0.213167	+	0.977016i	$0.431622\pi$
$644$	0	0
$645$	−17.5986	−0.692943
$646$	0	0
$647$	6.94767	0.273141	0.136571	−	0.990630i	$-0.456392\pi$
$647$	6.94767	0.273141	0.136571	+	0.990630i	$0.456392\pi$
$648$	0	0
$649$	−36.2217	−1.42183
$650$	0	0
$651$	−8.18814	−0.320919
$652$	0	0
$653$	29.1236	1.13970	0.569848	−	0.821750i	$-0.307002\pi$
$653$	29.1236	1.13970	0.569848	+	0.821750i	$0.307002\pi$
$654$	0	0
$655$	13.6234	0.532309
$656$	0	0
$657$	1.63139	0.0636465
$658$	0	0
$659$	−26.3513	−1.02650	−0.513250	−	0.858239i	$-0.671559\pi$
$659$	−26.3513	−1.02650	−0.513250	+	0.858239i	$0.671559\pi$
$660$	0	0
$661$	10.9094	0.424328	0.212164	−	0.977234i	$-0.431949\pi$
$661$	10.9094	0.424328	0.212164	+	0.977234i	$0.431949\pi$
$662$	0	0
$663$	1.01585	0.0394523
$664$	0	0
$665$	50.7173	1.96673
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−38.4119	−1.48509
$670$	0	0
$671$	71.1840	2.74803
$672$	0	0
$673$	−48.7774	−1.88023	−0.940115	−	0.340857i	$-0.889283\pi$
$673$	−48.7774	−1.88023	−0.940115	+	0.340857i	$0.889283\pi$
$674$	0	0
$675$	9.89367	0.380808
$676$	0	0
$677$	−14.9463	−0.574431	−0.287216	−	0.957866i	$-0.592730\pi$
$677$	−14.9463	−0.574431	−0.287216	+	0.957866i	$0.592730\pi$
$678$	0	0
$679$	50.6842	1.94508
$680$	0	0
$681$	−8.52177	−0.326555
$682$	0	0
$683$	2.82687	0.108167	0.0540836	−	0.998536i	$-0.482776\pi$
$683$	2.82687	0.108167	0.0540836	+	0.998536i	$0.482776\pi$
$684$	0	0
$685$	25.5163	0.974927
$686$	0	0
$687$	37.3520	1.42507
$688$	0	0
$689$	−0.757719	−0.0288668
$690$	0	0
$691$	−1.53295	−0.0583161	−0.0291580	−	0.999575i	$-0.509283\pi$
$691$	−1.53295	−0.0583161	−0.0291580	+	0.999575i	$0.509283\pi$
$692$	0	0
$693$	4.00000	0.151947
$694$	0	0
$695$	−28.9423	−1.09784
$696$	0	0
$697$	−0.250735	−0.00949726
$698$	0	0
$699$	−22.5686	−0.853623
$700$	0	0
$701$	1.87078	0.0706582	0.0353291	−	0.999376i	$-0.488752\pi$
$701$	1.87078	0.0706582	0.0353291	+	0.999376i	$0.488752\pi$
$702$	0	0
$703$	−44.9753	−1.69627
$704$	0	0
$705$	−5.39873	−0.203328
$706$	0	0
$707$	0.536083	0.0201615
$708$	0	0
$709$	39.7905	1.49436	0.747182	−	0.664619i	$-0.231406\pi$
$709$	39.7905	1.49436	0.747182	+	0.664619i	$0.231406\pi$
$710$	0	0
$711$	2.00512	0.0751978
$712$	0	0
$713$	0	0
$714$	0	0
$715$	16.5962	0.620662
$716$	0	0
$717$	−29.7141	−1.10970
$718$	0	0
$719$	21.9672	0.819238	0.409619	−	0.912257i	$-0.365662\pi$
$719$	21.9672	0.819238	0.409619	+	0.912257i	$0.365662\pi$
$720$	0	0
$721$	−46.4915	−1.73143
$722$	0	0
$723$	−35.5519	−1.32219
$724$	0	0
$725$	18.6566	0.692888
$726$	0	0
$727$	−30.3185	−1.12445	−0.562225	−	0.826984i	$-0.690055\pi$
$727$	−30.3185	−1.12445	−0.562225	+	0.826984i	$0.690055\pi$
$728$	0	0
$729$	28.3731	1.05086
$730$	0	0
$731$	−2.12080	−0.0784405
$732$	0	0
$733$	−10.2979	−0.380360	−0.190180	−	0.981749i	$-0.560907\pi$
$733$	−10.2979	−0.380360	−0.190180	+	0.981749i	$0.560907\pi$
$734$	0	0
$735$	30.5159	1.12560
$736$	0	0
$737$	−61.4423	−2.26326
$738$	0	0
$739$	37.8984	1.39412	0.697059	−	0.717014i	$-0.254492\pi$
$739$	37.8984	1.39412	0.697059	+	0.717014i	$0.254492\pi$
$740$	0	0
$741$	−19.4728	−0.715349
$742$	0	0
$743$	−16.6270	−0.609986	−0.304993	−	0.952355i	$-0.598654\pi$
$743$	−16.6270	−0.609986	−0.304993	+	0.952355i	$0.598654\pi$
$744$	0	0
$745$	14.3322	0.525092
$746$	0	0
$747$	−1.12635	−0.0412109
$748$	0	0
$749$	−39.5798	−1.44621
$750$	0	0
$751$	32.7327	1.19443	0.597217	−	0.802079i	$-0.296273\pi$
$751$	32.7327	1.19443	0.597217	+	0.802079i	$0.296273\pi$
$752$	0	0
$753$	48.0616	1.75146
$754$	0	0
$755$	−4.05412	−0.147545
$756$	0	0
$757$	6.84793	0.248892	0.124446	−	0.992226i	$-0.460285\pi$
$757$	6.84793	0.248892	0.124446	+	0.992226i	$0.460285\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−45.2528	−1.64041	−0.820206	−	0.572068i	$-0.806141\pi$
$761$	−45.2528	−1.64041	−0.820206	+	0.572068i	$0.806141\pi$
$762$	0	0
$763$	15.4835	0.560539
$764$	0	0
$765$	−0.110341	−0.00398938
$766$	0	0
$767$	−10.9424	−0.395109
$768$	0	0
$769$	−12.0773	−0.435519	−0.217760	−	0.976002i	$-0.569875\pi$
$769$	−12.0773	−0.435519	−0.217760	+	0.976002i	$0.569875\pi$
$770$	0	0
$771$	16.3327	0.588207
$772$	0	0
$773$	−12.2032	−0.438920	−0.219460	−	0.975621i	$-0.570430\pi$
$773$	−12.2032	−0.438920	−0.219460	+	0.975621i	$0.570430\pi$
$774$	0	0
$775$	−2.17552	−0.0781468
$776$	0	0
$777$	−45.5714	−1.63487
$778$	0	0
$779$	4.80632	0.172204
$780$	0	0
$781$	−42.7508	−1.52974
$782$	0	0
$783$	53.6730	1.91812
$784$	0	0
$785$	3.74173	0.133548
$786$	0	0
$787$	9.03717	0.322140	0.161070	−	0.986943i	$-0.448505\pi$
$787$	9.03717	0.322140	0.161070	+	0.986943i	$0.448505\pi$
$788$	0	0
$789$	−32.2239	−1.14720
$790$	0	0
$791$	79.9422	2.84242
$792$	0	0
$793$	21.5044	0.763645
$794$	0	0
$795$	−1.34387	−0.0476620
$796$	0	0
$797$	−25.3122	−0.896605	−0.448303	−	0.893882i	$-0.647971\pi$
$797$	−25.3122	−0.896605	−0.448303	+	0.893882i	$0.647971\pi$
$798$	0	0
$799$	−0.650598	−0.0230165
$800$	0	0
$801$	−1.96948	−0.0695882
$802$	0	0
$803$	−52.4444	−1.85072
$804$	0	0
$805$	0	0
$806$	0	0
$807$	15.2059	0.535274
$808$	0	0
$809$	−19.3627	−0.680755	−0.340377	−	0.940289i	$-0.610555\pi$
$809$	−19.3627	−0.680755	−0.340377	+	0.940289i	$0.610555\pi$
$810$	0	0
$811$	5.77977	0.202955	0.101478	−	0.994838i	$-0.467643\pi$
$811$	5.77977	0.202955	0.101478	+	0.994838i	$0.467643\pi$
$812$	0	0
$813$	41.1924	1.44468
$814$	0	0
$815$	−30.2287	−1.05886
$816$	0	0
$817$	40.6535	1.42229
$818$	0	0
$819$	1.20839	0.0422244
$820$	0	0
$821$	15.7745	0.550535	0.275268	−	0.961368i	$-0.411234\pi$
$821$	15.7745	0.550535	0.275268	+	0.961368i	$0.411234\pi$
$822$	0	0
$823$	−0.0603987	−0.00210537	−0.00105268	−	0.999999i	$-0.500335\pi$
$823$	−0.0603987	−0.00210537	−0.00105268	+	0.999999i	$0.500335\pi$
$824$	0	0
$825$	−17.3531	−0.604158
$826$	0	0
$827$	2.81262	0.0978043	0.0489022	−	0.998804i	$-0.484428\pi$
$827$	2.81262	0.0978043	0.0489022	+	0.998804i	$0.484428\pi$
$828$	0	0
$829$	−10.1515	−0.352576	−0.176288	−	0.984339i	$-0.556409\pi$
$829$	−10.1515	−0.352576	−0.176288	+	0.984339i	$0.556409\pi$
$830$	0	0
$831$	−41.5110	−1.44000
$832$	0	0
$833$	3.67746	0.127416
$834$	0	0
$835$	28.2062	0.976117
$836$	0	0
$837$	−6.25872	−0.216333
$838$	0	0
$839$	−3.20359	−0.110600	−0.0553001	−	0.998470i	$-0.517612\pi$
$839$	−3.20359	−0.110600	−0.0553001	+	0.998470i	$0.517612\pi$
$840$	0	0
$841$	72.2116	2.49006
$842$	0	0
$843$	17.2875	0.595414
$844$	0	0
$845$	−18.0427	−0.620689
$846$	0	0
$847$	−82.9238	−2.84930
$848$	0	0
$849$	10.8730	0.373159
$850$	0	0
$851$	0	0
$852$	0	0
$853$	25.2007	0.862856	0.431428	−	0.902147i	$-0.358010\pi$
$853$	25.2007	0.862856	0.431428	+	0.902147i	$0.358010\pi$
$854$	0	0
$855$	2.11512	0.0723355
$856$	0	0
$857$	47.5767	1.62519	0.812594	−	0.582830i	$-0.198055\pi$
$857$	47.5767	1.62519	0.812594	+	0.582830i	$0.198055\pi$
$858$	0	0
$859$	−39.6730	−1.35363	−0.676813	−	0.736155i	$-0.736639\pi$
$859$	−39.6730	−1.35363	−0.676813	+	0.736155i	$0.736639\pi$
$860$	0	0
$861$	4.87004	0.165970
$862$	0	0
$863$	16.7941	0.571676	0.285838	−	0.958278i	$-0.407728\pi$
$863$	16.7941	0.571676	0.285838	+	0.958278i	$0.407728\pi$
$864$	0	0
$865$	38.5604	1.31109
$866$	0	0
$867$	−28.3655	−0.963343
$868$	0	0
$869$	−64.4587	−2.18661
$870$	0	0
$871$	−18.5615	−0.628933
$872$	0	0
$873$	2.11374	0.0715391
$874$	0	0
$875$	50.4671	1.70610
$876$	0	0
$877$	−29.9037	−1.00978	−0.504888	−	0.863185i	$-0.668466\pi$
$877$	−29.9037	−1.00978	−0.504888	+	0.863185i	$0.668466\pi$
$878$	0	0
$879$	14.8608	0.501244
$880$	0	0
$881$	3.19957	0.107796	0.0538982	−	0.998546i	$-0.482835\pi$
$881$	3.19957	0.107796	0.0538982	+	0.998546i	$0.482835\pi$
$882$	0	0
$883$	−22.6597	−0.762559	−0.381280	−	0.924460i	$-0.624517\pi$
$883$	−22.6597	−0.762559	−0.381280	+	0.924460i	$0.624517\pi$
$884$	0	0
$885$	−19.4072	−0.652365
$886$	0	0
$887$	−13.5110	−0.453656	−0.226828	−	0.973935i	$-0.572836\pi$
$887$	−13.5110	−0.453656	−0.226828	+	0.973935i	$0.572836\pi$
$888$	0	0
$889$	25.0225	0.839227
$890$	0	0
$891$	−47.0324	−1.57565
$892$	0	0
$893$	12.4713	0.417336
$894$	0	0
$895$	−23.4216	−0.782899
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−11.8021	−0.393623
$900$	0	0
$901$	−0.161949	−0.00539530
$902$	0	0
$903$	41.1924	1.37080
$904$	0	0
$905$	11.9037	0.395692
$906$	0	0
$907$	53.0644	1.76198	0.880988	−	0.473139i	$-0.156879\pi$
$907$	53.0644	1.76198	0.880988	+	0.473139i	$0.156879\pi$
$908$	0	0
$909$	0.0223568	0.000741530 0
$910$	0	0
$911$	−17.4094	−0.576800	−0.288400	−	0.957510i	$-0.593123\pi$
$911$	−17.4094	−0.576800	−0.288400	+	0.957510i	$0.593123\pi$
$912$	0	0
$913$	36.2088	1.19834
$914$	0	0
$915$	38.1396	1.26086
$916$	0	0
$917$	−31.8877	−1.05303
$918$	0	0
$919$	30.9400	1.02062	0.510308	−	0.859992i	$-0.329531\pi$
$919$	30.9400	1.02062	0.510308	+	0.859992i	$0.329531\pi$
$920$	0	0
$921$	−36.2827	−1.19556
$922$	0	0
$923$	−12.9149	−0.425098
$924$	0	0
$925$	−12.1079	−0.398106
$926$	0	0
$927$	−1.93889	−0.0636814
$928$	0	0
$929$	−33.2939	−1.09234	−0.546169	−	0.837675i	$-0.683914\pi$
$929$	−33.2939	−1.09234	−0.546169	+	0.837675i	$0.683914\pi$
$930$	0	0
$931$	−70.4931	−2.31032
$932$	0	0
$933$	−10.0416	−0.328747
$934$	0	0
$935$	3.54714	0.116004
$936$	0	0
$937$	−5.97720	−0.195267	−0.0976333	−	0.995222i	$-0.531127\pi$
$937$	−5.97720	−0.195267	−0.0976333	+	0.995222i	$0.531127\pi$
$938$	0	0
$939$	23.8822	0.779366
$940$	0	0
$941$	−31.4445	−1.02506	−0.512530	−	0.858669i	$-0.671292\pi$
$941$	−31.4445	−1.02506	−0.512530	+	0.858669i	$0.671292\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	39.2804	1.27779
$946$	0	0
$947$	−8.82687	−0.286835	−0.143417	−	0.989662i	$-0.545809\pi$
$947$	−8.82687	−0.286835	−0.143417	+	0.989662i	$0.545809\pi$
$948$	0	0
$949$	−15.8433	−0.514295
$950$	0	0
$951$	14.3051	0.463875
$952$	0	0
$953$	36.0623	1.16817	0.584086	−	0.811692i	$-0.301453\pi$
$953$	36.0623	1.16817	0.584086	+	0.811692i	$0.301453\pi$
$954$	0	0
$955$	−7.75575	−0.250970
$956$	0	0
$957$	−94.1403	−3.04313
$958$	0	0
$959$	−59.7251	−1.92862
$960$	0	0
$961$	−29.6238	−0.955606
$962$	0	0
$963$	−1.65064	−0.0531911
$964$	0	0
$965$	−16.9415	−0.545365
$966$	0	0
$967$	−45.5767	−1.46565	−0.732823	−	0.680419i	$-0.761798\pi$
$967$	−45.5767	−1.46565	−0.732823	+	0.680419i	$0.761798\pi$
$968$	0	0
$969$	−4.16195	−0.133701
$970$	0	0
$971$	−37.0998	−1.19059	−0.595295	−	0.803507i	$-0.702965\pi$
$971$	−37.0998	−1.19059	−0.595295	+	0.803507i	$0.702965\pi$
$972$	0	0
$973$	67.7442	2.17178
$974$	0	0
$975$	−5.24232	−0.167888
$976$	0	0
$977$	19.1716	0.613354	0.306677	−	0.951814i	$-0.400783\pi$
$977$	19.1716	0.613354	0.306677	+	0.951814i	$0.400783\pi$
$978$	0	0
$979$	63.3132	2.02350
$980$	0	0
$981$	0.645723	0.0206164
$982$	0	0
$983$	14.7636	0.470887	0.235443	−	0.971888i	$-0.424346\pi$
$983$	14.7636	0.470887	0.235443	+	0.971888i	$0.424346\pi$
$984$	0	0
$985$	37.8244	1.20519
$986$	0	0
$987$	12.6366	0.402228
$988$	0	0
$989$	0	0
$990$	0	0
$991$	37.5163	1.19174	0.595872	−	0.803080i	$-0.296807\pi$
$991$	37.5163	1.19174	0.595872	+	0.803080i	$0.296807\pi$
$992$	0	0
$993$	−32.6566	−1.03633
$994$	0	0
$995$	−19.1455	−0.606954
$996$	0	0
$997$	29.0492	0.919998	0.459999	−	0.887919i	$-0.347850\pi$
$997$	29.0492	0.919998	0.459999	+	0.887919i	$0.347850\pi$
$998$	0	0
$999$	−34.8332	−1.10207

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8464.2.a.by.1.6		6
4.3	odd	2		4232.2.a.v.1.2	yes	6
23.22	odd	2	inner	8464.2.a.by.1.5		6
92.91	even	2		4232.2.a.v.1.1	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4232.2.a.v.1.1	✓	6	92.91	even	2
4232.2.a.v.1.2	yes	6	4.3	odd	2
8464.2.a.by.1.5		6	23.22	odd	2	inner
8464.2.a.by.1.6		6	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:	\( N \)	\(=\)	\( 8464 = 2^{4} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8464.a (trivial)

\(f(q)\)	\(=\)	\(q+1.68133 q^{3} +1.77357 q^{5} -4.15133 q^{7} -0.173127 q^{9} +5.56554 q^{11} +1.68133 q^{13} +2.98195 q^{15} +0.359354 q^{17} -6.88844 q^{19} -6.97976 q^{21} -1.85446 q^{25} -5.33508 q^{27} -10.0604 q^{29} +1.17313 q^{31} +9.35752 q^{33} -7.36266 q^{35} +6.52909 q^{37} +2.82687 q^{39} -0.697737 q^{41} -5.90169 q^{43} -0.307053 q^{45} -1.81047 q^{47} +10.2335 q^{49} +0.604193 q^{51} -0.450667 q^{53} +9.87086 q^{55} -11.5818 q^{57} -6.50820 q^{59} +12.7901 q^{61} +0.718708 q^{63} +2.98195 q^{65} -11.0398 q^{67} -7.68133 q^{71} -9.42306 q^{73} -3.11796 q^{75} -23.1044 q^{77} -11.5818 q^{79} -8.45065 q^{81} +6.50589 q^{83} +0.637339 q^{85} -16.9149 q^{87} +11.3759 q^{89} -6.97976 q^{91} +1.97241 q^{93} -12.2171 q^{95} -12.2091 q^{97} -0.963547 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{3} + 10 q^{9} - 4 q^{13} + 14 q^{25} - 40 q^{27} - 12 q^{29} - 4 q^{31} - 16 q^{35} + 28 q^{39} + 16 q^{41} - 28 q^{47} + 2 q^{49} + 28 q^{55} - 36 q^{59} - 32 q^{71} + 20 q^{73} - 80 q^{75}+ \cdots - 20 q^{95}+O(q^{100}) \) 6 * q - 4 * q^3 + 10 * q^9 - 4 * q^13 + 14 * q^25 - 40 * q^27 - 12 * q^29 - 4 * q^31 - 16 * q^35 + 28 * q^39 + 16 * q^41 - 28 * q^47 + 2 * q^49 + 28 * q^55 - 36 * q^59 - 32 * q^71 + 20 * q^73 - 80 * q^75 - 48 * q^77 + 62 * q^81 + 32 * q^85 - 28 * q^87 + 48 * q^93 - 20 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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