LMFDB - Embedded newform 7203.2.a.g.1.14

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7203,2,Mod(1,7203)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7203, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7203.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 7203 = 3 \cdot 7^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7203.a (trivial)

Newform invariants

comment:select newform

sage:traces = [18,3,-18,27] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	yes
Analytic conductor:	$57.5162445759$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 3 x^{17} - 27 x^{16} + 85 x^{15} + 287 x^{14} - 973 x^{13} - 1504 x^{12} + 5775 x^{11} + \cdots + 351 $ x^18 - 3x^17 - 27x^16 + 85x^15 + 287x^14 - 973x^13 - 1504x^12 + 5775x^11 + 3877x^10 - 18987x^9 - 3525x^8 + 34209x^7 - 3492x^6 - 30947x^5 + 8197x^4 + 11327x^3 - 3405x^2 - 1350*x + 351
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 147)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Root			$1.98596$ of defining polynomial
Character	$\chi$	$=$	7203.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.98596 q^{2} -1.00000 q^{3} +1.94405 q^{4} -3.52654 q^{5} -1.98596 q^{6} -0.111122 q^{8} +1.00000 q^{9} -7.00357 q^{10} -4.16832 q^{11} -1.94405 q^{12} -5.71756 q^{13} +3.52654 q^{15} -4.10878 q^{16} -3.10640 q^{17} +1.98596 q^{18} +2.60172 q^{19} -6.85576 q^{20} -8.27812 q^{22} -0.118359 q^{23} +0.111122 q^{24} +7.43648 q^{25} -11.3549 q^{26} -1.00000 q^{27} +5.45091 q^{29} +7.00357 q^{30} -8.12903 q^{31} -7.93763 q^{32} +4.16832 q^{33} -6.16919 q^{34} +1.94405 q^{36} +2.42275 q^{37} +5.16691 q^{38} +5.71756 q^{39} +0.391876 q^{40} -4.05753 q^{41} -10.3247 q^{43} -8.10340 q^{44} -3.52654 q^{45} -0.235057 q^{46} -4.02730 q^{47} +4.10878 q^{48} +14.7686 q^{50} +3.10640 q^{51} -11.1152 q^{52} +4.86574 q^{53} -1.98596 q^{54} +14.6997 q^{55} -2.60172 q^{57} +10.8253 q^{58} +9.28785 q^{59} +6.85576 q^{60} -6.09812 q^{61} -16.1439 q^{62} -7.54628 q^{64} +20.1632 q^{65} +8.27812 q^{66} +2.57419 q^{67} -6.03898 q^{68} +0.118359 q^{69} +2.06205 q^{71} -0.111122 q^{72} +9.63811 q^{73} +4.81149 q^{74} -7.43648 q^{75} +5.05786 q^{76} +11.3549 q^{78} +8.80338 q^{79} +14.4898 q^{80} +1.00000 q^{81} -8.05809 q^{82} +2.26826 q^{83} +10.9548 q^{85} -20.5045 q^{86} -5.45091 q^{87} +0.463192 q^{88} +9.72942 q^{89} -7.00357 q^{90} -0.230096 q^{92} +8.12903 q^{93} -7.99807 q^{94} -9.17505 q^{95} +7.93763 q^{96} -10.8774 q^{97} -4.16832 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 3 q^{2} - 18 q^{3} + 27 q^{4} + 2 q^{5} - 3 q^{6} + 3 q^{8} + 18 q^{9} + 2 q^{10} - 27 q^{12} - q^{13} - 2 q^{15} + 37 q^{16} + 5 q^{17} + 3 q^{18} - 3 q^{19} + 19 q^{20} + 4 q^{23} - 3 q^{24} + 34 q^{25}+ \cdots - 86 q^{97}+O(q^{100}) $ 18 * q + 3 * q^2 - 18 * q^3 + 27 * q^4 + 2 * q^5 - 3 * q^6 + 3 * q^8 + 18 * q^9 + 2 * q^10 - 27 * q^12 - q^13 - 2 * q^15 + 37 * q^16 + 5 * q^17 + 3 * q^18 - 3 * q^19 + 19 * q^20 + 4 * q^23 - 3 * q^24 + 34 * q^25 - 3 * q^26 - 18 * q^27 + 10 * q^29 - 2 * q^30 - 8 * q^31 - q^32 - 3 * q^34 + 27 * q^36 + 9 * q^37 - 5 * q^38 + q^39 + 14 * q^40 - q^41 + 4 * q^43 - 13 * q^44 + 2 * q^45 - 2 * q^46 + 21 * q^47 - 37 * q^48 + 118 * q^50 - 5 * q^51 - 108 * q^52 + 89 * q^53 - 3 * q^54 - 2 * q^55 + 3 * q^57 + 103 * q^58 - 17 * q^59 - 19 * q^60 - 77 * q^61 + 37 * q^62 + 35 * q^64 + 103 * q^65 - 11 * q^67 + 55 * q^68 - 4 * q^69 - 3 * q^71 + 3 * q^72 + 9 * q^73 + 110 * q^74 - 34 * q^75 - 121 * q^76 + 3 * q^78 - 7 * q^79 + 56 * q^80 + 18 * q^81 - q^82 + q^83 + 104 * q^85 - 20 * q^86 - 10 * q^87 - 17 * q^88 + 15 * q^89 + 2 * q^90 - 7 * q^92 + 8 * q^93 + 6 * q^94 - 19 * q^95 + q^96 - 86 * q^97

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 $

$2$	1.98596	1.40429	0.702144	−	0.712035i	$-0.252226\pi$
$2$	1.98596	1.40429	0.702144	+	0.712035i	$0.252226\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.94405	0.972023
$5$	−3.52654	−1.57712	−0.788558	−	0.614960i	$-0.789172\pi$
$5$	−3.52654	−1.57712	−0.788558	+	0.614960i	$0.789172\pi$
$6$	−1.98596	−0.810766
$7$	0	0
$7$	0	0
$8$	−0.111122	−0.0392876
$9$	1.00000	0.333333
$10$	−7.00357	−2.21472
$11$	−4.16832	−1.25679	−0.628397	−	0.777893i	$-0.716289\pi$
$11$	−4.16832	−1.25679	−0.628397	+	0.777893i	$0.716289\pi$
$12$	−1.94405	−0.561198
$13$	−5.71756	−1.58577	−0.792884	−	0.609373i	$-0.791421\pi$
$13$	−5.71756	−1.58577	−0.792884	+	0.609373i	$0.791421\pi$
$14$	0	0
$15$	3.52654	0.910548
$16$	−4.10878	−1.02719
$17$	−3.10640	−0.753412	−0.376706	−	0.926333i	$-0.622943\pi$
$17$	−3.10640	−0.753412	−0.376706	+	0.926333i	$0.622943\pi$
$18$	1.98596	0.468096
$19$	2.60172	0.596874	0.298437	−	0.954429i	$-0.403535\pi$
$19$	2.60172	0.596874	0.298437	+	0.954429i	$0.403535\pi$
$20$	−6.85576	−1.53299
$21$	0	0
$22$	−8.27812	−1.76490
$23$	−0.118359	−0.0246796	−0.0123398	−	0.999924i	$-0.503928\pi$
$23$	−0.118359	−0.0246796	−0.0123398	+	0.999924i	$0.503928\pi$
$24$	0.111122	0.0226827
$25$	7.43648	1.48730
$26$	−11.3549	−2.22687
$27$	−1.00000	−0.192450
$28$	0	0
$29$	5.45091	1.01221	0.506104	−	0.862472i	$-0.331085\pi$
$29$	5.45091	1.01221	0.506104	+	0.862472i	$0.331085\pi$
$30$	7.00357	1.27867
$31$	−8.12903	−1.46002	−0.730008	−	0.683438i	$-0.760484\pi$
$31$	−8.12903	−1.46002	−0.730008	+	0.683438i	$0.760484\pi$
$32$	−7.93763	−1.40319
$33$	4.16832	0.725611
$34$	−6.16919	−1.05801
$35$	0	0
$36$	1.94405	0.324008
$37$	2.42275	0.398298	0.199149	−	0.979969i	$-0.436182\pi$
$37$	2.42275	0.398298	0.199149	+	0.979969i	$0.436182\pi$
$38$	5.16691	0.838183
$39$	5.71756	0.915543
$40$	0.391876	0.0619610
$41$	−4.05753	−0.633679	−0.316840	−	0.948479i	$-0.602622\pi$
$41$	−4.05753	−0.633679	−0.316840	+	0.948479i	$0.602622\pi$
$42$	0	0
$43$	−10.3247	−1.57450	−0.787251	−	0.616633i	$-0.788496\pi$
$43$	−10.3247	−1.57450	−0.787251	+	0.616633i	$0.788496\pi$
$44$	−8.10340	−1.22163
$45$	−3.52654	−0.525705
$46$	−0.235057	−0.0346573
$47$	−4.02730	−0.587442	−0.293721	−	0.955891i	$-0.594894\pi$
$47$	−4.02730	−0.587442	−0.293721	+	0.955891i	$0.594894\pi$
$48$	4.10878	0.593051
$49$	0	0
$50$	14.7686	2.08859
$51$	3.10640	0.434983
$52$	−11.1152	−1.54140
$53$	4.86574	0.668361	0.334181	−	0.942509i	$-0.391540\pi$
$53$	4.86574	0.668361	0.334181	+	0.942509i	$0.391540\pi$
$54$	−1.98596	−0.270255
$55$	14.6997	1.98211
$56$	0	0
$57$	−2.60172	−0.344606
$58$	10.8253	1.42143
$59$	9.28785	1.20917	0.604587	−	0.796539i	$-0.293338\pi$
$59$	9.28785	1.20917	0.604587	+	0.796539i	$0.293338\pi$
$60$	6.85576	0.885074
$61$	−6.09812	−0.780784	−0.390392	−	0.920649i	$-0.627660\pi$
$61$	−6.09812	−0.780784	−0.390392	+	0.920649i	$0.627660\pi$
$62$	−16.1439	−2.05028
$63$	0	0
$64$	−7.54628	−0.943285
$65$	20.1632	2.50094
$66$	8.27812	1.01897
$67$	2.57419	0.314488	0.157244	−	0.987560i	$-0.449739\pi$
$67$	2.57419	0.314488	0.157244	+	0.987560i	$0.449739\pi$
$68$	−6.03898	−0.732334
$69$	0.118359	0.0142488
$70$	0	0
$71$	2.06205	0.244720	0.122360	−	0.992486i	$-0.460954\pi$
$71$	2.06205	0.244720	0.122360	+	0.992486i	$0.460954\pi$
$72$	−0.111122	−0.0130959
$73$	9.63811	1.12806	0.564028	−	0.825756i	$-0.309251\pi$
$73$	9.63811	1.12806	0.564028	+	0.825756i	$0.309251\pi$
$74$	4.81149	0.559325
$75$	−7.43648	−0.858690
$76$	5.05786	0.580176
$77$	0	0
$78$	11.3549	1.28569
$79$	8.80338	0.990458	0.495229	−	0.868763i	$-0.335084\pi$
$79$	8.80338	0.990458	0.495229	+	0.868763i	$0.335084\pi$
$80$	14.4898	1.62000
$81$	1.00000	0.111111
$82$	−8.05809	−0.889868
$83$	2.26826	0.248974	0.124487	−	0.992221i	$-0.460271\pi$
$83$	2.26826	0.248974	0.124487	+	0.992221i	$0.460271\pi$
$84$	0	0
$85$	10.9548	1.18822
$86$	−20.5045	−2.21105
$87$	−5.45091	−0.584399
$88$	0.463192	0.0493764
$89$	9.72942	1.03132	0.515658	−	0.856794i	$-0.327547\pi$
$89$	9.72942	1.03132	0.515658	+	0.856794i	$0.327547\pi$
$90$	−7.00357	−0.738241
$91$	0	0
$92$	−0.230096	−0.0239892
$93$	8.12903	0.842941
$94$	−7.99807	−0.824938
$95$	−9.17505	−0.941340
$96$	7.93763	0.810131
$97$	−10.8774	−1.10443	−0.552217	−	0.833701i	$-0.686218\pi$
$97$	−10.8774	−1.10443	−0.552217	+	0.833701i	$0.686218\pi$
$98$	0	0
$99$	−4.16832	−0.418931
$100$	14.4569	1.44569
$101$	−0.704625	−0.0701128	−0.0350564	−	0.999385i	$-0.511161\pi$
$101$	−0.704625	−0.0701128	−0.0350564	+	0.999385i	$0.511161\pi$
$102$	6.16919	0.610841
$103$	−7.32688	−0.721939	−0.360970	−	0.932578i	$-0.617554\pi$
$103$	−7.32688	−0.721939	−0.360970	+	0.932578i	$0.617554\pi$
$104$	0.635347	0.0623009
$105$	0	0
$106$	9.66318	0.938571
$107$	11.6278	1.12410	0.562050	−	0.827103i	$-0.310013\pi$
$107$	11.6278	1.12410	0.562050	+	0.827103i	$0.310013\pi$
$108$	−1.94405	−0.187066
$109$	−12.5490	−1.20197	−0.600986	−	0.799260i	$-0.705225\pi$
$109$	−12.5490	−1.20197	−0.600986	+	0.799260i	$0.705225\pi$
$110$	29.1931	2.78345
$111$	−2.42275	−0.229957
$112$	0	0
$113$	−11.4631	−1.07836	−0.539178	−	0.842192i	$-0.681265\pi$
$113$	−11.4631	−1.07836	−0.539178	+	0.842192i	$0.681265\pi$
$114$	−5.16691	−0.483925
$115$	0.417399	0.0389226
$116$	10.5968	0.983890
$117$	−5.71756	−0.528589
$118$	18.4453	1.69803
$119$	0	0
$120$	−0.391876	−0.0357732
$121$	6.37485	0.579532
$122$	−12.1106	−1.09644
$123$	4.05753	0.365855
$124$	−15.8032	−1.41917
$125$	−8.59233	−0.768522
$126$	0	0
$127$	−5.98431	−0.531021	−0.265511	−	0.964108i	$-0.585541\pi$
$127$	−5.98431	−0.531021	−0.265511	+	0.964108i	$0.585541\pi$
$128$	0.888628	0.0785444
$129$	10.3247	0.909039
$130$	40.0434	3.51204
$131$	−7.00691	−0.612197	−0.306098	−	0.952000i	$-0.599024\pi$
$131$	−7.00691	−0.612197	−0.306098	+	0.952000i	$0.599024\pi$
$132$	8.10340	0.705310
$133$	0	0
$134$	5.11225	0.441631
$135$	3.52654	0.303516
$136$	0.345189	0.0295997
$137$	−4.48500	−0.383180	−0.191590	−	0.981475i	$-0.561364\pi$
$137$	−4.48500	−0.383180	−0.191590	+	0.981475i	$0.561364\pi$
$138$	0.235057	0.0200094
$139$	4.72976	0.401173	0.200586	−	0.979676i	$-0.435715\pi$
$139$	4.72976	0.401173	0.200586	+	0.979676i	$0.435715\pi$
$140$	0	0
$141$	4.02730	0.339160
$142$	4.09515	0.343658
$143$	23.8326	1.99298
$144$	−4.10878	−0.342398
$145$	−19.2229	−1.59637
$146$	19.1409	1.58411
$147$	0	0
$148$	4.70994	0.387155
$149$	15.2896	1.25257	0.626285	−	0.779594i	$-0.284575\pi$
$149$	15.2896	1.25257	0.626285	+	0.779594i	$0.284575\pi$
$150$	−14.7686	−1.20585
$151$	−6.17019	−0.502123	−0.251061	−	0.967971i	$-0.580780\pi$
$151$	−6.17019	−0.502123	−0.251061	+	0.967971i	$0.580780\pi$
$152$	−0.289108	−0.0234497
$153$	−3.10640	−0.251137
$154$	0	0
$155$	28.6673	2.30262
$156$	11.1152	0.889929
$157$	0.837470	0.0668374	0.0334187	−	0.999441i	$-0.489361\pi$
$157$	0.837470	0.0668374	0.0334187	+	0.999441i	$0.489361\pi$
$158$	17.4832	1.39089
$159$	−4.86574	−0.385879
$160$	27.9924	2.21299
$161$	0	0
$162$	1.98596	0.156032
$163$	4.09767	0.320954	0.160477	−	0.987040i	$-0.448697\pi$
$163$	4.09767	0.320954	0.160477	+	0.987040i	$0.448697\pi$
$164$	−7.88802	−0.615951
$165$	−14.6997	−1.14437
$166$	4.50468	0.349631
$167$	4.68621	0.362630	0.181315	−	0.983425i	$-0.441965\pi$
$167$	4.68621	0.362630	0.181315	+	0.983425i	$0.441965\pi$
$168$	0	0
$169$	19.6905	1.51466
$170$	21.7559	1.66860
$171$	2.60172	0.198958
$172$	−20.0717	−1.53045
$173$	15.9360	1.21159	0.605794	−	0.795621i	$-0.292856\pi$
$173$	15.9360	1.21159	0.605794	+	0.795621i	$0.292856\pi$
$174$	−10.8253	−0.820664
$175$	0	0
$176$	17.1267	1.29097
$177$	−9.28785	−0.698117
$178$	19.3223	1.44826
$179$	−20.0238	−1.49665	−0.748325	−	0.663333i	$-0.769141\pi$
$179$	−20.0238	−1.49665	−0.748325	+	0.663333i	$0.769141\pi$
$180$	−6.85576	−0.510998
$181$	5.59510	0.415880	0.207940	−	0.978142i	$-0.433324\pi$
$181$	5.59510	0.415880	0.207940	+	0.978142i	$0.433324\pi$
$182$	0	0
$183$	6.09812	0.450786
$184$	0.0131523	0.000969602 0
$185$	−8.54393	−0.628162
$186$	16.1439	1.18373
$187$	12.9485	0.946884
$188$	−7.82926	−0.571008
$189$	0	0
$190$	−18.2213	−1.32191
$191$	−4.69789	−0.339927	−0.169964	−	0.985450i	$-0.554365\pi$
$191$	−4.69789	−0.339927	−0.169964	+	0.985450i	$0.554365\pi$
$192$	7.54628	0.544606
$193$	19.9171	1.43366	0.716831	−	0.697247i	$-0.245592\pi$
$193$	19.9171	1.43366	0.716831	+	0.697247i	$0.245592\pi$
$194$	−21.6021	−1.55094
$195$	−20.1632	−1.44392
$196$	0	0
$197$	−2.76013	−0.196651	−0.0983257	−	0.995154i	$-0.531349\pi$
$197$	−2.76013	−0.196651	−0.0983257	+	0.995154i	$0.531349\pi$
$198$	−8.27812	−0.588300
$199$	−16.7577	−1.18792	−0.593962	−	0.804493i	$-0.702437\pi$
$199$	−16.7577	−1.18792	−0.593962	+	0.804493i	$0.702437\pi$
$200$	−0.826356	−0.0584322
$201$	−2.57419	−0.181570
$202$	−1.39936	−0.0984585
$203$	0	0
$204$	6.03898	0.422813
$205$	14.3090	0.999386
$206$	−14.5509	−1.01381
$207$	−0.118359	−0.00822654
$208$	23.4922	1.62889
$209$	−10.8448	−0.750149
$210$	0	0
$211$	−16.6114	−1.14358	−0.571789	−	0.820401i	$-0.693750\pi$
$211$	−16.6114	−1.14358	−0.571789	+	0.820401i	$0.693750\pi$
$212$	9.45923	0.649663
$213$	−2.06205	−0.141289
$214$	23.0923	1.57856
$215$	36.4105	2.48317
$216$	0.111122	0.00756089
$217$	0	0
$218$	−24.9217	−1.68791
$219$	−9.63811	−0.651283
$220$	28.5769	1.92666
$221$	17.7610	1.19474
$222$	−4.81149	−0.322926
$223$	14.6457	0.980750	0.490375	−	0.871512i	$-0.336860\pi$
$223$	14.6457	0.980750	0.490375	+	0.871512i	$0.336860\pi$
$224$	0	0
$225$	7.43648	0.495765
$226$	−22.7653	−1.51432
$227$	29.8391	1.98049	0.990246	−	0.139330i	$-0.0444950\pi$
$227$	29.8391	1.98049	0.990246	+	0.139330i	$0.0444950\pi$
$228$	−5.05786	−0.334965
$229$	26.6518	1.76120	0.880600	−	0.473861i	$-0.157140\pi$
$229$	26.6518	1.76120	0.880600	+	0.473861i	$0.157140\pi$
$230$	0.828938	0.0546586
$231$	0	0
$232$	−0.605716	−0.0397672
$233$	13.6897	0.896841	0.448421	−	0.893823i	$-0.351987\pi$
$233$	13.6897	0.896841	0.448421	+	0.893823i	$0.351987\pi$
$234$	−11.3549	−0.742291
$235$	14.2024	0.926465
$236$	18.0560	1.17535
$237$	−8.80338	−0.571841
$238$	0	0
$239$	−24.8498	−1.60740	−0.803701	−	0.595034i	$-0.797139\pi$
$239$	−24.8498	−1.60740	−0.803701	+	0.595034i	$0.797139\pi$
$240$	−14.4898	−0.935310
$241$	−16.5102	−1.06351	−0.531757	−	0.846897i	$-0.678468\pi$
$241$	−16.5102	−1.06351	−0.531757	+	0.846897i	$0.678468\pi$
$242$	12.6602	0.813830
$243$	−1.00000	−0.0641500
$244$	−11.8550	−0.758940
$245$	0	0
$246$	8.05809	0.513765
$247$	−14.8755	−0.946504
$248$	0.903314	0.0573605
$249$	−2.26826	−0.143745
$250$	−17.0640	−1.07923
$251$	−10.5468	−0.665705	−0.332853	−	0.942979i	$-0.608011\pi$
$251$	−10.5468	−0.665705	−0.332853	+	0.942979i	$0.608011\pi$
$252$	0	0
$253$	0.493359	0.0310172
$254$	−11.8846	−0.745707
$255$	−10.9548	−0.686018
$256$	16.8573	1.05358
$257$	2.10588	0.131361	0.0656806	−	0.997841i	$-0.479078\pi$
$257$	2.10588	0.131361	0.0656806	+	0.997841i	$0.479078\pi$
$258$	20.5045	1.27655
$259$	0	0
$260$	39.1982	2.43097
$261$	5.45091	0.337403
$262$	−13.9155	−0.859700
$263$	−6.22719	−0.383985	−0.191993	−	0.981396i	$-0.561495\pi$
$263$	−6.22719	−0.383985	−0.191993	+	0.981396i	$0.561495\pi$
$264$	−0.463192	−0.0285075
$265$	−17.1592	−1.05408
$266$	0	0
$267$	−9.72942	−0.595431
$268$	5.00435	0.305690
$269$	19.8838	1.21234	0.606170	−	0.795335i	$-0.292705\pi$
$269$	19.8838	1.21234	0.606170	+	0.795335i	$0.292705\pi$
$270$	7.00357	0.426224
$271$	−5.79241	−0.351864	−0.175932	−	0.984402i	$-0.556294\pi$
$271$	−5.79241	−0.351864	−0.175932	+	0.984402i	$0.556294\pi$
$272$	12.7635	0.773901
$273$	0	0
$274$	−8.90705	−0.538094
$275$	−30.9976	−1.86922
$276$	0.230096	0.0138501
$277$	−6.35047	−0.381562	−0.190781	−	0.981633i	$-0.561102\pi$
$277$	−6.35047	−0.381562	−0.190781	+	0.981633i	$0.561102\pi$
$278$	9.39312	0.563362
$279$	−8.12903	−0.486672
$280$	0	0
$281$	−30.3985	−1.81342	−0.906711	−	0.421753i	$-0.861415\pi$
$281$	−30.3985	−1.81342	−0.906711	+	0.421753i	$0.861415\pi$
$282$	7.99807	0.476278
$283$	16.4389	0.977190	0.488595	−	0.872511i	$-0.337510\pi$
$283$	16.4389	0.977190	0.488595	+	0.872511i	$0.337510\pi$
$284$	4.00872	0.237874
$285$	9.17505	0.543483
$286$	47.3307	2.79872
$287$	0	0
$288$	−7.93763	−0.467729
$289$	−7.35029	−0.432370
$290$	−38.1759	−2.24176
$291$	10.8774	0.637645
$292$	18.7369	1.09650
$293$	1.83585	0.107251	0.0536256	−	0.998561i	$-0.482922\pi$
$293$	1.83585	0.107251	0.0536256	+	0.998561i	$0.482922\pi$
$294$	0	0
$295$	−32.7540	−1.90701
$296$	−0.269221	−0.0156482
$297$	4.16832	0.241870
$298$	30.3645	1.75897
$299$	0.676727	0.0391361
$300$	−14.4569	−0.834667
$301$	0	0
$302$	−12.2538	−0.705124
$303$	0.704625	0.0404796
$304$	−10.6899	−0.613106
$305$	21.5052	1.23139
$306$	−6.16919	−0.352669
$307$	−21.2566	−1.21318	−0.606588	−	0.795016i	$-0.707462\pi$
$307$	−21.2566	−1.21318	−0.606588	+	0.795016i	$0.707462\pi$
$308$	0	0
$309$	7.32688	0.416812
$310$	56.9322	3.23353
$311$	26.3448	1.49388	0.746940	−	0.664892i	$-0.231522\pi$
$311$	26.3448	1.49388	0.746940	+	0.664892i	$0.231522\pi$
$312$	−0.635347	−0.0359695
$313$	−24.4409	−1.38148	−0.690740	−	0.723103i	$-0.742715\pi$
$313$	−24.4409	−1.38148	−0.690740	+	0.723103i	$0.742715\pi$
$314$	1.66318	0.0938589
$315$	0	0
$316$	17.1142	0.962748
$317$	17.2881	0.970995	0.485498	−	0.874238i	$-0.338638\pi$
$317$	17.2881	0.970995	0.485498	+	0.874238i	$0.338638\pi$
$318$	−9.66318	−0.541884
$319$	−22.7211	−1.27214
$320$	26.6123	1.48767
$321$	−11.6278	−0.648999
$322$	0	0
$323$	−8.08197	−0.449693
$324$	1.94405	0.108003
$325$	−42.5185	−2.35850
$326$	8.13781	0.450712
$327$	12.5490	0.693959
$328$	0.450880	0.0248957
$329$	0	0
$330$	−29.1931	−1.60703
$331$	−7.32284	−0.402500	−0.201250	−	0.979540i	$-0.564500\pi$
$331$	−7.32284	−0.402500	−0.201250	+	0.979540i	$0.564500\pi$
$332$	4.40961	0.242009
$333$	2.42275	0.132766
$334$	9.30664	0.509237
$335$	−9.07800	−0.495984
$336$	0	0
$337$	−33.8858	−1.84588	−0.922939	−	0.384946i	$-0.874220\pi$
$337$	−33.8858	−1.84588	−0.922939	+	0.384946i	$0.874220\pi$
$338$	39.1047	2.12701
$339$	11.4631	0.622589
$340$	21.2967	1.15498
$341$	33.8844	1.83494
$342$	5.16691	0.279394
$343$	0	0
$344$	1.14730	0.0618583
$345$	−0.417399	−0.0224720
$346$	31.6482	1.70142
$347$	31.5084	1.69146	0.845731	−	0.533610i	$-0.179165\pi$
$347$	31.5084	1.69146	0.845731	+	0.533610i	$0.179165\pi$
$348$	−10.5968	−0.568049
$349$	−8.72164	−0.466859	−0.233429	−	0.972374i	$-0.574995\pi$
$349$	−8.72164	−0.466859	−0.233429	+	0.972374i	$0.574995\pi$
$350$	0	0
$351$	5.71756	0.305181
$352$	33.0866	1.76352
$353$	−10.6065	−0.564529	−0.282265	−	0.959337i	$-0.591086\pi$
$353$	−10.6065	−0.564529	−0.282265	+	0.959337i	$0.591086\pi$
$354$	−18.4453	−0.980357
$355$	−7.27190	−0.385952
$356$	18.9144	1.00246
$357$	0	0
$358$	−39.7665	−2.10173
$359$	14.3884	0.759390	0.379695	−	0.925112i	$-0.376029\pi$
$359$	14.3884	0.759390	0.379695	+	0.925112i	$0.376029\pi$
$360$	0.391876	0.0206537
$361$	−12.2311	−0.643741
$362$	11.1117	0.584015
$363$	−6.37485	−0.334593
$364$	0	0
$365$	−33.9892	−1.77907
$366$	12.1106	0.633033
$367$	11.6523	0.608243	0.304122	−	0.952633i	$-0.401637\pi$
$367$	11.6523	0.608243	0.304122	+	0.952633i	$0.401637\pi$
$368$	0.486312	0.0253508
$369$	−4.05753	−0.211226
$370$	−16.9679	−0.882120
$371$	0	0
$372$	15.8032	0.819358
$373$	14.7340	0.762898	0.381449	−	0.924390i	$-0.375425\pi$
$373$	14.7340	0.762898	0.381449	+	0.924390i	$0.375425\pi$
$374$	25.7151	1.32970
$375$	8.59233	0.443706
$376$	0.447522	0.0230792
$377$	−31.1659	−1.60513
$378$	0	0
$379$	14.0556	0.721989	0.360995	−	0.932568i	$-0.382437\pi$
$379$	14.0556	0.721989	0.360995	+	0.932568i	$0.382437\pi$
$380$	−17.8367	−0.915005
$381$	5.98431	0.306585
$382$	−9.32983	−0.477356
$383$	−23.1430	−1.18255	−0.591275	−	0.806470i	$-0.701375\pi$
$383$	−23.1430	−1.18255	−0.591275	+	0.806470i	$0.701375\pi$
$384$	−0.888628	−0.0453476
$385$	0	0
$386$	39.5545	2.01327
$387$	−10.3247	−0.524834
$388$	−21.1462	−1.07354
$389$	19.5845	0.992976	0.496488	−	0.868044i	$-0.334623\pi$
$389$	19.5845	0.992976	0.496488	+	0.868044i	$0.334623\pi$
$390$	−40.0434	−2.02768
$391$	0.367671	0.0185939
$392$	0	0
$393$	7.00691	0.353452
$394$	−5.48152	−0.276155
$395$	−31.0455	−1.56207
$396$	−8.10340	−0.407211
$397$	3.70821	0.186110	0.0930548	−	0.995661i	$-0.470337\pi$
$397$	3.70821	0.186110	0.0930548	+	0.995661i	$0.470337\pi$
$398$	−33.2802	−1.66819
$399$	0	0
$400$	−30.5548	−1.52774
$401$	−4.93680	−0.246532	−0.123266	−	0.992374i	$-0.539337\pi$
$401$	−4.93680	−0.246532	−0.123266	+	0.992374i	$0.539337\pi$
$402$	−5.11225	−0.254976
$403$	46.4782	2.31525
$404$	−1.36982	−0.0681512
$405$	−3.52654	−0.175235
$406$	0	0
$407$	−10.0988	−0.500579
$408$	−0.345189	−0.0170894
$409$	−27.4372	−1.35668	−0.678342	−	0.734746i	$-0.737301\pi$
$409$	−27.4372	−1.35668	−0.678342	+	0.734746i	$0.737301\pi$
$410$	28.4172	1.40342
$411$	4.48500	0.221229
$412$	−14.2438	−0.701742
$413$	0	0
$414$	−0.235057	−0.0115524
$415$	−7.99911	−0.392661
$416$	45.3839	2.22513
$417$	−4.72976	−0.231617
$418$	−21.5373	−1.05342
$419$	1.52279	0.0743932	0.0371966	−	0.999308i	$-0.488157\pi$
$419$	1.52279	0.0743932	0.0371966	+	0.999308i	$0.488157\pi$
$420$	0	0
$421$	−26.0481	−1.26951	−0.634754	−	0.772714i	$-0.718899\pi$
$421$	−26.0481	−1.26951	−0.634754	+	0.772714i	$0.718899\pi$
$422$	−32.9897	−1.60591
$423$	−4.02730	−0.195814
$424$	−0.540691	−0.0262583
$425$	−23.1007	−1.12055
$426$	−4.09515	−0.198411
$427$	0	0
$428$	22.6049	1.09265
$429$	−23.8326	−1.15065
$430$	72.3098	3.48709
$431$	8.48143	0.408536	0.204268	−	0.978915i	$-0.434519\pi$
$431$	8.48143	0.408536	0.204268	+	0.978915i	$0.434519\pi$
$432$	4.10878	0.197684
$433$	34.9258	1.67843	0.839215	−	0.543800i	$-0.183015\pi$
$433$	34.9258	1.67843	0.839215	+	0.543800i	$0.183015\pi$
$434$	0	0
$435$	19.2229	0.921665
$436$	−24.3957	−1.16834
$437$	−0.307937	−0.0147306
$438$	−19.1409	−0.914588
$439$	11.7717	0.561832	0.280916	−	0.959732i	$-0.409362\pi$
$439$	11.7717	0.561832	0.280916	+	0.959732i	$0.409362\pi$
$440$	−1.63346	−0.0778723
$441$	0	0
$442$	35.2728	1.67775
$443$	28.6386	1.36066	0.680330	−	0.732906i	$-0.261837\pi$
$443$	28.6386	1.36066	0.680330	+	0.732906i	$0.261837\pi$
$444$	−4.70994	−0.223524
$445$	−34.3112	−1.62651
$446$	29.0859	1.37726
$447$	−15.2896	−0.723172
$448$	0	0
$449$	−37.5359	−1.77143	−0.885714	−	0.464232i	$-0.846330\pi$
$449$	−37.5359	−1.77143	−0.885714	+	0.464232i	$0.846330\pi$
$450$	14.7686	0.696197
$451$	16.9131	0.796404
$452$	−22.2848	−1.04819
$453$	6.17019	0.289901
$454$	59.2594	2.78118
$455$	0	0
$456$	0.289108	0.0135387
$457$	−16.4790	−0.770855	−0.385428	−	0.922738i	$-0.625946\pi$
$457$	−16.4790	−0.770855	−0.385428	+	0.922738i	$0.625946\pi$
$458$	52.9294	2.47323
$459$	3.10640	0.144994
$460$	0.811442	0.0378337
$461$	−6.56643	−0.305829	−0.152914	−	0.988239i	$-0.548866\pi$
$461$	−6.56643	−0.305829	−0.152914	+	0.988239i	$0.548866\pi$
$462$	0	0
$463$	−17.2216	−0.800354	−0.400177	−	0.916438i	$-0.631051\pi$
$463$	−17.2216	−0.800354	−0.400177	+	0.916438i	$0.631051\pi$
$464$	−22.3966	−1.03974
$465$	−28.6673	−1.32942
$466$	27.1872	1.25942
$467$	9.74748	0.451060	0.225530	−	0.974236i	$-0.427589\pi$
$467$	9.74748	0.451060	0.225530	+	0.974236i	$0.427589\pi$
$468$	−11.1152	−0.513801
$469$	0	0
$470$	28.2055	1.30102
$471$	−0.837470	−0.0385886
$472$	−1.03208	−0.0475055
$473$	43.0366	1.97882
$474$	−17.4832	−0.803029
$475$	19.3476	0.887729
$476$	0	0
$477$	4.86574	0.222787
$478$	−49.3508	−2.25725
$479$	14.6514	0.669440	0.334720	−	0.942318i	$-0.391358\pi$
$479$	14.6514	0.669440	0.334720	+	0.942318i	$0.391358\pi$
$480$	−27.9924	−1.27767
$481$	−13.8522	−0.631608
$482$	−32.7886	−1.49348
$483$	0	0
$484$	12.3930	0.563319
$485$	38.3596	1.74182
$486$	−1.98596	−0.0900851
$487$	−31.6142	−1.43258	−0.716289	−	0.697804i	$-0.754161\pi$
$487$	−31.6142	−1.43258	−0.716289	+	0.697804i	$0.754161\pi$
$488$	0.677635	0.0306751
$489$	−4.09767	−0.185303
$490$	0	0
$491$	−16.2732	−0.734399	−0.367200	−	0.930142i	$-0.619683\pi$
$491$	−16.2732	−0.734399	−0.367200	+	0.930142i	$0.619683\pi$
$492$	7.88802	0.355619
$493$	−16.9327	−0.762611
$494$	−29.5421	−1.32916
$495$	14.6997	0.660704
$496$	33.4004	1.49972
$497$	0	0
$498$	−4.50468	−0.201860
$499$	−18.0542	−0.808218	−0.404109	−	0.914711i	$-0.632418\pi$
$499$	−18.0542	−0.808218	−0.404109	+	0.914711i	$0.632418\pi$
$500$	−16.7039	−0.747021
$501$	−4.68621	−0.209365
$502$	−20.9455	−0.934842
$503$	12.3552	0.550891	0.275446	−	0.961317i	$-0.411175\pi$
$503$	12.3552	0.550891	0.275446	+	0.961317i	$0.411175\pi$
$504$	0	0
$505$	2.48489	0.110576
$506$	0.979792	0.0435571
$507$	−19.6905	−0.874488
$508$	−11.6338	−0.516165
$509$	−5.12371	−0.227104	−0.113552	−	0.993532i	$-0.536223\pi$
$509$	−5.12371	−0.227104	−0.113552	+	0.993532i	$0.536223\pi$
$510$	−21.7559	−0.963367
$511$	0	0
$512$	31.7008	1.40099
$513$	−2.60172	−0.114869
$514$	4.18220	0.184469
$515$	25.8385	1.13858
$516$	20.0717	0.883607
$517$	16.7871	0.738294
$518$	0	0
$519$	−15.9360	−0.699511
$520$	−2.24058	−0.0982558
$521$	−22.7067	−0.994799	−0.497399	−	0.867522i	$-0.665712\pi$
$521$	−22.7067	−0.994799	−0.497399	+	0.867522i	$0.665712\pi$
$522$	10.8253	0.473811
$523$	−21.3782	−0.934805	−0.467403	−	0.884045i	$-0.654810\pi$
$523$	−21.3782	−0.934805	−0.467403	+	0.884045i	$0.654810\pi$
$524$	−13.6218	−0.595069
$525$	0	0
$526$	−12.3670	−0.539226
$527$	25.2520	1.09999
$528$	−17.1267	−0.745343
$529$	−22.9860	−0.999391
$530$	−34.0776	−1.48024
$531$	9.28785	0.403058
$532$	0	0
$533$	23.1992	1.00487
$534$	−19.3223	−0.836156
$535$	−41.0058	−1.77284
$536$	−0.286050	−0.0123555
$537$	20.0238	0.864091
$538$	39.4886	1.70247
$539$	0	0
$540$	6.85576	0.295025
$541$	1.26681	0.0544644	0.0272322	−	0.999629i	$-0.491331\pi$
$541$	1.26681	0.0544644	0.0272322	+	0.999629i	$0.491331\pi$
$542$	−11.5035	−0.494118
$543$	−5.59510	−0.240109
$544$	24.6574	1.05718
$545$	44.2544	1.89565
$546$	0	0
$547$	23.9982	1.02609	0.513044	−	0.858362i	$-0.328518\pi$
$547$	23.9982	1.02609	0.513044	+	0.858362i	$0.328518\pi$
$548$	−8.71905	−0.372459
$549$	−6.09812	−0.260261
$550$	−61.5600	−2.62493
$551$	14.1817	0.604162
$552$	−0.0131523	−0.000559800 0
$553$	0	0
$554$	−12.6118	−0.535823
$555$	8.54393	0.362670
$556$	9.19487	0.389949
$557$	−12.9557	−0.548950	−0.274475	−	0.961594i	$-0.588504\pi$
$557$	−12.9557	−0.548950	−0.274475	+	0.961594i	$0.588504\pi$
$558$	−16.1439	−0.683428
$559$	59.0321	2.49679
$560$	0	0
$561$	−12.9485	−0.546684
$562$	−60.3703	−2.54657
$563$	−31.9986	−1.34858	−0.674289	−	0.738467i	$-0.735550\pi$
$563$	−31.9986	−1.34858	−0.674289	+	0.738467i	$0.735550\pi$
$564$	7.82926	0.329671
$565$	40.4250	1.70069
$566$	32.6470	1.37225
$567$	0	0
$568$	−0.229139	−0.00961446
$569$	43.0360	1.80416	0.902081	−	0.431566i	$-0.142039\pi$
$569$	43.0360	1.80416	0.902081	+	0.431566i	$0.142039\pi$
$570$	18.2213	0.763207
$571$	−37.4553	−1.56746	−0.783728	−	0.621105i	$-0.786684\pi$
$571$	−37.4553	−1.56746	−0.783728	+	0.621105i	$0.786684\pi$
$572$	46.3317	1.93723
$573$	4.69789	0.196257
$574$	0	0
$575$	−0.880176	−0.0367059
$576$	−7.54628	−0.314428
$577$	−11.5447	−0.480613	−0.240306	−	0.970697i	$-0.577248\pi$
$577$	−11.5447	−0.480613	−0.240306	+	0.970697i	$0.577248\pi$
$578$	−14.5974	−0.607171
$579$	−19.9171	−0.827725
$580$	−37.3701	−1.55171
$581$	0	0
$582$	21.6021	0.895437
$583$	−20.2820	−0.839993
$584$	−1.07101	−0.0443185
$585$	20.1632	0.833646
$586$	3.64592	0.150612
$587$	7.80179	0.322014	0.161007	−	0.986953i	$-0.448526\pi$
$587$	7.80179	0.322014	0.161007	+	0.986953i	$0.448526\pi$
$588$	0	0
$589$	−21.1494	−0.871447
$590$	−65.0481	−2.67799
$591$	2.76013	0.113537
$592$	−9.95455	−0.409129
$593$	28.3443	1.16396	0.581981	−	0.813202i	$-0.302278\pi$
$593$	28.3443	1.16396	0.581981	+	0.813202i	$0.302278\pi$
$594$	8.27812	0.339655
$595$	0	0
$596$	29.7236	1.21753
$597$	16.7577	0.685848
$598$	1.34395	0.0549584
$599$	29.9724	1.22464	0.612320	−	0.790610i	$-0.290236\pi$
$599$	29.9724	1.22464	0.612320	+	0.790610i	$0.290236\pi$
$600$	0.826356	0.0337359
$601$	−22.8402	−0.931670	−0.465835	−	0.884872i	$-0.654246\pi$
$601$	−22.8402	−0.931670	−0.465835	+	0.884872i	$0.654246\pi$
$602$	0	0
$603$	2.57419	0.104829
$604$	−11.9951	−0.488075
$605$	−22.4812	−0.913990
$606$	1.39936	0.0568450
$607$	16.6281	0.674915	0.337457	−	0.941341i	$-0.390433\pi$
$607$	16.6281	0.674915	0.337457	+	0.941341i	$0.390433\pi$
$608$	−20.6515	−0.837527
$609$	0	0
$610$	42.7086	1.72922
$611$	23.0264	0.931547
$612$	−6.03898	−0.244111
$613$	32.0242	1.29345	0.646724	−	0.762724i	$-0.276139\pi$
$613$	32.0242	1.29345	0.646724	+	0.762724i	$0.276139\pi$
$614$	−42.2147	−1.70365
$615$	−14.3090	−0.576996
$616$	0	0
$617$	−18.9118	−0.761360	−0.380680	−	0.924707i	$-0.624310\pi$
$617$	−18.9118	−0.761360	−0.380680	+	0.924707i	$0.624310\pi$
$618$	14.5509	0.585324
$619$	31.6066	1.27038	0.635189	−	0.772357i	$-0.280922\pi$
$619$	31.6066	1.27038	0.635189	+	0.772357i	$0.280922\pi$
$620$	55.7306	2.23820
$621$	0.118359	0.00474960
$622$	52.3199	2.09784
$623$	0	0
$624$	−23.4922	−0.940441
$625$	−6.88119	−0.275248
$626$	−48.5386	−1.93999
$627$	10.8448	0.433098
$628$	1.62808	0.0649675
$629$	−7.52603	−0.300083
$630$	0	0
$631$	38.7774	1.54370	0.771852	−	0.635802i	$-0.219331\pi$
$631$	38.7774	1.54370	0.771852	+	0.635802i	$0.219331\pi$
$632$	−0.978249	−0.0389127
$633$	16.6114	0.660245
$634$	34.3335	1.36356
$635$	21.1039	0.837483
$636$	−9.45923	−0.375083
$637$	0	0
$638$	−45.1233	−1.78645
$639$	2.06205	0.0815734
$640$	−3.13378	−0.123874
$641$	31.3003	1.23629	0.618143	−	0.786066i	$-0.287885\pi$
$641$	31.3003	1.23629	0.618143	+	0.786066i	$0.287885\pi$
$642$	−23.0923	−0.911381
$643$	12.1692	0.479908	0.239954	−	0.970784i	$-0.422868\pi$
$643$	12.1692	0.479908	0.239954	+	0.970784i	$0.422868\pi$
$644$	0	0
$645$	−36.4105	−1.43366
$646$	−16.0505	−0.631498
$647$	0.765117	0.0300798	0.0150399	−	0.999887i	$-0.495212\pi$
$647$	0.765117	0.0300798	0.0150399	+	0.999887i	$0.495212\pi$
$648$	−0.111122	−0.00436528
$649$	−38.7147	−1.51968
$650$	−84.4402	−3.31202
$651$	0	0
$652$	7.96605	0.311975
$653$	−19.6609	−0.769392	−0.384696	−	0.923043i	$-0.625694\pi$
$653$	−19.6609	−0.769392	−0.384696	+	0.923043i	$0.625694\pi$
$654$	24.9217	0.974517
$655$	24.7102	0.965505
$656$	16.6715	0.650912
$657$	9.63811	0.376018
$658$	0	0
$659$	−5.59156	−0.217816	−0.108908	−	0.994052i	$-0.534735\pi$
$659$	−5.59156	−0.217816	−0.108908	+	0.994052i	$0.534735\pi$
$660$	−28.5769	−1.11236
$661$	17.1141	0.665663	0.332831	−	0.942986i	$-0.391996\pi$
$661$	17.1141	0.665663	0.332831	+	0.942986i	$0.391996\pi$
$662$	−14.5429	−0.565225
$663$	−17.7610	−0.689782
$664$	−0.252054	−0.00978158
$665$	0	0
$666$	4.81149	0.186442
$667$	−0.645166	−0.0249809
$668$	9.11021	0.352485
$669$	−14.6457	−0.566236
$670$	−18.0286	−0.696504
$671$	25.4189	0.981285
$672$	0	0
$673$	44.5719	1.71812	0.859060	−	0.511875i	$-0.171049\pi$
$673$	44.5719	1.71812	0.859060	+	0.511875i	$0.171049\pi$
$674$	−67.2959	−2.59214
$675$	−7.43648	−0.286230
$676$	38.2793	1.47228
$677$	26.0351	1.00061	0.500305	−	0.865849i	$-0.333221\pi$
$677$	26.0351	1.00061	0.500305	+	0.865849i	$0.333221\pi$
$678$	22.7653	0.874294
$679$	0	0
$680$	−1.21732	−0.0466822
$681$	−29.8391	−1.14344
$682$	67.2930	2.57678
$683$	−17.0152	−0.651070	−0.325535	−	0.945530i	$-0.605544\pi$
$683$	−17.0152	−0.651070	−0.325535	+	0.945530i	$0.605544\pi$
$684$	5.05786	0.193392
$685$	15.8165	0.604319
$686$	0	0
$687$	−26.6518	−1.01683
$688$	42.4219	1.61732
$689$	−27.8202	−1.05987
$690$	−0.828938	−0.0315571
$691$	−9.99238	−0.380128	−0.190064	−	0.981772i	$-0.560870\pi$
$691$	−9.99238	−0.380128	−0.190064	+	0.981772i	$0.560870\pi$
$692$	30.9802	1.17769
$693$	0	0
$694$	62.5746	2.37530
$695$	−16.6797	−0.632696
$696$	0.605716	0.0229596
$697$	12.6043	0.477422
$698$	−17.3208	−0.655604
$699$	−13.6897	−0.517792
$700$	0	0
$701$	26.4336	0.998382	0.499191	−	0.866492i	$-0.333631\pi$
$701$	26.4336	0.998382	0.499191	+	0.866492i	$0.333631\pi$
$702$	11.3549	0.428562
$703$	6.30331	0.237734
$704$	31.4553	1.18552
$705$	−14.2024	−0.534895
$706$	−21.0642	−0.792761
$707$	0	0
$708$	−18.0560	−0.678586
$709$	35.3387	1.32717	0.663586	−	0.748100i	$-0.269034\pi$
$709$	35.3387	1.32717	0.663586	+	0.748100i	$0.269034\pi$
$710$	−14.4417	−0.541988
$711$	8.80338	0.330153
$712$	−1.08115	−0.0405179
$713$	0.962146	0.0360327
$714$	0	0
$715$	−84.0466	−3.14317
$716$	−38.9272	−1.45478
$717$	24.8498	0.928034
$718$	28.5748	1.06640
$719$	6.55726	0.244545	0.122272	−	0.992497i	$-0.460982\pi$
$719$	6.55726	0.244545	0.122272	+	0.992497i	$0.460982\pi$
$720$	14.4898	0.540002
$721$	0	0
$722$	−24.2905	−0.903997
$723$	16.5102	0.614020
$724$	10.8771	0.404245
$725$	40.5356	1.50545
$726$	−12.6602	−0.469865
$727$	−10.9664	−0.406720	−0.203360	−	0.979104i	$-0.565186\pi$
$727$	−10.9664	−0.406720	−0.203360	+	0.979104i	$0.565186\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−67.5012	−2.49833
$731$	32.0726	1.18625
$732$	11.8550	0.438174
$733$	10.8463	0.400616	0.200308	−	0.979733i	$-0.435806\pi$
$733$	10.8463	0.400616	0.200308	+	0.979733i	$0.435806\pi$
$734$	23.1410	0.854148
$735$	0	0
$736$	0.939493	0.0346302
$737$	−10.7301	−0.395247
$738$	−8.05809	−0.296623
$739$	25.6956	0.945228	0.472614	−	0.881270i	$-0.343310\pi$
$739$	25.6956	0.945228	0.472614	+	0.881270i	$0.343310\pi$
$740$	−16.6098	−0.610588
$741$	14.8755	0.546464
$742$	0	0
$743$	−30.4544	−1.11726	−0.558632	−	0.829416i	$-0.688674\pi$
$743$	−30.4544	−1.11726	−0.558632	+	0.829416i	$0.688674\pi$
$744$	−0.903314	−0.0331171
$745$	−53.9193	−1.97545
$746$	29.2612	1.07133
$747$	2.26826	0.0829913
$748$	25.1724	0.920394
$749$	0	0
$750$	17.0640	0.623091
$751$	−29.3770	−1.07198	−0.535991	−	0.844224i	$-0.680062\pi$
$751$	−29.3770	−1.07198	−0.535991	+	0.844224i	$0.680062\pi$
$752$	16.5473	0.603417
$753$	10.5468	0.384345
$754$	−61.8944	−2.25406
$755$	21.7594	0.791906
$756$	0	0
$757$	34.3651	1.24902	0.624511	−	0.781016i	$-0.285298\pi$
$757$	34.3651	1.24902	0.624511	+	0.781016i	$0.285298\pi$
$758$	27.9139	1.01388
$759$	−0.493359	−0.0179078
$760$	1.01955	0.0369830
$761$	−2.85152	−0.103368	−0.0516838	−	0.998664i	$-0.516459\pi$
$761$	−2.85152	−0.103368	−0.0516838	+	0.998664i	$0.516459\pi$
$762$	11.8846	0.430534
$763$	0	0
$764$	−9.13292	−0.330417
$765$	10.9548	0.396073
$766$	−45.9610	−1.66064
$767$	−53.1039	−1.91747
$768$	−16.8573	−0.608287
$769$	−18.4814	−0.666456	−0.333228	−	0.942846i	$-0.608138\pi$
$769$	−18.4814	−0.666456	−0.333228	+	0.942846i	$0.608138\pi$
$770$	0	0
$771$	−2.10588	−0.0758414
$772$	38.7197	1.39355
$773$	−18.1065	−0.651244	−0.325622	−	0.945500i	$-0.605574\pi$
$773$	−18.1065	−0.651244	−0.325622	+	0.945500i	$0.605574\pi$
$774$	−20.5045	−0.737018
$775$	−60.4513	−2.17148
$776$	1.20872	0.0433905
$777$	0	0
$778$	38.8942	1.39442
$779$	−10.5565	−0.378227
$780$	−39.1982	−1.40352
$781$	−8.59527	−0.307563
$782$	0.730181	0.0261112
$783$	−5.45091	−0.194800
$784$	0	0
$785$	−2.95337	−0.105410
$786$	13.9155	0.496348
$787$	26.0136	0.927286	0.463643	−	0.886022i	$-0.346542\pi$
$787$	26.0136	0.927286	0.463643	+	0.886022i	$0.346542\pi$
$788$	−5.36583	−0.191150
$789$	6.22719	0.221694
$790$	−61.6551	−2.19359
$791$	0	0
$792$	0.463192	0.0164588
$793$	34.8664	1.23814
$794$	7.36436	0.261351
$795$	17.1592	0.608575
$796$	−32.5778	−1.15469
$797$	−33.3245	−1.18041	−0.590206	−	0.807252i	$-0.700954\pi$
$797$	−33.3245	−1.18041	−0.590206	+	0.807252i	$0.700954\pi$
$798$	0	0
$799$	12.5104	0.442586
$800$	−59.0280	−2.08696
$801$	9.72942	0.343772
$802$	−9.80429	−0.346202
$803$	−40.1747	−1.41773
$804$	−5.00435	−0.176490
$805$	0	0
$806$	92.3040	3.25127
$807$	−19.8838	−0.699944
$808$	0.0782993	0.00275456
$809$	−33.5506	−1.17958	−0.589788	−	0.807558i	$-0.700789\pi$
$809$	−33.5506	−1.17958	−0.589788	+	0.807558i	$0.700789\pi$
$810$	−7.00357	−0.246080
$811$	−29.0009	−1.01836	−0.509179	−	0.860661i	$-0.670051\pi$
$811$	−29.0009	−1.01836	−0.509179	+	0.860661i	$0.670051\pi$
$812$	0	0
$813$	5.79241	0.203149
$814$	−20.0558	−0.702956
$815$	−14.4506	−0.506182
$816$	−12.7635	−0.446812
$817$	−26.8619	−0.939780
$818$	−54.4893	−1.90517
$819$	0	0
$820$	27.8174	0.971426
$821$	−50.1067	−1.74874	−0.874368	−	0.485263i	$-0.838724\pi$
$821$	−50.1067	−1.74874	−0.874368	+	0.485263i	$0.838724\pi$
$822$	8.90705	0.310669
$823$	−54.4376	−1.89758	−0.948788	−	0.315913i	$-0.897689\pi$
$823$	−54.4376	−1.89758	−0.948788	+	0.315913i	$0.897689\pi$
$824$	0.814178	0.0283632
$825$	30.9976	1.07920
$826$	0	0
$827$	23.7589	0.826179	0.413090	−	0.910690i	$-0.364450\pi$
$827$	23.7589	0.826179	0.413090	+	0.910690i	$0.364450\pi$
$828$	−0.230096	−0.00799639
$829$	−50.3522	−1.74880	−0.874402	−	0.485202i	$-0.838746\pi$
$829$	−50.3522	−1.74880	−0.874402	+	0.485202i	$0.838746\pi$
$830$	−15.8859	−0.551409
$831$	6.35047	0.220295
$832$	43.1464	1.49583
$833$	0	0
$834$	−9.39312	−0.325257
$835$	−16.5261	−0.571910
$836$	−21.0827	−0.729162
$837$	8.12903	0.280980
$838$	3.02420	0.104469
$839$	13.0501	0.450540	0.225270	−	0.974296i	$-0.427674\pi$
$839$	13.0501	0.450540	0.225270	+	0.974296i	$0.427674\pi$
$840$	0	0
$841$	0.712440	0.0245669
$842$	−51.7306	−1.78275
$843$	30.3985	1.04698
$844$	−32.2934	−1.11158
$845$	−69.4395	−2.38879
$846$	−7.99807	−0.274979
$847$	0	0
$848$	−19.9923	−0.686537
$849$	−16.4389	−0.564181
$850$	−45.8771	−1.57357
$851$	−0.286755	−0.00982984
$852$	−4.00872	−0.137337
$853$	12.7386	0.436163	0.218081	−	0.975931i	$-0.430020\pi$
$853$	12.7386	0.436163	0.218081	+	0.975931i	$0.430020\pi$
$854$	0	0
$855$	−9.17505	−0.313780
$856$	−1.29210	−0.0441631
$857$	0.897232	0.0306489	0.0153244	−	0.999883i	$-0.495122\pi$
$857$	0.897232	0.0306489	0.0153244	+	0.999883i	$0.495122\pi$
$858$	−47.3307	−1.61584
$859$	15.7189	0.536322	0.268161	−	0.963374i	$-0.413584\pi$
$859$	15.7189	0.536322	0.268161	+	0.963374i	$0.413584\pi$
$860$	70.7836	2.41370
$861$	0	0
$862$	16.8438	0.573702
$863$	−24.6747	−0.839935	−0.419968	−	0.907539i	$-0.637959\pi$
$863$	−24.6747	−0.839935	−0.419968	+	0.907539i	$0.637959\pi$
$864$	7.93763	0.270044
$865$	−56.1988	−1.91082
$866$	69.3614	2.35700
$867$	7.35029	0.249629
$868$	0	0
$869$	−36.6953	−1.24480
$870$	38.1759	1.29428
$871$	−14.7181	−0.498705
$872$	1.39446	0.0472225
$873$	−10.8774	−0.368145
$874$	−0.611552	−0.0206860
$875$	0	0
$876$	−18.7369	−0.633062
$877$	12.9006	0.435622	0.217811	−	0.975991i	$-0.430108\pi$
$877$	12.9006	0.435622	0.217811	+	0.975991i	$0.430108\pi$
$878$	23.3781	0.788973
$879$	−1.83585	−0.0619215
$880$	−60.3979	−2.03601
$881$	−14.7712	−0.497654	−0.248827	−	0.968548i	$-0.580045\pi$
$881$	−14.7712	−0.497654	−0.248827	+	0.968548i	$0.580045\pi$
$882$	0	0
$883$	42.3259	1.42438	0.712190	−	0.701986i	$-0.247703\pi$
$883$	42.3259	1.42438	0.712190	+	0.701986i	$0.247703\pi$
$884$	34.5283	1.16131
$885$	32.7540	1.10101
$886$	56.8751	1.91076
$887$	−6.80265	−0.228411	−0.114205	−	0.993457i	$-0.536432\pi$
$887$	−6.80265	−0.228411	−0.114205	+	0.993457i	$0.536432\pi$
$888$	0.269221	0.00903447
$889$	0	0
$890$	−68.1407	−2.28408
$891$	−4.16832	−0.139644
$892$	28.4720	0.953312
$893$	−10.4779	−0.350629
$894$	−30.3645	−1.01554
$895$	70.6147	2.36039
$896$	0	0
$897$	−0.676727	−0.0225953
$898$	−74.5448	−2.48759
$899$	−44.3106	−1.47784
$900$	14.4569	0.481895
$901$	−15.1149	−0.503552
$902$	33.5887	1.11838
$903$	0	0
$904$	1.27380	0.0423660
$905$	−19.7313	−0.655892
$906$	12.2538	0.407104
$907$	14.0009	0.464891	0.232445	−	0.972609i	$-0.425327\pi$
$907$	14.0009	0.464891	0.232445	+	0.972609i	$0.425327\pi$
$908$	58.0086	1.92508
$909$	−0.704625	−0.0233709
$910$	0	0
$911$	24.6238	0.815822	0.407911	−	0.913022i	$-0.366257\pi$
$911$	24.6238	0.815822	0.407911	+	0.913022i	$0.366257\pi$
$912$	10.6899	0.353977
$913$	−9.45483	−0.312909
$914$	−32.7267	−1.08250
$915$	−21.5052	−0.710942
$916$	51.8123	1.71193
$917$	0	0
$918$	6.16919	0.203614
$919$	44.8940	1.48092	0.740459	−	0.672102i	$-0.234608\pi$
$919$	44.8940	1.48092	0.740459	+	0.672102i	$0.234608\pi$
$920$	−0.0463822	−0.00152918
$921$	21.2566	0.700428
$922$	−13.0407	−0.429472
$923$	−11.7899	−0.388069
$924$	0	0
$925$	18.0167	0.592387
$926$	−34.2014	−1.12393
$927$	−7.32688	−0.240646
$928$	−43.2673	−1.42032
$929$	30.9750	1.01626	0.508129	−	0.861281i	$-0.330337\pi$
$929$	30.9750	1.01626	0.508129	+	0.861281i	$0.330337\pi$
$930$	−56.9322	−1.86688
$931$	0	0
$932$	26.6134	0.871750
$933$	−26.3448	−0.862492
$934$	19.3581	0.633417
$935$	−45.6632	−1.49335
$936$	0.635347	0.0207670
$937$	46.3133	1.51299	0.756495	−	0.653999i	$-0.226910\pi$
$937$	46.3133	1.51299	0.756495	+	0.653999i	$0.226910\pi$
$938$	0	0
$939$	24.4409	0.797598
$940$	27.6102	0.900545
$941$	−25.1150	−0.818725	−0.409362	−	0.912372i	$-0.634249\pi$
$941$	−25.1150	−0.818725	−0.409362	+	0.912372i	$0.634249\pi$
$942$	−1.66318	−0.0541895
$943$	0.480246	0.0156390
$944$	−38.1617	−1.24206
$945$	0	0
$946$	85.4691	2.77884
$947$	17.2026	0.559010	0.279505	−	0.960144i	$-0.409830\pi$
$947$	17.2026	0.559010	0.279505	+	0.960144i	$0.409830\pi$
$948$	−17.1142	−0.555843
$949$	−55.1065	−1.78883
$950$	38.4236	1.24663
$951$	−17.2881	−0.560604
$952$	0	0
$953$	−15.8945	−0.514874	−0.257437	−	0.966295i	$-0.582878\pi$
$953$	−15.8945	−0.514874	−0.257437	+	0.966295i	$0.582878\pi$
$954$	9.66318	0.312857
$955$	16.5673	0.536105
$956$	−48.3092	−1.56243
$957$	22.7211	0.734470
$958$	29.0972	0.940086
$959$	0	0
$960$	−26.6123	−0.858907
$961$	35.0811	1.13165
$962$	−27.5100	−0.886959
$963$	11.6278	0.374700
$964$	−32.0966	−1.03376
$965$	−70.2383	−2.26105
$966$	0	0
$967$	−14.5849	−0.469018	−0.234509	−	0.972114i	$-0.575348\pi$
$967$	−14.5849	−0.469018	−0.234509	+	0.972114i	$0.575348\pi$
$968$	−0.708386	−0.0227684
$969$	8.08197	0.259630
$970$	76.1807	2.44602
$971$	−18.5281	−0.594596	−0.297298	−	0.954785i	$-0.596085\pi$
$971$	−18.5281	−0.594596	−0.297298	+	0.954785i	$0.596085\pi$
$972$	−1.94405	−0.0623553
$973$	0	0
$974$	−62.7847	−2.01175
$975$	42.5185	1.36168
$976$	25.0558	0.802017
$977$	7.82128	0.250225	0.125112	−	0.992143i	$-0.460071\pi$
$977$	7.82128	0.250225	0.125112	+	0.992143i	$0.460071\pi$
$978$	−8.13781	−0.260218
$979$	−40.5553	−1.29615
$980$	0	0
$981$	−12.5490	−0.400657
$982$	−32.3180	−1.03131
$983$	52.3256	1.66893	0.834463	−	0.551064i	$-0.185778\pi$
$983$	52.3256	1.66893	0.834463	+	0.551064i	$0.185778\pi$
$984$	−0.450880	−0.0143735
$985$	9.73372	0.310142
$986$	−33.6277	−1.07092
$987$	0	0
$988$	−28.9186	−0.920024
$989$	1.22202	0.0388581
$990$	29.1931	0.927818
$991$	−5.56223	−0.176690	−0.0883451	−	0.996090i	$-0.528158\pi$
$991$	−5.56223	−0.176690	−0.0883451	+	0.996090i	$0.528158\pi$
$992$	64.5252	2.04868
$993$	7.32284	0.232383
$994$	0	0
$995$	59.0968	1.87349
$996$	−4.40961	−0.139724
$997$	24.2559	0.768192	0.384096	−	0.923293i	$-0.374513\pi$
$997$	24.2559	0.768192	0.384096	+	0.923293i	$0.374513\pi$
$998$	−35.8550	−1.13497
$999$	−2.42275	−0.0766525

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7203.2.a.g.1.14		18
7.6	odd	2		7203.2.a.h.1.14		18
49.6	odd	14		147.2.i.b.85.2	yes	36
49.41	odd	14		147.2.i.b.64.2	✓	36
147.41	even	14		441.2.u.d.64.5		36
147.104	even	14		441.2.u.d.379.5		36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.2.i.b.64.2	✓	36	49.41	odd	14
147.2.i.b.85.2	yes	36	49.6	odd	14
441.2.u.d.64.5		36	147.41	even	14
441.2.u.d.379.5		36	147.104	even	14
7203.2.a.g.1.14		18	1.1	even	1	trivial
7203.2.a.h.1.14		18	7.6	odd	2

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:	\( N \)	\(=\)	\( 7203 = 3 \cdot 7^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7203.a (trivial)

\(f(q)\)	\(=\)	\(q+1.98596 q^{2} -1.00000 q^{3} +1.94405 q^{4} -3.52654 q^{5} -1.98596 q^{6} -0.111122 q^{8} +1.00000 q^{9} -7.00357 q^{10} -4.16832 q^{11} -1.94405 q^{12} -5.71756 q^{13} +3.52654 q^{15} -4.10878 q^{16} -3.10640 q^{17} +1.98596 q^{18} +2.60172 q^{19} -6.85576 q^{20} -8.27812 q^{22} -0.118359 q^{23} +0.111122 q^{24} +7.43648 q^{25} -11.3549 q^{26} -1.00000 q^{27} +5.45091 q^{29} +7.00357 q^{30} -8.12903 q^{31} -7.93763 q^{32} +4.16832 q^{33} -6.16919 q^{34} +1.94405 q^{36} +2.42275 q^{37} +5.16691 q^{38} +5.71756 q^{39} +0.391876 q^{40} -4.05753 q^{41} -10.3247 q^{43} -8.10340 q^{44} -3.52654 q^{45} -0.235057 q^{46} -4.02730 q^{47} +4.10878 q^{48} +14.7686 q^{50} +3.10640 q^{51} -11.1152 q^{52} +4.86574 q^{53} -1.98596 q^{54} +14.6997 q^{55} -2.60172 q^{57} +10.8253 q^{58} +9.28785 q^{59} +6.85576 q^{60} -6.09812 q^{61} -16.1439 q^{62} -7.54628 q^{64} +20.1632 q^{65} +8.27812 q^{66} +2.57419 q^{67} -6.03898 q^{68} +0.118359 q^{69} +2.06205 q^{71} -0.111122 q^{72} +9.63811 q^{73} +4.81149 q^{74} -7.43648 q^{75} +5.05786 q^{76} +11.3549 q^{78} +8.80338 q^{79} +14.4898 q^{80} +1.00000 q^{81} -8.05809 q^{82} +2.26826 q^{83} +10.9548 q^{85} -20.5045 q^{86} -5.45091 q^{87} +0.463192 q^{88} +9.72942 q^{89} -7.00357 q^{90} -0.230096 q^{92} +8.12903 q^{93} -7.99807 q^{94} -9.17505 q^{95} +7.93763 q^{96} -10.8774 q^{97} -4.16832 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 3 q^{2} - 18 q^{3} + 27 q^{4} + 2 q^{5} - 3 q^{6} + 3 q^{8} + 18 q^{9} + 2 q^{10} - 27 q^{12} - q^{13} - 2 q^{15} + 37 q^{16} + 5 q^{17} + 3 q^{18} - 3 q^{19} + 19 q^{20} + 4 q^{23} - 3 q^{24} + 34 q^{25}+ \cdots - 86 q^{97}+O(q^{100}) \) 18 * q + 3 * q^2 - 18 * q^3 + 27 * q^4 + 2 * q^5 - 3 * q^6 + 3 * q^8 + 18 * q^9 + 2 * q^10 - 27 * q^12 - q^13 - 2 * q^15 + 37 * q^16 + 5 * q^17 + 3 * q^18 - 3 * q^19 + 19 * q^20 + 4 * q^23 - 3 * q^24 + 34 * q^25 - 3 * q^26 - 18 * q^27 + 10 * q^29 - 2 * q^30 - 8 * q^31 - q^32 - 3 * q^34 + 27 * q^36 + 9 * q^37 - 5 * q^38 + q^39 + 14 * q^40 - q^41 + 4 * q^43 - 13 * q^44 + 2 * q^45 - 2 * q^46 + 21 * q^47 - 37 * q^48 + 118 * q^50 - 5 * q^51 - 108 * q^52 + 89 * q^53 - 3 * q^54 - 2 * q^55 + 3 * q^57 + 103 * q^58 - 17 * q^59 - 19 * q^60 - 77 * q^61 + 37 * q^62 + 35 * q^64 + 103 * q^65 - 11 * q^67 + 55 * q^68 - 4 * q^69 - 3 * q^71 + 3 * q^72 + 9 * q^73 + 110 * q^74 - 34 * q^75 - 121 * q^76 + 3 * q^78 - 7 * q^79 + 56 * q^80 + 18 * q^81 - q^82 + q^83 + 104 * q^85 - 20 * q^86 - 10 * q^87 - 17 * q^88 + 15 * q^89 + 2 * q^90 - 7 * q^92 + 8 * q^93 + 6 * q^94 - 19 * q^95 + q^96 - 86 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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