LMFDB - Embedded newform 7203.2.a.p.1.11

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 = [36,0,36,24] 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:	$1$
Dimension:	$36$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
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)

$f(q)$	$=$	\(q-1.21757 q^{2} +1.00000 q^{3} -0.517511 q^{4} -3.57513 q^{5} -1.21757 q^{6} +3.06526 q^{8} +1.00000 q^{9} +4.35299 q^{10} -4.28056 q^{11} -0.517511 q^{12} -4.08072 q^{13} -3.57513 q^{15} -2.69716 q^{16} -5.83413 q^{17} -1.21757 q^{18} -3.43868 q^{19} +1.85017 q^{20} +5.21190 q^{22} +7.62869 q^{23} +3.06526 q^{24} +7.78158 q^{25} +4.96858 q^{26} +1.00000 q^{27} +10.1967 q^{29} +4.35299 q^{30} +1.78887 q^{31} -2.84652 q^{32} -4.28056 q^{33} +7.10349 q^{34} -0.517511 q^{36} +7.88083 q^{37} +4.18685 q^{38} -4.08072 q^{39} -10.9587 q^{40} +2.00531 q^{41} +2.85285 q^{43} +2.21524 q^{44} -3.57513 q^{45} -9.28851 q^{46} +6.11940 q^{47} -2.69716 q^{48} -9.47465 q^{50} -5.83413 q^{51} +2.11182 q^{52} +1.26956 q^{53} -1.21757 q^{54} +15.3036 q^{55} -3.43868 q^{57} -12.4152 q^{58} +7.94737 q^{59} +1.85017 q^{60} -8.44502 q^{61} -2.17808 q^{62} +8.86017 q^{64} +14.5891 q^{65} +5.21190 q^{66} -6.77752 q^{67} +3.01923 q^{68} +7.62869 q^{69} +9.18136 q^{71} +3.06526 q^{72} -5.97461 q^{73} -9.59550 q^{74} +7.78158 q^{75} +1.77956 q^{76} +4.96858 q^{78} -6.86811 q^{79} +9.64270 q^{80} +1.00000 q^{81} -2.44161 q^{82} +0.480942 q^{83} +20.8578 q^{85} -3.47356 q^{86} +10.1967 q^{87} -13.1210 q^{88} +4.26669 q^{89} +4.35299 q^{90} -3.94794 q^{92} +1.78887 q^{93} -7.45082 q^{94} +12.2937 q^{95} -2.84652 q^{96} +1.03105 q^{97} -4.28056 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q + 36 q^{3} + 24 q^{4} - 24 q^{5} + 36 q^{9} - 24 q^{10} + 24 q^{12} - 48 q^{13} - 24 q^{15} + 4 q^{16} - 48 q^{17} - 24 q^{19} - 48 q^{20} - 16 q^{22} + 4 q^{23} + 16 q^{25} - 24 q^{26} + 36 q^{27}+ \cdots - 88 q^{97}+O(q^{100}) $ 36 * q + 36 * q^3 + 24 * q^4 - 24 * q^5 + 36 * q^9 - 24 * q^10 + 24 * q^12 - 48 * q^13 - 24 * q^15 + 4 * q^16 - 48 * q^17 - 24 * q^19 - 48 * q^20 - 16 * q^22 + 4 * q^23 + 16 * q^25 - 24 * q^26 + 36 * q^27 + 4 * q^29 - 24 * q^30 - 24 * q^31 + 20 * q^32 - 24 * q^34 + 24 * q^36 - 24 * q^37 - 44 * q^38 - 48 * q^39 - 64 * q^40 - 44 * q^41 - 44 * q^43 + 20 * q^44 - 24 * q^45 - 44 * q^46 - 44 * q^47 + 4 * q^48 + 44 * q^50 - 48 * q^51 - 80 * q^52 + 24 * q^53 - 40 * q^55 - 24 * q^57 + 8 * q^58 - 20 * q^59 - 48 * q^60 - 88 * q^61 - 24 * q^62 - 40 * q^64 + 40 * q^65 - 16 * q^66 - 40 * q^67 - 68 * q^68 + 4 * q^69 + 16 * q^71 - 88 * q^73 + 48 * q^74 + 16 * q^75 - 40 * q^76 - 24 * q^78 - 108 * q^79 - 32 * q^80 + 36 * q^81 - 48 * q^82 - 64 * q^83 - 20 * q^85 - 44 * q^86 + 4 * q^87 - 120 * q^88 - 40 * q^89 - 24 * q^90 - 28 * q^92 - 24 * q^93 - 24 * q^94 + 28 * q^95 + 20 * q^96 - 88 * 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 $

$2$	−1.21757	−0.860956	−0.430478	−	0.902601i	$-0.641655\pi$
$2$	−1.21757	−0.860956	−0.430478	+	0.902601i	$0.641655\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−0.517511	−0.258756
$5$	−3.57513	−1.59885	−0.799424	−	0.600767i	$-0.794862\pi$
$5$	−3.57513	−1.59885	−0.799424	+	0.600767i	$0.794862\pi$
$6$	−1.21757	−0.497073
$7$	0	0
$7$	0	0
$8$	3.06526	1.08373
$9$	1.00000	0.333333
$10$	4.35299	1.37654
$11$	−4.28056	−1.29064	−0.645319	−	0.763913i	$-0.723276\pi$
$11$	−4.28056	−1.29064	−0.645319	+	0.763913i	$0.723276\pi$
$12$	−0.517511	−0.149393
$13$	−4.08072	−1.13179	−0.565894	−	0.824478i	$-0.691469\pi$
$13$	−4.08072	−1.13179	−0.565894	+	0.824478i	$0.691469\pi$
$14$	0	0
$15$	−3.57513	−0.923095
$16$	−2.69716	−0.674290
$17$	−5.83413	−1.41498	−0.707492	−	0.706721i	$-0.750174\pi$
$17$	−5.83413	−1.41498	−0.707492	+	0.706721i	$0.750174\pi$
$18$	−1.21757	−0.286985
$19$	−3.43868	−0.788888	−0.394444	−	0.918920i	$-0.629063\pi$
$19$	−3.43868	−0.788888	−0.394444	+	0.918920i	$0.629063\pi$
$20$	1.85017	0.413711
$21$	0	0
$22$	5.21190	1.11118
$23$	7.62869	1.59069	0.795346	−	0.606155i	$-0.207289\pi$
$23$	7.62869	1.59069	0.795346	+	0.606155i	$0.207289\pi$
$24$	3.06526	0.625693
$25$	7.78158	1.55632
$26$	4.96858	0.974420
$27$	1.00000	0.192450
$28$	0	0
$29$	10.1967	1.89347	0.946735	−	0.322012i	$-0.104359\pi$
$29$	10.1967	1.89347	0.946735	+	0.322012i	$0.104359\pi$
$30$	4.35299	0.794744
$31$	1.78887	0.321290	0.160645	−	0.987012i	$-0.448643\pi$
$31$	1.78887	0.321290	0.160645	+	0.987012i	$0.448643\pi$
$32$	−2.84652	−0.503199
$33$	−4.28056	−0.745150
$34$	7.10349	1.21824
$35$	0	0
$36$	−0.517511	−0.0862519
$37$	7.88083	1.29560	0.647800	−	0.761810i	$-0.275689\pi$
$37$	7.88083	1.29560	0.647800	+	0.761810i	$0.275689\pi$
$38$	4.18685	0.679197
$39$	−4.08072	−0.653438
$40$	−10.9587	−1.73272
$41$	2.00531	0.313176	0.156588	−	0.987664i	$-0.449950\pi$
$41$	2.00531	0.313176	0.156588	+	0.987664i	$0.449950\pi$
$42$	0	0
$43$	2.85285	0.435056	0.217528	−	0.976054i	$-0.430201\pi$
$43$	2.85285	0.435056	0.217528	+	0.976054i	$0.430201\pi$
$44$	2.21524	0.333960
$45$	−3.57513	−0.532949
$46$	−9.28851	−1.36952
$47$	6.11940	0.892606	0.446303	−	0.894882i	$-0.352740\pi$
$47$	6.11940	0.892606	0.446303	+	0.894882i	$0.352740\pi$
$48$	−2.69716	−0.389301
$49$	0	0
$50$	−9.47465	−1.33992
$51$	−5.83413	−0.816942
$52$	2.11182	0.292857
$53$	1.26956	0.174388	0.0871939	−	0.996191i	$-0.472210\pi$
$53$	1.26956	0.174388	0.0871939	+	0.996191i	$0.472210\pi$
$54$	−1.21757	−0.165691
$55$	15.3036	2.06353
$56$	0	0
$57$	−3.43868	−0.455464
$58$	−12.4152	−1.63019
$59$	7.94737	1.03466	0.517329	−	0.855786i	$-0.326926\pi$
$59$	7.94737	1.03466	0.517329	+	0.855786i	$0.326926\pi$
$60$	1.85017	0.238856
$61$	−8.44502	−1.08127	−0.540637	−	0.841256i	$-0.681817\pi$
$61$	−8.44502	−1.08127	−0.540637	+	0.841256i	$0.681817\pi$
$62$	−2.17808	−0.276616
$63$	0	0
$64$	8.86017	1.10752
$65$	14.5891	1.80956
$66$	5.21190	0.641541
$67$	−6.77752	−0.828006	−0.414003	−	0.910275i	$-0.635870\pi$
$67$	−6.77752	−0.828006	−0.414003	+	0.910275i	$0.635870\pi$
$68$	3.01923	0.366135
$69$	7.62869	0.918387
$70$	0	0
$71$	9.18136	1.08963	0.544814	−	0.838557i	$-0.316600\pi$
$71$	9.18136	1.08963	0.544814	+	0.838557i	$0.316600\pi$
$72$	3.06526	0.361244
$73$	−5.97461	−0.699275	−0.349638	−	0.936885i	$-0.613695\pi$
$73$	−5.97461	−0.699275	−0.349638	+	0.936885i	$0.613695\pi$
$74$	−9.59550	−1.11545
$75$	7.78158	0.898539
$76$	1.77956	0.204129
$77$	0	0
$78$	4.96858	0.562581
$79$	−6.86811	−0.772722	−0.386361	−	0.922348i	$-0.626268\pi$
$79$	−6.86811	−0.772722	−0.386361	+	0.922348i	$0.626268\pi$
$80$	9.64270	1.07809
$81$	1.00000	0.111111
$82$	−2.44161	−0.269631
$83$	0.480942	0.0527902	0.0263951	−	0.999652i	$-0.491597\pi$
$83$	0.480942	0.0527902	0.0263951	+	0.999652i	$0.491597\pi$
$84$	0	0
$85$	20.8578	2.26235
$86$	−3.47356	−0.374564
$87$	10.1967	1.09320
$88$	−13.1210	−1.39871
$89$	4.26669	0.452268	0.226134	−	0.974096i	$-0.427391\pi$
$89$	4.26669	0.452268	0.226134	+	0.974096i	$0.427391\pi$
$90$	4.35299	0.458846
$91$	0	0
$92$	−3.94794	−0.411601
$93$	1.78887	0.185497
$94$	−7.45082	−0.768494
$95$	12.2937	1.26131
$96$	−2.84652	−0.290522
$97$	1.03105	0.104687	0.0523437	−	0.998629i	$-0.483331\pi$
$97$	1.03105	0.104687	0.0523437	+	0.998629i	$0.483331\pi$
$98$	0	0
$99$	−4.28056	−0.430213
$100$	−4.02705	−0.402705
$101$	−11.9846	−1.19251	−0.596257	−	0.802794i	$-0.703346\pi$
$101$	−11.9846	−1.19251	−0.596257	+	0.802794i	$0.703346\pi$
$102$	7.10349	0.703351
$103$	−8.99004	−0.885815	−0.442908	−	0.896567i	$-0.646053\pi$
$103$	−8.99004	−0.885815	−0.442908	+	0.896567i	$0.646053\pi$
$104$	−12.5085	−1.22656
$105$	0	0
$106$	−1.54579	−0.150140
$107$	−6.27103	−0.606244	−0.303122	−	0.952952i	$-0.598029\pi$
$107$	−6.27103	−0.606244	−0.303122	+	0.952952i	$0.598029\pi$
$108$	−0.517511	−0.0497975
$109$	10.1820	0.975257	0.487628	−	0.873051i	$-0.337862\pi$
$109$	10.1820	0.975257	0.487628	+	0.873051i	$0.337862\pi$
$110$	−18.6333	−1.77661
$111$	7.88083	0.748015
$112$	0	0
$113$	8.65989	0.814654	0.407327	−	0.913282i	$-0.366461\pi$
$113$	8.65989	0.814654	0.407327	+	0.913282i	$0.366461\pi$
$114$	4.18685	0.392135
$115$	−27.2736	−2.54328
$116$	−5.27688	−0.489946
$117$	−4.08072	−0.377263
$118$	−9.67651	−0.890795
$119$	0	0
$120$	−10.9587	−1.00039
$121$	7.32321	0.665746
$122$	10.2824	0.930929
$123$	2.00531	0.180812
$124$	−0.925758	−0.0831355
$125$	−9.94450	−0.889463
$126$	0	0
$127$	−17.9284	−1.59089	−0.795445	−	0.606026i	$-0.792763\pi$
$127$	−17.9284	−1.59089	−0.795445	+	0.606026i	$0.792763\pi$
$128$	−5.09488	−0.450328
$129$	2.85285	0.251180
$130$	−17.7633	−1.55795
$131$	−9.33404	−0.815519	−0.407760	−	0.913089i	$-0.633690\pi$
$131$	−9.33404	−0.815519	−0.407760	+	0.913089i	$0.633690\pi$
$132$	2.21524	0.192812
$133$	0	0
$134$	8.25214	0.712877
$135$	−3.57513	−0.307698
$136$	−17.8831	−1.53347
$137$	−17.5844	−1.50234	−0.751170	−	0.660109i	$-0.770510\pi$
$137$	−17.5844	−1.50234	−0.751170	+	0.660109i	$0.770510\pi$
$138$	−9.28851	−0.790690
$139$	9.03696	0.766505	0.383252	−	0.923644i	$-0.374804\pi$
$139$	9.03696	0.766505	0.383252	+	0.923644i	$0.374804\pi$
$140$	0	0
$141$	6.11940	0.515346
$142$	−11.1790	−0.938121
$143$	17.4678	1.46073
$144$	−2.69716	−0.224763
$145$	−36.4544	−3.02737
$146$	7.27454	0.602045
$147$	0	0
$148$	−4.07842	−0.335244
$149$	−11.8293	−0.969093	−0.484547	−	0.874765i	$-0.661015\pi$
$149$	−11.8293	−0.969093	−0.484547	+	0.874765i	$0.661015\pi$
$150$	−9.47465	−0.773602
$151$	11.1186	0.904823	0.452411	−	0.891809i	$-0.350564\pi$
$151$	11.1186	0.904823	0.452411	+	0.891809i	$0.350564\pi$
$152$	−10.5404	−0.854943
$153$	−5.83413	−0.471662
$154$	0	0
$155$	−6.39543	−0.513693
$156$	2.11182	0.169081
$157$	3.30716	0.263940	0.131970	−	0.991254i	$-0.457870\pi$
$157$	3.30716	0.263940	0.131970	+	0.991254i	$0.457870\pi$
$158$	8.36244	0.665280
$159$	1.26956	0.100683
$160$	10.1767	0.804539
$161$	0	0
$162$	−1.21757	−0.0956617
$163$	−1.67951	−0.131549	−0.0657745	−	0.997835i	$-0.520952\pi$
$163$	−1.67951	−0.131549	−0.0657745	+	0.997835i	$0.520952\pi$
$164$	−1.03777	−0.0810361
$165$	15.3036	1.19138
$166$	−0.585583	−0.0454500
$167$	20.5490	1.59013	0.795066	−	0.606522i	$-0.207436\pi$
$167$	20.5490	1.59013	0.795066	+	0.606522i	$0.207436\pi$
$168$	0	0
$169$	3.65229	0.280945
$170$	−25.3959	−1.94778
$171$	−3.43868	−0.262963
$172$	−1.47638	−0.112573
$173$	25.2306	1.91825	0.959125	−	0.282982i	$-0.0913238\pi$
$173$	25.2306	1.91825	0.959125	+	0.282982i	$0.0913238\pi$
$174$	−12.4152	−0.941193
$175$	0	0
$176$	11.5454	0.870264
$177$	7.94737	0.597361
$178$	−5.19501	−0.389383
$179$	−0.901383	−0.0673725	−0.0336863	−	0.999432i	$-0.510725\pi$
$179$	−0.901383	−0.0673725	−0.0336863	+	0.999432i	$0.510725\pi$
$180$	1.85017	0.137904
$181$	−13.7277	−1.02037	−0.510185	−	0.860065i	$-0.670423\pi$
$181$	−13.7277	−1.02037	−0.510185	+	0.860065i	$0.670423\pi$
$182$	0	0
$183$	−8.44502	−0.624274
$184$	23.3839	1.72389
$185$	−28.1750	−2.07147
$186$	−2.17808	−0.159704
$187$	24.9734	1.82623
$188$	−3.16686	−0.230967
$189$	0	0
$190$	−14.9686	−1.08593
$191$	17.1789	1.24302	0.621511	−	0.783405i	$-0.286519\pi$
$191$	17.1789	1.24302	0.621511	+	0.783405i	$0.286519\pi$
$192$	8.86017	0.639428
$193$	0.985028	0.0709039	0.0354519	−	0.999371i	$-0.488713\pi$
$193$	0.985028	0.0709039	0.0354519	+	0.999371i	$0.488713\pi$
$194$	−1.25538	−0.0901312
$195$	14.5891	1.04475
$196$	0	0
$197$	−6.14672	−0.437935	−0.218968	−	0.975732i	$-0.570269\pi$
$197$	−6.14672	−0.437935	−0.218968	+	0.975732i	$0.570269\pi$
$198$	5.21190	0.370394
$199$	−16.9486	−1.20146	−0.600728	−	0.799453i	$-0.705123\pi$
$199$	−16.9486	−1.20146	−0.600728	+	0.799453i	$0.705123\pi$
$200$	23.8525	1.68663
$201$	−6.77752	−0.478050
$202$	14.5922	1.02670
$203$	0	0
$204$	3.01923	0.211388
$205$	−7.16924	−0.500721
$206$	10.9460	0.762647
$207$	7.62869	0.530231
$208$	11.0064	0.763154
$209$	14.7195	1.01817
$210$	0	0
$211$	−20.6562	−1.42203	−0.711017	−	0.703175i	$-0.751765\pi$
$211$	−20.6562	−1.42203	−0.711017	+	0.703175i	$0.751765\pi$
$212$	−0.657013	−0.0451238
$213$	9.18136	0.629097
$214$	7.63545	0.521949
$215$	−10.1993	−0.695589
$216$	3.06526	0.208564
$217$	0	0
$218$	−12.3973	−0.839653
$219$	−5.97461	−0.403727
$220$	−7.91977	−0.533951
$221$	23.8075	1.60146
$222$	−9.59550	−0.644008
$223$	18.7904	1.25830	0.629150	−	0.777284i	$-0.283403\pi$
$223$	18.7904	1.25830	0.629150	+	0.777284i	$0.283403\pi$
$224$	0	0
$225$	7.78158	0.518772
$226$	−10.5441	−0.701381
$227$	−5.08160	−0.337277	−0.168639	−	0.985678i	$-0.553937\pi$
$227$	−5.08160	−0.337277	−0.168639	+	0.985678i	$0.553937\pi$
$228$	1.77956	0.117854
$229$	25.3656	1.67621	0.838103	−	0.545511i	$-0.183665\pi$
$229$	25.3656	1.67621	0.838103	+	0.545511i	$0.183665\pi$
$230$	33.2076	2.18965
$231$	0	0
$232$	31.2554	2.05202
$233$	−24.5975	−1.61144	−0.805718	−	0.592300i	$-0.798220\pi$
$233$	−24.5975	−1.61144	−0.805718	+	0.592300i	$0.798220\pi$
$234$	4.96858	0.324807
$235$	−21.8777	−1.42714
$236$	−4.11285	−0.267724
$237$	−6.86811	−0.446132
$238$	0	0
$239$	12.5318	0.810617	0.405309	−	0.914180i	$-0.367164\pi$
$239$	12.5318	0.810617	0.405309	+	0.914180i	$0.367164\pi$
$240$	9.64270	0.622434
$241$	7.30292	0.470422	0.235211	−	0.971944i	$-0.424422\pi$
$241$	7.30292	0.470422	0.235211	+	0.971944i	$0.424422\pi$
$242$	−8.91655	−0.573178
$243$	1.00000	0.0641500
$244$	4.37039	0.279786
$245$	0	0
$246$	−2.44161	−0.155671
$247$	14.0323	0.892854
$248$	5.48333	0.348192
$249$	0.480942	0.0304785
$250$	12.1082	0.765788
$251$	3.30634	0.208695	0.104347	−	0.994541i	$-0.466725\pi$
$251$	3.30634	0.208695	0.104347	+	0.994541i	$0.466725\pi$
$252$	0	0
$253$	−32.6551	−2.05301
$254$	21.8292	1.36969
$255$	20.8578	1.30617
$256$	−11.5170	−0.719809
$257$	12.4915	0.779200	0.389600	−	0.920984i	$-0.372613\pi$
$257$	12.4915	0.779200	0.389600	+	0.920984i	$0.372613\pi$
$258$	−3.47356	−0.216255
$259$	0	0
$260$	−7.55003	−0.468233
$261$	10.1967	0.631157
$262$	11.3649	0.702126
$263$	19.9543	1.23043	0.615217	−	0.788358i	$-0.289068\pi$
$263$	19.9543	1.23043	0.615217	+	0.788358i	$0.289068\pi$
$264$	−13.1210	−0.807543
$265$	−4.53885	−0.278820
$266$	0	0
$267$	4.26669	0.261117
$268$	3.50744	0.214251
$269$	25.3647	1.54651	0.773255	−	0.634095i	$-0.218627\pi$
$269$	25.3647	1.54651	0.773255	+	0.634095i	$0.218627\pi$
$270$	4.35299	0.264915
$271$	−21.5216	−1.30734	−0.653672	−	0.756778i	$-0.726772\pi$
$271$	−21.5216	−1.30734	−0.653672	+	0.756778i	$0.726772\pi$
$272$	15.7356	0.954110
$273$	0	0
$274$	21.4104	1.29345
$275$	−33.3095	−2.00864
$276$	−3.94794	−0.237638
$277$	−20.1066	−1.20809	−0.604045	−	0.796950i	$-0.706445\pi$
$277$	−20.1066	−1.20809	−0.604045	+	0.796950i	$0.706445\pi$
$278$	−11.0032	−0.659927
$279$	1.78887	0.107097
$280$	0	0
$281$	−12.7696	−0.761768	−0.380884	−	0.924623i	$-0.624380\pi$
$281$	−12.7696	−0.761768	−0.380884	+	0.924623i	$0.624380\pi$
$282$	−7.45082	−0.443690
$283$	8.17790	0.486126	0.243063	−	0.970011i	$-0.421848\pi$
$283$	8.17790	0.486126	0.243063	+	0.970011i	$0.421848\pi$
$284$	−4.75146	−0.281947
$285$	12.2937	0.728218
$286$	−21.2683	−1.25762
$287$	0	0
$288$	−2.84652	−0.167733
$289$	17.0371	1.00218
$290$	44.3860	2.60643
$291$	1.03105	0.0604413
$292$	3.09193	0.180941
$293$	−16.8403	−0.983819	−0.491909	−	0.870646i	$-0.663701\pi$
$293$	−16.8403	−0.983819	−0.491909	+	0.870646i	$0.663701\pi$
$294$	0	0
$295$	−28.4129	−1.65426
$296$	24.1568	1.40408
$297$	−4.28056	−0.248383
$298$	14.4030	0.834346
$299$	−31.1306	−1.80033
$300$	−4.02705	−0.232502
$301$	0	0
$302$	−13.5378	−0.779012
$303$	−11.9846	−0.688498
$304$	9.27467	0.531939
$305$	30.1921	1.72879
$306$	7.10349	0.406080
$307$	−26.2616	−1.49883	−0.749415	−	0.662100i	$-0.769665\pi$
$307$	−26.2616	−1.49883	−0.749415	+	0.662100i	$0.769665\pi$
$308$	0	0
$309$	−8.99004	−0.511426
$310$	7.78692	0.442267
$311$	−9.24413	−0.524186	−0.262093	−	0.965043i	$-0.584413\pi$
$311$	−9.24413	−0.524186	−0.262093	+	0.965043i	$0.584413\pi$
$312$	−12.5085	−0.708152
$313$	2.79778	0.158140	0.0790699	−	0.996869i	$-0.474805\pi$
$313$	2.79778	0.158140	0.0790699	+	0.996869i	$0.474805\pi$
$314$	−4.02672	−0.227241
$315$	0	0
$316$	3.55432	0.199946
$317$	9.40383	0.528172	0.264086	−	0.964499i	$-0.414930\pi$
$317$	9.40383	0.528172	0.264086	+	0.964499i	$0.414930\pi$
$318$	−1.54579	−0.0866834
$319$	−43.6474	−2.44379
$320$	−31.6763	−1.77076
$321$	−6.27103	−0.350015
$322$	0	0
$323$	20.0617	1.11626
$324$	−0.517511	−0.0287506
$325$	−31.7544	−1.76142
$326$	2.04492	0.113258
$327$	10.1820	0.563065
$328$	6.14678	0.339399
$329$	0	0
$330$	−18.6333	−1.02573
$331$	3.22397	0.177205	0.0886027	−	0.996067i	$-0.471760\pi$
$331$	3.22397	0.177205	0.0886027	+	0.996067i	$0.471760\pi$
$332$	−0.248893	−0.0136598
$333$	7.88083	0.431867
$334$	−25.0200	−1.36903
$335$	24.2305	1.32386
$336$	0	0
$337$	0.976474	0.0531919	0.0265960	−	0.999646i	$-0.491533\pi$
$337$	0.976474	0.0531919	0.0265960	+	0.999646i	$0.491533\pi$
$338$	−4.44693	−0.241881
$339$	8.65989	0.470341
$340$	−10.7941	−0.585395
$341$	−7.65735	−0.414669
$342$	4.18685	0.226399
$343$	0	0
$344$	8.74474	0.471485
$345$	−27.2736	−1.46836
$346$	−30.7202	−1.65153
$347$	10.3546	0.555866	0.277933	−	0.960600i	$-0.410351\pi$
$347$	10.3546	0.555866	0.277933	+	0.960600i	$0.410351\pi$
$348$	−5.27688	−0.282871
$349$	13.1827	0.705653	0.352826	−	0.935689i	$-0.385221\pi$
$349$	13.1827	0.705653	0.352826	+	0.935689i	$0.385221\pi$
$350$	0	0
$351$	−4.08072	−0.217813
$352$	12.1847	0.649448
$353$	−19.4610	−1.03581	−0.517903	−	0.855440i	$-0.673287\pi$
$353$	−19.4610	−1.03581	−0.517903	+	0.855440i	$0.673287\pi$
$354$	−9.67651	−0.514301
$355$	−32.8246	−1.74215
$356$	−2.20806	−0.117027
$357$	0	0
$358$	1.09750	0.0580048
$359$	−25.6891	−1.35582	−0.677908	−	0.735147i	$-0.737113\pi$
$359$	−25.6891	−1.35582	−0.677908	+	0.735147i	$0.737113\pi$
$360$	−10.9587	−0.577575
$361$	−7.17547	−0.377656
$362$	16.7145	0.878494
$363$	7.32321	0.384369
$364$	0	0
$365$	21.3600	1.11803
$366$	10.2824	0.537472
$367$	31.9898	1.66985	0.834927	−	0.550361i	$-0.185510\pi$
$367$	31.9898	1.66985	0.834927	+	0.550361i	$0.185510\pi$
$368$	−20.5758	−1.07259
$369$	2.00531	0.104392
$370$	34.3052	1.78344
$371$	0	0
$372$	−0.925758	−0.0479983
$373$	−25.2018	−1.30490	−0.652449	−	0.757833i	$-0.726258\pi$
$373$	−25.2018	−1.30490	−0.652449	+	0.757833i	$0.726258\pi$
$374$	−30.4069	−1.57231
$375$	−9.94450	−0.513532
$376$	18.7575	0.967346
$377$	−41.6097	−2.14301
$378$	0	0
$379$	−15.5156	−0.796984	−0.398492	−	0.917172i	$-0.630466\pi$
$379$	−15.5156	−0.796984	−0.398492	+	0.917172i	$0.630466\pi$
$380$	−6.36215	−0.326371
$381$	−17.9284	−0.918500
$382$	−20.9166	−1.07019
$383$	−21.6027	−1.10384	−0.551922	−	0.833896i	$-0.686105\pi$
$383$	−21.6027	−1.10384	−0.551922	+	0.833896i	$0.686105\pi$
$384$	−5.09488	−0.259997
$385$	0	0
$386$	−1.19935	−0.0610451
$387$	2.85285	0.145019
$388$	−0.533581	−0.0270885
$389$	−11.7710	−0.596811	−0.298406	−	0.954439i	$-0.596455\pi$
$389$	−11.7710	−0.596811	−0.298406	+	0.954439i	$0.596455\pi$
$390$	−17.7633	−0.899482
$391$	−44.5068	−2.25081
$392$	0	0
$393$	−9.33404	−0.470840
$394$	7.48409	0.377043
$395$	24.5544	1.23547
$396$	2.21524	0.111320
$397$	−4.58423	−0.230076	−0.115038	−	0.993361i	$-0.536699\pi$
$397$	−4.58423	−0.230076	−0.115038	+	0.993361i	$0.536699\pi$
$398$	20.6362	1.03440
$399$	0	0
$400$	−20.9882	−1.04941
$401$	32.5591	1.62593	0.812963	−	0.582316i	$-0.197853\pi$
$401$	32.5591	1.62593	0.812963	+	0.582316i	$0.197853\pi$
$402$	8.25214	0.411579
$403$	−7.29986	−0.363632
$404$	6.20218	0.308570
$405$	−3.57513	−0.177650
$406$	0	0
$407$	−33.7344	−1.67215
$408$	−17.8831	−0.885346
$409$	13.8885	0.686743	0.343371	−	0.939200i	$-0.388431\pi$
$409$	13.8885	0.686743	0.343371	+	0.939200i	$0.388431\pi$
$410$	8.72909	0.431099
$411$	−17.5844	−0.867377
$412$	4.65245	0.229210
$413$	0	0
$414$	−9.28851	−0.456505
$415$	−1.71943	−0.0844036
$416$	11.6159	0.569515
$417$	9.03696	0.442542
$418$	−17.9221	−0.876598
$419$	−14.7011	−0.718196	−0.359098	−	0.933300i	$-0.616916\pi$
$419$	−14.7011	−0.718196	−0.359098	+	0.933300i	$0.616916\pi$
$420$	0	0
$421$	−34.5466	−1.68370	−0.841849	−	0.539714i	$-0.818532\pi$
$421$	−34.5466	−1.68370	−0.841849	+	0.539714i	$0.818532\pi$
$422$	25.1505	1.22431
$423$	6.11940	0.297535
$424$	3.89154	0.188990
$425$	−45.3987	−2.20216
$426$	−11.1790	−0.541624
$427$	0	0
$428$	3.24533	0.156869
$429$	17.4678	0.843352
$430$	12.4185	0.598871
$431$	−18.3935	−0.885985	−0.442992	−	0.896525i	$-0.646083\pi$
$431$	−18.3935	−0.885985	−0.442992	+	0.896525i	$0.646083\pi$
$432$	−2.69716	−0.129767
$433$	6.26868	0.301253	0.150627	−	0.988591i	$-0.451871\pi$
$433$	6.26868	0.301253	0.150627	+	0.988591i	$0.451871\pi$
$434$	0	0
$435$	−36.4544	−1.74785
$436$	−5.26929	−0.252353
$437$	−26.2326	−1.25488
$438$	7.27454	0.347591
$439$	−7.46208	−0.356146	−0.178073	−	0.984017i	$-0.556986\pi$
$439$	−7.46208	−0.356146	−0.178073	+	0.984017i	$0.556986\pi$
$440$	46.9094	2.23632
$441$	0	0
$442$	−28.9874	−1.37879
$443$	0.715561	0.0339973	0.0169987	−	0.999856i	$-0.494589\pi$
$443$	0.715561	0.0339973	0.0169987	+	0.999856i	$0.494589\pi$
$444$	−4.07842	−0.193553
$445$	−15.2540	−0.723108
$446$	−22.8787	−1.08334
$447$	−11.8293	−0.559506
$448$	0	0
$449$	8.22950	0.388374	0.194187	−	0.980965i	$-0.437793\pi$
$449$	8.22950	0.388374	0.194187	+	0.980965i	$0.437793\pi$
$450$	−9.47465	−0.446639
$451$	−8.58384	−0.404197
$452$	−4.48159	−0.210796
$453$	11.1186	0.522400
$454$	6.18722	0.290381
$455$	0	0
$456$	−10.5404	−0.493602
$457$	0.681998	0.0319025	0.0159513	−	0.999873i	$-0.494922\pi$
$457$	0.681998	0.0319025	0.0159513	+	0.999873i	$0.494922\pi$
$458$	−30.8845	−1.44314
$459$	−5.83413	−0.272314
$460$	14.1144	0.658087
$461$	−24.7133	−1.15101	−0.575506	−	0.817798i	$-0.695195\pi$
$461$	−24.7133	−1.15101	−0.575506	+	0.817798i	$0.695195\pi$
$462$	0	0
$463$	30.1904	1.40307	0.701533	−	0.712637i	$-0.252499\pi$
$463$	30.1904	1.40307	0.701533	+	0.712637i	$0.252499\pi$
$464$	−27.5020	−1.27675
$465$	−6.39543	−0.296581
$466$	29.9493	1.38737
$467$	−27.6734	−1.28057	−0.640287	−	0.768136i	$-0.721185\pi$
$467$	−27.6734	−1.28057	−0.640287	+	0.768136i	$0.721185\pi$
$468$	2.11182	0.0976189
$469$	0	0
$470$	26.6377	1.22870
$471$	3.30716	0.152386
$472$	24.3607	1.12129
$473$	−12.2118	−0.561500
$474$	8.36244	0.384099
$475$	−26.7584	−1.22776
$476$	0	0
$477$	1.26956	0.0581293
$478$	−15.2585	−0.697906
$479$	−6.69003	−0.305675	−0.152838	−	0.988251i	$-0.548841\pi$
$479$	−6.69003	−0.305675	−0.152838	+	0.988251i	$0.548841\pi$
$480$	10.1767	0.464501
$481$	−32.1595	−1.46635
$482$	−8.89185	−0.405013
$483$	0	0
$484$	−3.78984	−0.172266
$485$	−3.68615	−0.167379
$486$	−1.21757	−0.0552303
$487$	26.6335	1.20688	0.603439	−	0.797409i	$-0.293797\pi$
$487$	26.6335	1.20688	0.603439	+	0.797409i	$0.293797\pi$
$488$	−25.8862	−1.17181
$489$	−1.67951	−0.0759499
$490$	0	0
$491$	11.3768	0.513426	0.256713	−	0.966488i	$-0.417360\pi$
$491$	11.3768	0.513426	0.256713	+	0.966488i	$0.417360\pi$
$492$	−1.03777	−0.0467862
$493$	−59.4886	−2.67923
$494$	−17.0854	−0.768707
$495$	15.3036	0.687845
$496$	−4.82485	−0.216642
$497$	0	0
$498$	−0.585583	−0.0262406
$499$	−7.16867	−0.320914	−0.160457	−	0.987043i	$-0.551297\pi$
$499$	−7.16867	−0.320914	−0.160457	+	0.987043i	$0.551297\pi$
$500$	5.14639	0.230154
$501$	20.5490	0.918064
$502$	−4.02572	−0.179677
$503$	−26.2372	−1.16986	−0.584930	−	0.811084i	$-0.698878\pi$
$503$	−26.2372	−1.16986	−0.584930	+	0.811084i	$0.698878\pi$
$504$	0	0
$505$	42.8466	1.90665
$506$	39.7600	1.76755
$507$	3.65229	0.162204
$508$	9.27816	0.411652
$509$	−4.79688	−0.212618	−0.106309	−	0.994333i	$-0.533903\pi$
$509$	−4.79688	−0.212618	−0.106309	+	0.994333i	$0.533903\pi$
$510$	−25.3959	−1.12455
$511$	0	0
$512$	24.2125	1.07005
$513$	−3.43868	−0.151821
$514$	−15.2094	−0.670856
$515$	32.1406	1.41628
$516$	−1.47638	−0.0649942
$517$	−26.1944	−1.15203
$518$	0	0
$519$	25.2306	1.10750
$520$	44.7194	1.96108
$521$	37.8526	1.65835	0.829175	−	0.558988i	$-0.188810\pi$
$521$	37.8526	1.65835	0.829175	+	0.558988i	$0.188810\pi$
$522$	−12.4152	−0.543398
$523$	−5.23475	−0.228900	−0.114450	−	0.993429i	$-0.536511\pi$
$523$	−5.23475	−0.228900	−0.114450	+	0.993429i	$0.536511\pi$
$524$	4.83047	0.211020
$525$	0	0
$526$	−24.2959	−1.05935
$527$	−10.4365	−0.454620
$528$	11.5454	0.502447
$529$	35.1970	1.53030
$530$	5.52640	0.240051
$531$	7.94737	0.344886
$532$	0	0
$533$	−8.18310	−0.354449
$534$	−5.19501	−0.224810
$535$	22.4198	0.969292
$536$	−20.7749	−0.897337
$537$	−0.901383	−0.0388976
$538$	−30.8834	−1.33148
$539$	0	0
$540$	1.85017	0.0796187
$541$	−25.9361	−1.11508	−0.557539	−	0.830151i	$-0.688254\pi$
$541$	−25.9361	−1.11508	−0.557539	+	0.830151i	$0.688254\pi$
$542$	26.2042	1.12556
$543$	−13.7277	−0.589111
$544$	16.6070	0.712019
$545$	−36.4019	−1.55929
$546$	0	0
$547$	−24.2422	−1.03652	−0.518260	−	0.855223i	$-0.673420\pi$
$547$	−24.2422	−1.03652	−0.518260	+	0.855223i	$0.673420\pi$
$548$	9.10015	0.388739
$549$	−8.44502	−0.360425
$550$	40.5568	1.72935
$551$	−35.0630	−1.49374
$552$	23.3839	0.995286
$553$	0	0
$554$	24.4813	1.04011
$555$	−28.1750	−1.19596
$556$	−4.67673	−0.198337
$557$	26.8953	1.13959	0.569796	−	0.821786i	$-0.307022\pi$
$557$	26.8953	1.13959	0.569796	+	0.821786i	$0.307022\pi$
$558$	−2.17808	−0.0922054
$559$	−11.6417	−0.492392
$560$	0	0
$561$	24.9734	1.05438
$562$	15.5479	0.655848
$563$	−16.1177	−0.679281	−0.339640	−	0.940555i	$-0.610305\pi$
$563$	−16.1177	−0.679281	−0.339640	+	0.940555i	$0.610305\pi$
$564$	−3.16686	−0.133349
$565$	−30.9603	−1.30251
$566$	−9.95721	−0.418533
$567$	0	0
$568$	28.1433	1.18086
$569$	−38.1714	−1.60023	−0.800113	−	0.599849i	$-0.795227\pi$
$569$	−38.1714	−1.60023	−0.800113	+	0.599849i	$0.795227\pi$
$570$	−14.9686	−0.626964
$571$	5.65881	0.236814	0.118407	−	0.992965i	$-0.462221\pi$
$571$	5.65881	0.236814	0.118407	+	0.992965i	$0.462221\pi$
$572$	−9.03977	−0.377972
$573$	17.1789	0.717659
$574$	0	0
$575$	59.3633	2.47562
$576$	8.86017	0.369174
$577$	7.83417	0.326141	0.163070	−	0.986614i	$-0.447860\pi$
$577$	7.83417	0.326141	0.163070	+	0.986614i	$0.447860\pi$
$578$	−20.7439	−0.862834
$579$	0.985028	0.0409364
$580$	18.8656	0.783350
$581$	0	0
$582$	−1.25538	−0.0520373
$583$	−5.43444	−0.225071
$584$	−18.3137	−0.757827
$585$	14.5891	0.603186
$586$	20.5043	0.847024
$587$	19.7426	0.814863	0.407432	−	0.913236i	$-0.366424\pi$
$587$	19.7426	0.814863	0.407432	+	0.913236i	$0.366424\pi$
$588$	0	0
$589$	−6.15134	−0.253461
$590$	34.5948	1.42425
$591$	−6.14672	−0.252842
$592$	−21.2558	−0.873610
$593$	15.8243	0.649825	0.324912	−	0.945744i	$-0.394665\pi$
$593$	15.8243	0.649825	0.324912	+	0.945744i	$0.394665\pi$
$594$	5.21190	0.213847
$595$	0	0
$596$	6.12179	0.250758
$597$	−16.9486	−0.693661
$598$	37.9038	1.55000
$599$	23.7065	0.968622	0.484311	−	0.874896i	$-0.339070\pi$
$599$	23.7065	0.968622	0.484311	+	0.874896i	$0.339070\pi$
$600$	23.8525	0.973776
$601$	26.9474	1.09921	0.549603	−	0.835426i	$-0.314779\pi$
$601$	26.9474	1.09921	0.549603	+	0.835426i	$0.314779\pi$
$602$	0	0
$603$	−6.77752	−0.276002
$604$	−5.75402	−0.234128
$605$	−26.1814	−1.06443
$606$	14.5922	0.592767
$607$	−29.9582	−1.21597	−0.607984	−	0.793949i	$-0.708021\pi$
$607$	−29.9582	−1.21597	−0.607984	+	0.793949i	$0.708021\pi$
$608$	9.78829	0.396967
$609$	0	0
$610$	−36.7611	−1.48841
$611$	−24.9715	−1.01024
$612$	3.01923	0.122045
$613$	−39.0642	−1.57779	−0.788894	−	0.614530i	$-0.789346\pi$
$613$	−39.0642	−1.57779	−0.788894	+	0.614530i	$0.789346\pi$
$614$	31.9755	1.29043
$615$	−7.16924	−0.289092
$616$	0	0
$617$	42.6265	1.71608	0.858039	−	0.513584i	$-0.171682\pi$
$617$	42.6265	1.71608	0.858039	+	0.513584i	$0.171682\pi$
$618$	10.9460	0.440315
$619$	8.50193	0.341721	0.170861	−	0.985295i	$-0.445345\pi$
$619$	8.50193	0.341721	0.170861	+	0.985295i	$0.445345\pi$
$620$	3.30971	0.132921
$621$	7.62869	0.306129
$622$	11.2554	0.451301
$623$	0	0
$624$	11.0064	0.440607
$625$	−3.35496	−0.134198
$626$	−3.40650	−0.136151
$627$	14.7195	0.587840
$628$	−1.71149	−0.0682961
$629$	−45.9778	−1.83325
$630$	0	0
$631$	−1.38729	−0.0552273	−0.0276137	−	0.999619i	$-0.508791\pi$
$631$	−1.38729	−0.0552273	−0.0276137	+	0.999619i	$0.508791\pi$
$632$	−21.0525	−0.837424
$633$	−20.6562	−0.821012
$634$	−11.4499	−0.454732
$635$	64.0965	2.54359
$636$	−0.657013	−0.0260523
$637$	0	0
$638$	53.1440	2.10399
$639$	9.18136	0.363209
$640$	18.2149	0.720006
$641$	18.3408	0.724417	0.362209	−	0.932097i	$-0.382023\pi$
$641$	18.3408	0.724417	0.362209	+	0.932097i	$0.382023\pi$
$642$	7.63545	0.301347
$643$	−33.6368	−1.32651	−0.663253	−	0.748395i	$-0.730825\pi$
$643$	−33.6368	−1.32651	−0.663253	+	0.748395i	$0.730825\pi$
$644$	0	0
$645$	−10.1993	−0.401598
$646$	−24.4266	−0.961054
$647$	−38.5595	−1.51593	−0.757965	−	0.652295i	$-0.773806\pi$
$647$	−38.5595	−1.51593	−0.757965	+	0.652295i	$0.773806\pi$
$648$	3.06526	0.120415
$649$	−34.0192	−1.33537
$650$	38.6634	1.51650
$651$	0	0
$652$	0.869163	0.0340391
$653$	−5.54967	−0.217175	−0.108588	−	0.994087i	$-0.534633\pi$
$653$	−5.54967	−0.217175	−0.108588	+	0.994087i	$0.534633\pi$
$654$	−12.3973	−0.484774
$655$	33.3705	1.30389
$656$	−5.40863	−0.211172
$657$	−5.97461	−0.233092
$658$	0	0
$659$	3.51279	0.136839	0.0684194	−	0.997657i	$-0.478204\pi$
$659$	3.51279	0.136839	0.0684194	+	0.997657i	$0.478204\pi$
$660$	−7.91977	−0.308277
$661$	4.08305	0.158812	0.0794061	−	0.996842i	$-0.474698\pi$
$661$	4.08305	0.158812	0.0794061	+	0.996842i	$0.474698\pi$
$662$	−3.92542	−0.152566
$663$	23.8075	0.924605
$664$	1.47421	0.0572105
$665$	0	0
$666$	−9.59550	−0.371818
$667$	77.7872	3.01193
$668$	−10.6344	−0.411456
$669$	18.7904	0.726480
$670$	−29.5025	−1.13978
$671$	36.1494	1.39553
$672$	0	0
$673$	−11.2784	−0.434750	−0.217375	−	0.976088i	$-0.569749\pi$
$673$	−11.2784	−0.434750	−0.217375	+	0.976088i	$0.569749\pi$
$674$	−1.18893	−0.0457959
$675$	7.78158	0.299513
$676$	−1.89010	−0.0726962
$677$	29.1736	1.12123	0.560616	−	0.828076i	$-0.310564\pi$
$677$	29.1736	1.12123	0.560616	+	0.828076i	$0.310564\pi$
$678$	−10.5441	−0.404942
$679$	0	0
$680$	63.9345	2.45178
$681$	−5.08160	−0.194727
$682$	9.32339	0.357011
$683$	40.7152	1.55792	0.778962	−	0.627071i	$-0.215746\pi$
$683$	40.7152	1.55792	0.778962	+	0.627071i	$0.215746\pi$
$684$	1.77956	0.0680430
$685$	62.8667	2.40201
$686$	0	0
$687$	25.3656	0.967758
$688$	−7.69460	−0.293354
$689$	−5.18073	−0.197370
$690$	33.2076	1.26419
$691$	33.8808	1.28889	0.644443	−	0.764653i	$-0.277089\pi$
$691$	33.8808	1.28889	0.644443	+	0.764653i	$0.277089\pi$
$692$	−13.0571	−0.496358
$693$	0	0
$694$	−12.6075	−0.478576
$695$	−32.3083	−1.22552
$696$	31.2554	1.18473
$697$	−11.6992	−0.443140
$698$	−16.0509	−0.607536
$699$	−24.5975	−0.930363
$700$	0	0
$701$	3.77985	0.142763	0.0713814	−	0.997449i	$-0.477259\pi$
$701$	3.77985	0.142763	0.0713814	+	0.997449i	$0.477259\pi$
$702$	4.96858	0.187527
$703$	−27.0997	−1.02208
$704$	−37.9265	−1.42941
$705$	−21.8777	−0.823960
$706$	23.6952	0.891782
$707$	0	0
$708$	−4.11285	−0.154570
$709$	40.3458	1.51522	0.757608	−	0.652709i	$-0.226368\pi$
$709$	40.3458	1.51522	0.757608	+	0.652709i	$0.226368\pi$
$710$	39.9664	1.49991
$711$	−6.86811	−0.257574
$712$	13.0785	0.490138
$713$	13.6467	0.511073
$714$	0	0
$715$	−62.4496	−2.33548
$716$	0.466476	0.0174330
$717$	12.5318	0.468010
$718$	31.2783	1.16730
$719$	−25.1260	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$719$	−25.1260	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$720$	9.64270	0.359362
$721$	0	0
$722$	8.73667	0.325145
$723$	7.30292	0.271599
$724$	7.10423	0.264027
$725$	79.3460	2.94684
$726$	−8.91655	−0.330924
$727$	43.8758	1.62726	0.813631	−	0.581382i	$-0.197488\pi$
$727$	43.8758	1.62726	0.813631	+	0.581382i	$0.197488\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−26.0074	−0.962578
$731$	−16.6439	−0.615598
$732$	4.37039	0.161534
$733$	6.65398	0.245770	0.122885	−	0.992421i	$-0.460785\pi$
$733$	6.65398	0.245770	0.122885	+	0.992421i	$0.460785\pi$
$734$	−38.9500	−1.43767
$735$	0	0
$736$	−21.7153	−0.800435
$737$	29.0116	1.06866
$738$	−2.44161	−0.0898770
$739$	−1.68170	−0.0618624	−0.0309312	−	0.999522i	$-0.509847\pi$
$739$	−1.68170	−0.0618624	−0.0309312	+	0.999522i	$0.509847\pi$
$740$	14.5809	0.536004
$741$	14.0323	0.515489
$742$	0	0
$743$	18.7457	0.687714	0.343857	−	0.939022i	$-0.388266\pi$
$743$	18.7457	0.687714	0.343857	+	0.939022i	$0.388266\pi$
$744$	5.48333	0.201029
$745$	42.2913	1.54943
$746$	30.6850	1.12346
$747$	0.480942	0.0175967
$748$	−12.9240	−0.472548
$749$	0	0
$750$	12.1082	0.442128
$751$	10.3704	0.378423	0.189211	−	0.981936i	$-0.439407\pi$
$751$	10.3704	0.378423	0.189211	+	0.981936i	$0.439407\pi$
$752$	−16.5050	−0.601875
$753$	3.30634	0.120490
$754$	50.6629	1.84504
$755$	−39.7506	−1.44667
$756$	0	0
$757$	−37.5256	−1.36389	−0.681946	−	0.731402i	$-0.738866\pi$
$757$	−37.5256	−1.36389	−0.681946	+	0.731402i	$0.738866\pi$
$758$	18.8914	0.686168
$759$	−32.6551	−1.18530
$760$	37.6835	1.36692
$761$	−5.02938	−0.182315	−0.0911574	−	0.995836i	$-0.529057\pi$
$761$	−5.02938	−0.182315	−0.0911574	+	0.995836i	$0.529057\pi$
$762$	21.8292	0.790788
$763$	0	0
$764$	−8.89028	−0.321639
$765$	20.8578	0.754115
$766$	26.3029	0.950361
$767$	−32.4310	−1.17102
$768$	−11.5170	−0.415582
$769$	−49.5153	−1.78557	−0.892785	−	0.450484i	$-0.851251\pi$
$769$	−49.5153	−1.78557	−0.892785	+	0.450484i	$0.851251\pi$
$770$	0	0
$771$	12.4915	0.449871
$772$	−0.509763	−0.0183468
$773$	−35.3310	−1.27077	−0.635384	−	0.772197i	$-0.719158\pi$
$773$	−35.3310	−1.27077	−0.635384	+	0.772197i	$0.719158\pi$
$774$	−3.47356	−0.124855
$775$	13.9202	0.500028
$776$	3.16044	0.113453
$777$	0	0
$778$	14.3320	0.513828
$779$	−6.89561	−0.247061
$780$	−7.55003	−0.270335
$781$	−39.3014	−1.40631
$782$	54.1904	1.93784
$783$	10.1967	0.364399
$784$	0	0
$785$	−11.8235	−0.422001
$786$	11.3649	0.405373
$787$	12.4323	0.443164	0.221582	−	0.975142i	$-0.428878\pi$
$787$	12.4323	0.443164	0.221582	+	0.975142i	$0.428878\pi$
$788$	3.18099	0.113318
$789$	19.9543	0.710392
$790$	−29.8968	−1.06368
$791$	0	0
$792$	−13.1210	−0.466235
$793$	34.4618	1.22377
$794$	5.58164	0.198085
$795$	−4.53885	−0.160977
$796$	8.77110	0.310884
$797$	−15.6882	−0.555705	−0.277852	−	0.960624i	$-0.589623\pi$
$797$	−15.6882	−0.555705	−0.277852	+	0.960624i	$0.589623\pi$
$798$	0	0
$799$	−35.7014	−1.26302
$800$	−22.1504	−0.783136
$801$	4.26669	0.150756
$802$	−39.6432	−1.39985
$803$	25.5747	0.902511
$804$	3.50744	0.123698
$805$	0	0
$806$	8.88813	0.313071
$807$	25.3647	0.892878
$808$	−36.7360	−1.29237
$809$	−35.7662	−1.25747	−0.628736	−	0.777619i	$-0.716427\pi$
$809$	−35.7662	−1.25747	−0.628736	+	0.777619i	$0.716427\pi$
$810$	4.35299	0.152949
$811$	−15.8131	−0.555271	−0.277636	−	0.960686i	$-0.589551\pi$
$811$	−15.8131	−0.555271	−0.277636	+	0.960686i	$0.589551\pi$
$812$	0	0
$813$	−21.5216	−0.754795
$814$	41.0741	1.43965
$815$	6.00446	0.210327
$816$	15.7356	0.550856
$817$	−9.81006	−0.343210
$818$	−16.9103	−0.591255
$819$	0	0
$820$	3.71016	0.129564
$821$	16.5924	0.579078	0.289539	−	0.957166i	$-0.406498\pi$
$821$	16.5924	0.579078	0.289539	+	0.957166i	$0.406498\pi$
$822$	21.4104	0.746773
$823$	−14.5684	−0.507824	−0.253912	−	0.967227i	$-0.581717\pi$
$823$	−14.5684	−0.507824	−0.253912	+	0.967227i	$0.581717\pi$
$824$	−27.5568	−0.959987
$825$	−33.3095	−1.15969
$826$	0	0
$827$	−17.8032	−0.619077	−0.309539	−	0.950887i	$-0.600175\pi$
$827$	−17.8032	−0.619077	−0.309539	+	0.950887i	$0.600175\pi$
$828$	−3.94794	−0.137200
$829$	−16.2079	−0.562923	−0.281462	−	0.959572i	$-0.590819\pi$
$829$	−16.2079	−0.562923	−0.281462	+	0.959572i	$0.590819\pi$
$830$	2.09354	0.0726677
$831$	−20.1066	−0.697491
$832$	−36.1559	−1.25348
$833$	0	0
$834$	−11.0032	−0.381009
$835$	−73.4656	−2.54238
$836$	−7.61750	−0.263457
$837$	1.78887	0.0618322
$838$	17.8997	0.618335
$839$	−37.2453	−1.28585	−0.642924	−	0.765930i	$-0.722279\pi$
$839$	−37.2453	−1.28585	−0.642924	+	0.765930i	$0.722279\pi$
$840$	0	0
$841$	74.9717	2.58523
$842$	42.0630	1.44959
$843$	−12.7696	−0.439807
$844$	10.6898	0.367959
$845$	−13.0574	−0.449189
$846$	−7.45082	−0.256165
$847$	0	0
$848$	−3.42421	−0.117588
$849$	8.17790	0.280665
$850$	55.2764	1.89596
$851$	60.1204	2.06090
$852$	−4.75146	−0.162782
$853$	−0.715212	−0.0244884	−0.0122442	−	0.999925i	$-0.503898\pi$
$853$	−0.715212	−0.0244884	−0.0122442	+	0.999925i	$0.503898\pi$
$854$	0	0
$855$	12.2937	0.420437
$856$	−19.2223	−0.657006
$857$	13.1835	0.450341	0.225170	−	0.974319i	$-0.427706\pi$
$857$	13.1835	0.450341	0.225170	+	0.974319i	$0.427706\pi$
$858$	−21.2683	−0.726089
$859$	−5.87896	−0.200588	−0.100294	−	0.994958i	$-0.531978\pi$
$859$	−5.87896	−0.200588	−0.100294	+	0.994958i	$0.531978\pi$
$860$	5.27827	0.179988
$861$	0	0
$862$	22.3955	0.762793
$863$	4.56267	0.155315	0.0776575	−	0.996980i	$-0.475256\pi$
$863$	4.56267	0.155315	0.0776575	+	0.996980i	$0.475256\pi$
$864$	−2.84652	−0.0968407
$865$	−90.2029	−3.06699
$866$	−7.63258	−0.259366
$867$	17.0371	0.578610
$868$	0	0
$869$	29.3994	0.997305
$870$	44.3860	1.50482
$871$	27.6572	0.937128
$872$	31.2104	1.05692
$873$	1.03105	0.0348958
$874$	31.9402	1.08039
$875$	0	0
$876$	3.09193	0.104467
$877$	2.65595	0.0896851	0.0448425	−	0.998994i	$-0.485721\pi$
$877$	2.65595	0.0896851	0.0448425	+	0.998994i	$0.485721\pi$
$878$	9.08565	0.306626
$879$	−16.8403	−0.568008
$880$	−41.2762	−1.39142
$881$	−45.8891	−1.54604	−0.773021	−	0.634380i	$-0.781255\pi$
$881$	−45.8891	−1.54604	−0.773021	+	0.634380i	$0.781255\pi$
$882$	0	0
$883$	23.4502	0.789163	0.394581	−	0.918861i	$-0.370890\pi$
$883$	23.4502	0.789163	0.394581	+	0.918861i	$0.370890\pi$
$884$	−12.3206	−0.414388
$885$	−28.4129	−0.955089
$886$	−0.871249	−0.0292702
$887$	−24.2960	−0.815781	−0.407890	−	0.913031i	$-0.633735\pi$
$887$	−24.2960	−0.815781	−0.407890	+	0.913031i	$0.633735\pi$
$888$	24.1568	0.810648
$889$	0	0
$890$	18.5729	0.622564
$891$	−4.28056	−0.143404
$892$	−9.72425	−0.325592
$893$	−21.0427	−0.704165
$894$	14.4030	0.481710
$895$	3.22256	0.107718
$896$	0	0
$897$	−31.1306	−1.03942
$898$	−10.0200	−0.334373
$899$	18.2404	0.608353
$900$	−4.02705	−0.134235
$901$	−7.40679	−0.246756
$902$	10.4515	0.347996
$903$	0	0
$904$	26.5448	0.882867
$905$	49.0783	1.63142
$906$	−13.5378	−0.449763
$907$	−41.7674	−1.38686	−0.693432	−	0.720522i	$-0.743902\pi$
$907$	−41.7674	−1.38686	−0.693432	+	0.720522i	$0.743902\pi$
$908$	2.62978	0.0872724
$909$	−11.9846	−0.397505
$910$	0	0
$911$	−16.3065	−0.540259	−0.270129	−	0.962824i	$-0.587066\pi$
$911$	−16.3065	−0.540259	−0.270129	+	0.962824i	$0.587066\pi$
$912$	9.27467	0.307115
$913$	−2.05870	−0.0681331
$914$	−0.830384	−0.0274666
$915$	30.1921	0.998119
$916$	−13.1270	−0.433728
$917$	0	0
$918$	7.10349	0.234450
$919$	−40.0868	−1.32234	−0.661171	−	0.750235i	$-0.729940\pi$
$919$	−40.0868	−1.32234	−0.661171	+	0.750235i	$0.729940\pi$
$920$	−83.6006	−2.75623
$921$	−26.2616	−0.865350
$922$	30.0903	0.990970
$923$	−37.4666	−1.23323
$924$	0	0
$925$	61.3252	2.01636
$926$	−36.7591	−1.20798
$927$	−8.99004	−0.295272
$928$	−29.0250	−0.952793
$929$	56.1756	1.84306	0.921532	−	0.388303i	$-0.126939\pi$
$929$	56.1756	1.84306	0.921532	+	0.388303i	$0.126939\pi$
$930$	7.78692	0.255343
$931$	0	0
$932$	12.7295	0.416968
$933$	−9.24413	−0.302639
$934$	33.6945	1.10252
$935$	−89.2831	−2.91987
$936$	−12.5085	−0.408852
$937$	18.2937	0.597629	0.298815	−	0.954311i	$-0.403409\pi$
$937$	18.2937	0.597629	0.298815	+	0.954311i	$0.403409\pi$
$938$	0	0
$939$	2.79778	0.0913020
$940$	11.3219	0.369281
$941$	27.0760	0.882652	0.441326	−	0.897347i	$-0.354508\pi$
$941$	27.0760	0.882652	0.441326	+	0.897347i	$0.354508\pi$
$942$	−4.02672	−0.131198
$943$	15.2979	0.498167
$944$	−21.4353	−0.697660
$945$	0	0
$946$	14.8688	0.483427
$947$	−15.8047	−0.513584	−0.256792	−	0.966467i	$-0.582666\pi$
$947$	−15.8047	−0.513584	−0.256792	+	0.966467i	$0.582666\pi$
$948$	3.55432	0.115439
$949$	24.3807	0.791432
$950$	32.5803	1.05704
$951$	9.40383	0.304940
$952$	0	0
$953$	11.7217	0.379702	0.189851	−	0.981813i	$-0.439200\pi$
$953$	11.7217	0.379702	0.189851	+	0.981813i	$0.439200\pi$
$954$	−1.54579	−0.0500467
$955$	−61.4169	−1.98740
$956$	−6.48537	−0.209752
$957$	−43.6474	−1.41092
$958$	8.14561	0.263173
$959$	0	0
$960$	−31.6763	−1.02235
$961$	−27.8000	−0.896773
$962$	39.1566	1.26246
$963$	−6.27103	−0.202081
$964$	−3.77934	−0.121724
$965$	−3.52161	−0.113365
$966$	0	0
$967$	−34.1130	−1.09700	−0.548500	−	0.836151i	$-0.684801\pi$
$967$	−34.1130	−1.09700	−0.548500	+	0.836151i	$0.684801\pi$
$968$	22.4475	0.721491
$969$	20.0617	0.644475
$970$	4.48816	0.144106
$971$	−0.807110	−0.0259014	−0.0129507	−	0.999916i	$-0.504122\pi$
$971$	−0.807110	−0.0259014	−0.0129507	+	0.999916i	$0.504122\pi$
$972$	−0.517511	−0.0165992
$973$	0	0
$974$	−32.4283	−1.03907
$975$	−31.7544	−1.01696
$976$	22.7776	0.729092
$977$	12.3748	0.395906	0.197953	−	0.980212i	$-0.436571\pi$
$977$	12.3748	0.395906	0.197953	+	0.980212i	$0.436571\pi$
$978$	2.04492	0.0653895
$979$	−18.2638	−0.583715
$980$	0	0
$981$	10.1820	0.325086
$982$	−13.8521	−0.442037
$983$	34.9832	1.11579	0.557896	−	0.829911i	$-0.311609\pi$
$983$	34.9832	1.11579	0.557896	+	0.829911i	$0.311609\pi$
$984$	6.14678	0.195952
$985$	21.9753	0.700192
$986$	72.4319	2.30670
$987$	0	0
$988$	−7.26187	−0.231031
$989$	21.7636	0.692041
$990$	−18.6333	−0.592204
$991$	−56.1530	−1.78376	−0.891879	−	0.452273i	$-0.850613\pi$
$991$	−56.1530	−1.78376	−0.891879	+	0.452273i	$0.850613\pi$
$992$	−5.09205	−0.161673
$993$	3.22397	0.102310
$994$	0	0
$995$	60.5936	1.92095
$996$	−0.248893	−0.00788647
$997$	15.2494	0.482955	0.241477	−	0.970406i	$-0.422368\pi$
$997$	15.2494	0.482955	0.241477	+	0.970406i	$0.422368\pi$
$998$	8.72840	0.276293
$999$	7.88083	0.249338

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7203.2.a.p.1.11	yes	36
7.6	odd	2		7203.2.a.o.1.11	✓	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7203.2.a.o.1.11	✓	36	7.6	odd	2
7203.2.a.p.1.11	yes	36	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:	\( N \)	\(=\)	\( 7203 = 3 \cdot 7^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7203.a (trivial)

\(f(q)\)	\(=\)	\(q-1.21757 q^{2} +1.00000 q^{3} -0.517511 q^{4} -3.57513 q^{5} -1.21757 q^{6} +3.06526 q^{8} +1.00000 q^{9} +4.35299 q^{10} -4.28056 q^{11} -0.517511 q^{12} -4.08072 q^{13} -3.57513 q^{15} -2.69716 q^{16} -5.83413 q^{17} -1.21757 q^{18} -3.43868 q^{19} +1.85017 q^{20} +5.21190 q^{22} +7.62869 q^{23} +3.06526 q^{24} +7.78158 q^{25} +4.96858 q^{26} +1.00000 q^{27} +10.1967 q^{29} +4.35299 q^{30} +1.78887 q^{31} -2.84652 q^{32} -4.28056 q^{33} +7.10349 q^{34} -0.517511 q^{36} +7.88083 q^{37} +4.18685 q^{38} -4.08072 q^{39} -10.9587 q^{40} +2.00531 q^{41} +2.85285 q^{43} +2.21524 q^{44} -3.57513 q^{45} -9.28851 q^{46} +6.11940 q^{47} -2.69716 q^{48} -9.47465 q^{50} -5.83413 q^{51} +2.11182 q^{52} +1.26956 q^{53} -1.21757 q^{54} +15.3036 q^{55} -3.43868 q^{57} -12.4152 q^{58} +7.94737 q^{59} +1.85017 q^{60} -8.44502 q^{61} -2.17808 q^{62} +8.86017 q^{64} +14.5891 q^{65} +5.21190 q^{66} -6.77752 q^{67} +3.01923 q^{68} +7.62869 q^{69} +9.18136 q^{71} +3.06526 q^{72} -5.97461 q^{73} -9.59550 q^{74} +7.78158 q^{75} +1.77956 q^{76} +4.96858 q^{78} -6.86811 q^{79} +9.64270 q^{80} +1.00000 q^{81} -2.44161 q^{82} +0.480942 q^{83} +20.8578 q^{85} -3.47356 q^{86} +10.1967 q^{87} -13.1210 q^{88} +4.26669 q^{89} +4.35299 q^{90} -3.94794 q^{92} +1.78887 q^{93} -7.45082 q^{94} +12.2937 q^{95} -2.84652 q^{96} +1.03105 q^{97} -4.28056 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 36 q^{3} + 24 q^{4} - 24 q^{5} + 36 q^{9} - 24 q^{10} + 24 q^{12} - 48 q^{13} - 24 q^{15} + 4 q^{16} - 48 q^{17} - 24 q^{19} - 48 q^{20} - 16 q^{22} + 4 q^{23} + 16 q^{25} - 24 q^{26} + 36 q^{27}+ \cdots - 88 q^{97}+O(q^{100}) \) 36 * q + 36 * q^3 + 24 * q^4 - 24 * q^5 + 36 * q^9 - 24 * q^10 + 24 * q^12 - 48 * q^13 - 24 * q^15 + 4 * q^16 - 48 * q^17 - 24 * q^19 - 48 * q^20 - 16 * q^22 + 4 * q^23 + 16 * q^25 - 24 * q^26 + 36 * q^27 + 4 * q^29 - 24 * q^30 - 24 * q^31 + 20 * q^32 - 24 * q^34 + 24 * q^36 - 24 * q^37 - 44 * q^38 - 48 * q^39 - 64 * q^40 - 44 * q^41 - 44 * q^43 + 20 * q^44 - 24 * q^45 - 44 * q^46 - 44 * q^47 + 4 * q^48 + 44 * q^50 - 48 * q^51 - 80 * q^52 + 24 * q^53 - 40 * q^55 - 24 * q^57 + 8 * q^58 - 20 * q^59 - 48 * q^60 - 88 * q^61 - 24 * q^62 - 40 * q^64 + 40 * q^65 - 16 * q^66 - 40 * q^67 - 68 * q^68 + 4 * q^69 + 16 * q^71 - 88 * q^73 + 48 * q^74 + 16 * q^75 - 40 * q^76 - 24 * q^78 - 108 * q^79 - 32 * q^80 + 36 * q^81 - 48 * q^82 - 64 * q^83 - 20 * q^85 - 44 * q^86 + 4 * q^87 - 120 * q^88 - 40 * q^89 - 24 * q^90 - 28 * q^92 - 24 * q^93 - 24 * q^94 + 28 * q^95 + 20 * q^96 - 88 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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