LMFDB - Embedded newform 538.4.b.a.537.17

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [538,4,Mod(537,538)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(538, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("538.537"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$N$	$=$	$538 = 2 \cdot 269$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	538.b (of order $2$ , degree $1$ , minimal)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

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

Self dual:	no
Analytic conductor:	$31.7430275831$
Analytic rank:	$0$
Dimension:	$68$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			537.17
Character	$\chi$	$=$	538.537
Dual form			538.4.b.a.537.52

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000i q^{2} -0.122336i q^{3} -4.00000 q^{4} +7.26358 q^{5} -0.244671 q^{6} +31.0626i q^{7} +8.00000i q^{8} +26.9850 q^{9} -14.5272i q^{10} -3.20731 q^{11} +0.489343i q^{12} -10.0275 q^{13} +62.1251 q^{14} -0.888595i q^{15} +16.0000 q^{16} -27.1710i q^{17} -53.9701i q^{18} +41.3494i q^{19} -29.0543 q^{20} +3.80006 q^{21} +6.41461i q^{22} -160.747 q^{23} +0.978685 q^{24} -72.2404 q^{25} +20.0551i q^{26} -6.60430i q^{27} -124.250i q^{28} +265.809i q^{29} -1.77719 q^{30} +85.9313i q^{31} -32.0000i q^{32} +0.392368i q^{33} -54.3420 q^{34} +225.625i q^{35} -107.940 q^{36} -320.328 q^{37} +82.6987 q^{38} +1.22673i q^{39} +58.1086i q^{40} -357.805 q^{41} -7.60012i q^{42} +263.079 q^{43} +12.8292 q^{44} +196.008 q^{45} +321.494i q^{46} +577.976 q^{47} -1.95737i q^{48} -621.883 q^{49} +144.481i q^{50} -3.32398 q^{51} +40.1102 q^{52} -482.084 q^{53} -13.2086 q^{54} -23.2965 q^{55} -248.501 q^{56} +5.05850 q^{57} +531.617 q^{58} -377.297i q^{59} +3.55438i q^{60} +695.295 q^{61} +171.863 q^{62} +838.225i q^{63} -64.0000 q^{64} -72.8359 q^{65} +0.784736 q^{66} +383.312 q^{67} +108.684i q^{68} +19.6651i q^{69} +451.251 q^{70} +1054.15i q^{71} +215.880i q^{72} +685.580 q^{73} +640.656i q^{74} +8.83758i q^{75} -165.397i q^{76} -99.6271i q^{77} +2.45345 q^{78} -707.008 q^{79} +116.217 q^{80} +727.788 q^{81} +715.609i q^{82} -66.6066i q^{83} -15.2002 q^{84} -197.359i q^{85} -526.158i q^{86} +32.5179 q^{87} -25.6584i q^{88} -1138.10 q^{89} -392.016i q^{90} -311.481i q^{91} +642.988 q^{92} +10.5125 q^{93} -1155.95i q^{94} +300.344i q^{95} -3.91474 q^{96} +29.1175 q^{97} +1243.77i q^{98} -86.5492 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$68 q - 272 q^{4} + 38 q^{5} - 4 q^{6} - 594 q^{9} + 18 q^{11} - 114 q^{13} + 8 q^{14} + 1088 q^{16} - 152 q^{20} - 20 q^{21} - 224 q^{23} + 16 q^{24} + 1098 q^{25} + 384 q^{30} - 600 q^{34} + 2376 q^{36}+ \cdots + 3370 q^{99}+O(q^{100})$ 68 * q - 272 * q^4 + 38 * q^5 - 4 * q^6 - 594 * q^9 + 18 * q^11 - 114 * q^13 + 8 * q^14 + 1088 * q^16 - 152 * q^20 - 20 * q^21 - 224 * q^23 + 16 * q^24 + 1098 * q^25 + 384 * q^30 - 600 * q^34 + 2376 * q^36 - 302 * q^37 + 436 * q^38 - 624 * q^41 - 654 * q^43 - 72 * q^44 - 1074 * q^45 + 692 * q^47 - 3052 * q^49 + 596 * q^51 + 456 * q^52 + 326 * q^53 + 1504 * q^54 + 828 * q^55 - 32 * q^56 - 1024 * q^57 - 100 * q^58 + 226 * q^61 + 312 * q^62 - 4352 * q^64 + 316 * q^65 + 2480 * q^66 - 1318 * q^67 + 608 * q^70 - 2028 * q^73 - 1152 * q^78 - 1444 * q^79 + 608 * q^80 + 3204 * q^81 + 80 * q^84 - 5600 * q^87 - 1284 * q^89 + 896 * q^92 - 2528 * q^93 - 64 * q^96 - 288 * q^97 + 3370 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/538\mathbb{Z}\right)^\times$ .

$n$	$271$
$\chi(n)$	$-1$

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

)

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

$2$		−	2.00000i		−	0.707107i
$2$		−	2.00000i		−	0.707107i
$3$		−	0.122336i		−	0.0235435i	−0.999931	−	0.0117718i	$-0.996253\pi$
$3$		−	0.122336i		−	0.0235435i	0.999931	−	0.0117718i	$-0.00374715\pi$
$4$	−4.00000			−0.500000
$5$	7.26358			0.649674			0.324837	−	0.945770i	$-0.394690\pi$
$5$	7.26358			0.649674			0.324837	+	0.945770i	$0.394690\pi$
$6$	−0.244671			−0.0166478
$7$			31.0626i			1.67722i	0.544731	+	0.838611i	$0.316632\pi$
$7$			31.0626i			1.67722i	−0.544731	+	0.838611i	$0.683368\pi$
$8$			8.00000i			0.353553i
$9$	26.9850			0.999446
$10$		−	14.5272i		−	0.459389i
$11$	−3.20731			−0.0879126			−0.0439563	−	0.999033i	$-0.513996\pi$
$11$	−3.20731			−0.0879126			−0.0439563	+	0.999033i	$0.513996\pi$
$12$			0.489343i			0.0117718i
$13$	−10.0275			−0.213934			−0.106967	−	0.994263i	$-0.534114\pi$
$13$	−10.0275			−0.213934			−0.106967	+	0.994263i	$0.534114\pi$
$14$	62.1251			1.18597
$15$		−	0.888595i		−	0.0152956i
$16$	16.0000			0.250000
$17$		−	27.1710i		−	0.387643i	−0.981037	−	0.193821i	$-0.937912\pi$
$17$		−	27.1710i		−	0.387643i	0.981037	−	0.193821i	$-0.0620883\pi$
$18$		−	53.9701i		−	0.706715i
$19$			41.3494i			0.499273i	0.968340	+	0.249637i	$0.0803112\pi$
$19$			41.3494i			0.499273i	−0.968340	+	0.249637i	$0.919689\pi$
$20$	−29.0543			−0.324837
$21$	3.80006			0.0394877
$22$			6.41461i			0.0621636i
$23$	−160.747			−1.45731			−0.728654	−	0.684882i	$-0.759854\pi$
$23$	−160.747			−1.45731			−0.728654	+	0.684882i	$0.759854\pi$
$24$	0.978685			0.00832389
$25$	−72.2404			−0.577923
$26$			20.0551i			0.151274i
$27$		−	6.60430i		−	0.0470740i
$28$		−	124.250i		−	0.838611i
$29$			265.809i			1.70205i	0.525127	+	0.851024i	$0.324018\pi$
$29$			265.809i			1.70205i	−0.525127	+	0.851024i	$0.675982\pi$
$30$	−1.77719			−0.0108156
$31$			85.9313i			0.497862i	0.968521	+	0.248931i	$0.0800792\pi$
$31$			85.9313i			0.497862i	−0.968521	+	0.248931i	$0.919921\pi$
$32$		−	32.0000i		−	0.176777i
$33$			0.392368i			0.00206977i
$34$	−54.3420			−0.274105
$35$			225.625i			1.08965i
$36$	−107.940			−0.499723
$37$	−320.328			−1.42329			−0.711644	−	0.702541i	$-0.752049\pi$
$37$	−320.328			−1.42329			−0.711644	+	0.702541i	$0.752049\pi$
$38$	82.6987			0.353040
$39$			1.22673i			0.00503676i
$40$			58.1086i			0.229695i
$41$	−357.805			−1.36292			−0.681460	−	0.731856i	$-0.738654\pi$
$41$	−357.805			−1.36292			−0.681460	+	0.731856i	$0.738654\pi$
$42$		−	7.60012i		−	0.0279220i
$43$	263.079			0.933004			0.466502	−	0.884520i	$-0.345514\pi$
$43$	263.079			0.933004			0.466502	+	0.884520i	$0.345514\pi$
$44$	12.8292			0.0439563
$45$	196.008			0.649314
$46$			321.494i			1.03047i
$47$	577.976			1.79375			0.896877	−	0.442280i	$-0.145830\pi$
$47$	577.976			1.79375			0.896877	+	0.442280i	$0.145830\pi$
$48$		−	1.95737i		−	0.00588588i
$49$	−621.883			−1.81307
$50$			144.481i			0.408653i
$51$	−3.32398			−0.00912648
$52$	40.1102			0.106967
$53$	−482.084			−1.24942			−0.624711	−	0.780856i	$-0.714783\pi$
$53$	−482.084			−1.24942			−0.624711	+	0.780856i	$0.714783\pi$
$54$	−13.2086			−0.0332863
$55$	−23.2965			−0.0571146
$56$	−248.501			−0.592987
$57$	5.05850			0.0117546
$58$	531.617			1.20353
$59$		−	377.297i		−	0.832539i	−0.909241	−	0.416270i	$-0.863337\pi$
$59$		−	377.297i		−	0.832539i	0.909241	−	0.416270i	$-0.136663\pi$
$60$			3.55438i			0.00764781i
$61$	695.295			1.45940			0.729700	−	0.683767i	$-0.239660\pi$
$61$	695.295			1.45940			0.729700	+	0.683767i	$0.239660\pi$
$62$	171.863			0.352041
$63$			838.225i			1.67629i
$64$	−64.0000			−0.125000
$65$	−72.8359			−0.138987
$66$	0.784736			0.00146355
$67$	383.312			0.698940			0.349470	−	0.936948i	$-0.386362\pi$
$67$	383.312			0.698940			0.349470	+	0.936948i	$0.386362\pi$
$68$			108.684i			0.193821i
$69$			19.6651i			0.0343102i
$70$	451.251			0.770497
$71$			1054.15i			1.76204i	0.473081	+	0.881019i	$0.343142\pi$
$71$			1054.15i			1.76204i	−0.473081	+	0.881019i	$0.656858\pi$
$72$			215.880i			0.353357i
$73$	685.580			1.09919			0.549596	−	0.835430i	$-0.314782\pi$
$73$	685.580			1.09919			0.549596	+	0.835430i	$0.314782\pi$
$74$			640.656i			1.00642i
$75$			8.83758i			0.0136063i
$76$		−	165.397i		−	0.249637i
$77$		−	99.6271i		−	0.147449i
$78$	2.45345			0.00356153
$79$	−707.008			−1.00689			−0.503447	−	0.864026i	$-0.667935\pi$
$79$	−707.008			−1.00689			−0.503447	+	0.864026i	$0.667935\pi$
$80$	116.217			0.162419
$81$	727.788			0.998337
$82$			715.609i			0.963729i
$83$		−	66.6066i		−	0.0880847i	−0.999030	−	0.0440423i	$-0.985976\pi$
$83$		−	66.6066i		−	0.0880847i	0.999030	−	0.0440423i	$-0.0140236\pi$
$84$	−15.2002			−0.0197438
$85$		−	197.359i		−	0.251842i
$86$		−	526.158i		−	0.659734i
$87$	32.5179			0.0400722
$88$		−	25.6584i		−	0.0310818i
$89$	−1138.10			−1.35549			−0.677746	−	0.735296i	$-0.737043\pi$
$89$	−1138.10			−1.35549			−0.677746	+	0.735296i	$0.737043\pi$
$90$		−	392.016i		−	0.459134i
$91$		−	311.481i		−	0.358815i
$92$	642.988			0.728654
$93$	10.5125			0.0117214
$94$		−	1155.95i		−	1.26838i
$95$			300.344i			0.324365i
$96$	−3.91474			−0.00416194
$97$	29.1175			0.0304787			0.0152393	−	0.999884i	$-0.495149\pi$
$97$	29.1175			0.0304787			0.0152393	+	0.999884i	$0.495149\pi$
$98$			1243.77i			1.28203i
$99$	−86.5492			−0.0878639
$100$	288.962			0.288962
$101$			691.301i			0.681059i	0.940234	+	0.340530i	$0.110606\pi$
$101$			691.301i			0.681059i	−0.940234	+	0.340530i	$0.889394\pi$
$102$			6.64796i			0.00645339i
$103$	495.942			0.474433			0.237217	−	0.971457i	$-0.423765\pi$
$103$	495.942			0.474433			0.237217	+	0.971457i	$0.423765\pi$
$104$		−	80.2204i		−	0.0756371i
$105$	27.6020			0.0256541
$106$			964.169i			0.883475i
$107$			1846.33i			1.66814i	0.551657	+	0.834071i	$0.313996\pi$
$107$			1846.33i			1.66814i	−0.551657	+	0.834071i	$0.686004\pi$
$108$			26.4172i			0.0235370i
$109$		−	1176.18i		−	1.03356i	−0.856118	−	0.516780i	$-0.827130\pi$
$109$		−	1176.18i		−	1.03356i	0.856118	−	0.516780i	$-0.172870\pi$
$110$			46.5930i			0.0403861i
$111$			39.1876i			0.0335092i
$112$			497.001i			0.419305i
$113$			1643.91i			1.36855i	0.729225	+	0.684274i	$0.239881\pi$
$113$			1643.91i			1.36855i	−0.729225	+	0.684274i	$0.760119\pi$
$114$		−	10.1170i		−	0.00831179i
$115$	−1167.60			−0.946776
$116$		−	1063.23i		−	0.851024i
$117$	−270.594			−0.213815
$118$	−754.593			−0.588694
$119$	844.000			0.650163
$120$	7.10876			0.00540782
$121$	−1320.71			−0.992271
$122$		−	1390.59i		−	1.03195i
$123$			43.7723i			0.0320879i
$124$		−	343.725i		−	0.248931i
$125$	−1432.67			−1.02514
$126$	1676.45			1.18532
$127$	1520.34			1.06227			0.531136	−	0.847287i	$-0.321765\pi$
$127$	1520.34			1.06227			0.531136	+	0.847287i	$0.321765\pi$
$128$			128.000i			0.0883883i
$129$		−	32.1840i		−	0.0219662i
$130$			145.672i			0.0982789i
$131$	1277.37			0.851939			0.425969	−	0.904738i	$-0.359933\pi$
$131$	1277.37			0.851939			0.425969	+	0.904738i	$0.359933\pi$
$132$		−	1.56947i		−	0.00103489i
$133$	−1284.42			−0.837392
$134$		−	766.623i		−	0.494225i
$135$		−	47.9708i		−	0.0305828i
$136$	217.368			0.137052
$137$			2021.91i			1.26090i	0.776230	+	0.630450i	$0.217129\pi$
$137$			2021.91i			1.26090i	−0.776230	+	0.630450i	$0.782871\pi$
$138$	39.3302			0.0242609
$139$		−	269.209i		−	0.164274i	−0.996621	−	0.0821368i	$-0.973826\pi$
$139$		−	269.209i		−	0.164274i	0.996621	−	0.0821368i	$-0.0261744\pi$
$140$		−	902.502i		−	0.544824i
$141$		−	70.7071i		−	0.0422313i
$142$	2108.30			1.24595
$143$	32.1614			0.0188075
$144$	431.761			0.249861
$145$			1930.72i			1.10578i
$146$		−	1371.16i		−	0.777247i
$147$			76.0785i			0.0426861i
$148$	1281.31			0.711644
$149$	280.768			0.154372			0.0771860	−	0.997017i	$-0.475406\pi$
$149$	280.768			0.154372			0.0771860	+	0.997017i	$0.475406\pi$
$150$	17.6752			0.00962114
$151$	2271.80			1.22435			0.612174	−	0.790723i	$-0.290295\pi$
$151$	2271.80			1.22435			0.612174	+	0.790723i	$0.290295\pi$
$152$	−330.795			−0.176520
$153$		−	733.210i		−	0.387428i
$154$	−199.254			−0.104262
$155$			624.169i			0.323448i
$156$		−	4.90691i		−	0.00251838i
$157$		−	162.060i		−	0.0823808i	−0.999151	−	0.0411904i	$-0.986885\pi$
$157$		−	162.060i		−	0.0823808i	0.999151	−	0.0411904i	$-0.0131150\pi$
$158$			1414.02i			0.711981i
$159$			58.9761i			0.0294158i
$160$		−	232.435i		−	0.114847i
$161$		−	4993.22i		−	2.44423i
$162$		−	1455.58i		−	0.705931i
$163$		−	2598.91i		−	1.24885i	−0.781085	−	0.624425i	$-0.785333\pi$
$163$		−	2598.91i		−	1.24885i	0.781085	−	0.624425i	$-0.214667\pi$
$164$	1431.22			0.681460
$165$			2.85000i			0.00134468i
$166$	−133.213			−0.0622853
$167$		−	436.551i		−	0.202283i	−0.994872	−	0.101142i	$-0.967750\pi$
$167$		−	436.551i		−	0.202283i	0.994872	−	0.101142i	$-0.0322495\pi$
$168$			30.4005i			0.0139610i
$169$	−2096.45			−0.954232
$170$	−394.717			−0.178079
$171$			1115.81i			0.498997i
$172$	−1052.32			−0.466502
$173$	4047.07			1.77857			0.889286	−	0.457351i	$-0.151202\pi$
$173$	4047.07			1.77857			0.889286	+	0.457351i	$0.151202\pi$
$174$		−	65.0357i		−	0.0283353i
$175$		−	2243.97i		−	0.969305i
$176$	−51.3169			−0.0219782
$177$	−46.1568			−0.0196009
$178$			2276.21i			0.958478i
$179$		−	1628.93i		−	0.680179i	−0.940393	−	0.340089i	$-0.889543\pi$
$179$		−	1628.93i		−	0.680179i	0.940393	−	0.340089i	$-0.110457\pi$
$180$	−784.032			−0.324657
$181$			3241.10i			1.33099i	0.746403	+	0.665495i	$0.231779\pi$
$181$			3241.10i			1.33099i	−0.746403	+	0.665495i	$0.768221\pi$
$182$	−622.963			−0.253720
$183$		−	85.0594i		−	0.0343594i
$184$		−	1285.98i		−	0.515236i
$185$	−2326.73			−0.924673
$186$		−	21.0249i		−	0.00828829i
$187$			87.1456i			0.0340787i
$188$	−2311.90			−0.896877
$189$	205.146			0.0789535
$190$	600.689			0.229361
$191$	−3075.70			−1.16518			−0.582591	−	0.812765i	$-0.697961\pi$
$191$	−3075.70			−1.16518			−0.582591	+	0.812765i	$0.697961\pi$
$192$			7.82948i			0.00294294i
$193$		−	1969.70i		−	0.734621i	−0.930098	−	0.367310i	$-0.880279\pi$
$193$		−	1969.70i		−	0.734621i	0.930098	−	0.367310i	$-0.119721\pi$
$194$		−	58.2350i		−	0.0215517i
$195$			8.91043i			0.00327225i
$196$	2487.53			0.906535
$197$			1090.38i			0.394347i	0.980369	+	0.197174i	$0.0631762\pi$
$197$			1090.38i			0.394347i	−0.980369	+	0.197174i	$0.936824\pi$
$198$			173.098i			0.0621292i
$199$	1851.20			0.659436			0.329718	−	0.944079i	$-0.393046\pi$
$199$	1851.20			0.659436			0.329718	+	0.944079i	$0.393046\pi$
$200$		−	577.923i		−	0.204327i
$201$		−	46.8927i		−	0.0164555i
$202$	1382.60			0.481582
$203$	−8256.70			−2.85471
$204$	13.2959			0.00456324
$205$	−2598.94			−0.885454
$206$		−	991.884i		−	0.335475i
$207$	−4337.77			−1.45650
$208$	−160.441			−0.0534835
$209$		−	132.620i		−	0.0438924i
$210$		−	55.2041i		−	0.0181402i
$211$	−2123.86			−0.692952			−0.346476	−	0.938059i	$-0.612622\pi$
$211$	−2123.86			−0.692952			−0.346476	+	0.938059i	$0.612622\pi$
$212$	1928.34			0.624711
$213$	128.960			0.0414846
$214$	3692.65			1.17955
$215$	1910.90			0.606149
$216$	52.8344			0.0166432
$217$	−2669.25			−0.835024
$218$	−2352.37			−0.730838
$219$		−	83.8709i		−	0.0258789i
$220$	93.1861			0.0285573
$221$			272.458i			0.0829300i
$222$	78.3751			0.0236946
$223$			2167.13i			0.650770i	0.945582	+	0.325385i	$0.105494\pi$
$223$			2167.13i			0.650770i	−0.945582	+	0.325385i	$0.894506\pi$
$224$	994.002			0.296494
$225$	−1949.41			−0.577603
$226$	3287.82			0.967709
$227$			4523.56i			1.32264i	0.750104	+	0.661320i	$0.230004\pi$
$227$			4523.56i			1.32264i	−0.750104	+	0.661320i	$0.769996\pi$
$228$	−20.2340			−0.00587732
$229$		−	493.569i		−	0.142428i	−0.997461	−	0.0712139i	$-0.977313\pi$
$229$		−	493.569i		−	0.142428i	0.997461	−	0.0712139i	$-0.0226873\pi$
$230$			2335.20i			0.669471i
$231$	−12.1880			−0.00347147
$232$	−2126.47			−0.601765
$233$	−417.669			−0.117435			−0.0587177	−	0.998275i	$-0.518701\pi$
$233$	−417.669			−0.117435			−0.0587177	+	0.998275i	$0.518701\pi$
$234$			541.188i			0.151190i
$235$	4198.17			1.16536
$236$			1509.19i			0.416270i
$237$			86.4923i			0.0237058i
$238$		−	1688.00i		−	0.459735i
$239$	−1920.77			−0.519850			−0.259925	−	0.965629i	$-0.583698\pi$
$239$	−1920.77			−0.519850			−0.259925	+	0.965629i	$0.583698\pi$
$240$		−	14.2175i		−	0.00382390i
$241$		−	3148.64i		−	0.841584i	−0.907157	−	0.420792i	$-0.861752\pi$
$241$		−	3148.64i		−	0.841584i	0.907157	−	0.420792i	$-0.138248\pi$
$242$			2641.43i			0.701642i
$243$		−	267.350i		−	0.0705783i
$244$	−2781.18			−0.729700
$245$	−4517.10			−1.17791
$246$	87.5445			0.0226896
$247$		−	414.633i		−	0.106812i
$248$	−687.450			−0.176021
$249$	−8.14837			−0.00207382
$250$			2865.34i			0.724881i
$251$			1308.03i			0.328934i	0.986383	+	0.164467i	$0.0525904\pi$
$251$			1308.03i			0.328934i	−0.986383	+	0.164467i	$0.947410\pi$
$252$		−	3352.90i		−	0.838146i
$253$	515.565			0.128116
$254$		−	3040.68i		−	0.751139i
$255$	−24.1440			−0.00592924
$256$	256.000			0.0625000
$257$		−	8167.52i		−	1.98240i	−0.132382	−	0.991199i	$-0.542263\pi$
$257$		−	8167.52i		−	1.98240i	0.132382	−	0.991199i	$-0.457737\pi$
$258$	−64.3679			−0.0155325
$259$		−	9950.22i		−	2.38717i
$260$	291.344			0.0694937
$261$			7172.85i			1.70110i
$262$		−	2554.73i		−	0.602412i
$263$	−185.643			−0.0435256			−0.0217628	−	0.999763i	$-0.506928\pi$
$263$	−185.643			−0.0435256			−0.0217628	+	0.999763i	$0.506928\pi$
$264$	−3.13894			−0.000731775
$265$	−3501.66			−0.811718
$266$			2568.83i			0.592125i
$267$			139.231i			0.0319131i
$268$	−1533.25			−0.349470
$269$	4197.58	+	1358.48i	0.951416	+	0.307910i
$269$	4197.58	+	1358.48i	0.951416	+	0.307910i
$270$	−95.9417			−0.0216253
$271$		−	5940.84i		−	1.33166i	−0.746103	−	0.665831i	$-0.768077\pi$
$271$		−	5940.84i		−	1.33166i	0.746103	−	0.665831i	$-0.231923\pi$
$272$		−	434.736i		−	0.0969107i
$273$	−38.1053			−0.00844776
$274$	4043.82			0.891591
$275$	231.697			0.0508068
$276$		−	78.6604i		−	0.0171551i
$277$			1194.15i			0.259022i	0.991578	+	0.129511i	$0.0413408\pi$
$277$			1194.15i			0.259022i	−0.991578	+	0.129511i	$0.958659\pi$
$278$	−538.418			−0.116159
$279$			2318.86i			0.497586i
$280$	−1805.00			−0.385249
$281$		−	6228.31i		−	1.32224i	−0.750280	−	0.661120i	$-0.770081\pi$
$281$		−	6228.31i		−	1.32224i	0.750280	−	0.661120i	$-0.229919\pi$
$282$	−141.414			−0.0298620
$283$	−5882.70			−1.23566			−0.617828	−	0.786314i	$-0.711987\pi$
$283$	−5882.70			−1.23566			−0.617828	+	0.786314i	$0.711987\pi$
$284$		−	4216.60i		−	0.881019i
$285$	36.7428			0.00763669
$286$		−	64.3228i		−	0.0132989i
$287$		−	11114.3i		−	2.28592i
$288$		−	863.521i		−	0.176679i
$289$	4174.74			0.849733
$290$	3861.44			0.781902
$291$		−	3.56211i		−	0.000717575i
$292$	−2742.32			−0.549596
$293$	−2627.25			−0.523841			−0.261921	−	0.965089i	$-0.584356\pi$
$293$	−2627.25			−0.523841			−0.261921	+	0.965089i	$0.584356\pi$
$294$	152.157			0.0301836
$295$		−	2740.52i		−	0.540879i
$296$		−	2562.62i		−	0.503208i
$297$			21.1820i			0.00413840i
$298$		−	561.537i		−	0.109158i
$299$	1611.90			0.311768
$300$		−	35.3503i		−	0.00680317i
$301$			8171.91i			1.56485i
$302$		−	4543.60i		−	0.865745i
$303$	84.5708			0.0160345
$304$			661.590i			0.124818i
$305$	5050.33			0.948135
$306$	−1466.42			−0.273953
$307$	2129.41			0.395870			0.197935	−	0.980215i	$-0.436577\pi$
$307$	2129.41			0.395870			0.197935	+	0.980215i	$0.436577\pi$
$308$			398.509i			0.0737245i
$309$		−	60.6714i		−	0.0111698i
$310$	1248.34			0.228712
$311$		−	6693.56i		−	1.22044i	−0.792232	−	0.610220i	$-0.791081\pi$
$311$		−	6693.56i		−	1.22044i	0.792232	−	0.610220i	$-0.208919\pi$
$312$	−9.81382			−0.00178076
$313$	6519.74			1.17737			0.588686	−	0.808362i	$-0.299646\pi$
$313$	6519.74			1.17737			0.588686	+	0.808362i	$0.299646\pi$
$314$	−324.120			−0.0582520
$315$			6088.51i			1.08904i
$316$	2828.03			0.503447
$317$		−	5992.74i		−	1.06178i	−0.847439	−	0.530892i	$-0.821857\pi$
$317$		−	5992.74i		−	1.06178i	0.847439	−	0.530892i	$-0.178143\pi$
$318$	117.952			0.0208001
$319$		−	852.529i		−	0.149632i
$320$	−464.869			−0.0812093
$321$	225.872			0.0392739
$322$	−9986.44			−1.72833
$323$	1123.50			0.193540
$324$	−2911.15			−0.499169
$325$	724.394			0.123637
$326$	−5197.82			−0.883070
$327$	−143.889			−0.0243336
$328$		−	2862.44i		−	0.481865i
$329$			17953.4i			3.00852i
$330$	5.69999			0.000950831
$331$	7466.81			1.23992			0.619959	−	0.784634i	$-0.287149\pi$
$331$	7466.81			1.23992			0.619959	+	0.784634i	$0.287149\pi$
$332$			266.427i			0.0440423i
$333$	−8644.07			−1.42250
$334$	−873.102			−0.143036
$335$	2784.21			0.454083
$336$	60.8010			0.00987192
$337$			308.183i			0.0498154i	0.999690	+	0.0249077i	$0.00792919\pi$
$337$			308.183i			0.0498154i	−0.999690	+	0.0249077i	$0.992071\pi$
$338$			4192.90i			0.674744i
$339$	201.109			0.0322204
$340$			789.434i			0.125921i
$341$		−	275.608i		−	0.0437683i
$342$	2231.63			0.352844
$343$		−	8662.83i		−	1.36370i
$344$			2104.63i			0.329867i
$345$			142.839i			0.0222904i
$346$		−	8094.14i		−	1.25764i
$347$	2712.41			0.419625			0.209812	−	0.977742i	$-0.432715\pi$
$347$	2712.41			0.419625			0.209812	+	0.977742i	$0.432715\pi$
$348$	−130.071			−0.0200361
$349$	−5057.24			−0.775668			−0.387834	−	0.921729i	$-0.626777\pi$
$349$	−5057.24			−0.775668			−0.387834	+	0.921729i	$0.626777\pi$
$350$	−4487.95			−0.685402
$351$			66.2249i			0.0100707i
$352$			102.634i			0.0155409i
$353$	7714.10			1.16312			0.581558	−	0.813505i	$-0.302443\pi$
$353$	7714.10			1.16312			0.581558	+	0.813505i	$0.302443\pi$
$354$			92.3137i			0.0138599i
$355$			7656.91i			1.14475i
$356$	4552.42			0.677746
$357$		−	103.251i		−	0.0153071i
$358$	−3257.86			−0.480959
$359$			12270.4i			1.80392i	0.431823	+	0.901959i	$0.357871\pi$
$359$			12270.4i			1.80392i	−0.431823	+	0.901959i	$0.642129\pi$
$360$			1568.06i			0.229567i
$361$	5149.23			0.750726
$362$	6482.20			0.941151
$363$			161.570i			0.0233616i
$364$			1245.93i			0.179407i
$365$	4979.76			0.714117
$366$	−170.119			−0.0242958
$367$			2970.62i			0.422520i	0.977430	+	0.211260i	$0.0677567\pi$
$367$			2970.62i			0.422520i	−0.977430	+	0.211260i	$0.932243\pi$
$368$	−2571.95			−0.364327
$369$	−9655.37			−1.36216
$370$			4653.46i			0.653843i
$371$		−	14974.8i		−	2.09556i
$372$	−42.0498			−0.00586071
$373$			9243.34i			1.28312i	0.767075	+	0.641558i	$0.221712\pi$
$373$			9243.34i			1.28312i	−0.767075	+	0.641558i	$0.778288\pi$
$374$	174.291			0.0240973
$375$			175.267i			0.0241353i
$376$			4623.81i			0.634188i
$377$		−	2665.41i		−	0.364126i
$378$		−	410.293i		−	0.0558285i
$379$		−	13810.3i		−	1.87174i	−0.352349	−	0.935869i	$-0.614617\pi$
$379$		−	13810.3i		−	1.87174i	0.352349	−	0.935869i	$-0.385383\pi$
$380$		−	1201.38i		−	0.162183i
$381$		−	185.992i		−	0.0250096i
$382$			6151.40i			0.823908i
$383$			13891.0i			1.85326i	0.375975	+	0.926630i	$0.377308\pi$
$383$			13891.0i			1.85326i	−0.375975	+	0.926630i	$0.622692\pi$
$384$	15.6590			0.00208097
$385$		−	723.650i		−	0.0957938i
$386$	−3939.39			−0.519455
$387$	7099.20			0.932487
$388$	−116.470			−0.0152393
$389$	−7854.08			−1.02370			−0.511848	−	0.859076i	$-0.671039\pi$
$389$	−7854.08			−1.02370			−0.511848	+	0.859076i	$0.671039\pi$
$390$	17.8209			0.00231383
$391$			4367.66i			0.564915i
$392$		−	4975.07i		−	0.641017i
$393$		−	156.267i		−	0.0200576i
$394$	2180.76			0.278846
$395$	−5135.41			−0.654153
$396$	346.197			0.0439319
$397$			9957.88i			1.25887i	0.777053	+	0.629436i	$0.216714\pi$
$397$			9957.88i			1.25887i	−0.777053	+	0.629436i	$0.783286\pi$
$398$		−	3702.39i		−	0.466292i
$399$			157.130i			0.0197151i
$400$	−1155.85			−0.144481
$401$		−	8394.48i		−	1.04539i	−0.852520	−	0.522694i	$-0.824927\pi$
$401$		−	8394.48i		−	1.04539i	0.852520	−	0.522694i	$-0.175073\pi$
$402$	−93.7854			−0.0116358
$403$		−	861.680i		−	0.106510i
$404$		−	2765.20i		−	0.340530i
$405$	5286.35			0.648594
$406$			16513.4i			2.01859i
$407$	1027.39			0.125125
$408$		−	26.5918i		−	0.00322670i
$409$		−	3999.23i		−	0.483495i	−0.970339	−	0.241747i	$-0.922279\pi$
$409$		−	3999.23i		−	0.483495i	0.970339	−	0.241747i	$-0.0777205\pi$
$410$			5197.88i			0.626110i
$411$	247.351			0.0296860
$412$	−1983.77			−0.237217
$413$	11719.8			1.39635
$414$			8675.53i			1.02990i
$415$		−	483.803i		−	0.0572264i
$416$			320.882i			0.0378185i
$417$	−32.9339			−0.00386758
$418$	−265.240			−0.0310366
$419$	12862.8			1.49974			0.749870	−	0.661585i	$-0.230116\pi$
$419$	12862.8			1.49974			0.749870	+	0.661585i	$0.230116\pi$
$420$	−110.408			−0.0128271
$421$	6298.92			0.729194			0.364597	−	0.931165i	$-0.381207\pi$
$421$	6298.92			0.729194			0.364597	+	0.931165i	$0.381207\pi$
$422$			4247.73i			0.489991i
$423$	15596.7			1.79276
$424$		−	3856.68i		−	0.441738i
$425$			1962.84i			0.224028i
$426$		−	257.921i		−	0.0293340i
$427$			21597.7i			2.44774i
$428$		−	7385.31i		−	0.834071i
$429$		−	3.93449i		−	0.000442795i
$430$		−	3821.79i		−	0.428612i
$431$		−	6881.85i		−	0.769112i	−0.923102	−	0.384556i	$-0.874355\pi$
$431$		−	6881.85i		−	0.769112i	0.923102	−	0.384556i	$-0.125645\pi$
$432$		−	105.669i		−	0.0117685i
$433$	−8331.33			−0.924661			−0.462330	−	0.886708i	$-0.652987\pi$
$433$	−8331.33			−0.924661			−0.462330	+	0.886708i	$0.652987\pi$
$434$			5338.49i			0.590451i
$435$	236.196			0.0260339
$436$			4704.74i			0.516780i
$437$		−	6646.79i		−	0.727595i
$438$	−167.742			−0.0182991
$439$	16612.5			1.80609			0.903045	−	0.429545i	$-0.141326\pi$
$439$	16612.5			1.80609			0.903045	+	0.429545i	$0.141326\pi$
$440$		−	186.372i		−	0.0201931i
$441$	−16781.5			−1.81207
$442$	544.917			0.0586404
$443$		−	4223.25i		−	0.452941i	−0.974018	−	0.226470i	$-0.927281\pi$
$443$		−	4223.25i		−	0.452941i	0.974018	−	0.226470i	$-0.0727186\pi$
$444$		−	156.750i		−	0.0167546i
$445$	−8266.71			−0.880629
$446$	4334.26			0.460164
$447$		−	34.3480i		−	0.00363446i
$448$		−	1988.00i		−	0.209653i
$449$	−7728.23			−0.812289			−0.406145	−	0.913809i	$-0.633127\pi$
$449$	−7728.23			−0.812289			−0.406145	+	0.913809i	$0.633127\pi$
$450$			3898.82i			0.408427i
$451$	1147.59			0.119818
$452$		−	6575.63i		−	0.684274i
$453$		−	277.922i		−	0.0288255i
$454$	9047.13			0.935248
$455$		−	2262.47i		−	0.233113i
$456$			40.4680i			0.00415590i
$457$	17690.5			1.81078			0.905388	−	0.424586i	$-0.139580\pi$
$457$	17690.5			1.81078			0.905388	+	0.424586i	$0.139580\pi$
$458$	−987.138			−0.100712
$459$	−179.445			−0.0182479
$460$	4670.40			0.473388
$461$		−	7931.13i		−	0.801279i	−0.916236	−	0.400639i	$-0.868788\pi$
$461$		−	7931.13i		−	0.801279i	0.916236	−	0.400639i	$-0.131212\pi$
$462$			24.3759i			0.00245470i
$463$			427.435i			0.0429040i	0.999770	+	0.0214520i	$0.00682891\pi$
$463$			427.435i			0.0429040i	−0.999770	+	0.0214520i	$0.993171\pi$
$464$			4252.94i			0.425512i
$465$	76.3581			0.00761510
$466$			835.339i			0.0830393i
$467$			13725.1i			1.36000i	0.733212	+	0.680000i	$0.238020\pi$
$467$			13725.1i			1.36000i	−0.733212	+	0.680000i	$0.761980\pi$
$468$	1082.38			0.106908
$469$			11906.6i			1.17228i
$470$		−	8396.35i		−	0.824031i
$471$	−19.8257			−0.00193953
$472$	3018.37			0.294347
$473$	−843.775			−0.0820229
$474$	172.985			0.0167625
$475$		−	2987.09i		−	0.288542i
$476$	−3376.00			−0.325081
$477$	−13009.1			−1.24873
$478$			3841.54i			0.367590i
$479$		−	7925.52i		−	0.756005i	−0.925805	−	0.378002i	$-0.876611\pi$
$479$		−	7925.52i		−	0.756005i	0.925805	−	0.378002i	$-0.123389\pi$
$480$	−28.4350			−0.00270391
$481$	3212.11			0.304489
$482$	−6297.28			−0.595090
$483$	−610.849			−0.0575457
$484$	5282.85			0.496136
$485$	211.497			0.0198012
$486$	−534.701			−0.0499064
$487$	21429.7			1.99399			0.996993	−	0.0774867i	$-0.0246895\pi$
$487$	21429.7			1.99399			0.996993	+	0.0774867i	$0.0246895\pi$
$488$			5562.36i			0.515976i
$489$	−317.940			−0.0294023
$490$			9034.20i			0.832905i
$491$	−5643.24			−0.518688			−0.259344	−	0.965785i	$-0.583506\pi$
$491$	−5643.24			−0.518688			−0.259344	+	0.965785i	$0.583506\pi$
$492$		−	175.089i		−	0.0160440i
$493$	7222.28			0.659787
$494$	−829.265			−0.0755271
$495$	−628.657			−0.0570829
$496$			1374.90i			0.124465i
$497$	−32744.6			−2.95533
$498$			16.2967i			0.00146641i
$499$			19020.6i			1.70637i	0.521604	+	0.853187i	$0.325334\pi$
$499$			19020.6i			1.70637i	−0.521604	+	0.853187i	$0.674666\pi$
$500$	5730.69			0.512568
$501$	−53.4058			−0.00476246
$502$	2616.07			0.232591
$503$		−	10333.1i		−	0.915965i	−0.888961	−	0.457982i	$-0.848572\pi$
$503$		−	10333.1i		−	0.915965i	0.888961	−	0.457982i	$-0.151428\pi$
$504$	−6705.80			−0.592659
$505$			5021.32i			0.442467i
$506$		−	1031.13i		−	0.0905915i
$507$			256.470i			0.0224660i
$508$	−6081.36			−0.531136
$509$			11741.1i			1.02243i	0.859453	+	0.511215i	$0.170805\pi$
$509$			11741.1i			1.02243i	−0.859453	+	0.511215i	$0.829195\pi$
$510$			48.2880i			0.00419260i
$511$			21295.9i			1.84359i
$512$		−	512.000i		−	0.0441942i
$513$	273.083			0.0235028
$514$	−16335.0			−1.40177
$515$	3602.31			0.308227
$516$			128.736i			0.0109831i
$517$	−1853.74			−0.157694
$518$	−19900.4			−1.68798
$519$		−	495.101i		−	0.0418739i
$520$		−	582.687i		−	0.0491395i
$521$		−	23455.7i		−	1.97238i	−0.165605	−	0.986192i	$-0.552958\pi$
$521$		−	23455.7i		−	1.97238i	0.165605	−	0.986192i	$-0.447042\pi$
$522$	14345.7			1.20286
$523$		−	4737.20i		−	0.396067i	−0.980195	−	0.198034i	$-0.936544\pi$
$523$		−	4737.20i		−	0.396067i	0.980195	−	0.198034i	$-0.0634555\pi$
$524$	−5109.46			−0.425969
$525$	−274.518			−0.0228208
$526$			371.286i			0.0307772i
$527$	2334.84			0.192993
$528$			6.27789i			0.000517443i
$529$	13672.6			1.12375
$530$			7003.32i			0.573971i
$531$		−	10181.4i		−	0.832078i
$532$	5137.67			0.418696
$533$	3587.90			0.291575
$534$	278.462			0.0225659
$535$			13410.9i			1.08375i
$536$			3066.49i			0.247112i
$537$	−199.276			−0.0160138
$538$	2716.95	−	8395.15i	0.217725	−	0.672752i
$539$	1994.57			0.159392
$540$			191.883i			0.0152914i
$541$		−	16504.9i		−	1.31165i	−0.754915	−	0.655823i	$-0.772322\pi$
$541$		−	16504.9i		−	1.31165i	0.754915	−	0.655823i	$-0.227678\pi$
$542$	−11881.7			−0.941627
$543$	396.502			0.0313362
$544$	−869.471			−0.0685262
$545$		−	8543.31i		−	0.671478i
$546$			76.2106i			0.00597347i
$547$	2430.36			0.189972			0.0949861	−	0.995479i	$-0.469719\pi$
$547$	2430.36			0.189972			0.0949861	+	0.995479i	$0.469719\pi$
$548$		−	8087.63i		−	0.630450i
$549$	18762.6			1.45859
$550$		−	463.394i		−	0.0359258i
$551$	−10991.0			−0.849787
$552$	−157.321			−0.0121305
$553$		−	21961.5i		−	1.68878i
$554$	2388.29			0.183157
$555$			284.642i			0.0217701i
$556$			1076.84i			0.0821368i
$557$			19699.4i			1.49854i	0.662263	+	0.749272i	$0.269596\pi$
$557$			19699.4i			1.49854i	−0.662263	+	0.749272i	$0.730404\pi$
$558$	4637.72			0.351846
$559$	−2638.04			−0.199601
$560$			3610.01i			0.272412i
$561$	10.6610			0.000802333
$562$	−12456.6			−0.934965
$563$	−10922.1			−0.817606			−0.408803	−	0.912623i	$-0.634054\pi$
$563$	−10922.1			−0.817606			−0.408803	+	0.912623i	$0.634054\pi$
$564$			282.828i			0.0211156i
$565$			11940.7i			0.889110i
$566$			11765.4i			0.873740i
$567$			22607.0i			1.67443i
$568$	−8433.21			−0.622975
$569$		−	2918.32i		−	0.215013i	−0.994204	−	0.107506i	$-0.965713\pi$
$569$		−	2918.32i		−	0.215013i	0.994204	−	0.107506i	$-0.0342866\pi$
$570$		−	73.4857i		−	0.00539996i
$571$		−	7609.77i		−	0.557722i	−0.960332	−	0.278861i	$-0.910043\pi$
$571$		−	7609.77i		−	0.557722i	0.960332	−	0.278861i	$-0.0899568\pi$
$572$	−128.646			−0.00940375
$573$			376.268i			0.0274325i
$574$	−22228.7			−1.61639
$575$	11612.4			0.842212
$576$	−1727.04			−0.124931
$577$			16744.6i			1.20813i	0.796937	+	0.604063i	$0.206452\pi$
$577$			16744.6i			1.20813i	−0.796937	+	0.604063i	$0.793548\pi$
$578$		−	8349.48i		−	0.600852i
$579$	−240.964			−0.0172956
$580$		−	7722.89i		−	0.552889i
$581$	2068.97			0.147737
$582$	−7.12421			−0.000507402
$583$	1546.19			0.109840
$584$			5484.64i			0.388623i
$585$	−1965.48			−0.138910
$586$			5254.50i			0.370412i
$587$	6105.40			0.429296			0.214648	−	0.976691i	$-0.431140\pi$
$587$	6105.40			0.429296			0.214648	+	0.976691i	$0.431140\pi$
$588$		−	304.314i		−	0.0213430i
$589$	−3553.20			−0.248569
$590$	−5481.05			−0.382460
$591$	133.392			0.00928432
$592$	−5125.25			−0.355822
$593$	−24079.3			−1.66749			−0.833743	−	0.552152i	$-0.813807\pi$
$593$	−24079.3			−1.66749			−0.833743	+	0.552152i	$0.813807\pi$
$594$	42.3640			0.00292629
$595$	6130.46			0.422394
$596$	−1123.07			−0.0771860
$597$		−	226.467i		−	0.0155254i
$598$		−	3223.80i		−	0.220453i
$599$	−13036.2			−0.889224			−0.444612	−	0.895723i	$-0.646658\pi$
$599$	−13036.2			−0.889224			−0.444612	+	0.895723i	$0.646658\pi$
$600$	−70.7006			−0.00481057
$601$			23316.0i			1.58250i	0.611495	+	0.791248i	$0.290568\pi$
$601$			23316.0i			1.58250i	−0.611495	+	0.791248i	$0.709432\pi$
$602$	16343.8			1.10652
$603$	10343.7			0.698552
$604$	−9087.20			−0.612174
$605$	−9593.11			−0.644653
$606$		−	169.142i		−	0.0113381i
$607$		−	5393.06i		−	0.360622i	−0.983610	−	0.180311i	$-0.942290\pi$
$607$		−	5393.06i		−	0.360622i	0.983610	−	0.180311i	$-0.0577104\pi$
$608$	1323.18			0.0882599
$609$			1010.09i			0.0672099i
$610$		−	10100.7i		−	0.670433i
$611$	−5795.68			−0.383745
$612$			2932.84i			0.193714i
$613$			3038.78i			0.200221i	0.994976	+	0.100110i	$0.0319196\pi$
$613$			3038.78i			0.200221i	−0.994976	+	0.100110i	$0.968080\pi$
$614$		−	4258.82i		−	0.279922i
$615$			317.943i			0.0208467i
$616$	797.017			0.0521311
$617$	13293.5			0.867385			0.433692	−	0.901061i	$-0.357210\pi$
$617$	13293.5			0.867385			0.433692	+	0.901061i	$0.357210\pi$
$618$	−121.343			−0.00789826
$619$	7323.81			0.475555			0.237778	−	0.971320i	$-0.423581\pi$
$619$	7323.81			0.475555			0.237778	+	0.971320i	$0.423581\pi$
$620$		−	2496.67i		−	0.161724i
$621$			1061.62i			0.0686013i
$622$	−13387.1			−0.862982
$623$		−	35352.5i		−	2.27346i
$624$			19.6276i			0.00125919i
$625$	−1376.27			−0.0880814
$626$		−	13039.5i		−	0.832527i
$627$	−16.2242			−0.00103338
$628$			648.240i			0.0411904i
$629$			8703.63i			0.551727i
$630$	12177.0			0.770070
$631$	−10301.6			−0.649922			−0.324961	−	0.945727i	$-0.605351\pi$
$631$	−10301.6			−0.649922			−0.324961	+	0.945727i	$0.605351\pi$
$632$		−	5656.06i		−	0.355991i
$633$			259.824i			0.0163145i
$634$	−11985.5			−0.750795
$635$	11043.1			0.690130
$636$		−	235.905i		−	0.0147079i
$637$	6235.97			0.387877
$638$	−1705.06			−0.105805
$639$			28446.3i			1.76106i
$640$			929.738i			0.0574236i
$641$	−28814.6			−1.77552			−0.887761	−	0.460304i	$-0.847740\pi$
$641$	−28814.6			−1.77552			−0.887761	+	0.460304i	$0.847740\pi$
$642$		−	451.743i		−	0.0277709i
$643$	21689.4			1.33024			0.665121	−	0.746736i	$-0.268380\pi$
$643$	21689.4			1.33024			0.665121	+	0.746736i	$0.268380\pi$
$644$			19972.9i			1.22211i
$645$		−	233.771i		−	0.0142709i
$646$		−	2247.00i		−	0.136853i
$647$			8098.38i			0.492087i	0.969259	+	0.246044i	$0.0791306\pi$
$647$			8098.38i			0.492087i	−0.969259	+	0.246044i	$0.920869\pi$
$648$			5822.30i			0.352966i
$649$			1210.11i			0.0731907i
$650$		−	1448.79i		−	0.0874249i
$651$			326.544i			0.0196594i
$652$			10395.6i			0.624425i
$653$	−8898.08			−0.533244			−0.266622	−	0.963801i	$-0.585908\pi$
$653$	−8898.08			−0.533244			−0.266622	+	0.963801i	$0.585908\pi$
$654$			287.779i			0.0172065i
$655$	9278.25			0.553483
$656$	−5724.87			−0.340730
$657$	18500.4			1.09858
$658$	35906.8			2.12735
$659$	29685.7			1.75477			0.877383	−	0.479790i	$-0.159287\pi$
$659$	29685.7			1.75477			0.877383	+	0.479790i	$0.159287\pi$
$660$		−	11.4000i		−	0.000672339i
$661$		−	20652.3i		−	1.21525i	−0.794224	−	0.607625i	$-0.792123\pi$
$661$		−	20652.3i		−	1.21525i	0.794224	−	0.607625i	$-0.207877\pi$
$662$		−	14933.6i		−	0.876755i
$663$	33.3314			0.00195246
$664$	532.853			0.0311426
$665$	−9329.47			−0.544032
$666$			17288.1i			1.00586i
$667$		−	42727.9i		−	2.48041i
$668$			1746.20i			0.101142i
$669$	265.117			0.0153214
$670$		−	5568.43i		−	0.321085i
$671$	−2230.02			−0.128300
$672$		−	121.602i		−	0.00698050i
$673$		−	10310.0i		−	0.590525i	−0.955416	−	0.295262i	$-0.904593\pi$
$673$		−	10310.0i		−	0.590525i	0.955416	−	0.295262i	$-0.0954070\pi$
$674$	616.366			0.0352248
$675$			477.097i			0.0272051i
$676$	8385.79			0.477116
$677$		−	23022.6i		−	1.30699i	−0.756932	−	0.653494i	$-0.773303\pi$
$677$		−	23022.6i		−	1.30699i	0.756932	−	0.653494i	$-0.226697\pi$
$678$		−	402.217i		−	0.0227833i
$679$			904.464i			0.0511195i
$680$	1578.87			0.0890395
$681$	553.393			0.0311396
$682$	−551.216			−0.0309489
$683$			16684.0i			0.934691i	0.884075	+	0.467346i	$0.154790\pi$
$683$			16684.0i			0.934691i	−0.884075	+	0.467346i	$0.845210\pi$
$684$		−	4463.25i		−	0.249498i
$685$			14686.3i			0.819174i
$686$	−17325.7			−0.964281
$687$	−60.3811			−0.00335325
$688$	4209.27			0.233251
$689$	4834.13			0.267294
$690$	285.678			0.0157617
$691$		−	778.602i		−	0.0428645i	−0.999770	−	0.0214323i	$-0.993177\pi$
$691$		−	778.602i		−	0.0428645i	0.999770	−	0.0214323i	$-0.00682263\pi$
$692$	−16188.3			−0.889286
$693$		−	2688.44i		−	0.147367i
$694$		−	5424.82i		−	0.296719i
$695$		−	1955.42i		−	0.106724i
$696$			260.143i			0.0141677i
$697$			9721.90i			0.528326i
$698$			10114.5i			0.548480i
$699$			51.0959i			0.00276484i
$700$			8975.89i			0.484653i
$701$		−	15688.2i		−	0.845271i	−0.906300	−	0.422636i	$-0.861105\pi$
$701$		−	15688.2i		−	0.845271i	0.906300	−	0.422636i	$-0.138895\pi$
$702$	132.450			0.00712108
$703$		−	13245.4i		−	0.710609i
$704$	205.268			0.0109891
$705$		−	513.586i		−	0.0274366i
$706$		−	15428.2i		−	0.822448i
$707$	−21473.6			−1.14229
$708$	184.627			0.00980045
$709$			20712.4i			1.09714i	0.836106	+	0.548568i	$0.184827\pi$
$709$			20712.4i			1.09714i	−0.836106	+	0.548568i	$0.815173\pi$
$710$	15313.8			0.809461
$711$	−19078.6			−1.00634
$712$		−	9104.84i		−	0.479239i
$713$		−	13813.2i		−	0.725538i
$714$	−206.503			−0.0108238
$715$	233.607			0.0122187
$716$			6515.72i			0.340089i
$717$			234.979i			0.0122391i
$718$	24540.8			1.27556
$719$		−	17274.8i		−	0.896024i	−0.894028	−	0.448012i	$-0.852132\pi$
$719$		−	17274.8i		−	0.896024i	0.894028	−	0.448012i	$-0.147868\pi$
$720$	3136.13			0.162329
$721$			15405.2i			0.795729i
$722$		−	10298.5i		−	0.530844i
$723$	−385.191			−0.0198138
$724$		−	12964.4i		−	0.665495i
$725$		−	19202.1i		−	0.983653i
$726$	323.141			0.0165191
$727$	−9015.71			−0.459937			−0.229968	−	0.973198i	$-0.573862\pi$
$727$	−9015.71			−0.459937			−0.229968	+	0.973198i	$0.573862\pi$
$728$	2491.85			0.126860
$729$	19617.6			0.996676
$730$		−	9959.53i		−	0.504957i
$731$		−	7148.12i		−	0.361673i
$732$			340.238i			0.0171797i
$733$			19576.5i			0.986458i	0.869899	+	0.493229i	$0.164184\pi$
$733$			19576.5i			0.986458i	−0.869899	+	0.493229i	$0.835816\pi$
$734$	5941.23			0.298767
$735$			552.602i			0.0277320i
$736$			5143.91i			0.257618i
$737$	−1229.40			−0.0614456
$738$			19310.7i			0.963195i
$739$		−	3744.57i		−	0.186396i	−0.995648	−	0.0931978i	$-0.970291\pi$
$739$		−	3744.57i		−	0.186396i	0.995648	−	0.0931978i	$-0.0297089\pi$
$740$	9306.92			0.462337
$741$	−50.7244			−0.00251472
$742$	−29949.6			−1.48178
$743$	33774.0			1.66763			0.833814	−	0.552046i	$-0.186153\pi$
$743$	33774.0			1.66763			0.833814	+	0.552046i	$0.186153\pi$
$744$			84.0997i			0.00414414i
$745$	2039.38			0.100292
$746$	18486.7			0.907300
$747$		−	1797.38i		−	0.0880359i
$748$		−	348.582i		−	0.0170394i
$749$	−57351.7			−2.79784
$750$	350.534			0.0170662
$751$	−8064.59			−0.391852			−0.195926	−	0.980619i	$-0.562771\pi$
$751$	−8064.59			−0.391852			−0.195926	+	0.980619i	$0.562771\pi$
$752$	9247.61			0.448438
$753$	160.019			0.00774426
$754$	−5330.82			−0.257476
$755$	16501.4			0.795428
$756$	−820.586			−0.0394767
$757$		−	5247.19i		−	0.251932i	−0.992035	−	0.125966i	$-0.959797\pi$
$757$		−	5247.19i		−	0.251932i	0.992035	−	0.125966i	$-0.0402030\pi$
$758$	−27620.6			−1.32352
$759$		−	63.0720i		−	0.00301630i
$760$	−2402.75			−0.114680
$761$		−	26230.0i		−	1.24946i	−0.780842	−	0.624729i	$-0.785210\pi$
$761$		−	26230.0i		−	1.24946i	0.780842	−	0.624729i	$-0.214790\pi$
$762$	−371.984			−0.0176845
$763$	36535.3			1.73351
$764$	12302.8			0.582591
$765$		−	5325.73i		−	0.251702i
$766$	27782.1			1.31045
$767$			3783.36i			0.178108i
$768$		−	31.3179i		−	0.00147147i
$769$	−22118.8			−1.03722			−0.518611	−	0.855010i	$-0.673551\pi$
$769$	−22118.8			−1.03722			−0.518611	+	0.855010i	$0.673551\pi$
$770$	−1447.30			−0.0677364
$771$	−999.180			−0.0466726
$772$			7878.78i			0.367310i
$773$	14551.5			0.677076			0.338538	−	0.940953i	$-0.390068\pi$
$773$	14551.5			0.677076			0.338538	+	0.940953i	$0.390068\pi$
$774$		−	14198.4i		−	0.659368i
$775$		−	6207.71i		−	0.287726i
$776$			232.940i			0.0107758i
$777$	−1217.27			−0.0562023
$778$			15708.2i			0.723862i
$779$		−	14795.0i		−	0.680469i
$780$		−	35.6417i		−	0.00163613i
$781$		−	3380.98i		−	0.154905i
$782$	8735.31			0.399455
$783$	1755.48			0.0801222
$784$	−9950.13			−0.453268
$785$		−	1177.14i		−	0.0535207i
$786$	−312.535			−0.0141829
$787$	−15917.0			−0.720940			−0.360470	−	0.932771i	$-0.617384\pi$
$787$	−15917.0			−0.720940			−0.360470	+	0.932771i	$0.617384\pi$
$788$		−	4361.52i		−	0.197174i
$789$			22.7108i			0.00102475i
$790$			10270.8i			0.462556i
$791$	−51064.0			−2.29536
$792$		−	692.394i		−	0.0310646i
$793$	−6972.10			−0.312215
$794$	19915.8			0.890156
$795$			428.378i			0.0191107i
$796$	−7404.79			−0.329718
$797$		−	27816.6i		−	1.23628i	−0.786068	−	0.618140i	$-0.787886\pi$
$797$		−	27816.6i		−	1.23628i	0.786068	−	0.618140i	$-0.212114\pi$
$798$	314.260			0.0139407
$799$		−	15704.2i		−	0.695336i
$800$			2311.69i			0.102163i
$801$	−30711.8			−1.35474
$802$	−16789.0			−0.739201
$803$	−2198.86			−0.0966329
$804$			187.571i			0.00822775i
$805$		−	36268.6i		−	1.58795i
$806$	−1723.36			−0.0753136
$807$	166.190	−	513.513i	0.00724928	−	0.0223997i
$808$	−5530.41			−0.240791
$809$		−	125.653i		−	0.00546073i	−0.999996	−	0.00273037i	$-0.999131\pi$
$809$		−	125.653i		−	0.00546073i	0.999996	−	0.00273037i	$-0.000869104\pi$
$810$		−	10572.7i		−	0.458625i
$811$	−23561.6			−1.02017			−0.510086	−	0.860123i	$-0.670387\pi$
$811$	−23561.6			−1.02017			−0.510086	+	0.860123i	$0.670387\pi$
$812$	33026.8			1.42736
$813$	−726.777			−0.0313520
$814$		−	2054.78i		−	0.0884767i
$815$		−	18877.4i		−	0.811345i
$816$	−53.1837			−0.00228162
$817$			10878.2i			0.465824i
$818$	−7998.47			−0.341882
$819$		−	8405.34i		−	0.358616i
$820$	10395.8			0.442727
$821$	−13635.4			−0.579631			−0.289816	−	0.957083i	$-0.593594\pi$
$821$	−13635.4			−0.579631			−0.289816	+	0.957083i	$0.593594\pi$
$822$		−	494.703i		−	0.0209912i
$823$	10832.5			0.458804			0.229402	−	0.973332i	$-0.426323\pi$
$823$	10832.5			0.458804			0.229402	+	0.973332i	$0.426323\pi$
$824$			3967.54i			0.167737i
$825$		−	28.3448i		−	0.00119617i
$826$		−	23439.6i		−	0.987371i
$827$	−32051.0			−1.34767			−0.673835	−	0.738882i	$-0.735354\pi$
$827$	−32051.0			−1.34767			−0.673835	+	0.738882i	$0.735354\pi$
$828$	17351.1			0.728250
$829$			8792.02i			0.368347i	0.982894	+	0.184173i	$0.0589608\pi$
$829$			8792.02i			0.368347i	−0.982894	+	0.184173i	$0.941039\pi$
$830$	−967.605			−0.0404651
$831$	146.087			0.00609830
$832$	641.763			0.0267417
$833$			16897.2i			0.702824i
$834$			65.8678i			0.00273479i
$835$		−	3170.92i		−	0.131418i
$836$			530.480i			0.0219462i
$837$	567.515			0.0234363
$838$		−	25725.7i		−	1.06048i
$839$		−	20675.3i		−	0.850765i	−0.905014	−	0.425382i	$-0.860140\pi$
$839$		−	20675.3i		−	0.850765i	0.905014	−	0.425382i	$-0.139860\pi$
$840$			220.816i			0.00907011i
$841$	−46265.2			−1.89697
$842$		−	12597.8i		−	0.515618i
$843$	−761.944			−0.0311302
$844$	8495.46			0.346476
$845$	−15227.7			−0.619940
$846$		−	31193.4i		−	1.26767i
$847$		−	41024.7i		−	1.66426i
$848$	−7713.35			−0.312356
$849$			719.665i			0.0290917i
$850$	3925.68			0.158412
$851$	51491.8			2.07417
$852$	−515.841			−0.0207423
$853$		−	31131.7i		−	1.24962i	−0.780776	−	0.624811i	$-0.785176\pi$
$853$		−	31131.7i		−	1.24962i	0.780776	−	0.624811i	$-0.214824\pi$
$854$	43195.3			1.73081
$855$			8104.80i			0.324185i
$856$	−14770.6			−0.589777
$857$			44684.1i			1.78107i	0.454910	+	0.890537i	$0.349671\pi$
$857$			44684.1i			1.78107i	−0.454910	+	0.890537i	$0.650329\pi$
$858$	−7.86898			−0.000313103
$859$	30638.1			1.21695			0.608474	−	0.793574i	$-0.291782\pi$
$859$	30638.1			1.21695			0.608474	+	0.793574i	$0.291782\pi$
$860$	−7643.58			−0.303075
$861$	−1359.68			−0.0538185
$862$	−13763.7			−0.543844
$863$	26688.4			1.05271			0.526353	−	0.850266i	$-0.323559\pi$
$863$	26688.4			1.05271			0.526353	+	0.850266i	$0.323559\pi$
$864$	−211.337			−0.00832158
$865$	29396.2			1.15549
$866$			16662.7i			0.653834i
$867$		−	510.719i		−	0.0200057i
$868$	10677.0			0.417512
$869$	2267.59			0.0885186
$870$		−	472.392i		−	0.0184087i
$871$	−3843.68			−0.149527
$872$	9409.48			0.365419
$873$	785.736			0.0304618
$874$	−13293.6			−0.514487
$875$		−	44502.5i		−	1.71938i
$876$			335.484i			0.0129394i
$877$	−22694.3			−0.873809			−0.436905	−	0.899508i	$-0.643925\pi$
$877$	−22694.3			−0.873809			−0.436905	+	0.899508i	$0.643925\pi$
$878$		−	33225.1i		−	1.27710i
$879$			321.406i			0.0123331i
$880$	−372.744			−0.0142786
$881$		−	8706.87i		−	0.332965i	−0.986044	−	0.166482i	$-0.946759\pi$
$881$		−	8706.87i		−	0.332965i	0.986044	−	0.166482i	$-0.0532409\pi$
$882$			33563.1i			1.28132i
$883$			21582.4i			0.822543i	0.911513	+	0.411272i	$0.134915\pi$
$883$			21582.4i			0.822543i	−0.911513	+	0.411272i	$0.865085\pi$
$884$		−	1089.83i		−	0.0414650i
$885$	−335.264			−0.0127342
$886$	−8446.50			−0.320277
$887$	15786.3			0.597579			0.298790	−	0.954319i	$-0.403417\pi$
$887$	15786.3			0.597579			0.298790	+	0.954319i	$0.403417\pi$
$888$	−313.500			−0.0118473
$889$			47225.7i			1.78166i
$890$			16533.4i			0.622699i
$891$	−2334.24			−0.0877665
$892$		−	8668.52i		−	0.325385i
$893$			23898.9i			0.895573i
$894$	−68.6960			−0.00256995
$895$		−	11831.9i		−	0.441895i
$896$	−3976.01			−0.148247
$897$		−	197.193i		−	0.00734011i
$898$			15456.5i			0.574375i
$899$	−22841.3			−0.847385
$900$	7797.64			0.288801
$901$			13098.7i			0.484330i
$902$		−	2295.18i		−	0.0847240i
$903$	999.717			0.0368422
$904$	−13151.3			−0.483855
$905$			23542.0i			0.864709i
$906$	−555.845			−0.0203827
$907$	9745.82			0.356786			0.178393	−	0.983959i	$-0.442910\pi$
$907$	9745.82			0.356786			0.178393	+	0.983959i	$0.442910\pi$
$908$		−	18094.3i		−	0.661320i
$909$			18654.8i			0.680682i
$910$	−4524.94			−0.164836
$911$			646.928i			0.0235276i	0.999931	+	0.0117638i	$0.00374463\pi$
$911$			646.928i			0.0235276i	−0.999931	+	0.0117638i	$0.996255\pi$
$912$	80.9360			0.00293866
$913$			213.628i			0.00774376i
$914$		−	35380.9i		−	1.28041i
$915$		−	617.836i		−	0.0223224i
$916$			1974.28i			0.0712139i
$917$			39678.3i			1.42889i
$918$			358.890i			0.0129032i
$919$			10915.2i			0.391795i	0.980624	+	0.195897i	$0.0627620\pi$
$919$			10915.2i			0.391795i	−0.980624	+	0.195897i	$0.937238\pi$
$920$		−	9340.79i		−	0.334736i
$921$		−	260.503i		−	0.00932016i
$922$	−15862.3			−0.566590
$923$		−	10570.6i		−	0.376960i
$924$	48.7518			0.00173573
$925$	23140.6			0.822551
$926$	854.869			0.0303377
$927$	13383.0			0.474170
$928$	8505.87			0.300882
$929$			31007.7i			1.09508i	0.836780	+	0.547540i	$0.184436\pi$
$929$			31007.7i			1.09508i	−0.836780	+	0.547540i	$0.815564\pi$
$930$		−	152.716i		−	0.00538469i
$931$		−	25714.5i		−	0.905218i
$932$	1670.68			0.0587177
$933$	−818.861			−0.0287335
$934$	27450.1			0.961665
$935$			632.989i			0.0221401i
$936$		−	2164.75i		−	0.0755952i
$937$			8644.64i			0.301396i	0.988580	+	0.150698i	$0.0481521\pi$
$937$			8644.64i			0.301396i	−0.988580	+	0.150698i	$0.951848\pi$
$938$	23813.3			0.828925
$939$		−	797.596i		−	0.0277195i
$940$	−16792.7			−0.582678
$941$			18563.7i			0.643102i	0.946892	+	0.321551i	$0.104204\pi$
$941$			18563.7i			0.643102i	−0.946892	+	0.321551i	$0.895796\pi$
$942$			39.6514i			0.00137146i
$943$	57516.0			1.98619
$944$		−	6036.74i		−	0.208135i
$945$	1490.10			0.0512940
$946$			1687.55i			0.0579989i
$947$			56236.7i			1.92972i	0.262758	+	0.964862i	$0.415368\pi$
$947$			56236.7i			1.92972i	−0.262758	+	0.964862i	$0.584632\pi$
$948$		−	345.969i		−	0.0118529i
$949$	−6874.69			−0.235155
$950$	−5974.19			−0.204030
$951$	−733.126			−0.0249981
$952$			6752.00i			0.229867i
$953$		−	18539.4i		−	0.630169i	−0.949064	−	0.315084i	$-0.897967\pi$
$953$		−	18539.4i		−	0.630169i	0.949064	−	0.315084i	$-0.102033\pi$
$954$			26018.1i			0.882985i
$955$	−22340.6			−0.756989
$956$	7683.08			0.259925
$957$	−104.295			−0.00352285
$958$	−15851.0			−0.534576
$959$	−62805.6			−2.11481
$960$			56.8701i			0.00191195i
$961$	22406.8			0.752134
$962$		−	6424.21i		−	0.215307i
$963$			49823.2i			1.66722i
$964$			12594.6i			0.420792i
$965$		−	14307.0i		−	0.477264i
$966$			1221.70i			0.0406910i
$967$			39149.7i			1.30193i	0.759106	+	0.650967i	$0.225637\pi$
$967$			39149.7i			1.30193i	−0.759106	+	0.650967i	$0.774363\pi$
$968$		−	10565.7i		−	0.350821i
$969$		−	137.444i		−	0.00455661i
$970$		−	422.994i		−	0.0140016i
$971$	−32454.4			−1.07262			−0.536308	−	0.844022i	$-0.680181\pi$
$971$	−32454.4			−1.07262			−0.536308	+	0.844022i	$0.680181\pi$
$972$			1069.40i			0.0352892i
$973$	8362.33			0.275523
$974$		−	42859.4i		−	1.40996i
$975$		−	88.6193i		−	0.00291086i
$976$	11124.7			0.364850
$977$	22864.2			0.748712			0.374356	−	0.927285i	$-0.377864\pi$
$977$	22864.2			0.748712			0.374356	+	0.927285i	$0.377864\pi$
$978$			635.879i			0.0207906i
$979$	3650.25			0.119165
$980$	18068.4			0.588953
$981$		−	31739.4i		−	1.03299i
$982$			11286.5i			0.366768i
$983$	25559.0			0.829304			0.414652	−	0.909980i	$-0.363903\pi$
$983$	25559.0			0.829304			0.414652	+	0.909980i	$0.363903\pi$
$984$	−350.178			−0.0113448
$985$			7920.06i			0.256197i
$986$		−	14444.6i		−	0.466540i
$987$	2196.34			0.0708312
$988$			1658.53i			0.0534058i
$989$	−42289.2			−1.35967
$990$			1257.31i			0.0403637i
$991$			8692.11i			0.278622i	0.990249	+	0.139311i	$0.0444887\pi$
$991$			8692.11i			0.278622i	−0.990249	+	0.139311i	$0.955511\pi$
$992$	2749.80			0.0880103
$993$		−	913.457i		−	0.0291920i
$994$			65489.3i			2.08973i
$995$	13446.3			0.428419
$996$	32.5935			0.00103691
$997$	−12135.9			−0.385505			−0.192752	−	0.981247i	$-0.561741\pi$
$997$	−12135.9			−0.385505			−0.192752	+	0.981247i	$0.561741\pi$
$998$	38041.3			1.20659
$999$			2115.54i			0.0669998i

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Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	538.4.b.a.537.17	✓	68
269.268	even	2	inner	538.4.b.a.537.52	yes	68

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
538.4.b.a.537.17	✓	68	1.1	even	1	trivial
538.4.b.a.537.52	yes	68	269.268	even	2	inner

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Elliptic curves over $\Q$
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Abelian varieties over $\F_{q}$
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$q$ -expansion

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Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	538.4.b.a.537.17

Level	$538$
Weight	$4$
Character	538.537
Analytic conductor	$31.743$
Analytic rank	$0$
Dimension	$68$
Inner twists	$2$