LMFDB - Embedded newform 538.4.b.a.537.10

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)

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);

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")

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			537.10
Character	$\chi$	$=$	538.537
Dual form			538.4.b.a.537.59

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.00000i q^{2} -4.97105i q^{3} -4.00000 q^{4} +0.491173 q^{5} -9.94210 q^{6} -17.1942i q^{7} +8.00000i q^{8} +2.28866 q^{9} -0.982346i q^{10} -19.6218 q^{11} +19.8842i q^{12} +14.1613 q^{13} -34.3883 q^{14} -2.44165i q^{15} +16.0000 q^{16} -109.956i q^{17} -4.57732i q^{18} -17.2762i q^{19} -1.96469 q^{20} -85.4731 q^{21} +39.2435i q^{22} -168.514 q^{23} +39.7684 q^{24} -124.759 q^{25} -28.3226i q^{26} -145.595i q^{27} +68.7767i q^{28} +23.6495i q^{29} -4.88329 q^{30} +259.469i q^{31} -32.0000i q^{32} +97.5408i q^{33} -219.911 q^{34} -8.44531i q^{35} -9.15464 q^{36} -54.3877 q^{37} -34.5525 q^{38} -70.3964i q^{39} +3.92938i q^{40} -165.067 q^{41} +170.946i q^{42} +549.924 q^{43} +78.4871 q^{44} +1.12413 q^{45} +337.028i q^{46} -236.507 q^{47} -79.5368i q^{48} +47.3607 q^{49} +249.517i q^{50} -546.595 q^{51} -56.6451 q^{52} +167.645 q^{53} -291.191 q^{54} -9.63768 q^{55} +137.553 q^{56} -85.8811 q^{57} +47.2991 q^{58} +254.511i q^{59} +9.76658i q^{60} -653.530 q^{61} +518.938 q^{62} -39.3516i q^{63} -64.0000 q^{64} +6.95564 q^{65} +195.082 q^{66} -24.6311 q^{67} +439.823i q^{68} +837.692i q^{69} -16.8906 q^{70} -8.37227i q^{71} +18.3093i q^{72} -945.701 q^{73} +108.775i q^{74} +620.182i q^{75} +69.1050i q^{76} +337.380i q^{77} -140.793 q^{78} +647.384 q^{79} +7.85877 q^{80} -661.968 q^{81} +330.134i q^{82} -260.105i q^{83} +341.892 q^{84} -54.0073i q^{85} -1099.85i q^{86} +117.563 q^{87} -156.974i q^{88} +1535.00 q^{89} -2.24826i q^{90} -243.491i q^{91} +674.056 q^{92} +1289.83 q^{93} +473.015i q^{94} -8.48562i q^{95} -159.074 q^{96} -1068.96 q^{97} -94.7214i q^{98} -44.9076 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$		−	4.97105i		−	0.956679i	−0.878175	−	0.478340i	$-0.841239\pi$
$3$		−	4.97105i		−	0.956679i	0.878175	−	0.478340i	$-0.158761\pi$
$4$	−4.00000			−0.500000
$5$	0.491173			0.0439318			0.0219659	−	0.999759i	$-0.493007\pi$
$5$	0.491173			0.0439318			0.0219659	+	0.999759i	$0.493007\pi$
$6$	−9.94210			−0.676474
$7$		−	17.1942i		−	0.928398i	−0.885731	−	0.464199i	$-0.846342\pi$
$7$		−	17.1942i		−	0.928398i	0.885731	−	0.464199i	$-0.153658\pi$
$8$			8.00000i			0.353553i
$9$	2.28866			0.0847652
$10$		−	0.982346i		−	0.0310645i
$11$	−19.6218			−0.537835			−0.268918	−	0.963163i	$-0.586666\pi$
$11$	−19.6218			−0.537835			−0.268918	+	0.963163i	$0.586666\pi$
$12$			19.8842i			0.478340i
$13$	14.1613			0.302126			0.151063	−	0.988524i	$-0.451730\pi$
$13$	14.1613			0.302126			0.151063	+	0.988524i	$0.451730\pi$
$14$	−34.3883			−0.656476
$15$		−	2.44165i		−	0.0420287i
$16$	16.0000			0.250000
$17$		−	109.956i		−	1.56872i	−0.620309	−	0.784358i	$-0.712993\pi$
$17$		−	109.956i		−	1.56872i	0.620309	−	0.784358i	$-0.287007\pi$
$18$		−	4.57732i		−	0.0599380i
$19$		−	17.2762i		−	0.208602i	−0.994546	−	0.104301i	$-0.966739\pi$
$19$		−	17.2762i		−	0.208602i	0.994546	−	0.104301i	$-0.0332606\pi$
$20$	−1.96469			−0.0219659
$21$	−85.4731			−0.888179
$22$			39.2435i			0.380307i
$23$	−168.514			−1.52772			−0.763861	−	0.645381i	$-0.776699\pi$
$23$	−168.514			−1.52772			−0.763861	+	0.645381i	$0.776699\pi$
$24$	39.7684			0.338237
$25$	−124.759			−0.998070
$26$		−	28.3226i		−	0.213635i
$27$		−	145.595i		−	1.03777i
$28$			68.7767i			0.464199i
$29$			23.6495i			0.151435i	0.997129	+	0.0757174i	$0.0241247\pi$
$29$			23.6495i			0.151435i	−0.997129	+	0.0757174i	$0.975875\pi$
$30$	−4.88329			−0.0297188
$31$			259.469i			1.50329i	0.659568	+	0.751645i	$0.270739\pi$
$31$			259.469i			1.50329i	−0.659568	+	0.751645i	$0.729261\pi$
$32$		−	32.0000i		−	0.176777i
$33$			97.5408i			0.514536i
$34$	−219.911			−1.10925
$35$		−	8.44531i		−	0.0407862i
$36$	−9.15464			−0.0423826
$37$	−54.3877			−0.241656			−0.120828	−	0.992673i	$-0.538555\pi$
$37$	−54.3877			−0.241656			−0.120828	+	0.992673i	$0.538555\pi$
$38$	−34.5525			−0.147504
$39$		−	70.3964i		−	0.289037i
$40$			3.92938i			0.0155323i
$41$	−165.067			−0.628759			−0.314380	−	0.949297i	$-0.601796\pi$
$41$	−165.067			−0.628759			−0.314380	+	0.949297i	$0.601796\pi$
$42$			170.946i			0.628037i
$43$	549.924			1.95029			0.975147	−	0.221561i	$-0.0711151\pi$
$43$	549.924			1.95029			0.975147	+	0.221561i	$0.0711151\pi$
$44$	78.4871			0.268918
$45$	1.12413			0.00372389
$46$			337.028i			1.08026i
$47$	−236.507			−0.734003			−0.367002	−	0.930220i	$-0.619616\pi$
$47$	−236.507			−0.734003			−0.367002	+	0.930220i	$0.619616\pi$
$48$		−	79.5368i		−	0.239170i
$49$	47.3607			0.138078
$50$			249.517i			0.705742i
$51$	−546.595			−1.50076
$52$	−56.6451			−0.151063
$53$	167.645			0.434486			0.217243	−	0.976118i	$-0.430294\pi$
$53$	167.645			0.434486			0.217243	+	0.976118i	$0.430294\pi$
$54$	−291.191			−0.733816
$55$	−9.63768			−0.0236281
$56$	137.553			0.328238
$57$	−85.8811			−0.199565
$58$	47.2991			0.107081
$59$			254.511i			0.561602i	0.959766	+	0.280801i	$0.0906001\pi$
$59$			254.511i			0.561602i	−0.959766	+	0.280801i	$0.909400\pi$
$60$			9.76658i			0.0210143i
$61$	−653.530			−1.37174			−0.685868	−	0.727726i	$-0.740577\pi$
$61$	−653.530			−1.37174			−0.685868	+	0.727726i	$0.740577\pi$
$62$	518.938			1.06299
$63$		−	39.3516i		−	0.0786958i
$64$	−64.0000			−0.125000
$65$	6.95564			0.0132729
$66$	195.082			0.363832
$67$	−24.6311			−0.0449129			−0.0224565	−	0.999748i	$-0.507149\pi$
$67$	−24.6311			−0.0449129			−0.0224565	+	0.999748i	$0.507149\pi$
$68$			439.823i			0.784358i
$69$			837.692i			1.46154i
$70$	−16.8906			−0.0288402
$71$		−	8.37227i		−	0.0139944i	−0.999976	−	0.00699722i	$-0.997773\pi$
$71$		−	8.37227i		−	0.0139944i	0.999976	−	0.00699722i	$-0.00222730\pi$
$72$			18.3093i			0.0299690i
$73$	−945.701			−1.51625			−0.758123	−	0.652112i	$-0.773883\pi$
$73$	−945.701			−1.51625			−0.758123	+	0.652112i	$0.773883\pi$
$74$			108.775i			0.170877i
$75$			620.182i			0.954833i
$76$			69.1050i			0.104301i
$77$			337.380i			0.499325i
$78$	−140.793			−0.204380
$79$	647.384			0.921980			0.460990	−	0.887405i	$-0.347494\pi$
$79$	647.384			0.921980			0.460990	+	0.887405i	$0.347494\pi$
$80$	7.85877			0.0109830
$81$	−661.968			−0.908050
$82$			330.134i			0.444600i
$83$		−	260.105i		−	0.343978i	−0.985099	−	0.171989i	$-0.944981\pi$
$83$		−	260.105i		−	0.343978i	0.985099	−	0.171989i	$-0.0550194\pi$
$84$	341.892			0.444089
$85$		−	54.0073i		−	0.0689166i
$86$		−	1099.85i		−	1.37907i
$87$	117.563			0.144875
$88$		−	156.974i		−	0.190153i
$89$	1535.00			1.82820			0.914102	−	0.405485i	$-0.132897\pi$
$89$	1535.00			1.82820			0.914102	+	0.405485i	$0.132897\pi$
$90$		−	2.24826i		−	0.00263319i
$91$		−	243.491i		−	0.280493i
$92$	674.056			0.763861
$93$	1289.83			1.43817
$94$			473.015i			0.519019i
$95$		−	8.48562i		−	0.00916428i
$96$	−159.074			−0.169119
$97$	−1068.96			−1.11894			−0.559468	−	0.828852i	$-0.688995\pi$
$97$	−1068.96			−1.11894			−0.559468	+	0.828852i	$0.688995\pi$
$98$		−	94.7214i		−	0.0976358i
$99$	−44.9076			−0.0455897
$100$	499.035			0.499035
$101$			1215.32i			1.19732i	0.801005	+	0.598658i	$0.204299\pi$
$101$			1215.32i			1.19732i	−0.801005	+	0.598658i	$0.795701\pi$
$102$			1093.19i			1.06120i
$103$	1809.33			1.73086			0.865431	−	0.501028i	$-0.167045\pi$
$103$	1809.33			1.73086			0.865431	+	0.501028i	$0.167045\pi$
$104$			113.290i			0.106818i
$105$	−41.9821			−0.0390193
$106$		−	335.289i		−	0.307228i
$107$		−	1524.20i		−	1.37710i	−0.725189	−	0.688550i	$-0.758248\pi$
$107$		−	1524.20i		−	1.37710i	0.725189	−	0.688550i	$-0.241752\pi$
$108$			582.382i			0.518886i
$109$			1803.26i			1.58459i	0.610135	+	0.792297i	$0.291115\pi$
$109$			1803.26i			1.58459i	−0.610135	+	0.792297i	$0.708885\pi$
$110$			19.2754i			0.0167076i
$111$			270.364i			0.231187i
$112$		−	275.107i		−	0.232099i
$113$			1149.08i			0.956603i	0.878196	+	0.478301i	$0.158747\pi$
$113$			1149.08i			0.956603i	−0.878196	+	0.478301i	$0.841253\pi$
$114$			171.762i			0.141114i
$115$	−82.7696			−0.0671157
$116$		−	94.5982i		−	0.0757174i
$117$	32.4104			0.0256097
$118$	509.022			0.397113
$119$	−1890.60			−1.45639
$120$	19.5332			0.0148594
$121$	−945.986			−0.710733
$122$			1307.06i			0.969964i
$123$			820.556i			0.601521i
$124$		−	1037.88i		−	0.751645i
$125$	−122.675			−0.0877789
$126$	−78.7032			−0.0556463
$127$	−2220.46			−1.55145			−0.775726	−	0.631070i	$-0.782616\pi$
$127$	−2220.46			−1.55145			−0.775726	+	0.631070i	$0.782616\pi$
$128$			128.000i			0.0883883i
$129$		−	2733.70i		−	1.86580i
$130$		−	13.9113i		−	0.00938538i
$131$	1206.79			0.804865			0.402433	−	0.915450i	$-0.368165\pi$
$131$	1206.79			0.804865			0.402433	+	0.915450i	$0.368165\pi$
$132$		−	390.163i		−	0.257268i
$133$	−297.051			−0.193666
$134$			49.2622i			0.0317582i
$135$		−	71.5125i		−	0.0455912i
$136$	879.645			0.554625
$137$			931.261i			0.580752i	0.956913	+	0.290376i	$0.0937803\pi$
$137$			931.261i			0.580752i	−0.956913	+	0.290376i	$0.906220\pi$
$138$	1675.38			1.03346
$139$		−	1688.27i		−	1.03019i	−0.857132	−	0.515097i	$-0.827756\pi$
$139$		−	1688.27i		−	1.03019i	0.857132	−	0.515097i	$-0.172244\pi$
$140$			33.7812i			0.0203931i
$141$			1175.69i			0.702205i
$142$	−16.7445			−0.00989556
$143$	−277.869			−0.162494
$144$	36.6186			0.0211913
$145$			11.6160i			0.00665281i
$146$			1891.40i			1.07215i
$147$		−	235.432i		−	0.132096i
$148$	217.551			0.120828
$149$	−2065.19			−1.13548			−0.567742	−	0.823207i	$-0.692183\pi$
$149$	−2065.19			−1.13548			−0.567742	+	0.823207i	$0.692183\pi$
$150$	1240.36			0.675169
$151$	763.903			0.411693			0.205846	−	0.978584i	$-0.434005\pi$
$151$	763.903			0.411693			0.205846	+	0.978584i	$0.434005\pi$
$152$	138.210			0.0737520
$153$		−	251.651i		−	0.132972i
$154$	674.760			0.353076
$155$			127.444i			0.0660423i
$156$			281.586i			0.144519i
$157$		−	100.674i		−	0.0511759i	−0.999673	−	0.0255880i	$-0.991854\pi$
$157$		−	100.674i		−	0.0511759i	0.999673	−	0.0255880i	$-0.00814579\pi$
$158$		−	1294.77i		−	0.651938i
$159$		−	833.370i		−	0.415664i
$160$		−	15.7175i		−	0.00776613i
$161$			2897.46i			1.41833i
$162$			1323.94i			0.642088i
$163$		−	2904.30i		−	1.39560i	−0.716295	−	0.697798i	$-0.754163\pi$
$163$		−	2904.30i		−	1.39560i	0.716295	−	0.697798i	$-0.245837\pi$
$164$	660.268			0.314380
$165$			47.9094i			0.0226045i
$166$	−520.209			−0.243229
$167$		−	1794.45i		−	0.831491i	−0.909481	−	0.415745i	$-0.863521\pi$
$167$		−	1794.45i		−	0.831491i	0.909481	−	0.415745i	$-0.136479\pi$
$168$		−	683.784i		−	0.314019i
$169$	−1996.46			−0.908720
$170$	−108.015			−0.0487314
$171$		−	39.5395i		−	0.0176822i
$172$	−2199.70			−0.975147
$173$	−3022.54			−1.32832			−0.664160	−	0.747590i	$-0.731211\pi$
$173$	−3022.54			−1.32832			−0.664160	+	0.747590i	$0.731211\pi$
$174$		−	235.126i		−	0.102442i
$175$			2145.12i			0.926606i
$176$	−313.948			−0.134459
$177$	1265.19			0.537273
$178$		−	3070.01i		−	1.29274i
$179$		−	2080.63i		−	0.868789i	−0.900723	−	0.434395i	$-0.856962\pi$
$179$		−	2080.63i		−	0.868789i	0.900723	−	0.434395i	$-0.143038\pi$
$180$	−4.49651			−0.00186195
$181$		−	2500.24i		−	1.02675i	−0.858165	−	0.513374i	$-0.828395\pi$
$181$		−	2500.24i		−	1.02675i	0.858165	−	0.513374i	$-0.171605\pi$
$182$	−486.983			−0.198338
$183$			3248.73i			1.31231i
$184$		−	1348.11i		−	0.540131i
$185$	−26.7137			−0.0106164
$186$		−	2579.67i		−	1.01694i
$187$			2157.52i			0.843710i
$188$	946.030			0.367002
$189$	−2503.39			−0.963465
$190$	−16.9712			−0.00648012
$191$	−3732.90			−1.41415			−0.707077	−	0.707136i	$-0.749987\pi$
$191$	−3732.90			−1.41415			−0.707077	+	0.707136i	$0.749987\pi$
$192$			318.147i			0.119585i
$193$			389.871i			0.145407i	0.997354	+	0.0727035i	$0.0231627\pi$
$193$			389.871i			0.145407i	−0.997354	+	0.0727035i	$0.976837\pi$
$194$			2137.93i			0.791208i
$195$		−	34.5768i		−	0.0126979i
$196$	−189.443			−0.0690389
$197$		−	4214.97i		−	1.52439i	−0.647349	−	0.762194i	$-0.724122\pi$
$197$		−	4214.97i		−	1.52439i	0.647349	−	0.762194i	$-0.275878\pi$
$198$			89.8151i			0.0322368i
$199$	3698.83			1.31760			0.658801	−	0.752317i	$-0.271064\pi$
$199$	3698.83			1.31760			0.658801	+	0.752317i	$0.271064\pi$
$200$		−	998.070i		−	0.352871i
$201$			122.442i			0.0429673i
$202$	2430.64			0.846630
$203$	406.634			0.140592
$204$	2186.38			0.750379
$205$	−81.0764			−0.0276226
$206$		−	3618.66i		−	1.22390i
$207$	−385.671			−0.129498
$208$	226.581			0.0755314
$209$			338.990i			0.112194i
$210$			83.9641i			0.0275908i
$211$	−3734.81			−1.21856			−0.609278	−	0.792957i	$-0.708540\pi$
$211$	−3734.81			−1.21856			−0.609278	+	0.792957i	$0.708540\pi$
$212$	−670.578			−0.217243
$213$	−41.6190			−0.0133882
$214$	−3048.39			−0.973757
$215$	270.108			0.0856800
$216$	1164.76			0.366908
$217$	4461.35			1.39565
$218$	3606.52			1.12048
$219$			4701.13i			1.45056i
$220$	38.5507			0.0118140
$221$		−	1557.11i		−	0.473949i
$222$	540.728			0.163474
$223$		−	5678.10i		−	1.70508i	−0.522658	−	0.852542i	$-0.675060\pi$
$223$		−	5678.10i		−	1.70508i	0.522658	−	0.852542i	$-0.324940\pi$
$224$	−550.213			−0.164119
$225$	−285.530			−0.0846016
$226$	2298.16			0.676420
$227$		−	5738.77i		−	1.67795i	−0.544166	−	0.838977i	$-0.683154\pi$
$227$		−	5738.77i		−	1.67795i	0.544166	−	0.838977i	$-0.316846\pi$
$228$	343.524			0.0997827
$229$			2013.25i			0.580958i	0.956881	+	0.290479i	$0.0938147\pi$
$229$			2013.25i			0.580958i	−0.956881	+	0.290479i	$0.906185\pi$
$230$			165.539i			0.0474579i
$231$	1677.13			0.477694
$232$	−189.196			−0.0535403
$233$	−18.4888			−0.00519846			−0.00259923	−	0.999997i	$-0.500827\pi$
$233$	−18.4888			−0.00519846			−0.00259923	+	0.999997i	$0.500827\pi$
$234$		−	64.8207i		−	0.0181088i
$235$	−116.166			−0.0322461
$236$		−	1018.04i		−	0.280801i
$237$		−	3218.18i		−	0.882039i
$238$			3781.19i			1.02982i
$239$	−3139.14			−0.849598			−0.424799	−	0.905288i	$-0.639655\pi$
$239$	−3139.14			−0.849598			−0.424799	+	0.905288i	$0.639655\pi$
$240$		−	39.0663i		−	0.0105072i
$241$			6179.50i			1.65169i	0.563899	+	0.825844i	$0.309301\pi$
$241$			6179.50i			1.65169i	−0.563899	+	0.825844i	$0.690699\pi$
$242$			1891.97i			0.502564i
$243$		−	640.399i		−	0.169060i
$244$	2614.12			0.685868
$245$	23.2623			0.00606601
$246$	1641.11			0.425339
$247$		−	244.654i		−	0.0630241i
$248$	−2075.75			−0.531493
$249$	−1292.99			−0.329077
$250$			245.350i			0.0620691i
$251$		−	2121.11i		−	0.533399i	−0.963780	−	0.266699i	$-0.914067\pi$
$251$		−	2121.11i		−	0.533399i	0.963780	−	0.266699i	$-0.0859331\pi$
$252$			157.406i			0.0393479i
$253$	3306.54			0.821663
$254$			4440.93i			1.09704i
$255$	−268.473			−0.0659310
$256$	256.000			0.0625000
$257$		−	7021.50i		−	1.70424i	−0.523348	−	0.852119i	$-0.675317\pi$
$257$		−	7021.50i		−	1.70424i	0.523348	−	0.852119i	$-0.324683\pi$
$258$	−5467.40			−1.31932
$259$			935.150i			0.224353i
$260$	−27.8226			−0.00663647
$261$			54.1258i			0.0128364i
$262$		−	2413.57i		−	0.569126i
$263$	3532.12			0.828137			0.414068	−	0.910246i	$-0.364107\pi$
$263$	3532.12			0.828137			0.414068	+	0.910246i	$0.364107\pi$
$264$	−780.326			−0.181916
$265$	82.3425			0.0190878
$266$			594.101i			0.136942i
$267$		−	7630.58i		−	1.74900i
$268$	98.5243			0.0224565
$269$	4225.82	−	1267.88i	0.957818	−	0.287375i
$269$	4225.82	−	1267.88i	0.957818	−	0.287375i
$270$	−143.025			−0.0322379
$271$			6518.86i			1.46123i	0.682791	+	0.730614i	$0.260766\pi$
$271$			6518.86i			1.46123i	−0.682791	+	0.730614i	$0.739234\pi$
$272$		−	1759.29i		−	0.392179i
$273$	−1210.41			−0.268341
$274$	1862.52			0.410653
$275$	2447.99			0.536797
$276$		−	3350.77i		−	0.730770i
$277$		−	5406.48i		−	1.17272i	−0.810050	−	0.586361i	$-0.800560\pi$
$277$		−	5406.48i		−	1.17272i	0.810050	−	0.586361i	$-0.199440\pi$
$278$	−3376.54			−0.728457
$279$			593.836i			0.127427i
$280$	67.5625			0.0144201
$281$		−	926.537i		−	0.196700i	−0.995152	−	0.0983498i	$-0.968644\pi$
$281$		−	926.537i		−	0.196700i	0.995152	−	0.0983498i	$-0.0313564\pi$
$282$	2351.38			0.496534
$283$	6646.88			1.39617			0.698085	−	0.716015i	$-0.254036\pi$
$283$	6646.88			1.39617			0.698085	+	0.716015i	$0.254036\pi$
$284$			33.4891i			0.00699722i
$285$	−42.1825			−0.00876728
$286$			555.739i			0.114900i
$287$			2838.19i			0.583739i
$288$		−	73.2371i		−	0.0149845i
$289$	−7177.25			−1.46087
$290$	23.2320			0.00470425
$291$			5313.87i			1.07046i
$292$	3782.80			0.758123
$293$	−9849.91			−1.96395			−0.981976	−	0.189005i	$-0.939474\pi$
$293$	−9849.91			−1.96395			−0.981976	+	0.189005i	$0.939474\pi$
$294$	−470.865			−0.0934061
$295$			125.009i			0.0246722i
$296$		−	435.101i		−	0.0854383i
$297$			2856.84i			0.558150i
$298$			4130.38i			0.802908i
$299$	−2386.37			−0.461564
$300$		−	2480.73i		−	0.477416i
$301$		−	9455.48i		−	1.81065i
$302$		−	1527.81i		−	0.291111i
$303$	6041.42			1.14545
$304$		−	276.420i		−	0.0521506i
$305$	−320.996			−0.0602629
$306$	−503.302			−0.0940258
$307$	−1277.12			−0.237424			−0.118712	−	0.992929i	$-0.537876\pi$
$307$	−1277.12			−0.237424			−0.118712	+	0.992929i	$0.537876\pi$
$308$		−	1349.52i		−	0.249662i
$309$		−	8994.28i		−	1.65588i
$310$	254.888			0.0466990
$311$		−	8103.30i		−	1.47748i	−0.673991	−	0.738739i	$-0.735421\pi$
$311$		−	8103.30i		−	1.47748i	0.673991	−	0.738739i	$-0.264579\pi$
$312$	563.172			0.102190
$313$	3877.65			0.700248			0.350124	−	0.936703i	$-0.386139\pi$
$313$	3877.65			0.700248			0.350124	+	0.936703i	$0.386139\pi$
$314$	−201.347			−0.0361869
$315$		−	19.3284i		−	0.00345725i
$316$	−2589.54			−0.460990
$317$			5698.56i			1.00966i	0.863218	+	0.504831i	$0.168445\pi$
$317$			5698.56i			1.00966i	−0.863218	+	0.504831i	$0.831555\pi$
$318$	−1666.74			−0.293919
$319$		−	464.046i		−	0.0814470i
$320$	−31.4351			−0.00549148
$321$	−7576.86			−1.31744
$322$	5794.92			1.00291
$323$	−1899.62			−0.327238
$324$	2647.87			0.454025
$325$	−1766.74			−0.301543
$326$	−5808.59			−0.986835
$327$	8964.09			1.51595
$328$		−	1320.54i		−	0.222300i
$329$			4066.55i			0.681447i
$330$	95.8188			0.0159838
$331$	5128.06			0.851553			0.425776	−	0.904828i	$-0.360001\pi$
$331$	5128.06			0.851553			0.425776	+	0.904828i	$0.360001\pi$
$332$			1040.42i			0.171989i
$333$	−124.475			−0.0204840
$334$	−3588.91			−0.587953
$335$	−12.0981			−0.00197311
$336$	−1367.57			−0.222045
$337$		−	2053.95i		−	0.332005i	−0.986125	−	0.166002i	$-0.946914\pi$
$337$		−	2053.95i		−	0.332005i	0.986125	−	0.166002i	$-0.0530860\pi$
$338$			3992.92i			0.642562i
$339$	5712.12			0.915162
$340$			216.029i			0.0344583i
$341$		−	5091.24i		−	0.808522i
$342$	−79.0789			−0.0125032
$343$		−	6711.93i		−	1.05659i
$344$			4399.39i			0.689533i
$345$			411.452i			0.0642081i
$346$			6045.08i			0.939264i
$347$	−3619.43			−0.559945			−0.279973	−	0.960008i	$-0.590325\pi$
$347$	−3619.43			−0.559945			−0.279973	+	0.960008i	$0.590325\pi$
$348$	−470.252			−0.0724373
$349$	−1481.94			−0.227296			−0.113648	−	0.993521i	$-0.536254\pi$
$349$	−1481.94			−0.227296			−0.113648	+	0.993521i	$0.536254\pi$
$350$	4290.24			0.655209
$351$		−	2061.82i		−	0.313538i
$352$			627.897i			0.0950767i
$353$	4019.35			0.606030			0.303015	−	0.952986i	$-0.402007\pi$
$353$	4019.35			0.606030			0.303015	+	0.952986i	$0.402007\pi$
$354$		−	2530.38i		−	0.379909i
$355$		−	4.11223i		−	0.000614802i
$356$	−6140.02			−0.914102
$357$			9398.25i			1.39330i
$358$	−4161.25			−0.614327
$359$		−	8085.22i		−	1.18864i	−0.804229	−	0.594320i	$-0.797421\pi$
$359$		−	8085.22i		−	1.18864i	0.804229	−	0.594320i	$-0.202579\pi$
$360$			8.99303i			0.00131659i
$361$	6560.53			0.956485
$362$	−5000.48			−0.726021
$363$			4702.54i			0.679944i
$364$			973.966i			0.140246i
$365$	−464.503			−0.0666115
$366$	6497.46			0.927944
$367$			1385.89i			0.197120i	0.995131	+	0.0985598i	$0.0314236\pi$
$367$			1385.89i			0.197120i	−0.995131	+	0.0985598i	$0.968576\pi$
$368$	−2696.22			−0.381931
$369$	−377.782			−0.0532969
$370$			53.4275i			0.00750693i
$371$		−	2882.51i		−	0.403376i
$372$	−5159.33			−0.719083
$373$			7270.97i			1.00932i	0.863318	+	0.504660i	$0.168382\pi$
$373$			7270.97i			1.00932i	−0.863318	+	0.504660i	$0.831618\pi$
$374$	4315.05			0.596593
$375$			609.822i			0.0839762i
$376$		−	1892.06i		−	0.259509i
$377$			334.908i			0.0457523i
$378$			5006.78i			0.681273i
$379$			3762.63i			0.509957i	0.966947	+	0.254978i	$0.0820683\pi$
$379$			3762.63i			0.509957i	−0.966947	+	0.254978i	$0.917932\pi$
$380$			33.9425i			0.00458214i
$381$			11038.0i			1.48424i
$382$			7465.81i			0.999958i
$383$		−	11400.3i		−	1.52096i	−0.649363	−	0.760478i	$-0.724965\pi$
$383$		−	11400.3i		−	1.52096i	0.649363	−	0.760478i	$-0.275035\pi$
$384$	636.294			0.0845593
$385$			165.712i			0.0219363i
$386$	779.743			0.102818
$387$	1258.59			0.165317
$388$	4275.86			0.559468
$389$	−653.370			−0.0851599			−0.0425799	−	0.999093i	$-0.513558\pi$
$389$	−653.370			−0.0851599			−0.0425799	+	0.999093i	$0.513558\pi$
$390$	−69.1537			−0.00897880
$391$			18529.1i			2.39656i
$392$			378.886i			0.0488179i
$393$		−	5998.99i		−	0.769998i
$394$	−8429.94			−1.07790
$395$	317.978			0.0405043
$396$	179.630			0.0227948
$397$			2854.42i			0.360855i	0.983588	+	0.180427i	$0.0577481\pi$
$397$			2854.42i			0.360855i	−0.983588	+	0.180427i	$0.942252\pi$
$398$		−	7397.66i		−	0.931686i
$399$			1476.65i			0.185276i
$400$	−1996.14			−0.249517
$401$		−	10514.9i		−	1.30944i	−0.755870	−	0.654722i	$-0.772786\pi$
$401$		−	10514.9i		−	1.30944i	0.755870	−	0.654722i	$-0.227214\pi$
$402$	244.885			0.0303824
$403$			3674.41i			0.454182i
$404$		−	4861.28i		−	0.598658i
$405$	−325.141			−0.0398923
$406$		−	813.268i		−	0.0994134i
$407$	1067.18			0.129971
$408$		−	4372.76i		−	0.530598i
$409$			11827.5i			1.42991i	0.699170	+	0.714955i	$0.253553\pi$
$409$			11827.5i			1.42991i	−0.699170	+	0.714955i	$0.746447\pi$
$410$			162.153i			0.0195321i
$411$	4629.34			0.555593
$412$	−7237.33			−0.865431
$413$	4376.11			0.521390
$414$			771.343i			0.0915687i
$415$		−	127.756i		−	0.0151116i
$416$		−	453.161i		−	0.0534088i
$417$	−8392.46			−0.985565
$418$	677.981			0.0793328
$419$	10214.1			1.19091			0.595454	−	0.803389i	$-0.296972\pi$
$419$	10214.1			1.19091			0.595454	+	0.803389i	$0.296972\pi$
$420$	167.928			0.0195097
$421$	−12928.3			−1.49664			−0.748322	−	0.663336i	$-0.769140\pi$
$421$	−12928.3			−1.49664			−0.748322	+	0.663336i	$0.769140\pi$
$422$			7469.63i			0.861649i
$423$	−541.285			−0.0622179
$424$			1341.16i			0.153614i
$425$			13717.9i			1.56569i
$426$			83.2379i			0.00946688i
$427$			11236.9i			1.27352i
$428$			6096.79i			0.688550i
$429$			1381.30i			0.155454i
$430$		−	540.216i		−	0.0605849i
$431$			809.747i			0.0904968i	0.998976	+	0.0452484i	$0.0144079\pi$
$431$			809.747i			0.0904968i	−0.998976	+	0.0452484i	$0.985592\pi$
$432$		−	2329.53i		−	0.259443i
$433$	14186.4			1.57449			0.787246	−	0.616639i	$-0.211506\pi$
$433$	14186.4			1.57449			0.787246	+	0.616639i	$0.211506\pi$
$434$		−	8922.70i		−	0.986874i
$435$	57.7438			0.00636461
$436$		−	7213.03i		−	0.792297i
$437$			2911.29i			0.318686i
$438$	9402.25			1.02570
$439$	6650.04			0.722982			0.361491	−	0.932376i	$-0.382268\pi$
$439$	6650.04			0.722982			0.361491	+	0.932376i	$0.382268\pi$
$440$		−	77.1015i		−	0.00835379i
$441$	108.393			0.0117042
$442$	−3114.23			−0.335133
$443$			12616.5i			1.35311i	0.736394	+	0.676553i	$0.236527\pi$
$443$			12616.5i			1.35311i	−0.736394	+	0.676553i	$0.763473\pi$
$444$		−	1081.46i		−	0.115594i
$445$	753.953			0.0803164
$446$	−11356.2			−1.20568
$447$			10266.2i			1.08629i
$448$			1100.43i			0.116050i
$449$	9702.08			1.01975			0.509877	−	0.860247i	$-0.329691\pi$
$449$	9702.08			1.01975			0.509877	+	0.860247i	$0.329691\pi$
$450$			571.061i			0.0598224i
$451$	3238.91			0.338169
$452$		−	4596.31i		−	0.478301i
$453$		−	3797.40i		−	0.393858i
$454$	−11477.5			−1.18649
$455$		−	119.596i		−	0.0123226i
$456$		−	687.049i		−	0.0705570i
$457$	−9734.10			−0.996372			−0.498186	−	0.867070i	$-0.666000\pi$
$457$	−9734.10			−0.996372			−0.498186	+	0.867070i	$0.666000\pi$
$458$	4026.50			0.410800
$459$	−16009.0			−1.62797
$460$	331.078			0.0335578
$461$		−	407.461i		−	0.0411656i	−0.999788	−	0.0205828i	$-0.993448\pi$
$461$		−	407.461i		−	0.0411656i	0.999788	−	0.0205828i	$-0.00655217\pi$
$462$		−	3354.27i		−	0.337780i
$463$		−	5639.04i		−	0.566023i	−0.959117	−	0.283011i	$-0.908667\pi$
$463$		−	5639.04i		−	0.566023i	0.959117	−	0.283011i	$-0.0913334\pi$
$464$			378.393i			0.0378587i
$465$	633.531			0.0631813
$466$			36.9776i			0.00367587i
$467$			4732.43i			0.468931i	0.972124	+	0.234466i	$0.0753340\pi$
$467$			4732.43i			0.468931i	−0.972124	+	0.234466i	$0.924666\pi$
$468$	−129.641			−0.0128049
$469$			423.511i			0.0416971i
$470$			232.332i			0.0228014i
$471$	−500.453			−0.0489590
$472$	−2036.09			−0.198556
$473$	−10790.5			−1.04894
$474$	−6436.36			−0.623696
$475$			2155.36i			0.208200i
$476$	7562.38			0.728196
$477$	383.681			0.0368293
$478$			6278.27i			0.600756i
$479$		−	104.634i		−	0.00998092i	−0.999988	−	0.00499046i	$-0.998411\pi$
$479$		−	104.634i		−	0.00998092i	0.999988	−	0.00499046i	$-0.00158852\pi$
$480$	−78.1327			−0.00742969
$481$	−770.199			−0.0730105
$482$	12359.0			1.16792
$483$	14403.4			1.35689
$484$	3783.94			0.355367
$485$	−525.046			−0.0491569
$486$	−1280.80			−0.119544
$487$	−1644.29			−0.152997			−0.0764987	−	0.997070i	$-0.524374\pi$
$487$	−1644.29			−0.152997			−0.0764987	+	0.997070i	$0.524374\pi$
$488$		−	5228.24i		−	0.484982i
$489$	−14437.4			−1.33514
$490$		−	46.5246i		−	0.00428932i
$491$	−16237.4			−1.49243			−0.746216	−	0.665703i	$-0.768132\pi$
$491$	−16237.4			−1.49243			−0.746216	+	0.665703i	$0.768132\pi$
$492$		−	3282.22i		−	0.300760i
$493$	2600.40			0.237558
$494$	−489.307			−0.0445647
$495$	−22.0574			−0.00200284
$496$			4151.50i			0.375823i
$497$	−143.954			−0.0129924
$498$			2585.99i			0.232692i
$499$		−	12847.2i		−	1.15254i	−0.817258	−	0.576271i	$-0.804507\pi$
$499$		−	12847.2i		−	1.15254i	0.817258	−	0.576271i	$-0.195493\pi$
$500$	490.699			0.0438895
$501$	−8920.31			−0.795470
$502$	−4242.22			−0.377170
$503$		−	10589.3i		−	0.938676i	−0.883019	−	0.469338i	$-0.844493\pi$
$503$		−	10589.3i		−	0.938676i	0.883019	−	0.469338i	$-0.155507\pi$
$504$	314.813			0.0278232
$505$			596.932i			0.0526003i
$506$		−	6613.09i		−	0.581003i
$507$			9924.49i			0.869353i
$508$	8881.85			0.775726
$509$			8224.29i			0.716180i	0.933687	+	0.358090i	$0.116572\pi$
$509$			8224.29i			0.716180i	−0.933687	+	0.358090i	$0.883428\pi$
$510$			536.946i			0.0466203i
$511$			16260.5i			1.40768i
$512$		−	512.000i		−	0.0441942i
$513$	−2515.34			−0.216482
$514$	−14043.0			−1.20508
$515$	888.695			0.0760400
$516$			10934.8i			0.932902i
$517$	4640.69			0.394773
$518$	1870.30			0.158641
$519$			15025.2i			1.27078i
$520$			55.6451i			0.00469269i
$521$		−	8452.55i		−	0.710773i	−0.934719	−	0.355387i	$-0.884349\pi$
$521$		−	8452.55i		−	0.710773i	0.934719	−	0.355387i	$-0.115651\pi$
$522$	108.252			0.00907671
$523$			22589.9i			1.88869i	0.328952	+	0.944347i	$0.393305\pi$
$523$			22589.9i			1.88869i	−0.328952	+	0.944347i	$0.606695\pi$
$524$	−4827.14			−0.402433
$525$	10663.5			0.886464
$526$		−	7064.24i		−	0.585581i
$527$	28530.1			2.35823
$528$			1560.65i			0.128634i
$529$	16230.0			1.33393
$530$		−	164.685i		−	0.0134971i
$531$			582.490i			0.0476043i
$532$	1188.20			0.0968329
$533$	−2337.56			−0.189964
$534$	−15261.2			−1.23673
$535$		−	748.645i		−	0.0604986i
$536$		−	197.049i		−	0.0158791i
$537$	−10342.9			−0.831152
$538$	−2535.76	−	8451.65i	−0.203205	−	0.677280i
$539$	−929.301			−0.0742631
$540$			286.050i			0.0227956i
$541$		−	2035.86i		−	0.161790i	−0.996723	−	0.0808949i	$-0.974222\pi$
$541$		−	2035.86i		−	0.161790i	0.996723	−	0.0808949i	$-0.0257778\pi$
$542$	13037.7			1.03324
$543$	−12428.8			−0.982269
$544$	−3518.58			−0.277312
$545$			885.712i			0.0696142i
$546$			2420.82i			0.189746i
$547$	20136.9			1.57403			0.787013	−	0.616936i	$-0.211626\pi$
$547$	20136.9			1.57403			0.787013	+	0.616936i	$0.211626\pi$
$548$		−	3725.04i		−	0.290376i
$549$	−1495.71			−0.116275
$550$		−	4895.97i		−	0.379573i
$551$	408.575			0.0315896
$552$	−6701.53			−0.516732
$553$		−	11131.2i		−	0.855964i
$554$	−10813.0			−0.829239
$555$			132.795i			0.0101565i
$556$			6753.07i			0.515097i
$557$			5028.14i			0.382494i	0.981542	+	0.191247i	$0.0612531\pi$
$557$			5028.14i			0.382494i	−0.981542	+	0.191247i	$0.938747\pi$
$558$	1187.67			0.0901043
$559$	7787.63			0.589234
$560$		−	135.125i		−	0.0101966i
$561$	10725.2			0.807160
$562$	−1853.07			−0.139088
$563$	14450.1			1.08170			0.540852	−	0.841118i	$-0.318102\pi$
$563$	14450.1			1.08170			0.540852	+	0.841118i	$0.318102\pi$
$564$		−	4702.76i		−	0.351103i
$565$			564.396i			0.0420253i
$566$		−	13293.8i		−	0.987241i
$567$			11382.0i			0.843031i
$568$	66.9781			0.00494778
$569$		−	14248.7i		−	1.04980i	−0.851165	−	0.524899i	$-0.824103\pi$
$569$		−	14248.7i		−	1.04980i	0.851165	−	0.524899i	$-0.175897\pi$
$570$			84.3649i			0.00619940i
$571$		−	13873.8i		−	1.01681i	−0.861118	−	0.508405i	$-0.830235\pi$
$571$		−	13873.8i		−	1.01681i	0.861118	−	0.508405i	$-0.169765\pi$
$572$	1111.48			0.0812469
$573$			18556.5i			1.35289i
$574$	5676.38			0.412765
$575$	21023.6			1.52477
$576$	−146.474			−0.0105957
$577$		−	20269.9i		−	1.46248i	−0.682122	−	0.731238i	$-0.738943\pi$
$577$		−	20269.9i		−	1.46248i	0.682122	−	0.731238i	$-0.261057\pi$
$578$			14354.5i			1.03299i
$579$	1938.07			0.139108
$580$		−	46.4641i		−	0.00332641i
$581$	−4472.28			−0.319348
$582$	10627.7			0.756932
$583$	−3289.48			−0.233682
$584$		−	7565.61i		−	0.536074i
$585$	15.9191			0.00112508
$586$			19699.8i			1.38872i
$587$	−2970.21			−0.208848			−0.104424	−	0.994533i	$-0.533300\pi$
$587$	−2970.21			−0.208848			−0.104424	+	0.994533i	$0.533300\pi$
$588$			941.730i			0.0660481i
$589$	4482.65			0.313590
$590$	250.018			0.0174459
$591$	−20952.8			−1.45835
$592$	−870.202			−0.0604140
$593$	−24536.3			−1.69913			−0.849567	−	0.527480i	$-0.823137\pi$
$593$	−24536.3			−1.69913			−0.849567	+	0.527480i	$0.823137\pi$
$594$	5713.68			0.394672
$595$	−928.610			−0.0639820
$596$	8260.77			0.567742
$597$		−	18387.1i		−	1.26052i
$598$			4772.75i			0.326375i
$599$	10498.0			0.716085			0.358043	−	0.933705i	$-0.383444\pi$
$599$	10498.0			0.716085			0.358043	+	0.933705i	$0.383444\pi$
$600$	−4961.46			−0.337584
$601$		−	6548.56i		−	0.444462i	−0.974994	−	0.222231i	$-0.928666\pi$
$601$		−	6548.56i		−	0.444462i	0.974994	−	0.222231i	$-0.0713339\pi$
$602$	−18911.0			−1.28032
$603$	−56.3722			−0.00380705
$604$	−3055.61			−0.205846
$605$	−464.643			−0.0312238
$606$		−	12082.8i		−	0.809953i
$607$			7950.59i			0.531639i	0.964023	+	0.265819i	$0.0856424\pi$
$607$			7950.59i			0.531639i	−0.964023	+	0.265819i	$0.914358\pi$
$608$	−552.840			−0.0368760
$609$		−	2021.40i		−	0.134501i
$610$			641.992i			0.0426123i
$611$	−3349.25			−0.221761
$612$			1006.60i			0.0664862i
$613$			3773.35i			0.248620i	0.992243	+	0.124310i	$0.0396718\pi$
$613$			3773.35i			0.248620i	−0.992243	+	0.124310i	$0.960328\pi$
$614$			2554.24i			0.167884i
$615$			403.035i			0.0264259i
$616$	−2699.04			−0.176538
$617$	43.4014			0.00283189			0.00141594	−	0.999999i	$-0.499549\pi$
$617$	43.4014			0.00283189			0.00141594	+	0.999999i	$0.499549\pi$
$618$	−17988.6			−1.17088
$619$	−14313.8			−0.929436			−0.464718	−	0.885459i	$-0.653844\pi$
$619$	−14313.8			−0.929436			−0.464718	+	0.885459i	$0.653844\pi$
$620$		−	509.777i		−	0.0330212i
$621$			24534.9i			1.58543i
$622$	−16206.6			−1.04474
$623$		−	26393.1i		−	1.69730i
$624$		−	1126.34i		−	0.0722593i
$625$	15534.6			0.994214
$626$		−	7755.29i		−	0.495150i
$627$	1685.14			0.107333
$628$			402.694i			0.0255880i
$629$			5980.23i			0.379090i
$630$	−38.6569			−0.00244465
$631$	−4542.03			−0.286554			−0.143277	−	0.989683i	$-0.545764\pi$
$631$	−4542.03			−0.286554			−0.143277	+	0.989683i	$0.545764\pi$
$632$			5179.07i			0.325969i
$633$			18565.9i			1.16577i
$634$	11397.1			0.713939
$635$	−1090.63			−0.0681581
$636$			3333.48i			0.207832i
$637$	670.688			0.0417168
$638$	−928.092			−0.0575917
$639$		−	19.1613i		−	0.00118624i
$640$			62.8701i			0.00388306i
$641$	25371.2			1.56334			0.781672	−	0.623690i	$-0.214367\pi$
$641$	25371.2			1.56334			0.781672	+	0.623690i	$0.214367\pi$
$642$			15153.7i			0.931573i
$643$	−21658.4			−1.32834			−0.664170	−	0.747582i	$-0.731215\pi$
$643$	−21658.4			−1.32834			−0.664170	+	0.747582i	$0.731215\pi$
$644$		−	11589.8i		−	0.709167i
$645$		−	1342.72i		−	0.0819682i
$646$			3799.24i			0.231392i
$647$		−	13175.0i		−	0.800563i	−0.916392	−	0.400281i	$-0.868912\pi$
$647$		−	13175.0i		−	0.800563i	0.916392	−	0.400281i	$-0.131088\pi$
$648$		−	5295.75i		−	0.321044i
$649$		−	4993.96i		−	0.302049i
$650$			3533.49i			0.213223i
$651$		−	22177.6i		−	1.33519i
$652$			11617.2i			0.697798i
$653$	−13695.5			−0.820747			−0.410373	−	0.911918i	$-0.634602\pi$
$653$	−13695.5			−0.820747			−0.410373	+	0.911918i	$0.634602\pi$
$654$		−	17928.2i		−	1.07194i
$655$	592.741			0.0353592
$656$	−2641.07			−0.157190
$657$	−2164.39			−0.128525
$658$	8133.09			0.481856
$659$	−23621.7			−1.39631			−0.698157	−	0.715944i	$-0.745996\pi$
$659$	−23621.7			−1.39631			−0.698157	+	0.715944i	$0.745996\pi$
$660$		−	191.638i		−	0.0113022i
$661$		−	18677.2i		−	1.09903i	−0.835484	−	0.549515i	$-0.814813\pi$
$661$		−	18677.2i		−	1.09903i	0.835484	−	0.549515i	$-0.185187\pi$
$662$		−	10256.1i		−	0.602139i
$663$	−7740.49			−0.453417
$664$	2080.84			0.121615
$665$	−145.903			−0.00850810
$666$			248.950i			0.0144844i
$667$		−	3985.28i		−	0.231350i
$668$			7177.81i			0.415745i
$669$	−28226.1			−1.63122
$670$			24.1962i			0.00139520i
$671$	12823.4			0.737768
$672$			2735.14i			0.157009i
$673$			274.854i			0.0157427i	0.999969	+	0.00787135i	$0.00250555\pi$
$673$			274.854i			0.0157427i	−0.999969	+	0.00787135i	$0.997494\pi$
$674$	−4107.90			−0.234763
$675$			18164.3i			1.03577i
$676$	7985.83			0.454360
$677$			6423.71i			0.364672i	0.983236	+	0.182336i	$0.0583659\pi$
$677$			6423.71i			0.364672i	−0.983236	+	0.182336i	$0.941634\pi$
$678$		−	11424.2i		−	0.647117i
$679$			18379.9i			1.03882i
$680$	432.058			0.0243657
$681$	−28527.7			−1.60526
$682$	−10182.5			−0.571712
$683$			5398.28i			0.302430i	0.988501	+	0.151215i	$0.0483185\pi$
$683$			5398.28i			0.302430i	−0.988501	+	0.151215i	$0.951681\pi$
$684$			158.158i			0.00884110i
$685$			457.410i			0.0255135i
$686$	−13423.9			−0.747121
$687$	10008.0			0.555791
$688$	8798.78			0.487573
$689$	2374.06			0.131269
$690$	822.903			0.0454020
$691$			34348.9i			1.89102i	0.325597	+	0.945509i	$0.394435\pi$
$691$			34348.9i			1.89102i	−0.325597	+	0.945509i	$0.605565\pi$
$692$	12090.2			0.664160
$693$			772.148i			0.0423254i
$694$			7238.86i			0.395941i
$695$		−	829.232i		−	0.0452583i
$696$			940.505i			0.0512209i
$697$			18150.0i			0.986344i
$698$			2963.88i			0.160723i
$699$			91.9087i			0.00497326i
$700$		−	8580.49i		−	0.463303i
$701$		−	30763.3i		−	1.65751i	−0.559614	−	0.828753i	$-0.689051\pi$
$701$		−	30763.3i		−	1.65751i	0.559614	−	0.828753i	$-0.310949\pi$
$702$	−4123.64			−0.221705
$703$			939.614i			0.0504100i
$704$	1255.79			0.0672294
$705$			577.467i			0.0308492i
$706$		−	8038.70i		−	0.428528i
$707$	20896.4			1.11158
$708$	−5060.75			−0.268637
$709$			14299.8i			0.757462i	0.925507	+	0.378731i	$0.123639\pi$
$709$			14299.8i			0.757462i	−0.925507	+	0.378731i	$0.876361\pi$
$710$	−8.22446			−0.000434730
$711$	1481.64			0.0781518
$712$			12280.0i			0.646368i
$713$		−	43724.2i		−	2.29661i
$714$	18796.5			0.985212
$715$	−136.482			−0.00713865
$716$			8322.50i			0.434395i
$717$			15604.8i			0.812792i
$718$	−16170.4			−0.840495
$719$			28014.5i			1.45308i	0.687123	+	0.726541i	$0.258873\pi$
$719$			28014.5i			1.45308i	−0.687123	+	0.726541i	$0.741127\pi$
$720$	17.9861			0.000930973
$721$		−	31110.0i		−	1.60693i
$722$		−	13121.1i		−	0.676337i
$723$	30718.6			1.58013
$724$			10001.0i			0.513374i
$725$		−	2950.49i		−	0.151143i
$726$	9405.09			0.480793
$727$	−5345.19			−0.272685			−0.136343	−	0.990662i	$-0.543535\pi$
$727$	−5345.19			−0.272685			−0.136343	+	0.990662i	$0.543535\pi$
$728$	1947.93			0.0991691
$729$	−21056.6			−1.06979
$730$			929.005i			0.0471014i
$731$		−	60467.2i		−	3.05946i
$732$		−	12994.9i		−	0.656155i
$733$		−	28680.8i		−	1.44523i	−0.691253	−	0.722613i	$-0.742941\pi$
$733$		−	28680.8i		−	1.44523i	0.691253	−	0.722613i	$-0.257059\pi$
$734$	2771.78			0.139385
$735$		−	115.638i		−	0.00580323i
$736$			5392.45i			0.270066i
$737$	483.305			0.0241557
$738$			755.564i			0.0376866i
$739$		−	20096.1i		−	1.00033i	−0.865929	−	0.500166i	$-0.833272\pi$
$739$		−	20096.1i		−	1.00033i	0.865929	−	0.500166i	$-0.166728\pi$
$740$	106.855			0.00530820
$741$	−1216.19			−0.0602938
$742$	−5765.02			−0.285230
$743$	29965.2			1.47957			0.739783	−	0.672846i	$-0.234928\pi$
$743$	29965.2			1.47957			0.739783	+	0.672846i	$0.234928\pi$
$744$			10318.7i			0.508469i
$745$	−1014.37			−0.0498839
$746$	14541.9			0.713697
$747$		−	595.291i		−	0.0291574i
$748$		−	8630.10i		−	0.421855i
$749$	−26207.3			−1.27850
$750$	1219.64			0.0593802
$751$	40077.2			1.94732			0.973659	−	0.228007i	$-0.0732209\pi$
$751$	40077.2			1.94732			0.973659	+	0.228007i	$0.0732209\pi$
$752$	−3784.12			−0.183501
$753$	−10544.1			−0.510292
$754$	669.816			0.0323518
$755$	375.209			0.0180864
$756$	10013.6			0.481733
$757$			4605.25i			0.221111i	0.993870	+	0.110555i	$0.0352629\pi$
$757$			4605.25i			0.221111i	−0.993870	+	0.110555i	$0.964737\pi$
$758$	7525.27			0.360594
$759$		−	16437.0i		−	0.786067i
$760$	67.8850			0.00324006
$761$		−	18346.2i		−	0.873913i	−0.899483	−	0.436957i	$-0.856056\pi$
$761$		−	18346.2i		−	0.873913i	0.899483	−	0.436957i	$-0.143944\pi$
$762$	22076.1			1.04952
$763$	31005.5			1.47113
$764$	14931.6			0.707077
$765$		−	123.604i		−	0.00584173i
$766$	−22800.5			−1.07548
$767$			3604.20i			0.169674i
$768$		−	1272.59i		−	0.0597924i
$769$	−42165.8			−1.97729			−0.988646	−	0.150264i	$-0.951988\pi$
$769$	−42165.8			−1.97729			−0.988646	+	0.150264i	$0.951988\pi$
$770$	331.424			0.0155113
$771$	−34904.2			−1.63041
$772$		−	1559.49i		−	0.0727035i
$773$	17171.4			0.798979			0.399490	−	0.916738i	$-0.369187\pi$
$773$	17171.4			0.798979			0.399490	+	0.916738i	$0.369187\pi$
$774$		−	2517.18i		−	0.116897i
$775$		−	32371.0i		−	1.50039i
$776$		−	8551.71i		−	0.395604i
$777$	4648.68			0.214634
$778$			1306.74i			0.0602171i
$779$			2851.74i			0.131161i
$780$			138.307i			0.00634897i
$781$			164.279i			0.00752670i
$782$	37058.1			1.69462
$783$	3443.27			0.157155
$784$	757.771			0.0345195
$785$		−	49.4481i		−	0.00224825i
$786$	−11998.0			−0.544471
$787$	38059.5			1.72385			0.861927	−	0.507032i	$-0.169257\pi$
$787$	38059.5			1.72385			0.861927	+	0.507032i	$0.169257\pi$
$788$			16859.9i			0.762194i
$789$		−	17558.4i		−	0.792261i
$790$		−	635.955i		−	0.0286409i
$791$	19757.4			0.888108
$792$		−	359.261i		−	0.0161184i
$793$	−9254.82			−0.414437
$794$	5708.85			0.255163
$795$		−	409.329i		−	0.0182609i
$796$	−14795.3			−0.658801
$797$		−	17406.8i		−	0.773629i	−0.922158	−	0.386814i	$-0.873575\pi$
$797$		−	17406.8i		−	0.773629i	0.922158	−	0.386814i	$-0.126425\pi$
$798$	2953.31			0.131010
$799$			26005.3i			1.15144i
$800$			3992.28i			0.176436i
$801$	3513.10			0.154968
$802$	−21029.7			−0.925917
$803$	18556.3			0.815490
$804$		−	489.769i		−	0.0214836i
$805$			1423.15i			0.0623100i
$806$	7348.83			0.321156
$807$	−6302.69	−	21006.8i	−0.274926	−	0.916325i
$808$	−9722.56			−0.423315
$809$		−	13456.0i		−	0.584782i	−0.956299	−	0.292391i	$-0.905549\pi$
$809$		−	13456.0i		−	0.584782i	0.956299	−	0.292391i	$-0.0944509\pi$
$810$			650.282i			0.0282081i
$811$	−13647.5			−0.590909			−0.295455	−	0.955357i	$-0.595471\pi$
$811$	−13647.5			−0.590909			−0.295455	+	0.955357i	$0.595471\pi$
$812$	−1626.54			−0.0702959
$813$	32405.6			1.39793
$814$		−	2134.36i		−	0.0919035i
$815$		−	1426.51i		−	0.0613111i
$816$	−8745.52			−0.375189
$817$		−	9500.62i		−	0.406835i
$818$	23655.0			1.01110
$819$		−	557.269i		−	0.0237760i
$820$	324.306			0.0138113
$821$	18572.7			0.789516			0.394758	−	0.918785i	$-0.370828\pi$
$821$	18572.7			0.789516			0.394758	+	0.918785i	$0.370828\pi$
$822$		−	9258.69i		−	0.392863i
$823$	15833.9			0.670637			0.335318	−	0.942105i	$-0.391156\pi$
$823$	15833.9			0.670637			0.335318	+	0.942105i	$0.391156\pi$
$824$			14474.7i			0.611952i
$825$		−	12169.1i		−	0.513542i
$826$		−	8752.21i		−	0.368678i
$827$	−20772.3			−0.873427			−0.436713	−	0.899601i	$-0.643858\pi$
$827$	−20772.3			−0.873427			−0.436713	+	0.899601i	$0.643858\pi$
$828$	1542.69			0.0647488
$829$			27114.8i			1.13599i	0.823032	+	0.567995i	$0.192281\pi$
$829$			27114.8i			1.13599i	−0.823032	+	0.567995i	$0.807719\pi$
$830$	−255.513			−0.0106855
$831$	−26875.9			−1.12192
$832$	−906.322			−0.0377657
$833$		−	5207.58i		−	0.216605i
$834$			16784.9i			0.696900i
$835$		−	881.387i		−	0.0365289i
$836$		−	1355.96i		−	0.0560968i
$837$	37777.5			1.56007
$838$		−	20428.2i		−	0.842100i
$839$			32083.4i			1.32019i	0.751181	+	0.660096i	$0.229485\pi$
$839$			32083.4i			1.32019i	−0.751181	+	0.660096i	$0.770515\pi$
$840$		−	335.856i		−	0.0137954i
$841$	23829.7			0.977067
$842$			25856.6i			1.05829i
$843$	−4605.86			−0.188178
$844$	14939.3			0.609278
$845$	−980.606			−0.0399218
$846$			1082.57i			0.0439947i
$847$			16265.4i			0.659843i
$848$	2682.31			0.108621
$849$		−	33042.0i		−	1.33569i
$850$	27435.9			1.10711
$851$	9165.08			0.369183
$852$	166.476			0.00669409
$853$		−	233.062i		−	0.00935507i	−0.999989	−	0.00467754i	$-0.998511\pi$
$853$		−	233.062i		−	0.00935507i	0.999989	−	0.00467754i	$-0.00148891\pi$
$854$	22473.8			0.900512
$855$		−	19.4207i		−	0.000776812i
$856$	12193.6			0.486878
$857$		−	24513.7i		−	0.977097i	−0.872537	−	0.488549i	$-0.837526\pi$
$857$		−	24513.7i		−	0.977097i	0.872537	−	0.488549i	$-0.162474\pi$
$858$	2762.61			0.109923
$859$	−1295.58			−0.0514604			−0.0257302	−	0.999669i	$-0.508191\pi$
$859$	−1295.58			−0.0514604			−0.0257302	+	0.999669i	$0.508191\pi$
$860$	−1080.43			−0.0428400
$861$	14108.8			0.558450
$862$	1619.49			0.0639909
$863$	−28028.1			−1.10555			−0.552774	−	0.833331i	$-0.686431\pi$
$863$	−28028.1			−1.10555			−0.552774	+	0.833331i	$0.686431\pi$
$864$	−4659.05			−0.183454
$865$	−1484.59			−0.0583556
$866$		−	28372.8i		−	1.11333i
$867$			35678.5i			1.39758i
$868$	−17845.4			−0.697826
$869$	−12702.8			−0.495873
$870$		−	115.488i		−	0.00450046i
$871$	−348.808			−0.0135693
$872$	−14426.1			−0.560239
$873$	−2446.50			−0.0948469
$874$	5822.58			0.225345
$875$			2109.29i			0.0814937i
$876$		−	18804.5i		−	0.725280i
$877$	18077.6			0.696053			0.348026	−	0.937485i	$-0.386852\pi$
$877$	18077.6			0.696053			0.348026	+	0.937485i	$0.386852\pi$
$878$		−	13300.1i		−	0.511225i
$879$			48964.4i			1.87887i
$880$	−154.203			−0.00590702
$881$		−	7175.06i		−	0.274386i	−0.990544	−	0.137193i	$-0.956192\pi$
$881$		−	7175.06i		−	0.274386i	0.990544	−	0.137193i	$-0.0438081\pi$
$882$		−	216.785i		−	0.00827612i
$883$		−	15751.7i		−	0.600324i	−0.953888	−	0.300162i	$-0.902959\pi$
$883$		−	15751.7i		−	0.600324i	0.953888	−	0.300162i	$-0.0970407\pi$
$884$			6228.45i			0.236975i
$885$	621.426			0.0236034
$886$	25232.9			0.956791
$887$	3456.14			0.130830			0.0654148	−	0.997858i	$-0.479163\pi$
$887$	3456.14			0.130830			0.0654148	+	0.997858i	$0.479163\pi$
$888$	−2162.91			−0.0817371
$889$			38179.0i			1.44036i
$890$		−	1507.91i		−	0.0567922i
$891$	12989.0			0.488381
$892$			22712.4i			0.852542i
$893$			4085.96i			0.153115i
$894$	20532.3			0.768125
$895$		−	1021.95i		−	0.0381675i
$896$	2200.85			0.0820595
$897$			11862.8i			0.441569i
$898$		−	19404.2i		−	0.721075i
$899$	−6136.32			−0.227651
$900$	1142.12			0.0423008
$901$		−	18433.5i		−	0.681585i
$902$		−	6477.81i		−	0.239121i
$903$	−47003.7			−1.73221
$904$	−9192.62			−0.338210
$905$		−	1228.05i		−	0.0451070i
$906$	−7594.80			−0.278499
$907$	−35237.2			−1.29000			−0.645001	−	0.764181i	$-0.723143\pi$
$907$	−35237.2			−1.29000			−0.645001	+	0.764181i	$0.723143\pi$
$908$			22955.1i			0.838977i
$909$			2781.45i			0.101491i
$910$	−239.193			−0.00871337
$911$		−	26327.4i		−	0.957483i	−0.877956	−	0.478742i	$-0.841093\pi$
$911$		−	26327.4i		−	0.957483i	0.877956	−	0.478742i	$-0.158907\pi$
$912$	−1374.10			−0.0498913
$913$			5103.71i			0.185004i
$914$			19468.2i			0.704542i
$915$			1595.69i			0.0576522i
$916$		−	8053.01i		−	0.290479i
$917$		−	20749.7i		−	0.747235i
$918$			32018.1i			1.15115i
$919$		−	15385.9i		−	0.552267i	−0.961119	−	0.276133i	$-0.910947\pi$
$919$		−	15385.9i		−	0.552267i	0.961119	−	0.276133i	$-0.0890532\pi$
$920$		−	662.156i		−	0.0237290i
$921$			6348.63i			0.227138i
$922$	−814.921			−0.0291085
$923$		−	118.562i		−	0.00422808i
$924$	−6708.53			−0.238847
$925$	6785.34			0.241190
$926$	−11278.1			−0.400239
$927$	4140.95			0.146717
$928$	756.785			0.0267702
$929$		−	22325.7i		−	0.788464i	−0.919011	−	0.394232i	$-0.871011\pi$
$929$		−	22325.7i		−	0.788464i	0.919011	−	0.394232i	$-0.128989\pi$
$930$		−	1267.06i		−	0.0446759i
$931$		−	818.215i		−	0.0288033i
$932$	73.9552			0.00259923
$933$	−40281.9			−1.41347
$934$	9464.87			0.331584
$935$			1059.72i			0.0370658i
$936$			259.283i			0.00905441i
$937$			28940.3i			1.00901i	0.863410	+	0.504503i	$0.168324\pi$
$937$			28940.3i			1.00901i	−0.863410	+	0.504503i	$0.831676\pi$
$938$	847.022			0.0294843
$939$		−	19276.0i		−	0.669912i
$940$	464.664			0.0161231
$941$			15079.6i			0.522401i	0.965285	+	0.261201i	$0.0841184\pi$
$941$			15079.6i			0.522401i	−0.965285	+	0.261201i	$0.915882\pi$
$942$			1000.91i			0.0346192i
$943$	27816.1			0.960569
$944$			4072.18i			0.140401i
$945$	−1229.60			−0.0423268
$946$			21581.0i			0.741710i
$947$		−	32265.4i		−	1.10717i	−0.832794	−	0.553583i	$-0.813260\pi$
$947$		−	32265.4i		−	1.10717i	0.832794	−	0.553583i	$-0.186740\pi$
$948$			12872.7i			0.441020i
$949$	−13392.3			−0.458096
$950$	4310.72			0.147219
$951$	28327.8			0.965923
$952$		−	15124.8i		−	0.514912i
$953$		−	25941.3i		−	0.881763i	−0.897565	−	0.440882i	$-0.854666\pi$
$953$		−	25941.3i		−	0.881763i	0.897565	−	0.440882i	$-0.145334\pi$
$954$		−	767.363i		−	0.0260422i
$955$	−1833.50			−0.0621264
$956$	12556.5			0.424799
$957$	−2306.80			−0.0779186
$958$	−209.269			−0.00705758
$959$	16012.2			0.539168
$960$			156.265i			0.00525359i
$961$	−37533.1			−1.25988
$962$			1540.40i			0.0516262i
$963$		−	3488.37i		−	0.116730i
$964$		−	24718.0i		−	0.825844i
$965$			191.494i			0.00638800i
$966$		−	28806.8i		−	0.959466i
$967$			37830.9i			1.25808i	0.777375	+	0.629038i	$0.216551\pi$
$967$			37830.9i			1.25808i	−0.777375	+	0.629038i	$0.783449\pi$
$968$		−	7567.89i		−	0.251282i
$969$			9443.11i			0.313061i
$970$			1050.09i			0.0347592i
$971$	−49352.0			−1.63108			−0.815541	−	0.578699i	$-0.803561\pi$
$971$	−49352.0			−1.63108			−0.815541	+	0.578699i	$0.803561\pi$
$972$			2561.59i			0.0845300i
$973$	−29028.4			−0.956430
$974$			3288.57i			0.108186i
$975$			8782.57i			0.288479i
$976$	−10456.5			−0.342934
$977$	−13029.6			−0.426667			−0.213334	−	0.976979i	$-0.568432\pi$
$977$	−13029.6			−0.426667			−0.213334	+	0.976979i	$0.568432\pi$
$978$			28874.8i			0.944084i
$979$	−30119.5			−0.983272
$980$	−93.0492			−0.00303301
$981$			4127.04i			0.134318i
$982$			32474.8i			1.05531i
$983$	1542.48			0.0500483			0.0250242	−	0.999687i	$-0.492034\pi$
$983$	1542.48			0.0500483			0.0250242	+	0.999687i	$0.492034\pi$
$984$	−6564.45			−0.212670
$985$		−	2070.28i		−	0.0669692i
$986$		−	5200.80i		−	0.167979i
$987$	20215.0			0.651926
$988$			978.615i			0.0315120i
$989$	−92669.9			−2.97951
$990$			44.1148i			0.00141622i
$991$			19079.4i			0.611581i	0.952099	+	0.305790i	$0.0989207\pi$
$991$			19079.4i			0.611581i	−0.952099	+	0.305790i	$0.901079\pi$
$992$	8303.01			0.265747
$993$		−	25491.9i		−	0.814663i
$994$			287.908i			0.00918702i
$995$	1816.76			0.0578847
$996$	5171.97			0.164538
$997$	−19438.7			−0.617481			−0.308740	−	0.951146i	$-0.599907\pi$
$997$	−19438.7			−0.617481			−0.308740	+	0.951146i	$0.599907\pi$
$998$	−25694.4			−0.814971
$999$			7918.59i			0.250784i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	538.4.b.a.537.10	✓	68
269.268	even	2	inner	538.4.b.a.537.59	yes	68

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
538.4.b.a.537.10	✓	68	1.1	even	1	trivial
538.4.b.a.537.59	yes	68	269.268	even	2	inner

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

\(f(q)\)	\(=\)	\(q-2.00000i q^{2} -4.97105i q^{3} -4.00000 q^{4} +0.491173 q^{5} -9.94210 q^{6} -17.1942i q^{7} +8.00000i q^{8} +2.28866 q^{9} -0.982346i q^{10} -19.6218 q^{11} +19.8842i q^{12} +14.1613 q^{13} -34.3883 q^{14} -2.44165i q^{15} +16.0000 q^{16} -109.956i q^{17} -4.57732i q^{18} -17.2762i q^{19} -1.96469 q^{20} -85.4731 q^{21} +39.2435i q^{22} -168.514 q^{23} +39.7684 q^{24} -124.759 q^{25} -28.3226i q^{26} -145.595i q^{27} +68.7767i q^{28} +23.6495i q^{29} -4.88329 q^{30} +259.469i q^{31} -32.0000i q^{32} +97.5408i q^{33} -219.911 q^{34} -8.44531i q^{35} -9.15464 q^{36} -54.3877 q^{37} -34.5525 q^{38} -70.3964i q^{39} +3.92938i q^{40} -165.067 q^{41} +170.946i q^{42} +549.924 q^{43} +78.4871 q^{44} +1.12413 q^{45} +337.028i q^{46} -236.507 q^{47} -79.5368i q^{48} +47.3607 q^{49} +249.517i q^{50} -546.595 q^{51} -56.6451 q^{52} +167.645 q^{53} -291.191 q^{54} -9.63768 q^{55} +137.553 q^{56} -85.8811 q^{57} +47.2991 q^{58} +254.511i q^{59} +9.76658i q^{60} -653.530 q^{61} +518.938 q^{62} -39.3516i q^{63} -64.0000 q^{64} +6.95564 q^{65} +195.082 q^{66} -24.6311 q^{67} +439.823i q^{68} +837.692i q^{69} -16.8906 q^{70} -8.37227i q^{71} +18.3093i q^{72} -945.701 q^{73} +108.775i q^{74} +620.182i q^{75} +69.1050i q^{76} +337.380i q^{77} -140.793 q^{78} +647.384 q^{79} +7.85877 q^{80} -661.968 q^{81} +330.134i q^{82} -260.105i q^{83} +341.892 q^{84} -54.0073i q^{85} -1099.85i q^{86} +117.563 q^{87} -156.974i q^{88} +1535.00 q^{89} -2.24826i q^{90} -243.491i q^{91} +674.056 q^{92} +1289.83 q^{93} +473.015i q^{94} -8.48562i q^{95} -159.074 q^{96} -1068.96 q^{97} -94.7214i q^{98} -44.9076 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 68 q - 272 q^{4} + 38 q^{5} - 4 q^{6} - 594 q^{9} + 18 q^{11} - 114 q^{13} + 8 q^{14} + 1088 q^{16} - 152 q^{20} - 20 q^{21} - 224 q^{23} + 16 q^{24} + 1098 q^{25} + 384 q^{30} - 600 q^{34} + 2376 q^{36}+ \cdots + 3370 q^{99}+O(q^{100}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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