LMFDB - Embedded newform 4006.2.a.f.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4006,2,Mod(1,4006)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4006, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4006.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4006 = 2 \cdot 2003 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4006.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$31.9880710497$
Analytic rank:	$1$
Dimension:	$31$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Character	$\chi$	$=$	4006.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +0.105946 q^{3} +1.00000 q^{4} +1.78036 q^{5} +0.105946 q^{6} +0.955066 q^{7} +1.00000 q^{8} -2.98878 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +0.105946 q^{3} +1.00000 q^{4} +1.78036 q^{5} +0.105946 q^{6} +0.955066 q^{7} +1.00000 q^{8} -2.98878 q^{9} +1.78036 q^{10} -2.10529 q^{11} +0.105946 q^{12} -3.36034 q^{13} +0.955066 q^{14} +0.188621 q^{15} +1.00000 q^{16} -2.54320 q^{17} -2.98878 q^{18} -5.77098 q^{19} +1.78036 q^{20} +0.101185 q^{21} -2.10529 q^{22} +0.486303 q^{23} +0.105946 q^{24} -1.83033 q^{25} -3.36034 q^{26} -0.634484 q^{27} +0.955066 q^{28} +1.06211 q^{29} +0.188621 q^{30} -5.55691 q^{31} +1.00000 q^{32} -0.223046 q^{33} -2.54320 q^{34} +1.70036 q^{35} -2.98878 q^{36} -4.18704 q^{37} -5.77098 q^{38} -0.356013 q^{39} +1.78036 q^{40} -3.88468 q^{41} +0.101185 q^{42} +5.14276 q^{43} -2.10529 q^{44} -5.32109 q^{45} +0.486303 q^{46} -6.72074 q^{47} +0.105946 q^{48} -6.08785 q^{49} -1.83033 q^{50} -0.269441 q^{51} -3.36034 q^{52} +9.60376 q^{53} -0.634484 q^{54} -3.74816 q^{55} +0.955066 q^{56} -0.611410 q^{57} +1.06211 q^{58} +2.80607 q^{59} +0.188621 q^{60} -5.55859 q^{61} -5.55691 q^{62} -2.85448 q^{63} +1.00000 q^{64} -5.98260 q^{65} -0.223046 q^{66} -2.20396 q^{67} -2.54320 q^{68} +0.0515216 q^{69} +1.70036 q^{70} +8.52744 q^{71} -2.98878 q^{72} -14.9106 q^{73} -4.18704 q^{74} -0.193915 q^{75} -5.77098 q^{76} -2.01069 q^{77} -0.356013 q^{78} +3.49342 q^{79} +1.78036 q^{80} +8.89911 q^{81} -3.88468 q^{82} -5.72119 q^{83} +0.101185 q^{84} -4.52780 q^{85} +5.14276 q^{86} +0.112525 q^{87} -2.10529 q^{88} +14.5721 q^{89} -5.32109 q^{90} -3.20934 q^{91} +0.486303 q^{92} -0.588730 q^{93} -6.72074 q^{94} -10.2744 q^{95} +0.105946 q^{96} +14.5307 q^{97} -6.08785 q^{98} +6.29223 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 31 q + 31 q^{2} - 13 q^{3} + 31 q^{4} - 23 q^{5} - 13 q^{6} - 18 q^{7} + 31 q^{8} + 20 q^{9}+O(q^{10}) $ 31 * q + 31 * q^2 - 13 * q^3 + 31 * q^4 - 23 * q^5 - 13 * q^6 - 18 * q^7 + 31 * q^8 + 20 * q^9	\( 31 q + 31 q^{2} - 13 q^{3} + 31 q^{4} - 23 q^{5} - 13 q^{6} - 18 q^{7} + 31 q^{8} + 20 q^{9} - 23 q^{10} - 32 q^{11} - 13 q^{12} - 8 q^{13} - 18 q^{14} - 14 q^{15} + 31 q^{16} - 30 q^{17} + 20 q^{18} - 38 q^{19} - 23 q^{20} - 16 q^{21} - 32 q^{22} - 24 q^{23} - 13 q^{24} + 40 q^{25} - 8 q^{26} - 28 q^{27} - 18 q^{28} - 7 q^{29} - 14 q^{30} - 32 q^{31} + 31 q^{32} - 9 q^{33} - 30 q^{34} - 14 q^{35} + 20 q^{36} + 11 q^{37} - 38 q^{38} - 9 q^{39} - 23 q^{40} - 76 q^{41} - 16 q^{42} - 33 q^{43} - 32 q^{44} - 40 q^{45} - 24 q^{46} - 96 q^{47} - 13 q^{48} + 15 q^{49} + 40 q^{50} - 55 q^{51} - 8 q^{52} - 28 q^{53} - 28 q^{54} - 52 q^{55} - 18 q^{56} - 21 q^{57} - 7 q^{58} - 72 q^{59} - 14 q^{60} - 9 q^{61} - 32 q^{62} - 54 q^{63} + 31 q^{64} - 38 q^{65} - 9 q^{66} - 4 q^{67} - 30 q^{68} - 17 q^{69} - 14 q^{70} - 61 q^{71} + 20 q^{72} - 62 q^{73} + 11 q^{74} - 63 q^{75} - 38 q^{76} - 9 q^{77} - 9 q^{78} - 30 q^{79} - 23 q^{80} - 13 q^{81} - 76 q^{82} - 90 q^{83} - 16 q^{84} + 26 q^{85} - 33 q^{86} - 34 q^{87} - 32 q^{88} - 99 q^{89} - 40 q^{90} - 47 q^{91} - 24 q^{92} - 6 q^{93} - 96 q^{94} - 24 q^{95} - 13 q^{96} - 46 q^{97} + 15 q^{98} - 48 q^{99}+O(q^{100}) \) 31 * q + 31 * q^2 - 13 * q^3 + 31 * q^4 - 23 * q^5 - 13 * q^6 - 18 * q^7 + 31 * q^8 + 20 * q^9 - 23 * q^10 - 32 * q^11 - 13 * q^12 - 8 * q^13 - 18 * q^14 - 14 * q^15 + 31 * q^16 - 30 * q^17 + 20 * q^18 - 38 * q^19 - 23 * q^20 - 16 * q^21 - 32 * q^22 - 24 * q^23 - 13 * q^24 + 40 * q^25 - 8 * q^26 - 28 * q^27 - 18 * q^28 - 7 * q^29 - 14 * q^30 - 32 * q^31 + 31 * q^32 - 9 * q^33 - 30 * q^34 - 14 * q^35 + 20 * q^36 + 11 * q^37 - 38 * q^38 - 9 * q^39 - 23 * q^40 - 76 * q^41 - 16 * q^42 - 33 * q^43 - 32 * q^44 - 40 * q^45 - 24 * q^46 - 96 * q^47 - 13 * q^48 + 15 * q^49 + 40 * q^50 - 55 * q^51 - 8 * q^52 - 28 * q^53 - 28 * q^54 - 52 * q^55 - 18 * q^56 - 21 * q^57 - 7 * q^58 - 72 * q^59 - 14 * q^60 - 9 * q^61 - 32 * q^62 - 54 * q^63 + 31 * q^64 - 38 * q^65 - 9 * q^66 - 4 * q^67 - 30 * q^68 - 17 * q^69 - 14 * q^70 - 61 * q^71 + 20 * q^72 - 62 * q^73 + 11 * q^74 - 63 * q^75 - 38 * q^76 - 9 * q^77 - 9 * q^78 - 30 * q^79 - 23 * q^80 - 13 * q^81 - 76 * q^82 - 90 * q^83 - 16 * q^84 + 26 * q^85 - 33 * q^86 - 34 * q^87 - 32 * q^88 - 99 * q^89 - 40 * q^90 - 47 * q^91 - 24 * q^92 - 6 * q^93 - 96 * q^94 - 24 * q^95 - 13 * q^96 - 46 * q^97 + 15 * q^98 - 48 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0.105946	0.0611677	0.0305838	−	0.999532i	$-0.490263\pi$
$3$	0.105946	0.0611677	0.0305838	+	0.999532i	$0.490263\pi$
$4$	1.00000	0.500000
$5$	1.78036	0.796200	0.398100	−	0.917342i	$-0.369670\pi$
$5$	1.78036	0.796200	0.398100	+	0.917342i	$0.369670\pi$
$6$	0.105946	0.0432521
$7$	0.955066	0.360981	0.180491	−	0.983577i	$-0.442231\pi$
$7$	0.955066	0.360981	0.180491	+	0.983577i	$0.442231\pi$
$8$	1.00000	0.353553
$9$	−2.98878	−0.996259
$10$	1.78036	0.562998
$11$	−2.10529	−0.634768	−0.317384	−	0.948297i	$-0.602804\pi$
$11$	−2.10529	−0.634768	−0.317384	+	0.948297i	$0.602804\pi$
$12$	0.105946	0.0305838
$13$	−3.36034	−0.931990	−0.465995	−	0.884787i	$-0.654303\pi$
$13$	−3.36034	−0.931990	−0.465995	+	0.884787i	$0.654303\pi$
$14$	0.955066	0.255252
$15$	0.188621	0.0487017
$16$	1.00000	0.250000
$17$	−2.54320	−0.616817	−0.308408	−	0.951254i	$-0.599796\pi$
$17$	−2.54320	−0.616817	−0.308408	+	0.951254i	$0.599796\pi$
$18$	−2.98878	−0.704461
$19$	−5.77098	−1.32395	−0.661977	−	0.749524i	$-0.730282\pi$
$19$	−5.77098	−1.32395	−0.661977	+	0.749524i	$0.730282\pi$
$20$	1.78036	0.398100
$21$	0.101185	0.0220804
$22$	−2.10529	−0.448849
$23$	0.486303	0.101401	0.0507006	−	0.998714i	$-0.483855\pi$
$23$	0.486303	0.101401	0.0507006	+	0.998714i	$0.483855\pi$
$24$	0.105946	0.0216260
$25$	−1.83033	−0.366066
$26$	−3.36034	−0.659016
$27$	−0.634484	−0.122106
$28$	0.955066	0.180491
$29$	1.06211	0.197228	0.0986141	−	0.995126i	$-0.468559\pi$
$29$	1.06211	0.197228	0.0986141	+	0.995126i	$0.468559\pi$
$30$	0.188621	0.0344373
$31$	−5.55691	−0.998051	−0.499025	−	0.866587i	$-0.666309\pi$
$31$	−5.55691	−0.998051	−0.499025	+	0.866587i	$0.666309\pi$
$32$	1.00000	0.176777
$33$	−0.223046	−0.0388273
$34$	−2.54320	−0.436155
$35$	1.70036	0.287413
$36$	−2.98878	−0.498129
$37$	−4.18704	−0.688345	−0.344173	−	0.938906i	$-0.611841\pi$
$37$	−4.18704	−0.688345	−0.344173	+	0.938906i	$0.611841\pi$
$38$	−5.77098	−0.936177
$39$	−0.356013	−0.0570076
$40$	1.78036	0.281499
$41$	−3.88468	−0.606685	−0.303342	−	0.952882i	$-0.598103\pi$
$41$	−3.88468	−0.606685	−0.303342	+	0.952882i	$0.598103\pi$
$42$	0.101185	0.0156132
$43$	5.14276	0.784263	0.392132	−	0.919909i	$-0.371738\pi$
$43$	5.14276	0.784263	0.392132	+	0.919909i	$0.371738\pi$
$44$	−2.10529	−0.317384
$45$	−5.32109	−0.793221
$46$	0.486303	0.0717014
$47$	−6.72074	−0.980321	−0.490160	−	0.871632i	$-0.663062\pi$
$47$	−6.72074	−0.980321	−0.490160	+	0.871632i	$0.663062\pi$
$48$	0.105946	0.0152919
$49$	−6.08785	−0.869693
$50$	−1.83033	−0.258848
$51$	−0.269441	−0.0377292
$52$	−3.36034	−0.465995
$53$	9.60376	1.31918	0.659589	−	0.751627i	$-0.270730\pi$
$53$	9.60376	1.31918	0.659589	+	0.751627i	$0.270730\pi$
$54$	−0.634484	−0.0863423
$55$	−3.74816	−0.505402
$56$	0.955066	0.127626
$57$	−0.611410	−0.0809832
$58$	1.06211	0.139461
$59$	2.80607	0.365319	0.182659	−	0.983176i	$-0.441529\pi$
$59$	2.80607	0.365319	0.182659	+	0.983176i	$0.441529\pi$
$60$	0.188621	0.0243508
$61$	−5.55859	−0.711704	−0.355852	−	0.934542i	$-0.615809\pi$
$61$	−5.55859	−0.711704	−0.355852	+	0.934542i	$0.615809\pi$
$62$	−5.55691	−0.705728
$63$	−2.85448	−0.359630
$64$	1.00000	0.125000
$65$	−5.98260	−0.742050
$66$	−0.223046	−0.0274550
$67$	−2.20396	−0.269256	−0.134628	−	0.990896i	$-0.542984\pi$
$67$	−2.20396	−0.269256	−0.134628	+	0.990896i	$0.542984\pi$
$68$	−2.54320	−0.308408
$69$	0.0515216	0.00620247
$70$	1.70036	0.203232
$71$	8.52744	1.01202	0.506011	−	0.862527i	$-0.331120\pi$
$71$	8.52744	1.01202	0.506011	+	0.862527i	$0.331120\pi$
$72$	−2.98878	−0.352231
$73$	−14.9106	−1.74515	−0.872576	−	0.488479i	$-0.837552\pi$
$73$	−14.9106	−1.74515	−0.872576	+	0.488479i	$0.837552\pi$
$74$	−4.18704	−0.486734
$75$	−0.193915	−0.0223914
$76$	−5.77098	−0.661977
$77$	−2.01069	−0.229139
$78$	−0.356013	−0.0403105
$79$	3.49342	0.393041	0.196520	−	0.980500i	$-0.437036\pi$
$79$	3.49342	0.393041	0.196520	+	0.980500i	$0.437036\pi$
$80$	1.78036	0.199050
$81$	8.89911	0.988790
$82$	−3.88468	−0.428991
$83$	−5.72119	−0.627982	−0.313991	−	0.949426i	$-0.601666\pi$
$83$	−5.72119	−0.627982	−0.313991	+	0.949426i	$0.601666\pi$
$84$	0.101185	0.0110402
$85$	−4.52780	−0.491109
$86$	5.14276	0.554558
$87$	0.112525	0.0120640
$88$	−2.10529	−0.224425
$89$	14.5721	1.54463	0.772317	−	0.635237i	$-0.219098\pi$
$89$	14.5721	1.54463	0.772317	+	0.635237i	$0.219098\pi$
$90$	−5.32109	−0.560892
$91$	−3.20934	−0.336431
$92$	0.486303	0.0507006
$93$	−0.588730	−0.0610484
$94$	−6.72074	−0.693192
$95$	−10.2744	−1.05413
$96$	0.105946	0.0108130
$97$	14.5307	1.47536	0.737682	−	0.675148i	$-0.235920\pi$
$97$	14.5307	1.47536	0.737682	+	0.675148i	$0.235920\pi$
$98$	−6.08785	−0.614966
$99$	6.29223	0.632393
$100$	−1.83033	−0.183033
$101$	−0.684403	−0.0681007	−0.0340503	−	0.999420i	$-0.510841\pi$
$101$	−0.684403	−0.0681007	−0.0340503	+	0.999420i	$0.510841\pi$
$102$	−0.269441	−0.0266786
$103$	1.84473	0.181766	0.0908832	−	0.995862i	$-0.471031\pi$
$103$	1.84473	0.181766	0.0908832	+	0.995862i	$0.471031\pi$
$104$	−3.36034	−0.329508
$105$	0.180145	0.0175804
$106$	9.60376	0.932800
$107$	−3.38827	−0.327557	−0.163778	−	0.986497i	$-0.552368\pi$
$107$	−3.38827	−0.327557	−0.163778	+	0.986497i	$0.552368\pi$
$108$	−0.634484	−0.0610532
$109$	−4.74817	−0.454792	−0.227396	−	0.973802i	$-0.573021\pi$
$109$	−4.74817	−0.454792	−0.227396	+	0.973802i	$0.573021\pi$
$110$	−3.74816	−0.357373
$111$	−0.443598	−0.0421045
$112$	0.955066	0.0902453
$113$	4.78070	0.449731	0.224865	−	0.974390i	$-0.427806\pi$
$113$	4.78070	0.449731	0.224865	+	0.974390i	$0.427806\pi$
$114$	−0.611410	−0.0572638
$115$	0.865792	0.0807356
$116$	1.06211	0.0986141
$117$	10.0433	0.928503
$118$	2.80607	0.258320
$119$	−2.42892	−0.222659
$120$	0.188621	0.0172186
$121$	−6.56776	−0.597069
$122$	−5.55859	−0.503251
$123$	−0.411564	−0.0371095
$124$	−5.55691	−0.499025
$125$	−12.1604	−1.08766
$126$	−2.85448	−0.254297
$127$	6.22670	0.552530	0.276265	−	0.961082i	$-0.410903\pi$
$127$	6.22670	0.552530	0.276265	+	0.961082i	$0.410903\pi$
$128$	1.00000	0.0883883
$129$	0.544852	0.0479716
$130$	−5.98260	−0.524708
$131$	−18.8671	−1.64842	−0.824212	−	0.566281i	$-0.808382\pi$
$131$	−18.8671	−1.64842	−0.824212	+	0.566281i	$0.808382\pi$
$132$	−0.223046	−0.0194137
$133$	−5.51167	−0.477923
$134$	−2.20396	−0.190393
$135$	−1.12961	−0.0972211
$136$	−2.54320	−0.218078
$137$	18.8590	1.61123	0.805617	−	0.592436i	$-0.201834\pi$
$137$	18.8590	1.61123	0.805617	+	0.592436i	$0.201834\pi$
$138$	0.0515216	0.00438581
$139$	10.2223	0.867044	0.433522	−	0.901143i	$-0.357271\pi$
$139$	10.2223	0.867044	0.433522	+	0.901143i	$0.357271\pi$
$140$	1.70036	0.143706
$141$	−0.712032	−0.0599639
$142$	8.52744	0.715607
$143$	7.07448	0.591598
$144$	−2.98878	−0.249065
$145$	1.89093	0.157033
$146$	−14.9106	−1.23401
$147$	−0.644980	−0.0531971
$148$	−4.18704	−0.344173
$149$	18.0980	1.48265	0.741324	−	0.671147i	$-0.234198\pi$
$149$	18.0980	1.48265	0.741324	+	0.671147i	$0.234198\pi$
$150$	−0.193915	−0.0158331
$151$	23.5963	1.92024	0.960121	−	0.279584i	$-0.0901967\pi$
$151$	23.5963	1.92024	0.960121	+	0.279584i	$0.0901967\pi$
$152$	−5.77098	−0.468089
$153$	7.60105	0.614509
$154$	−2.01069	−0.162026
$155$	−9.89328	−0.794647
$156$	−0.356013	−0.0285038
$157$	2.72556	0.217524	0.108762	−	0.994068i	$-0.465311\pi$
$157$	2.72556	0.217524	0.108762	+	0.994068i	$0.465311\pi$
$158$	3.49342	0.277922
$159$	1.01748	0.0806910
$160$	1.78036	0.140750
$161$	0.464451	0.0366039
$162$	8.89911	0.699180
$163$	−8.67853	−0.679755	−0.339877	−	0.940470i	$-0.610386\pi$
$163$	−8.67853	−0.679755	−0.339877	+	0.940470i	$0.610386\pi$
$164$	−3.88468	−0.303342
$165$	−0.397101	−0.0309143
$166$	−5.72119	−0.444050
$167$	−5.92499	−0.458489	−0.229245	−	0.973369i	$-0.573626\pi$
$167$	−5.92499	−0.458489	−0.229245	+	0.973369i	$0.573626\pi$
$168$	0.101185	0.00780659
$169$	−1.70814	−0.131395
$170$	−4.52780	−0.347267
$171$	17.2482	1.31900
$172$	5.14276	0.392132
$173$	2.60204	0.197829	0.0989147	−	0.995096i	$-0.468463\pi$
$173$	2.60204	0.197829	0.0989147	+	0.995096i	$0.468463\pi$
$174$	0.112525	0.00853053
$175$	−1.74809	−0.132143
$176$	−2.10529	−0.158692
$177$	0.297290	0.0223457
$178$	14.5721	1.09222
$179$	9.60559	0.717956	0.358978	−	0.933346i	$-0.383125\pi$
$179$	9.60559	0.717956	0.358978	+	0.933346i	$0.383125\pi$
$180$	−5.32109	−0.396610
$181$	−1.62009	−0.120420	−0.0602102	−	0.998186i	$-0.519177\pi$
$181$	−1.62009	−0.120420	−0.0602102	+	0.998186i	$0.519177\pi$
$182$	−3.20934	−0.237892
$183$	−0.588907	−0.0435333
$184$	0.486303	0.0358507
$185$	−7.45443	−0.548060
$186$	−0.588730	−0.0431678
$187$	5.35417	0.391536
$188$	−6.72074	−0.490160
$189$	−0.605974	−0.0440781
$190$	−10.2744	−0.745384
$191$	−19.9005	−1.43995	−0.719974	−	0.694001i	$-0.755846\pi$
$191$	−19.9005	−1.43995	−0.719974	+	0.694001i	$0.755846\pi$
$192$	0.105946	0.00764596
$193$	−8.25830	−0.594445	−0.297223	−	0.954808i	$-0.596060\pi$
$193$	−8.25830	−0.594445	−0.297223	+	0.954808i	$0.596060\pi$
$194$	14.5307	1.04324
$195$	−0.633829	−0.0453894
$196$	−6.08785	−0.434846
$197$	−8.75315	−0.623636	−0.311818	−	0.950142i	$-0.600938\pi$
$197$	−8.75315	−0.623636	−0.311818	+	0.950142i	$0.600938\pi$
$198$	6.29223	0.447170
$199$	−21.9469	−1.55577	−0.777886	−	0.628406i	$-0.783708\pi$
$199$	−21.9469	−1.55577	−0.777886	+	0.628406i	$0.783708\pi$
$200$	−1.83033	−0.129424
$201$	−0.233499	−0.0164698
$202$	−0.684403	−0.0481544
$203$	1.01438	0.0711956
$204$	−0.269441	−0.0188646
$205$	−6.91611	−0.483042
$206$	1.84473	0.128528
$207$	−1.45345	−0.101022
$208$	−3.36034	−0.232997
$209$	12.1496	0.840404
$210$	0.180145	0.0124312
$211$	7.73297	0.532359	0.266180	−	0.963923i	$-0.414239\pi$
$211$	7.73297	0.532359	0.266180	+	0.963923i	$0.414239\pi$
$212$	9.60376	0.659589
$213$	0.903444	0.0619030
$214$	−3.38827	−0.231618
$215$	9.15594	0.624430
$216$	−0.634484	−0.0431712
$217$	−5.30722	−0.360277
$218$	−4.74817	−0.321587
$219$	−1.57971	−0.106747
$220$	−3.74816	−0.252701
$221$	8.54601	0.574867
$222$	−0.443598	−0.0297724
$223$	−12.9455	−0.866897	−0.433449	−	0.901178i	$-0.642703\pi$
$223$	−12.9455	−0.866897	−0.433449	+	0.901178i	$0.642703\pi$
$224$	0.955066	0.0638130
$225$	5.47045	0.364697
$226$	4.78070	0.318008
$227$	−2.49395	−0.165529	−0.0827647	−	0.996569i	$-0.526375\pi$
$227$	−2.49395	−0.165529	−0.0827647	+	0.996569i	$0.526375\pi$
$228$	−0.611410	−0.0404916
$229$	−1.75782	−0.116160	−0.0580799	−	0.998312i	$-0.518498\pi$
$229$	−1.75782	−0.116160	−0.0580799	+	0.998312i	$0.518498\pi$
$230$	0.865792	0.0570887
$231$	−0.213024	−0.0140159
$232$	1.06211	0.0697307
$233$	5.40640	0.354185	0.177092	−	0.984194i	$-0.443331\pi$
$233$	5.40640	0.354185	0.177092	+	0.984194i	$0.443331\pi$
$234$	10.0433	0.656550
$235$	−11.9653	−0.780531
$236$	2.80607	0.182659
$237$	0.370113	0.0240414
$238$	−2.42892	−0.157444
$239$	1.22912	0.0795051	0.0397525	−	0.999210i	$-0.487343\pi$
$239$	1.22912	0.0795051	0.0397525	+	0.999210i	$0.487343\pi$
$240$	0.188621	0.0121754
$241$	12.4259	0.800424	0.400212	−	0.916423i	$-0.368937\pi$
$241$	12.4259	0.800424	0.400212	+	0.916423i	$0.368937\pi$
$242$	−6.56776	−0.422192
$243$	2.84627	0.182588
$244$	−5.55859	−0.355852
$245$	−10.8385	−0.692449
$246$	−0.411564	−0.0262404
$247$	19.3924	1.23391
$248$	−5.55691	−0.352864
$249$	−0.606134	−0.0384122
$250$	−12.1604	−0.769093
$251$	0.447538	0.0282483	0.0141242	−	0.999900i	$-0.495504\pi$
$251$	0.447538	0.0282483	0.0141242	+	0.999900i	$0.495504\pi$
$252$	−2.85448	−0.179815
$253$	−1.02381	−0.0643663
$254$	6.22670	0.390698
$255$	−0.479700	−0.0300400
$256$	1.00000	0.0625000
$257$	−7.98817	−0.498289	−0.249144	−	0.968466i	$-0.580149\pi$
$257$	−7.98817	−0.498289	−0.249144	+	0.968466i	$0.580149\pi$
$258$	0.544852	0.0339210
$259$	−3.99890	−0.248480
$260$	−5.98260	−0.371025
$261$	−3.17440	−0.196490
$262$	−18.8671	−1.16561
$263$	−1.54704	−0.0953946	−0.0476973	−	0.998862i	$-0.515188\pi$
$263$	−1.54704	−0.0953946	−0.0476973	+	0.998862i	$0.515188\pi$
$264$	−0.223046	−0.0137275
$265$	17.0981	1.05033
$266$	−5.51167	−0.337942
$267$	1.54384	0.0944817
$268$	−2.20396	−0.134628
$269$	−22.4773	−1.37046	−0.685232	−	0.728325i	$-0.740299\pi$
$269$	−22.4773	−1.37046	−0.685232	+	0.728325i	$0.740299\pi$
$270$	−1.12961	−0.0687457
$271$	4.35115	0.264314	0.132157	−	0.991229i	$-0.457810\pi$
$271$	4.35115	0.264314	0.132157	+	0.991229i	$0.457810\pi$
$272$	−2.54320	−0.154204
$273$	−0.340016	−0.0205787
$274$	18.8590	1.13932
$275$	3.85338	0.232367
$276$	0.0515216	0.00310124
$277$	−6.00059	−0.360541	−0.180270	−	0.983617i	$-0.557697\pi$
$277$	−6.00059	−0.360541	−0.180270	+	0.983617i	$0.557697\pi$
$278$	10.2223	0.613093
$279$	16.6084	0.994316
$280$	1.70036	0.101616
$281$	18.7791	1.12026	0.560132	−	0.828403i	$-0.310750\pi$
$281$	18.7791	1.12026	0.560132	+	0.828403i	$0.310750\pi$
$282$	−0.712032	−0.0424009
$283$	0.141302	0.00839954	0.00419977	−	0.999991i	$-0.498663\pi$
$283$	0.141302	0.00839954	0.00419977	+	0.999991i	$0.498663\pi$
$284$	8.52744	0.506011
$285$	−1.08853	−0.0644788
$286$	7.07448	0.418323
$287$	−3.71012	−0.219002
$288$	−2.98878	−0.176115
$289$	−10.5321	−0.619537
$290$	1.89093	0.111039
$291$	1.53946	0.0902446
$292$	−14.9106	−0.872576
$293$	5.08847	0.297271	0.148636	−	0.988892i	$-0.452512\pi$
$293$	5.08847	0.297271	0.148636	+	0.988892i	$0.452512\pi$
$294$	−0.644980	−0.0376160
$295$	4.99580	0.290867
$296$	−4.18704	−0.243367
$297$	1.33577	0.0775093
$298$	18.0980	1.04839
$299$	−1.63414	−0.0945048
$300$	−0.193915	−0.0111957
$301$	4.91167	0.283104
$302$	23.5963	1.35782
$303$	−0.0725094	−0.00416556
$304$	−5.77098	−0.330989
$305$	−9.89626	−0.566658
$306$	7.60105	0.434523
$307$	−14.9248	−0.851802	−0.425901	−	0.904770i	$-0.640043\pi$
$307$	−14.9248	−0.851802	−0.425901	+	0.904770i	$0.640043\pi$
$308$	−2.01069	−0.114570
$309$	0.195441	0.0111182
$310$	−9.89328	−0.561901
$311$	2.18581	0.123946	0.0619729	−	0.998078i	$-0.480261\pi$
$311$	2.18581	0.123946	0.0619729	+	0.998078i	$0.480261\pi$
$312$	−0.356013	−0.0201552
$313$	−31.5982	−1.78604	−0.893019	−	0.450019i	$-0.851417\pi$
$313$	−31.5982	−1.78604	−0.893019	+	0.450019i	$0.851417\pi$
$314$	2.72556	0.153812
$315$	−5.08199	−0.286338
$316$	3.49342	0.196520
$317$	−24.2145	−1.36002	−0.680010	−	0.733203i	$-0.738025\pi$
$317$	−24.2145	−1.36002	−0.680010	+	0.733203i	$0.738025\pi$
$318$	1.01748	0.0570572
$319$	−2.23604	−0.125194
$320$	1.78036	0.0995249
$321$	−0.358972	−0.0200359
$322$	0.464451	0.0258829
$323$	14.6768	0.816637
$324$	8.89911	0.494395
$325$	6.15053	0.341170
$326$	−8.67853	−0.480659
$327$	−0.503048	−0.0278186
$328$	−3.88468	−0.214495
$329$	−6.41875	−0.353877
$330$	−0.397101	−0.0218597
$331$	7.70093	0.423282	0.211641	−	0.977348i	$-0.432119\pi$
$331$	7.70093	0.423282	0.211641	+	0.977348i	$0.432119\pi$
$332$	−5.72119	−0.313991
$333$	12.5141	0.685770
$334$	−5.92499	−0.324201
$335$	−3.92383	−0.214382
$336$	0.101185	0.00552009
$337$	32.1482	1.75122	0.875611	−	0.483017i	$-0.160459\pi$
$337$	32.1482	1.75122	0.875611	+	0.483017i	$0.160459\pi$
$338$	−1.70814	−0.0929106
$339$	0.506494	0.0275090
$340$	−4.52780	−0.245555
$341$	11.6989	0.633531
$342$	17.2482	0.932674
$343$	−12.4998	−0.674924
$344$	5.14276	0.277279
$345$	0.0917268	0.00493841
$346$	2.60204	0.139886
$347$	−21.0435	−1.12967	−0.564836	−	0.825203i	$-0.691061\pi$
$347$	−21.0435	−1.12967	−0.564836	+	0.825203i	$0.691061\pi$
$348$	0.112525	0.00603199
$349$	7.27442	0.389391	0.194695	−	0.980864i	$-0.437628\pi$
$349$	7.27442	0.389391	0.194695	+	0.980864i	$0.437628\pi$
$350$	−1.74809	−0.0934392
$351$	2.13208	0.113802
$352$	−2.10529	−0.112212
$353$	−27.6561	−1.47199	−0.735993	−	0.676989i	$-0.763285\pi$
$353$	−27.6561	−1.47199	−0.735993	+	0.676989i	$0.763285\pi$
$354$	0.297290	0.0158008
$355$	15.1819	0.805771
$356$	14.5721	0.772317
$357$	−0.257334	−0.0136195
$358$	9.60559	0.507671
$359$	−19.9387	−1.05233	−0.526163	−	0.850384i	$-0.676370\pi$
$359$	−19.9387	−1.05233	−0.526163	+	0.850384i	$0.676370\pi$
$360$	−5.32109	−0.280446
$361$	14.3043	0.752855
$362$	−1.62009	−0.0851501
$363$	−0.695825	−0.0365213
$364$	−3.20934	−0.168215
$365$	−26.5461	−1.38949
$366$	−0.588907	−0.0307827
$367$	−3.60837	−0.188355	−0.0941777	−	0.995555i	$-0.530022\pi$
$367$	−3.60837	−0.188355	−0.0941777	+	0.995555i	$0.530022\pi$
$368$	0.486303	0.0253503
$369$	11.6104	0.604415
$370$	−7.45443	−0.387537
$371$	9.17223	0.476198
$372$	−0.588730	−0.0305242
$373$	8.47104	0.438614	0.219307	−	0.975656i	$-0.429620\pi$
$373$	8.47104	0.438614	0.219307	+	0.975656i	$0.429620\pi$
$374$	5.35417	0.276858
$375$	−1.28834	−0.0665297
$376$	−6.72074	−0.346596
$377$	−3.56903	−0.183815
$378$	−0.605974	−0.0311679
$379$	−8.08687	−0.415395	−0.207697	−	0.978193i	$-0.566597\pi$
$379$	−8.08687	−0.415395	−0.207697	+	0.978193i	$0.566597\pi$
$380$	−10.2744	−0.527066
$381$	0.659690	0.0337970
$382$	−19.9005	−1.01820
$383$	−12.0791	−0.617213	−0.308607	−	0.951190i	$-0.599863\pi$
$383$	−12.0791	−0.617213	−0.308607	+	0.951190i	$0.599863\pi$
$384$	0.105946	0.00540651
$385$	−3.57974	−0.182441
$386$	−8.25830	−0.420336
$387$	−15.3705	−0.781329
$388$	14.5307	0.737682
$389$	2.46996	0.125232	0.0626158	−	0.998038i	$-0.480056\pi$
$389$	2.46996	0.125232	0.0626158	+	0.998038i	$0.480056\pi$
$390$	−0.633829	−0.0320952
$391$	−1.23677	−0.0625459
$392$	−6.08785	−0.307483
$393$	−1.99888	−0.100830
$394$	−8.75315	−0.440977
$395$	6.21954	0.312939
$396$	6.29223	0.316197
$397$	−18.9899	−0.953076	−0.476538	−	0.879154i	$-0.658109\pi$
$397$	−18.9899	−0.953076	−0.476538	+	0.879154i	$0.658109\pi$
$398$	−21.9469	−1.10010
$399$	−0.583937	−0.0292334
$400$	−1.83033	−0.0915166
$401$	10.4098	0.519840	0.259920	−	0.965630i	$-0.416304\pi$
$401$	10.4098	0.519840	0.259920	+	0.965630i	$0.416304\pi$
$402$	−0.233499	−0.0116459
$403$	18.6731	0.930173
$404$	−0.684403	−0.0340503
$405$	15.8436	0.787274
$406$	1.01438	0.0503429
$407$	8.81493	0.436940
$408$	−0.269441	−0.0133393
$409$	2.74576	0.135769	0.0678847	−	0.997693i	$-0.478375\pi$
$409$	2.74576	0.135769	0.0678847	+	0.997693i	$0.478375\pi$
$410$	−6.91611	−0.341562
$411$	1.99803	0.0985555
$412$	1.84473	0.0908832
$413$	2.67998	0.131873
$414$	−1.45345	−0.0714332
$415$	−10.1858	−0.499999
$416$	−3.36034	−0.164754
$417$	1.08301	0.0530351
$418$	12.1496	0.594256
$419$	12.8786	0.629162	0.314581	−	0.949231i	$-0.398136\pi$
$419$	12.8786	0.629162	0.314581	+	0.949231i	$0.398136\pi$
$420$	0.180145	0.00879019
$421$	5.58042	0.271973	0.135987	−	0.990711i	$-0.456580\pi$
$421$	5.58042	0.271973	0.135987	+	0.990711i	$0.456580\pi$
$422$	7.73297	0.376435
$423$	20.0868	0.976653
$424$	9.60376	0.466400
$425$	4.65490	0.225796
$426$	0.903444	0.0437720
$427$	−5.30882	−0.256912
$428$	−3.38827	−0.163778
$429$	0.749509	0.0361866
$430$	9.15594	0.441539
$431$	−14.4810	−0.697525	−0.348763	−	0.937211i	$-0.613398\pi$
$431$	−14.4810	−0.697525	−0.348763	+	0.937211i	$0.613398\pi$
$432$	−0.634484	−0.0305266
$433$	8.46061	0.406591	0.203295	−	0.979117i	$-0.434835\pi$
$433$	8.46061	0.406591	0.203295	+	0.979117i	$0.434835\pi$
$434$	−5.30722	−0.254755
$435$	0.200335	0.00960534
$436$	−4.74817	−0.227396
$437$	−2.80645	−0.134251
$438$	−1.57971	−0.0754814
$439$	−19.5665	−0.933860	−0.466930	−	0.884294i	$-0.654640\pi$
$439$	−19.5665	−0.933860	−0.466930	+	0.884294i	$0.654640\pi$
$440$	−3.74816	−0.178687
$441$	18.1952	0.866439
$442$	8.54601	0.406492
$443$	1.21379	0.0576688	0.0288344	−	0.999584i	$-0.490820\pi$
$443$	1.21379	0.0576688	0.0288344	+	0.999584i	$0.490820\pi$
$444$	−0.443598	−0.0210522
$445$	25.9434	1.22984
$446$	−12.9455	−0.612989
$447$	1.91740	0.0906901
$448$	0.955066	0.0451226
$449$	−24.6320	−1.16246	−0.581228	−	0.813741i	$-0.697428\pi$
$449$	−24.6320	−1.16246	−0.581228	+	0.813741i	$0.697428\pi$
$450$	5.47045	0.257880
$451$	8.17837	0.385104
$452$	4.78070	0.224865
$453$	2.49993	0.117457
$454$	−2.49395	−0.117047
$455$	−5.71377	−0.267866
$456$	−0.611410	−0.0286319
$457$	28.1692	1.31770	0.658849	−	0.752275i	$-0.271044\pi$
$457$	28.1692	1.31770	0.658849	+	0.752275i	$0.271044\pi$
$458$	−1.75782	−0.0821373
$459$	1.61362	0.0753173
$460$	0.865792	0.0403678
$461$	25.4559	1.18560	0.592799	−	0.805350i	$-0.298023\pi$
$461$	25.4559	1.18560	0.592799	+	0.805350i	$0.298023\pi$
$462$	−0.213024	−0.00991075
$463$	−40.3560	−1.87550	−0.937751	−	0.347309i	$-0.887096\pi$
$463$	−40.3560	−1.87550	−0.937751	+	0.347309i	$0.887096\pi$
$464$	1.06211	0.0493070
$465$	−1.04815	−0.0486067
$466$	5.40640	0.250446
$467$	−16.0521	−0.742805	−0.371402	−	0.928472i	$-0.621123\pi$
$467$	−16.0521	−0.742805	−0.371402	+	0.928472i	$0.621123\pi$
$468$	10.0433	0.464251
$469$	−2.10492	−0.0971964
$470$	−11.9653	−0.551919
$471$	0.288761	0.0133054
$472$	2.80607	0.129160
$473$	−10.8270	−0.497825
$474$	0.370113	0.0169998
$475$	10.5628	0.484655
$476$	−2.42892	−0.111330
$477$	−28.7035	−1.31424
$478$	1.22912	0.0562186
$479$	−8.47762	−0.387352	−0.193676	−	0.981066i	$-0.562041\pi$
$479$	−8.47762	−0.387352	−0.193676	+	0.981066i	$0.562041\pi$
$480$	0.188621	0.00860932
$481$	14.0699	0.641531
$482$	12.4259	0.565985
$483$	0.0492065	0.00223898
$484$	−6.56776	−0.298535
$485$	25.8697	1.17468
$486$	2.84627	0.129110
$487$	19.4749	0.882490	0.441245	−	0.897387i	$-0.354537\pi$
$487$	19.4749	0.882490	0.441245	+	0.897387i	$0.354537\pi$
$488$	−5.55859	−0.251625
$489$	−0.919451	−0.0415790
$490$	−10.8385	−0.489635
$491$	−14.9190	−0.673287	−0.336644	−	0.941632i	$-0.609292\pi$
$491$	−14.9190	−0.673287	−0.336644	+	0.941632i	$0.609292\pi$
$492$	−0.411564	−0.0185547
$493$	−2.70115	−0.121654
$494$	19.3924	0.872507
$495$	11.2024	0.503511
$496$	−5.55691	−0.249513
$497$	8.14427	0.365321
$498$	−0.606134	−0.0271615
$499$	−43.9913	−1.96932	−0.984661	−	0.174476i	$-0.944177\pi$
$499$	−43.9913	−1.96932	−0.984661	+	0.174476i	$0.944177\pi$
$500$	−12.1604	−0.543831
$501$	−0.627726	−0.0280447
$502$	0.447538	0.0199746
$503$	−13.6717	−0.609590	−0.304795	−	0.952418i	$-0.598588\pi$
$503$	−13.6717	−0.609590	−0.304795	+	0.952418i	$0.598588\pi$
$504$	−2.85448	−0.127149
$505$	−1.21848	−0.0542217
$506$	−1.02381	−0.0455138
$507$	−0.180970	−0.00803715
$508$	6.22670	0.276265
$509$	−21.8131	−0.966850	−0.483425	−	0.875386i	$-0.660607\pi$
$509$	−21.8131	−0.966850	−0.483425	+	0.875386i	$0.660607\pi$
$510$	−0.479700	−0.0212415
$511$	−14.2406	−0.629967
$512$	1.00000	0.0441942
$513$	3.66160	0.161663
$514$	−7.98817	−0.352343
$515$	3.28427	0.144722
$516$	0.544852	0.0239858
$517$	14.1491	0.622277
$518$	−3.99890	−0.175702
$519$	0.275674	0.0121008
$520$	−5.98260	−0.262354
$521$	22.2827	0.976221	0.488110	−	0.872782i	$-0.337686\pi$
$521$	22.2827	0.976221	0.488110	+	0.872782i	$0.337686\pi$
$522$	−3.17440	−0.138940
$523$	44.0071	1.92430	0.962148	−	0.272527i	$-0.0878596\pi$
$523$	44.0071	1.92430	0.962148	+	0.272527i	$0.0878596\pi$
$524$	−18.8671	−0.824212
$525$	−0.185202	−0.00808288
$526$	−1.54704	−0.0674541
$527$	14.1323	0.615614
$528$	−0.223046	−0.00970683
$529$	−22.7635	−0.989718
$530$	17.0981	0.742695
$531$	−8.38671	−0.363952
$532$	−5.51167	−0.238961
$533$	13.0538	0.565424
$534$	1.54384	0.0668086
$535$	−6.03234	−0.260801
$536$	−2.20396	−0.0951964
$537$	1.01767	0.0439157
$538$	−22.4773	−0.969064
$539$	12.8167	0.552053
$540$	−1.12961	−0.0486106
$541$	26.7070	1.14822	0.574112	−	0.818777i	$-0.305347\pi$
$541$	26.7070	1.14822	0.574112	+	0.818777i	$0.305347\pi$
$542$	4.35115	0.186898
$543$	−0.171641	−0.00736584
$544$	−2.54320	−0.109039
$545$	−8.45344	−0.362106
$546$	−0.340016	−0.0145513
$547$	7.02491	0.300363	0.150182	−	0.988658i	$-0.452014\pi$
$547$	7.02491	0.300363	0.150182	+	0.988658i	$0.452014\pi$
$548$	18.8590	0.805617
$549$	16.6134	0.709041
$550$	3.85338	0.164309
$551$	−6.12940	−0.261121
$552$	0.0515216	0.00219291
$553$	3.33645	0.141880
$554$	−6.00059	−0.254941
$555$	−0.789763	−0.0335236
$556$	10.2223	0.433522
$557$	−7.62986	−0.323288	−0.161644	−	0.986849i	$-0.551680\pi$
$557$	−7.62986	−0.323288	−0.161644	+	0.986849i	$0.551680\pi$
$558$	16.6084	0.703088
$559$	−17.2814	−0.730925
$560$	1.70036	0.0718532
$561$	0.567250	0.0239493
$562$	18.7791	0.792146
$563$	17.8645	0.752898	0.376449	−	0.926437i	$-0.377145\pi$
$563$	17.8645	0.752898	0.376449	+	0.926437i	$0.377145\pi$
$564$	−0.712032	−0.0299820
$565$	8.51136	0.358075
$566$	0.141302	0.00593937
$567$	8.49923	0.356934
$568$	8.52744	0.357804
$569$	17.8426	0.748000	0.374000	−	0.927429i	$-0.377986\pi$
$569$	17.8426	0.748000	0.374000	+	0.927429i	$0.377986\pi$
$570$	−1.08853	−0.0455934
$571$	24.2791	1.01605	0.508024	−	0.861343i	$-0.330376\pi$
$571$	24.2791	1.01605	0.508024	+	0.861343i	$0.330376\pi$
$572$	7.07448	0.295799
$573$	−2.10837	−0.0880783
$574$	−3.71012	−0.154858
$575$	−0.890096	−0.0371196
$576$	−2.98878	−0.124532
$577$	11.1432	0.463895	0.231948	−	0.972728i	$-0.425490\pi$
$577$	11.1432	0.463895	0.231948	+	0.972728i	$0.425490\pi$
$578$	−10.5321	−0.438079
$579$	−0.874930	−0.0363608
$580$	1.89093	0.0785165
$581$	−5.46411	−0.226690
$582$	1.53946	0.0638126
$583$	−20.2187	−0.837372
$584$	−14.9106	−0.617004
$585$	17.8806	0.739273
$586$	5.08847	0.210203
$587$	10.3191	0.425917	0.212958	−	0.977061i	$-0.431690\pi$
$587$	10.3191	0.425917	0.212958	+	0.977061i	$0.431690\pi$
$588$	−0.644980	−0.0265985
$589$	32.0688	1.32137
$590$	4.99580	0.205674
$591$	−0.927357	−0.0381464
$592$	−4.18704	−0.172086
$593$	34.3285	1.40970	0.704851	−	0.709356i	$-0.251014\pi$
$593$	34.3285	1.40970	0.704851	+	0.709356i	$0.251014\pi$
$594$	1.33577	0.0548074
$595$	−4.32435	−0.177281
$596$	18.0980	0.741324
$597$	−2.32517	−0.0951629
$598$	−1.63414	−0.0668250
$599$	4.88502	0.199597	0.0997983	−	0.995008i	$-0.468180\pi$
$599$	4.88502	0.199597	0.0997983	+	0.995008i	$0.468180\pi$
$600$	−0.193915	−0.00791656
$601$	−22.8199	−0.930845	−0.465422	−	0.885089i	$-0.654098\pi$
$601$	−22.8199	−0.930845	−0.465422	+	0.885089i	$0.654098\pi$
$602$	4.91167	0.200185
$603$	6.58713	0.268249
$604$	23.5963	0.960121
$605$	−11.6930	−0.475386
$606$	−0.0725094	−0.00294549
$607$	−7.80132	−0.316646	−0.158323	−	0.987387i	$-0.550609\pi$
$607$	−7.80132	−0.316646	−0.158323	+	0.987387i	$0.550609\pi$
$608$	−5.77098	−0.234044
$609$	0.107469	0.00435487
$610$	−9.89626	−0.400688
$611$	22.5840	0.913649
$612$	7.60105	0.307254
$613$	−10.0075	−0.404198	−0.202099	−	0.979365i	$-0.564776\pi$
$613$	−10.0075	−0.404198	−0.202099	+	0.979365i	$0.564776\pi$
$614$	−14.9248	−0.602315
$615$	−0.732731	−0.0295466
$616$	−2.01069	−0.0810130
$617$	−37.3059	−1.50188	−0.750940	−	0.660371i	$-0.770399\pi$
$617$	−37.3059	−1.50188	−0.750940	+	0.660371i	$0.770399\pi$
$618$	0.195441	0.00786177
$619$	41.2627	1.65849	0.829244	−	0.558886i	$-0.188771\pi$
$619$	41.2627	1.65849	0.829244	+	0.558886i	$0.188771\pi$
$620$	−9.89328	−0.397324
$621$	−0.308551	−0.0123817
$622$	2.18581	0.0876429
$623$	13.9173	0.557584
$624$	−0.356013	−0.0142519
$625$	−12.4982	−0.499929
$626$	−31.5982	−1.26292
$627$	1.28719	0.0514056
$628$	2.72556	0.108762
$629$	10.6485	0.424583
$630$	−5.08199	−0.202471
$631$	−12.7313	−0.506827	−0.253413	−	0.967358i	$-0.581553\pi$
$631$	−12.7313	−0.506827	−0.253413	+	0.967358i	$0.581553\pi$
$632$	3.49342	0.138961
$633$	0.819273	0.0325632
$634$	−24.2145	−0.961679
$635$	11.0857	0.439924
$636$	1.01748	0.0403455
$637$	20.4572	0.810545
$638$	−2.23604	−0.0885257
$639$	−25.4866	−1.00823
$640$	1.78036	0.0703748
$641$	37.6686	1.48782	0.743911	−	0.668279i	$-0.232969\pi$
$641$	37.6686	1.48782	0.743911	+	0.668279i	$0.232969\pi$
$642$	−0.358972	−0.0141675
$643$	−18.8880	−0.744869	−0.372435	−	0.928058i	$-0.621477\pi$
$643$	−18.8880	−0.744869	−0.372435	+	0.928058i	$0.621477\pi$
$644$	0.464451	0.0183019
$645$	0.970031	0.0381949
$646$	14.6768	0.577450
$647$	−2.30344	−0.0905574	−0.0452787	−	0.998974i	$-0.514418\pi$
$647$	−2.30344	−0.0905574	−0.0452787	+	0.998974i	$0.514418\pi$
$648$	8.89911	0.349590
$649$	−5.90758	−0.231893
$650$	6.15053	0.241244
$651$	−0.562276	−0.0220373
$652$	−8.67853	−0.339877
$653$	10.5370	0.412344	0.206172	−	0.978516i	$-0.433899\pi$
$653$	10.5370	0.412344	0.206172	+	0.978516i	$0.433899\pi$
$654$	−0.503048	−0.0196707
$655$	−33.5901	−1.31247
$656$	−3.88468	−0.151671
$657$	44.5644	1.73862
$658$	−6.41875	−0.250229
$659$	−12.3931	−0.482765	−0.241382	−	0.970430i	$-0.577601\pi$
$659$	−12.3931	−0.482765	−0.241382	+	0.970430i	$0.577601\pi$
$660$	−0.397101	−0.0154571
$661$	24.8320	0.965851	0.482926	−	0.875661i	$-0.339574\pi$
$661$	24.8320	0.965851	0.482926	+	0.875661i	$0.339574\pi$
$662$	7.70093	0.299305
$663$	0.905411	0.0351632
$664$	−5.72119	−0.222025
$665$	−9.81274	−0.380522
$666$	12.5141	0.484913
$667$	0.516505	0.0199992
$668$	−5.92499	−0.229245
$669$	−1.37152	−0.0530261
$670$	−3.92383	−0.151591
$671$	11.7024	0.451767
$672$	0.101185	0.00390329
$673$	8.18910	0.315667	0.157833	−	0.987466i	$-0.449549\pi$
$673$	8.18910	0.315667	0.157833	+	0.987466i	$0.449549\pi$
$674$	32.1482	1.23830
$675$	1.16132	0.0446991
$676$	−1.70814	−0.0656977
$677$	28.6762	1.10211	0.551057	−	0.834467i	$-0.314225\pi$
$677$	28.6762	1.10211	0.551057	+	0.834467i	$0.314225\pi$
$678$	0.506494	0.0194518
$679$	13.8777	0.532579
$680$	−4.52780	−0.173633
$681$	−0.264223	−0.0101250
$682$	11.6989	0.447974
$683$	−16.7535	−0.641054	−0.320527	−	0.947239i	$-0.603860\pi$
$683$	−16.7535	−0.641054	−0.320527	+	0.947239i	$0.603860\pi$
$684$	17.2482	0.659500
$685$	33.5758	1.28286
$686$	−12.4998	−0.477243
$687$	−0.186233	−0.00710522
$688$	5.14276	0.196066
$689$	−32.2719	−1.22946
$690$	0.0917268	0.00349198
$691$	30.3319	1.15388	0.576940	−	0.816787i	$-0.304247\pi$
$691$	30.3319	1.15388	0.576940	+	0.816787i	$0.304247\pi$
$692$	2.60204	0.0989147
$693$	6.00950	0.228282
$694$	−21.0435	−0.798799
$695$	18.1993	0.690340
$696$	0.112525	0.00426526
$697$	9.87951	0.374213
$698$	7.27442	0.275341
$699$	0.572783	0.0216647
$700$	−1.74809	−0.0660715
$701$	20.0036	0.755525	0.377763	−	0.925902i	$-0.376694\pi$
$701$	20.0036	0.755525	0.377763	+	0.925902i	$0.376694\pi$
$702$	2.13208	0.0804701
$703$	24.1633	0.911338
$704$	−2.10529	−0.0793460
$705$	−1.26767	−0.0477433
$706$	−27.6561	−1.04085
$707$	−0.653650	−0.0245830
$708$	0.297290	0.0111729
$709$	−9.61069	−0.360937	−0.180469	−	0.983581i	$-0.557761\pi$
$709$	−9.61069	−0.360937	−0.180469	+	0.983581i	$0.557761\pi$
$710$	15.1819	0.569766
$711$	−10.4411	−0.391570
$712$	14.5721	0.546111
$713$	−2.70234	−0.101203
$714$	−0.257334	−0.00963047
$715$	12.5951	0.471030
$716$	9.60559	0.358978
$717$	0.130220	0.00486314
$718$	−19.9387	−0.744106
$719$	−24.7766	−0.924010	−0.462005	−	0.886877i	$-0.652870\pi$
$719$	−24.7766	−0.924010	−0.462005	+	0.886877i	$0.652870\pi$
$720$	−5.32109	−0.198305
$721$	1.76184	0.0656142
$722$	14.3043	0.532349
$723$	1.31647	0.0489600
$724$	−1.62009	−0.0602102
$725$	−1.94401	−0.0721986
$726$	−0.695825	−0.0258245
$727$	−17.6116	−0.653179	−0.326590	−	0.945166i	$-0.605899\pi$
$727$	−17.6116	−0.653179	−0.326590	+	0.945166i	$0.605899\pi$
$728$	−3.20934	−0.118946
$729$	−26.3958	−0.977621
$730$	−26.5461	−0.982517
$731$	−13.0791	−0.483747
$732$	−0.588907	−0.0217666
$733$	−2.52214	−0.0931575	−0.0465788	−	0.998915i	$-0.514832\pi$
$733$	−2.52214	−0.0931575	−0.0465788	+	0.998915i	$0.514832\pi$
$734$	−3.60837	−0.133187
$735$	−1.14829	−0.0423555
$736$	0.486303	0.0179254
$737$	4.63997	0.170915
$738$	11.6104	0.427386
$739$	−23.8491	−0.877305	−0.438652	−	0.898657i	$-0.644544\pi$
$739$	−23.8491	−0.877305	−0.438652	+	0.898657i	$0.644544\pi$
$740$	−7.45443	−0.274030
$741$	2.05454	0.0754755
$742$	9.17223	0.336723
$743$	−12.9187	−0.473940	−0.236970	−	0.971517i	$-0.576154\pi$
$743$	−12.9187	−0.473940	−0.236970	+	0.971517i	$0.576154\pi$
$744$	−0.588730	−0.0215839
$745$	32.2209	1.18048
$746$	8.47104	0.310147
$747$	17.0994	0.625633
$748$	5.35417	0.195768
$749$	−3.23603	−0.118242
$750$	−1.28834	−0.0470436
$751$	−25.6931	−0.937556	−0.468778	−	0.883316i	$-0.655306\pi$
$751$	−25.6931	−0.937556	−0.468778	+	0.883316i	$0.655306\pi$
$752$	−6.72074	−0.245080
$753$	0.0474146	0.00172789
$754$	−3.56903	−0.129977
$755$	42.0099	1.52890
$756$	−0.605974	−0.0220391
$757$	1.48705	0.0540479	0.0270240	−	0.999635i	$-0.491397\pi$
$757$	1.48705	0.0540479	0.0270240	+	0.999635i	$0.491397\pi$
$758$	−8.08687	−0.293728
$759$	−0.108468	−0.00393713
$760$	−10.2744	−0.372692
$761$	−15.4899	−0.561508	−0.280754	−	0.959780i	$-0.590585\pi$
$761$	−15.4899	−0.561508	−0.280754	+	0.959780i	$0.590585\pi$
$762$	0.659690	0.0238981
$763$	−4.53482	−0.164171
$764$	−19.9005	−0.719974
$765$	13.5326	0.489272
$766$	−12.0791	−0.436436
$767$	−9.42933	−0.340474
$768$	0.105946	0.00382298
$769$	21.9957	0.793186	0.396593	−	0.917995i	$-0.370192\pi$
$769$	21.9957	0.793186	0.396593	+	0.917995i	$0.370192\pi$
$770$	−3.57974	−0.129005
$771$	−0.846311	−0.0304791
$772$	−8.25830	−0.297223
$773$	14.2432	0.512292	0.256146	−	0.966638i	$-0.417547\pi$
$773$	14.2432	0.512292	0.256146	+	0.966638i	$0.417547\pi$
$774$	−15.3705	−0.552483
$775$	10.1710	0.365353
$776$	14.5307	0.521620
$777$	−0.423666	−0.0151989
$778$	2.46996	0.0885522
$779$	22.4184	0.803223
$780$	−0.633829	−0.0226947
$781$	−17.9527	−0.642399
$782$	−1.23677	−0.0442266
$783$	−0.673889	−0.0240828
$784$	−6.08785	−0.217423
$785$	4.85247	0.173192
$786$	−1.99888	−0.0712978
$787$	−15.8024	−0.563295	−0.281647	−	0.959518i	$-0.590881\pi$
$787$	−15.8024	−0.563295	−0.281647	+	0.959518i	$0.590881\pi$
$788$	−8.75315	−0.311818
$789$	−0.163902	−0.00583506
$790$	6.21954	0.221281
$791$	4.56589	0.162344
$792$	6.29223	0.223585
$793$	18.6787	0.663301
$794$	−18.9899	−0.673927
$795$	1.81147	0.0642462
$796$	−21.9469	−0.777886
$797$	53.4671	1.89390	0.946951	−	0.321378i	$-0.104146\pi$
$797$	53.4671	1.89390	0.946951	+	0.321378i	$0.104146\pi$
$798$	−0.583937	−0.0206711
$799$	17.0922	0.604678
$800$	−1.83033	−0.0647120
$801$	−43.5526	−1.53886
$802$	10.4098	0.367582
$803$	31.3911	1.10777
$804$	−0.233499	−0.00823489
$805$	0.826889	0.0291440
$806$	18.6731	0.657732
$807$	−2.38137	−0.0838280
$808$	−0.684403	−0.0240772
$809$	37.9987	1.33596	0.667981	−	0.744178i	$-0.267159\pi$
$809$	37.9987	1.33596	0.667981	+	0.744178i	$0.267159\pi$
$810$	15.8436	0.556687
$811$	9.93564	0.348888	0.174444	−	0.984667i	$-0.444187\pi$
$811$	9.93564	0.348888	0.174444	+	0.984667i	$0.444187\pi$
$812$	1.01438	0.0355978
$813$	0.460985	0.0161675
$814$	8.81493	0.308963
$815$	−15.4509	−0.541221
$816$	−0.269441	−0.00943231
$817$	−29.6788	−1.03833
$818$	2.74576	0.0960034
$819$	9.59201	0.335172
$820$	−6.91611	−0.241521
$821$	−37.0820	−1.29417	−0.647086	−	0.762417i	$-0.724012\pi$
$821$	−37.0820	−1.29417	−0.647086	+	0.762417i	$0.724012\pi$
$822$	1.99803	0.0696892
$823$	25.3049	0.882075	0.441038	−	0.897489i	$-0.354611\pi$
$823$	25.3049	0.882075	0.441038	+	0.897489i	$0.354611\pi$
$824$	1.84473	0.0642641
$825$	0.408248	0.0142134
$826$	2.67998	0.0932485
$827$	−22.2732	−0.774515	−0.387258	−	0.921971i	$-0.626578\pi$
$827$	−22.2732	−0.774515	−0.387258	+	0.921971i	$0.626578\pi$
$828$	−1.45345	−0.0505109
$829$	−11.0575	−0.384044	−0.192022	−	0.981391i	$-0.561505\pi$
$829$	−11.0575	−0.384044	−0.192022	+	0.981391i	$0.561505\pi$
$830$	−10.1858	−0.353553
$831$	−0.635736	−0.0220534
$832$	−3.36034	−0.116499
$833$	15.4826	0.536441
$834$	1.08301	0.0375014
$835$	−10.5486	−0.365049
$836$	12.1496	0.420202
$837$	3.52577	0.121868
$838$	12.8786	0.444885
$839$	−0.653986	−0.0225781	−0.0112890	−	0.999936i	$-0.503593\pi$
$839$	−0.653986	−0.0225781	−0.0112890	+	0.999936i	$0.503593\pi$
$840$	0.180145	0.00621560
$841$	−27.8719	−0.961101
$842$	5.58042	0.192314
$843$	1.98956	0.0685239
$844$	7.73297	0.266180
$845$	−3.04110	−0.104617
$846$	20.0868	0.690598
$847$	−6.27265	−0.215531
$848$	9.60376	0.329794
$849$	0.0149703	0.000513780 0
$850$	4.65490	0.159662
$851$	−2.03617	−0.0697990
$852$	0.903444	0.0309515
$853$	14.6112	0.500278	0.250139	−	0.968210i	$-0.419524\pi$
$853$	14.6112	0.500278	0.250139	+	0.968210i	$0.419524\pi$
$854$	−5.30882	−0.181664
$855$	30.7079	1.05019
$856$	−3.38827	−0.115809
$857$	34.6207	1.18262	0.591310	−	0.806444i	$-0.298611\pi$
$857$	34.6207	1.18262	0.591310	+	0.806444i	$0.298611\pi$
$858$	0.749509	0.0255878
$859$	−10.4056	−0.355036	−0.177518	−	0.984118i	$-0.556807\pi$
$859$	−10.4056	−0.355036	−0.177518	+	0.984118i	$0.556807\pi$
$860$	9.15594	0.312215
$861$	−0.393071	−0.0133958
$862$	−14.4810	−0.493225
$863$	−24.4717	−0.833026	−0.416513	−	0.909130i	$-0.636748\pi$
$863$	−24.4717	−0.833026	−0.416513	+	0.909130i	$0.636748\pi$
$864$	−0.634484	−0.0215856
$865$	4.63256	0.157512
$866$	8.46061	0.287503
$867$	−1.11583	−0.0378957
$868$	−5.30722	−0.180139
$869$	−7.35467	−0.249490
$870$	0.200335	0.00679200
$871$	7.40604	0.250944
$872$	−4.74817	−0.160793
$873$	−43.4289	−1.46984
$874$	−2.80645	−0.0949295
$875$	−11.6140	−0.392625
$876$	−1.57971	−0.0533734
$877$	13.9875	0.472325	0.236162	−	0.971714i	$-0.424110\pi$
$877$	13.9875	0.472325	0.236162	+	0.971714i	$0.424110\pi$
$878$	−19.5665	−0.660339
$879$	0.539100	0.0181834
$880$	−3.74816	−0.126351
$881$	37.1823	1.25270	0.626351	−	0.779541i	$-0.284548\pi$
$881$	37.1823	1.25270	0.626351	+	0.779541i	$0.284548\pi$
$882$	18.1952	0.612665
$883$	−17.1589	−0.577442	−0.288721	−	0.957413i	$-0.593230\pi$
$883$	−17.1589	−0.577442	−0.288721	+	0.957413i	$0.593230\pi$
$884$	8.54601	0.287433
$885$	0.529283	0.0177916
$886$	1.21379	0.0407780
$887$	−30.6524	−1.02921	−0.514603	−	0.857429i	$-0.672061\pi$
$887$	−30.6524	−1.02921	−0.514603	+	0.857429i	$0.672061\pi$
$888$	−0.443598	−0.0148862
$889$	5.94691	0.199453
$890$	25.9434	0.869626
$891$	−18.7352	−0.627652
$892$	−12.9455	−0.433449
$893$	38.7853	1.29790
$894$	1.91740	0.0641276
$895$	17.1014	0.571636
$896$	0.955066	0.0319065
$897$	−0.173130	−0.00578064
$898$	−24.6320	−0.821981
$899$	−5.90203	−0.196844
$900$	5.47045	0.182348
$901$	−24.4243	−0.813691
$902$	8.17837	0.272310
$903$	0.520370	0.0173168
$904$	4.78070	0.159004
$905$	−2.88434	−0.0958787
$906$	2.49993	0.0830545
$907$	−8.14380	−0.270410	−0.135205	−	0.990818i	$-0.543169\pi$
$907$	−8.14380	−0.270410	−0.135205	+	0.990818i	$0.543169\pi$
$908$	−2.49395	−0.0827647
$909$	2.04553	0.0678459
$910$	−5.71377	−0.189410
$911$	26.5986	0.881249	0.440625	−	0.897691i	$-0.354757\pi$
$911$	26.5986	0.881249	0.440625	+	0.897691i	$0.354757\pi$
$912$	−0.611410	−0.0202458
$913$	12.0448	0.398623
$914$	28.1692	0.931753
$915$	−1.04846	−0.0346612
$916$	−1.75782	−0.0580799
$917$	−18.0193	−0.595050
$918$	1.61362	0.0532574
$919$	−2.13805	−0.0705277	−0.0352639	−	0.999378i	$-0.511227\pi$
$919$	−2.13805	−0.0705277	−0.0352639	+	0.999378i	$0.511227\pi$
$920$	0.865792	0.0285443
$921$	−1.58121	−0.0521028
$922$	25.4559	0.838344
$923$	−28.6551	−0.943193
$924$	−0.213024	−0.00700796
$925$	7.66367	0.251980
$926$	−40.3560	−1.32618
$927$	−5.51348	−0.181086
$928$	1.06211	0.0348653
$929$	43.0661	1.41295	0.706476	−	0.707737i	$-0.250284\pi$
$929$	43.0661	1.41295	0.706476	+	0.707737i	$0.250284\pi$
$930$	−1.04815	−0.0343701
$931$	35.1329	1.15143
$932$	5.40640	0.177092
$933$	0.231577	0.00758147
$934$	−16.0521	−0.525242
$935$	9.53233	0.311740
$936$	10.0433	0.328275
$937$	8.44285	0.275816	0.137908	−	0.990445i	$-0.455962\pi$
$937$	8.44285	0.275816	0.137908	+	0.990445i	$0.455962\pi$
$938$	−2.10492	−0.0687282
$939$	−3.34769	−0.109248
$940$	−11.9653	−0.390266
$941$	−51.7609	−1.68736	−0.843678	−	0.536849i	$-0.819615\pi$
$941$	−51.7609	−1.68736	−0.843678	+	0.536849i	$0.819615\pi$
$942$	0.288761	0.00940835
$943$	−1.88913	−0.0615185
$944$	2.80607	0.0913297
$945$	−1.07885	−0.0350950
$946$	−10.8270	−0.352016
$947$	−10.6103	−0.344788	−0.172394	−	0.985028i	$-0.555150\pi$
$947$	−10.6103	−0.344788	−0.172394	+	0.985028i	$0.555150\pi$
$948$	0.370113	0.0120207
$949$	50.1046	1.62646
$950$	10.5628	0.342703
$951$	−2.56541	−0.0831893
$952$	−2.42892	−0.0787219
$953$	−21.2146	−0.687209	−0.343604	−	0.939115i	$-0.611648\pi$
$953$	−21.2146	−0.687209	−0.343604	+	0.939115i	$0.611648\pi$
$954$	−28.7035	−0.929309
$955$	−35.4300	−1.14649
$956$	1.22912	0.0397525
$957$	−0.236898	−0.00765784
$958$	−8.47762	−0.273899
$959$	18.0116	0.581625
$960$	0.188621	0.00608771
$961$	−0.120744	−0.00389496
$962$	14.0699	0.453631
$963$	10.1268	0.326331
$964$	12.4259	0.400212
$965$	−14.7027	−0.473297
$966$	0.0492065	0.00158319
$967$	−18.6299	−0.599098	−0.299549	−	0.954081i	$-0.596836\pi$
$967$	−18.6299	−0.599098	−0.299549	+	0.954081i	$0.596836\pi$
$968$	−6.56776	−0.211096
$969$	1.55494	0.0499518
$970$	25.8697	0.830627
$971$	38.0136	1.21991	0.609957	−	0.792435i	$-0.291187\pi$
$971$	38.0136	1.21991	0.609957	+	0.792435i	$0.291187\pi$
$972$	2.84627	0.0912942
$973$	9.76297	0.312986
$974$	19.4749	0.624015
$975$	0.651621	0.0208686
$976$	−5.55859	−0.177926
$977$	52.5378	1.68083	0.840416	−	0.541942i	$-0.182311\pi$
$977$	52.5378	1.68083	0.840416	+	0.541942i	$0.182311\pi$
$978$	−0.919451	−0.0294008
$979$	−30.6784	−0.980485
$980$	−10.8385	−0.346224
$981$	14.1912	0.453091
$982$	−14.9190	−0.476086
$983$	−10.6025	−0.338167	−0.169084	−	0.985602i	$-0.554081\pi$
$983$	−10.6025	−0.338167	−0.169084	+	0.985602i	$0.554081\pi$
$984$	−0.411564	−0.0131202
$985$	−15.5837	−0.496539
$986$	−2.70115	−0.0860221
$987$	−0.680038	−0.0216459
$988$	19.3924	0.616956
$989$	2.50094	0.0795252
$990$	11.2024	0.356036
$991$	52.4135	1.66497	0.832485	−	0.554047i	$-0.186917\pi$
$991$	52.4135	1.66497	0.832485	+	0.554047i	$0.186917\pi$
$992$	−5.55691	−0.176432
$993$	0.815879	0.0258911
$994$	8.14427	0.258321
$995$	−39.0732	−1.23870
$996$	−0.606134	−0.0192061
$997$	−11.7332	−0.371594	−0.185797	−	0.982588i	$-0.559487\pi$
$997$	−11.7332	−0.371594	−0.185797	+	0.982588i	$0.559487\pi$
$998$	−43.9913	−1.39252
$999$	2.65661	0.0840514

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4006.2.a.f.1.18	✓	31

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4006.2.a.f.1.18	✓	31	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4006 = 2 \cdot 2003 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4006.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +0.105946 q^{3} +1.00000 q^{4} +1.78036 q^{5} +0.105946 q^{6} +0.955066 q^{7} +1.00000 q^{8} -2.98878 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +0.105946 q^{3} +1.00000 q^{4} +1.78036 q^{5} +0.105946 q^{6} +0.955066 q^{7} +1.00000 q^{8} -2.98878 q^{9} +1.78036 q^{10} -2.10529 q^{11} +0.105946 q^{12} -3.36034 q^{13} +0.955066 q^{14} +0.188621 q^{15} +1.00000 q^{16} -2.54320 q^{17} -2.98878 q^{18} -5.77098 q^{19} +1.78036 q^{20} +0.101185 q^{21} -2.10529 q^{22} +0.486303 q^{23} +0.105946 q^{24} -1.83033 q^{25} -3.36034 q^{26} -0.634484 q^{27} +0.955066 q^{28} +1.06211 q^{29} +0.188621 q^{30} -5.55691 q^{31} +1.00000 q^{32} -0.223046 q^{33} -2.54320 q^{34} +1.70036 q^{35} -2.98878 q^{36} -4.18704 q^{37} -5.77098 q^{38} -0.356013 q^{39} +1.78036 q^{40} -3.88468 q^{41} +0.101185 q^{42} +5.14276 q^{43} -2.10529 q^{44} -5.32109 q^{45} +0.486303 q^{46} -6.72074 q^{47} +0.105946 q^{48} -6.08785 q^{49} -1.83033 q^{50} -0.269441 q^{51} -3.36034 q^{52} +9.60376 q^{53} -0.634484 q^{54} -3.74816 q^{55} +0.955066 q^{56} -0.611410 q^{57} +1.06211 q^{58} +2.80607 q^{59} +0.188621 q^{60} -5.55859 q^{61} -5.55691 q^{62} -2.85448 q^{63} +1.00000 q^{64} -5.98260 q^{65} -0.223046 q^{66} -2.20396 q^{67} -2.54320 q^{68} +0.0515216 q^{69} +1.70036 q^{70} +8.52744 q^{71} -2.98878 q^{72} -14.9106 q^{73} -4.18704 q^{74} -0.193915 q^{75} -5.77098 q^{76} -2.01069 q^{77} -0.356013 q^{78} +3.49342 q^{79} +1.78036 q^{80} +8.89911 q^{81} -3.88468 q^{82} -5.72119 q^{83} +0.101185 q^{84} -4.52780 q^{85} +5.14276 q^{86} +0.112525 q^{87} -2.10529 q^{88} +14.5721 q^{89} -5.32109 q^{90} -3.20934 q^{91} +0.486303 q^{92} -0.588730 q^{93} -6.72074 q^{94} -10.2744 q^{95} +0.105946 q^{96} +14.5307 q^{97} -6.08785 q^{98} +6.29223 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 31 q + 31 q^{2} - 13 q^{3} + 31 q^{4} - 23 q^{5} - 13 q^{6} - 18 q^{7} + 31 q^{8} + 20 q^{9}+O(q^{10}) \) 31 * q + 31 * q^2 - 13 * q^3 + 31 * q^4 - 23 * q^5 - 13 * q^6 - 18 * q^7 + 31 * q^8 + 20 * q^9	\( 31 q + 31 q^{2} - 13 q^{3} + 31 q^{4} - 23 q^{5} - 13 q^{6} - 18 q^{7} + 31 q^{8} + 20 q^{9} - 23 q^{10} - 32 q^{11} - 13 q^{12} - 8 q^{13} - 18 q^{14} - 14 q^{15} + 31 q^{16} - 30 q^{17} + 20 q^{18} - 38 q^{19} - 23 q^{20} - 16 q^{21} - 32 q^{22} - 24 q^{23} - 13 q^{24} + 40 q^{25} - 8 q^{26} - 28 q^{27} - 18 q^{28} - 7 q^{29} - 14 q^{30} - 32 q^{31} + 31 q^{32} - 9 q^{33} - 30 q^{34} - 14 q^{35} + 20 q^{36} + 11 q^{37} - 38 q^{38} - 9 q^{39} - 23 q^{40} - 76 q^{41} - 16 q^{42} - 33 q^{43} - 32 q^{44} - 40 q^{45} - 24 q^{46} - 96 q^{47} - 13 q^{48} + 15 q^{49} + 40 q^{50} - 55 q^{51} - 8 q^{52} - 28 q^{53} - 28 q^{54} - 52 q^{55} - 18 q^{56} - 21 q^{57} - 7 q^{58} - 72 q^{59} - 14 q^{60} - 9 q^{61} - 32 q^{62} - 54 q^{63} + 31 q^{64} - 38 q^{65} - 9 q^{66} - 4 q^{67} - 30 q^{68} - 17 q^{69} - 14 q^{70} - 61 q^{71} + 20 q^{72} - 62 q^{73} + 11 q^{74} - 63 q^{75} - 38 q^{76} - 9 q^{77} - 9 q^{78} - 30 q^{79} - 23 q^{80} - 13 q^{81} - 76 q^{82} - 90 q^{83} - 16 q^{84} + 26 q^{85} - 33 q^{86} - 34 q^{87} - 32 q^{88} - 99 q^{89} - 40 q^{90} - 47 q^{91} - 24 q^{92} - 6 q^{93} - 96 q^{94} - 24 q^{95} - 13 q^{96} - 46 q^{97} + 15 q^{98} - 48 q^{99}+O(q^{100}) \) 31 * q + 31 * q^2 - 13 * q^3 + 31 * q^4 - 23 * q^5 - 13 * q^6 - 18 * q^7 + 31 * q^8 + 20 * q^9 - 23 * q^10 - 32 * q^11 - 13 * q^12 - 8 * q^13 - 18 * q^14 - 14 * q^15 + 31 * q^16 - 30 * q^17 + 20 * q^18 - 38 * q^19 - 23 * q^20 - 16 * q^21 - 32 * q^22 - 24 * q^23 - 13 * q^24 + 40 * q^25 - 8 * q^26 - 28 * q^27 - 18 * q^28 - 7 * q^29 - 14 * q^30 - 32 * q^31 + 31 * q^32 - 9 * q^33 - 30 * q^34 - 14 * q^35 + 20 * q^36 + 11 * q^37 - 38 * q^38 - 9 * q^39 - 23 * q^40 - 76 * q^41 - 16 * q^42 - 33 * q^43 - 32 * q^44 - 40 * q^45 - 24 * q^46 - 96 * q^47 - 13 * q^48 + 15 * q^49 + 40 * q^50 - 55 * q^51 - 8 * q^52 - 28 * q^53 - 28 * q^54 - 52 * q^55 - 18 * q^56 - 21 * q^57 - 7 * q^58 - 72 * q^59 - 14 * q^60 - 9 * q^61 - 32 * q^62 - 54 * q^63 + 31 * q^64 - 38 * q^65 - 9 * q^66 - 4 * q^67 - 30 * q^68 - 17 * q^69 - 14 * q^70 - 61 * q^71 + 20 * q^72 - 62 * q^73 + 11 * q^74 - 63 * q^75 - 38 * q^76 - 9 * q^77 - 9 * q^78 - 30 * q^79 - 23 * q^80 - 13 * q^81 - 76 * q^82 - 90 * q^83 - 16 * q^84 + 26 * q^85 - 33 * q^86 - 34 * q^87 - 32 * q^88 - 99 * q^89 - 40 * q^90 - 47 * q^91 - 24 * q^92 - 6 * q^93 - 96 * q^94 - 24 * q^95 - 13 * q^96 - 46 * q^97 + 15 * q^98 - 48 * q^99

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