LMFDB - Embedded newform 8042.2.a.a.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8042,2,Mod(1,8042)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8042, 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("8042.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8042 = 2 \cdot 4021 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8042.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:	$64.2156933055$
Analytic rank:	$1$
Dimension:	$67$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Character	$\chi$	$=$	8042.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -2.16681 q^{3} +1.00000 q^{4} -2.63325 q^{5} -2.16681 q^{6} -3.58663 q^{7} +1.00000 q^{8} +1.69508 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.16681 q^{3} +1.00000 q^{4} -2.63325 q^{5} -2.16681 q^{6} -3.58663 q^{7} +1.00000 q^{8} +1.69508 q^{9} -2.63325 q^{10} -2.72032 q^{11} -2.16681 q^{12} -3.55957 q^{13} -3.58663 q^{14} +5.70576 q^{15} +1.00000 q^{16} +5.45332 q^{17} +1.69508 q^{18} +0.879136 q^{19} -2.63325 q^{20} +7.77156 q^{21} -2.72032 q^{22} +8.64820 q^{23} -2.16681 q^{24} +1.93400 q^{25} -3.55957 q^{26} +2.82751 q^{27} -3.58663 q^{28} -4.56127 q^{29} +5.70576 q^{30} -8.85929 q^{31} +1.00000 q^{32} +5.89442 q^{33} +5.45332 q^{34} +9.44449 q^{35} +1.69508 q^{36} +7.07454 q^{37} +0.879136 q^{38} +7.71294 q^{39} -2.63325 q^{40} -4.17143 q^{41} +7.77156 q^{42} +11.3694 q^{43} -2.72032 q^{44} -4.46358 q^{45} +8.64820 q^{46} +12.6680 q^{47} -2.16681 q^{48} +5.86390 q^{49} +1.93400 q^{50} -11.8163 q^{51} -3.55957 q^{52} -6.00130 q^{53} +2.82751 q^{54} +7.16328 q^{55} -3.58663 q^{56} -1.90492 q^{57} -4.56127 q^{58} +1.97871 q^{59} +5.70576 q^{60} +10.4498 q^{61} -8.85929 q^{62} -6.07964 q^{63} +1.00000 q^{64} +9.37325 q^{65} +5.89442 q^{66} -12.7025 q^{67} +5.45332 q^{68} -18.7390 q^{69} +9.44449 q^{70} -13.1015 q^{71} +1.69508 q^{72} +16.9049 q^{73} +7.07454 q^{74} -4.19063 q^{75} +0.879136 q^{76} +9.75677 q^{77} +7.71294 q^{78} +2.73471 q^{79} -2.63325 q^{80} -11.2119 q^{81} -4.17143 q^{82} +2.17661 q^{83} +7.77156 q^{84} -14.3599 q^{85} +11.3694 q^{86} +9.88343 q^{87} -2.72032 q^{88} -15.5102 q^{89} -4.46358 q^{90} +12.7669 q^{91} +8.64820 q^{92} +19.1964 q^{93} +12.6680 q^{94} -2.31499 q^{95} -2.16681 q^{96} +0.589255 q^{97} +5.86390 q^{98} -4.61117 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9}+O(q^{10}) $ 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9} - 20 q^{10} - 13 q^{11} - 11 q^{12} - 51 q^{13} - 40 q^{14} - 31 q^{15} + 67 q^{16} - 34 q^{17} + 24 q^{18} - 33 q^{19} - 20 q^{20} - 39 q^{21} - 13 q^{22} - 43 q^{23} - 11 q^{24} - 9 q^{25} - 51 q^{26} - 29 q^{27} - 40 q^{28} - 63 q^{29} - 31 q^{30} - 43 q^{31} + 67 q^{32} - 49 q^{33} - 34 q^{34} - 20 q^{35} + 24 q^{36} - 77 q^{37} - 33 q^{38} - 40 q^{39} - 20 q^{40} - 50 q^{41} - 39 q^{42} - 56 q^{43} - 13 q^{44} - 48 q^{45} - 43 q^{46} - 48 q^{47} - 11 q^{48} + q^{49} - 9 q^{50} - 18 q^{51} - 51 q^{52} - 91 q^{53} - 29 q^{54} - 58 q^{55} - 40 q^{56} - 65 q^{57} - 63 q^{58} - 17 q^{59} - 31 q^{60} - 45 q^{61} - 43 q^{62} - 67 q^{63} + 67 q^{64} - 65 q^{65} - 49 q^{66} - 112 q^{67} - 34 q^{68} - 57 q^{69} - 20 q^{70} - 75 q^{71} + 24 q^{72} - 79 q^{73} - 77 q^{74} - 5 q^{75} - 33 q^{76} - 85 q^{77} - 40 q^{78} - 80 q^{79} - 20 q^{80} - 77 q^{81} - 50 q^{82} - 22 q^{83} - 39 q^{84} - 134 q^{85} - 56 q^{86} - 49 q^{87} - 13 q^{88} - 77 q^{89} - 48 q^{90} - 17 q^{91} - 43 q^{92} - 97 q^{93} - 48 q^{94} - 73 q^{95} - 11 q^{96} - 87 q^{97} + q^{98} - 44 q^{99}+O(q^{100}) \) 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9 - 20 * q^10 - 13 * q^11 - 11 * q^12 - 51 * q^13 - 40 * q^14 - 31 * q^15 + 67 * q^16 - 34 * q^17 + 24 * q^18 - 33 * q^19 - 20 * q^20 - 39 * q^21 - 13 * q^22 - 43 * q^23 - 11 * q^24 - 9 * q^25 - 51 * q^26 - 29 * q^27 - 40 * q^28 - 63 * q^29 - 31 * q^30 - 43 * q^31 + 67 * q^32 - 49 * q^33 - 34 * q^34 - 20 * q^35 + 24 * q^36 - 77 * q^37 - 33 * q^38 - 40 * q^39 - 20 * q^40 - 50 * q^41 - 39 * q^42 - 56 * q^43 - 13 * q^44 - 48 * q^45 - 43 * q^46 - 48 * q^47 - 11 * q^48 + q^49 - 9 * q^50 - 18 * q^51 - 51 * q^52 - 91 * q^53 - 29 * q^54 - 58 * q^55 - 40 * q^56 - 65 * q^57 - 63 * q^58 - 17 * q^59 - 31 * q^60 - 45 * q^61 - 43 * q^62 - 67 * q^63 + 67 * q^64 - 65 * q^65 - 49 * q^66 - 112 * q^67 - 34 * q^68 - 57 * q^69 - 20 * q^70 - 75 * q^71 + 24 * q^72 - 79 * q^73 - 77 * q^74 - 5 * q^75 - 33 * q^76 - 85 * q^77 - 40 * q^78 - 80 * q^79 - 20 * q^80 - 77 * q^81 - 50 * q^82 - 22 * q^83 - 39 * q^84 - 134 * q^85 - 56 * q^86 - 49 * q^87 - 13 * q^88 - 77 * q^89 - 48 * q^90 - 17 * q^91 - 43 * q^92 - 97 * q^93 - 48 * q^94 - 73 * q^95 - 11 * q^96 - 87 * q^97 + q^98 - 44 * 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
$3$	−2.16681	−1.25101	−0.625505	−	0.780220i	$-0.715107\pi$
$3$	−2.16681	−1.25101	−0.625505	+	0.780220i	$0.715107\pi$
$4$	1.00000	0.500000
$5$	−2.63325	−1.17763	−0.588813	−	0.808270i	$-0.700404\pi$
$5$	−2.63325	−1.17763	−0.588813	+	0.808270i	$0.700404\pi$
$6$	−2.16681	−0.884598
$7$	−3.58663	−1.35562	−0.677809	−	0.735238i	$-0.737070\pi$
$7$	−3.58663	−1.35562	−0.677809	+	0.735238i	$0.737070\pi$
$8$	1.00000	0.353553
$9$	1.69508	0.565028
$10$	−2.63325	−0.832707
$11$	−2.72032	−0.820207	−0.410103	−	0.912039i	$-0.634507\pi$
$11$	−2.72032	−0.820207	−0.410103	+	0.912039i	$0.634507\pi$
$12$	−2.16681	−0.625505
$13$	−3.55957	−0.987248	−0.493624	−	0.869675i	$-0.664328\pi$
$13$	−3.55957	−0.987248	−0.493624	+	0.869675i	$0.664328\pi$
$14$	−3.58663	−0.958567
$15$	5.70576	1.47322
$16$	1.00000	0.250000
$17$	5.45332	1.32262	0.661312	−	0.750111i	$-0.270000\pi$
$17$	5.45332	1.32262	0.661312	+	0.750111i	$0.270000\pi$
$18$	1.69508	0.399535
$19$	0.879136	0.201688	0.100844	−	0.994902i	$-0.467846\pi$
$19$	0.879136	0.201688	0.100844	+	0.994902i	$0.467846\pi$
$20$	−2.63325	−0.588813
$21$	7.77156	1.69589
$22$	−2.72032	−0.579974
$23$	8.64820	1.80327	0.901637	−	0.432493i	$-0.142366\pi$
$23$	8.64820	1.80327	0.901637	+	0.432493i	$0.142366\pi$
$24$	−2.16681	−0.442299
$25$	1.93400	0.386801
$26$	−3.55957	−0.698090
$27$	2.82751	0.544155
$28$	−3.58663	−0.677809
$29$	−4.56127	−0.847007	−0.423504	−	0.905894i	$-0.639200\pi$
$29$	−4.56127	−0.847007	−0.423504	+	0.905894i	$0.639200\pi$
$30$	5.70576	1.04173
$31$	−8.85929	−1.59118	−0.795588	−	0.605838i	$-0.792838\pi$
$31$	−8.85929	−1.59118	−0.795588	+	0.605838i	$0.792838\pi$
$32$	1.00000	0.176777
$33$	5.89442	1.02609
$34$	5.45332	0.935236
$35$	9.44449	1.59641
$36$	1.69508	0.282514
$37$	7.07454	1.16305	0.581523	−	0.813530i	$-0.302457\pi$
$37$	7.07454	1.16305	0.581523	+	0.813530i	$0.302457\pi$
$38$	0.879136	0.142615
$39$	7.71294	1.23506
$40$	−2.63325	−0.416353
$41$	−4.17143	−0.651467	−0.325734	−	0.945462i	$-0.605611\pi$
$41$	−4.17143	−0.651467	−0.325734	+	0.945462i	$0.605611\pi$
$42$	7.77156	1.19918
$43$	11.3694	1.73382	0.866910	−	0.498465i	$-0.166103\pi$
$43$	11.3694	1.73382	0.866910	+	0.498465i	$0.166103\pi$
$44$	−2.72032	−0.410103
$45$	−4.46358	−0.665391
$46$	8.64820	1.27511
$47$	12.6680	1.84782	0.923910	−	0.382610i	$-0.124975\pi$
$47$	12.6680	1.84782	0.923910	+	0.382610i	$0.124975\pi$
$48$	−2.16681	−0.312753
$49$	5.86390	0.837700
$50$	1.93400	0.273510
$51$	−11.8163	−1.65462
$52$	−3.55957	−0.493624
$53$	−6.00130	−0.824342	−0.412171	−	0.911106i	$-0.635229\pi$
$53$	−6.00130	−0.824342	−0.412171	+	0.911106i	$0.635229\pi$
$54$	2.82751	0.384776
$55$	7.16328	0.965896
$56$	−3.58663	−0.479283
$57$	−1.90492	−0.252313
$58$	−4.56127	−0.598924
$59$	1.97871	0.257606	0.128803	−	0.991670i	$-0.458887\pi$
$59$	1.97871	0.257606	0.128803	+	0.991670i	$0.458887\pi$
$60$	5.70576	0.736611
$61$	10.4498	1.33796	0.668979	−	0.743281i	$-0.266731\pi$
$61$	10.4498	1.33796	0.668979	+	0.743281i	$0.266731\pi$
$62$	−8.85929	−1.12513
$63$	−6.07964	−0.765962
$64$	1.00000	0.125000
$65$	9.37325	1.16261
$66$	5.89442	0.725553
$67$	−12.7025	−1.55185	−0.775927	−	0.630823i	$-0.782717\pi$
$67$	−12.7025	−1.55185	−0.775927	+	0.630823i	$0.782717\pi$
$68$	5.45332	0.661312
$69$	−18.7390	−2.25592
$70$	9.44449	1.12883
$71$	−13.1015	−1.55486	−0.777430	−	0.628970i	$-0.783477\pi$
$71$	−13.1015	−1.55486	−0.777430	+	0.628970i	$0.783477\pi$
$72$	1.69508	0.199768
$73$	16.9049	1.97857	0.989283	−	0.146014i	$-0.0466443\pi$
$73$	16.9049	1.97857	0.989283	+	0.146014i	$0.0466443\pi$
$74$	7.07454	0.822398
$75$	−4.19063	−0.483892
$76$	0.879136	0.100844
$77$	9.75677	1.11189
$78$	7.71294	0.873318
$79$	2.73471	0.307679	0.153840	−	0.988096i	$-0.450836\pi$
$79$	2.73471	0.307679	0.153840	+	0.988096i	$0.450836\pi$
$80$	−2.63325	−0.294406
$81$	−11.2119	−1.24577
$82$	−4.17143	−0.460657
$83$	2.17661	0.238914	0.119457	−	0.992839i	$-0.461885\pi$
$83$	2.17661	0.238914	0.119457	+	0.992839i	$0.461885\pi$
$84$	7.77156	0.847946
$85$	−14.3599	−1.55756
$86$	11.3694	1.22600
$87$	9.88343	1.05961
$88$	−2.72032	−0.289987
$89$	−15.5102	−1.64408	−0.822039	−	0.569431i	$-0.807163\pi$
$89$	−15.5102	−1.64408	−0.822039	+	0.569431i	$0.807163\pi$
$90$	−4.46358	−0.470503
$91$	12.7669	1.33833
$92$	8.64820	0.901637
$93$	19.1964	1.99058
$94$	12.6680	1.30661
$95$	−2.31499	−0.237512
$96$	−2.16681	−0.221150
$97$	0.589255	0.0598298	0.0299149	−	0.999552i	$-0.490476\pi$
$97$	0.589255	0.0598298	0.0299149	+	0.999552i	$0.490476\pi$
$98$	5.86390	0.592344
$99$	−4.61117	−0.463440
$100$	1.93400	0.193400
$101$	8.04875	0.800881	0.400440	−	0.916323i	$-0.368857\pi$
$101$	8.04875	0.800881	0.400440	+	0.916323i	$0.368857\pi$
$102$	−11.8163	−1.16999
$103$	−13.3113	−1.31161	−0.655803	−	0.754932i	$-0.727670\pi$
$103$	−13.3113	−1.31161	−0.655803	+	0.754932i	$0.727670\pi$
$104$	−3.55957	−0.349045
$105$	−20.4645	−1.99713
$106$	−6.00130	−0.582898
$107$	−1.22465	−0.118392	−0.0591958	−	0.998246i	$-0.518854\pi$
$107$	−1.22465	−0.118392	−0.0591958	+	0.998246i	$0.518854\pi$
$108$	2.82751	0.272077
$109$	13.1327	1.25789	0.628945	−	0.777450i	$-0.283487\pi$
$109$	13.1327	1.25789	0.628945	+	0.777450i	$0.283487\pi$
$110$	7.16328	0.682992
$111$	−15.3292	−1.45498
$112$	−3.58663	−0.338905
$113$	4.94615	0.465295	0.232647	−	0.972561i	$-0.425261\pi$
$113$	4.94615	0.465295	0.232647	+	0.972561i	$0.425261\pi$
$114$	−1.90492	−0.178413
$115$	−22.7729	−2.12358
$116$	−4.56127	−0.423504
$117$	−6.03378	−0.557823
$118$	1.97871	0.182155
$119$	−19.5590	−1.79297
$120$	5.70576	0.520863
$121$	−3.59987	−0.327261
$122$	10.4498	0.946080
$123$	9.03870	0.814992
$124$	−8.85929	−0.795588
$125$	8.07353	0.722119
$126$	−6.07964	−0.541617
$127$	−1.61198	−0.143040	−0.0715202	−	0.997439i	$-0.522785\pi$
$127$	−1.61198	−0.143040	−0.0715202	+	0.997439i	$0.522785\pi$
$128$	1.00000	0.0883883
$129$	−24.6354	−2.16903
$130$	9.37325	0.822088
$131$	14.6187	1.27724	0.638622	−	0.769521i	$-0.279505\pi$
$131$	14.6187	1.27724	0.638622	+	0.769521i	$0.279505\pi$
$132$	5.89442	0.513044
$133$	−3.15314	−0.273411
$134$	−12.7025	−1.09733
$135$	−7.44554	−0.640810
$136$	5.45332	0.467618
$137$	−12.8442	−1.09736	−0.548678	−	0.836034i	$-0.684869\pi$
$137$	−12.8442	−1.09736	−0.548678	+	0.836034i	$0.684869\pi$
$138$	−18.7390	−1.59517
$139$	18.5575	1.57403	0.787014	−	0.616935i	$-0.211626\pi$
$139$	18.5575	1.57403	0.787014	+	0.616935i	$0.211626\pi$
$140$	9.44449	0.798205
$141$	−27.4492	−2.31164
$142$	−13.1015	−1.09945
$143$	9.68318	0.809748
$144$	1.69508	0.141257
$145$	12.0110	0.997457
$146$	16.9049	1.39906
$147$	−12.7060	−1.04797
$148$	7.07454	0.581523
$149$	−19.4969	−1.59725	−0.798624	−	0.601830i	$-0.794439\pi$
$149$	−19.4969	−1.59725	−0.798624	+	0.601830i	$0.794439\pi$
$150$	−4.19063	−0.342163
$151$	−12.6917	−1.03284	−0.516419	−	0.856336i	$-0.672735\pi$
$151$	−12.6917	−1.03284	−0.516419	+	0.856336i	$0.672735\pi$
$152$	0.879136	0.0713074
$153$	9.24383	0.747319
$154$	9.75677	0.786223
$155$	23.3287	1.87381
$156$	7.71294	0.617529
$157$	−8.38080	−0.668861	−0.334430	−	0.942420i	$-0.608544\pi$
$157$	−8.38080	−0.668861	−0.334430	+	0.942420i	$0.608544\pi$
$158$	2.73471	0.217562
$159$	13.0037	1.03126
$160$	−2.63325	−0.208177
$161$	−31.0179	−2.44455
$162$	−11.2119	−0.880893
$163$	−16.7291	−1.31033	−0.655163	−	0.755488i	$-0.727400\pi$
$163$	−16.7291	−1.31033	−0.655163	+	0.755488i	$0.727400\pi$
$164$	−4.17143	−0.325734
$165$	−15.5215	−1.20835
$166$	2.17661	0.168938
$167$	−13.3770	−1.03514	−0.517571	−	0.855640i	$-0.673164\pi$
$167$	−13.3770	−1.03514	−0.517571	+	0.855640i	$0.673164\pi$
$168$	7.77156	0.599589
$169$	−0.329429	−0.0253407
$170$	−14.3599	−1.10136
$171$	1.49021	0.113959
$172$	11.3694	0.866910
$173$	14.9931	1.13991	0.569953	−	0.821677i	$-0.306961\pi$
$173$	14.9931	1.13991	0.569953	+	0.821677i	$0.306961\pi$
$174$	9.88343	0.749261
$175$	−6.93656	−0.524354
$176$	−2.72032	−0.205052
$177$	−4.28749	−0.322267
$178$	−15.5102	−1.16254
$179$	12.2441	0.915166	0.457583	−	0.889167i	$-0.348715\pi$
$179$	12.2441	0.915166	0.457583	+	0.889167i	$0.348715\pi$
$180$	−4.46358	−0.332696
$181$	7.19185	0.534566	0.267283	−	0.963618i	$-0.413874\pi$
$181$	7.19185	0.534566	0.267283	+	0.963618i	$0.413874\pi$
$182$	12.7669	0.946343
$183$	−22.6428	−1.67380
$184$	8.64820	0.637554
$185$	−18.6290	−1.36963
$186$	19.1964	1.40755
$187$	−14.8348	−1.08483
$188$	12.6680	0.923910
$189$	−10.1412	−0.737666
$190$	−2.31499	−0.167947
$191$	11.1634	0.807753	0.403876	−	0.914814i	$-0.367663\pi$
$191$	11.1634	0.807753	0.403876	+	0.914814i	$0.367663\pi$
$192$	−2.16681	−0.156376
$193$	−0.336046	−0.0241891	−0.0120946	−	0.999927i	$-0.503850\pi$
$193$	−0.336046	−0.0241891	−0.0120946	+	0.999927i	$0.503850\pi$
$194$	0.589255	0.0423061
$195$	−20.3101	−1.45444
$196$	5.86390	0.418850
$197$	−9.84975	−0.701766	−0.350883	−	0.936419i	$-0.614118\pi$
$197$	−9.84975	−0.701766	−0.350883	+	0.936419i	$0.614118\pi$
$198$	−4.61117	−0.327701
$199$	−7.70799	−0.546405	−0.273203	−	0.961957i	$-0.588083\pi$
$199$	−7.70799	−0.546405	−0.273203	+	0.961957i	$0.588083\pi$
$200$	1.93400	0.136755
$201$	27.5239	1.94139
$202$	8.04875	0.566308
$203$	16.3596	1.14822
$204$	−11.8163	−0.827308
$205$	10.9844	0.767184
$206$	−13.3113	−0.927446
$207$	14.6594	1.01890
$208$	−3.55957	−0.246812
$209$	−2.39153	−0.165426
$210$	−20.4645	−1.41218
$211$	9.70765	0.668302	0.334151	−	0.942520i	$-0.391550\pi$
$211$	9.70765	0.668302	0.334151	+	0.942520i	$0.391550\pi$
$212$	−6.00130	−0.412171
$213$	28.3885	1.94515
$214$	−1.22465	−0.0837156
$215$	−29.9385	−2.04179
$216$	2.82751	0.192388
$217$	31.7750	2.15703
$218$	13.1327	0.889462
$219$	−36.6297	−2.47521
$220$	7.16328	0.482948
$221$	−19.4115	−1.30576
$222$	−15.3292	−1.02883
$223$	21.4091	1.43366	0.716831	−	0.697247i	$-0.245592\pi$
$223$	21.4091	1.43366	0.716831	+	0.697247i	$0.245592\pi$
$224$	−3.58663	−0.239642
$225$	3.27830	0.218553
$226$	4.94615	0.329013
$227$	9.09940	0.603949	0.301974	−	0.953316i	$-0.402354\pi$
$227$	9.09940	0.603949	0.301974	+	0.953316i	$0.402354\pi$
$228$	−1.90492	−0.126157
$229$	18.3060	1.20970	0.604848	−	0.796341i	$-0.293234\pi$
$229$	18.3060	1.20970	0.604848	+	0.796341i	$0.293234\pi$
$230$	−22.7729	−1.50160
$231$	−21.1411	−1.39098
$232$	−4.56127	−0.299462
$233$	−14.2533	−0.933766	−0.466883	−	0.884319i	$-0.654623\pi$
$233$	−14.2533	−0.933766	−0.466883	+	0.884319i	$0.654623\pi$
$234$	−6.03378	−0.394440
$235$	−33.3580	−2.17604
$236$	1.97871	0.128803
$237$	−5.92561	−0.384910
$238$	−19.5590	−1.26782
$239$	13.6487	0.882858	0.441429	−	0.897296i	$-0.354472\pi$
$239$	13.6487	0.882858	0.441429	+	0.897296i	$0.354472\pi$
$240$	5.70576	0.368305
$241$	−11.2223	−0.722890	−0.361445	−	0.932393i	$-0.617716\pi$
$241$	−11.2223	−0.722890	−0.361445	+	0.932393i	$0.617716\pi$
$242$	−3.59987	−0.231408
$243$	15.8117	1.01432
$244$	10.4498	0.668979
$245$	−15.4411	−0.986497
$246$	9.03870	0.576287
$247$	−3.12935	−0.199116
$248$	−8.85929	−0.562566
$249$	−4.71631	−0.298884
$250$	8.07353	0.510615
$251$	−0.508330	−0.0320855	−0.0160427	−	0.999871i	$-0.505107\pi$
$251$	−0.508330	−0.0320855	−0.0160427	+	0.999871i	$0.505107\pi$
$252$	−6.07964	−0.382981
$253$	−23.5259	−1.47906
$254$	−1.61198	−0.101145
$255$	31.1153	1.94852
$256$	1.00000	0.0625000
$257$	−14.1431	−0.882221	−0.441110	−	0.897453i	$-0.645415\pi$
$257$	−14.1431	−0.882221	−0.441110	+	0.897453i	$0.645415\pi$
$258$	−24.6354	−1.53373
$259$	−25.3737	−1.57665
$260$	9.37325	0.581304
$261$	−7.73174	−0.478583
$262$	14.6187	0.903148
$263$	−23.3660	−1.44081	−0.720405	−	0.693553i	$-0.756044\pi$
$263$	−23.3660	−1.44081	−0.720405	+	0.693553i	$0.756044\pi$
$264$	5.89442	0.362777
$265$	15.8029	0.970766
$266$	−3.15314	−0.193331
$267$	33.6077	2.05676
$268$	−12.7025	−0.775927
$269$	−20.8768	−1.27288	−0.636439	−	0.771327i	$-0.719593\pi$
$269$	−20.8768	−1.27288	−0.636439	+	0.771327i	$0.719593\pi$
$270$	−7.44554	−0.453121
$271$	9.58159	0.582040	0.291020	−	0.956717i	$-0.406005\pi$
$271$	9.58159	0.582040	0.291020	+	0.956717i	$0.406005\pi$
$272$	5.45332	0.330656
$273$	−27.6634	−1.67427
$274$	−12.8442	−0.775948
$275$	−5.26111	−0.317257
$276$	−18.7390	−1.12796
$277$	−25.4950	−1.53185	−0.765923	−	0.642933i	$-0.777717\pi$
$277$	−25.4950	−1.53185	−0.765923	+	0.642933i	$0.777717\pi$
$278$	18.5575	1.11301
$279$	−15.0172	−0.899059
$280$	9.44449	0.564416
$281$	−12.9909	−0.774974	−0.387487	−	0.921875i	$-0.626657\pi$
$281$	−12.9909	−0.774974	−0.387487	+	0.921875i	$0.626657\pi$
$282$	−27.4492	−1.63458
$283$	−28.8328	−1.71393	−0.856967	−	0.515371i	$-0.827654\pi$
$283$	−28.8328	−1.71393	−0.856967	+	0.515371i	$0.827654\pi$
$284$	−13.1015	−0.777430
$285$	5.01614	0.297131
$286$	9.68318	0.572578
$287$	14.9614	0.883141
$288$	1.69508	0.0998838
$289$	12.7387	0.749334
$290$	12.0110	0.705308
$291$	−1.27681	−0.0748477
$292$	16.9049	0.989283
$293$	4.93012	0.288021	0.144010	−	0.989576i	$-0.454000\pi$
$293$	4.93012	0.288021	0.144010	+	0.989576i	$0.454000\pi$
$294$	−12.7060	−0.741028
$295$	−5.21043	−0.303363
$296$	7.07454	0.411199
$297$	−7.69173	−0.446319
$298$	−19.4969	−1.12943
$299$	−30.7839	−1.78028
$300$	−4.19063	−0.241946
$301$	−40.7779	−2.35040
$302$	−12.6917	−0.730327
$303$	−17.4402	−1.00191
$304$	0.879136	0.0504219
$305$	−27.5169	−1.57561
$306$	9.24383	0.528435
$307$	17.8898	1.02103	0.510513	−	0.859870i	$-0.329456\pi$
$307$	17.8898	1.02103	0.510513	+	0.859870i	$0.329456\pi$
$308$	9.75677	0.555944
$309$	28.8432	1.64083
$310$	23.3287	1.32498
$311$	−20.9486	−1.18789	−0.593944	−	0.804506i	$-0.702430\pi$
$311$	−20.9486	−1.18789	−0.593944	+	0.804506i	$0.702430\pi$
$312$	7.71294	0.436659
$313$	3.40369	0.192388	0.0961940	−	0.995363i	$-0.469333\pi$
$313$	3.40369	0.192388	0.0961940	+	0.995363i	$0.469333\pi$
$314$	−8.38080	−0.472956
$315$	16.0092	0.902016
$316$	2.73471	0.153840
$317$	5.01900	0.281895	0.140948	−	0.990017i	$-0.454985\pi$
$317$	5.01900	0.281895	0.140948	+	0.990017i	$0.454985\pi$
$318$	13.0037	0.729212
$319$	12.4081	0.694721
$320$	−2.63325	−0.147203
$321$	2.65360	0.148109
$322$	−31.0179	−1.72856
$323$	4.79421	0.266757
$324$	−11.2119	−0.622886
$325$	−6.88423	−0.381869
$326$	−16.7291	−0.926540
$327$	−28.4562	−1.57363
$328$	−4.17143	−0.230328
$329$	−45.4355	−2.50494
$330$	−15.5215	−0.854430
$331$	−1.55646	−0.0855508	−0.0427754	−	0.999085i	$-0.513620\pi$
$331$	−1.55646	−0.0855508	−0.0427754	+	0.999085i	$0.513620\pi$
$332$	2.17661	0.119457
$333$	11.9919	0.657154
$334$	−13.3770	−0.731956
$335$	33.4488	1.82750
$336$	7.77156	0.423973
$337$	20.5517	1.11952	0.559760	−	0.828655i	$-0.310893\pi$
$337$	20.5517	1.11952	0.559760	+	0.828655i	$0.310893\pi$
$338$	−0.329429	−0.0179186
$339$	−10.7174	−0.582089
$340$	−14.3599	−0.778778
$341$	24.1001	1.30509
$342$	1.49021	0.0805813
$343$	4.07476	0.220016
$344$	11.3694	0.612998
$345$	49.3446	2.65662
$346$	14.9931	0.806035
$347$	28.3800	1.52352	0.761759	−	0.647861i	$-0.224336\pi$
$347$	28.3800	1.52352	0.761759	+	0.647861i	$0.224336\pi$
$348$	9.88343	0.529807
$349$	18.7188	1.00199	0.500997	−	0.865449i	$-0.332967\pi$
$349$	18.7188	1.00199	0.500997	+	0.865449i	$0.332967\pi$
$350$	−6.93656	−0.370774
$351$	−10.0647	−0.537216
$352$	−2.72032	−0.144993
$353$	−23.9757	−1.27610	−0.638048	−	0.769996i	$-0.720258\pi$
$353$	−23.9757	−1.27610	−0.638048	+	0.769996i	$0.720258\pi$
$354$	−4.28749	−0.227877
$355$	34.4995	1.83104
$356$	−15.5102	−0.822039
$357$	42.3808	2.24303
$358$	12.2441	0.647120
$359$	9.36182	0.494098	0.247049	−	0.969003i	$-0.420539\pi$
$359$	9.36182	0.494098	0.247049	+	0.969003i	$0.420539\pi$
$360$	−4.46358	−0.235251
$361$	−18.2271	−0.959322
$362$	7.19185	0.377995
$363$	7.80025	0.409407
$364$	12.7669	0.669166
$365$	−44.5147	−2.33001
$366$	−22.6428	−1.18356
$367$	−1.58772	−0.0828782	−0.0414391	−	0.999141i	$-0.513194\pi$
$367$	−1.58772	−0.0828782	−0.0414391	+	0.999141i	$0.513194\pi$
$368$	8.64820	0.450819
$369$	−7.07091	−0.368097
$370$	−18.6290	−0.968477
$371$	21.5244	1.11749
$372$	19.1964	0.995289
$373$	−35.4264	−1.83431	−0.917155	−	0.398532i	$-0.869520\pi$
$373$	−35.4264	−1.83431	−0.917155	+	0.398532i	$0.869520\pi$
$374$	−14.8348	−0.767087
$375$	−17.4938	−0.903378
$376$	12.6680	0.653303
$377$	16.2362	0.836206
$378$	−10.1412	−0.521609
$379$	34.8392	1.78957	0.894786	−	0.446495i	$-0.147328\pi$
$379$	34.8392	1.78957	0.894786	+	0.446495i	$0.147328\pi$
$380$	−2.31499	−0.118756
$381$	3.49287	0.178945
$382$	11.1634	0.571167
$383$	8.15572	0.416738	0.208369	−	0.978050i	$-0.433185\pi$
$383$	8.15572	0.416738	0.208369	+	0.978050i	$0.433185\pi$
$384$	−2.16681	−0.110575
$385$	−25.6920	−1.30939
$386$	−0.336046	−0.0171043
$387$	19.2721	0.979656
$388$	0.589255	0.0299149
$389$	−17.5055	−0.887566	−0.443783	−	0.896134i	$-0.646364\pi$
$389$	−17.5055	−0.887566	−0.443783	+	0.896134i	$0.646364\pi$
$390$	−20.3101	−1.02844
$391$	47.1614	2.38505
$392$	5.86390	0.296172
$393$	−31.6761	−1.59785
$394$	−9.84975	−0.496223
$395$	−7.20118	−0.362331
$396$	−4.61117	−0.231720
$397$	6.73925	0.338233	0.169117	−	0.985596i	$-0.445909\pi$
$397$	6.73925	0.338233	0.169117	+	0.985596i	$0.445909\pi$
$398$	−7.70799	−0.386367
$399$	6.83226	0.342041
$400$	1.93400	0.0967002
$401$	29.9563	1.49595	0.747973	−	0.663729i	$-0.231027\pi$
$401$	29.9563	1.49595	0.747973	+	0.663729i	$0.231027\pi$
$402$	27.5239	1.37277
$403$	31.5353	1.57089
$404$	8.04875	0.400440
$405$	29.5238	1.46705
$406$	16.3596	0.811913
$407$	−19.2450	−0.953939
$408$	−11.8163	−0.584995
$409$	0.596412	0.0294907	0.0147453	−	0.999891i	$-0.495306\pi$
$409$	0.596412	0.0294907	0.0147453	+	0.999891i	$0.495306\pi$
$410$	10.9844	0.542481
$411$	27.8311	1.37280
$412$	−13.3113	−0.655803
$413$	−7.09688	−0.349215
$414$	14.6594	0.720472
$415$	−5.73156	−0.281351
$416$	−3.55957	−0.174523
$417$	−40.2107	−1.96913
$418$	−2.39153	−0.116974
$419$	−21.1500	−1.03324	−0.516622	−	0.856213i	$-0.672811\pi$
$419$	−21.1500	−1.03324	−0.516622	+	0.856213i	$0.672811\pi$
$420$	−20.4645	−0.998563
$421$	−10.2422	−0.499174	−0.249587	−	0.968352i	$-0.580295\pi$
$421$	−10.2422	−0.499174	−0.249587	+	0.968352i	$0.580295\pi$
$422$	9.70765	0.472561
$423$	21.4733	1.04407
$424$	−6.00130	−0.291449
$425$	10.5467	0.511592
$426$	28.3885	1.37543
$427$	−37.4795	−1.81376
$428$	−1.22465	−0.0591958
$429$	−20.9816	−1.01300
$430$	−29.9385	−1.44376
$431$	−19.1207	−0.921010	−0.460505	−	0.887657i	$-0.652332\pi$
$431$	−19.1207	−0.921010	−0.460505	+	0.887657i	$0.652332\pi$
$432$	2.82751	0.136039
$433$	−21.6741	−1.04159	−0.520796	−	0.853681i	$-0.674365\pi$
$433$	−21.6741	−1.04159	−0.520796	+	0.853681i	$0.674365\pi$
$434$	31.7750	1.52525
$435$	−26.0255	−1.24783
$436$	13.1327	0.628945
$437$	7.60295	0.363698
$438$	−36.6297	−1.75024
$439$	12.3347	0.588703	0.294352	−	0.955697i	$-0.404896\pi$
$439$	12.3347	0.588703	0.294352	+	0.955697i	$0.404896\pi$
$440$	7.16328	0.341496
$441$	9.93981	0.473324
$442$	−19.4115	−0.923311
$443$	21.1311	1.00397	0.501984	−	0.864877i	$-0.332604\pi$
$443$	21.1311	1.00397	0.501984	+	0.864877i	$0.332604\pi$
$444$	−15.3292	−0.727492
$445$	40.8422	1.93611
$446$	21.4091	1.01375
$447$	42.2462	1.99818
$448$	−3.58663	−0.169452
$449$	−24.7970	−1.17024	−0.585121	−	0.810946i	$-0.698953\pi$
$449$	−24.7970	−1.17024	−0.585121	+	0.810946i	$0.698953\pi$
$450$	3.27830	0.154541
$451$	11.3476	0.534338
$452$	4.94615	0.232647
$453$	27.5006	1.29209
$454$	9.09940	0.427056
$455$	−33.6184	−1.57605
$456$	−1.90492	−0.0892063
$457$	−2.26738	−0.106064	−0.0530318	−	0.998593i	$-0.516888\pi$
$457$	−2.26738	−0.106064	−0.0530318	+	0.998593i	$0.516888\pi$
$458$	18.3060	0.855384
$459$	15.4193	0.719712
$460$	−22.7729	−1.06179
$461$	14.5354	0.676980	0.338490	−	0.940970i	$-0.390084\pi$
$461$	14.5354	0.676980	0.338490	+	0.940970i	$0.390084\pi$
$462$	−21.1411	−0.983573
$463$	9.21637	0.428321	0.214161	−	0.976798i	$-0.431298\pi$
$463$	9.21637	0.428321	0.214161	+	0.976798i	$0.431298\pi$
$464$	−4.56127	−0.211752
$465$	−50.5490	−2.34416
$466$	−14.2533	−0.660272
$467$	42.0625	1.94642	0.973211	−	0.229915i	$-0.0738448\pi$
$467$	42.0625	1.94642	0.973211	+	0.229915i	$0.0738448\pi$
$468$	−6.03378	−0.278911
$469$	45.5590	2.10372
$470$	−33.3580	−1.53869
$471$	18.1596	0.836752
$472$	1.97871	0.0910773
$473$	−30.9284	−1.42209
$474$	−5.92561	−0.272172
$475$	1.70025	0.0780130
$476$	−19.5590	−0.896486
$477$	−10.1727	−0.465776
$478$	13.6487	0.624275
$479$	10.9050	0.498260	0.249130	−	0.968470i	$-0.419855\pi$
$479$	10.9050	0.498260	0.249130	+	0.968470i	$0.419855\pi$
$480$	5.70576	0.260431
$481$	−25.1823	−1.14822
$482$	−11.2223	−0.511160
$483$	67.2100	3.05816
$484$	−3.59987	−0.163630
$485$	−1.55166	−0.0704571
$486$	15.8117	0.717232
$487$	−20.4181	−0.925233	−0.462617	−	0.886558i	$-0.653089\pi$
$487$	−20.4181	−0.925233	−0.462617	+	0.886558i	$0.653089\pi$
$488$	10.4498	0.473040
$489$	36.2489	1.63923
$490$	−15.4411	−0.697559
$491$	15.7205	0.709455	0.354728	−	0.934970i	$-0.384574\pi$
$491$	15.7205	0.709455	0.354728	+	0.934970i	$0.384574\pi$
$492$	9.03870	0.407496
$493$	−24.8741	−1.12027
$494$	−3.12935	−0.140796
$495$	12.1424	0.545758
$496$	−8.85929	−0.397794
$497$	46.9901	2.10780
$498$	−4.71631	−0.211343
$499$	5.41477	0.242399	0.121199	−	0.992628i	$-0.461326\pi$
$499$	5.41477	0.242399	0.121199	+	0.992628i	$0.461326\pi$
$500$	8.07353	0.361059
$501$	28.9854	1.29497
$502$	−0.508330	−0.0226879
$503$	7.93633	0.353863	0.176932	−	0.984223i	$-0.443383\pi$
$503$	7.93633	0.353863	0.176932	+	0.984223i	$0.443383\pi$
$504$	−6.07964	−0.270808
$505$	−21.1944	−0.943137
$506$	−23.5259	−1.04585
$507$	0.713811	0.0317014
$508$	−1.61198	−0.0715202
$509$	−12.2169	−0.541505	−0.270752	−	0.962649i	$-0.587272\pi$
$509$	−12.2169	−0.541505	−0.270752	+	0.962649i	$0.587272\pi$
$510$	31.1153	1.37781
$511$	−60.6315	−2.68218
$512$	1.00000	0.0441942
$513$	2.48577	0.109749
$514$	−14.1431	−0.623824
$515$	35.0521	1.54458
$516$	−24.6354	−1.08451
$517$	−34.4610	−1.51559
$518$	−25.3737	−1.11486
$519$	−32.4873	−1.42603
$520$	9.37325	0.411044
$521$	18.9290	0.829294	0.414647	−	0.909982i	$-0.363905\pi$
$521$	18.9290	0.829294	0.414647	+	0.909982i	$0.363905\pi$
$522$	−7.73174	−0.338409
$523$	30.5655	1.33653	0.668267	−	0.743921i	$-0.267036\pi$
$523$	30.5655	1.33653	0.668267	+	0.743921i	$0.267036\pi$
$524$	14.6187	0.638622
$525$	15.0302	0.655973
$526$	−23.3660	−1.01881
$527$	−48.3125	−2.10453
$528$	5.89442	0.256522
$529$	51.7914	2.25180
$530$	15.8029	0.686435
$531$	3.35407	0.145554
$532$	−3.15314	−0.136706
$533$	14.8485	0.643160
$534$	33.6077	1.45435
$535$	3.22482	0.139421
$536$	−12.7025	−0.548663
$537$	−26.5307	−1.14488
$538$	−20.8768	−0.900061
$539$	−15.9517	−0.687088
$540$	−7.44554	−0.320405
$541$	15.6498	0.672836	0.336418	−	0.941713i	$-0.390785\pi$
$541$	15.6498	0.672836	0.336418	+	0.941713i	$0.390785\pi$
$542$	9.58159	0.411565
$543$	−15.5834	−0.668748
$544$	5.45332	0.233809
$545$	−34.5818	−1.48132
$546$	−27.6634	−1.18389
$547$	−18.7379	−0.801175	−0.400587	−	0.916259i	$-0.631194\pi$
$547$	−18.7379	−0.801175	−0.400587	+	0.916259i	$0.631194\pi$
$548$	−12.8442	−0.548678
$549$	17.7133	0.755984
$550$	−5.26111	−0.224334
$551$	−4.00998	−0.170831
$552$	−18.7390	−0.797587
$553$	−9.80840	−0.417096
$554$	−25.4950	−1.08318
$555$	40.3656	1.71343
$556$	18.5575	0.787014
$557$	−17.0652	−0.723077	−0.361538	−	0.932357i	$-0.617748\pi$
$557$	−17.0652	−0.723077	−0.361538	+	0.932357i	$0.617748\pi$
$558$	−15.0172	−0.635731
$559$	−40.4703	−1.71171
$560$	9.44449	0.399102
$561$	32.1442	1.35713
$562$	−12.9909	−0.547990
$563$	24.5523	1.03476	0.517378	−	0.855757i	$-0.326908\pi$
$563$	24.5523	1.03476	0.517378	+	0.855757i	$0.326908\pi$
$564$	−27.4492	−1.15582
$565$	−13.0245	−0.547943
$566$	−28.8328	−1.21193
$567$	40.2131	1.68879
$568$	−13.1015	−0.549726
$569$	10.8278	0.453926	0.226963	−	0.973903i	$-0.427120\pi$
$569$	10.8278	0.453926	0.226963	+	0.973903i	$0.427120\pi$
$570$	5.01614	0.210103
$571$	−6.25324	−0.261690	−0.130845	−	0.991403i	$-0.541769\pi$
$571$	−6.25324	−0.261690	−0.130845	+	0.991403i	$0.541769\pi$
$572$	9.68318	0.404874
$573$	−24.1889	−1.01051
$574$	14.9614	0.624475
$575$	16.7257	0.697508
$576$	1.69508	0.0706285
$577$	−35.0404	−1.45875	−0.729376	−	0.684113i	$-0.760190\pi$
$577$	−35.0404	−1.45875	−0.729376	+	0.684113i	$0.760190\pi$
$578$	12.7387	0.529859
$579$	0.728149	0.0302608
$580$	12.0110	0.498728
$581$	−7.80669	−0.323876
$582$	−1.27681	−0.0529253
$583$	16.3254	0.676131
$584$	16.9049	0.699528
$585$	15.8884	0.656906
$586$	4.93012	0.203661
$587$	−7.32743	−0.302435	−0.151218	−	0.988500i	$-0.548319\pi$
$587$	−7.32743	−0.302435	−0.151218	+	0.988500i	$0.548319\pi$
$588$	−12.7060	−0.523986
$589$	−7.78853	−0.320921
$590$	−5.21043	−0.214510
$591$	21.3426	0.877916
$592$	7.07454	0.290762
$593$	−17.6775	−0.725930	−0.362965	−	0.931803i	$-0.618236\pi$
$593$	−17.6775	−0.725930	−0.362965	+	0.931803i	$0.618236\pi$
$594$	−7.69173	−0.315596
$595$	51.5038	2.11145
$596$	−19.4969	−0.798624
$597$	16.7018	0.683559
$598$	−30.7839	−1.25885
$599$	−29.0133	−1.18545	−0.592725	−	0.805405i	$-0.701948\pi$
$599$	−29.0133	−1.18545	−0.592725	+	0.805405i	$0.701948\pi$
$600$	−4.19063	−0.171082
$601$	−25.8170	−1.05310	−0.526548	−	0.850145i	$-0.676514\pi$
$601$	−25.8170	−1.05310	−0.526548	+	0.850145i	$0.676514\pi$
$602$	−40.7779	−1.66198
$603$	−21.5317	−0.876840
$604$	−12.6917	−0.516419
$605$	9.47935	0.385390
$606$	−17.4402	−0.708458
$607$	−10.4459	−0.423984	−0.211992	−	0.977271i	$-0.567995\pi$
$607$	−10.4459	−0.423984	−0.211992	+	0.977271i	$0.567995\pi$
$608$	0.879136	0.0356537
$609$	−35.4482	−1.43643
$610$	−27.5169	−1.11413
$611$	−45.0927	−1.82426
$612$	9.24383	0.373660
$613$	7.85116	0.317105	0.158553	−	0.987351i	$-0.449317\pi$
$613$	7.85116	0.317105	0.158553	+	0.987351i	$0.449317\pi$
$614$	17.8898	0.721974
$615$	−23.8012	−0.959756
$616$	9.75677	0.393111
$617$	−16.4732	−0.663187	−0.331594	−	0.943422i	$-0.607586\pi$
$617$	−16.4732	−0.663187	−0.331594	+	0.943422i	$0.607586\pi$
$618$	28.8432	1.16024
$619$	−25.8124	−1.03749	−0.518744	−	0.854930i	$-0.673600\pi$
$619$	−25.8124	−1.03749	−0.518744	+	0.854930i	$0.673600\pi$
$620$	23.3287	0.936904
$621$	24.4529	0.981261
$622$	−20.9486	−0.839964
$623$	55.6293	2.22874
$624$	7.71294	0.308765
$625$	−30.9296	−1.23719
$626$	3.40369	0.136039
$627$	5.18200	0.206949
$628$	−8.38080	−0.334430
$629$	38.5797	1.53827
$630$	16.0092	0.637822
$631$	7.97799	0.317599	0.158799	−	0.987311i	$-0.449238\pi$
$631$	7.97799	0.317599	0.158799	+	0.987311i	$0.449238\pi$
$632$	2.73471	0.108781
$633$	−21.0347	−0.836053
$634$	5.01900	0.199330
$635$	4.24476	0.168448
$636$	13.0037	0.515630
$637$	−20.8730	−0.827018
$638$	12.4081	0.491242
$639$	−22.2081	−0.878539
$640$	−2.63325	−0.104088
$641$	39.8863	1.57542	0.787708	−	0.616049i	$-0.211268\pi$
$641$	39.8863	1.57542	0.787708	+	0.616049i	$0.211268\pi$
$642$	2.65360	0.104729
$643$	15.7579	0.621431	0.310715	−	0.950503i	$-0.399431\pi$
$643$	15.7579	0.621431	0.310715	+	0.950503i	$0.399431\pi$
$644$	−31.0179	−1.22228
$645$	64.8712	2.55430
$646$	4.79421	0.188626
$647$	−18.6199	−0.732024	−0.366012	−	0.930610i	$-0.619277\pi$
$647$	−18.6199	−0.732024	−0.366012	+	0.930610i	$0.619277\pi$
$648$	−11.2119	−0.440447
$649$	−5.38271	−0.211290
$650$	−6.88423	−0.270022
$651$	−68.8505	−2.69846
$652$	−16.7291	−0.655163
$653$	−8.43123	−0.329939	−0.164970	−	0.986299i	$-0.552753\pi$
$653$	−8.43123	−0.329939	−0.164970	+	0.986299i	$0.552753\pi$
$654$	−28.4562	−1.11273
$655$	−38.4948	−1.50411
$656$	−4.17143	−0.162867
$657$	28.6552	1.11794
$658$	−45.4355	−1.77126
$659$	−11.9357	−0.464950	−0.232475	−	0.972602i	$-0.574682\pi$
$659$	−11.9357	−0.464950	−0.232475	+	0.972602i	$0.574682\pi$
$660$	−15.5215	−0.604173
$661$	−37.2554	−1.44907	−0.724533	−	0.689240i	$-0.757944\pi$
$661$	−37.2554	−1.44907	−0.724533	+	0.689240i	$0.757944\pi$
$662$	−1.55646	−0.0604936
$663$	42.0611	1.63352
$664$	2.17661	0.0844689
$665$	8.30299	0.321976
$666$	11.9919	0.464678
$667$	−39.4468	−1.52739
$668$	−13.3770	−0.517571
$669$	−46.3896	−1.79353
$670$	33.4488	1.29224
$671$	−28.4268	−1.09740
$672$	7.77156	0.299794
$673$	−16.8939	−0.651214	−0.325607	−	0.945505i	$-0.605569\pi$
$673$	−16.8939	−0.651214	−0.325607	+	0.945505i	$0.605569\pi$
$674$	20.5517	0.791621
$675$	5.46842	0.210480
$676$	−0.329429	−0.0126703
$677$	12.6099	0.484636	0.242318	−	0.970197i	$-0.422092\pi$
$677$	12.6099	0.484636	0.242318	+	0.970197i	$0.422092\pi$
$678$	−10.7174	−0.411599
$679$	−2.11344	−0.0811064
$680$	−14.3599	−0.550679
$681$	−19.7167	−0.755546
$682$	24.1001	0.922840
$683$	19.1663	0.733379	0.366690	−	0.930343i	$-0.380491\pi$
$683$	19.1663	0.733379	0.366690	+	0.930343i	$0.380491\pi$
$684$	1.49021	0.0569796
$685$	33.8221	1.29227
$686$	4.07476	0.155575
$687$	−39.6657	−1.51334
$688$	11.3694	0.433455
$689$	21.3621	0.813830
$690$	49.3446	1.87852
$691$	16.3248	0.621023	0.310512	−	0.950570i	$-0.399500\pi$
$691$	16.3248	0.621023	0.310512	+	0.950570i	$0.399500\pi$
$692$	14.9931	0.569953
$693$	16.5385	0.628247
$694$	28.3800	1.07729
$695$	−48.8666	−1.85362
$696$	9.88343	0.374630
$697$	−22.7481	−0.861646
$698$	18.7188	0.708517
$699$	30.8843	1.16815
$700$	−6.93656	−0.262177
$701$	10.4526	0.394789	0.197395	−	0.980324i	$-0.436752\pi$
$701$	10.4526	0.394789	0.197395	+	0.980324i	$0.436752\pi$
$702$	−10.0647	−0.379869
$703$	6.21948	0.234572
$704$	−2.72032	−0.102526
$705$	72.2807	2.72225
$706$	−23.9757	−0.902337
$707$	−28.8679	−1.08569
$708$	−4.28749	−0.161134
$709$	−48.7109	−1.82938	−0.914688	−	0.404160i	$-0.867564\pi$
$709$	−48.7109	−1.82938	−0.914688	+	0.404160i	$0.867564\pi$
$710$	34.4995	1.29474
$711$	4.63557	0.173847
$712$	−15.5102	−0.581269
$713$	−76.6170	−2.86933
$714$	42.3808	1.58606
$715$	−25.4982	−0.953579
$716$	12.2441	0.457583
$717$	−29.5741	−1.10447
$718$	9.36182	0.349380
$719$	−28.5575	−1.06501	−0.532507	−	0.846426i	$-0.678750\pi$
$719$	−28.5575	−1.06501	−0.532507	+	0.846426i	$0.678750\pi$
$720$	−4.46358	−0.166348
$721$	47.7429	1.77804
$722$	−18.2271	−0.678343
$723$	24.3166	0.904343
$724$	7.19185	0.267283
$725$	−8.82152	−0.327623
$726$	7.80025	0.289494
$727$	−23.3547	−0.866176	−0.433088	−	0.901352i	$-0.642576\pi$
$727$	−23.3547	−0.866176	−0.433088	+	0.901352i	$0.642576\pi$
$728$	12.7669	0.473172
$729$	−0.625106	−0.0231521
$730$	−44.5147	−1.64756
$731$	62.0010	2.29319
$732$	−22.6428	−0.836900
$733$	12.0568	0.445327	0.222663	−	0.974895i	$-0.428525\pi$
$733$	12.0568	0.445327	0.222663	+	0.974895i	$0.428525\pi$
$734$	−1.58772	−0.0586037
$735$	33.4580	1.23412
$736$	8.64820	0.318777
$737$	34.5548	1.27284
$738$	−7.07091	−0.260284
$739$	22.1837	0.816039	0.408020	−	0.912973i	$-0.366219\pi$
$739$	22.1837	0.816039	0.408020	+	0.912973i	$0.366219\pi$
$740$	−18.6290	−0.684817
$741$	6.78072	0.249096
$742$	21.5244	0.790187
$743$	−9.10897	−0.334176	−0.167088	−	0.985942i	$-0.553436\pi$
$743$	−9.10897	−0.334176	−0.167088	+	0.985942i	$0.553436\pi$
$744$	19.1964	0.703776
$745$	51.3402	1.88096
$746$	−35.4264	−1.29705
$747$	3.68954	0.134993
$748$	−14.8348	−0.542413
$749$	4.39238	0.160494
$750$	−17.4938	−0.638785
$751$	−28.8867	−1.05409	−0.527046	−	0.849837i	$-0.676700\pi$
$751$	−28.8867	−1.05409	−0.527046	+	0.849837i	$0.676700\pi$
$752$	12.6680	0.461955
$753$	1.10146	0.0401393
$754$	16.2362	0.591287
$755$	33.4205	1.21630
$756$	−10.1412	−0.368833
$757$	3.88043	0.141037	0.0705184	−	0.997510i	$-0.477535\pi$
$757$	3.88043	0.141037	0.0705184	+	0.997510i	$0.477535\pi$
$758$	34.8392	1.26542
$759$	50.9762	1.85032
$760$	−2.31499	−0.0839733
$761$	−27.8279	−1.00876	−0.504381	−	0.863481i	$-0.668279\pi$
$761$	−27.8279	−1.00876	−0.504381	+	0.863481i	$0.668279\pi$
$762$	3.49287	0.126533
$763$	−47.1023	−1.70522
$764$	11.1634	0.403876
$765$	−24.3413	−0.880062
$766$	8.15572	0.294678
$767$	−7.04335	−0.254321
$768$	−2.16681	−0.0781882
$769$	−16.5287	−0.596039	−0.298020	−	0.954560i	$-0.596326\pi$
$769$	−16.5287	−0.596039	−0.298020	+	0.954560i	$0.596326\pi$
$770$	−25.6920	−0.925876
$771$	30.6454	1.10367
$772$	−0.336046	−0.0120946
$773$	−39.3899	−1.41675	−0.708377	−	0.705834i	$-0.750572\pi$
$773$	−39.3899	−1.41675	−0.708377	+	0.705834i	$0.750572\pi$
$774$	19.2721	0.692722
$775$	−17.1339	−0.615468
$776$	0.589255	0.0211530
$777$	54.9802	1.97240
$778$	−17.5055	−0.627604
$779$	−3.66725	−0.131393
$780$	−20.3101	−0.727218
$781$	35.6402	1.27531
$782$	47.1614	1.68649
$783$	−12.8970	−0.460903
$784$	5.86390	0.209425
$785$	22.0687	0.787667
$786$	−31.6761	−1.12985
$787$	12.1422	0.432823	0.216411	−	0.976302i	$-0.430565\pi$
$787$	12.1422	0.432823	0.216411	+	0.976302i	$0.430565\pi$
$788$	−9.84975	−0.350883
$789$	50.6298	1.80247
$790$	−7.20118	−0.256207
$791$	−17.7400	−0.630762
$792$	−4.61117	−0.163851
$793$	−37.1968	−1.32090
$794$	6.73925	0.239167
$795$	−34.2420	−1.21444
$796$	−7.70799	−0.273203
$797$	−17.0440	−0.603729	−0.301864	−	0.953351i	$-0.597609\pi$
$797$	−17.0440	−0.603729	−0.301864	+	0.953351i	$0.597609\pi$
$798$	6.83226	0.241859
$799$	69.0827	2.44397
$800$	1.93400	0.0683774
$801$	−26.2911	−0.928950
$802$	29.9563	1.05779
$803$	−45.9866	−1.62283
$804$	27.5239	0.970693
$805$	81.6778	2.87877
$806$	31.5353	1.11078
$807$	45.2361	1.59238
$808$	8.04875	0.283154
$809$	30.6461	1.07746	0.538729	−	0.842479i	$-0.318905\pi$
$809$	30.6461	1.07746	0.538729	+	0.842479i	$0.318905\pi$
$810$	29.5238	1.03736
$811$	−8.13804	−0.285765	−0.142883	−	0.989740i	$-0.545637\pi$
$811$	−8.13804	−0.285765	−0.142883	+	0.989740i	$0.545637\pi$
$812$	16.3596	0.574109
$813$	−20.7615	−0.728139
$814$	−19.2450	−0.674537
$815$	44.0519	1.54307
$816$	−11.8163	−0.413654
$817$	9.99526	0.349690
$818$	0.596412	0.0208530
$819$	21.6409	0.756195
$820$	10.9844	0.383592
$821$	32.9040	1.14836	0.574178	−	0.818730i	$-0.305322\pi$
$821$	32.9040	1.14836	0.574178	+	0.818730i	$0.305322\pi$
$822$	27.8311	0.970720
$823$	−20.7529	−0.723401	−0.361701	−	0.932294i	$-0.617804\pi$
$823$	−20.7529	−0.723401	−0.361701	+	0.932294i	$0.617804\pi$
$824$	−13.3113	−0.463723
$825$	11.3998	0.396892
$826$	−7.09688	−0.246932
$827$	−29.5029	−1.02592	−0.512958	−	0.858414i	$-0.671450\pi$
$827$	−29.5029	−1.02592	−0.512958	+	0.858414i	$0.671450\pi$
$828$	14.6594	0.509450
$829$	−49.4402	−1.71713	−0.858564	−	0.512706i	$-0.828643\pi$
$829$	−49.4402	−1.71713	−0.858564	+	0.512706i	$0.828643\pi$
$830$	−5.73156	−0.198945
$831$	55.2429	1.91636
$832$	−3.55957	−0.123406
$833$	31.9777	1.10796
$834$	−40.2107	−1.39238
$835$	35.2249	1.21901
$836$	−2.39153	−0.0827128
$837$	−25.0498	−0.865846
$838$	−21.1500	−0.730614
$839$	−16.6412	−0.574517	−0.287258	−	0.957853i	$-0.592744\pi$
$839$	−16.6412	−0.574517	−0.287258	+	0.957853i	$0.592744\pi$
$840$	−20.4645	−0.706091
$841$	−8.19479	−0.282579
$842$	−10.2422	−0.352969
$843$	28.1490	0.969501
$844$	9.70765	0.334151
$845$	0.867468	0.0298418
$846$	21.4733	0.738269
$847$	12.9114	0.443641
$848$	−6.00130	−0.206086
$849$	62.4754	2.14415
$850$	10.5467	0.361750
$851$	61.1820	2.09729
$852$	28.3885	0.972573
$853$	48.5817	1.66341	0.831704	−	0.555220i	$-0.187366\pi$
$853$	48.5817	1.66341	0.831704	+	0.555220i	$0.187366\pi$
$854$	−37.4795	−1.28252
$855$	−3.92409	−0.134201
$856$	−1.22465	−0.0418578
$857$	23.1136	0.789545	0.394772	−	0.918779i	$-0.370823\pi$
$857$	23.1136	0.789545	0.394772	+	0.918779i	$0.370823\pi$
$858$	−20.9816	−0.716301
$859$	−55.2570	−1.88534	−0.942672	−	0.333720i	$-0.891696\pi$
$859$	−55.2570	−1.88534	−0.942672	+	0.333720i	$0.891696\pi$
$860$	−29.9385	−1.02089
$861$	−32.4185	−1.10482
$862$	−19.1207	−0.651253
$863$	−49.1212	−1.67211	−0.836053	−	0.548649i	$-0.815142\pi$
$863$	−49.1212	−1.67211	−0.836053	+	0.548649i	$0.815142\pi$
$864$	2.82751	0.0961939
$865$	−39.4806	−1.34238
$866$	−21.6741	−0.736517
$867$	−27.6023	−0.937425
$868$	31.7750	1.07851
$869$	−7.43929	−0.252361
$870$	−26.0255	−0.882348
$871$	45.2154	1.53206
$872$	13.1327	0.444731
$873$	0.998837	0.0338055
$874$	7.60295	0.257174
$875$	−28.9568	−0.978917
$876$	−36.6297	−1.23760
$877$	−54.6943	−1.84689	−0.923447	−	0.383726i	$-0.874641\pi$
$877$	−54.6943	−1.84689	−0.923447	+	0.383726i	$0.874641\pi$
$878$	12.3347	0.416276
$879$	−10.6827	−0.360317
$880$	7.16328	0.241474
$881$	−11.4378	−0.385350	−0.192675	−	0.981263i	$-0.561716\pi$
$881$	−11.4378	−0.385350	−0.192675	+	0.981263i	$0.561716\pi$
$882$	9.93981	0.334691
$883$	−5.37084	−0.180743	−0.0903715	−	0.995908i	$-0.528805\pi$
$883$	−5.37084	−0.180743	−0.0903715	+	0.995908i	$0.528805\pi$
$884$	−19.4115	−0.652879
$885$	11.2900	0.379510
$886$	21.1311	0.709912
$887$	38.9985	1.30944	0.654721	−	0.755871i	$-0.272786\pi$
$887$	38.9985	1.30944	0.654721	+	0.755871i	$0.272786\pi$
$888$	−15.3292	−0.514415
$889$	5.78159	0.193908
$890$	40.8422	1.36903
$891$	30.5001	1.02179
$892$	21.4091	0.716831
$893$	11.1369	0.372683
$894$	42.2462	1.41292
$895$	−32.2417	−1.07772
$896$	−3.58663	−0.119821
$897$	66.7030	2.22715
$898$	−24.7970	−0.827485
$899$	40.4097	1.34774
$900$	3.27830	0.109277
$901$	−32.7270	−1.09029
$902$	11.3476	0.377834
$903$	88.3581	2.94037
$904$	4.94615	0.164507
$905$	−18.9379	−0.629519
$906$	27.5006	0.913647
$907$	−14.1298	−0.469172	−0.234586	−	0.972095i	$-0.575373\pi$
$907$	−14.1298	−0.469172	−0.234586	+	0.972095i	$0.575373\pi$
$908$	9.09940	0.301974
$909$	13.6433	0.452520
$910$	−33.6184	−1.11444
$911$	−3.68240	−0.122003	−0.0610017	−	0.998138i	$-0.519430\pi$
$911$	−3.68240	−0.122003	−0.0610017	+	0.998138i	$0.519430\pi$
$912$	−1.90492	−0.0630784
$913$	−5.92107	−0.195959
$914$	−2.26738	−0.0749984
$915$	59.6240	1.97111
$916$	18.3060	0.604848
$917$	−52.4319	−1.73146
$918$	15.4193	0.508913
$919$	−33.3049	−1.09863	−0.549313	−	0.835617i	$-0.685111\pi$
$919$	−33.3049	−1.09863	−0.549313	+	0.835617i	$0.685111\pi$
$920$	−22.7729	−0.750800
$921$	−38.7639	−1.27731
$922$	14.5354	0.478697
$923$	46.6357	1.53503
$924$	−21.1411	−0.695491
$925$	13.6822	0.449868
$926$	9.21637	0.302869
$927$	−22.5638	−0.741094
$928$	−4.56127	−0.149731
$929$	−41.9157	−1.37521	−0.687605	−	0.726085i	$-0.741338\pi$
$929$	−41.9157	−1.37521	−0.687605	+	0.726085i	$0.741338\pi$
$930$	−50.5490	−1.65757
$931$	5.15517	0.168954
$932$	−14.2533	−0.466883
$933$	45.3918	1.48606
$934$	42.0625	1.37633
$935$	39.0636	1.27752
$936$	−6.03378	−0.197220
$937$	−20.6823	−0.675662	−0.337831	−	0.941207i	$-0.609693\pi$
$937$	−20.6823	−0.675662	−0.337831	+	0.941207i	$0.609693\pi$
$938$	45.5590	1.48755
$939$	−7.37517	−0.240680
$940$	−33.3580	−1.08802
$941$	−41.4672	−1.35179	−0.675896	−	0.736997i	$-0.736243\pi$
$941$	−41.4672	−1.35179	−0.675896	+	0.736997i	$0.736243\pi$
$942$	18.1596	0.591673
$943$	−36.0753	−1.17477
$944$	1.97871	0.0644014
$945$	26.7044	0.868694
$946$	−30.9284	−1.00557
$947$	−0.553610	−0.0179899	−0.00899495	−	0.999960i	$-0.502863\pi$
$947$	−0.553610	−0.0179899	−0.00899495	+	0.999960i	$0.502863\pi$
$948$	−5.92561	−0.192455
$949$	−60.1741	−1.95334
$950$	1.70025	0.0551635
$951$	−10.8752	−0.352654
$952$	−19.5590	−0.633912
$953$	47.4110	1.53579	0.767897	−	0.640573i	$-0.221303\pi$
$953$	47.4110	1.53579	0.767897	+	0.640573i	$0.221303\pi$
$954$	−10.1727	−0.329354
$955$	−29.3959	−0.951230
$956$	13.6487	0.441429
$957$	−26.8861	−0.869103
$958$	10.9050	0.352323
$959$	46.0675	1.48760
$960$	5.70576	0.184153
$961$	47.4871	1.53184
$962$	−25.1823	−0.811911
$963$	−2.07589	−0.0668946
$964$	−11.2223	−0.361445
$965$	0.884893	0.0284857
$966$	67.2100	2.16245
$967$	40.1457	1.29100	0.645500	−	0.763760i	$-0.276649\pi$
$967$	40.1457	1.29100	0.645500	+	0.763760i	$0.276649\pi$
$968$	−3.59987	−0.115704
$969$	−10.3882	−0.333716
$970$	−1.55166	−0.0498207
$971$	33.9983	1.09106	0.545528	−	0.838092i	$-0.316329\pi$
$971$	33.9983	1.09106	0.545528	+	0.838092i	$0.316329\pi$
$972$	15.8117	0.507159
$973$	−66.5589	−2.13378
$974$	−20.4181	−0.654239
$975$	14.9169	0.477722
$976$	10.4498	0.334490
$977$	6.00161	0.192008	0.0960042	−	0.995381i	$-0.469394\pi$
$977$	6.00161	0.192008	0.0960042	+	0.995381i	$0.469394\pi$
$978$	36.2489	1.15911
$979$	42.1927	1.34848
$980$	−15.4411	−0.493249
$981$	22.2611	0.710742
$982$	15.7205	0.501661
$983$	36.9547	1.17867	0.589336	−	0.807888i	$-0.299390\pi$
$983$	36.9547	1.17867	0.589336	+	0.807888i	$0.299390\pi$
$984$	9.03870	0.288143
$985$	25.9369	0.826417
$986$	−24.8741	−0.792152
$987$	98.4502	3.13370
$988$	−3.12935	−0.0995579
$989$	98.3250	3.12655
$990$	12.1424	0.385909
$991$	−0.448183	−0.0142370	−0.00711850	−	0.999975i	$-0.502266\pi$
$991$	−0.448183	−0.0142370	−0.00711850	+	0.999975i	$0.502266\pi$
$992$	−8.85929	−0.281283
$993$	3.37256	0.107025
$994$	46.9901	1.49044
$995$	20.2971	0.643460
$996$	−4.71631	−0.149442
$997$	−5.43259	−0.172052	−0.0860259	−	0.996293i	$-0.527417\pi$
$997$	−5.43259	−0.172052	−0.0860259	+	0.996293i	$0.527417\pi$
$998$	5.41477	0.171402
$999$	20.0033	0.632878

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8042.2.a.a.1.13	✓	67

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8042.2.a.a.1.13	✓	67	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:	\( N \)	\(=\)	\( 8042 = 2 \cdot 4021 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8042.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.16681 q^{3} +1.00000 q^{4} -2.63325 q^{5} -2.16681 q^{6} -3.58663 q^{7} +1.00000 q^{8} +1.69508 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.16681 q^{3} +1.00000 q^{4} -2.63325 q^{5} -2.16681 q^{6} -3.58663 q^{7} +1.00000 q^{8} +1.69508 q^{9} -2.63325 q^{10} -2.72032 q^{11} -2.16681 q^{12} -3.55957 q^{13} -3.58663 q^{14} +5.70576 q^{15} +1.00000 q^{16} +5.45332 q^{17} +1.69508 q^{18} +0.879136 q^{19} -2.63325 q^{20} +7.77156 q^{21} -2.72032 q^{22} +8.64820 q^{23} -2.16681 q^{24} +1.93400 q^{25} -3.55957 q^{26} +2.82751 q^{27} -3.58663 q^{28} -4.56127 q^{29} +5.70576 q^{30} -8.85929 q^{31} +1.00000 q^{32} +5.89442 q^{33} +5.45332 q^{34} +9.44449 q^{35} +1.69508 q^{36} +7.07454 q^{37} +0.879136 q^{38} +7.71294 q^{39} -2.63325 q^{40} -4.17143 q^{41} +7.77156 q^{42} +11.3694 q^{43} -2.72032 q^{44} -4.46358 q^{45} +8.64820 q^{46} +12.6680 q^{47} -2.16681 q^{48} +5.86390 q^{49} +1.93400 q^{50} -11.8163 q^{51} -3.55957 q^{52} -6.00130 q^{53} +2.82751 q^{54} +7.16328 q^{55} -3.58663 q^{56} -1.90492 q^{57} -4.56127 q^{58} +1.97871 q^{59} +5.70576 q^{60} +10.4498 q^{61} -8.85929 q^{62} -6.07964 q^{63} +1.00000 q^{64} +9.37325 q^{65} +5.89442 q^{66} -12.7025 q^{67} +5.45332 q^{68} -18.7390 q^{69} +9.44449 q^{70} -13.1015 q^{71} +1.69508 q^{72} +16.9049 q^{73} +7.07454 q^{74} -4.19063 q^{75} +0.879136 q^{76} +9.75677 q^{77} +7.71294 q^{78} +2.73471 q^{79} -2.63325 q^{80} -11.2119 q^{81} -4.17143 q^{82} +2.17661 q^{83} +7.77156 q^{84} -14.3599 q^{85} +11.3694 q^{86} +9.88343 q^{87} -2.72032 q^{88} -15.5102 q^{89} -4.46358 q^{90} +12.7669 q^{91} +8.64820 q^{92} +19.1964 q^{93} +12.6680 q^{94} -2.31499 q^{95} -2.16681 q^{96} +0.589255 q^{97} +5.86390 q^{98} -4.61117 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9}+O(q^{10}) \) 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9} - 20 q^{10} - 13 q^{11} - 11 q^{12} - 51 q^{13} - 40 q^{14} - 31 q^{15} + 67 q^{16} - 34 q^{17} + 24 q^{18} - 33 q^{19} - 20 q^{20} - 39 q^{21} - 13 q^{22} - 43 q^{23} - 11 q^{24} - 9 q^{25} - 51 q^{26} - 29 q^{27} - 40 q^{28} - 63 q^{29} - 31 q^{30} - 43 q^{31} + 67 q^{32} - 49 q^{33} - 34 q^{34} - 20 q^{35} + 24 q^{36} - 77 q^{37} - 33 q^{38} - 40 q^{39} - 20 q^{40} - 50 q^{41} - 39 q^{42} - 56 q^{43} - 13 q^{44} - 48 q^{45} - 43 q^{46} - 48 q^{47} - 11 q^{48} + q^{49} - 9 q^{50} - 18 q^{51} - 51 q^{52} - 91 q^{53} - 29 q^{54} - 58 q^{55} - 40 q^{56} - 65 q^{57} - 63 q^{58} - 17 q^{59} - 31 q^{60} - 45 q^{61} - 43 q^{62} - 67 q^{63} + 67 q^{64} - 65 q^{65} - 49 q^{66} - 112 q^{67} - 34 q^{68} - 57 q^{69} - 20 q^{70} - 75 q^{71} + 24 q^{72} - 79 q^{73} - 77 q^{74} - 5 q^{75} - 33 q^{76} - 85 q^{77} - 40 q^{78} - 80 q^{79} - 20 q^{80} - 77 q^{81} - 50 q^{82} - 22 q^{83} - 39 q^{84} - 134 q^{85} - 56 q^{86} - 49 q^{87} - 13 q^{88} - 77 q^{89} - 48 q^{90} - 17 q^{91} - 43 q^{92} - 97 q^{93} - 48 q^{94} - 73 q^{95} - 11 q^{96} - 87 q^{97} + q^{98} - 44 q^{99}+O(q^{100}) \) 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9 - 20 * q^10 - 13 * q^11 - 11 * q^12 - 51 * q^13 - 40 * q^14 - 31 * q^15 + 67 * q^16 - 34 * q^17 + 24 * q^18 - 33 * q^19 - 20 * q^20 - 39 * q^21 - 13 * q^22 - 43 * q^23 - 11 * q^24 - 9 * q^25 - 51 * q^26 - 29 * q^27 - 40 * q^28 - 63 * q^29 - 31 * q^30 - 43 * q^31 + 67 * q^32 - 49 * q^33 - 34 * q^34 - 20 * q^35 + 24 * q^36 - 77 * q^37 - 33 * q^38 - 40 * q^39 - 20 * q^40 - 50 * q^41 - 39 * q^42 - 56 * q^43 - 13 * q^44 - 48 * q^45 - 43 * q^46 - 48 * q^47 - 11 * q^48 + q^49 - 9 * q^50 - 18 * q^51 - 51 * q^52 - 91 * q^53 - 29 * q^54 - 58 * q^55 - 40 * q^56 - 65 * q^57 - 63 * q^58 - 17 * q^59 - 31 * q^60 - 45 * q^61 - 43 * q^62 - 67 * q^63 + 67 * q^64 - 65 * q^65 - 49 * q^66 - 112 * q^67 - 34 * q^68 - 57 * q^69 - 20 * q^70 - 75 * q^71 + 24 * q^72 - 79 * q^73 - 77 * q^74 - 5 * q^75 - 33 * q^76 - 85 * q^77 - 40 * q^78 - 80 * q^79 - 20 * q^80 - 77 * q^81 - 50 * q^82 - 22 * q^83 - 39 * q^84 - 134 * q^85 - 56 * q^86 - 49 * q^87 - 13 * q^88 - 77 * q^89 - 48 * q^90 - 17 * q^91 - 43 * q^92 - 97 * q^93 - 48 * q^94 - 73 * q^95 - 11 * q^96 - 87 * q^97 + q^98 - 44 * q^99

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