LMFDB - Embedded newform 8042.2.a.a.1.16

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			1.16
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)

$f(q)$	$=$	$q+1.00000 q^{2} -1.98933 q^{3} +1.00000 q^{4} +4.18312 q^{5} -1.98933 q^{6} -0.474573 q^{7} +1.00000 q^{8} +0.957423 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.98933 q^{3} +1.00000 q^{4} +4.18312 q^{5} -1.98933 q^{6} -0.474573 q^{7} +1.00000 q^{8} +0.957423 q^{9} +4.18312 q^{10} -1.64498 q^{11} -1.98933 q^{12} -2.31472 q^{13} -0.474573 q^{14} -8.32159 q^{15} +1.00000 q^{16} -2.88362 q^{17} +0.957423 q^{18} -6.87657 q^{19} +4.18312 q^{20} +0.944080 q^{21} -1.64498 q^{22} +3.42193 q^{23} -1.98933 q^{24} +12.4985 q^{25} -2.31472 q^{26} +4.06335 q^{27} -0.474573 q^{28} -9.60384 q^{29} -8.32159 q^{30} +4.42024 q^{31} +1.00000 q^{32} +3.27241 q^{33} -2.88362 q^{34} -1.98519 q^{35} +0.957423 q^{36} -1.78331 q^{37} -6.87657 q^{38} +4.60474 q^{39} +4.18312 q^{40} +1.06121 q^{41} +0.944080 q^{42} +7.16539 q^{43} -1.64498 q^{44} +4.00501 q^{45} +3.42193 q^{46} +10.3673 q^{47} -1.98933 q^{48} -6.77478 q^{49} +12.4985 q^{50} +5.73647 q^{51} -2.31472 q^{52} -8.42623 q^{53} +4.06335 q^{54} -6.88116 q^{55} -0.474573 q^{56} +13.6797 q^{57} -9.60384 q^{58} -10.5518 q^{59} -8.32159 q^{60} -0.448317 q^{61} +4.42024 q^{62} -0.454366 q^{63} +1.00000 q^{64} -9.68275 q^{65} +3.27241 q^{66} +3.74413 q^{67} -2.88362 q^{68} -6.80734 q^{69} -1.98519 q^{70} +5.11142 q^{71} +0.957423 q^{72} -7.11511 q^{73} -1.78331 q^{74} -24.8635 q^{75} -6.87657 q^{76} +0.780664 q^{77} +4.60474 q^{78} -3.62629 q^{79} +4.18312 q^{80} -10.9556 q^{81} +1.06121 q^{82} +1.55571 q^{83} +0.944080 q^{84} -12.0625 q^{85} +7.16539 q^{86} +19.1052 q^{87} -1.64498 q^{88} -5.45327 q^{89} +4.00501 q^{90} +1.09850 q^{91} +3.42193 q^{92} -8.79331 q^{93} +10.3673 q^{94} -28.7655 q^{95} -1.98933 q^{96} -17.9809 q^{97} -6.77478 q^{98} -1.57494 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.98933	−1.14854	−0.574269	−	0.818666i	$-0.694714\pi$
$3$	−1.98933	−1.14854	−0.574269	+	0.818666i	$0.694714\pi$
$4$	1.00000	0.500000
$5$	4.18312	1.87075	0.935373	−	0.353662i	$-0.115064\pi$
$5$	4.18312	1.87075	0.935373	+	0.353662i	$0.115064\pi$
$6$	−1.98933	−0.812139
$7$	−0.474573	−0.179372	−0.0896858	−	0.995970i	$-0.528586\pi$
$7$	−0.474573	−0.179372	−0.0896858	+	0.995970i	$0.528586\pi$
$8$	1.00000	0.353553
$9$	0.957423	0.319141
$10$	4.18312	1.32282
$11$	−1.64498	−0.495981	−0.247991	−	0.968762i	$-0.579770\pi$
$11$	−1.64498	−0.495981	−0.247991	+	0.968762i	$0.579770\pi$
$12$	−1.98933	−0.574269
$13$	−2.31472	−0.641988	−0.320994	−	0.947081i	$-0.604017\pi$
$13$	−2.31472	−0.641988	−0.320994	+	0.947081i	$0.604017\pi$
$14$	−0.474573	−0.126835
$15$	−8.32159	−2.14862
$16$	1.00000	0.250000
$17$	−2.88362	−0.699382	−0.349691	−	0.936865i	$-0.613713\pi$
$17$	−2.88362	−0.699382	−0.349691	+	0.936865i	$0.613713\pi$
$18$	0.957423	0.225667
$19$	−6.87657	−1.57759	−0.788796	−	0.614655i	$-0.789295\pi$
$19$	−6.87657	−1.57759	−0.788796	+	0.614655i	$0.789295\pi$
$20$	4.18312	0.935373
$21$	0.944080	0.206015
$22$	−1.64498	−0.350712
$23$	3.42193	0.713522	0.356761	−	0.934196i	$-0.383881\pi$
$23$	3.42193	0.713522	0.356761	+	0.934196i	$0.383881\pi$
$24$	−1.98933	−0.406070
$25$	12.4985	2.49969
$26$	−2.31472	−0.453954
$27$	4.06335	0.781993
$28$	−0.474573	−0.0896858
$29$	−9.60384	−1.78339	−0.891694	−	0.452638i	$-0.850483\pi$
$29$	−9.60384	−1.78339	−0.891694	+	0.452638i	$0.850483\pi$
$30$	−8.32159	−1.51931
$31$	4.42024	0.793899	0.396949	−	0.917840i	$-0.370069\pi$
$31$	4.42024	0.793899	0.396949	+	0.917840i	$0.370069\pi$
$32$	1.00000	0.176777
$33$	3.27241	0.569654
$34$	−2.88362	−0.494538
$35$	−1.98519	−0.335559
$36$	0.957423	0.159570
$37$	−1.78331	−0.293174	−0.146587	−	0.989198i	$-0.546829\pi$
$37$	−1.78331	−0.293174	−0.146587	+	0.989198i	$0.546829\pi$
$38$	−6.87657	−1.11553
$39$	4.60474	0.737348
$40$	4.18312	0.661409
$41$	1.06121	0.165733	0.0828664	−	0.996561i	$-0.473593\pi$
$41$	1.06121	0.165733	0.0828664	+	0.996561i	$0.473593\pi$
$42$	0.944080	0.145675
$43$	7.16539	1.09271	0.546356	−	0.837553i	$-0.316015\pi$
$43$	7.16539	1.09271	0.546356	+	0.837553i	$0.316015\pi$
$44$	−1.64498	−0.247991
$45$	4.00501	0.597032
$46$	3.42193	0.504536
$47$	10.3673	1.51223	0.756113	−	0.654441i	$-0.227096\pi$
$47$	10.3673	1.51223	0.756113	+	0.654441i	$0.227096\pi$
$48$	−1.98933	−0.287135
$49$	−6.77478	−0.967826
$50$	12.4985	1.76755
$51$	5.73647	0.803267
$52$	−2.31472	−0.320994
$53$	−8.42623	−1.15743	−0.578716	−	0.815529i	$-0.696446\pi$
$53$	−8.42623	−1.15743	−0.578716	+	0.815529i	$0.696446\pi$
$54$	4.06335	0.552953
$55$	−6.88116	−0.927856
$56$	−0.474573	−0.0634174
$57$	13.6797	1.81193
$58$	−9.60384	−1.26105
$59$	−10.5518	−1.37373	−0.686866	−	0.726784i	$-0.741014\pi$
$59$	−10.5518	−1.37373	−0.686866	+	0.726784i	$0.741014\pi$
$60$	−8.32159	−1.07431
$61$	−0.448317	−0.0574012	−0.0287006	−	0.999588i	$-0.509137\pi$
$61$	−0.448317	−0.0574012	−0.0287006	+	0.999588i	$0.509137\pi$
$62$	4.42024	0.561371
$63$	−0.454366	−0.0572448
$64$	1.00000	0.125000
$65$	−9.68275	−1.20100
$66$	3.27241	0.402806
$67$	3.74413	0.457418	0.228709	−	0.973495i	$-0.426550\pi$
$67$	3.74413	0.457418	0.228709	+	0.973495i	$0.426550\pi$
$68$	−2.88362	−0.349691
$69$	−6.80734	−0.819507
$70$	−1.98519	−0.237276
$71$	5.11142	0.606614	0.303307	−	0.952893i	$-0.401909\pi$
$71$	5.11142	0.606614	0.303307	+	0.952893i	$0.401909\pi$
$72$	0.957423	0.112833
$73$	−7.11511	−0.832761	−0.416380	−	0.909191i	$-0.636702\pi$
$73$	−7.11511	−0.832761	−0.416380	+	0.909191i	$0.636702\pi$
$74$	−1.78331	−0.207305
$75$	−24.8635	−2.87099
$76$	−6.87657	−0.788796
$77$	0.780664	0.0889650
$78$	4.60474	0.521384
$79$	−3.62629	−0.407990	−0.203995	−	0.978972i	$-0.565393\pi$
$79$	−3.62629	−0.407990	−0.203995	+	0.978972i	$0.565393\pi$
$80$	4.18312	0.467687
$81$	−10.9556	−1.21729
$82$	1.06121	0.117191
$83$	1.55571	0.170762	0.0853808	−	0.996348i	$-0.472789\pi$
$83$	1.55571	0.170762	0.0853808	+	0.996348i	$0.472789\pi$
$84$	0.944080	0.103008
$85$	−12.0625	−1.30837
$86$	7.16539	0.772664
$87$	19.1052	2.04829
$88$	−1.64498	−0.175356
$89$	−5.45327	−0.578046	−0.289023	−	0.957322i	$-0.593330\pi$
$89$	−5.45327	−0.578046	−0.289023	+	0.957322i	$0.593330\pi$
$90$	4.00501	0.422165
$91$	1.09850	0.115154
$92$	3.42193	0.356761
$93$	−8.79331	−0.911823
$94$	10.3673	1.06931
$95$	−28.7655	−2.95128
$96$	−1.98933	−0.203035
$97$	−17.9809	−1.82569	−0.912843	−	0.408311i	$-0.866118\pi$
$97$	−17.9809	−1.82569	−0.912843	+	0.408311i	$0.866118\pi$
$98$	−6.77478	−0.684356
$99$	−1.57494	−0.158288
$100$	12.4985	1.24985
$101$	−1.45358	−0.144637	−0.0723183	−	0.997382i	$-0.523040\pi$
$101$	−1.45358	−0.144637	−0.0723183	+	0.997382i	$0.523040\pi$
$102$	5.73647	0.567995
$103$	−10.3315	−1.01800	−0.508998	−	0.860768i	$-0.669984\pi$
$103$	−10.3315	−1.01800	−0.508998	+	0.860768i	$0.669984\pi$
$104$	−2.31472	−0.226977
$105$	3.94920	0.385402
$106$	−8.42623	−0.818427
$107$	−13.7672	−1.33093	−0.665464	−	0.746430i	$-0.731766\pi$
$107$	−13.7672	−1.33093	−0.665464	+	0.746430i	$0.731766\pi$
$108$	4.06335	0.390997
$109$	−18.1779	−1.74113	−0.870565	−	0.492054i	$-0.836246\pi$
$109$	−18.1779	−1.74113	−0.870565	+	0.492054i	$0.836246\pi$
$110$	−6.88116	−0.656093
$111$	3.54758	0.336722
$112$	−0.474573	−0.0448429
$113$	4.19917	0.395025	0.197512	−	0.980300i	$-0.436714\pi$
$113$	4.19917	0.395025	0.197512	+	0.980300i	$0.436714\pi$
$114$	13.6797	1.28122
$115$	14.3143	1.33482
$116$	−9.60384	−0.891694
$117$	−2.21617	−0.204885
$118$	−10.5518	−0.971375
$119$	1.36849	0.125449
$120$	−8.32159	−0.759654
$121$	−8.29403	−0.754002
$122$	−0.448317	−0.0405888
$123$	−2.11109	−0.190350
$124$	4.42024	0.396949
$125$	31.3669	2.80555
$126$	−0.454366	−0.0404782
$127$	−17.8711	−1.58581	−0.792904	−	0.609347i	$-0.791432\pi$
$127$	−17.8711	−1.58581	−0.792904	+	0.609347i	$0.791432\pi$
$128$	1.00000	0.0883883
$129$	−14.2543	−1.25502
$130$	−9.68275	−0.849233
$131$	−4.64025	−0.405421	−0.202710	−	0.979239i	$-0.564975\pi$
$131$	−4.64025	−0.405421	−0.202710	+	0.979239i	$0.564975\pi$
$132$	3.27241	0.284827
$133$	3.26343	0.282975
$134$	3.74413	0.323443
$135$	16.9975	1.46291
$136$	−2.88362	−0.247269
$137$	15.7299	1.34390	0.671950	−	0.740596i	$-0.265457\pi$
$137$	15.7299	1.34390	0.671950	+	0.740596i	$0.265457\pi$
$138$	−6.80734	−0.579479
$139$	16.6442	1.41174	0.705871	−	0.708341i	$-0.250556\pi$
$139$	16.6442	1.41174	0.705871	+	0.708341i	$0.250556\pi$
$140$	−1.98519	−0.167779
$141$	−20.6240	−1.73685
$142$	5.11142	0.428941
$143$	3.80768	0.318414
$144$	0.957423	0.0797852
$145$	−40.1740	−3.33627
$146$	−7.11511	−0.588851
$147$	13.4773	1.11159
$148$	−1.78331	−0.146587
$149$	5.33057	0.436697	0.218349	−	0.975871i	$-0.429933\pi$
$149$	5.33057	0.436697	0.218349	+	0.975871i	$0.429933\pi$
$150$	−24.8635	−2.03010
$151$	17.6667	1.43770	0.718849	−	0.695166i	$-0.244669\pi$
$151$	17.6667	1.43770	0.718849	+	0.695166i	$0.244669\pi$
$152$	−6.87657	−0.557763
$153$	−2.76085	−0.223201
$154$	0.780664	0.0629077
$155$	18.4904	1.48518
$156$	4.60474	0.368674
$157$	16.2883	1.29995	0.649975	−	0.759955i	$-0.274779\pi$
$157$	16.2883	1.29995	0.649975	+	0.759955i	$0.274779\pi$
$158$	−3.62629	−0.288492
$159$	16.7625	1.32935
$160$	4.18312	0.330704
$161$	−1.62395	−0.127986
$162$	−10.9556	−0.860754
$163$	−4.11200	−0.322077	−0.161038	−	0.986948i	$-0.551484\pi$
$163$	−4.11200	−0.322077	−0.161038	+	0.986948i	$0.551484\pi$
$164$	1.06121	0.0828664
$165$	13.6889	1.06568
$166$	1.55571	0.120747
$167$	−22.9167	−1.77335	−0.886673	−	0.462397i	$-0.846990\pi$
$167$	−22.9167	−1.77335	−0.886673	+	0.462397i	$0.846990\pi$
$168$	0.944080	0.0728374
$169$	−7.64206	−0.587851
$170$	−12.0625	−0.925155
$171$	−6.58378	−0.503474
$172$	7.16539	0.546356
$173$	−13.2232	−1.00534	−0.502671	−	0.864478i	$-0.667649\pi$
$173$	−13.2232	−1.00534	−0.502671	+	0.864478i	$0.667649\pi$
$174$	19.1052	1.44836
$175$	−5.93143	−0.448374
$176$	−1.64498	−0.123995
$177$	20.9911	1.57778
$178$	−5.45327	−0.408740
$179$	15.1931	1.13559	0.567794	−	0.823171i	$-0.307797\pi$
$179$	15.1931	1.13559	0.567794	+	0.823171i	$0.307797\pi$
$180$	4.00501	0.298516
$181$	12.6131	0.937528	0.468764	−	0.883323i	$-0.344699\pi$
$181$	12.6131	0.937528	0.468764	+	0.883323i	$0.344699\pi$
$182$	1.09850	0.0814265
$183$	0.891850	0.0659275
$184$	3.42193	0.252268
$185$	−7.45979	−0.548455
$186$	−8.79331	−0.644756
$187$	4.74352	0.346880
$188$	10.3673	0.756113
$189$	−1.92836	−0.140267
$190$	−28.7655	−2.08687
$191$	−14.8350	−1.07342	−0.536710	−	0.843767i	$-0.680333\pi$
$191$	−14.8350	−1.07342	−0.536710	+	0.843767i	$0.680333\pi$
$192$	−1.98933	−0.143567
$193$	−18.8113	−1.35407	−0.677033	−	0.735952i	$-0.736735\pi$
$193$	−18.8113	−1.35407	−0.677033	+	0.735952i	$0.736735\pi$
$194$	−17.9809	−1.29095
$195$	19.2622	1.37939
$196$	−6.77478	−0.483913
$197$	20.2467	1.44252	0.721258	−	0.692666i	$-0.243564\pi$
$197$	20.2467	1.44252	0.721258	+	0.692666i	$0.243564\pi$
$198$	−1.57494	−0.111926
$199$	13.5938	0.963642	0.481821	−	0.876270i	$-0.339976\pi$
$199$	13.5938	0.963642	0.481821	+	0.876270i	$0.339976\pi$
$200$	12.4985	0.883775
$201$	−7.44829	−0.525362
$202$	−1.45358	−0.102274
$203$	4.55772	0.319889
$204$	5.73647	0.401633
$205$	4.43915	0.310044
$206$	−10.3315	−0.719832
$207$	3.27623	0.227714
$208$	−2.31472	−0.160497
$209$	11.3118	0.782456
$210$	3.94920	0.272521
$211$	−13.3638	−0.920003	−0.460001	−	0.887918i	$-0.652151\pi$
$211$	−13.3638	−0.920003	−0.460001	+	0.887918i	$0.652151\pi$
$212$	−8.42623	−0.578716
$213$	−10.1683	−0.696720
$214$	−13.7672	−0.941108
$215$	29.9737	2.04419
$216$	4.06335	0.276476
$217$	−2.09773	−0.142403
$218$	−18.1779	−1.23116
$219$	14.1543	0.956458
$220$	−6.88116	−0.463928
$221$	6.67479	0.448995
$222$	3.54758	0.238098
$223$	−0.111513	−0.00746748	−0.00373374	−	0.999993i	$-0.501188\pi$
$223$	−0.111513	−0.00746748	−0.00373374	+	0.999993i	$0.501188\pi$
$224$	−0.474573	−0.0317087
$225$	11.9663	0.797754
$226$	4.19917	0.279325
$227$	−1.85493	−0.123116	−0.0615579	−	0.998104i	$-0.519607\pi$
$227$	−1.85493	−0.123116	−0.0615579	+	0.998104i	$0.519607\pi$
$228$	13.6797	0.905963
$229$	−5.21795	−0.344812	−0.172406	−	0.985026i	$-0.555154\pi$
$229$	−5.21795	−0.344812	−0.172406	+	0.985026i	$0.555154\pi$
$230$	14.3143	0.943859
$231$	−1.55300	−0.102180
$232$	−9.60384	−0.630523
$233$	−0.368052	−0.0241119	−0.0120559	−	0.999927i	$-0.503838\pi$
$233$	−0.368052	−0.0241119	−0.0120559	+	0.999927i	$0.503838\pi$
$234$	−2.21617	−0.144875
$235$	43.3676	2.82899
$236$	−10.5518	−0.686866
$237$	7.21388	0.468592
$238$	1.36849	0.0887060
$239$	−17.7774	−1.14992	−0.574962	−	0.818180i	$-0.694983\pi$
$239$	−17.7774	−1.14992	−0.574962	+	0.818180i	$0.694983\pi$
$240$	−8.32159	−0.537156
$241$	−26.0379	−1.67725	−0.838625	−	0.544710i	$-0.816640\pi$
$241$	−26.0379	−1.67725	−0.838625	+	0.544710i	$0.816640\pi$
$242$	−8.29403	−0.533160
$243$	9.60423	0.616111
$244$	−0.448317	−0.0287006
$245$	−28.3397	−1.81056
$246$	−2.11109	−0.134598
$247$	15.9173	1.01280
$248$	4.42024	0.280686
$249$	−3.09482	−0.196126
$250$	31.3669	1.98382
$251$	−2.18913	−0.138176	−0.0690882	−	0.997611i	$-0.522009\pi$
$251$	−2.18913	−0.138176	−0.0690882	+	0.997611i	$0.522009\pi$
$252$	−0.454366	−0.0286224
$253$	−5.62902	−0.353894
$254$	−17.8711	−1.12134
$255$	23.9963	1.50271
$256$	1.00000	0.0625000
$257$	−5.34519	−0.333424	−0.166712	−	0.986006i	$-0.553315\pi$
$257$	−5.34519	−0.333424	−0.166712	+	0.986006i	$0.553315\pi$
$258$	−14.2543	−0.887434
$259$	0.846309	0.0525871
$260$	−9.68275	−0.600499
$261$	−9.19493	−0.569152
$262$	−4.64025	−0.286676
$263$	−19.2592	−1.18757	−0.593787	−	0.804622i	$-0.702368\pi$
$263$	−19.2592	−1.18757	−0.593787	+	0.804622i	$0.702368\pi$
$264$	3.27241	0.201403
$265$	−35.2479	−2.16526
$266$	3.26343	0.200094
$267$	10.8483	0.663908
$268$	3.74413	0.228709
$269$	2.58412	0.157557	0.0787784	−	0.996892i	$-0.474898\pi$
$269$	2.58412	0.157557	0.0787784	+	0.996892i	$0.474898\pi$
$270$	16.9975	1.03443
$271$	−16.8995	−1.02657	−0.513287	−	0.858217i	$-0.671572\pi$
$271$	−16.8995	−1.02657	−0.513287	+	0.858217i	$0.671572\pi$
$272$	−2.88362	−0.174845
$273$	−2.18528	−0.132259
$274$	15.7299	0.950281
$275$	−20.5598	−1.23980
$276$	−6.80734	−0.409754
$277$	5.03892	0.302759	0.151380	−	0.988476i	$-0.451628\pi$
$277$	5.03892	0.302759	0.151380	+	0.988476i	$0.451628\pi$
$278$	16.6442	0.998252
$279$	4.23204	0.253366
$280$	−1.98519	−0.118638
$281$	−12.8954	−0.769273	−0.384636	−	0.923068i	$-0.625673\pi$
$281$	−12.8954	−0.769273	−0.384636	+	0.923068i	$0.625673\pi$
$282$	−20.6240	−1.22814
$283$	28.4974	1.69400	0.846998	−	0.531596i	$-0.178407\pi$
$283$	28.4974	1.69400	0.846998	+	0.531596i	$0.178407\pi$
$284$	5.11142	0.303307
$285$	57.2239	3.38965
$286$	3.80768	0.225153
$287$	−0.503620	−0.0297278
$288$	0.957423	0.0564167
$289$	−8.68471	−0.510865
$290$	−40.1740	−2.35910
$291$	35.7699	2.09687
$292$	−7.11511	−0.416380
$293$	−1.12299	−0.0656059	−0.0328029	−	0.999462i	$-0.510443\pi$
$293$	−1.12299	−0.0656059	−0.0328029	+	0.999462i	$0.510443\pi$
$294$	13.4773	0.786010
$295$	−44.1396	−2.56990
$296$	−1.78331	−0.103653
$297$	−6.68415	−0.387854
$298$	5.33057	0.308792
$299$	−7.92082	−0.458073
$300$	−24.8635	−1.43550
$301$	−3.40050	−0.196001
$302$	17.6667	1.01661
$303$	2.89165	0.166121
$304$	−6.87657	−0.394398
$305$	−1.87536	−0.107383
$306$	−2.76085	−0.157827
$307$	19.4889	1.11229	0.556145	−	0.831085i	$-0.312280\pi$
$307$	19.4889	1.11229	0.556145	+	0.831085i	$0.312280\pi$
$308$	0.780664	0.0444825
$309$	20.5528	1.16921
$310$	18.4904	1.05018
$311$	−1.97238	−0.111843	−0.0559217	−	0.998435i	$-0.517810\pi$
$311$	−1.97238	−0.111843	−0.0559217	+	0.998435i	$0.517810\pi$
$312$	4.60474	0.260692
$313$	−8.70211	−0.491872	−0.245936	−	0.969286i	$-0.579095\pi$
$313$	−8.70211	−0.491872	−0.245936	+	0.969286i	$0.579095\pi$
$314$	16.2883	0.919204
$315$	−1.90067	−0.107091
$316$	−3.62629	−0.203995
$317$	−7.49906	−0.421189	−0.210595	−	0.977573i	$-0.567540\pi$
$317$	−7.49906	−0.421189	−0.210595	+	0.977573i	$0.567540\pi$
$318$	16.7625	0.939995
$319$	15.7982	0.884527
$320$	4.18312	0.233843
$321$	27.3875	1.52862
$322$	−1.62395	−0.0904994
$323$	19.8294	1.10334
$324$	−10.9556	−0.608645
$325$	−28.9305	−1.60477
$326$	−4.11200	−0.227742
$327$	36.1618	1.99975
$328$	1.06121	0.0585954
$329$	−4.92004	−0.271250
$330$	13.6889	0.753548
$331$	−15.4599	−0.849753	−0.424876	−	0.905251i	$-0.639682\pi$
$331$	−15.4599	−0.849753	−0.424876	+	0.905251i	$0.639682\pi$
$332$	1.55571	0.0853808
$333$	−1.70738	−0.0935638
$334$	−22.9167	−1.25395
$335$	15.6621	0.855713
$336$	0.944080	0.0515038
$337$	22.7842	1.24114	0.620568	−	0.784153i	$-0.286902\pi$
$337$	22.7842	1.24114	0.620568	+	0.784153i	$0.286902\pi$
$338$	−7.64206	−0.415673
$339$	−8.35352	−0.453701
$340$	−12.0625	−0.654183
$341$	−7.27123	−0.393759
$342$	−6.58378	−0.356010
$343$	6.53713	0.352972
$344$	7.16539	0.386332
$345$	−28.4759	−1.53309
$346$	−13.2232	−0.710884
$347$	34.3711	1.84514	0.922569	−	0.385832i	$-0.126085\pi$
$347$	34.3711	1.84514	0.922569	+	0.385832i	$0.126085\pi$
$348$	19.1052	1.02415
$349$	−33.0556	−1.76942	−0.884712	−	0.466138i	$-0.845645\pi$
$349$	−33.0556	−1.76942	−0.884712	+	0.466138i	$0.845645\pi$
$350$	−5.93143	−0.317048
$351$	−9.40553	−0.502030
$352$	−1.64498	−0.0876780
$353$	−8.08837	−0.430500	−0.215250	−	0.976559i	$-0.569057\pi$
$353$	−8.08837	−0.430500	−0.215250	+	0.976559i	$0.569057\pi$
$354$	20.9911	1.11566
$355$	21.3817	1.13482
$356$	−5.45327	−0.289023
$357$	−2.72237	−0.144083
$358$	15.1931	0.802982
$359$	11.5420	0.609161	0.304581	−	0.952487i	$-0.401484\pi$
$359$	11.5420	0.609161	0.304581	+	0.952487i	$0.401484\pi$
$360$	4.00501	0.211083
$361$	28.2872	1.48880
$362$	12.6131	0.662932
$363$	16.4995	0.866001
$364$	1.09850	0.0575772
$365$	−29.7633	−1.55788
$366$	0.891850	0.0466177
$367$	22.6836	1.18407	0.592037	−	0.805911i	$-0.298324\pi$
$367$	22.6836	1.18407	0.592037	+	0.805911i	$0.298324\pi$
$368$	3.42193	0.178380
$369$	1.01602	0.0528921
$370$	−7.45979	−0.387816
$371$	3.99886	0.207610
$372$	−8.79331	−0.455912
$373$	−5.55121	−0.287431	−0.143715	−	0.989619i	$-0.545905\pi$
$373$	−5.55121	−0.287431	−0.143715	+	0.989619i	$0.545905\pi$
$374$	4.74352	0.245281
$375$	−62.3991	−3.22228
$376$	10.3673	0.534653
$377$	22.2302	1.14491
$378$	−1.92836	−0.0991840
$379$	23.3873	1.20132	0.600662	−	0.799503i	$-0.294904\pi$
$379$	23.3873	1.20132	0.600662	+	0.799503i	$0.294904\pi$
$380$	−28.7655	−1.47564
$381$	35.5516	1.82136
$382$	−14.8350	−0.759023
$383$	−11.6663	−0.596118	−0.298059	−	0.954548i	$-0.596339\pi$
$383$	−11.6663	−0.596118	−0.298059	+	0.954548i	$0.596339\pi$
$384$	−1.98933	−0.101517
$385$	3.26561	0.166431
$386$	−18.8113	−0.957470
$387$	6.86031	0.348729
$388$	−17.9809	−0.912843
$389$	16.0353	0.813022	0.406511	−	0.913646i	$-0.366745\pi$
$389$	16.0353	0.813022	0.406511	+	0.913646i	$0.366745\pi$
$390$	19.2622	0.975377
$391$	−9.86756	−0.499024
$392$	−6.77478	−0.342178
$393$	9.23098	0.465641
$394$	20.2467	1.02001
$395$	−15.1692	−0.763245
$396$	−1.57494	−0.0791440
$397$	27.3738	1.37385	0.686926	−	0.726727i	$-0.258960\pi$
$397$	27.3738	1.37385	0.686926	+	0.726727i	$0.258960\pi$
$398$	13.5938	0.681398
$399$	−6.49203	−0.325008
$400$	12.4985	0.624923
$401$	9.90184	0.494474	0.247237	−	0.968955i	$-0.420477\pi$
$401$	9.90184	0.494474	0.247237	+	0.968955i	$0.420477\pi$
$402$	−7.44829	−0.371487
$403$	−10.2316	−0.509674
$404$	−1.45358	−0.0723183
$405$	−45.8286	−2.27724
$406$	4.55772	0.226196
$407$	2.93351	0.145409
$408$	5.73647	0.283998
$409$	35.4042	1.75063	0.875313	−	0.483556i	$-0.160655\pi$
$409$	35.4042	1.75063	0.875313	+	0.483556i	$0.160655\pi$
$410$	4.43915	0.219234
$411$	−31.2920	−1.54352
$412$	−10.3315	−0.508998
$413$	5.00761	0.246409
$414$	3.27623	0.161018
$415$	6.50773	0.319452
$416$	−2.31472	−0.113489
$417$	−33.1107	−1.62144
$418$	11.3118	0.553280
$419$	6.61810	0.323315	0.161658	−	0.986847i	$-0.448316\pi$
$419$	6.61810	0.323315	0.161658	+	0.986847i	$0.448316\pi$
$420$	3.94920	0.192701
$421$	−21.5293	−1.04927	−0.524636	−	0.851326i	$-0.675799\pi$
$421$	−21.5293	−1.04927	−0.524636	+	0.851326i	$0.675799\pi$
$422$	−13.3638	−0.650540
$423$	9.92589	0.482613
$424$	−8.42623	−0.409214
$425$	−36.0409	−1.74824
$426$	−10.1683	−0.492656
$427$	0.212759	0.0102961
$428$	−13.7672	−0.665464
$429$	−7.57472	−0.365711
$430$	29.9737	1.44546
$431$	−24.9918	−1.20381	−0.601906	−	0.798567i	$-0.705592\pi$
$431$	−24.9918	−1.20381	−0.601906	+	0.798567i	$0.705592\pi$
$432$	4.06335	0.195498
$433$	17.6771	0.849506	0.424753	−	0.905309i	$-0.360361\pi$
$433$	17.6771	0.849506	0.424753	+	0.905309i	$0.360361\pi$
$434$	−2.09773	−0.100694
$435$	79.9192	3.83183
$436$	−18.1779	−0.870565
$437$	−23.5311	−1.12565
$438$	14.1543	0.676318
$439$	19.0271	0.908115	0.454058	−	0.890972i	$-0.349976\pi$
$439$	19.0271	0.908115	0.454058	+	0.890972i	$0.349976\pi$
$440$	−6.88116	−0.328046
$441$	−6.48633	−0.308873
$442$	6.67479	0.317487
$443$	17.8049	0.845938	0.422969	−	0.906144i	$-0.360988\pi$
$443$	17.8049	0.845938	0.422969	+	0.906144i	$0.360988\pi$
$444$	3.54758	0.168361
$445$	−22.8117	−1.08138
$446$	−0.111513	−0.00528031
$447$	−10.6042	−0.501564
$448$	−0.474573	−0.0224214
$449$	−29.3336	−1.38434	−0.692169	−	0.721735i	$-0.743345\pi$
$449$	−29.3336	−1.38434	−0.692169	+	0.721735i	$0.743345\pi$
$450$	11.9663	0.564097
$451$	−1.74567	−0.0822004
$452$	4.19917	0.197512
$453$	−35.1449	−1.65125
$454$	−1.85493	−0.0870560
$455$	4.59517	0.215425
$456$	13.6797	0.640612
$457$	−23.8034	−1.11348	−0.556738	−	0.830688i	$-0.687947\pi$
$457$	−23.8034	−1.11348	−0.556738	+	0.830688i	$0.687947\pi$
$458$	−5.21795	−0.243819
$459$	−11.7172	−0.546912
$460$	14.3143	0.667409
$461$	−2.65682	−0.123741	−0.0618703	−	0.998084i	$-0.519707\pi$
$461$	−2.65682	−0.123741	−0.0618703	+	0.998084i	$0.519707\pi$
$462$	−1.55300	−0.0722520
$463$	−9.37826	−0.435845	−0.217922	−	0.975966i	$-0.569928\pi$
$463$	−9.37826	−0.435845	−0.217922	+	0.975966i	$0.569928\pi$
$464$	−9.60384	−0.445847
$465$	−36.7834	−1.70579
$466$	−0.368052	−0.0170497
$467$	15.2842	0.707270	0.353635	−	0.935384i	$-0.384946\pi$
$467$	15.2842	0.707270	0.353635	+	0.935384i	$0.384946\pi$
$468$	−2.21617	−0.102442
$469$	−1.77686	−0.0820478
$470$	43.3676	2.00040
$471$	−32.4028	−1.49304
$472$	−10.5518	−0.485688
$473$	−11.7870	−0.541965
$474$	7.21388	0.331344
$475$	−85.9465	−3.94350
$476$	1.36849	0.0627246
$477$	−8.06746	−0.369384
$478$	−17.7774	−0.813119
$479$	−40.6094	−1.85549	−0.927746	−	0.373211i	$-0.878257\pi$
$479$	−40.6094	−1.85549	−0.927746	+	0.373211i	$0.878257\pi$
$480$	−8.32159	−0.379827
$481$	4.12786	0.188214
$482$	−26.0379	−1.18599
$483$	3.23058	0.146996
$484$	−8.29403	−0.377001
$485$	−75.2163	−3.41540
$486$	9.60423	0.435657
$487$	−13.7642	−0.623718	−0.311859	−	0.950128i	$-0.600952\pi$
$487$	−13.7642	−0.623718	−0.311859	+	0.950128i	$0.600952\pi$
$488$	−0.448317	−0.0202944
$489$	8.18011	0.369917
$490$	−28.3397	−1.28026
$491$	−3.16730	−0.142938	−0.0714691	−	0.997443i	$-0.522769\pi$
$491$	−3.16730	−0.142938	−0.0714691	+	0.997443i	$0.522769\pi$
$492$	−2.11109	−0.0951752
$493$	27.6939	1.24727
$494$	15.9173	0.716155
$495$	−6.58818	−0.296117
$496$	4.42024	0.198475
$497$	−2.42574	−0.108809
$498$	−3.09482	−0.138682
$499$	8.62864	0.386271	0.193135	−	0.981172i	$-0.438134\pi$
$499$	8.62864	0.386271	0.193135	+	0.981172i	$0.438134\pi$
$500$	31.3669	1.40277
$501$	45.5888	2.03676
$502$	−2.18913	−0.0977054
$503$	16.4282	0.732497	0.366248	−	0.930517i	$-0.380642\pi$
$503$	16.4282	0.732497	0.366248	+	0.930517i	$0.380642\pi$
$504$	−0.454366	−0.0202391
$505$	−6.08050	−0.270579
$506$	−5.62902	−0.250241
$507$	15.2026	0.675170
$508$	−17.8711	−0.792904
$509$	−16.7018	−0.740296	−0.370148	−	0.928973i	$-0.620693\pi$
$509$	−16.7018	−0.740296	−0.370148	+	0.928973i	$0.620693\pi$
$510$	23.9963	1.06258
$511$	3.37664	0.149374
$512$	1.00000	0.0441942
$513$	−27.9419	−1.23367
$514$	−5.34519	−0.235766
$515$	−43.2180	−1.90441
$516$	−14.2543	−0.627511
$517$	−17.0540	−0.750036
$518$	0.846309	0.0371847
$519$	26.3053	1.15467
$520$	−9.68275	−0.424617
$521$	16.0802	0.704485	0.352242	−	0.935909i	$-0.385419\pi$
$521$	16.0802	0.704485	0.352242	+	0.935909i	$0.385419\pi$
$522$	−9.19493	−0.402451
$523$	−9.00341	−0.393692	−0.196846	−	0.980434i	$-0.563070\pi$
$523$	−9.00341	−0.393692	−0.196846	+	0.980434i	$0.563070\pi$
$524$	−4.64025	−0.202710
$525$	11.7996	0.514975
$526$	−19.2592	−0.839742
$527$	−12.7463	−0.555238
$528$	3.27241	0.142413
$529$	−11.2904	−0.490887
$530$	−35.2479	−1.53107
$531$	−10.1026	−0.438414
$532$	3.26343	0.141488
$533$	−2.45640	−0.106398
$534$	10.8483	0.469454
$535$	−57.5899	−2.48983
$536$	3.74413	0.161722
$537$	−30.2241	−1.30427
$538$	2.58412	0.111410
$539$	11.1444	0.480024
$540$	16.9975	0.731455
$541$	27.9037	1.19967	0.599837	−	0.800122i	$-0.295232\pi$
$541$	27.9037	1.19967	0.599837	+	0.800122i	$0.295232\pi$
$542$	−16.8995	−0.725897
$543$	−25.0917	−1.07679
$544$	−2.88362	−0.123634
$545$	−76.0404	−3.25721
$546$	−2.18528	−0.0935215
$547$	−4.01884	−0.171833	−0.0859165	−	0.996302i	$-0.527382\pi$
$547$	−4.01884	−0.171833	−0.0859165	+	0.996302i	$0.527382\pi$
$548$	15.7299	0.671950
$549$	−0.429229	−0.0183191
$550$	−20.5598	−0.876672
$551$	66.0414	2.81346
$552$	−6.80734	−0.289740
$553$	1.72094	0.0731817
$554$	5.03892	0.214083
$555$	14.8400	0.629921
$556$	16.6442	0.705871
$557$	−14.4793	−0.613506	−0.306753	−	0.951789i	$-0.599243\pi$
$557$	−14.4793	−0.613506	−0.306753	+	0.951789i	$0.599243\pi$
$558$	4.23204	0.179156
$559$	−16.5859	−0.701508
$560$	−1.98519	−0.0838897
$561$	−9.43641	−0.398405
$562$	−12.8954	−0.543958
$563$	−5.31302	−0.223917	−0.111959	−	0.993713i	$-0.535712\pi$
$563$	−5.31302	−0.223917	−0.111959	+	0.993713i	$0.535712\pi$
$564$	−20.6240	−0.868425
$565$	17.5656	0.738991
$566$	28.4974	1.19784
$567$	5.19923	0.218347
$568$	5.11142	0.214471
$569$	−11.5685	−0.484977	−0.242488	−	0.970154i	$-0.577964\pi$
$569$	−11.5685	−0.484977	−0.242488	+	0.970154i	$0.577964\pi$
$570$	57.2239	2.39685
$571$	22.6878	0.949453	0.474727	−	0.880133i	$-0.342547\pi$
$571$	22.6878	0.949453	0.474727	+	0.880133i	$0.342547\pi$
$572$	3.80768	0.159207
$573$	29.5116	1.23286
$574$	−0.503620	−0.0210207
$575$	42.7689	1.78359
$576$	0.957423	0.0398926
$577$	34.5170	1.43696	0.718482	−	0.695546i	$-0.244837\pi$
$577$	34.5170	1.43696	0.718482	+	0.695546i	$0.244837\pi$
$578$	−8.68471	−0.361236
$579$	37.4218	1.55520
$580$	−40.1740	−1.66813
$581$	−0.738298	−0.0306298
$582$	35.7699	1.48271
$583$	13.8610	0.574064
$584$	−7.11511	−0.294425
$585$	−9.27048	−0.383287
$586$	−1.12299	−0.0463904
$587$	26.8394	1.10778	0.553891	−	0.832589i	$-0.313142\pi$
$587$	26.8394	1.10778	0.553891	+	0.832589i	$0.313142\pi$
$588$	13.4773	0.555793
$589$	−30.3961	−1.25245
$590$	−44.1396	−1.81720
$591$	−40.2773	−1.65679
$592$	−1.78331	−0.0732935
$593$	−45.0683	−1.85074	−0.925368	−	0.379071i	$-0.876244\pi$
$593$	−45.0683	−1.85074	−0.925368	+	0.379071i	$0.876244\pi$
$594$	−6.68415	−0.274254
$595$	5.72455	0.234684
$596$	5.33057	0.218349
$597$	−27.0426	−1.10678
$598$	−7.92082	−0.323906
$599$	−7.49872	−0.306389	−0.153195	−	0.988196i	$-0.548956\pi$
$599$	−7.49872	−0.306389	−0.153195	+	0.988196i	$0.548956\pi$
$600$	−24.8635	−1.01505
$601$	−29.5657	−1.20601	−0.603006	−	0.797737i	$-0.706030\pi$
$601$	−29.5657	−1.20601	−0.603006	+	0.797737i	$0.706030\pi$
$602$	−3.40050	−0.138594
$603$	3.58471	0.145981
$604$	17.6667	0.718849
$605$	−34.6949	−1.41055
$606$	2.89165	0.117465
$607$	19.7198	0.800401	0.400201	−	0.916428i	$-0.368940\pi$
$607$	19.7198	0.800401	0.400201	+	0.916428i	$0.368940\pi$
$608$	−6.87657	−0.278882
$609$	−9.06679	−0.367405
$610$	−1.87536	−0.0759313
$611$	−23.9974	−0.970832
$612$	−2.76085	−0.111601
$613$	2.56921	0.103770	0.0518848	−	0.998653i	$-0.483477\pi$
$613$	2.56921	0.103770	0.0518848	+	0.998653i	$0.483477\pi$
$614$	19.4889	0.786508
$615$	−8.83093	−0.356098
$616$	0.780664	0.0314539
$617$	15.2894	0.615528	0.307764	−	0.951463i	$-0.400419\pi$
$617$	15.2894	0.615528	0.307764	+	0.951463i	$0.400419\pi$
$618$	20.5528	0.826755
$619$	−23.4825	−0.943843	−0.471922	−	0.881640i	$-0.656439\pi$
$619$	−23.4825	−0.943843	−0.471922	+	0.881640i	$0.656439\pi$
$620$	18.4904	0.742592
$621$	13.9045	0.557969
$622$	−1.97238	−0.0790853
$623$	2.58797	0.103685
$624$	4.60474	0.184337
$625$	68.7193	2.74877
$626$	−8.70211	−0.347806
$627$	−22.5030	−0.898681
$628$	16.2883	0.649975
$629$	5.14239	0.205041
$630$	−1.90067	−0.0757244
$631$	36.0811	1.43637	0.718183	−	0.695854i	$-0.244974\pi$
$631$	36.0811	1.43637	0.718183	+	0.695854i	$0.244974\pi$
$632$	−3.62629	−0.144246
$633$	26.5850	1.05666
$634$	−7.49906	−0.297826
$635$	−74.7571	−2.96664
$636$	16.7625	0.664677
$637$	15.6817	0.621333
$638$	15.7982	0.625455
$639$	4.89379	0.193595
$640$	4.18312	0.165352
$641$	−11.2523	−0.444439	−0.222219	−	0.974997i	$-0.571330\pi$
$641$	−11.2523	−0.444439	−0.222219	+	0.974997i	$0.571330\pi$
$642$	27.3875	1.08090
$643$	−30.9620	−1.22102	−0.610510	−	0.792008i	$-0.709036\pi$
$643$	−30.9620	−1.22102	−0.610510	+	0.792008i	$0.709036\pi$
$644$	−1.62395	−0.0639928
$645$	−59.6274	−2.34783
$646$	19.8294	0.780179
$647$	4.12008	0.161977	0.0809886	−	0.996715i	$-0.474192\pi$
$647$	4.12008	0.161977	0.0809886	+	0.996715i	$0.474192\pi$
$648$	−10.9556	−0.430377
$649$	17.3576	0.681346
$650$	−28.9305	−1.13475
$651$	4.17306	0.163555
$652$	−4.11200	−0.161038
$653$	35.9180	1.40558	0.702790	−	0.711398i	$-0.251937\pi$
$653$	35.9180	1.40558	0.702790	+	0.711398i	$0.251937\pi$
$654$	36.1618	1.41404
$655$	−19.4107	−0.758440
$656$	1.06121	0.0414332
$657$	−6.81217	−0.265768
$658$	−4.92004	−0.191803
$659$	−32.7456	−1.27559	−0.637795	−	0.770206i	$-0.720153\pi$
$659$	−32.7456	−1.27559	−0.637795	+	0.770206i	$0.720153\pi$
$660$	13.6889	0.532839
$661$	44.4352	1.72833	0.864164	−	0.503210i	$-0.167848\pi$
$661$	44.4352	1.72833	0.864164	+	0.503210i	$0.167848\pi$
$662$	−15.4599	−0.600866
$663$	−13.2783	−0.515688
$664$	1.55571	0.0603734
$665$	13.6513	0.529375
$666$	−1.70738	−0.0661596
$667$	−32.8637	−1.27249
$668$	−22.9167	−0.886673
$669$	0.221836	0.00857669
$670$	15.6621	0.605080
$671$	0.737475	0.0284699
$672$	0.944080	0.0364187
$673$	8.06606	0.310924	0.155462	−	0.987842i	$-0.450313\pi$
$673$	8.06606	0.310924	0.155462	+	0.987842i	$0.450313\pi$
$674$	22.7842	0.877616
$675$	50.7857	1.95474
$676$	−7.64206	−0.293926
$677$	−26.4165	−1.01527	−0.507634	−	0.861573i	$-0.669480\pi$
$677$	−26.4165	−1.01527	−0.507634	+	0.861573i	$0.669480\pi$
$678$	−8.35352	−0.320815
$679$	8.53325	0.327476
$680$	−12.0625	−0.462577
$681$	3.69006	0.141403
$682$	−7.27123	−0.278430
$683$	−20.0634	−0.767706	−0.383853	−	0.923394i	$-0.625403\pi$
$683$	−20.0634	−0.767706	−0.383853	+	0.923394i	$0.625403\pi$
$684$	−6.58378	−0.251737
$685$	65.8002	2.51410
$686$	6.53713	0.249589
$687$	10.3802	0.396029
$688$	7.16539	0.273178
$689$	19.5044	0.743057
$690$	−28.4759	−1.08406
$691$	−10.7900	−0.410470	−0.205235	−	0.978713i	$-0.565796\pi$
$691$	−10.7900	−0.410470	−0.205235	+	0.978713i	$0.565796\pi$
$692$	−13.2232	−0.502671
$693$	0.747426	0.0283924
$694$	34.3711	1.30471
$695$	69.6246	2.64101
$696$	19.1052	0.724180
$697$	−3.06012	−0.115910
$698$	−33.0556	−1.25117
$699$	0.732176	0.0276934
$700$	−5.93143	−0.224187
$701$	1.91478	0.0723203	0.0361601	−	0.999346i	$-0.488487\pi$
$701$	1.91478	0.0723203	0.0361601	+	0.999346i	$0.488487\pi$
$702$	−9.40553	−0.354989
$703$	12.2630	0.462509
$704$	−1.64498	−0.0619977
$705$	−86.2724	−3.24921
$706$	−8.08837	−0.304410
$707$	0.689830	0.0259437
$708$	20.9911	0.788892
$709$	8.85290	0.332478	0.166239	−	0.986086i	$-0.446838\pi$
$709$	8.85290	0.332478	0.166239	+	0.986086i	$0.446838\pi$
$710$	21.3817	0.802440
$711$	−3.47189	−0.130206
$712$	−5.45327	−0.204370
$713$	15.1258	0.566464
$714$	−2.72237	−0.101882
$715$	15.9280	0.595672
$716$	15.1931	0.567794
$717$	35.3651	1.32073
$718$	11.5420	0.430742
$719$	−48.7524	−1.81816	−0.909078	−	0.416625i	$-0.863213\pi$
$719$	−48.7524	−1.81816	−0.909078	+	0.416625i	$0.863213\pi$
$720$	4.00501	0.149258
$721$	4.90306	0.182600
$722$	28.2872	1.05274
$723$	51.7979	1.92639
$724$	12.6131	0.468764
$725$	−120.033	−4.45792
$726$	16.4995	0.612355
$727$	−33.9372	−1.25866	−0.629330	−	0.777138i	$-0.716671\pi$
$727$	−33.9372	−1.25866	−0.629330	+	0.777138i	$0.716671\pi$
$728$	1.09850	0.0407132
$729$	13.7609	0.509662
$730$	−29.7633	−1.10159
$731$	−20.6623	−0.764223
$732$	0.891850	0.0329637
$733$	47.8091	1.76587	0.882935	−	0.469495i	$-0.155564\pi$
$733$	47.8091	1.76587	0.882935	+	0.469495i	$0.155564\pi$
$734$	22.6836	0.837267
$735$	56.3769	2.07949
$736$	3.42193	0.126134
$737$	−6.15903	−0.226871
$738$	1.01602	0.0374004
$739$	−34.5436	−1.27071	−0.635353	−	0.772222i	$-0.719146\pi$
$739$	−34.5436	−1.27071	−0.635353	+	0.772222i	$0.719146\pi$
$740$	−7.45979	−0.274227
$741$	−31.6648	−1.16323
$742$	3.99886	0.146803
$743$	19.7476	0.724470	0.362235	−	0.932087i	$-0.382014\pi$
$743$	19.7476	0.724470	0.362235	+	0.932087i	$0.382014\pi$
$744$	−8.79331	−0.322378
$745$	22.2984	0.816950
$746$	−5.55121	−0.203244
$747$	1.48947	0.0544970
$748$	4.74352	0.173440
$749$	6.53355	0.238731
$750$	−62.3991	−2.27849
$751$	50.2282	1.83285	0.916426	−	0.400205i	$-0.131061\pi$
$751$	50.2282	1.83285	0.916426	+	0.400205i	$0.131061\pi$
$752$	10.3673	0.378057
$753$	4.35489	0.158701
$754$	22.2302	0.809577
$755$	73.9020	2.68957
$756$	−1.92836	−0.0701337
$757$	−46.2342	−1.68041	−0.840205	−	0.542268i	$-0.817566\pi$
$757$	−46.2342	−1.68041	−0.840205	+	0.542268i	$0.817566\pi$
$758$	23.3873	0.849464
$759$	11.1980	0.406460
$760$	−28.7655	−1.04343
$761$	16.4378	0.595870	0.297935	−	0.954586i	$-0.403702\pi$
$761$	16.4378	0.595870	0.297935	+	0.954586i	$0.403702\pi$
$762$	35.5516	1.28790
$763$	8.62675	0.312309
$764$	−14.8350	−0.536710
$765$	−11.5489	−0.417553
$766$	−11.6663	−0.421519
$767$	24.4246	0.881920
$768$	−1.98933	−0.0717837
$769$	36.2035	1.30553	0.652766	−	0.757560i	$-0.273609\pi$
$769$	36.2035	1.30553	0.652766	+	0.757560i	$0.273609\pi$
$770$	3.26561	0.117684
$771$	10.6333	0.382950
$772$	−18.8113	−0.677033
$773$	−5.42466	−0.195111	−0.0975557	−	0.995230i	$-0.531102\pi$
$773$	−5.42466	−0.195111	−0.0975557	+	0.995230i	$0.531102\pi$
$774$	6.86031	0.246589
$775$	55.2462	1.98450
$776$	−17.9809	−0.645477
$777$	−1.68359	−0.0603983
$778$	16.0353	0.574893
$779$	−7.29746	−0.261459
$780$	19.2622	0.689696
$781$	−8.40821	−0.300870
$782$	−9.86756	−0.352863
$783$	−39.0238	−1.39460
$784$	−6.77478	−0.241956
$785$	68.1360	2.43188
$786$	9.23098	0.329258
$787$	−44.6895	−1.59301	−0.796504	−	0.604633i	$-0.793320\pi$
$787$	−44.6895	−1.59301	−0.796504	+	0.604633i	$0.793320\pi$
$788$	20.2467	0.721258
$789$	38.3129	1.36398
$790$	−15.1692	−0.539696
$791$	−1.99281	−0.0708562
$792$	−1.57494	−0.0559632
$793$	1.03773	0.0368509
$794$	27.3738	0.971460
$795$	70.1196	2.48688
$796$	13.5938	0.481821
$797$	−25.5059	−0.903464	−0.451732	−	0.892154i	$-0.649194\pi$
$797$	−25.5059	−0.903464	−0.451732	+	0.892154i	$0.649194\pi$
$798$	−6.49203	−0.229815
$799$	−29.8954	−1.05762
$800$	12.4985	0.441887
$801$	−5.22109	−0.184478
$802$	9.90184	0.349646
$803$	11.7042	0.413034
$804$	−7.44829	−0.262681
$805$	−6.79319	−0.239428
$806$	−10.2316	−0.360394
$807$	−5.14067	−0.180960
$808$	−1.45358	−0.0511368
$809$	−4.20177	−0.147727	−0.0738633	−	0.997268i	$-0.523533\pi$
$809$	−4.20177	−0.147727	−0.0738633	+	0.997268i	$0.523533\pi$
$810$	−45.8286	−1.61025
$811$	15.7122	0.551729	0.275865	−	0.961196i	$-0.411036\pi$
$811$	15.7122	0.551729	0.275865	+	0.961196i	$0.411036\pi$
$812$	4.55772	0.159945
$813$	33.6187	1.17906
$814$	2.93351	0.102820
$815$	−17.2010	−0.602524
$816$	5.73647	0.200817
$817$	−49.2733	−1.72385
$818$	35.4042	1.23788
$819$	1.05173	0.0367505
$820$	4.43915	0.155022
$821$	48.7730	1.70219	0.851094	−	0.525014i	$-0.175940\pi$
$821$	48.7730	1.70219	0.851094	+	0.525014i	$0.175940\pi$
$822$	−31.2920	−1.09143
$823$	−15.7202	−0.547971	−0.273985	−	0.961734i	$-0.588342\pi$
$823$	−15.7202	−0.547971	−0.273985	+	0.961734i	$0.588342\pi$
$824$	−10.3315	−0.359916
$825$	40.9001	1.42396
$826$	5.00761	0.174237
$827$	−12.9109	−0.448956	−0.224478	−	0.974479i	$-0.572068\pi$
$827$	−12.9109	−0.448956	−0.224478	+	0.974479i	$0.572068\pi$
$828$	3.27623	0.113857
$829$	8.42570	0.292637	0.146318	−	0.989238i	$-0.453258\pi$
$829$	8.42570	0.292637	0.146318	+	0.989238i	$0.453258\pi$
$830$	6.50773	0.225886
$831$	−10.0241	−0.347731
$832$	−2.31472	−0.0802485
$833$	19.5359	0.676880
$834$	−33.1107	−1.14653
$835$	−95.8632	−3.31748
$836$	11.3118	0.391228
$837$	17.9610	0.620823
$838$	6.61810	0.228619
$839$	15.3530	0.530043	0.265022	−	0.964242i	$-0.414621\pi$
$839$	15.3530	0.530043	0.265022	+	0.964242i	$0.414621\pi$
$840$	3.94920	0.136260
$841$	63.2337	2.18047
$842$	−21.5293	−0.741948
$843$	25.6531	0.883540
$844$	−13.3638	−0.460001
$845$	−31.9676	−1.09972
$846$	9.92589	0.341259
$847$	3.93612	0.135247
$848$	−8.42623	−0.289358
$849$	−56.6907	−1.94562
$850$	−36.0409	−1.23619
$851$	−6.10236	−0.209186
$852$	−10.1683	−0.348360
$853$	−39.6835	−1.35874	−0.679368	−	0.733798i	$-0.737746\pi$
$853$	−39.6835	−1.35874	−0.679368	+	0.733798i	$0.737746\pi$
$854$	0.212759	0.00728047
$855$	−27.5407	−0.941873
$856$	−13.7672	−0.470554
$857$	15.3071	0.522879	0.261440	−	0.965220i	$-0.415803\pi$
$857$	15.3071	0.522879	0.261440	+	0.965220i	$0.415803\pi$
$858$	−7.57472	−0.258597
$859$	13.8908	0.473948	0.236974	−	0.971516i	$-0.423844\pi$
$859$	13.8908	0.473948	0.236974	+	0.971516i	$0.423844\pi$
$860$	29.9737	1.02209
$861$	1.00186	0.0341435
$862$	−24.9918	−0.851223
$863$	28.9141	0.984248	0.492124	−	0.870525i	$-0.336221\pi$
$863$	28.9141	0.984248	0.492124	+	0.870525i	$0.336221\pi$
$864$	4.06335	0.138238
$865$	−55.3143	−1.88074
$866$	17.6771	0.600692
$867$	17.2767	0.586748
$868$	−2.09773	−0.0712014
$869$	5.96519	0.202355
$870$	79.9192	2.70951
$871$	−8.66661	−0.293657
$872$	−18.1779	−0.615582
$873$	−17.2153	−0.582651
$874$	−23.5311	−0.795952
$875$	−14.8859	−0.503235
$876$	14.1543	0.478229
$877$	25.1673	0.849840	0.424920	−	0.905231i	$-0.360302\pi$
$877$	25.1673	0.849840	0.424920	+	0.905231i	$0.360302\pi$
$878$	19.0271	0.642134
$879$	2.23400	0.0753509
$880$	−6.88116	−0.231964
$881$	8.64097	0.291122	0.145561	−	0.989349i	$-0.453501\pi$
$881$	8.64097	0.291122	0.145561	+	0.989349i	$0.453501\pi$
$882$	−6.48633	−0.218406
$883$	24.0934	0.810808	0.405404	−	0.914138i	$-0.367131\pi$
$883$	24.0934	0.810808	0.405404	+	0.914138i	$0.367131\pi$
$884$	6.67479	0.224497
$885$	87.8080	2.95163
$886$	17.8049	0.598169
$887$	−1.47342	−0.0494726	−0.0247363	−	0.999694i	$-0.507875\pi$
$887$	−1.47342	−0.0494726	−0.0247363	+	0.999694i	$0.507875\pi$
$888$	3.54758	0.119049
$889$	8.48116	0.284449
$890$	−22.8117	−0.764649
$891$	18.0218	0.603753
$892$	−0.111513	−0.00373374
$893$	−71.2914	−2.38568
$894$	−10.6042	−0.354659
$895$	63.5546	2.12440
$896$	−0.474573	−0.0158544
$897$	15.7571	0.526114
$898$	−29.3336	−0.978875
$899$	−42.4513	−1.41583
$900$	11.9663	0.398877
$901$	24.2981	0.809486
$902$	−1.74567	−0.0581244
$903$	6.76470	0.225115
$904$	4.19917	0.139662
$905$	52.7623	1.75388
$906$	−35.1449	−1.16761
$907$	21.4521	0.712304	0.356152	−	0.934428i	$-0.384089\pi$
$907$	21.4521	0.712304	0.356152	+	0.934428i	$0.384089\pi$
$908$	−1.85493	−0.0615579
$909$	−1.39169	−0.0461595
$910$	4.59517	0.152328
$911$	−38.0964	−1.26219	−0.631095	−	0.775705i	$-0.717394\pi$
$911$	−38.0964	−1.26219	−0.631095	+	0.775705i	$0.717394\pi$
$912$	13.6797	0.452981
$913$	−2.55912	−0.0846946
$914$	−23.8034	−0.787346
$915$	3.73071	0.123334
$916$	−5.21795	−0.172406
$917$	2.20214	0.0727210
$918$	−11.7172	−0.386725
$919$	−45.4482	−1.49920	−0.749599	−	0.661892i	$-0.769754\pi$
$919$	−45.4482	−1.49920	−0.749599	+	0.661892i	$0.769754\pi$
$920$	14.3143	0.471930
$921$	−38.7698	−1.27751
$922$	−2.65682	−0.0874978
$923$	−11.8315	−0.389439
$924$	−1.55300	−0.0510899
$925$	−22.2886	−0.732845
$926$	−9.37826	−0.308189
$927$	−9.89164	−0.324884
$928$	−9.60384	−0.315261
$929$	−16.3576	−0.536675	−0.268338	−	0.963325i	$-0.586474\pi$
$929$	−16.3576	−0.536675	−0.268338	+	0.963325i	$0.586474\pi$
$930$	−36.7834	−1.20618
$931$	46.5872	1.52683
$932$	−0.368052	−0.0120559
$933$	3.92371	0.128457
$934$	15.2842	0.500115
$935$	19.8427	0.648925
$936$	−2.21617	−0.0724377
$937$	22.9535	0.749858	0.374929	−	0.927053i	$-0.377667\pi$
$937$	22.9535	0.749858	0.374929	+	0.927053i	$0.377667\pi$
$938$	−1.77686	−0.0580165
$939$	17.3113	0.564934
$940$	43.3676	1.41450
$941$	40.6526	1.32524	0.662618	−	0.748957i	$-0.269445\pi$
$941$	40.6526	1.32524	0.662618	+	0.748957i	$0.269445\pi$
$942$	−32.4028	−1.05574
$943$	3.63138	0.118254
$944$	−10.5518	−0.343433
$945$	−8.06654	−0.262405
$946$	−11.7870	−0.383227
$947$	−13.3149	−0.432675	−0.216338	−	0.976319i	$-0.569411\pi$
$947$	−13.3149	−0.432675	−0.216338	+	0.976319i	$0.569411\pi$
$948$	7.21388	0.234296
$949$	16.4695	0.534623
$950$	−85.9465	−2.78847
$951$	14.9181	0.483752
$952$	1.36849	0.0443530
$953$	20.9200	0.677666	0.338833	−	0.940847i	$-0.389968\pi$
$953$	20.9200	0.677666	0.338833	+	0.940847i	$0.389968\pi$
$954$	−8.06746	−0.261194
$955$	−62.0564	−2.00810
$956$	−17.7774	−0.574962
$957$	−31.4277	−1.01591
$958$	−40.6094	−1.31203
$959$	−7.46500	−0.241057
$960$	−8.32159	−0.268578
$961$	−11.4615	−0.369725
$962$	4.12786	0.133088
$963$	−13.1810	−0.424753
$964$	−26.0379	−0.838625
$965$	−78.6898	−2.53312
$966$	3.23058	0.103942
$967$	−8.80297	−0.283085	−0.141542	−	0.989932i	$-0.545206\pi$
$967$	−8.80297	−0.283085	−0.141542	+	0.989932i	$0.545206\pi$
$968$	−8.29403	−0.266580
$969$	−39.4472	−1.26723
$970$	−75.2163	−2.41505
$971$	4.87576	0.156471	0.0782354	−	0.996935i	$-0.475071\pi$
$971$	4.87576	0.156471	0.0782354	+	0.996935i	$0.475071\pi$
$972$	9.60423	0.308056
$973$	−7.89888	−0.253226
$974$	−13.7642	−0.441035
$975$	57.5522	1.84314
$976$	−0.448317	−0.0143503
$977$	−15.2086	−0.486566	−0.243283	−	0.969955i	$-0.578224\pi$
$977$	−15.2086	−0.486566	−0.243283	+	0.969955i	$0.578224\pi$
$978$	8.18011	0.261571
$979$	8.97055	0.286700
$980$	−28.3397	−0.905278
$981$	−17.4040	−0.555666
$982$	−3.16730	−0.101073
$983$	−26.5822	−0.847842	−0.423921	−	0.905699i	$-0.639347\pi$
$983$	−26.5822	−0.847842	−0.423921	+	0.905699i	$0.639347\pi$
$984$	−2.11109	−0.0672991
$985$	84.6942	2.69858
$986$	27.6939	0.881953
$987$	9.78756	0.311542
$988$	15.9173	0.506398
$989$	24.5195	0.779674
$990$	−6.58818	−0.209386
$991$	−51.7593	−1.64419	−0.822094	−	0.569352i	$-0.807194\pi$
$991$	−51.7593	−1.64419	−0.822094	+	0.569352i	$0.807194\pi$
$992$	4.42024	0.140343
$993$	30.7548	0.975974
$994$	−2.42574	−0.0769399
$995$	56.8646	1.80273
$996$	−3.09482	−0.0980632
$997$	−53.7429	−1.70206	−0.851028	−	0.525120i	$-0.824020\pi$
$997$	−53.7429	−1.70206	−0.851028	+	0.525120i	$0.824020\pi$
$998$	8.62864	0.273135
$999$	−7.24622	−0.229260

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8042.2.a.a.1.16	✓	67

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8042.2.a.a.1.16	✓	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.98933 q^{3} +1.00000 q^{4} +4.18312 q^{5} -1.98933 q^{6} -0.474573 q^{7} +1.00000 q^{8} +0.957423 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.98933 q^{3} +1.00000 q^{4} +4.18312 q^{5} -1.98933 q^{6} -0.474573 q^{7} +1.00000 q^{8} +0.957423 q^{9} +4.18312 q^{10} -1.64498 q^{11} -1.98933 q^{12} -2.31472 q^{13} -0.474573 q^{14} -8.32159 q^{15} +1.00000 q^{16} -2.88362 q^{17} +0.957423 q^{18} -6.87657 q^{19} +4.18312 q^{20} +0.944080 q^{21} -1.64498 q^{22} +3.42193 q^{23} -1.98933 q^{24} +12.4985 q^{25} -2.31472 q^{26} +4.06335 q^{27} -0.474573 q^{28} -9.60384 q^{29} -8.32159 q^{30} +4.42024 q^{31} +1.00000 q^{32} +3.27241 q^{33} -2.88362 q^{34} -1.98519 q^{35} +0.957423 q^{36} -1.78331 q^{37} -6.87657 q^{38} +4.60474 q^{39} +4.18312 q^{40} +1.06121 q^{41} +0.944080 q^{42} +7.16539 q^{43} -1.64498 q^{44} +4.00501 q^{45} +3.42193 q^{46} +10.3673 q^{47} -1.98933 q^{48} -6.77478 q^{49} +12.4985 q^{50} +5.73647 q^{51} -2.31472 q^{52} -8.42623 q^{53} +4.06335 q^{54} -6.88116 q^{55} -0.474573 q^{56} +13.6797 q^{57} -9.60384 q^{58} -10.5518 q^{59} -8.32159 q^{60} -0.448317 q^{61} +4.42024 q^{62} -0.454366 q^{63} +1.00000 q^{64} -9.68275 q^{65} +3.27241 q^{66} +3.74413 q^{67} -2.88362 q^{68} -6.80734 q^{69} -1.98519 q^{70} +5.11142 q^{71} +0.957423 q^{72} -7.11511 q^{73} -1.78331 q^{74} -24.8635 q^{75} -6.87657 q^{76} +0.780664 q^{77} +4.60474 q^{78} -3.62629 q^{79} +4.18312 q^{80} -10.9556 q^{81} +1.06121 q^{82} +1.55571 q^{83} +0.944080 q^{84} -12.0625 q^{85} +7.16539 q^{86} +19.1052 q^{87} -1.64498 q^{88} -5.45327 q^{89} +4.00501 q^{90} +1.09850 q^{91} +3.42193 q^{92} -8.79331 q^{93} +10.3673 q^{94} -28.7655 q^{95} -1.98933 q^{96} -17.9809 q^{97} -6.77478 q^{98} -1.57494 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9}+O(q^{10}) \) 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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