LMFDB - Embedded newform 8042.2.a.b.1.12

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:	$82$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
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)

$f(q)$	$=$	$q-1.00000 q^{2} -2.79380 q^{3} +1.00000 q^{4} -0.886780 q^{5} +2.79380 q^{6} +2.62201 q^{7} -1.00000 q^{8} +4.80532 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.79380 q^{3} +1.00000 q^{4} -0.886780 q^{5} +2.79380 q^{6} +2.62201 q^{7} -1.00000 q^{8} +4.80532 q^{9} +0.886780 q^{10} +3.03259 q^{11} -2.79380 q^{12} -2.48346 q^{13} -2.62201 q^{14} +2.47749 q^{15} +1.00000 q^{16} -4.84175 q^{17} -4.80532 q^{18} +3.55000 q^{19} -0.886780 q^{20} -7.32538 q^{21} -3.03259 q^{22} +5.55378 q^{23} +2.79380 q^{24} -4.21362 q^{25} +2.48346 q^{26} -5.04370 q^{27} +2.62201 q^{28} +4.20868 q^{29} -2.47749 q^{30} +7.14850 q^{31} -1.00000 q^{32} -8.47245 q^{33} +4.84175 q^{34} -2.32515 q^{35} +4.80532 q^{36} -6.83685 q^{37} -3.55000 q^{38} +6.93830 q^{39} +0.886780 q^{40} +3.04708 q^{41} +7.32538 q^{42} -3.30178 q^{43} +3.03259 q^{44} -4.26126 q^{45} -5.55378 q^{46} -1.83368 q^{47} -2.79380 q^{48} -0.125046 q^{49} +4.21362 q^{50} +13.5269 q^{51} -2.48346 q^{52} -12.5213 q^{53} +5.04370 q^{54} -2.68924 q^{55} -2.62201 q^{56} -9.91799 q^{57} -4.20868 q^{58} -1.13641 q^{59} +2.47749 q^{60} -6.09911 q^{61} -7.14850 q^{62} +12.5996 q^{63} +1.00000 q^{64} +2.20229 q^{65} +8.47245 q^{66} -3.46115 q^{67} -4.84175 q^{68} -15.5162 q^{69} +2.32515 q^{70} -2.15439 q^{71} -4.80532 q^{72} +3.89521 q^{73} +6.83685 q^{74} +11.7720 q^{75} +3.55000 q^{76} +7.95149 q^{77} -6.93830 q^{78} +3.11508 q^{79} -0.886780 q^{80} -0.324871 q^{81} -3.04708 q^{82} -5.00884 q^{83} -7.32538 q^{84} +4.29357 q^{85} +3.30178 q^{86} -11.7582 q^{87} -3.03259 q^{88} +11.3563 q^{89} +4.26126 q^{90} -6.51168 q^{91} +5.55378 q^{92} -19.9715 q^{93} +1.83368 q^{94} -3.14807 q^{95} +2.79380 q^{96} -11.4987 q^{97} +0.125046 q^{98} +14.5726 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 82 q - 82 q^{2} - 13 q^{3} + 82 q^{4} + 3 q^{5} + 13 q^{6} - 37 q^{7} - 82 q^{8} + 91 q^{9}+O(q^{10}) $ 82 * q - 82 * q^2 - 13 * q^3 + 82 * q^4 + 3 * q^5 + 13 * q^6 - 37 * q^7 - 82 * q^8 + 91 * q^9	\( 82 q - 82 q^{2} - 13 q^{3} + 82 q^{4} + 3 q^{5} + 13 q^{6} - 37 q^{7} - 82 q^{8} + 91 q^{9} - 3 q^{10} - 16 q^{11} - 13 q^{12} - 42 q^{13} + 37 q^{14} - 9 q^{15} + 82 q^{16} + 3 q^{17} - 91 q^{18} - 42 q^{19} + 3 q^{20} - q^{21} + 16 q^{22} - 6 q^{23} + 13 q^{24} + 53 q^{25} + 42 q^{26} - 49 q^{27} - 37 q^{28} + 15 q^{29} + 9 q^{30} - 40 q^{31} - 82 q^{32} - 37 q^{33} - 3 q^{34} - 42 q^{35} + 91 q^{36} - 72 q^{37} + 42 q^{38} - 14 q^{39} - 3 q^{40} + 8 q^{41} + q^{42} - 93 q^{43} - 16 q^{44} - 11 q^{45} + 6 q^{46} + 7 q^{47} - 13 q^{48} + 61 q^{49} - 53 q^{50} - 70 q^{51} - 42 q^{52} + 18 q^{53} + 49 q^{54} - 62 q^{55} + 37 q^{56} - 51 q^{57} - 15 q^{58} - 47 q^{59} - 9 q^{60} - 14 q^{61} + 40 q^{62} - 100 q^{63} + 82 q^{64} + q^{65} + 37 q^{66} - 150 q^{67} + 3 q^{68} + 31 q^{69} + 42 q^{70} + 7 q^{71} - 91 q^{72} - 78 q^{73} + 72 q^{74} - 49 q^{75} - 42 q^{76} + 29 q^{77} + 14 q^{78} - 59 q^{79} + 3 q^{80} + 122 q^{81} - 8 q^{82} - 52 q^{83} - q^{84} - 108 q^{85} + 93 q^{86} - 49 q^{87} + 16 q^{88} + 38 q^{89} + 11 q^{90} - 69 q^{91} - 6 q^{92} - 63 q^{93} - 7 q^{94} + 5 q^{95} + 13 q^{96} - 74 q^{97} - 61 q^{98} - 89 q^{99}+O(q^{100}) \) 82 * q - 82 * q^2 - 13 * q^3 + 82 * q^4 + 3 * q^5 + 13 * q^6 - 37 * q^7 - 82 * q^8 + 91 * q^9 - 3 * q^10 - 16 * q^11 - 13 * q^12 - 42 * q^13 + 37 * q^14 - 9 * q^15 + 82 * q^16 + 3 * q^17 - 91 * q^18 - 42 * q^19 + 3 * q^20 - q^21 + 16 * q^22 - 6 * q^23 + 13 * q^24 + 53 * q^25 + 42 * q^26 - 49 * q^27 - 37 * q^28 + 15 * q^29 + 9 * q^30 - 40 * q^31 - 82 * q^32 - 37 * q^33 - 3 * q^34 - 42 * q^35 + 91 * q^36 - 72 * q^37 + 42 * q^38 - 14 * q^39 - 3 * q^40 + 8 * q^41 + q^42 - 93 * q^43 - 16 * q^44 - 11 * q^45 + 6 * q^46 + 7 * q^47 - 13 * q^48 + 61 * q^49 - 53 * q^50 - 70 * q^51 - 42 * q^52 + 18 * q^53 + 49 * q^54 - 62 * q^55 + 37 * q^56 - 51 * q^57 - 15 * q^58 - 47 * q^59 - 9 * q^60 - 14 * q^61 + 40 * q^62 - 100 * q^63 + 82 * q^64 + q^65 + 37 * q^66 - 150 * q^67 + 3 * q^68 + 31 * q^69 + 42 * q^70 + 7 * q^71 - 91 * q^72 - 78 * q^73 + 72 * q^74 - 49 * q^75 - 42 * q^76 + 29 * q^77 + 14 * q^78 - 59 * q^79 + 3 * q^80 + 122 * q^81 - 8 * q^82 - 52 * q^83 - q^84 - 108 * q^85 + 93 * q^86 - 49 * q^87 + 16 * q^88 + 38 * q^89 + 11 * q^90 - 69 * q^91 - 6 * q^92 - 63 * q^93 - 7 * q^94 + 5 * q^95 + 13 * q^96 - 74 * q^97 - 61 * q^98 - 89 * 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
$3$	−2.79380	−1.61300	−0.806501	−	0.591233i	$-0.798641\pi$
$3$	−2.79380	−1.61300	−0.806501	+	0.591233i	$0.798641\pi$
$4$	1.00000	0.500000
$5$	−0.886780	−0.396580	−0.198290	−	0.980143i	$-0.563539\pi$
$5$	−0.886780	−0.396580	−0.198290	+	0.980143i	$0.563539\pi$
$6$	2.79380	1.14056
$7$	2.62201	0.991028	0.495514	−	0.868600i	$-0.334980\pi$
$7$	2.62201	0.991028	0.495514	+	0.868600i	$0.334980\pi$
$8$	−1.00000	−0.353553
$9$	4.80532	1.60177
$10$	0.886780	0.280424
$11$	3.03259	0.914360	0.457180	−	0.889374i	$-0.348859\pi$
$11$	3.03259	0.914360	0.457180	+	0.889374i	$0.348859\pi$
$12$	−2.79380	−0.806501
$13$	−2.48346	−0.688789	−0.344395	−	0.938825i	$-0.611916\pi$
$13$	−2.48346	−0.688789	−0.344395	+	0.938825i	$0.611916\pi$
$14$	−2.62201	−0.700763
$15$	2.47749	0.639684
$16$	1.00000	0.250000
$17$	−4.84175	−1.17430	−0.587148	−	0.809479i	$-0.699750\pi$
$17$	−4.84175	−1.17430	−0.587148	+	0.809479i	$0.699750\pi$
$18$	−4.80532	−1.13262
$19$	3.55000	0.814426	0.407213	−	0.913333i	$-0.366501\pi$
$19$	3.55000	0.814426	0.407213	+	0.913333i	$0.366501\pi$
$20$	−0.886780	−0.198290
$21$	−7.32538	−1.59853
$22$	−3.03259	−0.646551
$23$	5.55378	1.15804	0.579022	−	0.815312i	$-0.303435\pi$
$23$	5.55378	1.15804	0.579022	+	0.815312i	$0.303435\pi$
$24$	2.79380	0.570282
$25$	−4.21362	−0.842724
$26$	2.48346	0.487047
$27$	−5.04370	−0.970660
$28$	2.62201	0.495514
$29$	4.20868	0.781532	0.390766	−	0.920490i	$-0.372210\pi$
$29$	4.20868	0.781532	0.390766	+	0.920490i	$0.372210\pi$
$30$	−2.47749	−0.452325
$31$	7.14850	1.28391	0.641954	−	0.766743i	$-0.278124\pi$
$31$	7.14850	1.28391	0.641954	+	0.766743i	$0.278124\pi$
$32$	−1.00000	−0.176777
$33$	−8.47245	−1.47486
$34$	4.84175	0.830353
$35$	−2.32515	−0.393022
$36$	4.80532	0.800886
$37$	−6.83685	−1.12397	−0.561986	−	0.827147i	$-0.689962\pi$
$37$	−6.83685	−1.12397	−0.561986	+	0.827147i	$0.689962\pi$
$38$	−3.55000	−0.575886
$39$	6.93830	1.11102
$40$	0.886780	0.140212
$41$	3.04708	0.475874	0.237937	−	0.971281i	$-0.423529\pi$
$41$	3.04708	0.475874	0.237937	+	0.971281i	$0.423529\pi$
$42$	7.32538	1.13033
$43$	−3.30178	−0.503516	−0.251758	−	0.967790i	$-0.581009\pi$
$43$	−3.30178	−0.503516	−0.251758	+	0.967790i	$0.581009\pi$
$44$	3.03259	0.457180
$45$	−4.26126	−0.635231
$46$	−5.55378	−0.818861
$47$	−1.83368	−0.267470	−0.133735	−	0.991017i	$-0.542697\pi$
$47$	−1.83368	−0.267470	−0.133735	+	0.991017i	$0.542697\pi$
$48$	−2.79380	−0.403250
$49$	−0.125046	−0.0178637
$50$	4.21362	0.595896
$51$	13.5269	1.89414
$52$	−2.48346	−0.344395
$53$	−12.5213	−1.71994	−0.859970	−	0.510345i	$-0.829518\pi$
$53$	−12.5213	−1.71994	−0.859970	+	0.510345i	$0.829518\pi$
$54$	5.04370	0.686360
$55$	−2.68924	−0.362617
$56$	−2.62201	−0.350381
$57$	−9.91799	−1.31367
$58$	−4.20868	−0.552627
$59$	−1.13641	−0.147948	−0.0739740	−	0.997260i	$-0.523568\pi$
$59$	−1.13641	−0.147948	−0.0739740	+	0.997260i	$0.523568\pi$
$60$	2.47749	0.319842
$61$	−6.09911	−0.780910	−0.390455	−	0.920622i	$-0.627682\pi$
$61$	−6.09911	−0.780910	−0.390455	+	0.920622i	$0.627682\pi$
$62$	−7.14850	−0.907860
$63$	12.5996	1.58740
$64$	1.00000	0.125000
$65$	2.20229	0.273160
$66$	8.47245	1.04289
$67$	−3.46115	−0.422847	−0.211423	−	0.977395i	$-0.567810\pi$
$67$	−3.46115	−0.422847	−0.211423	+	0.977395i	$0.567810\pi$
$68$	−4.84175	−0.587148
$69$	−15.5162	−1.86793
$70$	2.32515	0.277908
$71$	−2.15439	−0.255679	−0.127840	−	0.991795i	$-0.540804\pi$
$71$	−2.15439	−0.255679	−0.127840	+	0.991795i	$0.540804\pi$
$72$	−4.80532	−0.566312
$73$	3.89521	0.455900	0.227950	−	0.973673i	$-0.426798\pi$
$73$	3.89521	0.455900	0.227950	+	0.973673i	$0.426798\pi$
$74$	6.83685	0.794768
$75$	11.7720	1.35932
$76$	3.55000	0.407213
$77$	7.95149	0.906157
$78$	−6.93830	−0.785608
$79$	3.11508	0.350474	0.175237	−	0.984526i	$-0.443931\pi$
$79$	3.11508	0.350474	0.175237	+	0.984526i	$0.443931\pi$
$80$	−0.886780	−0.0991450
$81$	−0.324871	−0.0360968
$82$	−3.04708	−0.336494
$83$	−5.00884	−0.549792	−0.274896	−	0.961474i	$-0.588643\pi$
$83$	−5.00884	−0.549792	−0.274896	+	0.961474i	$0.588643\pi$
$84$	−7.32538	−0.799265
$85$	4.29357	0.465703
$86$	3.30178	0.356040
$87$	−11.7582	−1.26061
$88$	−3.03259	−0.323275
$89$	11.3563	1.20377	0.601884	−	0.798583i	$-0.294417\pi$
$89$	11.3563	1.20377	0.601884	+	0.798583i	$0.294417\pi$
$90$	4.26126	0.449176
$91$	−6.51168	−0.682609
$92$	5.55378	0.579022
$93$	−19.9715	−2.07095
$94$	1.83368	0.189130
$95$	−3.14807	−0.322985
$96$	2.79380	0.285141
$97$	−11.4987	−1.16752	−0.583759	−	0.811927i	$-0.698419\pi$
$97$	−11.4987	−1.16752	−0.583759	+	0.811927i	$0.698419\pi$
$98$	0.125046	0.0126315
$99$	14.5726	1.46460
$100$	−4.21362	−0.421362
$101$	−2.74390	−0.273028	−0.136514	−	0.990638i	$-0.543590\pi$
$101$	−2.74390	−0.273028	−0.136514	+	0.990638i	$0.543590\pi$
$102$	−13.5269	−1.33936
$103$	7.54518	0.743448	0.371724	−	0.928343i	$-0.378767\pi$
$103$	7.54518	0.743448	0.371724	+	0.928343i	$0.378767\pi$
$104$	2.48346	0.243524
$105$	6.49600	0.633945
$106$	12.5213	1.21618
$107$	2.72089	0.263038	0.131519	−	0.991314i	$-0.458015\pi$
$107$	2.72089	0.263038	0.131519	+	0.991314i	$0.458015\pi$
$108$	−5.04370	−0.485330
$109$	−12.0644	−1.15556	−0.577778	−	0.816194i	$-0.696080\pi$
$109$	−12.0644	−1.15556	−0.577778	+	0.816194i	$0.696080\pi$
$110$	2.68924	0.256409
$111$	19.1008	1.81297
$112$	2.62201	0.247757
$113$	4.67196	0.439501	0.219751	−	0.975556i	$-0.429476\pi$
$113$	4.67196	0.439501	0.219751	+	0.975556i	$0.429476\pi$
$114$	9.91799	0.928905
$115$	−4.92499	−0.459257
$116$	4.20868	0.390766
$117$	−11.9338	−1.10328
$118$	1.13641	0.104615
$119$	−12.6951	−1.16376
$120$	−2.47749	−0.226162
$121$	−1.80339	−0.163945
$122$	6.09911	0.552187
$123$	−8.51293	−0.767585
$124$	7.14850	0.641954
$125$	8.17046	0.730788
$126$	−12.5996	−1.12246
$127$	−16.1272	−1.43106	−0.715529	−	0.698584i	$-0.753814\pi$
$127$	−16.1272	−1.43106	−0.715529	+	0.698584i	$0.753814\pi$
$128$	−1.00000	−0.0883883
$129$	9.22451	0.812172
$130$	−2.20229	−0.193153
$131$	−8.49505	−0.742216	−0.371108	−	0.928590i	$-0.621022\pi$
$131$	−8.49505	−0.742216	−0.371108	+	0.928590i	$0.621022\pi$
$132$	−8.47245	−0.737432
$133$	9.30815	0.807119
$134$	3.46115	0.298998
$135$	4.47265	0.384944
$136$	4.84175	0.415177
$137$	−9.95240	−0.850291	−0.425146	−	0.905125i	$-0.639777\pi$
$137$	−9.95240	−0.850291	−0.425146	+	0.905125i	$0.639777\pi$
$138$	15.5162	1.32082
$139$	−9.70314	−0.823009	−0.411505	−	0.911408i	$-0.634997\pi$
$139$	−9.70314	−0.823009	−0.411505	+	0.911408i	$0.634997\pi$
$140$	−2.32515	−0.196511
$141$	5.12294	0.431430
$142$	2.15439	0.180793
$143$	−7.53133	−0.629802
$144$	4.80532	0.400443
$145$	−3.73217	−0.309940
$146$	−3.89521	−0.322370
$147$	0.349353	0.0288142
$148$	−6.83685	−0.561986
$149$	1.29208	0.105851	0.0529255	−	0.998598i	$-0.483145\pi$
$149$	1.29208	0.105851	0.0529255	+	0.998598i	$0.483145\pi$
$150$	−11.7720	−0.961181
$151$	10.0253	0.815851	0.407925	−	0.913015i	$-0.366252\pi$
$151$	10.0253	0.815851	0.407925	+	0.913015i	$0.366252\pi$
$152$	−3.55000	−0.287943
$153$	−23.2661	−1.88096
$154$	−7.95149	−0.640750
$155$	−6.33915	−0.509172
$156$	6.93830	0.555509
$157$	11.8337	0.944432	0.472216	−	0.881483i	$-0.343454\pi$
$157$	11.8337	0.944432	0.472216	+	0.881483i	$0.343454\pi$
$158$	−3.11508	−0.247823
$159$	34.9821	2.77426
$160$	0.886780	0.0701061
$161$	14.5621	1.14765
$162$	0.324871	0.0255243
$163$	2.58894	0.202781	0.101391	−	0.994847i	$-0.467671\pi$
$163$	2.58894	0.202781	0.101391	+	0.994847i	$0.467671\pi$
$164$	3.04708	0.237937
$165$	7.51320	0.584902
$166$	5.00884	0.388762
$167$	18.4541	1.42802	0.714009	−	0.700136i	$-0.246877\pi$
$167$	18.4541	1.42802	0.714009	+	0.700136i	$0.246877\pi$
$168$	7.32538	0.565165
$169$	−6.83240	−0.525570
$170$	−4.29357	−0.329302
$171$	17.0589	1.30452
$172$	−3.30178	−0.251758
$173$	−1.33836	−0.101753	−0.0508767	−	0.998705i	$-0.516202\pi$
$173$	−1.33836	−0.101753	−0.0508767	+	0.998705i	$0.516202\pi$
$174$	11.7582	0.891387
$175$	−11.0482	−0.835163
$176$	3.03259	0.228590
$177$	3.17490	0.238640
$178$	−11.3563	−0.851193
$179$	22.8316	1.70652	0.853258	−	0.521489i	$-0.174623\pi$
$179$	22.8316	1.70652	0.853258	+	0.521489i	$0.174623\pi$
$180$	−4.26126	−0.317616
$181$	4.72498	0.351205	0.175603	−	0.984461i	$-0.443813\pi$
$181$	4.72498	0.351205	0.175603	+	0.984461i	$0.443813\pi$
$182$	6.51168	0.482678
$183$	17.0397	1.25961
$184$	−5.55378	−0.409430
$185$	6.06278	0.445745
$186$	19.9715	1.46438
$187$	−14.6830	−1.07373
$188$	−1.83368	−0.133735
$189$	−13.2246	−0.961951
$190$	3.14807	0.228385
$191$	6.76562	0.489543	0.244772	−	0.969581i	$-0.421287\pi$
$191$	6.76562	0.489543	0.244772	+	0.969581i	$0.421287\pi$
$192$	−2.79380	−0.201625
$193$	−5.13844	−0.369873	−0.184936	−	0.982751i	$-0.559208\pi$
$193$	−5.13844	−0.369873	−0.184936	+	0.982751i	$0.559208\pi$
$194$	11.4987	0.825561
$195$	−6.15275	−0.440607
$196$	−0.125046	−0.00893185
$197$	−10.2964	−0.733587	−0.366793	−	0.930302i	$-0.619544\pi$
$197$	−10.2964	−0.733587	−0.366793	+	0.930302i	$0.619544\pi$
$198$	−14.5726	−1.03563
$199$	12.9955	0.921225	0.460613	−	0.887601i	$-0.347630\pi$
$199$	12.9955	0.921225	0.460613	+	0.887601i	$0.347630\pi$
$200$	4.21362	0.297948
$201$	9.66977	0.682053
$202$	2.74390	0.193060
$203$	11.0352	0.774520
$204$	13.5269	0.947071
$205$	−2.70209	−0.188722
$206$	−7.54518	−0.525697
$207$	26.6877	1.85492
$208$	−2.48346	−0.172197
$209$	10.7657	0.744679
$210$	−6.49600	−0.448267
$211$	6.80736	0.468638	0.234319	−	0.972160i	$-0.424714\pi$
$211$	6.80736	0.468638	0.234319	+	0.972160i	$0.424714\pi$
$212$	−12.5213	−0.859970
$213$	6.01894	0.412411
$214$	−2.72089	−0.185996
$215$	2.92795	0.199685
$216$	5.04370	0.343180
$217$	18.7435	1.27239
$218$	12.0644	0.817101
$219$	−10.8824	−0.735367
$220$	−2.68924	−0.181309
$221$	12.0243	0.808843
$222$	−19.1008	−1.28196
$223$	−8.65283	−0.579436	−0.289718	−	0.957112i	$-0.593562\pi$
$223$	−8.65283	−0.579436	−0.289718	+	0.957112i	$0.593562\pi$
$224$	−2.62201	−0.175191
$225$	−20.2478	−1.34985
$226$	−4.67196	−0.310774
$227$	7.09709	0.471051	0.235525	−	0.971868i	$-0.424319\pi$
$227$	7.09709	0.471051	0.235525	+	0.971868i	$0.424319\pi$
$228$	−9.91799	−0.656835
$229$	18.3506	1.21264	0.606320	−	0.795221i	$-0.292645\pi$
$229$	18.3506	1.21264	0.606320	+	0.795221i	$0.292645\pi$
$230$	4.92499	0.324744
$231$	−22.2149	−1.46163
$232$	−4.20868	−0.276313
$233$	17.1286	1.12213	0.561066	−	0.827771i	$-0.310391\pi$
$233$	17.1286	1.12213	0.561066	+	0.827771i	$0.310391\pi$
$234$	11.9338	0.780139
$235$	1.62607	0.106073
$236$	−1.13641	−0.0739740
$237$	−8.70291	−0.565315
$238$	12.6951	0.822903
$239$	−21.6573	−1.40089	−0.700446	−	0.713705i	$-0.747016\pi$
$239$	−21.6573	−1.40089	−0.700446	+	0.713705i	$0.747016\pi$
$240$	2.47749	0.159921
$241$	8.66815	0.558365	0.279182	−	0.960238i	$-0.409937\pi$
$241$	8.66815	0.558365	0.279182	+	0.960238i	$0.409937\pi$
$242$	1.80339	0.115927
$243$	16.0387	1.02888
$244$	−6.09911	−0.390455
$245$	0.110888	0.00708438
$246$	8.51293	0.542765
$247$	−8.81630	−0.560968
$248$	−7.14850	−0.453930
$249$	13.9937	0.886815
$250$	−8.17046	−0.516745
$251$	16.4218	1.03654	0.518268	−	0.855218i	$-0.326577\pi$
$251$	16.4218	1.03654	0.518268	+	0.855218i	$0.326577\pi$
$252$	12.5996	0.793701
$253$	16.8424	1.05887
$254$	16.1272	1.01191
$255$	−11.9954	−0.751179
$256$	1.00000	0.0625000
$257$	13.1522	0.820413	0.410207	−	0.911993i	$-0.365457\pi$
$257$	13.1522	0.820413	0.410207	+	0.911993i	$0.365457\pi$
$258$	−9.22451	−0.574293
$259$	−17.9263	−1.11389
$260$	2.20229	0.136580
$261$	20.2240	1.25184
$262$	8.49505	0.524826
$263$	1.27114	0.0783819	0.0391909	−	0.999232i	$-0.487522\pi$
$263$	1.27114	0.0783819	0.0391909	+	0.999232i	$0.487522\pi$
$264$	8.47245	0.521443
$265$	11.1037	0.682094
$266$	−9.30815	−0.570719
$267$	−31.7273	−1.94168
$268$	−3.46115	−0.211423
$269$	−16.5947	−1.01180	−0.505898	−	0.862593i	$-0.668839\pi$
$269$	−16.5947	−1.01180	−0.505898	+	0.862593i	$0.668839\pi$
$270$	−4.47265	−0.272197
$271$	−3.35907	−0.204049	−0.102024	−	0.994782i	$-0.532532\pi$
$271$	−3.35907	−0.204049	−0.102024	+	0.994782i	$0.532532\pi$
$272$	−4.84175	−0.293574
$273$	18.1923	1.10105
$274$	9.95240	0.601247
$275$	−12.7782	−0.770554
$276$	−15.5162	−0.933963
$277$	30.9050	1.85690	0.928452	−	0.371453i	$-0.121140\pi$
$277$	30.9050	1.85690	0.928452	+	0.371453i	$0.121140\pi$
$278$	9.70314	0.581955
$279$	34.3508	2.05653
$280$	2.32515	0.138954
$281$	−8.14494	−0.485886	−0.242943	−	0.970041i	$-0.578113\pi$
$281$	−8.14494	−0.485886	−0.242943	+	0.970041i	$0.578113\pi$
$282$	−5.12294	−0.305067
$283$	−19.0145	−1.13029	−0.565147	−	0.824990i	$-0.691181\pi$
$283$	−19.0145	−1.13029	−0.565147	+	0.824990i	$0.691181\pi$
$284$	−2.15439	−0.127840
$285$	8.79507	0.520975
$286$	7.53133	0.445337
$287$	7.98948	0.471604
$288$	−4.80532	−0.283156
$289$	6.44254	0.378973
$290$	3.73217	0.219161
$291$	32.1251	1.88321
$292$	3.89521	0.227950
$293$	−14.2791	−0.834193	−0.417096	−	0.908862i	$-0.636952\pi$
$293$	−14.2791	−0.834193	−0.417096	+	0.908862i	$0.636952\pi$
$294$	−0.349353	−0.0203747
$295$	1.00775	0.0586733
$296$	6.83685	0.397384
$297$	−15.2955	−0.887533
$298$	−1.29208	−0.0748480
$299$	−13.7926	−0.797648
$300$	11.7720	0.679658
$301$	−8.65730	−0.498999
$302$	−10.0253	−0.576894
$303$	7.66590	0.440394
$304$	3.55000	0.203606
$305$	5.40856	0.309694
$306$	23.2661	1.33004
$307$	−1.71575	−0.0979230	−0.0489615	−	0.998801i	$-0.515591\pi$
$307$	−1.71575	−0.0979230	−0.0489615	+	0.998801i	$0.515591\pi$
$308$	7.95149	0.453078
$309$	−21.0797	−1.19918
$310$	6.33915	0.360039
$311$	−7.19130	−0.407781	−0.203891	−	0.978994i	$-0.565359\pi$
$311$	−7.19130	−0.407781	−0.203891	+	0.978994i	$0.565359\pi$
$312$	−6.93830	−0.392804
$313$	−3.03697	−0.171660	−0.0858299	−	0.996310i	$-0.527354\pi$
$313$	−3.03697	−0.171660	−0.0858299	+	0.996310i	$0.527354\pi$
$314$	−11.8337	−0.667815
$315$	−11.1731	−0.629532
$316$	3.11508	0.175237
$317$	−14.8692	−0.835135	−0.417568	−	0.908646i	$-0.637117\pi$
$317$	−14.8692	−0.835135	−0.417568	+	0.908646i	$0.637117\pi$
$318$	−34.9821	−1.96170
$319$	12.7632	0.714602
$320$	−0.886780	−0.0495725
$321$	−7.60161	−0.424280
$322$	−14.5621	−0.811514
$323$	−17.1882	−0.956378
$324$	−0.324871	−0.0180484
$325$	10.4644	0.580459
$326$	−2.58894	−0.143388
$327$	33.7054	1.86391
$328$	−3.04708	−0.168247
$329$	−4.80794	−0.265070
$330$	−7.51320	−0.413588
$331$	−29.5660	−1.62510	−0.812549	−	0.582894i	$-0.801920\pi$
$331$	−29.5660	−1.62510	−0.812549	+	0.582894i	$0.801920\pi$
$332$	−5.00884	−0.274896
$333$	−32.8532	−1.80035
$334$	−18.4541	−1.00976
$335$	3.06928	0.167693
$336$	−7.32538	−0.399632
$337$	−22.6440	−1.23350	−0.616748	−	0.787161i	$-0.711550\pi$
$337$	−22.6440	−1.23350	−0.616748	+	0.787161i	$0.711550\pi$
$338$	6.83240	0.371634
$339$	−13.0525	−0.708916
$340$	4.29357	0.232851
$341$	21.6785	1.17396
$342$	−17.0589	−0.922438
$343$	−18.6820	−1.00873
$344$	3.30178	0.178020
$345$	13.7594	0.740782
$346$	1.33836	0.0719505
$347$	24.3286	1.30603	0.653014	−	0.757345i	$-0.273504\pi$
$347$	24.3286	1.30603	0.653014	+	0.757345i	$0.273504\pi$
$348$	−11.7582	−0.630306
$349$	−5.54334	−0.296728	−0.148364	−	0.988933i	$-0.547401\pi$
$349$	−5.54334	−0.296728	−0.148364	+	0.988933i	$0.547401\pi$
$350$	11.0482	0.590550
$351$	12.5258	0.668580
$352$	−3.03259	−0.161638
$353$	23.6236	1.25736	0.628679	−	0.777665i	$-0.283596\pi$
$353$	23.6236	1.25736	0.628679	+	0.777665i	$0.283596\pi$
$354$	−3.17490	−0.168744
$355$	1.91047	0.101397
$356$	11.3563	0.601884
$357$	35.4677	1.87715
$358$	−22.8316	−1.20669
$359$	25.3038	1.33548	0.667741	−	0.744394i	$-0.267261\pi$
$359$	25.3038	1.33548	0.667741	+	0.744394i	$0.267261\pi$
$360$	4.26126	0.224588
$361$	−6.39750	−0.336711
$362$	−4.72498	−0.248340
$363$	5.03832	0.264443
$364$	−6.51168	−0.341305
$365$	−3.45419	−0.180801
$366$	−17.0397	−0.890678
$367$	−28.3647	−1.48063	−0.740314	−	0.672261i	$-0.765323\pi$
$367$	−28.3647	−1.48063	−0.740314	+	0.672261i	$0.765323\pi$
$368$	5.55378	0.289511
$369$	14.6422	0.762242
$370$	−6.06278	−0.315189
$371$	−32.8311	−1.70451
$372$	−19.9715	−1.03547
$373$	−26.3256	−1.36309	−0.681544	−	0.731777i	$-0.738691\pi$
$373$	−26.3256	−1.36309	−0.681544	+	0.731777i	$0.738691\pi$
$374$	14.6830	0.759242
$375$	−22.8266	−1.17876
$376$	1.83368	0.0945650
$377$	−10.4521	−0.538311
$378$	13.2246	0.680202
$379$	−28.0543	−1.44105	−0.720525	−	0.693429i	$-0.756099\pi$
$379$	−28.0543	−1.44105	−0.720525	+	0.693429i	$0.756099\pi$
$380$	−3.14807	−0.161493
$381$	45.0562	2.30830
$382$	−6.76562	−0.346159
$383$	12.4894	0.638178	0.319089	−	0.947725i	$-0.396623\pi$
$383$	12.4894	0.638178	0.319089	+	0.947725i	$0.396623\pi$
$384$	2.79380	0.142571
$385$	−7.05123	−0.359364
$386$	5.13844	0.261539
$387$	−15.8661	−0.806519
$388$	−11.4987	−0.583759
$389$	−10.0015	−0.507094	−0.253547	−	0.967323i	$-0.581597\pi$
$389$	−10.0015	−0.507094	−0.253547	+	0.967323i	$0.581597\pi$
$390$	6.15275	0.311557
$391$	−26.8900	−1.35989
$392$	0.125046	0.00631577
$393$	23.7335	1.19719
$394$	10.2964	0.518724
$395$	−2.76239	−0.138991
$396$	14.5726	0.732299
$397$	5.42949	0.272498	0.136249	−	0.990675i	$-0.456495\pi$
$397$	5.42949	0.272498	0.136249	+	0.990675i	$0.456495\pi$
$398$	−12.9955	−0.651405
$399$	−26.0051	−1.30188
$400$	−4.21362	−0.210681
$401$	−27.6484	−1.38069	−0.690347	−	0.723478i	$-0.742542\pi$
$401$	−27.6484	−1.38069	−0.690347	+	0.723478i	$0.742542\pi$
$402$	−9.66977	−0.482284
$403$	−17.7530	−0.884342
$404$	−2.74390	−0.136514
$405$	0.288089	0.0143153
$406$	−11.0352	−0.547668
$407$	−20.7334	−1.02771
$408$	−13.5269	−0.669680
$409$	8.15914	0.403444	0.201722	−	0.979443i	$-0.435346\pi$
$409$	8.15914	0.403444	0.201722	+	0.979443i	$0.435346\pi$
$410$	2.70209	0.133447
$411$	27.8050	1.37152
$412$	7.54518	0.371724
$413$	−2.97968	−0.146621
$414$	−26.6877	−1.31163
$415$	4.44174	0.218037
$416$	2.48346	0.121762
$417$	27.1086	1.32751
$418$	−10.7657	−0.526567
$419$	−10.8196	−0.528571	−0.264285	−	0.964445i	$-0.585136\pi$
$419$	−10.8196	−0.528571	−0.264285	+	0.964445i	$0.585136\pi$
$420$	6.49600	0.316972
$421$	−2.12525	−0.103579	−0.0517893	−	0.998658i	$-0.516492\pi$
$421$	−2.12525	−0.103579	−0.0517893	+	0.998658i	$0.516492\pi$
$422$	−6.80736	−0.331377
$423$	−8.81143	−0.428426
$424$	12.5213	0.608090
$425$	20.4013	0.989608
$426$	−6.01894	−0.291619
$427$	−15.9919	−0.773904
$428$	2.72089	0.131519
$429$	21.0410	1.01587
$430$	−2.92795	−0.141198
$431$	13.7154	0.660646	0.330323	−	0.943868i	$-0.392842\pi$
$431$	13.7154	0.660646	0.330323	+	0.943868i	$0.392842\pi$
$432$	−5.04370	−0.242665
$433$	16.9526	0.814689	0.407345	−	0.913275i	$-0.366455\pi$
$433$	16.9526	0.814689	0.407345	+	0.913275i	$0.366455\pi$
$434$	−18.7435	−0.899715
$435$	10.4269	0.499934
$436$	−12.0644	−0.577778
$437$	19.7159	0.943141
$438$	10.8824	0.519983
$439$	−32.5865	−1.55527	−0.777634	−	0.628718i	$-0.783580\pi$
$439$	−32.5865	−1.55527	−0.777634	+	0.628718i	$0.783580\pi$
$440$	2.68924	0.128205
$441$	−0.600885	−0.0286136
$442$	−12.0243	−0.571938
$443$	−9.35635	−0.444533	−0.222267	−	0.974986i	$-0.571346\pi$
$443$	−9.35635	−0.444533	−0.222267	+	0.974986i	$0.571346\pi$
$444$	19.1008	0.906483
$445$	−10.0706	−0.477391
$446$	8.65283	0.409723
$447$	−3.60980	−0.170738
$448$	2.62201	0.123878
$449$	9.98519	0.471230	0.235615	−	0.971846i	$-0.424290\pi$
$449$	9.98519	0.471230	0.235615	+	0.971846i	$0.424290\pi$
$450$	20.2478	0.954490
$451$	9.24054	0.435120
$452$	4.67196	0.219751
$453$	−28.0088	−1.31597
$454$	−7.09709	−0.333083
$455$	5.77443	0.270709
$456$	9.91799	0.464452
$457$	8.29949	0.388234	0.194117	−	0.980978i	$-0.437816\pi$
$457$	8.29949	0.388234	0.194117	+	0.980978i	$0.437816\pi$
$458$	−18.3506	−0.857466
$459$	24.4203	1.13984
$460$	−4.92499	−0.229629
$461$	−34.9487	−1.62772	−0.813862	−	0.581058i	$-0.802639\pi$
$461$	−34.9487	−1.62772	−0.813862	+	0.581058i	$0.802639\pi$
$462$	22.2149	1.03353
$463$	37.2154	1.72954	0.864772	−	0.502164i	$-0.167463\pi$
$463$	37.2154	1.72954	0.864772	+	0.502164i	$0.167463\pi$
$464$	4.20868	0.195383
$465$	17.7103	0.821296
$466$	−17.1286	−0.793468
$467$	10.0642	0.465717	0.232858	−	0.972511i	$-0.425192\pi$
$467$	10.0642	0.465717	0.232858	+	0.972511i	$0.425192\pi$
$468$	−11.9338	−0.551642
$469$	−9.07519	−0.419053
$470$	−1.62607	−0.0750052
$471$	−33.0610	−1.52337
$472$	1.13641	0.0523075
$473$	−10.0129	−0.460395
$474$	8.70291	0.399738
$475$	−14.9584	−0.686336
$476$	−12.6951	−0.581880
$477$	−60.1691	−2.75495
$478$	21.6573	0.990581
$479$	−26.5425	−1.21276	−0.606379	−	0.795176i	$-0.707378\pi$
$479$	−26.5425	−1.21276	−0.606379	+	0.795176i	$0.707378\pi$
$480$	−2.47749	−0.113081
$481$	16.9791	0.774179
$482$	−8.66815	−0.394823
$483$	−40.6836	−1.85117
$484$	−1.80339	−0.0819724
$485$	10.1968	0.463015
$486$	−16.0387	−0.727531
$487$	−35.4643	−1.60704	−0.803521	−	0.595277i	$-0.797042\pi$
$487$	−35.4643	−1.60704	−0.803521	+	0.595277i	$0.797042\pi$
$488$	6.09911	0.276094
$489$	−7.23297	−0.327086
$490$	−0.110888	−0.00500942
$491$	42.0999	1.89994	0.949971	−	0.312338i	$-0.101112\pi$
$491$	42.0999	1.89994	0.949971	+	0.312338i	$0.101112\pi$
$492$	−8.51293	−0.383792
$493$	−20.3774	−0.917751
$494$	8.81630	0.396664
$495$	−12.9227	−0.580830
$496$	7.14850	0.320977
$497$	−5.64884	−0.253385
$498$	−13.9937	−0.627073
$499$	−24.4562	−1.09481	−0.547405	−	0.836868i	$-0.684384\pi$
$499$	−24.4562	−1.09481	−0.547405	+	0.836868i	$0.684384\pi$
$500$	8.17046	0.365394
$501$	−51.5570	−2.30340
$502$	−16.4218	−0.732942
$503$	3.06104	0.136485	0.0682425	−	0.997669i	$-0.478261\pi$
$503$	3.06104	0.136485	0.0682425	+	0.997669i	$0.478261\pi$
$504$	−12.5996	−0.561231
$505$	2.43323	0.108277
$506$	−16.8424	−0.748734
$507$	19.0884	0.847744
$508$	−16.1272	−0.715529
$509$	−11.3055	−0.501106	−0.250553	−	0.968103i	$-0.580613\pi$
$509$	−11.3055	−0.501106	−0.250553	+	0.968103i	$0.580613\pi$
$510$	11.9954	0.531164
$511$	10.2133	0.451809
$512$	−1.00000	−0.0441942
$513$	−17.9051	−0.790531
$514$	−13.1522	−0.580120
$515$	−6.69091	−0.294837
$516$	9.22451	0.406086
$517$	−5.56081	−0.244564
$518$	17.9263	0.787637
$519$	3.73910	0.164128
$520$	−2.20229	−0.0965767
$521$	−20.5995	−0.902481	−0.451241	−	0.892402i	$-0.649018\pi$
$521$	−20.5995	−0.902481	−0.451241	+	0.892402i	$0.649018\pi$
$522$	−20.2240	−0.885182
$523$	−32.8977	−1.43852	−0.719259	−	0.694742i	$-0.755518\pi$
$523$	−32.8977	−1.43852	−0.719259	+	0.694742i	$0.755518\pi$
$524$	−8.49505	−0.371108
$525$	30.8664	1.34712
$526$	−1.27114	−0.0554244
$527$	−34.6112	−1.50769
$528$	−8.47245	−0.368716
$529$	7.84452	0.341066
$530$	−11.1037	−0.482313
$531$	−5.46082	−0.236979
$532$	9.30815	0.403559
$533$	−7.56731	−0.327777
$534$	31.7273	1.37297
$535$	−2.41283	−0.104316
$536$	3.46115	0.149499
$537$	−63.7870	−2.75261
$538$	16.5947	0.715448
$539$	−0.379213	−0.0163339
$540$	4.47265	0.192472
$541$	14.2269	0.611661	0.305830	−	0.952086i	$-0.401066\pi$
$541$	14.2269	0.611661	0.305830	+	0.952086i	$0.401066\pi$
$542$	3.35907	0.144284
$543$	−13.2007	−0.566494
$544$	4.84175	0.207588
$545$	10.6984	0.458270
$546$	−18.1923	−0.778560
$547$	14.6440	0.626134	0.313067	−	0.949731i	$-0.398644\pi$
$547$	14.6440	0.626134	0.313067	+	0.949731i	$0.398644\pi$
$548$	−9.95240	−0.425146
$549$	−29.3081	−1.25084
$550$	12.7782	0.544864
$551$	14.9408	0.636500
$552$	15.5162	0.660412
$553$	8.16778	0.347330
$554$	−30.9050	−1.31303
$555$	−16.9382	−0.718987
$556$	−9.70314	−0.411505
$557$	3.48559	0.147689	0.0738445	−	0.997270i	$-0.476473\pi$
$557$	3.48559	0.147689	0.0738445	+	0.997270i	$0.476473\pi$
$558$	−34.3508	−1.45419
$559$	8.19985	0.346817
$560$	−2.32515	−0.0982555
$561$	41.0215	1.73193
$562$	8.14494	0.343574
$563$	2.38876	0.100674	0.0503371	−	0.998732i	$-0.483970\pi$
$563$	2.38876	0.100674	0.0503371	+	0.998732i	$0.483970\pi$
$564$	5.12294	0.215715
$565$	−4.14300	−0.174297
$566$	19.0145	0.799239
$567$	−0.851816	−0.0357729
$568$	2.15439	0.0903963
$569$	12.4627	0.522464	0.261232	−	0.965276i	$-0.415871\pi$
$569$	12.4627	0.522464	0.261232	+	0.965276i	$0.415871\pi$
$570$	−8.79507	−0.368385
$571$	−33.5473	−1.40391	−0.701955	−	0.712222i	$-0.747689\pi$
$571$	−33.5473	−1.40391	−0.701955	+	0.712222i	$0.747689\pi$
$572$	−7.53133	−0.314901
$573$	−18.9018	−0.789634
$574$	−7.98948	−0.333475
$575$	−23.4015	−0.975912
$576$	4.80532	0.200222
$577$	12.7586	0.531146	0.265573	−	0.964091i	$-0.414439\pi$
$577$	12.7586	0.531146	0.265573	+	0.964091i	$0.414439\pi$
$578$	−6.44254	−0.267974
$579$	14.3558	0.596605
$580$	−3.73217	−0.154970
$581$	−13.1333	−0.544859
$582$	−32.1251	−1.33163
$583$	−37.9721	−1.57264
$584$	−3.89521	−0.161185
$585$	10.5827	0.437540
$586$	14.2791	0.589863
$587$	17.6755	0.729548	0.364774	−	0.931096i	$-0.381146\pi$
$587$	17.6755	0.729548	0.364774	+	0.931096i	$0.381146\pi$
$588$	0.349353	0.0144071
$589$	25.3772	1.04565
$590$	−1.00775	−0.0414883
$591$	28.7660	1.18328
$592$	−6.83685	−0.280993
$593$	11.0454	0.453582	0.226791	−	0.973943i	$-0.427177\pi$
$593$	11.0454	0.453582	0.226791	+	0.973943i	$0.427177\pi$
$594$	15.2955	0.627581
$595$	11.2578	0.461524
$596$	1.29208	0.0529255
$597$	−36.3068	−1.48594
$598$	13.7926	0.564022
$599$	−20.4378	−0.835066	−0.417533	−	0.908662i	$-0.637105\pi$
$599$	−20.4378	−0.835066	−0.417533	+	0.908662i	$0.637105\pi$
$600$	−11.7720	−0.480590
$601$	−39.6407	−1.61698	−0.808489	−	0.588511i	$-0.799714\pi$
$601$	−39.6407	−1.61698	−0.808489	+	0.588511i	$0.799714\pi$
$602$	8.65730	0.352845
$603$	−16.6319	−0.677305
$604$	10.0253	0.407925
$605$	1.59921	0.0650173
$606$	−7.66590	−0.311406
$607$	16.5506	0.671769	0.335884	−	0.941903i	$-0.390965\pi$
$607$	16.5506	0.671769	0.335884	+	0.941903i	$0.390965\pi$
$608$	−3.55000	−0.143971
$609$	−30.8302	−1.24930
$610$	−5.40856	−0.218986
$611$	4.55389	0.184231
$612$	−23.2661	−0.940478
$613$	−30.9763	−1.25112	−0.625561	−	0.780175i	$-0.715130\pi$
$613$	−30.9763	−1.25112	−0.625561	+	0.780175i	$0.715130\pi$
$614$	1.71575	0.0692420
$615$	7.54910	0.304409
$616$	−7.95149	−0.320375
$617$	−5.47785	−0.220530	−0.110265	−	0.993902i	$-0.535170\pi$
$617$	−5.47785	−0.220530	−0.110265	+	0.993902i	$0.535170\pi$
$618$	21.0797	0.847950
$619$	−28.0123	−1.12591	−0.562955	−	0.826487i	$-0.690336\pi$
$619$	−28.0123	−1.12591	−0.562955	+	0.826487i	$0.690336\pi$
$620$	−6.33915	−0.254586
$621$	−28.0116	−1.12407
$622$	7.19130	0.288345
$623$	29.7764	1.19297
$624$	6.93830	0.277754
$625$	13.8227	0.552908
$626$	3.03697	0.121382
$627$	−30.0772	−1.20117
$628$	11.8337	0.472216
$629$	33.1023	1.31988
$630$	11.1731	0.445146
$631$	46.6084	1.85545	0.927725	−	0.373265i	$-0.121762\pi$
$631$	46.6084	1.85545	0.927725	+	0.373265i	$0.121762\pi$
$632$	−3.11508	−0.123911
$633$	−19.0184	−0.755913
$634$	14.8692	0.590530
$635$	14.3013	0.567529
$636$	34.9821	1.38713
$637$	0.310547	0.0123043
$638$	−12.7632	−0.505300
$639$	−10.3525	−0.409540
$640$	0.886780	0.0350531
$641$	11.4576	0.452547	0.226274	−	0.974064i	$-0.427346\pi$
$641$	11.4576	0.452547	0.226274	+	0.974064i	$0.427346\pi$
$642$	7.60161	0.300012
$643$	15.3108	0.603798	0.301899	−	0.953340i	$-0.402379\pi$
$643$	15.3108	0.603798	0.301899	+	0.953340i	$0.402379\pi$
$644$	14.5621	0.573827
$645$	−8.18011	−0.322091
$646$	17.1882	0.676261
$647$	−26.4115	−1.03834	−0.519171	−	0.854670i	$-0.673759\pi$
$647$	−26.4115	−1.03834	−0.519171	+	0.854670i	$0.673759\pi$
$648$	0.324871	0.0127621
$649$	−3.44627	−0.135278
$650$	−10.4644	−0.410447
$651$	−52.3655	−2.05236
$652$	2.58894	0.101391
$653$	−4.22137	−0.165195	−0.0825975	−	0.996583i	$-0.526322\pi$
$653$	−4.22137	−0.165195	−0.0825975	+	0.996583i	$0.526322\pi$
$654$	−33.7054	−1.31798
$655$	7.53324	0.294348
$656$	3.04708	0.118968
$657$	18.7177	0.730248
$658$	4.80794	0.187433
$659$	−3.64845	−0.142124	−0.0710618	−	0.997472i	$-0.522639\pi$
$659$	−3.64845	−0.142124	−0.0710618	+	0.997472i	$0.522639\pi$
$660$	7.51320	0.292451
$661$	−16.5102	−0.642174	−0.321087	−	0.947050i	$-0.604048\pi$
$661$	−16.5102	−0.642174	−0.321087	+	0.947050i	$0.604048\pi$
$662$	29.5660	1.14912
$663$	−33.5935	−1.30466
$664$	5.00884	0.194381
$665$	−8.25428	−0.320087
$666$	32.8532	1.27304
$667$	23.3741	0.905048
$668$	18.4541	0.714009
$669$	24.1743	0.934631
$670$	−3.06928	−0.118577
$671$	−18.4961	−0.714034
$672$	7.32538	0.282583
$673$	−8.17078	−0.314961	−0.157480	−	0.987522i	$-0.550337\pi$
$673$	−8.17078	−0.314961	−0.157480	+	0.987522i	$0.550337\pi$
$674$	22.6440	0.872213
$675$	21.2522	0.817999
$676$	−6.83240	−0.262785
$677$	−0.476855	−0.0183270	−0.00916352	−	0.999958i	$-0.502917\pi$
$677$	−0.476855	−0.0183270	−0.00916352	+	0.999958i	$0.502917\pi$
$678$	13.0525	0.501279
$679$	−30.1498	−1.15704
$680$	−4.29357	−0.164651
$681$	−19.8279	−0.759805
$682$	−21.6785	−0.830112
$683$	−6.13876	−0.234893	−0.117447	−	0.993079i	$-0.537471\pi$
$683$	−6.13876	−0.234893	−0.117447	+	0.993079i	$0.537471\pi$
$684$	17.0589	0.652262
$685$	8.82559	0.337209
$686$	18.6820	0.713281
$687$	−51.2678	−1.95599
$688$	−3.30178	−0.125879
$689$	31.0963	1.18468
$690$	−13.7594	−0.523812
$691$	45.1152	1.71626	0.858131	−	0.513430i	$-0.171625\pi$
$691$	45.1152	1.71626	0.858131	+	0.513430i	$0.171625\pi$
$692$	−1.33836	−0.0508767
$693$	38.2095	1.45146
$694$	−24.3286	−0.923502
$695$	8.60455	0.326389
$696$	11.7582	0.445694
$697$	−14.7532	−0.558817
$698$	5.54334	0.209819
$699$	−47.8539	−1.81000
$700$	−11.0482	−0.417582
$701$	−14.5253	−0.548612	−0.274306	−	0.961642i	$-0.588448\pi$
$701$	−14.5253	−0.548612	−0.274306	+	0.961642i	$0.588448\pi$
$702$	−12.5258	−0.472758
$703$	−24.2708	−0.915391
$704$	3.03259	0.114295
$705$	−4.54292	−0.171096
$706$	−23.6236	−0.889086
$707$	−7.19453	−0.270578
$708$	3.17490	0.119320
$709$	4.86442	0.182687	0.0913436	−	0.995819i	$-0.470884\pi$
$709$	4.86442	0.182687	0.0913436	+	0.995819i	$0.470884\pi$
$710$	−1.91047	−0.0716987
$711$	14.9690	0.561380
$712$	−11.3563	−0.425596
$713$	39.7012	1.48682
$714$	−35.4677	−1.32734
$715$	6.67863	0.249767
$716$	22.8316	0.853258
$717$	60.5061	2.25964
$718$	−25.3038	−0.944328
$719$	15.9779	0.595874	0.297937	−	0.954586i	$-0.403702\pi$
$719$	15.9779	0.595874	0.297937	+	0.954586i	$0.403702\pi$
$720$	−4.26126	−0.158808
$721$	19.7836	0.736778
$722$	6.39750	0.238090
$723$	−24.2171	−0.900643
$724$	4.72498	0.175603
$725$	−17.7338	−0.658616
$726$	−5.03832	−0.186990
$727$	−6.72530	−0.249428	−0.124714	−	0.992193i	$-0.539801\pi$
$727$	−6.72530	−0.249428	−0.124714	+	0.992193i	$0.539801\pi$
$728$	6.51168	0.241339
$729$	−43.8344	−1.62349
$730$	3.45419	0.127845
$731$	15.9864	0.591278
$732$	17.0397	0.629805
$733$	24.8427	0.917586	0.458793	−	0.888543i	$-0.348282\pi$
$733$	24.8427	0.917586	0.458793	+	0.888543i	$0.348282\pi$
$734$	28.3647	1.04696
$735$	−0.309799	−0.0114271
$736$	−5.55378	−0.204715
$737$	−10.4963	−0.386635
$738$	−14.6422	−0.538986
$739$	−38.4794	−1.41549	−0.707743	−	0.706470i	$-0.750287\pi$
$739$	−38.4794	−1.41549	−0.707743	+	0.706470i	$0.750287\pi$
$740$	6.06278	0.222872
$741$	24.6310	0.904841
$742$	32.8311	1.20527
$743$	−33.9723	−1.24632	−0.623162	−	0.782093i	$-0.714152\pi$
$743$	−33.9723	−1.24632	−0.623162	+	0.782093i	$0.714152\pi$
$744$	19.9715	0.732190
$745$	−1.14579	−0.0419784
$746$	26.3256	0.963849
$747$	−24.0691	−0.880642
$748$	−14.6830	−0.536865
$749$	7.13420	0.260678
$750$	22.8266	0.833510
$751$	−9.66485	−0.352675	−0.176338	−	0.984330i	$-0.556425\pi$
$751$	−9.66485	−0.352675	−0.176338	+	0.984330i	$0.556425\pi$
$752$	−1.83368	−0.0668676
$753$	−45.8793	−1.67193
$754$	10.4521	0.380643
$755$	−8.89027	−0.323550
$756$	−13.2246	−0.480976
$757$	−37.1462	−1.35010	−0.675052	−	0.737771i	$-0.735879\pi$
$757$	−37.1462	−1.35010	−0.675052	+	0.737771i	$0.735879\pi$
$758$	28.0543	1.01898
$759$	−47.0542	−1.70796
$760$	3.14807	0.114192
$761$	−8.66982	−0.314281	−0.157140	−	0.987576i	$-0.550227\pi$
$761$	−8.66982	−0.314281	−0.157140	+	0.987576i	$0.550227\pi$
$762$	−45.0562	−1.63221
$763$	−31.6329	−1.14519
$764$	6.76562	0.244772
$765$	20.6320	0.745950
$766$	−12.4894	−0.451260
$767$	2.82224	0.101905
$768$	−2.79380	−0.100813
$769$	27.3235	0.985310	0.492655	−	0.870225i	$-0.336026\pi$
$769$	27.3235	0.985310	0.492655	+	0.870225i	$0.336026\pi$
$770$	7.05123	0.254109
$771$	−36.7447	−1.32333
$772$	−5.13844	−0.184936
$773$	−9.58158	−0.344625	−0.172313	−	0.985042i	$-0.555124\pi$
$773$	−9.58158	−0.344625	−0.172313	+	0.985042i	$0.555124\pi$
$774$	15.8661	0.570295
$775$	−30.1211	−1.08198
$776$	11.4987	0.412780
$777$	50.0825	1.79670
$778$	10.0015	0.358570
$779$	10.8171	0.387564
$780$	−6.15275	−0.220304
$781$	−6.53339	−0.233783
$782$	26.8900	0.961586
$783$	−21.2273	−0.758602
$784$	−0.125046	−0.00446592
$785$	−10.4939	−0.374543
$786$	−23.7335	−0.846545
$787$	12.5924	0.448872	0.224436	−	0.974489i	$-0.427946\pi$
$787$	12.5924	0.448872	0.224436	+	0.974489i	$0.427946\pi$
$788$	−10.2964	−0.366793
$789$	−3.55131	−0.126430
$790$	2.76239	0.0982815
$791$	12.2499	0.435558
$792$	−14.5726	−0.517813
$793$	15.1469	0.537883
$794$	−5.42949	−0.192685
$795$	−31.0215	−1.10022
$796$	12.9955	0.460613
$797$	27.5645	0.976386	0.488193	−	0.872736i	$-0.337656\pi$
$797$	27.5645	0.976386	0.488193	+	0.872736i	$0.337656\pi$
$798$	26.0051	0.920570
$799$	8.87824	0.314089
$800$	4.21362	0.148974
$801$	54.5708	1.92816
$802$	27.6484	0.976299
$803$	11.8126	0.416857
$804$	9.66977	0.341026
$805$	−12.9134	−0.455137
$806$	17.7530	0.625324
$807$	46.3623	1.63203
$808$	2.74390	0.0965299
$809$	30.6942	1.07915	0.539576	−	0.841937i	$-0.318585\pi$
$809$	30.6942	1.07915	0.539576	+	0.841937i	$0.318585\pi$
$810$	−0.288089	−0.0101224
$811$	−34.1880	−1.20050	−0.600251	−	0.799811i	$-0.704933\pi$
$811$	−34.1880	−1.20050	−0.600251	+	0.799811i	$0.704933\pi$
$812$	11.0352	0.387260
$813$	9.38456	0.329131
$814$	20.7334	0.726704
$815$	−2.29582	−0.0804190
$816$	13.5269	0.473536
$817$	−11.7213	−0.410077
$818$	−8.15914	−0.285278
$819$	−31.2907	−1.09338
$820$	−2.70209	−0.0943610
$821$	−25.4020	−0.886535	−0.443267	−	0.896389i	$-0.646181\pi$
$821$	−25.4020	−0.886535	−0.443267	+	0.896389i	$0.646181\pi$
$822$	−27.8050	−0.969812
$823$	−46.4294	−1.61843	−0.809214	−	0.587514i	$-0.800107\pi$
$823$	−46.4294	−1.61843	−0.809214	+	0.587514i	$0.800107\pi$
$824$	−7.54518	−0.262849
$825$	35.6997	1.24290
$826$	2.97968	0.103676
$827$	−29.3160	−1.01942	−0.509708	−	0.860347i	$-0.670247\pi$
$827$	−29.3160	−1.01942	−0.509708	+	0.860347i	$0.670247\pi$
$828$	26.6877	0.927462
$829$	−8.13849	−0.282661	−0.141331	−	0.989962i	$-0.545138\pi$
$829$	−8.13849	−0.282661	−0.141331	+	0.989962i	$0.545138\pi$
$830$	−4.44174	−0.154175
$831$	−86.3425	−2.99519
$832$	−2.48346	−0.0860986
$833$	0.605441	0.0209773
$834$	−27.1086	−0.938695
$835$	−16.3647	−0.566324
$836$	10.7657	0.372339
$837$	−36.0549	−1.24624
$838$	10.8196	0.373756
$839$	45.5143	1.57133	0.785663	−	0.618654i	$-0.212322\pi$
$839$	45.5143	1.57133	0.785663	+	0.618654i	$0.212322\pi$
$840$	−6.49600	−0.224133
$841$	−11.2870	−0.389208
$842$	2.12525	0.0732411
$843$	22.7553	0.783735
$844$	6.80736	0.234319
$845$	6.05884	0.208430
$846$	8.81143	0.302943
$847$	−4.72852	−0.162474
$848$	−12.5213	−0.429985
$849$	53.1227	1.82317
$850$	−20.4013	−0.699759
$851$	−37.9704	−1.30161
$852$	6.01894	0.206206
$853$	−2.03563	−0.0696987	−0.0348494	−	0.999393i	$-0.511095\pi$
$853$	−2.03563	−0.0696987	−0.0348494	+	0.999393i	$0.511095\pi$
$854$	15.9919	0.547233
$855$	−15.1275	−0.517349
$856$	−2.72089	−0.0929980
$857$	42.5371	1.45304	0.726519	−	0.687146i	$-0.241137\pi$
$857$	42.5371	1.45304	0.726519	+	0.687146i	$0.241137\pi$
$858$	−21.0410	−0.718329
$859$	−40.9421	−1.39693	−0.698464	−	0.715645i	$-0.746133\pi$
$859$	−40.9421	−1.39693	−0.698464	+	0.715645i	$0.746133\pi$
$860$	2.92795	0.0998423
$861$	−22.3210	−0.760698
$862$	−13.7154	−0.467147
$863$	−37.3081	−1.26998	−0.634991	−	0.772519i	$-0.718996\pi$
$863$	−37.3081	−1.26998	−0.634991	+	0.772519i	$0.718996\pi$
$864$	5.04370	0.171590
$865$	1.18683	0.0403534
$866$	−16.9526	−0.576072
$867$	−17.9992	−0.611284
$868$	18.7435	0.636195
$869$	9.44677	0.320460
$870$	−10.4269	−0.353506
$871$	8.59565	0.291252
$872$	12.0644	0.408551
$873$	−55.2550	−1.87010
$874$	−19.7159	−0.666901
$875$	21.4230	0.724231
$876$	−10.8824	−0.367683
$877$	0.175871	0.00593873	0.00296937	−	0.999996i	$-0.499055\pi$
$877$	0.175871	0.00593873	0.00296937	+	0.999996i	$0.499055\pi$
$878$	32.5865	1.09974
$879$	39.8929	1.34555
$880$	−2.68924	−0.0906543
$881$	40.5644	1.36665	0.683325	−	0.730114i	$-0.260533\pi$
$881$	40.5644	1.36665	0.683325	+	0.730114i	$0.260533\pi$
$882$	0.600885	0.0202329
$883$	19.7224	0.663711	0.331855	−	0.943330i	$-0.392325\pi$
$883$	19.7224	0.663711	0.331855	+	0.943330i	$0.392325\pi$
$884$	12.0243	0.404421
$885$	−2.81544	−0.0946400
$886$	9.35635	0.314333
$887$	−31.1828	−1.04702	−0.523509	−	0.852020i	$-0.675377\pi$
$887$	−31.1828	−1.04702	−0.523509	+	0.852020i	$0.675377\pi$
$888$	−19.1008	−0.640981
$889$	−42.2857	−1.41822
$890$	10.0706	0.337566
$891$	−0.985201	−0.0330055
$892$	−8.65283	−0.289718
$893$	−6.50957	−0.217835
$894$	3.60980	0.120730
$895$	−20.2466	−0.676770
$896$	−2.62201	−0.0875953
$897$	38.5338	1.28661
$898$	−9.98519	−0.333210
$899$	30.0857	1.00342
$900$	−20.2478	−0.674926
$901$	60.6252	2.01972
$902$	−9.24054	−0.307676
$903$	24.1868	0.804886
$904$	−4.67196	−0.155387
$905$	−4.19002	−0.139281
$906$	28.0088	0.930530
$907$	−14.6419	−0.486175	−0.243088	−	0.970004i	$-0.578160\pi$
$907$	−14.6419	−0.486175	−0.243088	+	0.970004i	$0.578160\pi$
$908$	7.09709	0.235525
$909$	−13.1853	−0.437329
$910$	−5.77443	−0.191420
$911$	−10.9726	−0.363539	−0.181770	−	0.983341i	$-0.558183\pi$
$911$	−10.9726	−0.363539	−0.181770	+	0.983341i	$0.558183\pi$
$912$	−9.91799	−0.328417
$913$	−15.1898	−0.502708
$914$	−8.29949	−0.274523
$915$	−15.1104	−0.499536
$916$	18.3506	0.606320
$917$	−22.2741	−0.735556
$918$	−24.4203	−0.805991
$919$	38.9903	1.28617	0.643086	−	0.765794i	$-0.277654\pi$
$919$	38.9903	1.28617	0.643086	+	0.765794i	$0.277654\pi$
$920$	4.92499	0.162372
$921$	4.79346	0.157950
$922$	34.9487	1.15098
$923$	5.35036	0.176109
$924$	−22.2149	−0.730816
$925$	28.8079	0.947198
$926$	−37.2154	−1.22297
$927$	36.2570	1.19084
$928$	−4.20868	−0.138157
$929$	−19.5782	−0.642339	−0.321169	−	0.947022i	$-0.604076\pi$
$929$	−19.5782	−0.642339	−0.321169	+	0.947022i	$0.604076\pi$
$930$	−17.7103	−0.580744
$931$	−0.443913	−0.0145486
$932$	17.1286	0.561066
$933$	20.0911	0.657752
$934$	−10.0642	−0.329311
$935$	13.0206	0.425820
$936$	11.9338	0.390070
$937$	−24.3416	−0.795206	−0.397603	−	0.917558i	$-0.630158\pi$
$937$	−24.3416	−0.795206	−0.397603	+	0.917558i	$0.630158\pi$
$938$	9.07519	0.296315
$939$	8.48469	0.276888
$940$	1.62607	0.0530367
$941$	35.4482	1.15558	0.577789	−	0.816186i	$-0.303916\pi$
$941$	35.4482	1.15558	0.577789	+	0.816186i	$0.303916\pi$
$942$	33.0610	1.07719
$943$	16.9228	0.551083
$944$	−1.13641	−0.0369870
$945$	11.7274	0.381491
$946$	10.0129	0.325549
$947$	−17.4530	−0.567147	−0.283573	−	0.958951i	$-0.591520\pi$
$947$	−17.4530	−0.567147	−0.283573	+	0.958951i	$0.591520\pi$
$948$	−8.70291	−0.282658
$949$	−9.67361	−0.314019
$950$	14.9584	0.485313
$951$	41.5415	1.34707
$952$	12.6951	0.411452
$953$	−49.9444	−1.61786	−0.808928	−	0.587908i	$-0.799952\pi$
$953$	−49.9444	−1.61786	−0.808928	+	0.587908i	$0.799952\pi$
$954$	60.1691	1.94805
$955$	−5.99962	−0.194143
$956$	−21.6573	−0.700446
$957$	−35.6578	−1.15265
$958$	26.5425	0.857549
$959$	−26.0953	−0.842662
$960$	2.47749	0.0799605
$961$	20.1010	0.648421
$962$	−16.9791	−0.547427
$963$	13.0747	0.421327
$964$	8.66815	0.279182
$965$	4.55666	0.146684
$966$	40.6836	1.30897
$967$	44.4186	1.42840	0.714202	−	0.699939i	$-0.246790\pi$
$967$	44.4186	1.42840	0.714202	+	0.699939i	$0.246790\pi$
$968$	1.80339	0.0579633
$969$	48.0204	1.54264
$970$	−10.1968	−0.327401
$971$	−11.2305	−0.360405	−0.180203	−	0.983630i	$-0.557675\pi$
$971$	−11.2305	−0.360405	−0.180203	+	0.983630i	$0.557675\pi$
$972$	16.0387	0.514442
$973$	−25.4418	−0.815625
$974$	35.4643	1.13635
$975$	−29.2354	−0.936282
$976$	−6.09911	−0.195228
$977$	8.98533	0.287466	0.143733	−	0.989616i	$-0.454089\pi$
$977$	8.98533	0.287466	0.143733	+	0.989616i	$0.454089\pi$
$978$	7.23297	0.231285
$979$	34.4391	1.10068
$980$	0.110888	0.00354219
$981$	−57.9731	−1.85094
$982$	−42.0999	−1.34346
$983$	−6.71848	−0.214286	−0.107143	−	0.994244i	$-0.534170\pi$
$983$	−6.71848	−0.214286	−0.107143	+	0.994244i	$0.534170\pi$
$984$	8.51293	0.271382
$985$	9.13063	0.290926
$986$	20.3774	0.648948
$987$	13.4324	0.427559
$988$	−8.81630	−0.280484
$989$	−18.3374	−0.583094
$990$	12.9227	0.410709
$991$	7.26496	0.230779	0.115390	−	0.993320i	$-0.463188\pi$
$991$	7.26496	0.230779	0.115390	+	0.993320i	$0.463188\pi$
$992$	−7.14850	−0.226965
$993$	82.6016	2.62128
$994$	5.64884	0.179170
$995$	−11.5241	−0.365340
$996$	13.9937	0.443407
$997$	−31.0237	−0.982531	−0.491265	−	0.871010i	$-0.663466\pi$
$997$	−31.0237	−0.982531	−0.491265	+	0.871010i	$0.663466\pi$
$998$	24.4562	0.774147
$999$	34.4830	1.09099

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8042.2.a.b.1.12	✓	82

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8042.2.a.b.1.12	✓	82	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.79380 q^{3} +1.00000 q^{4} -0.886780 q^{5} +2.79380 q^{6} +2.62201 q^{7} -1.00000 q^{8} +4.80532 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.79380 q^{3} +1.00000 q^{4} -0.886780 q^{5} +2.79380 q^{6} +2.62201 q^{7} -1.00000 q^{8} +4.80532 q^{9} +0.886780 q^{10} +3.03259 q^{11} -2.79380 q^{12} -2.48346 q^{13} -2.62201 q^{14} +2.47749 q^{15} +1.00000 q^{16} -4.84175 q^{17} -4.80532 q^{18} +3.55000 q^{19} -0.886780 q^{20} -7.32538 q^{21} -3.03259 q^{22} +5.55378 q^{23} +2.79380 q^{24} -4.21362 q^{25} +2.48346 q^{26} -5.04370 q^{27} +2.62201 q^{28} +4.20868 q^{29} -2.47749 q^{30} +7.14850 q^{31} -1.00000 q^{32} -8.47245 q^{33} +4.84175 q^{34} -2.32515 q^{35} +4.80532 q^{36} -6.83685 q^{37} -3.55000 q^{38} +6.93830 q^{39} +0.886780 q^{40} +3.04708 q^{41} +7.32538 q^{42} -3.30178 q^{43} +3.03259 q^{44} -4.26126 q^{45} -5.55378 q^{46} -1.83368 q^{47} -2.79380 q^{48} -0.125046 q^{49} +4.21362 q^{50} +13.5269 q^{51} -2.48346 q^{52} -12.5213 q^{53} +5.04370 q^{54} -2.68924 q^{55} -2.62201 q^{56} -9.91799 q^{57} -4.20868 q^{58} -1.13641 q^{59} +2.47749 q^{60} -6.09911 q^{61} -7.14850 q^{62} +12.5996 q^{63} +1.00000 q^{64} +2.20229 q^{65} +8.47245 q^{66} -3.46115 q^{67} -4.84175 q^{68} -15.5162 q^{69} +2.32515 q^{70} -2.15439 q^{71} -4.80532 q^{72} +3.89521 q^{73} +6.83685 q^{74} +11.7720 q^{75} +3.55000 q^{76} +7.95149 q^{77} -6.93830 q^{78} +3.11508 q^{79} -0.886780 q^{80} -0.324871 q^{81} -3.04708 q^{82} -5.00884 q^{83} -7.32538 q^{84} +4.29357 q^{85} +3.30178 q^{86} -11.7582 q^{87} -3.03259 q^{88} +11.3563 q^{89} +4.26126 q^{90} -6.51168 q^{91} +5.55378 q^{92} -19.9715 q^{93} +1.83368 q^{94} -3.14807 q^{95} +2.79380 q^{96} -11.4987 q^{97} +0.125046 q^{98} +14.5726 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 82 q - 82 q^{2} - 13 q^{3} + 82 q^{4} + 3 q^{5} + 13 q^{6} - 37 q^{7} - 82 q^{8} + 91 q^{9}+O(q^{10}) \) 82 * q - 82 * q^2 - 13 * q^3 + 82 * q^4 + 3 * q^5 + 13 * q^6 - 37 * q^7 - 82 * q^8 + 91 * q^9	\( 82 q - 82 q^{2} - 13 q^{3} + 82 q^{4} + 3 q^{5} + 13 q^{6} - 37 q^{7} - 82 q^{8} + 91 q^{9} - 3 q^{10} - 16 q^{11} - 13 q^{12} - 42 q^{13} + 37 q^{14} - 9 q^{15} + 82 q^{16} + 3 q^{17} - 91 q^{18} - 42 q^{19} + 3 q^{20} - q^{21} + 16 q^{22} - 6 q^{23} + 13 q^{24} + 53 q^{25} + 42 q^{26} - 49 q^{27} - 37 q^{28} + 15 q^{29} + 9 q^{30} - 40 q^{31} - 82 q^{32} - 37 q^{33} - 3 q^{34} - 42 q^{35} + 91 q^{36} - 72 q^{37} + 42 q^{38} - 14 q^{39} - 3 q^{40} + 8 q^{41} + q^{42} - 93 q^{43} - 16 q^{44} - 11 q^{45} + 6 q^{46} + 7 q^{47} - 13 q^{48} + 61 q^{49} - 53 q^{50} - 70 q^{51} - 42 q^{52} + 18 q^{53} + 49 q^{54} - 62 q^{55} + 37 q^{56} - 51 q^{57} - 15 q^{58} - 47 q^{59} - 9 q^{60} - 14 q^{61} + 40 q^{62} - 100 q^{63} + 82 q^{64} + q^{65} + 37 q^{66} - 150 q^{67} + 3 q^{68} + 31 q^{69} + 42 q^{70} + 7 q^{71} - 91 q^{72} - 78 q^{73} + 72 q^{74} - 49 q^{75} - 42 q^{76} + 29 q^{77} + 14 q^{78} - 59 q^{79} + 3 q^{80} + 122 q^{81} - 8 q^{82} - 52 q^{83} - q^{84} - 108 q^{85} + 93 q^{86} - 49 q^{87} + 16 q^{88} + 38 q^{89} + 11 q^{90} - 69 q^{91} - 6 q^{92} - 63 q^{93} - 7 q^{94} + 5 q^{95} + 13 q^{96} - 74 q^{97} - 61 q^{98} - 89 q^{99}+O(q^{100}) \) 82 * q - 82 * q^2 - 13 * q^3 + 82 * q^4 + 3 * q^5 + 13 * q^6 - 37 * q^7 - 82 * q^8 + 91 * q^9 - 3 * q^10 - 16 * q^11 - 13 * q^12 - 42 * q^13 + 37 * q^14 - 9 * q^15 + 82 * q^16 + 3 * q^17 - 91 * q^18 - 42 * q^19 + 3 * q^20 - q^21 + 16 * q^22 - 6 * q^23 + 13 * q^24 + 53 * q^25 + 42 * q^26 - 49 * q^27 - 37 * q^28 + 15 * q^29 + 9 * q^30 - 40 * q^31 - 82 * q^32 - 37 * q^33 - 3 * q^34 - 42 * q^35 + 91 * q^36 - 72 * q^37 + 42 * q^38 - 14 * q^39 - 3 * q^40 + 8 * q^41 + q^42 - 93 * q^43 - 16 * q^44 - 11 * q^45 + 6 * q^46 + 7 * q^47 - 13 * q^48 + 61 * q^49 - 53 * q^50 - 70 * q^51 - 42 * q^52 + 18 * q^53 + 49 * q^54 - 62 * q^55 + 37 * q^56 - 51 * q^57 - 15 * q^58 - 47 * q^59 - 9 * q^60 - 14 * q^61 + 40 * q^62 - 100 * q^63 + 82 * q^64 + q^65 + 37 * q^66 - 150 * q^67 + 3 * q^68 + 31 * q^69 + 42 * q^70 + 7 * q^71 - 91 * q^72 - 78 * q^73 + 72 * q^74 - 49 * q^75 - 42 * q^76 + 29 * q^77 + 14 * q^78 - 59 * q^79 + 3 * q^80 + 122 * q^81 - 8 * q^82 - 52 * q^83 - q^84 - 108 * q^85 + 93 * q^86 - 49 * q^87 + 16 * q^88 + 38 * q^89 + 11 * q^90 - 69 * q^91 - 6 * q^92 - 63 * q^93 - 7 * q^94 + 5 * q^95 + 13 * q^96 - 74 * q^97 - 61 * q^98 - 89 * q^99

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