LMFDB - Embedded newform 8042.2.a.a.1.15

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			1.15
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)

$f(q)$	$=$	$q+1.00000 q^{2} -2.04025 q^{3} +1.00000 q^{4} +2.67295 q^{5} -2.04025 q^{6} +4.19458 q^{7} +1.00000 q^{8} +1.16263 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.04025 q^{3} +1.00000 q^{4} +2.67295 q^{5} -2.04025 q^{6} +4.19458 q^{7} +1.00000 q^{8} +1.16263 q^{9} +2.67295 q^{10} -2.39834 q^{11} -2.04025 q^{12} -5.33719 q^{13} +4.19458 q^{14} -5.45350 q^{15} +1.00000 q^{16} +0.801038 q^{17} +1.16263 q^{18} -3.01007 q^{19} +2.67295 q^{20} -8.55800 q^{21} -2.39834 q^{22} -0.421516 q^{23} -2.04025 q^{24} +2.14469 q^{25} -5.33719 q^{26} +3.74869 q^{27} +4.19458 q^{28} +0.238867 q^{29} -5.45350 q^{30} -2.73783 q^{31} +1.00000 q^{32} +4.89322 q^{33} +0.801038 q^{34} +11.2119 q^{35} +1.16263 q^{36} -7.32133 q^{37} -3.01007 q^{38} +10.8892 q^{39} +2.67295 q^{40} -9.55708 q^{41} -8.55800 q^{42} -2.46109 q^{43} -2.39834 q^{44} +3.10767 q^{45} -0.421516 q^{46} -12.9705 q^{47} -2.04025 q^{48} +10.5945 q^{49} +2.14469 q^{50} -1.63432 q^{51} -5.33719 q^{52} +8.03477 q^{53} +3.74869 q^{54} -6.41065 q^{55} +4.19458 q^{56} +6.14130 q^{57} +0.238867 q^{58} +3.43602 q^{59} -5.45350 q^{60} +6.03348 q^{61} -2.73783 q^{62} +4.87676 q^{63} +1.00000 q^{64} -14.2661 q^{65} +4.89322 q^{66} -9.91857 q^{67} +0.801038 q^{68} +0.860000 q^{69} +11.2119 q^{70} -9.43912 q^{71} +1.16263 q^{72} -0.400632 q^{73} -7.32133 q^{74} -4.37570 q^{75} -3.01007 q^{76} -10.0600 q^{77} +10.8892 q^{78} -1.01068 q^{79} +2.67295 q^{80} -11.1362 q^{81} -9.55708 q^{82} -11.1142 q^{83} -8.55800 q^{84} +2.14114 q^{85} -2.46109 q^{86} -0.487349 q^{87} -2.39834 q^{88} -7.36845 q^{89} +3.10767 q^{90} -22.3872 q^{91} -0.421516 q^{92} +5.58587 q^{93} -12.9705 q^{94} -8.04577 q^{95} -2.04025 q^{96} -16.8175 q^{97} +10.5945 q^{98} -2.78839 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9}+O(q^{10}) $ 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9} - 20 q^{10} - 13 q^{11} - 11 q^{12} - 51 q^{13} - 40 q^{14} - 31 q^{15} + 67 q^{16} - 34 q^{17} + 24 q^{18} - 33 q^{19} - 20 q^{20} - 39 q^{21} - 13 q^{22} - 43 q^{23} - 11 q^{24} - 9 q^{25} - 51 q^{26} - 29 q^{27} - 40 q^{28} - 63 q^{29} - 31 q^{30} - 43 q^{31} + 67 q^{32} - 49 q^{33} - 34 q^{34} - 20 q^{35} + 24 q^{36} - 77 q^{37} - 33 q^{38} - 40 q^{39} - 20 q^{40} - 50 q^{41} - 39 q^{42} - 56 q^{43} - 13 q^{44} - 48 q^{45} - 43 q^{46} - 48 q^{47} - 11 q^{48} + q^{49} - 9 q^{50} - 18 q^{51} - 51 q^{52} - 91 q^{53} - 29 q^{54} - 58 q^{55} - 40 q^{56} - 65 q^{57} - 63 q^{58} - 17 q^{59} - 31 q^{60} - 45 q^{61} - 43 q^{62} - 67 q^{63} + 67 q^{64} - 65 q^{65} - 49 q^{66} - 112 q^{67} - 34 q^{68} - 57 q^{69} - 20 q^{70} - 75 q^{71} + 24 q^{72} - 79 q^{73} - 77 q^{74} - 5 q^{75} - 33 q^{76} - 85 q^{77} - 40 q^{78} - 80 q^{79} - 20 q^{80} - 77 q^{81} - 50 q^{82} - 22 q^{83} - 39 q^{84} - 134 q^{85} - 56 q^{86} - 49 q^{87} - 13 q^{88} - 77 q^{89} - 48 q^{90} - 17 q^{91} - 43 q^{92} - 97 q^{93} - 48 q^{94} - 73 q^{95} - 11 q^{96} - 87 q^{97} + q^{98} - 44 q^{99}+O(q^{100}) \) 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9 - 20 * q^10 - 13 * q^11 - 11 * q^12 - 51 * q^13 - 40 * q^14 - 31 * q^15 + 67 * q^16 - 34 * q^17 + 24 * q^18 - 33 * q^19 - 20 * q^20 - 39 * q^21 - 13 * q^22 - 43 * q^23 - 11 * q^24 - 9 * q^25 - 51 * q^26 - 29 * q^27 - 40 * q^28 - 63 * q^29 - 31 * q^30 - 43 * q^31 + 67 * q^32 - 49 * q^33 - 34 * q^34 - 20 * q^35 + 24 * q^36 - 77 * q^37 - 33 * q^38 - 40 * q^39 - 20 * q^40 - 50 * q^41 - 39 * q^42 - 56 * q^43 - 13 * q^44 - 48 * q^45 - 43 * q^46 - 48 * q^47 - 11 * q^48 + q^49 - 9 * q^50 - 18 * q^51 - 51 * q^52 - 91 * q^53 - 29 * q^54 - 58 * q^55 - 40 * q^56 - 65 * q^57 - 63 * q^58 - 17 * q^59 - 31 * q^60 - 45 * q^61 - 43 * q^62 - 67 * q^63 + 67 * q^64 - 65 * q^65 - 49 * q^66 - 112 * q^67 - 34 * q^68 - 57 * q^69 - 20 * q^70 - 75 * q^71 + 24 * q^72 - 79 * q^73 - 77 * q^74 - 5 * q^75 - 33 * q^76 - 85 * q^77 - 40 * q^78 - 80 * q^79 - 20 * q^80 - 77 * q^81 - 50 * q^82 - 22 * q^83 - 39 * q^84 - 134 * q^85 - 56 * q^86 - 49 * q^87 - 13 * q^88 - 77 * q^89 - 48 * q^90 - 17 * q^91 - 43 * q^92 - 97 * q^93 - 48 * q^94 - 73 * q^95 - 11 * q^96 - 87 * q^97 + q^98 - 44 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−2.04025	−1.17794	−0.588970	−	0.808155i	$-0.700467\pi$
$3$	−2.04025	−1.17794	−0.588970	+	0.808155i	$0.700467\pi$
$4$	1.00000	0.500000
$5$	2.67295	1.19538	0.597691	−	0.801727i	$-0.296085\pi$
$5$	2.67295	1.19538	0.597691	+	0.801727i	$0.296085\pi$
$6$	−2.04025	−0.832930
$7$	4.19458	1.58540	0.792701	−	0.609611i	$-0.208674\pi$
$7$	4.19458	1.58540	0.792701	+	0.609611i	$0.208674\pi$
$8$	1.00000	0.353553
$9$	1.16263	0.387544
$10$	2.67295	0.845262
$11$	−2.39834	−0.723127	−0.361563	−	0.932348i	$-0.617757\pi$
$11$	−2.39834	−0.723127	−0.361563	+	0.932348i	$0.617757\pi$
$12$	−2.04025	−0.588970
$13$	−5.33719	−1.48027	−0.740135	−	0.672458i	$-0.765238\pi$
$13$	−5.33719	−1.48027	−0.740135	+	0.672458i	$0.765238\pi$
$14$	4.19458	1.12105
$15$	−5.45350	−1.40809
$16$	1.00000	0.250000
$17$	0.801038	0.194280	0.0971401	−	0.995271i	$-0.469030\pi$
$17$	0.801038	0.194280	0.0971401	+	0.995271i	$0.469030\pi$
$18$	1.16263	0.274035
$19$	−3.01007	−0.690557	−0.345278	−	0.938500i	$-0.612216\pi$
$19$	−3.01007	−0.690557	−0.345278	+	0.938500i	$0.612216\pi$
$20$	2.67295	0.597691
$21$	−8.55800	−1.86751
$22$	−2.39834	−0.511328
$23$	−0.421516	−0.0878922	−0.0439461	−	0.999034i	$-0.513993\pi$
$23$	−0.421516	−0.0878922	−0.0439461	+	0.999034i	$0.513993\pi$
$24$	−2.04025	−0.416465
$25$	2.14469	0.428937
$26$	−5.33719	−1.04671
$27$	3.74869	0.721436
$28$	4.19458	0.792701
$29$	0.238867	0.0443564	0.0221782	−	0.999754i	$-0.492940\pi$
$29$	0.238867	0.0443564	0.0221782	+	0.999754i	$0.492940\pi$
$30$	−5.45350	−0.995669
$31$	−2.73783	−0.491729	−0.245865	−	0.969304i	$-0.579072\pi$
$31$	−2.73783	−0.491729	−0.245865	+	0.969304i	$0.579072\pi$
$32$	1.00000	0.176777
$33$	4.89322	0.851800
$34$	0.801038	0.137377
$35$	11.2119	1.89516
$36$	1.16263	0.193772
$37$	−7.32133	−1.20362	−0.601809	−	0.798640i	$-0.705553\pi$
$37$	−7.32133	−1.20362	−0.601809	+	0.798640i	$0.705553\pi$
$38$	−3.01007	−0.488297
$39$	10.8892	1.74367
$40$	2.67295	0.422631
$41$	−9.55708	−1.49257	−0.746283	−	0.665629i	$-0.768163\pi$
$41$	−9.55708	−1.49257	−0.746283	+	0.665629i	$0.768163\pi$
$42$	−8.55800	−1.32053
$43$	−2.46109	−0.375313	−0.187657	−	0.982235i	$-0.560089\pi$
$43$	−2.46109	−0.375313	−0.187657	+	0.982235i	$0.560089\pi$
$44$	−2.39834	−0.361563
$45$	3.10767	0.463264
$46$	−0.421516	−0.0621492
$47$	−12.9705	−1.89194	−0.945969	−	0.324256i	$-0.894886\pi$
$47$	−12.9705	−1.89194	−0.945969	+	0.324256i	$0.894886\pi$
$48$	−2.04025	−0.294485
$49$	10.5945	1.51350
$50$	2.14469	0.303304
$51$	−1.63432	−0.228851
$52$	−5.33719	−0.740135
$53$	8.03477	1.10366	0.551830	−	0.833956i	$-0.313930\pi$
$53$	8.03477	1.10366	0.551830	+	0.833956i	$0.313930\pi$
$54$	3.74869	0.510133
$55$	−6.41065	−0.864412
$56$	4.19458	0.560524
$57$	6.14130	0.813435
$58$	0.238867	0.0313647
$59$	3.43602	0.447332	0.223666	−	0.974666i	$-0.428198\pi$
$59$	3.43602	0.447332	0.223666	+	0.974666i	$0.428198\pi$
$60$	−5.45350	−0.704044
$61$	6.03348	0.772508	0.386254	−	0.922392i	$-0.373769\pi$
$61$	6.03348	0.772508	0.386254	+	0.922392i	$0.373769\pi$
$62$	−2.73783	−0.347705
$63$	4.87676	0.614413
$64$	1.00000	0.125000
$65$	−14.2661	−1.76949
$66$	4.89322	0.602314
$67$	−9.91857	−1.21175	−0.605873	−	0.795561i	$-0.707176\pi$
$67$	−9.91857	−1.21175	−0.605873	+	0.795561i	$0.707176\pi$
$68$	0.801038	0.0971401
$69$	0.860000	0.103532
$70$	11.2119	1.34008
$71$	−9.43912	−1.12022	−0.560109	−	0.828419i	$-0.689241\pi$
$71$	−9.43912	−1.12022	−0.560109	+	0.828419i	$0.689241\pi$
$72$	1.16263	0.137018
$73$	−0.400632	−0.0468905	−0.0234452	−	0.999725i	$-0.507464\pi$
$73$	−0.400632	−0.0468905	−0.0234452	+	0.999725i	$0.507464\pi$
$74$	−7.32133	−0.851087
$75$	−4.37570	−0.505263
$76$	−3.01007	−0.345278
$77$	−10.0600	−1.14645
$78$	10.8892	1.23296
$79$	−1.01068	−0.113711	−0.0568553	−	0.998382i	$-0.518107\pi$
$79$	−1.01068	−0.113711	−0.0568553	+	0.998382i	$0.518107\pi$
$80$	2.67295	0.298845
$81$	−11.1362	−1.23735
$82$	−9.55708	−1.05540
$83$	−11.1142	−1.21994	−0.609971	−	0.792423i	$-0.708819\pi$
$83$	−11.1142	−1.21994	−0.609971	+	0.792423i	$0.708819\pi$
$84$	−8.55800	−0.933754
$85$	2.14114	0.232239
$86$	−2.46109	−0.265386
$87$	−0.487349	−0.0522492
$88$	−2.39834	−0.255664
$89$	−7.36845	−0.781054	−0.390527	−	0.920592i	$-0.627707\pi$
$89$	−7.36845	−0.781054	−0.390527	+	0.920592i	$0.627707\pi$
$90$	3.10767	0.327577
$91$	−22.3872	−2.34682
$92$	−0.421516	−0.0439461
$93$	5.58587	0.579228
$94$	−12.9705	−1.33780
$95$	−8.04577	−0.825479
$96$	−2.04025	−0.208232
$97$	−16.8175	−1.70756	−0.853779	−	0.520635i	$-0.825695\pi$
$97$	−16.8175	−1.70756	−0.853779	+	0.520635i	$0.825695\pi$
$98$	10.5945	1.07020
$99$	−2.78839	−0.280244
$100$	2.14469	0.214469
$101$	10.1920	1.01414	0.507070	−	0.861905i	$-0.330729\pi$
$101$	10.1920	1.01414	0.507070	+	0.861905i	$0.330729\pi$
$102$	−1.63432	−0.161822
$103$	10.7986	1.06402	0.532011	−	0.846737i	$-0.321436\pi$
$103$	10.7986	1.06402	0.532011	+	0.846737i	$0.321436\pi$
$104$	−5.33719	−0.523354
$105$	−22.8751	−2.23239
$106$	8.03477	0.780406
$107$	−7.01385	−0.678055	−0.339027	−	0.940777i	$-0.610098\pi$
$107$	−7.01385	−0.678055	−0.339027	+	0.940777i	$0.610098\pi$
$108$	3.74869	0.360718
$109$	−14.5679	−1.39535	−0.697676	−	0.716414i	$-0.745782\pi$
$109$	−14.5679	−1.39535	−0.697676	+	0.716414i	$0.745782\pi$
$110$	−6.41065	−0.611232
$111$	14.9374	1.41779
$112$	4.19458	0.396350
$113$	3.92168	0.368920	0.184460	−	0.982840i	$-0.440946\pi$
$113$	3.92168	0.368920	0.184460	+	0.982840i	$0.440946\pi$
$114$	6.14130	0.575185
$115$	−1.12669	−0.105065
$116$	0.238867	0.0221782
$117$	−6.20519	−0.573670
$118$	3.43602	0.316311
$119$	3.36002	0.308012
$120$	−5.45350	−0.497835
$121$	−5.24797	−0.477088
$122$	6.03348	0.546246
$123$	19.4989	1.75815
$124$	−2.73783	−0.245865
$125$	−7.63212	−0.682638
$126$	4.87676	0.434456
$127$	7.63448	0.677450	0.338725	−	0.940885i	$-0.390004\pi$
$127$	7.63448	0.677450	0.338725	+	0.940885i	$0.390004\pi$
$128$	1.00000	0.0883883
$129$	5.02125	0.442097
$130$	−14.2661	−1.25122
$131$	21.4281	1.87219	0.936093	−	0.351753i	$-0.114414\pi$
$131$	21.4281	1.87219	0.936093	+	0.351753i	$0.114414\pi$
$132$	4.89322	0.425900
$133$	−12.6260	−1.09481
$134$	−9.91857	−0.856834
$135$	10.0201	0.862392
$136$	0.801038	0.0686885
$137$	−11.1135	−0.949491	−0.474745	−	0.880123i	$-0.657460\pi$
$137$	−11.1135	−0.949491	−0.474745	+	0.880123i	$0.657460\pi$
$138$	0.860000	0.0732081
$139$	14.4888	1.22892	0.614462	−	0.788947i	$-0.289373\pi$
$139$	14.4888	1.22892	0.614462	+	0.788947i	$0.289373\pi$
$140$	11.2119	0.947580
$141$	26.4631	2.22859
$142$	−9.43912	−0.792113
$143$	12.8004	1.07042
$144$	1.16263	0.0968861
$145$	0.638480	0.0530229
$146$	−0.400632	−0.0331566
$147$	−21.6154	−1.78281
$148$	−7.32133	−0.601809
$149$	4.23832	0.347217	0.173608	−	0.984815i	$-0.444457\pi$
$149$	4.23832	0.347217	0.173608	+	0.984815i	$0.444457\pi$
$150$	−4.37570	−0.357275
$151$	−3.74351	−0.304643	−0.152321	−	0.988331i	$-0.548675\pi$
$151$	−3.74351	−0.304643	−0.152321	+	0.988331i	$0.548675\pi$
$152$	−3.01007	−0.244149
$153$	0.931314	0.0752922
$154$	−10.0600	−0.810660
$155$	−7.31810	−0.587804
$156$	10.8892	0.871835
$157$	6.92682	0.552820	0.276410	−	0.961040i	$-0.410855\pi$
$157$	6.92682	0.552820	0.276410	+	0.961040i	$0.410855\pi$
$158$	−1.01068	−0.0804056
$159$	−16.3930	−1.30005
$160$	2.67295	0.211316
$161$	−1.76808	−0.139344
$162$	−11.1362	−0.874941
$163$	12.6789	0.993089	0.496544	−	0.868011i	$-0.334602\pi$
$163$	12.6789	0.993089	0.496544	+	0.868011i	$0.334602\pi$
$164$	−9.55708	−0.746283
$165$	13.0794	1.01823
$166$	−11.1142	−0.862630
$167$	18.9367	1.46537	0.732685	−	0.680568i	$-0.238267\pi$
$167$	18.9367	1.46537	0.732685	+	0.680568i	$0.238267\pi$
$168$	−8.55800	−0.660264
$169$	15.4856	1.19120
$170$	2.14114	0.164218
$171$	−3.49960	−0.267621
$172$	−2.46109	−0.187657
$173$	−12.5582	−0.954779	−0.477390	−	0.878692i	$-0.658417\pi$
$173$	−12.5582	−0.954779	−0.477390	+	0.878692i	$0.658417\pi$
$174$	−0.487349	−0.0369458
$175$	8.99605	0.680038
$176$	−2.39834	−0.180782
$177$	−7.01036	−0.526931
$178$	−7.36845	−0.552288
$179$	22.5167	1.68298	0.841488	−	0.540276i	$-0.181680\pi$
$179$	22.5167	1.68298	0.841488	+	0.540276i	$0.181680\pi$
$180$	3.10767	0.231632
$181$	−3.08568	−0.229357	−0.114678	−	0.993403i	$-0.536584\pi$
$181$	−3.08568	−0.229357	−0.114678	+	0.993403i	$0.536584\pi$
$182$	−22.3872	−1.65945
$183$	−12.3098	−0.909969
$184$	−0.421516	−0.0310746
$185$	−19.5696	−1.43878
$186$	5.58587	0.409576
$187$	−1.92116	−0.140489
$188$	−12.9705	−0.945969
$189$	15.7242	1.14377
$190$	−8.04577	−0.583702
$191$	−7.37115	−0.533358	−0.266679	−	0.963785i	$-0.585926\pi$
$191$	−7.37115	−0.533358	−0.266679	+	0.963785i	$0.585926\pi$
$192$	−2.04025	−0.147243
$193$	24.3783	1.75479	0.877395	−	0.479768i	$-0.159279\pi$
$193$	24.3783	1.75479	0.877395	+	0.479768i	$0.159279\pi$
$194$	−16.8175	−1.20743
$195$	29.1064	2.08435
$196$	10.5945	0.756748
$197$	−1.53694	−0.109502	−0.0547511	−	0.998500i	$-0.517437\pi$
$197$	−1.53694	−0.109502	−0.0547511	+	0.998500i	$0.517437\pi$
$198$	−2.78839	−0.198162
$199$	−24.3425	−1.72560	−0.862798	−	0.505549i	$-0.831290\pi$
$199$	−24.3425	−1.72560	−0.862798	+	0.505549i	$0.831290\pi$
$200$	2.14469	0.151652
$201$	20.2364	1.42736
$202$	10.1920	0.717105
$203$	1.00194	0.0703227
$204$	−1.63432	−0.114425
$205$	−25.5456	−1.78419
$206$	10.7986	0.752377
$207$	−0.490069	−0.0340621
$208$	−5.33719	−0.370067
$209$	7.21916	0.499360
$210$	−22.8751	−1.57853
$211$	−1.08059	−0.0743909	−0.0371954	−	0.999308i	$-0.511842\pi$
$211$	−1.08059	−0.0743909	−0.0371954	+	0.999308i	$0.511842\pi$
$212$	8.03477	0.551830
$213$	19.2582	1.31955
$214$	−7.01385	−0.479457
$215$	−6.57839	−0.448642
$216$	3.74869	0.255066
$217$	−11.4840	−0.779588
$218$	−14.5679	−0.986662
$219$	0.817391	0.0552342
$220$	−6.41065	−0.432206
$221$	−4.27529	−0.287587
$222$	14.9374	1.00253
$223$	23.1324	1.54906	0.774531	−	0.632536i	$-0.217986\pi$
$223$	23.1324	1.54906	0.774531	+	0.632536i	$0.217986\pi$
$224$	4.19458	0.280262
$225$	2.49348	0.166232
$226$	3.92168	0.260866
$227$	−18.6940	−1.24076	−0.620381	−	0.784300i	$-0.713022\pi$
$227$	−18.6940	−1.24076	−0.620381	+	0.784300i	$0.713022\pi$
$228$	6.14130	0.406718
$229$	14.9675	0.989084	0.494542	−	0.869154i	$-0.335336\pi$
$229$	14.9675	0.989084	0.494542	+	0.869154i	$0.335336\pi$
$230$	−1.12669	−0.0742920
$231$	20.5250	1.35045
$232$	0.238867	0.0156824
$233$	5.44756	0.356882	0.178441	−	0.983951i	$-0.442895\pi$
$233$	5.44756	0.356882	0.178441	+	0.983951i	$0.442895\pi$
$234$	−6.20519	−0.405646
$235$	−34.6695	−2.26159
$236$	3.43602	0.223666
$237$	2.06205	0.133944
$238$	3.36002	0.217797
$239$	−19.3347	−1.25066	−0.625329	−	0.780361i	$-0.715035\pi$
$239$	−19.3347	−1.25066	−0.625329	+	0.780361i	$0.715035\pi$
$240$	−5.45350	−0.352022
$241$	9.45825	0.609260	0.304630	−	0.952471i	$-0.401467\pi$
$241$	9.45825	0.609260	0.304630	+	0.952471i	$0.401467\pi$
$242$	−5.24797	−0.337352
$243$	11.4746	0.736093
$244$	6.03348	0.386254
$245$	28.3186	1.80921
$246$	19.4989	1.24320
$247$	16.0653	1.02221
$248$	−2.73783	−0.173853
$249$	22.6758	1.43702
$250$	−7.63212	−0.482698
$251$	−15.5320	−0.980373	−0.490187	−	0.871618i	$-0.663071\pi$
$251$	−15.5320	−0.980373	−0.490187	+	0.871618i	$0.663071\pi$
$252$	4.87676	0.307207
$253$	1.01094	0.0635572
$254$	7.63448	0.479030
$255$	−4.36846	−0.273564
$256$	1.00000	0.0625000
$257$	9.63121	0.600779	0.300389	−	0.953817i	$-0.402883\pi$
$257$	9.63121	0.600779	0.300389	+	0.953817i	$0.402883\pi$
$258$	5.02125	0.312610
$259$	−30.7099	−1.90822
$260$	−14.2661	−0.884744
$261$	0.277714	0.0171901
$262$	21.4281	1.32384
$263$	−0.0567599	−0.00349997	−0.00174998	−	0.999998i	$-0.500557\pi$
$263$	−0.0567599	−0.00349997	−0.00174998	+	0.999998i	$0.500557\pi$
$264$	4.89322	0.301157
$265$	21.4766	1.31930
$266$	−12.6260	−0.774147
$267$	15.0335	0.920035
$268$	−9.91857	−0.605873
$269$	−2.65267	−0.161736	−0.0808682	−	0.996725i	$-0.525769\pi$
$269$	−2.65267	−0.161736	−0.0808682	+	0.996725i	$0.525769\pi$
$270$	10.0201	0.609803
$271$	−3.48178	−0.211503	−0.105752	−	0.994393i	$-0.533725\pi$
$271$	−3.48178	−0.211503	−0.105752	+	0.994393i	$0.533725\pi$
$272$	0.801038	0.0485701
$273$	45.6757	2.76442
$274$	−11.1135	−0.671391
$275$	−5.14369	−0.310176
$276$	0.860000	0.0517659
$277$	5.86663	0.352492	0.176246	−	0.984346i	$-0.443605\pi$
$277$	5.86663	0.352492	0.176246	+	0.984346i	$0.443605\pi$
$278$	14.4888	0.868980
$279$	−3.18310	−0.190567
$280$	11.2119	0.670040
$281$	−11.2021	−0.668264	−0.334132	−	0.942526i	$-0.608443\pi$
$281$	−11.2021	−0.668264	−0.334132	+	0.942526i	$0.608443\pi$
$282$	26.4631	1.57585
$283$	19.5530	1.16231	0.581154	−	0.813794i	$-0.302602\pi$
$283$	19.5530	1.16231	0.581154	+	0.813794i	$0.302602\pi$
$284$	−9.43912	−0.560109
$285$	16.4154	0.972365
$286$	12.8004	0.756903
$287$	−40.0879	−2.36631
$288$	1.16263	0.0685088
$289$	−16.3583	−0.962255
$290$	0.638480	0.0374928
$291$	34.3120	2.01140
$292$	−0.400632	−0.0234452
$293$	−15.1528	−0.885237	−0.442618	−	0.896710i	$-0.645950\pi$
$293$	−15.1528	−0.885237	−0.442618	+	0.896710i	$0.645950\pi$
$294$	−21.6154	−1.26064
$295$	9.18433	0.534732
$296$	−7.32133	−0.425544
$297$	−8.99064	−0.521690
$298$	4.23832	0.245519
$299$	2.24971	0.130104
$300$	−4.37570	−0.252631
$301$	−10.3232	−0.595022
$302$	−3.74351	−0.215415
$303$	−20.7942	−1.19460
$304$	−3.01007	−0.172639
$305$	16.1272	0.923442
$306$	0.931314	0.0532397
$307$	−19.2004	−1.09582	−0.547912	−	0.836536i	$-0.684577\pi$
$307$	−19.2004	−1.09582	−0.547912	+	0.836536i	$0.684577\pi$
$308$	−10.0600	−0.573223
$309$	−22.0320	−1.25336
$310$	−7.31810	−0.415640
$311$	31.8211	1.80441	0.902205	−	0.431308i	$-0.141948\pi$
$311$	31.8211	1.80441	0.902205	+	0.431308i	$0.141948\pi$
$312$	10.8892	0.616480
$313$	−3.36399	−0.190144	−0.0950719	−	0.995470i	$-0.530308\pi$
$313$	−3.36399	−0.190144	−0.0950719	+	0.995470i	$0.530308\pi$
$314$	6.92682	0.390903
$315$	13.0353	0.734458
$316$	−1.01068	−0.0568553
$317$	−8.00488	−0.449599	−0.224800	−	0.974405i	$-0.572173\pi$
$317$	−8.00488	−0.449599	−0.224800	+	0.974405i	$0.572173\pi$
$318$	−16.3930	−0.919272
$319$	−0.572883	−0.0320753
$320$	2.67295	0.149423
$321$	14.3100	0.798708
$322$	−1.76808	−0.0985314
$323$	−2.41118	−0.134162
$324$	−11.1362	−0.618677
$325$	−11.4466	−0.634943
$326$	12.6789	0.702220
$327$	29.7222	1.64364
$328$	−9.55708	−0.527702
$329$	−54.4057	−2.99948
$330$	13.0794	0.719995
$331$	−1.21229	−0.0666335	−0.0333168	−	0.999445i	$-0.510607\pi$
$331$	−1.21229	−0.0666335	−0.0333168	+	0.999445i	$0.510607\pi$
$332$	−11.1142	−0.609971
$333$	−8.51202	−0.466456
$334$	18.9367	1.03617
$335$	−26.5119	−1.44850
$336$	−8.55800	−0.466877
$337$	33.2630	1.81195	0.905975	−	0.423331i	$-0.139139\pi$
$337$	33.2630	1.81195	0.905975	+	0.423331i	$0.139139\pi$
$338$	15.4856	0.842305
$339$	−8.00122	−0.434566
$340$	2.14114	0.116120
$341$	6.56625	0.355582
$342$	−3.49960	−0.189237
$343$	15.0773	0.814098
$344$	−2.46109	−0.132693
$345$	2.29874	0.123760
$346$	−12.5582	−0.675131
$347$	−3.24867	−0.174398	−0.0871988	−	0.996191i	$-0.527792\pi$
$347$	−3.24867	−0.174398	−0.0871988	+	0.996191i	$0.527792\pi$
$348$	−0.487349	−0.0261246
$349$	−9.62179	−0.515043	−0.257521	−	0.966273i	$-0.582906\pi$
$349$	−9.62179	−0.515043	−0.257521	+	0.966273i	$0.582906\pi$
$350$	8.99605	0.480859
$351$	−20.0075	−1.06792
$352$	−2.39834	−0.127832
$353$	−3.20686	−0.170684	−0.0853419	−	0.996352i	$-0.527198\pi$
$353$	−3.20686	−0.170684	−0.0853419	+	0.996352i	$0.527198\pi$
$354$	−7.01036	−0.372596
$355$	−25.2303	−1.33909
$356$	−7.36845	−0.390527
$357$	−6.85528	−0.362820
$358$	22.5167	1.19004
$359$	−19.9591	−1.05340	−0.526701	−	0.850050i	$-0.676571\pi$
$359$	−19.9591	−1.05340	−0.526701	+	0.850050i	$0.676571\pi$
$360$	3.10767	0.163788
$361$	−9.93949	−0.523131
$362$	−3.08568	−0.162180
$363$	10.7072	0.561981
$364$	−22.3872	−1.17341
$365$	−1.07087	−0.0560520
$366$	−12.3098	−0.643445
$367$	14.0355	0.732647	0.366324	−	0.930488i	$-0.380616\pi$
$367$	14.0355	0.732647	0.366324	+	0.930488i	$0.380616\pi$
$368$	−0.421516	−0.0219731
$369$	−11.1114	−0.578435
$370$	−19.5696	−1.01737
$371$	33.7025	1.74974
$372$	5.58587	0.289614
$373$	−8.72325	−0.451673	−0.225836	−	0.974165i	$-0.572511\pi$
$373$	−8.72325	−0.451673	−0.225836	+	0.974165i	$0.572511\pi$
$374$	−1.92116	−0.0993409
$375$	15.5715	0.804107
$376$	−12.9705	−0.668901
$377$	−1.27488	−0.0656595
$378$	15.7242	0.808765
$379$	31.6337	1.62491	0.812457	−	0.583021i	$-0.198130\pi$
$379$	31.6337	1.62491	0.812457	+	0.583021i	$0.198130\pi$
$380$	−8.04577	−0.412739
$381$	−15.5763	−0.797996
$382$	−7.37115	−0.377141
$383$	−14.2137	−0.726287	−0.363144	−	0.931733i	$-0.618297\pi$
$383$	−14.2137	−0.726287	−0.363144	+	0.931733i	$0.618297\pi$
$384$	−2.04025	−0.104116
$385$	−26.8900	−1.37044
$386$	24.3783	1.24082
$387$	−2.86135	−0.145451
$388$	−16.8175	−0.853779
$389$	−14.0417	−0.711943	−0.355971	−	0.934497i	$-0.615850\pi$
$389$	−14.0417	−0.711943	−0.355971	+	0.934497i	$0.615850\pi$
$390$	29.1064	1.47386
$391$	−0.337651	−0.0170757
$392$	10.5945	0.535102
$393$	−43.7188	−2.20532
$394$	−1.53694	−0.0774298
$395$	−2.70151	−0.135928
$396$	−2.78839	−0.140122
$397$	−9.38410	−0.470974	−0.235487	−	0.971877i	$-0.575669\pi$
$397$	−9.38410	−0.470974	−0.235487	+	0.971877i	$0.575669\pi$
$398$	−24.3425	−1.22018
$399$	25.7602	1.28962
$400$	2.14469	0.107234
$401$	−30.7776	−1.53696	−0.768479	−	0.639875i	$-0.778986\pi$
$401$	−30.7776	−1.53696	−0.768479	+	0.639875i	$0.778986\pi$
$402$	20.2364	1.00930
$403$	14.6123	0.727892
$404$	10.1920	0.507070
$405$	−29.7665	−1.47911
$406$	1.00194	0.0497257
$407$	17.5590	0.870369
$408$	−1.63432	−0.0809109
$409$	−3.48340	−0.172243	−0.0861215	−	0.996285i	$-0.527447\pi$
$409$	−3.48340	−0.172243	−0.0861215	+	0.996285i	$0.527447\pi$
$410$	−25.5456	−1.26161
$411$	22.6744	1.11844
$412$	10.7986	0.532011
$413$	14.4127	0.709201
$414$	−0.490069	−0.0240856
$415$	−29.7078	−1.45830
$416$	−5.33719	−0.261677
$417$	−29.5608	−1.44760
$418$	7.21916	0.353101
$419$	29.5874	1.44544	0.722720	−	0.691141i	$-0.242892\pi$
$419$	29.5874	1.44544	0.722720	+	0.691141i	$0.242892\pi$
$420$	−22.8751	−1.11619
$421$	−6.68560	−0.325836	−0.162918	−	0.986640i	$-0.552091\pi$
$421$	−6.68560	−0.325836	−0.162918	+	0.986640i	$0.552091\pi$
$422$	−1.08059	−0.0526023
$423$	−15.0799	−0.733210
$424$	8.03477	0.390203
$425$	1.71798	0.0833340
$426$	19.2582	0.933062
$427$	25.3079	1.22473
$428$	−7.01385	−0.339027
$429$	−26.1160	−1.26089
$430$	−6.57839	−0.317238
$431$	9.98406	0.480915	0.240458	−	0.970660i	$-0.422703\pi$
$431$	9.98406	0.480915	0.240458	+	0.970660i	$0.422703\pi$
$432$	3.74869	0.180359
$433$	−30.0644	−1.44480	−0.722401	−	0.691475i	$-0.756961\pi$
$433$	−30.0644	−1.44480	−0.722401	+	0.691475i	$0.756961\pi$
$434$	−11.4840	−0.551252
$435$	−1.30266	−0.0624578
$436$	−14.5679	−0.697676
$437$	1.26879	0.0606946
$438$	0.817391	0.0390565
$439$	−36.2268	−1.72901	−0.864506	−	0.502622i	$-0.832369\pi$
$439$	−36.2268	−1.72901	−0.864506	+	0.502622i	$0.832369\pi$
$440$	−6.41065	−0.305616
$441$	12.3175	0.586547
$442$	−4.27529	−0.203355
$443$	−19.9284	−0.946827	−0.473413	−	0.880840i	$-0.656978\pi$
$443$	−19.9284	−0.946827	−0.473413	+	0.880840i	$0.656978\pi$
$444$	14.9374	0.708896
$445$	−19.6955	−0.933657
$446$	23.1324	1.09535
$447$	−8.64724	−0.409000
$448$	4.19458	0.198175
$449$	3.12974	0.147701	0.0738507	−	0.997269i	$-0.476471\pi$
$449$	3.12974	0.147701	0.0738507	+	0.997269i	$0.476471\pi$
$450$	2.49348	0.117544
$451$	22.9211	1.07931
$452$	3.92168	0.184460
$453$	7.63771	0.358851
$454$	−18.6940	−0.877352
$455$	−59.8401	−2.80535
$456$	6.14130	0.287593
$457$	10.8299	0.506601	0.253301	−	0.967388i	$-0.418484\pi$
$457$	10.8299	0.506601	0.253301	+	0.967388i	$0.418484\pi$
$458$	14.9675	0.699388
$459$	3.00285	0.140161
$460$	−1.12669	−0.0525324
$461$	31.1707	1.45177	0.725883	−	0.687819i	$-0.241432\pi$
$461$	31.1707	1.45177	0.725883	+	0.687819i	$0.241432\pi$
$462$	20.5250	0.954909
$463$	6.78995	0.315556	0.157778	−	0.987475i	$-0.449567\pi$
$463$	6.78995	0.315556	0.157778	+	0.987475i	$0.449567\pi$
$464$	0.238867	0.0110891
$465$	14.9308	0.692398
$466$	5.44756	0.252354
$467$	37.4714	1.73397	0.866984	−	0.498336i	$-0.166055\pi$
$467$	37.4714	1.73397	0.866984	+	0.498336i	$0.166055\pi$
$468$	−6.20519	−0.286835
$469$	−41.6042	−1.92110
$470$	−34.6695	−1.59918
$471$	−14.1325	−0.651190
$472$	3.43602	0.158156
$473$	5.90254	0.271399
$474$	2.06205	0.0947130
$475$	−6.45565	−0.296206
$476$	3.36002	0.154006
$477$	9.34149	0.427718
$478$	−19.3347	−0.884349
$479$	6.84565	0.312786	0.156393	−	0.987695i	$-0.450013\pi$
$479$	6.84565	0.312786	0.156393	+	0.987695i	$0.450013\pi$
$480$	−5.45350	−0.248917
$481$	39.0753	1.78168
$482$	9.45825	0.430812
$483$	3.60734	0.164139
$484$	−5.24797	−0.238544
$485$	−44.9524	−2.04118
$486$	11.4746	0.520496
$487$	−28.9894	−1.31364	−0.656818	−	0.754049i	$-0.728098\pi$
$487$	−28.9894	−1.31364	−0.656818	+	0.754049i	$0.728098\pi$
$488$	6.03348	0.273123
$489$	−25.8682	−1.16980
$490$	28.3186	1.27930
$491$	30.4910	1.37604	0.688020	−	0.725692i	$-0.258480\pi$
$491$	30.4910	1.37604	0.688020	+	0.725692i	$0.258480\pi$
$492$	19.4989	0.879077
$493$	0.191341	0.00861758
$494$	16.0653	0.722812
$495$	−7.45324	−0.334998
$496$	−2.73783	−0.122932
$497$	−39.5931	−1.77599
$498$	22.6758	1.01613
$499$	28.2435	1.26435	0.632176	−	0.774825i	$-0.282162\pi$
$499$	28.2435	1.26435	0.632176	+	0.774825i	$0.282162\pi$
$500$	−7.63212	−0.341319
$501$	−38.6358	−1.72612
$502$	−15.5320	−0.693228
$503$	−2.11175	−0.0941585	−0.0470792	−	0.998891i	$-0.514991\pi$
$503$	−2.11175	−0.0941585	−0.0470792	+	0.998891i	$0.514991\pi$
$504$	4.87676	0.217228
$505$	27.2427	1.21228
$506$	1.01094	0.0449417
$507$	−31.5945	−1.40316
$508$	7.63448	0.338725
$509$	−10.3413	−0.458371	−0.229186	−	0.973383i	$-0.573606\pi$
$509$	−10.3413	−0.458371	−0.229186	+	0.973383i	$0.573606\pi$
$510$	−4.36846	−0.193439
$511$	−1.68048	−0.0743402
$512$	1.00000	0.0441942
$513$	−11.2838	−0.498193
$514$	9.63121	0.424815
$515$	28.8643	1.27191
$516$	5.02125	0.221048
$517$	31.1076	1.36811
$518$	−30.7099	−1.34931
$519$	25.6218	1.12467
$520$	−14.2661	−0.625608
$521$	−7.62420	−0.334022	−0.167011	−	0.985955i	$-0.553412\pi$
$521$	−7.62420	−0.334022	−0.167011	+	0.985955i	$0.553412\pi$
$522$	0.277714	0.0121552
$523$	−26.0289	−1.13816	−0.569081	−	0.822281i	$-0.692701\pi$
$523$	−26.0289	−1.13816	−0.569081	+	0.822281i	$0.692701\pi$
$524$	21.4281	0.936093
$525$	−18.3542	−0.801044
$526$	−0.0567599	−0.00247485
$527$	−2.19311	−0.0955333
$528$	4.89322	0.212950
$529$	−22.8223	−0.992275
$530$	21.4766	0.932883
$531$	3.99483	0.173361
$532$	−12.6260	−0.547405
$533$	51.0079	2.20940
$534$	15.0335	0.650563
$535$	−18.7477	−0.810534
$536$	−9.91857	−0.428417
$537$	−45.9397	−1.98245
$538$	−2.65267	−0.114365
$539$	−25.4092	−1.09445
$540$	10.0201	0.431196
$541$	3.80091	0.163414	0.0817069	−	0.996656i	$-0.473963\pi$
$541$	3.80091	0.163414	0.0817069	+	0.996656i	$0.473963\pi$
$542$	−3.48178	−0.149555
$543$	6.29556	0.270168
$544$	0.801038	0.0343442
$545$	−38.9393	−1.66798
$546$	45.6757	1.95474
$547$	1.55100	0.0663159	0.0331580	−	0.999450i	$-0.489444\pi$
$547$	1.55100	0.0663159	0.0331580	+	0.999450i	$0.489444\pi$
$548$	−11.1135	−0.474745
$549$	7.01472	0.299381
$550$	−5.14369	−0.219327
$551$	−0.719005	−0.0306306
$552$	0.860000	0.0366040
$553$	−4.23938	−0.180277
$554$	5.86663	0.249249
$555$	39.9269	1.69480
$556$	14.4888	0.614462
$557$	−11.9035	−0.504366	−0.252183	−	0.967680i	$-0.581149\pi$
$557$	−11.9035	−0.504366	−0.252183	+	0.967680i	$0.581149\pi$
$558$	−3.18310	−0.134751
$559$	13.1353	0.555565
$560$	11.2119	0.473790
$561$	3.91966	0.165488
$562$	−11.2021	−0.472534
$563$	30.6587	1.29211	0.646055	−	0.763291i	$-0.276418\pi$
$563$	30.6587	1.29211	0.646055	+	0.763291i	$0.276418\pi$
$564$	26.4631	1.11430
$565$	10.4825	0.441001
$566$	19.5530	0.821875
$567$	−46.7116	−1.96170
$568$	−9.43912	−0.396057
$569$	14.3386	0.601105	0.300553	−	0.953765i	$-0.402829\pi$
$569$	14.3386	0.601105	0.300553	+	0.953765i	$0.402829\pi$
$570$	16.4154	0.687566
$571$	−34.5427	−1.44557	−0.722783	−	0.691076i	$-0.757137\pi$
$571$	−34.5427	−1.44557	−0.722783	+	0.691076i	$0.757137\pi$
$572$	12.8004	0.535211
$573$	15.0390	0.628264
$574$	−40.0879	−1.67324
$575$	−0.904020	−0.0377002
$576$	1.16263	0.0484431
$577$	−40.0807	−1.66858	−0.834290	−	0.551326i	$-0.814122\pi$
$577$	−40.0807	−1.66858	−0.834290	+	0.551326i	$0.814122\pi$
$578$	−16.3583	−0.680417
$579$	−49.7380	−2.06704
$580$	0.638480	0.0265114
$581$	−46.6194	−1.93410
$582$	34.3120	1.42228
$583$	−19.2701	−0.798086
$584$	−0.400632	−0.0165783
$585$	−16.5862	−0.685755
$586$	−15.1528	−0.625957
$587$	39.1026	1.61394	0.806969	−	0.590594i	$-0.201106\pi$
$587$	39.1026	1.61394	0.806969	+	0.590594i	$0.201106\pi$
$588$	−21.6154	−0.891405
$589$	8.24106	0.339567
$590$	9.18433	0.378113
$591$	3.13574	0.128987
$592$	−7.32133	−0.300905
$593$	11.9437	0.490471	0.245235	−	0.969464i	$-0.421135\pi$
$593$	11.9437	0.490471	0.245235	+	0.969464i	$0.421135\pi$
$594$	−8.99064	−0.368890
$595$	8.98117	0.368192
$596$	4.23832	0.173608
$597$	49.6649	2.03265
$598$	2.24971	0.0919976
$599$	−31.8588	−1.30171	−0.650857	−	0.759200i	$-0.725590\pi$
$599$	−31.8588	−1.30171	−0.650857	+	0.759200i	$0.725590\pi$
$600$	−4.37570	−0.178637
$601$	−11.2253	−0.457888	−0.228944	−	0.973440i	$-0.573527\pi$
$601$	−11.2253	−0.457888	−0.228944	+	0.973440i	$0.573527\pi$
$602$	−10.3232	−0.420744
$603$	−11.5317	−0.469605
$604$	−3.74351	−0.152321
$605$	−14.0276	−0.570302
$606$	−20.7942	−0.844707
$607$	−1.67512	−0.0679910	−0.0339955	−	0.999422i	$-0.510823\pi$
$607$	−1.67512	−0.0679910	−0.0339955	+	0.999422i	$0.510823\pi$
$608$	−3.01007	−0.122074
$609$	−2.04422	−0.0828360
$610$	16.1272	0.652972
$611$	69.2259	2.80058
$612$	0.931314	0.0376461
$613$	10.4102	0.420463	0.210231	−	0.977652i	$-0.432578\pi$
$613$	10.4102	0.420463	0.210231	+	0.977652i	$0.432578\pi$
$614$	−19.2004	−0.774864
$615$	52.1196	2.10166
$616$	−10.0600	−0.405330
$617$	19.3567	0.779271	0.389636	−	0.920969i	$-0.372601\pi$
$617$	19.3567	0.779271	0.389636	+	0.920969i	$0.372601\pi$
$618$	−22.0320	−0.886256
$619$	19.3738	0.778699	0.389349	−	0.921090i	$-0.372700\pi$
$619$	19.3738	0.778699	0.389349	+	0.921090i	$0.372700\pi$
$620$	−7.31810	−0.293902
$621$	−1.58014	−0.0634086
$622$	31.8211	1.27591
$623$	−30.9075	−1.23828
$624$	10.8892	0.435918
$625$	−31.1238	−1.24495
$626$	−3.36399	−0.134452
$627$	−14.7289	−0.588217
$628$	6.92682	0.276410
$629$	−5.86466	−0.233839
$630$	13.0353	0.519341
$631$	2.89056	0.115071	0.0575357	−	0.998343i	$-0.481676\pi$
$631$	2.89056	0.115071	0.0575357	+	0.998343i	$0.481676\pi$
$632$	−1.01068	−0.0402028
$633$	2.20468	0.0876280
$634$	−8.00488	−0.317915
$635$	20.4066	0.809812
$636$	−16.3930	−0.650023
$637$	−56.5447	−2.24038
$638$	−0.572883	−0.0226807
$639$	−10.9742	−0.434134
$640$	2.67295	0.105658
$641$	−14.9793	−0.591648	−0.295824	−	0.955243i	$-0.595594\pi$
$641$	−14.9793	−0.591648	−0.295824	+	0.955243i	$0.595594\pi$
$642$	14.3100	0.564772
$643$	−14.0359	−0.553523	−0.276762	−	0.960939i	$-0.589261\pi$
$643$	−14.0359	−0.553523	−0.276762	+	0.960939i	$0.589261\pi$
$644$	−1.76808	−0.0696722
$645$	13.4216	0.528474
$646$	−2.41118	−0.0948666
$647$	−12.3248	−0.484539	−0.242269	−	0.970209i	$-0.577892\pi$
$647$	−12.3248	−0.484539	−0.242269	+	0.970209i	$0.577892\pi$
$648$	−11.1362	−0.437471
$649$	−8.24075	−0.323478
$650$	−11.4466	−0.448972
$651$	23.4304	0.918308
$652$	12.6789	0.496544
$653$	−36.3293	−1.42167	−0.710837	−	0.703357i	$-0.751684\pi$
$653$	−36.3293	−1.42167	−0.710837	+	0.703357i	$0.751684\pi$
$654$	29.7222	1.16223
$655$	57.2765	2.23798
$656$	−9.55708	−0.373141
$657$	−0.465788	−0.0181721
$658$	−54.4057	−2.12095
$659$	−4.39841	−0.171338	−0.0856688	−	0.996324i	$-0.527303\pi$
$659$	−4.39841	−0.171338	−0.0856688	+	0.996324i	$0.527303\pi$
$660$	13.0794	0.509113
$661$	−26.4084	−1.02717	−0.513584	−	0.858039i	$-0.671682\pi$
$661$	−26.4084	−1.02717	−0.513584	+	0.858039i	$0.671682\pi$
$662$	−1.21229	−0.0471170
$663$	8.72268	0.338761
$664$	−11.1142	−0.431315
$665$	−33.7486	−1.30872
$666$	−8.51202	−0.329834
$667$	−0.100686	−0.00389858
$668$	18.9367	0.732685
$669$	−47.1960	−1.82470
$670$	−26.5119	−1.02424
$671$	−14.4703	−0.558621
$672$	−8.55800	−0.330132
$673$	−31.4133	−1.21089	−0.605447	−	0.795886i	$-0.707006\pi$
$673$	−31.4133	−1.21089	−0.605447	+	0.795886i	$0.707006\pi$
$674$	33.2630	1.28124
$675$	8.03977	0.309451
$676$	15.4856	0.595599
$677$	22.4385	0.862382	0.431191	−	0.902261i	$-0.358093\pi$
$677$	22.4385	0.862382	0.431191	+	0.902261i	$0.358093\pi$
$678$	−8.00122	−0.307285
$679$	−70.5423	−2.70717
$680$	2.14114	0.0821089
$681$	38.1405	1.46155
$682$	6.56625	0.251435
$683$	−30.9126	−1.18284	−0.591418	−	0.806365i	$-0.701432\pi$
$683$	−30.9126	−1.18284	−0.591418	+	0.806365i	$0.701432\pi$
$684$	−3.49960	−0.133811
$685$	−29.7059	−1.13500
$686$	15.0773	0.575654
$687$	−30.5376	−1.16508
$688$	−2.46109	−0.0938283
$689$	−42.8831	−1.63372
$690$	2.29874	0.0875116
$691$	−6.42503	−0.244420	−0.122210	−	0.992504i	$-0.538998\pi$
$691$	−6.42503	−0.244420	−0.122210	+	0.992504i	$0.538998\pi$
$692$	−12.5582	−0.477390
$693$	−11.6961	−0.444299
$694$	−3.24867	−0.123318
$695$	38.7279	1.46903
$696$	−0.487349	−0.0184729
$697$	−7.65559	−0.289976
$698$	−9.62179	−0.364190
$699$	−11.1144	−0.420386
$700$	8.99605	0.340019
$701$	12.7870	0.482958	0.241479	−	0.970406i	$-0.422367\pi$
$701$	12.7870	0.482958	0.241479	+	0.970406i	$0.422367\pi$
$702$	−20.0075	−0.755134
$703$	22.0377	0.831167
$704$	−2.39834	−0.0903908
$705$	70.7346	2.66402
$706$	−3.20686	−0.120692
$707$	42.7510	1.60782
$708$	−7.01036	−0.263465
$709$	−14.1053	−0.529736	−0.264868	−	0.964285i	$-0.585328\pi$
$709$	−14.1053	−0.529736	−0.264868	+	0.964285i	$0.585328\pi$
$710$	−25.2303	−0.946878
$711$	−1.17505	−0.0440679
$712$	−7.36845	−0.276144
$713$	1.15404	0.0432192
$714$	−6.85528	−0.256553
$715$	34.2149	1.27956
$716$	22.5167	0.841488
$717$	39.4477	1.47320
$718$	−19.9591	−0.744868
$719$	6.06752	0.226280	0.113140	−	0.993579i	$-0.463909\pi$
$719$	6.06752	0.226280	0.113140	+	0.993579i	$0.463909\pi$
$720$	3.10767	0.115816
$721$	45.2958	1.68690
$722$	−9.93949	−0.369910
$723$	−19.2972	−0.717672
$724$	−3.08568	−0.114678
$725$	0.512294	0.0190261
$726$	10.7072	0.397381
$727$	28.4746	1.05606	0.528032	−	0.849224i	$-0.322930\pi$
$727$	28.4746	1.05606	0.528032	+	0.849224i	$0.322930\pi$
$728$	−22.3872	−0.829727
$729$	9.99755	0.370280
$730$	−1.07087	−0.0396347
$731$	−1.97143	−0.0729159
$732$	−12.3098	−0.454984
$733$	−32.0738	−1.18467	−0.592337	−	0.805690i	$-0.701795\pi$
$733$	−32.0738	−1.18467	−0.592337	+	0.805690i	$0.701795\pi$
$734$	14.0355	0.518060
$735$	−57.7770	−2.13114
$736$	−0.421516	−0.0155373
$737$	23.7881	0.876246
$738$	−11.1114	−0.409016
$739$	15.0996	0.555449	0.277724	−	0.960661i	$-0.410420\pi$
$739$	15.0996	0.555449	0.277724	+	0.960661i	$0.410420\pi$
$740$	−19.5696	−0.719392
$741$	−32.7773	−1.20410
$742$	33.7025	1.23726
$743$	16.4345	0.602924	0.301462	−	0.953478i	$-0.402525\pi$
$743$	16.4345	0.602924	0.301462	+	0.953478i	$0.402525\pi$
$744$	5.58587	0.204788
$745$	11.3288	0.415056
$746$	−8.72325	−0.319381
$747$	−12.9218	−0.472782
$748$	−1.92116	−0.0702446
$749$	−29.4201	−1.07499
$750$	15.5715	0.568590
$751$	−3.26334	−0.119081	−0.0595404	−	0.998226i	$-0.518964\pi$
$751$	−3.26334	−0.119081	−0.0595404	+	0.998226i	$0.518964\pi$
$752$	−12.9705	−0.472985
$753$	31.6893	1.15482
$754$	−1.27488	−0.0464283
$755$	−10.0062	−0.364164
$756$	15.7242	0.571883
$757$	−30.7109	−1.11621	−0.558103	−	0.829771i	$-0.688471\pi$
$757$	−30.7109	−1.11621	−0.558103	+	0.829771i	$0.688471\pi$
$758$	31.6337	1.14899
$759$	−2.06257	−0.0748666
$760$	−8.04577	−0.291851
$761$	−41.6871	−1.51115	−0.755577	−	0.655059i	$-0.772644\pi$
$761$	−41.6871	−1.51115	−0.755577	+	0.655059i	$0.772644\pi$
$762$	−15.5763	−0.564269
$763$	−61.1062	−2.21219
$764$	−7.37115	−0.266679
$765$	2.48936	0.0900030
$766$	−14.2137	−0.513563
$767$	−18.3387	−0.662172
$768$	−2.04025	−0.0736213
$769$	6.83008	0.246299	0.123150	−	0.992388i	$-0.460701\pi$
$769$	6.83008	0.246299	0.123150	+	0.992388i	$0.460701\pi$
$770$	−26.8900	−0.969048
$771$	−19.6501	−0.707682
$772$	24.3783	0.877395
$773$	−13.6397	−0.490587	−0.245294	−	0.969449i	$-0.578884\pi$
$773$	−13.6397	−0.490587	−0.245294	+	0.969449i	$0.578884\pi$
$774$	−2.86135	−0.102849
$775$	−5.87179	−0.210921
$776$	−16.8175	−0.603713
$777$	62.6559	2.24777
$778$	−14.0417	−0.503420
$779$	28.7675	1.03070
$780$	29.1064	1.04218
$781$	22.6382	0.810059
$782$	−0.337651	−0.0120744
$783$	0.895438	0.0320003
$784$	10.5945	0.378374
$785$	18.5151	0.660831
$786$	−43.7188	−1.55940
$787$	−15.2235	−0.542660	−0.271330	−	0.962486i	$-0.587464\pi$
$787$	−15.2235	−0.542660	−0.271330	+	0.962486i	$0.587464\pi$
$788$	−1.53694	−0.0547511
$789$	0.115805	0.00412275
$790$	−2.70151	−0.0961153
$791$	16.4498	0.584887
$792$	−2.78839	−0.0990811
$793$	−32.2018	−1.14352
$794$	−9.38410	−0.333029
$795$	−43.8177	−1.55405
$796$	−24.3425	−0.862798
$797$	44.0125	1.55900	0.779501	−	0.626401i	$-0.215473\pi$
$797$	44.0125	1.55900	0.779501	+	0.626401i	$0.215473\pi$
$798$	25.7602	0.911900
$799$	−10.3898	−0.367566
$800$	2.14469	0.0758261
$801$	−8.56680	−0.302693
$802$	−30.7776	−1.08679
$803$	0.960852	0.0339077
$804$	20.2364	0.713682
$805$	−4.72600	−0.166570
$806$	14.6123	0.514697
$807$	5.41213	0.190516
$808$	10.1920	0.358552
$809$	39.5341	1.38995	0.694973	−	0.719036i	$-0.255416\pi$
$809$	39.5341	1.38995	0.694973	+	0.719036i	$0.255416\pi$
$810$	−29.7665	−1.04589
$811$	41.9015	1.47136	0.735680	−	0.677329i	$-0.236863\pi$
$811$	41.9015	1.47136	0.735680	+	0.677329i	$0.236863\pi$
$812$	1.00194	0.0351614
$813$	7.10372	0.249138
$814$	17.5590	0.615444
$815$	33.8901	1.18712
$816$	−1.63432	−0.0572127
$817$	7.40806	0.259175
$818$	−3.48340	−0.121794
$819$	−26.0282	−0.909498
$820$	−25.5456	−0.892093
$821$	22.5847	0.788211	0.394105	−	0.919065i	$-0.371054\pi$
$821$	22.5847	0.788211	0.394105	+	0.919065i	$0.371054\pi$
$822$	22.6744	0.790859
$823$	−16.9816	−0.591940	−0.295970	−	0.955197i	$-0.595643\pi$
$823$	−16.9816	−0.591940	−0.295970	+	0.955197i	$0.595643\pi$
$824$	10.7986	0.376189
$825$	10.4944	0.365369
$826$	14.4127	0.501481
$827$	27.4384	0.954127	0.477063	−	0.878869i	$-0.341701\pi$
$827$	27.4384	0.954127	0.477063	+	0.878869i	$0.341701\pi$
$828$	−0.490069	−0.0170311
$829$	−5.21012	−0.180955	−0.0904774	−	0.995899i	$-0.528839\pi$
$829$	−5.21012	−0.180955	−0.0904774	+	0.995899i	$0.528839\pi$
$830$	−29.7078	−1.03117
$831$	−11.9694	−0.415214
$832$	−5.33719	−0.185034
$833$	8.48658	0.294043
$834$	−29.5608	−1.02361
$835$	50.6171	1.75168
$836$	7.21916	0.249680
$837$	−10.2633	−0.354751
$838$	29.5874	1.02208
$839$	−28.1121	−0.970538	−0.485269	−	0.874365i	$-0.661278\pi$
$839$	−28.1121	−0.970538	−0.485269	+	0.874365i	$0.661278\pi$
$840$	−22.8751	−0.789267
$841$	−28.9429	−0.998033
$842$	−6.68560	−0.230401
$843$	22.8552	0.787175
$844$	−1.08059	−0.0371954
$845$	41.3923	1.42394
$846$	−15.0799	−0.518458
$847$	−22.0130	−0.756376
$848$	8.03477	0.275915
$849$	−39.8931	−1.36913
$850$	1.71798	0.0589261
$851$	3.08606	0.105789
$852$	19.2582	0.659775
$853$	33.5853	1.14994	0.574969	−	0.818175i	$-0.305014\pi$
$853$	33.5853	1.14994	0.574969	+	0.818175i	$0.305014\pi$
$854$	25.3079	0.866018
$855$	−9.35428	−0.319910
$856$	−7.01385	−0.239729
$857$	−1.17386	−0.0400983	−0.0200491	−	0.999799i	$-0.506382\pi$
$857$	−1.17386	−0.0400983	−0.0200491	+	0.999799i	$0.506382\pi$
$858$	−26.1160	−0.891587
$859$	30.1159	1.02754	0.513771	−	0.857927i	$-0.328248\pi$
$859$	30.1159	1.02754	0.513771	+	0.857927i	$0.328248\pi$
$860$	−6.57839	−0.224321
$861$	81.7895	2.78738
$862$	9.98406	0.340058
$863$	−19.8174	−0.674593	−0.337296	−	0.941398i	$-0.609512\pi$
$863$	−19.8174	−0.674593	−0.337296	+	0.941398i	$0.609512\pi$
$864$	3.74869	0.127533
$865$	−33.5674	−1.14133
$866$	−30.0644	−1.02163
$867$	33.3752	1.13348
$868$	−11.4840	−0.389794
$869$	2.42396	0.0822272
$870$	−1.30266	−0.0441643
$871$	52.9373	1.79371
$872$	−14.5679	−0.493331
$873$	−19.5526	−0.661755
$874$	1.26879	0.0429175
$875$	−32.0135	−1.08225
$876$	0.817391	0.0276171
$877$	21.6564	0.731285	0.365642	−	0.930755i	$-0.380849\pi$
$877$	21.6564	0.731285	0.365642	+	0.930755i	$0.380849\pi$
$878$	−36.2268	−1.22260
$879$	30.9156	1.04276
$880$	−6.41065	−0.216103
$881$	−31.7939	−1.07116	−0.535582	−	0.844484i	$-0.679908\pi$
$881$	−31.7939	−1.07116	−0.535582	+	0.844484i	$0.679908\pi$
$882$	12.3175	0.414752
$883$	52.6894	1.77314	0.886569	−	0.462596i	$-0.153082\pi$
$883$	52.6894	1.77314	0.886569	+	0.462596i	$0.153082\pi$
$884$	−4.27529	−0.143794
$885$	−18.7384	−0.629883
$886$	−19.9284	−0.669508
$887$	−5.01236	−0.168298	−0.0841492	−	0.996453i	$-0.526817\pi$
$887$	−5.01236	−0.168298	−0.0841492	+	0.996453i	$0.526817\pi$
$888$	14.9374	0.501265
$889$	32.0234	1.07403
$890$	−19.6955	−0.660195
$891$	26.7084	0.894763
$892$	23.1324	0.774531
$893$	39.0420	1.30649
$894$	−8.64724	−0.289207
$895$	60.1860	2.01180
$896$	4.19458	0.140131
$897$	−4.58998	−0.153255
$898$	3.12974	0.104441
$899$	−0.653977	−0.0218114
$900$	2.49348	0.0831161
$901$	6.43616	0.214419
$902$	22.9211	0.763190
$903$	21.0620	0.700901
$904$	3.92168	0.130433
$905$	−8.24787	−0.274169
$906$	7.63771	0.253746
$907$	−40.8132	−1.35518	−0.677590	−	0.735440i	$-0.736976\pi$
$907$	−40.8132	−1.35518	−0.677590	+	0.735440i	$0.736976\pi$
$908$	−18.6940	−0.620381
$909$	11.8495	0.393024
$910$	−59.8401	−1.98368
$911$	−15.3199	−0.507570	−0.253785	−	0.967261i	$-0.581676\pi$
$911$	−15.3199	−0.507570	−0.253785	+	0.967261i	$0.581676\pi$
$912$	6.14130	0.203359
$913$	26.6556	0.882173
$914$	10.8299	0.358221
$915$	−32.9036	−1.08776
$916$	14.9675	0.494542
$917$	89.8820	2.96817
$918$	3.00285	0.0991087
$919$	47.7432	1.57490	0.787452	−	0.616376i	$-0.211400\pi$
$919$	47.7432	1.57490	0.787452	+	0.616376i	$0.211400\pi$
$920$	−1.12669	−0.0371460
$921$	39.1736	1.29081
$922$	31.1707	1.02655
$923$	50.3784	1.65822
$924$	20.5250	0.675223
$925$	−15.7019	−0.516277
$926$	6.78995	0.223132
$927$	12.5549	0.412356
$928$	0.238867	0.00784118
$929$	16.8886	0.554097	0.277048	−	0.960856i	$-0.410644\pi$
$929$	16.8886	0.554097	0.277048	+	0.960856i	$0.410644\pi$
$930$	14.9308	0.489600
$931$	−31.8901	−1.04516
$932$	5.44756	0.178441
$933$	−64.9231	−2.12549
$934$	37.4714	1.22610
$935$	−5.13518	−0.167938
$936$	−6.20519	−0.202823
$937$	30.2579	0.988481	0.494240	−	0.869325i	$-0.335446\pi$
$937$	30.2579	0.988481	0.494240	+	0.869325i	$0.335446\pi$
$938$	−41.6042	−1.35843
$939$	6.86339	0.223978
$940$	−34.6695	−1.13079
$941$	−60.1500	−1.96083	−0.980417	−	0.196932i	$-0.936902\pi$
$941$	−60.1500	−1.96083	−0.980417	+	0.196932i	$0.936902\pi$
$942$	−14.1325	−0.460461
$943$	4.02846	0.131185
$944$	3.43602	0.111833
$945$	42.0300	1.36724
$946$	5.90254	0.191908
$947$	30.5954	0.994216	0.497108	−	0.867689i	$-0.334395\pi$
$947$	30.5954	0.994216	0.497108	+	0.867689i	$0.334395\pi$
$948$	2.06205	0.0669722
$949$	2.13825	0.0694105
$950$	−6.45565	−0.209449
$951$	16.3320	0.529601
$952$	3.36002	0.108899
$953$	−18.8637	−0.611055	−0.305527	−	0.952183i	$-0.598833\pi$
$953$	−18.8637	−0.611055	−0.305527	+	0.952183i	$0.598833\pi$
$954$	9.34149	0.302442
$955$	−19.7028	−0.637566
$956$	−19.3347	−0.625329
$957$	1.16883	0.0377828
$958$	6.84565	0.221173
$959$	−46.6165	−1.50532
$960$	−5.45350	−0.176011
$961$	−23.5043	−0.758202
$962$	39.0753	1.25984
$963$	−8.15454	−0.262776
$964$	9.45825	0.304630
$965$	65.1622	2.09764
$966$	3.60734	0.116064
$967$	29.8978	0.961450	0.480725	−	0.876871i	$-0.340374\pi$
$967$	29.8978	0.961450	0.480725	+	0.876871i	$0.340374\pi$
$968$	−5.24797	−0.168676
$969$	4.91942	0.158034
$970$	−44.9524	−1.44334
$971$	−41.1553	−1.32074	−0.660368	−	0.750942i	$-0.729600\pi$
$971$	−41.1553	−1.32074	−0.660368	+	0.750942i	$0.729600\pi$
$972$	11.4746	0.368047
$973$	60.7744	1.94834
$974$	−28.9894	−0.928881
$975$	23.3540	0.747925
$976$	6.03348	0.193127
$977$	15.3161	0.490005	0.245002	−	0.969522i	$-0.421211\pi$
$977$	15.3161	0.490005	0.245002	+	0.969522i	$0.421211\pi$
$978$	−25.8682	−0.827173
$979$	17.6720	0.564801
$980$	28.3186	0.904603
$981$	−16.9371	−0.540761
$982$	30.4910	0.973007
$983$	53.5638	1.70842	0.854210	−	0.519928i	$-0.174041\pi$
$983$	53.5638	1.70842	0.854210	+	0.519928i	$0.174041\pi$
$984$	19.4989	0.621601
$985$	−4.10816	−0.130897
$986$	0.191341	0.00609355
$987$	111.001	3.53321
$988$	16.0653	0.511105
$989$	1.03739	0.0329871
$990$	−7.45324	−0.236879
$991$	−0.354954	−0.0112755	−0.00563775	−	0.999984i	$-0.501795\pi$
$991$	−0.354954	−0.0112755	−0.00563775	+	0.999984i	$0.501795\pi$
$992$	−2.73783	−0.0869263
$993$	2.47338	0.0784904
$994$	−39.5931	−1.25582
$995$	−65.0665	−2.06275
$996$	22.6758	0.718510
$997$	6.46381	0.204711	0.102355	−	0.994748i	$-0.467362\pi$
$997$	6.46381	0.204711	0.102355	+	0.994748i	$0.467362\pi$
$998$	28.2435	0.894032
$999$	−27.4454	−0.868334

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8042.2.a.a.1.15	✓	67

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8042.2.a.a.1.15	✓	67	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8042 = 2 \cdot 4021 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8042.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.04025 q^{3} +1.00000 q^{4} +2.67295 q^{5} -2.04025 q^{6} +4.19458 q^{7} +1.00000 q^{8} +1.16263 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.04025 q^{3} +1.00000 q^{4} +2.67295 q^{5} -2.04025 q^{6} +4.19458 q^{7} +1.00000 q^{8} +1.16263 q^{9} +2.67295 q^{10} -2.39834 q^{11} -2.04025 q^{12} -5.33719 q^{13} +4.19458 q^{14} -5.45350 q^{15} +1.00000 q^{16} +0.801038 q^{17} +1.16263 q^{18} -3.01007 q^{19} +2.67295 q^{20} -8.55800 q^{21} -2.39834 q^{22} -0.421516 q^{23} -2.04025 q^{24} +2.14469 q^{25} -5.33719 q^{26} +3.74869 q^{27} +4.19458 q^{28} +0.238867 q^{29} -5.45350 q^{30} -2.73783 q^{31} +1.00000 q^{32} +4.89322 q^{33} +0.801038 q^{34} +11.2119 q^{35} +1.16263 q^{36} -7.32133 q^{37} -3.01007 q^{38} +10.8892 q^{39} +2.67295 q^{40} -9.55708 q^{41} -8.55800 q^{42} -2.46109 q^{43} -2.39834 q^{44} +3.10767 q^{45} -0.421516 q^{46} -12.9705 q^{47} -2.04025 q^{48} +10.5945 q^{49} +2.14469 q^{50} -1.63432 q^{51} -5.33719 q^{52} +8.03477 q^{53} +3.74869 q^{54} -6.41065 q^{55} +4.19458 q^{56} +6.14130 q^{57} +0.238867 q^{58} +3.43602 q^{59} -5.45350 q^{60} +6.03348 q^{61} -2.73783 q^{62} +4.87676 q^{63} +1.00000 q^{64} -14.2661 q^{65} +4.89322 q^{66} -9.91857 q^{67} +0.801038 q^{68} +0.860000 q^{69} +11.2119 q^{70} -9.43912 q^{71} +1.16263 q^{72} -0.400632 q^{73} -7.32133 q^{74} -4.37570 q^{75} -3.01007 q^{76} -10.0600 q^{77} +10.8892 q^{78} -1.01068 q^{79} +2.67295 q^{80} -11.1362 q^{81} -9.55708 q^{82} -11.1142 q^{83} -8.55800 q^{84} +2.14114 q^{85} -2.46109 q^{86} -0.487349 q^{87} -2.39834 q^{88} -7.36845 q^{89} +3.10767 q^{90} -22.3872 q^{91} -0.421516 q^{92} +5.58587 q^{93} -12.9705 q^{94} -8.04577 q^{95} -2.04025 q^{96} -16.8175 q^{97} +10.5945 q^{98} -2.78839 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9}+O(q^{10}) \) 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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