LMFDB - Embedded newform 8038.2.a.c.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8038,2,Mod(1,8038)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8038, 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("8038.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8038 = 2 \cdot 4019 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8038.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:	$64.1837531447$
Analytic rank:	$1$
Dimension:	$84$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Character	$\chi$	$=$	8038.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -3.18728 q^{3} +1.00000 q^{4} -3.96813 q^{5} +3.18728 q^{6} +3.34102 q^{7} -1.00000 q^{8} +7.15875 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -3.18728 q^{3} +1.00000 q^{4} -3.96813 q^{5} +3.18728 q^{6} +3.34102 q^{7} -1.00000 q^{8} +7.15875 q^{9} +3.96813 q^{10} -3.22823 q^{11} -3.18728 q^{12} +6.84974 q^{13} -3.34102 q^{14} +12.6475 q^{15} +1.00000 q^{16} +7.55331 q^{17} -7.15875 q^{18} +4.06098 q^{19} -3.96813 q^{20} -10.6488 q^{21} +3.22823 q^{22} -5.29353 q^{23} +3.18728 q^{24} +10.7460 q^{25} -6.84974 q^{26} -13.2551 q^{27} +3.34102 q^{28} -10.0428 q^{29} -12.6475 q^{30} +0.401830 q^{31} -1.00000 q^{32} +10.2893 q^{33} -7.55331 q^{34} -13.2576 q^{35} +7.15875 q^{36} +0.401957 q^{37} -4.06098 q^{38} -21.8320 q^{39} +3.96813 q^{40} -2.36209 q^{41} +10.6488 q^{42} -6.57932 q^{43} -3.22823 q^{44} -28.4068 q^{45} +5.29353 q^{46} +5.23165 q^{47} -3.18728 q^{48} +4.16245 q^{49} -10.7460 q^{50} -24.0745 q^{51} +6.84974 q^{52} -6.62813 q^{53} +13.2551 q^{54} +12.8100 q^{55} -3.34102 q^{56} -12.9435 q^{57} +10.0428 q^{58} +11.6193 q^{59} +12.6475 q^{60} -8.83445 q^{61} -0.401830 q^{62} +23.9176 q^{63} +1.00000 q^{64} -27.1806 q^{65} -10.2893 q^{66} -2.93285 q^{67} +7.55331 q^{68} +16.8719 q^{69} +13.2576 q^{70} +8.44310 q^{71} -7.15875 q^{72} -9.06558 q^{73} -0.401957 q^{74} -34.2506 q^{75} +4.06098 q^{76} -10.7856 q^{77} +21.8320 q^{78} -2.52200 q^{79} -3.96813 q^{80} +20.7715 q^{81} +2.36209 q^{82} +6.78106 q^{83} -10.6488 q^{84} -29.9725 q^{85} +6.57932 q^{86} +32.0094 q^{87} +3.22823 q^{88} +2.79050 q^{89} +28.4068 q^{90} +22.8851 q^{91} -5.29353 q^{92} -1.28074 q^{93} -5.23165 q^{94} -16.1145 q^{95} +3.18728 q^{96} -7.58015 q^{97} -4.16245 q^{98} -23.1101 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 84 q - 84 q^{2} - 19 q^{3} + 84 q^{4} - 32 q^{5} + 19 q^{6} + q^{7} - 84 q^{8} + 77 q^{9}+O(q^{10}) $ 84 * q - 84 * q^2 - 19 * q^3 + 84 * q^4 - 32 * q^5 + 19 * q^6 + q^7 - 84 * q^8 + 77 * q^9	\( 84 q - 84 q^{2} - 19 q^{3} + 84 q^{4} - 32 q^{5} + 19 q^{6} + q^{7} - 84 q^{8} + 77 q^{9} + 32 q^{10} - 6 q^{11} - 19 q^{12} - 29 q^{13} - q^{14} - 4 q^{15} + 84 q^{16} - 36 q^{17} - 77 q^{18} + 33 q^{19} - 32 q^{20} - 21 q^{21} + 6 q^{22} - 62 q^{23} + 19 q^{24} + 82 q^{25} + 29 q^{26} - 82 q^{27} + q^{28} - 51 q^{29} + 4 q^{30} + 39 q^{31} - 84 q^{32} - 32 q^{33} + 36 q^{34} - 34 q^{35} + 77 q^{36} - 32 q^{37} - 33 q^{38} + 29 q^{39} + 32 q^{40} - 38 q^{41} + 21 q^{42} - 6 q^{44} - 91 q^{45} + 62 q^{46} - 58 q^{47} - 19 q^{48} + 83 q^{49} - 82 q^{50} - q^{51} - 29 q^{52} - 106 q^{53} + 82 q^{54} + 32 q^{55} - q^{56} - 44 q^{57} + 51 q^{58} - 42 q^{59} - 4 q^{60} - 41 q^{61} - 39 q^{62} - 9 q^{63} + 84 q^{64} - 49 q^{65} + 32 q^{66} - 16 q^{67} - 36 q^{68} - 45 q^{69} + 34 q^{70} - 62 q^{71} - 77 q^{72} - 16 q^{73} + 32 q^{74} - 80 q^{75} + 33 q^{76} - 134 q^{77} - 29 q^{78} + 53 q^{79} - 32 q^{80} + 56 q^{81} + 38 q^{82} - 90 q^{83} - 21 q^{84} - 60 q^{85} - 3 q^{87} + 6 q^{88} - 54 q^{89} + 91 q^{90} + 33 q^{91} - 62 q^{92} - 69 q^{93} + 58 q^{94} - 47 q^{95} + 19 q^{96} - 31 q^{97} - 83 q^{98} + 23 q^{99}+O(q^{100}) \) 84 * q - 84 * q^2 - 19 * q^3 + 84 * q^4 - 32 * q^5 + 19 * q^6 + q^7 - 84 * q^8 + 77 * q^9 + 32 * q^10 - 6 * q^11 - 19 * q^12 - 29 * q^13 - q^14 - 4 * q^15 + 84 * q^16 - 36 * q^17 - 77 * q^18 + 33 * q^19 - 32 * q^20 - 21 * q^21 + 6 * q^22 - 62 * q^23 + 19 * q^24 + 82 * q^25 + 29 * q^26 - 82 * q^27 + q^28 - 51 * q^29 + 4 * q^30 + 39 * q^31 - 84 * q^32 - 32 * q^33 + 36 * q^34 - 34 * q^35 + 77 * q^36 - 32 * q^37 - 33 * q^38 + 29 * q^39 + 32 * q^40 - 38 * q^41 + 21 * q^42 - 6 * q^44 - 91 * q^45 + 62 * q^46 - 58 * q^47 - 19 * q^48 + 83 * q^49 - 82 * q^50 - q^51 - 29 * q^52 - 106 * q^53 + 82 * q^54 + 32 * q^55 - q^56 - 44 * q^57 + 51 * q^58 - 42 * q^59 - 4 * q^60 - 41 * q^61 - 39 * q^62 - 9 * q^63 + 84 * q^64 - 49 * q^65 + 32 * q^66 - 16 * q^67 - 36 * q^68 - 45 * q^69 + 34 * q^70 - 62 * q^71 - 77 * q^72 - 16 * q^73 + 32 * q^74 - 80 * q^75 + 33 * q^76 - 134 * q^77 - 29 * q^78 + 53 * q^79 - 32 * q^80 + 56 * q^81 + 38 * q^82 - 90 * q^83 - 21 * q^84 - 60 * q^85 - 3 * q^87 + 6 * q^88 - 54 * q^89 + 91 * q^90 + 33 * q^91 - 62 * q^92 - 69 * q^93 + 58 * q^94 - 47 * q^95 + 19 * q^96 - 31 * q^97 - 83 * q^98 + 23 * 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
$3$	−3.18728	−1.84018	−0.920088	−	0.391711i	$-0.871883\pi$
$3$	−3.18728	−1.84018	−0.920088	+	0.391711i	$0.871883\pi$
$4$	1.00000	0.500000
$5$	−3.96813	−1.77460	−0.887300	−	0.461192i	$-0.847422\pi$
$5$	−3.96813	−1.77460	−0.887300	+	0.461192i	$0.847422\pi$
$6$	3.18728	1.30120
$7$	3.34102	1.26279	0.631394	−	0.775462i	$-0.282483\pi$
$7$	3.34102	1.26279	0.631394	+	0.775462i	$0.282483\pi$
$8$	−1.00000	−0.353553
$9$	7.15875	2.38625
$10$	3.96813	1.25483
$11$	−3.22823	−0.973347	−0.486674	−	0.873584i	$-0.661790\pi$
$11$	−3.22823	−0.973347	−0.486674	+	0.873584i	$0.661790\pi$
$12$	−3.18728	−0.920088
$13$	6.84974	1.89978	0.949888	−	0.312591i	$-0.101197\pi$
$13$	6.84974	1.89978	0.949888	+	0.312591i	$0.101197\pi$
$14$	−3.34102	−0.892926
$15$	12.6475	3.26558
$16$	1.00000	0.250000
$17$	7.55331	1.83195	0.915973	−	0.401240i	$-0.131421\pi$
$17$	7.55331	1.83195	0.915973	+	0.401240i	$0.131421\pi$
$18$	−7.15875	−1.68733
$19$	4.06098	0.931652	0.465826	−	0.884876i	$-0.345757\pi$
$19$	4.06098	0.931652	0.465826	+	0.884876i	$0.345757\pi$
$20$	−3.96813	−0.887300
$21$	−10.6488	−2.32375
$22$	3.22823	0.688260
$23$	−5.29353	−1.10378	−0.551888	−	0.833918i	$-0.686093\pi$
$23$	−5.29353	−1.10378	−0.551888	+	0.833918i	$0.686093\pi$
$24$	3.18728	0.650601
$25$	10.7460	2.14921
$26$	−6.84974	−1.34334
$27$	−13.2551	−2.55095
$28$	3.34102	0.631394
$29$	−10.0428	−1.86491	−0.932455	−	0.361287i	$-0.882338\pi$
$29$	−10.0428	−1.86491	−0.932455	+	0.361287i	$0.882338\pi$
$30$	−12.6475	−2.30911
$31$	0.401830	0.0721708	0.0360854	−	0.999349i	$-0.488511\pi$
$31$	0.401830	0.0721708	0.0360854	+	0.999349i	$0.488511\pi$
$32$	−1.00000	−0.176777
$33$	10.2893	1.79113
$34$	−7.55331	−1.29538
$35$	−13.2576	−2.24095
$36$	7.15875	1.19313
$37$	0.401957	0.0660813	0.0330407	−	0.999454i	$-0.489481\pi$
$37$	0.401957	0.0660813	0.0330407	+	0.999454i	$0.489481\pi$
$38$	−4.06098	−0.658778
$39$	−21.8320	−3.49592
$40$	3.96813	0.627416
$41$	−2.36209	−0.368896	−0.184448	−	0.982842i	$-0.559050\pi$
$41$	−2.36209	−0.368896	−0.184448	+	0.982842i	$0.559050\pi$
$42$	10.6488	1.64314
$43$	−6.57932	−1.00334	−0.501668	−	0.865060i	$-0.667280\pi$
$43$	−6.57932	−1.00334	−0.501668	+	0.865060i	$0.667280\pi$
$44$	−3.22823	−0.486674
$45$	−28.4068	−4.23464
$46$	5.29353	0.780488
$47$	5.23165	0.763115	0.381558	−	0.924345i	$-0.375388\pi$
$47$	5.23165	0.763115	0.381558	+	0.924345i	$0.375388\pi$
$48$	−3.18728	−0.460044
$49$	4.16245	0.594635
$50$	−10.7460	−1.51972
$51$	−24.0745	−3.37110
$52$	6.84974	0.949888
$53$	−6.62813	−0.910444	−0.455222	−	0.890378i	$-0.650440\pi$
$53$	−6.62813	−0.910444	−0.455222	+	0.890378i	$0.650440\pi$
$54$	13.2551	1.80379
$55$	12.8100	1.72730
$56$	−3.34102	−0.446463
$57$	−12.9435	−1.71440
$58$	10.0428	1.31869
$59$	11.6193	1.51270	0.756352	−	0.654165i	$-0.226980\pi$
$59$	11.6193	1.51270	0.756352	+	0.654165i	$0.226980\pi$
$60$	12.6475	1.63279
$61$	−8.83445	−1.13114	−0.565568	−	0.824702i	$-0.691343\pi$
$61$	−8.83445	−1.13114	−0.565568	+	0.824702i	$0.691343\pi$
$62$	−0.401830	−0.0510324
$63$	23.9176	3.01333
$64$	1.00000	0.125000
$65$	−27.1806	−3.37134
$66$	−10.2893	−1.26652
$67$	−2.93285	−0.358305	−0.179152	−	0.983821i	$-0.557335\pi$
$67$	−2.93285	−0.358305	−0.179152	+	0.983821i	$0.557335\pi$
$68$	7.55331	0.915973
$69$	16.8719	2.03114
$70$	13.2576	1.58459
$71$	8.44310	1.00201	0.501006	−	0.865444i	$-0.332963\pi$
$71$	8.44310	1.00201	0.501006	+	0.865444i	$0.332963\pi$
$72$	−7.15875	−0.843667
$73$	−9.06558	−1.06105	−0.530523	−	0.847670i	$-0.678004\pi$
$73$	−9.06558	−1.06105	−0.530523	+	0.847670i	$0.678004\pi$
$74$	−0.401957	−0.0467265
$75$	−34.2506	−3.95492
$76$	4.06098	0.465826
$77$	−10.7856	−1.22913
$78$	21.8320	2.47199
$79$	−2.52200	−0.283747	−0.141873	−	0.989885i	$-0.545313\pi$
$79$	−2.52200	−0.283747	−0.141873	+	0.989885i	$0.545313\pi$
$80$	−3.96813	−0.443650
$81$	20.7715	2.30794
$82$	2.36209	0.260849
$83$	6.78106	0.744318	0.372159	−	0.928169i	$-0.378618\pi$
$83$	6.78106	0.744318	0.372159	+	0.928169i	$0.378618\pi$
$84$	−10.6488	−1.16188
$85$	−29.9725	−3.25097
$86$	6.57932	0.709466
$87$	32.0094	3.43176
$88$	3.22823	0.344130
$89$	2.79050	0.295792	0.147896	−	0.989003i	$-0.452750\pi$
$89$	2.79050	0.295792	0.147896	+	0.989003i	$0.452750\pi$
$90$	28.4068	2.99434
$91$	22.8851	2.39902
$92$	−5.29353	−0.551888
$93$	−1.28074	−0.132807
$94$	−5.23165	−0.539604
$95$	−16.1145	−1.65331
$96$	3.18728	0.325300
$97$	−7.58015	−0.769647	−0.384824	−	0.922990i	$-0.625738\pi$
$97$	−7.58015	−0.769647	−0.384824	+	0.922990i	$0.625738\pi$
$98$	−4.16245	−0.420470
$99$	−23.1101	−2.32265
$100$	10.7460	1.07460
$101$	−14.1118	−1.40418	−0.702091	−	0.712088i	$-0.747750\pi$
$101$	−14.1118	−1.40418	−0.702091	+	0.712088i	$0.747750\pi$
$102$	24.0745	2.38373
$103$	10.7392	1.05817	0.529085	−	0.848569i	$-0.322535\pi$
$103$	10.7392	1.05817	0.529085	+	0.848569i	$0.322535\pi$
$104$	−6.84974	−0.671672
$105$	42.2557	4.12374
$106$	6.62813	0.643781
$107$	9.69254	0.937013	0.468507	−	0.883460i	$-0.344792\pi$
$107$	9.69254	0.937013	0.468507	+	0.883460i	$0.344792\pi$
$108$	−13.2551	−1.27547
$109$	−19.3110	−1.84966	−0.924830	−	0.380382i	$-0.875793\pi$
$109$	−19.3110	−1.84966	−0.924830	+	0.380382i	$0.875793\pi$
$110$	−12.8100	−1.22139
$111$	−1.28115	−0.121601
$112$	3.34102	0.315697
$113$	3.84076	0.361308	0.180654	−	0.983547i	$-0.442179\pi$
$113$	3.84076	0.361308	0.180654	+	0.983547i	$0.442179\pi$
$114$	12.9435	1.21227
$115$	21.0054	1.95876
$116$	−10.0428	−0.932455
$117$	49.0356	4.53334
$118$	−11.6193	−1.06964
$119$	25.2358	2.31336
$120$	−12.6475	−1.15456
$121$	−0.578548	−0.0525953
$122$	8.83445	0.799833
$123$	7.52863	0.678834
$124$	0.401830	0.0360854
$125$	−22.8010	−2.03938
$126$	−23.9176	−2.13075
$127$	0.423607	0.0375891	0.0187945	−	0.999823i	$-0.494017\pi$
$127$	0.423607	0.0375891	0.0187945	+	0.999823i	$0.494017\pi$
$128$	−1.00000	−0.0883883
$129$	20.9701	1.84632
$130$	27.1806	2.38390
$131$	−0.921277	−0.0804924	−0.0402462	−	0.999190i	$-0.512814\pi$
$131$	−0.921277	−0.0804924	−0.0402462	+	0.999190i	$0.512814\pi$
$132$	10.2893	0.895565
$133$	13.5678	1.17648
$134$	2.93285	0.253360
$135$	52.5979	4.52691
$136$	−7.55331	−0.647691
$137$	14.3262	1.22397	0.611986	−	0.790869i	$-0.290371\pi$
$137$	14.3262	1.22397	0.611986	+	0.790869i	$0.290371\pi$
$138$	−16.8719	−1.43624
$139$	19.1451	1.62387	0.811934	−	0.583749i	$-0.198415\pi$
$139$	19.1451	1.62387	0.811934	+	0.583749i	$0.198415\pi$
$140$	−13.2576	−1.12047
$141$	−16.6747	−1.40427
$142$	−8.44310	−0.708529
$143$	−22.1125	−1.84914
$144$	7.15875	0.596563
$145$	39.8513	3.30947
$146$	9.06558	0.750273
$147$	−13.2669	−1.09423
$148$	0.401957	0.0330407
$149$	−16.8325	−1.37897	−0.689485	−	0.724300i	$-0.742163\pi$
$149$	−16.8325	−1.37897	−0.689485	+	0.724300i	$0.742163\pi$
$150$	34.2506	2.79655
$151$	−10.1909	−0.829320	−0.414660	−	0.909976i	$-0.636099\pi$
$151$	−10.1909	−0.829320	−0.414660	+	0.909976i	$0.636099\pi$
$152$	−4.06098	−0.329389
$153$	54.0722	4.37148
$154$	10.7856	0.869127
$155$	−1.59451	−0.128074
$156$	−21.8320	−1.74796
$157$	9.36291	0.747241	0.373621	−	0.927582i	$-0.378116\pi$
$157$	9.36291	0.747241	0.373621	+	0.927582i	$0.378116\pi$
$158$	2.52200	0.200639
$159$	21.1257	1.67538
$160$	3.96813	0.313708
$161$	−17.6858	−1.39384
$162$	−20.7715	−1.63196
$163$	4.28815	0.335874	0.167937	−	0.985798i	$-0.446290\pi$
$163$	4.28815	0.335874	0.167937	+	0.985798i	$0.446290\pi$
$164$	−2.36209	−0.184448
$165$	−40.8291	−3.17854
$166$	−6.78106	−0.526312
$167$	−20.1355	−1.55813	−0.779064	−	0.626944i	$-0.784305\pi$
$167$	−20.1355	−1.55813	−0.779064	+	0.626944i	$0.784305\pi$
$168$	10.6488	0.821571
$169$	33.9189	2.60915
$170$	29.9725	2.29878
$171$	29.0715	2.22316
$172$	−6.57932	−0.501668
$173$	−6.65189	−0.505733	−0.252867	−	0.967501i	$-0.581373\pi$
$173$	−6.65189	−0.505733	−0.252867	+	0.967501i	$0.581373\pi$
$174$	−32.0094	−2.42662
$175$	35.9028	2.71399
$176$	−3.22823	−0.243337
$177$	−37.0339	−2.78364
$178$	−2.79050	−0.209157
$179$	−13.7457	−1.02741	−0.513703	−	0.857968i	$-0.671727\pi$
$179$	−13.7457	−1.02741	−0.513703	+	0.857968i	$0.671727\pi$
$180$	−28.4068	−2.11732
$181$	−4.16777	−0.309788	−0.154894	−	0.987931i	$-0.549504\pi$
$181$	−4.16777	−0.309788	−0.154894	+	0.987931i	$0.549504\pi$
$182$	−22.8851	−1.69636
$183$	28.1579	2.08149
$184$	5.29353	0.390244
$185$	−1.59502	−0.117268
$186$	1.28074	0.0939087
$187$	−24.3838	−1.78312
$188$	5.23165	0.381558
$189$	−44.2856	−3.22130
$190$	16.1145	1.16907
$191$	−14.5076	−1.04974	−0.524868	−	0.851184i	$-0.675885\pi$
$191$	−14.5076	−1.04974	−0.524868	+	0.851184i	$0.675885\pi$
$192$	−3.18728	−0.230022
$193$	−17.2968	−1.24505	−0.622526	−	0.782599i	$-0.713893\pi$
$193$	−17.2968	−1.24505	−0.622526	+	0.782599i	$0.713893\pi$
$194$	7.58015	0.544223
$195$	86.6323	6.20387
$196$	4.16245	0.297318
$197$	19.9921	1.42438	0.712190	−	0.701986i	$-0.247703\pi$
$197$	19.9921	1.42438	0.712190	+	0.701986i	$0.247703\pi$
$198$	23.1101	1.64236
$199$	−15.6963	−1.11268	−0.556341	−	0.830954i	$-0.687795\pi$
$199$	−15.6963	−1.11268	−0.556341	+	0.830954i	$0.687795\pi$
$200$	−10.7460	−0.759859
$201$	9.34781	0.659344
$202$	14.1118	0.992906
$203$	−33.5534	−2.35499
$204$	−24.0745	−1.68555
$205$	9.37306	0.654643
$206$	−10.7392	−0.748239
$207$	−37.8950	−2.63389
$208$	6.84974	0.474944
$209$	−13.1098	−0.906821
$210$	−42.2557	−2.91592
$211$	26.4263	1.81926	0.909632	−	0.415414i	$-0.136363\pi$
$211$	26.4263	1.81926	0.909632	+	0.415414i	$0.136363\pi$
$212$	−6.62813	−0.455222
$213$	−26.9105	−1.84388
$214$	−9.69254	−0.662569
$215$	26.1076	1.78052
$216$	13.2551	0.901895
$217$	1.34252	0.0911364
$218$	19.3110	1.30791
$219$	28.8945	1.95251
$220$	12.8100	0.863651
$221$	51.7382	3.48029
$222$	1.28115	0.0859851
$223$	−22.9379	−1.53604	−0.768018	−	0.640428i	$-0.778757\pi$
$223$	−22.9379	−1.53604	−0.768018	+	0.640428i	$0.778757\pi$
$224$	−3.34102	−0.223232
$225$	76.9282	5.12855
$226$	−3.84076	−0.255483
$227$	16.6349	1.10410	0.552048	−	0.833813i	$-0.313847\pi$
$227$	16.6349	1.10410	0.552048	+	0.833813i	$0.313847\pi$
$228$	−12.9435	−0.857202
$229$	−12.2674	−0.810655	−0.405327	−	0.914172i	$-0.632842\pi$
$229$	−12.2674	−0.810655	−0.405327	+	0.914172i	$0.632842\pi$
$230$	−21.0054	−1.38505
$231$	34.3767	2.26182
$232$	10.0428	0.659345
$233$	−23.6418	−1.54883	−0.774414	−	0.632680i	$-0.781955\pi$
$233$	−23.6418	−1.54883	−0.774414	+	0.632680i	$0.781955\pi$
$234$	−49.0356	−3.20556
$235$	−20.7599	−1.35422
$236$	11.6193	0.756352
$237$	8.03831	0.522144
$238$	−25.2358	−1.63579
$239$	12.3024	0.795776	0.397888	−	0.917434i	$-0.369743\pi$
$239$	12.3024	0.795776	0.397888	+	0.917434i	$0.369743\pi$
$240$	12.6475	0.816395
$241$	−25.0886	−1.61610	−0.808049	−	0.589115i	$-0.799477\pi$
$241$	−25.0886	−1.61610	−0.808049	+	0.589115i	$0.799477\pi$
$242$	0.578548	0.0371905
$243$	−26.4391	−1.69607
$244$	−8.83445	−0.565568
$245$	−16.5171	−1.05524
$246$	−7.52863	−0.480008
$247$	27.8166	1.76993
$248$	−0.401830	−0.0255162
$249$	−21.6131	−1.36968
$250$	22.8010	1.44206
$251$	−9.60728	−0.606406	−0.303203	−	0.952926i	$-0.598056\pi$
$251$	−9.60728	−0.606406	−0.303203	+	0.952926i	$0.598056\pi$
$252$	23.9176	1.50666
$253$	17.0887	1.07436
$254$	−0.423607	−0.0265795
$255$	95.5307	5.98236
$256$	1.00000	0.0625000
$257$	10.8285	0.675466	0.337733	−	0.941242i	$-0.390340\pi$
$257$	10.8285	0.675466	0.337733	+	0.941242i	$0.390340\pi$
$258$	−20.9701	−1.30554
$259$	1.34295	0.0834467
$260$	−27.1806	−1.68567
$261$	−71.8942	−4.45014
$262$	0.921277	0.0569167
$263$	−10.7923	−0.665481	−0.332741	−	0.943018i	$-0.607973\pi$
$263$	−10.7923	−0.665481	−0.332741	+	0.943018i	$0.607973\pi$
$264$	−10.2893	−0.633260
$265$	26.3013	1.61567
$266$	−13.5678	−0.831897
$267$	−8.89409	−0.544309
$268$	−2.93285	−0.179152
$269$	13.0929	0.798288	0.399144	−	0.916888i	$-0.369307\pi$
$269$	13.0929	0.798288	0.399144	+	0.916888i	$0.369307\pi$
$270$	−52.5979	−3.20101
$271$	6.82927	0.414849	0.207424	−	0.978251i	$-0.433492\pi$
$271$	6.82927	0.414849	0.207424	+	0.978251i	$0.433492\pi$
$272$	7.55331	0.457987
$273$	−72.9414	−4.41461
$274$	−14.3262	−0.865479
$275$	−34.6906	−2.09192
$276$	16.8719	1.01557
$277$	−3.23914	−0.194621	−0.0973104	−	0.995254i	$-0.531024\pi$
$277$	−3.23914	−0.194621	−0.0973104	+	0.995254i	$0.531024\pi$
$278$	−19.1451	−1.14825
$279$	2.87660	0.172217
$280$	13.2576	0.792294
$281$	−6.48064	−0.386603	−0.193301	−	0.981139i	$-0.561920\pi$
$281$	−6.48064	−0.386603	−0.193301	+	0.981139i	$0.561920\pi$
$282$	16.6747	0.992967
$283$	−0.427917	−0.0254370	−0.0127185	−	0.999919i	$-0.504049\pi$
$283$	−0.427917	−0.0254370	−0.0127185	+	0.999919i	$0.504049\pi$
$284$	8.44310	0.501006
$285$	51.3613	3.04238
$286$	22.1125	1.30754
$287$	−7.89179	−0.465837
$288$	−7.15875	−0.421833
$289$	40.0525	2.35603
$290$	−39.8513	−2.34015
$291$	24.1600	1.41629
$292$	−9.06558	−0.530523
$293$	1.85853	0.108576	0.0542882	−	0.998525i	$-0.482711\pi$
$293$	1.85853	0.108576	0.0542882	+	0.998525i	$0.482711\pi$
$294$	13.2669	0.773740
$295$	−46.1068	−2.68444
$296$	−0.401957	−0.0233633
$297$	42.7905	2.48296
$298$	16.8325	0.975079
$299$	−36.2593	−2.09693
$300$	−34.2506	−1.97746
$301$	−21.9817	−1.26700
$302$	10.1909	0.586418
$303$	44.9784	2.58394
$304$	4.06098	0.232913
$305$	35.0562	2.00731
$306$	−54.0722	−3.09110
$307$	−5.77645	−0.329679	−0.164840	−	0.986320i	$-0.552711\pi$
$307$	−5.77645	−0.329679	−0.164840	+	0.986320i	$0.552711\pi$
$308$	−10.7856	−0.614566
$309$	−34.2290	−1.94722
$310$	1.59451	0.0905622
$311$	8.06052	0.457070	0.228535	−	0.973536i	$-0.426606\pi$
$311$	8.06052	0.457070	0.228535	+	0.973536i	$0.426606\pi$
$312$	21.8320	1.23600
$313$	25.9681	1.46781	0.733903	−	0.679255i	$-0.237697\pi$
$313$	25.9681	1.46781	0.733903	+	0.679255i	$0.237697\pi$
$314$	−9.36291	−0.528380
$315$	−94.9079	−5.34746
$316$	−2.52200	−0.141873
$317$	−0.161805	−0.00908786	−0.00454393	−	0.999990i	$-0.501446\pi$
$317$	−0.161805	−0.00908786	−0.00454393	+	0.999990i	$0.501446\pi$
$318$	−21.1257	−1.18467
$319$	32.4206	1.81520
$320$	−3.96813	−0.221825
$321$	−30.8928	−1.72427
$322$	17.6858	0.985591
$323$	30.6738	1.70674
$324$	20.7715	1.15397
$325$	73.6075	4.08301
$326$	−4.28815	−0.237499
$327$	61.5496	3.40370
$328$	2.36209	0.130424
$329$	17.4791	0.963653
$330$	40.8291	2.24757
$331$	3.29695	0.181217	0.0906084	−	0.995887i	$-0.471119\pi$
$331$	3.29695	0.181217	0.0906084	+	0.995887i	$0.471119\pi$
$332$	6.78106	0.372159
$333$	2.87751	0.157687
$334$	20.1355	1.10176
$335$	11.6379	0.635848
$336$	−10.6488	−0.580939
$337$	−3.70402	−0.201771	−0.100885	−	0.994898i	$-0.532168\pi$
$337$	−3.70402	−0.201771	−0.100885	+	0.994898i	$0.532168\pi$
$338$	−33.9189	−1.84495
$339$	−12.2416	−0.664870
$340$	−29.9725	−1.62549
$341$	−1.29720	−0.0702472
$342$	−29.0715	−1.57201
$343$	−9.48034	−0.511890
$344$	6.57932	0.354733
$345$	−66.9500	−3.60447
$346$	6.65189	0.357608
$347$	−23.5003	−1.26156	−0.630782	−	0.775960i	$-0.717266\pi$
$347$	−23.5003	−1.26156	−0.630782	+	0.775960i	$0.717266\pi$
$348$	32.0094	1.71588
$349$	−30.1176	−1.61216	−0.806079	−	0.591808i	$-0.798414\pi$
$349$	−30.1176	−1.61216	−0.806079	+	0.591808i	$0.798414\pi$
$350$	−35.9028	−1.91908
$351$	−90.7940	−4.84622
$352$	3.22823	0.172065
$353$	−18.5054	−0.984943	−0.492471	−	0.870329i	$-0.663906\pi$
$353$	−18.5054	−0.984943	−0.492471	+	0.870329i	$0.663906\pi$
$354$	37.0339	1.96833
$355$	−33.5033	−1.77817
$356$	2.79050	0.147896
$357$	−80.4335	−4.25699
$358$	13.7457	0.726485
$359$	24.5040	1.29327	0.646635	−	0.762800i	$-0.276176\pi$
$359$	24.5040	1.29327	0.646635	+	0.762800i	$0.276176\pi$
$360$	28.4068	1.49717
$361$	−2.50846	−0.132024
$362$	4.16777	0.219053
$363$	1.84400	0.0967846
$364$	22.8851	1.19951
$365$	35.9734	1.88293
$366$	−28.1579	−1.47183
$367$	17.5792	0.917624	0.458812	−	0.888533i	$-0.348275\pi$
$367$	17.5792	0.917624	0.458812	+	0.888533i	$0.348275\pi$
$368$	−5.29353	−0.275944
$369$	−16.9096	−0.880278
$370$	1.59502	0.0829210
$371$	−22.1448	−1.14970
$372$	−1.28074	−0.0664035
$373$	−17.9531	−0.929579	−0.464789	−	0.885421i	$-0.653870\pi$
$373$	−17.9531	−0.929579	−0.464789	+	0.885421i	$0.653870\pi$
$374$	24.3838	1.26086
$375$	72.6731	3.75283
$376$	−5.23165	−0.269802
$377$	−68.7909	−3.54291
$378$	44.2856	2.27781
$379$	24.9535	1.28178	0.640889	−	0.767634i	$-0.278566\pi$
$379$	24.9535	1.28178	0.640889	+	0.767634i	$0.278566\pi$
$380$	−16.1145	−0.826655
$381$	−1.35015	−0.0691705
$382$	14.5076	0.742276
$383$	−15.6066	−0.797459	−0.398730	−	0.917069i	$-0.630549\pi$
$383$	−15.6066	−0.797459	−0.398730	+	0.917069i	$0.630549\pi$
$384$	3.18728	0.162650
$385$	42.7986	2.18122
$386$	17.2968	0.880384
$387$	−47.0997	−2.39421
$388$	−7.58015	−0.384824
$389$	38.2658	1.94015	0.970075	−	0.242804i	$-0.0780672\pi$
$389$	38.2658	1.94015	0.970075	+	0.242804i	$0.0780672\pi$
$390$	−86.6323	−4.38680
$391$	−39.9836	−2.02206
$392$	−4.16245	−0.210235
$393$	2.93637	0.148120
$394$	−19.9921	−1.00719
$395$	10.0076	0.503537
$396$	−23.1101	−1.16132
$397$	−23.9783	−1.20344	−0.601718	−	0.798708i	$-0.705517\pi$
$397$	−23.9783	−1.20344	−0.601718	+	0.798708i	$0.705517\pi$
$398$	15.6963	0.786785
$399$	−43.2445	−2.16493
$400$	10.7460	0.537302
$401$	6.72372	0.335767	0.167883	−	0.985807i	$-0.446307\pi$
$401$	6.72372	0.335767	0.167883	+	0.985807i	$0.446307\pi$
$402$	−9.34781	−0.466227
$403$	2.75243	0.137108
$404$	−14.1118	−0.702091
$405$	−82.4238	−4.09567
$406$	33.5534	1.66523
$407$	−1.29761	−0.0643201
$408$	24.0745	1.19187
$409$	22.1936	1.09740	0.548701	−	0.836019i	$-0.315123\pi$
$409$	22.1936	1.09740	0.548701	+	0.836019i	$0.315123\pi$
$410$	−9.37306	−0.462902
$411$	−45.6617	−2.25233
$412$	10.7392	0.529085
$413$	38.8204	1.91022
$414$	37.8950	1.86244
$415$	−26.9081	−1.32087
$416$	−6.84974	−0.335836
$417$	−61.0209	−2.98821
$418$	13.1098	0.641219
$419$	37.5936	1.83657	0.918283	−	0.395925i	$-0.129576\pi$
$419$	37.5936	1.83657	0.918283	+	0.395925i	$0.129576\pi$
$420$	42.2557	2.06187
$421$	11.8976	0.579853	0.289926	−	0.957049i	$-0.406369\pi$
$421$	11.8976	0.579853	0.289926	+	0.957049i	$0.406369\pi$
$422$	−26.4263	−1.28641
$423$	37.4521	1.82098
$424$	6.62813	0.321891
$425$	81.1681	3.93723
$426$	26.9105	1.30382
$427$	−29.5161	−1.42838
$428$	9.69254	0.468507
$429$	70.4788	3.40275
$430$	−26.1076	−1.25902
$431$	−10.0045	−0.481898	−0.240949	−	0.970538i	$-0.577459\pi$
$431$	−10.0045	−0.481898	−0.240949	+	0.970538i	$0.577459\pi$
$432$	−13.2551	−0.637736
$433$	−35.1917	−1.69121	−0.845603	−	0.533812i	$-0.820759\pi$
$433$	−35.1917	−1.69121	−0.845603	+	0.533812i	$0.820759\pi$
$434$	−1.34252	−0.0644432
$435$	−127.017	−6.09001
$436$	−19.3110	−0.924830
$437$	−21.4969	−1.02834
$438$	−28.8945	−1.38063
$439$	22.7458	1.08560	0.542799	−	0.839863i	$-0.317365\pi$
$439$	22.7458	1.08560	0.542799	+	0.839863i	$0.317365\pi$
$440$	−12.8100	−0.610694
$441$	29.7979	1.41895
$442$	−51.7382	−2.46093
$443$	0.569085	0.0270380	0.0135190	−	0.999909i	$-0.495697\pi$
$443$	0.569085	0.0270380	0.0135190	+	0.999909i	$0.495697\pi$
$444$	−1.28115	−0.0608006
$445$	−11.0730	−0.524913
$446$	22.9379	1.08614
$447$	53.6498	2.53755
$448$	3.34102	0.157849
$449$	34.7150	1.63830	0.819150	−	0.573579i	$-0.194445\pi$
$449$	34.7150	1.63830	0.819150	+	0.573579i	$0.194445\pi$
$450$	−76.9282	−3.62643
$451$	7.62535	0.359064
$452$	3.84076	0.180654
$453$	32.4811	1.52610
$454$	−16.6349	−0.780713
$455$	−90.8112	−4.25729
$456$	12.9435	0.606134
$457$	24.7903	1.15964	0.579820	−	0.814744i	$-0.303123\pi$
$457$	24.7903	1.15964	0.579820	+	0.814744i	$0.303123\pi$
$458$	12.2674	0.573219
$459$	−100.120	−4.67319
$460$	21.0054	0.979381
$461$	1.24441	0.0579582	0.0289791	−	0.999580i	$-0.490774\pi$
$461$	1.24441	0.0579582	0.0289791	+	0.999580i	$0.490774\pi$
$462$	−34.3767	−1.59935
$463$	−30.5376	−1.41920	−0.709601	−	0.704604i	$-0.751125\pi$
$463$	−30.5376	−1.41920	−0.709601	+	0.704604i	$0.751125\pi$
$464$	−10.0428	−0.466227
$465$	5.08215	0.235679
$466$	23.6418	1.09519
$467$	−4.62672	−0.214099	−0.107050	−	0.994254i	$-0.534140\pi$
$467$	−4.62672	−0.214099	−0.107050	+	0.994254i	$0.534140\pi$
$468$	49.0356	2.26667
$469$	−9.79872	−0.452463
$470$	20.7599	0.957581
$471$	−29.8422	−1.37506
$472$	−11.6193	−0.534821
$473$	21.2395	0.976595
$474$	−8.03831	−0.369212
$475$	43.6394	2.00231
$476$	25.2358	1.15668
$477$	−47.4492	−2.17255
$478$	−12.3024	−0.562698
$479$	6.05603	0.276707	0.138354	−	0.990383i	$-0.455819\pi$
$479$	6.05603	0.276707	0.138354	+	0.990383i	$0.455819\pi$
$480$	−12.6475	−0.577278
$481$	2.75330	0.125540
$482$	25.0886	1.14275
$483$	56.3696	2.56491
$484$	−0.578548	−0.0262976
$485$	30.0790	1.36582
$486$	26.4391	1.19930
$487$	−11.6914	−0.529788	−0.264894	−	0.964278i	$-0.585337\pi$
$487$	−11.6914	−0.529788	−0.264894	+	0.964278i	$0.585337\pi$
$488$	8.83445	0.399917
$489$	−13.6675	−0.618067
$490$	16.5171	0.746167
$491$	21.9017	0.988412	0.494206	−	0.869345i	$-0.335459\pi$
$491$	21.9017	0.988412	0.494206	+	0.869345i	$0.335459\pi$
$492$	7.52863	0.339417
$493$	−75.8567	−3.41641
$494$	−27.8166	−1.25153
$495$	91.7037	4.12178
$496$	0.401830	0.0180427
$497$	28.2086	1.26533
$498$	21.6131	0.968508
$499$	−17.2657	−0.772917	−0.386459	−	0.922307i	$-0.626302\pi$
$499$	−17.2657	−0.772917	−0.386459	+	0.922307i	$0.626302\pi$
$500$	−22.8010	−1.01969
$501$	64.1773	2.86723
$502$	9.60728	0.428794
$503$	37.0158	1.65045	0.825226	−	0.564802i	$-0.191048\pi$
$503$	37.0158	1.65045	0.825226	+	0.564802i	$0.191048\pi$
$504$	−23.9176	−1.06537
$505$	55.9976	2.49186
$506$	−17.0887	−0.759686
$507$	−108.109	−4.80129
$508$	0.423607	0.0187945
$509$	−17.4429	−0.773142	−0.386571	−	0.922260i	$-0.626341\pi$
$509$	−17.4429	−0.773142	−0.386571	+	0.922260i	$0.626341\pi$
$510$	−95.5307	−4.23017
$511$	−30.2883	−1.33988
$512$	−1.00000	−0.0441942
$513$	−53.8287	−2.37659
$514$	−10.8285	−0.477626
$515$	−42.6147	−1.87783
$516$	20.9701	0.923158
$517$	−16.8890	−0.742776
$518$	−1.34295	−0.0590058
$519$	21.2014	0.930639
$520$	27.1806	1.19195
$521$	−12.6215	−0.552958	−0.276479	−	0.961020i	$-0.589168\pi$
$521$	−12.6215	−0.552958	−0.276479	+	0.961020i	$0.589168\pi$
$522$	71.8942	3.14672
$523$	9.46865	0.414035	0.207018	−	0.978337i	$-0.433624\pi$
$523$	9.46865	0.414035	0.207018	+	0.978337i	$0.433624\pi$
$524$	−0.921277	−0.0402462
$525$	−114.432	−4.99423
$526$	10.7923	0.470566
$527$	3.03514	0.132213
$528$	10.2893	0.447783
$529$	5.02143	0.218323
$530$	−26.3013	−1.14245
$531$	83.1796	3.60969
$532$	13.5678	0.588240
$533$	−16.1797	−0.700819
$534$	8.89409	0.384885
$535$	−38.4612	−1.66282
$536$	2.93285	0.126680
$537$	43.8115	1.89061
$538$	−13.0929	−0.564475
$539$	−13.4373	−0.578786
$540$	52.5979	2.26345
$541$	16.1311	0.693531	0.346765	−	0.937952i	$-0.387280\pi$
$541$	16.1311	0.693531	0.346765	+	0.937952i	$0.387280\pi$
$542$	−6.82927	−0.293342
$543$	13.2838	0.570064
$544$	−7.55331	−0.323845
$545$	76.6285	3.28241
$546$	72.9414	3.12160
$547$	−19.0153	−0.813037	−0.406518	−	0.913643i	$-0.633257\pi$
$547$	−19.0153	−0.813037	−0.406518	+	0.913643i	$0.633257\pi$
$548$	14.3262	0.611986
$549$	−63.2436	−2.69917
$550$	34.6906	1.47921
$551$	−40.7838	−1.73745
$552$	−16.8719	−0.718118
$553$	−8.42605	−0.358312
$554$	3.23914	0.137618
$555$	5.08376	0.215794
$556$	19.1451	0.811934
$557$	−11.0019	−0.466164	−0.233082	−	0.972457i	$-0.574881\pi$
$557$	−11.0019	−0.466164	−0.233082	+	0.972457i	$0.574881\pi$
$558$	−2.87660	−0.121776
$559$	−45.0666	−1.90611
$560$	−13.2576	−0.560236
$561$	77.7180	3.28126
$562$	6.48064	0.273369
$563$	17.4637	0.736009	0.368005	−	0.929824i	$-0.380041\pi$
$563$	17.4637	0.736009	0.368005	+	0.929824i	$0.380041\pi$
$564$	−16.6747	−0.702133
$565$	−15.2406	−0.641177
$566$	0.427917	0.0179867
$567$	69.3980	2.91444
$568$	−8.44310	−0.354265
$569$	−6.49516	−0.272291	−0.136146	−	0.990689i	$-0.543472\pi$
$569$	−6.49516	−0.272291	−0.136146	+	0.990689i	$0.543472\pi$
$570$	−51.3613	−2.15129
$571$	−9.92488	−0.415343	−0.207672	−	0.978199i	$-0.566589\pi$
$571$	−9.92488	−0.415343	−0.207672	+	0.978199i	$0.566589\pi$
$572$	−22.1125	−0.924571
$573$	46.2399	1.93170
$574$	7.89179	0.329397
$575$	−56.8844	−2.37224
$576$	7.15875	0.298281
$577$	−34.0806	−1.41879	−0.709397	−	0.704809i	$-0.751033\pi$
$577$	−34.0806	−1.41879	−0.709397	+	0.704809i	$0.751033\pi$
$578$	−40.0525	−1.66596
$579$	55.1298	2.29111
$580$	39.8513	1.65473
$581$	22.6557	0.939917
$582$	−24.1600	−1.00147
$583$	21.3971	0.886178
$584$	9.06558	0.375136
$585$	−194.579	−8.04487
$586$	−1.85853	−0.0767751
$587$	−19.1619	−0.790895	−0.395447	−	0.918489i	$-0.629410\pi$
$587$	−19.1619	−0.790895	−0.395447	+	0.918489i	$0.629410\pi$
$588$	−13.2669	−0.547117
$589$	1.63182	0.0672380
$590$	46.1068	1.89819
$591$	−63.7205	−2.62111
$592$	0.401957	0.0165203
$593$	−22.3688	−0.918576	−0.459288	−	0.888287i	$-0.651895\pi$
$593$	−22.3688	−0.918576	−0.459288	+	0.888287i	$0.651895\pi$
$594$	−42.7905	−1.75571
$595$	−100.139	−4.10529
$596$	−16.8325	−0.689485
$597$	50.0286	2.04753
$598$	36.2593	1.48275
$599$	−30.1124	−1.23036	−0.615180	−	0.788387i	$-0.710917\pi$
$599$	−30.1124	−1.23036	−0.615180	+	0.788387i	$0.710917\pi$
$600$	34.2506	1.39828
$601$	−3.81963	−0.155806	−0.0779029	−	0.996961i	$-0.524822\pi$
$601$	−3.81963	−0.155806	−0.0779029	+	0.996961i	$0.524822\pi$
$602$	21.9817	0.895906
$603$	−20.9955	−0.855005
$604$	−10.1909	−0.414660
$605$	2.29575	0.0933356
$606$	−44.9784	−1.82712
$607$	4.36929	0.177344	0.0886720	−	0.996061i	$-0.471738\pi$
$607$	4.36929	0.177344	0.0886720	+	0.996061i	$0.471738\pi$
$608$	−4.06098	−0.164694
$609$	106.944	4.33359
$610$	−35.0562	−1.41938
$611$	35.8355	1.44975
$612$	54.0722	2.18574
$613$	28.6175	1.15585	0.577926	−	0.816089i	$-0.303862\pi$
$613$	28.6175	1.15585	0.577926	+	0.816089i	$0.303862\pi$
$614$	5.77645	0.233119
$615$	−29.8746	−1.20466
$616$	10.7856	0.434564
$617$	−25.8568	−1.04096	−0.520478	−	0.853875i	$-0.674246\pi$
$617$	−25.8568	−1.04096	−0.520478	+	0.853875i	$0.674246\pi$
$618$	34.2290	1.37689
$619$	−32.2227	−1.29514	−0.647570	−	0.762006i	$-0.724215\pi$
$619$	−32.2227	−1.29514	−0.647570	+	0.762006i	$0.724215\pi$
$620$	−1.59451	−0.0640371
$621$	70.1662	2.81567
$622$	−8.06052	−0.323197
$623$	9.32312	0.373523
$624$	−21.8320	−0.873981
$625$	36.7471	1.46988
$626$	−25.9681	−1.03789
$627$	41.7845	1.66871
$628$	9.36291	0.373621
$629$	3.03610	0.121057
$630$	94.9079	3.78122
$631$	−9.34408	−0.371982	−0.185991	−	0.982551i	$-0.559550\pi$
$631$	−9.34408	−0.371982	−0.185991	+	0.982551i	$0.559550\pi$
$632$	2.52200	0.100320
$633$	−84.2282	−3.34777
$634$	0.161805	0.00642609
$635$	−1.68093	−0.0667056
$636$	21.1257	0.837689
$637$	28.5117	1.12967
$638$	−32.4206	−1.28354
$639$	60.4421	2.39105
$640$	3.96813	0.156854
$641$	9.88041	0.390253	0.195126	−	0.980778i	$-0.437488\pi$
$641$	9.88041	0.390253	0.195126	+	0.980778i	$0.437488\pi$
$642$	30.8928	1.21924
$643$	30.6746	1.20969	0.604843	−	0.796345i	$-0.293236\pi$
$643$	30.6746	1.20969	0.604843	+	0.796345i	$0.293236\pi$
$644$	−17.6858	−0.696918
$645$	−83.2121	−3.27647
$646$	−30.6738	−1.20685
$647$	−46.7926	−1.83961	−0.919803	−	0.392381i	$-0.871651\pi$
$647$	−46.7926	−1.83961	−0.919803	+	0.392381i	$0.871651\pi$
$648$	−20.7715	−0.815980
$649$	−37.5097	−1.47239
$650$	−73.6075	−2.88712
$651$	−4.27900	−0.167707
$652$	4.28815	0.167937
$653$	3.17147	0.124109	0.0620547	−	0.998073i	$-0.480235\pi$
$653$	3.17147	0.124109	0.0620547	+	0.998073i	$0.480235\pi$
$654$	−61.5496	−2.40678
$655$	3.65574	0.142842
$656$	−2.36209	−0.0922240
$657$	−64.8982	−2.53192
$658$	−17.4791	−0.681406
$659$	26.9171	1.04854	0.524271	−	0.851551i	$-0.324338\pi$
$659$	26.9171	1.04854	0.524271	+	0.851551i	$0.324338\pi$
$660$	−40.8291	−1.58927
$661$	36.4670	1.41840	0.709201	−	0.705006i	$-0.249056\pi$
$661$	36.4670	1.41840	0.709201	+	0.705006i	$0.249056\pi$
$662$	−3.29695	−0.128140
$663$	−164.904	−6.40434
$664$	−6.78106	−0.263156
$665$	−53.8389	−2.08778
$666$	−2.87751	−0.111501
$667$	53.1621	2.05844
$668$	−20.1355	−0.779064
$669$	73.1095	2.82658
$670$	−11.6379	−0.449612
$671$	28.5196	1.10099
$672$	10.6488	0.410786
$673$	−15.0735	−0.581042	−0.290521	−	0.956869i	$-0.593829\pi$
$673$	−15.0735	−0.581042	−0.290521	+	0.956869i	$0.593829\pi$
$674$	3.70402	0.142673
$675$	−142.440	−5.48251
$676$	33.9189	1.30457
$677$	−30.2645	−1.16316	−0.581580	−	0.813489i	$-0.697565\pi$
$677$	−30.2645	−1.16316	−0.581580	+	0.813489i	$0.697565\pi$
$678$	12.2416	0.470134
$679$	−25.3255	−0.971902
$680$	29.9725	1.14939
$681$	−53.0200	−2.03173
$682$	1.29720	0.0496723
$683$	18.7717	0.718279	0.359139	−	0.933284i	$-0.383070\pi$
$683$	18.7717	0.718279	0.359139	+	0.933284i	$0.383070\pi$
$684$	29.0715	1.11158
$685$	−56.8483	−2.17206
$686$	9.48034	0.361961
$687$	39.0997	1.49175
$688$	−6.57932	−0.250834
$689$	−45.4010	−1.72964
$690$	66.9500	2.54874
$691$	4.65219	0.176977	0.0884887	−	0.996077i	$-0.471796\pi$
$691$	4.65219	0.176977	0.0884887	+	0.996077i	$0.471796\pi$
$692$	−6.65189	−0.252867
$693$	−77.2113	−2.93302
$694$	23.5003	0.892060
$695$	−75.9703	−2.88172
$696$	−32.0094	−1.21331
$697$	−17.8416	−0.675797
$698$	30.1176	1.13997
$699$	75.3531	2.85012
$700$	35.9028	1.35700
$701$	−25.0303	−0.945380	−0.472690	−	0.881229i	$-0.656717\pi$
$701$	−25.0303	−0.945380	−0.472690	+	0.881229i	$0.656717\pi$
$702$	90.7940	3.42680
$703$	1.63234	0.0615648
$704$	−3.22823	−0.121668
$705$	66.1675	2.49201
$706$	18.5054	0.696460
$707$	−47.1480	−1.77318
$708$	−37.0339	−1.39182
$709$	−26.2475	−0.985747	−0.492873	−	0.870101i	$-0.664053\pi$
$709$	−26.2475	−0.985747	−0.492873	+	0.870101i	$0.664053\pi$
$710$	33.5033	1.25736
$711$	−18.0543	−0.677091
$712$	−2.79050	−0.104578
$713$	−2.12710	−0.0796604
$714$	80.4335	3.01015
$715$	87.7453	3.28149
$716$	−13.7457	−0.513703
$717$	−39.2112	−1.46437
$718$	−24.5040	−0.914479
$719$	9.26721	0.345609	0.172804	−	0.984956i	$-0.444717\pi$
$719$	9.26721	0.345609	0.172804	+	0.984956i	$0.444717\pi$
$720$	−28.4068	−1.05866
$721$	35.8801	1.33624
$722$	2.50846	0.0933553
$723$	79.9644	2.97391
$724$	−4.16777	−0.154894
$725$	−107.921	−4.00808
$726$	−1.84400	−0.0684371
$727$	−27.8788	−1.03397	−0.516984	−	0.855995i	$-0.672945\pi$
$727$	−27.8788	−1.03397	−0.516984	+	0.855995i	$0.672945\pi$
$728$	−22.8851	−0.848180
$729$	21.9546	0.813132
$730$	−35.9734	−1.33143
$731$	−49.6956	−1.83806
$732$	28.1579	1.04074
$733$	−47.0351	−1.73728	−0.868640	−	0.495443i	$-0.835006\pi$
$733$	−47.0351	−1.73728	−0.868640	+	0.495443i	$0.835006\pi$
$734$	−17.5792	−0.648859
$735$	52.6447	1.94183
$736$	5.29353	0.195122
$737$	9.46791	0.348755
$738$	16.9096	0.622450
$739$	−6.06038	−0.222935	−0.111467	−	0.993768i	$-0.535555\pi$
$739$	−6.06038	−0.222935	−0.111467	+	0.993768i	$0.535555\pi$
$740$	−1.59502	−0.0586340
$741$	−88.6594	−3.25698
$742$	22.1448	0.812960
$743$	−24.6509	−0.904354	−0.452177	−	0.891928i	$-0.649352\pi$
$743$	−24.6509	−0.904354	−0.452177	+	0.891928i	$0.649352\pi$
$744$	1.28074	0.0469543
$745$	66.7934	2.44712
$746$	17.9531	0.657311
$747$	48.5439	1.77613
$748$	−24.3838	−0.891560
$749$	32.3830	1.18325
$750$	−72.6731	−2.65365
$751$	−18.3409	−0.669268	−0.334634	−	0.942348i	$-0.608613\pi$
$751$	−18.3409	−0.669268	−0.334634	+	0.942348i	$0.608613\pi$
$752$	5.23165	0.190779
$753$	30.6211	1.11589
$754$	68.7909	2.50522
$755$	40.4386	1.47171
$756$	−44.2856	−1.61065
$757$	16.0860	0.584656	0.292328	−	0.956318i	$-0.405570\pi$
$757$	16.0860	0.584656	0.292328	+	0.956318i	$0.405570\pi$
$758$	−24.9535	−0.906354
$759$	−54.4665	−1.97701
$760$	16.1145	0.584533
$761$	−30.0974	−1.09103	−0.545516	−	0.838101i	$-0.683666\pi$
$761$	−30.0974	−1.09103	−0.545516	+	0.838101i	$0.683666\pi$
$762$	1.35015	0.0489109
$763$	−64.5186	−2.33573
$764$	−14.5076	−0.524868
$765$	−214.566	−7.75763
$766$	15.6066	0.563889
$767$	79.5891	2.87380
$768$	−3.18728	−0.115011
$769$	23.7052	0.854831	0.427416	−	0.904055i	$-0.359424\pi$
$769$	23.7052	0.854831	0.427416	+	0.904055i	$0.359424\pi$
$770$	−42.7986	−1.54235
$771$	−34.5136	−1.24298
$772$	−17.2968	−0.622526
$773$	6.86586	0.246948	0.123474	−	0.992348i	$-0.460596\pi$
$773$	6.86586	0.246948	0.123474	+	0.992348i	$0.460596\pi$
$774$	47.0997	1.69296
$775$	4.31808	0.155110
$776$	7.58015	0.272111
$777$	−4.28035	−0.153557
$778$	−38.2658	−1.37189
$779$	−9.59238	−0.343683
$780$	86.6323	3.10193
$781$	−27.2563	−0.975305
$782$	39.9836	1.42981
$783$	133.119	4.75728
$784$	4.16245	0.148659
$785$	−37.1532	−1.32606
$786$	−2.93637	−0.104737
$787$	−6.94816	−0.247675	−0.123838	−	0.992302i	$-0.539520\pi$
$787$	−6.94816	−0.247675	−0.123838	+	0.992302i	$0.539520\pi$
$788$	19.9921	0.712190
$789$	34.3980	1.22460
$790$	−10.0076	−0.356055
$791$	12.8321	0.456255
$792$	23.1101	0.821181
$793$	−60.5137	−2.14890
$794$	23.9783	0.850958
$795$	−83.8295	−2.97313
$796$	−15.6963	−0.556341
$797$	10.9365	0.387391	0.193696	−	0.981062i	$-0.437953\pi$
$797$	10.9365	0.387391	0.193696	+	0.981062i	$0.437953\pi$
$798$	43.2445	1.53084
$799$	39.5163	1.39799
$800$	−10.7460	−0.379930
$801$	19.9765	0.705834
$802$	−6.72372	−0.237423
$803$	29.2658	1.03277
$804$	9.34781	0.329672
$805$	70.1795	2.47350
$806$	−2.75243	−0.0969502
$807$	−41.7307	−1.46899
$808$	14.1118	0.496453
$809$	39.8664	1.40163	0.700814	−	0.713344i	$-0.252820\pi$
$809$	39.8664	1.40163	0.700814	+	0.713344i	$0.252820\pi$
$810$	82.4238	2.89608
$811$	24.9691	0.876785	0.438393	−	0.898784i	$-0.355548\pi$
$811$	24.9691	0.876785	0.438393	+	0.898784i	$0.355548\pi$
$812$	−33.5534	−1.17749
$813$	−21.7668	−0.763395
$814$	1.29761	0.0454812
$815$	−17.0159	−0.596042
$816$	−24.0745	−0.842776
$817$	−26.7185	−0.934761
$818$	−22.1936	−0.775980
$819$	163.829	5.72465
$820$	9.37306	0.327321
$821$	−25.6293	−0.894469	−0.447234	−	0.894417i	$-0.647591\pi$
$821$	−25.6293	−0.894469	−0.447234	+	0.894417i	$0.647591\pi$
$822$	45.6617	1.59263
$823$	14.3705	0.500924	0.250462	−	0.968126i	$-0.419417\pi$
$823$	14.3705	0.500924	0.250462	+	0.968126i	$0.419417\pi$
$824$	−10.7392	−0.374119
$825$	110.569	3.84951
$826$	−38.8204	−1.35073
$827$	−20.4760	−0.712022	−0.356011	−	0.934482i	$-0.615863\pi$
$827$	−20.4760	−0.712022	−0.356011	+	0.934482i	$0.615863\pi$
$828$	−37.8950	−1.31694
$829$	−9.07500	−0.315188	−0.157594	−	0.987504i	$-0.550374\pi$
$829$	−9.07500	−0.315188	−0.157594	+	0.987504i	$0.550374\pi$
$830$	26.9081	0.933994
$831$	10.3240	0.358137
$832$	6.84974	0.237472
$833$	31.4402	1.08934
$834$	61.0209	2.11298
$835$	79.9000	2.76505
$836$	−13.1098	−0.453410
$837$	−5.32629	−0.184104
$838$	−37.5936	−1.29865
$839$	28.1072	0.970369	0.485184	−	0.874412i	$-0.338753\pi$
$839$	28.1072	0.970369	0.485184	+	0.874412i	$0.338753\pi$
$840$	−42.2557	−1.45796
$841$	71.8587	2.47789
$842$	−11.8976	−0.410018
$843$	20.6556	0.711418
$844$	26.4263	0.909632
$845$	−134.595	−4.63020
$846$	−37.4521	−1.28763
$847$	−1.93294	−0.0664167
$848$	−6.62813	−0.227611
$849$	1.36389	0.0468086
$850$	−81.1681	−2.78404
$851$	−2.12777	−0.0729390
$852$	−26.9105	−0.921939
$853$	−24.8160	−0.849683	−0.424841	−	0.905268i	$-0.639670\pi$
$853$	−24.8160	−0.849683	−0.424841	+	0.905268i	$0.639670\pi$
$854$	29.5161	1.01002
$855$	−115.360	−3.94521
$856$	−9.69254	−0.331284
$857$	4.60124	0.157175	0.0785876	−	0.996907i	$-0.474959\pi$
$857$	4.60124	0.157175	0.0785876	+	0.996907i	$0.474959\pi$
$858$	−70.4788	−2.40611
$859$	21.9812	0.749990	0.374995	−	0.927027i	$-0.377644\pi$
$859$	21.9812	0.749990	0.374995	+	0.927027i	$0.377644\pi$
$860$	26.1076	0.890261
$861$	25.1533	0.857223
$862$	10.0045	0.340753
$863$	16.7933	0.571651	0.285825	−	0.958282i	$-0.407732\pi$
$863$	16.7933	0.571651	0.285825	+	0.958282i	$0.407732\pi$
$864$	13.2551	0.450948
$865$	26.3955	0.897475
$866$	35.1917	1.19586
$867$	−127.658	−4.33551
$868$	1.34252	0.0455682
$869$	8.14158	0.276184
$870$	127.017	4.30629
$871$	−20.0893	−0.680699
$872$	19.3110	0.653953
$873$	−54.2644	−1.83657
$874$	21.4969	0.727143
$875$	−76.1787	−2.57531
$876$	28.8945	0.976256
$877$	−20.4569	−0.690781	−0.345391	−	0.938459i	$-0.612254\pi$
$877$	−20.4569	−0.690781	−0.345391	+	0.938459i	$0.612254\pi$
$878$	−22.7458	−0.767633
$879$	−5.92365	−0.199800
$880$	12.8100	0.431826
$881$	38.1671	1.28588	0.642940	−	0.765916i	$-0.277714\pi$
$881$	38.1671	1.28588	0.642940	+	0.765916i	$0.277714\pi$
$882$	−29.7979	−1.00335
$883$	−22.8701	−0.769641	−0.384820	−	0.922992i	$-0.625737\pi$
$883$	−22.8701	−0.769641	−0.384820	+	0.922992i	$0.625737\pi$
$884$	51.7382	1.74014
$885$	146.955	4.93985
$886$	−0.569085	−0.0191188
$887$	3.48910	0.117153	0.0585763	−	0.998283i	$-0.481344\pi$
$887$	3.48910	0.117153	0.0585763	+	0.998283i	$0.481344\pi$
$888$	1.28115	0.0429926
$889$	1.41528	0.0474670
$890$	11.0730	0.371169
$891$	−67.0550	−2.24643
$892$	−22.9379	−0.768018
$893$	21.2456	0.710958
$894$	−53.6498	−1.79432
$895$	54.5449	1.82323
$896$	−3.34102	−0.111616
$897$	115.568	3.85872
$898$	−34.7150	−1.15845
$899$	−4.03551	−0.134592
$900$	76.9282	2.56427
$901$	−50.0643	−1.66788
$902$	−7.62535	−0.253896
$903$	70.0617	2.33151
$904$	−3.84076	−0.127742
$905$	16.5382	0.549749
$906$	−32.4811	−1.07911
$907$	23.8884	0.793203	0.396601	−	0.917991i	$-0.370190\pi$
$907$	23.8884	0.793203	0.396601	+	0.917991i	$0.370190\pi$
$908$	16.6349	0.552048
$909$	−101.023	−3.35073
$910$	90.8112	3.01036
$911$	−35.0321	−1.16067	−0.580333	−	0.814379i	$-0.697078\pi$
$911$	−35.0321	−1.16067	−0.580333	+	0.814379i	$0.697078\pi$
$912$	−12.9435	−0.428601
$913$	−21.8908	−0.724480
$914$	−24.7903	−0.819990
$915$	−111.734	−3.69381
$916$	−12.2674	−0.405327
$917$	−3.07801	−0.101645
$918$	100.120	3.30445
$919$	−10.6957	−0.352819	−0.176409	−	0.984317i	$-0.556448\pi$
$919$	−10.6957	−0.352819	−0.176409	+	0.984317i	$0.556448\pi$
$920$	−21.0054	−0.692527
$921$	18.4112	0.606668
$922$	−1.24441	−0.0409826
$923$	57.8330	1.90360
$924$	34.3767	1.13091
$925$	4.31944	0.142022
$926$	30.5376	1.00353
$927$	76.8796	2.52506
$928$	10.0428	0.329673
$929$	14.5272	0.476621	0.238310	−	0.971189i	$-0.423406\pi$
$929$	14.5272	0.476621	0.238310	+	0.971189i	$0.423406\pi$
$930$	−5.08215	−0.166650
$931$	16.9036	0.553993
$932$	−23.6418	−0.774414
$933$	−25.6911	−0.841089
$934$	4.62672	0.151391
$935$	96.7580	3.16433
$936$	−49.0356	−1.60278
$937$	9.20048	0.300567	0.150283	−	0.988643i	$-0.451981\pi$
$937$	9.20048	0.300567	0.150283	+	0.988643i	$0.451981\pi$
$938$	9.79872	0.319940
$939$	−82.7677	−2.70102
$940$	−20.7599	−0.677112
$941$	−51.7231	−1.68612	−0.843062	−	0.537816i	$-0.819249\pi$
$941$	−51.7231	−1.68612	−0.843062	+	0.537816i	$0.819249\pi$
$942$	29.8422	0.972312
$943$	12.5038	0.407179
$944$	11.6193	0.378176
$945$	175.731	5.71653
$946$	−21.2395	−0.690557
$947$	34.1912	1.11107	0.555533	−	0.831495i	$-0.312514\pi$
$947$	34.1912	1.11107	0.555533	+	0.831495i	$0.312514\pi$
$948$	8.03831	0.261072
$949$	−62.0969	−2.01575
$950$	−43.6394	−1.41585
$951$	0.515717	0.0167233
$952$	−25.2358	−0.817897
$953$	−33.5508	−1.08682	−0.543409	−	0.839468i	$-0.682867\pi$
$953$	−33.5508	−1.08682	−0.543409	+	0.839468i	$0.682867\pi$
$954$	47.4492	1.53622
$955$	57.5682	1.86286
$956$	12.3024	0.397888
$957$	−103.333	−3.34030
$958$	−6.05603	−0.195661
$959$	47.8643	1.54562
$960$	12.6475	0.408197
$961$	−30.8385	−0.994791
$962$	−2.75330	−0.0887700
$963$	69.3865	2.23595
$964$	−25.0886	−0.808049
$965$	68.6359	2.20947
$966$	−56.3696	−1.81366
$967$	36.6755	1.17940	0.589702	−	0.807621i	$-0.299245\pi$
$967$	36.6755	1.17940	0.589702	+	0.807621i	$0.299245\pi$
$968$	0.578548	0.0185952
$969$	−97.7660	−3.14070
$970$	−30.0790	−0.965778
$971$	−30.0201	−0.963390	−0.481695	−	0.876339i	$-0.659979\pi$
$971$	−30.0201	−0.963390	−0.481695	+	0.876339i	$0.659979\pi$
$972$	−26.4391	−0.848036
$973$	63.9643	2.05060
$974$	11.6914	0.374617
$975$	−234.608	−7.51346
$976$	−8.83445	−0.282784
$977$	6.79868	0.217509	0.108754	−	0.994069i	$-0.465314\pi$
$977$	6.79868	0.217509	0.108754	+	0.994069i	$0.465314\pi$
$978$	13.6675	0.437039
$979$	−9.00835	−0.287908
$980$	−16.5171	−0.527620
$981$	−138.243	−4.41375
$982$	−21.9017	−0.698913
$983$	−8.25014	−0.263139	−0.131569	−	0.991307i	$-0.542002\pi$
$983$	−8.25014	−0.263139	−0.131569	+	0.991307i	$0.542002\pi$
$984$	−7.52863	−0.240004
$985$	−79.3313	−2.52771
$986$	75.8567	2.41577
$987$	−55.7107	−1.77329
$988$	27.8166	0.884965
$989$	34.8278	1.10746
$990$	−91.7037	−2.91454
$991$	−35.3386	−1.12257	−0.561284	−	0.827623i	$-0.689692\pi$
$991$	−35.3386	−1.12257	−0.561284	+	0.827623i	$0.689692\pi$
$992$	−0.401830	−0.0127581
$993$	−10.5083	−0.333471
$994$	−28.2086	−0.894723
$995$	62.2850	1.97457
$996$	−21.6131	−0.684838
$997$	−26.4075	−0.836334	−0.418167	−	0.908370i	$-0.637327\pi$
$997$	−26.4075	−0.836334	−0.418167	+	0.908370i	$0.637327\pi$
$998$	17.2657	0.546535
$999$	−5.32798	−0.168570

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8038.2.a.c.1.6	✓	84

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8038.2.a.c.1.6	✓	84	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:	\( N \)	\(=\)	\( 8038 = 2 \cdot 4019 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8038.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -3.18728 q^{3} +1.00000 q^{4} -3.96813 q^{5} +3.18728 q^{6} +3.34102 q^{7} -1.00000 q^{8} +7.15875 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -3.18728 q^{3} +1.00000 q^{4} -3.96813 q^{5} +3.18728 q^{6} +3.34102 q^{7} -1.00000 q^{8} +7.15875 q^{9} +3.96813 q^{10} -3.22823 q^{11} -3.18728 q^{12} +6.84974 q^{13} -3.34102 q^{14} +12.6475 q^{15} +1.00000 q^{16} +7.55331 q^{17} -7.15875 q^{18} +4.06098 q^{19} -3.96813 q^{20} -10.6488 q^{21} +3.22823 q^{22} -5.29353 q^{23} +3.18728 q^{24} +10.7460 q^{25} -6.84974 q^{26} -13.2551 q^{27} +3.34102 q^{28} -10.0428 q^{29} -12.6475 q^{30} +0.401830 q^{31} -1.00000 q^{32} +10.2893 q^{33} -7.55331 q^{34} -13.2576 q^{35} +7.15875 q^{36} +0.401957 q^{37} -4.06098 q^{38} -21.8320 q^{39} +3.96813 q^{40} -2.36209 q^{41} +10.6488 q^{42} -6.57932 q^{43} -3.22823 q^{44} -28.4068 q^{45} +5.29353 q^{46} +5.23165 q^{47} -3.18728 q^{48} +4.16245 q^{49} -10.7460 q^{50} -24.0745 q^{51} +6.84974 q^{52} -6.62813 q^{53} +13.2551 q^{54} +12.8100 q^{55} -3.34102 q^{56} -12.9435 q^{57} +10.0428 q^{58} +11.6193 q^{59} +12.6475 q^{60} -8.83445 q^{61} -0.401830 q^{62} +23.9176 q^{63} +1.00000 q^{64} -27.1806 q^{65} -10.2893 q^{66} -2.93285 q^{67} +7.55331 q^{68} +16.8719 q^{69} +13.2576 q^{70} +8.44310 q^{71} -7.15875 q^{72} -9.06558 q^{73} -0.401957 q^{74} -34.2506 q^{75} +4.06098 q^{76} -10.7856 q^{77} +21.8320 q^{78} -2.52200 q^{79} -3.96813 q^{80} +20.7715 q^{81} +2.36209 q^{82} +6.78106 q^{83} -10.6488 q^{84} -29.9725 q^{85} +6.57932 q^{86} +32.0094 q^{87} +3.22823 q^{88} +2.79050 q^{89} +28.4068 q^{90} +22.8851 q^{91} -5.29353 q^{92} -1.28074 q^{93} -5.23165 q^{94} -16.1145 q^{95} +3.18728 q^{96} -7.58015 q^{97} -4.16245 q^{98} -23.1101 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q - 84 q^{2} - 19 q^{3} + 84 q^{4} - 32 q^{5} + 19 q^{6} + q^{7} - 84 q^{8} + 77 q^{9}+O(q^{10}) \) 84 * q - 84 * q^2 - 19 * q^3 + 84 * q^4 - 32 * q^5 + 19 * q^6 + q^7 - 84 * q^8 + 77 * q^9	\( 84 q - 84 q^{2} - 19 q^{3} + 84 q^{4} - 32 q^{5} + 19 q^{6} + q^{7} - 84 q^{8} + 77 q^{9} + 32 q^{10} - 6 q^{11} - 19 q^{12} - 29 q^{13} - q^{14} - 4 q^{15} + 84 q^{16} - 36 q^{17} - 77 q^{18} + 33 q^{19} - 32 q^{20} - 21 q^{21} + 6 q^{22} - 62 q^{23} + 19 q^{24} + 82 q^{25} + 29 q^{26} - 82 q^{27} + q^{28} - 51 q^{29} + 4 q^{30} + 39 q^{31} - 84 q^{32} - 32 q^{33} + 36 q^{34} - 34 q^{35} + 77 q^{36} - 32 q^{37} - 33 q^{38} + 29 q^{39} + 32 q^{40} - 38 q^{41} + 21 q^{42} - 6 q^{44} - 91 q^{45} + 62 q^{46} - 58 q^{47} - 19 q^{48} + 83 q^{49} - 82 q^{50} - q^{51} - 29 q^{52} - 106 q^{53} + 82 q^{54} + 32 q^{55} - q^{56} - 44 q^{57} + 51 q^{58} - 42 q^{59} - 4 q^{60} - 41 q^{61} - 39 q^{62} - 9 q^{63} + 84 q^{64} - 49 q^{65} + 32 q^{66} - 16 q^{67} - 36 q^{68} - 45 q^{69} + 34 q^{70} - 62 q^{71} - 77 q^{72} - 16 q^{73} + 32 q^{74} - 80 q^{75} + 33 q^{76} - 134 q^{77} - 29 q^{78} + 53 q^{79} - 32 q^{80} + 56 q^{81} + 38 q^{82} - 90 q^{83} - 21 q^{84} - 60 q^{85} - 3 q^{87} + 6 q^{88} - 54 q^{89} + 91 q^{90} + 33 q^{91} - 62 q^{92} - 69 q^{93} + 58 q^{94} - 47 q^{95} + 19 q^{96} - 31 q^{97} - 83 q^{98} + 23 q^{99}+O(q^{100}) \) 84 * q - 84 * q^2 - 19 * q^3 + 84 * q^4 - 32 * q^5 + 19 * q^6 + q^7 - 84 * q^8 + 77 * q^9 + 32 * q^10 - 6 * q^11 - 19 * q^12 - 29 * q^13 - q^14 - 4 * q^15 + 84 * q^16 - 36 * q^17 - 77 * q^18 + 33 * q^19 - 32 * q^20 - 21 * q^21 + 6 * q^22 - 62 * q^23 + 19 * q^24 + 82 * q^25 + 29 * q^26 - 82 * q^27 + q^28 - 51 * q^29 + 4 * q^30 + 39 * q^31 - 84 * q^32 - 32 * q^33 + 36 * q^34 - 34 * q^35 + 77 * q^36 - 32 * q^37 - 33 * q^38 + 29 * q^39 + 32 * q^40 - 38 * q^41 + 21 * q^42 - 6 * q^44 - 91 * q^45 + 62 * q^46 - 58 * q^47 - 19 * q^48 + 83 * q^49 - 82 * q^50 - q^51 - 29 * q^52 - 106 * q^53 + 82 * q^54 + 32 * q^55 - q^56 - 44 * q^57 + 51 * q^58 - 42 * q^59 - 4 * q^60 - 41 * q^61 - 39 * q^62 - 9 * q^63 + 84 * q^64 - 49 * q^65 + 32 * q^66 - 16 * q^67 - 36 * q^68 - 45 * q^69 + 34 * q^70 - 62 * q^71 - 77 * q^72 - 16 * q^73 + 32 * q^74 - 80 * q^75 + 33 * q^76 - 134 * q^77 - 29 * q^78 + 53 * q^79 - 32 * q^80 + 56 * q^81 + 38 * q^82 - 90 * q^83 - 21 * q^84 - 60 * q^85 - 3 * q^87 + 6 * q^88 - 54 * q^89 + 91 * q^90 + 33 * q^91 - 62 * q^92 - 69 * q^93 + 58 * q^94 - 47 * q^95 + 19 * q^96 - 31 * q^97 - 83 * q^98 + 23 * q^99

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