LMFDB - Embedded newform 8042.2.a.b.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8042, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8042.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$64.2156933055$
Analytic rank:	$1$
Dimension:	$82$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
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)

$f(q)$	$=$	$q-1.00000 q^{2} -2.21914 q^{3} +1.00000 q^{4} -2.28006 q^{5} +2.21914 q^{6} +5.09322 q^{7} -1.00000 q^{8} +1.92459 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.21914 q^{3} +1.00000 q^{4} -2.28006 q^{5} +2.21914 q^{6} +5.09322 q^{7} -1.00000 q^{8} +1.92459 q^{9} +2.28006 q^{10} +1.06576 q^{11} -2.21914 q^{12} -1.39165 q^{13} -5.09322 q^{14} +5.05978 q^{15} +1.00000 q^{16} +5.26765 q^{17} -1.92459 q^{18} -5.06744 q^{19} -2.28006 q^{20} -11.3026 q^{21} -1.06576 q^{22} +8.86350 q^{23} +2.21914 q^{24} +0.198674 q^{25} +1.39165 q^{26} +2.38648 q^{27} +5.09322 q^{28} -7.26091 q^{29} -5.05978 q^{30} +3.04346 q^{31} -1.00000 q^{32} -2.36508 q^{33} -5.26765 q^{34} -11.6129 q^{35} +1.92459 q^{36} -1.43872 q^{37} +5.06744 q^{38} +3.08828 q^{39} +2.28006 q^{40} -6.80445 q^{41} +11.3026 q^{42} -5.06719 q^{43} +1.06576 q^{44} -4.38819 q^{45} -8.86350 q^{46} +2.46518 q^{47} -2.21914 q^{48} +18.9409 q^{49} -0.198674 q^{50} -11.6897 q^{51} -1.39165 q^{52} -1.13148 q^{53} -2.38648 q^{54} -2.43000 q^{55} -5.09322 q^{56} +11.2454 q^{57} +7.26091 q^{58} -2.25547 q^{59} +5.05978 q^{60} -6.75058 q^{61} -3.04346 q^{62} +9.80239 q^{63} +1.00000 q^{64} +3.17305 q^{65} +2.36508 q^{66} -1.79330 q^{67} +5.26765 q^{68} -19.6694 q^{69} +11.6129 q^{70} -1.35796 q^{71} -1.92459 q^{72} -9.77724 q^{73} +1.43872 q^{74} -0.440887 q^{75} -5.06744 q^{76} +5.42817 q^{77} -3.08828 q^{78} -9.71494 q^{79} -2.28006 q^{80} -11.0697 q^{81} +6.80445 q^{82} +10.0963 q^{83} -11.3026 q^{84} -12.0106 q^{85} +5.06719 q^{86} +16.1130 q^{87} -1.06576 q^{88} -18.2234 q^{89} +4.38819 q^{90} -7.08800 q^{91} +8.86350 q^{92} -6.75388 q^{93} -2.46518 q^{94} +11.5541 q^{95} +2.21914 q^{96} -4.61706 q^{97} -18.9409 q^{98} +2.05116 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 82 q - 82 q^{2} - 13 q^{3} + 82 q^{4} + 3 q^{5} + 13 q^{6} - 37 q^{7} - 82 q^{8} + 91 q^{9}+O(q^{10}) $ 82 * q - 82 * q^2 - 13 * q^3 + 82 * q^4 + 3 * q^5 + 13 * q^6 - 37 * q^7 - 82 * q^8 + 91 * q^9	\( 82 q - 82 q^{2} - 13 q^{3} + 82 q^{4} + 3 q^{5} + 13 q^{6} - 37 q^{7} - 82 q^{8} + 91 q^{9} - 3 q^{10} - 16 q^{11} - 13 q^{12} - 42 q^{13} + 37 q^{14} - 9 q^{15} + 82 q^{16} + 3 q^{17} - 91 q^{18} - 42 q^{19} + 3 q^{20} - q^{21} + 16 q^{22} - 6 q^{23} + 13 q^{24} + 53 q^{25} + 42 q^{26} - 49 q^{27} - 37 q^{28} + 15 q^{29} + 9 q^{30} - 40 q^{31} - 82 q^{32} - 37 q^{33} - 3 q^{34} - 42 q^{35} + 91 q^{36} - 72 q^{37} + 42 q^{38} - 14 q^{39} - 3 q^{40} + 8 q^{41} + q^{42} - 93 q^{43} - 16 q^{44} - 11 q^{45} + 6 q^{46} + 7 q^{47} - 13 q^{48} + 61 q^{49} - 53 q^{50} - 70 q^{51} - 42 q^{52} + 18 q^{53} + 49 q^{54} - 62 q^{55} + 37 q^{56} - 51 q^{57} - 15 q^{58} - 47 q^{59} - 9 q^{60} - 14 q^{61} + 40 q^{62} - 100 q^{63} + 82 q^{64} + q^{65} + 37 q^{66} - 150 q^{67} + 3 q^{68} + 31 q^{69} + 42 q^{70} + 7 q^{71} - 91 q^{72} - 78 q^{73} + 72 q^{74} - 49 q^{75} - 42 q^{76} + 29 q^{77} + 14 q^{78} - 59 q^{79} + 3 q^{80} + 122 q^{81} - 8 q^{82} - 52 q^{83} - q^{84} - 108 q^{85} + 93 q^{86} - 49 q^{87} + 16 q^{88} + 38 q^{89} + 11 q^{90} - 69 q^{91} - 6 q^{92} - 63 q^{93} - 7 q^{94} + 5 q^{95} + 13 q^{96} - 74 q^{97} - 61 q^{98} - 89 q^{99}+O(q^{100}) \) 82 * q - 82 * q^2 - 13 * q^3 + 82 * q^4 + 3 * q^5 + 13 * q^6 - 37 * q^7 - 82 * q^8 + 91 * q^9 - 3 * q^10 - 16 * q^11 - 13 * q^12 - 42 * q^13 + 37 * q^14 - 9 * q^15 + 82 * q^16 + 3 * q^17 - 91 * q^18 - 42 * q^19 + 3 * q^20 - q^21 + 16 * q^22 - 6 * q^23 + 13 * q^24 + 53 * q^25 + 42 * q^26 - 49 * q^27 - 37 * q^28 + 15 * q^29 + 9 * q^30 - 40 * q^31 - 82 * q^32 - 37 * q^33 - 3 * q^34 - 42 * q^35 + 91 * q^36 - 72 * q^37 + 42 * q^38 - 14 * q^39 - 3 * q^40 + 8 * q^41 + q^42 - 93 * q^43 - 16 * q^44 - 11 * q^45 + 6 * q^46 + 7 * q^47 - 13 * q^48 + 61 * q^49 - 53 * q^50 - 70 * q^51 - 42 * q^52 + 18 * q^53 + 49 * q^54 - 62 * q^55 + 37 * q^56 - 51 * q^57 - 15 * q^58 - 47 * q^59 - 9 * q^60 - 14 * q^61 + 40 * q^62 - 100 * q^63 + 82 * q^64 + q^65 + 37 * q^66 - 150 * q^67 + 3 * q^68 + 31 * q^69 + 42 * q^70 + 7 * q^71 - 91 * q^72 - 78 * q^73 + 72 * q^74 - 49 * q^75 - 42 * q^76 + 29 * q^77 + 14 * q^78 - 59 * q^79 + 3 * q^80 + 122 * q^81 - 8 * q^82 - 52 * q^83 - q^84 - 108 * q^85 + 93 * q^86 - 49 * q^87 + 16 * q^88 + 38 * q^89 + 11 * q^90 - 69 * q^91 - 6 * q^92 - 63 * q^93 - 7 * q^94 + 5 * q^95 + 13 * q^96 - 74 * q^97 - 61 * q^98 - 89 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−2.21914	−1.28122	−0.640611	−	0.767865i	$-0.721319\pi$
$3$	−2.21914	−1.28122	−0.640611	+	0.767865i	$0.721319\pi$
$4$	1.00000	0.500000
$5$	−2.28006	−1.01967	−0.509837	−	0.860271i	$-0.670294\pi$
$5$	−2.28006	−1.01967	−0.509837	+	0.860271i	$0.670294\pi$
$6$	2.21914	0.905961
$7$	5.09322	1.92506	0.962529	−	0.271179i	$-0.0874137\pi$
$7$	5.09322	1.92506	0.962529	+	0.271179i	$0.0874137\pi$
$8$	−1.00000	−0.353553
$9$	1.92459	0.641531
$10$	2.28006	0.721018
$11$	1.06576	0.321340	0.160670	−	0.987008i	$-0.448635\pi$
$11$	1.06576	0.321340	0.160670	+	0.987008i	$0.448635\pi$
$12$	−2.21914	−0.640611
$13$	−1.39165	−0.385975	−0.192988	−	0.981201i	$-0.561818\pi$
$13$	−1.39165	−0.385975	−0.192988	+	0.981201i	$0.561818\pi$
$14$	−5.09322	−1.36122
$15$	5.05978	1.30643
$16$	1.00000	0.250000
$17$	5.26765	1.27759	0.638797	−	0.769375i	$-0.279432\pi$
$17$	5.26765	1.27759	0.638797	+	0.769375i	$0.279432\pi$
$18$	−1.92459	−0.453631
$19$	−5.06744	−1.16255	−0.581275	−	0.813707i	$-0.697446\pi$
$19$	−5.06744	−1.16255	−0.581275	+	0.813707i	$0.697446\pi$
$20$	−2.28006	−0.509837
$21$	−11.3026	−2.46643
$22$	−1.06576	−0.227221
$23$	8.86350	1.84817	0.924084	−	0.382190i	$-0.124830\pi$
$23$	8.86350	1.84817	0.924084	+	0.382190i	$0.124830\pi$
$24$	2.21914	0.452981
$25$	0.198674	0.0397349
$26$	1.39165	0.272926
$27$	2.38648	0.459278
$28$	5.09322	0.962529
$29$	−7.26091	−1.34832	−0.674159	−	0.738587i	$-0.735494\pi$
$29$	−7.26091	−1.34832	−0.674159	+	0.738587i	$0.735494\pi$
$30$	−5.05978	−0.923785
$31$	3.04346	0.546622	0.273311	−	0.961926i	$-0.411881\pi$
$31$	3.04346	0.546622	0.273311	+	0.961926i	$0.411881\pi$
$32$	−1.00000	−0.176777
$33$	−2.36508	−0.411708
$34$	−5.26765	−0.903395
$35$	−11.6129	−1.96293
$36$	1.92459	0.320766
$37$	−1.43872	−0.236524	−0.118262	−	0.992982i	$-0.537732\pi$
$37$	−1.43872	−0.236524	−0.118262	+	0.992982i	$0.537732\pi$
$38$	5.06744	0.822048
$39$	3.08828	0.494520
$40$	2.28006	0.360509
$41$	−6.80445	−1.06268	−0.531338	−	0.847160i	$-0.678311\pi$
$41$	−6.80445	−1.06268	−0.531338	+	0.847160i	$0.678311\pi$
$42$	11.3026	1.74403
$43$	−5.06719	−0.772739	−0.386370	−	0.922344i	$-0.626271\pi$
$43$	−5.06719	−0.772739	−0.386370	+	0.922344i	$0.626271\pi$
$44$	1.06576	0.160670
$45$	−4.38819	−0.654153
$46$	−8.86350	−1.30685
$47$	2.46518	0.359583	0.179792	−	0.983705i	$-0.442458\pi$
$47$	2.46518	0.359583	0.179792	+	0.983705i	$0.442458\pi$
$48$	−2.21914	−0.320306
$49$	18.9409	2.70585
$50$	−0.198674	−0.0280968
$51$	−11.6897	−1.63688
$52$	−1.39165	−0.192988
$53$	−1.13148	−0.155421	−0.0777104	−	0.996976i	$-0.524761\pi$
$53$	−1.13148	−0.155421	−0.0777104	+	0.996976i	$0.524761\pi$
$54$	−2.38648	−0.324759
$55$	−2.43000	−0.327662
$56$	−5.09322	−0.680611
$57$	11.2454	1.48949
$58$	7.26091	0.953404
$59$	−2.25547	−0.293637	−0.146818	−	0.989163i	$-0.546903\pi$
$59$	−2.25547	−0.293637	−0.146818	+	0.989163i	$0.546903\pi$
$60$	5.05978	0.653215
$61$	−6.75058	−0.864323	−0.432162	−	0.901796i	$-0.642249\pi$
$61$	−6.75058	−0.864323	−0.432162	+	0.901796i	$0.642249\pi$
$62$	−3.04346	−0.386520
$63$	9.80239	1.23498
$64$	1.00000	0.125000
$65$	3.17305	0.393569
$66$	2.36508	0.291121
$67$	−1.79330	−0.219086	−0.109543	−	0.993982i	$-0.534939\pi$
$67$	−1.79330	−0.219086	−0.109543	+	0.993982i	$0.534939\pi$
$68$	5.26765	0.638797
$69$	−19.6694	−2.36791
$70$	11.6129	1.38800
$71$	−1.35796	−0.161160	−0.0805798	−	0.996748i	$-0.525677\pi$
$71$	−1.35796	−0.161160	−0.0805798	+	0.996748i	$0.525677\pi$
$72$	−1.92459	−0.226816
$73$	−9.77724	−1.14434	−0.572170	−	0.820135i	$-0.693898\pi$
$73$	−9.77724	−1.14434	−0.572170	+	0.820135i	$0.693898\pi$
$74$	1.43872	0.167248
$75$	−0.440887	−0.0509092
$76$	−5.06744	−0.581275
$77$	5.42817	0.618597
$78$	−3.08828	−0.349679
$79$	−9.71494	−1.09302	−0.546508	−	0.837454i	$-0.684043\pi$
$79$	−9.71494	−1.09302	−0.546508	+	0.837454i	$0.684043\pi$
$80$	−2.28006	−0.254918
$81$	−11.0697	−1.22997
$82$	6.80445	0.751426
$83$	10.0963	1.10821	0.554105	−	0.832447i	$-0.313061\pi$
$83$	10.0963	1.10821	0.554105	+	0.832447i	$0.313061\pi$
$84$	−11.3026	−1.23321
$85$	−12.0106	−1.30273
$86$	5.06719	0.546409
$87$	16.1130	1.72749
$88$	−1.06576	−0.113611
$89$	−18.2234	−1.93168	−0.965840	−	0.259141i	$-0.916561\pi$
$89$	−18.2234	−1.93168	−0.965840	+	0.259141i	$0.916561\pi$
$90$	4.38819	0.462556
$91$	−7.08800	−0.743025
$92$	8.86350	0.924084
$93$	−6.75388	−0.700344
$94$	−2.46518	−0.254264
$95$	11.5541	1.18542
$96$	2.21914	0.226490
$97$	−4.61706	−0.468792	−0.234396	−	0.972141i	$-0.575311\pi$
$97$	−4.61706	−0.468792	−0.234396	+	0.972141i	$0.575311\pi$
$98$	−18.9409	−1.91332
$99$	2.05116	0.206149
$100$	0.198674	0.0198674
$101$	2.80720	0.279327	0.139663	−	0.990199i	$-0.455398\pi$
$101$	2.80720	0.279327	0.139663	+	0.990199i	$0.455398\pi$
$102$	11.6897	1.15745
$103$	−12.5806	−1.23960	−0.619799	−	0.784760i	$-0.712786\pi$
$103$	−12.5806	−1.23960	−0.619799	+	0.784760i	$0.712786\pi$
$104$	1.39165	0.136463
$105$	25.7706	2.51495
$106$	1.13148	0.109899
$107$	14.3866	1.39081	0.695403	−	0.718620i	$-0.255226\pi$
$107$	14.3866	1.39081	0.695403	+	0.718620i	$0.255226\pi$
$108$	2.38648	0.229639
$109$	−11.5706	−1.10826	−0.554130	−	0.832430i	$-0.686949\pi$
$109$	−11.5706	−1.10826	−0.554130	+	0.832430i	$0.686949\pi$
$110$	2.43000	0.231692
$111$	3.19273	0.303040
$112$	5.09322	0.481264
$113$	14.0965	1.32609	0.663044	−	0.748580i	$-0.269264\pi$
$113$	14.0965	1.32609	0.663044	+	0.748580i	$0.269264\pi$
$114$	−11.2454	−1.05323
$115$	−20.2093	−1.88453
$116$	−7.26091	−0.674159
$117$	−2.67837	−0.247615
$118$	2.25547	0.207633
$119$	26.8293	2.45944
$120$	−5.05978	−0.461892
$121$	−9.86415	−0.896741
$122$	6.75058	0.611169
$123$	15.1000	1.36153
$124$	3.04346	0.273311
$125$	10.9473	0.979157
$126$	−9.80239	−0.873266
$127$	11.4590	1.01682	0.508412	−	0.861114i	$-0.330233\pi$
$127$	11.4590	1.01682	0.508412	+	0.861114i	$0.330233\pi$
$128$	−1.00000	−0.0883883
$129$	11.2448	0.990051
$130$	−3.17305	−0.278295
$131$	10.7798	0.941840	0.470920	−	0.882176i	$-0.343922\pi$
$131$	10.7798	0.941840	0.470920	+	0.882176i	$0.343922\pi$
$132$	−2.36508	−0.205854
$133$	−25.8096	−2.23798
$134$	1.79330	0.154917
$135$	−5.44132	−0.468314
$136$	−5.26765	−0.451698
$137$	3.63920	0.310918	0.155459	−	0.987842i	$-0.450314\pi$
$137$	3.63920	0.310918	0.155459	+	0.987842i	$0.450314\pi$
$138$	19.6694	1.67437
$139$	7.06774	0.599478	0.299739	−	0.954021i	$-0.403100\pi$
$139$	7.06774	0.599478	0.299739	+	0.954021i	$0.403100\pi$
$140$	−11.6129	−0.981466
$141$	−5.47058	−0.460706
$142$	1.35796	0.113957
$143$	−1.48317	−0.124029
$144$	1.92459	0.160383
$145$	16.5553	1.37484
$146$	9.77724	0.809170
$147$	−42.0326	−3.46679
$148$	−1.43872	−0.118262
$149$	−21.0236	−1.72232	−0.861161	−	0.508332i	$-0.830262\pi$
$149$	−21.0236	−1.72232	−0.861161	+	0.508332i	$0.830262\pi$
$150$	0.440887	0.0359983
$151$	14.4462	1.17561	0.587807	−	0.809001i	$-0.299991\pi$
$151$	14.4462	1.17561	0.587807	+	0.809001i	$0.299991\pi$
$152$	5.06744	0.411024
$153$	10.1381	0.819616
$154$	−5.42817	−0.437414
$155$	−6.93928	−0.557376
$156$	3.08828	0.247260
$157$	2.32466	0.185528	0.0927639	−	0.995688i	$-0.470430\pi$
$157$	2.32466	0.185528	0.0927639	+	0.995688i	$0.470430\pi$
$158$	9.71494	0.772879
$159$	2.51092	0.199129
$160$	2.28006	0.180255
$161$	45.1438	3.55783
$162$	11.0697	0.869719
$163$	−2.49930	−0.195761	−0.0978803	−	0.995198i	$-0.531206\pi$
$163$	−2.49930	−0.195761	−0.0978803	+	0.995198i	$0.531206\pi$
$164$	−6.80445	−0.531338
$165$	5.39253	0.419808
$166$	−10.0963	−0.783623
$167$	8.83542	0.683706	0.341853	−	0.939754i	$-0.388946\pi$
$167$	8.83542	0.683706	0.341853	+	0.939754i	$0.388946\pi$
$168$	11.3026	0.872014
$169$	−11.0633	−0.851023
$170$	12.0106	0.921168
$171$	−9.75277	−0.745813
$172$	−5.06719	−0.386370
$173$	15.9158	1.21006	0.605028	−	0.796204i	$-0.293162\pi$
$173$	15.9158	1.21006	0.605028	+	0.796204i	$0.293162\pi$
$174$	−16.1130	−1.22152
$175$	1.01189	0.0764919
$176$	1.06576	0.0803349
$177$	5.00521	0.376214
$178$	18.2234	1.36590
$179$	−12.4761	−0.932505	−0.466253	−	0.884652i	$-0.654396\pi$
$179$	−12.4761	−0.932505	−0.466253	+	0.884652i	$0.654396\pi$
$180$	−4.38819	−0.327076
$181$	16.6603	1.23835	0.619175	−	0.785253i	$-0.287467\pi$
$181$	16.6603	1.23835	0.619175	+	0.785253i	$0.287467\pi$
$182$	7.08800	0.525398
$183$	14.9805	1.10739
$184$	−8.86350	−0.653426
$185$	3.28037	0.241178
$186$	6.75388	0.495218
$187$	5.61407	0.410542
$188$	2.46518	0.179792
$189$	12.1549	0.884137
$190$	−11.5541	−0.838221
$191$	−8.24635	−0.596685	−0.298342	−	0.954459i	$-0.596434\pi$
$191$	−8.24635	−0.596685	−0.298342	+	0.954459i	$0.596434\pi$
$192$	−2.21914	−0.160153
$193$	−21.6276	−1.55679	−0.778395	−	0.627775i	$-0.783966\pi$
$193$	−21.6276	−1.55679	−0.778395	+	0.627775i	$0.783966\pi$
$194$	4.61706	0.331486
$195$	−7.04146	−0.504249
$196$	18.9409	1.35292
$197$	−18.2167	−1.29789	−0.648944	−	0.760836i	$-0.724789\pi$
$197$	−18.2167	−1.29789	−0.648944	+	0.760836i	$0.724789\pi$
$198$	−2.05116	−0.145770
$199$	−15.6221	−1.10742	−0.553710	−	0.832710i	$-0.686788\pi$
$199$	−15.6221	−1.10742	−0.553710	+	0.832710i	$0.686788\pi$
$200$	−0.198674	−0.0140484
$201$	3.97959	0.280698
$202$	−2.80720	−0.197514
$203$	−36.9814	−2.59559
$204$	−11.6897	−0.818441
$205$	15.5146	1.08358
$206$	12.5806	0.876529
$207$	17.0586	1.18566
$208$	−1.39165	−0.0964938
$209$	−5.40069	−0.373574
$210$	−25.7706	−1.77834
$211$	15.5751	1.07224	0.536118	−	0.844143i	$-0.319890\pi$
$211$	15.5751	1.07224	0.536118	+	0.844143i	$0.319890\pi$
$212$	−1.13148	−0.0777104
$213$	3.01350	0.206481
$214$	−14.3866	−0.983448
$215$	11.5535	0.787942
$216$	−2.38648	−0.162379
$217$	15.5010	1.05228
$218$	11.5706	0.783658
$219$	21.6971	1.46615
$220$	−2.43000	−0.163831
$221$	−7.33075	−0.493119
$222$	−3.19273	−0.214282
$223$	−5.56766	−0.372838	−0.186419	−	0.982470i	$-0.559688\pi$
$223$	−5.56766	−0.372838	−0.186419	+	0.982470i	$0.559688\pi$
$224$	−5.09322	−0.340305
$225$	0.382367	0.0254912
$226$	−14.0965	−0.937686
$227$	−16.7805	−1.11376	−0.556881	−	0.830592i	$-0.688002\pi$
$227$	−16.7805	−1.11376	−0.556881	+	0.830592i	$0.688002\pi$
$228$	11.2454	0.744743
$229$	−1.05285	−0.0695745	−0.0347872	−	0.999395i	$-0.511075\pi$
$229$	−1.05285	−0.0695745	−0.0347872	+	0.999395i	$0.511075\pi$
$230$	20.2093	1.33256
$231$	−12.0459	−0.792561
$232$	7.26091	0.476702
$233$	−13.4736	−0.882684	−0.441342	−	0.897339i	$-0.645498\pi$
$233$	−13.4736	−0.882684	−0.441342	+	0.897339i	$0.645498\pi$
$234$	2.67837	0.175090
$235$	−5.62076	−0.366658
$236$	−2.25547	−0.146818
$237$	21.5588	1.40040
$238$	−26.8293	−1.73909
$239$	20.3105	1.31378	0.656890	−	0.753987i	$-0.271872\pi$
$239$	20.3105	1.31378	0.656890	+	0.753987i	$0.271872\pi$
$240$	5.05978	0.326607
$241$	−9.41464	−0.606450	−0.303225	−	0.952919i	$-0.598063\pi$
$241$	−9.41464	−0.606450	−0.303225	+	0.952919i	$0.598063\pi$
$242$	9.86415	0.634092
$243$	17.4058	1.11659
$244$	−6.75058	−0.432162
$245$	−43.1865	−2.75908
$246$	−15.1000	−0.962744
$247$	7.05212	0.448716
$248$	−3.04346	−0.193260
$249$	−22.4051	−1.41986
$250$	−10.9473	−0.692369
$251$	3.97491	0.250894	0.125447	−	0.992100i	$-0.459964\pi$
$251$	3.97491	0.250894	0.125447	+	0.992100i	$0.459964\pi$
$252$	9.80239	0.617492
$253$	9.44639	0.593890
$254$	−11.4590	−0.719003
$255$	26.6532	1.66909
$256$	1.00000	0.0625000
$257$	12.6711	0.790400	0.395200	−	0.918595i	$-0.370675\pi$
$257$	12.6711	0.790400	0.395200	+	0.918595i	$0.370675\pi$
$258$	−11.2448	−0.700072
$259$	−7.32773	−0.455323
$260$	3.17305	0.196784
$261$	−13.9743	−0.864988
$262$	−10.7798	−0.665981
$263$	22.3884	1.38053	0.690263	−	0.723559i	$-0.257495\pi$
$263$	22.3884	1.38053	0.690263	+	0.723559i	$0.257495\pi$
$264$	2.36508	0.145561
$265$	2.57984	0.158479
$266$	25.8096	1.58249
$267$	40.4404	2.47491
$268$	−1.79330	−0.109543
$269$	24.1252	1.47094	0.735470	−	0.677558i	$-0.236962\pi$
$269$	24.1252	1.47094	0.735470	+	0.677558i	$0.236962\pi$
$270$	5.44132	0.331148
$271$	24.2448	1.47276	0.736382	−	0.676566i	$-0.236533\pi$
$271$	24.2448	1.47276	0.736382	+	0.676566i	$0.236533\pi$
$272$	5.26765	0.319398
$273$	15.7293	0.951980
$274$	−3.63920	−0.219852
$275$	0.211740	0.0127684
$276$	−19.6694	−1.18396
$277$	−14.3591	−0.862754	−0.431377	−	0.902172i	$-0.641972\pi$
$277$	−14.3591	−0.862754	−0.431377	+	0.902172i	$0.641972\pi$
$278$	−7.06774	−0.423895
$279$	5.85743	0.350675
$280$	11.6129	0.694001
$281$	14.8007	0.882937	0.441469	−	0.897277i	$-0.354458\pi$
$281$	14.8007	0.882937	0.441469	+	0.897277i	$0.354458\pi$
$282$	5.47058	0.325769
$283$	12.2531	0.728370	0.364185	−	0.931327i	$-0.381348\pi$
$283$	12.2531	0.728370	0.364185	+	0.931327i	$0.381348\pi$
$284$	−1.35796	−0.0805798
$285$	−25.6401	−1.51879
$286$	1.48317	0.0877019
$287$	−34.6566	−2.04571
$288$	−1.92459	−0.113408
$289$	10.7482	0.632245
$290$	−16.5553	−0.972161
$291$	10.2459	0.600626
$292$	−9.77724	−0.572170
$293$	−1.02326	−0.0597795	−0.0298897	−	0.999553i	$-0.509516\pi$
$293$	−1.02326	−0.0597795	−0.0298897	+	0.999553i	$0.509516\pi$
$294$	42.0326	2.45139
$295$	5.14260	0.299414
$296$	1.43872	0.0836240
$297$	2.54342	0.147584
$298$	21.0236	1.21787
$299$	−12.3349	−0.713347
$300$	−0.440887	−0.0254546
$301$	−25.8083	−1.48757
$302$	−14.4462	−0.831285
$303$	−6.22957	−0.357879
$304$	−5.06744	−0.290638
$305$	15.3917	0.881328
$306$	−10.1381	−0.579556
$307$	33.7034	1.92356	0.961778	−	0.273832i	$-0.0882911\pi$
$307$	33.7034	1.92356	0.961778	+	0.273832i	$0.0882911\pi$
$308$	5.42817	0.309299
$309$	27.9180	1.58820
$310$	6.93928	0.394125
$311$	24.2937	1.37757	0.688785	−	0.724966i	$-0.258145\pi$
$311$	24.2937	1.37757	0.688785	+	0.724966i	$0.258145\pi$
$312$	−3.08828	−0.174839
$313$	6.24384	0.352923	0.176461	−	0.984308i	$-0.443535\pi$
$313$	6.24384	0.352923	0.176461	+	0.984308i	$0.443535\pi$
$314$	−2.32466	−0.131188
$315$	−22.3500	−1.25928
$316$	−9.71494	−0.546508
$317$	11.3465	0.637282	0.318641	−	0.947876i	$-0.396774\pi$
$317$	11.3465	0.637282	0.318641	+	0.947876i	$0.396774\pi$
$318$	−2.51092	−0.140805
$319$	−7.73841	−0.433268
$320$	−2.28006	−0.127459
$321$	−31.9259	−1.78193
$322$	−45.1438	−2.51577
$323$	−26.6935	−1.48527
$324$	−11.0697	−0.614984
$325$	−0.276486	−0.0153367
$326$	2.49930	0.138424
$327$	25.6767	1.41993
$328$	6.80445	0.375713
$329$	12.5557	0.692219
$330$	−5.39253	−0.296849
$331$	−5.20772	−0.286242	−0.143121	−	0.989705i	$-0.545714\pi$
$331$	−5.20772	−0.286242	−0.143121	+	0.989705i	$0.545714\pi$
$332$	10.0963	0.554105
$333$	−2.76895	−0.151738
$334$	−8.83542	−0.483453
$335$	4.08883	0.223397
$336$	−11.3026	−0.616607
$337$	17.3784	0.946660	0.473330	−	0.880885i	$-0.343052\pi$
$337$	17.3784	0.946660	0.473330	+	0.880885i	$0.343052\pi$
$338$	11.0633	0.601764
$339$	−31.2822	−1.69901
$340$	−12.0106	−0.651364
$341$	3.24361	0.175651
$342$	9.75277	0.527369
$343$	60.8178	3.28385
$344$	5.06719	0.273205
$345$	44.8474	2.41450
$346$	−15.9158	−0.855639
$347$	−5.94884	−0.319350	−0.159675	−	0.987170i	$-0.551045\pi$
$347$	−5.94884	−0.319350	−0.159675	+	0.987170i	$0.551045\pi$
$348$	16.1130	0.863747
$349$	−28.7674	−1.53988	−0.769942	−	0.638114i	$-0.779715\pi$
$349$	−28.7674	−1.53988	−0.769942	+	0.638114i	$0.779715\pi$
$350$	−1.01189	−0.0540880
$351$	−3.32115	−0.177270
$352$	−1.06576	−0.0568054
$353$	−15.4494	−0.822290	−0.411145	−	0.911570i	$-0.634871\pi$
$353$	−15.4494	−0.822290	−0.411145	+	0.911570i	$0.634871\pi$
$354$	−5.00521	−0.266024
$355$	3.09622	0.164330
$356$	−18.2234	−0.965840
$357$	−59.5381	−3.15109
$358$	12.4761	0.659381
$359$	−36.9671	−1.95105	−0.975525	−	0.219890i	$-0.929430\pi$
$359$	−36.9671	−1.95105	−0.975525	+	0.219890i	$0.929430\pi$
$360$	4.38819	0.231278
$361$	6.67897	0.351525
$362$	−16.6603	−0.875646
$363$	21.8900	1.14892
$364$	−7.08800	−0.371512
$365$	22.2927	1.16685
$366$	−14.9805	−0.783043
$367$	−18.0087	−0.940044	−0.470022	−	0.882655i	$-0.655754\pi$
$367$	−18.0087	−0.940044	−0.470022	+	0.882655i	$0.655754\pi$
$368$	8.86350	0.462042
$369$	−13.0958	−0.681740
$370$	−3.28037	−0.170538
$371$	−5.76288	−0.299194
$372$	−6.75388	−0.350172
$373$	−20.5298	−1.06299	−0.531495	−	0.847061i	$-0.678370\pi$
$373$	−20.5298	−1.06299	−0.531495	+	0.847061i	$0.678370\pi$
$374$	−5.61407	−0.290297
$375$	−24.2936	−1.25452
$376$	−2.46518	−0.127132
$377$	10.1047	0.520417
$378$	−12.1549	−0.625179
$379$	−3.98963	−0.204934	−0.102467	−	0.994736i	$-0.532674\pi$
$379$	−3.98963	−0.204934	−0.102467	+	0.994736i	$0.532674\pi$
$380$	11.5541	0.592711
$381$	−25.4292	−1.30278
$382$	8.24635	0.421920
$383$	−10.8773	−0.555804	−0.277902	−	0.960609i	$-0.589639\pi$
$383$	−10.8773	−0.555804	−0.277902	+	0.960609i	$0.589639\pi$
$384$	2.21914	0.113245
$385$	−12.3766	−0.630768
$386$	21.6276	1.10082
$387$	−9.75228	−0.495736
$388$	−4.61706	−0.234396
$389$	−26.9313	−1.36547	−0.682735	−	0.730666i	$-0.739210\pi$
$389$	−26.9313	−1.36547	−0.682735	+	0.730666i	$0.739210\pi$
$390$	7.04146	0.356558
$391$	46.6899	2.36121
$392$	−18.9409	−0.956661
$393$	−23.9220	−1.20671
$394$	18.2167	0.917746
$395$	22.1506	1.11452
$396$	2.05116	0.103075
$397$	−24.2008	−1.21460	−0.607302	−	0.794471i	$-0.707748\pi$
$397$	−24.2008	−1.21460	−0.607302	+	0.794471i	$0.707748\pi$
$398$	15.6221	0.783064
$399$	57.2752	2.86735
$400$	0.198674	0.00993372
$401$	−23.9552	−1.19627	−0.598134	−	0.801396i	$-0.704091\pi$
$401$	−23.9552	−1.19627	−0.598134	+	0.801396i	$0.704091\pi$
$402$	−3.97959	−0.198484
$403$	−4.23545	−0.210983
$404$	2.80720	0.139663
$405$	25.2396	1.25417
$406$	36.9814	1.83536
$407$	−1.53334	−0.0760047
$408$	11.6897	0.578725
$409$	2.47165	0.122215	0.0611077	−	0.998131i	$-0.480537\pi$
$409$	2.47165	0.122215	0.0611077	+	0.998131i	$0.480537\pi$
$410$	−15.5146	−0.766209
$411$	−8.07590	−0.398355
$412$	−12.5806	−0.619799
$413$	−11.4876	−0.565268
$414$	−17.0586	−0.838386
$415$	−23.0201	−1.13001
$416$	1.39165	0.0682314
$417$	−15.6843	−0.768065
$418$	5.40069	0.264157
$419$	−31.5645	−1.54203	−0.771014	−	0.636818i	$-0.780250\pi$
$419$	−31.5645	−1.54203	−0.771014	+	0.636818i	$0.780250\pi$
$420$	25.7706	1.25748
$421$	−35.9807	−1.75359	−0.876797	−	0.480861i	$-0.840324\pi$
$421$	−35.9807	−1.75359	−0.876797	+	0.480861i	$0.840324\pi$
$422$	−15.5751	−0.758186
$423$	4.74447	0.230684
$424$	1.13148	0.0549496
$425$	1.04655	0.0507650
$426$	−3.01350	−0.146004
$427$	−34.3822	−1.66387
$428$	14.3866	0.695403
$429$	3.29137	0.158909
$430$	−11.5535	−0.557159
$431$	−12.7627	−0.614757	−0.307378	−	0.951587i	$-0.599452\pi$
$431$	−12.7627	−0.614757	−0.307378	+	0.951587i	$0.599452\pi$
$432$	2.38648	0.114820
$433$	−9.23191	−0.443657	−0.221829	−	0.975086i	$-0.571203\pi$
$433$	−9.23191	−0.443657	−0.221829	+	0.975086i	$0.571203\pi$
$434$	−15.5010	−0.744074
$435$	−36.7386	−1.76148
$436$	−11.5706	−0.554130
$437$	−44.9153	−2.14859
$438$	−21.6971	−1.03673
$439$	22.2681	1.06280	0.531400	−	0.847121i	$-0.321666\pi$
$439$	22.2681	1.06280	0.531400	+	0.847121i	$0.321666\pi$
$440$	2.43000	0.115846
$441$	36.4536	1.73589
$442$	7.33075	0.348688
$443$	9.12249	0.433422	0.216711	−	0.976236i	$-0.430467\pi$
$443$	9.12249	0.433422	0.216711	+	0.976236i	$0.430467\pi$
$444$	3.19273	0.151520
$445$	41.5505	1.96968
$446$	5.56766	0.263636
$447$	46.6544	2.20668
$448$	5.09322	0.240632
$449$	11.5495	0.545055	0.272527	−	0.962148i	$-0.412140\pi$
$449$	11.5495	0.545055	0.272527	+	0.962148i	$0.412140\pi$
$450$	−0.382367	−0.0180250
$451$	−7.25193	−0.341480
$452$	14.0965	0.663044
$453$	−32.0581	−1.50622
$454$	16.7805	0.787548
$455$	16.1611	0.757643
$456$	−11.2454	−0.526613
$457$	−16.7133	−0.781817	−0.390908	−	0.920430i	$-0.627839\pi$
$457$	−16.7133	−0.781817	−0.390908	+	0.920430i	$0.627839\pi$
$458$	1.05285	0.0491966
$459$	12.5711	0.586771
$460$	−20.2093	−0.942264
$461$	−13.5210	−0.629734	−0.314867	−	0.949136i	$-0.601960\pi$
$461$	−13.5210	−0.629734	−0.314867	+	0.949136i	$0.601960\pi$
$462$	12.0459	0.560425
$463$	−18.9262	−0.879573	−0.439787	−	0.898102i	$-0.644946\pi$
$463$	−18.9262	−0.879573	−0.439787	+	0.898102i	$0.644946\pi$
$464$	−7.26091	−0.337079
$465$	15.3992	0.714123
$466$	13.4736	0.624152
$467$	21.0409	0.973659	0.486830	−	0.873497i	$-0.338153\pi$
$467$	21.0409	0.973659	0.486830	+	0.873497i	$0.338153\pi$
$468$	−2.67837	−0.123808
$469$	−9.13367	−0.421754
$470$	5.62076	0.259266
$471$	−5.15874	−0.237702
$472$	2.25547	0.103816
$473$	−5.40042	−0.248312
$474$	−21.5588	−0.990230
$475$	−1.00677	−0.0461938
$476$	26.8293	1.22972
$477$	−2.17764	−0.0997073
$478$	−20.3105	−0.928983
$479$	−26.6753	−1.21883	−0.609414	−	0.792852i	$-0.708595\pi$
$479$	−26.6753	−1.21883	−0.609414	+	0.792852i	$0.708595\pi$
$480$	−5.05978	−0.230946
$481$	2.00220	0.0912926
$482$	9.41464	0.428825
$483$	−100.181	−4.55837
$484$	−9.86415	−0.448370
$485$	10.5272	0.478015
$486$	−17.4058	−0.789545
$487$	−31.3367	−1.42000	−0.710001	−	0.704200i	$-0.751306\pi$
$487$	−31.3367	−1.42000	−0.710001	+	0.704200i	$0.751306\pi$
$488$	6.75058	0.305584
$489$	5.54631	0.250813
$490$	43.1865	1.95097
$491$	−37.5663	−1.69534	−0.847672	−	0.530520i	$-0.821997\pi$
$491$	−37.5663	−1.69534	−0.847672	+	0.530520i	$0.821997\pi$
$492$	15.1000	0.680763
$493$	−38.2480	−1.72260
$494$	−7.05212	−0.317290
$495$	−4.67677	−0.210205
$496$	3.04346	0.136656
$497$	−6.91637	−0.310242
$498$	22.4051	1.00399
$499$	−13.8857	−0.621609	−0.310805	−	0.950474i	$-0.600599\pi$
$499$	−13.8857	−0.621609	−0.310805	+	0.950474i	$0.600599\pi$
$500$	10.9473	0.489579
$501$	−19.6071	−0.875979
$502$	−3.97491	−0.177409
$503$	−3.85005	−0.171665	−0.0858326	−	0.996310i	$-0.527355\pi$
$503$	−3.85005	−0.171665	−0.0858326	+	0.996310i	$0.527355\pi$
$504$	−9.80239	−0.436633
$505$	−6.40058	−0.284822
$506$	−9.44639	−0.419943
$507$	24.5510	1.09035
$508$	11.4590	0.508412
$509$	36.8623	1.63389	0.816946	−	0.576714i	$-0.195665\pi$
$509$	36.8623	1.63389	0.816946	+	0.576714i	$0.195665\pi$
$510$	−26.6532	−1.18022
$511$	−49.7977	−2.20292
$512$	−1.00000	−0.0441942
$513$	−12.0933	−0.533934
$514$	−12.6711	−0.558897
$515$	28.6844	1.26399
$516$	11.2448	0.495026
$517$	2.62730	0.115548
$518$	7.32773	0.321962
$519$	−35.3194	−1.55035
$520$	−3.17305	−0.139148
$521$	27.5042	1.20498	0.602491	−	0.798126i	$-0.294175\pi$
$521$	27.5042	1.20498	0.602491	+	0.798126i	$0.294175\pi$
$522$	13.9743	0.611639
$523$	−27.9053	−1.22022	−0.610108	−	0.792318i	$-0.708874\pi$
$523$	−27.9053	−1.22022	−0.610108	+	0.792318i	$0.708874\pi$
$524$	10.7798	0.470920
$525$	−2.24554	−0.0980032
$526$	−22.3884	−0.976179
$527$	16.0319	0.698361
$528$	−2.36508	−0.102927
$529$	55.5616	2.41572
$530$	−2.57984	−0.112061
$531$	−4.34086	−0.188377
$532$	−25.8096	−1.11899
$533$	9.46944	0.410167
$534$	−40.4404	−1.75003
$535$	−32.8023	−1.41817
$536$	1.79330	0.0774587
$537$	27.6862	1.19475
$538$	−24.1252	−1.04011
$539$	20.1865	0.869496
$540$	−5.44132	−0.234157
$541$	−20.5568	−0.883806	−0.441903	−	0.897063i	$-0.645696\pi$
$541$	−20.5568	−0.883806	−0.441903	+	0.897063i	$0.645696\pi$
$542$	−24.2448	−1.04140
$543$	−36.9716	−1.58660
$544$	−5.26765	−0.225849
$545$	26.3816	1.13006
$546$	−15.7293	−0.673151
$547$	−8.74896	−0.374079	−0.187039	−	0.982352i	$-0.559889\pi$
$547$	−8.74896	−0.374079	−0.187039	+	0.982352i	$0.559889\pi$
$548$	3.63920	0.155459
$549$	−12.9921	−0.554490
$550$	−0.211740	−0.00902862
$551$	36.7942	1.56749
$552$	19.6694	0.837184
$553$	−49.4804	−2.10412
$554$	14.3591	0.610059
$555$	−7.27961	−0.309002
$556$	7.06774	0.299739
$557$	22.5769	0.956615	0.478308	−	0.878192i	$-0.341250\pi$
$557$	22.5769	0.956615	0.478308	+	0.878192i	$0.341250\pi$
$558$	−5.85743	−0.247965
$559$	7.05177	0.298258
$560$	−11.6129	−0.490733
$561$	−12.4584	−0.525995
$562$	−14.8007	−0.624331
$563$	−16.1965	−0.682603	−0.341302	−	0.939954i	$-0.610868\pi$
$563$	−16.1965	−0.682603	−0.341302	+	0.939954i	$0.610868\pi$
$564$	−5.47058	−0.230353
$565$	−32.1409	−1.35218
$566$	−12.2531	−0.515036
$567$	−56.3806	−2.36776
$568$	1.35796	0.0569785
$569$	−20.5271	−0.860542	−0.430271	−	0.902700i	$-0.641582\pi$
$569$	−20.5271	−0.860542	−0.430271	+	0.902700i	$0.641582\pi$
$570$	25.6401	1.07395
$571$	−26.4953	−1.10879	−0.554397	−	0.832252i	$-0.687051\pi$
$571$	−26.4953	−1.10879	−0.554397	+	0.832252i	$0.687051\pi$
$572$	−1.48317	−0.0620146
$573$	18.2998	0.764486
$574$	34.6566	1.44654
$575$	1.76095	0.0734367
$576$	1.92459	0.0801914
$577$	44.7975	1.86494	0.932471	−	0.361244i	$-0.117648\pi$
$577$	44.7975	1.86494	0.932471	+	0.361244i	$0.117648\pi$
$578$	−10.7482	−0.447065
$579$	47.9947	1.99459
$580$	16.5553	0.687422
$581$	51.4226	2.13337
$582$	−10.2459	−0.424707
$583$	−1.20589	−0.0499429
$584$	9.77724	0.404585
$585$	6.10684	0.252487
$586$	1.02326	0.0422705
$587$	10.1008	0.416905	0.208453	−	0.978032i	$-0.433157\pi$
$587$	10.1008	0.416905	0.208453	+	0.978032i	$0.433157\pi$
$588$	−42.0326	−1.73340
$589$	−15.4226	−0.635476
$590$	−5.14260	−0.211718
$591$	40.4255	1.66288
$592$	−1.43872	−0.0591311
$593$	−31.8229	−1.30681	−0.653405	−	0.757008i	$-0.726660\pi$
$593$	−31.8229	−1.30681	−0.653405	+	0.757008i	$0.726660\pi$
$594$	−2.54342	−0.104358
$595$	−61.1725	−2.50783
$596$	−21.0236	−0.861161
$597$	34.6676	1.41885
$598$	12.3349	0.504412
$599$	6.15030	0.251295	0.125647	−	0.992075i	$-0.459899\pi$
$599$	6.15030	0.251295	0.125647	+	0.992075i	$0.459899\pi$
$600$	0.440887	0.0179991
$601$	−2.91058	−0.118725	−0.0593625	−	0.998236i	$-0.518907\pi$
$601$	−2.91058	−0.118725	−0.0593625	+	0.998236i	$0.518907\pi$
$602$	25.8083	1.05187
$603$	−3.45137	−0.140551
$604$	14.4462	0.587807
$605$	22.4909	0.914383
$606$	6.22957	0.253059
$607$	−35.7597	−1.45144	−0.725720	−	0.687990i	$-0.758493\pi$
$607$	−35.7597	−1.45144	−0.725720	+	0.687990i	$0.758493\pi$
$608$	5.06744	0.205512
$609$	82.0671	3.32553
$610$	−15.3917	−0.623193
$611$	−3.43068	−0.138790
$612$	10.1381	0.409808
$613$	29.0316	1.17258	0.586288	−	0.810103i	$-0.300589\pi$
$613$	29.0316	1.17258	0.586288	+	0.810103i	$0.300589\pi$
$614$	−33.7034	−1.36016
$615$	−34.4290	−1.38831
$616$	−5.42817	−0.218707
$617$	31.5096	1.26853	0.634264	−	0.773117i	$-0.281303\pi$
$617$	31.5096	1.26853	0.634264	+	0.773117i	$0.281303\pi$
$618$	−27.9180	−1.12303
$619$	−14.8854	−0.598297	−0.299148	−	0.954207i	$-0.596703\pi$
$619$	−14.8854	−0.598297	−0.299148	+	0.954207i	$0.596703\pi$
$620$	−6.93928	−0.278688
$621$	21.1526	0.848823
$622$	−24.2937	−0.974088
$623$	−92.8160	−3.71859
$624$	3.08828	0.123630
$625$	−25.9539	−1.03816
$626$	−6.24384	−0.249554
$627$	11.9849	0.478631
$628$	2.32466	0.0927639
$629$	−7.57869	−0.302182
$630$	22.3500	0.890447
$631$	19.6843	0.783621	0.391810	−	0.920046i	$-0.371849\pi$
$631$	19.6843	0.783621	0.391810	+	0.920046i	$0.371849\pi$
$632$	9.71494	0.386440
$633$	−34.5635	−1.37377
$634$	−11.3465	−0.450626
$635$	−26.1273	−1.03683
$636$	2.51092	0.0995643
$637$	−26.3592	−1.04439
$638$	7.73841	0.306367
$639$	−2.61351	−0.103389
$640$	2.28006	0.0901273
$641$	−15.6343	−0.617518	−0.308759	−	0.951140i	$-0.599914\pi$
$641$	−15.6343	−0.617518	−0.308759	+	0.951140i	$0.599914\pi$
$642$	31.9259	1.26002
$643$	6.74033	0.265813	0.132906	−	0.991129i	$-0.457569\pi$
$643$	6.74033	0.265813	0.132906	+	0.991129i	$0.457569\pi$
$644$	45.1438	1.77891
$645$	−25.6389	−1.00953
$646$	26.6935	1.05024
$647$	−42.2456	−1.66085	−0.830423	−	0.557133i	$-0.811901\pi$
$647$	−42.2456	−1.66085	−0.830423	+	0.557133i	$0.811901\pi$
$648$	11.0697	0.434860
$649$	−2.40380	−0.0943572
$650$	0.276486	0.0108447
$651$	−34.3990	−1.34820
$652$	−2.49930	−0.0978803
$653$	26.9611	1.05507	0.527535	−	0.849533i	$-0.323116\pi$
$653$	26.9611	1.05507	0.527535	+	0.849533i	$0.323116\pi$
$654$	−25.6767	−1.00404
$655$	−24.5787	−0.960369
$656$	−6.80445	−0.265669
$657$	−18.8172	−0.734130
$658$	−12.5557	−0.489473
$659$	6.75586	0.263171	0.131585	−	0.991305i	$-0.457993\pi$
$659$	6.75586	0.263171	0.131585	+	0.991305i	$0.457993\pi$
$660$	5.39253	0.209904
$661$	−4.18624	−0.162826	−0.0814129	−	0.996680i	$-0.525943\pi$
$661$	−4.18624	−0.162826	−0.0814129	+	0.996680i	$0.525943\pi$
$662$	5.20772	0.202404
$663$	16.2680	0.631796
$664$	−10.0963	−0.391811
$665$	58.8475	2.28201
$666$	2.76895	0.107295
$667$	−64.3571	−2.49192
$668$	8.83542	0.341853
$669$	12.3554	0.477689
$670$	−4.08883	−0.157965
$671$	−7.19452	−0.277741
$672$	11.3026	0.436007
$673$	−42.6593	−1.64440	−0.822198	−	0.569202i	$-0.807252\pi$
$673$	−42.6593	−1.64440	−0.822198	+	0.569202i	$0.807252\pi$
$674$	−17.3784	−0.669390
$675$	0.474132	0.0182494
$676$	−11.0633	−0.425512
$677$	−7.47288	−0.287206	−0.143603	−	0.989635i	$-0.545869\pi$
$677$	−7.47288	−0.287206	−0.143603	+	0.989635i	$0.545869\pi$
$678$	31.2822	1.20138
$679$	−23.5157	−0.902451
$680$	12.0106	0.460584
$681$	37.2383	1.42698
$682$	−3.24361	−0.124204
$683$	−1.71648	−0.0656792	−0.0328396	−	0.999461i	$-0.510455\pi$
$683$	−1.71648	−0.0656792	−0.0328396	+	0.999461i	$0.510455\pi$
$684$	−9.75277	−0.372906
$685$	−8.29759	−0.317035
$686$	−60.8178	−2.32203
$687$	2.33643	0.0891404
$688$	−5.06719	−0.193185
$689$	1.57463	0.0599886
$690$	−44.8474	−1.70731
$691$	22.5283	0.857018	0.428509	−	0.903538i	$-0.359039\pi$
$691$	22.5283	0.857018	0.428509	+	0.903538i	$0.359039\pi$
$692$	15.9158	0.605028
$693$	10.4470	0.396850
$694$	5.94884	0.225815
$695$	−16.1149	−0.611272
$696$	−16.1130	−0.610761
$697$	−35.8435	−1.35767
$698$	28.7674	1.08886
$699$	29.8998	1.13092
$700$	1.01189	0.0382460
$701$	−30.2694	−1.14326	−0.571630	−	0.820511i	$-0.693689\pi$
$701$	−30.2694	−1.14326	−0.571630	+	0.820511i	$0.693689\pi$
$702$	3.32115	0.125349
$703$	7.29064	0.274972
$704$	1.06576	0.0401675
$705$	12.4733	0.469770
$706$	15.4494	0.581447
$707$	14.2977	0.537720
$708$	5.00521	0.188107
$709$	−3.98571	−0.149686	−0.0748432	−	0.997195i	$-0.523846\pi$
$709$	−3.98571	−0.149686	−0.0748432	+	0.997195i	$0.523846\pi$
$710$	−3.09622	−0.116199
$711$	−18.6973	−0.701204
$712$	18.2234	0.682952
$713$	26.9757	1.01025
$714$	59.5381	2.22816
$715$	3.38172	0.126469
$716$	−12.4761	−0.466253
$717$	−45.0720	−1.68324
$718$	36.9671	1.37960
$719$	26.0678	0.972166	0.486083	−	0.873913i	$-0.338425\pi$
$719$	26.0678	0.972166	0.486083	+	0.873913i	$0.338425\pi$
$720$	−4.38819	−0.163538
$721$	−64.0756	−2.38630
$722$	−6.67897	−0.248566
$723$	20.8924	0.776997
$724$	16.6603	0.619175
$725$	−1.44256	−0.0535752
$726$	−21.8900	−0.812412
$727$	23.3949	0.867669	0.433834	−	0.900993i	$-0.357160\pi$
$727$	23.3949	0.867669	0.433834	+	0.900993i	$0.357160\pi$
$728$	7.08800	0.262699
$729$	−5.41690	−0.200626
$730$	−22.2927	−0.825090
$731$	−26.6922	−0.987247
$732$	14.9805	0.553695
$733$	0.784664	0.0289822	0.0144911	−	0.999895i	$-0.495387\pi$
$733$	0.784664	0.0289822	0.0144911	+	0.999895i	$0.495387\pi$
$734$	18.0087	0.664712
$735$	95.8369	3.53500
$736$	−8.86350	−0.326713
$737$	−1.91123	−0.0704011
$738$	13.0958	0.482063
$739$	4.69092	0.172558	0.0862791	−	0.996271i	$-0.472502\pi$
$739$	4.69092	0.172558	0.0862791	+	0.996271i	$0.472502\pi$
$740$	3.28037	0.120589
$741$	−15.6497	−0.574905
$742$	5.76288	0.211562
$743$	−17.0685	−0.626182	−0.313091	−	0.949723i	$-0.601365\pi$
$743$	−17.0685	−0.626182	−0.313091	+	0.949723i	$0.601365\pi$
$744$	6.75388	0.247609
$745$	47.9351	1.75621
$746$	20.5298	0.751648
$747$	19.4312	0.710951
$748$	5.61407	0.205271
$749$	73.2742	2.67738
$750$	24.2936	0.887078
$751$	−23.9386	−0.873531	−0.436766	−	0.899575i	$-0.643876\pi$
$751$	−23.9386	−0.873531	−0.436766	+	0.899575i	$0.643876\pi$
$752$	2.46518	0.0898958
$753$	−8.82088	−0.321451
$754$	−10.1047	−0.367990
$755$	−32.9382	−1.19874
$756$	12.1549	0.442069
$757$	13.3873	0.486571	0.243285	−	0.969955i	$-0.421775\pi$
$757$	13.3873	0.486571	0.243285	+	0.969955i	$0.421775\pi$
$758$	3.98963	0.144910
$759$	−20.9629	−0.760905
$760$	−11.5541	−0.419110
$761$	51.7262	1.87507	0.937536	−	0.347889i	$-0.113101\pi$
$761$	51.7262	1.87507	0.937536	+	0.347889i	$0.113101\pi$
$762$	25.4292	0.921203
$763$	−58.9315	−2.13346
$764$	−8.24635	−0.298342
$765$	−23.1155	−0.835741
$766$	10.8773	0.393013
$767$	3.13883	0.113337
$768$	−2.21914	−0.0800764
$769$	28.4049	1.02431	0.512154	−	0.858894i	$-0.328848\pi$
$769$	28.4049	1.02431	0.512154	+	0.858894i	$0.328848\pi$
$770$	12.3766	0.446020
$771$	−28.1189	−1.01268
$772$	−21.6276	−0.778395
$773$	16.7382	0.602030	0.301015	−	0.953619i	$-0.402675\pi$
$773$	16.7382	0.602030	0.301015	+	0.953619i	$0.402675\pi$
$774$	9.75228	0.350539
$775$	0.604658	0.0217200
$776$	4.61706	0.165743
$777$	16.2613	0.583370
$778$	26.9313	0.965533
$779$	34.4812	1.23542
$780$	−7.04146	−0.252125
$781$	−1.44726	−0.0517870
$782$	−46.6899	−1.66963
$783$	−17.3280	−0.619253
$784$	18.9409	0.676462
$785$	−5.30036	−0.189178
$786$	23.9220	0.853270
$787$	−7.40849	−0.264084	−0.132042	−	0.991244i	$-0.542153\pi$
$787$	−7.40849	−0.264084	−0.132042	+	0.991244i	$0.542153\pi$
$788$	−18.2167	−0.648944
$789$	−49.6830	−1.76876
$790$	−22.1506	−0.788085
$791$	71.7967	2.55280
$792$	−2.05116	−0.0728848
$793$	9.39447	0.333607
$794$	24.2008	0.858854
$795$	−5.72504	−0.203046
$796$	−15.6221	−0.553710
$797$	−34.8257	−1.23359	−0.616795	−	0.787124i	$-0.711569\pi$
$797$	−34.8257	−1.23359	−0.616795	+	0.787124i	$0.711569\pi$
$798$	−57.2752	−2.02752
$799$	12.9857	0.459401
$800$	−0.198674	−0.00702420
$801$	−35.0727	−1.23923
$802$	23.9552	0.845889
$803$	−10.4202	−0.367722
$804$	3.97959	0.140349
$805$	−102.931	−3.62783
$806$	4.23545	0.149187
$807$	−53.5373	−1.88460
$808$	−2.80720	−0.0987568
$809$	−5.95982	−0.209536	−0.104768	−	0.994497i	$-0.533410\pi$
$809$	−5.95982	−0.209536	−0.104768	+	0.994497i	$0.533410\pi$
$810$	−25.2396	−0.886830
$811$	−6.10337	−0.214318	−0.107159	−	0.994242i	$-0.534175\pi$
$811$	−6.10337	−0.214318	−0.107159	+	0.994242i	$0.534175\pi$
$812$	−36.9814	−1.29779
$813$	−53.8026	−1.88694
$814$	1.53334	0.0537434
$815$	5.69856	0.199612
$816$	−11.6897	−0.409220
$817$	25.6777	0.898349
$818$	−2.47165	−0.0864193
$819$	−13.6415	−0.476673
$820$	15.5146	0.541792
$821$	9.73393	0.339717	0.169858	−	0.985469i	$-0.445669\pi$
$821$	9.73393	0.339717	0.169858	+	0.985469i	$0.445669\pi$
$822$	8.07590	0.281679
$823$	54.9153	1.91423	0.957114	−	0.289712i	$-0.0935594\pi$
$823$	54.9153	1.91423	0.957114	+	0.289712i	$0.0935594\pi$
$824$	12.5806	0.438264
$825$	−0.469881	−0.0163592
$826$	11.4876	0.399705
$827$	−21.9888	−0.764626	−0.382313	−	0.924033i	$-0.624872\pi$
$827$	−21.9888	−0.764626	−0.382313	+	0.924033i	$0.624872\pi$
$828$	17.0586	0.592829
$829$	−39.7237	−1.37966	−0.689831	−	0.723971i	$-0.742315\pi$
$829$	−39.7237	−1.37966	−0.689831	+	0.723971i	$0.742315\pi$
$830$	23.0201	0.799039
$831$	31.8649	1.10538
$832$	−1.39165	−0.0482469
$833$	99.7742	3.45697
$834$	15.6843	0.543104
$835$	−20.1453	−0.697157
$836$	−5.40069	−0.186787
$837$	7.26316	0.251052
$838$	31.5645	1.09038
$839$	−24.9466	−0.861254	−0.430627	−	0.902530i	$-0.641707\pi$
$839$	−24.9466	−0.861254	−0.430627	+	0.902530i	$0.641707\pi$
$840$	−25.7706	−0.889170
$841$	23.7208	0.817959
$842$	35.9807	1.23998
$843$	−32.8449	−1.13124
$844$	15.5751	0.536118
$845$	25.2250	0.867766
$846$	−4.74447	−0.163118
$847$	−50.2403	−1.72628
$848$	−1.13148	−0.0388552
$849$	−27.1913	−0.933204
$850$	−1.04655	−0.0358963
$851$	−12.7521	−0.437137
$852$	3.01350	0.103241
$853$	−56.9024	−1.94830	−0.974151	−	0.225896i	$-0.927469\pi$
$853$	−56.9024	−1.94830	−0.974151	+	0.225896i	$0.927469\pi$
$854$	34.3822	1.17654
$855$	22.2369	0.760486
$856$	−14.3866	−0.491724
$857$	−35.1196	−1.19966	−0.599831	−	0.800127i	$-0.704765\pi$
$857$	−35.1196	−1.19966	−0.599831	+	0.800127i	$0.704765\pi$
$858$	−3.29137	−0.112366
$859$	28.4488	0.970660	0.485330	−	0.874331i	$-0.338700\pi$
$859$	28.4488	0.970660	0.485330	+	0.874331i	$0.338700\pi$
$860$	11.5535	0.393971
$861$	76.9079	2.62102
$862$	12.7627	0.434699
$863$	−41.8341	−1.42405	−0.712024	−	0.702155i	$-0.752221\pi$
$863$	−41.8341	−1.42405	−0.712024	+	0.702155i	$0.752221\pi$
$864$	−2.38648	−0.0811897
$865$	−36.2890	−1.23386
$866$	9.23191	0.313713
$867$	−23.8517	−0.810047
$868$	15.5010	0.526139
$869$	−10.3538	−0.351229
$870$	36.7386	1.24556
$871$	2.49565	0.0845619
$872$	11.5706	0.391829
$873$	−8.88597	−0.300744
$874$	44.9153	1.51928
$875$	55.7571	1.88493
$876$	21.6971	0.733077
$877$	−47.6015	−1.60739	−0.803695	−	0.595042i	$-0.797136\pi$
$877$	−47.6015	−1.60739	−0.803695	+	0.595042i	$0.797136\pi$
$878$	−22.2681	−0.751512
$879$	2.27076	0.0765908
$880$	−2.43000	−0.0819154
$881$	37.2142	1.25378	0.626890	−	0.779108i	$-0.284328\pi$
$881$	37.2142	1.25378	0.626890	+	0.779108i	$0.284328\pi$
$882$	−36.4536	−1.22746
$883$	−47.2382	−1.58969	−0.794846	−	0.606812i	$-0.792448\pi$
$883$	−47.2382	−1.58969	−0.794846	+	0.606812i	$0.792448\pi$
$884$	−7.33075	−0.246560
$885$	−11.4122	−0.383616
$886$	−9.12249	−0.306476
$887$	32.0192	1.07510	0.537550	−	0.843232i	$-0.319350\pi$
$887$	32.0192	1.07510	0.537550	+	0.843232i	$0.319350\pi$
$888$	−3.19273	−0.107141
$889$	58.3633	1.95744
$890$	−41.5505	−1.39278
$891$	−11.7977	−0.395238
$892$	−5.56766	−0.186419
$893$	−12.4922	−0.418034
$894$	−46.6544	−1.56036
$895$	28.4462	0.950851
$896$	−5.09322	−0.170153
$897$	27.3729	0.913956
$898$	−11.5495	−0.385412
$899$	−22.0983	−0.737020
$900$	0.382367	0.0127456
$901$	−5.96025	−0.198565
$902$	7.25193	0.241463
$903$	57.2724	1.90591
$904$	−14.0965	−0.468843
$905$	−37.9865	−1.26271
$906$	32.0581	1.06506
$907$	22.8554	0.758901	0.379451	−	0.925212i	$-0.376113\pi$
$907$	22.8554	0.758901	0.379451	+	0.925212i	$0.376113\pi$
$908$	−16.7805	−0.556881
$909$	5.40271	0.179197
$910$	−16.1611	−0.535734
$911$	−18.8839	−0.625653	−0.312826	−	0.949810i	$-0.601276\pi$
$911$	−18.8839	−0.625653	−0.312826	+	0.949810i	$0.601276\pi$
$912$	11.2454	0.372372
$913$	10.7602	0.356112
$914$	16.7133	0.552828
$915$	−34.1564	−1.12918
$916$	−1.05285	−0.0347872
$917$	54.9042	1.81310
$918$	−12.5711	−0.414910
$919$	9.82218	0.324004	0.162002	−	0.986790i	$-0.448205\pi$
$919$	9.82218	0.324004	0.162002	+	0.986790i	$0.448205\pi$
$920$	20.2093	0.666281
$921$	−74.7927	−2.46450
$922$	13.5210	0.445289
$923$	1.88980	0.0622036
$924$	−12.0459	−0.396280
$925$	−0.285837	−0.00939827
$926$	18.9262	0.621952
$927$	−24.2125	−0.795241
$928$	7.26091	0.238351
$929$	−26.7968	−0.879173	−0.439587	−	0.898200i	$-0.644875\pi$
$929$	−26.7968	−0.879173	−0.439587	+	0.898200i	$0.644875\pi$
$930$	−15.3992	−0.504961
$931$	−95.9821	−3.14568
$932$	−13.4736	−0.441342
$933$	−53.9112	−1.76497
$934$	−21.0409	−0.688481
$935$	−12.8004	−0.418618
$936$	2.67837	0.0875452
$937$	11.5836	0.378420	0.189210	−	0.981937i	$-0.439407\pi$
$937$	11.5836	0.378420	0.189210	+	0.981937i	$0.439407\pi$
$938$	9.13367	0.298225
$939$	−13.8560	−0.452173
$940$	−5.62076	−0.183329
$941$	−39.3488	−1.28273	−0.641367	−	0.767234i	$-0.721632\pi$
$941$	−39.3488	−1.28273	−0.641367	+	0.767234i	$0.721632\pi$
$942$	5.15874	0.168081
$943$	−60.3113	−1.96400
$944$	−2.25547	−0.0734092
$945$	−27.7138	−0.901532
$946$	5.40042	0.175583
$947$	41.2835	1.34153	0.670767	−	0.741668i	$-0.265965\pi$
$947$	41.2835	1.34153	0.670767	+	0.741668i	$0.265965\pi$
$948$	21.5588	0.700199
$949$	13.6065	0.441687
$950$	1.00677	0.0326640
$951$	−25.1794	−0.816500
$952$	−26.8293	−0.869544
$953$	30.4959	0.987859	0.493929	−	0.869502i	$-0.335560\pi$
$953$	30.4959	0.987859	0.493929	+	0.869502i	$0.335560\pi$
$954$	2.17764	0.0705037
$955$	18.8022	0.608424
$956$	20.3105	0.656890
$957$	17.1726	0.555112
$958$	26.6753	0.861841
$959$	18.5353	0.598534
$960$	5.05978	0.163304
$961$	−21.7373	−0.701204
$962$	−2.00220	−0.0645536
$963$	27.6884	0.892245
$964$	−9.41464	−0.303225
$965$	49.3122	1.58742
$966$	100.181	3.22326
$967$	−47.6721	−1.53303	−0.766516	−	0.642225i	$-0.778011\pi$
$967$	−47.6721	−1.53303	−0.766516	+	0.642225i	$0.778011\pi$
$968$	9.86415	0.317046
$969$	59.2367	1.90296
$970$	−10.5272	−0.338007
$971$	−25.7850	−0.827481	−0.413740	−	0.910395i	$-0.635778\pi$
$971$	−25.7850	−0.827481	−0.413740	+	0.910395i	$0.635778\pi$
$972$	17.4058	0.558293
$973$	35.9976	1.15403
$974$	31.3367	1.00409
$975$	0.613562	0.0196497
$976$	−6.75058	−0.216081
$977$	−11.4678	−0.366886	−0.183443	−	0.983030i	$-0.558724\pi$
$977$	−11.4678	−0.366886	−0.183443	+	0.983030i	$0.558724\pi$
$978$	−5.54631	−0.177351
$979$	−19.4219	−0.620725
$980$	−43.1865	−1.37954
$981$	−22.2686	−0.710983
$982$	37.5663	1.19879
$983$	6.61669	0.211040	0.105520	−	0.994417i	$-0.466349\pi$
$983$	6.61669	0.211040	0.105520	+	0.994417i	$0.466349\pi$
$984$	−15.1000	−0.481372
$985$	41.5353	1.32342
$986$	38.2480	1.21806
$987$	−27.8629	−0.886886
$988$	7.05212	0.224358
$989$	−44.9130	−1.42815
$990$	4.67677	0.148638
$991$	19.5708	0.621688	0.310844	−	0.950461i	$-0.399388\pi$
$991$	19.5708	0.621688	0.310844	+	0.950461i	$0.399388\pi$
$992$	−3.04346	−0.0966300
$993$	11.5567	0.366740
$994$	6.91637	0.219374
$995$	35.6193	1.12921
$996$	−22.4051	−0.709932
$997$	53.1231	1.68243	0.841213	−	0.540705i	$-0.181842\pi$
$997$	53.1231	1.68243	0.841213	+	0.540705i	$0.181842\pi$
$998$	13.8857	0.439544
$999$	−3.43348	−0.108631

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8042.2.a.b.1.18	✓	82

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8042.2.a.b.1.18	✓	82	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.21914 q^{3} +1.00000 q^{4} -2.28006 q^{5} +2.21914 q^{6} +5.09322 q^{7} -1.00000 q^{8} +1.92459 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.21914 q^{3} +1.00000 q^{4} -2.28006 q^{5} +2.21914 q^{6} +5.09322 q^{7} -1.00000 q^{8} +1.92459 q^{9} +2.28006 q^{10} +1.06576 q^{11} -2.21914 q^{12} -1.39165 q^{13} -5.09322 q^{14} +5.05978 q^{15} +1.00000 q^{16} +5.26765 q^{17} -1.92459 q^{18} -5.06744 q^{19} -2.28006 q^{20} -11.3026 q^{21} -1.06576 q^{22} +8.86350 q^{23} +2.21914 q^{24} +0.198674 q^{25} +1.39165 q^{26} +2.38648 q^{27} +5.09322 q^{28} -7.26091 q^{29} -5.05978 q^{30} +3.04346 q^{31} -1.00000 q^{32} -2.36508 q^{33} -5.26765 q^{34} -11.6129 q^{35} +1.92459 q^{36} -1.43872 q^{37} +5.06744 q^{38} +3.08828 q^{39} +2.28006 q^{40} -6.80445 q^{41} +11.3026 q^{42} -5.06719 q^{43} +1.06576 q^{44} -4.38819 q^{45} -8.86350 q^{46} +2.46518 q^{47} -2.21914 q^{48} +18.9409 q^{49} -0.198674 q^{50} -11.6897 q^{51} -1.39165 q^{52} -1.13148 q^{53} -2.38648 q^{54} -2.43000 q^{55} -5.09322 q^{56} +11.2454 q^{57} +7.26091 q^{58} -2.25547 q^{59} +5.05978 q^{60} -6.75058 q^{61} -3.04346 q^{62} +9.80239 q^{63} +1.00000 q^{64} +3.17305 q^{65} +2.36508 q^{66} -1.79330 q^{67} +5.26765 q^{68} -19.6694 q^{69} +11.6129 q^{70} -1.35796 q^{71} -1.92459 q^{72} -9.77724 q^{73} +1.43872 q^{74} -0.440887 q^{75} -5.06744 q^{76} +5.42817 q^{77} -3.08828 q^{78} -9.71494 q^{79} -2.28006 q^{80} -11.0697 q^{81} +6.80445 q^{82} +10.0963 q^{83} -11.3026 q^{84} -12.0106 q^{85} +5.06719 q^{86} +16.1130 q^{87} -1.06576 q^{88} -18.2234 q^{89} +4.38819 q^{90} -7.08800 q^{91} +8.86350 q^{92} -6.75388 q^{93} -2.46518 q^{94} +11.5541 q^{95} +2.21914 q^{96} -4.61706 q^{97} -18.9409 q^{98} +2.05116 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 82 q - 82 q^{2} - 13 q^{3} + 82 q^{4} + 3 q^{5} + 13 q^{6} - 37 q^{7} - 82 q^{8} + 91 q^{9}+O(q^{10}) \) 82 * q - 82 * q^2 - 13 * q^3 + 82 * q^4 + 3 * q^5 + 13 * q^6 - 37 * q^7 - 82 * q^8 + 91 * q^9	\( 82 q - 82 q^{2} - 13 q^{3} + 82 q^{4} + 3 q^{5} + 13 q^{6} - 37 q^{7} - 82 q^{8} + 91 q^{9} - 3 q^{10} - 16 q^{11} - 13 q^{12} - 42 q^{13} + 37 q^{14} - 9 q^{15} + 82 q^{16} + 3 q^{17} - 91 q^{18} - 42 q^{19} + 3 q^{20} - q^{21} + 16 q^{22} - 6 q^{23} + 13 q^{24} + 53 q^{25} + 42 q^{26} - 49 q^{27} - 37 q^{28} + 15 q^{29} + 9 q^{30} - 40 q^{31} - 82 q^{32} - 37 q^{33} - 3 q^{34} - 42 q^{35} + 91 q^{36} - 72 q^{37} + 42 q^{38} - 14 q^{39} - 3 q^{40} + 8 q^{41} + q^{42} - 93 q^{43} - 16 q^{44} - 11 q^{45} + 6 q^{46} + 7 q^{47} - 13 q^{48} + 61 q^{49} - 53 q^{50} - 70 q^{51} - 42 q^{52} + 18 q^{53} + 49 q^{54} - 62 q^{55} + 37 q^{56} - 51 q^{57} - 15 q^{58} - 47 q^{59} - 9 q^{60} - 14 q^{61} + 40 q^{62} - 100 q^{63} + 82 q^{64} + q^{65} + 37 q^{66} - 150 q^{67} + 3 q^{68} + 31 q^{69} + 42 q^{70} + 7 q^{71} - 91 q^{72} - 78 q^{73} + 72 q^{74} - 49 q^{75} - 42 q^{76} + 29 q^{77} + 14 q^{78} - 59 q^{79} + 3 q^{80} + 122 q^{81} - 8 q^{82} - 52 q^{83} - q^{84} - 108 q^{85} + 93 q^{86} - 49 q^{87} + 16 q^{88} + 38 q^{89} + 11 q^{90} - 69 q^{91} - 6 q^{92} - 63 q^{93} - 7 q^{94} + 5 q^{95} + 13 q^{96} - 74 q^{97} - 61 q^{98} - 89 q^{99}+O(q^{100}) \) 82 * q - 82 * q^2 - 13 * q^3 + 82 * q^4 + 3 * q^5 + 13 * q^6 - 37 * q^7 - 82 * q^8 + 91 * q^9 - 3 * q^10 - 16 * q^11 - 13 * q^12 - 42 * q^13 + 37 * q^14 - 9 * q^15 + 82 * q^16 + 3 * q^17 - 91 * q^18 - 42 * q^19 + 3 * q^20 - q^21 + 16 * q^22 - 6 * q^23 + 13 * q^24 + 53 * q^25 + 42 * q^26 - 49 * q^27 - 37 * q^28 + 15 * q^29 + 9 * q^30 - 40 * q^31 - 82 * q^32 - 37 * q^33 - 3 * q^34 - 42 * q^35 + 91 * q^36 - 72 * q^37 + 42 * q^38 - 14 * q^39 - 3 * q^40 + 8 * q^41 + q^42 - 93 * q^43 - 16 * q^44 - 11 * q^45 + 6 * q^46 + 7 * q^47 - 13 * q^48 + 61 * q^49 - 53 * q^50 - 70 * q^51 - 42 * q^52 + 18 * q^53 + 49 * q^54 - 62 * q^55 + 37 * q^56 - 51 * q^57 - 15 * q^58 - 47 * q^59 - 9 * q^60 - 14 * q^61 + 40 * q^62 - 100 * q^63 + 82 * q^64 + q^65 + 37 * q^66 - 150 * q^67 + 3 * q^68 + 31 * q^69 + 42 * q^70 + 7 * q^71 - 91 * q^72 - 78 * q^73 + 72 * q^74 - 49 * q^75 - 42 * q^76 + 29 * q^77 + 14 * q^78 - 59 * q^79 + 3 * q^80 + 122 * q^81 - 8 * q^82 - 52 * q^83 - q^84 - 108 * q^85 + 93 * q^86 - 49 * q^87 + 16 * q^88 + 38 * q^89 + 11 * q^90 - 69 * q^91 - 6 * q^92 - 63 * q^93 - 7 * q^94 + 5 * q^95 + 13 * q^96 - 74 * q^97 - 61 * q^98 - 89 * q^99

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