LMFDB - Embedded newform 4030.2.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4030,2,Mod(1,4030)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4030, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4030.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4030 = 2 \cdot 5 \cdot 13 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4030.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:	$32.1797120146$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 14x^{6} - 6x^{5} + 54x^{4} + 46x^{3} - 32x^{2} - 43x - 11 $ x^8 - 14x^6 - 6x^5 + 54x^4 + 46x^3 - 32x^2 - 43x - 11
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.94234$ of defining polynomial
Character	$\chi$	$=$	4030.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -3.08723 q^{3} +1.00000 q^{4} +1.00000 q^{5} +3.08723 q^{6} +4.56306 q^{7} -1.00000 q^{8} +6.53101 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -3.08723 q^{3} +1.00000 q^{4} +1.00000 q^{5} +3.08723 q^{6} +4.56306 q^{7} -1.00000 q^{8} +6.53101 q^{9} -1.00000 q^{10} -3.21517 q^{11} -3.08723 q^{12} -1.00000 q^{13} -4.56306 q^{14} -3.08723 q^{15} +1.00000 q^{16} -4.97041 q^{17} -6.53101 q^{18} -7.48372 q^{19} +1.00000 q^{20} -14.0872 q^{21} +3.21517 q^{22} +8.03001 q^{23} +3.08723 q^{24} +1.00000 q^{25} +1.00000 q^{26} -10.9010 q^{27} +4.56306 q^{28} +4.31171 q^{29} +3.08723 q^{30} -1.00000 q^{31} -1.00000 q^{32} +9.92599 q^{33} +4.97041 q^{34} +4.56306 q^{35} +6.53101 q^{36} -5.02966 q^{37} +7.48372 q^{38} +3.08723 q^{39} -1.00000 q^{40} +7.36029 q^{41} +14.0872 q^{42} -1.97390 q^{43} -3.21517 q^{44} +6.53101 q^{45} -8.03001 q^{46} +3.07385 q^{47} -3.08723 q^{48} +13.8215 q^{49} -1.00000 q^{50} +15.3448 q^{51} -1.00000 q^{52} +3.25336 q^{53} +10.9010 q^{54} -3.21517 q^{55} -4.56306 q^{56} +23.1040 q^{57} -4.31171 q^{58} -4.07920 q^{59} -3.08723 q^{60} -4.76898 q^{61} +1.00000 q^{62} +29.8013 q^{63} +1.00000 q^{64} -1.00000 q^{65} -9.92599 q^{66} +12.0571 q^{67} -4.97041 q^{68} -24.7905 q^{69} -4.56306 q^{70} -8.72672 q^{71} -6.53101 q^{72} +5.98698 q^{73} +5.02966 q^{74} -3.08723 q^{75} -7.48372 q^{76} -14.6710 q^{77} -3.08723 q^{78} -15.0878 q^{79} +1.00000 q^{80} +14.0610 q^{81} -7.36029 q^{82} +11.1563 q^{83} -14.0872 q^{84} -4.97041 q^{85} +1.97390 q^{86} -13.3113 q^{87} +3.21517 q^{88} +2.41549 q^{89} -6.53101 q^{90} -4.56306 q^{91} +8.03001 q^{92} +3.08723 q^{93} -3.07385 q^{94} -7.48372 q^{95} +3.08723 q^{96} +13.0192 q^{97} -13.8215 q^{98} -20.9983 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} + q^{3} + 8 q^{4} + 8 q^{5} - q^{6} + 11 q^{7} - 8 q^{8} + 9 q^{9}+O(q^{10}) $ 8 * q - 8 * q^2 + q^3 + 8 * q^4 + 8 * q^5 - q^6 + 11 * q^7 - 8 * q^8 + 9 * q^9	$ 8 q - 8 q^{2} + q^{3} + 8 q^{4} + 8 q^{5} - q^{6} + 11 q^{7} - 8 q^{8} + 9 q^{9} - 8 q^{10} + q^{12} - 8 q^{13} - 11 q^{14} + q^{15} + 8 q^{16} + 7 q^{17} - 9 q^{18} - 2 q^{19} + 8 q^{20} + q^{21} + 8 q^{23} - q^{24} + 8 q^{25} + 8 q^{26} + 7 q^{27} + 11 q^{28} - q^{29} - q^{30} - 8 q^{31} - 8 q^{32} + 14 q^{33} - 7 q^{34} + 11 q^{35} + 9 q^{36} + q^{37} + 2 q^{38} - q^{39} - 8 q^{40} + 16 q^{41} - q^{42} + 3 q^{43} + 9 q^{45} - 8 q^{46} + 29 q^{47} + q^{48} + 11 q^{49} - 8 q^{50} + 11 q^{51} - 8 q^{52} + 22 q^{53} - 7 q^{54} - 11 q^{56} + 33 q^{57} + q^{58} - 8 q^{59} + q^{60} + 4 q^{61} + 8 q^{62} + 38 q^{63} + 8 q^{64} - 8 q^{65} - 14 q^{66} + 28 q^{67} + 7 q^{68} - 42 q^{69} - 11 q^{70} + 4 q^{71} - 9 q^{72} + 39 q^{73} - q^{74} + q^{75} - 2 q^{76} + 11 q^{77} + q^{78} - 16 q^{79} + 8 q^{80} + 32 q^{81} - 16 q^{82} + 25 q^{83} + q^{84} + 7 q^{85} - 3 q^{86} + 13 q^{87} + 21 q^{89} - 9 q^{90} - 11 q^{91} + 8 q^{92} - q^{93} - 29 q^{94} - 2 q^{95} - q^{96} + 28 q^{97} - 11 q^{98} + 23 q^{99}+O(q^{100}) $ 8 * q - 8 * q^2 + q^3 + 8 * q^4 + 8 * q^5 - q^6 + 11 * q^7 - 8 * q^8 + 9 * q^9 - 8 * q^10 + q^12 - 8 * q^13 - 11 * q^14 + q^15 + 8 * q^16 + 7 * q^17 - 9 * q^18 - 2 * q^19 + 8 * q^20 + q^21 + 8 * q^23 - q^24 + 8 * q^25 + 8 * q^26 + 7 * q^27 + 11 * q^28 - q^29 - q^30 - 8 * q^31 - 8 * q^32 + 14 * q^33 - 7 * q^34 + 11 * q^35 + 9 * q^36 + q^37 + 2 * q^38 - q^39 - 8 * q^40 + 16 * q^41 - q^42 + 3 * q^43 + 9 * q^45 - 8 * q^46 + 29 * q^47 + q^48 + 11 * q^49 - 8 * q^50 + 11 * q^51 - 8 * q^52 + 22 * q^53 - 7 * q^54 - 11 * q^56 + 33 * q^57 + q^58 - 8 * q^59 + q^60 + 4 * q^61 + 8 * q^62 + 38 * q^63 + 8 * q^64 - 8 * q^65 - 14 * q^66 + 28 * q^67 + 7 * q^68 - 42 * q^69 - 11 * q^70 + 4 * q^71 - 9 * q^72 + 39 * q^73 - q^74 + q^75 - 2 * q^76 + 11 * q^77 + q^78 - 16 * q^79 + 8 * q^80 + 32 * q^81 - 16 * q^82 + 25 * q^83 + q^84 + 7 * q^85 - 3 * q^86 + 13 * q^87 + 21 * q^89 - 9 * q^90 - 11 * q^91 + 8 * q^92 - q^93 - 29 * q^94 - 2 * q^95 - q^96 + 28 * q^97 - 11 * 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
$3$	−3.08723	−1.78241	−0.891207	−	0.453596i	$-0.850141\pi$
$3$	−3.08723	−1.78241	−0.891207	+	0.453596i	$0.850141\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	3.08723	1.26036
$7$	4.56306	1.72467	0.862336	−	0.506336i	$-0.169000\pi$
$7$	4.56306	1.72467	0.862336	+	0.506336i	$0.169000\pi$
$8$	−1.00000	−0.353553
$9$	6.53101	2.17700
$10$	−1.00000	−0.316228
$11$	−3.21517	−0.969412	−0.484706	−	0.874677i	$-0.661073\pi$
$11$	−3.21517	−0.969412	−0.484706	+	0.874677i	$0.661073\pi$
$12$	−3.08723	−0.891207
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−4.56306	−1.21953
$15$	−3.08723	−0.797120
$16$	1.00000	0.250000
$17$	−4.97041	−1.20550	−0.602751	−	0.797930i	$-0.705929\pi$
$17$	−4.97041	−1.20550	−0.602751	+	0.797930i	$0.705929\pi$
$18$	−6.53101	−1.53937
$19$	−7.48372	−1.71688	−0.858442	−	0.512911i	$-0.828567\pi$
$19$	−7.48372	−1.71688	−0.858442	+	0.512911i	$0.828567\pi$
$20$	1.00000	0.223607
$21$	−14.0872	−3.07408
$22$	3.21517	0.685477
$23$	8.03001	1.67437	0.837187	−	0.546917i	$-0.184199\pi$
$23$	8.03001	1.67437	0.837187	+	0.546917i	$0.184199\pi$
$24$	3.08723	0.630179
$25$	1.00000	0.200000
$26$	1.00000	0.196116
$27$	−10.9010	−2.09791
$28$	4.56306	0.862336
$29$	4.31171	0.800665	0.400333	−	0.916370i	$-0.368895\pi$
$29$	4.31171	0.800665	0.400333	+	0.916370i	$0.368895\pi$
$30$	3.08723	0.563649
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	−1.00000	−0.176777
$33$	9.92599	1.72789
$34$	4.97041	0.852418
$35$	4.56306	0.771297
$36$	6.53101	1.08850
$37$	−5.02966	−0.826870	−0.413435	−	0.910534i	$-0.635671\pi$
$37$	−5.02966	−0.826870	−0.413435	+	0.910534i	$0.635671\pi$
$38$	7.48372	1.21402
$39$	3.08723	0.494353
$40$	−1.00000	−0.158114
$41$	7.36029	1.14948	0.574742	−	0.818335i	$-0.305102\pi$
$41$	7.36029	1.14948	0.574742	+	0.818335i	$0.305102\pi$
$42$	14.0872	2.17370
$43$	−1.97390	−0.301017	−0.150509	−	0.988609i	$-0.548091\pi$
$43$	−1.97390	−0.301017	−0.150509	+	0.988609i	$0.548091\pi$
$44$	−3.21517	−0.484706
$45$	6.53101	0.973585
$46$	−8.03001	−1.18396
$47$	3.07385	0.448367	0.224183	−	0.974547i	$-0.428029\pi$
$47$	3.07385	0.448367	0.224183	+	0.974547i	$0.428029\pi$
$48$	−3.08723	−0.445604
$49$	13.8215	1.97450
$50$	−1.00000	−0.141421
$51$	15.3448	2.14870
$52$	−1.00000	−0.138675
$53$	3.25336	0.446884	0.223442	−	0.974717i	$-0.428271\pi$
$53$	3.25336	0.446884	0.223442	+	0.974717i	$0.428271\pi$
$54$	10.9010	1.48344
$55$	−3.21517	−0.433534
$56$	−4.56306	−0.609764
$57$	23.1040	3.06020
$58$	−4.31171	−0.566156
$59$	−4.07920	−0.531067	−0.265533	−	0.964102i	$-0.585548\pi$
$59$	−4.07920	−0.531067	−0.265533	+	0.964102i	$0.585548\pi$
$60$	−3.08723	−0.398560
$61$	−4.76898	−0.610605	−0.305303	−	0.952255i	$-0.598758\pi$
$61$	−4.76898	−0.610605	−0.305303	+	0.952255i	$0.598758\pi$
$62$	1.00000	0.127000
$63$	29.8013	3.75462
$64$	1.00000	0.125000
$65$	−1.00000	−0.124035
$66$	−9.92599	−1.22181
$67$	12.0571	1.47301	0.736506	−	0.676431i	$-0.236474\pi$
$67$	12.0571	1.47301	0.736506	+	0.676431i	$0.236474\pi$
$68$	−4.97041	−0.602751
$69$	−24.7905	−2.98443
$70$	−4.56306	−0.545389
$71$	−8.72672	−1.03567	−0.517835	−	0.855480i	$-0.673262\pi$
$71$	−8.72672	−1.03567	−0.517835	+	0.855480i	$0.673262\pi$
$72$	−6.53101	−0.769687
$73$	5.98698	0.700724	0.350362	−	0.936614i	$-0.386059\pi$
$73$	5.98698	0.700724	0.350362	+	0.936614i	$0.386059\pi$
$74$	5.02966	0.584686
$75$	−3.08723	−0.356483
$76$	−7.48372	−0.858442
$77$	−14.6710	−1.67192
$78$	−3.08723	−0.349560
$79$	−15.0878	−1.69752	−0.848758	−	0.528782i	$-0.822649\pi$
$79$	−15.0878	−1.69752	−0.848758	+	0.528782i	$0.822649\pi$
$80$	1.00000	0.111803
$81$	14.0610	1.56234
$82$	−7.36029	−0.812808
$83$	11.1563	1.22456	0.612282	−	0.790640i	$-0.290252\pi$
$83$	11.1563	1.22456	0.612282	+	0.790640i	$0.290252\pi$
$84$	−14.0872	−1.53704
$85$	−4.97041	−0.539117
$86$	1.97390	0.212851
$87$	−13.3113	−1.42712
$88$	3.21517	0.342739
$89$	2.41549	0.256041	0.128021	−	0.991771i	$-0.459138\pi$
$89$	2.41549	0.256041	0.128021	+	0.991771i	$0.459138\pi$
$90$	−6.53101	−0.688429
$91$	−4.56306	−0.478338
$92$	8.03001	0.837187
$93$	3.08723	0.320131
$94$	−3.07385	−0.317043
$95$	−7.48372	−0.767813
$96$	3.08723	0.315089
$97$	13.0192	1.32190	0.660951	−	0.750429i	$-0.270153\pi$
$97$	13.0192	1.32190	0.660951	+	0.750429i	$0.270153\pi$
$98$	−13.8215	−1.39618
$99$	−20.9983	−2.11041
$100$	1.00000	0.100000
$101$	2.95587	0.294120	0.147060	−	0.989128i	$-0.453019\pi$
$101$	2.95587	0.294120	0.147060	+	0.989128i	$0.453019\pi$
$102$	−15.3448	−1.51936
$103$	−13.6371	−1.34371	−0.671853	−	0.740684i	$-0.734501\pi$
$103$	−13.6371	−1.34371	−0.671853	+	0.740684i	$0.734501\pi$
$104$	1.00000	0.0980581
$105$	−14.0872	−1.37477
$106$	−3.25336	−0.315994
$107$	9.05773	0.875644	0.437822	−	0.899062i	$-0.355750\pi$
$107$	9.05773	0.875644	0.437822	+	0.899062i	$0.355750\pi$
$108$	−10.9010	−1.04895
$109$	−2.49872	−0.239334	−0.119667	−	0.992814i	$-0.538183\pi$
$109$	−2.49872	−0.239334	−0.119667	+	0.992814i	$0.538183\pi$
$110$	3.21517	0.306555
$111$	15.5277	1.47383
$112$	4.56306	0.431168
$113$	15.1957	1.42950	0.714748	−	0.699382i	$-0.246541\pi$
$113$	15.1957	1.42950	0.714748	+	0.699382i	$0.246541\pi$
$114$	−23.1040	−2.16389
$115$	8.03001	0.748803
$116$	4.31171	0.400333
$117$	−6.53101	−0.603792
$118$	4.07920	0.375521
$119$	−22.6803	−2.07910
$120$	3.08723	0.281825
$121$	−0.662654	−0.0602412
$122$	4.76898	0.431763
$123$	−22.7229	−2.04886
$124$	−1.00000	−0.0898027
$125$	1.00000	0.0894427
$126$	−29.8013	−2.65492
$127$	−0.0192539	−0.00170851	−0.000854255	−	1.00000i	$-0.500272\pi$
$127$	−0.0192539	−0.00170851	−0.000854255		1.00000i	$0.500272\pi$
$128$	−1.00000	−0.0883883
$129$	6.09390	0.536538
$130$	1.00000	0.0877058
$131$	16.1700	1.41278	0.706389	−	0.707823i	$-0.250323\pi$
$131$	16.1700	1.41278	0.706389	+	0.707823i	$0.250323\pi$
$132$	9.92599	0.863947
$133$	−34.1486	−2.96106
$134$	−12.0571	−1.04158
$135$	−10.9010	−0.938212
$136$	4.97041	0.426209
$137$	3.88087	0.331565	0.165783	−	0.986162i	$-0.446985\pi$
$137$	3.88087	0.331565	0.165783	+	0.986162i	$0.446985\pi$
$138$	24.7905	2.11031
$139$	−17.2203	−1.46061	−0.730304	−	0.683122i	$-0.760622\pi$
$139$	−17.2203	−1.46061	−0.730304	+	0.683122i	$0.760622\pi$
$140$	4.56306	0.385649
$141$	−9.48969	−0.799176
$142$	8.72672	0.732330
$143$	3.21517	0.268866
$144$	6.53101	0.544251
$145$	4.31171	0.358068
$146$	−5.98698	−0.495486
$147$	−42.6701	−3.51937
$148$	−5.02966	−0.413435
$149$	−0.518851	−0.0425059	−0.0212530	−	0.999774i	$-0.506766\pi$
$149$	−0.518851	−0.0425059	−0.0212530	+	0.999774i	$0.506766\pi$
$150$	3.08723	0.252072
$151$	3.84848	0.313185	0.156592	−	0.987663i	$-0.449949\pi$
$151$	3.84848	0.313185	0.156592	+	0.987663i	$0.449949\pi$
$152$	7.48372	0.607010
$153$	−32.4618	−2.62438
$154$	14.6710	1.18222
$155$	−1.00000	−0.0803219
$156$	3.08723	0.247176
$157$	−22.1060	−1.76425	−0.882126	−	0.471014i	$-0.843888\pi$
$157$	−22.1060	−1.76425	−0.882126	+	0.471014i	$0.843888\pi$
$158$	15.0878	1.20032
$159$	−10.0439	−0.796532
$160$	−1.00000	−0.0790569
$161$	36.6414	2.88775
$162$	−14.0610	−1.10474
$163$	8.19993	0.642268	0.321134	−	0.947034i	$-0.395936\pi$
$163$	8.19993	0.642268	0.321134	+	0.947034i	$0.395936\pi$
$164$	7.36029	0.574742
$165$	9.92599	0.772737
$166$	−11.1563	−0.865897
$167$	−13.7636	−1.06506	−0.532529	−	0.846411i	$-0.678758\pi$
$167$	−13.7636	−1.06506	−0.532529	+	0.846411i	$0.678758\pi$
$168$	14.0872	1.08685
$169$	1.00000	0.0769231
$170$	4.97041	0.381213
$171$	−48.8762	−3.73766
$172$	−1.97390	−0.150509
$173$	10.5642	0.803178	0.401589	−	0.915820i	$-0.368458\pi$
$173$	10.5642	0.803178	0.401589	+	0.915820i	$0.368458\pi$
$174$	13.3113	1.00912
$175$	4.56306	0.344935
$176$	−3.21517	−0.242353
$177$	12.5934	0.946581
$178$	−2.41549	−0.181049
$179$	−5.48985	−0.410331	−0.205165	−	0.978727i	$-0.565773\pi$
$179$	−5.48985	−0.410331	−0.205165	+	0.978727i	$0.565773\pi$
$180$	6.53101	0.486793
$181$	24.4447	1.81696	0.908481	−	0.417927i	$-0.137243\pi$
$181$	24.4447	1.81696	0.908481	+	0.417927i	$0.137243\pi$
$182$	4.56306	0.338236
$183$	14.7229	1.08835
$184$	−8.03001	−0.591980
$185$	−5.02966	−0.369788
$186$	−3.08723	−0.226367
$187$	15.9807	1.16863
$188$	3.07385	0.224183
$189$	−49.7421	−3.61820
$190$	7.48372	0.542926
$191$	22.1249	1.60090	0.800451	−	0.599399i	$-0.204594\pi$
$191$	22.1249	1.60090	0.800451	+	0.599399i	$0.204594\pi$
$192$	−3.08723	−0.222802
$193$	23.7807	1.71177	0.855887	−	0.517163i	$-0.173012\pi$
$193$	23.7807	1.71177	0.855887	+	0.517163i	$0.173012\pi$
$194$	−13.0192	−0.934726
$195$	3.08723	0.221081
$196$	13.8215	0.987248
$197$	6.56601	0.467809	0.233905	−	0.972260i	$-0.424850\pi$
$197$	6.56601	0.467809	0.233905	+	0.972260i	$0.424850\pi$
$198$	20.9983	1.49229
$199$	−3.00157	−0.212776	−0.106388	−	0.994325i	$-0.533929\pi$
$199$	−3.00157	−0.212776	−0.106388	+	0.994325i	$0.533929\pi$
$200$	−1.00000	−0.0707107
$201$	−37.2232	−2.62552
$202$	−2.95587	−0.207974
$203$	19.6746	1.38089
$204$	15.3448	1.07435
$205$	7.36029	0.514065
$206$	13.6371	0.950144
$207$	52.4441	3.64511
$208$	−1.00000	−0.0693375
$209$	24.0615	1.66437
$210$	14.0872	0.972110
$211$	−8.51518	−0.586210	−0.293105	−	0.956080i	$-0.594688\pi$
$211$	−8.51518	−0.586210	−0.293105	+	0.956080i	$0.594688\pi$
$212$	3.25336	0.223442
$213$	26.9414	1.84599
$214$	−9.05773	−0.619174
$215$	−1.97390	−0.134619
$216$	10.9010	0.741722
$217$	−4.56306	−0.309760
$218$	2.49872	0.169234
$219$	−18.4832	−1.24898
$220$	−3.21517	−0.216767
$221$	4.97041	0.334346
$222$	−15.5277	−1.04215
$223$	−12.2993	−0.823625	−0.411812	−	0.911269i	$-0.635104\pi$
$223$	−12.2993	−0.823625	−0.411812	+	0.911269i	$0.635104\pi$
$224$	−4.56306	−0.304882
$225$	6.53101	0.435400
$226$	−15.1957	−1.01081
$227$	3.23689	0.214840	0.107420	−	0.994214i	$-0.465741\pi$
$227$	3.23689	0.214840	0.107420	+	0.994214i	$0.465741\pi$
$228$	23.1040	1.53010
$229$	0.409560	0.0270645	0.0135322	−	0.999908i	$-0.495692\pi$
$229$	0.409560	0.0270645	0.0135322	+	0.999908i	$0.495692\pi$
$230$	−8.03001	−0.529483
$231$	45.2929	2.98005
$232$	−4.31171	−0.283078
$233$	19.8116	1.29790	0.648952	−	0.760830i	$-0.275208\pi$
$233$	19.8116	1.29790	0.648952	+	0.760830i	$0.275208\pi$
$234$	6.53101	0.426945
$235$	3.07385	0.200516
$236$	−4.07920	−0.265533
$237$	46.5797	3.02568
$238$	22.6803	1.47014
$239$	−6.55827	−0.424220	−0.212110	−	0.977246i	$-0.568033\pi$
$239$	−6.55827	−0.424220	−0.212110	+	0.977246i	$0.568033\pi$
$240$	−3.08723	−0.199280
$241$	−17.7867	−1.14574	−0.572872	−	0.819645i	$-0.694171\pi$
$241$	−17.7867	−1.14574	−0.572872	+	0.819645i	$0.694171\pi$
$242$	0.662654	0.0425970
$243$	−10.7066	−0.686826
$244$	−4.76898	−0.305303
$245$	13.8215	0.883022
$246$	22.7229	1.44876
$247$	7.48372	0.476178
$248$	1.00000	0.0635001
$249$	−34.4421	−2.18268
$250$	−1.00000	−0.0632456
$251$	11.2626	0.710890	0.355445	−	0.934697i	$-0.384329\pi$
$251$	11.2626	0.710890	0.355445	+	0.934697i	$0.384329\pi$
$252$	29.8013	1.87731
$253$	−25.8179	−1.62316
$254$	0.0192539	0.00120810
$255$	15.3448	0.960930
$256$	1.00000	0.0625000
$257$	3.31002	0.206474	0.103237	−	0.994657i	$-0.467080\pi$
$257$	3.31002	0.206474	0.103237	+	0.994657i	$0.467080\pi$
$258$	−6.09390	−0.379389
$259$	−22.9506	−1.42608
$260$	−1.00000	−0.0620174
$261$	28.1598	1.74305
$262$	−16.1700	−0.998986
$263$	10.3177	0.636218	0.318109	−	0.948054i	$-0.396952\pi$
$263$	10.3177	0.636218	0.318109	+	0.948054i	$0.396952\pi$
$264$	−9.92599	−0.610903
$265$	3.25336	0.199852
$266$	34.1486	2.09379
$267$	−7.45718	−0.456372
$268$	12.0571	0.736506
$269$	−9.71501	−0.592335	−0.296167	−	0.955136i	$-0.595709\pi$
$269$	−9.71501	−0.592335	−0.296167	+	0.955136i	$0.595709\pi$
$270$	10.9010	0.663416
$271$	27.8896	1.69417	0.847085	−	0.531457i	$-0.178355\pi$
$271$	27.8896	1.69417	0.847085	+	0.531457i	$0.178355\pi$
$272$	−4.97041	−0.301375
$273$	14.0872	0.852597
$274$	−3.88087	−0.234452
$275$	−3.21517	−0.193882
$276$	−24.7905	−1.49221
$277$	−1.32727	−0.0797479	−0.0398739	−	0.999205i	$-0.512696\pi$
$277$	−1.32727	−0.0797479	−0.0398739	+	0.999205i	$0.512696\pi$
$278$	17.2203	1.03281
$279$	−6.53101	−0.391001
$280$	−4.56306	−0.272695
$281$	−17.8894	−1.06719	−0.533595	−	0.845740i	$-0.679159\pi$
$281$	−17.8894	−1.06719	−0.533595	+	0.845740i	$0.679159\pi$
$282$	9.48969	0.565103
$283$	−16.6152	−0.987673	−0.493837	−	0.869555i	$-0.664406\pi$
$283$	−16.6152	−0.987673	−0.493837	+	0.869555i	$0.664406\pi$
$284$	−8.72672	−0.517835
$285$	23.1040	1.36856
$286$	−3.21517	−0.190117
$287$	33.5854	1.98248
$288$	−6.53101	−0.384843
$289$	7.70498	0.453234
$290$	−4.31171	−0.253193
$291$	−40.1934	−2.35618
$292$	5.98698	0.350362
$293$	−8.62081	−0.503633	−0.251817	−	0.967775i	$-0.581028\pi$
$293$	−8.62081	−0.503633	−0.251817	+	0.967775i	$0.581028\pi$
$294$	42.6701	2.48857
$295$	−4.07920	−0.237500
$296$	5.02966	0.292343
$297$	35.0487	2.03373
$298$	0.518851	0.0300562
$299$	−8.03001	−0.464388
$300$	−3.08723	−0.178241
$301$	−9.00702	−0.519156
$302$	−3.84848	−0.221455
$303$	−9.12547	−0.524244
$304$	−7.48372	−0.429221
$305$	−4.76898	−0.273071
$306$	32.4618	1.85572
$307$	−13.7062	−0.782257	−0.391128	−	0.920336i	$-0.627915\pi$
$307$	−13.7062	−0.782257	−0.391128	+	0.920336i	$0.627915\pi$
$308$	−14.6710	−0.835959
$309$	42.1010	2.39504
$310$	1.00000	0.0567962
$311$	−10.8891	−0.617463	−0.308732	−	0.951149i	$-0.599905\pi$
$311$	−10.8891	−0.617463	−0.308732	+	0.951149i	$0.599905\pi$
$312$	−3.08723	−0.174780
$313$	24.7248	1.39753	0.698764	−	0.715352i	$-0.253734\pi$
$313$	24.7248	1.39753	0.698764	+	0.715352i	$0.253734\pi$
$314$	22.1060	1.24751
$315$	29.8013	1.67912
$316$	−15.0878	−0.848758
$317$	32.2979	1.81403	0.907015	−	0.421099i	$-0.138355\pi$
$317$	32.2979	1.81403	0.907015	+	0.421099i	$0.138355\pi$
$318$	10.0439	0.563233
$319$	−13.8629	−0.776174
$320$	1.00000	0.0559017
$321$	−27.9633	−1.56076
$322$	−36.6414	−2.04195
$323$	37.1972	2.06971
$324$	14.0610	0.781168
$325$	−1.00000	−0.0554700
$326$	−8.19993	−0.454152
$327$	7.71412	0.426592
$328$	−7.36029	−0.406404
$329$	14.0261	0.773286
$330$	−9.92599	−0.546408
$331$	0.888538	0.0488385	0.0244192	−	0.999702i	$-0.492226\pi$
$331$	0.888538	0.0488385	0.0244192	+	0.999702i	$0.492226\pi$
$332$	11.1563	0.612282
$333$	−32.8487	−1.80010
$334$	13.7636	0.753110
$335$	12.0571	0.658751
$336$	−14.0872	−0.768521
$337$	26.2450	1.42966	0.714829	−	0.699300i	$-0.246505\pi$
$337$	26.2450	1.42966	0.714829	+	0.699300i	$0.246505\pi$
$338$	−1.00000	−0.0543928
$339$	−46.9128	−2.54795
$340$	−4.97041	−0.269558
$341$	3.21517	0.174111
$342$	48.8762	2.64292
$343$	31.1268	1.68069
$344$	1.97390	0.106426
$345$	−24.7905	−1.33468
$346$	−10.5642	−0.567933
$347$	12.6180	0.677370	0.338685	−	0.940900i	$-0.390018\pi$
$347$	12.6180	0.677370	0.338685	+	0.940900i	$0.390018\pi$
$348$	−13.3113	−0.713559
$349$	34.0490	1.82260	0.911300	−	0.411743i	$-0.135080\pi$
$349$	34.0490	1.82260	0.911300	+	0.411743i	$0.135080\pi$
$350$	−4.56306	−0.243906
$351$	10.9010	0.581854
$352$	3.21517	0.171369
$353$	−8.74282	−0.465333	−0.232667	−	0.972557i	$-0.574745\pi$
$353$	−8.74282	−0.465333	−0.232667	+	0.972557i	$0.574745\pi$
$354$	−12.5934	−0.669334
$355$	−8.72672	−0.463166
$356$	2.41549	0.128021
$357$	70.0192	3.70581
$358$	5.48985	0.290148
$359$	30.7987	1.62549	0.812746	−	0.582618i	$-0.197972\pi$
$359$	30.7987	1.62549	0.812746	+	0.582618i	$0.197972\pi$
$360$	−6.53101	−0.344214
$361$	37.0061	1.94769
$362$	−24.4447	−1.28479
$363$	2.04577	0.107375
$364$	−4.56306	−0.239169
$365$	5.98698	0.313373
$366$	−14.7229	−0.769581
$367$	−4.85090	−0.253215	−0.126608	−	0.991953i	$-0.540409\pi$
$367$	−4.85090	−0.253215	−0.126608	+	0.991953i	$0.540409\pi$
$368$	8.03001	0.418593
$369$	48.0701	2.50243
$370$	5.02966	0.261479
$371$	14.8453	0.770728
$372$	3.08723	0.160066
$373$	−12.9796	−0.672058	−0.336029	−	0.941852i	$-0.609084\pi$
$373$	−12.9796	−0.672058	−0.336029	+	0.941852i	$0.609084\pi$
$374$	−15.9807	−0.826344
$375$	−3.08723	−0.159424
$376$	−3.07385	−0.158522
$377$	−4.31171	−0.222065
$378$	49.7421	2.55846
$379$	−21.1723	−1.08755	−0.543774	−	0.839232i	$-0.683005\pi$
$379$	−21.1723	−1.08755	−0.543774	+	0.839232i	$0.683005\pi$
$380$	−7.48372	−0.383907
$381$	0.0594414	0.00304527
$382$	−22.1249	−1.13201
$383$	−35.6869	−1.82352	−0.911759	−	0.410726i	$-0.865275\pi$
$383$	−35.6869	−1.82352	−0.911759	+	0.410726i	$0.865275\pi$
$384$	3.08723	0.157545
$385$	−14.6710	−0.747704
$386$	−23.7807	−1.21041
$387$	−12.8916	−0.655315
$388$	13.0192	0.660951
$389$	−12.4822	−0.632872	−0.316436	−	0.948614i	$-0.602486\pi$
$389$	−12.4822	−0.632872	−0.316436	+	0.948614i	$0.602486\pi$
$390$	−3.08723	−0.156328
$391$	−39.9125	−2.01846
$392$	−13.8215	−0.698090
$393$	−49.9205	−2.51816
$394$	−6.56601	−0.330791
$395$	−15.0878	−0.759152
$396$	−20.9983	−1.05521
$397$	35.9686	1.80521	0.902607	−	0.430465i	$-0.141651\pi$
$397$	35.9686	1.80521	0.902607	+	0.430465i	$0.141651\pi$
$398$	3.00157	0.150455
$399$	105.425	5.27784
$400$	1.00000	0.0500000
$401$	8.94069	0.446477	0.223239	−	0.974764i	$-0.428337\pi$
$401$	8.94069	0.446477	0.223239	+	0.974764i	$0.428337\pi$
$402$	37.2232	1.85652
$403$	1.00000	0.0498135
$404$	2.95587	0.147060
$405$	14.0610	0.698698
$406$	−19.6746	−0.976433
$407$	16.1712	0.801578
$408$	−15.3448	−0.759681
$409$	17.9822	0.889165	0.444582	−	0.895738i	$-0.353352\pi$
$409$	17.9822	0.889165	0.444582	+	0.895738i	$0.353352\pi$
$410$	−7.36029	−0.363499
$411$	−11.9812	−0.590987
$412$	−13.6371	−0.671853
$413$	−18.6136	−0.915917
$414$	−52.4441	−2.57749
$415$	11.1563	0.547641
$416$	1.00000	0.0490290
$417$	53.1631	2.60341
$418$	−24.0615	−1.17688
$419$	−23.1337	−1.13016	−0.565078	−	0.825037i	$-0.691154\pi$
$419$	−23.1337	−1.13016	−0.565078	+	0.825037i	$0.691154\pi$
$420$	−14.0872	−0.687386
$421$	−7.46984	−0.364058	−0.182029	−	0.983293i	$-0.558266\pi$
$421$	−7.46984	−0.364058	−0.182029	+	0.983293i	$0.558266\pi$
$422$	8.51518	0.414513
$423$	20.0753	0.976096
$424$	−3.25336	−0.157997
$425$	−4.97041	−0.241100
$426$	−26.9414	−1.30532
$427$	−21.7611	−1.05309
$428$	9.05773	0.437822
$429$	−9.92599	−0.479231
$430$	1.97390	0.0951900
$431$	−11.9436	−0.575303	−0.287651	−	0.957735i	$-0.592874\pi$
$431$	−11.9436	−0.575303	−0.287651	+	0.957735i	$0.592874\pi$
$432$	−10.9010	−0.524477
$433$	−12.4126	−0.596510	−0.298255	−	0.954486i	$-0.596405\pi$
$433$	−12.4126	−0.596510	−0.298255	+	0.954486i	$0.596405\pi$
$434$	4.56306	0.219034
$435$	−13.3113	−0.638226
$436$	−2.49872	−0.119667
$437$	−60.0944	−2.87470
$438$	18.4832	0.883162
$439$	−17.4536	−0.833014	−0.416507	−	0.909132i	$-0.636746\pi$
$439$	−17.4536	−0.833014	−0.416507	+	0.909132i	$0.636746\pi$
$440$	3.21517	0.153277
$441$	90.2682	4.29848
$442$	−4.97041	−0.236418
$443$	20.0733	0.953710	0.476855	−	0.878982i	$-0.341777\pi$
$443$	20.0733	0.953710	0.476855	+	0.878982i	$0.341777\pi$
$444$	15.5277	0.736913
$445$	2.41549	0.114505
$446$	12.2993	0.582391
$447$	1.60181	0.0757632
$448$	4.56306	0.215584
$449$	−15.8399	−0.747529	−0.373765	−	0.927524i	$-0.621933\pi$
$449$	−15.8399	−0.747529	−0.373765	+	0.927524i	$0.621933\pi$
$450$	−6.53101	−0.307875
$451$	−23.6646	−1.11432
$452$	15.1957	0.714748
$453$	−11.8812	−0.558225
$454$	−3.23689	−0.151915
$455$	−4.56306	−0.213919
$456$	−23.1040	−1.08194
$457$	37.9181	1.77373	0.886867	−	0.462025i	$-0.152877\pi$
$457$	37.9181	1.77373	0.886867	+	0.462025i	$0.152877\pi$
$458$	−0.409560	−0.0191375
$459$	54.1826	2.52903
$460$	8.03001	0.374401
$461$	−3.56392	−0.165988	−0.0829941	−	0.996550i	$-0.526448\pi$
$461$	−3.56392	−0.165988	−0.0829941	+	0.996550i	$0.526448\pi$
$462$	−45.2929	−2.10721
$463$	−12.1243	−0.563465	−0.281732	−	0.959493i	$-0.590909\pi$
$463$	−12.1243	−0.563465	−0.281732	+	0.959493i	$0.590909\pi$
$464$	4.31171	0.200166
$465$	3.08723	0.143167
$466$	−19.8116	−0.917756
$467$	−26.7489	−1.23779	−0.618897	−	0.785473i	$-0.712420\pi$
$467$	−26.7489	−1.23779	−0.618897	+	0.785473i	$0.712420\pi$
$468$	−6.53101	−0.301896
$469$	55.0173	2.54046
$470$	−3.07385	−0.141786
$471$	68.2464	3.14463
$472$	4.07920	0.187760
$473$	6.34644	0.291810
$474$	−46.5797	−2.13948
$475$	−7.48372	−0.343377
$476$	−22.6803	−1.03955
$477$	21.2477	0.972866
$478$	6.55827	0.299968
$479$	27.5757	1.25997	0.629983	−	0.776609i	$-0.283062\pi$
$479$	27.5757	1.25997	0.629983	+	0.776609i	$0.283062\pi$
$480$	3.08723	0.140912
$481$	5.02966	0.229333
$482$	17.7867	0.810163
$483$	−113.121	−5.14716
$484$	−0.662654	−0.0301206
$485$	13.0192	0.591173
$486$	10.7066	0.485659
$487$	−16.1937	−0.733807	−0.366904	−	0.930259i	$-0.619582\pi$
$487$	−16.1937	−0.733807	−0.366904	+	0.930259i	$0.619582\pi$
$488$	4.76898	0.215881
$489$	−25.3151	−1.14479
$490$	−13.8215	−0.624391
$491$	26.9070	1.21429	0.607147	−	0.794590i	$-0.292314\pi$
$491$	26.9070	1.21429	0.607147	+	0.794590i	$0.292314\pi$
$492$	−22.7229	−1.02443
$493$	−21.4310	−0.965203
$494$	−7.48372	−0.336708
$495$	−20.9983	−0.943805
$496$	−1.00000	−0.0449013
$497$	−39.8205	−1.78619
$498$	34.4421	1.54339
$499$	37.5719	1.68195	0.840975	−	0.541074i	$-0.181982\pi$
$499$	37.5719	1.68195	0.840975	+	0.541074i	$0.181982\pi$
$500$	1.00000	0.0447214
$501$	42.4914	1.89838
$502$	−11.2626	−0.502675
$503$	−3.13781	−0.139908	−0.0699540	−	0.997550i	$-0.522285\pi$
$503$	−3.13781	−0.139908	−0.0699540	+	0.997550i	$0.522285\pi$
$504$	−29.8013	−1.32746
$505$	2.95587	0.131535
$506$	25.8179	1.14775
$507$	−3.08723	−0.137109
$508$	−0.0192539	−0.000854255 0
$509$	24.3219	1.07805	0.539024	−	0.842290i	$-0.318793\pi$
$509$	24.3219	1.07805	0.539024	+	0.842290i	$0.318793\pi$
$510$	−15.3448	−0.679480
$511$	27.3189	1.20852
$512$	−1.00000	−0.0441942
$513$	81.5803	3.60186
$514$	−3.31002	−0.145999
$515$	−13.6371	−0.600924
$516$	6.09390	0.268269
$517$	−9.88296	−0.434652
$518$	22.9506	1.00839
$519$	−32.6140	−1.43160
$520$	1.00000	0.0438529
$521$	38.6766	1.69445	0.847226	−	0.531232i	$-0.178271\pi$
$521$	38.6766	1.69445	0.847226	+	0.531232i	$0.178271\pi$
$522$	−28.1598	−1.23252
$523$	34.0244	1.48778	0.743892	−	0.668300i	$-0.232978\pi$
$523$	34.0244	1.48778	0.743892	+	0.668300i	$0.232978\pi$
$524$	16.1700	0.706389
$525$	−14.0872	−0.614816
$526$	−10.3177	−0.449874
$527$	4.97041	0.216514
$528$	9.92599	0.431973
$529$	41.4811	1.80353
$530$	−3.25336	−0.141317
$531$	−26.6413	−1.15613
$532$	−34.1486	−1.48053
$533$	−7.36029	−0.318809
$534$	7.45718	0.322704
$535$	9.05773	0.391600
$536$	−12.0571	−0.520789
$537$	16.9484	0.731379
$538$	9.71501	0.418844
$539$	−44.4385	−1.91410
$540$	−10.9010	−0.469106
$541$	33.2816	1.43089	0.715444	−	0.698670i	$-0.246224\pi$
$541$	33.2816	1.43089	0.715444	+	0.698670i	$0.246224\pi$
$542$	−27.8896	−1.19796
$543$	−75.4665	−3.23858
$544$	4.97041	0.213105
$545$	−2.49872	−0.107033
$546$	−14.0872	−0.602877
$547$	15.4486	0.660536	0.330268	−	0.943887i	$-0.392861\pi$
$547$	15.4486	0.660536	0.330268	+	0.943887i	$0.392861\pi$
$548$	3.88087	0.165783
$549$	−31.1462	−1.32929
$550$	3.21517	0.137095
$551$	−32.2677	−1.37465
$552$	24.7905	1.05515
$553$	−68.8467	−2.92766
$554$	1.32727	0.0563902
$555$	15.5277	0.659115
$556$	−17.2203	−0.730304
$557$	−6.55664	−0.277814	−0.138907	−	0.990305i	$-0.544359\pi$
$557$	−6.55664	−0.277814	−0.138907	+	0.990305i	$0.544359\pi$
$558$	6.53101	0.276480
$559$	1.97390	0.0834872
$560$	4.56306	0.192824
$561$	−49.3363	−2.08298
$562$	17.8894	0.754617
$563$	13.4097	0.565152	0.282576	−	0.959245i	$-0.408811\pi$
$563$	13.4097	0.565152	0.282576	+	0.959245i	$0.408811\pi$
$564$	−9.48969	−0.399588
$565$	15.1957	0.639290
$566$	16.6152	0.698391
$567$	64.1613	2.69452
$568$	8.72672	0.366165
$569$	−9.01973	−0.378127	−0.189063	−	0.981965i	$-0.560545\pi$
$569$	−9.01973	−0.378127	−0.189063	+	0.981965i	$0.560545\pi$
$570$	−23.1040	−0.967719
$571$	−10.1771	−0.425898	−0.212949	−	0.977063i	$-0.568307\pi$
$571$	−10.1771	−0.425898	−0.212949	+	0.977063i	$0.568307\pi$
$572$	3.21517	0.134433
$573$	−68.3047	−2.85347
$574$	−33.5854	−1.40183
$575$	8.03001	0.334875
$576$	6.53101	0.272125
$577$	−8.48115	−0.353075	−0.176537	−	0.984294i	$-0.556490\pi$
$577$	−8.48115	−0.353075	−0.176537	+	0.984294i	$0.556490\pi$
$578$	−7.70498	−0.320485
$579$	−73.4166	−3.05109
$580$	4.31171	0.179034
$581$	50.9068	2.11197
$582$	40.1934	1.66607
$583$	−10.4601	−0.433214
$584$	−5.98698	−0.247743
$585$	−6.53101	−0.270024
$586$	8.62081	0.356123
$587$	30.3243	1.25162	0.625809	−	0.779976i	$-0.284769\pi$
$587$	30.3243	1.25162	0.625809	+	0.779976i	$0.284769\pi$
$588$	−42.6701	−1.75969
$589$	7.48372	0.308361
$590$	4.07920	0.167938
$591$	−20.2708	−0.833830
$592$	−5.02966	−0.206718
$593$	−20.0303	−0.822547	−0.411273	−	0.911512i	$-0.634916\pi$
$593$	−20.0303	−0.822547	−0.411273	+	0.911512i	$0.634916\pi$
$594$	−35.0487	−1.43807
$595$	−22.6803	−0.929800
$596$	−0.518851	−0.0212530
$597$	9.26656	0.379255
$598$	8.03001	0.328372
$599$	−34.7036	−1.41795	−0.708976	−	0.705233i	$-0.750843\pi$
$599$	−34.7036	−1.41795	−0.708976	+	0.705233i	$0.750843\pi$
$600$	3.08723	0.126036
$601$	16.9332	0.690718	0.345359	−	0.938471i	$-0.387757\pi$
$601$	16.9332	0.690718	0.345359	+	0.938471i	$0.387757\pi$
$602$	9.00702	0.367099
$603$	78.7452	3.20675
$604$	3.84848	0.156592
$605$	−0.662654	−0.0269407
$606$	9.12547	0.370697
$607$	6.83045	0.277239	0.138620	−	0.990346i	$-0.455733\pi$
$607$	6.83045	0.277239	0.138620	+	0.990346i	$0.455733\pi$
$608$	7.48372	0.303505
$609$	−60.7400	−2.46131
$610$	4.76898	0.193090
$611$	−3.07385	−0.124355
$612$	−32.4618	−1.31219
$613$	34.7670	1.40423	0.702113	−	0.712065i	$-0.252240\pi$
$613$	34.7670	1.40423	0.702113	+	0.712065i	$0.252240\pi$
$614$	13.7062	0.553139
$615$	−22.7229	−0.916277
$616$	14.6710	0.591112
$617$	−24.5227	−0.987247	−0.493623	−	0.869676i	$-0.664328\pi$
$617$	−24.5227	−0.987247	−0.493623	+	0.869676i	$0.664328\pi$
$618$	−42.1010	−1.69355
$619$	30.6666	1.23260	0.616298	−	0.787513i	$-0.288632\pi$
$619$	30.6666	1.23260	0.616298	+	0.787513i	$0.288632\pi$
$620$	−1.00000	−0.0401610
$621$	−87.5355	−3.51268
$622$	10.8891	0.436613
$623$	11.0220	0.441588
$624$	3.08723	0.123588
$625$	1.00000	0.0400000
$626$	−24.7248	−0.988202
$627$	−74.2833	−2.96659
$628$	−22.1060	−0.882126
$629$	24.9995	0.996794
$630$	−29.8013	−1.18731
$631$	15.6583	0.623347	0.311673	−	0.950189i	$-0.399111\pi$
$631$	15.6583	0.623347	0.311673	+	0.950189i	$0.399111\pi$
$632$	15.0878	0.600162
$633$	26.2884	1.04487
$634$	−32.2979	−1.28271
$635$	−0.0192539	−0.000764069 0
$636$	−10.0439	−0.398266
$637$	−13.8215	−0.547627
$638$	13.8629	0.548838
$639$	−56.9942	−2.25466
$640$	−1.00000	−0.0395285
$641$	19.6311	0.775384	0.387692	−	0.921789i	$-0.373272\pi$
$641$	19.6311	0.775384	0.387692	+	0.921789i	$0.373272\pi$
$642$	27.9633	1.10362
$643$	39.8315	1.57080	0.785401	−	0.618987i	$-0.212457\pi$
$643$	39.8315	1.57080	0.785401	+	0.618987i	$0.212457\pi$
$644$	36.6414	1.44387
$645$	6.09390	0.239947
$646$	−37.1972	−1.46350
$647$	−29.9534	−1.17759	−0.588795	−	0.808282i	$-0.700398\pi$
$647$	−29.9534	−1.17759	−0.588795	+	0.808282i	$0.700398\pi$
$648$	−14.0610	−0.552369
$649$	13.1153	0.514822
$650$	1.00000	0.0392232
$651$	14.0872	0.552122
$652$	8.19993	0.321134
$653$	29.0487	1.13676	0.568381	−	0.822766i	$-0.307570\pi$
$653$	29.0487	1.13676	0.568381	+	0.822766i	$0.307570\pi$
$654$	−7.71412	−0.301646
$655$	16.1700	0.631814
$656$	7.36029	0.287371
$657$	39.1010	1.52548
$658$	−14.0261	−0.546796
$659$	−12.0778	−0.470484	−0.235242	−	0.971937i	$-0.575588\pi$
$659$	−12.0778	−0.470484	−0.235242	+	0.971937i	$0.575588\pi$
$660$	9.92599	0.386369
$661$	−37.1064	−1.44327	−0.721636	−	0.692273i	$-0.756609\pi$
$661$	−37.1064	−1.44327	−0.721636	+	0.692273i	$0.756609\pi$
$662$	−0.888538	−0.0345340
$663$	−15.3448	−0.595943
$664$	−11.1563	−0.432949
$665$	−34.1486	−1.32423
$666$	32.8487	1.27286
$667$	34.6231	1.34061
$668$	−13.7636	−0.532529
$669$	37.9709	1.46804
$670$	−12.0571	−0.465807
$671$	15.3331	0.591928
$672$	14.0872	0.543426
$673$	20.3136	0.783030	0.391515	−	0.920172i	$-0.371951\pi$
$673$	20.3136	0.783030	0.391515	+	0.920172i	$0.371951\pi$
$674$	−26.2450	−1.01092
$675$	−10.9010	−0.419581
$676$	1.00000	0.0384615
$677$	33.2322	1.27722	0.638608	−	0.769532i	$-0.279511\pi$
$677$	33.2322	1.27722	0.638608	+	0.769532i	$0.279511\pi$
$678$	46.9128	1.80168
$679$	59.4075	2.27985
$680$	4.97041	0.190607
$681$	−9.99305	−0.382934
$682$	−3.21517	−0.123115
$683$	−31.6581	−1.21136	−0.605681	−	0.795708i	$-0.707099\pi$
$683$	−31.6581	−1.21136	−0.605681	+	0.795708i	$0.707099\pi$
$684$	−48.8762	−1.86883
$685$	3.88087	0.148281
$686$	−31.1268	−1.18843
$687$	−1.26441	−0.0482401
$688$	−1.97390	−0.0752543
$689$	−3.25336	−0.123943
$690$	24.7905	0.943759
$691$	−23.2145	−0.883122	−0.441561	−	0.897231i	$-0.645575\pi$
$691$	−23.2145	−0.883122	−0.441561	+	0.897231i	$0.645575\pi$
$692$	10.5642	0.401589
$693$	−95.8165	−3.63977
$694$	−12.6180	−0.478973
$695$	−17.2203	−0.653204
$696$	13.3113	0.504562
$697$	−36.5837	−1.38570
$698$	−34.0490	−1.28877
$699$	−61.1631	−2.31340
$700$	4.56306	0.172467
$701$	−0.738392	−0.0278887	−0.0139443	−	0.999903i	$-0.504439\pi$
$701$	−0.738392	−0.0278887	−0.0139443	+	0.999903i	$0.504439\pi$
$702$	−10.9010	−0.411433
$703$	37.6405	1.41964
$704$	−3.21517	−0.121176
$705$	−9.48969	−0.357402
$706$	8.74282	0.329040
$707$	13.4878	0.507261
$708$	12.5934	0.473291
$709$	−19.3002	−0.724836	−0.362418	−	0.932016i	$-0.618049\pi$
$709$	−19.3002	−0.724836	−0.362418	+	0.932016i	$0.618049\pi$
$710$	8.72672	0.327508
$711$	−98.5388	−3.69549
$712$	−2.41549	−0.0905243
$713$	−8.03001	−0.300726
$714$	−70.0192	−2.62040
$715$	3.21517	0.120241
$716$	−5.48985	−0.205165
$717$	20.2469	0.756135
$718$	−30.7987	−1.14940
$719$	9.28150	0.346141	0.173071	−	0.984909i	$-0.444631\pi$
$719$	9.28150	0.346141	0.173071	+	0.984909i	$0.444631\pi$
$720$	6.53101	0.243396
$721$	−62.2270	−2.31745
$722$	−37.0061	−1.37722
$723$	54.9118	2.04219
$724$	24.4447	0.908481
$725$	4.31171	0.160133
$726$	−2.04577	−0.0759255
$727$	25.1892	0.934216	0.467108	−	0.884200i	$-0.345296\pi$
$727$	25.1892	0.934216	0.467108	+	0.884200i	$0.345296\pi$
$728$	4.56306	0.169118
$729$	−9.12947	−0.338129
$730$	−5.98698	−0.221588
$731$	9.81110	0.362877
$732$	14.7229	0.544176
$733$	−22.0934	−0.816040	−0.408020	−	0.912973i	$-0.633781\pi$
$733$	−22.0934	−0.816040	−0.408020	+	0.912973i	$0.633781\pi$
$734$	4.85090	0.179050
$735$	−42.6701	−1.57391
$736$	−8.03001	−0.295990
$737$	−38.7658	−1.42796
$738$	−48.0701	−1.76948
$739$	34.3474	1.26349	0.631745	−	0.775176i	$-0.282339\pi$
$739$	34.3474	1.26349	0.631745	+	0.775176i	$0.282339\pi$
$740$	−5.02966	−0.184894
$741$	−23.1040	−0.848746
$742$	−14.8453	−0.544987
$743$	43.2499	1.58668	0.793342	−	0.608776i	$-0.208339\pi$
$743$	43.2499	1.58668	0.793342	+	0.608776i	$0.208339\pi$
$744$	−3.08723	−0.113183
$745$	−0.518851	−0.0190092
$746$	12.9796	0.475217
$747$	72.8619	2.66588
$748$	15.9807	0.584314
$749$	41.3309	1.51020
$750$	3.08723	0.112730
$751$	29.6359	1.08143	0.540715	−	0.841206i	$-0.318154\pi$
$751$	29.6359	1.08143	0.540715	+	0.841206i	$0.318154\pi$
$752$	3.07385	0.112092
$753$	−34.7703	−1.26710
$754$	4.31171	0.157023
$755$	3.84848	0.140061
$756$	−49.7421	−1.80910
$757$	−40.2534	−1.46303	−0.731517	−	0.681823i	$-0.761187\pi$
$757$	−40.2534	−1.46303	−0.731517	+	0.681823i	$0.761187\pi$
$758$	21.1723	0.769012
$759$	79.7058	2.89314
$760$	7.48372	0.271463
$761$	7.07686	0.256536	0.128268	−	0.991740i	$-0.459058\pi$
$761$	7.07686	0.256536	0.128268	+	0.991740i	$0.459058\pi$
$762$	−0.0594414	−0.00215333
$763$	−11.4018	−0.412772
$764$	22.1249	0.800451
$765$	−32.4618	−1.17366
$766$	35.6869	1.28942
$767$	4.07920	0.147291
$768$	−3.08723	−0.111401
$769$	0.172015	0.00620301	0.00310150	−	0.999995i	$-0.499013\pi$
$769$	0.172015	0.00620301	0.00310150	+	0.999995i	$0.499013\pi$
$770$	14.6710	0.528707
$771$	−10.2188	−0.368021
$772$	23.7807	0.855887
$773$	23.4797	0.844507	0.422254	−	0.906478i	$-0.361239\pi$
$773$	23.4797	0.844507	0.422254	+	0.906478i	$0.361239\pi$
$774$	12.8916	0.463378
$775$	−1.00000	−0.0359211
$776$	−13.0192	−0.467363
$777$	70.8539	2.54187
$778$	12.4822	0.447508
$779$	−55.0823	−1.97353
$780$	3.08723	0.110541
$781$	28.0579	1.00399
$782$	39.9125	1.42727
$783$	−47.0022	−1.67972
$784$	13.8215	0.493624
$785$	−22.1060	−0.788997
$786$	49.9205	1.78061
$787$	−32.5051	−1.15868	−0.579341	−	0.815085i	$-0.696690\pi$
$787$	−32.5051	−1.15868	−0.579341	+	0.815085i	$0.696690\pi$
$788$	6.56601	0.233905
$789$	−31.8532	−1.13400
$790$	15.0878	0.536801
$791$	69.3390	2.46541
$792$	20.9983	0.746143
$793$	4.76898	0.169351
$794$	−35.9686	−1.27648
$795$	−10.0439	−0.356220
$796$	−3.00157	−0.106388
$797$	−28.5007	−1.00955	−0.504773	−	0.863252i	$-0.668424\pi$
$797$	−28.5007	−1.00955	−0.504773	+	0.863252i	$0.668424\pi$
$798$	−105.425	−3.73200
$799$	−15.2783	−0.540507
$800$	−1.00000	−0.0353553
$801$	15.7756	0.557403
$802$	−8.94069	−0.315707
$803$	−19.2492	−0.679289
$804$	−37.2232	−1.31276
$805$	36.6414	1.29144
$806$	−1.00000	−0.0352235
$807$	29.9925	1.05579
$808$	−2.95587	−0.103987
$809$	23.9358	0.841538	0.420769	−	0.907168i	$-0.361760\pi$
$809$	23.9358	0.841538	0.420769	+	0.907168i	$0.361760\pi$
$810$	−14.0610	−0.494054
$811$	50.6890	1.77993	0.889966	−	0.456026i	$-0.150728\pi$
$811$	50.6890	1.77993	0.889966	+	0.456026i	$0.150728\pi$
$812$	19.6746	0.690443
$813$	−86.1016	−3.01971
$814$	−16.1712	−0.566801
$815$	8.19993	0.287231
$816$	15.3448	0.537176
$817$	14.7721	0.516811
$818$	−17.9822	−0.628734
$819$	−29.8013	−1.04134
$820$	7.36029	0.257032
$821$	19.4593	0.679133	0.339566	−	0.940582i	$-0.389720\pi$
$821$	19.4593	0.679133	0.339566	+	0.940582i	$0.389720\pi$
$822$	11.9812	0.417891
$823$	−31.7014	−1.10504	−0.552521	−	0.833499i	$-0.686334\pi$
$823$	−31.7014	−1.10504	−0.552521	+	0.833499i	$0.686334\pi$
$824$	13.6371	0.475072
$825$	9.92599	0.345579
$826$	18.6136	0.647651
$827$	41.6560	1.44852	0.724260	−	0.689527i	$-0.242181\pi$
$827$	41.6560	1.44852	0.724260	+	0.689527i	$0.242181\pi$
$828$	52.4441	1.82256
$829$	39.8459	1.38391	0.691953	−	0.721942i	$-0.256750\pi$
$829$	39.8459	1.38391	0.691953	+	0.721942i	$0.256750\pi$
$830$	−11.1563	−0.387241
$831$	4.09759	0.142144
$832$	−1.00000	−0.0346688
$833$	−68.6984	−2.38026
$834$	−53.1631	−1.84089
$835$	−13.7636	−0.476309
$836$	24.0615	0.832183
$837$	10.9010	0.376795
$838$	23.1337	0.799141
$839$	42.3459	1.46194	0.730972	−	0.682408i	$-0.239067\pi$
$839$	42.3459	1.46194	0.730972	+	0.682408i	$0.239067\pi$
$840$	14.0872	0.486055
$841$	−10.4091	−0.358935
$842$	7.46984	0.257428
$843$	55.2286	1.90218
$844$	−8.51518	−0.293105
$845$	1.00000	0.0344010
$846$	−20.0753	−0.690204
$847$	−3.02373	−0.103896
$848$	3.25336	0.111721
$849$	51.2951	1.76044
$850$	4.97041	0.170484
$851$	−40.3882	−1.38449
$852$	26.9414	0.922997
$853$	−44.1156	−1.51049	−0.755244	−	0.655443i	$-0.772482\pi$
$853$	−44.1156	−1.51049	−0.755244	+	0.655443i	$0.772482\pi$
$854$	21.7611	0.744650
$855$	−48.8762	−1.67153
$856$	−9.05773	−0.309587
$857$	−15.7546	−0.538167	−0.269084	−	0.963117i	$-0.586721\pi$
$857$	−15.7546	−0.538167	−0.269084	+	0.963117i	$0.586721\pi$
$858$	9.92599	0.338868
$859$	24.1209	0.822995	0.411498	−	0.911411i	$-0.365006\pi$
$859$	24.1209	0.822995	0.411498	+	0.911411i	$0.365006\pi$
$860$	−1.97390	−0.0673095
$861$	−103.686	−3.53361
$862$	11.9436	0.406801
$863$	−24.3813	−0.829949	−0.414974	−	0.909833i	$-0.636209\pi$
$863$	−24.3813	−0.829949	−0.414974	+	0.909833i	$0.636209\pi$
$864$	10.9010	0.370861
$865$	10.5642	0.359192
$866$	12.4126	0.421796
$867$	−23.7871	−0.807851
$868$	−4.56306	−0.154880
$869$	48.5101	1.64559
$870$	13.3113	0.451294
$871$	−12.0571	−0.408540
$872$	2.49872	0.0846172
$873$	85.0287	2.87778
$874$	60.0944	2.03272
$875$	4.56306	0.154259
$876$	−18.4832	−0.624490
$877$	−38.9006	−1.31358	−0.656790	−	0.754073i	$-0.728086\pi$
$877$	−38.9006	−1.31358	−0.656790	+	0.754073i	$0.728086\pi$
$878$	17.4536	0.589030
$879$	26.6145	0.897683
$880$	−3.21517	−0.108384
$881$	−46.0325	−1.55087	−0.775437	−	0.631424i	$-0.782471\pi$
$881$	−46.0325	−1.55087	−0.775437	+	0.631424i	$0.782471\pi$
$882$	−90.2682	−3.03949
$883$	−14.8849	−0.500917	−0.250458	−	0.968127i	$-0.580581\pi$
$883$	−14.8849	−0.500917	−0.250458	+	0.968127i	$0.580581\pi$
$884$	4.97041	0.167173
$885$	12.5934	0.423324
$886$	−20.0733	−0.674375
$887$	26.9921	0.906305	0.453153	−	0.891433i	$-0.350299\pi$
$887$	26.9921	0.906305	0.453153	+	0.891433i	$0.350299\pi$
$888$	−15.5277	−0.521076
$889$	−0.0878568	−0.00294662
$890$	−2.41549	−0.0809674
$891$	−45.2087	−1.51455
$892$	−12.2993	−0.411812
$893$	−23.0038	−0.769793
$894$	−1.60181	−0.0535727
$895$	−5.48985	−0.183505
$896$	−4.56306	−0.152441
$897$	24.7905	0.827731
$898$	15.8399	0.528583
$899$	−4.31171	−0.143804
$900$	6.53101	0.217700
$901$	−16.1705	−0.538719
$902$	23.6646	0.787945
$903$	27.8068	0.925352
$904$	−15.1957	−0.505403
$905$	24.4447	0.812570
$906$	11.8812	0.394725
$907$	−32.9435	−1.09387	−0.546935	−	0.837175i	$-0.684206\pi$
$907$	−32.9435	−1.09387	−0.546935	+	0.837175i	$0.684206\pi$
$908$	3.23689	0.107420
$909$	19.3048	0.640301
$910$	4.56306	0.151264
$911$	−25.6798	−0.850809	−0.425405	−	0.905003i	$-0.639868\pi$
$911$	−25.6798	−0.850809	−0.425405	+	0.905003i	$0.639868\pi$
$912$	23.1040	0.765049
$913$	−35.8695	−1.18711
$914$	−37.9181	−1.25422
$915$	14.7229	0.486726
$916$	0.409560	0.0135322
$917$	73.7846	2.43658
$918$	−54.1826	−1.78829
$919$	12.5018	0.412395	0.206198	−	0.978510i	$-0.433891\pi$
$919$	12.5018	0.412395	0.206198	+	0.978510i	$0.433891\pi$
$920$	−8.03001	−0.264742
$921$	42.3144	1.39431
$922$	3.56392	0.117371
$923$	8.72672	0.287243
$924$	45.2929	1.49003
$925$	−5.02966	−0.165374
$926$	12.1243	0.398430
$927$	−89.0642	−2.92525
$928$	−4.31171	−0.141539
$929$	−17.5353	−0.575315	−0.287657	−	0.957733i	$-0.592876\pi$
$929$	−17.5353	−0.575315	−0.287657	+	0.957733i	$0.592876\pi$
$930$	−3.08723	−0.101234
$931$	−103.436	−3.38998
$932$	19.8116	0.648952
$933$	33.6171	1.10058
$934$	26.7489	0.875252
$935$	15.9807	0.522626
$936$	6.53101	0.213473
$937$	3.68841	0.120495	0.0602476	−	0.998183i	$-0.480811\pi$
$937$	3.68841	0.120495	0.0602476	+	0.998183i	$0.480811\pi$
$938$	−55.0173	−1.79638
$939$	−76.3312	−2.49098
$940$	3.07385	0.100258
$941$	10.7496	0.350426	0.175213	−	0.984531i	$-0.443939\pi$
$941$	10.7496	0.350426	0.175213	+	0.984531i	$0.443939\pi$
$942$	−68.2464	−2.22359
$943$	59.1032	1.92467
$944$	−4.07920	−0.132767
$945$	−49.7421	−1.61811
$946$	−6.34644	−0.206341
$947$	−27.2657	−0.886016	−0.443008	−	0.896518i	$-0.646089\pi$
$947$	−27.2657	−0.886016	−0.443008	+	0.896518i	$0.646089\pi$
$948$	46.5797	1.51284
$949$	−5.98698	−0.194346
$950$	7.48372	0.242804
$951$	−99.7111	−3.23335
$952$	22.6803	0.735071
$953$	36.5079	1.18261	0.591303	−	0.806449i	$-0.298614\pi$
$953$	36.5079	1.18261	0.591303	+	0.806449i	$0.298614\pi$
$954$	−21.2477	−0.687920
$955$	22.1249	0.715945
$956$	−6.55827	−0.212110
$957$	42.7980	1.38346
$958$	−27.5757	−0.890930
$959$	17.7086	0.571842
$960$	−3.08723	−0.0996400
$961$	1.00000	0.0322581
$962$	−5.02966	−0.162163
$963$	59.1561	1.90628
$964$	−17.7867	−0.572872
$965$	23.7807	0.765529
$966$	113.121	3.63959
$967$	−31.5740	−1.01535	−0.507676	−	0.861548i	$-0.669495\pi$
$967$	−31.5740	−1.01535	−0.507676	+	0.861548i	$0.669495\pi$
$968$	0.662654	0.0212985
$969$	−114.836	−3.68907
$970$	−13.0192	−0.418022
$971$	21.0323	0.674958	0.337479	−	0.941333i	$-0.390426\pi$
$971$	21.0323	0.674958	0.337479	+	0.941333i	$0.390426\pi$
$972$	−10.7066	−0.343413
$973$	−78.5773	−2.51907
$974$	16.1937	0.518880
$975$	3.08723	0.0988706
$976$	−4.76898	−0.152651
$977$	41.6187	1.33150	0.665751	−	0.746174i	$-0.268111\pi$
$977$	41.6187	1.33150	0.665751	+	0.746174i	$0.268111\pi$
$978$	25.3151	0.809487
$979$	−7.76622	−0.248209
$980$	13.8215	0.441511
$981$	−16.3191	−0.521030
$982$	−26.9070	−0.858635
$983$	−5.79313	−0.184772	−0.0923860	−	0.995723i	$-0.529449\pi$
$983$	−5.79313	−0.184772	−0.0923860	+	0.995723i	$0.529449\pi$
$984$	22.7229	0.724380
$985$	6.56601	0.209211
$986$	21.4310	0.682502
$987$	−43.3020	−1.37832
$988$	7.48372	0.238089
$989$	−15.8505	−0.504015
$990$	20.9983	0.667371
$991$	−42.9211	−1.36343	−0.681717	−	0.731616i	$-0.738766\pi$
$991$	−42.9211	−1.36343	−0.681717	+	0.731616i	$0.738766\pi$
$992$	1.00000	0.0317500
$993$	−2.74312	−0.0870504
$994$	39.8205	1.26303
$995$	−3.00157	−0.0951563
$996$	−34.4421	−1.09134
$997$	55.0370	1.74304	0.871519	−	0.490362i	$-0.163135\pi$
$997$	55.0370	1.74304	0.871519	+	0.490362i	$0.163135\pi$
$998$	−37.5719	−1.18932
$999$	54.8285	1.73470

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4030.2.a.l.1.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4030.2.a.l.1.1	✓	8	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:	\( N \)	\(=\)	\( 4030 = 2 \cdot 5 \cdot 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4030.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -3.08723 q^{3} +1.00000 q^{4} +1.00000 q^{5} +3.08723 q^{6} +4.56306 q^{7} -1.00000 q^{8} +6.53101 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -3.08723 q^{3} +1.00000 q^{4} +1.00000 q^{5} +3.08723 q^{6} +4.56306 q^{7} -1.00000 q^{8} +6.53101 q^{9} -1.00000 q^{10} -3.21517 q^{11} -3.08723 q^{12} -1.00000 q^{13} -4.56306 q^{14} -3.08723 q^{15} +1.00000 q^{16} -4.97041 q^{17} -6.53101 q^{18} -7.48372 q^{19} +1.00000 q^{20} -14.0872 q^{21} +3.21517 q^{22} +8.03001 q^{23} +3.08723 q^{24} +1.00000 q^{25} +1.00000 q^{26} -10.9010 q^{27} +4.56306 q^{28} +4.31171 q^{29} +3.08723 q^{30} -1.00000 q^{31} -1.00000 q^{32} +9.92599 q^{33} +4.97041 q^{34} +4.56306 q^{35} +6.53101 q^{36} -5.02966 q^{37} +7.48372 q^{38} +3.08723 q^{39} -1.00000 q^{40} +7.36029 q^{41} +14.0872 q^{42} -1.97390 q^{43} -3.21517 q^{44} +6.53101 q^{45} -8.03001 q^{46} +3.07385 q^{47} -3.08723 q^{48} +13.8215 q^{49} -1.00000 q^{50} +15.3448 q^{51} -1.00000 q^{52} +3.25336 q^{53} +10.9010 q^{54} -3.21517 q^{55} -4.56306 q^{56} +23.1040 q^{57} -4.31171 q^{58} -4.07920 q^{59} -3.08723 q^{60} -4.76898 q^{61} +1.00000 q^{62} +29.8013 q^{63} +1.00000 q^{64} -1.00000 q^{65} -9.92599 q^{66} +12.0571 q^{67} -4.97041 q^{68} -24.7905 q^{69} -4.56306 q^{70} -8.72672 q^{71} -6.53101 q^{72} +5.98698 q^{73} +5.02966 q^{74} -3.08723 q^{75} -7.48372 q^{76} -14.6710 q^{77} -3.08723 q^{78} -15.0878 q^{79} +1.00000 q^{80} +14.0610 q^{81} -7.36029 q^{82} +11.1563 q^{83} -14.0872 q^{84} -4.97041 q^{85} +1.97390 q^{86} -13.3113 q^{87} +3.21517 q^{88} +2.41549 q^{89} -6.53101 q^{90} -4.56306 q^{91} +8.03001 q^{92} +3.08723 q^{93} -3.07385 q^{94} -7.48372 q^{95} +3.08723 q^{96} +13.0192 q^{97} -13.8215 q^{98} -20.9983 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} + q^{3} + 8 q^{4} + 8 q^{5} - q^{6} + 11 q^{7} - 8 q^{8} + 9 q^{9}+O(q^{10}) \) 8 * q - 8 * q^2 + q^3 + 8 * q^4 + 8 * q^5 - q^6 + 11 * q^7 - 8 * q^8 + 9 * q^9	\( 8 q - 8 q^{2} + q^{3} + 8 q^{4} + 8 q^{5} - q^{6} + 11 q^{7} - 8 q^{8} + 9 q^{9} - 8 q^{10} + q^{12} - 8 q^{13} - 11 q^{14} + q^{15} + 8 q^{16} + 7 q^{17} - 9 q^{18} - 2 q^{19} + 8 q^{20} + q^{21} + 8 q^{23} - q^{24} + 8 q^{25} + 8 q^{26} + 7 q^{27} + 11 q^{28} - q^{29} - q^{30} - 8 q^{31} - 8 q^{32} + 14 q^{33} - 7 q^{34} + 11 q^{35} + 9 q^{36} + q^{37} + 2 q^{38} - q^{39} - 8 q^{40} + 16 q^{41} - q^{42} + 3 q^{43} + 9 q^{45} - 8 q^{46} + 29 q^{47} + q^{48} + 11 q^{49} - 8 q^{50} + 11 q^{51} - 8 q^{52} + 22 q^{53} - 7 q^{54} - 11 q^{56} + 33 q^{57} + q^{58} - 8 q^{59} + q^{60} + 4 q^{61} + 8 q^{62} + 38 q^{63} + 8 q^{64} - 8 q^{65} - 14 q^{66} + 28 q^{67} + 7 q^{68} - 42 q^{69} - 11 q^{70} + 4 q^{71} - 9 q^{72} + 39 q^{73} - q^{74} + q^{75} - 2 q^{76} + 11 q^{77} + q^{78} - 16 q^{79} + 8 q^{80} + 32 q^{81} - 16 q^{82} + 25 q^{83} + q^{84} + 7 q^{85} - 3 q^{86} + 13 q^{87} + 21 q^{89} - 9 q^{90} - 11 q^{91} + 8 q^{92} - q^{93} - 29 q^{94} - 2 q^{95} - q^{96} + 28 q^{97} - 11 q^{98} + 23 q^{99}+O(q^{100}) \) 8 * q - 8 * q^2 + q^3 + 8 * q^4 + 8 * q^5 - q^6 + 11 * q^7 - 8 * q^8 + 9 * q^9 - 8 * q^10 + q^12 - 8 * q^13 - 11 * q^14 + q^15 + 8 * q^16 + 7 * q^17 - 9 * q^18 - 2 * q^19 + 8 * q^20 + q^21 + 8 * q^23 - q^24 + 8 * q^25 + 8 * q^26 + 7 * q^27 + 11 * q^28 - q^29 - q^30 - 8 * q^31 - 8 * q^32 + 14 * q^33 - 7 * q^34 + 11 * q^35 + 9 * q^36 + q^37 + 2 * q^38 - q^39 - 8 * q^40 + 16 * q^41 - q^42 + 3 * q^43 + 9 * q^45 - 8 * q^46 + 29 * q^47 + q^48 + 11 * q^49 - 8 * q^50 + 11 * q^51 - 8 * q^52 + 22 * q^53 - 7 * q^54 - 11 * q^56 + 33 * q^57 + q^58 - 8 * q^59 + q^60 + 4 * q^61 + 8 * q^62 + 38 * q^63 + 8 * q^64 - 8 * q^65 - 14 * q^66 + 28 * q^67 + 7 * q^68 - 42 * q^69 - 11 * q^70 + 4 * q^71 - 9 * q^72 + 39 * q^73 - q^74 + q^75 - 2 * q^76 + 11 * q^77 + q^78 - 16 * q^79 + 8 * q^80 + 32 * q^81 - 16 * q^82 + 25 * q^83 + q^84 + 7 * q^85 - 3 * q^86 + 13 * q^87 + 21 * q^89 - 9 * q^90 - 11 * q^91 + 8 * q^92 - q^93 - 29 * q^94 - 2 * q^95 - q^96 + 28 * q^97 - 11 * q^98 + 23 * q^99

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