LMFDB - Embedded newform 8031.2.a.d.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8031,2,Mod(1,8031)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8031, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8031.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8031 = 3 \cdot 2677 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8031.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:	$64.1278578633$
Analytic rank:	$0$
Dimension:	$132$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Character	$\chi$	$=$	8031.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.55916 q^{2} +1.00000 q^{3} +4.54932 q^{4} -0.940256 q^{5} -2.55916 q^{6} +2.24269 q^{7} -6.52413 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.55916 q^{2} +1.00000 q^{3} +4.54932 q^{4} -0.940256 q^{5} -2.55916 q^{6} +2.24269 q^{7} -6.52413 q^{8} +1.00000 q^{9} +2.40627 q^{10} +3.39107 q^{11} +4.54932 q^{12} -0.371049 q^{13} -5.73942 q^{14} -0.940256 q^{15} +7.59768 q^{16} -2.95772 q^{17} -2.55916 q^{18} -2.82645 q^{19} -4.27752 q^{20} +2.24269 q^{21} -8.67832 q^{22} +3.66394 q^{23} -6.52413 q^{24} -4.11592 q^{25} +0.949574 q^{26} +1.00000 q^{27} +10.2027 q^{28} -3.36156 q^{29} +2.40627 q^{30} +9.96811 q^{31} -6.39545 q^{32} +3.39107 q^{33} +7.56929 q^{34} -2.10870 q^{35} +4.54932 q^{36} +6.89477 q^{37} +7.23335 q^{38} -0.371049 q^{39} +6.13435 q^{40} -4.49764 q^{41} -5.73942 q^{42} -7.61158 q^{43} +15.4271 q^{44} -0.940256 q^{45} -9.37662 q^{46} -5.33973 q^{47} +7.59768 q^{48} -1.97033 q^{49} +10.5333 q^{50} -2.95772 q^{51} -1.68802 q^{52} -4.36087 q^{53} -2.55916 q^{54} -3.18848 q^{55} -14.6316 q^{56} -2.82645 q^{57} +8.60277 q^{58} -4.97526 q^{59} -4.27752 q^{60} +12.6537 q^{61} -25.5100 q^{62} +2.24269 q^{63} +1.17165 q^{64} +0.348880 q^{65} -8.67832 q^{66} +7.43226 q^{67} -13.4556 q^{68} +3.66394 q^{69} +5.39652 q^{70} +6.49277 q^{71} -6.52413 q^{72} +12.3817 q^{73} -17.6448 q^{74} -4.11592 q^{75} -12.8584 q^{76} +7.60514 q^{77} +0.949574 q^{78} -0.841781 q^{79} -7.14376 q^{80} +1.00000 q^{81} +11.5102 q^{82} -11.7832 q^{83} +10.2027 q^{84} +2.78101 q^{85} +19.4793 q^{86} -3.36156 q^{87} -22.1238 q^{88} +1.39353 q^{89} +2.40627 q^{90} -0.832148 q^{91} +16.6684 q^{92} +9.96811 q^{93} +13.6653 q^{94} +2.65758 q^{95} -6.39545 q^{96} +4.08311 q^{97} +5.04239 q^{98} +3.39107 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 132 q + 4 q^{2} + 132 q^{3} + 156 q^{4} + 20 q^{5} + 4 q^{6} + 44 q^{7} + 9 q^{8} + 132 q^{9}+O(q^{10}) $ 132 * q + 4 * q^2 + 132 * q^3 + 156 * q^4 + 20 * q^5 + 4 * q^6 + 44 * q^7 + 9 * q^8 + 132 * q^9	\( 132 q + 4 q^{2} + 132 q^{3} + 156 q^{4} + 20 q^{5} + 4 q^{6} + 44 q^{7} + 9 q^{8} + 132 q^{9} + 40 q^{10} + 24 q^{11} + 156 q^{12} + 62 q^{13} + 25 q^{14} + 20 q^{15} + 192 q^{16} + 77 q^{17} + 4 q^{18} + 86 q^{19} + 26 q^{20} + 44 q^{21} + 52 q^{22} + 17 q^{23} + 9 q^{24} + 212 q^{25} + 13 q^{26} + 132 q^{27} + 95 q^{28} + 52 q^{29} + 40 q^{30} + 59 q^{31} - 8 q^{32} + 24 q^{33} + 41 q^{34} + 21 q^{35} + 156 q^{36} + 76 q^{37} + 2 q^{38} + 62 q^{39} + 91 q^{40} + 114 q^{41} + 25 q^{42} + 173 q^{43} + 44 q^{44} + 20 q^{45} + 48 q^{46} + 15 q^{47} + 192 q^{48} + 262 q^{49} - 9 q^{50} + 77 q^{51} + 144 q^{52} + 15 q^{53} + 4 q^{54} + 111 q^{55} + 66 q^{56} + 86 q^{57} + 33 q^{58} + 20 q^{59} + 26 q^{60} + 182 q^{61} + 16 q^{62} + 44 q^{63} + 255 q^{64} + 70 q^{65} + 52 q^{66} + 169 q^{67} + 128 q^{68} + 17 q^{69} + 2 q^{70} + 23 q^{71} + 9 q^{72} + 148 q^{73} + 57 q^{74} + 212 q^{75} + 143 q^{76} + 31 q^{77} + 13 q^{78} + 152 q^{79} + 27 q^{80} + 132 q^{81} + 67 q^{82} + 28 q^{83} + 95 q^{84} + 88 q^{85} - 10 q^{86} + 52 q^{87} + 130 q^{88} + 136 q^{89} + 40 q^{90} + 125 q^{91} + 59 q^{93} + 95 q^{94} + 2 q^{95} - 8 q^{96} + 147 q^{97} - 18 q^{98} + 24 q^{99}+O(q^{100}) \) 132 * q + 4 * q^2 + 132 * q^3 + 156 * q^4 + 20 * q^5 + 4 * q^6 + 44 * q^7 + 9 * q^8 + 132 * q^9 + 40 * q^10 + 24 * q^11 + 156 * q^12 + 62 * q^13 + 25 * q^14 + 20 * q^15 + 192 * q^16 + 77 * q^17 + 4 * q^18 + 86 * q^19 + 26 * q^20 + 44 * q^21 + 52 * q^22 + 17 * q^23 + 9 * q^24 + 212 * q^25 + 13 * q^26 + 132 * q^27 + 95 * q^28 + 52 * q^29 + 40 * q^30 + 59 * q^31 - 8 * q^32 + 24 * q^33 + 41 * q^34 + 21 * q^35 + 156 * q^36 + 76 * q^37 + 2 * q^38 + 62 * q^39 + 91 * q^40 + 114 * q^41 + 25 * q^42 + 173 * q^43 + 44 * q^44 + 20 * q^45 + 48 * q^46 + 15 * q^47 + 192 * q^48 + 262 * q^49 - 9 * q^50 + 77 * q^51 + 144 * q^52 + 15 * q^53 + 4 * q^54 + 111 * q^55 + 66 * q^56 + 86 * q^57 + 33 * q^58 + 20 * q^59 + 26 * q^60 + 182 * q^61 + 16 * q^62 + 44 * q^63 + 255 * q^64 + 70 * q^65 + 52 * q^66 + 169 * q^67 + 128 * q^68 + 17 * q^69 + 2 * q^70 + 23 * q^71 + 9 * q^72 + 148 * q^73 + 57 * q^74 + 212 * q^75 + 143 * q^76 + 31 * q^77 + 13 * q^78 + 152 * q^79 + 27 * q^80 + 132 * q^81 + 67 * q^82 + 28 * q^83 + 95 * q^84 + 88 * q^85 - 10 * q^86 + 52 * q^87 + 130 * q^88 + 136 * q^89 + 40 * q^90 + 125 * q^91 + 59 * q^93 + 95 * q^94 + 2 * q^95 - 8 * q^96 + 147 * q^97 - 18 * q^98 + 24 * 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 $

$2$	−2.55916	−1.80960	−0.904801	−	0.425834i	$-0.859981\pi$
$2$	−2.55916	−1.80960	−0.904801	+	0.425834i	$0.859981\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	4.54932	2.27466
$5$	−0.940256	−0.420495	−0.210248	−	0.977648i	$-0.567427\pi$
$5$	−0.940256	−0.420495	−0.210248	+	0.977648i	$0.567427\pi$
$6$	−2.55916	−1.04477
$7$	2.24269	0.847658	0.423829	−	0.905742i	$-0.360686\pi$
$7$	2.24269	0.847658	0.423829	+	0.905742i	$0.360686\pi$
$8$	−6.52413	−2.30663
$9$	1.00000	0.333333
$10$	2.40627	0.760929
$11$	3.39107	1.02245	0.511224	−	0.859448i	$-0.329192\pi$
$11$	3.39107	1.02245	0.511224	+	0.859448i	$0.329192\pi$
$12$	4.54932	1.31328
$13$	−0.371049	−0.102910	−0.0514552	−	0.998675i	$-0.516386\pi$
$13$	−0.371049	−0.102910	−0.0514552	+	0.998675i	$0.516386\pi$
$14$	−5.73942	−1.53392
$15$	−0.940256	−0.242773
$16$	7.59768	1.89942
$17$	−2.95772	−0.717352	−0.358676	−	0.933462i	$-0.616772\pi$
$17$	−2.95772	−0.717352	−0.358676	+	0.933462i	$0.616772\pi$
$18$	−2.55916	−0.603201
$19$	−2.82645	−0.648432	−0.324216	−	0.945983i	$-0.605100\pi$
$19$	−2.82645	−0.648432	−0.324216	+	0.945983i	$0.605100\pi$
$20$	−4.27752	−0.956484
$21$	2.24269	0.489396
$22$	−8.67832	−1.85022
$23$	3.66394	0.763984	0.381992	−	0.924166i	$-0.375238\pi$
$23$	3.66394	0.763984	0.381992	+	0.924166i	$0.375238\pi$
$24$	−6.52413	−1.33173
$25$	−4.11592	−0.823184
$26$	0.949574	0.186227
$27$	1.00000	0.192450
$28$	10.2027	1.92813
$29$	−3.36156	−0.624225	−0.312113	−	0.950045i	$-0.601037\pi$
$29$	−3.36156	−0.624225	−0.312113	+	0.950045i	$0.601037\pi$
$30$	2.40627	0.439322
$31$	9.96811	1.79033	0.895163	−	0.445739i	$-0.147059\pi$
$31$	9.96811	1.79033	0.895163	+	0.445739i	$0.147059\pi$
$32$	−6.39545	−1.13057
$33$	3.39107	0.590310
$34$	7.56929	1.29812
$35$	−2.10870	−0.356436
$36$	4.54932	0.758220
$37$	6.89477	1.13349	0.566746	−	0.823892i	$-0.308202\pi$
$37$	6.89477	1.13349	0.566746	+	0.823892i	$0.308202\pi$
$38$	7.23335	1.17340
$39$	−0.371049	−0.0594153
$40$	6.13435	0.969926
$41$	−4.49764	−0.702413	−0.351206	−	0.936298i	$-0.614228\pi$
$41$	−4.49764	−0.702413	−0.351206	+	0.936298i	$0.614228\pi$
$42$	−5.73942	−0.885612
$43$	−7.61158	−1.16076	−0.580378	−	0.814348i	$-0.697095\pi$
$43$	−7.61158	−1.16076	−0.580378	+	0.814348i	$0.697095\pi$
$44$	15.4271	2.32572
$45$	−0.940256	−0.140165
$46$	−9.37662	−1.38251
$47$	−5.33973	−0.778880	−0.389440	−	0.921052i	$-0.627331\pi$
$47$	−5.33973	−0.778880	−0.389440	+	0.921052i	$0.627331\pi$
$48$	7.59768	1.09663
$49$	−1.97033	−0.281475
$50$	10.5333	1.48964
$51$	−2.95772	−0.414164
$52$	−1.68802	−0.234086
$53$	−4.36087	−0.599011	−0.299506	−	0.954095i	$-0.596822\pi$
$53$	−4.36087	−0.599011	−0.299506	+	0.954095i	$0.596822\pi$
$54$	−2.55916	−0.348258
$55$	−3.18848	−0.429934
$56$	−14.6316	−1.95523
$57$	−2.82645	−0.374372
$58$	8.60277	1.12960
$59$	−4.97526	−0.647724	−0.323862	−	0.946104i	$-0.604981\pi$
$59$	−4.97526	−0.647724	−0.323862	+	0.946104i	$0.604981\pi$
$60$	−4.27752	−0.552226
$61$	12.6537	1.62014	0.810071	−	0.586332i	$-0.199428\pi$
$61$	12.6537	1.62014	0.810071	+	0.586332i	$0.199428\pi$
$62$	−25.5100	−3.23978
$63$	2.24269	0.282553
$64$	1.17165	0.146456
$65$	0.348880	0.0432733
$66$	−8.67832	−1.06823
$67$	7.43226	0.907996	0.453998	−	0.891003i	$-0.349997\pi$
$67$	7.43226	0.907996	0.453998	+	0.891003i	$0.349997\pi$
$68$	−13.4556	−1.63173
$69$	3.66394	0.441086
$70$	5.39652	0.645008
$71$	6.49277	0.770550	0.385275	−	0.922802i	$-0.374107\pi$
$71$	6.49277	0.770550	0.385275	+	0.922802i	$0.374107\pi$
$72$	−6.52413	−0.768876
$73$	12.3817	1.44917	0.724584	−	0.689187i	$-0.242032\pi$
$73$	12.3817	1.44917	0.724584	+	0.689187i	$0.242032\pi$
$74$	−17.6448	−2.05117
$75$	−4.11592	−0.475265
$76$	−12.8584	−1.47496
$77$	7.60514	0.866686
$78$	0.949574	0.107518
$79$	−0.841781	−0.0947078	−0.0473539	−	0.998878i	$-0.515079\pi$
$79$	−0.841781	−0.0947078	−0.0473539	+	0.998878i	$0.515079\pi$
$80$	−7.14376	−0.798697
$81$	1.00000	0.111111
$82$	11.5102	1.27109
$83$	−11.7832	−1.29337	−0.646686	−	0.762756i	$-0.723846\pi$
$83$	−11.7832	−1.29337	−0.646686	+	0.762756i	$0.723846\pi$
$84$	10.2027	1.11321
$85$	2.78101	0.301643
$86$	19.4793	2.10051
$87$	−3.36156	−0.360397
$88$	−22.1238	−2.35841
$89$	1.39353	0.147713	0.0738567	−	0.997269i	$-0.476469\pi$
$89$	1.39353	0.147713	0.0738567	+	0.997269i	$0.476469\pi$
$90$	2.40627	0.253643
$91$	−0.832148	−0.0872328
$92$	16.6684	1.73780
$93$	9.96811	1.03365
$94$	13.6653	1.40946
$95$	2.65758	0.272662
$96$	−6.39545	−0.652733
$97$	4.08311	0.414577	0.207289	−	0.978280i	$-0.433536\pi$
$97$	4.08311	0.414577	0.207289	+	0.978280i	$0.433536\pi$
$98$	5.04239	0.509359
$99$	3.39107	0.340816
$100$	−18.7246	−1.87246
$101$	15.8086	1.57302	0.786508	−	0.617580i	$-0.211887\pi$
$101$	15.8086	1.57302	0.786508	+	0.617580i	$0.211887\pi$
$102$	7.56929	0.749471
$103$	15.6738	1.54438	0.772191	−	0.635391i	$-0.219161\pi$
$103$	15.6738	1.54438	0.772191	+	0.635391i	$0.219161\pi$
$104$	2.42077	0.237376
$105$	−2.10870	−0.205788
$106$	11.1602	1.08397
$107$	4.89289	0.473014	0.236507	−	0.971630i	$-0.423997\pi$
$107$	4.89289	0.473014	0.236507	+	0.971630i	$0.423997\pi$
$108$	4.54932	0.437759
$109$	4.67266	0.447559	0.223780	−	0.974640i	$-0.428160\pi$
$109$	4.67266	0.447559	0.223780	+	0.974640i	$0.428160\pi$
$110$	8.15983	0.778010
$111$	6.89477	0.654422
$112$	17.0393	1.61006
$113$	−14.1871	−1.33461	−0.667304	−	0.744785i	$-0.732552\pi$
$113$	−14.1871	−1.33461	−0.667304	+	0.744785i	$0.732552\pi$
$114$	7.23335	0.677465
$115$	−3.44504	−0.321252
$116$	−15.2928	−1.41990
$117$	−0.371049	−0.0343034
$118$	12.7325	1.17212
$119$	−6.63326	−0.608070
$120$	6.13435	0.559987
$121$	0.499383	0.0453985
$122$	−32.3829	−2.93181
$123$	−4.49764	−0.405538
$124$	45.3481	4.07238
$125$	8.57129	0.766640
$126$	−5.73942	−0.511308
$127$	3.61375	0.320668	0.160334	−	0.987063i	$-0.448743\pi$
$127$	3.61375	0.320668	0.160334	+	0.987063i	$0.448743\pi$
$128$	9.79247	0.865540
$129$	−7.61158	−0.670162
$130$	−0.892842	−0.0783074
$131$	3.51203	0.306847	0.153424	−	0.988161i	$-0.450970\pi$
$131$	3.51203	0.306847	0.153424	+	0.988161i	$0.450970\pi$
$132$	15.4271	1.34276
$133$	−6.33886	−0.549649
$134$	−19.0204	−1.64311
$135$	−0.940256	−0.0809243
$136$	19.2966	1.65467
$137$	−17.4929	−1.49452	−0.747262	−	0.664530i	$-0.768632\pi$
$137$	−17.4929	−1.49452	−0.747262	+	0.664530i	$0.768632\pi$
$138$	−9.37662	−0.798191
$139$	3.99338	0.338714	0.169357	−	0.985555i	$-0.445831\pi$
$139$	3.99338	0.338714	0.169357	+	0.985555i	$0.445831\pi$
$140$	−9.59317	−0.810771
$141$	−5.33973	−0.449687
$142$	−16.6161	−1.39439
$143$	−1.25825	−0.105220
$144$	7.59768	0.633140
$145$	3.16072	0.262484
$146$	−31.6868	−2.62242
$147$	−1.97033	−0.162510
$148$	31.3665	2.57831
$149$	14.6120	1.19706	0.598532	−	0.801099i	$-0.295751\pi$
$149$	14.6120	1.19706	0.598532	+	0.801099i	$0.295751\pi$
$150$	10.5333	0.860041
$151$	19.5889	1.59412	0.797062	−	0.603898i	$-0.206386\pi$
$151$	19.5889	1.59412	0.797062	+	0.603898i	$0.206386\pi$
$152$	18.4401	1.49569
$153$	−2.95772	−0.239117
$154$	−19.4628	−1.56836
$155$	−9.37257	−0.752823
$156$	−1.68802	−0.135150
$157$	0.170215	0.0135847	0.00679233	−	0.999977i	$-0.497838\pi$
$157$	0.170215	0.0135847	0.00679233	+	0.999977i	$0.497838\pi$
$158$	2.15426	0.171383
$159$	−4.36087	−0.345839
$160$	6.01336	0.475398
$161$	8.21709	0.647597
$162$	−2.55916	−0.201067
$163$	4.21376	0.330047	0.165023	−	0.986290i	$-0.447230\pi$
$163$	4.21376	0.330047	0.165023	+	0.986290i	$0.447230\pi$
$164$	−20.4612	−1.59775
$165$	−3.18848	−0.248223
$166$	30.1551	2.34049
$167$	14.3524	1.11062	0.555312	−	0.831642i	$-0.312599\pi$
$167$	14.3524	1.11062	0.555312	+	0.831642i	$0.312599\pi$
$168$	−14.6316	−1.12885
$169$	−12.8623	−0.989409
$170$	−7.11707	−0.545854
$171$	−2.82645	−0.216144
$172$	−34.6275	−2.64032
$173$	−3.50680	−0.266617	−0.133309	−	0.991075i	$-0.542560\pi$
$173$	−3.50680	−0.266617	−0.133309	+	0.991075i	$0.542560\pi$
$174$	8.60277	0.652175
$175$	−9.23074	−0.697779
$176$	25.7643	1.94206
$177$	−4.97526	−0.373964
$178$	−3.56626	−0.267302
$179$	−20.3693	−1.52247	−0.761236	−	0.648475i	$-0.775407\pi$
$179$	−20.3693	−1.52247	−0.761236	+	0.648475i	$0.775407\pi$
$180$	−4.27752	−0.318828
$181$	7.41925	0.551468	0.275734	−	0.961234i	$-0.411079\pi$
$181$	7.41925	0.551468	0.275734	+	0.961234i	$0.411079\pi$
$182$	2.12960	0.157857
$183$	12.6537	0.935390
$184$	−23.9040	−1.76223
$185$	−6.48284	−0.476628
$186$	−25.5100	−1.87049
$187$	−10.0298	−0.733455
$188$	−24.2922	−1.77169
$189$	2.24269	0.163132
$190$	−6.80120	−0.493411
$191$	−3.67141	−0.265654	−0.132827	−	0.991139i	$-0.542405\pi$
$191$	−3.67141	−0.265654	−0.132827	+	0.991139i	$0.542405\pi$
$192$	1.17165	0.0845563
$193$	−11.4092	−0.821255	−0.410627	−	0.911803i	$-0.634690\pi$
$193$	−11.4092	−0.821255	−0.410627	+	0.911803i	$0.634690\pi$
$194$	−10.4494	−0.750220
$195$	0.348880	0.0249838
$196$	−8.96366	−0.640261
$197$	26.7934	1.90895	0.954477	−	0.298286i	$-0.0964149\pi$
$197$	26.7934	1.90895	0.954477	+	0.298286i	$0.0964149\pi$
$198$	−8.67832	−0.616741
$199$	16.5243	1.17138	0.585689	−	0.810536i	$-0.300824\pi$
$199$	16.5243	1.17138	0.585689	+	0.810536i	$0.300824\pi$
$200$	26.8528	1.89878
$201$	7.43226	0.524231
$202$	−40.4568	−2.84653
$203$	−7.53894	−0.529130
$204$	−13.4556	−0.942082
$205$	4.22893	0.295361
$206$	−40.1117	−2.79472
$207$	3.66394	0.254661
$208$	−2.81911	−0.195470
$209$	−9.58470	−0.662988
$210$	5.39652	0.372395
$211$	21.3753	1.47153	0.735767	−	0.677235i	$-0.236822\pi$
$211$	21.3753	1.47153	0.735767	+	0.677235i	$0.236822\pi$
$212$	−19.8390	−1.36255
$213$	6.49277	0.444877
$214$	−12.5217	−0.855967
$215$	7.15683	0.488092
$216$	−6.52413	−0.443911
$217$	22.3554	1.51758
$218$	−11.9581	−0.809905
$219$	12.3817	0.836677
$220$	−14.5054	−0.977954
$221$	1.09746	0.0738230
$222$	−17.6448	−1.18424
$223$	3.33092	0.223055	0.111528	−	0.993761i	$-0.464426\pi$
$223$	3.33092	0.223055	0.111528	+	0.993761i	$0.464426\pi$
$224$	−14.3430	−0.958334
$225$	−4.11592	−0.274395
$226$	36.3071	2.41511
$227$	20.6810	1.37265	0.686324	−	0.727296i	$-0.259223\pi$
$227$	20.6810	1.37265	0.686324	+	0.727296i	$0.259223\pi$
$228$	−12.8584	−0.851570
$229$	−14.7718	−0.976145	−0.488073	−	0.872803i	$-0.662300\pi$
$229$	−14.7718	−0.976145	−0.488073	+	0.872803i	$0.662300\pi$
$230$	8.81642	0.581338
$231$	7.60514	0.500381
$232$	21.9312	1.43986
$233$	7.33719	0.480675	0.240338	−	0.970689i	$-0.422742\pi$
$233$	7.33719	0.480675	0.240338	+	0.970689i	$0.422742\pi$
$234$	0.949574	0.0620756
$235$	5.02071	0.327515
$236$	−22.6341	−1.47335
$237$	−0.841781	−0.0546796
$238$	16.9756	1.10036
$239$	−0.812132	−0.0525324	−0.0262662	−	0.999655i	$-0.508362\pi$
$239$	−0.812132	−0.0525324	−0.0262662	+	0.999655i	$0.508362\pi$
$240$	−7.14376	−0.461128
$241$	11.7749	0.758490	0.379245	−	0.925296i	$-0.376184\pi$
$241$	11.7749	0.758490	0.379245	+	0.925296i	$0.376184\pi$
$242$	−1.27800	−0.0821532
$243$	1.00000	0.0641500
$244$	57.5658	3.68527
$245$	1.85261	0.118359
$246$	11.5102	0.733863
$247$	1.04875	0.0667304
$248$	−65.0333	−4.12962
$249$	−11.7832	−0.746729
$250$	−21.9353	−1.38731
$251$	16.0314	1.01189	0.505946	−	0.862565i	$-0.331144\pi$
$251$	16.0314	1.01189	0.505946	+	0.862565i	$0.331144\pi$
$252$	10.2027	0.642712
$253$	12.4247	0.781133
$254$	−9.24817	−0.580282
$255$	2.78101	0.174154
$256$	−27.4038	−1.71274
$257$	−23.7764	−1.48313	−0.741565	−	0.670881i	$-0.765916\pi$
$257$	−23.7764	−1.48313	−0.741565	+	0.670881i	$0.765916\pi$
$258$	19.4793	1.21273
$259$	15.4628	0.960814
$260$	1.58717	0.0984320
$261$	−3.36156	−0.208075
$262$	−8.98786	−0.555272
$263$	16.1426	0.995393	0.497697	−	0.867351i	$-0.334179\pi$
$263$	16.1426	0.995393	0.497697	+	0.867351i	$0.334179\pi$
$264$	−22.1238	−1.36163
$265$	4.10033	0.251881
$266$	16.2222	0.994646
$267$	1.39353	0.0852824
$268$	33.8118	2.06538
$269$	−28.0853	−1.71239	−0.856195	−	0.516653i	$-0.827178\pi$
$269$	−28.0853	−1.71239	−0.856195	+	0.516653i	$0.827178\pi$
$270$	2.40627	0.146441
$271$	2.32885	0.141468	0.0707339	−	0.997495i	$-0.477466\pi$
$271$	2.32885	0.141468	0.0707339	+	0.997495i	$0.477466\pi$
$272$	−22.4718	−1.36255
$273$	−0.832148	−0.0503639
$274$	44.7673	2.70449
$275$	−13.9574	−0.841662
$276$	16.6684	1.00332
$277$	3.47250	0.208642	0.104321	−	0.994544i	$-0.466733\pi$
$277$	3.47250	0.208642	0.104321	+	0.994544i	$0.466733\pi$
$278$	−10.2197	−0.612938
$279$	9.96811	0.596775
$280$	13.7575	0.822166
$281$	26.3828	1.57387	0.786933	−	0.617039i	$-0.211668\pi$
$281$	26.3828	1.57387	0.786933	+	0.617039i	$0.211668\pi$
$282$	13.6653	0.813754
$283$	−9.82820	−0.584226	−0.292113	−	0.956384i	$-0.594358\pi$
$283$	−9.82820	−0.584226	−0.292113	+	0.956384i	$0.594358\pi$
$284$	29.5377	1.75274
$285$	2.65758	0.157422
$286$	3.22008	0.190407
$287$	−10.0868	−0.595406
$288$	−6.39545	−0.376856
$289$	−8.25190	−0.485406
$290$	−8.08880	−0.474991
$291$	4.08311	0.239356
$292$	56.3283	3.29637
$293$	−2.67199	−0.156099	−0.0780497	−	0.996949i	$-0.524869\pi$
$293$	−2.67199	−0.156099	−0.0780497	+	0.996949i	$0.524869\pi$
$294$	5.04239	0.294078
$295$	4.67802	0.272365
$296$	−44.9824	−2.61455
$297$	3.39107	0.196770
$298$	−37.3946	−2.16621
$299$	−1.35950	−0.0786219
$300$	−18.7246	−1.08107
$301$	−17.0704	−0.983924
$302$	−50.1313	−2.88473
$303$	15.8086	0.908181
$304$	−21.4745	−1.23164
$305$	−11.8977	−0.681262
$306$	7.56929	0.432707
$307$	9.18913	0.524451	0.262226	−	0.965007i	$-0.415544\pi$
$307$	9.18913	0.524451	0.262226	+	0.965007i	$0.415544\pi$
$308$	34.5982	1.97142
$309$	15.6738	0.891649
$310$	23.9860	1.36231
$311$	22.6428	1.28396	0.641978	−	0.766723i	$-0.278114\pi$
$311$	22.6428	1.28396	0.641978	+	0.766723i	$0.278114\pi$
$312$	2.42077	0.137049
$313$	25.8501	1.46114	0.730568	−	0.682840i	$-0.239256\pi$
$313$	25.8501	1.46114	0.730568	+	0.682840i	$0.239256\pi$
$314$	−0.435609	−0.0245828
$315$	−2.10870	−0.118812
$316$	−3.82953	−0.215428
$317$	−2.37097	−0.133167	−0.0665834	−	0.997781i	$-0.521210\pi$
$317$	−2.37097	−0.133167	−0.0665834	+	0.997781i	$0.521210\pi$
$318$	11.1602	0.625832
$319$	−11.3993	−0.638237
$320$	−1.10165	−0.0615839
$321$	4.89289	0.273095
$322$	−21.0289	−1.17189
$323$	8.35984	0.465154
$324$	4.54932	0.252740
$325$	1.52721	0.0847141
$326$	−10.7837	−0.597254
$327$	4.67266	0.258399
$328$	29.3432	1.62021
$329$	−11.9754	−0.660224
$330$	8.15983	0.449184
$331$	−0.515764	−0.0283489	−0.0141745	−	0.999900i	$-0.504512\pi$
$331$	−0.515764	−0.0283489	−0.0141745	+	0.999900i	$0.504512\pi$
$332$	−53.6055	−2.94198
$333$	6.89477	0.377831
$334$	−36.7302	−2.00979
$335$	−6.98823	−0.381808
$336$	17.0393	0.929568
$337$	−23.1935	−1.26343	−0.631715	−	0.775201i	$-0.717649\pi$
$337$	−23.1935	−1.26343	−0.631715	+	0.775201i	$0.717649\pi$
$338$	32.9168	1.79044
$339$	−14.1871	−0.770536
$340$	12.6517	0.686136
$341$	33.8026	1.83051
$342$	7.23335	0.391135
$343$	−20.1177	−1.08625
$344$	49.6589	2.67743
$345$	−3.44504	−0.185475
$346$	8.97448	0.482471
$347$	31.0275	1.66564	0.832822	−	0.553540i	$-0.186724\pi$
$347$	31.0275	1.66564	0.832822	+	0.553540i	$0.186724\pi$
$348$	−15.2928	−0.819780
$349$	−26.5280	−1.42001	−0.710005	−	0.704196i	$-0.751307\pi$
$349$	−26.5280	−1.42001	−0.710005	+	0.704196i	$0.751307\pi$
$350$	23.6230	1.26270
$351$	−0.371049	−0.0198051
$352$	−21.6874	−1.15594
$353$	−2.11846	−0.112754	−0.0563772	−	0.998410i	$-0.517955\pi$
$353$	−2.11846	−0.112754	−0.0563772	+	0.998410i	$0.517955\pi$
$354$	12.7325	0.676726
$355$	−6.10486	−0.324012
$356$	6.33959	0.335998
$357$	−6.63326	−0.351069
$358$	52.1283	2.75507
$359$	22.2955	1.17671	0.588356	−	0.808602i	$-0.299775\pi$
$359$	22.2955	1.17671	0.588356	+	0.808602i	$0.299775\pi$
$360$	6.13435	0.323309
$361$	−11.0112	−0.579536
$362$	−18.9871	−0.997938
$363$	0.499383	0.0262108
$364$	−3.78571	−0.198425
$365$	−11.6420	−0.609368
$366$	−32.3829	−1.69268
$367$	−8.92930	−0.466105	−0.233053	−	0.972464i	$-0.574871\pi$
$367$	−8.92930	−0.466105	−0.233053	+	0.972464i	$0.574871\pi$
$368$	27.8374	1.45113
$369$	−4.49764	−0.234138
$370$	16.5907	0.862507
$371$	−9.78009	−0.507757
$372$	45.3481	2.35119
$373$	1.19990	0.0621284	0.0310642	−	0.999517i	$-0.490110\pi$
$373$	1.19990	0.0621284	0.0310642	+	0.999517i	$0.490110\pi$
$374$	25.6680	1.32726
$375$	8.57129	0.442620
$376$	34.8371	1.79659
$377$	1.24730	0.0642392
$378$	−5.73942	−0.295204
$379$	16.2681	0.835634	0.417817	−	0.908531i	$-0.362795\pi$
$379$	16.2681	0.835634	0.417817	+	0.908531i	$0.362795\pi$
$380$	12.0902	0.620215
$381$	3.61375	0.185138
$382$	9.39574	0.480728
$383$	4.08722	0.208847	0.104424	−	0.994533i	$-0.466700\pi$
$383$	4.08722	0.208847	0.104424	+	0.994533i	$0.466700\pi$
$384$	9.79247	0.499720
$385$	−7.15077	−0.364437
$386$	29.1981	1.48614
$387$	−7.61158	−0.386918
$388$	18.5754	0.943022
$389$	17.1494	0.869511	0.434756	−	0.900549i	$-0.356835\pi$
$389$	17.1494	0.869511	0.434756	+	0.900549i	$0.356835\pi$
$390$	−0.892842	−0.0452108
$391$	−10.8369	−0.548046
$392$	12.8547	0.649259
$393$	3.51203	0.177158
$394$	−68.5688	−3.45445
$395$	0.791489	0.0398242
$396$	15.4271	0.775240
$397$	−13.2304	−0.664017	−0.332008	−	0.943276i	$-0.607726\pi$
$397$	−13.2304	−0.664017	−0.332008	+	0.943276i	$0.607726\pi$
$398$	−42.2885	−2.11973
$399$	−6.33886	−0.317340
$400$	−31.2714	−1.56357
$401$	−23.0651	−1.15181	−0.575907	−	0.817515i	$-0.695351\pi$
$401$	−23.0651	−1.15181	−0.575907	+	0.817515i	$0.695351\pi$
$402$	−19.0204	−0.948650
$403$	−3.69865	−0.184243
$404$	71.9185	3.57808
$405$	−0.940256	−0.0467217
$406$	19.2934	0.957514
$407$	23.3807	1.15894
$408$	19.2966	0.955322
$409$	−33.6068	−1.66175	−0.830874	−	0.556460i	$-0.812159\pi$
$409$	−33.6068	−1.66175	−0.830874	+	0.556460i	$0.812159\pi$
$410$	−10.8225	−0.534486
$411$	−17.4929	−0.862863
$412$	71.3050	3.51294
$413$	−11.1580	−0.549049
$414$	−9.37662	−0.460836
$415$	11.0792	0.543857
$416$	2.37302	0.116347
$417$	3.99338	0.195557
$418$	24.5288	1.19974
$419$	−34.1121	−1.66648	−0.833242	−	0.552908i	$-0.813518\pi$
$419$	−34.1121	−1.66648	−0.833242	+	0.552908i	$0.813518\pi$
$420$	−9.59317	−0.468099
$421$	−21.3265	−1.03939	−0.519694	−	0.854352i	$-0.673954\pi$
$421$	−21.3265	−1.03939	−0.519694	+	0.854352i	$0.673954\pi$
$422$	−54.7028	−2.66289
$423$	−5.33973	−0.259627
$424$	28.4509	1.38170
$425$	12.1737	0.590513
$426$	−16.6161	−0.805051
$427$	28.3784	1.37333
$428$	22.2594	1.07595
$429$	−1.25825	−0.0607490
$430$	−18.3155	−0.883252
$431$	6.87553	0.331183	0.165591	−	0.986194i	$-0.447047\pi$
$431$	6.87553	0.331183	0.165591	+	0.986194i	$0.447047\pi$
$432$	7.59768	0.365544
$433$	−40.6366	−1.95287	−0.976436	−	0.215808i	$-0.930761\pi$
$433$	−40.6366	−1.95287	−0.976436	+	0.215808i	$0.930761\pi$
$434$	−57.2112	−2.74622
$435$	3.16072	0.151545
$436$	21.2574	1.01805
$437$	−10.3559	−0.495392
$438$	−31.6868	−1.51405
$439$	34.2359	1.63399	0.816994	−	0.576646i	$-0.195639\pi$
$439$	34.2359	1.63399	0.816994	+	0.576646i	$0.195639\pi$
$440$	20.8020	0.991698
$441$	−1.97033	−0.0938251
$442$	−2.80857	−0.133590
$443$	9.32682	0.443131	0.221565	−	0.975146i	$-0.428883\pi$
$443$	9.32682	0.443131	0.221565	+	0.975146i	$0.428883\pi$
$444$	31.3665	1.48859
$445$	−1.31027	−0.0621128
$446$	−8.52437	−0.403641
$447$	14.6120	0.691125
$448$	2.62764	0.124144
$449$	38.1329	1.79960	0.899802	−	0.436298i	$-0.143711\pi$
$449$	38.1329	1.79960	0.899802	+	0.436298i	$0.143711\pi$
$450$	10.5333	0.496545
$451$	−15.2518	−0.718180
$452$	−64.5416	−3.03578
$453$	19.5889	0.920368
$454$	−52.9261	−2.48395
$455$	0.782432	0.0366810
$456$	18.4401	0.863538
$457$	−15.6203	−0.730689	−0.365344	−	0.930872i	$-0.619049\pi$
$457$	−15.6203	−0.730689	−0.365344	+	0.930872i	$0.619049\pi$
$458$	37.8034	1.76643
$459$	−2.95772	−0.138055
$460$	−15.6726	−0.730738
$461$	−41.2157	−1.91960	−0.959802	−	0.280677i	$-0.909441\pi$
$461$	−41.2157	−1.91960	−0.959802	+	0.280677i	$0.909441\pi$
$462$	−19.4628	−0.905491
$463$	−29.4365	−1.36803	−0.684016	−	0.729467i	$-0.739768\pi$
$463$	−29.4365	−1.36803	−0.684016	+	0.729467i	$0.739768\pi$
$464$	−25.5400	−1.18567
$465$	−9.37257	−0.434643
$466$	−18.7771	−0.869831
$467$	−10.2328	−0.473519	−0.236759	−	0.971568i	$-0.576085\pi$
$467$	−10.2328	−0.473519	−0.236759	+	0.971568i	$0.576085\pi$
$468$	−1.68802	−0.0780287
$469$	16.6683	0.769670
$470$	−12.8488	−0.592672
$471$	0.170215	0.00784311
$472$	32.4593	1.49406
$473$	−25.8114	−1.18681
$474$	2.15426	0.0989483
$475$	11.6334	0.533779
$476$	−30.1768	−1.38315
$477$	−4.36087	−0.199670
$478$	2.07838	0.0950628
$479$	8.24917	0.376914	0.188457	−	0.982081i	$-0.439651\pi$
$479$	8.24917	0.376914	0.188457	+	0.982081i	$0.439651\pi$
$480$	6.01336	0.274471
$481$	−2.55829	−0.116648
$482$	−30.1340	−1.37256
$483$	8.21709	0.373891
$484$	2.27186	0.103266
$485$	−3.83917	−0.174328
$486$	−2.55916	−0.116086
$487$	−21.3051	−0.965427	−0.482713	−	0.875778i	$-0.660349\pi$
$487$	−21.3051	−0.965427	−0.482713	+	0.875778i	$0.660349\pi$
$488$	−82.5545	−3.73707
$489$	4.21376	0.190553
$490$	−4.74114	−0.214183
$491$	−4.76849	−0.215199	−0.107600	−	0.994194i	$-0.534316\pi$
$491$	−4.76849	−0.215199	−0.107600	+	0.994194i	$0.534316\pi$
$492$	−20.4612	−0.922462
$493$	9.94254	0.447789
$494$	−2.68392	−0.120755
$495$	−3.18848	−0.143311
$496$	75.7345	3.40058
$497$	14.5613	0.653163
$498$	30.1551	1.35128
$499$	38.8102	1.73738	0.868692	−	0.495353i	$-0.164961\pi$
$499$	38.8102	1.73738	0.868692	+	0.495353i	$0.164961\pi$
$500$	38.9936	1.74385
$501$	14.3524	0.641220
$502$	−41.0269	−1.83112
$503$	11.7784	0.525172	0.262586	−	0.964909i	$-0.415425\pi$
$503$	11.7784	0.525172	0.262586	+	0.964909i	$0.415425\pi$
$504$	−14.6316	−0.651744
$505$	−14.8641	−0.661445
$506$	−31.7968	−1.41354
$507$	−12.8623	−0.571236
$508$	16.4401	0.729411
$509$	17.2207	0.763294	0.381647	−	0.924308i	$-0.375357\pi$
$509$	17.2207	0.763294	0.381647	+	0.924308i	$0.375357\pi$
$510$	−7.11707	−0.315149
$511$	27.7683	1.22840
$512$	50.5459	2.23384
$513$	−2.82645	−0.124791
$514$	60.8477	2.68388
$515$	−14.7373	−0.649405
$516$	−34.6275	−1.52439
$517$	−18.1074	−0.796364
$518$	−39.5720	−1.73869
$519$	−3.50680	−0.153932
$520$	−2.27614	−0.0998154
$521$	23.2118	1.01693	0.508463	−	0.861084i	$-0.330214\pi$
$521$	23.2118	1.01693	0.508463	+	0.861084i	$0.330214\pi$
$522$	8.60277	0.376533
$523$	1.60921	0.0703659	0.0351829	−	0.999381i	$-0.488799\pi$
$523$	1.60921	0.0703659	0.0351829	+	0.999381i	$0.488799\pi$
$524$	15.9773	0.697973
$525$	−9.23074	−0.402863
$526$	−41.3115	−1.80127
$527$	−29.4829	−1.28429
$528$	25.7643	1.12125
$529$	−9.57555	−0.416328
$530$	−10.4934	−0.455805
$531$	−4.97526	−0.215908
$532$	−28.8375	−1.25026
$533$	1.66884	0.0722856
$534$	−3.56626	−0.154327
$535$	−4.60057	−0.198900
$536$	−48.4891	−2.09441
$537$	−20.3693	−0.878999
$538$	71.8748	3.09874
$539$	−6.68153	−0.287794
$540$	−4.27752	−0.184075
$541$	34.6296	1.48884	0.744422	−	0.667710i	$-0.232725\pi$
$541$	34.6296	1.48884	0.744422	+	0.667710i	$0.232725\pi$
$542$	−5.95992	−0.256000
$543$	7.41925	0.318390
$544$	18.9159	0.811015
$545$	−4.39349	−0.188197
$546$	2.12960	0.0911386
$547$	13.0333	0.557264	0.278632	−	0.960398i	$-0.410119\pi$
$547$	13.0333	0.557264	0.278632	+	0.960398i	$0.410119\pi$
$548$	−79.5810	−3.39953
$549$	12.6537	0.540047
$550$	35.7192	1.52307
$551$	9.50127	0.404768
$552$	−23.9040	−1.01742
$553$	−1.88786	−0.0802798
$554$	−8.88669	−0.377559
$555$	−6.48284	−0.275181
$556$	18.1672	0.770460
$557$	−16.1007	−0.682208	−0.341104	−	0.940026i	$-0.610801\pi$
$557$	−16.1007	−0.682208	−0.341104	+	0.940026i	$0.610801\pi$
$558$	−25.5100	−1.07993
$559$	2.82427	0.119454
$560$	−16.0213	−0.677022
$561$	−10.0298	−0.423460
$562$	−67.5179	−2.84807
$563$	−19.9812	−0.842105	−0.421053	−	0.907036i	$-0.638339\pi$
$563$	−19.9812	−0.842105	−0.421053	+	0.907036i	$0.638339\pi$
$564$	−24.2922	−1.02288
$565$	13.3395	0.561196
$566$	25.1520	1.05722
$567$	2.24269	0.0941843
$568$	−42.3597	−1.77737
$569$	22.5408	0.944961	0.472481	−	0.881341i	$-0.343359\pi$
$569$	22.5408	0.944961	0.472481	+	0.881341i	$0.343359\pi$
$570$	−6.80120	−0.284871
$571$	−15.7745	−0.660143	−0.330071	−	0.943956i	$-0.607073\pi$
$571$	−15.7745	−0.660143	−0.330071	+	0.943956i	$0.607073\pi$
$572$	−5.72420	−0.239341
$573$	−3.67141	−0.153375
$574$	25.8138	1.07745
$575$	−15.0805	−0.628899
$576$	1.17165	0.0488186
$577$	−18.7814	−0.781879	−0.390940	−	0.920416i	$-0.627850\pi$
$577$	−18.7814	−0.781879	−0.390940	+	0.920416i	$0.627850\pi$
$578$	21.1180	0.878391
$579$	−11.4092	−0.474152
$580$	14.3791	0.597061
$581$	−26.4261	−1.09634
$582$	−10.4494	−0.433140
$583$	−14.7880	−0.612458
$584$	−80.7798	−3.34269
$585$	0.348880	0.0144244
$586$	6.83806	0.282478
$587$	−1.01780	−0.0420090	−0.0210045	−	0.999779i	$-0.506686\pi$
$587$	−1.01780	−0.0420090	−0.0210045	+	0.999779i	$0.506686\pi$
$588$	−8.96366	−0.369655
$589$	−28.1744	−1.16090
$590$	−11.9718	−0.492872
$591$	26.7934	1.10213
$592$	52.3842	2.15298
$593$	−2.59125	−0.106410	−0.0532049	−	0.998584i	$-0.516944\pi$
$593$	−2.59125	−0.106410	−0.0532049	+	0.998584i	$0.516944\pi$
$594$	−8.67832	−0.356076
$595$	6.23696	0.255690
$596$	66.4748	2.72291
$597$	16.5243	0.676296
$598$	3.47918	0.142274
$599$	32.8837	1.34359	0.671795	−	0.740737i	$-0.265524\pi$
$599$	32.8837	1.34359	0.671795	+	0.740737i	$0.265524\pi$
$600$	26.8528	1.09626
$601$	19.9019	0.811815	0.405908	−	0.913914i	$-0.366955\pi$
$601$	19.9019	0.811815	0.405908	+	0.913914i	$0.366955\pi$
$602$	43.6861	1.78051
$603$	7.43226	0.302665
$604$	89.1163	3.62609
$605$	−0.469548	−0.0190898
$606$	−40.4568	−1.64345
$607$	28.0689	1.13928	0.569641	−	0.821894i	$-0.307082\pi$
$607$	28.0689	1.13928	0.569641	+	0.821894i	$0.307082\pi$
$608$	18.0764	0.733096
$609$	−7.53894	−0.305493
$610$	30.4482	1.23281
$611$	1.98130	0.0801548
$612$	−13.4556	−0.543911
$613$	−37.9297	−1.53197	−0.765983	−	0.642861i	$-0.777747\pi$
$613$	−37.9297	−1.53197	−0.765983	+	0.642861i	$0.777747\pi$
$614$	−23.5165	−0.949048
$615$	4.22893	0.170527
$616$	−49.6169	−1.99912
$617$	39.1783	1.57726	0.788629	−	0.614870i	$-0.210791\pi$
$617$	39.1783	1.57726	0.788629	+	0.614870i	$0.210791\pi$
$618$	−40.1117	−1.61353
$619$	−23.9782	−0.963765	−0.481882	−	0.876236i	$-0.660047\pi$
$619$	−23.9782	−0.963765	−0.481882	+	0.876236i	$0.660047\pi$
$620$	−42.6388	−1.71242
$621$	3.66394	0.147029
$622$	−57.9466	−2.32345
$623$	3.12525	0.125210
$624$	−2.81911	−0.112855
$625$	12.5204	0.500816
$626$	−66.1547	−2.64407
$627$	−9.58470	−0.382776
$628$	0.774364	0.0309005
$629$	−20.3928	−0.813113
$630$	5.39652	0.215003
$631$	14.0651	0.559924	0.279962	−	0.960011i	$-0.409678\pi$
$631$	14.0651	0.559924	0.279962	+	0.960011i	$0.409678\pi$
$632$	5.49189	0.218456
$633$	21.3753	0.849591
$634$	6.06770	0.240979
$635$	−3.39785	−0.134839
$636$	−19.8390	−0.786667
$637$	0.731087	0.0289667
$638$	29.1726	1.15496
$639$	6.49277	0.256850
$640$	−9.20742	−0.363955
$641$	−1.99096	−0.0786381	−0.0393190	−	0.999227i	$-0.512519\pi$
$641$	−1.99096	−0.0786381	−0.0393190	+	0.999227i	$0.512519\pi$
$642$	−12.5217	−0.494193
$643$	15.0786	0.594641	0.297321	−	0.954778i	$-0.403907\pi$
$643$	15.0786	0.594641	0.297321	+	0.954778i	$0.403907\pi$
$644$	37.3822	1.47306
$645$	7.15683	0.281800
$646$	−21.3942	−0.841744
$647$	−34.4876	−1.35585	−0.677923	−	0.735133i	$-0.737120\pi$
$647$	−34.4876	−1.35585	−0.677923	+	0.735133i	$0.737120\pi$
$648$	−6.52413	−0.256292
$649$	−16.8715	−0.662264
$650$	−3.90837	−0.153299
$651$	22.3554	0.876178
$652$	19.1697	0.750745
$653$	6.78375	0.265468	0.132734	−	0.991152i	$-0.457624\pi$
$653$	6.78375	0.265468	0.132734	+	0.991152i	$0.457624\pi$
$654$	−11.9581	−0.467599
$655$	−3.30220	−0.129028
$656$	−34.1716	−1.33418
$657$	12.3817	0.483056
$658$	30.6470	1.19474
$659$	4.80633	0.187228	0.0936139	−	0.995609i	$-0.470158\pi$
$659$	4.80633	0.187228	0.0936139	+	0.995609i	$0.470158\pi$
$660$	−14.5054	−0.564622
$661$	−31.1501	−1.21160	−0.605800	−	0.795617i	$-0.707147\pi$
$661$	−31.1501	−1.21160	−0.605800	+	0.795617i	$0.707147\pi$
$662$	1.31992	0.0513003
$663$	1.09746	0.0426217
$664$	76.8750	2.98333
$665$	5.96015	0.231125
$666$	−17.6448	−0.683724
$667$	−12.3165	−0.476898
$668$	65.2938	2.52629
$669$	3.33092	0.128781
$670$	17.8840	0.690920
$671$	42.9097	1.65651
$672$	−14.3430	−0.553295
$673$	−38.0840	−1.46803	−0.734014	−	0.679134i	$-0.762355\pi$
$673$	−38.0840	−1.46803	−0.734014	+	0.679134i	$0.762355\pi$
$674$	59.3560	2.28631
$675$	−4.11592	−0.158422
$676$	−58.5148	−2.25057
$677$	−20.0213	−0.769479	−0.384740	−	0.923025i	$-0.625709\pi$
$677$	−20.0213	−0.769479	−0.384740	+	0.923025i	$0.625709\pi$
$678$	36.3071	1.39436
$679$	9.15716	0.351420
$680$	−18.1437	−0.695779
$681$	20.6810	0.792498
$682$	−86.5064	−3.31250
$683$	−21.0632	−0.805962	−0.402981	−	0.915208i	$-0.632026\pi$
$683$	−21.0632	−0.805962	−0.402981	+	0.915208i	$0.632026\pi$
$684$	−12.8584	−0.491654
$685$	16.4478	0.628440
$686$	51.4845	1.96569
$687$	−14.7718	−0.563578
$688$	−57.8304	−2.20476
$689$	1.61809	0.0616445
$690$	8.81642	0.335635
$691$	42.2683	1.60796	0.803981	−	0.594654i	$-0.202711\pi$
$691$	42.2683	1.60796	0.803981	+	0.594654i	$0.202711\pi$
$692$	−15.9536	−0.606464
$693$	7.60514	0.288895
$694$	−79.4045	−3.01415
$695$	−3.75480	−0.142428
$696$	21.9312	0.831301
$697$	13.3027	0.503878
$698$	67.8895	2.56965
$699$	7.33719	0.277518
$700$	−41.9936	−1.58721
$701$	−1.62667	−0.0614385	−0.0307193	−	0.999528i	$-0.509780\pi$
$701$	−1.62667	−0.0614385	−0.0307193	+	0.999528i	$0.509780\pi$
$702$	0.949574	0.0358394
$703$	−19.4877	−0.734993
$704$	3.97314	0.149743
$705$	5.02071	0.189091
$706$	5.42150	0.204041
$707$	35.4539	1.33338
$708$	−22.6341	−0.850641
$709$	22.4813	0.844304	0.422152	−	0.906525i	$-0.361275\pi$
$709$	22.4813	0.844304	0.422152	+	0.906525i	$0.361275\pi$
$710$	15.6233	0.586333
$711$	−0.841781	−0.0315693
$712$	−9.09154	−0.340720
$713$	36.5226	1.36778
$714$	16.9756	0.635296
$715$	1.18308	0.0442447
$716$	−92.6664	−3.46311
$717$	−0.812132	−0.0303296
$718$	−57.0579	−2.12938
$719$	40.0774	1.49464	0.747318	−	0.664467i	$-0.231341\pi$
$719$	40.0774	1.49464	0.747318	+	0.664467i	$0.231341\pi$
$720$	−7.14376	−0.266232
$721$	35.1514	1.30911
$722$	28.1794	1.04873
$723$	11.7749	0.437914
$724$	33.7525	1.25440
$725$	13.8359	0.513852
$726$	−1.27800	−0.0474312
$727$	38.1146	1.41359	0.706797	−	0.707417i	$-0.250140\pi$
$727$	38.1146	1.41359	0.706797	+	0.707417i	$0.250140\pi$
$728$	5.42904	0.201214
$729$	1.00000	0.0370370
$730$	29.7937	1.10271
$731$	22.5129	0.832670
$732$	57.5658	2.12769
$733$	7.94150	0.293326	0.146663	−	0.989187i	$-0.453147\pi$
$733$	7.94150	0.293326	0.146663	+	0.989187i	$0.453147\pi$
$734$	22.8515	0.843465
$735$	1.85261	0.0683346
$736$	−23.4325	−0.863735
$737$	25.2034	0.928378
$738$	11.5102	0.423696
$739$	38.7267	1.42459	0.712293	−	0.701882i	$-0.247657\pi$
$739$	38.7267	1.42459	0.712293	+	0.701882i	$0.247657\pi$
$740$	−29.4925	−1.08417
$741$	1.04875	0.0385268
$742$	25.0289	0.918838
$743$	28.3617	1.04049	0.520246	−	0.854017i	$-0.325840\pi$
$743$	28.3617	1.04049	0.520246	+	0.854017i	$0.325840\pi$
$744$	−65.0333	−2.38424
$745$	−13.7390	−0.503359
$746$	−3.07074	−0.112428
$747$	−11.7832	−0.431124
$748$	−45.6290	−1.66836
$749$	10.9733	0.400954
$750$	−21.9353	−0.800966
$751$	2.58618	0.0943709	0.0471855	−	0.998886i	$-0.484975\pi$
$751$	2.58618	0.0943709	0.0471855	+	0.998886i	$0.484975\pi$
$752$	−40.5696	−1.47942
$753$	16.0314	0.584216
$754$	−3.19205	−0.116247
$755$	−18.4186	−0.670321
$756$	10.2027	0.371070
$757$	−18.4853	−0.671860	−0.335930	−	0.941887i	$-0.609051\pi$
$757$	−18.4853	−0.671860	−0.335930	+	0.941887i	$0.609051\pi$
$758$	−41.6326	−1.51216
$759$	12.4247	0.450988
$760$	−17.3384	−0.628931
$761$	28.0888	1.01822	0.509109	−	0.860702i	$-0.329975\pi$
$761$	28.0888	1.01822	0.509109	+	0.860702i	$0.329975\pi$
$762$	−9.24817	−0.335026
$763$	10.4793	0.379378
$764$	−16.7024	−0.604272
$765$	2.78101	0.100548
$766$	−10.4599	−0.377930
$767$	1.84606	0.0666575
$768$	−27.4038	−0.988850
$769$	8.49426	0.306311	0.153156	−	0.988202i	$-0.451056\pi$
$769$	8.49426	0.306311	0.153156	+	0.988202i	$0.451056\pi$
$770$	18.3000	0.659486
$771$	−23.7764	−0.856286
$772$	−51.9043	−1.86808
$773$	43.7365	1.57309	0.786547	−	0.617531i	$-0.211867\pi$
$773$	43.7365	1.57309	0.786547	+	0.617531i	$0.211867\pi$
$774$	19.4793	0.700168
$775$	−41.0279	−1.47377
$776$	−26.6388	−0.956275
$777$	15.4628	0.554726
$778$	−43.8882	−1.57347
$779$	12.7123	0.455467
$780$	1.58717	0.0568298
$781$	22.0175	0.787846
$782$	27.7334	0.991745
$783$	−3.36156	−0.120132
$784$	−14.9699	−0.534640
$785$	−0.160046	−0.00571228
$786$	−8.98786	−0.320586
$787$	30.1338	1.07415	0.537077	−	0.843533i	$-0.319528\pi$
$787$	30.1338	1.07415	0.537077	+	0.843533i	$0.319528\pi$
$788$	121.892	4.34222
$789$	16.1426	0.574690
$790$	−2.02555	−0.0720659
$791$	−31.8173	−1.13129
$792$	−22.1238	−0.786135
$793$	−4.69514	−0.166729
$794$	33.8589	1.20161
$795$	4.10033	0.145424
$796$	75.1745	2.66449
$797$	42.3258	1.49926	0.749628	−	0.661860i	$-0.230233\pi$
$797$	42.3258	1.49926	0.749628	+	0.661860i	$0.230233\pi$
$798$	16.2222	0.574259
$799$	15.7934	0.558732
$800$	26.3232	0.930664
$801$	1.39353	0.0492378
$802$	59.0273	2.08432
$803$	41.9872	1.48170
$804$	33.8118	1.19245
$805$	−7.72616	−0.272312
$806$	9.46546	0.333407
$807$	−28.0853	−0.988648
$808$	−103.137	−3.62836
$809$	45.4430	1.59769	0.798846	−	0.601536i	$-0.205444\pi$
$809$	45.4430	1.59769	0.798846	+	0.601536i	$0.205444\pi$
$810$	2.40627	0.0845477
$811$	6.93872	0.243651	0.121826	−	0.992552i	$-0.461125\pi$
$811$	6.93872	0.243651	0.121826	+	0.992552i	$0.461125\pi$
$812$	−34.2970	−1.20359
$813$	2.32885	0.0816765
$814$	−59.8349	−2.09721
$815$	−3.96201	−0.138783
$816$	−22.4718	−0.786671
$817$	21.5137	0.752671
$818$	86.0053	3.00710
$819$	−0.832148	−0.0290776
$820$	19.2388	0.671846
$821$	22.3360	0.779531	0.389765	−	0.920914i	$-0.372556\pi$
$821$	22.3360	0.779531	0.389765	+	0.920914i	$0.372556\pi$
$822$	44.7673	1.56144
$823$	−44.1601	−1.53932	−0.769662	−	0.638452i	$-0.779575\pi$
$823$	−44.1601	−1.53932	−0.769662	+	0.638452i	$0.779575\pi$
$824$	−102.258	−3.56231
$825$	−13.9574	−0.485934
$826$	28.5551	0.993560
$827$	−19.0224	−0.661474	−0.330737	−	0.943723i	$-0.607297\pi$
$827$	−19.0224	−0.661474	−0.330737	+	0.943723i	$0.607297\pi$
$828$	16.6684	0.579268
$829$	7.53900	0.261840	0.130920	−	0.991393i	$-0.458207\pi$
$829$	7.53900	0.261840	0.130920	+	0.991393i	$0.458207\pi$
$830$	−28.3535	−0.984164
$831$	3.47250	0.120460
$832$	−0.434737	−0.0150718
$833$	5.82768	0.201917
$834$	−10.2197	−0.353880
$835$	−13.4950	−0.467012
$836$	−43.6039	−1.50807
$837$	9.96811	0.344548
$838$	87.2984	3.01567
$839$	−27.7541	−0.958180	−0.479090	−	0.877766i	$-0.659033\pi$
$839$	−27.7541	−0.958180	−0.479090	+	0.877766i	$0.659033\pi$
$840$	13.7575	0.474678
$841$	−17.6999	−0.610343
$842$	54.5779	1.88088
$843$	26.3828	0.908672
$844$	97.2430	3.34724
$845$	12.0939	0.416042
$846$	13.6653	0.469821
$847$	1.11996	0.0384824
$848$	−33.1325	−1.13777
$849$	−9.82820	−0.337303
$850$	−31.1546	−1.06859
$851$	25.2620	0.865970
$852$	29.5377	1.01194
$853$	15.3939	0.527077	0.263539	−	0.964649i	$-0.415110\pi$
$853$	15.3939	0.527077	0.263539	+	0.964649i	$0.415110\pi$
$854$	−72.6250	−2.48518
$855$	2.65758	0.0908875
$856$	−31.9219	−1.09107
$857$	−44.6826	−1.52633	−0.763165	−	0.646204i	$-0.776356\pi$
$857$	−44.6826	−1.52633	−0.763165	+	0.646204i	$0.776356\pi$
$858$	3.22008	0.109932
$859$	−33.3891	−1.13922	−0.569610	−	0.821915i	$-0.692906\pi$
$859$	−33.3891	−1.13922	−0.569610	+	0.821915i	$0.692906\pi$
$860$	32.5587	1.11024
$861$	−10.0868	−0.343758
$862$	−17.5956	−0.599309
$863$	1.43141	0.0487257	0.0243628	−	0.999703i	$-0.492244\pi$
$863$	1.43141	0.0487257	0.0243628	+	0.999703i	$0.492244\pi$
$864$	−6.39545	−0.217578
$865$	3.29729	0.112111
$866$	103.996	3.53392
$867$	−8.25190	−0.280249
$868$	101.702	3.45199
$869$	−2.85454	−0.0968337
$870$	−8.08880	−0.274236
$871$	−2.75773	−0.0934421
$872$	−30.4850	−1.03235
$873$	4.08311	0.138192
$874$	26.5025	0.896462
$875$	19.2228	0.649849
$876$	56.3283	1.90316
$877$	−23.9388	−0.808357	−0.404179	−	0.914680i	$-0.632443\pi$
$877$	−23.9388	−0.808357	−0.404179	+	0.914680i	$0.632443\pi$
$878$	−87.6152	−2.95687
$879$	−2.67199	−0.0901240
$880$	−24.2250	−0.816625
$881$	−9.15577	−0.308466	−0.154233	−	0.988035i	$-0.549291\pi$
$881$	−9.15577	−0.308466	−0.154233	+	0.988035i	$0.549291\pi$
$882$	5.04239	0.169786
$883$	34.2289	1.15190	0.575948	−	0.817486i	$-0.304633\pi$
$883$	34.2289	1.15190	0.575948	+	0.817486i	$0.304633\pi$
$884$	4.99269	0.167922
$885$	4.67802	0.157250
$886$	−23.8689	−0.801890
$887$	12.0255	0.403776	0.201888	−	0.979409i	$-0.435292\pi$
$887$	12.0255	0.403776	0.201888	+	0.979409i	$0.435292\pi$
$888$	−44.9824	−1.50951
$889$	8.10453	0.271817
$890$	3.35320	0.112399
$891$	3.39107	0.113605
$892$	15.1534	0.507374
$893$	15.0925	0.505051
$894$	−37.3946	−1.25066
$895$	19.1523	0.640192
$896$	21.9615	0.733682
$897$	−1.35950	−0.0453924
$898$	−97.5884	−3.25657
$899$	−33.5084	−1.11757
$900$	−18.7246	−0.624155
$901$	12.8982	0.429702
$902$	39.0319	1.29962
$903$	−17.0704	−0.568069
$904$	92.5584	3.07845
$905$	−6.97599	−0.231890
$906$	−50.1313	−1.66550
$907$	−32.1397	−1.06718	−0.533590	−	0.845743i	$-0.679158\pi$
$907$	−32.1397	−1.06718	−0.533590	+	0.845743i	$0.679158\pi$
$908$	94.0846	3.12231
$909$	15.8086	0.524339
$910$	−2.00237	−0.0663780
$911$	12.7323	0.421840	0.210920	−	0.977503i	$-0.432354\pi$
$911$	12.7323	0.421840	0.210920	+	0.977503i	$0.432354\pi$
$912$	−21.4745	−0.711091
$913$	−39.9576	−1.32240
$914$	39.9750	1.32226
$915$	−11.8977	−0.393327
$916$	−67.2015	−2.22040
$917$	7.87640	0.260102
$918$	7.56929	0.249824
$919$	36.1030	1.19093	0.595463	−	0.803383i	$-0.296969\pi$
$919$	36.1030	1.19093	0.595463	+	0.803383i	$0.296969\pi$
$920$	22.4759	0.741008
$921$	9.18913	0.302792
$922$	105.478	3.47372
$923$	−2.40913	−0.0792975
$924$	34.5982	1.13820
$925$	−28.3783	−0.933073
$926$	75.3329	2.47559
$927$	15.6738	0.514794
$928$	21.4987	0.705728
$929$	−16.2884	−0.534405	−0.267202	−	0.963640i	$-0.586099\pi$
$929$	−16.2884	−0.534405	−0.267202	+	0.963640i	$0.586099\pi$
$930$	23.9860	0.786530
$931$	5.56903	0.182518
$932$	33.3792	1.09337
$933$	22.6428	0.741292
$934$	26.1875	0.856881
$935$	9.43062	0.308414
$936$	2.42077	0.0791253
$937$	−29.7918	−0.973256	−0.486628	−	0.873609i	$-0.661773\pi$
$937$	−29.7918	−0.973256	−0.486628	+	0.873609i	$0.661773\pi$
$938$	−42.6569	−1.39280
$939$	25.8501	0.843587
$940$	22.8408	0.744986
$941$	−41.8180	−1.36323	−0.681614	−	0.731712i	$-0.738722\pi$
$941$	−41.8180	−1.36323	−0.681614	+	0.731712i	$0.738722\pi$
$942$	−0.435609	−0.0141929
$943$	−16.4791	−0.536632
$944$	−37.8005	−1.23030
$945$	−2.10870	−0.0685962
$946$	66.0557	2.14766
$947$	−7.18190	−0.233380	−0.116690	−	0.993168i	$-0.537228\pi$
$947$	−7.18190	−0.233380	−0.116690	+	0.993168i	$0.537228\pi$
$948$	−3.82953	−0.124377
$949$	−4.59421	−0.149134
$950$	−29.7719	−0.965927
$951$	−2.37097	−0.0768839
$952$	43.2762	1.40259
$953$	−2.40080	−0.0777695	−0.0388848	−	0.999244i	$-0.512381\pi$
$953$	−2.40080	−0.0777695	−0.0388848	+	0.999244i	$0.512381\pi$
$954$	11.1602	0.361324
$955$	3.45206	0.111706
$956$	−3.69465	−0.119493
$957$	−11.3993	−0.368487
$958$	−21.1110	−0.682065
$959$	−39.2313	−1.26685
$960$	−1.10165	−0.0355555
$961$	68.3632	2.20527
$962$	6.54709	0.211087
$963$	4.89289	0.157671
$964$	53.5679	1.72531
$965$	10.7276	0.345334
$966$	−21.0289	−0.676593
$967$	17.8141	0.572864	0.286432	−	0.958101i	$-0.407531\pi$
$967$	17.8141	0.572864	0.286432	+	0.958101i	$0.407531\pi$
$968$	−3.25804	−0.104717
$969$	8.35984	0.268557
$970$	9.82506	0.315464
$971$	15.3397	0.492274	0.246137	−	0.969235i	$-0.420839\pi$
$971$	15.3397	0.492274	0.246137	+	0.969235i	$0.420839\pi$
$972$	4.54932	0.145920
$973$	8.95593	0.287114
$974$	54.5233	1.74704
$975$	1.52721	0.0489097
$976$	96.1389	3.07733
$977$	37.9203	1.21318	0.606588	−	0.795016i	$-0.292538\pi$
$977$	37.9203	1.21318	0.606588	+	0.795016i	$0.292538\pi$
$978$	−10.7837	−0.344825
$979$	4.72555	0.151029
$980$	8.42813	0.269227
$981$	4.67266	0.149186
$982$	12.2034	0.389425
$983$	−17.4036	−0.555090	−0.277545	−	0.960713i	$-0.589521\pi$
$983$	−17.4036	−0.555090	−0.277545	+	0.960713i	$0.589521\pi$
$984$	29.3432	0.935426
$985$	−25.1927	−0.802705
$986$	−25.4446	−0.810321
$987$	−11.9754	−0.381181
$988$	4.77110	0.151789
$989$	−27.8884	−0.886798
$990$	8.15983	0.259337
$991$	4.33897	0.137832	0.0689159	−	0.997622i	$-0.478046\pi$
$991$	4.33897	0.137832	0.0689159	+	0.997622i	$0.478046\pi$
$992$	−63.7506	−2.02408
$993$	−0.515764	−0.0163673
$994$	−37.2647	−1.18196
$995$	−15.5371	−0.492559
$996$	−53.6055	−1.69855
$997$	10.2686	0.325209	0.162604	−	0.986691i	$-0.448011\pi$
$997$	10.2686	0.325209	0.162604	+	0.986691i	$0.448011\pi$
$998$	−99.3217	−3.14397
$999$	6.89477	0.218141

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8031.2.a.d.1.11	✓	132

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8031.2.a.d.1.11	✓	132	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:	\( N \)	\(=\)	\( 8031 = 3 \cdot 2677 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8031.a (trivial)

\(f(q)\)	\(=\)	\(q-2.55916 q^{2} +1.00000 q^{3} +4.54932 q^{4} -0.940256 q^{5} -2.55916 q^{6} +2.24269 q^{7} -6.52413 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.55916 q^{2} +1.00000 q^{3} +4.54932 q^{4} -0.940256 q^{5} -2.55916 q^{6} +2.24269 q^{7} -6.52413 q^{8} +1.00000 q^{9} +2.40627 q^{10} +3.39107 q^{11} +4.54932 q^{12} -0.371049 q^{13} -5.73942 q^{14} -0.940256 q^{15} +7.59768 q^{16} -2.95772 q^{17} -2.55916 q^{18} -2.82645 q^{19} -4.27752 q^{20} +2.24269 q^{21} -8.67832 q^{22} +3.66394 q^{23} -6.52413 q^{24} -4.11592 q^{25} +0.949574 q^{26} +1.00000 q^{27} +10.2027 q^{28} -3.36156 q^{29} +2.40627 q^{30} +9.96811 q^{31} -6.39545 q^{32} +3.39107 q^{33} +7.56929 q^{34} -2.10870 q^{35} +4.54932 q^{36} +6.89477 q^{37} +7.23335 q^{38} -0.371049 q^{39} +6.13435 q^{40} -4.49764 q^{41} -5.73942 q^{42} -7.61158 q^{43} +15.4271 q^{44} -0.940256 q^{45} -9.37662 q^{46} -5.33973 q^{47} +7.59768 q^{48} -1.97033 q^{49} +10.5333 q^{50} -2.95772 q^{51} -1.68802 q^{52} -4.36087 q^{53} -2.55916 q^{54} -3.18848 q^{55} -14.6316 q^{56} -2.82645 q^{57} +8.60277 q^{58} -4.97526 q^{59} -4.27752 q^{60} +12.6537 q^{61} -25.5100 q^{62} +2.24269 q^{63} +1.17165 q^{64} +0.348880 q^{65} -8.67832 q^{66} +7.43226 q^{67} -13.4556 q^{68} +3.66394 q^{69} +5.39652 q^{70} +6.49277 q^{71} -6.52413 q^{72} +12.3817 q^{73} -17.6448 q^{74} -4.11592 q^{75} -12.8584 q^{76} +7.60514 q^{77} +0.949574 q^{78} -0.841781 q^{79} -7.14376 q^{80} +1.00000 q^{81} +11.5102 q^{82} -11.7832 q^{83} +10.2027 q^{84} +2.78101 q^{85} +19.4793 q^{86} -3.36156 q^{87} -22.1238 q^{88} +1.39353 q^{89} +2.40627 q^{90} -0.832148 q^{91} +16.6684 q^{92} +9.96811 q^{93} +13.6653 q^{94} +2.65758 q^{95} -6.39545 q^{96} +4.08311 q^{97} +5.04239 q^{98} +3.39107 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q + 4 q^{2} + 132 q^{3} + 156 q^{4} + 20 q^{5} + 4 q^{6} + 44 q^{7} + 9 q^{8} + 132 q^{9}+O(q^{10}) \) 132 * q + 4 * q^2 + 132 * q^3 + 156 * q^4 + 20 * q^5 + 4 * q^6 + 44 * q^7 + 9 * q^8 + 132 * q^9	\( 132 q + 4 q^{2} + 132 q^{3} + 156 q^{4} + 20 q^{5} + 4 q^{6} + 44 q^{7} + 9 q^{8} + 132 q^{9} + 40 q^{10} + 24 q^{11} + 156 q^{12} + 62 q^{13} + 25 q^{14} + 20 q^{15} + 192 q^{16} + 77 q^{17} + 4 q^{18} + 86 q^{19} + 26 q^{20} + 44 q^{21} + 52 q^{22} + 17 q^{23} + 9 q^{24} + 212 q^{25} + 13 q^{26} + 132 q^{27} + 95 q^{28} + 52 q^{29} + 40 q^{30} + 59 q^{31} - 8 q^{32} + 24 q^{33} + 41 q^{34} + 21 q^{35} + 156 q^{36} + 76 q^{37} + 2 q^{38} + 62 q^{39} + 91 q^{40} + 114 q^{41} + 25 q^{42} + 173 q^{43} + 44 q^{44} + 20 q^{45} + 48 q^{46} + 15 q^{47} + 192 q^{48} + 262 q^{49} - 9 q^{50} + 77 q^{51} + 144 q^{52} + 15 q^{53} + 4 q^{54} + 111 q^{55} + 66 q^{56} + 86 q^{57} + 33 q^{58} + 20 q^{59} + 26 q^{60} + 182 q^{61} + 16 q^{62} + 44 q^{63} + 255 q^{64} + 70 q^{65} + 52 q^{66} + 169 q^{67} + 128 q^{68} + 17 q^{69} + 2 q^{70} + 23 q^{71} + 9 q^{72} + 148 q^{73} + 57 q^{74} + 212 q^{75} + 143 q^{76} + 31 q^{77} + 13 q^{78} + 152 q^{79} + 27 q^{80} + 132 q^{81} + 67 q^{82} + 28 q^{83} + 95 q^{84} + 88 q^{85} - 10 q^{86} + 52 q^{87} + 130 q^{88} + 136 q^{89} + 40 q^{90} + 125 q^{91} + 59 q^{93} + 95 q^{94} + 2 q^{95} - 8 q^{96} + 147 q^{97} - 18 q^{98} + 24 q^{99}+O(q^{100}) \) 132 * q + 4 * q^2 + 132 * q^3 + 156 * q^4 + 20 * q^5 + 4 * q^6 + 44 * q^7 + 9 * q^8 + 132 * q^9 + 40 * q^10 + 24 * q^11 + 156 * q^12 + 62 * q^13 + 25 * q^14 + 20 * q^15 + 192 * q^16 + 77 * q^17 + 4 * q^18 + 86 * q^19 + 26 * q^20 + 44 * q^21 + 52 * q^22 + 17 * q^23 + 9 * q^24 + 212 * q^25 + 13 * q^26 + 132 * q^27 + 95 * q^28 + 52 * q^29 + 40 * q^30 + 59 * q^31 - 8 * q^32 + 24 * q^33 + 41 * q^34 + 21 * q^35 + 156 * q^36 + 76 * q^37 + 2 * q^38 + 62 * q^39 + 91 * q^40 + 114 * q^41 + 25 * q^42 + 173 * q^43 + 44 * q^44 + 20 * q^45 + 48 * q^46 + 15 * q^47 + 192 * q^48 + 262 * q^49 - 9 * q^50 + 77 * q^51 + 144 * q^52 + 15 * q^53 + 4 * q^54 + 111 * q^55 + 66 * q^56 + 86 * q^57 + 33 * q^58 + 20 * q^59 + 26 * q^60 + 182 * q^61 + 16 * q^62 + 44 * q^63 + 255 * q^64 + 70 * q^65 + 52 * q^66 + 169 * q^67 + 128 * q^68 + 17 * q^69 + 2 * q^70 + 23 * q^71 + 9 * q^72 + 148 * q^73 + 57 * q^74 + 212 * q^75 + 143 * q^76 + 31 * q^77 + 13 * q^78 + 152 * q^79 + 27 * q^80 + 132 * q^81 + 67 * q^82 + 28 * q^83 + 95 * q^84 + 88 * q^85 - 10 * q^86 + 52 * q^87 + 130 * q^88 + 136 * q^89 + 40 * q^90 + 125 * q^91 + 59 * q^93 + 95 * q^94 + 2 * q^95 - 8 * q^96 + 147 * q^97 - 18 * q^98 + 24 * q^99

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