LMFDB - Embedded newform 6045.2.a.bi.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6045,2,Mod(1,6045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6045, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6045.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6045 = 3 \cdot 5 \cdot 13 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6045.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:	$48.2695680219$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 4 x^{17} - 21 x^{16} + 97 x^{15} + 156 x^{14} - 935 x^{13} - 411 x^{12} + 4582 x^{11} + \cdots - 112 $ x^18 - 4x^17 - 21x^16 + 97x^15 + 156x^14 - 935x^13 - 411x^12 + 4582x^11 - 446x^10 - 12159x^9 + 4398x^8 + 17347x^7 - 7839x^6 - 12832x^5 + 4744x^4 + 5064x^3 - 746x^2 - 892*x - 112
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Root			$1.46257$ of defining polynomial
Character	$\chi$	$=$	6045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.46257 q^{2} +1.00000 q^{3} +0.139121 q^{4} +1.00000 q^{5} +1.46257 q^{6} +2.97636 q^{7} -2.72167 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.46257 q^{2} +1.00000 q^{3} +0.139121 q^{4} +1.00000 q^{5} +1.46257 q^{6} +2.97636 q^{7} -2.72167 q^{8} +1.00000 q^{9} +1.46257 q^{10} +5.96403 q^{11} +0.139121 q^{12} -1.00000 q^{13} +4.35314 q^{14} +1.00000 q^{15} -4.25889 q^{16} +5.00602 q^{17} +1.46257 q^{18} +4.39273 q^{19} +0.139121 q^{20} +2.97636 q^{21} +8.72282 q^{22} +8.46752 q^{23} -2.72167 q^{24} +1.00000 q^{25} -1.46257 q^{26} +1.00000 q^{27} +0.414073 q^{28} -8.29288 q^{29} +1.46257 q^{30} +1.00000 q^{31} -0.785589 q^{32} +5.96403 q^{33} +7.32168 q^{34} +2.97636 q^{35} +0.139121 q^{36} -8.83331 q^{37} +6.42469 q^{38} -1.00000 q^{39} -2.72167 q^{40} -10.8134 q^{41} +4.35314 q^{42} -4.75779 q^{43} +0.829719 q^{44} +1.00000 q^{45} +12.3844 q^{46} -6.54381 q^{47} -4.25889 q^{48} +1.85869 q^{49} +1.46257 q^{50} +5.00602 q^{51} -0.139121 q^{52} +0.293056 q^{53} +1.46257 q^{54} +5.96403 q^{55} -8.10066 q^{56} +4.39273 q^{57} -12.1289 q^{58} +12.2179 q^{59} +0.139121 q^{60} +2.37835 q^{61} +1.46257 q^{62} +2.97636 q^{63} +7.36879 q^{64} -1.00000 q^{65} +8.72282 q^{66} -7.20712 q^{67} +0.696441 q^{68} +8.46752 q^{69} +4.35314 q^{70} -9.06675 q^{71} -2.72167 q^{72} +11.6141 q^{73} -12.9194 q^{74} +1.00000 q^{75} +0.611119 q^{76} +17.7511 q^{77} -1.46257 q^{78} +10.1346 q^{79} -4.25889 q^{80} +1.00000 q^{81} -15.8154 q^{82} -8.37007 q^{83} +0.414073 q^{84} +5.00602 q^{85} -6.95862 q^{86} -8.29288 q^{87} -16.2321 q^{88} -0.845677 q^{89} +1.46257 q^{90} -2.97636 q^{91} +1.17801 q^{92} +1.00000 q^{93} -9.57080 q^{94} +4.39273 q^{95} -0.785589 q^{96} +9.88170 q^{97} +2.71847 q^{98} +5.96403 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 4 q^{2} + 18 q^{3} + 22 q^{4} + 18 q^{5} + 4 q^{6} + 8 q^{7} + 9 q^{8} + 18 q^{9}+O(q^{10}) $ 18 * q + 4 * q^2 + 18 * q^3 + 22 * q^4 + 18 * q^5 + 4 * q^6 + 8 * q^7 + 9 * q^8 + 18 * q^9	\( 18 q + 4 q^{2} + 18 q^{3} + 22 q^{4} + 18 q^{5} + 4 q^{6} + 8 q^{7} + 9 q^{8} + 18 q^{9} + 4 q^{10} + 6 q^{11} + 22 q^{12} - 18 q^{13} + 5 q^{14} + 18 q^{15} + 30 q^{16} + 18 q^{17} + 4 q^{18} + 12 q^{19} + 22 q^{20} + 8 q^{21} + 7 q^{22} + 32 q^{23} + 9 q^{24} + 18 q^{25} - 4 q^{26} + 18 q^{27} + 10 q^{28} + 7 q^{29} + 4 q^{30} + 18 q^{31} + 22 q^{32} + 6 q^{33} + 15 q^{34} + 8 q^{35} + 22 q^{36} + 3 q^{37} + 32 q^{38} - 18 q^{39} + 9 q^{40} + 4 q^{41} + 5 q^{42} + 14 q^{43} - 5 q^{44} + 18 q^{45} + 10 q^{46} + 23 q^{47} + 30 q^{48} + 28 q^{49} + 4 q^{50} + 18 q^{51} - 22 q^{52} + 35 q^{53} + 4 q^{54} + 6 q^{55} - 7 q^{56} + 12 q^{57} - 6 q^{58} + 28 q^{59} + 22 q^{60} + 19 q^{61} + 4 q^{62} + 8 q^{63} + 43 q^{64} - 18 q^{65} + 7 q^{66} + 34 q^{67} + 55 q^{68} + 32 q^{69} + 5 q^{70} - 8 q^{71} + 9 q^{72} + 22 q^{74} + 18 q^{75} + 2 q^{76} + 21 q^{77} - 4 q^{78} + 4 q^{79} + 30 q^{80} + 18 q^{81} + 29 q^{82} + 11 q^{83} + 10 q^{84} + 18 q^{85} - 22 q^{86} + 7 q^{87} - 31 q^{88} + 17 q^{89} + 4 q^{90} - 8 q^{91} + 33 q^{92} + 18 q^{93} - 14 q^{94} + 12 q^{95} + 22 q^{96} + 32 q^{97} + 20 q^{98} + 6 q^{99}+O(q^{100}) \) 18 * q + 4 * q^2 + 18 * q^3 + 22 * q^4 + 18 * q^5 + 4 * q^6 + 8 * q^7 + 9 * q^8 + 18 * q^9 + 4 * q^10 + 6 * q^11 + 22 * q^12 - 18 * q^13 + 5 * q^14 + 18 * q^15 + 30 * q^16 + 18 * q^17 + 4 * q^18 + 12 * q^19 + 22 * q^20 + 8 * q^21 + 7 * q^22 + 32 * q^23 + 9 * q^24 + 18 * q^25 - 4 * q^26 + 18 * q^27 + 10 * q^28 + 7 * q^29 + 4 * q^30 + 18 * q^31 + 22 * q^32 + 6 * q^33 + 15 * q^34 + 8 * q^35 + 22 * q^36 + 3 * q^37 + 32 * q^38 - 18 * q^39 + 9 * q^40 + 4 * q^41 + 5 * q^42 + 14 * q^43 - 5 * q^44 + 18 * q^45 + 10 * q^46 + 23 * q^47 + 30 * q^48 + 28 * q^49 + 4 * q^50 + 18 * q^51 - 22 * q^52 + 35 * q^53 + 4 * q^54 + 6 * q^55 - 7 * q^56 + 12 * q^57 - 6 * q^58 + 28 * q^59 + 22 * q^60 + 19 * q^61 + 4 * q^62 + 8 * q^63 + 43 * q^64 - 18 * q^65 + 7 * q^66 + 34 * q^67 + 55 * q^68 + 32 * q^69 + 5 * q^70 - 8 * q^71 + 9 * q^72 + 22 * q^74 + 18 * q^75 + 2 * q^76 + 21 * q^77 - 4 * q^78 + 4 * q^79 + 30 * q^80 + 18 * q^81 + 29 * q^82 + 11 * q^83 + 10 * q^84 + 18 * q^85 - 22 * q^86 + 7 * q^87 - 31 * q^88 + 17 * q^89 + 4 * q^90 - 8 * q^91 + 33 * q^92 + 18 * q^93 - 14 * q^94 + 12 * q^95 + 22 * q^96 + 32 * q^97 + 20 * q^98 + 6 * 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 $

$2$	1.46257	1.03420	0.517098	−	0.855926i	$-0.327012\pi$
$2$	1.46257	1.03420	0.517098	+	0.855926i	$0.327012\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0.139121	0.0695604
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	1.46257	0.597093
$7$	2.97636	1.12496	0.562478	−	0.826812i	$-0.309848\pi$
$7$	2.97636	1.12496	0.562478	+	0.826812i	$0.309848\pi$
$8$	−2.72167	−0.962257
$9$	1.00000	0.333333
$10$	1.46257	0.462506
$11$	5.96403	1.79822	0.899111	−	0.437721i	$-0.144214\pi$
$11$	5.96403	1.79822	0.899111	+	0.437721i	$0.144214\pi$
$12$	0.139121	0.0401607
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	4.35314	1.16343
$15$	1.00000	0.258199
$16$	−4.25889	−1.06472
$17$	5.00602	1.21414	0.607069	−	0.794649i	$-0.292345\pi$
$17$	5.00602	1.21414	0.607069	+	0.794649i	$0.292345\pi$
$18$	1.46257	0.344732
$19$	4.39273	1.00776	0.503880	−	0.863773i	$-0.331905\pi$
$19$	4.39273	1.00776	0.503880	+	0.863773i	$0.331905\pi$
$20$	0.139121	0.0311083
$21$	2.97636	0.649494
$22$	8.72282	1.85971
$23$	8.46752	1.76560	0.882800	−	0.469748i	$-0.155655\pi$
$23$	8.46752	1.76560	0.882800	+	0.469748i	$0.155655\pi$
$24$	−2.72167	−0.555559
$25$	1.00000	0.200000
$26$	−1.46257	−0.286834
$27$	1.00000	0.192450
$28$	0.414073	0.0782524
$29$	−8.29288	−1.53995	−0.769974	−	0.638075i	$-0.779731\pi$
$29$	−8.29288	−1.53995	−0.769974	+	0.638075i	$0.779731\pi$
$30$	1.46257	0.267028
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	−0.785589	−0.138874
$33$	5.96403	1.03820
$34$	7.32168	1.25566
$35$	2.97636	0.503096
$36$	0.139121	0.0231868
$37$	−8.83331	−1.45219	−0.726094	−	0.687596i	$-0.758666\pi$
$37$	−8.83331	−1.45219	−0.726094	+	0.687596i	$0.758666\pi$
$38$	6.42469	1.04222
$39$	−1.00000	−0.160128
$40$	−2.72167	−0.430334
$41$	−10.8134	−1.68877	−0.844387	−	0.535734i	$-0.820035\pi$
$41$	−10.8134	−1.68877	−0.844387	+	0.535734i	$0.820035\pi$
$42$	4.35314	0.671704
$43$	−4.75779	−0.725556	−0.362778	−	0.931876i	$-0.618172\pi$
$43$	−4.75779	−0.725556	−0.362778	+	0.931876i	$0.618172\pi$
$44$	0.829719	0.125085
$45$	1.00000	0.149071
$46$	12.3844	1.82598
$47$	−6.54381	−0.954513	−0.477256	−	0.878764i	$-0.658369\pi$
$47$	−6.54381	−0.954513	−0.477256	+	0.878764i	$0.658369\pi$
$48$	−4.25889	−0.614717
$49$	1.85869	0.265527
$50$	1.46257	0.206839
$51$	5.00602	0.700983
$52$	−0.139121	−0.0192926
$53$	0.293056	0.0402543	0.0201271	−	0.999797i	$-0.493593\pi$
$53$	0.293056	0.0402543	0.0201271	+	0.999797i	$0.493593\pi$
$54$	1.46257	0.199031
$55$	5.96403	0.804189
$56$	−8.10066	−1.08250
$57$	4.39273	0.581831
$58$	−12.1289	−1.59261
$59$	12.2179	1.59063	0.795316	−	0.606196i	$-0.207305\pi$
$59$	12.2179	1.59063	0.795316	+	0.606196i	$0.207305\pi$
$60$	0.139121	0.0179604
$61$	2.37835	0.304517	0.152258	−	0.988341i	$-0.451345\pi$
$61$	2.37835	0.304517	0.152258	+	0.988341i	$0.451345\pi$
$62$	1.46257	0.185747
$63$	2.97636	0.374986
$64$	7.36879	0.921099
$65$	−1.00000	−0.124035
$66$	8.72282	1.07371
$67$	−7.20712	−0.880490	−0.440245	−	0.897878i	$-0.645108\pi$
$67$	−7.20712	−0.880490	−0.440245	+	0.897878i	$0.645108\pi$
$68$	0.696441	0.0844559
$69$	8.46752	1.01937
$70$	4.35314	0.520300
$71$	−9.06675	−1.07602	−0.538012	−	0.842937i	$-0.680825\pi$
$71$	−9.06675	−1.07602	−0.538012	+	0.842937i	$0.680825\pi$
$72$	−2.72167	−0.320752
$73$	11.6141	1.35933	0.679665	−	0.733523i	$-0.262125\pi$
$73$	11.6141	1.35933	0.679665	+	0.733523i	$0.262125\pi$
$74$	−12.9194	−1.50185
$75$	1.00000	0.115470
$76$	0.611119	0.0701002
$77$	17.7511	2.02292
$78$	−1.46257	−0.165604
$79$	10.1346	1.14023	0.570116	−	0.821564i	$-0.306898\pi$
$79$	10.1346	1.14023	0.570116	+	0.821564i	$0.306898\pi$
$80$	−4.25889	−0.476158
$81$	1.00000	0.111111
$82$	−15.8154	−1.74652
$83$	−8.37007	−0.918735	−0.459367	−	0.888246i	$-0.651924\pi$
$83$	−8.37007	−0.918735	−0.459367	+	0.888246i	$0.651924\pi$
$84$	0.414073	0.0451790
$85$	5.00602	0.542979
$86$	−6.95862	−0.750367
$87$	−8.29288	−0.889090
$88$	−16.2321	−1.73035
$89$	−0.845677	−0.0896416	−0.0448208	−	0.998995i	$-0.514272\pi$
$89$	−0.845677	−0.0896416	−0.0448208	+	0.998995i	$0.514272\pi$
$90$	1.46257	0.154169
$91$	−2.97636	−0.312007
$92$	1.17801	0.122816
$93$	1.00000	0.103695
$94$	−9.57080	−0.987153
$95$	4.39273	0.450684
$96$	−0.785589	−0.0801789
$97$	9.88170	1.00334	0.501668	−	0.865061i	$-0.332720\pi$
$97$	9.88170	1.00334	0.501668	+	0.865061i	$0.332720\pi$
$98$	2.71847	0.274607
$99$	5.96403	0.599407
$100$	0.139121	0.0139121
$101$	−14.5634	−1.44911	−0.724556	−	0.689216i	$-0.757955\pi$
$101$	−14.5634	−1.44911	−0.724556	+	0.689216i	$0.757955\pi$
$102$	7.32168	0.724954
$103$	−14.5524	−1.43389	−0.716943	−	0.697132i	$-0.754459\pi$
$103$	−14.5524	−1.43389	−0.716943	+	0.697132i	$0.754459\pi$
$104$	2.72167	0.266882
$105$	2.97636	0.290463
$106$	0.428615	0.0416308
$107$	−17.9299	−1.73335	−0.866673	−	0.498877i	$-0.833746\pi$
$107$	−17.9299	−1.73335	−0.866673	+	0.498877i	$0.833746\pi$
$108$	0.139121	0.0133869
$109$	3.80742	0.364685	0.182342	−	0.983235i	$-0.441632\pi$
$109$	3.80742	0.364685	0.182342	+	0.983235i	$0.441632\pi$
$110$	8.72282	0.831689
$111$	−8.83331	−0.838421
$112$	−12.6760	−1.19777
$113$	6.39165	0.601276	0.300638	−	0.953738i	$-0.402800\pi$
$113$	6.39165	0.601276	0.300638	+	0.953738i	$0.402800\pi$
$114$	6.42469	0.601727
$115$	8.46752	0.789601
$116$	−1.15371	−0.107119
$117$	−1.00000	−0.0924500
$118$	17.8695	1.64502
$119$	14.8997	1.36585
$120$	−2.72167	−0.248454
$121$	24.5696	2.23360
$122$	3.47851	0.314930
$123$	−10.8134	−0.975014
$124$	0.139121	0.0124934
$125$	1.00000	0.0894427
$126$	4.35314	0.387808
$127$	18.4089	1.63353	0.816763	−	0.576973i	$-0.195766\pi$
$127$	18.4089	1.63353	0.816763	+	0.576973i	$0.195766\pi$
$128$	12.3486	1.09147
$129$	−4.75779	−0.418900
$130$	−1.46257	−0.128276
$131$	3.98428	0.348108	0.174054	−	0.984736i	$-0.444313\pi$
$131$	3.98428	0.348108	0.174054	+	0.984736i	$0.444313\pi$
$132$	0.829719	0.0722178
$133$	13.0743	1.13369
$134$	−10.5409	−0.910599
$135$	1.00000	0.0860663
$136$	−13.6248	−1.16831
$137$	20.4456	1.74678	0.873392	−	0.487017i	$-0.161915\pi$
$137$	20.4456	1.74678	0.873392	+	0.487017i	$0.161915\pi$
$138$	12.3844	1.05423
$139$	15.6927	1.33104	0.665519	−	0.746381i	$-0.268210\pi$
$139$	15.6927	1.33104	0.665519	+	0.746381i	$0.268210\pi$
$140$	0.414073	0.0349955
$141$	−6.54381	−0.551088
$142$	−13.2608	−1.11282
$143$	−5.96403	−0.498737
$144$	−4.25889	−0.354907
$145$	−8.29288	−0.688686
$146$	16.9865	1.40581
$147$	1.85869	0.153302
$148$	−1.22890	−0.101015
$149$	7.14445	0.585296	0.292648	−	0.956220i	$-0.405464\pi$
$149$	7.14445	0.585296	0.292648	+	0.956220i	$0.405464\pi$
$150$	1.46257	0.119419
$151$	−15.3524	−1.24936	−0.624681	−	0.780880i	$-0.714771\pi$
$151$	−15.3524	−1.24936	−0.624681	+	0.780880i	$0.714771\pi$
$152$	−11.9556	−0.969725
$153$	5.00602	0.404713
$154$	25.9622	2.09210
$155$	1.00000	0.0803219
$156$	−0.139121	−0.0111386
$157$	19.7416	1.57555	0.787776	−	0.615962i	$-0.211233\pi$
$157$	19.7416	1.57555	0.787776	+	0.615962i	$0.211233\pi$
$158$	14.8226	1.17922
$159$	0.293056	0.0232408
$160$	−0.785589	−0.0621063
$161$	25.2024	1.98622
$162$	1.46257	0.114911
$163$	−0.558383	−0.0437359	−0.0218680	−	0.999761i	$-0.506961\pi$
$163$	−0.558383	−0.0437359	−0.0218680	+	0.999761i	$0.506961\pi$
$164$	−1.50437	−0.117472
$165$	5.96403	0.464299
$166$	−12.2418	−0.950151
$167$	−13.9786	−1.08169	−0.540847	−	0.841121i	$-0.681896\pi$
$167$	−13.9786	−1.08169	−0.540847	+	0.841121i	$0.681896\pi$
$168$	−8.10066	−0.624980
$169$	1.00000	0.0769231
$170$	7.32168	0.561547
$171$	4.39273	0.335920
$172$	−0.661907	−0.0504700
$173$	3.46216	0.263223	0.131612	−	0.991301i	$-0.457985\pi$
$173$	3.46216	0.263223	0.131612	+	0.991301i	$0.457985\pi$
$174$	−12.1289	−0.919492
$175$	2.97636	0.224991
$176$	−25.4001	−1.91461
$177$	12.2179	0.918351
$178$	−1.23687	−0.0927070
$179$	17.7870	1.32946	0.664731	−	0.747083i	$-0.268546\pi$
$179$	17.7870	1.32946	0.664731	+	0.747083i	$0.268546\pi$
$180$	0.139121	0.0103694
$181$	0.822403	0.0611287	0.0305644	−	0.999533i	$-0.490270\pi$
$181$	0.822403	0.0611287	0.0305644	+	0.999533i	$0.490270\pi$
$182$	−4.35314	−0.322676
$183$	2.37835	0.175813
$184$	−23.0458	−1.69896
$185$	−8.83331	−0.649438
$186$	1.46257	0.107241
$187$	29.8560	2.18329
$188$	−0.910380	−0.0663963
$189$	2.97636	0.216498
$190$	6.42469	0.466096
$191$	−8.01806	−0.580167	−0.290083	−	0.957001i	$-0.593683\pi$
$191$	−8.01806	−0.580167	−0.290083	+	0.957001i	$0.593683\pi$
$192$	7.36879	0.531797
$193$	−11.9989	−0.863702	−0.431851	−	0.901945i	$-0.642139\pi$
$193$	−11.9989	−0.863702	−0.431851	+	0.901945i	$0.642139\pi$
$194$	14.4527	1.03764
$195$	−1.00000	−0.0716115
$196$	0.258583	0.0184702
$197$	21.4017	1.52481	0.762404	−	0.647102i	$-0.224019\pi$
$197$	21.4017	1.52481	0.762404	+	0.647102i	$0.224019\pi$
$198$	8.72282	0.619904
$199$	−4.49282	−0.318488	−0.159244	−	0.987239i	$-0.550906\pi$
$199$	−4.49282	−0.318488	−0.159244	+	0.987239i	$0.550906\pi$
$200$	−2.72167	−0.192451
$201$	−7.20712	−0.508351
$202$	−21.3000	−1.49866
$203$	−24.6825	−1.73238
$204$	0.696441	0.0487607
$205$	−10.8134	−0.755243
$206$	−21.2839	−1.48292
$207$	8.46752	0.588534
$208$	4.25889	0.295301
$209$	26.1983	1.81218
$210$	4.35314	0.300395
$211$	−9.16265	−0.630783	−0.315391	−	0.948962i	$-0.602136\pi$
$211$	−9.16265	−0.630783	−0.315391	+	0.948962i	$0.602136\pi$
$212$	0.0407701	0.00280010
$213$	−9.06675	−0.621243
$214$	−26.2237	−1.79262
$215$	−4.75779	−0.324479
$216$	−2.72167	−0.185186
$217$	2.97636	0.202048
$218$	5.56863	0.377156
$219$	11.6141	0.784810
$220$	0.829719	0.0559397
$221$	−5.00602	−0.336742
$222$	−12.9194	−0.867091
$223$	−18.6672	−1.25005	−0.625025	−	0.780604i	$-0.714911\pi$
$223$	−18.6672	−1.25005	−0.625025	+	0.780604i	$0.714911\pi$
$224$	−2.33819	−0.156227
$225$	1.00000	0.0666667
$226$	9.34826	0.621837
$227$	−16.1731	−1.07345	−0.536725	−	0.843757i	$-0.680339\pi$
$227$	−16.1731	−1.07345	−0.536725	+	0.843757i	$0.680339\pi$
$228$	0.611119	0.0404724
$229$	2.24917	0.148629	0.0743146	−	0.997235i	$-0.476323\pi$
$229$	2.24917	0.148629	0.0743146	+	0.997235i	$0.476323\pi$
$230$	12.3844	0.816601
$231$	17.7511	1.16793
$232$	22.5705	1.48183
$233$	0.701226	0.0459388	0.0229694	−	0.999736i	$-0.492688\pi$
$233$	0.701226	0.0459388	0.0229694	+	0.999736i	$0.492688\pi$
$234$	−1.46257	−0.0956114
$235$	−6.54381	−0.426871
$236$	1.69976	0.110645
$237$	10.1346	0.658313
$238$	21.7919	1.41256
$239$	8.63204	0.558360	0.279180	−	0.960239i	$-0.409937\pi$
$239$	8.63204	0.558360	0.279180	+	0.960239i	$0.409937\pi$
$240$	−4.25889	−0.274910
$241$	−25.8632	−1.66600	−0.832998	−	0.553276i	$-0.813377\pi$
$241$	−25.8632	−1.66600	−0.832998	+	0.553276i	$0.813377\pi$
$242$	35.9348	2.30998
$243$	1.00000	0.0641500
$244$	0.330878	0.0211823
$245$	1.85869	0.118747
$246$	−15.8154	−1.00836
$247$	−4.39273	−0.279503
$248$	−2.72167	−0.172826
$249$	−8.37007	−0.530432
$250$	1.46257	0.0925013
$251$	−28.4608	−1.79643	−0.898214	−	0.439558i	$-0.855135\pi$
$251$	−28.4608	−1.79643	−0.898214	+	0.439558i	$0.855135\pi$
$252$	0.414073	0.0260841
$253$	50.5005	3.17494
$254$	26.9244	1.68939
$255$	5.00602	0.313489
$256$	3.32312	0.207695
$257$	22.2519	1.38804	0.694019	−	0.719957i	$-0.255838\pi$
$257$	22.2519	1.38804	0.694019	+	0.719957i	$0.255838\pi$
$258$	−6.95862	−0.433225
$259$	−26.2911	−1.63365
$260$	−0.139121	−0.00862790
$261$	−8.29288	−0.513316
$262$	5.82729	0.360011
$263$	−1.97414	−0.121730	−0.0608652	−	0.998146i	$-0.519386\pi$
$263$	−1.97414	−0.121730	−0.0608652	+	0.998146i	$0.519386\pi$
$264$	−16.2321	−0.999018
$265$	0.293056	0.0180023
$266$	19.1222	1.17245
$267$	−0.845677	−0.0517546
$268$	−1.00266	−0.0612472
$269$	15.4973	0.944886	0.472443	−	0.881361i	$-0.343372\pi$
$269$	15.4973	0.944886	0.472443	+	0.881361i	$0.343372\pi$
$270$	1.46257	0.0890094
$271$	−21.5758	−1.31064	−0.655318	−	0.755353i	$-0.727466\pi$
$271$	−21.5758	−1.31064	−0.655318	+	0.755353i	$0.727466\pi$
$272$	−21.3201	−1.29272
$273$	−2.97636	−0.180137
$274$	29.9032	1.80652
$275$	5.96403	0.359644
$276$	1.17801	0.0709077
$277$	−16.3116	−0.980067	−0.490034	−	0.871703i	$-0.663016\pi$
$277$	−16.3116	−0.980067	−0.490034	+	0.871703i	$0.663016\pi$
$278$	22.9517	1.37655
$279$	1.00000	0.0598684
$280$	−8.10066	−0.484107
$281$	−8.70176	−0.519103	−0.259552	−	0.965729i	$-0.583575\pi$
$281$	−8.70176	−0.519103	−0.259552	+	0.965729i	$0.583575\pi$
$282$	−9.57080	−0.569933
$283$	23.3001	1.38505	0.692524	−	0.721395i	$-0.256499\pi$
$283$	23.3001	1.38505	0.692524	+	0.721395i	$0.256499\pi$
$284$	−1.26137	−0.0748487
$285$	4.39273	0.260203
$286$	−8.72282	−0.515791
$287$	−32.1846	−1.89980
$288$	−0.785589	−0.0462913
$289$	8.06027	0.474133
$290$	−12.1289	−0.712236
$291$	9.88170	0.579276
$292$	1.61576	0.0945555
$293$	−6.89027	−0.402534	−0.201267	−	0.979536i	$-0.564506\pi$
$293$	−6.89027	−0.402534	−0.201267	+	0.979536i	$0.564506\pi$
$294$	2.71847	0.158545
$295$	12.2179	0.711352
$296$	24.0414	1.39738
$297$	5.96403	0.346068
$298$	10.4493	0.605310
$299$	−8.46752	−0.489690
$300$	0.139121	0.00803214
$301$	−14.1609	−0.816220
$302$	−22.4540	−1.29208
$303$	−14.5634	−0.836645
$304$	−18.7081	−1.07298
$305$	2.37835	0.136184
$306$	7.32168	0.418552
$307$	−3.94989	−0.225432	−0.112716	−	0.993627i	$-0.535955\pi$
$307$	−3.94989	−0.225432	−0.112716	+	0.993627i	$0.535955\pi$
$308$	2.46954	0.140715
$309$	−14.5524	−0.827854
$310$	1.46257	0.0830686
$311$	−6.60927	−0.374777	−0.187389	−	0.982286i	$-0.560002\pi$
$311$	−6.60927	−0.374777	−0.187389	+	0.982286i	$0.560002\pi$
$312$	2.72167	0.154084
$313$	−31.7558	−1.79495	−0.897473	−	0.441069i	$-0.854599\pi$
$313$	−31.7558	−1.79495	−0.897473	+	0.441069i	$0.854599\pi$
$314$	28.8735	1.62943
$315$	2.97636	0.167699
$316$	1.40993	0.0793150
$317$	11.4062	0.640634	0.320317	−	0.947310i	$-0.396211\pi$
$317$	11.4062	0.640634	0.320317	+	0.947310i	$0.396211\pi$
$318$	0.428615	0.0240356
$319$	−49.4589	−2.76917
$320$	7.36879	0.411928
$321$	−17.9299	−1.00075
$322$	36.8603	2.05414
$323$	21.9901	1.22356
$324$	0.139121	0.00772893
$325$	−1.00000	−0.0554700
$326$	−0.816676	−0.0452315
$327$	3.80742	0.210551
$328$	29.4306	1.62503
$329$	−19.4767	−1.07379
$330$	8.72282	0.480176
$331$	23.7724	1.30665	0.653325	−	0.757078i	$-0.273374\pi$
$331$	23.7724	1.30665	0.653325	+	0.757078i	$0.273374\pi$
$332$	−1.16445	−0.0639075
$333$	−8.83331	−0.484063
$334$	−20.4447	−1.11868
$335$	−7.20712	−0.393767
$336$	−12.6760	−0.691530
$337$	−14.3693	−0.782747	−0.391374	−	0.920232i	$-0.628000\pi$
$337$	−14.3693	−0.782747	−0.391374	+	0.920232i	$0.628000\pi$
$338$	1.46257	0.0795535
$339$	6.39165	0.347147
$340$	0.696441	0.0377698
$341$	5.96403	0.322970
$342$	6.42469	0.347407
$343$	−15.3024	−0.826250
$344$	12.9492	0.698171
$345$	8.46752	0.455876
$346$	5.06366	0.272224
$347$	−4.32069	−0.231947	−0.115973	−	0.993252i	$-0.536999\pi$
$347$	−4.32069	−0.231947	−0.115973	+	0.993252i	$0.536999\pi$
$348$	−1.15371	−0.0618454
$349$	0.408767	0.0218808	0.0109404	−	0.999940i	$-0.496517\pi$
$349$	0.408767	0.0218808	0.0109404	+	0.999940i	$0.496517\pi$
$350$	4.35314	0.232685
$351$	−1.00000	−0.0533761
$352$	−4.68528	−0.249726
$353$	−0.449093	−0.0239028	−0.0119514	−	0.999929i	$-0.503804\pi$
$353$	−0.449093	−0.0239028	−0.0119514	+	0.999929i	$0.503804\pi$
$354$	17.8695	0.949755
$355$	−9.06675	−0.481213
$356$	−0.117651	−0.00623550
$357$	14.8997	0.788576
$358$	26.0148	1.37492
$359$	−21.7426	−1.14753	−0.573766	−	0.819019i	$-0.694518\pi$
$359$	−21.7426	−1.14753	−0.573766	+	0.819019i	$0.694518\pi$
$360$	−2.72167	−0.143445
$361$	0.296062	0.0155822
$362$	1.20282	0.0632191
$363$	24.5696	1.28957
$364$	−0.414073	−0.0217033
$365$	11.6141	0.607911
$366$	3.47851	0.181825
$367$	−26.9904	−1.40889	−0.704443	−	0.709761i	$-0.748803\pi$
$367$	−26.9904	−1.40889	−0.704443	+	0.709761i	$0.748803\pi$
$368$	−36.0622	−1.87987
$369$	−10.8134	−0.562925
$370$	−12.9194	−0.671646
$371$	0.872238	0.0452843
$372$	0.139121	0.00721307
$373$	−22.5549	−1.16785	−0.583924	−	0.811808i	$-0.698483\pi$
$373$	−22.5549	−1.16785	−0.583924	+	0.811808i	$0.698483\pi$
$374$	43.6667	2.25795
$375$	1.00000	0.0516398
$376$	17.8101	0.918486
$377$	8.29288	0.427105
$378$	4.35314	0.223901
$379$	−21.0375	−1.08063	−0.540313	−	0.841464i	$-0.681694\pi$
$379$	−21.0375	−1.08063	−0.540313	+	0.841464i	$0.681694\pi$
$380$	0.611119	0.0313498
$381$	18.4089	0.943117
$382$	−11.7270	−0.600006
$383$	17.2515	0.881512	0.440756	−	0.897627i	$-0.354710\pi$
$383$	17.2515	0.881512	0.440756	+	0.897627i	$0.354710\pi$
$384$	12.3486	0.630161
$385$	17.7511	0.904678
$386$	−17.5493	−0.893237
$387$	−4.75779	−0.241852
$388$	1.37475	0.0697923
$389$	−36.4164	−1.84638	−0.923192	−	0.384340i	$-0.874429\pi$
$389$	−36.4164	−1.84638	−0.923192	+	0.384340i	$0.874429\pi$
$390$	−1.46257	−0.0740603
$391$	42.3886	2.14368
$392$	−5.05875	−0.255506
$393$	3.98428	0.200980
$394$	31.3015	1.57695
$395$	10.1346	0.509927
$396$	0.829719	0.0416950
$397$	−18.3011	−0.918506	−0.459253	−	0.888305i	$-0.651883\pi$
$397$	−18.3011	−0.918506	−0.459253	+	0.888305i	$0.651883\pi$
$398$	−6.57108	−0.329378
$399$	13.0743	0.654535
$400$	−4.25889	−0.212944
$401$	−6.43276	−0.321237	−0.160618	−	0.987017i	$-0.551349\pi$
$401$	−6.43276	−0.321237	−0.160618	+	0.987017i	$0.551349\pi$
$402$	−10.5409	−0.525735
$403$	−1.00000	−0.0498135
$404$	−2.02607	−0.100801
$405$	1.00000	0.0496904
$406$	−36.1000	−1.79161
$407$	−52.6821	−2.61135
$408$	−13.6248	−0.674526
$409$	−1.29503	−0.0640353	−0.0320176	−	0.999487i	$-0.510193\pi$
$409$	−1.29503	−0.0640353	−0.0320176	+	0.999487i	$0.510193\pi$
$410$	−15.8154	−0.781068
$411$	20.4456	1.00851
$412$	−2.02453	−0.0997416
$413$	36.3647	1.78939
$414$	12.3844	0.608659
$415$	−8.37007	−0.410871
$416$	0.785589	0.0385167
$417$	15.6927	0.768475
$418$	38.3170	1.87415
$419$	31.3367	1.53090	0.765450	−	0.643496i	$-0.222517\pi$
$419$	31.3367	1.53090	0.765450	+	0.643496i	$0.222517\pi$
$420$	0.414073	0.0202047
$421$	16.4896	0.803652	0.401826	−	0.915716i	$-0.368376\pi$
$421$	16.4896	0.803652	0.401826	+	0.915716i	$0.368376\pi$
$422$	−13.4011	−0.652353
$423$	−6.54381	−0.318171
$424$	−0.797602	−0.0387350
$425$	5.00602	0.242828
$426$	−13.2608	−0.642487
$427$	7.07882	0.342568
$428$	−2.49442	−0.120572
$429$	−5.96403	−0.287946
$430$	−6.95862	−0.335574
$431$	17.0472	0.821135	0.410567	−	0.911830i	$-0.365331\pi$
$431$	17.0472	0.821135	0.410567	+	0.911830i	$0.365331\pi$
$432$	−4.25889	−0.204906
$433$	9.42596	0.452983	0.226491	−	0.974013i	$-0.427274\pi$
$433$	9.42596	0.452983	0.226491	+	0.974013i	$0.427274\pi$
$434$	4.35314	0.208957
$435$	−8.29288	−0.397613
$436$	0.529691	0.0253676
$437$	37.1955	1.77930
$438$	16.9865	0.811647
$439$	−2.20162	−0.105077	−0.0525387	−	0.998619i	$-0.516731\pi$
$439$	−2.20162	−0.105077	−0.0525387	+	0.998619i	$0.516731\pi$
$440$	−16.2321	−0.773836
$441$	1.85869	0.0885092
$442$	−7.32168	−0.348257
$443$	−33.3465	−1.58434	−0.792170	−	0.610301i	$-0.791049\pi$
$443$	−33.3465	−1.58434	−0.792170	+	0.610301i	$0.791049\pi$
$444$	−1.22890	−0.0583209
$445$	−0.845677	−0.0400890
$446$	−27.3022	−1.29280
$447$	7.14445	0.337921
$448$	21.9321	1.03620
$449$	−1.71464	−0.0809189	−0.0404595	−	0.999181i	$-0.512882\pi$
$449$	−1.71464	−0.0809189	−0.0404595	+	0.999181i	$0.512882\pi$
$450$	1.46257	0.0689464
$451$	−64.4916	−3.03679
$452$	0.889211	0.0418250
$453$	−15.3524	−0.721319
$454$	−23.6544	−1.11016
$455$	−2.97636	−0.139534
$456$	−11.9556	−0.559871
$457$	17.0582	0.797949	0.398975	−	0.916962i	$-0.369366\pi$
$457$	17.0582	0.797949	0.398975	+	0.916962i	$0.369366\pi$
$458$	3.28957	0.153712
$459$	5.00602	0.233661
$460$	1.17801	0.0549249
$461$	18.6067	0.866602	0.433301	−	0.901249i	$-0.357349\pi$
$461$	18.6067	0.866602	0.433301	+	0.901249i	$0.357349\pi$
$462$	25.9622	1.20787
$463$	18.0696	0.839763	0.419882	−	0.907579i	$-0.362072\pi$
$463$	18.0696	0.839763	0.419882	+	0.907579i	$0.362072\pi$
$464$	35.3184	1.63962
$465$	1.00000	0.0463739
$466$	1.02559	0.0475097
$467$	32.8767	1.52135	0.760676	−	0.649132i	$-0.224868\pi$
$467$	32.8767	1.52135	0.760676	+	0.649132i	$0.224868\pi$
$468$	−0.139121	−0.00643086
$469$	−21.4510	−0.990513
$470$	−9.57080	−0.441468
$471$	19.7416	0.909645
$472$	−33.2530	−1.53060
$473$	−28.3756	−1.30471
$474$	14.8226	0.680825
$475$	4.39273	0.201552
$476$	2.07286	0.0950093
$477$	0.293056	0.0134181
$478$	12.6250	0.577454
$479$	9.14903	0.418030	0.209015	−	0.977912i	$-0.432974\pi$
$479$	9.14903	0.418030	0.209015	+	0.977912i	$0.432974\pi$
$480$	−0.785589	−0.0358571
$481$	8.83331	0.402764
$482$	−37.8268	−1.72297
$483$	25.2024	1.14675
$484$	3.41814	0.155370
$485$	9.88170	0.448705
$486$	1.46257	0.0663437
$487$	27.6498	1.25293	0.626466	−	0.779449i	$-0.284501\pi$
$487$	27.6498	1.25293	0.626466	+	0.779449i	$0.284501\pi$
$488$	−6.47309	−0.293023
$489$	−0.558383	−0.0252510
$490$	2.71847	0.122808
$491$	0.827521	0.0373455	0.0186727	−	0.999826i	$-0.494056\pi$
$491$	0.827521	0.0373455	0.0186727	+	0.999826i	$0.494056\pi$
$492$	−1.50437	−0.0678223
$493$	−41.5143	−1.86971
$494$	−6.42469	−0.289060
$495$	5.96403	0.268063
$496$	−4.25889	−0.191230
$497$	−26.9859	−1.21048
$498$	−12.2418	−0.548570
$499$	34.5946	1.54867	0.774333	−	0.632778i	$-0.218085\pi$
$499$	34.5946	1.54867	0.774333	+	0.632778i	$0.218085\pi$
$500$	0.139121	0.00622167
$501$	−13.9786	−0.624517
$502$	−41.6260	−1.85786
$503$	−4.89696	−0.218345	−0.109172	−	0.994023i	$-0.534820\pi$
$503$	−4.89696	−0.218345	−0.109172	+	0.994023i	$0.534820\pi$
$504$	−8.10066	−0.360832
$505$	−14.5634	−0.648062
$506$	73.8607	3.28351
$507$	1.00000	0.0444116
$508$	2.56106	0.113629
$509$	−25.6310	−1.13607	−0.568036	−	0.823004i	$-0.692297\pi$
$509$	−25.6310	−1.13607	−0.568036	+	0.823004i	$0.692297\pi$
$510$	7.32168	0.324209
$511$	34.5678	1.52919
$512$	−19.8369	−0.876673
$513$	4.39273	0.193944
$514$	32.5451	1.43550
$515$	−14.5524	−0.641253
$516$	−0.661907	−0.0291388
$517$	−39.0275	−1.71643
$518$	−38.4526	−1.68951
$519$	3.46216	0.151972
$520$	2.72167	0.119353
$521$	0.0627846	0.00275064	0.00137532	−	0.999999i	$-0.499562\pi$
$521$	0.0627846	0.00275064	0.00137532	+	0.999999i	$0.499562\pi$
$522$	−12.1289	−0.530869
$523$	−13.5580	−0.592851	−0.296426	−	0.955056i	$-0.595795\pi$
$523$	−13.5580	−0.592851	−0.296426	+	0.955056i	$0.595795\pi$
$524$	0.554295	0.0242145
$525$	2.97636	0.129899
$526$	−2.88732	−0.125893
$527$	5.00602	0.218066
$528$	−25.4001	−1.10540
$529$	48.6989	2.11735
$530$	0.428615	0.0186179
$531$	12.2179	0.530210
$532$	1.81891	0.0788597
$533$	10.8134	0.468382
$534$	−1.23687	−0.0535244
$535$	−17.9299	−0.775176
$536$	19.6154	0.847258
$537$	17.7870	0.767565
$538$	22.6659	0.977197
$539$	11.0853	0.477477
$540$	0.139121	0.00598680
$541$	18.4866	0.794803	0.397401	−	0.917645i	$-0.369912\pi$
$541$	18.4866	0.794803	0.397401	+	0.917645i	$0.369912\pi$
$542$	−31.5562	−1.35545
$543$	0.822403	0.0352927
$544$	−3.93268	−0.168612
$545$	3.80742	0.163092
$546$	−4.35314	−0.186297
$547$	10.7189	0.458307	0.229154	−	0.973390i	$-0.426404\pi$
$547$	10.7189	0.458307	0.229154	+	0.973390i	$0.426404\pi$
$548$	2.84441	0.121507
$549$	2.37835	0.101506
$550$	8.72282	0.371942
$551$	−36.4283	−1.55190
$552$	−23.0458	−0.980895
$553$	30.1642	1.28271
$554$	−23.8569	−1.01358
$555$	−8.83331	−0.374953
$556$	2.18318	0.0925875
$557$	−30.1283	−1.27658	−0.638289	−	0.769797i	$-0.720357\pi$
$557$	−30.1283	−1.27658	−0.638289	+	0.769797i	$0.720357\pi$
$558$	1.46257	0.0619157
$559$	4.75779	0.201233
$560$	−12.6760	−0.535657
$561$	29.8560	1.26052
$562$	−12.7270	−0.536854
$563$	−30.3725	−1.28005	−0.640024	−	0.768355i	$-0.721076\pi$
$563$	−30.3725	−1.28005	−0.640024	+	0.768355i	$0.721076\pi$
$564$	−0.910380	−0.0383339
$565$	6.39165	0.268899
$566$	34.0781	1.43241
$567$	2.97636	0.124995
$568$	24.6767	1.03541
$569$	−19.5014	−0.817542	−0.408771	−	0.912637i	$-0.634043\pi$
$569$	−19.5014	−0.817542	−0.408771	+	0.912637i	$0.634043\pi$
$570$	6.42469	0.269101
$571$	−2.03442	−0.0851380	−0.0425690	−	0.999094i	$-0.513554\pi$
$571$	−2.03442	−0.0851380	−0.0425690	+	0.999094i	$0.513554\pi$
$572$	−0.829719	−0.0346923
$573$	−8.01806	−0.334959
$574$	−47.0723	−1.96476
$575$	8.46752	0.353120
$576$	7.36879	0.307033
$577$	−41.9943	−1.74824	−0.874122	−	0.485706i	$-0.838563\pi$
$577$	−41.9943	−1.74824	−0.874122	+	0.485706i	$0.838563\pi$
$578$	11.7887	0.490347
$579$	−11.9989	−0.498659
$580$	−1.15371	−0.0479052
$581$	−24.9123	−1.03354
$582$	14.4527	0.599084
$583$	1.74779	0.0723861
$584$	−31.6098	−1.30802
$585$	−1.00000	−0.0413449
$586$	−10.0775	−0.416298
$587$	22.8787	0.944304	0.472152	−	0.881517i	$-0.343477\pi$
$587$	22.8787	0.944304	0.472152	+	0.881517i	$0.343477\pi$
$588$	0.258583	0.0106638
$589$	4.39273	0.180999
$590$	17.8695	0.735677
$591$	21.4017	0.880348
$592$	37.6201	1.54618
$593$	3.09666	0.127164	0.0635822	−	0.997977i	$-0.479747\pi$
$593$	3.09666	0.127164	0.0635822	+	0.997977i	$0.479747\pi$
$594$	8.72282	0.357902
$595$	14.8997	0.610828
$596$	0.993940	0.0407134
$597$	−4.49282	−0.183879
$598$	−12.3844	−0.506435
$599$	−10.6834	−0.436511	−0.218256	−	0.975892i	$-0.570037\pi$
$599$	−10.6834	−0.436511	−0.218256	+	0.975892i	$0.570037\pi$
$600$	−2.72167	−0.111112
$601$	1.74814	0.0713083	0.0356542	−	0.999364i	$-0.488649\pi$
$601$	1.74814	0.0713083	0.0356542	+	0.999364i	$0.488649\pi$
$602$	−20.7113	−0.844131
$603$	−7.20712	−0.293497
$604$	−2.13584	−0.0869060
$605$	24.5696	0.998896
$606$	−21.3000	−0.865254
$607$	−13.0508	−0.529715	−0.264857	−	0.964288i	$-0.585325\pi$
$607$	−13.0508	−0.529715	−0.264857	+	0.964288i	$0.585325\pi$
$608$	−3.45088	−0.139952
$609$	−24.6825	−1.00019
$610$	3.47851	0.140841
$611$	6.54381	0.264734
$612$	0.696441	0.0281520
$613$	18.5170	0.747896	0.373948	−	0.927450i	$-0.378004\pi$
$613$	18.5170	0.747896	0.373948	+	0.927450i	$0.378004\pi$
$614$	−5.77701	−0.233141
$615$	−10.8134	−0.436040
$616$	−48.3126	−1.94657
$617$	−36.0791	−1.45249	−0.726245	−	0.687436i	$-0.758736\pi$
$617$	−36.0791	−1.45249	−0.726245	+	0.687436i	$0.758736\pi$
$618$	−21.2839	−0.856163
$619$	21.1784	0.851233	0.425616	−	0.904904i	$-0.360057\pi$
$619$	21.1784	0.851233	0.425616	+	0.904904i	$0.360057\pi$
$620$	0.139121	0.00558722
$621$	8.46752	0.339790
$622$	−9.66654	−0.387593
$623$	−2.51704	−0.100843
$624$	4.25889	0.170492
$625$	1.00000	0.0400000
$626$	−46.4452	−1.85633
$627$	26.1983	1.04626
$628$	2.74647	0.109596
$629$	−44.2198	−1.76316
$630$	4.35314	0.173433
$631$	−20.9746	−0.834987	−0.417494	−	0.908680i	$-0.637091\pi$
$631$	−20.9746	−0.834987	−0.417494	+	0.908680i	$0.637091\pi$
$632$	−27.5831	−1.09720
$633$	−9.16265	−0.364183
$634$	16.6823	0.662540
$635$	18.4089	0.730535
$636$	0.0407701	0.00161664
$637$	−1.85869	−0.0736441
$638$	−72.3373	−2.86386
$639$	−9.06675	−0.358675
$640$	12.3486	0.488120
$641$	39.0717	1.54324	0.771620	−	0.636084i	$-0.219447\pi$
$641$	39.0717	1.54324	0.771620	+	0.636084i	$0.219447\pi$
$642$	−26.2237	−1.03497
$643$	7.19771	0.283850	0.141925	−	0.989877i	$-0.454671\pi$
$643$	7.19771	0.283850	0.141925	+	0.989877i	$0.454671\pi$
$644$	3.50617	0.138162
$645$	−4.75779	−0.187338
$646$	32.1621	1.26540
$647$	−8.61910	−0.338852	−0.169426	−	0.985543i	$-0.554191\pi$
$647$	−8.61910	−0.338852	−0.169426	+	0.985543i	$0.554191\pi$
$648$	−2.72167	−0.106917
$649$	72.8677	2.86031
$650$	−1.46257	−0.0573668
$651$	2.97636	0.116653
$652$	−0.0776826	−0.00304229
$653$	−16.0104	−0.626535	−0.313267	−	0.949665i	$-0.601424\pi$
$653$	−16.0104	−0.626535	−0.313267	+	0.949665i	$0.601424\pi$
$654$	5.56863	0.217751
$655$	3.98428	0.155679
$656$	46.0532	1.79807
$657$	11.6141	0.453110
$658$	−28.4861	−1.11050
$659$	35.0819	1.36660	0.683298	−	0.730140i	$-0.260545\pi$
$659$	35.0819	1.36660	0.683298	+	0.730140i	$0.260545\pi$
$660$	0.829719	0.0322968
$661$	−44.5253	−1.73183	−0.865917	−	0.500188i	$-0.833264\pi$
$661$	−44.5253	−1.73183	−0.865917	+	0.500188i	$0.833264\pi$
$662$	34.7689	1.35133
$663$	−5.00602	−0.194418
$664$	22.7806	0.884058
$665$	13.0743	0.507000
$666$	−12.9194	−0.500615
$667$	−70.2201	−2.71893
$668$	−1.94471	−0.0752431
$669$	−18.6672	−0.721717
$670$	−10.5409	−0.407232
$671$	14.1845	0.547588
$672$	−2.33819	−0.0901978
$673$	−6.40394	−0.246854	−0.123427	−	0.992354i	$-0.539388\pi$
$673$	−6.40394	−0.246854	−0.123427	+	0.992354i	$0.539388\pi$
$674$	−21.0162	−0.809514
$675$	1.00000	0.0384900
$676$	0.139121	0.00535080
$677$	−3.54990	−0.136434	−0.0682169	−	0.997671i	$-0.521731\pi$
$677$	−3.54990	−0.136434	−0.0682169	+	0.997671i	$0.521731\pi$
$678$	9.34826	0.359018
$679$	29.4115	1.12871
$680$	−13.6248	−0.522485
$681$	−16.1731	−0.619756
$682$	8.72282	0.334014
$683$	8.91620	0.341169	0.170584	−	0.985343i	$-0.445434\pi$
$683$	8.91620	0.341169	0.170584	+	0.985343i	$0.445434\pi$
$684$	0.611119	0.0233667
$685$	20.4456	0.781186
$686$	−22.3808	−0.854504
$687$	2.24917	0.0858111
$688$	20.2629	0.772516
$689$	−0.293056	−0.0111645
$690$	12.3844	0.471465
$691$	38.2667	1.45574	0.727868	−	0.685717i	$-0.240511\pi$
$691$	38.2667	1.45574	0.727868	+	0.685717i	$0.240511\pi$
$692$	0.481658	0.0183099
$693$	17.7511	0.674307
$694$	−6.31933	−0.239878
$695$	15.6927	0.595258
$696$	22.5705	0.855532
$697$	−54.1323	−2.05041
$698$	0.597852	0.0226290
$699$	0.701226	0.0265228
$700$	0.414073	0.0156505
$701$	1.29555	0.0489322	0.0244661	−	0.999701i	$-0.492211\pi$
$701$	1.29555	0.0489322	0.0244661	+	0.999701i	$0.492211\pi$
$702$	−1.46257	−0.0552013
$703$	−38.8023	−1.46346
$704$	43.9477	1.65634
$705$	−6.54381	−0.246454
$706$	−0.656832	−0.0247202
$707$	−43.3458	−1.63019
$708$	1.69976	0.0638808
$709$	25.8751	0.971760	0.485880	−	0.874026i	$-0.338499\pi$
$709$	25.8751	0.971760	0.485880	+	0.874026i	$0.338499\pi$
$710$	−13.2608	−0.497668
$711$	10.1346	0.380077
$712$	2.30166	0.0862582
$713$	8.46752	0.317111
$714$	21.7919	0.815542
$715$	−5.96403	−0.223042
$716$	2.47454	0.0924778
$717$	8.63204	0.322369
$718$	−31.8002	−1.18677
$719$	15.1417	0.564689	0.282345	−	0.959313i	$-0.408888\pi$
$719$	15.1417	0.564689	0.282345	+	0.959313i	$0.408888\pi$
$720$	−4.25889	−0.158719
$721$	−43.3130	−1.61306
$722$	0.433012	0.0161150
$723$	−25.8632	−0.961863
$724$	0.114413	0.00425214
$725$	−8.29288	−0.307990
$726$	35.9348	1.33367
$727$	49.1071	1.82128	0.910640	−	0.413200i	$-0.135589\pi$
$727$	49.1071	1.82128	0.910640	+	0.413200i	$0.135589\pi$
$728$	8.10066	0.300231
$729$	1.00000	0.0370370
$730$	16.9865	0.628699
$731$	−23.8176	−0.880926
$732$	0.330878	0.0122296
$733$	24.6757	0.911417	0.455709	−	0.890129i	$-0.349386\pi$
$733$	24.6757	0.911417	0.455709	+	0.890129i	$0.349386\pi$
$734$	−39.4754	−1.45706
$735$	1.85869	0.0685589
$736$	−6.65200	−0.245196
$737$	−42.9835	−1.58332
$738$	−15.8154	−0.582174
$739$	−11.2945	−0.415474	−0.207737	−	0.978185i	$-0.566610\pi$
$739$	−11.2945	−0.415474	−0.207737	+	0.978185i	$0.566610\pi$
$740$	−1.22890	−0.0451751
$741$	−4.39273	−0.161371
$742$	1.27571	0.0468329
$743$	13.3701	0.490503	0.245251	−	0.969459i	$-0.421129\pi$
$743$	13.3701	0.490503	0.245251	+	0.969459i	$0.421129\pi$
$744$	−2.72167	−0.0997814
$745$	7.14445	0.261752
$746$	−32.9882	−1.20778
$747$	−8.37007	−0.306245
$748$	4.15359	0.151870
$749$	−53.3657	−1.94994
$750$	1.46257	0.0534056
$751$	−41.8835	−1.52835	−0.764175	−	0.645009i	$-0.776854\pi$
$751$	−41.8835	−1.52835	−0.764175	+	0.645009i	$0.776854\pi$
$752$	27.8693	1.01629
$753$	−28.4608	−1.03717
$754$	12.1289	0.441710
$755$	−15.3524	−0.558731
$756$	0.414073	0.0150597
$757$	−2.70314	−0.0982474	−0.0491237	−	0.998793i	$-0.515643\pi$
$757$	−2.70314	−0.0982474	−0.0491237	+	0.998793i	$0.515643\pi$
$758$	−30.7689	−1.11758
$759$	50.5005	1.83305
$760$	−11.9556	−0.433674
$761$	21.8455	0.791899	0.395949	−	0.918272i	$-0.370416\pi$
$761$	21.8455	0.791899	0.395949	+	0.918272i	$0.370416\pi$
$762$	26.9244	0.975367
$763$	11.3322	0.410255
$764$	−1.11548	−0.0403566
$765$	5.00602	0.180993
$766$	25.2316	0.911656
$767$	−12.2179	−0.441162
$768$	3.32312	0.119913
$769$	−31.7281	−1.14415	−0.572073	−	0.820203i	$-0.693861\pi$
$769$	−31.7281	−1.14415	−0.572073	+	0.820203i	$0.693861\pi$
$770$	25.9622	0.935614
$771$	22.2519	0.801384
$772$	−1.66930	−0.0600794
$773$	14.4256	0.518852	0.259426	−	0.965763i	$-0.416467\pi$
$773$	14.4256	0.518852	0.259426	+	0.965763i	$0.416467\pi$
$774$	−6.95862	−0.250122
$775$	1.00000	0.0359211
$776$	−26.8948	−0.965466
$777$	−26.2911	−0.943187
$778$	−53.2616	−1.90952
$779$	−47.5005	−1.70188
$780$	−0.139121	−0.00498132
$781$	−54.0743	−1.93493
$782$	61.9965	2.21699
$783$	−8.29288	−0.296363
$784$	−7.91596	−0.282713
$785$	19.7416	0.704608
$786$	5.82729	0.207853
$787$	−0.356684	−0.0127144	−0.00635721	−	0.999980i	$-0.502024\pi$
$787$	−0.356684	−0.0127144	−0.00635721	+	0.999980i	$0.502024\pi$
$788$	2.97742	0.106066
$789$	−1.97414	−0.0702811
$790$	14.8226	0.527365
$791$	19.0238	0.676410
$792$	−16.2321	−0.576783
$793$	−2.37835	−0.0844577
$794$	−26.7667	−0.949915
$795$	0.293056	0.0103936
$796$	−0.625044	−0.0221541
$797$	−13.9673	−0.494747	−0.247373	−	0.968920i	$-0.579567\pi$
$797$	−13.9673	−0.494747	−0.247373	+	0.968920i	$0.579567\pi$
$798$	19.1222	0.676917
$799$	−32.7585	−1.15891
$800$	−0.785589	−0.0277748
$801$	−0.845677	−0.0298805
$802$	−9.40839	−0.332222
$803$	69.2669	2.44438
$804$	−1.00266	−0.0353611
$805$	25.2024	0.888266
$806$	−1.46257	−0.0515169
$807$	15.4973	0.545530
$808$	39.6368	1.39442
$809$	2.28504	0.0803377	0.0401688	−	0.999193i	$-0.487210\pi$
$809$	2.28504	0.0803377	0.0401688	+	0.999193i	$0.487210\pi$
$810$	1.46257	0.0513896
$811$	−17.6830	−0.620934	−0.310467	−	0.950584i	$-0.600485\pi$
$811$	−17.6830	−0.620934	−0.310467	+	0.950584i	$0.600485\pi$
$812$	−3.43385	−0.120505
$813$	−21.5758	−0.756697
$814$	−77.0514	−2.70065
$815$	−0.558383	−0.0195593
$816$	−21.3201	−0.746352
$817$	−20.8997	−0.731187
$818$	−1.89408	−0.0662250
$819$	−2.97636	−0.104002
$820$	−1.50437	−0.0525349
$821$	48.4711	1.69165	0.845827	−	0.533457i	$-0.179107\pi$
$821$	48.4711	1.69165	0.845827	+	0.533457i	$0.179107\pi$
$822$	29.9032	1.04299
$823$	−29.1293	−1.01538	−0.507691	−	0.861539i	$-0.669501\pi$
$823$	−29.1293	−1.01538	−0.507691	+	0.861539i	$0.669501\pi$
$824$	39.6067	1.37977
$825$	5.96403	0.207641
$826$	53.1861	1.85058
$827$	25.6520	0.892006	0.446003	−	0.895031i	$-0.352847\pi$
$827$	25.6520	0.892006	0.446003	+	0.895031i	$0.352847\pi$
$828$	1.17801	0.0409386
$829$	9.69906	0.336862	0.168431	−	0.985713i	$-0.446130\pi$
$829$	9.69906	0.336862	0.168431	+	0.985713i	$0.446130\pi$
$830$	−12.2418	−0.424921
$831$	−16.3116	−0.565842
$832$	−7.36879	−0.255467
$833$	9.30466	0.322387
$834$	22.9517	0.794754
$835$	−13.9786	−0.483749
$836$	3.64473	0.126056
$837$	1.00000	0.0345651
$838$	45.8323	1.58325
$839$	−16.1450	−0.557386	−0.278693	−	0.960380i	$-0.589901\pi$
$839$	−16.1450	−0.557386	−0.278693	+	0.960380i	$0.589901\pi$
$840$	−8.10066	−0.279499
$841$	39.7718	1.37144
$842$	24.1172	0.831134
$843$	−8.70176	−0.299705
$844$	−1.27471	−0.0438775
$845$	1.00000	0.0344010
$846$	−9.57080	−0.329051
$847$	73.1279	2.51270
$848$	−1.24809	−0.0428596
$849$	23.3001	0.799658
$850$	7.32168	0.251131
$851$	−74.7963	−2.56398
$852$	−1.26137	−0.0432139
$853$	35.4709	1.21450	0.607251	−	0.794510i	$-0.292272\pi$
$853$	35.4709	1.21450	0.607251	+	0.794510i	$0.292272\pi$
$854$	10.3533	0.354282
$855$	4.39273	0.150228
$856$	48.7992	1.66792
$857$	35.6471	1.21768	0.608841	−	0.793292i	$-0.291635\pi$
$857$	35.6471	1.21768	0.608841	+	0.793292i	$0.291635\pi$
$858$	−8.72282	−0.297792
$859$	33.9236	1.15746	0.578728	−	0.815520i	$-0.303549\pi$
$859$	33.9236	1.15746	0.578728	+	0.815520i	$0.303549\pi$
$860$	−0.661907	−0.0225709
$861$	−32.1846	−1.09685
$862$	24.9328	0.849214
$863$	16.1850	0.550943	0.275472	−	0.961309i	$-0.411166\pi$
$863$	16.1850	0.550943	0.275472	+	0.961309i	$0.411166\pi$
$864$	−0.785589	−0.0267263
$865$	3.46216	0.117717
$866$	13.7862	0.468473
$867$	8.06027	0.273741
$868$	0.414073	0.0140545
$869$	60.4431	2.05039
$870$	−12.1289	−0.411210
$871$	7.20712	0.244204
$872$	−10.3626	−0.350920
$873$	9.88170	0.334445
$874$	54.4012	1.84015
$875$	2.97636	0.100619
$876$	1.61576	0.0545916
$877$	28.0594	0.947500	0.473750	−	0.880659i	$-0.342900\pi$
$877$	28.0594	0.947500	0.473750	+	0.880659i	$0.342900\pi$
$878$	−3.22002	−0.108671
$879$	−6.89027	−0.232403
$880$	−25.4001	−0.856237
$881$	−35.0764	−1.18175	−0.590876	−	0.806762i	$-0.701218\pi$
$881$	−35.0764	−1.18175	−0.590876	+	0.806762i	$0.701218\pi$
$882$	2.71847	0.0915358
$883$	−15.4011	−0.518289	−0.259144	−	0.965839i	$-0.583441\pi$
$883$	−15.4011	−0.518289	−0.259144	+	0.965839i	$0.583441\pi$
$884$	−0.696441	−0.0234239
$885$	12.2179	0.410699
$886$	−48.7717	−1.63852
$887$	41.4800	1.39276	0.696380	−	0.717673i	$-0.254793\pi$
$887$	41.4800	1.39276	0.696380	+	0.717673i	$0.254793\pi$
$888$	24.0414	0.806776
$889$	54.7914	1.83765
$890$	−1.23687	−0.0414598
$891$	5.96403	0.199802
$892$	−2.59700	−0.0869540
$893$	−28.7452	−0.961921
$894$	10.4493	0.349476
$895$	17.7870	0.594553
$896$	36.7538	1.22786
$897$	−8.46752	−0.282722
$898$	−2.50779	−0.0836860
$899$	−8.29288	−0.276583
$900$	0.139121	0.00463736
$901$	1.46704	0.0488743
$902$	−94.3236	−3.14063
$903$	−14.1609	−0.471245
$904$	−17.3960	−0.578582
$905$	0.822403	0.0273376
$906$	−22.4540	−0.745985
$907$	16.8558	0.559688	0.279844	−	0.960045i	$-0.409717\pi$
$907$	16.8558	0.559688	0.279844	+	0.960045i	$0.409717\pi$
$908$	−2.25002	−0.0746695
$909$	−14.5634	−0.483037
$910$	−4.35314	−0.144305
$911$	4.01004	0.132859	0.0664293	−	0.997791i	$-0.478839\pi$
$911$	4.01004	0.132859	0.0664293	+	0.997791i	$0.478839\pi$
$912$	−18.7081	−0.619488
$913$	−49.9193	−1.65209
$914$	24.9489	0.825236
$915$	2.37835	0.0786258
$916$	0.312906	0.0103387
$917$	11.8586	0.391606
$918$	7.32168	0.241651
$919$	−55.0278	−1.81520	−0.907600	−	0.419835i	$-0.862088\pi$
$919$	−55.0278	−1.81520	−0.907600	+	0.419835i	$0.862088\pi$
$920$	−23.0458	−0.759798
$921$	−3.94989	−0.130153
$922$	27.2137	0.896236
$923$	9.06675	0.298436
$924$	2.46954	0.0812419
$925$	−8.83331	−0.290438
$926$	26.4281	0.868480
$927$	−14.5524	−0.477962
$928$	6.51480	0.213859
$929$	16.7774	0.550447	0.275224	−	0.961380i	$-0.411248\pi$
$929$	16.7774	0.550447	0.275224	+	0.961380i	$0.411248\pi$
$930$	1.46257	0.0479597
$931$	8.16473	0.267588
$932$	0.0975550	0.00319552
$933$	−6.60927	−0.216378
$934$	48.0846	1.57337
$935$	29.8560	0.976397
$936$	2.72167	0.0889606
$937$	−30.9733	−1.01185	−0.505927	−	0.862576i	$-0.668850\pi$
$937$	−30.9733	−1.01185	−0.505927	+	0.862576i	$0.668850\pi$
$938$	−31.3736	−1.02438
$939$	−31.7558	−1.03631
$940$	−0.910380	−0.0296933
$941$	21.2003	0.691109	0.345555	−	0.938399i	$-0.387691\pi$
$941$	21.2003	0.691109	0.345555	+	0.938399i	$0.387691\pi$
$942$	28.8735	0.940751
$943$	−91.5630	−2.98170
$944$	−52.0345	−1.69358
$945$	2.97636	0.0968209
$946$	−41.5014	−1.34933
$947$	27.0942	0.880443	0.440222	−	0.897889i	$-0.354900\pi$
$947$	27.0942	0.880443	0.440222	+	0.897889i	$0.354900\pi$
$948$	1.40993	0.0457925
$949$	−11.6141	−0.377010
$950$	6.42469	0.208444
$951$	11.4062	0.369870
$952$	−40.5521	−1.31430
$953$	−8.15337	−0.264113	−0.132057	−	0.991242i	$-0.542158\pi$
$953$	−8.15337	−0.264113	−0.132057	+	0.991242i	$0.542158\pi$
$954$	0.428615	0.0138769
$955$	−8.01806	−0.259458
$956$	1.20090	0.0388397
$957$	−49.4589	−1.59878
$958$	13.3811	0.432325
$959$	60.8534	1.96506
$960$	7.36879	0.237827
$961$	1.00000	0.0322581
$962$	12.9194	0.416537
$963$	−17.9299	−0.577782
$964$	−3.59811	−0.115887
$965$	−11.9989	−0.386259
$966$	36.8603	1.18596
$967$	−12.2015	−0.392373	−0.196186	−	0.980567i	$-0.562856\pi$
$967$	−12.2015	−0.392373	−0.196186	+	0.980567i	$0.562856\pi$
$968$	−66.8704	−2.14930
$969$	21.9901	0.706424
$970$	14.4527	0.464049
$971$	19.1082	0.613210	0.306605	−	0.951837i	$-0.400807\pi$
$971$	19.1082	0.613210	0.306605	+	0.951837i	$0.400807\pi$
$972$	0.139121	0.00446230
$973$	46.7071	1.49736
$974$	40.4399	1.29578
$975$	−1.00000	−0.0320256
$976$	−10.1291	−0.324225
$977$	15.4466	0.494181	0.247090	−	0.968992i	$-0.420526\pi$
$977$	15.4466	0.494181	0.247090	+	0.968992i	$0.420526\pi$
$978$	−0.816676	−0.0261144
$979$	−5.04364	−0.161195
$980$	0.258583	0.00826012
$981$	3.80742	0.121562
$982$	1.21031	0.0386225
$983$	−33.5436	−1.06987	−0.534937	−	0.844892i	$-0.679665\pi$
$983$	−33.5436	−1.06987	−0.534937	+	0.844892i	$0.679665\pi$
$984$	29.4306	0.938214
$985$	21.4017	0.681915
$986$	−60.7177	−1.93365
$987$	−19.4767	−0.619950
$988$	−0.611119	−0.0194423
$989$	−40.2867	−1.28104
$990$	8.72282	0.277230
$991$	−49.6718	−1.57788	−0.788938	−	0.614473i	$-0.789369\pi$
$991$	−49.6718	−1.57788	−0.788938	+	0.614473i	$0.789369\pi$
$992$	−0.785589	−0.0249425
$993$	23.7724	0.754394
$994$	−39.4688	−1.25187
$995$	−4.49282	−0.142432
$996$	−1.16445	−0.0368970
$997$	−15.1959	−0.481257	−0.240629	−	0.970617i	$-0.577354\pi$
$997$	−15.1959	−0.481257	−0.240629	+	0.970617i	$0.577354\pi$
$998$	50.5971	1.60162
$999$	−8.83331	−0.279474

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6045.2.a.bi.1.13	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6045.2.a.bi.1.13	✓	18	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:	\( N \)	\(=\)	\( 6045 = 3 \cdot 5 \cdot 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6045.a (trivial)

\(f(q)\)	\(=\)	\(q+1.46257 q^{2} +1.00000 q^{3} +0.139121 q^{4} +1.00000 q^{5} +1.46257 q^{6} +2.97636 q^{7} -2.72167 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.46257 q^{2} +1.00000 q^{3} +0.139121 q^{4} +1.00000 q^{5} +1.46257 q^{6} +2.97636 q^{7} -2.72167 q^{8} +1.00000 q^{9} +1.46257 q^{10} +5.96403 q^{11} +0.139121 q^{12} -1.00000 q^{13} +4.35314 q^{14} +1.00000 q^{15} -4.25889 q^{16} +5.00602 q^{17} +1.46257 q^{18} +4.39273 q^{19} +0.139121 q^{20} +2.97636 q^{21} +8.72282 q^{22} +8.46752 q^{23} -2.72167 q^{24} +1.00000 q^{25} -1.46257 q^{26} +1.00000 q^{27} +0.414073 q^{28} -8.29288 q^{29} +1.46257 q^{30} +1.00000 q^{31} -0.785589 q^{32} +5.96403 q^{33} +7.32168 q^{34} +2.97636 q^{35} +0.139121 q^{36} -8.83331 q^{37} +6.42469 q^{38} -1.00000 q^{39} -2.72167 q^{40} -10.8134 q^{41} +4.35314 q^{42} -4.75779 q^{43} +0.829719 q^{44} +1.00000 q^{45} +12.3844 q^{46} -6.54381 q^{47} -4.25889 q^{48} +1.85869 q^{49} +1.46257 q^{50} +5.00602 q^{51} -0.139121 q^{52} +0.293056 q^{53} +1.46257 q^{54} +5.96403 q^{55} -8.10066 q^{56} +4.39273 q^{57} -12.1289 q^{58} +12.2179 q^{59} +0.139121 q^{60} +2.37835 q^{61} +1.46257 q^{62} +2.97636 q^{63} +7.36879 q^{64} -1.00000 q^{65} +8.72282 q^{66} -7.20712 q^{67} +0.696441 q^{68} +8.46752 q^{69} +4.35314 q^{70} -9.06675 q^{71} -2.72167 q^{72} +11.6141 q^{73} -12.9194 q^{74} +1.00000 q^{75} +0.611119 q^{76} +17.7511 q^{77} -1.46257 q^{78} +10.1346 q^{79} -4.25889 q^{80} +1.00000 q^{81} -15.8154 q^{82} -8.37007 q^{83} +0.414073 q^{84} +5.00602 q^{85} -6.95862 q^{86} -8.29288 q^{87} -16.2321 q^{88} -0.845677 q^{89} +1.46257 q^{90} -2.97636 q^{91} +1.17801 q^{92} +1.00000 q^{93} -9.57080 q^{94} +4.39273 q^{95} -0.785589 q^{96} +9.88170 q^{97} +2.71847 q^{98} +5.96403 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 4 q^{2} + 18 q^{3} + 22 q^{4} + 18 q^{5} + 4 q^{6} + 8 q^{7} + 9 q^{8} + 18 q^{9}+O(q^{10}) \) 18 * q + 4 * q^2 + 18 * q^3 + 22 * q^4 + 18 * q^5 + 4 * q^6 + 8 * q^7 + 9 * q^8 + 18 * q^9	\( 18 q + 4 q^{2} + 18 q^{3} + 22 q^{4} + 18 q^{5} + 4 q^{6} + 8 q^{7} + 9 q^{8} + 18 q^{9} + 4 q^{10} + 6 q^{11} + 22 q^{12} - 18 q^{13} + 5 q^{14} + 18 q^{15} + 30 q^{16} + 18 q^{17} + 4 q^{18} + 12 q^{19} + 22 q^{20} + 8 q^{21} + 7 q^{22} + 32 q^{23} + 9 q^{24} + 18 q^{25} - 4 q^{26} + 18 q^{27} + 10 q^{28} + 7 q^{29} + 4 q^{30} + 18 q^{31} + 22 q^{32} + 6 q^{33} + 15 q^{34} + 8 q^{35} + 22 q^{36} + 3 q^{37} + 32 q^{38} - 18 q^{39} + 9 q^{40} + 4 q^{41} + 5 q^{42} + 14 q^{43} - 5 q^{44} + 18 q^{45} + 10 q^{46} + 23 q^{47} + 30 q^{48} + 28 q^{49} + 4 q^{50} + 18 q^{51} - 22 q^{52} + 35 q^{53} + 4 q^{54} + 6 q^{55} - 7 q^{56} + 12 q^{57} - 6 q^{58} + 28 q^{59} + 22 q^{60} + 19 q^{61} + 4 q^{62} + 8 q^{63} + 43 q^{64} - 18 q^{65} + 7 q^{66} + 34 q^{67} + 55 q^{68} + 32 q^{69} + 5 q^{70} - 8 q^{71} + 9 q^{72} + 22 q^{74} + 18 q^{75} + 2 q^{76} + 21 q^{77} - 4 q^{78} + 4 q^{79} + 30 q^{80} + 18 q^{81} + 29 q^{82} + 11 q^{83} + 10 q^{84} + 18 q^{85} - 22 q^{86} + 7 q^{87} - 31 q^{88} + 17 q^{89} + 4 q^{90} - 8 q^{91} + 33 q^{92} + 18 q^{93} - 14 q^{94} + 12 q^{95} + 22 q^{96} + 32 q^{97} + 20 q^{98} + 6 q^{99}+O(q^{100}) \) 18 * q + 4 * q^2 + 18 * q^3 + 22 * q^4 + 18 * q^5 + 4 * q^6 + 8 * q^7 + 9 * q^8 + 18 * q^9 + 4 * q^10 + 6 * q^11 + 22 * q^12 - 18 * q^13 + 5 * q^14 + 18 * q^15 + 30 * q^16 + 18 * q^17 + 4 * q^18 + 12 * q^19 + 22 * q^20 + 8 * q^21 + 7 * q^22 + 32 * q^23 + 9 * q^24 + 18 * q^25 - 4 * q^26 + 18 * q^27 + 10 * q^28 + 7 * q^29 + 4 * q^30 + 18 * q^31 + 22 * q^32 + 6 * q^33 + 15 * q^34 + 8 * q^35 + 22 * q^36 + 3 * q^37 + 32 * q^38 - 18 * q^39 + 9 * q^40 + 4 * q^41 + 5 * q^42 + 14 * q^43 - 5 * q^44 + 18 * q^45 + 10 * q^46 + 23 * q^47 + 30 * q^48 + 28 * q^49 + 4 * q^50 + 18 * q^51 - 22 * q^52 + 35 * q^53 + 4 * q^54 + 6 * q^55 - 7 * q^56 + 12 * q^57 - 6 * q^58 + 28 * q^59 + 22 * q^60 + 19 * q^61 + 4 * q^62 + 8 * q^63 + 43 * q^64 - 18 * q^65 + 7 * q^66 + 34 * q^67 + 55 * q^68 + 32 * q^69 + 5 * q^70 - 8 * q^71 + 9 * q^72 + 22 * q^74 + 18 * q^75 + 2 * q^76 + 21 * q^77 - 4 * q^78 + 4 * q^79 + 30 * q^80 + 18 * q^81 + 29 * q^82 + 11 * q^83 + 10 * q^84 + 18 * q^85 - 22 * q^86 + 7 * q^87 - 31 * q^88 + 17 * q^89 + 4 * q^90 - 8 * q^91 + 33 * q^92 + 18 * q^93 - 14 * q^94 + 12 * q^95 + 22 * q^96 + 32 * q^97 + 20 * q^98 + 6 * q^99

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