LMFDB - Embedded newform 6045.2.a.bg.1.16

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:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} - 27 x^{14} + 51 x^{13} + 294 x^{12} - 517 x^{11} - 1657 x^{10} + 2678 x^{9} + \cdots - 428 $ x^16 - 2x^15 - 27x^14 + 51x^13 + 294x^12 - 517x^11 - 1657x^10 + 2678x^9 + 5124x^8 - 7583x^7 - 8358x^6 + 11579x^5 + 6027x^4 - 8572x^3 - 780x^2 + 2178*x - 428
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Root			$-2.78228$ 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)

$f(q)$	$=$	$q+2.78228 q^{2} -1.00000 q^{3} +5.74107 q^{4} -1.00000 q^{5} -2.78228 q^{6} +2.63079 q^{7} +10.4087 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+2.78228 q^{2} -1.00000 q^{3} +5.74107 q^{4} -1.00000 q^{5} -2.78228 q^{6} +2.63079 q^{7} +10.4087 q^{8} +1.00000 q^{9} -2.78228 q^{10} -2.15059 q^{11} -5.74107 q^{12} -1.00000 q^{13} +7.31960 q^{14} +1.00000 q^{15} +17.4778 q^{16} +2.72792 q^{17} +2.78228 q^{18} +5.12096 q^{19} -5.74107 q^{20} -2.63079 q^{21} -5.98355 q^{22} -1.65770 q^{23} -10.4087 q^{24} +1.00000 q^{25} -2.78228 q^{26} -1.00000 q^{27} +15.1036 q^{28} +0.589015 q^{29} +2.78228 q^{30} +1.00000 q^{31} +27.8106 q^{32} +2.15059 q^{33} +7.58983 q^{34} -2.63079 q^{35} +5.74107 q^{36} +7.51372 q^{37} +14.2479 q^{38} +1.00000 q^{39} -10.4087 q^{40} -2.50392 q^{41} -7.31960 q^{42} -12.6815 q^{43} -12.3467 q^{44} -1.00000 q^{45} -4.61218 q^{46} +3.79567 q^{47} -17.4778 q^{48} -0.0789236 q^{49} +2.78228 q^{50} -2.72792 q^{51} -5.74107 q^{52} -7.11031 q^{53} -2.78228 q^{54} +2.15059 q^{55} +27.3831 q^{56} -5.12096 q^{57} +1.63880 q^{58} +3.51642 q^{59} +5.74107 q^{60} +4.04143 q^{61} +2.78228 q^{62} +2.63079 q^{63} +42.4213 q^{64} +1.00000 q^{65} +5.98355 q^{66} +2.52332 q^{67} +15.6612 q^{68} +1.65770 q^{69} -7.31960 q^{70} -6.17389 q^{71} +10.4087 q^{72} +4.94101 q^{73} +20.9053 q^{74} -1.00000 q^{75} +29.3998 q^{76} -5.65776 q^{77} +2.78228 q^{78} +0.514769 q^{79} -17.4778 q^{80} +1.00000 q^{81} -6.96659 q^{82} +3.63128 q^{83} -15.1036 q^{84} -2.72792 q^{85} -35.2835 q^{86} -0.589015 q^{87} -22.3849 q^{88} -2.32826 q^{89} -2.78228 q^{90} -2.63079 q^{91} -9.51696 q^{92} -1.00000 q^{93} +10.5606 q^{94} -5.12096 q^{95} -27.8106 q^{96} +7.90981 q^{97} -0.219587 q^{98} -2.15059 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 2 q^{2} - 16 q^{3} + 26 q^{4} - 16 q^{5} + 2 q^{6} - 2 q^{7} - 9 q^{8} + 16 q^{9}+O(q^{10}) $ 16 * q - 2 * q^2 - 16 * q^3 + 26 * q^4 - 16 * q^5 + 2 * q^6 - 2 * q^7 - 9 * q^8 + 16 * q^9	\( 16 q - 2 q^{2} - 16 q^{3} + 26 q^{4} - 16 q^{5} + 2 q^{6} - 2 q^{7} - 9 q^{8} + 16 q^{9} + 2 q^{10} + 3 q^{11} - 26 q^{12} - 16 q^{13} - 5 q^{14} + 16 q^{15} + 38 q^{16} - 13 q^{17} - 2 q^{18} - 26 q^{20} + 2 q^{21} + q^{22} - 15 q^{23} + 9 q^{24} + 16 q^{25} + 2 q^{26} - 16 q^{27} + 8 q^{28} - 4 q^{29} - 2 q^{30} + 16 q^{31} - 30 q^{32} - 3 q^{33} + 29 q^{34} + 2 q^{35} + 26 q^{36} + 12 q^{37} + 16 q^{39} + 9 q^{40} - 12 q^{41} + 5 q^{42} - 7 q^{43} - 13 q^{44} - 16 q^{45} + 14 q^{46} + 17 q^{47} - 38 q^{48} + 16 q^{49} - 2 q^{50} + 13 q^{51} - 26 q^{52} - 36 q^{53} + 2 q^{54} - 3 q^{55} + 41 q^{56} + 16 q^{58} + 53 q^{59} + 26 q^{60} + 34 q^{61} - 2 q^{62} - 2 q^{63} + 79 q^{64} + 16 q^{65} - q^{66} - 13 q^{67} - 39 q^{68} + 15 q^{69} + 5 q^{70} - 11 q^{71} - 9 q^{72} + 34 q^{73} - 12 q^{74} - 16 q^{75} + 86 q^{76} - 32 q^{77} - 2 q^{78} - 7 q^{79} - 38 q^{80} + 16 q^{81} + 27 q^{82} - 28 q^{83} - 8 q^{84} + 13 q^{85} + 38 q^{86} + 4 q^{87} + 23 q^{88} - 8 q^{89} + 2 q^{90} + 2 q^{91} - 71 q^{92} - 16 q^{93} + 66 q^{94} + 30 q^{96} + 4 q^{97} + 22 q^{98} + 3 q^{99}+O(q^{100}) \) 16 * q - 2 * q^2 - 16 * q^3 + 26 * q^4 - 16 * q^5 + 2 * q^6 - 2 * q^7 - 9 * q^8 + 16 * q^9 + 2 * q^10 + 3 * q^11 - 26 * q^12 - 16 * q^13 - 5 * q^14 + 16 * q^15 + 38 * q^16 - 13 * q^17 - 2 * q^18 - 26 * q^20 + 2 * q^21 + q^22 - 15 * q^23 + 9 * q^24 + 16 * q^25 + 2 * q^26 - 16 * q^27 + 8 * q^28 - 4 * q^29 - 2 * q^30 + 16 * q^31 - 30 * q^32 - 3 * q^33 + 29 * q^34 + 2 * q^35 + 26 * q^36 + 12 * q^37 + 16 * q^39 + 9 * q^40 - 12 * q^41 + 5 * q^42 - 7 * q^43 - 13 * q^44 - 16 * q^45 + 14 * q^46 + 17 * q^47 - 38 * q^48 + 16 * q^49 - 2 * q^50 + 13 * q^51 - 26 * q^52 - 36 * q^53 + 2 * q^54 - 3 * q^55 + 41 * q^56 + 16 * q^58 + 53 * q^59 + 26 * q^60 + 34 * q^61 - 2 * q^62 - 2 * q^63 + 79 * q^64 + 16 * q^65 - q^66 - 13 * q^67 - 39 * q^68 + 15 * q^69 + 5 * q^70 - 11 * q^71 - 9 * q^72 + 34 * q^73 - 12 * q^74 - 16 * q^75 + 86 * q^76 - 32 * q^77 - 2 * q^78 - 7 * q^79 - 38 * q^80 + 16 * q^81 + 27 * q^82 - 28 * q^83 - 8 * q^84 + 13 * q^85 + 38 * q^86 + 4 * q^87 + 23 * q^88 - 8 * q^89 + 2 * q^90 + 2 * q^91 - 71 * q^92 - 16 * q^93 + 66 * q^94 + 30 * q^96 + 4 * q^97 + 22 * q^98 + 3 * 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 $

$2$	2.78228	1.96737	0.983684	−	0.179906i	$-0.0575792\pi$
$2$	2.78228	1.96737	0.983684	+	0.179906i	$0.0575792\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	5.74107	2.87054
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−2.78228	−1.13586
$7$	2.63079	0.994347	0.497173	−	0.867651i	$-0.334371\pi$
$7$	2.63079	0.994347	0.497173	+	0.867651i	$0.334371\pi$
$8$	10.4087	3.68003
$9$	1.00000	0.333333
$10$	−2.78228	−0.879834
$11$	−2.15059	−0.648428	−0.324214	−	0.945984i	$-0.605100\pi$
$11$	−2.15059	−0.648428	−0.324214	+	0.945984i	$0.605100\pi$
$12$	−5.74107	−1.65730
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	7.31960	1.95625
$15$	1.00000	0.258199
$16$	17.4778	4.36944
$17$	2.72792	0.661618	0.330809	−	0.943698i	$-0.392678\pi$
$17$	2.72792	0.661618	0.330809	+	0.943698i	$0.392678\pi$
$18$	2.78228	0.655789
$19$	5.12096	1.17483	0.587415	−	0.809286i	$-0.300146\pi$
$19$	5.12096	1.17483	0.587415	+	0.809286i	$0.300146\pi$
$20$	−5.74107	−1.28374
$21$	−2.63079	−0.574086
$22$	−5.98355	−1.27570
$23$	−1.65770	−0.345654	−0.172827	−	0.984952i	$-0.555290\pi$
$23$	−1.65770	−0.345654	−0.172827	+	0.984952i	$0.555290\pi$
$24$	−10.4087	−2.12467
$25$	1.00000	0.200000
$26$	−2.78228	−0.545650
$27$	−1.00000	−0.192450
$28$	15.1036	2.85431
$29$	0.589015	0.109377	0.0546886	−	0.998503i	$-0.482583\pi$
$29$	0.589015	0.109377	0.0546886	+	0.998503i	$0.482583\pi$
$30$	2.78228	0.507972
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	27.8106	4.91626
$33$	2.15059	0.374370
$34$	7.58983	1.30165
$35$	−2.63079	−0.444685
$36$	5.74107	0.956845
$37$	7.51372	1.23525	0.617624	−	0.786473i	$-0.288095\pi$
$37$	7.51372	1.23525	0.617624	+	0.786473i	$0.288095\pi$
$38$	14.2479	2.31132
$39$	1.00000	0.160128
$40$	−10.4087	−1.64576
$41$	−2.50392	−0.391046	−0.195523	−	0.980699i	$-0.562640\pi$
$41$	−2.50392	−0.391046	−0.195523	+	0.980699i	$0.562640\pi$
$42$	−7.31960	−1.12944
$43$	−12.6815	−1.93391	−0.966956	−	0.254944i	$-0.917943\pi$
$43$	−12.6815	−1.93391	−0.966956	+	0.254944i	$0.917943\pi$
$44$	−12.3467	−1.86134
$45$	−1.00000	−0.149071
$46$	−4.61218	−0.680028
$47$	3.79567	0.553655	0.276828	−	0.960920i	$-0.410717\pi$
$47$	3.79567	0.553655	0.276828	+	0.960920i	$0.410717\pi$
$48$	−17.4778	−2.52270
$49$	−0.0789236	−0.0112748
$50$	2.78228	0.393474
$51$	−2.72792	−0.381985
$52$	−5.74107	−0.796143
$53$	−7.11031	−0.976676	−0.488338	−	0.872654i	$-0.662397\pi$
$53$	−7.11031	−0.976676	−0.488338	+	0.872654i	$0.662397\pi$
$54$	−2.78228	−0.378620
$55$	2.15059	0.289986
$56$	27.3831	3.65923
$57$	−5.12096	−0.678288
$58$	1.63880	0.215185
$59$	3.51642	0.457798	0.228899	−	0.973450i	$-0.426487\pi$
$59$	3.51642	0.457798	0.228899	+	0.973450i	$0.426487\pi$
$60$	5.74107	0.741169
$61$	4.04143	0.517453	0.258726	−	0.965951i	$-0.416697\pi$
$61$	4.04143	0.517453	0.258726	+	0.965951i	$0.416697\pi$
$62$	2.78228	0.353350
$63$	2.63079	0.331449
$64$	42.4213	5.30266
$65$	1.00000	0.124035
$66$	5.98355	0.736524
$67$	2.52332	0.308273	0.154136	−	0.988050i	$-0.450740\pi$
$67$	2.52332	0.308273	0.154136	+	0.988050i	$0.450740\pi$
$68$	15.6612	1.89920
$69$	1.65770	0.199563
$70$	−7.31960	−0.874860
$71$	−6.17389	−0.732706	−0.366353	−	0.930476i	$-0.619394\pi$
$71$	−6.17389	−0.732706	−0.366353	+	0.930476i	$0.619394\pi$
$72$	10.4087	1.22668
$73$	4.94101	0.578302	0.289151	−	0.957284i	$-0.406627\pi$
$73$	4.94101	0.578302	0.289151	+	0.957284i	$0.406627\pi$
$74$	20.9053	2.43019
$75$	−1.00000	−0.115470
$76$	29.3998	3.37239
$77$	−5.65776	−0.644762
$78$	2.78228	0.315031
$79$	0.514769	0.0579160	0.0289580	−	0.999581i	$-0.490781\pi$
$79$	0.514769	0.0579160	0.0289580	+	0.999581i	$0.490781\pi$
$80$	−17.4778	−1.95407
$81$	1.00000	0.111111
$82$	−6.96659	−0.769331
$83$	3.63128	0.398585	0.199293	−	0.979940i	$-0.436136\pi$
$83$	3.63128	0.398585	0.199293	+	0.979940i	$0.436136\pi$
$84$	−15.1036	−1.64794
$85$	−2.72792	−0.295885
$86$	−35.2835	−3.80472
$87$	−0.589015	−0.0631490
$88$	−22.3849	−2.38624
$89$	−2.32826	−0.246795	−0.123397	−	0.992357i	$-0.539379\pi$
$89$	−2.32826	−0.246795	−0.123397	+	0.992357i	$0.539379\pi$
$90$	−2.78228	−0.293278
$91$	−2.63079	−0.275782
$92$	−9.51696	−0.992212
$93$	−1.00000	−0.103695
$94$	10.5606	1.08924
$95$	−5.12096	−0.525400
$96$	−27.8106	−2.83841
$97$	7.90981	0.803120	0.401560	−	0.915833i	$-0.368468\pi$
$97$	7.90981	0.803120	0.401560	+	0.915833i	$0.368468\pi$
$98$	−0.219587	−0.0221817
$99$	−2.15059	−0.216143
$100$	5.74107	0.574107
$101$	−3.05362	−0.303847	−0.151923	−	0.988392i	$-0.548547\pi$
$101$	−3.05362	−0.303847	−0.151923	+	0.988392i	$0.548547\pi$
$102$	−7.58983	−0.751505
$103$	13.8565	1.36532	0.682661	−	0.730735i	$-0.260823\pi$
$103$	13.8565	1.36532	0.682661	+	0.730735i	$0.260823\pi$
$104$	−10.4087	−1.02066
$105$	2.63079	0.256739
$106$	−19.7829	−1.92148
$107$	19.5357	1.88859	0.944294	−	0.329103i	$-0.106746\pi$
$107$	19.5357	1.88859	0.944294	+	0.329103i	$0.106746\pi$
$108$	−5.74107	−0.552435
$109$	9.96636	0.954604	0.477302	−	0.878739i	$-0.341615\pi$
$109$	9.96636	0.954604	0.477302	+	0.878739i	$0.341615\pi$
$110$	5.98355	0.570509
$111$	−7.51372	−0.713171
$112$	45.9804	4.34474
$113$	−3.69579	−0.347671	−0.173835	−	0.984775i	$-0.555616\pi$
$113$	−3.69579	−0.347671	−0.173835	+	0.984775i	$0.555616\pi$
$114$	−14.2479	−1.33444
$115$	1.65770	0.154581
$116$	3.38158	0.313971
$117$	−1.00000	−0.0924500
$118$	9.78365	0.900658
$119$	7.17660	0.657878
$120$	10.4087	0.950180
$121$	−6.37495	−0.579541
$122$	11.2444	1.01802
$123$	2.50392	0.225771
$124$	5.74107	0.515563
$125$	−1.00000	−0.0894427
$126$	7.31960	0.652082
$127$	−7.62224	−0.676364	−0.338182	−	0.941081i	$-0.609812\pi$
$127$	−7.62224	−0.676364	−0.338182	+	0.941081i	$0.609812\pi$
$128$	62.4066	5.51602
$129$	12.6815	1.11654
$130$	2.78228	0.244022
$131$	2.97639	0.260049	0.130024	−	0.991511i	$-0.458494\pi$
$131$	2.97639	0.260049	0.130024	+	0.991511i	$0.458494\pi$
$132$	12.3467	1.07464
$133$	13.4722	1.16819
$134$	7.02058	0.606486
$135$	1.00000	0.0860663
$136$	28.3941	2.43477
$137$	−18.8578	−1.61113	−0.805564	−	0.592508i	$-0.798138\pi$
$137$	−18.8578	−1.61113	−0.805564	+	0.592508i	$0.798138\pi$
$138$	4.61218	0.392615
$139$	17.9414	1.52177	0.760883	−	0.648889i	$-0.224766\pi$
$139$	17.9414	1.52177	0.760883	+	0.648889i	$0.224766\pi$
$140$	−15.1036	−1.27649
$141$	−3.79567	−0.319653
$142$	−17.1775	−1.44150
$143$	2.15059	0.179842
$144$	17.4778	1.45648
$145$	−0.589015	−0.0489150
$146$	13.7473	1.13773
$147$	0.0789236	0.00650951
$148$	43.1368	3.54583
$149$	3.69162	0.302429	0.151215	−	0.988501i	$-0.451682\pi$
$149$	3.69162	0.302429	0.151215	+	0.988501i	$0.451682\pi$
$150$	−2.78228	−0.227172
$151$	−4.17413	−0.339685	−0.169843	−	0.985471i	$-0.554326\pi$
$151$	−4.17413	−0.339685	−0.169843	+	0.985471i	$0.554326\pi$
$152$	53.3026	4.32341
$153$	2.72792	0.220539
$154$	−15.7415	−1.26848
$155$	−1.00000	−0.0803219
$156$	5.74107	0.459654
$157$	−16.3565	−1.30539	−0.652693	−	0.757622i	$-0.726361\pi$
$157$	−16.3565	−1.30539	−0.652693	+	0.757622i	$0.726361\pi$
$158$	1.43223	0.113942
$159$	7.11031	0.563884
$160$	−27.8106	−2.19862
$161$	−4.36106	−0.343700
$162$	2.78228	0.218596
$163$	−19.3542	−1.51594	−0.757969	−	0.652290i	$-0.773808\pi$
$163$	−19.3542	−1.51594	−0.757969	+	0.652290i	$0.773808\pi$
$164$	−14.3752	−1.12251
$165$	−2.15059	−0.167423
$166$	10.1032	0.784164
$167$	8.01889	0.620520	0.310260	−	0.950652i	$-0.399584\pi$
$167$	8.01889	0.620520	0.310260	+	0.950652i	$0.399584\pi$
$168$	−27.3831	−2.11266
$169$	1.00000	0.0769231
$170$	−7.58983	−0.582114
$171$	5.12096	0.391610
$172$	−72.8054	−5.55136
$173$	−4.17586	−0.317485	−0.158742	−	0.987320i	$-0.550744\pi$
$173$	−4.17586	−0.317485	−0.158742	+	0.987320i	$0.550744\pi$
$174$	−1.63880	−0.124237
$175$	2.63079	0.198869
$176$	−37.5875	−2.83327
$177$	−3.51642	−0.264310
$178$	−6.47786	−0.485536
$179$	−19.8024	−1.48010	−0.740052	−	0.672549i	$-0.765199\pi$
$179$	−19.8024	−1.48010	−0.740052	+	0.672549i	$0.765199\pi$
$180$	−5.74107	−0.427914
$181$	−5.41160	−0.402241	−0.201121	−	0.979566i	$-0.564458\pi$
$181$	−5.41160	−0.402241	−0.201121	+	0.979566i	$0.564458\pi$
$182$	−7.31960	−0.542565
$183$	−4.04143	−0.298751
$184$	−17.2545	−1.27202
$185$	−7.51372	−0.552420
$186$	−2.78228	−0.204007
$187$	−5.86664	−0.429012
$188$	21.7912	1.58929
$189$	−2.63079	−0.191362
$190$	−14.2479	−1.03365
$191$	−23.6379	−1.71038	−0.855190	−	0.518315i	$-0.826559\pi$
$191$	−23.6379	−1.71038	−0.855190	+	0.518315i	$0.826559\pi$
$192$	−42.4213	−3.06149
$193$	−5.66974	−0.408117	−0.204058	−	0.978959i	$-0.565413\pi$
$193$	−5.66974	−0.408117	−0.204058	+	0.978959i	$0.565413\pi$
$194$	22.0073	1.58003
$195$	−1.00000	−0.0716115
$196$	−0.453106	−0.0323647
$197$	15.2885	1.08926	0.544630	−	0.838676i	$-0.316670\pi$
$197$	15.2885	1.08926	0.544630	+	0.838676i	$0.316670\pi$
$198$	−5.98355	−0.425232
$199$	19.2162	1.36220	0.681101	−	0.732190i	$-0.261502\pi$
$199$	19.2162	1.36220	0.681101	+	0.732190i	$0.261502\pi$
$200$	10.4087	0.736006
$201$	−2.52332	−0.177981
$202$	−8.49603	−0.597779
$203$	1.54958	0.108759
$204$	−15.6612	−1.09650
$205$	2.50392	0.174881
$206$	38.5526	2.68609
$207$	−1.65770	−0.115218
$208$	−17.4778	−1.21186
$209$	−11.0131	−0.761792
$210$	7.31960	0.505100
$211$	−19.2985	−1.32857	−0.664283	−	0.747481i	$-0.731263\pi$
$211$	−19.2985	−1.32857	−0.664283	+	0.747481i	$0.731263\pi$
$212$	−40.8208	−2.80358
$213$	6.17389	0.423028
$214$	54.3538	3.71555
$215$	12.6815	0.864872
$216$	−10.4087	−0.708222
$217$	2.63079	0.178590
$218$	27.7292	1.87806
$219$	−4.94101	−0.333883
$220$	12.3467	0.832415
$221$	−2.72792	−0.183500
$222$	−20.9053	−1.40307
$223$	−8.37975	−0.561150	−0.280575	−	0.959832i	$-0.590525\pi$
$223$	−8.37975	−0.561150	−0.280575	+	0.959832i	$0.590525\pi$
$224$	73.1639	4.88847
$225$	1.00000	0.0666667
$226$	−10.2827	−0.683996
$227$	−8.26046	−0.548266	−0.274133	−	0.961692i	$-0.588391\pi$
$227$	−8.26046	−0.548266	−0.274133	+	0.961692i	$0.588391\pi$
$228$	−29.3998	−1.94705
$229$	23.4296	1.54827	0.774136	−	0.633019i	$-0.218185\pi$
$229$	23.4296	1.54827	0.774136	+	0.633019i	$0.218185\pi$
$230$	4.61218	0.304118
$231$	5.65776	0.372254
$232$	6.13088	0.402512
$233$	−22.8184	−1.49488	−0.747441	−	0.664328i	$-0.768718\pi$
$233$	−22.8184	−1.49488	−0.747441	+	0.664328i	$0.768718\pi$
$234$	−2.78228	−0.181883
$235$	−3.79567	−0.247602
$236$	20.1880	1.31413
$237$	−0.514769	−0.0334378
$238$	19.9673	1.29429
$239$	18.0654	1.16855	0.584277	−	0.811555i	$-0.301378\pi$
$239$	18.0654	1.16855	0.584277	+	0.811555i	$0.301378\pi$
$240$	17.4778	1.12818
$241$	−10.4422	−0.672642	−0.336321	−	0.941747i	$-0.609183\pi$
$241$	−10.4422	−0.672642	−0.336321	+	0.941747i	$0.609183\pi$
$242$	−17.7369	−1.14017
$243$	−1.00000	−0.0641500
$244$	23.2022	1.48537
$245$	0.0789236	0.00504225
$246$	6.96659	0.444174
$247$	−5.12096	−0.325839
$248$	10.4087	0.660953
$249$	−3.63128	−0.230123
$250$	−2.78228	−0.175967
$251$	13.5945	0.858079	0.429039	−	0.903286i	$-0.358852\pi$
$251$	13.5945	0.858079	0.429039	+	0.903286i	$0.358852\pi$
$252$	15.1036	0.951436
$253$	3.56503	0.224132
$254$	−21.2072	−1.33066
$255$	2.72792	0.170829
$256$	88.7900	5.54937
$257$	−22.6320	−1.41175	−0.705873	−	0.708338i	$-0.749445\pi$
$257$	−22.6320	−1.41175	−0.705873	+	0.708338i	$0.749445\pi$
$258$	35.2835	2.19665
$259$	19.7671	1.22827
$260$	5.74107	0.356046
$261$	0.589015	0.0364591
$262$	8.28116	0.511612
$263$	−28.5152	−1.75832	−0.879160	−	0.476527i	$-0.841895\pi$
$263$	−28.5152	−1.75832	−0.879160	+	0.476527i	$0.841895\pi$
$264$	22.3849	1.37769
$265$	7.11031	0.436783
$266$	37.4834	2.29825
$267$	2.32826	0.142487
$268$	14.4866	0.884908
$269$	1.04245	0.0635592	0.0317796	−	0.999495i	$-0.489883\pi$
$269$	1.04245	0.0635592	0.0317796	+	0.999495i	$0.489883\pi$
$270$	2.78228	0.169324
$271$	−6.00205	−0.364598	−0.182299	−	0.983243i	$-0.558354\pi$
$271$	−6.00205	−0.364598	−0.182299	+	0.983243i	$0.558354\pi$
$272$	47.6779	2.89090
$273$	2.63079	0.159223
$274$	−52.4676	−3.16968
$275$	−2.15059	−0.129686
$276$	9.51696	0.572854
$277$	9.45753	0.568248	0.284124	−	0.958788i	$-0.408297\pi$
$277$	9.45753	0.568248	0.284124	+	0.958788i	$0.408297\pi$
$278$	49.9179	2.99387
$279$	1.00000	0.0598684
$280$	−27.3831	−1.63646
$281$	20.8572	1.24424	0.622118	−	0.782923i	$-0.286272\pi$
$281$	20.8572	1.24424	0.622118	+	0.782923i	$0.286272\pi$
$282$	−10.5606	−0.628875
$283$	1.74986	0.104019	0.0520093	−	0.998647i	$-0.483437\pi$
$283$	1.74986	0.104019	0.0520093	+	0.998647i	$0.483437\pi$
$284$	−35.4447	−2.10326
$285$	5.12096	0.303340
$286$	5.98355	0.353814
$287$	−6.58729	−0.388835
$288$	27.8106	1.63875
$289$	−9.55845	−0.562262
$290$	−1.63880	−0.0962338
$291$	−7.90981	−0.463681
$292$	28.3667	1.66004
$293$	11.6967	0.683326	0.341663	−	0.939822i	$-0.389010\pi$
$293$	11.6967	0.683326	0.341663	+	0.939822i	$0.389010\pi$
$294$	0.219587	0.0128066
$295$	−3.51642	−0.204734
$296$	78.2081	4.54575
$297$	2.15059	0.124790
$298$	10.2711	0.594990
$299$	1.65770	0.0958672
$300$	−5.74107	−0.331461
$301$	−33.3624	−1.92298
$302$	−11.6136	−0.668286
$303$	3.05362	0.175426
$304$	89.5029	5.13335
$305$	−4.04143	−0.231412
$306$	7.58983	0.433882
$307$	23.6285	1.34855	0.674275	−	0.738480i	$-0.264456\pi$
$307$	23.6285	1.34855	0.674275	+	0.738480i	$0.264456\pi$
$308$	−32.4816	−1.85081
$309$	−13.8565	−0.788269
$310$	−2.78228	−0.158023
$311$	−25.1057	−1.42361	−0.711806	−	0.702376i	$-0.752122\pi$
$311$	−25.1057	−1.42361	−0.711806	+	0.702376i	$0.752122\pi$
$312$	10.4087	0.589277
$313$	−8.83500	−0.499384	−0.249692	−	0.968325i	$-0.580329\pi$
$313$	−8.83500	−0.499384	−0.249692	+	0.968325i	$0.580329\pi$
$314$	−45.5082	−2.56818
$315$	−2.63079	−0.148228
$316$	2.95533	0.166250
$317$	10.3691	0.582388	0.291194	−	0.956664i	$-0.405947\pi$
$317$	10.3691	0.582388	0.291194	+	0.956664i	$0.405947\pi$
$318$	19.7829	1.10937
$319$	−1.26673	−0.0709233
$320$	−42.4213	−2.37142
$321$	−19.5357	−1.09038
$322$	−12.1337	−0.676184
$323$	13.9696	0.777288
$324$	5.74107	0.318948
$325$	−1.00000	−0.0554700
$326$	−53.8488	−2.98241
$327$	−9.96636	−0.551141
$328$	−26.0625	−1.43906
$329$	9.98563	0.550525
$330$	−5.98355	−0.329383
$331$	27.1658	1.49317	0.746583	−	0.665293i	$-0.231693\pi$
$331$	27.1658	1.49317	0.746583	+	0.665293i	$0.231693\pi$
$332$	20.8475	1.14415
$333$	7.51372	0.411750
$334$	22.3108	1.22079
$335$	−2.52332	−0.137864
$336$	−45.9804	−2.50844
$337$	6.41544	0.349472	0.174736	−	0.984615i	$-0.444093\pi$
$337$	6.41544	0.349472	0.174736	+	0.984615i	$0.444093\pi$
$338$	2.78228	0.151336
$339$	3.69579	0.200728
$340$	−15.6612	−0.849347
$341$	−2.15059	−0.116461
$342$	14.2479	0.770440
$343$	−18.6232	−1.00556
$344$	−131.998	−7.11686
$345$	−1.65770	−0.0892475
$346$	−11.6184	−0.624609
$347$	13.0981	0.703141	0.351570	−	0.936161i	$-0.385648\pi$
$347$	13.0981	0.703141	0.351570	+	0.936161i	$0.385648\pi$
$348$	−3.38158	−0.181271
$349$	−8.93591	−0.478328	−0.239164	−	0.970979i	$-0.576873\pi$
$349$	−8.93591	−0.478328	−0.239164	+	0.970979i	$0.576873\pi$
$350$	7.31960	0.391249
$351$	1.00000	0.0533761
$352$	−59.8092	−3.18784
$353$	3.57626	0.190345	0.0951726	−	0.995461i	$-0.469660\pi$
$353$	3.57626	0.190345	0.0951726	+	0.995461i	$0.469660\pi$
$354$	−9.78365	−0.519995
$355$	6.17389	0.327676
$356$	−13.3667	−0.708433
$357$	−7.17660	−0.379826
$358$	−55.0959	−2.91191
$359$	−18.8391	−0.994292	−0.497146	−	0.867667i	$-0.665619\pi$
$359$	−18.8391	−0.994292	−0.497146	+	0.867667i	$0.665619\pi$
$360$	−10.4087	−0.548587
$361$	7.22425	0.380224
$362$	−15.0566	−0.791356
$363$	6.37495	0.334598
$364$	−15.1036	−0.791642
$365$	−4.94101	−0.258624
$366$	−11.2444	−0.587754
$367$	−27.4244	−1.43154	−0.715770	−	0.698336i	$-0.753924\pi$
$367$	−27.4244	−1.43154	−0.715770	+	0.698336i	$0.753924\pi$
$368$	−28.9728	−1.51031
$369$	−2.50392	−0.130349
$370$	−20.9053	−1.08681
$371$	−18.7058	−0.971155
$372$	−5.74107	−0.297661
$373$	26.0743	1.35008	0.675038	−	0.737783i	$-0.264127\pi$
$373$	26.0743	1.35008	0.675038	+	0.737783i	$0.264127\pi$
$374$	−16.3226	−0.844023
$375$	1.00000	0.0516398
$376$	39.5080	2.03747
$377$	−0.589015	−0.0303358
$378$	−7.31960	−0.376480
$379$	−1.79697	−0.0923043	−0.0461522	−	0.998934i	$-0.514696\pi$
$379$	−1.79697	−0.0923043	−0.0461522	+	0.998934i	$0.514696\pi$
$380$	−29.3998	−1.50818
$381$	7.62224	0.390499
$382$	−65.7672	−3.36495
$383$	−6.42834	−0.328473	−0.164236	−	0.986421i	$-0.552516\pi$
$383$	−6.42834	−0.328473	−0.164236	+	0.986421i	$0.552516\pi$
$384$	−62.4066	−3.18467
$385$	5.65776	0.288346
$386$	−15.7748	−0.802915
$387$	−12.6815	−0.644637
$388$	45.4108	2.30538
$389$	−8.39540	−0.425664	−0.212832	−	0.977089i	$-0.568269\pi$
$389$	−8.39540	−0.425664	−0.212832	+	0.977089i	$0.568269\pi$
$390$	−2.78228	−0.140886
$391$	−4.52207	−0.228691
$392$	−0.821492	−0.0414916
$393$	−2.97639	−0.150139
$394$	42.5368	2.14297
$395$	−0.514769	−0.0259008
$396$	−12.3467	−0.620445
$397$	3.22255	0.161735	0.0808677	−	0.996725i	$-0.474231\pi$
$397$	3.22255	0.161735	0.0808677	+	0.996725i	$0.474231\pi$
$398$	53.4649	2.67995
$399$	−13.4722	−0.674453
$400$	17.4778	0.873888
$401$	−19.0171	−0.949666	−0.474833	−	0.880076i	$-0.657492\pi$
$401$	−19.0171	−0.949666	−0.474833	+	0.880076i	$0.657492\pi$
$402$	−7.02058	−0.350155
$403$	−1.00000	−0.0498135
$404$	−17.5311	−0.872203
$405$	−1.00000	−0.0496904
$406$	4.31135	0.213969
$407$	−16.1590	−0.800970
$408$	−28.3941	−1.40572
$409$	−20.3960	−1.00852	−0.504258	−	0.863553i	$-0.668234\pi$
$409$	−20.3960	−1.00852	−0.504258	+	0.863553i	$0.668234\pi$
$410$	6.96659	0.344055
$411$	18.8578	0.930185
$412$	79.5512	3.91920
$413$	9.25097	0.455210
$414$	−4.61218	−0.226676
$415$	−3.63128	−0.178253
$416$	−27.8106	−1.36353
$417$	−17.9414	−0.878592
$418$	−30.6415	−1.49873
$419$	0.432651	0.0211364	0.0105682	−	0.999944i	$-0.496636\pi$
$419$	0.432651	0.0211364	0.0105682	+	0.999944i	$0.496636\pi$
$420$	15.1036	0.736979
$421$	11.7038	0.570406	0.285203	−	0.958467i	$-0.407939\pi$
$421$	11.7038	0.570406	0.285203	+	0.958467i	$0.407939\pi$
$422$	−53.6939	−2.61378
$423$	3.79567	0.184552
$424$	−74.0091	−3.59420
$425$	2.72792	0.132324
$426$	17.1775	0.832251
$427$	10.6322	0.514527
$428$	112.156	5.42126
$429$	−2.15059	−0.103832
$430$	35.2835	1.70152
$431$	5.47002	0.263482	0.131741	−	0.991284i	$-0.457943\pi$
$431$	5.47002	0.263482	0.131741	+	0.991284i	$0.457943\pi$
$432$	−17.4778	−0.840899
$433$	2.14804	0.103228	0.0516141	−	0.998667i	$-0.483563\pi$
$433$	2.14804	0.103228	0.0516141	+	0.998667i	$0.483563\pi$
$434$	7.31960	0.351352
$435$	0.589015	0.0282411
$436$	57.2176	2.74022
$437$	−8.48901	−0.406084
$438$	−13.7473	−0.656870
$439$	3.09216	0.147581	0.0737903	−	0.997274i	$-0.476490\pi$
$439$	3.09216	0.147581	0.0737903	+	0.997274i	$0.476490\pi$
$440$	22.3849	1.06716
$441$	−0.0789236	−0.00375827
$442$	−7.58983	−0.361012
$443$	−19.8565	−0.943409	−0.471704	−	0.881757i	$-0.656361\pi$
$443$	−19.8565	−0.943409	−0.471704	+	0.881757i	$0.656361\pi$
$444$	−43.1368	−2.04718
$445$	2.32826	0.110370
$446$	−23.3148	−1.10399
$447$	−3.69162	−0.174608
$448$	111.602	5.27268
$449$	0.190312	0.00898139	0.00449069	−	0.999990i	$-0.498571\pi$
$449$	0.190312	0.00898139	0.00449069	+	0.999990i	$0.498571\pi$
$450$	2.78228	0.131158
$451$	5.38490	0.253565
$452$	−21.2178	−0.998001
$453$	4.17413	0.196118
$454$	−22.9829	−1.07864
$455$	2.63079	0.123334
$456$	−53.3026	−2.49612
$457$	24.9769	1.16837	0.584185	−	0.811620i	$-0.301414\pi$
$457$	24.9769	1.16837	0.584185	+	0.811620i	$0.301414\pi$
$458$	65.1877	3.04602
$459$	−2.72792	−0.127328
$460$	9.51696	0.443731
$461$	−26.4145	−1.23025	−0.615124	−	0.788431i	$-0.710894\pi$
$461$	−26.4145	−1.23025	−0.615124	+	0.788431i	$0.710894\pi$
$462$	15.7415	0.732360
$463$	7.16607	0.333036	0.166518	−	0.986038i	$-0.446748\pi$
$463$	7.16607	0.333036	0.166518	+	0.986038i	$0.446748\pi$
$464$	10.2947	0.477917
$465$	1.00000	0.0463739
$466$	−63.4871	−2.94098
$467$	−15.8242	−0.732254	−0.366127	−	0.930565i	$-0.619316\pi$
$467$	−15.8242	−0.732254	−0.366127	+	0.930565i	$0.619316\pi$
$468$	−5.74107	−0.265381
$469$	6.63834	0.306530
$470$	−10.5606	−0.487125
$471$	16.3565	0.753666
$472$	36.6013	1.68471
$473$	27.2728	1.25400
$474$	−1.43223	−0.0657845
$475$	5.12096	0.234966
$476$	41.2014	1.88846
$477$	−7.11031	−0.325559
$478$	50.2630	2.29897
$479$	−41.0398	−1.87516	−0.937578	−	0.347775i	$-0.886937\pi$
$479$	−41.0398	−1.87516	−0.937578	+	0.347775i	$0.886937\pi$
$480$	27.8106	1.26937
$481$	−7.51372	−0.342596
$482$	−29.0531	−1.32333
$483$	4.36106	0.198435
$484$	−36.5991	−1.66359
$485$	−7.90981	−0.359166
$486$	−2.78228	−0.126207
$487$	−2.76415	−0.125255	−0.0626277	−	0.998037i	$-0.519948\pi$
$487$	−2.76415	−0.125255	−0.0626277	+	0.998037i	$0.519948\pi$
$488$	42.0661	1.90424
$489$	19.3542	0.875227
$490$	0.219587	0.00991995
$491$	−22.2831	−1.00562	−0.502810	−	0.864397i	$-0.667700\pi$
$491$	−22.2831	−1.00562	−0.502810	+	0.864397i	$0.667700\pi$
$492$	14.3752	0.648082
$493$	1.60678	0.0723660
$494$	−14.2479	−0.641045
$495$	2.15059	0.0966619
$496$	17.4778	0.784775
$497$	−16.2422	−0.728563
$498$	−10.1032	−0.452737
$499$	−16.7353	−0.749176	−0.374588	−	0.927191i	$-0.622216\pi$
$499$	−16.7353	−0.749176	−0.374588	+	0.927191i	$0.622216\pi$
$500$	−5.74107	−0.256749
$501$	−8.01889	−0.358257
$502$	37.8238	1.68816
$503$	−3.11360	−0.138829	−0.0694144	−	0.997588i	$-0.522113\pi$
$503$	−3.11360	−0.138829	−0.0694144	+	0.997588i	$0.522113\pi$
$504$	27.3831	1.21974
$505$	3.05362	0.135884
$506$	9.91891	0.440949
$507$	−1.00000	−0.0444116
$508$	−43.7598	−1.94153
$509$	23.2486	1.03048	0.515238	−	0.857047i	$-0.327704\pi$
$509$	23.2486	1.03048	0.515238	+	0.857047i	$0.327704\pi$
$510$	7.58983	0.336083
$511$	12.9988	0.575032
$512$	122.225	5.40164
$513$	−5.12096	−0.226096
$514$	−62.9686	−2.77742
$515$	−13.8565	−0.610590
$516$	72.8054	3.20508
$517$	−8.16294	−0.359006
$518$	54.9975	2.41645
$519$	4.17586	0.183300
$520$	10.4087	0.456452
$521$	24.4725	1.07216	0.536080	−	0.844167i	$-0.319904\pi$
$521$	24.4725	1.07216	0.536080	+	0.844167i	$0.319904\pi$
$522$	1.63880	0.0717284
$523$	39.4629	1.72559	0.862795	−	0.505554i	$-0.168712\pi$
$523$	39.4629	1.72559	0.862795	+	0.505554i	$0.168712\pi$
$524$	17.0877	0.746479
$525$	−2.63079	−0.114817
$526$	−79.3371	−3.45926
$527$	2.72792	0.118830
$528$	37.5875	1.63579
$529$	−20.2520	−0.880523
$530$	19.7829	0.859313
$531$	3.51642	0.152599
$532$	77.3448	3.35332
$533$	2.50392	0.108457
$534$	6.47786	0.280324
$535$	−19.5357	−0.844602
$536$	26.2645	1.13445
$537$	19.8024	0.854539
$538$	2.90038	0.125044
$539$	0.169733	0.00731090
$540$	5.74107	0.247056
$541$	30.3701	1.30571	0.652855	−	0.757483i	$-0.273571\pi$
$541$	30.3701	1.30571	0.652855	+	0.757483i	$0.273571\pi$
$542$	−16.6994	−0.717299
$543$	5.41160	0.232234
$544$	75.8651	3.25269
$545$	−9.96636	−0.426912
$546$	7.31960	0.313250
$547$	−3.29214	−0.140762	−0.0703808	−	0.997520i	$-0.522421\pi$
$547$	−3.29214	−0.140762	−0.0703808	+	0.997520i	$0.522421\pi$
$548$	−108.264	−4.62480
$549$	4.04143	0.172484
$550$	−5.98355	−0.255139
$551$	3.01632	0.128500
$552$	17.2545	0.734400
$553$	1.35425	0.0575886
$554$	26.3135	1.11795
$555$	7.51372	0.318940
$556$	103.003	4.36828
$557$	−22.6428	−0.959407	−0.479703	−	0.877431i	$-0.659256\pi$
$557$	−22.6428	−0.959407	−0.479703	+	0.877431i	$0.659256\pi$
$558$	2.78228	0.117783
$559$	12.6815	0.536371
$560$	−45.9804	−1.94303
$561$	5.86664	0.247690
$562$	58.0305	2.44787
$563$	28.4885	1.20065	0.600323	−	0.799758i	$-0.295039\pi$
$563$	28.4885	1.20065	0.600323	+	0.799758i	$0.295039\pi$
$564$	−21.7912	−0.917576
$565$	3.69579	0.155483
$566$	4.86860	0.204643
$567$	2.63079	0.110483
$568$	−64.2622	−2.69638
$569$	−38.2093	−1.60182	−0.800909	−	0.598786i	$-0.795650\pi$
$569$	−38.2093	−1.60182	−0.800909	+	0.598786i	$0.795650\pi$
$570$	14.2479	0.596781
$571$	−21.5137	−0.900321	−0.450161	−	0.892948i	$-0.648633\pi$
$571$	−21.5137	−0.900321	−0.450161	+	0.892948i	$0.648633\pi$
$572$	12.3467	0.516242
$573$	23.6379	0.987488
$574$	−18.3277	−0.764982
$575$	−1.65770	−0.0691308
$576$	42.4213	1.76755
$577$	−31.7931	−1.32356	−0.661781	−	0.749697i	$-0.730199\pi$
$577$	−31.7931	−1.32356	−0.661781	+	0.749697i	$0.730199\pi$
$578$	−26.5943	−1.10618
$579$	5.66974	0.235626
$580$	−3.38158	−0.140412
$581$	9.55316	0.396332
$582$	−22.0073	−0.912232
$583$	15.2914	0.633304
$584$	51.4295	2.12817
$585$	1.00000	0.0413449
$586$	32.5434	1.34435
$587$	−31.6860	−1.30782	−0.653911	−	0.756572i	$-0.726873\pi$
$587$	−31.6860	−1.30782	−0.653911	+	0.756572i	$0.726873\pi$
$588$	0.453106	0.0186858
$589$	5.12096	0.211006
$590$	−9.78365	−0.402786
$591$	−15.2885	−0.628884
$592$	131.323	5.39735
$593$	−22.9313	−0.941676	−0.470838	−	0.882220i	$-0.656048\pi$
$593$	−22.9313	−0.941676	−0.470838	+	0.882220i	$0.656048\pi$
$594$	5.98355	0.245508
$595$	−7.17660	−0.294212
$596$	21.1939	0.868134
$597$	−19.2162	−0.786467
$598$	4.61218	0.188606
$599$	−28.2355	−1.15367	−0.576836	−	0.816860i	$-0.695713\pi$
$599$	−28.2355	−1.15367	−0.576836	+	0.816860i	$0.695713\pi$
$600$	−10.4087	−0.424933
$601$	14.7669	0.602355	0.301178	−	0.953568i	$-0.402620\pi$
$601$	14.7669	0.602355	0.301178	+	0.953568i	$0.402620\pi$
$602$	−92.8236	−3.78321
$603$	2.52332	0.102758
$604$	−23.9640	−0.975079
$605$	6.37495	0.259179
$606$	8.49603	0.345128
$607$	42.6830	1.73245	0.866226	−	0.499653i	$-0.166539\pi$
$607$	42.6830	1.73245	0.866226	+	0.499653i	$0.166539\pi$
$608$	142.417	5.77577
$609$	−1.54958	−0.0627920
$610$	−11.2444	−0.455272
$611$	−3.79567	−0.153556
$612$	15.6612	0.633066
$613$	−40.2993	−1.62767	−0.813837	−	0.581093i	$-0.802625\pi$
$613$	−40.2993	−1.62767	−0.813837	+	0.581093i	$0.802625\pi$
$614$	65.7411	2.65309
$615$	−2.50392	−0.100968
$616$	−58.8900	−2.37275
$617$	40.6458	1.63634	0.818169	−	0.574977i	$-0.194989\pi$
$617$	40.6458	1.63634	0.818169	+	0.574977i	$0.194989\pi$
$618$	−38.5526	−1.55081
$619$	−37.7583	−1.51763	−0.758817	−	0.651304i	$-0.774222\pi$
$619$	−37.7583	−1.51763	−0.758817	+	0.651304i	$0.774222\pi$
$620$	−5.74107	−0.230567
$621$	1.65770	0.0665211
$622$	−69.8509	−2.80077
$623$	−6.12516	−0.245399
$624$	17.4778	0.699670
$625$	1.00000	0.0400000
$626$	−24.5814	−0.982472
$627$	11.0131	0.439821
$628$	−93.9036	−3.74716
$629$	20.4968	0.817263
$630$	−7.31960	−0.291620
$631$	44.4515	1.76959	0.884793	−	0.465984i	$-0.154299\pi$
$631$	44.4515	1.76959	0.884793	+	0.465984i	$0.154299\pi$
$632$	5.35808	0.213133
$633$	19.2985	0.767048
$634$	28.8498	1.14577
$635$	7.62224	0.302479
$636$	40.8208	1.61865
$637$	0.0789236	0.00312707
$638$	−3.52440	−0.139532
$639$	−6.17389	−0.244235
$640$	−62.4066	−2.46684
$641$	12.3139	0.486372	0.243186	−	0.969980i	$-0.421808\pi$
$641$	12.3139	0.486372	0.243186	+	0.969980i	$0.421808\pi$
$642$	−54.3538	−2.14517
$643$	−24.0212	−0.947305	−0.473652	−	0.880712i	$-0.657065\pi$
$643$	−24.0212	−0.947305	−0.473652	+	0.880712i	$0.657065\pi$
$644$	−25.0372	−0.986603
$645$	−12.6815	−0.499334
$646$	38.8672	1.52921
$647$	−15.2664	−0.600186	−0.300093	−	0.953910i	$-0.597018\pi$
$647$	−15.2664	−0.600186	−0.300093	+	0.953910i	$0.597018\pi$
$648$	10.4087	0.408892
$649$	−7.56238	−0.296849
$650$	−2.78228	−0.109130
$651$	−2.63079	−0.103109
$652$	−111.114	−4.35156
$653$	−33.9234	−1.32752	−0.663762	−	0.747944i	$-0.731041\pi$
$653$	−33.9234	−1.32752	−0.663762	+	0.747944i	$0.731041\pi$
$654$	−27.7292	−1.08430
$655$	−2.97639	−0.116297
$656$	−43.7628	−1.70865
$657$	4.94101	0.192767
$658$	27.7828	1.08309
$659$	23.3838	0.910904	0.455452	−	0.890260i	$-0.349478\pi$
$659$	23.3838	0.910904	0.455452	+	0.890260i	$0.349478\pi$
$660$	−12.3467	−0.480595
$661$	−35.0900	−1.36484	−0.682421	−	0.730959i	$-0.739073\pi$
$661$	−35.0900	−1.36484	−0.682421	+	0.730959i	$0.739073\pi$
$662$	75.5827	2.93760
$663$	2.72792	0.105944
$664$	37.7969	1.46681
$665$	−13.4722	−0.522429
$666$	20.9053	0.810063
$667$	−0.976408	−0.0378067
$668$	46.0370	1.78123
$669$	8.37975	0.323980
$670$	−7.02058	−0.271229
$671$	−8.69148	−0.335531
$672$	−73.1639	−2.82236
$673$	23.4167	0.902647	0.451323	−	0.892360i	$-0.350952\pi$
$673$	23.4167	0.902647	0.451323	+	0.892360i	$0.350952\pi$
$674$	17.8496	0.687539
$675$	−1.00000	−0.0384900
$676$	5.74107	0.220810
$677$	−9.85839	−0.378889	−0.189444	−	0.981891i	$-0.560669\pi$
$677$	−9.85839	−0.378889	−0.189444	+	0.981891i	$0.560669\pi$
$678$	10.2827	0.394905
$679$	20.8091	0.798579
$680$	−28.3941	−1.08886
$681$	8.26046	0.316542
$682$	−5.98355	−0.229122
$683$	26.4046	1.01034	0.505171	−	0.863019i	$-0.331429\pi$
$683$	26.4046	1.01034	0.505171	+	0.863019i	$0.331429\pi$
$684$	29.3998	1.12413
$685$	18.8578	0.720519
$686$	−51.8149	−1.97830
$687$	−23.4296	−0.893895
$688$	−221.644	−8.45011
$689$	7.11031	0.270881
$690$	−4.61218	−0.175583
$691$	−13.9247	−0.529719	−0.264860	−	0.964287i	$-0.585326\pi$
$691$	−13.9247	−0.529719	−0.264860	+	0.964287i	$0.585326\pi$
$692$	−23.9739	−0.911351
$693$	−5.65776	−0.214921
$694$	36.4425	1.38334
$695$	−17.9414	−0.680555
$696$	−6.13088	−0.232390
$697$	−6.83048	−0.258723
$698$	−24.8622	−0.941047
$699$	22.8184	0.863071
$700$	15.1036	0.570862
$701$	15.5701	0.588076	0.294038	−	0.955794i	$-0.405001\pi$
$701$	15.5701	0.588076	0.294038	+	0.955794i	$0.405001\pi$
$702$	2.78228	0.105010
$703$	38.4775	1.45121
$704$	−91.2308	−3.43839
$705$	3.79567	0.142953
$706$	9.95015	0.374479
$707$	−8.03345	−0.302129
$708$	−20.1880	−0.758711
$709$	−19.4245	−0.729501	−0.364751	−	0.931105i	$-0.618846\pi$
$709$	−19.4245	−0.729501	−0.364751	+	0.931105i	$0.618846\pi$
$710$	17.1775	0.644659
$711$	0.514769	0.0193053
$712$	−24.2341	−0.908212
$713$	−1.65770	−0.0620813
$714$	−19.9673	−0.747257
$715$	−2.15059	−0.0804276
$716$	−113.687	−4.24869
$717$	−18.0654	−0.674665
$718$	−52.4157	−1.95614
$719$	24.0922	0.898486	0.449243	−	0.893410i	$-0.351694\pi$
$719$	24.0922	0.898486	0.449243	+	0.893410i	$0.351694\pi$
$720$	−17.4778	−0.651358
$721$	36.4536	1.35760
$722$	20.0999	0.748040
$723$	10.4422	0.388350
$724$	−31.0684	−1.15465
$725$	0.589015	0.0218755
$726$	17.7369	0.658278
$727$	−3.97535	−0.147438	−0.0737188	−	0.997279i	$-0.523487\pi$
$727$	−3.97535	−0.147438	−0.0737188	+	0.997279i	$0.523487\pi$
$728$	−27.3831	−1.01489
$729$	1.00000	0.0370370
$730$	−13.7473	−0.508809
$731$	−34.5941	−1.27951
$732$	−23.2022	−0.857577
$733$	0.222714	0.00822611	0.00411306	−	0.999992i	$-0.498691\pi$
$733$	0.222714	0.00822611	0.00411306	+	0.999992i	$0.498691\pi$
$734$	−76.3023	−2.81637
$735$	−0.0789236	−0.00291114
$736$	−46.1016	−1.69933
$737$	−5.42664	−0.199893
$738$	−6.96659	−0.256444
$739$	−14.3958	−0.529559	−0.264780	−	0.964309i	$-0.585299\pi$
$739$	−14.3958	−0.529559	−0.264780	+	0.964309i	$0.585299\pi$
$740$	−43.1368	−1.58574
$741$	5.12096	0.188123
$742$	−52.0446	−1.91062
$743$	−14.2833	−0.524002	−0.262001	−	0.965068i	$-0.584382\pi$
$743$	−14.2833	−0.524002	−0.262001	+	0.965068i	$0.584382\pi$
$744$	−10.4087	−0.381602
$745$	−3.69162	−0.135250
$746$	72.5460	2.65610
$747$	3.63128	0.132862
$748$	−33.6808	−1.23149
$749$	51.3944	1.87791
$750$	2.78228	0.101594
$751$	3.48111	0.127027	0.0635137	−	0.997981i	$-0.479769\pi$
$751$	3.48111	0.127027	0.0635137	+	0.997981i	$0.479769\pi$
$752$	66.3398	2.41916
$753$	−13.5945	−0.495412
$754$	−1.63880	−0.0596817
$755$	4.17413	0.151912
$756$	−15.1036	−0.549312
$757$	−31.9510	−1.16128	−0.580640	−	0.814161i	$-0.697198\pi$
$757$	−31.9510	−1.16128	−0.580640	+	0.814161i	$0.697198\pi$
$758$	−4.99968	−0.181597
$759$	−3.56503	−0.129402
$760$	−53.3026	−1.93349
$761$	−0.845819	−0.0306609	−0.0153305	−	0.999882i	$-0.504880\pi$
$761$	−0.845819	−0.0306609	−0.0153305	+	0.999882i	$0.504880\pi$
$762$	21.2072	0.768255
$763$	26.2194	0.949207
$764$	−135.707	−4.90971
$765$	−2.72792	−0.0986282
$766$	−17.8854	−0.646226
$767$	−3.51642	−0.126970
$768$	−88.7900	−3.20393
$769$	−18.6725	−0.673349	−0.336674	−	0.941621i	$-0.609302\pi$
$769$	−18.6725	−0.673349	−0.336674	+	0.941621i	$0.609302\pi$
$770$	15.7415	0.567283
$771$	22.6320	0.815072
$772$	−32.5504	−1.17151
$773$	−49.2905	−1.77286	−0.886428	−	0.462866i	$-0.846821\pi$
$773$	−49.2905	−1.77286	−0.886428	+	0.462866i	$0.846821\pi$
$774$	−35.2835	−1.26824
$775$	1.00000	0.0359211
$776$	82.3309	2.95551
$777$	−19.7671	−0.709139
$778$	−23.3583	−0.837437
$779$	−12.8225	−0.459412
$780$	−5.74107	−0.205563
$781$	13.2775	0.475107
$782$	−12.5817	−0.449919
$783$	−0.589015	−0.0210497
$784$	−1.37941	−0.0492646
$785$	16.3565	0.583787
$786$	−8.28116	−0.295379
$787$	−1.71400	−0.0610975	−0.0305487	−	0.999533i	$-0.509725\pi$
$787$	−1.71400	−0.0610975	−0.0305487	+	0.999533i	$0.509725\pi$
$788$	87.7723	3.12676
$789$	28.5152	1.01517
$790$	−1.43223	−0.0509565
$791$	−9.72286	−0.345705
$792$	−22.3849	−0.795412
$793$	−4.04143	−0.143516
$794$	8.96604	0.318193
$795$	−7.11031	−0.252177
$796$	110.322	3.91025
$797$	−53.5002	−1.89507	−0.947537	−	0.319646i	$-0.896436\pi$
$797$	−53.5002	−1.89507	−0.947537	+	0.319646i	$0.896436\pi$
$798$	−37.4834	−1.32690
$799$	10.3543	0.366308
$800$	27.8106	0.983253
$801$	−2.32826	−0.0822649
$802$	−52.9107	−1.86834
$803$	−10.6261	−0.374987
$804$	−14.4866	−0.510902
$805$	4.36106	0.153707
$806$	−2.78228	−0.0980016
$807$	−1.04245	−0.0366959
$808$	−31.7843	−1.11817
$809$	−35.8524	−1.26050	−0.630252	−	0.776391i	$-0.717048\pi$
$809$	−35.8524	−1.26050	−0.630252	+	0.776391i	$0.717048\pi$
$810$	−2.78228	−0.0977593
$811$	26.9515	0.946395	0.473198	−	0.880956i	$-0.343100\pi$
$811$	26.9515	0.946395	0.473198	+	0.880956i	$0.343100\pi$
$812$	8.89623	0.312196
$813$	6.00205	0.210501
$814$	−44.9587	−1.57580
$815$	19.3542	0.677948
$816$	−47.6779	−1.66906
$817$	−64.9415	−2.27202
$818$	−56.7472	−1.98412
$819$	−2.63079	−0.0919274
$820$	14.3752	0.502002
$821$	42.1006	1.46932	0.734661	−	0.678434i	$-0.237341\pi$
$821$	42.1006	1.46932	0.734661	+	0.678434i	$0.237341\pi$
$822$	52.4676	1.83002
$823$	−31.9714	−1.11445	−0.557226	−	0.830361i	$-0.688134\pi$
$823$	−31.9714	−1.11445	−0.557226	+	0.830361i	$0.688134\pi$
$824$	144.228	5.02443
$825$	2.15059	0.0748740
$826$	25.7388	0.895566
$827$	48.0527	1.67096	0.835478	−	0.549524i	$-0.185191\pi$
$827$	48.0527	1.67096	0.835478	+	0.549524i	$0.185191\pi$
$828$	−9.51696	−0.330737
$829$	23.4669	0.815038	0.407519	−	0.913197i	$-0.366394\pi$
$829$	23.4669	0.815038	0.407519	+	0.913197i	$0.366394\pi$
$830$	−10.1032	−0.350689
$831$	−9.45753	−0.328078
$832$	−42.4213	−1.47069
$833$	−0.215297	−0.00745961
$834$	−49.9179	−1.72851
$835$	−8.01889	−0.277505
$836$	−63.2270	−2.18675
$837$	−1.00000	−0.0345651
$838$	1.20376	0.0415831
$839$	−55.4831	−1.91549	−0.957745	−	0.287620i	$-0.907136\pi$
$839$	−55.4831	−1.91549	−0.957745	+	0.287620i	$0.907136\pi$
$840$	27.3831	0.944808
$841$	−28.6531	−0.988037
$842$	32.5631	1.12220
$843$	−20.8572	−0.718360
$844$	−110.794	−3.81370
$845$	−1.00000	−0.0344010
$846$	10.5606	0.363081
$847$	−16.7712	−0.576265
$848$	−124.272	−4.26753
$849$	−1.74986	−0.0600551
$850$	7.58983	0.260329
$851$	−12.4555	−0.426969
$852$	35.4447	1.21432
$853$	8.38464	0.287085	0.143542	−	0.989644i	$-0.454151\pi$
$853$	8.38464	0.287085	0.143542	+	0.989644i	$0.454151\pi$
$854$	29.5817	1.01226
$855$	−5.12096	−0.175133
$856$	203.341	6.95006
$857$	10.4483	0.356907	0.178454	−	0.983948i	$-0.442891\pi$
$857$	10.4483	0.356907	0.178454	+	0.983948i	$0.442891\pi$
$858$	−5.98355	−0.204275
$859$	14.9953	0.511634	0.255817	−	0.966725i	$-0.417656\pi$
$859$	14.9953	0.511634	0.255817	+	0.966725i	$0.417656\pi$
$860$	72.8054	2.48264
$861$	6.58729	0.224494
$862$	15.2191	0.518365
$863$	−18.2939	−0.622731	−0.311365	−	0.950290i	$-0.600786\pi$
$863$	−18.2939	−0.622731	−0.311365	+	0.950290i	$0.600786\pi$
$864$	−27.8106	−0.946135
$865$	4.17586	0.141983
$866$	5.97645	0.203088
$867$	9.55845	0.324622
$868$	15.1036	0.512649
$869$	−1.10706	−0.0375544
$870$	1.63880	0.0555606
$871$	−2.52332	−0.0854995
$872$	103.737	3.51297
$873$	7.90981	0.267707
$874$	−23.6188	−0.798917
$875$	−2.63079	−0.0889371
$876$	−28.3667	−0.958422
$877$	41.5694	1.40370	0.701850	−	0.712325i	$-0.252358\pi$
$877$	41.5694	1.40370	0.701850	+	0.712325i	$0.252358\pi$
$878$	8.60324	0.290345
$879$	−11.6967	−0.394519
$880$	37.5875	1.26708
$881$	−28.5692	−0.962519	−0.481260	−	0.876578i	$-0.659821\pi$
$881$	−28.5692	−0.962519	−0.481260	+	0.876578i	$0.659821\pi$
$882$	−0.219587	−0.00739389
$883$	−18.5498	−0.624251	−0.312126	−	0.950041i	$-0.601041\pi$
$883$	−18.5498	−0.624251	−0.312126	+	0.950041i	$0.601041\pi$
$884$	−15.6612	−0.526743
$885$	3.51642	0.118203
$886$	−55.2462	−1.85603
$887$	37.8548	1.27104	0.635520	−	0.772085i	$-0.280786\pi$
$887$	37.8548	1.27104	0.635520	+	0.772085i	$0.280786\pi$
$888$	−78.2081	−2.62449
$889$	−20.0525	−0.672541
$890$	6.47786	0.217138
$891$	−2.15059	−0.0720476
$892$	−48.1088	−1.61080
$893$	19.4375	0.650451
$894$	−10.2711	−0.343517
$895$	19.8024	0.661923
$896$	164.179	5.48483
$897$	−1.65770	−0.0553489
$898$	0.529502	0.0176697
$899$	0.589015	0.0196447
$900$	5.74107	0.191369
$901$	−19.3964	−0.646187
$902$	14.9823	0.498856
$903$	33.3624	1.11023
$904$	−38.4684	−1.27944
$905$	5.41160	0.179888
$906$	11.6136	0.385835
$907$	−41.5318	−1.37904	−0.689520	−	0.724267i	$-0.742179\pi$
$907$	−41.5318	−1.37904	−0.689520	+	0.724267i	$0.742179\pi$
$908$	−47.4239	−1.57382
$909$	−3.05362	−0.101282
$910$	7.31960	0.242642
$911$	−11.1040	−0.367892	−0.183946	−	0.982936i	$-0.558887\pi$
$911$	−11.1040	−0.367892	−0.183946	+	0.982936i	$0.558887\pi$
$912$	−89.5029	−2.96374
$913$	−7.80941	−0.258454
$914$	69.4927	2.29861
$915$	4.04143	0.133606
$916$	134.511	4.44437
$917$	7.83028	0.258579
$918$	−7.58983	−0.250502
$919$	−20.8880	−0.689031	−0.344516	−	0.938781i	$-0.611957\pi$
$919$	−20.8880	−0.689031	−0.344516	+	0.938781i	$0.611957\pi$
$920$	17.2545	0.568864
$921$	−23.6285	−0.778586
$922$	−73.4926	−2.42035
$923$	6.17389	0.203216
$924$	32.4816	1.06857
$925$	7.51372	0.247050
$926$	19.9380	0.655204
$927$	13.8565	0.455107
$928$	16.3808	0.537727
$929$	2.69332	0.0883649	0.0441825	−	0.999023i	$-0.485932\pi$
$929$	2.69332	0.0883649	0.0441825	+	0.999023i	$0.485932\pi$
$930$	2.78228	0.0912345
$931$	−0.404165	−0.0132460
$932$	−131.002	−4.29111
$933$	25.1057	0.821922
$934$	−44.0272	−1.44061
$935$	5.86664	0.191860
$936$	−10.4087	−0.340219
$937$	−29.0596	−0.949337	−0.474669	−	0.880165i	$-0.657432\pi$
$937$	−29.0596	−0.949337	−0.474669	+	0.880165i	$0.657432\pi$
$938$	18.4697	0.603057
$939$	8.83500	0.288319
$940$	−21.7912	−0.710751
$941$	−24.5306	−0.799673	−0.399837	−	0.916586i	$-0.630933\pi$
$941$	−24.5306	−0.799673	−0.399837	+	0.916586i	$0.630933\pi$
$942$	45.5082	1.48274
$943$	4.15074	0.135167
$944$	61.4591	2.00032
$945$	2.63079	0.0855797
$946$	75.8804	2.46708
$947$	46.7558	1.51936	0.759679	−	0.650298i	$-0.225356\pi$
$947$	46.7558	1.51936	0.759679	+	0.650298i	$0.225356\pi$
$948$	−2.95533	−0.0959845
$949$	−4.94101	−0.160392
$950$	14.2479	0.462264
$951$	−10.3691	−0.336242
$952$	74.6991	2.42101
$953$	47.3179	1.53278	0.766389	−	0.642377i	$-0.222052\pi$
$953$	47.3179	1.53278	0.766389	+	0.642377i	$0.222052\pi$
$954$	−19.7829	−0.640494
$955$	23.6379	0.764905
$956$	103.715	3.35437
$957$	1.26673	0.0409476
$958$	−114.184	−3.68912
$959$	−49.6109	−1.60202
$960$	42.4213	1.36914
$961$	1.00000	0.0322581
$962$	−20.9053	−0.674013
$963$	19.5357	0.629529
$964$	−59.9495	−1.93084
$965$	5.66974	0.182515
$966$	12.1337	0.390395
$967$	−51.4923	−1.65588	−0.827940	−	0.560816i	$-0.810488\pi$
$967$	−51.4923	−1.65588	−0.827940	+	0.560816i	$0.810488\pi$
$968$	−66.3550	−2.13273
$969$	−13.9696	−0.448767
$970$	−22.0073	−0.706612
$971$	−44.6852	−1.43402	−0.717008	−	0.697065i	$-0.754489\pi$
$971$	−44.6852	−1.43402	−0.717008	+	0.697065i	$0.754489\pi$
$972$	−5.74107	−0.184145
$973$	47.2000	1.51316
$974$	−7.69062	−0.246424
$975$	1.00000	0.0320256
$976$	70.6352	2.26098
$977$	−39.1392	−1.25217	−0.626087	−	0.779753i	$-0.715345\pi$
$977$	−39.1392	−1.25217	−0.626087	+	0.779753i	$0.715345\pi$
$978$	53.8488	1.72189
$979$	5.00713	0.160029
$980$	0.453106	0.0144739
$981$	9.96636	0.318201
$982$	−61.9977	−1.97842
$983$	13.2894	0.423865	0.211932	−	0.977284i	$-0.432024\pi$
$983$	13.2894	0.423865	0.211932	+	0.977284i	$0.432024\pi$
$984$	26.0625	0.830843
$985$	−15.2885	−0.487132
$986$	4.47052	0.142370
$987$	−9.98563	−0.317846
$988$	−29.3998	−0.935333
$989$	21.0221	0.668464
$990$	5.98355	0.190170
$991$	−0.650089	−0.0206508	−0.0103254	−	0.999947i	$-0.503287\pi$
$991$	−0.650089	−0.0206508	−0.0103254	+	0.999947i	$0.503287\pi$
$992$	27.8106	0.882987
$993$	−27.1658	−0.862079
$994$	−45.1904	−1.43335
$995$	−19.2162	−0.609195
$996$	−20.8475	−0.660577
$997$	−44.3151	−1.40347	−0.701737	−	0.712436i	$-0.747592\pi$
$997$	−44.3151	−1.40347	−0.701737	+	0.712436i	$0.747592\pi$
$998$	−46.5623	−1.47390
$999$	−7.51372	−0.237724

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6045.2.a.bg.1.16	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6045.2.a.bg.1.16	✓	16	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)

\(f(q)\)	\(=\)	\(q+2.78228 q^{2} -1.00000 q^{3} +5.74107 q^{4} -1.00000 q^{5} -2.78228 q^{6} +2.63079 q^{7} +10.4087 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+2.78228 q^{2} -1.00000 q^{3} +5.74107 q^{4} -1.00000 q^{5} -2.78228 q^{6} +2.63079 q^{7} +10.4087 q^{8} +1.00000 q^{9} -2.78228 q^{10} -2.15059 q^{11} -5.74107 q^{12} -1.00000 q^{13} +7.31960 q^{14} +1.00000 q^{15} +17.4778 q^{16} +2.72792 q^{17} +2.78228 q^{18} +5.12096 q^{19} -5.74107 q^{20} -2.63079 q^{21} -5.98355 q^{22} -1.65770 q^{23} -10.4087 q^{24} +1.00000 q^{25} -2.78228 q^{26} -1.00000 q^{27} +15.1036 q^{28} +0.589015 q^{29} +2.78228 q^{30} +1.00000 q^{31} +27.8106 q^{32} +2.15059 q^{33} +7.58983 q^{34} -2.63079 q^{35} +5.74107 q^{36} +7.51372 q^{37} +14.2479 q^{38} +1.00000 q^{39} -10.4087 q^{40} -2.50392 q^{41} -7.31960 q^{42} -12.6815 q^{43} -12.3467 q^{44} -1.00000 q^{45} -4.61218 q^{46} +3.79567 q^{47} -17.4778 q^{48} -0.0789236 q^{49} +2.78228 q^{50} -2.72792 q^{51} -5.74107 q^{52} -7.11031 q^{53} -2.78228 q^{54} +2.15059 q^{55} +27.3831 q^{56} -5.12096 q^{57} +1.63880 q^{58} +3.51642 q^{59} +5.74107 q^{60} +4.04143 q^{61} +2.78228 q^{62} +2.63079 q^{63} +42.4213 q^{64} +1.00000 q^{65} +5.98355 q^{66} +2.52332 q^{67} +15.6612 q^{68} +1.65770 q^{69} -7.31960 q^{70} -6.17389 q^{71} +10.4087 q^{72} +4.94101 q^{73} +20.9053 q^{74} -1.00000 q^{75} +29.3998 q^{76} -5.65776 q^{77} +2.78228 q^{78} +0.514769 q^{79} -17.4778 q^{80} +1.00000 q^{81} -6.96659 q^{82} +3.63128 q^{83} -15.1036 q^{84} -2.72792 q^{85} -35.2835 q^{86} -0.589015 q^{87} -22.3849 q^{88} -2.32826 q^{89} -2.78228 q^{90} -2.63079 q^{91} -9.51696 q^{92} -1.00000 q^{93} +10.5606 q^{94} -5.12096 q^{95} -27.8106 q^{96} +7.90981 q^{97} -0.219587 q^{98} -2.15059 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 2 q^{2} - 16 q^{3} + 26 q^{4} - 16 q^{5} + 2 q^{6} - 2 q^{7} - 9 q^{8} + 16 q^{9}+O(q^{10}) \) 16 * q - 2 * q^2 - 16 * q^3 + 26 * q^4 - 16 * q^5 + 2 * q^6 - 2 * q^7 - 9 * q^8 + 16 * q^9	\( 16 q - 2 q^{2} - 16 q^{3} + 26 q^{4} - 16 q^{5} + 2 q^{6} - 2 q^{7} - 9 q^{8} + 16 q^{9} + 2 q^{10} + 3 q^{11} - 26 q^{12} - 16 q^{13} - 5 q^{14} + 16 q^{15} + 38 q^{16} - 13 q^{17} - 2 q^{18} - 26 q^{20} + 2 q^{21} + q^{22} - 15 q^{23} + 9 q^{24} + 16 q^{25} + 2 q^{26} - 16 q^{27} + 8 q^{28} - 4 q^{29} - 2 q^{30} + 16 q^{31} - 30 q^{32} - 3 q^{33} + 29 q^{34} + 2 q^{35} + 26 q^{36} + 12 q^{37} + 16 q^{39} + 9 q^{40} - 12 q^{41} + 5 q^{42} - 7 q^{43} - 13 q^{44} - 16 q^{45} + 14 q^{46} + 17 q^{47} - 38 q^{48} + 16 q^{49} - 2 q^{50} + 13 q^{51} - 26 q^{52} - 36 q^{53} + 2 q^{54} - 3 q^{55} + 41 q^{56} + 16 q^{58} + 53 q^{59} + 26 q^{60} + 34 q^{61} - 2 q^{62} - 2 q^{63} + 79 q^{64} + 16 q^{65} - q^{66} - 13 q^{67} - 39 q^{68} + 15 q^{69} + 5 q^{70} - 11 q^{71} - 9 q^{72} + 34 q^{73} - 12 q^{74} - 16 q^{75} + 86 q^{76} - 32 q^{77} - 2 q^{78} - 7 q^{79} - 38 q^{80} + 16 q^{81} + 27 q^{82} - 28 q^{83} - 8 q^{84} + 13 q^{85} + 38 q^{86} + 4 q^{87} + 23 q^{88} - 8 q^{89} + 2 q^{90} + 2 q^{91} - 71 q^{92} - 16 q^{93} + 66 q^{94} + 30 q^{96} + 4 q^{97} + 22 q^{98} + 3 q^{99}+O(q^{100}) \) 16 * q - 2 * q^2 - 16 * q^3 + 26 * q^4 - 16 * q^5 + 2 * q^6 - 2 * q^7 - 9 * q^8 + 16 * q^9 + 2 * q^10 + 3 * q^11 - 26 * q^12 - 16 * q^13 - 5 * q^14 + 16 * q^15 + 38 * q^16 - 13 * q^17 - 2 * q^18 - 26 * q^20 + 2 * q^21 + q^22 - 15 * q^23 + 9 * q^24 + 16 * q^25 + 2 * q^26 - 16 * q^27 + 8 * q^28 - 4 * q^29 - 2 * q^30 + 16 * q^31 - 30 * q^32 - 3 * q^33 + 29 * q^34 + 2 * q^35 + 26 * q^36 + 12 * q^37 + 16 * q^39 + 9 * q^40 - 12 * q^41 + 5 * q^42 - 7 * q^43 - 13 * q^44 - 16 * q^45 + 14 * q^46 + 17 * q^47 - 38 * q^48 + 16 * q^49 - 2 * q^50 + 13 * q^51 - 26 * q^52 - 36 * q^53 + 2 * q^54 - 3 * q^55 + 41 * q^56 + 16 * q^58 + 53 * q^59 + 26 * q^60 + 34 * q^61 - 2 * q^62 - 2 * q^63 + 79 * q^64 + 16 * q^65 - q^66 - 13 * q^67 - 39 * q^68 + 15 * q^69 + 5 * q^70 - 11 * q^71 - 9 * q^72 + 34 * q^73 - 12 * q^74 - 16 * q^75 + 86 * q^76 - 32 * q^77 - 2 * q^78 - 7 * q^79 - 38 * q^80 + 16 * q^81 + 27 * q^82 - 28 * q^83 - 8 * q^84 + 13 * q^85 + 38 * q^86 + 4 * q^87 + 23 * q^88 - 8 * q^89 + 2 * q^90 + 2 * q^91 - 71 * q^92 - 16 * q^93 + 66 * q^94 + 30 * q^96 + 4 * q^97 + 22 * q^98 + 3 * q^99

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