LMFDB - Embedded newform 8007.2.a.f.1.22

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8007,2,Mod(1,8007)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8007, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8007.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8007 = 3 \cdot 17 \cdot 157 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8007.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:	$63.9362168984$
Analytic rank:	$1$
Dimension:	$48$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.22
Character	$\chi$	$=$	8007.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.240024 q^{2} -1.00000 q^{3} -1.94239 q^{4} -0.991325 q^{5} +0.240024 q^{6} +2.53606 q^{7} +0.946267 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.240024 q^{2} -1.00000 q^{3} -1.94239 q^{4} -0.991325 q^{5} +0.240024 q^{6} +2.53606 q^{7} +0.946267 q^{8} +1.00000 q^{9} +0.237941 q^{10} +0.968009 q^{11} +1.94239 q^{12} +2.48904 q^{13} -0.608715 q^{14} +0.991325 q^{15} +3.65765 q^{16} -1.00000 q^{17} -0.240024 q^{18} -5.03842 q^{19} +1.92554 q^{20} -2.53606 q^{21} -0.232345 q^{22} -1.80653 q^{23} -0.946267 q^{24} -4.01728 q^{25} -0.597429 q^{26} -1.00000 q^{27} -4.92602 q^{28} +5.21307 q^{29} -0.237941 q^{30} +6.87873 q^{31} -2.77046 q^{32} -0.968009 q^{33} +0.240024 q^{34} -2.51406 q^{35} -1.94239 q^{36} +2.49140 q^{37} +1.20934 q^{38} -2.48904 q^{39} -0.938057 q^{40} -1.02216 q^{41} +0.608715 q^{42} -9.96365 q^{43} -1.88025 q^{44} -0.991325 q^{45} +0.433611 q^{46} -3.98622 q^{47} -3.65765 q^{48} -0.568389 q^{49} +0.964241 q^{50} +1.00000 q^{51} -4.83469 q^{52} -5.76082 q^{53} +0.240024 q^{54} -0.959611 q^{55} +2.39979 q^{56} +5.03842 q^{57} -1.25126 q^{58} +0.764881 q^{59} -1.92554 q^{60} -6.15129 q^{61} -1.65106 q^{62} +2.53606 q^{63} -6.65033 q^{64} -2.46745 q^{65} +0.232345 q^{66} +0.538859 q^{67} +1.94239 q^{68} +1.80653 q^{69} +0.603434 q^{70} +9.03013 q^{71} +0.946267 q^{72} -2.45264 q^{73} -0.597996 q^{74} +4.01728 q^{75} +9.78657 q^{76} +2.45493 q^{77} +0.597429 q^{78} +7.60120 q^{79} -3.62592 q^{80} +1.00000 q^{81} +0.245344 q^{82} +2.86258 q^{83} +4.92602 q^{84} +0.991325 q^{85} +2.39151 q^{86} -5.21307 q^{87} +0.915995 q^{88} -0.260961 q^{89} +0.237941 q^{90} +6.31237 q^{91} +3.50899 q^{92} -6.87873 q^{93} +0.956787 q^{94} +4.99471 q^{95} +2.77046 q^{96} +11.8513 q^{97} +0.136427 q^{98} +0.968009 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9}+O(q^{10}) $ 48 * q - q^2 - 48 * q^3 + 45 * q^4 + q^5 + q^6 - 13 * q^7 - 6 * q^8 + 48 * q^9	\( 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9} - 20 q^{10} + 5 q^{11} - 45 q^{12} - 8 q^{13} + 4 q^{14} - q^{15} + 39 q^{16} - 48 q^{17} - q^{18} - 6 q^{19} + 6 q^{20} + 13 q^{21} - 35 q^{22} - 8 q^{23} + 6 q^{24} + 13 q^{25} + 17 q^{26} - 48 q^{27} - 38 q^{28} + q^{29} + 20 q^{30} - 21 q^{31} - 3 q^{32} - 5 q^{33} + q^{34} + 19 q^{35} + 45 q^{36} - 58 q^{37} - 14 q^{38} + 8 q^{39} - 54 q^{40} - 3 q^{41} - 4 q^{42} - 33 q^{43} + 2 q^{44} + q^{45} - 26 q^{46} + 9 q^{47} - 39 q^{48} + 11 q^{49} + 4 q^{50} + 48 q^{51} - 31 q^{52} - 33 q^{53} + q^{54} - 21 q^{55} + 6 q^{57} - 55 q^{58} + 77 q^{59} - 6 q^{60} - 29 q^{61} - 46 q^{62} - 13 q^{63} + 24 q^{64} - 49 q^{65} + 35 q^{66} - 44 q^{67} - 45 q^{68} + 8 q^{69} + 4 q^{70} + 22 q^{71} - 6 q^{72} - 63 q^{73} - 16 q^{74} - 13 q^{75} - 46 q^{76} - 30 q^{77} - 17 q^{78} - 46 q^{79} - 14 q^{80} + 48 q^{81} - 75 q^{82} + 11 q^{83} + 38 q^{84} - q^{85} + 8 q^{86} - q^{87} - 116 q^{88} + 10 q^{89} - 20 q^{90} - 67 q^{91} - 64 q^{92} + 21 q^{93} - 16 q^{94} - 8 q^{95} + 3 q^{96} - 96 q^{97} - 46 q^{98} + 5 q^{99}+O(q^{100}) \) 48 * q - q^2 - 48 * q^3 + 45 * q^4 + q^5 + q^6 - 13 * q^7 - 6 * q^8 + 48 * q^9 - 20 * q^10 + 5 * q^11 - 45 * q^12 - 8 * q^13 + 4 * q^14 - q^15 + 39 * q^16 - 48 * q^17 - q^18 - 6 * q^19 + 6 * q^20 + 13 * q^21 - 35 * q^22 - 8 * q^23 + 6 * q^24 + 13 * q^25 + 17 * q^26 - 48 * q^27 - 38 * q^28 + q^29 + 20 * q^30 - 21 * q^31 - 3 * q^32 - 5 * q^33 + q^34 + 19 * q^35 + 45 * q^36 - 58 * q^37 - 14 * q^38 + 8 * q^39 - 54 * q^40 - 3 * q^41 - 4 * q^42 - 33 * q^43 + 2 * q^44 + q^45 - 26 * q^46 + 9 * q^47 - 39 * q^48 + 11 * q^49 + 4 * q^50 + 48 * q^51 - 31 * q^52 - 33 * q^53 + q^54 - 21 * q^55 + 6 * q^57 - 55 * q^58 + 77 * q^59 - 6 * q^60 - 29 * q^61 - 46 * q^62 - 13 * q^63 + 24 * q^64 - 49 * q^65 + 35 * q^66 - 44 * q^67 - 45 * q^68 + 8 * q^69 + 4 * q^70 + 22 * q^71 - 6 * q^72 - 63 * q^73 - 16 * q^74 - 13 * q^75 - 46 * q^76 - 30 * q^77 - 17 * q^78 - 46 * q^79 - 14 * q^80 + 48 * q^81 - 75 * q^82 + 11 * q^83 + 38 * q^84 - q^85 + 8 * q^86 - q^87 - 116 * q^88 + 10 * q^89 - 20 * q^90 - 67 * q^91 - 64 * q^92 + 21 * q^93 - 16 * q^94 - 8 * q^95 + 3 * q^96 - 96 * q^97 - 46 * q^98 + 5 * 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 $

$2$	−0.240024	−0.169722	−0.0848612	−	0.996393i	$-0.527045\pi$
$2$	−0.240024	−0.169722	−0.0848612	+	0.996393i	$0.527045\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.94239	−0.971194
$5$	−0.991325	−0.443334	−0.221667	−	0.975122i	$-0.571150\pi$
$5$	−0.991325	−0.443334	−0.221667	+	0.975122i	$0.571150\pi$
$6$	0.240024	0.0979893
$7$	2.53606	0.958541	0.479271	−	0.877667i	$-0.340901\pi$
$7$	2.53606	0.958541	0.479271	+	0.877667i	$0.340901\pi$
$8$	0.946267	0.334556
$9$	1.00000	0.333333
$10$	0.237941	0.0752437
$11$	0.968009	0.291866	0.145933	−	0.989294i	$-0.453382\pi$
$11$	0.968009	0.291866	0.145933	+	0.989294i	$0.453382\pi$
$12$	1.94239	0.560719
$13$	2.48904	0.690336	0.345168	−	0.938541i	$-0.387822\pi$
$13$	2.48904	0.690336	0.345168	+	0.938541i	$0.387822\pi$
$14$	−0.608715	−0.162686
$15$	0.991325	0.255959
$16$	3.65765	0.914413
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	−0.240024	−0.0565741
$19$	−5.03842	−1.15589	−0.577946	−	0.816075i	$-0.696146\pi$
$19$	−5.03842	−1.15589	−0.577946	+	0.816075i	$0.696146\pi$
$20$	1.92554	0.430563
$21$	−2.53606	−0.553414
$22$	−0.232345	−0.0495361
$23$	−1.80653	−0.376688	−0.188344	−	0.982103i	$-0.560312\pi$
$23$	−1.80653	−0.376688	−0.188344	+	0.982103i	$0.560312\pi$
$24$	−0.946267	−0.193156
$25$	−4.01728	−0.803455
$26$	−0.597429	−0.117166
$27$	−1.00000	−0.192450
$28$	−4.92602	−0.930930
$29$	5.21307	0.968043	0.484022	−	0.875056i	$-0.339176\pi$
$29$	5.21307	0.968043	0.484022	+	0.875056i	$0.339176\pi$
$30$	−0.237941	−0.0434420
$31$	6.87873	1.23546	0.617729	−	0.786391i	$-0.288053\pi$
$31$	6.87873	1.23546	0.617729	+	0.786391i	$0.288053\pi$
$32$	−2.77046	−0.489752
$33$	−0.968009	−0.168509
$34$	0.240024	0.0411637
$35$	−2.51406	−0.424954
$36$	−1.94239	−0.323731
$37$	2.49140	0.409584	0.204792	−	0.978805i	$-0.434348\pi$
$37$	2.49140	0.409584	0.204792	+	0.978805i	$0.434348\pi$
$38$	1.20934	0.196181
$39$	−2.48904	−0.398566
$40$	−0.938057	−0.148320
$41$	−1.02216	−0.159635	−0.0798176	−	0.996809i	$-0.525434\pi$
$41$	−1.02216	−0.159635	−0.0798176	+	0.996809i	$0.525434\pi$
$42$	0.608715	0.0939268
$43$	−9.96365	−1.51944	−0.759722	−	0.650249i	$-0.774665\pi$
$43$	−9.96365	−1.51944	−0.759722	+	0.650249i	$0.774665\pi$
$44$	−1.88025	−0.283458
$45$	−0.991325	−0.147778
$46$	0.433611	0.0639324
$47$	−3.98622	−0.581450	−0.290725	−	0.956807i	$-0.593896\pi$
$47$	−3.98622	−0.581450	−0.290725	+	0.956807i	$0.593896\pi$
$48$	−3.65765	−0.527936
$49$	−0.568389	−0.0811984
$50$	0.964241	0.136364
$51$	1.00000	0.140028
$52$	−4.83469	−0.670451
$53$	−5.76082	−0.791309	−0.395654	−	0.918399i	$-0.629482\pi$
$53$	−5.76082	−0.791309	−0.395654	+	0.918399i	$0.629482\pi$
$54$	0.240024	0.0326631
$55$	−0.959611	−0.129394
$56$	2.39979	0.320686
$57$	5.03842	0.667355
$58$	−1.25126	−0.164299
$59$	0.764881	0.0995791	0.0497895	−	0.998760i	$-0.484145\pi$
$59$	0.764881	0.0995791	0.0497895	+	0.998760i	$0.484145\pi$
$60$	−1.92554	−0.248586
$61$	−6.15129	−0.787592	−0.393796	−	0.919198i	$-0.628838\pi$
$61$	−6.15129	−0.787592	−0.393796	+	0.919198i	$0.628838\pi$
$62$	−1.65106	−0.209685
$63$	2.53606	0.319514
$64$	−6.65033	−0.831291
$65$	−2.46745	−0.306049
$66$	0.232345	0.0285997
$67$	0.538859	0.0658321	0.0329160	−	0.999458i	$-0.489521\pi$
$67$	0.538859	0.0658321	0.0329160	+	0.999458i	$0.489521\pi$
$68$	1.94239	0.235549
$69$	1.80653	0.217481
$70$	0.603434	0.0721242
$71$	9.03013	1.07168	0.535840	−	0.844320i	$-0.319995\pi$
$71$	9.03013	1.07168	0.535840	+	0.844320i	$0.319995\pi$
$72$	0.946267	0.111519
$73$	−2.45264	−0.287060	−0.143530	−	0.989646i	$-0.545845\pi$
$73$	−2.45264	−0.287060	−0.143530	+	0.989646i	$0.545845\pi$
$74$	−0.597996	−0.0695156
$75$	4.01728	0.463875
$76$	9.78657	1.12260
$77$	2.45493	0.279765
$78$	0.597429	0.0676456
$79$	7.60120	0.855202	0.427601	−	0.903968i	$-0.359359\pi$
$79$	7.60120	0.855202	0.427601	+	0.903968i	$0.359359\pi$
$80$	−3.62592	−0.405390
$81$	1.00000	0.111111
$82$	0.245344	0.0270937
$83$	2.86258	0.314209	0.157105	−	0.987582i	$-0.449784\pi$
$83$	2.86258	0.314209	0.157105	+	0.987582i	$0.449784\pi$
$84$	4.92602	0.537473
$85$	0.991325	0.107524
$86$	2.39151	0.257884
$87$	−5.21307	−0.558900
$88$	0.915995	0.0976454
$89$	−0.260961	−0.0276618	−0.0138309	−	0.999904i	$-0.504403\pi$
$89$	−0.260961	−0.0276618	−0.0138309	+	0.999904i	$0.504403\pi$
$90$	0.237941	0.0250812
$91$	6.31237	0.661716
$92$	3.50899	0.365837
$93$	−6.87873	−0.713292
$94$	0.956787	0.0986851
$95$	4.99471	0.512446
$96$	2.77046	0.282759
$97$	11.8513	1.20332	0.601659	−	0.798753i	$-0.294507\pi$
$97$	11.8513	1.20332	0.601659	+	0.798753i	$0.294507\pi$
$98$	0.136427	0.0137812
$99$	0.968009	0.0972886
$100$	7.80311	0.780311
$101$	−8.69500	−0.865185	−0.432592	−	0.901590i	$-0.642401\pi$
$101$	−8.69500	−0.865185	−0.432592	+	0.901590i	$0.642401\pi$
$102$	−0.240024	−0.0237659
$103$	−8.35617	−0.823358	−0.411679	−	0.911329i	$-0.635058\pi$
$103$	−8.35617	−0.823358	−0.411679	+	0.911329i	$0.635058\pi$
$104$	2.35530	0.230956
$105$	2.51406	0.245347
$106$	1.38273	0.134303
$107$	3.98996	0.385724	0.192862	−	0.981226i	$-0.438223\pi$
$107$	3.98996	0.385724	0.192862	+	0.981226i	$0.438223\pi$
$108$	1.94239	0.186906
$109$	12.8957	1.23519	0.617594	−	0.786497i	$-0.288107\pi$
$109$	12.8957	1.23519	0.617594	+	0.786497i	$0.288107\pi$
$110$	0.230329	0.0219610
$111$	−2.49140	−0.236474
$112$	9.27603	0.876502
$113$	−8.03772	−0.756125	−0.378062	−	0.925780i	$-0.623410\pi$
$113$	−8.03772	−0.756125	−0.378062	+	0.925780i	$0.623410\pi$
$114$	−1.20934	−0.113265
$115$	1.79086	0.166999
$116$	−10.1258	−0.940158
$117$	2.48904	0.230112
$118$	−0.183590	−0.0169008
$119$	−2.53606	−0.232480
$120$	0.938057	0.0856325
$121$	−10.0630	−0.914814
$122$	1.47645	0.133672
$123$	1.02216	0.0921655
$124$	−13.3612	−1.19987
$125$	8.93905	0.799533
$126$	−0.608715	−0.0542286
$127$	−22.2503	−1.97439	−0.987197	−	0.159505i	$-0.949010\pi$
$127$	−22.2503	−1.97439	−0.987197	+	0.159505i	$0.949010\pi$
$128$	7.13715	0.630841
$129$	9.96365	0.877251
$130$	0.592246	0.0519434
$131$	20.5053	1.79155	0.895777	−	0.444503i	$-0.146620\pi$
$131$	20.5053	1.79155	0.895777	+	0.444503i	$0.146620\pi$
$132$	1.88025	0.163655
$133$	−12.7777	−1.10797
$134$	−0.129339	−0.0111732
$135$	0.991325	0.0853196
$136$	−0.946267	−0.0811417
$137$	7.65833	0.654295	0.327148	−	0.944973i	$-0.393913\pi$
$137$	7.65833	0.654295	0.327148	+	0.944973i	$0.393913\pi$
$138$	−0.433611	−0.0369114
$139$	−7.18251	−0.609212	−0.304606	−	0.952478i	$-0.598525\pi$
$139$	−7.18251	−0.609212	−0.304606	+	0.952478i	$0.598525\pi$
$140$	4.88328	0.412713
$141$	3.98622	0.335700
$142$	−2.16745	−0.181888
$143$	2.40942	0.201486
$144$	3.65765	0.304804
$145$	−5.16785	−0.429166
$146$	0.588692	0.0487205
$147$	0.568389	0.0468799
$148$	−4.83927	−0.397786
$149$	15.5811	1.27645	0.638227	−	0.769848i	$-0.279668\pi$
$149$	15.5811	1.27645	0.638227	+	0.769848i	$0.279668\pi$
$150$	−0.964241	−0.0787300
$151$	−2.46873	−0.200902	−0.100451	−	0.994942i	$-0.532029\pi$
$151$	−2.46873	−0.200902	−0.100451	+	0.994942i	$0.532029\pi$
$152$	−4.76769	−0.386711
$153$	−1.00000	−0.0808452
$154$	−0.589242	−0.0474824
$155$	−6.81906	−0.547720
$156$	4.83469	0.387085
$157$	−1.00000	−0.0798087
$157$	−1.00000	−0.0798087
$158$	−1.82447	−0.145147
$159$	5.76082	0.456862
$160$	2.74642	0.217124
$161$	−4.58148	−0.361071
$162$	−0.240024	−0.0188580
$163$	−2.73288	−0.214056	−0.107028	−	0.994256i	$-0.534133\pi$
$163$	−2.73288	−0.214056	−0.107028	+	0.994256i	$0.534133\pi$
$164$	1.98544	0.155037
$165$	0.959611	0.0747056
$166$	−0.687088	−0.0533283
$167$	−3.20254	−0.247820	−0.123910	−	0.992293i	$-0.539543\pi$
$167$	−3.20254	−0.247820	−0.123910	+	0.992293i	$0.539543\pi$
$168$	−2.39979	−0.185148
$169$	−6.80466	−0.523436
$170$	−0.237941	−0.0182493
$171$	−5.03842	−0.385298
$172$	19.3533	1.47567
$173$	−8.49128	−0.645580	−0.322790	−	0.946471i	$-0.604621\pi$
$173$	−8.49128	−0.645580	−0.322790	+	0.946471i	$0.604621\pi$
$174$	1.25126	0.0948579
$175$	−10.1881	−0.770145
$176$	3.54064	0.266886
$177$	−0.764881	−0.0574920
$178$	0.0626367	0.00469482
$179$	23.2801	1.74004	0.870019	−	0.493018i	$-0.164106\pi$
$179$	23.2801	1.74004	0.870019	+	0.493018i	$0.164106\pi$
$180$	1.92554	0.143521
$181$	8.46143	0.628933	0.314467	−	0.949269i	$-0.398174\pi$
$181$	8.46143	0.628933	0.314467	+	0.949269i	$0.398174\pi$
$182$	−1.51512	−0.112308
$183$	6.15129	0.454716
$184$	−1.70946	−0.126023
$185$	−2.46979	−0.181583
$186$	1.65106	0.121062
$187$	−0.968009	−0.0707878
$188$	7.74279	0.564701
$189$	−2.53606	−0.184471
$190$	−1.19885	−0.0869736
$191$	−19.8888	−1.43911	−0.719553	−	0.694438i	$-0.755653\pi$
$191$	−19.8888	−1.43911	−0.719553	+	0.694438i	$0.755653\pi$
$192$	6.65033	0.479946
$193$	16.1007	1.15895	0.579476	−	0.814989i	$-0.303257\pi$
$193$	16.1007	1.15895	0.579476	+	0.814989i	$0.303257\pi$
$194$	−2.84459	−0.204230
$195$	2.46745	0.176698
$196$	1.10403	0.0788594
$197$	3.62394	0.258195	0.129097	−	0.991632i	$-0.458792\pi$
$197$	3.62394	0.258195	0.129097	+	0.991632i	$0.458792\pi$
$198$	−0.232345	−0.0165120
$199$	20.1136	1.42582	0.712908	−	0.701257i	$-0.247378\pi$
$199$	20.1136	1.42582	0.712908	+	0.701257i	$0.247378\pi$
$200$	−3.80141	−0.268801
$201$	−0.538859	−0.0380082
$202$	2.08701	0.146841
$203$	13.2207	0.927910
$204$	−1.94239	−0.135994
$205$	1.01330	0.0707717
$206$	2.00568	0.139742
$207$	−1.80653	−0.125563
$208$	9.10405	0.631252
$209$	−4.87723	−0.337365
$210$	−0.603434	−0.0416409
$211$	−25.9012	−1.78311	−0.891555	−	0.452913i	$-0.850385\pi$
$211$	−25.9012	−1.78311	−0.891555	+	0.452913i	$0.850385\pi$
$212$	11.1897	0.768515
$213$	−9.03013	−0.618734
$214$	−0.957685	−0.0654660
$215$	9.87722	0.673621
$216$	−0.946267	−0.0643853
$217$	17.4449	1.18424
$218$	−3.09529	−0.209639
$219$	2.45264	0.165734
$220$	1.86394	0.125667
$221$	−2.48904	−0.167431
$222$	0.597996	0.0401349
$223$	−5.20612	−0.348628	−0.174314	−	0.984690i	$-0.555771\pi$
$223$	−5.20612	−0.348628	−0.174314	+	0.984690i	$0.555771\pi$
$224$	−7.02605	−0.469448
$225$	−4.01728	−0.267818
$226$	1.92924	0.128331
$227$	10.0078	0.664241	0.332120	−	0.943237i	$-0.392236\pi$
$227$	10.0078	0.664241	0.332120	+	0.943237i	$0.392236\pi$
$228$	−9.78657	−0.648131
$229$	−12.7031	−0.839446	−0.419723	−	0.907652i	$-0.637873\pi$
$229$	−12.7031	−0.839446	−0.419723	+	0.907652i	$0.637873\pi$
$230$	−0.429849	−0.0283434
$231$	−2.45493	−0.161523
$232$	4.93296	0.323865
$233$	−7.43985	−0.487401	−0.243700	−	0.969851i	$-0.578361\pi$
$233$	−7.43985	−0.487401	−0.243700	+	0.969851i	$0.578361\pi$
$234$	−0.597429	−0.0390552
$235$	3.95164	0.257776
$236$	−1.48570	−0.0967106
$237$	−7.60120	−0.493751
$238$	0.608715	0.0394571
$239$	21.9317	1.41864	0.709320	−	0.704886i	$-0.249002\pi$
$239$	21.9317	1.41864	0.709320	+	0.704886i	$0.249002\pi$
$240$	3.62592	0.234052
$241$	−0.669110	−0.0431012	−0.0215506	−	0.999768i	$-0.506860\pi$
$241$	−0.669110	−0.0431012	−0.0215506	+	0.999768i	$0.506860\pi$
$242$	2.41535	0.155264
$243$	−1.00000	−0.0641500
$244$	11.9482	0.764905
$245$	0.563458	0.0359980
$246$	−0.245344	−0.0156425
$247$	−12.5408	−0.797955
$248$	6.50912	0.413329
$249$	−2.86258	−0.181409
$250$	−2.14558	−0.135699
$251$	−20.6937	−1.30617	−0.653087	−	0.757283i	$-0.726527\pi$
$251$	−20.6937	−1.30617	−0.653087	+	0.757283i	$0.726527\pi$
$252$	−4.92602	−0.310310
$253$	−1.74874	−0.109942
$254$	5.34060	0.335099
$255$	−0.991325	−0.0620792
$256$	11.5876	0.724223
$257$	−1.50443	−0.0938436	−0.0469218	−	0.998899i	$-0.514941\pi$
$257$	−1.50443	−0.0938436	−0.0469218	+	0.998899i	$0.514941\pi$
$258$	−2.39151	−0.148889
$259$	6.31835	0.392603
$260$	4.79275	0.297234
$261$	5.21307	0.322681
$262$	−4.92175	−0.304067
$263$	−28.9952	−1.78792	−0.893960	−	0.448146i	$-0.852084\pi$
$263$	−28.9952	−1.78792	−0.893960	+	0.448146i	$0.852084\pi$
$264$	−0.915995	−0.0563756
$265$	5.71084	0.350814
$266$	3.06696	0.188047
$267$	0.260961	0.0159705
$268$	−1.04667	−0.0639358
$269$	5.53574	0.337520	0.168760	−	0.985657i	$-0.446024\pi$
$269$	5.53574	0.337520	0.168760	+	0.985657i	$0.446024\pi$
$270$	−0.237941	−0.0144807
$271$	19.9638	1.21272	0.606358	−	0.795192i	$-0.292630\pi$
$271$	19.9638	1.21272	0.606358	+	0.795192i	$0.292630\pi$
$272$	−3.65765	−0.221778
$273$	−6.31237	−0.382042
$274$	−1.83818	−0.111049
$275$	−3.88876	−0.234501
$276$	−3.50899	−0.211216
$277$	16.8330	1.01140	0.505700	−	0.862710i	$-0.331234\pi$
$277$	16.8330	1.01140	0.505700	+	0.862710i	$0.331234\pi$
$278$	1.72397	0.103397
$279$	6.87873	0.411819
$280$	−2.37897	−0.142171
$281$	−16.0062	−0.954851	−0.477425	−	0.878672i	$-0.658430\pi$
$281$	−16.0062	−0.954851	−0.477425	+	0.878672i	$0.658430\pi$
$282$	−0.956787	−0.0569759
$283$	−14.3986	−0.855907	−0.427954	−	0.903801i	$-0.640765\pi$
$283$	−14.3986	−0.855907	−0.427954	+	0.903801i	$0.640765\pi$
$284$	−17.5400	−1.04081
$285$	−4.99471	−0.295861
$286$	−0.578317	−0.0341966
$287$	−2.59227	−0.153017
$288$	−2.77046	−0.163251
$289$	1.00000	0.0588235
$290$	1.24041	0.0728391
$291$	−11.8513	−0.694736
$292$	4.76398	0.278791
$293$	−21.0986	−1.23260	−0.616298	−	0.787513i	$-0.711368\pi$
$293$	−21.0986	−1.23260	−0.616298	+	0.787513i	$0.711368\pi$
$294$	−0.136427	−0.00795657
$295$	−0.758246	−0.0441468
$296$	2.35753	0.137029
$297$	−0.968009	−0.0561696
$298$	−3.73983	−0.216643
$299$	−4.49654	−0.260042
$300$	−7.80311	−0.450513
$301$	−25.2684	−1.45645
$302$	0.592554	0.0340977
$303$	8.69500	0.499515
$304$	−18.4288	−1.05696
$305$	6.09792	0.349166
$306$	0.240024	0.0137212
$307$	−22.6024	−1.28999	−0.644994	−	0.764187i	$-0.723140\pi$
$307$	−22.6024	−1.28999	−0.644994	+	0.764187i	$0.723140\pi$
$308$	−4.76843	−0.271707
$309$	8.35617	0.475366
$310$	1.63674	0.0929603
$311$	7.89681	0.447787	0.223893	−	0.974614i	$-0.428123\pi$
$311$	7.89681	0.447787	0.223893	+	0.974614i	$0.428123\pi$
$312$	−2.35530	−0.133343
$313$	−3.90937	−0.220970	−0.110485	−	0.993878i	$-0.535240\pi$
$313$	−3.90937	−0.220970	−0.110485	+	0.993878i	$0.535240\pi$
$314$	0.240024	0.0135453
$315$	−2.51406	−0.141651
$316$	−14.7645	−0.830567
$317$	−16.1235	−0.905586	−0.452793	−	0.891616i	$-0.649572\pi$
$317$	−16.1235	−0.905586	−0.452793	+	0.891616i	$0.649572\pi$
$318$	−1.38273	−0.0775398
$319$	5.04630	0.282539
$320$	6.59263	0.368539
$321$	−3.98996	−0.222698
$322$	1.09966	0.0612819
$323$	5.03842	0.280345
$324$	−1.94239	−0.107910
$325$	−9.99917	−0.554654
$326$	0.655956	0.0363300
$327$	−12.8957	−0.713137
$328$	−0.967240	−0.0534069
$329$	−10.1093	−0.557344
$330$	−0.230329	−0.0126792
$331$	19.4464	1.06887	0.534436	−	0.845209i	$-0.320524\pi$
$331$	19.4464	1.06887	0.534436	+	0.845209i	$0.320524\pi$
$332$	−5.56025	−0.305158
$333$	2.49140	0.136528
$334$	0.768686	0.0420606
$335$	−0.534184	−0.0291856
$336$	−9.27603	−0.506049
$337$	−35.7471	−1.94727	−0.973635	−	0.228109i	$-0.926746\pi$
$337$	−35.7471	−1.94727	−0.973635	+	0.228109i	$0.926746\pi$
$338$	1.63328	0.0888387
$339$	8.03772	0.436549
$340$	−1.92554	−0.104427
$341$	6.65868	0.360588
$342$	1.20934	0.0653936
$343$	−19.1939	−1.03637
$344$	−9.42827	−0.508338
$345$	−1.79086	−0.0964167
$346$	2.03811	0.109569
$347$	−14.9314	−0.801559	−0.400780	−	0.916174i	$-0.631261\pi$
$347$	−14.9314	−0.801559	−0.400780	+	0.916174i	$0.631261\pi$
$348$	10.1258	0.542801
$349$	−25.4274	−1.36110	−0.680550	−	0.732702i	$-0.738259\pi$
$349$	−25.4274	−1.36110	−0.680550	+	0.732702i	$0.738259\pi$
$350$	2.44538	0.130711
$351$	−2.48904	−0.132855
$352$	−2.68183	−0.142942
$353$	13.7431	0.731470	0.365735	−	0.930719i	$-0.380818\pi$
$353$	13.7431	0.731470	0.365735	+	0.930719i	$0.380818\pi$
$354$	0.183590	0.00975768
$355$	−8.95179	−0.475112
$356$	0.506887	0.0268650
$357$	2.53606	0.134223
$358$	−5.58778	−0.295323
$359$	11.5161	0.607798	0.303899	−	0.952704i	$-0.401711\pi$
$359$	11.5161	0.607798	0.303899	+	0.952704i	$0.401711\pi$
$360$	−0.938057	−0.0494400
$361$	6.38566	0.336088
$362$	−2.03094	−0.106744
$363$	10.0630	0.528168
$364$	−12.2611	−0.642655
$365$	2.43136	0.127263
$366$	−1.47645	−0.0771755
$367$	14.6221	0.763265	0.381633	−	0.924314i	$-0.375362\pi$
$367$	14.6221	0.763265	0.381633	+	0.924314i	$0.375362\pi$
$368$	−6.60767	−0.344448
$369$	−1.02216	−0.0532118
$370$	0.592808	0.0308186
$371$	−14.6098	−0.758502
$372$	13.3612	0.692745
$373$	−27.4428	−1.42093	−0.710467	−	0.703731i	$-0.751516\pi$
$373$	−27.4428	−1.42093	−0.710467	+	0.703731i	$0.751516\pi$
$374$	0.232345	0.0120143
$375$	−8.93905	−0.461610
$376$	−3.77203	−0.194527
$377$	12.9756	0.668276
$378$	0.608715	0.0313089
$379$	−11.9824	−0.615492	−0.307746	−	0.951469i	$-0.599575\pi$
$379$	−11.9824	−0.615492	−0.307746	+	0.951469i	$0.599575\pi$
$380$	−9.70166	−0.497685
$381$	22.2503	1.13992
$382$	4.77379	0.244248
$383$	−24.6762	−1.26089	−0.630446	−	0.776233i	$-0.717128\pi$
$383$	−24.6762	−1.26089	−0.630446	+	0.776233i	$0.717128\pi$
$384$	−7.13715	−0.364216
$385$	−2.43363	−0.124029
$386$	−3.86455	−0.196700
$387$	−9.96365	−0.506481
$388$	−23.0198	−1.16865
$389$	28.9910	1.46990	0.734950	−	0.678121i	$-0.237206\pi$
$389$	28.9910	1.46990	0.734950	+	0.678121i	$0.237206\pi$
$390$	−0.592246	−0.0299896
$391$	1.80653	0.0913603
$392$	−0.537847	−0.0271654
$393$	−20.5053	−1.03435
$394$	−0.869830	−0.0438214
$395$	−7.53526	−0.379140
$396$	−1.88025	−0.0944861
$397$	−24.1759	−1.21335	−0.606677	−	0.794948i	$-0.707498\pi$
$397$	−24.1759	−1.21335	−0.606677	+	0.794948i	$0.707498\pi$
$398$	−4.82774	−0.241993
$399$	12.7777	0.639687
$400$	−14.6938	−0.734690
$401$	−27.0335	−1.34999	−0.674995	−	0.737822i	$-0.735854\pi$
$401$	−27.0335	−1.34999	−0.674995	+	0.737822i	$0.735854\pi$
$402$	0.129339	0.00645084
$403$	17.1215	0.852881
$404$	16.8891	0.840262
$405$	−0.991325	−0.0492593
$406$	−3.17328	−0.157487
$407$	2.41170	0.119544
$408$	0.946267	0.0468472
$409$	−6.84701	−0.338563	−0.169281	−	0.985568i	$-0.554145\pi$
$409$	−6.84701	−0.338563	−0.169281	+	0.985568i	$0.554145\pi$
$410$	−0.243215	−0.0120115
$411$	−7.65833	−0.377758
$412$	16.2309	0.799641
$413$	1.93979	0.0954506
$414$	0.433611	0.0213108
$415$	−2.83775	−0.139300
$416$	−6.89579	−0.338094
$417$	7.18251	0.351729
$418$	1.17065	0.0572585
$419$	−13.1885	−0.644299	−0.322149	−	0.946689i	$-0.604405\pi$
$419$	−13.1885	−0.644299	−0.322149	+	0.946689i	$0.604405\pi$
$420$	−4.88328	−0.238280
$421$	−30.3604	−1.47968	−0.739838	−	0.672785i	$-0.765098\pi$
$421$	−30.3604	−1.47968	−0.739838	+	0.672785i	$0.765098\pi$
$422$	6.21689	0.302634
$423$	−3.98622	−0.193817
$424$	−5.45127	−0.264737
$425$	4.01728	0.194866
$426$	2.16745	0.105013
$427$	−15.6000	−0.754939
$428$	−7.75006	−0.374613
$429$	−2.40942	−0.116328
$430$	−2.37077	−0.114328
$431$	−12.8162	−0.617334	−0.308667	−	0.951170i	$-0.599883\pi$
$431$	−12.8162	−0.617334	−0.308667	+	0.951170i	$0.599883\pi$
$432$	−3.65765	−0.175979
$433$	24.2000	1.16298	0.581490	−	0.813554i	$-0.302470\pi$
$433$	24.2000	1.16298	0.581490	+	0.813554i	$0.302470\pi$
$434$	−4.18719	−0.200991
$435$	5.16785	0.247779
$436$	−25.0486	−1.19961
$437$	9.10207	0.435411
$438$	−0.588692	−0.0281288
$439$	−0.594555	−0.0283766	−0.0141883	−	0.999899i	$-0.504516\pi$
$439$	−0.594555	−0.0283766	−0.0141883	+	0.999899i	$0.504516\pi$
$440$	−0.908048	−0.0432895
$441$	−0.568389	−0.0270661
$442$	0.597429	0.0284168
$443$	20.0073	0.950578	0.475289	−	0.879830i	$-0.342344\pi$
$443$	20.0073	0.950578	0.475289	+	0.879830i	$0.342344\pi$
$444$	4.83927	0.229662
$445$	0.258697	0.0122634
$446$	1.24959	0.0591699
$447$	−15.5811	−0.736961
$448$	−16.8656	−0.796827
$449$	−30.4016	−1.43474	−0.717371	−	0.696691i	$-0.754655\pi$
$449$	−30.4016	−1.43474	−0.717371	+	0.696691i	$0.754655\pi$
$450$	0.964241	0.0454548
$451$	−0.989465	−0.0465921
$452$	15.6124	0.734344
$453$	2.46873	0.115991
$454$	−2.40211	−0.112737
$455$	−6.25761	−0.293361
$456$	4.76769	0.223267
$457$	−18.1442	−0.848750	−0.424375	−	0.905487i	$-0.639506\pi$
$457$	−18.1442	−0.848750	−0.424375	+	0.905487i	$0.639506\pi$
$458$	3.04905	0.142473
$459$	1.00000	0.0466760
$460$	−3.47855	−0.162188
$461$	−14.3459	−0.668153	−0.334077	−	0.942546i	$-0.608424\pi$
$461$	−14.3459	−0.668153	−0.334077	+	0.942546i	$0.608424\pi$
$462$	0.589242	0.0274140
$463$	−13.9429	−0.647981	−0.323990	−	0.946060i	$-0.605025\pi$
$463$	−13.9429	−0.647981	−0.323990	+	0.946060i	$0.605025\pi$
$464$	19.0676	0.885191
$465$	6.81906	0.316226
$466$	1.78574	0.0827228
$467$	5.68107	0.262888	0.131444	−	0.991324i	$-0.458039\pi$
$467$	5.68107	0.262888	0.131444	+	0.991324i	$0.458039\pi$
$468$	−4.83469	−0.223484
$469$	1.36658	0.0631028
$470$	−0.948487	−0.0437504
$471$	1.00000	0.0460776
$472$	0.723782	0.0333147
$473$	−9.64491	−0.443473
$474$	1.82447	0.0838006
$475$	20.2407	0.928708
$476$	4.92602	0.225784
$477$	−5.76082	−0.263770
$478$	−5.26412	−0.240775
$479$	−30.6261	−1.39934	−0.699672	−	0.714464i	$-0.746671\pi$
$479$	−30.6261	−1.39934	−0.699672	+	0.714464i	$0.746671\pi$
$480$	−2.74642	−0.125356
$481$	6.20121	0.282751
$482$	0.160602	0.00731524
$483$	4.58148	0.208465
$484$	19.5462	0.888463
$485$	−11.7485	−0.533471
$486$	0.240024	0.0108877
$487$	21.1475	0.958285	0.479143	−	0.877737i	$-0.340948\pi$
$487$	21.1475	0.958285	0.479143	+	0.877737i	$0.340948\pi$
$488$	−5.82076	−0.263493
$489$	2.73288	0.123585
$490$	−0.135243	−0.00610966
$491$	8.48807	0.383061	0.191531	−	0.981487i	$-0.438655\pi$
$491$	8.48807	0.383061	0.191531	+	0.981487i	$0.438655\pi$
$492$	−1.98544	−0.0895106
$493$	−5.21307	−0.234785
$494$	3.01010	0.135431
$495$	−0.959611	−0.0431313
$496$	25.1600	1.12972
$497$	22.9010	1.02725
$498$	0.687088	0.0307891
$499$	1.38700	0.0620906	0.0310453	−	0.999518i	$-0.490116\pi$
$499$	1.38700	0.0620906	0.0310453	+	0.999518i	$0.490116\pi$
$500$	−17.3631	−0.776502
$501$	3.20254	0.143079
$502$	4.96698	0.221687
$503$	12.6444	0.563785	0.281893	−	0.959446i	$-0.409038\pi$
$503$	12.6444	0.563785	0.281893	+	0.959446i	$0.409038\pi$
$504$	2.39979	0.106895
$505$	8.61957	0.383566
$506$	0.419739	0.0186597
$507$	6.80466	0.302206
$508$	43.2187	1.91752
$509$	6.76191	0.299716	0.149858	−	0.988708i	$-0.452118\pi$
$509$	6.76191	0.299716	0.149858	+	0.988708i	$0.452118\pi$
$510$	0.237941	0.0105362
$511$	−6.22005	−0.275159
$512$	−17.0556	−0.753758
$513$	5.03842	0.222452
$514$	0.361098	0.0159274
$515$	8.28368	0.365023
$516$	−19.3533	−0.851981
$517$	−3.85870	−0.169705
$518$	−1.51655	−0.0666336
$519$	8.49128	0.372726
$520$	−2.33487	−0.102391
$521$	−36.0732	−1.58040	−0.790198	−	0.612852i	$-0.790022\pi$
$521$	−36.0732	−1.58040	−0.790198	+	0.612852i	$0.790022\pi$
$522$	−1.25126	−0.0547662
$523$	−1.59381	−0.0696926	−0.0348463	−	0.999393i	$-0.511094\pi$
$523$	−1.59381	−0.0696926	−0.0348463	+	0.999393i	$0.511094\pi$
$524$	−39.8292	−1.73995
$525$	10.1881	0.444643
$526$	6.95954	0.303450
$527$	−6.87873	−0.299642
$528$	−3.54064	−0.154087
$529$	−19.7364	−0.858106
$530$	−1.37074	−0.0595410
$531$	0.764881	0.0331930
$532$	24.8193	1.07605
$533$	−2.54421	−0.110202
$534$	−0.0626367	−0.00271056
$535$	−3.95535	−0.171005
$536$	0.509904	0.0220245
$537$	−23.2801	−1.00461
$538$	−1.32871	−0.0572847
$539$	−0.550205	−0.0236990
$540$	−1.92554	−0.0828619
$541$	6.57918	0.282861	0.141431	−	0.989948i	$-0.454830\pi$
$541$	6.57918	0.282861	0.141431	+	0.989948i	$0.454830\pi$
$542$	−4.79179	−0.205825
$543$	−8.46143	−0.363115
$544$	2.77046	0.118782
$545$	−12.7839	−0.547601
$546$	1.51512	0.0648411
$547$	13.8876	0.593791	0.296896	−	0.954910i	$-0.404049\pi$
$547$	13.8876	0.593791	0.296896	+	0.954910i	$0.404049\pi$
$548$	−14.8755	−0.635448
$549$	−6.15129	−0.262531
$550$	0.933394	0.0398001
$551$	−26.2656	−1.11895
$552$	1.70946	0.0727595
$553$	19.2771	0.819747
$554$	−4.04033	−0.171657
$555$	2.46979	0.104837
$556$	13.9512	0.591664
$557$	−14.2308	−0.602979	−0.301489	−	0.953470i	$-0.597484\pi$
$557$	−14.2308	−0.602979	−0.301489	+	0.953470i	$0.597484\pi$
$558$	−1.65106	−0.0698949
$559$	−24.8000	−1.04893
$560$	−9.19556	−0.388583
$561$	0.968009	0.0408694
$562$	3.84187	0.162060
$563$	17.6032	0.741887	0.370943	−	0.928656i	$-0.379034\pi$
$563$	17.6032	0.741887	0.370943	+	0.928656i	$0.379034\pi$
$564$	−7.74279	−0.326030
$565$	7.96799	0.335216
$566$	3.45600	0.145267
$567$	2.53606	0.106505
$568$	8.54491	0.358537
$569$	25.1751	1.05540	0.527698	−	0.849432i	$-0.323055\pi$
$569$	25.1751	1.05540	0.527698	+	0.849432i	$0.323055\pi$
$570$	1.19885	0.0502142
$571$	24.7789	1.03697	0.518483	−	0.855088i	$-0.326497\pi$
$571$	24.7789	1.03697	0.518483	+	0.855088i	$0.326497\pi$
$572$	−4.68002	−0.195682
$573$	19.8888	0.830868
$574$	0.622207	0.0259704
$575$	7.25734	0.302652
$576$	−6.65033	−0.277097
$577$	3.99043	0.166124	0.0830620	−	0.996544i	$-0.473530\pi$
$577$	3.99043	0.166124	0.0830620	+	0.996544i	$0.473530\pi$
$578$	−0.240024	−0.00998367
$579$	−16.1007	−0.669122
$580$	10.0380	0.416804
$581$	7.25969	0.301183
$582$	2.84459	0.117912
$583$	−5.57652	−0.230956
$584$	−2.32085	−0.0960376
$585$	−2.46745	−0.102016
$586$	5.06417	0.209199
$587$	−1.46024	−0.0602705	−0.0301352	−	0.999546i	$-0.509594\pi$
$587$	−1.46024	−0.0602705	−0.0301352	+	0.999546i	$0.509594\pi$
$588$	−1.10403	−0.0455295
$589$	−34.6579	−1.42806
$590$	0.181997	0.00749269
$591$	−3.62394	−0.149069
$592$	9.11268	0.374529
$593$	−4.74474	−0.194843	−0.0974216	−	0.995243i	$-0.531060\pi$
$593$	−4.74474	−0.194843	−0.0974216	+	0.995243i	$0.531060\pi$
$594$	0.232345	0.00953324
$595$	2.51406	0.103066
$596$	−30.2646	−1.23968
$597$	−20.1136	−0.823196
$598$	1.07928	0.0441349
$599$	−2.37282	−0.0969510	−0.0484755	−	0.998824i	$-0.515436\pi$
$599$	−2.37282	−0.0969510	−0.0484755	+	0.998824i	$0.515436\pi$
$600$	3.80141	0.155192
$601$	−15.4996	−0.632244	−0.316122	−	0.948719i	$-0.602381\pi$
$601$	−15.4996	−0.632244	−0.316122	+	0.948719i	$0.602381\pi$
$602$	6.06503	0.247192
$603$	0.538859	0.0219440
$604$	4.79524	0.195115
$605$	9.97566	0.405568
$606$	−2.08701	−0.0847788
$607$	21.3333	0.865893	0.432946	−	0.901420i	$-0.357474\pi$
$607$	21.3333	0.865893	0.432946	+	0.901420i	$0.357474\pi$
$608$	13.9587	0.566101
$609$	−13.2207	−0.535729
$610$	−1.46365	−0.0592613
$611$	−9.92188	−0.401396
$612$	1.94239	0.0785164
$613$	−27.8282	−1.12397	−0.561985	−	0.827148i	$-0.689962\pi$
$613$	−27.8282	−1.12397	−0.561985	+	0.827148i	$0.689962\pi$
$614$	5.42512	0.218940
$615$	−1.01330	−0.0408601
$616$	2.32302	0.0935971
$617$	13.8149	0.556168	0.278084	−	0.960557i	$-0.410301\pi$
$617$	13.8149	0.556168	0.278084	+	0.960557i	$0.410301\pi$
$618$	−2.00568	−0.0806803
$619$	−23.3186	−0.937253	−0.468626	−	0.883396i	$-0.655251\pi$
$619$	−23.3186	−0.937253	−0.468626	+	0.883396i	$0.655251\pi$
$620$	13.2453	0.531942
$621$	1.80653	0.0724937
$622$	−1.89542	−0.0759995
$623$	−0.661812	−0.0265150
$624$	−9.10405	−0.364454
$625$	11.2249	0.448995
$626$	0.938340	0.0375036
$627$	4.87723	0.194778
$628$	1.94239	0.0775097
$629$	−2.49140	−0.0993388
$630$	0.603434	0.0240414
$631$	−45.2232	−1.80031	−0.900154	−	0.435572i	$-0.856546\pi$
$631$	−45.2232	−1.80031	−0.900154	+	0.435572i	$0.856546\pi$
$632$	7.19276	0.286113
$633$	25.9012	1.02948
$634$	3.87002	0.153698
$635$	22.0573	0.875316
$636$	−11.1897	−0.443702
$637$	−1.41474	−0.0560542
$638$	−1.21123	−0.0479531
$639$	9.03013	0.357226
$640$	−7.07523	−0.279673
$641$	45.6453	1.80288	0.901440	−	0.432904i	$-0.142511\pi$
$641$	45.6453	1.80288	0.901440	+	0.432904i	$0.142511\pi$
$642$	0.957685	0.0377968
$643$	−29.8558	−1.17740	−0.588700	−	0.808352i	$-0.700360\pi$
$643$	−29.8558	−1.17740	−0.588700	+	0.808352i	$0.700360\pi$
$644$	8.89901	0.350670
$645$	−9.87722	−0.388915
$646$	−1.20934	−0.0475808
$647$	−18.3694	−0.722177	−0.361088	−	0.932532i	$-0.617595\pi$
$647$	−18.3694	−0.722177	−0.361088	+	0.932532i	$0.617595\pi$
$648$	0.946267	0.0371729
$649$	0.740412	0.0290637
$650$	2.40004	0.0941373
$651$	−17.4449	−0.683719
$652$	5.30831	0.207890
$653$	6.73493	0.263558	0.131779	−	0.991279i	$-0.457931\pi$
$653$	6.73493	0.263558	0.131779	+	0.991279i	$0.457931\pi$
$654$	3.09529	0.121035
$655$	−20.3274	−0.794257
$656$	−3.73872	−0.145973
$657$	−2.45264	−0.0956866
$658$	2.42647	0.0945937
$659$	−1.03915	−0.0404796	−0.0202398	−	0.999795i	$-0.506443\pi$
$659$	−1.03915	−0.0404796	−0.0202398	+	0.999795i	$0.506443\pi$
$660$	−1.86394	−0.0725537
$661$	−17.2076	−0.669298	−0.334649	−	0.942343i	$-0.608618\pi$
$661$	−17.2076	−0.669298	−0.334649	+	0.942343i	$0.608618\pi$
$662$	−4.66761	−0.181412
$663$	2.48904	0.0966664
$664$	2.70877	0.105121
$665$	12.6669	0.491201
$666$	−0.597996	−0.0231719
$667$	−9.41759	−0.364650
$668$	6.22058	0.240682
$669$	5.20612	0.201280
$670$	0.128217	0.00495345
$671$	−5.95450	−0.229871
$672$	7.02605	0.271036
$673$	−0.609968	−0.0235125	−0.0117563	−	0.999931i	$-0.503742\pi$
$673$	−0.609968	−0.0235125	−0.0117563	+	0.999931i	$0.503742\pi$
$674$	8.58016	0.330495
$675$	4.01728	0.154625
$676$	13.2173	0.508358
$677$	49.1829	1.89025	0.945126	−	0.326707i	$-0.105939\pi$
$677$	49.1829	1.89025	0.945126	+	0.326707i	$0.105939\pi$
$678$	−1.92924	−0.0740921
$679$	30.0556	1.15343
$680$	0.938057	0.0359729
$681$	−10.0078	−0.383500
$682$	−1.59824	−0.0611998
$683$	1.15275	0.0441089	0.0220545	−	0.999757i	$-0.492979\pi$
$683$	1.15275	0.0441089	0.0220545	+	0.999757i	$0.492979\pi$
$684$	9.78657	0.374199
$685$	−7.59189	−0.290071
$686$	4.60699	0.175896
$687$	12.7031	0.484654
$688$	−36.4436	−1.38940
$689$	−14.3389	−0.546269
$690$	0.429849	0.0163641
$691$	16.4839	0.627076	0.313538	−	0.949576i	$-0.398486\pi$
$691$	16.4839	0.627076	0.313538	+	0.949576i	$0.398486\pi$
$692$	16.4934	0.626984
$693$	2.45493	0.0932551
$694$	3.58389	0.136043
$695$	7.12020	0.270084
$696$	−4.93296	−0.186983
$697$	1.02216	0.0387172
$698$	6.10319	0.231009
$699$	7.43985	0.281401
$700$	19.7892	0.747960
$701$	−4.92799	−0.186128	−0.0930639	−	0.995660i	$-0.529666\pi$
$701$	−4.92799	−0.186128	−0.0930639	+	0.995660i	$0.529666\pi$
$702$	0.597429	0.0225485
$703$	−12.5527	−0.473435
$704$	−6.43758	−0.242625
$705$	−3.95164	−0.148827
$706$	−3.29866	−0.124147
$707$	−22.0511	−0.829315
$708$	1.48570	0.0558359
$709$	−26.2907	−0.987368	−0.493684	−	0.869641i	$-0.664350\pi$
$709$	−26.2907	−0.987368	−0.493684	+	0.869641i	$0.664350\pi$
$710$	2.14864	0.0806371
$711$	7.60120	0.285067
$712$	−0.246938	−0.00925441
$713$	−12.4267	−0.465382
$714$	−0.608715	−0.0227806
$715$	−2.38851	−0.0893254
$716$	−45.2191	−1.68992
$717$	−21.9317	−0.819053
$718$	−2.76415	−0.103157
$719$	29.1849	1.08841	0.544206	−	0.838952i	$-0.316831\pi$
$719$	29.1849	1.08841	0.544206	+	0.838952i	$0.316831\pi$
$720$	−3.62592	−0.135130
$721$	−21.1918	−0.789223
$722$	−1.53271	−0.0570416
$723$	0.669110	0.0248845
$724$	−16.4354	−0.610816
$725$	−20.9424	−0.777779
$726$	−2.41535	−0.0896420
$727$	−42.1030	−1.56151	−0.780756	−	0.624836i	$-0.785166\pi$
$727$	−42.1030	−1.56151	−0.780756	+	0.624836i	$0.785166\pi$
$728$	5.97318	0.221381
$729$	1.00000	0.0370370
$730$	−0.583585	−0.0215994
$731$	9.96365	0.368519
$732$	−11.9482	−0.441618
$733$	27.8306	1.02794	0.513972	−	0.857807i	$-0.328173\pi$
$733$	27.8306	1.02794	0.513972	+	0.857807i	$0.328173\pi$
$734$	−3.50964	−0.129543
$735$	−0.563458	−0.0207834
$736$	5.00492	0.184484
$737$	0.521620	0.0192141
$738$	0.245344	0.00903123
$739$	11.3089	0.416004	0.208002	−	0.978128i	$-0.433304\pi$
$739$	11.3089	0.416004	0.208002	+	0.978128i	$0.433304\pi$
$740$	4.79729	0.176352
$741$	12.5408	0.460699
$742$	3.50669	0.128735
$743$	−36.1091	−1.32471	−0.662357	−	0.749188i	$-0.730444\pi$
$743$	−36.1091	−1.32471	−0.662357	+	0.749188i	$0.730444\pi$
$744$	−6.50912	−0.238636
$745$	−15.4459	−0.565895
$746$	6.58692	0.241164
$747$	2.86258	0.104736
$748$	1.88025	0.0687487
$749$	10.1188	0.369733
$750$	2.14558	0.0783456
$751$	−38.0646	−1.38900	−0.694498	−	0.719494i	$-0.744374\pi$
$751$	−38.0646	−1.38900	−0.694498	+	0.719494i	$0.744374\pi$
$752$	−14.5802	−0.531685
$753$	20.6937	0.754120
$754$	−3.11444	−0.113421
$755$	2.44731	0.0890669
$756$	4.92602	0.179158
$757$	32.7274	1.18950	0.594749	−	0.803912i	$-0.297251\pi$
$757$	32.7274	1.18950	0.594749	+	0.803912i	$0.297251\pi$
$758$	2.87605	0.104463
$759$	1.74874	0.0634752
$760$	4.72633	0.171442
$761$	2.74772	0.0996046	0.0498023	−	0.998759i	$-0.484141\pi$
$761$	2.74772	0.0996046	0.0498023	+	0.998759i	$0.484141\pi$
$762$	−5.34060	−0.193469
$763$	32.7044	1.18398
$764$	38.6319	1.39765
$765$	0.991325	0.0358414
$766$	5.92286	0.214002
$767$	1.90382	0.0687430
$768$	−11.5876	−0.418130
$769$	−47.6072	−1.71676	−0.858379	−	0.513016i	$-0.828528\pi$
$769$	−47.6072	−1.71676	−0.858379	+	0.513016i	$0.828528\pi$
$770$	0.584130	0.0210506
$771$	1.50443	0.0541806
$772$	−31.2738	−1.12557
$773$	0.0699166	0.00251473	0.00125736	−	0.999999i	$-0.499600\pi$
$773$	0.0699166	0.00251473	0.00125736	+	0.999999i	$0.499600\pi$
$774$	2.39151	0.0859612
$775$	−27.6338	−0.992634
$776$	11.2145	0.402577
$777$	−6.31835	−0.226670
$778$	−6.95852	−0.249475
$779$	5.15009	0.184521
$780$	−4.79275	−0.171608
$781$	8.74125	0.312786
$782$	−0.433611	−0.0155059
$783$	−5.21307	−0.186300
$784$	−2.07897	−0.0742488
$785$	0.991325	0.0353819
$786$	4.92175	0.175553
$787$	9.29281	0.331253	0.165626	−	0.986189i	$-0.447035\pi$
$787$	9.29281	0.331253	0.165626	+	0.986189i	$0.447035\pi$
$788$	−7.03909	−0.250757
$789$	28.9952	1.03226
$790$	1.80864	0.0643485
$791$	−20.3842	−0.724777
$792$	0.915995	0.0325485
$793$	−15.3108	−0.543703
$794$	5.80279	0.205933
$795$	−5.71084	−0.202543
$796$	−39.0685	−1.38474
$797$	−8.05133	−0.285193	−0.142596	−	0.989781i	$-0.545545\pi$
$797$	−8.05133	−0.285193	−0.142596	+	0.989781i	$0.545545\pi$
$798$	−3.06696	−0.108569
$799$	3.98622	0.141022
$800$	11.1297	0.393494
$801$	−0.260961	−0.00922059
$802$	6.48869	0.229124
$803$	−2.37418	−0.0837830
$804$	1.04667	0.0369133
$805$	4.54173	0.160075
$806$	−4.10956	−0.144753
$807$	−5.53574	−0.194867
$808$	−8.22779	−0.289453
$809$	−15.4751	−0.544076	−0.272038	−	0.962286i	$-0.587698\pi$
$809$	−15.4751	−0.544076	−0.272038	+	0.962286i	$0.587698\pi$
$810$	0.237941	0.00836041
$811$	16.3627	0.574573	0.287286	−	0.957845i	$-0.407247\pi$
$811$	16.3627	0.574573	0.287286	+	0.957845i	$0.407247\pi$
$812$	−25.6797	−0.901181
$813$	−19.9638	−0.700162
$814$	−0.578865	−0.0202892
$815$	2.70917	0.0948981
$816$	3.65765	0.128043
$817$	50.2011	1.75631
$818$	1.64344	0.0574617
$819$	6.31237	0.220572
$820$	−1.96822	−0.0687331
$821$	37.9202	1.32342	0.661711	−	0.749759i	$-0.269830\pi$
$821$	37.9202	1.32342	0.661711	+	0.749759i	$0.269830\pi$
$822$	1.83818	0.0641139
$823$	−37.4816	−1.30653	−0.653263	−	0.757131i	$-0.726600\pi$
$823$	−37.4816	−1.30653	−0.653263	+	0.757131i	$0.726600\pi$
$824$	−7.90717	−0.275459
$825$	3.88876	0.135389
$826$	−0.465595	−0.0162001
$827$	0.707432	0.0245998	0.0122999	−	0.999924i	$-0.496085\pi$
$827$	0.707432	0.0245998	0.0122999	+	0.999924i	$0.496085\pi$
$828$	3.50899	0.121946
$829$	−26.3553	−0.915359	−0.457679	−	0.889117i	$-0.651319\pi$
$829$	−26.3553	−0.915359	−0.457679	+	0.889117i	$0.651319\pi$
$830$	0.681127	0.0236423
$831$	−16.8330	−0.583932
$832$	−16.5530	−0.573870
$833$	0.568389	0.0196935
$834$	−1.72397	−0.0596963
$835$	3.17476	0.109867
$836$	9.47349	0.327647
$837$	−6.87873	−0.237764
$838$	3.16554	0.109352
$839$	−43.2798	−1.49419	−0.747093	−	0.664720i	$-0.768551\pi$
$839$	−43.2798	−1.49419	−0.747093	+	0.664720i	$0.768551\pi$
$840$	2.37897	0.0820823
$841$	−1.82387	−0.0628920
$842$	7.28722	0.251134
$843$	16.0062	0.551283
$844$	50.3101	1.73175
$845$	6.74563	0.232057
$846$	0.956787	0.0328950
$847$	−25.5203	−0.876887
$848$	−21.0711	−0.723583
$849$	14.3986	0.494158
$850$	−0.964241	−0.0330732
$851$	−4.50080	−0.154286
$852$	17.5400	0.600911
$853$	33.4424	1.14505	0.572523	−	0.819889i	$-0.305965\pi$
$853$	33.4424	1.14505	0.572523	+	0.819889i	$0.305965\pi$
$854$	3.74438	0.128130
$855$	4.99471	0.170815
$856$	3.77557	0.129046
$857$	−37.6711	−1.28682	−0.643410	−	0.765521i	$-0.722481\pi$
$857$	−37.6711	−1.28682	−0.643410	+	0.765521i	$0.722481\pi$
$858$	0.578317	0.0197434
$859$	28.5567	0.974343	0.487172	−	0.873306i	$-0.338029\pi$
$859$	28.5567	0.974343	0.487172	+	0.873306i	$0.338029\pi$
$860$	−19.1854	−0.654216
$861$	2.59227	0.0883444
$862$	3.07619	0.104775
$863$	1.18690	0.0404027	0.0202013	−	0.999796i	$-0.493569\pi$
$863$	1.18690	0.0404027	0.0202013	+	0.999796i	$0.493569\pi$
$864$	2.77046	0.0942528
$865$	8.41762	0.286208
$866$	−5.80858	−0.197384
$867$	−1.00000	−0.0339618
$868$	−33.8848	−1.15012
$869$	7.35803	0.249604
$870$	−1.24041	−0.0420537
$871$	1.34124	0.0454463
$872$	12.2028	0.413240
$873$	11.8513	0.401106
$874$	−2.18471	−0.0738990
$875$	22.6700	0.766385
$876$	−4.76398	−0.160960
$877$	29.7271	1.00381	0.501907	−	0.864922i	$-0.332632\pi$
$877$	29.7271	1.00381	0.501907	+	0.864922i	$0.332632\pi$
$878$	0.142707	0.00481614
$879$	21.0986	0.711639
$880$	−3.50992	−0.118319
$881$	41.8281	1.40922	0.704612	−	0.709593i	$-0.251121\pi$
$881$	41.8281	1.40922	0.704612	+	0.709593i	$0.251121\pi$
$882$	0.136427	0.00459373
$883$	6.76538	0.227673	0.113837	−	0.993499i	$-0.463686\pi$
$883$	6.76538	0.227673	0.113837	+	0.993499i	$0.463686\pi$
$884$	4.83469	0.162608
$885$	0.758246	0.0254881
$886$	−4.80224	−0.161334
$887$	−7.59763	−0.255104	−0.127552	−	0.991832i	$-0.540712\pi$
$887$	−7.59763	−0.255104	−0.127552	+	0.991832i	$0.540712\pi$
$888$	−2.35753	−0.0791136
$889$	−56.4281	−1.89254
$890$	−0.0620933	−0.00208137
$891$	0.968009	0.0324295
$892$	10.1123	0.338585
$893$	20.0842	0.672094
$894$	3.73983	0.125079
$895$	−23.0782	−0.771418
$896$	18.1003	0.604687
$897$	4.49654	0.150135
$898$	7.29711	0.243508
$899$	35.8593	1.19598
$900$	7.80311	0.260104
$901$	5.76082	0.191921
$902$	0.237495	0.00790772
$903$	25.2684	0.840881
$904$	−7.60582	−0.252966
$905$	−8.38802	−0.278827
$906$	−0.592554	−0.0196863
$907$	5.19380	0.172457	0.0862286	−	0.996275i	$-0.472518\pi$
$907$	5.19380	0.172457	0.0862286	+	0.996275i	$0.472518\pi$
$908$	−19.4390	−0.645107
$909$	−8.69500	−0.288395
$910$	1.50197	0.0497899
$911$	35.0858	1.16244	0.581222	−	0.813745i	$-0.302575\pi$
$911$	35.0858	1.16244	0.581222	+	0.813745i	$0.302575\pi$
$912$	18.4288	0.610238
$913$	2.77101	0.0917069
$914$	4.35504	0.144052
$915$	−6.09792	−0.201591
$916$	24.6744	0.815265
$917$	52.0027	1.71728
$918$	−0.240024	−0.00792196
$919$	−10.8085	−0.356539	−0.178270	−	0.983982i	$-0.557050\pi$
$919$	−10.8085	−0.356539	−0.178270	+	0.983982i	$0.557050\pi$
$920$	1.69463	0.0558703
$921$	22.6024	0.744775
$922$	3.44335	0.113401
$923$	22.4764	0.739819
$924$	4.76843	0.156870
$925$	−10.0087	−0.329083
$926$	3.34662	0.109977
$927$	−8.35617	−0.274453
$928$	−14.4426	−0.474101
$929$	12.8050	0.420117	0.210059	−	0.977689i	$-0.432635\pi$
$929$	12.8050	0.420117	0.210059	+	0.977689i	$0.432635\pi$
$930$	−1.63674	−0.0536707
$931$	2.86378	0.0938566
$932$	14.4511	0.473361
$933$	−7.89681	−0.258530
$934$	−1.36359	−0.0446180
$935$	0.959611	0.0313826
$936$	2.35530	0.0769854
$937$	51.6603	1.68767	0.843834	−	0.536604i	$-0.180293\pi$
$937$	51.6603	1.68767	0.843834	+	0.536604i	$0.180293\pi$
$938$	−0.328012	−0.0107100
$939$	3.90937	0.127577
$940$	−7.67562	−0.250351
$941$	−33.8917	−1.10484	−0.552418	−	0.833567i	$-0.686295\pi$
$941$	−33.8917	−1.10484	−0.552418	+	0.833567i	$0.686295\pi$
$942$	−0.240024	−0.00782039
$943$	1.84657	0.0601327
$944$	2.79767	0.0910564
$945$	2.51406	0.0817824
$946$	2.31501	0.0752674
$947$	−4.92203	−0.159945	−0.0799723	−	0.996797i	$-0.525483\pi$
$947$	−4.92203	−0.159945	−0.0799723	+	0.996797i	$0.525483\pi$
$948$	14.7645	0.479528
$949$	−6.10473	−0.198168
$950$	−4.85825	−0.157622
$951$	16.1235	0.522840
$952$	−2.39979	−0.0777777
$953$	49.6887	1.60957	0.804787	−	0.593564i	$-0.202279\pi$
$953$	49.6887	1.60957	0.804787	+	0.593564i	$0.202279\pi$
$954$	1.38273	0.0447676
$955$	19.7163	0.638004
$956$	−42.5998	−1.37778
$957$	−5.04630	−0.163124
$958$	7.35100	0.237500
$959$	19.4220	0.627169
$960$	−6.59263	−0.212776
$961$	16.3170	0.526354
$962$	−1.48844	−0.0479892
$963$	3.98996	0.128575
$964$	1.29967	0.0418596
$965$	−15.9610	−0.513803
$966$	−1.09966	−0.0353811
$967$	−28.6202	−0.920364	−0.460182	−	0.887825i	$-0.652216\pi$
$967$	−28.6202	−0.920364	−0.460182	+	0.887825i	$0.652216\pi$
$968$	−9.52224	−0.306056
$969$	−5.03842	−0.161857
$970$	2.81991	0.0905420
$971$	6.93039	0.222407	0.111203	−	0.993798i	$-0.464529\pi$
$971$	6.93039	0.222407	0.111203	+	0.993798i	$0.464529\pi$
$972$	1.94239	0.0623021
$973$	−18.2153	−0.583955
$974$	−5.07590	−0.162642
$975$	9.99917	0.320230
$976$	−22.4993	−0.720184
$977$	22.6649	0.725113	0.362557	−	0.931962i	$-0.381904\pi$
$977$	22.6649	0.725113	0.362557	+	0.931962i	$0.381904\pi$
$978$	−0.655956	−0.0209751
$979$	−0.252612	−0.00807352
$980$	−1.09445	−0.0349610
$981$	12.8957	0.411730
$982$	−2.03734	−0.0650140
$983$	46.2607	1.47549	0.737743	−	0.675081i	$-0.235892\pi$
$983$	46.2607	1.47549	0.737743	+	0.675081i	$0.235892\pi$
$984$	0.967240	0.0308345
$985$	−3.59250	−0.114466
$986$	1.25126	0.0398483
$987$	10.1093	0.321783
$988$	24.3592	0.774969
$989$	17.9997	0.572356
$990$	0.230329	0.00732035
$991$	−29.5112	−0.937455	−0.468728	−	0.883343i	$-0.655287\pi$
$991$	−29.5112	−0.937455	−0.468728	+	0.883343i	$0.655287\pi$
$992$	−19.0572	−0.605068
$993$	−19.4464	−0.617114
$994$	−5.49678	−0.174347
$995$	−19.9391	−0.632113
$996$	5.56025	0.176183
$997$	−19.4844	−0.617079	−0.308539	−	0.951212i	$-0.599840\pi$
$997$	−19.4844	−0.617079	−0.308539	+	0.951212i	$0.599840\pi$
$998$	−0.332913	−0.0105382
$999$	−2.49140	−0.0788245

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8007.2.a.f.1.22	✓	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8007.2.a.f.1.22	✓	48	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:	\( N \)	\(=\)	\( 8007 = 3 \cdot 17 \cdot 157 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8007.a (trivial)

\(f(q)\)	\(=\)	\(q-0.240024 q^{2} -1.00000 q^{3} -1.94239 q^{4} -0.991325 q^{5} +0.240024 q^{6} +2.53606 q^{7} +0.946267 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.240024 q^{2} -1.00000 q^{3} -1.94239 q^{4} -0.991325 q^{5} +0.240024 q^{6} +2.53606 q^{7} +0.946267 q^{8} +1.00000 q^{9} +0.237941 q^{10} +0.968009 q^{11} +1.94239 q^{12} +2.48904 q^{13} -0.608715 q^{14} +0.991325 q^{15} +3.65765 q^{16} -1.00000 q^{17} -0.240024 q^{18} -5.03842 q^{19} +1.92554 q^{20} -2.53606 q^{21} -0.232345 q^{22} -1.80653 q^{23} -0.946267 q^{24} -4.01728 q^{25} -0.597429 q^{26} -1.00000 q^{27} -4.92602 q^{28} +5.21307 q^{29} -0.237941 q^{30} +6.87873 q^{31} -2.77046 q^{32} -0.968009 q^{33} +0.240024 q^{34} -2.51406 q^{35} -1.94239 q^{36} +2.49140 q^{37} +1.20934 q^{38} -2.48904 q^{39} -0.938057 q^{40} -1.02216 q^{41} +0.608715 q^{42} -9.96365 q^{43} -1.88025 q^{44} -0.991325 q^{45} +0.433611 q^{46} -3.98622 q^{47} -3.65765 q^{48} -0.568389 q^{49} +0.964241 q^{50} +1.00000 q^{51} -4.83469 q^{52} -5.76082 q^{53} +0.240024 q^{54} -0.959611 q^{55} +2.39979 q^{56} +5.03842 q^{57} -1.25126 q^{58} +0.764881 q^{59} -1.92554 q^{60} -6.15129 q^{61} -1.65106 q^{62} +2.53606 q^{63} -6.65033 q^{64} -2.46745 q^{65} +0.232345 q^{66} +0.538859 q^{67} +1.94239 q^{68} +1.80653 q^{69} +0.603434 q^{70} +9.03013 q^{71} +0.946267 q^{72} -2.45264 q^{73} -0.597996 q^{74} +4.01728 q^{75} +9.78657 q^{76} +2.45493 q^{77} +0.597429 q^{78} +7.60120 q^{79} -3.62592 q^{80} +1.00000 q^{81} +0.245344 q^{82} +2.86258 q^{83} +4.92602 q^{84} +0.991325 q^{85} +2.39151 q^{86} -5.21307 q^{87} +0.915995 q^{88} -0.260961 q^{89} +0.237941 q^{90} +6.31237 q^{91} +3.50899 q^{92} -6.87873 q^{93} +0.956787 q^{94} +4.99471 q^{95} +2.77046 q^{96} +11.8513 q^{97} +0.136427 q^{98} +0.968009 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9}+O(q^{10}) \) 48 * q - q^2 - 48 * q^3 + 45 * q^4 + q^5 + q^6 - 13 * q^7 - 6 * q^8 + 48 * q^9	\( 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9} - 20 q^{10} + 5 q^{11} - 45 q^{12} - 8 q^{13} + 4 q^{14} - q^{15} + 39 q^{16} - 48 q^{17} - q^{18} - 6 q^{19} + 6 q^{20} + 13 q^{21} - 35 q^{22} - 8 q^{23} + 6 q^{24} + 13 q^{25} + 17 q^{26} - 48 q^{27} - 38 q^{28} + q^{29} + 20 q^{30} - 21 q^{31} - 3 q^{32} - 5 q^{33} + q^{34} + 19 q^{35} + 45 q^{36} - 58 q^{37} - 14 q^{38} + 8 q^{39} - 54 q^{40} - 3 q^{41} - 4 q^{42} - 33 q^{43} + 2 q^{44} + q^{45} - 26 q^{46} + 9 q^{47} - 39 q^{48} + 11 q^{49} + 4 q^{50} + 48 q^{51} - 31 q^{52} - 33 q^{53} + q^{54} - 21 q^{55} + 6 q^{57} - 55 q^{58} + 77 q^{59} - 6 q^{60} - 29 q^{61} - 46 q^{62} - 13 q^{63} + 24 q^{64} - 49 q^{65} + 35 q^{66} - 44 q^{67} - 45 q^{68} + 8 q^{69} + 4 q^{70} + 22 q^{71} - 6 q^{72} - 63 q^{73} - 16 q^{74} - 13 q^{75} - 46 q^{76} - 30 q^{77} - 17 q^{78} - 46 q^{79} - 14 q^{80} + 48 q^{81} - 75 q^{82} + 11 q^{83} + 38 q^{84} - q^{85} + 8 q^{86} - q^{87} - 116 q^{88} + 10 q^{89} - 20 q^{90} - 67 q^{91} - 64 q^{92} + 21 q^{93} - 16 q^{94} - 8 q^{95} + 3 q^{96} - 96 q^{97} - 46 q^{98} + 5 q^{99}+O(q^{100}) \) 48 * q - q^2 - 48 * q^3 + 45 * q^4 + q^5 + q^6 - 13 * q^7 - 6 * q^8 + 48 * q^9 - 20 * q^10 + 5 * q^11 - 45 * q^12 - 8 * q^13 + 4 * q^14 - q^15 + 39 * q^16 - 48 * q^17 - q^18 - 6 * q^19 + 6 * q^20 + 13 * q^21 - 35 * q^22 - 8 * q^23 + 6 * q^24 + 13 * q^25 + 17 * q^26 - 48 * q^27 - 38 * q^28 + q^29 + 20 * q^30 - 21 * q^31 - 3 * q^32 - 5 * q^33 + q^34 + 19 * q^35 + 45 * q^36 - 58 * q^37 - 14 * q^38 + 8 * q^39 - 54 * q^40 - 3 * q^41 - 4 * q^42 - 33 * q^43 + 2 * q^44 + q^45 - 26 * q^46 + 9 * q^47 - 39 * q^48 + 11 * q^49 + 4 * q^50 + 48 * q^51 - 31 * q^52 - 33 * q^53 + q^54 - 21 * q^55 + 6 * q^57 - 55 * q^58 + 77 * q^59 - 6 * q^60 - 29 * q^61 - 46 * q^62 - 13 * q^63 + 24 * q^64 - 49 * q^65 + 35 * q^66 - 44 * q^67 - 45 * q^68 + 8 * q^69 + 4 * q^70 + 22 * q^71 - 6 * q^72 - 63 * q^73 - 16 * q^74 - 13 * q^75 - 46 * q^76 - 30 * q^77 - 17 * q^78 - 46 * q^79 - 14 * q^80 + 48 * q^81 - 75 * q^82 + 11 * q^83 + 38 * q^84 - q^85 + 8 * q^86 - q^87 - 116 * q^88 + 10 * q^89 - 20 * q^90 - 67 * q^91 - 64 * q^92 + 21 * q^93 - 16 * q^94 - 8 * q^95 + 3 * q^96 - 96 * q^97 - 46 * q^98 + 5 * q^99

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