LMFDB - Embedded newform 8007.2.a.f.1.29

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			1.29
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)

$f(q)$	$=$	$q+0.522212 q^{2} -1.00000 q^{3} -1.72729 q^{4} -2.49100 q^{5} -0.522212 q^{6} +5.20591 q^{7} -1.94644 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.522212 q^{2} -1.00000 q^{3} -1.72729 q^{4} -2.49100 q^{5} -0.522212 q^{6} +5.20591 q^{7} -1.94644 q^{8} +1.00000 q^{9} -1.30083 q^{10} +3.79213 q^{11} +1.72729 q^{12} +1.98902 q^{13} +2.71859 q^{14} +2.49100 q^{15} +2.43813 q^{16} -1.00000 q^{17} +0.522212 q^{18} -4.44935 q^{19} +4.30269 q^{20} -5.20591 q^{21} +1.98030 q^{22} +1.01413 q^{23} +1.94644 q^{24} +1.20509 q^{25} +1.03869 q^{26} -1.00000 q^{27} -8.99213 q^{28} -8.84805 q^{29} +1.30083 q^{30} -7.21004 q^{31} +5.16610 q^{32} -3.79213 q^{33} -0.522212 q^{34} -12.9679 q^{35} -1.72729 q^{36} -11.2380 q^{37} -2.32350 q^{38} -1.98902 q^{39} +4.84859 q^{40} +3.60771 q^{41} -2.71859 q^{42} +3.48456 q^{43} -6.55012 q^{44} -2.49100 q^{45} +0.529593 q^{46} -0.328790 q^{47} -2.43813 q^{48} +20.1015 q^{49} +0.629315 q^{50} +1.00000 q^{51} -3.43562 q^{52} +7.74323 q^{53} -0.522212 q^{54} -9.44620 q^{55} -10.1330 q^{56} +4.44935 q^{57} -4.62056 q^{58} +9.57031 q^{59} -4.30269 q^{60} -4.50698 q^{61} -3.76517 q^{62} +5.20591 q^{63} -2.17846 q^{64} -4.95466 q^{65} -1.98030 q^{66} +1.51522 q^{67} +1.72729 q^{68} -1.01413 q^{69} -6.77201 q^{70} +0.798424 q^{71} -1.94644 q^{72} -2.42446 q^{73} -5.86862 q^{74} -1.20509 q^{75} +7.68533 q^{76} +19.7415 q^{77} -1.03869 q^{78} -0.0708311 q^{79} -6.07340 q^{80} +1.00000 q^{81} +1.88399 q^{82} -6.45036 q^{83} +8.99213 q^{84} +2.49100 q^{85} +1.81968 q^{86} +8.84805 q^{87} -7.38114 q^{88} -9.71172 q^{89} -1.30083 q^{90} +10.3547 q^{91} -1.75171 q^{92} +7.21004 q^{93} -0.171698 q^{94} +11.0833 q^{95} -5.16610 q^{96} +6.02464 q^{97} +10.4972 q^{98} +3.79213 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.522212	0.369260	0.184630	−	0.982808i	$-0.440891\pi$
$2$	0.522212	0.369260	0.184630	+	0.982808i	$0.440891\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.72729	−0.863647
$5$	−2.49100	−1.11401	−0.557005	−	0.830509i	$-0.688050\pi$
$5$	−2.49100	−1.11401	−0.557005	+	0.830509i	$0.688050\pi$
$6$	−0.522212	−0.213192
$7$	5.20591	1.96765	0.983824	−	0.179138i	$-0.0573309\pi$
$7$	5.20591	1.96765	0.983824	+	0.179138i	$0.0573309\pi$
$8$	−1.94644	−0.688170
$9$	1.00000	0.333333
$10$	−1.30083	−0.411359
$11$	3.79213	1.14337	0.571684	−	0.820474i	$-0.306290\pi$
$11$	3.79213	1.14337	0.571684	+	0.820474i	$0.306290\pi$
$12$	1.72729	0.498627
$13$	1.98902	0.551655	0.275828	−	0.961207i	$-0.411048\pi$
$13$	1.98902	0.551655	0.275828	+	0.961207i	$0.411048\pi$
$14$	2.71859	0.726574
$15$	2.49100	0.643174
$16$	2.43813	0.609533
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0.522212	0.123087
$19$	−4.44935	−1.02075	−0.510375	−	0.859952i	$-0.670493\pi$
$19$	−4.44935	−1.02075	−0.510375	+	0.859952i	$0.670493\pi$
$20$	4.30269	0.962112
$21$	−5.20591	−1.13602
$22$	1.98030	0.422200
$23$	1.01413	0.211461	0.105731	−	0.994395i	$-0.466282\pi$
$23$	1.01413	0.211461	0.105731	+	0.994395i	$0.466282\pi$
$24$	1.94644	0.397315
$25$	1.20509	0.241019
$26$	1.03869	0.203704
$27$	−1.00000	−0.192450
$28$	−8.99213	−1.69935
$29$	−8.84805	−1.64304	−0.821521	−	0.570179i	$-0.806874\pi$
$29$	−8.84805	−1.64304	−0.821521	+	0.570179i	$0.806874\pi$
$30$	1.30083	0.237498
$31$	−7.21004	−1.29496	−0.647481	−	0.762082i	$-0.724177\pi$
$31$	−7.21004	−1.29496	−0.647481	+	0.762082i	$0.724177\pi$
$32$	5.16610	0.913246
$33$	−3.79213	−0.660124
$34$	−0.522212	−0.0895587
$35$	−12.9679	−2.19198
$36$	−1.72729	−0.287882
$37$	−11.2380	−1.84751	−0.923757	−	0.382979i	$-0.874898\pi$
$37$	−11.2380	−1.84751	−0.923757	+	0.382979i	$0.874898\pi$
$38$	−2.32350	−0.376922
$39$	−1.98902	−0.318498
$40$	4.84859	0.766629
$41$	3.60771	0.563429	0.281715	−	0.959498i	$-0.409097\pi$
$41$	3.60771	0.563429	0.281715	+	0.959498i	$0.409097\pi$
$42$	−2.71859	−0.419487
$43$	3.48456	0.531390	0.265695	−	0.964057i	$-0.414399\pi$
$43$	3.48456	0.531390	0.265695	+	0.964057i	$0.414399\pi$
$44$	−6.55012	−0.987467
$45$	−2.49100	−0.371337
$46$	0.529593	0.0780842
$47$	−0.328790	−0.0479590	−0.0239795	−	0.999712i	$-0.507634\pi$
$47$	−0.328790	−0.0479590	−0.0239795	+	0.999712i	$0.507634\pi$
$48$	−2.43813	−0.351914
$49$	20.1015	2.87164
$50$	0.629315	0.0889986
$51$	1.00000	0.140028
$52$	−3.43562	−0.476435
$53$	7.74323	1.06361	0.531807	−	0.846865i	$-0.321513\pi$
$53$	7.74323	1.06361	0.531807	+	0.846865i	$0.321513\pi$
$54$	−0.522212	−0.0710641
$55$	−9.44620	−1.27372
$56$	−10.1330	−1.35408
$57$	4.44935	0.589330
$58$	−4.62056	−0.606709
$59$	9.57031	1.24595	0.622974	−	0.782242i	$-0.285924\pi$
$59$	9.57031	1.24595	0.622974	+	0.782242i	$0.285924\pi$
$60$	−4.30269	−0.555475
$61$	−4.50698	−0.577060	−0.288530	−	0.957471i	$-0.593166\pi$
$61$	−4.50698	−0.577060	−0.288530	+	0.957471i	$0.593166\pi$
$62$	−3.76517	−0.478178
$63$	5.20591	0.655883
$64$	−2.17846	−0.272308
$65$	−4.95466	−0.614549
$66$	−1.98030	−0.243757
$67$	1.51522	0.185114	0.0925568	−	0.995707i	$-0.470496\pi$
$67$	1.51522	0.185114	0.0925568	+	0.995707i	$0.470496\pi$
$68$	1.72729	0.209465
$69$	−1.01413	−0.122087
$70$	−6.77201	−0.809410
$71$	0.798424	0.0947556	0.0473778	−	0.998877i	$-0.484914\pi$
$71$	0.798424	0.0947556	0.0473778	+	0.998877i	$0.484914\pi$
$72$	−1.94644	−0.229390
$73$	−2.42446	−0.283761	−0.141881	−	0.989884i	$-0.545315\pi$
$73$	−2.42446	−0.283761	−0.141881	+	0.989884i	$0.545315\pi$
$74$	−5.86862	−0.682213
$75$	−1.20509	−0.139152
$76$	7.68533	0.881568
$77$	19.7415	2.24975
$78$	−1.03869	−0.117609
$79$	−0.0708311	−0.00796912	−0.00398456	−	0.999992i	$-0.501268\pi$
$79$	−0.0708311	−0.00796912	−0.00398456	+	0.999992i	$0.501268\pi$
$80$	−6.07340	−0.679026
$81$	1.00000	0.111111
$82$	1.88399	0.208052
$83$	−6.45036	−0.708019	−0.354010	−	0.935242i	$-0.615182\pi$
$83$	−6.45036	−0.708019	−0.354010	+	0.935242i	$0.615182\pi$
$84$	8.99213	0.981122
$85$	2.49100	0.270187
$86$	1.81968	0.196221
$87$	8.84805	0.948610
$88$	−7.38114	−0.786832
$89$	−9.71172	−1.02944	−0.514720	−	0.857358i	$-0.672104\pi$
$89$	−9.71172	−1.02944	−0.514720	+	0.857358i	$0.672104\pi$
$90$	−1.30083	−0.137120
$91$	10.3547	1.08546
$92$	−1.75171	−0.182628
$93$	7.21004	0.747647
$94$	−0.171698	−0.0177093
$95$	11.0833	1.13713
$96$	−5.16610	−0.527263
$97$	6.02464	0.611710	0.305855	−	0.952078i	$-0.401058\pi$
$97$	6.02464	0.611710	0.305855	+	0.952078i	$0.401058\pi$
$98$	10.4972	1.06038
$99$	3.79213	0.381123
$100$	−2.08155	−0.208155
$101$	16.1285	1.60485	0.802424	−	0.596754i	$-0.203543\pi$
$101$	16.1285	1.60485	0.802424	+	0.596754i	$0.203543\pi$
$102$	0.522212	0.0517067
$103$	−11.4708	−1.13025	−0.565125	−	0.825005i	$-0.691172\pi$
$103$	−11.4708	−1.13025	−0.565125	+	0.825005i	$0.691172\pi$
$104$	−3.87151	−0.379633
$105$	12.9679	1.26554
$106$	4.04361	0.392750
$107$	18.1871	1.75821	0.879107	−	0.476625i	$-0.158140\pi$
$107$	18.1871	1.75821	0.879107	+	0.476625i	$0.158140\pi$
$108$	1.72729	0.166209
$109$	−16.0166	−1.53411	−0.767054	−	0.641583i	$-0.778278\pi$
$109$	−16.0166	−1.53411	−0.767054	+	0.641583i	$0.778278\pi$
$110$	−4.93292	−0.470336
$111$	11.2380	1.06666
$112$	12.6927	1.19935
$113$	−16.9067	−1.59044	−0.795222	−	0.606318i	$-0.792646\pi$
$113$	−16.9067	−1.59044	−0.795222	+	0.606318i	$0.792646\pi$
$114$	2.32350	0.217616
$115$	−2.52621	−0.235570
$116$	15.2832	1.41901
$117$	1.98902	0.183885
$118$	4.99773	0.460079
$119$	−5.20591	−0.477225
$120$	−4.84859	−0.442613
$121$	3.38022	0.307292
$122$	−2.35360	−0.213085
$123$	−3.60771	−0.325296
$124$	12.4539	1.11839
$125$	9.45312	0.845513
$126$	2.71859	0.242191
$127$	6.58334	0.584177	0.292089	−	0.956391i	$-0.405650\pi$
$127$	6.58334	0.584177	0.292089	+	0.956391i	$0.405650\pi$
$128$	−11.4698	−1.01380
$129$	−3.48456	−0.306798
$130$	−2.58738	−0.226928
$131$	−9.63710	−0.841997	−0.420999	−	0.907061i	$-0.638320\pi$
$131$	−9.63710	−0.841997	−0.420999	+	0.907061i	$0.638320\pi$
$132$	6.55012	0.570114
$133$	−23.1629	−2.00848
$134$	0.791266	0.0683550
$135$	2.49100	0.214391
$136$	1.94644	0.166906
$137$	−14.6891	−1.25498	−0.627489	−	0.778625i	$-0.715917\pi$
$137$	−14.6891	−1.25498	−0.627489	+	0.778625i	$0.715917\pi$
$138$	−0.529593	−0.0450819
$139$	−0.760234	−0.0644822	−0.0322411	−	0.999480i	$-0.510264\pi$
$139$	−0.760234	−0.0644822	−0.0322411	+	0.999480i	$0.510264\pi$
$140$	22.3994	1.89310
$141$	0.328790	0.0276891
$142$	0.416947	0.0349894
$143$	7.54262	0.630745
$144$	2.43813	0.203178
$145$	22.0405	1.83036
$146$	−1.26608	−0.104782
$147$	−20.1015	−1.65794
$148$	19.4113	1.59560
$149$	−2.99363	−0.245248	−0.122624	−	0.992453i	$-0.539131\pi$
$149$	−2.99363	−0.245248	−0.122624	+	0.992453i	$0.539131\pi$
$150$	−0.629315	−0.0513834
$151$	−2.14812	−0.174812	−0.0874058	−	0.996173i	$-0.527858\pi$
$151$	−2.14812	−0.174812	−0.0874058	+	0.996173i	$0.527858\pi$
$152$	8.66038	0.702450
$153$	−1.00000	−0.0808452
$154$	10.3092	0.830742
$155$	17.9602	1.44260
$156$	3.43562	0.275070
$157$	−1.00000	−0.0798087
$157$	−1.00000	−0.0798087
$158$	−0.0369889	−0.00294268
$159$	−7.74323	−0.614078
$160$	−12.8688	−1.01737
$161$	5.27948	0.416081
$162$	0.522212	0.0410289
$163$	−6.26047	−0.490358	−0.245179	−	0.969478i	$-0.578847\pi$
$163$	−6.26047	−0.490358	−0.245179	+	0.969478i	$0.578847\pi$
$164$	−6.23157	−0.486604
$165$	9.44620	0.735385
$166$	−3.36846	−0.261443
$167$	−11.7678	−0.910621	−0.455311	−	0.890333i	$-0.650472\pi$
$167$	−11.7678	−0.910621	−0.455311	+	0.890333i	$0.650472\pi$
$168$	10.1330	0.781777
$169$	−9.04380	−0.695677
$170$	1.30083	0.0997693
$171$	−4.44935	−0.340250
$172$	−6.01886	−0.458934
$173$	19.4467	1.47850	0.739252	−	0.673429i	$-0.235179\pi$
$173$	19.4467	1.47850	0.739252	+	0.673429i	$0.235179\pi$
$174$	4.62056	0.350284
$175$	6.27361	0.474240
$176$	9.24571	0.696922
$177$	−9.57031	−0.719349
$178$	−5.07158	−0.380131
$179$	21.8141	1.63047	0.815233	−	0.579133i	$-0.196609\pi$
$179$	21.8141	1.63047	0.815233	+	0.579133i	$0.196609\pi$
$180$	4.30269	0.320704
$181$	11.0565	0.821823	0.410912	−	0.911675i	$-0.365210\pi$
$181$	11.0565	0.821823	0.410912	+	0.911675i	$0.365210\pi$
$182$	5.40733	0.400818
$183$	4.50698	0.333165
$184$	−1.97395	−0.145521
$185$	27.9939	2.05815
$186$	3.76517	0.276076
$187$	−3.79213	−0.277308
$188$	0.567917	0.0414196
$189$	−5.20591	−0.378674
$190$	5.78785	0.419895
$191$	−8.95026	−0.647618	−0.323809	−	0.946122i	$-0.604964\pi$
$191$	−8.95026	−0.647618	−0.323809	+	0.946122i	$0.604964\pi$
$192$	2.17846	0.157217
$193$	6.05971	0.436187	0.218094	−	0.975928i	$-0.430016\pi$
$193$	6.05971	0.436187	0.218094	+	0.975928i	$0.430016\pi$
$194$	3.14614	0.225880
$195$	4.95466	0.354810
$196$	−34.7211	−2.48008
$197$	8.51967	0.607001	0.303501	−	0.952831i	$-0.401845\pi$
$197$	8.51967	0.607001	0.303501	+	0.952831i	$0.401845\pi$
$198$	1.98030	0.140733
$199$	−4.14330	−0.293710	−0.146855	−	0.989158i	$-0.546915\pi$
$199$	−4.14330	−0.293710	−0.146855	+	0.989158i	$0.546915\pi$
$200$	−2.34564	−0.165862
$201$	−1.51522	−0.106875
$202$	8.42252	0.592606
$203$	−46.0621	−3.23293
$204$	−1.72729	−0.120935
$205$	−8.98681	−0.627666
$206$	−5.99019	−0.417356
$207$	1.01413	0.0704871
$208$	4.84950	0.336252
$209$	−16.8725	−1.16709
$210$	6.77201	0.467313
$211$	−21.8940	−1.50725	−0.753623	−	0.657307i	$-0.771695\pi$
$211$	−21.8940	−1.50725	−0.753623	+	0.657307i	$0.771695\pi$
$212$	−13.3748	−0.918588
$213$	−0.798424	−0.0547071
$214$	9.49753	0.649238
$215$	−8.68005	−0.591974
$216$	1.94644	0.132438
$217$	−37.5348	−2.54803
$218$	−8.36404	−0.566485
$219$	2.42446	0.163830
$220$	16.3164	1.10005
$221$	−1.98902	−0.133796
$222$	5.86862	0.393876
$223$	−18.6656	−1.24994	−0.624971	−	0.780648i	$-0.714889\pi$
$223$	−18.6656	−1.24994	−0.624971	+	0.780648i	$0.714889\pi$
$224$	26.8942	1.79695
$225$	1.20509	0.0803396
$226$	−8.82887	−0.587287
$227$	6.96732	0.462437	0.231219	−	0.972902i	$-0.425729\pi$
$227$	6.96732	0.462437	0.231219	+	0.972902i	$0.425729\pi$
$228$	−7.68533	−0.508973
$229$	−10.7510	−0.710444	−0.355222	−	0.934782i	$-0.615595\pi$
$229$	−10.7510	−0.710444	−0.355222	+	0.934782i	$0.615595\pi$
$230$	−1.31922	−0.0869866
$231$	−19.7415	−1.29889
$232$	17.2222	1.13069
$233$	−21.1681	−1.38677	−0.693385	−	0.720568i	$-0.743881\pi$
$233$	−21.1681	−1.38677	−0.693385	+	0.720568i	$0.743881\pi$
$234$	1.03869	0.0679014
$235$	0.819017	0.0534268
$236$	−16.5307	−1.07606
$237$	0.0708311	0.00460097
$238$	−2.71859	−0.176220
$239$	−19.0843	−1.23446	−0.617229	−	0.786784i	$-0.711745\pi$
$239$	−19.0843	−1.23446	−0.617229	+	0.786784i	$0.711745\pi$
$240$	6.07340	0.392036
$241$	−20.0543	−1.29181	−0.645907	−	0.763416i	$-0.723521\pi$
$241$	−20.0543	−1.29181	−0.645907	+	0.763416i	$0.723521\pi$
$242$	1.76519	0.113471
$243$	−1.00000	−0.0641500
$244$	7.78488	0.498376
$245$	−50.0728	−3.19903
$246$	−1.88399	−0.120119
$247$	−8.84984	−0.563102
$248$	14.0339	0.891154
$249$	6.45036	0.408775
$250$	4.93654	0.312214
$251$	16.9195	1.06795	0.533975	−	0.845501i	$-0.320698\pi$
$251$	16.9195	1.06795	0.533975	+	0.845501i	$0.320698\pi$
$252$	−8.99213	−0.566451
$253$	3.84572	0.241778
$254$	3.43790	0.215713
$255$	−2.49100	−0.155993
$256$	−1.63276	−0.102047
$257$	4.62699	0.288623	0.144312	−	0.989532i	$-0.453903\pi$
$257$	4.62699	0.288623	0.144312	+	0.989532i	$0.453903\pi$
$258$	−1.81968	−0.113288
$259$	−58.5039	−3.63526
$260$	8.55815	0.530754
$261$	−8.84805	−0.547680
$262$	−5.03261	−0.310916
$263$	−11.4847	−0.708178	−0.354089	−	0.935212i	$-0.615209\pi$
$263$	−11.4847	−0.708178	−0.354089	+	0.935212i	$0.615209\pi$
$264$	7.38114	0.454278
$265$	−19.2884	−1.18488
$266$	−12.0959	−0.741650
$267$	9.71172	0.594348
$268$	−2.61723	−0.159873
$269$	26.0351	1.58739	0.793695	−	0.608316i	$-0.208155\pi$
$269$	26.0351	1.58739	0.793695	+	0.608316i	$0.208155\pi$
$270$	1.30083	0.0791661
$271$	−21.6310	−1.31399	−0.656996	−	0.753894i	$-0.728173\pi$
$271$	−21.6310	−1.31399	−0.656996	+	0.753894i	$0.728173\pi$
$272$	−2.43813	−0.147834
$273$	−10.3547	−0.626692
$274$	−7.67085	−0.463413
$275$	4.56987	0.275573
$276$	1.75171	0.105440
$277$	−13.8130	−0.829942	−0.414971	−	0.909835i	$-0.636208\pi$
$277$	−13.8130	−0.829942	−0.414971	+	0.909835i	$0.636208\pi$
$278$	−0.397004	−0.0238107
$279$	−7.21004	−0.431654
$280$	25.2413	1.50846
$281$	−10.6326	−0.634286	−0.317143	−	0.948378i	$-0.602724\pi$
$281$	−10.6326	−0.634286	−0.317143	+	0.948378i	$0.602724\pi$
$282$	0.171698	0.0102245
$283$	18.4505	1.09677	0.548385	−	0.836226i	$-0.315243\pi$
$283$	18.4505	1.09677	0.548385	+	0.836226i	$0.315243\pi$
$284$	−1.37911	−0.0818354
$285$	−11.0833	−0.656520
$286$	3.93885	0.232909
$287$	18.7814	1.10863
$288$	5.16610	0.304415
$289$	1.00000	0.0588235
$290$	11.5098	0.675880
$291$	−6.02464	−0.353171
$292$	4.18775	0.245069
$293$	18.0697	1.05564	0.527820	−	0.849356i	$-0.323010\pi$
$293$	18.0697	1.05564	0.527820	+	0.849356i	$0.323010\pi$
$294$	−10.4972	−0.612211
$295$	−23.8397	−1.38800
$296$	21.8741	1.27140
$297$	−3.79213	−0.220041
$298$	−1.56331	−0.0905601
$299$	2.01713	0.116654
$300$	2.08155	0.120178
$301$	18.1403	1.04559
$302$	−1.12178	−0.0645510
$303$	−16.1285	−0.926559
$304$	−10.8481	−0.622181
$305$	11.2269	0.642850
$306$	−0.522212	−0.0298529
$307$	−25.7624	−1.47034	−0.735170	−	0.677883i	$-0.762898\pi$
$307$	−25.7624	−1.47034	−0.735170	+	0.677883i	$0.762898\pi$
$308$	−34.0993	−1.94299
$309$	11.4708	0.652550
$310$	9.37906	0.532695
$311$	−1.44216	−0.0817771	−0.0408886	−	0.999164i	$-0.513019\pi$
$311$	−1.44216	−0.0817771	−0.0408886	+	0.999164i	$0.513019\pi$
$312$	3.87151	0.219181
$313$	−19.1869	−1.08451	−0.542253	−	0.840215i	$-0.682429\pi$
$313$	−19.1869	−1.08451	−0.542253	+	0.840215i	$0.682429\pi$
$314$	−0.522212	−0.0294702
$315$	−12.9679	−0.730660
$316$	0.122346	0.00688251
$317$	16.6819	0.936950	0.468475	−	0.883477i	$-0.344804\pi$
$317$	16.6819	0.936950	0.468475	+	0.883477i	$0.344804\pi$
$318$	−4.04361	−0.226755
$319$	−33.5529	−1.87860
$320$	5.42656	0.303354
$321$	−18.1871	−1.01510
$322$	2.75701	0.153642
$323$	4.44935	0.247568
$324$	−1.72729	−0.0959608
$325$	2.39696	0.132959
$326$	−3.26929	−0.181069
$327$	16.0166	0.885717
$328$	−7.02218	−0.387735
$329$	−1.71165	−0.0943664
$330$	4.93292	0.271548
$331$	−4.91999	−0.270427	−0.135213	−	0.990816i	$-0.543172\pi$
$331$	−4.91999	−0.270427	−0.135213	+	0.990816i	$0.543172\pi$
$332$	11.1417	0.611479
$333$	−11.2380	−0.615838
$334$	−6.14530	−0.336256
$335$	−3.77442	−0.206218
$336$	−12.6927	−0.692443
$337$	−14.7265	−0.802205	−0.401102	−	0.916033i	$-0.631373\pi$
$337$	−14.7265	−0.802205	−0.401102	+	0.916033i	$0.631373\pi$
$338$	−4.72278	−0.256886
$339$	16.9067	0.918243
$340$	−4.30269	−0.233346
$341$	−27.3414	−1.48062
$342$	−2.32350	−0.125641
$343$	68.2050	3.68272
$344$	−6.78248	−0.365687
$345$	2.52621	0.136006
$346$	10.1553	0.545952
$347$	−1.32420	−0.0710867	−0.0355434	−	0.999368i	$-0.511316\pi$
$347$	−1.32420	−0.0710867	−0.0355434	+	0.999368i	$0.511316\pi$
$348$	−15.2832	−0.819264
$349$	1.58381	0.0847795	0.0423897	−	0.999101i	$-0.486503\pi$
$349$	1.58381	0.0847795	0.0423897	+	0.999101i	$0.486503\pi$
$350$	3.27616	0.175118
$351$	−1.98902	−0.106166
$352$	19.5905	1.04418
$353$	0.214568	0.0114203	0.00571016	−	0.999984i	$-0.498182\pi$
$353$	0.214568	0.0114203	0.00571016	+	0.999984i	$0.498182\pi$
$354$	−4.99773	−0.265627
$355$	−1.98888	−0.105559
$356$	16.7750	0.889073
$357$	5.20591	0.275526
$358$	11.3916	0.602066
$359$	−0.503239	−0.0265599	−0.0132800	−	0.999912i	$-0.504227\pi$
$359$	−0.503239	−0.0265599	−0.0132800	+	0.999912i	$0.504227\pi$
$360$	4.84859	0.255543
$361$	0.796677	0.0419304
$362$	5.77384	0.303466
$363$	−3.38022	−0.177415
$364$	−17.8855	−0.937457
$365$	6.03933	0.316113
$366$	2.35360	0.123025
$367$	17.4872	0.912826	0.456413	−	0.889768i	$-0.349134\pi$
$367$	17.4872	0.912826	0.456413	+	0.889768i	$0.349134\pi$
$368$	2.47259	0.128893
$369$	3.60771	0.187810
$370$	14.6187	0.759992
$371$	40.3106	2.09282
$372$	−12.4539	−0.645703
$373$	21.3453	1.10522	0.552610	−	0.833440i	$-0.313632\pi$
$373$	21.3453	1.10522	0.552610	+	0.833440i	$0.313632\pi$
$374$	−1.98030	−0.102399
$375$	−9.45312	−0.488157
$376$	0.639970	0.0330039
$377$	−17.5989	−0.906392
$378$	−2.71859	−0.139829
$379$	−37.6086	−1.93182	−0.965912	−	0.258870i	$-0.916650\pi$
$379$	−37.6086	−1.93182	−0.965912	+	0.258870i	$0.916650\pi$
$380$	−19.1442	−0.982075
$381$	−6.58334	−0.337275
$382$	−4.67394	−0.239139
$383$	7.85675	0.401461	0.200731	−	0.979646i	$-0.435668\pi$
$383$	7.85675	0.401461	0.200731	+	0.979646i	$0.435668\pi$
$384$	11.4698	0.585317
$385$	−49.1760	−2.50624
$386$	3.16445	0.161066
$387$	3.48456	0.177130
$388$	−10.4063	−0.528302
$389$	−31.0418	−1.57388	−0.786940	−	0.617029i	$-0.788336\pi$
$389$	−31.0418	−1.57388	−0.786940	+	0.617029i	$0.788336\pi$
$390$	2.58738	0.131017
$391$	−1.01413	−0.0512869
$392$	−39.1263	−1.97618
$393$	9.63710	0.486127
$394$	4.44908	0.224141
$395$	0.176440	0.00887768
$396$	−6.55012	−0.329156
$397$	4.96984	0.249429	0.124714	−	0.992193i	$-0.460199\pi$
$397$	4.96984	0.249429	0.124714	+	0.992193i	$0.460199\pi$
$398$	−2.16368	−0.108455
$399$	23.1629	1.15959
$400$	2.93818	0.146909
$401$	15.5267	0.775367	0.387684	−	0.921793i	$-0.373275\pi$
$401$	15.5267	0.775367	0.387684	+	0.921793i	$0.373275\pi$
$402$	−0.791266	−0.0394648
$403$	−14.3409	−0.714372
$404$	−27.8587	−1.38602
$405$	−2.49100	−0.123779
$406$	−24.0542	−1.19379
$407$	−42.6159	−2.11239
$408$	−1.94644	−0.0963631
$409$	21.7623	1.07607	0.538037	−	0.842921i	$-0.319166\pi$
$409$	21.7623	1.07607	0.538037	+	0.842921i	$0.319166\pi$
$410$	−4.69302	−0.231772
$411$	14.6891	0.724562
$412$	19.8134	0.976138
$413$	49.8221	2.45159
$414$	0.529593	0.0260281
$415$	16.0679	0.788741
$416$	10.2755	0.503797
$417$	0.760234	0.0372288
$418$	−8.81102	−0.430961
$419$	29.1199	1.42260	0.711299	−	0.702889i	$-0.248107\pi$
$419$	29.1199	1.42260	0.711299	+	0.702889i	$0.248107\pi$
$420$	−22.3994	−1.09298
$421$	29.3418	1.43003	0.715017	−	0.699107i	$-0.246419\pi$
$421$	29.3418	1.43003	0.715017	+	0.699107i	$0.246419\pi$
$422$	−11.4333	−0.556565
$423$	−0.328790	−0.0159863
$424$	−15.0717	−0.731948
$425$	−1.20509	−0.0584556
$426$	−0.416947	−0.0202012
$427$	−23.4629	−1.13545
$428$	−31.4145	−1.51848
$429$	−7.54262	−0.364161
$430$	−4.53283	−0.218592
$431$	−2.24604	−0.108188	−0.0540940	−	0.998536i	$-0.517227\pi$
$431$	−2.24604	−0.108188	−0.0540940	+	0.998536i	$0.517227\pi$
$432$	−2.43813	−0.117305
$433$	2.63146	0.126460	0.0632298	−	0.997999i	$-0.479860\pi$
$433$	2.63146	0.126460	0.0632298	+	0.997999i	$0.479860\pi$
$434$	−19.6011	−0.940885
$435$	−22.0405	−1.05676
$436$	27.6653	1.32493
$437$	−4.51223	−0.215849
$438$	1.26608	0.0604957
$439$	−33.1159	−1.58053	−0.790267	−	0.612762i	$-0.790058\pi$
$439$	−33.1159	−1.58053	−0.790267	+	0.612762i	$0.790058\pi$
$440$	18.3864	0.876539
$441$	20.1015	0.957213
$442$	−1.03869	−0.0494055
$443$	−20.3229	−0.965572	−0.482786	−	0.875738i	$-0.660375\pi$
$443$	−20.3229	−0.965572	−0.482786	+	0.875738i	$0.660375\pi$
$444$	−19.4113	−0.921220
$445$	24.1919	1.14681
$446$	−9.74742	−0.461554
$447$	2.99363	0.141594
$448$	−11.3409	−0.535806
$449$	−19.2412	−0.908047	−0.454023	−	0.890990i	$-0.650012\pi$
$449$	−19.2412	−0.908047	−0.454023	+	0.890990i	$0.650012\pi$
$450$	0.629315	0.0296662
$451$	13.6809	0.644208
$452$	29.2028	1.37358
$453$	2.14812	0.100928
$454$	3.63842	0.170760
$455$	−25.7935	−1.20922
$456$	−8.66038	−0.405560
$457$	33.2178	1.55387	0.776933	−	0.629584i	$-0.216775\pi$
$457$	33.2178	1.55387	0.776933	+	0.629584i	$0.216775\pi$
$458$	−5.61429	−0.262338
$459$	1.00000	0.0466760
$460$	4.36350	0.203449
$461$	11.3985	0.530883	0.265441	−	0.964127i	$-0.414482\pi$
$461$	11.3985	0.530883	0.265441	+	0.964127i	$0.414482\pi$
$462$	−10.3092	−0.479629
$463$	24.7190	1.14879	0.574395	−	0.818578i	$-0.305237\pi$
$463$	24.7190	1.14879	0.574395	+	0.818578i	$0.305237\pi$
$464$	−21.5727	−1.00149
$465$	−17.9602	−0.832886
$466$	−11.0543	−0.512078
$467$	−12.7157	−0.588413	−0.294207	−	0.955742i	$-0.595055\pi$
$467$	−12.7157	−0.588413	−0.294207	+	0.955742i	$0.595055\pi$
$468$	−3.43562	−0.158812
$469$	7.88809	0.364238
$470$	0.427701	0.0197284
$471$	1.00000	0.0460776
$472$	−18.6280	−0.857424
$473$	13.2139	0.607575
$474$	0.0369889	0.00169896
$475$	−5.36188	−0.246020
$476$	8.99213	0.412154
$477$	7.74323	0.354538
$478$	−9.96604	−0.455836
$479$	−29.0324	−1.32652	−0.663262	−	0.748387i	$-0.730829\pi$
$479$	−29.0324	−1.32652	−0.663262	+	0.748387i	$0.730829\pi$
$480$	12.8688	0.587377
$481$	−22.3526	−1.01919
$482$	−10.4726	−0.477015
$483$	−5.27948	−0.240225
$484$	−5.83863	−0.265392
$485$	−15.0074	−0.681451
$486$	−0.522212	−0.0236880
$487$	−16.4916	−0.747306	−0.373653	−	0.927568i	$-0.621895\pi$
$487$	−16.4916	−0.747306	−0.373653	+	0.927568i	$0.621895\pi$
$488$	8.77256	0.397115
$489$	6.26047	0.283108
$490$	−26.1486	−1.18128
$491$	−21.8575	−0.986416	−0.493208	−	0.869912i	$-0.664176\pi$
$491$	−21.8575	−0.986416	−0.493208	+	0.869912i	$0.664176\pi$
$492$	6.23157	0.280941
$493$	8.84805	0.398496
$494$	−4.62150	−0.207931
$495$	−9.44620	−0.424575
$496$	−17.5790	−0.789323
$497$	4.15652	0.186446
$498$	3.36846	0.150944
$499$	7.19177	0.321948	0.160974	−	0.986959i	$-0.448537\pi$
$499$	7.19177	0.321948	0.160974	+	0.986959i	$0.448537\pi$
$500$	−16.3283	−0.730225
$501$	11.7678	0.525747
$502$	8.83557	0.394351
$503$	−43.3914	−1.93473	−0.967364	−	0.253390i	$-0.918454\pi$
$503$	−43.3914	−1.93473	−0.967364	+	0.253390i	$0.918454\pi$
$504$	−10.1330	−0.451359
$505$	−40.1762	−1.78782
$506$	2.00828	0.0892790
$507$	9.04380	0.401649
$508$	−11.3714	−0.504523
$509$	0.935730	0.0414755	0.0207378	−	0.999785i	$-0.493398\pi$
$509$	0.935730	0.0414755	0.0207378	+	0.999785i	$0.493398\pi$
$510$	−1.30083	−0.0576018
$511$	−12.6215	−0.558342
$512$	22.0870	0.976117
$513$	4.44935	0.196443
$514$	2.41627	0.106577
$515$	28.5738	1.25911
$516$	6.01886	0.264966
$517$	−1.24681	−0.0548348
$518$	−30.5515	−1.34235
$519$	−19.4467	−0.853615
$520$	9.64394	0.422915
$521$	−17.3502	−0.760125	−0.380062	−	0.924961i	$-0.624097\pi$
$521$	−17.3502	−0.760125	−0.380062	+	0.924961i	$0.624097\pi$
$522$	−4.62056	−0.202236
$523$	17.7575	0.776480	0.388240	−	0.921558i	$-0.373083\pi$
$523$	17.7575	0.776480	0.388240	+	0.921558i	$0.373083\pi$
$524$	16.6461	0.727189
$525$	−6.27361	−0.273803
$526$	−5.99746	−0.261502
$527$	7.21004	0.314074
$528$	−9.24571	−0.402368
$529$	−21.9715	−0.955284
$530$	−10.0727	−0.437528
$531$	9.57031	0.415316
$532$	40.0091	1.73461
$533$	7.17580	0.310819
$534$	5.07158	0.219469
$535$	−45.3041	−1.95867
$536$	−2.94928	−0.127390
$537$	−21.8141	−0.941350
$538$	13.5959	0.586160
$539$	76.2273	3.28334
$540$	−4.30269	−0.185158
$541$	0.123900	0.00532689	0.00266345	−	0.999996i	$-0.499152\pi$
$541$	0.123900	0.00532689	0.00266345	+	0.999996i	$0.499152\pi$
$542$	−11.2960	−0.485205
$543$	−11.0565	−0.474480
$544$	−5.16610	−0.221495
$545$	39.8973	1.70901
$546$	−5.40733	−0.231412
$547$	−38.8959	−1.66307	−0.831533	−	0.555475i	$-0.812537\pi$
$547$	−38.8959	−1.66307	−0.831533	+	0.555475i	$0.812537\pi$
$548$	25.3725	1.08386
$549$	−4.50698	−0.192353
$550$	2.38644	0.101758
$551$	39.3680	1.67713
$552$	1.97395	0.0840168
$553$	−0.368740	−0.0156804
$554$	−7.21331	−0.306464
$555$	−27.9939	−1.18827
$556$	1.31315	0.0556899
$557$	20.3271	0.861289	0.430644	−	0.902522i	$-0.358286\pi$
$557$	20.3271	0.861289	0.430644	+	0.902522i	$0.358286\pi$
$558$	−3.76517	−0.159393
$559$	6.93086	0.293144
$560$	−31.6175	−1.33608
$561$	3.79213	0.160104
$562$	−5.55246	−0.234217
$563$	0.687516	0.0289753	0.0144877	−	0.999895i	$-0.495388\pi$
$563$	0.687516	0.0289753	0.0144877	+	0.999895i	$0.495388\pi$
$564$	−0.567917	−0.0239136
$565$	42.1145	1.77177
$566$	9.63509	0.404993
$567$	5.20591	0.218628
$568$	−1.55408	−0.0652079
$569$	−13.4798	−0.565104	−0.282552	−	0.959252i	$-0.591181\pi$
$569$	−13.4798	−0.565104	−0.282552	+	0.959252i	$0.591181\pi$
$570$	−5.78785	−0.242427
$571$	−33.1731	−1.38825	−0.694125	−	0.719854i	$-0.744209\pi$
$571$	−33.1731	−1.38825	−0.694125	+	0.719854i	$0.744209\pi$
$572$	−13.0283	−0.544741
$573$	8.95026	0.373903
$574$	9.80788	0.409373
$575$	1.22212	0.0509661
$576$	−2.17846	−0.0907693
$577$	18.5904	0.773928	0.386964	−	0.922095i	$-0.373524\pi$
$577$	18.5904	0.773928	0.386964	+	0.922095i	$0.373524\pi$
$578$	0.522212	0.0217212
$579$	−6.05971	−0.251833
$580$	−38.0704	−1.58079
$581$	−33.5800	−1.39313
$582$	−3.14614	−0.130412
$583$	29.3633	1.21610
$584$	4.71906	0.195276
$585$	−4.95466	−0.204850
$586$	9.43620	0.389806
$587$	−48.0234	−1.98214	−0.991068	−	0.133355i	$-0.957425\pi$
$587$	−48.0234	−1.98214	−0.991068	+	0.133355i	$0.957425\pi$
$588$	34.7211	1.43188
$589$	32.0800	1.32183
$590$	−12.4494	−0.512532
$591$	−8.51967	−0.350452
$592$	−27.3997	−1.12612
$593$	−39.2854	−1.61326	−0.806628	−	0.591059i	$-0.798710\pi$
$593$	−39.2854	−1.61326	−0.806628	+	0.591059i	$0.798710\pi$
$594$	−1.98030	−0.0812525
$595$	12.9679	0.531633
$596$	5.17088	0.211807
$597$	4.14330	0.169574
$598$	1.05337	0.0430755
$599$	−9.01560	−0.368367	−0.184184	−	0.982892i	$-0.558964\pi$
$599$	−9.01560	−0.368367	−0.184184	+	0.982892i	$0.558964\pi$
$600$	2.34564	0.0957604
$601$	43.2024	1.76226	0.881132	−	0.472871i	$-0.156782\pi$
$601$	43.2024	1.76226	0.881132	+	0.472871i	$0.156782\pi$
$602$	9.47309	0.386094
$603$	1.51522	0.0617045
$604$	3.71044	0.150976
$605$	−8.42013	−0.342327
$606$	−8.42252	−0.342141
$607$	14.1323	0.573610	0.286805	−	0.957989i	$-0.407407\pi$
$607$	14.1323	0.573610	0.286805	+	0.957989i	$0.407407\pi$
$608$	−22.9858	−0.932196
$609$	46.0621	1.86653
$610$	5.86283	0.237379
$611$	−0.653971	−0.0264568
$612$	1.72729	0.0698217
$613$	4.12819	0.166736	0.0833681	−	0.996519i	$-0.473432\pi$
$613$	4.12819	0.166736	0.0833681	+	0.996519i	$0.473432\pi$
$614$	−13.4535	−0.542938
$615$	8.98681	0.362383
$616$	−38.4255	−1.54821
$617$	20.8254	0.838399	0.419199	−	0.907894i	$-0.362311\pi$
$617$	20.8254	0.838399	0.419199	+	0.907894i	$0.362311\pi$
$618$	5.99019	0.240961
$619$	−13.7296	−0.551840	−0.275920	−	0.961181i	$-0.588982\pi$
$619$	−13.7296	−0.551840	−0.275920	+	0.961181i	$0.588982\pi$
$620$	−31.0226	−1.24590
$621$	−1.01413	−0.0406957
$622$	−0.753112	−0.0301970
$623$	−50.5583	−2.02558
$624$	−4.84950	−0.194135
$625$	−29.5732	−1.18293
$626$	−10.0196	−0.400465
$627$	16.8725	0.673822
$628$	1.72729	0.0689265
$629$	11.2380	0.448088
$630$	−6.77201	−0.269803
$631$	−11.2146	−0.446445	−0.223222	−	0.974768i	$-0.571658\pi$
$631$	−11.2146	−0.446445	−0.223222	+	0.974768i	$0.571658\pi$
$632$	0.137868	0.00548411
$633$	21.8940	0.870209
$634$	8.71150	0.345978
$635$	−16.3991	−0.650779
$636$	13.3748	0.530347
$637$	39.9822	1.58415
$638$	−17.5217	−0.693693
$639$	0.798424	0.0315852
$640$	28.5714	1.12938
$641$	12.7651	0.504189	0.252095	−	0.967703i	$-0.418881\pi$
$641$	12.7651	0.504189	0.252095	+	0.967703i	$0.418881\pi$
$642$	−9.49753	−0.374838
$643$	−35.8234	−1.41274	−0.706369	−	0.707843i	$-0.749668\pi$
$643$	−35.8234	−1.41274	−0.706369	+	0.707843i	$0.749668\pi$
$644$	−9.11921	−0.359347
$645$	8.68005	0.341777
$646$	2.32350	0.0914170
$647$	40.8681	1.60669	0.803346	−	0.595512i	$-0.203051\pi$
$647$	40.8681	1.60669	0.803346	+	0.595512i	$0.203051\pi$
$648$	−1.94644	−0.0764634
$649$	36.2918	1.42458
$650$	1.25172	0.0490965
$651$	37.5348	1.47111
$652$	10.8137	0.423496
$653$	−4.36228	−0.170709	−0.0853547	−	0.996351i	$-0.527202\pi$
$653$	−4.36228	−0.170709	−0.0853547	+	0.996351i	$0.527202\pi$
$654$	8.36404	0.327060
$655$	24.0060	0.937994
$656$	8.79607	0.343429
$657$	−2.42446	−0.0945870
$658$	−0.893846	−0.0348457
$659$	−18.8712	−0.735116	−0.367558	−	0.930001i	$-0.619806\pi$
$659$	−18.8712	−0.735116	−0.367558	+	0.930001i	$0.619806\pi$
$660$	−16.3164	−0.635113
$661$	14.0194	0.545290	0.272645	−	0.962115i	$-0.412102\pi$
$661$	14.0194	0.545290	0.272645	+	0.962115i	$0.412102\pi$
$662$	−2.56928	−0.0998578
$663$	1.98902	0.0772472
$664$	12.5552	0.487238
$665$	57.6988	2.23746
$666$	−5.86862	−0.227404
$667$	−8.97309	−0.347440
$668$	20.3265	0.786455
$669$	18.6656	0.721655
$670$	−1.97105	−0.0761482
$671$	−17.0910	−0.659792
$672$	−26.8942	−1.03747
$673$	15.3836	0.592994	0.296497	−	0.955034i	$-0.404181\pi$
$673$	15.3836	0.592994	0.296497	+	0.955034i	$0.404181\pi$
$674$	−7.69037	−0.296222
$675$	−1.20509	−0.0463841
$676$	15.6213	0.600819
$677$	−6.08494	−0.233863	−0.116932	−	0.993140i	$-0.537306\pi$
$677$	−6.08494	−0.233863	−0.116932	+	0.993140i	$0.537306\pi$
$678$	8.82887	0.339071
$679$	31.3637	1.20363
$680$	−4.84859	−0.185935
$681$	−6.96732	−0.266988
$682$	−14.2780	−0.546733
$683$	15.3846	0.588674	0.294337	−	0.955702i	$-0.404901\pi$
$683$	15.3846	0.588674	0.294337	+	0.955702i	$0.404901\pi$
$684$	7.68533	0.293856
$685$	36.5907	1.39806
$686$	35.6175	1.35988
$687$	10.7510	0.410175
$688$	8.49582	0.323900
$689$	15.4015	0.586749
$690$	1.31922	0.0502217
$691$	24.9157	0.947839	0.473920	−	0.880568i	$-0.342839\pi$
$691$	24.9157	0.947839	0.473920	+	0.880568i	$0.342839\pi$
$692$	−33.5901	−1.27691
$693$	19.7415	0.749916
$694$	−0.691513	−0.0262495
$695$	1.89374	0.0718338
$696$	−17.2222	−0.652805
$697$	−3.60771	−0.136652
$698$	0.827086	0.0313057
$699$	21.1681	0.800652
$700$	−10.8364	−0.409576
$701$	−11.0105	−0.415861	−0.207930	−	0.978144i	$-0.566673\pi$
$701$	−11.0105	−0.415861	−0.207930	+	0.978144i	$0.566673\pi$
$702$	−1.03869	−0.0392029
$703$	50.0017	1.88585
$704$	−8.26101	−0.311349
$705$	−0.819017	−0.0308460
$706$	0.112050	0.00421707
$707$	83.9636	3.15778
$708$	16.5307	0.621263
$709$	8.85382	0.332512	0.166256	−	0.986083i	$-0.446832\pi$
$709$	8.85382	0.332512	0.166256	+	0.986083i	$0.446832\pi$
$710$	−1.03862	−0.0389786
$711$	−0.0708311	−0.00265637
$712$	18.9033	0.708430
$713$	−7.31194	−0.273834
$714$	2.71859	0.101741
$715$	−18.7887	−0.702657
$716$	−37.6795	−1.40815
$717$	19.0843	0.712715
$718$	−0.262798	−0.00980752
$719$	1.50252	0.0560346	0.0280173	−	0.999607i	$-0.491081\pi$
$719$	1.50252	0.0560346	0.0280173	+	0.999607i	$0.491081\pi$
$720$	−6.07340	−0.226342
$721$	−59.7159	−2.22394
$722$	0.416034	0.0154832
$723$	20.0543	0.745829
$724$	−19.0978	−0.709765
$725$	−10.6627	−0.396004
$726$	−1.76519	−0.0655124
$727$	−8.16987	−0.303004	−0.151502	−	0.988457i	$-0.548411\pi$
$727$	−8.16987	−0.303004	−0.151502	+	0.988457i	$0.548411\pi$
$728$	−20.1547	−0.746983
$729$	1.00000	0.0370370
$730$	3.15381	0.116728
$731$	−3.48456	−0.128881
$732$	−7.78488	−0.287737
$733$	−49.6438	−1.83363	−0.916817	−	0.399307i	$-0.869251\pi$
$733$	−49.6438	−1.83363	−0.916817	+	0.399307i	$0.869251\pi$
$734$	9.13205	0.337070
$735$	50.0728	1.84696
$736$	5.23911	0.193116
$737$	5.74590	0.211653
$738$	1.88399	0.0693506
$739$	−14.2807	−0.525324	−0.262662	−	0.964888i	$-0.584600\pi$
$739$	−14.2807	−0.525324	−0.262662	+	0.964888i	$0.584600\pi$
$740$	−48.3536	−1.77751
$741$	8.84984	0.325107
$742$	21.0507	0.772794
$743$	−30.7706	−1.12887	−0.564433	−	0.825479i	$-0.690905\pi$
$743$	−30.7706	−1.12887	−0.564433	+	0.825479i	$0.690905\pi$
$744$	−14.0339	−0.514508
$745$	7.45713	0.273208
$746$	11.1468	0.408113
$747$	−6.45036	−0.236006
$748$	6.55012	0.239496
$749$	94.6803	3.45954
$750$	−4.93654	−0.180257
$751$	−33.7472	−1.23145	−0.615727	−	0.787960i	$-0.711138\pi$
$751$	−33.7472	−1.23145	−0.615727	+	0.787960i	$0.711138\pi$
$752$	−0.801635	−0.0292326
$753$	−16.9195	−0.616581
$754$	−9.19039	−0.334694
$755$	5.35098	0.194742
$756$	8.99213	0.327041
$757$	−1.48260	−0.0538862	−0.0269431	−	0.999637i	$-0.508577\pi$
$757$	−1.48260	−0.0538862	−0.0269431	+	0.999637i	$0.508577\pi$
$758$	−19.6397	−0.713345
$759$	−3.84572	−0.139591
$760$	−21.5730	−0.782536
$761$	−29.5931	−1.07275	−0.536374	−	0.843980i	$-0.680206\pi$
$761$	−29.5931	−1.07275	−0.536374	+	0.843980i	$0.680206\pi$
$762$	−3.43790	−0.124542
$763$	−83.3807	−3.01858
$764$	15.4597	0.559314
$765$	2.49100	0.0900624
$766$	4.10289	0.148244
$767$	19.0355	0.687334
$768$	1.63276	0.0589170
$769$	−14.9783	−0.540131	−0.270065	−	0.962842i	$-0.587045\pi$
$769$	−14.9783	−0.540131	−0.270065	+	0.962842i	$0.587045\pi$
$770$	−25.6803	−0.925455
$771$	−4.62699	−0.166637
$772$	−10.4669	−0.376712
$773$	−20.1276	−0.723940	−0.361970	−	0.932190i	$-0.617896\pi$
$773$	−20.1276	−0.723940	−0.361970	+	0.932190i	$0.617896\pi$
$774$	1.81968	0.0654071
$775$	−8.68878	−0.312110
$776$	−11.7266	−0.420961
$777$	58.5039	2.09882
$778$	−16.2104	−0.581171
$779$	−16.0519	−0.575120
$780$	−8.55815	−0.306431
$781$	3.02773	0.108341
$782$	−0.529593	−0.0189382
$783$	8.84805	0.316203
$784$	49.0101	1.75036
$785$	2.49100	0.0889077
$786$	5.03261	0.179507
$787$	−5.71390	−0.203679	−0.101839	−	0.994801i	$-0.532473\pi$
$787$	−5.71390	−0.203679	−0.101839	+	0.994801i	$0.532473\pi$
$788$	−14.7160	−0.524235
$789$	11.4847	0.408867
$790$	0.0921394	0.00327817
$791$	−88.0145	−3.12943
$792$	−7.38114	−0.262277
$793$	−8.96447	−0.318338
$794$	2.59531	0.0921041
$795$	19.2884	0.684090
$796$	7.15669	0.253662
$797$	14.2418	0.504470	0.252235	−	0.967666i	$-0.418834\pi$
$797$	14.2418	0.504470	0.252235	+	0.967666i	$0.418834\pi$
$798$	12.0959	0.428192
$799$	0.328790	0.0116318
$800$	6.22564	0.220110
$801$	−9.71172	−0.343147
$802$	8.10824	0.286312
$803$	−9.19384	−0.324444
$804$	2.61723	0.0923026
$805$	−13.1512	−0.463519
$806$	−7.48901	−0.263789
$807$	−26.0351	−0.916480
$808$	−31.3932	−1.10441
$809$	−26.6439	−0.936751	−0.468375	−	0.883530i	$-0.655160\pi$
$809$	−26.6439	−0.936751	−0.468375	+	0.883530i	$0.655160\pi$
$810$	−1.30083	−0.0457066
$811$	42.3718	1.48788	0.743938	−	0.668249i	$-0.232956\pi$
$811$	42.3718	1.48788	0.743938	+	0.668249i	$0.232956\pi$
$812$	79.5628	2.79211
$813$	21.6310	0.758634
$814$	−22.2545	−0.780021
$815$	15.5948	0.546264
$816$	2.43813	0.0853517
$817$	−15.5040	−0.542417
$818$	11.3645	0.397351
$819$	10.3547	0.361821
$820$	15.5229	0.542082
$821$	−11.9910	−0.418488	−0.209244	−	0.977863i	$-0.567100\pi$
$821$	−11.9910	−0.418488	−0.209244	+	0.977863i	$0.567100\pi$
$822$	7.67085	0.267552
$823$	−39.3510	−1.37169	−0.685845	−	0.727748i	$-0.740567\pi$
$823$	−39.3510	−1.37169	−0.685845	+	0.727748i	$0.740567\pi$
$824$	22.3272	0.777805
$825$	−4.56987	−0.159102
$826$	26.0177	0.905273
$827$	27.0540	0.940759	0.470379	−	0.882464i	$-0.344117\pi$
$827$	27.0540	0.940759	0.470379	+	0.882464i	$0.344117\pi$
$828$	−1.75171	−0.0608760
$829$	−27.6852	−0.961546	−0.480773	−	0.876845i	$-0.659644\pi$
$829$	−27.6852	−0.961546	−0.480773	+	0.876845i	$0.659644\pi$
$830$	8.39084	0.291250
$831$	13.8130	0.479167
$832$	−4.33301	−0.150220
$833$	−20.1015	−0.696475
$834$	0.397004	0.0137471
$835$	29.3137	1.01444
$836$	29.1437	1.00796
$837$	7.21004	0.249216
$838$	15.2068	0.525309
$839$	3.36480	0.116166	0.0580829	−	0.998312i	$-0.481501\pi$
$839$	3.36480	0.116166	0.0580829	+	0.998312i	$0.481501\pi$
$840$	−25.2413	−0.870907
$841$	49.2879	1.69958
$842$	15.3227	0.528054
$843$	10.6326	0.366205
$844$	37.8174	1.30173
$845$	22.5281	0.774991
$846$	−0.171698	−0.00590311
$847$	17.5971	0.604643
$848$	18.8790	0.648309
$849$	−18.4505	−0.633220
$850$	−0.629315	−0.0215853
$851$	−11.3968	−0.390678
$852$	1.37911	0.0472477
$853$	−32.7232	−1.12042	−0.560210	−	0.828350i	$-0.689280\pi$
$853$	−32.7232	−1.12042	−0.560210	+	0.828350i	$0.689280\pi$
$854$	−12.2526	−0.419276
$855$	11.0833	0.379042
$856$	−35.4001	−1.20995
$857$	−40.0228	−1.36715	−0.683576	−	0.729880i	$-0.739576\pi$
$857$	−40.0228	−1.36715	−0.683576	+	0.729880i	$0.739576\pi$
$858$	−3.93885	−0.134470
$859$	−45.8602	−1.56473	−0.782364	−	0.622821i	$-0.785986\pi$
$859$	−45.8602	−1.56473	−0.782364	+	0.622821i	$0.785986\pi$
$860$	14.9930	0.511257
$861$	−18.7814	−0.640068
$862$	−1.17291	−0.0399495
$863$	3.56646	0.121404	0.0607018	−	0.998156i	$-0.480666\pi$
$863$	3.56646	0.121404	0.0607018	+	0.998156i	$0.480666\pi$
$864$	−5.16610	−0.175754
$865$	−48.4417	−1.64707
$866$	1.37418	0.0466965
$867$	−1.00000	−0.0339618
$868$	64.8337	2.20060
$869$	−0.268600	−0.00911165
$870$	−11.5098	−0.390220
$871$	3.01380	0.102119
$872$	31.1753	1.05573
$873$	6.02464	0.203903
$874$	−2.35634	−0.0797044
$875$	49.2121	1.66367
$876$	−4.18775	−0.141491
$877$	27.2368	0.919722	0.459861	−	0.887991i	$-0.347899\pi$
$877$	27.2368	0.919722	0.459861	+	0.887991i	$0.347899\pi$
$878$	−17.2935	−0.583628
$879$	−18.0697	−0.609474
$880$	−23.0311	−0.776378
$881$	6.92647	0.233359	0.116679	−	0.993170i	$-0.462775\pi$
$881$	6.92647	0.233359	0.116679	+	0.993170i	$0.462775\pi$
$882$	10.4972	0.353460
$883$	−7.49222	−0.252133	−0.126067	−	0.992022i	$-0.540235\pi$
$883$	−7.49222	−0.252133	−0.126067	+	0.992022i	$0.540235\pi$
$884$	3.43562	0.115553
$885$	23.8397	0.801362
$886$	−10.6129	−0.356547
$887$	16.3091	0.547605	0.273803	−	0.961786i	$-0.411719\pi$
$887$	16.3091	0.547605	0.273803	+	0.961786i	$0.411719\pi$
$888$	−21.8741	−0.734046
$889$	34.2723	1.14945
$890$	12.6333	0.423470
$891$	3.79213	0.127041
$892$	32.2410	1.07951
$893$	1.46290	0.0489541
$894$	1.56331	0.0522849
$895$	−54.3391	−1.81636
$896$	−59.7108	−1.99480
$897$	−2.01713	−0.0673500
$898$	−10.0480	−0.335305
$899$	63.7948	2.12768
$900$	−2.08155	−0.0693851
$901$	−7.74323	−0.257965
$902$	7.14433	0.237880
$903$	−18.1403	−0.603671
$904$	32.9078	1.09450
$905$	−27.5418	−0.915520
$906$	1.12178	0.0372685
$907$	−5.00073	−0.166047	−0.0830233	−	0.996548i	$-0.526458\pi$
$907$	−5.00073	−0.166047	−0.0830233	+	0.996548i	$0.526458\pi$
$908$	−12.0346	−0.399383
$909$	16.1285	0.534949
$910$	−13.4697	−0.446515
$911$	−9.84715	−0.326251	−0.163125	−	0.986605i	$-0.552158\pi$
$911$	−9.84715	−0.326251	−0.163125	+	0.986605i	$0.552158\pi$
$912$	10.8481	0.359216
$913$	−24.4606	−0.809527
$914$	17.3468	0.573780
$915$	−11.2269	−0.371150
$916$	18.5701	0.613573
$917$	−50.1698	−1.65675
$918$	0.522212	0.0172356
$919$	26.6248	0.878270	0.439135	−	0.898421i	$-0.355285\pi$
$919$	26.6248	0.878270	0.439135	+	0.898421i	$0.355285\pi$
$920$	4.91711	0.162112
$921$	25.7624	0.848901
$922$	5.95245	0.196034
$923$	1.58808	0.0522724
$924$	34.0993	1.12178
$925$	−13.5428	−0.445286
$926$	12.9086	0.424202
$927$	−11.4708	−0.376750
$928$	−45.7099	−1.50050
$929$	−23.4936	−0.770799	−0.385399	−	0.922750i	$-0.625936\pi$
$929$	−23.4936	−0.770799	−0.385399	+	0.922750i	$0.625936\pi$
$930$	−9.37906	−0.307551
$931$	−89.4384	−2.93122
$932$	36.5636	1.19768
$933$	1.44216	0.0472141
$934$	−6.64031	−0.217277
$935$	9.44620	0.308924
$936$	−3.87151	−0.126544
$937$	9.16463	0.299396	0.149698	−	0.988732i	$-0.452170\pi$
$937$	9.16463	0.299396	0.149698	+	0.988732i	$0.452170\pi$
$938$	4.11926	0.134499
$939$	19.1869	0.626140
$940$	−1.41468	−0.0461419
$941$	33.1940	1.08209	0.541046	−	0.840993i	$-0.318028\pi$
$941$	33.1940	1.08209	0.541046	+	0.840993i	$0.318028\pi$
$942$	0.522212	0.0170146
$943$	3.65869	0.119143
$944$	23.3337	0.759447
$945$	12.9679	0.421847
$946$	6.90046	0.224353
$947$	39.4199	1.28097	0.640487	−	0.767969i	$-0.278733\pi$
$947$	39.4199	1.28097	0.640487	+	0.767969i	$0.278733\pi$
$948$	−0.122346	−0.00397362
$949$	−4.82229	−0.156538
$950$	−2.80004	−0.0908453
$951$	−16.6819	−0.540948
$952$	10.1330	0.328412
$953$	−20.3173	−0.658141	−0.329070	−	0.944305i	$-0.606735\pi$
$953$	−20.3173	−0.658141	−0.329070	+	0.944305i	$0.606735\pi$
$954$	4.04361	0.130917
$955$	22.2951	0.721453
$956$	32.9641	1.06614
$957$	33.5529	1.08461
$958$	−15.1611	−0.489832
$959$	−76.4703	−2.46936
$960$	−5.42656	−0.175141
$961$	20.9847	0.676927
$962$	−11.6728	−0.376346
$963$	18.1871	0.586071
$964$	34.6398	1.11567
$965$	−15.0947	−0.485917
$966$	−2.75701	−0.0887053
$967$	50.5603	1.62591	0.812955	−	0.582327i	$-0.197858\pi$
$967$	50.5603	1.62591	0.812955	+	0.582327i	$0.197858\pi$
$968$	−6.57939	−0.211470
$969$	−4.44935	−0.142934
$970$	−7.83705	−0.251633
$971$	−36.0127	−1.15570	−0.577851	−	0.816142i	$-0.696109\pi$
$971$	−36.0127	−1.15570	−0.577851	+	0.816142i	$0.696109\pi$
$972$	1.72729	0.0554030
$973$	−3.95771	−0.126878
$974$	−8.61213	−0.275950
$975$	−2.39696	−0.0767640
$976$	−10.9886	−0.351737
$977$	−16.9909	−0.543589	−0.271794	−	0.962355i	$-0.587617\pi$
$977$	−16.9909	−0.543589	−0.271794	+	0.962355i	$0.587617\pi$
$978$	3.26929	0.104541
$979$	−36.8281	−1.17703
$980$	86.4905	2.76284
$981$	−16.0166	−0.511369
$982$	−11.4143	−0.364244
$983$	11.7320	0.374192	0.187096	−	0.982342i	$-0.440092\pi$
$983$	11.7320	0.374192	0.187096	+	0.982342i	$0.440092\pi$
$984$	7.02218	0.223859
$985$	−21.2225	−0.676206
$986$	4.62056	0.147149
$987$	1.71165	0.0544825
$988$	15.2863	0.486321
$989$	3.53381	0.112368
$990$	−4.93292	−0.156779
$991$	−29.8970	−0.949709	−0.474855	−	0.880064i	$-0.657499\pi$
$991$	−29.8970	−0.949709	−0.474855	+	0.880064i	$0.657499\pi$
$992$	−37.2478	−1.18262
$993$	4.91999	0.156131
$994$	2.17059	0.0688469
$995$	10.3210	0.327196
$996$	−11.1417	−0.353037
$997$	47.7503	1.51227	0.756133	−	0.654417i	$-0.227086\pi$
$997$	47.7503	1.51227	0.756133	+	0.654417i	$0.227086\pi$
$998$	3.75563	0.118882
$999$	11.2380	0.355554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8007.2.a.f.1.29	✓	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8007.2.a.f.1.29	✓	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)

\(f(q)\)	\(=\)	\(q+0.522212 q^{2} -1.00000 q^{3} -1.72729 q^{4} -2.49100 q^{5} -0.522212 q^{6} +5.20591 q^{7} -1.94644 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.522212 q^{2} -1.00000 q^{3} -1.72729 q^{4} -2.49100 q^{5} -0.522212 q^{6} +5.20591 q^{7} -1.94644 q^{8} +1.00000 q^{9} -1.30083 q^{10} +3.79213 q^{11} +1.72729 q^{12} +1.98902 q^{13} +2.71859 q^{14} +2.49100 q^{15} +2.43813 q^{16} -1.00000 q^{17} +0.522212 q^{18} -4.44935 q^{19} +4.30269 q^{20} -5.20591 q^{21} +1.98030 q^{22} +1.01413 q^{23} +1.94644 q^{24} +1.20509 q^{25} +1.03869 q^{26} -1.00000 q^{27} -8.99213 q^{28} -8.84805 q^{29} +1.30083 q^{30} -7.21004 q^{31} +5.16610 q^{32} -3.79213 q^{33} -0.522212 q^{34} -12.9679 q^{35} -1.72729 q^{36} -11.2380 q^{37} -2.32350 q^{38} -1.98902 q^{39} +4.84859 q^{40} +3.60771 q^{41} -2.71859 q^{42} +3.48456 q^{43} -6.55012 q^{44} -2.49100 q^{45} +0.529593 q^{46} -0.328790 q^{47} -2.43813 q^{48} +20.1015 q^{49} +0.629315 q^{50} +1.00000 q^{51} -3.43562 q^{52} +7.74323 q^{53} -0.522212 q^{54} -9.44620 q^{55} -10.1330 q^{56} +4.44935 q^{57} -4.62056 q^{58} +9.57031 q^{59} -4.30269 q^{60} -4.50698 q^{61} -3.76517 q^{62} +5.20591 q^{63} -2.17846 q^{64} -4.95466 q^{65} -1.98030 q^{66} +1.51522 q^{67} +1.72729 q^{68} -1.01413 q^{69} -6.77201 q^{70} +0.798424 q^{71} -1.94644 q^{72} -2.42446 q^{73} -5.86862 q^{74} -1.20509 q^{75} +7.68533 q^{76} +19.7415 q^{77} -1.03869 q^{78} -0.0708311 q^{79} -6.07340 q^{80} +1.00000 q^{81} +1.88399 q^{82} -6.45036 q^{83} +8.99213 q^{84} +2.49100 q^{85} +1.81968 q^{86} +8.84805 q^{87} -7.38114 q^{88} -9.71172 q^{89} -1.30083 q^{90} +10.3547 q^{91} -1.75171 q^{92} +7.21004 q^{93} -0.171698 q^{94} +11.0833 q^{95} -5.16610 q^{96} +6.02464 q^{97} +10.4972 q^{98} +3.79213 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9}+O(q^{10}) \) 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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