LMFDB - Embedded newform 8007.2.a.f.1.19

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			1.19
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)

$f(q)$	$=$	$q-0.857233 q^{2} -1.00000 q^{3} -1.26515 q^{4} +0.241794 q^{5} +0.857233 q^{6} -3.40893 q^{7} +2.79900 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.857233 q^{2} -1.00000 q^{3} -1.26515 q^{4} +0.241794 q^{5} +0.857233 q^{6} -3.40893 q^{7} +2.79900 q^{8} +1.00000 q^{9} -0.207274 q^{10} +4.52587 q^{11} +1.26515 q^{12} -3.66863 q^{13} +2.92225 q^{14} -0.241794 q^{15} +0.130910 q^{16} -1.00000 q^{17} -0.857233 q^{18} +3.02291 q^{19} -0.305906 q^{20} +3.40893 q^{21} -3.87973 q^{22} -6.34192 q^{23} -2.79900 q^{24} -4.94154 q^{25} +3.14487 q^{26} -1.00000 q^{27} +4.31281 q^{28} +5.43257 q^{29} +0.207274 q^{30} +2.05869 q^{31} -5.71021 q^{32} -4.52587 q^{33} +0.857233 q^{34} -0.824258 q^{35} -1.26515 q^{36} +9.39150 q^{37} -2.59134 q^{38} +3.66863 q^{39} +0.676781 q^{40} +0.671979 q^{41} -2.92225 q^{42} -6.04121 q^{43} -5.72591 q^{44} +0.241794 q^{45} +5.43650 q^{46} -0.151884 q^{47} -0.130910 q^{48} +4.62079 q^{49} +4.23605 q^{50} +1.00000 q^{51} +4.64137 q^{52} -8.36573 q^{53} +0.857233 q^{54} +1.09433 q^{55} -9.54158 q^{56} -3.02291 q^{57} -4.65698 q^{58} +1.97010 q^{59} +0.305906 q^{60} +0.358392 q^{61} -1.76478 q^{62} -3.40893 q^{63} +4.63316 q^{64} -0.887052 q^{65} +3.87973 q^{66} +1.12443 q^{67} +1.26515 q^{68} +6.34192 q^{69} +0.706582 q^{70} -4.67774 q^{71} +2.79900 q^{72} +0.743140 q^{73} -8.05070 q^{74} +4.94154 q^{75} -3.82444 q^{76} -15.4284 q^{77} -3.14487 q^{78} +14.4989 q^{79} +0.0316532 q^{80} +1.00000 q^{81} -0.576043 q^{82} +4.57707 q^{83} -4.31281 q^{84} -0.241794 q^{85} +5.17872 q^{86} -5.43257 q^{87} +12.6679 q^{88} +1.89104 q^{89} -0.207274 q^{90} +12.5061 q^{91} +8.02349 q^{92} -2.05869 q^{93} +0.130200 q^{94} +0.730922 q^{95} +5.71021 q^{96} -1.97519 q^{97} -3.96110 q^{98} +4.52587 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.857233	−0.606155	−0.303078	−	0.952966i	$-0.598014\pi$
$2$	−0.857233	−0.606155	−0.303078	+	0.952966i	$0.598014\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.26515	−0.632576
$5$	0.241794	0.108134	0.0540668	−	0.998537i	$-0.482782\pi$
$5$	0.241794	0.108134	0.0540668	+	0.998537i	$0.482782\pi$
$6$	0.857233	0.349964
$7$	−3.40893	−1.28845	−0.644227	−	0.764834i	$-0.722821\pi$
$7$	−3.40893	−1.28845	−0.644227	+	0.764834i	$0.722821\pi$
$8$	2.79900	0.989595
$9$	1.00000	0.333333
$10$	−0.207274	−0.0655458
$11$	4.52587	1.36460	0.682301	−	0.731072i	$-0.260979\pi$
$11$	4.52587	1.36460	0.682301	+	0.731072i	$0.260979\pi$
$12$	1.26515	0.365218
$13$	−3.66863	−1.01749	−0.508747	−	0.860916i	$-0.669891\pi$
$13$	−3.66863	−1.01749	−0.508747	+	0.860916i	$0.669891\pi$
$14$	2.92225	0.781003
$15$	−0.241794	−0.0624309
$16$	0.130910	0.0327274
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	−0.857233	−0.202052
$19$	3.02291	0.693504	0.346752	−	0.937957i	$-0.387285\pi$
$19$	3.02291	0.693504	0.346752	+	0.937957i	$0.387285\pi$
$20$	−0.305906	−0.0684027
$21$	3.40893	0.743889
$22$	−3.87973	−0.827161
$23$	−6.34192	−1.32238	−0.661191	−	0.750218i	$-0.729949\pi$
$23$	−6.34192	−1.32238	−0.661191	+	0.750218i	$0.729949\pi$
$24$	−2.79900	−0.571343
$25$	−4.94154	−0.988307
$26$	3.14487	0.616760
$27$	−1.00000	−0.192450
$28$	4.31281	0.815044
$29$	5.43257	1.00880	0.504402	−	0.863469i	$-0.331713\pi$
$29$	5.43257	1.00880	0.504402	+	0.863469i	$0.331713\pi$
$30$	0.207274	0.0378429
$31$	2.05869	0.369752	0.184876	−	0.982762i	$-0.440812\pi$
$31$	2.05869	0.369752	0.184876	+	0.982762i	$0.440812\pi$
$32$	−5.71021	−1.00943
$33$	−4.52587	−0.787853
$34$	0.857233	0.147014
$35$	−0.824258	−0.139325
$36$	−1.26515	−0.210859
$37$	9.39150	1.54395	0.771976	−	0.635651i	$-0.219268\pi$
$37$	9.39150	1.54395	0.771976	+	0.635651i	$0.219268\pi$
$38$	−2.59134	−0.420371
$39$	3.66863	0.587451
$40$	0.676781	0.107008
$41$	0.671979	0.104945	0.0524727	−	0.998622i	$-0.483290\pi$
$41$	0.671979	0.104945	0.0524727	+	0.998622i	$0.483290\pi$
$42$	−2.92225	−0.450912
$43$	−6.04121	−0.921275	−0.460638	−	0.887588i	$-0.652379\pi$
$43$	−6.04121	−0.921275	−0.460638	+	0.887588i	$0.652379\pi$
$44$	−5.72591	−0.863214
$45$	0.241794	0.0360445
$46$	5.43650	0.801569
$47$	−0.151884	−0.0221546	−0.0110773	−	0.999939i	$-0.503526\pi$
$47$	−0.151884	−0.0221546	−0.0110773	+	0.999939i	$0.503526\pi$
$48$	−0.130910	−0.0188952
$49$	4.62079	0.660113
$50$	4.23605	0.599068
$51$	1.00000	0.140028
$52$	4.64137	0.643642
$53$	−8.36573	−1.14912	−0.574561	−	0.818462i	$-0.694827\pi$
$53$	−8.36573	−1.14912	−0.574561	+	0.818462i	$0.694827\pi$
$54$	0.857233	0.116655
$55$	1.09433	0.147559
$56$	−9.54158	−1.27505
$57$	−3.02291	−0.400395
$58$	−4.65698	−0.611492
$59$	1.97010	0.256486	0.128243	−	0.991743i	$-0.459066\pi$
$59$	1.97010	0.256486	0.128243	+	0.991743i	$0.459066\pi$
$60$	0.305906	0.0394923
$61$	0.358392	0.0458874	0.0229437	−	0.999737i	$-0.492696\pi$
$61$	0.358392	0.0458874	0.0229437	+	0.999737i	$0.492696\pi$
$62$	−1.76478	−0.224127
$63$	−3.40893	−0.429485
$64$	4.63316	0.579146
$65$	−0.887052	−0.110025
$66$	3.87973	0.477561
$67$	1.12443	0.137371	0.0686853	−	0.997638i	$-0.478120\pi$
$67$	1.12443	0.137371	0.0686853	+	0.997638i	$0.478120\pi$
$68$	1.26515	0.153422
$69$	6.34192	0.763477
$70$	0.706582	0.0844527
$71$	−4.67774	−0.555145	−0.277573	−	0.960705i	$-0.589530\pi$
$71$	−4.67774	−0.555145	−0.277573	+	0.960705i	$0.589530\pi$
$72$	2.79900	0.329865
$73$	0.743140	0.0869779	0.0434890	−	0.999054i	$-0.486153\pi$
$73$	0.743140	0.0869779	0.0434890	+	0.999054i	$0.486153\pi$
$74$	−8.05070	−0.935875
$75$	4.94154	0.570599
$76$	−3.82444	−0.438693
$77$	−15.4284	−1.75823
$78$	−3.14487	−0.356086
$79$	14.4989	1.63125	0.815624	−	0.578582i	$-0.196394\pi$
$79$	14.4989	1.63125	0.815624	+	0.578582i	$0.196394\pi$
$80$	0.0316532	0.00353893
$81$	1.00000	0.111111
$82$	−0.576043	−0.0636133
$83$	4.57707	0.502399	0.251200	−	0.967935i	$-0.419175\pi$
$83$	4.57707	0.502399	0.251200	+	0.967935i	$0.419175\pi$
$84$	−4.31281	−0.470566
$85$	−0.241794	−0.0262262
$86$	5.17872	0.558436
$87$	−5.43257	−0.582433
$88$	12.6679	1.35040
$89$	1.89104	0.200449	0.100225	−	0.994965i	$-0.468044\pi$
$89$	1.89104	0.200449	0.100225	+	0.994965i	$0.468044\pi$
$90$	−0.207274	−0.0218486
$91$	12.5061	1.31099
$92$	8.02349	0.836506
$93$	−2.05869	−0.213476
$94$	0.130200	0.0134291
$95$	0.730922	0.0749910
$96$	5.71021	0.582796
$97$	−1.97519	−0.200550	−0.100275	−	0.994960i	$-0.531972\pi$
$97$	−1.97519	−0.200550	−0.100275	+	0.994960i	$0.531972\pi$
$98$	−3.96110	−0.400131
$99$	4.52587	0.454867
$100$	6.25179	0.625179
$101$	5.48102	0.545381	0.272691	−	0.962102i	$-0.412086\pi$
$101$	5.48102	0.545381	0.272691	+	0.962102i	$0.412086\pi$
$102$	−0.857233	−0.0848787
$103$	7.38351	0.727519	0.363760	−	0.931493i	$-0.381493\pi$
$103$	7.38351	0.727519	0.363760	+	0.931493i	$0.381493\pi$
$104$	−10.2685	−1.00691
$105$	0.824258	0.0804394
$106$	7.17139	0.696547
$107$	−14.7394	−1.42491	−0.712454	−	0.701719i	$-0.752416\pi$
$107$	−14.7394	−1.42491	−0.712454	+	0.701719i	$0.752416\pi$
$108$	1.26515	0.121739
$109$	−1.57664	−0.151015	−0.0755073	−	0.997145i	$-0.524058\pi$
$109$	−1.57664	−0.151015	−0.0755073	+	0.997145i	$0.524058\pi$
$110$	−0.938095	−0.0894438
$111$	−9.39150	−0.891402
$112$	−0.446262	−0.0421678
$113$	4.29677	0.404206	0.202103	−	0.979364i	$-0.435222\pi$
$113$	4.29677	0.404206	0.202103	+	0.979364i	$0.435222\pi$
$114$	2.59134	0.242701
$115$	−1.53344	−0.142994
$116$	−6.87303	−0.638145
$117$	−3.66863	−0.339165
$118$	−1.68884	−0.155470
$119$	3.40893	0.312496
$120$	−0.676781	−0.0617813
$121$	9.48352	0.862138
$122$	−0.307225	−0.0278149
$123$	−0.671979	−0.0605903
$124$	−2.60456	−0.233896
$125$	−2.40380	−0.215003
$126$	2.92225	0.260334
$127$	−15.9253	−1.41314	−0.706572	−	0.707641i	$-0.749760\pi$
$127$	−15.9253	−1.41314	−0.706572	+	0.707641i	$0.749760\pi$
$128$	7.44872	0.658380
$129$	6.04121	0.531899
$130$	0.760411	0.0666924
$131$	−4.32209	−0.377623	−0.188812	−	0.982013i	$-0.560464\pi$
$131$	−4.32209	−0.377623	−0.188812	+	0.982013i	$0.560464\pi$
$132$	5.72591	0.498377
$133$	−10.3049	−0.893547
$134$	−0.963897	−0.0832680
$135$	−0.241794	−0.0208103
$136$	−2.79900	−0.240012
$137$	7.49284	0.640157	0.320078	−	0.947391i	$-0.396291\pi$
$137$	7.49284	0.640157	0.320078	+	0.947391i	$0.396291\pi$
$138$	−5.43650	−0.462786
$139$	15.9522	1.35304	0.676522	−	0.736422i	$-0.263486\pi$
$139$	15.9522	1.35304	0.676522	+	0.736422i	$0.263486\pi$
$140$	1.04281	0.0881337
$141$	0.151884	0.0127910
$142$	4.00991	0.336504
$143$	−16.6037	−1.38847
$144$	0.130910	0.0109091
$145$	1.31356	0.109086
$146$	−0.637044	−0.0527221
$147$	−4.62079	−0.381117
$148$	−11.8817	−0.976667
$149$	−4.20165	−0.344213	−0.172106	−	0.985078i	$-0.555057\pi$
$149$	−4.20165	−0.344213	−0.172106	+	0.985078i	$0.555057\pi$
$150$	−4.23605	−0.345872
$151$	11.6665	0.949410	0.474705	−	0.880145i	$-0.342555\pi$
$151$	11.6665	0.949410	0.474705	+	0.880145i	$0.342555\pi$
$152$	8.46112	0.686287
$153$	−1.00000	−0.0808452
$154$	13.2257	1.06576
$155$	0.497779	0.0399826
$156$	−4.64137	−0.371607
$157$	−1.00000	−0.0798087
$157$	−1.00000	−0.0798087
$158$	−12.4289	−0.988790
$159$	8.36573	0.663446
$160$	−1.38070	−0.109154
$161$	21.6191	1.70383
$162$	−0.857233	−0.0673506
$163$	17.3613	1.35984	0.679920	−	0.733286i	$-0.262015\pi$
$163$	17.3613	1.35984	0.679920	+	0.733286i	$0.262015\pi$
$164$	−0.850155	−0.0663859
$165$	−1.09433	−0.0851934
$166$	−3.92362	−0.304532
$167$	−3.28213	−0.253979	−0.126989	−	0.991904i	$-0.540531\pi$
$167$	−3.28213	−0.253979	−0.126989	+	0.991904i	$0.540531\pi$
$168$	9.54158	0.736149
$169$	0.458830	0.0352946
$170$	0.207274	0.0158972
$171$	3.02291	0.231168
$172$	7.64304	0.582776
$173$	19.1960	1.45945	0.729723	−	0.683743i	$-0.239649\pi$
$173$	19.1960	1.45945	0.729723	+	0.683743i	$0.239649\pi$
$174$	4.65698	0.353045
$175$	16.8453	1.27339
$176$	0.592480	0.0446599
$177$	−1.97010	−0.148082
$178$	−1.62106	−0.121503
$179$	−12.5693	−0.939470	−0.469735	−	0.882807i	$-0.655651\pi$
$179$	−12.5693	−0.939470	−0.469735	+	0.882807i	$0.655651\pi$
$180$	−0.305906	−0.0228009
$181$	1.55001	0.115211	0.0576056	−	0.998339i	$-0.481653\pi$
$181$	1.55001	0.115211	0.0576056	+	0.998339i	$0.481653\pi$
$182$	−10.7206	−0.794666
$183$	−0.358392	−0.0264931
$184$	−17.7510	−1.30862
$185$	2.27081	0.166953
$186$	1.76478	0.129400
$187$	−4.52587	−0.330965
$188$	0.192156	0.0140144
$189$	3.40893	0.247963
$190$	−0.626571	−0.0454562
$191$	24.8936	1.80123	0.900617	−	0.434613i	$-0.143115\pi$
$191$	24.8936	1.80123	0.900617	+	0.434613i	$0.143115\pi$
$192$	−4.63316	−0.334370
$193$	−0.656214	−0.0472353	−0.0236177	−	0.999721i	$-0.507518\pi$
$193$	−0.656214	−0.0472353	−0.0236177	+	0.999721i	$0.507518\pi$
$194$	1.69320	0.121564
$195$	0.887052	0.0635231
$196$	−5.84600	−0.417571
$197$	9.55795	0.680975	0.340488	−	0.940249i	$-0.389408\pi$
$197$	9.55795	0.680975	0.340488	+	0.940249i	$0.389408\pi$
$198$	−3.87973	−0.275720
$199$	4.21395	0.298719	0.149359	−	0.988783i	$-0.452279\pi$
$199$	4.21395	0.298719	0.149359	+	0.988783i	$0.452279\pi$
$200$	−13.8313	−0.978023
$201$	−1.12443	−0.0793110
$202$	−4.69851	−0.330586
$203$	−18.5193	−1.29980
$204$	−1.26515	−0.0885783
$205$	0.162480	0.0113481
$206$	−6.32939	−0.440990
$207$	−6.34192	−0.440794
$208$	−0.480259	−0.0333000
$209$	13.6813	0.946356
$210$	−0.706582	−0.0487588
$211$	14.7664	1.01656	0.508281	−	0.861191i	$-0.330281\pi$
$211$	14.7664	1.01656	0.508281	+	0.861191i	$0.330281\pi$
$212$	10.5839	0.726906
$213$	4.67774	0.320513
$214$	12.6351	0.863716
$215$	−1.46073	−0.0996208
$216$	−2.79900	−0.190448
$217$	−7.01793	−0.476408
$218$	1.35155	0.0915383
$219$	−0.743140	−0.0502167
$220$	−1.38449	−0.0933424
$221$	3.66863	0.246779
$222$	8.05070	0.540328
$223$	6.48515	0.434277	0.217139	−	0.976141i	$-0.430328\pi$
$223$	6.48515	0.434277	0.217139	+	0.976141i	$0.430328\pi$
$224$	19.4657	1.30061
$225$	−4.94154	−0.329436
$226$	−3.68334	−0.245012
$227$	14.3079	0.949648	0.474824	−	0.880081i	$-0.342512\pi$
$227$	14.3079	0.949648	0.474824	+	0.880081i	$0.342512\pi$
$228$	3.82444	0.253280
$229$	11.7106	0.773861	0.386931	−	0.922109i	$-0.373535\pi$
$229$	11.7106	0.773861	0.386931	+	0.922109i	$0.373535\pi$
$230$	1.31451	0.0866765
$231$	15.4284	1.01511
$232$	15.2058	0.998307
$233$	−25.2716	−1.65560	−0.827798	−	0.561026i	$-0.810407\pi$
$233$	−25.2716	−1.65560	−0.827798	+	0.561026i	$0.810407\pi$
$234$	3.14487	0.205587
$235$	−0.0367247	−0.00239565
$236$	−2.49248	−0.162247
$237$	−14.4989	−0.941802
$238$	−2.92225	−0.189421
$239$	10.0441	0.649698	0.324849	−	0.945766i	$-0.394687\pi$
$239$	10.0441	0.649698	0.324849	+	0.945766i	$0.394687\pi$
$240$	−0.0316532	−0.00204320
$241$	−17.3874	−1.12002	−0.560010	−	0.828486i	$-0.689203\pi$
$241$	−17.3874	−1.12002	−0.560010	+	0.828486i	$0.689203\pi$
$242$	−8.12958	−0.522590
$243$	−1.00000	−0.0641500
$244$	−0.453420	−0.0290272
$245$	1.11728	0.0713804
$246$	0.576043	0.0367271
$247$	−11.0899	−0.705636
$248$	5.76227	0.365905
$249$	−4.57707	−0.290060
$250$	2.06062	0.130325
$251$	−30.6012	−1.93153	−0.965765	−	0.259418i	$-0.916469\pi$
$251$	−30.6012	−1.93153	−0.965765	+	0.259418i	$0.916469\pi$
$252$	4.31281	0.271681
$253$	−28.7027	−1.80452
$254$	13.6517	0.856585
$255$	0.241794	0.0151417
$256$	−15.6516	−0.978226
$257$	1.86778	0.116509	0.0582544	−	0.998302i	$-0.481447\pi$
$257$	1.86778	0.116509	0.0582544	+	0.998302i	$0.481447\pi$
$258$	−5.17872	−0.322413
$259$	−32.0149	−1.98931
$260$	1.12226	0.0695993
$261$	5.43257	0.336268
$262$	3.70504	0.228898
$263$	2.20651	0.136059	0.0680296	−	0.997683i	$-0.478329\pi$
$263$	2.20651	0.136059	0.0680296	+	0.997683i	$0.478329\pi$
$264$	−12.6679	−0.779655
$265$	−2.02278	−0.124259
$266$	8.83370	0.541629
$267$	−1.89104	−0.115729
$268$	−1.42257	−0.0868973
$269$	−4.73650	−0.288789	−0.144395	−	0.989520i	$-0.546123\pi$
$269$	−4.73650	−0.288789	−0.144395	+	0.989520i	$0.546123\pi$
$270$	0.207274	0.0126143
$271$	−27.8344	−1.69082	−0.845408	−	0.534120i	$-0.820643\pi$
$271$	−27.8344	−1.69082	−0.845408	+	0.534120i	$0.820643\pi$
$272$	−0.130910	−0.00793757
$273$	−12.5061	−0.756903
$274$	−6.42311	−0.388034
$275$	−22.3648	−1.34865
$276$	−8.02349	−0.482957
$277$	16.8515	1.01251	0.506253	−	0.862385i	$-0.331030\pi$
$277$	16.8515	1.01251	0.506253	+	0.862385i	$0.331030\pi$
$278$	−13.6747	−0.820155
$279$	2.05869	0.123251
$280$	−2.30710	−0.137875
$281$	−7.80481	−0.465596	−0.232798	−	0.972525i	$-0.574788\pi$
$281$	−7.80481	−0.465596	−0.232798	+	0.972525i	$0.574788\pi$
$282$	−0.130200	−0.00775330
$283$	−6.52256	−0.387726	−0.193863	−	0.981029i	$-0.562102\pi$
$283$	−6.52256	−0.387726	−0.193863	+	0.981029i	$0.562102\pi$
$284$	5.91804	0.351171
$285$	−0.730922	−0.0432961
$286$	14.2333	0.841631
$287$	−2.29073	−0.135217
$288$	−5.71021	−0.336477
$289$	1.00000	0.0588235
$290$	−1.12603	−0.0661228
$291$	1.97519	0.115787
$292$	−0.940184	−0.0550201
$293$	−27.3131	−1.59565	−0.797825	−	0.602889i	$-0.794016\pi$
$293$	−27.3131	−1.59565	−0.797825	+	0.602889i	$0.794016\pi$
$294$	3.96110	0.231016
$295$	0.476360	0.0277347
$296$	26.2868	1.52789
$297$	−4.52587	−0.262618
$298$	3.60180	0.208646
$299$	23.2661	1.34552
$300$	−6.25179	−0.360947
$301$	20.5940	1.18702
$302$	−10.0009	−0.575490
$303$	−5.48102	−0.314876
$304$	0.395729	0.0226966
$305$	0.0866570	0.00496196
$306$	0.857233	0.0490048
$307$	9.25543	0.528235	0.264118	−	0.964490i	$-0.414919\pi$
$307$	9.25543	0.528235	0.264118	+	0.964490i	$0.414919\pi$
$308$	19.5192	1.11221
$309$	−7.38351	−0.420033
$310$	−0.426713	−0.0242357
$311$	10.6691	0.604991	0.302495	−	0.953151i	$-0.402180\pi$
$311$	10.6691	0.604991	0.302495	+	0.953151i	$0.402180\pi$
$312$	10.2685	0.581338
$313$	−2.68328	−0.151668	−0.0758341	−	0.997120i	$-0.524162\pi$
$313$	−2.68328	−0.151668	−0.0758341	+	0.997120i	$0.524162\pi$
$314$	0.857233	0.0483765
$315$	−0.824258	−0.0464417
$316$	−18.3432	−1.03189
$317$	3.79941	0.213396	0.106698	−	0.994291i	$-0.465972\pi$
$317$	3.79941	0.213396	0.106698	+	0.994291i	$0.465972\pi$
$318$	−7.17139	−0.402151
$319$	24.5871	1.37662
$320$	1.12027	0.0626251
$321$	14.7394	0.822671
$322$	−18.5327	−1.03278
$323$	−3.02291	−0.168199
$324$	−1.26515	−0.0702862
$325$	18.1287	1.00560
$326$	−14.8827	−0.824274
$327$	1.57664	0.0871883
$328$	1.88087	0.103853
$329$	0.517762	0.0285451
$330$	0.938095	0.0516404
$331$	−23.4474	−1.28879	−0.644393	−	0.764694i	$-0.722890\pi$
$331$	−23.4474	−1.28879	−0.644393	+	0.764694i	$0.722890\pi$
$332$	−5.79069	−0.317805
$333$	9.39150	0.514651
$334$	2.81355	0.153951
$335$	0.271880	0.0148544
$336$	0.446262	0.0243456
$337$	−20.3377	−1.10786	−0.553932	−	0.832562i	$-0.686873\pi$
$337$	−20.3377	−1.10786	−0.553932	+	0.832562i	$0.686873\pi$
$338$	−0.393324	−0.0213940
$339$	−4.29677	−0.233369
$340$	0.305906	0.0165901
$341$	9.31738	0.504564
$342$	−2.59134	−0.140124
$343$	8.11055	0.437928
$344$	−16.9093	−0.911689
$345$	1.53344	0.0825575
$346$	−16.4555	−0.884651
$347$	2.87697	0.154444	0.0772218	−	0.997014i	$-0.475395\pi$
$347$	2.87697	0.154444	0.0772218	+	0.997014i	$0.475395\pi$
$348$	6.87303	0.368433
$349$	−19.2949	−1.03283	−0.516415	−	0.856338i	$-0.672734\pi$
$349$	−19.2949	−1.03283	−0.516415	+	0.856338i	$0.672734\pi$
$350$	−14.4404	−0.771871
$351$	3.66863	0.195817
$352$	−25.8437	−1.37747
$353$	−31.2499	−1.66326	−0.831632	−	0.555327i	$-0.812593\pi$
$353$	−31.2499	−1.66326	−0.831632	+	0.555327i	$0.812593\pi$
$354$	1.68884	0.0897608
$355$	−1.13105	−0.0600298
$356$	−2.39245	−0.126799
$357$	−3.40893	−0.180420
$358$	10.7748	0.569465
$359$	4.06561	0.214575	0.107287	−	0.994228i	$-0.465784\pi$
$359$	4.06561	0.214575	0.107287	+	0.994228i	$0.465784\pi$
$360$	0.676781	0.0356695
$361$	−9.86200	−0.519053
$362$	−1.32872	−0.0698359
$363$	−9.48352	−0.497755
$364$	−15.8221	−0.829303
$365$	0.179687	0.00940523
$366$	0.307225	0.0160589
$367$	−34.9341	−1.82354	−0.911772	−	0.410696i	$-0.865286\pi$
$367$	−34.9341	−1.82354	−0.911772	+	0.410696i	$0.865286\pi$
$368$	−0.830219	−0.0432781
$369$	0.671979	0.0349818
$370$	−1.94661	−0.101200
$371$	28.5182	1.48059
$372$	2.60456	0.135040
$373$	−25.7863	−1.33516	−0.667581	−	0.744537i	$-0.732670\pi$
$373$	−25.7863	−1.33516	−0.667581	+	0.744537i	$0.732670\pi$
$374$	3.87973	0.200616
$375$	2.40380	0.124132
$376$	−0.425123	−0.0219240
$377$	−19.9301	−1.02645
$378$	−2.92225	−0.150304
$379$	−15.7885	−0.811002	−0.405501	−	0.914095i	$-0.632903\pi$
$379$	−15.7885	−0.811002	−0.405501	+	0.914095i	$0.632903\pi$
$380$	−0.924727	−0.0474375
$381$	15.9253	0.815880
$382$	−21.3396	−1.09183
$383$	29.7876	1.52208	0.761038	−	0.648707i	$-0.224690\pi$
$383$	29.7876	1.52208	0.761038	+	0.648707i	$0.224690\pi$
$384$	−7.44872	−0.380116
$385$	−3.73049	−0.190123
$386$	0.562528	0.0286319
$387$	−6.04121	−0.307092
$388$	2.49891	0.126863
$389$	−8.45717	−0.428796	−0.214398	−	0.976746i	$-0.568779\pi$
$389$	−8.45717	−0.428796	−0.214398	+	0.976746i	$0.568779\pi$
$390$	−0.760411	−0.0385049
$391$	6.34192	0.320725
$392$	12.9336	0.653244
$393$	4.32209	0.218021
$394$	−8.19339	−0.412777
$395$	3.50574	0.176393
$396$	−5.72591	−0.287738
$397$	−29.7287	−1.49204	−0.746020	−	0.665924i	$-0.768038\pi$
$397$	−29.7287	−1.49204	−0.746020	+	0.665924i	$0.768038\pi$
$398$	−3.61234	−0.181070
$399$	10.3049	0.515890
$400$	−0.646895	−0.0323447
$401$	−15.9131	−0.794663	−0.397332	−	0.917675i	$-0.630064\pi$
$401$	−15.9131	−0.794663	−0.397332	+	0.917675i	$0.630064\pi$
$402$	0.963897	0.0480748
$403$	−7.55257	−0.376221
$404$	−6.93431	−0.344995
$405$	0.241794	0.0120148
$406$	15.8753	0.787879
$407$	42.5047	2.10688
$408$	2.79900	0.138571
$409$	0.0427216	0.00211245	0.00105622	−	0.999999i	$-0.499664\pi$
$409$	0.0427216	0.00211245	0.00105622	+	0.999999i	$0.499664\pi$
$410$	−0.139284	−0.00687873
$411$	−7.49284	−0.369595
$412$	−9.34126	−0.460211
$413$	−6.71595	−0.330470
$414$	5.43650	0.267190
$415$	1.10671	0.0543262
$416$	20.9486	1.02709
$417$	−15.9522	−0.781181
$418$	−11.7281	−0.573639
$419$	13.9571	0.681851	0.340926	−	0.940090i	$-0.389260\pi$
$419$	13.9571	0.681851	0.340926	+	0.940090i	$0.389260\pi$
$420$	−1.04281	−0.0508840
$421$	25.2133	1.22882	0.614411	−	0.788986i	$-0.289394\pi$
$421$	25.2133	1.22882	0.614411	+	0.788986i	$0.289394\pi$
$422$	−12.6583	−0.616195
$423$	−0.151884	−0.00738486
$424$	−23.4157	−1.13716
$425$	4.94154	0.239700
$426$	−4.00991	−0.194281
$427$	−1.22173	−0.0591237
$428$	18.6475	0.901362
$429$	16.6037	0.801636
$430$	1.25218	0.0603857
$431$	−26.5718	−1.27992	−0.639959	−	0.768409i	$-0.721049\pi$
$431$	−26.5718	−1.27992	−0.639959	+	0.768409i	$0.721049\pi$
$432$	−0.130910	−0.00629840
$433$	−15.8640	−0.762378	−0.381189	−	0.924497i	$-0.624485\pi$
$433$	−15.8640	−0.762378	−0.381189	+	0.924497i	$0.624485\pi$
$434$	6.01601	0.288778
$435$	−1.31356	−0.0629806
$436$	1.99469	0.0955281
$437$	−19.1711	−0.917077
$438$	0.637044	0.0304391
$439$	−11.6157	−0.554389	−0.277194	−	0.960814i	$-0.589405\pi$
$439$	−11.6157	−0.554389	−0.277194	+	0.960814i	$0.589405\pi$
$440$	3.06302	0.146024
$441$	4.62079	0.220038
$442$	−3.14487	−0.149586
$443$	7.59717	0.360952	0.180476	−	0.983579i	$-0.442236\pi$
$443$	7.59717	0.360952	0.180476	+	0.983579i	$0.442236\pi$
$444$	11.8817	0.563879
$445$	0.457241	0.0216753
$446$	−5.55928	−0.263240
$447$	4.20165	0.198731
$448$	−15.7941	−0.746202
$449$	6.98900	0.329831	0.164916	−	0.986308i	$-0.447265\pi$
$449$	6.98900	0.329831	0.164916	+	0.986308i	$0.447265\pi$
$450$	4.23605	0.199689
$451$	3.04129	0.143209
$452$	−5.43607	−0.255691
$453$	−11.6665	−0.548142
$454$	−12.2652	−0.575635
$455$	3.02390	0.141763
$456$	−8.46112	−0.396228
$457$	32.2045	1.50647	0.753233	−	0.657754i	$-0.228493\pi$
$457$	32.2045	1.50647	0.753233	+	0.657754i	$0.228493\pi$
$458$	−10.0388	−0.469080
$459$	1.00000	0.0466760
$460$	1.94003	0.0904544
$461$	1.95480	0.0910443	0.0455221	−	0.998963i	$-0.485505\pi$
$461$	1.95480	0.0910443	0.0455221	+	0.998963i	$0.485505\pi$
$462$	−13.2257	−0.615316
$463$	−1.49937	−0.0696814	−0.0348407	−	0.999393i	$-0.511092\pi$
$463$	−1.49937	−0.0696814	−0.0348407	+	0.999393i	$0.511092\pi$
$464$	0.711177	0.0330155
$465$	−0.497779	−0.0230840
$466$	21.6636	1.00355
$467$	−17.9307	−0.829733	−0.414867	−	0.909882i	$-0.636172\pi$
$467$	−17.9307	−0.829733	−0.414867	+	0.909882i	$0.636172\pi$
$468$	4.64137	0.214547
$469$	−3.83309	−0.176996
$470$	0.0314816	0.00145214
$471$	1.00000	0.0460776
$472$	5.51432	0.253817
$473$	−27.3417	−1.25717
$474$	12.4289	0.570878
$475$	−14.9378	−0.685395
$476$	−4.31281	−0.197677
$477$	−8.36573	−0.383041
$478$	−8.61012	−0.393818
$479$	−2.97262	−0.135823	−0.0679113	−	0.997691i	$-0.521633\pi$
$479$	−2.97262	−0.135823	−0.0679113	+	0.997691i	$0.521633\pi$
$480$	1.38070	0.0630198
$481$	−34.4539	−1.57096
$482$	14.9051	0.678907
$483$	−21.6191	−0.983705
$484$	−11.9981	−0.545367
$485$	−0.477588	−0.0216862
$486$	0.857233	0.0388849
$487$	−16.6735	−0.755548	−0.377774	−	0.925898i	$-0.623310\pi$
$487$	−16.6735	−0.755548	−0.377774	+	0.925898i	$0.623310\pi$
$488$	1.00314	0.0454099
$489$	−17.3613	−0.785104
$490$	−0.957770	−0.0432676
$491$	−34.2413	−1.54529	−0.772645	−	0.634839i	$-0.781067\pi$
$491$	−34.2413	−1.54529	−0.772645	+	0.634839i	$0.781067\pi$
$492$	0.850155	0.0383279
$493$	−5.43257	−0.244671
$494$	9.50667	0.427725
$495$	1.09433	0.0491864
$496$	0.269503	0.0121010
$497$	15.9461	0.715279
$498$	3.92362	0.175822
$499$	−17.4042	−0.779118	−0.389559	−	0.921001i	$-0.627373\pi$
$499$	−17.4042	−0.779118	−0.389559	+	0.921001i	$0.627373\pi$
$500$	3.04118	0.136005
$501$	3.28213	0.146635
$502$	26.2324	1.17081
$503$	−9.10033	−0.405764	−0.202882	−	0.979203i	$-0.565031\pi$
$503$	−9.10033	−0.405764	−0.202882	+	0.979203i	$0.565031\pi$
$504$	−9.54158	−0.425016
$505$	1.32528	0.0589740
$506$	24.6049	1.09382
$507$	−0.458830	−0.0203774
$508$	20.1480	0.893921
$509$	31.7793	1.40859	0.704296	−	0.709906i	$-0.251263\pi$
$509$	31.7793	1.40859	0.704296	+	0.709906i	$0.251263\pi$
$510$	−0.207274	−0.00917824
$511$	−2.53331	−0.112067
$512$	−1.48035	−0.0654230
$513$	−3.02291	−0.133465
$514$	−1.60112	−0.0706224
$515$	1.78529	0.0786693
$516$	−7.64304	−0.336466
$517$	−0.687408	−0.0302322
$518$	27.4443	1.20583
$519$	−19.1960	−0.842612
$520$	−2.48286	−0.108880
$521$	9.90072	0.433758	0.216879	−	0.976198i	$-0.430412\pi$
$521$	9.90072	0.433758	0.216879	+	0.976198i	$0.430412\pi$
$522$	−4.65698	−0.203831
$523$	−42.1632	−1.84367	−0.921834	−	0.387584i	$-0.873310\pi$
$523$	−42.1632	−1.84367	−0.921834	+	0.387584i	$0.873310\pi$
$524$	5.46810	0.238875
$525$	−16.8453	−0.735191
$526$	−1.89149	−0.0824731
$527$	−2.05869	−0.0896780
$528$	−0.592480	−0.0257844
$529$	17.2199	0.748693
$530$	1.73400	0.0753201
$531$	1.97010	0.0854953
$532$	13.0372	0.565236
$533$	−2.46524	−0.106781
$534$	1.62106	0.0701501
$535$	−3.56389	−0.154080
$536$	3.14727	0.135941
$537$	12.5693	0.542403
$538$	4.06028	0.175051
$539$	20.9131	0.900792
$540$	0.305906	0.0131641
$541$	−24.3145	−1.04536	−0.522682	−	0.852528i	$-0.675068\pi$
$541$	−24.3145	−1.04536	−0.522682	+	0.852528i	$0.675068\pi$
$542$	23.8605	1.02490
$543$	−1.55001	−0.0665173
$544$	5.71021	0.244823
$545$	−0.381222	−0.0163297
$546$	10.7206	0.458801
$547$	31.7220	1.35634	0.678168	−	0.734907i	$-0.262774\pi$
$547$	31.7220	1.35634	0.678168	+	0.734907i	$0.262774\pi$
$548$	−9.47958	−0.404947
$549$	0.358392	0.0152958
$550$	19.1718	0.817489
$551$	16.4222	0.699609
$552$	17.7510	0.755533
$553$	−49.4256	−2.10179
$554$	−14.4456	−0.613736
$555$	−2.27081	−0.0963904
$556$	−20.1819	−0.855903
$557$	−10.6754	−0.452330	−0.226165	−	0.974089i	$-0.572619\pi$
$557$	−10.6754	−0.452330	−0.226165	+	0.974089i	$0.572619\pi$
$558$	−1.76478	−0.0747091
$559$	22.1629	0.937392
$560$	−0.107903	−0.00455975
$561$	4.52587	0.191082
$562$	6.69054	0.282224
$563$	12.3621	0.521000	0.260500	−	0.965474i	$-0.416113\pi$
$563$	12.3621	0.521000	0.260500	+	0.965474i	$0.416113\pi$
$564$	−0.192156	−0.00809124
$565$	1.03893	0.0437083
$566$	5.59136	0.235022
$567$	−3.40893	−0.143162
$568$	−13.0930	−0.549369
$569$	0.563472	0.0236220	0.0118110	−	0.999930i	$-0.496240\pi$
$569$	0.563472	0.0236220	0.0118110	+	0.999930i	$0.496240\pi$
$570$	0.626571	0.0262442
$571$	0.335604	0.0140446	0.00702230	−	0.999975i	$-0.497765\pi$
$571$	0.335604	0.0140446	0.00702230	+	0.999975i	$0.497765\pi$
$572$	21.0062	0.878315
$573$	−24.8936	−1.03994
$574$	1.96369	0.0819627
$575$	31.3388	1.30692
$576$	4.63316	0.193049
$577$	18.9547	0.789093	0.394547	−	0.918876i	$-0.370902\pi$
$577$	18.9547	0.789093	0.394547	+	0.918876i	$0.370902\pi$
$578$	−0.857233	−0.0356562
$579$	0.656214	0.0272713
$580$	−1.66186	−0.0690048
$581$	−15.6029	−0.647318
$582$	−1.69320	−0.0701852
$583$	−37.8622	−1.56809
$584$	2.08004	0.0860729
$585$	−0.887052	−0.0366751
$586$	23.4137	0.967212
$587$	−11.9679	−0.493967	−0.246984	−	0.969020i	$-0.579439\pi$
$587$	−11.9679	−0.493967	−0.246984	+	0.969020i	$0.579439\pi$
$588$	5.84600	0.241085
$589$	6.22325	0.256424
$590$	−0.408351	−0.0168116
$591$	−9.55795	−0.393161
$592$	1.22944	0.0505296
$593$	−6.06945	−0.249242	−0.124621	−	0.992204i	$-0.539772\pi$
$593$	−6.06945	−0.249242	−0.124621	+	0.992204i	$0.539772\pi$
$594$	3.87973	0.159187
$595$	0.824258	0.0337913
$596$	5.31573	0.217741
$597$	−4.21395	−0.172465
$598$	−19.9445	−0.815592
$599$	2.80080	0.114438	0.0572188	−	0.998362i	$-0.481777\pi$
$599$	2.80080	0.114438	0.0572188	+	0.998362i	$0.481777\pi$
$600$	13.8313	0.564662
$601$	−1.92608	−0.0785665	−0.0392833	−	0.999228i	$-0.512507\pi$
$601$	−1.92608	−0.0785665	−0.0392833	+	0.999228i	$0.512507\pi$
$602$	−17.6539	−0.719519
$603$	1.12443	0.0457902
$604$	−14.7599	−0.600573
$605$	2.29306	0.0932260
$606$	4.69851	0.190864
$607$	−20.5281	−0.833210	−0.416605	−	0.909088i	$-0.636780\pi$
$607$	−20.5281	−0.833210	−0.416605	+	0.909088i	$0.636780\pi$
$608$	−17.2615	−0.700045
$609$	18.5193	0.750438
$610$	−0.0742852	−0.00300772
$611$	0.557206	0.0225422
$612$	1.26515	0.0511407
$613$	11.7315	0.473833	0.236916	−	0.971530i	$-0.423863\pi$
$613$	11.7315	0.473833	0.236916	+	0.971530i	$0.423863\pi$
$614$	−7.93406	−0.320193
$615$	−0.162480	−0.00655184
$616$	−43.1840	−1.73993
$617$	25.9547	1.04490	0.522448	−	0.852671i	$-0.325019\pi$
$617$	25.9547	1.04490	0.522448	+	0.852671i	$0.325019\pi$
$618$	6.32939	0.254606
$619$	−20.4435	−0.821694	−0.410847	−	0.911704i	$-0.634767\pi$
$619$	−20.4435	−0.821694	−0.410847	+	0.911704i	$0.634767\pi$
$620$	−0.629766	−0.0252920
$621$	6.34192	0.254492
$622$	−9.14593	−0.366718
$623$	−6.44640	−0.258270
$624$	0.480259	0.0192257
$625$	24.1265	0.965058
$626$	2.30020	0.0919345
$627$	−13.6813	−0.546379
$628$	1.26515	0.0504850
$629$	−9.39150	−0.374464
$630$	0.706582	0.0281509
$631$	9.03129	0.359530	0.179765	−	0.983710i	$-0.442466\pi$
$631$	9.03129	0.359530	0.179765	+	0.983710i	$0.442466\pi$
$632$	40.5822	1.61427
$633$	−14.7664	−0.586912
$634$	−3.25698	−0.129351
$635$	−3.85065	−0.152808
$636$	−10.5839	−0.419680
$637$	−16.9520	−0.671661
$638$	−21.0769	−0.834443
$639$	−4.67774	−0.185048
$640$	1.80106	0.0711930
$641$	10.3769	0.409863	0.204931	−	0.978776i	$-0.434303\pi$
$641$	10.3769	0.409863	0.204931	+	0.978776i	$0.434303\pi$
$642$	−12.6351	−0.498667
$643$	−1.03564	−0.0408416	−0.0204208	−	0.999791i	$-0.506501\pi$
$643$	−1.03564	−0.0408416	−0.0204208	+	0.999791i	$0.506501\pi$
$644$	−27.3515	−1.07780
$645$	1.46073	0.0575161
$646$	2.59134	0.101955
$647$	7.27403	0.285972	0.142986	−	0.989725i	$-0.454330\pi$
$647$	7.27403	0.285972	0.142986	+	0.989725i	$0.454330\pi$
$648$	2.79900	0.109955
$649$	8.91644	0.350001
$650$	−15.5405	−0.609548
$651$	7.01793	0.275054
$652$	−21.9646	−0.860202
$653$	−25.1804	−0.985386	−0.492693	−	0.870203i	$-0.663987\pi$
$653$	−25.1804	−0.985386	−0.492693	+	0.870203i	$0.663987\pi$
$654$	−1.35155	−0.0528496
$655$	−1.04506	−0.0408337
$656$	0.0879685	0.00343459
$657$	0.743140	0.0289926
$658$	−0.443843	−0.0173028
$659$	14.0170	0.546023	0.273012	−	0.962011i	$-0.411980\pi$
$659$	14.0170	0.546023	0.273012	+	0.962011i	$0.411980\pi$
$660$	1.38449	0.0538912
$661$	−16.6106	−0.646076	−0.323038	−	0.946386i	$-0.604704\pi$
$661$	−16.6106	−0.646076	−0.323038	+	0.946386i	$0.604704\pi$
$662$	20.0999	0.781205
$663$	−3.66863	−0.142478
$664$	12.8112	0.497171
$665$	−2.49166	−0.0966225
$666$	−8.05070	−0.311958
$667$	−34.4529	−1.33402
$668$	4.15239	0.160661
$669$	−6.48515	−0.250730
$670$	−0.233064	−0.00900406
$671$	1.62204	0.0626180
$672$	−19.4657	−0.750906
$673$	41.8185	1.61198	0.805992	−	0.591926i	$-0.201632\pi$
$673$	41.8185	1.61198	0.805992	+	0.591926i	$0.201632\pi$
$674$	17.4341	0.671538
$675$	4.94154	0.190200
$676$	−0.580489	−0.0223265
$677$	−34.9775	−1.34429	−0.672147	−	0.740417i	$-0.734628\pi$
$677$	−34.9775	−1.34429	−0.672147	+	0.740417i	$0.734628\pi$
$678$	3.68334	0.141458
$679$	6.73327	0.258399
$680$	−0.676781	−0.0259533
$681$	−14.3079	−0.548280
$682$	−7.98716	−0.305844
$683$	30.8653	1.18103	0.590514	−	0.807028i	$-0.298925\pi$
$683$	30.8653	1.18103	0.590514	+	0.807028i	$0.298925\pi$
$684$	−3.82444	−0.146231
$685$	1.81172	0.0692224
$686$	−6.95263	−0.265453
$687$	−11.7106	−0.446789
$688$	−0.790852	−0.0301510
$689$	30.6908	1.16923
$690$	−1.31451	−0.0500427
$691$	−7.18879	−0.273475	−0.136737	−	0.990607i	$-0.543662\pi$
$691$	−7.18879	−0.273475	−0.136737	+	0.990607i	$0.543662\pi$
$692$	−24.2859	−0.923210
$693$	−15.4284	−0.586075
$694$	−2.46623	−0.0936169
$695$	3.85714	0.146310
$696$	−15.2058	−0.576373
$697$	−0.671979	−0.0254530
$698$	16.5402	0.626056
$699$	25.2716	0.955859
$700$	−21.3119	−0.805514
$701$	−35.6772	−1.34751	−0.673755	−	0.738955i	$-0.735320\pi$
$701$	−35.6772	−1.34751	−0.673755	+	0.738955i	$0.735320\pi$
$702$	−3.14487	−0.118695
$703$	28.3897	1.07074
$704$	20.9691	0.790303
$705$	0.0367247	0.00138313
$706$	26.7885	1.00820
$707$	−18.6844	−0.702699
$708$	2.49248	0.0936731
$709$	−3.61747	−0.135857	−0.0679285	−	0.997690i	$-0.521639\pi$
$709$	−3.61747	−0.135857	−0.0679285	+	0.997690i	$0.521639\pi$
$710$	0.969572	0.0363874
$711$	14.4989	0.543749
$712$	5.29300	0.198364
$713$	−13.0561	−0.488953
$714$	2.92225	0.109362
$715$	−4.01468	−0.150141
$716$	15.9020	0.594286
$717$	−10.0441	−0.375103
$718$	−3.48517	−0.130066
$719$	16.7119	0.623250	0.311625	−	0.950205i	$-0.399127\pi$
$719$	16.7119	0.623250	0.311625	+	0.950205i	$0.399127\pi$
$720$	0.0316532	0.00117964
$721$	−25.1699	−0.937375
$722$	8.45403	0.314627
$723$	17.3874	0.646644
$724$	−1.96100	−0.0728798
$725$	−26.8453	−0.997008
$726$	8.12958	0.301717
$727$	−36.7966	−1.36471	−0.682356	−	0.731020i	$-0.739044\pi$
$727$	−36.7966	−1.36471	−0.682356	+	0.731020i	$0.739044\pi$
$728$	35.0045	1.29735
$729$	1.00000	0.0370370
$730$	−0.154033	−0.00570103
$731$	6.04121	0.223442
$732$	0.453420	0.0167589
$733$	−2.92572	−0.108064	−0.0540320	−	0.998539i	$-0.517207\pi$
$733$	−2.92572	−0.108064	−0.0540320	+	0.998539i	$0.517207\pi$
$734$	29.9467	1.10535
$735$	−1.11728	−0.0412115
$736$	36.2137	1.33485
$737$	5.08901	0.187456
$738$	−0.576043	−0.0212044
$739$	−10.9595	−0.403151	−0.201575	−	0.979473i	$-0.564606\pi$
$739$	−10.9595	−0.403151	−0.201575	+	0.979473i	$0.564606\pi$
$740$	−2.87292	−0.105610
$741$	11.0899	0.407399
$742$	−24.4467	−0.897468
$743$	22.6789	0.832008	0.416004	−	0.909363i	$-0.363430\pi$
$743$	22.6789	0.832008	0.416004	+	0.909363i	$0.363430\pi$
$744$	−5.76227	−0.211255
$745$	−1.01593	−0.0372210
$746$	22.1048	0.809316
$747$	4.57707	0.167466
$748$	5.72591	0.209360
$749$	50.2455	1.83593
$750$	−2.06062	−0.0752432
$751$	−7.56336	−0.275991	−0.137995	−	0.990433i	$-0.544066\pi$
$751$	−7.56336	−0.275991	−0.137995	+	0.990433i	$0.544066\pi$
$752$	−0.0198831	−0.000725062 0
$753$	30.6012	1.11517
$754$	17.0847	0.622189
$755$	2.82090	0.102663
$756$	−4.31281	−0.156855
$757$	−40.5615	−1.47423	−0.737116	−	0.675766i	$-0.763813\pi$
$757$	−40.5615	−1.47423	−0.737116	+	0.675766i	$0.763813\pi$
$758$	13.5345	0.491593
$759$	28.7027	1.04184
$760$	2.04585	0.0742107
$761$	3.32068	0.120374	0.0601872	−	0.998187i	$-0.480830\pi$
$761$	3.32068	0.120374	0.0601872	+	0.998187i	$0.480830\pi$
$762$	−13.6517	−0.494550
$763$	5.37465	0.194575
$764$	−31.4941	−1.13942
$765$	−0.241794	−0.00874208
$766$	−25.5349	−0.922615
$767$	−7.22758	−0.260973
$768$	15.6516	0.564779
$769$	−17.8214	−0.642657	−0.321329	−	0.946968i	$-0.604129\pi$
$769$	−17.8214	−0.642657	−0.321329	+	0.946968i	$0.604129\pi$
$770$	3.19790	0.115244
$771$	−1.86778	−0.0672664
$772$	0.830210	0.0298799
$773$	−12.8445	−0.461985	−0.230992	−	0.972956i	$-0.574197\pi$
$773$	−12.8445	−0.461985	−0.230992	+	0.972956i	$0.574197\pi$
$774$	5.17872	0.186145
$775$	−10.1731	−0.365429
$776$	−5.52854	−0.198463
$777$	32.0149	1.14853
$778$	7.24977	0.259917
$779$	2.03133	0.0727801
$780$	−1.12226	−0.0401832
$781$	−21.1708	−0.757552
$782$	−5.43650	−0.194409
$783$	−5.43257	−0.194144
$784$	0.604906	0.0216038
$785$	−0.241794	−0.00863000
$786$	−3.70504	−0.132154
$787$	−39.9600	−1.42442	−0.712210	−	0.701966i	$-0.752306\pi$
$787$	−39.9600	−1.42442	−0.712210	+	0.701966i	$0.752306\pi$
$788$	−12.0922	−0.430768
$789$	−2.20651	−0.0785539
$790$	−3.00523	−0.106921
$791$	−14.6474	−0.520801
$792$	12.6679	0.450134
$793$	−1.31481	−0.0466901
$794$	25.4844	0.904408
$795$	2.02278	0.0717408
$796$	−5.33128	−0.188962
$797$	−15.5761	−0.551733	−0.275866	−	0.961196i	$-0.588965\pi$
$797$	−15.5761	−0.551733	−0.275866	+	0.961196i	$0.588965\pi$
$798$	−8.83370	−0.312709
$799$	0.151884	0.00537327
$800$	28.2172	0.997629
$801$	1.89104	0.0668165
$802$	13.6413	0.481690
$803$	3.36335	0.118690
$804$	1.42257	0.0501702
$805$	5.22738	0.184241
$806$	6.47432	0.228048
$807$	4.73650	0.166733
$808$	15.3413	0.539707
$809$	3.58732	0.126124	0.0630618	−	0.998010i	$-0.479913\pi$
$809$	3.58732	0.126124	0.0630618	+	0.998010i	$0.479913\pi$
$810$	−0.207274	−0.00728286
$811$	0.477963	0.0167836	0.00839178	−	0.999965i	$-0.497329\pi$
$811$	0.477963	0.0167836	0.00839178	+	0.999965i	$0.497329\pi$
$812$	23.4297	0.822220
$813$	27.8344	0.976194
$814$	−36.4365	−1.27710
$815$	4.19785	0.147044
$816$	0.130910	0.00458276
$817$	−18.2620	−0.638908
$818$	−0.0366224	−0.00128047
$819$	12.5061	0.436998
$820$	−0.205562	−0.00717855
$821$	−50.6974	−1.76935	−0.884675	−	0.466207i	$-0.845620\pi$
$821$	−50.6974	−1.76935	−0.884675	+	0.466207i	$0.845620\pi$
$822$	6.42311	0.224032
$823$	−26.6583	−0.929250	−0.464625	−	0.885508i	$-0.653811\pi$
$823$	−26.6583	−0.929250	−0.464625	+	0.885508i	$0.653811\pi$
$824$	20.6664	0.719949
$825$	22.3648	0.778641
$826$	5.75713	0.200316
$827$	−48.6942	−1.69326	−0.846632	−	0.532179i	$-0.821373\pi$
$827$	−48.6942	−1.69326	−0.846632	+	0.532179i	$0.821373\pi$
$828$	8.02349	0.278835
$829$	20.6776	0.718164	0.359082	−	0.933306i	$-0.383090\pi$
$829$	20.6776	0.718164	0.359082	+	0.933306i	$0.383090\pi$
$830$	−0.948708	−0.0329301
$831$	−16.8515	−0.584571
$832$	−16.9974	−0.589277
$833$	−4.62079	−0.160101
$834$	13.6747	0.473517
$835$	−0.793600	−0.0274637
$836$	−17.3089	−0.598642
$837$	−2.05869	−0.0711588
$838$	−11.9645	−0.413308
$839$	−1.97127	−0.0680558	−0.0340279	−	0.999421i	$-0.510834\pi$
$839$	−1.97127	−0.0680558	−0.0340279	+	0.999421i	$0.510834\pi$
$840$	2.30710	0.0796024
$841$	0.512862	0.0176849
$842$	−21.6137	−0.744857
$843$	7.80481	0.268812
$844$	−18.6817	−0.643052
$845$	0.110942	0.00381653
$846$	0.130200	0.00447637
$847$	−32.3286	−1.11082
$848$	−1.09516	−0.0376078
$849$	6.52256	0.223854
$850$	−4.23605	−0.145295
$851$	−59.5601	−2.04169
$852$	−5.91804	−0.202749
$853$	−24.2132	−0.829046	−0.414523	−	0.910039i	$-0.636051\pi$
$853$	−24.2132	−0.829046	−0.414523	+	0.910039i	$0.636051\pi$
$854$	1.04731	0.0358382
$855$	0.730922	0.0249970
$856$	−41.2554	−1.41008
$857$	9.75325	0.333165	0.166582	−	0.986028i	$-0.446727\pi$
$857$	9.75325	0.333165	0.166582	+	0.986028i	$0.446727\pi$
$858$	−14.2333	−0.485916
$859$	−8.41048	−0.286962	−0.143481	−	0.989653i	$-0.545830\pi$
$859$	−8.41048	−0.286962	−0.143481	+	0.989653i	$0.545830\pi$
$860$	1.84804	0.0630177
$861$	2.29073	0.0780678
$862$	22.7782	0.775829
$863$	17.9550	0.611195	0.305598	−	0.952161i	$-0.401144\pi$
$863$	17.9550	0.611195	0.305598	+	0.952161i	$0.401144\pi$
$864$	5.71021	0.194265
$865$	4.64148	0.157815
$866$	13.5992	0.462119
$867$	−1.00000	−0.0339618
$868$	8.87875	0.301364
$869$	65.6200	2.22600
$870$	1.12603	0.0381760
$871$	−4.12511	−0.139774
$872$	−4.41300	−0.149443
$873$	−1.97519	−0.0668499
$874$	16.4341	0.555891
$875$	8.19439	0.277021
$876$	0.940184	0.0317659
$877$	1.70283	0.0575004	0.0287502	−	0.999587i	$-0.490847\pi$
$877$	1.70283	0.0575004	0.0287502	+	0.999587i	$0.490847\pi$
$878$	9.95739	0.336046
$879$	27.3131	0.921249
$880$	0.143258	0.00482923
$881$	−33.1265	−1.11606	−0.558031	−	0.829820i	$-0.688443\pi$
$881$	−33.1265	−1.11606	−0.558031	+	0.829820i	$0.688443\pi$
$882$	−3.96110	−0.133377
$883$	33.4640	1.12615	0.563077	−	0.826405i	$-0.309618\pi$
$883$	33.4640	1.12615	0.563077	+	0.826405i	$0.309618\pi$
$884$	−4.64137	−0.156106
$885$	−0.476360	−0.0160126
$886$	−6.51255	−0.218793
$887$	13.1954	0.443059	0.221530	−	0.975154i	$-0.428895\pi$
$887$	13.1954	0.443059	0.221530	+	0.975154i	$0.428895\pi$
$888$	−26.2868	−0.882126
$889$	54.2883	1.82077
$890$	−0.391962	−0.0131386
$891$	4.52587	0.151622
$892$	−8.20469	−0.274713
$893$	−0.459132	−0.0153643
$894$	−3.60180	−0.120462
$895$	−3.03917	−0.101588
$896$	−25.3922	−0.848293
$897$	−23.2661	−0.776834
$898$	−5.99120	−0.199929
$899$	11.1840	0.373007
$900$	6.25179	0.208393
$901$	8.36573	0.278703
$902$	−2.60709	−0.0868068
$903$	−20.5940	−0.685327
$904$	12.0267	0.400000
$905$	0.374783	0.0124582
$906$	10.0009	0.332259
$907$	−51.5280	−1.71096	−0.855480	−	0.517836i	$-0.826738\pi$
$907$	−51.5280	−1.71096	−0.855480	+	0.517836i	$0.826738\pi$
$908$	−18.1017	−0.600724
$909$	5.48102	0.181794
$910$	−2.59219	−0.0859301
$911$	−7.85423	−0.260222	−0.130111	−	0.991499i	$-0.541533\pi$
$911$	−7.85423	−0.260222	−0.130111	+	0.991499i	$0.541533\pi$
$912$	−0.395729	−0.0131039
$913$	20.7152	0.685575
$914$	−27.6068	−0.913152
$915$	−0.0866570	−0.00286479
$916$	−14.8157	−0.489526
$917$	14.7337	0.486550
$918$	−0.857233	−0.0282929
$919$	27.1275	0.894853	0.447427	−	0.894321i	$-0.352341\pi$
$919$	27.1275	0.894853	0.447427	+	0.894321i	$0.352341\pi$
$920$	−4.29209	−0.141506
$921$	−9.25543	−0.304977
$922$	−1.67572	−0.0551870
$923$	17.1609	0.564857
$924$	−19.5192	−0.642135
$925$	−46.4084	−1.52590
$926$	1.28531	0.0422378
$927$	7.38351	0.242506
$928$	−31.0212	−1.01832
$929$	11.8559	0.388979	0.194489	−	0.980905i	$-0.437695\pi$
$929$	11.8559	0.388979	0.194489	+	0.980905i	$0.437695\pi$
$930$	0.426713	0.0139925
$931$	13.9683	0.457791
$932$	31.9724	1.04729
$933$	−10.6691	−0.349291
$934$	15.3708	0.502947
$935$	−1.09433	−0.0357884
$936$	−10.2685	−0.335636
$937$	41.9932	1.37186	0.685929	−	0.727669i	$-0.259396\pi$
$937$	41.9932	1.37186	0.685929	+	0.727669i	$0.259396\pi$
$938$	3.28585	0.107287
$939$	2.68328	0.0875657
$940$	0.0464623	0.00151543
$941$	8.05144	0.262470	0.131235	−	0.991351i	$-0.458106\pi$
$941$	8.05144	0.262470	0.131235	+	0.991351i	$0.458106\pi$
$942$	−0.857233	−0.0279302
$943$	−4.26163	−0.138778
$944$	0.257906	0.00839412
$945$	0.824258	0.0268131
$946$	23.4382	0.762043
$947$	−11.8260	−0.384292	−0.192146	−	0.981366i	$-0.561545\pi$
$947$	−11.8260	−0.384292	−0.192146	+	0.981366i	$0.561545\pi$
$948$	18.3432	0.595761
$949$	−2.72630	−0.0884995
$950$	12.8052	0.415456
$951$	−3.79941	−0.123204
$952$	9.54158	0.309244
$953$	15.0486	0.487472	0.243736	−	0.969842i	$-0.421627\pi$
$953$	15.0486	0.487472	0.243736	+	0.969842i	$0.421627\pi$
$954$	7.17139	0.232182
$955$	6.01911	0.194774
$956$	−12.7073	−0.410983
$957$	−24.5871	−0.794789
$958$	2.54823	0.0823296
$959$	−25.5426	−0.824812
$960$	−1.12027	−0.0361566
$961$	−26.7618	−0.863283
$962$	29.5350	0.952248
$963$	−14.7394	−0.474970
$964$	21.9977	0.708498
$965$	−0.158669	−0.00510772
$966$	18.5327	0.596278
$967$	36.3070	1.16755	0.583777	−	0.811914i	$-0.301574\pi$
$967$	36.3070	1.16755	0.583777	+	0.811914i	$0.301574\pi$
$968$	26.5443	0.853167
$969$	3.02291	0.0971099
$970$	0.409405	0.0131452
$971$	11.0138	0.353451	0.176726	−	0.984260i	$-0.443449\pi$
$971$	11.0138	0.353451	0.176726	+	0.984260i	$0.443449\pi$
$972$	1.26515	0.0405797
$973$	−54.3798	−1.74334
$974$	14.2931	0.457980
$975$	−18.1287	−0.580582
$976$	0.0469169	0.00150177
$977$	−40.6258	−1.29973	−0.649867	−	0.760048i	$-0.725176\pi$
$977$	−40.6258	−1.29973	−0.649867	+	0.760048i	$0.725176\pi$
$978$	14.8827	0.475895
$979$	8.55858	0.273534
$980$	−1.41353	−0.0451535
$981$	−1.57664	−0.0503382
$982$	29.3528	0.936686
$983$	−35.9967	−1.14812	−0.574059	−	0.818814i	$-0.694632\pi$
$983$	−35.9967	−1.14812	−0.574059	+	0.818814i	$0.694632\pi$
$984$	−1.88087	−0.0599598
$985$	2.31105	0.0736363
$986$	4.65698	0.148309
$987$	−0.517762	−0.0164805
$988$	14.0305	0.446368
$989$	38.3128	1.21828
$990$	−0.938095	−0.0298146
$991$	−12.1862	−0.387107	−0.193554	−	0.981090i	$-0.562001\pi$
$991$	−12.1862	−0.387107	−0.193554	+	0.981090i	$0.562001\pi$
$992$	−11.7556	−0.373240
$993$	23.4474	0.744081
$994$	−13.6695	−0.433570
$995$	1.01891	0.0323015
$996$	5.79069	0.183485
$997$	46.9573	1.48715	0.743576	−	0.668651i	$-0.233128\pi$
$997$	46.9573	1.48715	0.743576	+	0.668651i	$0.233128\pi$
$998$	14.9194	0.472267
$999$	−9.39150	−0.297134

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8007.2.a.f.1.19	✓	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8007.2.a.f.1.19	✓	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)

\(f(q)\)	\(=\)	\(q-0.857233 q^{2} -1.00000 q^{3} -1.26515 q^{4} +0.241794 q^{5} +0.857233 q^{6} -3.40893 q^{7} +2.79900 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.857233 q^{2} -1.00000 q^{3} -1.26515 q^{4} +0.241794 q^{5} +0.857233 q^{6} -3.40893 q^{7} +2.79900 q^{8} +1.00000 q^{9} -0.207274 q^{10} +4.52587 q^{11} +1.26515 q^{12} -3.66863 q^{13} +2.92225 q^{14} -0.241794 q^{15} +0.130910 q^{16} -1.00000 q^{17} -0.857233 q^{18} +3.02291 q^{19} -0.305906 q^{20} +3.40893 q^{21} -3.87973 q^{22} -6.34192 q^{23} -2.79900 q^{24} -4.94154 q^{25} +3.14487 q^{26} -1.00000 q^{27} +4.31281 q^{28} +5.43257 q^{29} +0.207274 q^{30} +2.05869 q^{31} -5.71021 q^{32} -4.52587 q^{33} +0.857233 q^{34} -0.824258 q^{35} -1.26515 q^{36} +9.39150 q^{37} -2.59134 q^{38} +3.66863 q^{39} +0.676781 q^{40} +0.671979 q^{41} -2.92225 q^{42} -6.04121 q^{43} -5.72591 q^{44} +0.241794 q^{45} +5.43650 q^{46} -0.151884 q^{47} -0.130910 q^{48} +4.62079 q^{49} +4.23605 q^{50} +1.00000 q^{51} +4.64137 q^{52} -8.36573 q^{53} +0.857233 q^{54} +1.09433 q^{55} -9.54158 q^{56} -3.02291 q^{57} -4.65698 q^{58} +1.97010 q^{59} +0.305906 q^{60} +0.358392 q^{61} -1.76478 q^{62} -3.40893 q^{63} +4.63316 q^{64} -0.887052 q^{65} +3.87973 q^{66} +1.12443 q^{67} +1.26515 q^{68} +6.34192 q^{69} +0.706582 q^{70} -4.67774 q^{71} +2.79900 q^{72} +0.743140 q^{73} -8.05070 q^{74} +4.94154 q^{75} -3.82444 q^{76} -15.4284 q^{77} -3.14487 q^{78} +14.4989 q^{79} +0.0316532 q^{80} +1.00000 q^{81} -0.576043 q^{82} +4.57707 q^{83} -4.31281 q^{84} -0.241794 q^{85} +5.17872 q^{86} -5.43257 q^{87} +12.6679 q^{88} +1.89104 q^{89} -0.207274 q^{90} +12.5061 q^{91} +8.02349 q^{92} -2.05869 q^{93} +0.130200 q^{94} +0.730922 q^{95} +5.71021 q^{96} -1.97519 q^{97} -3.96110 q^{98} +4.52587 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9}+O(q^{10}) \) 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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