LMFDB - Embedded newform 8043.2.a.u.1.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8043,2,Mod(1,8043)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8043, 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("8043.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8043 = 3 \cdot 7 \cdot 383 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8043.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:	$64.2236783457$
Analytic rank:	$0$
Dimension:	$53$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
Character	$\chi$	$=$	8043.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.969084 q^{2} +1.00000 q^{3} -1.06088 q^{4} -0.461084 q^{5} -0.969084 q^{6} +1.00000 q^{7} +2.96625 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.969084 q^{2} +1.00000 q^{3} -1.06088 q^{4} -0.461084 q^{5} -0.969084 q^{6} +1.00000 q^{7} +2.96625 q^{8} +1.00000 q^{9} +0.446829 q^{10} +4.32229 q^{11} -1.06088 q^{12} +5.50622 q^{13} -0.969084 q^{14} -0.461084 q^{15} -0.752789 q^{16} +5.02002 q^{17} -0.969084 q^{18} -3.51824 q^{19} +0.489153 q^{20} +1.00000 q^{21} -4.18867 q^{22} -5.24053 q^{23} +2.96625 q^{24} -4.78740 q^{25} -5.33599 q^{26} +1.00000 q^{27} -1.06088 q^{28} +9.33643 q^{29} +0.446829 q^{30} +9.55936 q^{31} -5.20298 q^{32} +4.32229 q^{33} -4.86482 q^{34} -0.461084 q^{35} -1.06088 q^{36} +9.90949 q^{37} +3.40947 q^{38} +5.50622 q^{39} -1.36769 q^{40} +3.87811 q^{41} -0.969084 q^{42} +9.77414 q^{43} -4.58542 q^{44} -0.461084 q^{45} +5.07851 q^{46} -12.0051 q^{47} -0.752789 q^{48} +1.00000 q^{49} +4.63939 q^{50} +5.02002 q^{51} -5.84142 q^{52} +9.35265 q^{53} -0.969084 q^{54} -1.99294 q^{55} +2.96625 q^{56} -3.51824 q^{57} -9.04779 q^{58} +1.62695 q^{59} +0.489153 q^{60} -10.7868 q^{61} -9.26382 q^{62} +1.00000 q^{63} +6.54770 q^{64} -2.53883 q^{65} -4.18867 q^{66} +5.24868 q^{67} -5.32562 q^{68} -5.24053 q^{69} +0.446829 q^{70} +11.0121 q^{71} +2.96625 q^{72} -5.20146 q^{73} -9.60312 q^{74} -4.78740 q^{75} +3.73242 q^{76} +4.32229 q^{77} -5.33599 q^{78} -4.78206 q^{79} +0.347099 q^{80} +1.00000 q^{81} -3.75822 q^{82} -9.19519 q^{83} -1.06088 q^{84} -2.31465 q^{85} -9.47197 q^{86} +9.33643 q^{87} +12.8210 q^{88} +5.96287 q^{89} +0.446829 q^{90} +5.50622 q^{91} +5.55955 q^{92} +9.55936 q^{93} +11.6339 q^{94} +1.62221 q^{95} -5.20298 q^{96} +14.7926 q^{97} -0.969084 q^{98} +4.32229 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 53 q + 11 q^{2} + 53 q^{3} + 63 q^{4} + 24 q^{5} + 11 q^{6} + 53 q^{7} + 30 q^{8} + 53 q^{9}+O(q^{10}) $ 53 * q + 11 * q^2 + 53 * q^3 + 63 * q^4 + 24 * q^5 + 11 * q^6 + 53 * q^7 + 30 * q^8 + 53 * q^9	\( 53 q + 11 q^{2} + 53 q^{3} + 63 q^{4} + 24 q^{5} + 11 q^{6} + 53 q^{7} + 30 q^{8} + 53 q^{9} + 2 q^{10} + 46 q^{11} + 63 q^{12} + 32 q^{13} + 11 q^{14} + 24 q^{15} + 67 q^{16} + 46 q^{17} + 11 q^{18} + 14 q^{19} + 53 q^{20} + 53 q^{21} + 13 q^{22} + 68 q^{23} + 30 q^{24} + 71 q^{25} + 11 q^{26} + 53 q^{27} + 63 q^{28} + 55 q^{29} + 2 q^{30} - 2 q^{31} + 51 q^{32} + 46 q^{33} - 7 q^{34} + 24 q^{35} + 63 q^{36} + 53 q^{37} + 16 q^{38} + 32 q^{39} - 20 q^{40} + 38 q^{41} + 11 q^{42} + 36 q^{43} + 70 q^{44} + 24 q^{45} + 4 q^{46} + 51 q^{47} + 67 q^{48} + 53 q^{49} + 32 q^{50} + 46 q^{51} + 10 q^{52} + 104 q^{53} + 11 q^{54} + 11 q^{55} + 30 q^{56} + 14 q^{57} + 4 q^{58} + 36 q^{59} + 53 q^{60} + 3 q^{61} + 25 q^{62} + 53 q^{63} + 82 q^{64} + 46 q^{65} + 13 q^{66} + 54 q^{67} + 88 q^{68} + 68 q^{69} + 2 q^{70} + 101 q^{71} + 30 q^{72} + q^{73} + 32 q^{74} + 71 q^{75} - 35 q^{76} + 46 q^{77} + 11 q^{78} + 14 q^{79} + 39 q^{80} + 53 q^{81} - 29 q^{82} + 38 q^{83} + 63 q^{84} + 16 q^{85} + 23 q^{86} + 55 q^{87} - 8 q^{88} + 52 q^{89} + 2 q^{90} + 32 q^{91} + 76 q^{92} - 2 q^{93} - 53 q^{94} + 46 q^{95} + 51 q^{96} - 3 q^{97} + 11 q^{98} + 46 q^{99}+O(q^{100}) \) 53 * q + 11 * q^2 + 53 * q^3 + 63 * q^4 + 24 * q^5 + 11 * q^6 + 53 * q^7 + 30 * q^8 + 53 * q^9 + 2 * q^10 + 46 * q^11 + 63 * q^12 + 32 * q^13 + 11 * q^14 + 24 * q^15 + 67 * q^16 + 46 * q^17 + 11 * q^18 + 14 * q^19 + 53 * q^20 + 53 * q^21 + 13 * q^22 + 68 * q^23 + 30 * q^24 + 71 * q^25 + 11 * q^26 + 53 * q^27 + 63 * q^28 + 55 * q^29 + 2 * q^30 - 2 * q^31 + 51 * q^32 + 46 * q^33 - 7 * q^34 + 24 * q^35 + 63 * q^36 + 53 * q^37 + 16 * q^38 + 32 * q^39 - 20 * q^40 + 38 * q^41 + 11 * q^42 + 36 * q^43 + 70 * q^44 + 24 * q^45 + 4 * q^46 + 51 * q^47 + 67 * q^48 + 53 * q^49 + 32 * q^50 + 46 * q^51 + 10 * q^52 + 104 * q^53 + 11 * q^54 + 11 * q^55 + 30 * q^56 + 14 * q^57 + 4 * q^58 + 36 * q^59 + 53 * q^60 + 3 * q^61 + 25 * q^62 + 53 * q^63 + 82 * q^64 + 46 * q^65 + 13 * q^66 + 54 * q^67 + 88 * q^68 + 68 * q^69 + 2 * q^70 + 101 * q^71 + 30 * q^72 + q^73 + 32 * q^74 + 71 * q^75 - 35 * q^76 + 46 * q^77 + 11 * q^78 + 14 * q^79 + 39 * q^80 + 53 * q^81 - 29 * q^82 + 38 * q^83 + 63 * q^84 + 16 * q^85 + 23 * q^86 + 55 * q^87 - 8 * q^88 + 52 * q^89 + 2 * q^90 + 32 * q^91 + 76 * q^92 - 2 * q^93 - 53 * q^94 + 46 * q^95 + 51 * q^96 - 3 * q^97 + 11 * q^98 + 46 * 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 $

$2$	−0.969084	−0.685246	−0.342623	−	0.939473i	$-0.611315\pi$
$2$	−0.969084	−0.685246	−0.342623	+	0.939473i	$0.611315\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.06088	−0.530438
$5$	−0.461084	−0.206203	−0.103102	−	0.994671i	$-0.532877\pi$
$5$	−0.461084	−0.206203	−0.103102	+	0.994671i	$0.532877\pi$
$6$	−0.969084	−0.395627
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	2.96625	1.04873
$9$	1.00000	0.333333
$10$	0.446829	0.141300
$11$	4.32229	1.30322	0.651610	−	0.758554i	$-0.274094\pi$
$11$	4.32229	1.30322	0.651610	+	0.758554i	$0.274094\pi$
$12$	−1.06088	−0.306249
$13$	5.50622	1.52715	0.763575	−	0.645719i	$-0.223442\pi$
$13$	5.50622	1.52715	0.763575	+	0.645719i	$0.223442\pi$
$14$	−0.969084	−0.258999
$15$	−0.461084	−0.119051
$16$	−0.752789	−0.188197
$17$	5.02002	1.21753	0.608766	−	0.793349i	$-0.291665\pi$
$17$	5.02002	1.21753	0.608766	+	0.793349i	$0.291665\pi$
$18$	−0.969084	−0.228415
$19$	−3.51824	−0.807140	−0.403570	−	0.914949i	$-0.632231\pi$
$19$	−3.51824	−0.807140	−0.403570	+	0.914949i	$0.632231\pi$
$20$	0.489153	0.109378
$21$	1.00000	0.218218
$22$	−4.18867	−0.893027
$23$	−5.24053	−1.09273	−0.546363	−	0.837549i	$-0.683988\pi$
$23$	−5.24053	−1.09273	−0.546363	+	0.837549i	$0.683988\pi$
$24$	2.96625	0.605482
$25$	−4.78740	−0.957480
$26$	−5.33599	−1.04647
$27$	1.00000	0.192450
$28$	−1.06088	−0.200487
$29$	9.33643	1.73373	0.866866	−	0.498541i	$-0.166131\pi$
$29$	9.33643	1.73373	0.866866	+	0.498541i	$0.166131\pi$
$30$	0.446829	0.0815795
$31$	9.55936	1.71691	0.858456	−	0.512888i	$-0.171424\pi$
$31$	9.55936	1.71691	0.858456	+	0.512888i	$0.171424\pi$
$32$	−5.20298	−0.919765
$33$	4.32229	0.752415
$34$	−4.86482	−0.834309
$35$	−0.461084	−0.0779375
$36$	−1.06088	−0.176813
$37$	9.90949	1.62911	0.814555	−	0.580087i	$-0.196981\pi$
$37$	9.90949	1.62911	0.814555	+	0.580087i	$0.196981\pi$
$38$	3.40947	0.553089
$39$	5.50622	0.881701
$40$	−1.36769	−0.216251
$41$	3.87811	0.605660	0.302830	−	0.953045i	$-0.402069\pi$
$41$	3.87811	0.605660	0.302830	+	0.953045i	$0.402069\pi$
$42$	−0.969084	−0.149533
$43$	9.77414	1.49054	0.745271	−	0.666761i	$-0.232320\pi$
$43$	9.77414	1.49054	0.745271	+	0.666761i	$0.232320\pi$
$44$	−4.58542	−0.691278
$45$	−0.461084	−0.0687344
$46$	5.07851	0.748785
$47$	−12.0051	−1.75112	−0.875560	−	0.483109i	$-0.839507\pi$
$47$	−12.0051	−1.75112	−0.875560	+	0.483109i	$0.839507\pi$
$48$	−0.752789	−0.108656
$49$	1.00000	0.142857
$50$	4.63939	0.656109
$51$	5.02002	0.702943
$52$	−5.84142	−0.810059
$53$	9.35265	1.28469	0.642343	−	0.766418i	$-0.277963\pi$
$53$	9.35265	1.28469	0.642343	+	0.766418i	$0.277963\pi$
$54$	−0.969084	−0.131876
$55$	−1.99294	−0.268728
$56$	2.96625	0.396381
$57$	−3.51824	−0.466003
$58$	−9.04779	−1.18803
$59$	1.62695	0.211811	0.105905	−	0.994376i	$-0.466226\pi$
$59$	1.62695	0.211811	0.105905	+	0.994376i	$0.466226\pi$
$60$	0.489153	0.0631494
$61$	−10.7868	−1.38111	−0.690553	−	0.723281i	$-0.742633\pi$
$61$	−10.7868	−1.38111	−0.690553	+	0.723281i	$0.742633\pi$
$62$	−9.26382	−1.17651
$63$	1.00000	0.125988
$64$	6.54770	0.818462
$65$	−2.53883	−0.314903
$66$	−4.18867	−0.515589
$67$	5.24868	0.641228	0.320614	−	0.947210i	$-0.396111\pi$
$67$	5.24868	0.641228	0.320614	+	0.947210i	$0.396111\pi$
$68$	−5.32562	−0.645826
$69$	−5.24053	−0.630885
$70$	0.446829	0.0534063
$71$	11.0121	1.30690	0.653449	−	0.756971i	$-0.273322\pi$
$71$	11.0121	1.30690	0.653449	+	0.756971i	$0.273322\pi$
$72$	2.96625	0.349575
$73$	−5.20146	−0.608784	−0.304392	−	0.952547i	$-0.598453\pi$
$73$	−5.20146	−0.608784	−0.304392	+	0.952547i	$0.598453\pi$
$74$	−9.60312	−1.11634
$75$	−4.78740	−0.552801
$76$	3.73242	0.428138
$77$	4.32229	0.492571
$78$	−5.33599	−0.604182
$79$	−4.78206	−0.538024	−0.269012	−	0.963137i	$-0.586697\pi$
$79$	−4.78206	−0.538024	−0.269012	+	0.963137i	$0.586697\pi$
$80$	0.347099	0.0388069
$81$	1.00000	0.111111
$82$	−3.75822	−0.415026
$83$	−9.19519	−1.00930	−0.504651	−	0.863323i	$-0.668379\pi$
$83$	−9.19519	−1.00930	−0.504651	+	0.863323i	$0.668379\pi$
$84$	−1.06088	−0.115751
$85$	−2.31465	−0.251059
$86$	−9.47197	−1.02139
$87$	9.33643	1.00097
$88$	12.8210	1.36672
$89$	5.96287	0.632063	0.316032	−	0.948749i	$-0.397649\pi$
$89$	5.96287	0.632063	0.316032	+	0.948749i	$0.397649\pi$
$90$	0.446829	0.0471000
$91$	5.50622	0.577208
$92$	5.55955	0.579623
$93$	9.55936	0.991259
$94$	11.6339	1.19995
$95$	1.62221	0.166435
$96$	−5.20298	−0.531027
$97$	14.7926	1.50196	0.750979	−	0.660326i	$-0.229582\pi$
$97$	14.7926	1.50196	0.750979	+	0.660326i	$0.229582\pi$
$98$	−0.969084	−0.0978923
$99$	4.32229	0.434407
$100$	5.07884	0.507884
$101$	−11.1905	−1.11349	−0.556746	−	0.830683i	$-0.687950\pi$
$101$	−11.1905	−1.11349	−0.556746	+	0.830683i	$0.687950\pi$
$102$	−4.86482	−0.481689
$103$	0.335454	0.0330533	0.0165266	−	0.999863i	$-0.494739\pi$
$103$	0.335454	0.0330533	0.0165266	+	0.999863i	$0.494739\pi$
$104$	16.3328	1.60156
$105$	−0.461084	−0.0449972
$106$	−9.06350	−0.880325
$107$	−2.89729	−0.280091	−0.140046	−	0.990145i	$-0.544725\pi$
$107$	−2.89729	−0.280091	−0.140046	+	0.990145i	$0.544725\pi$
$108$	−1.06088	−0.102083
$109$	0.0198558	0.00190184	0.000950918	−	1.00000i	$-0.499697\pi$
$109$	0.0198558	0.00190184	0.000950918		1.00000i	$0.499697\pi$
$110$	1.93133	0.184145
$111$	9.90949	0.940567
$112$	−0.752789	−0.0711319
$113$	7.72510	0.726716	0.363358	−	0.931650i	$-0.381630\pi$
$113$	7.72510	0.726716	0.363358	+	0.931650i	$0.381630\pi$
$114$	3.40947	0.319326
$115$	2.41632	0.225323
$116$	−9.90480	−0.919637
$117$	5.50622	0.509050
$118$	−1.57665	−0.145142
$119$	5.02002	0.460184
$120$	−1.36769	−0.124852
$121$	7.68222	0.698384
$122$	10.4533	0.946398
$123$	3.87811	0.349678
$124$	−10.1413	−0.910715
$125$	4.51282	0.403639
$126$	−0.969084	−0.0863329
$127$	−12.0398	−1.06836	−0.534180	−	0.845371i	$-0.679380\pi$
$127$	−12.0398	−1.06836	−0.534180	+	0.845371i	$0.679380\pi$
$128$	4.06068	0.358917
$129$	9.77414	0.860565
$130$	2.46034	0.215786
$131$	−20.5610	−1.79642	−0.898211	−	0.439565i	$-0.855133\pi$
$131$	−20.5610	−1.79642	−0.898211	+	0.439565i	$0.855133\pi$
$132$	−4.58542	−0.399109
$133$	−3.51824	−0.305070
$134$	−5.08641	−0.439399
$135$	−0.461084	−0.0396838
$136$	14.8906	1.27686
$137$	−14.6234	−1.24936	−0.624679	−	0.780882i	$-0.714770\pi$
$137$	−14.6234	−1.24936	−0.624679	+	0.780882i	$0.714770\pi$
$138$	5.07851	0.432311
$139$	1.61302	0.136815	0.0684075	−	0.997657i	$-0.478208\pi$
$139$	1.61302	0.136815	0.0684075	+	0.997657i	$0.478208\pi$
$140$	0.489153	0.0413410
$141$	−12.0051	−1.01101
$142$	−10.6717	−0.895546
$143$	23.7995	1.99021
$144$	−0.752789	−0.0627324
$145$	−4.30488	−0.357501
$146$	5.04065	0.417167
$147$	1.00000	0.0824786
$148$	−10.5127	−0.864142
$149$	−3.53148	−0.289310	−0.144655	−	0.989482i	$-0.546207\pi$
$149$	−3.53148	−0.289310	−0.144655	+	0.989482i	$0.546207\pi$
$150$	4.63939	0.378805
$151$	−15.6375	−1.27256	−0.636281	−	0.771458i	$-0.719528\pi$
$151$	−15.6375	−1.27256	−0.636281	+	0.771458i	$0.719528\pi$
$152$	−10.4360	−0.846469
$153$	5.02002	0.405844
$154$	−4.18867	−0.337532
$155$	−4.40767	−0.354032
$156$	−5.84142	−0.467688
$157$	−21.6759	−1.72992	−0.864961	−	0.501839i	$-0.832657\pi$
$157$	−21.6759	−1.72992	−0.864961	+	0.501839i	$0.832657\pi$
$158$	4.63422	0.368679
$159$	9.35265	0.741713
$160$	2.39901	0.189658
$161$	−5.24053	−0.413011
$162$	−0.969084	−0.0761384
$163$	−3.63012	−0.284333	−0.142166	−	0.989843i	$-0.545407\pi$
$163$	−3.63012	−0.284333	−0.142166	+	0.989843i	$0.545407\pi$
$164$	−4.11420	−0.321265
$165$	−1.99294	−0.155150
$166$	8.91091	0.691621
$167$	−19.3789	−1.49959	−0.749793	−	0.661672i	$-0.769847\pi$
$167$	−19.3789	−1.49959	−0.749793	+	0.661672i	$0.769847\pi$
$168$	2.96625	0.228851
$169$	17.3184	1.33219
$170$	2.24309	0.172037
$171$	−3.51824	−0.269047
$172$	−10.3692	−0.790641
$173$	1.87775	0.142762	0.0713812	−	0.997449i	$-0.477259\pi$
$173$	1.87775	0.142762	0.0713812	+	0.997449i	$0.477259\pi$
$174$	−9.04779	−0.685911
$175$	−4.78740	−0.361894
$176$	−3.25378	−0.245263
$177$	1.62695	0.122289
$178$	−5.77853	−0.433119
$179$	−11.9345	−0.892026	−0.446013	−	0.895026i	$-0.647156\pi$
$179$	−11.9345	−0.892026	−0.446013	+	0.895026i	$0.647156\pi$
$180$	0.489153	0.0364593
$181$	−23.4523	−1.74319	−0.871596	−	0.490225i	$-0.836915\pi$
$181$	−23.4523	−1.74319	−0.871596	+	0.490225i	$0.836915\pi$
$182$	−5.33599	−0.395530
$183$	−10.7868	−0.797382
$184$	−15.5447	−1.14597
$185$	−4.56911	−0.335927
$186$	−9.26382	−0.679256
$187$	21.6980	1.58671
$188$	12.7359	0.928861
$189$	1.00000	0.0727393
$190$	−1.57205	−0.114049
$191$	7.38179	0.534128	0.267064	−	0.963679i	$-0.413947\pi$
$191$	7.38179	0.534128	0.267064	+	0.963679i	$0.413947\pi$
$192$	6.54770	0.472539
$193$	6.49300	0.467376	0.233688	−	0.972312i	$-0.424921\pi$
$193$	6.49300	0.467376	0.233688	+	0.972312i	$0.424921\pi$
$194$	−14.3352	−1.02921
$195$	−2.53883	−0.181809
$196$	−1.06088	−0.0757769
$197$	13.0777	0.931747	0.465874	−	0.884851i	$-0.345740\pi$
$197$	13.0777	0.931747	0.465874	+	0.884851i	$0.345740\pi$
$198$	−4.18867	−0.297676
$199$	−24.1920	−1.71493	−0.857463	−	0.514546i	$-0.827961\pi$
$199$	−24.1920	−1.71493	−0.857463	+	0.514546i	$0.827961\pi$
$200$	−14.2006	−1.00413
$201$	5.24868	0.370213
$202$	10.8445	0.763016
$203$	9.33643	0.655289
$204$	−5.32562	−0.372868
$205$	−1.78814	−0.124889
$206$	−0.325083	−0.0226496
$207$	−5.24053	−0.364242
$208$	−4.14502	−0.287405
$209$	−15.2069	−1.05188
$210$	0.446829	0.0308342
$211$	−25.5996	−1.76235	−0.881174	−	0.472792i	$-0.843246\pi$
$211$	−25.5996	−1.76235	−0.881174	+	0.472792i	$0.843246\pi$
$212$	−9.92200	−0.681446
$213$	11.0121	0.754537
$214$	2.80771	0.191931
$215$	−4.50670	−0.307355
$216$	2.96625	0.201827
$217$	9.55936	0.648931
$218$	−0.0192419	−0.00130323
$219$	−5.20146	−0.351482
$220$	2.11426	0.142544
$221$	27.6413	1.85936
$222$	−9.60312	−0.644519
$223$	5.13317	0.343743	0.171871	−	0.985119i	$-0.445019\pi$
$223$	5.13317	0.343743	0.171871	+	0.985119i	$0.445019\pi$
$224$	−5.20298	−0.347638
$225$	−4.78740	−0.319160
$226$	−7.48627	−0.497979
$227$	−6.88314	−0.456850	−0.228425	−	0.973562i	$-0.573358\pi$
$227$	−6.88314	−0.456850	−0.228425	+	0.973562i	$0.573358\pi$
$228$	3.73242	0.247186
$229$	28.0231	1.85182	0.925910	−	0.377744i	$-0.123300\pi$
$229$	28.0231	1.85182	0.925910	+	0.377744i	$0.123300\pi$
$230$	−2.34162	−0.154402
$231$	4.32229	0.284386
$232$	27.6942	1.81821
$233$	−19.2923	−1.26388	−0.631939	−	0.775018i	$-0.717741\pi$
$233$	−19.2923	−1.26388	−0.631939	+	0.775018i	$0.717741\pi$
$234$	−5.33599	−0.348824
$235$	5.53535	0.361086
$236$	−1.72599	−0.112352
$237$	−4.78206	−0.310628
$238$	−4.86482	−0.315339
$239$	21.3950	1.38393	0.691965	−	0.721932i	$-0.256745\pi$
$239$	21.3950	1.38393	0.691965	+	0.721932i	$0.256745\pi$
$240$	0.347099	0.0224052
$241$	13.6368	0.878422	0.439211	−	0.898384i	$-0.355258\pi$
$241$	13.6368	0.878422	0.439211	+	0.898384i	$0.355258\pi$
$242$	−7.44472	−0.478565
$243$	1.00000	0.0641500
$244$	11.4434	0.732592
$245$	−0.461084	−0.0294576
$246$	−3.75822	−0.239615
$247$	−19.3722	−1.23262
$248$	28.3554	1.80057
$249$	−9.19519	−0.582721
$250$	−4.37330	−0.276592
$251$	18.9292	1.19480	0.597399	−	0.801944i	$-0.296201\pi$
$251$	18.9292	1.19480	0.597399	+	0.801944i	$0.296201\pi$
$252$	−1.06088	−0.0668289
$253$	−22.6511	−1.42406
$254$	11.6676	0.732089
$255$	−2.31465	−0.144949
$256$	−17.0305	−1.06441
$257$	29.1927	1.82099	0.910495	−	0.413521i	$-0.135701\pi$
$257$	29.1927	1.82099	0.910495	+	0.413521i	$0.135701\pi$
$258$	−9.47197	−0.589699
$259$	9.90949	0.615745
$260$	2.69338	0.167037
$261$	9.33643	0.577911
$262$	19.9253	1.23099
$263$	−8.82358	−0.544085	−0.272043	−	0.962285i	$-0.587699\pi$
$263$	−8.82358	−0.544085	−0.272043	+	0.962285i	$0.587699\pi$
$264$	12.8210	0.789077
$265$	−4.31236	−0.264906
$266$	3.40947	0.209048
$267$	5.96287	0.364922
$268$	−5.56820	−0.340132
$269$	14.1948	0.865475	0.432737	−	0.901520i	$-0.357548\pi$
$269$	14.1948	0.865475	0.432737	+	0.901520i	$0.357548\pi$
$270$	0.446829	0.0271932
$271$	−7.71380	−0.468580	−0.234290	−	0.972167i	$-0.575277\pi$
$271$	−7.71380	−0.468580	−0.234290	+	0.972167i	$0.575277\pi$
$272$	−3.77901	−0.229136
$273$	5.50622	0.333251
$274$	14.1713	0.856117
$275$	−20.6926	−1.24781
$276$	5.55955	0.334646
$277$	−8.09674	−0.486486	−0.243243	−	0.969965i	$-0.578211\pi$
$277$	−8.09674	−0.486486	−0.243243	+	0.969965i	$0.578211\pi$
$278$	−1.56316	−0.0937519
$279$	9.55936	0.572304
$280$	−1.36769	−0.0817351
$281$	−5.76636	−0.343992	−0.171996	−	0.985098i	$-0.555022\pi$
$281$	−5.76636	−0.343992	−0.171996	+	0.985098i	$0.555022\pi$
$282$	11.6339	0.692790
$283$	28.0252	1.66592	0.832962	−	0.553330i	$-0.186643\pi$
$283$	28.0252	1.66592	0.832962	+	0.553330i	$0.186643\pi$
$284$	−11.6825	−0.693228
$285$	1.62221	0.0960912
$286$	−23.0637	−1.36379
$287$	3.87811	0.228918
$288$	−5.20298	−0.306588
$289$	8.20057	0.482387
$290$	4.17179	0.244976
$291$	14.7926	0.867156
$292$	5.51810	0.322922
$293$	23.7636	1.38828	0.694142	−	0.719838i	$-0.255784\pi$
$293$	23.7636	1.38828	0.694142	+	0.719838i	$0.255784\pi$
$294$	−0.969084	−0.0565181
$295$	−0.750160	−0.0436760
$296$	29.3940	1.70849
$297$	4.32229	0.250805
$298$	3.42230	0.198249
$299$	−28.8555	−1.66876
$300$	5.07884	0.293227
$301$	9.77414	0.563372
$302$	15.1540	0.872017
$303$	−11.1905	−0.642875
$304$	2.64849	0.151902
$305$	4.97362	0.284789
$306$	−4.86482	−0.278103
$307$	−18.7582	−1.07059	−0.535293	−	0.844667i	$-0.679799\pi$
$307$	−18.7582	−1.07059	−0.535293	+	0.844667i	$0.679799\pi$
$308$	−4.58542	−0.261278
$309$	0.335454	0.0190833
$310$	4.27140	0.242599
$311$	9.10962	0.516559	0.258279	−	0.966070i	$-0.416844\pi$
$311$	9.10962	0.516559	0.258279	+	0.966070i	$0.416844\pi$
$312$	16.3328	0.924663
$313$	−33.1480	−1.87363	−0.936817	−	0.349820i	$-0.886243\pi$
$313$	−33.1480	−1.87363	−0.936817	+	0.349820i	$0.886243\pi$
$314$	21.0057	1.18542
$315$	−0.461084	−0.0259792
$316$	5.07317	0.285388
$317$	17.1804	0.964945	0.482472	−	0.875911i	$-0.339739\pi$
$317$	17.1804	0.964945	0.482472	+	0.875911i	$0.339739\pi$
$318$	−9.06350	−0.508256
$319$	40.3548	2.25944
$320$	−3.01904	−0.168770
$321$	−2.89729	−0.161711
$322$	5.07851	0.283014
$323$	−17.6616	−0.982720
$324$	−1.06088	−0.0589376
$325$	−26.3605	−1.46222
$326$	3.51789	0.194838
$327$	0.0198558	0.00109803
$328$	11.5034	0.635171
$329$	−12.0051	−0.661861
$330$	1.93133	0.106316
$331$	−10.7801	−0.592528	−0.296264	−	0.955106i	$-0.595741\pi$
$331$	−10.7801	−0.592528	−0.296264	+	0.955106i	$0.595741\pi$
$332$	9.75495	0.535373
$333$	9.90949	0.543036
$334$	18.7798	1.02759
$335$	−2.42008	−0.132223
$336$	−0.752789	−0.0410680
$337$	3.15540	0.171885	0.0859427	−	0.996300i	$-0.472610\pi$
$337$	3.15540	0.171885	0.0859427	+	0.996300i	$0.472610\pi$
$338$	−16.7830	−0.912876
$339$	7.72510	0.419570
$340$	2.45556	0.133171
$341$	41.3183	2.23751
$342$	3.40947	0.184363
$343$	1.00000	0.0539949
$344$	28.9925	1.56317
$345$	2.41632	0.130090
$346$	−1.81969	−0.0978273
$347$	10.6911	0.573929	0.286964	−	0.957941i	$-0.407354\pi$
$347$	10.6911	0.573929	0.286964	+	0.957941i	$0.407354\pi$
$348$	−9.90480	−0.530953
$349$	10.9252	0.584811	0.292405	−	0.956294i	$-0.405544\pi$
$349$	10.9252	0.584811	0.292405	+	0.956294i	$0.405544\pi$
$350$	4.63939	0.247986
$351$	5.50622	0.293900
$352$	−22.4888	−1.19866
$353$	3.03460	0.161516	0.0807578	−	0.996734i	$-0.474266\pi$
$353$	3.03460	0.161516	0.0807578	+	0.996734i	$0.474266\pi$
$354$	−1.57665	−0.0837980
$355$	−5.07751	−0.269486
$356$	−6.32587	−0.335271
$357$	5.02002	0.265687
$358$	11.5655	0.611257
$359$	3.37715	0.178239	0.0891195	−	0.996021i	$-0.471595\pi$
$359$	3.37715	0.178239	0.0891195	+	0.996021i	$0.471595\pi$
$360$	−1.36769	−0.0720836
$361$	−6.62197	−0.348525
$362$	22.7272	1.19452
$363$	7.68222	0.403212
$364$	−5.84142	−0.306173
$365$	2.39831	0.125533
$366$	10.4533	0.546403
$367$	12.7006	0.662967	0.331484	−	0.943461i	$-0.392451\pi$
$367$	12.7006	0.662967	0.331484	+	0.943461i	$0.392451\pi$
$368$	3.94501	0.205648
$369$	3.87811	0.201887
$370$	4.42785	0.230193
$371$	9.35265	0.485565
$372$	−10.1413	−0.525802
$373$	33.2490	1.72157	0.860783	−	0.508972i	$-0.169974\pi$
$373$	33.2490	1.72157	0.860783	+	0.508972i	$0.169974\pi$
$374$	−21.0272	−1.08729
$375$	4.51282	0.233041
$376$	−35.6100	−1.83645
$377$	51.4084	2.64767
$378$	−0.969084	−0.0498443
$379$	−23.3520	−1.19951	−0.599755	−	0.800184i	$-0.704735\pi$
$379$	−23.3520	−1.19951	−0.599755	+	0.800184i	$0.704735\pi$
$380$	−1.72096	−0.0882834
$381$	−12.0398	−0.616818
$382$	−7.15358	−0.366009
$383$	1.00000	0.0510976
$383$	1.00000	0.0510976
$384$	4.06068	0.207221
$385$	−1.99294	−0.101570
$386$	−6.29226	−0.320268
$387$	9.77414	0.496848
$388$	−15.6931	−0.796696
$389$	20.9468	1.06205	0.531023	−	0.847357i	$-0.321808\pi$
$389$	20.9468	1.06205	0.531023	+	0.847357i	$0.321808\pi$
$390$	2.46034	0.124584
$391$	−26.3075	−1.33043
$392$	2.96625	0.149818
$393$	−20.5610	−1.03716
$394$	−12.6734	−0.638476
$395$	2.20493	0.110942
$396$	−4.58542	−0.230426
$397$	−4.11684	−0.206618	−0.103309	−	0.994649i	$-0.532943\pi$
$397$	−4.11684	−0.206618	−0.103309	+	0.994649i	$0.532943\pi$
$398$	23.4441	1.17515
$399$	−3.51824	−0.176132
$400$	3.60390	0.180195
$401$	20.9324	1.04532	0.522658	−	0.852543i	$-0.324941\pi$
$401$	20.9324	1.04532	0.522658	+	0.852543i	$0.324941\pi$
$402$	−5.08641	−0.253687
$403$	52.6359	2.62198
$404$	11.8717	0.590639
$405$	−0.461084	−0.0229115
$406$	−9.04779	−0.449034
$407$	42.8317	2.12309
$408$	14.8906	0.737195
$409$	−7.92718	−0.391974	−0.195987	−	0.980607i	$-0.562791\pi$
$409$	−7.92718	−0.391974	−0.195987	+	0.980607i	$0.562791\pi$
$410$	1.73286	0.0855796
$411$	−14.6234	−0.721317
$412$	−0.355875	−0.0175327
$413$	1.62695	0.0800569
$414$	5.07851	0.249595
$415$	4.23976	0.208121
$416$	−28.6487	−1.40462
$417$	1.61302	0.0789901
$418$	14.7367	0.720798
$419$	−21.3547	−1.04325	−0.521623	−	0.853176i	$-0.674673\pi$
$419$	−21.3547	−1.04325	−0.521623	+	0.853176i	$0.674673\pi$
$420$	0.489153	0.0238682
$421$	−15.5229	−0.756539	−0.378270	−	0.925695i	$-0.623481\pi$
$421$	−15.5229	−0.756539	−0.378270	+	0.925695i	$0.623481\pi$
$422$	24.8081	1.20764
$423$	−12.0051	−0.583707
$424$	27.7423	1.34728
$425$	−24.0328	−1.16576
$426$	−10.6717	−0.517044
$427$	−10.7868	−0.522009
$428$	3.07366	0.148571
$429$	23.7995	1.14905
$430$	4.36737	0.210613
$431$	29.6234	1.42691	0.713455	−	0.700701i	$-0.247129\pi$
$431$	29.6234	1.42691	0.713455	+	0.700701i	$0.247129\pi$
$432$	−0.752789	−0.0362186
$433$	12.2527	0.588827	0.294413	−	0.955678i	$-0.404876\pi$
$433$	12.2527	0.588827	0.294413	+	0.955678i	$0.404876\pi$
$434$	−9.26382	−0.444678
$435$	−4.30488	−0.206403
$436$	−0.0210645	−0.00100881
$437$	18.4374	0.881982
$438$	5.04065	0.240851
$439$	24.2360	1.15672	0.578361	−	0.815781i	$-0.303692\pi$
$439$	24.2360	1.15672	0.578361	+	0.815781i	$0.303692\pi$
$440$	−5.91156	−0.281822
$441$	1.00000	0.0476190
$442$	−26.7867	−1.27412
$443$	21.9677	1.04371	0.521857	−	0.853033i	$-0.325239\pi$
$443$	21.9677	1.04371	0.521857	+	0.853033i	$0.325239\pi$
$444$	−10.5127	−0.498912
$445$	−2.74939	−0.130333
$446$	−4.97448	−0.235548
$447$	−3.53148	−0.167033
$448$	6.54770	0.309350
$449$	3.09098	0.145872	0.0729362	−	0.997337i	$-0.476763\pi$
$449$	3.09098	0.145872	0.0729362	+	0.997337i	$0.476763\pi$
$450$	4.63939	0.218703
$451$	16.7624	0.789308
$452$	−8.19537	−0.385478
$453$	−15.6375	−0.734714
$454$	6.67034	0.313055
$455$	−2.53883	−0.119022
$456$	−10.4360	−0.488709
$457$	−14.9003	−0.697009	−0.348504	−	0.937307i	$-0.613310\pi$
$457$	−14.9003	−0.697009	−0.348504	+	0.937307i	$0.613310\pi$
$458$	−27.1568	−1.26895
$459$	5.02002	0.234314
$460$	−2.56342	−0.119520
$461$	22.1201	1.03024	0.515118	−	0.857119i	$-0.327748\pi$
$461$	22.1201	1.03024	0.515118	+	0.857119i	$0.327748\pi$
$462$	−4.18867	−0.194874
$463$	−36.9815	−1.71868	−0.859338	−	0.511408i	$-0.829124\pi$
$463$	−36.9815	−1.71868	−0.859338	+	0.511408i	$0.829124\pi$
$464$	−7.02836	−0.326284
$465$	−4.40767	−0.204401
$466$	18.6958	0.866067
$467$	−15.6366	−0.723575	−0.361788	−	0.932261i	$-0.617833\pi$
$467$	−15.6366	−0.723575	−0.361788	+	0.932261i	$0.617833\pi$
$468$	−5.84142	−0.270020
$469$	5.24868	0.242362
$470$	−5.36422	−0.247433
$471$	−21.6759	−0.998771
$472$	4.82593	0.222131
$473$	42.2467	1.94251
$474$	4.63422	0.212857
$475$	16.8432	0.772821
$476$	−5.32562	−0.244099
$477$	9.35265	0.428228
$478$	−20.7336	−0.948332
$479$	−29.7121	−1.35758	−0.678790	−	0.734332i	$-0.737495\pi$
$479$	−29.7121	−1.35758	−0.678790	+	0.734332i	$0.737495\pi$
$480$	2.39901	0.109499
$481$	54.5638	2.48789
$482$	−13.2152	−0.601935
$483$	−5.24053	−0.238452
$484$	−8.14989	−0.370450
$485$	−6.82062	−0.309708
$486$	−0.969084	−0.0439585
$487$	−5.67644	−0.257224	−0.128612	−	0.991695i	$-0.541052\pi$
$487$	−5.67644	−0.257224	−0.128612	+	0.991695i	$0.541052\pi$
$488$	−31.9963	−1.44840
$489$	−3.63012	−0.164160
$490$	0.446829	0.0201857
$491$	10.5137	0.474478	0.237239	−	0.971451i	$-0.423757\pi$
$491$	10.5137	0.474478	0.237239	+	0.971451i	$0.423757\pi$
$492$	−4.11420	−0.185482
$493$	46.8690	2.11088
$494$	18.7733	0.844651
$495$	−1.99294	−0.0895761
$496$	−7.19618	−0.323118
$497$	11.0121	0.493961
$498$	8.91091	0.399307
$499$	34.9514	1.56464	0.782320	−	0.622876i	$-0.214036\pi$
$499$	34.9514	1.56464	0.782320	+	0.622876i	$0.214036\pi$
$500$	−4.78754	−0.214105
$501$	−19.3789	−0.865787
$502$	−18.3440	−0.818731
$503$	−19.6882	−0.877855	−0.438927	−	0.898523i	$-0.644641\pi$
$503$	−19.6882	−0.877855	−0.438927	+	0.898523i	$0.644641\pi$
$504$	2.96625	0.132127
$505$	5.15974	0.229606
$506$	21.9508	0.975832
$507$	17.3184	0.769139
$508$	12.7727	0.566699
$509$	−17.9929	−0.797523	−0.398762	−	0.917055i	$-0.630560\pi$
$509$	−17.9929	−0.797523	−0.398762	+	0.917055i	$0.630560\pi$
$510$	2.24309	0.0993257
$511$	−5.20146	−0.230099
$512$	8.38266	0.370465
$513$	−3.51824	−0.155334
$514$	−28.2902	−1.24783
$515$	−0.154673	−0.00681569
$516$	−10.3692	−0.456477
$517$	−51.8895	−2.28210
$518$	−9.60312	−0.421937
$519$	1.87775	0.0824239
$520$	−7.53080	−0.330247
$521$	−27.0305	−1.18423	−0.592115	−	0.805854i	$-0.701707\pi$
$521$	−27.0305	−1.18423	−0.592115	+	0.805854i	$0.701707\pi$
$522$	−9.04779	−0.396011
$523$	−10.6423	−0.465354	−0.232677	−	0.972554i	$-0.574748\pi$
$523$	−10.6423	−0.465354	−0.232677	+	0.972554i	$0.574748\pi$
$524$	21.8127	0.952891
$525$	−4.78740	−0.208939
$526$	8.55079	0.372832
$527$	47.9881	2.09040
$528$	−3.25378	−0.141602
$529$	4.46310	0.194048
$530$	4.17904	0.181526
$531$	1.62695	0.0706036
$532$	3.73242	0.161821
$533$	21.3537	0.924933
$534$	−5.77853	−0.250061
$535$	1.33589	0.0577557
$536$	15.5689	0.672473
$537$	−11.9345	−0.515011
$538$	−13.7560	−0.593063
$539$	4.32229	0.186174
$540$	0.489153	0.0210498
$541$	−5.70158	−0.245130	−0.122565	−	0.992460i	$-0.539112\pi$
$541$	−5.70158	−0.245130	−0.122565	+	0.992460i	$0.539112\pi$
$542$	7.47532	0.321093
$543$	−23.4523	−1.00643
$544$	−26.1190	−1.11984
$545$	−0.00915518	−0.000392165 0
$546$	−5.33599	−0.228359
$547$	−18.9209	−0.808998	−0.404499	−	0.914538i	$-0.632554\pi$
$547$	−18.9209	−0.808998	−0.404499	+	0.914538i	$0.632554\pi$
$548$	15.5136	0.662707
$549$	−10.7868	−0.460369
$550$	20.0528	0.855055
$551$	−32.8478	−1.39936
$552$	−15.5447	−0.661626
$553$	−4.78206	−0.203354
$554$	7.84642	0.333362
$555$	−4.56911	−0.193948
$556$	−1.71122	−0.0725718
$557$	−28.0928	−1.19033	−0.595166	−	0.803603i	$-0.702914\pi$
$557$	−28.0928	−1.19033	−0.595166	+	0.803603i	$0.702914\pi$
$558$	−9.26382	−0.392169
$559$	53.8186	2.27628
$560$	0.347099	0.0146676
$561$	21.6980	0.916090
$562$	5.58808	0.235719
$563$	−1.30554	−0.0550219	−0.0275109	−	0.999622i	$-0.508758\pi$
$563$	−1.30554	−0.0550219	−0.0275109	+	0.999622i	$0.508758\pi$
$564$	12.7359	0.536278
$565$	−3.56192	−0.149851
$566$	−27.1588	−1.14157
$567$	1.00000	0.0419961
$568$	32.6646	1.37058
$569$	23.5382	0.986774	0.493387	−	0.869810i	$-0.335759\pi$
$569$	23.5382	0.986774	0.493387	+	0.869810i	$0.335759\pi$
$570$	−1.57205	−0.0658461
$571$	−33.0401	−1.38268	−0.691342	−	0.722527i	$-0.742980\pi$
$571$	−33.0401	−1.38268	−0.691342	+	0.722527i	$0.742980\pi$
$572$	−25.2483	−1.05569
$573$	7.38179	0.308379
$574$	−3.75822	−0.156865
$575$	25.0885	1.04626
$576$	6.54770	0.272821
$577$	−10.8699	−0.452520	−0.226260	−	0.974067i	$-0.572650\pi$
$577$	−10.8699	−0.452520	−0.226260	+	0.974067i	$0.572650\pi$
$578$	−7.94704	−0.330553
$579$	6.49300	0.269840
$580$	4.56695	0.189632
$581$	−9.19519	−0.381481
$582$	−14.3352	−0.594215
$583$	40.4249	1.67423
$584$	−15.4288	−0.638448
$585$	−2.53883	−0.104968
$586$	−23.0289	−0.951316
$587$	−3.48535	−0.143856	−0.0719278	−	0.997410i	$-0.522915\pi$
$587$	−3.48535	−0.143856	−0.0719278	+	0.997410i	$0.522915\pi$
$588$	−1.06088	−0.0437498
$589$	−33.6321	−1.38579
$590$	0.726968	0.0299288
$591$	13.0777	0.537945
$592$	−7.45975	−0.306594
$593$	20.5790	0.845079	0.422540	−	0.906344i	$-0.361139\pi$
$593$	20.5790	0.845079	0.422540	+	0.906344i	$0.361139\pi$
$594$	−4.18867	−0.171863
$595$	−2.31465	−0.0948914
$596$	3.74646	0.153461
$597$	−24.1920	−0.990113
$598$	27.9634	1.14351
$599$	−2.26869	−0.0926963	−0.0463482	−	0.998925i	$-0.514758\pi$
$599$	−2.26869	−0.0926963	−0.0463482	+	0.998925i	$0.514758\pi$
$600$	−14.2006	−0.579738
$601$	23.6250	0.963682	0.481841	−	0.876259i	$-0.339968\pi$
$601$	23.6250	0.963682	0.481841	+	0.876259i	$0.339968\pi$
$602$	−9.47197	−0.386049
$603$	5.24868	0.213743
$604$	16.5894	0.675015
$605$	−3.54215	−0.144009
$606$	10.8445	0.440528
$607$	−16.0639	−0.652013	−0.326006	−	0.945368i	$-0.605703\pi$
$607$	−16.0639	−0.652013	−0.326006	+	0.945368i	$0.605703\pi$
$608$	18.3053	0.742379
$609$	9.33643	0.378331
$610$	−4.81985	−0.195150
$611$	−66.1026	−2.67422
$612$	−5.32562	−0.215275
$613$	−19.8065	−0.799975	−0.399988	−	0.916521i	$-0.630986\pi$
$613$	−19.8065	−0.799975	−0.399988	+	0.916521i	$0.630986\pi$
$614$	18.1782	0.733614
$615$	−1.78814	−0.0721047
$616$	12.8210	0.516572
$617$	−11.2482	−0.452835	−0.226418	−	0.974030i	$-0.572701\pi$
$617$	−11.2482	−0.452835	−0.226418	+	0.974030i	$0.572701\pi$
$618$	−0.325083	−0.0130768
$619$	45.6557	1.83506	0.917528	−	0.397672i	$-0.130182\pi$
$619$	45.6557	1.83506	0.917528	+	0.397672i	$0.130182\pi$
$620$	4.67599	0.187792
$621$	−5.24053	−0.210295
$622$	−8.82798	−0.353970
$623$	5.96287	0.238898
$624$	−4.14502	−0.165934
$625$	21.8562	0.874249
$626$	32.1232	1.28390
$627$	−15.2069	−0.607304
$628$	22.9954	0.917617
$629$	49.7458	1.98349
$630$	0.446829	0.0178021
$631$	21.3913	0.851576	0.425788	−	0.904823i	$-0.359997\pi$
$631$	21.3913	0.851576	0.425788	+	0.904823i	$0.359997\pi$
$632$	−14.1848	−0.564240
$633$	−25.5996	−1.01749
$634$	−16.6492	−0.661225
$635$	5.55137	0.220299
$636$	−9.92200	−0.393433
$637$	5.50622	0.218164
$638$	−39.1072	−1.54827
$639$	11.0121	0.435632
$640$	−1.87232	−0.0740098
$641$	−4.18347	−0.165237	−0.0826185	−	0.996581i	$-0.526328\pi$
$641$	−4.18347	−0.165237	−0.0826185	+	0.996581i	$0.526328\pi$
$642$	2.80771	0.110812
$643$	34.0241	1.34178	0.670891	−	0.741556i	$-0.265912\pi$
$643$	34.0241	1.34178	0.670891	+	0.741556i	$0.265912\pi$
$644$	5.55955	0.219077
$645$	−4.50670	−0.177451
$646$	17.1156	0.673405
$647$	−14.2401	−0.559836	−0.279918	−	0.960024i	$-0.590307\pi$
$647$	−14.2401	−0.559836	−0.279918	+	0.960024i	$0.590307\pi$
$648$	2.96625	0.116525
$649$	7.03215	0.276036
$650$	25.5455	1.00198
$651$	9.55936	0.374661
$652$	3.85111	0.150821
$653$	12.3075	0.481630	0.240815	−	0.970571i	$-0.422585\pi$
$653$	12.3075	0.481630	0.240815	+	0.970571i	$0.422585\pi$
$654$	−0.0192419	−0.000752418 0
$655$	9.48035	0.370428
$656$	−2.91940	−0.113984
$657$	−5.20146	−0.202928
$658$	11.6339	0.453538
$659$	19.8325	0.772563	0.386282	−	0.922381i	$-0.373759\pi$
$659$	19.8325	0.772563	0.386282	+	0.922381i	$0.373759\pi$
$660$	2.11426	0.0822976
$661$	−15.3206	−0.595901	−0.297951	−	0.954581i	$-0.596303\pi$
$661$	−15.3206	−0.595901	−0.297951	+	0.954581i	$0.596303\pi$
$662$	10.4468	0.406027
$663$	27.6413	1.07350
$664$	−27.2752	−1.05848
$665$	1.62221	0.0629065
$666$	−9.60312	−0.372113
$667$	−48.9278	−1.89449
$668$	20.5586	0.795438
$669$	5.13317	0.198460
$670$	2.34526	0.0906055
$671$	−46.6237	−1.79989
$672$	−5.20298	−0.200709
$673$	6.65943	0.256702	0.128351	−	0.991729i	$-0.459032\pi$
$673$	6.65943	0.256702	0.128351	+	0.991729i	$0.459032\pi$
$674$	−3.05784	−0.117784
$675$	−4.78740	−0.184267
$676$	−18.3727	−0.706643
$677$	−27.3632	−1.05165	−0.525826	−	0.850592i	$-0.676244\pi$
$677$	−27.3632	−1.05165	−0.525826	+	0.850592i	$0.676244\pi$
$678$	−7.48627	−0.287508
$679$	14.7926	0.567687
$680$	−6.86582	−0.263292
$681$	−6.88314	−0.263763
$682$	−40.0409	−1.53325
$683$	−24.6691	−0.943939	−0.471969	−	0.881615i	$-0.656457\pi$
$683$	−24.6691	−0.943939	−0.471969	+	0.881615i	$0.656457\pi$
$684$	3.73242	0.142713
$685$	6.74260	0.257621
$686$	−0.969084	−0.0369998
$687$	28.0231	1.06915
$688$	−7.35787	−0.280516
$689$	51.4977	1.96191
$690$	−2.34162	−0.0891440
$691$	6.83859	0.260152	0.130076	−	0.991504i	$-0.458478\pi$
$691$	6.83859	0.260152	0.130076	+	0.991504i	$0.458478\pi$
$692$	−1.99206	−0.0757266
$693$	4.32229	0.164190
$694$	−10.3606	−0.393282
$695$	−0.743740	−0.0282117
$696$	27.6942	1.04974
$697$	19.4682	0.737411
$698$	−10.5874	−0.400739
$699$	−19.2923	−0.729700
$700$	5.07884	0.191962
$701$	−21.0301	−0.794295	−0.397147	−	0.917755i	$-0.630000\pi$
$701$	−21.0301	−0.794295	−0.397147	+	0.917755i	$0.630000\pi$
$702$	−5.33599	−0.201394
$703$	−34.8640	−1.31492
$704$	28.3011	1.06664
$705$	5.53535	0.208473
$706$	−2.94079	−0.110678
$707$	−11.1905	−0.420861
$708$	−1.72599	−0.0648667
$709$	−8.18925	−0.307554	−0.153777	−	0.988106i	$-0.549144\pi$
$709$	−8.18925	−0.307554	−0.153777	+	0.988106i	$0.549144\pi$
$710$	4.92053	0.184664
$711$	−4.78206	−0.179341
$712$	17.6874	0.662862
$713$	−50.0960	−1.87611
$714$	−4.86482	−0.182061
$715$	−10.9736	−0.410388
$716$	12.6610	0.473165
$717$	21.3950	0.799012
$718$	−3.27274	−0.122138
$719$	28.4301	1.06026	0.530132	−	0.847915i	$-0.322142\pi$
$719$	28.4301	1.06026	0.530132	+	0.847915i	$0.322142\pi$
$720$	0.347099	0.0129356
$721$	0.335454	0.0124930
$722$	6.41725	0.238825
$723$	13.6368	0.507157
$724$	24.8799	0.924656
$725$	−44.6972	−1.66001
$726$	−7.44472	−0.276299
$727$	28.3566	1.05169	0.525845	−	0.850581i	$-0.323749\pi$
$727$	28.3566	1.05169	0.525845	+	0.850581i	$0.323749\pi$
$728$	16.3328	0.605334
$729$	1.00000	0.0370370
$730$	−2.32416	−0.0860211
$731$	49.0664	1.81479
$732$	11.4434	0.422962
$733$	−26.5915	−0.982181	−0.491090	−	0.871109i	$-0.663401\pi$
$733$	−26.5915	−0.982181	−0.491090	+	0.871109i	$0.663401\pi$
$734$	−12.3080	−0.454296
$735$	−0.461084	−0.0170073
$736$	27.2663	1.00505
$737$	22.6863	0.835662
$738$	−3.75822	−0.138342
$739$	11.2222	0.412814	0.206407	−	0.978466i	$-0.433823\pi$
$739$	11.2222	0.412814	0.206407	+	0.978466i	$0.433823\pi$
$740$	4.84726	0.178189
$741$	−19.3722	−0.711656
$742$	−9.06350	−0.332732
$743$	−5.92672	−0.217430	−0.108715	−	0.994073i	$-0.534674\pi$
$743$	−5.92672	−0.217430	−0.108715	+	0.994073i	$0.534674\pi$
$744$	28.3554	1.03956
$745$	1.62831	0.0596567
$746$	−32.2210	−1.17970
$747$	−9.19519	−0.336434
$748$	−23.0189	−0.841654
$749$	−2.89729	−0.105864
$750$	−4.37330	−0.159690
$751$	10.3525	0.377769	0.188885	−	0.981999i	$-0.439513\pi$
$751$	10.3525	0.377769	0.188885	+	0.981999i	$0.439513\pi$
$752$	9.03729	0.329556
$753$	18.9292	0.689817
$754$	−49.8191	−1.81430
$755$	7.21020	0.262406
$756$	−1.06088	−0.0385837
$757$	−33.2931	−1.21006	−0.605029	−	0.796203i	$-0.706839\pi$
$757$	−33.2931	−1.21006	−0.605029	+	0.796203i	$0.706839\pi$
$758$	22.6300	0.821959
$759$	−22.6511	−0.822182
$760$	4.81186	0.174545
$761$	35.2838	1.27904	0.639519	−	0.768775i	$-0.279134\pi$
$761$	35.2838	1.27904	0.639519	+	0.768775i	$0.279134\pi$
$762$	11.6676	0.422672
$763$	0.0198558	0.000718827 0
$764$	−7.83117	−0.283322
$765$	−2.31465	−0.0836864
$766$	−0.969084	−0.0350144
$767$	8.95833	0.323467
$768$	−17.0305	−0.614537
$769$	9.91472	0.357534	0.178767	−	0.983891i	$-0.442789\pi$
$769$	9.91472	0.357534	0.178767	+	0.983891i	$0.442789\pi$
$770$	1.93133	0.0696002
$771$	29.1927	1.05135
$772$	−6.88827	−0.247914
$773$	−4.19965	−0.151051	−0.0755255	−	0.997144i	$-0.524063\pi$
$773$	−4.19965	−0.151051	−0.0755255	+	0.997144i	$0.524063\pi$
$774$	−9.47197	−0.340463
$775$	−45.7645	−1.64391
$776$	43.8784	1.57514
$777$	9.90949	0.355501
$778$	−20.2992	−0.727763
$779$	−13.6441	−0.488852
$780$	2.69338	0.0964386
$781$	47.5976	1.70318
$782$	25.4942	0.911671
$783$	9.33643	0.333657
$784$	−0.752789	−0.0268853
$785$	9.99440	0.356715
$786$	19.9253	0.710713
$787$	2.51545	0.0896660	0.0448330	−	0.998994i	$-0.485724\pi$
$787$	2.51545	0.0896660	0.0448330	+	0.998994i	$0.485724\pi$
$788$	−13.8738	−0.494234
$789$	−8.82358	−0.314128
$790$	−2.13676	−0.0760227
$791$	7.72510	0.274673
$792$	12.8210	0.455574
$793$	−59.3944	−2.10916
$794$	3.98957	0.141584
$795$	−4.31236	−0.152944
$796$	25.6647	0.909662
$797$	−2.97719	−0.105458	−0.0527288	−	0.998609i	$-0.516792\pi$
$797$	−2.97719	−0.105458	−0.0527288	+	0.998609i	$0.516792\pi$
$798$	3.40947	0.120694
$799$	−60.2657	−2.13205
$800$	24.9087	0.880657
$801$	5.96287	0.210688
$802$	−20.2853	−0.716298
$803$	−22.4822	−0.793380
$804$	−5.56820	−0.196375
$805$	2.41632	0.0851642
$806$	−51.0086	−1.79670
$807$	14.1948	0.499682
$808$	−33.1937	−1.16775
$809$	26.0261	0.915027	0.457514	−	0.889203i	$-0.348740\pi$
$809$	26.0261	0.915027	0.457514	+	0.889203i	$0.348740\pi$
$810$	0.446829	0.0157000
$811$	−49.3323	−1.73229	−0.866146	−	0.499792i	$-0.833410\pi$
$811$	−49.3323	−1.73229	−0.866146	+	0.499792i	$0.833410\pi$
$812$	−9.90480	−0.347590
$813$	−7.71380	−0.270535
$814$	−41.5075	−1.45484
$815$	1.67379	0.0586303
$816$	−3.77901	−0.132292
$817$	−34.3878	−1.20308
$818$	7.68210	0.268598
$819$	5.50622	0.192403
$820$	1.89699	0.0662459
$821$	6.98969	0.243942	0.121971	−	0.992534i	$-0.461079\pi$
$821$	6.98969	0.243942	0.121971	+	0.992534i	$0.461079\pi$
$822$	14.1713	0.494279
$823$	48.8382	1.70239	0.851197	−	0.524846i	$-0.175877\pi$
$823$	48.8382	1.70239	0.851197	+	0.524846i	$0.175877\pi$
$824$	0.995040	0.0346638
$825$	−20.6926	−0.720422
$826$	−1.57665	−0.0548587
$827$	3.95608	0.137566	0.0687831	−	0.997632i	$-0.478088\pi$
$827$	3.95608	0.137566	0.0687831	+	0.997632i	$0.478088\pi$
$828$	5.55955	0.193208
$829$	−43.5508	−1.51258	−0.756291	−	0.654235i	$-0.772991\pi$
$829$	−43.5508	−1.51258	−0.756291	+	0.654235i	$0.772991\pi$
$830$	−4.10868	−0.142614
$831$	−8.09674	−0.280873
$832$	36.0531	1.24991
$833$	5.02002	0.173933
$834$	−1.56316	−0.0541277
$835$	8.93532	0.309219
$836$	16.1326	0.557958
$837$	9.55936	0.330420
$838$	20.6945	0.714879
$839$	−3.54516	−0.122393	−0.0611963	−	0.998126i	$-0.519492\pi$
$839$	−3.54516	−0.122393	−0.0611963	+	0.998126i	$0.519492\pi$
$840$	−1.36769	−0.0471898
$841$	58.1690	2.00583
$842$	15.0430	0.518415
$843$	−5.76636	−0.198604
$844$	27.1580	0.934816
$845$	−7.98526	−0.274701
$846$	11.6339	0.399983
$847$	7.68222	0.263964
$848$	−7.04057	−0.241774
$849$	28.0252	0.961822
$850$	23.2898	0.798835
$851$	−51.9309	−1.78017
$852$	−11.6825	−0.400235
$853$	−14.3958	−0.492903	−0.246451	−	0.969155i	$-0.579265\pi$
$853$	−14.3958	−0.492903	−0.246451	+	0.969155i	$0.579265\pi$
$854$	10.4533	0.357705
$855$	1.62221	0.0554783
$856$	−8.59406	−0.293739
$857$	−34.1357	−1.16605	−0.583027	−	0.812453i	$-0.698132\pi$
$857$	−34.1357	−1.16605	−0.583027	+	0.812453i	$0.698132\pi$
$858$	−23.0637	−0.787382
$859$	28.5536	0.974235	0.487118	−	0.873336i	$-0.338048\pi$
$859$	28.5536	0.974235	0.487118	+	0.873336i	$0.338048\pi$
$860$	4.78105	0.163033
$861$	3.87811	0.132166
$862$	−28.7076	−0.977784
$863$	52.5149	1.78763	0.893813	−	0.448439i	$-0.148020\pi$
$863$	52.5149	1.78763	0.893813	+	0.448439i	$0.148020\pi$
$864$	−5.20298	−0.177009
$865$	−0.865799	−0.0294381
$866$	−11.8739	−0.403491
$867$	8.20057	0.278506
$868$	−10.1413	−0.344218
$869$	−20.6695	−0.701164
$870$	4.17179	0.141437
$871$	28.9004	0.979252
$872$	0.0588971	0.00199451
$873$	14.7926	0.500653
$874$	−17.8674	−0.604375
$875$	4.51282	0.152561
$876$	5.51810	0.186439
$877$	11.6087	0.392000	0.196000	−	0.980604i	$-0.437205\pi$
$877$	11.6087	0.392000	0.196000	+	0.980604i	$0.437205\pi$
$878$	−23.4868	−0.792640
$879$	23.7636	0.801526
$880$	1.50026	0.0505739
$881$	15.7863	0.531853	0.265926	−	0.963993i	$-0.414322\pi$
$881$	15.7863	0.531853	0.265926	+	0.963993i	$0.414322\pi$
$882$	−0.969084	−0.0326308
$883$	−29.7306	−1.00052	−0.500258	−	0.865876i	$-0.666761\pi$
$883$	−29.7306	−1.00052	−0.500258	+	0.865876i	$0.666761\pi$
$884$	−29.3240	−0.986273
$885$	−0.750160	−0.0252164
$886$	−21.2885	−0.715201
$887$	−10.0849	−0.338617	−0.169308	−	0.985563i	$-0.554153\pi$
$887$	−10.0849	−0.338617	−0.169308	+	0.985563i	$0.554153\pi$
$888$	29.3940	0.986397
$889$	−12.0398	−0.403802
$890$	2.66439	0.0893105
$891$	4.32229	0.144802
$892$	−5.44566	−0.182334
$893$	42.2368	1.41340
$894$	3.42230	0.114459
$895$	5.50281	0.183939
$896$	4.06068	0.135658
$897$	−28.8555	−0.963456
$898$	−2.99542	−0.0999584
$899$	89.2503	2.97666
$900$	5.07884	0.169295
$901$	46.9505	1.56415
$902$	−16.2441	−0.540870
$903$	9.77414	0.325263
$904$	22.9145	0.762126
$905$	10.8135	0.359452
$906$	15.1540	0.503459
$907$	−57.8274	−1.92013	−0.960063	−	0.279784i	$-0.909737\pi$
$907$	−57.8274	−1.92013	−0.960063	+	0.279784i	$0.909737\pi$
$908$	7.30216	0.242331
$909$	−11.1905	−0.371164
$910$	2.46034	0.0815595
$911$	−4.73601	−0.156911	−0.0784556	−	0.996918i	$-0.524999\pi$
$911$	−4.73601	−0.156911	−0.0784556	+	0.996918i	$0.524999\pi$
$912$	2.64849	0.0877004
$913$	−39.7443	−1.31534
$914$	14.4397	0.477622
$915$	4.97362	0.164423
$916$	−29.7291	−0.982276
$917$	−20.5610	−0.678984
$918$	−4.86482	−0.160563
$919$	−22.1397	−0.730320	−0.365160	−	0.930945i	$-0.618986\pi$
$919$	−22.1397	−0.730320	−0.365160	+	0.930945i	$0.618986\pi$
$920$	7.16741	0.236303
$921$	−18.7582	−0.618103
$922$	−21.4362	−0.705965
$923$	60.6351	1.99583
$924$	−4.58542	−0.150849
$925$	−47.4407	−1.55984
$926$	35.8382	1.17772
$927$	0.335454	0.0110178
$928$	−48.5772	−1.59463
$929$	−15.5111	−0.508903	−0.254451	−	0.967086i	$-0.581895\pi$
$929$	−15.5111	−0.508903	−0.254451	+	0.967086i	$0.581895\pi$
$930$	4.27140	0.140065
$931$	−3.51824	−0.115306
$932$	20.4667	0.670409
$933$	9.10962	0.298235
$934$	15.1532	0.495827
$935$	−10.0046	−0.327185
$936$	16.3328	0.533854
$937$	26.4028	0.862541	0.431271	−	0.902223i	$-0.358065\pi$
$937$	26.4028	0.862541	0.431271	+	0.902223i	$0.358065\pi$
$938$	−5.08641	−0.166077
$939$	−33.1480	−1.08174
$940$	−5.87232	−0.191534
$941$	26.6002	0.867143	0.433571	−	0.901119i	$-0.357253\pi$
$941$	26.6002	0.867143	0.433571	+	0.901119i	$0.357253\pi$
$942$	21.0057	0.684404
$943$	−20.3234	−0.661820
$944$	−1.22475	−0.0398622
$945$	−0.461084	−0.0149991
$946$	−40.9406	−1.33109
$947$	9.92155	0.322407	0.161204	−	0.986921i	$-0.448462\pi$
$947$	9.92155	0.322407	0.161204	+	0.986921i	$0.448462\pi$
$948$	5.07317	0.164769
$949$	−28.6403	−0.929705
$950$	−16.3225	−0.529572
$951$	17.1804	0.557111
$952$	14.8906	0.482607
$953$	43.5749	1.41153	0.705765	−	0.708446i	$-0.250604\pi$
$953$	43.5749	1.41153	0.705765	+	0.708446i	$0.250604\pi$
$954$	−9.06350	−0.293442
$955$	−3.40363	−0.110139
$956$	−22.6975	−0.734089
$957$	40.3548	1.30449
$958$	28.7935	0.930276
$959$	−14.6234	−0.472213
$960$	−3.01904	−0.0974391
$961$	60.3813	1.94778
$962$	−52.8769	−1.70482
$963$	−2.89729	−0.0933637
$964$	−14.4669	−0.465949
$965$	−2.99382	−0.0963744
$966$	5.07851	0.163398
$967$	−23.9023	−0.768645	−0.384322	−	0.923199i	$-0.625565\pi$
$967$	−23.9023	−0.768645	−0.384322	+	0.923199i	$0.625565\pi$
$968$	22.7874	0.732414
$969$	−17.6616	−0.567374
$970$	6.60975	0.212226
$971$	−30.5030	−0.978887	−0.489443	−	0.872035i	$-0.662800\pi$
$971$	−30.5030	−0.978887	−0.489443	+	0.872035i	$0.662800\pi$
$972$	−1.06088	−0.0340276
$973$	1.61302	0.0517112
$974$	5.50095	0.176262
$975$	−26.3605	−0.844211
$976$	8.12018	0.259921
$977$	26.3390	0.842659	0.421330	−	0.906908i	$-0.361564\pi$
$977$	26.3390	0.842659	0.421330	+	0.906908i	$0.361564\pi$
$978$	3.51789	0.112490
$979$	25.7733	0.823718
$980$	0.489153	0.0156254
$981$	0.0198558	0.000633946 0
$982$	−10.1887	−0.325134
$983$	49.7784	1.58768	0.793842	−	0.608124i	$-0.208078\pi$
$983$	49.7784	1.58768	0.793842	+	0.608124i	$0.208078\pi$
$984$	11.5034	0.366716
$985$	−6.02992	−0.192129
$986$	−45.4200	−1.44647
$987$	−12.0051	−0.382126
$988$	20.5515	0.653831
$989$	−51.2216	−1.62875
$990$	1.93133	0.0613816
$991$	−7.17992	−0.228078	−0.114039	−	0.993476i	$-0.536379\pi$
$991$	−7.17992	−0.228078	−0.114039	+	0.993476i	$0.536379\pi$
$992$	−49.7371	−1.57915
$993$	−10.7801	−0.342096
$994$	−10.6717	−0.338485
$995$	11.1546	0.353623
$996$	9.75495	0.309098
$997$	−28.8406	−0.913390	−0.456695	−	0.889623i	$-0.650967\pi$
$997$	−28.8406	−0.913390	−0.456695	+	0.889623i	$0.650967\pi$
$998$	−33.8709	−1.07216
$999$	9.90949	0.313522

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8043.2.a.u.1.17	✓	53

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8043.2.a.u.1.17	✓	53	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:	\( N \)	\(=\)	\( 8043 = 3 \cdot 7 \cdot 383 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8043.a (trivial)

\(f(q)\)	\(=\)	\(q-0.969084 q^{2} +1.00000 q^{3} -1.06088 q^{4} -0.461084 q^{5} -0.969084 q^{6} +1.00000 q^{7} +2.96625 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.969084 q^{2} +1.00000 q^{3} -1.06088 q^{4} -0.461084 q^{5} -0.969084 q^{6} +1.00000 q^{7} +2.96625 q^{8} +1.00000 q^{9} +0.446829 q^{10} +4.32229 q^{11} -1.06088 q^{12} +5.50622 q^{13} -0.969084 q^{14} -0.461084 q^{15} -0.752789 q^{16} +5.02002 q^{17} -0.969084 q^{18} -3.51824 q^{19} +0.489153 q^{20} +1.00000 q^{21} -4.18867 q^{22} -5.24053 q^{23} +2.96625 q^{24} -4.78740 q^{25} -5.33599 q^{26} +1.00000 q^{27} -1.06088 q^{28} +9.33643 q^{29} +0.446829 q^{30} +9.55936 q^{31} -5.20298 q^{32} +4.32229 q^{33} -4.86482 q^{34} -0.461084 q^{35} -1.06088 q^{36} +9.90949 q^{37} +3.40947 q^{38} +5.50622 q^{39} -1.36769 q^{40} +3.87811 q^{41} -0.969084 q^{42} +9.77414 q^{43} -4.58542 q^{44} -0.461084 q^{45} +5.07851 q^{46} -12.0051 q^{47} -0.752789 q^{48} +1.00000 q^{49} +4.63939 q^{50} +5.02002 q^{51} -5.84142 q^{52} +9.35265 q^{53} -0.969084 q^{54} -1.99294 q^{55} +2.96625 q^{56} -3.51824 q^{57} -9.04779 q^{58} +1.62695 q^{59} +0.489153 q^{60} -10.7868 q^{61} -9.26382 q^{62} +1.00000 q^{63} +6.54770 q^{64} -2.53883 q^{65} -4.18867 q^{66} +5.24868 q^{67} -5.32562 q^{68} -5.24053 q^{69} +0.446829 q^{70} +11.0121 q^{71} +2.96625 q^{72} -5.20146 q^{73} -9.60312 q^{74} -4.78740 q^{75} +3.73242 q^{76} +4.32229 q^{77} -5.33599 q^{78} -4.78206 q^{79} +0.347099 q^{80} +1.00000 q^{81} -3.75822 q^{82} -9.19519 q^{83} -1.06088 q^{84} -2.31465 q^{85} -9.47197 q^{86} +9.33643 q^{87} +12.8210 q^{88} +5.96287 q^{89} +0.446829 q^{90} +5.50622 q^{91} +5.55955 q^{92} +9.55936 q^{93} +11.6339 q^{94} +1.62221 q^{95} -5.20298 q^{96} +14.7926 q^{97} -0.969084 q^{98} +4.32229 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 53 q + 11 q^{2} + 53 q^{3} + 63 q^{4} + 24 q^{5} + 11 q^{6} + 53 q^{7} + 30 q^{8} + 53 q^{9}+O(q^{10}) \) 53 * q + 11 * q^2 + 53 * q^3 + 63 * q^4 + 24 * q^5 + 11 * q^6 + 53 * q^7 + 30 * q^8 + 53 * q^9	\( 53 q + 11 q^{2} + 53 q^{3} + 63 q^{4} + 24 q^{5} + 11 q^{6} + 53 q^{7} + 30 q^{8} + 53 q^{9} + 2 q^{10} + 46 q^{11} + 63 q^{12} + 32 q^{13} + 11 q^{14} + 24 q^{15} + 67 q^{16} + 46 q^{17} + 11 q^{18} + 14 q^{19} + 53 q^{20} + 53 q^{21} + 13 q^{22} + 68 q^{23} + 30 q^{24} + 71 q^{25} + 11 q^{26} + 53 q^{27} + 63 q^{28} + 55 q^{29} + 2 q^{30} - 2 q^{31} + 51 q^{32} + 46 q^{33} - 7 q^{34} + 24 q^{35} + 63 q^{36} + 53 q^{37} + 16 q^{38} + 32 q^{39} - 20 q^{40} + 38 q^{41} + 11 q^{42} + 36 q^{43} + 70 q^{44} + 24 q^{45} + 4 q^{46} + 51 q^{47} + 67 q^{48} + 53 q^{49} + 32 q^{50} + 46 q^{51} + 10 q^{52} + 104 q^{53} + 11 q^{54} + 11 q^{55} + 30 q^{56} + 14 q^{57} + 4 q^{58} + 36 q^{59} + 53 q^{60} + 3 q^{61} + 25 q^{62} + 53 q^{63} + 82 q^{64} + 46 q^{65} + 13 q^{66} + 54 q^{67} + 88 q^{68} + 68 q^{69} + 2 q^{70} + 101 q^{71} + 30 q^{72} + q^{73} + 32 q^{74} + 71 q^{75} - 35 q^{76} + 46 q^{77} + 11 q^{78} + 14 q^{79} + 39 q^{80} + 53 q^{81} - 29 q^{82} + 38 q^{83} + 63 q^{84} + 16 q^{85} + 23 q^{86} + 55 q^{87} - 8 q^{88} + 52 q^{89} + 2 q^{90} + 32 q^{91} + 76 q^{92} - 2 q^{93} - 53 q^{94} + 46 q^{95} + 51 q^{96} - 3 q^{97} + 11 q^{98} + 46 q^{99}+O(q^{100}) \) 53 * q + 11 * q^2 + 53 * q^3 + 63 * q^4 + 24 * q^5 + 11 * q^6 + 53 * q^7 + 30 * q^8 + 53 * q^9 + 2 * q^10 + 46 * q^11 + 63 * q^12 + 32 * q^13 + 11 * q^14 + 24 * q^15 + 67 * q^16 + 46 * q^17 + 11 * q^18 + 14 * q^19 + 53 * q^20 + 53 * q^21 + 13 * q^22 + 68 * q^23 + 30 * q^24 + 71 * q^25 + 11 * q^26 + 53 * q^27 + 63 * q^28 + 55 * q^29 + 2 * q^30 - 2 * q^31 + 51 * q^32 + 46 * q^33 - 7 * q^34 + 24 * q^35 + 63 * q^36 + 53 * q^37 + 16 * q^38 + 32 * q^39 - 20 * q^40 + 38 * q^41 + 11 * q^42 + 36 * q^43 + 70 * q^44 + 24 * q^45 + 4 * q^46 + 51 * q^47 + 67 * q^48 + 53 * q^49 + 32 * q^50 + 46 * q^51 + 10 * q^52 + 104 * q^53 + 11 * q^54 + 11 * q^55 + 30 * q^56 + 14 * q^57 + 4 * q^58 + 36 * q^59 + 53 * q^60 + 3 * q^61 + 25 * q^62 + 53 * q^63 + 82 * q^64 + 46 * q^65 + 13 * q^66 + 54 * q^67 + 88 * q^68 + 68 * q^69 + 2 * q^70 + 101 * q^71 + 30 * q^72 + q^73 + 32 * q^74 + 71 * q^75 - 35 * q^76 + 46 * q^77 + 11 * q^78 + 14 * q^79 + 39 * q^80 + 53 * q^81 - 29 * q^82 + 38 * q^83 + 63 * q^84 + 16 * q^85 + 23 * q^86 + 55 * q^87 - 8 * q^88 + 52 * q^89 + 2 * q^90 + 32 * q^91 + 76 * q^92 - 2 * q^93 - 53 * q^94 + 46 * q^95 + 51 * q^96 - 3 * q^97 + 11 * q^98 + 46 * q^99

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