LMFDB - Embedded newform 8047.2.a.e.1.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8047,2,Mod(1,8047)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8047, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8047.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8047 = 13 \cdot 619 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8047.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:	$64.2556185065$
Analytic rank:	$0$
Dimension:	$168$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
Character	$\chi$	$=$	8047.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.39466 q^{2} -2.35552 q^{3} +3.73441 q^{4} +0.950970 q^{5} +5.64068 q^{6} -0.373318 q^{7} -4.15331 q^{8} +2.54848 q^{9} +O(q^{10})$	\(q-2.39466 q^{2} -2.35552 q^{3} +3.73441 q^{4} +0.950970 q^{5} +5.64068 q^{6} -0.373318 q^{7} -4.15331 q^{8} +2.54848 q^{9} -2.27725 q^{10} -3.65398 q^{11} -8.79647 q^{12} +1.00000 q^{13} +0.893969 q^{14} -2.24003 q^{15} +2.47697 q^{16} +3.69797 q^{17} -6.10275 q^{18} +3.28094 q^{19} +3.55131 q^{20} +0.879357 q^{21} +8.75004 q^{22} -7.28867 q^{23} +9.78322 q^{24} -4.09566 q^{25} -2.39466 q^{26} +1.06356 q^{27} -1.39412 q^{28} +0.134238 q^{29} +5.36412 q^{30} -4.73608 q^{31} +2.37512 q^{32} +8.60702 q^{33} -8.85540 q^{34} -0.355014 q^{35} +9.51706 q^{36} +10.4308 q^{37} -7.85675 q^{38} -2.35552 q^{39} -3.94968 q^{40} +8.57544 q^{41} -2.10576 q^{42} -4.99431 q^{43} -13.6454 q^{44} +2.42353 q^{45} +17.4539 q^{46} -8.75834 q^{47} -5.83456 q^{48} -6.86063 q^{49} +9.80771 q^{50} -8.71066 q^{51} +3.73441 q^{52} +0.507298 q^{53} -2.54688 q^{54} -3.47482 q^{55} +1.55051 q^{56} -7.72833 q^{57} -0.321454 q^{58} +5.49953 q^{59} -8.36518 q^{60} -11.3538 q^{61} +11.3413 q^{62} -0.951392 q^{63} -10.6415 q^{64} +0.950970 q^{65} -20.6109 q^{66} +7.66320 q^{67} +13.8097 q^{68} +17.1686 q^{69} +0.850138 q^{70} -1.28381 q^{71} -10.5846 q^{72} -10.9061 q^{73} -24.9781 q^{74} +9.64740 q^{75} +12.2524 q^{76} +1.36409 q^{77} +5.64068 q^{78} -10.8737 q^{79} +2.35553 q^{80} -10.1507 q^{81} -20.5353 q^{82} +10.3151 q^{83} +3.28388 q^{84} +3.51666 q^{85} +11.9597 q^{86} -0.316200 q^{87} +15.1761 q^{88} +14.3317 q^{89} -5.80353 q^{90} -0.373318 q^{91} -27.2188 q^{92} +11.1559 q^{93} +20.9733 q^{94} +3.12008 q^{95} -5.59464 q^{96} -12.3722 q^{97} +16.4289 q^{98} -9.31208 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 168 q + 11 q^{2} + 26 q^{3} + 181 q^{4} + 41 q^{5} + 11 q^{6} + 12 q^{7} + 27 q^{8} + 220 q^{9}+O(q^{10}) $ 168 * q + 11 * q^2 + 26 * q^3 + 181 * q^4 + 41 * q^5 + 11 * q^6 + 12 * q^7 + 27 * q^8 + 220 * q^9	\( 168 q + 11 q^{2} + 26 q^{3} + 181 q^{4} + 41 q^{5} + 11 q^{6} + 12 q^{7} + 27 q^{8} + 220 q^{9} + 11 q^{10} + 23 q^{11} + 78 q^{12} + 168 q^{13} + 47 q^{14} + 10 q^{15} + 203 q^{16} + 147 q^{17} + 13 q^{18} + 17 q^{19} + 81 q^{20} + 13 q^{21} + 20 q^{22} + 85 q^{23} + 14 q^{24} + 225 q^{25} + 11 q^{26} + 89 q^{27} + 12 q^{28} + 137 q^{29} + 26 q^{30} + 13 q^{31} + 60 q^{32} + 78 q^{33} - 2 q^{34} + 77 q^{35} + 278 q^{36} + 41 q^{37} + 68 q^{38} + 26 q^{39} + 11 q^{40} + 107 q^{41} + 43 q^{42} + 27 q^{43} + 39 q^{44} + 88 q^{45} - 23 q^{46} + 112 q^{47} + 127 q^{48} + 236 q^{49} + 14 q^{50} + 55 q^{51} + 181 q^{52} + 149 q^{53} + 3 q^{54} + 40 q^{55} + 134 q^{56} + 55 q^{57} - q^{58} + 44 q^{59} - 13 q^{60} + 81 q^{61} + 106 q^{62} + 34 q^{63} + 197 q^{64} + 41 q^{65} - 20 q^{66} - q^{67} + 278 q^{68} + 75 q^{69} - 42 q^{70} + 48 q^{71} - 34 q^{72} + 107 q^{73} + 74 q^{74} + 93 q^{75} + 20 q^{76} + 206 q^{77} + 11 q^{78} + 14 q^{79} + 115 q^{80} + 328 q^{81} + 48 q^{82} + 62 q^{83} - 11 q^{84} + 6 q^{85} + 27 q^{86} + 51 q^{87} + 31 q^{88} + 173 q^{89} - 21 q^{90} + 12 q^{91} + 179 q^{92} + 73 q^{93} + 17 q^{94} + 90 q^{95} - 33 q^{96} + 110 q^{97} - 13 q^{98} + 24 q^{99}+O(q^{100}) \) 168 * q + 11 * q^2 + 26 * q^3 + 181 * q^4 + 41 * q^5 + 11 * q^6 + 12 * q^7 + 27 * q^8 + 220 * q^9 + 11 * q^10 + 23 * q^11 + 78 * q^12 + 168 * q^13 + 47 * q^14 + 10 * q^15 + 203 * q^16 + 147 * q^17 + 13 * q^18 + 17 * q^19 + 81 * q^20 + 13 * q^21 + 20 * q^22 + 85 * q^23 + 14 * q^24 + 225 * q^25 + 11 * q^26 + 89 * q^27 + 12 * q^28 + 137 * q^29 + 26 * q^30 + 13 * q^31 + 60 * q^32 + 78 * q^33 - 2 * q^34 + 77 * q^35 + 278 * q^36 + 41 * q^37 + 68 * q^38 + 26 * q^39 + 11 * q^40 + 107 * q^41 + 43 * q^42 + 27 * q^43 + 39 * q^44 + 88 * q^45 - 23 * q^46 + 112 * q^47 + 127 * q^48 + 236 * q^49 + 14 * q^50 + 55 * q^51 + 181 * q^52 + 149 * q^53 + 3 * q^54 + 40 * q^55 + 134 * q^56 + 55 * q^57 - q^58 + 44 * q^59 - 13 * q^60 + 81 * q^61 + 106 * q^62 + 34 * q^63 + 197 * q^64 + 41 * q^65 - 20 * q^66 - q^67 + 278 * q^68 + 75 * q^69 - 42 * q^70 + 48 * q^71 - 34 * q^72 + 107 * q^73 + 74 * q^74 + 93 * q^75 + 20 * q^76 + 206 * q^77 + 11 * q^78 + 14 * q^79 + 115 * q^80 + 328 * q^81 + 48 * q^82 + 62 * q^83 - 11 * q^84 + 6 * q^85 + 27 * q^86 + 51 * q^87 + 31 * q^88 + 173 * q^89 - 21 * q^90 + 12 * q^91 + 179 * q^92 + 73 * q^93 + 17 * q^94 + 90 * q^95 - 33 * q^96 + 110 * q^97 - 13 * q^98 + 24 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.39466	−1.69328	−0.846641	−	0.532165i	$-0.821379\pi$
$2$	−2.39466	−1.69328	−0.846641	+	0.532165i	$0.821379\pi$
$3$	−2.35552	−1.35996	−0.679980	−	0.733230i	$-0.738012\pi$
$3$	−2.35552	−1.35996	−0.679980	+	0.733230i	$0.738012\pi$
$4$	3.73441	1.86720
$5$	0.950970	0.425287	0.212643	−	0.977130i	$-0.431793\pi$
$5$	0.950970	0.425287	0.212643	+	0.977130i	$0.431793\pi$
$6$	5.64068	2.30280
$7$	−0.373318	−0.141101	−0.0705504	−	0.997508i	$-0.522476\pi$
$7$	−0.373318	−0.141101	−0.0705504	+	0.997508i	$0.522476\pi$
$8$	−4.15331	−1.46842
$9$	2.54848	0.849493
$10$	−2.27725	−0.720131
$11$	−3.65398	−1.10172	−0.550858	−	0.834599i	$-0.685699\pi$
$11$	−3.65398	−1.10172	−0.550858	+	0.834599i	$0.685699\pi$
$12$	−8.79647	−2.53932
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0.893969	0.238923
$15$	−2.24003	−0.578374
$16$	2.47697	0.619243
$17$	3.69797	0.896890	0.448445	−	0.893810i	$-0.351978\pi$
$17$	3.69797	0.896890	0.448445	+	0.893810i	$0.351978\pi$
$18$	−6.10275	−1.43843
$19$	3.28094	0.752700	0.376350	−	0.926478i	$-0.377179\pi$
$19$	3.28094	0.752700	0.376350	+	0.926478i	$0.377179\pi$
$20$	3.55131	0.794097
$21$	0.879357	0.191892
$22$	8.75004	1.86551
$23$	−7.28867	−1.51979	−0.759896	−	0.650044i	$-0.774750\pi$
$23$	−7.28867	−1.51979	−0.759896	+	0.650044i	$0.774750\pi$
$24$	9.78322	1.99699
$25$	−4.09566	−0.819131
$26$	−2.39466	−0.469632
$27$	1.06356	0.204683
$28$	−1.39412	−0.263464
$29$	0.134238	0.0249273	0.0124637	−	0.999922i	$-0.496033\pi$
$29$	0.134238	0.0249273	0.0124637	+	0.999922i	$0.496033\pi$
$30$	5.36412	0.979349
$31$	−4.73608	−0.850624	−0.425312	−	0.905047i	$-0.639836\pi$
$31$	−4.73608	−0.850624	−0.425312	+	0.905047i	$0.639836\pi$
$32$	2.37512	0.419865
$33$	8.60702	1.49829
$34$	−8.85540	−1.51869
$35$	−0.355014	−0.0600083
$36$	9.51706	1.58618
$37$	10.4308	1.71481	0.857403	−	0.514646i	$-0.172076\pi$
$37$	10.4308	1.71481	0.857403	+	0.514646i	$0.172076\pi$
$38$	−7.85675	−1.27453
$39$	−2.35552	−0.377185
$40$	−3.94968	−0.624499
$41$	8.57544	1.33926	0.669629	−	0.742695i	$-0.266453\pi$
$41$	8.57544	1.33926	0.669629	+	0.742695i	$0.266453\pi$
$42$	−2.10576	−0.324926
$43$	−4.99431	−0.761625	−0.380813	−	0.924652i	$-0.624356\pi$
$43$	−4.99431	−0.761625	−0.380813	+	0.924652i	$0.624356\pi$
$44$	−13.6454	−2.05713
$45$	2.42353	0.361278
$46$	17.4539	2.57344
$47$	−8.75834	−1.27754	−0.638768	−	0.769400i	$-0.720556\pi$
$47$	−8.75834	−1.27754	−0.638768	+	0.769400i	$0.720556\pi$
$48$	−5.83456	−0.842146
$49$	−6.86063	−0.980091
$50$	9.80771	1.38702
$51$	−8.71066	−1.21974
$52$	3.73441	0.517869
$53$	0.507298	0.0696827	0.0348413	−	0.999393i	$-0.488907\pi$
$53$	0.507298	0.0696827	0.0348413	+	0.999393i	$0.488907\pi$
$54$	−2.54688	−0.346586
$55$	−3.47482	−0.468545
$56$	1.55051	0.207195
$57$	−7.72833	−1.02364
$58$	−0.321454	−0.0422090
$59$	5.49953	0.715978	0.357989	−	0.933726i	$-0.383463\pi$
$59$	5.49953	0.715978	0.357989	+	0.933726i	$0.383463\pi$
$60$	−8.36518	−1.07994
$61$	−11.3538	−1.45371	−0.726854	−	0.686792i	$-0.759019\pi$
$61$	−11.3538	−1.45371	−0.726854	+	0.686792i	$0.759019\pi$
$62$	11.3413	1.44035
$63$	−0.951392	−0.119864
$64$	−10.6415	−1.33019
$65$	0.950970	0.117953
$66$	−20.6109	−2.53703
$67$	7.66320	0.936209	0.468104	−	0.883673i	$-0.344937\pi$
$67$	7.66320	0.936209	0.468104	+	0.883673i	$0.344937\pi$
$68$	13.8097	1.67468
$69$	17.1686	2.06686
$70$	0.850138	0.101611
$71$	−1.28381	−0.152361	−0.0761803	−	0.997094i	$-0.524272\pi$
$71$	−1.28381	−0.152361	−0.0761803	+	0.997094i	$0.524272\pi$
$72$	−10.5846	−1.24741
$73$	−10.9061	−1.27647	−0.638234	−	0.769843i	$-0.720335\pi$
$73$	−10.9061	−1.27647	−0.638234	+	0.769843i	$0.720335\pi$
$74$	−24.9781	−2.90365
$75$	9.64740	1.11399
$76$	12.2524	1.40544
$77$	1.36409	0.155453
$78$	5.64068	0.638681
$79$	−10.8737	−1.22338	−0.611692	−	0.791096i	$-0.709511\pi$
$79$	−10.8737	−1.22338	−0.611692	+	0.791096i	$0.709511\pi$
$80$	2.35553	0.263356
$81$	−10.1507	−1.12785
$82$	−20.5353	−2.26774
$83$	10.3151	1.13222	0.566112	−	0.824328i	$-0.308447\pi$
$83$	10.3151	1.13222	0.566112	+	0.824328i	$0.308447\pi$
$84$	3.28388	0.358300
$85$	3.51666	0.381436
$86$	11.9597	1.28965
$87$	−0.316200	−0.0339002
$88$	15.1761	1.61778
$89$	14.3317	1.51916	0.759578	−	0.650416i	$-0.225405\pi$
$89$	14.3317	1.51916	0.759578	+	0.650416i	$0.225405\pi$
$90$	−5.80353	−0.611746
$91$	−0.373318	−0.0391343
$92$	−27.2188	−2.83776
$93$	11.1559	1.15682
$94$	20.9733	2.16323
$95$	3.12008	0.320113
$96$	−5.59464	−0.571000
$97$	−12.3722	−1.25620	−0.628101	−	0.778132i	$-0.716168\pi$
$97$	−12.3722	−1.25620	−0.628101	+	0.778132i	$0.716168\pi$
$98$	16.4289	1.65957
$99$	−9.31208	−0.935900
$100$	−15.2948	−1.52948
$101$	10.9556	1.09013	0.545063	−	0.838395i	$-0.316506\pi$
$101$	10.9556	1.09013	0.545063	+	0.838395i	$0.316506\pi$
$102$	20.8591	2.06536
$103$	−0.891441	−0.0878363	−0.0439182	−	0.999035i	$-0.513984\pi$
$103$	−0.891441	−0.0878363	−0.0439182	+	0.999035i	$0.513984\pi$
$104$	−4.15331	−0.407266
$105$	0.836243	0.0816090
$106$	−1.21481	−0.117992
$107$	−5.52392	−0.534017	−0.267009	−	0.963694i	$-0.586035\pi$
$107$	−5.52392	−0.534017	−0.267009	+	0.963694i	$0.586035\pi$
$108$	3.97178	0.382185
$109$	8.94737	0.857003	0.428501	−	0.903541i	$-0.359042\pi$
$109$	8.94737	0.857003	0.428501	+	0.903541i	$0.359042\pi$
$110$	8.32103	0.793379
$111$	−24.5699	−2.33207
$112$	−0.924697	−0.0873757
$113$	13.8137	1.29948	0.649742	−	0.760155i	$-0.274877\pi$
$113$	13.8137	1.29948	0.649742	+	0.760155i	$0.274877\pi$
$114$	18.5067	1.73332
$115$	−6.93131	−0.646348
$116$	0.501298	0.0465443
$117$	2.54848	0.235607
$118$	−13.1695	−1.21235
$119$	−1.38052	−0.126552
$120$	9.30355	0.849294
$121$	2.35154	0.213776
$122$	27.1886	2.46154
$123$	−20.1996	−1.82134
$124$	−17.6864	−1.58829
$125$	−8.64970	−0.773653
$126$	2.27826	0.202964
$127$	9.10141	0.807619	0.403810	−	0.914843i	$-0.367686\pi$
$127$	9.10141	0.807619	0.403810	+	0.914843i	$0.367686\pi$
$128$	20.7327	1.83253
$129$	11.7642	1.03578
$130$	−2.27725	−0.199728
$131$	−2.48516	−0.217129	−0.108565	−	0.994089i	$-0.534625\pi$
$131$	−2.48516	−0.217129	−0.108565	+	0.994089i	$0.534625\pi$
$132$	32.1421	2.79761
$133$	−1.22483	−0.106207
$134$	−18.3508	−1.58527
$135$	1.01142	0.0870491
$136$	−15.3588	−1.31701
$137$	−17.2218	−1.47136	−0.735679	−	0.677330i	$-0.763137\pi$
$137$	−17.2218	−1.47136	−0.735679	+	0.677330i	$0.763137\pi$
$138$	−41.1130	−3.49977
$139$	13.1946	1.11915	0.559575	−	0.828780i	$-0.310964\pi$
$139$	13.1946	1.11915	0.559575	+	0.828780i	$0.310964\pi$
$140$	−1.32577	−0.112048
$141$	20.6305	1.73740
$142$	3.07430	0.257989
$143$	−3.65398	−0.305561
$144$	6.31251	0.526043
$145$	0.127656	0.0106013
$146$	26.1165	2.16142
$147$	16.1604	1.33288
$148$	38.9527	3.20189
$149$	7.02167	0.575237	0.287619	−	0.957745i	$-0.407136\pi$
$149$	7.02167	0.575237	0.287619	+	0.957745i	$0.407136\pi$
$150$	−23.1023	−1.88629
$151$	6.09853	0.496292	0.248146	−	0.968723i	$-0.420179\pi$
$151$	6.09853	0.496292	0.248146	+	0.968723i	$0.420179\pi$
$152$	−13.6268	−1.10528
$153$	9.42421	0.761902
$154$	−3.26654	−0.263225
$155$	−4.50387	−0.361759
$156$	−8.79647	−0.704281
$157$	8.36582	0.667665	0.333832	−	0.942632i	$-0.391658\pi$
$157$	8.36582	0.667665	0.333832	+	0.942632i	$0.391658\pi$
$158$	26.0388	2.07153
$159$	−1.19495	−0.0947657
$160$	2.25867	0.178563
$161$	2.72099	0.214444
$162$	24.3075	1.90978
$163$	−14.6624	−1.14845	−0.574224	−	0.818698i	$-0.694696\pi$
$163$	−14.6624	−1.14845	−0.574224	+	0.818698i	$0.694696\pi$
$164$	32.0242	2.50067
$165$	8.18502	0.637203
$166$	−24.7011	−1.91718
$167$	−6.15869	−0.476573	−0.238287	−	0.971195i	$-0.576586\pi$
$167$	−6.15869	−0.476573	−0.238287	+	0.971195i	$0.576586\pi$
$168$	−3.65225	−0.281777
$169$	1.00000	0.0769231
$170$	−8.42122	−0.645878
$171$	8.36142	0.639414
$172$	−18.6508	−1.42211
$173$	19.7349	1.50042	0.750208	−	0.661202i	$-0.229954\pi$
$173$	19.7349	1.50042	0.750208	+	0.661202i	$0.229954\pi$
$174$	0.757191	0.0574025
$175$	1.52898	0.115580
$176$	−9.05080	−0.682230
$177$	−12.9543	−0.973701
$178$	−34.3196	−2.57236
$179$	−24.3540	−1.82030	−0.910152	−	0.414274i	$-0.864035\pi$
$179$	−24.3540	−1.82030	−0.910152	+	0.414274i	$0.864035\pi$
$180$	9.05044	0.674580
$181$	8.13609	0.604751	0.302376	−	0.953189i	$-0.402220\pi$
$181$	8.13609	0.604751	0.302376	+	0.953189i	$0.402220\pi$
$182$	0.893969	0.0662654
$183$	26.7442	1.97699
$184$	30.2721	2.23169
$185$	9.91934	0.729285
$186$	−26.7147	−1.95881
$187$	−13.5123	−0.988118
$188$	−32.7072	−2.38542
$189$	−0.397047	−0.0288810
$190$	−7.47154	−0.542042
$191$	7.64133	0.552907	0.276454	−	0.961027i	$-0.410841\pi$
$191$	7.64133	0.552907	0.276454	+	0.961027i	$0.410841\pi$
$192$	25.0664	1.80901
$193$	7.23575	0.520841	0.260420	−	0.965495i	$-0.416139\pi$
$193$	7.23575	0.520841	0.260420	+	0.965495i	$0.416139\pi$
$194$	29.6271	2.12710
$195$	−2.24003	−0.160412
$196$	−25.6204	−1.83003
$197$	−4.51271	−0.321517	−0.160759	−	0.986994i	$-0.551394\pi$
$197$	−4.51271	−0.321517	−0.160759	+	0.986994i	$0.551394\pi$
$198$	22.2993	1.58474
$199$	−14.8271	−1.05107	−0.525533	−	0.850773i	$-0.676134\pi$
$199$	−14.8271	−1.05107	−0.525533	+	0.850773i	$0.676134\pi$
$200$	17.0105	1.20283
$201$	−18.0508	−1.27321
$202$	−26.2350	−1.84589
$203$	−0.0501133	−0.00351726
$204$	−32.5291	−2.27749
$205$	8.15499	0.569569
$206$	2.13470	0.148732
$207$	−18.5750	−1.29105
$208$	2.47697	0.171747
$209$	−11.9885	−0.829261
$210$	−2.00252	−0.138187
$211$	−7.39412	−0.509032	−0.254516	−	0.967069i	$-0.581916\pi$
$211$	−7.39412	−0.509032	−0.254516	+	0.967069i	$0.581916\pi$
$212$	1.89446	0.130112
$213$	3.02405	0.207204
$214$	13.2279	0.904242
$215$	−4.74944	−0.323909
$216$	−4.41732	−0.300561
$217$	1.76806	0.120024
$218$	−21.4259	−1.45115
$219$	25.6896	1.73595
$220$	−12.9764	−0.874869
$221$	3.69797	0.248753
$222$	58.8365	3.94885
$223$	9.15753	0.613233	0.306617	−	0.951833i	$-0.400803\pi$
$223$	9.15753	0.613233	0.306617	+	0.951833i	$0.400803\pi$
$224$	−0.886673	−0.0592433
$225$	−10.4377	−0.695846
$226$	−33.0791	−2.20039
$227$	−4.02320	−0.267029	−0.133515	−	0.991047i	$-0.542626\pi$
$227$	−4.02320	−0.267029	−0.133515	+	0.991047i	$0.542626\pi$
$228$	−28.8607	−1.91135
$229$	−24.5494	−1.62227	−0.811134	−	0.584860i	$-0.801150\pi$
$229$	−24.5494	−1.62227	−0.811134	+	0.584860i	$0.801150\pi$
$230$	16.5981	1.09445
$231$	−3.21315	−0.211410
$232$	−0.557531	−0.0366037
$233$	16.1848	1.06030	0.530149	−	0.847904i	$-0.322136\pi$
$233$	16.1848	1.06030	0.530149	+	0.847904i	$0.322136\pi$
$234$	−6.10275	−0.398949
$235$	−8.32893	−0.543319
$236$	20.5375	1.33688
$237$	25.6132	1.66375
$238$	3.30588	0.214288
$239$	−0.476069	−0.0307944	−0.0153972	−	0.999881i	$-0.504901\pi$
$239$	−0.476069	−0.0307944	−0.0153972	+	0.999881i	$0.504901\pi$
$240$	−5.54849	−0.358154
$241$	−4.10281	−0.264285	−0.132143	−	0.991231i	$-0.542186\pi$
$241$	−4.10281	−0.264285	−0.132143	+	0.991231i	$0.542186\pi$
$242$	−5.63114	−0.361984
$243$	20.7195	1.32915
$244$	−42.3998	−2.71437
$245$	−6.52426	−0.416820
$246$	48.3713	3.08404
$247$	3.28094	0.208761
$248$	19.6704	1.24907
$249$	−24.2973	−1.53978
$250$	20.7131	1.31001
$251$	4.34522	0.274268	0.137134	−	0.990553i	$-0.456211\pi$
$251$	4.34522	0.274268	0.137134	+	0.990553i	$0.456211\pi$
$252$	−3.55288	−0.223811
$253$	26.6326	1.67438
$254$	−21.7948	−1.36753
$255$	−8.28358	−0.518738
$256$	−28.3646	−1.77279
$257$	−17.6779	−1.10272	−0.551359	−	0.834268i	$-0.685891\pi$
$257$	−17.6779	−1.10272	−0.551359	+	0.834268i	$0.685891\pi$
$258$	−28.1713	−1.75387
$259$	−3.89399	−0.241960
$260$	3.55131	0.220243
$261$	0.342102	0.0211756
$262$	5.95111	0.367661
$263$	27.1667	1.67517	0.837585	−	0.546307i	$-0.183967\pi$
$263$	27.1667	1.67517	0.837585	+	0.546307i	$0.183967\pi$
$264$	−35.7476	−2.20012
$265$	0.482425	0.0296351
$266$	2.93306	0.179838
$267$	−33.7586	−2.06599
$268$	28.6175	1.74809
$269$	−16.7211	−1.01950	−0.509751	−	0.860322i	$-0.670262\pi$
$269$	−16.7211	−1.01950	−0.509751	+	0.860322i	$0.670262\pi$
$270$	−2.42201	−0.147399
$271$	−25.1509	−1.52781	−0.763903	−	0.645331i	$-0.776719\pi$
$271$	−25.1509	−1.52781	−0.763903	+	0.645331i	$0.776719\pi$
$272$	9.15978	0.555393
$273$	0.879357	0.0532211
$274$	41.2404	2.49142
$275$	14.9654	0.902449
$276$	64.1146	3.85924
$277$	22.0821	1.32679	0.663393	−	0.748272i	$-0.269116\pi$
$277$	22.0821	1.32679	0.663393	+	0.748272i	$0.269116\pi$
$278$	−31.5966	−1.89504
$279$	−12.0698	−0.722600
$280$	1.47448	0.0881173
$281$	−19.5672	−1.16728	−0.583640	−	0.812013i	$-0.698372\pi$
$281$	−19.5672	−1.16728	−0.583640	+	0.812013i	$0.698372\pi$
$282$	−49.4030	−2.94190
$283$	21.0922	1.25380	0.626900	−	0.779099i	$-0.284323\pi$
$283$	21.0922	1.25380	0.626900	+	0.779099i	$0.284323\pi$
$284$	−4.79428	−0.284488
$285$	−7.34942	−0.435342
$286$	8.75004	0.517401
$287$	−3.20136	−0.188970
$288$	6.05294	0.356673
$289$	−3.32499	−0.195588
$290$	−0.305693	−0.0179509
$291$	29.1429	1.70839
$292$	−40.7280	−2.38342
$293$	18.9323	1.10604	0.553019	−	0.833169i	$-0.313476\pi$
$293$	18.9323	1.10604	0.553019	+	0.833169i	$0.313476\pi$
$294$	−38.6986	−2.25695
$295$	5.22989	0.304496
$296$	−43.3222	−2.51805
$297$	−3.88624	−0.225503
$298$	−16.8145	−0.974039
$299$	−7.28867	−0.421515
$300$	36.0273	2.08004
$301$	1.86446	0.107466
$302$	−14.6039	−0.840362
$303$	−25.8062	−1.48253
$304$	8.12681	0.466104
$305$	−10.7972	−0.618243
$306$	−22.5678	−1.29012
$307$	−7.44228	−0.424753	−0.212377	−	0.977188i	$-0.568120\pi$
$307$	−7.44228	−0.424753	−0.212377	+	0.977188i	$0.568120\pi$
$308$	5.09408	0.290262
$309$	2.09981	0.119454
$310$	10.7852	0.612560
$311$	0.850278	0.0482149	0.0241074	−	0.999709i	$-0.492326\pi$
$311$	0.850278	0.0482149	0.0241074	+	0.999709i	$0.492326\pi$
$312$	9.78322	0.553866
$313$	8.86440	0.501045	0.250523	−	0.968111i	$-0.419398\pi$
$313$	8.86440	0.501045	0.250523	+	0.968111i	$0.419398\pi$
$314$	−20.0333	−1.13054
$315$	−0.904746	−0.0509767
$316$	−40.6067	−2.28430
$317$	−9.04289	−0.507899	−0.253950	−	0.967217i	$-0.581730\pi$
$317$	−9.04289	−0.507899	−0.253950	+	0.967217i	$0.581730\pi$
$318$	2.86150	0.160465
$319$	−0.490501	−0.0274628
$320$	−10.1198	−0.565714
$321$	13.0117	0.726243
$322$	−6.51585	−0.363114
$323$	12.1328	0.675089
$324$	−37.9068	−2.10593
$325$	−4.09566	−0.227186
$326$	35.1115	1.94465
$327$	−21.0757	−1.16549
$328$	−35.6165	−1.96659
$329$	3.26964	0.180261
$330$	−19.6004	−1.07896
$331$	−28.1246	−1.54587	−0.772933	−	0.634487i	$-0.781211\pi$
$331$	−28.1246	−1.54587	−0.772933	+	0.634487i	$0.781211\pi$
$332$	38.5206	2.11409
$333$	26.5826	1.45672
$334$	14.7480	0.806973
$335$	7.28748	0.398157
$336$	2.17814	0.118827
$337$	7.54498	0.411001	0.205501	−	0.978657i	$-0.434118\pi$
$337$	7.54498	0.411001	0.205501	+	0.978657i	$0.434118\pi$
$338$	−2.39466	−0.130252
$339$	−32.5385	−1.76725
$340$	13.1326	0.712218
$341$	17.3055	0.937146
$342$	−20.0228	−1.08271
$343$	5.17442	0.279392
$344$	20.7429	1.11838
$345$	16.3268	0.879008
$346$	−47.2584	−2.54063
$347$	−10.6903	−0.573883	−0.286942	−	0.957948i	$-0.592639\pi$
$347$	−10.6903	−0.573883	−0.286942	+	0.957948i	$0.592639\pi$
$348$	−1.18082	−0.0632985
$349$	−20.1681	−1.07957	−0.539787	−	0.841802i	$-0.681495\pi$
$349$	−20.1681	−1.07957	−0.539787	+	0.841802i	$0.681495\pi$
$350$	−3.66139	−0.195710
$351$	1.06356	0.0567689
$352$	−8.67862	−0.462572
$353$	−12.1193	−0.645045	−0.322522	−	0.946562i	$-0.604531\pi$
$353$	−12.1193	−0.645045	−0.322522	+	0.946562i	$0.604531\pi$
$354$	31.0211	1.64875
$355$	−1.22087	−0.0647969
$356$	53.5204	2.83657
$357$	3.25184	0.172106
$358$	58.3196	3.08229
$359$	−34.0307	−1.79607	−0.898036	−	0.439923i	$-0.855006\pi$
$359$	−34.0307	−1.79607	−0.898036	+	0.439923i	$0.855006\pi$
$360$	−10.0657	−0.530508
$361$	−8.23541	−0.433443
$362$	−19.4832	−1.02401
$363$	−5.53910	−0.290728
$364$	−1.39412	−0.0730717
$365$	−10.3714	−0.542865
$366$	−64.0433	−3.34760
$367$	−16.5917	−0.866079	−0.433040	−	0.901375i	$-0.642559\pi$
$367$	−16.5917	−0.866079	−0.433040	+	0.901375i	$0.642559\pi$
$368$	−18.0538	−0.941121
$369$	21.8543	1.13769
$370$	−23.7535	−1.23488
$371$	−0.189383	−0.00983228
$372$	41.6607	2.16001
$373$	−11.7873	−0.610321	−0.305160	−	0.952301i	$-0.598710\pi$
$373$	−11.7873	−0.610321	−0.305160	+	0.952301i	$0.598710\pi$
$374$	32.3574	1.67316
$375$	20.3745	1.05214
$376$	36.3762	1.87596
$377$	0.134238	0.00691359
$378$	0.950794	0.0489036
$379$	−26.4599	−1.35915	−0.679576	−	0.733605i	$-0.737836\pi$
$379$	−26.4599	−1.35915	−0.679576	+	0.733605i	$0.737836\pi$
$380$	11.6516	0.597717
$381$	−21.4386	−1.09833
$382$	−18.2984	−0.936228
$383$	−1.68789	−0.0862473	−0.0431237	−	0.999070i	$-0.513731\pi$
$383$	−1.68789	−0.0862473	−0.0431237	+	0.999070i	$0.513731\pi$
$384$	−48.8362	−2.49216
$385$	1.29721	0.0661121
$386$	−17.3272	−0.881930
$387$	−12.7279	−0.646995
$388$	−46.2027	−2.34558
$389$	−11.7205	−0.594253	−0.297126	−	0.954838i	$-0.596028\pi$
$389$	−11.7205	−0.594253	−0.297126	+	0.954838i	$0.596028\pi$
$390$	5.36412	0.271623
$391$	−26.9533	−1.36309
$392$	28.4944	1.43918
$393$	5.85384	0.295287
$394$	10.8064	0.544419
$395$	−10.3405	−0.520289
$396$	−34.7751	−1.74751
$397$	5.57557	0.279830	0.139915	−	0.990164i	$-0.455317\pi$
$397$	5.57557	0.279830	0.139915	+	0.990164i	$0.455317\pi$
$398$	35.5059	1.77975
$399$	2.88512	0.144437
$400$	−10.1448	−0.507241
$401$	5.78679	0.288979	0.144489	−	0.989506i	$-0.453846\pi$
$401$	5.78679	0.288979	0.144489	+	0.989506i	$0.453846\pi$
$402$	43.2256	2.15590
$403$	−4.73608	−0.235921
$404$	40.9128	2.03549
$405$	−9.65301	−0.479662
$406$	0.120004	0.00595572
$407$	−38.1137	−1.88923
$408$	36.1781	1.79108
$409$	0.975005	0.0482109	0.0241055	−	0.999709i	$-0.492326\pi$
$409$	0.975005	0.0482109	0.0241055	+	0.999709i	$0.492326\pi$
$410$	−19.5284	−0.964441
$411$	40.5663	2.00099
$412$	−3.32900	−0.164008
$413$	−2.05307	−0.101025
$414$	44.4809	2.18612
$415$	9.80931	0.481520
$416$	2.37512	0.116450
$417$	−31.0801	−1.52200
$418$	28.7084	1.40417
$419$	−16.2031	−0.791573	−0.395787	−	0.918342i	$-0.629528\pi$
$419$	−16.2031	−0.791573	−0.395787	+	0.918342i	$0.629528\pi$
$420$	3.12287	0.152380
$421$	2.80805	0.136856	0.0684279	−	0.997656i	$-0.478202\pi$
$421$	2.80805	0.136856	0.0684279	+	0.997656i	$0.478202\pi$
$422$	17.7064	0.861935
$423$	−22.3205	−1.08526
$424$	−2.10697	−0.102323
$425$	−15.1456	−0.734671
$426$	−7.24157	−0.350855
$427$	4.23858	0.205119
$428$	−20.6286	−0.997119
$429$	8.60702	0.415551
$430$	11.3733	0.548469
$431$	21.9407	1.05685	0.528423	−	0.848981i	$-0.322783\pi$
$431$	21.9407	1.05685	0.528423	+	0.848981i	$0.322783\pi$
$432$	2.63442	0.126749
$433$	13.5725	0.652255	0.326127	−	0.945326i	$-0.394256\pi$
$433$	13.5725	0.652255	0.326127	+	0.945326i	$0.394256\pi$
$434$	−4.23391	−0.203234
$435$	−0.300697	−0.0144173
$436$	33.4131	1.60020
$437$	−23.9137	−1.14395
$438$	−61.5180	−2.93944
$439$	−27.9746	−1.33516	−0.667578	−	0.744540i	$-0.732669\pi$
$439$	−27.9746	−1.33516	−0.667578	+	0.744540i	$0.732669\pi$
$440$	14.4320	0.688020
$441$	−17.4842	−0.832580
$442$	−8.85540	−0.421208
$443$	34.2791	1.62865	0.814325	−	0.580409i	$-0.197107\pi$
$443$	34.2791	1.62865	0.814325	+	0.580409i	$0.197107\pi$
$444$	−91.7539	−4.35445
$445$	13.6290	0.646077
$446$	−21.9292	−1.03838
$447$	−16.5397	−0.782300
$448$	3.97268	0.187691
$449$	1.54605	0.0729626	0.0364813	−	0.999334i	$-0.488385\pi$
$449$	1.54605	0.0729626	0.0364813	+	0.999334i	$0.488385\pi$
$450$	24.9948	1.17826
$451$	−31.3345	−1.47548
$452$	51.5859	2.42640
$453$	−14.3652	−0.674937
$454$	9.63421	0.452156
$455$	−0.355014	−0.0166433
$456$	32.0982	1.50314
$457$	37.6731	1.76227	0.881136	−	0.472863i	$-0.156779\pi$
$457$	37.6731	1.76227	0.881136	+	0.472863i	$0.156779\pi$
$458$	58.7875	2.74696
$459$	3.93304	0.183578
$460$	−25.8843	−1.20686
$461$	5.56927	0.259387	0.129693	−	0.991554i	$-0.458601\pi$
$461$	5.56927	0.259387	0.129693	+	0.991554i	$0.458601\pi$
$462$	7.69441	0.357976
$463$	−22.7519	−1.05737	−0.528686	−	0.848818i	$-0.677315\pi$
$463$	−22.7519	−1.05737	−0.528686	+	0.848818i	$0.677315\pi$
$464$	0.332503	0.0154361
$465$	10.6090	0.491979
$466$	−38.7570	−1.79538
$467$	−32.3841	−1.49856	−0.749278	−	0.662255i	$-0.769599\pi$
$467$	−32.3841	−1.49856	−0.749278	+	0.662255i	$0.769599\pi$
$468$	9.51706	0.439926
$469$	−2.86081	−0.132100
$470$	19.9450	0.919992
$471$	−19.7059	−0.907998
$472$	−22.8413	−1.05135
$473$	18.2491	0.839094
$474$	−61.3348	−2.81720
$475$	−13.4376	−0.616560
$476$	−5.15542	−0.236298
$477$	1.29284	0.0591950
$478$	1.14003	0.0521435
$479$	9.62357	0.439712	0.219856	−	0.975532i	$-0.429441\pi$
$479$	9.62357	0.439712	0.219856	+	0.975532i	$0.429441\pi$
$480$	−5.32034	−0.242839
$481$	10.4308	0.475602
$482$	9.82484	0.447509
$483$	−6.40934	−0.291635
$484$	8.78161	0.399164
$485$	−11.7656	−0.534246
$486$	−49.6161	−2.25063
$487$	26.9789	1.22253	0.611266	−	0.791425i	$-0.290660\pi$
$487$	26.9789	1.22253	0.611266	+	0.791425i	$0.290660\pi$
$488$	47.1560	2.13465
$489$	34.5376	1.56184
$490$	15.6234	0.705793
$491$	32.4306	1.46357	0.731787	−	0.681534i	$-0.238687\pi$
$491$	32.4306	1.46357	0.731787	+	0.681534i	$0.238687\pi$
$492$	−75.4336	−3.40081
$493$	0.496407	0.0223571
$494$	−7.85675	−0.353492
$495$	−8.85552	−0.398026
$496$	−11.7311	−0.526743
$497$	0.479270	0.0214982
$498$	58.1839	2.60728
$499$	23.3563	1.04557	0.522785	−	0.852464i	$-0.324893\pi$
$499$	23.3563	1.04557	0.522785	+	0.852464i	$0.324893\pi$
$500$	−32.3015	−1.44457
$501$	14.5069	0.648121
$502$	−10.4053	−0.464413
$503$	−24.1327	−1.07602	−0.538012	−	0.842937i	$-0.680824\pi$
$503$	−24.1327	−1.07602	−0.538012	+	0.842937i	$0.680824\pi$
$504$	3.95143	0.176011
$505$	10.4185	0.463616
$506$	−63.7761	−2.83519
$507$	−2.35552	−0.104612
$508$	33.9883	1.50799
$509$	7.10843	0.315076	0.157538	−	0.987513i	$-0.449644\pi$
$509$	7.10843	0.315076	0.157538	+	0.987513i	$0.449644\pi$
$510$	19.8364	0.878369
$511$	4.07145	0.180111
$512$	26.4584	1.16931
$513$	3.48950	0.154065
$514$	42.3327	1.86721
$515$	−0.847734	−0.0373556
$516$	43.9323	1.93401
$517$	32.0028	1.40748
$518$	9.32478	0.409707
$519$	−46.4859	−2.04051
$520$	−3.94968	−0.173205
$521$	−1.88613	−0.0826329	−0.0413164	−	0.999146i	$-0.513155\pi$
$521$	−1.88613	−0.0826329	−0.0413164	+	0.999146i	$0.513155\pi$
$522$	−0.819219	−0.0358562
$523$	28.0915	1.22836	0.614178	−	0.789168i	$-0.289488\pi$
$523$	28.0915	1.22836	0.614178	+	0.789168i	$0.289488\pi$
$524$	−9.28058	−0.405424
$525$	−3.60154	−0.157184
$526$	−65.0551	−2.83654
$527$	−17.5139	−0.762917
$528$	21.3193	0.927805
$529$	30.1247	1.30977
$530$	−1.15524	−0.0501806
$531$	14.0154	0.608218
$532$	−4.57403	−0.198309
$533$	8.57544	0.371444
$534$	80.8405	3.49831
$535$	−5.25308	−0.227111
$536$	−31.8277	−1.37475
$537$	57.3664	2.47554
$538$	40.0413	1.72630
$539$	25.0686	1.07978
$540$	3.77705	0.162538
$541$	−12.8458	−0.552285	−0.276143	−	0.961117i	$-0.589056\pi$
$541$	−12.8458	−0.552285	−0.276143	+	0.961117i	$0.589056\pi$
$542$	60.2278	2.58701
$543$	−19.1647	−0.822438
$544$	8.78312	0.376573
$545$	8.50869	0.364472
$546$	−2.10576	−0.0901184
$547$	30.2283	1.29247	0.646233	−	0.763140i	$-0.276343\pi$
$547$	30.2283	1.29247	0.646233	+	0.763140i	$0.276343\pi$
$548$	−64.3132	−2.74732
$549$	−28.9350	−1.23492
$550$	−35.8371	−1.52810
$551$	0.440426	0.0187628
$552$	−71.3066	−3.03501
$553$	4.05933	0.172620
$554$	−52.8792	−2.24662
$555$	−23.3652	−0.991798
$556$	49.2739	2.08968
$557$	−6.11178	−0.258964	−0.129482	−	0.991582i	$-0.541331\pi$
$557$	−6.11178	−0.258964	−0.129482	+	0.991582i	$0.541331\pi$
$558$	28.9031	1.22356
$559$	−4.99431	−0.211237
$560$	−0.879360	−0.0371597
$561$	31.8285	1.34380
$562$	46.8568	1.97653
$563$	38.9400	1.64113	0.820563	−	0.571556i	$-0.193660\pi$
$563$	38.9400	1.64113	0.820563	+	0.571556i	$0.193660\pi$
$564$	77.0425	3.24408
$565$	13.1364	0.552653
$566$	−50.5087	−2.12304
$567$	3.78943	0.159141
$568$	5.33208	0.223729
$569$	4.77731	0.200275	0.100138	−	0.994974i	$-0.468072\pi$
$569$	4.77731	0.200275	0.100138	+	0.994974i	$0.468072\pi$
$570$	17.5994	0.737156
$571$	−14.2565	−0.596615	−0.298307	−	0.954470i	$-0.596422\pi$
$571$	−14.2565	−0.596615	−0.298307	+	0.954470i	$0.596422\pi$
$572$	−13.6454	−0.570544
$573$	−17.9993	−0.751932
$574$	7.66618	0.319980
$575$	29.8519	1.24491
$576$	−27.1198	−1.12999
$577$	10.0399	0.417964	0.208982	−	0.977919i	$-0.432985\pi$
$577$	10.0399	0.417964	0.208982	+	0.977919i	$0.432985\pi$
$578$	7.96222	0.331185
$579$	−17.0440	−0.708323
$580$	0.476719	0.0197947
$581$	−3.85079	−0.159758
$582$	−69.7873	−2.89278
$583$	−1.85365	−0.0767705
$584$	45.2966	1.87439
$585$	2.42353	0.100201
$586$	−45.3365	−1.87283
$587$	9.09964	0.375582	0.187791	−	0.982209i	$-0.439867\pi$
$587$	9.09964	0.375582	0.187791	+	0.982209i	$0.439867\pi$
$588$	60.3494	2.48877
$589$	−15.5388	−0.640265
$590$	−12.5238	−0.515597
$591$	10.6298	0.437251
$592$	25.8367	1.06188
$593$	−17.2129	−0.706851	−0.353425	−	0.935463i	$-0.614983\pi$
$593$	−17.2129	−0.706851	−0.353425	+	0.935463i	$0.614983\pi$
$594$	9.30623	0.381839
$595$	−1.31283	−0.0538209
$596$	26.2217	1.07408
$597$	34.9256	1.42941
$598$	17.4539	0.713743
$599$	1.14447	0.0467619	0.0233809	−	0.999727i	$-0.492557\pi$
$599$	1.14447	0.0467619	0.0233809	+	0.999727i	$0.492557\pi$
$600$	−40.0687	−1.63580
$601$	3.60742	0.147150	0.0735748	−	0.997290i	$-0.476559\pi$
$601$	3.60742	0.147150	0.0735748	+	0.997290i	$0.476559\pi$
$602$	−4.46476	−0.181970
$603$	19.5295	0.795303
$604$	22.7744	0.926677
$605$	2.23625	0.0909163
$606$	61.7972	2.51034
$607$	−16.5866	−0.673231	−0.336616	−	0.941642i	$-0.609282\pi$
$607$	−16.5866	−0.673231	−0.336616	+	0.941642i	$0.609282\pi$
$608$	7.79263	0.316033
$609$	0.118043	0.00478334
$610$	25.8555	1.04686
$611$	−8.75834	−0.354325
$612$	35.1938	1.42263
$613$	−16.2334	−0.655662	−0.327831	−	0.944736i	$-0.606318\pi$
$613$	−16.2334	−0.655662	−0.327831	+	0.944736i	$0.606318\pi$
$614$	17.8217	0.719227
$615$	−19.2093	−0.774592
$616$	−5.66551	−0.228270
$617$	46.2532	1.86209	0.931043	−	0.364910i	$-0.118900\pi$
$617$	46.2532	1.86209	0.931043	+	0.364910i	$0.118900\pi$
$618$	−5.02833	−0.202269
$619$	−1.00000	−0.0401934
$619$	−1.00000	−0.0401934
$620$	−16.8193	−0.675478
$621$	−7.75197	−0.311076
$622$	−2.03613	−0.0816413
$623$	−5.35027	−0.214354
$624$	−5.83456	−0.233569
$625$	12.2527	0.490107
$626$	−21.2272	−0.848411
$627$	28.2391	1.12776
$628$	31.2414	1.24667
$629$	38.5727	1.53799
$630$	2.16656	0.0863178
$631$	7.71844	0.307266	0.153633	−	0.988128i	$-0.450903\pi$
$631$	7.71844	0.307266	0.153633	+	0.988128i	$0.450903\pi$
$632$	45.1618	1.79644
$633$	17.4170	0.692264
$634$	21.6547	0.860016
$635$	8.65517	0.343470
$636$	−4.46243	−0.176947
$637$	−6.86063	−0.271828
$638$	1.17458	0.0465023
$639$	−3.27177	−0.129429
$640$	19.7162	0.779350
$641$	1.48489	0.0586498	0.0293249	−	0.999570i	$-0.490664\pi$
$641$	1.48489	0.0586498	0.0293249	+	0.999570i	$0.490664\pi$
$642$	−31.1586	−1.22973
$643$	13.7018	0.540345	0.270173	−	0.962812i	$-0.412919\pi$
$643$	13.7018	0.540345	0.270173	+	0.962812i	$0.412919\pi$
$644$	10.1613	0.400410
$645$	11.1874	0.440504
$646$	−29.0541	−1.14312
$647$	37.2763	1.46548	0.732742	−	0.680507i	$-0.238240\pi$
$647$	37.2763	1.46548	0.732742	+	0.680507i	$0.238240\pi$
$648$	42.1590	1.65616
$649$	−20.0951	−0.788803
$650$	9.80771	0.384690
$651$	−4.16470	−0.163228
$652$	−54.7554	−2.14439
$653$	−1.67646	−0.0656049	−0.0328024	−	0.999462i	$-0.510443\pi$
$653$	−1.67646	−0.0656049	−0.0328024	+	0.999462i	$0.510443\pi$
$654$	50.4692	1.97350
$655$	−2.36331	−0.0923422
$656$	21.2411	0.829327
$657$	−27.7941	−1.08435
$658$	−7.82969	−0.305233
$659$	49.2327	1.91783	0.958917	−	0.283686i	$-0.0915574\pi$
$659$	49.2327	1.91783	0.958917	+	0.283686i	$0.0915574\pi$
$660$	30.5662	1.18979
$661$	4.78802	0.186233	0.0931163	−	0.995655i	$-0.470317\pi$
$661$	4.78802	0.186233	0.0931163	+	0.995655i	$0.470317\pi$
$662$	67.3489	2.61759
$663$	−8.71066	−0.338294
$664$	−42.8417	−1.66258
$665$	−1.16478	−0.0451683
$666$	−63.6563	−2.46663
$667$	−0.978414	−0.0378843
$668$	−22.9990	−0.889859
$669$	−21.5707	−0.833973
$670$	−17.4510	−0.674193
$671$	41.4866	1.60157
$672$	2.08858	0.0805686
$673$	41.0926	1.58400	0.792002	−	0.610518i	$-0.209039\pi$
$673$	41.0926	1.58400	0.792002	+	0.610518i	$0.209039\pi$
$674$	−18.0677	−0.695940
$675$	−4.35600	−0.167662
$676$	3.73441	0.143631
$677$	27.5568	1.05909	0.529547	−	0.848281i	$-0.322362\pi$
$677$	27.5568	1.05909	0.529547	+	0.848281i	$0.322362\pi$
$678$	77.9186	2.99245
$679$	4.61874	0.177251
$680$	−14.6058	−0.560107
$681$	9.47674	0.363149
$682$	−41.4408	−1.58685
$683$	−11.6443	−0.445556	−0.222778	−	0.974869i	$-0.571512\pi$
$683$	−11.6443	−0.445556	−0.222778	+	0.974869i	$0.571512\pi$
$684$	31.2249	1.19391
$685$	−16.3774	−0.625749
$686$	−12.3910	−0.473090
$687$	57.8266	2.20622
$688$	−12.3708	−0.471631
$689$	0.507298	0.0193265
$690$	−39.0973	−1.48841
$691$	−27.1666	−1.03347	−0.516733	−	0.856146i	$-0.672852\pi$
$691$	−27.1666	−1.03347	−0.516733	+	0.856146i	$0.672852\pi$
$692$	73.6980	2.80158
$693$	3.47636	0.132056
$694$	25.5996	0.971746
$695$	12.5477	0.475960
$696$	1.31328	0.0497796
$697$	31.7118	1.20117
$698$	48.2958	1.82802
$699$	−38.1235	−1.44196
$700$	5.70983	0.215811
$701$	36.1699	1.36612	0.683058	−	0.730364i	$-0.260649\pi$
$701$	36.1699	1.36612	0.683058	+	0.730364i	$0.260649\pi$
$702$	−2.54688	−0.0961257
$703$	34.2227	1.29073
$704$	38.8840	1.46549
$705$	19.6190	0.738893
$706$	29.0216	1.09224
$707$	−4.08993	−0.153818
$708$	−48.3764	−1.81810
$709$	23.9086	0.897908	0.448954	−	0.893555i	$-0.351797\pi$
$709$	23.9086	0.897908	0.448954	+	0.893555i	$0.351797\pi$
$710$	2.92357	0.109719
$711$	−27.7113	−1.03926
$712$	−59.5240	−2.23076
$713$	34.5197	1.29277
$714$	−7.78706	−0.291423
$715$	−3.47482	−0.129951
$716$	−90.9477	−3.39888
$717$	1.12139	0.0418791
$718$	81.4920	3.04125
$719$	5.45538	0.203451	0.101726	−	0.994812i	$-0.467564\pi$
$719$	5.45538	0.203451	0.101726	+	0.994812i	$0.467564\pi$
$720$	6.00301	0.223719
$721$	0.332791	0.0123938
$722$	19.7210	0.733940
$723$	9.66426	0.359418
$724$	30.3835	1.12919
$725$	−0.549791	−0.0204187
$726$	13.2643	0.492284
$727$	−21.7091	−0.805146	−0.402573	−	0.915388i	$-0.631884\pi$
$727$	−21.7091	−0.805146	−0.402573	+	0.915388i	$0.631884\pi$
$728$	1.55051	0.0574655
$729$	−18.3531	−0.679744
$730$	24.8360	0.919223
$731$	−18.4688	−0.683094
$732$	99.8736	3.69144
$733$	−50.2797	−1.85712	−0.928561	−	0.371179i	$-0.878954\pi$
$733$	−50.2797	−1.85712	−0.928561	+	0.371179i	$0.878954\pi$
$734$	39.7315	1.46652
$735$	15.3680	0.566858
$736$	−17.3114	−0.638108
$737$	−28.0011	−1.03144
$738$	−52.3338	−1.92643
$739$	47.5268	1.74830	0.874151	−	0.485655i	$-0.161419\pi$
$739$	47.5268	1.74830	0.874151	+	0.485655i	$0.161419\pi$
$740$	37.0428	1.36172
$741$	−7.72833	−0.283907
$742$	0.453509	0.0166488
$743$	10.9509	0.401750	0.200875	−	0.979617i	$-0.435622\pi$
$743$	10.9509	0.401750	0.200875	+	0.979617i	$0.435622\pi$
$744$	−46.3341	−1.69869
$745$	6.67740	0.244641
$746$	28.2265	1.03345
$747$	26.2877	0.961817
$748$	−50.4604	−1.84502
$749$	2.06218	0.0753503
$750$	−48.7902	−1.78156
$751$	−34.0791	−1.24356	−0.621782	−	0.783191i	$-0.713591\pi$
$751$	−34.0791	−1.24356	−0.621782	+	0.783191i	$0.713591\pi$
$752$	−21.6942	−0.791105
$753$	−10.2353	−0.372994
$754$	−0.321454	−0.0117067
$755$	5.79953	0.211066
$756$	−1.48274	−0.0539266
$757$	39.8845	1.44963	0.724814	−	0.688945i	$-0.241926\pi$
$757$	39.8845	1.44963	0.724814	+	0.688945i	$0.241926\pi$
$758$	63.3624	2.30143
$759$	−62.7337	−2.27709
$760$	−12.9587	−0.470060
$761$	10.4536	0.378944	0.189472	−	0.981886i	$-0.439322\pi$
$761$	10.4536	0.378944	0.189472	+	0.981886i	$0.439322\pi$
$762$	51.3381	1.85978
$763$	−3.34021	−0.120924
$764$	28.5358	1.03239
$765$	8.96215	0.324027
$766$	4.04193	0.146041
$767$	5.49953	0.198576
$768$	66.8135	2.41093
$769$	−13.1511	−0.474241	−0.237121	−	0.971480i	$-0.576204\pi$
$769$	−13.1511	−0.474241	−0.237121	+	0.971480i	$0.576204\pi$
$770$	−3.10639	−0.111946
$771$	41.6407	1.49965
$772$	27.0212	0.972515
$773$	−4.06230	−0.146111	−0.0730554	−	0.997328i	$-0.523275\pi$
$773$	−4.06230	−0.146111	−0.0730554	+	0.997328i	$0.523275\pi$
$774$	30.4790	1.09555
$775$	19.3973	0.696773
$776$	51.3855	1.84463
$777$	9.17236	0.329057
$778$	28.0666	1.00624
$779$	28.1355	1.00806
$780$	−8.36518	−0.299522
$781$	4.69102	0.167858
$782$	64.5441	2.30809
$783$	0.142770	0.00510220
$784$	−16.9936	−0.606914
$785$	7.95565	0.283949
$786$	−14.0180	−0.500004
$787$	36.6929	1.30796	0.653980	−	0.756512i	$-0.273098\pi$
$787$	36.6929	1.30796	0.653980	+	0.756512i	$0.273098\pi$
$788$	−16.8523	−0.600338
$789$	−63.9917	−2.27817
$790$	24.7621	0.880996
$791$	−5.15690	−0.183358
$792$	38.6760	1.37429
$793$	−11.3538	−0.403186
$794$	−13.3516	−0.473831
$795$	−1.13636	−0.0403026
$796$	−55.3704	−1.96255
$797$	−35.5489	−1.25921	−0.629604	−	0.776917i	$-0.716783\pi$
$797$	−35.5489	−1.25921	−0.629604	+	0.776917i	$0.716783\pi$
$798$	−6.90889	−0.244572
$799$	−32.3881	−1.14581
$800$	−9.72766	−0.343925
$801$	36.5240	1.29051
$802$	−13.8574	−0.489322
$803$	39.8508	1.40630
$804$	−67.4091	−2.37734
$805$	2.58758	0.0912002
$806$	11.3413	0.399480
$807$	39.3868	1.38648
$808$	−45.5022	−1.60076
$809$	26.4672	0.930538	0.465269	−	0.885169i	$-0.345958\pi$
$809$	26.4672	0.930538	0.465269	+	0.885169i	$0.345958\pi$
$810$	23.1157	0.812202
$811$	−47.1653	−1.65620	−0.828099	−	0.560583i	$-0.810577\pi$
$811$	−47.1653	−1.65620	−0.828099	+	0.560583i	$0.810577\pi$
$812$	−0.187143	−0.00656744
$813$	59.2434	2.07776
$814$	91.2695	3.19899
$815$	−13.9435	−0.488420
$816$	−21.5761	−0.755313
$817$	−16.3860	−0.573275
$818$	−2.33481	−0.0816346
$819$	−0.951392	−0.0332443
$820$	30.4540	1.06350
$821$	−41.3415	−1.44283	−0.721415	−	0.692503i	$-0.756508\pi$
$821$	−41.3415	−1.44283	−0.721415	+	0.692503i	$0.756508\pi$
$822$	−97.1426	−3.38824
$823$	18.2446	0.635966	0.317983	−	0.948096i	$-0.396995\pi$
$823$	18.2446	0.635966	0.317983	+	0.948096i	$0.396995\pi$
$824$	3.70244	0.128980
$825$	−35.2514	−1.22730
$826$	4.91641	0.171064
$827$	−17.0148	−0.591661	−0.295831	−	0.955240i	$-0.595596\pi$
$827$	−17.0148	−0.591661	−0.295831	+	0.955240i	$0.595596\pi$
$828$	−69.3667	−2.41066
$829$	0.626920	0.0217738	0.0108869	−	0.999941i	$-0.496535\pi$
$829$	0.626920	0.0217738	0.0108869	+	0.999941i	$0.496535\pi$
$830$	−23.4900	−0.815350
$831$	−52.0149	−1.80438
$832$	−10.6415	−0.368929
$833$	−25.3704	−0.879034
$834$	74.4264	2.57717
$835$	−5.85673	−0.202680
$836$	−44.7699	−1.54840
$837$	−5.03712	−0.174108
$838$	38.8009	1.34036
$839$	14.3701	0.496111	0.248055	−	0.968746i	$-0.420209\pi$
$839$	14.3701	0.496111	0.248055	+	0.968746i	$0.420209\pi$
$840$	−3.47318	−0.119836
$841$	−28.9820	−0.999379
$842$	−6.72432	−0.231735
$843$	46.0909	1.58745
$844$	−27.6126	−0.950466
$845$	0.950970	0.0327144
$846$	53.4500	1.83765
$847$	−0.877871	−0.0301640
$848$	1.25656	0.0431505
$849$	−49.6831	−1.70512
$850$	36.2687	1.24400
$851$	−76.0263	−2.60615
$852$	11.2930	0.386892
$853$	50.1434	1.71688	0.858439	−	0.512916i	$-0.171435\pi$
$853$	50.1434	1.71688	0.858439	+	0.512916i	$0.171435\pi$
$854$	−10.1500	−0.347325
$855$	7.95146	0.271934
$856$	22.9426	0.784161
$857$	−19.6877	−0.672519	−0.336259	−	0.941769i	$-0.609162\pi$
$857$	−19.6877	−0.672519	−0.336259	+	0.941769i	$0.609162\pi$
$858$	−20.6109	−0.703644
$859$	−41.8476	−1.42782	−0.713910	−	0.700237i	$-0.753078\pi$
$859$	−41.8476	−1.42782	−0.713910	+	0.700237i	$0.753078\pi$
$860$	−17.7363	−0.604804
$861$	7.54088	0.256992
$862$	−52.5406	−1.78954
$863$	49.9578	1.70058	0.850291	−	0.526313i	$-0.176426\pi$
$863$	49.9578	1.70058	0.850291	+	0.526313i	$0.176426\pi$
$864$	2.52609	0.0859394
$865$	18.7673	0.638107
$866$	−32.5016	−1.10445
$867$	7.83208	0.265991
$868$	6.60265	0.224109
$869$	39.7321	1.34782
$870$	0.720067	0.0244125
$871$	7.66320	0.259658
$872$	−37.1612	−1.25844
$873$	−31.5302	−1.06714
$874$	57.2652	1.93703
$875$	3.22908	0.109163
$876$	95.9356	3.24136
$877$	26.7188	0.902228	0.451114	−	0.892466i	$-0.351027\pi$
$877$	26.7188	0.902228	0.451114	+	0.892466i	$0.351027\pi$
$878$	66.9898	2.26080
$879$	−44.5955	−1.50417
$880$	−8.60704	−0.290143
$881$	−49.8208	−1.67850	−0.839252	−	0.543742i	$-0.817007\pi$
$881$	−49.8208	−1.67850	−0.839252	+	0.543742i	$0.817007\pi$
$882$	41.8687	1.40979
$883$	−15.4300	−0.519263	−0.259631	−	0.965708i	$-0.583601\pi$
$883$	−15.4300	−0.519263	−0.259631	+	0.965708i	$0.583601\pi$
$884$	13.8097	0.464472
$885$	−12.3191	−0.414102
$886$	−82.0869	−2.75776
$887$	56.4188	1.89436	0.947180	−	0.320704i	$-0.103919\pi$
$887$	56.4188	1.89436	0.947180	+	0.320704i	$0.103919\pi$
$888$	102.046	3.42445
$889$	−3.39772	−0.113956
$890$	−32.6369	−1.09399
$891$	37.0904	1.24257
$892$	34.1979	1.14503
$893$	−28.7356	−0.961601
$894$	39.6069	1.32465
$895$	−23.1599	−0.774151
$896$	−7.73987	−0.258571
$897$	17.1686	0.573243
$898$	−3.70227	−0.123546
$899$	−0.635760	−0.0212038
$900$	−38.9786	−1.29929
$901$	1.87597	0.0624977
$902$	75.0354	2.49841
$903$	−4.39178	−0.146149
$904$	−57.3726	−1.90818
$905$	7.73719	0.257193
$906$	34.3999	1.14286
$907$	35.7228	1.18616	0.593078	−	0.805145i	$-0.297913\pi$
$907$	35.7228	1.18616	0.593078	+	0.805145i	$0.297913\pi$
$908$	−15.0243	−0.498598
$909$	27.9202	0.926055
$910$	0.850138	0.0281818
$911$	56.5632	1.87402	0.937012	−	0.349297i	$-0.113580\pi$
$911$	56.5632	1.87402	0.937012	+	0.349297i	$0.113580\pi$
$912$	−19.1429	−0.633884
$913$	−37.6910	−1.24739
$914$	−90.2143	−2.98402
$915$	25.4329	0.840787
$916$	−91.6774	−3.02910
$917$	0.927753	0.0306371
$918$	−9.41829	−0.310850
$919$	39.3279	1.29731	0.648653	−	0.761084i	$-0.275333\pi$
$919$	39.3279	1.29731	0.648653	+	0.761084i	$0.275333\pi$
$920$	28.7879	0.949109
$921$	17.5304	0.577648
$922$	−13.3365	−0.439215
$923$	−1.28381	−0.0422572
$924$	−11.9992	−0.394745
$925$	−42.7208	−1.40465
$926$	54.4831	1.79043
$927$	−2.27182	−0.0746164
$928$	0.318830	0.0104661
$929$	−11.3453	−0.372226	−0.186113	−	0.982528i	$-0.559589\pi$
$929$	−11.3453	−0.372226	−0.186113	+	0.982528i	$0.559589\pi$
$930$	−25.4049	−0.833058
$931$	−22.5094	−0.737714
$932$	60.4404	1.97979
$933$	−2.00285	−0.0655703
$934$	77.5489	2.53748
$935$	−12.8498	−0.420234
$936$	−10.5846	−0.345970
$937$	8.57460	0.280120	0.140060	−	0.990143i	$-0.455270\pi$
$937$	8.57460	0.280120	0.140060	+	0.990143i	$0.455270\pi$
$938$	6.85066	0.223682
$939$	−20.8803	−0.681402
$940$	−31.1036	−1.01449
$941$	43.4204	1.41546	0.707732	−	0.706481i	$-0.249719\pi$
$941$	43.4204	1.41546	0.707732	+	0.706481i	$0.249719\pi$
$942$	47.1889	1.53750
$943$	−62.5035	−2.03540
$944$	13.6222	0.443364
$945$	−0.377580	−0.0122827
$946$	−43.7004	−1.42082
$947$	33.3147	1.08258	0.541291	−	0.840835i	$-0.317936\pi$
$947$	33.3147	1.08258	0.541291	+	0.840835i	$0.317936\pi$
$948$	95.6499	3.10656
$949$	−10.9061	−0.354028
$950$	32.1785	1.04401
$951$	21.3007	0.690723
$952$	5.73373	0.185831
$953$	−5.36449	−0.173773	−0.0868865	−	0.996218i	$-0.527692\pi$
$953$	−5.36449	−0.173773	−0.0868865	+	0.996218i	$0.527692\pi$
$954$	−3.09591	−0.100234
$955$	7.26668	0.235144
$956$	−1.77784	−0.0574993
$957$	1.15539	0.0373483
$958$	−23.0452	−0.744557
$959$	6.42920	0.207610
$960$	23.8374	0.769349
$961$	−8.56959	−0.276438
$962$	−24.9781	−0.805327
$963$	−14.0776	−0.453644
$964$	−15.3216	−0.493474
$965$	6.88099	0.221507
$966$	15.3482	0.493821
$967$	−28.7360	−0.924086	−0.462043	−	0.886858i	$-0.652883\pi$
$967$	−28.7360	−0.924086	−0.462043	+	0.886858i	$0.652883\pi$
$968$	−9.76669	−0.313913
$969$	−28.5792	−0.918095
$970$	28.1745	0.904630
$971$	34.5726	1.10949	0.554744	−	0.832021i	$-0.312816\pi$
$971$	34.5726	1.10949	0.554744	+	0.832021i	$0.312816\pi$
$972$	77.3749	2.48180
$973$	−4.92577	−0.157913
$974$	−64.6055	−2.07009
$975$	9.64740	0.308964
$976$	−28.1231	−0.900199
$977$	33.6191	1.07557	0.537786	−	0.843082i	$-0.319261\pi$
$977$	33.6191	1.07557	0.537786	+	0.843082i	$0.319261\pi$
$978$	−82.7059	−2.64464
$979$	−52.3677	−1.67368
$980$	−24.3642	−0.778287
$981$	22.8022	0.728018
$982$	−77.6604	−2.47824
$983$	42.1431	1.34416	0.672079	−	0.740480i	$-0.265402\pi$
$983$	42.1431	1.34416	0.672079	+	0.740480i	$0.265402\pi$
$984$	83.8954	2.67449
$985$	−4.29145	−0.136737
$986$	−1.18873	−0.0378568
$987$	−7.70171	−0.245148
$988$	12.2524	0.389800
$989$	36.4019	1.15751
$990$	21.2060	0.673970
$991$	−5.87445	−0.186608	−0.0933040	−	0.995638i	$-0.529743\pi$
$991$	−5.87445	−0.186608	−0.0933040	+	0.995638i	$0.529743\pi$
$992$	−11.2487	−0.357148
$993$	66.2480	2.10232
$994$	−1.14769	−0.0364025
$995$	−14.1001	−0.447004
$996$	−90.7361	−2.87508
$997$	−24.0211	−0.760755	−0.380378	−	0.924831i	$-0.624206\pi$
$997$	−24.0211	−0.760755	−0.380378	+	0.924831i	$0.624206\pi$
$998$	−55.9304	−1.77045
$999$	11.0938	0.350992

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8047.2.a.e.1.17	✓	168

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8047.2.a.e.1.17	✓	168	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:	\( N \)	\(=\)	\( 8047 = 13 \cdot 619 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8047.a (trivial)

\(f(q)\)	\(=\)	\(q-2.39466 q^{2} -2.35552 q^{3} +3.73441 q^{4} +0.950970 q^{5} +5.64068 q^{6} -0.373318 q^{7} -4.15331 q^{8} +2.54848 q^{9} +O(q^{10})\)	\(q-2.39466 q^{2} -2.35552 q^{3} +3.73441 q^{4} +0.950970 q^{5} +5.64068 q^{6} -0.373318 q^{7} -4.15331 q^{8} +2.54848 q^{9} -2.27725 q^{10} -3.65398 q^{11} -8.79647 q^{12} +1.00000 q^{13} +0.893969 q^{14} -2.24003 q^{15} +2.47697 q^{16} +3.69797 q^{17} -6.10275 q^{18} +3.28094 q^{19} +3.55131 q^{20} +0.879357 q^{21} +8.75004 q^{22} -7.28867 q^{23} +9.78322 q^{24} -4.09566 q^{25} -2.39466 q^{26} +1.06356 q^{27} -1.39412 q^{28} +0.134238 q^{29} +5.36412 q^{30} -4.73608 q^{31} +2.37512 q^{32} +8.60702 q^{33} -8.85540 q^{34} -0.355014 q^{35} +9.51706 q^{36} +10.4308 q^{37} -7.85675 q^{38} -2.35552 q^{39} -3.94968 q^{40} +8.57544 q^{41} -2.10576 q^{42} -4.99431 q^{43} -13.6454 q^{44} +2.42353 q^{45} +17.4539 q^{46} -8.75834 q^{47} -5.83456 q^{48} -6.86063 q^{49} +9.80771 q^{50} -8.71066 q^{51} +3.73441 q^{52} +0.507298 q^{53} -2.54688 q^{54} -3.47482 q^{55} +1.55051 q^{56} -7.72833 q^{57} -0.321454 q^{58} +5.49953 q^{59} -8.36518 q^{60} -11.3538 q^{61} +11.3413 q^{62} -0.951392 q^{63} -10.6415 q^{64} +0.950970 q^{65} -20.6109 q^{66} +7.66320 q^{67} +13.8097 q^{68} +17.1686 q^{69} +0.850138 q^{70} -1.28381 q^{71} -10.5846 q^{72} -10.9061 q^{73} -24.9781 q^{74} +9.64740 q^{75} +12.2524 q^{76} +1.36409 q^{77} +5.64068 q^{78} -10.8737 q^{79} +2.35553 q^{80} -10.1507 q^{81} -20.5353 q^{82} +10.3151 q^{83} +3.28388 q^{84} +3.51666 q^{85} +11.9597 q^{86} -0.316200 q^{87} +15.1761 q^{88} +14.3317 q^{89} -5.80353 q^{90} -0.373318 q^{91} -27.2188 q^{92} +11.1559 q^{93} +20.9733 q^{94} +3.12008 q^{95} -5.59464 q^{96} -12.3722 q^{97} +16.4289 q^{98} -9.31208 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 168 q + 11 q^{2} + 26 q^{3} + 181 q^{4} + 41 q^{5} + 11 q^{6} + 12 q^{7} + 27 q^{8} + 220 q^{9}+O(q^{10}) \) 168 * q + 11 * q^2 + 26 * q^3 + 181 * q^4 + 41 * q^5 + 11 * q^6 + 12 * q^7 + 27 * q^8 + 220 * q^9	\( 168 q + 11 q^{2} + 26 q^{3} + 181 q^{4} + 41 q^{5} + 11 q^{6} + 12 q^{7} + 27 q^{8} + 220 q^{9} + 11 q^{10} + 23 q^{11} + 78 q^{12} + 168 q^{13} + 47 q^{14} + 10 q^{15} + 203 q^{16} + 147 q^{17} + 13 q^{18} + 17 q^{19} + 81 q^{20} + 13 q^{21} + 20 q^{22} + 85 q^{23} + 14 q^{24} + 225 q^{25} + 11 q^{26} + 89 q^{27} + 12 q^{28} + 137 q^{29} + 26 q^{30} + 13 q^{31} + 60 q^{32} + 78 q^{33} - 2 q^{34} + 77 q^{35} + 278 q^{36} + 41 q^{37} + 68 q^{38} + 26 q^{39} + 11 q^{40} + 107 q^{41} + 43 q^{42} + 27 q^{43} + 39 q^{44} + 88 q^{45} - 23 q^{46} + 112 q^{47} + 127 q^{48} + 236 q^{49} + 14 q^{50} + 55 q^{51} + 181 q^{52} + 149 q^{53} + 3 q^{54} + 40 q^{55} + 134 q^{56} + 55 q^{57} - q^{58} + 44 q^{59} - 13 q^{60} + 81 q^{61} + 106 q^{62} + 34 q^{63} + 197 q^{64} + 41 q^{65} - 20 q^{66} - q^{67} + 278 q^{68} + 75 q^{69} - 42 q^{70} + 48 q^{71} - 34 q^{72} + 107 q^{73} + 74 q^{74} + 93 q^{75} + 20 q^{76} + 206 q^{77} + 11 q^{78} + 14 q^{79} + 115 q^{80} + 328 q^{81} + 48 q^{82} + 62 q^{83} - 11 q^{84} + 6 q^{85} + 27 q^{86} + 51 q^{87} + 31 q^{88} + 173 q^{89} - 21 q^{90} + 12 q^{91} + 179 q^{92} + 73 q^{93} + 17 q^{94} + 90 q^{95} - 33 q^{96} + 110 q^{97} - 13 q^{98} + 24 q^{99}+O(q^{100}) \) 168 * q + 11 * q^2 + 26 * q^3 + 181 * q^4 + 41 * q^5 + 11 * q^6 + 12 * q^7 + 27 * q^8 + 220 * q^9 + 11 * q^10 + 23 * q^11 + 78 * q^12 + 168 * q^13 + 47 * q^14 + 10 * q^15 + 203 * q^16 + 147 * q^17 + 13 * q^18 + 17 * q^19 + 81 * q^20 + 13 * q^21 + 20 * q^22 + 85 * q^23 + 14 * q^24 + 225 * q^25 + 11 * q^26 + 89 * q^27 + 12 * q^28 + 137 * q^29 + 26 * q^30 + 13 * q^31 + 60 * q^32 + 78 * q^33 - 2 * q^34 + 77 * q^35 + 278 * q^36 + 41 * q^37 + 68 * q^38 + 26 * q^39 + 11 * q^40 + 107 * q^41 + 43 * q^42 + 27 * q^43 + 39 * q^44 + 88 * q^45 - 23 * q^46 + 112 * q^47 + 127 * q^48 + 236 * q^49 + 14 * q^50 + 55 * q^51 + 181 * q^52 + 149 * q^53 + 3 * q^54 + 40 * q^55 + 134 * q^56 + 55 * q^57 - q^58 + 44 * q^59 - 13 * q^60 + 81 * q^61 + 106 * q^62 + 34 * q^63 + 197 * q^64 + 41 * q^65 - 20 * q^66 - q^67 + 278 * q^68 + 75 * q^69 - 42 * q^70 + 48 * q^71 - 34 * q^72 + 107 * q^73 + 74 * q^74 + 93 * q^75 + 20 * q^76 + 206 * q^77 + 11 * q^78 + 14 * q^79 + 115 * q^80 + 328 * q^81 + 48 * q^82 + 62 * q^83 - 11 * q^84 + 6 * q^85 + 27 * q^86 + 51 * q^87 + 31 * q^88 + 173 * q^89 - 21 * q^90 + 12 * q^91 + 179 * q^92 + 73 * q^93 + 17 * q^94 + 90 * q^95 - 33 * q^96 + 110 * q^97 - 13 * q^98 + 24 * q^99

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