LMFDB - Embedded newform 8045.2.a.d.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8045,2,Mod(1,8045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8045, 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("8045.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8045 = 5 \cdot 1609 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8045.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:	$64.2396484261$
Analytic rank:	$0$
Dimension:	$141$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Character	$\chi$	$=$	8045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.49958 q^{2} -1.75258 q^{3} +4.24791 q^{4} -1.00000 q^{5} +4.38073 q^{6} +1.57689 q^{7} -5.61884 q^{8} +0.0715494 q^{9} +O(q^{10})$	\(q-2.49958 q^{2} -1.75258 q^{3} +4.24791 q^{4} -1.00000 q^{5} +4.38073 q^{6} +1.57689 q^{7} -5.61884 q^{8} +0.0715494 q^{9} +2.49958 q^{10} -4.88328 q^{11} -7.44482 q^{12} -0.233454 q^{13} -3.94156 q^{14} +1.75258 q^{15} +5.54892 q^{16} -3.51214 q^{17} -0.178844 q^{18} +3.67225 q^{19} -4.24791 q^{20} -2.76363 q^{21} +12.2062 q^{22} -3.37391 q^{23} +9.84748 q^{24} +1.00000 q^{25} +0.583539 q^{26} +5.13235 q^{27} +6.69847 q^{28} -5.12512 q^{29} -4.38073 q^{30} -1.31768 q^{31} -2.63231 q^{32} +8.55836 q^{33} +8.77888 q^{34} -1.57689 q^{35} +0.303936 q^{36} -6.46150 q^{37} -9.17908 q^{38} +0.409148 q^{39} +5.61884 q^{40} +0.336812 q^{41} +6.90791 q^{42} +0.351535 q^{43} -20.7437 q^{44} -0.0715494 q^{45} +8.43337 q^{46} -4.06641 q^{47} -9.72495 q^{48} -4.51343 q^{49} -2.49958 q^{50} +6.15532 q^{51} -0.991694 q^{52} -5.96160 q^{53} -12.8287 q^{54} +4.88328 q^{55} -8.86027 q^{56} -6.43592 q^{57} +12.8107 q^{58} +6.53582 q^{59} +7.44482 q^{60} +12.1400 q^{61} +3.29364 q^{62} +0.112825 q^{63} -4.51816 q^{64} +0.233454 q^{65} -21.3923 q^{66} -10.1050 q^{67} -14.9193 q^{68} +5.91306 q^{69} +3.94156 q^{70} +0.174239 q^{71} -0.402024 q^{72} -13.7376 q^{73} +16.1510 q^{74} -1.75258 q^{75} +15.5994 q^{76} -7.70038 q^{77} -1.02270 q^{78} -10.0836 q^{79} -5.54892 q^{80} -9.20953 q^{81} -0.841890 q^{82} -17.1431 q^{83} -11.7396 q^{84} +3.51214 q^{85} -0.878691 q^{86} +8.98221 q^{87} +27.4384 q^{88} +2.50107 q^{89} +0.178844 q^{90} -0.368131 q^{91} -14.3321 q^{92} +2.30934 q^{93} +10.1643 q^{94} -3.67225 q^{95} +4.61335 q^{96} -1.58121 q^{97} +11.2817 q^{98} -0.349396 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 141 q - 8 q^{2} + 11 q^{3} + 158 q^{4} - 141 q^{5} + 23 q^{6} + 29 q^{7} - 21 q^{8} + 160 q^{9}+O(q^{10}) $ 141 * q - 8 * q^2 + 11 * q^3 + 158 * q^4 - 141 * q^5 + 23 * q^6 + 29 * q^7 - 21 * q^8 + 160 * q^9	\( 141 q - 8 q^{2} + 11 q^{3} + 158 q^{4} - 141 q^{5} + 23 q^{6} + 29 q^{7} - 21 q^{8} + 160 q^{9} + 8 q^{10} + 32 q^{11} + 17 q^{12} + 35 q^{13} + 18 q^{14} - 11 q^{15} + 188 q^{16} - 13 q^{17} - 16 q^{18} + 152 q^{19} - 158 q^{20} + 40 q^{21} + 14 q^{22} - 77 q^{23} + 69 q^{24} + 141 q^{25} + 27 q^{26} + 38 q^{27} + 67 q^{28} + 22 q^{29} - 23 q^{30} + 86 q^{31} - 65 q^{32} + 51 q^{33} + 79 q^{34} - 29 q^{35} + 191 q^{36} + 45 q^{37} - 9 q^{38} + 55 q^{39} + 21 q^{40} + 36 q^{41} + 6 q^{42} + 132 q^{43} + 74 q^{44} - 160 q^{45} + 72 q^{46} - 16 q^{47} + 22 q^{48} + 212 q^{49} - 8 q^{50} + 82 q^{51} + 106 q^{52} - 28 q^{53} + 86 q^{54} - 32 q^{55} + 27 q^{56} + 10 q^{57} + 23 q^{58} + 71 q^{59} - 17 q^{60} + 116 q^{61} - 36 q^{62} + 45 q^{63} + 237 q^{64} - 35 q^{65} + 69 q^{66} + 99 q^{67} - 7 q^{68} + 45 q^{69} - 18 q^{70} + 34 q^{71} - 53 q^{72} + 125 q^{73} + 50 q^{74} + 11 q^{75} + 271 q^{76} - 31 q^{77} + 2 q^{78} + 101 q^{79} - 188 q^{80} + 221 q^{81} + 67 q^{82} + 67 q^{83} + 141 q^{84} + 13 q^{85} + 48 q^{86} - 21 q^{87} + 71 q^{88} + 79 q^{89} + 16 q^{90} + 228 q^{91} - 198 q^{92} - 12 q^{93} + 114 q^{94} - 152 q^{95} + 129 q^{96} + 98 q^{97} - 31 q^{98} + 195 q^{99}+O(q^{100}) \) 141 * q - 8 * q^2 + 11 * q^3 + 158 * q^4 - 141 * q^5 + 23 * q^6 + 29 * q^7 - 21 * q^8 + 160 * q^9 + 8 * q^10 + 32 * q^11 + 17 * q^12 + 35 * q^13 + 18 * q^14 - 11 * q^15 + 188 * q^16 - 13 * q^17 - 16 * q^18 + 152 * q^19 - 158 * q^20 + 40 * q^21 + 14 * q^22 - 77 * q^23 + 69 * q^24 + 141 * q^25 + 27 * q^26 + 38 * q^27 + 67 * q^28 + 22 * q^29 - 23 * q^30 + 86 * q^31 - 65 * q^32 + 51 * q^33 + 79 * q^34 - 29 * q^35 + 191 * q^36 + 45 * q^37 - 9 * q^38 + 55 * q^39 + 21 * q^40 + 36 * q^41 + 6 * q^42 + 132 * q^43 + 74 * q^44 - 160 * q^45 + 72 * q^46 - 16 * q^47 + 22 * q^48 + 212 * q^49 - 8 * q^50 + 82 * q^51 + 106 * q^52 - 28 * q^53 + 86 * q^54 - 32 * q^55 + 27 * q^56 + 10 * q^57 + 23 * q^58 + 71 * q^59 - 17 * q^60 + 116 * q^61 - 36 * q^62 + 45 * q^63 + 237 * q^64 - 35 * q^65 + 69 * q^66 + 99 * q^67 - 7 * q^68 + 45 * q^69 - 18 * q^70 + 34 * q^71 - 53 * q^72 + 125 * q^73 + 50 * q^74 + 11 * q^75 + 271 * q^76 - 31 * q^77 + 2 * q^78 + 101 * q^79 - 188 * q^80 + 221 * q^81 + 67 * q^82 + 67 * q^83 + 141 * q^84 + 13 * q^85 + 48 * q^86 - 21 * q^87 + 71 * q^88 + 79 * q^89 + 16 * q^90 + 228 * q^91 - 198 * q^92 - 12 * q^93 + 114 * q^94 - 152 * q^95 + 129 * q^96 + 98 * q^97 - 31 * q^98 + 195 * 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 $

$2$	−2.49958	−1.76747	−0.883736	−	0.467986i	$-0.844980\pi$
$2$	−2.49958	−1.76747	−0.883736	+	0.467986i	$0.844980\pi$
$3$	−1.75258	−1.01185	−0.505927	−	0.862576i	$-0.668850\pi$
$3$	−1.75258	−1.01185	−0.505927	+	0.862576i	$0.668850\pi$
$4$	4.24791	2.12396
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	4.38073	1.78842
$7$	1.57689	0.596007	0.298004	−	0.954565i	$-0.403679\pi$
$7$	1.57689	0.596007	0.298004	+	0.954565i	$0.403679\pi$
$8$	−5.61884	−1.98656
$9$	0.0715494	0.0238498
$10$	2.49958	0.790437
$11$	−4.88328	−1.47236	−0.736182	−	0.676783i	$-0.763373\pi$
$11$	−4.88328	−1.47236	−0.736182	+	0.676783i	$0.763373\pi$
$12$	−7.44482	−2.14913
$13$	−0.233454	−0.0647486	−0.0323743	−	0.999476i	$-0.510307\pi$
$13$	−0.233454	−0.0647486	−0.0323743	+	0.999476i	$0.510307\pi$
$14$	−3.94156	−1.05343
$15$	1.75258	0.452515
$16$	5.54892	1.38723
$17$	−3.51214	−0.851819	−0.425910	−	0.904766i	$-0.640046\pi$
$17$	−3.51214	−0.851819	−0.425910	+	0.904766i	$0.640046\pi$
$18$	−0.178844	−0.0421539
$19$	3.67225	0.842471	0.421236	−	0.906951i	$-0.361597\pi$
$19$	3.67225	0.842471	0.421236	+	0.906951i	$0.361597\pi$
$20$	−4.24791	−0.949862
$21$	−2.76363	−0.603073
$22$	12.2062	2.60236
$23$	−3.37391	−0.703510	−0.351755	−	0.936092i	$-0.614415\pi$
$23$	−3.37391	−0.703510	−0.351755	+	0.936092i	$0.614415\pi$
$24$	9.84748	2.01011
$25$	1.00000	0.200000
$26$	0.583539	0.114441
$27$	5.13235	0.987722
$28$	6.69847	1.26589
$29$	−5.12512	−0.951711	−0.475856	−	0.879523i	$-0.657862\pi$
$29$	−5.12512	−0.951711	−0.475856	+	0.879523i	$0.657862\pi$
$30$	−4.38073	−0.799808
$31$	−1.31768	−0.236662	−0.118331	−	0.992974i	$-0.537754\pi$
$31$	−1.31768	−0.236662	−0.118331	+	0.992974i	$0.537754\pi$
$32$	−2.63231	−0.465331
$33$	8.55836	1.48982
$34$	8.77888	1.50557
$35$	−1.57689	−0.266543
$36$	0.303936	0.0506559
$37$	−6.46150	−1.06226	−0.531132	−	0.847289i	$-0.678233\pi$
$37$	−6.46150	−1.06226	−0.531132	+	0.847289i	$0.678233\pi$
$38$	−9.17908	−1.48904
$39$	0.409148	0.0655162
$40$	5.61884	0.888416
$41$	0.336812	0.0526013	0.0263006	−	0.999654i	$-0.491627\pi$
$41$	0.336812	0.0526013	0.0263006	+	0.999654i	$0.491627\pi$
$42$	6.90791	1.06591
$43$	0.351535	0.0536086	0.0268043	−	0.999641i	$-0.491467\pi$
$43$	0.351535	0.0536086	0.0268043	+	0.999641i	$0.491467\pi$
$44$	−20.7437	−3.12724
$45$	−0.0715494	−0.0106660
$46$	8.43337	1.24343
$47$	−4.06641	−0.593147	−0.296573	−	0.955010i	$-0.595844\pi$
$47$	−4.06641	−0.593147	−0.296573	+	0.955010i	$0.595844\pi$
$48$	−9.72495	−1.40368
$49$	−4.51343	−0.644775
$50$	−2.49958	−0.353494
$51$	6.15532	0.861917
$52$	−0.991694	−0.137523
$53$	−5.96160	−0.818888	−0.409444	−	0.912335i	$-0.634277\pi$
$53$	−5.96160	−0.818888	−0.409444	+	0.912335i	$0.634277\pi$
$54$	−12.8287	−1.74577
$55$	4.88328	0.658462
$56$	−8.86027	−1.18400
$57$	−6.43592	−0.852458
$58$	12.8107	1.68212
$59$	6.53582	0.850892	0.425446	−	0.904984i	$-0.360117\pi$
$59$	6.53582	0.850892	0.425446	+	0.904984i	$0.360117\pi$
$60$	7.44482	0.961122
$61$	12.1400	1.55437	0.777184	−	0.629274i	$-0.216648\pi$
$61$	12.1400	1.55437	0.777184	+	0.629274i	$0.216648\pi$
$62$	3.29364	0.418293
$63$	0.112825	0.0142147
$64$	−4.51816	−0.564770
$65$	0.233454	0.0289565
$66$	−21.3923	−2.63321
$67$	−10.1050	−1.23452	−0.617260	−	0.786759i	$-0.711757\pi$
$67$	−10.1050	−1.23452	−0.617260	+	0.786759i	$0.711757\pi$
$68$	−14.9193	−1.80923
$69$	5.91306	0.711849
$70$	3.94156	0.471106
$71$	0.174239	0.0206783	0.0103392	−	0.999947i	$-0.496709\pi$
$71$	0.174239	0.0206783	0.0103392	+	0.999947i	$0.496709\pi$
$72$	−0.402024	−0.0473790
$73$	−13.7376	−1.60786	−0.803930	−	0.594724i	$-0.797261\pi$
$73$	−13.7376	−1.60786	−0.803930	+	0.594724i	$0.797261\pi$
$74$	16.1510	1.87752
$75$	−1.75258	−0.202371
$76$	15.5994	1.78937
$77$	−7.70038	−0.877540
$78$	−1.02270	−0.115798
$79$	−10.0836	−1.13449	−0.567246	−	0.823548i	$-0.691991\pi$
$79$	−10.0836	−1.13449	−0.567246	+	0.823548i	$0.691991\pi$
$80$	−5.54892	−0.620388
$81$	−9.20953	−1.02328
$82$	−0.841890	−0.0929712
$83$	−17.1431	−1.88170	−0.940850	−	0.338823i	$-0.889971\pi$
$83$	−17.1431	−1.88170	−0.940850	+	0.338823i	$0.889971\pi$
$84$	−11.7396	−1.28090
$85$	3.51214	0.380945
$86$	−0.878691	−0.0947517
$87$	8.98221	0.962994
$88$	27.4384	2.92494
$89$	2.50107	0.265113	0.132556	−	0.991175i	$-0.457681\pi$
$89$	2.50107	0.265113	0.132556	+	0.991175i	$0.457681\pi$
$90$	0.178844	0.0188518
$91$	−0.368131	−0.0385906
$92$	−14.3321	−1.49422
$93$	2.30934	0.239467
$94$	10.1643	1.04837
$95$	−3.67225	−0.376765
$96$	4.61335	0.470848
$97$	−1.58121	−0.160548	−0.0802740	−	0.996773i	$-0.525580\pi$
$97$	−1.58121	−0.160548	−0.0802740	+	0.996773i	$0.525580\pi$
$98$	11.2817	1.13962
$99$	−0.349396	−0.0351156
$100$	4.24791	0.424791
$101$	8.19557	0.815489	0.407745	−	0.913096i	$-0.366315\pi$
$101$	8.19557	0.815489	0.407745	+	0.913096i	$0.366315\pi$
$102$	−15.3857	−1.52341
$103$	5.19593	0.511970	0.255985	−	0.966681i	$-0.417600\pi$
$103$	5.19593	0.511970	0.255985	+	0.966681i	$0.417600\pi$
$104$	1.31174	0.128627
$105$	2.76363	0.269702
$106$	14.9015	1.44736
$107$	−13.1164	−1.26801	−0.634007	−	0.773328i	$-0.718591\pi$
$107$	−13.1164	−1.26801	−0.634007	+	0.773328i	$0.718591\pi$
$108$	21.8018	2.09788
$109$	17.6580	1.69133	0.845666	−	0.533712i	$-0.179204\pi$
$109$	17.6580	1.69133	0.845666	+	0.533712i	$0.179204\pi$
$110$	−12.2062	−1.16381
$111$	11.3243	1.07486
$112$	8.75002	0.826799
$113$	−7.19094	−0.676466	−0.338233	−	0.941062i	$-0.609829\pi$
$113$	−7.19094	−0.676466	−0.338233	+	0.941062i	$0.609829\pi$
$114$	16.0871	1.50670
$115$	3.37391	0.314619
$116$	−21.7711	−2.02139
$117$	−0.0167035	−0.00154424
$118$	−16.3368	−1.50393
$119$	−5.53825	−0.507691
$120$	−9.84748	−0.898948
$121$	12.8464	1.16786
$122$	−30.3449	−2.74730
$123$	−0.590292	−0.0532248
$124$	−5.59737	−0.502659
$125$	−1.00000	−0.0894427
$126$	−0.282016	−0.0251240
$127$	8.54095	0.757887	0.378944	−	0.925420i	$-0.376287\pi$
$127$	8.54095	0.757887	0.378944	+	0.925420i	$0.376287\pi$
$128$	16.5581	1.46355
$129$	−0.616095	−0.0542441
$130$	−0.583539	−0.0511797
$131$	−13.9655	−1.22017	−0.610085	−	0.792336i	$-0.708865\pi$
$131$	−13.9655	−1.22017	−0.610085	+	0.792336i	$0.708865\pi$
$132$	36.3551	3.16431
$133$	5.79072	0.502119
$134$	25.2582	2.18198
$135$	−5.13235	−0.441723
$136$	19.7341	1.69219
$137$	4.37830	0.374063	0.187032	−	0.982354i	$-0.440113\pi$
$137$	4.37830	0.374063	0.187032	+	0.982354i	$0.440113\pi$
$138$	−14.7802	−1.25817
$139$	−17.3019	−1.46753	−0.733765	−	0.679404i	$-0.762238\pi$
$139$	−17.3019	−1.46753	−0.733765	+	0.679404i	$0.762238\pi$
$140$	−6.69847	−0.566124
$141$	7.12672	0.600178
$142$	−0.435524	−0.0365483
$143$	1.14002	0.0953336
$144$	0.397022	0.0330852
$145$	5.12512	0.425618
$146$	34.3382	2.84185
$147$	7.91016	0.652419
$148$	−27.4479	−2.25620
$149$	5.24526	0.429709	0.214854	−	0.976646i	$-0.431072\pi$
$149$	5.24526	0.429709	0.214854	+	0.976646i	$0.431072\pi$
$150$	4.38073	0.357685
$151$	6.43739	0.523867	0.261934	−	0.965086i	$-0.415640\pi$
$151$	6.43739	0.523867	0.261934	+	0.965086i	$0.415640\pi$
$152$	−20.6337	−1.67362
$153$	−0.251292	−0.0203157
$154$	19.2477	1.55103
$155$	1.31768	0.105838
$156$	1.73803	0.139153
$157$	−12.4222	−0.991403	−0.495702	−	0.868493i	$-0.665089\pi$
$157$	−12.4222	−0.991403	−0.495702	+	0.868493i	$0.665089\pi$
$158$	25.2048	2.00518
$159$	10.4482	0.828596
$160$	2.63231	0.208103
$161$	−5.32028	−0.419297
$162$	23.0200	1.80862
$163$	−4.92495	−0.385752	−0.192876	−	0.981223i	$-0.561781\pi$
$163$	−4.92495	−0.385752	−0.192876	+	0.981223i	$0.561781\pi$
$164$	1.43075	0.111723
$165$	−8.55836	−0.666267
$166$	42.8506	3.32585
$167$	−11.6989	−0.905287	−0.452643	−	0.891692i	$-0.649519\pi$
$167$	−11.6989	−0.905287	−0.452643	+	0.891692i	$0.649519\pi$
$168$	15.5284	1.19804
$169$	−12.9455	−0.995808
$170$	−8.77888	−0.673310
$171$	0.262747	0.0200928
$172$	1.49329	0.113862
$173$	9.13788	0.694740	0.347370	−	0.937728i	$-0.387075\pi$
$173$	9.13788	0.694740	0.347370	+	0.937728i	$0.387075\pi$
$174$	−22.4518	−1.70206
$175$	1.57689	0.119201
$176$	−27.0969	−2.04251
$177$	−11.4546	−0.860979
$178$	−6.25162	−0.468579
$179$	14.8574	1.11050	0.555248	−	0.831685i	$-0.312623\pi$
$179$	14.8574	1.11050	0.555248	+	0.831685i	$0.312623\pi$
$180$	−0.303936	−0.0226540
$181$	10.4079	0.773610	0.386805	−	0.922162i	$-0.373579\pi$
$181$	10.4079	0.773610	0.386805	+	0.922162i	$0.373579\pi$
$182$	0.920174	0.0682079
$183$	−21.2764	−1.57279
$184$	18.9575	1.39756
$185$	6.46150	0.475059
$186$	−5.77238	−0.423252
$187$	17.1508	1.25419
$188$	−17.2737	−1.25982
$189$	8.09314	0.588690
$190$	9.17908	0.665920
$191$	−8.70878	−0.630145	−0.315073	−	0.949068i	$-0.602029\pi$
$191$	−8.70878	−0.630145	−0.315073	+	0.949068i	$0.602029\pi$
$192$	7.91846	0.571466
$193$	18.4211	1.32598	0.662990	−	0.748628i	$-0.269287\pi$
$193$	18.4211	1.32598	0.662990	+	0.748628i	$0.269287\pi$
$194$	3.95237	0.283764
$195$	−0.409148	−0.0292997
$196$	−19.1726	−1.36947
$197$	2.16143	0.153995	0.0769976	−	0.997031i	$-0.475467\pi$
$197$	2.16143	0.153995	0.0769976	+	0.997031i	$0.475467\pi$
$198$	0.873344	0.0620658
$199$	12.0088	0.851284	0.425642	−	0.904892i	$-0.360048\pi$
$199$	12.0088	0.851284	0.425642	+	0.904892i	$0.360048\pi$
$200$	−5.61884	−0.397312
$201$	17.7098	1.24915
$202$	−20.4855	−1.44135
$203$	−8.08174	−0.567227
$204$	26.1473	1.83067
$205$	−0.336812	−0.0235240
$206$	−12.9876	−0.904892
$207$	−0.241402	−0.0167786
$208$	−1.29542	−0.0898212
$209$	−17.9326	−1.24042
$210$	−6.90791	−0.476691
$211$	1.79485	0.123563	0.0617813	−	0.998090i	$-0.480322\pi$
$211$	1.79485	0.123563	0.0617813	+	0.998090i	$0.480322\pi$
$212$	−25.3243	−1.73928
$213$	−0.305368	−0.0209234
$214$	32.7856	2.24118
$215$	−0.351535	−0.0239745
$216$	−28.8379	−1.96217
$217$	−2.07783	−0.141052
$218$	−44.1377	−2.98938
$219$	24.0762	1.62692
$220$	20.7437	1.39854
$221$	0.819925	0.0551541
$222$	−28.3061	−1.89978
$223$	−11.2475	−0.753191	−0.376595	−	0.926378i	$-0.622905\pi$
$223$	−11.2475	−0.753191	−0.376595	+	0.926378i	$0.622905\pi$
$224$	−4.15086	−0.277341
$225$	0.0715494	0.00476996
$226$	17.9743	1.19563
$227$	−24.0517	−1.59636	−0.798182	−	0.602416i	$-0.794205\pi$
$227$	−24.0517	−1.59636	−0.798182	+	0.602416i	$0.794205\pi$
$228$	−27.3392	−1.81058
$229$	−2.28020	−0.150680	−0.0753399	−	0.997158i	$-0.524004\pi$
$229$	−2.28020	−0.150680	−0.0753399	+	0.997158i	$0.524004\pi$
$230$	−8.43337	−0.556080
$231$	13.4956	0.887943
$232$	28.7972	1.89063
$233$	−4.40843	−0.288806	−0.144403	−	0.989519i	$-0.546126\pi$
$233$	−4.40843	−0.288806	−0.144403	+	0.989519i	$0.546126\pi$
$234$	0.0417518	0.00272940
$235$	4.06641	0.265263
$236$	27.7636	1.80726
$237$	17.6723	1.14794
$238$	13.8433	0.897328
$239$	−7.22522	−0.467361	−0.233680	−	0.972313i	$-0.575077\pi$
$239$	−7.22522	−0.467361	−0.233680	+	0.972313i	$0.575077\pi$
$240$	9.72495	0.627743
$241$	0.598223	0.0385350	0.0192675	−	0.999814i	$-0.493867\pi$
$241$	0.598223	0.0385350	0.0192675	+	0.999814i	$0.493867\pi$
$242$	−32.1107	−2.06416
$243$	0.743406	0.0476895
$244$	51.5696	3.30141
$245$	4.51343	0.288352
$246$	1.47548	0.0940734
$247$	−0.857302	−0.0545488
$248$	7.40381	0.470143
$249$	30.0447	1.90401
$250$	2.49958	0.158087
$251$	−3.90537	−0.246505	−0.123252	−	0.992375i	$-0.539332\pi$
$251$	−3.90537	−0.246505	−0.123252	+	0.992375i	$0.539332\pi$
$252$	0.479272	0.0301913
$253$	16.4758	1.03582
$254$	−21.3488	−1.33954
$255$	−6.15532	−0.385461
$256$	−32.3521	−2.02201
$257$	−11.3508	−0.708041	−0.354020	−	0.935238i	$-0.615186\pi$
$257$	−11.3508	−0.708041	−0.354020	+	0.935238i	$0.615186\pi$
$258$	1.53998	0.0958749
$259$	−10.1891	−0.633117
$260$	0.991694	0.0615022
$261$	−0.366700	−0.0226981
$262$	34.9079	2.15661
$263$	7.11975	0.439022	0.219511	−	0.975610i	$-0.429554\pi$
$263$	7.11975	0.439022	0.219511	+	0.975610i	$0.429554\pi$
$264$	−48.0880	−2.95961
$265$	5.96160	0.366218
$266$	−14.4744	−0.887481
$267$	−4.38333	−0.268255
$268$	−42.9250	−2.62206
$269$	−21.4877	−1.31013	−0.655064	−	0.755573i	$-0.727358\pi$
$269$	−21.4877	−1.31013	−0.655064	+	0.755573i	$0.727358\pi$
$270$	12.8287	0.780732
$271$	−15.2721	−0.927716	−0.463858	−	0.885910i	$-0.653535\pi$
$271$	−15.2721	−0.927716	−0.463858	+	0.885910i	$0.653535\pi$
$272$	−19.4886	−1.18167
$273$	0.645181	0.0390481
$274$	−10.9439	−0.661147
$275$	−4.88328	−0.294473
$276$	25.1182	1.51194
$277$	−6.47162	−0.388842	−0.194421	−	0.980918i	$-0.562283\pi$
$277$	−6.47162	−0.388842	−0.194421	+	0.980918i	$0.562283\pi$
$278$	43.2476	2.59382
$279$	−0.0942790	−0.00564434
$280$	8.86027	0.529502
$281$	−9.47135	−0.565013	−0.282507	−	0.959265i	$-0.591166\pi$
$281$	−9.47135	−0.565013	−0.282507	+	0.959265i	$0.591166\pi$
$282$	−17.8138	−1.06080
$283$	−8.95125	−0.532096	−0.266048	−	0.963960i	$-0.585718\pi$
$283$	−8.95125	−0.532096	−0.266048	+	0.963960i	$0.585718\pi$
$284$	0.740150	0.0439198
$285$	6.43592	0.381231
$286$	−2.84958	−0.168499
$287$	0.531115	0.0313507
$288$	−0.188340	−0.0110981
$289$	−4.66486	−0.274404
$290$	−12.8107	−0.752268
$291$	2.77121	0.162451
$292$	−58.3559	−3.41502
$293$	−33.1503	−1.93666	−0.968331	−	0.249672i	$-0.919677\pi$
$293$	−33.1503	−1.93666	−0.968331	+	0.249672i	$0.919677\pi$
$294$	−19.7721	−1.15313
$295$	−6.53582	−0.380530
$296$	36.3061	2.11025
$297$	−25.0627	−1.45429
$298$	−13.1110	−0.759498
$299$	0.787655	0.0455513
$300$	−7.44482	−0.429827
$301$	0.554331	0.0319511
$302$	−16.0908	−0.925921
$303$	−14.3634	−0.825157
$304$	20.3770	1.16870
$305$	−12.1400	−0.695134
$306$	0.628124	0.0359075
$307$	−29.5736	−1.68785	−0.843927	−	0.536458i	$-0.819762\pi$
$307$	−29.5736	−1.68785	−0.843927	+	0.536458i	$0.819762\pi$
$308$	−32.7105	−1.86386
$309$	−9.10629	−0.518039
$310$	−3.29364	−0.187066
$311$	32.7259	1.85572	0.927859	−	0.372932i	$-0.121647\pi$
$311$	32.7259	1.85572	0.927859	+	0.372932i	$0.121647\pi$
$312$	−2.29894	−0.130152
$313$	−30.1842	−1.70611	−0.853056	−	0.521819i	$-0.825254\pi$
$313$	−30.1842	−1.70611	−0.853056	+	0.521819i	$0.825254\pi$
$314$	31.0504	1.75228
$315$	−0.112825	−0.00635699
$316$	−42.8342	−2.40961
$317$	25.1455	1.41231	0.706155	−	0.708058i	$-0.250428\pi$
$317$	25.1455	1.41231	0.706155	+	0.708058i	$0.250428\pi$
$318$	−26.1161	−1.46452
$319$	25.0274	1.40127
$320$	4.51816	0.252573
$321$	22.9876	1.28305
$322$	13.2985	0.741095
$323$	−12.8974	−0.717633
$324$	−39.1213	−2.17340
$325$	−0.233454	−0.0129497
$326$	12.3103	0.681805
$327$	−30.9472	−1.71138
$328$	−1.89249	−0.104496
$329$	−6.41227	−0.353520
$330$	21.3923	1.17761
$331$	−4.59614	−0.252627	−0.126313	−	0.991990i	$-0.540314\pi$
$331$	−4.59614	−0.252627	−0.126313	+	0.991990i	$0.540314\pi$
$332$	−72.8224	−3.99665
$333$	−0.462316	−0.0253348
$334$	29.2423	1.60007
$335$	10.1050	0.552094
$336$	−15.3351	−0.836601
$337$	4.84552	0.263952	0.131976	−	0.991253i	$-0.457868\pi$
$337$	4.84552	0.263952	0.131976	+	0.991253i	$0.457868\pi$
$338$	32.3583	1.76006
$339$	12.6027	0.684486
$340$	14.9193	0.809110
$341$	6.43459	0.348452
$342$	−0.656758	−0.0355134
$343$	−18.1554	−0.980298
$344$	−1.97522	−0.106497
$345$	−5.91306	−0.318349
$346$	−22.8409	−1.22793
$347$	36.5716	1.96327	0.981634	−	0.190771i	$-0.0610989\pi$
$347$	36.5716	1.96327	0.981634	+	0.190771i	$0.0610989\pi$
$348$	38.1556	2.04536
$349$	−31.8107	−1.70279	−0.851395	−	0.524525i	$-0.824243\pi$
$349$	−31.8107	−1.70279	−0.851395	+	0.524525i	$0.824243\pi$
$350$	−3.94156	−0.210685
$351$	−1.19817	−0.0639536
$352$	12.8543	0.685138
$353$	25.6645	1.36599	0.682993	−	0.730425i	$-0.260678\pi$
$353$	25.6645	1.36599	0.682993	+	0.730425i	$0.260678\pi$
$354$	28.6317	1.52176
$355$	−0.174239	−0.00924762
$356$	10.6243	0.563087
$357$	9.70625	0.513709
$358$	−37.1374	−1.96277
$359$	−25.1846	−1.32919	−0.664596	−	0.747203i	$-0.731396\pi$
$359$	−25.1846	−1.32919	−0.664596	+	0.747203i	$0.731396\pi$
$360$	0.402024	0.0211886
$361$	−5.51461	−0.290243
$362$	−26.0153	−1.36733
$363$	−22.5145	−1.18170
$364$	−1.56379	−0.0819648
$365$	13.7376	0.719057
$366$	53.1820	2.77987
$367$	9.08643	0.474308	0.237154	−	0.971472i	$-0.423785\pi$
$367$	9.08643	0.474308	0.237154	+	0.971472i	$0.423785\pi$
$368$	−18.7216	−0.975930
$369$	0.0240987	0.00125453
$370$	−16.1510	−0.839653
$371$	−9.40077	−0.488063
$372$	9.80987	0.508618
$373$	−18.6758	−0.966994	−0.483497	−	0.875346i	$-0.660634\pi$
$373$	−18.6758	−0.966994	−0.483497	+	0.875346i	$0.660634\pi$
$374$	−42.8698	−2.21674
$375$	1.75258	0.0905030
$376$	22.8485	1.17832
$377$	1.19648	0.0616220
$378$	−20.2295	−1.04049
$379$	−7.11499	−0.365472	−0.182736	−	0.983162i	$-0.558495\pi$
$379$	−7.11499	−0.365472	−0.182736	+	0.983162i	$0.558495\pi$
$380$	−15.5994	−0.800231
$381$	−14.9687	−0.766872
$382$	21.7683	1.11376
$383$	20.0236	1.02316	0.511579	−	0.859236i	$-0.329061\pi$
$383$	20.0236	1.02316	0.511579	+	0.859236i	$0.329061\pi$
$384$	−29.0195	−1.48090
$385$	7.70038	0.392448
$386$	−46.0451	−2.34363
$387$	0.0251521	0.00127855
$388$	−6.71685	−0.340997
$389$	−21.8750	−1.10910	−0.554552	−	0.832149i	$-0.687110\pi$
$389$	−21.8750	−1.10910	−0.554552	+	0.832149i	$0.687110\pi$
$390$	1.02270	0.0517864
$391$	11.8497	0.599263
$392$	25.3602	1.28088
$393$	24.4757	1.23463
$394$	−5.40266	−0.272182
$395$	10.0836	0.507361
$396$	−1.48420	−0.0745840
$397$	7.99940	0.401478	0.200739	−	0.979645i	$-0.435666\pi$
$397$	7.99940	0.401478	0.200739	+	0.979645i	$0.435666\pi$
$398$	−30.0171	−1.50462
$399$	−10.1487	−0.508071
$400$	5.54892	0.277446
$401$	21.1239	1.05488	0.527439	−	0.849593i	$-0.323152\pi$
$401$	21.1239	1.05488	0.527439	+	0.849593i	$0.323152\pi$
$402$	−44.2671	−2.20784
$403$	0.307618	0.0153235
$404$	34.8140	1.73206
$405$	9.20953	0.457625
$406$	20.2010	1.00256
$407$	31.5533	1.56404
$408$	−34.5857	−1.71225
$409$	1.72141	0.0851183	0.0425592	−	0.999094i	$-0.486449\pi$
$409$	1.72141	0.0851183	0.0425592	+	0.999094i	$0.486449\pi$
$410$	0.841890	0.0415780
$411$	−7.67334	−0.378498
$412$	22.0718	1.08740
$413$	10.3063	0.507138
$414$	0.603403	0.0296556
$415$	17.1431	0.841522
$416$	0.614525	0.0301296
$417$	30.3230	1.48493
$418$	44.8240	2.19242
$419$	12.9452	0.632417	0.316208	−	0.948690i	$-0.397590\pi$
$419$	12.9452	0.632417	0.316208	+	0.948690i	$0.397590\pi$
$420$	11.7396	0.572836
$421$	−2.31905	−0.113024	−0.0565118	−	0.998402i	$-0.517998\pi$
$421$	−2.31905	−0.113024	−0.0565118	+	0.998402i	$0.517998\pi$
$422$	−4.48637	−0.218393
$423$	−0.290949	−0.0141464
$424$	33.4972	1.62677
$425$	−3.51214	−0.170364
$426$	0.763291	0.0369816
$427$	19.1434	0.926414
$428$	−55.7174	−2.69320
$429$	−1.99799	−0.0964637
$430$	0.878691	0.0423742
$431$	5.60455	0.269962	0.134981	−	0.990848i	$-0.456903\pi$
$431$	5.60455	0.269962	0.134981	+	0.990848i	$0.456903\pi$
$432$	28.4790	1.37020
$433$	7.75163	0.372520	0.186260	−	0.982501i	$-0.440363\pi$
$433$	7.75163	0.372520	0.186260	+	0.982501i	$0.440363\pi$
$434$	5.19370	0.249306
$435$	−8.98221	−0.430664
$436$	75.0097	3.59231
$437$	−12.3898	−0.592686
$438$	−60.1805	−2.87554
$439$	12.4193	0.592742	0.296371	−	0.955073i	$-0.404223\pi$
$439$	12.4193	0.592742	0.296371	+	0.955073i	$0.404223\pi$
$440$	−27.4384	−1.30807
$441$	−0.322933	−0.0153778
$442$	−2.04947	−0.0974833
$443$	−16.0965	−0.764765	−0.382383	−	0.924004i	$-0.624896\pi$
$443$	−16.0965	−0.764765	−0.382383	+	0.924004i	$0.624896\pi$
$444$	48.1047	2.28295
$445$	−2.50107	−0.118562
$446$	28.1141	1.33124
$447$	−9.19276	−0.434803
$448$	−7.12463	−0.336607
$449$	−7.38856	−0.348688	−0.174344	−	0.984685i	$-0.555780\pi$
$449$	−7.38856	−0.348688	−0.174344	+	0.984685i	$0.555780\pi$
$450$	−0.178844	−0.00843077
$451$	−1.64475	−0.0774483
$452$	−30.5465	−1.43678
$453$	−11.2821	−0.530078
$454$	60.1191	2.82153
$455$	0.368131	0.0172583
$456$	36.1624	1.69346
$457$	−15.5835	−0.728967	−0.364483	−	0.931210i	$-0.618754\pi$
$457$	−15.5835	−0.728967	−0.364483	+	0.931210i	$0.618754\pi$
$458$	5.69955	0.266322
$459$	−18.0256	−0.841361
$460$	14.3321	0.668237
$461$	−12.3076	−0.573223	−0.286612	−	0.958047i	$-0.592529\pi$
$461$	−12.3076	−0.573223	−0.286612	+	0.958047i	$0.592529\pi$
$462$	−33.7333	−1.56941
$463$	21.2410	0.987151	0.493576	−	0.869703i	$-0.335690\pi$
$463$	21.2410	0.987151	0.493576	+	0.869703i	$0.335690\pi$
$464$	−28.4389	−1.32024
$465$	−2.30934	−0.107093
$466$	11.0192	0.510456
$467$	−7.20720	−0.333509	−0.166755	−	0.985998i	$-0.553329\pi$
$467$	−7.20720	−0.333509	−0.166755	+	0.985998i	$0.553329\pi$
$468$	−0.0709551	−0.00327990
$469$	−15.9344	−0.735783
$470$	−10.1643	−0.468845
$471$	21.7710	1.00316
$472$	−36.7237	−1.69035
$473$	−1.71664	−0.0789314
$474$	−44.1735	−2.02895
$475$	3.67225	0.168494
$476$	−23.5260	−1.07831
$477$	−0.426549	−0.0195303
$478$	18.0600	0.826047
$479$	−1.92832	−0.0881070	−0.0440535	−	0.999029i	$-0.514027\pi$
$479$	−1.92832	−0.0881070	−0.0440535	+	0.999029i	$0.514027\pi$
$480$	−4.61335	−0.210570
$481$	1.50847	0.0687801
$482$	−1.49531	−0.0681094
$483$	9.32424	0.424267
$484$	54.5705	2.48048
$485$	1.58121	0.0717992
$486$	−1.85820	−0.0842899
$487$	−18.2563	−0.827273	−0.413637	−	0.910442i	$-0.635742\pi$
$487$	−18.2563	−0.827273	−0.413637	+	0.910442i	$0.635742\pi$
$488$	−68.2127	−3.08784
$489$	8.63138	0.390325
$490$	−11.2817	−0.509654
$491$	43.9517	1.98351	0.991756	−	0.128141i	$-0.0409011\pi$
$491$	43.9517	1.98351	0.991756	+	0.128141i	$0.0409011\pi$
$492$	−2.50751	−0.113047
$493$	18.0002	0.810686
$494$	2.14290	0.0964135
$495$	0.349396	0.0157042
$496$	−7.31169	−0.328304
$497$	0.274754	0.0123244
$498$	−75.0993	−3.36528
$499$	−10.2585	−0.459234	−0.229617	−	0.973281i	$-0.573747\pi$
$499$	−10.2585	−0.459234	−0.229617	+	0.973281i	$0.573747\pi$
$500$	−4.24791	−0.189972
$501$	20.5033	0.916018
$502$	9.76178	0.435690
$503$	−16.9166	−0.754275	−0.377137	−	0.926157i	$-0.623092\pi$
$503$	−16.9166	−0.754275	−0.377137	+	0.926157i	$0.623092\pi$
$504$	−0.633947	−0.0282383
$505$	−8.19557	−0.364698
$506$	−41.1825	−1.83079
$507$	22.6881	1.00761
$508$	36.2812	1.60972
$509$	−9.79760	−0.434271	−0.217136	−	0.976141i	$-0.569671\pi$
$509$	−9.79760	−0.434271	−0.217136	+	0.976141i	$0.569671\pi$
$510$	15.3857	0.681292
$511$	−21.6626	−0.958296
$512$	47.7505	2.11029
$513$	18.8473	0.832127
$514$	28.3722	1.25144
$515$	−5.19593	−0.228960
$516$	−2.61711	−0.115212
$517$	19.8574	0.873329
$518$	25.4684	1.11902
$519$	−16.0149	−0.702976
$520$	−1.31174	−0.0575237
$521$	−23.1465	−1.01407	−0.507034	−	0.861926i	$-0.669258\pi$
$521$	−23.1465	−1.01407	−0.507034	+	0.861926i	$0.669258\pi$
$522$	0.916596	0.0401183
$523$	−11.1630	−0.488122	−0.244061	−	0.969760i	$-0.578480\pi$
$523$	−11.1630	−0.488122	−0.244061	+	0.969760i	$0.578480\pi$
$524$	−59.3241	−2.59159
$525$	−2.76363	−0.120615
$526$	−17.7964	−0.775959
$527$	4.62787	0.201593
$528$	47.4897	2.06672
$529$	−11.6167	−0.505074
$530$	−14.9015	−0.647280
$531$	0.467634	0.0202936
$532$	24.5984	1.06648
$533$	−0.0786304	−0.00340586
$534$	10.9565	0.474134
$535$	13.1164	0.567073
$536$	56.7782	2.45245
$537$	−26.0389	−1.12366
$538$	53.7103	2.31561
$539$	22.0403	0.949345
$540$	−21.8018	−0.938199
$541$	2.13822	0.0919292	0.0459646	−	0.998943i	$-0.485364\pi$
$541$	2.13822	0.0919292	0.0459646	+	0.998943i	$0.485364\pi$
$542$	38.1739	1.63971
$543$	−18.2406	−0.782781
$544$	9.24505	0.396378
$545$	−17.6580	−0.756387
$546$	−1.61268	−0.0690164
$547$	28.4656	1.21710	0.608552	−	0.793514i	$-0.291751\pi$
$547$	28.4656	1.21710	0.608552	+	0.793514i	$0.291751\pi$
$548$	18.5986	0.794494
$549$	0.868610	0.0370714
$550$	12.2062	0.520473
$551$	−18.8207	−0.801789
$552$	−33.2245	−1.41413
$553$	−15.9007	−0.676166
$554$	16.1764	0.687267
$555$	−11.3243	−0.480690
$556$	−73.4970	−3.11697
$557$	34.8716	1.47756	0.738778	−	0.673949i	$-0.235403\pi$
$557$	34.8716	1.47756	0.738778	+	0.673949i	$0.235403\pi$
$558$	0.235658	0.00997621
$559$	−0.0820674	−0.00347108
$560$	−8.75002	−0.369756
$561$	−30.0582	−1.26906
$562$	23.6744	0.998644
$563$	20.3103	0.855977	0.427989	−	0.903784i	$-0.359222\pi$
$563$	20.3103	0.855977	0.427989	+	0.903784i	$0.359222\pi$
$564$	30.2737	1.27475
$565$	7.19094	0.302525
$566$	22.3744	0.940465
$567$	−14.5224	−0.609883
$568$	−0.979018	−0.0410787
$569$	−17.3941	−0.729198	−0.364599	−	0.931165i	$-0.618794\pi$
$569$	−17.3941	−0.729198	−0.364599	+	0.931165i	$0.618794\pi$
$570$	−16.0871	−0.673815
$571$	−23.3953	−0.979063	−0.489531	−	0.871986i	$-0.662832\pi$
$571$	−23.3953	−0.979063	−0.489531	+	0.871986i	$0.662832\pi$
$572$	4.84272	0.202484
$573$	15.2629	0.637615
$574$	−1.32757	−0.0554115
$575$	−3.37391	−0.140702
$576$	−0.323272	−0.0134697
$577$	11.4381	0.476176	0.238088	−	0.971244i	$-0.423479\pi$
$577$	11.4381	0.476176	0.238088	+	0.971244i	$0.423479\pi$
$578$	11.6602	0.485001
$579$	−32.2845	−1.34170
$580$	21.7711	0.903994
$581$	−27.0327	−1.12151
$582$	−6.92686	−0.287128
$583$	29.1122	1.20570
$584$	77.1891	3.19411
$585$	0.0167035	0.000690606 0
$586$	82.8619	3.42299
$587$	−8.93819	−0.368918	−0.184459	−	0.982840i	$-0.559053\pi$
$587$	−8.93819	−0.368918	−0.184459	+	0.982840i	$0.559053\pi$
$588$	33.6016	1.38571
$589$	−4.83883	−0.199381
$590$	16.3368	0.672577
$591$	−3.78808	−0.155821
$592$	−35.8543	−1.47360
$593$	−10.6527	−0.437455	−0.218727	−	0.975786i	$-0.570191\pi$
$593$	−10.6527	−0.437455	−0.218727	+	0.975786i	$0.570191\pi$
$594$	62.6464	2.57041
$595$	5.53825	0.227046
$596$	22.2814	0.912682
$597$	−21.0465	−0.861375
$598$	−1.96881	−0.0805106
$599$	40.9612	1.67363	0.836815	−	0.547486i	$-0.184415\pi$
$599$	40.9612	1.67363	0.836815	+	0.547486i	$0.184415\pi$
$600$	9.84748	0.402022
$601$	−3.64537	−0.148698	−0.0743489	−	0.997232i	$-0.523688\pi$
$601$	−3.64537	−0.148698	−0.0743489	+	0.997232i	$0.523688\pi$
$602$	−1.38560	−0.0564727
$603$	−0.723005	−0.0294431
$604$	27.3455	1.11267
$605$	−12.8464	−0.522282
$606$	35.9025	1.45844
$607$	−24.8859	−1.01009	−0.505044	−	0.863094i	$-0.668524\pi$
$607$	−24.8859	−1.01009	−0.505044	+	0.863094i	$0.668524\pi$
$608$	−9.66650	−0.392028
$609$	14.1639	0.573951
$610$	30.3449	1.22863
$611$	0.949321	0.0384054
$612$	−1.06746	−0.0431497
$613$	7.69988	0.310995	0.155498	−	0.987836i	$-0.450302\pi$
$613$	7.69988	0.310995	0.155498	+	0.987836i	$0.450302\pi$
$614$	73.9216	2.98323
$615$	0.590292	0.0238029
$616$	43.2672	1.74328
$617$	−12.5735	−0.506188	−0.253094	−	0.967442i	$-0.581448\pi$
$617$	−12.5735	−0.506188	−0.253094	+	0.967442i	$0.581448\pi$
$618$	22.7619	0.915619
$619$	47.5432	1.91092	0.955461	−	0.295118i	$-0.0953590\pi$
$619$	47.5432	1.91092	0.955461	+	0.295118i	$0.0953590\pi$
$620$	5.59737	0.224796
$621$	−17.3161	−0.694872
$622$	−81.8012	−3.27993
$623$	3.94390	0.158009
$624$	2.27033	0.0908860
$625$	1.00000	0.0400000
$626$	75.4479	3.01550
$627$	31.4284	1.25513
$628$	−52.7686	−2.10570
$629$	22.6937	0.904857
$630$	0.282016	0.0112358
$631$	9.47911	0.377357	0.188679	−	0.982039i	$-0.439580\pi$
$631$	9.47911	0.377357	0.188679	+	0.982039i	$0.439580\pi$
$632$	56.6580	2.25374
$633$	−3.14562	−0.125027
$634$	−62.8531	−2.49622
$635$	−8.54095	−0.338937
$636$	44.3830	1.75990
$637$	1.05368	0.0417483
$638$	−62.5581	−2.47670
$639$	0.0124667	0.000493174 0
$640$	−16.5581	−0.654518
$641$	−43.1150	−1.70294	−0.851470	−	0.524403i	$-0.824289\pi$
$641$	−43.1150	−1.70294	−0.851470	+	0.524403i	$0.824289\pi$
$642$	−57.4595	−2.26775
$643$	17.2640	0.680827	0.340414	−	0.940276i	$-0.389433\pi$
$643$	17.2640	0.680827	0.340414	+	0.940276i	$0.389433\pi$
$644$	−22.6001	−0.890568
$645$	0.616095	0.0242587
$646$	32.2382	1.26840
$647$	48.4440	1.90453	0.952265	−	0.305271i	$-0.0987471\pi$
$647$	48.4440	1.90453	0.952265	+	0.305271i	$0.0987471\pi$
$648$	51.7468	2.03281
$649$	−31.9163	−1.25282
$650$	0.583539	0.0228883
$651$	3.64157	0.142724
$652$	−20.9207	−0.819319
$653$	−23.5910	−0.923189	−0.461595	−	0.887091i	$-0.652722\pi$
$653$	−23.5910	−0.923189	−0.461595	+	0.887091i	$0.652722\pi$
$654$	77.3550	3.02482
$655$	13.9655	0.545676
$656$	1.86895	0.0729701
$657$	−0.982915	−0.0383472
$658$	16.0280	0.624836
$659$	11.5078	0.448280	0.224140	−	0.974557i	$-0.428043\pi$
$659$	11.5078	0.448280	0.224140	+	0.974557i	$0.428043\pi$
$660$	−36.3551	−1.41512
$661$	−8.18442	−0.318337	−0.159169	−	0.987251i	$-0.550881\pi$
$661$	−8.18442	−0.318337	−0.159169	+	0.987251i	$0.550881\pi$
$662$	11.4884	0.446510
$663$	−1.43699	−0.0558080
$664$	96.3243	3.73811
$665$	−5.79072	−0.224554
$666$	1.15560	0.0447785
$667$	17.2917	0.669538
$668$	−49.6958	−1.92279
$669$	19.7123	0.762120
$670$	−25.2582	−0.975810
$671$	−59.2830	−2.28860
$672$	7.27473	0.280629
$673$	18.4691	0.711933	0.355967	−	0.934499i	$-0.384152\pi$
$673$	18.4691	0.711933	0.355967	+	0.934499i	$0.384152\pi$
$674$	−12.1118	−0.466528
$675$	5.13235	0.197544
$676$	−54.9913	−2.11505
$677$	48.1717	1.85139	0.925694	−	0.378273i	$-0.123482\pi$
$677$	48.1717	1.85139	0.925694	+	0.378273i	$0.123482\pi$
$678$	−31.5015	−1.20981
$679$	−2.49339	−0.0956877
$680$	−19.7341	−0.756770
$681$	42.1525	1.61529
$682$	−16.0838	−0.615880
$683$	7.41122	0.283583	0.141791	−	0.989897i	$-0.454714\pi$
$683$	7.41122	0.283583	0.141791	+	0.989897i	$0.454714\pi$
$684$	1.11613	0.0426761
$685$	−4.37830	−0.167286
$686$	45.3808	1.73265
$687$	3.99624	0.152466
$688$	1.95064	0.0743675
$689$	1.39176	0.0530219
$690$	14.7802	0.562672
$691$	−13.3120	−0.506413	−0.253206	−	0.967412i	$-0.581485\pi$
$691$	−13.3120	−0.506413	−0.253206	+	0.967412i	$0.581485\pi$
$692$	38.8169	1.47560
$693$	−0.550958	−0.0209292
$694$	−91.4138	−3.47002
$695$	17.3019	0.656299
$696$	−50.4695	−1.91304
$697$	−1.18293	−0.0448068
$698$	79.5136	3.00963
$699$	7.72614	0.292229
$700$	6.69847	0.253179
$701$	9.03056	0.341079	0.170540	−	0.985351i	$-0.445449\pi$
$701$	9.03056	0.341079	0.170540	+	0.985351i	$0.445449\pi$
$702$	2.99493	0.113036
$703$	−23.7282	−0.894926
$704$	22.0635	0.831548
$705$	−7.12672	−0.268408
$706$	−64.1506	−2.41434
$707$	12.9235	0.486038
$708$	−48.6580	−1.82868
$709$	50.3766	1.89193	0.945967	−	0.324263i	$-0.105116\pi$
$709$	50.3766	1.89193	0.945967	+	0.324263i	$0.105116\pi$
$710$	0.435524	0.0163449
$711$	−0.721475	−0.0270574
$712$	−14.0531	−0.526662
$713$	4.44573	0.166494
$714$	−24.2616	−0.907966
$715$	−1.14002	−0.0426345
$716$	63.1130	2.35864
$717$	12.6628	0.472901
$718$	62.9509	2.34931
$719$	21.0047	0.783344	0.391672	−	0.920105i	$-0.371897\pi$
$719$	21.0047	0.783344	0.391672	+	0.920105i	$0.371897\pi$
$720$	−0.397022	−0.0147961
$721$	8.19339	0.305138
$722$	13.7842	0.512995
$723$	−1.04844	−0.0389918
$724$	44.2116	1.64311
$725$	−5.12512	−0.190342
$726$	56.2767	2.08863
$727$	−17.1906	−0.637565	−0.318783	−	0.947828i	$-0.603274\pi$
$727$	−17.1906	−0.637565	−0.318783	+	0.947828i	$0.603274\pi$
$728$	2.06847	0.0766626
$729$	26.3257	0.975026
$730$	−34.3382	−1.27091
$731$	−1.23464	−0.0456648
$732$	−90.3801	−3.34054
$733$	−30.5674	−1.12903	−0.564517	−	0.825421i	$-0.690938\pi$
$733$	−30.5674	−1.12903	−0.564517	+	0.825421i	$0.690938\pi$
$734$	−22.7123	−0.838326
$735$	−7.91016	−0.291771
$736$	8.88119	0.327365
$737$	49.3455	1.81766
$738$	−0.0602368	−0.00221735
$739$	49.8340	1.83317	0.916586	−	0.399837i	$-0.130933\pi$
$739$	49.8340	1.83317	0.916586	+	0.399837i	$0.130933\pi$
$740$	27.4479	1.00900
$741$	1.50249	0.0551955
$742$	23.4980	0.862638
$743$	−15.0355	−0.551600	−0.275800	−	0.961215i	$-0.588943\pi$
$743$	−15.0355	−0.551600	−0.275800	+	0.961215i	$0.588943\pi$
$744$	−12.9758	−0.475716
$745$	−5.24526	−0.192172
$746$	46.6816	1.70913
$747$	−1.22658	−0.0448782
$748$	72.8549	2.66384
$749$	−20.6831	−0.755745
$750$	−4.38073	−0.159962
$751$	−29.5693	−1.07900	−0.539500	−	0.841986i	$-0.681387\pi$
$751$	−29.5693	−1.07900	−0.539500	+	0.841986i	$0.681387\pi$
$752$	−22.5642	−0.822831
$753$	6.84448	0.249427
$754$	−2.99071	−0.108915
$755$	−6.43739	−0.234281
$756$	34.3789	1.25035
$757$	−12.0742	−0.438843	−0.219422	−	0.975630i	$-0.570417\pi$
$757$	−12.0742	−0.438843	−0.219422	+	0.975630i	$0.570417\pi$
$758$	17.7845	0.645962
$759$	−28.8752	−1.04810
$760$	20.6337	0.748465
$761$	−13.3237	−0.482985	−0.241493	−	0.970403i	$-0.577637\pi$
$761$	−13.3237	−0.482985	−0.241493	+	0.970403i	$0.577637\pi$
$762$	37.4156	1.35542
$763$	27.8447	1.00805
$764$	−36.9941	−1.33840
$765$	0.251292	0.00908547
$766$	−50.0506	−1.80840
$767$	−1.52582	−0.0550941
$768$	56.6998	2.04598
$769$	17.9986	0.649045	0.324523	−	0.945878i	$-0.394796\pi$
$769$	17.9986	0.649045	0.324523	+	0.945878i	$0.394796\pi$
$770$	−19.2477	−0.693640
$771$	19.8932	0.716434
$772$	78.2512	2.81632
$773$	−24.1296	−0.867882	−0.433941	−	0.900941i	$-0.642877\pi$
$773$	−24.1296	−0.867882	−0.433941	+	0.900941i	$0.642877\pi$
$774$	−0.0628698	−0.00225981
$775$	−1.31768	−0.0473324
$776$	8.88458	0.318938
$777$	17.8572	0.640622
$778$	54.6783	1.96031
$779$	1.23686	0.0443151
$780$	−1.73803	−0.0622313
$781$	−0.850856	−0.0304460
$782$	−29.6192	−1.05918
$783$	−26.3039	−0.940026
$784$	−25.0447	−0.894452
$785$	12.4222	0.443369
$786$	−61.1789	−2.18218
$787$	39.0918	1.39347	0.696736	−	0.717328i	$-0.254635\pi$
$787$	39.0918	1.39347	0.696736	+	0.717328i	$0.254635\pi$
$788$	9.18155	0.327079
$789$	−12.4780	−0.444227
$790$	−25.2048	−0.896745
$791$	−11.3393	−0.403179
$792$	1.96320	0.0697592
$793$	−2.83414	−0.100643
$794$	−19.9952	−0.709601
$795$	−10.4482	−0.370559
$796$	51.0125	1.80809
$797$	15.7004	0.556138	0.278069	−	0.960561i	$-0.410306\pi$
$797$	15.7004	0.556138	0.278069	+	0.960561i	$0.410306\pi$
$798$	25.3675	0.898001
$799$	14.2818	0.505254
$800$	−2.63231	−0.0930663
$801$	0.178950	0.00632288
$802$	−52.8009	−1.86447
$803$	67.0844	2.36736
$804$	75.2297	2.65315
$805$	5.32028	0.187515
$806$	−0.768915	−0.0270839
$807$	37.6590	1.32566
$808$	−46.0495	−1.62002
$809$	−26.1079	−0.917905	−0.458953	−	0.888461i	$-0.651775\pi$
$809$	−26.1079	−0.917905	−0.458953	+	0.888461i	$0.651775\pi$
$810$	−23.0200	−0.808839
$811$	−10.1861	−0.357683	−0.178842	−	0.983878i	$-0.557235\pi$
$811$	−10.1861	−0.357683	−0.178842	+	0.983878i	$0.557235\pi$
$812$	−34.3305	−1.20476
$813$	26.7657	0.938714
$814$	−78.8701	−2.76440
$815$	4.92495	0.172513
$816$	34.1554	1.19568
$817$	1.29092	0.0451637
$818$	−4.30281	−0.150444
$819$	−0.0263396	−0.000920379 0
$820$	−1.43075	−0.0499639
$821$	9.05325	0.315961	0.157980	−	0.987442i	$-0.449502\pi$
$821$	9.05325	0.315961	0.157980	+	0.987442i	$0.449502\pi$
$822$	19.1801	0.668984
$823$	2.29954	0.0801571	0.0400785	−	0.999197i	$-0.487239\pi$
$823$	2.29954	0.0801571	0.0400785	+	0.999197i	$0.487239\pi$
$824$	−29.1951	−1.01706
$825$	8.55836	0.297964
$826$	−25.7613	−0.896351
$827$	15.9665	0.555210	0.277605	−	0.960695i	$-0.410459\pi$
$827$	15.9665	0.555210	0.277605	+	0.960695i	$0.410459\pi$
$828$	−1.02545	−0.0356369
$829$	21.9354	0.761846	0.380923	−	0.924607i	$-0.375606\pi$
$829$	21.9354	0.761846	0.380923	+	0.924607i	$0.375606\pi$
$830$	−42.8506	−1.48737
$831$	11.3421	0.393452
$832$	1.05479	0.0365681
$833$	15.8518	0.549232
$834$	−75.7950	−2.62456
$835$	11.6989	0.404856
$836$	−76.1761	−2.63461
$837$	−6.76279	−0.233756
$838$	−32.3577	−1.11778
$839$	30.0324	1.03683	0.518417	−	0.855128i	$-0.326521\pi$
$839$	30.0324	1.03683	0.518417	+	0.855128i	$0.326521\pi$
$840$	−15.5284	−0.535779
$841$	−2.73311	−0.0942453
$842$	5.79666	0.199766
$843$	16.5993	0.571711
$844$	7.62436	0.262441
$845$	12.9455	0.445339
$846$	0.727252	0.0250034
$847$	20.2574	0.696052
$848$	−33.0804	−1.13599
$849$	15.6878	0.538404
$850$	8.77888	0.301113
$851$	21.8005	0.747313
$852$	−1.29717	−0.0444405
$853$	11.0695	0.379013	0.189506	−	0.981879i	$-0.439311\pi$
$853$	11.0695	0.379013	0.189506	+	0.981879i	$0.439311\pi$
$854$	−47.8505	−1.63741
$855$	−0.262747	−0.00898576
$856$	73.6991	2.51898
$857$	−25.5425	−0.872515	−0.436258	−	0.899822i	$-0.643696\pi$
$857$	−25.5425	−0.872515	−0.436258	+	0.899822i	$0.643696\pi$
$858$	4.99413	0.170497
$859$	19.4666	0.664193	0.332097	−	0.943245i	$-0.392244\pi$
$859$	19.4666	0.664193	0.332097	+	0.943245i	$0.392244\pi$
$860$	−1.49329	−0.0509207
$861$	−0.930824	−0.0317224
$862$	−14.0090	−0.477149
$863$	13.5471	0.461149	0.230575	−	0.973055i	$-0.425939\pi$
$863$	13.5471	0.461149	0.230575	+	0.973055i	$0.425939\pi$
$864$	−13.5100	−0.459618
$865$	−9.13788	−0.310697
$866$	−19.3758	−0.658418
$867$	8.17557	0.277657
$868$	−8.82643	−0.299588
$869$	49.2410	1.67039
$870$	22.4518	0.761186
$871$	2.35905	0.0799334
$872$	−99.2175	−3.35993
$873$	−0.113135	−0.00382904
$874$	30.9694	1.04756
$875$	−1.57689	−0.0533085
$876$	102.274	3.45551
$877$	−49.3730	−1.66721	−0.833604	−	0.552363i	$-0.813726\pi$
$877$	−49.3730	−1.66721	−0.833604	+	0.552363i	$0.813726\pi$
$878$	−31.0431	−1.04766
$879$	58.0987	1.95962
$880$	27.0969	0.913438
$881$	−35.8929	−1.20926	−0.604631	−	0.796506i	$-0.706679\pi$
$881$	−35.8929	−1.20926	−0.604631	+	0.796506i	$0.706679\pi$
$882$	0.807198	0.0271798
$883$	44.4528	1.49596	0.747978	−	0.663724i	$-0.231025\pi$
$883$	44.4528	1.49596	0.747978	+	0.663724i	$0.231025\pi$
$884$	3.48297	0.117145
$885$	11.4546	0.385041
$886$	40.2344	1.35170
$887$	16.8867	0.567001	0.283500	−	0.958972i	$-0.408504\pi$
$887$	16.8867	0.567001	0.283500	+	0.958972i	$0.408504\pi$
$888$	−63.6295	−2.13527
$889$	13.4681	0.451706
$890$	6.25162	0.209555
$891$	44.9727	1.50664
$892$	−47.7785	−1.59974
$893$	−14.9329	−0.499709
$894$	22.9781	0.768501
$895$	−14.8574	−0.496629
$896$	26.1103	0.872285
$897$	−1.38043	−0.0460913
$898$	18.4683	0.616296
$899$	6.75326	0.225234
$900$	0.303936	0.0101312
$901$	20.9380	0.697545
$902$	4.11119	0.136888
$903$	−0.971512	−0.0323299
$904$	40.4047	1.34384
$905$	−10.4079	−0.345969
$906$	28.2004	0.936897
$907$	27.7586	0.921708	0.460854	−	0.887476i	$-0.347543\pi$
$907$	27.7586	0.921708	0.460854	+	0.887476i	$0.347543\pi$
$908$	−102.169	−3.39061
$909$	0.586388	0.0194493
$910$	−0.920174	−0.0305035
$911$	32.0146	1.06069	0.530346	−	0.847781i	$-0.322062\pi$
$911$	32.0146	1.06069	0.530346	+	0.847781i	$0.322062\pi$
$912$	−35.7124	−1.18256
$913$	83.7146	2.77055
$914$	38.9523	1.28843
$915$	21.2764	0.703375
$916$	−9.68609	−0.320037
$917$	−22.0220	−0.727230
$918$	45.0563	1.48708
$919$	−11.4239	−0.376841	−0.188420	−	0.982088i	$-0.560337\pi$
$919$	−11.4239	−0.376841	−0.188420	+	0.982088i	$0.560337\pi$
$920$	−18.9575	−0.625009
$921$	51.8302	1.70786
$922$	30.7639	1.01316
$923$	−0.0406768	−0.00133889
$924$	57.3280	1.88595
$925$	−6.46150	−0.212453
$926$	−53.0935	−1.74476
$927$	0.371765	0.0122104
$928$	13.4909	0.442861
$929$	−17.3658	−0.569754	−0.284877	−	0.958564i	$-0.591953\pi$
$929$	−17.3658	−0.569754	−0.284877	+	0.958564i	$0.591953\pi$
$930$	5.77238	0.189284
$931$	−16.5744	−0.543205
$932$	−18.7266	−0.613410
$933$	−57.3549	−1.87772
$934$	18.0150	0.589468
$935$	−17.1508	−0.560890
$936$	0.0938544	0.00306773
$937$	−44.4650	−1.45261	−0.726304	−	0.687374i	$-0.758763\pi$
$937$	−44.4650	−1.45261	−0.726304	+	0.687374i	$0.758763\pi$
$938$	39.8294	1.30047
$939$	52.9003	1.72634
$940$	17.2737	0.563407
$941$	21.5881	0.703753	0.351877	−	0.936046i	$-0.385544\pi$
$941$	21.5881	0.703753	0.351877	+	0.936046i	$0.385544\pi$
$942$	−54.4185	−1.77305
$943$	−1.13638	−0.0370055
$944$	36.2668	1.18038
$945$	−8.09314	−0.263270
$946$	4.29089	0.139509
$947$	−57.6970	−1.87490	−0.937450	−	0.348120i	$-0.886820\pi$
$947$	−57.6970	−1.87490	−0.937450	+	0.348120i	$0.886820\pi$
$948$	75.0705	2.43818
$949$	3.20709	0.104107
$950$	−9.17908	−0.297809
$951$	−44.0695	−1.42905
$952$	31.1185	1.00856
$953$	30.1314	0.976053	0.488027	−	0.872829i	$-0.337717\pi$
$953$	30.1314	0.976053	0.488027	+	0.872829i	$0.337717\pi$
$954$	1.06619	0.0345193
$955$	8.70878	0.281810
$956$	−30.6921	−0.992653
$957$	−43.8626	−1.41788
$958$	4.81998	0.155727
$959$	6.90409	0.222945
$960$	−7.91846	−0.255567
$961$	−29.2637	−0.943991
$962$	−3.77053	−0.121567
$963$	−0.938473	−0.0302419
$964$	2.54120	0.0818465
$965$	−18.4211	−0.592997
$966$	−23.3067	−0.749880
$967$	26.3654	0.847856	0.423928	−	0.905696i	$-0.360651\pi$
$967$	26.3654	0.847856	0.423928	+	0.905696i	$0.360651\pi$
$968$	−72.1820	−2.32002
$969$	22.6039	0.726140
$970$	−3.95237	−0.126903
$971$	−16.1127	−0.517082	−0.258541	−	0.966000i	$-0.583242\pi$
$971$	−16.1127	−0.517082	−0.258541	+	0.966000i	$0.583242\pi$
$972$	3.15792	0.101290
$973$	−27.2832	−0.874658
$974$	45.6332	1.46218
$975$	0.409148	0.0131032
$976$	67.3639	2.15627
$977$	61.6424	1.97212	0.986058	−	0.166403i	$-0.0532154\pi$
$977$	61.6424	1.97212	0.986058	+	0.166403i	$0.0532154\pi$
$978$	−21.5748	−0.689888
$979$	−12.2134	−0.390342
$980$	19.1726	0.612447
$981$	1.26342	0.0403379
$982$	−109.861	−3.50580
$983$	−40.3396	−1.28663	−0.643316	−	0.765600i	$-0.722442\pi$
$983$	−40.3396	−1.28663	−0.643316	+	0.765600i	$0.722442\pi$
$984$	3.31675	0.105734
$985$	−2.16143	−0.0688688
$986$	−44.9929	−1.43286
$987$	11.2380	0.357711
$988$	−3.64174	−0.115859
$989$	−1.18605	−0.0377142
$990$	−0.873344	−0.0277567
$991$	−43.8269	−1.39221	−0.696103	−	0.717942i	$-0.745084\pi$
$991$	−43.8269	−1.39221	−0.696103	+	0.717942i	$0.745084\pi$
$992$	3.46854	0.110126
$993$	8.05512	0.255622
$994$	−0.686771	−0.0217831
$995$	−12.0088	−0.380706
$996$	127.627	4.04403
$997$	5.84432	0.185091	0.0925457	−	0.995708i	$-0.470500\pi$
$997$	5.84432	0.185091	0.0925457	+	0.995708i	$0.470500\pi$
$998$	25.6420	0.811684
$999$	−33.1627	−1.04922

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8045.2.a.d.1.14	✓	141

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8045.2.a.d.1.14	✓	141	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:	\( N \)	\(=\)	\( 8045 = 5 \cdot 1609 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8045.a (trivial)

\(f(q)\)	\(=\)	\(q-2.49958 q^{2} -1.75258 q^{3} +4.24791 q^{4} -1.00000 q^{5} +4.38073 q^{6} +1.57689 q^{7} -5.61884 q^{8} +0.0715494 q^{9} +O(q^{10})\)	\(q-2.49958 q^{2} -1.75258 q^{3} +4.24791 q^{4} -1.00000 q^{5} +4.38073 q^{6} +1.57689 q^{7} -5.61884 q^{8} +0.0715494 q^{9} +2.49958 q^{10} -4.88328 q^{11} -7.44482 q^{12} -0.233454 q^{13} -3.94156 q^{14} +1.75258 q^{15} +5.54892 q^{16} -3.51214 q^{17} -0.178844 q^{18} +3.67225 q^{19} -4.24791 q^{20} -2.76363 q^{21} +12.2062 q^{22} -3.37391 q^{23} +9.84748 q^{24} +1.00000 q^{25} +0.583539 q^{26} +5.13235 q^{27} +6.69847 q^{28} -5.12512 q^{29} -4.38073 q^{30} -1.31768 q^{31} -2.63231 q^{32} +8.55836 q^{33} +8.77888 q^{34} -1.57689 q^{35} +0.303936 q^{36} -6.46150 q^{37} -9.17908 q^{38} +0.409148 q^{39} +5.61884 q^{40} +0.336812 q^{41} +6.90791 q^{42} +0.351535 q^{43} -20.7437 q^{44} -0.0715494 q^{45} +8.43337 q^{46} -4.06641 q^{47} -9.72495 q^{48} -4.51343 q^{49} -2.49958 q^{50} +6.15532 q^{51} -0.991694 q^{52} -5.96160 q^{53} -12.8287 q^{54} +4.88328 q^{55} -8.86027 q^{56} -6.43592 q^{57} +12.8107 q^{58} +6.53582 q^{59} +7.44482 q^{60} +12.1400 q^{61} +3.29364 q^{62} +0.112825 q^{63} -4.51816 q^{64} +0.233454 q^{65} -21.3923 q^{66} -10.1050 q^{67} -14.9193 q^{68} +5.91306 q^{69} +3.94156 q^{70} +0.174239 q^{71} -0.402024 q^{72} -13.7376 q^{73} +16.1510 q^{74} -1.75258 q^{75} +15.5994 q^{76} -7.70038 q^{77} -1.02270 q^{78} -10.0836 q^{79} -5.54892 q^{80} -9.20953 q^{81} -0.841890 q^{82} -17.1431 q^{83} -11.7396 q^{84} +3.51214 q^{85} -0.878691 q^{86} +8.98221 q^{87} +27.4384 q^{88} +2.50107 q^{89} +0.178844 q^{90} -0.368131 q^{91} -14.3321 q^{92} +2.30934 q^{93} +10.1643 q^{94} -3.67225 q^{95} +4.61335 q^{96} -1.58121 q^{97} +11.2817 q^{98} -0.349396 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 141 q - 8 q^{2} + 11 q^{3} + 158 q^{4} - 141 q^{5} + 23 q^{6} + 29 q^{7} - 21 q^{8} + 160 q^{9}+O(q^{10}) \) 141 * q - 8 * q^2 + 11 * q^3 + 158 * q^4 - 141 * q^5 + 23 * q^6 + 29 * q^7 - 21 * q^8 + 160 * q^9	\( 141 q - 8 q^{2} + 11 q^{3} + 158 q^{4} - 141 q^{5} + 23 q^{6} + 29 q^{7} - 21 q^{8} + 160 q^{9} + 8 q^{10} + 32 q^{11} + 17 q^{12} + 35 q^{13} + 18 q^{14} - 11 q^{15} + 188 q^{16} - 13 q^{17} - 16 q^{18} + 152 q^{19} - 158 q^{20} + 40 q^{21} + 14 q^{22} - 77 q^{23} + 69 q^{24} + 141 q^{25} + 27 q^{26} + 38 q^{27} + 67 q^{28} + 22 q^{29} - 23 q^{30} + 86 q^{31} - 65 q^{32} + 51 q^{33} + 79 q^{34} - 29 q^{35} + 191 q^{36} + 45 q^{37} - 9 q^{38} + 55 q^{39} + 21 q^{40} + 36 q^{41} + 6 q^{42} + 132 q^{43} + 74 q^{44} - 160 q^{45} + 72 q^{46} - 16 q^{47} + 22 q^{48} + 212 q^{49} - 8 q^{50} + 82 q^{51} + 106 q^{52} - 28 q^{53} + 86 q^{54} - 32 q^{55} + 27 q^{56} + 10 q^{57} + 23 q^{58} + 71 q^{59} - 17 q^{60} + 116 q^{61} - 36 q^{62} + 45 q^{63} + 237 q^{64} - 35 q^{65} + 69 q^{66} + 99 q^{67} - 7 q^{68} + 45 q^{69} - 18 q^{70} + 34 q^{71} - 53 q^{72} + 125 q^{73} + 50 q^{74} + 11 q^{75} + 271 q^{76} - 31 q^{77} + 2 q^{78} + 101 q^{79} - 188 q^{80} + 221 q^{81} + 67 q^{82} + 67 q^{83} + 141 q^{84} + 13 q^{85} + 48 q^{86} - 21 q^{87} + 71 q^{88} + 79 q^{89} + 16 q^{90} + 228 q^{91} - 198 q^{92} - 12 q^{93} + 114 q^{94} - 152 q^{95} + 129 q^{96} + 98 q^{97} - 31 q^{98} + 195 q^{99}+O(q^{100}) \) 141 * q - 8 * q^2 + 11 * q^3 + 158 * q^4 - 141 * q^5 + 23 * q^6 + 29 * q^7 - 21 * q^8 + 160 * q^9 + 8 * q^10 + 32 * q^11 + 17 * q^12 + 35 * q^13 + 18 * q^14 - 11 * q^15 + 188 * q^16 - 13 * q^17 - 16 * q^18 + 152 * q^19 - 158 * q^20 + 40 * q^21 + 14 * q^22 - 77 * q^23 + 69 * q^24 + 141 * q^25 + 27 * q^26 + 38 * q^27 + 67 * q^28 + 22 * q^29 - 23 * q^30 + 86 * q^31 - 65 * q^32 + 51 * q^33 + 79 * q^34 - 29 * q^35 + 191 * q^36 + 45 * q^37 - 9 * q^38 + 55 * q^39 + 21 * q^40 + 36 * q^41 + 6 * q^42 + 132 * q^43 + 74 * q^44 - 160 * q^45 + 72 * q^46 - 16 * q^47 + 22 * q^48 + 212 * q^49 - 8 * q^50 + 82 * q^51 + 106 * q^52 - 28 * q^53 + 86 * q^54 - 32 * q^55 + 27 * q^56 + 10 * q^57 + 23 * q^58 + 71 * q^59 - 17 * q^60 + 116 * q^61 - 36 * q^62 + 45 * q^63 + 237 * q^64 - 35 * q^65 + 69 * q^66 + 99 * q^67 - 7 * q^68 + 45 * q^69 - 18 * q^70 + 34 * q^71 - 53 * q^72 + 125 * q^73 + 50 * q^74 + 11 * q^75 + 271 * q^76 - 31 * q^77 + 2 * q^78 + 101 * q^79 - 188 * q^80 + 221 * q^81 + 67 * q^82 + 67 * q^83 + 141 * q^84 + 13 * q^85 + 48 * q^86 - 21 * q^87 + 71 * q^88 + 79 * q^89 + 16 * q^90 + 228 * q^91 - 198 * q^92 - 12 * q^93 + 114 * q^94 - 152 * q^95 + 129 * q^96 + 98 * q^97 - 31 * q^98 + 195 * q^99

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