LMFDB - Embedded newform 8045.2.a.b.1.2

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:	$1$
Dimension:	$126$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
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)

$f(q)$	$=$	$q-2.68479 q^{2} -1.35137 q^{3} +5.20811 q^{4} -1.00000 q^{5} +3.62815 q^{6} -0.768058 q^{7} -8.61310 q^{8} -1.17380 q^{9} +O(q^{10})$	\(q-2.68479 q^{2} -1.35137 q^{3} +5.20811 q^{4} -1.00000 q^{5} +3.62815 q^{6} -0.768058 q^{7} -8.61310 q^{8} -1.17380 q^{9} +2.68479 q^{10} -3.18004 q^{11} -7.03808 q^{12} -1.21168 q^{13} +2.06208 q^{14} +1.35137 q^{15} +12.7082 q^{16} +6.93109 q^{17} +3.15141 q^{18} -6.61206 q^{19} -5.20811 q^{20} +1.03793 q^{21} +8.53775 q^{22} -2.29373 q^{23} +11.6395 q^{24} +1.00000 q^{25} +3.25311 q^{26} +5.64035 q^{27} -4.00013 q^{28} -4.88278 q^{29} -3.62815 q^{30} +3.29380 q^{31} -16.8926 q^{32} +4.29741 q^{33} -18.6085 q^{34} +0.768058 q^{35} -6.11327 q^{36} -7.70629 q^{37} +17.7520 q^{38} +1.63743 q^{39} +8.61310 q^{40} +2.03938 q^{41} -2.78663 q^{42} +1.14944 q^{43} -16.5620 q^{44} +1.17380 q^{45} +6.15820 q^{46} +5.59980 q^{47} -17.1734 q^{48} -6.41009 q^{49} -2.68479 q^{50} -9.36646 q^{51} -6.31057 q^{52} +10.6886 q^{53} -15.1432 q^{54} +3.18004 q^{55} +6.61536 q^{56} +8.93535 q^{57} +13.1092 q^{58} +3.48522 q^{59} +7.03808 q^{60} -4.70154 q^{61} -8.84318 q^{62} +0.901546 q^{63} +19.9367 q^{64} +1.21168 q^{65} -11.5377 q^{66} -12.0711 q^{67} +36.0978 q^{68} +3.09968 q^{69} -2.06208 q^{70} -7.70478 q^{71} +10.1101 q^{72} +8.35165 q^{73} +20.6898 q^{74} -1.35137 q^{75} -34.4363 q^{76} +2.44246 q^{77} -4.39616 q^{78} +10.1650 q^{79} -12.7082 q^{80} -4.10080 q^{81} -5.47532 q^{82} +10.4686 q^{83} +5.40565 q^{84} -6.93109 q^{85} -3.08600 q^{86} +6.59844 q^{87} +27.3900 q^{88} +5.68024 q^{89} -3.15141 q^{90} +0.930642 q^{91} -11.9460 q^{92} -4.45115 q^{93} -15.0343 q^{94} +6.61206 q^{95} +22.8281 q^{96} -9.51962 q^{97} +17.2097 q^{98} +3.73273 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 126 q + 5 q^{2} - 9 q^{3} + 109 q^{4} - 126 q^{5} - 21 q^{6} - 23 q^{7} + 12 q^{8} + 109 q^{9}+O(q^{10}) $ 126 * q + 5 * q^2 - 9 * q^3 + 109 * q^4 - 126 * q^5 - 21 * q^6 - 23 * q^7 + 12 * q^8 + 109 * q^9	\( 126 q + 5 q^{2} - 9 q^{3} + 109 q^{4} - 126 q^{5} - 21 q^{6} - 23 q^{7} + 12 q^{8} + 109 q^{9} - 5 q^{10} - 44 q^{11} - 11 q^{12} - 35 q^{13} - 14 q^{14} + 9 q^{15} + 75 q^{16} + 11 q^{17} - 15 q^{18} - 130 q^{19} - 109 q^{20} - 44 q^{21} - 14 q^{22} + 75 q^{23} - 63 q^{24} + 126 q^{25} - 43 q^{26} - 42 q^{27} - 77 q^{28} - 24 q^{29} + 21 q^{30} - 78 q^{31} + 24 q^{32} - 29 q^{33} - 57 q^{34} + 23 q^{35} + 50 q^{36} - 31 q^{37} - 3 q^{38} - 57 q^{39} - 12 q^{40} - 38 q^{41} - 10 q^{42} - 100 q^{43} - 90 q^{44} - 109 q^{45} - 96 q^{46} + 12 q^{47} - 22 q^{48} + 65 q^{49} + 5 q^{50} - 74 q^{51} - 112 q^{52} + 20 q^{53} - 90 q^{54} + 44 q^{55} - 57 q^{56} + 6 q^{57} - 35 q^{58} - 97 q^{59} + 11 q^{60} - 102 q^{61} - 16 q^{62} - 15 q^{63} + 4 q^{64} + 35 q^{65} - 83 q^{66} - 121 q^{67} + 41 q^{68} - 71 q^{69} + 14 q^{70} - 32 q^{71} - 32 q^{72} - 85 q^{73} - 42 q^{74} - 9 q^{75} - 275 q^{76} + 13 q^{77} + 10 q^{78} - 97 q^{79} - 75 q^{80} + 86 q^{81} - 55 q^{82} - 73 q^{83} - 111 q^{84} - 11 q^{85} - 56 q^{86} - q^{87} - 37 q^{88} - 67 q^{89} + 15 q^{90} - 180 q^{91} + 98 q^{92} - 44 q^{93} - 86 q^{94} + 130 q^{95} - 179 q^{96} - 50 q^{97} + 18 q^{98} - 217 q^{99}+O(q^{100}) \) 126 * q + 5 * q^2 - 9 * q^3 + 109 * q^4 - 126 * q^5 - 21 * q^6 - 23 * q^7 + 12 * q^8 + 109 * q^9 - 5 * q^10 - 44 * q^11 - 11 * q^12 - 35 * q^13 - 14 * q^14 + 9 * q^15 + 75 * q^16 + 11 * q^17 - 15 * q^18 - 130 * q^19 - 109 * q^20 - 44 * q^21 - 14 * q^22 + 75 * q^23 - 63 * q^24 + 126 * q^25 - 43 * q^26 - 42 * q^27 - 77 * q^28 - 24 * q^29 + 21 * q^30 - 78 * q^31 + 24 * q^32 - 29 * q^33 - 57 * q^34 + 23 * q^35 + 50 * q^36 - 31 * q^37 - 3 * q^38 - 57 * q^39 - 12 * q^40 - 38 * q^41 - 10 * q^42 - 100 * q^43 - 90 * q^44 - 109 * q^45 - 96 * q^46 + 12 * q^47 - 22 * q^48 + 65 * q^49 + 5 * q^50 - 74 * q^51 - 112 * q^52 + 20 * q^53 - 90 * q^54 + 44 * q^55 - 57 * q^56 + 6 * q^57 - 35 * q^58 - 97 * q^59 + 11 * q^60 - 102 * q^61 - 16 * q^62 - 15 * q^63 + 4 * q^64 + 35 * q^65 - 83 * q^66 - 121 * q^67 + 41 * q^68 - 71 * q^69 + 14 * q^70 - 32 * q^71 - 32 * q^72 - 85 * q^73 - 42 * q^74 - 9 * q^75 - 275 * q^76 + 13 * q^77 + 10 * q^78 - 97 * q^79 - 75 * q^80 + 86 * q^81 - 55 * q^82 - 73 * q^83 - 111 * q^84 - 11 * q^85 - 56 * q^86 - q^87 - 37 * q^88 - 67 * q^89 + 15 * q^90 - 180 * q^91 + 98 * q^92 - 44 * q^93 - 86 * q^94 + 130 * q^95 - 179 * q^96 - 50 * q^97 + 18 * q^98 - 217 * 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 $

$2$	−2.68479	−1.89843	−0.949217	−	0.314621i	$-0.898122\pi$
$2$	−2.68479	−1.89843	−0.949217	+	0.314621i	$0.898122\pi$
$3$	−1.35137	−0.780214	−0.390107	−	0.920770i	$-0.627562\pi$
$3$	−1.35137	−0.780214	−0.390107	+	0.920770i	$0.627562\pi$
$4$	5.20811	2.60405
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	3.62815	1.48118
$7$	−0.768058	−0.290299	−0.145149	−	0.989410i	$-0.546366\pi$
$7$	−0.768058	−0.290299	−0.145149	+	0.989410i	$0.546366\pi$
$8$	−8.61310	−3.04519
$9$	−1.17380	−0.391266
$10$	2.68479	0.849006
$11$	−3.18004	−0.958818	−0.479409	−	0.877592i	$-0.659149\pi$
$11$	−3.18004	−0.958818	−0.479409	+	0.877592i	$0.659149\pi$
$12$	−7.03808	−2.03172
$13$	−1.21168	−0.336060	−0.168030	−	0.985782i	$-0.553741\pi$
$13$	−1.21168	−0.336060	−0.168030	+	0.985782i	$0.553741\pi$
$14$	2.06208	0.551113
$15$	1.35137	0.348922
$16$	12.7082	3.17704
$17$	6.93109	1.68104	0.840518	−	0.541784i	$-0.182251\pi$
$17$	6.93109	1.68104	0.840518	+	0.541784i	$0.182251\pi$
$18$	3.15141	0.742794
$19$	−6.61206	−1.51691	−0.758456	−	0.651725i	$-0.774046\pi$
$19$	−6.61206	−1.51691	−0.758456	+	0.651725i	$0.774046\pi$
$20$	−5.20811	−1.16457
$21$	1.03793	0.226495
$22$	8.53775	1.82025
$23$	−2.29373	−0.478277	−0.239138	−	0.970986i	$-0.576865\pi$
$23$	−2.29373	−0.478277	−0.239138	+	0.970986i	$0.576865\pi$
$24$	11.6395	2.37590
$25$	1.00000	0.200000
$26$	3.25311	0.637988
$27$	5.64035	1.08549
$28$	−4.00013	−0.755953
$29$	−4.88278	−0.906709	−0.453355	−	0.891330i	$-0.649773\pi$
$29$	−4.88278	−0.906709	−0.453355	+	0.891330i	$0.649773\pi$
$30$	−3.62815	−0.662406
$31$	3.29380	0.591585	0.295792	−	0.955252i	$-0.404416\pi$
$31$	3.29380	0.591585	0.295792	+	0.955252i	$0.404416\pi$
$32$	−16.8926	−2.98622
$33$	4.29741	0.748083
$34$	−18.6085	−3.19134
$35$	0.768058	0.129825
$36$	−6.11327	−1.01888
$37$	−7.70629	−1.26691	−0.633453	−	0.773781i	$-0.718363\pi$
$37$	−7.70629	−1.26691	−0.633453	+	0.773781i	$0.718363\pi$
$38$	17.7520	2.87976
$39$	1.63743	0.262199
$40$	8.61310	1.36185
$41$	2.03938	0.318498	0.159249	−	0.987238i	$-0.449093\pi$
$41$	2.03938	0.318498	0.159249	+	0.987238i	$0.449093\pi$
$42$	−2.78663	−0.429986
$43$	1.14944	0.175288	0.0876439	−	0.996152i	$-0.472066\pi$
$43$	1.14944	0.175288	0.0876439	+	0.996152i	$0.472066\pi$
$44$	−16.5620	−2.49681
$45$	1.17380	0.174980
$46$	6.15820	0.907977
$47$	5.59980	0.816815	0.408407	−	0.912800i	$-0.366084\pi$
$47$	5.59980	0.816815	0.408407	+	0.912800i	$0.366084\pi$
$48$	−17.1734	−2.47877
$49$	−6.41009	−0.915727
$50$	−2.68479	−0.379687
$51$	−9.36646	−1.31157
$52$	−6.31057	−0.875118
$53$	10.6886	1.46819	0.734094	−	0.679048i	$-0.237607\pi$
$53$	10.6886	1.46819	0.734094	+	0.679048i	$0.237607\pi$
$54$	−15.1432	−2.06072
$55$	3.18004	0.428797
$56$	6.61536	0.884015
$57$	8.93535	1.18352
$58$	13.1092	1.72133
$59$	3.48522	0.453737	0.226869	−	0.973925i	$-0.427151\pi$
$59$	3.48522	0.453737	0.226869	+	0.973925i	$0.427151\pi$
$60$	7.03808	0.908612
$61$	−4.70154	−0.601971	−0.300985	−	0.953629i	$-0.597316\pi$
$61$	−4.70154	−0.601971	−0.300985	+	0.953629i	$0.597316\pi$
$62$	−8.84318	−1.12308
$63$	0.901546	0.113584
$64$	19.9367	2.49209
$65$	1.21168	0.150291
$66$	−11.5377	−1.42019
$67$	−12.0711	−1.47472	−0.737362	−	0.675498i	$-0.763929\pi$
$67$	−12.0711	−1.47472	−0.737362	+	0.675498i	$0.763929\pi$
$68$	36.0978	4.37751
$69$	3.09968	0.373158
$70$	−2.06208	−0.246465
$71$	−7.70478	−0.914389	−0.457195	−	0.889367i	$-0.651146\pi$
$71$	−7.70478	−0.914389	−0.457195	+	0.889367i	$0.651146\pi$
$72$	10.1101	1.19148
$73$	8.35165	0.977487	0.488743	−	0.872428i	$-0.337455\pi$
$73$	8.35165	0.977487	0.488743	+	0.872428i	$0.337455\pi$
$74$	20.6898	2.40514
$75$	−1.35137	−0.156043
$76$	−34.4363	−3.95012
$77$	2.44246	0.278344
$78$	−4.39616	−0.497767
$79$	10.1650	1.14366	0.571828	−	0.820374i	$-0.306234\pi$
$79$	10.1650	1.14366	0.571828	+	0.820374i	$0.306234\pi$
$80$	−12.7082	−1.42082
$81$	−4.10080	−0.455644
$82$	−5.47532	−0.604648
$83$	10.4686	1.14908	0.574540	−	0.818477i	$-0.305181\pi$
$83$	10.4686	1.14908	0.574540	+	0.818477i	$0.305181\pi$
$84$	5.40565	0.589805
$85$	−6.93109	−0.751782
$86$	−3.08600	−0.332772
$87$	6.59844	0.707427
$88$	27.3900	2.91978
$89$	5.68024	0.602104	0.301052	−	0.953608i	$-0.402662\pi$
$89$	5.68024	0.602104	0.301052	+	0.953608i	$0.402662\pi$
$90$	−3.15141	−0.332187
$91$	0.930642	0.0975577
$92$	−11.9460	−1.24546
$93$	−4.45115	−0.461562
$94$	−15.0343	−1.55067
$95$	6.61206	0.678383
$96$	22.8281	2.32989
$97$	−9.51962	−0.966571	−0.483285	−	0.875463i	$-0.660557\pi$
$97$	−9.51962	−0.966571	−0.483285	+	0.875463i	$0.660557\pi$
$98$	17.2097	1.73845
$99$	3.73273	0.375153
$100$	5.20811	0.520811
$101$	3.05434	0.303918	0.151959	−	0.988387i	$-0.451442\pi$
$101$	3.05434	0.303918	0.151959	+	0.988387i	$0.451442\pi$
$102$	25.1470	2.48992
$103$	−0.684095	−0.0674058	−0.0337029	−	0.999432i	$-0.510730\pi$
$103$	−0.684095	−0.0674058	−0.0337029	+	0.999432i	$0.510730\pi$
$104$	10.4363	1.02337
$105$	−1.03793	−0.101292
$106$	−28.6966	−2.78726
$107$	7.78337	0.752447	0.376224	−	0.926529i	$-0.377222\pi$
$107$	7.78337	0.752447	0.376224	+	0.926529i	$0.377222\pi$
$108$	29.3755	2.82666
$109$	4.67251	0.447545	0.223773	−	0.974641i	$-0.428163\pi$
$109$	4.67251	0.447545	0.223773	+	0.974641i	$0.428163\pi$
$110$	−8.53775	−0.814042
$111$	10.4140	0.988458
$112$	−9.76061	−0.922291
$113$	12.5159	1.17740	0.588700	−	0.808352i	$-0.299640\pi$
$113$	12.5159	1.17740	0.588700	+	0.808352i	$0.299640\pi$
$114$	−23.9895	−2.24683
$115$	2.29373	0.213892
$116$	−25.4300	−2.36112
$117$	1.42227	0.131489
$118$	−9.35710	−0.861390
$119$	−5.32348	−0.488002
$120$	−11.6395	−1.06253
$121$	−0.887343	−0.0806675
$122$	12.6227	1.14280
$123$	−2.75596	−0.248497
$124$	17.1545	1.54052
$125$	−1.00000	−0.0894427
$126$	−2.42046	−0.215632
$127$	14.9548	1.32702	0.663511	−	0.748166i	$-0.269065\pi$
$127$	14.9548	1.32702	0.663511	+	0.748166i	$0.269065\pi$
$128$	−19.7408	−1.74486
$129$	−1.55332	−0.136762
$130$	−3.25311	−0.285317
$131$	13.9022	1.21464	0.607319	−	0.794458i	$-0.292245\pi$
$131$	13.9022	1.21464	0.607319	+	0.794458i	$0.292245\pi$
$132$	22.3814	1.94805
$133$	5.07845	0.440357
$134$	32.4085	2.79967
$135$	−5.64035	−0.485444
$136$	−59.6982	−5.11907
$137$	13.4156	1.14617	0.573085	−	0.819496i	$-0.305746\pi$
$137$	13.4156	1.14617	0.573085	+	0.819496i	$0.305746\pi$
$138$	−8.32201	−0.708416
$139$	−0.0946286	−0.00802629	−0.00401315	−	0.999992i	$-0.501277\pi$
$139$	−0.0946286	−0.00802629	−0.00401315	+	0.999992i	$0.501277\pi$
$140$	4.00013	0.338073
$141$	−7.56740	−0.637290
$142$	20.6857	1.73591
$143$	3.85320	0.322220
$144$	−14.9168	−1.24307
$145$	4.88278	0.405493
$146$	−22.4224	−1.85569
$147$	8.66240	0.714463
$148$	−40.1352	−3.29909
$149$	−0.141544	−0.0115957	−0.00579787	−	0.999983i	$-0.501846\pi$
$149$	−0.141544	−0.0115957	−0.00579787	+	0.999983i	$0.501846\pi$
$150$	3.62815	0.296237
$151$	−4.37632	−0.356140	−0.178070	−	0.984018i	$-0.556985\pi$
$151$	−4.37632	−0.356140	−0.178070	+	0.984018i	$0.556985\pi$
$152$	56.9504	4.61929
$153$	−8.13570	−0.657733
$154$	−6.55748	−0.528417
$155$	−3.29380	−0.264565
$156$	8.52791	0.682779
$157$	6.84300	0.546131	0.273066	−	0.961995i	$-0.411962\pi$
$157$	6.84300	0.546131	0.273066	+	0.961995i	$0.411962\pi$
$158$	−27.2910	−2.17116
$159$	−14.4442	−1.14550
$160$	16.8926	1.33548
$161$	1.76172	0.138843
$162$	11.0098	0.865011
$163$	−16.2289	−1.27115	−0.635574	−	0.772040i	$-0.719237\pi$
$163$	−16.2289	−1.27115	−0.635574	+	0.772040i	$0.719237\pi$
$164$	10.6213	0.829386
$165$	−4.29741	−0.334553
$166$	−28.1060	−2.18145
$167$	−22.4631	−1.73825	−0.869123	−	0.494597i	$-0.835316\pi$
$167$	−22.4631	−1.73825	−0.869123	+	0.494597i	$0.835316\pi$
$168$	−8.93980	−0.689721
$169$	−11.5318	−0.887064
$170$	18.6085	1.42721
$171$	7.76124	0.593516
$172$	5.98640	0.456459
$173$	23.2545	1.76801	0.884004	−	0.467480i	$-0.154838\pi$
$173$	23.2545	1.76801	0.884004	+	0.467480i	$0.154838\pi$
$174$	−17.7154	−1.34300
$175$	−0.768058	−0.0580597
$176$	−40.4125	−3.04621
$177$	−4.70982	−0.354012
$178$	−15.2503	−1.14306
$179$	−22.3183	−1.66814	−0.834072	−	0.551655i	$-0.813996\pi$
$179$	−22.3183	−1.66814	−0.834072	+	0.551655i	$0.813996\pi$
$180$	6.11327	0.455656
$181$	−23.8273	−1.77107	−0.885536	−	0.464571i	$-0.846208\pi$
$181$	−23.8273	−1.77107	−0.885536	+	0.464571i	$0.846208\pi$
$182$	−2.49858	−0.185207
$183$	6.35352	0.469666
$184$	19.7562	1.45644
$185$	7.70629	0.566578
$186$	11.9504	0.876246
$187$	−22.0411	−1.61181
$188$	29.1644	2.12703
$189$	−4.33211	−0.315115
$190$	−17.7520	−1.28787
$191$	15.4525	1.11810	0.559052	−	0.829132i	$-0.311165\pi$
$191$	15.4525	1.11810	0.559052	+	0.829132i	$0.311165\pi$
$192$	−26.9419	−1.94437
$193$	5.89131	0.424066	0.212033	−	0.977263i	$-0.431992\pi$
$193$	5.89131	0.424066	0.212033	+	0.977263i	$0.431992\pi$
$194$	25.5582	1.83497
$195$	−1.63743	−0.117259
$196$	−33.3844	−2.38460
$197$	−24.4882	−1.74471	−0.872356	−	0.488870i	$-0.837409\pi$
$197$	−24.4882	−1.74471	−0.872356	+	0.488870i	$0.837409\pi$
$198$	−10.0216	−0.712204
$199$	−13.6744	−0.969353	−0.484677	−	0.874693i	$-0.661063\pi$
$199$	−13.6744	−0.969353	−0.484677	+	0.874693i	$0.661063\pi$
$200$	−8.61310	−0.609038
$201$	16.3126	1.15060
$202$	−8.20026	−0.576969
$203$	3.75026	0.263216
$204$	−48.7816	−3.41539
$205$	−2.03938	−0.142437
$206$	1.83665	0.127966
$207$	2.69238	0.187134
$208$	−15.3983	−1.06768
$209$	21.0266	1.45444
$210$	2.78663	0.192296
$211$	21.8715	1.50570	0.752849	−	0.658193i	$-0.228679\pi$
$211$	21.8715	1.50570	0.752849	+	0.658193i	$0.228679\pi$
$212$	55.6672	3.82324
$213$	10.4120	0.713419
$214$	−20.8967	−1.42847
$215$	−1.14944	−0.0783911
$216$	−48.5809	−3.30551
$217$	−2.52983	−0.171736
$218$	−12.5447	−0.849635
$219$	−11.2862	−0.762649
$220$	16.5620	1.11661
$221$	−8.39827	−0.564929
$222$	−27.9595	−1.87652
$223$	19.7453	1.32224	0.661121	−	0.750280i	$-0.270081\pi$
$223$	19.7453	1.32224	0.661121	+	0.750280i	$0.270081\pi$
$224$	12.9745	0.866894
$225$	−1.17380	−0.0782533
$226$	−33.6027	−2.23522
$227$	−24.1096	−1.60021	−0.800104	−	0.599862i	$-0.795222\pi$
$227$	−24.1096	−1.60021	−0.800104	+	0.599862i	$0.795222\pi$
$228$	46.5362	3.08194
$229$	5.37273	0.355040	0.177520	−	0.984117i	$-0.443193\pi$
$229$	5.37273	0.355040	0.177520	+	0.984117i	$0.443193\pi$
$230$	−6.15820	−0.406060
$231$	−3.30066	−0.217168
$232$	42.0559	2.76110
$233$	−3.54894	−0.232499	−0.116249	−	0.993220i	$-0.537087\pi$
$233$	−3.54894	−0.232499	−0.116249	+	0.993220i	$0.537087\pi$
$234$	−3.81850	−0.249623
$235$	−5.59980	−0.365291
$236$	18.1514	1.18156
$237$	−13.7367	−0.892296
$238$	14.2924	0.926441
$239$	−18.1031	−1.17099	−0.585497	−	0.810675i	$-0.699100\pi$
$239$	−18.1031	−1.17099	−0.585497	+	0.810675i	$0.699100\pi$
$240$	17.1734	1.10854
$241$	8.42433	0.542659	0.271329	−	0.962487i	$-0.412537\pi$
$241$	8.42433	0.542659	0.271329	+	0.962487i	$0.412537\pi$
$242$	2.38233	0.153142
$243$	−11.3793	−0.729985
$244$	−24.4861	−1.56756
$245$	6.41009	0.409525
$246$	7.39918	0.471754
$247$	8.01172	0.509773
$248$	−28.3699	−1.80149
$249$	−14.1470	−0.896528
$250$	2.68479	0.169801
$251$	−2.97473	−0.187763	−0.0938817	−	0.995583i	$-0.529928\pi$
$251$	−2.97473	−0.187763	−0.0938817	+	0.995583i	$0.529928\pi$
$252$	4.69535	0.295779
$253$	7.29417	0.458580
$254$	−40.1505	−2.51927
$255$	9.36646	0.586551
$256$	13.1265	0.820408
$257$	19.0074	1.18565	0.592824	−	0.805332i	$-0.298013\pi$
$257$	19.0074	1.18565	0.592824	+	0.805332i	$0.298013\pi$
$258$	4.17033	0.259634
$259$	5.91888	0.367781
$260$	6.31057	0.391365
$261$	5.73140	0.354765
$262$	−37.3244	−2.30591
$263$	−14.5292	−0.895910	−0.447955	−	0.894056i	$-0.647848\pi$
$263$	−14.5292	−0.895910	−0.447955	+	0.894056i	$0.647848\pi$
$264$	−37.0140	−2.27806
$265$	−10.6886	−0.656593
$266$	−13.6346	−0.835990
$267$	−7.67611	−0.469770
$268$	−62.8678	−3.84026
$269$	28.8564	1.75941	0.879704	−	0.475521i	$-0.157741\pi$
$269$	28.8564	1.75941	0.879704	+	0.475521i	$0.157741\pi$
$270$	15.1432	0.921583
$271$	26.4094	1.60425	0.802127	−	0.597153i	$-0.203702\pi$
$271$	26.4094	1.60425	0.802127	+	0.597153i	$0.203702\pi$
$272$	88.0814	5.34072
$273$	−1.25764	−0.0761159
$274$	−36.0180	−2.17593
$275$	−3.18004	−0.191764
$276$	16.1435	0.971724
$277$	22.1839	1.33290	0.666449	−	0.745550i	$-0.267813\pi$
$277$	22.1839	1.33290	0.666449	+	0.745550i	$0.267813\pi$
$278$	0.254058	0.0152374
$279$	−3.86626	−0.231467
$280$	−6.61536	−0.395343
$281$	−17.7588	−1.05940	−0.529700	−	0.848185i	$-0.677695\pi$
$281$	−17.7588	−1.05940	−0.529700	+	0.848185i	$0.677695\pi$
$282$	20.3169	1.20985
$283$	16.8539	1.00186	0.500931	−	0.865487i	$-0.332991\pi$
$283$	16.8539	1.00186	0.500931	+	0.865487i	$0.332991\pi$
$284$	−40.1273	−2.38112
$285$	−8.93535	−0.529284
$286$	−10.3450	−0.611714
$287$	−1.56636	−0.0924595
$288$	19.8285	1.16841
$289$	31.0400	1.82588
$290$	−13.1092	−0.769801
$291$	12.8645	0.754132
$292$	43.4963	2.54543
$293$	29.5021	1.72353	0.861767	−	0.507305i	$-0.169358\pi$
$293$	29.5021	1.72353	0.861767	+	0.507305i	$0.169358\pi$
$294$	−23.2567	−1.35636
$295$	−3.48522	−0.202917
$296$	66.3750	3.85797
$297$	−17.9365	−1.04078
$298$	0.380017	0.0220138
$299$	2.77928	0.160730
$300$	−7.03808	−0.406344
$301$	−0.882836	−0.0508858
$302$	11.7495	0.676109
$303$	−4.12754	−0.237121
$304$	−84.0272	−4.81929
$305$	4.70154	0.269210
$306$	21.8427	1.24866
$307$	13.3336	0.760988	0.380494	−	0.924783i	$-0.375754\pi$
$307$	13.3336	0.760988	0.380494	+	0.924783i	$0.375754\pi$
$308$	12.7206	0.724822
$309$	0.924465	0.0525910
$310$	8.84318	0.502259
$311$	16.4973	0.935474	0.467737	−	0.883868i	$-0.345070\pi$
$311$	16.4973	0.935474	0.467737	+	0.883868i	$0.345070\pi$
$312$	−14.1033	−0.798445
$313$	19.4135	1.09731	0.548657	−	0.836048i	$-0.315139\pi$
$313$	19.4135	1.09731	0.548657	+	0.836048i	$0.315139\pi$
$314$	−18.3720	−1.03679
$315$	−0.901546	−0.0507963
$316$	52.9406	2.97814
$317$	13.9788	0.785130	0.392565	−	0.919724i	$-0.371588\pi$
$317$	13.9788	0.785130	0.392565	+	0.919724i	$0.371588\pi$
$318$	38.7797	2.17466
$319$	15.5274	0.869369
$320$	−19.9367	−1.11450
$321$	−10.5182	−0.587070
$322$	−4.72985	−0.263584
$323$	−45.8288	−2.54998
$324$	−21.3574	−1.18652
$325$	−1.21168	−0.0672120
$326$	43.5713	2.41319
$327$	−6.31429	−0.349181
$328$	−17.5654	−0.969887
$329$	−4.30097	−0.237120
$330$	11.5377	0.635127
$331$	−21.1579	−1.16294	−0.581470	−	0.813568i	$-0.697522\pi$
$331$	−21.1579	−1.16294	−0.581470	+	0.813568i	$0.697522\pi$
$332$	54.5217	2.99226
$333$	9.04563	0.495698
$334$	60.3087	3.29994
$335$	12.0711	0.659517
$336$	13.1902	0.719584
$337$	8.35624	0.455193	0.227597	−	0.973756i	$-0.426913\pi$
$337$	8.35624	0.455193	0.227597	+	0.973756i	$0.426913\pi$
$338$	30.9606	1.68403
$339$	−16.9136	−0.918623
$340$	−36.0978	−1.95768
$341$	−10.4744	−0.567222
$342$	−20.8373	−1.12675
$343$	10.2997	0.556133
$344$	−9.90023	−0.533785
$345$	−3.09968	−0.166881
$346$	−62.4335	−3.35645
$347$	−7.75514	−0.416318	−0.208159	−	0.978095i	$-0.566747\pi$
$347$	−7.75514	−0.416318	−0.208159	+	0.978095i	$0.566747\pi$
$348$	34.3654	1.84218
$349$	−11.1164	−0.595047	−0.297523	−	0.954715i	$-0.596161\pi$
$349$	−11.1164	−0.595047	−0.297523	+	0.954715i	$0.596161\pi$
$350$	2.06208	0.110223
$351$	−6.83430	−0.364788
$352$	53.7191	2.86324
$353$	−0.236057	−0.0125640	−0.00628202	−	0.999980i	$-0.502000\pi$
$353$	−0.236057	−0.0125640	−0.00628202	+	0.999980i	$0.502000\pi$
$354$	12.6449	0.672069
$355$	7.70478	0.408927
$356$	29.5833	1.56791
$357$	7.19399	0.380746
$358$	59.9199	3.16686
$359$	−35.1332	−1.85426	−0.927131	−	0.374739i	$-0.877732\pi$
$359$	−35.1332	−1.85426	−0.927131	+	0.374739i	$0.877732\pi$
$360$	−10.1101	−0.532846
$361$	24.7194	1.30102
$362$	63.9714	3.36226
$363$	1.19913	0.0629379
$364$	4.84688	0.254046
$365$	−8.35165	−0.437145
$366$	−17.0579	−0.891630
$367$	−32.0961	−1.67540	−0.837700	−	0.546130i	$-0.816100\pi$
$367$	−32.0961	−1.67540	−0.837700	+	0.546130i	$0.816100\pi$
$368$	−29.1492	−1.51951
$369$	−2.39382	−0.124618
$370$	−20.6898	−1.07561
$371$	−8.20944	−0.426213
$372$	−23.1821	−1.20193
$373$	−38.4303	−1.98985	−0.994923	−	0.100639i	$-0.967911\pi$
$373$	−38.4303	−1.98985	−0.994923	+	0.100639i	$0.967911\pi$
$374$	59.1759	3.05991
$375$	1.35137	0.0697844
$376$	−48.2316	−2.48736
$377$	5.91637	0.304709
$378$	11.6308	0.598225
$379$	16.1547	0.829812	0.414906	−	0.909864i	$-0.363814\pi$
$379$	16.1547	0.829812	0.414906	+	0.909864i	$0.363814\pi$
$380$	34.4363	1.76655
$381$	−20.2095	−1.03536
$382$	−41.4868	−2.12265
$383$	15.3135	0.782485	0.391242	−	0.920288i	$-0.372045\pi$
$383$	15.3135	0.782485	0.391242	+	0.920288i	$0.372045\pi$
$384$	26.6772	1.36136
$385$	−2.44246	−0.124479
$386$	−15.8170	−0.805062
$387$	−1.34921	−0.0685842
$388$	−49.5792	−2.51700
$389$	−10.0049	−0.507268	−0.253634	−	0.967300i	$-0.581626\pi$
$389$	−10.0049	−0.507268	−0.253634	+	0.967300i	$0.581626\pi$
$390$	4.39616	0.222608
$391$	−15.8981	−0.804000
$392$	55.2107	2.78856
$393$	−18.7870	−0.947678
$394$	65.7457	3.31222
$395$	−10.1650	−0.511459
$396$	19.4405	0.976919
$397$	0.443754	0.0222714	0.0111357	−	0.999938i	$-0.496455\pi$
$397$	0.443754	0.0222714	0.0111357	+	0.999938i	$0.496455\pi$
$398$	36.7130	1.84025
$399$	−6.86286	−0.343573
$400$	12.7082	0.635408
$401$	19.5774	0.977648	0.488824	−	0.872383i	$-0.337426\pi$
$401$	19.5774	0.977648	0.488824	+	0.872383i	$0.337426\pi$
$402$	−43.7959	−2.18434
$403$	−3.99104	−0.198808
$404$	15.9073	0.791419
$405$	4.10080	0.203770
$406$	−10.0687	−0.499699
$407$	24.5063	1.21473
$408$	80.6743	3.99397
$409$	−9.98928	−0.493938	−0.246969	−	0.969023i	$-0.579435\pi$
$409$	−9.98928	−0.493938	−0.246969	+	0.969023i	$0.579435\pi$
$410$	5.47532	0.270407
$411$	−18.1294	−0.894258
$412$	−3.56284	−0.175528
$413$	−2.67685	−0.131719
$414$	−7.22849	−0.355261
$415$	−10.4686	−0.513884
$416$	20.4684	1.00355
$417$	0.127878	0.00626223
$418$	−56.4521	−2.76116
$419$	26.1800	1.27898	0.639489	−	0.768800i	$-0.279146\pi$
$419$	26.1800	1.27898	0.639489	+	0.768800i	$0.279146\pi$
$420$	−5.40565	−0.263769
$421$	3.44452	0.167875	0.0839377	−	0.996471i	$-0.473250\pi$
$421$	3.44452	0.167875	0.0839377	+	0.996471i	$0.473250\pi$
$422$	−58.7205	−2.85847
$423$	−6.57304	−0.319592
$424$	−92.0617	−4.47091
$425$	6.93109	0.336207
$426$	−27.9541	−1.35438
$427$	3.61106	0.174751
$428$	40.5366	1.95941
$429$	−5.20709	−0.251401
$430$	3.08600	0.148820
$431$	18.9507	0.912824	0.456412	−	0.889768i	$-0.349134\pi$
$431$	18.9507	0.912824	0.456412	+	0.889768i	$0.349134\pi$
$432$	71.6785	3.44863
$433$	−19.5328	−0.938687	−0.469343	−	0.883016i	$-0.655509\pi$
$433$	−19.5328	−0.938687	−0.469343	+	0.883016i	$0.655509\pi$
$434$	6.79207	0.326030
$435$	−6.59844	−0.316371
$436$	24.3349	1.16543
$437$	15.1663	0.725503
$438$	30.3010	1.44784
$439$	37.5389	1.79163	0.895816	−	0.444424i	$-0.146592\pi$
$439$	37.5389	1.79163	0.895816	+	0.444424i	$0.146592\pi$
$440$	−27.3900	−1.30577
$441$	7.52415	0.358293
$442$	22.5476	1.07248
$443$	16.8817	0.802072	0.401036	−	0.916062i	$-0.368650\pi$
$443$	16.8817	0.802072	0.401036	+	0.916062i	$0.368650\pi$
$444$	54.2375	2.57400
$445$	−5.68024	−0.269269
$446$	−53.0120	−2.51019
$447$	0.191278	0.00904716
$448$	−15.3126	−0.723451
$449$	17.9340	0.846359	0.423179	−	0.906046i	$-0.360914\pi$
$449$	17.9340	0.846359	0.423179	+	0.906046i	$0.360914\pi$
$450$	3.15141	0.148559
$451$	−6.48532	−0.305382
$452$	65.1843	3.06601
$453$	5.91403	0.277865
$454$	64.7291	3.03789
$455$	−0.930642	−0.0436292
$456$	−76.9610	−3.60403
$457$	23.5235	1.10038	0.550192	−	0.835038i	$-0.314555\pi$
$457$	23.5235	1.10038	0.550192	+	0.835038i	$0.314555\pi$
$458$	−14.4247	−0.674021
$459$	39.0937	1.82474
$460$	11.9460	0.556986
$461$	33.4430	1.55759	0.778797	−	0.627276i	$-0.215830\pi$
$461$	33.4430	1.55759	0.778797	+	0.627276i	$0.215830\pi$
$462$	8.86159	0.412278
$463$	1.85644	0.0862761	0.0431381	−	0.999069i	$-0.486264\pi$
$463$	1.85644	0.0862761	0.0431381	+	0.999069i	$0.486264\pi$
$464$	−62.0512	−2.88065
$465$	4.45115	0.206417
$466$	9.52817	0.441384
$467$	−17.7874	−0.823100	−0.411550	−	0.911387i	$-0.635013\pi$
$467$	−17.7874	−0.823100	−0.411550	+	0.911387i	$0.635013\pi$
$468$	7.40734	0.342404
$469$	9.27133	0.428110
$470$	15.0343	0.693480
$471$	−9.24743	−0.426099
$472$	−30.0186	−1.38172
$473$	−3.65526	−0.168069
$474$	36.8803	1.69397
$475$	−6.61206	−0.303382
$476$	−27.7252	−1.27078
$477$	−12.5462	−0.574452
$478$	48.6031	2.22305
$479$	−17.3001	−0.790462	−0.395231	−	0.918582i	$-0.629335\pi$
$479$	−17.3001	−0.790462	−0.395231	+	0.918582i	$0.629335\pi$
$480$	−22.8281	−1.04196
$481$	9.33757	0.425756
$482$	−22.6176	−1.03020
$483$	−2.38074	−0.108327
$484$	−4.62138	−0.210063
$485$	9.51962	0.432264
$486$	30.5512	1.38583
$487$	5.22179	0.236622	0.118311	−	0.992977i	$-0.462252\pi$
$487$	5.22179	0.236622	0.118311	+	0.992977i	$0.462252\pi$
$488$	40.4949	1.83312
$489$	21.9313	0.991768
$490$	−17.2097	−0.777457
$491$	−34.7001	−1.56599	−0.782996	−	0.622027i	$-0.786309\pi$
$491$	−34.7001	−1.56599	−0.782996	+	0.622027i	$0.786309\pi$
$492$	−14.3533	−0.647098
$493$	−33.8430	−1.52421
$494$	−21.5098	−0.967771
$495$	−3.73273	−0.167774
$496$	41.8582	1.87949
$497$	5.91772	0.265446
$498$	37.9817	1.70200
$499$	−0.699327	−0.0313062	−0.0156531	−	0.999877i	$-0.504983\pi$
$499$	−0.699327	−0.0313062	−0.0156531	+	0.999877i	$0.504983\pi$
$500$	−5.20811	−0.232914
$501$	30.3559	1.35620
$502$	7.98654	0.356457
$503$	−15.0466	−0.670893	−0.335446	−	0.942059i	$-0.608887\pi$
$503$	−15.0466	−0.670893	−0.335446	+	0.942059i	$0.608887\pi$
$504$	−7.76510	−0.345885
$505$	−3.05434	−0.135916
$506$	−19.5833	−0.870585
$507$	15.5838	0.692099
$508$	77.8861	3.45564
$509$	−6.10303	−0.270512	−0.135256	−	0.990811i	$-0.543186\pi$
$509$	−6.10303	−0.270512	−0.135256	+	0.990811i	$0.543186\pi$
$510$	−25.1470	−1.11353
$511$	−6.41455	−0.283763
$512$	4.23964	0.187368
$513$	−37.2943	−1.64659
$514$	−51.0309	−2.25087
$515$	0.684095	0.0301448
$516$	−8.08984	−0.356135
$517$	−17.8076	−0.783177
$518$	−15.8910	−0.698208
$519$	−31.4254	−1.37942
$520$	−10.4363	−0.457664
$521$	20.5519	0.900397	0.450199	−	0.892928i	$-0.351353\pi$
$521$	20.5519	0.900397	0.450199	+	0.892928i	$0.351353\pi$
$522$	−15.3876	−0.673498
$523$	−9.59529	−0.419573	−0.209786	−	0.977747i	$-0.567277\pi$
$523$	−9.59529	−0.419573	−0.209786	+	0.977747i	$0.567277\pi$
$524$	72.4040	3.16298
$525$	1.03793	0.0452990
$526$	39.0079	1.70083
$527$	22.8296	0.994475
$528$	54.6122	2.37669
$529$	−17.7388	−0.771251
$530$	28.6966	1.24650
$531$	−4.09095	−0.177532
$532$	26.4491	1.14671
$533$	−2.47108	−0.107034
$534$	20.6088	0.891828
$535$	−7.78337	−0.336505
$536$	103.970	4.49082
$537$	30.1602	1.30151
$538$	−77.4735	−3.34012
$539$	20.3843	0.878016
$540$	−29.3755	−1.26412
$541$	−40.9246	−1.75948	−0.879742	−	0.475451i	$-0.842285\pi$
$541$	−40.9246	−1.75948	−0.879742	+	0.475451i	$0.842285\pi$
$542$	−70.9036	−3.04557
$543$	32.1995	1.38181
$544$	−117.084	−5.01993
$545$	−4.67251	−0.200148
$546$	3.37650	0.144501
$547$	−43.9389	−1.87869	−0.939345	−	0.342973i	$-0.888566\pi$
$547$	−43.9389	−1.87869	−0.939345	+	0.342973i	$0.888566\pi$
$548$	69.8698	2.98469
$549$	5.51867	0.235531
$550$	8.53775	0.364051
$551$	32.2852	1.37540
$552$	−26.6979	−1.13634
$553$	−7.80734	−0.332002
$554$	−59.5590	−2.53042
$555$	−10.4140	−0.442052
$556$	−0.492836	−0.0209009
$557$	12.3665	0.523985	0.261993	−	0.965070i	$-0.415620\pi$
$557$	12.3665	0.523985	0.261993	+	0.965070i	$0.415620\pi$
$558$	10.3801	0.439425
$559$	−1.39275	−0.0589072
$560$	9.76061	0.412461
$561$	29.7857	1.25755
$562$	47.6786	2.01120
$563$	4.31452	0.181835	0.0909177	−	0.995858i	$-0.471020\pi$
$563$	4.31452	0.181835	0.0909177	+	0.995858i	$0.471020\pi$
$564$	−39.4118	−1.65954
$565$	−12.5159	−0.526549
$566$	−45.2493	−1.90197
$567$	3.14965	0.132273
$568$	66.3621	2.78449
$569$	−24.8419	−1.04143	−0.520713	−	0.853732i	$-0.674334\pi$
$569$	−24.8419	−1.04143	−0.520713	+	0.853732i	$0.674334\pi$
$570$	23.9895	1.00481
$571$	−29.5222	−1.23546	−0.617732	−	0.786389i	$-0.711948\pi$
$571$	−29.5222	−1.23546	−0.617732	+	0.786389i	$0.711948\pi$
$572$	20.0679	0.839079
$573$	−20.8821	−0.872361
$574$	4.20536	0.175528
$575$	−2.29373	−0.0956553
$576$	−23.4017	−0.975072
$577$	20.7375	0.863315	0.431658	−	0.902038i	$-0.357929\pi$
$577$	20.7375	0.863315	0.431658	+	0.902038i	$0.357929\pi$
$578$	−83.3359	−3.46632
$579$	−7.96134	−0.330862
$580$	25.4300	1.05592
$581$	−8.04050	−0.333576
$582$	−34.5386	−1.43167
$583$	−33.9901	−1.40772
$584$	−71.9336	−2.97663
$585$	−1.42227	−0.0588037
$586$	−79.2071	−3.27202
$587$	−34.0310	−1.40461	−0.702304	−	0.711877i	$-0.747845\pi$
$587$	−34.0310	−1.40461	−0.702304	+	0.711877i	$0.747845\pi$
$588$	45.1147	1.86050
$589$	−21.7788	−0.897381
$590$	9.35710	0.385226
$591$	33.0926	1.36125
$592$	−97.9328	−4.02501
$593$	−9.75256	−0.400490	−0.200245	−	0.979746i	$-0.564174\pi$
$593$	−9.75256	−0.400490	−0.200245	+	0.979746i	$0.564174\pi$
$594$	48.1559	1.97586
$595$	5.32348	0.218241
$596$	−0.737177	−0.0301959
$597$	18.4792	0.756303
$598$	−7.46178	−0.305135
$599$	−22.3618	−0.913679	−0.456840	−	0.889549i	$-0.651019\pi$
$599$	−22.3618	−0.913679	−0.456840	+	0.889549i	$0.651019\pi$
$600$	11.6395	0.475180
$601$	29.7319	1.21279	0.606395	−	0.795164i	$-0.292615\pi$
$601$	29.7319	1.21279	0.606395	+	0.795164i	$0.292615\pi$
$602$	2.37023	0.0966034
$603$	14.1691	0.577010
$604$	−22.7924	−0.927408
$605$	0.887343	0.0360756
$606$	11.0816	0.450159
$607$	−14.1919	−0.576031	−0.288016	−	0.957626i	$-0.592996\pi$
$607$	−14.1919	−0.576031	−0.288016	+	0.957626i	$0.592996\pi$
$608$	111.695	4.52982
$609$	−5.06798	−0.205365
$610$	−12.6227	−0.511077
$611$	−6.78517	−0.274499
$612$	−42.3716	−1.71277
$613$	−24.5744	−0.992551	−0.496275	−	0.868165i	$-0.665299\pi$
$613$	−24.5744	−0.992551	−0.496275	+	0.868165i	$0.665299\pi$
$614$	−35.7979	−1.44469
$615$	2.75596	0.111131
$616$	−21.0371	−0.847610
$617$	−42.1149	−1.69548	−0.847740	−	0.530412i	$-0.822037\pi$
$617$	−42.1149	−1.69548	−0.847740	+	0.530412i	$0.822037\pi$
$618$	−2.48200	−0.0998405
$619$	11.7439	0.472028	0.236014	−	0.971750i	$-0.424159\pi$
$619$	11.7439	0.472028	0.236014	+	0.971750i	$0.424159\pi$
$620$	−17.1545	−0.688941
$621$	−12.9375	−0.519162
$622$	−44.2917	−1.77594
$623$	−4.36275	−0.174790
$624$	20.8087	0.833016
$625$	1.00000	0.0400000
$626$	−52.1211	−2.08318
$627$	−28.4148	−1.13478
$628$	35.6391	1.42215
$629$	−53.4130	−2.12971
$630$	2.42046	0.0964335
$631$	43.2918	1.72342	0.861710	−	0.507402i	$-0.169394\pi$
$631$	43.2918	1.72342	0.861710	+	0.507402i	$0.169394\pi$
$632$	−87.5525	−3.48265
$633$	−29.5565	−1.17477
$634$	−37.5303	−1.49052
$635$	−14.9548	−0.593463
$636$	−75.2270	−2.98294
$637$	7.76698	0.307739
$638$	−41.6879	−1.65044
$639$	9.04387	0.357770
$640$	19.7408	0.780325
$641$	18.3707	0.725598	0.362799	−	0.931867i	$-0.381821\pi$
$641$	18.3707	0.725598	0.362799	+	0.931867i	$0.381821\pi$
$642$	28.2392	1.11451
$643$	−20.7074	−0.816619	−0.408309	−	0.912844i	$-0.633881\pi$
$643$	−20.7074	−0.816619	−0.408309	+	0.912844i	$0.633881\pi$
$644$	9.17523	0.361555
$645$	1.55332	0.0611618
$646$	123.041	4.84097
$647$	30.4531	1.19723	0.598617	−	0.801036i	$-0.295717\pi$
$647$	30.4531	1.19723	0.598617	+	0.801036i	$0.295717\pi$
$648$	35.3206	1.38752
$649$	−11.0831	−0.435052
$650$	3.25311	0.127598
$651$	3.41874	0.133991
$652$	−84.5221	−3.31014
$653$	21.8753	0.856046	0.428023	−	0.903768i	$-0.359210\pi$
$653$	21.8753	0.856046	0.428023	+	0.903768i	$0.359210\pi$
$654$	16.9526	0.662897
$655$	−13.9022	−0.543203
$656$	25.9168	1.01188
$657$	−9.80316	−0.382458
$658$	11.5472	0.450157
$659$	−32.5612	−1.26841	−0.634203	−	0.773167i	$-0.718672\pi$
$659$	−32.5612	−1.26841	−0.634203	+	0.773167i	$0.718672\pi$
$660$	−22.3814	−0.871194
$661$	29.4444	1.14525	0.572627	−	0.819816i	$-0.305925\pi$
$661$	29.4444	1.14525	0.572627	+	0.819816i	$0.305925\pi$
$662$	56.8044	2.20777
$663$	11.3492	0.440765
$664$	−90.1672	−3.49917
$665$	−5.07845	−0.196934
$666$	−24.2856	−0.941050
$667$	11.1998	0.433658
$668$	−116.990	−4.52648
$669$	−26.6832	−1.03163
$670$	−32.4085	−1.25205
$671$	14.9511	0.577181
$672$	−17.5333	−0.676363
$673$	−9.76048	−0.376239	−0.188119	−	0.982146i	$-0.560239\pi$
$673$	−9.76048	−0.376239	−0.188119	+	0.982146i	$0.560239\pi$
$674$	−22.4348	−0.864155
$675$	5.64035	0.217097
$676$	−60.0590	−2.30996
$677$	−28.2834	−1.08702	−0.543510	−	0.839403i	$-0.682905\pi$
$677$	−28.2834	−1.08702	−0.543510	+	0.839403i	$0.682905\pi$
$678$	45.4096	1.74395
$679$	7.31162	0.280594
$680$	59.6982	2.28932
$681$	32.5809	1.24850
$682$	28.1217	1.07683
$683$	34.1494	1.30669	0.653346	−	0.757060i	$-0.273365\pi$
$683$	34.1494	1.30669	0.653346	+	0.757060i	$0.273365\pi$
$684$	40.4213	1.54555
$685$	−13.4156	−0.512583
$686$	−27.6526	−1.05578
$687$	−7.26055	−0.277007
$688$	14.6073	0.556897
$689$	−12.9511	−0.493399
$690$	8.32201	0.316813
$691$	0.310326	0.0118054	0.00590268	−	0.999983i	$-0.498121\pi$
$691$	0.310326	0.0118054	0.00590268	+	0.999983i	$0.498121\pi$
$692$	121.112	4.60399
$693$	−2.86695	−0.108907
$694$	20.8209	0.790352
$695$	0.0946286	0.00358947
$696$	−56.8330	−2.15425
$697$	14.1351	0.535406
$698$	29.8452	1.12966
$699$	4.79593	0.181399
$700$	−4.00013	−0.151191
$701$	18.8228	0.710927	0.355463	−	0.934690i	$-0.384323\pi$
$701$	18.8228	0.710927	0.355463	+	0.934690i	$0.384323\pi$
$702$	18.3487	0.692526
$703$	50.9545	1.92178
$704$	−63.3996	−2.38946
$705$	7.56740	0.285005
$706$	0.633764	0.0238520
$707$	−2.34591	−0.0882270
$708$	−24.5293	−0.921866
$709$	26.4443	0.993136	0.496568	−	0.867998i	$-0.334593\pi$
$709$	26.4443	0.993136	0.496568	+	0.867998i	$0.334593\pi$
$710$	−20.6857	−0.776322
$711$	−11.9317	−0.447474
$712$	−48.9245	−1.83352
$713$	−7.55511	−0.282941
$714$	−19.3144	−0.722822
$715$	−3.85320	−0.144101
$716$	−116.236	−4.34394
$717$	24.4640	0.913625
$718$	94.3254	3.52019
$719$	−0.919844	−0.0343044	−0.0171522	−	0.999853i	$-0.505460\pi$
$719$	−0.919844	−0.0343044	−0.0171522	+	0.999853i	$0.505460\pi$
$720$	14.9168	0.555918
$721$	0.525424	0.0195678
$722$	−66.3664	−2.46990
$723$	−11.3844	−0.423390
$724$	−124.095	−4.61197
$725$	−4.88278	−0.181342
$726$	−3.21941	−0.119484
$727$	5.00266	0.185538	0.0927692	−	0.995688i	$-0.470428\pi$
$727$	5.00266	0.185538	0.0927692	+	0.995688i	$0.470428\pi$
$728$	−8.01571	−0.297082
$729$	27.6801	1.02519
$730$	22.4224	0.829892
$731$	7.96686	0.294665
$732$	33.0898	1.22304
$733$	15.4916	0.572196	0.286098	−	0.958200i	$-0.407642\pi$
$733$	15.4916	0.572196	0.286098	+	0.958200i	$0.407642\pi$
$734$	86.1713	3.18064
$735$	−8.66240	−0.319517
$736$	38.7471	1.42824
$737$	38.3867	1.41399
$738$	6.42692	0.236578
$739$	−22.7620	−0.837312	−0.418656	−	0.908145i	$-0.637499\pi$
$739$	−22.7620	−0.837312	−0.418656	+	0.908145i	$0.637499\pi$
$740$	40.1352	1.47540
$741$	−10.8268	−0.397732
$742$	22.0406	0.809137
$743$	−35.5736	−1.30507	−0.652534	−	0.757759i	$-0.726294\pi$
$743$	−35.5736	−1.30507	−0.652534	+	0.757759i	$0.726294\pi$
$744$	38.3382	1.40555
$745$	0.141544	0.00518578
$746$	103.177	3.77759
$747$	−12.2880	−0.449596
$748$	−114.793	−4.19723
$749$	−5.97808	−0.218434
$750$	−3.62815	−0.132481
$751$	−26.3727	−0.962352	−0.481176	−	0.876624i	$-0.659790\pi$
$751$	−26.3727	−0.962352	−0.481176	+	0.876624i	$0.659790\pi$
$752$	71.1632	2.59505
$753$	4.01996	0.146496
$754$	−15.8842	−0.578469
$755$	4.37632	0.159271
$756$	−22.5621	−0.820576
$757$	−24.6762	−0.896872	−0.448436	−	0.893815i	$-0.648019\pi$
$757$	−24.6762	−0.896872	−0.448436	+	0.893815i	$0.648019\pi$
$758$	−43.3721	−1.57534
$759$	−9.85712	−0.357791
$760$	−56.9504	−2.06581
$761$	−38.1455	−1.38277	−0.691387	−	0.722485i	$-0.743000\pi$
$761$	−38.1455	−1.38277	−0.691387	+	0.722485i	$0.743000\pi$
$762$	54.2582	1.96557
$763$	−3.58876	−0.129922
$764$	80.4783	2.91160
$765$	8.13570	0.294147
$766$	−41.1136	−1.48550
$767$	−4.22298	−0.152483
$768$	−17.7388	−0.640094
$769$	28.0928	1.01305	0.506525	−	0.862225i	$-0.330930\pi$
$769$	28.0928	1.01305	0.506525	+	0.862225i	$0.330930\pi$
$770$	6.55748	0.236315
$771$	−25.6860	−0.925059
$772$	30.6826	1.10429
$773$	−33.9391	−1.22070	−0.610352	−	0.792130i	$-0.708972\pi$
$773$	−33.9391	−1.22070	−0.610352	+	0.792130i	$0.708972\pi$
$774$	3.62235	0.130203
$775$	3.29380	0.118317
$776$	81.9934	2.94339
$777$	−7.99859	−0.286948
$778$	26.8610	0.963014
$779$	−13.4845	−0.483133
$780$	−8.52791	−0.305348
$781$	24.5015	0.876733
$782$	42.6830	1.52634
$783$	−27.5406	−0.984220
$784$	−81.4605	−2.90930
$785$	−6.84300	−0.244237
$786$	50.4391	1.79910
$787$	−48.6860	−1.73547	−0.867734	−	0.497029i	$-0.834424\pi$
$787$	−48.6860	−1.73547	−0.867734	+	0.497029i	$0.834424\pi$
$788$	−127.537	−4.54333
$789$	19.6343	0.699001
$790$	27.2910	0.970971
$791$	−9.61296	−0.341797
$792$	−32.1504	−1.14241
$793$	5.69677	0.202298
$794$	−1.19139	−0.0422808
$795$	14.4442	0.512283
$796$	−71.2178	−2.52425
$797$	46.1176	1.63357	0.816784	−	0.576944i	$-0.195755\pi$
$797$	46.1176	1.63357	0.816784	+	0.576944i	$0.195755\pi$
$798$	18.4254	0.652251
$799$	38.8127	1.37309
$800$	−16.8926	−0.597243
$801$	−6.66746	−0.235583
$802$	−52.5612	−1.85600
$803$	−26.5586	−0.937232
$804$	84.9576	2.99622
$805$	−1.76172	−0.0620925
$806$	10.7151	0.377424
$807$	−38.9957	−1.37271
$808$	−26.3073	−0.925489
$809$	−29.6789	−1.04345	−0.521727	−	0.853113i	$-0.674712\pi$
$809$	−29.6789	−1.04345	−0.521727	+	0.853113i	$0.674712\pi$
$810$	−11.0098	−0.386845
$811$	−34.9143	−1.22601	−0.613004	−	0.790080i	$-0.710039\pi$
$811$	−34.9143	−1.22601	−0.613004	+	0.790080i	$0.710039\pi$
$812$	19.5317	0.685430
$813$	−35.6888	−1.25166
$814$	−65.7943	−2.30609
$815$	16.2289	0.568475
$816$	−119.031	−4.16690
$817$	−7.60016	−0.265896
$818$	26.8191	0.937709
$819$	−1.09239	−0.0381711
$820$	−10.6213	−0.370913
$821$	−30.2872	−1.05703	−0.528515	−	0.848924i	$-0.677251\pi$
$821$	−30.2872	−1.05703	−0.528515	+	0.848924i	$0.677251\pi$
$822$	48.6737	1.69769
$823$	−48.6825	−1.69697	−0.848483	−	0.529223i	$-0.822484\pi$
$823$	−48.6825	−1.69697	−0.848483	+	0.529223i	$0.822484\pi$
$824$	5.89218	0.205264
$825$	4.29741	0.149617
$826$	7.18679	0.250060
$827$	18.1170	0.629988	0.314994	−	0.949094i	$-0.397997\pi$
$827$	18.1170	0.629988	0.314994	+	0.949094i	$0.397997\pi$
$828$	14.0222	0.487306
$829$	−25.7900	−0.895723	−0.447862	−	0.894103i	$-0.647814\pi$
$829$	−25.7900	−0.895723	−0.447862	+	0.894103i	$0.647814\pi$
$830$	28.1060	0.975575
$831$	−29.9786	−1.03995
$832$	−24.1570	−0.837493
$833$	−44.4289	−1.53937
$834$	−0.343327	−0.0118884
$835$	22.4631	0.777367
$836$	109.509	3.78745
$837$	18.5782	0.642156
$838$	−70.2880	−2.42806
$839$	−20.9352	−0.722764	−0.361382	−	0.932418i	$-0.617695\pi$
$839$	−20.9352	−0.722764	−0.361382	+	0.932418i	$0.617695\pi$
$840$	8.93980	0.308452
$841$	−5.15847	−0.177878
$842$	−9.24781	−0.318701
$843$	23.9987	0.826558
$844$	113.909	3.92092
$845$	11.5318	0.396707
$846$	17.6472	0.606725
$847$	0.681531	0.0234177
$848$	135.832	4.66449
$849$	−22.7759	−0.781667
$850$	−18.6085	−0.638267
$851$	17.6762	0.605932
$852$	54.2269	1.85778
$853$	3.58490	0.122745	0.0613724	−	0.998115i	$-0.480452\pi$
$853$	3.58490	0.122745	0.0613724	+	0.998115i	$0.480452\pi$
$854$	−9.69494	−0.331754
$855$	−7.76124	−0.265429
$856$	−67.0390	−2.29134
$857$	24.4192	0.834144	0.417072	−	0.908873i	$-0.363056\pi$
$857$	24.4192	0.834144	0.417072	+	0.908873i	$0.363056\pi$
$858$	13.9800	0.477268
$859$	−4.64192	−0.158380	−0.0791902	−	0.996860i	$-0.525233\pi$
$859$	−4.64192	−0.158380	−0.0791902	+	0.996860i	$0.525233\pi$
$860$	−5.98640	−0.204135
$861$	2.11674	0.0721382
$862$	−50.8788	−1.73294
$863$	36.5930	1.24564	0.622820	−	0.782365i	$-0.285987\pi$
$863$	36.5930	1.24564	0.622820	+	0.782365i	$0.285987\pi$
$864$	−95.2800	−3.24149
$865$	−23.2545	−0.790677
$866$	52.4415	1.78204
$867$	−41.9465	−1.42458
$868$	−13.1756	−0.447210
$869$	−32.3252	−1.09656
$870$	17.7154	0.600610
$871$	14.6264	0.495596
$872$	−40.2448	−1.36286
$873$	11.1741	0.378187
$874$	−40.7184	−1.37732
$875$	0.768058	0.0259651
$876$	−58.7796	−1.98598
$877$	40.7928	1.37747	0.688737	−	0.725011i	$-0.258165\pi$
$877$	40.7928	1.37747	0.688737	+	0.725011i	$0.258165\pi$
$878$	−100.784	−3.40130
$879$	−39.8683	−1.34472
$880$	40.4125	1.36230
$881$	2.99568	0.100927	0.0504634	−	0.998726i	$-0.483930\pi$
$881$	2.99568	0.100927	0.0504634	+	0.998726i	$0.483930\pi$
$882$	−20.2008	−0.680196
$883$	22.2254	0.747946	0.373973	−	0.927440i	$-0.377995\pi$
$883$	22.2254	0.747946	0.373973	+	0.927440i	$0.377995\pi$
$884$	−43.7391	−1.47110
$885$	4.70982	0.158319
$886$	−45.3238	−1.52268
$887$	−37.0489	−1.24398	−0.621990	−	0.783025i	$-0.713676\pi$
$887$	−37.0489	−1.24398	−0.621990	+	0.783025i	$0.713676\pi$
$888$	−89.6972	−3.01004
$889$	−11.4861	−0.385233
$890$	15.2503	0.511190
$891$	13.0407	0.436880
$892$	102.836	3.44319
$893$	−37.0262	−1.23904
$894$	−0.513543	−0.0171754
$895$	22.3183	0.746017
$896$	15.1621	0.506530
$897$	−3.75583	−0.125403
$898$	−48.1491	−1.60676
$899$	−16.0829	−0.536395
$900$	−6.11327	−0.203776
$901$	74.0834	2.46807
$902$	17.4117	0.579747
$903$	1.19304	0.0397018
$904$	−107.801	−3.58541
$905$	23.8273	0.792047
$906$	−15.8779	−0.527509
$907$	−13.9374	−0.462785	−0.231392	−	0.972860i	$-0.574328\pi$
$907$	−13.9374	−0.462785	−0.231392	+	0.972860i	$0.574328\pi$
$908$	−125.565	−4.16703
$909$	−3.58518	−0.118913
$910$	2.49858	0.0828271
$911$	−31.7747	−1.05274	−0.526372	−	0.850255i	$-0.676448\pi$
$911$	−31.7747	−1.05274	−0.526372	+	0.850255i	$0.676448\pi$
$912$	113.552	3.76008
$913$	−33.2906	−1.10176
$914$	−63.1557	−2.08901
$915$	−6.35352	−0.210041
$916$	27.9818	0.924544
$917$	−10.6777	−0.352608
$918$	−104.959	−3.46415
$919$	14.5586	0.480242	0.240121	−	0.970743i	$-0.422813\pi$
$919$	14.5586	0.480242	0.240121	+	0.970743i	$0.422813\pi$
$920$	−19.7562	−0.651341
$921$	−18.0186	−0.593733
$922$	−89.7874	−2.95699
$923$	9.33574	0.307290
$924$	−17.1902	−0.565516
$925$	−7.70629	−0.253381
$926$	−4.98416	−0.163790
$927$	0.802990	0.0263736
$928$	82.4828	2.70763
$929$	−21.8656	−0.717388	−0.358694	−	0.933455i	$-0.616778\pi$
$929$	−21.8656	−0.717388	−0.358694	+	0.933455i	$0.616778\pi$
$930$	−11.9504	−0.391869
$931$	42.3839	1.38908
$932$	−18.4833	−0.605439
$933$	−22.2939	−0.729869
$934$	47.7553	1.56260
$935$	22.0411	0.720822
$936$	−12.2502	−0.400409
$937$	−39.7474	−1.29849	−0.649246	−	0.760578i	$-0.724915\pi$
$937$	−39.7474	−1.29849	−0.649246	+	0.760578i	$0.724915\pi$
$938$	−24.8916	−0.812739
$939$	−26.2348	−0.856140
$940$	−29.1644	−0.951236
$941$	−22.0701	−0.719465	−0.359732	−	0.933056i	$-0.617132\pi$
$941$	−22.0701	−0.719465	−0.359732	+	0.933056i	$0.617132\pi$
$942$	24.8274	0.808921
$943$	−4.67780	−0.152330
$944$	44.2908	1.44154
$945$	4.33211	0.140924
$946$	9.81362	0.319068
$947$	20.5061	0.666359	0.333180	−	0.942863i	$-0.391878\pi$
$947$	20.5061	0.666359	0.333180	+	0.942863i	$0.391878\pi$
$948$	−71.5423	−2.32359
$949$	−10.1195	−0.328494
$950$	17.7520	0.575951
$951$	−18.8906	−0.612570
$952$	45.8516	1.48606
$953$	42.7285	1.38411	0.692055	−	0.721845i	$-0.256705\pi$
$953$	42.7285	1.38411	0.692055	+	0.721845i	$0.256705\pi$
$954$	33.6840	1.09056
$955$	−15.4525	−0.500031
$956$	−94.2830	−3.04933
$957$	−20.9833	−0.678294
$958$	46.4472	1.50064
$959$	−10.3039	−0.332732
$960$	26.9419	0.869547
$961$	−20.1509	−0.650028
$962$	−25.0694	−0.808271
$963$	−9.13612	−0.294407
$964$	43.8748	1.41311
$965$	−5.89131	−0.189648
$966$	6.39178	0.205652
$967$	−58.4435	−1.87942	−0.939708	−	0.341979i	$-0.888903\pi$
$967$	−58.4435	−1.87942	−0.939708	+	0.341979i	$0.888903\pi$
$968$	7.64277	0.245648
$969$	61.9317	1.98953
$970$	−25.5582	−0.820624
$971$	11.2311	0.360424	0.180212	−	0.983628i	$-0.442322\pi$
$971$	11.2311	0.360424	0.180212	+	0.983628i	$0.442322\pi$
$972$	−59.2649	−1.90092
$973$	0.0726803	0.00233002
$974$	−14.0194	−0.449211
$975$	1.63743	0.0524397
$976$	−59.7480	−1.91249
$977$	−32.2341	−1.03126	−0.515630	−	0.856811i	$-0.672442\pi$
$977$	−32.2341	−1.03126	−0.515630	+	0.856811i	$0.672442\pi$
$978$	−58.8810	−1.88281
$979$	−18.0634	−0.577309
$980$	33.3844	1.06643
$981$	−5.48459	−0.175109
$982$	93.1624	2.97293
$983$	17.6268	0.562208	0.281104	−	0.959677i	$-0.409299\pi$
$983$	17.6268	0.562208	0.281104	+	0.959677i	$0.409299\pi$
$984$	23.7374	0.756719
$985$	24.4882	0.780259
$986$	90.8613	2.89361
$987$	5.81220	0.185004
$988$	41.7259	1.32748
$989$	−2.63651	−0.0838360
$990$	10.0216	0.318507
$991$	−9.80696	−0.311528	−0.155764	−	0.987794i	$-0.549784\pi$
$991$	−9.80696	−0.311528	−0.155764	+	0.987794i	$0.549784\pi$
$992$	−55.6409	−1.76660
$993$	28.5921	0.907343
$994$	−15.8878	−0.503932
$995$	13.6744	0.433508
$996$	−73.6789	−2.33461
$997$	−16.5522	−0.524212	−0.262106	−	0.965039i	$-0.584417\pi$
$997$	−16.5522	−0.524212	−0.262106	+	0.965039i	$0.584417\pi$
$998$	1.87755	0.0594327
$999$	−43.4661	−1.37521

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8045.2.a.b.1.2	✓	126

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8045.2.a.b.1.2	✓	126	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)

\(f(q)\)	\(=\)	\(q-2.68479 q^{2} -1.35137 q^{3} +5.20811 q^{4} -1.00000 q^{5} +3.62815 q^{6} -0.768058 q^{7} -8.61310 q^{8} -1.17380 q^{9} +O(q^{10})\)	\(q-2.68479 q^{2} -1.35137 q^{3} +5.20811 q^{4} -1.00000 q^{5} +3.62815 q^{6} -0.768058 q^{7} -8.61310 q^{8} -1.17380 q^{9} +2.68479 q^{10} -3.18004 q^{11} -7.03808 q^{12} -1.21168 q^{13} +2.06208 q^{14} +1.35137 q^{15} +12.7082 q^{16} +6.93109 q^{17} +3.15141 q^{18} -6.61206 q^{19} -5.20811 q^{20} +1.03793 q^{21} +8.53775 q^{22} -2.29373 q^{23} +11.6395 q^{24} +1.00000 q^{25} +3.25311 q^{26} +5.64035 q^{27} -4.00013 q^{28} -4.88278 q^{29} -3.62815 q^{30} +3.29380 q^{31} -16.8926 q^{32} +4.29741 q^{33} -18.6085 q^{34} +0.768058 q^{35} -6.11327 q^{36} -7.70629 q^{37} +17.7520 q^{38} +1.63743 q^{39} +8.61310 q^{40} +2.03938 q^{41} -2.78663 q^{42} +1.14944 q^{43} -16.5620 q^{44} +1.17380 q^{45} +6.15820 q^{46} +5.59980 q^{47} -17.1734 q^{48} -6.41009 q^{49} -2.68479 q^{50} -9.36646 q^{51} -6.31057 q^{52} +10.6886 q^{53} -15.1432 q^{54} +3.18004 q^{55} +6.61536 q^{56} +8.93535 q^{57} +13.1092 q^{58} +3.48522 q^{59} +7.03808 q^{60} -4.70154 q^{61} -8.84318 q^{62} +0.901546 q^{63} +19.9367 q^{64} +1.21168 q^{65} -11.5377 q^{66} -12.0711 q^{67} +36.0978 q^{68} +3.09968 q^{69} -2.06208 q^{70} -7.70478 q^{71} +10.1101 q^{72} +8.35165 q^{73} +20.6898 q^{74} -1.35137 q^{75} -34.4363 q^{76} +2.44246 q^{77} -4.39616 q^{78} +10.1650 q^{79} -12.7082 q^{80} -4.10080 q^{81} -5.47532 q^{82} +10.4686 q^{83} +5.40565 q^{84} -6.93109 q^{85} -3.08600 q^{86} +6.59844 q^{87} +27.3900 q^{88} +5.68024 q^{89} -3.15141 q^{90} +0.930642 q^{91} -11.9460 q^{92} -4.45115 q^{93} -15.0343 q^{94} +6.61206 q^{95} +22.8281 q^{96} -9.51962 q^{97} +17.2097 q^{98} +3.73273 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 126 q + 5 q^{2} - 9 q^{3} + 109 q^{4} - 126 q^{5} - 21 q^{6} - 23 q^{7} + 12 q^{8} + 109 q^{9}+O(q^{10}) \) 126 * q + 5 * q^2 - 9 * q^3 + 109 * q^4 - 126 * q^5 - 21 * q^6 - 23 * q^7 + 12 * q^8 + 109 * q^9	\( 126 q + 5 q^{2} - 9 q^{3} + 109 q^{4} - 126 q^{5} - 21 q^{6} - 23 q^{7} + 12 q^{8} + 109 q^{9} - 5 q^{10} - 44 q^{11} - 11 q^{12} - 35 q^{13} - 14 q^{14} + 9 q^{15} + 75 q^{16} + 11 q^{17} - 15 q^{18} - 130 q^{19} - 109 q^{20} - 44 q^{21} - 14 q^{22} + 75 q^{23} - 63 q^{24} + 126 q^{25} - 43 q^{26} - 42 q^{27} - 77 q^{28} - 24 q^{29} + 21 q^{30} - 78 q^{31} + 24 q^{32} - 29 q^{33} - 57 q^{34} + 23 q^{35} + 50 q^{36} - 31 q^{37} - 3 q^{38} - 57 q^{39} - 12 q^{40} - 38 q^{41} - 10 q^{42} - 100 q^{43} - 90 q^{44} - 109 q^{45} - 96 q^{46} + 12 q^{47} - 22 q^{48} + 65 q^{49} + 5 q^{50} - 74 q^{51} - 112 q^{52} + 20 q^{53} - 90 q^{54} + 44 q^{55} - 57 q^{56} + 6 q^{57} - 35 q^{58} - 97 q^{59} + 11 q^{60} - 102 q^{61} - 16 q^{62} - 15 q^{63} + 4 q^{64} + 35 q^{65} - 83 q^{66} - 121 q^{67} + 41 q^{68} - 71 q^{69} + 14 q^{70} - 32 q^{71} - 32 q^{72} - 85 q^{73} - 42 q^{74} - 9 q^{75} - 275 q^{76} + 13 q^{77} + 10 q^{78} - 97 q^{79} - 75 q^{80} + 86 q^{81} - 55 q^{82} - 73 q^{83} - 111 q^{84} - 11 q^{85} - 56 q^{86} - q^{87} - 37 q^{88} - 67 q^{89} + 15 q^{90} - 180 q^{91} + 98 q^{92} - 44 q^{93} - 86 q^{94} + 130 q^{95} - 179 q^{96} - 50 q^{97} + 18 q^{98} - 217 q^{99}+O(q^{100}) \) 126 * q + 5 * q^2 - 9 * q^3 + 109 * q^4 - 126 * q^5 - 21 * q^6 - 23 * q^7 + 12 * q^8 + 109 * q^9 - 5 * q^10 - 44 * q^11 - 11 * q^12 - 35 * q^13 - 14 * q^14 + 9 * q^15 + 75 * q^16 + 11 * q^17 - 15 * q^18 - 130 * q^19 - 109 * q^20 - 44 * q^21 - 14 * q^22 + 75 * q^23 - 63 * q^24 + 126 * q^25 - 43 * q^26 - 42 * q^27 - 77 * q^28 - 24 * q^29 + 21 * q^30 - 78 * q^31 + 24 * q^32 - 29 * q^33 - 57 * q^34 + 23 * q^35 + 50 * q^36 - 31 * q^37 - 3 * q^38 - 57 * q^39 - 12 * q^40 - 38 * q^41 - 10 * q^42 - 100 * q^43 - 90 * q^44 - 109 * q^45 - 96 * q^46 + 12 * q^47 - 22 * q^48 + 65 * q^49 + 5 * q^50 - 74 * q^51 - 112 * q^52 + 20 * q^53 - 90 * q^54 + 44 * q^55 - 57 * q^56 + 6 * q^57 - 35 * q^58 - 97 * q^59 + 11 * q^60 - 102 * q^61 - 16 * q^62 - 15 * q^63 + 4 * q^64 + 35 * q^65 - 83 * q^66 - 121 * q^67 + 41 * q^68 - 71 * q^69 + 14 * q^70 - 32 * q^71 - 32 * q^72 - 85 * q^73 - 42 * q^74 - 9 * q^75 - 275 * q^76 + 13 * q^77 + 10 * q^78 - 97 * q^79 - 75 * q^80 + 86 * q^81 - 55 * q^82 - 73 * q^83 - 111 * q^84 - 11 * q^85 - 56 * q^86 - q^87 - 37 * q^88 - 67 * q^89 + 15 * q^90 - 180 * q^91 + 98 * q^92 - 44 * q^93 - 86 * q^94 + 130 * q^95 - 179 * q^96 - 50 * q^97 + 18 * q^98 - 217 * q^99

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