LMFDB - Embedded newform 8042.2.a.c.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8042,2,Mod(1,8042)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8042, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8042.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8042 = 2 \cdot 4021 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8042.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:	$64.2156933055$
Analytic rank:	$0$
Dimension:	$86$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Character	$\chi$	$=$	8042.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -3.00414 q^{3} +1.00000 q^{4} -0.143261 q^{5} +3.00414 q^{6} -0.146700 q^{7} -1.00000 q^{8} +6.02484 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -3.00414 q^{3} +1.00000 q^{4} -0.143261 q^{5} +3.00414 q^{6} -0.146700 q^{7} -1.00000 q^{8} +6.02484 q^{9} +0.143261 q^{10} +0.117687 q^{11} -3.00414 q^{12} -6.33848 q^{13} +0.146700 q^{14} +0.430375 q^{15} +1.00000 q^{16} +4.19606 q^{17} -6.02484 q^{18} +2.12556 q^{19} -0.143261 q^{20} +0.440707 q^{21} -0.117687 q^{22} +6.31069 q^{23} +3.00414 q^{24} -4.97948 q^{25} +6.33848 q^{26} -9.08704 q^{27} -0.146700 q^{28} +1.74846 q^{29} -0.430375 q^{30} -5.62193 q^{31} -1.00000 q^{32} -0.353548 q^{33} -4.19606 q^{34} +0.0210164 q^{35} +6.02484 q^{36} +4.37146 q^{37} -2.12556 q^{38} +19.0417 q^{39} +0.143261 q^{40} -4.63468 q^{41} -0.440707 q^{42} +1.74631 q^{43} +0.117687 q^{44} -0.863123 q^{45} -6.31069 q^{46} -3.49996 q^{47} -3.00414 q^{48} -6.97848 q^{49} +4.97948 q^{50} -12.6055 q^{51} -6.33848 q^{52} +11.4668 q^{53} +9.08704 q^{54} -0.0168599 q^{55} +0.146700 q^{56} -6.38546 q^{57} -1.74846 q^{58} +6.01860 q^{59} +0.430375 q^{60} +1.18916 q^{61} +5.62193 q^{62} -0.883845 q^{63} +1.00000 q^{64} +0.908056 q^{65} +0.353548 q^{66} +14.6051 q^{67} +4.19606 q^{68} -18.9582 q^{69} -0.0210164 q^{70} -7.58579 q^{71} -6.02484 q^{72} +4.40295 q^{73} -4.37146 q^{74} +14.9590 q^{75} +2.12556 q^{76} -0.0172647 q^{77} -19.0417 q^{78} -9.92100 q^{79} -0.143261 q^{80} +9.22419 q^{81} +4.63468 q^{82} -15.2838 q^{83} +0.440707 q^{84} -0.601130 q^{85} -1.74631 q^{86} -5.25263 q^{87} -0.117687 q^{88} +8.97895 q^{89} +0.863123 q^{90} +0.929857 q^{91} +6.31069 q^{92} +16.8891 q^{93} +3.49996 q^{94} -0.304509 q^{95} +3.00414 q^{96} -6.41299 q^{97} +6.97848 q^{98} +0.709045 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 86 q - 86 q^{2} + 12 q^{3} + 86 q^{4} - 4 q^{5} - 12 q^{6} + 35 q^{7} - 86 q^{8} + 72 q^{9}+O(q^{10}) $ 86 * q - 86 * q^2 + 12 * q^3 + 86 * q^4 - 4 * q^5 - 12 * q^6 + 35 * q^7 - 86 * q^8 + 72 * q^9	\( 86 q - 86 q^{2} + 12 q^{3} + 86 q^{4} - 4 q^{5} - 12 q^{6} + 35 q^{7} - 86 q^{8} + 72 q^{9} + 4 q^{10} + 13 q^{11} + 12 q^{12} + 45 q^{13} - 35 q^{14} + 17 q^{15} + 86 q^{16} + 5 q^{17} - 72 q^{18} + 47 q^{19} - 4 q^{20} + 15 q^{21} - 13 q^{22} + 6 q^{23} - 12 q^{24} + 112 q^{25} - 45 q^{26} + 51 q^{27} + 35 q^{28} - 14 q^{29} - 17 q^{30} + 24 q^{31} - 86 q^{32} + 43 q^{33} - 5 q^{34} + 42 q^{35} + 72 q^{36} + 61 q^{37} - 47 q^{38} + 20 q^{39} + 4 q^{40} - 16 q^{41} - 15 q^{42} + 72 q^{43} + 13 q^{44} + 6 q^{45} - 6 q^{46} + 11 q^{47} + 12 q^{48} + 89 q^{49} - 112 q^{50} + 56 q^{51} + 45 q^{52} - 7 q^{53} - 51 q^{54} + 48 q^{55} - 35 q^{56} + 65 q^{57} + 14 q^{58} + 24 q^{59} + 17 q^{60} + 31 q^{61} - 24 q^{62} + 98 q^{63} + 86 q^{64} - 9 q^{65} - 43 q^{66} + 157 q^{67} + 5 q^{68} + q^{69} - 42 q^{70} - 11 q^{71} - 72 q^{72} + 74 q^{73} - 61 q^{74} + 76 q^{75} + 47 q^{76} - 13 q^{77} - 20 q^{78} + 57 q^{79} - 4 q^{80} + 34 q^{81} + 16 q^{82} + 65 q^{83} + 15 q^{84} + 102 q^{85} - 72 q^{86} + 49 q^{87} - 13 q^{88} - 34 q^{89} - 6 q^{90} + 91 q^{91} + 6 q^{92} + 57 q^{93} - 11 q^{94} - 13 q^{95} - 12 q^{96} + 64 q^{97} - 89 q^{98} + 56 q^{99}+O(q^{100}) \) 86 * q - 86 * q^2 + 12 * q^3 + 86 * q^4 - 4 * q^5 - 12 * q^6 + 35 * q^7 - 86 * q^8 + 72 * q^9 + 4 * q^10 + 13 * q^11 + 12 * q^12 + 45 * q^13 - 35 * q^14 + 17 * q^15 + 86 * q^16 + 5 * q^17 - 72 * q^18 + 47 * q^19 - 4 * q^20 + 15 * q^21 - 13 * q^22 + 6 * q^23 - 12 * q^24 + 112 * q^25 - 45 * q^26 + 51 * q^27 + 35 * q^28 - 14 * q^29 - 17 * q^30 + 24 * q^31 - 86 * q^32 + 43 * q^33 - 5 * q^34 + 42 * q^35 + 72 * q^36 + 61 * q^37 - 47 * q^38 + 20 * q^39 + 4 * q^40 - 16 * q^41 - 15 * q^42 + 72 * q^43 + 13 * q^44 + 6 * q^45 - 6 * q^46 + 11 * q^47 + 12 * q^48 + 89 * q^49 - 112 * q^50 + 56 * q^51 + 45 * q^52 - 7 * q^53 - 51 * q^54 + 48 * q^55 - 35 * q^56 + 65 * q^57 + 14 * q^58 + 24 * q^59 + 17 * q^60 + 31 * q^61 - 24 * q^62 + 98 * q^63 + 86 * q^64 - 9 * q^65 - 43 * q^66 + 157 * q^67 + 5 * q^68 + q^69 - 42 * q^70 - 11 * q^71 - 72 * q^72 + 74 * q^73 - 61 * q^74 + 76 * q^75 + 47 * q^76 - 13 * q^77 - 20 * q^78 + 57 * q^79 - 4 * q^80 + 34 * q^81 + 16 * q^82 + 65 * q^83 + 15 * q^84 + 102 * q^85 - 72 * q^86 + 49 * q^87 - 13 * q^88 - 34 * q^89 - 6 * q^90 + 91 * q^91 + 6 * q^92 + 57 * q^93 - 11 * q^94 - 13 * q^95 - 12 * q^96 + 64 * q^97 - 89 * q^98 + 56 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−3.00414	−1.73444	−0.867220	−	0.497926i	$-0.834095\pi$
$3$	−3.00414	−1.73444	−0.867220	+	0.497926i	$0.834095\pi$
$4$	1.00000	0.500000
$5$	−0.143261	−0.0640682	−0.0320341	−	0.999487i	$-0.510199\pi$
$5$	−0.143261	−0.0640682	−0.0320341	+	0.999487i	$0.510199\pi$
$6$	3.00414	1.22643
$7$	−0.146700	−0.0554474	−0.0277237	−	0.999616i	$-0.508826\pi$
$7$	−0.146700	−0.0554474	−0.0277237	+	0.999616i	$0.508826\pi$
$8$	−1.00000	−0.353553
$9$	6.02484	2.00828
$10$	0.143261	0.0453030
$11$	0.117687	0.0354839	0.0177420	−	0.999843i	$-0.494352\pi$
$11$	0.117687	0.0354839	0.0177420	+	0.999843i	$0.494352\pi$
$12$	−3.00414	−0.867220
$13$	−6.33848	−1.75798	−0.878990	−	0.476841i	$-0.841782\pi$
$13$	−6.33848	−1.75798	−0.878990	+	0.476841i	$0.841782\pi$
$14$	0.146700	0.0392073
$15$	0.430375	0.111122
$16$	1.00000	0.250000
$17$	4.19606	1.01769	0.508847	−	0.860857i	$-0.330072\pi$
$17$	4.19606	1.01769	0.508847	+	0.860857i	$0.330072\pi$
$18$	−6.02484	−1.42007
$19$	2.12556	0.487636	0.243818	−	0.969821i	$-0.421600\pi$
$19$	2.12556	0.487636	0.243818	+	0.969821i	$0.421600\pi$
$20$	−0.143261	−0.0320341
$21$	0.440707	0.0961702
$22$	−0.117687	−0.0250909
$23$	6.31069	1.31587	0.657935	−	0.753074i	$-0.271430\pi$
$23$	6.31069	1.31587	0.657935	+	0.753074i	$0.271430\pi$
$24$	3.00414	0.613217
$25$	−4.97948	−0.995895
$26$	6.33848	1.24308
$27$	−9.08704	−1.74880
$28$	−0.146700	−0.0277237
$29$	1.74846	0.324682	0.162341	−	0.986735i	$-0.448096\pi$
$29$	1.74846	0.324682	0.162341	+	0.986735i	$0.448096\pi$
$30$	−0.430375	−0.0785754
$31$	−5.62193	−1.00973	−0.504864	−	0.863199i	$-0.668457\pi$
$31$	−5.62193	−1.00973	−0.504864	+	0.863199i	$0.668457\pi$
$32$	−1.00000	−0.176777
$33$	−0.353548	−0.0615447
$34$	−4.19606	−0.719618
$35$	0.0210164	0.00355242
$36$	6.02484	1.00414
$37$	4.37146	0.718663	0.359332	−	0.933210i	$-0.383005\pi$
$37$	4.37146	0.718663	0.359332	+	0.933210i	$0.383005\pi$
$38$	−2.12556	−0.344811
$39$	19.0417	3.04911
$40$	0.143261	0.0226515
$41$	−4.63468	−0.723816	−0.361908	−	0.932214i	$-0.617875\pi$
$41$	−4.63468	−0.723816	−0.361908	+	0.932214i	$0.617875\pi$
$42$	−0.440707	−0.0680026
$43$	1.74631	0.266310	0.133155	−	0.991095i	$-0.457489\pi$
$43$	1.74631	0.266310	0.133155	+	0.991095i	$0.457489\pi$
$44$	0.117687	0.0177420
$45$	−0.863123	−0.128667
$46$	−6.31069	−0.930461
$47$	−3.49996	−0.510522	−0.255261	−	0.966872i	$-0.582161\pi$
$47$	−3.49996	−0.510522	−0.255261	+	0.966872i	$0.582161\pi$
$48$	−3.00414	−0.433610
$49$	−6.97848	−0.996926
$50$	4.97948	0.704204
$51$	−12.6055	−1.76513
$52$	−6.33848	−0.878990
$53$	11.4668	1.57509	0.787546	−	0.616256i	$-0.211351\pi$
$53$	11.4668	1.57509	0.787546	+	0.616256i	$0.211351\pi$
$54$	9.08704	1.23659
$55$	−0.0168599	−0.00227339
$56$	0.146700	0.0196036
$57$	−6.38546	−0.845775
$58$	−1.74846	−0.229585
$59$	6.01860	0.783555	0.391778	−	0.920060i	$-0.371860\pi$
$59$	6.01860	0.783555	0.391778	+	0.920060i	$0.371860\pi$
$60$	0.430375	0.0555612
$61$	1.18916	0.152256	0.0761281	−	0.997098i	$-0.475744\pi$
$61$	1.18916	0.152256	0.0761281	+	0.997098i	$0.475744\pi$
$62$	5.62193	0.713986
$63$	−0.883845	−0.111354
$64$	1.00000	0.125000
$65$	0.908056	0.112631
$66$	0.353548	0.0435187
$67$	14.6051	1.78430	0.892151	−	0.451737i	$-0.149196\pi$
$67$	14.6051	1.78430	0.892151	+	0.451737i	$0.149196\pi$
$68$	4.19606	0.508847
$69$	−18.9582	−2.28230
$70$	−0.0210164	−0.00251194
$71$	−7.58579	−0.900268	−0.450134	−	0.892961i	$-0.648624\pi$
$71$	−7.58579	−0.900268	−0.450134	+	0.892961i	$0.648624\pi$
$72$	−6.02484	−0.710034
$73$	4.40295	0.515327	0.257663	−	0.966235i	$-0.417047\pi$
$73$	4.40295	0.515327	0.257663	+	0.966235i	$0.417047\pi$
$74$	−4.37146	−0.508172
$75$	14.9590	1.72732
$76$	2.12556	0.243818
$77$	−0.0172647	−0.00196749
$78$	−19.0417	−2.15605
$79$	−9.92100	−1.11620	−0.558100	−	0.829774i	$-0.688469\pi$
$79$	−9.92100	−1.11620	−0.558100	+	0.829774i	$0.688469\pi$
$80$	−0.143261	−0.0160170
$81$	9.22419	1.02491
$82$	4.63468	0.511815
$83$	−15.2838	−1.67761	−0.838805	−	0.544432i	$-0.816745\pi$
$83$	−15.2838	−1.67761	−0.838805	+	0.544432i	$0.816745\pi$
$84$	0.440707	0.0480851
$85$	−0.601130	−0.0652017
$86$	−1.74631	−0.188310
$87$	−5.25263	−0.563141
$88$	−0.117687	−0.0125455
$89$	8.97895	0.951767	0.475883	−	0.879508i	$-0.342128\pi$
$89$	8.97895	0.951767	0.475883	+	0.879508i	$0.342128\pi$
$90$	0.863123	0.0909812
$91$	0.929857	0.0974755
$92$	6.31069	0.657935
$93$	16.8891	1.75131
$94$	3.49996	0.360994
$95$	−0.304509	−0.0312419
$96$	3.00414	0.306608
$97$	−6.41299	−0.651140	−0.325570	−	0.945518i	$-0.605556\pi$
$97$	−6.41299	−0.651140	−0.325570	+	0.945518i	$0.605556\pi$
$98$	6.97848	0.704933
$99$	0.709045	0.0712617
$100$	−4.97948	−0.497948
$101$	−4.75450	−0.473090	−0.236545	−	0.971620i	$-0.576015\pi$
$101$	−4.75450	−0.473090	−0.236545	+	0.971620i	$0.576015\pi$
$102$	12.6055	1.24813
$103$	2.09799	0.206721	0.103360	−	0.994644i	$-0.467040\pi$
$103$	2.09799	0.206721	0.103360	+	0.994644i	$0.467040\pi$
$104$	6.33848	0.621540
$105$	−0.0631361	−0.00616145
$106$	−11.4668	−1.11376
$107$	13.9416	1.34778	0.673891	−	0.738831i	$-0.264622\pi$
$107$	13.9416	1.34778	0.673891	+	0.738831i	$0.264622\pi$
$108$	−9.08704	−0.874401
$109$	15.3448	1.46976	0.734881	−	0.678196i	$-0.237238\pi$
$109$	15.3448	1.46976	0.734881	+	0.678196i	$0.237238\pi$
$110$	0.0168599	0.00160753
$111$	−13.1325	−1.24648
$112$	−0.146700	−0.0138619
$113$	−4.43684	−0.417382	−0.208691	−	0.977982i	$-0.566920\pi$
$113$	−4.43684	−0.417382	−0.208691	+	0.977982i	$0.566920\pi$
$114$	6.38546	0.598053
$115$	−0.904075	−0.0843054
$116$	1.74846	0.162341
$117$	−38.1884	−3.53052
$118$	−6.01860	−0.554057
$119$	−0.615562	−0.0564285
$120$	−0.430375	−0.0392877
$121$	−10.9861	−0.998741
$122$	−1.18916	−0.107661
$123$	13.9232	1.25542
$124$	−5.62193	−0.504864
$125$	1.42967	0.127873
$126$	0.883845	0.0787392
$127$	17.6214	1.56365	0.781825	−	0.623498i	$-0.214289\pi$
$127$	17.6214	1.56365	0.781825	+	0.623498i	$0.214289\pi$
$128$	−1.00000	−0.0883883
$129$	−5.24616	−0.461899
$130$	−0.908056	−0.0796418
$131$	−8.25962	−0.721646	−0.360823	−	0.932634i	$-0.617504\pi$
$131$	−8.25962	−0.721646	−0.360823	+	0.932634i	$0.617504\pi$
$132$	−0.353548	−0.0307724
$133$	−0.311819	−0.0270382
$134$	−14.6051	−1.26169
$135$	1.30182	0.112043
$136$	−4.19606	−0.359809
$137$	17.7800	1.51905	0.759524	−	0.650479i	$-0.225432\pi$
$137$	17.7800	1.51905	0.759524	+	0.650479i	$0.225432\pi$
$138$	18.9582	1.61383
$139$	−18.3371	−1.55534	−0.777668	−	0.628676i	$-0.783597\pi$
$139$	−18.3371	−1.55534	−0.777668	+	0.628676i	$0.783597\pi$
$140$	0.0210164	0.00177621
$141$	10.5144	0.885470
$142$	7.58579	0.636586
$143$	−0.745956	−0.0623800
$144$	6.02484	0.502070
$145$	−0.250486	−0.0208018
$146$	−4.40295	−0.364391
$147$	20.9643	1.72911
$148$	4.37146	0.359332
$149$	−7.59093	−0.621873	−0.310936	−	0.950431i	$-0.600643\pi$
$149$	−7.59093	−0.621873	−0.310936	+	0.950431i	$0.600643\pi$
$150$	−14.9590	−1.22140
$151$	−12.6027	−1.02560	−0.512798	−	0.858509i	$-0.671391\pi$
$151$	−12.6027	−1.02560	−0.512798	+	0.858509i	$0.671391\pi$
$152$	−2.12556	−0.172405
$153$	25.2806	2.04381
$154$	0.0172647	0.00139123
$155$	0.805402	0.0646914
$156$	19.0417	1.52455
$157$	−23.0055	−1.83604	−0.918020	−	0.396535i	$-0.870213\pi$
$157$	−23.0055	−1.83604	−0.918020	+	0.396535i	$0.870213\pi$
$158$	9.92100	0.789272
$159$	−34.4480	−2.73190
$160$	0.143261	0.0113258
$161$	−0.925780	−0.0729617
$162$	−9.22419	−0.724721
$163$	−7.96081	−0.623539	−0.311769	−	0.950158i	$-0.600922\pi$
$163$	−7.96081	−0.623539	−0.311769	+	0.950158i	$0.600922\pi$
$164$	−4.63468	−0.361908
$165$	0.0506495	0.00394306
$166$	15.2838	1.18625
$167$	22.0414	1.70561	0.852806	−	0.522228i	$-0.174899\pi$
$167$	22.0414	1.70561	0.852806	+	0.522228i	$0.174899\pi$
$168$	−0.440707	−0.0340013
$169$	27.1764	2.09049
$170$	0.601130	0.0461046
$171$	12.8061	0.979310
$172$	1.74631	0.133155
$173$	−3.46386	−0.263352	−0.131676	−	0.991293i	$-0.542036\pi$
$173$	−3.46386	−0.263352	−0.131676	+	0.991293i	$0.542036\pi$
$174$	5.25263	0.398200
$175$	0.730490	0.0552199
$176$	0.117687	0.00887098
$177$	−18.0807	−1.35903
$178$	−8.97895	−0.673001
$179$	3.47610	0.259816	0.129908	−	0.991526i	$-0.458532\pi$
$179$	3.47610	0.259816	0.129908	+	0.991526i	$0.458532\pi$
$180$	−0.863123	−0.0643334
$181$	1.99572	0.148341	0.0741705	−	0.997246i	$-0.476369\pi$
$181$	1.99572	0.148341	0.0741705	+	0.997246i	$0.476369\pi$
$182$	−0.929857	−0.0689256
$183$	−3.57240	−0.264079
$184$	−6.31069	−0.465231
$185$	−0.626258	−0.0460434
$186$	−16.8891	−1.23837
$187$	0.493821	0.0361118
$188$	−3.49996	−0.255261
$189$	1.33307	0.0969666
$190$	0.304509	0.0220914
$191$	−15.1161	−1.09376	−0.546881	−	0.837211i	$-0.684185\pi$
$191$	−15.1161	−1.09376	−0.546881	+	0.837211i	$0.684185\pi$
$192$	−3.00414	−0.216805
$193$	−19.1486	−1.37835	−0.689173	−	0.724596i	$-0.742026\pi$
$193$	−19.1486	−1.37835	−0.689173	+	0.724596i	$0.742026\pi$
$194$	6.41299	0.460426
$195$	−2.72793	−0.195351
$196$	−6.97848	−0.498463
$197$	−14.4147	−1.02701	−0.513503	−	0.858088i	$-0.671653\pi$
$197$	−14.4147	−1.02701	−0.513503	+	0.858088i	$0.671653\pi$
$198$	−0.709045	−0.0503896
$199$	0.101937	0.00722611	0.00361306	−	0.999993i	$-0.498850\pi$
$199$	0.101937	0.00722611	0.00361306	+	0.999993i	$0.498850\pi$
$200$	4.97948	0.352102
$201$	−43.8759	−3.09476
$202$	4.75450	0.334525
$203$	−0.256500	−0.0180028
$204$	−12.6055	−0.882564
$205$	0.663968	0.0463736
$206$	−2.09799	−0.146174
$207$	38.0209	2.64264
$208$	−6.33848	−0.439495
$209$	0.250150	0.0173032
$210$	0.0631361	0.00435680
$211$	−9.12493	−0.628186	−0.314093	−	0.949392i	$-0.601700\pi$
$211$	−9.12493	−0.628186	−0.314093	+	0.949392i	$0.601700\pi$
$212$	11.4668	0.787546
$213$	22.7888	1.56146
$214$	−13.9416	−0.953025
$215$	−0.250178	−0.0170620
$216$	9.08704	0.618295
$217$	0.824738	0.0559869
$218$	−15.3448	−1.03928
$219$	−13.2271	−0.893803
$220$	−0.0168599	−0.00113670
$221$	−26.5966	−1.78908
$222$	13.1325	0.881393
$223$	7.75136	0.519070	0.259535	−	0.965734i	$-0.416431\pi$
$223$	7.75136	0.519070	0.259535	+	0.965734i	$0.416431\pi$
$224$	0.146700	0.00980182
$225$	−30.0006	−2.00004
$226$	4.43684	0.295134
$227$	16.3899	1.08783	0.543917	−	0.839139i	$-0.316941\pi$
$227$	16.3899	1.08783	0.543917	+	0.839139i	$0.316941\pi$
$228$	−6.38546	−0.422888
$229$	−2.65714	−0.175589	−0.0877943	−	0.996139i	$-0.527982\pi$
$229$	−2.65714	−0.175589	−0.0877943	+	0.996139i	$0.527982\pi$
$230$	0.904075	0.0596129
$231$	0.0518655	0.00341250
$232$	−1.74846	−0.114792
$233$	2.56307	0.167912	0.0839562	−	0.996469i	$-0.473244\pi$
$233$	2.56307	0.167912	0.0839562	+	0.996469i	$0.473244\pi$
$234$	38.1884	2.49645
$235$	0.501407	0.0327082
$236$	6.01860	0.391778
$237$	29.8040	1.93598
$238$	0.615562	0.0399010
$239$	−0.605731	−0.0391815	−0.0195908	−	0.999808i	$-0.506236\pi$
$239$	−0.605731	−0.0391815	−0.0195908	+	0.999808i	$0.506236\pi$
$240$	0.430375	0.0277806
$241$	−28.9977	−1.86790	−0.933952	−	0.357397i	$-0.883664\pi$
$241$	−28.9977	−1.86790	−0.933952	+	0.357397i	$0.883664\pi$
$242$	10.9861	0.706216
$243$	−0.449626	−0.0288435
$244$	1.18916	0.0761281
$245$	0.999742	0.0638712
$246$	−13.9232	−0.887713
$247$	−13.4728	−0.857254
$248$	5.62193	0.356993
$249$	45.9145	2.90971
$250$	−1.42967	−0.0904201
$251$	10.1312	0.639479	0.319739	−	0.947506i	$-0.396405\pi$
$251$	10.1312	0.639479	0.319739	+	0.947506i	$0.396405\pi$
$252$	−0.883845	−0.0556770
$253$	0.742686	0.0466923
$254$	−17.6214	−1.10567
$255$	1.80588	0.113088
$256$	1.00000	0.0625000
$257$	−1.29014	−0.0804765	−0.0402382	−	0.999190i	$-0.512812\pi$
$257$	−1.29014	−0.0804765	−0.0402382	+	0.999190i	$0.512812\pi$
$258$	5.24616	0.326612
$259$	−0.641293	−0.0398480
$260$	0.908056	0.0563153
$261$	10.5342	0.652052
$262$	8.25962	0.510281
$263$	3.29653	0.203273	0.101636	−	0.994822i	$-0.467592\pi$
$263$	3.29653	0.203273	0.101636	+	0.994822i	$0.467592\pi$
$264$	0.353548	0.0217593
$265$	−1.64275	−0.100913
$266$	0.311819	0.0191189
$267$	−26.9740	−1.65078
$268$	14.6051	0.892151
$269$	−2.96748	−0.180931	−0.0904653	−	0.995900i	$-0.528835\pi$
$269$	−2.96748	−0.180931	−0.0904653	+	0.995900i	$0.528835\pi$
$270$	−1.30182	−0.0792260
$271$	6.81422	0.413935	0.206967	−	0.978348i	$-0.433641\pi$
$271$	6.81422	0.413935	0.206967	+	0.978348i	$0.433641\pi$
$272$	4.19606	0.254423
$273$	−2.79342	−0.169065
$274$	−17.7800	−1.07413
$275$	−0.586019	−0.0353383
$276$	−18.9582	−1.14115
$277$	19.2518	1.15673	0.578364	−	0.815779i	$-0.303691\pi$
$277$	19.2518	1.15673	0.578364	+	0.815779i	$0.303691\pi$
$278$	18.3371	1.09979
$279$	−33.8712	−2.02782
$280$	−0.0210164	−0.00125597
$281$	−20.5585	−1.22642	−0.613209	−	0.789920i	$-0.710122\pi$
$281$	−20.5585	−1.22642	−0.613209	+	0.789920i	$0.710122\pi$
$282$	−10.5144	−0.626122
$283$	10.3320	0.614171	0.307086	−	0.951682i	$-0.400646\pi$
$283$	10.3320	0.614171	0.307086	+	0.951682i	$0.400646\pi$
$284$	−7.58579	−0.450134
$285$	0.914786	0.0541873
$286$	0.745956	0.0441093
$287$	0.679909	0.0401337
$288$	−6.02484	−0.355017
$289$	0.606890	0.0356994
$290$	0.250486	0.0147091
$291$	19.2655	1.12936
$292$	4.40295	0.257663
$293$	−19.2625	−1.12533	−0.562665	−	0.826685i	$-0.690224\pi$
$293$	−19.2625	−1.12533	−0.562665	+	0.826685i	$0.690224\pi$
$294$	−20.9643	−1.22266
$295$	−0.862229	−0.0502009
$296$	−4.37146	−0.254086
$297$	−1.06943	−0.0620544
$298$	7.59093	0.439731
$299$	−40.0002	−2.31327
$300$	14.9590	0.863660
$301$	−0.256184	−0.0147662
$302$	12.6027	0.725206
$303$	14.2832	0.820547
$304$	2.12556	0.121909
$305$	−0.170360	−0.00975477
$306$	−25.2806	−1.44519
$307$	24.0421	1.37216	0.686078	−	0.727528i	$-0.259331\pi$
$307$	24.0421	1.37216	0.686078	+	0.727528i	$0.259331\pi$
$308$	−0.0172647	−0.000983747 0
$309$	−6.30264	−0.358545
$310$	−0.805402	−0.0457438
$311$	−1.21572	−0.0689370	−0.0344685	−	0.999406i	$-0.510974\pi$
$311$	−1.21572	−0.0689370	−0.0344685	+	0.999406i	$0.510974\pi$
$312$	−19.0417	−1.07802
$313$	−22.7088	−1.28358	−0.641789	−	0.766881i	$-0.721808\pi$
$313$	−22.7088	−1.28358	−0.641789	+	0.766881i	$0.721808\pi$
$314$	23.0055	1.29828
$315$	0.126620	0.00713425
$316$	−9.92100	−0.558100
$317$	−11.0177	−0.618815	−0.309407	−	0.950930i	$-0.600131\pi$
$317$	−11.0177	−0.618815	−0.309407	+	0.950930i	$0.600131\pi$
$318$	34.4480	1.93175
$319$	0.205771	0.0115210
$320$	−0.143261	−0.00800852
$321$	−41.8823	−2.33764
$322$	0.925780	0.0515917
$323$	8.91895	0.496264
$324$	9.22419	0.512455
$325$	31.5623	1.75076
$326$	7.96081	0.440909
$327$	−46.0978	−2.54921
$328$	4.63468	0.255908
$329$	0.513445	0.0283072
$330$	−0.0506495	−0.00278816
$331$	10.9201	0.600224	0.300112	−	0.953904i	$-0.402976\pi$
$331$	10.9201	0.600224	0.300112	+	0.953904i	$0.402976\pi$
$332$	−15.2838	−0.838805
$333$	26.3373	1.44328
$334$	−22.0414	−1.20605
$335$	−2.09234	−0.114317
$336$	0.440707	0.0240426
$337$	−8.71803	−0.474901	−0.237451	−	0.971400i	$-0.576312\pi$
$337$	−8.71803	−0.474901	−0.237451	+	0.971400i	$0.576312\pi$
$338$	−27.1764	−1.47820
$339$	13.3289	0.723925
$340$	−0.601130	−0.0326009
$341$	−0.661627	−0.0358291
$342$	−12.8061	−0.692477
$343$	2.05065	0.110724
$344$	−1.74631	−0.0941548
$345$	2.71597	0.146223
$346$	3.46386	0.186218
$347$	2.70409	0.145163	0.0725816	−	0.997362i	$-0.476876\pi$
$347$	2.70409	0.145163	0.0725816	+	0.997362i	$0.476876\pi$
$348$	−5.25263	−0.281570
$349$	−11.6485	−0.623530	−0.311765	−	0.950159i	$-0.600920\pi$
$349$	−11.6485	−0.623530	−0.311765	+	0.950159i	$0.600920\pi$
$350$	−0.730490	−0.0390463
$351$	57.5981	3.07436
$352$	−0.117687	−0.00627273
$353$	−13.2598	−0.705749	−0.352875	−	0.935671i	$-0.614796\pi$
$353$	−13.2598	−0.705749	−0.352875	+	0.935671i	$0.614796\pi$
$354$	18.0807	0.960979
$355$	1.08675	0.0576785
$356$	8.97895	0.475883
$357$	1.84923	0.0978718
$358$	−3.47610	−0.183718
$359$	10.4909	0.553686	0.276843	−	0.960915i	$-0.410712\pi$
$359$	10.4909	0.553686	0.276843	+	0.960915i	$0.410712\pi$
$360$	0.863123	0.0454906
$361$	−14.4820	−0.762211
$362$	−1.99572	−0.104893
$363$	33.0039	1.73226
$364$	0.929857	0.0487377
$365$	−0.630771	−0.0330160
$366$	3.57240	0.186732
$367$	7.05660	0.368351	0.184176	−	0.982893i	$-0.441038\pi$
$367$	7.05660	0.368351	0.184176	+	0.982893i	$0.441038\pi$
$368$	6.31069	0.328968
$369$	−27.9232	−1.45363
$370$	0.626258	0.0325576
$371$	−1.68219	−0.0873348
$372$	16.8891	0.875656
$373$	3.96851	0.205481	0.102741	−	0.994708i	$-0.467239\pi$
$373$	3.96851	0.205481	0.102741	+	0.994708i	$0.467239\pi$
$374$	−0.493821	−0.0255349
$375$	−4.29492	−0.221789
$376$	3.49996	0.180497
$377$	−11.0826	−0.570783
$378$	−1.33307	−0.0685657
$379$	32.0302	1.64528	0.822640	−	0.568562i	$-0.192500\pi$
$379$	32.0302	1.64528	0.822640	+	0.568562i	$0.192500\pi$
$380$	−0.304509	−0.0156210
$381$	−52.9372	−2.71206
$382$	15.1161	0.773406
$383$	−1.91383	−0.0977920	−0.0488960	−	0.998804i	$-0.515570\pi$
$383$	−1.91383	−0.0977920	−0.0488960	+	0.998804i	$0.515570\pi$
$384$	3.00414	0.153304
$385$	0.00247335	0.000126054 0
$386$	19.1486	0.974638
$387$	10.5213	0.534825
$388$	−6.41299	−0.325570
$389$	13.9441	0.706995	0.353498	−	0.935435i	$-0.384992\pi$
$389$	13.9441	0.706995	0.353498	+	0.935435i	$0.384992\pi$
$390$	2.72793	0.138134
$391$	26.4800	1.33915
$392$	6.97848	0.352466
$393$	24.8130	1.25165
$394$	14.4147	0.726204
$395$	1.42129	0.0715128
$396$	0.709045	0.0356308
$397$	−14.6579	−0.735657	−0.367828	−	0.929894i	$-0.619899\pi$
$397$	−14.6579	−0.735657	−0.367828	+	0.929894i	$0.619899\pi$
$398$	−0.101937	−0.00510963
$399$	0.936749	0.0468961
$400$	−4.97948	−0.248974
$401$	−35.0260	−1.74912	−0.874558	−	0.484920i	$-0.838849\pi$
$401$	−35.0260	−1.74912	−0.874558	+	0.484920i	$0.838849\pi$
$402$	43.8759	2.18833
$403$	35.6345	1.77508
$404$	−4.75450	−0.236545
$405$	−1.32147	−0.0656641
$406$	0.256500	0.0127299
$407$	0.514463	0.0255010
$408$	12.6055	0.624067
$409$	18.8063	0.929911	0.464956	−	0.885334i	$-0.346070\pi$
$409$	18.8063	0.929911	0.464956	+	0.885334i	$0.346070\pi$
$410$	−0.663968	−0.0327911
$411$	−53.4136	−2.63470
$412$	2.09799	0.103360
$413$	−0.882930	−0.0434461
$414$	−38.0209	−1.86863
$415$	2.18956	0.107481
$416$	6.33848	0.310770
$417$	55.0873	2.69763
$418$	−0.250150	−0.0122352
$419$	−8.86558	−0.433112	−0.216556	−	0.976270i	$-0.569482\pi$
$419$	−8.86558	−0.433112	−0.216556	+	0.976270i	$0.569482\pi$
$420$	−0.0631361	−0.00308073
$421$	−7.60121	−0.370460	−0.185230	−	0.982695i	$-0.559303\pi$
$421$	−7.60121	−0.370460	−0.185230	+	0.982695i	$0.559303\pi$
$422$	9.12493	0.444194
$423$	−21.0867	−1.02527
$424$	−11.4668	−0.556879
$425$	−20.8942	−1.01352
$426$	−22.7888	−1.10412
$427$	−0.174450	−0.00844222
$428$	13.9416	0.673891
$429$	2.24096	0.108194
$430$	0.250178	0.0120647
$431$	15.4499	0.744193	0.372097	−	0.928194i	$-0.378639\pi$
$431$	15.4499	0.744193	0.372097	+	0.928194i	$0.378639\pi$
$432$	−9.08704	−0.437200
$433$	12.4640	0.598983	0.299491	−	0.954099i	$-0.403183\pi$
$433$	12.4640	0.598983	0.299491	+	0.954099i	$0.403183\pi$
$434$	−0.824738	−0.0395887
$435$	0.752495	0.0360794
$436$	15.3448	0.734881
$437$	13.4137	0.641666
$438$	13.2271	0.632014
$439$	27.3398	1.30486	0.652429	−	0.757849i	$-0.273750\pi$
$439$	27.3398	1.30486	0.652429	+	0.757849i	$0.273750\pi$
$440$	0.0168599	0.000803765 0
$441$	−42.0442	−2.00211
$442$	26.5966	1.26507
$443$	−20.7905	−0.987787	−0.493894	−	0.869522i	$-0.664427\pi$
$443$	−20.7905	−0.987787	−0.493894	+	0.869522i	$0.664427\pi$
$444$	−13.1325	−0.623239
$445$	−1.28633	−0.0609779
$446$	−7.75136	−0.367038
$447$	22.8042	1.07860
$448$	−0.146700	−0.00693093
$449$	6.33018	0.298740	0.149370	−	0.988781i	$-0.452276\pi$
$449$	6.33018	0.298740	0.149370	+	0.988781i	$0.452276\pi$
$450$	30.0006	1.41424
$451$	−0.545441	−0.0256838
$452$	−4.43684	−0.208691
$453$	37.8604	1.77883
$454$	−16.3899	−0.769215
$455$	−0.133212	−0.00624507
$456$	6.38546	0.299027
$457$	9.74286	0.455752	0.227876	−	0.973690i	$-0.426822\pi$
$457$	9.74286	0.455752	0.227876	+	0.973690i	$0.426822\pi$
$458$	2.65714	0.124160
$459$	−38.1297	−1.77974
$460$	−0.904075	−0.0421527
$461$	20.5671	0.957904	0.478952	−	0.877841i	$-0.341017\pi$
$461$	20.5671	0.957904	0.478952	+	0.877841i	$0.341017\pi$
$462$	−0.0518655	−0.00241300
$463$	−12.2838	−0.570875	−0.285437	−	0.958397i	$-0.592139\pi$
$463$	−12.2838	−0.570875	−0.285437	+	0.958397i	$0.592139\pi$
$464$	1.74846	0.0811704
$465$	−2.41954	−0.112203
$466$	−2.56307	−0.118732
$467$	−18.8945	−0.874333	−0.437167	−	0.899381i	$-0.644018\pi$
$467$	−18.8945	−0.874333	−0.437167	+	0.899381i	$0.644018\pi$
$468$	−38.1884	−1.76526
$469$	−2.14258	−0.0989350
$470$	−0.501407	−0.0231282
$471$	69.1117	3.18450
$472$	−6.01860	−0.277029
$473$	0.205518	0.00944973
$474$	−29.8040	−1.36894
$475$	−10.5842	−0.485634
$476$	−0.615562	−0.0282142
$477$	69.0859	3.16323
$478$	0.605731	0.0277055
$479$	15.8799	0.725571	0.362785	−	0.931873i	$-0.381826\pi$
$479$	15.8799	0.725571	0.362785	+	0.931873i	$0.381826\pi$
$480$	−0.430375	−0.0196438
$481$	−27.7084	−1.26339
$482$	28.9977	1.32081
$483$	2.78117	0.126548
$484$	−10.9861	−0.499370
$485$	0.918730	0.0417174
$486$	0.449626	0.0203954
$487$	41.5460	1.88263	0.941313	−	0.337534i	$-0.109593\pi$
$487$	41.5460	1.88263	0.941313	+	0.337534i	$0.109593\pi$
$488$	−1.18916	−0.0538307
$489$	23.9154	1.08149
$490$	−0.999742	−0.0451638
$491$	22.1488	0.999559	0.499780	−	0.866153i	$-0.333414\pi$
$491$	22.1488	0.999559	0.499780	+	0.866153i	$0.333414\pi$
$492$	13.9232	0.627708
$493$	7.33665	0.330426
$494$	13.4728	0.606170
$495$	−0.101578	−0.00456561
$496$	−5.62193	−0.252432
$497$	1.11284	0.0499176
$498$	−45.9145	−2.05748
$499$	42.4407	1.89991	0.949953	−	0.312393i	$-0.101131\pi$
$499$	42.4407	1.89991	0.949953	+	0.312393i	$0.101131\pi$
$500$	1.42967	0.0639367
$501$	−66.2153	−2.95828
$502$	−10.1312	−0.452180
$503$	−27.8906	−1.24358	−0.621791	−	0.783183i	$-0.713595\pi$
$503$	−27.8906	−1.24358	−0.621791	+	0.783183i	$0.713595\pi$
$504$	0.883845	0.0393696
$505$	0.681133	0.0303100
$506$	−0.742686	−0.0330164
$507$	−81.6416	−3.62583
$508$	17.6214	0.781825
$509$	−7.03088	−0.311638	−0.155819	−	0.987786i	$-0.549802\pi$
$509$	−7.03088	−0.311638	−0.155819	+	0.987786i	$0.549802\pi$
$510$	−1.80588	−0.0799656
$511$	−0.645914	−0.0285736
$512$	−1.00000	−0.0441942
$513$	−19.3150	−0.852779
$514$	1.29014	0.0569055
$515$	−0.300559	−0.0132442
$516$	−5.24616	−0.230949
$517$	−0.411900	−0.0181153
$518$	0.641293	0.0281768
$519$	10.4059	0.456769
$520$	−0.908056	−0.0398209
$521$	14.6755	0.642947	0.321474	−	0.946919i	$-0.395822\pi$
$521$	14.6755	0.642947	0.321474	+	0.946919i	$0.395822\pi$
$522$	−10.5342	−0.461070
$523$	28.2656	1.23597	0.617984	−	0.786191i	$-0.287950\pi$
$523$	28.2656	1.23597	0.617984	+	0.786191i	$0.287950\pi$
$524$	−8.25962	−0.360823
$525$	−2.19449	−0.0957755
$526$	−3.29653	−0.143736
$527$	−23.5899	−1.02759
$528$	−0.353548	−0.0153862
$529$	16.8249	0.731516
$530$	1.64275	0.0713564
$531$	36.2611	1.57360
$532$	−0.311819	−0.0135191
$533$	29.3769	1.27245
$534$	26.9740	1.16728
$535$	−1.99728	−0.0863499
$536$	−14.6051	−0.630846
$537$	−10.4427	−0.450635
$538$	2.96748	0.127937
$539$	−0.821275	−0.0353748
$540$	1.30182	0.0560213
$541$	25.6007	1.10066	0.550330	−	0.834947i	$-0.314502\pi$
$541$	25.6007	1.10066	0.550330	+	0.834947i	$0.314502\pi$
$542$	−6.81422	−0.292696
$543$	−5.99543	−0.257288
$544$	−4.19606	−0.179904
$545$	−2.19830	−0.0941650
$546$	2.79342	0.119547
$547$	39.3113	1.68083	0.840415	−	0.541944i	$-0.182311\pi$
$547$	39.3113	1.68083	0.840415	+	0.541944i	$0.182311\pi$
$548$	17.7800	0.759524
$549$	7.16449	0.305773
$550$	0.586019	0.0249879
$551$	3.71646	0.158326
$552$	18.9582	0.806914
$553$	1.45541	0.0618904
$554$	−19.2518	−0.817930
$555$	1.88137	0.0798595
$556$	−18.3371	−0.777668
$557$	17.3929	0.736962	0.368481	−	0.929635i	$-0.379878\pi$
$557$	17.3929	0.736962	0.368481	+	0.929635i	$0.379878\pi$
$558$	33.8712	1.43388
$559$	−11.0690	−0.468168
$560$	0.0210164	0.000888104 0
$561$	−1.48351	−0.0626337
$562$	20.5585	0.867209
$563$	3.86127	0.162733	0.0813665	−	0.996684i	$-0.474072\pi$
$563$	3.86127	0.162733	0.0813665	+	0.996684i	$0.474072\pi$
$564$	10.5144	0.442735
$565$	0.635624	0.0267409
$566$	−10.3320	−0.434285
$567$	−1.35319	−0.0568287
$568$	7.58579	0.318293
$569$	5.94360	0.249169	0.124584	−	0.992209i	$-0.460240\pi$
$569$	5.94360	0.249169	0.124584	+	0.992209i	$0.460240\pi$
$570$	−0.914786	−0.0383162
$571$	28.0852	1.17533	0.587664	−	0.809105i	$-0.300048\pi$
$571$	28.0852	1.17533	0.587664	+	0.809105i	$0.300048\pi$
$572$	−0.745956	−0.0311900
$573$	45.4108	1.89706
$574$	−0.679909	−0.0283788
$575$	−31.4240	−1.31047
$576$	6.02484	0.251035
$577$	−7.17816	−0.298831	−0.149415	−	0.988775i	$-0.547739\pi$
$577$	−7.17816	−0.298831	−0.149415	+	0.988775i	$0.547739\pi$
$578$	−0.606890	−0.0252433
$579$	57.5250	2.39066
$580$	−0.250486	−0.0104009
$581$	2.24213	0.0930192
$582$	−19.2655	−0.798581
$583$	1.34950	0.0558904
$584$	−4.40295	−0.182196
$585$	5.47089	0.226194
$586$	19.2625	0.795729
$587$	−17.7193	−0.731352	−0.365676	−	0.930742i	$-0.619162\pi$
$587$	−17.7193	−0.731352	−0.365676	+	0.930742i	$0.619162\pi$
$588$	20.9643	0.864554
$589$	−11.9497	−0.492380
$590$	0.862229	0.0354974
$591$	43.3038	1.78128
$592$	4.37146	0.179666
$593$	40.0397	1.64424	0.822118	−	0.569318i	$-0.192793\pi$
$593$	40.0397	1.64424	0.822118	+	0.569318i	$0.192793\pi$
$594$	1.06943	0.0438791
$595$	0.0881859	0.00361527
$596$	−7.59093	−0.310936
$597$	−0.306232	−0.0125333
$598$	40.0002	1.63573
$599$	28.0194	1.14484	0.572421	−	0.819960i	$-0.306004\pi$
$599$	28.0194	1.14484	0.572421	+	0.819960i	$0.306004\pi$
$600$	−14.9590	−0.610700
$601$	10.7073	0.436759	0.218379	−	0.975864i	$-0.429923\pi$
$601$	10.7073	0.436759	0.218379	+	0.975864i	$0.429923\pi$
$602$	0.256184	0.0104413
$603$	87.9937	3.58338
$604$	−12.6027	−0.512798
$605$	1.57388	0.0639875
$606$	−14.2832	−0.580214
$607$	20.9526	0.850441	0.425221	−	0.905090i	$-0.360196\pi$
$607$	20.9526	0.850441	0.425221	+	0.905090i	$0.360196\pi$
$608$	−2.12556	−0.0862027
$609$	0.770561	0.0312247
$610$	0.170360	0.00689767
$611$	22.1845	0.897487
$612$	25.2806	1.02191
$613$	47.7736	1.92956	0.964779	−	0.263061i	$-0.0847320\pi$
$613$	47.7736	1.92956	0.964779	+	0.263061i	$0.0847320\pi$
$614$	−24.0421	−0.970261
$615$	−1.99465	−0.0804321
$616$	0.0172647	0.000695614 0
$617$	−24.1784	−0.973386	−0.486693	−	0.873573i	$-0.661797\pi$
$617$	−24.1784	−0.973386	−0.486693	+	0.873573i	$0.661797\pi$
$618$	6.30264	0.253530
$619$	13.4611	0.541046	0.270523	−	0.962713i	$-0.412803\pi$
$619$	13.4611	0.541046	0.270523	+	0.962713i	$0.412803\pi$
$620$	0.805402	0.0323457
$621$	−57.3455	−2.30120
$622$	1.21572	0.0487458
$623$	−1.31721	−0.0527730
$624$	19.0417	0.762277
$625$	24.6926	0.987703
$626$	22.7088	0.907627
$627$	−0.751485	−0.0300114
$628$	−23.0055	−0.918020
$629$	18.3429	0.731379
$630$	−0.126620	−0.00504468
$631$	26.8916	1.07054	0.535269	−	0.844682i	$-0.320210\pi$
$631$	26.8916	1.07054	0.535269	+	0.844682i	$0.320210\pi$
$632$	9.92100	0.394636
$633$	27.4125	1.08955
$634$	11.0177	0.437568
$635$	−2.52446	−0.100180
$636$	−34.4480	−1.36595
$637$	44.2330	1.75257
$638$	−0.205771	−0.00814656
$639$	−45.7032	−1.80799
$640$	0.143261	0.00566288
$641$	−2.72993	−0.107826	−0.0539128	−	0.998546i	$-0.517169\pi$
$641$	−2.72993	−0.107826	−0.0539128	+	0.998546i	$0.517169\pi$
$642$	41.8823	1.65296
$643$	31.9989	1.26191	0.630956	−	0.775818i	$-0.282663\pi$
$643$	31.9989	1.26191	0.630956	+	0.775818i	$0.282663\pi$
$644$	−0.925780	−0.0364808
$645$	0.751569	0.0295930
$646$	−8.91895	−0.350912
$647$	11.6185	0.456770	0.228385	−	0.973571i	$-0.426656\pi$
$647$	11.6185	0.456770	0.228385	+	0.973571i	$0.426656\pi$
$648$	−9.22419	−0.362361
$649$	0.708310	0.0278036
$650$	−31.5623	−1.23798
$651$	−2.47763	−0.0971058
$652$	−7.96081	−0.311769
$653$	42.3683	1.65800	0.829000	−	0.559249i	$-0.188910\pi$
$653$	42.3683	1.65800	0.829000	+	0.559249i	$0.188910\pi$
$654$	46.0978	1.80257
$655$	1.18328	0.0462346
$656$	−4.63468	−0.180954
$657$	26.5271	1.03492
$658$	−0.513445	−0.0200162
$659$	40.4560	1.57594	0.787972	−	0.615711i	$-0.211131\pi$
$659$	40.4560	1.57594	0.787972	+	0.615711i	$0.211131\pi$
$660$	0.0506495	0.00197153
$661$	3.39615	0.132095	0.0660475	−	0.997816i	$-0.478961\pi$
$661$	3.39615	0.132095	0.0660475	+	0.997816i	$0.478961\pi$
$662$	−10.9201	−0.424423
$663$	79.9000	3.10306
$664$	15.2838	0.593125
$665$	0.0446715	0.00173229
$666$	−26.3373	−1.02055
$667$	11.0340	0.427239
$668$	22.0414	0.852806
$669$	−23.2862	−0.900295
$670$	2.09234	0.0808343
$671$	0.139948	0.00540265
$672$	−0.440707	−0.0170007
$673$	5.56443	0.214493	0.107246	−	0.994232i	$-0.465797\pi$
$673$	5.56443	0.214493	0.107246	+	0.994232i	$0.465797\pi$
$674$	8.71803	0.335806
$675$	45.2487	1.74162
$676$	27.1764	1.04525
$677$	25.3145	0.972914	0.486457	−	0.873704i	$-0.338289\pi$
$677$	25.3145	0.972914	0.486457	+	0.873704i	$0.338289\pi$
$678$	−13.3289	−0.511892
$679$	0.940787	0.0361041
$680$	0.601130	0.0230523
$681$	−49.2374	−1.88678
$682$	0.661627	0.0253350
$683$	−32.8936	−1.25864	−0.629320	−	0.777147i	$-0.716666\pi$
$683$	−32.8936	−1.25864	−0.629320	+	0.777147i	$0.716666\pi$
$684$	12.8061	0.489655
$685$	−2.54718	−0.0973226
$686$	−2.05065	−0.0782940
$687$	7.98241	0.304548
$688$	1.74631	0.0665775
$689$	−72.6824	−2.76898
$690$	−2.71597	−0.103395
$691$	26.8369	1.02092	0.510461	−	0.859901i	$-0.329475\pi$
$691$	26.8369	1.02092	0.510461	+	0.859901i	$0.329475\pi$
$692$	−3.46386	−0.131676
$693$	−0.104017	−0.00395128
$694$	−2.70409	−0.102646
$695$	2.62699	0.0996475
$696$	5.25263	0.199100
$697$	−19.4474	−0.736623
$698$	11.6485	0.440903
$699$	−7.69982	−0.291234
$700$	0.730490	0.0276099
$701$	−8.25374	−0.311740	−0.155870	−	0.987778i	$-0.549818\pi$
$701$	−8.25374	−0.311740	−0.155870	+	0.987778i	$0.549818\pi$
$702$	−57.5981	−2.17390
$703$	9.29178	0.350446
$704$	0.117687	0.00443549
$705$	−1.50630	−0.0567304
$706$	13.2598	0.499040
$707$	0.697486	0.0262317
$708$	−18.0807	−0.679515
$709$	4.80417	0.180424	0.0902121	−	0.995923i	$-0.471245\pi$
$709$	4.80417	0.180424	0.0902121	+	0.995923i	$0.471245\pi$
$710$	−1.08675	−0.0407849
$711$	−59.7724	−2.24164
$712$	−8.97895	−0.336500
$713$	−35.4783	−1.32867
$714$	−1.84923	−0.0692058
$715$	0.106866	0.00399657
$716$	3.47610	0.129908
$717$	1.81970	0.0679580
$718$	−10.4909	−0.391515
$719$	42.3968	1.58113	0.790567	−	0.612376i	$-0.209786\pi$
$719$	42.3968	1.58113	0.790567	+	0.612376i	$0.209786\pi$
$720$	−0.863123	−0.0321667
$721$	−0.307775	−0.0114621
$722$	14.4820	0.538965
$723$	87.1130	3.23977
$724$	1.99572	0.0741705
$725$	−8.70643	−0.323349
$726$	−33.0039	−1.22489
$727$	−14.1494	−0.524773	−0.262387	−	0.964963i	$-0.584510\pi$
$727$	−14.1494	−0.524773	−0.262387	+	0.964963i	$0.584510\pi$
$728$	−0.929857	−0.0344628
$729$	−26.3218	−0.974883
$730$	0.630771	0.0233459
$731$	7.32762	0.271022
$732$	−3.57240	−0.132040
$733$	−20.2727	−0.748790	−0.374395	−	0.927269i	$-0.622150\pi$
$733$	−20.2727	−0.748790	−0.374395	+	0.927269i	$0.622150\pi$
$734$	−7.05660	−0.260464
$735$	−3.00336	−0.110781
$736$	−6.31069	−0.232615
$737$	1.71883	0.0633140
$738$	27.9232	1.02787
$739$	−5.82019	−0.214099	−0.107050	−	0.994254i	$-0.534140\pi$
$739$	−5.82019	−0.214099	−0.107050	+	0.994254i	$0.534140\pi$
$740$	−0.626258	−0.0230217
$741$	40.4742	1.48686
$742$	1.68219	0.0617550
$743$	16.1719	0.593291	0.296645	−	0.954988i	$-0.404132\pi$
$743$	16.1719	0.593291	0.296645	+	0.954988i	$0.404132\pi$
$744$	−16.8891	−0.619183
$745$	1.08748	0.0398423
$746$	−3.96851	−0.145297
$747$	−92.0822	−3.36911
$748$	0.493821	0.0180559
$749$	−2.04523	−0.0747310
$750$	4.29492	0.156828
$751$	−11.3104	−0.412723	−0.206361	−	0.978476i	$-0.566162\pi$
$751$	−11.3104	−0.412723	−0.206361	+	0.978476i	$0.566162\pi$
$752$	−3.49996	−0.127631
$753$	−30.4357	−1.10914
$754$	11.0826	0.403605
$755$	1.80548	0.0657081
$756$	1.33307	0.0484833
$757$	−18.8954	−0.686763	−0.343382	−	0.939196i	$-0.611572\pi$
$757$	−18.8954	−0.686763	−0.343382	+	0.939196i	$0.611572\pi$
$758$	−32.0302	−1.16339
$759$	−2.23113	−0.0809849
$760$	0.304509	0.0110457
$761$	24.6449	0.893378	0.446689	−	0.894689i	$-0.352603\pi$
$761$	24.6449	0.893378	0.446689	+	0.894689i	$0.352603\pi$
$762$	52.9372	1.91771
$763$	−2.25108	−0.0814946
$764$	−15.1161	−0.546881
$765$	−3.62171	−0.130943
$766$	1.91383	0.0691494
$767$	−38.1488	−1.37747
$768$	−3.00414	−0.108402
$769$	−27.6468	−0.996968	−0.498484	−	0.866899i	$-0.666110\pi$
$769$	−27.6468	−0.996968	−0.498484	+	0.866899i	$0.666110\pi$
$770$	−0.00247335	−8.91334e−5 0
$771$	3.87575	0.139582
$772$	−19.1486	−0.689173
$773$	0.540811	0.0194516	0.00972582	−	0.999953i	$-0.496904\pi$
$773$	0.540811	0.0194516	0.00972582	+	0.999953i	$0.496904\pi$
$774$	−10.5213	−0.378179
$775$	27.9943	1.00558
$776$	6.41299	0.230213
$777$	1.92653	0.0691140
$778$	−13.9441	−0.499921
$779$	−9.85128	−0.352959
$780$	−2.72793	−0.0976754
$781$	−0.892748	−0.0319450
$782$	−26.4800	−0.946924
$783$	−15.8884	−0.567804
$784$	−6.97848	−0.249231
$785$	3.29579	0.117632
$786$	−24.8130	−0.885052
$787$	−32.7652	−1.16795	−0.583977	−	0.811770i	$-0.698504\pi$
$787$	−32.7652	−1.16795	−0.583977	+	0.811770i	$0.698504\pi$
$788$	−14.4147	−0.513503
$789$	−9.90324	−0.352565
$790$	−1.42129	−0.0505672
$791$	0.650884	0.0231428
$792$	−0.709045	−0.0251948
$793$	−7.53746	−0.267663
$794$	14.6579	0.520188
$795$	4.93504	0.175028
$796$	0.101937	0.00361306
$797$	−4.04701	−0.143352	−0.0716762	−	0.997428i	$-0.522835\pi$
$797$	−4.04701	−0.143352	−0.0716762	+	0.997428i	$0.522835\pi$
$798$	−0.936749	−0.0331605
$799$	−14.6860	−0.519555
$800$	4.97948	0.176051
$801$	54.0967	1.91141
$802$	35.0260	1.23681
$803$	0.518170	0.0182858
$804$	−43.8759	−1.54738
$805$	0.132628	0.00467452
$806$	−35.6345	−1.25517
$807$	8.91472	0.313813
$808$	4.75450	0.167263
$809$	−19.8640	−0.698380	−0.349190	−	0.937052i	$-0.613543\pi$
$809$	−19.8640	−0.698380	−0.349190	+	0.937052i	$0.613543\pi$
$810$	1.32147	0.0464316
$811$	38.7045	1.35910	0.679550	−	0.733629i	$-0.262175\pi$
$811$	38.7045	1.35910	0.679550	+	0.733629i	$0.262175\pi$
$812$	−0.256500	−0.00900138
$813$	−20.4709	−0.717945
$814$	−0.514463	−0.0180319
$815$	1.14047	0.0399490
$816$	−12.6055	−0.441282
$817$	3.71188	0.129862
$818$	−18.8063	−0.657547
$819$	5.60224	0.195758
$820$	0.663968	0.0231868
$821$	46.1297	1.60994	0.804969	−	0.593317i	$-0.202182\pi$
$821$	46.1297	1.60994	0.804969	+	0.593317i	$0.202182\pi$
$822$	53.4136	1.86301
$823$	9.16276	0.319394	0.159697	−	0.987166i	$-0.448948\pi$
$823$	9.16276	0.319394	0.159697	+	0.987166i	$0.448948\pi$
$824$	−2.09799	−0.0730869
$825$	1.76048	0.0612921
$826$	0.882930	0.0307211
$827$	−46.9732	−1.63342	−0.816709	−	0.577050i	$-0.804204\pi$
$827$	−46.9732	−1.63342	−0.816709	+	0.577050i	$0.804204\pi$
$828$	38.0209	1.32132
$829$	4.56661	0.158605	0.0793025	−	0.996851i	$-0.474731\pi$
$829$	4.56661	0.158605	0.0793025	+	0.996851i	$0.474731\pi$
$830$	−2.18956	−0.0760008
$831$	−57.8350	−2.00627
$832$	−6.33848	−0.219747
$833$	−29.2821	−1.01456
$834$	−55.0873	−1.90752
$835$	−3.15766	−0.109275
$836$	0.250150	0.00865162
$837$	51.0867	1.76582
$838$	8.86558	0.306256
$839$	−22.8508	−0.788897	−0.394449	−	0.918918i	$-0.629064\pi$
$839$	−22.8508	−0.788897	−0.394449	+	0.918918i	$0.629064\pi$
$840$	0.0631361	0.00217840
$841$	−25.9429	−0.894582
$842$	7.60121	0.261955
$843$	61.7606	2.12715
$844$	−9.12493	−0.314093
$845$	−3.89331	−0.133934
$846$	21.0867	0.724977
$847$	1.61167	0.0553776
$848$	11.4668	0.393773
$849$	−31.0386	−1.06524
$850$	20.8942	0.716664
$851$	27.5869	0.945668
$852$	22.7888	0.780730
$853$	−33.1947	−1.13657	−0.568283	−	0.822833i	$-0.692392\pi$
$853$	−33.1947	−1.13657	−0.568283	+	0.822833i	$0.692392\pi$
$854$	0.174450	0.00596955
$855$	−1.83462	−0.0627426
$856$	−13.9416	−0.476513
$857$	49.9132	1.70500	0.852501	−	0.522726i	$-0.175085\pi$
$857$	49.9132	1.70500	0.852501	+	0.522726i	$0.175085\pi$
$858$	−2.24096	−0.0765050
$859$	42.3128	1.44370	0.721848	−	0.692052i	$-0.243293\pi$
$859$	42.3128	1.44370	0.721848	+	0.692052i	$0.243293\pi$
$860$	−0.250178	−0.00853100
$861$	−2.04254	−0.0696096
$862$	−15.4499	−0.526224
$863$	−54.1283	−1.84255	−0.921275	−	0.388911i	$-0.872851\pi$
$863$	−54.1283	−1.84255	−0.921275	+	0.388911i	$0.872851\pi$
$864$	9.08704	0.309147
$865$	0.496235	0.0168725
$866$	−12.4640	−0.423545
$867$	−1.82318	−0.0619184
$868$	0.824738	0.0279934
$869$	−1.16757	−0.0396071
$870$	−0.752495	−0.0255120
$871$	−92.5745	−3.13677
$872$	−15.3448	−0.519639
$873$	−38.6372	−1.30767
$874$	−13.4137	−0.453726
$875$	−0.209732	−0.00709025
$876$	−13.2271	−0.446902
$877$	3.55012	0.119879	0.0599395	−	0.998202i	$-0.480909\pi$
$877$	3.55012	0.119879	0.0599395	+	0.998202i	$0.480909\pi$
$878$	−27.3398	−0.922675
$879$	57.8673	1.95182
$880$	−0.0168599	−0.000568348 0
$881$	−1.70024	−0.0572825	−0.0286413	−	0.999590i	$-0.509118\pi$
$881$	−1.70024	−0.0572825	−0.0286413	+	0.999590i	$0.509118\pi$
$882$	42.0442	1.41570
$883$	37.5465	1.26354	0.631770	−	0.775156i	$-0.282329\pi$
$883$	37.5465	1.26354	0.631770	+	0.775156i	$0.282329\pi$
$884$	−26.5966	−0.894542
$885$	2.59026	0.0870705
$886$	20.7905	0.698471
$887$	−45.2411	−1.51905	−0.759524	−	0.650479i	$-0.774568\pi$
$887$	−45.2411	−1.51905	−0.759524	+	0.650479i	$0.774568\pi$
$888$	13.1325	0.440696
$889$	−2.58507	−0.0867004
$890$	1.28633	0.0431179
$891$	1.08557	0.0363679
$892$	7.75136	0.259535
$893$	−7.43937	−0.248949
$894$	−22.8042	−0.762686
$895$	−0.497989	−0.0166459
$896$	0.146700	0.00490091
$897$	120.166	4.01223
$898$	−6.33018	−0.211241
$899$	−9.82974	−0.327840
$900$	−30.0006	−1.00002
$901$	48.1155	1.60296
$902$	0.545441	0.0181612
$903$	0.769613	0.0256111
$904$	4.43684	0.147567
$905$	−0.285909	−0.00950393
$906$	−37.8604	−1.25783
$907$	−7.19189	−0.238803	−0.119401	−	0.992846i	$-0.538098\pi$
$907$	−7.19189	−0.238803	−0.119401	+	0.992846i	$0.538098\pi$
$908$	16.3899	0.543917
$909$	−28.6451	−0.950098
$910$	0.133212	0.00441593
$911$	−14.4244	−0.477900	−0.238950	−	0.971032i	$-0.576803\pi$
$911$	−14.4244	−0.477900	−0.238950	+	0.971032i	$0.576803\pi$
$912$	−6.38546	−0.211444
$913$	−1.79870	−0.0595282
$914$	−9.74286	−0.322265
$915$	0.511784	0.0169191
$916$	−2.65714	−0.0877943
$917$	1.21169	0.0400135
$918$	38.1297	1.25847
$919$	−18.5273	−0.611158	−0.305579	−	0.952167i	$-0.598850\pi$
$919$	−18.5273	−0.611158	−0.305579	+	0.952167i	$0.598850\pi$
$920$	0.904075	0.0298065
$921$	−72.2258	−2.37992
$922$	−20.5671	−0.677340
$923$	48.0824	1.58265
$924$	0.0518655	0.00170625
$925$	−21.7676	−0.715713
$926$	12.2838	0.403670
$927$	12.6400	0.415154
$928$	−1.74846	−0.0573961
$929$	−38.0357	−1.24791	−0.623956	−	0.781460i	$-0.714475\pi$
$929$	−38.0357	−1.24791	−0.623956	+	0.781460i	$0.714475\pi$
$930$	2.41954	0.0793398
$931$	−14.8332	−0.486137
$932$	2.56307	0.0839562
$933$	3.65218	0.119567
$934$	18.8945	0.618247
$935$	−0.0707451	−0.00231361
$936$	38.1884	1.24823
$937$	19.1756	0.626440	0.313220	−	0.949681i	$-0.398592\pi$
$937$	19.1756	0.626440	0.313220	+	0.949681i	$0.398592\pi$
$938$	2.14258	0.0699576
$939$	68.2204	2.22629
$940$	0.501407	0.0163541
$941$	−51.1645	−1.66792	−0.833958	−	0.551828i	$-0.813930\pi$
$941$	−51.1645	−1.66792	−0.833958	+	0.551828i	$0.813930\pi$
$942$	−69.1117	−2.25178
$943$	−29.2481	−0.952448
$944$	6.01860	0.195889
$945$	−0.190977	−0.00621247
$946$	−0.205518	−0.00668197
$947$	−44.9508	−1.46071	−0.730353	−	0.683070i	$-0.760644\pi$
$947$	−44.9508	−1.46071	−0.730353	+	0.683070i	$0.760644\pi$
$948$	29.8040	0.967990
$949$	−27.9081	−0.905934
$950$	10.5842	0.343395
$951$	33.0986	1.07330
$952$	0.615562	0.0199505
$953$	6.97623	0.225982	0.112991	−	0.993596i	$-0.463957\pi$
$953$	6.97623	0.225982	0.112991	+	0.993596i	$0.463957\pi$
$954$	−69.0859	−2.23674
$955$	2.16554	0.0700753
$956$	−0.605731	−0.0195908
$957$	−0.618165	−0.0199824
$958$	−15.8799	−0.513056
$959$	−2.60833	−0.0842273
$960$	0.430375	0.0138903
$961$	0.606098	0.0195516
$962$	27.7084	0.893355
$963$	83.9957	2.70672
$964$	−28.9977	−0.933952
$965$	2.74324	0.0883082
$966$	−2.78117	−0.0894827
$967$	−31.2281	−1.00423	−0.502115	−	0.864801i	$-0.667444\pi$
$967$	−31.2281	−1.00423	−0.502115	+	0.864801i	$0.667444\pi$
$968$	10.9861	0.353108
$969$	−26.7938	−0.860740
$970$	−0.918730	−0.0294986
$971$	−11.1372	−0.357410	−0.178705	−	0.983903i	$-0.557191\pi$
$971$	−11.1372	−0.357410	−0.178705	+	0.983903i	$0.557191\pi$
$972$	−0.449626	−0.0144218
$973$	2.69006	0.0862394
$974$	−41.5460	−1.33122
$975$	−94.8176	−3.03659
$976$	1.18916	0.0380640
$977$	16.9926	0.543642	0.271821	−	0.962348i	$-0.412374\pi$
$977$	16.9926	0.543642	0.271821	+	0.962348i	$0.412374\pi$
$978$	−23.9154	−0.764729
$979$	1.05670	0.0337724
$980$	0.999742	0.0319356
$981$	92.4498	2.95170
$982$	−22.1488	−0.706795
$983$	31.8893	1.01711	0.508556	−	0.861029i	$-0.330179\pi$
$983$	31.8893	1.01711	0.508556	+	0.861029i	$0.330179\pi$
$984$	−13.9232	−0.443856
$985$	2.06507	0.0657985
$986$	−7.33665	−0.233647
$987$	−1.54246	−0.0490970
$988$	−13.4728	−0.428627
$989$	11.0204	0.350430
$990$	0.101578	0.00322837
$991$	−50.8613	−1.61566	−0.807831	−	0.589414i	$-0.799359\pi$
$991$	−50.8613	−1.61566	−0.807831	+	0.589414i	$0.799359\pi$
$992$	5.62193	0.178496
$993$	−32.8056	−1.04105
$994$	−1.11284	−0.0352970
$995$	−0.0146036	−0.000462964 0
$996$	45.9145	1.45486
$997$	54.0886	1.71300	0.856502	−	0.516144i	$-0.172633\pi$
$997$	54.0886	1.71300	0.856502	+	0.516144i	$0.172633\pi$
$998$	−42.4407	−1.34344
$999$	−39.7236	−1.25680

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8042.2.a.c.1.3	✓	86

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8042.2.a.c.1.3	✓	86	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:	\( N \)	\(=\)	\( 8042 = 2 \cdot 4021 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8042.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -3.00414 q^{3} +1.00000 q^{4} -0.143261 q^{5} +3.00414 q^{6} -0.146700 q^{7} -1.00000 q^{8} +6.02484 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -3.00414 q^{3} +1.00000 q^{4} -0.143261 q^{5} +3.00414 q^{6} -0.146700 q^{7} -1.00000 q^{8} +6.02484 q^{9} +0.143261 q^{10} +0.117687 q^{11} -3.00414 q^{12} -6.33848 q^{13} +0.146700 q^{14} +0.430375 q^{15} +1.00000 q^{16} +4.19606 q^{17} -6.02484 q^{18} +2.12556 q^{19} -0.143261 q^{20} +0.440707 q^{21} -0.117687 q^{22} +6.31069 q^{23} +3.00414 q^{24} -4.97948 q^{25} +6.33848 q^{26} -9.08704 q^{27} -0.146700 q^{28} +1.74846 q^{29} -0.430375 q^{30} -5.62193 q^{31} -1.00000 q^{32} -0.353548 q^{33} -4.19606 q^{34} +0.0210164 q^{35} +6.02484 q^{36} +4.37146 q^{37} -2.12556 q^{38} +19.0417 q^{39} +0.143261 q^{40} -4.63468 q^{41} -0.440707 q^{42} +1.74631 q^{43} +0.117687 q^{44} -0.863123 q^{45} -6.31069 q^{46} -3.49996 q^{47} -3.00414 q^{48} -6.97848 q^{49} +4.97948 q^{50} -12.6055 q^{51} -6.33848 q^{52} +11.4668 q^{53} +9.08704 q^{54} -0.0168599 q^{55} +0.146700 q^{56} -6.38546 q^{57} -1.74846 q^{58} +6.01860 q^{59} +0.430375 q^{60} +1.18916 q^{61} +5.62193 q^{62} -0.883845 q^{63} +1.00000 q^{64} +0.908056 q^{65} +0.353548 q^{66} +14.6051 q^{67} +4.19606 q^{68} -18.9582 q^{69} -0.0210164 q^{70} -7.58579 q^{71} -6.02484 q^{72} +4.40295 q^{73} -4.37146 q^{74} +14.9590 q^{75} +2.12556 q^{76} -0.0172647 q^{77} -19.0417 q^{78} -9.92100 q^{79} -0.143261 q^{80} +9.22419 q^{81} +4.63468 q^{82} -15.2838 q^{83} +0.440707 q^{84} -0.601130 q^{85} -1.74631 q^{86} -5.25263 q^{87} -0.117687 q^{88} +8.97895 q^{89} +0.863123 q^{90} +0.929857 q^{91} +6.31069 q^{92} +16.8891 q^{93} +3.49996 q^{94} -0.304509 q^{95} +3.00414 q^{96} -6.41299 q^{97} +6.97848 q^{98} +0.709045 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 86 q - 86 q^{2} + 12 q^{3} + 86 q^{4} - 4 q^{5} - 12 q^{6} + 35 q^{7} - 86 q^{8} + 72 q^{9}+O(q^{10}) \) 86 * q - 86 * q^2 + 12 * q^3 + 86 * q^4 - 4 * q^5 - 12 * q^6 + 35 * q^7 - 86 * q^8 + 72 * q^9	\( 86 q - 86 q^{2} + 12 q^{3} + 86 q^{4} - 4 q^{5} - 12 q^{6} + 35 q^{7} - 86 q^{8} + 72 q^{9} + 4 q^{10} + 13 q^{11} + 12 q^{12} + 45 q^{13} - 35 q^{14} + 17 q^{15} + 86 q^{16} + 5 q^{17} - 72 q^{18} + 47 q^{19} - 4 q^{20} + 15 q^{21} - 13 q^{22} + 6 q^{23} - 12 q^{24} + 112 q^{25} - 45 q^{26} + 51 q^{27} + 35 q^{28} - 14 q^{29} - 17 q^{30} + 24 q^{31} - 86 q^{32} + 43 q^{33} - 5 q^{34} + 42 q^{35} + 72 q^{36} + 61 q^{37} - 47 q^{38} + 20 q^{39} + 4 q^{40} - 16 q^{41} - 15 q^{42} + 72 q^{43} + 13 q^{44} + 6 q^{45} - 6 q^{46} + 11 q^{47} + 12 q^{48} + 89 q^{49} - 112 q^{50} + 56 q^{51} + 45 q^{52} - 7 q^{53} - 51 q^{54} + 48 q^{55} - 35 q^{56} + 65 q^{57} + 14 q^{58} + 24 q^{59} + 17 q^{60} + 31 q^{61} - 24 q^{62} + 98 q^{63} + 86 q^{64} - 9 q^{65} - 43 q^{66} + 157 q^{67} + 5 q^{68} + q^{69} - 42 q^{70} - 11 q^{71} - 72 q^{72} + 74 q^{73} - 61 q^{74} + 76 q^{75} + 47 q^{76} - 13 q^{77} - 20 q^{78} + 57 q^{79} - 4 q^{80} + 34 q^{81} + 16 q^{82} + 65 q^{83} + 15 q^{84} + 102 q^{85} - 72 q^{86} + 49 q^{87} - 13 q^{88} - 34 q^{89} - 6 q^{90} + 91 q^{91} + 6 q^{92} + 57 q^{93} - 11 q^{94} - 13 q^{95} - 12 q^{96} + 64 q^{97} - 89 q^{98} + 56 q^{99}+O(q^{100}) \) 86 * q - 86 * q^2 + 12 * q^3 + 86 * q^4 - 4 * q^5 - 12 * q^6 + 35 * q^7 - 86 * q^8 + 72 * q^9 + 4 * q^10 + 13 * q^11 + 12 * q^12 + 45 * q^13 - 35 * q^14 + 17 * q^15 + 86 * q^16 + 5 * q^17 - 72 * q^18 + 47 * q^19 - 4 * q^20 + 15 * q^21 - 13 * q^22 + 6 * q^23 - 12 * q^24 + 112 * q^25 - 45 * q^26 + 51 * q^27 + 35 * q^28 - 14 * q^29 - 17 * q^30 + 24 * q^31 - 86 * q^32 + 43 * q^33 - 5 * q^34 + 42 * q^35 + 72 * q^36 + 61 * q^37 - 47 * q^38 + 20 * q^39 + 4 * q^40 - 16 * q^41 - 15 * q^42 + 72 * q^43 + 13 * q^44 + 6 * q^45 - 6 * q^46 + 11 * q^47 + 12 * q^48 + 89 * q^49 - 112 * q^50 + 56 * q^51 + 45 * q^52 - 7 * q^53 - 51 * q^54 + 48 * q^55 - 35 * q^56 + 65 * q^57 + 14 * q^58 + 24 * q^59 + 17 * q^60 + 31 * q^61 - 24 * q^62 + 98 * q^63 + 86 * q^64 - 9 * q^65 - 43 * q^66 + 157 * q^67 + 5 * q^68 + q^69 - 42 * q^70 - 11 * q^71 - 72 * q^72 + 74 * q^73 - 61 * q^74 + 76 * q^75 + 47 * q^76 - 13 * q^77 - 20 * q^78 + 57 * q^79 - 4 * q^80 + 34 * q^81 + 16 * q^82 + 65 * q^83 + 15 * q^84 + 102 * q^85 - 72 * q^86 + 49 * q^87 - 13 * q^88 - 34 * q^89 - 6 * q^90 + 91 * q^91 + 6 * q^92 + 57 * q^93 - 11 * q^94 - 13 * q^95 - 12 * q^96 + 64 * q^97 - 89 * q^98 + 56 * q^99

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