LMFDB - Embedded newform 4009.2.a.e.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4009,2,Mod(1,4009)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4009, 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("4009.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4009 = 19 \cdot 211 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4009.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:	$32.0120261703$
Analytic rank:	$0$
Dimension:	$82$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Character	$\chi$	$=$	4009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.62201 q^{2} -2.46509 q^{3} +4.87496 q^{4} -1.74208 q^{5} +6.46350 q^{6} -2.61324 q^{7} -7.53817 q^{8} +3.07667 q^{9} +O(q^{10})$	\(q-2.62201 q^{2} -2.46509 q^{3} +4.87496 q^{4} -1.74208 q^{5} +6.46350 q^{6} -2.61324 q^{7} -7.53817 q^{8} +3.07667 q^{9} +4.56776 q^{10} +1.71187 q^{11} -12.0172 q^{12} -1.39296 q^{13} +6.85194 q^{14} +4.29439 q^{15} +10.0153 q^{16} +5.91699 q^{17} -8.06707 q^{18} +1.00000 q^{19} -8.49257 q^{20} +6.44186 q^{21} -4.48856 q^{22} -3.75998 q^{23} +18.5823 q^{24} -1.96515 q^{25} +3.65237 q^{26} -0.188994 q^{27} -12.7394 q^{28} +9.69159 q^{29} -11.2599 q^{30} +5.25599 q^{31} -11.1839 q^{32} -4.21992 q^{33} -15.5144 q^{34} +4.55247 q^{35} +14.9986 q^{36} +2.87922 q^{37} -2.62201 q^{38} +3.43378 q^{39} +13.1321 q^{40} -2.71761 q^{41} -16.8906 q^{42} +4.20331 q^{43} +8.34531 q^{44} -5.35981 q^{45} +9.85871 q^{46} -8.83905 q^{47} -24.6886 q^{48} -0.170998 q^{49} +5.15265 q^{50} -14.5859 q^{51} -6.79063 q^{52} -1.48254 q^{53} +0.495546 q^{54} -2.98222 q^{55} +19.6990 q^{56} -2.46509 q^{57} -25.4115 q^{58} +6.77185 q^{59} +20.9349 q^{60} -8.79139 q^{61} -13.7813 q^{62} -8.04006 q^{63} +9.29365 q^{64} +2.42665 q^{65} +11.0647 q^{66} +1.51857 q^{67} +28.8451 q^{68} +9.26868 q^{69} -11.9366 q^{70} +15.1559 q^{71} -23.1925 q^{72} -5.06143 q^{73} -7.54936 q^{74} +4.84427 q^{75} +4.87496 q^{76} -4.47353 q^{77} -9.00341 q^{78} +13.3272 q^{79} -17.4474 q^{80} -8.76412 q^{81} +7.12560 q^{82} +11.0999 q^{83} +31.4038 q^{84} -10.3079 q^{85} -11.0211 q^{86} -23.8907 q^{87} -12.9044 q^{88} -7.03692 q^{89} +14.0535 q^{90} +3.64014 q^{91} -18.3297 q^{92} -12.9565 q^{93} +23.1761 q^{94} -1.74208 q^{95} +27.5692 q^{96} -2.39455 q^{97} +0.448359 q^{98} +5.26687 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 82 q + 15 q^{2} + 12 q^{3} + 89 q^{4} + 9 q^{5} + 9 q^{6} + 14 q^{7} + 42 q^{8} + 92 q^{9}+O(q^{10}) $ 82 * q + 15 * q^2 + 12 * q^3 + 89 * q^4 + 9 * q^5 + 9 * q^6 + 14 * q^7 + 42 * q^8 + 92 * q^9	\( 82 q + 15 q^{2} + 12 q^{3} + 89 q^{4} + 9 q^{5} + 9 q^{6} + 14 q^{7} + 42 q^{8} + 92 q^{9} + 4 q^{10} + 41 q^{11} + 26 q^{12} + 13 q^{13} + 22 q^{14} + 41 q^{15} + 87 q^{16} + 12 q^{17} + 24 q^{18} + 82 q^{19} + 26 q^{20} + 29 q^{21} + 2 q^{22} + 59 q^{23} + 16 q^{24} + 67 q^{25} + 24 q^{26} + 42 q^{27} - 2 q^{28} + 101 q^{29} - 22 q^{30} + 48 q^{31} + 69 q^{32} + 3 q^{33} + q^{34} + 38 q^{35} + 82 q^{36} + 16 q^{37} + 15 q^{38} + 82 q^{39} + 20 q^{40} + 86 q^{41} - q^{42} + 9 q^{43} + 82 q^{44} - 8 q^{45} + 43 q^{46} + 24 q^{47} + 34 q^{48} + 76 q^{49} + 82 q^{50} + 57 q^{51} - 22 q^{52} + 39 q^{53} + 17 q^{54} - 21 q^{55} + 50 q^{56} + 12 q^{57} + 33 q^{58} + 79 q^{59} + 87 q^{60} + 4 q^{61} + 40 q^{62} + 44 q^{63} + 90 q^{64} + 66 q^{65} - 39 q^{66} + 33 q^{67} - 9 q^{68} + 60 q^{69} + 30 q^{70} + 168 q^{71} + 15 q^{72} - 28 q^{73} + 35 q^{74} + 55 q^{75} + 89 q^{76} + 19 q^{77} - 41 q^{78} + 121 q^{79} + 64 q^{80} + 110 q^{81} + 41 q^{82} + 28 q^{84} + 17 q^{85} + 80 q^{86} + 29 q^{87} + 49 q^{88} + 83 q^{89} - 42 q^{90} + 38 q^{91} + 71 q^{92} - q^{93} + 89 q^{94} + 9 q^{95} + 35 q^{96} - 23 q^{97} + 135 q^{98} + 93 q^{99}+O(q^{100}) \) 82 * q + 15 * q^2 + 12 * q^3 + 89 * q^4 + 9 * q^5 + 9 * q^6 + 14 * q^7 + 42 * q^8 + 92 * q^9 + 4 * q^10 + 41 * q^11 + 26 * q^12 + 13 * q^13 + 22 * q^14 + 41 * q^15 + 87 * q^16 + 12 * q^17 + 24 * q^18 + 82 * q^19 + 26 * q^20 + 29 * q^21 + 2 * q^22 + 59 * q^23 + 16 * q^24 + 67 * q^25 + 24 * q^26 + 42 * q^27 - 2 * q^28 + 101 * q^29 - 22 * q^30 + 48 * q^31 + 69 * q^32 + 3 * q^33 + q^34 + 38 * q^35 + 82 * q^36 + 16 * q^37 + 15 * q^38 + 82 * q^39 + 20 * q^40 + 86 * q^41 - q^42 + 9 * q^43 + 82 * q^44 - 8 * q^45 + 43 * q^46 + 24 * q^47 + 34 * q^48 + 76 * q^49 + 82 * q^50 + 57 * q^51 - 22 * q^52 + 39 * q^53 + 17 * q^54 - 21 * q^55 + 50 * q^56 + 12 * q^57 + 33 * q^58 + 79 * q^59 + 87 * q^60 + 4 * q^61 + 40 * q^62 + 44 * q^63 + 90 * q^64 + 66 * q^65 - 39 * q^66 + 33 * q^67 - 9 * q^68 + 60 * q^69 + 30 * q^70 + 168 * q^71 + 15 * q^72 - 28 * q^73 + 35 * q^74 + 55 * q^75 + 89 * q^76 + 19 * q^77 - 41 * q^78 + 121 * q^79 + 64 * q^80 + 110 * q^81 + 41 * q^82 + 28 * q^84 + 17 * q^85 + 80 * q^86 + 29 * q^87 + 49 * q^88 + 83 * q^89 - 42 * q^90 + 38 * q^91 + 71 * q^92 - q^93 + 89 * q^94 + 9 * q^95 + 35 * q^96 - 23 * q^97 + 135 * q^98 + 93 * 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 $

$2$	−2.62201	−1.85404	−0.927022	−	0.375007i	$-0.877640\pi$
$2$	−2.62201	−1.85404	−0.927022	+	0.375007i	$0.877640\pi$
$3$	−2.46509	−1.42322	−0.711610	−	0.702575i	$-0.752034\pi$
$3$	−2.46509	−1.42322	−0.711610	+	0.702575i	$0.752034\pi$
$4$	4.87496	2.43748
$5$	−1.74208	−0.779083	−0.389541	−	0.921009i	$-0.627366\pi$
$5$	−1.74208	−0.779083	−0.389541	+	0.921009i	$0.627366\pi$
$6$	6.46350	2.63871
$7$	−2.61324	−0.987710	−0.493855	−	0.869544i	$-0.664413\pi$
$7$	−2.61324	−0.987710	−0.493855	+	0.869544i	$0.664413\pi$
$8$	−7.53817	−2.66515
$9$	3.07667	1.02556
$10$	4.56776	1.44445
$11$	1.71187	0.516149	0.258075	−	0.966125i	$-0.416912\pi$
$11$	1.71187	0.516149	0.258075	+	0.966125i	$0.416912\pi$
$12$	−12.0172	−3.46907
$13$	−1.39296	−0.386338	−0.193169	−	0.981165i	$-0.561877\pi$
$13$	−1.39296	−0.386338	−0.193169	+	0.981165i	$0.561877\pi$
$14$	6.85194	1.83126
$15$	4.29439	1.10881
$16$	10.0153	2.50382
$17$	5.91699	1.43508	0.717540	−	0.696517i	$-0.245268\pi$
$17$	5.91699	1.43508	0.717540	+	0.696517i	$0.245268\pi$
$18$	−8.06707	−1.90143
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−8.49257	−1.89900
$21$	6.44186	1.40573
$22$	−4.48856	−0.956964
$23$	−3.75998	−0.784009	−0.392005	−	0.919963i	$-0.628218\pi$
$23$	−3.75998	−0.784009	−0.392005	+	0.919963i	$0.628218\pi$
$24$	18.5823	3.79309
$25$	−1.96515	−0.393030
$26$	3.65237	0.716288
$27$	−0.188994	−0.0363720
$28$	−12.7394	−2.40752
$29$	9.69159	1.79968	0.899842	−	0.436216i	$-0.143682\pi$
$29$	9.69159	1.79968	0.899842	+	0.436216i	$0.143682\pi$
$30$	−11.2599	−2.05578
$31$	5.25599	0.944004	0.472002	−	0.881597i	$-0.343531\pi$
$31$	5.25599	0.944004	0.472002	+	0.881597i	$0.343531\pi$
$32$	−11.1839	−1.97704
$33$	−4.21992	−0.734594
$34$	−15.5144	−2.66070
$35$	4.55247	0.769508
$36$	14.9986	2.49977
$37$	2.87922	0.473341	0.236671	−	0.971590i	$-0.423944\pi$
$37$	2.87922	0.473341	0.236671	+	0.971590i	$0.423944\pi$
$38$	−2.62201	−0.425347
$39$	3.43378	0.549845
$40$	13.1321	2.07637
$41$	−2.71761	−0.424419	−0.212209	−	0.977224i	$-0.568066\pi$
$41$	−2.71761	−0.424419	−0.212209	+	0.977224i	$0.568066\pi$
$42$	−16.8906	−2.60628
$43$	4.20331	0.640999	0.320500	−	0.947249i	$-0.396149\pi$
$43$	4.20331	0.640999	0.320500	+	0.947249i	$0.396149\pi$
$44$	8.34531	1.25810
$45$	−5.35981	−0.798993
$46$	9.85871	1.45359
$47$	−8.83905	−1.28931	−0.644654	−	0.764474i	$-0.722999\pi$
$47$	−8.83905	−1.28931	−0.644654	+	0.764474i	$0.722999\pi$
$48$	−24.6886	−3.56349
$49$	−0.170998	−0.0244283
$50$	5.15265	0.728695
$51$	−14.5859	−2.04244
$52$	−6.79063	−0.941691
$53$	−1.48254	−0.203643	−0.101822	−	0.994803i	$-0.532467\pi$
$53$	−1.48254	−0.203643	−0.101822	+	0.994803i	$0.532467\pi$
$54$	0.495546	0.0674353
$55$	−2.98222	−0.402123
$56$	19.6990	2.63239
$57$	−2.46509	−0.326509
$58$	−25.4115	−3.33669
$59$	6.77185	0.881620	0.440810	−	0.897600i	$-0.354691\pi$
$59$	6.77185	0.881620	0.440810	+	0.897600i	$0.354691\pi$
$60$	20.9349	2.70269
$61$	−8.79139	−1.12562	−0.562811	−	0.826585i	$-0.690280\pi$
$61$	−8.79139	−1.12562	−0.562811	+	0.826585i	$0.690280\pi$
$62$	−13.7813	−1.75023
$63$	−8.04006	−1.01295
$64$	9.29365	1.16171
$65$	2.42665	0.300989
$66$	11.0647	1.36197
$67$	1.51857	0.185523	0.0927615	−	0.995688i	$-0.470431\pi$
$67$	1.51857	0.185523	0.0927615	+	0.995688i	$0.470431\pi$
$68$	28.8451	3.49798
$69$	9.26868	1.11582
$70$	−11.9366	−1.42670
$71$	15.1559	1.79867	0.899334	−	0.437261i	$-0.144052\pi$
$71$	15.1559	1.79867	0.899334	+	0.437261i	$0.144052\pi$
$72$	−23.1925	−2.73326
$73$	−5.06143	−0.592395	−0.296198	−	0.955127i	$-0.595719\pi$
$73$	−5.06143	−0.592395	−0.296198	+	0.955127i	$0.595719\pi$
$74$	−7.54936	−0.877595
$75$	4.84427	0.559369
$76$	4.87496	0.559196
$77$	−4.47353	−0.509806
$78$	−9.00341	−1.01944
$79$	13.3272	1.49943	0.749713	−	0.661763i	$-0.230191\pi$
$79$	13.3272	1.49943	0.749713	+	0.661763i	$0.230191\pi$
$80$	−17.4474	−1.95068
$81$	−8.76412	−0.973791
$82$	7.12560	0.786891
$83$	11.0999	1.21837	0.609186	−	0.793027i	$-0.291496\pi$
$83$	11.0999	1.21837	0.609186	+	0.793027i	$0.291496\pi$
$84$	31.4038	3.42643
$85$	−10.3079	−1.11805
$86$	−11.0211	−1.18844
$87$	−23.8907	−2.56135
$88$	−12.9044	−1.37561
$89$	−7.03692	−0.745912	−0.372956	−	0.927849i	$-0.621656\pi$
$89$	−7.03692	−0.745912	−0.372956	+	0.927849i	$0.621656\pi$
$90$	14.0535	1.48137
$91$	3.64014	0.381590
$92$	−18.3297	−1.91101
$93$	−12.9565	−1.34353
$94$	23.1761	2.39043
$95$	−1.74208	−0.178734
$96$	27.5692	2.81377
$97$	−2.39455	−0.243129	−0.121565	−	0.992584i	$-0.538791\pi$
$97$	−2.39455	−0.243129	−0.121565	+	0.992584i	$0.538791\pi$
$98$	0.448359	0.0452911
$99$	5.26687	0.529340
$100$	−9.58002	−0.958002
$101$	0.850558	0.0846336	0.0423168	−	0.999104i	$-0.486526\pi$
$101$	0.850558	0.0846336	0.0423168	+	0.999104i	$0.486526\pi$
$102$	38.2445	3.78677
$103$	−5.77261	−0.568792	−0.284396	−	0.958707i	$-0.591793\pi$
$103$	−5.77261	−0.568792	−0.284396	+	0.958707i	$0.591793\pi$
$104$	10.5004	1.02965
$105$	−11.2222	−1.09518
$106$	3.88725	0.377563
$107$	−14.1897	−1.37177	−0.685884	−	0.727710i	$-0.740584\pi$
$107$	−14.1897	−1.37177	−0.685884	+	0.727710i	$0.740584\pi$
$108$	−0.921339	−0.0886559
$109$	−11.0396	−1.05740	−0.528699	−	0.848809i	$-0.677320\pi$
$109$	−11.0396	−1.05740	−0.528699	+	0.848809i	$0.677320\pi$
$110$	7.81943	0.745554
$111$	−7.09754	−0.673669
$112$	−26.1723	−2.47305
$113$	10.8613	1.02175	0.510874	−	0.859656i	$-0.329322\pi$
$113$	10.8613	1.02175	0.510874	+	0.859656i	$0.329322\pi$
$114$	6.46350	0.605362
$115$	6.55019	0.610808
$116$	47.2461	4.38669
$117$	−4.28568	−0.396212
$118$	−17.7559	−1.63456
$119$	−15.4625	−1.41744
$120$	−32.3718	−2.95513
$121$	−8.06949	−0.733590
$122$	23.0512	2.08695
$123$	6.69914	0.604041
$124$	25.6227	2.30099
$125$	12.1339	1.08529
$126$	21.0811	1.87806
$127$	−0.534549	−0.0474336	−0.0237168	−	0.999719i	$-0.507550\pi$
$127$	−0.534549	−0.0474336	−0.0237168	+	0.999719i	$0.507550\pi$
$128$	−2.00038	−0.176811
$129$	−10.3615	−0.912283
$130$	−6.36272	−0.558048
$131$	−1.16308	−0.101619	−0.0508094	−	0.998708i	$-0.516180\pi$
$131$	−1.16308	−0.101619	−0.0508094	+	0.998708i	$0.516180\pi$
$132$	−20.5719	−1.79056
$133$	−2.61324	−0.226596
$134$	−3.98171	−0.343968
$135$	0.329244	0.0283368
$136$	−44.6033	−3.82470
$137$	16.3587	1.39762	0.698809	−	0.715309i	$-0.253714\pi$
$137$	16.3587	1.39762	0.698809	+	0.715309i	$0.253714\pi$
$138$	−24.3026	−2.06878
$139$	−12.7365	−1.08030	−0.540150	−	0.841569i	$-0.681632\pi$
$139$	−12.7365	−1.08030	−0.540150	+	0.841569i	$0.681632\pi$
$140$	22.1931	1.87566
$141$	21.7891	1.83497
$142$	−39.7389	−3.33481
$143$	−2.38458	−0.199408
$144$	30.8137	2.56781
$145$	−16.8835	−1.40210
$146$	13.2711	1.09833
$147$	0.421526	0.0347669
$148$	14.0361	1.15376
$149$	−13.1868	−1.08030	−0.540150	−	0.841569i	$-0.681633\pi$
$149$	−13.1868	−1.08030	−0.540150	+	0.841569i	$0.681633\pi$
$150$	−12.7018	−1.03709
$151$	14.1210	1.14915	0.574574	−	0.818453i	$-0.305168\pi$
$151$	14.1210	1.14915	0.574574	+	0.818453i	$0.305168\pi$
$152$	−7.53817	−0.611426
$153$	18.2046	1.47176
$154$	11.7297	0.945203
$155$	−9.15637	−0.735457
$156$	16.7395	1.34023
$157$	−10.1092	−0.806804	−0.403402	−	0.915023i	$-0.632172\pi$
$157$	−10.1092	−0.806804	−0.403402	+	0.915023i	$0.632172\pi$
$158$	−34.9441	−2.78000
$159$	3.65461	0.289829
$160$	19.4832	1.54028
$161$	9.82571	0.774374
$162$	22.9796	1.80545
$163$	−10.0834	−0.789789	−0.394895	−	0.918726i	$-0.629219\pi$
$163$	−10.0834	−0.789789	−0.394895	+	0.918726i	$0.629219\pi$
$164$	−13.2482	−1.03451
$165$	7.35145	0.572310
$166$	−29.1041	−2.25892
$167$	2.95621	0.228758	0.114379	−	0.993437i	$-0.463512\pi$
$167$	2.95621	0.228758	0.114379	+	0.993437i	$0.463512\pi$
$168$	−48.5599	−3.74647
$169$	−11.0597	−0.850743
$170$	27.0274	2.07291
$171$	3.07667	0.235279
$172$	20.4910	1.56242
$173$	−9.05376	−0.688345	−0.344172	−	0.938906i	$-0.611840\pi$
$173$	−9.05376	−0.688345	−0.344172	+	0.938906i	$0.611840\pi$
$174$	62.6416	4.74885
$175$	5.13540	0.388200
$176$	17.1449	1.29235
$177$	−16.6932	−1.25474
$178$	18.4509	1.38295
$179$	−8.42588	−0.629780	−0.314890	−	0.949128i	$-0.601968\pi$
$179$	−8.42588	−0.629780	−0.314890	+	0.949128i	$0.601968\pi$
$180$	−26.1288	−1.94753
$181$	−1.23538	−0.0918254	−0.0459127	−	0.998945i	$-0.514620\pi$
$181$	−1.23538	−0.0918254	−0.0459127	+	0.998945i	$0.514620\pi$
$182$	−9.54450	−0.707485
$183$	21.6716	1.60201
$184$	28.3434	2.08950
$185$	−5.01584	−0.368772
$186$	33.9721	2.49096
$187$	10.1291	0.740716
$188$	−43.0900	−3.14266
$189$	0.493887	0.0359250
$190$	4.56776	0.331380
$191$	3.48042	0.251834	0.125917	−	0.992041i	$-0.459813\pi$
$191$	3.48042	0.251834	0.125917	+	0.992041i	$0.459813\pi$
$192$	−22.9097	−1.65336
$193$	−25.4876	−1.83464	−0.917320	−	0.398151i	$-0.869652\pi$
$193$	−25.4876	−1.83464	−0.917320	+	0.398151i	$0.869652\pi$
$194$	6.27854	0.450773
$195$	−5.98192	−0.428374
$196$	−0.833608	−0.0595434
$197$	21.7866	1.55223	0.776116	−	0.630590i	$-0.217187\pi$
$197$	21.7866	1.55223	0.776116	+	0.630590i	$0.217187\pi$
$198$	−13.8098	−0.981420
$199$	−8.28745	−0.587482	−0.293741	−	0.955885i	$-0.594900\pi$
$199$	−8.28745	−0.587482	−0.293741	+	0.955885i	$0.594900\pi$
$200$	14.8136	1.04748
$201$	−3.74341	−0.264040
$202$	−2.23017	−0.156914
$203$	−25.3264	−1.77757
$204$	−71.1057	−4.97839
$205$	4.73429	0.330657
$206$	15.1358	1.05456
$207$	−11.5682	−0.804046
$208$	−13.9509	−0.967321
$209$	1.71187	0.118413
$210$	29.4249	2.03051
$211$	−1.00000	−0.0688428
$211$	−1.00000	−0.0688428
$212$	−7.22734	−0.496376
$213$	−37.3605	−2.55990
$214$	37.2056	2.54332
$215$	−7.32252	−0.499391
$216$	1.42467	0.0969367
$217$	−13.7352	−0.932403
$218$	28.9459	1.96046
$219$	12.4769	0.843109
$220$	−14.5382	−0.980166
$221$	−8.24214	−0.554427
$222$	18.6098	1.24901
$223$	5.72985	0.383699	0.191850	−	0.981424i	$-0.438551\pi$
$223$	5.72985	0.383699	0.191850	+	0.981424i	$0.438551\pi$
$224$	29.2260	1.95275
$225$	−6.04612	−0.403075
$226$	−28.4786	−1.89437
$227$	−15.5729	−1.03361	−0.516803	−	0.856104i	$-0.672878\pi$
$227$	−15.5729	−1.03361	−0.516803	+	0.856104i	$0.672878\pi$
$228$	−12.0172	−0.795859
$229$	3.65481	0.241517	0.120758	−	0.992682i	$-0.461467\pi$
$229$	3.65481	0.241517	0.120758	+	0.992682i	$0.461467\pi$
$230$	−17.1747	−1.13246
$231$	11.0277	0.725567
$232$	−73.0569	−4.79642
$233$	−14.3863	−0.942475	−0.471237	−	0.882006i	$-0.656192\pi$
$233$	−14.3863	−0.942475	−0.471237	+	0.882006i	$0.656192\pi$
$234$	11.2371	0.734594
$235$	15.3984	1.00448
$236$	33.0125	2.14893
$237$	−32.8527	−2.13401
$238$	40.5429	2.62800
$239$	−21.5699	−1.39524	−0.697621	−	0.716467i	$-0.745758\pi$
$239$	−21.5699	−1.39524	−0.697621	+	0.716467i	$0.745758\pi$
$240$	43.0095	2.77625
$241$	−23.9872	−1.54515	−0.772576	−	0.634923i	$-0.781032\pi$
$241$	−23.9872	−1.54515	−0.772576	+	0.634923i	$0.781032\pi$
$242$	21.1583	1.36011
$243$	22.1713	1.42229
$244$	−42.8577	−2.74368
$245$	0.297893	0.0190317
$246$	−17.5652	−1.11992
$247$	−1.39296	−0.0886321
$248$	−39.6206	−2.51591
$249$	−27.3623	−1.73401
$250$	−31.8152	−2.01217
$251$	29.7886	1.88024	0.940120	−	0.340844i	$-0.110713\pi$
$251$	29.7886	1.88024	0.940120	+	0.340844i	$0.110713\pi$
$252$	−39.1949	−2.46905
$253$	−6.43661	−0.404666
$254$	1.40160	0.0879439
$255$	25.4098	1.59123
$256$	−13.3423	−0.833892
$257$	11.2581	0.702263	0.351132	−	0.936326i	$-0.385797\pi$
$257$	11.2581	0.702263	0.351132	+	0.936326i	$0.385797\pi$
$258$	27.1681	1.69141
$259$	−7.52409	−0.467524
$260$	11.8298	0.733655
$261$	29.8178	1.84568
$262$	3.04961	0.188406
$263$	12.4485	0.767609	0.383804	−	0.923414i	$-0.374614\pi$
$263$	12.4485	0.767609	0.383804	+	0.923414i	$0.374614\pi$
$264$	31.8105	1.95780
$265$	2.58271	0.158655
$266$	6.85194	0.420119
$267$	17.3467	1.06160
$268$	7.40296	0.452208
$269$	−12.3977	−0.755902	−0.377951	−	0.925826i	$-0.623371\pi$
$269$	−12.3977	−0.755902	−0.377951	+	0.925826i	$0.623371\pi$
$270$	−0.863281	−0.0525376
$271$	19.3659	1.17639	0.588197	−	0.808717i	$-0.299838\pi$
$271$	19.3659	1.17639	0.588197	+	0.808717i	$0.299838\pi$
$272$	59.2603	3.59318
$273$	−8.97327	−0.543087
$274$	−42.8927	−2.59124
$275$	−3.36409	−0.202862
$276$	45.1844	2.71978
$277$	−24.3453	−1.46277	−0.731383	−	0.681967i	$-0.761125\pi$
$277$	−24.3453	−1.46277	−0.731383	+	0.681967i	$0.761125\pi$
$278$	33.3954	2.00292
$279$	16.1710	0.968130
$280$	−34.3173	−2.05085
$281$	−3.95475	−0.235921	−0.117960	−	0.993018i	$-0.537636\pi$
$281$	−3.95475	−0.235921	−0.117960	+	0.993018i	$0.537636\pi$
$282$	−57.1312	−3.40211
$283$	−14.0758	−0.836722	−0.418361	−	0.908281i	$-0.637395\pi$
$283$	−14.0758	−0.836722	−0.418361	+	0.908281i	$0.637395\pi$
$284$	73.8841	4.38422
$285$	4.29439	0.254378
$286$	6.25239	0.369712
$287$	7.10174	0.419203
$288$	−34.4090	−2.02757
$289$	18.0108	1.05946
$290$	44.2689	2.59956
$291$	5.90277	0.346027
$292$	−24.6742	−1.44395
$293$	19.9666	1.16646	0.583231	−	0.812307i	$-0.301788\pi$
$293$	19.9666	1.16646	0.583231	+	0.812307i	$0.301788\pi$
$294$	−1.10525	−0.0644593
$295$	−11.7971	−0.686855
$296$	−21.7041	−1.26152
$297$	−0.323535	−0.0187734
$298$	34.5758	2.00292
$299$	5.23751	0.302893
$300$	23.6156	1.36345
$301$	−10.9843	−0.633122
$302$	−37.0254	−2.13057
$303$	−2.09670	−0.120452
$304$	10.0153	0.574416
$305$	15.3153	0.876953
$306$	−47.7327	−2.72870
$307$	27.7375	1.58307	0.791533	−	0.611127i	$-0.209283\pi$
$307$	27.7375	1.58307	0.791533	+	0.611127i	$0.209283\pi$
$308$	−21.8083	−1.24264
$309$	14.2300	0.809516
$310$	24.0081	1.36357
$311$	23.4672	1.33070	0.665351	−	0.746530i	$-0.268282\pi$
$311$	23.4672	1.33070	0.665351	+	0.746530i	$0.268282\pi$
$312$	−25.8844	−1.46542
$313$	12.3488	0.697995	0.348997	−	0.937124i	$-0.386522\pi$
$313$	12.3488	0.697995	0.348997	+	0.937124i	$0.386522\pi$
$314$	26.5065	1.49585
$315$	14.0064	0.789174
$316$	64.9695	3.65482
$317$	25.8521	1.45200	0.725998	−	0.687697i	$-0.241378\pi$
$317$	25.8521	1.45200	0.725998	+	0.687697i	$0.241378\pi$
$318$	−9.58242	−0.537356
$319$	16.5908	0.928906
$320$	−16.1903	−0.905065
$321$	34.9789	1.95233
$322$	−25.7631	−1.43572
$323$	5.91699	0.329230
$324$	−42.7247	−2.37359
$325$	2.73738	0.151843
$326$	26.4387	1.46430
$327$	27.2135	1.50491
$328$	20.4858	1.13114
$329$	23.0985	1.27346
$330$	−19.2756	−1.06109
$331$	−0.682382	−0.0375071	−0.0187536	−	0.999824i	$-0.505970\pi$
$331$	−0.682382	−0.0375071	−0.0187536	+	0.999824i	$0.505970\pi$
$332$	54.1115	2.96976
$333$	8.85841	0.485438
$334$	−7.75121	−0.424128
$335$	−2.64547	−0.144538
$336$	64.5170	3.51969
$337$	−3.40784	−0.185637	−0.0928184	−	0.995683i	$-0.529588\pi$
$337$	−3.40784	−0.185637	−0.0928184	+	0.995683i	$0.529588\pi$
$338$	28.9986	1.57731
$339$	−26.7742	−1.45417
$340$	−50.2504	−2.72521
$341$	8.99760	0.487247
$342$	−8.06707	−0.436217
$343$	18.7395	1.01184
$344$	−31.6853	−1.70836
$345$	−16.1468	−0.869314
$346$	23.7391	1.27622
$347$	16.2965	0.874842	0.437421	−	0.899257i	$-0.355892\pi$
$347$	16.2965	0.874842	0.437421	+	0.899257i	$0.355892\pi$
$348$	−116.466	−6.24322
$349$	19.5903	1.04865	0.524323	−	0.851519i	$-0.324318\pi$
$349$	19.5903	1.04865	0.524323	+	0.851519i	$0.324318\pi$
$350$	−13.4651	−0.719740
$351$	0.263262	0.0140519
$352$	−19.1453	−1.02045
$353$	14.9073	0.793437	0.396719	−	0.917940i	$-0.370149\pi$
$353$	14.9073	0.793437	0.396719	+	0.917940i	$0.370149\pi$
$354$	43.7699	2.32634
$355$	−26.4027	−1.40131
$356$	−34.3047	−1.81814
$357$	38.1164	2.01733
$358$	22.0928	1.16764
$359$	−6.11331	−0.322648	−0.161324	−	0.986901i	$-0.551576\pi$
$359$	−6.11331	−0.322648	−0.161324	+	0.986901i	$0.551576\pi$
$360$	40.4031	2.12943
$361$	1.00000	0.0526316
$362$	3.23919	0.170248
$363$	19.8920	1.04406
$364$	17.7455	0.930118
$365$	8.81742	0.461525
$366$	−56.8232	−2.97019
$367$	−3.41883	−0.178462	−0.0892308	−	0.996011i	$-0.528441\pi$
$367$	−3.41883	−0.178462	−0.0892308	+	0.996011i	$0.528441\pi$
$368$	−37.6572	−1.96302
$369$	−8.36117	−0.435265
$370$	13.1516	0.683719
$371$	3.87424	0.201140
$372$	−63.1624	−3.27482
$373$	0.272719	0.0141208	0.00706042	−	0.999975i	$-0.497753\pi$
$373$	0.272719	0.0141208	0.00706042	+	0.999975i	$0.497753\pi$
$374$	−26.5587	−1.37332
$375$	−29.9111	−1.54460
$376$	66.6303	3.43620
$377$	−13.5000	−0.695287
$378$	−1.29498	−0.0666065
$379$	27.5757	1.41647	0.708233	−	0.705978i	$-0.249492\pi$
$379$	27.5757	1.41647	0.708233	+	0.705978i	$0.249492\pi$
$380$	−8.49257	−0.435660
$381$	1.31771	0.0675084
$382$	−9.12570	−0.466912
$383$	−20.1529	−1.02976	−0.514882	−	0.857261i	$-0.672164\pi$
$383$	−20.1529	−1.02976	−0.514882	+	0.857261i	$0.672164\pi$
$384$	4.93113	0.251640
$385$	7.79326	0.397181
$386$	66.8289	3.40150
$387$	12.9322	0.657381
$388$	−11.6733	−0.592623
$389$	12.3565	0.626501	0.313251	−	0.949670i	$-0.398582\pi$
$389$	12.3565	0.626501	0.313251	+	0.949670i	$0.398582\pi$
$390$	15.6847	0.794225
$391$	−22.2477	−1.12512
$392$	1.28901	0.0651050
$393$	2.86710	0.144626
$394$	−57.1248	−2.87791
$395$	−23.2171	−1.16818
$396$	25.6758	1.29026
$397$	−7.28873	−0.365811	−0.182905	−	0.983131i	$-0.558550\pi$
$397$	−7.28873	−0.365811	−0.182905	+	0.983131i	$0.558550\pi$
$398$	21.7298	1.08922
$399$	6.44186	0.322496
$400$	−19.6815	−0.984077
$401$	29.9916	1.49771	0.748855	−	0.662733i	$-0.230604\pi$
$401$	29.9916	1.49771	0.748855	+	0.662733i	$0.230604\pi$
$402$	9.81528	0.489542
$403$	−7.32140	−0.364705
$404$	4.14643	0.206293
$405$	15.2678	0.758663
$406$	66.4062	3.29569
$407$	4.92887	0.244315
$408$	109.951	5.44339
$409$	13.2077	0.653077	0.326538	−	0.945184i	$-0.394118\pi$
$409$	13.2077	0.653077	0.326538	+	0.945184i	$0.394118\pi$
$410$	−12.4134	−0.613053
$411$	−40.3256	−1.98912
$412$	−28.1412	−1.38642
$413$	−17.6964	−0.870785
$414$	30.3320	1.49074
$415$	−19.3369	−0.949213
$416$	15.5787	0.763808
$417$	31.3967	1.53750
$418$	−4.48856	−0.219543
$419$	7.34948	0.359045	0.179523	−	0.983754i	$-0.442545\pi$
$419$	7.34948	0.359045	0.179523	+	0.983754i	$0.442545\pi$
$420$	−54.7080	−2.66948
$421$	28.8538	1.40625	0.703125	−	0.711067i	$-0.251788\pi$
$421$	28.8538	1.40625	0.703125	+	0.711067i	$0.251788\pi$
$422$	2.62201	0.127638
$423$	−27.1948	−1.32226
$424$	11.1757	0.542739
$425$	−11.6278	−0.564030
$426$	97.9598	4.74617
$427$	22.9740	1.11179
$428$	−69.1741	−3.34366
$429$	5.87820	0.283802
$430$	19.1997	0.925894
$431$	−1.81656	−0.0875006	−0.0437503	−	0.999042i	$-0.513931\pi$
$431$	−1.81656	−0.0875006	−0.0437503	+	0.999042i	$0.513931\pi$
$432$	−1.89283	−0.0910689
$433$	−1.72590	−0.0829415	−0.0414707	−	0.999140i	$-0.513204\pi$
$433$	−1.72590	−0.0829415	−0.0414707	+	0.999140i	$0.513204\pi$
$434$	36.0138	1.72872
$435$	41.6195	1.99550
$436$	−53.8174	−2.57738
$437$	−3.75998	−0.179864
$438$	−32.7145	−1.56316
$439$	−17.7013	−0.844839	−0.422419	−	0.906401i	$-0.638819\pi$
$439$	−17.7013	−0.844839	−0.422419	+	0.906401i	$0.638819\pi$
$440$	22.4805	1.07172
$441$	−0.526104	−0.0250526
$442$	21.6110	1.02793
$443$	−3.93981	−0.187186	−0.0935930	−	0.995611i	$-0.529835\pi$
$443$	−3.93981	−0.187186	−0.0935930	+	0.995611i	$0.529835\pi$
$444$	−34.6002	−1.64205
$445$	12.2589	0.581127
$446$	−15.0237	−0.711395
$447$	32.5065	1.53751
$448$	−24.2865	−1.14743
$449$	−13.9891	−0.660187	−0.330093	−	0.943948i	$-0.607080\pi$
$449$	−13.9891	−0.660187	−0.330093	+	0.943948i	$0.607080\pi$
$450$	15.8530	0.747318
$451$	−4.65220	−0.219063
$452$	52.9485	2.49049
$453$	−34.8094	−1.63549
$454$	40.8322	1.91635
$455$	−6.34142	−0.297290
$456$	18.5823	0.870195
$457$	24.9517	1.16719	0.583595	−	0.812045i	$-0.301645\pi$
$457$	24.9517	1.16719	0.583595	+	0.812045i	$0.301645\pi$
$458$	−9.58297	−0.447783
$459$	−1.11828	−0.0521967
$460$	31.9319	1.48883
$461$	−28.9490	−1.34829	−0.674144	−	0.738600i	$-0.735487\pi$
$461$	−28.9490	−1.34829	−0.674144	+	0.738600i	$0.735487\pi$
$462$	−28.9147	−1.34523
$463$	22.9853	1.06822	0.534109	−	0.845415i	$-0.320647\pi$
$463$	22.9853	1.06822	0.534109	+	0.845415i	$0.320647\pi$
$464$	97.0640	4.50608
$465$	22.5713	1.04672
$466$	37.7209	1.74739
$467$	−9.96987	−0.461351	−0.230675	−	0.973031i	$-0.574094\pi$
$467$	−9.96987	−0.461351	−0.230675	+	0.973031i	$0.574094\pi$
$468$	−20.8925	−0.965757
$469$	−3.96838	−0.183243
$470$	−40.3747	−1.86235
$471$	24.9202	1.14826
$472$	−51.0474	−2.34965
$473$	7.19554	0.330852
$474$	86.1403	3.95656
$475$	−1.96515	−0.0901673
$476$	−75.3789	−3.45499
$477$	−4.56130	−0.208847
$478$	56.5566	2.58684
$479$	21.3386	0.974987	0.487493	−	0.873127i	$-0.337911\pi$
$479$	21.3386	0.974987	0.487493	+	0.873127i	$0.337911\pi$
$480$	−48.0278	−2.19216
$481$	−4.01065	−0.182870
$482$	62.8948	2.86478
$483$	−24.2213	−1.10211
$484$	−39.3384	−1.78811
$485$	4.17150	0.189418
$486$	−58.1335	−2.63699
$487$	20.8613	0.945315	0.472657	−	0.881246i	$-0.343295\pi$
$487$	20.8613	0.945315	0.472657	+	0.881246i	$0.343295\pi$
$488$	66.2710	2.99995
$489$	24.8564	1.12404
$490$	−0.781079	−0.0352855
$491$	34.4849	1.55628	0.778141	−	0.628090i	$-0.216163\pi$
$491$	34.4849	1.55628	0.778141	+	0.628090i	$0.216163\pi$
$492$	32.6580	1.47234
$493$	57.3451	2.58269
$494$	3.65237	0.164328
$495$	−9.17532	−0.412400
$496$	52.6402	2.36362
$497$	−39.6058	−1.77656
$498$	71.7442	3.21494
$499$	−20.0148	−0.895987	−0.447994	−	0.894037i	$-0.647861\pi$
$499$	−20.0148	−0.895987	−0.447994	+	0.894037i	$0.647861\pi$
$500$	59.1520	2.64536
$501$	−7.28732	−0.325573
$502$	−78.1061	−3.48605
$503$	−8.41685	−0.375289	−0.187644	−	0.982237i	$-0.560085\pi$
$503$	−8.41685	−0.375289	−0.187644	+	0.982237i	$0.560085\pi$
$504$	60.6074	2.69967
$505$	−1.48174	−0.0659366
$506$	16.8769	0.750268
$507$	27.2630	1.21079
$508$	−2.60590	−0.115618
$509$	−12.6310	−0.559857	−0.279929	−	0.960021i	$-0.590311\pi$
$509$	−12.6310	−0.559857	−0.279929	+	0.960021i	$0.590311\pi$
$510$	−66.6250	−2.95020
$511$	13.2267	0.585115
$512$	38.9844	1.72288
$513$	−0.188994	−0.00834431
$514$	−29.5190	−1.30203
$515$	10.0564	0.443136
$516$	−50.5121	−2.22367
$517$	−15.1313	−0.665476
$518$	19.7283	0.866810
$519$	22.3183	0.979666
$520$	−18.2925	−0.802181
$521$	−22.3186	−0.977796	−0.488898	−	0.872341i	$-0.662601\pi$
$521$	−22.3186	−0.977796	−0.488898	+	0.872341i	$0.662601\pi$
$522$	−78.1827	−3.42197
$523$	−19.2607	−0.842212	−0.421106	−	0.907011i	$-0.638358\pi$
$523$	−19.2607	−0.842212	−0.421106	+	0.907011i	$0.638358\pi$
$524$	−5.66996	−0.247693
$525$	−12.6592	−0.552494
$526$	−32.6402	−1.42318
$527$	31.0997	1.35472
$528$	−42.2637	−1.83929
$529$	−8.86257	−0.385329
$530$	−6.77191	−0.294153
$531$	20.8347	0.904151
$532$	−12.7394	−0.552323
$533$	3.78552	0.163969
$534$	−45.4832	−1.96825
$535$	24.7196	1.06872
$536$	−11.4472	−0.494446
$537$	20.7705	0.896315
$538$	32.5070	1.40147
$539$	−0.292727	−0.0126087
$540$	1.60505	0.0690703
$541$	1.14613	0.0492758	0.0246379	−	0.999696i	$-0.492157\pi$
$541$	1.14613	0.0492758	0.0246379	+	0.999696i	$0.492157\pi$
$542$	−50.7777	−2.18109
$543$	3.04533	0.130688
$544$	−66.1747	−2.83722
$545$	19.2318	0.823801
$546$	23.5280	1.00691
$547$	−26.0713	−1.11473	−0.557364	−	0.830268i	$-0.688187\pi$
$547$	−26.0713	−1.11473	−0.557364	+	0.830268i	$0.688187\pi$
$548$	79.7479	3.40666
$549$	−27.0482	−1.15439
$550$	8.82069	0.376116
$551$	9.69159	0.412876
$552$	−69.8689	−2.97382
$553$	−34.8271	−1.48100
$554$	63.8337	2.71203
$555$	12.3645	0.524844
$556$	−62.0901	−2.63321
$557$	12.1205	0.513564	0.256782	−	0.966469i	$-0.417338\pi$
$557$	12.1205	0.513564	0.256782	+	0.966469i	$0.417338\pi$
$558$	−42.4005	−1.79495
$559$	−5.85506	−0.247643
$560$	45.5943	1.92671
$561$	−24.9692	−1.05420
$562$	10.3694	0.437408
$563$	21.6328	0.911714	0.455857	−	0.890053i	$-0.349333\pi$
$563$	21.6328	0.911714	0.455857	+	0.890053i	$0.349333\pi$
$564$	106.221	4.47270
$565$	−18.9213	−0.796026
$566$	36.9071	1.55132
$567$	22.9027	0.961823
$568$	−114.247	−4.79372
$569$	23.2365	0.974123	0.487062	−	0.873368i	$-0.338069\pi$
$569$	23.2365	0.974123	0.487062	+	0.873368i	$0.338069\pi$
$570$	−11.2599	−0.471627
$571$	24.4517	1.02327	0.511636	−	0.859202i	$-0.329040\pi$
$571$	24.4517	1.02327	0.511636	+	0.859202i	$0.329040\pi$
$572$	−11.6247	−0.486053
$573$	−8.57954	−0.358416
$574$	−18.6209	−0.777220
$575$	7.38892	0.308139
$576$	28.5935	1.19140
$577$	40.3839	1.68121	0.840603	−	0.541652i	$-0.182201\pi$
$577$	40.3839	1.68121	0.840603	+	0.541652i	$0.182201\pi$
$578$	−47.2244	−1.96428
$579$	62.8293	2.61110
$580$	−82.3065	−3.41759
$581$	−29.0067	−1.20340
$582$	−15.4772	−0.641549
$583$	−2.53793	−0.105110
$584$	38.1539	1.57882
$585$	7.46601	0.308682
$586$	−52.3527	−2.16267
$587$	−22.6157	−0.933451	−0.466726	−	0.884402i	$-0.654566\pi$
$587$	−22.6157	−0.933451	−0.466726	+	0.884402i	$0.654566\pi$
$588$	2.05492	0.0847434
$589$	5.25599	0.216569
$590$	30.9322	1.27346
$591$	−53.7059	−2.20917
$592$	28.8362	1.18516
$593$	−20.4653	−0.840410	−0.420205	−	0.907429i	$-0.638042\pi$
$593$	−20.4653	−0.840410	−0.420205	+	0.907429i	$0.638042\pi$
$594$	0.848312	0.0348067
$595$	26.9369	1.10431
$596$	−64.2848	−2.63321
$597$	20.4293	0.836116
$598$	−13.7328	−0.561577
$599$	−24.2560	−0.991073	−0.495537	−	0.868587i	$-0.665029\pi$
$599$	−24.2560	−0.991073	−0.495537	+	0.868587i	$0.665029\pi$
$600$	−36.5170	−1.49080
$601$	1.10470	0.0450617	0.0225308	−	0.999746i	$-0.492828\pi$
$601$	1.10470	0.0450617	0.0225308	+	0.999746i	$0.492828\pi$
$602$	28.8009	1.17384
$603$	4.67214	0.190264
$604$	68.8391	2.80102
$605$	14.0577	0.571527
$606$	5.49758	0.223324
$607$	13.6629	0.554560	0.277280	−	0.960789i	$-0.410567\pi$
$607$	13.6629	0.554560	0.277280	+	0.960789i	$0.410567\pi$
$608$	−11.1839	−0.453565
$609$	62.4319	2.52987
$610$	−40.1570	−1.62591
$611$	12.3125	0.498109
$612$	88.7467	3.58737
$613$	−29.4308	−1.18870	−0.594350	−	0.804206i	$-0.702591\pi$
$613$	−29.4308	−1.18870	−0.594350	+	0.804206i	$0.702591\pi$
$614$	−72.7282	−2.93507
$615$	−11.6705	−0.470598
$616$	33.7222	1.35871
$617$	21.7303	0.874828	0.437414	−	0.899260i	$-0.355894\pi$
$617$	21.7303	0.874828	0.437414	+	0.899260i	$0.355894\pi$
$618$	−37.3112	−1.50088
$619$	−34.7272	−1.39581	−0.697903	−	0.716192i	$-0.745883\pi$
$619$	−34.7272	−1.39581	−0.697903	+	0.716192i	$0.745883\pi$
$620$	−44.6369	−1.79266
$621$	0.710615	0.0285160
$622$	−61.5313	−2.46718
$623$	18.3891	0.736745
$624$	34.3902	1.37671
$625$	−11.3124	−0.452497
$626$	−32.3787	−1.29411
$627$	−4.21992	−0.168528
$628$	−49.2820	−1.96657
$629$	17.0363	0.679283
$630$	−36.7251	−1.46316
$631$	−17.7040	−0.704785	−0.352392	−	0.935852i	$-0.614632\pi$
$631$	−17.7040	−0.704785	−0.352392	+	0.935852i	$0.614632\pi$
$632$	−100.463	−3.99619
$633$	2.46509	0.0979785
$634$	−67.7844	−2.69206
$635$	0.931228	0.0369547
$636$	17.8160	0.706452
$637$	0.238194	0.00943759
$638$	−43.5013	−1.72223
$639$	46.6295	1.84464
$640$	3.48483	0.137750
$641$	−2.16283	−0.0854267	−0.0427133	−	0.999087i	$-0.513600\pi$
$641$	−2.16283	−0.0854267	−0.0427133	+	0.999087i	$0.513600\pi$
$642$	−91.7150	−3.61970
$643$	−27.8411	−1.09795	−0.548974	−	0.835840i	$-0.684981\pi$
$643$	−27.8411	−1.09795	−0.548974	+	0.835840i	$0.684981\pi$
$644$	47.8999	1.88752
$645$	18.0507	0.710744
$646$	−15.5144	−0.610407
$647$	−2.35647	−0.0926425	−0.0463213	−	0.998927i	$-0.514750\pi$
$647$	−2.35647	−0.0926425	−0.0463213	+	0.998927i	$0.514750\pi$
$648$	66.0654	2.59529
$649$	11.5926	0.455048
$650$	−7.17745	−0.281523
$651$	33.8584	1.32701
$652$	−49.1559	−1.92509
$653$	−14.1678	−0.554427	−0.277214	−	0.960808i	$-0.589411\pi$
$653$	−14.1678	−0.554427	−0.277214	+	0.960808i	$0.589411\pi$
$654$	−71.3542	−2.79017
$655$	2.02618	0.0791694
$656$	−27.2176	−1.06267
$657$	−15.5723	−0.607535
$658$	−60.5647	−2.36106
$659$	−8.66898	−0.337696	−0.168848	−	0.985642i	$-0.554005\pi$
$659$	−8.66898	−0.337696	−0.168848	+	0.985642i	$0.554005\pi$
$660$	35.8380	1.39499
$661$	15.7771	0.613660	0.306830	−	0.951764i	$-0.400732\pi$
$661$	15.7771	0.613660	0.306830	+	0.951764i	$0.400732\pi$
$662$	1.78922	0.0695398
$663$	20.3176	0.789071
$664$	−83.6730	−3.24714
$665$	4.55247	0.176537
$666$	−23.2269	−0.900023
$667$	−36.4402	−1.41097
$668$	14.4114	0.557593
$669$	−14.1246	−0.546088
$670$	6.93647	0.267979
$671$	−15.0498	−0.580990
$672$	−72.0448	−2.77919
$673$	44.2789	1.70682	0.853412	−	0.521236i	$-0.174529\pi$
$673$	44.2789	1.70682	0.853412	+	0.521236i	$0.174529\pi$
$674$	8.93540	0.344179
$675$	0.371403	0.0142953
$676$	−53.9153	−2.07367
$677$	38.3728	1.47479	0.737394	−	0.675463i	$-0.236056\pi$
$677$	38.3728	1.47479	0.737394	+	0.675463i	$0.236056\pi$
$678$	70.2022	2.69610
$679$	6.25752	0.240141
$680$	77.7026	2.97976
$681$	38.3885	1.47105
$682$	−23.5918	−0.903378
$683$	−20.8073	−0.796168	−0.398084	−	0.917349i	$-0.630325\pi$
$683$	−20.8073	−0.796168	−0.398084	+	0.917349i	$0.630325\pi$
$684$	14.9986	0.573487
$685$	−28.4982	−1.08886
$686$	−49.1352	−1.87599
$687$	−9.00944	−0.343732
$688$	42.0974	1.60495
$689$	2.06513	0.0786751
$690$	42.3371	1.61175
$691$	6.37831	0.242642	0.121321	−	0.992613i	$-0.461287\pi$
$691$	6.37831	0.242642	0.121321	+	0.992613i	$0.461287\pi$
$692$	−44.1367	−1.67782
$693$	−13.7636	−0.522835
$694$	−42.7296	−1.62199
$695$	22.1881	0.841642
$696$	180.092	6.82636
$697$	−16.0800	−0.609075
$698$	−51.3661	−1.94424
$699$	35.4634	1.34135
$700$	25.0349	0.946229
$701$	−40.2701	−1.52098	−0.760490	−	0.649349i	$-0.775041\pi$
$701$	−40.2701	−1.52098	−0.760490	+	0.649349i	$0.775041\pi$
$702$	−0.690277	−0.0260528
$703$	2.87922	0.108592
$704$	15.9096	0.599614
$705$	−37.9583	−1.42959
$706$	−39.0872	−1.47107
$707$	−2.22271	−0.0835935
$708$	−81.3787	−3.05840
$709$	29.7261	1.11639	0.558193	−	0.829711i	$-0.311495\pi$
$709$	29.7261	1.11639	0.558193	+	0.829711i	$0.311495\pi$
$710$	69.2283	2.59809
$711$	41.0034	1.53775
$712$	53.0455	1.98797
$713$	−19.7624	−0.740108
$714$	−99.9418	−3.74023
$715$	4.15413	0.155356
$716$	−41.0758	−1.53507
$717$	53.1718	1.98574
$718$	16.0292	0.598204
$719$	−15.3262	−0.571570	−0.285785	−	0.958294i	$-0.592254\pi$
$719$	−15.3262	−0.571570	−0.285785	+	0.958294i	$0.592254\pi$
$720$	−53.6800	−2.00053
$721$	15.0852	0.561801
$722$	−2.62201	−0.0975812
$723$	59.1306	2.19909
$724$	−6.02244	−0.223822
$725$	−19.0454	−0.707330
$726$	−52.1571	−1.93573
$727$	51.9545	1.92689	0.963444	−	0.267911i	$-0.0863333\pi$
$727$	51.9545	1.92689	0.963444	+	0.267911i	$0.0863333\pi$
$728$	−27.4400	−1.01699
$729$	−28.3619	−1.05044
$730$	−23.1194	−0.855687
$731$	24.8710	0.919886
$732$	105.648	3.90486
$733$	3.74708	0.138402	0.0692008	−	0.997603i	$-0.477955\pi$
$733$	3.74708	0.138402	0.0692008	+	0.997603i	$0.477955\pi$
$734$	8.96422	0.330875
$735$	−0.734332	−0.0270863
$736$	42.0510	1.55002
$737$	2.59960	0.0957576
$738$	21.9231	0.807001
$739$	46.1211	1.69659	0.848296	−	0.529522i	$-0.177629\pi$
$739$	46.1211	1.69659	0.848296	+	0.529522i	$0.177629\pi$
$740$	−24.4520	−0.898873
$741$	3.43378	0.126143
$742$	−10.1583	−0.372923
$743$	10.9446	0.401519	0.200760	−	0.979641i	$-0.435659\pi$
$743$	10.9446	0.401519	0.200760	+	0.979641i	$0.435659\pi$
$744$	97.6683	3.58069
$745$	22.9724	0.841644
$746$	−0.715072	−0.0261807
$747$	34.1507	1.24951
$748$	49.3791	1.80548
$749$	37.0810	1.35491
$750$	78.4272	2.86376
$751$	1.34853	0.0492084	0.0246042	−	0.999697i	$-0.492167\pi$
$751$	1.34853	0.0492084	0.0246042	+	0.999697i	$0.492167\pi$
$752$	−88.5256	−3.22820
$753$	−73.4316	−2.67600
$754$	35.3973	1.28909
$755$	−24.5999	−0.895281
$756$	2.40768	0.0875664
$757$	−7.45270	−0.270873	−0.135436	−	0.990786i	$-0.543244\pi$
$757$	−7.45270	−0.270873	−0.135436	+	0.990786i	$0.543244\pi$
$758$	−72.3038	−2.62619
$759$	15.8668	0.575929
$760$	13.1321	0.476352
$761$	−2.55473	−0.0926089	−0.0463045	−	0.998927i	$-0.514744\pi$
$761$	−2.55473	−0.0926089	−0.0463045	+	0.998927i	$0.514744\pi$
$762$	−3.45506	−0.125164
$763$	28.8490	1.04440
$764$	16.9669	0.613840
$765$	−31.7139	−1.14662
$766$	52.8411	1.90923
$767$	−9.43294	−0.340604
$768$	32.8899	1.18681
$769$	7.62557	0.274985	0.137493	−	0.990503i	$-0.456096\pi$
$769$	7.62557	0.274985	0.137493	+	0.990503i	$0.456096\pi$
$770$	−20.4340	−0.736391
$771$	−27.7523	−0.999475
$772$	−124.251	−4.47189
$773$	−50.7500	−1.82535	−0.912676	−	0.408683i	$-0.865988\pi$
$773$	−50.7500	−1.82535	−0.912676	+	0.408683i	$0.865988\pi$
$774$	−33.9084	−1.21881
$775$	−10.3288	−0.371022
$776$	18.0505	0.647976
$777$	18.5475	0.665390
$778$	−32.3990	−1.16156
$779$	−2.71761	−0.0973683
$780$	−29.1616	−1.04415
$781$	25.9449	0.928382
$782$	58.3339	2.08602
$783$	−1.83166	−0.0654581
$784$	−1.71259	−0.0611641
$785$	17.6111	0.628567
$786$	−7.51756	−0.268143
$787$	19.1941	0.684197	0.342098	−	0.939664i	$-0.388862\pi$
$787$	19.1941	0.684197	0.342098	+	0.939664i	$0.388862\pi$
$788$	106.209	3.78353
$789$	−30.6867	−1.09248
$790$	60.8755	2.16585
$791$	−28.3832	−1.00919
$792$	−39.7026	−1.41077
$793$	12.2461	0.434871
$794$	19.1112	0.678229
$795$	−6.36662	−0.225801
$796$	−40.4010	−1.43197
$797$	−38.0214	−1.34679	−0.673394	−	0.739284i	$-0.735164\pi$
$797$	−38.0214	−1.34679	−0.673394	+	0.739284i	$0.735164\pi$
$798$	−16.8906	−0.597922
$799$	−52.3006	−1.85026
$800$	21.9780	0.777038
$801$	−21.6503	−0.764975
$802$	−78.6385	−2.77682
$803$	−8.66453	−0.305765
$804$	−18.2490	−0.643592
$805$	−17.1172	−0.603301
$806$	19.1968	0.676179
$807$	30.5615	1.07581
$808$	−6.41165	−0.225561
$809$	30.2932	1.06505	0.532527	−	0.846413i	$-0.321243\pi$
$809$	30.2932	1.06505	0.532527	+	0.846413i	$0.321243\pi$
$810$	−40.0324	−1.40660
$811$	−5.31147	−0.186511	−0.0932555	−	0.995642i	$-0.529727\pi$
$811$	−5.31147	−0.186511	−0.0932555	+	0.995642i	$0.529727\pi$
$812$	−123.465	−4.33278
$813$	−47.7387	−1.67427
$814$	−12.9236	−0.452970
$815$	17.5660	0.615311
$816$	−146.082	−5.11389
$817$	4.20331	0.147055
$818$	−34.6307	−1.21083
$819$	11.1995	0.391342
$820$	23.0795	0.805969
$821$	−19.1630	−0.668793	−0.334397	−	0.942432i	$-0.608532\pi$
$821$	−19.1630	−0.668793	−0.334397	+	0.942432i	$0.608532\pi$
$822$	105.734	3.68791
$823$	17.5966	0.613379	0.306689	−	0.951810i	$-0.400779\pi$
$823$	17.5966	0.613379	0.306689	+	0.951810i	$0.400779\pi$
$824$	43.5149	1.51591
$825$	8.29279	0.288718
$826$	46.4003	1.61447
$827$	−0.467499	−0.0162565	−0.00812827	−	0.999967i	$-0.502587\pi$
$827$	−0.467499	−0.0162565	−0.00812827	+	0.999967i	$0.502587\pi$
$828$	−56.3945	−1.95984
$829$	50.7873	1.76392	0.881959	−	0.471327i	$-0.156225\pi$
$829$	50.7873	1.76392	0.881959	+	0.471327i	$0.156225\pi$
$830$	50.7017	1.75988
$831$	60.0133	2.08184
$832$	−12.9457	−0.448812
$833$	−1.01179	−0.0350566
$834$	−82.3226	−2.85060
$835$	−5.14995	−0.178221
$836$	8.34531	0.288629
$837$	−0.993354	−0.0343353
$838$	−19.2704	−0.665686
$839$	46.8658	1.61799	0.808993	−	0.587818i	$-0.200013\pi$
$839$	46.8658	1.61799	0.808993	+	0.587818i	$0.200013\pi$
$840$	84.5952	2.91881
$841$	64.9270	2.23886
$842$	−75.6551	−2.60725
$843$	9.74882	0.335767
$844$	−4.87496	−0.167803
$845$	19.2668	0.662799
$846$	71.3052	2.45152
$847$	21.0875	0.724574
$848$	−14.8481	−0.509886
$849$	34.6982	1.19084
$850$	30.4882	1.04574
$851$	−10.8258	−0.371104
$852$	−182.131	−6.23970
$853$	−24.2256	−0.829467	−0.414734	−	0.909943i	$-0.636125\pi$
$853$	−24.2256	−0.829467	−0.414734	+	0.909943i	$0.636125\pi$
$854$	−60.2381	−2.06131
$855$	−5.35981	−0.183302
$856$	106.964	3.65596
$857$	−19.8224	−0.677121	−0.338561	−	0.940945i	$-0.609940\pi$
$857$	−19.8224	−0.677121	−0.338561	+	0.940945i	$0.609940\pi$
$858$	−15.4127	−0.526181
$859$	−36.3961	−1.24182	−0.620910	−	0.783882i	$-0.713237\pi$
$859$	−36.3961	−1.24182	−0.620910	+	0.783882i	$0.713237\pi$
$860$	−35.6969	−1.21726
$861$	−17.5064	−0.596618
$862$	4.76304	0.162230
$863$	29.0808	0.989923	0.494962	−	0.868915i	$-0.335182\pi$
$863$	29.0808	0.989923	0.494962	+	0.868915i	$0.335182\pi$
$864$	2.11369	0.0719090
$865$	15.7724	0.536277
$866$	4.52534	0.153777
$867$	−44.3981	−1.50784
$868$	−66.9583	−2.27271
$869$	22.8145	0.773928
$870$	−109.127	−3.69975
$871$	−2.11531	−0.0716746
$872$	83.2181	2.81812
$873$	−7.36723	−0.249343
$874$	9.85871	0.333476
$875$	−31.7086	−1.07195
$876$	60.8242	2.05506
$877$	5.00898	0.169141	0.0845707	−	0.996417i	$-0.473048\pi$
$877$	5.00898	0.169141	0.0845707	+	0.996417i	$0.473048\pi$
$878$	46.4131	1.56637
$879$	−49.2195	−1.66013
$880$	−29.8678	−1.00684
$881$	−37.6285	−1.26774	−0.633868	−	0.773441i	$-0.718534\pi$
$881$	−37.6285	−1.26774	−0.633868	+	0.773441i	$0.718534\pi$
$882$	1.37945	0.0464486
$883$	−38.8695	−1.30806	−0.654031	−	0.756467i	$-0.726924\pi$
$883$	−38.8695	−1.30806	−0.654031	+	0.756467i	$0.726924\pi$
$884$	−40.1801	−1.35140
$885$	29.0810	0.977546
$886$	10.3302	0.347051
$887$	−43.8901	−1.47368	−0.736842	−	0.676065i	$-0.763684\pi$
$887$	−43.8901	−1.47368	−0.736842	+	0.676065i	$0.763684\pi$
$888$	53.5025	1.79543
$889$	1.39690	0.0468506
$890$	−32.1430	−1.07744
$891$	−15.0031	−0.502622
$892$	27.9328	0.935258
$893$	−8.83905	−0.295788
$894$	−85.2326	−2.85060
$895$	14.6786	0.490650
$896$	5.22748	0.174638
$897$	−12.9109	−0.431083
$898$	36.6796	1.22401
$899$	50.9390	1.69891
$900$	−29.4746	−0.982485
$901$	−8.77220	−0.292244
$902$	12.1981	0.406153
$903$	27.0772	0.901072
$904$	−81.8746	−2.72311
$905$	2.15214	0.0715395
$906$	91.2708	3.03227
$907$	40.6506	1.34978	0.674891	−	0.737917i	$-0.264191\pi$
$907$	40.6506	1.34978	0.674891	+	0.737917i	$0.264191\pi$
$908$	−75.9170	−2.51939
$909$	2.61688	0.0867965
$910$	16.6273	0.551189
$911$	29.1957	0.967296	0.483648	−	0.875263i	$-0.339312\pi$
$911$	29.1957	0.967296	0.483648	+	0.875263i	$0.339312\pi$
$912$	−24.6886	−0.817520
$913$	19.0016	0.628862
$914$	−65.4237	−2.16402
$915$	−37.7537	−1.24810
$916$	17.8170	0.588692
$917$	3.03940	0.100370
$918$	2.93214	0.0967750
$919$	−8.45800	−0.279004	−0.139502	−	0.990222i	$-0.544550\pi$
$919$	−8.45800	−0.279004	−0.139502	+	0.990222i	$0.544550\pi$
$920$	−49.3764	−1.62789
$921$	−68.3755	−2.25305
$922$	75.9046	2.49979
$923$	−21.1115	−0.694895
$924$	53.7593	1.76855
$925$	−5.65811	−0.186037
$926$	−60.2678	−1.98052
$927$	−17.7604	−0.583328
$928$	−108.389	−3.55805
$929$	50.9830	1.67270	0.836349	−	0.548198i	$-0.184686\pi$
$929$	50.9830	1.67270	0.836349	+	0.548198i	$0.184686\pi$
$930$	−59.1822	−1.94066
$931$	−0.170998	−0.00560424
$932$	−70.1323	−2.29726
$933$	−57.8488	−1.89388
$934$	26.1411	0.855364
$935$	−17.6458	−0.577079
$936$	32.3062	1.05596
$937$	−54.3238	−1.77468	−0.887341	−	0.461113i	$-0.847450\pi$
$937$	−54.3238	−1.77468	−0.887341	+	0.461113i	$0.847450\pi$
$938$	10.4052	0.339740
$939$	−30.4409	−0.993400
$940$	75.0663	2.44839
$941$	−31.6411	−1.03147	−0.515736	−	0.856748i	$-0.672481\pi$
$941$	−31.6411	−1.03147	−0.515736	+	0.856748i	$0.672481\pi$
$942$	−65.3410	−2.12892
$943$	10.2181	0.332748
$944$	67.8220	2.20742
$945$	−0.860391	−0.0279885
$946$	−18.8668	−0.613413
$947$	−35.4017	−1.15040	−0.575201	−	0.818012i	$-0.695076\pi$
$947$	−35.4017	−1.15040	−0.575201	+	0.818012i	$0.695076\pi$
$948$	−160.156	−5.20161
$949$	7.05038	0.228865
$950$	5.15265	0.167174
$951$	−63.7276	−2.06651
$952$	116.559	3.77770
$953$	−12.9547	−0.419642	−0.209821	−	0.977740i	$-0.567288\pi$
$953$	−12.9547	−0.419642	−0.209821	+	0.977740i	$0.567288\pi$
$954$	11.9598	0.387212
$955$	−6.06317	−0.196200
$956$	−105.152	−3.40087
$957$	−40.8978	−1.32204
$958$	−55.9502	−1.80767
$959$	−42.7491	−1.38044
$960$	39.9106	1.28811
$961$	−3.37452	−0.108856
$962$	10.5160	0.339049
$963$	−43.6570	−1.40683
$964$	−116.937	−3.76627
$965$	44.4015	1.42934
$966$	63.5084	2.04335
$967$	37.5637	1.20797	0.603984	−	0.796996i	$-0.293579\pi$
$967$	37.5637	1.20797	0.603984	+	0.796996i	$0.293579\pi$
$968$	60.8292	1.95512
$969$	−14.5859	−0.468567
$970$	−10.9377	−0.351189
$971$	−31.9958	−1.02679	−0.513397	−	0.858151i	$-0.671613\pi$
$971$	−31.9958	−1.02679	−0.513397	+	0.858151i	$0.671613\pi$
$972$	108.084	3.46680
$973$	33.2836	1.06702
$974$	−54.6985	−1.75265
$975$	−6.74789	−0.216106
$976$	−88.0483	−2.81836
$977$	−44.7976	−1.43320	−0.716601	−	0.697483i	$-0.754303\pi$
$977$	−44.7976	−1.43320	−0.716601	+	0.697483i	$0.754303\pi$
$978$	−65.1738	−2.08403
$979$	−12.0463	−0.385002
$980$	1.45221	0.0463893
$981$	−33.9651	−1.08442
$982$	−90.4199	−2.88541
$983$	31.7869	1.01385	0.506923	−	0.861991i	$-0.330783\pi$
$983$	31.7869	1.01385	0.506923	+	0.861991i	$0.330783\pi$
$984$	−50.4993	−1.60986
$985$	−37.9541	−1.20932
$986$	−150.360	−4.78842
$987$	−56.9400	−1.81242
$988$	−6.79063	−0.216039
$989$	−15.8044	−0.502550
$990$	24.0578	0.764607
$991$	28.5521	0.906986	0.453493	−	0.891260i	$-0.350178\pi$
$991$	28.5521	0.906986	0.453493	+	0.891260i	$0.350178\pi$
$992$	−58.7823	−1.86634
$993$	1.68213	0.0533809
$994$	103.847	3.29383
$995$	14.4374	0.457697
$996$	−133.390	−4.22662
$997$	26.8723	0.851053	0.425527	−	0.904946i	$-0.360089\pi$
$997$	26.8723	0.851053	0.425527	+	0.904946i	$0.360089\pi$
$998$	52.4792	1.66120
$999$	−0.544157	−0.0172164

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4009.2.a.e.1.4	✓	82

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4009.2.a.e.1.4	✓	82	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:	\( N \)	\(=\)	\( 4009 = 19 \cdot 211 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4009.a (trivial)

\(f(q)\)	\(=\)	\(q-2.62201 q^{2} -2.46509 q^{3} +4.87496 q^{4} -1.74208 q^{5} +6.46350 q^{6} -2.61324 q^{7} -7.53817 q^{8} +3.07667 q^{9} +O(q^{10})\)	\(q-2.62201 q^{2} -2.46509 q^{3} +4.87496 q^{4} -1.74208 q^{5} +6.46350 q^{6} -2.61324 q^{7} -7.53817 q^{8} +3.07667 q^{9} +4.56776 q^{10} +1.71187 q^{11} -12.0172 q^{12} -1.39296 q^{13} +6.85194 q^{14} +4.29439 q^{15} +10.0153 q^{16} +5.91699 q^{17} -8.06707 q^{18} +1.00000 q^{19} -8.49257 q^{20} +6.44186 q^{21} -4.48856 q^{22} -3.75998 q^{23} +18.5823 q^{24} -1.96515 q^{25} +3.65237 q^{26} -0.188994 q^{27} -12.7394 q^{28} +9.69159 q^{29} -11.2599 q^{30} +5.25599 q^{31} -11.1839 q^{32} -4.21992 q^{33} -15.5144 q^{34} +4.55247 q^{35} +14.9986 q^{36} +2.87922 q^{37} -2.62201 q^{38} +3.43378 q^{39} +13.1321 q^{40} -2.71761 q^{41} -16.8906 q^{42} +4.20331 q^{43} +8.34531 q^{44} -5.35981 q^{45} +9.85871 q^{46} -8.83905 q^{47} -24.6886 q^{48} -0.170998 q^{49} +5.15265 q^{50} -14.5859 q^{51} -6.79063 q^{52} -1.48254 q^{53} +0.495546 q^{54} -2.98222 q^{55} +19.6990 q^{56} -2.46509 q^{57} -25.4115 q^{58} +6.77185 q^{59} +20.9349 q^{60} -8.79139 q^{61} -13.7813 q^{62} -8.04006 q^{63} +9.29365 q^{64} +2.42665 q^{65} +11.0647 q^{66} +1.51857 q^{67} +28.8451 q^{68} +9.26868 q^{69} -11.9366 q^{70} +15.1559 q^{71} -23.1925 q^{72} -5.06143 q^{73} -7.54936 q^{74} +4.84427 q^{75} +4.87496 q^{76} -4.47353 q^{77} -9.00341 q^{78} +13.3272 q^{79} -17.4474 q^{80} -8.76412 q^{81} +7.12560 q^{82} +11.0999 q^{83} +31.4038 q^{84} -10.3079 q^{85} -11.0211 q^{86} -23.8907 q^{87} -12.9044 q^{88} -7.03692 q^{89} +14.0535 q^{90} +3.64014 q^{91} -18.3297 q^{92} -12.9565 q^{93} +23.1761 q^{94} -1.74208 q^{95} +27.5692 q^{96} -2.39455 q^{97} +0.448359 q^{98} +5.26687 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 82 q + 15 q^{2} + 12 q^{3} + 89 q^{4} + 9 q^{5} + 9 q^{6} + 14 q^{7} + 42 q^{8} + 92 q^{9}+O(q^{10}) \) 82 * q + 15 * q^2 + 12 * q^3 + 89 * q^4 + 9 * q^5 + 9 * q^6 + 14 * q^7 + 42 * q^8 + 92 * q^9	\( 82 q + 15 q^{2} + 12 q^{3} + 89 q^{4} + 9 q^{5} + 9 q^{6} + 14 q^{7} + 42 q^{8} + 92 q^{9} + 4 q^{10} + 41 q^{11} + 26 q^{12} + 13 q^{13} + 22 q^{14} + 41 q^{15} + 87 q^{16} + 12 q^{17} + 24 q^{18} + 82 q^{19} + 26 q^{20} + 29 q^{21} + 2 q^{22} + 59 q^{23} + 16 q^{24} + 67 q^{25} + 24 q^{26} + 42 q^{27} - 2 q^{28} + 101 q^{29} - 22 q^{30} + 48 q^{31} + 69 q^{32} + 3 q^{33} + q^{34} + 38 q^{35} + 82 q^{36} + 16 q^{37} + 15 q^{38} + 82 q^{39} + 20 q^{40} + 86 q^{41} - q^{42} + 9 q^{43} + 82 q^{44} - 8 q^{45} + 43 q^{46} + 24 q^{47} + 34 q^{48} + 76 q^{49} + 82 q^{50} + 57 q^{51} - 22 q^{52} + 39 q^{53} + 17 q^{54} - 21 q^{55} + 50 q^{56} + 12 q^{57} + 33 q^{58} + 79 q^{59} + 87 q^{60} + 4 q^{61} + 40 q^{62} + 44 q^{63} + 90 q^{64} + 66 q^{65} - 39 q^{66} + 33 q^{67} - 9 q^{68} + 60 q^{69} + 30 q^{70} + 168 q^{71} + 15 q^{72} - 28 q^{73} + 35 q^{74} + 55 q^{75} + 89 q^{76} + 19 q^{77} - 41 q^{78} + 121 q^{79} + 64 q^{80} + 110 q^{81} + 41 q^{82} + 28 q^{84} + 17 q^{85} + 80 q^{86} + 29 q^{87} + 49 q^{88} + 83 q^{89} - 42 q^{90} + 38 q^{91} + 71 q^{92} - q^{93} + 89 q^{94} + 9 q^{95} + 35 q^{96} - 23 q^{97} + 135 q^{98} + 93 q^{99}+O(q^{100}) \) 82 * q + 15 * q^2 + 12 * q^3 + 89 * q^4 + 9 * q^5 + 9 * q^6 + 14 * q^7 + 42 * q^8 + 92 * q^9 + 4 * q^10 + 41 * q^11 + 26 * q^12 + 13 * q^13 + 22 * q^14 + 41 * q^15 + 87 * q^16 + 12 * q^17 + 24 * q^18 + 82 * q^19 + 26 * q^20 + 29 * q^21 + 2 * q^22 + 59 * q^23 + 16 * q^24 + 67 * q^25 + 24 * q^26 + 42 * q^27 - 2 * q^28 + 101 * q^29 - 22 * q^30 + 48 * q^31 + 69 * q^32 + 3 * q^33 + q^34 + 38 * q^35 + 82 * q^36 + 16 * q^37 + 15 * q^38 + 82 * q^39 + 20 * q^40 + 86 * q^41 - q^42 + 9 * q^43 + 82 * q^44 - 8 * q^45 + 43 * q^46 + 24 * q^47 + 34 * q^48 + 76 * q^49 + 82 * q^50 + 57 * q^51 - 22 * q^52 + 39 * q^53 + 17 * q^54 - 21 * q^55 + 50 * q^56 + 12 * q^57 + 33 * q^58 + 79 * q^59 + 87 * q^60 + 4 * q^61 + 40 * q^62 + 44 * q^63 + 90 * q^64 + 66 * q^65 - 39 * q^66 + 33 * q^67 - 9 * q^68 + 60 * q^69 + 30 * q^70 + 168 * q^71 + 15 * q^72 - 28 * q^73 + 35 * q^74 + 55 * q^75 + 89 * q^76 + 19 * q^77 - 41 * q^78 + 121 * q^79 + 64 * q^80 + 110 * q^81 + 41 * q^82 + 28 * q^84 + 17 * q^85 + 80 * q^86 + 29 * q^87 + 49 * q^88 + 83 * q^89 - 42 * q^90 + 38 * q^91 + 71 * q^92 - q^93 + 89 * q^94 + 9 * q^95 + 35 * q^96 - 23 * q^97 + 135 * q^98 + 93 * q^99

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