LMFDB - Embedded newform 409.2.a.b.1.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [409,2,Mod(1,409)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(409, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("409.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 409 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	409.a (trivial)

Newform invariants

comment:select newform

sage:traces = [20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$3.26588144267$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 5 x^{19} - 19 x^{18} + 126 x^{17} + 100 x^{16} - 1283 x^{15} + 247 x^{14} + 6767 x^{13} + \cdots - 4 $ x^20 - 5x^19 - 19x^18 + 126x^17 + 100x^16 - 1283x^15 + 247x^14 + 6767x^13 - 4554x^12 - 19689x^11 + 18771x^10 + 31011x^9 - 35515x^8 - 23548x^7 + 31466x^6 + 5354x^5 - 10552x^4 + 1129x^3 + 523x^2 - 54*x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-0.210883$ of defining polynomial
Character	$\chi$	$=$	409.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.210883 q^{2} -0.465479 q^{3} -1.95553 q^{4} +3.93021 q^{5} +0.0981616 q^{6} +3.14994 q^{7} +0.834154 q^{8} -2.78333 q^{9} -0.828815 q^{10} -1.62625 q^{11} +0.910257 q^{12} -0.227937 q^{13} -0.664268 q^{14} -1.82943 q^{15} +3.73515 q^{16} +3.43456 q^{17} +0.586957 q^{18} -1.46643 q^{19} -7.68564 q^{20} -1.46623 q^{21} +0.342949 q^{22} +8.43301 q^{23} -0.388281 q^{24} +10.4466 q^{25} +0.0480681 q^{26} +2.69202 q^{27} -6.15979 q^{28} -4.57096 q^{29} +0.385796 q^{30} +7.83221 q^{31} -2.45599 q^{32} +0.756986 q^{33} -0.724290 q^{34} +12.3799 q^{35} +5.44288 q^{36} +4.91791 q^{37} +0.309244 q^{38} +0.106100 q^{39} +3.27840 q^{40} -11.6975 q^{41} +0.309203 q^{42} -0.0242311 q^{43} +3.18018 q^{44} -10.9391 q^{45} -1.77838 q^{46} +13.2179 q^{47} -1.73863 q^{48} +2.92211 q^{49} -2.20300 q^{50} -1.59871 q^{51} +0.445738 q^{52} -9.86106 q^{53} -0.567701 q^{54} -6.39151 q^{55} +2.62753 q^{56} +0.682590 q^{57} +0.963938 q^{58} -8.69395 q^{59} +3.57750 q^{60} -8.81199 q^{61} -1.65168 q^{62} -8.76732 q^{63} -6.95237 q^{64} -0.895841 q^{65} -0.159636 q^{66} -4.79208 q^{67} -6.71638 q^{68} -3.92539 q^{69} -2.61071 q^{70} -1.78500 q^{71} -2.32172 q^{72} +5.16567 q^{73} -1.03710 q^{74} -4.86265 q^{75} +2.86764 q^{76} -5.12259 q^{77} -0.0223747 q^{78} -15.4468 q^{79} +14.6799 q^{80} +7.09691 q^{81} +2.46681 q^{82} +10.7852 q^{83} +2.86725 q^{84} +13.4985 q^{85} +0.00510993 q^{86} +2.12768 q^{87} -1.35654 q^{88} +9.64179 q^{89} +2.30686 q^{90} -0.717988 q^{91} -16.4910 q^{92} -3.64573 q^{93} -2.78742 q^{94} -5.76336 q^{95} +1.14321 q^{96} -0.0996286 q^{97} -0.616223 q^{98} +4.52639 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 5 q^{2} + q^{3} + 23 q^{4} + 8 q^{5} - 4 q^{6} + 5 q^{7} + 12 q^{8} + 27 q^{9} - 5 q^{10} + 30 q^{11} - 5 q^{12} - 7 q^{13} + 17 q^{14} + 26 q^{15} + 29 q^{16} + 6 q^{17} - 4 q^{18} + q^{19} + 14 q^{20}+ \cdots + 43 q^{99}+O(q^{100}) $ 20 * q + 5 * q^2 + q^3 + 23 * q^4 + 8 * q^5 - 4 * q^6 + 5 * q^7 + 12 * q^8 + 27 * q^9 - 5 * q^10 + 30 * q^11 - 5 * q^12 - 7 * q^13 + 17 * q^14 + 26 * q^15 + 29 * q^16 + 6 * q^17 - 4 * q^18 + q^19 + 14 * q^20 - 3 * q^21 + q^22 + 37 * q^23 - 17 * q^24 + 14 * q^25 + 11 * q^26 - 5 * q^27 - 6 * q^28 + 12 * q^29 - 6 * q^30 + 10 * q^31 + 24 * q^32 - 22 * q^33 - 11 * q^34 + 12 * q^35 + 21 * q^36 - 4 * q^37 - 7 * q^38 + 31 * q^39 - 34 * q^40 - 7 * q^41 - 35 * q^42 - q^43 + 58 * q^44 - 10 * q^45 - 22 * q^46 + 51 * q^47 - 58 * q^48 + 5 * q^49 + q^50 + 6 * q^51 - 44 * q^52 + 10 * q^53 - 49 * q^54 - 5 * q^55 + 31 * q^56 - 30 * q^57 - 22 * q^58 + 43 * q^59 - 26 * q^60 - 21 * q^61 - 26 * q^62 + 8 * q^63 - 4 * q^64 - 10 * q^65 - 37 * q^66 + 4 * q^67 - 16 * q^68 - 19 * q^69 - 66 * q^70 + 97 * q^71 - 43 * q^72 - 36 * q^73 + 17 * q^74 - 19 * q^75 - 43 * q^76 - 20 * q^77 - 71 * q^78 + 16 * q^79 - 23 * q^80 + 16 * q^81 - 67 * q^82 + 28 * q^83 - 73 * q^84 - 30 * q^85 + 37 * q^86 - 14 * q^87 - 48 * q^88 + 2 * q^89 - 121 * q^90 - 25 * q^91 + 14 * q^92 - 27 * q^93 - 60 * q^94 + 61 * q^95 - 88 * q^96 - 36 * q^97 - 54 * q^98 + 43 * q^99

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 $

$2$	−0.210883	−0.149117	−0.0745584	−	0.997217i	$-0.523755\pi$
$2$	−0.210883	−0.149117	−0.0745584	+	0.997217i	$0.523755\pi$
$3$	−0.465479	−0.268744	−0.134372	−	0.990931i	$-0.542902\pi$
$3$	−0.465479	−0.268744	−0.134372	+	0.990931i	$0.542902\pi$
$4$	−1.95553	−0.977764
$5$	3.93021	1.75764	0.878822	−	0.477150i	$-0.158330\pi$
$5$	3.93021	1.75764	0.878822	+	0.477150i	$0.158330\pi$
$6$	0.0981616	0.0400743
$7$	3.14994	1.19056	0.595282	−	0.803517i	$-0.297040\pi$
$7$	3.14994	1.19056	0.595282	+	0.803517i	$0.297040\pi$
$8$	0.834154	0.294918
$9$	−2.78333	−0.927776
$10$	−0.828815	−0.262094
$11$	−1.62625	−0.490333	−0.245167	−	0.969481i	$-0.578843\pi$
$11$	−1.62625	−0.490333	−0.245167	+	0.969481i	$0.578843\pi$
$12$	0.910257	0.262769
$13$	−0.227937	−0.0632184	−0.0316092	−	0.999500i	$-0.510063\pi$
$13$	−0.227937	−0.0632184	−0.0316092	+	0.999500i	$0.510063\pi$
$14$	−0.664268	−0.177533
$15$	−1.82943	−0.472357
$16$	3.73515	0.933787
$17$	3.43456	0.833003	0.416501	−	0.909135i	$-0.363256\pi$
$17$	3.43456	0.833003	0.416501	+	0.909135i	$0.363256\pi$
$18$	0.586957	0.138347
$19$	−1.46643	−0.336421	−0.168211	−	0.985751i	$-0.553799\pi$
$19$	−1.46643	−0.336421	−0.168211	+	0.985751i	$0.553799\pi$
$20$	−7.68564	−1.71856
$21$	−1.46623	−0.319958
$22$	0.342949	0.0731170
$23$	8.43301	1.75840	0.879202	−	0.476449i	$-0.158076\pi$
$23$	8.43301	1.75840	0.879202	+	0.476449i	$0.158076\pi$
$24$	−0.388281	−0.0792575
$25$	10.4466	2.08931
$26$	0.0480681	0.00942693
$27$	2.69202	0.518079
$28$	−6.15979	−1.16409
$29$	−4.57096	−0.848806	−0.424403	−	0.905473i	$-0.639516\pi$
$29$	−4.57096	−0.848806	−0.424403	+	0.905473i	$0.639516\pi$
$30$	0.385796	0.0704363
$31$	7.83221	1.40671	0.703354	−	0.710840i	$-0.251685\pi$
$31$	7.83221	1.40671	0.703354	+	0.710840i	$0.251685\pi$
$32$	−2.45599	−0.434161
$33$	0.756986	0.131774
$34$	−0.724290	−0.124215
$35$	12.3799	2.09259
$36$	5.44288	0.907147
$37$	4.91791	0.808500	0.404250	−	0.914649i	$-0.367533\pi$
$37$	4.91791	0.808500	0.404250	+	0.914649i	$0.367533\pi$
$38$	0.309244	0.0501660
$39$	0.106100	0.0169896
$40$	3.27840	0.518361
$41$	−11.6975	−1.82684	−0.913422	−	0.407013i	$-0.866570\pi$
$41$	−11.6975	−1.82684	−0.913422	+	0.407013i	$0.866570\pi$
$42$	0.309203	0.0477111
$43$	−0.0242311	−0.00369521	−0.00184761	−	0.999998i	$-0.500588\pi$
$43$	−0.0242311	−0.00369521	−0.00184761	+	0.999998i	$0.500588\pi$
$44$	3.18018	0.479430
$45$	−10.9391	−1.63070
$46$	−1.77838	−0.262208
$47$	13.2179	1.92802	0.964011	−	0.265861i	$-0.0856561\pi$
$47$	13.2179	1.92802	0.964011	+	0.265861i	$0.0856561\pi$
$48$	−1.73863	−0.250950
$49$	2.92211	0.417444
$50$	−2.20300	−0.311551
$51$	−1.59871	−0.223865
$52$	0.445738	0.0618127
$53$	−9.86106	−1.35452	−0.677260	−	0.735743i	$-0.736833\pi$
$53$	−9.86106	−1.35452	−0.677260	+	0.735743i	$0.736833\pi$
$54$	−0.567701	−0.0772543
$55$	−6.39151	−0.861831
$56$	2.62753	0.351119
$57$	0.682590	0.0904113
$58$	0.963938	0.126571
$59$	−8.69395	−1.13186	−0.565928	−	0.824455i	$-0.691482\pi$
$59$	−8.69395	−1.13186	−0.565928	+	0.824455i	$0.691482\pi$
$60$	3.57750	0.461854
$61$	−8.81199	−1.12826	−0.564130	−	0.825686i	$-0.690788\pi$
$61$	−8.81199	−1.12826	−0.564130	+	0.825686i	$0.690788\pi$
$62$	−1.65168	−0.209764
$63$	−8.76732	−1.10458
$64$	−6.95237	−0.869046
$65$	−0.895841	−0.111115
$66$	−0.159636	−0.0196498
$67$	−4.79208	−0.585446	−0.292723	−	0.956197i	$-0.594561\pi$
$67$	−4.79208	−0.585446	−0.292723	+	0.956197i	$0.594561\pi$
$68$	−6.71638	−0.814480
$69$	−3.92539	−0.472561
$70$	−2.61071	−0.312040
$71$	−1.78500	−0.211841	−0.105920	−	0.994375i	$-0.533779\pi$
$71$	−1.78500	−0.211841	−0.105920	+	0.994375i	$0.533779\pi$
$72$	−2.32172	−0.273618
$73$	5.16567	0.604597	0.302298	−	0.953213i	$-0.402246\pi$
$73$	5.16567	0.604597	0.302298	+	0.953213i	$0.402246\pi$
$74$	−1.03710	−0.120561
$75$	−4.86265	−0.561490
$76$	2.86764	0.328940
$77$	−5.12259	−0.583774
$78$	−0.0223747	−0.00253343
$79$	−15.4468	−1.73790	−0.868951	−	0.494899i	$-0.835205\pi$
$79$	−15.4468	−1.73790	−0.868951	+	0.494899i	$0.835205\pi$
$80$	14.6799	1.64126
$81$	7.09691	0.788546
$82$	2.46681	0.272413
$83$	10.7852	1.18383	0.591913	−	0.806002i	$-0.298373\pi$
$83$	10.7852	1.18383	0.591913	+	0.806002i	$0.298373\pi$
$84$	2.86725	0.312843
$85$	13.4985	1.46412
$86$	0.00510993	0.000551018 0
$87$	2.12768	0.228112
$88$	−1.35654	−0.144608
$89$	9.64179	1.02203	0.511014	−	0.859572i	$-0.329270\pi$
$89$	9.64179	1.02203	0.511014	+	0.859572i	$0.329270\pi$
$90$	2.30686	0.243165
$91$	−0.717988	−0.0752656
$92$	−16.4910	−1.71930
$93$	−3.64573	−0.378045
$94$	−2.78742	−0.287501
$95$	−5.76336	−0.591308
$96$	1.14321	0.116678
$97$	−0.0996286	−0.0101158	−0.00505788	−	0.999987i	$-0.501610\pi$
$97$	−0.0996286	−0.0101158	−0.00505788	+	0.999987i	$0.501610\pi$
$98$	−0.616223	−0.0622480
$99$	4.52639	0.454920
$100$	−20.4285	−2.04285
$101$	3.10440	0.308899	0.154450	−	0.988001i	$-0.450640\pi$
$101$	3.10440	0.308899	0.154450	+	0.988001i	$0.450640\pi$
$102$	0.337142	0.0333820
$103$	−3.96285	−0.390472	−0.195236	−	0.980756i	$-0.562547\pi$
$103$	−3.96285	−0.390472	−0.195236	+	0.980756i	$0.562547\pi$
$104$	−0.190135	−0.0186442
$105$	−5.76259	−0.562371
$106$	2.07953	0.201982
$107$	0.0231899	0.00224185	0.00112093	−	0.999999i	$-0.499643\pi$
$107$	0.0231899	0.00224185	0.00112093	+	0.999999i	$0.499643\pi$
$108$	−5.26432	−0.506559
$109$	−20.0715	−1.92250	−0.961249	−	0.275681i	$-0.911097\pi$
$109$	−20.0715	−1.92250	−0.961249	+	0.275681i	$0.911097\pi$
$110$	1.34786	0.128514
$111$	−2.28919	−0.217280
$112$	11.7655	1.11173
$113$	−1.37218	−0.129084	−0.0645419	−	0.997915i	$-0.520559\pi$
$113$	−1.37218	−0.129084	−0.0645419	+	0.997915i	$0.520559\pi$
$114$	−0.143947	−0.0134818
$115$	33.1435	3.09065
$116$	8.93864	0.829932
$117$	0.634424	0.0586526
$118$	1.83341	0.168779
$119$	10.8186	0.991744
$120$	−1.52603	−0.139306
$121$	−8.35530	−0.759573
$122$	1.85830	0.168242
$123$	5.44495	0.490954
$124$	−15.3161	−1.37543
$125$	21.4061	1.91462
$126$	1.84888	0.164711
$127$	−6.36447	−0.564755	−0.282378	−	0.959303i	$-0.591123\pi$
$127$	−6.36447	−0.564755	−0.282378	+	0.959303i	$0.591123\pi$
$128$	6.37811	0.563751
$129$	0.0112791	0.000993067 0
$130$	0.188918	0.0165692
$131$	1.24267	0.108572	0.0542861	−	0.998525i	$-0.482712\pi$
$131$	1.24267	0.108572	0.0542861	+	0.998525i	$0.482712\pi$
$132$	−1.48031	−0.128844
$133$	−4.61915	−0.400531
$134$	1.01057	0.0872998
$135$	10.5802	0.910598
$136$	2.86495	0.245667
$137$	−14.0313	−1.19877	−0.599386	−	0.800460i	$-0.704588\pi$
$137$	−14.0313	−1.19877	−0.599386	+	0.800460i	$0.704588\pi$
$138$	0.827798	0.0704668
$139$	2.08333	0.176706	0.0883530	−	0.996089i	$-0.471840\pi$
$139$	2.08333	0.176706	0.0883530	+	0.996089i	$0.471840\pi$
$140$	−24.2093	−2.04606
$141$	−6.15264	−0.518145
$142$	0.376426	0.0315890
$143$	0.370683	0.0309981
$144$	−10.3961	−0.866346
$145$	−17.9648	−1.49190
$146$	−1.08935	−0.0901555
$147$	−1.36018	−0.112186
$148$	−9.61712	−0.790522
$149$	−6.12739	−0.501975	−0.250988	−	0.967990i	$-0.580755\pi$
$149$	−6.12739	−0.501975	−0.250988	+	0.967990i	$0.580755\pi$
$150$	1.02545	0.0837277
$151$	−5.90580	−0.480607	−0.240303	−	0.970698i	$-0.577247\pi$
$151$	−5.90580	−0.480607	−0.240303	+	0.970698i	$0.577247\pi$
$152$	−1.22322	−0.0992166
$153$	−9.55951	−0.772840
$154$	1.08027	0.0870505
$155$	30.7822	2.47249
$156$	−0.207482	−0.0166118
$157$	16.6845	1.33157	0.665783	−	0.746145i	$-0.268098\pi$
$157$	16.6845	1.33157	0.665783	+	0.746145i	$0.268098\pi$
$158$	3.25747	0.259150
$159$	4.59012	0.364020
$160$	−9.65254	−0.763101
$161$	26.5635	2.09349
$162$	−1.49662	−0.117585
$163$	−16.8844	−1.32249	−0.661245	−	0.750170i	$-0.729972\pi$
$163$	−16.8844	−1.32249	−0.661245	+	0.750170i	$0.729972\pi$
$164$	22.8748	1.78622
$165$	2.97511	0.231612
$166$	−2.27441	−0.176528
$167$	−10.6104	−0.821061	−0.410530	−	0.911847i	$-0.634656\pi$
$167$	−10.6104	−0.821061	−0.410530	+	0.911847i	$0.634656\pi$
$168$	−1.22306	−0.0943612
$169$	−12.9480	−0.996003
$170$	−2.84661	−0.218325
$171$	4.08155	0.312124
$172$	0.0473846	0.00361304
$173$	−20.2866	−1.54237	−0.771183	−	0.636614i	$-0.780335\pi$
$173$	−20.2866	−1.54237	−0.771183	+	0.636614i	$0.780335\pi$
$174$	−0.448693	−0.0340153
$175$	32.9060	2.48746
$176$	−6.07429	−0.457867
$177$	4.04685	0.304180
$178$	−2.03329	−0.152402
$179$	19.8818	1.48604	0.743019	−	0.669270i	$-0.233393\pi$
$179$	19.8818	1.48604	0.743019	+	0.669270i	$0.233393\pi$
$180$	21.3917	1.59444
$181$	−8.92116	−0.663104	−0.331552	−	0.943437i	$-0.607572\pi$
$181$	−8.92116	−0.663104	−0.331552	+	0.943437i	$0.607572\pi$
$182$	0.151412	0.0112234
$183$	4.10179	0.303213
$184$	7.03443	0.518585
$185$	19.3284	1.42105
$186$	0.768823	0.0563728
$187$	−5.58546	−0.408449
$188$	−25.8479	−1.88515
$189$	8.47969	0.616807
$190$	1.21539	0.0881740
$191$	14.0469	1.01640	0.508199	−	0.861240i	$-0.330311\pi$
$191$	14.0469	1.01640	0.508199	+	0.861240i	$0.330311\pi$
$192$	3.23618	0.233551
$193$	−2.56573	−0.184685	−0.0923425	−	0.995727i	$-0.529435\pi$
$193$	−2.56573	−0.184685	−0.0923425	+	0.995727i	$0.529435\pi$
$194$	0.0210100	0.00150843
$195$	0.416995	0.0298616
$196$	−5.71427	−0.408162
$197$	14.2539	1.01555	0.507776	−	0.861489i	$-0.330468\pi$
$197$	14.2539	1.01555	0.507776	+	0.861489i	$0.330468\pi$
$198$	−0.954540	−0.0678362
$199$	−2.78595	−0.197491	−0.0987454	−	0.995113i	$-0.531483\pi$
$199$	−2.78595	−0.197491	−0.0987454	+	0.995113i	$0.531483\pi$
$200$	8.71403	0.616175
$201$	2.23061	0.157335
$202$	−0.654666	−0.0460621
$203$	−14.3982	−1.01056
$204$	3.12633	0.218887
$205$	−45.9737	−3.21094
$206$	0.835699	0.0582259
$207$	−23.4718	−1.63141
$208$	−0.851379	−0.0590325
$209$	2.38478	0.164958
$210$	1.21523	0.0838590
$211$	−10.8396	−0.746231	−0.373115	−	0.927785i	$-0.621710\pi$
$211$	−10.8396	−0.746231	−0.373115	+	0.927785i	$0.621710\pi$
$212$	19.2836	1.32440
$213$	0.830880	0.0569310
$214$	−0.00489035	−0.000334298 0
$215$	−0.0952334	−0.00649486
$216$	2.24556	0.152791
$217$	24.6710	1.67478
$218$	4.23273	0.286677
$219$	−2.40451	−0.162482
$220$	12.4988	0.842668
$221$	−0.782864	−0.0526611
$222$	0.482750	0.0324001
$223$	4.20807	0.281793	0.140897	−	0.990024i	$-0.455002\pi$
$223$	4.20807	0.281793	0.140897	+	0.990024i	$0.455002\pi$
$224$	−7.73621	−0.516897
$225$	−29.0762	−1.93841
$226$	0.289369	0.0192486
$227$	18.1139	1.20226	0.601130	−	0.799151i	$-0.294717\pi$
$227$	18.1139	1.20226	0.601130	+	0.799151i	$0.294717\pi$
$228$	−1.33482	−0.0884009
$229$	−10.3201	−0.681972	−0.340986	−	0.940068i	$-0.610761\pi$
$229$	−10.3201	−0.681972	−0.340986	+	0.940068i	$0.610761\pi$
$230$	−6.98940	−0.460868
$231$	2.38446	0.156886
$232$	−3.81288	−0.250328
$233$	9.57148	0.627048	0.313524	−	0.949580i	$-0.398490\pi$
$233$	9.57148	0.627048	0.313524	+	0.949580i	$0.398490\pi$
$234$	−0.133789	−0.00874608
$235$	51.9490	3.38878
$236$	17.0013	1.10669
$237$	7.19016	0.467051
$238$	−2.28147	−0.147886
$239$	22.1946	1.43565	0.717824	−	0.696225i	$-0.245138\pi$
$239$	22.1946	1.43565	0.717824	+	0.696225i	$0.245138\pi$
$240$	−6.83319	−0.441081
$241$	21.1582	1.36292	0.681461	−	0.731855i	$-0.261345\pi$
$241$	21.1582	1.36292	0.681461	+	0.731855i	$0.261345\pi$
$242$	1.76199	0.113265
$243$	−11.3795	−0.729996
$244$	17.2321	1.10317
$245$	11.4845	0.733718
$246$	−1.14825	−0.0732095
$247$	0.334253	0.0212680
$248$	6.53327	0.414863
$249$	−5.02027	−0.318147
$250$	−4.51418	−0.285502
$251$	−16.6302	−1.04969	−0.524844	−	0.851199i	$-0.675876\pi$
$251$	−16.6302	−1.04969	−0.524844	+	0.851199i	$0.675876\pi$
$252$	17.1447	1.08002
$253$	−13.7142	−0.862204
$254$	1.34216	0.0842145
$255$	−6.28328	−0.393475
$256$	12.5597	0.784982
$257$	13.7086	0.855117	0.427559	−	0.903988i	$-0.359374\pi$
$257$	13.7086	0.855117	0.427559	+	0.903988i	$0.359374\pi$
$258$	−0.00237857	−0.000148083 0
$259$	15.4911	0.962572
$260$	1.75184	0.108645
$261$	12.7225	0.787502
$262$	−0.262057	−0.0161899
$263$	−9.13520	−0.563301	−0.281650	−	0.959517i	$-0.590882\pi$
$263$	−9.13520	−0.563301	−0.281650	+	0.959517i	$0.590882\pi$
$264$	0.631443	0.0388626
$265$	−38.7560	−2.38076
$266$	0.974100	0.0597259
$267$	−4.48805	−0.274664
$268$	9.37105	0.572428
$269$	21.3210	1.29997	0.649983	−	0.759949i	$-0.274776\pi$
$269$	21.3210	1.29997	0.649983	+	0.759949i	$0.274776\pi$
$270$	−2.23118	−0.135786
$271$	−11.7560	−0.714125	−0.357062	−	0.934081i	$-0.616222\pi$
$271$	−11.7560	−0.714125	−0.357062	+	0.934081i	$0.616222\pi$
$272$	12.8286	0.777847
$273$	0.334208	0.0202272
$274$	2.95895	0.178757
$275$	−16.9887	−1.02446
$276$	7.67621	0.462054
$277$	−30.9171	−1.85763	−0.928814	−	0.370546i	$-0.879171\pi$
$277$	−30.9171	−1.85763	−0.928814	+	0.370546i	$0.879171\pi$
$278$	−0.439340	−0.0263498
$279$	−21.7996	−1.30511
$280$	10.3268	0.617142
$281$	6.30130	0.375904	0.187952	−	0.982178i	$-0.439815\pi$
$281$	6.30130	0.375904	0.187952	+	0.982178i	$0.439815\pi$
$282$	1.29749	0.0772642
$283$	18.2263	1.08344	0.541722	−	0.840558i	$-0.317773\pi$
$283$	18.2263	1.08344	0.541722	+	0.840558i	$0.317773\pi$
$284$	3.49062	0.207130
$285$	2.68272	0.158911
$286$	−0.0781708	−0.00462234
$287$	−36.8464	−2.17498
$288$	6.83582	0.402805
$289$	−5.20381	−0.306106
$290$	3.78848	0.222467
$291$	0.0463750	0.00271855
$292$	−10.1016	−0.591153
$293$	10.9586	0.640209	0.320104	−	0.947382i	$-0.396282\pi$
$293$	10.9586	0.640209	0.320104	+	0.947382i	$0.396282\pi$
$294$	0.286839	0.0167288
$295$	−34.1690	−1.98940
$296$	4.10230	0.238441
$297$	−4.37790	−0.254031
$298$	1.29216	0.0748529
$299$	−1.92220	−0.111164
$300$	9.50905	0.549005
$301$	−0.0763265	−0.00439939
$302$	1.24543	0.0716666
$303$	−1.44503	−0.0830150
$304$	−5.47732	−0.314146
$305$	−34.6330	−1.98308
$306$	2.01594	0.115243
$307$	18.7053	1.06757	0.533784	−	0.845621i	$-0.320770\pi$
$307$	18.7053	1.06757	0.533784	+	0.845621i	$0.320770\pi$
$308$	10.0174	0.570793
$309$	1.84463	0.104937
$310$	−6.49145	−0.368690
$311$	−1.19994	−0.0680426	−0.0340213	−	0.999421i	$-0.510831\pi$
$311$	−1.19994	−0.0680426	−0.0340213	+	0.999421i	$0.510831\pi$
$312$	0.0885037	0.00501054
$313$	−5.53314	−0.312752	−0.156376	−	0.987698i	$-0.549981\pi$
$313$	−5.53314	−0.312752	−0.156376	+	0.987698i	$0.549981\pi$
$314$	−3.51847	−0.198559
$315$	−34.4574	−1.94145
$316$	30.2067	1.69926
$317$	−28.3479	−1.59217	−0.796087	−	0.605182i	$-0.793100\pi$
$317$	−28.3479	−1.59217	−0.796087	+	0.605182i	$0.793100\pi$
$318$	−0.967977	−0.0542815
$319$	7.43353	0.416198
$320$	−27.3243	−1.52747
$321$	−0.0107944	−0.000602485 0
$322$	−5.60178	−0.312175
$323$	−5.03652	−0.280240
$324$	−13.8782	−0.771012
$325$	−2.38116	−0.132083
$326$	3.56064	0.197205
$327$	9.34285	0.516661
$328$	−9.75753	−0.538769
$329$	41.6354	2.29544
$330$	−0.627401	−0.0345373
$331$	26.6467	1.46463	0.732317	−	0.680964i	$-0.238439\pi$
$331$	26.6467	1.46463	0.732317	+	0.680964i	$0.238439\pi$
$332$	−21.0907	−1.15750
$333$	−13.6882	−0.750107
$334$	2.23756	0.122434
$335$	−18.8339	−1.02900
$336$	−5.47659	−0.298772
$337$	9.19382	0.500820	0.250410	−	0.968140i	$-0.419435\pi$
$337$	9.19382	0.500820	0.250410	+	0.968140i	$0.419435\pi$
$338$	2.73052	0.148521
$339$	0.638721	0.0346905
$340$	−26.3968	−1.43157
$341$	−12.7372	−0.689755
$342$	−0.860729	−0.0465429
$343$	−12.8451	−0.693570
$344$	−0.0202125	−0.00108978
$345$	−15.4276	−0.830594
$346$	4.27811	0.229993
$347$	16.3109	0.875614	0.437807	−	0.899069i	$-0.355755\pi$
$347$	16.3109	0.875614	0.437807	+	0.899069i	$0.355755\pi$
$348$	−4.16075	−0.223040
$349$	−2.37467	−0.127113	−0.0635566	−	0.997978i	$-0.520244\pi$
$349$	−2.37467	−0.127113	−0.0635566	+	0.997978i	$0.520244\pi$
$350$	−6.93932	−0.370922
$351$	−0.613611	−0.0327521
$352$	3.99405	0.212884
$353$	−4.14047	−0.220375	−0.110187	−	0.993911i	$-0.535145\pi$
$353$	−4.14047	−0.220375	−0.110187	+	0.993911i	$0.535145\pi$
$354$	−0.853412	−0.0453583
$355$	−7.01543	−0.372340
$356$	−18.8548	−0.999302
$357$	−5.03585	−0.266526
$358$	−4.19274	−0.221593
$359$	−2.76532	−0.145948	−0.0729740	−	0.997334i	$-0.523249\pi$
$359$	−2.76532	−0.145948	−0.0729740	+	0.997334i	$0.523249\pi$
$360$	−9.12487	−0.480923
$361$	−16.8496	−0.886821
$362$	1.88132	0.0988800
$363$	3.88922	0.204131
$364$	1.40405	0.0735920
$365$	20.3022	1.06267
$366$	−0.864999	−0.0452142
$367$	16.0284	0.836674	0.418337	−	0.908292i	$-0.362613\pi$
$367$	16.0284	0.836674	0.418337	+	0.908292i	$0.362613\pi$
$368$	31.4985	1.64198
$369$	32.5580	1.69490
$370$	−4.07604	−0.211903
$371$	−31.0617	−1.61264
$372$	7.12933	0.369639
$373$	8.14927	0.421953	0.210976	−	0.977491i	$-0.432336\pi$
$373$	8.14927	0.421953	0.210976	+	0.977491i	$0.432336\pi$
$374$	1.17788	0.0609066
$375$	−9.96409	−0.514543
$376$	11.0257	0.568608
$377$	1.04189	0.0536601
$378$	−1.78822	−0.0919763
$379$	−4.81160	−0.247155	−0.123578	−	0.992335i	$-0.539437\pi$
$379$	−4.81160	−0.247155	−0.123578	+	0.992335i	$0.539437\pi$
$380$	11.2704	0.578160
$381$	2.96253	0.151775
$382$	−2.96225	−0.151562
$383$	−8.13904	−0.415886	−0.207943	−	0.978141i	$-0.566677\pi$
$383$	−8.13904	−0.415886	−0.207943	+	0.978141i	$0.566677\pi$
$384$	−2.96888	−0.151505
$385$	−20.1329	−1.02607
$386$	0.541068	0.0275396
$387$	0.0674432	0.00342833
$388$	0.194827	0.00989082
$389$	20.0557	1.01686	0.508431	−	0.861103i	$-0.330226\pi$
$389$	20.0557	1.01686	0.508431	+	0.861103i	$0.330226\pi$
$390$	−0.0879372	−0.00445287
$391$	28.9637	1.46476
$392$	2.43749	0.123112
$393$	−0.578435	−0.0291782
$394$	−3.00591	−0.151436
$395$	−60.7092	−3.05461
$396$	−8.85149	−0.444804
$397$	−6.19867	−0.311102	−0.155551	−	0.987828i	$-0.549715\pi$
$397$	−6.19867	−0.311102	−0.155551	+	0.987828i	$0.549715\pi$
$398$	0.587510	0.0294492
$399$	2.15012	0.107640
$400$	39.0194	1.95097
$401$	−32.1589	−1.60594	−0.802970	−	0.596020i	$-0.796748\pi$
$401$	−32.1589	−1.60594	−0.802970	+	0.596020i	$0.796748\pi$
$402$	−0.470398	−0.0234613
$403$	−1.78525	−0.0889298
$404$	−6.07074	−0.302031
$405$	27.8923	1.38598
$406$	3.03634	0.150691
$407$	−7.99777	−0.396435
$408$	−1.33357	−0.0660217
$409$	1.00000	0.0494468
$409$	1.00000	0.0494468
$410$	9.69507	0.478805
$411$	6.53126	0.322163
$412$	7.74947	0.381789
$413$	−27.3854	−1.34755
$414$	4.94981	0.243270
$415$	42.3880	2.08074
$416$	0.559811	0.0274470
$417$	−0.969748	−0.0474888
$418$	−0.502909	−0.0245981
$419$	15.4478	0.754674	0.377337	−	0.926076i	$-0.376840\pi$
$419$	15.4478	0.754674	0.377337	+	0.926076i	$0.376840\pi$
$420$	11.2689	0.549867
$421$	17.5930	0.857428	0.428714	−	0.903440i	$-0.358967\pi$
$421$	17.5930	0.857428	0.428714	+	0.903440i	$0.358967\pi$
$422$	2.28589	0.111276
$423$	−36.7897	−1.78877
$424$	−8.22564	−0.399472
$425$	35.8793	1.74040
$426$	−0.175219	−0.00848937
$427$	−27.7572	−1.34327
$428$	−0.0453485	−0.00219200
$429$	−0.172545	−0.00833057
$430$	0.0200831	0.000968493 0
$431$	40.0514	1.92921	0.964603	−	0.263706i	$-0.0849448\pi$
$431$	40.0514	1.92921	0.964603	+	0.263706i	$0.0849448\pi$
$432$	10.0551	0.483776
$433$	16.1485	0.776049	0.388025	−	0.921649i	$-0.373158\pi$
$433$	16.1485	0.776049	0.388025	+	0.921649i	$0.373158\pi$
$434$	−5.20269	−0.249737
$435$	8.36225	0.400939
$436$	39.2503	1.87975
$437$	−12.3664	−0.591564
$438$	0.507071	0.0242288
$439$	−18.1075	−0.864226	−0.432113	−	0.901819i	$-0.642232\pi$
$439$	−18.1075	−0.864226	−0.432113	+	0.901819i	$0.642232\pi$
$440$	−5.33150	−0.254169
$441$	−8.13320	−0.387295
$442$	0.165093	0.00785266
$443$	−23.9859	−1.13960	−0.569802	−	0.821782i	$-0.692980\pi$
$443$	−23.9859	−1.13960	−0.569802	+	0.821782i	$0.692980\pi$
$444$	4.47657	0.212448
$445$	37.8943	1.79636
$446$	−0.887410	−0.0420201
$447$	2.85217	0.134903
$448$	−21.8995	−1.03466
$449$	−0.573472	−0.0270638	−0.0135319	−	0.999908i	$-0.504307\pi$
$449$	−0.573472	−0.0270638	−0.0135319	+	0.999908i	$0.504307\pi$
$450$	6.13168	0.289050
$451$	19.0231	0.895763
$452$	2.68334	0.126213
$453$	2.74902	0.129160
$454$	−3.81991	−0.179277
$455$	−2.82184	−0.132290
$456$	0.569385	0.0266639
$457$	−7.96285	−0.372486	−0.186243	−	0.982504i	$-0.559631\pi$
$457$	−7.96285	−0.372486	−0.186243	+	0.982504i	$0.559631\pi$
$458$	2.17634	0.101694
$459$	9.24589	0.431561
$460$	−64.8131	−3.02193
$461$	−14.3699	−0.669275	−0.334637	−	0.942347i	$-0.608614\pi$
$461$	−14.3699	−0.669275	−0.334637	+	0.942347i	$0.608614\pi$
$462$	−0.502842	−0.0233943
$463$	−12.1041	−0.562524	−0.281262	−	0.959631i	$-0.590753\pi$
$463$	−12.1041	−0.562524	−0.281262	+	0.959631i	$0.590753\pi$
$464$	−17.0732	−0.792604
$465$	−14.3285	−0.664468
$466$	−2.01846	−0.0935035
$467$	22.2189	1.02817	0.514085	−	0.857740i	$-0.328132\pi$
$467$	22.2189	1.02817	0.514085	+	0.857740i	$0.328132\pi$
$468$	−1.24063	−0.0573484
$469$	−15.0948	−0.697011
$470$	−10.9552	−0.505324
$471$	−7.76627	−0.357851
$472$	−7.25209	−0.333805
$473$	0.0394059	0.00181188
$474$	−1.51628	−0.0696452
$475$	−15.3191	−0.702888
$476$	−21.1562	−0.969691
$477$	27.4466	1.25669
$478$	−4.68046	−0.214079
$479$	−24.9655	−1.14070	−0.570352	−	0.821401i	$-0.693193\pi$
$479$	−24.9655	−1.14070	−0.570352	+	0.821401i	$0.693193\pi$
$480$	4.49306	0.205079
$481$	−1.12098	−0.0511121
$482$	−4.46191	−0.203234
$483$	−12.3647	−0.562615
$484$	16.3390	0.742683
$485$	−0.391561	−0.0177799
$486$	2.39975	0.108855
$487$	13.7316	0.622237	0.311119	−	0.950371i	$-0.399296\pi$
$487$	13.7316	0.622237	0.311119	+	0.950371i	$0.399296\pi$
$488$	−7.35055	−0.332744
$489$	7.85934	0.355412
$490$	−2.42189	−0.109410
$491$	−28.0957	−1.26794	−0.633971	−	0.773357i	$-0.718576\pi$
$491$	−28.0957	−1.26794	−0.633971	+	0.773357i	$0.718576\pi$
$492$	−10.6477	−0.480038
$493$	−15.6992	−0.707057
$494$	−0.0704883	−0.00317142
$495$	17.7897	0.799587
$496$	29.2545	1.31356
$497$	−5.62264	−0.252210
$498$	1.05869	0.0474410
$499$	−5.81559	−0.260341	−0.130171	−	0.991492i	$-0.541553\pi$
$499$	−5.81559	−0.260341	−0.130171	+	0.991492i	$0.541553\pi$
$500$	−41.8602	−1.87205
$501$	4.93894	0.220655
$502$	3.50702	0.156526
$503$	−10.5757	−0.471545	−0.235773	−	0.971808i	$-0.575762\pi$
$503$	−10.5757	−0.471545	−0.235773	+	0.971808i	$0.575762\pi$
$504$	−7.31329	−0.325760
$505$	12.2010	0.542935
$506$	2.89209	0.128569
$507$	6.02704	0.267670
$508$	12.4459	0.552198
$509$	−8.60097	−0.381231	−0.190616	−	0.981665i	$-0.561048\pi$
$509$	−8.60097	−0.381231	−0.190616	+	0.981665i	$0.561048\pi$
$510$	1.32504	0.0586737
$511$	16.2716	0.719811
$512$	−15.4048	−0.680805
$513$	−3.94764	−0.174293
$514$	−2.89091	−0.127512
$515$	−15.5748	−0.686310
$516$	−0.0220565	−0.000970985 0
$517$	−21.4956	−0.945374
$518$	−3.26681	−0.143536
$519$	9.44301	0.414502
$520$	−0.747269	−0.0327699
$521$	17.4050	0.762527	0.381263	−	0.924466i	$-0.375489\pi$
$521$	17.4050	0.762527	0.381263	+	0.924466i	$0.375489\pi$
$522$	−2.68296	−0.117430
$523$	14.8430	0.649037	0.324519	−	0.945879i	$-0.394798\pi$
$523$	14.8430	0.649037	0.324519	+	0.945879i	$0.394798\pi$
$524$	−2.43007	−0.106158
$525$	−15.3170	−0.668491
$526$	1.92646	0.0839976
$527$	26.9002	1.17179
$528$	2.82745	0.123049
$529$	48.1157	2.09199
$530$	8.17299	0.355012
$531$	24.1981	1.05011
$532$	9.03288	0.391625
$533$	2.66630	0.115490
$534$	0.946454	0.0409571
$535$	0.0911411	0.00394038
$536$	−3.99733	−0.172658
$537$	−9.25458	−0.399364
$538$	−4.49624	−0.193847
$539$	−4.75209	−0.204687
$540$	−20.6899	−0.890350
$541$	14.0776	0.605242	0.302621	−	0.953111i	$-0.402138\pi$
$541$	14.0776	0.605242	0.302621	+	0.953111i	$0.402138\pi$
$542$	2.47914	0.106488
$543$	4.15261	0.178206
$544$	−8.43523	−0.361658
$545$	−78.8851	−3.37907
$546$	−0.0704789	−0.00301622
$547$	1.12385	0.0480522	0.0240261	−	0.999711i	$-0.492352\pi$
$547$	1.12385	0.0480522	0.0240261	+	0.999711i	$0.492352\pi$
$548$	27.4385	1.17212
$549$	24.5267	1.04677
$550$	3.58263	0.152764
$551$	6.70297	0.285556
$552$	−3.27438	−0.139367
$553$	−48.6565	−2.06908
$554$	6.51989	0.277004
$555$	−8.99698	−0.381900
$556$	−4.07402	−0.172777
$557$	−32.8052	−1.39000	−0.695001	−	0.719009i	$-0.744596\pi$
$557$	−32.8052	−1.39000	−0.695001	+	0.719009i	$0.744596\pi$
$558$	4.59717	0.194614
$559$	0.00552317	0.000233605 0
$560$	46.2408	1.95403
$561$	2.59991	0.109768
$562$	−1.32884	−0.0560536
$563$	−3.10850	−0.131008	−0.0655038	−	0.997852i	$-0.520865\pi$
$563$	−3.10850	−0.131008	−0.0655038	+	0.997852i	$0.520865\pi$
$564$	12.0317	0.506624
$565$	−5.39295	−0.226883
$566$	−3.84362	−0.161560
$567$	22.3548	0.938815
$568$	−1.48897	−0.0624756
$569$	6.98241	0.292718	0.146359	−	0.989232i	$-0.453245\pi$
$569$	6.98241	0.292718	0.146359	+	0.989232i	$0.453245\pi$
$570$	−0.565741	−0.0236963
$571$	−23.2019	−0.970968	−0.485484	−	0.874246i	$-0.661357\pi$
$571$	−23.2019	−0.970968	−0.485484	+	0.874246i	$0.661357\pi$
$572$	−0.724882	−0.0303088
$573$	−6.53853	−0.273151
$574$	7.77029	0.324326
$575$	88.0959	3.67385
$576$	19.3507	0.806281
$577$	12.8872	0.536500	0.268250	−	0.963349i	$-0.413555\pi$
$577$	12.8872	0.536500	0.268250	+	0.963349i	$0.413555\pi$
$578$	1.09739	0.0456456
$579$	1.19429	0.0496330
$580$	35.1307	1.45872
$581$	33.9726	1.40942
$582$	−0.00977970	−0.000405382 0
$583$	16.0366	0.664167
$584$	4.30897	0.178306
$585$	2.49342	0.103090
$586$	−2.31098	−0.0954659
$587$	−42.5802	−1.75747	−0.878736	−	0.477308i	$-0.841613\pi$
$587$	−42.5802	−1.75747	−0.878736	+	0.477308i	$0.841613\pi$
$588$	2.65987	0.109691
$589$	−11.4854	−0.473246
$590$	7.20567	0.296653
$591$	−6.63491	−0.272924
$592$	18.3691	0.754967
$593$	31.4923	1.29323	0.646617	−	0.762815i	$-0.276183\pi$
$593$	31.4923	1.29323	0.646617	+	0.762815i	$0.276183\pi$
$594$	0.923225	0.0378804
$595$	42.5196	1.74313
$596$	11.9823	0.490813
$597$	1.29680	0.0530745
$598$	0.405359	0.0165764
$599$	5.46454	0.223275	0.111637	−	0.993749i	$-0.464390\pi$
$599$	5.46454	0.223275	0.111637	+	0.993749i	$0.464390\pi$
$600$	−4.05620	−0.165594
$601$	−40.9777	−1.67152	−0.835758	−	0.549098i	$-0.814971\pi$
$601$	−40.9777	−1.67152	−0.835758	+	0.549098i	$0.814971\pi$
$602$	0.0160960	0.000656023 0
$603$	13.3379	0.543163
$604$	11.5490	0.469920
$605$	−32.8381	−1.33506
$606$	0.304733	0.0123789
$607$	29.4394	1.19491	0.597453	−	0.801904i	$-0.296179\pi$
$607$	29.4394	1.19491	0.597453	+	0.801904i	$0.296179\pi$
$608$	3.60152	0.146061
$609$	6.70208	0.271582
$610$	7.30350	0.295710
$611$	−3.01284	−0.121887
$612$	18.6939	0.755656
$613$	37.2544	1.50469	0.752346	−	0.658768i	$-0.228922\pi$
$613$	37.2544	1.50469	0.752346	+	0.658768i	$0.228922\pi$
$614$	−3.94463	−0.159192
$615$	21.3998	0.862923
$616$	−4.27303	−0.172165
$617$	37.4244	1.50665	0.753326	−	0.657648i	$-0.228448\pi$
$617$	37.4244	1.50665	0.753326	+	0.657648i	$0.228448\pi$
$618$	−0.389000	−0.0156479
$619$	3.61457	0.145282	0.0726408	−	0.997358i	$-0.476857\pi$
$619$	3.61457	0.145282	0.0726408	+	0.997358i	$0.476857\pi$
$620$	−60.1956	−2.41751
$621$	22.7018	0.910993
$622$	0.253048	0.0101463
$623$	30.3711	1.21679
$624$	0.396299	0.0158647
$625$	31.8977	1.27591
$626$	1.16685	0.0466365
$627$	−1.11006	−0.0443317
$628$	−32.6270	−1.30196
$629$	16.8909	0.673483
$630$	7.26648	0.289503
$631$	−37.8129	−1.50531	−0.752654	−	0.658416i	$-0.771227\pi$
$631$	−37.8129	−1.50531	−0.752654	+	0.658416i	$0.771227\pi$
$632$	−12.8850	−0.512538
$633$	5.04562	0.200545
$634$	5.97808	0.237420
$635$	−25.0137	−0.992638
$636$	−8.97610	−0.355926
$637$	−0.666058	−0.0263902
$638$	−1.56761	−0.0620621
$639$	4.96825	0.196541
$640$	25.0673	0.990873
$641$	22.4458	0.886558	0.443279	−	0.896384i	$-0.353815\pi$
$641$	22.4458	0.886558	0.443279	+	0.896384i	$0.353815\pi$
$642$	0.00227636	8.98406e−5 0
$643$	−13.7064	−0.540528	−0.270264	−	0.962786i	$-0.587111\pi$
$643$	−13.7064	−0.540528	−0.270264	+	0.962786i	$0.587111\pi$
$644$	−51.9456	−2.04694
$645$	0.0443291	0.00174546
$646$	1.06212	0.0417885
$647$	46.4281	1.82528	0.912638	−	0.408768i	$-0.134042\pi$
$647$	46.4281	1.82528	0.912638	+	0.408768i	$0.134042\pi$
$648$	5.91991	0.232556
$649$	14.1386	0.554987
$650$	0.502146	0.0196958
$651$	−11.4838	−0.450087
$652$	33.0180	1.29308
$653$	8.77512	0.343397	0.171698	−	0.985150i	$-0.445075\pi$
$653$	8.77512	0.343397	0.171698	+	0.985150i	$0.445075\pi$
$654$	−1.97025	−0.0770428
$655$	4.88394	0.190831
$656$	−43.6919	−1.70588
$657$	−14.3778	−0.560930
$658$	−8.78021	−0.342288
$659$	−41.7359	−1.62580	−0.812900	−	0.582403i	$-0.802113\pi$
$659$	−41.7359	−1.62580	−0.812900	+	0.582403i	$0.802113\pi$
$660$	−5.81792	−0.226462
$661$	−27.6636	−1.07599	−0.537995	−	0.842948i	$-0.680818\pi$
$661$	−27.6636	−1.07599	−0.537995	+	0.842948i	$0.680818\pi$
$662$	−5.61933	−0.218402
$663$	0.364407	0.0141524
$664$	8.99649	0.349132
$665$	−18.1542	−0.703991
$666$	2.88660	0.111854
$667$	−38.5469	−1.49254
$668$	20.7490	0.802804
$669$	−1.95877	−0.0757303
$670$	3.97175	0.153442
$671$	14.3305	0.553223
$672$	3.60104	0.138913
$673$	−2.37413	−0.0915161	−0.0457581	−	0.998953i	$-0.514570\pi$
$673$	−2.37413	−0.0915161	−0.0457581	+	0.998953i	$0.514570\pi$
$674$	−1.93882	−0.0746806
$675$	28.1223	1.08243
$676$	25.3203	0.973856
$677$	−10.6760	−0.410312	−0.205156	−	0.978729i	$-0.565770\pi$
$677$	−10.6760	−0.410312	−0.205156	+	0.978729i	$0.565770\pi$
$678$	−0.134695	−0.00517294
$679$	−0.313824	−0.0120435
$680$	11.2599	0.431796
$681$	−8.43162	−0.323101
$682$	2.68605	0.102854
$683$	29.1410	1.11505	0.557525	−	0.830160i	$-0.311751\pi$
$683$	29.1410	1.11505	0.557525	+	0.830160i	$0.311751\pi$
$684$	−7.98158	−0.305183
$685$	−55.1458	−2.10701
$686$	2.70881	0.103423
$687$	4.80379	0.183276
$688$	−0.0905068	−0.00345054
$689$	2.24770	0.0856306
$690$	3.25342	0.123856
$691$	4.39099	0.167041	0.0835205	−	0.996506i	$-0.473384\pi$
$691$	4.39099	0.167041	0.0835205	+	0.996506i	$0.473384\pi$
$692$	39.6711	1.50807
$693$	14.2579	0.541611
$694$	−3.43969	−0.130569
$695$	8.18794	0.310586
$696$	1.77482	0.0672742
$697$	−40.1758	−1.52177
$698$	0.500777	0.0189547
$699$	−4.45532	−0.168516
$700$	−64.3486	−2.43215
$701$	38.3125	1.44704	0.723522	−	0.690301i	$-0.242522\pi$
$701$	38.3125	1.44704	0.723522	+	0.690301i	$0.242522\pi$
$702$	0.129400	0.00488389
$703$	−7.21175	−0.271996
$704$	11.3063	0.426122
$705$	−24.1811	−0.910715
$706$	0.873154	0.0328616
$707$	9.77867	0.367765
$708$	−7.91373	−0.297416
$709$	−39.1970	−1.47207	−0.736037	−	0.676942i	$-0.763305\pi$
$709$	−39.1970	−1.47207	−0.736037	+	0.676942i	$0.763305\pi$
$710$	1.47944	0.0555222
$711$	42.9935	1.61238
$712$	8.04274	0.301414
$713$	66.0492	2.47356
$714$	1.06198	0.0397434
$715$	1.45686	0.0544836
$716$	−38.8795	−1.45300
$717$	−10.3311	−0.385822
$718$	0.583159	0.0217633
$719$	23.9975	0.894954	0.447477	−	0.894295i	$-0.352323\pi$
$719$	23.9975	0.894954	0.447477	+	0.894295i	$0.352323\pi$
$720$	−40.8590	−1.52273
$721$	−12.4827	−0.464882
$722$	3.55329	0.132240
$723$	−9.84871	−0.366277
$724$	17.4456	0.648360
$725$	−47.7508	−1.77342
$726$	−0.820170	−0.0304394
$727$	−6.12887	−0.227307	−0.113654	−	0.993520i	$-0.536255\pi$
$727$	−6.12887	−0.227307	−0.113654	+	0.993520i	$0.536255\pi$
$728$	−0.598913	−0.0221972
$729$	−15.9938	−0.592363
$730$	−4.28139	−0.158461
$731$	−0.0832232	−0.00307812
$732$	−8.02118	−0.296471
$733$	33.1518	1.22449	0.612244	−	0.790669i	$-0.290267\pi$
$733$	33.1518	1.22449	0.612244	+	0.790669i	$0.290267\pi$
$734$	−3.38011	−0.124762
$735$	−5.34580	−0.197183
$736$	−20.7114	−0.763431
$737$	7.79313	0.287064
$738$	−6.86594	−0.252739
$739$	18.4020	0.676929	0.338465	−	0.940979i	$-0.390092\pi$
$739$	18.4020	0.676929	0.338465	+	0.940979i	$0.390092\pi$
$740$	−37.7973	−1.38946
$741$	−0.155588	−0.00571566
$742$	6.55039	0.240472
$743$	6.26452	0.229823	0.114911	−	0.993376i	$-0.463342\pi$
$743$	6.26452	0.229823	0.114911	+	0.993376i	$0.463342\pi$
$744$	−3.04110	−0.111492
$745$	−24.0819	−0.882293
$746$	−1.71854	−0.0629203
$747$	−30.0187	−1.09833
$748$	10.9225	0.399367
$749$	0.0730467	0.00266907
$750$	2.10126	0.0767270
$751$	15.4802	0.564879	0.282440	−	0.959285i	$-0.408856\pi$
$751$	15.4802	0.564879	0.282440	+	0.959285i	$0.408856\pi$
$752$	49.3707	1.80036
$753$	7.74099	0.282098
$754$	−0.219717	−0.00800163
$755$	−23.2110	−0.844736
$756$	−16.5823	−0.603091
$757$	−1.63473	−0.0594151	−0.0297076	−	0.999559i	$-0.509458\pi$
$757$	−1.63473	−0.0594151	−0.0297076	+	0.999559i	$0.509458\pi$
$758$	1.01468	0.0368550
$759$	6.38367	0.231713
$760$	−4.80753	−0.174387
$761$	−45.6149	−1.65354	−0.826769	−	0.562541i	$-0.809824\pi$
$761$	−45.6149	−1.65354	−0.826769	+	0.562541i	$0.809824\pi$
$762$	−0.624747	−0.0226322
$763$	−63.2239	−2.28886
$764$	−27.4691	−0.993797
$765$	−37.5709	−1.35838
$766$	1.71639	0.0620155
$767$	1.98167	0.0715541
$768$	−5.84628	−0.210959
$769$	−17.7146	−0.638803	−0.319402	−	0.947619i	$-0.603482\pi$
$769$	−17.7146	−0.638803	−0.319402	+	0.947619i	$0.603482\pi$
$770$	4.24568	0.153004
$771$	−6.38105	−0.229808
$772$	5.01735	0.180578
$773$	−2.72077	−0.0978595	−0.0489297	−	0.998802i	$-0.515581\pi$
$773$	−2.72077	−0.0978595	−0.0489297	+	0.998802i	$0.515581\pi$
$774$	−0.0142226	−0.000511222 0
$775$	81.8196	2.93905
$776$	−0.0831056	−0.00298332
$777$	−7.21079	−0.258686
$778$	−4.22940	−0.151631
$779$	17.1535	0.614589
$780$	−0.815446	−0.0291976
$781$	2.90286	0.103873
$782$	−6.10795	−0.218420
$783$	−12.3051	−0.439748
$784$	10.9145	0.389804
$785$	65.5735	2.34042
$786$	0.121982	0.00435095
$787$	−40.4764	−1.44283	−0.721414	−	0.692504i	$-0.756507\pi$
$787$	−40.4764	−1.44283	−0.721414	+	0.692504i	$0.756507\pi$
$788$	−27.8740	−0.992969
$789$	4.25224	0.151384
$790$	12.8025	0.455494
$791$	−4.32228	−0.153683
$792$	3.77571	0.134164
$793$	2.00858	0.0713268
$794$	1.30719	0.0463906
$795$	18.0401	0.639817
$796$	5.44801	0.193099
$797$	−21.9576	−0.777779	−0.388890	−	0.921284i	$-0.627141\pi$
$797$	−21.9576	−0.777779	−0.388890	+	0.921284i	$0.627141\pi$
$798$	−0.453423	−0.0160510
$799$	45.3975	1.60605
$800$	−25.6566	−0.907098
$801$	−26.8363	−0.948214
$802$	6.78177	0.239473
$803$	−8.40069	−0.296454
$804$	−4.36202	−0.153837
$805$	104.400	3.67962
$806$	0.376480	0.0132609
$807$	−9.92449	−0.349359
$808$	2.58955	0.0911000
$809$	−4.43284	−0.155850	−0.0779252	−	0.996959i	$-0.524830\pi$
$809$	−4.43284	−0.155850	−0.0779252	+	0.996959i	$0.524830\pi$
$810$	−5.88202	−0.206673
$811$	−6.94412	−0.243841	−0.121921	−	0.992540i	$-0.538905\pi$
$811$	−6.94412	−0.243841	−0.121921	+	0.992540i	$0.538905\pi$
$812$	28.1562	0.988087
$813$	5.47216	0.191917
$814$	1.68659	0.0591151
$815$	−66.3593	−2.32447
$816$	−5.97144	−0.209042
$817$	0.0355331	0.00124315
$818$	−0.210883	−0.00737335
$819$	1.99840	0.0698297
$820$	89.9029	3.13954
$821$	−16.2448	−0.566948	−0.283474	−	0.958980i	$-0.591487\pi$
$821$	−16.2448	−0.566948	−0.283474	+	0.958980i	$0.591487\pi$
$822$	−1.37733	−0.0480399
$823$	−30.5558	−1.06511	−0.532554	−	0.846396i	$-0.678768\pi$
$823$	−30.5558	−1.06511	−0.532554	+	0.846396i	$0.678768\pi$
$824$	−3.30563	−0.115157
$825$	7.90789	0.275318
$826$	5.77512	0.200942
$827$	15.0576	0.523605	0.261802	−	0.965121i	$-0.415683\pi$
$827$	15.0576	0.523605	0.261802	+	0.965121i	$0.415683\pi$
$828$	45.8999	1.59513
$829$	−45.5012	−1.58032	−0.790161	−	0.612900i	$-0.790003\pi$
$829$	−45.5012	−1.58032	−0.790161	+	0.612900i	$0.790003\pi$
$830$	−8.93891	−0.310274
$831$	14.3913	0.499227
$832$	1.58470	0.0549397
$833$	10.0362	0.347732
$834$	0.204503	0.00708137
$835$	−41.7013	−1.44313
$836$	−4.66350	−0.161291
$837$	21.0845	0.728786
$838$	−3.25768	−0.112535
$839$	5.93023	0.204734	0.102367	−	0.994747i	$-0.467358\pi$
$839$	5.93023	0.204734	0.102367	+	0.994747i	$0.467358\pi$
$840$	−4.80689	−0.165853
$841$	−8.10634	−0.279529
$842$	−3.71006	−0.127857
$843$	−2.93312	−0.101022
$844$	21.1972	0.729638
$845$	−50.8885	−1.75062
$846$	7.75831	0.266736
$847$	−26.3187	−0.904321
$848$	−36.8325	−1.26483
$849$	−8.48397	−0.291169
$850$	−7.56633	−0.259523
$851$	41.4728	1.42167
$852$	−1.62481	−0.0556651
$853$	39.1926	1.34193	0.670964	−	0.741490i	$-0.265880\pi$
$853$	39.1926	1.34193	0.670964	+	0.741490i	$0.265880\pi$
$854$	5.85353	0.200304
$855$	16.0413	0.548602
$856$	0.0193439	0.000661162 0
$857$	13.5917	0.464284	0.232142	−	0.972682i	$-0.425427\pi$
$857$	13.5917	0.464284	0.232142	+	0.972682i	$0.425427\pi$
$858$	0.0363869	0.00124223
$859$	15.3820	0.524826	0.262413	−	0.964956i	$-0.415482\pi$
$859$	15.3820	0.524826	0.262413	+	0.964956i	$0.415482\pi$
$860$	0.186232	0.00635044
$861$	17.1512	0.584513
$862$	−8.44615	−0.287677
$863$	33.6795	1.14646	0.573232	−	0.819393i	$-0.305689\pi$
$863$	33.6795	1.14646	0.573232	+	0.819393i	$0.305689\pi$
$864$	−6.61156	−0.224930
$865$	−79.7308	−2.71093
$866$	−3.40545	−0.115722
$867$	2.42226	0.0822644
$868$	−48.2448	−1.63754
$869$	25.1204	0.852151
$870$	−1.76346	−0.0597868
$871$	1.09229	0.0370109
$872$	−16.7427	−0.566979
$873$	0.277299	0.00938516
$874$	2.60786	0.0882122
$875$	67.4279	2.27948
$876$	4.70209	0.158869
$877$	−6.38799	−0.215707	−0.107854	−	0.994167i	$-0.534398\pi$
$877$	−6.38799	−0.215707	−0.107854	+	0.994167i	$0.534398\pi$
$878$	3.81857	0.128871
$879$	−5.10100	−0.172053
$880$	−23.8732	−0.804767
$881$	44.1449	1.48728	0.743640	−	0.668580i	$-0.233097\pi$
$881$	44.1449	1.48728	0.743640	+	0.668580i	$0.233097\pi$
$882$	1.71515	0.0577522
$883$	−54.6120	−1.83784	−0.918921	−	0.394443i	$-0.870938\pi$
$883$	−54.6120	−1.83784	−0.918921	+	0.394443i	$0.870938\pi$
$884$	1.53091	0.0514901
$885$	15.9050	0.534640
$886$	5.05821	0.169934
$887$	40.0445	1.34456	0.672282	−	0.740295i	$-0.265314\pi$
$887$	40.0445	1.34456	0.672282	+	0.740295i	$0.265314\pi$
$888$	−1.90953	−0.0640797
$889$	−20.0477	−0.672378
$890$	−7.99126	−0.267868
$891$	−11.5414	−0.386650
$892$	−8.22900	−0.275527
$893$	−19.3830	−0.648628
$894$	−0.601474	−0.0201163
$895$	78.1398	2.61193
$896$	20.0907	0.671182
$897$	0.894742	0.0298746
$898$	0.120935	0.00403567
$899$	−35.8007	−1.19402
$900$	56.8593	1.89531
$901$	−33.8684	−1.12832
$902$	−4.01165	−0.133573
$903$	0.0355284	0.00118231
$904$	−1.14461	−0.0380691
$905$	−35.0620	−1.16550
$906$	−0.579723	−0.0192600
$907$	8.32073	0.276285	0.138143	−	0.990412i	$-0.455887\pi$
$907$	8.32073	0.276285	0.138143	+	0.990412i	$0.455887\pi$
$908$	−35.4222	−1.17553
$909$	−8.64057	−0.286590
$910$	0.595079	0.0197267
$911$	39.5875	1.31159	0.655796	−	0.754938i	$-0.272333\pi$
$911$	39.5875	1.31159	0.655796	+	0.754938i	$0.272333\pi$
$912$	2.54958	0.0844249
$913$	−17.5394	−0.580470
$914$	1.67923	0.0555440
$915$	16.1209	0.532941
$916$	20.1813	0.666808
$917$	3.91432	0.129262
$918$	−1.94980	−0.0643531
$919$	−44.1664	−1.45691	−0.728457	−	0.685091i	$-0.759762\pi$
$919$	−44.1664	−1.45691	−0.728457	+	0.685091i	$0.759762\pi$
$920$	27.6468	0.911487
$921$	−8.70692	−0.286903
$922$	3.03038	0.0998001
$923$	0.406868	0.0133922
$924$	−4.66288	−0.153397
$925$	51.3752	1.68921
$926$	2.55254	0.0838817
$927$	11.0299	0.362270
$928$	11.2262	0.368519
$929$	−8.41670	−0.276143	−0.138071	−	0.990422i	$-0.544090\pi$
$929$	−8.41670	−0.276143	−0.138071	+	0.990422i	$0.544090\pi$
$930$	3.02163	0.0990833
$931$	−4.28506	−0.140437
$932$	−18.7173	−0.613105
$933$	0.558549	0.0182861
$934$	−4.68559	−0.153317
$935$	−21.9520	−0.717908
$936$	0.529208	0.0172977
$937$	−17.8263	−0.582359	−0.291179	−	0.956668i	$-0.594048\pi$
$937$	−17.8263	−0.582359	−0.291179	+	0.956668i	$0.594048\pi$
$938$	3.18323	0.103936
$939$	2.57556	0.0840503
$940$	−101.588	−3.31342
$941$	−17.4626	−0.569263	−0.284632	−	0.958637i	$-0.591871\pi$
$941$	−17.4626	−0.569263	−0.284632	+	0.958637i	$0.591871\pi$
$942$	1.63778	0.0533616
$943$	−98.6453	−3.21233
$944$	−32.4732	−1.05691
$945$	33.3270	1.08413
$946$	−0.00831003	−0.000270183 0
$947$	−17.2483	−0.560495	−0.280247	−	0.959928i	$-0.590417\pi$
$947$	−17.2483	−0.560495	−0.280247	+	0.959928i	$0.590417\pi$
$948$	−14.0606	−0.456666
$949$	−1.17745	−0.0382216
$950$	3.23054	0.104812
$951$	13.1953	0.427888
$952$	9.02442	0.292483
$953$	40.9211	1.32556	0.662782	−	0.748812i	$-0.269375\pi$
$953$	40.9211	1.32556	0.662782	+	0.748812i	$0.269375\pi$
$954$	−5.78802	−0.187394
$955$	55.2073	1.78646
$956$	−43.4021	−1.40373
$957$	−3.46015	−0.111851
$958$	5.26480	0.170098
$959$	−44.1976	−1.42721
$960$	12.7189	0.410500
$961$	30.3436	0.978825
$962$	0.236395	0.00762167
$963$	−0.0645451	−0.00207994
$964$	−41.3755	−1.33262
$965$	−10.0838	−0.324610
$966$	2.60751	0.0838953
$967$	−30.6822	−0.986672	−0.493336	−	0.869839i	$-0.664223\pi$
$967$	−30.6822	−0.986672	−0.493336	+	0.869839i	$0.664223\pi$
$968$	−6.96961	−0.224012
$969$	2.34440	0.0753128
$970$	0.0825736	0.00265128
$971$	32.9595	1.05772	0.528860	−	0.848709i	$-0.322620\pi$
$971$	32.9595	1.05772	0.528860	+	0.848709i	$0.322620\pi$
$972$	22.2530	0.713764
$973$	6.56237	0.210380
$974$	−2.89576	−0.0927860
$975$	1.10838	0.0354965
$976$	−32.9141	−1.05355
$977$	−30.9423	−0.989933	−0.494967	−	0.868912i	$-0.664820\pi$
$977$	−30.9423	−0.989933	−0.494967	+	0.868912i	$0.664820\pi$
$978$	−1.65740	−0.0529979
$979$	−15.6800	−0.501135
$980$	−22.4583	−0.717403
$981$	55.8655	1.78365
$982$	5.92491	0.189071
$983$	−22.0276	−0.702571	−0.351286	−	0.936268i	$-0.614255\pi$
$983$	−22.0276	−0.702571	−0.351286	+	0.936268i	$0.614255\pi$
$984$	4.54192	0.144791
$985$	56.0210	1.78498
$986$	3.31070	0.105434
$987$	−19.3804	−0.616886
$988$	−0.653641	−0.0207951
$989$	−0.204341	−0.00649767
$990$	−3.75154	−0.119232
$991$	−17.4230	−0.553461	−0.276730	−	0.960948i	$-0.589251\pi$
$991$	−17.4230	−0.553461	−0.276730	+	0.960948i	$0.589251\pi$
$992$	−19.2358	−0.610738
$993$	−12.4035	−0.393612
$994$	1.18572	0.0376088
$995$	−10.9494	−0.347118
$996$	9.81728	0.311072
$997$	42.5229	1.34671	0.673356	−	0.739318i	$-0.264852\pi$
$997$	42.5229	1.34671	0.673356	+	0.739318i	$0.264852\pi$
$998$	1.22641	0.0388213
$999$	13.2391	0.418867

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	409.2.a.b.1.8	✓	20
3.2	odd	2		3681.2.a.i.1.13		20
4.3	odd	2		6544.2.a.i.1.11		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
409.2.a.b.1.8	✓	20	1.1	even	1	trivial
3681.2.a.i.1.13		20	3.2	odd	2
6544.2.a.i.1.11		20	4.3	odd	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 409 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	409.a (trivial)

\(f(q)\)	\(=\)	\(q-0.210883 q^{2} -0.465479 q^{3} -1.95553 q^{4} +3.93021 q^{5} +0.0981616 q^{6} +3.14994 q^{7} +0.834154 q^{8} -2.78333 q^{9} -0.828815 q^{10} -1.62625 q^{11} +0.910257 q^{12} -0.227937 q^{13} -0.664268 q^{14} -1.82943 q^{15} +3.73515 q^{16} +3.43456 q^{17} +0.586957 q^{18} -1.46643 q^{19} -7.68564 q^{20} -1.46623 q^{21} +0.342949 q^{22} +8.43301 q^{23} -0.388281 q^{24} +10.4466 q^{25} +0.0480681 q^{26} +2.69202 q^{27} -6.15979 q^{28} -4.57096 q^{29} +0.385796 q^{30} +7.83221 q^{31} -2.45599 q^{32} +0.756986 q^{33} -0.724290 q^{34} +12.3799 q^{35} +5.44288 q^{36} +4.91791 q^{37} +0.309244 q^{38} +0.106100 q^{39} +3.27840 q^{40} -11.6975 q^{41} +0.309203 q^{42} -0.0242311 q^{43} +3.18018 q^{44} -10.9391 q^{45} -1.77838 q^{46} +13.2179 q^{47} -1.73863 q^{48} +2.92211 q^{49} -2.20300 q^{50} -1.59871 q^{51} +0.445738 q^{52} -9.86106 q^{53} -0.567701 q^{54} -6.39151 q^{55} +2.62753 q^{56} +0.682590 q^{57} +0.963938 q^{58} -8.69395 q^{59} +3.57750 q^{60} -8.81199 q^{61} -1.65168 q^{62} -8.76732 q^{63} -6.95237 q^{64} -0.895841 q^{65} -0.159636 q^{66} -4.79208 q^{67} -6.71638 q^{68} -3.92539 q^{69} -2.61071 q^{70} -1.78500 q^{71} -2.32172 q^{72} +5.16567 q^{73} -1.03710 q^{74} -4.86265 q^{75} +2.86764 q^{76} -5.12259 q^{77} -0.0223747 q^{78} -15.4468 q^{79} +14.6799 q^{80} +7.09691 q^{81} +2.46681 q^{82} +10.7852 q^{83} +2.86725 q^{84} +13.4985 q^{85} +0.00510993 q^{86} +2.12768 q^{87} -1.35654 q^{88} +9.64179 q^{89} +2.30686 q^{90} -0.717988 q^{91} -16.4910 q^{92} -3.64573 q^{93} -2.78742 q^{94} -5.76336 q^{95} +1.14321 q^{96} -0.0996286 q^{97} -0.616223 q^{98} +4.52639 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 5 q^{2} + q^{3} + 23 q^{4} + 8 q^{5} - 4 q^{6} + 5 q^{7} + 12 q^{8} + 27 q^{9} - 5 q^{10} + 30 q^{11} - 5 q^{12} - 7 q^{13} + 17 q^{14} + 26 q^{15} + 29 q^{16} + 6 q^{17} - 4 q^{18} + q^{19} + 14 q^{20}+ \cdots + 43 q^{99}+O(q^{100}) \) 20 * q + 5 * q^2 + q^3 + 23 * q^4 + 8 * q^5 - 4 * q^6 + 5 * q^7 + 12 * q^8 + 27 * q^9 - 5 * q^10 + 30 * q^11 - 5 * q^12 - 7 * q^13 + 17 * q^14 + 26 * q^15 + 29 * q^16 + 6 * q^17 - 4 * q^18 + q^19 + 14 * q^20 - 3 * q^21 + q^22 + 37 * q^23 - 17 * q^24 + 14 * q^25 + 11 * q^26 - 5 * q^27 - 6 * q^28 + 12 * q^29 - 6 * q^30 + 10 * q^31 + 24 * q^32 - 22 * q^33 - 11 * q^34 + 12 * q^35 + 21 * q^36 - 4 * q^37 - 7 * q^38 + 31 * q^39 - 34 * q^40 - 7 * q^41 - 35 * q^42 - q^43 + 58 * q^44 - 10 * q^45 - 22 * q^46 + 51 * q^47 - 58 * q^48 + 5 * q^49 + q^50 + 6 * q^51 - 44 * q^52 + 10 * q^53 - 49 * q^54 - 5 * q^55 + 31 * q^56 - 30 * q^57 - 22 * q^58 + 43 * q^59 - 26 * q^60 - 21 * q^61 - 26 * q^62 + 8 * q^63 - 4 * q^64 - 10 * q^65 - 37 * q^66 + 4 * q^67 - 16 * q^68 - 19 * q^69 - 66 * q^70 + 97 * q^71 - 43 * q^72 - 36 * q^73 + 17 * q^74 - 19 * q^75 - 43 * q^76 - 20 * q^77 - 71 * q^78 + 16 * q^79 - 23 * q^80 + 16 * q^81 - 67 * q^82 + 28 * q^83 - 73 * q^84 - 30 * q^85 + 37 * q^86 - 14 * q^87 - 48 * q^88 + 2 * q^89 - 121 * q^90 - 25 * q^91 + 14 * q^92 - 27 * q^93 - 60 * q^94 + 61 * q^95 - 88 * q^96 - 36 * q^97 - 54 * q^98 + 43 * q^99

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