LMFDB - Embedded newform 867.4.a.p.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [867,4,Mod(1,867)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(867, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("867.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 867 = 3 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	867.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:	$51.1546559750$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 36x^{6} + 116x^{5} + 431x^{4} - 860x^{3} - 1756x^{2} + 480x + 544 $ x^8 - 4x^7 - 36x^6 + 116x^5 + 431x^4 - 860x^3 - 1756x^2 + 480*x + 544
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 51)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.61897$ of defining polynomial
Character	$\chi$	$=$	867.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.61897 q^{2} -3.00000 q^{3} -1.14098 q^{4} -13.9560 q^{5} +7.85692 q^{6} +16.4910 q^{7} +23.9400 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-2.61897 q^{2} -3.00000 q^{3} -1.14098 q^{4} -13.9560 q^{5} +7.85692 q^{6} +16.4910 q^{7} +23.9400 q^{8} +9.00000 q^{9} +36.5503 q^{10} +22.9767 q^{11} +3.42293 q^{12} +6.55333 q^{13} -43.1894 q^{14} +41.8679 q^{15} -53.5704 q^{16} -23.5708 q^{18} +46.8958 q^{19} +15.9234 q^{20} -49.4729 q^{21} -60.1755 q^{22} +129.200 q^{23} -71.8199 q^{24} +69.7687 q^{25} -17.1630 q^{26} -27.0000 q^{27} -18.8158 q^{28} +179.411 q^{29} -109.651 q^{30} -294.609 q^{31} -51.2204 q^{32} -68.9302 q^{33} -230.147 q^{35} -10.2688 q^{36} +251.367 q^{37} -122.819 q^{38} -19.6600 q^{39} -334.105 q^{40} -65.9139 q^{41} +129.568 q^{42} -261.618 q^{43} -26.2159 q^{44} -125.604 q^{45} -338.372 q^{46} -132.800 q^{47} +160.711 q^{48} -71.0479 q^{49} -182.722 q^{50} -7.47720 q^{52} -532.952 q^{53} +70.7123 q^{54} -320.663 q^{55} +394.793 q^{56} -140.687 q^{57} -469.874 q^{58} +845.784 q^{59} -47.7702 q^{60} -936.707 q^{61} +771.572 q^{62} +148.419 q^{63} +562.708 q^{64} -91.4580 q^{65} +180.527 q^{66} -592.758 q^{67} -387.601 q^{69} +602.750 q^{70} +250.698 q^{71} +215.460 q^{72} +679.648 q^{73} -658.323 q^{74} -209.306 q^{75} -53.5070 q^{76} +378.909 q^{77} +51.4890 q^{78} +982.157 q^{79} +747.626 q^{80} +81.0000 q^{81} +172.627 q^{82} +632.122 q^{83} +56.4474 q^{84} +685.170 q^{86} -538.234 q^{87} +550.063 q^{88} +634.869 q^{89} +328.953 q^{90} +108.071 q^{91} -147.415 q^{92} +883.826 q^{93} +347.800 q^{94} -654.476 q^{95} +153.661 q^{96} -700.085 q^{97} +186.073 q^{98} +206.791 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{2} - 24 q^{3} + 24 q^{4} + 24 q^{5} + 12 q^{6} + 28 q^{7} - 48 q^{8} + 72 q^{9}+O(q^{10}) $ 8 * q - 4 * q^2 - 24 * q^3 + 24 * q^4 + 24 * q^5 + 12 * q^6 + 28 * q^7 - 48 * q^8 + 72 * q^9	\( 8 q - 4 q^{2} - 24 q^{3} + 24 q^{4} + 24 q^{5} + 12 q^{6} + 28 q^{7} - 48 q^{8} + 72 q^{9} - 72 q^{10} + 12 q^{11} - 72 q^{12} - 60 q^{13} + 56 q^{14} - 72 q^{15} - 100 q^{16} - 36 q^{18} - 140 q^{19} + 224 q^{20} - 84 q^{21} + 484 q^{22} - 96 q^{23} + 144 q^{24} + 220 q^{25} + 500 q^{26} - 216 q^{27} + 348 q^{28} + 380 q^{29} + 216 q^{30} + 124 q^{31} - 448 q^{32} - 36 q^{33} - 592 q^{35} + 216 q^{36} + 1260 q^{37} + 84 q^{38} + 180 q^{39} - 836 q^{40} + 284 q^{41} - 168 q^{42} - 76 q^{43} - 1460 q^{44} + 216 q^{45} + 1056 q^{46} - 184 q^{47} + 300 q^{48} - 536 q^{49} - 360 q^{50} - 1580 q^{52} + 1064 q^{53} + 108 q^{54} - 2148 q^{55} + 244 q^{56} + 420 q^{57} - 532 q^{58} + 1224 q^{59} - 672 q^{60} + 324 q^{61} + 3068 q^{62} + 252 q^{63} - 876 q^{64} + 652 q^{65} - 1452 q^{66} - 624 q^{67} + 288 q^{69} - 1932 q^{70} + 2636 q^{71} - 432 q^{72} + 1640 q^{73} - 120 q^{74} - 660 q^{75} - 1172 q^{76} + 504 q^{77} - 1500 q^{78} + 3788 q^{79} + 1740 q^{80} + 648 q^{81} + 1916 q^{82} + 2032 q^{83} - 1044 q^{84} + 1612 q^{86} - 1140 q^{87} + 4576 q^{88} + 1304 q^{89} - 648 q^{90} - 2872 q^{91} - 1744 q^{92} - 372 q^{93} - 208 q^{94} - 748 q^{95} + 1344 q^{96} + 2376 q^{97} + 604 q^{98} + 108 q^{99}+O(q^{100}) \) 8 * q - 4 * q^2 - 24 * q^3 + 24 * q^4 + 24 * q^5 + 12 * q^6 + 28 * q^7 - 48 * q^8 + 72 * q^9 - 72 * q^10 + 12 * q^11 - 72 * q^12 - 60 * q^13 + 56 * q^14 - 72 * q^15 - 100 * q^16 - 36 * q^18 - 140 * q^19 + 224 * q^20 - 84 * q^21 + 484 * q^22 - 96 * q^23 + 144 * q^24 + 220 * q^25 + 500 * q^26 - 216 * q^27 + 348 * q^28 + 380 * q^29 + 216 * q^30 + 124 * q^31 - 448 * q^32 - 36 * q^33 - 592 * q^35 + 216 * q^36 + 1260 * q^37 + 84 * q^38 + 180 * q^39 - 836 * q^40 + 284 * q^41 - 168 * q^42 - 76 * q^43 - 1460 * q^44 + 216 * q^45 + 1056 * q^46 - 184 * q^47 + 300 * q^48 - 536 * q^49 - 360 * q^50 - 1580 * q^52 + 1064 * q^53 + 108 * q^54 - 2148 * q^55 + 244 * q^56 + 420 * q^57 - 532 * q^58 + 1224 * q^59 - 672 * q^60 + 324 * q^61 + 3068 * q^62 + 252 * q^63 - 876 * q^64 + 652 * q^65 - 1452 * q^66 - 624 * q^67 + 288 * q^69 - 1932 * q^70 + 2636 * q^71 - 432 * q^72 + 1640 * q^73 - 120 * q^74 - 660 * q^75 - 1172 * q^76 + 504 * q^77 - 1500 * q^78 + 3788 * q^79 + 1740 * q^80 + 648 * q^81 + 1916 * q^82 + 2032 * q^83 - 1044 * q^84 + 1612 * q^86 - 1140 * q^87 + 4576 * q^88 + 1304 * q^89 - 648 * q^90 - 2872 * q^91 - 1744 * q^92 - 372 * q^93 - 208 * q^94 - 748 * q^95 + 1344 * q^96 + 2376 * q^97 + 604 * q^98 + 108 * 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 $

$2$	−2.61897	−0.925947	−0.462974	−	0.886372i	$-0.653218\pi$
$2$	−2.61897	−0.925947	−0.462974	+	0.886372i	$0.653218\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−1.14098	−0.142622
$5$	−13.9560	−1.24826	−0.624129	−	0.781321i	$-0.714546\pi$
$5$	−13.9560	−1.24826	−0.624129	+	0.781321i	$0.714546\pi$
$6$	7.85692	0.534596
$7$	16.4910	0.890429	0.445214	−	0.895424i	$-0.353127\pi$
$7$	16.4910	0.890429	0.445214	+	0.895424i	$0.353127\pi$
$8$	23.9400	1.05801
$9$	9.00000	0.333333
$10$	36.5503	1.15582
$11$	22.9767	0.629795	0.314898	−	0.949126i	$-0.398030\pi$
$11$	22.9767	0.629795	0.314898	+	0.949126i	$0.398030\pi$
$12$	3.42293	0.0823429
$13$	6.55333	0.139813	0.0699064	−	0.997554i	$-0.477730\pi$
$13$	6.55333	0.139813	0.0699064	+	0.997554i	$0.477730\pi$
$14$	−43.1894	−0.824490
$15$	41.8679	0.720682
$16$	−53.5704	−0.837037
$17$	0	0
$17$	0	0
$18$	−23.5708	−0.308649
$19$	46.8958	0.566244	0.283122	−	0.959084i	$-0.408630\pi$
$19$	46.8958	0.566244	0.283122	+	0.959084i	$0.408630\pi$
$20$	15.9234	0.178029
$21$	−49.4729	−0.514089
$22$	−60.1755	−0.583157
$23$	129.200	1.17131	0.585655	−	0.810560i	$-0.300837\pi$
$23$	129.200	1.17131	0.585655	+	0.810560i	$0.300837\pi$
$24$	−71.8199	−0.610841
$25$	69.7687	0.558150
$26$	−17.1630	−0.129459
$27$	−27.0000	−0.192450
$28$	−18.8158	−0.126995
$29$	179.411	1.14882	0.574411	−	0.818567i	$-0.305231\pi$
$29$	179.411	1.14882	0.574411	+	0.818567i	$0.305231\pi$
$30$	−109.651	−0.667314
$31$	−294.609	−1.70688	−0.853440	−	0.521192i	$-0.825488\pi$
$31$	−294.609	−1.70688	−0.853440	+	0.521192i	$0.825488\pi$
$32$	−51.2204	−0.282956
$33$	−68.9302	−0.363613
$34$	0	0
$35$	−230.147	−1.11149
$36$	−10.2688	−0.0475407
$37$	251.367	1.11688	0.558438	−	0.829546i	$-0.311401\pi$
$37$	251.367	1.11688	0.558438	+	0.829546i	$0.311401\pi$
$38$	−122.819	−0.524312
$39$	−19.6600	−0.0807210
$40$	−334.105	−1.32067
$41$	−65.9139	−0.251074	−0.125537	−	0.992089i	$-0.540065\pi$
$41$	−65.9139	−0.251074	−0.125537	+	0.992089i	$0.540065\pi$
$42$	129.568	0.476019
$43$	−261.618	−0.927821	−0.463911	−	0.885882i	$-0.653554\pi$
$43$	−261.618	−0.927821	−0.463911	+	0.885882i	$0.653554\pi$
$44$	−26.2159	−0.0898227
$45$	−125.604	−0.416086
$46$	−338.372	−1.08457
$47$	−132.800	−0.412147	−0.206073	−	0.978537i	$-0.566069\pi$
$47$	−132.800	−0.412147	−0.206073	+	0.978537i	$0.566069\pi$
$48$	160.711	0.483263
$49$	−71.0479	−0.207137
$50$	−182.722	−0.516817
$51$	0	0
$52$	−7.47720	−0.0199404
$53$	−532.952	−1.38126	−0.690628	−	0.723210i	$-0.742666\pi$
$53$	−532.952	−1.38126	−0.690628	+	0.723210i	$0.742666\pi$
$54$	70.7123	0.178199
$55$	−320.663	−0.786148
$56$	394.793	0.942080
$57$	−140.687	−0.326921
$58$	−469.874	−1.06375
$59$	845.784	1.86630	0.933150	−	0.359488i	$-0.117049\pi$
$59$	845.784	1.86630	0.933150	+	0.359488i	$0.117049\pi$
$60$	−47.7702	−0.102785
$61$	−936.707	−1.96611	−0.983057	−	0.183299i	$-0.941322\pi$
$61$	−936.707	−1.96611	−0.983057	+	0.183299i	$0.941322\pi$
$62$	771.572	1.58048
$63$	148.419	0.296810
$64$	562.708	1.09904
$65$	−91.4580	−0.174523
$66$	180.527	0.336686
$67$	−592.758	−1.08085	−0.540425	−	0.841392i	$-0.681736\pi$
$67$	−592.758	−1.08085	−0.540425	+	0.841392i	$0.681736\pi$
$68$	0	0
$69$	−387.601	−0.676256
$70$	602.750	1.02918
$71$	250.698	0.419047	0.209523	−	0.977804i	$-0.432809\pi$
$71$	250.698	0.419047	0.209523	+	0.977804i	$0.432809\pi$
$72$	215.460	0.352669
$73$	679.648	1.08968	0.544841	−	0.838539i	$-0.316590\pi$
$73$	679.648	1.08968	0.544841	+	0.838539i	$0.316590\pi$
$74$	−658.323	−1.03417
$75$	−209.306	−0.322248
$76$	−53.5070	−0.0807589
$77$	378.909	0.560788
$78$	51.4890	0.0747434
$79$	982.157	1.39875	0.699375	−	0.714755i	$-0.253462\pi$
$79$	982.157	1.39875	0.699375	+	0.714755i	$0.253462\pi$
$80$	747.626	1.04484
$81$	81.0000	0.111111
$82$	172.627	0.232481
$83$	632.122	0.835956	0.417978	−	0.908457i	$-0.362739\pi$
$83$	632.122	0.835956	0.417978	+	0.908457i	$0.362739\pi$
$84$	56.4474	0.0733205
$85$	0	0
$86$	685.170	0.859113
$87$	−538.234	−0.663273
$88$	550.063	0.666328
$89$	634.869	0.756134	0.378067	−	0.925778i	$-0.376589\pi$
$89$	634.869	0.756134	0.378067	+	0.925778i	$0.376589\pi$
$90$	328.953	0.385274
$91$	108.071	0.124493
$92$	−147.415	−0.167055
$93$	883.826	0.985467
$94$	347.800	0.381626
$95$	−654.476	−0.706819
$96$	153.661	0.163365
$97$	−700.085	−0.732813	−0.366407	−	0.930455i	$-0.619412\pi$
$97$	−700.085	−0.732813	−0.366407	+	0.930455i	$0.619412\pi$
$98$	186.073	0.191798
$99$	206.791	0.209932
$100$	−79.6045	−0.0796045
$101$	−381.929	−0.376271	−0.188136	−	0.982143i	$-0.560244\pi$
$101$	−381.929	−0.376271	−0.188136	+	0.982143i	$0.560244\pi$
$102$	0	0
$103$	−181.240	−0.173380	−0.0866901	−	0.996235i	$-0.527629\pi$
$103$	−181.240	−0.173380	−0.0866901	+	0.996235i	$0.527629\pi$
$104$	156.887	0.147923
$105$	690.442	0.641716
$106$	1395.79	1.27897
$107$	−249.908	−0.225790	−0.112895	−	0.993607i	$-0.536012\pi$
$107$	−249.908	−0.225790	−0.112895	+	0.993607i	$0.536012\pi$
$108$	30.8064	0.0274476
$109$	1835.46	1.61289	0.806445	−	0.591308i	$-0.201388\pi$
$109$	1835.46	1.61289	0.806445	+	0.591308i	$0.201388\pi$
$110$	839.807	0.727931
$111$	−754.100	−0.644829
$112$	−883.427	−0.745322
$113$	−316.816	−0.263748	−0.131874	−	0.991266i	$-0.542099\pi$
$113$	−316.816	−0.263748	−0.131874	+	0.991266i	$0.542099\pi$
$114$	368.457	0.302712
$115$	−1803.11	−1.46210
$116$	−204.704	−0.163847
$117$	58.9800	0.0466043
$118$	−2215.09	−1.72809
$119$	0	0
$120$	1002.32	0.762487
$121$	−803.069	−0.603358
$122$	2453.21	1.82052
$123$	197.742	0.144958
$124$	336.141	0.243439
$125$	770.805	0.551543
$126$	−388.705	−0.274830
$127$	837.160	0.584929	0.292464	−	0.956276i	$-0.405525\pi$
$127$	837.160	0.584929	0.292464	+	0.956276i	$0.405525\pi$
$128$	−1063.95	−0.734696
$129$	784.853	0.535678
$130$	239.526	0.161599
$131$	2374.50	1.58367	0.791837	−	0.610732i	$-0.209125\pi$
$131$	2374.50	1.58367	0.791837	+	0.610732i	$0.209125\pi$
$132$	78.6478	0.0518592
$133$	773.357	0.504200
$134$	1552.42	1.00081
$135$	376.811	0.240227
$136$	0	0
$137$	−855.800	−0.533693	−0.266846	−	0.963739i	$-0.585982\pi$
$137$	−855.800	−0.533693	−0.266846	+	0.963739i	$0.585982\pi$
$138$	1015.12	0.626178
$139$	−1741.24	−1.06252	−0.531260	−	0.847209i	$-0.678281\pi$
$139$	−1741.24	−1.06252	−0.531260	+	0.847209i	$0.678281\pi$
$140$	262.593	0.158522
$141$	398.401	0.237953
$142$	−656.571	−0.388015
$143$	150.574	0.0880535
$144$	−482.133	−0.279012
$145$	−2503.86	−1.43403
$146$	−1779.98	−1.00899
$147$	213.144	0.119590
$148$	−286.803	−0.159291
$149$	−2459.05	−1.35204	−0.676018	−	0.736885i	$-0.736296\pi$
$149$	−2459.05	−1.35204	−0.676018	+	0.736885i	$0.736296\pi$
$150$	548.167	0.298385
$151$	1113.99	0.600367	0.300183	−	0.953882i	$-0.402952\pi$
$151$	1113.99	0.600367	0.300183	+	0.953882i	$0.402952\pi$
$152$	1122.68	0.599090
$153$	0	0
$154$	−992.352	−0.519260
$155$	4111.54	2.13063
$156$	22.4316	0.0115126
$157$	−869.235	−0.441863	−0.220932	−	0.975289i	$-0.570910\pi$
$157$	−869.235	−0.441863	−0.220932	+	0.975289i	$0.570910\pi$
$158$	−2572.24	−1.29517
$159$	1598.86	0.797469
$160$	714.830	0.353202
$161$	2130.64	1.04297
$162$	−212.137	−0.102883
$163$	2769.97	1.33105	0.665524	−	0.746377i	$-0.268208\pi$
$163$	2769.97	1.33105	0.665524	+	0.746377i	$0.268208\pi$
$164$	75.2063	0.0358087
$165$	961.988	0.453883
$166$	−1655.51	−0.774051
$167$	3142.74	1.45624	0.728121	−	0.685449i	$-0.240394\pi$
$167$	3142.74	1.45624	0.728121	+	0.685449i	$0.240394\pi$
$168$	−1184.38	−0.543910
$169$	−2154.05	−0.980452
$170$	0	0
$171$	422.062	0.188748
$172$	298.499	0.132328
$173$	−1375.39	−0.604445	−0.302222	−	0.953237i	$-0.597729\pi$
$173$	−1375.39	−0.604445	−0.302222	+	0.953237i	$0.597729\pi$
$174$	1409.62	0.614155
$175$	1150.55	0.496993
$176$	−1230.87	−0.527162
$177$	−2537.35	−1.07751
$178$	−1662.70	−0.700140
$179$	−1384.96	−0.578306	−0.289153	−	0.957283i	$-0.593374\pi$
$179$	−1384.96	−0.578306	−0.289153	+	0.957283i	$0.593374\pi$
$180$	143.311	0.0593431
$181$	3118.29	1.28056	0.640278	−	0.768143i	$-0.278819\pi$
$181$	3118.29	1.28056	0.640278	+	0.768143i	$0.278819\pi$
$182$	−283.035	−0.115274
$183$	2810.12	1.13514
$184$	3093.05	1.23926
$185$	−3508.06	−1.39415
$186$	−2314.72	−0.912490
$187$	0	0
$188$	151.522	0.0587812
$189$	−445.256	−0.171363
$190$	1714.05	0.654477
$191$	3256.98	1.23386	0.616930	−	0.787018i	$-0.288376\pi$
$191$	3256.98	1.23386	0.616930	+	0.787018i	$0.288376\pi$
$192$	−1688.12	−0.634530
$193$	−2026.34	−0.755747	−0.377873	−	0.925857i	$-0.623345\pi$
$193$	−2026.34	−0.755747	−0.377873	+	0.925857i	$0.623345\pi$
$194$	1833.50	0.678546
$195$	274.374	0.100761
$196$	81.0640	0.0295423
$197$	2777.72	1.00459	0.502295	−	0.864696i	$-0.332489\pi$
$197$	2777.72	1.00459	0.502295	+	0.864696i	$0.332489\pi$
$198$	−541.580	−0.194386
$199$	−54.7239	−0.0194938	−0.00974691	−	0.999952i	$-0.503103\pi$
$199$	−54.7239	−0.0194938	−0.00974691	+	0.999952i	$0.503103\pi$
$200$	1670.26	0.590527
$201$	1778.27	0.624029
$202$	1000.26	0.348407
$203$	2958.67	1.02294
$204$	0	0
$205$	919.892	0.313405
$206$	474.664	0.160541
$207$	1162.80	0.390437
$208$	−351.064	−0.117029
$209$	1077.51	0.356618
$210$	−1808.25	−0.594195
$211$	1891.52	0.617145	0.308572	−	0.951201i	$-0.400149\pi$
$211$	1891.52	0.617145	0.308572	+	0.951201i	$0.400149\pi$
$212$	608.086	0.196998
$213$	−752.093	−0.241937
$214$	654.503	0.209070
$215$	3651.12	1.15816
$216$	−646.379	−0.203614
$217$	−4858.38	−1.51985
$218$	−4807.02	−1.49345
$219$	−2038.94	−0.629128
$220$	365.868	0.112122
$221$	0	0
$222$	1974.97	0.597077
$223$	−2648.31	−0.795265	−0.397632	−	0.917545i	$-0.630168\pi$
$223$	−2648.31	−0.795265	−0.397632	+	0.917545i	$0.630168\pi$
$224$	−844.675	−0.251952
$225$	627.918	0.186050
$226$	829.733	0.244217
$227$	5745.61	1.67996	0.839978	−	0.542621i	$-0.182568\pi$
$227$	5745.61	1.67996	0.839978	+	0.542621i	$0.182568\pi$
$228$	160.521	0.0466261
$229$	−6317.09	−1.82290	−0.911452	−	0.411406i	$-0.865038\pi$
$229$	−6317.09	−1.82290	−0.911452	+	0.411406i	$0.865038\pi$
$230$	4722.31	1.35383
$231$	−1136.73	−0.323771
$232$	4295.10	1.21546
$233$	1153.33	0.324280	0.162140	−	0.986768i	$-0.448160\pi$
$233$	1153.33	0.324280	0.162140	+	0.986768i	$0.448160\pi$
$234$	−154.467	−0.0431531
$235$	1853.35	0.514466
$236$	−965.019	−0.266175
$237$	−2946.47	−0.807569
$238$	0	0
$239$	−1828.39	−0.494847	−0.247424	−	0.968907i	$-0.579584\pi$
$239$	−1828.39	−0.494847	−0.247424	+	0.968907i	$0.579584\pi$
$240$	−2242.88	−0.603238
$241$	3399.35	0.908595	0.454298	−	0.890850i	$-0.349890\pi$
$241$	3399.35	0.908595	0.454298	+	0.890850i	$0.349890\pi$
$242$	2103.22	0.558677
$243$	−243.000	−0.0641500
$244$	1068.76	0.280411
$245$	991.541	0.258560
$246$	−517.881	−0.134223
$247$	307.324	0.0791682
$248$	−7052.92	−1.80589
$249$	−1896.37	−0.482640
$250$	−2018.72	−0.510700
$251$	813.958	0.204688	0.102344	−	0.994749i	$-0.467366\pi$
$251$	813.958	0.204688	0.102344	+	0.994749i	$0.467366\pi$
$252$	−169.342	−0.0423316
$253$	2968.60	0.737686
$254$	−2192.50	−0.541613
$255$	0	0
$256$	−1715.20	−0.418749
$257$	931.357	0.226056	0.113028	−	0.993592i	$-0.463945\pi$
$257$	931.357	0.226056	0.113028	+	0.993592i	$0.463945\pi$
$258$	−2055.51	−0.496009
$259$	4145.28	0.994498
$260$	104.351	0.0248908
$261$	1614.70	0.382941
$262$	−6218.76	−1.46640
$263$	3196.74	0.749503	0.374752	−	0.927125i	$-0.377728\pi$
$263$	3196.74	0.749503	0.374752	+	0.927125i	$0.377728\pi$
$264$	−1650.19	−0.384705
$265$	7437.86	1.72417
$266$	−2025.40	−0.466862
$267$	−1904.61	−0.436554
$268$	676.323	0.154153
$269$	−3556.08	−0.806016	−0.403008	−	0.915196i	$-0.632035\pi$
$269$	−3556.08	−0.806016	−0.403008	+	0.915196i	$0.632035\pi$
$270$	−986.858	−0.222438
$271$	3988.16	0.893961	0.446980	−	0.894544i	$-0.352499\pi$
$271$	3988.16	0.893961	0.446980	+	0.894544i	$0.352499\pi$
$272$	0	0
$273$	−324.212	−0.0718763
$274$	2241.32	0.494171
$275$	1603.06	0.351520
$276$	442.244	0.0964491
$277$	6595.01	1.43053	0.715263	−	0.698856i	$-0.246307\pi$
$277$	6595.01	1.43053	0.715263	+	0.698856i	$0.246307\pi$
$278$	4560.27	0.983838
$279$	−2651.48	−0.568960
$280$	−5509.72	−1.17596
$281$	6871.56	1.45880	0.729400	−	0.684088i	$-0.239799\pi$
$281$	6871.56	1.45880	0.729400	+	0.684088i	$0.239799\pi$
$282$	−1043.40	−0.220332
$283$	−5931.04	−1.24581	−0.622904	−	0.782298i	$-0.714047\pi$
$283$	−5931.04	−1.24581	−0.622904	+	0.782298i	$0.714047\pi$
$284$	−286.040	−0.0597653
$285$	1963.43	0.408082
$286$	−394.350	−0.0815329
$287$	−1086.98	−0.223563
$288$	−460.984	−0.0943185
$289$	0	0
$290$	6557.53	1.32783
$291$	2100.26	0.423090
$292$	−775.462	−0.155413
$293$	−2624.91	−0.523375	−0.261687	−	0.965153i	$-0.584279\pi$
$293$	−2624.91	−0.523375	−0.261687	+	0.965153i	$0.584279\pi$
$294$	−558.218	−0.110734
$295$	−11803.7	−2.32962
$296$	6017.71	1.18166
$297$	−620.372	−0.121204
$298$	6440.19	1.25191
$299$	846.693	0.163764
$300$	238.813	0.0459597
$301$	−4314.33	−0.826159
$302$	−2917.52	−0.555908
$303$	1145.79	0.217240
$304$	−2512.22	−0.473967
$305$	13072.6	2.45422
$306$	0	0
$307$	−5618.04	−1.04443	−0.522213	−	0.852815i	$-0.674893\pi$
$307$	−5618.04	−1.04443	−0.522213	+	0.852815i	$0.674893\pi$
$308$	−432.326	−0.0799807
$309$	543.721	0.100101
$310$	−10768.0	−1.97285
$311$	−10610.8	−1.93468	−0.967338	−	0.253489i	$-0.918422\pi$
$311$	−10610.8	−1.93468	−0.967338	+	0.253489i	$0.918422\pi$
$312$	−470.660	−0.0854034
$313$	831.078	0.150081	0.0750404	−	0.997180i	$-0.476091\pi$
$313$	831.078	0.150081	0.0750404	+	0.997180i	$0.476091\pi$
$314$	2276.50	0.409142
$315$	−2071.33	−0.370495
$316$	−1120.62	−0.199493
$317$	921.313	0.163237	0.0816185	−	0.996664i	$-0.473991\pi$
$317$	921.313	0.163237	0.0816185	+	0.996664i	$0.473991\pi$
$318$	−4187.36	−0.738414
$319$	4122.29	0.723523
$320$	−7853.13	−1.37188
$321$	749.725	0.130360
$322$	−5580.09	−0.965733
$323$	0	0
$324$	−92.4191	−0.0158469
$325$	457.218	0.0780365
$326$	−7254.47	−1.23248
$327$	−5506.38	−0.931203
$328$	−1577.98	−0.265638
$329$	−2190.00	−0.366987
$330$	−2519.42	−0.420271
$331$	11586.6	1.92404	0.962018	−	0.272986i	$-0.0880111\pi$
$331$	11586.6	1.92404	0.962018	+	0.272986i	$0.0880111\pi$
$332$	−721.236	−0.119226
$333$	2262.30	0.372292
$334$	−8230.75	−1.34840
$335$	8272.50	1.34918
$336$	2650.28	0.430312
$337$	−5207.29	−0.841719	−0.420860	−	0.907126i	$-0.638272\pi$
$337$	−5207.29	−0.841719	−0.420860	+	0.907126i	$0.638272\pi$
$338$	5641.41	0.907847
$339$	950.448	0.152275
$340$	0	0
$341$	−6769.15	−1.07498
$342$	−1105.37	−0.174771
$343$	−6828.05	−1.07487
$344$	−6263.12	−0.981642
$345$	5409.34	0.844143
$346$	3602.11	0.559684
$347$	2621.36	0.405539	0.202769	−	0.979227i	$-0.435006\pi$
$347$	2621.36	0.405539	0.202769	+	0.979227i	$0.435006\pi$
$348$	614.112	0.0945973
$349$	−9672.01	−1.48347	−0.741735	−	0.670693i	$-0.765997\pi$
$349$	−9672.01	−1.48347	−0.741735	+	0.670693i	$0.765997\pi$
$350$	−3013.27	−0.460189
$351$	−176.940	−0.0269070
$352$	−1176.88	−0.178204
$353$	2811.04	0.423843	0.211921	−	0.977287i	$-0.432028\pi$
$353$	2811.04	0.423843	0.211921	+	0.977287i	$0.432028\pi$
$354$	6645.26	0.997716
$355$	−3498.73	−0.523079
$356$	−724.370	−0.107841
$357$	0	0
$358$	3627.17	0.535481
$359$	−2873.82	−0.422491	−0.211245	−	0.977433i	$-0.567752\pi$
$359$	−2873.82	−0.422491	−0.211245	+	0.977433i	$0.567752\pi$
$360$	−3006.95	−0.440222
$361$	−4659.78	−0.679368
$362$	−8166.72	−1.18573
$363$	2409.21	0.348349
$364$	−123.306	−0.0177555
$365$	−9485.14	−1.36021
$366$	−7359.63	−1.05108
$367$	−818.419	−0.116406	−0.0582032	−	0.998305i	$-0.518537\pi$
$367$	−818.419	−0.116406	−0.0582032	+	0.998305i	$0.518537\pi$
$368$	−6921.31	−0.980430
$369$	−593.226	−0.0836913
$370$	9187.52	1.29091
$371$	−8788.90	−1.22991
$372$	−1008.42	−0.140549
$373$	9572.68	1.32883	0.664416	−	0.747363i	$-0.268680\pi$
$373$	9572.68	1.32883	0.664416	+	0.747363i	$0.268680\pi$
$374$	0	0
$375$	−2312.42	−0.318434
$376$	−3179.23	−0.436055
$377$	1175.74	0.160620
$378$	1166.11	0.158673
$379$	684.001	0.0927038	0.0463519	−	0.998925i	$-0.485240\pi$
$379$	684.001	0.0927038	0.0463519	+	0.998925i	$0.485240\pi$
$380$	746.741	0.100808
$381$	−2511.48	−0.337709
$382$	−8529.96	−1.14249
$383$	9056.16	1.20822	0.604110	−	0.796901i	$-0.293529\pi$
$383$	9056.16	1.20822	0.604110	+	0.796901i	$0.293529\pi$
$384$	3191.86	0.424177
$385$	−5288.04	−0.700008
$386$	5306.93	0.699782
$387$	−2354.56	−0.309274
$388$	798.781	0.104515
$389$	12957.0	1.68880	0.844401	−	0.535712i	$-0.179957\pi$
$389$	12957.0	1.68880	0.844401	+	0.535712i	$0.179957\pi$
$390$	−718.579	−0.0932991
$391$	0	0
$392$	−1700.88	−0.219152
$393$	−7123.51	−0.914335
$394$	−7274.77	−0.930197
$395$	−13706.9	−1.74600
$396$	−235.943	−0.0299409
$397$	4380.63	0.553798	0.276899	−	0.960899i	$-0.410693\pi$
$397$	4380.63	0.553798	0.276899	+	0.960899i	$0.410693\pi$
$398$	143.320	0.0180503
$399$	−2320.07	−0.291100
$400$	−3737.54	−0.467192
$401$	−455.645	−0.0567427	−0.0283714	−	0.999597i	$-0.509032\pi$
$401$	−455.645	−0.0567427	−0.0283714	+	0.999597i	$0.509032\pi$
$402$	−4657.25	−0.577817
$403$	−1930.67	−0.238644
$404$	435.772	0.0536645
$405$	−1130.43	−0.138695
$406$	−7748.67	−0.947192
$407$	5775.59	0.703403
$408$	0	0
$409$	9457.75	1.14341	0.571706	−	0.820459i	$-0.306282\pi$
$409$	9457.75	1.14341	0.571706	+	0.820459i	$0.306282\pi$
$410$	−2409.17	−0.290197
$411$	2567.40	0.308128
$412$	206.791	0.0247278
$413$	13947.8	1.66181
$414$	−3045.35	−0.361524
$415$	−8821.86	−1.04349
$416$	−335.665	−0.0395608
$417$	5223.73	0.613447
$418$	−2821.98	−0.330209
$419$	−7548.46	−0.880111	−0.440056	−	0.897971i	$-0.645041\pi$
$419$	−7548.46	−0.880111	−0.440056	+	0.897971i	$0.645041\pi$
$420$	−787.778	−0.0915229
$421$	−2790.54	−0.323047	−0.161523	−	0.986869i	$-0.551641\pi$
$421$	−2790.54	−0.323047	−0.161523	+	0.986869i	$0.551641\pi$
$422$	−4953.84	−0.571443
$423$	−1195.20	−0.137382
$424$	−12758.9	−1.46138
$425$	0	0
$426$	1969.71	0.224021
$427$	−15447.2	−1.75068
$428$	285.139	0.0322026
$429$	−451.723	−0.0508377
$430$	−9562.20	−1.07240
$431$	−1094.38	−0.122307	−0.0611535	−	0.998128i	$-0.519478\pi$
$431$	−1094.38	−0.122307	−0.0611535	+	0.998128i	$0.519478\pi$
$432$	1446.40	0.161088
$433$	−6041.44	−0.670515	−0.335258	−	0.942126i	$-0.608823\pi$
$433$	−6041.44	−0.670515	−0.335258	+	0.942126i	$0.608823\pi$
$434$	12724.0	1.40730
$435$	7511.57	0.827936
$436$	−2094.22	−0.230034
$437$	6058.95	0.663247
$438$	5339.94	0.582539
$439$	572.128	0.0622008	0.0311004	−	0.999516i	$-0.490099\pi$
$439$	572.128	0.0622008	0.0311004	+	0.999516i	$0.490099\pi$
$440$	−7676.65	−0.831750
$441$	−639.431	−0.0690456
$442$	0	0
$443$	5357.07	0.574542	0.287271	−	0.957849i	$-0.407252\pi$
$443$	5357.07	0.574542	0.287271	+	0.957849i	$0.407252\pi$
$444$	860.410	0.0919668
$445$	−8860.20	−0.943851
$446$	6935.86	0.736373
$447$	7377.16	0.780598
$448$	9279.60	0.978616
$449$	10944.7	1.15036	0.575182	−	0.818026i	$-0.304931\pi$
$449$	10944.7	1.15036	0.575182	+	0.818026i	$0.304931\pi$
$450$	−1644.50	−0.172272
$451$	−1514.49	−0.158125
$452$	361.480	0.0376163
$453$	−3341.98	−0.346622
$454$	−15047.6	−1.55555
$455$	−1508.23	−0.155400
$456$	−3368.05	−0.345885
$457$	4124.68	0.422198	0.211099	−	0.977465i	$-0.432296\pi$
$457$	4124.68	0.422198	0.211099	+	0.977465i	$0.432296\pi$
$458$	16544.3	1.68791
$459$	0	0
$460$	2057.31	0.208527
$461$	2778.11	0.280671	0.140336	−	0.990104i	$-0.455182\pi$
$461$	2778.11	0.280671	0.140336	+	0.990104i	$0.455182\pi$
$462$	2977.06	0.299795
$463$	6414.52	0.643862	0.321931	−	0.946763i	$-0.395668\pi$
$463$	6414.52	0.643862	0.321931	+	0.946763i	$0.395668\pi$
$464$	−9611.13	−0.961607
$465$	−12334.6	−1.23012
$466$	−3020.54	−0.300266
$467$	15377.5	1.52373	0.761867	−	0.647733i	$-0.224283\pi$
$467$	15377.5	1.52373	0.761867	+	0.647733i	$0.224283\pi$
$468$	−67.2948	−0.00664680
$469$	−9775.15	−0.962419
$470$	−4853.89	−0.476368
$471$	2607.70	0.255110
$472$	20248.0	1.97456
$473$	−6011.12	−0.584338
$474$	7716.73	0.747766
$475$	3271.86	0.316049
$476$	0	0
$477$	−4796.57	−0.460419
$478$	4788.50	0.458202
$479$	−3609.87	−0.344341	−0.172170	−	0.985067i	$-0.555078\pi$
$479$	−3609.87	−0.344341	−0.172170	+	0.985067i	$0.555078\pi$
$480$	−2144.49	−0.203921
$481$	1647.29	0.156154
$482$	−8902.81	−0.841311
$483$	−6391.92	−0.602158
$484$	916.283	0.0860521
$485$	9770.36	0.914741
$486$	636.411	0.0593995
$487$	−5320.06	−0.495020	−0.247510	−	0.968885i	$-0.579612\pi$
$487$	−5320.06	−0.495020	−0.247510	+	0.968885i	$0.579612\pi$
$488$	−22424.7	−2.08016
$489$	−8309.91	−0.768480
$490$	−2596.82	−0.239413
$491$	14135.4	1.29923	0.649613	−	0.760265i	$-0.274931\pi$
$491$	14135.4	1.29923	0.649613	+	0.760265i	$0.274931\pi$
$492$	−225.619	−0.0206741
$493$	0	0
$494$	−804.873	−0.0733055
$495$	−2885.96	−0.262049
$496$	15782.3	1.42872
$497$	4134.25	0.373131
$498$	4966.53	0.446899
$499$	11991.6	1.07579	0.537895	−	0.843012i	$-0.319220\pi$
$499$	11991.6	1.07579	0.537895	+	0.843012i	$0.319220\pi$
$500$	−879.471	−0.0786622
$501$	−9428.21	−0.840761
$502$	−2131.74	−0.189530
$503$	4959.32	0.439613	0.219806	−	0.975544i	$-0.429457\pi$
$503$	4959.32	0.439613	0.219806	+	0.975544i	$0.429457\pi$
$504$	3553.14	0.314027
$505$	5330.19	0.469684
$506$	−7774.70	−0.683058
$507$	6462.16	0.566064
$508$	−955.180	−0.0834237
$509$	13266.9	1.15529	0.577647	−	0.816286i	$-0.303971\pi$
$509$	13266.9	1.15529	0.577647	+	0.816286i	$0.303971\pi$
$510$	0	0
$511$	11208.1	0.970284
$512$	13003.7	1.12244
$513$	−1266.19	−0.108974
$514$	−2439.20	−0.209316
$515$	2529.38	0.216423
$516$	−895.498	−0.0763995
$517$	−3051.32	−0.259568
$518$	−10856.4	−0.920853
$519$	4126.17	0.348976
$520$	−2189.50	−0.184646
$521$	17873.0	1.50294	0.751470	−	0.659767i	$-0.229345\pi$
$521$	17873.0	1.50294	0.751470	+	0.659767i	$0.229345\pi$
$522$	−4228.86	−0.354583
$523$	12040.0	1.00664	0.503319	−	0.864101i	$-0.332112\pi$
$523$	12040.0	1.00664	0.503319	+	0.864101i	$0.332112\pi$
$524$	−2709.25	−0.225867
$525$	−3451.66	−0.286939
$526$	−8372.17	−0.694000
$527$	0	0
$528$	3692.62	0.304357
$529$	4525.73	0.371968
$530$	−19479.6	−1.59649
$531$	7612.05	0.622100
$532$	−882.382	−0.0719100
$533$	−431.956	−0.0351034
$534$	4988.11	0.404226
$535$	3487.71	0.281844
$536$	−14190.6	−1.14355
$537$	4154.88	0.333885
$538$	9313.29	0.746328
$539$	−1632.45	−0.130454
$540$	−429.932	−0.0342617
$541$	22193.6	1.76373	0.881865	−	0.471502i	$-0.156288\pi$
$541$	22193.6	1.76373	0.881865	+	0.471502i	$0.156288\pi$
$542$	−10444.9	−0.827760
$543$	−9354.87	−0.739330
$544$	0	0
$545$	−25615.6	−2.01331
$546$	849.104	0.0665536
$547$	−264.096	−0.0206434	−0.0103217	−	0.999947i	$-0.503286\pi$
$547$	−264.096	−0.0206434	−0.0103217	+	0.999947i	$0.503286\pi$
$548$	976.447	0.0761163
$549$	−8430.36	−0.655371
$550$	−4198.37	−0.325489
$551$	8413.64	0.650513
$552$	−9279.16	−0.715484
$553$	16196.7	1.24549
$554$	−17272.2	−1.32459
$555$	10524.2	0.804913
$556$	1986.72	0.151539
$557$	−12977.6	−0.987211	−0.493605	−	0.869686i	$-0.664321\pi$
$557$	−12977.6	−0.987211	−0.493605	+	0.869686i	$0.664321\pi$
$558$	6944.15	0.526827
$559$	−1714.47	−0.129721
$560$	12329.1	0.930354
$561$	0	0
$562$	−17996.4	−1.35077
$563$	1456.99	0.109067	0.0545335	−	0.998512i	$-0.482633\pi$
$563$	1456.99	0.109067	0.0545335	+	0.998512i	$0.482633\pi$
$564$	−454.566	−0.0339374
$565$	4421.47	0.329226
$566$	15533.2	1.15355
$567$	1335.77	0.0989365
$568$	6001.70	0.443355
$569$	6684.94	0.492526	0.246263	−	0.969203i	$-0.420797\pi$
$569$	6684.94	0.492526	0.246263	+	0.969203i	$0.420797\pi$
$570$	−5142.16	−0.377862
$571$	1472.61	0.107928	0.0539639	−	0.998543i	$-0.482814\pi$
$571$	1472.61	0.107928	0.0539639	+	0.998543i	$0.482814\pi$
$572$	−171.802	−0.0125584
$573$	−9770.95	−0.712369
$574$	2846.79	0.207008
$575$	9014.14	0.653767
$576$	5064.37	0.366346
$577$	−1141.54	−0.0823623	−0.0411812	−	0.999152i	$-0.513112\pi$
$577$	−1141.54	−0.0823623	−0.0411812	+	0.999152i	$0.513112\pi$
$578$	0	0
$579$	6079.02	0.436331
$580$	2856.84	0.204524
$581$	10424.3	0.744360
$582$	−5500.51	−0.391759
$583$	−12245.5	−0.869909
$584$	16270.8	1.15289
$585$	−823.122	−0.0581742
$586$	6874.56	0.484617
$587$	6439.45	0.452785	0.226392	−	0.974036i	$-0.427307\pi$
$587$	6439.45	0.452785	0.226392	+	0.974036i	$0.427307\pi$
$588$	−243.192	−0.0170562
$589$	−13815.9	−0.966510
$590$	30913.6	2.15711
$591$	−8333.15	−0.580000
$592$	−13465.8	−0.934866
$593$	−14614.3	−1.01203	−0.506017	−	0.862524i	$-0.668883\pi$
$593$	−14614.3	−1.01203	−0.506017	+	0.862524i	$0.668883\pi$
$594$	1624.74	0.112229
$595$	0	0
$596$	2805.72	0.192830
$597$	164.172	0.0112548
$598$	−2217.47	−0.151637
$599$	−1567.04	−0.106891	−0.0534455	−	0.998571i	$-0.517020\pi$
$599$	−1567.04	−0.106891	−0.0534455	+	0.998571i	$0.517020\pi$
$600$	−5010.78	−0.340941
$601$	−2516.77	−0.170817	−0.0854087	−	0.996346i	$-0.527220\pi$
$601$	−2516.77	−0.170817	−0.0854087	+	0.996346i	$0.527220\pi$
$602$	11299.1	0.764979
$603$	−5334.82	−0.360283
$604$	−1271.04	−0.0856255
$605$	11207.6	0.753146
$606$	−3000.79	−0.201153
$607$	21900.4	1.46443	0.732217	−	0.681071i	$-0.238486\pi$
$607$	21900.4	1.46443	0.732217	+	0.681071i	$0.238486\pi$
$608$	−2402.02	−0.160222
$609$	−8876.00	−0.590597
$610$	−34236.9	−2.27248
$611$	−870.284	−0.0576235
$612$	0	0
$613$	13213.1	0.870589	0.435294	−	0.900288i	$-0.356644\pi$
$613$	13213.1	0.870589	0.435294	+	0.900288i	$0.356644\pi$
$614$	14713.5	0.967082
$615$	−2759.68	−0.180945
$616$	9071.07	0.593318
$617$	−14870.0	−0.970251	−0.485126	−	0.874444i	$-0.661226\pi$
$617$	−14870.0	−0.970251	−0.485126	+	0.874444i	$0.661226\pi$
$618$	−1423.99	−0.0926883
$619$	−16392.4	−1.06440	−0.532202	−	0.846617i	$-0.678635\pi$
$619$	−16392.4	−1.06440	−0.532202	+	0.846617i	$0.678635\pi$
$620$	−4691.17	−0.303874
$621$	−3488.41	−0.225419
$622$	27789.5	1.79141
$623$	10469.6	0.673284
$624$	1053.19	0.0675665
$625$	−19478.4	−1.24662
$626$	−2176.57	−0.138967
$627$	−3232.54	−0.205893
$628$	991.776	0.0630194
$629$	0	0
$630$	5424.75	0.343059
$631$	−9666.81	−0.609872	−0.304936	−	0.952373i	$-0.598635\pi$
$631$	−9666.81	−0.609872	−0.304936	+	0.952373i	$0.598635\pi$
$632$	23512.8	1.47989
$633$	−5674.56	−0.356309
$634$	−2412.90	−0.151149
$635$	−11683.4	−0.730142
$636$	−1824.26	−0.113737
$637$	−465.600	−0.0289604
$638$	−10796.2	−0.669944
$639$	2256.28	0.139682
$640$	14848.5	0.917091
$641$	−15876.0	−0.978258	−0.489129	−	0.872212i	$-0.662685\pi$
$641$	−15876.0	−0.978258	−0.489129	+	0.872212i	$0.662685\pi$
$642$	−1963.51	−0.120706
$643$	16556.8	1.01545	0.507726	−	0.861519i	$-0.330486\pi$
$643$	16556.8	1.01545	0.507726	+	0.861519i	$0.330486\pi$
$644$	−2431.01	−0.148750
$645$	−10953.4	−0.668664
$646$	0	0
$647$	−6262.34	−0.380522	−0.190261	−	0.981733i	$-0.560934\pi$
$647$	−6262.34	−0.380522	−0.190261	+	0.981733i	$0.560934\pi$
$648$	1939.14	0.117556
$649$	19433.4	1.17539
$650$	−1197.44	−0.0722577
$651$	14575.1	0.877488
$652$	−3160.47	−0.189837
$653$	4980.21	0.298454	0.149227	−	0.988803i	$-0.452321\pi$
$653$	4980.21	0.298454	0.149227	+	0.988803i	$0.452321\pi$
$654$	14421.1	0.862245
$655$	−33138.5	−1.97683
$656$	3531.03	0.210158
$657$	6116.83	0.363227
$658$	5735.57	0.339811
$659$	−24997.0	−1.47761	−0.738806	−	0.673918i	$-0.764610\pi$
$659$	−24997.0	−1.47761	−0.738806	+	0.673918i	$0.764610\pi$
$660$	−1097.60	−0.0647337
$661$	9132.23	0.537371	0.268686	−	0.963228i	$-0.413411\pi$
$661$	9132.23	0.537371	0.268686	+	0.963228i	$0.413411\pi$
$662$	−30344.9	−1.78156
$663$	0	0
$664$	15133.0	0.884448
$665$	−10792.9	−0.629372
$666$	−5924.90	−0.344723
$667$	23180.0	1.34563
$668$	−3585.79	−0.207692
$669$	7944.94	0.459146
$670$	−21665.5	−1.24927
$671$	−21522.5	−1.23825
$672$	2534.02	0.145464
$673$	23959.7	1.37233	0.686166	−	0.727445i	$-0.259293\pi$
$673$	23959.7	1.37233	0.686166	+	0.727445i	$0.259293\pi$
$674$	13637.8	0.779388
$675$	−1883.76	−0.107416
$676$	2457.72	0.139834
$677$	6059.51	0.343997	0.171999	−	0.985097i	$-0.444978\pi$
$677$	6059.51	0.343997	0.171999	+	0.985097i	$0.444978\pi$
$678$	−2489.20	−0.140999
$679$	−11545.1	−0.652518
$680$	0	0
$681$	−17236.8	−0.969923
$682$	17728.2	0.995379
$683$	−13661.0	−0.765337	−0.382668	−	0.923886i	$-0.624995\pi$
$683$	−13661.0	−0.765337	−0.382668	+	0.923886i	$0.624995\pi$
$684$	−481.563	−0.0269196
$685$	11943.5	0.666186
$686$	17882.5	0.995272
$687$	18951.3	1.05245
$688$	14014.9	0.776621
$689$	−3492.61	−0.193117
$690$	−14166.9	−0.781632
$691$	−4623.31	−0.254528	−0.127264	−	0.991869i	$-0.540620\pi$
$691$	−4623.31	−0.254528	−0.127264	+	0.991869i	$0.540620\pi$
$692$	1569.29	0.0862071
$693$	3410.18	0.186929
$694$	−6865.27	−0.375507
$695$	24300.7	1.32630
$696$	−12885.3	−0.701748
$697$	0	0
$698$	25330.7	1.37361
$699$	−3459.99	−0.187223
$700$	−1312.75	−0.0708821
$701$	323.946	0.0174540	0.00872702	−	0.999962i	$-0.497222\pi$
$701$	323.946	0.0174540	0.00872702	+	0.999962i	$0.497222\pi$
$702$	463.401	0.0249145
$703$	11788.0	0.632424
$704$	12929.2	0.692170
$705$	−5560.06	−0.297027
$706$	−7362.04	−0.392456
$707$	−6298.38	−0.335043
$708$	2895.06	0.153676
$709$	34798.4	1.84327	0.921637	−	0.388053i	$-0.126852\pi$
$709$	34798.4	1.84327	0.921637	+	0.388053i	$0.126852\pi$
$710$	9163.07	0.484343
$711$	8839.41	0.466250
$712$	15198.7	0.799996
$713$	−38063.5	−1.99928
$714$	0	0
$715$	−2101.41	−0.109914
$716$	1580.21	0.0824792
$717$	5485.16	0.285700
$718$	7526.45	0.391204
$719$	−16523.2	−0.857039	−0.428520	−	0.903532i	$-0.640965\pi$
$719$	−16523.2	−0.857039	−0.428520	+	0.903532i	$0.640965\pi$
$720$	6728.63	0.348280
$721$	−2988.83	−0.154383
$722$	12203.9	0.629059
$723$	−10198.1	−0.524578
$724$	−3557.90	−0.182636
$725$	12517.3	0.641215
$726$	−6309.65	−0.322552
$727$	−20666.2	−1.05429	−0.527144	−	0.849776i	$-0.676737\pi$
$727$	−20666.2	−1.05429	−0.527144	+	0.849776i	$0.676737\pi$
$728$	2587.21	0.131715
$729$	729.000	0.0370370
$730$	24841.3	1.25948
$731$	0	0
$732$	−3206.28	−0.161896
$733$	26854.1	1.35318	0.676589	−	0.736360i	$-0.263457\pi$
$733$	26854.1	1.35318	0.676589	+	0.736360i	$0.263457\pi$
$734$	2143.42	0.107786
$735$	−2974.62	−0.149280
$736$	−6617.70	−0.331429
$737$	−13619.6	−0.680714
$738$	1553.64	0.0774937
$739$	9166.76	0.456299	0.228149	−	0.973626i	$-0.426733\pi$
$739$	9166.76	0.456299	0.228149	+	0.973626i	$0.426733\pi$
$740$	4002.61	0.198837
$741$	−921.971	−0.0457078
$742$	23017.9	1.13883
$743$	−5004.64	−0.247109	−0.123555	−	0.992338i	$-0.539429\pi$
$743$	−5004.64	−0.247109	−0.123555	+	0.992338i	$0.539429\pi$
$744$	21158.8	1.04263
$745$	34318.4	1.68769
$746$	−25070.6	−1.23043
$747$	5689.10	0.278652
$748$	0	0
$749$	−4121.23	−0.201050
$750$	6056.16	0.294853
$751$	24581.5	1.19440	0.597199	−	0.802093i	$-0.296280\pi$
$751$	24581.5	1.19440	0.597199	+	0.802093i	$0.296280\pi$
$752$	7114.16	0.344982
$753$	−2441.87	−0.118176
$754$	−3079.24	−0.148726
$755$	−15546.8	−0.749413
$756$	508.027	0.0244402
$757$	13742.9	0.659836	0.329918	−	0.944010i	$-0.392979\pi$
$757$	13742.9	0.659836	0.329918	+	0.944010i	$0.392979\pi$
$758$	−1791.38	−0.0858388
$759$	−8905.81	−0.425903
$760$	−15668.1	−0.747820
$761$	24971.1	1.18949	0.594746	−	0.803914i	$-0.297253\pi$
$761$	24971.1	1.18949	0.594746	+	0.803914i	$0.297253\pi$
$762$	6577.50	0.312700
$763$	30268.5	1.43616
$764$	−3716.14	−0.175976
$765$	0	0
$766$	−23717.8	−1.11875
$767$	5542.70	0.260933
$768$	5145.59	0.241765
$769$	−830.637	−0.0389513	−0.0194756	−	0.999810i	$-0.506200\pi$
$769$	−830.637	−0.0389513	−0.0194756	+	0.999810i	$0.506200\pi$
$770$	13849.2	0.648171
$771$	−2794.07	−0.130514
$772$	2312.01	0.107786
$773$	22381.9	1.04142	0.520712	−	0.853733i	$-0.325667\pi$
$773$	22381.9	1.04142	0.520712	+	0.853733i	$0.325667\pi$
$774$	6166.53	0.286371
$775$	−20554.5	−0.952694
$776$	−16760.0	−0.775322
$777$	−12435.8	−0.574174
$778$	−33933.9	−1.56374
$779$	−3091.09	−0.142169
$780$	−313.054	−0.0143707
$781$	5760.22	0.263914
$782$	0	0
$783$	−4844.11	−0.221091
$784$	3806.06	0.173381
$785$	12131.0	0.551559
$786$	18656.3	0.846625
$787$	18310.4	0.829345	0.414672	−	0.909971i	$-0.363896\pi$
$787$	18310.4	0.829345	0.414672	+	0.909971i	$0.363896\pi$
$788$	−3169.31	−0.143277
$789$	−9590.22	−0.432726
$790$	35898.1	1.61671
$791$	−5224.61	−0.234849
$792$	4950.57	0.222109
$793$	−6138.55	−0.274888
$794$	−11472.8	−0.512787
$795$	−22313.6	−0.995447
$796$	62.4386	0.00278025
$797$	34256.7	1.52250	0.761251	−	0.648457i	$-0.224586\pi$
$797$	34256.7	1.52250	0.761251	+	0.648457i	$0.224586\pi$
$798$	6076.21	0.269543
$799$	0	0
$800$	−3573.58	−0.157932
$801$	5713.82	0.252045
$802$	1193.32	0.0525407
$803$	15616.1	0.686277
$804$	−2028.97	−0.0890002
$805$	−29735.1	−1.30189
$806$	5056.37	0.220971
$807$	10668.3	0.465353
$808$	−9143.38	−0.398098
$809$	−21646.5	−0.940731	−0.470366	−	0.882472i	$-0.655878\pi$
$809$	−21646.5	−0.940731	−0.470366	+	0.882472i	$0.655878\pi$
$810$	2960.57	0.128425
$811$	601.339	0.0260368	0.0130184	−	0.999915i	$-0.495856\pi$
$811$	601.339	0.0260368	0.0130184	+	0.999915i	$0.495856\pi$
$812$	−3375.77	−0.145894
$813$	−11964.5	−0.516129
$814$	−15126.1	−0.651314
$815$	−38657.6	−1.66149
$816$	0	0
$817$	−12268.8	−0.525373
$818$	−24769.6	−1.05874
$819$	972.637	0.0414978
$820$	−1049.58	−0.0446985
$821$	−2987.72	−0.127006	−0.0635032	−	0.997982i	$-0.520227\pi$
$821$	−2987.72	−0.127006	−0.0635032	+	0.997982i	$0.520227\pi$
$822$	−6723.95	−0.285310
$823$	26515.0	1.12303	0.561515	−	0.827467i	$-0.310219\pi$
$823$	26515.0	1.12303	0.561515	+	0.827467i	$0.310219\pi$
$824$	−4338.89	−0.183438
$825$	−4809.18	−0.202950
$826$	−36528.9	−1.53874
$827$	23718.6	0.997313	0.498657	−	0.866800i	$-0.333827\pi$
$827$	23718.6	0.997313	0.498657	+	0.866800i	$0.333827\pi$
$828$	−1326.73	−0.0556849
$829$	29178.6	1.22245	0.611227	−	0.791455i	$-0.290676\pi$
$829$	29178.6	1.22245	0.611227	+	0.791455i	$0.290676\pi$
$830$	23104.2	0.966216
$831$	−19785.0	−0.825914
$832$	3687.61	0.153660
$833$	0	0
$834$	−13680.8	−0.568019
$835$	−43859.9	−1.81777
$836$	−1229.42	−0.0508616
$837$	7954.43	0.328489
$838$	19769.2	0.814936
$839$	−27390.5	−1.12709	−0.563543	−	0.826087i	$-0.690562\pi$
$839$	−27390.5	−1.12709	−0.563543	+	0.826087i	$0.690562\pi$
$840$	16529.2	0.678941
$841$	7799.42	0.319792
$842$	7308.36	0.299124
$843$	−20614.7	−0.842238
$844$	−2158.18	−0.0880184
$845$	30061.9	1.22386
$846$	3130.20	0.127209
$847$	−13243.4	−0.537247
$848$	28550.4	1.15616
$849$	17793.1	0.719268
$850$	0	0
$851$	32476.7	1.30821
$852$	858.120	0.0345055
$853$	18446.1	0.740425	0.370212	−	0.928947i	$-0.379285\pi$
$853$	18446.1	0.740425	0.370212	+	0.928947i	$0.379285\pi$
$854$	40455.8	1.62104
$855$	−5890.28	−0.235606
$856$	−5982.80	−0.238888
$857$	−37812.2	−1.50717	−0.753583	−	0.657353i	$-0.771676\pi$
$857$	−37812.2	−1.50717	−0.753583	+	0.657353i	$0.771676\pi$
$858$	1183.05	0.0470730
$859$	−4355.74	−0.173010	−0.0865051	−	0.996251i	$-0.527570\pi$
$859$	−4355.74	−0.173010	−0.0865051	+	0.996251i	$0.527570\pi$
$860$	−4165.85	−0.165179
$861$	3260.95	0.129074
$862$	2866.15	0.113250
$863$	7368.67	0.290652	0.145326	−	0.989384i	$-0.453577\pi$
$863$	7368.67	0.290652	0.145326	+	0.989384i	$0.453577\pi$
$864$	1382.95	0.0544548
$865$	19194.9	0.754503
$866$	15822.4	0.620862
$867$	0	0
$868$	5543.30	0.216765
$869$	22566.8	0.880927
$870$	−19672.6	−0.766625
$871$	−3884.54	−0.151117
$872$	43940.9	1.70645
$873$	−6300.77	−0.244271
$874$	−15868.2	−0.614132
$875$	12711.3	0.491110
$876$	2326.39	0.0897275
$877$	−9775.74	−0.376401	−0.188200	−	0.982131i	$-0.560265\pi$
$877$	−9775.74	−0.376401	−0.188200	+	0.982131i	$0.560265\pi$
$878$	−1498.39	−0.0575947
$879$	7874.72	0.302170
$880$	17178.0	0.658035
$881$	5851.55	0.223773	0.111886	−	0.993721i	$-0.464311\pi$
$881$	5851.55	0.223773	0.111886	+	0.993721i	$0.464311\pi$
$882$	1674.65	0.0639325
$883$	8886.47	0.338679	0.169340	−	0.985558i	$-0.445837\pi$
$883$	8886.47	0.338679	0.169340	+	0.985558i	$0.445837\pi$
$884$	0	0
$885$	35411.2	1.34501
$886$	−14030.0	−0.531996
$887$	18647.2	0.705877	0.352939	−	0.935647i	$-0.385182\pi$
$887$	18647.2	0.705877	0.352939	+	0.935647i	$0.385182\pi$
$888$	−18053.1	−0.682234
$889$	13805.6	0.520837
$890$	23204.6	0.873956
$891$	1861.12	0.0699773
$892$	3021.66	0.113422
$893$	−6227.77	−0.233376
$894$	−19320.6	−0.722793
$895$	19328.4	0.721876
$896$	−17545.6	−0.654195
$897$	−2540.08	−0.0945493
$898$	−28663.9	−1.06518
$899$	−52856.1	−1.96090
$900$	−716.440	−0.0265348
$901$	0	0
$902$	3966.40	0.146416
$903$	12943.0	0.476983
$904$	−7584.57	−0.279048
$905$	−43518.7	−1.59847
$906$	8752.55	0.320953
$907$	5647.26	0.206741	0.103370	−	0.994643i	$-0.467037\pi$
$907$	5647.26	0.206741	0.103370	+	0.994643i	$0.467037\pi$
$908$	−6555.61	−0.239599
$909$	−3437.36	−0.125424
$910$	3950.02	0.143892
$911$	−226.380	−0.00823306	−0.00411653	−	0.999992i	$-0.501310\pi$
$911$	−226.380	−0.00823306	−0.00411653	+	0.999992i	$0.501310\pi$
$912$	7536.67	0.273645
$913$	14524.1	0.526482
$914$	−10802.4	−0.390933
$915$	−39217.9	−1.41694
$916$	7207.65	0.259986
$917$	39157.9	1.41015
$918$	0	0
$919$	−1341.34	−0.0481466	−0.0240733	−	0.999710i	$-0.507664\pi$
$919$	−1341.34	−0.0481466	−0.0240733	+	0.999710i	$0.507664\pi$
$920$	−43166.5	−1.54691
$921$	16854.1	0.602999
$922$	−7275.80	−0.259887
$923$	1642.90	0.0585882
$924$	1296.98	0.0461769
$925$	17537.5	0.623384
$926$	−16799.5	−0.596182
$927$	−1631.16	−0.0577934
$928$	−9189.53	−0.325066
$929$	−3269.44	−0.115465	−0.0577324	−	0.998332i	$-0.518387\pi$
$929$	−3269.44	−0.115465	−0.0577324	+	0.998332i	$0.518387\pi$
$930$	32304.1	1.13902
$931$	−3331.85	−0.117290
$932$	−1315.92	−0.0462494
$933$	31832.5	1.11699
$934$	−40273.2	−1.41090
$935$	0	0
$936$	1411.98	0.0493077
$937$	−14892.7	−0.519236	−0.259618	−	0.965711i	$-0.583597\pi$
$937$	−14892.7	−0.519236	−0.259618	+	0.965711i	$0.583597\pi$
$938$	25600.9	0.891149
$939$	−2493.23	−0.0866492
$940$	−2114.63	−0.0733742
$941$	−18900.2	−0.654760	−0.327380	−	0.944893i	$-0.606166\pi$
$941$	−18900.2	−0.654760	−0.327380	+	0.944893i	$0.606166\pi$
$942$	−6829.51	−0.236218
$943$	−8516.11	−0.294085
$944$	−45308.9	−1.56216
$945$	6213.98	0.213905
$946$	15743.0	0.541066
$947$	22282.0	0.764591	0.382296	−	0.924040i	$-0.375134\pi$
$947$	22282.0	0.764591	0.382296	+	0.924040i	$0.375134\pi$
$948$	3361.85	0.115177
$949$	4453.96	0.152352
$950$	−8568.91	−0.292645
$951$	−2763.94	−0.0942449
$952$	0	0
$953$	−11984.3	−0.407355	−0.203677	−	0.979038i	$-0.565289\pi$
$953$	−11984.3	−0.407355	−0.203677	+	0.979038i	$0.565289\pi$
$954$	12562.1	0.426324
$955$	−45454.3	−1.54018
$956$	2086.15	0.0705761
$957$	−12366.9	−0.417726
$958$	9454.16	0.318841
$959$	−14113.0	−0.475215
$960$	23559.4	0.792058
$961$	57003.2	1.91344
$962$	−4314.21	−0.144590
$963$	−2249.17	−0.0752634
$964$	−3878.58	−0.129586
$965$	28279.5	0.943368
$966$	16740.3	0.557566
$967$	40295.9	1.34005	0.670025	−	0.742338i	$-0.266283\pi$
$967$	40295.9	1.34005	0.670025	+	0.742338i	$0.266283\pi$
$968$	−19225.5	−0.638357
$969$	0	0
$970$	−25588.3	−0.847001
$971$	8158.11	0.269625	0.134813	−	0.990871i	$-0.456957\pi$
$971$	8158.11	0.269625	0.134813	+	0.990871i	$0.456957\pi$
$972$	277.257	0.00914921
$973$	−28714.8	−0.946099
$974$	13933.1	0.458363
$975$	−1371.65	−0.0450544
$976$	50179.7	1.64571
$977$	24635.9	0.806727	0.403364	−	0.915040i	$-0.367841\pi$
$977$	24635.9	0.806727	0.403364	+	0.915040i	$0.367841\pi$
$978$	21763.4	0.711572
$979$	14587.2	0.476210
$980$	−1131.33	−0.0368764
$981$	16519.1	0.537630
$982$	−37020.1	−1.20301
$983$	−22082.3	−0.716497	−0.358249	−	0.933626i	$-0.616626\pi$
$983$	−22082.3	−0.716497	−0.358249	+	0.933626i	$0.616626\pi$
$984$	4733.94	0.153366
$985$	−38765.7	−1.25399
$986$	0	0
$987$	6570.01	0.211880
$988$	−350.649	−0.0112911
$989$	−33801.1	−1.08677
$990$	7558.26	0.242644
$991$	−41034.4	−1.31534	−0.657669	−	0.753307i	$-0.728457\pi$
$991$	−41034.4	−1.31534	−0.657669	+	0.753307i	$0.728457\pi$
$992$	15090.0	0.482971
$993$	−34759.7	−1.11084
$994$	−10827.5	−0.345500
$995$	763.724	0.0243333
$996$	2163.71	0.0688350
$997$	−50908.7	−1.61715	−0.808573	−	0.588396i	$-0.799760\pi$
$997$	−50908.7	−1.61715	−0.808573	+	0.588396i	$0.799760\pi$
$998$	−31405.7	−0.996124
$999$	−6786.90	−0.214943

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	867.4.a.p.1.3		8
17.2	even	8		51.4.e.a.4.3	✓	16
17.9	even	8		51.4.e.a.13.6	yes	16
17.16	even	2		867.4.a.q.1.3		8
51.2	odd	8		153.4.f.b.55.6		16
51.26	odd	8		153.4.f.b.64.3		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
51.4.e.a.4.3	✓	16	17.2	even	8
51.4.e.a.13.6	yes	16	17.9	even	8
153.4.f.b.55.6		16	51.2	odd	8
153.4.f.b.64.3		16	51.26	odd	8
867.4.a.p.1.3		8	1.1	even	1	trivial
867.4.a.q.1.3		8	17.16	even	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 867 = 3 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	867.a (trivial)

\(f(q)\)	\(=\)	\(q-2.61897 q^{2} -3.00000 q^{3} -1.14098 q^{4} -13.9560 q^{5} +7.85692 q^{6} +16.4910 q^{7} +23.9400 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-2.61897 q^{2} -3.00000 q^{3} -1.14098 q^{4} -13.9560 q^{5} +7.85692 q^{6} +16.4910 q^{7} +23.9400 q^{8} +9.00000 q^{9} +36.5503 q^{10} +22.9767 q^{11} +3.42293 q^{12} +6.55333 q^{13} -43.1894 q^{14} +41.8679 q^{15} -53.5704 q^{16} -23.5708 q^{18} +46.8958 q^{19} +15.9234 q^{20} -49.4729 q^{21} -60.1755 q^{22} +129.200 q^{23} -71.8199 q^{24} +69.7687 q^{25} -17.1630 q^{26} -27.0000 q^{27} -18.8158 q^{28} +179.411 q^{29} -109.651 q^{30} -294.609 q^{31} -51.2204 q^{32} -68.9302 q^{33} -230.147 q^{35} -10.2688 q^{36} +251.367 q^{37} -122.819 q^{38} -19.6600 q^{39} -334.105 q^{40} -65.9139 q^{41} +129.568 q^{42} -261.618 q^{43} -26.2159 q^{44} -125.604 q^{45} -338.372 q^{46} -132.800 q^{47} +160.711 q^{48} -71.0479 q^{49} -182.722 q^{50} -7.47720 q^{52} -532.952 q^{53} +70.7123 q^{54} -320.663 q^{55} +394.793 q^{56} -140.687 q^{57} -469.874 q^{58} +845.784 q^{59} -47.7702 q^{60} -936.707 q^{61} +771.572 q^{62} +148.419 q^{63} +562.708 q^{64} -91.4580 q^{65} +180.527 q^{66} -592.758 q^{67} -387.601 q^{69} +602.750 q^{70} +250.698 q^{71} +215.460 q^{72} +679.648 q^{73} -658.323 q^{74} -209.306 q^{75} -53.5070 q^{76} +378.909 q^{77} +51.4890 q^{78} +982.157 q^{79} +747.626 q^{80} +81.0000 q^{81} +172.627 q^{82} +632.122 q^{83} +56.4474 q^{84} +685.170 q^{86} -538.234 q^{87} +550.063 q^{88} +634.869 q^{89} +328.953 q^{90} +108.071 q^{91} -147.415 q^{92} +883.826 q^{93} +347.800 q^{94} -654.476 q^{95} +153.661 q^{96} -700.085 q^{97} +186.073 q^{98} +206.791 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{2} - 24 q^{3} + 24 q^{4} + 24 q^{5} + 12 q^{6} + 28 q^{7} - 48 q^{8} + 72 q^{9}+O(q^{10}) \) 8 * q - 4 * q^2 - 24 * q^3 + 24 * q^4 + 24 * q^5 + 12 * q^6 + 28 * q^7 - 48 * q^8 + 72 * q^9	\( 8 q - 4 q^{2} - 24 q^{3} + 24 q^{4} + 24 q^{5} + 12 q^{6} + 28 q^{7} - 48 q^{8} + 72 q^{9} - 72 q^{10} + 12 q^{11} - 72 q^{12} - 60 q^{13} + 56 q^{14} - 72 q^{15} - 100 q^{16} - 36 q^{18} - 140 q^{19} + 224 q^{20} - 84 q^{21} + 484 q^{22} - 96 q^{23} + 144 q^{24} + 220 q^{25} + 500 q^{26} - 216 q^{27} + 348 q^{28} + 380 q^{29} + 216 q^{30} + 124 q^{31} - 448 q^{32} - 36 q^{33} - 592 q^{35} + 216 q^{36} + 1260 q^{37} + 84 q^{38} + 180 q^{39} - 836 q^{40} + 284 q^{41} - 168 q^{42} - 76 q^{43} - 1460 q^{44} + 216 q^{45} + 1056 q^{46} - 184 q^{47} + 300 q^{48} - 536 q^{49} - 360 q^{50} - 1580 q^{52} + 1064 q^{53} + 108 q^{54} - 2148 q^{55} + 244 q^{56} + 420 q^{57} - 532 q^{58} + 1224 q^{59} - 672 q^{60} + 324 q^{61} + 3068 q^{62} + 252 q^{63} - 876 q^{64} + 652 q^{65} - 1452 q^{66} - 624 q^{67} + 288 q^{69} - 1932 q^{70} + 2636 q^{71} - 432 q^{72} + 1640 q^{73} - 120 q^{74} - 660 q^{75} - 1172 q^{76} + 504 q^{77} - 1500 q^{78} + 3788 q^{79} + 1740 q^{80} + 648 q^{81} + 1916 q^{82} + 2032 q^{83} - 1044 q^{84} + 1612 q^{86} - 1140 q^{87} + 4576 q^{88} + 1304 q^{89} - 648 q^{90} - 2872 q^{91} - 1744 q^{92} - 372 q^{93} - 208 q^{94} - 748 q^{95} + 1344 q^{96} + 2376 q^{97} + 604 q^{98} + 108 q^{99}+O(q^{100}) \) 8 * q - 4 * q^2 - 24 * q^3 + 24 * q^4 + 24 * q^5 + 12 * q^6 + 28 * q^7 - 48 * q^8 + 72 * q^9 - 72 * q^10 + 12 * q^11 - 72 * q^12 - 60 * q^13 + 56 * q^14 - 72 * q^15 - 100 * q^16 - 36 * q^18 - 140 * q^19 + 224 * q^20 - 84 * q^21 + 484 * q^22 - 96 * q^23 + 144 * q^24 + 220 * q^25 + 500 * q^26 - 216 * q^27 + 348 * q^28 + 380 * q^29 + 216 * q^30 + 124 * q^31 - 448 * q^32 - 36 * q^33 - 592 * q^35 + 216 * q^36 + 1260 * q^37 + 84 * q^38 + 180 * q^39 - 836 * q^40 + 284 * q^41 - 168 * q^42 - 76 * q^43 - 1460 * q^44 + 216 * q^45 + 1056 * q^46 - 184 * q^47 + 300 * q^48 - 536 * q^49 - 360 * q^50 - 1580 * q^52 + 1064 * q^53 + 108 * q^54 - 2148 * q^55 + 244 * q^56 + 420 * q^57 - 532 * q^58 + 1224 * q^59 - 672 * q^60 + 324 * q^61 + 3068 * q^62 + 252 * q^63 - 876 * q^64 + 652 * q^65 - 1452 * q^66 - 624 * q^67 + 288 * q^69 - 1932 * q^70 + 2636 * q^71 - 432 * q^72 + 1640 * q^73 - 120 * q^74 - 660 * q^75 - 1172 * q^76 + 504 * q^77 - 1500 * q^78 + 3788 * q^79 + 1740 * q^80 + 648 * q^81 + 1916 * q^82 + 2032 * q^83 - 1044 * q^84 + 1612 * q^86 - 1140 * q^87 + 4576 * q^88 + 1304 * q^89 - 648 * q^90 - 2872 * q^91 - 1744 * q^92 - 372 * q^93 - 208 * q^94 - 748 * q^95 + 1344 * q^96 + 2376 * q^97 + 604 * q^98 + 108 * q^99

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