LMFDB - Embedded newform 363.8.a.m.1.10

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [363,8,Mod(1,363)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(363, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 8, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("363.1"); S:= CuspForms(chi, 8); N := Newforms(S);

Level:	$ N $	$=$	$ 363 = 3 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	363.a (trivial)

Newform invariants

comment:select newform

sage:traces = [10,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$113.395764251$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 883x^{8} + 270112x^{6} - 33122976x^{4} + 1371591936x^{2} - 94618368 $ x^10 - 883x^8 + 270112x^6 - 33122976x^4 + 1371591936x^2 - 94618368
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{2}\cdot 11^{4} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$18.5524$ of defining polynomial
Character	$\chi$	$=$	363.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+18.5524 q^{2} +27.0000 q^{3} +216.192 q^{4} -234.289 q^{5} +500.915 q^{6} -468.897 q^{7} +1636.18 q^{8} +729.000 q^{9} -4346.62 q^{10} +5837.19 q^{12} -3588.75 q^{13} -8699.17 q^{14} -6325.79 q^{15} +2682.47 q^{16} +625.901 q^{17} +13524.7 q^{18} +26359.3 q^{19} -50651.4 q^{20} -12660.2 q^{21} +78104.0 q^{23} +44176.8 q^{24} -23233.9 q^{25} -66579.9 q^{26} +19683.0 q^{27} -101372. q^{28} -149494. q^{29} -117359. q^{30} -311481. q^{31} -159665. q^{32} +11612.0 q^{34} +109857. q^{35} +157604. q^{36} -204907. q^{37} +489029. q^{38} -96896.2 q^{39} -383338. q^{40} -760746. q^{41} -234878. q^{42} +98326.7 q^{43} -170796. q^{45} +1.44902e6 q^{46} -588443. q^{47} +72426.6 q^{48} -603679. q^{49} -431044. q^{50} +16899.3 q^{51} -775859. q^{52} -458970. q^{53} +365167. q^{54} -767199. q^{56} +711701. q^{57} -2.77347e6 q^{58} -738903. q^{59} -1.36759e6 q^{60} +2.12197e6 q^{61} -5.77872e6 q^{62} -341826. q^{63} -3.30552e6 q^{64} +840803. q^{65} +3.34623e6 q^{67} +135315. q^{68} +2.10881e6 q^{69} +2.03812e6 q^{70} -2.89651e6 q^{71} +1.19277e6 q^{72} -338231. q^{73} -3.80152e6 q^{74} -627314. q^{75} +5.69867e6 q^{76} -1.79766e6 q^{78} -7.85461e6 q^{79} -628471. q^{80} +531441. q^{81} -1.41137e7 q^{82} -8.90762e6 q^{83} -2.73704e6 q^{84} -146642. q^{85} +1.82420e6 q^{86} -4.03633e6 q^{87} +7.44444e6 q^{89} -3.16869e6 q^{90} +1.68275e6 q^{91} +1.68855e7 q^{92} -8.40998e6 q^{93} -1.09170e7 q^{94} -6.17568e6 q^{95} -4.31094e6 q^{96} +80295.7 q^{97} -1.11997e7 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 270 q^{3} + 486 q^{4} - 946 q^{5} + 7290 q^{9} + 13122 q^{12} - 16948 q^{14} - 25542 q^{15} - 35374 q^{16} - 48738 q^{20} + 2108 q^{23} - 76684 q^{25} - 367002 q^{26} + 196830 q^{27} - 664572 q^{31}+ \cdots + 15434810 q^{97}+O(q^{100}) $ 10 * q + 270 * q^3 + 486 * q^4 - 946 * q^5 + 7290 * q^9 + 13122 * q^12 - 16948 * q^14 - 25542 * q^15 - 35374 * q^16 - 48738 * q^20 + 2108 * q^23 - 76684 * q^25 - 367002 * q^26 + 196830 * q^27 - 664572 * q^31 + 68938 * q^34 + 354294 * q^36 - 1031062 * q^37 + 1077608 * q^38 - 457596 * q^42 - 689634 * q^45 - 219544 * q^47 - 955098 * q^48 - 51026 * q^49 - 6176094 * q^53 - 6137980 * q^56 - 4487498 * q^58 - 3815652 * q^59 - 1315926 * q^60 - 9266602 * q^64 - 14109384 * q^67 + 56916 * q^69 - 15425108 * q^70 - 5276156 * q^71 - 2070468 * q^75 - 9909054 * q^78 + 934666 * q^80 + 5314410 * q^81 - 26188378 * q^82 + 3673636 * q^86 - 9677426 * q^89 - 28234212 * q^91 + 19508076 * q^92 - 17943444 * q^93 + 15434810 * q^97

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 $

$2$	18.5524	1.63982	0.819909	−	0.572494i	$-0.194024\pi$
$2$	18.5524	1.63982	0.819909	+	0.572494i	$0.194024\pi$
$3$	27.0000	0.577350
$3$	27.0000	0.577350
$4$	216.192	1.68900
$5$	−234.289	−0.838216	−0.419108	−	0.907936i	$-0.637657\pi$
$5$	−234.289	−0.838216	−0.419108	+	0.907936i	$0.637657\pi$
$6$	500.915	0.946749
$7$	−468.897	−0.516695	−0.258347	−	0.966052i	$-0.583178\pi$
$7$	−468.897	−0.516695	−0.258347	+	0.966052i	$0.583178\pi$
$8$	1636.18	1.12984
$9$	729.000	0.333333
$10$	−4346.62	−1.37452
$11$	0	0
$11$	0	0
$12$	5837.19	0.975146
$13$	−3588.75	−0.453045	−0.226522	−	0.974006i	$-0.572736\pi$
$13$	−3588.75	−0.453045	−0.226522	+	0.974006i	$0.572736\pi$
$14$	−8699.17	−0.847285
$15$	−6325.79	−0.483944
$16$	2682.47	0.163725
$17$	625.901	0.0308983	0.0154492	−	0.999881i	$-0.495082\pi$
$17$	625.901	0.0308983	0.0154492	+	0.999881i	$0.495082\pi$
$18$	13524.7	0.546606
$19$	26359.3	0.881650	0.440825	−	0.897593i	$-0.354686\pi$
$19$	26359.3	0.881650	0.440825	+	0.897593i	$0.354686\pi$
$20$	−50651.4	−1.41575
$21$	−12660.2	−0.298314
$22$	0	0
$23$	78104.0	1.33852	0.669261	−	0.743027i	$-0.266611\pi$
$23$	78104.0	1.33852	0.669261	+	0.743027i	$0.266611\pi$
$24$	44176.8	0.652312
$25$	−23233.9	−0.297393
$26$	−66579.9	−0.742911
$27$	19683.0	0.192450
$28$	−101372.	−0.872699
$29$	−149494.	−1.13823	−0.569115	−	0.822258i	$-0.692714\pi$
$29$	−149494.	−1.13823	−0.569115	+	0.822258i	$0.692714\pi$
$30$	−117359.	−0.793581
$31$	−311481.	−1.87787	−0.938935	−	0.344096i	$-0.888185\pi$
$31$	−311481.	−1.87787	−0.938935	+	0.344096i	$0.888185\pi$
$32$	−159665.	−0.861358
$33$	0	0
$34$	11612.0	0.0506676
$35$	109857.	0.433102
$36$	157604.	0.563001
$37$	−204907.	−0.665045	−0.332522	−	0.943095i	$-0.607900\pi$
$37$	−204907.	−0.665045	−0.332522	+	0.943095i	$0.607900\pi$
$38$	489029.	1.44575
$39$	−96896.2	−0.261566
$40$	−383338.	−0.947048
$41$	−760746.	−1.72384	−0.861918	−	0.507048i	$-0.830737\pi$
$41$	−760746.	−1.72384	−0.861918	+	0.507048i	$0.830737\pi$
$42$	−234878.	−0.489180
$43$	98326.7	0.188596	0.0942979	−	0.995544i	$-0.469939\pi$
$43$	98326.7	0.188596	0.0942979	+	0.995544i	$0.469939\pi$
$44$	0	0
$45$	−170796.	−0.279405
$46$	1.44902e6	2.19493
$47$	−588443.	−0.826726	−0.413363	−	0.910566i	$-0.635646\pi$
$47$	−588443.	−0.826726	−0.413363	+	0.910566i	$0.635646\pi$
$48$	72426.6	0.0945265
$49$	−603679.	−0.733026
$50$	−431044.	−0.487671
$51$	16899.3	0.0178391
$52$	−775859.	−0.765193
$53$	−458970.	−0.423466	−0.211733	−	0.977328i	$-0.567911\pi$
$53$	−458970.	−0.423466	−0.211733	+	0.977328i	$0.567911\pi$
$54$	365167.	0.315583
$55$	0	0
$56$	−767199.	−0.583781
$57$	711701.	0.509021
$58$	−2.77347e6	−1.86649
$59$	−738903.	−0.468387	−0.234194	−	0.972190i	$-0.575245\pi$
$59$	−738903.	−0.468387	−0.234194	+	0.972190i	$0.575245\pi$
$60$	−1.36759e6	−0.817383
$61$	2.12197e6	1.19697	0.598487	−	0.801132i	$-0.295769\pi$
$61$	2.12197e6	1.19697	0.598487	+	0.801132i	$0.295769\pi$
$62$	−5.77872e6	−3.07936
$63$	−341826.	−0.172232
$64$	−3.30552e6	−1.57619
$65$	840803.	0.379750
$66$	0	0
$67$	3.34623e6	1.35924	0.679618	−	0.733567i	$-0.262146\pi$
$67$	3.34623e6	1.35924	0.679618	+	0.733567i	$0.262146\pi$
$68$	135315.	0.0521873
$69$	2.10881e6	0.772796
$70$	2.03812e6	0.710208
$71$	−2.89651e6	−0.960440	−0.480220	−	0.877148i	$-0.659443\pi$
$71$	−2.89651e6	−0.960440	−0.480220	+	0.877148i	$0.659443\pi$
$72$	1.19277e6	0.376612
$73$	−338231.	−0.101762	−0.0508808	−	0.998705i	$-0.516203\pi$
$73$	−338231.	−0.101762	−0.0508808	+	0.998705i	$0.516203\pi$
$74$	−3.80152e6	−1.09055
$75$	−627314.	−0.171700
$76$	5.69867e6	1.48911
$77$	0	0
$78$	−1.79766e6	−0.428920
$79$	−7.85461e6	−1.79238	−0.896189	−	0.443672i	$-0.853676\pi$
$79$	−7.85461e6	−1.79238	−0.896189	+	0.443672i	$0.853676\pi$
$80$	−628471.	−0.137237
$81$	531441.	0.111111
$82$	−1.41137e7	−2.82678
$83$	−8.90762e6	−1.70997	−0.854985	−	0.518653i	$-0.826434\pi$
$83$	−8.90762e6	−1.70997	−0.854985	+	0.518653i	$0.826434\pi$
$84$	−2.73704e6	−0.503853
$85$	−146642.	−0.0258995
$86$	1.82420e6	0.309263
$87$	−4.03633e6	−0.657157
$88$	0	0
$89$	7.44444e6	1.11935	0.559676	−	0.828711i	$-0.310925\pi$
$89$	7.44444e6	1.11935	0.559676	+	0.828711i	$0.310925\pi$
$90$	−3.16869e6	−0.458174
$91$	1.68275e6	0.234086
$92$	1.68855e7	2.26077
$93$	−8.40998e6	−1.08419
$94$	−1.09170e7	−1.35568
$95$	−6.17568e6	−0.739013
$96$	−4.31094e6	−0.497305
$97$	80295.7	0.00893287	0.00446644	−	0.999990i	$-0.498578\pi$
$97$	80295.7	0.00893287	0.00446644	+	0.999990i	$0.498578\pi$
$98$	−1.11997e7	−1.20203
$99$	0	0
$100$	−5.02298e6	−0.502298
$101$	8.62184e6	0.832675	0.416337	−	0.909210i	$-0.363314\pi$
$101$	8.62184e6	0.832675	0.416337	+	0.909210i	$0.363314\pi$
$102$	313524.	0.0292529
$103$	1.90067e7	1.71386	0.856932	−	0.515429i	$-0.172367\pi$
$103$	1.90067e7	1.71386	0.856932	+	0.515429i	$0.172367\pi$
$104$	−5.87183e6	−0.511867
$105$	2.96614e6	0.250052
$106$	−8.51501e6	−0.694408
$107$	1.63674e7	1.29163	0.645814	−	0.763494i	$-0.276518\pi$
$107$	1.63674e7	1.29163	0.645814	+	0.763494i	$0.276518\pi$
$108$	4.25531e6	0.325049
$109$	6.35346e6	0.469913	0.234957	−	0.972006i	$-0.424505\pi$
$109$	6.35346e6	0.469913	0.234957	+	0.972006i	$0.424505\pi$
$110$	0	0
$111$	−5.53249e6	−0.383964
$112$	−1.25780e6	−0.0845957
$113$	3.73822e6	0.243719	0.121860	−	0.992547i	$-0.461114\pi$
$113$	3.73822e6	0.243719	0.121860	+	0.992547i	$0.461114\pi$
$114$	1.32038e7	0.834701
$115$	−1.82989e7	−1.12197
$116$	−3.23194e7	−1.92247
$117$	−2.61620e6	−0.151015
$118$	−1.37084e7	−0.768070
$119$	−293483.	−0.0159650
$120$	−1.03501e7	−0.546778
$121$	0	0
$122$	3.93677e7	1.96282
$123$	−2.05401e7	−0.995257
$124$	−6.73397e7	−3.17172
$125$	2.37472e7	1.08750
$126$	−6.34170e6	−0.282428
$127$	−1.23971e7	−0.537042	−0.268521	−	0.963274i	$-0.586535\pi$
$127$	−1.23971e7	−0.537042	−0.268521	+	0.963274i	$0.586535\pi$
$128$	−4.08883e7	−1.72331
$129$	2.65482e6	0.108886
$130$	1.55989e7	0.622720
$131$	1.85775e7	0.722002	0.361001	−	0.932565i	$-0.382435\pi$
$131$	1.85775e7	0.722002	0.361001	+	0.932565i	$0.382435\pi$
$132$	0	0
$133$	−1.23598e7	−0.455544
$134$	6.20807e7	2.22890
$135$	−4.61150e6	−0.161315
$136$	1.02409e6	0.0349100
$137$	1.02260e7	0.339770	0.169885	−	0.985464i	$-0.445660\pi$
$137$	1.02260e7	0.339770	0.169885	+	0.985464i	$0.445660\pi$
$138$	3.91235e7	1.26725
$139$	3.95243e7	1.24828	0.624140	−	0.781312i	$-0.285449\pi$
$139$	3.95243e7	1.24828	0.624140	+	0.781312i	$0.285449\pi$
$140$	2.37503e7	0.731510
$141$	−1.58880e7	−0.477311
$142$	−5.37372e7	−1.57495
$143$	0	0
$144$	1.95552e6	0.0545749
$145$	3.50247e7	0.954083
$146$	−6.27501e6	−0.166870
$147$	−1.62993e7	−0.423213
$148$	−4.42993e7	−1.12326
$149$	827328.	0.0204892	0.0102446	−	0.999948i	$-0.496739\pi$
$149$	827328.	0.0204892	0.0102446	+	0.999948i	$0.496739\pi$
$150$	−1.16382e7	−0.281557
$151$	−3.45316e7	−0.816202	−0.408101	−	0.912937i	$-0.633809\pi$
$151$	−3.45316e7	−0.816202	−0.408101	+	0.912937i	$0.633809\pi$
$152$	4.31285e7	0.996121
$153$	456282.	0.0102994
$154$	0	0
$155$	7.29764e7	1.57406
$156$	−2.09482e7	−0.441785
$157$	6.55426e6	0.135168	0.0675841	−	0.997714i	$-0.478471\pi$
$157$	6.55426e6	0.135168	0.0675841	+	0.997714i	$0.478471\pi$
$158$	−1.45722e8	−2.93917
$159$	−1.23922e7	−0.244488
$160$	3.74076e7	0.722005
$161$	−3.66227e7	−0.691608
$162$	9.85952e6	0.182202
$163$	2.80135e7	0.506653	0.253326	−	0.967381i	$-0.418475\pi$
$163$	2.80135e7	0.506653	0.253326	+	0.967381i	$0.418475\pi$
$164$	−1.64467e8	−2.91156
$165$	0	0
$166$	−1.65258e8	−2.80404
$167$	5.82636e7	0.968032	0.484016	−	0.875059i	$-0.339178\pi$
$167$	5.82636e7	0.968032	0.484016	+	0.875059i	$0.339178\pi$
$168$	−2.07144e7	−0.337046
$169$	−4.98694e7	−0.794750
$170$	−2.72055e6	−0.0424704
$171$	1.92159e7	0.293883
$172$	2.12575e7	0.318539
$173$	5.93699e7	0.871775	0.435888	−	0.900001i	$-0.356434\pi$
$173$	5.93699e7	0.871775	0.435888	+	0.900001i	$0.356434\pi$
$174$	−7.48837e7	−1.07762
$175$	1.08943e7	0.153662
$176$	0	0
$177$	−1.99504e7	−0.270424
$178$	1.38112e8	1.83553
$179$	−6.78540e7	−0.884280	−0.442140	−	0.896946i	$-0.645781\pi$
$179$	−6.78540e7	−0.884280	−0.442140	+	0.896946i	$0.645781\pi$
$180$	−3.69248e7	−0.471916
$181$	−3.36683e7	−0.422033	−0.211017	−	0.977482i	$-0.567677\pi$
$181$	−3.36683e7	−0.422033	−0.211017	+	0.977482i	$0.567677\pi$
$182$	3.12191e7	0.383858
$183$	5.72932e7	0.691073
$184$	1.27792e8	1.51231
$185$	4.80074e7	0.557451
$186$	−1.56025e8	−1.77787
$187$	0	0
$188$	−1.27217e8	−1.39634
$189$	−9.22930e6	−0.0994380
$190$	−1.14574e8	−1.21185
$191$	−9.16013e7	−0.951229	−0.475614	−	0.879654i	$-0.657774\pi$
$191$	−9.16013e7	−0.951229	−0.475614	+	0.879654i	$0.657774\pi$
$192$	−8.92491e7	−0.910017
$193$	4.36787e7	0.437340	0.218670	−	0.975799i	$-0.429828\pi$
$193$	4.36787e7	0.437340	0.218670	+	0.975799i	$0.429828\pi$
$194$	1.48968e6	0.0146483
$195$	2.27017e7	0.219249
$196$	−1.30511e8	−1.23808
$197$	9.17788e7	0.855285	0.427642	−	0.903948i	$-0.359344\pi$
$197$	9.17788e7	0.855285	0.427642	+	0.903948i	$0.359344\pi$
$198$	0	0
$199$	−2.08686e8	−1.87718	−0.938592	−	0.345030i	$-0.887869\pi$
$199$	−2.08686e8	−1.87718	−0.938592	+	0.345030i	$0.887869\pi$
$200$	−3.80147e7	−0.336006
$201$	9.03483e7	0.784755
$202$	1.59956e8	1.36543
$203$	7.00971e7	0.588118
$204$	3.65350e6	0.0301303
$205$	1.78234e8	1.44495
$206$	3.52620e8	2.81043
$207$	5.69378e7	0.446174
$208$	−9.62669e6	−0.0741746
$209$	0	0
$210$	5.50291e7	0.410039
$211$	4.58052e7	0.335680	0.167840	−	0.985814i	$-0.446321\pi$
$211$	4.58052e7	0.335680	0.167840	+	0.985814i	$0.446321\pi$
$212$	−9.92258e7	−0.715235
$213$	−7.82056e7	−0.554510
$214$	3.03656e8	2.11804
$215$	−2.30368e7	−0.158084
$216$	3.22049e7	0.217437
$217$	1.46052e8	0.970285
$218$	1.17872e8	0.770572
$219$	−9.13225e6	−0.0587521
$220$	0	0
$221$	−2.24620e6	−0.0139983
$222$	−1.02641e8	−0.629631
$223$	−7.05467e7	−0.426000	−0.213000	−	0.977052i	$-0.568323\pi$
$223$	−7.05467e7	−0.426000	−0.213000	+	0.977052i	$0.568323\pi$
$224$	7.48662e7	0.445059
$225$	−1.69375e7	−0.0991311
$226$	6.93530e7	0.399655
$227$	−2.18807e7	−0.124157	−0.0620785	−	0.998071i	$-0.519773\pi$
$227$	−2.18807e7	−0.124157	−0.0620785	+	0.998071i	$0.519773\pi$
$228$	1.53864e8	0.859737
$229$	1.88464e8	1.03706	0.518531	−	0.855059i	$-0.326479\pi$
$229$	1.88464e8	1.03706	0.518531	+	0.855059i	$0.326479\pi$
$230$	−3.39488e8	−1.83983
$231$	0	0
$232$	−2.44598e8	−1.28601
$233$	−3.37569e8	−1.74830	−0.874151	−	0.485655i	$-0.838581\pi$
$233$	−3.37569e8	−1.74830	−0.874151	+	0.485655i	$0.838581\pi$
$234$	−4.85368e7	−0.247637
$235$	1.37865e8	0.692976
$236$	−1.59745e8	−0.791107
$237$	−2.12074e8	−1.03483
$238$	−5.44482e6	−0.0261797
$239$	1.48471e8	0.703477	0.351738	−	0.936098i	$-0.385591\pi$
$239$	1.48471e8	0.703477	0.351738	+	0.936098i	$0.385591\pi$
$240$	−1.69687e7	−0.0792337
$241$	1.82232e8	0.838622	0.419311	−	0.907843i	$-0.362272\pi$
$241$	1.82232e8	0.838622	0.419311	+	0.907843i	$0.362272\pi$
$242$	0	0
$243$	1.43489e7	0.0641500
$244$	4.58753e8	2.02169
$245$	1.41435e8	0.614435
$246$	−3.81069e8	−1.63204
$247$	−9.45969e7	−0.399427
$248$	−5.09638e8	−2.12169
$249$	−2.40506e8	−0.987252
$250$	4.40568e8	1.78330
$251$	2.25050e8	0.898299	0.449150	−	0.893457i	$-0.351727\pi$
$251$	2.25050e8	0.898299	0.449150	+	0.893457i	$0.351727\pi$
$252$	−7.39001e7	−0.290900
$253$	0	0
$254$	−2.29997e8	−0.880651
$255$	−3.95932e6	−0.0149531
$256$	−3.35471e8	−1.24973
$257$	−4.61804e8	−1.69704	−0.848520	−	0.529163i	$-0.822506\pi$
$257$	−4.61804e8	−1.69704	−0.848520	+	0.529163i	$0.822506\pi$
$258$	4.92534e7	0.178553
$259$	9.60803e7	0.343625
$260$	1.81775e8	0.641398
$261$	−1.08981e8	−0.379410
$262$	3.44658e8	1.18395
$263$	−9.62818e7	−0.326362	−0.163181	−	0.986596i	$-0.552175\pi$
$263$	−9.62818e7	−0.326362	−0.163181	+	0.986596i	$0.552175\pi$
$264$	0	0
$265$	1.07531e8	0.354956
$266$	−2.29304e8	−0.747009
$267$	2.01000e8	0.646259
$268$	7.23430e8	2.29575
$269$	−5.75588e8	−1.80293	−0.901465	−	0.432852i	$-0.857507\pi$
$269$	−5.75588e8	−1.80293	−0.901465	+	0.432852i	$0.857507\pi$
$270$	−8.55545e7	−0.264527
$271$	−4.02597e8	−1.22879	−0.614395	−	0.788999i	$-0.710600\pi$
$271$	−4.02597e8	−1.22879	−0.614395	+	0.788999i	$0.710600\pi$
$272$	1.67896e6	0.00505882
$273$	4.54343e7	0.135150
$274$	1.89718e8	0.557161
$275$	0	0
$276$	4.55908e8	1.30525
$277$	3.79298e8	1.07226	0.536131	−	0.844135i	$-0.319885\pi$
$277$	3.79298e8	1.07226	0.536131	+	0.844135i	$0.319885\pi$
$278$	7.33271e8	2.04695
$279$	−2.27069e8	−0.625956
$280$	1.79746e8	0.489335
$281$	5.10144e8	1.37158	0.685789	−	0.727800i	$-0.259457\pi$
$281$	5.10144e8	1.37158	0.685789	+	0.727800i	$0.259457\pi$
$282$	−2.94760e8	−0.782702
$283$	−3.47516e8	−0.911427	−0.455714	−	0.890126i	$-0.650616\pi$
$283$	−3.47516e8	−0.911427	−0.455714	+	0.890126i	$0.650616\pi$
$284$	−6.26202e8	−1.62218
$285$	−1.66743e8	−0.426670
$286$	0	0
$287$	3.56711e8	0.890697
$288$	−1.16396e8	−0.287119
$289$	−4.09947e8	−0.999045
$290$	6.49792e8	1.56452
$291$	2.16798e6	0.00515740
$292$	−7.31230e7	−0.171876
$293$	6.99250e6	0.0162404	0.00812019	−	0.999967i	$-0.497415\pi$
$293$	6.99250e6	0.0162404	0.00812019	+	0.999967i	$0.497415\pi$
$294$	−3.02392e8	−0.693992
$295$	1.73116e8	0.392610
$296$	−3.35265e8	−0.751392
$297$	0	0
$298$	1.53489e7	0.0335986
$299$	−2.80295e8	−0.606411
$300$	−1.35620e8	−0.290002
$301$	−4.61051e7	−0.0974465
$302$	−6.40645e8	−1.33842
$303$	2.32790e8	0.480745
$304$	7.07079e7	0.144348
$305$	−4.97153e8	−1.00332
$306$	8.46513e6	0.0168892
$307$	9.53166e8	1.88011	0.940057	−	0.341016i	$-0.110771\pi$
$307$	9.53166e8	1.88011	0.940057	+	0.341016i	$0.110771\pi$
$308$	0	0
$309$	5.13181e8	0.989500
$310$	1.35389e9	2.58117
$311$	−5.15371e8	−0.971536	−0.485768	−	0.874088i	$-0.661460\pi$
$311$	−5.15371e8	−0.971536	−0.485768	+	0.874088i	$0.661460\pi$
$312$	−1.58539e8	−0.295526
$313$	7.41904e7	0.136755	0.0683774	−	0.997660i	$-0.478218\pi$
$313$	7.41904e7	0.136755	0.0683774	+	0.997660i	$0.478218\pi$
$314$	1.21597e8	0.221651
$315$	8.00859e7	0.144367
$316$	−1.69811e9	−3.02733
$317$	−7.41912e8	−1.30811	−0.654056	−	0.756446i	$-0.726934\pi$
$317$	−7.41912e8	−1.30811	−0.654056	+	0.756446i	$0.726934\pi$
$318$	−2.29905e8	−0.400916
$319$	0	0
$320$	7.74446e8	1.32119
$321$	4.41921e8	0.745722
$322$	−6.79440e8	−1.13411
$323$	1.64983e7	0.0272415
$324$	1.14893e8	0.187667
$325$	8.33804e7	0.134733
$326$	5.19718e8	0.830818
$327$	1.71543e8	0.271305
$328$	−1.24472e9	−1.94765
$329$	2.75919e8	0.427165
$330$	0	0
$331$	−9.64868e8	−1.46241	−0.731207	−	0.682156i	$-0.761042\pi$
$331$	−9.64868e8	−1.46241	−0.731207	+	0.682156i	$0.761042\pi$
$332$	−1.92576e9	−2.88814
$333$	−1.49377e8	−0.221682
$334$	1.08093e9	1.58740
$335$	−7.83985e8	−1.13933
$336$	−3.39606e7	−0.0488414
$337$	−1.08105e8	−0.153866	−0.0769329	−	0.997036i	$-0.524513\pi$
$337$	−1.08105e8	−0.153866	−0.0769329	+	0.997036i	$0.524513\pi$
$338$	−9.25198e8	−1.30325
$339$	1.00932e8	0.140712
$340$	−3.17028e7	−0.0437442
$341$	0	0
$342$	3.56502e8	0.481915
$343$	6.69220e8	0.895446
$344$	1.60880e8	0.213083
$345$	−4.94070e8	−0.647771
$346$	1.10145e9	1.42955
$347$	8.06717e8	1.03650	0.518248	−	0.855230i	$-0.326584\pi$
$347$	8.06717e8	1.03650	0.518248	+	0.855230i	$0.326584\pi$
$348$	−8.72623e8	−1.10994
$349$	−1.27856e9	−1.61002	−0.805012	−	0.593259i	$-0.797841\pi$
$349$	−1.27856e9	−1.61002	−0.805012	+	0.593259i	$0.797841\pi$
$350$	2.02115e8	0.251977
$351$	−7.06373e7	−0.0871885
$352$	0	0
$353$	−5.03843e8	−0.609655	−0.304827	−	0.952408i	$-0.598599\pi$
$353$	−5.03843e8	−0.609655	−0.304827	+	0.952408i	$0.598599\pi$
$354$	−3.70128e8	−0.443445
$355$	6.78618e8	0.805056
$356$	1.60943e9	1.89059
$357$	−7.92405e6	−0.00921739
$358$	−1.25886e9	−1.45006
$359$	−9.78486e8	−1.11615	−0.558077	−	0.829789i	$-0.688461\pi$
$359$	−9.78486e8	−1.11615	−0.558077	+	0.829789i	$0.688461\pi$
$360$	−2.79453e8	−0.315683
$361$	−1.99059e8	−0.222693
$362$	−6.24629e8	−0.692058
$363$	0	0
$364$	3.63798e8	0.395372
$365$	7.92438e7	0.0852982
$366$	1.06293e9	1.13323
$367$	−5.35617e8	−0.565618	−0.282809	−	0.959176i	$-0.591266\pi$
$367$	−5.35617e8	−0.565618	−0.282809	+	0.959176i	$0.591266\pi$
$368$	2.09511e8	0.219149
$369$	−5.54584e8	−0.574612
$370$	8.90653e8	0.914119
$371$	2.15210e8	0.218803
$372$	−1.81817e9	−1.83120
$373$	1.40133e9	1.39817	0.699084	−	0.715039i	$-0.253591\pi$
$373$	1.40133e9	1.39817	0.699084	+	0.715039i	$0.253591\pi$
$374$	0	0
$375$	6.41175e8	0.627866
$376$	−9.62798e8	−0.934066
$377$	5.36495e8	0.515669
$378$	−1.71226e8	−0.163060
$379$	−1.85262e9	−1.74803	−0.874013	−	0.485902i	$-0.838491\pi$
$379$	−1.85262e9	−1.74803	−0.874013	+	0.485902i	$0.838491\pi$
$380$	−1.33513e9	−1.24819
$381$	−3.34723e8	−0.310062
$382$	−1.69943e9	−1.55984
$383$	1.83220e9	1.66639	0.833195	−	0.552979i	$-0.186509\pi$
$383$	1.83220e9	1.66639	0.833195	+	0.552979i	$0.186509\pi$
$384$	−1.10398e9	−0.994956
$385$	0	0
$386$	8.10346e8	0.717158
$387$	7.16802e7	0.0628653
$388$	1.73593e7	0.0150876
$389$	−6.91249e8	−0.595403	−0.297702	−	0.954659i	$-0.596220\pi$
$389$	−6.91249e8	−0.595403	−0.297702	+	0.954659i	$0.596220\pi$
$390$	4.21171e8	0.359528
$391$	4.88854e7	0.0413581
$392$	−9.87726e8	−0.828200
$393$	5.01593e8	0.416848
$394$	1.70272e9	1.40251
$395$	1.84025e9	1.50240
$396$	0	0
$397$	1.09921e9	0.881683	0.440841	−	0.897585i	$-0.354680\pi$
$397$	1.09921e9	0.881683	0.440841	+	0.897585i	$0.354680\pi$
$398$	−3.87162e9	−3.07824
$399$	−3.33714e8	−0.263008
$400$	−6.23240e7	−0.0486906
$401$	1.38884e9	1.07559	0.537793	−	0.843077i	$-0.319258\pi$
$401$	1.38884e9	1.07559	0.537793	+	0.843077i	$0.319258\pi$
$402$	1.67618e9	1.28685
$403$	1.11783e9	0.850759
$404$	1.86398e9	1.40639
$405$	−1.24511e8	−0.0931351
$406$	1.30047e9	0.964405
$407$	0	0
$408$	2.76503e7	0.0201553
$409$	−2.44277e8	−0.176543	−0.0882714	−	0.996096i	$-0.528134\pi$
$409$	−2.44277e8	−0.176543	−0.0882714	+	0.996096i	$0.528134\pi$
$410$	3.30667e9	2.36945
$411$	2.76103e8	0.196166
$412$	4.10910e9	2.89472
$413$	3.46469e8	0.242013
$414$	1.05633e9	0.731644
$415$	2.08695e9	1.43332
$416$	5.72996e8	0.390234
$417$	1.06716e9	0.720695
$418$	0	0
$419$	2.09745e8	0.139297	0.0696487	−	0.997572i	$-0.477812\pi$
$419$	2.09745e8	0.139297	0.0696487	+	0.997572i	$0.477812\pi$
$420$	6.41257e8	0.422338
$421$	9.19954e8	0.600868	0.300434	−	0.953803i	$-0.402869\pi$
$421$	9.19954e8	0.600868	0.300434	+	0.953803i	$0.402869\pi$
$422$	8.49797e8	0.550454
$423$	−4.28975e8	−0.275575
$424$	−7.50957e8	−0.478448
$425$	−1.45421e7	−0.00918895
$426$	−1.45090e9	−0.909295
$427$	−9.94985e8	−0.618471
$428$	3.53851e9	2.18156
$429$	0	0
$430$	−4.27389e8	−0.259229
$431$	2.10688e9	1.26756	0.633780	−	0.773513i	$-0.281502\pi$
$431$	2.10688e9	1.26756	0.633780	+	0.773513i	$0.281502\pi$
$432$	5.27990e7	0.0315088
$433$	1.12524e9	0.666095	0.333048	−	0.942910i	$-0.391923\pi$
$433$	1.12524e9	0.666095	0.333048	+	0.942910i	$0.391923\pi$
$434$	2.70962e9	1.59109
$435$	9.45666e8	0.550840
$436$	1.37357e9	0.793684
$437$	2.05877e9	1.18011
$438$	−1.69425e8	−0.0963427
$439$	1.06709e9	0.601968	0.300984	−	0.953629i	$-0.402685\pi$
$439$	1.06709e9	0.601968	0.300984	+	0.953629i	$0.402685\pi$
$440$	0	0
$441$	−4.40082e8	−0.244342
$442$	−4.16725e7	−0.0229547
$443$	1.44691e9	0.790733	0.395366	−	0.918523i	$-0.370618\pi$
$443$	1.44691e9	0.790733	0.395366	+	0.918523i	$0.370618\pi$
$444$	−1.19608e9	−0.648515
$445$	−1.74415e9	−0.938260
$446$	−1.30881e9	−0.698563
$447$	2.23379e7	0.0118295
$448$	1.54995e9	0.814412
$449$	3.55718e9	1.85457	0.927287	−	0.374351i	$-0.122134\pi$
$449$	3.55718e9	1.85457	0.927287	+	0.374351i	$0.122134\pi$
$450$	−3.14231e8	−0.162557
$451$	0	0
$452$	8.08174e8	0.411643
$453$	−9.32354e8	−0.471234
$454$	−4.05940e8	−0.203595
$455$	−3.94250e8	−0.196215
$456$	1.16447e9	0.575110
$457$	3.99312e9	1.95707	0.978533	−	0.206092i	$-0.0660745\pi$
$457$	3.99312e9	1.95707	0.978533	+	0.206092i	$0.0660745\pi$
$458$	3.49646e9	1.70059
$459$	1.23196e7	0.00594638
$460$	−3.95607e9	−1.89501
$461$	6.63378e8	0.315361	0.157681	−	0.987490i	$-0.449598\pi$
$461$	6.63378e8	0.315361	0.157681	+	0.987490i	$0.449598\pi$
$462$	0	0
$463$	−2.93044e9	−1.37214	−0.686072	−	0.727534i	$-0.740666\pi$
$463$	−2.93044e9	−1.37214	−0.686072	+	0.727534i	$0.740666\pi$
$464$	−4.01012e8	−0.186356
$465$	1.97036e9	0.908784
$466$	−6.26271e9	−2.86689
$467$	−2.59060e8	−0.117704	−0.0588520	−	0.998267i	$-0.518744\pi$
$467$	−2.59060e8	−0.117704	−0.0588520	+	0.998267i	$0.518744\pi$
$468$	−5.65601e8	−0.255064
$469$	−1.56904e9	−0.702310
$470$	2.55774e9	1.13635
$471$	1.76965e8	0.0780394
$472$	−1.20898e9	−0.529201
$473$	0	0
$474$	−3.93449e9	−1.69693
$475$	−6.12428e8	−0.262197
$476$	−6.34488e7	−0.0269649
$477$	−3.34589e8	−0.141155
$478$	2.75450e9	1.15357
$479$	−4.01670e9	−1.66992	−0.834959	−	0.550312i	$-0.814509\pi$
$479$	−4.01670e9	−1.66992	−0.834959	+	0.550312i	$0.814509\pi$
$480$	1.01001e9	0.416850
$481$	7.35360e8	0.301295
$482$	3.38085e9	1.37519
$483$	−9.88813e8	−0.399300
$484$	0	0
$485$	−1.88124e7	−0.00748768
$486$	2.66207e8	0.105194
$487$	−1.80670e9	−0.708817	−0.354408	−	0.935091i	$-0.615318\pi$
$487$	−1.80670e9	−0.708817	−0.354408	+	0.935091i	$0.615318\pi$
$488$	3.47192e9	1.35239
$489$	7.56364e8	0.292516
$490$	2.62396e9	1.00756
$491$	6.04401e8	0.230431	0.115215	−	0.993341i	$-0.463244\pi$
$491$	6.04401e8	0.230431	0.115215	+	0.993341i	$0.463244\pi$
$492$	−4.44062e9	−1.68099
$493$	−9.35683e7	−0.0351694
$494$	−1.75500e9	−0.654987
$495$	0	0
$496$	−8.35536e8	−0.307454
$497$	1.35816e9	0.496254
$498$	−4.46196e9	−1.61891
$499$	−2.62714e9	−0.946523	−0.473261	−	0.880922i	$-0.656923\pi$
$499$	−2.62714e9	−0.946523	−0.473261	+	0.880922i	$0.656923\pi$
$500$	5.13396e9	1.83678
$501$	1.57312e9	0.558893
$502$	4.17522e9	1.47305
$503$	1.77277e8	0.0621103	0.0310551	−	0.999518i	$-0.490113\pi$
$503$	1.77277e8	0.0621103	0.0310551	+	0.999518i	$0.490113\pi$
$504$	−5.59288e8	−0.194594
$505$	−2.02000e9	−0.697961
$506$	0	0
$507$	−1.34647e9	−0.458849
$508$	−2.68017e9	−0.907065
$509$	5.30178e9	1.78201	0.891004	−	0.453995i	$-0.150002\pi$
$509$	5.30178e9	1.78201	0.891004	+	0.453995i	$0.150002\pi$
$510$	−7.34550e7	−0.0245203
$511$	1.58596e8	0.0525797
$512$	−9.90086e8	−0.326008
$513$	5.18830e8	0.169674
$514$	−8.56759e9	−2.78284
$515$	−4.45305e9	−1.43659
$516$	5.73952e8	0.183908
$517$	0	0
$518$	1.78252e9	0.563483
$519$	1.60299e9	0.503320
$520$	1.37570e9	0.429055
$521$	−5.93733e9	−1.83933	−0.919663	−	0.392708i	$-0.871538\pi$
$521$	−5.93733e9	−1.83933	−0.919663	+	0.392708i	$0.871538\pi$
$522$	−2.02186e9	−0.622163
$523$	4.12003e9	1.25934	0.629672	−	0.776861i	$-0.283189\pi$
$523$	4.12003e9	1.25934	0.629672	+	0.776861i	$0.283189\pi$
$524$	4.01632e9	1.21946
$525$	2.94146e8	0.0887166
$526$	−1.78626e9	−0.535174
$527$	−1.94956e8	−0.0580230
$528$	0	0
$529$	2.69541e9	0.791643
$530$	1.99497e9	0.582064
$531$	−5.38660e8	−0.156129
$532$	−2.67209e9	−0.769415
$533$	2.73012e9	0.780975
$534$	3.72903e9	1.05975
$535$	−3.83471e9	−1.08266
$536$	5.47504e9	1.53571
$537$	−1.83206e9	−0.510539
$538$	−1.06786e10	−2.95648
$539$	0	0
$540$	−9.96971e8	−0.272461
$541$	−4.74225e9	−1.28764	−0.643819	−	0.765178i	$-0.722651\pi$
$541$	−4.74225e9	−1.28764	−0.643819	+	0.765178i	$0.722651\pi$
$542$	−7.46914e9	−2.01499
$543$	−9.09045e8	−0.243661
$544$	−9.99343e7	−0.0266145
$545$	−1.48854e9	−0.393889
$546$	8.42916e8	0.221621
$547$	9.68431e8	0.252996	0.126498	−	0.991967i	$-0.459626\pi$
$547$	9.68431e8	0.252996	0.126498	+	0.991967i	$0.459626\pi$
$548$	2.21079e9	0.573872
$549$	1.54692e9	0.398991
$550$	0	0
$551$	−3.94055e9	−1.00352
$552$	3.45039e9	0.873134
$553$	3.68300e9	0.926113
$554$	7.03689e9	1.75831
$555$	1.29620e9	0.321845
$556$	8.54484e9	2.10835
$557$	−5.05809e9	−1.24021	−0.620103	−	0.784521i	$-0.712909\pi$
$557$	−5.05809e9	−1.24021	−0.620103	+	0.784521i	$0.712909\pi$
$558$	−4.21269e9	−1.02645
$559$	−3.52870e8	−0.0854424
$560$	2.94688e8	0.0709095
$561$	0	0
$562$	9.46441e9	2.24914
$563$	−5.54853e8	−0.131038	−0.0655192	−	0.997851i	$-0.520870\pi$
$563$	−5.54853e8	−0.131038	−0.0655192	+	0.997851i	$0.520870\pi$
$564$	−3.43485e9	−0.806179
$565$	−8.75822e8	−0.204290
$566$	−6.44726e9	−1.49457
$567$	−2.49191e8	−0.0574105
$568$	−4.73920e9	−1.08514
$569$	9.26007e8	0.210728	0.105364	−	0.994434i	$-0.466399\pi$
$569$	9.26007e8	0.210728	0.105364	+	0.994434i	$0.466399\pi$
$570$	−3.09349e9	−0.699660
$571$	−1.56780e9	−0.352423	−0.176211	−	0.984352i	$-0.556384\pi$
$571$	−1.56780e9	−0.352423	−0.176211	+	0.984352i	$0.556384\pi$
$572$	0	0
$573$	−2.47324e9	−0.549192
$574$	6.61786e9	1.46058
$575$	−1.81466e9	−0.398068
$576$	−2.40972e9	−0.525398
$577$	−3.87215e9	−0.839145	−0.419573	−	0.907722i	$-0.637820\pi$
$577$	−3.87215e9	−0.839145	−0.419573	+	0.907722i	$0.637820\pi$
$578$	−7.60551e9	−1.63825
$579$	1.17933e9	0.252498
$580$	7.57206e9	1.61145
$581$	4.17676e9	0.883533
$582$	4.02213e7	0.00845719
$583$	0	0
$584$	−5.53407e8	−0.114974
$585$	6.12945e8	0.126583
$586$	1.29728e8	0.0266313
$587$	8.31706e9	1.69721	0.848607	−	0.529024i	$-0.177442\pi$
$587$	8.31706e9	1.69721	0.848607	+	0.529024i	$0.177442\pi$
$588$	−3.52379e9	−0.714807
$589$	−8.21041e9	−1.65562
$590$	3.21173e9	0.643809
$591$	2.47803e9	0.493799
$592$	−5.49656e8	−0.108884
$593$	8.76659e9	1.72639	0.863196	−	0.504869i	$-0.168459\pi$
$593$	8.76659e9	1.72639	0.863196	+	0.504869i	$0.168459\pi$
$594$	0	0
$595$	6.87598e7	0.0133821
$596$	1.78862e8	0.0346064
$597$	−5.63451e9	−1.08379
$598$	−5.20016e9	−0.994403
$599$	−3.04049e9	−0.578029	−0.289015	−	0.957325i	$-0.593328\pi$
$599$	−3.04049e9	−0.578029	−0.289015	+	0.957325i	$0.593328\pi$
$600$	−1.02640e9	−0.193993
$601$	−1.40462e9	−0.263935	−0.131968	−	0.991254i	$-0.542129\pi$
$601$	−1.40462e9	−0.263935	−0.131968	+	0.991254i	$0.542129\pi$
$602$	−8.55361e8	−0.159794
$603$	2.43941e9	0.453078
$604$	−7.46547e9	−1.37857
$605$	0	0
$606$	4.31881e9	0.788334
$607$	−2.66663e9	−0.483952	−0.241976	−	0.970282i	$-0.577796\pi$
$607$	−2.66663e9	−0.483952	−0.241976	+	0.970282i	$0.577796\pi$
$608$	−4.20865e9	−0.759416
$609$	1.89262e9	0.339550
$610$	−9.22340e9	−1.64527
$611$	2.11177e9	0.374544
$612$	9.86446e7	0.0173958
$613$	−6.58322e9	−1.15432	−0.577161	−	0.816630i	$-0.695840\pi$
$613$	−6.58322e9	−1.15432	−0.577161	+	0.816630i	$0.695840\pi$
$614$	1.76835e10	3.08305
$615$	4.81232e9	0.834241
$616$	0	0
$617$	5.06268e8	0.0867725	0.0433862	−	0.999058i	$-0.486185\pi$
$617$	5.06268e8	0.0867725	0.0433862	+	0.999058i	$0.486185\pi$
$618$	9.52075e9	1.62260
$619$	3.10872e9	0.526823	0.263411	−	0.964684i	$-0.415152\pi$
$619$	3.10872e9	0.526823	0.263411	+	0.964684i	$0.415152\pi$
$620$	1.57769e10	2.65859
$621$	1.53732e9	0.257599
$622$	−9.56139e9	−1.59314
$623$	−3.49067e9	−0.578364
$624$	−2.59921e8	−0.0428247
$625$	−3.74856e9	−0.614164
$626$	1.37641e9	0.224253
$627$	0	0
$628$	1.41698e9	0.228299
$629$	−1.28252e8	−0.0205488
$630$	1.48579e9	0.236736
$631$	4.92193e9	0.779889	0.389944	−	0.920838i	$-0.372494\pi$
$631$	4.92193e9	0.779889	0.389944	+	0.920838i	$0.372494\pi$
$632$	−1.28515e10	−2.02510
$633$	1.23674e9	0.193805
$634$	−1.37643e10	−2.14507
$635$	2.90451e9	0.450158
$636$	−2.67910e9	−0.412941
$637$	2.16645e9	0.332094
$638$	0	0
$639$	−2.11155e9	−0.320147
$640$	9.57967e9	1.44451
$641$	−3.96263e9	−0.594265	−0.297133	−	0.954836i	$-0.596030\pi$
$641$	−3.96263e9	−0.594265	−0.297133	+	0.954836i	$0.596030\pi$
$642$	8.19870e9	1.22285
$643$	−9.86915e9	−1.46400	−0.732000	−	0.681304i	$-0.761413\pi$
$643$	−9.86915e9	−1.46400	−0.732000	+	0.681304i	$0.761413\pi$
$644$	−7.91755e9	−1.16813
$645$	−6.21995e8	−0.0912699
$646$	3.06084e8	0.0446711
$647$	−3.13058e8	−0.0454423	−0.0227211	−	0.999742i	$-0.507233\pi$
$647$	−3.13058e8	−0.0454423	−0.0227211	+	0.999742i	$0.507233\pi$
$648$	8.69532e8	0.125537
$649$	0	0
$650$	1.54691e9	0.220937
$651$	3.94341e9	0.560195
$652$	6.05629e9	0.855737
$653$	6.18723e9	0.869561	0.434780	−	0.900537i	$-0.356826\pi$
$653$	6.18723e9	0.869561	0.434780	+	0.900537i	$0.356826\pi$
$654$	3.18255e9	0.444890
$655$	−4.35250e9	−0.605194
$656$	−2.04067e9	−0.282235
$657$	−2.46571e8	−0.0339205
$658$	5.11896e9	0.700473
$659$	−6.01078e9	−0.818148	−0.409074	−	0.912501i	$-0.634148\pi$
$659$	−6.01078e9	−0.818148	−0.409074	+	0.912501i	$0.634148\pi$
$660$	0	0
$661$	−3.51253e9	−0.473059	−0.236529	−	0.971624i	$-0.576010\pi$
$661$	−3.51253e9	−0.473059	−0.236529	+	0.971624i	$0.576010\pi$
$662$	−1.79006e10	−2.39809
$663$	−6.06474e7	−0.00808193
$664$	−1.45745e10	−1.93199
$665$	2.89576e9	0.381844
$666$	−2.77131e9	−0.363517
$667$	−1.16761e10	−1.52355
$668$	1.25961e10	1.63501
$669$	−1.90476e9	−0.245951
$670$	−1.45448e10	−1.86830
$671$	0	0
$672$	2.02139e9	0.256955
$673$	−6.16678e9	−0.779840	−0.389920	−	0.920849i	$-0.627497\pi$
$673$	−6.16678e9	−0.779840	−0.389920	+	0.920849i	$0.627497\pi$
$674$	−2.00561e9	−0.252312
$675$	−4.57312e8	−0.0572334
$676$	−1.07814e10	−1.34233
$677$	−1.93674e9	−0.239889	−0.119944	−	0.992781i	$-0.538272\pi$
$677$	−1.93674e9	−0.239889	−0.119944	+	0.992781i	$0.538272\pi$
$678$	1.87253e9	0.230741
$679$	−3.76504e7	−0.00461557
$680$	−2.39932e8	−0.0292622
$681$	−5.90780e8	−0.0716821
$682$	0	0
$683$	−1.27287e10	−1.52866	−0.764331	−	0.644824i	$-0.776931\pi$
$683$	−1.27287e10	−1.52866	−0.764331	+	0.644824i	$0.776931\pi$
$684$	4.15433e9	0.496369
$685$	−2.39584e9	−0.284801
$686$	1.24156e10	1.46837
$687$	5.08853e9	0.598748
$688$	2.63758e8	0.0308778
$689$	1.64713e9	0.191849
$690$	−9.16618e9	−1.06223
$691$	−4.60960e9	−0.531483	−0.265742	−	0.964044i	$-0.585617\pi$
$691$	−4.60960e9	−0.531483	−0.265742	+	0.964044i	$0.585617\pi$
$692$	1.28353e10	1.47243
$693$	0	0
$694$	1.49665e10	1.69967
$695$	−9.26009e9	−1.04633
$696$	−6.60416e9	−0.742480
$697$	−4.76152e8	−0.0532636
$698$	−2.37204e10	−2.64014
$699$	−9.11435e9	−1.00938
$700$	2.35526e9	0.259535
$701$	5.67606e9	0.622349	0.311175	−	0.950353i	$-0.399278\pi$
$701$	5.67606e9	0.622349	0.311175	+	0.950353i	$0.399278\pi$
$702$	−1.31049e9	−0.142973
$703$	−5.40121e9	−0.586337
$704$	0	0
$705$	3.72237e9	0.400090
$706$	−9.34751e9	−0.999722
$707$	−4.04276e9	−0.430239
$708$	−4.31311e9	−0.456746
$709$	1.18615e10	1.24991	0.624955	−	0.780661i	$-0.285117\pi$
$709$	1.18615e10	1.24991	0.624955	+	0.780661i	$0.285117\pi$
$710$	1.25900e10	1.32015
$711$	−5.72601e9	−0.597460
$712$	1.21804e10	1.26469
$713$	−2.43279e10	−2.51357
$714$	−1.47010e8	−0.0151148
$715$	0	0
$716$	−1.46695e10	−1.49355
$717$	4.00872e9	0.406153
$718$	−1.81533e10	−1.83029
$719$	−5.79741e9	−0.581678	−0.290839	−	0.956772i	$-0.593934\pi$
$719$	−5.79741e9	−0.581678	−0.290839	+	0.956772i	$0.593934\pi$
$720$	−4.58155e8	−0.0457456
$721$	−8.91219e9	−0.885545
$722$	−3.69303e9	−0.365177
$723$	4.92028e9	0.484179
$724$	−7.27883e9	−0.712815
$725$	3.47332e9	0.338502
$726$	0	0
$727$	1.17506e10	1.13420	0.567100	−	0.823649i	$-0.308065\pi$
$727$	1.17506e10	1.13420	0.567100	+	0.823649i	$0.308065\pi$
$728$	2.75328e9	0.264479
$729$	3.87420e8	0.0370370
$730$	1.47016e9	0.139874
$731$	6.15428e7	0.00582729
$732$	1.23863e10	1.16722
$733$	−1.45394e10	−1.36358	−0.681791	−	0.731547i	$-0.738799\pi$
$733$	−1.45394e10	−1.36358	−0.681791	+	0.731547i	$0.738799\pi$
$734$	−9.93699e9	−0.927510
$735$	3.81875e9	0.354744
$736$	−1.24704e10	−1.15295
$737$	0	0
$738$	−1.02889e10	−0.942259
$739$	8.28187e9	0.754871	0.377435	−	0.926036i	$-0.376806\pi$
$739$	8.28187e9	0.754871	0.377435	+	0.926036i	$0.376806\pi$
$740$	1.03788e10	0.941536
$741$	−2.55412e9	−0.230609
$742$	3.99266e9	0.358797
$743$	1.23435e10	1.10402	0.552011	−	0.833837i	$-0.313860\pi$
$743$	1.23435e10	1.10402	0.552011	+	0.833837i	$0.313860\pi$
$744$	−1.37602e10	−1.22496
$745$	−1.93834e8	−0.0171744
$746$	2.59981e10	2.29274
$747$	−6.49366e9	−0.569990
$748$	0	0
$749$	−7.67465e9	−0.667378
$750$	1.18953e10	1.02959
$751$	−1.14760e10	−0.988664	−0.494332	−	0.869273i	$-0.664587\pi$
$751$	−1.14760e10	−0.988664	−0.494332	+	0.869273i	$0.664587\pi$
$752$	−1.57848e9	−0.135356
$753$	6.07635e9	0.518633
$754$	9.95328e9	0.845603
$755$	8.09036e9	0.684154
$756$	−1.99530e9	−0.167951
$757$	−1.85393e10	−1.55331	−0.776654	−	0.629928i	$-0.783084\pi$
$757$	−1.85393e10	−1.55331	−0.776654	+	0.629928i	$0.783084\pi$
$758$	−3.43705e10	−2.86644
$759$	0	0
$760$	−1.01045e10	−0.834965
$761$	4.22288e9	0.347346	0.173673	−	0.984803i	$-0.444436\pi$
$761$	4.22288e9	0.347346	0.173673	+	0.984803i	$0.444436\pi$
$762$	−6.20992e9	−0.508444
$763$	−2.97912e9	−0.242802
$764$	−1.98035e10	−1.60663
$765$	−1.06902e8	−0.00863315
$766$	3.39917e10	2.73258
$767$	2.65173e9	0.212200
$768$	−9.05771e9	−0.721529
$769$	1.47984e10	1.17347	0.586734	−	0.809780i	$-0.300413\pi$
$769$	1.47984e10	1.17347	0.586734	+	0.809780i	$0.300413\pi$
$770$	0	0
$771$	−1.24687e10	−0.979787
$772$	9.44300e9	0.738668
$773$	5.12045e9	0.398731	0.199365	−	0.979925i	$-0.436112\pi$
$773$	5.12045e9	0.398731	0.199365	+	0.979925i	$0.436112\pi$
$774$	1.32984e9	0.103088
$775$	7.23690e9	0.558466
$776$	1.31378e8	0.0100927
$777$	2.59417e9	0.198392
$778$	−1.28243e10	−0.976352
$779$	−2.00527e10	−1.51982
$780$	4.90792e9	0.370311
$781$	0	0
$782$	9.06942e8	0.0678197
$783$	−2.94248e9	−0.219052
$784$	−1.61935e9	−0.120015
$785$	−1.53559e9	−0.113300
$786$	9.30577e9	0.683555
$787$	−2.15704e9	−0.157742	−0.0788709	−	0.996885i	$-0.525131\pi$
$787$	−2.15704e9	−0.157742	−0.0788709	+	0.996885i	$0.525131\pi$
$788$	1.98419e10	1.44458
$789$	−2.59961e9	−0.188425
$790$	3.41410e10	2.46366
$791$	−1.75284e9	−0.125929
$792$	0	0
$793$	−7.61521e9	−0.542283
$794$	2.03929e10	1.44580
$795$	2.90335e9	0.204934
$796$	−4.51162e10	−3.17057
$797$	1.14528e10	0.801321	0.400661	−	0.916227i	$-0.368781\pi$
$797$	1.14528e10	0.801321	0.400661	+	0.916227i	$0.368781\pi$
$798$	−6.19121e9	−0.431286
$799$	−3.68307e8	−0.0255444
$800$	3.70963e9	0.256162
$801$	5.42700e9	0.373118
$802$	2.57662e10	1.76377
$803$	0	0
$804$	1.95326e10	1.32545
$805$	8.58028e9	0.579717
$806$	2.07384e10	1.39509
$807$	−1.55409e10	−1.04092
$808$	1.41069e10	0.940786
$809$	−1.32729e10	−0.881348	−0.440674	−	0.897667i	$-0.645261\pi$
$809$	−1.32729e10	−0.881348	−0.440674	+	0.897667i	$0.645261\pi$
$810$	−2.30997e9	−0.152725
$811$	−5.89483e9	−0.388060	−0.194030	−	0.980996i	$-0.562156\pi$
$811$	−5.89483e9	−0.388060	−0.194030	+	0.980996i	$0.562156\pi$
$812$	1.51545e10	0.993331
$813$	−1.08701e10	−0.709442
$814$	0	0
$815$	−6.56324e9	−0.424684
$816$	4.53319e7	0.00292071
$817$	2.59182e9	0.166275
$818$	−4.53192e9	−0.289498
$819$	1.22673e9	0.0780287
$820$	3.85328e10	2.44052
$821$	2.07684e10	1.30979	0.654895	−	0.755720i	$-0.272713\pi$
$821$	2.07684e10	1.30979	0.654895	+	0.755720i	$0.272713\pi$
$822$	5.12238e9	0.321677
$823$	−3.08800e10	−1.93098	−0.965491	−	0.260437i	$-0.916133\pi$
$823$	−3.08800e10	−1.93098	−0.965491	+	0.260437i	$0.916133\pi$
$824$	3.10984e10	1.93639
$825$	0	0
$826$	6.42784e9	0.396858
$827$	−9.93544e9	−0.610827	−0.305413	−	0.952220i	$-0.598795\pi$
$827$	−9.93544e9	−0.610827	−0.305413	+	0.952220i	$0.598795\pi$
$828$	1.23095e10	0.753589
$829$	−6.96802e9	−0.424784	−0.212392	−	0.977185i	$-0.568125\pi$
$829$	−6.96802e9	−0.424784	−0.212392	+	0.977185i	$0.568125\pi$
$830$	3.87180e10	2.35039
$831$	1.02410e10	0.619071
$832$	1.18627e10	0.714087
$833$	−3.77843e8	−0.0226493
$834$	1.97983e10	1.18181
$835$	−1.36505e10	−0.811420
$836$	0	0
$837$	−6.13088e9	−0.361396
$838$	3.89128e9	0.228422
$839$	−3.81998e9	−0.223303	−0.111652	−	0.993747i	$-0.535614\pi$
$839$	−3.81998e9	−0.223303	−0.111652	+	0.993747i	$0.535614\pi$
$840$	4.85314e9	0.282518
$841$	5.09849e9	0.295567
$842$	1.70674e10	0.985313
$843$	1.37739e10	0.791881
$844$	9.90272e9	0.566965
$845$	1.16838e10	0.666173
$846$	−7.95852e9	−0.451893
$847$	0	0
$848$	−1.23117e9	−0.0693319
$849$	−9.38293e9	−0.526213
$850$	−2.69791e8	−0.0150682
$851$	−1.60041e10	−0.890178
$852$	−1.69075e10	−0.936568
$853$	−2.33127e10	−1.28609	−0.643044	−	0.765829i	$-0.722329\pi$
$853$	−2.33127e10	−1.28609	−0.643044	+	0.765829i	$0.722329\pi$
$854$	−1.84594e10	−1.01418
$855$	−4.50207e9	−0.246338
$856$	2.67801e10	1.45933
$857$	−3.57453e9	−0.193993	−0.0969965	−	0.995285i	$-0.530924\pi$
$857$	−3.57453e9	−0.193993	−0.0969965	+	0.995285i	$0.530924\pi$
$858$	0	0
$859$	−1.03216e10	−0.555612	−0.277806	−	0.960637i	$-0.589607\pi$
$859$	−1.03216e10	−0.555612	−0.277806	+	0.960637i	$0.589607\pi$
$860$	−4.98038e9	−0.267004
$861$	9.63120e9	0.514244
$862$	3.90877e10	2.07857
$863$	1.36465e10	0.722741	0.361370	−	0.932422i	$-0.382309\pi$
$863$	1.36465e10	0.722741	0.361370	+	0.932422i	$0.382309\pi$
$864$	−3.14268e9	−0.165768
$865$	−1.39097e10	−0.730736
$866$	2.08759e10	1.09227
$867$	−1.10686e10	−0.576799
$868$	3.15754e10	1.63881
$869$	0	0
$870$	1.75444e10	0.903277
$871$	−1.20088e10	−0.615794
$872$	1.03954e10	0.530925
$873$	5.85355e7	0.00297762
$874$	3.81951e10	1.93516
$875$	−1.11350e10	−0.561904
$876$	−1.97432e9	−0.0992324
$877$	3.05799e10	1.53087	0.765434	−	0.643514i	$-0.222524\pi$
$877$	3.05799e10	1.53087	0.765434	+	0.643514i	$0.222524\pi$
$878$	1.97970e10	0.987118
$879$	1.88798e8	0.00937638
$880$	0	0
$881$	2.02865e10	0.999519	0.499759	−	0.866164i	$-0.333422\pi$
$881$	2.02865e10	0.999519	0.499759	+	0.866164i	$0.333422\pi$
$882$	−8.16458e9	−0.400676
$883$	−6.43226e9	−0.314414	−0.157207	−	0.987566i	$-0.550249\pi$
$883$	−6.43226e9	−0.314414	−0.157207	+	0.987566i	$0.550249\pi$
$884$	−4.85611e8	−0.0236432
$885$	4.67414e9	0.226673
$886$	2.68438e10	1.29666
$887$	9.22503e9	0.443849	0.221924	−	0.975064i	$-0.428766\pi$
$887$	9.22503e9	0.443849	0.221924	+	0.975064i	$0.428766\pi$
$888$	−9.05214e9	−0.433816
$889$	5.81298e9	0.277487
$890$	−3.23581e10	−1.53857
$891$	0	0
$892$	−1.52516e10	−0.719515
$893$	−1.55109e10	−0.728883
$894$	4.14421e8	0.0193982
$895$	1.58974e10	0.741218
$896$	1.91724e10	0.890428
$897$	−7.56798e9	−0.350111
$898$	6.59944e10	3.04116
$899$	4.65644e10	2.13745
$900$	−3.66175e9	−0.167433
$901$	−2.87270e8	−0.0130844
$902$	0	0
$903$	−1.24484e9	−0.0562608
$904$	6.11639e9	0.275363
$905$	7.88811e9	0.353755
$906$	−1.72974e10	−0.772738
$907$	−3.68996e10	−1.64209	−0.821043	−	0.570866i	$-0.806607\pi$
$907$	−3.68996e10	−1.64209	−0.821043	+	0.570866i	$0.806607\pi$
$908$	−4.73044e9	−0.209701
$909$	6.28532e9	0.277558
$910$	−7.31429e9	−0.321756
$911$	−1.20467e10	−0.527902	−0.263951	−	0.964536i	$-0.585026\pi$
$911$	−1.20467e10	−0.527902	−0.263951	+	0.964536i	$0.585026\pi$
$912$	1.90911e9	0.0833393
$913$	0	0
$914$	7.40820e10	3.20923
$915$	−1.34231e10	−0.579269
$916$	4.07445e10	1.75160
$917$	−8.71095e9	−0.373055
$918$	2.28559e8	0.00975098
$919$	−3.75192e10	−1.59459	−0.797296	−	0.603589i	$-0.793737\pi$
$919$	−3.75192e10	−1.59459	−0.797296	+	0.603589i	$0.793737\pi$
$920$	−2.99402e10	−1.26764
$921$	2.57355e10	1.08548
$922$	1.23073e10	0.517135
$923$	1.03948e10	0.435122
$924$	0	0
$925$	4.76078e9	0.197780
$926$	−5.43668e10	−2.25006
$927$	1.38559e10	0.571288
$928$	2.38689e10	0.980424
$929$	2.44185e10	0.999228	0.499614	−	0.866248i	$-0.333475\pi$
$929$	2.44185e10	0.999228	0.499614	+	0.866248i	$0.333475\pi$
$930$	3.65550e10	1.49024
$931$	−1.59125e10	−0.646273
$932$	−7.29797e10	−2.95288
$933$	−1.39150e10	−0.560917
$934$	−4.80619e9	−0.193013
$935$	0	0
$936$	−4.28057e9	−0.170622
$937$	−3.66303e9	−0.145463	−0.0727313	−	0.997352i	$-0.523172\pi$
$937$	−3.66303e9	−0.145463	−0.0727313	+	0.997352i	$0.523172\pi$
$938$	−2.91095e10	−1.15166
$939$	2.00314e9	0.0789555
$940$	2.98054e10	1.17044
$941$	−9.93516e9	−0.388697	−0.194348	−	0.980933i	$-0.562259\pi$
$941$	−9.93516e9	−0.388697	−0.194348	+	0.980933i	$0.562259\pi$
$942$	3.28313e9	0.127970
$943$	−5.94173e10	−2.30739
$944$	−1.98208e9	−0.0766866
$945$	2.16232e9	0.0833505
$946$	0	0
$947$	−6.70589e9	−0.256585	−0.128293	−	0.991736i	$-0.540950\pi$
$947$	−6.70589e9	−0.256585	−0.128293	+	0.991736i	$0.540950\pi$
$948$	−4.58488e10	−1.74783
$949$	1.21383e9	0.0461026
$950$	−1.13620e10	−0.429955
$951$	−2.00316e10	−0.755239
$952$	−4.80191e8	−0.0180378
$953$	−1.42513e10	−0.533371	−0.266686	−	0.963784i	$-0.585929\pi$
$953$	−1.42513e10	−0.533371	−0.266686	+	0.963784i	$0.585929\pi$
$954$	−6.20744e9	−0.231469
$955$	2.14611e10	0.797336
$956$	3.20983e10	1.18817
$957$	0	0
$958$	−7.45195e10	−2.73836
$959$	−4.79496e9	−0.175558
$960$	2.09100e10	0.762791
$961$	6.95077e10	2.52639
$962$	1.36427e10	0.494069
$963$	1.19319e10	0.430543
$964$	3.93972e10	1.41643
$965$	−1.02334e10	−0.366586
$966$	−1.83449e10	−0.654779
$967$	4.27895e10	1.52175	0.760876	−	0.648897i	$-0.224769\pi$
$967$	4.27895e10	1.52175	0.760876	+	0.648897i	$0.224769\pi$
$968$	0	0
$969$	4.45455e8	0.0157279
$970$	−3.49015e8	−0.0122784
$971$	1.36249e10	0.477601	0.238801	−	0.971069i	$-0.423246\pi$
$971$	1.36249e10	0.477601	0.238801	+	0.971069i	$0.423246\pi$
$972$	3.10212e9	0.108350
$973$	−1.85328e10	−0.644980
$974$	−3.35186e10	−1.16233
$975$	2.25127e9	0.0777879
$976$	5.69211e9	0.195974
$977$	−1.49822e10	−0.513978	−0.256989	−	0.966414i	$-0.582730\pi$
$977$	−1.49822e10	−0.513978	−0.256989	+	0.966414i	$0.582730\pi$
$978$	1.40324e10	0.479673
$979$	0	0
$980$	3.05772e10	1.03778
$981$	4.63167e9	0.156638
$982$	1.12131e10	0.377864
$983$	−6.43998e9	−0.216246	−0.108123	−	0.994138i	$-0.534484\pi$
$983$	−6.43998e9	−0.216246	−0.108123	+	0.994138i	$0.534484\pi$
$984$	−3.36073e10	−1.12448
$985$	−2.15027e10	−0.716913
$986$	−1.73592e9	−0.0576713
$987$	7.44981e9	0.246624
$988$	−2.04511e10	−0.674633
$989$	7.67971e9	0.252440
$990$	0	0
$991$	−4.33001e10	−1.41329	−0.706645	−	0.707569i	$-0.749792\pi$
$991$	−4.33001e10	−1.41329	−0.706645	+	0.707569i	$0.749792\pi$
$992$	4.97325e10	1.61752
$993$	−2.60514e10	−0.844325
$994$	2.51972e10	0.813767
$995$	4.88927e10	1.57349
$996$	−5.19955e10	−1.66747
$997$	5.43254e9	0.173608	0.0868041	−	0.996225i	$-0.472335\pi$
$997$	5.43254e9	0.173608	0.0868041	+	0.996225i	$0.472335\pi$
$998$	−4.87398e10	−1.55212
$999$	−4.03319e9	−0.127988

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	363.8.a.m.1.10	yes	10
11.10	odd	2	inner	363.8.a.m.1.1	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
363.8.a.m.1.1	✓	10	11.10	odd	2	inner
363.8.a.m.1.10	yes	10	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 363 = 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	363.a (trivial)

\(f(q)\)	\(=\)	\(q+18.5524 q^{2} +27.0000 q^{3} +216.192 q^{4} -234.289 q^{5} +500.915 q^{6} -468.897 q^{7} +1636.18 q^{8} +729.000 q^{9} -4346.62 q^{10} +5837.19 q^{12} -3588.75 q^{13} -8699.17 q^{14} -6325.79 q^{15} +2682.47 q^{16} +625.901 q^{17} +13524.7 q^{18} +26359.3 q^{19} -50651.4 q^{20} -12660.2 q^{21} +78104.0 q^{23} +44176.8 q^{24} -23233.9 q^{25} -66579.9 q^{26} +19683.0 q^{27} -101372. q^{28} -149494. q^{29} -117359. q^{30} -311481. q^{31} -159665. q^{32} +11612.0 q^{34} +109857. q^{35} +157604. q^{36} -204907. q^{37} +489029. q^{38} -96896.2 q^{39} -383338. q^{40} -760746. q^{41} -234878. q^{42} +98326.7 q^{43} -170796. q^{45} +1.44902e6 q^{46} -588443. q^{47} +72426.6 q^{48} -603679. q^{49} -431044. q^{50} +16899.3 q^{51} -775859. q^{52} -458970. q^{53} +365167. q^{54} -767199. q^{56} +711701. q^{57} -2.77347e6 q^{58} -738903. q^{59} -1.36759e6 q^{60} +2.12197e6 q^{61} -5.77872e6 q^{62} -341826. q^{63} -3.30552e6 q^{64} +840803. q^{65} +3.34623e6 q^{67} +135315. q^{68} +2.10881e6 q^{69} +2.03812e6 q^{70} -2.89651e6 q^{71} +1.19277e6 q^{72} -338231. q^{73} -3.80152e6 q^{74} -627314. q^{75} +5.69867e6 q^{76} -1.79766e6 q^{78} -7.85461e6 q^{79} -628471. q^{80} +531441. q^{81} -1.41137e7 q^{82} -8.90762e6 q^{83} -2.73704e6 q^{84} -146642. q^{85} +1.82420e6 q^{86} -4.03633e6 q^{87} +7.44444e6 q^{89} -3.16869e6 q^{90} +1.68275e6 q^{91} +1.68855e7 q^{92} -8.40998e6 q^{93} -1.09170e7 q^{94} -6.17568e6 q^{95} -4.31094e6 q^{96} +80295.7 q^{97} -1.11997e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 270 q^{3} + 486 q^{4} - 946 q^{5} + 7290 q^{9} + 13122 q^{12} - 16948 q^{14} - 25542 q^{15} - 35374 q^{16} - 48738 q^{20} + 2108 q^{23} - 76684 q^{25} - 367002 q^{26} + 196830 q^{27} - 664572 q^{31}+ \cdots + 15434810 q^{97}+O(q^{100}) \) 10 * q + 270 * q^3 + 486 * q^4 - 946 * q^5 + 7290 * q^9 + 13122 * q^12 - 16948 * q^14 - 25542 * q^15 - 35374 * q^16 - 48738 * q^20 + 2108 * q^23 - 76684 * q^25 - 367002 * q^26 + 196830 * q^27 - 664572 * q^31 + 68938 * q^34 + 354294 * q^36 - 1031062 * q^37 + 1077608 * q^38 - 457596 * q^42 - 689634 * q^45 - 219544 * q^47 - 955098 * q^48 - 51026 * q^49 - 6176094 * q^53 - 6137980 * q^56 - 4487498 * q^58 - 3815652 * q^59 - 1315926 * q^60 - 9266602 * q^64 - 14109384 * q^67 + 56916 * q^69 - 15425108 * q^70 - 5276156 * q^71 - 2070468 * q^75 - 9909054 * q^78 + 934666 * q^80 + 5314410 * q^81 - 26188378 * q^82 + 3673636 * q^86 - 9677426 * q^89 - 28234212 * q^91 + 19508076 * q^92 - 17943444 * q^93 + 15434810 * q^97

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