LMFDB - Embedded newform 147.8.a.m.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [147,8,Mod(1,147)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("147.1"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(147, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 147 = 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	147.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,-14,162] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$45.9205987462$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 553x^{4} + 1386x^{3} + 69169x^{2} - 44864x - 1106944 $ x^6 - 2x^5 - 553x^4 + 1386x^3 + 69169x^2 - 44864*x - 1106944
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4}\cdot 7^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$4.50858$ of defining polynomial
Character	$\chi$	$=$	147.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+11.9730 q^{2} +27.0000 q^{3} +15.3529 q^{4} +147.782 q^{5} +323.271 q^{6} -1348.72 q^{8} +729.000 q^{9} +1769.40 q^{10} -6362.41 q^{11} +414.529 q^{12} -6191.39 q^{13} +3990.11 q^{15} -18113.5 q^{16} -15339.7 q^{17} +8728.32 q^{18} -11942.0 q^{19} +2268.88 q^{20} -76177.1 q^{22} +78573.4 q^{23} -36415.6 q^{24} -56285.5 q^{25} -74129.5 q^{26} +19683.0 q^{27} -10830.8 q^{29} +47773.7 q^{30} -46357.4 q^{31} -44235.9 q^{32} -171785. q^{33} -183663. q^{34} +11192.3 q^{36} +357314. q^{37} -142982. q^{38} -167167. q^{39} -199317. q^{40} -352362. q^{41} -618076. q^{43} -97681.5 q^{44} +107733. q^{45} +940760. q^{46} -1.00174e6 q^{47} -489063. q^{48} -673906. q^{50} -414173. q^{51} -95055.9 q^{52} +723424. q^{53} +235665. q^{54} -940249. q^{55} -322435. q^{57} -129677. q^{58} -1.61679e6 q^{59} +61259.9 q^{60} -1.01852e6 q^{61} -555038. q^{62} +1.78889e6 q^{64} -914976. q^{65} -2.05678e6 q^{66} +3.83846e6 q^{67} -235510. q^{68} +2.12148e6 q^{69} -344495. q^{71} -983220. q^{72} +2.73444e6 q^{73} +4.27812e6 q^{74} -1.51971e6 q^{75} -183345. q^{76} -2.00150e6 q^{78} -8.30571e6 q^{79} -2.67684e6 q^{80} +531441. q^{81} -4.21883e6 q^{82} +9.63245e6 q^{83} -2.26694e6 q^{85} -7.40023e6 q^{86} -292432. q^{87} +8.58113e6 q^{88} -9.22066e6 q^{89} +1.28989e6 q^{90} +1.20633e6 q^{92} -1.25165e6 q^{93} -1.19938e7 q^{94} -1.76482e6 q^{95} -1.19437e6 q^{96} -1.32776e7 q^{97} -4.63819e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 14 q^{2} + 162 q^{3} + 438 q^{4} - 500 q^{5} - 378 q^{6} - 1218 q^{8} + 4374 q^{9} - 6912 q^{10} - 1204 q^{11} + 11826 q^{12} - 17576 q^{13} - 13500 q^{15} + 15522 q^{16} - 60836 q^{17} - 10206 q^{18}+ \cdots - 877716 q^{99}+O(q^{100}) $ 6 * q - 14 * q^2 + 162 * q^3 + 438 * q^4 - 500 * q^5 - 378 * q^6 - 1218 * q^8 + 4374 * q^9 - 6912 * q^10 - 1204 * q^11 + 11826 * q^12 - 17576 * q^13 - 13500 * q^15 + 15522 * q^16 - 60836 * q^17 - 10206 * q^18 + 17280 * q^19 - 43924 * q^20 - 244452 * q^22 + 47708 * q^23 - 32886 * q^24 - 126954 * q^25 - 225228 * q^26 + 118098 * q^27 - 85880 * q^29 - 186624 * q^30 - 344872 * q^31 - 399678 * q^32 - 32508 * q^33 + 216304 * q^34 + 319302 * q^36 - 535896 * q^37 + 1154872 * q^38 - 474552 * q^39 - 817048 * q^40 + 696316 * q^41 - 162720 * q^43 + 1504868 * q^44 - 364500 * q^45 + 216540 * q^46 - 2280672 * q^47 + 419094 * q^48 + 3654866 * q^50 - 1642572 * q^51 - 5978752 * q^52 + 5321348 * q^53 - 275562 * q^54 - 1716328 * q^55 + 466560 * q^57 + 2346000 * q^58 - 2702888 * q^59 - 1185948 * q^60 - 7891216 * q^61 - 8107760 * q^62 - 4370970 * q^64 + 3217860 * q^65 - 6600204 * q^66 + 565920 * q^67 - 21007124 * q^68 + 1288116 * q^69 - 4370788 * q^71 - 887922 * q^72 - 11994056 * q^73 - 278200 * q^74 - 3427758 * q^75 - 3409936 * q^76 - 6081156 * q^78 - 16816944 * q^79 - 14397828 * q^80 + 3188646 * q^81 - 29067128 * q^82 - 2431304 * q^83 - 2530620 * q^85 + 16351456 * q^86 - 2318760 * q^87 - 23615724 * q^88 - 26982380 * q^89 - 5038848 * q^90 + 22208116 * q^92 - 9311544 * q^93 + 1880792 * q^94 - 11963336 * q^95 - 10791306 * q^96 - 14260040 * q^97 - 877716 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	11.9730	1.05827	0.529137	−	0.848536i	$-0.322516\pi$
$2$	11.9730	1.05827	0.529137	+	0.848536i	$0.322516\pi$
$3$	27.0000	0.577350
$3$	27.0000	0.577350
$4$	15.3529	0.119945
$5$	147.782	0.528721	0.264360	−	0.964424i	$-0.414839\pi$
$5$	147.782	0.528721	0.264360	+	0.964424i	$0.414839\pi$
$6$	323.271	0.610995
$7$	0	0
$7$	0	0
$8$	−1348.72	−0.931340
$9$	729.000	0.333333
$10$	1769.40	0.559532
$11$	−6362.41	−1.44128	−0.720638	−	0.693312i	$-0.756151\pi$
$11$	−6362.41	−1.44128	−0.720638	+	0.693312i	$0.756151\pi$
$12$	414.529	0.0692501
$13$	−6191.39	−0.781603	−0.390802	−	0.920475i	$-0.627802\pi$
$13$	−6191.39	−0.781603	−0.390802	+	0.920475i	$0.627802\pi$
$14$	0	0
$15$	3990.11	0.305257
$16$	−18113.5	−1.10556
$17$	−15339.7	−0.757263	−0.378632	−	0.925547i	$-0.623605\pi$
$17$	−15339.7	−0.757263	−0.378632	+	0.925547i	$0.623605\pi$
$18$	8728.32	0.352758
$19$	−11942.0	−0.399430	−0.199715	−	0.979854i	$-0.564002\pi$
$19$	−11942.0	−0.399430	−0.199715	+	0.979854i	$0.564002\pi$
$20$	2268.88	0.0634172
$21$	0	0
$22$	−76177.1	−1.52526
$23$	78573.4	1.34657	0.673284	−	0.739384i	$-0.264883\pi$
$23$	78573.4	1.34657	0.673284	+	0.739384i	$0.264883\pi$
$24$	−36415.6	−0.537709
$25$	−56285.5	−0.720454
$26$	−74129.5	−0.827151
$27$	19683.0	0.192450
$28$	0	0
$29$	−10830.8	−0.0824646	−0.0412323	−	0.999150i	$-0.513128\pi$
$29$	−10830.8	−0.0824646	−0.0412323	+	0.999150i	$0.513128\pi$
$30$	47773.7	0.323046
$31$	−46357.4	−0.279482	−0.139741	−	0.990188i	$-0.544627\pi$
$31$	−46357.4	−0.279482	−0.139741	+	0.990188i	$0.544627\pi$
$32$	−44235.9	−0.238644
$33$	−171785.	−0.832121
$34$	−183663.	−0.801393
$35$	0	0
$36$	11192.3	0.0399815
$37$	357314.	1.15969	0.579847	−	0.814725i	$-0.303112\pi$
$37$	357314.	1.15969	0.579847	+	0.814725i	$0.303112\pi$
$38$	−142982.	−0.422706
$39$	−167167.	−0.451259
$40$	−199317.	−0.492419
$41$	−352362.	−0.798445	−0.399222	−	0.916854i	$-0.630720\pi$
$41$	−352362.	−0.798445	−0.399222	+	0.916854i	$0.630720\pi$
$42$	0	0
$43$	−618076.	−1.18550	−0.592751	−	0.805385i	$-0.701958\pi$
$43$	−618076.	−1.18550	−0.592751	+	0.805385i	$0.701958\pi$
$44$	−97681.5	−0.172873
$45$	107733.	0.176240
$46$	940760.	1.42504
$47$	−1.00174e6	−1.40738	−0.703692	−	0.710505i	$-0.748466\pi$
$47$	−1.00174e6	−1.40738	−0.703692	+	0.710505i	$0.748466\pi$
$48$	−489063.	−0.638294
$49$	0	0
$50$	−673906.	−0.762438
$51$	−414173.	−0.437206
$52$	−95055.9	−0.0937491
$53$	723424.	0.667463	0.333732	−	0.942668i	$-0.391692\pi$
$53$	723424.	0.667463	0.333732	+	0.942668i	$0.391692\pi$
$54$	235665.	0.203665
$55$	−940249.	−0.762033
$56$	0	0
$57$	−322435.	−0.230611
$58$	−129677.	−0.0872702
$59$	−1.61679e6	−1.02488	−0.512439	−	0.858724i	$-0.671258\pi$
$59$	−1.61679e6	−1.02488	−0.512439	+	0.858724i	$0.671258\pi$
$60$	61259.9	0.0366140
$61$	−1.01852e6	−0.574531	−0.287265	−	0.957851i	$-0.592746\pi$
$61$	−1.01852e6	−0.574531	−0.287265	+	0.957851i	$0.592746\pi$
$62$	−555038.	−0.295768
$63$	0	0
$64$	1.78889e6	0.853008
$65$	−914976.	−0.413250
$66$	−2.05678e6	−0.880612
$67$	3.83846e6	1.55917	0.779587	−	0.626294i	$-0.215429\pi$
$67$	3.83846e6	1.55917	0.779587	+	0.626294i	$0.215429\pi$
$68$	−235510.	−0.0908297
$69$	2.12148e6	0.777441
$70$	0	0
$71$	−344495.	−0.114229	−0.0571147	−	0.998368i	$-0.518190\pi$
$71$	−344495.	−0.114229	−0.0571147	+	0.998368i	$0.518190\pi$
$72$	−983220.	−0.310447
$73$	2.73444e6	0.822693	0.411347	−	0.911479i	$-0.365059\pi$
$73$	2.73444e6	0.822693	0.411347	+	0.911479i	$0.365059\pi$
$74$	4.27812e6	1.22727
$75$	−1.51971e6	−0.415954
$76$	−183345.	−0.0479094
$77$	0	0
$78$	−2.00150e6	−0.477556
$79$	−8.30571e6	−1.89532	−0.947659	−	0.319286i	$-0.896557\pi$
$79$	−8.30571e6	−1.89532	−0.947659	+	0.319286i	$0.896557\pi$
$80$	−2.67684e6	−0.584532
$81$	531441.	0.111111
$82$	−4.21883e6	−0.844974
$83$	9.63245e6	1.84911	0.924556	−	0.381046i	$-0.124436\pi$
$83$	9.63245e6	1.84911	0.924556	+	0.381046i	$0.124436\pi$
$84$	0	0
$85$	−2.26694e6	−0.400381
$86$	−7.40023e6	−1.25459
$87$	−292432.	−0.0476110
$88$	8.58113e6	1.34232
$89$	−9.22066e6	−1.38643	−0.693214	−	0.720732i	$-0.743806\pi$
$89$	−9.22066e6	−1.38643	−0.693214	+	0.720732i	$0.743806\pi$
$90$	1.28989e6	0.186511
$91$	0	0
$92$	1.20633e6	0.161514
$93$	−1.25165e6	−0.161359
$94$	−1.19938e7	−1.48940
$95$	−1.76482e6	−0.211187
$96$	−1.19437e6	−0.137781
$97$	−1.32776e7	−1.47713	−0.738564	−	0.674183i	$-0.764496\pi$
$97$	−1.32776e7	−1.47713	−0.738564	+	0.674183i	$0.764496\pi$
$98$	0	0
$99$	−4.63819e6	−0.480425
$100$	−864146.	−0.0864146
$101$	3.77157e6	0.364248	0.182124	−	0.983276i	$-0.441703\pi$
$101$	3.77157e6	0.364248	0.182124	+	0.983276i	$0.441703\pi$
$102$	−4.95890e6	−0.462684
$103$	2.02815e7	1.82882	0.914410	−	0.404790i	$-0.132655\pi$
$103$	2.02815e7	1.82882	0.914410	+	0.404790i	$0.132655\pi$
$104$	8.35048e6	0.727938
$105$	0	0
$106$	8.66156e6	0.706359
$107$	2.24485e6	0.177151	0.0885757	−	0.996069i	$-0.471768\pi$
$107$	2.24485e6	0.177151	0.0885757	+	0.996069i	$0.471768\pi$
$108$	302191.	0.0230834
$109$	1.63686e7	1.21065	0.605324	−	0.795979i	$-0.293044\pi$
$109$	1.63686e7	1.21065	0.605324	+	0.795979i	$0.293044\pi$
$110$	−1.12576e7	−0.806440
$111$	9.64746e6	0.669550
$112$	0	0
$113$	−1.76531e7	−1.15092	−0.575462	−	0.817828i	$-0.695178\pi$
$113$	−1.76531e7	−1.15092	−0.575462	+	0.817828i	$0.695178\pi$
$114$	−3.86051e6	−0.244049
$115$	1.16117e7	0.711958
$116$	−166284.	−0.00989119
$117$	−4.51352e6	−0.260534
$118$	−1.93578e7	−1.08460
$119$	0	0
$120$	−5.38156e6	−0.284298
$121$	2.09930e7	1.07728
$122$	−1.21947e7	−0.608011
$123$	−9.51376e6	−0.460982
$124$	−711721.	−0.0335223
$125$	−1.98634e7	−0.909640
$126$	0	0
$127$	2.50963e6	0.108717	0.0543584	−	0.998521i	$-0.482689\pi$
$127$	2.50963e6	0.108717	0.0543584	+	0.998521i	$0.482689\pi$
$128$	2.70805e7	1.14136
$129$	−1.66881e7	−0.684450
$130$	−1.09550e7	−0.437332
$131$	−2.38551e7	−0.927112	−0.463556	−	0.886068i	$-0.653427\pi$
$131$	−2.38551e7	−0.927112	−0.463556	+	0.886068i	$0.653427\pi$
$132$	−2.63740e6	−0.0998084
$133$	0	0
$134$	4.59579e7	1.65003
$135$	2.90879e6	0.101752
$136$	2.06891e7	0.705270
$137$	−1.65732e7	−0.550660	−0.275330	−	0.961350i	$-0.588787\pi$
$137$	−1.65732e7	−0.550660	−0.275330	+	0.961350i	$0.588787\pi$
$138$	2.54005e7	0.822746
$139$	−4.23290e7	−1.33686	−0.668431	−	0.743774i	$-0.733034\pi$
$139$	−4.23290e7	−1.33686	−0.668431	+	0.743774i	$0.733034\pi$
$140$	0	0
$141$	−2.70470e7	−0.812554
$142$	−4.12464e6	−0.120886
$143$	3.93921e7	1.12651
$144$	−1.32047e7	−0.368519
$145$	−1.60060e6	−0.0436008
$146$	3.27394e7	0.870635
$147$	0	0
$148$	5.48580e6	0.139099
$149$	4.63550e7	1.14801	0.574003	−	0.818853i	$-0.305390\pi$
$149$	4.63550e7	1.14801	0.574003	+	0.818853i	$0.305390\pi$
$150$	−1.81955e7	−0.440194
$151$	−6.41314e6	−0.151583	−0.0757916	−	0.997124i	$-0.524148\pi$
$151$	−6.41314e6	−0.151583	−0.0757916	+	0.997124i	$0.524148\pi$
$152$	1.61065e7	0.372005
$153$	−1.11827e7	−0.252421
$154$	0	0
$155$	−6.85079e6	−0.147768
$156$	−2.56651e6	−0.0541261
$157$	7.37494e7	1.52093	0.760466	−	0.649378i	$-0.224971\pi$
$157$	7.37494e7	1.52093	0.760466	+	0.649378i	$0.224971\pi$
$158$	−9.94443e7	−2.00577
$159$	1.95324e7	0.385360
$160$	−6.53727e6	−0.126176
$161$	0	0
$162$	6.36295e6	0.117586
$163$	6.32095e7	1.14321	0.571604	−	0.820529i	$-0.306321\pi$
$163$	6.32095e7	1.14321	0.571604	+	0.820529i	$0.306321\pi$
$164$	−5.40978e6	−0.0957692
$165$	−2.53867e7	−0.439960
$166$	1.15329e8	1.95687
$167$	1.01813e8	1.69159	0.845794	−	0.533510i	$-0.179127\pi$
$167$	1.01813e8	1.69159	0.845794	+	0.533510i	$0.179127\pi$
$168$	0	0
$169$	−2.44152e7	−0.389096
$170$	−2.71421e7	−0.423713
$171$	−8.70573e6	−0.133143
$172$	−9.48927e6	−0.142195
$173$	1.35596e7	0.199106	0.0995531	−	0.995032i	$-0.468259\pi$
$173$	1.35596e7	0.199106	0.0995531	+	0.995032i	$0.468259\pi$
$174$	−3.50129e6	−0.0503854
$175$	0	0
$176$	1.15245e8	1.59341
$177$	−4.36534e7	−0.591713
$178$	−1.10399e8	−1.46722
$179$	4.47771e7	0.583540	0.291770	−	0.956489i	$-0.405756\pi$
$179$	4.47771e7	0.583540	0.291770	+	0.956489i	$0.405756\pi$
$180$	1.65402e6	0.0211391
$181$	9.90123e7	1.24112	0.620561	−	0.784158i	$-0.286905\pi$
$181$	9.90123e7	1.24112	0.620561	+	0.784158i	$0.286905\pi$
$182$	0	0
$183$	−2.74999e7	−0.331705
$184$	−1.05974e8	−1.25411
$185$	5.28045e7	0.613155
$186$	−1.49860e7	−0.170762
$187$	9.75977e7	1.09143
$188$	−1.53796e7	−0.168808
$189$	0	0
$190$	−2.11302e7	−0.223494
$191$	9.30277e7	0.966041	0.483020	−	0.875609i	$-0.339540\pi$
$191$	9.30277e7	0.966041	0.483020	+	0.875609i	$0.339540\pi$
$192$	4.82999e7	0.492484
$193$	−1.21860e8	−1.22015	−0.610073	−	0.792345i	$-0.708860\pi$
$193$	−1.21860e8	−1.22015	−0.610073	+	0.792345i	$0.708860\pi$
$194$	−1.58973e8	−1.56321
$195$	−2.47043e7	−0.238590
$196$	0	0
$197$	1.33984e8	1.24859	0.624296	−	0.781188i	$-0.285386\pi$
$197$	1.33984e8	1.24859	0.624296	+	0.781188i	$0.285386\pi$
$198$	−5.55331e7	−0.508422
$199$	−2.67278e6	−0.0240424	−0.0120212	−	0.999928i	$-0.503827\pi$
$199$	−2.67278e6	−0.0240424	−0.0120212	+	0.999928i	$0.503827\pi$
$200$	7.59136e7	0.670988
$201$	1.03638e8	0.900190
$202$	4.51570e7	0.385474
$203$	0	0
$204$	−6.35877e6	−0.0524406
$205$	−5.20727e7	−0.422155
$206$	2.42831e8	1.93539
$207$	5.72800e7	0.448856
$208$	1.12147e8	0.864108
$209$	7.59800e7	0.575688
$210$	0	0
$211$	−9.35784e7	−0.685784	−0.342892	−	0.939375i	$-0.611406\pi$
$211$	−9.35784e7	−0.685784	−0.342892	+	0.939375i	$0.611406\pi$
$212$	1.11067e7	0.0800586
$213$	−9.30135e6	−0.0659504
$214$	2.68776e7	0.187475
$215$	−9.13406e7	−0.626800
$216$	−2.65469e7	−0.179236
$217$	0	0
$218$	1.95981e8	1.28120
$219$	7.38298e7	0.474982
$220$	−1.44356e7	−0.0914017
$221$	9.49743e7	0.591880
$222$	1.15509e8	0.708567
$223$	−1.41723e8	−0.855804	−0.427902	−	0.903825i	$-0.640747\pi$
$223$	−1.41723e8	−0.855804	−0.427902	+	0.903825i	$0.640747\pi$
$224$	0	0
$225$	−4.10321e7	−0.240151
$226$	−2.11361e8	−1.21799
$227$	−1.90934e8	−1.08341	−0.541704	−	0.840569i	$-0.682221\pi$
$227$	−1.90934e8	−1.08341	−0.541704	+	0.840569i	$0.682221\pi$
$228$	−4.95031e6	−0.0276605
$229$	−2.55329e8	−1.40500	−0.702500	−	0.711683i	$-0.747933\pi$
$229$	−2.55329e8	−1.40500	−0.702500	+	0.711683i	$0.747933\pi$
$230$	1.39027e8	0.753447
$231$	0	0
$232$	1.46078e7	0.0768026
$233$	1.72507e8	0.893430	0.446715	−	0.894676i	$-0.352594\pi$
$233$	1.72507e8	0.893430	0.446715	+	0.894676i	$0.352594\pi$
$234$	−5.40404e7	−0.275717
$235$	−1.48039e8	−0.744113
$236$	−2.48224e7	−0.122929
$237$	−2.24254e8	−1.09426
$238$	0	0
$239$	−7.15156e7	−0.338851	−0.169425	−	0.985543i	$-0.554191\pi$
$239$	−7.15156e7	−0.338851	−0.169425	+	0.985543i	$0.554191\pi$
$240$	−7.22748e7	−0.337479
$241$	−2.43260e8	−1.11946	−0.559732	−	0.828674i	$-0.689096\pi$
$241$	−2.43260e8	−1.11946	−0.559732	+	0.828674i	$0.689096\pi$
$242$	2.51350e8	1.14005
$243$	1.43489e7	0.0641500
$244$	−1.56372e7	−0.0689119
$245$	0	0
$246$	−1.13908e8	−0.487846
$247$	7.39377e7	0.312195
$248$	6.25234e7	0.260292
$249$	2.60076e8	1.06759
$250$	−2.37825e8	−0.962649
$251$	−4.15842e8	−1.65986	−0.829928	−	0.557871i	$-0.811618\pi$
$251$	−4.15842e8	−1.65986	−0.829928	+	0.557871i	$0.811618\pi$
$252$	0	0
$253$	−4.99916e8	−1.94077
$254$	3.00478e7	0.115052
$255$	−6.12073e7	−0.231160
$256$	9.52581e7	0.354864
$257$	−4.63963e8	−1.70497	−0.852487	−	0.522749i	$-0.824906\pi$
$257$	−4.63963e8	−1.70497	−0.852487	+	0.522749i	$0.824906\pi$
$258$	−1.99806e8	−0.724336
$259$	0	0
$260$	−1.40475e7	−0.0495671
$261$	−7.89565e6	−0.0274882
$262$	−2.85618e8	−0.981138
$263$	1.90290e8	0.645018	0.322509	−	0.946566i	$-0.395474\pi$
$263$	1.90290e8	0.645018	0.322509	+	0.946566i	$0.395474\pi$
$264$	2.31691e8	0.774987
$265$	1.06909e8	0.352902
$266$	0	0
$267$	−2.48958e8	−0.800454
$268$	5.89315e7	0.187015
$269$	2.61320e8	0.818540	0.409270	−	0.912413i	$-0.365783\pi$
$269$	2.61320e8	0.818540	0.409270	+	0.912413i	$0.365783\pi$
$270$	3.48270e7	0.107682
$271$	−5.98252e7	−0.182596	−0.0912981	−	0.995824i	$-0.529102\pi$
$271$	−5.98252e7	−0.182596	−0.0912981	+	0.995824i	$0.529102\pi$
$272$	2.77856e8	0.837199
$273$	0	0
$274$	−1.98431e8	−0.582749
$275$	3.58111e8	1.03837
$276$	3.25709e7	0.0932499
$277$	−3.40657e8	−0.963027	−0.481514	−	0.876439i	$-0.659913\pi$
$277$	−3.40657e8	−0.963027	−0.481514	+	0.876439i	$0.659913\pi$
$278$	−5.06806e8	−1.41477
$279$	−3.37945e7	−0.0931605
$280$	0	0
$281$	4.33869e8	1.16650	0.583252	−	0.812291i	$-0.301780\pi$
$281$	4.33869e8	1.16650	0.583252	+	0.812291i	$0.301780\pi$
$282$	−3.23834e8	−0.859905
$283$	−4.62429e8	−1.21281	−0.606405	−	0.795156i	$-0.707389\pi$
$283$	−4.62429e8	−1.21281	−0.606405	+	0.795156i	$0.707389\pi$
$284$	−5.28900e6	−0.0137012
$285$	−4.76500e7	−0.121929
$286$	4.71642e8	1.19215
$287$	0	0
$288$	−3.22480e7	−0.0795479
$289$	−1.75031e8	−0.426552
$290$	−1.91640e7	−0.0461416
$291$	−3.58495e8	−0.852820
$292$	4.19816e7	0.0986776
$293$	−5.23279e8	−1.21534	−0.607668	−	0.794191i	$-0.707895\pi$
$293$	−5.23279e8	−1.21534	−0.607668	+	0.794191i	$0.707895\pi$
$294$	0	0
$295$	−2.38933e8	−0.541874
$296$	−4.81917e8	−1.08007
$297$	−1.25231e8	−0.277374
$298$	5.55008e8	1.21491
$299$	−4.86478e8	−1.05248
$300$	−2.33319e7	−0.0498915
$301$	0	0
$302$	−7.67846e7	−0.160417
$303$	1.01832e8	0.210299
$304$	2.16311e8	0.441593
$305$	−1.50518e8	−0.303766
$306$	−1.33890e8	−0.267131
$307$	−5.56846e8	−1.09838	−0.549188	−	0.835699i	$-0.685063\pi$
$307$	−5.56846e8	−1.09838	−0.549188	+	0.835699i	$0.685063\pi$
$308$	0	0
$309$	5.47602e8	1.05587
$310$	−8.20246e7	−0.156379
$311$	−2.00327e8	−0.377641	−0.188820	−	0.982012i	$-0.560466\pi$
$311$	−2.00327e8	−0.377641	−0.188820	+	0.982012i	$0.560466\pi$
$312$	2.25463e8	0.420275
$313$	5.63849e7	0.103934	0.0519670	−	0.998649i	$-0.483451\pi$
$313$	5.63849e7	0.103934	0.0519670	+	0.998649i	$0.483451\pi$
$314$	8.83002e8	1.60956
$315$	0	0
$316$	−1.27517e8	−0.227333
$317$	2.66478e8	0.469844	0.234922	−	0.972014i	$-0.424517\pi$
$317$	2.66478e8	0.469844	0.234922	+	0.972014i	$0.424517\pi$
$318$	2.33862e8	0.407817
$319$	6.89099e7	0.118854
$320$	2.64365e8	0.451003
$321$	6.06110e7	0.102278
$322$	0	0
$323$	1.83188e8	0.302473
$324$	8.15917e6	0.0133272
$325$	3.48485e8	0.563109
$326$	7.56807e8	1.20983
$327$	4.41951e8	0.698968
$328$	4.75239e8	0.743624
$329$	0	0
$330$	−3.03955e8	−0.465598
$331$	−7.10038e8	−1.07618	−0.538088	−	0.842889i	$-0.680853\pi$
$331$	−7.10038e8	−1.07618	−0.538088	+	0.842889i	$0.680853\pi$
$332$	1.47886e8	0.221791
$333$	2.60482e8	0.386565
$334$	1.21900e9	1.79016
$335$	5.67255e8	0.824368
$336$	0	0
$337$	1.99282e8	0.283638	0.141819	−	0.989893i	$-0.454705\pi$
$337$	1.99282e8	0.283638	0.141819	+	0.989893i	$0.454705\pi$
$338$	−2.92324e8	−0.411771
$339$	−4.76634e8	−0.664486
$340$	−3.48041e7	−0.0480236
$341$	2.94945e8	0.402810
$342$	−1.04234e8	−0.140902
$343$	0	0
$344$	8.33615e8	1.10411
$345$	3.13517e8	0.411049
$346$	1.62349e8	0.210709
$347$	−2.12282e8	−0.272747	−0.136373	−	0.990658i	$-0.543545\pi$
$347$	−2.12282e8	−0.272747	−0.136373	+	0.990658i	$0.543545\pi$
$348$	−4.48968e6	−0.00571068
$349$	−1.06480e9	−1.34084	−0.670421	−	0.741981i	$-0.733887\pi$
$349$	−1.06480e9	−1.34084	−0.670421	+	0.741981i	$0.733887\pi$
$350$	0	0
$351$	−1.21865e8	−0.150420
$352$	2.81447e8	0.343951
$353$	−7.14274e7	−0.0864278	−0.0432139	−	0.999066i	$-0.513760\pi$
$353$	−7.14274e7	−0.0864278	−0.0432139	+	0.999066i	$0.513760\pi$
$354$	−5.22662e8	−0.626195
$355$	−5.09101e7	−0.0603955
$356$	−1.41564e8	−0.166295
$357$	0	0
$358$	5.36117e8	0.617545
$359$	−778009.	−0.000887471 0	−0.000443735	−	1.00000i	$-0.500141\pi$
$359$	−778009.	−0.000887471 0	−0.000443735		1.00000i	$0.500141\pi$
$360$	−1.45302e8	−0.164140
$361$	−7.51260e8	−0.840456
$362$	1.18548e9	1.31345
$363$	5.66812e8	0.621965
$364$	0	0
$365$	4.04101e8	0.434975
$366$	−3.29257e8	−0.351035
$367$	1.79377e9	1.89424	0.947122	−	0.320874i	$-0.103976\pi$
$367$	1.79377e9	1.89424	0.947122	+	0.320874i	$0.103976\pi$
$368$	−1.42324e9	−1.48871
$369$	−2.56872e8	−0.266148
$370$	6.32229e8	0.648886
$371$	0	0
$372$	−1.92165e7	−0.0193541
$373$	3.13767e8	0.313059	0.156529	−	0.987673i	$-0.449969\pi$
$373$	3.13767e8	0.313059	0.156529	+	0.987673i	$0.449969\pi$
$374$	1.16854e9	1.15503
$375$	−5.36313e8	−0.525181
$376$	1.35107e9	1.31075
$377$	6.70577e7	0.0644546
$378$	0	0
$379$	1.58359e9	1.49419	0.747096	−	0.664716i	$-0.231448\pi$
$379$	1.58359e9	1.49419	0.747096	+	0.664716i	$0.231448\pi$
$380$	−2.70951e7	−0.0253307
$381$	6.77600e7	0.0627676
$382$	1.11382e9	1.02234
$383$	6.11227e8	0.555913	0.277956	−	0.960594i	$-0.410343\pi$
$383$	6.11227e8	0.555913	0.277956	+	0.960594i	$0.410343\pi$
$384$	7.31175e8	0.658964
$385$	0	0
$386$	−1.45904e9	−1.29125
$387$	−4.50578e8	−0.395168
$388$	−2.03850e8	−0.177174
$389$	−1.22334e9	−1.05371	−0.526856	−	0.849954i	$-0.676629\pi$
$389$	−1.22334e9	−1.05371	−0.526856	+	0.849954i	$0.676629\pi$
$390$	−2.95785e8	−0.252494
$391$	−1.20530e9	−1.01971
$392$	0	0
$393$	−6.44088e8	−0.535268
$394$	1.60419e9	1.32135
$395$	−1.22743e9	−1.00209
$396$	−7.12098e7	−0.0576244
$397$	−2.35711e9	−1.89066	−0.945328	−	0.326120i	$-0.894259\pi$
$397$	−2.35711e9	−1.89066	−0.945328	+	0.326120i	$0.894259\pi$
$398$	−3.20013e7	−0.0254435
$399$	0	0
$400$	1.01952e9	0.796504
$401$	1.83192e9	1.41873	0.709367	−	0.704840i	$-0.248981\pi$
$401$	1.83192e9	1.41873	0.709367	+	0.704840i	$0.248981\pi$
$402$	1.24086e9	0.952648
$403$	2.87017e8	0.218444
$404$	5.79046e7	0.0436896
$405$	7.85374e7	0.0587468
$406$	0	0
$407$	−2.27337e9	−1.67144
$408$	5.58606e8	0.407188
$409$	−3.21133e8	−0.232089	−0.116044	−	0.993244i	$-0.537021\pi$
$409$	−3.21133e8	−0.232089	−0.116044	+	0.993244i	$0.537021\pi$
$410$	−6.23467e8	−0.446755
$411$	−4.47475e8	−0.317924
$412$	3.11381e8	0.219357
$413$	0	0
$414$	6.85814e8	0.475013
$415$	1.42350e9	0.977665
$416$	2.73882e8	0.186525
$417$	−1.14288e9	−0.771837
$418$	9.09709e8	0.609236
$419$	−1.07921e9	−0.716735	−0.358368	−	0.933581i	$-0.616667\pi$
$419$	−1.07921e9	−0.716735	−0.358368	+	0.933581i	$0.616667\pi$
$420$	0	0
$421$	2.69627e8	0.176107	0.0880535	−	0.996116i	$-0.471935\pi$
$421$	2.69627e8	0.176107	0.0880535	+	0.996116i	$0.471935\pi$
$422$	−1.12042e9	−0.725747
$423$	−7.30269e8	−0.469128
$424$	−9.75700e8	−0.621635
$425$	8.63405e8	0.545574
$426$	−1.11365e8	−0.0697936
$427$	0	0
$428$	3.44650e7	0.0212483
$429$	1.06359e9	0.650388
$430$	−1.09362e9	−0.663327
$431$	−1.66399e9	−1.00111	−0.500554	−	0.865705i	$-0.666870\pi$
$431$	−1.66399e9	−1.00111	−0.500554	+	0.865705i	$0.666870\pi$
$432$	−3.56527e8	−0.212765
$433$	−8.52786e8	−0.504815	−0.252408	−	0.967621i	$-0.581222\pi$
$433$	−8.52786e8	−0.504815	−0.252408	+	0.967621i	$0.581222\pi$
$434$	0	0
$435$	−4.32161e7	−0.0251729
$436$	2.51305e8	0.145211
$437$	−9.38325e8	−0.537859
$438$	8.83965e8	0.502661
$439$	6.18483e8	0.348901	0.174450	−	0.984666i	$-0.444185\pi$
$439$	6.18483e8	0.348901	0.174450	+	0.984666i	$0.444185\pi$
$440$	1.26814e9	0.709711
$441$	0	0
$442$	1.13713e9	0.626371
$443$	−8.31055e8	−0.454168	−0.227084	−	0.973875i	$-0.572919\pi$
$443$	−8.31055e8	−0.454168	−0.227084	+	0.973875i	$0.572919\pi$
$444$	1.48117e8	0.0803089
$445$	−1.36265e9	−0.733033
$446$	−1.69685e9	−0.905675
$447$	1.25158e9	0.662802
$448$	0	0
$449$	−9.02353e8	−0.470451	−0.235225	−	0.971941i	$-0.575583\pi$
$449$	−9.02353e8	−0.470451	−0.235225	+	0.971941i	$0.575583\pi$
$450$	−4.91278e8	−0.254146
$451$	2.24187e9	1.15078
$452$	−2.71027e8	−0.138047
$453$	−1.73155e8	−0.0875166
$454$	−2.28605e9	−1.14654
$455$	0	0
$456$	4.34875e8	0.214777
$457$	4.81860e6	0.00236164	0.00118082	−	0.999999i	$-0.499624\pi$
$457$	4.81860e6	0.00236164	0.00118082	+	0.999999i	$0.499624\pi$
$458$	−3.05706e9	−1.48688
$459$	−3.01932e8	−0.145735
$460$	1.78274e8	0.0853956
$461$	−8.95231e8	−0.425580	−0.212790	−	0.977098i	$-0.568255\pi$
$461$	−8.95231e8	−0.425580	−0.212790	+	0.977098i	$0.568255\pi$
$462$	0	0
$463$	−2.52287e9	−1.18130	−0.590652	−	0.806926i	$-0.701129\pi$
$463$	−2.52287e9	−1.18130	−0.590652	+	0.806926i	$0.701129\pi$
$464$	1.96183e8	0.0911694
$465$	−1.84971e8	−0.0853138
$466$	2.06543e9	0.945494
$467$	3.69100e9	1.67701	0.838504	−	0.544895i	$-0.183430\pi$
$467$	3.69100e9	1.67701	0.838504	+	0.544895i	$0.183430\pi$
$468$	−6.92957e7	−0.0312497
$469$	0	0
$470$	−1.77247e9	−0.787476
$471$	1.99123e9	0.878110
$472$	2.18061e9	0.954509
$473$	3.93245e9	1.70864
$474$	−2.68500e9	−1.15803
$475$	6.72162e8	0.287771
$476$	0	0
$477$	5.27376e8	0.222488
$478$	−8.56257e8	−0.358597
$479$	1.28903e9	0.535907	0.267953	−	0.963432i	$-0.413653\pi$
$479$	1.28903e9	0.535907	0.267953	+	0.963432i	$0.413653\pi$
$480$	−1.76506e8	−0.0728477
$481$	−2.21227e9	−0.906421
$482$	−2.91255e9	−1.18470
$483$	0	0
$484$	3.22304e8	0.129213
$485$	−1.96219e9	−0.780989
$486$	1.71800e8	0.0678883
$487$	2.88487e9	1.13181	0.565907	−	0.824469i	$-0.308526\pi$
$487$	2.88487e9	1.13181	0.565907	+	0.824469i	$0.308526\pi$
$488$	1.37370e9	0.535083
$489$	1.70666e9	0.660032
$490$	0	0
$491$	1.04090e8	0.0396849	0.0198425	−	0.999803i	$-0.493684\pi$
$491$	1.04090e8	0.0396849	0.0198425	+	0.999803i	$0.493684\pi$
$492$	−1.46064e8	−0.0552924
$493$	1.66142e8	0.0624474
$494$	8.85257e8	0.330388
$495$	−6.85442e8	−0.254011
$496$	8.39693e8	0.308983
$497$	0	0
$498$	3.11389e9	1.12980
$499$	−4.00550e9	−1.44313	−0.721565	−	0.692347i	$-0.756577\pi$
$499$	−4.00550e9	−1.44313	−0.721565	+	0.692347i	$0.756577\pi$
$500$	−3.04962e8	−0.109106
$501$	2.74894e9	0.976639
$502$	−4.97888e9	−1.75658
$503$	−1.59670e9	−0.559417	−0.279708	−	0.960085i	$-0.590238\pi$
$503$	−1.59670e9	−0.559417	−0.279708	+	0.960085i	$0.590238\pi$
$504$	0	0
$505$	5.57370e8	0.192586
$506$	−5.98550e9	−2.05387
$507$	−6.59211e8	−0.224645
$508$	3.85301e7	0.0130400
$509$	−3.43182e7	−0.0115349	−0.00576743	−	0.999983i	$-0.501836\pi$
$509$	−3.43182e7	−0.0115349	−0.00576743	+	0.999983i	$0.501836\pi$
$510$	−7.32836e8	−0.244631
$511$	0	0
$512$	−2.32578e9	−0.765816
$513$	−2.35055e8	−0.0768703
$514$	−5.55503e9	−1.80433
$515$	2.99725e9	0.966935
$516$	−2.56210e8	−0.0820962
$517$	6.37348e9	2.02843
$518$	0	0
$519$	3.66108e8	0.114954
$520$	1.23405e9	0.384876
$521$	−1.29223e9	−0.400321	−0.200161	−	0.979763i	$-0.564146\pi$
$521$	−1.29223e9	−0.400321	−0.200161	+	0.979763i	$0.564146\pi$
$522$	−9.45347e7	−0.0290901
$523$	−2.58606e8	−0.0790465	−0.0395233	−	0.999219i	$-0.512584\pi$
$523$	−2.58606e8	−0.0790465	−0.0395233	+	0.999219i	$0.512584\pi$
$524$	−3.66246e8	−0.111202
$525$	0	0
$526$	2.27835e9	0.682605
$527$	7.11111e8	0.211641
$528$	3.11162e9	0.919958
$529$	2.76895e9	0.813243
$530$	1.28002e9	0.373467
$531$	−1.17864e9	−0.341626
$532$	0	0
$533$	2.18161e9	0.624067
$534$	−2.98078e9	−0.847100
$535$	3.31749e8	0.0936636
$536$	−5.17702e9	−1.45212
$537$	1.20898e9	0.336907
$538$	3.12879e9	0.866240
$539$	0	0
$540$	4.46584e7	0.0122047
$541$	−3.55359e8	−0.0964888	−0.0482444	−	0.998836i	$-0.515363\pi$
$541$	−3.55359e8	−0.0964888	−0.0482444	+	0.998836i	$0.515363\pi$
$542$	−7.16288e8	−0.193237
$543$	2.67333e9	0.716562
$544$	6.78567e8	0.180716
$545$	2.41898e9	0.640095
$546$	0	0
$547$	−3.16657e9	−0.827244	−0.413622	−	0.910449i	$-0.635736\pi$
$547$	−3.16657e9	−0.827244	−0.413622	+	0.910449i	$0.635736\pi$
$548$	−2.54446e8	−0.0660487
$549$	−7.42498e8	−0.191510
$550$	4.28767e9	1.09888
$551$	1.29342e8	0.0329388
$552$	−2.86129e9	−0.724062
$553$	0	0
$554$	−4.07869e9	−1.01915
$555$	1.42572e9	0.354005
$556$	−6.49874e8	−0.160349
$557$	6.34922e9	1.55678	0.778390	−	0.627781i	$-0.216037\pi$
$557$	6.34922e9	1.55678	0.778390	+	0.627781i	$0.216037\pi$
$558$	−4.04622e8	−0.0985894
$559$	3.82675e9	0.926593
$560$	0	0
$561$	2.63514e9	0.630135
$562$	5.19472e9	1.23448
$563$	−1.17095e9	−0.276540	−0.138270	−	0.990395i	$-0.544154\pi$
$563$	−1.17095e9	−0.276540	−0.138270	+	0.990395i	$0.544154\pi$
$564$	−4.15250e8	−0.0974614
$565$	−2.60881e9	−0.608518
$566$	−5.53667e9	−1.28349
$567$	0	0
$568$	4.64628e8	0.106386
$569$	−5.09242e9	−1.15886	−0.579431	−	0.815021i	$-0.696725\pi$
$569$	−5.09242e9	−1.15886	−0.579431	+	0.815021i	$0.696725\pi$
$570$	−5.70514e8	−0.129034
$571$	−6.92793e9	−1.55732	−0.778659	−	0.627447i	$-0.784100\pi$
$571$	−6.92793e9	−1.55732	−0.778659	+	0.627447i	$0.784100\pi$
$572$	6.04784e8	0.135118
$573$	2.51175e9	0.557744
$574$	0	0
$575$	−4.42254e9	−0.970140
$576$	1.30410e9	0.284336
$577$	1.02827e9	0.222839	0.111419	−	0.993773i	$-0.464460\pi$
$577$	1.02827e9	0.222839	0.111419	+	0.993773i	$0.464460\pi$
$578$	−2.09564e9	−0.451409
$579$	−3.29023e9	−0.704452
$580$	−2.45738e7	−0.00522968
$581$	0	0
$582$	−4.29226e9	−0.902518
$583$	−4.60272e9	−0.961998
$584$	−3.68800e9	−0.766207
$585$	−6.67017e8	−0.137750
$586$	−6.26522e9	−1.28616
$587$	−2.89305e9	−0.590368	−0.295184	−	0.955440i	$-0.595381\pi$
$587$	−2.89305e9	−0.590368	−0.295184	+	0.955440i	$0.595381\pi$
$588$	0	0
$589$	5.53601e8	0.111633
$590$	−2.86074e9	−0.573451
$591$	3.61756e9	0.720875
$592$	−6.47218e9	−1.28211
$593$	7.60849e9	1.49833	0.749165	−	0.662384i	$-0.230455\pi$
$593$	7.60849e9	1.49833	0.749165	+	0.662384i	$0.230455\pi$
$594$	−1.49939e9	−0.293537
$595$	0	0
$596$	7.11684e8	0.137697
$597$	−7.21652e7	−0.0138809
$598$	−5.82461e9	−1.11381
$599$	3.85051e9	0.732023	0.366011	−	0.930610i	$-0.380723\pi$
$599$	3.85051e9	0.732023	0.366011	+	0.930610i	$0.380723\pi$
$600$	2.04967e9	0.387395
$601$	−2.41853e9	−0.454456	−0.227228	−	0.973842i	$-0.572966\pi$
$601$	−2.41853e9	−0.454456	−0.227228	+	0.973842i	$0.572966\pi$
$602$	0	0
$603$	2.79823e9	0.519725
$604$	−9.84604e7	−0.0181816
$605$	3.10239e9	0.569578
$606$	1.21924e9	0.222554
$607$	−3.68586e9	−0.668926	−0.334463	−	0.942409i	$-0.608555\pi$
$607$	−3.68586e9	−0.668926	−0.334463	+	0.942409i	$0.608555\pi$
$608$	5.28266e8	0.0953213
$609$	0	0
$610$	−1.80216e9	−0.321468
$611$	6.20216e9	1.10002
$612$	−1.71687e8	−0.0302766
$613$	−1.07967e10	−1.89312	−0.946558	−	0.322532i	$-0.895466\pi$
$613$	−1.07967e10	−1.89312	−0.946558	+	0.322532i	$0.895466\pi$
$614$	−6.66712e9	−1.16238
$615$	−1.40596e9	−0.243731
$616$	0	0
$617$	1.77134e9	0.303601	0.151801	−	0.988411i	$-0.451493\pi$
$617$	1.77134e9	0.303601	0.151801	+	0.988411i	$0.451493\pi$
$618$	6.55644e9	1.11740
$619$	8.68180e8	0.147127	0.0735635	−	0.997291i	$-0.476563\pi$
$619$	8.68180e8	0.147127	0.0735635	+	0.997291i	$0.476563\pi$
$620$	−1.05180e8	−0.0177240
$621$	1.54656e9	0.259147
$622$	−2.39852e9	−0.399647
$623$	0	0
$624$	3.02798e9	0.498893
$625$	1.46184e9	0.239508
$626$	6.75097e8	0.109991
$627$	2.05146e9	0.332374
$628$	1.13227e9	0.182428
$629$	−5.48110e9	−0.878194
$630$	0	0
$631$	−6.66494e9	−1.05607	−0.528036	−	0.849222i	$-0.677071\pi$
$631$	−6.66494e9	−1.05607	−0.528036	+	0.849222i	$0.677071\pi$
$632$	1.12021e10	1.76518
$633$	−2.52662e9	−0.395937
$634$	3.19054e9	0.497224
$635$	3.70878e8	0.0574808
$636$	2.99880e8	0.0462219
$637$	0	0
$638$	8.25059e8	0.125780
$639$	−2.51137e8	−0.0380765
$640$	4.00202e9	0.603461
$641$	−3.56030e9	−0.533928	−0.266964	−	0.963706i	$-0.586021\pi$
$641$	−3.56030e9	−0.533928	−0.266964	+	0.963706i	$0.586021\pi$
$642$	7.25696e8	0.108239
$643$	6.88842e9	1.02184	0.510918	−	0.859629i	$-0.329306\pi$
$643$	6.88842e9	1.02184	0.510918	+	0.859629i	$0.329306\pi$
$644$	0	0
$645$	−2.46620e9	−0.361883
$646$	2.19331e9	0.320100
$647$	1.02798e10	1.49217	0.746085	−	0.665851i	$-0.231931\pi$
$647$	1.02798e10	1.49217	0.746085	+	0.665851i	$0.231931\pi$
$648$	−7.16767e8	−0.103482
$649$	1.02867e10	1.47713
$650$	4.17242e9	0.595924
$651$	0	0
$652$	9.70449e8	0.137122
$653$	8.94969e9	1.25780	0.628901	−	0.777486i	$-0.283505\pi$
$653$	8.94969e9	1.25780	0.628901	+	0.777486i	$0.283505\pi$
$654$	5.29148e9	0.739700
$655$	−3.52536e9	−0.490183
$656$	6.38249e9	0.882727
$657$	1.99340e9	0.274231
$658$	0	0
$659$	8.16275e9	1.11106	0.555530	−	0.831496i	$-0.312515\pi$
$659$	8.16275e9	1.11106	0.555530	+	0.831496i	$0.312515\pi$
$660$	−3.89760e8	−0.0527708
$661$	1.09427e8	0.0147374	0.00736868	−	0.999973i	$-0.497654\pi$
$661$	1.09427e8	0.0147374	0.00736868	+	0.999973i	$0.497654\pi$
$662$	−8.50129e9	−1.13889
$663$	2.56431e9	0.341722
$664$	−1.29915e10	−1.72215
$665$	0	0
$666$	3.11875e9	0.409091
$667$	−8.51012e8	−0.111044
$668$	1.56312e9	0.202897
$669$	−3.82653e9	−0.494099
$670$	6.79174e9	0.872408
$671$	6.48021e9	0.828057
$672$	0	0
$673$	1.03359e10	1.30705	0.653527	−	0.756903i	$-0.273289\pi$
$673$	1.03359e10	1.30705	0.653527	+	0.756903i	$0.273289\pi$
$674$	2.38601e9	0.300167
$675$	−1.10787e9	−0.138651
$676$	−3.74845e8	−0.0466700
$677$	−1.42026e9	−0.175917	−0.0879586	−	0.996124i	$-0.528034\pi$
$677$	−1.42026e9	−0.175917	−0.0879586	+	0.996124i	$0.528034\pi$
$678$	−5.70674e9	−0.703209
$679$	0	0
$680$	3.05748e9	0.372891
$681$	−5.15521e9	−0.625506
$682$	3.53137e9	0.426283
$683$	2.33921e9	0.280929	0.140464	−	0.990086i	$-0.455140\pi$
$683$	2.33921e9	0.280929	0.140464	+	0.990086i	$0.455140\pi$
$684$	−1.33658e8	−0.0159698
$685$	−2.44922e9	−0.291145
$686$	0	0
$687$	−6.89389e9	−0.811178
$688$	1.11955e10	1.31064
$689$	−4.47900e9	−0.521691
$690$	3.75374e9	0.435003
$691$	4.07953e9	0.470367	0.235184	−	0.971951i	$-0.424431\pi$
$691$	4.07953e9	0.470367	0.235184	+	0.971951i	$0.424431\pi$
$692$	2.08179e8	0.0238817
$693$	0	0
$694$	−2.54165e9	−0.288641
$695$	−6.25547e9	−0.706827
$696$	3.94410e8	0.0443420
$697$	5.40514e9	0.604633
$698$	−1.27488e10	−1.41898
$699$	4.65769e9	0.515822
$700$	0	0
$701$	4.48674e9	0.491947	0.245973	−	0.969277i	$-0.420892\pi$
$701$	4.48674e9	0.491947	0.245973	+	0.969277i	$0.420892\pi$
$702$	−1.45909e9	−0.159185
$703$	−4.26705e9	−0.463216
$704$	−1.13816e10	−1.22942
$705$	−3.99706e9	−0.429614
$706$	−8.55201e8	−0.0914643
$707$	0	0
$708$	−6.70206e8	−0.0709728
$709$	−1.02918e10	−1.08450	−0.542248	−	0.840218i	$-0.682427\pi$
$709$	−1.02918e10	−1.08450	−0.542248	+	0.840218i	$0.682427\pi$
$710$	−6.09547e8	−0.0639150
$711$	−6.05486e9	−0.631772
$712$	1.24361e10	1.29124
$713$	−3.64246e9	−0.376341
$714$	0	0
$715$	5.82145e9	0.595607
$716$	6.87459e8	0.0699925
$717$	−1.93092e9	−0.195636
$718$	−9.31511e6	−0.000939188 0
$719$	1.38302e10	1.38764	0.693820	−	0.720149i	$-0.255927\pi$
$719$	1.38302e10	1.38764	0.693820	+	0.720149i	$0.255927\pi$
$720$	−1.95142e9	−0.194844
$721$	0	0
$722$	−8.99484e9	−0.889433
$723$	−6.56801e9	−0.646323
$724$	1.52013e9	0.148866
$725$	6.09617e8	0.0594120
$726$	6.78645e9	0.658210
$727$	1.90270e10	1.83654	0.918270	−	0.395954i	$-0.129586\pi$
$727$	1.90270e10	1.83654	0.918270	+	0.395954i	$0.129586\pi$
$728$	0	0
$729$	3.87420e8	0.0370370
$730$	4.83830e9	0.460323
$731$	9.48114e9	0.897738
$732$	−4.22204e8	−0.0397863
$733$	−9.26816e8	−0.0869220	−0.0434610	−	0.999055i	$-0.513838\pi$
$733$	−9.26816e8	−0.0869220	−0.0434610	+	0.999055i	$0.513838\pi$
$734$	2.14768e10	2.00463
$735$	0	0
$736$	−3.47576e9	−0.321350
$737$	−2.44218e10	−2.24720
$738$	−3.07553e9	−0.281658
$739$	3.58717e9	0.326961	0.163481	−	0.986547i	$-0.447728\pi$
$739$	3.58717e9	0.326961	0.163481	+	0.986547i	$0.447728\pi$
$740$	8.10703e8	0.0735446
$741$	1.99632e9	0.180246
$742$	0	0
$743$	2.20176e10	1.96928	0.984642	−	0.174583i	$-0.0558577\pi$
$743$	2.20176e10	1.96928	0.984642	+	0.174583i	$0.0558577\pi$
$744$	1.68813e9	0.150280
$745$	6.85043e9	0.606975
$746$	3.75673e9	0.331302
$747$	7.02205e9	0.616371
$748$	1.49841e9	0.130911
$749$	0	0
$750$	−6.42128e9	−0.555786
$751$	−1.05025e10	−0.904797	−0.452398	−	0.891816i	$-0.649431\pi$
$751$	−1.05025e10	−0.904797	−0.452398	+	0.891816i	$0.649431\pi$
$752$	1.81450e10	1.55594
$753$	−1.12277e10	−0.958318
$754$	8.02882e8	0.0682106
$755$	−9.47746e8	−0.0801453
$756$	0	0
$757$	1.58187e9	0.132536	0.0662682	−	0.997802i	$-0.478891\pi$
$757$	1.58187e9	0.132536	0.0662682	+	0.997802i	$0.478891\pi$
$758$	1.89604e10	1.58126
$759$	−1.34977e10	−1.12051
$760$	2.38025e9	0.196687
$761$	−1.67401e10	−1.37693	−0.688466	−	0.725269i	$-0.741715\pi$
$761$	−1.67401e10	−1.37693	−0.688466	+	0.725269i	$0.741715\pi$
$762$	8.11291e8	0.0664254
$763$	0	0
$764$	1.42825e9	0.115871
$765$	−1.65260e9	−0.133460
$766$	7.31822e9	0.588308
$767$	1.00102e10	0.801047
$768$	2.57197e9	0.204881
$769$	−2.13180e10	−1.69046	−0.845229	−	0.534405i	$-0.820536\pi$
$769$	−2.13180e10	−1.69046	−0.845229	+	0.534405i	$0.820536\pi$
$770$	0	0
$771$	−1.25270e10	−0.984367
$772$	−1.87091e9	−0.146350
$773$	−1.65142e10	−1.28597	−0.642984	−	0.765880i	$-0.722304\pi$
$773$	−1.65142e10	−1.28597	−0.642984	+	0.765880i	$0.722304\pi$
$774$	−5.39477e9	−0.418196
$775$	2.60925e9	0.201354
$776$	1.79078e10	1.37571
$777$	0	0
$778$	−1.46470e10	−1.11512
$779$	4.20791e9	0.318923
$780$	−3.79284e8	−0.0286176
$781$	2.19181e9	0.164636
$782$	−1.44310e10	−1.07913
$783$	−2.13183e8	−0.0158703
$784$	0	0
$785$	1.08988e10	0.804149
$786$	−7.71167e9	−0.566461
$787$	9.18131e9	0.671418	0.335709	−	0.941966i	$-0.391024\pi$
$787$	9.18131e9	0.671418	0.335709	+	0.941966i	$0.391024\pi$
$788$	2.05704e9	0.149762
$789$	5.13784e9	0.372401
$790$	−1.46961e10	−1.06049
$791$	0	0
$792$	6.25565e9	0.447439
$793$	6.30602e9	0.449055
$794$	−2.82217e10	−2.00083
$795$	2.88654e9	0.203748
$796$	−4.10350e7	−0.00288376
$797$	1.31379e10	0.919225	0.459612	−	0.888120i	$-0.347988\pi$
$797$	1.31379e10	0.919225	0.459612	+	0.888120i	$0.347988\pi$
$798$	0	0
$799$	1.53664e10	1.06576
$800$	2.48984e9	0.171932
$801$	−6.72186e9	−0.462143
$802$	2.19336e10	1.50141
$803$	−1.73976e10	−1.18573
$804$	1.59115e9	0.107973
$805$	0	0
$806$	3.43645e9	0.231173
$807$	7.05564e9	0.472584
$808$	−5.08681e9	−0.339239
$809$	−1.85815e10	−1.23384	−0.616922	−	0.787024i	$-0.711621\pi$
$809$	−1.85815e10	−1.23384	−0.616922	+	0.787024i	$0.711621\pi$
$810$	9.40329e8	0.0621702
$811$	−2.24767e9	−0.147965	−0.0739826	−	0.997260i	$-0.523571\pi$
$811$	−2.24767e9	−0.147965	−0.0739826	+	0.997260i	$0.523571\pi$
$812$	0	0
$813$	−1.61528e9	−0.105422
$814$	−2.72191e10	−1.76884
$815$	9.34122e9	0.604438
$816$	7.50211e9	0.483357
$817$	7.38108e9	0.473525
$818$	−3.84493e9	−0.245614
$819$	0	0
$820$	−7.99468e8	−0.0506352
$821$	5.85982e9	0.369559	0.184779	−	0.982780i	$-0.440843\pi$
$821$	5.85982e9	0.369559	0.184779	+	0.982780i	$0.440843\pi$
$822$	−5.35763e9	−0.336450
$823$	1.47824e10	0.924372	0.462186	−	0.886783i	$-0.347065\pi$
$823$	1.47824e10	0.924372	0.462186	+	0.886783i	$0.347065\pi$
$824$	−2.73542e10	−1.70325
$825$	9.66900e9	0.599505
$826$	0	0
$827$	−2.98930e10	−1.83781	−0.918905	−	0.394478i	$-0.870925\pi$
$827$	−2.98930e10	−1.83781	−0.918905	+	0.394478i	$0.870925\pi$
$828$	8.79415e8	0.0538378
$829$	−2.15141e10	−1.31154	−0.655772	−	0.754959i	$-0.727657\pi$
$829$	−2.15141e10	−1.31154	−0.655772	+	0.754959i	$0.727657\pi$
$830$	1.70436e10	1.03464
$831$	−9.19775e9	−0.556004
$832$	−1.10757e10	−0.666713
$833$	0	0
$834$	−1.36838e10	−0.816816
$835$	1.50461e10	0.894378
$836$	1.16651e9	0.0690507
$837$	−9.12453e8	−0.0537863
$838$	−1.29214e10	−0.758502
$839$	−1.24647e10	−0.728644	−0.364322	−	0.931273i	$-0.618699\pi$
$839$	−1.24647e10	−0.728644	−0.364322	+	0.931273i	$0.618699\pi$
$840$	0	0
$841$	−1.71326e10	−0.993200
$842$	3.22825e9	0.186370
$843$	1.17145e10	0.673482
$844$	−1.43670e9	−0.0822561
$845$	−3.60813e9	−0.205723
$846$	−8.74351e9	−0.496466
$847$	0	0
$848$	−1.31037e10	−0.737919
$849$	−1.24856e10	−0.700216
$850$	1.03376e10	0.577367
$851$	2.80753e10	1.56161
$852$	−1.42803e8	−0.00791040
$853$	−2.55888e10	−1.41165	−0.705826	−	0.708385i	$-0.749424\pi$
$853$	−2.55888e10	−1.41165	−0.705826	+	0.708385i	$0.749424\pi$
$854$	0	0
$855$	−1.28655e9	−0.0703956
$856$	−3.02769e9	−0.164988
$857$	2.05773e10	1.11675	0.558375	−	0.829589i	$-0.311425\pi$
$857$	2.05773e10	1.11675	0.558375	+	0.829589i	$0.311425\pi$
$858$	1.27343e10	0.688289
$859$	3.87736e9	0.208718	0.104359	−	0.994540i	$-0.466721\pi$
$859$	3.87736e9	0.208718	0.104359	+	0.994540i	$0.466721\pi$
$860$	−1.40234e9	−0.0751813
$861$	0	0
$862$	−1.99230e10	−1.05945
$863$	6.63817e9	0.351569	0.175784	−	0.984429i	$-0.443754\pi$
$863$	6.63817e9	0.351569	0.175784	+	0.984429i	$0.443754\pi$
$864$	−8.70695e8	−0.0459270
$865$	2.00386e9	0.105272
$866$	−1.02104e10	−0.534233
$867$	−4.72583e9	−0.246270
$868$	0	0
$869$	5.28443e10	2.73167
$870$	−5.17427e8	−0.0266398
$871$	−2.37654e10	−1.21866
$872$	−2.20767e10	−1.12752
$873$	−9.67936e9	−0.492376
$874$	−1.12346e10	−0.569202
$875$	0	0
$876$	1.13350e9	0.0569716
$877$	5.21042e9	0.260840	0.130420	−	0.991459i	$-0.458367\pi$
$877$	5.21042e9	0.260840	0.130420	+	0.991459i	$0.458367\pi$
$878$	7.40510e9	0.369233
$879$	−1.41285e10	−0.701675
$880$	1.70312e10	0.842471
$881$	−3.14280e10	−1.54846	−0.774231	−	0.632903i	$-0.781863\pi$
$881$	−3.14280e10	−1.54846	−0.774231	+	0.632903i	$0.781863\pi$
$882$	0	0
$883$	−1.28989e10	−0.630507	−0.315254	−	0.949007i	$-0.602090\pi$
$883$	−1.28989e10	−0.630507	−0.315254	+	0.949007i	$0.602090\pi$
$884$	1.45813e9	0.0709928
$885$	−6.45118e9	−0.312851
$886$	−9.95023e9	−0.480634
$887$	−1.56991e10	−0.755337	−0.377668	−	0.925941i	$-0.623274\pi$
$887$	−1.56991e10	−0.755337	−0.377668	+	0.925941i	$0.623274\pi$
$888$	−1.30118e10	−0.623578
$889$	0	0
$890$	−1.63150e10	−0.775750
$891$	−3.38124e9	−0.160142
$892$	−2.17587e9	−0.102649
$893$	1.19628e10	0.562151
$894$	1.49852e10	0.701426
$895$	6.61725e9	0.308530
$896$	0	0
$897$	−1.31349e10	−0.607650
$898$	−1.08039e10	−0.497866
$899$	5.02088e8	0.0230473
$900$	−6.29963e8	−0.0288049
$901$	−1.10971e10	−0.505446
$902$	2.68419e10	1.21784
$903$	0	0
$904$	2.38092e10	1.07190
$905$	1.46322e10	0.656207
$906$	−2.07318e9	−0.0926166
$907$	8.82395e9	0.392679	0.196339	−	0.980536i	$-0.437095\pi$
$907$	8.82395e9	0.392679	0.196339	+	0.980536i	$0.437095\pi$
$908$	−2.93139e9	−0.129949
$909$	2.74947e9	0.121416
$910$	0	0
$911$	7.83674e9	0.343417	0.171708	−	0.985148i	$-0.445071\pi$
$911$	7.83674e9	0.343417	0.171708	+	0.985148i	$0.445071\pi$
$912$	5.84041e9	0.254954
$913$	−6.12855e10	−2.66508
$914$	5.76932e7	0.00249927
$915$	−4.06399e9	−0.175380
$916$	−3.92005e9	−0.168522
$917$	0	0
$918$	−3.61504e9	−0.154228
$919$	1.08422e10	0.460801	0.230401	−	0.973096i	$-0.425996\pi$
$919$	1.08422e10	0.460801	0.230401	+	0.973096i	$0.425996\pi$
$920$	−1.56610e10	−0.663075
$921$	−1.50348e10	−0.634147
$922$	−1.07186e10	−0.450381
$923$	2.13290e9	0.0892821
$924$	0	0
$925$	−2.01116e10	−0.835506
$926$	−3.02064e10	−1.25014
$927$	1.47852e10	0.609606
$928$	4.79110e8	0.0196796
$929$	1.27560e10	0.521989	0.260994	−	0.965340i	$-0.415950\pi$
$929$	1.27560e10	0.521989	0.260994	+	0.965340i	$0.415950\pi$
$930$	−2.21466e9	−0.0902854
$931$	0	0
$932$	2.64848e9	0.107162
$933$	−5.40884e9	−0.218031
$934$	4.41924e10	1.77474
$935$	1.44232e10	0.577059
$936$	6.08750e9	0.242646
$937$	2.41827e10	0.960321	0.480160	−	0.877181i	$-0.340578\pi$
$937$	2.41827e10	0.960321	0.480160	+	0.877181i	$0.340578\pi$
$938$	0	0
$939$	1.52239e9	0.0600064
$940$	−2.27283e9	−0.0892524
$941$	1.31783e10	0.515581	0.257790	−	0.966201i	$-0.417006\pi$
$941$	1.31783e10	0.515581	0.257790	+	0.966201i	$0.417006\pi$
$942$	2.38411e10	0.929282
$943$	−2.76862e10	−1.07516
$944$	2.92857e10	1.13306
$945$	0	0
$946$	4.70833e10	1.80821
$947$	4.31517e10	1.65110	0.825550	−	0.564329i	$-0.190865\pi$
$947$	4.31517e10	1.65110	0.825550	+	0.564329i	$0.190865\pi$
$948$	−3.44295e9	−0.131251
$949$	−1.69300e10	−0.643020
$950$	8.04781e9	0.304540
$951$	7.19490e9	0.271265
$952$	0	0
$953$	−3.33513e10	−1.24821	−0.624105	−	0.781340i	$-0.714536\pi$
$953$	−3.33513e10	−1.24821	−0.624105	+	0.781340i	$0.714536\pi$
$954$	6.31428e9	0.235453
$955$	1.37478e10	0.510766
$956$	−1.09797e9	−0.0406433
$957$	1.86057e9	0.0686205
$958$	1.54336e10	0.567136
$959$	0	0
$960$	7.13786e9	0.260387
$961$	−2.53636e10	−0.921890
$962$	−2.64875e10	−0.959242
$963$	1.63650e9	0.0590504
$964$	−3.73474e9	−0.134274
$965$	−1.80088e10	−0.645117
$966$	0	0
$967$	−4.08789e10	−1.45381	−0.726903	−	0.686740i	$-0.759041\pi$
$967$	−4.08789e10	−1.45381	−0.726903	+	0.686740i	$0.759041\pi$
$968$	−2.83138e10	−1.00331
$969$	4.94607e9	0.174633
$970$	−2.34933e10	−0.826500
$971$	2.93692e10	1.02950	0.514749	−	0.857341i	$-0.327885\pi$
$971$	2.93692e10	1.02950	0.514749	+	0.857341i	$0.327885\pi$
$972$	2.20298e8	0.00769445
$973$	0	0
$974$	3.45406e10	1.19777
$975$	9.40910e9	0.325111
$976$	1.84488e10	0.635177
$977$	2.34233e10	0.803556	0.401778	−	0.915737i	$-0.368392\pi$
$977$	2.34233e10	0.803556	0.401778	+	0.915737i	$0.368392\pi$
$978$	2.04338e10	0.698495
$979$	5.86656e10	1.99822
$980$	0	0
$981$	1.19327e10	0.403549
$982$	1.24628e9	0.0419975
$983$	1.13192e10	0.380082	0.190041	−	0.981776i	$-0.439138\pi$
$983$	1.13192e10	0.380082	0.190041	+	0.981776i	$0.439138\pi$
$984$	1.28314e10	0.429331
$985$	1.98004e10	0.660157
$986$	1.98922e9	0.0660865
$987$	0	0
$988$	1.13516e9	0.0374462
$989$	−4.85644e10	−1.59636
$990$	−8.20680e9	−0.268813
$991$	−3.15390e10	−1.02942	−0.514708	−	0.857366i	$-0.672100\pi$
$991$	−3.15390e10	−1.02942	−0.514708	+	0.857366i	$0.672100\pi$
$992$	2.05066e9	0.0666965
$993$	−1.91710e10	−0.621331
$994$	0	0
$995$	−3.94989e8	−0.0127117
$996$	3.99293e9	0.128051
$997$	1.60961e10	0.514385	0.257193	−	0.966360i	$-0.417203\pi$
$997$	1.60961e10	0.514385	0.257193	+	0.966360i	$0.417203\pi$
$998$	−4.79579e10	−1.52723
$999$	7.03300e9	0.223183

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	147.8.a.m.1.5	yes	6
3.2	odd	2		441.8.a.z.1.2		6
7.2	even	3		147.8.e.o.67.2		12
7.3	odd	6		147.8.e.p.79.2		12
7.4	even	3		147.8.e.o.79.2		12
7.5	odd	6		147.8.e.p.67.2		12
7.6	odd	2		147.8.a.l.1.5	✓	6
21.20	even	2		441.8.a.y.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.8.a.l.1.5	✓	6	7.6	odd	2
147.8.a.m.1.5	yes	6	1.1	even	1	trivial
147.8.e.o.67.2		12	7.2	even	3
147.8.e.o.79.2		12	7.4	even	3
147.8.e.p.67.2		12	7.5	odd	6
147.8.e.p.79.2		12	7.3	odd	6
441.8.a.y.1.2		6	21.20	even	2
441.8.a.z.1.2		6	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 147 = 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	147.a (trivial)

\(f(q)\)	\(=\)	\(q+11.9730 q^{2} +27.0000 q^{3} +15.3529 q^{4} +147.782 q^{5} +323.271 q^{6} -1348.72 q^{8} +729.000 q^{9} +1769.40 q^{10} -6362.41 q^{11} +414.529 q^{12} -6191.39 q^{13} +3990.11 q^{15} -18113.5 q^{16} -15339.7 q^{17} +8728.32 q^{18} -11942.0 q^{19} +2268.88 q^{20} -76177.1 q^{22} +78573.4 q^{23} -36415.6 q^{24} -56285.5 q^{25} -74129.5 q^{26} +19683.0 q^{27} -10830.8 q^{29} +47773.7 q^{30} -46357.4 q^{31} -44235.9 q^{32} -171785. q^{33} -183663. q^{34} +11192.3 q^{36} +357314. q^{37} -142982. q^{38} -167167. q^{39} -199317. q^{40} -352362. q^{41} -618076. q^{43} -97681.5 q^{44} +107733. q^{45} +940760. q^{46} -1.00174e6 q^{47} -489063. q^{48} -673906. q^{50} -414173. q^{51} -95055.9 q^{52} +723424. q^{53} +235665. q^{54} -940249. q^{55} -322435. q^{57} -129677. q^{58} -1.61679e6 q^{59} +61259.9 q^{60} -1.01852e6 q^{61} -555038. q^{62} +1.78889e6 q^{64} -914976. q^{65} -2.05678e6 q^{66} +3.83846e6 q^{67} -235510. q^{68} +2.12148e6 q^{69} -344495. q^{71} -983220. q^{72} +2.73444e6 q^{73} +4.27812e6 q^{74} -1.51971e6 q^{75} -183345. q^{76} -2.00150e6 q^{78} -8.30571e6 q^{79} -2.67684e6 q^{80} +531441. q^{81} -4.21883e6 q^{82} +9.63245e6 q^{83} -2.26694e6 q^{85} -7.40023e6 q^{86} -292432. q^{87} +8.58113e6 q^{88} -9.22066e6 q^{89} +1.28989e6 q^{90} +1.20633e6 q^{92} -1.25165e6 q^{93} -1.19938e7 q^{94} -1.76482e6 q^{95} -1.19437e6 q^{96} -1.32776e7 q^{97} -4.63819e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 14 q^{2} + 162 q^{3} + 438 q^{4} - 500 q^{5} - 378 q^{6} - 1218 q^{8} + 4374 q^{9} - 6912 q^{10} - 1204 q^{11} + 11826 q^{12} - 17576 q^{13} - 13500 q^{15} + 15522 q^{16} - 60836 q^{17} - 10206 q^{18}+ \cdots - 877716 q^{99}+O(q^{100}) \) 6 * q - 14 * q^2 + 162 * q^3 + 438 * q^4 - 500 * q^5 - 378 * q^6 - 1218 * q^8 + 4374 * q^9 - 6912 * q^10 - 1204 * q^11 + 11826 * q^12 - 17576 * q^13 - 13500 * q^15 + 15522 * q^16 - 60836 * q^17 - 10206 * q^18 + 17280 * q^19 - 43924 * q^20 - 244452 * q^22 + 47708 * q^23 - 32886 * q^24 - 126954 * q^25 - 225228 * q^26 + 118098 * q^27 - 85880 * q^29 - 186624 * q^30 - 344872 * q^31 - 399678 * q^32 - 32508 * q^33 + 216304 * q^34 + 319302 * q^36 - 535896 * q^37 + 1154872 * q^38 - 474552 * q^39 - 817048 * q^40 + 696316 * q^41 - 162720 * q^43 + 1504868 * q^44 - 364500 * q^45 + 216540 * q^46 - 2280672 * q^47 + 419094 * q^48 + 3654866 * q^50 - 1642572 * q^51 - 5978752 * q^52 + 5321348 * q^53 - 275562 * q^54 - 1716328 * q^55 + 466560 * q^57 + 2346000 * q^58 - 2702888 * q^59 - 1185948 * q^60 - 7891216 * q^61 - 8107760 * q^62 - 4370970 * q^64 + 3217860 * q^65 - 6600204 * q^66 + 565920 * q^67 - 21007124 * q^68 + 1288116 * q^69 - 4370788 * q^71 - 887922 * q^72 - 11994056 * q^73 - 278200 * q^74 - 3427758 * q^75 - 3409936 * q^76 - 6081156 * q^78 - 16816944 * q^79 - 14397828 * q^80 + 3188646 * q^81 - 29067128 * q^82 - 2431304 * q^83 - 2530620 * q^85 + 16351456 * q^86 - 2318760 * q^87 - 23615724 * q^88 - 26982380 * q^89 - 5038848 * q^90 + 22208116 * q^92 - 9311544 * q^93 + 1880792 * q^94 - 11963336 * q^95 - 10791306 * q^96 - 14260040 * q^97 - 877716 * q^99

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