LMFDB - Embedded newform 387.8.a.b.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [387,8,Mod(1,387)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(387, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("387.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 387 = 3^{2} \cdot 43 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	387.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$120.893004862$
Analytic rank:	$0$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 2 x^{10} - 977 x^{9} + 2592 x^{8} + 344686 x^{7} - 1160956 x^{6} - 53409536 x^{5} + \cdots + 238240894976 $ x^11 - 2x^10 - 977x^9 + 2592x^8 + 344686x^7 - 1160956x^6 - 53409536x^5 + 209758592x^4 + 3410917248x^3 - 14180732672x^2 - 60918607872x + 238240894976
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{4}\cdot 7 $
Twist minimal:	no (minimal twist has level 43)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-14.3182$ of defining polynomial
Character	$\chi$	$=$	387.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-12.3182 q^{2} +23.7379 q^{4} -330.186 q^{5} -467.397 q^{7} +1284.32 q^{8} +O(q^{10})$	\(q-12.3182 q^{2} +23.7379 q^{4} -330.186 q^{5} -467.397 q^{7} +1284.32 q^{8} +4067.29 q^{10} +359.251 q^{11} -11439.7 q^{13} +5757.49 q^{14} -18859.0 q^{16} +19000.7 q^{17} -3671.03 q^{19} -7837.90 q^{20} -4425.32 q^{22} +3785.73 q^{23} +30897.5 q^{25} +140917. q^{26} -11095.0 q^{28} -115539. q^{29} +15139.6 q^{31} +67915.2 q^{32} -234054. q^{34} +154328. q^{35} -357942. q^{37} +45220.5 q^{38} -424064. q^{40} -540673. q^{41} +79507.0 q^{43} +8527.85 q^{44} -46633.3 q^{46} -289961. q^{47} -605083. q^{49} -380601. q^{50} -271555. q^{52} +329115. q^{53} -118619. q^{55} -600288. q^{56} +1.42323e6 q^{58} -2.44911e6 q^{59} -1.25749e6 q^{61} -186492. q^{62} +1.57735e6 q^{64} +3.77723e6 q^{65} -1.70199e6 q^{67} +451036. q^{68} -1.90104e6 q^{70} +3.75585e6 q^{71} -5.83440e6 q^{73} +4.40920e6 q^{74} -87142.4 q^{76} -167913. q^{77} +3.48456e6 q^{79} +6.22696e6 q^{80} +6.66012e6 q^{82} -5.17957e6 q^{83} -6.27376e6 q^{85} -979383. q^{86} +461394. q^{88} -5.98803e6 q^{89} +5.34690e6 q^{91} +89865.1 q^{92} +3.57180e6 q^{94} +1.21212e6 q^{95} -5.18336e6 q^{97} +7.45353e6 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + 24 q^{2} + 602 q^{4} + 752 q^{5} - 12 q^{7} + 3810 q^{8}+O(q^{10}) $ 11 * q + 24 * q^2 + 602 * q^4 + 752 * q^5 - 12 * q^7 + 3810 * q^8	\( 11 q + 24 q^{2} + 602 q^{4} + 752 q^{5} - 12 q^{7} + 3810 q^{8} - 1333 q^{10} - 1333 q^{11} - 17967 q^{13} + 22352 q^{14} - 34406 q^{16} + 63095 q^{17} - 54524 q^{19} + 280995 q^{20} - 289358 q^{22} + 138139 q^{23} + 3455 q^{25} + 132946 q^{26} - 12704 q^{28} + 308658 q^{29} - 209523 q^{31} + 644934 q^{32} + 762435 q^{34} + 578892 q^{35} - 298472 q^{37} + 369707 q^{38} + 2633173 q^{40} + 1346735 q^{41} + 874577 q^{43} - 3134292 q^{44} + 3588111 q^{46} - 499284 q^{47} + 2544563 q^{49} - 3049745 q^{50} + 983088 q^{52} + 2210495 q^{53} - 1855072 q^{55} + 469976 q^{56} + 4397067 q^{58} + 5824216 q^{59} - 4453034 q^{61} - 1002789 q^{62} + 4757538 q^{64} + 2162872 q^{65} - 6859513 q^{67} + 9397005 q^{68} + 845078 q^{70} + 10726554 q^{71} - 4456898 q^{73} - 1046637 q^{74} + 5861267 q^{76} + 17019816 q^{77} - 15541320 q^{79} + 15680911 q^{80} + 20233655 q^{82} + 11146767 q^{83} - 12471976 q^{85} + 1908168 q^{86} - 24463544 q^{88} + 13531356 q^{89} - 19746448 q^{91} + 26023161 q^{92} + 20288857 q^{94} + 12291624 q^{95} - 10999901 q^{97} - 29909168 q^{98}+O(q^{100}) \) 11 * q + 24 * q^2 + 602 * q^4 + 752 * q^5 - 12 * q^7 + 3810 * q^8 - 1333 * q^10 - 1333 * q^11 - 17967 * q^13 + 22352 * q^14 - 34406 * q^16 + 63095 * q^17 - 54524 * q^19 + 280995 * q^20 - 289358 * q^22 + 138139 * q^23 + 3455 * q^25 + 132946 * q^26 - 12704 * q^28 + 308658 * q^29 - 209523 * q^31 + 644934 * q^32 + 762435 * q^34 + 578892 * q^35 - 298472 * q^37 + 369707 * q^38 + 2633173 * q^40 + 1346735 * q^41 + 874577 * q^43 - 3134292 * q^44 + 3588111 * q^46 - 499284 * q^47 + 2544563 * q^49 - 3049745 * q^50 + 983088 * q^52 + 2210495 * q^53 - 1855072 * q^55 + 469976 * q^56 + 4397067 * q^58 + 5824216 * q^59 - 4453034 * q^61 - 1002789 * q^62 + 4757538 * q^64 + 2162872 * q^65 - 6859513 * q^67 + 9397005 * q^68 + 845078 * q^70 + 10726554 * q^71 - 4456898 * q^73 - 1046637 * q^74 + 5861267 * q^76 + 17019816 * q^77 - 15541320 * q^79 + 15680911 * q^80 + 20233655 * q^82 + 11146767 * q^83 - 12471976 * q^85 + 1908168 * q^86 - 24463544 * q^88 + 13531356 * q^89 - 19746448 * q^91 + 26023161 * q^92 + 20288857 * q^94 + 12291624 * q^95 - 10999901 * q^97 - 29909168 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−12.3182	−1.08878	−0.544392	−	0.838831i	$-0.683240\pi$
$2$	−12.3182	−1.08878	−0.544392	+	0.838831i	$0.683240\pi$
$3$	0	0
$3$	0	0
$4$	23.7379	0.185452
$5$	−330.186	−1.18131	−0.590654	−	0.806925i	$-0.701130\pi$
$5$	−330.186	−1.18131	−0.590654	+	0.806925i	$0.701130\pi$
$6$	0	0
$7$	−467.397	−0.515042	−0.257521	−	0.966273i	$-0.582906\pi$
$7$	−467.397	−0.515042	−0.257521	+	0.966273i	$0.582906\pi$
$8$	1284.32	0.886867
$9$	0	0
$10$	4067.29	1.28619
$11$	359.251	0.0813811	0.0406906	−	0.999172i	$-0.487044\pi$
$11$	359.251	0.0813811	0.0406906	+	0.999172i	$0.487044\pi$
$12$	0	0
$13$	−11439.7	−1.44416	−0.722078	−	0.691812i	$-0.756813\pi$
$13$	−11439.7	−1.44416	−0.722078	+	0.691812i	$0.756813\pi$
$14$	5757.49	0.560770
$15$	0	0
$16$	−18859.0	−1.15106
$17$	19000.7	0.937990	0.468995	−	0.883201i	$-0.344616\pi$
$17$	19000.7	0.937990	0.468995	+	0.883201i	$0.344616\pi$
$18$	0	0
$19$	−3671.03	−0.122786	−0.0613932	−	0.998114i	$-0.519554\pi$
$19$	−3671.03	−0.122786	−0.0613932	+	0.998114i	$0.519554\pi$
$20$	−7837.90	−0.219076
$21$	0	0
$22$	−4425.32	−0.0886065
$23$	3785.73	0.0648787	0.0324393	−	0.999474i	$-0.489672\pi$
$23$	3785.73	0.0648787	0.0324393	+	0.999474i	$0.489672\pi$
$24$	0	0
$25$	30897.5	0.395488
$26$	140917.	1.57237
$27$	0	0
$28$	−11095.0	−0.0955157
$29$	−115539.	−0.879702	−0.439851	−	0.898071i	$-0.644969\pi$
$29$	−115539.	−0.879702	−0.439851	+	0.898071i	$0.644969\pi$
$30$	0	0
$31$	15139.6	0.0912741	0.0456371	−	0.998958i	$-0.485468\pi$
$31$	15139.6	0.0912741	0.0456371	+	0.998958i	$0.485468\pi$
$32$	67915.2	0.366389
$33$	0	0
$34$	−234054.	−1.02127
$35$	154328.	0.608424
$36$	0	0
$37$	−357942.	−1.16173	−0.580867	−	0.813998i	$-0.697286\pi$
$37$	−357942.	−1.16173	−0.580867	+	0.813998i	$0.697286\pi$
$38$	45220.5	0.133688
$39$	0	0
$40$	−424064.	−1.04766
$41$	−540673.	−1.22516	−0.612578	−	0.790410i	$-0.709867\pi$
$41$	−540673.	−1.22516	−0.612578	+	0.790410i	$0.709867\pi$
$42$	0	0
$43$	79507.0	0.152499
$43$	79507.0	0.152499
$44$	8527.85	0.0150923
$45$	0	0
$46$	−46633.3	−0.0706389
$47$	−289961.	−0.407378	−0.203689	−	0.979036i	$-0.565293\pi$
$47$	−289961.	−0.407378	−0.203689	+	0.979036i	$0.565293\pi$
$48$	0	0
$49$	−605083.	−0.734731
$50$	−380601.	−0.430601
$51$	0	0
$52$	−271555.	−0.267822
$53$	329115.	0.303656	0.151828	−	0.988407i	$-0.451484\pi$
$53$	329115.	0.303656	0.151828	+	0.988407i	$0.451484\pi$
$54$	0	0
$55$	−118619.	−0.0961361
$56$	−600288.	−0.456774
$57$	0	0
$58$	1.42323e6	0.957806
$59$	−2.44911e6	−1.55248	−0.776240	−	0.630437i	$-0.782876\pi$
$59$	−2.44911e6	−1.55248	−0.776240	+	0.630437i	$0.782876\pi$
$60$	0	0
$61$	−1.25749e6	−0.709333	−0.354666	−	0.934993i	$-0.615406\pi$
$61$	−1.25749e6	−0.709333	−0.354666	+	0.934993i	$0.615406\pi$
$62$	−186492.	−0.0993779
$63$	0	0
$64$	1.57735e6	0.752141
$65$	3.77723e6	1.70599
$66$	0	0
$67$	−1.70199e6	−0.691347	−0.345673	−	0.938355i	$-0.612350\pi$
$67$	−1.70199e6	−0.691347	−0.345673	+	0.938355i	$0.612350\pi$
$68$	451036.	0.173952
$69$	0	0
$70$	−1.90104e6	−0.662442
$71$	3.75585e6	1.24539	0.622693	−	0.782466i	$-0.286039\pi$
$71$	3.75585e6	1.24539	0.622693	+	0.782466i	$0.286039\pi$
$72$	0	0
$73$	−5.83440e6	−1.75536	−0.877680	−	0.479247i	$-0.840910\pi$
$73$	−5.83440e6	−1.75536	−0.877680	+	0.479247i	$0.840910\pi$
$74$	4.40920e6	1.26488
$75$	0	0
$76$	−87142.4	−0.0227710
$77$	−167913.	−0.0419147
$78$	0	0
$79$	3.48456e6	0.795158	0.397579	−	0.917568i	$-0.369851\pi$
$79$	3.48456e6	0.795158	0.397579	+	0.917568i	$0.369851\pi$
$80$	6.22696e6	1.35976
$81$	0	0
$82$	6.66012e6	1.33393
$83$	−5.17957e6	−0.994307	−0.497154	−	0.867663i	$-0.665621\pi$
$83$	−5.17957e6	−0.994307	−0.497154	+	0.867663i	$0.665621\pi$
$84$	0	0
$85$	−6.27376e6	−1.10806
$86$	−979383.	−0.166038
$87$	0	0
$88$	461394.	0.0721743
$89$	−5.98803e6	−0.900365	−0.450183	−	0.892936i	$-0.648641\pi$
$89$	−5.98803e6	−0.900365	−0.450183	+	0.892936i	$0.648641\pi$
$90$	0	0
$91$	5.34690e6	0.743801
$92$	89865.1	0.0120319
$93$	0	0
$94$	3.57180e6	0.443547
$95$	1.21212e6	0.145049
$96$	0	0
$97$	−5.18336e6	−0.576647	−0.288324	−	0.957533i	$-0.593098\pi$
$97$	−5.18336e6	−0.576647	−0.288324	+	0.957533i	$0.593098\pi$
$98$	7.45353e6	0.799964
$99$	0	0
$100$	733440.	0.0733440
$101$	930045.	0.0898213	0.0449106	−	0.998991i	$-0.485700\pi$
$101$	930045.	0.0898213	0.0449106	+	0.998991i	$0.485700\pi$
$102$	0	0
$103$	−1.70172e7	−1.53447	−0.767233	−	0.641368i	$-0.778367\pi$
$103$	−1.70172e7	−1.53447	−0.767233	+	0.641368i	$0.778367\pi$
$104$	−1.46923e7	−1.28077
$105$	0	0
$106$	−4.05410e6	−0.330616
$107$	−1.27256e7	−1.00423	−0.502116	−	0.864800i	$-0.667445\pi$
$107$	−1.27256e7	−1.00423	−0.502116	+	0.864800i	$0.667445\pi$
$108$	0	0
$109$	−1.67891e7	−1.24175	−0.620874	−	0.783910i	$-0.713222\pi$
$109$	−1.67891e7	−1.24175	−0.620874	+	0.783910i	$0.713222\pi$
$110$	1.46118e6	0.104672
$111$	0	0
$112$	8.81463e6	0.592845
$113$	−1.36793e7	−0.891842	−0.445921	−	0.895072i	$-0.647124\pi$
$113$	−1.36793e7	−0.891842	−0.445921	+	0.895072i	$0.647124\pi$
$114$	0	0
$115$	−1.24999e6	−0.0766417
$116$	−2.74265e6	−0.163143
$117$	0	0
$118$	3.01686e7	1.69032
$119$	−8.88087e6	−0.483105
$120$	0	0
$121$	−1.93581e7	−0.993377
$122$	1.54900e7	0.772311
$123$	0	0
$124$	359381.	0.0169270
$125$	1.55938e7	0.714115
$126$	0	0
$127$	1.29034e6	0.0558972	0.0279486	−	0.999609i	$-0.491103\pi$
$127$	1.29034e6	0.0558972	0.0279486	+	0.999609i	$0.491103\pi$
$128$	−2.81233e7	−1.18531
$129$	0	0
$130$	−4.65287e7	−1.85746
$131$	3.96207e7	1.53983	0.769915	−	0.638146i	$-0.220299\pi$
$131$	3.96207e7	1.53983	0.769915	+	0.638146i	$0.220299\pi$
$132$	0	0
$133$	1.71583e6	0.0632402
$134$	2.09655e7	0.752728
$135$	0	0
$136$	2.44030e7	0.831873
$137$	1.20603e7	0.400715	0.200358	−	0.979723i	$-0.435790\pi$
$137$	1.20603e7	0.400715	0.200358	+	0.979723i	$0.435790\pi$
$138$	0	0
$139$	3.45946e7	1.09259	0.546294	−	0.837594i	$-0.316038\pi$
$139$	3.45946e7	1.09259	0.546294	+	0.837594i	$0.316038\pi$
$140$	3.66341e6	0.112833
$141$	0	0
$142$	−4.62653e7	−1.35596
$143$	−4.10973e6	−0.117527
$144$	0	0
$145$	3.81493e7	1.03920
$146$	7.18693e7	1.91121
$147$	0	0
$148$	−8.49678e6	−0.215446
$149$	−3.46074e7	−0.857072	−0.428536	−	0.903525i	$-0.640970\pi$
$149$	−3.46074e7	−0.857072	−0.428536	+	0.903525i	$0.640970\pi$
$150$	0	0
$151$	−2.74859e7	−0.649666	−0.324833	−	0.945771i	$-0.605308\pi$
$151$	−2.74859e7	−0.649666	−0.324833	+	0.945771i	$0.605308\pi$
$152$	−4.71478e6	−0.108895
$153$	0	0
$154$	2.06838e6	0.0456361
$155$	−4.99887e6	−0.107823
$156$	0	0
$157$	−3.91834e7	−0.808078	−0.404039	−	0.914742i	$-0.632394\pi$
$157$	−3.91834e7	−0.808078	−0.404039	+	0.914742i	$0.632394\pi$
$158$	−4.29235e7	−0.865756
$159$	0	0
$160$	−2.24246e7	−0.432818
$161$	−1.76944e6	−0.0334153
$162$	0	0
$163$	4.81506e7	0.870854	0.435427	−	0.900224i	$-0.356597\pi$
$163$	4.81506e7	0.870854	0.435427	+	0.900224i	$0.356597\pi$
$164$	−1.28344e7	−0.227208
$165$	0	0
$166$	6.38030e7	1.08259
$167$	5.80547e7	0.964561	0.482280	−	0.876017i	$-0.339809\pi$
$167$	5.80547e7	0.964561	0.482280	+	0.876017i	$0.339809\pi$
$168$	0	0
$169$	6.81189e7	1.08559
$170$	7.72813e7	1.20643
$171$	0	0
$172$	1.88733e6	0.0282812
$173$	3.04236e7	0.446735	0.223367	−	0.974734i	$-0.428295\pi$
$173$	3.04236e7	0.446735	0.223367	+	0.974734i	$0.428295\pi$
$174$	0	0
$175$	−1.44414e7	−0.203693
$176$	−6.77510e6	−0.0936745
$177$	0	0
$178$	7.37617e7	0.980304
$179$	−6.08790e6	−0.0793382	−0.0396691	−	0.999213i	$-0.512630\pi$
$179$	−6.08790e6	−0.0793382	−0.0396691	+	0.999213i	$0.512630\pi$
$180$	0	0
$181$	1.51213e8	1.89546	0.947728	−	0.319079i	$-0.103373\pi$
$181$	1.51213e8	1.89546	0.947728	+	0.319079i	$0.103373\pi$
$182$	−6.58641e7	−0.809840
$183$	0	0
$184$	4.86209e6	0.0575388
$185$	1.18187e8	1.37237
$186$	0	0
$187$	6.82602e6	0.0763347
$188$	−6.88307e6	−0.0755491
$189$	0	0
$190$	−1.49311e7	−0.157927
$191$	−1.15042e8	−1.19465	−0.597323	−	0.802000i	$-0.703769\pi$
$191$	−1.15042e8	−1.19465	−0.597323	+	0.802000i	$0.703769\pi$
$192$	0	0
$193$	5.16665e7	0.517319	0.258659	−	0.965969i	$-0.416719\pi$
$193$	5.16665e7	0.517319	0.258659	+	0.965969i	$0.416719\pi$
$194$	6.38496e7	0.627845
$195$	0	0
$196$	−1.43634e7	−0.136257
$197$	7.84505e7	0.731078	0.365539	−	0.930796i	$-0.380885\pi$
$197$	7.84505e7	0.731078	0.365539	+	0.930796i	$0.380885\pi$
$198$	0	0
$199$	−7.08100e7	−0.636955	−0.318478	−	0.947930i	$-0.603172\pi$
$199$	−7.08100e7	−0.636955	−0.318478	+	0.947930i	$0.603172\pi$
$200$	3.96823e7	0.350745
$201$	0	0
$202$	−1.14565e7	−0.0977960
$203$	5.40026e7	0.453084
$204$	0	0
$205$	1.78522e8	1.44729
$206$	2.09621e8	1.67070
$207$	0	0
$208$	2.15741e8	1.66231
$209$	−1.31882e6	−0.00999250
$210$	0	0
$211$	−1.44130e8	−1.05625	−0.528124	−	0.849167i	$-0.677104\pi$
$211$	−1.44130e8	−1.05625	−0.528124	+	0.849167i	$0.677104\pi$
$212$	7.81248e6	0.0563136
$213$	0	0
$214$	1.56756e8	1.09339
$215$	−2.62521e7	−0.180148
$216$	0	0
$217$	−7.07619e6	−0.0470101
$218$	2.06811e8	1.35200
$219$	0	0
$220$	−2.81577e6	−0.0178286
$221$	−2.17363e8	−1.35460
$222$	0	0
$223$	1.64343e8	0.992396	0.496198	−	0.868209i	$-0.334729\pi$
$223$	1.64343e8	0.992396	0.496198	+	0.868209i	$0.334729\pi$
$224$	−3.17434e7	−0.188706
$225$	0	0
$226$	1.68504e8	0.971024
$227$	3.36542e8	1.90963	0.954814	−	0.297204i	$-0.0960543\pi$
$227$	3.36542e8	1.90963	0.954814	+	0.297204i	$0.0960543\pi$
$228$	0	0
$229$	−5.78696e7	−0.318439	−0.159220	−	0.987243i	$-0.550898\pi$
$229$	−5.78696e7	−0.318439	−0.159220	+	0.987243i	$0.550898\pi$
$230$	1.53976e7	0.0834463
$231$	0	0
$232$	−1.48389e8	−0.780179
$233$	2.71239e8	1.40477	0.702386	−	0.711797i	$-0.252118\pi$
$233$	2.71239e8	1.40477	0.702386	+	0.711797i	$0.252118\pi$
$234$	0	0
$235$	9.57411e7	0.481239
$236$	−5.81366e7	−0.287911
$237$	0	0
$238$	1.09396e8	0.525997
$239$	1.16728e8	0.553075	0.276537	−	0.961003i	$-0.410813\pi$
$239$	1.16728e8	0.553075	0.276537	+	0.961003i	$0.410813\pi$
$240$	0	0
$241$	−4.09158e8	−1.88292	−0.941460	−	0.337126i	$-0.890545\pi$
$241$	−4.09158e8	−1.88292	−0.941460	+	0.337126i	$0.890545\pi$
$242$	2.38457e8	1.08157
$243$	0	0
$244$	−2.98501e7	−0.131547
$245$	1.99790e8	0.867944
$246$	0	0
$247$	4.19956e7	0.177323
$248$	1.94441e7	0.0809480
$249$	0	0
$250$	−1.92088e8	−0.777517
$251$	3.43101e8	1.36950	0.684752	−	0.728776i	$-0.259910\pi$
$251$	3.43101e8	1.36950	0.684752	+	0.728776i	$0.259910\pi$
$252$	0	0
$253$	1.36003e6	0.00527990
$254$	−1.58946e7	−0.0608601
$255$	0	0
$256$	1.44527e8	0.538405
$257$	−7.47286e7	−0.274613	−0.137307	−	0.990529i	$-0.543845\pi$
$257$	−7.47286e7	−0.274613	−0.137307	+	0.990529i	$0.543845\pi$
$258$	0	0
$259$	1.67301e8	0.598343
$260$	8.96634e7	0.316380
$261$	0	0
$262$	−4.88056e8	−1.67654
$263$	−2.57577e8	−0.873097	−0.436549	−	0.899681i	$-0.643799\pi$
$263$	−2.57577e8	−0.873097	−0.436549	+	0.899681i	$0.643799\pi$
$264$	0	0
$265$	−1.08669e8	−0.358711
$266$	−2.11359e7	−0.0688550
$267$	0	0
$268$	−4.04017e7	−0.128212
$269$	3.45824e8	1.08323	0.541617	−	0.840625i	$-0.317812\pi$
$269$	3.45824e8	1.08323	0.541617	+	0.840625i	$0.317812\pi$
$270$	0	0
$271$	−2.24146e8	−0.684129	−0.342065	−	0.939676i	$-0.611126\pi$
$271$	−2.24146e8	−0.684129	−0.342065	+	0.939676i	$0.611126\pi$
$272$	−3.58333e8	−1.07968
$273$	0	0
$274$	−1.48561e8	−0.436293
$275$	1.11000e7	0.0321852
$276$	0	0
$277$	2.45845e8	0.694995	0.347497	−	0.937681i	$-0.387032\pi$
$277$	2.45845e8	0.694995	0.347497	+	0.937681i	$0.387032\pi$
$278$	−4.26143e8	−1.18959
$279$	0	0
$280$	1.98206e8	0.539591
$281$	−4.79601e8	−1.28946	−0.644731	−	0.764410i	$-0.723030\pi$
$281$	−4.79601e8	−1.28946	−0.644731	+	0.764410i	$0.723030\pi$
$282$	0	0
$283$	5.71921e8	1.49997	0.749987	−	0.661452i	$-0.230060\pi$
$283$	5.71921e8	1.49997	0.749987	+	0.661452i	$0.230060\pi$
$284$	8.91559e7	0.230959
$285$	0	0
$286$	5.06245e7	0.127962
$287$	2.52709e8	0.631007
$288$	0	0
$289$	−4.93122e7	−0.120174
$290$	−4.69930e8	−1.13146
$291$	0	0
$292$	−1.38496e8	−0.325535
$293$	−4.33629e8	−1.00712	−0.503560	−	0.863960i	$-0.667977\pi$
$293$	−4.33629e8	−1.00712	−0.503560	+	0.863960i	$0.667977\pi$
$294$	0	0
$295$	8.08661e8	1.83396
$296$	−4.59713e8	−1.03030
$297$	0	0
$298$	4.26301e8	0.933167
$299$	−4.33077e7	−0.0936949
$300$	0	0
$301$	−3.71614e7	−0.0785432
$302$	3.38576e8	0.707346
$303$	0	0
$304$	6.92318e7	0.141335
$305$	4.15205e8	0.837940
$306$	0	0
$307$	−5.42296e8	−1.06968	−0.534838	−	0.844955i	$-0.679627\pi$
$307$	−5.42296e8	−1.06968	−0.534838	+	0.844955i	$0.679627\pi$
$308$	−3.98589e6	−0.00777317
$309$	0	0
$310$	6.15770e7	0.117396
$311$	8.11532e8	1.52983	0.764917	−	0.644129i	$-0.222780\pi$
$311$	8.11532e8	1.52983	0.764917	+	0.644129i	$0.222780\pi$
$312$	0	0
$313$	−8.40388e8	−1.54908	−0.774541	−	0.632524i	$-0.782019\pi$
$313$	−8.40388e8	−1.54908	−0.774541	+	0.632524i	$0.782019\pi$
$314$	4.82669e8	0.879823
$315$	0	0
$316$	8.27160e7	0.147464
$317$	−3.45739e8	−0.609595	−0.304797	−	0.952417i	$-0.598589\pi$
$317$	−3.45739e8	−0.609595	−0.304797	+	0.952417i	$0.598589\pi$
$318$	0	0
$319$	−4.15075e7	−0.0715911
$320$	−5.20820e8	−0.888510
$321$	0	0
$322$	2.17963e7	0.0363820
$323$	−6.97521e7	−0.115172
$324$	0	0
$325$	−3.53459e8	−0.571146
$326$	−5.93129e8	−0.948173
$327$	0	0
$328$	−6.94398e8	−1.08655
$329$	1.35527e8	0.209817
$330$	0	0
$331$	5.89707e8	0.893796	0.446898	−	0.894585i	$-0.352529\pi$
$331$	5.89707e8	0.893796	0.446898	+	0.894585i	$0.352529\pi$
$332$	−1.22952e8	−0.184396
$333$	0	0
$334$	−7.15129e8	−1.05020
$335$	5.61974e8	0.816693
$336$	0	0
$337$	4.92765e8	0.701351	0.350676	−	0.936497i	$-0.385952\pi$
$337$	4.92765e8	0.701351	0.350676	+	0.936497i	$0.385952\pi$
$338$	−8.39101e8	−1.18197
$339$	0	0
$340$	−1.48926e8	−0.205491
$341$	5.43891e6	0.00742799
$342$	0	0
$343$	6.67736e8	0.893460
$344$	1.02113e8	0.135246
$345$	0	0
$346$	−3.74764e8	−0.486398
$347$	−9.83108e8	−1.26313	−0.631565	−	0.775323i	$-0.717587\pi$
$347$	−9.83108e8	−1.26313	−0.631565	+	0.775323i	$0.717587\pi$
$348$	0	0
$349$	1.25957e9	1.58611	0.793053	−	0.609153i	$-0.208491\pi$
$349$	1.25957e9	1.58611	0.793053	+	0.609153i	$0.208491\pi$
$350$	1.77892e8	0.221778
$351$	0	0
$352$	2.43986e7	0.0298171
$353$	8.97531e7	0.108602	0.0543010	−	0.998525i	$-0.482707\pi$
$353$	8.97531e7	0.108602	0.0543010	+	0.998525i	$0.482707\pi$
$354$	0	0
$355$	−1.24013e9	−1.47118
$356$	−1.42143e8	−0.166975
$357$	0	0
$358$	7.49920e7	0.0863822
$359$	−1.81494e8	−0.207029	−0.103515	−	0.994628i	$-0.533009\pi$
$359$	−1.81494e8	−0.207029	−0.103515	+	0.994628i	$0.533009\pi$
$360$	0	0
$361$	−8.80395e8	−0.984923
$362$	−1.86267e9	−2.06374
$363$	0	0
$364$	1.26924e8	0.137940
$365$	1.92643e9	2.07362
$366$	0	0
$367$	1.07414e9	1.13430	0.567150	−	0.823614i	$-0.308046\pi$
$367$	1.07414e9	1.13430	0.567150	+	0.823614i	$0.308046\pi$
$368$	−7.13949e7	−0.0746792
$369$	0	0
$370$	−1.45585e9	−1.49421
$371$	−1.53827e8	−0.156396
$372$	0	0
$373$	−1.39148e9	−1.38834	−0.694169	−	0.719812i	$-0.744228\pi$
$373$	−1.39148e9	−1.38834	−0.694169	+	0.719812i	$0.744228\pi$
$374$	−8.40842e7	−0.0831120
$375$	0	0
$376$	−3.72404e8	−0.361290
$377$	1.32173e9	1.27043
$378$	0	0
$379$	−1.85979e8	−0.175480	−0.0877400	−	0.996143i	$-0.527964\pi$
$379$	−1.85979e8	−0.175480	−0.0877400	+	0.996143i	$0.527964\pi$
$380$	2.87732e7	0.0268996
$381$	0	0
$382$	1.41711e9	1.30071
$383$	−8.12564e8	−0.739029	−0.369515	−	0.929225i	$-0.620476\pi$
$383$	−8.12564e8	−0.739029	−0.369515	+	0.929225i	$0.620476\pi$
$384$	0	0
$385$	5.54424e7	0.0495142
$386$	−6.36437e8	−0.563249
$387$	0	0
$388$	−1.23042e8	−0.106940
$389$	3.04526e8	0.262301	0.131151	−	0.991362i	$-0.458133\pi$
$389$	3.04526e8	0.262301	0.131151	+	0.991362i	$0.458133\pi$
$390$	0	0
$391$	7.19315e7	0.0608556
$392$	−7.77121e8	−0.651609
$393$	0	0
$394$	−9.66368e8	−0.795986
$395$	−1.15055e9	−0.939326
$396$	0	0
$397$	−2.15202e9	−1.72616	−0.863078	−	0.505071i	$-0.831466\pi$
$397$	−2.15202e9	−1.72616	−0.863078	+	0.505071i	$0.831466\pi$
$398$	8.72251e8	0.693507
$399$	0	0
$400$	−5.82694e8	−0.455230
$401$	−1.32277e9	−1.02442	−0.512211	−	0.858860i	$-0.671173\pi$
$401$	−1.32277e9	−1.02442	−0.512211	+	0.858860i	$0.671173\pi$
$402$	0	0
$403$	−1.73193e8	−0.131814
$404$	2.20773e7	0.0166575
$405$	0	0
$406$	−6.65215e8	−0.493311
$407$	−1.28591e8	−0.0945433
$408$	0	0
$409$	5.69273e8	0.411423	0.205712	−	0.978613i	$-0.434049\pi$
$409$	5.69273e8	0.411423	0.205712	+	0.978613i	$0.434049\pi$
$410$	−2.19907e9	−1.57578
$411$	0	0
$412$	−4.03951e8	−0.284570
$413$	1.14471e9	0.799594
$414$	0	0
$415$	1.71022e9	1.17458
$416$	−7.76931e8	−0.529122
$417$	0	0
$418$	1.62455e7	0.0108797
$419$	2.17734e9	1.44603	0.723016	−	0.690831i	$-0.242755\pi$
$419$	2.17734e9	1.44603	0.723016	+	0.690831i	$0.242755\pi$
$420$	0	0
$421$	−2.30881e8	−0.150800	−0.0754001	−	0.997153i	$-0.524023\pi$
$421$	−2.30881e8	−0.150800	−0.0754001	+	0.997153i	$0.524023\pi$
$422$	1.77542e9	1.15003
$423$	0	0
$424$	4.22689e8	0.269302
$425$	5.87074e8	0.370964
$426$	0	0
$427$	5.87747e8	0.365336
$428$	−3.02078e8	−0.186237
$429$	0	0
$430$	3.23378e8	0.196142
$431$	−2.33302e9	−1.40361	−0.701807	−	0.712367i	$-0.747623\pi$
$431$	−2.33302e9	−1.40361	−0.701807	+	0.712367i	$0.747623\pi$
$432$	0	0
$433$	−2.85667e8	−0.169104	−0.0845518	−	0.996419i	$-0.526946\pi$
$433$	−2.85667e8	−0.169104	−0.0845518	+	0.996419i	$0.526946\pi$
$434$	8.71659e7	0.0511838
$435$	0	0
$436$	−3.98536e8	−0.230285
$437$	−1.38975e7	−0.00796622
$438$	0	0
$439$	2.53362e9	1.42927	0.714636	−	0.699496i	$-0.246592\pi$
$439$	2.53362e9	1.42927	0.714636	+	0.699496i	$0.246592\pi$
$440$	−1.52346e8	−0.0852600
$441$	0	0
$442$	2.67752e9	1.47487
$443$	6.54414e8	0.357635	0.178817	−	0.983882i	$-0.442773\pi$
$443$	6.54414e8	0.357635	0.178817	+	0.983882i	$0.442773\pi$
$444$	0	0
$445$	1.97716e9	1.06361
$446$	−2.02441e9	−1.08051
$447$	0	0
$448$	−7.37251e8	−0.387385
$449$	−2.92747e9	−1.52627	−0.763133	−	0.646241i	$-0.776340\pi$
$449$	−2.92747e9	−1.52627	−0.763133	+	0.646241i	$0.776340\pi$
$450$	0	0
$451$	−1.94237e8	−0.0997046
$452$	−3.24716e8	−0.165394
$453$	0	0
$454$	−4.14559e9	−2.07917
$455$	−1.76547e9	−0.878658
$456$	0	0
$457$	−2.45680e9	−1.20410	−0.602050	−	0.798459i	$-0.705649\pi$
$457$	−2.45680e9	−1.20410	−0.602050	+	0.798459i	$0.705649\pi$
$458$	7.12849e8	0.346712
$459$	0	0
$460$	−2.96721e7	−0.0142134
$461$	−1.25968e9	−0.598832	−0.299416	−	0.954123i	$-0.596792\pi$
$461$	−1.25968e9	−0.598832	−0.299416	+	0.954123i	$0.596792\pi$
$462$	0	0
$463$	1.60779e9	0.752829	0.376414	−	0.926451i	$-0.377157\pi$
$463$	1.60779e9	0.752829	0.376414	+	0.926451i	$0.377157\pi$
$464$	2.17895e9	1.01259
$465$	0	0
$466$	−3.34117e9	−1.52949
$467$	−1.74456e9	−0.792643	−0.396321	−	0.918112i	$-0.629713\pi$
$467$	−1.74456e9	−0.792643	−0.396321	+	0.918112i	$0.629713\pi$
$468$	0	0
$469$	7.95507e8	0.356073
$470$	−1.17936e9	−0.523966
$471$	0	0
$472$	−3.14544e9	−1.37684
$473$	2.85630e7	0.0124105
$474$	0	0
$475$	−1.13426e8	−0.0485605
$476$	−2.10813e8	−0.0895928
$477$	0	0
$478$	−1.43788e9	−0.602179
$479$	−2.90237e9	−1.20664	−0.603322	−	0.797498i	$-0.706156\pi$
$479$	−2.90237e9	−1.20664	−0.603322	+	0.797498i	$0.706156\pi$
$480$	0	0
$481$	4.09476e9	1.67773
$482$	5.04009e9	2.05009
$483$	0	0
$484$	−4.59520e8	−0.184224
$485$	1.71147e9	0.681198
$486$	0	0
$487$	9.92687e8	0.389458	0.194729	−	0.980857i	$-0.437617\pi$
$487$	9.92687e8	0.389458	0.194729	+	0.980857i	$0.437617\pi$
$488$	−1.61502e9	−0.629084
$489$	0	0
$490$	−2.46105e9	−0.945004
$491$	−4.20047e9	−1.60145	−0.800723	−	0.599034i	$-0.795551\pi$
$491$	−4.20047e9	−1.60145	−0.800723	+	0.599034i	$0.795551\pi$
$492$	0	0
$493$	−2.19532e9	−0.825152
$494$	−5.17310e8	−0.193066
$495$	0	0
$496$	−2.85517e8	−0.105062
$497$	−1.75547e9	−0.641427
$498$	0	0
$499$	3.12871e9	1.12723	0.563616	−	0.826037i	$-0.309410\pi$
$499$	3.12871e9	1.12723	0.563616	+	0.826037i	$0.309410\pi$
$500$	3.70165e8	0.132434
$501$	0	0
$502$	−4.22638e9	−1.49110
$503$	−3.86209e8	−0.135311	−0.0676556	−	0.997709i	$-0.521552\pi$
$503$	−3.86209e8	−0.135311	−0.0676556	+	0.997709i	$0.521552\pi$
$504$	0	0
$505$	−3.07087e8	−0.106107
$506$	−1.67531e7	−0.00574867
$507$	0	0
$508$	3.06299e7	0.0103663
$509$	1.34663e9	0.452621	0.226311	−	0.974055i	$-0.427334\pi$
$509$	1.34663e9	0.452621	0.226311	+	0.974055i	$0.427334\pi$
$510$	0	0
$511$	2.72698e9	0.904085
$512$	1.81947e9	0.599102
$513$	0	0
$514$	9.20522e8	0.298994
$515$	5.61883e9	1.81268
$516$	0	0
$517$	−1.04169e8	−0.0331529
$518$	−2.06085e9	−0.651466
$519$	0	0
$520$	4.85118e9	1.51299
$521$	4.59610e9	1.42383	0.711914	−	0.702267i	$-0.247829\pi$
$521$	4.59610e9	1.42383	0.711914	+	0.702267i	$0.247829\pi$
$522$	0	0
$523$	−3.76968e9	−1.15226	−0.576128	−	0.817359i	$-0.695437\pi$
$523$	−3.76968e9	−1.15226	−0.576128	+	0.817359i	$0.695437\pi$
$524$	9.40511e8	0.285565
$525$	0	0
$526$	3.17289e9	0.950615
$527$	2.87662e8	0.0856142
$528$	0	0
$529$	−3.39049e9	−0.995791
$530$	1.33860e9	0.390559
$531$	0	0
$532$	4.07301e7	0.0117280
$533$	6.18515e9	1.76932
$534$	0	0
$535$	4.20180e9	1.18631
$536$	−2.18591e9	−0.613133
$537$	0	0
$538$	−4.25993e9	−1.17941
$539$	−2.17377e8	−0.0597933
$540$	0	0
$541$	6.00863e9	1.63149	0.815746	−	0.578410i	$-0.196327\pi$
$541$	6.00863e9	1.63149	0.815746	+	0.578410i	$0.196327\pi$
$542$	2.76107e9	0.744869
$543$	0	0
$544$	1.29044e9	0.343669
$545$	5.54350e9	1.46689
$546$	0	0
$547$	2.57778e9	0.673427	0.336714	−	0.941607i	$-0.390685\pi$
$547$	2.57778e9	0.673427	0.336714	+	0.941607i	$0.390685\pi$
$548$	2.86286e8	0.0743135
$549$	0	0
$550$	−1.36731e8	−0.0350428
$551$	4.24147e8	0.108015
$552$	0	0
$553$	−1.62867e9	−0.409540
$554$	−3.02836e9	−0.756700
$555$	0	0
$556$	8.21202e8	0.202623
$557$	1.73000e9	0.424181	0.212091	−	0.977250i	$-0.431973\pi$
$557$	1.73000e9	0.424181	0.212091	+	0.977250i	$0.431973\pi$
$558$	0	0
$559$	−9.09538e8	−0.220232
$560$	−2.91046e9	−0.700332
$561$	0	0
$562$	5.90782e9	1.40395
$563$	4.68663e9	1.10683	0.553416	−	0.832905i	$-0.313324\pi$
$563$	4.68663e9	1.10683	0.553416	+	0.832905i	$0.313324\pi$
$564$	0	0
$565$	4.51669e9	1.05354
$566$	−7.04504e9	−1.63315
$567$	0	0
$568$	4.82372e9	1.10449
$569$	−8.45042e9	−1.92303	−0.961513	−	0.274759i	$-0.911402\pi$
$569$	−8.45042e9	−1.92303	−0.961513	+	0.274759i	$0.911402\pi$
$570$	0	0
$571$	3.74129e8	0.0840998	0.0420499	−	0.999116i	$-0.486611\pi$
$571$	3.74129e8	0.0840998	0.0420499	+	0.999116i	$0.486611\pi$
$572$	−9.75563e7	−0.0217956
$573$	0	0
$574$	−3.11292e9	−0.687031
$575$	1.16969e8	0.0256587
$576$	0	0
$577$	5.59761e7	0.0121307	0.00606537	−	0.999982i	$-0.498069\pi$
$577$	5.59761e7	0.0121307	0.00606537	+	0.999982i	$0.498069\pi$
$578$	6.07437e8	0.130844
$579$	0	0
$580$	9.05583e8	0.192722
$581$	2.42092e9	0.512111
$582$	0	0
$583$	1.18235e8	0.0247119
$584$	−7.49324e9	−1.55677
$585$	0	0
$586$	5.34153e9	1.09654
$587$	4.41966e9	0.901894	0.450947	−	0.892551i	$-0.351086\pi$
$587$	4.41966e9	0.901894	0.450947	+	0.892551i	$0.351086\pi$
$588$	0	0
$589$	−5.55778e7	−0.0112072
$590$	−9.96124e9	−1.99678
$591$	0	0
$592$	6.75042e9	1.33723
$593$	−4.30470e9	−0.847718	−0.423859	−	0.905728i	$-0.639325\pi$
$593$	−4.30470e9	−0.847718	−0.423859	+	0.905728i	$0.639325\pi$
$594$	0	0
$595$	2.93234e9	0.570695
$596$	−8.21506e8	−0.158946
$597$	0	0
$598$	5.33472e8	0.102014
$599$	2.79184e9	0.530758	0.265379	−	0.964144i	$-0.414503\pi$
$599$	2.79184e9	0.530758	0.265379	+	0.964144i	$0.414503\pi$
$600$	0	0
$601$	9.53719e9	1.79209	0.896045	−	0.443963i	$-0.146428\pi$
$601$	9.53719e9	1.79209	0.896045	+	0.443963i	$0.146428\pi$
$602$	4.57761e8	0.0855167
$603$	0	0
$604$	−6.52455e8	−0.120482
$605$	6.39177e9	1.17348
$606$	0	0
$607$	−1.03426e10	−1.87703	−0.938515	−	0.345240i	$-0.887798\pi$
$607$	−1.03426e10	−1.87703	−0.938515	+	0.345240i	$0.887798\pi$
$608$	−2.49319e8	−0.0449876
$609$	0	0
$610$	−5.11457e9	−0.912336
$611$	3.31708e9	0.588317
$612$	0	0
$613$	−2.76574e7	−0.00484954	−0.00242477	−	0.999997i	$-0.500772\pi$
$613$	−2.76574e7	−0.00484954	−0.00242477	+	0.999997i	$0.500772\pi$
$614$	6.68011e9	1.16465
$615$	0	0
$616$	−2.15654e8	−0.0371728
$617$	−5.52656e9	−0.947234	−0.473617	−	0.880731i	$-0.657052\pi$
$617$	−5.52656e9	−0.947234	−0.473617	+	0.880731i	$0.657052\pi$
$618$	0	0
$619$	−1.07166e10	−1.81609	−0.908046	−	0.418871i	$-0.862426\pi$
$619$	−1.07166e10	−1.81609	−0.908046	+	0.418871i	$0.862426\pi$
$620$	−1.18662e8	−0.0199960
$621$	0	0
$622$	−9.99661e9	−1.66566
$623$	2.79879e9	0.463726
$624$	0	0
$625$	−7.56273e9	−1.23908
$626$	1.03521e10	1.68662
$627$	0	0
$628$	−9.30130e8	−0.149860
$629$	−6.80115e9	−1.08970
$630$	0	0
$631$	−6.12871e7	−0.00971105	−0.00485552	−	0.999988i	$-0.501546\pi$
$631$	−6.12871e7	−0.00971105	−0.00485552	+	0.999988i	$0.501546\pi$
$632$	4.47530e9	0.705199
$633$	0	0
$634$	4.25888e9	0.663717
$635$	−4.26051e8	−0.0660318
$636$	0	0
$637$	6.92198e9	1.06107
$638$	5.11297e8	0.0779473
$639$	0	0
$640$	9.28591e9	1.40021
$641$	−9.04843e9	−1.35697	−0.678485	−	0.734614i	$-0.737363\pi$
$641$	−9.04843e9	−1.35697	−0.678485	+	0.734614i	$0.737363\pi$
$642$	0	0
$643$	−6.45538e9	−0.957599	−0.478799	−	0.877924i	$-0.658928\pi$
$643$	−6.45538e9	−0.957599	−0.478799	+	0.877924i	$0.658928\pi$
$644$	−4.20027e7	−0.00619693
$645$	0	0
$646$	8.59220e8	0.125398
$647$	−2.44098e8	−0.0354322	−0.0177161	−	0.999843i	$-0.505640\pi$
$647$	−2.44098e8	−0.0354322	−0.0177161	+	0.999843i	$0.505640\pi$
$648$	0	0
$649$	−8.79845e8	−0.126343
$650$	4.35397e9	0.621855
$651$	0	0
$652$	1.14299e9	0.161502
$653$	−3.48013e9	−0.489102	−0.244551	−	0.969636i	$-0.578641\pi$
$653$	−3.48013e9	−0.489102	−0.244551	+	0.969636i	$0.578641\pi$
$654$	0	0
$655$	−1.30822e10	−1.81901
$656$	1.01965e10	1.41023
$657$	0	0
$658$	−1.66945e9	−0.228446
$659$	9.27568e9	1.26254	0.631272	−	0.775561i	$-0.282533\pi$
$659$	9.27568e9	1.26254	0.631272	+	0.775561i	$0.282533\pi$
$660$	0	0
$661$	−3.16224e9	−0.425882	−0.212941	−	0.977065i	$-0.568304\pi$
$661$	−3.16224e9	−0.425882	−0.212941	+	0.977065i	$0.568304\pi$
$662$	−7.26412e9	−0.973151
$663$	0	0
$664$	−6.65223e9	−0.881819
$665$	−5.66542e8	−0.0747062
$666$	0	0
$667$	−4.37399e8	−0.0570739
$668$	1.37809e9	0.178880
$669$	0	0
$670$	−6.92250e9	−0.889203
$671$	−4.51754e8	−0.0577263
$672$	0	0
$673$	−2.80981e9	−0.355323	−0.177662	−	0.984092i	$-0.556853\pi$
$673$	−2.80981e9	−0.355323	−0.177662	+	0.984092i	$0.556853\pi$
$674$	−6.06998e9	−0.763620
$675$	0	0
$676$	1.61700e9	0.201324
$677$	1.54447e10	1.91302	0.956511	−	0.291696i	$-0.0942195\pi$
$677$	1.54447e10	1.91302	0.956511	+	0.291696i	$0.0942195\pi$
$678$	0	0
$679$	2.42269e9	0.296998
$680$	−8.05752e9	−0.982698
$681$	0	0
$682$	−6.69975e7	−0.00808748
$683$	8.31895e9	0.999070	0.499535	−	0.866294i	$-0.333504\pi$
$683$	8.31895e9	0.999070	0.499535	+	0.866294i	$0.333504\pi$
$684$	0	0
$685$	−3.98214e9	−0.473368
$686$	−8.22530e9	−0.972786
$687$	0	0
$688$	−1.49942e9	−0.175535
$689$	−3.76498e9	−0.438526
$690$	0	0
$691$	9.72394e9	1.12116	0.560582	−	0.828099i	$-0.310577\pi$
$691$	9.72394e9	1.12116	0.560582	+	0.828099i	$0.310577\pi$
$692$	7.22192e8	0.0828479
$693$	0	0
$694$	1.21101e10	1.37528
$695$	−1.14226e10	−1.29068
$696$	0	0
$697$	−1.02732e10	−1.14918
$698$	−1.55156e10	−1.72693
$699$	0	0
$700$	−3.42808e8	−0.0377753
$701$	2.48982e8	0.0272995	0.0136498	−	0.999907i	$-0.495655\pi$
$701$	2.48982e8	0.0272995	0.0136498	+	0.999907i	$0.495655\pi$
$702$	0	0
$703$	1.31402e9	0.142645
$704$	5.66666e8	0.0612101
$705$	0	0
$706$	−1.10560e9	−0.118244
$707$	−4.34701e8	−0.0462618
$708$	0	0
$709$	7.54233e9	0.794774	0.397387	−	0.917651i	$-0.369917\pi$
$709$	7.54233e9	0.794774	0.397387	+	0.917651i	$0.369917\pi$
$710$	1.52761e10	1.60180
$711$	0	0
$712$	−7.69055e9	−0.798505
$713$	5.73143e7	0.00592174
$714$	0	0
$715$	1.35697e9	0.138836
$716$	−1.44514e8	−0.0147134
$717$	0	0
$718$	2.23568e9	0.225410
$719$	−6.70726e9	−0.672967	−0.336484	−	0.941689i	$-0.609238\pi$
$719$	−6.70726e9	−0.672967	−0.336484	+	0.941689i	$0.609238\pi$
$720$	0	0
$721$	7.95378e9	0.790315
$722$	1.08449e10	1.07237
$723$	0	0
$724$	3.58947e9	0.351516
$725$	−3.56986e9	−0.347911
$726$	0	0
$727$	1.58709e10	1.53190	0.765951	−	0.642899i	$-0.222269\pi$
$727$	1.58709e10	1.53190	0.765951	+	0.642899i	$0.222269\pi$
$728$	6.86713e9	0.659653
$729$	0	0
$730$	−2.37302e10	−2.25773
$731$	1.51069e9	0.143042
$732$	0	0
$733$	−2.20320e9	−0.206629	−0.103314	−	0.994649i	$-0.532945\pi$
$733$	−2.20320e9	−0.206629	−0.103314	+	0.994649i	$0.532945\pi$
$734$	−1.32314e10	−1.23501
$735$	0	0
$736$	2.57108e8	0.0237708
$737$	−6.11443e8	−0.0562626
$738$	0	0
$739$	−1.48017e10	−1.34913	−0.674567	−	0.738214i	$-0.735670\pi$
$739$	−1.48017e10	−1.34913	−0.674567	+	0.738214i	$0.735670\pi$
$740$	2.80551e9	0.254508
$741$	0	0
$742$	1.89487e9	0.170281
$743$	2.07320e10	1.85430	0.927152	−	0.374686i	$-0.122250\pi$
$743$	2.07320e10	1.85430	0.927152	+	0.374686i	$0.122250\pi$
$744$	0	0
$745$	1.14269e10	1.01247
$746$	1.71405e10	1.51160
$747$	0	0
$748$	1.62035e8	0.0141564
$749$	5.94790e9	0.517222
$750$	0	0
$751$	−2.94907e9	−0.254065	−0.127033	−	0.991899i	$-0.540545\pi$
$751$	−2.94907e9	−0.254065	−0.127033	+	0.991899i	$0.540545\pi$
$752$	5.46837e9	0.468917
$753$	0	0
$754$	−1.62814e10	−1.38322
$755$	9.07543e9	0.767455
$756$	0	0
$757$	−9.04947e9	−0.758207	−0.379103	−	0.925354i	$-0.623768\pi$
$757$	−9.04947e9	−0.758207	−0.379103	+	0.925354i	$0.623768\pi$
$758$	2.29093e9	0.191060
$759$	0	0
$760$	1.55675e9	0.128639
$761$	−2.07707e10	−1.70846	−0.854230	−	0.519896i	$-0.825971\pi$
$761$	−2.07707e10	−1.70846	−0.854230	+	0.519896i	$0.825971\pi$
$762$	0	0
$763$	7.84716e9	0.639553
$764$	−2.73085e9	−0.221550
$765$	0	0
$766$	1.00093e10	0.804644
$767$	2.80172e10	2.24202
$768$	0	0
$769$	−1.18493e9	−0.0939617	−0.0469808	−	0.998896i	$-0.514960\pi$
$769$	−1.18493e9	−0.0939617	−0.0469808	+	0.998896i	$0.514960\pi$
$770$	−6.82951e8	−0.0539103
$771$	0	0
$772$	1.22645e9	0.0959378
$773$	1.66232e9	0.129446	0.0647228	−	0.997903i	$-0.479384\pi$
$773$	1.66232e9	0.129446	0.0647228	+	0.997903i	$0.479384\pi$
$774$	0	0
$775$	4.67775e8	0.0360978
$776$	−6.65710e9	−0.511410
$777$	0	0
$778$	−3.75121e9	−0.285590
$779$	1.98483e9	0.150433
$780$	0	0
$781$	1.34929e9	0.101351
$782$	−8.86066e8	−0.0662586
$783$	0	0
$784$	1.14112e10	0.845719
$785$	1.29378e10	0.954589
$786$	0	0
$787$	2.65010e10	1.93799	0.968993	−	0.247088i	$-0.0794736\pi$
$787$	2.65010e10	1.93799	0.968993	+	0.247088i	$0.0794736\pi$
$788$	1.86225e9	0.135580
$789$	0	0
$790$	1.41727e10	1.02272
$791$	6.39365e9	0.459336
$792$	0	0
$793$	1.43853e10	1.02439
$794$	2.65090e10	1.87941
$795$	0	0
$796$	−1.68088e9	−0.118125
$797$	2.48128e7	0.00173609	0.000868045	−	1.00000i	$-0.499724\pi$
$797$	2.48128e7	0.00173609	0.000868045		1.00000i	$0.499724\pi$
$798$	0	0
$799$	−5.50947e9	−0.382117
$800$	2.09841e9	0.144902
$801$	0	0
$802$	1.62941e10	1.11537
$803$	−2.09601e9	−0.142853
$804$	0	0
$805$	5.84243e8	0.0394737
$806$	2.13342e9	0.143517
$807$	0	0
$808$	1.19448e9	0.0796596
$809$	2.21936e8	0.0147369	0.00736847	−	0.999973i	$-0.497655\pi$
$809$	2.21936e8	0.0147369	0.00736847	+	0.999973i	$0.497655\pi$
$810$	0	0
$811$	1.47954e10	0.973987	0.486994	−	0.873405i	$-0.338093\pi$
$811$	1.47954e10	0.973987	0.486994	+	0.873405i	$0.338093\pi$
$812$	1.28191e9	0.0840253
$813$	0	0
$814$	1.58401e9	0.102937
$815$	−1.58986e10	−1.02875
$816$	0	0
$817$	−2.91873e8	−0.0187248
$818$	−7.01241e9	−0.447951
$819$	0	0
$820$	4.23774e9	0.268402
$821$	2.55026e10	1.60836	0.804181	−	0.594384i	$-0.202604\pi$
$821$	2.55026e10	1.60836	0.804181	+	0.594384i	$0.202604\pi$
$822$	0	0
$823$	1.87610e9	0.117316	0.0586578	−	0.998278i	$-0.481318\pi$
$823$	1.87610e9	0.117316	0.0586578	+	0.998278i	$0.481318\pi$
$824$	−2.18555e10	−1.36087
$825$	0	0
$826$	−1.41007e10	−0.870585
$827$	2.34241e10	1.44010	0.720052	−	0.693920i	$-0.244118\pi$
$827$	2.34241e10	1.44010	0.720052	+	0.693920i	$0.244118\pi$
$828$	0	0
$829$	2.88836e10	1.76080	0.880402	−	0.474228i	$-0.157273\pi$
$829$	2.88836e10	1.76080	0.880402	+	0.474228i	$0.157273\pi$
$830$	−2.10668e10	−1.27887
$831$	0	0
$832$	−1.80445e10	−1.08621
$833$	−1.14970e10	−0.689171
$834$	0	0
$835$	−1.91688e10	−1.13944
$836$	−3.13060e7	−0.00185313
$837$	0	0
$838$	−2.68209e10	−1.57442
$839$	−8.35473e9	−0.488388	−0.244194	−	0.969726i	$-0.578523\pi$
$839$	−8.35473e9	−0.488388	−0.244194	+	0.969726i	$0.578523\pi$
$840$	0	0
$841$	−3.90062e9	−0.226124
$842$	2.84404e9	0.164189
$843$	0	0
$844$	−3.42134e9	−0.195883
$845$	−2.24919e10	−1.28241
$846$	0	0
$847$	9.04793e9	0.511631
$848$	−6.20676e9	−0.349526
$849$	0	0
$850$	−7.23169e9	−0.403899
$851$	−1.35507e9	−0.0753718
$852$	0	0
$853$	−2.54143e10	−1.40203	−0.701014	−	0.713147i	$-0.747269\pi$
$853$	−2.54143e10	−1.40203	−0.701014	+	0.713147i	$0.747269\pi$
$854$	−7.23998e9	−0.397773
$855$	0	0
$856$	−1.63437e10	−0.890620
$857$	2.93738e10	1.59414	0.797071	−	0.603885i	$-0.206381\pi$
$857$	2.93738e10	1.59414	0.797071	+	0.603885i	$0.206381\pi$
$858$	0	0
$859$	−7.87923e9	−0.424138	−0.212069	−	0.977255i	$-0.568020\pi$
$859$	−7.87923e9	−0.424138	−0.212069	+	0.977255i	$0.568020\pi$
$860$	−6.23168e8	−0.0334088
$861$	0	0
$862$	2.87386e10	1.52823
$863$	1.41785e10	0.750916	0.375458	−	0.926839i	$-0.377485\pi$
$863$	1.41785e10	0.750916	0.375458	+	0.926839i	$0.377485\pi$
$864$	0	0
$865$	−1.00454e10	−0.527731
$866$	3.51890e9	0.184117
$867$	0	0
$868$	−1.67974e8	−0.00871811
$869$	1.25183e9	0.0647108
$870$	0	0
$871$	1.94703e10	0.998412
$872$	−2.15625e10	−1.10127
$873$	0	0
$874$	1.71192e8	0.00867350
$875$	−7.28852e9	−0.367800
$876$	0	0
$877$	−9.31500e9	−0.466320	−0.233160	−	0.972438i	$-0.574907\pi$
$877$	−9.31500e9	−0.466320	−0.233160	+	0.972438i	$0.574907\pi$
$878$	−3.12096e10	−1.55617
$879$	0	0
$880$	2.23704e9	0.110658
$881$	−1.57869e10	−0.777826	−0.388913	−	0.921275i	$-0.627149\pi$
$881$	−1.57869e10	−0.777826	−0.388913	+	0.921275i	$0.627149\pi$
$882$	0	0
$883$	−9.82559e9	−0.480282	−0.240141	−	0.970738i	$-0.577194\pi$
$883$	−9.82559e9	−0.480282	−0.240141	+	0.970738i	$0.577194\pi$
$884$	−5.15973e9	−0.251214
$885$	0	0
$886$	−8.06120e9	−0.389387
$887$	5.02494e9	0.241768	0.120884	−	0.992667i	$-0.461427\pi$
$887$	5.02494e9	0.241768	0.120884	+	0.992667i	$0.461427\pi$
$888$	0	0
$889$	−6.03100e8	−0.0287895
$890$	−2.43550e10	−1.15804
$891$	0	0
$892$	3.90116e9	0.184042
$893$	1.06446e9	0.0500205
$894$	0	0
$895$	2.01014e9	0.0937228
$896$	1.31448e10	0.610484
$897$	0	0
$898$	3.60611e10	1.66178
$899$	−1.74921e9	−0.0802940
$900$	0	0
$901$	6.25341e9	0.284826
$902$	2.39265e9	0.108557
$903$	0	0
$904$	−1.75686e10	−0.790945
$905$	−4.99283e10	−2.23912
$906$	0	0
$907$	−3.57868e9	−0.159257	−0.0796283	−	0.996825i	$-0.525373\pi$
$907$	−3.57868e9	−0.159257	−0.0796283	+	0.996825i	$0.525373\pi$
$908$	7.98879e9	0.354144
$909$	0	0
$910$	2.17474e10	0.956670
$911$	2.81588e10	1.23396	0.616978	−	0.786980i	$-0.288357\pi$
$911$	2.81588e10	1.23396	0.616978	+	0.786980i	$0.288357\pi$
$912$	0	0
$913$	−1.86077e9	−0.0809178
$914$	3.02633e10	1.31100
$915$	0	0
$916$	−1.37370e9	−0.0590552
$917$	−1.85186e10	−0.793078
$918$	0	0
$919$	−4.74290e9	−0.201576	−0.100788	−	0.994908i	$-0.532136\pi$
$919$	−4.74290e9	−0.201576	−0.100788	+	0.994908i	$0.532136\pi$
$920$	−1.60539e9	−0.0679710
$921$	0	0
$922$	1.55169e10	0.652000
$923$	−4.29659e10	−1.79853
$924$	0	0
$925$	−1.10595e10	−0.459452
$926$	−1.98051e10	−0.819668
$927$	0	0
$928$	−7.84685e9	−0.322313
$929$	2.37371e10	0.971344	0.485672	−	0.874141i	$-0.338575\pi$
$929$	2.37371e10	0.971344	0.485672	+	0.874141i	$0.338575\pi$
$930$	0	0
$931$	2.22128e9	0.0902150
$932$	6.43862e9	0.260518
$933$	0	0
$934$	2.14899e10	0.863018
$935$	−2.25385e9	−0.0901748
$936$	0	0
$937$	−1.59383e10	−0.632926	−0.316463	−	0.948605i	$-0.602495\pi$
$937$	−1.59383e10	−0.632926	−0.316463	+	0.948605i	$0.602495\pi$
$938$	−9.79921e9	−0.387687
$939$	0	0
$940$	2.27269e9	0.0892468
$941$	4.38313e10	1.71483	0.857415	−	0.514626i	$-0.172069\pi$
$941$	4.38313e10	1.71483	0.857415	+	0.514626i	$0.172069\pi$
$942$	0	0
$943$	−2.04684e9	−0.0794865
$944$	4.61877e10	1.78700
$945$	0	0
$946$	−3.51844e8	−0.0135124
$947$	3.68371e10	1.40948	0.704742	−	0.709464i	$-0.251063\pi$
$947$	3.68371e10	1.40948	0.704742	+	0.709464i	$0.251063\pi$
$948$	0	0
$949$	6.67440e10	2.53501
$950$	1.39720e9	0.0528720
$951$	0	0
$952$	−1.14059e10	−0.428450
$953$	−3.58699e10	−1.34247	−0.671236	−	0.741244i	$-0.734236\pi$
$953$	−3.58699e10	−1.34247	−0.671236	+	0.741244i	$0.734236\pi$
$954$	0	0
$955$	3.79852e10	1.41125
$956$	2.77088e9	0.102569
$957$	0	0
$958$	3.57520e10	1.31377
$959$	−5.63695e9	−0.206385
$960$	0	0
$961$	−2.72834e10	−0.991669
$962$	−5.04401e10	−1.82668
$963$	0	0
$964$	−9.71254e9	−0.349191
$965$	−1.70595e10	−0.611112
$966$	0	0
$967$	−6.33260e9	−0.225211	−0.112605	−	0.993640i	$-0.535920\pi$
$967$	−6.33260e9	−0.225211	−0.112605	+	0.993640i	$0.535920\pi$
$968$	−2.48620e10	−0.880994
$969$	0	0
$970$	−2.10822e10	−0.741678
$971$	3.71924e10	1.30373	0.651864	−	0.758336i	$-0.273987\pi$
$971$	3.71924e10	1.30373	0.651864	+	0.758336i	$0.273987\pi$
$972$	0	0
$973$	−1.61694e10	−0.562729
$974$	−1.22281e10	−0.424036
$975$	0	0
$976$	2.37149e10	0.816484
$977$	−3.55570e10	−1.21982	−0.609908	−	0.792472i	$-0.708794\pi$
$977$	−3.55570e10	−1.21982	−0.609908	+	0.792472i	$0.708794\pi$
$978$	0	0
$979$	−2.15121e9	−0.0732727
$980$	4.74258e9	0.160962
$981$	0	0
$982$	5.17422e10	1.74363
$983$	2.71260e10	0.910854	0.455427	−	0.890273i	$-0.349487\pi$
$983$	2.71260e10	0.910854	0.455427	+	0.890273i	$0.349487\pi$
$984$	0	0
$985$	−2.59032e10	−0.863628
$986$	2.70424e10	0.898413
$987$	0	0
$988$	9.96886e8	0.0328849
$989$	3.00992e8	0.00989390
$990$	0	0
$991$	−1.89518e10	−0.618575	−0.309287	−	0.950969i	$-0.600091\pi$
$991$	−1.89518e10	−0.618575	−0.309287	+	0.950969i	$0.600091\pi$
$992$	1.02821e9	0.0334418
$993$	0	0
$994$	2.16243e10	0.698376
$995$	2.33804e10	0.752440
$996$	0	0
$997$	8.54283e9	0.273004	0.136502	−	0.990640i	$-0.456414\pi$
$997$	8.54283e9	0.273004	0.136502	+	0.990640i	$0.456414\pi$
$998$	−3.85401e10	−1.22731
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	387.8.a.b.1.3		11
3.2	odd	2		43.8.a.a.1.9	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.8.a.a.1.9	✓	11	3.2	odd	2
387.8.a.b.1.3		11	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 387 = 3^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	387.a (trivial)

\(f(q)\)	\(=\)	\(q-12.3182 q^{2} +23.7379 q^{4} -330.186 q^{5} -467.397 q^{7} +1284.32 q^{8} +O(q^{10})\)	\(q-12.3182 q^{2} +23.7379 q^{4} -330.186 q^{5} -467.397 q^{7} +1284.32 q^{8} +4067.29 q^{10} +359.251 q^{11} -11439.7 q^{13} +5757.49 q^{14} -18859.0 q^{16} +19000.7 q^{17} -3671.03 q^{19} -7837.90 q^{20} -4425.32 q^{22} +3785.73 q^{23} +30897.5 q^{25} +140917. q^{26} -11095.0 q^{28} -115539. q^{29} +15139.6 q^{31} +67915.2 q^{32} -234054. q^{34} +154328. q^{35} -357942. q^{37} +45220.5 q^{38} -424064. q^{40} -540673. q^{41} +79507.0 q^{43} +8527.85 q^{44} -46633.3 q^{46} -289961. q^{47} -605083. q^{49} -380601. q^{50} -271555. q^{52} +329115. q^{53} -118619. q^{55} -600288. q^{56} +1.42323e6 q^{58} -2.44911e6 q^{59} -1.25749e6 q^{61} -186492. q^{62} +1.57735e6 q^{64} +3.77723e6 q^{65} -1.70199e6 q^{67} +451036. q^{68} -1.90104e6 q^{70} +3.75585e6 q^{71} -5.83440e6 q^{73} +4.40920e6 q^{74} -87142.4 q^{76} -167913. q^{77} +3.48456e6 q^{79} +6.22696e6 q^{80} +6.66012e6 q^{82} -5.17957e6 q^{83} -6.27376e6 q^{85} -979383. q^{86} +461394. q^{88} -5.98803e6 q^{89} +5.34690e6 q^{91} +89865.1 q^{92} +3.57180e6 q^{94} +1.21212e6 q^{95} -5.18336e6 q^{97} +7.45353e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 24 q^{2} + 602 q^{4} + 752 q^{5} - 12 q^{7} + 3810 q^{8}+O(q^{10}) \) 11 * q + 24 * q^2 + 602 * q^4 + 752 * q^5 - 12 * q^7 + 3810 * q^8	\( 11 q + 24 q^{2} + 602 q^{4} + 752 q^{5} - 12 q^{7} + 3810 q^{8} - 1333 q^{10} - 1333 q^{11} - 17967 q^{13} + 22352 q^{14} - 34406 q^{16} + 63095 q^{17} - 54524 q^{19} + 280995 q^{20} - 289358 q^{22} + 138139 q^{23} + 3455 q^{25} + 132946 q^{26} - 12704 q^{28} + 308658 q^{29} - 209523 q^{31} + 644934 q^{32} + 762435 q^{34} + 578892 q^{35} - 298472 q^{37} + 369707 q^{38} + 2633173 q^{40} + 1346735 q^{41} + 874577 q^{43} - 3134292 q^{44} + 3588111 q^{46} - 499284 q^{47} + 2544563 q^{49} - 3049745 q^{50} + 983088 q^{52} + 2210495 q^{53} - 1855072 q^{55} + 469976 q^{56} + 4397067 q^{58} + 5824216 q^{59} - 4453034 q^{61} - 1002789 q^{62} + 4757538 q^{64} + 2162872 q^{65} - 6859513 q^{67} + 9397005 q^{68} + 845078 q^{70} + 10726554 q^{71} - 4456898 q^{73} - 1046637 q^{74} + 5861267 q^{76} + 17019816 q^{77} - 15541320 q^{79} + 15680911 q^{80} + 20233655 q^{82} + 11146767 q^{83} - 12471976 q^{85} + 1908168 q^{86} - 24463544 q^{88} + 13531356 q^{89} - 19746448 q^{91} + 26023161 q^{92} + 20288857 q^{94} + 12291624 q^{95} - 10999901 q^{97} - 29909168 q^{98}+O(q^{100}) \) 11 * q + 24 * q^2 + 602 * q^4 + 752 * q^5 - 12 * q^7 + 3810 * q^8 - 1333 * q^10 - 1333 * q^11 - 17967 * q^13 + 22352 * q^14 - 34406 * q^16 + 63095 * q^17 - 54524 * q^19 + 280995 * q^20 - 289358 * q^22 + 138139 * q^23 + 3455 * q^25 + 132946 * q^26 - 12704 * q^28 + 308658 * q^29 - 209523 * q^31 + 644934 * q^32 + 762435 * q^34 + 578892 * q^35 - 298472 * q^37 + 369707 * q^38 + 2633173 * q^40 + 1346735 * q^41 + 874577 * q^43 - 3134292 * q^44 + 3588111 * q^46 - 499284 * q^47 + 2544563 * q^49 - 3049745 * q^50 + 983088 * q^52 + 2210495 * q^53 - 1855072 * q^55 + 469976 * q^56 + 4397067 * q^58 + 5824216 * q^59 - 4453034 * q^61 - 1002789 * q^62 + 4757538 * q^64 + 2162872 * q^65 - 6859513 * q^67 + 9397005 * q^68 + 845078 * q^70 + 10726554 * q^71 - 4456898 * q^73 - 1046637 * q^74 + 5861267 * q^76 + 17019816 * q^77 - 15541320 * q^79 + 15680911 * q^80 + 20233655 * q^82 + 11146767 * q^83 - 12471976 * q^85 + 1908168 * q^86 - 24463544 * q^88 + 13531356 * q^89 - 19746448 * q^91 + 26023161 * q^92 + 20288857 * q^94 + 12291624 * q^95 - 10999901 * q^97 - 29909168 * q^98

Introduction

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