LMFDB - Embedded newform 405.8.a.j.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [405,8,Mod(1,405)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(405, 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("405.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 405 = 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	405.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:	$126.515935321$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 7 x^{14} - 1479 x^{13} + 9623 x^{12} + 858424 x^{11} - 5043114 x^{10} - 248945154 x^{9} + \cdots + 784812676793472 $ x^15 - 7x^14 - 1479x^13 + 9623x^12 + 858424x^11 - 5043114x^10 - 248945154x^9 + 1250661006x^8 + 38068075521x^7 - 145949082795x^6 - 2962070172819x^5 + 6595509983499x^4 + 100284834644406x^3 - 44138883317076x^2 - 768967569531540x + 784812676793472
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{33} $
Twist minimal:	no (minimal twist has level 45)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$-5.12335$ of defining polynomial
Character	$\chi$	$=$	405.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+6.12335 q^{2} -90.5045 q^{4} +125.000 q^{5} +726.455 q^{7} -1337.98 q^{8} +O(q^{10})$	\(q+6.12335 q^{2} -90.5045 q^{4} +125.000 q^{5} +726.455 q^{7} -1337.98 q^{8} +765.419 q^{10} -8730.50 q^{11} +452.485 q^{13} +4448.34 q^{14} +3391.65 q^{16} -17929.5 q^{17} +45296.7 q^{19} -11313.1 q^{20} -53459.9 q^{22} -31705.4 q^{23} +15625.0 q^{25} +2770.73 q^{26} -65747.4 q^{28} -197432. q^{29} +147830. q^{31} +192030. q^{32} -109789. q^{34} +90806.8 q^{35} +97678.1 q^{37} +277368. q^{38} -167248. q^{40} -226957. q^{41} +355547. q^{43} +790150. q^{44} -194144. q^{46} +377653. q^{47} -295807. q^{49} +95677.4 q^{50} -40952.0 q^{52} +1.68457e6 q^{53} -1.09131e6 q^{55} -971982. q^{56} -1.20895e6 q^{58} -1.05675e6 q^{59} -1.24101e6 q^{61} +905215. q^{62} +741735. q^{64} +56560.7 q^{65} -3.36923e6 q^{67} +1.62270e6 q^{68} +556042. q^{70} -303064. q^{71} +6.38027e6 q^{73} +598117. q^{74} -4.09956e6 q^{76} -6.34231e6 q^{77} +4.39582e6 q^{79} +423957. q^{80} -1.38974e6 q^{82} -5.86217e6 q^{83} -2.24119e6 q^{85} +2.17714e6 q^{86} +1.16812e7 q^{88} +4.78941e6 q^{89} +328710. q^{91} +2.86949e6 q^{92} +2.31250e6 q^{94} +5.66209e6 q^{95} -9.88594e6 q^{97} -1.81133e6 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 8 q^{2} + 1088 q^{4} + 1875 q^{5} + 1289 q^{7} + 4434 q^{8}+O(q^{10}) $ 15 * q + 8 * q^2 + 1088 * q^4 + 1875 * q^5 + 1289 * q^7 + 4434 * q^8	\( 15 q + 8 q^{2} + 1088 q^{4} + 1875 q^{5} + 1289 q^{7} + 4434 q^{8} + 1000 q^{10} + 1658 q^{11} + 9874 q^{13} + 7695 q^{14} + 62612 q^{16} - 3598 q^{17} + 21376 q^{19} + 136000 q^{20} - 52519 q^{22} + 96441 q^{23} + 234375 q^{25} + 126146 q^{26} + 174449 q^{28} + 259297 q^{29} + 373568 q^{31} + 1134550 q^{32} + 423851 q^{34} + 161125 q^{35} + 517872 q^{37} + 690059 q^{38} + 554250 q^{40} + 520501 q^{41} + 1898836 q^{43} + 1277707 q^{44} + 3154677 q^{46} + 2259041 q^{47} + 4316308 q^{49} + 125000 q^{50} + 5398554 q^{52} - 102274 q^{53} + 207250 q^{55} - 504621 q^{56} + 3190987 q^{58} - 1680874 q^{59} - 1066457 q^{61} - 274110 q^{62} + 6541980 q^{64} + 1234250 q^{65} + 6522389 q^{67} + 1420717 q^{68} + 961875 q^{70} - 32786 q^{71} + 5359102 q^{73} - 4045556 q^{74} + 4649241 q^{76} + 2586078 q^{77} + 9319346 q^{79} + 7826500 q^{80} + 7460620 q^{82} - 12758277 q^{83} - 449750 q^{85} - 20044675 q^{86} + 6691143 q^{88} - 18776241 q^{89} + 9244102 q^{91} - 13862829 q^{92} + 25905119 q^{94} + 2672000 q^{95} + 2788224 q^{97} - 1679531 q^{98}+O(q^{100}) \) 15 * q + 8 * q^2 + 1088 * q^4 + 1875 * q^5 + 1289 * q^7 + 4434 * q^8 + 1000 * q^10 + 1658 * q^11 + 9874 * q^13 + 7695 * q^14 + 62612 * q^16 - 3598 * q^17 + 21376 * q^19 + 136000 * q^20 - 52519 * q^22 + 96441 * q^23 + 234375 * q^25 + 126146 * q^26 + 174449 * q^28 + 259297 * q^29 + 373568 * q^31 + 1134550 * q^32 + 423851 * q^34 + 161125 * q^35 + 517872 * q^37 + 690059 * q^38 + 554250 * q^40 + 520501 * q^41 + 1898836 * q^43 + 1277707 * q^44 + 3154677 * q^46 + 2259041 * q^47 + 4316308 * q^49 + 125000 * q^50 + 5398554 * q^52 - 102274 * q^53 + 207250 * q^55 - 504621 * q^56 + 3190987 * q^58 - 1680874 * q^59 - 1066457 * q^61 - 274110 * q^62 + 6541980 * q^64 + 1234250 * q^65 + 6522389 * q^67 + 1420717 * q^68 + 961875 * q^70 - 32786 * q^71 + 5359102 * q^73 - 4045556 * q^74 + 4649241 * q^76 + 2586078 * q^77 + 9319346 * q^79 + 7826500 * q^80 + 7460620 * q^82 - 12758277 * q^83 - 449750 * q^85 - 20044675 * q^86 + 6691143 * q^88 - 18776241 * q^89 + 9244102 * q^91 - 13862829 * q^92 + 25905119 * q^94 + 2672000 * q^95 + 2788224 * q^97 - 1679531 * 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 $

$2$	6.12335	0.541233	0.270617	−	0.962687i	$-0.412772\pi$
$2$	6.12335	0.541233	0.270617	+	0.962687i	$0.412772\pi$
$3$	0	0
$3$	0	0
$4$	−90.5045	−0.707067
$5$	125.000	0.447214
$5$	125.000	0.447214
$6$	0	0
$7$	726.455	0.800507	0.400254	−	0.916404i	$-0.368922\pi$
$7$	726.455	0.800507	0.400254	+	0.916404i	$0.368922\pi$
$8$	−1337.98	−0.923921
$9$	0	0
$10$	765.419	0.242047
$11$	−8730.50	−1.97772	−0.988860	−	0.148852i	$-0.952442\pi$
$11$	−8730.50	−1.97772	−0.988860	+	0.148852i	$0.952442\pi$
$12$	0	0
$13$	452.485	0.0571219	0.0285610	−	0.999592i	$-0.490908\pi$
$13$	452.485	0.0571219	0.0285610	+	0.999592i	$0.490908\pi$
$14$	4448.34	0.433261
$15$	0	0
$16$	3391.65	0.207010
$17$	−17929.5	−0.885110	−0.442555	−	0.896741i	$-0.645928\pi$
$17$	−17929.5	−0.885110	−0.442555	+	0.896741i	$0.645928\pi$
$18$	0	0
$19$	45296.7	1.51506	0.757529	−	0.652801i	$-0.226406\pi$
$19$	45296.7	1.51506	0.757529	+	0.652801i	$0.226406\pi$
$20$	−11313.1	−0.316210
$21$	0	0
$22$	−53459.9	−1.07041
$23$	−31705.4	−0.543358	−0.271679	−	0.962388i	$-0.587579\pi$
$23$	−31705.4	−0.543358	−0.271679	+	0.962388i	$0.587579\pi$
$24$	0	0
$25$	15625.0	0.200000
$26$	2770.73	0.0309163
$27$	0	0
$28$	−65747.4	−0.566012
$29$	−197432.	−1.50323	−0.751614	−	0.659603i	$-0.770724\pi$
$29$	−197432.	−1.50323	−0.751614	+	0.659603i	$0.770724\pi$
$30$	0	0
$31$	147830.	0.891244	0.445622	−	0.895221i	$-0.352983\pi$
$31$	147830.	0.891244	0.445622	+	0.895221i	$0.352983\pi$
$32$	192030.	1.03596
$33$	0	0
$34$	−109789.	−0.479051
$35$	90806.8	0.357998
$36$	0	0
$37$	97678.1	0.317023	0.158512	−	0.987357i	$-0.449330\pi$
$37$	97678.1	0.317023	0.158512	+	0.987357i	$0.449330\pi$
$38$	277368.	0.820000
$39$	0	0
$40$	−167248.	−0.413190
$41$	−226957.	−0.514281	−0.257140	−	0.966374i	$-0.582780\pi$
$41$	−226957.	−0.514281	−0.257140	+	0.966374i	$0.582780\pi$
$42$	0	0
$43$	355547.	0.681957	0.340979	−	0.940071i	$-0.389242\pi$
$43$	355547.	0.681957	0.340979	+	0.940071i	$0.389242\pi$
$44$	790150.	1.39838
$45$	0	0
$46$	−194144.	−0.294083
$47$	377653.	0.530580	0.265290	−	0.964169i	$-0.414532\pi$
$47$	377653.	0.530580	0.265290	+	0.964169i	$0.414532\pi$
$48$	0	0
$49$	−295807.	−0.359188
$50$	95677.4	0.108247
$51$	0	0
$52$	−40952.0	−0.0403890
$53$	1.68457e6	1.55426	0.777131	−	0.629339i	$-0.216674\pi$
$53$	1.68457e6	1.55426	0.777131	+	0.629339i	$0.216674\pi$
$54$	0	0
$55$	−1.09131e6	−0.884463
$56$	−971982.	−0.739606
$57$	0	0
$58$	−1.20895e6	−0.813597
$59$	−1.05675e6	−0.669869	−0.334935	−	0.942241i	$-0.608714\pi$
$59$	−1.05675e6	−0.669869	−0.334935	+	0.942241i	$0.608714\pi$
$60$	0	0
$61$	−1.24101e6	−0.700035	−0.350017	−	0.936743i	$-0.613824\pi$
$61$	−1.24101e6	−0.700035	−0.350017	+	0.936743i	$0.613824\pi$
$62$	905215.	0.482371
$63$	0	0
$64$	741735.	0.353687
$65$	56560.7	0.0255457
$66$	0	0
$67$	−3.36923e6	−1.36858	−0.684289	−	0.729211i	$-0.739887\pi$
$67$	−3.36923e6	−1.36858	−0.684289	+	0.729211i	$0.739887\pi$
$68$	1.62270e6	0.625832
$69$	0	0
$70$	556042.	0.193760
$71$	−303064.	−0.100492	−0.0502459	−	0.998737i	$-0.516000\pi$
$71$	−303064.	−0.100492	−0.0502459	+	0.998737i	$0.516000\pi$
$72$	0	0
$73$	6.38027e6	1.91959	0.959796	−	0.280697i	$-0.0905657\pi$
$73$	6.38027e6	1.91959	0.959796	+	0.280697i	$0.0905657\pi$
$74$	598117.	0.171583
$75$	0	0
$76$	−4.09956e6	−1.07125
$77$	−6.34231e6	−1.58318
$78$	0	0
$79$	4.39582e6	1.00310	0.501551	−	0.865128i	$-0.332763\pi$
$79$	4.39582e6	1.00310	0.501551	+	0.865128i	$0.332763\pi$
$80$	423957.	0.0925777
$81$	0	0
$82$	−1.38974e6	−0.278346
$83$	−5.86217e6	−1.12534	−0.562671	−	0.826681i	$-0.690226\pi$
$83$	−5.86217e6	−1.12534	−0.562671	+	0.826681i	$0.690226\pi$
$84$	0	0
$85$	−2.24119e6	−0.395833
$86$	2.17714e6	0.369098
$87$	0	0
$88$	1.16812e7	1.82726
$89$	4.78941e6	0.720141	0.360070	−	0.932925i	$-0.382753\pi$
$89$	4.78941e6	0.720141	0.360070	+	0.932925i	$0.382753\pi$
$90$	0	0
$91$	328710.	0.0457265
$92$	2.86949e6	0.384191
$93$	0	0
$94$	2.31250e6	0.287167
$95$	5.66209e6	0.677555
$96$	0	0
$97$	−9.88594e6	−1.09981	−0.549904	−	0.835228i	$-0.685336\pi$
$97$	−9.88594e6	−1.09981	−0.549904	+	0.835228i	$0.685336\pi$
$98$	−1.81133e6	−0.194404
$99$	0	0
$100$	−1.41413e6	−0.141413
$101$	1.75479e7	1.69473	0.847363	−	0.531014i	$-0.178189\pi$
$101$	1.75479e7	1.69473	0.847363	+	0.531014i	$0.178189\pi$
$102$	0	0
$103$	4.16783e6	0.375820	0.187910	−	0.982186i	$-0.439829\pi$
$103$	4.16783e6	0.375820	0.187910	+	0.982186i	$0.439829\pi$
$104$	−605417.	−0.0527761
$105$	0	0
$106$	1.03152e7	0.841218
$107$	4.30105e6	0.339415	0.169708	−	0.985494i	$-0.445718\pi$
$107$	4.30105e6	0.339415	0.169708	+	0.985494i	$0.445718\pi$
$108$	0	0
$109$	2.01834e7	1.49280	0.746399	−	0.665499i	$-0.231781\pi$
$109$	2.01834e7	1.49280	0.746399	+	0.665499i	$0.231781\pi$
$110$	−6.68249e6	−0.478701
$111$	0	0
$112$	2.46388e6	0.165713
$113$	2.34077e6	0.152610	0.0763051	−	0.997085i	$-0.475688\pi$
$113$	2.34077e6	0.152610	0.0763051	+	0.997085i	$0.475688\pi$
$114$	0	0
$115$	−3.96318e6	−0.242997
$116$	1.78685e7	1.06288
$117$	0	0
$118$	−6.47085e6	−0.362556
$119$	−1.30250e7	−0.708537
$120$	0	0
$121$	5.67344e7	2.91137
$122$	−7.59912e6	−0.378882
$123$	0	0
$124$	−1.33793e7	−0.630169
$125$	1.95312e6	0.0894427
$126$	0	0
$127$	−6.24033e6	−0.270330	−0.135165	−	0.990823i	$-0.543156\pi$
$127$	−6.24033e6	−0.270330	−0.135165	+	0.990823i	$0.543156\pi$
$128$	−2.00379e7	−0.844535
$129$	0	0
$130$	346341.	0.0138262
$131$	2.08434e7	0.810063	0.405032	−	0.914303i	$-0.367261\pi$
$131$	2.08434e7	0.810063	0.405032	+	0.914303i	$0.367261\pi$
$132$	0	0
$133$	3.29060e7	1.21282
$134$	−2.06310e7	−0.740719
$135$	0	0
$136$	2.39893e7	0.817771
$137$	3.22404e7	1.07122	0.535610	−	0.844466i	$-0.320082\pi$
$137$	3.22404e7	1.07122	0.535610	+	0.844466i	$0.320082\pi$
$138$	0	0
$139$	1.05708e7	0.333852	0.166926	−	0.985969i	$-0.446616\pi$
$139$	1.05708e7	0.333852	0.166926	+	0.985969i	$0.446616\pi$
$140$	−8.21843e6	−0.253128
$141$	0	0
$142$	−1.85577e6	−0.0543895
$143$	−3.95042e6	−0.112971
$144$	0	0
$145$	−2.46790e7	−0.672264
$146$	3.90687e7	1.03895
$147$	0	0
$148$	−8.84031e6	−0.224157
$149$	4.66859e7	1.15620	0.578101	−	0.815965i	$-0.303794\pi$
$149$	4.66859e7	1.15620	0.578101	+	0.815965i	$0.303794\pi$
$150$	0	0
$151$	−2.28827e7	−0.540863	−0.270431	−	0.962739i	$-0.587166\pi$
$151$	−2.28827e7	−0.540863	−0.270431	+	0.962739i	$0.587166\pi$
$152$	−6.06062e7	−1.39979
$153$	0	0
$154$	−3.88362e7	−0.856869
$155$	1.84787e7	0.398576
$156$	0	0
$157$	6.34688e7	1.30892	0.654458	−	0.756099i	$-0.272897\pi$
$157$	6.34688e7	1.30892	0.654458	+	0.756099i	$0.272897\pi$
$158$	2.69171e7	0.542912
$159$	0	0
$160$	2.40037e7	0.463296
$161$	−2.30326e7	−0.434962
$162$	0	0
$163$	7.27533e7	1.31582	0.657909	−	0.753097i	$-0.271441\pi$
$163$	7.27533e7	1.31582	0.657909	+	0.753097i	$0.271441\pi$
$164$	2.05406e7	0.363631
$165$	0	0
$166$	−3.58961e7	−0.609073
$167$	3.47433e7	0.577249	0.288624	−	0.957442i	$-0.406802\pi$
$167$	3.47433e7	0.577249	0.288624	+	0.957442i	$0.406802\pi$
$168$	0	0
$169$	−6.25438e7	−0.996737
$170$	−1.37236e7	−0.214238
$171$	0	0
$172$	−3.21786e7	−0.482189
$173$	6.71674e6	0.0986273	0.0493136	−	0.998783i	$-0.484297\pi$
$173$	6.71674e6	0.0986273	0.0493136	+	0.998783i	$0.484297\pi$
$174$	0	0
$175$	1.13509e7	0.160101
$176$	−2.96108e7	−0.409408
$177$	0	0
$178$	2.93273e7	0.389764
$179$	5.85397e7	0.762895	0.381448	−	0.924390i	$-0.375426\pi$
$179$	5.85397e7	0.762895	0.381448	+	0.924390i	$0.375426\pi$
$180$	0	0
$181$	8.51883e7	1.06784	0.533918	−	0.845536i	$-0.320719\pi$
$181$	8.51883e7	1.06784	0.533918	+	0.845536i	$0.320719\pi$
$182$	2.01281e6	0.0247487
$183$	0	0
$184$	4.24213e7	0.502020
$185$	1.22098e7	0.141777
$186$	0	0
$187$	1.56533e8	1.75050
$188$	−3.41793e7	−0.375155
$189$	0	0
$190$	3.46710e7	0.366715
$191$	1.09798e7	0.114019	0.0570094	−	0.998374i	$-0.481843\pi$
$191$	1.09798e7	0.114019	0.0570094	+	0.998374i	$0.481843\pi$
$192$	0	0
$193$	5.21430e6	0.0522091	0.0261045	−	0.999659i	$-0.491690\pi$
$193$	5.21430e6	0.0522091	0.0261045	+	0.999659i	$0.491690\pi$
$194$	−6.05351e7	−0.595252
$195$	0	0
$196$	2.67719e7	0.253970
$197$	−8.37710e7	−0.780660	−0.390330	−	0.920675i	$-0.627639\pi$
$197$	−8.37710e7	−0.780660	−0.390330	+	0.920675i	$0.627639\pi$
$198$	0	0
$199$	805325.	0.00724411	0.00362206	−	0.999993i	$-0.498847\pi$
$199$	805325.	0.00724411	0.00362206	+	0.999993i	$0.498847\pi$
$200$	−2.09059e7	−0.184784
$201$	0	0
$202$	1.07452e8	0.917242
$203$	−1.43425e8	−1.20335
$204$	0	0
$205$	−2.83696e7	−0.229993
$206$	2.55211e7	0.203406
$207$	0	0
$208$	1.53467e6	0.0118248
$209$	−3.95463e8	−2.99636
$210$	0	0
$211$	−4.90762e7	−0.359652	−0.179826	−	0.983698i	$-0.557553\pi$
$211$	−4.90762e7	−0.359652	−0.179826	+	0.983698i	$0.557553\pi$
$212$	−1.52462e8	−1.09897
$213$	0	0
$214$	2.63369e7	0.183703
$215$	4.44433e7	0.304980
$216$	0	0
$217$	1.07392e8	0.713447
$218$	1.23590e8	0.807951
$219$	0	0
$220$	9.87687e7	0.625374
$221$	−8.11284e6	−0.0505591
$222$	0	0
$223$	−1.47952e8	−0.893418	−0.446709	−	0.894679i	$-0.647404\pi$
$223$	−1.47952e8	−0.893418	−0.446709	+	0.894679i	$0.647404\pi$
$224$	1.39501e8	0.829295
$225$	0	0
$226$	1.43334e7	0.0825977
$227$	3.12868e6	0.0177529	0.00887647	−	0.999961i	$-0.497174\pi$
$227$	3.12868e6	0.0177529	0.00887647	+	0.999961i	$0.497174\pi$
$228$	0	0
$229$	8.18463e7	0.450376	0.225188	−	0.974315i	$-0.427700\pi$
$229$	8.18463e7	0.450376	0.225188	+	0.974315i	$0.427700\pi$
$230$	−2.42679e7	−0.131518
$231$	0	0
$232$	2.64160e8	1.38886
$233$	−2.67507e8	−1.38545	−0.692723	−	0.721204i	$-0.743589\pi$
$233$	−2.67507e8	−1.38545	−0.692723	+	0.721204i	$0.743589\pi$
$234$	0	0
$235$	4.72067e7	0.237282
$236$	9.56407e7	0.473642
$237$	0	0
$238$	−7.97565e7	−0.383484
$239$	1.98031e8	0.938299	0.469150	−	0.883119i	$-0.344560\pi$
$239$	1.98031e8	0.938299	0.469150	+	0.883119i	$0.344560\pi$
$240$	0	0
$241$	4.25193e8	1.95671	0.978354	−	0.206937i	$-0.0663495\pi$
$241$	4.25193e8	1.95671	0.978354	+	0.206937i	$0.0663495\pi$
$242$	3.47405e8	1.57573
$243$	0	0
$244$	1.12317e8	0.494971
$245$	−3.69758e7	−0.160634
$246$	0	0
$247$	2.04961e7	0.0865430
$248$	−1.97794e8	−0.823439
$249$	0	0
$250$	1.19597e7	0.0484094
$251$	2.82458e8	1.12744	0.563722	−	0.825964i	$-0.309369\pi$
$251$	2.82458e8	1.12744	0.563722	+	0.825964i	$0.309369\pi$
$252$	0	0
$253$	2.76804e8	1.07461
$254$	−3.82117e7	−0.146312
$255$	0	0
$256$	−2.17641e8	−0.810777
$257$	2.51487e8	0.924166	0.462083	−	0.886837i	$-0.347102\pi$
$257$	2.51487e8	0.924166	0.462083	+	0.886837i	$0.347102\pi$
$258$	0	0
$259$	7.09587e7	0.253779
$260$	−5.11900e6	−0.0180625
$261$	0	0
$262$	1.27631e8	0.438433
$263$	−5.64808e7	−0.191450	−0.0957250	−	0.995408i	$-0.530517\pi$
$263$	−5.64808e7	−0.191450	−0.0957250	+	0.995408i	$0.530517\pi$
$264$	0	0
$265$	2.10572e8	0.695087
$266$	2.01495e8	0.656416
$267$	0	0
$268$	3.04931e8	0.967675
$269$	−6.07411e7	−0.190261	−0.0951305	−	0.995465i	$-0.530327\pi$
$269$	−6.07411e7	−0.190261	−0.0951305	+	0.995465i	$0.530327\pi$
$270$	0	0
$271$	1.76283e8	0.538045	0.269023	−	0.963134i	$-0.413299\pi$
$271$	1.76283e8	0.538045	0.269023	+	0.963134i	$0.413299\pi$
$272$	−6.08107e7	−0.183227
$273$	0	0
$274$	1.97419e8	0.579779
$275$	−1.36414e8	−0.395544
$276$	0	0
$277$	−3.76696e8	−1.06491	−0.532454	−	0.846459i	$-0.678730\pi$
$277$	−3.76696e8	−1.06491	−0.532454	+	0.846459i	$0.678730\pi$
$278$	6.47285e7	0.180692
$279$	0	0
$280$	−1.21498e8	−0.330762
$281$	−3.22561e8	−0.867241	−0.433620	−	0.901096i	$-0.642764\pi$
$281$	−3.22561e8	−0.867241	−0.433620	+	0.901096i	$0.642764\pi$
$282$	0	0
$283$	−4.36729e8	−1.14541	−0.572703	−	0.819763i	$-0.694105\pi$
$283$	−4.36729e8	−1.14541	−0.572703	+	0.819763i	$0.694105\pi$
$284$	2.74287e7	0.0710544
$285$	0	0
$286$	−2.41898e7	−0.0611437
$287$	−1.64874e8	−0.411685
$288$	0	0
$289$	−8.88716e7	−0.216581
$290$	−1.51118e8	−0.363852
$291$	0	0
$292$	−5.77443e8	−1.35728
$293$	−6.55918e8	−1.52340	−0.761698	−	0.647932i	$-0.775634\pi$
$293$	−6.55918e8	−1.52340	−0.761698	+	0.647932i	$0.775634\pi$
$294$	0	0
$295$	−1.32094e8	−0.299575
$296$	−1.30691e8	−0.292904
$297$	0	0
$298$	2.85874e8	0.625775
$299$	−1.43462e7	−0.0310377
$300$	0	0
$301$	2.58288e8	0.545912
$302$	−1.40119e8	−0.292733
$303$	0	0
$304$	1.53631e8	0.313632
$305$	−1.55126e8	−0.313065
$306$	0	0
$307$	1.46092e8	0.288166	0.144083	−	0.989566i	$-0.453977\pi$
$307$	1.46092e8	0.288166	0.144083	+	0.989566i	$0.453977\pi$
$308$	5.74008e8	1.11941
$309$	0	0
$310$	1.13152e8	0.215723
$311$	2.29483e8	0.432603	0.216301	−	0.976327i	$-0.430601\pi$
$311$	2.29483e8	0.432603	0.216301	+	0.976327i	$0.430601\pi$
$312$	0	0
$313$	1.61723e8	0.298102	0.149051	−	0.988829i	$-0.452378\pi$
$313$	1.61723e8	0.298102	0.149051	+	0.988829i	$0.452378\pi$
$314$	3.88642e8	0.708428
$315$	0	0
$316$	−3.97841e8	−0.709259
$317$	5.90430e8	1.04102	0.520512	−	0.853854i	$-0.325741\pi$
$317$	5.90430e8	1.04102	0.520512	+	0.853854i	$0.325741\pi$
$318$	0	0
$319$	1.72368e9	2.97296
$320$	9.27169e7	0.158174
$321$	0	0
$322$	−1.41036e8	−0.235416
$323$	−8.12148e8	−1.34099
$324$	0	0
$325$	7.07008e6	0.0114244
$326$	4.45494e8	0.712164
$327$	0	0
$328$	3.03664e8	0.475155
$329$	2.74348e8	0.424733
$330$	0	0
$331$	4.99156e8	0.756551	0.378275	−	0.925693i	$-0.376517\pi$
$331$	4.99156e8	0.756551	0.378275	+	0.925693i	$0.376517\pi$
$332$	5.30553e8	0.795692
$333$	0	0
$334$	2.12745e8	0.312426
$335$	−4.21154e8	−0.612046
$336$	0	0
$337$	−5.33772e8	−0.759715	−0.379858	−	0.925045i	$-0.624027\pi$
$337$	−5.33772e8	−0.759715	−0.379858	+	0.925045i	$0.624027\pi$
$338$	−3.82978e8	−0.539467
$339$	0	0
$340$	2.02838e8	0.279880
$341$	−1.29063e9	−1.76263
$342$	0	0
$343$	−8.13157e8	−1.08804
$344$	−4.75715e8	−0.630074
$345$	0	0
$346$	4.11290e7	0.0533803
$347$	7.92483e8	1.01821	0.509104	−	0.860705i	$-0.329977\pi$
$347$	7.92483e8	1.01821	0.509104	+	0.860705i	$0.329977\pi$
$348$	0	0
$349$	−1.26526e9	−1.59328	−0.796638	−	0.604457i	$-0.793390\pi$
$349$	−1.26526e9	−1.59328	−0.796638	+	0.604457i	$0.793390\pi$
$350$	6.95053e7	0.0866522
$351$	0	0
$352$	−1.67652e9	−2.04884
$353$	9.20539e8	1.11386	0.556930	−	0.830559i	$-0.311979\pi$
$353$	9.20539e8	1.11386	0.556930	+	0.830559i	$0.311979\pi$
$354$	0	0
$355$	−3.78830e7	−0.0449413
$356$	−4.33464e8	−0.509188
$357$	0	0
$358$	3.58459e8	0.412904
$359$	1.05612e9	1.20471	0.602353	−	0.798230i	$-0.294230\pi$
$359$	1.05612e9	1.20471	0.602353	+	0.798230i	$0.294230\pi$
$360$	0	0
$361$	1.15792e9	1.29540
$362$	5.21638e8	0.577948
$363$	0	0
$364$	−2.97497e7	−0.0323317
$365$	7.97534e8	0.858468
$366$	0	0
$367$	−1.34008e9	−1.41514	−0.707571	−	0.706643i	$-0.750209\pi$
$367$	−1.34008e9	−1.41514	−0.707571	+	0.706643i	$0.750209\pi$
$368$	−1.07534e8	−0.112481
$369$	0	0
$370$	7.47647e7	0.0767345
$371$	1.22377e9	1.24420
$372$	0	0
$373$	−1.17492e9	−1.17227	−0.586134	−	0.810214i	$-0.699351\pi$
$373$	−1.17492e9	−1.17227	−0.586134	+	0.810214i	$0.699351\pi$
$374$	9.58510e8	0.947428
$375$	0	0
$376$	−5.05293e8	−0.490214
$377$	−8.93351e7	−0.0858673
$378$	0	0
$379$	−1.77122e9	−1.67123	−0.835613	−	0.549319i	$-0.814887\pi$
$379$	−1.77122e9	−1.67123	−0.835613	+	0.549319i	$0.814887\pi$
$380$	−5.12445e8	−0.479076
$381$	0	0
$382$	6.72331e7	0.0617108
$383$	1.89701e9	1.72534	0.862670	−	0.505767i	$-0.168791\pi$
$383$	1.89701e9	1.72534	0.862670	+	0.505767i	$0.168791\pi$
$384$	0	0
$385$	−7.92789e8	−0.708019
$386$	3.19290e7	0.0282573
$387$	0	0
$388$	8.94722e8	0.777638
$389$	−1.55533e9	−1.33967	−0.669836	−	0.742509i	$-0.733636\pi$
$389$	−1.55533e9	−1.33967	−0.669836	+	0.742509i	$0.733636\pi$
$390$	0	0
$391$	5.68463e8	0.480932
$392$	3.95784e8	0.331861
$393$	0	0
$394$	−5.12959e8	−0.422519
$395$	5.49477e8	0.448600
$396$	0	0
$397$	1.47275e9	1.18131	0.590653	−	0.806926i	$-0.298870\pi$
$397$	1.47275e9	1.18131	0.590653	+	0.806926i	$0.298870\pi$
$398$	4.93129e6	0.00392075
$399$	0	0
$400$	5.29946e7	0.0414020
$401$	−1.28588e9	−0.995851	−0.497926	−	0.867220i	$-0.665905\pi$
$401$	−1.28588e9	−0.995851	−0.497926	+	0.867220i	$0.665905\pi$
$402$	0	0
$403$	6.68909e7	0.0509095
$404$	−1.58816e9	−1.19828
$405$	0	0
$406$	−8.78245e8	−0.651290
$407$	−8.52778e8	−0.626983
$408$	0	0
$409$	1.02432e9	0.740293	0.370146	−	0.928973i	$-0.379308\pi$
$409$	1.02432e9	0.740293	0.370146	+	0.928973i	$0.379308\pi$
$410$	−1.73717e8	−0.124480
$411$	0	0
$412$	−3.77208e8	−0.265730
$413$	−7.67681e8	−0.536235
$414$	0	0
$415$	−7.32771e8	−0.503268
$416$	8.68907e7	0.0591761
$417$	0	0
$418$	−2.42156e9	−1.62173
$419$	−9.07692e8	−0.602823	−0.301411	−	0.953494i	$-0.597458\pi$
$419$	−9.07692e8	−0.602823	−0.301411	+	0.953494i	$0.597458\pi$
$420$	0	0
$421$	−1.86524e8	−0.121828	−0.0609140	−	0.998143i	$-0.519402\pi$
$421$	−1.86524e8	−0.121828	−0.0609140	+	0.998143i	$0.519402\pi$
$422$	−3.00511e8	−0.194656
$423$	0	0
$424$	−2.25393e9	−1.43602
$425$	−2.80148e8	−0.177022
$426$	0	0
$427$	−9.01535e8	−0.560383
$428$	−3.89265e8	−0.239989
$429$	0	0
$430$	2.72142e8	0.165066
$431$	1.07539e9	0.646986	0.323493	−	0.946231i	$-0.395143\pi$
$431$	1.07539e9	0.646986	0.323493	+	0.946231i	$0.395143\pi$
$432$	0	0
$433$	−1.01325e9	−0.599804	−0.299902	−	0.953970i	$-0.596954\pi$
$433$	−1.01325e9	−0.599804	−0.299902	+	0.953970i	$0.596954\pi$
$434$	6.57598e8	0.386141
$435$	0	0
$436$	−1.82669e9	−1.05551
$437$	−1.43615e9	−0.823219
$438$	0	0
$439$	9.03879e8	0.509899	0.254950	−	0.966954i	$-0.417941\pi$
$439$	9.03879e8	0.509899	0.254950	+	0.966954i	$0.417941\pi$
$440$	1.46015e9	0.817174
$441$	0	0
$442$	−4.96778e7	−0.0273643
$443$	−6.59044e8	−0.360165	−0.180082	−	0.983652i	$-0.557636\pi$
$443$	−6.59044e8	−0.360165	−0.180082	+	0.983652i	$0.557636\pi$
$444$	0	0
$445$	5.98677e8	0.322057
$446$	−9.05964e8	−0.483547
$447$	0	0
$448$	5.38837e8	0.283129
$449$	7.44129e8	0.387959	0.193980	−	0.981006i	$-0.437860\pi$
$449$	7.44129e8	0.387959	0.193980	+	0.981006i	$0.437860\pi$
$450$	0	0
$451$	1.98145e9	1.01710
$452$	−2.11850e8	−0.107906
$453$	0	0
$454$	1.91580e7	0.00960848
$455$	4.10887e7	0.0204495
$456$	0	0
$457$	6.79576e8	0.333067	0.166533	−	0.986036i	$-0.446743\pi$
$457$	6.79576e8	0.333067	0.166533	+	0.986036i	$0.446743\pi$
$458$	5.01174e8	0.243758
$459$	0	0
$460$	3.58686e8	0.171815
$461$	1.55945e9	0.741341	0.370671	−	0.928764i	$-0.379128\pi$
$461$	1.55945e9	0.741341	0.370671	+	0.928764i	$0.379128\pi$
$462$	0	0
$463$	−1.82436e9	−0.854237	−0.427118	−	0.904196i	$-0.640471\pi$
$463$	−1.82436e9	−0.854237	−0.427118	+	0.904196i	$0.640471\pi$
$464$	−6.69621e8	−0.311183
$465$	0	0
$466$	−1.63804e9	−0.749849
$467$	1.27437e9	0.579013	0.289506	−	0.957176i	$-0.406509\pi$
$467$	1.27437e9	0.579013	0.289506	+	0.957176i	$0.406509\pi$
$468$	0	0
$469$	−2.44759e9	−1.09556
$470$	2.89063e8	0.128425
$471$	0	0
$472$	1.41391e9	0.618906
$473$	−3.10410e9	−1.34872
$474$	0	0
$475$	7.07762e8	0.303012
$476$	1.17882e9	0.500983
$477$	0	0
$478$	1.21262e9	0.507839
$479$	−2.59252e9	−1.07782	−0.538912	−	0.842362i	$-0.681165\pi$
$479$	−2.59252e9	−1.07782	−0.538912	+	0.842362i	$0.681165\pi$
$480$	0	0
$481$	4.41979e7	0.0181090
$482$	2.60361e9	1.05904
$483$	0	0
$484$	−5.13472e9	−2.05853
$485$	−1.23574e9	−0.491849
$486$	0	0
$487$	1.27925e9	0.501883	0.250942	−	0.968002i	$-0.419260\pi$
$487$	1.27925e9	0.501883	0.250942	+	0.968002i	$0.419260\pi$
$488$	1.66044e9	0.646777
$489$	0	0
$490$	−2.26416e8	−0.0869403
$491$	−2.61255e8	−0.0996048	−0.0498024	−	0.998759i	$-0.515859\pi$
$491$	−2.61255e8	−0.0996048	−0.0498024	+	0.998759i	$0.515859\pi$
$492$	0	0
$493$	3.53986e9	1.33052
$494$	1.25505e8	0.0468400
$495$	0	0
$496$	5.01388e8	0.184496
$497$	−2.20162e8	−0.0804444
$498$	0	0
$499$	−5.37559e9	−1.93675	−0.968376	−	0.249495i	$-0.919735\pi$
$499$	−5.37559e9	−1.93675	−0.968376	+	0.249495i	$0.919735\pi$
$500$	−1.76767e8	−0.0632420
$501$	0	0
$502$	1.72959e9	0.610211
$503$	1.52082e9	0.532831	0.266416	−	0.963858i	$-0.414161\pi$
$503$	1.52082e9	0.532831	0.266416	+	0.963858i	$0.414161\pi$
$504$	0	0
$505$	2.19348e9	0.757905
$506$	1.69497e9	0.581614
$507$	0	0
$508$	5.64778e8	0.191141
$509$	1.82883e9	0.614697	0.307349	−	0.951597i	$-0.400558\pi$
$509$	1.82883e9	0.614697	0.307349	+	0.951597i	$0.400558\pi$
$510$	0	0
$511$	4.63498e9	1.53665
$512$	1.23216e9	0.405715
$513$	0	0
$514$	1.53995e9	0.500189
$515$	5.20979e8	0.168072
$516$	0	0
$517$	−3.29710e9	−1.04934
$518$	4.34505e8	0.137354
$519$	0	0
$520$	−7.56771e7	−0.0236022
$521$	−1.68457e9	−0.521864	−0.260932	−	0.965357i	$-0.584030\pi$
$521$	−1.68457e9	−0.521864	−0.260932	+	0.965357i	$0.584030\pi$
$522$	0	0
$523$	−1.32473e9	−0.404921	−0.202460	−	0.979290i	$-0.564894\pi$
$523$	−1.32473e9	−0.404921	−0.202460	+	0.979290i	$0.564894\pi$
$524$	−1.88642e9	−0.572769
$525$	0	0
$526$	−3.45852e8	−0.103619
$527$	−2.65052e9	−0.788848
$528$	0	0
$529$	−2.39959e9	−0.704762
$530$	1.28940e9	0.376204
$531$	0	0
$532$	−2.97814e9	−0.857541
$533$	−1.02695e8	−0.0293767
$534$	0	0
$535$	5.37631e8	0.151791
$536$	4.50797e9	1.26446
$537$	0	0
$538$	−3.71939e8	−0.102976
$539$	2.58254e9	0.710373
$540$	0	0
$541$	1.32550e9	0.359905	0.179953	−	0.983675i	$-0.442406\pi$
$541$	1.32550e9	0.359905	0.179953	+	0.983675i	$0.442406\pi$
$542$	1.07945e9	0.291208
$543$	0	0
$544$	−3.44300e9	−0.916940
$545$	2.52292e9	0.667599
$546$	0	0
$547$	−3.68225e9	−0.961960	−0.480980	−	0.876731i	$-0.659719\pi$
$547$	−3.68225e9	−0.961960	−0.480980	+	0.876731i	$0.659719\pi$
$548$	−2.91790e9	−0.757423
$549$	0	0
$550$	−8.35311e8	−0.214081
$551$	−8.94303e9	−2.27748
$552$	0	0
$553$	3.19336e9	0.802990
$554$	−2.30665e9	−0.576364
$555$	0	0
$556$	−9.56701e8	−0.236056
$557$	−4.00634e9	−0.982323	−0.491162	−	0.871069i	$-0.663428\pi$
$557$	−4.00634e9	−0.982323	−0.491162	+	0.871069i	$0.663428\pi$
$558$	0	0
$559$	1.60880e8	0.0389547
$560$	3.07985e8	0.0741091
$561$	0	0
$562$	−1.97516e9	−0.469380
$563$	−3.48943e9	−0.824090	−0.412045	−	0.911164i	$-0.635185\pi$
$563$	−3.48943e9	−0.824090	−0.412045	+	0.911164i	$0.635185\pi$
$564$	0	0
$565$	2.92596e8	0.0682494
$566$	−2.67425e9	−0.619932
$567$	0	0
$568$	4.05494e8	0.0928465
$569$	−3.06362e9	−0.697175	−0.348588	−	0.937276i	$-0.613339\pi$
$569$	−3.06362e9	−0.697175	−0.348588	+	0.937276i	$0.613339\pi$
$570$	0	0
$571$	−5.90584e7	−0.0132756	−0.00663782	−	0.999978i	$-0.502113\pi$
$571$	−5.90584e7	−0.0132756	−0.00663782	+	0.999978i	$0.502113\pi$
$572$	3.57531e8	0.0798781
$573$	0	0
$574$	−1.00958e9	−0.222818
$575$	−4.95397e8	−0.108672
$576$	0	0
$577$	−8.98217e9	−1.94655	−0.973276	−	0.229640i	$-0.926245\pi$
$577$	−8.98217e9	−1.94655	−0.973276	+	0.229640i	$0.926245\pi$
$578$	−5.44192e8	−0.117221
$579$	0	0
$580$	2.23356e9	0.475336
$581$	−4.25860e9	−0.900845
$582$	0	0
$583$	−1.47072e10	−3.07389
$584$	−8.53668e9	−1.77355
$585$	0	0
$586$	−4.01642e9	−0.824512
$587$	4.80145e9	0.979804	0.489902	−	0.871777i	$-0.337032\pi$
$587$	4.80145e9	0.979804	0.489902	+	0.871777i	$0.337032\pi$
$588$	0	0
$589$	6.69622e9	1.35029
$590$	−8.08857e8	−0.162140
$591$	0	0
$592$	3.31290e8	0.0656270
$593$	−1.99199e8	−0.0392280	−0.0196140	−	0.999808i	$-0.506244\pi$
$593$	−1.99199e8	−0.0392280	−0.0196140	+	0.999808i	$0.506244\pi$
$594$	0	0
$595$	−1.62812e9	−0.316867
$596$	−4.22528e9	−0.817512
$597$	0	0
$598$	−8.78471e7	−0.0167986
$599$	−6.90150e9	−1.31205	−0.656024	−	0.754740i	$-0.727763\pi$
$599$	−6.90150e9	−1.31205	−0.656024	+	0.754740i	$0.727763\pi$
$600$	0	0
$601$	7.76192e9	1.45851	0.729253	−	0.684244i	$-0.239868\pi$
$601$	7.76192e9	1.45851	0.729253	+	0.684244i	$0.239868\pi$
$602$	1.58159e9	0.295465
$603$	0	0
$604$	2.07098e9	0.382426
$605$	7.09180e9	1.30201
$606$	0	0
$607$	3.86567e9	0.701559	0.350779	−	0.936458i	$-0.385917\pi$
$607$	3.86567e9	0.701559	0.350779	+	0.936458i	$0.385917\pi$
$608$	8.69832e9	1.56954
$609$	0	0
$610$	−9.49890e8	−0.169441
$611$	1.70883e8	0.0303077
$612$	0	0
$613$	−6.89667e8	−0.120928	−0.0604641	−	0.998170i	$-0.519258\pi$
$613$	−6.89667e8	−0.120928	−0.0604641	+	0.998170i	$0.519258\pi$
$614$	8.94575e8	0.155965
$615$	0	0
$616$	8.48589e9	1.46273
$617$	−1.07235e9	−0.183797	−0.0918986	−	0.995768i	$-0.529294\pi$
$617$	−1.07235e9	−0.183797	−0.0918986	+	0.995768i	$0.529294\pi$
$618$	0	0
$619$	9.09074e9	1.54057	0.770286	−	0.637699i	$-0.220114\pi$
$619$	9.09074e9	1.54057	0.770286	+	0.637699i	$0.220114\pi$
$620$	−1.67241e9	−0.281820
$621$	0	0
$622$	1.40521e9	0.234139
$623$	3.47929e9	0.576478
$624$	0	0
$625$	2.44141e8	0.0400000
$626$	9.90285e8	0.161343
$627$	0	0
$628$	−5.74421e9	−0.925490
$629$	−1.75132e9	−0.280600
$630$	0	0
$631$	−2.57012e9	−0.407240	−0.203620	−	0.979050i	$-0.565271\pi$
$631$	−2.57012e9	−0.407240	−0.203620	+	0.979050i	$0.565271\pi$
$632$	−5.88152e9	−0.926786
$633$	0	0
$634$	3.61541e9	0.563437
$635$	−7.80041e8	−0.120895
$636$	0	0
$637$	−1.33848e8	−0.0205175
$638$	1.05547e10	1.60907
$639$	0	0
$640$	−2.50474e9	−0.377687
$641$	5.47924e9	0.821708	0.410854	−	0.911701i	$-0.365231\pi$
$641$	5.47924e9	0.821708	0.410854	+	0.911701i	$0.365231\pi$
$642$	0	0
$643$	7.02618e9	1.04227	0.521136	−	0.853474i	$-0.325509\pi$
$643$	7.02618e9	1.04227	0.521136	+	0.853474i	$0.325509\pi$
$644$	2.08455e9	0.307547
$645$	0	0
$646$	−4.97307e9	−0.725790
$647$	−1.34234e9	−0.194849	−0.0974246	−	0.995243i	$-0.531060\pi$
$647$	−1.34234e9	−0.194849	−0.0974246	+	0.995243i	$0.531060\pi$
$648$	0	0
$649$	9.22595e9	1.32481
$650$	4.32926e7	0.00618325
$651$	0	0
$652$	−6.58450e9	−0.930371
$653$	6.07751e9	0.854141	0.427071	−	0.904218i	$-0.359546\pi$
$653$	6.07751e9	0.854141	0.427071	+	0.904218i	$0.359546\pi$
$654$	0	0
$655$	2.60542e9	0.362271
$656$	−7.69760e8	−0.106461
$657$	0	0
$658$	1.67993e9	0.229880
$659$	1.07205e10	1.45920	0.729599	−	0.683875i	$-0.239707\pi$
$659$	1.07205e10	1.45920	0.729599	+	0.683875i	$0.239707\pi$
$660$	0	0
$661$	4.32257e8	0.0582153	0.0291076	−	0.999576i	$-0.490733\pi$
$661$	4.32257e8	0.0582153	0.0291076	+	0.999576i	$0.490733\pi$
$662$	3.05651e9	0.409470
$663$	0	0
$664$	7.84346e9	1.03973
$665$	4.11325e9	0.542388
$666$	0	0
$667$	6.25967e9	0.816791
$668$	−3.14442e9	−0.408153
$669$	0	0
$670$	−2.57888e9	−0.331260
$671$	1.08346e10	1.38447
$672$	0	0
$673$	1.04864e10	1.32609	0.663046	−	0.748578i	$-0.269263\pi$
$673$	1.04864e10	1.32609	0.663046	+	0.748578i	$0.269263\pi$
$674$	−3.26847e9	−0.411183
$675$	0	0
$676$	5.66050e9	0.704760
$677$	1.22653e10	1.51921	0.759605	−	0.650385i	$-0.225392\pi$
$677$	1.22653e10	1.51921	0.759605	+	0.650385i	$0.225392\pi$
$678$	0	0
$679$	−7.18168e9	−0.880404
$680$	2.99867e9	0.365718
$681$	0	0
$682$	−7.90298e9	−0.953994
$683$	−1.03494e10	−1.24292	−0.621459	−	0.783446i	$-0.713460\pi$
$683$	−1.03494e10	−1.24292	−0.621459	+	0.783446i	$0.713460\pi$
$684$	0	0
$685$	4.03005e9	0.479064
$686$	−4.97925e9	−0.588883
$687$	0	0
$688$	1.20589e9	0.141172
$689$	7.62245e8	0.0887824
$690$	0	0
$691$	3.23844e9	0.373390	0.186695	−	0.982418i	$-0.440222\pi$
$691$	3.23844e9	0.373390	0.186695	+	0.982418i	$0.440222\pi$
$692$	−6.07895e8	−0.0697361
$693$	0	0
$694$	4.85265e9	0.551088
$695$	1.32134e9	0.149303
$696$	0	0
$697$	4.06923e9	0.455195
$698$	−7.74763e9	−0.862333
$699$	0	0
$700$	−1.02730e9	−0.113202
$701$	4.36859e9	0.478992	0.239496	−	0.970897i	$-0.423018\pi$
$701$	4.36859e9	0.478992	0.239496	+	0.970897i	$0.423018\pi$
$702$	0	0
$703$	4.42450e9	0.480309
$704$	−6.47571e9	−0.699493
$705$	0	0
$706$	5.63679e9	0.602858
$707$	1.27477e10	1.35664
$708$	0	0
$709$	1.08067e10	1.13876	0.569381	−	0.822074i	$-0.307183\pi$
$709$	1.08067e10	1.13876	0.569381	+	0.822074i	$0.307183\pi$
$710$	−2.31971e8	−0.0243237
$711$	0	0
$712$	−6.40814e9	−0.665353
$713$	−4.68701e9	−0.484265
$714$	0	0
$715$	−4.93803e8	−0.0505222
$716$	−5.29811e9	−0.539418
$717$	0	0
$718$	6.46697e9	0.652027
$719$	−2.73109e9	−0.274021	−0.137011	−	0.990570i	$-0.543749\pi$
$719$	−2.73109e9	−0.274021	−0.137011	+	0.990570i	$0.543749\pi$
$720$	0	0
$721$	3.02774e9	0.300847
$722$	7.09037e9	0.701114
$723$	0	0
$724$	−7.70992e9	−0.755032
$725$	−3.08488e9	−0.300646
$726$	0	0
$727$	7.05493e9	0.680961	0.340480	−	0.940252i	$-0.389410\pi$
$727$	7.05493e9	0.680961	0.340480	+	0.940252i	$0.389410\pi$
$728$	−4.39808e8	−0.0422477
$729$	0	0
$730$	4.88358e9	0.464631
$731$	−6.37478e9	−0.603607
$732$	0	0
$733$	1.59257e10	1.49360	0.746802	−	0.665047i	$-0.231588\pi$
$733$	1.59257e10	1.49360	0.746802	+	0.665047i	$0.231588\pi$
$734$	−8.20579e9	−0.765921
$735$	0	0
$736$	−6.08839e9	−0.562898
$737$	2.94151e10	2.70666
$738$	0	0
$739$	1.57729e10	1.43766	0.718829	−	0.695187i	$-0.244678\pi$
$739$	1.57729e10	1.43766	0.718829	+	0.695187i	$0.244678\pi$
$740$	−1.10504e9	−0.100246
$741$	0	0
$742$	7.49355e9	0.673401
$743$	−5.56895e9	−0.498095	−0.249048	−	0.968491i	$-0.580118\pi$
$743$	−5.56895e9	−0.498095	−0.249048	+	0.968491i	$0.580118\pi$
$744$	0	0
$745$	5.83574e9	0.517069
$746$	−7.19445e9	−0.634471
$747$	0	0
$748$	−1.41670e10	−1.23772
$749$	3.12452e9	0.271704
$750$	0	0
$751$	1.25384e9	0.108019	0.0540097	−	0.998540i	$-0.482800\pi$
$751$	1.25384e9	0.108019	0.0540097	+	0.998540i	$0.482800\pi$
$752$	1.28087e9	0.109835
$753$	0	0
$754$	−5.47031e8	−0.0464742
$755$	−2.86033e9	−0.241881
$756$	0	0
$757$	1.00239e8	0.00839850	0.00419925	−	0.999991i	$-0.498663\pi$
$757$	1.00239e8	0.00839850	0.00419925	+	0.999991i	$0.498663\pi$
$758$	−1.08458e10	−0.904523
$759$	0	0
$760$	−7.57577e9	−0.626007
$761$	−1.73316e10	−1.42558	−0.712792	−	0.701376i	$-0.752569\pi$
$761$	−1.73316e10	−1.42558	−0.712792	+	0.701376i	$0.752569\pi$
$762$	0	0
$763$	1.46623e10	1.19499
$764$	−9.93720e8	−0.0806189
$765$	0	0
$766$	1.16161e10	0.933811
$767$	−4.78164e8	−0.0382642
$768$	0	0
$769$	−4.59975e9	−0.364747	−0.182374	−	0.983229i	$-0.558378\pi$
$769$	−4.59975e9	−0.364747	−0.182374	+	0.983229i	$0.558378\pi$
$770$	−4.85453e9	−0.383203
$771$	0	0
$772$	−4.71918e8	−0.0369153
$773$	1.76165e10	1.37180	0.685901	−	0.727695i	$-0.259408\pi$
$773$	1.76165e10	1.37180	0.685901	+	0.727695i	$0.259408\pi$
$774$	0	0
$775$	2.30984e9	0.178249
$776$	1.32272e10	1.01614
$777$	0	0
$778$	−9.52383e9	−0.725075
$779$	−1.02804e10	−0.779165
$780$	0	0
$781$	2.64590e9	0.198744
$782$	3.48090e9	0.260296
$783$	0	0
$784$	−1.00327e9	−0.0743555
$785$	7.93360e9	0.585365
$786$	0	0
$787$	1.03787e10	0.758979	0.379489	−	0.925196i	$-0.376100\pi$
$787$	1.03787e10	0.758979	0.379489	+	0.925196i	$0.376100\pi$
$788$	7.58165e9	0.551979
$789$	0	0
$790$	3.36464e9	0.242797
$791$	1.70046e9	0.122166
$792$	0	0
$793$	−5.61537e8	−0.0399873
$794$	9.01818e9	0.639362
$795$	0	0
$796$	−7.28856e7	−0.00512207
$797$	1.87790e10	1.31392	0.656960	−	0.753925i	$-0.271842\pi$
$797$	1.87790e10	1.31392	0.656960	+	0.753925i	$0.271842\pi$
$798$	0	0
$799$	−6.77113e9	−0.469621
$800$	3.00047e9	0.207192
$801$	0	0
$802$	−7.87389e9	−0.538988
$803$	−5.57029e10	−3.79641
$804$	0	0
$805$	−2.87907e9	−0.194521
$806$	4.09596e8	0.0275539
$807$	0	0
$808$	−2.34787e10	−1.56579
$809$	−1.78937e10	−1.18818	−0.594088	−	0.804400i	$-0.702487\pi$
$809$	−1.78937e10	−1.18818	−0.594088	+	0.804400i	$0.702487\pi$
$810$	0	0
$811$	7.84603e9	0.516508	0.258254	−	0.966077i	$-0.416853\pi$
$811$	7.84603e9	0.516508	0.258254	+	0.966077i	$0.416853\pi$
$812$	1.29807e10	0.850845
$813$	0	0
$814$	−5.22186e9	−0.339344
$815$	9.09416e9	0.588452
$816$	0	0
$817$	1.61051e10	1.03320
$818$	6.27226e9	0.400671
$819$	0	0
$820$	2.56758e9	0.162621
$821$	−1.39659e10	−0.880784	−0.440392	−	0.897806i	$-0.645161\pi$
$821$	−1.39659e10	−0.880784	−0.440392	+	0.897806i	$0.645161\pi$
$822$	0	0
$823$	1.98420e9	0.124076	0.0620378	−	0.998074i	$-0.480240\pi$
$823$	1.98420e9	0.124076	0.0620378	+	0.998074i	$0.480240\pi$
$824$	−5.57648e9	−0.347228
$825$	0	0
$826$	−4.70078e9	−0.290228
$827$	2.14836e10	1.32080	0.660402	−	0.750912i	$-0.270386\pi$
$827$	2.14836e10	1.32080	0.660402	+	0.750912i	$0.270386\pi$
$828$	0	0
$829$	−2.80722e10	−1.71134	−0.855670	−	0.517523i	$-0.826854\pi$
$829$	−2.80722e10	−1.71134	−0.855670	+	0.517523i	$0.826854\pi$
$830$	−4.48701e9	−0.272386
$831$	0	0
$832$	3.35624e8	0.0202033
$833$	5.30367e9	0.317921
$834$	0	0
$835$	4.34291e9	0.258153
$836$	3.57912e10	2.11863
$837$	0	0
$838$	−5.55812e9	−0.326268
$839$	1.44740e10	0.846098	0.423049	−	0.906107i	$-0.360960\pi$
$839$	1.44740e10	0.846098	0.423049	+	0.906107i	$0.360960\pi$
$840$	0	0
$841$	2.17296e10	1.25969
$842$	−1.14215e9	−0.0659374
$843$	0	0
$844$	4.44162e9	0.254298
$845$	−7.81797e9	−0.445754
$846$	0	0
$847$	4.12150e10	2.33058
$848$	5.71349e9	0.321748
$849$	0	0
$850$	−1.71545e9	−0.0958101
$851$	−3.09693e9	−0.172257
$852$	0	0
$853$	2.48277e9	0.136966	0.0684832	−	0.997652i	$-0.478184\pi$
$853$	2.48277e9	0.136966	0.0684832	+	0.997652i	$0.478184\pi$
$854$	−5.52042e9	−0.303298
$855$	0	0
$856$	−5.75472e9	−0.313593
$857$	−6.25581e9	−0.339509	−0.169754	−	0.985486i	$-0.554297\pi$
$857$	−6.25581e9	−0.339509	−0.169754	+	0.985486i	$0.554297\pi$
$858$	0	0
$859$	−2.45116e10	−1.31946	−0.659730	−	0.751503i	$-0.729329\pi$
$859$	−2.45116e10	−1.31946	−0.659730	+	0.751503i	$0.729329\pi$
$860$	−4.02232e9	−0.215642
$861$	0	0
$862$	6.58498e9	0.350170
$863$	−1.70869e10	−0.904954	−0.452477	−	0.891776i	$-0.649460\pi$
$863$	−1.70869e10	−0.904954	−0.452477	+	0.891776i	$0.649460\pi$
$864$	0	0
$865$	8.39592e8	0.0441075
$866$	−6.20449e9	−0.324634
$867$	0	0
$868$	−9.71944e9	−0.504455
$869$	−3.83777e10	−1.98385
$870$	0	0
$871$	−1.52453e9	−0.0781757
$872$	−2.70049e10	−1.37923
$873$	0	0
$874$	−8.79407e9	−0.445554
$875$	1.41886e9	0.0715996
$876$	0	0
$877$	−2.18320e10	−1.09294	−0.546469	−	0.837479i	$-0.684029\pi$
$877$	−2.18320e10	−1.09294	−0.546469	+	0.837479i	$0.684029\pi$
$878$	5.53477e9	0.275974
$879$	0	0
$880$	−3.70135e9	−0.183093
$881$	−2.86992e10	−1.41401	−0.707007	−	0.707207i	$-0.749955\pi$
$881$	−2.86992e10	−1.41401	−0.707007	+	0.707207i	$0.749955\pi$
$882$	0	0
$883$	2.42249e10	1.18413	0.592065	−	0.805890i	$-0.298313\pi$
$883$	2.42249e10	1.18413	0.592065	+	0.805890i	$0.298313\pi$
$884$	7.34249e8	0.0357487
$885$	0	0
$886$	−4.03556e9	−0.194933
$887$	1.98794e10	0.956467	0.478233	−	0.878233i	$-0.341277\pi$
$887$	1.98794e10	0.956467	0.478233	+	0.878233i	$0.341277\pi$
$888$	0	0
$889$	−4.53332e9	−0.216401
$890$	3.66591e9	0.174308
$891$	0	0
$892$	1.33903e10	0.631706
$893$	1.71065e10	0.803859
$894$	0	0
$895$	7.31747e9	0.341177
$896$	−1.45566e10	−0.676056
$897$	0	0
$898$	4.55657e9	0.209976
$899$	−2.91864e10	−1.33974
$900$	0	0
$901$	−3.02036e10	−1.37569
$902$	1.21331e10	0.550490
$903$	0	0
$904$	−3.13190e9	−0.141000
$905$	1.06485e10	0.477551
$906$	0	0
$907$	−2.58368e10	−1.14977	−0.574887	−	0.818233i	$-0.694954\pi$
$907$	−2.58368e10	−1.14977	−0.574887	+	0.818233i	$0.694954\pi$
$908$	−2.83160e8	−0.0125525
$909$	0	0
$910$	2.51601e8	0.0110680
$911$	3.28878e10	1.44119	0.720594	−	0.693358i	$-0.243869\pi$
$911$	3.28878e10	1.44119	0.720594	+	0.693358i	$0.243869\pi$
$912$	0	0
$913$	5.11796e10	2.22561
$914$	4.16128e9	0.180267
$915$	0	0
$916$	−7.40746e9	−0.318446
$917$	1.51418e10	0.648462
$918$	0	0
$919$	2.78641e10	1.18424	0.592122	−	0.805849i	$-0.298291\pi$
$919$	2.78641e10	1.18424	0.592122	+	0.805849i	$0.298291\pi$
$920$	5.30266e9	0.224510
$921$	0	0
$922$	9.54906e9	0.401238
$923$	−1.37132e8	−0.00574028
$924$	0	0
$925$	1.52622e9	0.0634046
$926$	−1.11712e10	−0.462341
$927$	0	0
$928$	−3.79129e10	−1.55729
$929$	−2.46897e10	−1.01032	−0.505161	−	0.863025i	$-0.668567\pi$
$929$	−2.46897e10	−1.01032	−0.505161	+	0.863025i	$0.668567\pi$
$930$	0	0
$931$	−1.33991e10	−0.544191
$932$	2.42106e10	0.979602
$933$	0	0
$934$	7.80345e9	0.313381
$935$	1.95667e10	0.782847
$936$	0	0
$937$	−2.07461e10	−0.823851	−0.411926	−	0.911217i	$-0.635144\pi$
$937$	−2.07461e10	−0.823851	−0.411926	+	0.911217i	$0.635144\pi$
$938$	−1.49875e10	−0.592951
$939$	0	0
$940$	−4.27242e9	−0.167775
$941$	4.26569e10	1.66888	0.834441	−	0.551097i	$-0.185791\pi$
$941$	4.26569e10	1.66888	0.834441	+	0.551097i	$0.185791\pi$
$942$	0	0
$943$	7.19577e9	0.279439
$944$	−3.58413e9	−0.138670
$945$	0	0
$946$	−1.90075e10	−0.729972
$947$	−1.21136e10	−0.463497	−0.231748	−	0.972776i	$-0.574445\pi$
$947$	−1.21136e10	−0.463497	−0.231748	+	0.972776i	$0.574445\pi$
$948$	0	0
$949$	2.88698e9	0.109651
$950$	4.33387e9	0.164000
$951$	0	0
$952$	1.74272e10	0.654632
$953$	−5.08378e10	−1.90266	−0.951331	−	0.308170i	$-0.900283\pi$
$953$	−5.08378e10	−1.90266	−0.951331	+	0.308170i	$0.900283\pi$
$954$	0	0
$955$	1.37247e9	0.0509908
$956$	−1.79227e10	−0.663440
$957$	0	0
$958$	−1.58749e10	−0.583354
$959$	2.34212e10	0.857519
$960$	0	0
$961$	−5.65892e9	−0.205685
$962$	2.70639e8	0.00980117
$963$	0	0
$964$	−3.84819e10	−1.38352
$965$	6.51788e8	0.0233486
$966$	0	0
$967$	3.45096e10	1.22729	0.613644	−	0.789583i	$-0.289703\pi$
$967$	3.45096e10	1.22729	0.613644	+	0.789583i	$0.289703\pi$
$968$	−7.59095e10	−2.68988
$969$	0	0
$970$	−7.56689e9	−0.266205
$971$	−3.51565e9	−0.123236	−0.0616181	−	0.998100i	$-0.519626\pi$
$971$	−3.51565e9	−0.123236	−0.0616181	+	0.998100i	$0.519626\pi$
$972$	0	0
$973$	7.67917e9	0.267251
$974$	7.83328e9	0.271636
$975$	0	0
$976$	−4.20906e9	−0.144914
$977$	−5.12073e10	−1.75671	−0.878357	−	0.478005i	$-0.841360\pi$
$977$	−5.12073e10	−1.75671	−0.878357	+	0.478005i	$0.841360\pi$
$978$	0	0
$979$	−4.18140e10	−1.42424
$980$	3.34648e9	0.113579
$981$	0	0
$982$	−1.59976e9	−0.0539094
$983$	4.75142e10	1.59546	0.797730	−	0.603015i	$-0.206034\pi$
$983$	4.75142e10	1.59546	0.797730	+	0.603015i	$0.206034\pi$
$984$	0	0
$985$	−1.04714e10	−0.349122
$986$	2.16758e10	0.720122
$987$	0	0
$988$	−1.85499e9	−0.0611917
$989$	−1.12728e10	−0.370547
$990$	0	0
$991$	9.03493e9	0.294895	0.147447	−	0.989070i	$-0.452894\pi$
$991$	9.03493e9	0.294895	0.147447	+	0.989070i	$0.452894\pi$
$992$	2.83878e10	0.923294
$993$	0	0
$994$	−1.34813e9	−0.0435392
$995$	1.00666e8	0.00323967
$996$	0	0
$997$	−5.28444e10	−1.68875	−0.844376	−	0.535751i	$-0.820029\pi$
$997$	−5.28444e10	−1.68875	−0.844376	+	0.535751i	$0.820029\pi$
$998$	−3.29166e10	−1.04823
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.8.a.j.1.10		15
3.2	odd	2		405.8.a.i.1.6		15
9.2	odd	6		45.8.e.b.31.10	yes	30
9.4	even	3		135.8.e.b.46.6		30
9.5	odd	6		45.8.e.b.16.10	✓	30
9.7	even	3		135.8.e.b.91.6		30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.8.e.b.16.10	✓	30	9.5	odd	6
45.8.e.b.31.10	yes	30	9.2	odd	6
135.8.e.b.46.6		30	9.4	even	3
135.8.e.b.91.6		30	9.7	even	3
405.8.a.i.1.6		15	3.2	odd	2
405.8.a.j.1.10		15	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:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	405.a (trivial)

\(f(q)\)	\(=\)	\(q+6.12335 q^{2} -90.5045 q^{4} +125.000 q^{5} +726.455 q^{7} -1337.98 q^{8} +O(q^{10})\)	\(q+6.12335 q^{2} -90.5045 q^{4} +125.000 q^{5} +726.455 q^{7} -1337.98 q^{8} +765.419 q^{10} -8730.50 q^{11} +452.485 q^{13} +4448.34 q^{14} +3391.65 q^{16} -17929.5 q^{17} +45296.7 q^{19} -11313.1 q^{20} -53459.9 q^{22} -31705.4 q^{23} +15625.0 q^{25} +2770.73 q^{26} -65747.4 q^{28} -197432. q^{29} +147830. q^{31} +192030. q^{32} -109789. q^{34} +90806.8 q^{35} +97678.1 q^{37} +277368. q^{38} -167248. q^{40} -226957. q^{41} +355547. q^{43} +790150. q^{44} -194144. q^{46} +377653. q^{47} -295807. q^{49} +95677.4 q^{50} -40952.0 q^{52} +1.68457e6 q^{53} -1.09131e6 q^{55} -971982. q^{56} -1.20895e6 q^{58} -1.05675e6 q^{59} -1.24101e6 q^{61} +905215. q^{62} +741735. q^{64} +56560.7 q^{65} -3.36923e6 q^{67} +1.62270e6 q^{68} +556042. q^{70} -303064. q^{71} +6.38027e6 q^{73} +598117. q^{74} -4.09956e6 q^{76} -6.34231e6 q^{77} +4.39582e6 q^{79} +423957. q^{80} -1.38974e6 q^{82} -5.86217e6 q^{83} -2.24119e6 q^{85} +2.17714e6 q^{86} +1.16812e7 q^{88} +4.78941e6 q^{89} +328710. q^{91} +2.86949e6 q^{92} +2.31250e6 q^{94} +5.66209e6 q^{95} -9.88594e6 q^{97} -1.81133e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 8 q^{2} + 1088 q^{4} + 1875 q^{5} + 1289 q^{7} + 4434 q^{8}+O(q^{10}) \) 15 * q + 8 * q^2 + 1088 * q^4 + 1875 * q^5 + 1289 * q^7 + 4434 * q^8	\( 15 q + 8 q^{2} + 1088 q^{4} + 1875 q^{5} + 1289 q^{7} + 4434 q^{8} + 1000 q^{10} + 1658 q^{11} + 9874 q^{13} + 7695 q^{14} + 62612 q^{16} - 3598 q^{17} + 21376 q^{19} + 136000 q^{20} - 52519 q^{22} + 96441 q^{23} + 234375 q^{25} + 126146 q^{26} + 174449 q^{28} + 259297 q^{29} + 373568 q^{31} + 1134550 q^{32} + 423851 q^{34} + 161125 q^{35} + 517872 q^{37} + 690059 q^{38} + 554250 q^{40} + 520501 q^{41} + 1898836 q^{43} + 1277707 q^{44} + 3154677 q^{46} + 2259041 q^{47} + 4316308 q^{49} + 125000 q^{50} + 5398554 q^{52} - 102274 q^{53} + 207250 q^{55} - 504621 q^{56} + 3190987 q^{58} - 1680874 q^{59} - 1066457 q^{61} - 274110 q^{62} + 6541980 q^{64} + 1234250 q^{65} + 6522389 q^{67} + 1420717 q^{68} + 961875 q^{70} - 32786 q^{71} + 5359102 q^{73} - 4045556 q^{74} + 4649241 q^{76} + 2586078 q^{77} + 9319346 q^{79} + 7826500 q^{80} + 7460620 q^{82} - 12758277 q^{83} - 449750 q^{85} - 20044675 q^{86} + 6691143 q^{88} - 18776241 q^{89} + 9244102 q^{91} - 13862829 q^{92} + 25905119 q^{94} + 2672000 q^{95} + 2788224 q^{97} - 1679531 q^{98}+O(q^{100}) \) 15 * q + 8 * q^2 + 1088 * q^4 + 1875 * q^5 + 1289 * q^7 + 4434 * q^8 + 1000 * q^10 + 1658 * q^11 + 9874 * q^13 + 7695 * q^14 + 62612 * q^16 - 3598 * q^17 + 21376 * q^19 + 136000 * q^20 - 52519 * q^22 + 96441 * q^23 + 234375 * q^25 + 126146 * q^26 + 174449 * q^28 + 259297 * q^29 + 373568 * q^31 + 1134550 * q^32 + 423851 * q^34 + 161125 * q^35 + 517872 * q^37 + 690059 * q^38 + 554250 * q^40 + 520501 * q^41 + 1898836 * q^43 + 1277707 * q^44 + 3154677 * q^46 + 2259041 * q^47 + 4316308 * q^49 + 125000 * q^50 + 5398554 * q^52 - 102274 * q^53 + 207250 * q^55 - 504621 * q^56 + 3190987 * q^58 - 1680874 * q^59 - 1066457 * q^61 - 274110 * q^62 + 6541980 * q^64 + 1234250 * q^65 + 6522389 * q^67 + 1420717 * q^68 + 961875 * q^70 - 32786 * q^71 + 5359102 * q^73 - 4045556 * q^74 + 4649241 * q^76 + 2586078 * q^77 + 9319346 * q^79 + 7826500 * q^80 + 7460620 * q^82 - 12758277 * q^83 - 449750 * q^85 - 20044675 * q^86 + 6691143 * q^88 - 18776241 * q^89 + 9244102 * q^91 - 13862829 * q^92 + 25905119 * q^94 + 2672000 * q^95 + 2788224 * q^97 - 1679531 * q^98

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