LMFDB - Embedded newform 405.8.a.j.1.5

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			1.5
Root			$13.6120$ 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)

$f(q)$	$=$	$q-12.6120 q^{2} +31.0613 q^{4} +125.000 q^{5} +1431.12 q^{7} +1222.59 q^{8} +O(q^{10})$	\(q-12.6120 q^{2} +31.0613 q^{4} +125.000 q^{5} +1431.12 q^{7} +1222.59 q^{8} -1576.49 q^{10} -6971.24 q^{11} +1073.61 q^{13} -18049.2 q^{14} -19395.0 q^{16} -33422.4 q^{17} -5977.86 q^{19} +3882.67 q^{20} +87921.0 q^{22} +79760.1 q^{23} +15625.0 q^{25} -13540.4 q^{26} +44452.5 q^{28} +208029. q^{29} +117587. q^{31} +88118.4 q^{32} +421521. q^{34} +178890. q^{35} -216973. q^{37} +75392.4 q^{38} +152823. q^{40} +337598. q^{41} +176702. q^{43} -216536. q^{44} -1.00593e6 q^{46} -1.16825e6 q^{47} +1.22456e6 q^{49} -197062. q^{50} +33347.9 q^{52} -1.37587e6 q^{53} -871405. q^{55} +1.74967e6 q^{56} -2.62365e6 q^{58} -1.63977e6 q^{59} +1.29789e6 q^{61} -1.48300e6 q^{62} +1.37122e6 q^{64} +134202. q^{65} +2.83724e6 q^{67} -1.03814e6 q^{68} -2.25615e6 q^{70} +137496. q^{71} -1.27242e6 q^{73} +2.73645e6 q^{74} -185680. q^{76} -9.97668e6 q^{77} +1.77628e6 q^{79} -2.42438e6 q^{80} -4.25776e6 q^{82} +4.30071e6 q^{83} -4.17780e6 q^{85} -2.22855e6 q^{86} -8.52294e6 q^{88} -310100. q^{89} +1.53647e6 q^{91} +2.47745e6 q^{92} +1.47339e7 q^{94} -747232. q^{95} +1.03719e7 q^{97} -1.54441e7 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−12.6120	−1.11475	−0.557375	−	0.830261i	$-0.688191\pi$
$2$	−12.6120	−1.11475	−0.557375	+	0.830261i	$0.688191\pi$
$3$	0	0
$3$	0	0
$4$	31.0613	0.242667
$5$	125.000	0.447214
$5$	125.000	0.447214
$6$	0	0
$7$	1431.12	1.57700	0.788502	−	0.615032i	$-0.210857\pi$
$7$	1431.12	1.57700	0.788502	+	0.615032i	$0.210857\pi$
$8$	1222.59	0.844237
$9$	0	0
$10$	−1576.49	−0.498531
$11$	−6971.24	−1.57919	−0.789597	−	0.613625i	$-0.789711\pi$
$11$	−6971.24	−1.57919	−0.789597	+	0.613625i	$0.789711\pi$
$12$	0	0
$13$	1073.61	0.135533	0.0677667	−	0.997701i	$-0.478413\pi$
$13$	1073.61	0.135533	0.0677667	+	0.997701i	$0.478413\pi$
$14$	−18049.2	−1.75796
$15$	0	0
$16$	−19395.0	−1.18378
$17$	−33422.4	−1.64993	−0.824966	−	0.565183i	$-0.808806\pi$
$17$	−33422.4	−1.64993	−0.824966	+	0.565183i	$0.808806\pi$
$18$	0	0
$19$	−5977.86	−0.199944	−0.0999719	−	0.994990i	$-0.531875\pi$
$19$	−5977.86	−0.199944	−0.0999719	+	0.994990i	$0.531875\pi$
$20$	3882.67	0.108524
$21$	0	0
$22$	87921.0	1.76041
$23$	79760.1	1.36690	0.683452	−	0.729995i	$-0.260478\pi$
$23$	79760.1	1.36690	0.683452	+	0.729995i	$0.260478\pi$
$24$	0	0
$25$	15625.0	0.200000
$26$	−13540.4	−0.151086
$27$	0	0
$28$	44452.5	0.382686
$29$	208029.	1.58391	0.791956	−	0.610578i	$-0.209063\pi$
$29$	208029.	1.58391	0.791956	+	0.610578i	$0.209063\pi$
$30$	0	0
$31$	117587.	0.708915	0.354458	−	0.935072i	$-0.384666\pi$
$31$	117587.	0.708915	0.354458	+	0.935072i	$0.384666\pi$
$32$	88118.4	0.475381
$33$	0	0
$34$	421521.	1.83926
$35$	178890.	0.705258
$36$	0	0
$37$	−216973.	−0.704204	−0.352102	−	0.935962i	$-0.614533\pi$
$37$	−216973.	−0.704204	−0.352102	+	0.935962i	$0.614533\pi$
$38$	75392.4	0.222887
$39$	0	0
$40$	152823.	0.377554
$41$	337598.	0.764990	0.382495	−	0.923958i	$-0.375065\pi$
$41$	337598.	0.764990	0.382495	+	0.923958i	$0.375065\pi$
$42$	0	0
$43$	176702.	0.338923	0.169461	−	0.985537i	$-0.445797\pi$
$43$	176702.	0.338923	0.169461	+	0.985537i	$0.445797\pi$
$44$	−216536.	−0.383218
$45$	0	0
$46$	−1.00593e6	−1.52376
$47$	−1.16825e6	−1.64132	−0.820662	−	0.571415i	$-0.806395\pi$
$47$	−1.16825e6	−1.64132	−0.820662	+	0.571415i	$0.806395\pi$
$48$	0	0
$49$	1.22456e6	1.48694
$50$	−197062.	−0.222950
$51$	0	0
$52$	33347.9	0.0328894
$53$	−1.37587e6	−1.26944	−0.634719	−	0.772743i	$-0.718884\pi$
$53$	−1.37587e6	−1.26944	−0.634719	+	0.772743i	$0.718884\pi$
$54$	0	0
$55$	−871405.	−0.706237
$56$	1.74967e6	1.33137
$57$	0	0
$58$	−2.62365e6	−1.76567
$59$	−1.63977e6	−1.03944	−0.519721	−	0.854336i	$-0.673964\pi$
$59$	−1.63977e6	−1.03944	−0.519721	+	0.854336i	$0.673964\pi$
$60$	0	0
$61$	1.29789e6	0.732121	0.366060	−	0.930591i	$-0.380706\pi$
$61$	1.29789e6	0.732121	0.366060	+	0.930591i	$0.380706\pi$
$62$	−1.48300e6	−0.790263
$63$	0	0
$64$	1.37122e6	0.653849
$65$	134202.	0.0606124
$66$	0	0
$67$	2.83724e6	1.15248	0.576240	−	0.817280i	$-0.304519\pi$
$67$	2.83724e6	1.15248	0.576240	+	0.817280i	$0.304519\pi$
$68$	−1.03814e6	−0.400383
$69$	0	0
$70$	−2.25615e6	−0.786186
$71$	137496.	0.0455915	0.0227958	−	0.999740i	$-0.492743\pi$
$71$	137496.	0.0455915	0.0227958	+	0.999740i	$0.492743\pi$
$72$	0	0
$73$	−1.27242e6	−0.382825	−0.191412	−	0.981510i	$-0.561307\pi$
$73$	−1.27242e6	−0.382825	−0.191412	+	0.981510i	$0.561307\pi$
$74$	2.73645e6	0.785011
$75$	0	0
$76$	−185680.	−0.0485197
$77$	−9.97668e6	−2.49040
$78$	0	0
$79$	1.77628e6	0.405338	0.202669	−	0.979247i	$-0.435039\pi$
$79$	1.77628e6	0.405338	0.202669	+	0.979247i	$0.435039\pi$
$80$	−2.42438e6	−0.529402
$81$	0	0
$82$	−4.25776e6	−0.852772
$83$	4.30071e6	0.825595	0.412797	−	0.910823i	$-0.364552\pi$
$83$	4.30071e6	0.825595	0.412797	+	0.910823i	$0.364552\pi$
$84$	0	0
$85$	−4.17780e6	−0.737872
$86$	−2.22855e6	−0.377814
$87$	0	0
$88$	−8.52294e6	−1.33321
$89$	−310100.	−0.0466270	−0.0233135	−	0.999728i	$-0.507422\pi$
$89$	−310100.	−0.0466270	−0.0233135	+	0.999728i	$0.507422\pi$
$90$	0	0
$91$	1.53647e6	0.213737
$92$	2.47745e6	0.331702
$93$	0	0
$94$	1.47339e7	1.82966
$95$	−747232.	−0.0894176
$96$	0	0
$97$	1.03719e7	1.15387	0.576936	−	0.816789i	$-0.304248\pi$
$97$	1.03719e7	1.15387	0.576936	+	0.816789i	$0.304248\pi$
$98$	−1.54441e7	−1.65757
$99$	0	0
$100$	485333.	0.0485333
$101$	−1.28730e7	−1.24324	−0.621621	−	0.783318i	$-0.713526\pi$
$101$	−1.28730e7	−1.24324	−0.621621	+	0.783318i	$0.713526\pi$
$102$	0	0
$103$	1.32810e7	1.19757	0.598784	−	0.800910i	$-0.295651\pi$
$103$	1.32810e7	1.19757	0.598784	+	0.800910i	$0.295651\pi$
$104$	1.31259e6	0.114422
$105$	0	0
$106$	1.73524e7	1.41510
$107$	1.19778e6	0.0945224	0.0472612	−	0.998883i	$-0.484951\pi$
$107$	1.19778e6	0.0945224	0.0472612	+	0.998883i	$0.484951\pi$
$108$	0	0
$109$	1.00980e7	0.746862	0.373431	−	0.927658i	$-0.378181\pi$
$109$	1.00980e7	0.746862	0.373431	+	0.927658i	$0.378181\pi$
$110$	1.09901e7	0.787278
$111$	0	0
$112$	−2.77566e7	−1.86682
$113$	1.41709e7	0.923898	0.461949	−	0.886907i	$-0.347150\pi$
$113$	1.41709e7	0.923898	0.461949	+	0.886907i	$0.347150\pi$
$114$	0	0
$115$	9.97001e6	0.611298
$116$	6.46166e6	0.384363
$117$	0	0
$118$	2.06807e7	1.15872
$119$	−4.78314e7	−2.60195
$120$	0	0
$121$	2.91110e7	1.49386
$122$	−1.63689e7	−0.816132
$123$	0	0
$124$	3.65241e6	0.172030
$125$	1.95312e6	0.0894427
$126$	0	0
$127$	−2.40565e7	−1.04212	−0.521062	−	0.853519i	$-0.674464\pi$
$127$	−2.40565e7	−1.04212	−0.521062	+	0.853519i	$0.674464\pi$
$128$	−2.85729e7	−1.20426
$129$	0	0
$130$	−1.69255e6	−0.0675677
$131$	3.34444e6	0.129979	0.0649895	−	0.997886i	$-0.479299\pi$
$131$	3.34444e6	0.129979	0.0649895	+	0.997886i	$0.479299\pi$
$132$	0	0
$133$	−8.55503e6	−0.315312
$134$	−3.57831e7	−1.28473
$135$	0	0
$136$	−4.08617e7	−1.39293
$137$	−2.44031e7	−0.810817	−0.405409	−	0.914136i	$-0.632871\pi$
$137$	−2.44031e7	−0.810817	−0.405409	+	0.914136i	$0.632871\pi$
$138$	0	0
$139$	−1.57132e7	−0.496265	−0.248132	−	0.968726i	$-0.579817\pi$
$139$	−1.57132e7	−0.496265	−0.248132	+	0.968726i	$0.579817\pi$
$140$	5.55656e6	0.171142
$141$	0	0
$142$	−1.73409e6	−0.0508232
$143$	−7.48442e6	−0.214034
$144$	0	0
$145$	2.60036e7	0.708347
$146$	1.60477e7	0.426754
$147$	0	0
$148$	−6.73945e6	−0.170887
$149$	3.14497e7	0.778868	0.389434	−	0.921054i	$-0.372671\pi$
$149$	3.14497e7	0.778868	0.389434	+	0.921054i	$0.372671\pi$
$150$	0	0
$151$	−4.00563e7	−0.946784	−0.473392	−	0.880852i	$-0.656971\pi$
$151$	−4.00563e7	−0.946784	−0.473392	+	0.880852i	$0.656971\pi$
$152$	−7.30844e6	−0.168800
$153$	0	0
$154$	1.25825e8	2.77617
$155$	1.46984e7	0.317036
$156$	0	0
$157$	1.67196e7	0.344809	0.172404	−	0.985026i	$-0.444846\pi$
$157$	1.67196e7	0.344809	0.172404	+	0.985026i	$0.444846\pi$
$158$	−2.24024e7	−0.451850
$159$	0	0
$160$	1.10148e7	0.212597
$161$	1.14146e8	2.15561
$162$	0	0
$163$	4.79060e7	0.866429	0.433215	−	0.901291i	$-0.357379\pi$
$163$	4.79060e7	0.866429	0.433215	+	0.901291i	$0.357379\pi$
$164$	1.04862e7	0.185638
$165$	0	0
$166$	−5.42404e7	−0.920332
$167$	−1.01156e8	−1.68067	−0.840336	−	0.542065i	$-0.817643\pi$
$167$	−1.01156e8	−1.68067	−0.840336	+	0.542065i	$0.817643\pi$
$168$	0	0
$169$	−6.15959e7	−0.981631
$170$	5.26902e7	0.822542
$171$	0	0
$172$	5.48858e6	0.0822452
$173$	2.69669e7	0.395977	0.197989	−	0.980204i	$-0.436559\pi$
$173$	2.69669e7	0.395977	0.197989	+	0.980204i	$0.436559\pi$
$174$	0	0
$175$	2.23612e7	0.315401
$176$	1.35208e8	1.86942
$177$	0	0
$178$	3.91097e6	0.0519774
$179$	−5.11783e6	−0.0666961	−0.0333481	−	0.999444i	$-0.510617\pi$
$179$	−5.11783e6	−0.0666961	−0.0333481	+	0.999444i	$0.510617\pi$
$180$	0	0
$181$	1.25537e8	1.57361	0.786804	−	0.617203i	$-0.211734\pi$
$181$	1.25537e8	1.57361	0.786804	+	0.617203i	$0.211734\pi$
$182$	−1.93779e7	−0.238263
$183$	0	0
$184$	9.75136e7	1.15399
$185$	−2.71216e7	−0.314930
$186$	0	0
$187$	2.32995e8	2.60556
$188$	−3.62875e7	−0.398294
$189$	0	0
$190$	9.42405e6	0.0996782
$191$	4.66169e7	0.484090	0.242045	−	0.970265i	$-0.422182\pi$
$191$	4.66169e7	0.484090	0.242045	+	0.970265i	$0.422182\pi$
$192$	0	0
$193$	−8.53747e6	−0.0854828	−0.0427414	−	0.999086i	$-0.513609\pi$
$193$	−8.53747e6	−0.0854828	−0.0427414	+	0.999086i	$0.513609\pi$
$194$	−1.30810e8	−1.28628
$195$	0	0
$196$	3.80365e7	0.360831
$197$	1.08071e8	1.00711	0.503555	−	0.863963i	$-0.332025\pi$
$197$	1.08071e8	1.00711	0.503555	+	0.863963i	$0.332025\pi$
$198$	0	0
$199$	1.88254e7	0.169340	0.0846699	−	0.996409i	$-0.473016\pi$
$199$	1.88254e7	0.169340	0.0846699	+	0.996409i	$0.473016\pi$
$200$	1.91029e7	0.168847
$201$	0	0
$202$	1.62354e8	1.38590
$203$	2.97715e8	2.49784
$204$	0	0
$205$	4.21997e7	0.342114
$206$	−1.67499e8	−1.33499
$207$	0	0
$208$	−2.08228e7	−0.160442
$209$	4.16731e7	0.315750
$210$	0	0
$211$	4.24310e7	0.310953	0.155477	−	0.987840i	$-0.450309\pi$
$211$	4.24310e7	0.310953	0.155477	+	0.987840i	$0.450309\pi$
$212$	−4.27363e7	−0.308050
$213$	0	0
$214$	−1.51064e7	−0.105369
$215$	2.20877e7	0.151571
$216$	0	0
$217$	1.68281e8	1.11796
$218$	−1.27355e8	−0.832564
$219$	0	0
$220$	−2.70670e7	−0.171380
$221$	−3.58827e7	−0.223621
$222$	0	0
$223$	1.87281e8	1.13091	0.565453	−	0.824781i	$-0.308701\pi$
$223$	1.87281e8	1.13091	0.565453	+	0.824781i	$0.308701\pi$
$224$	1.26108e8	0.749677
$225$	0	0
$226$	−1.78723e8	−1.02991
$227$	1.47364e8	0.836181	0.418091	−	0.908405i	$-0.362699\pi$
$227$	1.47364e8	0.836181	0.418091	+	0.908405i	$0.362699\pi$
$228$	0	0
$229$	−1.30904e8	−0.720328	−0.360164	−	0.932889i	$-0.617279\pi$
$229$	−1.30904e8	−0.720328	−0.360164	+	0.932889i	$0.617279\pi$
$230$	−1.25741e8	−0.681445
$231$	0	0
$232$	2.54333e8	1.33720
$233$	8.28507e7	0.429092	0.214546	−	0.976714i	$-0.431173\pi$
$233$	8.28507e7	0.429092	0.214546	+	0.976714i	$0.431173\pi$
$234$	0	0
$235$	−1.46032e8	−0.734022
$236$	−5.09334e7	−0.252238
$237$	0	0
$238$	6.03247e8	2.90052
$239$	1.27999e8	0.606478	0.303239	−	0.952914i	$-0.401932\pi$
$239$	1.27999e8	0.606478	0.303239	+	0.952914i	$0.401932\pi$
$240$	0	0
$241$	−2.31548e7	−0.106557	−0.0532784	−	0.998580i	$-0.516967\pi$
$241$	−2.31548e7	−0.106557	−0.0532784	+	0.998580i	$0.516967\pi$
$242$	−3.67147e8	−1.66528
$243$	0	0
$244$	4.03141e7	0.177661
$245$	1.53070e8	0.664980
$246$	0	0
$247$	−6.41791e6	−0.0270991
$248$	1.43760e8	0.598492
$249$	0	0
$250$	−2.46327e7	−0.0997062
$251$	−1.34624e8	−0.537358	−0.268679	−	0.963230i	$-0.586587\pi$
$251$	−1.34624e8	−0.537358	−0.268679	+	0.963230i	$0.586587\pi$
$252$	0	0
$253$	−5.56027e8	−2.15861
$254$	3.03399e8	1.16171
$255$	0	0
$256$	1.84844e8	0.688598
$257$	3.51419e7	0.129140	0.0645698	−	0.997913i	$-0.479432\pi$
$257$	3.51419e7	0.129140	0.0645698	+	0.997913i	$0.479432\pi$
$258$	0	0
$259$	−3.10514e8	−1.11053
$260$	4.16849e6	0.0147086
$261$	0	0
$262$	−4.21799e7	−0.144894
$263$	4.47688e7	0.151751	0.0758753	−	0.997117i	$-0.475825\pi$
$263$	4.47688e7	0.151751	0.0758753	+	0.997117i	$0.475825\pi$
$264$	0	0
$265$	−1.71983e8	−0.567710
$266$	1.07896e8	0.351494
$267$	0	0
$268$	8.81283e7	0.279669
$269$	1.46369e8	0.458475	0.229238	−	0.973370i	$-0.426377\pi$
$269$	1.46369e8	0.458475	0.229238	+	0.973370i	$0.426377\pi$
$270$	0	0
$271$	9.72043e7	0.296683	0.148342	−	0.988936i	$-0.452607\pi$
$271$	9.72043e7	0.296683	0.148342	+	0.988936i	$0.452607\pi$
$272$	6.48228e8	1.95316
$273$	0	0
$274$	3.07771e8	0.903858
$275$	−1.08926e8	−0.315839
$276$	0	0
$277$	2.12829e8	0.601662	0.300831	−	0.953677i	$-0.402736\pi$
$277$	2.12829e8	0.601662	0.300831	+	0.953677i	$0.402736\pi$
$278$	1.98174e8	0.553211
$279$	0	0
$280$	2.18708e8	0.595405
$281$	1.27812e8	0.343637	0.171818	−	0.985129i	$-0.445036\pi$
$281$	1.27812e8	0.343637	0.171818	+	0.985129i	$0.445036\pi$
$282$	0	0
$283$	−3.06655e8	−0.804263	−0.402131	−	0.915582i	$-0.631730\pi$
$283$	−3.06655e8	−0.804263	−0.402131	+	0.915582i	$0.631730\pi$
$284$	4.27079e6	0.0110635
$285$	0	0
$286$	9.43932e7	0.238594
$287$	4.83142e8	1.20639
$288$	0	0
$289$	7.06716e8	1.72227
$290$	−3.27957e8	−0.789630
$291$	0	0
$292$	−3.95230e7	−0.0928987
$293$	−3.40751e8	−0.791408	−0.395704	−	0.918378i	$-0.629499\pi$
$293$	−3.40751e8	−0.791408	−0.395704	+	0.918378i	$0.629499\pi$
$294$	0	0
$295$	−2.04971e8	−0.464853
$296$	−2.65268e8	−0.594515
$297$	0	0
$298$	−3.96642e8	−0.868243
$299$	8.56316e7	0.185261
$300$	0	0
$301$	2.52881e8	0.534483
$302$	5.05188e8	1.05543
$303$	0	0
$304$	1.15941e8	0.236689
$305$	1.62236e8	0.327414
$306$	0	0
$307$	4.37777e8	0.863513	0.431756	−	0.901990i	$-0.357894\pi$
$307$	4.37777e8	0.863513	0.431756	+	0.901990i	$0.357894\pi$
$308$	−3.09889e8	−0.604336
$309$	0	0
$310$	−1.85376e8	−0.353416
$311$	−3.28386e8	−0.619047	−0.309524	−	0.950892i	$-0.600170\pi$
$311$	−3.28386e8	−0.619047	−0.309524	+	0.950892i	$0.600170\pi$
$312$	0	0
$313$	6.45482e8	1.18981	0.594907	−	0.803795i	$-0.297189\pi$
$313$	6.45482e8	1.18981	0.594907	+	0.803795i	$0.297189\pi$
$314$	−2.10867e8	−0.384375
$315$	0	0
$316$	5.51737e7	0.0983619
$317$	−8.36660e8	−1.47517	−0.737584	−	0.675255i	$-0.764034\pi$
$317$	−8.36660e8	−1.47517	−0.737584	+	0.675255i	$0.764034\pi$
$318$	0	0
$319$	−1.45022e9	−2.50131
$320$	1.71403e8	0.292410
$321$	0	0
$322$	−1.43961e9	−2.40297
$323$	1.99794e8	0.329894
$324$	0	0
$325$	1.67752e7	0.0271067
$326$	−6.04188e8	−0.965851
$327$	0	0
$328$	4.12742e8	0.645833
$329$	−1.67191e9	−2.58837
$330$	0	0
$331$	4.31471e8	0.653964	0.326982	−	0.945031i	$-0.393968\pi$
$331$	4.31471e8	0.653964	0.326982	+	0.945031i	$0.393968\pi$
$332$	1.33586e8	0.200344
$333$	0	0
$334$	1.27577e9	1.87353
$335$	3.54655e8	0.515405
$336$	0	0
$337$	8.98413e8	1.27871	0.639354	−	0.768913i	$-0.279202\pi$
$337$	8.98413e8	1.27871	0.639354	+	0.768913i	$0.279202\pi$
$338$	7.76844e8	1.09427
$339$	0	0
$340$	−1.29768e8	−0.179057
$341$	−8.19729e8	−1.11951
$342$	0	0
$343$	5.73904e8	0.767909
$344$	2.16033e8	0.286131
$345$	0	0
$346$	−3.40106e8	−0.441415
$347$	7.65439e8	0.983461	0.491731	−	0.870747i	$-0.336364\pi$
$347$	7.65439e8	0.983461	0.491731	+	0.870747i	$0.336364\pi$
$348$	0	0
$349$	1.41434e9	1.78100	0.890500	−	0.454984i	$-0.150355\pi$
$349$	1.41434e9	1.78100	0.890500	+	0.454984i	$0.150355\pi$
$350$	−2.82019e8	−0.351593
$351$	0	0
$352$	−6.14294e8	−0.750719
$353$	−4.74414e8	−0.574045	−0.287022	−	0.957924i	$-0.592665\pi$
$353$	−4.74414e8	−0.574045	−0.287022	+	0.957924i	$0.592665\pi$
$354$	0	0
$355$	1.71869e7	0.0203892
$356$	−9.63213e6	−0.0113148
$357$	0	0
$358$	6.45459e7	0.0743495
$359$	1.27893e9	1.45887	0.729433	−	0.684053i	$-0.239784\pi$
$359$	1.27893e9	1.45887	0.729433	+	0.684053i	$0.239784\pi$
$360$	0	0
$361$	−8.58137e8	−0.960022
$362$	−1.58327e9	−1.75418
$363$	0	0
$364$	4.77248e7	0.0518668
$365$	−1.59052e8	−0.171204
$366$	0	0
$367$	−1.94029e8	−0.204897	−0.102448	−	0.994738i	$-0.532668\pi$
$367$	−1.94029e8	−0.204897	−0.102448	+	0.994738i	$0.532668\pi$
$368$	−1.54695e9	−1.61811
$369$	0	0
$370$	3.42056e8	0.351068
$371$	−1.96903e9	−2.00191
$372$	0	0
$373$	7.85140e8	0.783369	0.391685	−	0.920100i	$-0.371892\pi$
$373$	7.85140e8	0.783369	0.391685	+	0.920100i	$0.371892\pi$
$374$	−2.93853e9	−2.90455
$375$	0	0
$376$	−1.42829e9	−1.38567
$377$	2.23343e8	0.214673
$378$	0	0
$379$	−1.25080e8	−0.118018	−0.0590092	−	0.998257i	$-0.518794\pi$
$379$	−1.25080e8	−0.118018	−0.0590092	+	0.998257i	$0.518794\pi$
$380$	−2.32100e7	−0.0216987
$381$	0	0
$382$	−5.87929e8	−0.539639
$383$	1.30841e9	1.19000	0.595000	−	0.803726i	$-0.297152\pi$
$383$	1.30841e9	1.19000	0.595000	+	0.803726i	$0.297152\pi$
$384$	0	0
$385$	−1.24709e9	−1.11374
$386$	1.07674e8	0.0952919
$387$	0	0
$388$	3.22165e8	0.280006
$389$	1.66130e9	1.43095	0.715476	−	0.698637i	$-0.246210\pi$
$389$	1.66130e9	1.43095	0.715476	+	0.698637i	$0.246210\pi$
$390$	0	0
$391$	−2.66577e9	−2.25530
$392$	1.49713e9	1.25533
$393$	0	0
$394$	−1.36298e9	−1.12267
$395$	2.22035e8	0.181272
$396$	0	0
$397$	1.12109e9	0.899239	0.449619	−	0.893220i	$-0.351560\pi$
$397$	1.12109e9	0.899239	0.449619	+	0.893220i	$0.351560\pi$
$398$	−2.37425e8	−0.188772
$399$	0	0
$400$	−3.03048e8	−0.236756
$401$	6.21873e8	0.481611	0.240805	−	0.970573i	$-0.422588\pi$
$401$	6.21873e8	0.481611	0.240805	+	0.970573i	$0.422588\pi$
$402$	0	0
$403$	1.26243e8	0.0960817
$404$	−3.99853e8	−0.301693
$405$	0	0
$406$	−3.75476e9	−2.78446
$407$	1.51257e9	1.11208
$408$	0	0
$409$	5.20000e8	0.375813	0.187907	−	0.982187i	$-0.439830\pi$
$409$	5.20000e8	0.375813	0.187907	+	0.982187i	$0.439830\pi$
$410$	−5.32221e8	−0.381371
$411$	0	0
$412$	4.12525e8	0.290610
$413$	−2.34670e9	−1.63920
$414$	0	0
$415$	5.37589e8	0.369217
$416$	9.46051e7	0.0644300
$417$	0	0
$418$	−5.25579e8	−0.351982
$419$	1.89386e9	1.25776	0.628882	−	0.777501i	$-0.283513\pi$
$419$	1.89386e9	1.25776	0.628882	+	0.777501i	$0.283513\pi$
$420$	0	0
$421$	−1.01376e9	−0.662138	−0.331069	−	0.943606i	$-0.607409\pi$
$421$	−1.01376e9	−0.662138	−0.331069	+	0.943606i	$0.607409\pi$
$422$	−5.35138e8	−0.346635
$423$	0	0
$424$	−1.68212e9	−1.07171
$425$	−5.22224e8	−0.329986
$426$	0	0
$427$	1.85743e9	1.15456
$428$	3.72047e7	0.0229374
$429$	0	0
$430$	−2.78569e8	−0.168964
$431$	2.47784e9	1.49074	0.745372	−	0.666649i	$-0.232272\pi$
$431$	2.47784e9	1.49074	0.745372	+	0.666649i	$0.232272\pi$
$432$	0	0
$433$	−2.67876e9	−1.58572	−0.792858	−	0.609406i	$-0.791408\pi$
$433$	−2.67876e9	−1.58572	−0.792858	+	0.609406i	$0.791408\pi$
$434$	−2.12236e9	−1.24625
$435$	0	0
$436$	3.13656e8	0.181239
$437$	−4.76794e8	−0.273304
$438$	0	0
$439$	−5.47128e8	−0.308648	−0.154324	−	0.988020i	$-0.549320\pi$
$439$	−5.47128e8	−0.308648	−0.154324	+	0.988020i	$0.549320\pi$
$440$	−1.06537e9	−0.596232
$441$	0	0
$442$	4.52551e8	0.249281
$443$	−2.52027e7	−0.0137731	−0.00688657	−	0.999976i	$-0.502192\pi$
$443$	−2.52027e7	−0.0137731	−0.00688657	+	0.999976i	$0.502192\pi$
$444$	0	0
$445$	−3.87625e7	−0.0208522
$446$	−2.36198e9	−1.26068
$447$	0	0
$448$	1.96238e9	1.03112
$449$	1.98471e9	1.03475	0.517375	−	0.855759i	$-0.326909\pi$
$449$	1.98471e9	1.03475	0.517375	+	0.855759i	$0.326909\pi$
$450$	0	0
$451$	−2.35347e9	−1.20807
$452$	4.40168e8	0.224199
$453$	0	0
$454$	−1.85855e9	−0.932133
$455$	1.92059e8	0.0955860
$456$	0	0
$457$	2.56277e9	1.25604	0.628019	−	0.778198i	$-0.283866\pi$
$457$	2.56277e9	1.25604	0.628019	+	0.778198i	$0.283866\pi$
$458$	1.65096e9	0.802985
$459$	0	0
$460$	3.09682e8	0.148342
$461$	2.73737e8	0.130131	0.0650655	−	0.997881i	$-0.479274\pi$
$461$	2.73737e8	0.130131	0.0650655	+	0.997881i	$0.479274\pi$
$462$	0	0
$463$	−1.92450e9	−0.901124	−0.450562	−	0.892745i	$-0.648776\pi$
$463$	−1.92450e9	−0.901124	−0.450562	+	0.892745i	$0.648776\pi$
$464$	−4.03473e9	−1.87500
$465$	0	0
$466$	−1.04491e9	−0.478330
$467$	−2.58760e9	−1.17568	−0.587839	−	0.808978i	$-0.700021\pi$
$467$	−2.58760e9	−1.17568	−0.587839	+	0.808978i	$0.700021\pi$
$468$	0	0
$469$	4.06042e9	1.81747
$470$	1.84174e9	0.818251
$471$	0	0
$472$	−2.00476e9	−0.877535
$473$	−1.23183e9	−0.535225
$474$	0	0
$475$	−9.34040e7	−0.0399888
$476$	−1.48571e9	−0.631406
$477$	0	0
$478$	−1.61432e9	−0.676072
$479$	3.80025e9	1.57993	0.789964	−	0.613153i	$-0.210099\pi$
$479$	3.80025e9	1.57993	0.789964	+	0.613153i	$0.210099\pi$
$480$	0	0
$481$	−2.32945e8	−0.0954432
$482$	2.92027e8	0.118784
$483$	0	0
$484$	9.04227e8	0.362509
$485$	1.29649e9	0.516027
$486$	0	0
$487$	1.83003e9	0.717971	0.358985	−	0.933343i	$-0.383123\pi$
$487$	1.83003e9	0.717971	0.358985	+	0.933343i	$0.383123\pi$
$488$	1.58678e9	0.618084
$489$	0	0
$490$	−1.93051e9	−0.741287
$491$	3.85697e9	1.47049	0.735243	−	0.677803i	$-0.237068\pi$
$491$	3.85697e9	1.47049	0.735243	+	0.677803i	$0.237068\pi$
$492$	0	0
$493$	−6.95283e9	−2.61335
$494$	8.09424e7	0.0302087
$495$	0	0
$496$	−2.28061e9	−0.839199
$497$	1.96773e8	0.0718980
$498$	0	0
$499$	2.01869e9	0.727305	0.363653	−	0.931535i	$-0.381530\pi$
$499$	2.01869e9	0.727305	0.363653	+	0.931535i	$0.381530\pi$
$500$	6.06667e7	0.0217048
$501$	0	0
$502$	1.69787e9	0.599020
$503$	2.07443e9	0.726794	0.363397	−	0.931634i	$-0.381617\pi$
$503$	2.07443e9	0.726794	0.363397	+	0.931634i	$0.381617\pi$
$504$	0	0
$505$	−1.60913e9	−0.555995
$506$	7.01258e9	2.40631
$507$	0	0
$508$	−7.47226e8	−0.252888
$509$	2.01193e9	0.676239	0.338119	−	0.941103i	$-0.390209\pi$
$509$	2.01193e9	0.676239	0.338119	+	0.941103i	$0.390209\pi$
$510$	0	0
$511$	−1.82098e9	−0.603716
$512$	1.32609e9	0.436645
$513$	0	0
$514$	−4.43208e8	−0.143958
$515$	1.66013e9	0.535569
$516$	0	0
$517$	8.14417e9	2.59197
$518$	3.91618e9	1.23797
$519$	0	0
$520$	1.64073e8	0.0511712
$521$	6.62149e8	0.205127	0.102564	−	0.994726i	$-0.467295\pi$
$521$	6.62149e8	0.205127	0.102564	+	0.994726i	$0.467295\pi$
$522$	0	0
$523$	−5.03481e9	−1.53896	−0.769479	−	0.638672i	$-0.779484\pi$
$523$	−5.03481e9	−1.53896	−0.769479	+	0.638672i	$0.779484\pi$
$524$	1.03883e8	0.0315416
$525$	0	0
$526$	−5.64622e8	−0.169164
$527$	−3.93004e9	−1.16966
$528$	0	0
$529$	2.95685e9	0.868428
$530$	2.16905e9	0.632854
$531$	0	0
$532$	−2.65730e8	−0.0765157
$533$	3.62450e8	0.103682
$534$	0	0
$535$	1.49723e8	0.0422717
$536$	3.46877e9	0.972967
$537$	0	0
$538$	−1.84600e9	−0.511085
$539$	−8.53670e9	−2.34817
$540$	0	0
$541$	3.53003e9	0.958491	0.479246	−	0.877681i	$-0.340910\pi$
$541$	3.53003e9	0.958491	0.479246	+	0.877681i	$0.340910\pi$
$542$	−1.22594e9	−0.330727
$543$	0	0
$544$	−2.94512e9	−0.784346
$545$	1.26224e9	0.334007
$546$	0	0
$547$	−8.78242e8	−0.229434	−0.114717	−	0.993398i	$-0.536596\pi$
$547$	−8.78242e8	−0.229434	−0.114717	+	0.993398i	$0.536596\pi$
$548$	−7.57993e8	−0.196758
$549$	0	0
$550$	1.37377e9	0.352081
$551$	−1.24357e9	−0.316693
$552$	0	0
$553$	2.54207e9	0.639219
$554$	−2.68419e9	−0.670702
$555$	0	0
$556$	−4.88073e8	−0.120427
$557$	6.76885e9	1.65967	0.829834	−	0.558010i	$-0.188435\pi$
$557$	6.76885e9	1.65967	0.829834	+	0.558010i	$0.188435\pi$
$558$	0	0
$559$	1.89709e8	0.0459354
$560$	−3.46958e9	−0.834870
$561$	0	0
$562$	−1.61196e9	−0.383069
$563$	−7.48410e9	−1.76750	−0.883752	−	0.467956i	$-0.844991\pi$
$563$	−7.48410e9	−1.76750	−0.883752	+	0.467956i	$0.844991\pi$
$564$	0	0
$565$	1.77137e9	0.413180
$566$	3.86752e9	0.896551
$567$	0	0
$568$	1.68100e8	0.0384901
$569$	−7.88625e9	−1.79464	−0.897321	−	0.441379i	$-0.854490\pi$
$569$	−7.88625e9	−1.79464	−0.897321	+	0.441379i	$0.854490\pi$
$570$	0	0
$571$	−1.60301e9	−0.360338	−0.180169	−	0.983636i	$-0.557665\pi$
$571$	−1.60301e9	−0.360338	−0.180169	+	0.983636i	$0.557665\pi$
$572$	−2.32476e8	−0.0519388
$573$	0	0
$574$	−6.09337e9	−1.34483
$575$	1.24625e9	0.273381
$576$	0	0
$577$	−4.95491e9	−1.07379	−0.536896	−	0.843648i	$-0.680403\pi$
$577$	−4.95491e9	−1.07379	−0.536896	+	0.843648i	$0.680403\pi$
$578$	−8.91306e9	−1.91990
$579$	0	0
$580$	8.07708e8	0.171892
$581$	6.15483e9	1.30197
$582$	0	0
$583$	9.59151e9	2.00469
$584$	−1.55564e9	−0.323195
$585$	0	0
$586$	4.29753e9	0.882221
$587$	5.84547e9	1.19285	0.596426	−	0.802668i	$-0.296587\pi$
$587$	5.84547e9	1.19285	0.596426	+	0.802668i	$0.296587\pi$
$588$	0	0
$589$	−7.02919e8	−0.141743
$590$	2.58508e9	0.518194
$591$	0	0
$592$	4.20819e9	0.833623
$593$	−8.41164e9	−1.65649	−0.828245	−	0.560365i	$-0.810661\pi$
$593$	−8.41164e9	−1.65649	−0.828245	+	0.560365i	$0.810661\pi$
$594$	0	0
$595$	−5.97892e9	−1.16363
$596$	9.76868e8	0.189005
$597$	0	0
$598$	−1.07998e9	−0.206520
$599$	9.31303e9	1.77050	0.885252	−	0.465111i	$-0.153986\pi$
$599$	9.31303e9	1.77050	0.885252	+	0.465111i	$0.153986\pi$
$600$	0	0
$601$	−4.22114e9	−0.793176	−0.396588	−	0.917997i	$-0.629806\pi$
$601$	−4.22114e9	−0.793176	−0.396588	+	0.917997i	$0.629806\pi$
$602$	−3.18932e9	−0.595814
$603$	0	0
$604$	−1.24420e9	−0.229753
$605$	3.63888e9	0.668073
$606$	0	0
$607$	−7.08773e9	−1.28631	−0.643157	−	0.765735i	$-0.722376\pi$
$607$	−7.08773e9	−1.28631	−0.643157	+	0.765735i	$0.722376\pi$
$608$	−5.26759e8	−0.0950494
$609$	0	0
$610$	−2.04611e9	−0.364985
$611$	−1.25425e9	−0.222454
$612$	0	0
$613$	7.93893e9	1.39204	0.696018	−	0.718024i	$-0.254953\pi$
$613$	7.93893e9	1.39204	0.696018	+	0.718024i	$0.254953\pi$
$614$	−5.52122e9	−0.962600
$615$	0	0
$616$	−1.21973e10	−2.10249
$617$	1.03857e10	1.78008	0.890040	−	0.455883i	$-0.150676\pi$
$617$	1.03857e10	1.78008	0.890040	+	0.455883i	$0.150676\pi$
$618$	0	0
$619$	−4.65335e9	−0.788586	−0.394293	−	0.918985i	$-0.629010\pi$
$619$	−4.65335e9	−0.788586	−0.394293	+	0.918985i	$0.629010\pi$
$620$	4.56552e8	0.0769342
$621$	0	0
$622$	4.14159e9	0.690083
$623$	−4.43791e8	−0.0735309
$624$	0	0
$625$	2.44141e8	0.0400000
$626$	−8.14079e9	−1.32634
$627$	0	0
$628$	5.19334e8	0.0836736
$629$	7.25173e9	1.16189
$630$	0	0
$631$	−7.07388e9	−1.12087	−0.560434	−	0.828199i	$-0.689366\pi$
$631$	−7.07388e9	−1.12087	−0.560434	+	0.828199i	$0.689366\pi$
$632$	2.17166e9	0.342201
$633$	0	0
$634$	1.05519e10	1.64444
$635$	−3.00706e9	−0.466052
$636$	0	0
$637$	1.31471e9	0.201530
$638$	1.82901e10	2.78833
$639$	0	0
$640$	−3.57162e9	−0.538561
$641$	−1.19963e9	−0.179905	−0.0899526	−	0.995946i	$-0.528672\pi$
$641$	−1.19963e9	−0.179905	−0.0899526	+	0.995946i	$0.528672\pi$
$642$	0	0
$643$	9.48268e9	1.40667	0.703336	−	0.710858i	$-0.251693\pi$
$643$	9.48268e9	1.40667	0.703336	+	0.710858i	$0.251693\pi$
$644$	3.54553e9	0.523096
$645$	0	0
$646$	−2.51979e9	−0.367749
$647$	1.33646e10	1.93996	0.969979	−	0.243189i	$-0.0781936\pi$
$647$	1.33646e10	1.93996	0.969979	+	0.243189i	$0.0781936\pi$
$648$	0	0
$649$	1.14312e10	1.64148
$650$	−2.11568e8	−0.0302172
$651$	0	0
$652$	1.48802e9	0.210253
$653$	8.29370e9	1.16561	0.582804	−	0.812613i	$-0.301956\pi$
$653$	8.29370e9	1.16561	0.582804	+	0.812613i	$0.301956\pi$
$654$	0	0
$655$	4.18055e8	0.0581284
$656$	−6.54772e9	−0.905580
$657$	0	0
$658$	2.10860e10	2.88539
$659$	−3.21939e9	−0.438203	−0.219101	−	0.975702i	$-0.570313\pi$
$659$	−3.21939e9	−0.438203	−0.219101	+	0.975702i	$0.570313\pi$
$660$	0	0
$661$	1.29884e9	0.174924	0.0874620	−	0.996168i	$-0.472124\pi$
$661$	1.29884e9	0.174924	0.0874620	+	0.996168i	$0.472124\pi$
$662$	−5.44169e9	−0.729006
$663$	0	0
$664$	5.25799e9	0.696998
$665$	−1.06938e9	−0.141012
$666$	0	0
$667$	1.65924e10	2.16506
$668$	−3.14203e9	−0.407843
$669$	0	0
$670$	−4.47289e9	−0.574548
$671$	−9.04789e9	−1.15616
$672$	0	0
$673$	5.18453e9	0.655627	0.327814	−	0.944742i	$-0.393688\pi$
$673$	5.18453e9	0.655627	0.327814	+	0.944742i	$0.393688\pi$
$674$	−1.13307e10	−1.42544
$675$	0	0
$676$	−1.91325e9	−0.238209
$677$	7.51491e9	0.930815	0.465407	−	0.885097i	$-0.345908\pi$
$677$	7.51491e9	0.930815	0.465407	+	0.885097i	$0.345908\pi$
$678$	0	0
$679$	1.48434e10	1.81966
$680$	−5.10771e9	−0.622939
$681$	0	0
$682$	1.03384e10	1.24798
$683$	−1.22009e10	−1.46527	−0.732635	−	0.680622i	$-0.761710\pi$
$683$	−1.22009e10	−1.46527	−0.732635	+	0.680622i	$0.761710\pi$
$684$	0	0
$685$	−3.05039e9	−0.362609
$686$	−7.23804e9	−0.856026
$687$	0	0
$688$	−3.42713e9	−0.401210
$689$	−1.47715e9	−0.172051
$690$	0	0
$691$	−1.03000e10	−1.18758	−0.593792	−	0.804618i	$-0.702370\pi$
$691$	−1.03000e10	−1.18758	−0.593792	+	0.804618i	$0.702370\pi$
$692$	8.37629e8	0.0960904
$693$	0	0
$694$	−9.65368e9	−1.09631
$695$	−1.96415e9	−0.221936
$696$	0	0
$697$	−1.12833e10	−1.26218
$698$	−1.78375e10	−1.98537
$699$	0	0
$700$	6.94570e8	0.0765372
$701$	−2.60923e9	−0.286088	−0.143044	−	0.989716i	$-0.545689\pi$
$701$	−2.60923e9	−0.286088	−0.143044	+	0.989716i	$0.545689\pi$
$702$	0	0
$703$	1.29703e9	0.140801
$704$	−9.55911e9	−1.03256
$705$	0	0
$706$	5.98328e9	0.639916
$707$	−1.84228e10	−1.96060
$708$	0	0
$709$	1.00647e10	1.06057	0.530287	−	0.847818i	$-0.322084\pi$
$709$	1.00647e10	1.06057	0.530287	+	0.847818i	$0.322084\pi$
$710$	−2.16761e8	−0.0227288
$711$	0	0
$712$	−3.79124e8	−0.0393642
$713$	9.37877e9	0.969019
$714$	0	0
$715$	−9.35553e8	−0.0957188
$716$	−1.58967e8	−0.0161849
$717$	0	0
$718$	−1.61298e10	−1.62627
$719$	−5.30945e9	−0.532719	−0.266360	−	0.963874i	$-0.585821\pi$
$719$	−5.30945e9	−0.532719	−0.266360	+	0.963874i	$0.585821\pi$
$720$	0	0
$721$	1.90067e10	1.88857
$722$	1.08228e10	1.07018
$723$	0	0
$724$	3.89934e9	0.381862
$725$	3.25046e9	0.316783
$726$	0	0
$727$	4.38547e9	0.423298	0.211649	−	0.977346i	$-0.432117\pi$
$727$	4.38547e9	0.423298	0.211649	+	0.977346i	$0.432117\pi$
$728$	1.87847e9	0.180445
$729$	0	0
$730$	2.00596e9	0.190850
$731$	−5.90578e9	−0.559199
$732$	0	0
$733$	4.12062e9	0.386454	0.193227	−	0.981154i	$-0.438105\pi$
$733$	4.12062e9	0.386454	0.193227	+	0.981154i	$0.438105\pi$
$734$	2.44708e9	0.228409
$735$	0	0
$736$	7.02833e9	0.649800
$737$	−1.97791e10	−1.81999
$738$	0	0
$739$	−1.08437e10	−0.988375	−0.494187	−	0.869355i	$-0.664534\pi$
$739$	−1.08437e10	−0.988375	−0.494187	+	0.869355i	$0.664534\pi$
$740$	−8.42432e8	−0.0764229
$741$	0	0
$742$	2.48333e10	2.23163
$743$	−7.70759e9	−0.689379	−0.344689	−	0.938717i	$-0.612016\pi$
$743$	−7.70759e9	−0.689379	−0.344689	+	0.938717i	$0.612016\pi$
$744$	0	0
$745$	3.93121e9	0.348321
$746$	−9.90215e9	−0.873260
$747$	0	0
$748$	7.23715e9	0.632283
$749$	1.71417e9	0.149062
$750$	0	0
$751$	−5.57048e9	−0.479902	−0.239951	−	0.970785i	$-0.577131\pi$
$751$	−5.57048e9	−0.479902	−0.239951	+	0.970785i	$0.577131\pi$
$752$	2.26583e10	1.94296
$753$	0	0
$754$	−2.81679e9	−0.239307
$755$	−5.00703e9	−0.423415
$756$	0	0
$757$	−1.94118e10	−1.62641	−0.813206	−	0.581976i	$-0.802280\pi$
$757$	−1.94118e10	−1.62641	−0.813206	+	0.581976i	$0.802280\pi$
$758$	1.57750e9	0.131561
$759$	0	0
$760$	−9.13555e8	−0.0754896
$761$	−1.20471e10	−0.990916	−0.495458	−	0.868632i	$-0.665000\pi$
$761$	−1.20471e10	−0.990916	−0.495458	+	0.868632i	$0.665000\pi$
$762$	0	0
$763$	1.44514e10	1.17780
$764$	1.44798e9	0.117472
$765$	0	0
$766$	−1.65016e10	−1.32655
$767$	−1.76048e9	−0.140879
$768$	0	0
$769$	4.74718e9	0.376438	0.188219	−	0.982127i	$-0.439729\pi$
$769$	4.74718e9	0.376438	0.188219	+	0.982127i	$0.439729\pi$
$770$	1.57282e10	1.24154
$771$	0	0
$772$	−2.65185e8	−0.0207438
$773$	−5.53817e9	−0.431259	−0.215629	−	0.976475i	$-0.569180\pi$
$773$	−5.53817e9	−0.431259	−0.215629	+	0.976475i	$0.569180\pi$
$774$	0	0
$775$	1.83730e9	0.141783
$776$	1.26806e10	0.974142
$777$	0	0
$778$	−2.09523e10	−1.59515
$779$	−2.01811e9	−0.152955
$780$	0	0
$781$	−9.58514e8	−0.0719979
$782$	3.36206e10	2.51409
$783$	0	0
$784$	−2.37504e10	−1.76021
$785$	2.08996e9	0.154203
$786$	0	0
$787$	1.71637e10	1.25516	0.627579	−	0.778553i	$-0.284046\pi$
$787$	1.71637e10	1.25516	0.627579	+	0.778553i	$0.284046\pi$
$788$	3.35682e9	0.244392
$789$	0	0
$790$	−2.80030e9	−0.202073
$791$	2.02803e10	1.45699
$792$	0	0
$793$	1.39343e9	0.0992269
$794$	−1.41392e10	−1.00243
$795$	0	0
$796$	5.84743e8	0.0410931
$797$	1.74350e10	1.21988	0.609940	−	0.792448i	$-0.291193\pi$
$797$	1.74350e10	1.21988	0.609940	+	0.792448i	$0.291193\pi$
$798$	0	0
$799$	3.90457e10	2.70807
$800$	1.37685e9	0.0950761
$801$	0	0
$802$	−7.84303e9	−0.536875
$803$	8.87033e9	0.604555
$804$	0	0
$805$	1.42683e10	0.964020
$806$	−1.59217e9	−0.107107
$807$	0	0
$808$	−1.57384e10	−1.04959
$809$	−2.60264e10	−1.72820	−0.864101	−	0.503318i	$-0.832112\pi$
$809$	−2.60264e10	−1.72820	−0.864101	+	0.503318i	$0.832112\pi$
$810$	0	0
$811$	−9.11296e9	−0.599911	−0.299955	−	0.953953i	$-0.596972\pi$
$811$	−9.11296e9	−0.599911	−0.299955	+	0.953953i	$0.596972\pi$
$812$	9.24741e9	0.606141
$813$	0	0
$814$	−1.90764e10	−1.23969
$815$	5.98825e9	0.387479
$816$	0	0
$817$	−1.05630e9	−0.0677655
$818$	−6.55822e9	−0.418937
$819$	0	0
$820$	1.31078e9	0.0830196
$821$	−2.68831e10	−1.69542	−0.847711	−	0.530458i	$-0.822020\pi$
$821$	−2.68831e10	−1.69542	−0.847711	+	0.530458i	$0.822020\pi$
$822$	0	0
$823$	2.99988e9	0.187588	0.0937940	−	0.995592i	$-0.470100\pi$
$823$	2.99988e9	0.187588	0.0937940	+	0.995592i	$0.470100\pi$
$824$	1.62372e10	1.01103
$825$	0	0
$826$	2.95965e10	1.82730
$827$	5.56356e9	0.342045	0.171023	−	0.985267i	$-0.445293\pi$
$827$	5.56356e9	0.342045	0.171023	+	0.985267i	$0.445293\pi$
$828$	0	0
$829$	9.11555e9	0.555702	0.277851	−	0.960624i	$-0.410378\pi$
$829$	9.11555e9	0.555702	0.277851	+	0.960624i	$0.410378\pi$
$830$	−6.78005e9	−0.411585
$831$	0	0
$832$	1.47216e9	0.0886184
$833$	−4.09277e10	−2.45335
$834$	0	0
$835$	−1.26445e10	−0.751620
$836$	1.29442e9	0.0766220
$837$	0	0
$838$	−2.38853e10	−1.40209
$839$	6.66284e9	0.389486	0.194743	−	0.980854i	$-0.437613\pi$
$839$	6.66284e9	0.389486	0.194743	+	0.980854i	$0.437613\pi$
$840$	0	0
$841$	2.60262e10	1.50878
$842$	1.27855e10	0.738118
$843$	0	0
$844$	1.31796e9	0.0754580
$845$	−7.69948e9	−0.438999
$846$	0	0
$847$	4.16614e10	2.35582
$848$	2.66850e10	1.50273
$849$	0	0
$850$	6.58627e9	0.367852
$851$	−1.73057e10	−0.962580
$852$	0	0
$853$	−1.70353e9	−0.0939785	−0.0469893	−	0.998895i	$-0.514963\pi$
$853$	−1.70353e9	−0.0939785	−0.0469893	+	0.998895i	$0.514963\pi$
$854$	−2.34259e10	−1.28704
$855$	0	0
$856$	1.46439e9	0.0797993
$857$	−3.12984e10	−1.69859	−0.849296	−	0.527917i	$-0.822973\pi$
$857$	−3.12984e10	−1.69859	−0.849296	+	0.527917i	$0.822973\pi$
$858$	0	0
$859$	−7.05289e9	−0.379657	−0.189828	−	0.981817i	$-0.560793\pi$
$859$	−7.05289e9	−0.379657	−0.189828	+	0.981817i	$0.560793\pi$
$860$	6.86073e8	0.0367812
$861$	0	0
$862$	−3.12504e10	−1.66181
$863$	−8.75820e9	−0.463850	−0.231925	−	0.972734i	$-0.574502\pi$
$863$	−8.75820e9	−0.463850	−0.231925	+	0.972734i	$0.574502\pi$
$864$	0	0
$865$	3.37087e9	0.177086
$866$	3.37843e10	1.76768
$867$	0	0
$868$	5.22704e9	0.271292
$869$	−1.23829e10	−0.640107
$870$	0	0
$871$	3.04610e9	0.156200
$872$	1.23456e10	0.630529
$873$	0	0
$874$	6.01331e9	0.304666
$875$	2.79516e9	0.141052
$876$	0	0
$877$	−6.43645e9	−0.322216	−0.161108	−	0.986937i	$-0.551507\pi$
$877$	−6.43645e9	−0.322216	−0.161108	+	0.986937i	$0.551507\pi$
$878$	6.90035e9	0.344065
$879$	0	0
$880$	1.69009e10	0.836029
$881$	−1.02831e10	−0.506649	−0.253325	−	0.967381i	$-0.581524\pi$
$881$	−1.02831e10	−0.506649	−0.253325	+	0.967381i	$0.581524\pi$
$882$	0	0
$883$	2.72426e10	1.33164	0.665820	−	0.746112i	$-0.268082\pi$
$883$	2.72426e10	1.33164	0.665820	+	0.746112i	$0.268082\pi$
$884$	−1.11457e9	−0.0542653
$885$	0	0
$886$	3.17855e8	0.0153536
$887$	3.39938e10	1.63556	0.817781	−	0.575530i	$-0.195204\pi$
$887$	3.39938e10	1.63556	0.817781	+	0.575530i	$0.195204\pi$
$888$	0	0
$889$	−3.44277e10	−1.64343
$890$	4.88871e8	0.0232450
$891$	0	0
$892$	5.81719e9	0.274433
$893$	6.98364e9	0.328172
$894$	0	0
$895$	−6.39729e8	−0.0298274
$896$	−4.08913e10	−1.89912
$897$	0	0
$898$	−2.50311e10	−1.15349
$899$	2.44616e10	1.12286
$900$	0	0
$901$	4.59848e10	2.09448
$902$	2.96819e10	1.34669
$903$	0	0
$904$	1.73252e10	0.779989
$905$	1.56921e10	0.703739
$906$	0	0
$907$	3.34086e10	1.48673	0.743367	−	0.668884i	$-0.233228\pi$
$907$	3.34086e10	1.48673	0.743367	+	0.668884i	$0.233228\pi$
$908$	4.57732e9	0.202913
$909$	0	0
$910$	−2.42224e9	−0.106554
$911$	−2.35548e10	−1.03220	−0.516101	−	0.856528i	$-0.672617\pi$
$911$	−2.35548e10	−1.03220	−0.516101	+	0.856528i	$0.672617\pi$
$912$	0	0
$913$	−2.99813e10	−1.30378
$914$	−3.23215e10	−1.40017
$915$	0	0
$916$	−4.06607e9	−0.174799
$917$	4.78629e9	0.204978
$918$	0	0
$919$	−4.30271e10	−1.82868	−0.914340	−	0.404946i	$-0.867290\pi$
$919$	−4.30271e10	−1.82868	−0.914340	+	0.404946i	$0.867290\pi$
$920$	1.21892e10	0.516081
$921$	0	0
$922$	−3.45236e9	−0.145063
$923$	1.47617e8	0.00617918
$924$	0	0
$925$	−3.39020e9	−0.140841
$926$	2.42717e10	1.00453
$927$	0	0
$928$	1.83312e10	0.752961
$929$	−1.15316e10	−0.471883	−0.235941	−	0.971767i	$-0.575817\pi$
$929$	−1.15316e10	−0.471883	−0.235941	+	0.971767i	$0.575817\pi$
$930$	0	0
$931$	−7.32024e9	−0.297305
$932$	2.57345e9	0.104126
$933$	0	0
$934$	3.26347e10	1.31059
$935$	2.91244e10	1.16524
$936$	0	0
$937$	2.06024e10	0.818144	0.409072	−	0.912502i	$-0.365852\pi$
$937$	2.06024e10	0.818144	0.409072	+	0.912502i	$0.365852\pi$
$938$	−5.12099e10	−2.02602
$939$	0	0
$940$	−4.53593e9	−0.178123
$941$	4.21622e10	1.64953	0.824764	−	0.565477i	$-0.191308\pi$
$941$	4.21622e10	1.64953	0.824764	+	0.565477i	$0.191308\pi$
$942$	0	0
$943$	2.69268e10	1.04567
$944$	3.18034e10	1.23047
$945$	0	0
$946$	1.55358e10	0.596642
$947$	4.89497e10	1.87295	0.936473	−	0.350739i	$-0.114069\pi$
$947$	4.89497e10	1.87295	0.936473	+	0.350739i	$0.114069\pi$
$948$	0	0
$949$	−1.36609e9	−0.0518855
$950$	1.17801e9	0.0445774
$951$	0	0
$952$	−5.84780e10	−2.19666
$953$	4.73024e10	1.77034	0.885172	−	0.465264i	$-0.154041\pi$
$953$	4.73024e10	1.77034	0.885172	+	0.465264i	$0.154041\pi$
$954$	0	0
$955$	5.82711e9	0.216492
$956$	3.97583e9	0.147172
$957$	0	0
$958$	−4.79285e10	−1.76122
$959$	−3.49238e10	−1.27866
$960$	0	0
$961$	−1.36859e10	−0.497440
$962$	2.93789e9	0.106395
$963$	0	0
$964$	−7.19218e8	−0.0258578
$965$	−1.06718e9	−0.0382291
$966$	0	0
$967$	−2.89581e10	−1.02986	−0.514929	−	0.857233i	$-0.672182\pi$
$967$	−2.89581e10	−1.02986	−0.514929	+	0.857233i	$0.672182\pi$
$968$	3.55907e10	1.26117
$969$	0	0
$970$	−1.63513e10	−0.575241
$971$	−1.49551e10	−0.524230	−0.262115	−	0.965037i	$-0.584420\pi$
$971$	−1.49551e10	−0.524230	−0.262115	+	0.965037i	$0.584420\pi$
$972$	0	0
$973$	−2.24875e10	−0.782611
$974$	−2.30802e10	−0.800358
$975$	0	0
$976$	−2.51726e10	−0.866670
$977$	3.61050e10	1.23862	0.619308	−	0.785148i	$-0.287413\pi$
$977$	3.61050e10	1.23862	0.619308	+	0.785148i	$0.287413\pi$
$978$	0	0
$979$	2.16178e9	0.0736331
$980$	4.75456e9	0.161369
$981$	0	0
$982$	−4.86439e10	−1.63922
$983$	−1.58879e10	−0.533494	−0.266747	−	0.963767i	$-0.585949\pi$
$983$	−1.58879e10	−0.533494	−0.266747	+	0.963767i	$0.585949\pi$
$984$	0	0
$985$	1.35088e10	0.450393
$986$	8.76887e10	2.91323
$987$	0	0
$988$	−1.99349e8	−0.00657604
$989$	1.40937e10	0.463275
$990$	0	0
$991$	−1.47680e10	−0.482017	−0.241009	−	0.970523i	$-0.577478\pi$
$991$	−1.47680e10	−0.482017	−0.241009	+	0.970523i	$0.577478\pi$
$992$	1.03616e10	0.337004
$993$	0	0
$994$	−2.48169e9	−0.0801483
$995$	2.35318e9	0.0757311
$996$	0	0
$997$	−3.62017e10	−1.15690	−0.578450	−	0.815718i	$-0.696342\pi$
$997$	−3.62017e10	−1.15690	−0.578450	+	0.815718i	$0.696342\pi$
$998$	−2.54596e10	−0.810763
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.8.a.j.1.5		15
3.2	odd	2		405.8.a.i.1.11		15
9.2	odd	6		45.8.e.b.31.5	yes	30
9.4	even	3		135.8.e.b.46.11		30
9.5	odd	6		45.8.e.b.16.5	✓	30
9.7	even	3		135.8.e.b.91.11		30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.8.e.b.16.5	✓	30	9.5	odd	6
45.8.e.b.31.5	yes	30	9.2	odd	6
135.8.e.b.46.11		30	9.4	even	3
135.8.e.b.91.11		30	9.7	even	3
405.8.a.i.1.11		15	3.2	odd	2
405.8.a.j.1.5		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)

\(f(q)\)	\(=\)	\(q-12.6120 q^{2} +31.0613 q^{4} +125.000 q^{5} +1431.12 q^{7} +1222.59 q^{8} +O(q^{10})\)	\(q-12.6120 q^{2} +31.0613 q^{4} +125.000 q^{5} +1431.12 q^{7} +1222.59 q^{8} -1576.49 q^{10} -6971.24 q^{11} +1073.61 q^{13} -18049.2 q^{14} -19395.0 q^{16} -33422.4 q^{17} -5977.86 q^{19} +3882.67 q^{20} +87921.0 q^{22} +79760.1 q^{23} +15625.0 q^{25} -13540.4 q^{26} +44452.5 q^{28} +208029. q^{29} +117587. q^{31} +88118.4 q^{32} +421521. q^{34} +178890. q^{35} -216973. q^{37} +75392.4 q^{38} +152823. q^{40} +337598. q^{41} +176702. q^{43} -216536. q^{44} -1.00593e6 q^{46} -1.16825e6 q^{47} +1.22456e6 q^{49} -197062. q^{50} +33347.9 q^{52} -1.37587e6 q^{53} -871405. q^{55} +1.74967e6 q^{56} -2.62365e6 q^{58} -1.63977e6 q^{59} +1.29789e6 q^{61} -1.48300e6 q^{62} +1.37122e6 q^{64} +134202. q^{65} +2.83724e6 q^{67} -1.03814e6 q^{68} -2.25615e6 q^{70} +137496. q^{71} -1.27242e6 q^{73} +2.73645e6 q^{74} -185680. q^{76} -9.97668e6 q^{77} +1.77628e6 q^{79} -2.42438e6 q^{80} -4.25776e6 q^{82} +4.30071e6 q^{83} -4.17780e6 q^{85} -2.22855e6 q^{86} -8.52294e6 q^{88} -310100. q^{89} +1.53647e6 q^{91} +2.47745e6 q^{92} +1.47339e7 q^{94} -747232. q^{95} +1.03719e7 q^{97} -1.54441e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 8 q^{2} + 1088 q^{4} + 1875 q^{5} + 1289 q^{7} + 4434 q^{8}+O(q^{10}) \) 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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