LMFDB - Embedded newform 405.8.a.g.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:	$1$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 2 x^{13} - 1221 x^{12} + 3034 x^{11} + 559330 x^{10} - 1662468 x^{9} - 119658132 x^{8} + \cdots + 11075674978368 $ x^14 - 2x^13 - 1221x^12 + 3034x^11 + 559330x^10 - 1662468x^9 - 119658132x^8 + 388985472x^7 + 12117415968x^6 - 35096050656x^5 - 526695898608x^4 + 532226461536x^3 + 9146880114336x^2 + 18387947059776*x + 11075674978368
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{28}\cdot 13 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$9.32475$ 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)

$f(q)$	$=$	$q-12.0568 q^{2} +17.3664 q^{4} -125.000 q^{5} -1464.81 q^{7} +1333.89 q^{8} +O(q^{10})$	\(q-12.0568 q^{2} +17.3664 q^{4} -125.000 q^{5} -1464.81 q^{7} +1333.89 q^{8} +1507.10 q^{10} -2372.99 q^{11} +8200.03 q^{13} +17660.9 q^{14} -18305.3 q^{16} -7892.52 q^{17} -54469.0 q^{19} -2170.80 q^{20} +28610.7 q^{22} -11797.4 q^{23} +15625.0 q^{25} -98866.2 q^{26} -25438.5 q^{28} +166268. q^{29} +218261. q^{31} +49965.9 q^{32} +95158.5 q^{34} +183101. q^{35} +31534.9 q^{37} +656722. q^{38} -166736. q^{40} +253965. q^{41} -422476. q^{43} -41210.3 q^{44} +142238. q^{46} -809587. q^{47} +1.32213e6 q^{49} -188387. q^{50} +142405. q^{52} +585700. q^{53} +296624. q^{55} -1.95389e6 q^{56} -2.00466e6 q^{58} -430929. q^{59} +3.02231e6 q^{61} -2.63153e6 q^{62} +1.74065e6 q^{64} -1.02500e6 q^{65} +1.21837e6 q^{67} -137065. q^{68} -2.20762e6 q^{70} -5.34587e6 q^{71} +5.01076e6 q^{73} -380210. q^{74} -945931. q^{76} +3.47598e6 q^{77} +8.39634e6 q^{79} +2.28816e6 q^{80} -3.06200e6 q^{82} +2.53600e6 q^{83} +986564. q^{85} +5.09371e6 q^{86} -3.16530e6 q^{88} +4.09287e6 q^{89} -1.20115e7 q^{91} -204878. q^{92} +9.76103e6 q^{94} +6.80863e6 q^{95} +4.05445e6 q^{97} -1.59406e7 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 16 q^{2} + 714 q^{4} - 1750 q^{5} + 1538 q^{7} + 126 q^{8}+O(q^{10}) $ 14 * q - 16 * q^2 + 714 * q^4 - 1750 * q^5 + 1538 * q^7 + 126 * q^8	\( 14 q - 16 q^{2} + 714 q^{4} - 1750 q^{5} + 1538 q^{7} + 126 q^{8} + 2000 q^{10} - 10648 q^{11} + 17268 q^{13} - 9180 q^{14} + 34122 q^{16} - 3886 q^{17} + 22986 q^{19} - 89250 q^{20} - 49706 q^{22} - 155040 q^{23} + 218750 q^{25} - 7804 q^{26} + 256160 q^{28} - 195836 q^{29} + 92786 q^{31} - 157778 q^{32} + 787348 q^{34} - 192250 q^{35} + 876406 q^{37} - 329320 q^{38} - 15750 q^{40} - 795164 q^{41} + 730350 q^{43} - 2360876 q^{44} - 225654 q^{46} - 2687842 q^{47} + 1663586 q^{49} - 250000 q^{50} + 3875836 q^{52} - 2533750 q^{53} + 1331000 q^{55} - 2055276 q^{56} - 318934 q^{58} - 2283340 q^{59} + 2600400 q^{61} - 5702022 q^{62} + 474098 q^{64} - 2158500 q^{65} + 2422160 q^{67} - 1114364 q^{68} + 1147500 q^{70} - 6395324 q^{71} - 540774 q^{73} - 260516 q^{74} - 2417042 q^{76} + 1384890 q^{77} - 307384 q^{79} - 4265250 q^{80} - 14044738 q^{82} - 10991322 q^{83} + 485750 q^{85} - 2847712 q^{86} - 19226592 q^{88} - 2094000 q^{89} - 9496256 q^{91} - 39213378 q^{92} - 28132682 q^{94} - 2873250 q^{95} - 12859994 q^{97} - 28336478 q^{98}+O(q^{100}) \) 14 * q - 16 * q^2 + 714 * q^4 - 1750 * q^5 + 1538 * q^7 + 126 * q^8 + 2000 * q^10 - 10648 * q^11 + 17268 * q^13 - 9180 * q^14 + 34122 * q^16 - 3886 * q^17 + 22986 * q^19 - 89250 * q^20 - 49706 * q^22 - 155040 * q^23 + 218750 * q^25 - 7804 * q^26 + 256160 * q^28 - 195836 * q^29 + 92786 * q^31 - 157778 * q^32 + 787348 * q^34 - 192250 * q^35 + 876406 * q^37 - 329320 * q^38 - 15750 * q^40 - 795164 * q^41 + 730350 * q^43 - 2360876 * q^44 - 225654 * q^46 - 2687842 * q^47 + 1663586 * q^49 - 250000 * q^50 + 3875836 * q^52 - 2533750 * q^53 + 1331000 * q^55 - 2055276 * q^56 - 318934 * q^58 - 2283340 * q^59 + 2600400 * q^61 - 5702022 * q^62 + 474098 * q^64 - 2158500 * q^65 + 2422160 * q^67 - 1114364 * q^68 + 1147500 * q^70 - 6395324 * q^71 - 540774 * q^73 - 260516 * q^74 - 2417042 * q^76 + 1384890 * q^77 - 307384 * q^79 - 4265250 * q^80 - 14044738 * q^82 - 10991322 * q^83 + 485750 * q^85 - 2847712 * q^86 - 19226592 * q^88 - 2094000 * q^89 - 9496256 * q^91 - 39213378 * q^92 - 28132682 * q^94 - 2873250 * q^95 - 12859994 * q^97 - 28336478 * 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 $

$2$	−12.0568	−1.06568	−0.532840	−	0.846216i	$-0.678875\pi$
$2$	−12.0568	−1.06568	−0.532840	+	0.846216i	$0.678875\pi$
$3$	0	0
$3$	0	0
$4$	17.3664	0.135675
$5$	−125.000	−0.447214
$5$	−125.000	−0.447214
$6$	0	0
$7$	−1464.81	−1.61413	−0.807065	−	0.590463i	$-0.798945\pi$
$7$	−1464.81	−1.61413	−0.807065	+	0.590463i	$0.798945\pi$
$8$	1333.89	0.921094
$9$	0	0
$10$	1507.10	0.476587
$11$	−2372.99	−0.537554	−0.268777	−	0.963202i	$-0.586619\pi$
$11$	−2372.99	−0.537554	−0.268777	+	0.963202i	$0.586619\pi$
$12$	0	0
$13$	8200.03	1.03518	0.517588	−	0.855630i	$-0.326830\pi$
$13$	8200.03	1.03518	0.517588	+	0.855630i	$0.326830\pi$
$14$	17660.9	1.72015
$15$	0	0
$16$	−18305.3	−1.11727
$17$	−7892.52	−0.389623	−0.194811	−	0.980841i	$-0.562409\pi$
$17$	−7892.52	−0.389623	−0.194811	+	0.980841i	$0.562409\pi$
$18$	0	0
$19$	−54469.0	−1.82185	−0.910923	−	0.412576i	$-0.864629\pi$
$19$	−54469.0	−1.82185	−0.910923	+	0.412576i	$0.864629\pi$
$20$	−2170.80	−0.0606757
$21$	0	0
$22$	28610.7	0.572860
$23$	−11797.4	−0.202180	−0.101090	−	0.994877i	$-0.532233\pi$
$23$	−11797.4	−0.202180	−0.101090	+	0.994877i	$0.532233\pi$
$24$	0	0
$25$	15625.0	0.200000
$26$	−98866.2	−1.10317
$27$	0	0
$28$	−25438.5	−0.218997
$29$	166268.	1.26595	0.632973	−	0.774174i	$-0.281834\pi$
$29$	166268.	1.26595	0.632973	+	0.774174i	$0.281834\pi$
$30$	0	0
$31$	218261.	1.31586	0.657930	−	0.753079i	$-0.271432\pi$
$31$	218261.	1.31586	0.657930	+	0.753079i	$0.271432\pi$
$32$	49965.9	0.269556
$33$	0	0
$34$	95158.5	0.415213
$35$	183101.	0.721861
$36$	0	0
$37$	31534.9	0.102349	0.0511747	−	0.998690i	$-0.483703\pi$
$37$	31534.9	0.102349	0.0511747	+	0.998690i	$0.483703\pi$
$38$	656722.	1.94151
$39$	0	0
$40$	−166736.	−0.411926
$41$	253965.	0.575480	0.287740	−	0.957709i	$-0.407096\pi$
$41$	253965.	0.575480	0.287740	+	0.957709i	$0.407096\pi$
$42$	0	0
$43$	−422476.	−0.810331	−0.405165	−	0.914243i	$-0.632786\pi$
$43$	−422476.	−0.810331	−0.405165	+	0.914243i	$0.632786\pi$
$44$	−41210.3	−0.0729326
$45$	0	0
$46$	142238.	0.215459
$47$	−809587.	−1.13742	−0.568710	−	0.822538i	$-0.692557\pi$
$47$	−809587.	−1.13742	−0.568710	+	0.822538i	$0.692557\pi$
$48$	0	0
$49$	1.32213e6	1.60542
$50$	−188387.	−0.213136
$51$	0	0
$52$	142405.	0.140447
$53$	585700.	0.540393	0.270197	−	0.962805i	$-0.412911\pi$
$53$	585700.	0.540393	0.270197	+	0.962805i	$0.412911\pi$
$54$	0	0
$55$	296624.	0.240401
$56$	−1.95389e6	−1.48677
$57$	0	0
$58$	−2.00466e6	−1.34909
$59$	−430929.	−0.273164	−0.136582	−	0.990629i	$-0.543612\pi$
$59$	−430929.	−0.273164	−0.136582	+	0.990629i	$0.543612\pi$
$60$	0	0
$61$	3.02231e6	1.70484	0.852421	−	0.522856i	$-0.175133\pi$
$61$	3.02231e6	1.70484	0.852421	+	0.522856i	$0.175133\pi$
$62$	−2.63153e6	−1.40229
$63$	0	0
$64$	1.74065e6	0.830007
$65$	−1.02500e6	−0.462944
$66$	0	0
$67$	1.21837e6	0.494898	0.247449	−	0.968901i	$-0.420408\pi$
$67$	1.21837e6	0.494898	0.247449	+	0.968901i	$0.420408\pi$
$68$	−137065.	−0.0528621
$69$	0	0
$70$	−2.20762e6	−0.769273
$71$	−5.34587e6	−1.77262	−0.886308	−	0.463097i	$-0.846738\pi$
$71$	−5.34587e6	−1.77262	−0.886308	+	0.463097i	$0.846738\pi$
$72$	0	0
$73$	5.01076e6	1.50756	0.753778	−	0.657129i	$-0.228229\pi$
$73$	5.01076e6	1.50756	0.753778	+	0.657129i	$0.228229\pi$
$74$	−380210.	−0.109072
$75$	0	0
$76$	−945931.	−0.247179
$77$	3.47598e6	0.867681
$78$	0	0
$79$	8.39634e6	1.91600	0.957999	−	0.286771i	$-0.0925820\pi$
$79$	8.39634e6	1.91600	0.957999	+	0.286771i	$0.0925820\pi$
$80$	2.28816e6	0.499657
$81$	0	0
$82$	−3.06200e6	−0.613278
$83$	2.53600e6	0.486829	0.243415	−	0.969922i	$-0.421732\pi$
$83$	2.53600e6	0.486829	0.243415	+	0.969922i	$0.421732\pi$
$84$	0	0
$85$	986564.	0.174245
$86$	5.09371e6	0.863554
$87$	0	0
$88$	−3.16530e6	−0.495138
$89$	4.09287e6	0.615408	0.307704	−	0.951482i	$-0.400439\pi$
$89$	4.09287e6	0.615408	0.307704	+	0.951482i	$0.400439\pi$
$90$	0	0
$91$	−1.20115e7	−1.67091
$92$	−204878.	−0.0274307
$93$	0	0
$94$	9.76103e6	1.21213
$95$	6.80863e6	0.814755
$96$	0	0
$97$	4.05445e6	0.451057	0.225528	−	0.974237i	$-0.427589\pi$
$97$	4.05445e6	0.451057	0.225528	+	0.974237i	$0.427589\pi$
$98$	−1.59406e7	−1.71086
$99$	0	0
$100$	271350.	0.0271350
$101$	−1.58730e7	−1.53297	−0.766485	−	0.642263i	$-0.777996\pi$
$101$	−1.58730e7	−1.53297	−0.766485	+	0.642263i	$0.777996\pi$
$102$	0	0
$103$	543717.	0.0490278	0.0245139	−	0.999699i	$-0.492196\pi$
$103$	543717.	0.0490278	0.0245139	+	0.999699i	$0.492196\pi$
$104$	1.09379e7	0.953494
$105$	0	0
$106$	−7.06167e6	−0.575887
$107$	−6.97154e6	−0.550155	−0.275078	−	0.961422i	$-0.588704\pi$
$107$	−6.97154e6	−0.550155	−0.275078	+	0.961422i	$0.588704\pi$
$108$	0	0
$109$	−8.57815e6	−0.634455	−0.317227	−	0.948350i	$-0.602752\pi$
$109$	−8.57815e6	−0.634455	−0.317227	+	0.948350i	$0.602752\pi$
$110$	−3.57634e6	−0.256191
$111$	0	0
$112$	2.68138e7	1.80341
$113$	2.69619e7	1.75783	0.878913	−	0.476982i	$-0.158269\pi$
$113$	2.69619e7	1.75783	0.878913	+	0.476982i	$0.158269\pi$
$114$	0	0
$115$	1.47467e6	0.0904175
$116$	2.88748e6	0.171757
$117$	0	0
$118$	5.19562e6	0.291106
$119$	1.15610e7	0.628902
$120$	0	0
$121$	−1.38561e7	−0.711036
$122$	−3.64393e7	−1.81682
$123$	0	0
$124$	3.79040e6	0.178529
$125$	−1.95312e6	−0.0894427
$126$	0	0
$127$	3.28609e7	1.42353	0.711766	−	0.702417i	$-0.247896\pi$
$127$	3.28609e7	1.42353	0.711766	+	0.702417i	$0.247896\pi$
$128$	−2.73823e7	−1.15408
$129$	0	0
$130$	1.23583e7	0.493351
$131$	1.61293e7	0.626852	0.313426	−	0.949613i	$-0.398523\pi$
$131$	1.61293e7	0.626852	0.313426	+	0.949613i	$0.398523\pi$
$132$	0	0
$133$	7.97868e7	2.94070
$134$	−1.46896e7	−0.527403
$135$	0	0
$136$	−1.05277e7	−0.358879
$137$	3.07783e7	1.02264	0.511319	−	0.859391i	$-0.329157\pi$
$137$	3.07783e7	1.02264	0.511319	+	0.859391i	$0.329157\pi$
$138$	0	0
$139$	4.00517e7	1.26494	0.632468	−	0.774586i	$-0.282042\pi$
$139$	4.00517e7	1.26494	0.632468	+	0.774586i	$0.282042\pi$
$140$	3.17981e6	0.0979385
$141$	0	0
$142$	6.44541e7	1.88904
$143$	−1.94586e7	−0.556462
$144$	0	0
$145$	−2.07835e7	−0.566149
$146$	−6.04137e7	−1.60657
$147$	0	0
$148$	547648.	0.0138863
$149$	−4.78745e6	−0.118564	−0.0592819	−	0.998241i	$-0.518881\pi$
$149$	−4.78745e6	−0.118564	−0.0592819	+	0.998241i	$0.518881\pi$
$150$	0	0
$151$	−2.69743e7	−0.637575	−0.318787	−	0.947826i	$-0.603276\pi$
$151$	−2.69743e7	−0.637575	−0.318787	+	0.947826i	$0.603276\pi$
$152$	−7.26555e7	−1.67809
$153$	0	0
$154$	−4.19092e7	−0.924671
$155$	−2.72826e7	−0.588470
$156$	0	0
$157$	−3.53831e7	−0.729705	−0.364853	−	0.931065i	$-0.618881\pi$
$157$	−3.53831e7	−0.729705	−0.364853	+	0.931065i	$0.618881\pi$
$158$	−1.01233e8	−2.04184
$159$	0	0
$160$	−6.24573e6	−0.120549
$161$	1.72809e7	0.326344
$162$	0	0
$163$	−6.24588e7	−1.12963	−0.564816	−	0.825217i	$-0.691053\pi$
$163$	−6.24588e7	−1.12963	−0.564816	+	0.825217i	$0.691053\pi$
$164$	4.41046e6	0.0780782
$165$	0	0
$166$	−3.05761e7	−0.518804
$167$	−2.51216e7	−0.417387	−0.208694	−	0.977981i	$-0.566921\pi$
$167$	−2.51216e7	−0.417387	−0.208694	+	0.977981i	$0.566921\pi$
$168$	0	0
$169$	4.49204e6	0.0715880
$170$	−1.18948e7	−0.185689
$171$	0	0
$172$	−7.33689e6	−0.109942
$173$	−8.98965e7	−1.32002	−0.660011	−	0.751256i	$-0.729448\pi$
$173$	−8.98965e7	−1.32002	−0.660011	+	0.751256i	$0.729448\pi$
$174$	0	0
$175$	−2.28877e7	−0.322826
$176$	4.34383e7	0.600591
$177$	0	0
$178$	−4.93469e7	−0.655828
$179$	−5.25681e7	−0.685073	−0.342537	−	0.939504i	$-0.611286\pi$
$179$	−5.25681e7	−0.685073	−0.342537	+	0.939504i	$0.611286\pi$
$180$	0	0
$181$	−8.62531e6	−0.108118	−0.0540592	−	0.998538i	$-0.517216\pi$
$181$	−8.62531e6	−0.108118	−0.0540592	+	0.998538i	$0.517216\pi$
$182$	1.44820e8	1.78065
$183$	0	0
$184$	−1.57363e7	−0.186226
$185$	−3.94186e6	−0.0457721
$186$	0	0
$187$	1.87289e7	0.209443
$188$	−1.40596e7	−0.154320
$189$	0	0
$190$	−8.20902e7	−0.868268
$191$	−1.42593e8	−1.48075	−0.740377	−	0.672192i	$-0.765353\pi$
$191$	−1.42593e8	−1.48075	−0.740377	+	0.672192i	$0.765353\pi$
$192$	0	0
$193$	7.63321e7	0.764287	0.382144	−	0.924103i	$-0.375186\pi$
$193$	7.63321e7	0.764287	0.382144	+	0.924103i	$0.375186\pi$
$194$	−4.88837e7	−0.480682
$195$	0	0
$196$	2.29606e7	0.217815
$197$	6.44339e7	0.600458	0.300229	−	0.953867i	$-0.402937\pi$
$197$	6.44339e7	0.600458	0.300229	+	0.953867i	$0.402937\pi$
$198$	0	0
$199$	−1.25118e8	−1.12547	−0.562733	−	0.826639i	$-0.690250\pi$
$199$	−1.25118e8	−1.12547	−0.562733	+	0.826639i	$0.690250\pi$
$200$	2.08420e7	0.184219
$201$	0	0
$202$	1.91377e8	1.63366
$203$	−2.43551e8	−2.04340
$204$	0	0
$205$	−3.17456e7	−0.257362
$206$	−6.55549e6	−0.0522480
$207$	0	0
$208$	−1.50104e8	−1.15657
$209$	1.29254e8	0.979340
$210$	0	0
$211$	4.48581e7	0.328740	0.164370	−	0.986399i	$-0.447441\pi$
$211$	4.48581e7	0.328740	0.164370	+	0.986399i	$0.447441\pi$
$212$	1.01715e7	0.0733179
$213$	0	0
$214$	8.40544e7	0.586290
$215$	5.28095e7	0.362391
$216$	0	0
$217$	−3.19711e8	−2.12397
$218$	1.03425e8	0.676126
$219$	0	0
$220$	5.15129e6	0.0326165
$221$	−6.47189e7	−0.403328
$222$	0	0
$223$	8.80554e7	0.531727	0.265864	−	0.964011i	$-0.414343\pi$
$223$	8.80554e7	0.531727	0.265864	+	0.964011i	$0.414343\pi$
$224$	−7.31906e7	−0.435098
$225$	0	0
$226$	−3.25074e8	−1.87328
$227$	−3.50410e8	−1.98832	−0.994159	−	0.107924i	$-0.965580\pi$
$227$	−3.50410e8	−1.98832	−0.994159	+	0.107924i	$0.965580\pi$
$228$	0	0
$229$	7.66227e7	0.421632	0.210816	−	0.977526i	$-0.432388\pi$
$229$	7.66227e7	0.421632	0.210816	+	0.977526i	$0.432388\pi$
$230$	−1.77798e7	−0.0963561
$231$	0	0
$232$	2.21783e8	1.16606
$233$	2.70243e8	1.39961	0.699807	−	0.714332i	$-0.253269\pi$
$233$	2.70243e8	1.39961	0.699807	+	0.714332i	$0.253269\pi$
$234$	0	0
$235$	1.01198e8	0.508670
$236$	−7.48369e6	−0.0370616
$237$	0	0
$238$	−1.39389e8	−0.670208
$239$	−3.99003e8	−1.89053	−0.945266	−	0.326301i	$-0.894198\pi$
$239$	−3.99003e8	−1.89053	−0.945266	+	0.326301i	$0.894198\pi$
$240$	0	0
$241$	−1.53549e8	−0.706624	−0.353312	−	0.935506i	$-0.614945\pi$
$241$	−1.53549e8	−0.706624	−0.353312	+	0.935506i	$0.614945\pi$
$242$	1.67060e8	0.757737
$243$	0	0
$244$	5.24866e7	0.231304
$245$	−1.65266e8	−0.717964
$246$	0	0
$247$	−4.46648e8	−1.88593
$248$	2.91135e8	1.21203
$249$	0	0
$250$	2.35484e7	0.0953174
$251$	−4.14335e8	−1.65384	−0.826920	−	0.562319i	$-0.809909\pi$
$251$	−4.14335e8	−1.65384	−0.826920	+	0.562319i	$0.809909\pi$
$252$	0	0
$253$	2.79950e7	0.108682
$254$	−3.96198e8	−1.51703
$255$	0	0
$256$	1.07340e8	0.399872
$257$	−2.13133e8	−0.783223	−0.391611	−	0.920131i	$-0.628082\pi$
$257$	−2.13133e8	−0.783223	−0.391611	+	0.920131i	$0.628082\pi$
$258$	0	0
$259$	−4.61927e7	−0.165205
$260$	−1.78006e7	−0.0628100
$261$	0	0
$262$	−1.94467e8	−0.668024
$263$	1.25701e8	0.426083	0.213042	−	0.977043i	$-0.431663\pi$
$263$	1.25701e8	0.426083	0.213042	+	0.977043i	$0.431663\pi$
$264$	0	0
$265$	−7.32125e7	−0.241671
$266$	−9.61974e8	−3.13384
$267$	0	0
$268$	2.11586e7	0.0671453
$269$	1.58826e8	0.497494	0.248747	−	0.968569i	$-0.419981\pi$
$269$	1.58826e8	0.497494	0.248747	+	0.968569i	$0.419981\pi$
$270$	0	0
$271$	−1.74004e8	−0.531088	−0.265544	−	0.964099i	$-0.585551\pi$
$271$	−1.74004e8	−0.531088	−0.265544	+	0.964099i	$0.585551\pi$
$272$	1.44475e8	0.435313
$273$	0	0
$274$	−3.71087e8	−1.08981
$275$	−3.70780e7	−0.107511
$276$	0	0
$277$	3.39080e7	0.0958568	0.0479284	−	0.998851i	$-0.484738\pi$
$277$	3.39080e7	0.0958568	0.0479284	+	0.998851i	$0.484738\pi$
$278$	−4.82895e8	−1.34802
$279$	0	0
$280$	2.44237e8	0.664902
$281$	2.83986e8	0.763528	0.381764	−	0.924260i	$-0.375317\pi$
$281$	2.83986e8	0.763528	0.381764	+	0.924260i	$0.375317\pi$
$282$	0	0
$283$	3.03738e8	0.796613	0.398306	−	0.917252i	$-0.369598\pi$
$283$	3.03738e8	0.796613	0.398306	+	0.917252i	$0.369598\pi$
$284$	−9.28386e7	−0.240500
$285$	0	0
$286$	2.34609e8	0.593011
$287$	−3.72010e8	−0.928899
$288$	0	0
$289$	−3.48047e8	−0.848194
$290$	2.50582e8	0.603334
$291$	0	0
$292$	8.70189e7	0.204538
$293$	3.72423e8	0.864968	0.432484	−	0.901642i	$-0.357637\pi$
$293$	3.72423e8	0.864968	0.432484	+	0.901642i	$0.357637\pi$
$294$	0	0
$295$	5.38661e7	0.122163
$296$	4.20640e7	0.0942735
$297$	0	0
$298$	5.77213e7	0.126351
$299$	−9.67387e7	−0.209291
$300$	0	0
$301$	6.18847e8	1.30798
$302$	3.25224e8	0.679451
$303$	0	0
$304$	9.97072e8	2.03549
$305$	−3.77788e8	−0.762429
$306$	0	0
$307$	−2.07872e8	−0.410026	−0.205013	−	0.978759i	$-0.565724\pi$
$307$	−2.07872e8	−0.410026	−0.205013	+	0.978759i	$0.565724\pi$
$308$	6.03653e7	0.117723
$309$	0	0
$310$	3.28941e8	0.627122
$311$	3.32636e8	0.627058	0.313529	−	0.949579i	$-0.398489\pi$
$311$	3.32636e8	0.627058	0.313529	+	0.949579i	$0.398489\pi$
$312$	0	0
$313$	−4.70114e8	−0.866560	−0.433280	−	0.901259i	$-0.642644\pi$
$313$	−4.70114e8	−0.866560	−0.433280	+	0.901259i	$0.642644\pi$
$314$	4.26607e8	0.777633
$315$	0	0
$316$	1.45814e8	0.259953
$317$	−3.35718e8	−0.591926	−0.295963	−	0.955199i	$-0.595641\pi$
$317$	−3.35718e8	−0.591926	−0.295963	+	0.955199i	$0.595641\pi$
$318$	0	0
$319$	−3.94552e8	−0.680514
$320$	−2.17581e8	−0.371190
$321$	0	0
$322$	−2.08352e8	−0.347779
$323$	4.29897e8	0.709833
$324$	0	0
$325$	1.28126e8	0.207035
$326$	7.53053e8	1.20383
$327$	0	0
$328$	3.38760e8	0.530071
$329$	1.18589e9	1.83594
$330$	0	0
$331$	−3.20156e8	−0.485248	−0.242624	−	0.970120i	$-0.578008\pi$
$331$	−3.20156e8	−0.485248	−0.242624	+	0.970120i	$0.578008\pi$
$332$	4.40413e7	0.0660506
$333$	0	0
$334$	3.02886e8	0.444801
$335$	−1.52296e8	−0.221325
$336$	0	0
$337$	−6.42996e8	−0.915174	−0.457587	−	0.889165i	$-0.651286\pi$
$337$	−6.42996e8	−0.915174	−0.457587	+	0.889165i	$0.651286\pi$
$338$	−5.41597e7	−0.0762900
$339$	0	0
$340$	1.71331e7	0.0236406
$341$	−5.17931e8	−0.707345
$342$	0	0
$343$	−7.30334e8	−0.977220
$344$	−5.63535e8	−0.746391
$345$	0	0
$346$	1.08386e9	1.40672
$347$	1.50909e9	1.93893	0.969464	−	0.245232i	$-0.0788640\pi$
$347$	1.50909e9	1.93893	0.969464	+	0.245232i	$0.0788640\pi$
$348$	0	0
$349$	−7.56335e8	−0.952413	−0.476206	−	0.879333i	$-0.657989\pi$
$349$	−7.56335e8	−0.952413	−0.476206	+	0.879333i	$0.657989\pi$
$350$	2.75952e8	0.344029
$351$	0	0
$352$	−1.18569e8	−0.144901
$353$	−1.13004e9	−1.36735	−0.683676	−	0.729785i	$-0.739620\pi$
$353$	−1.13004e9	−1.36735	−0.683676	+	0.729785i	$0.739620\pi$
$354$	0	0
$355$	6.68234e8	0.792738
$356$	7.10784e7	0.0834955
$357$	0	0
$358$	6.33803e8	0.730069
$359$	−4.23115e8	−0.482645	−0.241322	−	0.970445i	$-0.577581\pi$
$359$	−4.23115e8	−0.482645	−0.241322	+	0.970445i	$0.577581\pi$
$360$	0	0
$361$	2.07300e9	2.31913
$362$	1.03994e8	0.115220
$363$	0	0
$364$	−2.08597e8	−0.226700
$365$	−6.26345e8	−0.674200
$366$	0	0
$367$	1.21616e9	1.28428	0.642139	−	0.766589i	$-0.278047\pi$
$367$	1.21616e9	1.28428	0.642139	+	0.766589i	$0.278047\pi$
$368$	2.15954e8	0.225889
$369$	0	0
$370$	4.75262e7	0.0487784
$371$	−8.57940e8	−0.872265
$372$	0	0
$373$	−5.89256e8	−0.587927	−0.293963	−	0.955817i	$-0.594974\pi$
$373$	−5.89256e8	−0.587927	−0.293963	+	0.955817i	$0.594974\pi$
$374$	−2.25810e8	−0.223199
$375$	0	0
$376$	−1.07990e9	−1.04767
$377$	1.36340e9	1.31048
$378$	0	0
$379$	−1.75648e8	−0.165732	−0.0828660	−	0.996561i	$-0.526407\pi$
$379$	−1.75648e8	−0.165732	−0.0828660	+	0.996561i	$0.526407\pi$
$380$	1.18241e8	0.110542
$381$	0	0
$382$	1.71922e9	1.57801
$383$	−5.75288e8	−0.523226	−0.261613	−	0.965173i	$-0.584254\pi$
$383$	−5.75288e8	−0.523226	−0.261613	+	0.965173i	$0.584254\pi$
$384$	0	0
$385$	−4.34498e8	−0.388039
$386$	−9.20321e8	−0.814486
$387$	0	0
$388$	7.04113e7	0.0611971
$389$	4.85671e8	0.418330	0.209165	−	0.977880i	$-0.432925\pi$
$389$	4.85671e8	0.418330	0.209165	+	0.977880i	$0.432925\pi$
$390$	0	0
$391$	9.31108e7	0.0787737
$392$	1.76357e9	1.47874
$393$	0	0
$394$	−7.76866e8	−0.639896
$395$	−1.04954e9	−0.856860
$396$	0	0
$397$	2.04690e9	1.64183	0.820917	−	0.571048i	$-0.193463\pi$
$397$	2.04690e9	1.64183	0.820917	+	0.571048i	$0.193463\pi$
$398$	1.50852e9	1.19939
$399$	0	0
$400$	−2.86020e8	−0.223453
$401$	4.27245e8	0.330881	0.165440	−	0.986220i	$-0.447095\pi$
$401$	4.27245e8	0.330881	0.165440	+	0.986220i	$0.447095\pi$
$402$	0	0
$403$	1.78975e9	1.36215
$404$	−2.75656e8	−0.207986
$405$	0	0
$406$	2.93645e9	2.17761
$407$	−7.48321e7	−0.0550183
$408$	0	0
$409$	−1.83873e9	−1.32888	−0.664440	−	0.747342i	$-0.731330\pi$
$409$	−1.83873e9	−1.32888	−0.664440	+	0.747342i	$0.731330\pi$
$410$	3.82750e8	0.274266
$411$	0	0
$412$	9.44241e6	0.00665185
$413$	6.31230e8	0.440922
$414$	0	0
$415$	−3.17000e8	−0.217717
$416$	4.09722e8	0.279037
$417$	0	0
$418$	−1.55840e9	−1.04366
$419$	2.78603e9	1.85028	0.925138	−	0.379630i	$-0.123949\pi$
$419$	2.78603e9	1.85028	0.925138	+	0.379630i	$0.123949\pi$
$420$	0	0
$421$	−1.62779e9	−1.06319	−0.531597	−	0.846998i	$-0.678408\pi$
$421$	−1.62779e9	−1.06319	−0.531597	+	0.846998i	$0.678408\pi$
$422$	−5.40846e8	−0.350332
$423$	0	0
$424$	7.81258e8	0.497753
$425$	−1.23321e8	−0.0779245
$426$	0	0
$427$	−4.42711e9	−2.75184
$428$	−1.21071e8	−0.0746423
$429$	0	0
$430$	−6.36713e8	−0.386193
$431$	−3.35057e8	−0.201581	−0.100790	−	0.994908i	$-0.532137\pi$
$431$	−3.35057e8	−0.201581	−0.100790	+	0.994908i	$0.532137\pi$
$432$	0	0
$433$	1.24576e9	0.737440	0.368720	−	0.929541i	$-0.379796\pi$
$433$	1.24576e9	0.737440	0.368720	+	0.929541i	$0.379796\pi$
$434$	3.85469e9	2.26347
$435$	0	0
$436$	−1.48972e8	−0.0860797
$437$	6.42590e8	0.368340
$438$	0	0
$439$	9.53732e8	0.538023	0.269011	−	0.963137i	$-0.413303\pi$
$439$	9.53732e8	0.538023	0.269011	+	0.963137i	$0.413303\pi$
$440$	3.95663e8	0.221432
$441$	0	0
$442$	7.80303e8	0.429819
$443$	−1.97472e9	−1.07918	−0.539589	−	0.841928i	$-0.681420\pi$
$443$	−1.97472e9	−1.07918	−0.539589	+	0.841928i	$0.681420\pi$
$444$	0	0
$445$	−5.11609e8	−0.275219
$446$	−1.06167e9	−0.566651
$447$	0	0
$448$	−2.54972e9	−1.33974
$449$	2.13232e9	1.11171	0.555854	−	0.831280i	$-0.312391\pi$
$449$	2.13232e9	1.11171	0.555854	+	0.831280i	$0.312391\pi$
$450$	0	0
$451$	−6.02656e8	−0.309351
$452$	4.68231e8	0.238493
$453$	0	0
$454$	4.22482e9	2.11891
$455$	1.50144e9	0.747253
$456$	0	0
$457$	1.55327e9	0.761272	0.380636	−	0.924725i	$-0.375705\pi$
$457$	1.55327e9	0.761272	0.380636	+	0.924725i	$0.375705\pi$
$458$	−9.23824e8	−0.449325
$459$	0	0
$460$	2.56097e7	0.0122674
$461$	−3.45071e9	−1.64042	−0.820211	−	0.572061i	$-0.806144\pi$
$461$	−3.45071e9	−1.64042	−0.820211	+	0.572061i	$0.806144\pi$
$462$	0	0
$463$	1.12948e9	0.528866	0.264433	−	0.964404i	$-0.414815\pi$
$463$	1.12948e9	0.528866	0.264433	+	0.964404i	$0.414815\pi$
$464$	−3.04359e9	−1.41440
$465$	0	0
$466$	−3.25826e9	−1.49154
$467$	1.41287e9	0.641936	0.320968	−	0.947090i	$-0.395992\pi$
$467$	1.41287e9	0.641936	0.320968	+	0.947090i	$0.395992\pi$
$468$	0	0
$469$	−1.78468e9	−0.798830
$470$	−1.22013e9	−0.542080
$471$	0	0
$472$	−5.74811e8	−0.251610
$473$	1.00253e9	0.435596
$474$	0	0
$475$	−8.51078e8	−0.364369
$476$	2.00774e8	0.0853262
$477$	0	0
$478$	4.81070e9	2.01470
$479$	2.73876e9	1.13862	0.569311	−	0.822122i	$-0.307210\pi$
$479$	2.73876e9	1.13862	0.569311	+	0.822122i	$0.307210\pi$
$480$	0	0
$481$	2.58587e8	0.105950
$482$	1.85131e9	0.753035
$483$	0	0
$484$	−2.40630e8	−0.0964698
$485$	−5.06807e8	−0.201719
$486$	0	0
$487$	−2.99714e9	−1.17586	−0.587930	−	0.808912i	$-0.700057\pi$
$487$	−2.99714e9	−1.17586	−0.587930	+	0.808912i	$0.700057\pi$
$488$	4.03142e9	1.57032
$489$	0	0
$490$	1.99258e9	0.765120
$491$	−3.88172e9	−1.47992	−0.739961	−	0.672650i	$-0.765156\pi$
$491$	−3.88172e9	−1.47992	−0.739961	+	0.672650i	$0.765156\pi$
$492$	0	0
$493$	−1.31227e9	−0.493242
$494$	5.38514e9	2.00980
$495$	0	0
$496$	−3.99533e9	−1.47017
$497$	7.83070e9	2.86123
$498$	0	0
$499$	−2.56257e9	−0.923259	−0.461629	−	0.887073i	$-0.652735\pi$
$499$	−2.56257e9	−0.923259	−0.461629	+	0.887073i	$0.652735\pi$
$500$	−3.39188e7	−0.0121351
$501$	0	0
$502$	4.99555e9	1.76247
$503$	2.51809e9	0.882233	0.441117	−	0.897450i	$-0.354583\pi$
$503$	2.51809e9	0.882233	0.441117	+	0.897450i	$0.354583\pi$
$504$	0	0
$505$	1.98412e9	0.685565
$506$	−3.37530e8	−0.115821
$507$	0	0
$508$	5.70676e8	0.193138
$509$	−2.07925e9	−0.698868	−0.349434	−	0.936961i	$-0.613626\pi$
$509$	−2.07925e9	−0.698868	−0.349434	+	0.936961i	$0.613626\pi$
$510$	0	0
$511$	−7.33982e9	−2.43339
$512$	2.21076e9	0.727943
$513$	0	0
$514$	2.56970e9	0.834665
$515$	−6.79647e7	−0.0219259
$516$	0	0
$517$	1.92114e9	0.611424
$518$	5.56936e8	0.176056
$519$	0	0
$520$	−1.36724e9	−0.426416
$521$	4.38332e9	1.35791	0.678956	−	0.734179i	$-0.262433\pi$
$521$	4.38332e9	1.35791	0.678956	+	0.734179i	$0.262433\pi$
$522$	0	0
$523$	1.75148e9	0.535363	0.267682	−	0.963507i	$-0.413743\pi$
$523$	1.75148e9	0.535363	0.267682	+	0.963507i	$0.413743\pi$
$524$	2.80108e8	0.0850482
$525$	0	0
$526$	−1.51556e9	−0.454069
$527$	−1.72263e9	−0.512689
$528$	0	0
$529$	−3.26565e9	−0.959123
$530$	8.82709e8	0.257544
$531$	0	0
$532$	1.38561e9	0.398979
$533$	2.08252e9	0.595722
$534$	0	0
$535$	8.71442e8	0.246037
$536$	1.62516e9	0.455848
$537$	0	0
$538$	−1.91493e9	−0.530169
$539$	−3.13740e9	−0.862997
$540$	0	0
$541$	2.73146e9	0.741659	0.370829	−	0.928701i	$-0.379073\pi$
$541$	2.73146e9	0.741659	0.370829	+	0.928701i	$0.379073\pi$
$542$	2.09793e9	0.565970
$543$	0	0
$544$	−3.94356e8	−0.105025
$545$	1.07227e9	0.283737
$546$	0	0
$547$	5.63318e9	1.47163	0.735814	−	0.677184i	$-0.236800\pi$
$547$	5.63318e9	1.47163	0.735814	+	0.677184i	$0.236800\pi$
$548$	5.34508e8	0.138746
$549$	0	0
$550$	4.47042e8	0.114572
$551$	−9.05645e9	−2.30636
$552$	0	0
$553$	−1.22990e10	−3.09267
$554$	−4.08822e8	−0.102153
$555$	0	0
$556$	6.95553e8	0.171620
$557$	−2.58915e7	−0.00634840	−0.00317420	−	0.999995i	$-0.501010\pi$
$557$	−2.58915e7	−0.00634840	−0.00317420	+	0.999995i	$0.501010\pi$
$558$	0	0
$559$	−3.46432e9	−0.838834
$560$	−3.35173e9	−0.806512
$561$	0	0
$562$	−3.42397e9	−0.813677
$563$	−9.99739e8	−0.236106	−0.118053	−	0.993007i	$-0.537665\pi$
$563$	−9.99739e8	−0.236106	−0.118053	+	0.993007i	$0.537665\pi$
$564$	0	0
$565$	−3.37024e9	−0.786124
$566$	−3.66211e9	−0.848934
$567$	0	0
$568$	−7.13079e9	−1.63275
$569$	−3.24256e9	−0.737896	−0.368948	−	0.929450i	$-0.620282\pi$
$569$	−3.24256e9	−0.737896	−0.368948	+	0.929450i	$0.620282\pi$
$570$	0	0
$571$	3.00024e9	0.674419	0.337209	−	0.941430i	$-0.390517\pi$
$571$	3.00024e9	0.674419	0.337209	+	0.941430i	$0.390517\pi$
$572$	−3.37926e8	−0.0754980
$573$	0	0
$574$	4.48526e9	0.989910
$575$	−1.84334e8	−0.0404359
$576$	0	0
$577$	7.38365e9	1.60013	0.800066	−	0.599911i	$-0.204798\pi$
$577$	7.38365e9	1.60013	0.800066	+	0.599911i	$0.204798\pi$
$578$	4.19633e9	0.903904
$579$	0	0
$580$	−3.60934e8	−0.0768122
$581$	−3.71477e9	−0.785805
$582$	0	0
$583$	−1.38986e9	−0.290490
$584$	6.68379e9	1.38860
$585$	0	0
$586$	−4.49023e9	−0.921779
$587$	−4.30762e9	−0.879030	−0.439515	−	0.898235i	$-0.644850\pi$
$587$	−4.30762e9	−0.879030	−0.439515	+	0.898235i	$0.644850\pi$
$588$	0	0
$589$	−1.18884e10	−2.39730
$590$	−6.49453e8	−0.130186
$591$	0	0
$592$	−5.77256e8	−0.114352
$593$	−2.59556e9	−0.511139	−0.255570	−	0.966791i	$-0.582263\pi$
$593$	−2.59556e9	−0.511139	−0.255570	+	0.966791i	$0.582263\pi$
$594$	0	0
$595$	−1.44513e9	−0.281253
$596$	−8.31408e7	−0.0160861
$597$	0	0
$598$	1.16636e9	0.223038
$599$	−6.31359e9	−1.20028	−0.600139	−	0.799895i	$-0.704888\pi$
$599$	−6.31359e9	−1.20028	−0.600139	+	0.799895i	$0.704888\pi$
$600$	0	0
$601$	−1.05517e10	−1.98271	−0.991356	−	0.131197i	$-0.958118\pi$
$601$	−1.05517e10	−1.98271	−0.991356	+	0.131197i	$0.958118\pi$
$602$	−7.46132e9	−1.39389
$603$	0	0
$604$	−4.68447e8	−0.0865030
$605$	1.73201e9	0.317985
$606$	0	0
$607$	6.75373e9	1.22570	0.612849	−	0.790200i	$-0.290023\pi$
$607$	6.75373e9	1.22570	0.612849	+	0.790200i	$0.290023\pi$
$608$	−2.72159e9	−0.491089
$609$	0	0
$610$	4.55492e9	0.812505
$611$	−6.63864e9	−1.17743
$612$	0	0
$613$	−3.91725e9	−0.686863	−0.343431	−	0.939178i	$-0.611589\pi$
$613$	−3.91725e9	−0.686863	−0.343431	+	0.939178i	$0.611589\pi$
$614$	2.50627e9	0.436956
$615$	0	0
$616$	4.63657e9	0.799216
$617$	9.90538e9	1.69775	0.848874	−	0.528596i	$-0.177281\pi$
$617$	9.90538e9	1.69775	0.848874	+	0.528596i	$0.177281\pi$
$618$	0	0
$619$	2.38591e9	0.404331	0.202166	−	0.979351i	$-0.435202\pi$
$619$	2.38591e9	0.404331	0.202166	+	0.979351i	$0.435202\pi$
$620$	−4.73800e8	−0.0798407
$621$	0	0
$622$	−4.01053e9	−0.668244
$623$	−5.99528e9	−0.993348
$624$	0	0
$625$	2.44141e8	0.0400000
$626$	5.66808e9	0.923476
$627$	0	0
$628$	−6.14478e8	−0.0990028
$629$	−2.48890e8	−0.0398777
$630$	0	0
$631$	1.02507e10	1.62425	0.812124	−	0.583485i	$-0.198311\pi$
$631$	1.02507e10	1.62425	0.812124	+	0.583485i	$0.198311\pi$
$632$	1.11998e10	1.76482
$633$	0	0
$634$	4.04769e9	0.630804
$635$	−4.10762e9	−0.636623
$636$	0	0
$637$	1.08415e10	1.66189
$638$	4.75704e9	0.725211
$639$	0	0
$640$	3.42279e9	0.516119
$641$	−3.13604e9	−0.470304	−0.235152	−	0.971959i	$-0.575559\pi$
$641$	−3.13604e9	−0.470304	−0.235152	+	0.971959i	$0.575559\pi$
$642$	0	0
$643$	1.15029e10	1.70635	0.853175	−	0.521625i	$-0.174674\pi$
$643$	1.15029e10	1.70635	0.853175	+	0.521625i	$0.174674\pi$
$644$	3.00107e8	0.0442767
$645$	0	0
$646$	−5.18319e9	−0.756455
$647$	4.90238e8	0.0711610	0.0355805	−	0.999367i	$-0.488672\pi$
$647$	4.90238e8	0.0711610	0.0355805	+	0.999367i	$0.488672\pi$
$648$	0	0
$649$	1.02259e9	0.146840
$650$	−1.54478e9	−0.220633
$651$	0	0
$652$	−1.08468e9	−0.153263
$653$	−1.10590e10	−1.55424	−0.777120	−	0.629352i	$-0.783320\pi$
$653$	−1.10590e10	−1.55424	−0.777120	+	0.629352i	$0.783320\pi$
$654$	0	0
$655$	−2.01616e9	−0.280337
$656$	−4.64890e9	−0.642965
$657$	0	0
$658$	−1.42981e10	−1.95653
$659$	3.05338e9	0.415606	0.207803	−	0.978171i	$-0.433369\pi$
$659$	3.05338e9	0.415606	0.207803	+	0.978171i	$0.433369\pi$
$660$	0	0
$661$	5.14758e9	0.693264	0.346632	−	0.938001i	$-0.387325\pi$
$661$	5.14758e9	0.693264	0.346632	+	0.938001i	$0.387325\pi$
$662$	3.86006e9	0.517120
$663$	0	0
$664$	3.38274e9	0.448416
$665$	−9.97335e9	−1.31512
$666$	0	0
$667$	−1.96152e9	−0.255949
$668$	−4.36271e8	−0.0566290
$669$	0	0
$670$	1.83620e9	0.235862
$671$	−7.17191e9	−0.916444
$672$	0	0
$673$	1.03759e10	1.31212	0.656059	−	0.754710i	$-0.272222\pi$
$673$	1.03759e10	1.31212	0.656059	+	0.754710i	$0.272222\pi$
$674$	7.75248e9	0.975283
$675$	0	0
$676$	7.80106e7	0.00971271
$677$	−1.37331e10	−1.70102	−0.850508	−	0.525961i	$-0.823706\pi$
$677$	−1.37331e10	−1.70102	−0.850508	+	0.525961i	$0.823706\pi$
$678$	0	0
$679$	−5.93901e9	−0.728064
$680$	1.31597e9	0.160496
$681$	0	0
$682$	6.24459e9	0.753804
$683$	8.46133e9	1.01617	0.508085	−	0.861307i	$-0.330354\pi$
$683$	8.46133e9	1.01617	0.508085	+	0.861307i	$0.330354\pi$
$684$	0	0
$685$	−3.84728e9	−0.457338
$686$	8.80549e9	1.04140
$687$	0	0
$688$	7.73355e9	0.905356
$689$	4.80276e9	0.559402
$690$	0	0
$691$	1.38236e10	1.59385	0.796926	−	0.604077i	$-0.206458\pi$
$691$	1.38236e10	1.59385	0.796926	+	0.604077i	$0.206458\pi$
$692$	−1.56118e9	−0.179094
$693$	0	0
$694$	−1.81948e10	−2.06628
$695$	−5.00646e9	−0.565697
$696$	0	0
$697$	−2.00442e9	−0.224220
$698$	9.11898e9	1.01497
$699$	0	0
$700$	−3.97477e8	−0.0437994
$701$	5.52975e9	0.606307	0.303154	−	0.952942i	$-0.401960\pi$
$701$	5.52975e9	0.606307	0.303154	+	0.952942i	$0.401960\pi$
$702$	0	0
$703$	−1.71767e9	−0.186465
$704$	−4.13055e9	−0.446173
$705$	0	0
$706$	1.36246e10	1.45716
$707$	2.32509e10	2.47441
$708$	0	0
$709$	5.99768e9	0.632006	0.316003	−	0.948758i	$-0.397659\pi$
$709$	5.99768e9	0.632006	0.316003	+	0.948758i	$0.397659\pi$
$710$	−8.05677e9	−0.844805
$711$	0	0
$712$	5.45943e9	0.566849
$713$	−2.57490e9	−0.266040
$714$	0	0
$715$	2.43233e9	0.248857
$716$	−9.12919e8	−0.0929473
$717$	0	0
$718$	5.10141e9	0.514345
$719$	−1.66927e10	−1.67484	−0.837422	−	0.546557i	$-0.815938\pi$
$719$	−1.66927e10	−1.67484	−0.837422	+	0.546557i	$0.815938\pi$
$720$	0	0
$721$	−7.96443e8	−0.0791373
$722$	−2.49938e10	−2.47145
$723$	0	0
$724$	−1.49791e8	−0.0146690
$725$	2.59794e9	0.253189
$726$	0	0
$727$	9.44607e8	0.0911760	0.0455880	−	0.998960i	$-0.485484\pi$
$727$	9.44607e8	0.0911760	0.0455880	+	0.998960i	$0.485484\pi$
$728$	−1.60220e10	−1.53906
$729$	0	0
$730$	7.55171e9	0.718481
$731$	3.33440e9	0.315723
$732$	0	0
$733$	4.41352e9	0.413925	0.206962	−	0.978349i	$-0.433642\pi$
$733$	4.41352e9	0.413925	0.206962	+	0.978349i	$0.433642\pi$
$734$	−1.46630e10	−1.36863
$735$	0	0
$736$	−5.89465e8	−0.0544987
$737$	−2.89117e9	−0.266034
$738$	0	0
$739$	8.97799e9	0.818321	0.409160	−	0.912463i	$-0.365822\pi$
$739$	8.97799e9	0.818321	0.409160	+	0.912463i	$0.365822\pi$
$740$	−6.84560e7	−0.00621012
$741$	0	0
$742$	1.03440e10	0.929556
$743$	7.16921e8	0.0641225	0.0320612	−	0.999486i	$-0.489793\pi$
$743$	7.16921e8	0.0641225	0.0320612	+	0.999486i	$0.489793\pi$
$744$	0	0
$745$	5.98431e8	0.0530234
$746$	7.10454e9	0.626542
$747$	0	0
$748$	3.25253e8	0.0284162
$749$	1.02120e10	0.888022
$750$	0	0
$751$	−2.15502e9	−0.185657	−0.0928287	−	0.995682i	$-0.529591\pi$
$751$	−2.15502e9	−0.185657	−0.0928287	+	0.995682i	$0.529591\pi$
$752$	1.48197e10	1.27080
$753$	0	0
$754$	−1.64383e10	−1.39655
$755$	3.37179e9	0.285132
$756$	0	0
$757$	−1.95505e10	−1.63803	−0.819014	−	0.573773i	$-0.805479\pi$
$757$	−1.95505e10	−1.63803	−0.819014	+	0.573773i	$0.805479\pi$
$758$	2.11775e9	0.176617
$759$	0	0
$760$	9.08194e9	0.750466
$761$	−2.21298e10	−1.82025	−0.910125	−	0.414334i	$-0.864015\pi$
$761$	−2.21298e10	−1.82025	−0.910125	+	0.414334i	$0.864015\pi$
$762$	0	0
$763$	1.25654e10	1.02409
$764$	−2.47634e9	−0.200901
$765$	0	0
$766$	6.93613e9	0.557592
$767$	−3.53363e9	−0.282773
$768$	0	0
$769$	−1.44804e10	−1.14826	−0.574129	−	0.818765i	$-0.694659\pi$
$769$	−1.44804e10	−1.14826	−0.574129	+	0.818765i	$0.694659\pi$
$770$	5.23866e9	0.413526
$771$	0	0
$772$	1.32561e9	0.103695
$773$	−1.28374e10	−0.999653	−0.499826	−	0.866126i	$-0.666603\pi$
$773$	−1.28374e10	−0.999653	−0.499826	+	0.866126i	$0.666603\pi$
$774$	0	0
$775$	3.41032e9	0.263172
$776$	5.40818e9	0.415466
$777$	0	0
$778$	−5.85564e9	−0.445806
$779$	−1.38332e10	−1.04844
$780$	0	0
$781$	1.26857e10	0.952876
$782$	−1.12262e9	−0.0839477
$783$	0	0
$784$	−2.42020e10	−1.79368
$785$	4.42289e9	0.326334
$786$	0	0
$787$	−8.56090e9	−0.626048	−0.313024	−	0.949745i	$-0.601342\pi$
$787$	−8.56090e9	−0.626048	−0.313024	+	0.949745i	$0.601342\pi$
$788$	1.11898e9	0.0814671
$789$	0	0
$790$	1.26541e10	0.913139
$791$	−3.94941e10	−2.83736
$792$	0	0
$793$	2.47830e10	1.76481
$794$	−2.46790e10	−1.74967
$795$	0	0
$796$	−2.17284e9	−0.152698
$797$	−2.34604e7	−0.00164147	−0.000820733	−	1.00000i	$-0.500261\pi$
$797$	−2.34604e7	−0.00164147	−0.000820733		1.00000i	$0.500261\pi$
$798$	0	0
$799$	6.38968e9	0.443165
$800$	7.80717e8	0.0539111
$801$	0	0
$802$	−5.15120e9	−0.352613
$803$	−1.18905e10	−0.810392
$804$	0	0
$805$	−2.16011e9	−0.145946
$806$	−2.15786e10	−1.45161
$807$	0	0
$808$	−2.11728e10	−1.41201
$809$	−6.34015e9	−0.420998	−0.210499	−	0.977594i	$-0.567509\pi$
$809$	−6.34015e9	−0.420998	−0.210499	+	0.977594i	$0.567509\pi$
$810$	0	0
$811$	−1.94688e10	−1.28164	−0.640820	−	0.767691i	$-0.721405\pi$
$811$	−1.94688e10	−1.28164	−0.640820	+	0.767691i	$0.721405\pi$
$812$	−4.22961e9	−0.277239
$813$	0	0
$814$	9.02235e8	0.0586319
$815$	7.80735e9	0.505187
$816$	0	0
$817$	2.30118e10	1.47630
$818$	2.21692e10	1.41616
$819$	0	0
$820$	−5.51307e8	−0.0349176
$821$	−4.29792e9	−0.271055	−0.135527	−	0.990774i	$-0.543273\pi$
$821$	−4.29792e9	−0.271055	−0.135527	+	0.990774i	$0.543273\pi$
$822$	0	0
$823$	−2.40020e10	−1.50089	−0.750443	−	0.660936i	$-0.770160\pi$
$823$	−2.40020e10	−1.50089	−0.750443	+	0.660936i	$0.770160\pi$
$824$	7.25257e8	0.0451593
$825$	0	0
$826$	−7.61061e9	−0.469882
$827$	−2.64939e10	−1.62883	−0.814417	−	0.580281i	$-0.802943\pi$
$827$	−2.64939e10	−1.62883	−0.814417	+	0.580281i	$0.802943\pi$
$828$	0	0
$829$	−1.08670e10	−0.662475	−0.331237	−	0.943547i	$-0.607466\pi$
$829$	−1.08670e10	−0.662475	−0.331237	+	0.943547i	$0.607466\pi$
$830$	3.82201e9	0.232016
$831$	0	0
$832$	1.42734e10	0.859203
$833$	−1.04349e10	−0.625506
$834$	0	0
$835$	3.14020e9	0.186661
$836$	2.24469e9	0.132872
$837$	0	0
$838$	−3.35906e10	−1.97180
$839$	2.02927e10	1.18624	0.593120	−	0.805114i	$-0.297896\pi$
$839$	2.02927e10	1.18624	0.593120	+	0.805114i	$0.297896\pi$
$840$	0	0
$841$	1.03951e10	0.602621
$842$	1.96260e10	1.13302
$843$	0	0
$844$	7.79025e8	0.0446018
$845$	−5.61505e8	−0.0320151
$846$	0	0
$847$	2.02965e10	1.14770
$848$	−1.07214e10	−0.603764
$849$	0	0
$850$	1.48685e9	0.0830427
$851$	−3.72028e8	−0.0206930
$852$	0	0
$853$	−9.91606e9	−0.547038	−0.273519	−	0.961867i	$-0.588188\pi$
$853$	−9.91606e9	−0.547038	−0.273519	+	0.961867i	$0.588188\pi$
$854$	5.33768e10	2.93258
$855$	0	0
$856$	−9.29924e9	−0.506745
$857$	−1.62876e10	−0.883945	−0.441973	−	0.897028i	$-0.645721\pi$
$857$	−1.62876e10	−0.883945	−0.441973	+	0.897028i	$0.645721\pi$
$858$	0	0
$859$	−1.20162e10	−0.646834	−0.323417	−	0.946257i	$-0.604832\pi$
$859$	−1.20162e10	−0.646834	−0.323417	+	0.946257i	$0.604832\pi$
$860$	9.17111e8	0.0491674
$861$	0	0
$862$	4.03972e9	0.214820
$863$	2.81898e10	1.49298	0.746489	−	0.665397i	$-0.231738\pi$
$863$	2.81898e10	1.49298	0.746489	+	0.665397i	$0.231738\pi$
$864$	0	0
$865$	1.12371e10	0.590332
$866$	−1.50199e10	−0.785875
$867$	0	0
$868$	−5.55223e9	−0.288170
$869$	−1.99244e10	−1.02995
$870$	0	0
$871$	9.99064e9	0.512306
$872$	−1.14423e10	−0.584393
$873$	0	0
$874$	−7.74758e9	−0.392533
$875$	2.86096e9	0.144372
$876$	0	0
$877$	−5.00888e9	−0.250751	−0.125375	−	0.992109i	$-0.540014\pi$
$877$	−5.00888e9	−0.250751	−0.125375	+	0.992109i	$0.540014\pi$
$878$	−1.14990e10	−0.573360
$879$	0	0
$880$	−5.42979e9	−0.268593
$881$	2.73101e10	1.34557	0.672787	−	0.739837i	$-0.265097\pi$
$881$	2.73101e10	1.34557	0.672787	+	0.739837i	$0.265097\pi$
$882$	0	0
$883$	1.74787e10	0.854374	0.427187	−	0.904163i	$-0.359505\pi$
$883$	1.74787e10	0.854374	0.427187	+	0.904163i	$0.359505\pi$
$884$	−1.12393e9	−0.0547215
$885$	0	0
$886$	2.38088e10	1.15006
$887$	1.03162e10	0.496350	0.248175	−	0.968715i	$-0.420169\pi$
$887$	1.03162e10	0.496350	0.248175	+	0.968715i	$0.420169\pi$
$888$	0	0
$889$	−4.81351e10	−2.29776
$890$	6.16836e9	0.293295
$891$	0	0
$892$	1.52921e9	0.0721421
$893$	4.40974e10	2.07221
$894$	0	0
$895$	6.57102e9	0.306374
$896$	4.01099e10	1.86283
$897$	0	0
$898$	−2.57090e10	−1.18473
$899$	3.62898e10	1.66581
$900$	0	0
$901$	−4.62265e9	−0.210549
$902$	7.26611e9	0.329670
$903$	0	0
$904$	3.59641e10	1.61912
$905$	1.07816e9	0.0483520
$906$	0	0
$907$	−4.73650e9	−0.210781	−0.105391	−	0.994431i	$-0.533609\pi$
$907$	−4.73650e9	−0.210781	−0.105391	+	0.994431i	$0.533609\pi$
$908$	−6.08536e9	−0.269765
$909$	0	0
$910$	−1.81025e10	−0.796333
$911$	3.13815e10	1.37518	0.687589	−	0.726100i	$-0.258669\pi$
$911$	3.13815e10	1.37518	0.687589	+	0.726100i	$0.258669\pi$
$912$	0	0
$913$	−6.01792e9	−0.261697
$914$	−1.87274e10	−0.811273
$915$	0	0
$916$	1.33066e9	0.0572049
$917$	−2.36263e10	−1.01182
$918$	0	0
$919$	−2.15885e9	−0.0917527	−0.0458764	−	0.998947i	$-0.514608\pi$
$919$	−2.15885e9	−0.0917527	−0.0458764	+	0.998947i	$0.514608\pi$
$920$	1.96704e9	0.0832830
$921$	0	0
$922$	4.16046e10	1.74817
$923$	−4.38363e10	−1.83497
$924$	0	0
$925$	4.92733e8	0.0204699
$926$	−1.36179e10	−0.563602
$927$	0	0
$928$	8.30772e9	0.341243
$929$	−1.14185e10	−0.467254	−0.233627	−	0.972326i	$-0.575059\pi$
$929$	−1.14185e10	−0.467254	−0.233627	+	0.972326i	$0.575059\pi$
$930$	0	0
$931$	−7.20150e10	−2.92482
$932$	4.69315e9	0.189893
$933$	0	0
$934$	−1.70346e10	−0.684099
$935$	−2.34111e9	−0.0936658
$936$	0	0
$937$	3.92128e10	1.55718	0.778591	−	0.627531i	$-0.215935\pi$
$937$	3.92128e10	1.55718	0.778591	+	0.627531i	$0.215935\pi$
$938$	2.15175e10	0.851298
$939$	0	0
$940$	1.75745e9	0.0690138
$941$	−3.34263e9	−0.130775	−0.0653876	−	0.997860i	$-0.520828\pi$
$941$	−3.34263e9	−0.130775	−0.0653876	+	0.997860i	$0.520828\pi$
$942$	0	0
$943$	−2.99611e9	−0.116350
$944$	7.88829e9	0.305197
$945$	0	0
$946$	−1.20873e10	−0.464206
$947$	2.47701e10	0.947770	0.473885	−	0.880587i	$-0.342851\pi$
$947$	2.47701e10	0.947770	0.473885	+	0.880587i	$0.342851\pi$
$948$	0	0
$949$	4.10884e10	1.56059
$950$	1.02613e10	0.388301
$951$	0	0
$952$	1.54211e10	0.579278
$953$	−1.62449e10	−0.607983	−0.303991	−	0.952675i	$-0.598319\pi$
$953$	−1.62449e10	−0.607983	−0.303991	+	0.952675i	$0.598319\pi$
$954$	0	0
$955$	1.78242e10	0.662213
$956$	−6.92925e9	−0.256498
$957$	0	0
$958$	−3.30207e10	−1.21341
$959$	−4.50843e10	−1.65067
$960$	0	0
$961$	2.01251e10	0.731488
$962$	−3.11773e9	−0.112908
$963$	0	0
$964$	−2.66660e9	−0.0958712
$965$	−9.54151e9	−0.341800
$966$	0	0
$967$	1.85625e10	0.660150	0.330075	−	0.943955i	$-0.392926\pi$
$967$	1.85625e10	0.660150	0.330075	+	0.943955i	$0.392926\pi$
$968$	−1.84824e10	−0.654931
$969$	0	0
$970$	6.11047e9	0.214968
$971$	3.52896e10	1.23703	0.618515	−	0.785773i	$-0.287735\pi$
$971$	3.52896e10	1.23703	0.618515	+	0.785773i	$0.287735\pi$
$972$	0	0
$973$	−5.86681e10	−2.04177
$974$	3.61359e10	1.25309
$975$	0	0
$976$	−5.53243e10	−1.90476
$977$	−3.39159e10	−1.16352	−0.581759	−	0.813361i	$-0.697635\pi$
$977$	−3.39159e10	−1.16352	−0.581759	+	0.813361i	$0.697635\pi$
$978$	0	0
$979$	−9.71235e9	−0.330815
$980$	−2.87008e9	−0.0974097
$981$	0	0
$982$	4.68011e10	1.57712
$983$	−3.99312e10	−1.34084	−0.670418	−	0.741984i	$-0.733885\pi$
$983$	−3.99312e10	−1.34084	−0.670418	+	0.741984i	$0.733885\pi$
$984$	0	0
$985$	−8.05424e9	−0.268533
$986$	1.58218e10	0.525638
$987$	0	0
$988$	−7.75666e9	−0.255874
$989$	4.98410e9	0.163832
$990$	0	0
$991$	2.11416e10	0.690050	0.345025	−	0.938594i	$-0.387870\pi$
$991$	2.11416e10	0.690050	0.345025	+	0.938594i	$0.387870\pi$
$992$	1.09056e10	0.354698
$993$	0	0
$994$	−9.44131e10	−3.04916
$995$	1.56397e10	0.503324
$996$	0	0
$997$	1.17905e9	0.0376791	0.0188395	−	0.999823i	$-0.494003\pi$
$997$	1.17905e9	0.0376791	0.0188395	+	0.999823i	$0.494003\pi$
$998$	3.08964e10	0.983899
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.8.a.g.1.5	✓	14
3.2	odd	2		405.8.a.h.1.10	yes	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
405.8.a.g.1.5	✓	14	1.1	even	1	trivial
405.8.a.h.1.10	yes	14	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	405.a (trivial)

\(f(q)\)	\(=\)	\(q-12.0568 q^{2} +17.3664 q^{4} -125.000 q^{5} -1464.81 q^{7} +1333.89 q^{8} +O(q^{10})\)	\(q-12.0568 q^{2} +17.3664 q^{4} -125.000 q^{5} -1464.81 q^{7} +1333.89 q^{8} +1507.10 q^{10} -2372.99 q^{11} +8200.03 q^{13} +17660.9 q^{14} -18305.3 q^{16} -7892.52 q^{17} -54469.0 q^{19} -2170.80 q^{20} +28610.7 q^{22} -11797.4 q^{23} +15625.0 q^{25} -98866.2 q^{26} -25438.5 q^{28} +166268. q^{29} +218261. q^{31} +49965.9 q^{32} +95158.5 q^{34} +183101. q^{35} +31534.9 q^{37} +656722. q^{38} -166736. q^{40} +253965. q^{41} -422476. q^{43} -41210.3 q^{44} +142238. q^{46} -809587. q^{47} +1.32213e6 q^{49} -188387. q^{50} +142405. q^{52} +585700. q^{53} +296624. q^{55} -1.95389e6 q^{56} -2.00466e6 q^{58} -430929. q^{59} +3.02231e6 q^{61} -2.63153e6 q^{62} +1.74065e6 q^{64} -1.02500e6 q^{65} +1.21837e6 q^{67} -137065. q^{68} -2.20762e6 q^{70} -5.34587e6 q^{71} +5.01076e6 q^{73} -380210. q^{74} -945931. q^{76} +3.47598e6 q^{77} +8.39634e6 q^{79} +2.28816e6 q^{80} -3.06200e6 q^{82} +2.53600e6 q^{83} +986564. q^{85} +5.09371e6 q^{86} -3.16530e6 q^{88} +4.09287e6 q^{89} -1.20115e7 q^{91} -204878. q^{92} +9.76103e6 q^{94} +6.80863e6 q^{95} +4.05445e6 q^{97} -1.59406e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 16 q^{2} + 714 q^{4} - 1750 q^{5} + 1538 q^{7} + 126 q^{8}+O(q^{10}) \) 14 * q - 16 * q^2 + 714 * q^4 - 1750 * q^5 + 1538 * q^7 + 126 * q^8	\( 14 q - 16 q^{2} + 714 q^{4} - 1750 q^{5} + 1538 q^{7} + 126 q^{8} + 2000 q^{10} - 10648 q^{11} + 17268 q^{13} - 9180 q^{14} + 34122 q^{16} - 3886 q^{17} + 22986 q^{19} - 89250 q^{20} - 49706 q^{22} - 155040 q^{23} + 218750 q^{25} - 7804 q^{26} + 256160 q^{28} - 195836 q^{29} + 92786 q^{31} - 157778 q^{32} + 787348 q^{34} - 192250 q^{35} + 876406 q^{37} - 329320 q^{38} - 15750 q^{40} - 795164 q^{41} + 730350 q^{43} - 2360876 q^{44} - 225654 q^{46} - 2687842 q^{47} + 1663586 q^{49} - 250000 q^{50} + 3875836 q^{52} - 2533750 q^{53} + 1331000 q^{55} - 2055276 q^{56} - 318934 q^{58} - 2283340 q^{59} + 2600400 q^{61} - 5702022 q^{62} + 474098 q^{64} - 2158500 q^{65} + 2422160 q^{67} - 1114364 q^{68} + 1147500 q^{70} - 6395324 q^{71} - 540774 q^{73} - 260516 q^{74} - 2417042 q^{76} + 1384890 q^{77} - 307384 q^{79} - 4265250 q^{80} - 14044738 q^{82} - 10991322 q^{83} + 485750 q^{85} - 2847712 q^{86} - 19226592 q^{88} - 2094000 q^{89} - 9496256 q^{91} - 39213378 q^{92} - 28132682 q^{94} - 2873250 q^{95} - 12859994 q^{97} - 28336478 q^{98}+O(q^{100}) \) 14 * q - 16 * q^2 + 714 * q^4 - 1750 * q^5 + 1538 * q^7 + 126 * q^8 + 2000 * q^10 - 10648 * q^11 + 17268 * q^13 - 9180 * q^14 + 34122 * q^16 - 3886 * q^17 + 22986 * q^19 - 89250 * q^20 - 49706 * q^22 - 155040 * q^23 + 218750 * q^25 - 7804 * q^26 + 256160 * q^28 - 195836 * q^29 + 92786 * q^31 - 157778 * q^32 + 787348 * q^34 - 192250 * q^35 + 876406 * q^37 - 329320 * q^38 - 15750 * q^40 - 795164 * q^41 + 730350 * q^43 - 2360876 * q^44 - 225654 * q^46 - 2687842 * q^47 + 1663586 * q^49 - 250000 * q^50 + 3875836 * q^52 - 2533750 * q^53 + 1331000 * q^55 - 2055276 * q^56 - 318934 * q^58 - 2283340 * q^59 + 2600400 * q^61 - 5702022 * q^62 + 474098 * q^64 - 2158500 * q^65 + 2422160 * q^67 - 1114364 * q^68 + 1147500 * q^70 - 6395324 * q^71 - 540774 * q^73 - 260516 * q^74 - 2417042 * q^76 + 1384890 * q^77 - 307384 * q^79 - 4265250 * q^80 - 14044738 * q^82 - 10991322 * q^83 + 485750 * q^85 - 2847712 * q^86 - 19226592 * q^88 - 2094000 * q^89 - 9496256 * q^91 - 39213378 * q^92 - 28132682 * q^94 - 2873250 * q^95 - 12859994 * q^97 - 28336478 * q^98

Introduction

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