LMFDB - Embedded newform 405.8.a.i.1.1

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			1.1
Root			$-21.0996$ 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)

$f(q)$	$=$	$q-22.0996 q^{2} +360.392 q^{4} -125.000 q^{5} -1217.00 q^{7} -5135.77 q^{8} +O(q^{10})$	\(q-22.0996 q^{2} +360.392 q^{4} -125.000 q^{5} -1217.00 q^{7} -5135.77 q^{8} +2762.45 q^{10} +4240.51 q^{11} +6565.14 q^{13} +26895.1 q^{14} +67368.2 q^{16} +52.6752 q^{17} +29264.7 q^{19} -45049.0 q^{20} -93713.6 q^{22} -71911.8 q^{23} +15625.0 q^{25} -145087. q^{26} -438596. q^{28} -3018.73 q^{29} +167138. q^{31} -831431. q^{32} -1164.10 q^{34} +152125. q^{35} +332127. q^{37} -646738. q^{38} +641971. q^{40} -258257. q^{41} +359598. q^{43} +1.52825e6 q^{44} +1.58922e6 q^{46} -886155. q^{47} +657540. q^{49} -345306. q^{50} +2.36602e6 q^{52} +552041. q^{53} -530064. q^{55} +6.25021e6 q^{56} +66712.7 q^{58} +842213. q^{59} -1.65464e6 q^{61} -3.69367e6 q^{62} +9.75115e6 q^{64} -820643. q^{65} -1.30236e6 q^{67} +18983.7 q^{68} -3.36189e6 q^{70} -3.50388e6 q^{71} -2.14571e6 q^{73} -7.33987e6 q^{74} +1.05468e7 q^{76} -5.16070e6 q^{77} +3.08591e6 q^{79} -8.42102e6 q^{80} +5.70738e6 q^{82} +7.81178e6 q^{83} -6584.40 q^{85} -7.94698e6 q^{86} -2.17783e7 q^{88} +1.32762e6 q^{89} -7.98976e6 q^{91} -2.59164e7 q^{92} +1.95837e7 q^{94} -3.65809e6 q^{95} +4.44129e6 q^{97} -1.45314e7 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−22.0996	−1.95335	−0.976673	−	0.214731i	$-0.931112\pi$
$2$	−22.0996	−1.95335	−0.976673	+	0.214731i	$0.931112\pi$
$3$	0	0
$3$	0	0
$4$	360.392	2.81556
$5$	−125.000	−0.447214
$5$	−125.000	−0.447214
$6$	0	0
$7$	−1217.00	−1.34106	−0.670528	−	0.741885i	$-0.733932\pi$
$7$	−1217.00	−1.34106	−0.670528	+	0.741885i	$0.733932\pi$
$8$	−5135.77	−3.54642
$9$	0	0
$10$	2762.45	0.873563
$11$	4240.51	0.960603	0.480302	−	0.877103i	$-0.340527\pi$
$11$	4240.51	0.960603	0.480302	+	0.877103i	$0.340527\pi$
$12$	0	0
$13$	6565.14	0.828786	0.414393	−	0.910098i	$-0.363994\pi$
$13$	6565.14	0.828786	0.414393	+	0.910098i	$0.363994\pi$
$14$	26895.1	2.61954
$15$	0	0
$16$	67368.2	4.11183
$17$	52.6752	0.00260037	0.00130019	−	0.999999i	$-0.499586\pi$
$17$	52.6752	0.00260037	0.00130019	+	0.999999i	$0.499586\pi$
$18$	0	0
$19$	29264.7	0.978829	0.489414	−	0.872051i	$-0.337211\pi$
$19$	29264.7	0.978829	0.489414	+	0.872051i	$0.337211\pi$
$20$	−45049.0	−1.25916
$21$	0	0
$22$	−93713.6	−1.87639
$23$	−71911.8	−1.23240	−0.616201	−	0.787589i	$-0.711329\pi$
$23$	−71911.8	−1.23240	−0.616201	+	0.787589i	$0.711329\pi$
$24$	0	0
$25$	15625.0	0.200000
$26$	−145087.	−1.61891
$27$	0	0
$28$	−438596.	−3.77582
$29$	−3018.73	−0.0229843	−0.0114922	−	0.999934i	$-0.503658\pi$
$29$	−3018.73	−0.0229843	−0.0114922	+	0.999934i	$0.503658\pi$
$30$	0	0
$31$	167138.	1.00765	0.503823	−	0.863807i	$-0.331926\pi$
$31$	167138.	1.00765	0.503823	+	0.863807i	$0.331926\pi$
$32$	−831431.	−4.48540
$33$	0	0
$34$	−1164.10	−0.00507942
$35$	152125.	0.599738
$36$	0	0
$37$	332127.	1.07795	0.538974	−	0.842322i	$-0.318812\pi$
$37$	332127.	1.07795	0.538974	+	0.842322i	$0.318812\pi$
$38$	−646738.	−1.91199
$39$	0	0
$40$	641971.	1.58601
$41$	−258257.	−0.585207	−0.292603	−	0.956234i	$-0.594522\pi$
$41$	−258257.	−0.585207	−0.292603	+	0.956234i	$0.594522\pi$
$42$	0	0
$43$	359598.	0.689728	0.344864	−	0.938653i	$-0.387925\pi$
$43$	359598.	0.689728	0.344864	+	0.938653i	$0.387925\pi$
$44$	1.52825e6	2.70464
$45$	0	0
$46$	1.58922e6	2.40731
$47$	−886155.	−1.24499	−0.622497	−	0.782623i	$-0.713882\pi$
$47$	−886155.	−1.24499	−0.622497	+	0.782623i	$0.713882\pi$
$48$	0	0
$49$	657540.	0.798429
$50$	−345306.	−0.390669
$51$	0	0
$52$	2.36602e6	2.33350
$53$	552041.	0.509338	0.254669	−	0.967028i	$-0.418034\pi$
$53$	552041.	0.509338	0.254669	+	0.967028i	$0.418034\pi$
$54$	0	0
$55$	−530064.	−0.429595
$56$	6.25021e6	4.75595
$57$	0	0
$58$	66712.7	0.0448963
$59$	842213.	0.533876	0.266938	−	0.963714i	$-0.413988\pi$
$59$	842213.	0.533876	0.266938	+	0.963714i	$0.413988\pi$
$60$	0	0
$61$	−1.65464e6	−0.933358	−0.466679	−	0.884427i	$-0.654550\pi$
$61$	−1.65464e6	−0.933358	−0.466679	+	0.884427i	$0.654550\pi$
$62$	−3.69367e6	−1.96828
$63$	0	0
$64$	9.75115e6	4.64971
$65$	−820643.	−0.370644
$66$	0	0
$67$	−1.30236e6	−0.529015	−0.264507	−	0.964384i	$-0.585209\pi$
$67$	−1.30236e6	−0.529015	−0.264507	+	0.964384i	$0.585209\pi$
$68$	18983.7	0.00732150
$69$	0	0
$70$	−3.36189e6	−1.17150
$71$	−3.50388e6	−1.16184	−0.580919	−	0.813961i	$-0.697307\pi$
$71$	−3.50388e6	−1.16184	−0.580919	+	0.813961i	$0.697307\pi$
$72$	0	0
$73$	−2.14571e6	−0.645568	−0.322784	−	0.946473i	$-0.604619\pi$
$73$	−2.14571e6	−0.645568	−0.322784	+	0.946473i	$0.604619\pi$
$74$	−7.33987e6	−2.10561
$75$	0	0
$76$	1.05468e7	2.75595
$77$	−5.16070e6	−1.28822
$78$	0	0
$79$	3.08591e6	0.704188	0.352094	−	0.935965i	$-0.385470\pi$
$79$	3.08591e6	0.704188	0.352094	+	0.935965i	$0.385470\pi$
$80$	−8.42102e6	−1.83886
$81$	0	0
$82$	5.70738e6	1.14311
$83$	7.81178e6	1.49960	0.749802	−	0.661663i	$-0.230149\pi$
$83$	7.81178e6	1.49960	0.749802	+	0.661663i	$0.230149\pi$
$84$	0	0
$85$	−6584.40	−0.00116292
$86$	−7.94698e6	−1.34728
$87$	0	0
$88$	−2.17783e7	−3.40670
$89$	1.32762e6	0.199622	0.0998108	−	0.995006i	$-0.468176\pi$
$89$	1.32762e6	0.199622	0.0998108	+	0.995006i	$0.468176\pi$
$90$	0	0
$91$	−7.98976e6	−1.11145
$92$	−2.59164e7	−3.46990
$93$	0	0
$94$	1.95837e7	2.43190
$95$	−3.65809e6	−0.437746
$96$	0	0
$97$	4.44129e6	0.494092	0.247046	−	0.969004i	$-0.420540\pi$
$97$	4.44129e6	0.494092	0.247046	+	0.969004i	$0.420540\pi$
$98$	−1.45314e7	−1.55961
$99$	0	0
$100$	5.63112e6	0.563112
$101$	5.91783e6	0.571528	0.285764	−	0.958300i	$-0.407753\pi$
$101$	5.91783e6	0.571528	0.285764	+	0.958300i	$0.407753\pi$
$102$	0	0
$103$	−1.86874e7	−1.68507	−0.842536	−	0.538640i	$-0.818938\pi$
$103$	−1.86874e7	−1.68507	−0.842536	+	0.538640i	$0.818938\pi$
$104$	−3.37170e7	−2.93922
$105$	0	0
$106$	−1.21999e7	−0.994913
$107$	8.05747e6	0.635851	0.317925	−	0.948116i	$-0.397014\pi$
$107$	8.05747e6	0.635851	0.317925	+	0.948116i	$0.397014\pi$
$108$	0	0
$109$	−2.03900e7	−1.50808	−0.754040	−	0.656829i	$-0.771897\pi$
$109$	−2.03900e7	−1.50808	−0.754040	+	0.656829i	$0.771897\pi$
$110$	1.17142e7	0.839147
$111$	0	0
$112$	−8.19869e7	−5.51418
$113$	−7.61400e6	−0.496408	−0.248204	−	0.968708i	$-0.579840\pi$
$113$	−7.61400e6	−0.496408	−0.248204	+	0.968708i	$0.579840\pi$
$114$	0	0
$115$	8.98897e6	0.551147
$116$	−1.08793e6	−0.0647137
$117$	0	0
$118$	−1.86126e7	−1.04284
$119$	−64105.6	−0.00348724
$120$	0	0
$121$	−1.50522e6	−0.0772414
$122$	3.65668e7	1.82317
$123$	0	0
$124$	6.02350e7	2.83709
$125$	−1.95312e6	−0.0894427
$126$	0	0
$127$	−2.31228e7	−1.00168	−0.500838	−	0.865541i	$-0.666975\pi$
$127$	−2.31228e7	−1.00168	−0.500838	+	0.865541i	$0.666975\pi$
$128$	−1.09073e8	−4.59710
$129$	0	0
$130$	1.81359e7	0.723997
$131$	−1.89703e7	−0.737268	−0.368634	−	0.929575i	$-0.620175\pi$
$131$	−1.89703e7	−0.737268	−0.368634	+	0.929575i	$0.620175\pi$
$132$	0	0
$133$	−3.56151e7	−1.31266
$134$	2.87815e7	1.03335
$135$	0	0
$136$	−270528.	−0.00922201
$137$	4.83720e7	1.60721	0.803603	−	0.595165i	$-0.202913\pi$
$137$	4.83720e7	1.60721	0.803603	+	0.595165i	$0.202913\pi$
$138$	0	0
$139$	3.99532e7	1.26183	0.630914	−	0.775853i	$-0.282680\pi$
$139$	3.99532e7	1.26183	0.630914	+	0.775853i	$0.282680\pi$
$140$	5.48245e7	1.68860
$141$	0	0
$142$	7.74344e7	2.26947
$143$	2.78396e7	0.796135
$144$	0	0
$145$	377341.	0.0102789
$146$	4.74194e7	1.26102
$147$	0	0
$148$	1.19696e8	3.03503
$149$	−4.99679e7	−1.23748	−0.618742	−	0.785594i	$-0.712357\pi$
$149$	−4.99679e7	−1.23748	−0.618742	+	0.785594i	$0.712357\pi$
$150$	0	0
$151$	−1.51982e7	−0.359230	−0.179615	−	0.983737i	$-0.557485\pi$
$151$	−1.51982e7	−0.359230	−0.179615	+	0.983737i	$0.557485\pi$
$152$	−1.50297e8	−3.47134
$153$	0	0
$154$	1.14049e8	2.51634
$155$	−2.08922e7	−0.450633
$156$	0	0
$157$	7.91530e7	1.63237	0.816185	−	0.577791i	$-0.196085\pi$
$157$	7.91530e7	1.63237	0.816185	+	0.577791i	$0.196085\pi$
$158$	−6.81974e7	−1.37552
$159$	0	0
$160$	1.03929e8	2.00593
$161$	8.75164e7	1.65272
$162$	0	0
$163$	−7.75660e7	−1.40286	−0.701431	−	0.712738i	$-0.747455\pi$
$163$	−7.75660e7	−1.40286	−0.701431	+	0.712738i	$0.747455\pi$
$164$	−9.30739e7	−1.64769
$165$	0	0
$166$	−1.72637e8	−2.92924
$167$	4.28667e7	0.712218	0.356109	−	0.934444i	$-0.384103\pi$
$167$	4.28667e7	0.712218	0.356109	+	0.934444i	$0.384103\pi$
$168$	0	0
$169$	−1.96474e7	−0.313113
$170$	145513.	0.00227159
$171$	0	0
$172$	1.29596e8	1.94197
$173$	−9.49431e7	−1.39413	−0.697063	−	0.717010i	$-0.745510\pi$
$173$	−9.49431e7	−1.39413	−0.697063	+	0.717010i	$0.745510\pi$
$174$	0	0
$175$	−1.90156e7	−0.268211
$176$	2.85676e8	3.94983
$177$	0	0
$178$	−2.93398e7	−0.389930
$179$	−4.58166e7	−0.597086	−0.298543	−	0.954396i	$-0.596501\pi$
$179$	−4.58166e7	−0.597086	−0.298543	+	0.954396i	$0.596501\pi$
$180$	0	0
$181$	4.93404e7	0.618482	0.309241	−	0.950984i	$-0.399925\pi$
$181$	4.93404e7	0.618482	0.309241	+	0.950984i	$0.399925\pi$
$182$	1.76571e8	2.17104
$183$	0	0
$184$	3.69322e8	4.37062
$185$	−4.15159e7	−0.482073
$186$	0	0
$187$	223370.	0.00249792
$188$	−3.19363e8	−3.50535
$189$	0	0
$190$	8.08423e7	0.855069
$191$	−2.41141e7	−0.250411	−0.125206	−	0.992131i	$-0.539959\pi$
$191$	−2.41141e7	−0.250411	−0.125206	+	0.992131i	$0.539959\pi$
$192$	0	0
$193$	−1.36257e7	−0.136429	−0.0682146	−	0.997671i	$-0.521730\pi$
$193$	−1.36257e7	−0.136429	−0.0682146	+	0.997671i	$0.521730\pi$
$194$	−9.81506e7	−0.965133
$195$	0	0
$196$	2.36972e8	2.24802
$197$	−1.39225e8	−1.29744	−0.648719	−	0.761028i	$-0.724695\pi$
$197$	−1.39225e8	−1.29744	−0.648719	+	0.761028i	$0.724695\pi$
$198$	0	0
$199$	−5.59168e7	−0.502987	−0.251493	−	0.967859i	$-0.580922\pi$
$199$	−5.59168e7	−0.502987	−0.251493	+	0.967859i	$0.580922\pi$
$200$	−8.02463e7	−0.709284
$201$	0	0
$202$	−1.30782e8	−1.11639
$203$	3.67379e6	0.0308232
$204$	0	0
$205$	3.22822e7	0.261712
$206$	4.12984e8	3.29153
$207$	0	0
$208$	4.42282e8	3.40782
$209$	1.24097e8	0.940266
$210$	0	0
$211$	1.84693e8	1.35351	0.676754	−	0.736209i	$-0.263386\pi$
$211$	1.84693e8	1.35351	0.676754	+	0.736209i	$0.263386\pi$
$212$	1.98951e8	1.43407
$213$	0	0
$214$	−1.78067e8	−1.24204
$215$	−4.49498e7	−0.308456
$216$	0	0
$217$	−2.03406e8	−1.35131
$218$	4.50610e8	2.94580
$219$	0	0
$220$	−1.91031e8	−1.20955
$221$	345820.	0.00215515
$222$	0	0
$223$	1.50600e8	0.909406	0.454703	−	0.890643i	$-0.349745\pi$
$223$	1.50600e8	0.909406	0.454703	+	0.890643i	$0.349745\pi$
$224$	1.01185e9	6.01517
$225$	0	0
$226$	1.68266e8	0.969656
$227$	−1.97015e7	−0.111791	−0.0558957	−	0.998437i	$-0.517801\pi$
$227$	−1.97015e7	−0.111791	−0.0558957	+	0.998437i	$0.517801\pi$
$228$	0	0
$229$	−1.46510e8	−0.806202	−0.403101	−	0.915155i	$-0.632068\pi$
$229$	−1.46510e8	−0.806202	−0.403101	+	0.915155i	$0.632068\pi$
$230$	−1.98653e8	−1.07658
$231$	0	0
$232$	1.55035e7	0.0815120
$233$	2.27880e8	1.18021	0.590107	−	0.807325i	$-0.299086\pi$
$233$	2.27880e8	1.18021	0.590107	+	0.807325i	$0.299086\pi$
$234$	0	0
$235$	1.10769e8	0.556778
$236$	3.03527e8	1.50316
$237$	0	0
$238$	1.41671e6	0.00681179
$239$	3.51646e8	1.66615	0.833073	−	0.553163i	$-0.186580\pi$
$239$	3.51646e8	1.66615	0.833073	+	0.553163i	$0.186580\pi$
$240$	0	0
$241$	1.04073e8	0.478937	0.239468	−	0.970904i	$-0.423027\pi$
$241$	1.04073e8	0.478937	0.239468	+	0.970904i	$0.423027\pi$
$242$	3.32647e7	0.150879
$243$	0	0
$244$	−5.96318e8	−2.62793
$245$	−8.21925e7	−0.357068
$246$	0	0
$247$	1.92127e8	0.811240
$248$	−8.58379e8	−3.57354
$249$	0	0
$250$	4.31633e7	0.174713
$251$	2.56926e8	1.02553	0.512767	−	0.858528i	$-0.328621\pi$
$251$	2.56926e8	1.02553	0.512767	+	0.858528i	$0.328621\pi$
$252$	0	0
$253$	−3.04943e8	−1.18385
$254$	5.11004e8	1.95662
$255$	0	0
$256$	1.16233e9	4.33001
$257$	1.98477e8	0.729365	0.364683	−	0.931132i	$-0.381177\pi$
$257$	1.98477e8	0.729365	0.364683	+	0.931132i	$0.381177\pi$
$258$	0	0
$259$	−4.04198e8	−1.44559
$260$	−2.95753e8	−1.04357
$261$	0	0
$262$	4.19237e8	1.44014
$263$	−4.26251e8	−1.44484	−0.722421	−	0.691454i	$-0.756971\pi$
$263$	−4.26251e8	−1.44484	−0.722421	+	0.691454i	$0.756971\pi$
$264$	0	0
$265$	−6.90051e7	−0.227783
$266$	7.87079e8	2.56409
$267$	0	0
$268$	−4.69358e8	−1.48947
$269$	1.48074e8	0.463817	0.231908	−	0.972738i	$-0.425503\pi$
$269$	1.48074e8	0.463817	0.231908	+	0.972738i	$0.425503\pi$
$270$	0	0
$271$	4.10831e8	1.25392	0.626961	−	0.779050i	$-0.284298\pi$
$271$	4.10831e8	1.25392	0.626961	+	0.779050i	$0.284298\pi$
$272$	3.54863e6	0.0106923
$273$	0	0
$274$	−1.06900e9	−3.13943
$275$	6.62580e7	0.192121
$276$	0	0
$277$	4.71135e8	1.33188	0.665942	−	0.746004i	$-0.268030\pi$
$277$	4.71135e8	1.33188	0.665942	+	0.746004i	$0.268030\pi$
$278$	−8.82950e8	−2.46479
$279$	0	0
$280$	−7.81277e8	−2.12692
$281$	2.57091e8	0.691217	0.345609	−	0.938379i	$-0.387673\pi$
$281$	2.57091e8	0.691217	0.345609	+	0.938379i	$0.387673\pi$
$282$	0	0
$283$	−1.17784e7	−0.0308910	−0.0154455	−	0.999881i	$-0.504917\pi$
$283$	−1.17784e7	−0.0308910	−0.0154455	+	0.999881i	$0.504917\pi$
$284$	−1.26277e9	−3.27123
$285$	0	0
$286$	−6.15243e8	−1.55513
$287$	3.14299e8	0.784795
$288$	0	0
$289$	−4.10336e8	−0.999993
$290$	−8.33909e6	−0.0200782
$291$	0	0
$292$	−7.73298e8	−1.81764
$293$	6.65321e8	1.54524	0.772618	−	0.634871i	$-0.218947\pi$
$293$	6.65321e8	1.54524	0.772618	+	0.634871i	$0.218947\pi$
$294$	0	0
$295$	−1.05277e8	−0.238756
$296$	−1.70573e9	−3.82286
$297$	0	0
$298$	1.10427e9	2.41723
$299$	−4.72111e8	−1.02140
$300$	0	0
$301$	−4.37630e8	−0.924964
$302$	3.35874e8	0.701701
$303$	0	0
$304$	1.97151e9	4.02477
$305$	2.06830e8	0.417410
$306$	0	0
$307$	6.64306e8	1.31034	0.655170	−	0.755481i	$-0.272597\pi$
$307$	6.64306e8	1.31034	0.655170	+	0.755481i	$0.272597\pi$
$308$	−1.85987e9	−3.62707
$309$	0	0
$310$	4.61709e8	0.880242
$311$	−3.80556e8	−0.717392	−0.358696	−	0.933454i	$-0.616779\pi$
$311$	−3.80556e8	−0.717392	−0.358696	+	0.933454i	$0.616779\pi$
$312$	0	0
$313$	1.87533e7	0.0345678	0.0172839	−	0.999851i	$-0.494498\pi$
$313$	1.87533e7	0.0345678	0.0172839	+	0.999851i	$0.494498\pi$
$314$	−1.74925e9	−3.18858
$315$	0	0
$316$	1.11214e9	1.98269
$317$	3.28823e8	0.579769	0.289885	−	0.957062i	$-0.406383\pi$
$317$	3.28823e8	0.579769	0.289885	+	0.957062i	$0.406383\pi$
$318$	0	0
$319$	−1.28010e7	−0.0220788
$320$	−1.21889e9	−2.07941
$321$	0	0
$322$	−1.93408e9	−3.22833
$323$	1.54153e6	0.00254532
$324$	0	0
$325$	1.02580e8	0.165757
$326$	1.71418e9	2.74027
$327$	0	0
$328$	1.32635e9	2.07539
$329$	1.07845e9	1.66960
$330$	0	0
$331$	−6.56524e8	−0.995068	−0.497534	−	0.867445i	$-0.665761\pi$
$331$	−6.56524e8	−0.995068	−0.497534	+	0.867445i	$0.665761\pi$
$332$	2.81530e9	4.22223
$333$	0	0
$334$	−9.47337e8	−1.39121
$335$	1.62794e8	0.236582
$336$	0	0
$337$	1.35247e8	0.192496	0.0962482	−	0.995357i	$-0.469316\pi$
$337$	1.35247e8	0.192496	0.0962482	+	0.995357i	$0.469316\pi$
$338$	4.34200e8	0.611619
$339$	0	0
$340$	−2.37297e6	−0.00327428
$341$	7.08749e8	0.967948
$342$	0	0
$343$	2.02025e8	0.270318
$344$	−1.84681e9	−2.44607
$345$	0	0
$346$	2.09820e9	2.72321
$347$	6.03177e8	0.774982	0.387491	−	0.921873i	$-0.373342\pi$
$347$	6.03177e8	0.774982	0.387491	+	0.921873i	$0.373342\pi$
$348$	0	0
$349$	−6.08708e8	−0.766514	−0.383257	−	0.923642i	$-0.625198\pi$
$349$	−6.08708e8	−0.766514	−0.383257	+	0.923642i	$0.625198\pi$
$350$	4.20237e8	0.523909
$351$	0	0
$352$	−3.52569e9	−4.30869
$353$	7.02991e8	0.850625	0.425313	−	0.905047i	$-0.360164\pi$
$353$	7.02991e8	0.850625	0.425313	+	0.905047i	$0.360164\pi$
$354$	0	0
$355$	4.37986e8	0.519590
$356$	4.78462e8	0.562047
$357$	0	0
$358$	1.01253e9	1.16632
$359$	1.20890e9	1.37898	0.689490	−	0.724295i	$-0.257835\pi$
$359$	1.20890e9	1.37898	0.689490	+	0.724295i	$0.257835\pi$
$360$	0	0
$361$	−3.74479e7	−0.0418941
$362$	−1.09040e9	−1.20811
$363$	0	0
$364$	−2.87945e9	−3.12935
$365$	2.68214e8	0.288707
$366$	0	0
$367$	1.14576e9	1.20994	0.604968	−	0.796249i	$-0.293186\pi$
$367$	1.14576e9	1.20994	0.604968	+	0.796249i	$0.293186\pi$
$368$	−4.84456e9	−5.06742
$369$	0	0
$370$	9.17484e8	0.941656
$371$	−6.71833e8	−0.683050
$372$	0	0
$373$	−1.20011e9	−1.19740	−0.598700	−	0.800973i	$-0.704316\pi$
$373$	−1.20011e9	−1.19740	−0.598700	+	0.800973i	$0.704316\pi$
$374$	−4.93639e6	−0.00487931
$375$	0	0
$376$	4.55108e9	4.41527
$377$	−1.98184e7	−0.0190491
$378$	0	0
$379$	1.17904e9	1.11248	0.556238	−	0.831023i	$-0.312244\pi$
$379$	1.17904e9	1.11248	0.556238	+	0.831023i	$0.312244\pi$
$380$	−1.31835e9	−1.23250
$381$	0	0
$382$	5.32912e8	0.489140
$383$	−1.01942e9	−0.927168	−0.463584	−	0.886053i	$-0.653437\pi$
$383$	−1.01942e9	−0.927168	−0.463584	+	0.886053i	$0.653437\pi$
$384$	0	0
$385$	6.45087e8	0.576110
$386$	3.01122e8	0.266494
$387$	0	0
$388$	1.60060e9	1.39115
$389$	3.21679e8	0.277076	0.138538	−	0.990357i	$-0.455760\pi$
$389$	3.21679e8	0.277076	0.138538	+	0.990357i	$0.455760\pi$
$390$	0	0
$391$	−3.78797e6	−0.00320470
$392$	−3.37697e9	−2.83156
$393$	0	0
$394$	3.07682e9	2.53435
$395$	−3.85739e8	−0.314923
$396$	0	0
$397$	−1.41656e9	−1.13624	−0.568119	−	0.822947i	$-0.692329\pi$
$397$	−1.41656e9	−1.13624	−0.568119	+	0.822947i	$0.692329\pi$
$398$	1.23574e9	0.982507
$399$	0	0
$400$	1.05263e9	0.822365
$401$	4.77425e8	0.369743	0.184871	−	0.982763i	$-0.440813\pi$
$401$	4.77425e8	0.369743	0.184871	+	0.982763i	$0.440813\pi$
$402$	0	0
$403$	1.09728e9	0.835123
$404$	2.13274e9	1.60917
$405$	0	0
$406$	−8.11892e7	−0.0602084
$407$	1.40839e9	1.03548
$408$	0	0
$409$	1.15909e9	0.837694	0.418847	−	0.908057i	$-0.362434\pi$
$409$	1.15909e9	0.837694	0.418847	+	0.908057i	$0.362434\pi$
$410$	−7.13423e8	−0.511215
$411$	0	0
$412$	−6.73479e9	−4.74443
$413$	−1.02497e9	−0.715956
$414$	0	0
$415$	−9.76472e8	−0.670643
$416$	−5.45846e9	−3.71744
$417$	0	0
$418$	−2.74250e9	−1.83667
$419$	−1.69756e8	−0.112740	−0.0563699	−	0.998410i	$-0.517953\pi$
$419$	−1.69756e8	−0.112740	−0.0563699	+	0.998410i	$0.517953\pi$
$420$	0	0
$421$	4.41380e8	0.288287	0.144144	−	0.989557i	$-0.453957\pi$
$421$	4.41380e8	0.288287	0.144144	+	0.989557i	$0.453957\pi$
$422$	−4.08163e9	−2.64387
$423$	0	0
$424$	−2.83515e9	−1.80633
$425$	823051.	0.000520074 0
$426$	0	0
$427$	2.01369e9	1.25168
$428$	2.90385e9	1.79028
$429$	0	0
$430$	9.93372e8	0.602521
$431$	−8.25340e8	−0.496549	−0.248275	−	0.968690i	$-0.579864\pi$
$431$	−8.25340e8	−0.496549	−0.248275	+	0.968690i	$0.579864\pi$
$432$	0	0
$433$	−1.67378e9	−0.990808	−0.495404	−	0.868663i	$-0.664980\pi$
$433$	−1.67378e9	−0.990808	−0.495404	+	0.868663i	$0.664980\pi$
$434$	4.49519e9	2.63957
$435$	0	0
$436$	−7.34838e9	−4.24609
$437$	−2.10448e9	−1.20631
$438$	0	0
$439$	−1.44324e9	−0.814166	−0.407083	−	0.913391i	$-0.633454\pi$
$439$	−1.44324e9	−0.814166	−0.407083	+	0.913391i	$0.633454\pi$
$440$	2.72229e9	1.52352
$441$	0	0
$442$	−7.64249e6	−0.00420976
$443$	−2.27112e9	−1.24116	−0.620579	−	0.784144i	$-0.713103\pi$
$443$	−2.27112e9	−1.24116	−0.620579	+	0.784144i	$0.713103\pi$
$444$	0	0
$445$	−1.65952e8	−0.0892735
$446$	−3.32820e9	−1.77638
$447$	0	0
$448$	−1.18671e10	−6.23552
$449$	5.44080e8	0.283662	0.141831	−	0.989891i	$-0.454701\pi$
$449$	5.44080e8	0.283662	0.141831	+	0.989891i	$0.454701\pi$
$450$	0	0
$451$	−1.09514e9	−0.562152
$452$	−2.74403e9	−1.39767
$453$	0	0
$454$	4.35394e8	0.218367
$455$	9.98721e8	0.497055
$456$	0	0
$457$	−3.28019e8	−0.160765	−0.0803826	−	0.996764i	$-0.525614\pi$
$457$	−3.28019e8	−0.160765	−0.0803826	+	0.996764i	$0.525614\pi$
$458$	3.23782e9	1.57479
$459$	0	0
$460$	3.23955e9	1.55179
$461$	1.76163e9	0.837453	0.418727	−	0.908112i	$-0.362477\pi$
$461$	1.76163e9	0.837453	0.418727	+	0.908112i	$0.362477\pi$
$462$	0	0
$463$	3.78338e9	1.77152	0.885760	−	0.464143i	$-0.153638\pi$
$463$	3.78338e9	1.77152	0.885760	+	0.464143i	$0.153638\pi$
$464$	−2.03366e8	−0.0945075
$465$	0	0
$466$	−5.03606e9	−2.30537
$467$	3.20085e9	1.45431	0.727154	−	0.686475i	$-0.240843\pi$
$467$	3.20085e9	1.45431	0.727154	+	0.686475i	$0.240843\pi$
$468$	0	0
$469$	1.58496e9	0.709438
$470$	−2.44796e9	−1.08758
$471$	0	0
$472$	−4.32541e9	−1.89335
$473$	1.52488e9	0.662555
$474$	0	0
$475$	4.57261e8	0.195766
$476$	−2.31032e7	−0.00981854
$477$	0	0
$478$	−7.77123e9	−3.25456
$479$	3.17484e9	1.31992	0.659959	−	0.751301i	$-0.270574\pi$
$479$	3.17484e9	1.31992	0.659959	+	0.751301i	$0.270574\pi$
$480$	0	0
$481$	2.18046e9	0.893389
$482$	−2.29997e9	−0.935530
$483$	0	0
$484$	−5.42468e8	−0.217478
$485$	−5.55161e8	−0.220965
$486$	0	0
$487$	4.35441e9	1.70835	0.854177	−	0.519982i	$-0.174061\pi$
$487$	4.35441e9	1.70835	0.854177	+	0.519982i	$0.174061\pi$
$488$	8.49783e9	3.31008
$489$	0	0
$490$	1.81642e9	0.697478
$491$	1.42469e9	0.543168	0.271584	−	0.962415i	$-0.412452\pi$
$491$	1.42469e9	0.543168	0.271584	+	0.962415i	$0.412452\pi$
$492$	0	0
$493$	−159012.	−5.97677e−5 0
$494$	−4.24593e9	−1.58463
$495$	0	0
$496$	1.12597e10	4.14327
$497$	4.26422e9	1.55809
$498$	0	0
$499$	−1.28434e9	−0.462729	−0.231364	−	0.972867i	$-0.574319\pi$
$499$	−1.28434e9	−0.462729	−0.231364	+	0.972867i	$0.574319\pi$
$500$	−7.03890e8	−0.251831
$501$	0	0
$502$	−5.67796e9	−2.00322
$503$	3.92093e9	1.37373	0.686864	−	0.726786i	$-0.258987\pi$
$503$	3.92093e9	1.37373	0.686864	+	0.726786i	$0.258987\pi$
$504$	0	0
$505$	−7.39729e8	−0.255595
$506$	6.73911e9	2.31247
$507$	0	0
$508$	−8.33327e9	−2.82028
$509$	2.88717e9	0.970420	0.485210	−	0.874398i	$-0.338743\pi$
$509$	2.88717e9	0.970420	0.485210	+	0.874398i	$0.338743\pi$
$510$	0	0
$511$	2.61133e9	0.865742
$512$	−1.17256e10	−3.86092
$513$	0	0
$514$	−4.38627e9	−1.42470
$515$	2.33592e9	0.753587
$516$	0	0
$517$	−3.75775e9	−1.19594
$518$	8.93260e9	2.82373
$519$	0	0
$520$	4.21463e9	1.31446
$521$	1.18611e9	0.367447	0.183723	−	0.982978i	$-0.441185\pi$
$521$	1.18611e9	0.367447	0.183723	+	0.982978i	$0.441185\pi$
$522$	0	0
$523$	−5.52692e9	−1.68938	−0.844689	−	0.535257i	$-0.820215\pi$
$523$	−5.52692e9	−1.68938	−0.844689	+	0.535257i	$0.820215\pi$
$524$	−6.83676e9	−2.07582
$525$	0	0
$526$	9.41997e9	2.82228
$527$	8.80401e6	0.00262025
$528$	0	0
$529$	1.76648e9	0.518815
$530$	1.52499e9	0.444939
$531$	0	0
$532$	−1.28354e10	−3.69588
$533$	−1.69550e9	−0.485011
$534$	0	0
$535$	−1.00718e9	−0.284361
$536$	6.68859e9	1.87611
$537$	0	0
$538$	−3.27238e9	−0.905995
$539$	2.78831e9	0.766973
$540$	0	0
$541$	4.83250e9	1.31214	0.656072	−	0.754698i	$-0.272217\pi$
$541$	4.83250e9	1.31214	0.656072	+	0.754698i	$0.272217\pi$
$542$	−9.07920e9	−2.44934
$543$	0	0
$544$	−4.37958e7	−0.0116637
$545$	2.54875e9	0.674433
$546$	0	0
$547$	−2.40012e9	−0.627014	−0.313507	−	0.949586i	$-0.601504\pi$
$547$	−2.40012e9	−0.627014	−0.313507	+	0.949586i	$0.601504\pi$
$548$	1.74329e10	4.52519
$549$	0	0
$550$	−1.46428e9	−0.375278
$551$	−8.83423e7	−0.0224977
$552$	0	0
$553$	−3.75555e9	−0.944355
$554$	−1.04119e10	−2.60163
$555$	0	0
$556$	1.43988e10	3.55275
$557$	1.24483e9	0.305224	0.152612	−	0.988286i	$-0.451232\pi$
$557$	1.24483e9	0.305224	0.152612	+	0.988286i	$0.451232\pi$
$558$	0	0
$559$	2.36081e9	0.571637
$560$	1.02484e10	2.46602
$561$	0	0
$562$	−5.68160e9	−1.35019
$563$	4.36341e9	1.03050	0.515249	−	0.857041i	$-0.327700\pi$
$563$	4.36341e9	1.03050	0.515249	+	0.857041i	$0.327700\pi$
$564$	0	0
$565$	9.51750e8	0.222000
$566$	2.60297e8	0.0603408
$567$	0	0
$568$	1.79951e10	4.12037
$569$	3.07131e9	0.698924	0.349462	−	0.936950i	$-0.386364\pi$
$569$	3.07131e9	0.698924	0.349462	+	0.936950i	$0.386364\pi$
$570$	0	0
$571$	8.21466e9	1.84656	0.923279	−	0.384129i	$-0.125498\pi$
$571$	8.21466e9	1.84656	0.923279	+	0.384129i	$0.125498\pi$
$572$	1.00332e10	2.24157
$573$	0	0
$574$	−6.94587e9	−1.53298
$575$	−1.12362e9	−0.246480
$576$	0	0
$577$	−3.20849e9	−0.695321	−0.347661	−	0.937620i	$-0.613024\pi$
$577$	−3.20849e9	−0.695321	−0.347661	+	0.937620i	$0.613024\pi$
$578$	9.06826e9	1.95333
$579$	0	0
$580$	1.35991e8	0.0289409
$581$	−9.50691e9	−2.01105
$582$	0	0
$583$	2.34094e9	0.489272
$584$	1.10199e10	2.28945
$585$	0	0
$586$	−1.47033e10	−3.01838
$587$	3.86828e9	0.789377	0.394689	−	0.918815i	$-0.370853\pi$
$587$	3.86828e9	0.789377	0.394689	+	0.918815i	$0.370853\pi$
$588$	0	0
$589$	4.89123e9	0.986313
$590$	2.32657e9	0.466374
$591$	0	0
$592$	2.23748e10	4.43234
$593$	−8.07068e9	−1.58935	−0.794674	−	0.607037i	$-0.792358\pi$
$593$	−8.07068e9	−1.58935	−0.794674	+	0.607037i	$0.792358\pi$
$594$	0	0
$595$	8.01320e6	0.00155954
$596$	−1.80080e10	−3.48421
$597$	0	0
$598$	1.04335e10	1.99514
$599$	−7.70886e9	−1.46553	−0.732767	−	0.680479i	$-0.761772\pi$
$599$	−7.70886e9	−1.46553	−0.732767	+	0.680479i	$0.761772\pi$
$600$	0	0
$601$	−7.10328e9	−1.33475	−0.667373	−	0.744724i	$-0.732581\pi$
$601$	−7.10328e9	−1.33475	−0.667373	+	0.744724i	$0.732581\pi$
$602$	9.67145e9	1.80677
$603$	0	0
$604$	−5.47731e9	−1.01144
$605$	1.88152e8	0.0345434
$606$	0	0
$607$	−4.25447e9	−0.772121	−0.386061	−	0.922473i	$-0.626164\pi$
$607$	−4.25447e9	−0.772121	−0.386061	+	0.922473i	$0.626164\pi$
$608$	−2.43316e10	−4.39044
$609$	0	0
$610$	−4.57085e9	−0.815347
$611$	−5.81773e9	−1.03183
$612$	0	0
$613$	8.50210e9	1.49078	0.745392	−	0.666627i	$-0.232263\pi$
$613$	8.50210e9	1.49078	0.745392	+	0.666627i	$0.232263\pi$
$614$	−1.46809e10	−2.55955
$615$	0	0
$616$	2.65041e10	4.56858
$617$	−3.79681e9	−0.650761	−0.325380	−	0.945583i	$-0.605492\pi$
$617$	−3.79681e9	−0.650761	−0.325380	+	0.945583i	$0.605492\pi$
$618$	0	0
$619$	3.84381e9	0.651396	0.325698	−	0.945474i	$-0.394401\pi$
$619$	3.84381e9	0.651396	0.325698	+	0.945474i	$0.394401\pi$
$620$	−7.52938e9	−1.26879
$621$	0	0
$622$	8.41012e9	1.40132
$623$	−1.61571e9	−0.267703
$624$	0	0
$625$	2.44141e8	0.0400000
$626$	−4.14440e8	−0.0675230
$627$	0	0
$628$	2.85261e10	4.59604
$629$	1.74949e7	0.00280307
$630$	0	0
$631$	1.16248e10	1.84196	0.920982	−	0.389605i	$-0.127388\pi$
$631$	1.16248e10	1.84196	0.920982	+	0.389605i	$0.127388\pi$
$632$	−1.58485e10	−2.49735
$633$	0	0
$634$	−7.26686e9	−1.13249
$635$	2.89035e9	0.447963
$636$	0	0
$637$	4.31685e9	0.661727
$638$	2.82896e8	0.0431275
$639$	0	0
$640$	1.36342e10	2.05588
$641$	1.11576e10	1.67328	0.836641	−	0.547752i	$-0.184516\pi$
$641$	1.11576e10	1.67328	0.836641	+	0.547752i	$0.184516\pi$
$642$	0	0
$643$	−8.45168e8	−0.125373	−0.0626866	−	0.998033i	$-0.519967\pi$
$643$	−8.45168e8	−0.125373	−0.0626866	+	0.998033i	$0.519967\pi$
$644$	3.15402e10	4.65333
$645$	0	0
$646$	−3.40671e7	−0.00497189
$647$	−9.67362e9	−1.40418	−0.702092	−	0.712086i	$-0.747751\pi$
$647$	−9.67362e9	−1.40418	−0.702092	+	0.712086i	$0.747751\pi$
$648$	0	0
$649$	3.57142e9	0.512843
$650$	−2.26698e9	−0.323781
$651$	0	0
$652$	−2.79542e10	−3.94984
$653$	8.54426e9	1.20082	0.600410	−	0.799692i	$-0.295004\pi$
$653$	8.54426e9	1.20082	0.600410	+	0.799692i	$0.295004\pi$
$654$	0	0
$655$	2.37129e9	0.329716
$656$	−1.73983e10	−2.40627
$657$	0	0
$658$	−2.38333e10	−3.26132
$659$	−6.30845e9	−0.858665	−0.429332	−	0.903147i	$-0.641251\pi$
$659$	−6.30845e9	−0.858665	−0.429332	+	0.903147i	$0.641251\pi$
$660$	0	0
$661$	−8.14119e9	−1.09643	−0.548217	−	0.836336i	$-0.684693\pi$
$661$	−8.14119e9	−1.09643	−0.548217	+	0.836336i	$0.684693\pi$
$662$	1.45089e10	1.94371
$663$	0	0
$664$	−4.01194e10	−5.31822
$665$	4.45189e9	0.587041
$666$	0	0
$667$	2.17082e8	0.0283259
$668$	1.54488e10	2.00529
$669$	0	0
$670$	−3.59769e9	−0.462128
$671$	−7.01651e9	−0.896587
$672$	0	0
$673$	1.41804e10	1.79323	0.896617	−	0.442807i	$-0.146017\pi$
$673$	1.41804e10	1.79323	0.896617	+	0.442807i	$0.146017\pi$
$674$	−2.98890e9	−0.376012
$675$	0	0
$676$	−7.08077e9	−0.881590
$677$	−1.33769e9	−0.165690	−0.0828449	−	0.996562i	$-0.526401\pi$
$677$	−1.33769e9	−0.165690	−0.0828449	+	0.996562i	$0.526401\pi$
$678$	0	0
$679$	−5.40504e9	−0.662605
$680$	3.38160e7	0.00412421
$681$	0	0
$682$	−1.56631e10	−1.89074
$683$	5.85440e9	0.703088	0.351544	−	0.936171i	$-0.385657\pi$
$683$	5.85440e9	0.703088	0.351544	+	0.936171i	$0.385657\pi$
$684$	0	0
$685$	−6.04649e9	−0.718765
$686$	−4.46467e9	−0.528025
$687$	0	0
$688$	2.42255e10	2.83604
$689$	3.62423e9	0.422132
$690$	0	0
$691$	−6.31213e9	−0.727785	−0.363892	−	0.931441i	$-0.618552\pi$
$691$	−6.31213e9	−0.727785	−0.363892	+	0.931441i	$0.618552\pi$
$692$	−3.42167e10	−3.92525
$693$	0	0
$694$	−1.33300e10	−1.51381
$695$	−4.99415e9	−0.564307
$696$	0	0
$697$	−1.36038e7	−0.00152175
$698$	1.34522e10	1.49727
$699$	0	0
$700$	−6.85306e9	−0.755165
$701$	−4.35585e9	−0.477596	−0.238798	−	0.971069i	$-0.576753\pi$
$701$	−4.35585e9	−0.477596	−0.238798	+	0.971069i	$0.576753\pi$
$702$	0	0
$703$	9.71960e9	1.05513
$704$	4.13499e10	4.46653
$705$	0	0
$706$	−1.55358e10	−1.66157
$707$	−7.20198e9	−0.766451
$708$	0	0
$709$	−7.66462e9	−0.807660	−0.403830	−	0.914834i	$-0.632321\pi$
$709$	−7.66462e9	−0.807660	−0.403830	+	0.914834i	$0.632321\pi$
$710$	−9.67930e9	−1.01494
$711$	0	0
$712$	−6.81832e9	−0.707942
$713$	−1.20192e10	−1.24183
$714$	0	0
$715$	−3.47995e9	−0.356042
$716$	−1.65119e10	−1.68113
$717$	0	0
$718$	−2.67161e10	−2.69363
$719$	1.92682e9	0.193326	0.0966631	−	0.995317i	$-0.469183\pi$
$719$	1.92682e9	0.193326	0.0966631	+	0.995317i	$0.469183\pi$
$720$	0	0
$721$	2.27425e10	2.25977
$722$	8.27584e8	0.0818337
$723$	0	0
$724$	1.77819e10	1.74138
$725$	−4.71677e7	−0.00459686
$726$	0	0
$727$	−1.94843e10	−1.88067	−0.940337	−	0.340244i	$-0.889490\pi$
$727$	−1.94843e10	−1.88067	−0.940337	+	0.340244i	$0.889490\pi$
$728$	4.10336e10	3.94166
$729$	0	0
$730$	−5.92742e9	−0.563944
$731$	1.89419e7	0.00179355
$732$	0	0
$733$	−2.00549e9	−0.188086	−0.0940431	−	0.995568i	$-0.529979\pi$
$733$	−2.00549e9	−0.188086	−0.0940431	+	0.995568i	$0.529979\pi$
$734$	−2.53208e10	−2.36343
$735$	0	0
$736$	5.97896e10	5.52782
$737$	−5.52265e9	−0.508173
$738$	0	0
$739$	−7.15205e9	−0.651891	−0.325945	−	0.945389i	$-0.605683\pi$
$739$	−7.15205e9	−0.651891	−0.325945	+	0.945389i	$0.605683\pi$
$740$	−1.49620e10	−1.35731
$741$	0	0
$742$	1.48472e10	1.33423
$743$	9.89988e9	0.885460	0.442730	−	0.896655i	$-0.354010\pi$
$743$	9.89988e9	0.885460	0.442730	+	0.896655i	$0.354010\pi$
$744$	0	0
$745$	6.24599e9	0.553419
$746$	2.65219e10	2.33894
$747$	0	0
$748$	8.05008e7	0.00703306
$749$	−9.80592e9	−0.852711
$750$	0	0
$751$	1.33004e9	0.114584	0.0572922	−	0.998357i	$-0.481753\pi$
$751$	1.33004e9	0.114584	0.0572922	+	0.998357i	$0.481753\pi$
$752$	−5.96986e10	−5.11920
$753$	0	0
$754$	4.37979e8	0.0372094
$755$	1.89978e9	0.160653
$756$	0	0
$757$	−3.71511e9	−0.311269	−0.155635	−	0.987815i	$-0.549742\pi$
$757$	−3.71511e9	−0.311269	−0.155635	+	0.987815i	$0.549742\pi$
$758$	−2.60563e10	−2.17305
$759$	0	0
$760$	1.87871e10	1.55243
$761$	1.67203e10	1.37530	0.687652	−	0.726041i	$-0.258642\pi$
$761$	1.67203e10	1.37530	0.687652	+	0.726041i	$0.258642\pi$
$762$	0	0
$763$	2.48146e10	2.02242
$764$	−8.69053e9	−0.705049
$765$	0	0
$766$	2.25288e10	1.81108
$767$	5.52925e9	0.442469
$768$	0	0
$769$	1.99386e9	0.158108	0.0790538	−	0.996870i	$-0.474810\pi$
$769$	1.99386e9	0.158108	0.0790538	+	0.996870i	$0.474810\pi$
$770$	−1.42562e10	−1.12534
$771$	0	0
$772$	−4.91058e9	−0.384125
$773$	−1.59270e10	−1.24024	−0.620120	−	0.784507i	$-0.712916\pi$
$773$	−1.59270e10	−1.24024	−0.620120	+	0.784507i	$0.712916\pi$
$774$	0	0
$775$	2.61152e9	0.201529
$776$	−2.28094e10	−1.75226
$777$	0	0
$778$	−7.10897e9	−0.541225
$779$	−7.55783e9	−0.572817
$780$	0	0
$781$	−1.48583e10	−1.11607
$782$	8.37126e7	0.00625989
$783$	0	0
$784$	4.42973e10	3.28300
$785$	−9.89412e9	−0.730018
$786$	0	0
$787$	1.76202e10	1.28854	0.644272	−	0.764797i	$-0.277161\pi$
$787$	1.76202e10	1.28854	0.644272	+	0.764797i	$0.277161\pi$
$788$	−5.01757e10	−3.65302
$789$	0	0
$790$	8.52468e9	0.615153
$791$	9.26622e9	0.665710
$792$	0	0
$793$	−1.08629e10	−0.773554
$794$	3.13055e10	2.21946
$795$	0	0
$796$	−2.01520e10	−1.41619
$797$	6.49516e9	0.454450	0.227225	−	0.973842i	$-0.427035\pi$
$797$	6.49516e9	0.454450	0.227225	+	0.973842i	$0.427035\pi$
$798$	0	0
$799$	−4.66784e7	−0.00323744
$800$	−1.29911e10	−0.897080
$801$	0	0
$802$	−1.05509e10	−0.722235
$803$	−9.09893e9	−0.620134
$804$	0	0
$805$	−1.09396e10	−0.739119
$806$	−2.42495e10	−1.63128
$807$	0	0
$808$	−3.03926e10	−2.02688
$809$	−8.91269e9	−0.591819	−0.295909	−	0.955216i	$-0.595623\pi$
$809$	−8.91269e9	−0.591819	−0.295909	+	0.955216i	$0.595623\pi$
$810$	0	0
$811$	−6.28373e9	−0.413661	−0.206830	−	0.978377i	$-0.566315\pi$
$811$	−6.28373e9	−0.413661	−0.206830	+	0.978377i	$0.566315\pi$
$812$	1.32400e9	0.0867847
$813$	0	0
$814$	−3.11248e10	−2.02265
$815$	9.69575e9	0.627379
$816$	0	0
$817$	1.05235e10	0.675126
$818$	−2.56154e10	−1.63631
$819$	0	0
$820$	1.16342e10	0.736868
$821$	−1.43501e10	−0.905014	−0.452507	−	0.891761i	$-0.649470\pi$
$821$	−1.43501e10	−0.905014	−0.452507	+	0.891761i	$0.649470\pi$
$822$	0	0
$823$	7.27905e9	0.455172	0.227586	−	0.973758i	$-0.426917\pi$
$823$	7.27905e9	0.455172	0.227586	+	0.973758i	$0.426917\pi$
$824$	9.59741e10	5.97598
$825$	0	0
$826$	2.26515e10	1.39851
$827$	3.46957e9	0.213307	0.106654	−	0.994296i	$-0.465986\pi$
$827$	3.46957e9	0.213307	0.106654	+	0.994296i	$0.465986\pi$
$828$	0	0
$829$	8.26309e9	0.503734	0.251867	−	0.967762i	$-0.418955\pi$
$829$	8.26309e9	0.503734	0.251867	+	0.967762i	$0.418955\pi$
$830$	2.15796e10	1.31000
$831$	0	0
$832$	6.40177e10	3.85362
$833$	3.46361e7	0.00207621
$834$	0	0
$835$	−5.35834e9	−0.318514
$836$	4.47237e10	2.64738
$837$	0	0
$838$	3.75155e9	0.220220
$839$	1.08198e10	0.632488	0.316244	−	0.948678i	$-0.397578\pi$
$839$	1.08198e10	0.632488	0.316244	+	0.948678i	$0.397578\pi$
$840$	0	0
$841$	−1.72408e10	−0.999472
$842$	−9.75432e9	−0.563125
$843$	0	0
$844$	6.65617e10	3.81089
$845$	2.45593e9	0.140029
$846$	0	0
$847$	1.83184e9	0.103585
$848$	3.71900e10	2.09431
$849$	0	0
$850$	−1.81891e7	−0.00101588
$851$	−2.38838e10	−1.32847
$852$	0	0
$853$	1.13530e9	0.0626307	0.0313153	−	0.999510i	$-0.490030\pi$
$853$	1.13530e9	0.0626307	0.0313153	+	0.999510i	$0.490030\pi$
$854$	−4.45017e10	−2.44497
$855$	0	0
$856$	−4.13813e10	−2.25499
$857$	−1.30642e9	−0.0709008	−0.0354504	−	0.999371i	$-0.511287\pi$
$857$	−1.30642e9	−0.0709008	−0.0354504	+	0.999371i	$0.511287\pi$
$858$	0	0
$859$	2.32648e10	1.25234	0.626171	−	0.779686i	$-0.284621\pi$
$859$	2.32648e10	1.25234	0.626171	+	0.779686i	$0.284621\pi$
$860$	−1.61995e10	−0.868477
$861$	0	0
$862$	1.82397e10	0.969933
$863$	−2.47805e10	−1.31242	−0.656209	−	0.754579i	$-0.727841\pi$
$863$	−2.47805e10	−1.31242	−0.656209	+	0.754579i	$0.727841\pi$
$864$	0	0
$865$	1.18679e10	0.623472
$866$	3.69897e10	1.93539
$867$	0	0
$868$	−7.33059e10	−3.80469
$869$	1.30859e10	0.676446
$870$	0	0
$871$	−8.55015e9	−0.438440
$872$	1.04718e11	5.34828
$873$	0	0
$874$	4.65081e10	2.35634
$875$	2.37695e9	0.119948
$876$	0	0
$877$	1.12067e10	0.561022	0.280511	−	0.959851i	$-0.409496\pi$
$877$	1.12067e10	0.561022	0.280511	+	0.959851i	$0.409496\pi$
$878$	3.18950e10	1.59035
$879$	0	0
$880$	−3.57094e10	−1.76642
$881$	3.56338e10	1.75569	0.877843	−	0.478948i	$-0.158982\pi$
$881$	3.56338e10	1.75569	0.877843	+	0.478948i	$0.158982\pi$
$882$	0	0
$883$	−1.75955e10	−0.860078	−0.430039	−	0.902810i	$-0.641500\pi$
$883$	−1.75955e10	−0.860078	−0.430039	+	0.902810i	$0.641500\pi$
$884$	1.24631e8	0.00606796
$885$	0	0
$886$	5.01909e10	2.42441
$887$	−1.43123e10	−0.688613	−0.344307	−	0.938857i	$-0.611886\pi$
$887$	−1.43123e10	−0.688613	−0.344307	+	0.938857i	$0.611886\pi$
$888$	0	0
$889$	2.81404e10	1.34330
$890$	3.66747e9	0.174382
$891$	0	0
$892$	5.42750e10	2.56049
$893$	−2.59331e10	−1.21864
$894$	0	0
$895$	5.72707e9	0.267025
$896$	1.32742e11	6.16496
$897$	0	0
$898$	−1.20239e10	−0.554089
$899$	−5.04543e8	−0.0231600
$900$	0	0
$901$	2.90789e7	0.00132447
$902$	2.42022e10	1.09808
$903$	0	0
$904$	3.91037e10	1.76047
$905$	−6.16755e9	−0.276594
$906$	0	0
$907$	−1.31304e10	−0.584322	−0.292161	−	0.956369i	$-0.594374\pi$
$907$	−1.31304e10	−0.584322	−0.292161	+	0.956369i	$0.594374\pi$
$908$	−7.10025e9	−0.314755
$909$	0	0
$910$	−2.20713e10	−0.970920
$911$	1.52716e10	0.669224	0.334612	−	0.942356i	$-0.391395\pi$
$911$	1.52716e10	0.669224	0.334612	+	0.942356i	$0.391395\pi$
$912$	0	0
$913$	3.31259e10	1.44052
$914$	7.24908e9	0.314030
$915$	0	0
$916$	−5.28011e10	−2.26991
$917$	2.30869e10	0.988718
$918$	0	0
$919$	−3.98954e10	−1.69558	−0.847791	−	0.530330i	$-0.822068\pi$
$919$	−3.98954e10	−1.69558	−0.847791	+	0.530330i	$0.822068\pi$
$920$	−4.61652e10	−1.95460
$921$	0	0
$922$	−3.89312e10	−1.63584
$923$	−2.30035e10	−0.962915
$924$	0	0
$925$	5.18948e9	0.215590
$926$	−8.36111e10	−3.46039
$927$	0	0
$928$	2.50987e9	0.103094
$929$	3.88607e10	1.59021	0.795106	−	0.606470i	$-0.207415\pi$
$929$	3.88607e10	1.59021	0.795106	+	0.606470i	$0.207415\pi$
$930$	0	0
$931$	1.92427e10	0.781525
$932$	8.21262e10	3.32297
$933$	0	0
$934$	−7.07375e10	−2.84077
$935$	−2.79213e7	−0.00111711
$936$	0	0
$937$	−1.02710e10	−0.407872	−0.203936	−	0.978984i	$-0.565373\pi$
$937$	−1.02710e10	−0.407872	−0.203936	+	0.978984i	$0.565373\pi$
$938$	−3.50270e10	−1.38578
$939$	0	0
$940$	3.99204e10	1.56764
$941$	−3.43314e9	−0.134316	−0.0671579	−	0.997742i	$-0.521393\pi$
$941$	−3.43314e9	−0.134316	−0.0671579	+	0.997742i	$0.521393\pi$
$942$	0	0
$943$	1.85718e10	0.721210
$944$	5.67384e10	2.19520
$945$	0	0
$946$	−3.36993e10	−1.29420
$947$	9.51925e9	0.364232	0.182116	−	0.983277i	$-0.441705\pi$
$947$	9.51925e9	0.364232	0.182116	+	0.983277i	$0.441705\pi$
$948$	0	0
$949$	−1.40869e10	−0.535037
$950$	−1.01053e10	−0.382398
$951$	0	0
$952$	3.29232e8	0.0123672
$953$	−2.76653e10	−1.03541	−0.517703	−	0.855561i	$-0.673213\pi$
$953$	−2.76653e10	−1.03541	−0.517703	+	0.855561i	$0.673213\pi$
$954$	0	0
$955$	3.01426e9	0.111987
$956$	1.26730e11	4.69114
$957$	0	0
$958$	−7.01626e10	−2.57826
$959$	−5.88686e10	−2.15535
$960$	0	0
$961$	4.22336e8	0.0153506
$962$	−4.81873e10	−1.74510
$963$	0	0
$964$	3.75071e10	1.34848
$965$	1.70321e9	0.0610130
$966$	0	0
$967$	4.18214e10	1.48732	0.743662	−	0.668556i	$-0.233087\pi$
$967$	4.18214e10	1.48732	0.743662	+	0.668556i	$0.233087\pi$
$968$	7.73044e9	0.273930
$969$	0	0
$970$	1.22688e10	0.431621
$971$	−2.39112e10	−0.838173	−0.419086	−	0.907946i	$-0.637650\pi$
$971$	−2.39112e10	−0.838173	−0.419086	+	0.907946i	$0.637650\pi$
$972$	0	0
$973$	−4.86230e10	−1.69218
$974$	−9.62307e10	−3.33701
$975$	0	0
$976$	−1.11470e11	−3.83781
$977$	−2.41698e10	−0.829168	−0.414584	−	0.910011i	$-0.636073\pi$
$977$	−2.41698e10	−0.829168	−0.414584	+	0.910011i	$0.636073\pi$
$978$	0	0
$979$	5.62977e9	0.191757
$980$	−2.96215e10	−1.00535
$981$	0	0
$982$	−3.14850e10	−1.06100
$983$	−1.44160e10	−0.484070	−0.242035	−	0.970268i	$-0.577815\pi$
$983$	−1.44160e10	−0.484070	−0.242035	+	0.970268i	$0.577815\pi$
$984$	0	0
$985$	1.74032e10	0.580232
$986$	3.51411e6	0.000116747 0
$987$	0	0
$988$	6.92410e10	2.28410
$989$	−2.58594e10	−0.850023
$990$	0	0
$991$	1.67451e10	0.546552	0.273276	−	0.961936i	$-0.411893\pi$
$991$	1.67451e10	0.546552	0.273276	+	0.961936i	$0.411893\pi$
$992$	−1.38963e11	−4.51970
$993$	0	0
$994$	−9.42375e10	−3.04349
$995$	6.98960e9	0.224942
$996$	0	0
$997$	1.15614e10	0.369468	0.184734	−	0.982789i	$-0.440858\pi$
$997$	1.15614e10	0.369468	0.184734	+	0.982789i	$0.440858\pi$
$998$	2.83833e10	0.903870
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.8.a.i.1.1		15
3.2	odd	2		405.8.a.j.1.15		15
9.2	odd	6		135.8.e.b.91.1		30
9.4	even	3		45.8.e.b.16.15	✓	30
9.5	odd	6		135.8.e.b.46.1		30
9.7	even	3		45.8.e.b.31.15	yes	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.8.e.b.16.15	✓	30	9.4	even	3
45.8.e.b.31.15	yes	30	9.7	even	3
135.8.e.b.46.1		30	9.5	odd	6
135.8.e.b.91.1		30	9.2	odd	6
405.8.a.i.1.1		15	1.1	even	1	trivial
405.8.a.j.1.15		15	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)

\(f(q)\)	\(=\)	\(q-22.0996 q^{2} +360.392 q^{4} -125.000 q^{5} -1217.00 q^{7} -5135.77 q^{8} +O(q^{10})\)	\(q-22.0996 q^{2} +360.392 q^{4} -125.000 q^{5} -1217.00 q^{7} -5135.77 q^{8} +2762.45 q^{10} +4240.51 q^{11} +6565.14 q^{13} +26895.1 q^{14} +67368.2 q^{16} +52.6752 q^{17} +29264.7 q^{19} -45049.0 q^{20} -93713.6 q^{22} -71911.8 q^{23} +15625.0 q^{25} -145087. q^{26} -438596. q^{28} -3018.73 q^{29} +167138. q^{31} -831431. q^{32} -1164.10 q^{34} +152125. q^{35} +332127. q^{37} -646738. q^{38} +641971. q^{40} -258257. q^{41} +359598. q^{43} +1.52825e6 q^{44} +1.58922e6 q^{46} -886155. q^{47} +657540. q^{49} -345306. q^{50} +2.36602e6 q^{52} +552041. q^{53} -530064. q^{55} +6.25021e6 q^{56} +66712.7 q^{58} +842213. q^{59} -1.65464e6 q^{61} -3.69367e6 q^{62} +9.75115e6 q^{64} -820643. q^{65} -1.30236e6 q^{67} +18983.7 q^{68} -3.36189e6 q^{70} -3.50388e6 q^{71} -2.14571e6 q^{73} -7.33987e6 q^{74} +1.05468e7 q^{76} -5.16070e6 q^{77} +3.08591e6 q^{79} -8.42102e6 q^{80} +5.70738e6 q^{82} +7.81178e6 q^{83} -6584.40 q^{85} -7.94698e6 q^{86} -2.17783e7 q^{88} +1.32762e6 q^{89} -7.98976e6 q^{91} -2.59164e7 q^{92} +1.95837e7 q^{94} -3.65809e6 q^{95} +4.44129e6 q^{97} -1.45314e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - 8 q^{2} + 1088 q^{4} - 1875 q^{5} + 1289 q^{7} - 4434 q^{8}+O(q^{10}) \) 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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