LMFDB - Embedded newform 405.8.a.i.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(405, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("405.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

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

Embedding invariants

Embedding label			1.10
Root			$10.0738$ 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)

$f(q)$	$=$	$q+9.07378 q^{2} -45.6666 q^{4} -125.000 q^{5} -490.864 q^{7} -1575.81 q^{8} +O(q^{10})$	\(q+9.07378 q^{2} -45.6666 q^{4} -125.000 q^{5} -490.864 q^{7} -1575.81 q^{8} -1134.22 q^{10} -6239.55 q^{11} -11466.2 q^{13} -4453.99 q^{14} -8453.24 q^{16} +15499.6 q^{17} +12634.8 q^{19} +5708.32 q^{20} -56616.3 q^{22} -114011. q^{23} +15625.0 q^{25} -104042. q^{26} +22416.1 q^{28} -209514. q^{29} -272894. q^{31} +125001. q^{32} +140640. q^{34} +61358.0 q^{35} +410589. q^{37} +114646. q^{38} +196976. q^{40} +643846. q^{41} -50823.4 q^{43} +284939. q^{44} -1.03451e6 q^{46} +123731. q^{47} -582596. q^{49} +141778. q^{50} +523622. q^{52} -1.32097e6 q^{53} +779944. q^{55} +773509. q^{56} -1.90109e6 q^{58} +1.00402e6 q^{59} -54206.8 q^{61} -2.47618e6 q^{62} +2.21625e6 q^{64} +1.43327e6 q^{65} +722215. q^{67} -707815. q^{68} +556749. q^{70} -121397. q^{71} -2.75261e6 q^{73} +3.72559e6 q^{74} -576989. q^{76} +3.06277e6 q^{77} -960694. q^{79} +1.05666e6 q^{80} +5.84212e6 q^{82} +3.64417e6 q^{83} -1.93746e6 q^{85} -461161. q^{86} +9.83235e6 q^{88} +2.14726e6 q^{89} +5.62834e6 q^{91} +5.20647e6 q^{92} +1.12271e6 q^{94} -1.57935e6 q^{95} +224178. q^{97} -5.28634e6 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	9.07378	0.802016	0.401008	−	0.916075i	$-0.368660\pi$
$2$	9.07378	0.802016	0.401008	+	0.916075i	$0.368660\pi$
$3$	0	0
$3$	0	0
$4$	−45.6666	−0.356770
$5$	−125.000	−0.447214
$5$	−125.000	−0.447214
$6$	0	0
$7$	−490.864	−0.540901	−0.270451	−	0.962734i	$-0.587173\pi$
$7$	−490.864	−0.540901	−0.270451	+	0.962734i	$0.587173\pi$
$8$	−1575.81	−1.08815
$9$	0	0
$10$	−1134.22	−0.358673
$11$	−6239.55	−1.41344	−0.706722	−	0.707491i	$-0.749827\pi$
$11$	−6239.55	−1.41344	−0.706722	+	0.707491i	$0.749827\pi$
$12$	0	0
$13$	−11466.2	−1.44750	−0.723749	−	0.690064i	$-0.757582\pi$
$13$	−11466.2	−1.44750	−0.723749	+	0.690064i	$0.757582\pi$
$14$	−4453.99	−0.433812
$15$	0	0
$16$	−8453.24	−0.515945
$17$	15499.6	0.765157	0.382578	−	0.923923i	$-0.375036\pi$
$17$	15499.6	0.765157	0.382578	+	0.923923i	$0.375036\pi$
$18$	0	0
$19$	12634.8	0.422602	0.211301	−	0.977421i	$-0.432230\pi$
$19$	12634.8	0.422602	0.211301	+	0.977421i	$0.432230\pi$
$20$	5708.32	0.159552
$21$	0	0
$22$	−56616.3	−1.13361
$23$	−114011.	−1.95388	−0.976940	−	0.213516i	$-0.931509\pi$
$23$	−114011.	−1.95388	−0.976940	+	0.213516i	$0.931509\pi$
$24$	0	0
$25$	15625.0	0.200000
$26$	−104042.	−1.16092
$27$	0	0
$28$	22416.1	0.192977
$29$	−209514.	−1.59522	−0.797610	−	0.603174i	$-0.793903\pi$
$29$	−209514.	−1.59522	−0.797610	+	0.603174i	$0.793903\pi$
$30$	0	0
$31$	−272894.	−1.64524	−0.822618	−	0.568594i	$-0.807488\pi$
$31$	−272894.	−1.64524	−0.822618	+	0.568594i	$0.807488\pi$
$32$	125001.	0.674355
$33$	0	0
$34$	140640.	0.613668
$35$	61358.0	0.241898
$36$	0	0
$37$	410589.	1.33260	0.666302	−	0.745682i	$-0.267876\pi$
$37$	410589.	1.33260	0.666302	+	0.745682i	$0.267876\pi$
$38$	114646.	0.338934
$39$	0	0
$40$	196976.	0.486636
$41$	643846.	1.45894	0.729472	−	0.684010i	$-0.239766\pi$
$41$	643846.	1.45894	0.729472	+	0.684010i	$0.239766\pi$
$42$	0	0
$43$	−50823.4	−0.0974820	−0.0487410	−	0.998811i	$-0.515521\pi$
$43$	−50823.4	−0.0974820	−0.0487410	+	0.998811i	$0.515521\pi$
$44$	284939.	0.504275
$45$	0	0
$46$	−1.03451e6	−1.56704
$47$	123731.	0.173835	0.0869175	−	0.996216i	$-0.472298\pi$
$47$	123731.	0.173835	0.0869175	+	0.996216i	$0.472298\pi$
$48$	0	0
$49$	−582596.	−0.707426
$50$	141778.	0.160403
$51$	0	0
$52$	523622.	0.516424
$53$	−1.32097e6	−1.21878	−0.609392	−	0.792869i	$-0.708586\pi$
$53$	−1.32097e6	−1.21878	−0.609392	+	0.792869i	$0.708586\pi$
$54$	0	0
$55$	779944.	0.632112
$56$	773509.	0.588583
$57$	0	0
$58$	−1.90109e6	−1.27939
$59$	1.00402e6	0.636442	0.318221	−	0.948017i	$-0.396915\pi$
$59$	1.00402e6	0.636442	0.318221	+	0.948017i	$0.396915\pi$
$60$	0	0
$61$	−54206.8	−0.0305773	−0.0152886	−	0.999883i	$-0.504867\pi$
$61$	−54206.8	−0.0305773	−0.0152886	+	0.999883i	$0.504867\pi$
$62$	−2.47618e6	−1.31951
$63$	0	0
$64$	2.21625e6	1.05679
$65$	1.43327e6	0.647340
$66$	0	0
$67$	722215.	0.293363	0.146681	−	0.989184i	$-0.453141\pi$
$67$	722215.	0.293363	0.146681	+	0.989184i	$0.453141\pi$
$68$	−707815.	−0.272985
$69$	0	0
$70$	556749.	0.194006
$71$	−121397.	−0.0402534	−0.0201267	−	0.999797i	$-0.506407\pi$
$71$	−121397.	−0.0402534	−0.0201267	+	0.999797i	$0.506407\pi$
$72$	0	0
$73$	−2.75261e6	−0.828161	−0.414081	−	0.910240i	$-0.635897\pi$
$73$	−2.75261e6	−0.828161	−0.414081	+	0.910240i	$0.635897\pi$
$74$	3.72559e6	1.06877
$75$	0	0
$76$	−576989.	−0.150772
$77$	3.06277e6	0.764534
$78$	0	0
$79$	−960694.	−0.219225	−0.109613	−	0.993974i	$-0.534961\pi$
$79$	−960694.	−0.219225	−0.109613	+	0.993974i	$0.534961\pi$
$80$	1.05666e6	0.230738
$81$	0	0
$82$	5.84212e6	1.17010
$83$	3.64417e6	0.699561	0.349780	−	0.936832i	$-0.386256\pi$
$83$	3.64417e6	0.699561	0.349780	+	0.936832i	$0.386256\pi$
$84$	0	0
$85$	−1.93746e6	−0.342188
$86$	−461161.	−0.0781822
$87$	0	0
$88$	9.83235e6	1.53804
$89$	2.14726e6	0.322863	0.161432	−	0.986884i	$-0.448389\pi$
$89$	2.14726e6	0.322863	0.161432	+	0.986884i	$0.448389\pi$
$90$	0	0
$91$	5.62834e6	0.782953
$92$	5.20647e6	0.697086
$93$	0	0
$94$	1.12271e6	0.139418
$95$	−1.57935e6	−0.188993
$96$	0	0
$97$	224178.	0.0249397	0.0124699	−	0.999922i	$-0.496031\pi$
$97$	224178.	0.0249397	0.0124699	+	0.999922i	$0.496031\pi$
$98$	−5.28634e6	−0.567367
$99$	0	0
$100$	−713540.	−0.0713540
$101$	−5.57998e6	−0.538900	−0.269450	−	0.963014i	$-0.586842\pi$
$101$	−5.57998e6	−0.538900	−0.269450	+	0.963014i	$0.586842\pi$
$102$	0	0
$103$	−9.90812e6	−0.893431	−0.446716	−	0.894676i	$-0.647406\pi$
$103$	−9.90812e6	−0.893431	−0.446716	+	0.894676i	$0.647406\pi$
$104$	1.80686e7	1.57510
$105$	0	0
$106$	−1.19862e7	−0.977484
$107$	7.11157e6	0.561206	0.280603	−	0.959824i	$-0.409466\pi$
$107$	7.11157e6	0.561206	0.280603	+	0.959824i	$0.409466\pi$
$108$	0	0
$109$	−1.63384e7	−1.20842	−0.604210	−	0.796825i	$-0.706511\pi$
$109$	−1.63384e7	−1.20842	−0.604210	+	0.796825i	$0.706511\pi$
$110$	7.07703e6	0.506964
$111$	0	0
$112$	4.14939e6	0.279075
$113$	−1.40252e6	−0.0914395	−0.0457198	−	0.998954i	$-0.514558\pi$
$113$	−1.40252e6	−0.0914395	−0.0457198	+	0.998954i	$0.514558\pi$
$114$	0	0
$115$	1.42513e7	0.873801
$116$	9.56780e6	0.569127
$117$	0	0
$118$	9.11023e6	0.510437
$119$	−7.60822e6	−0.413874
$120$	0	0
$121$	1.94448e7	0.997826
$122$	−491860.	−0.0245235
$123$	0	0
$124$	1.24621e7	0.586971
$125$	−1.95312e6	−0.0894427
$126$	0	0
$127$	3.37654e7	1.46271	0.731356	−	0.681996i	$-0.238888\pi$
$127$	3.37654e7	1.46271	0.731356	+	0.681996i	$0.238888\pi$
$128$	4.10960e6	0.173207
$129$	0	0
$130$	1.30052e7	0.519178
$131$	2.84327e7	1.10501	0.552507	−	0.833508i	$-0.313671\pi$
$131$	2.84327e7	1.10501	0.552507	+	0.833508i	$0.313671\pi$
$132$	0	0
$133$	−6.20198e6	−0.228586
$134$	6.55322e6	0.235282
$135$	0	0
$136$	−2.44245e7	−0.832606
$137$	−1.62248e7	−0.539086	−0.269543	−	0.962988i	$-0.586873\pi$
$137$	−1.62248e7	−0.539086	−0.269543	+	0.962988i	$0.586873\pi$
$138$	0	0
$139$	5.26247e7	1.66203	0.831013	−	0.556253i	$-0.187762\pi$
$139$	5.26247e7	1.66203	0.831013	+	0.556253i	$0.187762\pi$
$140$	−2.80201e6	−0.0863021
$141$	0	0
$142$	−1.10153e6	−0.0322839
$143$	7.15439e7	2.04596
$144$	0	0
$145$	2.61893e7	0.713404
$146$	−2.49766e7	−0.664199
$147$	0	0
$148$	−1.87502e7	−0.475433
$149$	−5.07039e7	−1.25571	−0.627855	−	0.778331i	$-0.716067\pi$
$149$	−5.07039e7	−1.25571	−0.627855	+	0.778331i	$0.716067\pi$
$150$	0	0
$151$	1.46682e7	0.346702	0.173351	−	0.984860i	$-0.444541\pi$
$151$	1.46682e7	0.346702	0.173351	+	0.984860i	$0.444541\pi$
$152$	−1.99101e7	−0.459855
$153$	0	0
$154$	2.77909e7	0.613169
$155$	3.41118e7	0.735772
$156$	0	0
$157$	2.75475e7	0.568111	0.284055	−	0.958808i	$-0.408320\pi$
$157$	2.75475e7	0.568111	0.284055	+	0.958808i	$0.408320\pi$
$158$	−8.71713e6	−0.175822
$159$	0	0
$160$	−1.56251e7	−0.301581
$161$	5.59637e7	1.05686
$162$	0	0
$163$	−7.23116e7	−1.30783	−0.653915	−	0.756568i	$-0.726874\pi$
$163$	−7.23116e7	−1.30783	−0.653915	+	0.756568i	$0.726874\pi$
$164$	−2.94023e7	−0.520508
$165$	0	0
$166$	3.30664e7	0.561059
$167$	−9.36755e7	−1.55639	−0.778195	−	0.628023i	$-0.783864\pi$
$167$	−9.36755e7	−1.55639	−0.778195	+	0.628023i	$0.783864\pi$
$168$	0	0
$169$	6.87252e7	1.09525
$170$	−1.75800e7	−0.274441
$171$	0	0
$172$	2.32093e6	0.0347787
$173$	−8.99290e6	−0.132050	−0.0660250	−	0.997818i	$-0.521032\pi$
$173$	−8.99290e6	−0.132050	−0.0660250	+	0.997818i	$0.521032\pi$
$174$	0	0
$175$	−7.66975e6	−0.108180
$176$	5.27444e7	0.729260
$177$	0	0
$178$	1.94837e7	0.258942
$179$	−7.86643e6	−0.102516	−0.0512581	−	0.998685i	$-0.516323\pi$
$179$	−7.86643e6	−0.102516	−0.0512581	+	0.998685i	$0.516323\pi$
$180$	0	0
$181$	9.96349e7	1.24893	0.624463	−	0.781054i	$-0.285318\pi$
$181$	9.96349e7	1.24893	0.624463	+	0.781054i	$0.285318\pi$
$182$	5.10703e7	0.627941
$183$	0	0
$184$	1.79659e8	2.12612
$185$	−5.13236e7	−0.595959
$186$	0	0
$187$	−9.67108e7	−1.08151
$188$	−5.65038e6	−0.0620191
$189$	0	0
$190$	−1.43307e7	−0.151576
$191$	2.04236e7	0.212087	0.106044	−	0.994361i	$-0.466182\pi$
$191$	2.04236e7	0.212087	0.106044	+	0.994361i	$0.466182\pi$
$192$	0	0
$193$	7.49373e7	0.750322	0.375161	−	0.926960i	$-0.377587\pi$
$193$	7.49373e7	0.750322	0.375161	+	0.926960i	$0.377587\pi$
$194$	2.03414e6	0.0200021
$195$	0	0
$196$	2.66051e7	0.252388
$197$	−1.29377e8	−1.20566	−0.602831	−	0.797869i	$-0.705961\pi$
$197$	−1.29377e8	−1.20566	−0.602831	+	0.797869i	$0.705961\pi$
$198$	0	0
$199$	−4.86056e7	−0.437220	−0.218610	−	0.975812i	$-0.570152\pi$
$199$	−4.86056e7	−0.437220	−0.218610	+	0.975812i	$0.570152\pi$
$200$	−2.46221e7	−0.217630
$201$	0	0
$202$	−5.06315e7	−0.432206
$203$	1.02843e8	0.862857
$204$	0	0
$205$	−8.04808e7	−0.652460
$206$	−8.99041e7	−0.716546
$207$	0	0
$208$	9.69266e7	0.746829
$209$	−7.88356e7	−0.597325
$210$	0	0
$211$	−1.06138e8	−0.777826	−0.388913	−	0.921275i	$-0.627149\pi$
$211$	−1.06138e8	−0.777826	−0.388913	+	0.921275i	$0.627149\pi$
$212$	6.03240e7	0.434825
$213$	0	0
$214$	6.45288e7	0.450096
$215$	6.35293e6	0.0435953
$216$	0	0
$217$	1.33954e8	0.889911
$218$	−1.48251e8	−0.969173
$219$	0	0
$220$	−3.56174e7	−0.225519
$221$	−1.77722e8	−1.10756
$222$	0	0
$223$	3.15677e8	1.90623	0.953117	−	0.302603i	$-0.0978556\pi$
$223$	3.15677e8	1.90623	0.953117	+	0.302603i	$0.0978556\pi$
$224$	−6.13585e7	−0.364760
$225$	0	0
$226$	−1.27261e7	−0.0733360
$227$	1.63542e8	0.927982	0.463991	−	0.885840i	$-0.346417\pi$
$227$	1.63542e8	0.927982	0.463991	+	0.885840i	$0.346417\pi$
$228$	0	0
$229$	1.06123e8	0.583965	0.291983	−	0.956424i	$-0.405685\pi$
$229$	1.06123e8	0.583965	0.291983	+	0.956424i	$0.405685\pi$
$230$	1.29313e8	0.700803
$231$	0	0
$232$	3.30155e8	1.73584
$233$	−1.31661e8	−0.681885	−0.340943	−	0.940084i	$-0.610746\pi$
$233$	−1.31661e8	−0.681885	−0.340943	+	0.940084i	$0.610746\pi$
$234$	0	0
$235$	−1.54664e7	−0.0777414
$236$	−4.58500e7	−0.227064
$237$	0	0
$238$	−6.90353e7	−0.331934
$239$	−1.05811e8	−0.501348	−0.250674	−	0.968072i	$-0.580652\pi$
$239$	−1.05811e8	−0.501348	−0.250674	+	0.968072i	$0.580652\pi$
$240$	0	0
$241$	1.46606e8	0.674671	0.337335	−	0.941385i	$-0.390474\pi$
$241$	1.46606e8	0.674671	0.337335	+	0.941385i	$0.390474\pi$
$242$	1.76438e8	0.800273
$243$	0	0
$244$	2.47544e6	0.0109091
$245$	7.28244e7	0.316370
$246$	0	0
$247$	−1.44873e8	−0.611715
$248$	4.30030e8	1.79027
$249$	0	0
$250$	−1.77222e7	−0.0717345
$251$	−4.94911e7	−0.197546	−0.0987732	−	0.995110i	$-0.531492\pi$
$251$	−4.94911e7	−0.197546	−0.0987732	+	0.995110i	$0.531492\pi$
$252$	0	0
$253$	7.11375e8	2.76170
$254$	3.06380e8	1.17312
$255$	0	0
$256$	−2.46390e8	−0.917874
$257$	−2.07344e8	−0.761947	−0.380974	−	0.924586i	$-0.624411\pi$
$257$	−2.07344e8	−0.761947	−0.380974	+	0.924586i	$0.624411\pi$
$258$	0	0
$259$	−2.01543e8	−0.720807
$260$	−6.54527e7	−0.230952
$261$	0	0
$262$	2.57992e8	0.886240
$263$	−5.23689e8	−1.77512	−0.887561	−	0.460689i	$-0.847602\pi$
$263$	−5.23689e8	−1.77512	−0.887561	+	0.460689i	$0.847602\pi$
$264$	0	0
$265$	1.65121e8	0.545056
$266$	−5.62754e7	−0.183330
$267$	0	0
$268$	−3.29811e7	−0.104663
$269$	−3.70195e8	−1.15957	−0.579785	−	0.814769i	$-0.696864\pi$
$269$	−3.70195e8	−1.15957	−0.579785	+	0.814769i	$0.696864\pi$
$270$	0	0
$271$	8.64816e7	0.263956	0.131978	−	0.991253i	$-0.457867\pi$
$271$	8.64816e7	0.263956	0.131978	+	0.991253i	$0.457867\pi$
$272$	−1.31022e8	−0.394779
$273$	0	0
$274$	−1.47220e8	−0.432355
$275$	−9.74930e7	−0.282689
$276$	0	0
$277$	3.05826e7	0.0864560	0.0432280	−	0.999065i	$-0.486236\pi$
$277$	3.05826e7	0.0864560	0.0432280	+	0.999065i	$0.486236\pi$
$278$	4.77505e8	1.33297
$279$	0	0
$280$	−9.66887e7	−0.263222
$281$	−1.67165e8	−0.449442	−0.224721	−	0.974423i	$-0.572147\pi$
$281$	−1.67165e8	−0.449442	−0.224721	+	0.974423i	$0.572147\pi$
$282$	0	0
$283$	3.27259e8	0.858300	0.429150	−	0.903233i	$-0.358813\pi$
$283$	3.27259e8	0.858300	0.429150	+	0.903233i	$0.358813\pi$
$284$	5.54378e6	0.0143612
$285$	0	0
$286$	6.49174e8	1.64089
$287$	−3.16041e8	−0.789145
$288$	0	0
$289$	−1.70100e8	−0.414535
$290$	2.37636e8	0.572162
$291$	0	0
$292$	1.25702e8	0.295463
$293$	−4.05327e8	−0.941388	−0.470694	−	0.882297i	$-0.655996\pi$
$293$	−4.05327e8	−0.941388	−0.470694	+	0.882297i	$0.655996\pi$
$294$	0	0
$295$	−1.25502e8	−0.284626
$296$	−6.47011e8	−1.45007
$297$	0	0
$298$	−4.60076e8	−1.00710
$299$	1.30727e9	2.82824
$300$	0	0
$301$	2.49474e7	0.0527282
$302$	1.33096e8	0.278061
$303$	0	0
$304$	−1.06805e8	−0.218039
$305$	6.77585e6	0.0136746
$306$	0	0
$307$	1.52290e8	0.300392	0.150196	−	0.988656i	$-0.452010\pi$
$307$	1.52290e8	0.300392	0.150196	+	0.988656i	$0.452010\pi$
$308$	−1.39866e8	−0.272763
$309$	0	0
$310$	3.09523e8	0.590101
$311$	−5.16803e7	−0.0974235	−0.0487118	−	0.998813i	$-0.515512\pi$
$311$	−5.16803e7	−0.0974235	−0.0487118	+	0.998813i	$0.515512\pi$
$312$	0	0
$313$	−2.53194e8	−0.466710	−0.233355	−	0.972392i	$-0.574970\pi$
$313$	−2.53194e8	−0.466710	−0.233355	+	0.972392i	$0.574970\pi$
$314$	2.49960e8	0.455634
$315$	0	0
$316$	4.38716e7	0.0782130
$317$	−8.91131e8	−1.57121	−0.785604	−	0.618729i	$-0.787648\pi$
$317$	−8.91131e8	−1.57121	−0.785604	+	0.618729i	$0.787648\pi$
$318$	0	0
$319$	1.30727e9	2.25476
$320$	−2.77031e8	−0.472610
$321$	0	0
$322$	5.07802e8	0.847616
$323$	1.95835e8	0.323357
$324$	0	0
$325$	−1.79159e8	−0.289499
$326$	−6.56139e8	−1.04890
$327$	0	0
$328$	−1.01458e9	−1.58755
$329$	−6.07353e7	−0.0940276
$330$	0	0
$331$	−1.05200e9	−1.59447	−0.797234	−	0.603670i	$-0.793704\pi$
$331$	−1.05200e9	−1.59447	−0.797234	+	0.603670i	$0.793704\pi$
$332$	−1.66417e8	−0.249582
$333$	0	0
$334$	−8.49991e8	−1.24825
$335$	−9.02769e7	−0.131196
$336$	0	0
$337$	1.14312e9	1.62700	0.813499	−	0.581567i	$-0.197560\pi$
$337$	1.14312e9	1.62700	0.813499	+	0.581567i	$0.197560\pi$
$338$	6.23597e8	0.878407
$339$	0	0
$340$	8.84769e7	0.122083
$341$	1.70274e9	2.32545
$342$	0	0
$343$	6.90223e8	0.923549
$344$	8.00882e7	0.106075
$345$	0	0
$346$	−8.15995e7	−0.105906
$347$	4.91728e8	0.631788	0.315894	−	0.948794i	$-0.397696\pi$
$347$	4.91728e8	0.631788	0.315894	+	0.948794i	$0.397696\pi$
$348$	0	0
$349$	−1.01563e9	−1.27893	−0.639464	−	0.768821i	$-0.720844\pi$
$349$	−1.01563e9	−1.27893	−0.639464	+	0.768821i	$0.720844\pi$
$350$	−6.95936e7	−0.0867623
$351$	0	0
$352$	−7.79950e8	−0.953164
$353$	2.85962e8	0.346016	0.173008	−	0.984920i	$-0.444651\pi$
$353$	2.85962e8	0.346016	0.173008	+	0.984920i	$0.444651\pi$
$354$	0	0
$355$	1.51746e7	0.0180019
$356$	−9.80578e7	−0.115188
$357$	0	0
$358$	−7.13782e7	−0.0822196
$359$	−2.80619e8	−0.320100	−0.160050	−	0.987109i	$-0.551166\pi$
$359$	−2.80619e8	−0.320100	−0.160050	+	0.987109i	$0.551166\pi$
$360$	0	0
$361$	−7.34233e8	−0.821407
$362$	9.04065e8	1.00166
$363$	0	0
$364$	−2.57027e8	−0.279334
$365$	3.44076e8	0.370365
$366$	0	0
$367$	−1.86695e8	−0.197152	−0.0985759	−	0.995130i	$-0.531429\pi$
$367$	−1.86695e8	−0.197152	−0.0985759	+	0.995130i	$0.531429\pi$
$368$	9.63759e8	1.00809
$369$	0	0
$370$	−4.65699e8	−0.477968
$371$	6.48415e8	0.659241
$372$	0	0
$373$	−1.57359e9	−1.57004	−0.785018	−	0.619473i	$-0.787347\pi$
$373$	−1.57359e9	−1.57004	−0.785018	+	0.619473i	$0.787347\pi$
$374$	−8.77532e8	−0.867386
$375$	0	0
$376$	−1.94977e8	−0.189159
$377$	2.40233e9	2.30908
$378$	0	0
$379$	−2.04212e7	−0.0192683	−0.00963415	−	0.999954i	$-0.503067\pi$
$379$	−2.04212e7	−0.0192683	−0.00963415	+	0.999954i	$0.503067\pi$
$380$	7.21236e7	0.0674272
$381$	0	0
$382$	1.85319e8	0.170097
$383$	6.34517e8	0.577095	0.288548	−	0.957466i	$-0.406828\pi$
$383$	6.34517e8	0.577095	0.288548	+	0.957466i	$0.406828\pi$
$384$	0	0
$385$	−3.82846e8	−0.341910
$386$	6.79964e8	0.601770
$387$	0	0
$388$	−1.02374e7	−0.00889774
$389$	1.05418e9	0.908011	0.454006	−	0.890999i	$-0.349995\pi$
$389$	1.05418e9	0.908011	0.454006	+	0.890999i	$0.349995\pi$
$390$	0	0
$391$	−1.76712e9	−1.49502
$392$	9.18061e8	0.769786
$393$	0	0
$394$	−1.17394e9	−0.966961
$395$	1.20087e8	0.0980405
$396$	0	0
$397$	6.40301e8	0.513591	0.256795	−	0.966466i	$-0.417333\pi$
$397$	6.40301e8	0.513591	0.256795	+	0.966466i	$0.417333\pi$
$398$	−4.41036e8	−0.350658
$399$	0	0
$400$	−1.32082e8	−0.103189
$401$	8.63189e8	0.668499	0.334249	−	0.942485i	$-0.391517\pi$
$401$	8.63189e8	0.668499	0.334249	+	0.942485i	$0.391517\pi$
$402$	0	0
$403$	3.12906e9	2.38148
$404$	2.54819e8	0.192263
$405$	0	0
$406$	9.33175e8	0.692025
$407$	−2.56189e9	−1.88356
$408$	0	0
$409$	9.31703e8	0.673358	0.336679	−	0.941620i	$-0.390696\pi$
$409$	9.31703e8	0.673358	0.336679	+	0.941620i	$0.390696\pi$
$410$	−7.30265e8	−0.523283
$411$	0	0
$412$	4.52470e8	0.318749
$413$	−4.92836e8	−0.344252
$414$	0	0
$415$	−4.55521e8	−0.312853
$416$	−1.43329e9	−0.976127
$417$	0	0
$418$	−7.15337e8	−0.479064
$419$	−1.92437e9	−1.27803	−0.639014	−	0.769195i	$-0.720657\pi$
$419$	−1.92437e9	−1.27803	−0.639014	+	0.769195i	$0.720657\pi$
$420$	0	0
$421$	−2.30629e9	−1.50635	−0.753175	−	0.657820i	$-0.771479\pi$
$421$	−2.30629e9	−1.50635	−0.753175	+	0.657820i	$0.771479\pi$
$422$	−9.63073e8	−0.623829
$423$	0	0
$424$	2.08160e9	1.32622
$425$	2.42182e8	0.153031
$426$	0	0
$427$	2.66081e7	0.0165393
$428$	−3.24761e8	−0.200222
$429$	0	0
$430$	5.76451e7	0.0349641
$431$	−2.75111e9	−1.65515	−0.827576	−	0.561353i	$-0.810281\pi$
$431$	−2.75111e9	−1.65515	−0.827576	+	0.561353i	$0.810281\pi$
$432$	0	0
$433$	−4.20977e8	−0.249202	−0.124601	−	0.992207i	$-0.539765\pi$
$433$	−4.20977e8	−0.249202	−0.124601	+	0.992207i	$0.539765\pi$
$434$	1.21547e9	0.713723
$435$	0	0
$436$	7.46121e8	0.431128
$437$	−1.44050e9	−0.825713
$438$	0	0
$439$	3.33004e8	0.187855	0.0939276	−	0.995579i	$-0.470058\pi$
$439$	3.33004e8	0.187855	0.0939276	+	0.995579i	$0.470058\pi$
$440$	−1.22904e9	−0.687833
$441$	0	0
$442$	−1.61261e9	−0.888283
$443$	1.55294e9	0.848677	0.424338	−	0.905504i	$-0.360507\pi$
$443$	1.55294e9	0.848677	0.424338	+	0.905504i	$0.360507\pi$
$444$	0	0
$445$	−2.68407e8	−0.144389
$446$	2.86438e9	1.52883
$447$	0	0
$448$	−1.08788e9	−0.571619
$449$	−2.78767e9	−1.45338	−0.726689	−	0.686966i	$-0.758942\pi$
$449$	−2.78767e9	−1.45338	−0.726689	+	0.686966i	$0.758942\pi$
$450$	0	0
$451$	−4.01731e9	−2.06214
$452$	6.40482e7	0.0326229
$453$	0	0
$454$	1.48395e9	0.744257
$455$	−7.03543e8	−0.350147
$456$	0	0
$457$	6.72250e8	0.329476	0.164738	−	0.986337i	$-0.447322\pi$
$457$	6.72250e8	0.329476	0.164738	+	0.986337i	$0.447322\pi$
$458$	9.62940e8	0.468349
$459$	0	0
$460$	−6.50809e8	−0.311746
$461$	−2.67024e9	−1.26940	−0.634698	−	0.772761i	$-0.718875\pi$
$461$	−2.67024e9	−1.26940	−0.634698	+	0.772761i	$0.718875\pi$
$462$	0	0
$463$	−2.45920e9	−1.15149	−0.575746	−	0.817629i	$-0.695288\pi$
$463$	−2.45920e9	−1.15149	−0.575746	+	0.817629i	$0.695288\pi$
$464$	1.77108e9	0.823046
$465$	0	0
$466$	−1.19466e9	−0.546883
$467$	−8.95274e8	−0.406768	−0.203384	−	0.979099i	$-0.565194\pi$
$467$	−8.95274e8	−0.406768	−0.203384	+	0.979099i	$0.565194\pi$
$468$	0	0
$469$	−3.54509e8	−0.158680
$470$	−1.40339e8	−0.0623498
$471$	0	0
$472$	−1.58214e9	−0.692546
$473$	3.17115e8	0.137785
$474$	0	0
$475$	1.97419e8	0.0845204
$476$	3.47441e8	0.147658
$477$	0	0
$478$	−9.60107e8	−0.402089
$479$	3.39003e9	1.40938	0.704691	−	0.709514i	$-0.251086\pi$
$479$	3.39003e9	1.40938	0.704691	+	0.709514i	$0.251086\pi$
$480$	0	0
$481$	−4.70789e9	−1.92894
$482$	1.33027e9	0.541097
$483$	0	0
$484$	−8.87977e8	−0.355994
$485$	−2.80222e7	−0.0111534
$486$	0	0
$487$	5.37566e8	0.210902	0.105451	−	0.994424i	$-0.466371\pi$
$487$	5.37566e8	0.210902	0.105451	+	0.994424i	$0.466371\pi$
$488$	8.54197e7	0.0332727
$489$	0	0
$490$	6.60793e8	0.253734
$491$	−1.23890e9	−0.472337	−0.236168	−	0.971712i	$-0.575892\pi$
$491$	−1.23890e9	−0.472337	−0.236168	+	0.971712i	$0.575892\pi$
$492$	0	0
$493$	−3.24740e9	−1.22059
$494$	−1.31455e9	−0.490606
$495$	0	0
$496$	2.30684e9	0.848852
$497$	5.95893e7	0.0217731
$498$	0	0
$499$	1.70904e9	0.615743	0.307871	−	0.951428i	$-0.400383\pi$
$499$	1.70904e9	0.615743	0.307871	+	0.951428i	$0.400383\pi$
$500$	8.91925e7	0.0319105
$501$	0	0
$502$	−4.49071e8	−0.158435
$503$	−2.02866e9	−0.710757	−0.355378	−	0.934723i	$-0.615648\pi$
$503$	−2.02866e9	−0.710757	−0.355378	+	0.934723i	$0.615648\pi$
$504$	0	0
$505$	6.97498e8	0.241003
$506$	6.45485e9	2.21493
$507$	0	0
$508$	−1.54195e9	−0.521852
$509$	5.25156e8	0.176513	0.0882564	−	0.996098i	$-0.471871\pi$
$509$	5.25156e8	0.176513	0.0882564	+	0.996098i	$0.471871\pi$
$510$	0	0
$511$	1.35116e9	0.447953
$512$	−2.76172e9	−0.909357
$513$	0	0
$514$	−1.88139e9	−0.611094
$515$	1.23852e9	0.399555
$516$	0	0
$517$	−7.72028e8	−0.245706
$518$	−1.82876e9	−0.578099
$519$	0	0
$520$	−2.25857e9	−0.704405
$521$	2.31213e9	0.716274	0.358137	−	0.933669i	$-0.383412\pi$
$521$	2.31213e9	0.716274	0.358137	+	0.933669i	$0.383412\pi$
$522$	0	0
$523$	3.20590e9	0.979929	0.489965	−	0.871742i	$-0.337010\pi$
$523$	3.20590e9	0.979929	0.489965	+	0.871742i	$0.337010\pi$
$524$	−1.29842e9	−0.394236
$525$	0	0
$526$	−4.75184e9	−1.42368
$527$	−4.22976e9	−1.25886
$528$	0	0
$529$	9.59359e9	2.81764
$530$	1.49827e9	0.437144
$531$	0	0
$532$	2.83223e8	0.0815526
$533$	−7.38247e9	−2.11182
$534$	0	0
$535$	−8.88947e8	−0.250979
$536$	−1.13807e9	−0.319223
$537$	0	0
$538$	−3.35906e9	−0.929995
$539$	3.63513e9	0.999907
$540$	0	0
$541$	−2.34273e9	−0.636110	−0.318055	−	0.948072i	$-0.603030\pi$
$541$	−2.34273e9	−0.636110	−0.318055	+	0.948072i	$0.603030\pi$
$542$	7.84715e8	0.211697
$543$	0	0
$544$	1.93747e9	0.515987
$545$	2.04231e9	0.540422
$546$	0	0
$547$	−7.83283e8	−0.204627	−0.102314	−	0.994752i	$-0.532625\pi$
$547$	−7.83283e8	−0.204627	−0.102314	+	0.994752i	$0.532625\pi$
$548$	7.40931e8	0.192330
$549$	0	0
$550$	−8.84629e8	−0.226721
$551$	−2.64718e9	−0.674143
$552$	0	0
$553$	4.71570e8	0.118579
$554$	2.77500e8	0.0693391
$555$	0	0
$556$	−2.40319e9	−0.592961
$557$	−4.77251e9	−1.17018	−0.585091	−	0.810967i	$-0.698941\pi$
$557$	−4.77251e9	−1.17018	−0.585091	+	0.810967i	$0.698941\pi$
$558$	0	0
$559$	5.82752e8	0.141105
$560$	−5.18674e8	−0.124806
$561$	0	0
$562$	−1.51682e9	−0.360460
$563$	1.41884e9	0.335084	0.167542	−	0.985865i	$-0.446417\pi$
$563$	1.41884e9	0.335084	0.167542	+	0.985865i	$0.446417\pi$
$564$	0	0
$565$	1.75315e8	0.0408930
$566$	2.96948e9	0.688371
$567$	0	0
$568$	1.91299e8	0.0438018
$569$	−4.84761e9	−1.10315	−0.551576	−	0.834125i	$-0.685973\pi$
$569$	−4.84761e9	−1.10315	−0.551576	+	0.834125i	$0.685973\pi$
$570$	0	0
$571$	4.84041e9	1.08807	0.544034	−	0.839063i	$-0.316896\pi$
$571$	4.84041e9	1.08807	0.544034	+	0.839063i	$0.316896\pi$
$572$	−3.26717e9	−0.729936
$573$	0	0
$574$	−2.86769e9	−0.632907
$575$	−1.78142e9	−0.390776
$576$	0	0
$577$	5.14909e9	1.11587	0.557937	−	0.829883i	$-0.311593\pi$
$577$	5.14909e9	1.11587	0.557937	+	0.829883i	$0.311593\pi$
$578$	−1.54345e9	−0.332464
$579$	0	0
$580$	−1.19597e9	−0.254521
$581$	−1.78879e9	−0.378393
$582$	0	0
$583$	8.24224e9	1.72268
$584$	4.33760e9	0.901165
$585$	0	0
$586$	−3.67784e9	−0.755008
$587$	−3.28703e9	−0.670766	−0.335383	−	0.942082i	$-0.608866\pi$
$587$	−3.28703e9	−0.670766	−0.335383	+	0.942082i	$0.608866\pi$
$588$	0	0
$589$	−3.44797e9	−0.695280
$590$	−1.13878e9	−0.228274
$591$	0	0
$592$	−3.47081e9	−0.687550
$593$	−3.95848e9	−0.779537	−0.389769	−	0.920913i	$-0.627445\pi$
$593$	−3.95848e9	−0.779537	−0.389769	+	0.920913i	$0.627445\pi$
$594$	0	0
$595$	9.51027e8	0.185090
$596$	2.31547e9	0.447999
$597$	0	0
$598$	1.18619e10	2.26829
$599$	6.82646e9	1.29778	0.648891	−	0.760881i	$-0.275233\pi$
$599$	6.82646e9	1.29778	0.648891	+	0.760881i	$0.275233\pi$
$600$	0	0
$601$	−9.12039e8	−0.171377	−0.0856885	−	0.996322i	$-0.527309\pi$
$601$	−9.12039e8	−0.171377	−0.0856885	+	0.996322i	$0.527309\pi$
$602$	2.26367e8	0.0422888
$603$	0	0
$604$	−6.69844e8	−0.123693
$605$	−2.43060e9	−0.446241
$606$	0	0
$607$	5.49386e9	0.997051	0.498525	−	0.866875i	$-0.333875\pi$
$607$	5.49386e9	0.997051	0.498525	+	0.866875i	$0.333875\pi$
$608$	1.57937e9	0.284984
$609$	0	0
$610$	6.14825e7	0.0109672
$611$	−1.41873e9	−0.251626
$612$	0	0
$613$	−8.13874e9	−1.42707	−0.713535	−	0.700619i	$-0.752907\pi$
$613$	−8.13874e9	−1.42707	−0.713535	+	0.700619i	$0.752907\pi$
$614$	1.38185e9	0.240919
$615$	0	0
$616$	−4.82635e9	−0.831929
$617$	−4.05867e8	−0.0695643	−0.0347821	−	0.999395i	$-0.511074\pi$
$617$	−4.05867e8	−0.0695643	−0.0347821	+	0.999395i	$0.511074\pi$
$618$	0	0
$619$	−7.20851e9	−1.22160	−0.610799	−	0.791786i	$-0.709152\pi$
$619$	−7.20851e9	−1.22160	−0.610799	+	0.791786i	$0.709152\pi$
$620$	−1.55777e9	−0.262502
$621$	0	0
$622$	−4.68936e8	−0.0781353
$623$	−1.05401e9	−0.174637
$624$	0	0
$625$	2.44141e8	0.0400000
$626$	−2.29742e9	−0.374309
$627$	0	0
$628$	−1.25800e9	−0.202685
$629$	6.36398e9	1.01965
$630$	0	0
$631$	−3.72119e9	−0.589629	−0.294815	−	0.955555i	$-0.595258\pi$
$631$	−3.72119e9	−0.589629	−0.294815	+	0.955555i	$0.595258\pi$
$632$	1.51387e9	0.238550
$633$	0	0
$634$	−8.08592e9	−1.26013
$635$	−4.22068e9	−0.654145
$636$	0	0
$637$	6.68016e9	1.02400
$638$	1.18619e10	1.80835
$639$	0	0
$640$	−5.13699e8	−0.0774603
$641$	2.77948e9	0.416832	0.208416	−	0.978040i	$-0.433169\pi$
$641$	2.77948e9	0.416832	0.208416	+	0.978040i	$0.433169\pi$
$642$	0	0
$643$	1.05943e10	1.57158	0.785788	−	0.618495i	$-0.212257\pi$
$643$	1.05943e10	1.57158	0.785788	+	0.618495i	$0.212257\pi$
$644$	−2.55567e9	−0.377055
$645$	0	0
$646$	1.77697e9	0.259337
$647$	1.33949e9	0.194435	0.0972176	−	0.995263i	$-0.469006\pi$
$647$	1.33949e9	0.194435	0.0972176	+	0.995263i	$0.469006\pi$
$648$	0	0
$649$	−6.26461e9	−0.899576
$650$	−1.62565e9	−0.232183
$651$	0	0
$652$	3.30222e9	0.466594
$653$	5.85182e9	0.822422	0.411211	−	0.911540i	$-0.365106\pi$
$653$	5.85182e9	0.822422	0.411211	+	0.911540i	$0.365106\pi$
$654$	0	0
$655$	−3.55408e9	−0.494178
$656$	−5.44259e9	−0.752735
$657$	0	0
$658$	−5.51098e8	−0.0754116
$659$	−1.06877e9	−0.145475	−0.0727373	−	0.997351i	$-0.523173\pi$
$659$	−1.06877e9	−0.145475	−0.0727373	+	0.997351i	$0.523173\pi$
$660$	0	0
$661$	1.36682e10	1.84080	0.920398	−	0.390984i	$-0.127865\pi$
$661$	1.36682e10	1.84080	0.920398	+	0.390984i	$0.127865\pi$
$662$	−9.54557e9	−1.27879
$663$	0	0
$664$	−5.74253e9	−0.761228
$665$	7.75248e8	0.102227
$666$	0	0
$667$	2.38868e10	3.11687
$668$	4.27784e9	0.555273
$669$	0	0
$670$	−8.19152e8	−0.105221
$671$	3.38226e8	0.0432193
$672$	0	0
$673$	−7.16987e9	−0.906689	−0.453345	−	0.891335i	$-0.649769\pi$
$673$	−7.16987e9	−0.906689	−0.453345	+	0.891335i	$0.649769\pi$
$674$	1.03724e10	1.30488
$675$	0	0
$676$	−3.13844e9	−0.390752
$677$	1.00897e10	1.24974	0.624871	−	0.780728i	$-0.285152\pi$
$677$	1.00897e10	1.24974	0.624871	+	0.780728i	$0.285152\pi$
$678$	0	0
$679$	−1.10041e8	−0.0134899
$680$	3.05306e9	0.372353
$681$	0	0
$682$	1.54503e10	1.86505
$683$	−7.27134e9	−0.873257	−0.436628	−	0.899642i	$-0.643828\pi$
$683$	−7.27134e9	−0.873257	−0.436628	+	0.899642i	$0.643828\pi$
$684$	0	0
$685$	2.02810e9	0.241086
$686$	6.26293e9	0.740701
$687$	0	0
$688$	4.29623e8	0.0502954
$689$	1.51465e10	1.76419
$690$	0	0
$691$	6.90630e9	0.796292	0.398146	−	0.917322i	$-0.369654\pi$
$691$	6.90630e9	0.796292	0.398146	+	0.917322i	$0.369654\pi$
$692$	4.10675e8	0.0471115
$693$	0	0
$694$	4.46183e9	0.506704
$695$	−6.57809e9	−0.743281
$696$	0	0
$697$	9.97939e9	1.11632
$698$	−9.21559e9	−1.02572
$699$	0	0
$700$	3.50251e8	0.0385955
$701$	1.17284e10	1.28596	0.642979	−	0.765884i	$-0.277698\pi$
$701$	1.17284e10	1.28596	0.642979	+	0.765884i	$0.277698\pi$
$702$	0	0
$703$	5.18772e9	0.563161
$704$	−1.38284e10	−1.49371
$705$	0	0
$706$	2.59475e9	0.277511
$707$	2.73901e9	0.291492
$708$	0	0
$709$	1.71360e10	1.80571	0.902853	−	0.429949i	$-0.141468\pi$
$709$	1.71360e10	1.80571	0.902853	+	0.429949i	$0.141468\pi$
$710$	1.37691e8	0.0144378
$711$	0	0
$712$	−3.38367e9	−0.351324
$713$	3.11128e10	3.21459
$714$	0	0
$715$	−8.94299e9	−0.914980
$716$	3.59233e8	0.0365747
$717$	0	0
$718$	−2.54627e9	−0.256726
$719$	−3.74748e9	−0.376001	−0.188000	−	0.982169i	$-0.560201\pi$
$719$	−3.74748e9	−0.376001	−0.188000	+	0.982169i	$0.560201\pi$
$720$	0	0
$721$	4.86354e9	0.483258
$722$	−6.66227e9	−0.658782
$723$	0	0
$724$	−4.54999e9	−0.445579
$725$	−3.27366e9	−0.319044
$726$	0	0
$727$	−3.44286e9	−0.332315	−0.166157	−	0.986099i	$-0.553136\pi$
$727$	−3.44286e9	−0.332315	−0.166157	+	0.986099i	$0.553136\pi$
$728$	−8.86921e9	−0.851972
$729$	0	0
$730$	3.12207e9	0.297039
$731$	−7.87745e8	−0.0745890
$732$	0	0
$733$	−6.25369e9	−0.586506	−0.293253	−	0.956035i	$-0.594738\pi$
$733$	−6.25369e9	−0.586506	−0.293253	+	0.956035i	$0.594738\pi$
$734$	−1.69403e9	−0.158119
$735$	0	0
$736$	−1.42514e10	−1.31761
$737$	−4.50630e9	−0.414652
$738$	0	0
$739$	1.47730e10	1.34652	0.673262	−	0.739404i	$-0.264892\pi$
$739$	1.47730e10	1.34652	0.673262	+	0.739404i	$0.264892\pi$
$740$	2.34377e9	0.212620
$741$	0	0
$742$	5.88357e9	0.528722
$743$	−2.30715e9	−0.206355	−0.103177	−	0.994663i	$-0.532901\pi$
$743$	−2.30715e9	−0.206355	−0.103177	+	0.994663i	$0.532901\pi$
$744$	0	0
$745$	6.33798e9	0.561570
$746$	−1.42784e10	−1.25919
$747$	0	0
$748$	4.41645e9	0.385849
$749$	−3.49082e9	−0.303557
$750$	0	0
$751$	−1.42740e10	−1.22972	−0.614860	−	0.788637i	$-0.710787\pi$
$751$	−1.42740e10	−1.22972	−0.614860	+	0.788637i	$0.710787\pi$
$752$	−1.04593e9	−0.0896893
$753$	0	0
$754$	2.17982e10	1.85192
$755$	−1.83352e9	−0.155050
$756$	0	0
$757$	−2.00906e10	−1.68328	−0.841641	−	0.540037i	$-0.818410\pi$
$757$	−2.00906e10	−1.68328	−0.841641	+	0.540037i	$0.818410\pi$
$758$	−1.85297e8	−0.0154535
$759$	0	0
$760$	2.48876e9	0.205653
$761$	7.14277e9	0.587517	0.293758	−	0.955880i	$-0.405094\pi$
$761$	7.14277e9	0.587517	0.293758	+	0.955880i	$0.405094\pi$
$762$	0	0
$763$	8.01995e9	0.653636
$764$	−9.32674e8	−0.0756664
$765$	0	0
$766$	5.75747e9	0.462840
$767$	−1.15123e10	−0.921249
$768$	0	0
$769$	9.99239e9	0.792368	0.396184	−	0.918171i	$-0.370334\pi$
$769$	9.99239e9	0.792368	0.396184	+	0.918171i	$0.370334\pi$
$770$	−3.47386e9	−0.274217
$771$	0	0
$772$	−3.42213e9	−0.267692
$773$	1.35395e10	1.05432	0.527162	−	0.849765i	$-0.323256\pi$
$773$	1.35395e10	1.05432	0.527162	+	0.849765i	$0.323256\pi$
$774$	0	0
$775$	−4.26397e9	−0.329047
$776$	−3.53262e8	−0.0271382
$777$	0	0
$778$	9.56540e9	0.728240
$779$	8.13489e9	0.616553
$780$	0	0
$781$	7.57461e8	0.0568960
$782$	−1.60345e10	−1.19903
$783$	0	0
$784$	4.92482e9	0.364993
$785$	−3.44343e9	−0.254067
$786$	0	0
$787$	−5.54882e9	−0.405778	−0.202889	−	0.979202i	$-0.565033\pi$
$787$	−5.54882e9	−0.405778	−0.202889	+	0.979202i	$0.565033\pi$
$788$	5.90821e9	0.430144
$789$	0	0
$790$	1.08964e9	0.0786300
$791$	6.88446e8	0.0494598
$792$	0	0
$793$	6.21546e8	0.0442606
$794$	5.80995e9	0.411908
$795$	0	0
$796$	2.21965e9	0.155987
$797$	1.76485e10	1.23482	0.617408	−	0.786643i	$-0.288183\pi$
$797$	1.76485e10	1.23482	0.617408	+	0.786643i	$0.288183\pi$
$798$	0	0
$799$	1.91779e9	0.133011
$800$	1.95314e9	0.134871
$801$	0	0
$802$	7.83238e9	0.536147
$803$	1.71751e10	1.17056
$804$	0	0
$805$	−6.99546e9	−0.472640
$806$	2.83924e10	1.90998
$807$	0	0
$808$	8.79300e9	0.586405
$809$	6.10924e9	0.405665	0.202832	−	0.979213i	$-0.434985\pi$
$809$	6.10924e9	0.405665	0.202832	+	0.979213i	$0.434985\pi$
$810$	0	0
$811$	1.72578e10	1.13609	0.568044	−	0.822998i	$-0.307700\pi$
$811$	1.72578e10	1.13609	0.568044	+	0.822998i	$0.307700\pi$
$812$	−4.69649e9	−0.307841
$813$	0	0
$814$	−2.32460e10	−1.51065
$815$	9.03894e9	0.584879
$816$	0	0
$817$	−6.42145e8	−0.0411961
$818$	8.45406e9	0.540044
$819$	0	0
$820$	3.67528e9	0.232778
$821$	−1.22071e10	−0.769860	−0.384930	−	0.922946i	$-0.625774\pi$
$821$	−1.22071e10	−0.769860	−0.384930	+	0.922946i	$0.625774\pi$
$822$	0	0
$823$	9.86793e9	0.617059	0.308529	−	0.951215i	$-0.400163\pi$
$823$	9.86793e9	0.617059	0.308529	+	0.951215i	$0.400163\pi$
$824$	1.56133e10	0.972188
$825$	0	0
$826$	−4.47188e9	−0.276096
$827$	−1.36923e10	−0.841795	−0.420898	−	0.907108i	$-0.638285\pi$
$827$	−1.36923e10	−0.841795	−0.420898	+	0.907108i	$0.638285\pi$
$828$	0	0
$829$	3.15751e10	1.92488	0.962441	−	0.271492i	$-0.0875170\pi$
$829$	3.15751e10	1.92488	0.962441	+	0.271492i	$0.0875170\pi$
$830$	−4.13330e9	−0.250913
$831$	0	0
$832$	−2.54119e10	−1.52970
$833$	−9.03002e9	−0.541292
$834$	0	0
$835$	1.17094e10	0.696039
$836$	3.60015e9	0.213108
$837$	0	0
$838$	−1.74613e10	−1.02500
$839$	2.47567e10	1.44719	0.723594	−	0.690226i	$-0.242489\pi$
$839$	2.47567e10	1.44719	0.723594	+	0.690226i	$0.242489\pi$
$840$	0	0
$841$	2.66463e10	1.54473
$842$	−2.09267e10	−1.20812
$843$	0	0
$844$	4.84696e9	0.277505
$845$	−8.59065e9	−0.489810
$846$	0	0
$847$	−9.54475e9	−0.539725
$848$	1.11665e10	0.628825
$849$	0	0
$850$	2.19750e9	0.122734
$851$	−4.68115e10	−2.60375
$852$	0	0
$853$	2.92760e10	1.61506	0.807532	−	0.589823i	$-0.200803\pi$
$853$	2.92760e10	1.61506	0.807532	+	0.589823i	$0.200803\pi$
$854$	2.41436e8	0.0132648
$855$	0	0
$856$	−1.12065e10	−0.610677
$857$	−1.41527e10	−0.768078	−0.384039	−	0.923317i	$-0.625467\pi$
$857$	−1.41527e10	−0.768078	−0.384039	+	0.923317i	$0.625467\pi$
$858$	0	0
$859$	−1.78767e10	−0.962302	−0.481151	−	0.876638i	$-0.659781\pi$
$859$	−1.78767e10	−0.962302	−0.481151	+	0.876638i	$0.659781\pi$
$860$	−2.90117e8	−0.0155535
$861$	0	0
$862$	−2.49630e10	−1.32746
$863$	−2.49936e10	−1.32370	−0.661852	−	0.749635i	$-0.730229\pi$
$863$	−2.49936e10	−1.32370	−0.661852	+	0.749635i	$0.730229\pi$
$864$	0	0
$865$	1.12411e9	0.0590545
$866$	−3.81985e9	−0.199864
$867$	0	0
$868$	−6.11722e9	−0.317494
$869$	5.99430e9	0.309863
$870$	0	0
$871$	−8.28106e9	−0.424642
$872$	2.57463e10	1.31494
$873$	0	0
$874$	−1.30708e10	−0.662236
$875$	9.58719e8	0.0483797
$876$	0	0
$877$	3.80409e10	1.90437	0.952187	−	0.305516i	$-0.0988290\pi$
$877$	3.80409e10	1.90437	0.952187	+	0.305516i	$0.0988290\pi$
$878$	3.02160e9	0.150663
$879$	0	0
$880$	−6.59305e9	−0.326135
$881$	−1.03097e10	−0.507962	−0.253981	−	0.967209i	$-0.581740\pi$
$881$	−1.03097e10	−0.507962	−0.253981	+	0.967209i	$0.581740\pi$
$882$	0	0
$883$	2.85539e10	1.39574	0.697868	−	0.716226i	$-0.254132\pi$
$883$	2.85539e10	1.39574	0.697868	+	0.716226i	$0.254132\pi$
$884$	8.11595e9	0.395145
$885$	0	0
$886$	1.40911e10	0.680652
$887$	−3.70718e10	−1.78366	−0.891829	−	0.452373i	$-0.850578\pi$
$887$	−3.70718e10	−1.78366	−0.891829	+	0.452373i	$0.850578\pi$
$888$	0	0
$889$	−1.65742e10	−0.791183
$890$	−2.43547e9	−0.115802
$891$	0	0
$892$	−1.44159e10	−0.680087
$893$	1.56332e9	0.0734630
$894$	0	0
$895$	9.83304e8	0.0458466
$896$	−2.01725e9	−0.0936876
$897$	0	0
$898$	−2.52947e10	−1.16563
$899$	5.71752e10	2.62451
$900$	0	0
$901$	−2.04745e10	−0.932560
$902$	−3.64522e10	−1.65387
$903$	0	0
$904$	2.21010e9	0.0995001
$905$	−1.24544e10	−0.558537
$906$	0	0
$907$	7.67630e9	0.341607	0.170803	−	0.985305i	$-0.445364\pi$
$907$	7.67630e9	0.341607	0.170803	+	0.985305i	$0.445364\pi$
$908$	−7.46842e9	−0.331076
$909$	0	0
$910$	−6.38379e9	−0.280824
$911$	2.65141e10	1.16188	0.580941	−	0.813945i	$-0.302685\pi$
$911$	2.65141e10	1.16188	0.580941	+	0.813945i	$0.302685\pi$
$912$	0	0
$913$	−2.27380e10	−0.988790
$914$	6.09985e9	0.264245
$915$	0	0
$916$	−4.84629e9	−0.208341
$917$	−1.39566e10	−0.597704
$918$	0	0
$919$	1.07609e10	0.457345	0.228672	−	0.973503i	$-0.426562\pi$
$919$	1.07609e10	0.457345	0.228672	+	0.973503i	$0.426562\pi$
$920$	−2.24574e10	−0.950828
$921$	0	0
$922$	−2.42292e10	−1.01808
$923$	1.39196e9	0.0582668
$924$	0	0
$925$	6.41545e9	0.266521
$926$	−2.23143e10	−0.923515
$927$	0	0
$928$	−2.61895e10	−1.07574
$929$	−6.65162e9	−0.272190	−0.136095	−	0.990696i	$-0.543455\pi$
$929$	−6.65162e9	−0.272190	−0.136095	+	0.990696i	$0.543455\pi$
$930$	0	0
$931$	−7.36099e9	−0.298960
$932$	6.01251e9	0.243276
$933$	0	0
$934$	−8.12352e9	−0.326235
$935$	1.20888e10	0.483665
$936$	0	0
$937$	−2.58854e10	−1.02794	−0.513968	−	0.857810i	$-0.671825\pi$
$937$	−2.58854e10	−1.02794	−0.513968	+	0.857810i	$0.671825\pi$
$938$	−3.21674e9	−0.127264
$939$	0	0
$940$	7.06298e8	0.0277358
$941$	1.10482e10	0.432245	0.216122	−	0.976366i	$-0.430659\pi$
$941$	1.10482e10	0.432245	0.216122	+	0.976366i	$0.430659\pi$
$942$	0	0
$943$	−7.34053e10	−2.85060
$944$	−8.48720e9	−0.328369
$945$	0	0
$946$	2.87743e9	0.110506
$947$	−1.57580e10	−0.602944	−0.301472	−	0.953475i	$-0.597478\pi$
$947$	−1.57580e10	−0.602944	−0.301472	+	0.953475i	$0.597478\pi$
$948$	0	0
$949$	3.15620e10	1.19876
$950$	1.79134e9	0.0677867
$951$	0	0
$952$	1.19891e10	0.450358
$953$	1.31991e10	0.493991	0.246996	−	0.969017i	$-0.420557\pi$
$953$	1.31991e10	0.493991	0.246996	+	0.969017i	$0.420557\pi$
$954$	0	0
$955$	−2.55295e9	−0.0948483
$956$	4.83204e9	0.178866
$957$	0	0
$958$	3.07603e10	1.13035
$959$	7.96418e9	0.291592
$960$	0	0
$961$	4.69586e10	1.70680
$962$	−4.27184e10	−1.54704
$963$	0	0
$964$	−6.69499e9	−0.240702
$965$	−9.36716e9	−0.335554
$966$	0	0
$967$	2.58458e10	0.919172	0.459586	−	0.888133i	$-0.347998\pi$
$967$	2.58458e10	0.919172	0.459586	+	0.888133i	$0.347998\pi$
$968$	−3.06414e10	−1.08579
$969$	0	0
$970$	−2.54267e8	−0.00894519
$971$	5.22524e10	1.83163	0.915817	−	0.401596i	$-0.131544\pi$
$971$	5.22524e10	1.83163	0.915817	+	0.401596i	$0.131544\pi$
$972$	0	0
$973$	−2.58316e10	−0.898992
$974$	4.87776e9	0.169147
$975$	0	0
$976$	4.58223e8	0.0157762
$977$	−3.14583e10	−1.07921	−0.539603	−	0.841920i	$-0.681426\pi$
$977$	−3.14583e10	−1.07921	−0.539603	+	0.841920i	$0.681426\pi$
$978$	0	0
$979$	−1.33979e10	−0.456350
$980$	−3.32564e9	−0.112871
$981$	0	0
$982$	−1.12415e10	−0.378822
$983$	5.64727e9	0.189627	0.0948137	−	0.995495i	$-0.469774\pi$
$983$	5.64727e9	0.189627	0.0948137	+	0.995495i	$0.469774\pi$
$984$	0	0
$985$	1.61721e10	0.539189
$986$	−2.94661e10	−0.978936
$987$	0	0
$988$	6.61587e9	0.218242
$989$	5.79441e9	0.190468
$990$	0	0
$991$	3.88542e9	0.126818	0.0634089	−	0.997988i	$-0.479803\pi$
$991$	3.88542e9	0.126818	0.0634089	+	0.997988i	$0.479803\pi$
$992$	−3.41121e10	−1.10947
$993$	0	0
$994$	5.40700e8	0.0174624
$995$	6.07570e9	0.195531
$996$	0	0
$997$	−6.08165e9	−0.194352	−0.0971758	−	0.995267i	$-0.530981\pi$
$997$	−6.08165e9	−0.194352	−0.0971758	+	0.995267i	$0.530981\pi$
$998$	1.55074e10	0.493836
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.8.a.i.1.10		15
3.2	odd	2		405.8.a.j.1.6		15
9.2	odd	6		135.8.e.b.91.10		30
9.4	even	3		45.8.e.b.16.6	✓	30
9.5	odd	6		135.8.e.b.46.10		30
9.7	even	3		45.8.e.b.31.6	yes	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.8.e.b.16.6	✓	30	9.4	even	3
45.8.e.b.31.6	yes	30	9.7	even	3
135.8.e.b.46.10		30	9.5	odd	6
135.8.e.b.91.10		30	9.2	odd	6
405.8.a.i.1.10		15	1.1	even	1	trivial
405.8.a.j.1.6		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)

\(f(q)\)	\(=\)	\(q+9.07378 q^{2} -45.6666 q^{4} -125.000 q^{5} -490.864 q^{7} -1575.81 q^{8} +O(q^{10})\)	\(q+9.07378 q^{2} -45.6666 q^{4} -125.000 q^{5} -490.864 q^{7} -1575.81 q^{8} -1134.22 q^{10} -6239.55 q^{11} -11466.2 q^{13} -4453.99 q^{14} -8453.24 q^{16} +15499.6 q^{17} +12634.8 q^{19} +5708.32 q^{20} -56616.3 q^{22} -114011. q^{23} +15625.0 q^{25} -104042. q^{26} +22416.1 q^{28} -209514. q^{29} -272894. q^{31} +125001. q^{32} +140640. q^{34} +61358.0 q^{35} +410589. q^{37} +114646. q^{38} +196976. q^{40} +643846. q^{41} -50823.4 q^{43} +284939. q^{44} -1.03451e6 q^{46} +123731. q^{47} -582596. q^{49} +141778. q^{50} +523622. q^{52} -1.32097e6 q^{53} +779944. q^{55} +773509. q^{56} -1.90109e6 q^{58} +1.00402e6 q^{59} -54206.8 q^{61} -2.47618e6 q^{62} +2.21625e6 q^{64} +1.43327e6 q^{65} +722215. q^{67} -707815. q^{68} +556749. q^{70} -121397. q^{71} -2.75261e6 q^{73} +3.72559e6 q^{74} -576989. q^{76} +3.06277e6 q^{77} -960694. q^{79} +1.05666e6 q^{80} +5.84212e6 q^{82} +3.64417e6 q^{83} -1.93746e6 q^{85} -461161. q^{86} +9.83235e6 q^{88} +2.14726e6 q^{89} +5.62834e6 q^{91} +5.20647e6 q^{92} +1.12271e6 q^{94} -1.57935e6 q^{95} +224178. q^{97} -5.28634e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - 8 q^{2} + 1088 q^{4} - 1875 q^{5} + 1289 q^{7} - 4434 q^{8}+O(q^{10}) \) 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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