LMFDB - Embedded newform 240.8.h.b.191.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [240,8,Mod(191,240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(240, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1, 0]))

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

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

chi := DirichletCharacter("240.191");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 240 = 2^{4} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	240.h (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$74.9724061162$
Analytic rank:	$0$
Dimension:	$36$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			191.1
Character	$\chi$	$=$	240.191
Dual form			240.8.h.b.191.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-45.6703 - 10.0611i) q^{3} -125.000i q^{5} -760.932i q^{7} +(1984.55 + 918.983i) q^{9} -3804.34 q^{11} +10503.7 q^{13} +(-1257.63 + 5708.79i) q^{15} -12061.0i q^{17} +30050.7i q^{19} +(-7655.78 + 34752.0i) q^{21} +50110.0 q^{23} -15625.0 q^{25} +(-81389.0 - 61936.9i) q^{27} +151965. i q^{29} +135147. i q^{31} +(173745. + 38275.6i) q^{33} -95116.5 q^{35} -79956.1 q^{37} +(-479707. - 105678. i) q^{39} +201072. i q^{41} -198815. i q^{43} +(114873. - 248069. i) q^{45} -828817. q^{47} +244525. q^{49} +(-121346. + 550829. i) q^{51} +908729. i q^{53} +475542. i q^{55} +(302342. - 1.37242e6i) q^{57} +566260. q^{59} +2.64404e6 q^{61} +(699284. - 1.51011e6i) q^{63} -1.31296e6i q^{65} +589032. i q^{67} +(-2.28854e6 - 504160. i) q^{69} -2.36264e6 q^{71} +3.01222e6 q^{73} +(713598. + 157204. i) q^{75} +2.89484e6i q^{77} -2.54546e6i q^{79} +(3.09391e6 + 3.64754e6i) q^{81} +7.29920e6 q^{83} -1.50762e6 q^{85} +(1.52893e6 - 6.94031e6i) q^{87} -9.16320e6i q^{89} -7.99261e6i q^{91} +(1.35972e6 - 6.17221e6i) q^{93} +3.75634e6 q^{95} +1.23848e7 q^{97} +(-7.54989e6 - 3.49612e6i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q + 3324 q^{9} - 432 q^{13} - 22032 q^{21} - 562500 q^{25} - 386232 q^{33} + 1200048 q^{37} - 1710000 q^{45} - 3237972 q^{49} - 9331200 q^{57} + 11262360 q^{61} - 11352552 q^{69} + 37210296 q^{73} - 12564252 q^{81}+ \cdots + 94537320 q^{97}+O(q^{100}) $ 36 * q + 3324 * q^9 - 432 * q^13 - 22032 * q^21 - 562500 * q^25 - 386232 * q^33 + 1200048 * q^37 - 1710000 * q^45 - 3237972 * q^49 - 9331200 * q^57 + 11262360 * q^61 - 11352552 * q^69 + 37210296 * q^73 - 12564252 * q^81 + 5271000 * q^85 + 5594400 * q^93 + 94537320 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/240\mathbb{Z}\right)^\times$.

$n$	$31$	$97$	$161$	$181$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	−45.6703	−	10.0611i	−0.976583	−	0.215139i
$3$	−45.6703	−	10.0611i	−0.976583	−	0.215139i
$4$		0			0
$5$		−	125.000i		−	0.447214i
$5$		−	125.000i		−	0.447214i
$6$		0			0
$7$		−	760.932i		−	0.838499i	−0.907871	−	0.419250i	$-0.862293\pi$
$7$		−	760.932i		−	0.838499i	0.907871	−	0.419250i	$-0.137707\pi$
$8$		0			0
$9$	1984.55	+	918.983i	0.907430	+	0.420203i
$10$		0			0
$11$	−3804.34			−0.861796			−0.430898	−	0.902401i	$-0.641803\pi$
$11$	−3804.34			−0.861796			−0.430898	+	0.902401i	$0.641803\pi$
$12$		0			0
$13$	10503.7			1.32599			0.662996	−	0.748623i	$-0.269285\pi$
$13$	10503.7			1.32599			0.662996	+	0.748623i	$0.269285\pi$
$14$		0			0
$15$	−1257.63	+	5708.79i	−0.0962131	+	0.436741i
$16$		0			0
$17$		−	12061.0i		−	0.595404i	−0.954659	−	0.297702i	$-0.903780\pi$
$17$		−	12061.0i		−	0.595404i	0.954659	−	0.297702i	$-0.0962202\pi$
$18$		0			0
$19$			30050.7i			1.00512i	0.864543	+	0.502559i	$0.167608\pi$
$19$			30050.7i			1.00512i	−0.864543	+	0.502559i	$0.832392\pi$
$20$		0			0
$21$	−7655.78	+	34752.0i	−0.180394	+	0.818865i
$22$		0			0
$23$	50110.0			0.858770			0.429385	−	0.903122i	$-0.358730\pi$
$23$	50110.0			0.858770			0.429385	+	0.903122i	$0.358730\pi$
$24$		0			0
$25$	−15625.0			−0.200000
$26$		0			0
$27$	−81389.0	−	61936.9i	−0.795779	−	0.605587i
$28$		0			0
$29$			151965.i			1.15705i	0.815665	+	0.578525i	$0.196371\pi$
$29$			151965.i			1.15705i	−0.815665	+	0.578525i	$0.803629\pi$
$30$		0			0
$31$			135147.i			0.814781i	0.913254	+	0.407391i	$0.133561\pi$
$31$			135147.i			0.814781i	−0.913254	+	0.407391i	$0.866439\pi$
$32$		0			0
$33$	173745.	+	38275.6i	0.841616	+	0.185406i
$34$		0			0
$35$	−95116.5			−0.374988
$36$		0			0
$37$	−79956.1			−0.259505			−0.129752	−	0.991546i	$-0.541418\pi$
$37$	−79956.1			−0.259505			−0.129752	+	0.991546i	$0.541418\pi$
$38$		0			0
$39$	−479707.	−	105678.i	−1.29494	−	0.285273i
$40$		0			0
$41$			201072.i			0.455626i	0.973705	+	0.227813i	$0.0731575\pi$
$41$			201072.i			0.455626i	−0.973705	+	0.227813i	$0.926842\pi$
$42$		0			0
$43$		−	198815.i		−	0.381338i	−0.981654	−	0.190669i	$-0.938934\pi$
$43$		−	198815.i		−	0.381338i	0.981654	−	0.190669i	$-0.0610657\pi$
$44$		0			0
$45$	114873.	−	248069.i	0.187920	−	0.405815i
$46$		0			0
$47$	−828817.			−1.16444			−0.582218	−	0.813032i	$-0.697815\pi$
$47$	−828817.			−1.16444			−0.582218	+	0.813032i	$0.697815\pi$
$48$		0			0
$49$	244525.			0.296919
$50$		0			0
$51$	−121346.	+	550829.i	−0.128095	+	0.581462i
$52$		0			0
$53$			908729.i			0.838434i	0.907886	+	0.419217i	$0.137695\pi$
$53$			908729.i			0.838434i	−0.907886	+	0.419217i	$0.862305\pi$
$54$		0			0
$55$			475542.i			0.385407i
$56$		0			0
$57$	302342.	−	1.37242e6i	0.216240	−	0.981582i
$58$		0			0
$59$	566260.			0.358950			0.179475	−	0.983763i	$-0.442560\pi$
$59$	566260.			0.358950			0.179475	+	0.983763i	$0.442560\pi$
$60$		0			0
$61$	2.64404e6			1.49147			0.745734	−	0.666243i	$-0.232099\pi$
$61$	2.64404e6			1.49147			0.745734	+	0.666243i	$0.232099\pi$
$62$		0			0
$63$	699284.	−	1.51011e6i	0.352340	−	0.760880i
$64$		0			0
$65$		−	1.31296e6i		−	0.593002i
$66$		0			0
$67$			589032.i			0.239264i	0.992818	+	0.119632i	$0.0381715\pi$
$67$			589032.i			0.239264i	−0.992818	+	0.119632i	$0.961829\pi$
$68$		0			0
$69$	−2.28854e6	−	504160.i	−0.838661	−	0.184755i
$70$		0			0
$71$	−2.36264e6			−0.783416			−0.391708	−	0.920090i	$-0.628116\pi$
$71$	−2.36264e6			−0.783416			−0.391708	+	0.920090i	$0.628116\pi$
$72$		0			0
$73$	3.01222e6			0.906269			0.453134	−	0.891442i	$-0.350306\pi$
$73$	3.01222e6			0.906269			0.453134	+	0.891442i	$0.350306\pi$
$74$		0			0
$75$	713598.	+	157204.i	0.195317	+	0.0430278i
$76$		0			0
$77$			2.89484e6i			0.722615i
$78$		0			0
$79$		−	2.54546e6i		−	0.580859i	−0.956896	−	0.290430i	$-0.906202\pi$
$79$		−	2.54546e6i		−	0.580859i	0.956896	−	0.290430i	$-0.0937982\pi$
$80$		0			0
$81$	3.09391e6	+	3.64754e6i	0.646860	+	0.762609i
$82$		0			0
$83$	7.29920e6			1.40121			0.700603	−	0.713552i	$-0.252915\pi$
$83$	7.29920e6			1.40121			0.700603	+	0.713552i	$0.252915\pi$
$84$		0			0
$85$	−1.50762e6			−0.266273
$86$		0			0
$87$	1.52893e6	−	6.94031e6i	0.248927	−	1.12996i
$88$		0			0
$89$		−	9.16320e6i		−	1.37779i	−0.724862	−	0.688894i	$-0.758097\pi$
$89$		−	9.16320e6i		−	1.37779i	0.724862	−	0.688894i	$-0.241903\pi$
$90$		0			0
$91$		−	7.99261e6i		−	1.11184i
$92$		0			0
$93$	1.35972e6	−	6.17221e6i	0.175291	−	0.795702i
$94$		0			0
$95$	3.75634e6			0.449503
$96$		0			0
$97$	1.23848e7			1.37781			0.688903	−	0.724853i	$-0.258092\pi$
$97$	1.23848e7			1.37781			0.688903	+	0.724853i	$0.258092\pi$
$98$		0			0
$99$	−7.54989e6	−	3.49612e6i	−0.782020	−	0.362129i
$100$		0			0
$101$		−	6.04572e6i		−	0.583879i	−0.956437	−	0.291940i	$-0.905699\pi$
$101$		−	6.04572e6i		−	0.583879i	0.956437	−	0.291940i	$-0.0943006\pi$
$102$		0			0
$103$		−	1.15425e7i		−	1.04080i	−0.853921	−	0.520402i	$-0.825782\pi$
$103$		−	1.15425e7i		−	1.04080i	0.853921	−	0.520402i	$-0.174218\pi$
$104$		0			0
$105$	4.34400e6	+	956973.i	0.366207	+	0.0806746i
$106$		0			0
$107$	6.45345e6			0.509271			0.254635	−	0.967037i	$-0.418045\pi$
$107$	6.45345e6			0.509271			0.254635	+	0.967037i	$0.418045\pi$
$108$		0			0
$109$	2.06544e7			1.52764			0.763818	−	0.645431i	$-0.223322\pi$
$109$	2.06544e7			1.52764			0.763818	+	0.645431i	$0.223322\pi$
$110$		0			0
$111$	3.65162e6	+	804443.i	0.253428	+	0.0558296i
$112$		0			0
$113$		−	8.45467e6i		−	0.551217i	−0.961270	−	0.275608i	$-0.911121\pi$
$113$		−	8.45467e6i		−	0.551217i	0.961270	−	0.275608i	$-0.0888793\pi$
$114$		0			0
$115$		−	6.26375e6i		−	0.384054i
$116$		0			0
$117$	2.08451e7	+	9.65273e6i	1.20325	+	0.557185i
$118$		0			0
$119$	−9.17760e6			−0.499246
$120$		0			0
$121$	−5.01420e6			−0.257308
$122$		0			0
$123$	2.02300e6	−	9.18303e6i	0.0980230	−	0.444957i
$124$		0			0
$125$			1.95312e6i			0.0894427i
$126$		0			0
$127$			4.25522e7i			1.84335i	0.387958	+	0.921677i	$0.373181\pi$
$127$			4.25522e7i			1.84335i	−0.387958	+	0.921677i	$0.626819\pi$
$128$		0			0
$129$	−2.00029e6	+	9.07995e6i	−0.0820407	+	0.372409i
$130$		0			0
$131$	−3.33225e7			−1.29505			−0.647527	−	0.762043i	$-0.724197\pi$
$131$	−3.33225e7			−1.29505			−0.647527	+	0.762043i	$0.724197\pi$
$132$		0			0
$133$	2.28665e7			0.842791
$134$		0			0
$135$	−7.74211e6	+	1.01736e7i	−0.270827	+	0.355883i
$136$		0			0
$137$		−	3.60823e7i		−	1.19887i	−0.800424	−	0.599435i	$-0.795392\pi$
$137$		−	3.60823e7i		−	1.19887i	0.800424	−	0.599435i	$-0.204608\pi$
$138$		0			0
$139$		−	4.37391e7i		−	1.38140i	−0.723143	−	0.690698i	$-0.757303\pi$
$139$		−	4.37391e7i		−	1.38140i	0.723143	−	0.690698i	$-0.242697\pi$
$140$		0			0
$141$	3.78523e7	+	8.33877e6i	1.13717	+	0.250516i
$142$		0			0
$143$	−3.99596e7			−1.14273
$144$		0			0
$145$	1.89957e7			0.517448
$146$		0			0
$147$	−1.11675e7	−	2.46019e6i	−0.289966	−	0.0638789i
$148$		0			0
$149$			5.07431e7i			1.25668i	0.777939	+	0.628340i	$0.216265\pi$
$149$			5.07431e7i			1.25668i	−0.777939	+	0.628340i	$0.783735\pi$
$150$		0			0
$151$		−	7.03294e7i		−	1.66233i	−0.556026	−	0.831165i	$-0.687674\pi$
$151$		−	7.03294e7i		−	1.66233i	0.556026	−	0.831165i	$-0.312326\pi$
$152$		0			0
$153$	1.10838e7	−	2.39357e7i	0.250190	−	0.540288i
$154$		0			0
$155$	1.68934e7			0.364381
$156$		0			0
$157$	6.90904e7			1.42485			0.712424	−	0.701749i	$-0.247597\pi$
$157$	6.90904e7			1.42485			0.712424	+	0.701749i	$0.247597\pi$
$158$		0			0
$159$	9.14278e6	−	4.15019e7i	0.180380	−	0.818801i
$160$		0			0
$161$		−	3.81303e7i		−	0.720078i
$162$		0			0
$163$		−	6.12629e6i		−	0.110800i	−0.998464	−	0.0554001i	$-0.982357\pi$
$163$		−	6.12629e6i		−	0.110800i	0.998464	−	0.0554001i	$-0.0176434\pi$
$164$		0			0
$165$	4.78446e6	−	2.17181e7i	0.0829161	−	0.376382i
$166$		0			0
$167$	1.81268e7			0.301172			0.150586	−	0.988597i	$-0.451884\pi$
$167$	1.81268e7			0.301172			0.150586	+	0.988597i	$0.451884\pi$
$168$		0			0
$169$	4.75794e7			0.758255
$170$		0			0
$171$	−2.76161e7	+	5.96371e7i	−0.422353	+	0.912075i
$172$		0			0
$173$			9.22004e7i			1.35385i	0.736050	+	0.676927i	$0.236689\pi$
$173$			9.22004e7i			1.35385i	−0.736050	+	0.676927i	$0.763311\pi$
$174$		0			0
$175$			1.18896e7i			0.167700i
$176$		0			0
$177$	−2.58613e7	−	5.69718e6i	−0.350545	−	0.0772242i
$178$		0			0
$179$	3.36489e7			0.438516			0.219258	−	0.975667i	$-0.429636\pi$
$179$	3.36489e7			0.438516			0.219258	+	0.975667i	$0.429636\pi$
$180$		0			0
$181$	−8.48527e6			−0.106363			−0.0531815	−	0.998585i	$-0.516936\pi$
$181$	−8.48527e6			−0.106363			−0.0531815	+	0.998585i	$0.516936\pi$
$182$		0			0
$183$	−1.20754e8	−	2.66019e7i	−1.45654	−	0.320873i
$184$		0			0
$185$			9.99451e6i			0.116054i
$186$		0			0
$187$			4.58841e7i			0.513117i
$188$		0			0
$189$	−4.71298e7	+	6.19315e7i	−0.507784	+	0.667261i
$190$		0			0
$191$	1.64623e8			1.70952			0.854759	−	0.519026i	$-0.173705\pi$
$191$	1.64623e8			1.70952			0.854759	+	0.519026i	$0.173705\pi$
$192$		0			0
$193$	−1.48271e8			−1.48459			−0.742294	−	0.670074i	$-0.766262\pi$
$193$	−1.48271e8			−1.48459			−0.742294	+	0.670074i	$0.766262\pi$
$194$		0			0
$195$	−1.32098e7	+	5.99634e7i	−0.127578	+	0.579116i
$196$		0			0
$197$		−	1.04866e8i		−	0.977241i	−0.872496	−	0.488621i	$-0.837500\pi$
$197$		−	1.04866e8i		−	0.977241i	0.872496	−	0.488621i	$-0.162500\pi$
$198$		0			0
$199$		−	1.40662e8i		−	1.26529i	−0.774442	−	0.632645i	$-0.781969\pi$
$199$		−	1.40662e8i		−	1.26529i	0.774442	−	0.632645i	$-0.218031\pi$
$200$		0			0
$201$	5.92629e6	−	2.69013e7i	0.0514750	−	0.233661i
$202$		0			0
$203$	1.15635e8			0.970185
$204$		0			0
$205$	2.51340e7			0.203762
$206$		0			0
$207$	9.94458e7	+	4.60502e7i	0.779274	+	0.360857i
$208$		0			0
$209$		−	1.14323e8i		−	0.866207i
$210$		0			0
$211$			9.98157e7i			0.731493i	0.930714	+	0.365747i	$0.119186\pi$
$211$			9.98157e7i			0.731493i	−0.930714	+	0.365747i	$0.880814\pi$
$212$		0			0
$213$	1.07902e8	+	2.37706e7i	0.765071	+	0.168543i
$214$		0			0
$215$	−2.48519e7			−0.170540
$216$		0			0
$217$	1.02838e8			0.683193
$218$		0			0
$219$	−1.37569e8	−	3.03061e7i	−0.885047	−	0.194974i
$220$		0			0
$221$		−	1.26685e8i		−	0.789501i
$222$		0			0
$223$		−	1.79936e7i		−	0.108655i	−0.998523	−	0.0543277i	$-0.982698\pi$
$223$		−	1.79936e7i		−	0.108655i	0.998523	−	0.0543277i	$-0.0173016\pi$
$224$		0			0
$225$	−3.10086e7	−	1.43591e7i	−0.181486	−	0.0840405i
$226$		0			0
$227$	2.98422e8			1.69332			0.846662	−	0.532131i	$-0.178609\pi$
$227$	2.98422e8			1.69332			0.846662	+	0.532131i	$0.178609\pi$
$228$		0			0
$229$	−3.46069e8			−1.90431			−0.952156	−	0.305613i	$-0.901139\pi$
$229$	−3.46069e8			−1.90431			−0.952156	+	0.305613i	$0.901139\pi$
$230$		0			0
$231$	2.91252e7	−	1.32208e8i	0.155463	−	0.705694i
$232$		0			0
$233$		−	4.55950e7i		−	0.236141i	−0.993005	−	0.118071i	$-0.962329\pi$
$233$		−	4.55950e7i		−	0.236141i	0.993005	−	0.118071i	$-0.0376709\pi$
$234$		0			0
$235$			1.03602e8i			0.520752i
$236$		0			0
$237$	−2.56100e7	+	1.16252e8i	−0.124966	+	0.567257i
$238$		0			0
$239$	−1.71629e8			−0.813203			−0.406602	−	0.913606i	$-0.633286\pi$
$239$	−1.71629e8			−0.813203			−0.406602	+	0.913606i	$0.633286\pi$
$240$		0			0
$241$	−2.30295e8			−1.05980			−0.529900	−	0.848060i	$-0.677771\pi$
$241$	−2.30295e8			−1.05980			−0.529900	+	0.848060i	$0.677771\pi$
$242$		0			0
$243$	−1.04602e8	−	1.97712e8i	−0.467645	−	0.883916i
$244$		0			0
$245$		−	3.05657e7i		−	0.132786i
$246$		0			0
$247$			3.15644e8i			1.33278i
$248$		0			0
$249$	−3.33356e8	−	7.34376e7i	−1.36839	−	0.301454i
$250$		0			0
$251$	−1.92142e8			−0.766947			−0.383473	−	0.923552i	$-0.625272\pi$
$251$	−1.92142e8			−0.766947			−0.383473	+	0.923552i	$0.625272\pi$
$252$		0			0
$253$	−1.90635e8			−0.740084
$254$		0			0
$255$	6.88536e7	+	1.51683e7i	0.260038	+	0.0572857i
$256$		0			0
$257$			4.15493e8i			1.52686i	0.645893	+	0.763428i	$0.276485\pi$
$257$			4.15493e8i			1.52686i	−0.645893	+	0.763428i	$0.723515\pi$
$258$		0			0
$259$			6.08411e7i			0.217595i
$260$		0			0
$261$	−1.39654e8	+	3.01583e8i	−0.486195	+	1.04994i
$262$		0			0
$263$	1.51351e8			0.513025			0.256513	−	0.966541i	$-0.417426\pi$
$263$	1.51351e8			0.513025			0.256513	+	0.966541i	$0.417426\pi$
$264$		0			0
$265$	1.13591e8			0.374959
$266$		0			0
$267$	−9.21915e7	+	4.18486e8i	−0.296416	+	1.34552i
$268$		0			0
$269$			8.24085e7i			0.258130i	0.991636	+	0.129065i	$0.0411976\pi$
$269$			8.24085e7i			0.258130i	−0.991636	+	0.129065i	$0.958802\pi$
$270$		0			0
$271$			1.59569e8i			0.487032i	0.969897	+	0.243516i	$0.0783008\pi$
$271$			1.59569e8i			0.487032i	−0.969897	+	0.243516i	$0.921699\pi$
$272$		0			0
$273$	−8.04141e7	+	3.65025e8i	−0.239201	+	1.08581i
$274$		0			0
$275$	5.94427e7			0.172359
$276$		0			0
$277$	−2.29446e8			−0.648638			−0.324319	−	0.945948i	$-0.605135\pi$
$277$	−2.29446e8			−0.648638			−0.324319	+	0.945948i	$0.605135\pi$
$278$		0			0
$279$	−1.24198e8	+	2.68206e8i	−0.342373	+	0.739357i
$280$		0			0
$281$		−	2.03011e8i		−	0.545817i	−0.962040	−	0.272909i	$-0.912014\pi$
$281$		−	2.03011e8i		−	0.545817i	0.962040	−	0.272909i	$-0.0879856\pi$
$282$		0			0
$283$			3.76831e8i			0.988313i	0.869373	+	0.494156i	$0.164523\pi$
$283$			3.76831e8i			0.988313i	−0.869373	+	0.494156i	$0.835477\pi$
$284$		0			0
$285$	−1.71553e8	−	3.77928e7i	−0.438977	−	0.0967056i
$286$		0			0
$287$	1.53002e8			0.382042
$288$		0			0
$289$	2.64871e8			0.645494
$290$		0			0
$291$	−5.65618e8	−	1.24604e8i	−1.34554	−	0.296420i
$292$		0			0
$293$		−	7.41770e7i		−	0.172279i	−0.996283	−	0.0861396i	$-0.972547\pi$
$293$		−	7.41770e7i		−	0.172279i	0.996283	−	0.0861396i	$-0.0274531\pi$
$294$		0			0
$295$		−	7.07825e7i		−	0.160527i
$296$		0			0
$297$	3.09631e8	+	2.35629e8i	0.685799	+	0.521892i
$298$		0			0
$299$	5.26341e8			1.13872
$300$		0			0
$301$	−1.51285e8			−0.319752
$302$		0			0
$303$	−6.08263e7	+	2.76110e8i	−0.125615	+	0.570207i
$304$		0			0
$305$		−	3.30505e8i		−	0.667005i
$306$		0			0
$307$			2.00392e8i			0.395272i	0.980276	+	0.197636i	$0.0633263\pi$
$307$			2.00392e8i			0.395272i	−0.980276	+	0.197636i	$0.936674\pi$
$308$		0			0
$309$	−1.16130e8	+	5.27149e8i	−0.223918	+	1.01643i
$310$		0			0
$311$	−3.17968e7			−0.0599407			−0.0299703	−	0.999551i	$-0.509541\pi$
$311$	−3.17968e7			−0.0599407			−0.0299703	+	0.999551i	$0.509541\pi$
$312$		0			0
$313$	−3.85295e8			−0.710213			−0.355106	−	0.934826i	$-0.615555\pi$
$313$	−3.85295e8			−0.710213			−0.355106	+	0.934826i	$0.615555\pi$
$314$		0			0
$315$	−1.88763e8	−	8.74104e7i	−0.340276	−	0.157571i
$316$		0			0
$317$		−	6.29377e8i		−	1.10969i	−0.831952	−	0.554847i	$-0.812777\pi$
$317$		−	6.29377e8i		−	1.10969i	0.831952	−	0.554847i	$-0.187223\pi$
$318$		0			0
$319$		−	5.78128e8i		−	0.997140i
$320$		0			0
$321$	−2.94731e8	−	6.49286e7i	−0.497345	−	0.109564i
$322$		0			0
$323$	3.62442e8			0.598452
$324$		0			0
$325$	−1.64120e8			−0.265198
$326$		0			0
$327$	−9.43293e8	−	2.07805e8i	−1.49186	−	0.328654i
$328$		0			0
$329$			6.30673e8i			0.976379i
$330$		0			0
$331$			9.39863e8i			1.42451i	0.701919	+	0.712257i	$0.252327\pi$
$331$			9.39863e8i			1.42451i	−0.701919	+	0.712257i	$0.747673\pi$
$332$		0			0
$333$	−1.58677e8	−	7.34783e7i	−0.235483	−	0.109045i
$334$		0			0
$335$	7.36291e7			0.107002
$336$		0			0
$337$	−5.21667e8			−0.742487			−0.371243	−	0.928536i	$-0.621068\pi$
$337$	−5.21667e8			−0.742487			−0.371243	+	0.928536i	$0.621068\pi$
$338$		0			0
$339$	−8.50630e7	+	3.86127e8i	−0.118588	+	0.538309i
$340$		0			0
$341$		−	5.14145e8i		−	0.702175i
$342$		0			0
$343$		−	8.12727e8i		−	1.08747i
$344$		0			0
$345$	−6.30199e7	+	2.86067e8i	−0.0826249	+	0.375060i
$346$		0			0
$347$	1.09409e9			1.40573			0.702863	−	0.711325i	$-0.251905\pi$
$347$	1.09409e9			1.40573			0.702863	+	0.711325i	$0.251905\pi$
$348$		0			0
$349$	7.37051e8			0.928130			0.464065	−	0.885801i	$-0.346390\pi$
$349$	7.37051e8			0.928130			0.464065	+	0.885801i	$0.346390\pi$
$350$		0			0
$351$	−8.54887e8	−	6.50567e8i	−1.05520	−	0.803003i
$352$		0			0
$353$		−	1.02817e9i		−	1.24410i	−0.782979	−	0.622048i	$-0.786301\pi$
$353$		−	1.02817e9i		−	1.24410i	0.782979	−	0.622048i	$-0.213699\pi$
$354$		0			0
$355$			2.95330e8i			0.350354i
$356$		0			0
$357$	4.19144e8	+	9.23364e7i	0.487555	+	0.107407i
$358$		0			0
$359$	1.03362e9			1.17904			0.589521	−	0.807753i	$-0.299317\pi$
$359$	1.03362e9			1.17904			0.589521	+	0.807753i	$0.299317\pi$
$360$		0			0
$361$	−9.17367e6			−0.0102628
$362$		0			0
$363$	2.29000e8	+	5.04482e7i	0.251283	+	0.0553570i
$364$		0			0
$365$		−	3.76528e8i		−	0.405296i
$366$		0			0
$367$		−	1.56202e9i		−	1.64951i	−0.565487	−	0.824757i	$-0.691312\pi$
$367$		−	1.56202e9i		−	1.64951i	0.565487	−	0.824757i	$-0.308688\pi$
$368$		0			0
$369$	−1.84782e8	+	3.99038e8i	−0.191455	+	0.413449i
$370$		0			0
$371$	6.91481e8			0.703026
$372$		0			0
$373$	−2.43288e8			−0.242739			−0.121370	−	0.992607i	$-0.538729\pi$
$373$	−2.43288e8			−0.242739			−0.121370	+	0.992607i	$0.538729\pi$
$374$		0			0
$375$	1.96505e7	−	8.91998e7i	0.0192426	−	0.0873483i
$376$		0			0
$377$			1.59620e9i			1.53424i
$378$		0			0
$379$		−	1.70107e8i		−	0.160503i	−0.996775	−	0.0802517i	$-0.974428\pi$
$379$		−	1.70107e8i		−	0.160503i	0.996775	−	0.0802517i	$-0.0255724\pi$
$380$		0			0
$381$	4.28120e8	−	1.94337e9i	0.396578	−	1.80019i
$382$		0			0
$383$	9.90159e8			0.900553			0.450276	−	0.892889i	$-0.351325\pi$
$383$	9.90159e8			0.900553			0.450276	+	0.892889i	$0.351325\pi$
$384$		0			0
$385$	3.61855e8			0.323163
$386$		0			0
$387$	1.82708e8	−	3.94559e8i	0.160239	−	0.346038i
$388$		0			0
$389$		−	5.23780e8i		−	0.451155i	−0.974225	−	0.225577i	$-0.927573\pi$
$389$		−	5.23780e8i		−	0.451155i	0.974225	−	0.225577i	$-0.0724269\pi$
$390$		0			0
$391$		−	6.04376e8i		−	0.511315i
$392$		0			0
$393$	1.52185e9	+	3.35259e8i	1.26473	+	0.278617i
$394$		0			0
$395$	−3.18182e8			−0.259768
$396$		0			0
$397$	−9.50812e7			−0.0762654			−0.0381327	−	0.999273i	$-0.512141\pi$
$397$	−9.50812e7			−0.0762654			−0.0381327	+	0.999273i	$0.512141\pi$
$398$		0			0
$399$	−1.04432e9	−	2.30062e8i	−0.823056	−	0.181317i
$400$		0			0
$401$		−	1.36022e9i		−	1.05342i	−0.850044	−	0.526712i	$-0.823425\pi$
$401$		−	1.36022e9i		−	1.05342i	0.850044	−	0.526712i	$-0.176575\pi$
$402$		0			0
$403$			1.41955e9i			1.08039i
$404$		0			0
$405$	4.55942e8	−	3.86739e8i	0.341049	−	0.289284i
$406$		0			0
$407$	3.04180e8			0.223640
$408$		0			0
$409$	1.30616e9			0.943987			0.471994	−	0.881602i	$-0.343535\pi$
$409$	1.30616e9			0.943987			0.471994	+	0.881602i	$0.343535\pi$
$410$		0			0
$411$	−3.63026e8	+	1.64789e9i	−0.257924	+	1.17080i
$412$		0			0
$413$		−	4.30886e8i		−	0.300979i
$414$		0			0
$415$		−	9.12400e8i		−	0.626638i
$416$		0			0
$417$	−4.40062e8	+	1.99758e9i	−0.297192	+	1.34905i
$418$		0			0
$419$	1.67879e9			1.11493			0.557464	−	0.830201i	$-0.311775\pi$
$419$	1.67879e9			1.11493			0.557464	+	0.830201i	$0.311775\pi$
$420$		0			0
$421$	1.27287e9			0.831373			0.415686	−	0.909508i	$-0.363541\pi$
$421$	1.27287e9			0.831373			0.415686	+	0.909508i	$0.363541\pi$
$422$		0			0
$423$	−1.64483e9	−	7.61668e8i	−1.05665	−	0.489299i
$424$		0			0
$425$			1.88453e8i			0.119081i
$426$		0			0
$427$		−	2.01194e9i		−	1.25060i
$428$		0			0
$429$	1.82497e9	+	4.02036e8i	1.11598	+	0.245847i
$430$		0			0
$431$	−3.52146e8			−0.211862			−0.105931	−	0.994374i	$-0.533782\pi$
$431$	−3.52146e8			−0.211862			−0.105931	+	0.994374i	$0.533782\pi$
$432$		0			0
$433$	2.89022e9			1.71090			0.855448	−	0.517889i	$-0.173282\pi$
$433$	2.89022e9			1.71090			0.855448	+	0.517889i	$0.173282\pi$
$434$		0			0
$435$	−8.67538e8	−	1.91117e8i	−0.505331	−	0.111323i
$436$		0			0
$437$			1.50584e9i			0.863165i
$438$		0			0
$439$			1.32877e9i			0.749588i	0.927108	+	0.374794i	$0.122287\pi$
$439$			1.32877e9i			0.749588i	−0.927108	+	0.374794i	$0.877713\pi$
$440$		0			0
$441$	4.85273e8	+	2.24715e8i	0.269433	+	0.124766i
$442$		0			0
$443$	−2.31758e9			−1.26655			−0.633274	−	0.773928i	$-0.718289\pi$
$443$	−2.31758e9			−1.26655			−0.633274	+	0.773928i	$0.718289\pi$
$444$		0			0
$445$	−1.14540e9			−0.616165
$446$		0			0
$447$	5.10529e8	−	2.31745e9i	0.270361	−	1.22725i
$448$		0			0
$449$		−	2.97362e9i		−	1.55033i	−0.631761	−	0.775164i	$-0.717667\pi$
$449$		−	2.97362e9i		−	1.55033i	0.631761	−	0.775164i	$-0.282333\pi$
$450$		0			0
$451$		−	7.64946e8i		−	0.392657i
$452$		0			0
$453$	−7.07588e8	+	3.21196e9i	−0.357632	+	1.62340i
$454$		0			0
$455$	−9.99076e8			−0.497232
$456$		0			0
$457$	2.71785e9			1.33205			0.666023	−	0.745931i	$-0.267995\pi$
$457$	2.71785e9			1.33205			0.666023	+	0.745931i	$0.267995\pi$
$458$		0			0
$459$	−7.47021e8	+	9.81633e8i	−0.360569	+	0.473810i
$460$		0			0
$461$		−	3.77424e9i		−	1.79422i	−0.441805	−	0.897111i	$-0.645662\pi$
$461$		−	3.77424e9i		−	1.79422i	0.441805	−	0.897111i	$-0.354338\pi$
$462$		0			0
$463$		−	1.58236e9i		−	0.740922i	−0.928848	−	0.370461i	$-0.879200\pi$
$463$		−	1.58236e9i		−	0.740922i	0.928848	−	0.370461i	$-0.120800\pi$
$464$		0			0
$465$	−7.71526e8	−	1.69965e8i	−0.355849	−	0.0783926i
$466$		0			0
$467$	9.70261e8			0.440838			0.220419	−	0.975405i	$-0.429257\pi$
$467$	9.70261e8			0.440838			0.220419	+	0.975405i	$0.429257\pi$
$468$		0			0
$469$	4.48214e8			0.200623
$470$		0			0
$471$	−3.15538e9	−	6.95122e8i	−1.39148	−	0.306541i
$472$		0			0
$473$			7.56360e8i			0.328636i
$474$		0			0
$475$		−	4.69542e8i		−	0.201024i
$476$		0			0
$477$	−8.35107e8	+	1.80342e9i	−0.352312	+	0.760820i
$478$		0			0
$479$	1.46718e9			0.609971			0.304985	−	0.952357i	$-0.401348\pi$
$479$	1.46718e9			0.609971			0.304985	+	0.952357i	$0.401348\pi$
$480$		0			0
$481$	−8.39835e8			−0.344101
$482$		0			0
$483$	−3.83631e8	+	1.74142e9i	−0.154917	+	0.703216i
$484$		0			0
$485$		−	1.54810e9i		−	0.616174i
$486$		0			0
$487$		−	1.12821e9i		−	0.442628i	−0.975203	−	0.221314i	$-0.928966\pi$
$487$		−	1.12821e9i		−	0.442628i	0.975203	−	0.221314i	$-0.0710345\pi$
$488$		0			0
$489$	−6.16369e7	+	2.79789e8i	−0.0238375	+	0.108206i
$490$		0			0
$491$	−2.30073e8			−0.0877164			−0.0438582	−	0.999038i	$-0.513965\pi$
$491$	−2.30073e8			−0.0877164			−0.0438582	+	0.999038i	$0.513965\pi$
$492$		0			0
$493$	1.83285e9			0.688912
$494$		0			0
$495$	−4.37015e8	+	9.43737e8i	−0.161949	+	0.349730i
$496$		0			0
$497$			1.79781e9i			0.656894i
$498$		0			0
$499$			5.08620e9i			1.83249i	0.400619	+	0.916245i	$0.368795\pi$
$499$			5.08620e9i			1.83249i	−0.400619	+	0.916245i	$0.631205\pi$
$500$		0			0
$501$	−8.27858e8	−	1.82375e8i	−0.294120	−	0.0647939i
$502$		0			0
$503$	3.65029e9			1.27891			0.639455	−	0.768829i	$-0.279160\pi$
$503$	3.65029e9			1.27891			0.639455	+	0.768829i	$0.279160\pi$
$504$		0			0
$505$	−7.55714e8			−0.261119
$506$		0			0
$507$	−2.17296e9	−	4.78699e8i	−0.740499	−	0.163130i
$508$		0			0
$509$			3.21215e9i			1.07965i	0.841777	+	0.539826i	$0.181510\pi$
$509$			3.21215e9i			1.07965i	−0.841777	+	0.539826i	$0.818490\pi$
$510$		0			0
$511$		−	2.29210e9i		−	0.759906i
$512$		0			0
$513$	1.86125e9	−	2.44580e9i	0.608686	−	0.799853i
$514$		0			0
$515$	−1.44281e9			−0.465462
$516$		0			0
$517$	3.15310e9			1.00351
$518$		0			0
$519$	9.27634e8	−	4.21082e9i	0.291267	−	1.32215i
$520$		0			0
$521$		−	6.03429e9i		−	1.86936i	−0.355485	−	0.934682i	$-0.615684\pi$
$521$		−	6.03429e9i		−	1.86936i	0.355485	−	0.934682i	$-0.384316\pi$
$522$		0			0
$523$			1.74287e9i			0.532733i	0.963872	+	0.266367i	$0.0858232\pi$
$523$			1.74287e9i			0.532733i	−0.963872	+	0.266367i	$0.914177\pi$
$524$		0			0
$525$	1.19622e8	−	5.43000e8i	0.0360788	−	0.163773i
$526$		0			0
$527$	1.63001e9			0.485124
$528$		0			0
$529$	−8.93814e8			−0.262514
$530$		0			0
$531$	1.12377e9	+	5.20383e8i	0.325722	+	0.150832i
$532$		0			0
$533$			2.11200e9i			0.604157i
$534$		0			0
$535$		−	8.06681e8i		−	0.227753i
$536$		0			0
$537$	−1.53675e9	−	3.38543e8i	−0.428247	−	0.0943419i
$538$		0			0
$539$	−9.30257e8			−0.255884
$540$		0			0
$541$	−1.86005e9			−0.505050			−0.252525	−	0.967590i	$-0.581261\pi$
$541$	−1.86005e9			−0.505050			−0.252525	+	0.967590i	$0.581261\pi$
$542$		0			0
$543$	3.87525e8	+	8.53708e7i	0.103872	+	0.0228828i
$544$		0			0
$545$		−	2.58180e9i		−	0.683180i
$546$		0			0
$547$			4.73873e9i			1.23796i	0.785407	+	0.618979i	$0.212454\pi$
$547$			4.73873e9i			1.23796i	−0.785407	+	0.618979i	$0.787546\pi$
$548$		0			0
$549$	5.24724e9	+	2.42983e9i	1.35340	+	0.626719i
$550$		0			0
$551$	−4.56667e9			−1.16297
$552$		0			0
$553$	−1.93692e9			−0.487050
$554$		0			0
$555$	1.00555e8	−	4.56452e8i	0.0249678	−	0.113336i
$556$		0			0
$557$			2.20149e9i			0.539789i	0.962890	+	0.269895i	$0.0869888\pi$
$557$			2.20149e9i			0.539789i	−0.962890	+	0.269895i	$0.913011\pi$
$558$		0			0
$559$		−	2.08830e9i		−	0.505651i
$560$		0			0
$561$	4.61642e8	−	2.09554e9i	0.110392	−	0.501102i
$562$		0			0
$563$	7.38148e9			1.74327			0.871634	−	0.490158i	$-0.163061\pi$
$563$	7.38148e9			1.74327			0.871634	+	0.490158i	$0.163061\pi$
$564$		0			0
$565$	−1.05683e9			−0.246512
$566$		0			0
$567$	2.77553e9	−	2.35425e9i	0.639447	−	0.542391i
$568$		0			0
$569$			7.58120e9i			1.72522i	0.505867	+	0.862611i	$0.331173\pi$
$569$			7.58120e9i			1.72522i	−0.505867	+	0.862611i	$0.668827\pi$
$570$		0			0
$571$			2.97462e9i			0.668659i	0.942456	+	0.334330i	$0.108510\pi$
$571$			2.97462e9i			0.668659i	−0.942456	+	0.334330i	$0.891490\pi$
$572$		0			0
$573$	−7.51837e9	−	1.65628e9i	−1.66949	−	0.367784i
$574$		0			0
$575$	−7.82969e8			−0.171754
$576$		0			0
$577$	−6.91714e9			−1.49903			−0.749516	−	0.661986i	$-0.769714\pi$
$577$	−6.91714e9			−1.49903			−0.749516	+	0.661986i	$0.769714\pi$
$578$		0			0
$579$	6.77158e9	+	1.49176e9i	1.44982	+	0.319393i
$580$		0			0
$581$		−	5.55419e9i		−	1.17491i
$582$		0			0
$583$		−	3.45711e9i		−	0.722559i
$584$		0			0
$585$	1.20659e9	−	2.60564e9i	0.249181	−	0.538108i
$586$		0			0
$587$	7.49059e9			1.52856			0.764280	−	0.644884i	$-0.223095\pi$
$587$	7.49059e9			1.52856			0.764280	+	0.644884i	$0.223095\pi$
$588$		0			0
$589$	−4.06127e9			−0.818951
$590$		0			0
$591$	−1.05506e9	+	4.78925e9i	−0.210243	+	0.954358i
$592$		0			0
$593$			2.65792e8i			0.0523419i	0.999657	+	0.0261710i	$0.00833143\pi$
$593$			2.65792e8i			0.0523419i	−0.999657	+	0.0261710i	$0.991669\pi$
$594$		0			0
$595$			1.14720e9i			0.223270i
$596$		0			0
$597$	−1.41521e9	+	6.42406e9i	−0.272214	+	1.23566i
$598$		0			0
$599$	5.46914e9			1.03974			0.519870	−	0.854245i	$-0.325980\pi$
$599$	5.46914e9			1.03974			0.519870	+	0.854245i	$0.325980\pi$
$600$		0			0
$601$	−8.59551e9			−1.61514			−0.807571	−	0.589770i	$-0.799218\pi$
$601$	−8.59551e9			−1.61514			−0.807571	+	0.589770i	$0.799218\pi$
$602$		0			0
$603$	−5.41311e8	+	1.16896e9i	−0.100539	+	0.217115i
$604$		0			0
$605$			6.26775e8i			0.115072i
$606$		0			0
$607$		−	5.12787e9i		−	0.930630i	−0.885145	−	0.465315i	$-0.845941\pi$
$607$		−	5.12787e9i		−	0.930630i	0.885145	−	0.465315i	$-0.154059\pi$
$608$		0			0
$609$	−5.28110e9	−	1.16341e9i	−0.947467	−	0.208725i
$610$		0			0
$611$	−8.70565e9			−1.54403
$612$		0			0
$613$	−2.23066e9			−0.391131			−0.195565	−	0.980691i	$-0.562654\pi$
$613$	−2.23066e9			−0.391131			−0.195565	+	0.980691i	$0.562654\pi$
$614$		0			0
$615$	−1.14788e9	−	2.52875e8i	−0.198991	−	0.0438372i
$616$		0			0
$617$			1.07297e10i			1.83902i	0.393061	+	0.919512i	$0.371416\pi$
$617$			1.07297e10i			1.83902i	−0.393061	+	0.919512i	$0.628584\pi$
$618$		0			0
$619$			1.54216e9i			0.261343i	0.991426	+	0.130672i	$0.0417134\pi$
$619$			1.54216e9i			0.261343i	−0.991426	+	0.130672i	$0.958287\pi$
$620$		0			0
$621$	−4.07840e9	−	3.10366e9i	−0.683392	−	0.520060i
$622$		0			0
$623$	−6.97257e9			−1.15527
$624$		0			0
$625$	2.44141e8			0.0400000
$626$		0			0
$627$	−1.15021e9	+	5.22116e9i	−0.186355	+	0.845923i
$628$		0			0
$629$			9.64350e8i			0.154510i
$630$		0			0
$631$			6.82215e9i			1.08098i	0.841350	+	0.540491i	$0.181762\pi$
$631$			6.82215e9i			1.08098i	−0.841350	+	0.540491i	$0.818238\pi$
$632$		0			0
$633$	1.00425e9	−	4.55861e9i	0.157373	−	0.714364i
$634$		0			0
$635$	5.31902e9			0.824373
$636$		0			0
$637$	2.56842e9			0.393712
$638$		0			0
$639$	−4.68877e9	−	2.17122e9i	−0.710896	−	0.329194i
$640$		0			0
$641$		−	8.91076e9i		−	1.33632i	−0.744016	−	0.668162i	$-0.767081\pi$
$641$		−	8.91076e9i		−	1.33632i	0.744016	−	0.668162i	$-0.232919\pi$
$642$		0			0
$643$		−	1.99236e8i		−	0.0295550i	−0.999891	−	0.0147775i	$-0.995296\pi$
$643$		−	1.99236e8i		−	0.0295550i	0.999891	−	0.0147775i	$-0.00470399\pi$
$644$		0			0
$645$	1.13499e9	+	2.50037e8i	0.166546	+	0.0366897i
$646$		0			0
$647$	1.24392e9			0.180562			0.0902810	−	0.995916i	$-0.471223\pi$
$647$	1.24392e9			0.180562			0.0902810	+	0.995916i	$0.471223\pi$
$648$		0			0
$649$	−2.15424e9			−0.309342
$650$		0			0
$651$	−4.69663e9	−	1.03466e9i	−0.667195	−	0.146982i
$652$		0			0
$653$		−	5.19203e8i		−	0.0729695i	−0.999334	−	0.0364848i	$-0.988384\pi$
$653$		−	5.19203e8i		−	0.0729695i	0.999334	−	0.0364848i	$-0.0116160\pi$
$654$		0			0
$655$			4.16531e9i			0.579165i
$656$		0			0
$657$	5.97791e9	+	2.76818e9i	0.822376	+	0.380816i
$658$		0			0
$659$	−5.86727e9			−0.798615			−0.399307	−	0.916817i	$-0.630749\pi$
$659$	−5.86727e9			−0.798615			−0.399307	+	0.916817i	$0.630749\pi$
$660$		0			0
$661$	1.08269e10			1.45814			0.729069	−	0.684441i	$-0.239954\pi$
$661$	1.08269e10			1.45814			0.729069	+	0.684441i	$0.239954\pi$
$662$		0			0
$663$	−1.27459e9	+	5.78575e9i	−0.169853	+	0.771014i
$664$		0			0
$665$		−	2.85832e9i		−	0.376908i
$666$		0			0
$667$			7.61499e9i			0.993639i
$668$		0			0
$669$	−1.81035e8	+	8.21773e8i	−0.0233760	+	0.106111i
$670$		0			0
$671$	−1.00588e10			−1.28534
$672$		0			0
$673$	−3.35910e9			−0.424786			−0.212393	−	0.977184i	$-0.568126\pi$
$673$	−3.35910e9			−0.424786			−0.212393	+	0.977184i	$0.568126\pi$
$674$		0			0
$675$	1.27170e9	+	9.67764e8i	0.159156	+	0.121117i
$676$		0			0
$677$			1.33115e10i			1.64879i	0.566014	+	0.824396i	$0.308485\pi$
$677$			1.33115e10i			1.64879i	−0.566014	+	0.824396i	$0.691515\pi$
$678$		0			0
$679$		−	9.42400e9i		−	1.15529i
$680$		0			0
$681$	−1.36290e10	−	3.00244e9i	−1.65367	−	0.364300i
$682$		0			0
$683$	1.48174e9			0.177951			0.0889755	−	0.996034i	$-0.471641\pi$
$683$	1.48174e9			0.177951			0.0889755	+	0.996034i	$0.471641\pi$
$684$		0			0
$685$	−4.51028e9			−0.536151
$686$		0			0
$687$	1.58051e10	+	3.48182e9i	1.85972	+	0.409692i
$688$		0			0
$689$			9.54503e9i			1.11176i
$690$		0			0
$691$			1.14439e10i			1.31948i	0.751495	+	0.659739i	$0.229333\pi$
$691$			1.14439e10i			1.31948i	−0.751495	+	0.659739i	$0.770667\pi$
$692$		0			0
$693$	−2.66031e9	+	5.74496e9i	−0.303645	+	0.655723i
$694$		0			0
$695$	−5.46739e9			−0.617779
$696$		0			0
$697$	2.42513e9			0.271282
$698$		0			0
$699$	−4.58734e8	+	2.08234e9i	−0.0508032	+	0.230612i
$700$		0			0
$701$		−	5.96485e9i		−	0.654014i	−0.945022	−	0.327007i	$-0.893960\pi$
$701$		−	5.96485e9i		−	0.654014i	0.945022	−	0.327007i	$-0.106040\pi$
$702$		0			0
$703$		−	2.40274e9i		−	0.260833i
$704$		0			0
$705$	1.04235e9	−	4.73154e9i	0.112034	−	0.508558i
$706$		0			0
$707$	−4.60038e9			−0.489582
$708$		0			0
$709$	−8.65286e9			−0.911796			−0.455898	−	0.890032i	$-0.650682\pi$
$709$	−8.65286e9			−0.911796			−0.455898	+	0.890032i	$0.650682\pi$
$710$		0			0
$711$	2.33923e9	−	5.05159e9i	0.244079	−	0.527089i
$712$		0			0
$713$			6.77222e9i			0.699710i
$714$		0			0
$715$			4.99495e9i			0.511046i
$716$		0			0
$717$	7.83837e9	+	1.72677e9i	0.794161	+	0.174952i
$718$		0			0
$719$	−4.30863e9			−0.432303			−0.216151	−	0.976360i	$-0.569350\pi$
$719$	−4.30863e9			−0.432303			−0.216151	+	0.976360i	$0.569350\pi$
$720$		0			0
$721$	−8.78305e9			−0.872714
$722$		0			0
$723$	1.05176e10	+	2.31701e9i	1.03498	+	0.228004i
$724$		0			0
$725$		−	2.37446e9i		−	0.231410i
$726$		0			0
$727$			1.00091e10i			0.966103i	0.875592	+	0.483051i	$0.160472\pi$
$727$			1.00091e10i			0.966103i	−0.875592	+	0.483051i	$0.839528\pi$
$728$		0			0
$729$	2.78800e9	+	1.00820e10i	0.266530	+	0.963827i
$730$		0			0
$731$	−2.39791e9			−0.227050
$732$		0			0
$733$	−1.13721e10			−1.06653			−0.533267	−	0.845947i	$-0.679036\pi$
$733$	−1.13721e10			−1.06653			−0.533267	+	0.845947i	$0.679036\pi$
$734$		0			0
$735$	−3.07523e8	+	1.39594e9i	−0.0285675	+	0.129677i
$736$		0			0
$737$		−	2.24088e9i		−	0.206197i
$738$		0			0
$739$		−	1.19662e10i		−	1.09069i	−0.838213	−	0.545343i	$-0.816399\pi$
$739$		−	1.19662e10i		−	1.09069i	0.838213	−	0.545343i	$-0.183601\pi$
$740$		0			0
$741$	3.17571e9	−	1.44155e10i	0.286733	−	1.30157i
$742$		0			0
$743$	−1.28719e10			−1.15128			−0.575639	−	0.817704i	$-0.695247\pi$
$743$	−1.28719e10			−1.15128			−0.575639	+	0.817704i	$0.695247\pi$
$744$		0			0
$745$	6.34288e9			0.562005
$746$		0			0
$747$	1.44856e10	+	6.70784e9i	1.27150	+	0.588790i
$748$		0			0
$749$		−	4.91064e9i		−	0.427023i
$750$		0			0
$751$		−	1.61524e10i		−	1.39154i	−0.718264	−	0.695771i	$-0.755063\pi$
$751$		−	1.61524e10i		−	1.39154i	0.718264	−	0.695771i	$-0.244937\pi$
$752$		0			0
$753$	8.77520e9	+	1.93316e9i	0.748988	+	0.165000i
$754$		0			0
$755$	−8.79117e9			−0.743417
$756$		0			0
$757$	7.64165e9			0.640253			0.320126	−	0.947375i	$-0.396275\pi$
$757$	7.64165e9			0.640253			0.320126	+	0.947375i	$0.396275\pi$
$758$		0			0
$759$	8.70636e9	+	1.91799e9i	0.722754	+	0.159221i
$760$		0			0
$761$			3.58231e9i			0.294657i	0.989088	+	0.147329i	$0.0470675\pi$
$761$			3.58231e9i			0.294657i	−0.989088	+	0.147329i	$0.952933\pi$
$762$		0			0
$763$		−	1.57166e10i		−	1.28092i
$764$		0			0
$765$	−2.99196e9	−	1.38548e9i	−0.241624	−	0.111889i
$766$		0			0
$767$	5.94783e9			0.475965
$768$		0			0
$769$	8.09590e9			0.641982			0.320991	−	0.947082i	$-0.395984\pi$
$769$	8.09590e9			0.641982			0.320991	+	0.947082i	$0.395984\pi$
$770$		0			0
$771$	4.18030e9	−	1.89757e10i	0.328487	−	1.49110i
$772$		0			0
$773$			1.82909e10i			1.42432i	0.702019	+	0.712158i	$0.252282\pi$
$773$			1.82909e10i			1.42432i	−0.702019	+	0.712158i	$0.747718\pi$
$774$		0			0
$775$		−	2.11167e9i		−	0.162956i
$776$		0			0
$777$	6.12126e8	−	2.77863e9i	0.0468131	−	0.212499i
$778$		0			0
$779$	−6.04236e9			−0.457958
$780$		0			0
$781$	8.98826e9			0.675145
$782$		0			0
$783$	9.41227e9	−	1.23683e10i	0.700693	−	0.920756i
$784$		0			0
$785$		−	8.63630e9i		−	0.637212i
$786$		0			0
$787$		−	5.79422e9i		−	0.423725i	−0.977300	−	0.211862i	$-0.932047\pi$
$787$		−	5.79422e9i		−	0.423725i	0.977300	−	0.211862i	$-0.0679528\pi$
$788$		0			0
$789$	−6.91222e9	−	1.52275e9i	−0.501012	−	0.110372i
$790$		0			0
$791$	−6.43343e9			−0.462195
$792$		0			0
$793$	2.77723e10			1.97768
$794$		0			0
$795$	−5.18774e9	−	1.14285e9i	−0.366179	−	0.0806684i
$796$		0			0
$797$		−	2.41098e9i		−	0.168690i	−0.996437	−	0.0843449i	$-0.973120\pi$
$797$		−	2.41098e9i		−	0.168690i	0.996437	−	0.0843449i	$-0.0268798\pi$
$798$		0			0
$799$			9.99635e9i			0.693311i
$800$		0			0
$801$	8.42083e9	−	1.81848e10i	0.578950	−	1.25025i
$802$		0			0
$803$	−1.14595e10			−0.781019
$804$		0			0
$805$	−4.76629e9			−0.322029
$806$		0			0
$807$	8.29117e8	−	3.76362e9i	0.0555339	−	0.252086i
$808$		0			0
$809$			7.58192e9i			0.503454i	0.967798	+	0.251727i	$0.0809984\pi$
$809$			7.58192e9i			0.503454i	−0.967798	+	0.251727i	$0.919002\pi$
$810$		0			0
$811$			1.14466e10i			0.753537i	0.926307	+	0.376769i	$0.122965\pi$
$811$			1.14466e10i			0.753537i	−0.926307	+	0.376769i	$0.877035\pi$
$812$		0			0
$813$	1.60544e9	−	7.28758e9i	0.104780	−	0.475627i
$814$		0			0
$815$	−7.65786e8			−0.0495514
$816$		0			0
$817$	5.97454e9			0.383290
$818$		0			0
$819$	7.34507e9	−	1.58617e10i	0.467199	−	1.00892i
$820$		0			0
$821$		−	2.91315e10i		−	1.83723i	−0.395159	−	0.918613i	$-0.629310\pi$
$821$		−	2.91315e10i		−	1.83723i	0.395159	−	0.918613i	$-0.370690\pi$
$822$		0			0
$823$			1.04276e9i			0.0652053i	0.999468	+	0.0326027i	$0.0103796\pi$
$823$			1.04276e9i			0.0652053i	−0.999468	+	0.0326027i	$0.989620\pi$
$824$		0			0
$825$	−2.71477e9	−	5.98057e8i	−0.168323	−	0.0370812i
$826$		0			0
$827$	2.22845e10			1.37004			0.685022	−	0.728523i	$-0.259793\pi$
$827$	2.22845e10			1.37004			0.685022	+	0.728523i	$0.259793\pi$
$828$		0			0
$829$	−1.93468e9			−0.117942			−0.0589710	−	0.998260i	$-0.518782\pi$
$829$	−1.93468e9			−0.117942			−0.0589710	+	0.998260i	$0.518782\pi$
$830$		0			0
$831$	1.04789e10	+	2.30847e9i	0.633449	+	0.139547i
$832$		0			0
$833$		−	2.94922e9i		−	0.176787i
$834$		0			0
$835$		−	2.26586e9i		−	0.134688i
$836$		0			0
$837$	8.37059e9	−	1.09995e10i	0.493421	−	0.648386i
$838$		0			0
$839$	1.54591e10			0.903688			0.451844	−	0.892097i	$-0.350766\pi$
$839$	1.54591e10			0.903688			0.451844	+	0.892097i	$0.350766\pi$
$840$		0			0
$841$	−5.84362e9			−0.338763
$842$		0			0
$843$	−2.04250e9	+	9.27156e9i	−0.117427	+	0.533036i
$844$		0			0
$845$		−	5.94742e9i		−	0.339102i
$846$		0			0
$847$			3.81547e9i			0.215752i
$848$		0			0
$849$	3.79132e9	−	1.72100e10i	0.212625	−	0.965170i
$850$		0			0
$851$	−4.00660e9			−0.222855
$852$		0			0
$853$	−2.16552e9			−0.119465			−0.0597324	−	0.998214i	$-0.519025\pi$
$853$	−2.16552e9			−0.119465			−0.0597324	+	0.998214i	$0.519025\pi$
$854$		0			0
$855$	7.45464e9	+	3.45201e9i	0.407892	+	0.188882i
$856$		0			0
$857$			2.42895e10i			1.31821i	0.752050	+	0.659106i	$0.229065\pi$
$857$			2.42895e10i			1.31821i	−0.752050	+	0.659106i	$0.770935\pi$
$858$		0			0
$859$		−	1.48431e10i		−	0.799002i	−0.916733	−	0.399501i	$-0.869183\pi$
$859$		−	1.48431e10i		−	0.799002i	0.916733	−	0.399501i	$-0.130817\pi$
$860$		0			0
$861$	−6.98766e9	−	1.53937e9i	−0.373096	−	0.0821922i
$862$		0			0
$863$	−3.51327e9			−0.186069			−0.0930344	−	0.995663i	$-0.529657\pi$
$863$	−3.51327e9			−0.186069			−0.0930344	+	0.995663i	$0.529657\pi$
$864$		0			0
$865$	1.15251e10			0.605462
$866$		0			0
$867$	−1.20967e10	−	2.66488e9i	−0.630378	−	0.138871i
$868$		0			0
$869$			9.68377e9i			0.500582i
$870$		0			0
$871$			6.18703e9i			0.317262i
$872$		0			0
$873$	2.45783e10	+	1.13814e10i	1.25026	+	0.578958i
$874$		0			0
$875$	1.48620e9			0.0749977
$876$		0			0
$877$	3.19417e10			1.59904			0.799521	−	0.600638i	$-0.205087\pi$
$877$	3.19417e10			1.59904			0.799521	+	0.600638i	$0.205087\pi$
$878$		0			0
$879$	−7.46300e8	+	3.38769e9i	−0.0370640	+	0.168245i
$880$		0			0
$881$			3.24270e10i			1.59769i	0.601540	+	0.798843i	$0.294554\pi$
$881$			3.24270e10i			1.59769i	−0.601540	+	0.798843i	$0.705446\pi$
$882$		0			0
$883$		−	9.58112e9i		−	0.468332i	−0.972197	−	0.234166i	$-0.924764\pi$
$883$		−	9.58112e9i		−	0.468332i	0.972197	−	0.234166i	$-0.0752359\pi$
$884$		0			0
$885$	−7.12147e8	+	3.23266e9i	−0.0345357	+	0.156768i
$886$		0			0
$887$	−2.36771e10			−1.13919			−0.569593	−	0.821927i	$-0.692899\pi$
$887$	−2.36771e10			−1.13919			−0.569593	+	0.821927i	$0.692899\pi$
$888$		0			0
$889$	3.23793e10			1.54565
$890$		0			0
$891$	−1.17703e10	−	1.38764e10i	−0.557461	−	0.657213i
$892$		0			0
$893$		−	2.49065e10i		−	1.17040i
$894$		0			0
$895$		−	4.20611e9i		−	0.196110i
$896$		0			0
$897$	−2.40381e10	−	5.29555e9i	−1.11206	−	0.244984i
$898$		0			0
$899$	−2.05377e10			−0.942742
$900$		0			0
$901$	1.09602e10			0.499207
$902$		0			0
$903$	6.90923e9	+	1.52209e9i	0.312264	+	0.0687911i
$904$		0			0
$905$			1.06066e9i			0.0475670i
$906$		0			0
$907$		−	3.26154e10i		−	1.45143i	−0.687994	−	0.725716i	$-0.741509\pi$
$907$		−	3.26154e10i		−	1.45143i	0.687994	−	0.725716i	$-0.258491\pi$
$908$		0			0
$909$	5.55591e9	−	1.19980e10i	0.245347	−	0.529830i
$910$		0			0
$911$	−2.12092e10			−0.929416			−0.464708	−	0.885464i	$-0.653841\pi$
$911$	−2.12092e10			−0.929416			−0.464708	+	0.885464i	$0.653841\pi$
$912$		0			0
$913$	−2.77686e10			−1.20755
$914$		0			0
$915$	−3.32523e9	+	1.50943e10i	−0.143499	+	0.651386i
$916$		0			0
$917$			2.53561e10i			1.08590i
$918$		0			0
$919$		−	1.41019e10i		−	0.599338i	−0.954043	−	0.299669i	$-0.903124\pi$
$919$		−	1.41019e10i		−	0.599338i	0.954043	−	0.299669i	$-0.0968763\pi$
$920$		0			0
$921$	2.01615e9	−	9.15195e9i	0.0850383	−	0.386016i
$922$		0			0
$923$	−2.48164e10			−1.03880
$924$		0			0
$925$	1.24931e9			0.0519010
$926$		0			0
$927$	1.06073e10	−	2.29066e10i	0.437349	−	0.944457i
$928$		0			0
$929$			2.75245e10i			1.12633i	0.826346	+	0.563163i	$0.190416\pi$
$929$			2.75245e10i			1.12633i	−0.826346	+	0.563163i	$0.809584\pi$
$930$		0			0
$931$			7.34817e9i			0.298439i
$932$		0			0
$933$	1.45217e9	+	3.19909e8i	0.0585371	+	0.0128956i
$934$		0			0
$935$	5.73551e9			0.229473
$936$		0			0
$937$	7.69338e9			0.305512			0.152756	−	0.988264i	$-0.451185\pi$
$937$	7.69338e9			0.305512			0.152756	+	0.988264i	$0.451185\pi$
$938$		0			0
$939$	1.75965e10	+	3.87648e9i	0.693582	+	0.152795i
$940$		0			0
$941$		−	3.48056e10i		−	1.36171i	−0.732417	−	0.680856i	$-0.761608\pi$
$941$		−	3.48056e10i		−	1.36171i	0.732417	−	0.680856i	$-0.238392\pi$
$942$		0			0
$943$			1.00757e10i			0.391278i
$944$		0			0
$945$	7.74144e9	+	5.89122e9i	0.298408	+	0.227088i
$946$		0			0
$947$	2.56634e6			9.81950e−5			4.90975e−5	−	1.00000i	$-0.499984\pi$
$947$	2.56634e6			9.81950e−5			4.90975e−5		1.00000i	$0.499984\pi$
$948$		0			0
$949$	3.16395e10			1.20171
$950$		0			0
$951$	−6.33220e9	+	2.87438e10i	−0.238739	+	1.08371i
$952$		0			0
$953$		−	6.45497e9i		−	0.241585i	−0.992678	−	0.120792i	$-0.961456\pi$
$953$		−	6.45497e9i		−	0.241585i	0.992678	−	0.120792i	$-0.0385435\pi$
$954$		0			0
$955$		−	2.05779e10i		−	0.764519i
$956$		0			0
$957$	−5.81658e9	+	2.64033e10i	−0.214524	+	0.973791i
$958$		0			0
$959$	−2.74562e10			−1.00525
$960$		0			0
$961$	9.24786e9			0.336132
$962$		0			0
$963$	1.28072e10	+	5.93061e9i	0.462128	+	0.213997i
$964$		0			0
$965$			1.85339e10i			0.663928i
$966$		0			0
$967$			3.54915e10i			1.26221i	0.775697	+	0.631106i	$0.217399\pi$
$967$			3.54915e10i			1.26221i	−0.775697	+	0.631106i	$0.782601\pi$
$968$		0			0
$969$	−1.65528e10	−	3.64655e9i	−0.584438	−	0.128750i
$970$		0			0
$971$	−1.97987e10			−0.694015			−0.347008	−	0.937862i	$-0.612802\pi$
$971$	−1.97987e10			−0.694015			−0.347008	+	0.937862i	$0.612802\pi$
$972$		0			0
$973$	−3.32825e10			−1.15830
$974$		0			0
$975$	7.49543e9	+	1.65123e9i	0.258988	+	0.0570545i
$976$		0			0
$977$			9.36552e9i			0.321293i	0.987012	+	0.160646i	$0.0513579\pi$
$977$			9.36552e9i			0.321293i	−0.987012	+	0.160646i	$0.948642\pi$
$978$		0			0
$979$			3.48599e10i			1.18737i
$980$		0			0
$981$	4.09897e10	+	1.89810e10i	1.38622	+	0.641917i
$982$		0			0
$983$	3.16871e10			1.06401			0.532005	−	0.846741i	$-0.321439\pi$
$983$	3.16871e10			1.06401			0.532005	+	0.846741i	$0.321439\pi$
$984$		0			0
$985$	−1.31082e10			−0.437036
$986$		0			0
$987$	6.34524e9	−	2.88030e10i	0.210057	−	0.953516i
$988$		0			0
$989$		−	9.96263e9i		−	0.327482i
$990$		0			0
$991$			8.01370e9i			0.261562i	0.991411	+	0.130781i	$0.0417485\pi$
$991$			8.01370e9i			0.261562i	−0.991411	+	0.130781i	$0.958251\pi$
$992$		0			0
$993$	9.45602e9	−	4.29238e10i	0.306469	−	1.39116i
$994$		0			0
$995$	−1.75827e10			−0.565855
$996$		0			0
$997$	−4.56357e10			−1.45838			−0.729191	−	0.684310i	$-0.760104\pi$
$997$	−4.56357e10			−1.45838			−0.729191	+	0.684310i	$0.760104\pi$
$998$		0			0
$999$	6.50755e9	+	4.95223e9i	0.206509	+	0.157153i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	240.8.h.b.191.1	✓	36
3.2	odd	2	inner	240.8.h.b.191.35	yes	36
4.3	odd	2	inner	240.8.h.b.191.36	yes	36
12.11	even	2	inner	240.8.h.b.191.2	yes	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
240.8.h.b.191.1	✓	36	1.1	even	1	trivial
240.8.h.b.191.2	yes	36	12.11	even	2	inner
240.8.h.b.191.35	yes	36	3.2	odd	2	inner
240.8.h.b.191.36	yes	36	4.3	odd	2	inner

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:	\( N \)	\(=\)	\( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	240.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-45.6703 - 10.0611i) q^{3} -125.000i q^{5} -760.932i q^{7} +(1984.55 + 918.983i) q^{9} -3804.34 q^{11} +10503.7 q^{13} +(-1257.63 + 5708.79i) q^{15} -12061.0i q^{17} +30050.7i q^{19} +(-7655.78 + 34752.0i) q^{21} +50110.0 q^{23} -15625.0 q^{25} +(-81389.0 - 61936.9i) q^{27} +151965. i q^{29} +135147. i q^{31} +(173745. + 38275.6i) q^{33} -95116.5 q^{35} -79956.1 q^{37} +(-479707. - 105678. i) q^{39} +201072. i q^{41} -198815. i q^{43} +(114873. - 248069. i) q^{45} -828817. q^{47} +244525. q^{49} +(-121346. + 550829. i) q^{51} +908729. i q^{53} +475542. i q^{55} +(302342. - 1.37242e6i) q^{57} +566260. q^{59} +2.64404e6 q^{61} +(699284. - 1.51011e6i) q^{63} -1.31296e6i q^{65} +589032. i q^{67} +(-2.28854e6 - 504160. i) q^{69} -2.36264e6 q^{71} +3.01222e6 q^{73} +(713598. + 157204. i) q^{75} +2.89484e6i q^{77} -2.54546e6i q^{79} +(3.09391e6 + 3.64754e6i) q^{81} +7.29920e6 q^{83} -1.50762e6 q^{85} +(1.52893e6 - 6.94031e6i) q^{87} -9.16320e6i q^{89} -7.99261e6i q^{91} +(1.35972e6 - 6.17221e6i) q^{93} +3.75634e6 q^{95} +1.23848e7 q^{97} +(-7.54989e6 - 3.49612e6i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 3324 q^{9} - 432 q^{13} - 22032 q^{21} - 562500 q^{25} - 386232 q^{33} + 1200048 q^{37} - 1710000 q^{45} - 3237972 q^{49} - 9331200 q^{57} + 11262360 q^{61} - 11352552 q^{69} + 37210296 q^{73} - 12564252 q^{81}+ \cdots + 94537320 q^{97}+O(q^{100}) \) 36 * q + 3324 * q^9 - 432 * q^13 - 22032 * q^21 - 562500 * q^25 - 386232 * q^33 + 1200048 * q^37 - 1710000 * q^45 - 3237972 * q^49 - 9331200 * q^57 + 11262360 * q^61 - 11352552 * q^69 + 37210296 * q^73 - 12564252 * q^81 + 5271000 * q^85 + 5594400 * q^93 + 94537320 * q^97

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