LMFDB - Embedded newform 405.8.a.f.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("405.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$126.515935321$
Analytic rank:	$1$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 5 x^{12} - 1170 x^{11} + 4622 x^{10} + 503384 x^{9} - 1392714 x^{8} - 97100172 x^{7} + \cdots + 4693741072256 $ x^13 - 5x^12 - 1170x^11 + 4622x^10 + 503384x^9 - 1392714x^8 - 97100172x^7 + 151526988x^6 + 8205832521x^5 - 4072720601x^4 - 282543812090x^3 - 118496553138x^2 + 3000776721014x + 4693741072256
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{24} $
Twist minimal:	no (minimal twist has level 45)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Root			$-20.5744$ 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)

$f(q)$	$=$	$q+21.5744 q^{2} +337.457 q^{4} -125.000 q^{5} +77.0394 q^{7} +4518.92 q^{8} +O(q^{10})$	\(q+21.5744 q^{2} +337.457 q^{4} -125.000 q^{5} +77.0394 q^{7} +4518.92 q^{8} -2696.81 q^{10} -3711.79 q^{11} -9904.64 q^{13} +1662.08 q^{14} +54298.6 q^{16} -12228.4 q^{17} -42221.9 q^{19} -42182.1 q^{20} -80079.7 q^{22} -83769.2 q^{23} +15625.0 q^{25} -213687. q^{26} +25997.5 q^{28} -77925.9 q^{29} +255661. q^{31} +593042. q^{32} -263821. q^{34} -9629.93 q^{35} -505096. q^{37} -910913. q^{38} -564864. q^{40} +362971. q^{41} -72126.8 q^{43} -1.25257e6 q^{44} -1.80727e6 q^{46} +521445. q^{47} -817608. q^{49} +337101. q^{50} -3.34239e6 q^{52} +1.72594e6 q^{53} +463973. q^{55} +348135. q^{56} -1.68121e6 q^{58} -667127. q^{59} +444389. q^{61} +5.51574e6 q^{62} +5.84433e6 q^{64} +1.23808e6 q^{65} +2.99810e6 q^{67} -4.12656e6 q^{68} -207760. q^{70} -4.05089e6 q^{71} -1.33628e6 q^{73} -1.08972e7 q^{74} -1.42481e7 q^{76} -285954. q^{77} -621602. q^{79} -6.78733e6 q^{80} +7.83089e6 q^{82} +1.29991e6 q^{83} +1.52855e6 q^{85} -1.55609e6 q^{86} -1.67732e7 q^{88} -416509. q^{89} -763048. q^{91} -2.82685e7 q^{92} +1.12499e7 q^{94} +5.27773e6 q^{95} -1.63057e7 q^{97} -1.76394e7 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q + 8 q^{2} + 704 q^{4} - 1625 q^{5} - 1455 q^{7} + 1236 q^{8}+O(q^{10}) $ 13 * q + 8 * q^2 + 704 * q^4 - 1625 * q^5 - 1455 * q^7 + 1236 * q^8	\( 13 q + 8 q^{2} + 704 q^{4} - 1625 q^{5} - 1455 q^{7} + 1236 q^{8} - 1000 q^{10} + 1658 q^{11} - 13568 q^{13} + 12351 q^{14} + 52076 q^{16} + 6944 q^{17} - 45812 q^{19} - 88000 q^{20} - 17993 q^{22} + 96441 q^{23} + 203125 q^{25} + 126146 q^{26} - 216945 q^{28} + 39043 q^{29} - 158520 q^{31} + 725206 q^{32} - 441617 q^{34} + 181875 q^{35} - 505438 q^{37} + 615041 q^{38} - 154500 q^{40} + 1578883 q^{41} - 1082090 q^{43} + 498211 q^{44} - 2312547 q^{46} + 1690139 q^{47} - 1054816 q^{49} + 125000 q^{50} - 4100644 q^{52} - 102274 q^{53} - 207250 q^{55} + 389331 q^{56} - 5780521 q^{58} - 2908966 q^{59} - 3091451 q^{61} + 5212476 q^{62} - 5659352 q^{64} + 1696000 q^{65} - 1849533 q^{67} - 563369 q^{68} - 1543875 q^{70} - 2617958 q^{71} + 5310946 q^{73} - 11485004 q^{74} - 14421739 q^{76} + 1719660 q^{77} - 3632166 q^{79} - 6509500 q^{80} - 7347658 q^{82} + 1424115 q^{83} - 868000 q^{85} - 23193379 q^{86} - 20229447 q^{88} - 2816067 q^{89} - 16929702 q^{91} - 27930183 q^{92} - 11693189 q^{94} + 5726500 q^{95} - 12062476 q^{97} - 29617805 q^{98}+O(q^{100}) \) 13 * q + 8 * q^2 + 704 * q^4 - 1625 * q^5 - 1455 * q^7 + 1236 * q^8 - 1000 * q^10 + 1658 * q^11 - 13568 * q^13 + 12351 * q^14 + 52076 * q^16 + 6944 * q^17 - 45812 * q^19 - 88000 * q^20 - 17993 * q^22 + 96441 * q^23 + 203125 * q^25 + 126146 * q^26 - 216945 * q^28 + 39043 * q^29 - 158520 * q^31 + 725206 * q^32 - 441617 * q^34 + 181875 * q^35 - 505438 * q^37 + 615041 * q^38 - 154500 * q^40 + 1578883 * q^41 - 1082090 * q^43 + 498211 * q^44 - 2312547 * q^46 + 1690139 * q^47 - 1054816 * q^49 + 125000 * q^50 - 4100644 * q^52 - 102274 * q^53 - 207250 * q^55 + 389331 * q^56 - 5780521 * q^58 - 2908966 * q^59 - 3091451 * q^61 + 5212476 * q^62 - 5659352 * q^64 + 1696000 * q^65 - 1849533 * q^67 - 563369 * q^68 - 1543875 * q^70 - 2617958 * q^71 + 5310946 * q^73 - 11485004 * q^74 - 14421739 * q^76 + 1719660 * q^77 - 3632166 * q^79 - 6509500 * q^80 - 7347658 * q^82 + 1424115 * q^83 - 868000 * q^85 - 23193379 * q^86 - 20229447 * q^88 - 2816067 * q^89 - 16929702 * q^91 - 27930183 * q^92 - 11693189 * q^94 + 5726500 * q^95 - 12062476 * q^97 - 29617805 * 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 $

$2$	21.5744	1.90693	0.953465	−	0.301504i	$-0.0974887\pi$
$2$	21.5744	1.90693	0.953465	+	0.301504i	$0.0974887\pi$
$3$	0	0
$3$	0	0
$4$	337.457	2.63638
$5$	−125.000	−0.447214
$5$	−125.000	−0.447214
$6$	0	0
$7$	77.0394	0.0848926	0.0424463	−	0.999099i	$-0.486485\pi$
$7$	77.0394	0.0848926	0.0424463	+	0.999099i	$0.486485\pi$
$8$	4518.92	3.12046
$9$	0	0
$10$	−2696.81	−0.852805
$11$	−3711.79	−0.840831	−0.420415	−	0.907332i	$-0.638116\pi$
$11$	−3711.79	−0.840831	−0.420415	+	0.907332i	$0.638116\pi$
$12$	0	0
$13$	−9904.64	−1.25037	−0.625183	−	0.780478i	$-0.714976\pi$
$13$	−9904.64	−1.25037	−0.625183	+	0.780478i	$0.714976\pi$
$14$	1662.08	0.161884
$15$	0	0
$16$	54298.6	3.31413
$17$	−12228.4	−0.603669	−0.301835	−	0.953360i	$-0.597599\pi$
$17$	−12228.4	−0.603669	−0.301835	+	0.953360i	$0.597599\pi$
$18$	0	0
$19$	−42221.9	−1.41221	−0.706106	−	0.708106i	$-0.749550\pi$
$19$	−42221.9	−1.41221	−0.706106	+	0.708106i	$0.749550\pi$
$20$	−42182.1	−1.17903
$21$	0	0
$22$	−80079.7	−1.60340
$23$	−83769.2	−1.43561	−0.717806	−	0.696243i	$-0.754853\pi$
$23$	−83769.2	−1.43561	−0.717806	+	0.696243i	$0.754853\pi$
$24$	0	0
$25$	15625.0	0.200000
$26$	−213687.	−2.38436
$27$	0	0
$28$	25997.5	0.223809
$29$	−77925.9	−0.593320	−0.296660	−	0.954983i	$-0.595873\pi$
$29$	−77925.9	−0.593320	−0.296660	+	0.954983i	$0.595873\pi$
$30$	0	0
$31$	255661.	1.54134	0.770670	−	0.637234i	$-0.219922\pi$
$31$	255661.	1.54134	0.770670	+	0.637234i	$0.219922\pi$
$32$	593042.	3.19934
$33$	0	0
$34$	−263821.	−1.15116
$35$	−9629.93	−0.0379651
$36$	0	0
$37$	−505096.	−1.63934	−0.819668	−	0.572839i	$-0.805842\pi$
$37$	−505096.	−1.63934	−0.819668	+	0.572839i	$0.805842\pi$
$38$	−910913.	−2.69299
$39$	0	0
$40$	−564864.	−1.39551
$41$	362971.	0.822485	0.411242	−	0.911526i	$-0.365095\pi$
$41$	362971.	0.822485	0.411242	+	0.911526i	$0.365095\pi$
$42$	0	0
$43$	−72126.8	−0.138343	−0.0691714	−	0.997605i	$-0.522036\pi$
$43$	−72126.8	−0.138343	−0.0691714	+	0.997605i	$0.522036\pi$
$44$	−1.25257e6	−2.21675
$45$	0	0
$46$	−1.80727e6	−2.73761
$47$	521445.	0.732599	0.366299	−	0.930497i	$-0.380625\pi$
$47$	521445.	0.732599	0.366299	+	0.930497i	$0.380625\pi$
$48$	0	0
$49$	−817608.	−0.992793
$50$	337101.	0.381386
$51$	0	0
$52$	−3.34239e6	−3.29644
$53$	1.72594e6	1.59243	0.796216	−	0.605013i	$-0.206832\pi$
$53$	1.72594e6	1.59243	0.796216	+	0.605013i	$0.206832\pi$
$54$	0	0
$55$	463973.	0.376031
$56$	348135.	0.264904
$57$	0	0
$58$	−1.68121e6	−1.13142
$59$	−667127.	−0.422889	−0.211444	−	0.977390i	$-0.567817\pi$
$59$	−667127.	−0.422889	−0.211444	+	0.977390i	$0.567817\pi$
$60$	0	0
$61$	444389.	0.250674	0.125337	−	0.992114i	$-0.459999\pi$
$61$	444389.	0.250674	0.125337	+	0.992114i	$0.459999\pi$
$62$	5.51574e6	2.93923
$63$	0	0
$64$	5.84433e6	2.78679
$65$	1.23808e6	0.559180
$66$	0	0
$67$	2.99810e6	1.21782	0.608912	−	0.793238i	$-0.291606\pi$
$67$	2.99810e6	1.21782	0.608912	+	0.793238i	$0.291606\pi$
$68$	−4.12656e6	−1.59150
$69$	0	0
$70$	−207760.	−0.0723968
$71$	−4.05089e6	−1.34322	−0.671608	−	0.740906i	$-0.734396\pi$
$71$	−4.05089e6	−1.34322	−0.671608	+	0.740906i	$0.734396\pi$
$72$	0	0
$73$	−1.33628e6	−0.402037	−0.201019	−	0.979587i	$-0.564425\pi$
$73$	−1.33628e6	−0.402037	−0.201019	+	0.979587i	$0.564425\pi$
$74$	−1.08972e7	−3.12610
$75$	0	0
$76$	−1.42481e7	−3.72313
$77$	−285954.	−0.0713803
$78$	0	0
$79$	−621602.	−0.141846	−0.0709231	−	0.997482i	$-0.522594\pi$
$79$	−621602.	−0.141846	−0.0709231	+	0.997482i	$0.522594\pi$
$80$	−6.78733e6	−1.48212
$81$	0	0
$82$	7.83089e6	1.56842
$83$	1.29991e6	0.249540	0.124770	−	0.992186i	$-0.460181\pi$
$83$	1.29991e6	0.249540	0.124770	+	0.992186i	$0.460181\pi$
$84$	0	0
$85$	1.52855e6	0.269969
$86$	−1.55609e6	−0.263810
$87$	0	0
$88$	−1.67732e7	−2.62378
$89$	−416509.	−0.0626266	−0.0313133	−	0.999510i	$-0.509969\pi$
$89$	−416509.	−0.0626266	−0.0313133	+	0.999510i	$0.509969\pi$
$90$	0	0
$91$	−763048.	−0.106147
$92$	−2.82685e7	−3.78482
$93$	0	0
$94$	1.12499e7	1.39701
$95$	5.27773e6	0.631560
$96$	0	0
$97$	−1.63057e7	−1.81401	−0.907004	−	0.421121i	$-0.861637\pi$
$97$	−1.63057e7	−1.81401	−0.907004	+	0.421121i	$0.861637\pi$
$98$	−1.76394e7	−1.89319
$99$	0	0
$100$	5.27276e6	0.527276
$101$	−9.40384e6	−0.908198	−0.454099	−	0.890951i	$-0.650039\pi$
$101$	−9.40384e6	−0.908198	−0.454099	+	0.890951i	$0.650039\pi$
$102$	0	0
$103$	8.74294e6	0.788365	0.394182	−	0.919032i	$-0.371028\pi$
$103$	8.74294e6	0.788365	0.394182	+	0.919032i	$0.371028\pi$
$104$	−4.47582e7	−3.90172
$105$	0	0
$106$	3.72363e7	3.03666
$107$	−6.08896e6	−0.480507	−0.240254	−	0.970710i	$-0.577231\pi$
$107$	−6.08896e6	−0.480507	−0.240254	+	0.970710i	$0.577231\pi$
$108$	0	0
$109$	7.84384e6	0.580144	0.290072	−	0.957005i	$-0.406321\pi$
$109$	7.84384e6	0.580144	0.290072	+	0.957005i	$0.406321\pi$
$110$	1.00100e7	0.717064
$111$	0	0
$112$	4.18313e6	0.281345
$113$	1.22394e7	0.797968	0.398984	−	0.916958i	$-0.369363\pi$
$113$	1.22394e7	0.797968	0.398984	+	0.916958i	$0.369363\pi$
$114$	0	0
$115$	1.04712e7	0.642025
$116$	−2.62966e7	−1.56422
$117$	0	0
$118$	−1.43929e7	−0.806420
$119$	−942070.	−0.0512471
$120$	0	0
$121$	−5.70982e6	−0.293004
$122$	9.58745e6	0.478017
$123$	0	0
$124$	8.62745e7	4.06356
$125$	−1.95312e6	−0.0894427
$126$	0	0
$127$	−3.21340e7	−1.39204	−0.696021	−	0.718021i	$-0.745048\pi$
$127$	−3.21340e7	−1.39204	−0.696021	+	0.718021i	$0.745048\pi$
$128$	5.01788e7	2.11488
$129$	0	0
$130$	2.67109e7	1.06632
$131$	1.78530e6	0.0693844	0.0346922	−	0.999398i	$-0.488955\pi$
$131$	1.78530e6	0.0693844	0.0346922	+	0.999398i	$0.488955\pi$
$132$	0	0
$133$	−3.25275e6	−0.119886
$134$	6.46824e7	2.32231
$135$	0	0
$136$	−5.52592e7	−1.88373
$137$	−1.68069e7	−0.558427	−0.279214	−	0.960229i	$-0.590074\pi$
$137$	−1.68069e7	−0.558427	−0.279214	+	0.960229i	$0.590074\pi$
$138$	0	0
$139$	5.26365e7	1.66240	0.831199	−	0.555975i	$-0.187655\pi$
$139$	5.26365e7	1.66240	0.831199	+	0.555975i	$0.187655\pi$
$140$	−3.24968e6	−0.100091
$141$	0	0
$142$	−8.73957e7	−2.56142
$143$	3.67639e7	1.05135
$144$	0	0
$145$	9.74073e6	0.265341
$146$	−2.88294e7	−0.766656
$147$	0	0
$148$	−1.70448e8	−4.32192
$149$	2.77766e7	0.687903	0.343951	−	0.938987i	$-0.388234\pi$
$149$	2.77766e7	0.687903	0.343951	+	0.938987i	$0.388234\pi$
$150$	0	0
$151$	−4.67006e7	−1.10383	−0.551917	−	0.833899i	$-0.686103\pi$
$151$	−4.67006e7	−1.10383	−0.551917	+	0.833899i	$0.686103\pi$
$152$	−1.90797e8	−4.40676
$153$	0	0
$154$	−6.16929e6	−0.136117
$155$	−3.19576e7	−0.689308
$156$	0	0
$157$	5.48151e7	1.13045	0.565225	−	0.824937i	$-0.308789\pi$
$157$	5.48151e7	1.13045	0.565225	+	0.824937i	$0.308789\pi$
$158$	−1.34107e7	−0.270491
$159$	0	0
$160$	−7.41302e7	−1.43079
$161$	−6.45353e6	−0.121873
$162$	0	0
$163$	−2.57329e7	−0.465406	−0.232703	−	0.972548i	$-0.574757\pi$
$163$	−2.57329e7	−0.465406	−0.232703	+	0.972548i	$0.574757\pi$
$164$	1.22487e8	2.16838
$165$	0	0
$166$	2.80448e7	0.475855
$167$	5.07965e6	0.0843968	0.0421984	−	0.999109i	$-0.486564\pi$
$167$	5.07965e6	0.0843968	0.0421984	+	0.999109i	$0.486564\pi$
$168$	0	0
$169$	3.53534e7	0.563413
$170$	3.29777e7	0.514812
$171$	0	0
$172$	−2.43397e7	−0.364725
$173$	2.97037e7	0.436164	0.218082	−	0.975930i	$-0.430020\pi$
$173$	2.97037e7	0.436164	0.218082	+	0.975930i	$0.430020\pi$
$174$	0	0
$175$	1.20374e6	0.0169785
$176$	−2.01545e8	−2.78662
$177$	0	0
$178$	−8.98594e6	−0.119425
$179$	2.19545e7	0.286114	0.143057	−	0.989714i	$-0.454307\pi$
$179$	2.19545e7	0.286114	0.143057	+	0.989714i	$0.454307\pi$
$180$	0	0
$181$	−6.53992e7	−0.819780	−0.409890	−	0.912135i	$-0.634433\pi$
$181$	−6.53992e7	−0.819780	−0.409890	+	0.912135i	$0.634433\pi$
$182$	−1.64623e7	−0.202414
$183$	0	0
$184$	−3.78546e8	−4.47978
$185$	6.31370e7	0.733133
$186$	0	0
$187$	4.53893e7	0.507584
$188$	1.75965e8	1.93141
$189$	0	0
$190$	1.13864e8	1.20434
$191$	−1.44580e8	−1.50139	−0.750693	−	0.660652i	$-0.770280\pi$
$191$	−1.44580e8	−1.50139	−0.750693	+	0.660652i	$0.770280\pi$
$192$	0	0
$193$	1.24343e8	1.24500	0.622501	−	0.782619i	$-0.286117\pi$
$193$	1.24343e8	1.24500	0.622501	+	0.782619i	$0.286117\pi$
$194$	−3.51787e8	−3.45919
$195$	0	0
$196$	−2.75907e8	−2.61738
$197$	1.17681e7	0.109667	0.0548334	−	0.998496i	$-0.482537\pi$
$197$	1.17681e7	0.109667	0.0548334	+	0.998496i	$0.482537\pi$
$198$	0	0
$199$	−5.37697e7	−0.483673	−0.241836	−	0.970317i	$-0.577750\pi$
$199$	−5.37697e7	−0.483673	−0.241836	+	0.970317i	$0.577750\pi$
$200$	7.06081e7	0.624093
$201$	0	0
$202$	−2.02883e8	−1.73187
$203$	−6.00336e6	−0.0503684
$204$	0	0
$205$	−4.53713e7	−0.367826
$206$	1.88624e8	1.50336
$207$	0	0
$208$	−5.37808e8	−4.14387
$209$	1.56719e8	1.18743
$210$	0	0
$211$	−6.08207e7	−0.445720	−0.222860	−	0.974850i	$-0.571539\pi$
$211$	−6.08207e7	−0.445720	−0.222860	+	0.974850i	$0.571539\pi$
$212$	5.82431e8	4.19826
$213$	0	0
$214$	−1.31366e8	−0.916294
$215$	9.01584e6	0.0618688
$216$	0	0
$217$	1.96960e7	0.130848
$218$	1.69226e8	1.10629
$219$	0	0
$220$	1.56571e8	0.991361
$221$	1.21118e8	0.754807
$222$	0	0
$223$	4.85052e7	0.292901	0.146451	−	0.989218i	$-0.453215\pi$
$223$	4.85052e7	0.292901	0.146451	+	0.989218i	$0.453215\pi$
$224$	4.56876e7	0.271600
$225$	0	0
$226$	2.64058e8	1.52167
$227$	2.26620e8	1.28590	0.642952	−	0.765907i	$-0.277709\pi$
$227$	2.26620e8	1.28590	0.642952	+	0.765907i	$0.277709\pi$
$228$	0	0
$229$	−9.24623e7	−0.508792	−0.254396	−	0.967100i	$-0.581877\pi$
$229$	−9.24623e7	−0.508792	−0.254396	+	0.967100i	$0.581877\pi$
$230$	2.25909e8	1.22430
$231$	0	0
$232$	−3.52140e8	−1.85143
$233$	−7.26107e7	−0.376058	−0.188029	−	0.982163i	$-0.560210\pi$
$233$	−7.26107e7	−0.376058	−0.188029	+	0.982163i	$0.560210\pi$
$234$	0	0
$235$	−6.51807e7	−0.327628
$236$	−2.25126e8	−1.11490
$237$	0	0
$238$	−2.03246e7	−0.0977245
$239$	3.12153e8	1.47902	0.739512	−	0.673143i	$-0.235056\pi$
$239$	3.12153e8	1.47902	0.739512	+	0.673143i	$0.235056\pi$
$240$	0	0
$241$	3.15792e8	1.45326	0.726628	−	0.687032i	$-0.241087\pi$
$241$	3.15792e8	1.45326	0.726628	+	0.687032i	$0.241087\pi$
$242$	−1.23186e8	−0.558738
$243$	0	0
$244$	1.49962e8	0.660872
$245$	1.02201e8	0.443991
$246$	0	0
$247$	4.18192e8	1.76578
$248$	1.15531e9	4.80970
$249$	0	0
$250$	−4.21376e7	−0.170561
$251$	−2.13552e8	−0.852403	−0.426202	−	0.904628i	$-0.640149\pi$
$251$	−2.13552e8	−0.852403	−0.426202	+	0.904628i	$0.640149\pi$
$252$	0	0
$253$	3.10933e8	1.20711
$254$	−6.93274e8	−2.65453
$255$	0	0
$256$	3.34505e8	1.24613
$257$	−3.20976e8	−1.17952	−0.589762	−	0.807577i	$-0.700779\pi$
$257$	−3.20976e8	−1.17952	−0.589762	+	0.807577i	$0.700779\pi$
$258$	0	0
$259$	−3.89123e7	−0.139167
$260$	4.17799e8	1.47421
$261$	0	0
$262$	3.85169e7	0.132311
$263$	2.06317e8	0.699344	0.349672	−	0.936872i	$-0.386293\pi$
$263$	2.06317e8	0.699344	0.349672	+	0.936872i	$0.386293\pi$
$264$	0	0
$265$	−2.15743e8	−0.712157
$266$	−7.01762e7	−0.228615
$267$	0	0
$268$	1.01173e9	3.21065
$269$	9.36790e7	0.293433	0.146717	−	0.989179i	$-0.453129\pi$
$269$	9.36790e7	0.293433	0.146717	+	0.989179i	$0.453129\pi$
$270$	0	0
$271$	6.73891e7	0.205682	0.102841	−	0.994698i	$-0.467207\pi$
$271$	6.73891e7	0.205682	0.102841	+	0.994698i	$0.467207\pi$
$272$	−6.63987e8	−2.00064
$273$	0	0
$274$	−3.62601e8	−1.06488
$275$	−5.79966e7	−0.168166
$276$	0	0
$277$	−4.66769e8	−1.31954	−0.659770	−	0.751467i	$-0.729346\pi$
$277$	−4.66769e8	−1.31954	−0.659770	+	0.751467i	$0.729346\pi$
$278$	1.13560e9	3.17008
$279$	0	0
$280$	−4.35168e7	−0.118469
$281$	4.16221e8	1.11906	0.559528	−	0.828811i	$-0.310982\pi$
$281$	4.16221e8	1.11906	0.559528	+	0.828811i	$0.310982\pi$
$282$	0	0
$283$	−1.63023e8	−0.427560	−0.213780	−	0.976882i	$-0.568578\pi$
$283$	−1.63023e8	−0.427560	−0.213780	+	0.976882i	$0.568578\pi$
$284$	−1.36700e9	−3.54123
$285$	0	0
$286$	7.93161e8	2.00484
$287$	2.79630e7	0.0698229
$288$	0	0
$289$	−2.60804e8	−0.635583
$290$	2.10151e8	0.505986
$291$	0	0
$292$	−4.50935e8	−1.05992
$293$	4.09573e8	0.951249	0.475624	−	0.879648i	$-0.342222\pi$
$293$	4.09573e8	0.951249	0.475624	+	0.879648i	$0.342222\pi$
$294$	0	0
$295$	8.33908e7	0.189122
$296$	−2.28249e9	−5.11549
$297$	0	0
$298$	5.99265e8	1.31178
$299$	8.29704e8	1.79504
$300$	0	0
$301$	−5.55660e6	−0.0117443
$302$	−1.00754e9	−2.10493
$303$	0	0
$304$	−2.29259e9	−4.68025
$305$	−5.55486e7	−0.112105
$306$	0	0
$307$	−5.65974e7	−0.111638	−0.0558191	−	0.998441i	$-0.517777\pi$
$307$	−5.65974e7	−0.111638	−0.0558191	+	0.998441i	$0.517777\pi$
$308$	−9.64970e7	−0.188186
$309$	0	0
$310$	−6.89468e8	−1.31446
$311$	−3.53547e8	−0.666477	−0.333239	−	0.942843i	$-0.608141\pi$
$311$	−3.53547e8	−0.666477	−0.333239	+	0.942843i	$0.608141\pi$
$312$	0	0
$313$	−4.61856e8	−0.851337	−0.425669	−	0.904879i	$-0.639961\pi$
$313$	−4.61856e8	−0.851337	−0.425669	+	0.904879i	$0.639961\pi$
$314$	1.18261e9	2.15569
$315$	0	0
$316$	−2.09764e8	−0.373961
$317$	1.36670e8	0.240971	0.120486	−	0.992715i	$-0.461555\pi$
$317$	1.36670e8	0.240971	0.120486	+	0.992715i	$0.461555\pi$
$318$	0	0
$319$	2.89244e8	0.498881
$320$	−7.30541e8	−1.24629
$321$	0	0
$322$	−1.39231e8	−0.232403
$323$	5.16307e8	0.852509
$324$	0	0
$325$	−1.54760e8	−0.250073
$326$	−5.55172e8	−0.887496
$327$	0	0
$328$	1.64023e9	2.56653
$329$	4.01718e7	0.0621922
$330$	0	0
$331$	4.82675e8	0.731571	0.365786	−	0.930699i	$-0.380800\pi$
$331$	4.82675e8	0.731571	0.365786	+	0.930699i	$0.380800\pi$
$332$	4.38663e8	0.657882
$333$	0	0
$334$	1.09591e8	0.160939
$335$	−3.74763e8	−0.544627
$336$	0	0
$337$	−3.59281e8	−0.511364	−0.255682	−	0.966761i	$-0.582300\pi$
$337$	−3.59281e8	−0.511364	−0.255682	+	0.966761i	$0.582300\pi$
$338$	7.62729e8	1.07439
$339$	0	0
$340$	5.15821e8	0.711742
$341$	−9.48959e8	−1.29601
$342$	0	0
$343$	−1.26433e8	−0.169173
$344$	−3.25935e8	−0.431694
$345$	0	0
$346$	6.40842e8	0.831734
$347$	−5.08730e8	−0.653633	−0.326817	−	0.945088i	$-0.605976\pi$
$347$	−5.08730e8	−0.653633	−0.326817	+	0.945088i	$0.605976\pi$
$348$	0	0
$349$	−9.75598e8	−1.22852	−0.614259	−	0.789104i	$-0.710545\pi$
$349$	−9.75598e8	−1.22852	−0.614259	+	0.789104i	$0.710545\pi$
$350$	2.59700e7	0.0323768
$351$	0	0
$352$	−2.20124e9	−2.69010
$353$	1.31639e9	1.59284	0.796420	−	0.604744i	$-0.206724\pi$
$353$	1.31639e9	1.59284	0.796420	+	0.604744i	$0.206724\pi$
$354$	0	0
$355$	5.06361e8	0.600705
$356$	−1.40554e8	−0.165108
$357$	0	0
$358$	4.73657e8	0.545599
$359$	−6.14819e8	−0.701320	−0.350660	−	0.936503i	$-0.614043\pi$
$359$	−6.14819e8	−0.701320	−0.350660	+	0.936503i	$0.614043\pi$
$360$	0	0
$361$	8.88814e8	0.994342
$362$	−1.41095e9	−1.56326
$363$	0	0
$364$	−2.57496e8	−0.279843
$365$	1.67034e8	0.179796
$366$	0	0
$367$	9.79465e8	1.03433	0.517163	−	0.855887i	$-0.326988\pi$
$367$	9.79465e8	1.03433	0.517163	+	0.855887i	$0.326988\pi$
$368$	−4.54855e9	−4.75780
$369$	0	0
$370$	1.36215e9	1.39803
$371$	1.32966e8	0.135186
$372$	0	0
$373$	−1.47376e9	−1.47044	−0.735219	−	0.677830i	$-0.762920\pi$
$373$	−1.47376e9	−1.47044	−0.735219	+	0.677830i	$0.762920\pi$
$374$	9.79249e8	0.967926
$375$	0	0
$376$	2.35637e9	2.28605
$377$	7.71828e8	0.741866
$378$	0	0
$379$	−8.48418e6	−0.00800521	−0.00400260	−	0.999992i	$-0.501274\pi$
$379$	−8.48418e6	−0.00800521	−0.00400260	+	0.999992i	$0.501274\pi$
$380$	1.78101e9	1.66503
$381$	0	0
$382$	−3.11924e9	−2.86304
$383$	1.28101e9	1.16508	0.582540	−	0.812802i	$-0.302059\pi$
$383$	1.28101e9	1.16508	0.582540	+	0.812802i	$0.302059\pi$
$384$	0	0
$385$	3.57442e7	0.0319222
$386$	2.68263e9	2.37413
$387$	0	0
$388$	−5.50248e9	−4.78242
$389$	−7.63558e8	−0.657686	−0.328843	−	0.944385i	$-0.606659\pi$
$389$	−7.63558e8	−0.657686	−0.328843	+	0.944385i	$0.606659\pi$
$390$	0	0
$391$	1.02437e9	0.866635
$392$	−3.69470e9	−3.09798
$393$	0	0
$394$	2.53891e8	0.209127
$395$	7.77002e7	0.0634355
$396$	0	0
$397$	−2.17372e9	−1.74356	−0.871781	−	0.489897i	$-0.837035\pi$
$397$	−2.17372e9	−1.74356	−0.871781	+	0.489897i	$0.837035\pi$
$398$	−1.16005e9	−0.922330
$399$	0	0
$400$	8.48416e8	0.662825
$401$	1.31239e8	0.101638	0.0508192	−	0.998708i	$-0.483817\pi$
$401$	1.31239e8	0.101638	0.0508192	+	0.998708i	$0.483817\pi$
$402$	0	0
$403$	−2.53223e9	−1.92724
$404$	−3.17339e9	−2.39436
$405$	0	0
$406$	−1.29519e8	−0.0960491
$407$	1.87481e9	1.37840
$408$	0	0
$409$	−9.88205e8	−0.714193	−0.357096	−	0.934068i	$-0.616233\pi$
$409$	−9.88205e8	−0.714193	−0.357096	+	0.934068i	$0.616233\pi$
$410$	−9.78861e8	−0.701419
$411$	0	0
$412$	2.95036e9	2.07843
$413$	−5.13951e7	−0.0359001
$414$	0	0
$415$	−1.62489e8	−0.111598
$416$	−5.87387e9	−4.00034
$417$	0	0
$418$	3.38112e9	2.26435
$419$	4.60420e8	0.305777	0.152889	−	0.988243i	$-0.451142\pi$
$419$	4.60420e8	0.305777	0.152889	+	0.988243i	$0.451142\pi$
$420$	0	0
$421$	−4.87183e7	−0.0318204	−0.0159102	−	0.999873i	$-0.505065\pi$
$421$	−4.87183e7	−0.0318204	−0.0159102	+	0.999873i	$0.505065\pi$
$422$	−1.31217e9	−0.849958
$423$	0	0
$424$	7.79939e9	4.96913
$425$	−1.91069e8	−0.120734
$426$	0	0
$427$	3.42355e7	0.0212804
$428$	−2.05476e9	−1.26680
$429$	0	0
$430$	1.94512e8	0.117979
$431$	1.83386e9	1.10331	0.551653	−	0.834074i	$-0.313997\pi$
$431$	1.83386e9	1.10331	0.551653	+	0.834074i	$0.313997\pi$
$432$	0	0
$433$	1.55904e9	0.922889	0.461444	−	0.887169i	$-0.347331\pi$
$433$	1.55904e9	0.922889	0.461444	+	0.887169i	$0.347331\pi$
$434$	4.24930e8	0.249519
$435$	0	0
$436$	2.64696e9	1.52948
$437$	3.53689e9	2.02739
$438$	0	0
$439$	1.11681e9	0.630021	0.315011	−	0.949088i	$-0.397992\pi$
$439$	1.11681e9	0.630021	0.315011	+	0.949088i	$0.397992\pi$
$440$	2.09666e9	1.17339
$441$	0	0
$442$	2.61306e9	1.43936
$443$	−3.35260e9	−1.83218	−0.916091	−	0.400970i	$-0.868673\pi$
$443$	−3.35260e9	−1.83218	−0.916091	+	0.400970i	$0.868673\pi$
$444$	0	0
$445$	5.20636e7	0.0280075
$446$	1.04647e9	0.558542
$447$	0	0
$448$	4.50243e8	0.236578
$449$	−2.10749e9	−1.09876	−0.549381	−	0.835572i	$-0.685136\pi$
$449$	−2.10749e9	−1.09876	−0.549381	+	0.835572i	$0.685136\pi$
$450$	0	0
$451$	−1.34727e9	−0.691570
$452$	4.13027e9	2.10375
$453$	0	0
$454$	4.88921e9	2.45213
$455$	9.53809e7	0.0474703
$456$	0	0
$457$	1.16279e9	0.569896	0.284948	−	0.958543i	$-0.408024\pi$
$457$	1.16279e9	0.569896	0.284948	+	0.958543i	$0.408024\pi$
$458$	−1.99482e9	−0.970231
$459$	0	0
$460$	3.53356e9	1.69262
$461$	2.14726e9	1.02078	0.510389	−	0.859943i	$-0.329501\pi$
$461$	2.14726e9	1.02078	0.510389	+	0.859943i	$0.329501\pi$
$462$	0	0
$463$	−2.88368e8	−0.135025	−0.0675125	−	0.997718i	$-0.521506\pi$
$463$	−2.88368e8	−0.135025	−0.0675125	+	0.997718i	$0.521506\pi$
$464$	−4.23127e9	−1.96634
$465$	0	0
$466$	−1.56654e9	−0.717116
$467$	2.41472e9	1.09713	0.548565	−	0.836108i	$-0.315174\pi$
$467$	2.41472e9	1.09713	0.548565	+	0.836108i	$0.315174\pi$
$468$	0	0
$469$	2.30972e8	0.103384
$470$	−1.40624e9	−0.624764
$471$	0	0
$472$	−3.01469e9	−1.31961
$473$	2.67719e8	0.116323
$474$	0	0
$475$	−6.59717e8	−0.282442
$476$	−3.17908e8	−0.135107
$477$	0	0
$478$	6.73454e9	2.82040
$479$	−3.74097e7	−0.0155529	−0.00777643	−	0.999970i	$-0.502475\pi$
$479$	−3.74097e7	−0.0155529	−0.00777643	+	0.999970i	$0.502475\pi$
$480$	0	0
$481$	5.00280e9	2.04977
$482$	6.81304e9	2.77126
$483$	0	0
$484$	−1.92682e9	−0.772470
$485$	2.03822e9	0.811249
$486$	0	0
$487$	−2.82893e9	−1.10987	−0.554933	−	0.831895i	$-0.687256\pi$
$487$	−2.82893e9	−1.10987	−0.554933	+	0.831895i	$0.687256\pi$
$488$	2.00816e9	0.782219
$489$	0	0
$490$	2.20493e9	0.846659
$491$	−1.37275e9	−0.523366	−0.261683	−	0.965154i	$-0.584277\pi$
$491$	−1.37275e9	−0.523366	−0.261683	+	0.965154i	$0.584277\pi$
$492$	0	0
$493$	9.52910e8	0.358169
$494$	9.02227e9	3.36722
$495$	0	0
$496$	1.38820e10	5.10819
$497$	−3.12078e8	−0.114029
$498$	0	0
$499$	1.08803e9	0.392004	0.196002	−	0.980604i	$-0.437204\pi$
$499$	1.08803e9	0.392004	0.196002	+	0.980604i	$0.437204\pi$
$500$	−6.59095e8	−0.235805
$501$	0	0
$502$	−4.60726e9	−1.62547
$503$	3.89442e9	1.36444	0.682220	−	0.731147i	$-0.261015\pi$
$503$	3.89442e9	1.36444	0.682220	+	0.731147i	$0.261015\pi$
$504$	0	0
$505$	1.17548e9	0.406158
$506$	6.70821e9	2.30187
$507$	0	0
$508$	−1.08439e10	−3.66995
$509$	3.40268e9	1.14369	0.571845	−	0.820362i	$-0.306228\pi$
$509$	3.40268e9	1.14369	0.571845	+	0.820362i	$0.306228\pi$
$510$	0	0
$511$	−1.02946e8	−0.0341300
$512$	7.93885e8	0.261404
$513$	0	0
$514$	−6.92489e9	−2.24927
$515$	−1.09287e9	−0.352567
$516$	0	0
$517$	−1.93549e9	−0.615992
$518$	−8.39512e8	−0.265383
$519$	0	0
$520$	5.59478e9	1.74490
$521$	4.16130e9	1.28913	0.644566	−	0.764549i	$-0.277038\pi$
$521$	4.16130e9	1.28913	0.644566	+	0.764549i	$0.277038\pi$
$522$	0	0
$523$	−1.03759e9	−0.317153	−0.158576	−	0.987347i	$-0.550690\pi$
$523$	−1.03759e9	−0.317153	−0.158576	+	0.987347i	$0.550690\pi$
$524$	6.02462e8	0.182924
$525$	0	0
$526$	4.45118e9	1.33360
$527$	−3.12633e9	−0.930460
$528$	0	0
$529$	3.61245e9	1.06098
$530$	−4.65453e9	−1.35803
$531$	0	0
$532$	−1.09766e9	−0.316066
$533$	−3.59509e9	−1.02841
$534$	0	0
$535$	7.61120e8	0.214889
$536$	1.35482e10	3.80018
$537$	0	0
$538$	2.02107e9	0.559557
$539$	3.03479e9	0.834771
$540$	0	0
$541$	−6.63214e9	−1.80079	−0.900395	−	0.435073i	$-0.856722\pi$
$541$	−6.63214e9	−1.80079	−0.900395	+	0.435073i	$0.856722\pi$
$542$	1.45388e9	0.392222
$543$	0	0
$544$	−7.25197e9	−1.93134
$545$	−9.80480e8	−0.259448
$546$	0	0
$547$	−4.40357e9	−1.15040	−0.575201	−	0.818012i	$-0.695076\pi$
$547$	−4.40357e9	−1.15040	−0.575201	+	0.818012i	$0.695076\pi$
$548$	−5.67162e9	−1.47223
$549$	0	0
$550$	−1.25125e9	−0.320681
$551$	3.29018e9	0.837893
$552$	0	0
$553$	−4.78878e7	−0.0120417
$554$	−1.00703e10	−2.51627
$555$	0	0
$556$	1.77625e10	4.38271
$557$	3.69019e9	0.904805	0.452403	−	0.891814i	$-0.350567\pi$
$557$	3.69019e9	0.904805	0.452403	+	0.891814i	$0.350567\pi$
$558$	0	0
$559$	7.14389e8	0.172979
$560$	−5.22892e8	−0.125821
$561$	0	0
$562$	8.97974e9	2.13396
$563$	7.42508e9	1.75357	0.876783	−	0.480887i	$-0.159685\pi$
$563$	7.42508e9	1.75357	0.876783	+	0.480887i	$0.159685\pi$
$564$	0	0
$565$	−1.52992e9	−0.356862
$566$	−3.51714e9	−0.815328
$567$	0	0
$568$	−1.83056e10	−4.19146
$569$	2.37122e7	0.00539609	0.00269804	−	0.999996i	$-0.499141\pi$
$569$	2.37122e7	0.00539609	0.00269804	+	0.999996i	$0.499141\pi$
$570$	0	0
$571$	−7.30904e9	−1.64299	−0.821494	−	0.570218i	$-0.806859\pi$
$571$	−7.30904e9	−1.64299	−0.821494	+	0.570218i	$0.806859\pi$
$572$	1.24062e10	2.77175
$573$	0	0
$574$	6.03287e8	0.133147
$575$	−1.30889e9	−0.287122
$576$	0	0
$577$	1.06324e8	0.0230418	0.0115209	−	0.999934i	$-0.496333\pi$
$577$	1.06324e8	0.0230418	0.0115209	+	0.999934i	$0.496333\pi$
$578$	−5.62671e9	−1.21201
$579$	0	0
$580$	3.28708e9	0.699539
$581$	1.00144e8	0.0211841
$582$	0	0
$583$	−6.40633e9	−1.33897
$584$	−6.03852e9	−1.25454
$585$	0	0
$586$	8.83630e9	1.81396
$587$	−1.99776e9	−0.407671	−0.203835	−	0.979005i	$-0.565341\pi$
$587$	−1.99776e9	−0.407671	−0.203835	+	0.979005i	$0.565341\pi$
$588$	0	0
$589$	−1.07945e10	−2.17670
$590$	1.79911e9	0.360642
$591$	0	0
$592$	−2.74260e10	−5.43297
$593$	−4.01518e9	−0.790703	−0.395352	−	0.918530i	$-0.629377\pi$
$593$	−4.01518e9	−0.790703	−0.395352	+	0.918530i	$0.629377\pi$
$594$	0	0
$595$	1.17759e8	0.0229184
$596$	9.37340e9	1.81357
$597$	0	0
$598$	1.79004e10	3.42301
$599$	7.47026e9	1.42018	0.710088	−	0.704113i	$-0.248655\pi$
$599$	7.47026e9	1.42018	0.710088	+	0.704113i	$0.248655\pi$
$600$	0	0
$601$	−3.72264e9	−0.699504	−0.349752	−	0.936842i	$-0.613734\pi$
$601$	−3.72264e9	−0.699504	−0.349752	+	0.936842i	$0.613734\pi$
$602$	−1.19881e8	−0.0223955
$603$	0	0
$604$	−1.57594e10	−2.91012
$605$	7.13728e8	0.131035
$606$	0	0
$607$	2.65818e6	0.000482419 0	0.000241209	−	1.00000i	$-0.499923\pi$
$607$	2.65818e6	0.000482419 0	0.000241209		1.00000i	$0.499923\pi$
$608$	−2.50393e10	−4.51815
$609$	0	0
$610$	−1.19843e9	−0.213776
$611$	−5.16473e9	−0.916016
$612$	0	0
$613$	8.86304e9	1.55407	0.777036	−	0.629456i	$-0.216722\pi$
$613$	8.86304e9	1.55407	0.777036	+	0.629456i	$0.216722\pi$
$614$	−1.22106e9	−0.212886
$615$	0	0
$616$	−1.29220e9	−0.222740
$617$	−5.11855e9	−0.877301	−0.438650	−	0.898658i	$-0.644543\pi$
$617$	−5.11855e9	−0.877301	−0.438650	+	0.898658i	$0.644543\pi$
$618$	0	0
$619$	−8.12696e9	−1.37724	−0.688622	−	0.725120i	$-0.741784\pi$
$619$	−8.12696e9	−1.37724	−0.688622	+	0.725120i	$0.741784\pi$
$620$	−1.07843e10	−1.81728
$621$	0	0
$622$	−7.62757e9	−1.27093
$623$	−3.20876e7	−0.00531654
$624$	0	0
$625$	2.44141e8	0.0400000
$626$	−9.96429e9	−1.62344
$627$	0	0
$628$	1.84977e10	2.98030
$629$	6.17653e9	0.989617
$630$	0	0
$631$	−5.79171e9	−0.917708	−0.458854	−	0.888512i	$-0.651740\pi$
$631$	−5.79171e9	−0.917708	−0.458854	+	0.888512i	$0.651740\pi$
$632$	−2.80897e9	−0.442626
$633$	0	0
$634$	2.94858e9	0.459516
$635$	4.01675e9	0.622540
$636$	0	0
$637$	8.09811e9	1.24135
$638$	6.24028e9	0.951331
$639$	0	0
$640$	−6.27234e9	−0.945802
$641$	8.78823e9	1.31795	0.658974	−	0.752165i	$-0.270991\pi$
$641$	8.78823e9	1.31795	0.658974	+	0.752165i	$0.270991\pi$
$642$	0	0
$643$	−1.03553e10	−1.53612	−0.768058	−	0.640381i	$-0.778777\pi$
$643$	−1.03553e10	−1.53612	−0.768058	+	0.640381i	$0.778777\pi$
$644$	−2.17779e9	−0.321303
$645$	0	0
$646$	1.11390e10	1.62567
$647$	−5.32602e9	−0.773104	−0.386552	−	0.922268i	$-0.626334\pi$
$647$	−5.32602e9	−0.773104	−0.386552	+	0.922268i	$0.626334\pi$
$648$	0	0
$649$	2.47623e9	0.355578
$650$	−3.33886e9	−0.476872
$651$	0	0
$652$	−8.68373e9	−1.22699
$653$	−1.11639e10	−1.56898	−0.784491	−	0.620140i	$-0.787076\pi$
$653$	−1.11639e10	−1.56898	−0.784491	+	0.620140i	$0.787076\pi$
$654$	0	0
$655$	−2.23163e8	−0.0310297
$656$	1.97088e10	2.72582
$657$	0	0
$658$	8.66685e8	0.118596
$659$	1.17456e10	1.59874	0.799370	−	0.600840i	$-0.205167\pi$
$659$	1.17456e10	1.59874	0.799370	+	0.600840i	$0.205167\pi$
$660$	0	0
$661$	1.98137e9	0.266846	0.133423	−	0.991059i	$-0.457403\pi$
$661$	1.98137e9	0.266846	0.133423	+	0.991059i	$0.457403\pi$
$662$	1.04134e10	1.39506
$663$	0	0
$664$	5.87418e9	0.778680
$665$	4.06593e8	0.0536148
$666$	0	0
$667$	6.52779e9	0.851777
$668$	1.71416e9	0.222502
$669$	0	0
$670$	−8.08530e9	−1.03857
$671$	−1.64948e9	−0.210774
$672$	0	0
$673$	1.11459e9	0.140949	0.0704747	−	0.997514i	$-0.477549\pi$
$673$	1.11459e9	0.140949	0.0704747	+	0.997514i	$0.477549\pi$
$674$	−7.75129e9	−0.975135
$675$	0	0
$676$	1.19302e10	1.48537
$677$	1.70825e9	0.211589	0.105794	−	0.994388i	$-0.466261\pi$
$677$	1.70825e9	0.211589	0.105794	+	0.994388i	$0.466261\pi$
$678$	0	0
$679$	−1.25618e9	−0.153996
$680$	6.90740e9	0.842429
$681$	0	0
$682$	−2.04733e10	−2.47139
$683$	3.51300e9	0.421897	0.210948	−	0.977497i	$-0.432345\pi$
$683$	3.51300e9	0.421897	0.210948	+	0.977497i	$0.432345\pi$
$684$	0	0
$685$	2.10087e9	0.249736
$686$	−2.72773e9	−0.322602
$687$	0	0
$688$	−3.91638e9	−0.458486
$689$	−1.70948e10	−1.99112
$690$	0	0
$691$	1.50845e10	1.73924	0.869619	−	0.493724i	$-0.164365\pi$
$691$	1.50845e10	1.73924	0.869619	+	0.493724i	$0.164365\pi$
$692$	1.00237e10	1.14989
$693$	0	0
$694$	−1.09756e10	−1.24643
$695$	−6.57956e9	−0.743447
$696$	0	0
$697$	−4.43856e9	−0.496509
$698$	−2.10480e10	−2.34270
$699$	0	0
$700$	4.06211e8	0.0447619
$701$	2.94496e9	0.322898	0.161449	−	0.986881i	$-0.448383\pi$
$701$	2.94496e9	0.322898	0.161449	+	0.986881i	$0.448383\pi$
$702$	0	0
$703$	2.13261e10	2.31509
$704$	−2.16929e10	−2.34322
$705$	0	0
$706$	2.84003e10	3.03743
$707$	−7.24466e8	−0.0770993
$708$	0	0
$709$	−2.86510e9	−0.301911	−0.150955	−	0.988541i	$-0.548235\pi$
$709$	−2.86510e9	−0.301911	−0.150955	+	0.988541i	$0.548235\pi$
$710$	1.09245e10	1.14550
$711$	0	0
$712$	−1.88217e9	−0.195424
$713$	−2.14165e10	−2.21277
$714$	0	0
$715$	−4.59549e9	−0.470176
$716$	7.40871e9	0.754305
$717$	0	0
$718$	−1.32644e10	−1.33737
$719$	−8.01601e9	−0.804280	−0.402140	−	0.915578i	$-0.631733\pi$
$719$	−8.01601e9	−0.804280	−0.402140	+	0.915578i	$0.631733\pi$
$720$	0	0
$721$	6.73551e8	0.0669263
$722$	1.91757e10	1.89614
$723$	0	0
$724$	−2.20694e10	−2.16125
$725$	−1.21759e9	−0.118664
$726$	0	0
$727$	−1.13939e9	−0.109977	−0.0549884	−	0.998487i	$-0.517512\pi$
$727$	−1.13939e9	−0.109977	−0.0549884	+	0.998487i	$0.517512\pi$
$728$	−3.44815e9	−0.331227
$729$	0	0
$730$	3.60368e9	0.342859
$731$	8.81996e8	0.0835134
$732$	0	0
$733$	3.62791e9	0.340246	0.170123	−	0.985423i	$-0.445583\pi$
$733$	3.62791e9	0.340246	0.170123	+	0.985423i	$0.445583\pi$
$734$	2.11314e10	1.97239
$735$	0	0
$736$	−4.96786e10	−4.59301
$737$	−1.11283e10	−1.02398
$738$	0	0
$739$	−2.09850e9	−0.191273	−0.0956364	−	0.995416i	$-0.530489\pi$
$739$	−2.09850e9	−0.191273	−0.0956364	+	0.995416i	$0.530489\pi$
$740$	2.13060e10	1.93282
$741$	0	0
$742$	2.86866e9	0.257790
$743$	1.34432e10	1.20238	0.601189	−	0.799107i	$-0.294694\pi$
$743$	1.34432e10	1.20238	0.601189	+	0.799107i	$0.294694\pi$
$744$	0	0
$745$	−3.47207e9	−0.307639
$746$	−3.17956e10	−2.80402
$747$	0	0
$748$	1.53169e10	1.33818
$749$	−4.69090e8	−0.0407915
$750$	0	0
$751$	7.36560e9	0.634554	0.317277	−	0.948333i	$-0.397232\pi$
$751$	7.36560e9	0.634554	0.317277	+	0.948333i	$0.397232\pi$
$752$	2.83138e10	2.42792
$753$	0	0
$754$	1.66518e10	1.41469
$755$	5.83758e9	0.493649
$756$	0	0
$757$	−2.10411e10	−1.76292	−0.881459	−	0.472261i	$-0.843438\pi$
$757$	−2.10411e10	−1.76292	−0.881459	+	0.472261i	$0.843438\pi$
$758$	−1.83041e8	−0.0152654
$759$	0	0
$760$	2.38496e10	1.97076
$761$	−1.25985e10	−1.03627	−0.518135	−	0.855299i	$-0.673374\pi$
$761$	−1.25985e10	−1.03627	−0.518135	+	0.855299i	$0.673374\pi$
$762$	0	0
$763$	6.04285e8	0.0492499
$764$	−4.87896e10	−3.95822
$765$	0	0
$766$	2.76370e10	2.22172
$767$	6.60765e9	0.528766
$768$	0	0
$769$	−4.38316e9	−0.347573	−0.173786	−	0.984783i	$-0.555600\pi$
$769$	−4.38316e9	−0.347573	−0.173786	+	0.984783i	$0.555600\pi$
$770$	7.71162e8	0.0608735
$771$	0	0
$772$	4.19603e10	3.28230
$773$	−3.99464e9	−0.311064	−0.155532	−	0.987831i	$-0.549709\pi$
$773$	−3.99464e9	−0.311064	−0.155532	+	0.987831i	$0.549709\pi$
$774$	0	0
$775$	3.99470e9	0.308268
$776$	−7.36843e10	−5.66055
$777$	0	0
$778$	−1.64734e10	−1.25416
$779$	−1.53253e10	−1.16152
$780$	0	0
$781$	1.50360e10	1.12942
$782$	2.21001e10	1.65261
$783$	0	0
$784$	−4.43950e10	−3.29024
$785$	−6.85189e9	−0.505553
$786$	0	0
$787$	5.11304e9	0.373910	0.186955	−	0.982368i	$-0.440138\pi$
$787$	5.11304e9	0.373910	0.186955	+	0.982368i	$0.440138\pi$
$788$	3.97123e9	0.289123
$789$	0	0
$790$	1.67634e9	0.120967
$791$	9.42916e8	0.0677416
$792$	0	0
$793$	−4.40151e9	−0.313434
$794$	−4.68968e10	−3.32485
$795$	0	0
$796$	−1.81449e10	−1.27515
$797$	−2.45500e10	−1.71770	−0.858850	−	0.512227i	$-0.828821\pi$
$797$	−2.45500e10	−1.71770	−0.858850	+	0.512227i	$0.828821\pi$
$798$	0	0
$799$	−6.37645e9	−0.442248
$800$	9.26628e9	0.639868
$801$	0	0
$802$	2.83141e9	0.193817
$803$	4.95997e9	0.338045
$804$	0	0
$805$	8.06691e8	0.0545032
$806$	−5.46315e10	−3.67511
$807$	0	0
$808$	−4.24952e10	−2.83400
$809$	1.35721e10	0.901215	0.450607	−	0.892722i	$-0.351207\pi$
$809$	1.35721e10	0.901215	0.450607	+	0.892722i	$0.351207\pi$
$810$	0	0
$811$	−9.12272e9	−0.600553	−0.300276	−	0.953852i	$-0.597079\pi$
$811$	−9.12272e9	−0.600553	−0.300276	+	0.953852i	$0.597079\pi$
$812$	−2.02588e9	−0.132790
$813$	0	0
$814$	4.04480e10	2.62852
$815$	3.21661e9	0.208136
$816$	0	0
$817$	3.04533e9	0.195369
$818$	−2.13200e10	−1.36192
$819$	0	0
$820$	−1.53109e10	−0.969731
$821$	−4.23260e9	−0.266935	−0.133468	−	0.991053i	$-0.542611\pi$
$821$	−4.23260e9	−0.266935	−0.133468	+	0.991053i	$0.542611\pi$
$822$	0	0
$823$	−4.77048e9	−0.298306	−0.149153	−	0.988814i	$-0.547655\pi$
$823$	−4.77048e9	−0.298306	−0.149153	+	0.988814i	$0.547655\pi$
$824$	3.95086e10	2.46006
$825$	0	0
$826$	−1.10882e9	−0.0684590
$827$	−1.45501e10	−0.894532	−0.447266	−	0.894401i	$-0.647602\pi$
$827$	−1.45501e10	−0.894532	−0.447266	+	0.894401i	$0.647602\pi$
$828$	0	0
$829$	−2.11062e10	−1.28668	−0.643339	−	0.765582i	$-0.722451\pi$
$829$	−2.11062e10	−1.28668	−0.643339	+	0.765582i	$0.722451\pi$
$830$	−3.50560e9	−0.212809
$831$	0	0
$832$	−5.78859e10	−3.48451
$833$	9.99805e9	0.599319
$834$	0	0
$835$	−6.34956e8	−0.0377434
$836$	5.28857e10	3.13052
$837$	0	0
$838$	9.93331e9	0.583096
$839$	−4.46737e9	−0.261147	−0.130574	−	0.991439i	$-0.541682\pi$
$839$	−4.46737e9	−0.261147	−0.130574	+	0.991439i	$0.541682\pi$
$840$	0	0
$841$	−1.11774e10	−0.647972
$842$	−1.05107e9	−0.0606792
$843$	0	0
$844$	−2.05243e10	−1.17509
$845$	−4.41917e9	−0.251966
$846$	0	0
$847$	−4.39881e8	−0.0248739
$848$	9.37163e10	5.27752
$849$	0	0
$850$	−4.12221e9	−0.230231
$851$	4.23115e10	2.35345
$852$	0	0
$853$	−1.36806e10	−0.754717	−0.377359	−	0.926067i	$-0.623168\pi$
$853$	−1.36806e10	−0.754717	−0.377359	+	0.926067i	$0.623168\pi$
$854$	7.38611e8	0.0405801
$855$	0	0
$856$	−2.75155e10	−1.49941
$857$	1.74184e10	0.945310	0.472655	−	0.881248i	$-0.343296\pi$
$857$	1.74184e10	0.945310	0.472655	+	0.881248i	$0.343296\pi$
$858$	0	0
$859$	1.66986e9	0.0898887	0.0449443	−	0.998989i	$-0.485689\pi$
$859$	1.66986e9	0.0898887	0.0449443	+	0.998989i	$0.485689\pi$
$860$	3.04246e9	0.163110
$861$	0	0
$862$	3.95646e10	2.10393
$863$	−1.39002e9	−0.0736180	−0.0368090	−	0.999322i	$-0.511719\pi$
$863$	−1.39002e9	−0.0736180	−0.0368090	+	0.999322i	$0.511719\pi$
$864$	0	0
$865$	−3.71297e9	−0.195058
$866$	3.36354e10	1.75988
$867$	0	0
$868$	6.64654e9	0.344966
$869$	2.30725e9	0.119269
$870$	0	0
$871$	−2.96951e10	−1.52273
$872$	3.54456e10	1.81032
$873$	0	0
$874$	7.63065e10	3.86609
$875$	−1.50468e8	−0.00759302
$876$	0	0
$877$	−3.06068e10	−1.53221	−0.766106	−	0.642714i	$-0.777808\pi$
$877$	−3.06068e10	−1.53221	−0.766106	+	0.642714i	$0.777808\pi$
$878$	2.40947e10	1.20141
$879$	0	0
$880$	2.51931e10	1.24621
$881$	−1.39630e10	−0.687960	−0.343980	−	0.938977i	$-0.611775\pi$
$881$	−1.39630e10	−0.687960	−0.343980	+	0.938977i	$0.611775\pi$
$882$	0	0
$883$	−2.45301e10	−1.19905	−0.599525	−	0.800356i	$-0.704644\pi$
$883$	−2.45301e10	−1.19905	−0.599525	+	0.800356i	$0.704644\pi$
$884$	4.08721e10	1.98996
$885$	0	0
$886$	−7.23305e10	−3.49384
$887$	−3.21493e10	−1.54681	−0.773407	−	0.633909i	$-0.781449\pi$
$887$	−3.21493e10	−1.54681	−0.773407	+	0.633909i	$0.781449\pi$
$888$	0	0
$889$	−2.47559e9	−0.118174
$890$	1.12324e9	0.0534083
$891$	0	0
$892$	1.63684e10	0.772200
$893$	−2.20164e10	−1.03458
$894$	0	0
$895$	−2.74432e9	−0.127954
$896$	3.86574e9	0.179537
$897$	0	0
$898$	−4.54679e10	−2.09526
$899$	−1.99226e10	−0.914507
$900$	0	0
$901$	−2.11056e10	−0.961302
$902$	−2.90666e10	−1.31878
$903$	0	0
$904$	5.53088e10	2.49003
$905$	8.17490e9	0.366617
$906$	0	0
$907$	−2.35375e10	−1.04745	−0.523726	−	0.851887i	$-0.675459\pi$
$907$	−2.35375e10	−1.04745	−0.523726	+	0.851887i	$0.675459\pi$
$908$	7.64746e10	3.39013
$909$	0	0
$910$	2.05779e9	0.0905225
$911$	−4.13487e10	−1.81196	−0.905978	−	0.423324i	$-0.860863\pi$
$911$	−4.13487e10	−1.81196	−0.905978	+	0.423324i	$0.860863\pi$
$912$	0	0
$913$	−4.82498e9	−0.209821
$914$	2.50866e10	1.08675
$915$	0	0
$916$	−3.12020e10	−1.34137
$917$	1.37539e8	0.00589023
$918$	0	0
$919$	2.43059e9	0.103302	0.0516509	−	0.998665i	$-0.483552\pi$
$919$	2.43059e9	0.103302	0.0516509	+	0.998665i	$0.483552\pi$
$920$	4.73182e10	2.00342
$921$	0	0
$922$	4.63260e10	1.94655
$923$	4.01226e10	1.67951
$924$	0	0
$925$	−7.89213e9	−0.327867
$926$	−6.22138e9	−0.257483
$927$	0	0
$928$	−4.62133e10	−1.89823
$929$	1.55843e10	0.637721	0.318861	−	0.947802i	$-0.396700\pi$
$929$	1.55843e10	0.637721	0.318861	+	0.947802i	$0.396700\pi$
$930$	0	0
$931$	3.45209e10	1.40203
$932$	−2.45030e10	−0.991432
$933$	0	0
$934$	5.20963e10	2.09215
$935$	−5.67366e9	−0.226998
$936$	0	0
$937$	6.21376e9	0.246755	0.123377	−	0.992360i	$-0.460627\pi$
$937$	6.21376e9	0.246755	0.123377	+	0.992360i	$0.460627\pi$
$938$	4.98309e9	0.197146
$939$	0	0
$940$	−2.19957e10	−0.863753
$941$	−2.50671e10	−0.980711	−0.490356	−	0.871522i	$-0.663133\pi$
$941$	−2.50671e10	−0.980711	−0.490356	+	0.871522i	$0.663133\pi$
$942$	0	0
$943$	−3.04058e10	−1.18077
$944$	−3.62241e10	−1.40151
$945$	0	0
$946$	5.77589e9	0.221820
$947$	−2.08614e10	−0.798211	−0.399105	−	0.916905i	$-0.630679\pi$
$947$	−2.08614e10	−0.798211	−0.399105	+	0.916905i	$0.630679\pi$
$948$	0	0
$949$	1.32353e10	0.502693
$950$	−1.42330e10	−0.538598
$951$	0	0
$952$	−4.25714e9	−0.159915
$953$	3.28254e10	1.22853	0.614263	−	0.789101i	$-0.289453\pi$
$953$	3.28254e10	1.22853	0.614263	+	0.789101i	$0.289453\pi$
$954$	0	0
$955$	1.80725e10	0.671440
$956$	1.05338e11	3.89927
$957$	0	0
$958$	−8.07094e8	−0.0296582
$959$	−1.29480e9	−0.0474064
$960$	0	0
$961$	3.78499e10	1.37573
$962$	1.07933e11	3.90877
$963$	0	0
$964$	1.06566e11	3.83133
$965$	−1.55428e10	−0.556781
$966$	0	0
$967$	3.06244e10	1.08912	0.544560	−	0.838722i	$-0.316697\pi$
$967$	3.06244e10	1.08912	0.544560	+	0.838722i	$0.316697\pi$
$968$	−2.58022e10	−0.914309
$969$	0	0
$970$	4.39734e10	1.54700
$971$	−5.35879e10	−1.87845	−0.939224	−	0.343305i	$-0.888454\pi$
$971$	−5.35879e10	−1.87845	−0.939224	+	0.343305i	$0.888454\pi$
$972$	0	0
$973$	4.05508e9	0.141125
$974$	−6.10326e10	−2.11644
$975$	0	0
$976$	2.41297e10	0.830764
$977$	−3.26601e9	−0.112044	−0.0560218	−	0.998430i	$-0.517842\pi$
$977$	−3.26601e9	−0.112044	−0.0560218	+	0.998430i	$0.517842\pi$
$978$	0	0
$979$	1.54599e9	0.0526584
$980$	3.44884e10	1.17053
$981$	0	0
$982$	−2.96163e10	−0.998022
$983$	9.45925e9	0.317629	0.158814	−	0.987308i	$-0.449233\pi$
$983$	9.45925e9	0.317629	0.158814	+	0.987308i	$0.449233\pi$
$984$	0	0
$985$	−1.47101e9	−0.0490445
$986$	2.05585e10	0.683003
$987$	0	0
$988$	1.41122e11	4.65527
$989$	6.04200e9	0.198607
$990$	0	0
$991$	−4.44721e10	−1.45154	−0.725771	−	0.687937i	$-0.758517\pi$
$991$	−4.44721e10	−1.45154	−0.725771	+	0.687937i	$0.758517\pi$
$992$	1.51618e11	4.93127
$993$	0	0
$994$	−6.73291e9	−0.217446
$995$	6.72121e9	0.216305
$996$	0	0
$997$	−8.65709e9	−0.276655	−0.138328	−	0.990387i	$-0.544173\pi$
$997$	−8.65709e9	−0.276655	−0.138328	+	0.990387i	$0.544173\pi$
$998$	2.34737e10	0.747524
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.8.a.f.1.13		13
3.2	odd	2		405.8.a.e.1.1		13
9.2	odd	6		45.8.e.a.31.13	yes	26
9.4	even	3		135.8.e.a.46.1		26
9.5	odd	6		45.8.e.a.16.13	✓	26
9.7	even	3		135.8.e.a.91.1		26

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.8.e.a.16.13	✓	26	9.5	odd	6
45.8.e.a.31.13	yes	26	9.2	odd	6
135.8.e.a.46.1		26	9.4	even	3
135.8.e.a.91.1		26	9.7	even	3
405.8.a.e.1.1		13	3.2	odd	2
405.8.a.f.1.13		13	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+21.5744 q^{2} +337.457 q^{4} -125.000 q^{5} +77.0394 q^{7} +4518.92 q^{8} +O(q^{10})\)	\(q+21.5744 q^{2} +337.457 q^{4} -125.000 q^{5} +77.0394 q^{7} +4518.92 q^{8} -2696.81 q^{10} -3711.79 q^{11} -9904.64 q^{13} +1662.08 q^{14} +54298.6 q^{16} -12228.4 q^{17} -42221.9 q^{19} -42182.1 q^{20} -80079.7 q^{22} -83769.2 q^{23} +15625.0 q^{25} -213687. q^{26} +25997.5 q^{28} -77925.9 q^{29} +255661. q^{31} +593042. q^{32} -263821. q^{34} -9629.93 q^{35} -505096. q^{37} -910913. q^{38} -564864. q^{40} +362971. q^{41} -72126.8 q^{43} -1.25257e6 q^{44} -1.80727e6 q^{46} +521445. q^{47} -817608. q^{49} +337101. q^{50} -3.34239e6 q^{52} +1.72594e6 q^{53} +463973. q^{55} +348135. q^{56} -1.68121e6 q^{58} -667127. q^{59} +444389. q^{61} +5.51574e6 q^{62} +5.84433e6 q^{64} +1.23808e6 q^{65} +2.99810e6 q^{67} -4.12656e6 q^{68} -207760. q^{70} -4.05089e6 q^{71} -1.33628e6 q^{73} -1.08972e7 q^{74} -1.42481e7 q^{76} -285954. q^{77} -621602. q^{79} -6.78733e6 q^{80} +7.83089e6 q^{82} +1.29991e6 q^{83} +1.52855e6 q^{85} -1.55609e6 q^{86} -1.67732e7 q^{88} -416509. q^{89} -763048. q^{91} -2.82685e7 q^{92} +1.12499e7 q^{94} +5.27773e6 q^{95} -1.63057e7 q^{97} -1.76394e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q + 8 q^{2} + 704 q^{4} - 1625 q^{5} - 1455 q^{7} + 1236 q^{8}+O(q^{10}) \) 13 * q + 8 * q^2 + 704 * q^4 - 1625 * q^5 - 1455 * q^7 + 1236 * q^8	\( 13 q + 8 q^{2} + 704 q^{4} - 1625 q^{5} - 1455 q^{7} + 1236 q^{8} - 1000 q^{10} + 1658 q^{11} - 13568 q^{13} + 12351 q^{14} + 52076 q^{16} + 6944 q^{17} - 45812 q^{19} - 88000 q^{20} - 17993 q^{22} + 96441 q^{23} + 203125 q^{25} + 126146 q^{26} - 216945 q^{28} + 39043 q^{29} - 158520 q^{31} + 725206 q^{32} - 441617 q^{34} + 181875 q^{35} - 505438 q^{37} + 615041 q^{38} - 154500 q^{40} + 1578883 q^{41} - 1082090 q^{43} + 498211 q^{44} - 2312547 q^{46} + 1690139 q^{47} - 1054816 q^{49} + 125000 q^{50} - 4100644 q^{52} - 102274 q^{53} - 207250 q^{55} + 389331 q^{56} - 5780521 q^{58} - 2908966 q^{59} - 3091451 q^{61} + 5212476 q^{62} - 5659352 q^{64} + 1696000 q^{65} - 1849533 q^{67} - 563369 q^{68} - 1543875 q^{70} - 2617958 q^{71} + 5310946 q^{73} - 11485004 q^{74} - 14421739 q^{76} + 1719660 q^{77} - 3632166 q^{79} - 6509500 q^{80} - 7347658 q^{82} + 1424115 q^{83} - 868000 q^{85} - 23193379 q^{86} - 20229447 q^{88} - 2816067 q^{89} - 16929702 q^{91} - 27930183 q^{92} - 11693189 q^{94} + 5726500 q^{95} - 12062476 q^{97} - 29617805 q^{98}+O(q^{100}) \) 13 * q + 8 * q^2 + 704 * q^4 - 1625 * q^5 - 1455 * q^7 + 1236 * q^8 - 1000 * q^10 + 1658 * q^11 - 13568 * q^13 + 12351 * q^14 + 52076 * q^16 + 6944 * q^17 - 45812 * q^19 - 88000 * q^20 - 17993 * q^22 + 96441 * q^23 + 203125 * q^25 + 126146 * q^26 - 216945 * q^28 + 39043 * q^29 - 158520 * q^31 + 725206 * q^32 - 441617 * q^34 + 181875 * q^35 - 505438 * q^37 + 615041 * q^38 - 154500 * q^40 + 1578883 * q^41 - 1082090 * q^43 + 498211 * q^44 - 2312547 * q^46 + 1690139 * q^47 - 1054816 * q^49 + 125000 * q^50 - 4100644 * q^52 - 102274 * q^53 - 207250 * q^55 + 389331 * q^56 - 5780521 * q^58 - 2908966 * q^59 - 3091451 * q^61 + 5212476 * q^62 - 5659352 * q^64 + 1696000 * q^65 - 1849533 * q^67 - 563369 * q^68 - 1543875 * q^70 - 2617958 * q^71 + 5310946 * q^73 - 11485004 * q^74 - 14421739 * q^76 + 1719660 * q^77 - 3632166 * q^79 - 6509500 * q^80 - 7347658 * q^82 + 1424115 * q^83 - 868000 * q^85 - 23193379 * q^86 - 20229447 * q^88 - 2816067 * q^89 - 16929702 * q^91 - 27930183 * q^92 - 11693189 * q^94 + 5726500 * q^95 - 12062476 * q^97 - 29617805 * q^98

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