LMFDB - Embedded newform 387.8.a.d.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [387,8,Mod(1,387)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(387, 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("387.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 387 = 3^{2} \cdot 43 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	387.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:	$120.893004862$
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} - 3 x^{12} - 1279 x^{11} + 3765 x^{10} + 598742 x^{9} - 1518614 x^{8} - 124677082 x^{7} + \cdots + 1551032970660 $ x^13 - 3x^12 - 1279x^11 + 3765x^10 + 598742x^9 - 1518614x^8 - 124677082x^7 + 193428526x^6 + 11160446785x^5 + 1754605765x^4 - 349352939351x^3 - 481872751923x^2 + 2098464001560x + 1551032970660
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{3} $
Twist minimal:	no (minimal twist has level 43)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$7.08185$ of defining polynomial
Character	$\chi$	$=$	387.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.08185 q^{2} -62.6836 q^{4} -164.718 q^{5} +1314.61 q^{7} +1541.08 q^{8} +O(q^{10})$	\(q-8.08185 q^{2} -62.6836 q^{4} -164.718 q^{5} +1314.61 q^{7} +1541.08 q^{8} +1331.23 q^{10} -8636.39 q^{11} +8425.12 q^{13} -10624.5 q^{14} -4431.25 q^{16} -6309.90 q^{17} +16664.4 q^{19} +10325.1 q^{20} +69798.0 q^{22} -44105.5 q^{23} -50992.9 q^{25} -68090.6 q^{26} -82404.5 q^{28} +23876.9 q^{29} +27160.3 q^{31} -161445. q^{32} +50995.7 q^{34} -216540. q^{35} -114297. q^{37} -134680. q^{38} -253844. q^{40} +792610. q^{41} -79507.0 q^{43} +541361. q^{44} +356454. q^{46} -490927. q^{47} +904653. q^{49} +412117. q^{50} -528117. q^{52} +1.64370e6 q^{53} +1.42257e6 q^{55} +2.02591e6 q^{56} -192970. q^{58} +625984. q^{59} -1.22257e6 q^{61} -219505. q^{62} +1.87198e6 q^{64} -1.38777e6 q^{65} +1.75201e6 q^{67} +395528. q^{68} +1.75005e6 q^{70} -1.15151e6 q^{71} +2.26726e6 q^{73} +923731. q^{74} -1.04459e6 q^{76} -1.13535e7 q^{77} +4.91491e6 q^{79} +729909. q^{80} -6.40576e6 q^{82} +9.34319e6 q^{83} +1.03936e6 q^{85} +642564. q^{86} -1.33093e7 q^{88} -9.44626e6 q^{89} +1.10757e7 q^{91} +2.76469e6 q^{92} +3.96760e6 q^{94} -2.74494e6 q^{95} -6.08401e6 q^{97} -7.31127e6 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 16 q^{2} + 922 q^{4} - 998 q^{5} + 1360 q^{7} - 3870 q^{8}+O(q^{10}) $ 13 * q - 16 * q^2 + 922 * q^4 - 998 * q^5 + 1360 * q^7 - 3870 * q^8	\( 13 q - 16 q^{2} + 922 q^{4} - 998 q^{5} + 1360 q^{7} - 3870 q^{8} + 4667 q^{10} - 1620 q^{11} + 13550 q^{13} - 44160 q^{14} + 114026 q^{16} - 110880 q^{17} + 105058 q^{19} - 167251 q^{20} + 201504 q^{22} - 160184 q^{23} + 270149 q^{25} - 272104 q^{26} + 208172 q^{28} - 285546 q^{29} - 99616 q^{31} - 200126 q^{32} - 80941 q^{34} + 187104 q^{35} + 176038 q^{37} - 652165 q^{38} - 895387 q^{40} + 410260 q^{41} - 1033591 q^{43} + 2177076 q^{44} - 3975765 q^{46} + 424556 q^{47} - 1561359 q^{49} + 4063801 q^{50} - 4172312 q^{52} - 3992458 q^{53} + 406960 q^{55} - 1559556 q^{56} - 4052005 q^{58} - 2248836 q^{59} + 6210394 q^{61} - 885317 q^{62} - 3096318 q^{64} - 5600420 q^{65} - 1993648 q^{67} - 9327135 q^{68} - 1105098 q^{70} - 4978064 q^{71} + 8224814 q^{73} + 3613563 q^{74} + 10687121 q^{76} - 17261892 q^{77} + 6945708 q^{79} - 15822799 q^{80} - 508449 q^{82} - 22937328 q^{83} - 575532 q^{85} + 1272112 q^{86} + 11202656 q^{88} - 9291302 q^{89} + 25581108 q^{91} + 14388137 q^{92} - 24645805 q^{94} - 30750464 q^{95} + 10001852 q^{97} + 32304856 q^{98}+O(q^{100}) \) 13 * q - 16 * q^2 + 922 * q^4 - 998 * q^5 + 1360 * q^7 - 3870 * q^8 + 4667 * q^10 - 1620 * q^11 + 13550 * q^13 - 44160 * q^14 + 114026 * q^16 - 110880 * q^17 + 105058 * q^19 - 167251 * q^20 + 201504 * q^22 - 160184 * q^23 + 270149 * q^25 - 272104 * q^26 + 208172 * q^28 - 285546 * q^29 - 99616 * q^31 - 200126 * q^32 - 80941 * q^34 + 187104 * q^35 + 176038 * q^37 - 652165 * q^38 - 895387 * q^40 + 410260 * q^41 - 1033591 * q^43 + 2177076 * q^44 - 3975765 * q^46 + 424556 * q^47 - 1561359 * q^49 + 4063801 * q^50 - 4172312 * q^52 - 3992458 * q^53 + 406960 * q^55 - 1559556 * q^56 - 4052005 * q^58 - 2248836 * q^59 + 6210394 * q^61 - 885317 * q^62 - 3096318 * q^64 - 5600420 * q^65 - 1993648 * q^67 - 9327135 * q^68 - 1105098 * q^70 - 4978064 * q^71 + 8224814 * q^73 + 3613563 * q^74 + 10687121 * q^76 - 17261892 * q^77 + 6945708 * q^79 - 15822799 * q^80 - 508449 * q^82 - 22937328 * q^83 - 575532 * q^85 + 1272112 * q^86 + 11202656 * q^88 - 9291302 * q^89 + 25581108 * q^91 + 14388137 * q^92 - 24645805 * q^94 - 30750464 * q^95 + 10001852 * q^97 + 32304856 * 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 $

$2$	−8.08185	−0.714342	−0.357171	−	0.934039i	$-0.616259\pi$
$2$	−8.08185	−0.714342	−0.357171	+	0.934039i	$0.616259\pi$
$3$	0	0
$3$	0	0
$4$	−62.6836	−0.489716
$5$	−164.718	−0.589314	−0.294657	−	0.955603i	$-0.595205\pi$
$5$	−164.718	−0.589314	−0.294657	+	0.955603i	$0.595205\pi$
$6$	0	0
$7$	1314.61	1.44862	0.724308	−	0.689476i	$-0.242159\pi$
$7$	1314.61	1.44862	0.724308	+	0.689476i	$0.242159\pi$
$8$	1541.08	1.06417
$9$	0	0
$10$	1331.23	0.420972
$11$	−8636.39	−1.95640	−0.978201	−	0.207662i	$-0.933415\pi$
$11$	−8636.39	−1.95640	−0.978201	+	0.207662i	$0.933415\pi$
$12$	0	0
$13$	8425.12	1.06359	0.531795	−	0.846873i	$-0.321518\pi$
$13$	8425.12	1.06359	0.531795	+	0.846873i	$0.321518\pi$
$14$	−10624.5	−1.03481
$15$	0	0
$16$	−4431.25	−0.270462
$17$	−6309.90	−0.311495	−0.155748	−	0.987797i	$-0.549779\pi$
$17$	−6309.90	−0.311495	−0.155748	+	0.987797i	$0.549779\pi$
$18$	0	0
$19$	16664.4	0.557382	0.278691	−	0.960381i	$-0.410099\pi$
$19$	16664.4	0.557382	0.278691	+	0.960381i	$0.410099\pi$
$20$	10325.1	0.288597
$21$	0	0
$22$	69798.0	1.39754
$23$	−44105.5	−0.755867	−0.377933	−	0.925833i	$-0.623365\pi$
$23$	−44105.5	−0.755867	−0.377933	+	0.925833i	$0.623365\pi$
$24$	0	0
$25$	−50992.9	−0.652709
$26$	−68090.6	−0.759767
$27$	0	0
$28$	−82404.5	−0.709411
$29$	23876.9	0.181796	0.0908981	−	0.995860i	$-0.471026\pi$
$29$	23876.9	0.181796	0.0908981	+	0.995860i	$0.471026\pi$
$30$	0	0
$31$	27160.3	0.163745	0.0818725	−	0.996643i	$-0.473910\pi$
$31$	27160.3	0.163745	0.0818725	+	0.996643i	$0.473910\pi$
$32$	−161445.	−0.870964
$33$	0	0
$34$	50995.7	0.222514
$35$	−216540.	−0.853690
$36$	0	0
$37$	−114297.	−0.370961	−0.185481	−	0.982648i	$-0.559384\pi$
$37$	−114297.	−0.370961	−0.185481	+	0.982648i	$0.559384\pi$
$38$	−134680.	−0.398161
$39$	0	0
$40$	−253844.	−0.627128
$41$	792610.	1.79604	0.898020	−	0.439954i	$-0.145005\pi$
$41$	792610.	1.79604	0.898020	+	0.439954i	$0.145005\pi$
$42$	0	0
$43$	−79507.0	−0.152499
$43$	−79507.0	−0.152499
$44$	541361.	0.958081
$45$	0	0
$46$	356454.	0.539947
$47$	−490927.	−0.689722	−0.344861	−	0.938654i	$-0.612074\pi$
$47$	−490927.	−0.689722	−0.344861	+	0.938654i	$0.612074\pi$
$48$	0	0
$49$	904653.	1.09849
$50$	412117.	0.466257
$51$	0	0
$52$	−528117.	−0.520857
$53$	1.64370e6	1.51655	0.758275	−	0.651935i	$-0.226042\pi$
$53$	1.64370e6	1.51655	0.758275	+	0.651935i	$0.226042\pi$
$54$	0	0
$55$	1.42257e6	1.15293
$56$	2.02591e6	1.54157
$57$	0	0
$58$	−192970.	−0.129865
$59$	625984.	0.396809	0.198404	−	0.980120i	$-0.436424\pi$
$59$	625984.	0.396809	0.198404	+	0.980120i	$0.436424\pi$
$60$	0	0
$61$	−1.22257e6	−0.689635	−0.344817	−	0.938670i	$-0.612059\pi$
$61$	−1.22257e6	−0.689635	−0.344817	+	0.938670i	$0.612059\pi$
$62$	−219505.	−0.116970
$63$	0	0
$64$	1.87198e6	0.892628
$65$	−1.38777e6	−0.626789
$66$	0	0
$67$	1.75201e6	0.711665	0.355833	−	0.934550i	$-0.384197\pi$
$67$	1.75201e6	0.711665	0.355833	+	0.934550i	$0.384197\pi$
$68$	395528.	0.152544
$69$	0	0
$70$	1.75005e6	0.609826
$71$	−1.15151e6	−0.381823	−0.190911	−	0.981607i	$-0.561144\pi$
$71$	−1.15151e6	−0.381823	−0.190911	+	0.981607i	$0.561144\pi$
$72$	0	0
$73$	2.26726e6	0.682137	0.341068	−	0.940038i	$-0.389211\pi$
$73$	2.26726e6	0.682137	0.341068	+	0.940038i	$0.389211\pi$
$74$	923731.	0.264993
$75$	0	0
$76$	−1.04459e6	−0.272959
$77$	−1.13535e7	−2.83407
$78$	0	0
$79$	4.91491e6	1.12156	0.560778	−	0.827966i	$-0.310502\pi$
$79$	4.91491e6	1.12156	0.560778	+	0.827966i	$0.310502\pi$
$80$	729909.	0.159387
$81$	0	0
$82$	−6.40576e6	−1.28299
$83$	9.34319e6	1.79358	0.896792	−	0.442452i	$-0.145891\pi$
$83$	9.34319e6	1.79358	0.896792	+	0.442452i	$0.145891\pi$
$84$	0	0
$85$	1.03936e6	0.183569
$86$	642564.	0.108936
$87$	0	0
$88$	−1.33093e7	−2.08194
$89$	−9.44626e6	−1.42035	−0.710174	−	0.704026i	$-0.751384\pi$
$89$	−9.44626e6	−1.42035	−0.710174	+	0.704026i	$0.751384\pi$
$90$	0	0
$91$	1.10757e7	1.54073
$92$	2.76469e6	0.370160
$93$	0	0
$94$	3.96760e6	0.492697
$95$	−2.74494e6	−0.328473
$96$	0	0
$97$	−6.08401e6	−0.676845	−0.338422	−	0.940994i	$-0.609893\pi$
$97$	−6.08401e6	−0.676845	−0.338422	+	0.940994i	$0.609893\pi$
$98$	−7.31127e6	−0.784697
$99$	0	0
$100$	3.19642e6	0.319642
$101$	−8.43016e6	−0.814163	−0.407081	−	0.913392i	$-0.633453\pi$
$101$	−8.43016e6	−0.814163	−0.407081	+	0.913392i	$0.633453\pi$
$102$	0	0
$103$	−1.11690e7	−1.00712	−0.503561	−	0.863959i	$-0.667977\pi$
$103$	−1.11690e7	−1.00712	−0.503561	+	0.863959i	$0.667977\pi$
$104$	1.29838e7	1.13184
$105$	0	0
$106$	−1.32841e7	−1.08333
$107$	−9.97447e6	−0.787131	−0.393565	−	0.919297i	$-0.628758\pi$
$107$	−9.97447e6	−0.787131	−0.393565	+	0.919297i	$0.628758\pi$
$108$	0	0
$109$	−1.68276e7	−1.24460	−0.622301	−	0.782778i	$-0.713802\pi$
$109$	−1.68276e7	−1.24460	−0.622301	+	0.782778i	$0.713802\pi$
$110$	−1.14970e7	−0.823589
$111$	0	0
$112$	−5.82536e6	−0.391796
$113$	1.00006e7	0.652006	0.326003	−	0.945369i	$-0.394298\pi$
$113$	1.00006e7	0.652006	0.326003	+	0.945369i	$0.394298\pi$
$114$	0	0
$115$	7.26498e6	0.445443
$116$	−1.49669e6	−0.0890285
$117$	0	0
$118$	−5.05911e6	−0.283457
$119$	−8.29505e6	−0.451237
$120$	0	0
$121$	5.51001e7	2.82751
$122$	9.88063e6	0.492635
$123$	0	0
$124$	−1.70250e6	−0.0801886
$125$	2.12681e7	0.973965
$126$	0	0
$127$	1.37203e7	0.594362	0.297181	−	0.954821i	$-0.403954\pi$
$127$	1.37203e7	0.594362	0.297181	+	0.954821i	$0.403954\pi$
$128$	5.53594e6	0.233322
$129$	0	0
$130$	1.12158e7	0.447741
$131$	2.22841e7	0.866054	0.433027	−	0.901381i	$-0.357446\pi$
$131$	2.22841e7	0.866054	0.433027	+	0.901381i	$0.357446\pi$
$132$	0	0
$133$	2.19072e7	0.807433
$134$	−1.41595e7	−0.508372
$135$	0	0
$136$	−9.72405e6	−0.331483
$137$	1.65210e6	0.0548928	0.0274464	−	0.999623i	$-0.491262\pi$
$137$	1.65210e6	0.0548928	0.0274464	+	0.999623i	$0.491262\pi$
$138$	0	0
$139$	167822.	0.00530027	0.00265014	−	0.999996i	$-0.499156\pi$
$139$	167822.	0.00530027	0.00265014	+	0.999996i	$0.499156\pi$
$140$	1.35735e7	0.418066
$141$	0	0
$142$	9.30630e6	0.272752
$143$	−7.27626e7	−2.08081
$144$	0	0
$145$	−3.93296e6	−0.107135
$146$	−1.83237e7	−0.487279
$147$	0	0
$148$	7.16455e6	0.181666
$149$	3.24074e7	0.802587	0.401294	−	0.915949i	$-0.368561\pi$
$149$	3.24074e7	0.802587	0.401294	+	0.915949i	$0.368561\pi$
$150$	0	0
$151$	−2.03174e7	−0.480229	−0.240114	−	0.970745i	$-0.577185\pi$
$151$	−2.03174e7	−0.480229	−0.240114	+	0.970745i	$0.577185\pi$
$152$	2.56812e7	0.593147
$153$	0	0
$154$	9.17571e7	2.02450
$155$	−4.47379e6	−0.0964973
$156$	0	0
$157$	−7.75567e7	−1.59945	−0.799725	−	0.600366i	$-0.795022\pi$
$157$	−7.75567e7	−1.59945	−0.799725	+	0.600366i	$0.795022\pi$
$158$	−3.97216e7	−0.801174
$159$	0	0
$160$	2.65930e7	0.513271
$161$	−5.79814e7	−1.09496
$162$	0	0
$163$	−8.55135e7	−1.54660	−0.773300	−	0.634040i	$-0.781395\pi$
$163$	−8.55135e7	−1.54660	−0.773300	+	0.634040i	$0.781395\pi$
$164$	−4.96837e7	−0.879550
$165$	0	0
$166$	−7.55103e7	−1.28123
$167$	5.30786e7	0.881885	0.440942	−	0.897535i	$-0.354644\pi$
$167$	5.30786e7	0.881885	0.440942	+	0.897535i	$0.354644\pi$
$168$	0	0
$169$	8.23412e6	0.131224
$170$	−8.39993e6	−0.131131
$171$	0	0
$172$	4.98379e6	0.0746810
$173$	−7.43872e7	−1.09229	−0.546144	−	0.837691i	$-0.683905\pi$
$173$	−7.43872e7	−1.09229	−0.546144	+	0.837691i	$0.683905\pi$
$174$	0	0
$175$	−6.70357e7	−0.945525
$176$	3.82700e7	0.529133
$177$	0	0
$178$	7.63433e7	1.01461
$179$	−1.45355e8	−1.89428	−0.947139	−	0.320824i	$-0.896040\pi$
$179$	−1.45355e8	−1.89428	−0.947139	+	0.320824i	$0.896040\pi$
$180$	0	0
$181$	−4.70208e7	−0.589406	−0.294703	−	0.955589i	$-0.595221\pi$
$181$	−4.70208e7	−0.589406	−0.294703	+	0.955589i	$0.595221\pi$
$182$	−8.95125e7	−1.10061
$183$	0	0
$184$	−6.79699e7	−0.804368
$185$	1.88268e7	0.218613
$186$	0	0
$187$	5.44948e7	0.609410
$188$	3.07731e7	0.337768
$189$	0	0
$190$	2.21842e7	0.234642
$191$	6.43802e7	0.668552	0.334276	−	0.942475i	$-0.391508\pi$
$191$	6.43802e7	0.668552	0.334276	+	0.942475i	$0.391508\pi$
$192$	0	0
$193$	−1.29604e8	−1.29768	−0.648842	−	0.760923i	$-0.724746\pi$
$193$	−1.29604e8	−1.29768	−0.648842	+	0.760923i	$0.724746\pi$
$194$	4.91701e7	0.483499
$195$	0	0
$196$	−5.67070e7	−0.537948
$197$	−8.33653e7	−0.776879	−0.388440	−	0.921474i	$-0.626986\pi$
$197$	−8.33653e7	−0.776879	−0.388440	+	0.921474i	$0.626986\pi$
$198$	0	0
$199$	−1.89130e8	−1.70127	−0.850636	−	0.525755i	$-0.823783\pi$
$199$	−1.89130e8	−1.70127	−0.850636	+	0.525755i	$0.823783\pi$
$200$	−7.85840e7	−0.694591
$201$	0	0
$202$	6.81313e7	0.581590
$203$	3.13888e7	0.263353
$204$	0	0
$205$	−1.30557e8	−1.05843
$206$	9.02659e7	0.719430
$207$	0	0
$208$	−3.73338e7	−0.287661
$209$	−1.43921e8	−1.09046
$210$	0	0
$211$	1.41316e8	1.03563	0.517814	−	0.855493i	$-0.326746\pi$
$211$	1.41316e8	1.03563	0.517814	+	0.855493i	$0.326746\pi$
$212$	−1.03033e8	−0.742679
$213$	0	0
$214$	8.06122e7	0.562280
$215$	1.30963e7	0.0898696
$216$	0	0
$217$	3.57051e7	0.237204
$218$	1.35998e8	0.889070
$219$	0	0
$220$	−8.91720e7	−0.564611
$221$	−5.31617e7	−0.331303
$222$	0	0
$223$	−1.10823e8	−0.669211	−0.334605	−	0.942358i	$-0.608603\pi$
$223$	−1.10823e8	−0.669211	−0.334605	+	0.942358i	$0.608603\pi$
$224$	−2.12237e8	−1.26169
$225$	0	0
$226$	−8.08234e7	−0.465755
$227$	1.43855e8	0.816269	0.408134	−	0.912922i	$-0.366180\pi$
$227$	1.43855e8	0.816269	0.408134	+	0.912922i	$0.366180\pi$
$228$	0	0
$229$	−3.52277e8	−1.93847	−0.969236	−	0.246132i	$-0.920840\pi$
$229$	−3.52277e8	−1.93847	−0.969236	+	0.246132i	$0.920840\pi$
$230$	−5.87145e7	−0.318198
$231$	0	0
$232$	3.67961e7	0.193461
$233$	−3.78934e6	−0.0196254	−0.00981269	−	0.999952i	$-0.503124\pi$
$233$	−3.78934e6	−0.0196254	−0.00981269	+	0.999952i	$0.503124\pi$
$234$	0	0
$235$	8.08647e7	0.406463
$236$	−3.92390e7	−0.194324
$237$	0	0
$238$	6.70394e7	0.322337
$239$	9.27614e6	0.0439516	0.0219758	−	0.999759i	$-0.493004\pi$
$239$	9.27614e6	0.0439516	0.0219758	+	0.999759i	$0.493004\pi$
$240$	0	0
$241$	3.57012e8	1.64294	0.821472	−	0.570249i	$-0.193153\pi$
$241$	3.57012e8	1.64294	0.821472	+	0.570249i	$0.193153\pi$
$242$	−4.45311e8	−2.01981
$243$	0	0
$244$	7.66351e7	0.337725
$245$	−1.49013e8	−0.647355
$246$	0	0
$247$	1.40400e8	0.592826
$248$	4.18561e7	0.174252
$249$	0	0
$250$	−1.71885e8	−0.695744
$251$	−4.81876e8	−1.92343	−0.961716	−	0.274047i	$-0.911637\pi$
$251$	−4.81876e8	−1.92343	−0.961716	+	0.274047i	$0.911637\pi$
$252$	0	0
$253$	3.80912e8	1.47878
$254$	−1.10886e8	−0.424577
$255$	0	0
$256$	−2.84354e8	−1.05930
$257$	1.70067e6	0.00624962	0.00312481	−	0.999995i	$-0.499005\pi$
$257$	1.70067e6	0.00624962	0.00312481	+	0.999995i	$0.499005\pi$
$258$	0	0
$259$	−1.50256e8	−0.537380
$260$	8.69906e7	0.306948
$261$	0	0
$262$	−1.80097e8	−0.618658
$263$	3.62647e8	1.22925	0.614623	−	0.788821i	$-0.289308\pi$
$263$	3.62647e8	1.22925	0.614623	+	0.788821i	$0.289308\pi$
$264$	0	0
$265$	−2.70747e8	−0.893724
$266$	−1.77051e8	−0.576783
$267$	0	0
$268$	−1.09823e8	−0.348514
$269$	−3.91885e8	−1.22751	−0.613756	−	0.789496i	$-0.710342\pi$
$269$	−3.91885e8	−1.22751	−0.613756	+	0.789496i	$0.710342\pi$
$270$	0	0
$271$	5.27272e6	0.0160932	0.00804660	−	0.999968i	$-0.497439\pi$
$271$	5.27272e6	0.0160932	0.00804660	+	0.999968i	$0.497439\pi$
$272$	2.79608e7	0.0842477
$273$	0	0
$274$	−1.33521e7	−0.0392122
$275$	4.40394e8	1.27696
$276$	0	0
$277$	5.31633e8	1.50291	0.751455	−	0.659784i	$-0.229352\pi$
$277$	5.31633e8	1.50291	0.751455	+	0.659784i	$0.229352\pi$
$278$	−1.35632e6	−0.00378620
$279$	0	0
$280$	−3.33705e8	−0.908468
$281$	−5.83130e8	−1.56781	−0.783905	−	0.620881i	$-0.786775\pi$
$281$	−5.83130e8	−1.56781	−0.783905	+	0.620881i	$0.786775\pi$
$282$	0	0
$283$	−2.16632e8	−0.568159	−0.284079	−	0.958801i	$-0.591688\pi$
$283$	−2.16632e8	−0.568159	−0.284079	+	0.958801i	$0.591688\pi$
$284$	7.21806e7	0.186985
$285$	0	0
$286$	5.88057e8	1.48641
$287$	1.04197e9	2.60177
$288$	0	0
$289$	−3.70524e8	−0.902971
$290$	3.17856e7	0.0765311
$291$	0	0
$292$	−1.42120e8	−0.334053
$293$	9.62588e7	0.223565	0.111782	−	0.993733i	$-0.464344\pi$
$293$	9.62588e7	0.223565	0.111782	+	0.993733i	$0.464344\pi$
$294$	0	0
$295$	−1.03111e8	−0.233845
$296$	−1.76140e8	−0.394764
$297$	0	0
$298$	−2.61912e8	−0.573321
$299$	−3.71594e8	−0.803932
$300$	0	0
$301$	−1.04521e8	−0.220912
$302$	1.64202e8	0.343047
$303$	0	0
$304$	−7.38444e7	−0.150751
$305$	2.01380e8	0.406411
$306$	0	0
$307$	7.18187e8	1.41662	0.708310	−	0.705902i	$-0.249458\pi$
$307$	7.18187e8	1.41662	0.708310	+	0.705902i	$0.249458\pi$
$308$	7.11677e8	1.38789
$309$	0	0
$310$	3.61565e7	0.0689320
$311$	1.67567e8	0.315883	0.157942	−	0.987448i	$-0.449514\pi$
$311$	1.67567e8	0.315883	0.157942	+	0.987448i	$0.449514\pi$
$312$	0	0
$313$	6.19823e8	1.14252	0.571258	−	0.820770i	$-0.306455\pi$
$313$	6.19823e8	1.14252	0.571258	+	0.820770i	$0.306455\pi$
$314$	6.26802e8	1.14255
$315$	0	0
$316$	−3.08084e8	−0.549244
$317$	−2.63030e8	−0.463765	−0.231883	−	0.972744i	$-0.574489\pi$
$317$	−2.63030e8	−0.463765	−0.231883	+	0.972744i	$0.574489\pi$
$318$	0	0
$319$	−2.06210e8	−0.355666
$320$	−3.08349e8	−0.526038
$321$	0	0
$322$	4.68598e8	0.782176
$323$	−1.05151e8	−0.173622
$324$	0	0
$325$	−4.29621e8	−0.694215
$326$	6.91108e8	1.10480
$327$	0	0
$328$	1.22147e9	1.91129
$329$	−6.45377e8	−0.999143
$330$	0	0
$331$	−9.80269e8	−1.48575	−0.742877	−	0.669427i	$-0.766540\pi$
$331$	−9.80269e8	−1.48575	−0.742877	+	0.669427i	$0.766540\pi$
$332$	−5.85665e8	−0.878347
$333$	0	0
$334$	−4.28973e8	−0.629967
$335$	−2.88589e8	−0.419394
$336$	0	0
$337$	−6.28527e8	−0.894580	−0.447290	−	0.894389i	$-0.647611\pi$
$337$	−6.28527e8	−0.894580	−0.447290	+	0.894389i	$0.647611\pi$
$338$	−6.65470e7	−0.0937389
$339$	0	0
$340$	−6.51507e7	−0.0898965
$341$	−2.34567e8	−0.320351
$342$	0	0
$343$	1.06628e8	0.142673
$344$	−1.22526e8	−0.162284
$345$	0	0
$346$	6.01187e8	0.780267
$347$	−1.41484e9	−1.81783	−0.908915	−	0.416982i	$-0.863088\pi$
$347$	−1.41484e9	−1.81783	−0.908915	+	0.416982i	$0.863088\pi$
$348$	0	0
$349$	1.08261e9	1.36327	0.681637	−	0.731691i	$-0.261269\pi$
$349$	1.08261e9	1.36327	0.681637	+	0.731691i	$0.261269\pi$
$350$	5.41773e8	0.675428
$351$	0	0
$352$	1.39430e9	1.70395
$353$	1.04347e9	1.26261	0.631306	−	0.775533i	$-0.282519\pi$
$353$	1.04347e9	1.26261	0.631306	+	0.775533i	$0.282519\pi$
$354$	0	0
$355$	1.89674e8	0.225014
$356$	5.92126e8	0.695567
$357$	0	0
$358$	1.17474e9	1.35316
$359$	−7.36087e8	−0.839650	−0.419825	−	0.907605i	$-0.637909\pi$
$359$	−7.36087e8	−0.839650	−0.419825	+	0.907605i	$0.637909\pi$
$360$	0	0
$361$	−6.16168e8	−0.689325
$362$	3.80015e8	0.421037
$363$	0	0
$364$	−6.94267e8	−0.754522
$365$	−3.73459e8	−0.401993
$366$	0	0
$367$	5.85636e8	0.618439	0.309219	−	0.950991i	$-0.399932\pi$
$367$	5.85636e8	0.618439	0.309219	+	0.950991i	$0.399932\pi$
$368$	1.95443e8	0.204433
$369$	0	0
$370$	−1.52155e8	−0.156164
$371$	2.16082e9	2.19690
$372$	0	0
$373$	−1.55931e9	−1.55579	−0.777896	−	0.628393i	$-0.783713\pi$
$373$	−1.55931e9	−1.55579	−0.777896	+	0.628393i	$0.783713\pi$
$374$	−4.40419e8	−0.435327
$375$	0	0
$376$	−7.56556e8	−0.733979
$377$	2.01166e8	0.193357
$378$	0	0
$379$	−2.07398e7	−0.0195689	−0.00978446	−	0.999952i	$-0.503115\pi$
$379$	−2.07398e7	−0.0195689	−0.00978446	+	0.999952i	$0.503115\pi$
$380$	1.72063e8	0.160859
$381$	0	0
$382$	−5.20311e8	−0.477575
$383$	−1.16713e9	−1.06151	−0.530755	−	0.847525i	$-0.678092\pi$
$383$	−1.16713e9	−1.06151	−0.530755	+	0.847525i	$0.678092\pi$
$384$	0	0
$385$	1.87013e9	1.67016
$386$	1.04744e9	0.926990
$387$	0	0
$388$	3.81368e8	0.331462
$389$	1.12210e9	0.966512	0.483256	−	0.875479i	$-0.339454\pi$
$389$	1.12210e9	0.966512	0.483256	+	0.875479i	$0.339454\pi$
$390$	0	0
$391$	2.78301e8	0.235449
$392$	1.39414e9	1.16897
$393$	0	0
$394$	6.73746e8	0.554957
$395$	−8.09575e8	−0.660948
$396$	0	0
$397$	6.25123e8	0.501417	0.250708	−	0.968063i	$-0.419337\pi$
$397$	6.25123e8	0.501417	0.250708	+	0.968063i	$0.419337\pi$
$398$	1.52852e9	1.21529
$399$	0	0
$400$	2.25962e8	0.176533
$401$	−6.80350e8	−0.526899	−0.263449	−	0.964673i	$-0.584860\pi$
$401$	−6.80350e8	−0.526899	−0.263449	+	0.964673i	$0.584860\pi$
$402$	0	0
$403$	2.28828e8	0.174158
$404$	5.28433e8	0.398708
$405$	0	0
$406$	−2.53680e8	−0.188124
$407$	9.87113e8	0.725749
$408$	0	0
$409$	1.45254e9	1.04977	0.524887	−	0.851172i	$-0.324108\pi$
$409$	1.45254e9	1.04977	0.524887	+	0.851172i	$0.324108\pi$
$410$	1.05515e9	0.756082
$411$	0	0
$412$	7.00111e8	0.493204
$413$	8.22925e8	0.574824
$414$	0	0
$415$	−1.53899e9	−1.05698
$416$	−1.36019e9	−0.926349
$417$	0	0
$418$	1.16315e9	0.778963
$419$	2.07021e9	1.37488	0.687441	−	0.726240i	$-0.258734\pi$
$419$	2.07021e9	1.37488	0.687441	+	0.726240i	$0.258734\pi$
$420$	0	0
$421$	−1.86742e9	−1.21971	−0.609853	−	0.792514i	$-0.708772\pi$
$421$	−1.86742e9	−1.21971	−0.609853	+	0.792514i	$0.708772\pi$
$422$	−1.14210e9	−0.739792
$423$	0	0
$424$	2.53307e9	1.61386
$425$	3.21760e8	0.203316
$426$	0	0
$427$	−1.60720e9	−0.999016
$428$	6.25236e8	0.385470
$429$	0	0
$430$	−1.05842e8	−0.0641976
$431$	−1.03410e9	−0.622144	−0.311072	−	0.950386i	$-0.600688\pi$
$431$	−1.03410e9	−0.622144	−0.311072	+	0.950386i	$0.600688\pi$
$432$	0	0
$433$	1.88745e9	1.11730	0.558648	−	0.829405i	$-0.311320\pi$
$433$	1.88745e9	1.11730	0.558648	+	0.829405i	$0.311320\pi$
$434$	−2.88564e8	−0.169444
$435$	0	0
$436$	1.05482e9	0.609501
$437$	−7.34993e8	−0.421307
$438$	0	0
$439$	−2.19197e9	−1.23654	−0.618270	−	0.785965i	$-0.712166\pi$
$439$	−2.19197e9	−1.23654	−0.618270	+	0.785965i	$0.712166\pi$
$440$	2.19229e9	1.22691
$441$	0	0
$442$	4.29645e8	0.236664
$443$	1.23285e9	0.673745	0.336873	−	0.941550i	$-0.390631\pi$
$443$	1.23285e9	0.673745	0.336873	+	0.941550i	$0.390631\pi$
$444$	0	0
$445$	1.55597e9	0.837031
$446$	8.95655e8	0.478045
$447$	0	0
$448$	2.46092e9	1.29308
$449$	−2.06378e9	−1.07597	−0.537985	−	0.842954i	$-0.680814\pi$
$449$	−2.06378e9	−1.07597	−0.537985	+	0.842954i	$0.680814\pi$
$450$	0	0
$451$	−6.84529e9	−3.51378
$452$	−6.26874e8	−0.319298
$453$	0	0
$454$	−1.16261e9	−0.583095
$455$	−1.82438e9	−0.907976
$456$	0	0
$457$	−3.67119e9	−1.79928	−0.899642	−	0.436628i	$-0.856173\pi$
$457$	−3.67119e9	−1.79928	−0.899642	+	0.436628i	$0.856173\pi$
$458$	2.84705e9	1.38473
$459$	0	0
$460$	−4.55395e8	−0.218140
$461$	−1.50027e9	−0.713206	−0.356603	−	0.934256i	$-0.616065\pi$
$461$	−1.50027e9	−0.713206	−0.356603	+	0.934256i	$0.616065\pi$
$462$	0	0
$463$	1.00012e9	0.468293	0.234147	−	0.972201i	$-0.424770\pi$
$463$	1.00012e9	0.468293	0.234147	+	0.972201i	$0.424770\pi$
$464$	−1.05805e8	−0.0491690
$465$	0	0
$466$	3.06249e7	0.0140192
$467$	1.13114e9	0.513933	0.256967	−	0.966420i	$-0.417277\pi$
$467$	1.13114e9	0.513933	0.256967	+	0.966420i	$0.417277\pi$
$468$	0	0
$469$	2.30321e9	1.03093
$470$	−6.53536e8	−0.290354
$471$	0	0
$472$	9.64690e8	0.422271
$473$	6.86654e8	0.298348
$474$	0	0
$475$	−8.49768e8	−0.363808
$476$	5.19964e8	0.220978
$477$	0	0
$478$	−7.49684e7	−0.0313964
$479$	−4.27056e8	−0.177546	−0.0887730	−	0.996052i	$-0.528295\pi$
$479$	−4.27056e8	−0.177546	−0.0887730	+	0.996052i	$0.528295\pi$
$480$	0	0
$481$	−9.62965e8	−0.394551
$482$	−2.88532e9	−1.17362
$483$	0	0
$484$	−3.45387e9	−1.38467
$485$	1.00215e9	0.398874
$486$	0	0
$487$	1.86591e9	0.732049	0.366025	−	0.930605i	$-0.380719\pi$
$487$	1.86591e9	0.732049	0.366025	+	0.930605i	$0.380719\pi$
$488$	−1.88407e9	−0.733886
$489$	0	0
$490$	1.20430e9	0.462433
$491$	2.67497e9	1.01985	0.509923	−	0.860220i	$-0.329674\pi$
$491$	2.67497e9	1.01985	0.509923	+	0.860220i	$0.329674\pi$
$492$	0	0
$493$	−1.50661e8	−0.0566287
$494$	−1.13469e9	−0.423481
$495$	0	0
$496$	−1.20354e8	−0.0442868
$497$	−1.51378e9	−0.553115
$498$	0	0
$499$	−2.01727e9	−0.726794	−0.363397	−	0.931634i	$-0.618383\pi$
$499$	−2.01727e9	−0.726794	−0.363397	+	0.931634i	$0.618383\pi$
$500$	−1.33316e9	−0.476966
$501$	0	0
$502$	3.89445e9	1.37399
$503$	−2.78344e9	−0.975201	−0.487601	−	0.873067i	$-0.662128\pi$
$503$	−2.78344e9	−0.975201	−0.487601	+	0.873067i	$0.662128\pi$
$504$	0	0
$505$	1.38860e9	0.479798
$506$	−3.07848e9	−1.05635
$507$	0	0
$508$	−8.60039e8	−0.291068
$509$	6.34132e8	0.213141	0.106571	−	0.994305i	$-0.466013\pi$
$509$	6.34132e8	0.213141	0.106571	+	0.994305i	$0.466013\pi$
$510$	0	0
$511$	2.98056e9	0.988154
$512$	1.58950e9	0.523380
$513$	0	0
$514$	−1.37446e7	−0.00446437
$515$	1.83973e9	0.593512
$516$	0	0
$517$	4.23984e9	1.34937
$518$	1.21434e9	0.383873
$519$	0	0
$520$	−2.13866e9	−0.667007
$521$	2.55557e9	0.791692	0.395846	−	0.918317i	$-0.370451\pi$
$521$	2.55557e9	0.791692	0.395846	+	0.918317i	$0.370451\pi$
$522$	0	0
$523$	1.86691e9	0.570647	0.285323	−	0.958431i	$-0.407899\pi$
$523$	1.86691e9	0.570647	0.285323	+	0.958431i	$0.407899\pi$
$524$	−1.39685e9	−0.424120
$525$	0	0
$526$	−2.93086e9	−0.878101
$527$	−1.71379e8	−0.0510058
$528$	0	0
$529$	−1.45953e9	−0.428666
$530$	2.18814e9	0.638425
$531$	0	0
$532$	−1.37322e9	−0.395413
$533$	6.67783e9	1.91025
$534$	0	0
$535$	1.64298e9	0.463867
$536$	2.69999e9	0.757330
$537$	0	0
$538$	3.16716e9	0.876863
$539$	−7.81294e9	−2.14909
$540$	0	0
$541$	−2.51318e8	−0.0682391	−0.0341195	−	0.999418i	$-0.510863\pi$
$541$	−2.51318e8	−0.0682391	−0.0341195	+	0.999418i	$0.510863\pi$
$542$	−4.26134e7	−0.0114960
$543$	0	0
$544$	1.01870e9	0.271301
$545$	2.77182e9	0.733461
$546$	0	0
$547$	1.95845e9	0.511631	0.255815	−	0.966726i	$-0.417656\pi$
$547$	1.95845e9	0.511631	0.255815	+	0.966726i	$0.417656\pi$
$548$	−1.03560e8	−0.0268819
$549$	0	0
$550$	−3.55920e9	−0.912186
$551$	3.97895e8	0.101330
$552$	0	0
$553$	6.46118e9	1.62470
$554$	−4.29658e9	−1.07359
$555$	0	0
$556$	−1.05197e7	−0.00259563
$557$	9.85537e7	0.0241646	0.0120823	−	0.999927i	$-0.496154\pi$
$557$	9.85537e7	0.0241646	0.0120823	+	0.999927i	$0.496154\pi$
$558$	0	0
$559$	−6.69856e8	−0.162196
$560$	9.59544e8	0.230891
$561$	0	0
$562$	4.71277e9	1.11995
$563$	8.32439e8	0.196595	0.0982976	−	0.995157i	$-0.468660\pi$
$563$	8.32439e8	0.196595	0.0982976	+	0.995157i	$0.468660\pi$
$564$	0	0
$565$	−1.64728e9	−0.384236
$566$	1.75079e9	0.405859
$567$	0	0
$568$	−1.77456e9	−0.406323
$569$	4.82012e8	0.109689	0.0548447	−	0.998495i	$-0.482534\pi$
$569$	4.82012e8	0.109689	0.0548447	+	0.998495i	$0.482534\pi$
$570$	0	0
$571$	−3.91868e9	−0.880874	−0.440437	−	0.897784i	$-0.645176\pi$
$571$	−3.91868e9	−0.880874	−0.440437	+	0.897784i	$0.645176\pi$
$572$	4.56103e9	1.01901
$573$	0	0
$574$	−8.42107e9	−1.85856
$575$	2.24907e9	0.493361
$576$	0	0
$577$	−6.04095e9	−1.30915	−0.654576	−	0.755996i	$-0.727153\pi$
$577$	−6.04095e9	−1.30915	−0.654576	+	0.755996i	$0.727153\pi$
$578$	2.99452e9	0.645030
$579$	0	0
$580$	2.46532e8	0.0524658
$581$	1.22826e10	2.59822
$582$	0	0
$583$	−1.41956e10	−2.96698
$584$	3.49402e9	0.725907
$585$	0	0
$586$	−7.77949e8	−0.159702
$587$	−9.41828e9	−1.92193	−0.960967	−	0.276665i	$-0.910771\pi$
$587$	−9.41828e9	−1.92193	−0.960967	+	0.276665i	$0.910771\pi$
$588$	0	0
$589$	4.52611e8	0.0912686
$590$	8.33329e8	0.167045
$591$	0	0
$592$	5.06479e8	0.100331
$593$	−2.77653e9	−0.546778	−0.273389	−	0.961904i	$-0.588145\pi$
$593$	−2.77653e9	−0.546778	−0.273389	+	0.961904i	$0.588145\pi$
$594$	0	0
$595$	1.36635e9	0.265920
$596$	−2.03141e9	−0.393040
$597$	0	0
$598$	3.00317e9	0.574282
$599$	−2.48861e9	−0.473110	−0.236555	−	0.971618i	$-0.576018\pi$
$599$	−2.48861e9	−0.473110	−0.236555	+	0.971618i	$0.576018\pi$
$600$	0	0
$601$	9.96150e8	0.187182	0.0935910	−	0.995611i	$-0.470165\pi$
$601$	9.96150e8	0.187182	0.0935910	+	0.995611i	$0.470165\pi$
$602$	8.44720e8	0.157807
$603$	0	0
$604$	1.27357e9	0.235176
$605$	−9.07599e9	−1.66629
$606$	0	0
$607$	4.07704e9	0.739920	0.369960	−	0.929048i	$-0.379371\pi$
$607$	4.07704e9	0.739920	0.369960	+	0.929048i	$0.379371\pi$
$608$	−2.69039e9	−0.485460
$609$	0	0
$610$	−1.62752e9	−0.290317
$611$	−4.13612e9	−0.733582
$612$	0	0
$613$	7.37622e9	1.29337	0.646684	−	0.762758i	$-0.276155\pi$
$613$	7.37622e9	1.29337	0.646684	+	0.762758i	$0.276155\pi$
$614$	−5.80428e9	−1.01195
$615$	0	0
$616$	−1.74966e10	−3.01593
$617$	3.57832e9	0.613312	0.306656	−	0.951820i	$-0.400790\pi$
$617$	3.57832e9	0.613312	0.306656	+	0.951820i	$0.400790\pi$
$618$	0	0
$619$	−2.12273e9	−0.359731	−0.179865	−	0.983691i	$-0.557566\pi$
$619$	−2.12273e9	−0.359731	−0.179865	+	0.983691i	$0.557566\pi$
$620$	2.80434e8	0.0472563
$621$	0	0
$622$	−1.35425e9	−0.225649
$623$	−1.24181e10	−2.05754
$624$	0	0
$625$	4.80577e8	0.0787377
$626$	−5.00932e9	−0.816147
$627$	0	0
$628$	4.86154e9	0.783277
$629$	7.21203e8	0.115553
$630$	0	0
$631$	−1.03343e10	−1.63749	−0.818744	−	0.574159i	$-0.805329\pi$
$631$	−1.03343e10	−1.63749	−0.818744	+	0.574159i	$0.805329\pi$
$632$	7.57425e9	1.19352
$633$	0	0
$634$	2.12577e9	0.331287
$635$	−2.25999e9	−0.350266
$636$	0	0
$637$	7.62181e9	1.16834
$638$	1.66656e9	0.254067
$639$	0	0
$640$	−9.11871e8	−0.137500
$641$	−4.06584e9	−0.609744	−0.304872	−	0.952393i	$-0.598614\pi$
$641$	−4.06584e9	−0.609744	−0.304872	+	0.952393i	$0.598614\pi$
$642$	0	0
$643$	1.27525e10	1.89172	0.945860	−	0.324574i	$-0.105221\pi$
$643$	1.27525e10	1.89172	0.945860	+	0.324574i	$0.105221\pi$
$644$	3.63449e9	0.536220
$645$	0	0
$646$	8.49815e8	0.124025
$647$	−6.06984e9	−0.881074	−0.440537	−	0.897734i	$-0.645212\pi$
$647$	−6.06984e9	−0.881074	−0.440537	+	0.897734i	$0.645212\pi$
$648$	0	0
$649$	−5.40625e9	−0.776318
$650$	3.47213e9	0.495907
$651$	0	0
$652$	5.36030e9	0.757395
$653$	−8.96474e9	−1.25992	−0.629958	−	0.776629i	$-0.716928\pi$
$653$	−8.96474e9	−1.25992	−0.629958	+	0.776629i	$0.716928\pi$
$654$	0	0
$655$	−3.67059e9	−0.510378
$656$	−3.51226e9	−0.485761
$657$	0	0
$658$	5.21584e9	0.713730
$659$	−7.66829e9	−1.04376	−0.521879	−	0.853020i	$-0.674769\pi$
$659$	−7.66829e9	−1.04376	−0.521879	+	0.853020i	$0.674769\pi$
$660$	0	0
$661$	9.45702e9	1.27365	0.636824	−	0.771010i	$-0.280248\pi$
$661$	9.45702e9	1.27365	0.636824	+	0.771010i	$0.280248\pi$
$662$	7.92239e9	1.06134
$663$	0	0
$664$	1.43986e10	1.90867
$665$	−3.60852e9	−0.475832
$666$	0	0
$667$	−1.05310e9	−0.137414
$668$	−3.32716e9	−0.431873
$669$	0	0
$670$	2.33233e9	0.299591
$671$	1.05586e10	1.34920
$672$	0	0
$673$	7.72168e9	0.976470	0.488235	−	0.872712i	$-0.337641\pi$
$673$	7.72168e9	0.976470	0.488235	+	0.872712i	$0.337641\pi$
$674$	5.07966e9	0.639036
$675$	0	0
$676$	−5.16145e8	−0.0642626
$677$	3.61144e9	0.447321	0.223661	−	0.974667i	$-0.428199\pi$
$677$	3.61144e9	0.447321	0.223661	+	0.974667i	$0.428199\pi$
$678$	0	0
$679$	−7.99810e9	−0.980489
$680$	1.60173e9	0.195347
$681$	0	0
$682$	1.89573e9	0.228840
$683$	−1.07988e10	−1.29689	−0.648444	−	0.761262i	$-0.724580\pi$
$683$	−1.07988e10	−1.29689	−0.648444	+	0.761262i	$0.724580\pi$
$684$	0	0
$685$	−2.72132e8	−0.0323491
$686$	−8.61751e8	−0.101917
$687$	0	0
$688$	3.52316e8	0.0412451
$689$	1.38484e10	1.61299
$690$	0	0
$691$	3.51258e9	0.404998	0.202499	−	0.979282i	$-0.435094\pi$
$691$	3.51258e9	0.404998	0.202499	+	0.979282i	$0.435094\pi$
$692$	4.66286e9	0.534911
$693$	0	0
$694$	1.14345e10	1.29855
$695$	−2.76434e7	−0.00312353
$696$	0	0
$697$	−5.00129e9	−0.559458
$698$	−8.74949e9	−0.973843
$699$	0	0
$700$	4.20204e9	0.463039
$701$	2.46256e9	0.270006	0.135003	−	0.990845i	$-0.456896\pi$
$701$	2.46256e9	0.270006	0.135003	+	0.990845i	$0.456896\pi$
$702$	0	0
$703$	−1.90469e9	−0.206767
$704$	−1.61671e10	−1.74634
$705$	0	0
$706$	−8.43320e9	−0.901937
$707$	−1.10824e10	−1.17941
$708$	0	0
$709$	−1.34920e10	−1.42172	−0.710859	−	0.703335i	$-0.751693\pi$
$709$	−1.34920e10	−1.42172	−0.710859	+	0.703335i	$0.751693\pi$
$710$	−1.53292e9	−0.160737
$711$	0	0
$712$	−1.45574e10	−1.51149
$713$	−1.19792e9	−0.123769
$714$	0	0
$715$	1.19853e10	1.22625
$716$	9.11137e9	0.927658
$717$	0	0
$718$	5.94894e9	0.599797
$719$	−7.00395e9	−0.702736	−0.351368	−	0.936237i	$-0.614283\pi$
$719$	−7.00395e9	−0.702736	−0.351368	+	0.936237i	$0.614283\pi$
$720$	0	0
$721$	−1.46828e10	−1.45893
$722$	4.97978e9	0.492414
$723$	0	0
$724$	2.94743e9	0.288642
$725$	−1.21755e9	−0.118660
$726$	0	0
$727$	8.00714e9	0.772871	0.386436	−	0.922316i	$-0.373706\pi$
$727$	8.00714e9	0.772871	0.386436	+	0.922316i	$0.373706\pi$
$728$	1.70686e10	1.63960
$729$	0	0
$730$	3.01824e9	0.287160
$731$	5.01681e8	0.0475026
$732$	0	0
$733$	9.27809e9	0.870151	0.435075	−	0.900394i	$-0.356722\pi$
$733$	9.27809e9	0.870151	0.435075	+	0.900394i	$0.356722\pi$
$734$	−4.73303e9	−0.441777
$735$	0	0
$736$	7.12062e9	0.658332
$737$	−1.51311e10	−1.39230
$738$	0	0
$739$	−1.50075e10	−1.36789	−0.683947	−	0.729532i	$-0.739738\pi$
$739$	−1.50075e10	−1.36789	−0.683947	+	0.729532i	$0.739738\pi$
$740$	−1.18013e9	−0.107058
$741$	0	0
$742$	−1.74634e10	−1.56934
$743$	−2.52291e9	−0.225653	−0.112827	−	0.993615i	$-0.535990\pi$
$743$	−2.52291e9	−0.225653	−0.112827	+	0.993615i	$0.535990\pi$
$744$	0	0
$745$	−5.33809e9	−0.472976
$746$	1.26021e10	1.11137
$747$	0	0
$748$	−3.41593e9	−0.298438
$749$	−1.31125e10	−1.14025
$750$	0	0
$751$	−9.71269e9	−0.836758	−0.418379	−	0.908273i	$-0.637402\pi$
$751$	−9.71269e9	−0.836758	−0.418379	+	0.908273i	$0.637402\pi$
$752$	2.17542e9	0.186544
$753$	0	0
$754$	−1.62579e9	−0.138123
$755$	3.34664e9	0.283006
$756$	0	0
$757$	−1.39407e10	−1.16802	−0.584008	−	0.811748i	$-0.698516\pi$
$757$	−1.39407e10	−1.16802	−0.584008	+	0.811748i	$0.698516\pi$
$758$	1.67616e8	0.0139789
$759$	0	0
$760$	−4.23016e9	−0.349550
$761$	−9.33520e9	−0.767852	−0.383926	−	0.923364i	$-0.625428\pi$
$761$	−9.33520e9	−0.767852	−0.383926	+	0.923364i	$0.625428\pi$
$762$	0	0
$763$	−2.21217e10	−1.80295
$764$	−4.03558e9	−0.327401
$765$	0	0
$766$	9.43258e9	0.758281
$767$	5.27399e9	0.422042
$768$	0	0
$769$	−1.02873e10	−0.815754	−0.407877	−	0.913037i	$-0.633731\pi$
$769$	−1.02873e10	−0.815754	−0.407877	+	0.913037i	$0.633731\pi$
$770$	−1.51141e10	−1.19307
$771$	0	0
$772$	8.12408e9	0.635497
$773$	6.15947e9	0.479640	0.239820	−	0.970817i	$-0.422912\pi$
$773$	6.15947e9	0.479640	0.239820	+	0.970817i	$0.422912\pi$
$774$	0	0
$775$	−1.38498e9	−0.106878
$776$	−9.37594e9	−0.720275
$777$	0	0
$778$	−9.06863e9	−0.690420
$779$	1.32084e10	1.00108
$780$	0	0
$781$	9.94486e9	0.746999
$782$	−2.24919e9	−0.168191
$783$	0	0
$784$	−4.00875e9	−0.297100
$785$	1.27750e10	0.942579
$786$	0	0
$787$	6.80084e9	0.497337	0.248669	−	0.968589i	$-0.420007\pi$
$787$	6.80084e9	0.497337	0.248669	+	0.968589i	$0.420007\pi$
$788$	5.22564e9	0.380450
$789$	0	0
$790$	6.54287e9	0.472143
$791$	1.31469e10	0.944506
$792$	0	0
$793$	−1.03003e10	−0.733489
$794$	−5.05215e9	−0.358183
$795$	0	0
$796$	1.18553e10	0.833140
$797$	1.73727e10	1.21552	0.607762	−	0.794119i	$-0.292067\pi$
$797$	1.73727e10	1.21552	0.607762	+	0.794119i	$0.292067\pi$
$798$	0	0
$799$	3.09770e9	0.214845
$800$	8.23255e9	0.568486
$801$	0	0
$802$	5.49849e9	0.376386
$803$	−1.95810e10	−1.33453
$804$	0	0
$805$	9.55061e9	0.645276
$806$	−1.84936e9	−0.124408
$807$	0	0
$808$	−1.29915e10	−0.866404
$809$	−9.63683e9	−0.639903	−0.319952	−	0.947434i	$-0.603667\pi$
$809$	−9.63683e9	−0.639903	−0.319952	+	0.947434i	$0.603667\pi$
$810$	0	0
$811$	9.66741e9	0.636410	0.318205	−	0.948022i	$-0.396920\pi$
$811$	9.66741e9	0.636410	0.318205	+	0.948022i	$0.396920\pi$
$812$	−1.96756e9	−0.128968
$813$	0	0
$814$	−7.97770e9	−0.518433
$815$	1.40856e10	0.911433
$816$	0	0
$817$	−1.32494e9	−0.0850000
$818$	−1.17392e10	−0.749897
$819$	0	0
$820$	8.18381e9	0.518331
$821$	−4.33448e9	−0.273361	−0.136680	−	0.990615i	$-0.543643\pi$
$821$	−4.33448e9	−0.273361	−0.136680	+	0.990615i	$0.543643\pi$
$822$	0	0
$823$	7.05830e9	0.441368	0.220684	−	0.975345i	$-0.429171\pi$
$823$	7.05830e9	0.441368	0.220684	+	0.975345i	$0.429171\pi$
$824$	−1.72122e10	−1.07175
$825$	0	0
$826$	−6.65076e9	−0.410621
$827$	−1.97938e10	−1.21691	−0.608457	−	0.793587i	$-0.708211\pi$
$827$	−1.97938e10	−1.21691	−0.608457	+	0.793587i	$0.708211\pi$
$828$	0	0
$829$	1.55516e10	0.948055	0.474027	−	0.880510i	$-0.342800\pi$
$829$	1.55516e10	0.948055	0.474027	+	0.880510i	$0.342800\pi$
$830$	1.24379e10	0.755048
$831$	0	0
$832$	1.57716e10	0.949390
$833$	−5.70827e9	−0.342174
$834$	0	0
$835$	−8.74302e9	−0.519707
$836$	9.02147e9	0.534017
$837$	0	0
$838$	−1.67311e10	−0.982135
$839$	4.02189e9	0.235106	0.117553	−	0.993067i	$-0.462495\pi$
$839$	4.02189e9	0.235106	0.117553	+	0.993067i	$0.462495\pi$
$840$	0	0
$841$	−1.66798e10	−0.966950
$842$	1.50922e10	0.871287
$843$	0	0
$844$	−8.85822e9	−0.507163
$845$	−1.35631e9	−0.0773323
$846$	0	0
$847$	7.24351e10	4.09597
$848$	−7.28365e9	−0.410170
$849$	0	0
$850$	−2.60042e9	−0.145237
$851$	5.04112e9	0.280397
$852$	0	0
$853$	−7.65536e9	−0.422322	−0.211161	−	0.977451i	$-0.567725\pi$
$853$	−7.65536e9	−0.422322	−0.211161	+	0.977451i	$0.567725\pi$
$854$	1.29892e10	0.713639
$855$	0	0
$856$	−1.53714e10	−0.837638
$857$	1.60019e10	0.868437	0.434218	−	0.900808i	$-0.357025\pi$
$857$	1.60019e10	0.868437	0.434218	+	0.900808i	$0.357025\pi$
$858$	0	0
$859$	−9.96999e9	−0.536684	−0.268342	−	0.963324i	$-0.586476\pi$
$859$	−9.96999e9	−0.536684	−0.268342	+	0.963324i	$0.586476\pi$
$860$	−8.20921e8	−0.0440106
$861$	0	0
$862$	8.35742e9	0.444423
$863$	−2.21580e10	−1.17353	−0.586763	−	0.809759i	$-0.699598\pi$
$863$	−2.21580e10	−1.17353	−0.586763	+	0.809759i	$0.699598\pi$
$864$	0	0
$865$	1.22529e10	0.643701
$866$	−1.52541e10	−0.798131
$867$	0	0
$868$	−2.23813e9	−0.116162
$869$	−4.24471e10	−2.19421
$870$	0	0
$871$	1.47609e10	0.756920
$872$	−2.59327e10	−1.32446
$873$	0	0
$874$	5.94011e9	0.300957
$875$	2.79592e10	1.41090
$876$	0	0
$877$	−8.03090e9	−0.402037	−0.201018	−	0.979587i	$-0.564425\pi$
$877$	−8.03090e9	−0.402037	−0.201018	+	0.979587i	$0.564425\pi$
$878$	1.77152e10	0.883313
$879$	0	0
$880$	−6.30378e9	−0.311825
$881$	−1.58502e10	−0.780944	−0.390472	−	0.920615i	$-0.627688\pi$
$881$	−1.58502e10	−0.780944	−0.390472	+	0.920615i	$0.627688\pi$
$882$	0	0
$883$	−1.53480e10	−0.750222	−0.375111	−	0.926980i	$-0.622395\pi$
$883$	−1.53480e10	−0.750222	−0.375111	+	0.926980i	$0.622395\pi$
$884$	3.33237e9	0.162245
$885$	0	0
$886$	−9.96368e9	−0.481284
$887$	−6.55736e9	−0.315498	−0.157749	−	0.987479i	$-0.550424\pi$
$887$	−6.55736e9	−0.315498	−0.157749	+	0.987479i	$0.550424\pi$
$888$	0	0
$889$	1.80368e10	0.861002
$890$	−1.25751e10	−0.597926
$891$	0	0
$892$	6.94679e9	0.327723
$893$	−8.18102e9	−0.384439
$894$	0	0
$895$	2.39426e10	1.11632
$896$	7.27760e9	0.337995
$897$	0	0
$898$	1.66791e10	0.768611
$899$	6.48503e8	0.0297682
$900$	0	0
$901$	−1.03716e10	−0.472398
$902$	5.53226e10	2.51004
$903$	0	0
$904$	1.54117e10	0.693842
$905$	7.74518e9	0.347345
$906$	0	0
$907$	−3.93411e10	−1.75074	−0.875368	−	0.483457i	$-0.839381\pi$
$907$	−3.93411e10	−1.75074	−0.875368	+	0.483457i	$0.839381\pi$
$908$	−9.01733e9	−0.399740
$909$	0	0
$910$	1.47443e10	0.648605
$911$	−3.45238e10	−1.51288	−0.756440	−	0.654064i	$-0.773063\pi$
$911$	−3.45238e10	−1.51288	−0.756440	+	0.654064i	$0.773063\pi$
$912$	0	0
$913$	−8.06914e10	−3.50897
$914$	2.96700e10	1.28530
$915$	0	0
$916$	2.20820e10	0.949301
$917$	2.92948e10	1.25458
$918$	0	0
$919$	−9.91457e9	−0.421376	−0.210688	−	0.977553i	$-0.567570\pi$
$919$	−9.91457e9	−0.421376	−0.210688	+	0.977553i	$0.567570\pi$
$920$	1.11959e10	0.474025
$921$	0	0
$922$	1.21249e10	0.509473
$923$	−9.70157e9	−0.406103
$924$	0	0
$925$	5.82833e9	0.242130
$926$	−8.08281e9	−0.334521
$927$	0	0
$928$	−3.85481e9	−0.158338
$929$	4.62667e10	1.89327	0.946636	−	0.322304i	$-0.104457\pi$
$929$	4.62667e10	1.89327	0.946636	+	0.322304i	$0.104457\pi$
$930$	0	0
$931$	1.50755e10	0.612278
$932$	2.37530e8	0.00961086
$933$	0	0
$934$	−9.14169e9	−0.367124
$935$	−8.97629e9	−0.359134
$936$	0	0
$937$	−2.62108e10	−1.04086	−0.520430	−	0.853904i	$-0.674228\pi$
$937$	−2.62108e10	−1.04086	−0.520430	+	0.853904i	$0.674228\pi$
$938$	−1.86142e10	−0.736436
$939$	0	0
$940$	−5.06889e9	−0.199052
$941$	−2.90508e10	−1.13657	−0.568283	−	0.822833i	$-0.692392\pi$
$941$	−2.90508e10	−1.13657	−0.568283	+	0.822833i	$0.692392\pi$
$942$	0	0
$943$	−3.49584e10	−1.35757
$944$	−2.77390e9	−0.107322
$945$	0	0
$946$	−5.54943e9	−0.213123
$947$	−6.74455e9	−0.258064	−0.129032	−	0.991640i	$-0.541187\pi$
$947$	−6.74455e9	−0.258064	−0.129032	+	0.991640i	$0.541187\pi$
$948$	0	0
$949$	1.91019e10	0.725514
$950$	6.86770e9	0.259883
$951$	0	0
$952$	−1.27833e10	−0.480191
$953$	−1.83022e10	−0.684981	−0.342490	−	0.939521i	$-0.611270\pi$
$953$	−1.83022e10	−0.684981	−0.342490	+	0.939521i	$0.611270\pi$
$954$	0	0
$955$	−1.06046e10	−0.393987
$956$	−5.81462e8	−0.0215238
$957$	0	0
$958$	3.45141e9	0.126828
$959$	2.17187e9	0.0795186
$960$	0	0
$961$	−2.67749e10	−0.973188
$962$	7.78254e9	0.281844
$963$	0	0
$964$	−2.23788e10	−0.804576
$965$	2.13482e10	0.764744
$966$	0	0
$967$	−2.88943e9	−0.102759	−0.0513795	−	0.998679i	$-0.516362\pi$
$967$	−2.88943e9	−0.102759	−0.0513795	+	0.998679i	$0.516362\pi$
$968$	8.49135e10	3.00894
$969$	0	0
$970$	−8.09922e9	−0.284933
$971$	5.32082e10	1.86514	0.932569	−	0.360991i	$-0.117562\pi$
$971$	5.32082e10	1.86514	0.932569	+	0.360991i	$0.117562\pi$
$972$	0	0
$973$	2.20621e8	0.00767806
$974$	−1.50800e10	−0.522933
$975$	0	0
$976$	5.41751e9	0.186520
$977$	1.19053e10	0.408423	0.204211	−	0.978927i	$-0.434537\pi$
$977$	1.19053e10	0.408423	0.204211	+	0.978927i	$0.434537\pi$
$978$	0	0
$979$	8.15816e10	2.77877
$980$	9.34067e9	0.317020
$981$	0	0
$982$	−2.16187e10	−0.728518
$983$	4.93240e10	1.65623	0.828116	−	0.560557i	$-0.189413\pi$
$983$	4.93240e10	1.65623	0.828116	+	0.560557i	$0.189413\pi$
$984$	0	0
$985$	1.37318e10	0.457826
$986$	1.21762e9	0.0404522
$987$	0	0
$988$	−8.80078e9	−0.290317
$989$	3.50669e9	0.115269
$990$	0	0
$991$	3.40025e10	1.10982	0.554911	−	0.831910i	$-0.312752\pi$
$991$	3.40025e10	1.10982	0.554911	+	0.831910i	$0.312752\pi$
$992$	−4.38489e9	−0.142616
$993$	0	0
$994$	1.22341e10	0.395113
$995$	3.11531e10	1.00258
$996$	0	0
$997$	−4.30970e10	−1.37725	−0.688627	−	0.725116i	$-0.741786\pi$
$997$	−4.30970e10	−1.37725	−0.688627	+	0.725116i	$0.741786\pi$
$998$	1.63033e10	0.519179
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	387.8.a.d.1.5		13
3.2	odd	2		43.8.a.b.1.9	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.8.a.b.1.9	✓	13	3.2	odd	2
387.8.a.d.1.5		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:	\( N \)	\(=\)	\( 387 = 3^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	387.a (trivial)

\(f(q)\)	\(=\)	\(q-8.08185 q^{2} -62.6836 q^{4} -164.718 q^{5} +1314.61 q^{7} +1541.08 q^{8} +O(q^{10})\)	\(q-8.08185 q^{2} -62.6836 q^{4} -164.718 q^{5} +1314.61 q^{7} +1541.08 q^{8} +1331.23 q^{10} -8636.39 q^{11} +8425.12 q^{13} -10624.5 q^{14} -4431.25 q^{16} -6309.90 q^{17} +16664.4 q^{19} +10325.1 q^{20} +69798.0 q^{22} -44105.5 q^{23} -50992.9 q^{25} -68090.6 q^{26} -82404.5 q^{28} +23876.9 q^{29} +27160.3 q^{31} -161445. q^{32} +50995.7 q^{34} -216540. q^{35} -114297. q^{37} -134680. q^{38} -253844. q^{40} +792610. q^{41} -79507.0 q^{43} +541361. q^{44} +356454. q^{46} -490927. q^{47} +904653. q^{49} +412117. q^{50} -528117. q^{52} +1.64370e6 q^{53} +1.42257e6 q^{55} +2.02591e6 q^{56} -192970. q^{58} +625984. q^{59} -1.22257e6 q^{61} -219505. q^{62} +1.87198e6 q^{64} -1.38777e6 q^{65} +1.75201e6 q^{67} +395528. q^{68} +1.75005e6 q^{70} -1.15151e6 q^{71} +2.26726e6 q^{73} +923731. q^{74} -1.04459e6 q^{76} -1.13535e7 q^{77} +4.91491e6 q^{79} +729909. q^{80} -6.40576e6 q^{82} +9.34319e6 q^{83} +1.03936e6 q^{85} +642564. q^{86} -1.33093e7 q^{88} -9.44626e6 q^{89} +1.10757e7 q^{91} +2.76469e6 q^{92} +3.96760e6 q^{94} -2.74494e6 q^{95} -6.08401e6 q^{97} -7.31127e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 16 q^{2} + 922 q^{4} - 998 q^{5} + 1360 q^{7} - 3870 q^{8}+O(q^{10}) \) 13 * q - 16 * q^2 + 922 * q^4 - 998 * q^5 + 1360 * q^7 - 3870 * q^8	\( 13 q - 16 q^{2} + 922 q^{4} - 998 q^{5} + 1360 q^{7} - 3870 q^{8} + 4667 q^{10} - 1620 q^{11} + 13550 q^{13} - 44160 q^{14} + 114026 q^{16} - 110880 q^{17} + 105058 q^{19} - 167251 q^{20} + 201504 q^{22} - 160184 q^{23} + 270149 q^{25} - 272104 q^{26} + 208172 q^{28} - 285546 q^{29} - 99616 q^{31} - 200126 q^{32} - 80941 q^{34} + 187104 q^{35} + 176038 q^{37} - 652165 q^{38} - 895387 q^{40} + 410260 q^{41} - 1033591 q^{43} + 2177076 q^{44} - 3975765 q^{46} + 424556 q^{47} - 1561359 q^{49} + 4063801 q^{50} - 4172312 q^{52} - 3992458 q^{53} + 406960 q^{55} - 1559556 q^{56} - 4052005 q^{58} - 2248836 q^{59} + 6210394 q^{61} - 885317 q^{62} - 3096318 q^{64} - 5600420 q^{65} - 1993648 q^{67} - 9327135 q^{68} - 1105098 q^{70} - 4978064 q^{71} + 8224814 q^{73} + 3613563 q^{74} + 10687121 q^{76} - 17261892 q^{77} + 6945708 q^{79} - 15822799 q^{80} - 508449 q^{82} - 22937328 q^{83} - 575532 q^{85} + 1272112 q^{86} + 11202656 q^{88} - 9291302 q^{89} + 25581108 q^{91} + 14388137 q^{92} - 24645805 q^{94} - 30750464 q^{95} + 10001852 q^{97} + 32304856 q^{98}+O(q^{100}) \) 13 * q - 16 * q^2 + 922 * q^4 - 998 * q^5 + 1360 * q^7 - 3870 * q^8 + 4667 * q^10 - 1620 * q^11 + 13550 * q^13 - 44160 * q^14 + 114026 * q^16 - 110880 * q^17 + 105058 * q^19 - 167251 * q^20 + 201504 * q^22 - 160184 * q^23 + 270149 * q^25 - 272104 * q^26 + 208172 * q^28 - 285546 * q^29 - 99616 * q^31 - 200126 * q^32 - 80941 * q^34 + 187104 * q^35 + 176038 * q^37 - 652165 * q^38 - 895387 * q^40 + 410260 * q^41 - 1033591 * q^43 + 2177076 * q^44 - 3975765 * q^46 + 424556 * q^47 - 1561359 * q^49 + 4063801 * q^50 - 4172312 * q^52 - 3992458 * q^53 + 406960 * q^55 - 1559556 * q^56 - 4052005 * q^58 - 2248836 * q^59 + 6210394 * q^61 - 885317 * q^62 - 3096318 * q^64 - 5600420 * q^65 - 1993648 * q^67 - 9327135 * q^68 - 1105098 * q^70 - 4978064 * q^71 + 8224814 * q^73 + 3613563 * q^74 + 10687121 * q^76 - 17261892 * q^77 + 6945708 * q^79 - 15822799 * q^80 - 508449 * q^82 - 22937328 * q^83 - 575532 * q^85 + 1272112 * q^86 + 11202656 * q^88 - 9291302 * q^89 + 25581108 * q^91 + 14388137 * q^92 - 24645805 * q^94 - 30750464 * q^95 + 10001852 * q^97 + 32304856 * q^98

Introduction

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