LMFDB - Embedded newform 387.8.a.b.1.10

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:	$0$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 2 x^{10} - 977 x^{9} + 2592 x^{8} + 344686 x^{7} - 1160956 x^{6} - 53409536 x^{5} + \cdots + 238240894976 $ x^11 - 2x^10 - 977x^9 + 2592x^8 + 344686x^7 - 1160956x^6 - 53409536x^5 + 209758592x^4 + 3410917248x^3 - 14180732672x^2 - 60918607872x + 238240894976
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{4}\cdot 7 $
Twist minimal:	no (minimal twist has level 43)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$14.1572$ 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)

$f(q)$	$=$	$q+16.1572 q^{2} +133.057 q^{4} +272.935 q^{5} +1101.65 q^{7} +81.7025 q^{8} +O(q^{10})$	\(q+16.1572 q^{2} +133.057 q^{4} +272.935 q^{5} +1101.65 q^{7} +81.7025 q^{8} +4409.88 q^{10} -685.243 q^{11} +13204.0 q^{13} +17799.7 q^{14} -15711.2 q^{16} +1155.38 q^{17} -12969.6 q^{19} +36315.9 q^{20} -11071.6 q^{22} +106988. q^{23} -3631.41 q^{25} +213341. q^{26} +146582. q^{28} -71411.7 q^{29} +229489. q^{31} -264307. q^{32} +18667.8 q^{34} +300680. q^{35} -279218. q^{37} -209553. q^{38} +22299.5 q^{40} -345820. q^{41} +79507.0 q^{43} -91176.2 q^{44} +1.72863e6 q^{46} +735243. q^{47} +390095. q^{49} -58673.6 q^{50} +1.75688e6 q^{52} +714374. q^{53} -187027. q^{55} +90007.8 q^{56} -1.15382e6 q^{58} +915243. q^{59} -575.793 q^{61} +3.70792e6 q^{62} -2.25945e6 q^{64} +3.60384e6 q^{65} -3.55836e6 q^{67} +153731. q^{68} +4.85816e6 q^{70} +1.66804e6 q^{71} -1.03269e6 q^{73} -4.51140e6 q^{74} -1.72569e6 q^{76} -754900. q^{77} +1.58783e6 q^{79} -4.28813e6 q^{80} -5.58750e6 q^{82} +8.26293e6 q^{83} +315344. q^{85} +1.28461e6 q^{86} -55986.1 q^{88} -7.66713e6 q^{89} +1.45462e7 q^{91} +1.42354e7 q^{92} +1.18795e7 q^{94} -3.53986e6 q^{95} -1.26607e7 q^{97} +6.30287e6 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + 24 q^{2} + 602 q^{4} + 752 q^{5} - 12 q^{7} + 3810 q^{8}+O(q^{10}) $ 11 * q + 24 * q^2 + 602 * q^4 + 752 * q^5 - 12 * q^7 + 3810 * q^8	\( 11 q + 24 q^{2} + 602 q^{4} + 752 q^{5} - 12 q^{7} + 3810 q^{8} - 1333 q^{10} - 1333 q^{11} - 17967 q^{13} + 22352 q^{14} - 34406 q^{16} + 63095 q^{17} - 54524 q^{19} + 280995 q^{20} - 289358 q^{22} + 138139 q^{23} + 3455 q^{25} + 132946 q^{26} - 12704 q^{28} + 308658 q^{29} - 209523 q^{31} + 644934 q^{32} + 762435 q^{34} + 578892 q^{35} - 298472 q^{37} + 369707 q^{38} + 2633173 q^{40} + 1346735 q^{41} + 874577 q^{43} - 3134292 q^{44} + 3588111 q^{46} - 499284 q^{47} + 2544563 q^{49} - 3049745 q^{50} + 983088 q^{52} + 2210495 q^{53} - 1855072 q^{55} + 469976 q^{56} + 4397067 q^{58} + 5824216 q^{59} - 4453034 q^{61} - 1002789 q^{62} + 4757538 q^{64} + 2162872 q^{65} - 6859513 q^{67} + 9397005 q^{68} + 845078 q^{70} + 10726554 q^{71} - 4456898 q^{73} - 1046637 q^{74} + 5861267 q^{76} + 17019816 q^{77} - 15541320 q^{79} + 15680911 q^{80} + 20233655 q^{82} + 11146767 q^{83} - 12471976 q^{85} + 1908168 q^{86} - 24463544 q^{88} + 13531356 q^{89} - 19746448 q^{91} + 26023161 q^{92} + 20288857 q^{94} + 12291624 q^{95} - 10999901 q^{97} - 29909168 q^{98}+O(q^{100}) \) 11 * q + 24 * q^2 + 602 * q^4 + 752 * q^5 - 12 * q^7 + 3810 * q^8 - 1333 * q^10 - 1333 * q^11 - 17967 * q^13 + 22352 * q^14 - 34406 * q^16 + 63095 * q^17 - 54524 * q^19 + 280995 * q^20 - 289358 * q^22 + 138139 * q^23 + 3455 * q^25 + 132946 * q^26 - 12704 * q^28 + 308658 * q^29 - 209523 * q^31 + 644934 * q^32 + 762435 * q^34 + 578892 * q^35 - 298472 * q^37 + 369707 * q^38 + 2633173 * q^40 + 1346735 * q^41 + 874577 * q^43 - 3134292 * q^44 + 3588111 * q^46 - 499284 * q^47 + 2544563 * q^49 - 3049745 * q^50 + 983088 * q^52 + 2210495 * q^53 - 1855072 * q^55 + 469976 * q^56 + 4397067 * q^58 + 5824216 * q^59 - 4453034 * q^61 - 1002789 * q^62 + 4757538 * q^64 + 2162872 * q^65 - 6859513 * q^67 + 9397005 * q^68 + 845078 * q^70 + 10726554 * q^71 - 4456898 * q^73 - 1046637 * q^74 + 5861267 * q^76 + 17019816 * q^77 - 15541320 * q^79 + 15680911 * q^80 + 20233655 * q^82 + 11146767 * q^83 - 12471976 * q^85 + 1908168 * q^86 - 24463544 * q^88 + 13531356 * q^89 - 19746448 * q^91 + 26023161 * q^92 + 20288857 * q^94 + 12291624 * q^95 - 10999901 * q^97 - 29909168 * 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 $

$2$	16.1572	1.42811	0.714056	−	0.700088i	$-0.246856\pi$
$2$	16.1572	1.42811	0.714056	+	0.700088i	$0.246856\pi$
$3$	0	0
$3$	0	0
$4$	133.057	1.03951
$5$	272.935	0.976482	0.488241	−	0.872709i	$-0.337639\pi$
$5$	272.935	0.976482	0.488241	+	0.872709i	$0.337639\pi$
$6$	0	0
$7$	1101.65	1.21395	0.606976	−	0.794720i	$-0.292382\pi$
$7$	1101.65	1.21395	0.606976	+	0.794720i	$0.292382\pi$
$8$	81.7025	0.0564184
$9$	0	0
$10$	4409.88	1.39453
$11$	−685.243	−0.155228	−0.0776140	−	0.996983i	$-0.524730\pi$
$11$	−685.243	−0.155228	−0.0776140	+	0.996983i	$0.524730\pi$
$12$	0	0
$13$	13204.0	1.66688	0.833440	−	0.552609i	$-0.186368\pi$
$13$	13204.0	1.66688	0.833440	+	0.552609i	$0.186368\pi$
$14$	17799.7	1.73366
$15$	0	0
$16$	−15711.2	−0.958934
$17$	1155.38	0.0570367	0.0285184	−	0.999593i	$-0.490921\pi$
$17$	1155.38	0.0570367	0.0285184	+	0.999593i	$0.490921\pi$
$18$	0	0
$19$	−12969.6	−0.433800	−0.216900	−	0.976194i	$-0.569594\pi$
$19$	−12969.6	−0.433800	−0.216900	+	0.976194i	$0.569594\pi$
$20$	36315.9	1.01506
$21$	0	0
$22$	−11071.6	−0.221683
$23$	106988.	1.83352	0.916761	−	0.399436i	$-0.130794\pi$
$23$	106988.	1.83352	0.916761	+	0.399436i	$0.130794\pi$
$24$	0	0
$25$	−3631.41	−0.0464820
$26$	213341.	2.38049
$27$	0	0
$28$	146582.	1.26191
$29$	−71411.7	−0.543721	−0.271861	−	0.962337i	$-0.587639\pi$
$29$	−71411.7	−0.543721	−0.271861	+	0.962337i	$0.587639\pi$
$30$	0	0
$31$	229489.	1.38356	0.691778	−	0.722111i	$-0.256828\pi$
$31$	229489.	1.38356	0.691778	+	0.722111i	$0.256828\pi$
$32$	−264307.	−1.42588
$33$	0	0
$34$	18667.8	0.0814549
$35$	300680.	1.18540
$36$	0	0
$37$	−279218.	−0.906229	−0.453115	−	0.891452i	$-0.649687\pi$
$37$	−279218.	−0.906229	−0.453115	+	0.891452i	$0.649687\pi$
$38$	−209553.	−0.619515
$39$	0	0
$40$	22299.5	0.0550915
$41$	−345820.	−0.783622	−0.391811	−	0.920046i	$-0.628151\pi$
$41$	−345820.	−0.783622	−0.391811	+	0.920046i	$0.628151\pi$
$42$	0	0
$43$	79507.0	0.152499
$43$	79507.0	0.152499
$44$	−91176.2	−0.161360
$45$	0	0
$46$	1.72863e6	2.61848
$47$	735243.	1.03297	0.516486	−	0.856296i	$-0.327240\pi$
$47$	735243.	1.03297	0.516486	+	0.856296i	$0.327240\pi$
$48$	0	0
$49$	390095.	0.473680
$50$	−58673.6	−0.0663816
$51$	0	0
$52$	1.75688e6	1.73273
$53$	714374.	0.659113	0.329557	−	0.944136i	$-0.393101\pi$
$53$	714374.	0.659113	0.329557	+	0.944136i	$0.393101\pi$
$54$	0	0
$55$	−187027.	−0.151577
$56$	90007.8	0.0684892
$57$	0	0
$58$	−1.15382e6	−0.776495
$59$	915243.	0.580169	0.290084	−	0.957001i	$-0.406317\pi$
$59$	915243.	0.580169	0.290084	+	0.957001i	$0.406317\pi$
$60$	0	0
$61$	−575.793	−0.000324797 0	−0.000162399	−	1.00000i	$-0.500052\pi$
$61$	−575.793	−0.000324797 0	−0.000162399		1.00000i	$0.500052\pi$
$62$	3.70792e6	1.97587
$63$	0	0
$64$	−2.25945e6	−1.07739
$65$	3.60384e6	1.62768
$66$	0	0
$67$	−3.55836e6	−1.44540	−0.722700	−	0.691162i	$-0.757099\pi$
$67$	−3.55836e6	−1.44540	−0.722700	+	0.691162i	$0.757099\pi$
$68$	153731.	0.0592900
$69$	0	0
$70$	4.85816e6	1.69289
$71$	1.66804e6	0.553099	0.276549	−	0.961000i	$-0.410809\pi$
$71$	1.66804e6	0.553099	0.276549	+	0.961000i	$0.410809\pi$
$72$	0	0
$73$	−1.03269e6	−0.310700	−0.155350	−	0.987860i	$-0.549650\pi$
$73$	−1.03269e6	−0.310700	−0.155350	+	0.987860i	$0.549650\pi$
$74$	−4.51140e6	−1.29420
$75$	0	0
$76$	−1.72569e6	−0.450937
$77$	−754900.	−0.188439
$78$	0	0
$79$	1.58783e6	0.362334	0.181167	−	0.983452i	$-0.442013\pi$
$79$	1.58783e6	0.362334	0.181167	+	0.983452i	$0.442013\pi$
$80$	−4.28813e6	−0.936382
$81$	0	0
$82$	−5.58750e6	−1.11910
$83$	8.26293e6	1.58621	0.793105	−	0.609085i	$-0.208463\pi$
$83$	8.26293e6	1.58621	0.793105	+	0.609085i	$0.208463\pi$
$84$	0	0
$85$	315344.	0.0556954
$86$	1.28461e6	0.217785
$87$	0	0
$88$	−55986.1	−0.00875771
$89$	−7.66713e6	−1.15284	−0.576418	−	0.817155i	$-0.695550\pi$
$89$	−7.66713e6	−1.15284	−0.576418	+	0.817155i	$0.695550\pi$
$90$	0	0
$91$	1.45462e7	2.02351
$92$	1.42354e7	1.90596
$93$	0	0
$94$	1.18795e7	1.47520
$95$	−3.53986e6	−0.423598
$96$	0	0
$97$	−1.26607e7	−1.40850	−0.704251	−	0.709951i	$-0.748717\pi$
$97$	−1.26607e7	−1.40850	−0.704251	+	0.709951i	$0.748717\pi$
$98$	6.30287e6	0.676468
$99$	0	0
$100$	−483183.	−0.0483183
$101$	7.42880e6	0.717454	0.358727	−	0.933443i	$-0.383211\pi$
$101$	7.42880e6	0.717454	0.358727	+	0.933443i	$0.383211\pi$
$102$	0	0
$103$	1.57866e7	1.42350	0.711752	−	0.702431i	$-0.247902\pi$
$103$	1.57866e7	1.42350	0.711752	+	0.702431i	$0.247902\pi$
$104$	1.07880e6	0.0940427
$105$	0	0
$106$	1.15423e7	0.941288
$107$	1.40430e6	0.110819	0.0554097	−	0.998464i	$-0.482354\pi$
$107$	1.40430e6	0.110819	0.0554097	+	0.998464i	$0.482354\pi$
$108$	0	0
$109$	−6.10839e6	−0.451787	−0.225894	−	0.974152i	$-0.572530\pi$
$109$	−6.10839e6	−0.451787	−0.225894	+	0.974152i	$0.572530\pi$
$110$	−3.02184e6	−0.216470
$111$	0	0
$112$	−1.73083e7	−1.16410
$113$	8.10527e6	0.528437	0.264218	−	0.964463i	$-0.414886\pi$
$113$	8.10527e6	0.528437	0.264218	+	0.964463i	$0.414886\pi$
$114$	0	0
$115$	2.92007e7	1.79040
$116$	−9.50180e6	−0.565201
$117$	0	0
$118$	1.47878e7	0.828546
$119$	1.27283e6	0.0692398
$120$	0	0
$121$	−1.90176e7	−0.975904
$122$	−9303.23	−0.000463847 0
$123$	0	0
$124$	3.05351e7	1.43821
$125$	−2.23142e7	−1.02187
$126$	0	0
$127$	3.81416e7	1.65229	0.826145	−	0.563457i	$-0.190529\pi$
$127$	3.81416e7	1.65229	0.826145	+	0.563457i	$0.190529\pi$
$128$	−2.67514e6	−0.112749
$129$	0	0
$130$	5.82282e7	2.32451
$131$	−476451.	−0.0185169	−0.00925845	−	0.999957i	$-0.502947\pi$
$131$	−476451.	−0.0185169	−0.00925845	+	0.999957i	$0.502947\pi$
$132$	0	0
$133$	−1.42880e7	−0.526612
$134$	−5.74933e7	−2.06419
$135$	0	0
$136$	94397.6	0.00321792
$137$	5.02778e7	1.67053	0.835265	−	0.549848i	$-0.185314\pi$
$137$	5.02778e7	1.67053	0.835265	+	0.549848i	$0.185314\pi$
$138$	0	0
$139$	−5.83450e7	−1.84269	−0.921344	−	0.388748i	$-0.872908\pi$
$139$	−5.83450e7	−1.84269	−0.921344	+	0.388748i	$0.872908\pi$
$140$	4.00075e7	1.23223
$141$	0	0
$142$	2.69510e7	0.789887
$143$	−9.04796e6	−0.258747
$144$	0	0
$145$	−1.94908e7	−0.530934
$146$	−1.66855e7	−0.443714
$147$	0	0
$148$	−3.71519e7	−0.942030
$149$	6.73082e7	1.66693	0.833463	−	0.552575i	$-0.186355\pi$
$149$	6.73082e7	1.66693	0.833463	+	0.552575i	$0.186355\pi$
$150$	0	0
$151$	5.87220e7	1.38797	0.693987	−	0.719988i	$-0.255853\pi$
$151$	5.87220e7	1.38797	0.693987	+	0.719988i	$0.255853\pi$
$152$	−1.05965e6	−0.0244743
$153$	0	0
$154$	−1.21971e7	−0.269113
$155$	6.26357e7	1.35102
$156$	0	0
$157$	−5.66713e6	−0.116873	−0.0584365	−	0.998291i	$-0.518612\pi$
$157$	−5.66713e6	−0.116873	−0.0584365	+	0.998291i	$0.518612\pi$
$158$	2.56550e7	0.517454
$159$	0	0
$160$	−7.21387e7	−1.39235
$161$	1.17863e8	2.22581
$162$	0	0
$163$	2.44165e7	0.441598	0.220799	−	0.975319i	$-0.429134\pi$
$163$	2.44165e7	0.441598	0.220799	+	0.975319i	$0.429134\pi$
$164$	−4.60136e7	−0.814579
$165$	0	0
$166$	1.33506e8	2.26529
$167$	−1.01135e7	−0.168033	−0.0840164	−	0.996464i	$-0.526775\pi$
$167$	−1.01135e7	−0.168033	−0.0840164	+	0.996464i	$0.526775\pi$
$168$	0	0
$169$	1.11598e8	1.77849
$170$	5.09510e6	0.0795392
$171$	0	0
$172$	1.05789e7	0.158523
$173$	4.24908e7	0.623927	0.311963	−	0.950094i	$-0.399013\pi$
$173$	4.24908e7	0.623927	0.311963	+	0.950094i	$0.399013\pi$
$174$	0	0
$175$	−4.00055e6	−0.0564270
$176$	1.07660e7	0.148853
$177$	0	0
$178$	−1.23880e8	−1.64638
$179$	−8.27387e7	−1.07826	−0.539129	−	0.842223i	$-0.681247\pi$
$179$	−8.27387e7	−1.07826	−0.539129	+	0.842223i	$0.681247\pi$
$180$	0	0
$181$	1.19734e7	0.150087	0.0750435	−	0.997180i	$-0.476090\pi$
$181$	1.19734e7	0.150087	0.0750435	+	0.997180i	$0.476090\pi$
$182$	2.35027e8	2.88980
$183$	0	0
$184$	8.74116e6	0.103444
$185$	−7.62085e7	−0.884917
$186$	0	0
$187$	−791718.	−0.00885370
$188$	9.78290e7	1.07378
$189$	0	0
$190$	−5.71944e7	−0.604945
$191$	−1.61956e8	−1.68183	−0.840914	−	0.541169i	$-0.817982\pi$
$191$	−1.61956e8	−1.68183	−0.840914	+	0.541169i	$0.817982\pi$
$192$	0	0
$193$	−8.69257e7	−0.870357	−0.435178	−	0.900344i	$-0.643315\pi$
$193$	−8.69257e7	−0.870357	−0.435178	+	0.900344i	$0.643315\pi$
$194$	−2.04562e8	−2.01150
$195$	0	0
$196$	5.19048e7	0.492393
$197$	1.25816e8	1.17247	0.586237	−	0.810139i	$-0.300609\pi$
$197$	1.25816e8	1.17247	0.586237	+	0.810139i	$0.300609\pi$
$198$	0	0
$199$	−1.43871e8	−1.29416	−0.647079	−	0.762423i	$-0.724009\pi$
$199$	−1.43871e8	−1.29416	−0.647079	+	0.762423i	$0.724009\pi$
$200$	−296695.	−0.00262244
$201$	0	0
$202$	1.20029e8	1.02460
$203$	−7.86708e7	−0.660051
$204$	0	0
$205$	−9.43864e7	−0.765193
$206$	2.55068e8	2.03292
$207$	0	0
$208$	−2.07451e8	−1.59843
$209$	8.88733e6	0.0673379
$210$	0	0
$211$	−1.83659e8	−1.34593	−0.672966	−	0.739673i	$-0.734980\pi$
$211$	−1.83659e8	−1.34593	−0.672966	+	0.739673i	$0.734980\pi$
$212$	9.50523e7	0.685152
$213$	0	0
$214$	2.26896e7	0.158262
$215$	2.17003e7	0.148912
$216$	0	0
$217$	2.52817e8	1.67957
$218$	−9.86948e7	−0.645203
$219$	0	0
$220$	−2.48852e7	−0.157566
$221$	1.52557e7	0.0950734
$222$	0	0
$223$	6.03715e7	0.364557	0.182278	−	0.983247i	$-0.441653\pi$
$223$	6.03715e7	0.364557	0.182278	+	0.983247i	$0.441653\pi$
$224$	−2.91175e8	−1.73095
$225$	0	0
$226$	1.30959e8	0.754667
$227$	−5.49050e7	−0.311545	−0.155773	−	0.987793i	$-0.549787\pi$
$227$	−5.49050e7	−0.311545	−0.155773	+	0.987793i	$0.549787\pi$
$228$	0	0
$229$	−1.05868e8	−0.582560	−0.291280	−	0.956638i	$-0.594081\pi$
$229$	−1.05868e8	−0.582560	−0.291280	+	0.956638i	$0.594081\pi$
$230$	4.71803e8	2.55690
$231$	0	0
$232$	−5.83451e6	−0.0306758
$233$	−4.21892e7	−0.218502	−0.109251	−	0.994014i	$-0.534845\pi$
$233$	−4.21892e7	−0.218502	−0.109251	+	0.994014i	$0.534845\pi$
$234$	0	0
$235$	2.00674e8	1.00868
$236$	1.21779e8	0.603089
$237$	0	0
$238$	2.05654e7	0.0988823
$239$	−1.65597e8	−0.784621	−0.392310	−	0.919833i	$-0.628324\pi$
$239$	−1.65597e8	−0.784621	−0.392310	+	0.919833i	$0.628324\pi$
$240$	0	0
$241$	−1.17817e8	−0.542185	−0.271093	−	0.962553i	$-0.587385\pi$
$241$	−1.17817e8	−0.542185	−0.271093	+	0.962553i	$0.587385\pi$
$242$	−3.07272e8	−1.39370
$243$	0	0
$244$	−76613.1	−0.000337628 0
$245$	1.06471e8	0.462540
$246$	0	0
$247$	−1.71251e8	−0.723092
$248$	1.87498e7	0.0780579
$249$	0	0
$250$	−3.60536e8	−1.45935
$251$	1.68271e8	0.671662	0.335831	−	0.941922i	$-0.390983\pi$
$251$	1.68271e8	0.671662	0.335831	+	0.941922i	$0.390983\pi$
$252$	0	0
$253$	−7.33125e7	−0.284614
$254$	6.16264e8	2.35966
$255$	0	0
$256$	2.45986e8	0.916371
$257$	−4.08556e8	−1.50136	−0.750681	−	0.660665i	$-0.770274\pi$
$257$	−4.08556e8	−1.50136	−0.750681	+	0.660665i	$0.770274\pi$
$258$	0	0
$259$	−3.07602e8	−1.10012
$260$	4.79515e8	1.69198
$261$	0	0
$262$	−7.69813e6	−0.0264442
$263$	−2.68865e8	−0.911357	−0.455679	−	0.890144i	$-0.650603\pi$
$263$	−2.68865e8	−0.911357	−0.455679	+	0.890144i	$0.650603\pi$
$264$	0	0
$265$	1.94978e8	0.643613
$266$	−2.30855e8	−0.752061
$267$	0	0
$268$	−4.73464e8	−1.50250
$269$	4.68785e8	1.46839	0.734194	−	0.678939i	$-0.237560\pi$
$269$	4.68785e8	1.46839	0.734194	+	0.678939i	$0.237560\pi$
$270$	0	0
$271$	7.19335e7	0.219553	0.109776	−	0.993956i	$-0.464987\pi$
$271$	7.19335e7	0.219553	0.109776	+	0.993956i	$0.464987\pi$
$272$	−1.81524e7	−0.0546944
$273$	0	0
$274$	8.12351e8	2.38570
$275$	2.48840e6	0.00721532
$276$	0	0
$277$	−1.86677e8	−0.527729	−0.263864	−	0.964560i	$-0.584997\pi$
$277$	−1.86677e8	−0.527729	−0.263864	+	0.964560i	$0.584997\pi$
$278$	−9.42695e8	−2.63157
$279$	0	0
$280$	2.45663e7	0.0668785
$281$	−4.97318e8	−1.33709	−0.668547	−	0.743670i	$-0.733083\pi$
$281$	−4.97318e8	−1.33709	−0.668547	+	0.743670i	$0.733083\pi$
$282$	0	0
$283$	−5.31540e8	−1.39407	−0.697033	−	0.717039i	$-0.745497\pi$
$283$	−5.31540e8	−1.39407	−0.697033	+	0.717039i	$0.745497\pi$
$284$	2.21944e8	0.574949
$285$	0	0
$286$	−1.46190e8	−0.369519
$287$	−3.80973e8	−0.951279
$288$	0	0
$289$	−4.09004e8	−0.996747
$290$	−3.14917e8	−0.758234
$291$	0	0
$292$	−1.37407e8	−0.322974
$293$	6.12847e8	1.42336	0.711680	−	0.702503i	$-0.247934\pi$
$293$	6.12847e8	1.42336	0.711680	+	0.702503i	$0.247934\pi$
$294$	0	0
$295$	2.49802e8	0.566525
$296$	−2.28128e7	−0.0511280
$297$	0	0
$298$	1.08752e9	2.38056
$299$	1.41267e9	3.05626
$300$	0	0
$301$	8.75891e7	0.185126
$302$	9.48785e8	1.98218
$303$	0	0
$304$	2.03768e8	0.415985
$305$	−157154.	−0.000317159 0
$306$	0	0
$307$	−5.15243e8	−1.01631	−0.508157	−	0.861264i	$-0.669673\pi$
$307$	−5.15243e8	−1.01631	−0.508157	+	0.861264i	$0.669673\pi$
$308$	−1.00444e8	−0.195884
$309$	0	0
$310$	1.01202e9	1.92940
$311$	−6.25714e8	−1.17954	−0.589772	−	0.807570i	$-0.700782\pi$
$311$	−6.25714e8	−1.17954	−0.589772	+	0.807570i	$0.700782\pi$
$312$	0	0
$313$	−6.49995e8	−1.19813	−0.599066	−	0.800699i	$-0.704461\pi$
$313$	−6.49995e8	−1.19813	−0.599066	+	0.800699i	$0.704461\pi$
$314$	−9.15652e7	−0.166908
$315$	0	0
$316$	2.11271e8	0.376648
$317$	−7.98060e8	−1.40711	−0.703555	−	0.710641i	$-0.748405\pi$
$317$	−7.98060e8	−1.40711	−0.703555	+	0.710641i	$0.748405\pi$
$318$	0	0
$319$	4.89343e7	0.0844008
$320$	−6.16683e8	−1.05205
$321$	0	0
$322$	1.90435e9	3.17870
$323$	−1.49849e7	−0.0247425
$324$	0	0
$325$	−4.79492e7	−0.0774800
$326$	3.94504e8	0.630652
$327$	0	0
$328$	−2.82543e7	−0.0442106
$329$	8.09983e8	1.25398
$330$	0	0
$331$	8.36375e8	1.26766	0.633831	−	0.773472i	$-0.281482\pi$
$331$	8.36375e8	1.26766	0.633831	+	0.773472i	$0.281482\pi$
$332$	1.09944e9	1.64887
$333$	0	0
$334$	−1.63406e8	−0.239970
$335$	−9.71202e8	−1.41141
$336$	0	0
$337$	1.03465e9	1.47262	0.736310	−	0.676644i	$-0.236566\pi$
$337$	1.03465e9	1.47262	0.736310	+	0.676644i	$0.236566\pi$
$338$	1.80311e9	2.53989
$339$	0	0
$340$	4.19587e7	0.0578956
$341$	−1.57256e8	−0.214767
$342$	0	0
$343$	−4.77509e8	−0.638928
$344$	6.49592e6	0.00860372
$345$	0	0
$346$	6.86534e8	0.891037
$347$	−1.89330e8	−0.243258	−0.121629	−	0.992576i	$-0.538812\pi$
$347$	−1.89330e8	−0.243258	−0.121629	+	0.992576i	$0.538812\pi$
$348$	0	0
$349$	3.29547e8	0.414981	0.207490	−	0.978237i	$-0.433470\pi$
$349$	3.29547e8	0.414981	0.207490	+	0.978237i	$0.433470\pi$
$350$	−6.46379e7	−0.0805841
$351$	0	0
$352$	1.81115e8	0.221337
$353$	1.18738e9	1.43674	0.718371	−	0.695661i	$-0.244888\pi$
$353$	1.18738e9	1.43674	0.718371	+	0.695661i	$0.244888\pi$
$354$	0	0
$355$	4.55267e8	0.540091
$356$	−1.02016e9	−1.19838
$357$	0	0
$358$	−1.33683e9	−1.53987
$359$	7.45557e7	0.0850453	0.0425226	−	0.999096i	$-0.486461\pi$
$359$	7.45557e7	0.0850453	0.0425226	+	0.999096i	$0.486461\pi$
$360$	0	0
$361$	−7.25661e8	−0.811818
$362$	1.93458e8	0.214341
$363$	0	0
$364$	1.93548e9	2.10345
$365$	−2.81858e8	−0.303393
$366$	0	0
$367$	−1.32724e9	−1.40158	−0.700789	−	0.713368i	$-0.747169\pi$
$367$	−1.32724e9	−1.40158	−0.700789	+	0.713368i	$0.747169\pi$
$368$	−1.68090e9	−1.75823
$369$	0	0
$370$	−1.23132e9	−1.26376
$371$	7.86992e8	0.800132
$372$	0	0
$373$	1.54991e8	0.154641	0.0773207	−	0.997006i	$-0.475363\pi$
$373$	1.54991e8	0.154641	0.0773207	+	0.997006i	$0.475363\pi$
$374$	−1.27920e7	−0.0126441
$375$	0	0
$376$	6.00712e7	0.0582786
$377$	−9.42921e8	−0.906318
$378$	0	0
$379$	−1.15983e9	−1.09435	−0.547174	−	0.837019i	$-0.684296\pi$
$379$	−1.15983e9	−1.09435	−0.547174	+	0.837019i	$0.684296\pi$
$380$	−4.71002e8	−0.440332
$381$	0	0
$382$	−2.61677e9	−2.40184
$383$	8.33044e8	0.757656	0.378828	−	0.925467i	$-0.376327\pi$
$383$	8.33044e8	0.757656	0.378828	+	0.925467i	$0.376327\pi$
$384$	0	0
$385$	−2.06039e8	−0.184008
$386$	−1.40448e9	−1.24297
$387$	0	0
$388$	−1.68459e9	−1.46415
$389$	3.11695e8	0.268477	0.134238	−	0.990949i	$-0.457141\pi$
$389$	3.11695e8	0.268477	0.134238	+	0.990949i	$0.457141\pi$
$390$	0	0
$391$	1.23612e8	0.104578
$392$	3.18718e7	0.0267242
$393$	0	0
$394$	2.03284e9	1.67443
$395$	4.33375e8	0.353813
$396$	0	0
$397$	−2.87056e8	−0.230250	−0.115125	−	0.993351i	$-0.536727\pi$
$397$	−2.87056e8	−0.230250	−0.115125	+	0.993351i	$0.536727\pi$
$398$	−2.32456e9	−1.84820
$399$	0	0
$400$	5.70537e7	0.0445732
$401$	−8.00363e8	−0.619843	−0.309921	−	0.950762i	$-0.600303\pi$
$401$	−8.00363e8	−0.619843	−0.309921	+	0.950762i	$0.600303\pi$
$402$	0	0
$403$	3.03018e9	2.30622
$404$	9.88452e8	0.745797
$405$	0	0
$406$	−1.27110e9	−0.942627
$407$	1.91332e8	0.140672
$408$	0	0
$409$	−1.90519e9	−1.37691	−0.688456	−	0.725278i	$-0.741711\pi$
$409$	−1.90519e9	−1.37691	−0.688456	+	0.725278i	$0.741711\pi$
$410$	−1.52502e9	−1.09278
$411$	0	0
$412$	2.10052e9	1.47974
$413$	1.00828e9	0.704297
$414$	0	0
$415$	2.25524e9	1.54891
$416$	−3.48992e9	−2.37678
$417$	0	0
$418$	1.43595e8	0.0961660
$419$	−1.79896e9	−1.19474	−0.597369	−	0.801967i	$-0.703787\pi$
$419$	−1.79896e9	−1.19474	−0.597369	+	0.801967i	$0.703787\pi$
$420$	0	0
$421$	3.47444e7	0.0226933	0.0113467	−	0.999936i	$-0.496388\pi$
$421$	3.47444e7	0.0226933	0.0113467	+	0.999936i	$0.496388\pi$
$422$	−2.96742e9	−1.92214
$423$	0	0
$424$	5.83661e7	0.0371861
$425$	−4.19567e6	−0.00265118
$426$	0	0
$427$	−634324.	−0.000394288 0
$428$	1.86851e8	0.115197
$429$	0	0
$430$	3.50616e8	0.212663
$431$	−1.91702e9	−1.15334	−0.576669	−	0.816978i	$-0.695648\pi$
$431$	−1.91702e9	−1.15334	−0.576669	+	0.816978i	$0.695648\pi$
$432$	0	0
$433$	−7.40677e8	−0.438451	−0.219226	−	0.975674i	$-0.570353\pi$
$433$	−7.40677e8	−0.438451	−0.219226	+	0.975674i	$0.570353\pi$
$434$	4.08483e9	2.39861
$435$	0	0
$436$	−8.12763e8	−0.469636
$437$	−1.38759e9	−0.795381
$438$	0	0
$439$	2.96692e9	1.67371	0.836856	−	0.547423i	$-0.184391\pi$
$439$	2.96692e9	1.67371	0.836856	+	0.547423i	$0.184391\pi$
$440$	−1.52806e7	−0.00855175
$441$	0	0
$442$	2.46490e8	0.135776
$443$	−1.34329e9	−0.734102	−0.367051	−	0.930201i	$-0.619633\pi$
$443$	−1.34329e9	−0.734102	−0.367051	+	0.930201i	$0.619633\pi$
$444$	0	0
$445$	−2.09263e9	−1.12572
$446$	9.75438e8	0.520628
$447$	0	0
$448$	−2.48913e9	−1.30790
$449$	3.37789e9	1.76110	0.880549	−	0.473954i	$-0.157174\pi$
$449$	3.37789e9	1.76110	0.880549	+	0.473954i	$0.157174\pi$
$450$	0	0
$451$	2.36971e8	0.121640
$452$	1.07846e9	0.549313
$453$	0	0
$454$	−8.87114e8	−0.444922
$455$	3.97018e9	1.97593
$456$	0	0
$457$	2.50731e7	0.0122886	0.00614429	−	0.999981i	$-0.498044\pi$
$457$	2.50731e7	0.0122886	0.00614429	+	0.999981i	$0.498044\pi$
$458$	−1.71054e9	−0.831961
$459$	0	0
$460$	3.88535e9	1.86113
$461$	7.94770e8	0.377823	0.188911	−	0.981994i	$-0.439504\pi$
$461$	7.94770e8	0.377823	0.188911	+	0.981994i	$0.439504\pi$
$462$	0	0
$463$	−3.63682e9	−1.70290	−0.851449	−	0.524438i	$-0.824276\pi$
$463$	−3.63682e9	−1.70290	−0.851449	+	0.524438i	$0.824276\pi$
$464$	1.12196e9	0.521392
$465$	0	0
$466$	−6.81662e8	−0.312046
$467$	2.11944e8	0.0962967	0.0481483	−	0.998840i	$-0.484668\pi$
$467$	2.11944e8	0.0962967	0.0481483	+	0.998840i	$0.484668\pi$
$468$	0	0
$469$	−3.92008e9	−1.75465
$470$	3.24233e9	1.44051
$471$	0	0
$472$	7.47777e7	0.0327322
$473$	−5.44816e7	−0.0236721
$474$	0	0
$475$	4.70979e7	0.0201639
$476$	1.69359e8	0.0719752
$477$	0	0
$478$	−2.67559e9	−1.12053
$479$	−2.49114e9	−1.03568	−0.517838	−	0.855479i	$-0.673263\pi$
$479$	−2.49114e9	−1.03568	−0.517838	+	0.855479i	$0.673263\pi$
$480$	0	0
$481$	−3.68681e9	−1.51058
$482$	−1.90360e9	−0.774302
$483$	0	0
$484$	−2.53042e9	−1.01446
$485$	−3.45556e9	−1.37538
$486$	0	0
$487$	−6.93003e8	−0.271884	−0.135942	−	0.990717i	$-0.543406\pi$
$487$	−6.93003e8	−0.271884	−0.135942	+	0.990717i	$0.543406\pi$
$488$	−47043.7	−1.83245e−5 0
$489$	0	0
$490$	1.72027e9	0.660559
$491$	2.43700e9	0.929118	0.464559	−	0.885542i	$-0.346213\pi$
$491$	2.43700e9	0.929118	0.464559	+	0.885542i	$0.346213\pi$
$492$	0	0
$493$	−8.25078e7	−0.0310121
$494$	−2.76694e9	−1.03266
$495$	0	0
$496$	−3.60555e9	−1.32674
$497$	1.83760e9	0.671435
$498$	0	0
$499$	−3.89667e9	−1.40392	−0.701959	−	0.712217i	$-0.747691\pi$
$499$	−3.89667e9	−1.40392	−0.701959	+	0.712217i	$0.747691\pi$
$500$	−2.96905e9	−1.06224
$501$	0	0
$502$	2.71879e9	0.959209
$503$	−4.29376e9	−1.50435	−0.752176	−	0.658962i	$-0.770996\pi$
$503$	−4.29376e9	−1.50435	−0.752176	+	0.658962i	$0.770996\pi$
$504$	0	0
$505$	2.02758e9	0.700581
$506$	−1.18453e9	−0.406461
$507$	0	0
$508$	5.07500e9	1.71756
$509$	−3.04510e9	−1.02350	−0.511751	−	0.859134i	$-0.671003\pi$
$509$	−3.04510e9	−1.02350	−0.511751	+	0.859134i	$0.671003\pi$
$510$	0	0
$511$	−1.13767e9	−0.377175
$512$	4.31688e9	1.42143
$513$	0	0
$514$	−6.60114e9	−2.14411
$515$	4.30872e9	1.39003
$516$	0	0
$517$	−5.03820e8	−0.160346
$518$	−4.97000e9	−1.57109
$519$	0	0
$520$	2.94443e8	0.0918310
$521$	−6.13864e9	−1.90169	−0.950846	−	0.309665i	$-0.899783\pi$
$521$	−6.13864e9	−1.90169	−0.950846	+	0.309665i	$0.899783\pi$
$522$	0	0
$523$	−5.32103e9	−1.62645	−0.813223	−	0.581953i	$-0.802289\pi$
$523$	−5.32103e9	−1.62645	−0.813223	+	0.581953i	$0.802289\pi$
$524$	−6.33949e7	−0.0192484
$525$	0	0
$526$	−4.34411e9	−1.30152
$527$	2.65148e8	0.0789134
$528$	0	0
$529$	8.04153e9	2.36180
$530$	3.15030e9	0.919151
$531$	0	0
$532$	−1.90111e9	−0.547416
$533$	−4.56621e9	−1.30620
$534$	0	0
$535$	3.83282e8	0.108213
$536$	−2.90727e8	−0.0815471
$537$	0	0
$538$	7.57428e9	2.09702
$539$	−2.67310e8	−0.0735284
$540$	0	0
$541$	2.71071e9	0.736026	0.368013	−	0.929821i	$-0.380038\pi$
$541$	2.71071e9	0.736026	0.368013	+	0.929821i	$0.380038\pi$
$542$	1.16225e9	0.313546
$543$	0	0
$544$	−3.05376e8	−0.0813277
$545$	−1.66720e9	−0.441163
$546$	0	0
$547$	−2.01310e9	−0.525908	−0.262954	−	0.964808i	$-0.584697\pi$
$547$	−2.01310e9	−0.525908	−0.262954	+	0.964808i	$0.584697\pi$
$548$	6.68980e9	1.73652
$549$	0	0
$550$	4.02057e7	0.0103043
$551$	9.26181e8	0.235866
$552$	0	0
$553$	1.74924e9	0.439856
$554$	−3.01618e9	−0.753656
$555$	0	0
$556$	−7.76319e9	−1.91548
$557$	−1.55537e9	−0.381364	−0.190682	−	0.981652i	$-0.561070\pi$
$557$	−1.55537e9	−0.381364	−0.190682	+	0.981652i	$0.561070\pi$
$558$	0	0
$559$	1.04981e9	0.254197
$560$	−4.72403e9	−1.13672
$561$	0	0
$562$	−8.03529e9	−1.90952
$563$	5.00376e9	1.18173	0.590863	−	0.806772i	$-0.298787\pi$
$563$	5.00376e9	1.18173	0.590863	+	0.806772i	$0.298787\pi$
$564$	0	0
$565$	2.21221e9	0.516009
$566$	−8.58822e9	−1.99088
$567$	0	0
$568$	1.36283e8	0.0312049
$569$	4.14184e9	0.942542	0.471271	−	0.881988i	$-0.343795\pi$
$569$	4.14184e9	0.942542	0.471271	+	0.881988i	$0.343795\pi$
$570$	0	0
$571$	7.78342e9	1.74962	0.874810	−	0.484465i	$-0.160986\pi$
$571$	7.78342e9	1.74962	0.874810	+	0.484465i	$0.160986\pi$
$572$	−1.20389e9	−0.268969
$573$	0	0
$574$	−6.15548e9	−1.35853
$575$	−3.88516e8	−0.0852259
$576$	0	0
$577$	−2.29128e9	−0.496551	−0.248275	−	0.968690i	$-0.579864\pi$
$577$	−2.29128e9	−0.496551	−0.248275	+	0.968690i	$0.579864\pi$
$578$	−6.60838e9	−1.42347
$579$	0	0
$580$	−2.59338e9	−0.551909
$581$	9.10288e9	1.92558
$582$	0	0
$583$	−4.89520e8	−0.102313
$584$	−8.43735e7	−0.0175292
$585$	0	0
$586$	9.90191e9	2.03272
$587$	−4.43578e9	−0.905184	−0.452592	−	0.891718i	$-0.649501\pi$
$587$	−4.43578e9	−0.905184	−0.452592	+	0.891718i	$0.649501\pi$
$588$	0	0
$589$	−2.97638e9	−0.600186
$590$	4.03611e9	0.809061
$591$	0	0
$592$	4.38685e9	0.869014
$593$	−6.57596e9	−1.29499	−0.647496	−	0.762068i	$-0.724184\pi$
$593$	−6.57596e9	−1.29499	−0.647496	+	0.762068i	$0.724184\pi$
$594$	0	0
$595$	3.47400e8	0.0676115
$596$	8.95581e9	1.73278
$597$	0	0
$598$	2.28248e10	4.36469
$599$	9.99449e9	1.90006	0.950029	−	0.312162i	$-0.101053\pi$
$599$	9.99449e9	1.90006	0.950029	+	0.312162i	$0.101053\pi$
$600$	0	0
$601$	−8.94449e8	−0.168072	−0.0840359	−	0.996463i	$-0.526781\pi$
$601$	−8.94449e8	−0.168072	−0.0840359	+	0.996463i	$0.526781\pi$
$602$	1.41520e9	0.264381
$603$	0	0
$604$	7.81335e9	1.44281
$605$	−5.19057e9	−0.952953
$606$	0	0
$607$	−6.24872e9	−1.13405	−0.567023	−	0.823702i	$-0.691905\pi$
$607$	−6.24872e9	−1.13405	−0.567023	+	0.823702i	$0.691905\pi$
$608$	3.42796e9	0.618548
$609$	0	0
$610$	−2.53918e6	−0.000452938 0
$611$	9.70817e9	1.72184
$612$	0	0
$613$	4.93146e9	0.864696	0.432348	−	0.901707i	$-0.357685\pi$
$613$	4.93146e9	0.864696	0.432348	+	0.901707i	$0.357685\pi$
$614$	−8.32491e9	−1.45141
$615$	0	0
$616$	−6.16772e7	−0.0106314
$617$	7.71513e9	1.32235	0.661173	−	0.750234i	$-0.270059\pi$
$617$	7.71513e9	1.32235	0.661173	+	0.750234i	$0.270059\pi$
$618$	0	0
$619$	1.04934e10	1.77827	0.889137	−	0.457641i	$-0.151305\pi$
$619$	1.04934e10	1.77827	0.889137	+	0.457641i	$0.151305\pi$
$620$	8.33410e9	1.40439
$621$	0	0
$622$	−1.01098e10	−1.68452
$623$	−8.44651e9	−1.39949
$624$	0	0
$625$	−5.80662e9	−0.951357
$626$	−1.05021e10	−1.71107
$627$	0	0
$628$	−7.54049e8	−0.121490
$629$	−3.22604e8	−0.0516883
$630$	0	0
$631$	5.59798e9	0.887010	0.443505	−	0.896272i	$-0.353735\pi$
$631$	5.59798e9	0.887010	0.443505	+	0.896272i	$0.353735\pi$
$632$	1.29730e8	0.0204423
$633$	0	0
$634$	−1.28945e10	−2.00951
$635$	1.04102e10	1.61343
$636$	0	0
$637$	5.15083e9	0.789567
$638$	7.90644e8	0.120534
$639$	0	0
$640$	−7.30139e8	−0.110097
$641$	−1.08513e10	−1.62734	−0.813670	−	0.581327i	$-0.802534\pi$
$641$	−1.08513e10	−1.62734	−0.813670	+	0.581327i	$0.802534\pi$
$642$	0	0
$643$	−4.25867e9	−0.631736	−0.315868	−	0.948803i	$-0.602296\pi$
$643$	−4.25867e9	−0.631736	−0.315868	+	0.948803i	$0.602296\pi$
$644$	1.56825e10	2.31374
$645$	0	0
$646$	−2.42114e8	−0.0353351
$647$	−3.03007e9	−0.439833	−0.219917	−	0.975519i	$-0.570579\pi$
$647$	−3.03007e9	−0.439833	−0.219917	+	0.975519i	$0.570579\pi$
$648$	0	0
$649$	−6.27164e8	−0.0900585
$650$	−7.74727e8	−0.110650
$651$	0	0
$652$	3.24878e9	0.459044
$653$	3.93756e9	0.553389	0.276695	−	0.960958i	$-0.410761\pi$
$653$	3.93756e9	0.553389	0.276695	+	0.960958i	$0.410761\pi$
$654$	0	0
$655$	−1.30040e8	−0.0180814
$656$	5.43323e9	0.751441
$657$	0	0
$658$	1.30871e10	1.79082
$659$	−9.30694e7	−0.0126680	−0.00633400	−	0.999980i	$-0.502016\pi$
$659$	−9.30694e7	−0.0126680	−0.00633400	+	0.999980i	$0.502016\pi$
$660$	0	0
$661$	1.17439e10	1.58164	0.790818	−	0.612051i	$-0.209655\pi$
$661$	1.17439e10	1.58164	0.790818	+	0.612051i	$0.209655\pi$
$662$	1.35135e10	1.81036
$663$	0	0
$664$	6.75102e8	0.0894914
$665$	−3.89970e9	−0.514227
$666$	0	0
$667$	−7.64016e9	−0.996925
$668$	−1.34567e9	−0.174671
$669$	0	0
$670$	−1.56919e10	−2.01565
$671$	394558.	5.04176e−5 0
$672$	0	0
$673$	1.30594e9	0.165146	0.0825732	−	0.996585i	$-0.473686\pi$
$673$	1.30594e9	0.165146	0.0825732	+	0.996585i	$0.473686\pi$
$674$	1.67172e10	2.10307
$675$	0	0
$676$	1.48488e10	1.84875
$677$	−1.71912e9	−0.212935	−0.106467	−	0.994316i	$-0.533954\pi$
$677$	−1.71912e9	−0.212935	−0.106467	+	0.994316i	$0.533954\pi$
$678$	0	0
$679$	−1.39477e10	−1.70985
$680$	2.57644e7	0.00314224
$681$	0	0
$682$	−2.54082e9	−0.306711
$683$	4.52474e9	0.543402	0.271701	−	0.962382i	$-0.412414\pi$
$683$	4.52474e9	0.543402	0.271701	+	0.962382i	$0.412414\pi$
$684$	0	0
$685$	1.37226e10	1.63124
$686$	−7.71523e9	−0.912461
$687$	0	0
$688$	−1.24915e9	−0.146236
$689$	9.43261e9	1.09866
$690$	0	0
$691$	1.43457e10	1.65405	0.827025	−	0.562165i	$-0.190031\pi$
$691$	1.43457e10	1.65405	0.827025	+	0.562165i	$0.190031\pi$
$692$	5.65368e9	0.648575
$693$	0	0
$694$	−3.05905e9	−0.347399
$695$	−1.59244e10	−1.79935
$696$	0	0
$697$	−3.99554e8	−0.0446952
$698$	5.32457e9	0.592639
$699$	0	0
$700$	−5.32300e8	−0.0586561
$701$	−1.24050e7	−0.00136014	−0.000680069	−	1.00000i	$-0.500216\pi$
$701$	−1.24050e7	−0.00136014	−0.000680069		1.00000i	$0.500216\pi$
$702$	0	0
$703$	3.62135e9	0.393122
$704$	1.54827e9	0.167241
$705$	0	0
$706$	1.91848e10	2.05183
$707$	8.18396e9	0.870954
$708$	0	0
$709$	−1.13921e8	−0.0120044	−0.00600220	−	0.999982i	$-0.501911\pi$
$709$	−1.13921e8	−0.0120044	−0.00600220	+	0.999982i	$0.501911\pi$
$710$	7.35586e9	0.771311
$711$	0	0
$712$	−6.26424e8	−0.0650412
$713$	2.45525e10	2.53678
$714$	0	0
$715$	−2.46951e9	−0.252662
$716$	−1.10089e10	−1.12086
$717$	0	0
$718$	1.20461e9	0.121454
$719$	6.47748e8	0.0649912	0.0324956	−	0.999472i	$-0.489655\pi$
$719$	6.47748e8	0.0649912	0.0324956	+	0.999472i	$0.489655\pi$
$720$	0	0
$721$	1.73914e10	1.72807
$722$	−1.17247e10	−1.15937
$723$	0	0
$724$	1.59314e9	0.156016
$725$	2.59325e8	0.0252733
$726$	0	0
$727$	−5.68079e9	−0.548325	−0.274163	−	0.961683i	$-0.588401\pi$
$727$	−5.68079e9	−0.548325	−0.274163	+	0.961683i	$0.588401\pi$
$728$	1.18846e9	0.114163
$729$	0	0
$730$	−4.55405e9	−0.433279
$731$	9.18610e7	0.00869802
$732$	0	0
$733$	−1.04141e10	−0.976697	−0.488348	−	0.872649i	$-0.662400\pi$
$733$	−1.04141e10	−0.976697	−0.488348	+	0.872649i	$0.662400\pi$
$734$	−2.14445e10	−2.00161
$735$	0	0
$736$	−2.82776e10	−2.61439
$737$	2.43834e9	0.224367
$738$	0	0
$739$	7.74337e9	0.705788	0.352894	−	0.935663i	$-0.385198\pi$
$739$	7.74337e9	0.705788	0.352894	+	0.935663i	$0.385198\pi$
$740$	−1.01401e10	−0.919876
$741$	0	0
$742$	1.27156e10	1.14268
$743$	5.39810e9	0.482815	0.241407	−	0.970424i	$-0.422391\pi$
$743$	5.39810e9	0.482815	0.241407	+	0.970424i	$0.422391\pi$
$744$	0	0
$745$	1.83708e10	1.62772
$746$	2.50423e9	0.220845
$747$	0	0
$748$	−1.05343e8	−0.00920347
$749$	1.54705e9	0.134529
$750$	0	0
$751$	−1.23730e10	−1.06595	−0.532973	−	0.846132i	$-0.678925\pi$
$751$	−1.23730e10	−1.06595	−0.532973	+	0.846132i	$0.678925\pi$
$752$	−1.15515e10	−0.990552
$753$	0	0
$754$	−1.52350e10	−1.29432
$755$	1.60273e10	1.35533
$756$	0	0
$757$	2.74811e9	0.230249	0.115125	−	0.993351i	$-0.463273\pi$
$757$	2.74811e9	0.230249	0.115125	+	0.993351i	$0.463273\pi$
$758$	−1.87396e10	−1.56285
$759$	0	0
$760$	−2.89215e8	−0.0238987
$761$	6.98646e9	0.574660	0.287330	−	0.957832i	$-0.407232\pi$
$761$	6.98646e9	0.574660	0.287330	+	0.957832i	$0.407232\pi$
$762$	0	0
$763$	−6.72933e9	−0.548448
$764$	−2.15494e10	−1.74827
$765$	0	0
$766$	1.34597e10	1.08202
$767$	1.20849e10	0.967072
$768$	0	0
$769$	1.64020e10	1.30064	0.650318	−	0.759662i	$-0.274636\pi$
$769$	1.64020e10	1.30064	0.650318	+	0.759662i	$0.274636\pi$
$770$	−3.32902e9	−0.262784
$771$	0	0
$772$	−1.15660e10	−0.904741
$773$	6.42318e9	0.500175	0.250087	−	0.968223i	$-0.419541\pi$
$773$	6.42318e9	0.500175	0.250087	+	0.968223i	$0.419541\pi$
$774$	0	0
$775$	−8.33370e8	−0.0643105
$776$	−1.03441e9	−0.0794654
$777$	0	0
$778$	5.03613e9	0.383415
$779$	4.48515e9	0.339935
$780$	0	0
$781$	−1.14301e9	−0.0858564
$782$	1.99722e9	0.149349
$783$	0	0
$784$	−6.12886e9	−0.454227
$785$	−1.54676e9	−0.114124
$786$	0	0
$787$	8.36518e9	0.611735	0.305868	−	0.952074i	$-0.401053\pi$
$787$	8.36518e9	0.611735	0.305868	+	0.952074i	$0.401053\pi$
$788$	1.67406e10	1.21879
$789$	0	0
$790$	7.00214e9	0.505285
$791$	8.92919e9	0.641497
$792$	0	0
$793$	−7.60279e6	−0.000541398 0
$794$	−4.63804e9	−0.328823
$795$	0	0
$796$	−1.91430e10	−1.34528
$797$	−1.20395e10	−0.842376	−0.421188	−	0.906973i	$-0.638387\pi$
$797$	−1.20395e10	−0.842376	−0.421188	+	0.906973i	$0.638387\pi$
$798$	0	0
$799$	8.49487e8	0.0589173
$800$	9.59808e8	0.0662780
$801$	0	0
$802$	−1.29317e10	−0.885206
$803$	7.07645e8	0.0482293
$804$	0	0
$805$	3.21690e10	2.17346
$806$	4.89594e10	3.29354
$807$	0	0
$808$	6.06952e8	0.0404776
$809$	2.68334e10	1.78178	0.890892	−	0.454215i	$-0.150080\pi$
$809$	2.68334e10	1.78178	0.890892	+	0.454215i	$0.150080\pi$
$810$	0	0
$811$	7.68247e9	0.505741	0.252870	−	0.967500i	$-0.418625\pi$
$811$	7.68247e9	0.505741	0.252870	+	0.967500i	$0.418625\pi$
$812$	−1.04677e10	−0.686127
$813$	0	0
$814$	3.09141e9	0.200896
$815$	6.66412e9	0.431213
$816$	0	0
$817$	−1.03117e9	−0.0661538
$818$	−3.07826e10	−1.96639
$819$	0	0
$820$	−1.25587e10	−0.795422
$821$	−1.61885e10	−1.02096	−0.510478	−	0.859891i	$-0.670531\pi$
$821$	−1.61885e10	−1.02096	−0.510478	+	0.859891i	$0.670531\pi$
$822$	0	0
$823$	2.11454e10	1.32226	0.661131	−	0.750271i	$-0.270077\pi$
$823$	2.11454e10	1.32226	0.661131	+	0.750271i	$0.270077\pi$
$824$	1.28981e9	0.0803118
$825$	0	0
$826$	1.62910e10	1.00582
$827$	−2.52181e10	−1.55040	−0.775198	−	0.631718i	$-0.782350\pi$
$827$	−2.52181e10	−1.55040	−0.775198	+	0.631718i	$0.782350\pi$
$828$	0	0
$829$	1.48168e10	0.903260	0.451630	−	0.892205i	$-0.350843\pi$
$829$	1.48168e10	0.903260	0.451630	+	0.892205i	$0.350843\pi$
$830$	3.64385e10	2.21201
$831$	0	0
$832$	−2.98338e10	−1.79588
$833$	4.50709e8	0.0270171
$834$	0	0
$835$	−2.76033e9	−0.164081
$836$	1.18252e9	0.0699981
$837$	0	0
$838$	−2.90662e10	−1.70622
$839$	1.45298e10	0.849361	0.424680	−	0.905343i	$-0.360386\pi$
$839$	1.45298e10	0.849361	0.424680	+	0.905343i	$0.360386\pi$
$840$	0	0
$841$	−1.21503e10	−0.704367
$842$	5.61375e8	0.0324086
$843$	0	0
$844$	−2.44370e10	−1.39910
$845$	3.04589e10	1.73667
$846$	0	0
$847$	−2.09508e10	−1.18470
$848$	−1.12237e10	−0.632046
$849$	0	0
$850$	−6.77904e7	−0.00378619
$851$	−2.98729e10	−1.66159
$852$	0	0
$853$	−1.93358e10	−1.06669	−0.533347	−	0.845896i	$-0.679066\pi$
$853$	−1.93358e10	−1.06669	−0.533347	+	0.845896i	$0.679066\pi$
$854$	−1.02489e7	−0.000563088 0
$855$	0	0
$856$	1.14735e8	0.00625224
$857$	−3.62840e9	−0.196916	−0.0984582	−	0.995141i	$-0.531391\pi$
$857$	−3.62840e9	−0.196916	−0.0984582	+	0.995141i	$0.531391\pi$
$858$	0	0
$859$	5.66924e9	0.305175	0.152588	−	0.988290i	$-0.451239\pi$
$859$	5.66924e9	0.305175	0.152588	+	0.988290i	$0.451239\pi$
$860$	2.88736e9	0.154795
$861$	0	0
$862$	−3.09738e10	−1.64710
$863$	7.37525e9	0.390606	0.195303	−	0.980743i	$-0.437431\pi$
$863$	7.37525e9	0.390606	0.195303	+	0.980743i	$0.437431\pi$
$864$	0	0
$865$	1.15972e10	0.609253
$866$	−1.19673e10	−0.626158
$867$	0	0
$868$	3.36391e10	1.74592
$869$	−1.08805e9	−0.0562444
$870$	0	0
$871$	−4.69847e10	−2.40931
$872$	−4.99071e8	−0.0254891
$873$	0	0
$874$	−2.24196e10	−1.13589
$875$	−2.45825e10	−1.24050
$876$	0	0
$877$	−1.97900e10	−0.990711	−0.495355	−	0.868690i	$-0.664962\pi$
$877$	−1.97900e10	−0.990711	−0.495355	+	0.868690i	$0.664962\pi$
$878$	4.79373e10	2.39025
$879$	0	0
$880$	2.93841e9	0.145353
$881$	−5.73994e9	−0.282808	−0.141404	−	0.989952i	$-0.545162\pi$
$881$	−5.73994e9	−0.282808	−0.141404	+	0.989952i	$0.545162\pi$
$882$	0	0
$883$	−3.59778e9	−0.175862	−0.0879311	−	0.996127i	$-0.528026\pi$
$883$	−3.59778e9	−0.175862	−0.0879311	+	0.996127i	$0.528026\pi$
$884$	2.02987e9	0.0988293
$885$	0	0
$886$	−2.17039e10	−1.04838
$887$	2.01902e9	0.0971422	0.0485711	−	0.998820i	$-0.484533\pi$
$887$	2.01902e9	0.0971422	0.0485711	+	0.998820i	$0.484533\pi$
$888$	0	0
$889$	4.20188e10	2.00580
$890$	−3.38111e10	−1.60766
$891$	0	0
$892$	8.03284e9	0.378959
$893$	−9.53581e9	−0.448103
$894$	0	0
$895$	−2.25823e10	−1.05290
$896$	−2.94707e9	−0.136871
$897$	0	0
$898$	5.45775e10	2.51505
$899$	−1.63882e10	−0.752268
$900$	0	0
$901$	8.25375e8	0.0375937
$902$	3.82879e9	0.173716
$903$	0	0
$904$	6.62221e8	0.0298135
$905$	3.26797e9	0.146557
$906$	0	0
$907$	9.54457e9	0.424748	0.212374	−	0.977188i	$-0.431881\pi$
$907$	9.54457e9	0.424748	0.212374	+	0.977188i	$0.431881\pi$
$908$	−7.30548e9	−0.323853
$909$	0	0
$910$	6.41472e10	2.82184
$911$	−1.73423e10	−0.759964	−0.379982	−	0.924994i	$-0.624070\pi$
$911$	−1.73423e10	−0.759964	−0.379982	+	0.924994i	$0.624070\pi$
$912$	0	0
$913$	−5.66212e9	−0.246224
$914$	4.05113e8	0.0175495
$915$	0	0
$916$	−1.40864e10	−0.605574
$917$	−5.24883e8	−0.0224786
$918$	0	0
$919$	1.56927e8	0.00666949	0.00333475	−	0.999994i	$-0.498939\pi$
$919$	1.56927e8	0.00666949	0.00333475	+	0.999994i	$0.498939\pi$
$920$	2.38577e9	0.101012
$921$	0	0
$922$	1.28413e10	0.539574
$923$	2.20249e10	0.921949
$924$	0	0
$925$	1.01396e9	0.0421234
$926$	−5.87610e10	−2.43193
$927$	0	0
$928$	1.88746e10	0.775283
$929$	9.87406e9	0.404055	0.202028	−	0.979380i	$-0.435247\pi$
$929$	9.87406e9	0.404055	0.202028	+	0.979380i	$0.435247\pi$
$930$	0	0
$931$	−5.05938e9	−0.205482
$932$	−5.61356e9	−0.227134
$933$	0	0
$934$	3.42443e9	0.137522
$935$	−2.16088e8	−0.00864548
$936$	0	0
$937$	−1.57119e10	−0.623935	−0.311968	−	0.950093i	$-0.600988\pi$
$937$	−1.57119e10	−0.623935	−0.311968	+	0.950093i	$0.600988\pi$
$938$	−6.33377e10	−2.50583
$939$	0	0
$940$	2.67010e10	1.04853
$941$	−9.28175e9	−0.363133	−0.181567	−	0.983379i	$-0.558117\pi$
$941$	−9.28175e9	−0.363133	−0.181567	+	0.983379i	$0.558117\pi$
$942$	0	0
$943$	−3.69984e10	−1.43679
$944$	−1.43795e10	−0.556344
$945$	0	0
$946$	−8.80273e8	−0.0338064
$947$	3.17945e10	1.21654	0.608270	−	0.793730i	$-0.291864\pi$
$947$	3.17945e10	1.21654	0.608270	+	0.793730i	$0.291864\pi$
$948$	0	0
$949$	−1.36357e10	−0.517899
$950$	7.60973e8	0.0287963
$951$	0	0
$952$	1.03993e8	0.00390640
$953$	1.99879e10	0.748070	0.374035	−	0.927415i	$-0.377974\pi$
$953$	1.99879e10	0.748070	0.374035	+	0.927415i	$0.377974\pi$
$954$	0	0
$955$	−4.42036e10	−1.64228
$956$	−2.20338e10	−0.815618
$957$	0	0
$958$	−4.02500e10	−1.47906
$959$	5.53887e10	2.02794
$960$	0	0
$961$	2.51527e10	0.914225
$962$	−5.95687e10	−2.15727
$963$	0	0
$964$	−1.56763e10	−0.563605
$965$	−2.37251e10	−0.849888
$966$	0	0
$967$	5.44278e10	1.93566	0.967828	−	0.251611i	$-0.0809604\pi$
$967$	5.44278e10	1.93566	0.967828	+	0.251611i	$0.0809604\pi$
$968$	−1.55379e9	−0.0550589
$969$	0	0
$970$	−5.58323e10	−1.96419
$971$	−2.25661e10	−0.791024	−0.395512	−	0.918461i	$-0.629433\pi$
$971$	−2.25661e10	−0.791024	−0.395512	+	0.918461i	$0.629433\pi$
$972$	0	0
$973$	−6.42759e10	−2.23693
$974$	−1.11970e10	−0.388281
$975$	0	0
$976$	9.04638e6	0.000311459 0
$977$	4.26254e9	0.146230	0.0731152	−	0.997324i	$-0.476706\pi$
$977$	4.26254e9	0.146230	0.0731152	+	0.997324i	$0.476706\pi$
$978$	0	0
$979$	5.25385e9	0.178953
$980$	1.41666e10	0.480813
$981$	0	0
$982$	3.93753e10	1.32688
$983$	1.23804e10	0.415717	0.207858	−	0.978159i	$-0.433351\pi$
$983$	1.23804e10	0.415717	0.207858	+	0.978159i	$0.433351\pi$
$984$	0	0
$985$	3.43396e10	1.14490
$986$	−1.33310e9	−0.0442887
$987$	0	0
$988$	−2.27861e10	−0.751658
$989$	8.50627e9	0.279610
$990$	0	0
$991$	3.54785e10	1.15800	0.578998	−	0.815329i	$-0.303444\pi$
$991$	3.54785e10	1.15800	0.578998	+	0.815329i	$0.303444\pi$
$992$	−6.06557e10	−1.97279
$993$	0	0
$994$	2.96906e10	0.958885
$995$	−3.92674e10	−1.26372
$996$	0	0
$997$	−1.77312e10	−0.566636	−0.283318	−	0.959026i	$-0.591435\pi$
$997$	−1.77312e10	−0.566636	−0.283318	+	0.959026i	$0.591435\pi$
$998$	−6.29595e10	−2.00495
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	387.8.a.b.1.10		11
3.2	odd	2		43.8.a.a.1.2	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.8.a.a.1.2	✓	11	3.2	odd	2
387.8.a.b.1.10		11	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)

\(f(q)\)	\(=\)	\(q+16.1572 q^{2} +133.057 q^{4} +272.935 q^{5} +1101.65 q^{7} +81.7025 q^{8} +O(q^{10})\)	\(q+16.1572 q^{2} +133.057 q^{4} +272.935 q^{5} +1101.65 q^{7} +81.7025 q^{8} +4409.88 q^{10} -685.243 q^{11} +13204.0 q^{13} +17799.7 q^{14} -15711.2 q^{16} +1155.38 q^{17} -12969.6 q^{19} +36315.9 q^{20} -11071.6 q^{22} +106988. q^{23} -3631.41 q^{25} +213341. q^{26} +146582. q^{28} -71411.7 q^{29} +229489. q^{31} -264307. q^{32} +18667.8 q^{34} +300680. q^{35} -279218. q^{37} -209553. q^{38} +22299.5 q^{40} -345820. q^{41} +79507.0 q^{43} -91176.2 q^{44} +1.72863e6 q^{46} +735243. q^{47} +390095. q^{49} -58673.6 q^{50} +1.75688e6 q^{52} +714374. q^{53} -187027. q^{55} +90007.8 q^{56} -1.15382e6 q^{58} +915243. q^{59} -575.793 q^{61} +3.70792e6 q^{62} -2.25945e6 q^{64} +3.60384e6 q^{65} -3.55836e6 q^{67} +153731. q^{68} +4.85816e6 q^{70} +1.66804e6 q^{71} -1.03269e6 q^{73} -4.51140e6 q^{74} -1.72569e6 q^{76} -754900. q^{77} +1.58783e6 q^{79} -4.28813e6 q^{80} -5.58750e6 q^{82} +8.26293e6 q^{83} +315344. q^{85} +1.28461e6 q^{86} -55986.1 q^{88} -7.66713e6 q^{89} +1.45462e7 q^{91} +1.42354e7 q^{92} +1.18795e7 q^{94} -3.53986e6 q^{95} -1.26607e7 q^{97} +6.30287e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 24 q^{2} + 602 q^{4} + 752 q^{5} - 12 q^{7} + 3810 q^{8}+O(q^{10}) \) 11 * q + 24 * q^2 + 602 * q^4 + 752 * q^5 - 12 * q^7 + 3810 * q^8	\( 11 q + 24 q^{2} + 602 q^{4} + 752 q^{5} - 12 q^{7} + 3810 q^{8} - 1333 q^{10} - 1333 q^{11} - 17967 q^{13} + 22352 q^{14} - 34406 q^{16} + 63095 q^{17} - 54524 q^{19} + 280995 q^{20} - 289358 q^{22} + 138139 q^{23} + 3455 q^{25} + 132946 q^{26} - 12704 q^{28} + 308658 q^{29} - 209523 q^{31} + 644934 q^{32} + 762435 q^{34} + 578892 q^{35} - 298472 q^{37} + 369707 q^{38} + 2633173 q^{40} + 1346735 q^{41} + 874577 q^{43} - 3134292 q^{44} + 3588111 q^{46} - 499284 q^{47} + 2544563 q^{49} - 3049745 q^{50} + 983088 q^{52} + 2210495 q^{53} - 1855072 q^{55} + 469976 q^{56} + 4397067 q^{58} + 5824216 q^{59} - 4453034 q^{61} - 1002789 q^{62} + 4757538 q^{64} + 2162872 q^{65} - 6859513 q^{67} + 9397005 q^{68} + 845078 q^{70} + 10726554 q^{71} - 4456898 q^{73} - 1046637 q^{74} + 5861267 q^{76} + 17019816 q^{77} - 15541320 q^{79} + 15680911 q^{80} + 20233655 q^{82} + 11146767 q^{83} - 12471976 q^{85} + 1908168 q^{86} - 24463544 q^{88} + 13531356 q^{89} - 19746448 q^{91} + 26023161 q^{92} + 20288857 q^{94} + 12291624 q^{95} - 10999901 q^{97} - 29909168 q^{98}+O(q^{100}) \) 11 * q + 24 * q^2 + 602 * q^4 + 752 * q^5 - 12 * q^7 + 3810 * q^8 - 1333 * q^10 - 1333 * q^11 - 17967 * q^13 + 22352 * q^14 - 34406 * q^16 + 63095 * q^17 - 54524 * q^19 + 280995 * q^20 - 289358 * q^22 + 138139 * q^23 + 3455 * q^25 + 132946 * q^26 - 12704 * q^28 + 308658 * q^29 - 209523 * q^31 + 644934 * q^32 + 762435 * q^34 + 578892 * q^35 - 298472 * q^37 + 369707 * q^38 + 2633173 * q^40 + 1346735 * q^41 + 874577 * q^43 - 3134292 * q^44 + 3588111 * q^46 - 499284 * q^47 + 2544563 * q^49 - 3049745 * q^50 + 983088 * q^52 + 2210495 * q^53 - 1855072 * q^55 + 469976 * q^56 + 4397067 * q^58 + 5824216 * q^59 - 4453034 * q^61 - 1002789 * q^62 + 4757538 * q^64 + 2162872 * q^65 - 6859513 * q^67 + 9397005 * q^68 + 845078 * q^70 + 10726554 * q^71 - 4456898 * q^73 - 1046637 * q^74 + 5861267 * q^76 + 17019816 * q^77 - 15541320 * q^79 + 15680911 * q^80 + 20233655 * q^82 + 11146767 * q^83 - 12471976 * q^85 + 1908168 * q^86 - 24463544 * q^88 + 13531356 * q^89 - 19746448 * q^91 + 26023161 * q^92 + 20288857 * q^94 + 12291624 * q^95 - 10999901 * q^97 - 29909168 * q^98

Introduction

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