LMFDB - Embedded newform 387.8.a.b.1.4

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			1.4
Root			$-10.3097$ 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)

$f(q)$	$=$	$q-8.30971 q^{2} -58.9487 q^{4} +187.284 q^{5} -1501.21 q^{7} +1553.49 q^{8} +O(q^{10})$	\(q-8.30971 q^{2} -58.9487 q^{4} +187.284 q^{5} -1501.21 q^{7} +1553.49 q^{8} -1556.27 q^{10} -1942.03 q^{11} -4434.78 q^{13} +12474.6 q^{14} -5363.63 q^{16} -31834.8 q^{17} +8145.26 q^{19} -11040.1 q^{20} +16137.7 q^{22} -11917.4 q^{23} -43049.8 q^{25} +36851.8 q^{26} +88494.3 q^{28} +115330. q^{29} -100315. q^{31} -154276. q^{32} +264538. q^{34} -281152. q^{35} -455058. q^{37} -67684.8 q^{38} +290943. q^{40} -396428. q^{41} +79507.0 q^{43} +114480. q^{44} +99029.9 q^{46} -249124. q^{47} +1.43009e6 q^{49} +357732. q^{50} +261424. q^{52} +363774. q^{53} -363711. q^{55} -2.33211e6 q^{56} -958356. q^{58} +705627. q^{59} -831030. q^{61} +833586. q^{62} +1.96854e6 q^{64} -830562. q^{65} -2.27120e6 q^{67} +1.87662e6 q^{68} +2.33629e6 q^{70} -4.30808e6 q^{71} -3.27796e6 q^{73} +3.78140e6 q^{74} -480152. q^{76} +2.91540e6 q^{77} -8.41456e6 q^{79} -1.00452e6 q^{80} +3.29420e6 q^{82} +3.46468e6 q^{83} -5.96213e6 q^{85} -660680. q^{86} -3.01693e6 q^{88} -5.16792e6 q^{89} +6.65753e6 q^{91} +702513. q^{92} +2.07015e6 q^{94} +1.52547e6 q^{95} +1.54280e7 q^{97} -1.18836e7 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−8.30971	−0.734482	−0.367241	−	0.930126i	$-0.619698\pi$
$2$	−8.30971	−0.734482	−0.367241	+	0.930126i	$0.619698\pi$
$3$	0	0
$3$	0	0
$4$	−58.9487	−0.460536
$5$	187.284	0.670047	0.335023	−	0.942210i	$-0.391256\pi$
$5$	187.284	0.670047	0.335023	+	0.942210i	$0.391256\pi$
$6$	0	0
$7$	−1501.21	−1.65424	−0.827119	−	0.562027i	$-0.810022\pi$
$7$	−1501.21	−1.65424	−0.827119	+	0.562027i	$0.810022\pi$
$8$	1553.49	1.07274
$9$	0	0
$10$	−1556.27	−0.492137
$11$	−1942.03	−0.439929	−0.219964	−	0.975508i	$-0.570594\pi$
$11$	−1942.03	−0.439929	−0.219964	+	0.975508i	$0.570594\pi$
$12$	0	0
$13$	−4434.78	−0.559849	−0.279924	−	0.960022i	$-0.590309\pi$
$13$	−4434.78	−0.559849	−0.279924	+	0.960022i	$0.590309\pi$
$14$	12474.6	1.21501
$15$	0	0
$16$	−5363.63	−0.327370
$17$	−31834.8	−1.57156	−0.785779	−	0.618508i	$-0.787738\pi$
$17$	−31834.8	−1.57156	−0.785779	+	0.618508i	$0.787738\pi$
$18$	0	0
$19$	8145.26	0.272438	0.136219	−	0.990679i	$-0.456505\pi$
$19$	8145.26	0.272438	0.136219	+	0.990679i	$0.456505\pi$
$20$	−11040.1	−0.308581
$21$	0	0
$22$	16137.7	0.323120
$23$	−11917.4	−0.204236	−0.102118	−	0.994772i	$-0.532562\pi$
$23$	−11917.4	−0.204236	−0.102118	+	0.994772i	$0.532562\pi$
$24$	0	0
$25$	−43049.8	−0.551038
$26$	36851.8	0.411199
$27$	0	0
$28$	88494.3	0.761837
$29$	115330.	0.878108	0.439054	−	0.898461i	$-0.355314\pi$
$29$	115330.	0.878108	0.439054	+	0.898461i	$0.355314\pi$
$30$	0	0
$31$	−100315.	−0.604781	−0.302391	−	0.953184i	$-0.597785\pi$
$31$	−100315.	−0.604781	−0.302391	+	0.953184i	$0.597785\pi$
$32$	−154276.	−0.832290
$33$	0	0
$34$	264538.	1.15428
$35$	−281152.	−1.10842
$36$	0	0
$37$	−455058.	−1.47693	−0.738466	−	0.674291i	$-0.764449\pi$
$37$	−455058.	−1.47693	−0.738466	+	0.674291i	$0.764449\pi$
$38$	−67684.8	−0.200101
$39$	0	0
$40$	290943.	0.718784
$41$	−396428.	−0.898298	−0.449149	−	0.893457i	$-0.648273\pi$
$41$	−396428.	−0.898298	−0.449149	+	0.893457i	$0.648273\pi$
$42$	0	0
$43$	79507.0	0.152499
$43$	79507.0	0.152499
$44$	114480.	0.202603
$45$	0	0
$46$	99029.9	0.150008
$47$	−249124.	−0.350004	−0.175002	−	0.984568i	$-0.555993\pi$
$47$	−249124.	−0.350004	−0.175002	+	0.984568i	$0.555993\pi$
$48$	0	0
$49$	1.43009e6	1.73650
$50$	357732.	0.404727
$51$	0	0
$52$	261424.	0.257831
$53$	363774.	0.335634	0.167817	−	0.985818i	$-0.446328\pi$
$53$	363774.	0.335634	0.167817	+	0.985818i	$0.446328\pi$
$54$	0	0
$55$	−363711.	−0.294773
$56$	−2.33211e6	−1.77456
$57$	0	0
$58$	−958356.	−0.644954
$59$	705627.	0.447294	0.223647	−	0.974670i	$-0.428204\pi$
$59$	705627.	0.447294	0.223647	+	0.974670i	$0.428204\pi$
$60$	0	0
$61$	−831030.	−0.468773	−0.234386	−	0.972144i	$-0.575308\pi$
$61$	−831030.	−0.468773	−0.234386	+	0.972144i	$0.575308\pi$
$62$	833586.	0.444201
$63$	0	0
$64$	1.96854e6	0.938672
$65$	−830562.	−0.375125
$66$	0	0
$67$	−2.27120e6	−0.922557	−0.461279	−	0.887255i	$-0.652609\pi$
$67$	−2.27120e6	−0.922557	−0.461279	+	0.887255i	$0.652609\pi$
$68$	1.87662e6	0.723759
$69$	0	0
$70$	2.33629e6	0.814112
$71$	−4.30808e6	−1.42850	−0.714248	−	0.699893i	$-0.753231\pi$
$71$	−4.30808e6	−1.42850	−0.714248	+	0.699893i	$0.753231\pi$
$72$	0	0
$73$	−3.27796e6	−0.986220	−0.493110	−	0.869967i	$-0.664140\pi$
$73$	−3.27796e6	−0.986220	−0.493110	+	0.869967i	$0.664140\pi$
$74$	3.78140e6	1.08478
$75$	0	0
$76$	−480152.	−0.125467
$77$	2.91540e6	0.727747
$78$	0	0
$79$	−8.41456e6	−1.92016	−0.960078	−	0.279732i	$-0.909754\pi$
$79$	−8.41456e6	−1.92016	−0.960078	+	0.279732i	$0.909754\pi$
$80$	−1.00452e6	−0.219353
$81$	0	0
$82$	3.29420e6	0.659784
$83$	3.46468e6	0.665104	0.332552	−	0.943085i	$-0.392090\pi$
$83$	3.46468e6	0.665104	0.332552	+	0.943085i	$0.392090\pi$
$84$	0	0
$85$	−5.96213e6	−1.05302
$86$	−660680.	−0.112007
$87$	0	0
$88$	−3.01693e6	−0.471928
$89$	−5.16792e6	−0.777053	−0.388526	−	0.921438i	$-0.627016\pi$
$89$	−5.16792e6	−0.777053	−0.388526	+	0.921438i	$0.627016\pi$
$90$	0	0
$91$	6.65753e6	0.926123
$92$	702513.	0.0940583
$93$	0	0
$94$	2.07015e6	0.257071
$95$	1.52547e6	0.182546
$96$	0	0
$97$	1.54280e7	1.71637	0.858183	−	0.513344i	$-0.171593\pi$
$97$	1.54280e7	1.71637	0.858183	+	0.513344i	$0.171593\pi$
$98$	−1.18836e7	−1.27543
$99$	0	0
$100$	2.53773e6	0.253773
$101$	−4.87285e6	−0.470606	−0.235303	−	0.971922i	$-0.575608\pi$
$101$	−4.87285e6	−0.470606	−0.235303	+	0.971922i	$0.575608\pi$
$102$	0	0
$103$	−1.16758e7	−1.05283	−0.526413	−	0.850229i	$-0.676463\pi$
$103$	−1.16758e7	−1.05283	−0.526413	+	0.850229i	$0.676463\pi$
$104$	−6.88939e6	−0.600570
$105$	0	0
$106$	−3.02286e6	−0.246517
$107$	1.74276e7	1.37529	0.687646	−	0.726046i	$-0.258644\pi$
$107$	1.74276e7	1.37529	0.687646	+	0.726046i	$0.258644\pi$
$108$	0	0
$109$	1.50763e6	0.111507	0.0557535	−	0.998445i	$-0.482244\pi$
$109$	1.50763e6	0.111507	0.0557535	+	0.998445i	$0.482244\pi$
$110$	3.02234e6	0.216505
$111$	0	0
$112$	8.05193e6	0.541548
$113$	−2.05709e7	−1.34116	−0.670579	−	0.741838i	$-0.733954\pi$
$113$	−2.05709e7	−1.34116	−0.670579	+	0.741838i	$0.733954\pi$
$114$	0	0
$115$	−2.23193e6	−0.136848
$116$	−6.79852e6	−0.404400
$117$	0	0
$118$	−5.86356e6	−0.328529
$119$	4.77906e7	2.59973
$120$	0	0
$121$	−1.57157e7	−0.806463
$122$	6.90562e6	0.344305
$123$	0	0
$124$	5.91341e6	0.278524
$125$	−2.26941e7	−1.03927
$126$	0	0
$127$	7.41435e6	0.321188	0.160594	−	0.987021i	$-0.448659\pi$
$127$	7.41435e6	0.321188	0.160594	+	0.987021i	$0.448659\pi$
$128$	3.38940e6	0.142853
$129$	0	0
$130$	6.90174e6	0.275522
$131$	−4.53106e7	−1.76096	−0.880482	−	0.474079i	$-0.842781\pi$
$131$	−4.53106e7	−1.76096	−0.880482	+	0.474079i	$0.842781\pi$
$132$	0	0
$133$	−1.22277e7	−0.450677
$134$	1.88730e7	0.677602
$135$	0	0
$136$	−4.94550e7	−1.68587
$137$	4.06972e7	1.35220	0.676102	−	0.736808i	$-0.263668\pi$
$137$	4.06972e7	1.35220	0.676102	+	0.736808i	$0.263668\pi$
$138$	0	0
$139$	−4.72006e7	−1.49072	−0.745360	−	0.666662i	$-0.767722\pi$
$139$	−4.72006e7	−1.49072	−0.745360	+	0.666662i	$0.767722\pi$
$140$	1.65735e7	0.510466
$141$	0	0
$142$	3.57989e7	1.04920
$143$	8.61250e6	0.246294
$144$	0	0
$145$	2.15994e7	0.588373
$146$	2.72389e7	0.724360
$147$	0	0
$148$	2.68250e7	0.680181
$149$	6.71032e7	1.66185	0.830924	−	0.556386i	$-0.187812\pi$
$149$	6.71032e7	1.66185	0.830924	+	0.556386i	$0.187812\pi$
$150$	0	0
$151$	5.69247e7	1.34549	0.672747	−	0.739873i	$-0.265114\pi$
$151$	5.69247e7	1.34549	0.672747	+	0.739873i	$0.265114\pi$
$152$	1.26536e7	0.292254
$153$	0	0
$154$	−2.42261e7	−0.534517
$155$	−1.87873e7	−0.405232
$156$	0	0
$157$	7.10156e7	1.46455	0.732277	−	0.681007i	$-0.238458\pi$
$157$	7.10156e7	1.46455	0.732277	+	0.681007i	$0.238458\pi$
$158$	6.99226e7	1.41032
$159$	0	0
$160$	−2.88935e7	−0.557673
$161$	1.78905e7	0.337856
$162$	0	0
$163$	4.61695e7	0.835024	0.417512	−	0.908671i	$-0.362902\pi$
$163$	4.61695e7	0.835024	0.417512	+	0.908671i	$0.362902\pi$
$164$	2.33689e7	0.413699
$165$	0	0
$166$	−2.87905e7	−0.488507
$167$	4.64887e7	0.772395	0.386198	−	0.922416i	$-0.373788\pi$
$167$	4.64887e7	0.772395	0.386198	+	0.922416i	$0.373788\pi$
$168$	0	0
$169$	−4.30812e7	−0.686570
$170$	4.95436e7	0.773422
$171$	0	0
$172$	−4.68683e6	−0.0702311
$173$	3.94243e7	0.578898	0.289449	−	0.957193i	$-0.406528\pi$
$173$	3.94243e7	0.578898	0.289449	+	0.957193i	$0.406528\pi$
$174$	0	0
$175$	6.46268e7	0.911548
$176$	1.04163e7	0.144019
$177$	0	0
$178$	4.29439e7	0.570731
$179$	−2.63135e7	−0.342921	−0.171460	−	0.985191i	$-0.554849\pi$
$179$	−2.63135e7	−0.342921	−0.171460	+	0.985191i	$0.554849\pi$
$180$	0	0
$181$	−3.37886e7	−0.423541	−0.211771	−	0.977319i	$-0.567923\pi$
$181$	−3.37886e7	−0.423541	−0.211771	+	0.977319i	$0.567923\pi$
$182$	−5.53222e7	−0.680220
$183$	0	0
$184$	−1.85135e7	−0.219092
$185$	−8.52249e7	−0.989613
$186$	0	0
$187$	6.18242e7	0.691374
$188$	1.46855e7	0.161189
$189$	0	0
$190$	−1.26763e7	−0.134077
$191$	1.27960e8	1.32880	0.664398	−	0.747379i	$-0.268688\pi$
$191$	1.27960e8	1.32880	0.664398	+	0.747379i	$0.268688\pi$
$192$	0	0
$193$	−4.75377e7	−0.475979	−0.237990	−	0.971268i	$-0.576488\pi$
$193$	−4.75377e7	−0.475979	−0.237990	+	0.971268i	$0.576488\pi$
$194$	−1.28203e8	−1.26064
$195$	0	0
$196$	−8.43016e7	−0.799723
$197$	−1.75594e7	−0.163636	−0.0818178	−	0.996647i	$-0.526073\pi$
$197$	−1.75594e7	−0.163636	−0.0818178	+	0.996647i	$0.526073\pi$
$198$	0	0
$199$	1.25178e8	1.12601	0.563004	−	0.826454i	$-0.309646\pi$
$199$	1.25178e8	1.12601	0.563004	+	0.826454i	$0.309646\pi$
$200$	−6.68775e7	−0.591119
$201$	0	0
$202$	4.04919e7	0.345652
$203$	−1.73134e8	−1.45260
$204$	0	0
$205$	−7.42445e7	−0.601902
$206$	9.70225e7	0.773281
$207$	0	0
$208$	2.37865e7	0.183278
$209$	−1.58184e7	−0.119853
$210$	0	0
$211$	8.23349e7	0.603386	0.301693	−	0.953405i	$-0.402448\pi$
$211$	8.23349e7	0.603386	0.301693	+	0.953405i	$0.402448\pi$
$212$	−2.14440e7	−0.154572
$213$	0	0
$214$	−1.44819e8	−1.01013
$215$	1.48904e7	0.102181
$216$	0	0
$217$	1.50593e8	1.00045
$218$	−1.25280e7	−0.0818999
$219$	0	0
$220$	2.14403e7	0.135754
$221$	1.41180e8	0.879834
$222$	0	0
$223$	2.64599e8	1.59779	0.798897	−	0.601468i	$-0.205417\pi$
$223$	2.64599e8	1.59779	0.798897	+	0.601468i	$0.205417\pi$
$224$	2.31601e8	1.37681
$225$	0	0
$226$	1.70939e8	0.985056
$227$	4.70117e7	0.266757	0.133378	−	0.991065i	$-0.457417\pi$
$227$	4.70117e7	0.266757	0.133378	+	0.991065i	$0.457417\pi$
$228$	0	0
$229$	−1.76441e7	−0.0970900	−0.0485450	−	0.998821i	$-0.515458\pi$
$229$	−1.76441e7	−0.0970900	−0.0485450	+	0.998821i	$0.515458\pi$
$230$	1.85467e7	0.100512
$231$	0	0
$232$	1.79163e8	0.941979
$233$	2.96873e8	1.53754	0.768768	−	0.639528i	$-0.220870\pi$
$233$	2.96873e8	1.53754	0.768768	+	0.639528i	$0.220870\pi$
$234$	0	0
$235$	−4.66568e7	−0.234519
$236$	−4.15957e7	−0.205995
$237$	0	0
$238$	−3.97126e8	−1.90946
$239$	−7.90520e7	−0.374559	−0.187280	−	0.982307i	$-0.559967\pi$
$239$	−7.90520e7	−0.374559	−0.187280	+	0.982307i	$0.559967\pi$
$240$	0	0
$241$	2.33364e8	1.07393	0.536963	−	0.843606i	$-0.319571\pi$
$241$	2.33364e8	1.07393	0.536963	+	0.843606i	$0.319571\pi$
$242$	1.30593e8	0.592332
$243$	0	0
$244$	4.89881e7	0.215887
$245$	2.67832e8	1.16354
$246$	0	0
$247$	−3.61224e7	−0.152524
$248$	−1.55838e8	−0.648772
$249$	0	0
$250$	1.88581e8	0.763323
$251$	8.80712e7	0.351541	0.175770	−	0.984431i	$-0.443758\pi$
$251$	8.80712e7	0.351541	0.175770	+	0.984431i	$0.443758\pi$
$252$	0	0
$253$	2.31439e7	0.0898495
$254$	−6.16111e7	−0.235907
$255$	0	0
$256$	−2.80138e8	−1.04359
$257$	1.29253e8	0.474979	0.237489	−	0.971390i	$-0.423675\pi$
$257$	1.29253e8	0.474979	0.237489	+	0.971390i	$0.423675\pi$
$258$	0	0
$259$	6.83136e8	2.44320
$260$	4.89605e7	0.172759
$261$	0	0
$262$	3.76518e8	1.29340
$263$	1.06033e8	0.359415	0.179708	−	0.983720i	$-0.442485\pi$
$263$	1.06033e8	0.359415	0.179708	+	0.983720i	$0.442485\pi$
$264$	0	0
$265$	6.81289e7	0.224890
$266$	1.01609e8	0.331014
$267$	0	0
$268$	1.33884e8	0.424871
$269$	2.84361e8	0.890711	0.445356	−	0.895354i	$-0.353077\pi$
$269$	2.84361e8	0.890711	0.445356	+	0.895354i	$0.353077\pi$
$270$	0	0
$271$	−3.12787e8	−0.954677	−0.477339	−	0.878719i	$-0.658398\pi$
$271$	−3.12787e8	−0.954677	−0.477339	+	0.878719i	$0.658398\pi$
$272$	1.70750e8	0.514481
$273$	0	0
$274$	−3.38182e8	−0.993170
$275$	8.36042e7	0.242417
$276$	0	0
$277$	−1.35018e8	−0.381692	−0.190846	−	0.981620i	$-0.561123\pi$
$277$	−1.35018e8	−0.381692	−0.190846	+	0.981620i	$0.561123\pi$
$278$	3.92224e8	1.09491
$279$	0	0
$280$	−4.36767e8	−1.18904
$281$	−2.08009e8	−0.559255	−0.279628	−	0.960109i	$-0.590211\pi$
$281$	−2.08009e8	−0.559255	−0.279628	+	0.960109i	$0.590211\pi$
$282$	0	0
$283$	3.32152e8	0.871134	0.435567	−	0.900156i	$-0.356548\pi$
$283$	3.32152e8	0.871134	0.435567	+	0.900156i	$0.356548\pi$
$284$	2.53955e8	0.657875
$285$	0	0
$286$	−7.15674e7	−0.180898
$287$	5.95121e8	1.48600
$288$	0	0
$289$	6.03113e8	1.46979
$290$	−1.79484e8	−0.432149
$291$	0	0
$292$	1.93231e8	0.454190
$293$	−2.71797e8	−0.631260	−0.315630	−	0.948882i	$-0.602216\pi$
$293$	−2.71797e8	−0.631260	−0.315630	+	0.948882i	$0.602216\pi$
$294$	0	0
$295$	1.32152e8	0.299708
$296$	−7.06927e8	−1.58436
$297$	0	0
$298$	−5.57608e8	−1.22060
$299$	5.28509e7	0.114341
$300$	0	0
$301$	−1.19357e8	−0.252269
$302$	−4.73028e8	−0.988241
$303$	0	0
$304$	−4.36881e7	−0.0891879
$305$	−1.55638e8	−0.314100
$306$	0	0
$307$	−2.68192e7	−0.0529007	−0.0264503	−	0.999650i	$-0.508420\pi$
$307$	−2.68192e7	−0.0529007	−0.0264503	+	0.999650i	$0.508420\pi$
$308$	−1.71859e8	−0.335154
$309$	0	0
$310$	1.56117e8	0.297635
$311$	−8.62107e8	−1.62517	−0.812587	−	0.582840i	$-0.801942\pi$
$311$	−8.62107e8	−1.62517	−0.812587	+	0.582840i	$0.801942\pi$
$312$	0	0
$313$	1.68619e7	0.0310814	0.0155407	−	0.999879i	$-0.495053\pi$
$313$	1.68619e7	0.0310814	0.0155407	+	0.999879i	$0.495053\pi$
$314$	−5.90120e8	−1.07569
$315$	0	0
$316$	4.96027e8	0.884302
$317$	6.67505e8	1.17692	0.588461	−	0.808526i	$-0.299734\pi$
$317$	6.67505e8	1.17692	0.588461	+	0.808526i	$0.299734\pi$
$318$	0	0
$319$	−2.23974e8	−0.386305
$320$	3.68675e8	0.628954
$321$	0	0
$322$	−1.48665e8	−0.248149
$323$	−2.59302e8	−0.428152
$324$	0	0
$325$	1.90917e8	0.308498
$326$	−3.83656e8	−0.613310
$327$	0	0
$328$	−6.15847e8	−0.963638
$329$	3.73987e8	0.578989
$330$	0	0
$331$	−1.47144e8	−0.223021	−0.111510	−	0.993763i	$-0.535569\pi$
$331$	−1.47144e8	−0.223021	−0.111510	+	0.993763i	$0.535569\pi$
$332$	−2.04238e8	−0.306304
$333$	0	0
$334$	−3.86308e8	−0.567310
$335$	−4.25359e8	−0.618156
$336$	0	0
$337$	−1.09036e9	−1.55190	−0.775951	−	0.630793i	$-0.782730\pi$
$337$	−1.09036e9	−1.55190	−0.775951	+	0.630793i	$0.782730\pi$
$338$	3.57993e8	0.504273
$339$	0	0
$340$	3.51460e8	0.484953
$341$	1.94814e8	0.266061
$342$	0	0
$343$	−9.10548e8	−1.21835
$344$	1.23513e8	0.163591
$345$	0	0
$346$	−3.27604e8	−0.425190
$347$	−2.16834e8	−0.278596	−0.139298	−	0.990251i	$-0.544485\pi$
$347$	−2.16834e8	−0.278596	−0.139298	+	0.990251i	$0.544485\pi$
$348$	0	0
$349$	−1.24901e9	−1.57281	−0.786404	−	0.617712i	$-0.788060\pi$
$349$	−1.24901e9	−1.57281	−0.786404	+	0.617712i	$0.788060\pi$
$350$	−5.37030e8	−0.669515
$351$	0	0
$352$	2.99610e8	0.366149
$353$	4.43185e8	0.536258	0.268129	−	0.963383i	$-0.413595\pi$
$353$	4.43185e8	0.536258	0.268129	+	0.963383i	$0.413595\pi$
$354$	0	0
$355$	−8.06832e8	−0.957159
$356$	3.04642e8	0.357861
$357$	0	0
$358$	2.18658e8	0.251869
$359$	−1.42062e9	−1.62049	−0.810247	−	0.586089i	$-0.800667\pi$
$359$	−1.42062e9	−1.62049	−0.810247	+	0.586089i	$0.800667\pi$
$360$	0	0
$361$	−8.27527e8	−0.925778
$362$	2.80774e8	0.311083
$363$	0	0
$364$	−3.92453e8	−0.426513
$365$	−6.13908e8	−0.660813
$366$	0	0
$367$	−1.61585e9	−1.70636	−0.853180	−	0.521617i	$-0.825329\pi$
$367$	−1.61585e9	−1.70636	−0.853180	+	0.521617i	$0.825329\pi$
$368$	6.39203e7	0.0668608
$369$	0	0
$370$	7.08194e8	0.726852
$371$	−5.46100e8	−0.555218
$372$	0	0
$373$	1.89477e9	1.89049	0.945245	−	0.326361i	$-0.105822\pi$
$373$	1.89477e9	1.89049	0.945245	+	0.326361i	$0.105822\pi$
$374$	−5.13741e8	−0.507801
$375$	0	0
$376$	−3.87011e8	−0.375462
$377$	−5.11462e8	−0.491607
$378$	0	0
$379$	−6.09382e8	−0.574979	−0.287490	−	0.957784i	$-0.592821\pi$
$379$	−6.09382e8	−0.574979	−0.287490	+	0.957784i	$0.592821\pi$
$380$	−8.99246e7	−0.0840690
$381$	0	0
$382$	−1.06331e9	−0.975977
$383$	−1.23247e9	−1.12094	−0.560470	−	0.828175i	$-0.689380\pi$
$383$	−1.23247e9	−1.12094	−0.560470	+	0.828175i	$0.689380\pi$
$384$	0	0
$385$	5.46007e8	0.487625
$386$	3.95025e8	0.349598
$387$	0	0
$388$	−9.09463e8	−0.790449
$389$	1.81226e9	1.56098	0.780491	−	0.625167i	$-0.214969\pi$
$389$	1.81226e9	1.56098	0.780491	+	0.625167i	$0.214969\pi$
$390$	0	0
$391$	3.79387e8	0.320969
$392$	2.22162e9	1.86281
$393$	0	0
$394$	1.45914e8	0.120187
$395$	−1.57591e9	−1.28659
$396$	0	0
$397$	1.53157e9	1.22848	0.614242	−	0.789117i	$-0.289462\pi$
$397$	1.53157e9	1.22848	0.614242	+	0.789117i	$0.289462\pi$
$398$	−1.04019e9	−0.827033
$399$	0	0
$400$	2.30903e8	0.180393
$401$	1.62261e9	1.25663	0.628317	−	0.777958i	$-0.283744\pi$
$401$	1.62261e9	1.25663	0.628317	+	0.777958i	$0.283744\pi$
$402$	0	0
$403$	4.44874e8	0.338586
$404$	2.87248e8	0.216731
$405$	0	0
$406$	1.43869e9	1.06691
$407$	8.83737e8	0.649745
$408$	0	0
$409$	1.42089e8	0.102690	0.0513452	−	0.998681i	$-0.483649\pi$
$409$	1.42089e8	0.102690	0.0513452	+	0.998681i	$0.483649\pi$
$410$	6.16950e8	0.442086
$411$	0	0
$412$	6.88273e8	0.484864
$413$	−1.05929e9	−0.739931
$414$	0	0
$415$	6.48877e8	0.445650
$416$	6.84183e8	0.465956
$417$	0	0
$418$	1.31446e8	0.0880300
$419$	2.04493e8	0.135809	0.0679047	−	0.997692i	$-0.478369\pi$
$419$	2.04493e8	0.135809	0.0679047	+	0.997692i	$0.478369\pi$
$420$	0	0
$421$	−1.14148e9	−0.745557	−0.372778	−	0.927920i	$-0.621595\pi$
$421$	−1.14148e9	−0.745557	−0.372778	+	0.927920i	$0.621595\pi$
$422$	−6.84180e8	−0.443176
$423$	0	0
$424$	5.65119e8	0.360047
$425$	1.37048e9	0.865987
$426$	0	0
$427$	1.24755e9	0.775462
$428$	−1.02734e9	−0.633372
$429$	0	0
$430$	−1.23735e8	−0.0750502
$431$	−2.13524e9	−1.28463	−0.642314	−	0.766442i	$-0.722025\pi$
$431$	−2.13524e9	−1.28463	−0.642314	+	0.766442i	$0.722025\pi$
$432$	0	0
$433$	1.78226e9	1.05503	0.527514	−	0.849547i	$-0.323124\pi$
$433$	1.78226e9	1.05503	0.527514	+	0.849547i	$0.323124\pi$
$434$	−1.25139e9	−0.734814
$435$	0	0
$436$	−8.88728e7	−0.0513530
$437$	−9.70700e7	−0.0556417
$438$	0	0
$439$	−8.74976e8	−0.493595	−0.246797	−	0.969067i	$-0.579378\pi$
$439$	−8.74976e8	−0.493595	−0.246797	+	0.969067i	$0.579378\pi$
$440$	−5.65022e8	−0.316214
$441$	0	0
$442$	−1.17317e9	−0.646222
$443$	−2.15446e9	−1.17740	−0.588702	−	0.808350i	$-0.700361\pi$
$443$	−2.15446e9	−1.17740	−0.588702	+	0.808350i	$0.700361\pi$
$444$	0	0
$445$	−9.67867e8	−0.520661
$446$	−2.19874e9	−1.17355
$447$	0	0
$448$	−2.95519e9	−1.55279
$449$	−2.98017e9	−1.55374	−0.776870	−	0.629661i	$-0.783194\pi$
$449$	−2.98017e9	−1.55374	−0.776870	+	0.629661i	$0.783194\pi$
$450$	0	0
$451$	7.69876e8	0.395187
$452$	1.21263e9	0.617652
$453$	0	0
$454$	−3.90654e8	−0.195928
$455$	1.24685e9	0.620545
$456$	0	0
$457$	2.85176e9	1.39768	0.698838	−	0.715280i	$-0.253701\pi$
$457$	2.85176e9	1.39768	0.698838	+	0.715280i	$0.253701\pi$
$458$	1.46617e8	0.0713108
$459$	0	0
$460$	1.31569e8	0.0630234
$461$	−1.65358e8	−0.0786091	−0.0393045	−	0.999227i	$-0.512514\pi$
$461$	−1.65358e8	−0.0786091	−0.0393045	+	0.999227i	$0.512514\pi$
$462$	0	0
$463$	−1.65271e9	−0.773862	−0.386931	−	0.922109i	$-0.626465\pi$
$463$	−1.65271e9	−0.773862	−0.386931	+	0.922109i	$0.626465\pi$
$464$	−6.18585e8	−0.287466
$465$	0	0
$466$	−2.46693e9	−1.12929
$467$	−5.62317e8	−0.255489	−0.127744	−	0.991807i	$-0.540774\pi$
$467$	−5.62317e8	−0.255489	−0.127744	+	0.991807i	$0.540774\pi$
$468$	0	0
$469$	3.40954e9	1.52613
$470$	3.87705e8	0.172250
$471$	0	0
$472$	1.09618e9	0.479829
$473$	−1.54405e8	−0.0670885
$474$	0	0
$475$	−3.50652e8	−0.150123
$476$	−2.81719e9	−1.19727
$477$	0	0
$478$	6.56900e8	0.275107
$479$	−2.55353e9	−1.06162	−0.530808	−	0.847492i	$-0.678111\pi$
$479$	−2.55353e9	−1.06162	−0.530808	+	0.847492i	$0.678111\pi$
$480$	0	0
$481$	2.01808e9	0.826858
$482$	−1.93919e9	−0.788779
$483$	0	0
$484$	9.26418e8	0.371405
$485$	2.88942e9	1.15005
$486$	0	0
$487$	−8.52710e8	−0.334542	−0.167271	−	0.985911i	$-0.553495\pi$
$487$	−8.52710e8	−0.334542	−0.167271	+	0.985911i	$0.553495\pi$
$488$	−1.29100e9	−0.502870
$489$	0	0
$490$	−2.22561e9	−0.854598
$491$	−4.91989e9	−1.87573	−0.937864	−	0.347002i	$-0.887200\pi$
$491$	−4.91989e9	−1.87573	−0.937864	+	0.347002i	$0.887200\pi$
$492$	0	0
$493$	−3.67149e9	−1.38000
$494$	3.00167e8	0.112026
$495$	0	0
$496$	5.38050e8	0.197987
$497$	6.46732e9	2.36307
$498$	0	0
$499$	−2.15629e9	−0.776882	−0.388441	−	0.921474i	$-0.626986\pi$
$499$	−2.15629e9	−0.776882	−0.388441	+	0.921474i	$0.626986\pi$
$500$	1.33778e9	0.478620
$501$	0	0
$502$	−7.31847e8	−0.258200
$503$	4.90685e9	1.71915	0.859577	−	0.511007i	$-0.170727\pi$
$503$	4.90685e9	1.71915	0.859577	+	0.511007i	$0.170727\pi$
$504$	0	0
$505$	−9.12604e8	−0.315328
$506$	−1.92320e8	−0.0659928
$507$	0	0
$508$	−4.37066e8	−0.147919
$509$	1.30025e9	0.437035	0.218517	−	0.975833i	$-0.429878\pi$
$509$	1.30025e9	0.437035	0.218517	+	0.975833i	$0.429878\pi$
$510$	0	0
$511$	4.92090e9	1.63144
$512$	1.89402e9	0.623649
$513$	0	0
$514$	−1.07405e9	−0.348863
$515$	−2.18669e9	−0.705442
$516$	0	0
$517$	4.83807e8	0.153977
$518$	−5.67667e9	−1.79448
$519$	0	0
$520$	−1.29027e9	−0.402410
$521$	3.07871e9	0.953756	0.476878	−	0.878970i	$-0.341768\pi$
$521$	3.07871e9	0.953756	0.476878	+	0.878970i	$0.341768\pi$
$522$	0	0
$523$	−3.18094e9	−0.972298	−0.486149	−	0.873876i	$-0.661599\pi$
$523$	−3.18094e9	−0.972298	−0.486149	+	0.873876i	$0.661599\pi$
$524$	2.67100e9	0.810988
$525$	0	0
$526$	−8.81105e8	−0.263984
$527$	3.19349e9	0.950449
$528$	0	0
$529$	−3.26280e9	−0.958288
$530$	−5.66132e8	−0.165178
$531$	0	0
$532$	7.20808e8	0.207553
$533$	1.75807e9	0.502911
$534$	0	0
$535$	3.26391e9	0.921510
$536$	−3.52828e9	−0.989662
$537$	0	0
$538$	−2.36296e9	−0.654211
$539$	−2.77728e9	−0.763938
$540$	0	0
$541$	1.47730e9	0.401123	0.200561	−	0.979681i	$-0.435723\pi$
$541$	1.47730e9	0.401123	0.200561	+	0.979681i	$0.435723\pi$
$542$	2.59917e9	0.701193
$543$	0	0
$544$	4.91135e9	1.30799
$545$	2.82355e8	0.0747149
$546$	0	0
$547$	1.95085e9	0.509646	0.254823	−	0.966988i	$-0.417983\pi$
$547$	1.95085e9	0.509646	0.254823	+	0.966988i	$0.417983\pi$
$548$	−2.39904e9	−0.622739
$549$	0	0
$550$	−6.94727e8	−0.178051
$551$	9.39389e8	0.239230
$552$	0	0
$553$	1.26320e10	3.17640
$554$	1.12196e9	0.280346
$555$	0	0
$556$	2.78241e9	0.686531
$557$	−3.18198e9	−0.780197	−0.390099	−	0.920773i	$-0.627559\pi$
$557$	−3.18198e9	−0.780197	−0.390099	+	0.920773i	$0.627559\pi$
$558$	0	0
$559$	−3.52596e8	−0.0853761
$560$	1.50799e9	0.362862
$561$	0	0
$562$	1.72850e9	0.410763
$563$	5.93733e9	1.40221	0.701103	−	0.713060i	$-0.252691\pi$
$563$	5.93733e9	1.40221	0.701103	+	0.713060i	$0.252691\pi$
$564$	0	0
$565$	−3.85260e9	−0.898638
$566$	−2.76009e9	−0.639832
$567$	0	0
$568$	−6.69255e9	−1.53240
$569$	7.94607e9	1.80825	0.904127	−	0.427265i	$-0.140523\pi$
$569$	7.94607e9	1.80825	0.904127	+	0.427265i	$0.140523\pi$
$570$	0	0
$571$	7.59574e9	1.70743	0.853717	−	0.520737i	$-0.174343\pi$
$571$	7.59574e9	1.70743	0.853717	+	0.520737i	$0.174343\pi$
$572$	−5.07695e8	−0.113427
$573$	0	0
$574$	−4.94529e9	−1.09144
$575$	5.13041e8	0.112542
$576$	0	0
$577$	−7.50510e9	−1.62645	−0.813226	−	0.581948i	$-0.802291\pi$
$577$	−7.50510e9	−1.62645	−0.813226	+	0.581948i	$0.802291\pi$
$578$	−5.01170e9	−1.07954
$579$	0	0
$580$	−1.27325e9	−0.270967
$581$	−5.20120e9	−1.10024
$582$	0	0
$583$	−7.06461e8	−0.147655
$584$	−5.09228e9	−1.05795
$585$	0	0
$586$	2.25856e9	0.463649
$587$	−6.28252e8	−0.128204	−0.0641019	−	0.997943i	$-0.520418\pi$
$587$	−6.28252e8	−0.128204	−0.0641019	+	0.997943i	$0.520418\pi$
$588$	0	0
$589$	−8.17089e8	−0.164765
$590$	−1.09815e9	−0.220130
$591$	0	0
$592$	2.44076e9	0.483503
$593$	4.09806e9	0.807024	0.403512	−	0.914974i	$-0.367789\pi$
$593$	4.09806e9	0.807024	0.403512	+	0.914974i	$0.367789\pi$
$594$	0	0
$595$	8.95041e9	1.74194
$596$	−3.95564e9	−0.765341
$597$	0	0
$598$	−4.39176e8	−0.0839817
$599$	−3.70675e8	−0.0704692	−0.0352346	−	0.999379i	$-0.511218\pi$
$599$	−3.70675e8	−0.0704692	−0.0352346	+	0.999379i	$0.511218\pi$
$600$	0	0
$601$	−5.04786e9	−0.948520	−0.474260	−	0.880385i	$-0.657284\pi$
$601$	−5.04786e9	−0.948520	−0.474260	+	0.880385i	$0.657284\pi$
$602$	9.91819e8	0.185287
$603$	0	0
$604$	−3.35564e9	−0.619649
$605$	−2.94329e9	−0.540367
$606$	0	0
$607$	1.88528e9	0.342148	0.171074	−	0.985258i	$-0.445276\pi$
$607$	1.88528e9	0.342148	0.171074	+	0.985258i	$0.445276\pi$
$608$	−1.25662e9	−0.226747
$609$	0	0
$610$	1.29331e9	0.230700
$611$	1.10481e9	0.195949
$612$	0	0
$613$	−5.95226e9	−1.04369	−0.521843	−	0.853041i	$-0.674755\pi$
$613$	−5.95226e9	−1.04369	−0.521843	+	0.853041i	$0.674755\pi$
$614$	2.22860e8	0.0388546
$615$	0	0
$616$	4.52904e9	0.780682
$617$	5.79027e9	0.992433	0.496216	−	0.868199i	$-0.334722\pi$
$617$	5.79027e9	0.992433	0.496216	+	0.868199i	$0.334722\pi$
$618$	0	0
$619$	9.93652e7	0.0168390	0.00841951	−	0.999965i	$-0.497320\pi$
$619$	9.93652e7	0.0168390	0.00841951	+	0.999965i	$0.497320\pi$
$620$	1.10749e9	0.186624
$621$	0	0
$622$	7.16386e9	1.19366
$623$	7.75812e9	1.28543
$624$	0	0
$625$	−8.86962e8	−0.145320
$626$	−1.40117e8	−0.0228287
$627$	0	0
$628$	−4.18628e9	−0.674480
$629$	1.44866e10	2.32108
$630$	0	0
$631$	7.60492e9	1.20501	0.602507	−	0.798114i	$-0.294169\pi$
$631$	7.60492e9	1.20501	0.602507	+	0.798114i	$0.294169\pi$
$632$	−1.30719e10	−2.05982
$633$	0	0
$634$	−5.54678e9	−0.864427
$635$	1.38859e9	0.215211
$636$	0	0
$637$	−6.34212e9	−0.972179
$638$	1.86116e9	0.283734
$639$	0	0
$640$	6.34780e8	0.0957180
$641$	7.05623e9	1.05820	0.529102	−	0.848558i	$-0.322529\pi$
$641$	7.05623e9	1.05820	0.529102	+	0.848558i	$0.322529\pi$
$642$	0	0
$643$	−1.00382e10	−1.48908	−0.744539	−	0.667579i	$-0.767330\pi$
$643$	−1.00382e10	−1.48908	−0.744539	+	0.667579i	$0.767330\pi$
$644$	−1.05462e9	−0.155595
$645$	0	0
$646$	2.15473e9	0.314470
$647$	1.05830e10	1.53618	0.768092	−	0.640339i	$-0.221206\pi$
$647$	1.05830e10	1.53618	0.768092	+	0.640339i	$0.221206\pi$
$648$	0	0
$649$	−1.37035e9	−0.196778
$650$	−1.58646e9	−0.226586
$651$	0	0
$652$	−2.72163e9	−0.384559
$653$	−5.20110e9	−0.730969	−0.365484	−	0.930818i	$-0.619097\pi$
$653$	−5.20110e9	−0.730969	−0.365484	+	0.930818i	$0.619097\pi$
$654$	0	0
$655$	−8.48594e9	−1.17993
$656$	2.12629e9	0.294076
$657$	0	0
$658$	−3.10772e9	−0.425257
$659$	4.76603e9	0.648721	0.324361	−	0.945934i	$-0.394851\pi$
$659$	4.76603e9	0.648721	0.324361	+	0.945934i	$0.394851\pi$
$660$	0	0
$661$	5.23533e9	0.705081	0.352541	−	0.935796i	$-0.385318\pi$
$661$	5.23533e9	0.705081	0.352541	+	0.935796i	$0.385318\pi$
$662$	1.22273e9	0.163805
$663$	0	0
$664$	5.38234e9	0.713482
$665$	−2.29006e9	−0.301974
$666$	0	0
$667$	−1.37443e9	−0.179341
$668$	−2.74044e9	−0.355716
$669$	0	0
$670$	3.53461e9	0.454025
$671$	1.61389e9	0.206227
$672$	0	0
$673$	−5.60424e8	−0.0708703	−0.0354351	−	0.999372i	$-0.511282\pi$
$673$	−5.60424e8	−0.0708703	−0.0354351	+	0.999372i	$0.511282\pi$
$674$	9.06056e9	1.13984
$675$	0	0
$676$	2.53958e9	0.316190
$677$	3.94896e9	0.489128	0.244564	−	0.969633i	$-0.421355\pi$
$677$	3.94896e9	0.489128	0.244564	+	0.969633i	$0.421355\pi$
$678$	0	0
$679$	−2.31607e10	−2.83928
$680$	−9.26211e9	−1.12961
$681$	0	0
$682$	−1.61885e9	−0.195417
$683$	−1.30516e10	−1.56744	−0.783720	−	0.621115i	$-0.786680\pi$
$683$	−1.30516e10	−1.56744	−0.783720	+	0.621115i	$0.786680\pi$
$684$	0	0
$685$	7.62192e9	0.906040
$686$	7.56639e9	0.894858
$687$	0	0
$688$	−4.26446e8	−0.0499234
$689$	−1.61326e9	−0.187904
$690$	0	0
$691$	2.94742e9	0.339835	0.169918	−	0.985458i	$-0.445650\pi$
$691$	2.94742e9	0.339835	0.169918	+	0.985458i	$0.445650\pi$
$692$	−2.32401e9	−0.266604
$693$	0	0
$694$	1.80183e9	0.204624
$695$	−8.83991e9	−0.998852
$696$	0	0
$697$	1.26202e10	1.41173
$698$	1.03789e10	1.15520
$699$	0	0
$700$	−3.80966e9	−0.419801
$701$	7.95719e9	0.872462	0.436231	−	0.899835i	$-0.356313\pi$
$701$	7.95719e9	0.872462	0.436231	+	0.899835i	$0.356313\pi$
$702$	0	0
$703$	−3.70656e9	−0.402372
$704$	−3.82297e9	−0.412949
$705$	0	0
$706$	−3.68274e9	−0.393872
$707$	7.31516e9	0.778495
$708$	0	0
$709$	4.19832e9	0.442398	0.221199	−	0.975229i	$-0.429003\pi$
$709$	4.19832e9	0.442398	0.221199	+	0.975229i	$0.429003\pi$
$710$	6.70455e9	0.703016
$711$	0	0
$712$	−8.02831e9	−0.833574
$713$	1.19549e9	0.123518
$714$	0	0
$715$	1.61298e9	0.165028
$716$	1.55115e9	0.157927
$717$	0	0
$718$	1.18049e10	1.19022
$719$	−6.55620e9	−0.657811	−0.328906	−	0.944363i	$-0.606680\pi$
$719$	−6.55620e9	−0.657811	−0.328906	+	0.944363i	$0.606680\pi$
$720$	0	0
$721$	1.75278e10	1.74162
$722$	6.87651e9	0.679967
$723$	0	0
$724$	1.99179e9	0.195056
$725$	−4.96492e9	−0.483870
$726$	0	0
$727$	−1.72739e9	−0.166732	−0.0833662	−	0.996519i	$-0.526567\pi$
$727$	−1.72739e9	−0.166732	−0.0833662	+	0.996519i	$0.526567\pi$
$728$	1.03424e10	0.993487
$729$	0	0
$730$	5.10140e9	0.485355
$731$	−2.53109e9	−0.239660
$732$	0	0
$733$	1.67067e10	1.56685	0.783426	−	0.621486i	$-0.213471\pi$
$733$	1.67067e10	1.56685	0.783426	+	0.621486i	$0.213471\pi$
$734$	1.34273e10	1.25329
$735$	0	0
$736$	1.83857e9	0.169984
$737$	4.41075e9	0.405860
$738$	0	0
$739$	1.02748e10	0.936522	0.468261	−	0.883590i	$-0.344881\pi$
$739$	1.02748e10	0.936522	0.468261	+	0.883590i	$0.344881\pi$
$740$	5.02389e9	0.455753
$741$	0	0
$742$	4.53794e9	0.407798
$743$	−1.44275e10	−1.29042	−0.645209	−	0.764006i	$-0.723230\pi$
$743$	−1.44275e10	−1.29042	−0.645209	+	0.764006i	$0.723230\pi$
$744$	0	0
$745$	1.25673e10	1.11352
$746$	−1.57450e10	−1.38853
$747$	0	0
$748$	−3.64445e9	−0.318403
$749$	−2.61625e10	−2.27506
$750$	0	0
$751$	−7.52453e9	−0.648246	−0.324123	−	0.946015i	$-0.605069\pi$
$751$	−7.52453e9	−0.648246	−0.324123	+	0.946015i	$0.605069\pi$
$752$	1.33621e9	0.114581
$753$	0	0
$754$	4.25010e9	0.361077
$755$	1.06611e10	0.901543
$756$	0	0
$757$	−2.21226e10	−1.85353	−0.926767	−	0.375636i	$-0.877424\pi$
$757$	−2.21226e10	−1.85353	−0.926767	+	0.375636i	$0.877424\pi$
$758$	5.06379e9	0.422312
$759$	0	0
$760$	2.36981e9	0.195824
$761$	−4.50264e9	−0.370357	−0.185178	−	0.982705i	$-0.559286\pi$
$761$	−4.50264e9	−0.370357	−0.185178	+	0.982705i	$0.559286\pi$
$762$	0	0
$763$	−2.26327e9	−0.184459
$764$	−7.54309e9	−0.611959
$765$	0	0
$766$	1.02415e10	0.823310
$767$	−3.12930e9	−0.250417
$768$	0	0
$769$	−3.77262e9	−0.299158	−0.149579	−	0.988750i	$-0.547792\pi$
$769$	−3.77262e9	−0.299158	−0.149579	+	0.988750i	$0.547792\pi$
$770$	−4.53716e9	−0.358151
$771$	0	0
$772$	2.80229e9	0.219206
$773$	6.76884e9	0.527091	0.263546	−	0.964647i	$-0.415108\pi$
$773$	6.76884e9	0.527091	0.263546	+	0.964647i	$0.415108\pi$
$774$	0	0
$775$	4.31853e9	0.333257
$776$	2.39673e10	1.84121
$777$	0	0
$778$	−1.50594e10	−1.14651
$779$	−3.22901e9	−0.244730
$780$	0	0
$781$	8.36643e9	0.628437
$782$	−3.15259e9	−0.235746
$783$	0	0
$784$	−7.67045e9	−0.568479
$785$	1.33001e10	0.981319
$786$	0	0
$787$	−6.17492e9	−0.451564	−0.225782	−	0.974178i	$-0.572494\pi$
$787$	−6.17492e9	−0.451564	−0.225782	+	0.974178i	$0.572494\pi$
$788$	1.03510e9	0.0753602
$789$	0	0
$790$	1.30954e10	0.944980
$791$	3.08813e10	2.21859
$792$	0	0
$793$	3.68544e9	0.262442
$794$	−1.27269e10	−0.902300
$795$	0	0
$796$	−7.37906e9	−0.518568
$797$	−7.98475e8	−0.0558672	−0.0279336	−	0.999610i	$-0.508893\pi$
$797$	−7.98475e8	−0.0558672	−0.0279336	+	0.999610i	$0.508893\pi$
$798$	0	0
$799$	7.93079e9	0.550051
$800$	6.64157e9	0.458623
$801$	0	0
$802$	−1.34834e10	−0.922974
$803$	6.36591e9	0.433867
$804$	0	0
$805$	3.35059e9	0.226379
$806$	−3.69677e9	−0.248685
$807$	0	0
$808$	−7.56992e9	−0.504837
$809$	−1.09645e10	−0.728064	−0.364032	−	0.931386i	$-0.618600\pi$
$809$	−1.09645e10	−0.728064	−0.364032	+	0.931386i	$0.618600\pi$
$810$	0	0
$811$	−1.61849e10	−1.06546	−0.532729	−	0.846286i	$-0.678834\pi$
$811$	−1.61849e10	−1.06546	−0.532729	+	0.846286i	$0.678834\pi$
$812$	1.02060e10	0.668975
$813$	0	0
$814$	−7.34360e9	−0.477226
$815$	8.64680e9	0.559505
$816$	0	0
$817$	6.47605e8	0.0415464
$818$	−1.18072e9	−0.0754243
$819$	0	0
$820$	4.37661e9	0.277198
$821$	−1.75120e10	−1.10442	−0.552212	−	0.833704i	$-0.686216\pi$
$821$	−1.75120e10	−1.10442	−0.552212	+	0.833704i	$0.686216\pi$
$822$	0	0
$823$	9.36396e9	0.585545	0.292773	−	0.956182i	$-0.405422\pi$
$823$	9.36396e9	0.585545	0.292773	+	0.956182i	$0.405422\pi$
$824$	−1.81382e10	−1.12940
$825$	0	0
$826$	8.80242e9	0.543466
$827$	−1.31384e10	−0.807743	−0.403872	−	0.914816i	$-0.632336\pi$
$827$	−1.31384e10	−0.807743	−0.403872	+	0.914816i	$0.632336\pi$
$828$	0	0
$829$	5.48336e9	0.334277	0.167138	−	0.985933i	$-0.446547\pi$
$829$	5.48336e9	0.334277	0.167138	+	0.985933i	$0.446547\pi$
$830$	−5.39199e9	−0.327322
$831$	0	0
$832$	−8.73004e9	−0.525514
$833$	−4.55264e10	−2.72902
$834$	0	0
$835$	8.70657e9	0.517541
$836$	9.32472e8	0.0551968
$837$	0	0
$838$	−1.69928e9	−0.0997495
$839$	2.31219e10	1.35162	0.675812	−	0.737074i	$-0.263793\pi$
$839$	2.31219e10	1.35162	0.675812	+	0.737074i	$0.263793\pi$
$840$	0	0
$841$	−3.94896e9	−0.228927
$842$	9.48536e9	0.547598
$843$	0	0
$844$	−4.85353e9	−0.277881
$845$	−8.06841e9	−0.460034
$846$	0	0
$847$	2.35925e10	1.33408
$848$	−1.95115e9	−0.109876
$849$	0	0
$850$	−1.13883e10	−0.636052
$851$	5.42309e9	0.301643
$852$	0	0
$853$	−2.88007e10	−1.58885	−0.794423	−	0.607365i	$-0.792227\pi$
$853$	−2.88007e10	−1.58885	−0.794423	+	0.607365i	$0.792227\pi$
$854$	−1.03668e10	−0.569563
$855$	0	0
$856$	2.70736e10	1.47533
$857$	9.62309e9	0.522254	0.261127	−	0.965304i	$-0.415906\pi$
$857$	9.62309e9	0.522254	0.261127	+	0.965304i	$0.415906\pi$
$858$	0	0
$859$	−2.79141e10	−1.50261	−0.751306	−	0.659954i	$-0.770576\pi$
$859$	−2.79141e10	−1.50261	−0.751306	+	0.659954i	$0.770576\pi$
$860$	−8.77767e8	−0.0470581
$861$	0	0
$862$	1.77433e10	0.943535
$863$	1.26248e10	0.668633	0.334317	−	0.942461i	$-0.391495\pi$
$863$	1.26248e10	0.668633	0.334317	+	0.942461i	$0.391495\pi$
$864$	0	0
$865$	7.38352e9	0.387889
$866$	−1.48101e10	−0.774898
$867$	0	0
$868$	−8.87727e9	−0.460745
$869$	1.63414e10	0.844732
$870$	0	0
$871$	1.00723e10	0.516492
$872$	2.34209e9	0.119618
$873$	0	0
$874$	8.06624e8	0.0408678
$875$	3.40685e10	1.71920
$876$	0	0
$877$	−1.54386e10	−0.772876	−0.386438	−	0.922315i	$-0.626295\pi$
$877$	−1.54386e10	−0.772876	−0.386438	+	0.922315i	$0.626295\pi$
$878$	7.27080e9	0.362536
$879$	0	0
$880$	1.95081e9	0.0964997
$881$	−7.19197e9	−0.354350	−0.177175	−	0.984179i	$-0.556696\pi$
$881$	−7.19197e9	−0.354350	−0.177175	+	0.984179i	$0.556696\pi$
$882$	0	0
$883$	−2.42226e10	−1.18402	−0.592009	−	0.805931i	$-0.701665\pi$
$883$	−2.42226e10	−1.18402	−0.592009	+	0.805931i	$0.701665\pi$
$884$	−8.32238e9	−0.405196
$885$	0	0
$886$	1.79029e10	0.864781
$887$	−2.30169e9	−0.110742	−0.0553712	−	0.998466i	$-0.517634\pi$
$887$	−2.30169e9	−0.110742	−0.0553712	+	0.998466i	$0.517634\pi$
$888$	0	0
$889$	−1.11305e10	−0.531322
$890$	8.04269e9	0.382416
$891$	0	0
$892$	−1.55977e10	−0.735842
$893$	−2.02918e9	−0.0953542
$894$	0	0
$895$	−4.92810e9	−0.229773
$896$	−5.08821e9	−0.236312
$897$	0	0
$898$	2.47643e10	1.14119
$899$	−1.15692e10	−0.531063
$900$	0	0
$901$	−1.15806e10	−0.527468
$902$	−6.39745e9	−0.290258
$903$	0	0
$904$	−3.19568e10	−1.43871
$905$	−6.32806e9	−0.283792
$906$	0	0
$907$	−3.00170e10	−1.33580	−0.667899	−	0.744252i	$-0.732806\pi$
$907$	−3.00170e10	−1.33580	−0.667899	+	0.744252i	$0.732806\pi$
$908$	−2.77128e9	−0.122851
$909$	0	0
$910$	−1.03609e10	−0.455779
$911$	−1.56381e10	−0.685283	−0.342642	−	0.939466i	$-0.611322\pi$
$911$	−1.56381e10	−0.685283	−0.342642	+	0.939466i	$0.611322\pi$
$912$	0	0
$913$	−6.72852e9	−0.292598
$914$	−2.36973e10	−1.02657
$915$	0	0
$916$	1.04009e9	0.0447135
$917$	6.80207e10	2.91305
$918$	0	0
$919$	2.98731e10	1.26963	0.634813	−	0.772666i	$-0.281077\pi$
$919$	2.98731e10	1.26963	0.634813	+	0.772666i	$0.281077\pi$
$920$	−3.46728e9	−0.146802
$921$	0	0
$922$	1.37408e9	0.0577370
$923$	1.91054e10	0.799742
$924$	0	0
$925$	1.95901e10	0.813845
$926$	1.37336e10	0.568387
$927$	0	0
$928$	−1.77926e10	−0.730840
$929$	2.76307e10	1.13067	0.565337	−	0.824860i	$-0.308746\pi$
$929$	2.76307e10	1.13067	0.565337	+	0.824860i	$0.308746\pi$
$930$	0	0
$931$	1.16484e10	0.473089
$932$	−1.75003e10	−0.708091
$933$	0	0
$934$	4.67269e9	0.187652
$935$	1.15787e10	0.463253
$936$	0	0
$937$	3.33320e10	1.32365	0.661825	−	0.749658i	$-0.269782\pi$
$937$	3.33320e10	1.32365	0.661825	+	0.749658i	$0.269782\pi$
$938$	−2.83323e10	−1.12091
$939$	0	0
$940$	2.75036e9	0.108004
$941$	−1.77120e10	−0.692955	−0.346477	−	0.938058i	$-0.612622\pi$
$941$	−1.77120e10	−0.692955	−0.346477	+	0.938058i	$0.612622\pi$
$942$	0	0
$943$	4.72438e9	0.183465
$944$	−3.78472e9	−0.146431
$945$	0	0
$946$	1.28306e9	0.0492753
$947$	3.48632e10	1.33396	0.666980	−	0.745076i	$-0.267587\pi$
$947$	3.48632e10	1.33396	0.666980	+	0.745076i	$0.267587\pi$
$948$	0	0
$949$	1.45370e10	0.552134
$950$	2.91382e9	0.110263
$951$	0	0
$952$	7.42423e10	2.78883
$953$	−2.76617e10	−1.03527	−0.517635	−	0.855602i	$-0.673187\pi$
$953$	−2.76617e10	−1.03527	−0.517635	+	0.855602i	$0.673187\pi$
$954$	0	0
$955$	2.39649e10	0.890355
$956$	4.66001e9	0.172498
$957$	0	0
$958$	2.12191e10	0.779737
$959$	−6.10950e10	−2.23687
$960$	0	0
$961$	−1.74496e10	−0.634239
$962$	−1.67697e10	−0.607312
$963$	0	0
$964$	−1.37565e10	−0.494582
$965$	−8.90304e9	−0.318928
$966$	0	0
$967$	2.41546e9	0.0859028	0.0429514	−	0.999077i	$-0.486324\pi$
$967$	2.41546e9	0.0859028	0.0429514	+	0.999077i	$0.486324\pi$
$968$	−2.44141e10	−0.865123
$969$	0	0
$970$	−2.40103e10	−0.844687
$971$	3.04413e10	1.06708	0.533539	−	0.845776i	$-0.320862\pi$
$971$	3.04413e10	1.06708	0.533539	+	0.845776i	$0.320862\pi$
$972$	0	0
$973$	7.08580e10	2.46601
$974$	7.08578e9	0.245715
$975$	0	0
$976$	4.45734e9	0.153462
$977$	−2.20394e10	−0.756082	−0.378041	−	0.925789i	$-0.623402\pi$
$977$	−2.20394e10	−0.756082	−0.378041	+	0.925789i	$0.623402\pi$
$978$	0	0
$979$	1.00363e10	0.341848
$980$	−1.57883e10	−0.535852
$981$	0	0
$982$	4.08829e10	1.37769
$983$	−9.14954e9	−0.307229	−0.153614	−	0.988131i	$-0.549091\pi$
$983$	−9.14954e9	−0.307229	−0.153614	+	0.988131i	$0.549091\pi$
$984$	0	0
$985$	−3.28859e9	−0.109644
$986$	3.05090e10	1.01358
$987$	0	0
$988$	2.12937e9	0.0702428
$989$	−9.47514e8	−0.0311457
$990$	0	0
$991$	2.26100e10	0.737976	0.368988	−	0.929434i	$-0.379704\pi$
$991$	2.26100e10	0.737976	0.368988	+	0.929434i	$0.379704\pi$
$992$	1.54762e10	0.503354
$993$	0	0
$994$	−5.37416e10	−1.73563
$995$	2.34438e10	0.754478
$996$	0	0
$997$	−5.32639e10	−1.70216	−0.851080	−	0.525036i	$-0.824052\pi$
$997$	−5.32639e10	−1.70216	−0.851080	+	0.525036i	$0.824052\pi$
$998$	1.79181e10	0.570606
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	387.8.a.b.1.4		11
3.2	odd	2		43.8.a.a.1.8	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.8.a.a.1.8	✓	11	3.2	odd	2
387.8.a.b.1.4		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)

\(f(q)\)	\(=\)	\(q-8.30971 q^{2} -58.9487 q^{4} +187.284 q^{5} -1501.21 q^{7} +1553.49 q^{8} +O(q^{10})\)	\(q-8.30971 q^{2} -58.9487 q^{4} +187.284 q^{5} -1501.21 q^{7} +1553.49 q^{8} -1556.27 q^{10} -1942.03 q^{11} -4434.78 q^{13} +12474.6 q^{14} -5363.63 q^{16} -31834.8 q^{17} +8145.26 q^{19} -11040.1 q^{20} +16137.7 q^{22} -11917.4 q^{23} -43049.8 q^{25} +36851.8 q^{26} +88494.3 q^{28} +115330. q^{29} -100315. q^{31} -154276. q^{32} +264538. q^{34} -281152. q^{35} -455058. q^{37} -67684.8 q^{38} +290943. q^{40} -396428. q^{41} +79507.0 q^{43} +114480. q^{44} +99029.9 q^{46} -249124. q^{47} +1.43009e6 q^{49} +357732. q^{50} +261424. q^{52} +363774. q^{53} -363711. q^{55} -2.33211e6 q^{56} -958356. q^{58} +705627. q^{59} -831030. q^{61} +833586. q^{62} +1.96854e6 q^{64} -830562. q^{65} -2.27120e6 q^{67} +1.87662e6 q^{68} +2.33629e6 q^{70} -4.30808e6 q^{71} -3.27796e6 q^{73} +3.78140e6 q^{74} -480152. q^{76} +2.91540e6 q^{77} -8.41456e6 q^{79} -1.00452e6 q^{80} +3.29420e6 q^{82} +3.46468e6 q^{83} -5.96213e6 q^{85} -660680. q^{86} -3.01693e6 q^{88} -5.16792e6 q^{89} +6.65753e6 q^{91} +702513. q^{92} +2.07015e6 q^{94} +1.52547e6 q^{95} +1.54280e7 q^{97} -1.18836e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 24 q^{2} + 602 q^{4} + 752 q^{5} - 12 q^{7} + 3810 q^{8}+O(q^{10}) \) 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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