LMFDB - Embedded newform 387.8.a.d.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(387, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("387.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 387 = 3^{2} \cdot 43 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	387.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$120.893004862$
Analytic rank:	$1$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 3 x^{12} - 1279 x^{11} + 3765 x^{10} + 598742 x^{9} - 1518614 x^{8} - 124677082 x^{7} + \cdots + 1551032970660 $ x^13 - 3x^12 - 1279x^11 + 3765x^10 + 598742x^9 - 1518614x^8 - 124677082x^7 + 193428526x^6 + 11160446785x^5 + 1754605765x^4 - 349352939351x^3 - 481872751923x^2 + 2098464001560x + 1551032970660
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{3} $
Twist minimal:	no (minimal twist has level 43)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-4.71679$ 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)

$f(q)$	$=$	$q+3.71679 q^{2} -114.185 q^{4} -385.350 q^{5} -1546.77 q^{7} -900.153 q^{8} +O(q^{10})$	\(q+3.71679 q^{2} -114.185 q^{4} -385.350 q^{5} -1546.77 q^{7} -900.153 q^{8} -1432.27 q^{10} +1662.15 q^{11} +12725.9 q^{13} -5749.00 q^{14} +11270.1 q^{16} -23803.1 q^{17} +30546.0 q^{19} +44001.4 q^{20} +6177.85 q^{22} +63784.8 q^{23} +70369.8 q^{25} +47299.5 q^{26} +176618. q^{28} -190003. q^{29} +143977. q^{31} +157108. q^{32} -88471.1 q^{34} +596047. q^{35} -53873.0 q^{37} +113533. q^{38} +346874. q^{40} +355277. q^{41} -79507.0 q^{43} -189793. q^{44} +237075. q^{46} +39435.7 q^{47} +1.56894e6 q^{49} +261550. q^{50} -1.45311e6 q^{52} -1.01320e6 q^{53} -640508. q^{55} +1.39233e6 q^{56} -706202. q^{58} -1.24979e6 q^{59} -751833. q^{61} +535133. q^{62} -858631. q^{64} -4.90393e6 q^{65} -3.22613e6 q^{67} +2.71797e6 q^{68} +2.21538e6 q^{70} -1.10265e6 q^{71} +4.53962e6 q^{73} -200235. q^{74} -3.48791e6 q^{76} -2.57095e6 q^{77} +3.63589e6 q^{79} -4.34292e6 q^{80} +1.32049e6 q^{82} +149479. q^{83} +9.17253e6 q^{85} -295511. q^{86} -1.49618e6 q^{88} +8.46943e6 q^{89} -1.96840e7 q^{91} -7.28329e6 q^{92} +146574. q^{94} -1.17709e7 q^{95} +5.99834e6 q^{97} +5.83143e6 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 16 q^{2} + 922 q^{4} - 998 q^{5} + 1360 q^{7} - 3870 q^{8}+O(q^{10}) $ 13 * q - 16 * q^2 + 922 * q^4 - 998 * q^5 + 1360 * q^7 - 3870 * q^8	\( 13 q - 16 q^{2} + 922 q^{4} - 998 q^{5} + 1360 q^{7} - 3870 q^{8} + 4667 q^{10} - 1620 q^{11} + 13550 q^{13} - 44160 q^{14} + 114026 q^{16} - 110880 q^{17} + 105058 q^{19} - 167251 q^{20} + 201504 q^{22} - 160184 q^{23} + 270149 q^{25} - 272104 q^{26} + 208172 q^{28} - 285546 q^{29} - 99616 q^{31} - 200126 q^{32} - 80941 q^{34} + 187104 q^{35} + 176038 q^{37} - 652165 q^{38} - 895387 q^{40} + 410260 q^{41} - 1033591 q^{43} + 2177076 q^{44} - 3975765 q^{46} + 424556 q^{47} - 1561359 q^{49} + 4063801 q^{50} - 4172312 q^{52} - 3992458 q^{53} + 406960 q^{55} - 1559556 q^{56} - 4052005 q^{58} - 2248836 q^{59} + 6210394 q^{61} - 885317 q^{62} - 3096318 q^{64} - 5600420 q^{65} - 1993648 q^{67} - 9327135 q^{68} - 1105098 q^{70} - 4978064 q^{71} + 8224814 q^{73} + 3613563 q^{74} + 10687121 q^{76} - 17261892 q^{77} + 6945708 q^{79} - 15822799 q^{80} - 508449 q^{82} - 22937328 q^{83} - 575532 q^{85} + 1272112 q^{86} + 11202656 q^{88} - 9291302 q^{89} + 25581108 q^{91} + 14388137 q^{92} - 24645805 q^{94} - 30750464 q^{95} + 10001852 q^{97} + 32304856 q^{98}+O(q^{100}) \) 13 * q - 16 * q^2 + 922 * q^4 - 998 * q^5 + 1360 * q^7 - 3870 * q^8 + 4667 * q^10 - 1620 * q^11 + 13550 * q^13 - 44160 * q^14 + 114026 * q^16 - 110880 * q^17 + 105058 * q^19 - 167251 * q^20 + 201504 * q^22 - 160184 * q^23 + 270149 * q^25 - 272104 * q^26 + 208172 * q^28 - 285546 * q^29 - 99616 * q^31 - 200126 * q^32 - 80941 * q^34 + 187104 * q^35 + 176038 * q^37 - 652165 * q^38 - 895387 * q^40 + 410260 * q^41 - 1033591 * q^43 + 2177076 * q^44 - 3975765 * q^46 + 424556 * q^47 - 1561359 * q^49 + 4063801 * q^50 - 4172312 * q^52 - 3992458 * q^53 + 406960 * q^55 - 1559556 * q^56 - 4052005 * q^58 - 2248836 * q^59 + 6210394 * q^61 - 885317 * q^62 - 3096318 * q^64 - 5600420 * q^65 - 1993648 * q^67 - 9327135 * q^68 - 1105098 * q^70 - 4978064 * q^71 + 8224814 * q^73 + 3613563 * q^74 + 10687121 * q^76 - 17261892 * q^77 + 6945708 * q^79 - 15822799 * q^80 - 508449 * q^82 - 22937328 * q^83 - 575532 * q^85 + 1272112 * q^86 + 11202656 * q^88 - 9291302 * q^89 + 25581108 * q^91 + 14388137 * q^92 - 24645805 * q^94 - 30750464 * q^95 + 10001852 * q^97 + 32304856 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	3.71679	0.328521	0.164260	−	0.986417i	$-0.447476\pi$
$2$	3.71679	0.328521	0.164260	+	0.986417i	$0.447476\pi$
$3$	0	0
$3$	0	0
$4$	−114.185	−0.892074
$5$	−385.350	−1.37867	−0.689336	−	0.724442i	$-0.742097\pi$
$5$	−385.350	−1.37867	−0.689336	+	0.724442i	$0.742097\pi$
$6$	0	0
$7$	−1546.77	−1.70444	−0.852219	−	0.523185i	$-0.824744\pi$
$7$	−1546.77	−1.70444	−0.852219	+	0.523185i	$0.824744\pi$
$8$	−900.153	−0.621586
$9$	0	0
$10$	−1432.27	−0.452922
$11$	1662.15	0.376526	0.188263	−	0.982119i	$-0.439714\pi$
$11$	1662.15	0.376526	0.188263	+	0.982119i	$0.439714\pi$
$12$	0	0
$13$	12725.9	1.60652	0.803261	−	0.595628i	$-0.203097\pi$
$13$	12725.9	1.60652	0.803261	+	0.595628i	$0.203097\pi$
$14$	−5749.00	−0.559944
$15$	0	0
$16$	11270.1	0.687870
$17$	−23803.1	−1.17507	−0.587533	−	0.809200i	$-0.699901\pi$
$17$	−23803.1	−1.17507	−0.587533	+	0.809200i	$0.699901\pi$
$18$	0	0
$19$	30546.0	1.02169	0.510843	−	0.859674i	$-0.329333\pi$
$19$	30546.0	1.02169	0.510843	+	0.859674i	$0.329333\pi$
$20$	44001.4	1.22988
$21$	0	0
$22$	6177.85	0.123697
$23$	63784.8	1.09312	0.546562	−	0.837419i	$-0.315936\pi$
$23$	63784.8	1.09312	0.546562	+	0.837419i	$0.315936\pi$
$24$	0	0
$25$	70369.8	0.900734
$26$	47299.5	0.527776
$27$	0	0
$28$	176618.	1.52049
$29$	−190003.	−1.44666	−0.723332	−	0.690500i	$-0.757391\pi$
$29$	−190003.	−1.44666	−0.723332	+	0.690500i	$0.757391\pi$
$30$	0	0
$31$	143977.	0.868015	0.434008	−	0.900909i	$-0.357099\pi$
$31$	143977.	0.868015	0.434008	+	0.900909i	$0.357099\pi$
$32$	157108.	0.847566
$33$	0	0
$34$	−88471.1	−0.386034
$35$	596047.	2.34986
$36$	0	0
$37$	−53873.0	−0.174850	−0.0874249	−	0.996171i	$-0.527864\pi$
$37$	−53873.0	−0.174850	−0.0874249	+	0.996171i	$0.527864\pi$
$38$	113533.	0.335645
$39$	0	0
$40$	346874.	0.856963
$41$	355277.	0.805051	0.402526	−	0.915409i	$-0.368132\pi$
$41$	355277.	0.805051	0.402526	+	0.915409i	$0.368132\pi$
$42$	0	0
$43$	−79507.0	−0.152499
$43$	−79507.0	−0.152499
$44$	−189793.	−0.335889
$45$	0	0
$46$	237075.	0.359114
$47$	39435.7	0.0554047	0.0277024	−	0.999616i	$-0.491181\pi$
$47$	39435.7	0.0554047	0.0277024	+	0.999616i	$0.491181\pi$
$48$	0	0
$49$	1.56894e6	1.90511
$50$	261550.	0.295910
$51$	0	0
$52$	−1.45311e6	−1.43314
$53$	−1.01320e6	−0.934823	−0.467411	−	0.884040i	$-0.654813\pi$
$53$	−1.01320e6	−0.934823	−0.467411	+	0.884040i	$0.654813\pi$
$54$	0	0
$55$	−640508.	−0.519105
$56$	1.39233e6	1.05946
$57$	0	0
$58$	−706202.	−0.475260
$59$	−1.24979e6	−0.792236	−0.396118	−	0.918200i	$-0.629643\pi$
$59$	−1.24979e6	−0.792236	−0.396118	+	0.918200i	$0.629643\pi$
$60$	0	0
$61$	−751833.	−0.424098	−0.212049	−	0.977259i	$-0.568014\pi$
$61$	−751833.	−0.424098	−0.212049	+	0.977259i	$0.568014\pi$
$62$	535133.	0.285161
$63$	0	0
$64$	−858631.	−0.409427
$65$	−4.90393e6	−2.21486
$66$	0	0
$67$	−3.22613e6	−1.31045	−0.655224	−	0.755435i	$-0.727426\pi$
$67$	−3.22613e6	−1.31045	−0.655224	+	0.755435i	$0.727426\pi$
$68$	2.71797e6	1.04825
$69$	0	0
$70$	2.21538e6	0.771978
$71$	−1.10265e6	−0.365623	−0.182811	−	0.983148i	$-0.558520\pi$
$71$	−1.10265e6	−0.365623	−0.182811	+	0.983148i	$0.558520\pi$
$72$	0	0
$73$	4.53962e6	1.36581	0.682904	−	0.730508i	$-0.260717\pi$
$73$	4.53962e6	1.36581	0.682904	+	0.730508i	$0.260717\pi$
$74$	−200235.	−0.0574418
$75$	0	0
$76$	−3.48791e6	−0.911419
$77$	−2.57095e6	−0.641765
$78$	0	0
$79$	3.63589e6	0.829691	0.414846	−	0.909892i	$-0.363836\pi$
$79$	3.63589e6	0.829691	0.414846	+	0.909892i	$0.363836\pi$
$80$	−4.34292e6	−0.948347
$81$	0	0
$82$	1.32049e6	0.264476
$83$	149479.	0.0286951	0.0143476	−	0.999897i	$-0.495433\pi$
$83$	149479.	0.0286951	0.0143476	+	0.999897i	$0.495433\pi$
$84$	0	0
$85$	9.17253e6	1.62003
$86$	−295511.	−0.0500990
$87$	0	0
$88$	−1.49618e6	−0.234043
$89$	8.46943e6	1.27347	0.636736	−	0.771082i	$-0.280284\pi$
$89$	8.46943e6	1.27347	0.636736	+	0.771082i	$0.280284\pi$
$90$	0	0
$91$	−1.96840e7	−2.73822
$92$	−7.28329e6	−0.975148
$93$	0	0
$94$	146574.	0.0182016
$95$	−1.17709e7	−1.40857
$96$	0	0
$97$	5.99834e6	0.667313	0.333657	−	0.942695i	$-0.391717\pi$
$97$	5.99834e6	0.667313	0.333657	+	0.942695i	$0.391717\pi$
$98$	5.83143e6	0.625869
$99$	0	0
$100$	−8.03521e6	−0.803521
$101$	8.50960e6	0.821834	0.410917	−	0.911673i	$-0.365209\pi$
$101$	8.50960e6	0.821834	0.410917	+	0.911673i	$0.365209\pi$
$102$	0	0
$103$	1.74827e6	0.157644	0.0788222	−	0.996889i	$-0.474884\pi$
$103$	1.74827e6	0.157644	0.0788222	+	0.996889i	$0.474884\pi$
$104$	−1.14552e7	−0.998591
$105$	0	0
$106$	−3.76585e6	−0.307109
$107$	−1.06723e7	−0.842196	−0.421098	−	0.907015i	$-0.638355\pi$
$107$	−1.06723e7	−0.842196	−0.421098	+	0.907015i	$0.638355\pi$
$108$	0	0
$109$	5.16184e6	0.381779	0.190889	−	0.981612i	$-0.438863\pi$
$109$	5.16184e6	0.381779	0.190889	+	0.981612i	$0.438863\pi$
$110$	−2.38063e6	−0.170537
$111$	0	0
$112$	−1.74321e7	−1.17243
$113$	−1.76860e7	−1.15307	−0.576534	−	0.817073i	$-0.695595\pi$
$113$	−1.76860e7	−1.15307	−0.576534	+	0.817073i	$0.695595\pi$
$114$	0	0
$115$	−2.45795e7	−1.50706
$116$	2.16956e7	1.29053
$117$	0	0
$118$	−4.64520e6	−0.260266
$119$	3.68178e7	2.00283
$120$	0	0
$121$	−1.67244e7	−0.858228
$122$	−2.79440e6	−0.139325
$123$	0	0
$124$	−1.64401e7	−0.774334
$125$	2.98845e6	0.136855
$126$	0	0
$127$	4.09391e7	1.77347	0.886737	−	0.462274i	$-0.152966\pi$
$127$	4.09391e7	1.77347	0.886737	+	0.462274i	$0.152966\pi$
$128$	−2.33012e7	−0.982071
$129$	0	0
$130$	−1.82269e7	−0.727629
$131$	−4.15303e7	−1.61404	−0.807021	−	0.590522i	$-0.798922\pi$
$131$	−4.15303e7	−1.61404	−0.807021	+	0.590522i	$0.798922\pi$
$132$	0	0
$133$	−4.72476e7	−1.74140
$134$	−1.19908e7	−0.430509
$135$	0	0
$136$	2.14264e7	0.730404
$137$	3.50608e7	1.16493	0.582466	−	0.812855i	$-0.302088\pi$
$137$	3.50608e7	1.16493	0.582466	+	0.812855i	$0.302088\pi$
$138$	0	0
$139$	5.57867e7	1.76189	0.880945	−	0.473219i	$-0.156908\pi$
$139$	5.57867e7	1.76189	0.880945	+	0.473219i	$0.156908\pi$
$140$	−6.80599e7	−2.09625
$141$	0	0
$142$	−4.09831e6	−0.120115
$143$	2.11523e7	0.604897
$144$	0	0
$145$	7.32178e7	1.99447
$146$	1.68728e7	0.448697
$147$	0	0
$148$	6.15151e6	0.155979
$149$	1.78918e7	0.443099	0.221550	−	0.975149i	$-0.428888\pi$
$149$	1.78918e7	0.443099	0.221550	+	0.975149i	$0.428888\pi$
$150$	0	0
$151$	−2.85564e7	−0.674970	−0.337485	−	0.941331i	$-0.609576\pi$
$151$	−2.85564e7	−0.674970	−0.337485	+	0.941331i	$0.609576\pi$
$152$	−2.74961e7	−0.635065
$153$	0	0
$154$	−9.55568e6	−0.210833
$155$	−5.54816e7	−1.19671
$156$	0	0
$157$	6.75273e7	1.39261	0.696307	−	0.717744i	$-0.254825\pi$
$157$	6.75273e7	1.39261	0.696307	+	0.717744i	$0.254825\pi$
$158$	1.35139e7	0.272571
$159$	0	0
$160$	−6.05416e7	−1.16851
$161$	−9.86601e7	−1.86316
$162$	0	0
$163$	−7.20769e7	−1.30359	−0.651793	−	0.758397i	$-0.725983\pi$
$163$	−7.20769e7	−1.30359	−0.651793	+	0.758397i	$0.725983\pi$
$164$	−4.05675e7	−0.718165
$165$	0	0
$166$	555583.	0.00942694
$167$	−2.87709e7	−0.478019	−0.239010	−	0.971017i	$-0.576823\pi$
$167$	−2.87709e7	−0.478019	−0.239010	+	0.971017i	$0.576823\pi$
$168$	0	0
$169$	9.91998e7	1.58091
$170$	3.40924e7	0.532213
$171$	0	0
$172$	9.07854e6	0.136040
$173$	1.24594e8	1.82951	0.914755	−	0.404010i	$-0.132384\pi$
$173$	1.24594e8	1.82951	0.914755	+	0.404010i	$0.132384\pi$
$174$	0	0
$175$	−1.08846e8	−1.53525
$176$	1.87325e7	0.259001
$177$	0	0
$178$	3.14791e7	0.418362
$179$	1.45448e7	0.189550	0.0947749	−	0.995499i	$-0.469787\pi$
$179$	1.45448e7	0.189550	0.0947749	+	0.995499i	$0.469787\pi$
$180$	0	0
$181$	−4.71999e7	−0.591651	−0.295826	−	0.955242i	$-0.595595\pi$
$181$	−4.71999e7	−0.591651	−0.295826	+	0.955242i	$0.595595\pi$
$182$	−7.31612e7	−0.899562
$183$	0	0
$184$	−5.74160e7	−0.679471
$185$	2.07600e7	0.241060
$186$	0	0
$187$	−3.95642e7	−0.442442
$188$	−4.50298e6	−0.0494251
$189$	0	0
$190$	−4.37501e7	−0.462744
$191$	5.66800e6	0.0588590	0.0294295	−	0.999567i	$-0.490631\pi$
$191$	5.66800e6	0.0588590	0.0294295	+	0.999567i	$0.490631\pi$
$192$	0	0
$193$	1.30205e8	1.30369	0.651847	−	0.758350i	$-0.273994\pi$
$193$	1.30205e8	1.30369	0.651847	+	0.758350i	$0.273994\pi$
$194$	2.22946e7	0.219226
$195$	0	0
$196$	−1.79150e8	−1.69950
$197$	−1.88254e8	−1.75434	−0.877169	−	0.480182i	$-0.840571\pi$
$197$	−1.88254e8	−1.75434	−0.877169	+	0.480182i	$0.840571\pi$
$198$	0	0
$199$	1.14630e8	1.03113	0.515564	−	0.856851i	$-0.327583\pi$
$199$	1.14630e8	1.03113	0.515564	+	0.856851i	$0.327583\pi$
$200$	−6.33436e7	−0.559884
$201$	0	0
$202$	3.16284e7	0.269990
$203$	2.93890e8	2.46575
$204$	0	0
$205$	−1.36906e8	−1.10990
$206$	6.49796e6	0.0517895
$207$	0	0
$208$	1.43422e8	1.10508
$209$	5.07720e7	0.384691
$210$	0	0
$211$	3.12083e7	0.228708	0.114354	−	0.993440i	$-0.463520\pi$
$211$	3.12083e7	0.228708	0.114354	+	0.993440i	$0.463520\pi$
$212$	1.15693e8	0.833931
$213$	0	0
$214$	−3.96666e7	−0.276679
$215$	3.06380e7	0.210245
$216$	0	0
$217$	−2.22699e8	−1.47948
$218$	1.91855e7	0.125422
$219$	0	0
$220$	7.31367e7	0.463080
$221$	−3.02916e8	−1.88777
$222$	0	0
$223$	−1.46395e8	−0.884015	−0.442007	−	0.897011i	$-0.645733\pi$
$223$	−1.46395e8	−0.884015	−0.442007	+	0.897011i	$0.645733\pi$
$224$	−2.43009e8	−1.44462
$225$	0	0
$226$	−6.57352e7	−0.378807
$227$	2.31019e7	0.131086	0.0655432	−	0.997850i	$-0.479122\pi$
$227$	2.31019e7	0.131086	0.0655432	+	0.997850i	$0.479122\pi$
$228$	0	0
$229$	−2.46033e8	−1.35385	−0.676924	−	0.736053i	$-0.736687\pi$
$229$	−2.46033e8	−1.35385	−0.676924	+	0.736053i	$0.736687\pi$
$230$	−9.13568e7	−0.495100
$231$	0	0
$232$	1.71032e8	0.899226
$233$	−2.56892e8	−1.33047	−0.665235	−	0.746634i	$-0.731669\pi$
$233$	−2.56892e8	−1.33047	−0.665235	+	0.746634i	$0.731669\pi$
$234$	0	0
$235$	−1.51965e7	−0.0763849
$236$	1.42708e8	0.706733
$237$	0	0
$238$	1.36844e8	0.657971
$239$	−2.98201e8	−1.41292	−0.706458	−	0.707755i	$-0.749708\pi$
$239$	−2.98201e8	−1.41292	−0.706458	+	0.707755i	$0.749708\pi$
$240$	0	0
$241$	−2.29672e8	−1.05694	−0.528468	−	0.848953i	$-0.677233\pi$
$241$	−2.29672e8	−1.05694	−0.528468	+	0.848953i	$0.677233\pi$
$242$	−6.21612e7	−0.281946
$243$	0	0
$244$	8.58484e7	0.378327
$245$	−6.04592e8	−2.62652
$246$	0	0
$247$	3.88726e8	1.64136
$248$	−1.29601e8	−0.539546
$249$	0	0
$250$	1.11075e7	0.0449598
$251$	−1.54752e8	−0.617702	−0.308851	−	0.951111i	$-0.599944\pi$
$251$	−1.54752e8	−0.617702	−0.308851	+	0.951111i	$0.599944\pi$
$252$	0	0
$253$	1.06020e8	0.411589
$254$	1.52162e8	0.582623
$255$	0	0
$256$	2.32992e7	0.0867961
$257$	−2.73313e8	−1.00437	−0.502185	−	0.864760i	$-0.667470\pi$
$257$	−2.73313e8	−1.00437	−0.502185	+	0.864760i	$0.667470\pi$
$258$	0	0
$259$	8.33289e7	0.298021
$260$	5.59957e8	1.97582
$261$	0	0
$262$	−1.54359e8	−0.530247
$263$	−2.20336e8	−0.746863	−0.373431	−	0.927658i	$-0.621819\pi$
$263$	−2.20336e8	−0.746863	−0.373431	+	0.927658i	$0.621819\pi$
$264$	0	0
$265$	3.90437e8	1.28881
$266$	−1.75609e8	−0.572087
$267$	0	0
$268$	3.68377e8	1.16902
$269$	1.74740e8	0.547342	0.273671	−	0.961823i	$-0.411762\pi$
$269$	1.74740e8	0.547342	0.273671	+	0.961823i	$0.411762\pi$
$270$	0	0
$271$	−5.89750e8	−1.80001	−0.900005	−	0.435879i	$-0.856438\pi$
$271$	−5.89750e8	−1.80001	−0.900005	+	0.435879i	$0.856438\pi$
$272$	−2.68262e8	−0.808292
$273$	0	0
$274$	1.30314e8	0.382704
$275$	1.16965e8	0.339149
$276$	0	0
$277$	−3.46288e8	−0.978944	−0.489472	−	0.872019i	$-0.662810\pi$
$277$	−3.46288e8	−0.978944	−0.489472	+	0.872019i	$0.662810\pi$
$278$	2.07347e8	0.578818
$279$	0	0
$280$	−5.36533e8	−1.46064
$281$	2.47819e8	0.666289	0.333145	−	0.942876i	$-0.391890\pi$
$281$	2.47819e8	0.666289	0.333145	+	0.942876i	$0.391890\pi$
$282$	0	0
$283$	2.38221e8	0.624779	0.312390	−	0.949954i	$-0.398871\pi$
$283$	2.38221e8	0.624779	0.312390	+	0.949954i	$0.398871\pi$
$284$	1.25906e8	0.326162
$285$	0	0
$286$	7.86186e7	0.198721
$287$	−5.49530e8	−1.37216
$288$	0	0
$289$	1.56248e8	0.380779
$290$	2.72135e8	0.655227
$291$	0	0
$292$	−5.18359e8	−1.21840
$293$	−6.02688e8	−1.39977	−0.699884	−	0.714257i	$-0.746765\pi$
$293$	−6.02688e8	−1.39977	−0.699884	+	0.714257i	$0.746765\pi$
$294$	0	0
$295$	4.81606e8	1.09223
$296$	4.84939e7	0.108684
$297$	0	0
$298$	6.64999e7	0.145567
$299$	8.11718e8	1.75613
$300$	0	0
$301$	1.22979e8	0.259924
$302$	−1.06138e8	−0.221742
$303$	0	0
$304$	3.44256e8	0.702787
$305$	2.89719e8	0.584692
$306$	0	0
$307$	−5.50528e8	−1.08591	−0.542957	−	0.839761i	$-0.682695\pi$
$307$	−5.50528e8	−1.08591	−0.542957	+	0.839761i	$0.682695\pi$
$308$	2.93565e8	0.572502
$309$	0	0
$310$	−2.06213e8	−0.393143
$311$	8.54076e8	1.61003	0.805017	−	0.593251i	$-0.202156\pi$
$311$	8.54076e8	1.61003	0.805017	+	0.593251i	$0.202156\pi$
$312$	0	0
$313$	6.27627e8	1.15690	0.578451	−	0.815717i	$-0.303657\pi$
$313$	6.27627e8	1.15690	0.578451	+	0.815717i	$0.303657\pi$
$314$	2.50985e8	0.457503
$315$	0	0
$316$	−4.15166e8	−0.740146
$317$	8.96686e7	0.158100	0.0790502	−	0.996871i	$-0.474811\pi$
$317$	8.96686e7	0.158100	0.0790502	+	0.996871i	$0.474811\pi$
$318$	0	0
$319$	−3.15813e8	−0.544706
$320$	3.30874e8	0.564465
$321$	0	0
$322$	−3.66699e8	−0.612088
$323$	−7.27090e8	−1.20055
$324$	0	0
$325$	8.95519e8	1.44705
$326$	−2.67895e8	−0.428255
$327$	0	0
$328$	−3.19803e8	−0.500408
$329$	−6.09977e7	−0.0944339
$330$	0	0
$331$	2.00092e8	0.303271	0.151635	−	0.988436i	$-0.451546\pi$
$331$	2.00092e8	0.303271	0.151635	+	0.988436i	$0.451546\pi$
$332$	−1.70684e7	−0.0255982
$333$	0	0
$334$	−1.06935e8	−0.157039
$335$	1.24319e9	1.80668
$336$	0	0
$337$	4.67534e8	0.665439	0.332719	−	0.943026i	$-0.392034\pi$
$337$	4.67534e8	0.665439	0.332719	+	0.943026i	$0.392034\pi$
$338$	3.68705e8	0.519362
$339$	0	0
$340$	−1.04737e9	−1.44519
$341$	2.39311e8	0.326830
$342$	0	0
$343$	−1.15296e9	−1.54271
$344$	7.15684e7	0.0947910
$345$	0	0
$346$	4.63089e8	0.601032
$347$	1.38116e8	0.177456	0.0887279	−	0.996056i	$-0.471720\pi$
$347$	1.38116e8	0.177456	0.0887279	+	0.996056i	$0.471720\pi$
$348$	0	0
$349$	−6.09602e7	−0.0767639	−0.0383820	−	0.999263i	$-0.512220\pi$
$349$	−6.09602e7	−0.0767639	−0.0383820	+	0.999263i	$0.512220\pi$
$350$	−4.04556e8	−0.504360
$351$	0	0
$352$	2.61136e8	0.319130
$353$	8.21666e7	0.0994223	0.0497111	−	0.998764i	$-0.484170\pi$
$353$	8.21666e7	0.0994223	0.0497111	+	0.998764i	$0.484170\pi$
$354$	0	0
$355$	4.24906e8	0.504073
$356$	−9.67086e8	−1.13603
$357$	0	0
$358$	5.40601e7	0.0622711
$359$	1.75626e8	0.200336	0.100168	−	0.994971i	$-0.468062\pi$
$359$	1.75626e8	0.200336	0.100168	+	0.994971i	$0.468062\pi$
$360$	0	0
$361$	3.91890e7	0.0438418
$362$	−1.75432e8	−0.194370
$363$	0	0
$364$	2.24762e9	2.44269
$365$	−1.74934e9	−1.88300
$366$	0	0
$367$	3.51295e7	0.0370972	0.0185486	−	0.999828i	$-0.494095\pi$
$367$	3.51295e7	0.0370972	0.0185486	+	0.999828i	$0.494095\pi$
$368$	7.18858e8	0.751927
$369$	0	0
$370$	7.71605e7	0.0791934
$371$	1.56718e9	1.59335
$372$	0	0
$373$	−1.67183e9	−1.66806	−0.834031	−	0.551718i	$-0.813972\pi$
$373$	−1.67183e9	−1.66806	−0.834031	+	0.551718i	$0.813972\pi$
$374$	−1.47052e8	−0.145352
$375$	0	0
$376$	−3.54981e7	−0.0344388
$377$	−2.41796e9	−2.32410
$378$	0	0
$379$	−1.75757e9	−1.65835	−0.829175	−	0.558989i	$-0.811189\pi$
$379$	−1.75757e9	−1.65835	−0.829175	+	0.558989i	$0.811189\pi$
$380$	1.34407e9	1.25655
$381$	0	0
$382$	2.10668e7	0.0193364
$383$	2.17759e8	0.198053	0.0990263	−	0.995085i	$-0.468427\pi$
$383$	2.17759e8	0.198053	0.0990263	+	0.995085i	$0.468427\pi$
$384$	0	0
$385$	9.90716e8	0.884783
$386$	4.83943e8	0.428291
$387$	0	0
$388$	−6.84923e8	−0.595293
$389$	−4.98336e8	−0.429239	−0.214619	−	0.976698i	$-0.568851\pi$
$389$	−4.98336e8	−0.429239	−0.214619	+	0.976698i	$0.568851\pi$
$390$	0	0
$391$	−1.51827e9	−1.28449
$392$	−1.41229e9	−1.18419
$393$	0	0
$394$	−6.99702e8	−0.576337
$395$	−1.40109e9	−1.14387
$396$	0	0
$397$	−9.41907e8	−0.755512	−0.377756	−	0.925905i	$-0.623304\pi$
$397$	−9.41907e8	−0.755512	−0.377756	+	0.925905i	$0.623304\pi$
$398$	4.26056e8	0.338747
$399$	0	0
$400$	7.93073e8	0.619588
$401$	1.13370e9	0.877994	0.438997	−	0.898488i	$-0.355334\pi$
$401$	1.13370e9	0.877994	0.438997	+	0.898488i	$0.355334\pi$
$402$	0	0
$403$	1.83224e9	1.39449
$404$	−9.71672e8	−0.733137
$405$	0	0
$406$	1.09233e9	0.810051
$407$	−8.95447e7	−0.0658354
$408$	0	0
$409$	2.01366e9	1.45531	0.727653	−	0.685946i	$-0.240611\pi$
$409$	2.01366e9	1.45531	0.727653	+	0.685946i	$0.240611\pi$
$410$	−5.08851e8	−0.364626
$411$	0	0
$412$	−1.99627e8	−0.140631
$413$	1.93313e9	1.35032
$414$	0	0
$415$	−5.76019e7	−0.0395611
$416$	1.99934e9	1.36163
$417$	0	0
$418$	1.88709e8	0.126379
$419$	2.42088e9	1.60777	0.803886	−	0.594784i	$-0.202762\pi$
$419$	2.42088e9	1.60777	0.803886	+	0.594784i	$0.202762\pi$
$420$	0	0
$421$	6.39130e8	0.417448	0.208724	−	0.977975i	$-0.433069\pi$
$421$	6.39130e8	0.417448	0.208724	+	0.977975i	$0.433069\pi$
$422$	1.15995e8	0.0751355
$423$	0	0
$424$	9.12034e8	0.581073
$425$	−1.67502e9	−1.05842
$426$	0	0
$427$	1.16291e9	0.722850
$428$	1.21862e9	0.751301
$429$	0	0
$430$	1.13875e8	0.0690700
$431$	2.22935e9	1.34125	0.670623	−	0.741798i	$-0.266027\pi$
$431$	2.22935e9	1.34125	0.670623	+	0.741798i	$0.266027\pi$
$432$	0	0
$433$	−6.57831e8	−0.389409	−0.194705	−	0.980862i	$-0.562375\pi$
$433$	−6.57831e8	−0.389409	−0.194705	+	0.980862i	$0.562375\pi$
$434$	−8.27725e8	−0.486040
$435$	0	0
$436$	−5.89407e8	−0.340575
$437$	1.94837e9	1.11683
$438$	0	0
$439$	−1.25699e9	−0.709097	−0.354549	−	0.935038i	$-0.615365\pi$
$439$	−1.25699e9	−0.709097	−0.354549	+	0.935038i	$0.615365\pi$
$440$	5.76555e8	0.322668
$441$	0	0
$442$	−1.12587e9	−0.620171
$443$	−6.21522e8	−0.339659	−0.169830	−	0.985473i	$-0.554322\pi$
$443$	−6.21522e8	−0.339659	−0.169830	+	0.985473i	$0.554322\pi$
$444$	0	0
$445$	−3.26370e9	−1.75570
$446$	−5.44120e8	−0.290417
$447$	0	0
$448$	1.32810e9	0.697843
$449$	−3.37102e8	−0.175752	−0.0878758	−	0.996131i	$-0.528008\pi$
$449$	−3.37102e8	−0.175752	−0.0878758	+	0.996131i	$0.528008\pi$
$450$	0	0
$451$	5.90522e8	0.303122
$452$	2.01948e9	1.02862
$453$	0	0
$454$	8.58650e7	0.0430646
$455$	7.58522e9	3.77510
$456$	0	0
$457$	3.14261e9	1.54022	0.770112	−	0.637909i	$-0.220200\pi$
$457$	3.14261e9	1.54022	0.770112	+	0.637909i	$0.220200\pi$
$458$	−9.14455e8	−0.444767
$459$	0	0
$460$	2.80662e9	1.34441
$461$	−6.04340e8	−0.287295	−0.143647	−	0.989629i	$-0.545883\pi$
$461$	−6.04340e8	−0.287295	−0.143647	+	0.989629i	$0.545883\pi$
$462$	0	0
$463$	−1.62669e8	−0.0761677	−0.0380839	−	0.999275i	$-0.512125\pi$
$463$	−1.62669e8	−0.0761677	−0.0380839	+	0.999275i	$0.512125\pi$
$464$	−2.14135e9	−0.995117
$465$	0	0
$466$	−9.54814e8	−0.437087
$467$	−1.96986e9	−0.895005	−0.447502	−	0.894283i	$-0.647686\pi$
$467$	−1.96986e9	−0.895005	−0.447502	+	0.894283i	$0.647686\pi$
$468$	0	0
$469$	4.99006e9	2.23358
$470$	−5.64824e7	−0.0250940
$471$	0	0
$472$	1.12500e9	0.492443
$473$	−1.32152e8	−0.0574196
$474$	0	0
$475$	2.14952e9	0.920267
$476$	−4.20406e9	−1.78667
$477$	0	0
$478$	−1.10835e9	−0.464173
$479$	2.56454e9	1.06619	0.533097	−	0.846054i	$-0.321028\pi$
$479$	2.56454e9	1.06619	0.533097	+	0.846054i	$0.321028\pi$
$480$	0	0
$481$	−6.85582e8	−0.280900
$482$	−8.53643e8	−0.347225
$483$	0	0
$484$	1.90969e9	0.765603
$485$	−2.31146e9	−0.920005
$486$	0	0
$487$	−2.97468e9	−1.16705	−0.583523	−	0.812096i	$-0.698326\pi$
$487$	−2.97468e9	−1.16705	−0.583523	+	0.812096i	$0.698326\pi$
$488$	6.76764e8	0.263614
$489$	0	0
$490$	−2.24714e9	−0.862868
$491$	−1.23197e9	−0.469696	−0.234848	−	0.972032i	$-0.575459\pi$
$491$	−1.23197e9	−0.469696	−0.234848	+	0.972032i	$0.575459\pi$
$492$	0	0
$493$	4.52266e9	1.69993
$494$	1.44481e9	0.539221
$495$	0	0
$496$	1.62263e9	0.597082
$497$	1.70554e9	0.623181
$498$	0	0
$499$	2.78394e9	1.00301	0.501507	−	0.865153i	$-0.332779\pi$
$499$	2.78394e9	1.00301	0.501507	+	0.865153i	$0.332779\pi$
$500$	−3.41238e8	−0.122085
$501$	0	0
$502$	−5.75181e8	−0.202928
$503$	−3.98467e8	−0.139606	−0.0698030	−	0.997561i	$-0.522237\pi$
$503$	−3.98467e8	−0.139606	−0.0698030	+	0.997561i	$0.522237\pi$
$504$	0	0
$505$	−3.27918e9	−1.13304
$506$	3.94052e8	0.135216
$507$	0	0
$508$	−4.67465e9	−1.58207
$509$	−2.69989e9	−0.907473	−0.453736	−	0.891136i	$-0.649909\pi$
$509$	−2.69989e9	−0.907473	−0.453736	+	0.891136i	$0.649909\pi$
$510$	0	0
$511$	−7.02173e9	−2.32794
$512$	3.06915e9	1.01059
$513$	0	0
$514$	−1.01585e9	−0.329956
$515$	−6.73697e8	−0.217340
$516$	0	0
$517$	6.55478e7	0.0208613
$518$	3.09716e8	0.0979060
$519$	0	0
$520$	4.41428e9	1.37673
$521$	−3.82871e8	−0.118610	−0.0593049	−	0.998240i	$-0.518888\pi$
$521$	−3.82871e8	−0.118610	−0.0593049	+	0.998240i	$0.518888\pi$
$522$	0	0
$523$	2.76861e9	0.846264	0.423132	−	0.906068i	$-0.360931\pi$
$523$	2.76861e9	0.846264	0.423132	+	0.906068i	$0.360931\pi$
$524$	4.74215e9	1.43985
$525$	0	0
$526$	−8.18943e8	−0.245360
$527$	−3.42710e9	−1.01997
$528$	0	0
$529$	6.63671e8	0.194921
$530$	1.45117e9	0.423402
$531$	0	0
$532$	5.39499e9	1.55346
$533$	4.52122e9	1.29333
$534$	0	0
$535$	4.11256e9	1.16111
$536$	2.90401e9	0.814556
$537$	0	0
$538$	6.49471e8	0.179813
$539$	2.60781e9	0.717323
$540$	0	0
$541$	−3.30164e9	−0.896477	−0.448238	−	0.893914i	$-0.647948\pi$
$541$	−3.30164e9	−0.896477	−0.448238	+	0.893914i	$0.647948\pi$
$542$	−2.19198e9	−0.591341
$543$	0	0
$544$	−3.73966e9	−0.995945
$545$	−1.98912e9	−0.526347
$546$	0	0
$547$	6.45168e9	1.68546	0.842728	−	0.538340i	$-0.180948\pi$
$547$	6.45168e9	1.68546	0.842728	+	0.538340i	$0.180948\pi$
$548$	−4.00344e9	−1.03921
$549$	0	0
$550$	4.34734e8	0.111418
$551$	−5.80385e9	−1.47804
$552$	0	0
$553$	−5.62388e9	−1.41416
$554$	−1.28708e9	−0.321604
$555$	0	0
$556$	−6.37003e9	−1.57174
$557$	−6.11372e9	−1.49904	−0.749519	−	0.661983i	$-0.769715\pi$
$557$	−6.11372e9	−1.49904	−0.749519	+	0.661983i	$0.769715\pi$
$558$	0	0
$559$	−1.01180e9	−0.244992
$560$	6.71748e9	1.61640
$561$	0	0
$562$	9.21092e8	0.218890
$563$	4.59129e9	1.08432	0.542158	−	0.840277i	$-0.317608\pi$
$563$	4.59129e9	1.08432	0.542158	+	0.840277i	$0.317608\pi$
$564$	0	0
$565$	6.81531e9	1.58970
$566$	8.85416e8	0.205253
$567$	0	0
$568$	9.92552e8	0.227266
$569$	4.85419e9	1.10465	0.552324	−	0.833630i	$-0.313741\pi$
$569$	4.85419e9	1.10465	0.552324	+	0.833630i	$0.313741\pi$
$570$	0	0
$571$	−3.97419e9	−0.893351	−0.446675	−	0.894696i	$-0.647392\pi$
$571$	−3.97419e9	−0.893351	−0.446675	+	0.894696i	$0.647392\pi$
$572$	−2.41528e9	−0.539612
$573$	0	0
$574$	−2.04249e9	−0.450783
$575$	4.48852e9	0.984614
$576$	0	0
$577$	2.46945e9	0.535161	0.267581	−	0.963535i	$-0.413776\pi$
$577$	2.46945e9	0.535161	0.267581	+	0.963535i	$0.413776\pi$
$578$	5.80743e8	0.125094
$579$	0	0
$580$	−8.36041e9	−1.77922
$581$	−2.31209e8	−0.0489090
$582$	0	0
$583$	−1.68408e9	−0.351985
$584$	−4.08635e9	−0.848967
$585$	0	0
$586$	−2.24006e9	−0.459853
$587$	−8.02057e9	−1.63671	−0.818355	−	0.574713i	$-0.805114\pi$
$587$	−8.02057e9	−1.63671	−0.818355	+	0.574713i	$0.805114\pi$
$588$	0	0
$589$	4.39793e9	0.886839
$590$	1.79003e9	0.358821
$591$	0	0
$592$	−6.07152e8	−0.120274
$593$	−1.99128e9	−0.392141	−0.196070	−	0.980590i	$-0.562818\pi$
$593$	−1.99128e9	−0.392141	−0.196070	+	0.980590i	$0.562818\pi$
$594$	0	0
$595$	−1.41878e10	−2.76124
$596$	−2.04298e9	−0.395278
$597$	0	0
$598$	3.01699e9	0.576925
$599$	−4.99536e9	−0.949670	−0.474835	−	0.880075i	$-0.657492\pi$
$599$	−4.99536e9	−0.949670	−0.474835	+	0.880075i	$0.657492\pi$
$600$	0	0
$601$	−4.19069e9	−0.787454	−0.393727	−	0.919227i	$-0.628814\pi$
$601$	−4.19069e9	−0.787454	−0.393727	+	0.919227i	$0.628814\pi$
$602$	4.57086e8	0.0853906
$603$	0	0
$604$	3.26073e9	0.602124
$605$	6.44477e9	1.18321
$606$	0	0
$607$	−2.91949e9	−0.529842	−0.264921	−	0.964270i	$-0.585346\pi$
$607$	−2.91949e9	−0.529842	−0.264921	+	0.964270i	$0.585346\pi$
$608$	4.79903e9	0.865946
$609$	0	0
$610$	1.07682e9	0.192084
$611$	5.01854e8	0.0890088
$612$	0	0
$613$	−3.83809e9	−0.672983	−0.336491	−	0.941687i	$-0.609240\pi$
$613$	−3.83809e9	−0.672983	−0.336491	+	0.941687i	$0.609240\pi$
$614$	−2.04620e9	−0.356745
$615$	0	0
$616$	2.31425e9	0.398912
$617$	5.20036e9	0.891323	0.445661	−	0.895202i	$-0.352969\pi$
$617$	5.20036e9	0.891323	0.445661	+	0.895202i	$0.352969\pi$
$618$	0	0
$619$	−6.96863e9	−1.18095	−0.590473	−	0.807058i	$-0.701059\pi$
$619$	−6.96863e9	−1.18095	−0.590473	+	0.807058i	$0.701059\pi$
$620$	6.33519e9	1.06755
$621$	0	0
$622$	3.17442e9	0.528930
$623$	−1.31002e10	−2.17055
$624$	0	0
$625$	−6.64924e9	−1.08941
$626$	2.33276e9	0.380067
$627$	0	0
$628$	−7.71064e9	−1.24232
$629$	1.28234e9	0.205460
$630$	0	0
$631$	1.04787e10	1.66037	0.830183	−	0.557490i	$-0.188236\pi$
$631$	1.04787e10	1.66037	0.830183	+	0.557490i	$0.188236\pi$
$632$	−3.27286e9	−0.515724
$633$	0	0
$634$	3.33279e8	0.0519393
$635$	−1.57759e10	−2.44504
$636$	0	0
$637$	1.99662e10	3.06060
$638$	−1.17381e9	−0.178947
$639$	0	0
$640$	8.97911e9	1.35395
$641$	2.92164e9	0.438151	0.219076	−	0.975708i	$-0.429696\pi$
$641$	2.92164e9	0.438151	0.219076	+	0.975708i	$0.429696\pi$
$642$	0	0
$643$	7.68989e9	1.14073	0.570364	−	0.821392i	$-0.306802\pi$
$643$	7.68989e9	1.14073	0.570364	+	0.821392i	$0.306802\pi$
$644$	1.12655e10	1.66208
$645$	0	0
$646$	−2.70244e9	−0.394405
$647$	−1.38869e8	−0.0201577	−0.0100789	−	0.999949i	$-0.503208\pi$
$647$	−1.38869e8	−0.0201577	−0.0100789	+	0.999949i	$0.503208\pi$
$648$	0	0
$649$	−2.07733e9	−0.298297
$650$	3.32846e9	0.475386
$651$	0	0
$652$	8.23013e9	1.16289
$653$	−4.91204e9	−0.690344	−0.345172	−	0.938539i	$-0.612179\pi$
$653$	−4.91204e9	−0.690344	−0.345172	+	0.938539i	$0.612179\pi$
$654$	0	0
$655$	1.60037e10	2.22523
$656$	4.00399e9	0.553771
$657$	0	0
$658$	−2.26716e8	−0.0310235
$659$	1.44098e9	0.196137	0.0980686	−	0.995180i	$-0.468734\pi$
$659$	1.44098e9	0.196137	0.0980686	+	0.995180i	$0.468734\pi$
$660$	0	0
$661$	6.29813e9	0.848216	0.424108	−	0.905612i	$-0.360588\pi$
$661$	6.29813e9	0.848216	0.424108	+	0.905612i	$0.360588\pi$
$662$	7.43698e8	0.0996309
$663$	0	0
$664$	−1.34554e8	−0.0178365
$665$	1.82069e10	2.40082
$666$	0	0
$667$	−1.21193e10	−1.58138
$668$	3.28522e9	0.426429
$669$	0	0
$670$	4.62067e9	0.593531
$671$	−1.24966e9	−0.159684
$672$	0	0
$673$	−1.12010e10	−1.41646	−0.708229	−	0.705983i	$-0.750505\pi$
$673$	−1.12010e10	−1.41646	−0.708229	+	0.705983i	$0.750505\pi$
$674$	1.73772e9	0.218611
$675$	0	0
$676$	−1.13272e10	−1.41029
$677$	−1.35096e10	−1.67333	−0.836666	−	0.547714i	$-0.815498\pi$
$677$	−1.35096e10	−1.67333	−0.836666	+	0.547714i	$0.815498\pi$
$678$	0	0
$679$	−9.27802e9	−1.13739
$680$	−8.25667e9	−1.00699
$681$	0	0
$682$	8.89468e8	0.107371
$683$	−7.42709e9	−0.891962	−0.445981	−	0.895042i	$-0.647145\pi$
$683$	−7.42709e9	−0.891962	−0.445981	+	0.895042i	$0.647145\pi$
$684$	0	0
$685$	−1.35107e10	−1.60606
$686$	−4.28530e9	−0.506812
$687$	0	0
$688$	−8.96049e8	−0.104899
$689$	−1.28939e10	−1.50181
$690$	0	0
$691$	6.66846e9	0.768869	0.384434	−	0.923152i	$-0.374397\pi$
$691$	6.66846e9	0.768869	0.384434	+	0.923152i	$0.374397\pi$
$692$	−1.42268e10	−1.63206
$693$	0	0
$694$	5.13347e8	0.0582979
$695$	−2.14974e10	−2.42907
$696$	0	0
$697$	−8.45669e9	−0.945988
$698$	−2.26576e8	−0.0252185
$699$	0	0
$700$	1.24286e10	1.36955
$701$	8.02719e9	0.880137	0.440069	−	0.897964i	$-0.354954\pi$
$701$	8.02719e9	0.880137	0.440069	+	0.897964i	$0.354954\pi$
$702$	0	0
$703$	−1.64561e9	−0.178642
$704$	−1.42717e9	−0.154160
$705$	0	0
$706$	3.05396e8	0.0326623
$707$	−1.31624e10	−1.40077
$708$	0	0
$709$	6.52080e9	0.687130	0.343565	−	0.939129i	$-0.388365\pi$
$709$	6.52080e9	0.687130	0.343565	+	0.939129i	$0.388365\pi$
$710$	1.57929e9	0.165599
$711$	0	0
$712$	−7.62378e9	−0.791572
$713$	9.18354e9	0.948849
$714$	0	0
$715$	−8.15104e9	−0.833953
$716$	−1.66081e9	−0.169092
$717$	0	0
$718$	6.52765e8	0.0658145
$719$	−1.19598e10	−1.19997	−0.599986	−	0.800010i	$-0.704827\pi$
$719$	−1.19598e10	−1.19997	−0.599986	+	0.800010i	$0.704827\pi$
$720$	0	0
$721$	−2.70417e9	−0.268695
$722$	1.45657e8	0.0144030
$723$	0	0
$724$	5.38954e9	0.527797
$725$	−1.33705e10	−1.30306
$726$	0	0
$727$	−3.08388e9	−0.297665	−0.148832	−	0.988862i	$-0.547551\pi$
$727$	−3.08388e9	−0.297665	−0.148832	+	0.988862i	$0.547551\pi$
$728$	1.77186e10	1.70204
$729$	0	0
$730$	−6.50195e9	−0.618605
$731$	1.89251e9	0.179196
$732$	0	0
$733$	−1.53686e10	−1.44135	−0.720676	−	0.693272i	$-0.756168\pi$
$733$	−1.53686e10	−1.44135	−0.720676	+	0.693272i	$0.756168\pi$
$734$	1.30569e8	0.0121872
$735$	0	0
$736$	1.00211e10	0.926495
$737$	−5.36229e9	−0.493417
$738$	0	0
$739$	−8.64423e9	−0.787899	−0.393949	−	0.919132i	$-0.628892\pi$
$739$	−8.64423e9	−0.787899	−0.393949	+	0.919132i	$0.628892\pi$
$740$	−2.37049e9	−0.215044
$741$	0	0
$742$	5.82489e9	0.523448
$743$	−6.82591e9	−0.610520	−0.305260	−	0.952269i	$-0.598743\pi$
$743$	−6.82591e9	−0.610520	−0.305260	+	0.952269i	$0.598743\pi$
$744$	0	0
$745$	−6.89460e9	−0.610888
$746$	−6.21385e9	−0.547993
$747$	0	0
$748$	4.51766e9	0.394691
$749$	1.65075e10	1.43547
$750$	0	0
$751$	−1.00999e10	−0.870113	−0.435057	−	0.900403i	$-0.643272\pi$
$751$	−1.00999e10	−0.870113	−0.435057	+	0.900403i	$0.643272\pi$
$752$	4.44442e8	0.0381112
$753$	0	0
$754$	−8.98705e9	−0.763515
$755$	1.10042e10	0.930562
$756$	0	0
$757$	−1.25390e10	−1.05057	−0.525287	−	0.850925i	$-0.676042\pi$
$757$	−1.25390e10	−1.05057	−0.525287	+	0.850925i	$0.676042\pi$
$758$	−6.53253e9	−0.544803
$759$	0	0
$760$	1.05956e10	0.875546
$761$	−1.68387e10	−1.38504	−0.692518	−	0.721400i	$-0.743499\pi$
$761$	−1.68387e10	−1.38504	−0.692518	+	0.721400i	$0.743499\pi$
$762$	0	0
$763$	−7.98415e9	−0.650718
$764$	−6.47203e8	−0.0525066
$765$	0	0
$766$	8.09365e8	0.0650644
$767$	−1.59047e10	−1.27274
$768$	0	0
$769$	5.50473e9	0.436510	0.218255	−	0.975892i	$-0.429964\pi$
$769$	5.50473e9	0.436510	0.218255	+	0.975892i	$0.429964\pi$
$770$	3.68228e9	0.290670
$771$	0	0
$772$	−1.48675e10	−1.16299
$773$	1.58271e9	0.123246	0.0616230	−	0.998099i	$-0.480372\pi$
$773$	1.58271e9	0.123246	0.0616230	+	0.998099i	$0.480372\pi$
$774$	0	0
$775$	1.01316e10	0.781851
$776$	−5.39942e9	−0.414793
$777$	0	0
$778$	−1.85221e9	−0.141014
$779$	1.08523e10	0.822509
$780$	0	0
$781$	−1.83276e9	−0.137666
$782$	−5.64311e9	−0.421983
$783$	0	0
$784$	1.76821e10	1.31047
$785$	−2.60217e10	−1.91996
$786$	0	0
$787$	−9.16094e9	−0.669929	−0.334964	−	0.942231i	$-0.608724\pi$
$787$	−9.16094e9	−0.669929	−0.334964	+	0.942231i	$0.608724\pi$
$788$	2.14959e10	1.56500
$789$	0	0
$790$	−5.20757e9	−0.375786
$791$	2.73561e10	1.96534
$792$	0	0
$793$	−9.56774e9	−0.681323
$794$	−3.50087e9	−0.248202
$795$	0	0
$796$	−1.30891e10	−0.919842
$797$	1.42782e10	0.999006	0.499503	−	0.866312i	$-0.333516\pi$
$797$	1.42782e10	0.999006	0.499503	+	0.866312i	$0.333516\pi$
$798$	0	0
$799$	−9.38691e8	−0.0651042
$800$	1.10557e10	0.763431
$801$	0	0
$802$	4.21371e9	0.288440
$803$	7.54551e9	0.514262
$804$	0	0
$805$	3.80187e10	2.56869
$806$	6.81004e9	0.458118
$807$	0	0
$808$	−7.65994e9	−0.510841
$809$	−8.21284e9	−0.545348	−0.272674	−	0.962106i	$-0.587908\pi$
$809$	−8.21284e9	−0.545348	−0.272674	+	0.962106i	$0.587908\pi$
$810$	0	0
$811$	−1.87787e10	−1.23621	−0.618104	−	0.786096i	$-0.712099\pi$
$811$	−1.87787e10	−1.23621	−0.618104	+	0.786096i	$0.712099\pi$
$812$	−3.35580e10	−2.19963
$813$	0	0
$814$	−3.32819e8	−0.0216283
$815$	2.77748e10	1.79721
$816$	0	0
$817$	−2.42862e9	−0.155806
$818$	7.48434e9	0.478098
$819$	0	0
$820$	1.56327e10	0.990114
$821$	1.50345e10	0.948173	0.474086	−	0.880478i	$-0.342778\pi$
$821$	1.50345e10	0.948173	0.474086	+	0.880478i	$0.342778\pi$
$822$	0	0
$823$	9.26651e9	0.579451	0.289726	−	0.957110i	$-0.406436\pi$
$823$	9.26651e9	0.579451	0.289726	+	0.957110i	$0.406436\pi$
$824$	−1.57371e9	−0.0979896
$825$	0	0
$826$	7.18504e9	0.443608
$827$	2.93861e10	1.80665	0.903323	−	0.428962i	$-0.141121\pi$
$827$	2.93861e10	1.80665	0.903323	+	0.428962i	$0.141121\pi$
$828$	0	0
$829$	3.66600e9	0.223487	0.111743	−	0.993737i	$-0.464357\pi$
$829$	3.66600e9	0.223487	0.111743	+	0.993737i	$0.464357\pi$
$830$	−2.14094e8	−0.0129967
$831$	0	0
$832$	−1.09268e10	−0.657753
$833$	−3.73457e10	−2.23863
$834$	0	0
$835$	1.10869e10	0.659031
$836$	−5.79742e9	−0.343173
$837$	0	0
$838$	8.99791e9	0.528187
$839$	1.20233e9	0.0702840	0.0351420	−	0.999382i	$-0.488812\pi$
$839$	1.20233e9	0.0702840	0.0351420	+	0.999382i	$0.488812\pi$
$840$	0	0
$841$	1.88513e10	1.09284
$842$	2.37551e9	0.137140
$843$	0	0
$844$	−3.56354e9	−0.204025
$845$	−3.82267e10	−2.17956
$846$	0	0
$847$	2.58688e10	1.46280
$848$	−1.14188e10	−0.643037
$849$	0	0
$850$	−6.22570e9	−0.347714
$851$	−3.43628e9	−0.191133
$852$	0	0
$853$	2.76893e10	1.52753	0.763767	−	0.645491i	$-0.223347\pi$
$853$	2.76893e10	1.52753	0.763767	+	0.645491i	$0.223347\pi$
$854$	4.32229e9	0.237471
$855$	0	0
$856$	9.60666e9	0.523497
$857$	3.11448e10	1.69025	0.845127	−	0.534565i	$-0.179525\pi$
$857$	3.11448e10	1.69025	0.845127	+	0.534565i	$0.179525\pi$
$858$	0	0
$859$	1.06707e10	0.574405	0.287202	−	0.957870i	$-0.407275\pi$
$859$	1.06707e10	0.574405	0.287202	+	0.957870i	$0.407275\pi$
$860$	−3.49842e9	−0.187554
$861$	0	0
$862$	8.28604e9	0.440627
$863$	−2.01685e10	−1.06816	−0.534079	−	0.845435i	$-0.679341\pi$
$863$	−2.01685e10	−1.06816	−0.534079	+	0.845435i	$0.679341\pi$
$864$	0	0
$865$	−4.80122e10	−2.52229
$866$	−2.44502e9	−0.127929
$867$	0	0
$868$	2.54290e10	1.31980
$869$	6.04338e9	0.312400
$870$	0	0
$871$	−4.10553e10	−2.10526
$872$	−4.64644e9	−0.237308
$873$	0	0
$874$	7.24169e9	0.366902
$875$	−4.62244e9	−0.233261
$876$	0	0
$877$	−8.78810e9	−0.439943	−0.219971	−	0.975506i	$-0.570596\pi$
$877$	−8.78810e9	−0.439943	−0.219971	+	0.975506i	$0.570596\pi$
$878$	−4.67197e9	−0.232953
$879$	0	0
$880$	−7.21857e9	−0.357077
$881$	2.67005e10	1.31554	0.657769	−	0.753220i	$-0.271500\pi$
$881$	2.67005e10	1.31554	0.657769	+	0.753220i	$0.271500\pi$
$882$	0	0
$883$	2.75848e10	1.34836	0.674181	−	0.738566i	$-0.264497\pi$
$883$	2.75848e10	1.34836	0.674181	+	0.738566i	$0.264497\pi$
$884$	3.45886e10	1.68403
$885$	0	0
$886$	−2.31007e9	−0.111585
$887$	−1.57163e10	−0.756165	−0.378083	−	0.925772i	$-0.623416\pi$
$887$	−1.57163e10	−0.756165	−0.378083	+	0.925772i	$0.623416\pi$
$888$	0	0
$889$	−6.33231e10	−3.02278
$890$	−1.21305e10	−0.576784
$891$	0	0
$892$	1.67162e10	0.788606
$893$	1.20460e9	0.0566062
$894$	0	0
$895$	−5.60486e9	−0.261327
$896$	3.60415e10	1.67388
$897$	0	0
$898$	−1.25294e9	−0.0577381
$899$	−2.73561e10	−1.25573
$900$	0	0
$901$	2.41173e10	1.09848
$902$	2.19485e9	0.0995821
$903$	0	0
$904$	1.59201e10	0.716731
$905$	1.81885e10	0.815692
$906$	0	0
$907$	−2.21424e10	−0.985369	−0.492685	−	0.870208i	$-0.663984\pi$
$907$	−2.21424e10	−0.985369	−0.492685	+	0.870208i	$0.663984\pi$
$908$	−2.63790e9	−0.116939
$909$	0	0
$910$	2.81927e10	1.24020
$911$	−2.20181e10	−0.964862	−0.482431	−	0.875934i	$-0.660246\pi$
$911$	−2.20181e10	−0.964862	−0.482431	+	0.875934i	$0.660246\pi$
$912$	0	0
$913$	2.48456e8	0.0108044
$914$	1.16804e10	0.505996
$915$	0	0
$916$	2.80934e10	1.20773
$917$	6.42376e10	2.75104
$918$	0	0
$919$	1.01632e10	0.431943	0.215972	−	0.976400i	$-0.430708\pi$
$919$	1.01632e10	0.431943	0.215972	+	0.976400i	$0.430708\pi$
$920$	2.21253e10	0.936766
$921$	0	0
$922$	−2.24620e9	−0.0943824
$923$	−1.40322e10	−0.587381
$924$	0	0
$925$	−3.79103e9	−0.157493
$926$	−6.04606e8	−0.0250227
$927$	0	0
$928$	−2.98510e10	−1.22614
$929$	8.04262e9	0.329111	0.164556	−	0.986368i	$-0.447381\pi$
$929$	8.04262e9	0.329111	0.164556	+	0.986368i	$0.447381\pi$
$930$	0	0
$931$	4.79249e10	1.94643
$932$	2.93333e10	1.18688
$933$	0	0
$934$	−7.32154e9	−0.294028
$935$	1.52461e10	0.609983
$936$	0	0
$937$	−3.08374e10	−1.22459	−0.612293	−	0.790631i	$-0.709753\pi$
$937$	−3.08374e10	−1.22459	−0.612293	+	0.790631i	$0.709753\pi$
$938$	1.85470e10	0.733777
$939$	0	0
$940$	1.73522e9	0.0681409
$941$	−8.87704e8	−0.0347300	−0.0173650	−	0.999849i	$-0.505528\pi$
$941$	−8.87704e8	−0.0347300	−0.0173650	+	0.999849i	$0.505528\pi$
$942$	0	0
$943$	2.26613e10	0.880021
$944$	−1.40852e10	−0.544955
$945$	0	0
$946$	−4.91182e8	−0.0188635
$947$	8.02900e9	0.307211	0.153605	−	0.988132i	$-0.450912\pi$
$947$	8.02900e9	0.307211	0.153605	+	0.988132i	$0.450912\pi$
$948$	0	0
$949$	5.77707e10	2.19420
$950$	7.98931e9	0.302327
$951$	0	0
$952$	−3.31416e10	−1.24493
$953$	6.27853e9	0.234981	0.117491	−	0.993074i	$-0.462515\pi$
$953$	6.27853e9	0.234981	0.117491	+	0.993074i	$0.462515\pi$
$954$	0	0
$955$	−2.18417e9	−0.0811472
$956$	3.40502e10	1.26043
$957$	0	0
$958$	9.53187e9	0.350267
$959$	−5.42309e10	−1.98555
$960$	0	0
$961$	−6.78322e9	−0.246549
$962$	−2.54816e9	−0.0922815
$963$	0	0
$964$	2.62252e10	0.942865
$965$	−5.01744e10	−1.79737
$966$	0	0
$967$	−1.81180e10	−0.644343	−0.322172	−	0.946681i	$-0.604413\pi$
$967$	−1.81180e10	−0.644343	−0.322172	+	0.946681i	$0.604413\pi$
$968$	1.50546e10	0.533463
$969$	0	0
$970$	−8.59121e9	−0.302241
$971$	2.56716e10	0.899883	0.449942	−	0.893058i	$-0.351445\pi$
$971$	2.56716e10	0.899883	0.449942	+	0.893058i	$0.351445\pi$
$972$	0	0
$973$	−8.62889e10	−3.00303
$974$	−1.10562e10	−0.383399
$975$	0	0
$976$	−8.47320e9	−0.291725
$977$	−2.60682e10	−0.894295	−0.447148	−	0.894460i	$-0.647560\pi$
$977$	−2.60682e10	−0.894295	−0.447148	+	0.894460i	$0.647560\pi$
$978$	0	0
$979$	1.40774e10	0.479495
$980$	6.90356e10	2.34305
$981$	0	0
$982$	−4.57899e9	−0.154305
$983$	−1.74429e10	−0.585708	−0.292854	−	0.956157i	$-0.594605\pi$
$983$	−1.74429e10	−0.585708	−0.292854	+	0.956157i	$0.594605\pi$
$984$	0	0
$985$	7.25439e10	2.41866
$986$	1.68098e10	0.558461
$987$	0	0
$988$	−4.43868e10	−1.46421
$989$	−5.07134e9	−0.166700
$990$	0	0
$991$	1.57746e10	0.514874	0.257437	−	0.966295i	$-0.417122\pi$
$991$	1.57746e10	0.514874	0.257437	+	0.966295i	$0.417122\pi$
$992$	2.26199e10	0.735700
$993$	0	0
$994$	6.33913e9	0.204728
$995$	−4.41727e10	−1.42159
$996$	0	0
$997$	2.18743e9	0.0699040	0.0349520	−	0.999389i	$-0.488872\pi$
$997$	2.18743e9	0.0699040	0.0349520	+	0.999389i	$0.488872\pi$
$998$	1.03473e10	0.329511
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	387.8.a.d.1.8		13
3.2	odd	2		43.8.a.b.1.6	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.8.a.b.1.6	✓	13	3.2	odd	2
387.8.a.d.1.8		13	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 387 = 3^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	387.a (trivial)

\(f(q)\)	\(=\)	\(q+3.71679 q^{2} -114.185 q^{4} -385.350 q^{5} -1546.77 q^{7} -900.153 q^{8} +O(q^{10})\)	\(q+3.71679 q^{2} -114.185 q^{4} -385.350 q^{5} -1546.77 q^{7} -900.153 q^{8} -1432.27 q^{10} +1662.15 q^{11} +12725.9 q^{13} -5749.00 q^{14} +11270.1 q^{16} -23803.1 q^{17} +30546.0 q^{19} +44001.4 q^{20} +6177.85 q^{22} +63784.8 q^{23} +70369.8 q^{25} +47299.5 q^{26} +176618. q^{28} -190003. q^{29} +143977. q^{31} +157108. q^{32} -88471.1 q^{34} +596047. q^{35} -53873.0 q^{37} +113533. q^{38} +346874. q^{40} +355277. q^{41} -79507.0 q^{43} -189793. q^{44} +237075. q^{46} +39435.7 q^{47} +1.56894e6 q^{49} +261550. q^{50} -1.45311e6 q^{52} -1.01320e6 q^{53} -640508. q^{55} +1.39233e6 q^{56} -706202. q^{58} -1.24979e6 q^{59} -751833. q^{61} +535133. q^{62} -858631. q^{64} -4.90393e6 q^{65} -3.22613e6 q^{67} +2.71797e6 q^{68} +2.21538e6 q^{70} -1.10265e6 q^{71} +4.53962e6 q^{73} -200235. q^{74} -3.48791e6 q^{76} -2.57095e6 q^{77} +3.63589e6 q^{79} -4.34292e6 q^{80} +1.32049e6 q^{82} +149479. q^{83} +9.17253e6 q^{85} -295511. q^{86} -1.49618e6 q^{88} +8.46943e6 q^{89} -1.96840e7 q^{91} -7.28329e6 q^{92} +146574. q^{94} -1.17709e7 q^{95} +5.99834e6 q^{97} +5.83143e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 16 q^{2} + 922 q^{4} - 998 q^{5} + 1360 q^{7} - 3870 q^{8}+O(q^{10}) \) 13 * q - 16 * q^2 + 922 * q^4 - 998 * q^5 + 1360 * q^7 - 3870 * q^8	\( 13 q - 16 q^{2} + 922 q^{4} - 998 q^{5} + 1360 q^{7} - 3870 q^{8} + 4667 q^{10} - 1620 q^{11} + 13550 q^{13} - 44160 q^{14} + 114026 q^{16} - 110880 q^{17} + 105058 q^{19} - 167251 q^{20} + 201504 q^{22} - 160184 q^{23} + 270149 q^{25} - 272104 q^{26} + 208172 q^{28} - 285546 q^{29} - 99616 q^{31} - 200126 q^{32} - 80941 q^{34} + 187104 q^{35} + 176038 q^{37} - 652165 q^{38} - 895387 q^{40} + 410260 q^{41} - 1033591 q^{43} + 2177076 q^{44} - 3975765 q^{46} + 424556 q^{47} - 1561359 q^{49} + 4063801 q^{50} - 4172312 q^{52} - 3992458 q^{53} + 406960 q^{55} - 1559556 q^{56} - 4052005 q^{58} - 2248836 q^{59} + 6210394 q^{61} - 885317 q^{62} - 3096318 q^{64} - 5600420 q^{65} - 1993648 q^{67} - 9327135 q^{68} - 1105098 q^{70} - 4978064 q^{71} + 8224814 q^{73} + 3613563 q^{74} + 10687121 q^{76} - 17261892 q^{77} + 6945708 q^{79} - 15822799 q^{80} - 508449 q^{82} - 22937328 q^{83} - 575532 q^{85} + 1272112 q^{86} + 11202656 q^{88} - 9291302 q^{89} + 25581108 q^{91} + 14388137 q^{92} - 24645805 q^{94} - 30750464 q^{95} + 10001852 q^{97} + 32304856 q^{98}+O(q^{100}) \) 13 * q - 16 * q^2 + 922 * q^4 - 998 * q^5 + 1360 * q^7 - 3870 * q^8 + 4667 * q^10 - 1620 * q^11 + 13550 * q^13 - 44160 * q^14 + 114026 * q^16 - 110880 * q^17 + 105058 * q^19 - 167251 * q^20 + 201504 * q^22 - 160184 * q^23 + 270149 * q^25 - 272104 * q^26 + 208172 * q^28 - 285546 * q^29 - 99616 * q^31 - 200126 * q^32 - 80941 * q^34 + 187104 * q^35 + 176038 * q^37 - 652165 * q^38 - 895387 * q^40 + 410260 * q^41 - 1033591 * q^43 + 2177076 * q^44 - 3975765 * q^46 + 424556 * q^47 - 1561359 * q^49 + 4063801 * q^50 - 4172312 * q^52 - 3992458 * q^53 + 406960 * q^55 - 1559556 * q^56 - 4052005 * q^58 - 2248836 * q^59 + 6210394 * q^61 - 885317 * q^62 - 3096318 * q^64 - 5600420 * q^65 - 1993648 * q^67 - 9327135 * q^68 - 1105098 * q^70 - 4978064 * q^71 + 8224814 * q^73 + 3613563 * q^74 + 10687121 * q^76 - 17261892 * q^77 + 6945708 * q^79 - 15822799 * q^80 - 508449 * q^82 - 22937328 * q^83 - 575532 * q^85 + 1272112 * q^86 + 11202656 * q^88 - 9291302 * q^89 + 25581108 * q^91 + 14388137 * q^92 - 24645805 * q^94 - 30750464 * q^95 + 10001852 * q^97 + 32304856 * q^98

Introduction

L-functions

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