LMFDB - Embedded newform 387.8.a.a.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("387.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$120.893004862$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - x^{9} - 825 x^{8} + 431 x^{7} + 229838 x^{6} - 1804 x^{5} - 25242488 x^{4} - 2085744 x^{3} + \cdots - 5193030528 $ x^10 - x^9 - 825x^8 + 431x^7 + 229838x^6 - 1804x^5 - 25242488x^4 - 2085744x^3 + 933060896x^2 - 813446208x - 5193030528
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 129)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$18.9565$ 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)

$f(q)$	$=$	$q+18.9565 q^{2} +231.347 q^{4} -143.919 q^{5} +769.785 q^{7} +1959.10 q^{8} +O(q^{10})$	\(q+18.9565 q^{2} +231.347 q^{4} -143.919 q^{5} +769.785 q^{7} +1959.10 q^{8} -2728.19 q^{10} -1650.52 q^{11} -9347.27 q^{13} +14592.4 q^{14} +7525.07 q^{16} +14775.0 q^{17} -51568.7 q^{19} -33295.2 q^{20} -31288.0 q^{22} -10878.3 q^{23} -57412.4 q^{25} -177191. q^{26} +178088. q^{28} +30724.3 q^{29} +133061. q^{31} -108116. q^{32} +280082. q^{34} -110786. q^{35} -228078. q^{37} -977560. q^{38} -281951. q^{40} -43747.2 q^{41} -79507.0 q^{43} -381843. q^{44} -206214. q^{46} -334602. q^{47} -230974. q^{49} -1.08834e6 q^{50} -2.16246e6 q^{52} +455391. q^{53} +237541. q^{55} +1.50808e6 q^{56} +582425. q^{58} +1.39056e6 q^{59} +175277. q^{61} +2.52236e6 q^{62} -3.01270e6 q^{64} +1.34525e6 q^{65} -3.52977e6 q^{67} +3.41816e6 q^{68} -2.10012e6 q^{70} +1.29468e6 q^{71} -2.41753e6 q^{73} -4.32355e6 q^{74} -1.19303e7 q^{76} -1.27055e6 q^{77} +5.01210e6 q^{79} -1.08300e6 q^{80} -829291. q^{82} -8.40027e6 q^{83} -2.12641e6 q^{85} -1.50717e6 q^{86} -3.23353e6 q^{88} +3.28424e6 q^{89} -7.19539e6 q^{91} -2.51667e6 q^{92} -6.34287e6 q^{94} +7.42170e6 q^{95} -1.76546e7 q^{97} -4.37845e6 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + q^{2} + 371 q^{4} + 122 q^{5} - 2052 q^{7} + 927 q^{8}+O(q^{10}) $ 10 * q + q^2 + 371 * q^4 + 122 * q^5 - 2052 * q^7 + 927 * q^8	\( 10 q + q^{2} + 371 q^{4} + 122 q^{5} - 2052 q^{7} + 927 q^{8} - 10032 q^{10} + 8888 q^{11} - 16432 q^{13} + 28408 q^{14} - 26669 q^{16} + 48122 q^{17} - 56146 q^{19} + 88940 q^{20} - 100626 q^{22} + 236336 q^{23} - 135016 q^{25} + 166748 q^{26} - 259060 q^{28} + 248818 q^{29} - 430970 q^{31} - 69493 q^{32} + 445522 q^{34} - 298982 q^{35} - 261254 q^{37} - 257662 q^{38} - 671432 q^{40} + 126814 q^{41} - 795070 q^{43} + 620022 q^{44} - 809038 q^{46} - 627080 q^{47} - 1256116 q^{49} + 83117 q^{50} - 3674204 q^{52} + 1612384 q^{53} - 4732974 q^{55} + 4301484 q^{56} - 7268516 q^{58} + 3442492 q^{59} - 5217214 q^{61} + 2500324 q^{62} - 4657369 q^{64} + 1224166 q^{65} - 6810926 q^{67} + 3563486 q^{68} - 2745858 q^{70} + 13935120 q^{71} - 13743720 q^{73} + 1752692 q^{74} - 15817594 q^{76} + 7685750 q^{77} - 9007608 q^{79} + 13641024 q^{80} - 6329026 q^{82} + 21779128 q^{83} - 13177392 q^{85} - 79507 q^{86} - 13214750 q^{88} + 1895364 q^{89} - 16439838 q^{91} - 9614510 q^{92} + 3404276 q^{94} + 16861514 q^{95} - 20434472 q^{97} - 35731457 q^{98}+O(q^{100}) \) 10 * q + q^2 + 371 * q^4 + 122 * q^5 - 2052 * q^7 + 927 * q^8 - 10032 * q^10 + 8888 * q^11 - 16432 * q^13 + 28408 * q^14 - 26669 * q^16 + 48122 * q^17 - 56146 * q^19 + 88940 * q^20 - 100626 * q^22 + 236336 * q^23 - 135016 * q^25 + 166748 * q^26 - 259060 * q^28 + 248818 * q^29 - 430970 * q^31 - 69493 * q^32 + 445522 * q^34 - 298982 * q^35 - 261254 * q^37 - 257662 * q^38 - 671432 * q^40 + 126814 * q^41 - 795070 * q^43 + 620022 * q^44 - 809038 * q^46 - 627080 * q^47 - 1256116 * q^49 + 83117 * q^50 - 3674204 * q^52 + 1612384 * q^53 - 4732974 * q^55 + 4301484 * q^56 - 7268516 * q^58 + 3442492 * q^59 - 5217214 * q^61 + 2500324 * q^62 - 4657369 * q^64 + 1224166 * q^65 - 6810926 * q^67 + 3563486 * q^68 - 2745858 * q^70 + 13935120 * q^71 - 13743720 * q^73 + 1752692 * q^74 - 15817594 * q^76 + 7685750 * q^77 - 9007608 * q^79 + 13641024 * q^80 - 6329026 * q^82 + 21779128 * q^83 - 13177392 * q^85 - 79507 * q^86 - 13214750 * q^88 + 1895364 * q^89 - 16439838 * q^91 - 9614510 * q^92 + 3404276 * q^94 + 16861514 * q^95 - 20434472 * q^97 - 35731457 * 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 $

$2$	18.9565	1.67553	0.837765	−	0.546031i	$-0.183862\pi$
$2$	18.9565	1.67553	0.837765	+	0.546031i	$0.183862\pi$
$3$	0	0
$3$	0	0
$4$	231.347	1.80740
$5$	−143.919	−0.514899	−0.257450	−	0.966292i	$-0.582882\pi$
$5$	−143.919	−0.514899	−0.257450	+	0.966292i	$0.582882\pi$
$6$	0	0
$7$	769.785	0.848255	0.424127	−	0.905603i	$-0.360581\pi$
$7$	769.785	0.848255	0.424127	+	0.905603i	$0.360581\pi$
$8$	1959.10	1.35282
$9$	0	0
$10$	−2728.19	−0.862729
$11$	−1650.52	−0.373893	−0.186946	−	0.982370i	$-0.559859\pi$
$11$	−1650.52	−0.373893	−0.186946	+	0.982370i	$0.559859\pi$
$12$	0	0
$13$	−9347.27	−1.18000	−0.590001	−	0.807402i	$-0.700873\pi$
$13$	−9347.27	−1.18000	−0.590001	+	0.807402i	$0.700873\pi$
$14$	14592.4	1.42128
$15$	0	0
$16$	7525.07	0.459294
$17$	14775.0	0.729386	0.364693	−	0.931128i	$-0.381174\pi$
$17$	14775.0	0.729386	0.364693	+	0.931128i	$0.381174\pi$
$18$	0	0
$19$	−51568.7	−1.72484	−0.862420	−	0.506194i	$-0.831052\pi$
$19$	−51568.7	−1.72484	−0.862420	+	0.506194i	$0.831052\pi$
$20$	−33295.2	−0.930629
$21$	0	0
$22$	−31288.0	−0.626468
$23$	−10878.3	−0.186429	−0.0932147	−	0.995646i	$-0.529714\pi$
$23$	−10878.3	−0.186429	−0.0932147	+	0.995646i	$0.529714\pi$
$24$	0	0
$25$	−57412.4	−0.734879
$26$	−177191.	−1.97713
$27$	0	0
$28$	178088.	1.53313
$29$	30724.3	0.233932	0.116966	−	0.993136i	$-0.462683\pi$
$29$	30724.3	0.233932	0.116966	+	0.993136i	$0.462683\pi$
$30$	0	0
$31$	133061.	0.802203	0.401102	−	0.916034i	$-0.368627\pi$
$31$	133061.	0.802203	0.401102	+	0.916034i	$0.368627\pi$
$32$	−108116.	−0.583262
$33$	0	0
$34$	280082.	1.22211
$35$	−110786.	−0.436766
$36$	0	0
$37$	−228078.	−0.740248	−0.370124	−	0.928982i	$-0.620685\pi$
$37$	−228078.	−0.740248	−0.370124	+	0.928982i	$0.620685\pi$
$38$	−977560.	−2.89002
$39$	0	0
$40$	−281951.	−0.696567
$41$	−43747.2	−0.0991303	−0.0495651	−	0.998771i	$-0.515784\pi$
$41$	−43747.2	−0.0991303	−0.0495651	+	0.998771i	$0.515784\pi$
$42$	0	0
$43$	−79507.0	−0.152499
$43$	−79507.0	−0.152499
$44$	−381843.	−0.675773
$45$	0	0
$46$	−206214.	−0.312368
$47$	−334602.	−0.470096	−0.235048	−	0.971984i	$-0.575525\pi$
$47$	−334602.	−0.470096	−0.235048	+	0.971984i	$0.575525\pi$
$48$	0	0
$49$	−230974.	−0.280464
$50$	−1.08834e6	−1.23131
$51$	0	0
$52$	−2.16246e6	−2.13274
$53$	455391.	0.420164	0.210082	−	0.977684i	$-0.432627\pi$
$53$	455391.	0.420164	0.210082	+	0.977684i	$0.432627\pi$
$54$	0	0
$55$	237541.	0.192517
$56$	1.50808e6	1.14754
$57$	0	0
$58$	582425.	0.391960
$59$	1.39056e6	0.881470	0.440735	−	0.897637i	$-0.354718\pi$
$59$	1.39056e6	0.881470	0.440735	+	0.897637i	$0.354718\pi$
$60$	0	0
$61$	175277.	0.0988712	0.0494356	−	0.998777i	$-0.484258\pi$
$61$	175277.	0.0988712	0.0494356	+	0.998777i	$0.484258\pi$
$62$	2.52236e6	1.34412
$63$	0	0
$64$	−3.01270e6	−1.43657
$65$	1.34525e6	0.607583
$66$	0	0
$67$	−3.52977e6	−1.43379	−0.716893	−	0.697184i	$-0.754436\pi$
$67$	−3.52977e6	−1.43379	−0.716893	+	0.697184i	$0.754436\pi$
$68$	3.41816e6	1.31829
$69$	0	0
$70$	−2.10012e6	−0.731814
$71$	1.29468e6	0.429297	0.214649	−	0.976691i	$-0.431139\pi$
$71$	1.29468e6	0.429297	0.214649	+	0.976691i	$0.431139\pi$
$72$	0	0
$73$	−2.41753e6	−0.727346	−0.363673	−	0.931527i	$-0.618478\pi$
$73$	−2.41753e6	−0.727346	−0.363673	+	0.931527i	$0.618478\pi$
$74$	−4.32355e6	−1.24031
$75$	0	0
$76$	−1.19303e7	−3.11747
$77$	−1.27055e6	−0.317156
$78$	0	0
$79$	5.01210e6	1.14373	0.571867	−	0.820346i	$-0.306219\pi$
$79$	5.01210e6	1.14373	0.571867	+	0.820346i	$0.306219\pi$
$80$	−1.08300e6	−0.236490
$81$	0	0
$82$	−829291.	−0.166096
$83$	−8.40027e6	−1.61257	−0.806287	−	0.591524i	$-0.798526\pi$
$83$	−8.40027e6	−1.61257	−0.806287	+	0.591524i	$0.798526\pi$
$84$	0	0
$85$	−2.12641e6	−0.375560
$86$	−1.50717e6	−0.255516
$87$	0	0
$88$	−3.23353e6	−0.505810
$89$	3.28424e6	0.493821	0.246911	−	0.969038i	$-0.420585\pi$
$89$	3.28424e6	0.493821	0.246911	+	0.969038i	$0.420585\pi$
$90$	0	0
$91$	−7.19539e6	−1.00094
$92$	−2.51667e6	−0.336953
$93$	0	0
$94$	−6.34287e6	−0.787659
$95$	7.42170e6	0.888118
$96$	0	0
$97$	−1.76546e7	−1.96406	−0.982032	−	0.188713i	$-0.939568\pi$
$97$	−1.76546e7	−1.96406	−0.982032	+	0.188713i	$0.939568\pi$
$98$	−4.37845e6	−0.469926
$99$	0	0
$100$	−1.32822e7	−1.32822
$101$	2.63164e6	0.254157	0.127078	−	0.991893i	$-0.459440\pi$
$101$	2.63164e6	0.254157	0.127078	+	0.991893i	$0.459440\pi$
$102$	0	0
$103$	−2.33018e6	−0.210116	−0.105058	−	0.994466i	$-0.533503\pi$
$103$	−2.33018e6	−0.210116	−0.105058	+	0.994466i	$0.533503\pi$
$104$	−1.83122e7	−1.59633
$105$	0	0
$106$	8.63259e6	0.703997
$107$	−2.18495e7	−1.72424	−0.862122	−	0.506701i	$-0.830865\pi$
$107$	−2.18495e7	−1.72424	−0.862122	+	0.506701i	$0.830865\pi$
$108$	0	0
$109$	1.02421e7	0.757523	0.378761	−	0.925494i	$-0.376350\pi$
$109$	1.02421e7	0.757523	0.378761	+	0.925494i	$0.376350\pi$
$110$	4.50293e6	0.322568
$111$	0	0
$112$	5.79269e6	0.389598
$113$	5.10421e6	0.332777	0.166389	−	0.986060i	$-0.446789\pi$
$113$	5.10421e6	0.332777	0.166389	+	0.986060i	$0.446789\pi$
$114$	0	0
$115$	1.56559e6	0.0959924
$116$	7.10799e6	0.422809
$117$	0	0
$118$	2.63601e7	1.47693
$119$	1.13736e7	0.618705
$120$	0	0
$121$	−1.67629e7	−0.860204
$122$	3.32262e6	0.165662
$123$	0	0
$124$	3.07832e7	1.44990
$125$	1.95064e7	0.893288
$126$	0	0
$127$	1.56386e7	0.677462	0.338731	−	0.940883i	$-0.390002\pi$
$127$	1.56386e7	0.677462	0.338731	+	0.940883i	$0.390002\pi$
$128$	−4.32713e7	−1.82375
$129$	0	0
$130$	2.55011e7	1.01802
$131$	9.76189e6	0.379389	0.189694	−	0.981843i	$-0.439250\pi$
$131$	9.76189e6	0.379389	0.189694	+	0.981843i	$0.439250\pi$
$132$	0	0
$133$	−3.96968e7	−1.46310
$134$	−6.69118e7	−2.40235
$135$	0	0
$136$	2.89457e7	0.986730
$137$	−2.43782e7	−0.809990	−0.404995	−	0.914319i	$-0.632727\pi$
$137$	−2.43782e7	−0.809990	−0.404995	+	0.914319i	$0.632727\pi$
$138$	0	0
$139$	−1.66515e7	−0.525898	−0.262949	−	0.964810i	$-0.584695\pi$
$139$	−1.66515e7	−0.525898	−0.262949	+	0.964810i	$0.584695\pi$
$140$	−2.56301e7	−0.789410
$141$	0	0
$142$	2.45425e7	0.719301
$143$	1.54279e7	0.441194
$144$	0	0
$145$	−4.42181e6	−0.120451
$146$	−4.58277e7	−1.21869
$147$	0	0
$148$	−5.27652e7	−1.33792
$149$	4.70326e7	1.16479	0.582394	−	0.812907i	$-0.302116\pi$
$149$	4.70326e7	1.16479	0.582394	+	0.812907i	$0.302116\pi$
$150$	0	0
$151$	3.28240e7	0.775840	0.387920	−	0.921693i	$-0.373194\pi$
$151$	3.28240e7	0.775840	0.387920	+	0.921693i	$0.373194\pi$
$152$	−1.01028e8	−2.33340
$153$	0	0
$154$	−2.40851e7	−0.531404
$155$	−1.91499e7	−0.413054
$156$	0	0
$157$	−5.68580e6	−0.117258	−0.0586291	−	0.998280i	$-0.518673\pi$
$157$	−5.68580e6	−0.117258	−0.0586291	+	0.998280i	$0.518673\pi$
$158$	9.50117e7	1.91636
$159$	0	0
$160$	1.55599e7	0.300321
$161$	−8.37397e6	−0.158140
$162$	0	0
$163$	−9.10027e7	−1.64588	−0.822939	−	0.568130i	$-0.807667\pi$
$163$	−9.10027e7	−1.64588	−0.822939	+	0.568130i	$0.807667\pi$
$164$	−1.01208e7	−0.179168
$165$	0	0
$166$	−1.59239e8	−2.70192
$167$	2.20885e7	0.366994	0.183497	−	0.983020i	$-0.441258\pi$
$167$	2.20885e7	0.366994	0.183497	+	0.983020i	$0.441258\pi$
$168$	0	0
$169$	2.46229e7	0.392407
$170$	−4.03091e7	−0.629263
$171$	0	0
$172$	−1.83937e7	−0.275626
$173$	−5.10352e7	−0.749390	−0.374695	−	0.927148i	$-0.622253\pi$
$173$	−5.10352e7	−0.749390	−0.374695	+	0.927148i	$0.622253\pi$
$174$	0	0
$175$	−4.41952e7	−0.623364
$176$	−1.24203e7	−0.171727
$177$	0	0
$178$	6.22575e7	0.827412
$179$	7.08774e7	0.923681	0.461841	−	0.886963i	$-0.347189\pi$
$179$	7.08774e7	0.923681	0.461841	+	0.886963i	$0.347189\pi$
$180$	0	0
$181$	5.27758e7	0.661546	0.330773	−	0.943710i	$-0.392691\pi$
$181$	5.27758e7	0.661546	0.330773	+	0.943710i	$0.392691\pi$
$182$	−1.36399e8	−1.67711
$183$	0	0
$184$	−2.13117e7	−0.252206
$185$	3.28247e7	0.381153
$186$	0	0
$187$	−2.43865e7	−0.272712
$188$	−7.74093e7	−0.849651
$189$	0	0
$190$	1.40689e8	1.48807
$191$	1.64986e8	1.71329	0.856643	−	0.515910i	$-0.172546\pi$
$191$	1.64986e8	1.71329	0.856643	+	0.515910i	$0.172546\pi$
$192$	0	0
$193$	−8.60695e7	−0.861785	−0.430893	−	0.902403i	$-0.641801\pi$
$193$	−8.60695e7	−0.861785	−0.430893	+	0.902403i	$0.641801\pi$
$194$	−3.34668e8	−3.29085
$195$	0	0
$196$	−5.34353e7	−0.506911
$197$	−7.53049e7	−0.701764	−0.350882	−	0.936420i	$-0.614118\pi$
$197$	−7.53049e7	−0.701764	−0.350882	+	0.936420i	$0.614118\pi$
$198$	0	0
$199$	−2.16331e7	−0.194596	−0.0972980	−	0.995255i	$-0.531020\pi$
$199$	−2.16331e7	−0.194596	−0.0972980	+	0.995255i	$0.531020\pi$
$200$	−1.12476e8	−0.994160
$201$	0	0
$202$	4.98866e7	0.425847
$203$	2.36511e7	0.198434
$204$	0	0
$205$	6.29604e6	0.0510421
$206$	−4.41719e7	−0.352055
$207$	0	0
$208$	−7.03389e7	−0.541968
$209$	8.51153e7	0.644905
$210$	0	0
$211$	2.54654e8	1.86622	0.933110	−	0.359592i	$-0.117084\pi$
$211$	2.54654e8	1.86622	0.933110	+	0.359592i	$0.117084\pi$
$212$	1.05353e8	0.759404
$213$	0	0
$214$	−4.14190e8	−2.88902
$215$	1.14425e7	0.0785214
$216$	0	0
$217$	1.02428e8	0.680472
$218$	1.94154e8	1.26925
$219$	0	0
$220$	5.49544e7	0.347955
$221$	−1.38106e8	−0.860678
$222$	0	0
$223$	1.85306e8	1.11898	0.559491	−	0.828837i	$-0.310997\pi$
$223$	1.85306e8	1.11898	0.559491	+	0.828837i	$0.310997\pi$
$224$	−8.32257e7	−0.494754
$225$	0	0
$226$	9.67577e7	0.557578
$227$	3.56468e7	0.202269	0.101135	−	0.994873i	$-0.467753\pi$
$227$	3.56468e7	0.202269	0.101135	+	0.994873i	$0.467753\pi$
$228$	0	0
$229$	1.83730e8	1.01101	0.505506	−	0.862823i	$-0.331306\pi$
$229$	1.83730e8	1.01101	0.505506	+	0.862823i	$0.331306\pi$
$230$	2.96781e7	0.160838
$231$	0	0
$232$	6.01919e7	0.316468
$233$	1.89314e8	0.980476	0.490238	−	0.871589i	$-0.336910\pi$
$233$	1.89314e8	0.980476	0.490238	+	0.871589i	$0.336910\pi$
$234$	0	0
$235$	4.81555e7	0.242052
$236$	3.21702e8	1.59317
$237$	0	0
$238$	2.15603e8	1.03666
$239$	−1.04601e8	−0.495615	−0.247807	−	0.968809i	$-0.579710\pi$
$239$	−1.04601e8	−0.495615	−0.247807	+	0.968809i	$0.579710\pi$
$240$	0	0
$241$	3.86413e7	0.177825	0.0889123	−	0.996039i	$-0.471661\pi$
$241$	3.86413e7	0.177825	0.0889123	+	0.996039i	$0.471661\pi$
$242$	−3.17766e8	−1.44130
$243$	0	0
$244$	4.05498e7	0.178700
$245$	3.32415e7	0.144411
$246$	0	0
$247$	4.82027e8	2.03532
$248$	2.60679e8	1.08524
$249$	0	0
$250$	3.69772e8	1.49673
$251$	−1.19863e8	−0.478439	−0.239219	−	0.970966i	$-0.576891\pi$
$251$	−1.19863e8	−0.478439	−0.239219	+	0.970966i	$0.576891\pi$
$252$	0	0
$253$	1.79549e7	0.0697046
$254$	2.96452e8	1.13511
$255$	0	0
$256$	−4.34644e8	−1.61918
$257$	2.41961e8	0.889160	0.444580	−	0.895739i	$-0.353353\pi$
$257$	2.41961e8	0.889160	0.444580	+	0.895739i	$0.353353\pi$
$258$	0	0
$259$	−1.75571e8	−0.627919
$260$	3.11219e8	1.09814
$261$	0	0
$262$	1.85051e8	0.635677
$263$	−3.53107e7	−0.119691	−0.0598455	−	0.998208i	$-0.519061\pi$
$263$	−3.53107e7	−0.119691	−0.0598455	+	0.998208i	$0.519061\pi$
$264$	0	0
$265$	−6.55392e7	−0.216342
$266$	−7.52511e8	−2.45147
$267$	0	0
$268$	−8.16601e8	−2.59142
$269$	2.79047e8	0.874066	0.437033	−	0.899446i	$-0.356029\pi$
$269$	2.79047e8	0.874066	0.437033	+	0.899446i	$0.356029\pi$
$270$	0	0
$271$	−5.00030e8	−1.52617	−0.763085	−	0.646298i	$-0.776316\pi$
$271$	−5.00030e8	−1.52617	−0.763085	+	0.646298i	$0.776316\pi$
$272$	1.11183e8	0.335003
$273$	0	0
$274$	−4.62124e8	−1.35716
$275$	9.47604e7	0.274766
$276$	0	0
$277$	4.43897e7	0.125488	0.0627441	−	0.998030i	$-0.480015\pi$
$277$	4.43897e7	0.125488	0.0627441	+	0.998030i	$0.480015\pi$
$278$	−3.15653e8	−0.881157
$279$	0	0
$280$	−2.17041e8	−0.590866
$281$	4.73373e8	1.27272	0.636358	−	0.771394i	$-0.280440\pi$
$281$	4.73373e8	1.27272	0.636358	+	0.771394i	$0.280440\pi$
$282$	0	0
$283$	1.46131e8	0.383257	0.191629	−	0.981468i	$-0.438623\pi$
$283$	1.46131e8	0.383257	0.191629	+	0.981468i	$0.438623\pi$
$284$	2.99521e8	0.775912
$285$	0	0
$286$	2.92458e8	0.739234
$287$	−3.36759e7	−0.0840877
$288$	0	0
$289$	−1.92037e8	−0.467996
$290$	−8.38218e7	−0.201820
$291$	0	0
$292$	−5.59288e8	−1.31461
$293$	2.52767e8	0.587061	0.293531	−	0.955950i	$-0.405170\pi$
$293$	2.52767e8	0.587061	0.293531	+	0.955950i	$0.405170\pi$
$294$	0	0
$295$	−2.00127e8	−0.453868
$296$	−4.46827e8	−1.00142
$297$	0	0
$298$	8.91571e8	1.95164
$299$	1.01683e8	0.219987
$300$	0	0
$301$	−6.12033e7	−0.129358
$302$	6.22227e8	1.29994
$303$	0	0
$304$	−3.88058e8	−0.792208
$305$	−2.52256e7	−0.0509087
$306$	0	0
$307$	3.33761e8	0.658341	0.329171	−	0.944270i	$-0.393231\pi$
$307$	3.33761e8	0.658341	0.329171	+	0.944270i	$0.393231\pi$
$308$	−2.93937e8	−0.573228
$309$	0	0
$310$	−3.63015e8	−0.692084
$311$	−7.28569e8	−1.37344	−0.686720	−	0.726922i	$-0.740950\pi$
$311$	−7.28569e8	−1.37344	−0.686720	+	0.726922i	$0.740950\pi$
$312$	0	0
$313$	−5.59471e8	−1.03127	−0.515635	−	0.856809i	$-0.672444\pi$
$313$	−5.59471e8	−1.03127	−0.515635	+	0.856809i	$0.672444\pi$
$314$	−1.07783e8	−0.196469
$315$	0	0
$316$	1.15954e9	2.06718
$317$	−4.03517e8	−0.711466	−0.355733	−	0.934588i	$-0.615769\pi$
$317$	−4.03517e8	−0.711466	−0.355733	+	0.934588i	$0.615769\pi$
$318$	0	0
$319$	−5.07112e7	−0.0874654
$320$	4.33583e8	0.739687
$321$	0	0
$322$	−1.58741e8	−0.264968
$323$	−7.61930e8	−1.25807
$324$	0	0
$325$	5.36649e8	0.867159
$326$	−1.72509e9	−2.75772
$327$	0	0
$328$	−8.57049e7	−0.134106
$329$	−2.57572e8	−0.398761
$330$	0	0
$331$	−2.09854e8	−0.318067	−0.159033	−	0.987273i	$-0.550838\pi$
$331$	−2.09854e8	−0.318067	−0.159033	+	0.987273i	$0.550838\pi$
$332$	−1.94338e9	−2.91457
$333$	0	0
$334$	4.18720e8	0.614909
$335$	5.07999e8	0.738255
$336$	0	0
$337$	−2.40639e8	−0.342500	−0.171250	−	0.985228i	$-0.554781\pi$
$337$	−2.40639e8	−0.342500	−0.171250	+	0.985228i	$0.554781\pi$
$338$	4.66764e8	0.657489
$339$	0	0
$340$	−4.91938e8	−0.678788
$341$	−2.19620e8	−0.299938
$342$	0	0
$343$	−8.11751e8	−1.08616
$344$	−1.55762e8	−0.206303
$345$	0	0
$346$	−9.67446e8	−1.25563
$347$	−1.35452e9	−1.74034	−0.870168	−	0.492755i	$-0.835990\pi$
$347$	−1.35452e9	−1.74034	−0.870168	+	0.492755i	$0.835990\pi$
$348$	0	0
$349$	−2.59226e8	−0.326430	−0.163215	−	0.986591i	$-0.552186\pi$
$349$	−2.59226e8	−0.326430	−0.163215	+	0.986591i	$0.552186\pi$
$350$	−8.37784e8	−1.04447
$351$	0	0
$352$	1.78447e8	0.218077
$353$	−4.41102e7	−0.0533738	−0.0266869	−	0.999644i	$-0.508496\pi$
$353$	−4.41102e7	−0.0533738	−0.0266869	+	0.999644i	$0.508496\pi$
$354$	0	0
$355$	−1.86329e8	−0.221045
$356$	7.59799e8	0.892532
$357$	0	0
$358$	1.34358e9	1.54766
$359$	1.48397e9	1.69275	0.846377	−	0.532585i	$-0.178779\pi$
$359$	1.48397e9	1.69275	0.846377	+	0.532585i	$0.178779\pi$
$360$	0	0
$361$	1.76546e9	1.97507
$362$	1.00044e9	1.10844
$363$	0	0
$364$	−1.66463e9	−1.80910
$365$	3.47927e8	0.374510
$366$	0	0
$367$	−9.88203e8	−1.04355	−0.521777	−	0.853082i	$-0.674731\pi$
$367$	−9.88203e8	−1.04355	−0.521777	+	0.853082i	$0.674731\pi$
$368$	−8.18602e7	−0.0856259
$369$	0	0
$370$	6.22240e8	0.638634
$371$	3.50553e8	0.356406
$372$	0	0
$373$	3.89901e8	0.389021	0.194511	−	0.980900i	$-0.437688\pi$
$373$	3.89901e8	0.389021	0.194511	+	0.980900i	$0.437688\pi$
$374$	−4.62282e8	−0.456937
$375$	0	0
$376$	−6.55518e8	−0.635956
$377$	−2.87189e8	−0.276040
$378$	0	0
$379$	8.94238e8	0.843754	0.421877	−	0.906653i	$-0.361371\pi$
$379$	8.94238e8	0.843754	0.421877	+	0.906653i	$0.361371\pi$
$380$	1.71699e9	1.60518
$381$	0	0
$382$	3.12754e9	2.87066
$383$	−1.97434e9	−1.79567	−0.897833	−	0.440336i	$-0.854859\pi$
$383$	−1.97434e9	−1.79567	−0.897833	+	0.440336i	$0.854859\pi$
$384$	0	0
$385$	1.82855e8	0.163303
$386$	−1.63157e9	−1.44395
$387$	0	0
$388$	−4.08433e9	−3.54985
$389$	5.84773e8	0.503690	0.251845	−	0.967768i	$-0.418963\pi$
$389$	5.84773e8	0.503690	0.251845	+	0.967768i	$0.418963\pi$
$390$	0	0
$391$	−1.60728e8	−0.135979
$392$	−4.52501e8	−0.379418
$393$	0	0
$394$	−1.42751e9	−1.17583
$395$	−7.21335e8	−0.588908
$396$	0	0
$397$	6.30832e8	0.505996	0.252998	−	0.967467i	$-0.418583\pi$
$397$	6.30832e8	0.505996	0.252998	+	0.967467i	$0.418583\pi$
$398$	−4.10088e8	−0.326051
$399$	0	0
$400$	−4.32032e8	−0.337525
$401$	1.91609e9	1.48392	0.741961	−	0.670443i	$-0.233896\pi$
$401$	1.91609e9	1.48392	0.741961	+	0.670443i	$0.233896\pi$
$402$	0	0
$403$	−1.24376e9	−0.946602
$404$	6.08823e8	0.459363
$405$	0	0
$406$	4.48342e8	0.332482
$407$	3.76448e8	0.276773
$408$	0	0
$409$	2.52604e8	0.182561	0.0912807	−	0.995825i	$-0.470904\pi$
$409$	2.52604e8	0.182561	0.0912807	+	0.995825i	$0.470904\pi$
$410$	1.19351e8	0.0855226
$411$	0	0
$412$	−5.39080e8	−0.379763
$413$	1.07043e9	0.747711
$414$	0	0
$415$	1.20896e9	0.830313
$416$	1.01059e9	0.688250
$417$	0	0
$418$	1.61348e9	1.08056
$419$	−1.23901e9	−0.822859	−0.411429	−	0.911442i	$-0.634970\pi$
$419$	−1.23901e9	−0.822859	−0.411429	+	0.911442i	$0.634970\pi$
$420$	0	0
$421$	1.03896e9	0.678595	0.339297	−	0.940679i	$-0.389811\pi$
$421$	1.03896e9	0.678595	0.339297	+	0.940679i	$0.389811\pi$
$422$	4.82735e9	3.12691
$423$	0	0
$424$	8.92154e8	0.568407
$425$	−8.48271e8	−0.536011
$426$	0	0
$427$	1.34925e8	0.0838679
$428$	−5.05483e9	−3.11640
$429$	0	0
$430$	2.16910e8	0.131565
$431$	2.60636e9	1.56807	0.784033	−	0.620719i	$-0.213159\pi$
$431$	2.60636e9	1.56807	0.784033	+	0.620719i	$0.213159\pi$
$432$	0	0
$433$	1.60251e9	0.948621	0.474310	−	0.880358i	$-0.342697\pi$
$433$	1.60251e9	0.948621	0.474310	+	0.880358i	$0.342697\pi$
$434$	1.94168e9	1.14015
$435$	0	0
$436$	2.36948e9	1.36915
$437$	5.60981e8	0.321561
$438$	0	0
$439$	−1.73718e9	−0.979982	−0.489991	−	0.871728i	$-0.663000\pi$
$439$	−1.73718e9	−0.979982	−0.489991	+	0.871728i	$0.663000\pi$
$440$	4.65365e8	0.260441
$441$	0	0
$442$	−2.61801e9	−1.44209
$443$	2.25578e9	1.23277	0.616386	−	0.787444i	$-0.288596\pi$
$443$	2.25578e9	1.23277	0.616386	+	0.787444i	$0.288596\pi$
$444$	0	0
$445$	−4.72663e8	−0.254268
$446$	3.51275e9	1.87489
$447$	0	0
$448$	−2.31913e9	−1.21857
$449$	5.99192e8	0.312395	0.156197	−	0.987726i	$-0.450076\pi$
$449$	5.99192e8	0.312395	0.156197	+	0.987726i	$0.450076\pi$
$450$	0	0
$451$	7.22056e7	0.0370641
$452$	1.18084e9	0.601462
$453$	0	0
$454$	6.75736e8	0.338908
$455$	1.03555e9	0.515385
$456$	0	0
$457$	−9.10080e8	−0.446039	−0.223020	−	0.974814i	$-0.571591\pi$
$457$	−9.10080e8	−0.446039	−0.223020	+	0.974814i	$0.571591\pi$
$458$	3.48287e9	1.69398
$459$	0	0
$460$	3.62196e8	0.173497
$461$	2.53597e9	1.20557	0.602784	−	0.797905i	$-0.294058\pi$
$461$	2.53597e9	1.20557	0.602784	+	0.797905i	$0.294058\pi$
$462$	0	0
$463$	2.56143e9	1.19936	0.599678	−	0.800241i	$-0.295295\pi$
$463$	2.56143e9	1.19936	0.599678	+	0.800241i	$0.295295\pi$
$464$	2.31203e8	0.107444
$465$	0	0
$466$	3.58872e9	1.64282
$467$	−6.32049e8	−0.287172	−0.143586	−	0.989638i	$-0.545863\pi$
$467$	−6.32049e8	−0.287172	−0.143586	+	0.989638i	$0.545863\pi$
$468$	0	0
$469$	−2.71716e9	−1.21621
$470$	9.12858e8	0.405565
$471$	0	0
$472$	2.72424e9	1.19247
$473$	1.31228e8	0.0570181
$474$	0	0
$475$	2.96068e9	1.26755
$476$	2.63125e9	1.11825
$477$	0	0
$478$	−1.98287e9	−0.830418
$479$	−1.71787e9	−0.714194	−0.357097	−	0.934067i	$-0.616233\pi$
$479$	−1.71787e9	−0.714194	−0.357097	+	0.934067i	$0.616233\pi$
$480$	0	0
$481$	2.13191e9	0.873495
$482$	7.32502e8	0.297951
$483$	0	0
$484$	−3.87806e9	−1.55473
$485$	2.54082e9	1.01130
$486$	0	0
$487$	4.04905e9	1.58855	0.794276	−	0.607557i	$-0.207850\pi$
$487$	4.04905e9	1.58855	0.794276	+	0.607557i	$0.207850\pi$
$488$	3.43384e8	0.133755
$489$	0	0
$490$	6.30142e8	0.241965
$491$	−4.58209e9	−1.74694	−0.873471	−	0.486877i	$-0.838136\pi$
$491$	−4.58209e9	−1.74694	−0.873471	+	0.486877i	$0.838136\pi$
$492$	0	0
$493$	4.53954e8	0.170627
$494$	9.13752e9	3.41023
$495$	0	0
$496$	1.00129e9	0.368447
$497$	9.96625e8	0.364153
$498$	0	0
$499$	−4.19526e9	−1.51150	−0.755748	−	0.654863i	$-0.772726\pi$
$499$	−4.19526e9	−1.51150	−0.755748	+	0.654863i	$0.772726\pi$
$500$	4.51274e9	1.61453
$501$	0	0
$502$	−2.27217e9	−0.801638
$503$	2.30167e9	0.806410	0.403205	−	0.915110i	$-0.367896\pi$
$503$	2.30167e9	0.806410	0.403205	+	0.915110i	$0.367896\pi$
$504$	0	0
$505$	−3.78742e8	−0.130865
$506$	3.40361e8	0.116792
$507$	0	0
$508$	3.61794e9	1.22444
$509$	−2.78240e9	−0.935206	−0.467603	−	0.883939i	$-0.654882\pi$
$509$	−2.78240e9	−0.935206	−0.467603	+	0.883939i	$0.654882\pi$
$510$	0	0
$511$	−1.86098e9	−0.616975
$512$	−2.70060e9	−0.889231
$513$	0	0
$514$	4.58673e9	1.48981
$515$	3.35356e8	0.108188
$516$	0	0
$517$	5.52268e8	0.175765
$518$	−3.32820e9	−1.05210
$519$	0	0
$520$	2.63547e9	0.821951
$521$	5.58417e7	0.0172992	0.00864960	−	0.999963i	$-0.497247\pi$
$521$	5.58417e7	0.0172992	0.00864960	+	0.999963i	$0.497247\pi$
$522$	0	0
$523$	1.86687e9	0.570636	0.285318	−	0.958433i	$-0.407901\pi$
$523$	1.86687e9	0.570636	0.285318	+	0.958433i	$0.407901\pi$
$524$	2.25838e9	0.685707
$525$	0	0
$526$	−6.69366e8	−0.200546
$527$	1.96598e9	0.585116
$528$	0	0
$529$	−3.28649e9	−0.965244
$530$	−1.24239e9	−0.362487
$531$	0	0
$532$	−9.18374e9	−2.64441
$533$	4.08917e8	0.116974
$534$	0	0
$535$	3.14456e9	0.887812
$536$	−6.91515e9	−1.93966
$537$	0	0
$538$	5.28974e9	1.46452
$539$	3.81228e8	0.104863
$540$	0	0
$541$	3.82374e9	1.03824	0.519120	−	0.854702i	$-0.326260\pi$
$541$	3.82374e9	1.03824	0.519120	+	0.854702i	$0.326260\pi$
$542$	−9.47879e9	−2.55714
$543$	0	0
$544$	−1.59741e9	−0.425423
$545$	−1.47403e9	−0.390048
$546$	0	0
$547$	4.29227e8	0.112132	0.0560662	−	0.998427i	$-0.482144\pi$
$547$	4.29227e8	0.112132	0.0560662	+	0.998427i	$0.482144\pi$
$548$	−5.63983e9	−1.46398
$549$	0	0
$550$	1.79632e9	0.460378
$551$	−1.58441e9	−0.403495
$552$	0	0
$553$	3.85824e9	0.970177
$554$	8.41471e8	0.210259
$555$	0	0
$556$	−3.85227e9	−0.950507
$557$	−8.11843e9	−1.99058	−0.995289	−	0.0969563i	$-0.969089\pi$
$557$	−8.11843e9	−1.99058	−0.995289	+	0.0969563i	$0.969089\pi$
$558$	0	0
$559$	7.43173e8	0.179949
$560$	−8.33676e8	−0.200604
$561$	0	0
$562$	8.97348e9	2.13247
$563$	−7.41509e9	−1.75120	−0.875602	−	0.483033i	$-0.839535\pi$
$563$	−7.41509e9	−1.75120	−0.875602	+	0.483033i	$0.839535\pi$
$564$	0	0
$565$	−7.34591e8	−0.171347
$566$	2.77013e9	0.642159
$567$	0	0
$568$	2.53640e9	0.580763
$569$	−1.11079e9	−0.252777	−0.126388	−	0.991981i	$-0.540339\pi$
$569$	−1.11079e9	−0.252777	−0.126388	+	0.991981i	$0.540339\pi$
$570$	0	0
$571$	−3.78299e9	−0.850372	−0.425186	−	0.905106i	$-0.639791\pi$
$571$	−3.78299e9	−0.850372	−0.425186	+	0.905106i	$0.639791\pi$
$572$	3.56919e9	0.797414
$573$	0	0
$574$	−6.38376e8	−0.140891
$575$	6.24551e8	0.137003
$576$	0	0
$577$	4.92543e9	1.06740	0.533702	−	0.845673i	$-0.320801\pi$
$577$	4.92543e9	1.06740	0.533702	+	0.845673i	$0.320801\pi$
$578$	−3.64034e9	−0.784141
$579$	0	0
$580$	−1.02297e9	−0.217704
$581$	−6.46640e9	−1.36787
$582$	0	0
$583$	−7.51632e8	−0.157096
$584$	−4.73617e9	−0.983970
$585$	0	0
$586$	4.79156e9	0.983639
$587$	−5.34056e9	−1.08982	−0.544908	−	0.838496i	$-0.683436\pi$
$587$	−5.34056e9	−1.08982	−0.544908	+	0.838496i	$0.683436\pi$
$588$	0	0
$589$	−6.86178e9	−1.38367
$590$	−3.79371e9	−0.760469
$591$	0	0
$592$	−1.71630e9	−0.339992
$593$	−2.42758e9	−0.478060	−0.239030	−	0.971012i	$-0.576829\pi$
$593$	−2.42758e9	−0.478060	−0.239030	+	0.971012i	$0.576829\pi$
$594$	0	0
$595$	−1.63687e9	−0.318571
$596$	1.08809e10	2.10524
$597$	0	0
$598$	1.92754e9	0.368595
$599$	7.08555e9	1.34704	0.673519	−	0.739170i	$-0.264782\pi$
$599$	7.08555e9	1.34704	0.673519	+	0.739170i	$0.264782\pi$
$600$	0	0
$601$	−2.48308e9	−0.466584	−0.233292	−	0.972407i	$-0.574950\pi$
$601$	−2.48308e9	−0.466584	−0.233292	+	0.972407i	$0.574950\pi$
$602$	−1.16020e9	−0.216743
$603$	0	0
$604$	7.59374e9	1.40225
$605$	2.41250e9	0.442919
$606$	0	0
$607$	3.11843e9	0.565947	0.282973	−	0.959128i	$-0.408679\pi$
$607$	3.11843e9	0.565947	0.282973	+	0.959128i	$0.408679\pi$
$608$	5.57538e9	1.00603
$609$	0	0
$610$	−4.78188e8	−0.0852990
$611$	3.12762e9	0.554714
$612$	0	0
$613$	6.98736e9	1.22518	0.612592	−	0.790399i	$-0.290127\pi$
$613$	6.98736e9	1.22518	0.612592	+	0.790399i	$0.290127\pi$
$614$	6.32692e9	1.10307
$615$	0	0
$616$	−2.48912e9	−0.429056
$617$	−1.50314e9	−0.257632	−0.128816	−	0.991668i	$-0.541118\pi$
$617$	−1.50314e9	−0.257632	−0.128816	+	0.991668i	$0.541118\pi$
$618$	0	0
$619$	−8.74008e9	−1.48115	−0.740573	−	0.671976i	$-0.765446\pi$
$619$	−8.74008e9	−1.48115	−0.740573	+	0.671976i	$0.765446\pi$
$620$	−4.43028e9	−0.746553
$621$	0	0
$622$	−1.38111e10	−2.30124
$623$	2.52816e9	0.418886
$624$	0	0
$625$	1.67801e9	0.274926
$626$	−1.06056e10	−1.72792
$627$	0	0
$628$	−1.31539e9	−0.211932
$629$	−3.36986e9	−0.539927
$630$	0	0
$631$	9.74784e9	1.54456	0.772281	−	0.635281i	$-0.219116\pi$
$631$	9.74784e9	1.54456	0.772281	+	0.635281i	$0.219116\pi$
$632$	9.81918e9	1.54727
$633$	0	0
$634$	−7.64924e9	−1.19208
$635$	−2.25069e9	−0.348824
$636$	0	0
$637$	2.15898e9	0.330949
$638$	−9.61304e8	−0.146551
$639$	0	0
$640$	6.22754e9	0.939046
$641$	−1.55930e9	−0.233844	−0.116922	−	0.993141i	$-0.537303\pi$
$641$	−1.55930e9	−0.233844	−0.116922	+	0.993141i	$0.537303\pi$
$642$	0	0
$643$	2.38058e9	0.353138	0.176569	−	0.984288i	$-0.443500\pi$
$643$	2.38058e9	0.353138	0.176569	+	0.984288i	$0.443500\pi$
$644$	−1.93729e9	−0.285822
$645$	0	0
$646$	−1.44435e10	−2.10794
$647$	−2.99528e9	−0.434783	−0.217392	−	0.976084i	$-0.569755\pi$
$647$	−2.99528e9	−0.434783	−0.217392	+	0.976084i	$0.569755\pi$
$648$	0	0
$649$	−2.29515e9	−0.329575
$650$	1.01730e10	1.45295
$651$	0	0
$652$	−2.10532e10	−2.97476
$653$	−3.06264e9	−0.430427	−0.215214	−	0.976567i	$-0.569045\pi$
$653$	−3.06264e9	−0.430427	−0.215214	+	0.976567i	$0.569045\pi$
$654$	0	0
$655$	−1.40492e9	−0.195347
$656$	−3.29201e8	−0.0455299
$657$	0	0
$658$	−4.88265e9	−0.668135
$659$	4.08197e9	0.555611	0.277806	−	0.960637i	$-0.410393\pi$
$659$	4.08197e9	0.555611	0.277806	+	0.960637i	$0.410393\pi$
$660$	0	0
$661$	−5.40059e9	−0.727338	−0.363669	−	0.931528i	$-0.618476\pi$
$661$	−5.40059e9	−0.727338	−0.363669	+	0.931528i	$0.618476\pi$
$662$	−3.97808e9	−0.532930
$663$	0	0
$664$	−1.64569e10	−2.18153
$665$	5.71311e9	0.753351
$666$	0	0
$667$	−3.34229e8	−0.0436118
$668$	5.11011e9	0.663305
$669$	0	0
$670$	9.62987e9	1.23697
$671$	−2.89298e8	−0.0369672
$672$	0	0
$673$	1.26468e10	1.59929	0.799646	−	0.600472i	$-0.205021\pi$
$673$	1.26468e10	1.59929	0.799646	+	0.600472i	$0.205021\pi$
$674$	−4.56166e9	−0.573870
$675$	0	0
$676$	5.69645e9	0.709236
$677$	−1.10240e10	−1.36546	−0.682731	−	0.730669i	$-0.739208\pi$
$677$	−1.10240e10	−1.36546	−0.682731	+	0.730669i	$0.739208\pi$
$678$	0	0
$679$	−1.35902e10	−1.66603
$680$	−4.16583e9	−0.508066
$681$	0	0
$682$	−4.16321e9	−0.502555
$683$	9.01963e9	1.08322	0.541610	−	0.840630i	$-0.317815\pi$
$683$	9.01963e9	1.08322	0.541610	+	0.840630i	$0.317815\pi$
$684$	0	0
$685$	3.50848e9	0.417063
$686$	−1.53879e10	−1.81989
$687$	0	0
$688$	−5.98296e8	−0.0700417
$689$	−4.25666e9	−0.495794
$690$	0	0
$691$	−2.52976e9	−0.291680	−0.145840	−	0.989308i	$-0.546589\pi$
$691$	−2.52976e9	−0.291680	−0.145840	+	0.989308i	$0.546589\pi$
$692$	−1.18068e10	−1.35445
$693$	0	0
$694$	−2.56769e10	−2.91598
$695$	2.39646e9	0.270784
$696$	0	0
$697$	−6.46366e8	−0.0723043
$698$	−4.91401e9	−0.546943
$699$	0	0
$700$	−1.02244e10	−1.12667
$701$	8.95135e9	0.981467	0.490733	−	0.871310i	$-0.336729\pi$
$701$	8.95135e9	0.981467	0.490733	+	0.871310i	$0.336729\pi$
$702$	0	0
$703$	1.17617e10	1.27681
$704$	4.97252e9	0.537121
$705$	0	0
$706$	−8.36174e8	−0.0894293
$707$	2.02580e9	0.215590
$708$	0	0
$709$	2.12538e9	0.223962	0.111981	−	0.993710i	$-0.464280\pi$
$709$	2.12538e9	0.223962	0.111981	+	0.993710i	$0.464280\pi$
$710$	−3.53213e9	−0.370367
$711$	0	0
$712$	6.43414e9	0.668052
$713$	−1.44748e9	−0.149554
$714$	0	0
$715$	−2.22036e9	−0.227171
$716$	1.63973e10	1.66946
$717$	0	0
$718$	2.81308e10	2.83626
$719$	−1.38505e10	−1.38968	−0.694842	−	0.719163i	$-0.744526\pi$
$719$	−1.38505e10	−1.38968	−0.694842	+	0.719163i	$0.744526\pi$
$720$	0	0
$721$	−1.79373e9	−0.178232
$722$	3.34669e10	3.30929
$723$	0	0
$724$	1.22095e10	1.19568
$725$	−1.76396e9	−0.171912
$726$	0	0
$727$	−8.10792e9	−0.782599	−0.391299	−	0.920263i	$-0.627974\pi$
$727$	−8.10792e9	−0.782599	−0.391299	+	0.920263i	$0.627974\pi$
$728$	−1.40965e10	−1.35410
$729$	0	0
$730$	6.59547e9	0.627503
$731$	−1.17472e9	−0.111230
$732$	0	0
$733$	1.61158e10	1.51143	0.755716	−	0.654899i	$-0.227289\pi$
$733$	1.61158e10	1.51143	0.755716	+	0.654899i	$0.227289\pi$
$734$	−1.87328e10	−1.74851
$735$	0	0
$736$	1.17612e9	0.108737
$737$	5.82595e9	0.536082
$738$	0	0
$739$	−1.66293e10	−1.51572	−0.757860	−	0.652417i	$-0.773755\pi$
$739$	−1.66293e10	−1.51572	−0.757860	+	0.652417i	$0.773755\pi$
$740$	7.59390e9	0.688896
$741$	0	0
$742$	6.64524e9	0.597168
$743$	−1.15115e10	−1.02960	−0.514801	−	0.857310i	$-0.672134\pi$
$743$	−1.15115e10	−1.02960	−0.514801	+	0.857310i	$0.672134\pi$
$744$	0	0
$745$	−6.76887e9	−0.599748
$746$	7.39114e9	0.651817
$747$	0	0
$748$	−5.64175e9	−0.492900
$749$	−1.68194e10	−1.46260
$750$	0	0
$751$	−1.34568e10	−1.15932	−0.579660	−	0.814858i	$-0.696815\pi$
$751$	−1.34568e10	−1.15932	−0.579660	+	0.814858i	$0.696815\pi$
$752$	−2.51791e9	−0.215912
$753$	0	0
$754$	−5.44408e9	−0.462514
$755$	−4.72399e9	−0.399479
$756$	0	0
$757$	−9.56204e9	−0.801152	−0.400576	−	0.916264i	$-0.631190\pi$
$757$	−9.56204e9	−0.801152	−0.400576	+	0.916264i	$0.631190\pi$
$758$	1.69516e10	1.41373
$759$	0	0
$760$	1.45398e10	1.20147
$761$	−1.11141e10	−0.914172	−0.457086	−	0.889423i	$-0.651107\pi$
$761$	−1.11141e10	−0.914172	−0.457086	+	0.889423i	$0.651107\pi$
$762$	0	0
$763$	7.88420e9	0.642572
$764$	3.81690e10	3.09659
$765$	0	0
$766$	−3.74264e10	−3.00869
$767$	−1.29979e10	−1.04014
$768$	0	0
$769$	6.30892e9	0.500280	0.250140	−	0.968210i	$-0.419523\pi$
$769$	6.30892e9	0.500280	0.250140	+	0.968210i	$0.419523\pi$
$770$	3.46629e9	0.273620
$771$	0	0
$772$	−1.99119e10	−1.55759
$773$	−1.63601e10	−1.27396	−0.636981	−	0.770879i	$-0.719817\pi$
$773$	−1.63601e10	−1.27396	−0.636981	+	0.770879i	$0.719817\pi$
$774$	0	0
$775$	−7.63934e9	−0.589522
$776$	−3.45870e10	−2.65703
$777$	0	0
$778$	1.10852e10	0.843948
$779$	2.25598e9	0.170984
$780$	0	0
$781$	−2.13690e9	−0.160511
$782$	−3.04683e9	−0.227837
$783$	0	0
$784$	−1.73810e9	−0.128816
$785$	8.18293e8	0.0603761
$786$	0	0
$787$	−2.42010e10	−1.76979	−0.884895	−	0.465791i	$-0.845770\pi$
$787$	−2.42010e10	−1.76979	−0.884895	+	0.465791i	$0.845770\pi$
$788$	−1.74216e10	−1.26837
$789$	0	0
$790$	−1.36740e10	−0.986732
$791$	3.92914e9	0.282280
$792$	0	0
$793$	−1.63836e9	−0.116668
$794$	1.19583e10	0.847811
$795$	0	0
$796$	−5.00477e9	−0.351713
$797$	−4.58354e9	−0.320699	−0.160349	−	0.987060i	$-0.551262\pi$
$797$	−4.58354e9	−0.320699	−0.160349	+	0.987060i	$0.551262\pi$
$798$	0	0
$799$	−4.94376e9	−0.342881
$800$	6.20718e9	0.428627
$801$	0	0
$802$	3.63223e10	2.48635
$803$	3.99018e9	0.271949
$804$	0	0
$805$	1.20517e9	0.0814260
$806$	−2.35772e10	−1.58606
$807$	0	0
$808$	5.15564e9	0.343829
$809$	−2.49689e10	−1.65798	−0.828989	−	0.559265i	$-0.811083\pi$
$809$	−2.49689e10	−1.65798	−0.828989	+	0.559265i	$0.811083\pi$
$810$	0	0
$811$	−1.17317e10	−0.772302	−0.386151	−	0.922436i	$-0.626196\pi$
$811$	−1.17317e10	−0.772302	−0.386151	+	0.922436i	$0.626196\pi$
$812$	5.47162e9	0.358649
$813$	0	0
$814$	7.13611e9	0.463742
$815$	1.30970e10	0.847461
$816$	0	0
$817$	4.10007e9	0.263036
$818$	4.78848e9	0.305887
$819$	0	0
$820$	1.45657e9	0.0922535
$821$	−1.93348e10	−1.21938	−0.609689	−	0.792641i	$-0.708706\pi$
$821$	−1.93348e10	−1.21938	−0.609689	+	0.792641i	$0.708706\pi$
$822$	0	0
$823$	−1.62602e10	−1.01678	−0.508389	−	0.861128i	$-0.669759\pi$
$823$	−1.62602e10	−1.01678	−0.508389	+	0.861128i	$0.669759\pi$
$824$	−4.56504e9	−0.284249
$825$	0	0
$826$	2.02916e10	1.25281
$827$	−1.84019e10	−1.13134	−0.565672	−	0.824631i	$-0.691383\pi$
$827$	−1.84019e10	−1.13134	−0.565672	+	0.824631i	$0.691383\pi$
$828$	0	0
$829$	3.70182e9	0.225670	0.112835	−	0.993614i	$-0.464007\pi$
$829$	3.70182e9	0.225670	0.112835	+	0.993614i	$0.464007\pi$
$830$	2.29175e10	1.39121
$831$	0	0
$832$	2.81605e10	1.69515
$833$	−3.41266e9	−0.204567
$834$	0	0
$835$	−3.17895e9	−0.188965
$836$	1.96912e10	1.16560
$837$	0	0
$838$	−2.34872e10	−1.37872
$839$	7.64145e9	0.446693	0.223347	−	0.974739i	$-0.428302\pi$
$839$	7.64145e9	0.446693	0.223347	+	0.974739i	$0.428302\pi$
$840$	0	0
$841$	−1.63059e10	−0.945276
$842$	1.96950e10	1.13701
$843$	0	0
$844$	5.89136e10	3.37300
$845$	−3.54370e9	−0.202050
$846$	0	0
$847$	−1.29039e10	−0.729672
$848$	3.42685e9	0.192979
$849$	0	0
$850$	−1.60802e10	−0.898102
$851$	2.48111e9	0.138004
$852$	0	0
$853$	4.47770e9	0.247021	0.123510	−	0.992343i	$-0.460585\pi$
$853$	4.47770e9	0.247021	0.123510	+	0.992343i	$0.460585\pi$
$854$	2.55771e9	0.140523
$855$	0	0
$856$	−4.28053e10	−2.33260
$857$	1.03626e10	0.562390	0.281195	−	0.959651i	$-0.409269\pi$
$857$	1.03626e10	0.562390	0.281195	+	0.959651i	$0.409269\pi$
$858$	0	0
$859$	−2.54167e10	−1.36818	−0.684089	−	0.729399i	$-0.739800\pi$
$859$	−2.54167e10	−1.36818	−0.684089	+	0.729399i	$0.739800\pi$
$860$	2.64720e9	0.141920
$861$	0	0
$862$	4.94074e10	2.62734
$863$	−6.46303e9	−0.342293	−0.171147	−	0.985246i	$-0.554747\pi$
$863$	−6.46303e9	−0.342293	−0.171147	+	0.985246i	$0.554747\pi$
$864$	0	0
$865$	7.34491e9	0.385861
$866$	3.03779e10	1.58944
$867$	0	0
$868$	2.36965e10	1.22989
$869$	−8.27258e9	−0.427634
$870$	0	0
$871$	3.29937e10	1.69187
$872$	2.00652e10	1.02479
$873$	0	0
$874$	1.06342e10	0.538785
$875$	1.50157e10	0.757735
$876$	0	0
$877$	−3.36965e10	−1.68689	−0.843443	−	0.537219i	$-0.819475\pi$
$877$	−3.36965e10	−1.68689	−0.843443	+	0.537219i	$0.819475\pi$
$878$	−3.29307e10	−1.64199
$879$	0	0
$880$	1.78751e9	0.0884219
$881$	7.24785e9	0.357103	0.178551	−	0.983931i	$-0.442859\pi$
$881$	7.24785e9	0.357103	0.178551	+	0.983931i	$0.442859\pi$
$882$	0	0
$883$	−2.61111e10	−1.27633	−0.638165	−	0.769899i	$-0.720306\pi$
$883$	−2.61111e10	−1.27633	−0.638165	+	0.769899i	$0.720306\pi$
$884$	−3.19505e10	−1.55559
$885$	0	0
$886$	4.27615e10	2.06555
$887$	−7.41092e9	−0.356566	−0.178283	−	0.983979i	$-0.557054\pi$
$887$	−7.41092e9	−0.356566	−0.178283	+	0.983979i	$0.557054\pi$
$888$	0	0
$889$	1.20384e10	0.574660
$890$	−8.96002e9	−0.426034
$891$	0	0
$892$	4.28701e10	2.02245
$893$	1.72550e10	0.810840
$894$	0	0
$895$	−1.02006e10	−0.475603
$896$	−3.33096e10	−1.54700
$897$	0	0
$898$	1.13585e10	0.523426
$899$	4.08821e9	0.187661
$900$	0	0
$901$	6.72842e9	0.306462
$902$	1.36876e9	0.0621020
$903$	0	0
$904$	9.99963e9	0.450189
$905$	−7.59543e9	−0.340629
$906$	0	0
$907$	−3.75015e9	−0.166887	−0.0834437	−	0.996512i	$-0.526592\pi$
$907$	−3.75015e9	−0.166887	−0.0834437	+	0.996512i	$0.526592\pi$
$908$	8.24678e9	0.365581
$909$	0	0
$910$	1.96304e10	0.863542
$911$	4.01781e10	1.76066	0.880328	−	0.474365i	$-0.157322\pi$
$911$	4.01781e10	1.76066	0.880328	+	0.474365i	$0.157322\pi$
$912$	0	0
$913$	1.38648e10	0.602930
$914$	−1.72519e10	−0.747352
$915$	0	0
$916$	4.25054e10	1.82730
$917$	7.51455e9	0.321818
$918$	0	0
$919$	−3.75667e10	−1.59661	−0.798304	−	0.602255i	$-0.794269\pi$
$919$	−3.75667e10	−1.59661	−0.798304	+	0.602255i	$0.794269\pi$
$920$	3.06715e9	0.129861
$921$	0	0
$922$	4.80731e10	2.01996
$923$	−1.21017e10	−0.506572
$924$	0	0
$925$	1.30945e10	0.543993
$926$	4.85555e10	2.00956
$927$	0	0
$928$	−3.32178e9	−0.136444
$929$	1.26889e10	0.519241	0.259620	−	0.965711i	$-0.416403\pi$
$929$	1.26889e10	0.519241	0.259620	+	0.965711i	$0.416403\pi$
$930$	0	0
$931$	1.19110e10	0.483756
$932$	4.37972e10	1.77211
$933$	0	0
$934$	−1.19814e10	−0.481165
$935$	3.50968e9	0.140419
$936$	0	0
$937$	−1.41607e10	−0.562338	−0.281169	−	0.959658i	$-0.590722\pi$
$937$	−1.41607e10	−0.562338	−0.281169	+	0.959658i	$0.590722\pi$
$938$	−5.15077e10	−2.03780
$939$	0	0
$940$	1.11406e10	0.437484
$941$	−2.02950e9	−0.0794007	−0.0397004	−	0.999212i	$-0.512640\pi$
$941$	−2.02950e9	−0.0794007	−0.0397004	+	0.999212i	$0.512640\pi$
$942$	0	0
$943$	4.75896e8	0.0184808
$944$	1.04641e10	0.404854
$945$	0	0
$946$	2.48762e9	0.0955355
$947$	3.78908e10	1.44980	0.724901	−	0.688853i	$-0.241885\pi$
$947$	3.78908e10	1.44980	0.724901	+	0.688853i	$0.241885\pi$
$948$	0	0
$949$	2.25973e10	0.858271
$950$	5.61241e10	2.12381
$951$	0	0
$952$	2.22820e10	0.836998
$953$	4.24474e10	1.58864	0.794321	−	0.607499i	$-0.207827\pi$
$953$	4.24474e10	1.58864	0.794321	+	0.607499i	$0.207827\pi$
$954$	0	0
$955$	−2.37445e10	−0.882169
$956$	−2.41992e10	−0.895774
$957$	0	0
$958$	−3.25647e10	−1.19665
$959$	−1.87660e10	−0.687078
$960$	0	0
$961$	−9.80743e9	−0.356470
$962$	4.04134e10	1.46357
$963$	0	0
$964$	8.93955e9	0.321400
$965$	1.23870e10	0.443732
$966$	0	0
$967$	−5.12086e10	−1.82117	−0.910585	−	0.413322i	$-0.864369\pi$
$967$	−5.12086e10	−1.82117	−0.910585	+	0.413322i	$0.864369\pi$
$968$	−3.28402e10	−1.16370
$969$	0	0
$970$	4.81650e10	1.69446
$971$	−3.91136e10	−1.37107	−0.685536	−	0.728039i	$-0.740432\pi$
$971$	−3.91136e10	−1.37107	−0.685536	+	0.728039i	$0.740432\pi$
$972$	0	0
$973$	−1.28181e10	−0.446095
$974$	7.67556e10	2.66167
$975$	0	0
$976$	1.31897e9	0.0454109
$977$	−7.63663e9	−0.261982	−0.130991	−	0.991384i	$-0.541816\pi$
$977$	−7.63663e9	−0.261982	−0.130991	+	0.991384i	$0.541816\pi$
$978$	0	0
$979$	−5.42071e9	−0.184636
$980$	7.69033e9	0.261008
$981$	0	0
$982$	−8.68602e10	−2.92705
$983$	2.86639e10	0.962495	0.481247	−	0.876585i	$-0.340184\pi$
$983$	2.86639e10	0.962495	0.481247	+	0.876585i	$0.340184\pi$
$984$	0	0
$985$	1.08378e10	0.361338
$986$	8.60535e9	0.285890
$987$	0	0
$988$	1.11515e11	3.67863
$989$	8.64903e8	0.0284302
$990$	0	0
$991$	4.27195e10	1.39434	0.697169	−	0.716907i	$-0.254443\pi$
$991$	4.27195e10	1.39434	0.697169	+	0.716907i	$0.254443\pi$
$992$	−1.43859e10	−0.467894
$993$	0	0
$994$	1.88925e10	0.610150
$995$	3.11341e9	0.100197
$996$	0	0
$997$	−4.61415e10	−1.47455	−0.737274	−	0.675594i	$-0.763887\pi$
$997$	−4.61415e10	−1.47455	−0.737274	+	0.675594i	$0.763887\pi$
$998$	−7.95272e10	−2.53255
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	387.8.a.a.1.10		10
3.2	odd	2		129.8.a.a.1.1	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
129.8.a.a.1.1	✓	10	3.2	odd	2
387.8.a.a.1.10		10	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)

\(f(q)\)	\(=\)	\(q+18.9565 q^{2} +231.347 q^{4} -143.919 q^{5} +769.785 q^{7} +1959.10 q^{8} +O(q^{10})\)	\(q+18.9565 q^{2} +231.347 q^{4} -143.919 q^{5} +769.785 q^{7} +1959.10 q^{8} -2728.19 q^{10} -1650.52 q^{11} -9347.27 q^{13} +14592.4 q^{14} +7525.07 q^{16} +14775.0 q^{17} -51568.7 q^{19} -33295.2 q^{20} -31288.0 q^{22} -10878.3 q^{23} -57412.4 q^{25} -177191. q^{26} +178088. q^{28} +30724.3 q^{29} +133061. q^{31} -108116. q^{32} +280082. q^{34} -110786. q^{35} -228078. q^{37} -977560. q^{38} -281951. q^{40} -43747.2 q^{41} -79507.0 q^{43} -381843. q^{44} -206214. q^{46} -334602. q^{47} -230974. q^{49} -1.08834e6 q^{50} -2.16246e6 q^{52} +455391. q^{53} +237541. q^{55} +1.50808e6 q^{56} +582425. q^{58} +1.39056e6 q^{59} +175277. q^{61} +2.52236e6 q^{62} -3.01270e6 q^{64} +1.34525e6 q^{65} -3.52977e6 q^{67} +3.41816e6 q^{68} -2.10012e6 q^{70} +1.29468e6 q^{71} -2.41753e6 q^{73} -4.32355e6 q^{74} -1.19303e7 q^{76} -1.27055e6 q^{77} +5.01210e6 q^{79} -1.08300e6 q^{80} -829291. q^{82} -8.40027e6 q^{83} -2.12641e6 q^{85} -1.50717e6 q^{86} -3.23353e6 q^{88} +3.28424e6 q^{89} -7.19539e6 q^{91} -2.51667e6 q^{92} -6.34287e6 q^{94} +7.42170e6 q^{95} -1.76546e7 q^{97} -4.37845e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + q^{2} + 371 q^{4} + 122 q^{5} - 2052 q^{7} + 927 q^{8}+O(q^{10}) \) 10 * q + q^2 + 371 * q^4 + 122 * q^5 - 2052 * q^7 + 927 * q^8	\( 10 q + q^{2} + 371 q^{4} + 122 q^{5} - 2052 q^{7} + 927 q^{8} - 10032 q^{10} + 8888 q^{11} - 16432 q^{13} + 28408 q^{14} - 26669 q^{16} + 48122 q^{17} - 56146 q^{19} + 88940 q^{20} - 100626 q^{22} + 236336 q^{23} - 135016 q^{25} + 166748 q^{26} - 259060 q^{28} + 248818 q^{29} - 430970 q^{31} - 69493 q^{32} + 445522 q^{34} - 298982 q^{35} - 261254 q^{37} - 257662 q^{38} - 671432 q^{40} + 126814 q^{41} - 795070 q^{43} + 620022 q^{44} - 809038 q^{46} - 627080 q^{47} - 1256116 q^{49} + 83117 q^{50} - 3674204 q^{52} + 1612384 q^{53} - 4732974 q^{55} + 4301484 q^{56} - 7268516 q^{58} + 3442492 q^{59} - 5217214 q^{61} + 2500324 q^{62} - 4657369 q^{64} + 1224166 q^{65} - 6810926 q^{67} + 3563486 q^{68} - 2745858 q^{70} + 13935120 q^{71} - 13743720 q^{73} + 1752692 q^{74} - 15817594 q^{76} + 7685750 q^{77} - 9007608 q^{79} + 13641024 q^{80} - 6329026 q^{82} + 21779128 q^{83} - 13177392 q^{85} - 79507 q^{86} - 13214750 q^{88} + 1895364 q^{89} - 16439838 q^{91} - 9614510 q^{92} + 3404276 q^{94} + 16861514 q^{95} - 20434472 q^{97} - 35731457 q^{98}+O(q^{100}) \) 10 * q + q^2 + 371 * q^4 + 122 * q^5 - 2052 * q^7 + 927 * q^8 - 10032 * q^10 + 8888 * q^11 - 16432 * q^13 + 28408 * q^14 - 26669 * q^16 + 48122 * q^17 - 56146 * q^19 + 88940 * q^20 - 100626 * q^22 + 236336 * q^23 - 135016 * q^25 + 166748 * q^26 - 259060 * q^28 + 248818 * q^29 - 430970 * q^31 - 69493 * q^32 + 445522 * q^34 - 298982 * q^35 - 261254 * q^37 - 257662 * q^38 - 671432 * q^40 + 126814 * q^41 - 795070 * q^43 + 620022 * q^44 - 809038 * q^46 - 627080 * q^47 - 1256116 * q^49 + 83117 * q^50 - 3674204 * q^52 + 1612384 * q^53 - 4732974 * q^55 + 4301484 * q^56 - 7268516 * q^58 + 3442492 * q^59 - 5217214 * q^61 + 2500324 * q^62 - 4657369 * q^64 + 1224166 * q^65 - 6810926 * q^67 + 3563486 * q^68 - 2745858 * q^70 + 13935120 * q^71 - 13743720 * q^73 + 1752692 * q^74 - 15817594 * q^76 + 7685750 * q^77 - 9007608 * q^79 + 13641024 * q^80 - 6329026 * q^82 + 21779128 * q^83 - 13177392 * q^85 - 79507 * q^86 - 13214750 * q^88 + 1895364 * q^89 - 16439838 * q^91 - 9614510 * q^92 + 3404276 * q^94 + 16861514 * q^95 - 20434472 * q^97 - 35731457 * q^98

Introduction

L-functions

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