LMFDB - Embedded newform 531.8.a.e.1.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [531,8,Mod(1,531)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(531, 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("531.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 531 = 3^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	531.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:	$165.876448532$
Analytic rank:	$1$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 6 x^{17} - 1798 x^{16} + 11087 x^{15} + 1326765 x^{14} - 8403720 x^{13} - 518334228 x^{12} + \cdots + 51\!\cdots\!48 $ x^18 - 6x^17 - 1798x^16 + 11087x^15 + 1326765x^14 - 8403720x^13 - 518334228x^12 + 3375594921x^11 + 115310342333x^10 - 774932111214x^9 - 14600047830166x^8 + 102185027148481x^7 + 988557475638619x^6 - 7379206238519716x^5 - 30152342836849520x^4 + 260578770749067175x^3 + 182609347488069978x^2 - 3481290425710753600*x + 5164646074739714048
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{5} $
Twist minimal:	no (minimal twist has level 177)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
Root			$-20.5204$ of defining polynomial
Character	$\chi$	$=$	531.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+19.5204 q^{2} +253.046 q^{4} +49.6011 q^{5} -119.919 q^{7} +2440.94 q^{8} +O(q^{10})$	\(q+19.5204 q^{2} +253.046 q^{4} +49.6011 q^{5} -119.919 q^{7} +2440.94 q^{8} +968.233 q^{10} +5494.59 q^{11} -12895.9 q^{13} -2340.86 q^{14} +15258.3 q^{16} -25621.3 q^{17} -17560.8 q^{19} +12551.3 q^{20} +107256. q^{22} -63260.9 q^{23} -75664.7 q^{25} -251733. q^{26} -30344.9 q^{28} +29622.4 q^{29} +166521. q^{31} -14593.0 q^{32} -500138. q^{34} -5948.10 q^{35} +107247. q^{37} -342794. q^{38} +121073. q^{40} -520929. q^{41} -943358. q^{43} +1.39038e6 q^{44} -1.23488e6 q^{46} +789984. q^{47} -809163. q^{49} -1.47701e6 q^{50} -3.26325e6 q^{52} +930361. q^{53} +272538. q^{55} -292714. q^{56} +578240. q^{58} -205379. q^{59} +2.19795e6 q^{61} +3.25055e6 q^{62} -2.23792e6 q^{64} -639650. q^{65} +1.13372e6 q^{67} -6.48336e6 q^{68} -116109. q^{70} -2.46554e6 q^{71} +5.09118e6 q^{73} +2.09351e6 q^{74} -4.44368e6 q^{76} -658903. q^{77} +4.40371e6 q^{79} +756827. q^{80} -1.01687e7 q^{82} -172296. q^{83} -1.27084e6 q^{85} -1.84147e7 q^{86} +1.34120e7 q^{88} -8.28035e6 q^{89} +1.54646e6 q^{91} -1.60079e7 q^{92} +1.54208e7 q^{94} -871035. q^{95} -6.60133e6 q^{97} -1.57952e7 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - 24 q^{2} + 1358 q^{4} - 678 q^{5} + 3081 q^{7} - 4107 q^{8}+O(q^{10}) $ 18 * q - 24 * q^2 + 1358 * q^4 - 678 * q^5 + 3081 * q^7 - 4107 * q^8	\( 18 q - 24 q^{2} + 1358 q^{4} - 678 q^{5} + 3081 q^{7} - 4107 q^{8} + 3609 q^{10} - 15070 q^{11} + 13662 q^{13} - 20861 q^{14} + 60482 q^{16} - 71919 q^{17} + 56231 q^{19} - 143053 q^{20} + 274198 q^{22} - 150029 q^{23} + 399672 q^{25} - 182846 q^{26} + 434150 q^{28} - 591285 q^{29} + 426733 q^{31} - 1205630 q^{32} + 403548 q^{34} - 912879 q^{35} + 7703 q^{37} + 417859 q^{38} + 618020 q^{40} - 770959 q^{41} + 793050 q^{43} - 2591274 q^{44} - 4068019 q^{46} - 1410373 q^{47} + 1637427 q^{49} - 1021549 q^{50} - 3749190 q^{52} - 1037934 q^{53} + 331974 q^{55} + 391748 q^{56} + 653724 q^{58} - 3696822 q^{59} - 1374623 q^{61} - 5251718 q^{62} + 5077197 q^{64} - 3257170 q^{65} - 2436904 q^{67} - 14119909 q^{68} + 5185580 q^{70} - 14289172 q^{71} + 5482515 q^{73} - 14934154 q^{74} + 3822912 q^{76} - 23157109 q^{77} + 19786414 q^{79} - 31978143 q^{80} + 9749509 q^{82} - 30227337 q^{83} + 9946981 q^{85} - 44295864 q^{86} + 39970897 q^{88} - 31061677 q^{89} + 26377785 q^{91} - 4719698 q^{92} + 44488296 q^{94} - 15534599 q^{95} + 12084118 q^{97} - 42274744 q^{98}+O(q^{100}) \) 18 * q - 24 * q^2 + 1358 * q^4 - 678 * q^5 + 3081 * q^7 - 4107 * q^8 + 3609 * q^10 - 15070 * q^11 + 13662 * q^13 - 20861 * q^14 + 60482 * q^16 - 71919 * q^17 + 56231 * q^19 - 143053 * q^20 + 274198 * q^22 - 150029 * q^23 + 399672 * q^25 - 182846 * q^26 + 434150 * q^28 - 591285 * q^29 + 426733 * q^31 - 1205630 * q^32 + 403548 * q^34 - 912879 * q^35 + 7703 * q^37 + 417859 * q^38 + 618020 * q^40 - 770959 * q^41 + 793050 * q^43 - 2591274 * q^44 - 4068019 * q^46 - 1410373 * q^47 + 1637427 * q^49 - 1021549 * q^50 - 3749190 * q^52 - 1037934 * q^53 + 331974 * q^55 + 391748 * q^56 + 653724 * q^58 - 3696822 * q^59 - 1374623 * q^61 - 5251718 * q^62 + 5077197 * q^64 - 3257170 * q^65 - 2436904 * q^67 - 14119909 * q^68 + 5185580 * q^70 - 14289172 * q^71 + 5482515 * q^73 - 14934154 * q^74 + 3822912 * q^76 - 23157109 * q^77 + 19786414 * q^79 - 31978143 * q^80 + 9749509 * q^82 - 30227337 * q^83 + 9946981 * q^85 - 44295864 * q^86 + 39970897 * q^88 - 31061677 * q^89 + 26377785 * q^91 - 4719698 * q^92 + 44488296 * q^94 - 15534599 * q^95 + 12084118 * q^97 - 42274744 * 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 $

$2$	19.5204	1.72538	0.862688	−	0.505737i	$-0.168779\pi$
$2$	19.5204	1.72538	0.862688	+	0.505737i	$0.168779\pi$
$3$	0	0
$3$	0	0
$4$	253.046	1.97692
$5$	49.6011	0.177458	0.0887292	−	0.996056i	$-0.471719\pi$
$5$	49.6011	0.177458	0.0887292	+	0.996056i	$0.471719\pi$
$6$	0	0
$7$	−119.919	−0.132143	−0.0660714	−	0.997815i	$-0.521047\pi$
$7$	−119.919	−0.132143	−0.0660714	+	0.997815i	$0.521047\pi$
$8$	2440.94	1.68555
$9$	0	0
$10$	968.233	0.306182
$11$	5494.59	1.24469	0.622344	−	0.782744i	$-0.286180\pi$
$11$	5494.59	1.24469	0.622344	+	0.782744i	$0.286180\pi$
$12$	0	0
$13$	−12895.9	−1.62798	−0.813990	−	0.580879i	$-0.802709\pi$
$13$	−12895.9	−1.62798	−0.813990	+	0.580879i	$0.802709\pi$
$14$	−2340.86	−0.227996
$15$	0	0
$16$	15258.3	0.931291
$17$	−25621.3	−1.26482	−0.632411	−	0.774633i	$-0.717935\pi$
$17$	−25621.3	−1.26482	−0.632411	+	0.774633i	$0.717935\pi$
$18$	0	0
$19$	−17560.8	−0.587363	−0.293681	−	0.955903i	$-0.594880\pi$
$19$	−17560.8	−0.587363	−0.293681	+	0.955903i	$0.594880\pi$
$20$	12551.3	0.350821
$21$	0	0
$22$	107256.	2.14755
$23$	−63260.9	−1.08415	−0.542074	−	0.840331i	$-0.682361\pi$
$23$	−63260.9	−1.08415	−0.542074	+	0.840331i	$0.682361\pi$
$24$	0	0
$25$	−75664.7	−0.968509
$26$	−251733.	−2.80888
$27$	0	0
$28$	−30344.9	−0.261236
$29$	29622.4	0.225542	0.112771	−	0.993621i	$-0.464027\pi$
$29$	29622.4	0.225542	0.112771	+	0.993621i	$0.464027\pi$
$30$	0	0
$31$	166521.	1.00393	0.501963	−	0.864889i	$-0.332611\pi$
$31$	166521.	1.00393	0.501963	+	0.864889i	$0.332611\pi$
$32$	−14593.0	−0.0787264
$33$	0	0
$34$	−500138.	−2.18229
$35$	−5948.10	−0.0234498
$36$	0	0
$37$	107247.	0.348080	0.174040	−	0.984739i	$-0.444318\pi$
$37$	107247.	0.348080	0.174040	+	0.984739i	$0.444318\pi$
$38$	−342794.	−1.01342
$39$	0	0
$40$	121073.	0.299115
$41$	−520929.	−1.18042	−0.590208	−	0.807251i	$-0.700954\pi$
$41$	−520929.	−1.18042	−0.590208	+	0.807251i	$0.700954\pi$
$42$	0	0
$43$	−943358.	−1.80941	−0.904704	−	0.426040i	$-0.859908\pi$
$43$	−943358.	−1.80941	−0.904704	+	0.426040i	$0.859908\pi$
$44$	1.39038e6	2.46065
$45$	0	0
$46$	−1.23488e6	−1.87056
$47$	789984.	1.10988	0.554939	−	0.831891i	$-0.312741\pi$
$47$	789984.	1.10988	0.554939	+	0.831891i	$0.312741\pi$
$48$	0	0
$49$	−809163.	−0.982538
$50$	−1.47701e6	−1.67104
$51$	0	0
$52$	−3.26325e6	−3.21839
$53$	930361.	0.858392	0.429196	−	0.903211i	$-0.358797\pi$
$53$	930361.	0.858392	0.429196	+	0.903211i	$0.358797\pi$
$54$	0	0
$55$	272538.	0.220880
$56$	−292714.	−0.222734
$57$	0	0
$58$	578240.	0.389144
$59$	−205379.	−0.130189
$59$	−205379.	−0.130189
$60$	0	0
$61$	2.19795e6	1.23983	0.619915	−	0.784669i	$-0.287167\pi$
$61$	2.19795e6	1.23983	0.619915	+	0.784669i	$0.287167\pi$
$62$	3.25055e6	1.73215
$63$	0	0
$64$	−2.23792e6	−1.06712
$65$	−639650.	−0.288899
$66$	0	0
$67$	1.13372e6	0.460513	0.230257	−	0.973130i	$-0.426043\pi$
$67$	1.13372e6	0.460513	0.230257	+	0.973130i	$0.426043\pi$
$68$	−6.48336e6	−2.50045
$69$	0	0
$70$	−116109.	−0.0404598
$71$	−2.46554e6	−0.817537	−0.408769	−	0.912638i	$-0.634042\pi$
$71$	−2.46554e6	−0.817537	−0.408769	+	0.912638i	$0.634042\pi$
$72$	0	0
$73$	5.09118e6	1.53175	0.765876	−	0.642988i	$-0.222306\pi$
$73$	5.09118e6	1.53175	0.765876	+	0.642988i	$0.222306\pi$
$74$	2.09351e6	0.600569
$75$	0	0
$76$	−4.44368e6	−1.16117
$77$	−658903.	−0.164477
$78$	0	0
$79$	4.40371e6	1.00490	0.502452	−	0.864605i	$-0.332431\pi$
$79$	4.40371e6	1.00490	0.502452	+	0.864605i	$0.332431\pi$
$80$	756827.	0.165265
$81$	0	0
$82$	−1.01687e7	−2.03666
$83$	−172296.	−0.0330751	−0.0165375	−	0.999863i	$-0.505264\pi$
$83$	−172296.	−0.0330751	−0.0165375	+	0.999863i	$0.505264\pi$
$84$	0	0
$85$	−1.27084e6	−0.224453
$86$	−1.84147e7	−3.12191
$87$	0	0
$88$	1.34120e7	2.09799
$89$	−8.28035e6	−1.24504	−0.622521	−	0.782603i	$-0.713891\pi$
$89$	−8.28035e6	−1.24504	−0.622521	+	0.782603i	$0.713891\pi$
$90$	0	0
$91$	1.54646e6	0.215126
$92$	−1.60079e7	−2.14327
$93$	0	0
$94$	1.54208e7	1.91496
$95$	−871035.	−0.104232
$96$	0	0
$97$	−6.60133e6	−0.734396	−0.367198	−	0.930143i	$-0.619683\pi$
$97$	−6.60133e6	−0.734396	−0.367198	+	0.930143i	$0.619683\pi$
$98$	−1.57952e7	−1.69525
$99$	0	0
$100$	−1.91466e7	−1.91466
$101$	1.77940e6	0.171849	0.0859247	−	0.996302i	$-0.472616\pi$
$101$	1.77940e6	0.171849	0.0859247	+	0.996302i	$0.472616\pi$
$102$	0	0
$103$	−6.76389e6	−0.609911	−0.304955	−	0.952367i	$-0.598642\pi$
$103$	−6.76389e6	−0.609911	−0.304955	+	0.952367i	$0.598642\pi$
$104$	−3.14781e7	−2.74405
$105$	0	0
$106$	1.81610e7	1.48105
$107$	−2.49224e7	−1.96674	−0.983371	−	0.181607i	$-0.941870\pi$
$107$	−2.49224e7	−1.96674	−0.983371	+	0.181607i	$0.941870\pi$
$108$	0	0
$109$	−2.15957e7	−1.59726	−0.798630	−	0.601823i	$-0.794441\pi$
$109$	−2.15957e7	−1.59726	−0.798630	+	0.601823i	$0.794441\pi$
$110$	5.32004e6	0.381101
$111$	0	0
$112$	−1.82975e6	−0.123063
$113$	1.84076e7	1.20011	0.600056	−	0.799958i	$-0.295145\pi$
$113$	1.84076e7	1.20011	0.600056	+	0.799958i	$0.295145\pi$
$114$	0	0
$115$	−3.13781e6	−0.192391
$116$	7.49582e6	0.445878
$117$	0	0
$118$	−4.00908e6	−0.224625
$119$	3.07247e6	0.167137
$120$	0	0
$121$	1.07033e7	0.549249
$122$	4.29047e7	2.13917
$123$	0	0
$124$	4.21373e7	1.98468
$125$	−7.62814e6	−0.349328
$126$	0	0
$127$	2.65579e7	1.15049	0.575243	−	0.817983i	$-0.304908\pi$
$127$	2.65579e7	1.15049	0.575243	+	0.817983i	$0.304908\pi$
$128$	−4.18172e7	−1.76246
$129$	0	0
$130$	−1.24862e7	−0.498459
$131$	−3.58406e7	−1.39292	−0.696460	−	0.717595i	$-0.745243\pi$
$131$	−3.58406e7	−1.39292	−0.696460	+	0.717595i	$0.745243\pi$
$132$	0	0
$133$	2.10587e6	0.0776158
$134$	2.21306e7	0.794558
$135$	0	0
$136$	−6.25400e7	−2.13193
$137$	−3.94491e7	−1.31074	−0.655368	−	0.755310i	$-0.727487\pi$
$137$	−3.94491e7	−1.31074	−0.655368	+	0.755310i	$0.727487\pi$
$138$	0	0
$139$	−1.10784e7	−0.349886	−0.174943	−	0.984579i	$-0.555974\pi$
$139$	−1.10784e7	−0.349886	−0.174943	+	0.984579i	$0.555974\pi$
$140$	−1.50514e6	−0.0463584
$141$	0	0
$142$	−4.81283e7	−1.41056
$143$	−7.08575e7	−2.02633
$144$	0	0
$145$	1.46930e6	0.0400243
$146$	9.93819e7	2.64285
$147$	0	0
$148$	2.71384e7	0.688127
$149$	−2.08170e7	−0.515546	−0.257773	−	0.966206i	$-0.582989\pi$
$149$	−2.08170e7	−0.515546	−0.257773	+	0.966206i	$0.582989\pi$
$150$	0	0
$151$	−2.65440e7	−0.627403	−0.313702	−	0.949522i	$-0.601569\pi$
$151$	−2.65440e7	−0.627403	−0.313702	+	0.949522i	$0.601569\pi$
$152$	−4.28648e7	−0.990031
$153$	0	0
$154$	−1.28620e7	−0.283784
$155$	8.25961e6	0.178155
$156$	0	0
$157$	5.98029e7	1.23331	0.616656	−	0.787232i	$-0.288487\pi$
$157$	5.98029e7	1.23331	0.616656	+	0.787232i	$0.288487\pi$
$158$	8.59622e7	1.73384
$159$	0	0
$160$	−723831.	−0.0139707
$161$	7.58616e6	0.143262
$162$	0	0
$163$	9.55079e7	1.72736	0.863679	−	0.504042i	$-0.168154\pi$
$163$	9.55079e7	1.72736	0.863679	+	0.504042i	$0.168154\pi$
$164$	−1.31819e8	−2.33359
$165$	0	0
$166$	−3.36328e6	−0.0570669
$167$	1.67858e7	0.278890	0.139445	−	0.990230i	$-0.455468\pi$
$167$	1.67858e7	0.278890	0.139445	+	0.990230i	$0.455468\pi$
$168$	0	0
$169$	1.03555e8	1.65032
$170$	−2.48074e7	−0.387266
$171$	0	0
$172$	−2.38713e8	−3.57706
$173$	7.07459e7	1.03882	0.519410	−	0.854525i	$-0.326152\pi$
$173$	7.07459e7	1.03882	0.519410	+	0.854525i	$0.326152\pi$
$174$	0	0
$175$	9.07361e6	0.127981
$176$	8.38379e7	1.15917
$177$	0	0
$178$	−1.61636e8	−2.14816
$179$	9.77537e6	0.127394	0.0636968	−	0.997969i	$-0.479711\pi$
$179$	9.77537e6	0.127394	0.0636968	+	0.997969i	$0.479711\pi$
$180$	0	0
$181$	9.13710e7	1.14534	0.572668	−	0.819787i	$-0.305908\pi$
$181$	9.13710e7	1.14534	0.572668	+	0.819787i	$0.305908\pi$
$182$	3.01874e7	0.371173
$183$	0	0
$184$	−1.54416e8	−1.82739
$185$	5.31957e6	0.0617698
$186$	0	0
$187$	−1.40778e8	−1.57431
$188$	1.99902e8	2.19414
$189$	0	0
$190$	−1.70029e7	−0.179840
$191$	1.44165e8	1.49707	0.748536	−	0.663094i	$-0.230757\pi$
$191$	1.44165e8	1.49707	0.748536	+	0.663094i	$0.230757\pi$
$192$	0	0
$193$	5.46350e7	0.547042	0.273521	−	0.961866i	$-0.411812\pi$
$193$	5.46350e7	0.547042	0.273521	+	0.961866i	$0.411812\pi$
$194$	−1.28861e8	−1.26711
$195$	0	0
$196$	−2.04755e8	−1.94240
$197$	−1.55456e8	−1.44869	−0.724347	−	0.689436i	$-0.757859\pi$
$197$	−1.55456e8	−1.44869	−0.724347	+	0.689436i	$0.757859\pi$
$198$	0	0
$199$	−2.03195e7	−0.182780	−0.0913898	−	0.995815i	$-0.529131\pi$
$199$	−2.03195e7	−0.182780	−0.0913898	+	0.995815i	$0.529131\pi$
$200$	−1.84693e8	−1.63247
$201$	0	0
$202$	3.47345e7	0.296505
$203$	−3.55227e6	−0.0298037
$204$	0	0
$205$	−2.58386e7	−0.209475
$206$	−1.32034e8	−1.05233
$207$	0	0
$208$	−1.96769e8	−1.51612
$209$	−9.64893e7	−0.731084
$210$	0	0
$211$	−1.34211e8	−0.983557	−0.491779	−	0.870720i	$-0.663653\pi$
$211$	−1.34211e8	−0.983557	−0.491779	+	0.870720i	$0.663653\pi$
$212$	2.35424e8	1.69697
$213$	0	0
$214$	−4.86496e8	−3.39337
$215$	−4.67916e7	−0.321095
$216$	0	0
$217$	−1.99689e7	−0.132662
$218$	−4.21557e8	−2.75587
$219$	0	0
$220$	6.89645e7	0.436663
$221$	3.30409e8	2.05911
$222$	0	0
$223$	1.58205e8	0.955328	0.477664	−	0.878543i	$-0.341484\pi$
$223$	1.58205e8	0.955328	0.477664	+	0.878543i	$0.341484\pi$
$224$	1.74998e6	0.0104031
$225$	0	0
$226$	3.59323e8	2.07064
$227$	−2.48289e8	−1.40886	−0.704428	−	0.709775i	$-0.748796\pi$
$227$	−2.48289e8	−1.40886	−0.704428	+	0.709775i	$0.748796\pi$
$228$	0	0
$229$	5.03600e6	0.0277116	0.0138558	−	0.999904i	$-0.495589\pi$
$229$	5.03600e6	0.0277116	0.0138558	+	0.999904i	$0.495589\pi$
$230$	−6.12513e7	−0.331947
$231$	0	0
$232$	7.23065e7	0.380162
$233$	−2.76994e8	−1.43458	−0.717291	−	0.696774i	$-0.754618\pi$
$233$	−2.76994e8	−1.43458	−0.717291	+	0.696774i	$0.754618\pi$
$234$	0	0
$235$	3.91841e7	0.196957
$236$	−5.19703e7	−0.257373
$237$	0	0
$238$	5.99758e7	0.288374
$239$	−3.80453e8	−1.80264	−0.901319	−	0.433157i	$-0.857400\pi$
$239$	−3.80453e8	−1.80264	−0.901319	+	0.433157i	$0.857400\pi$
$240$	0	0
$241$	6.73805e7	0.310080	0.155040	−	0.987908i	$-0.450449\pi$
$241$	6.73805e7	0.310080	0.155040	+	0.987908i	$0.450449\pi$
$242$	2.08933e8	0.947661
$243$	0	0
$244$	5.56181e8	2.45105
$245$	−4.01354e7	−0.174360
$246$	0	0
$247$	2.26462e8	0.956215
$248$	4.06467e8	1.69217
$249$	0	0
$250$	−1.48904e8	−0.602722
$251$	1.44862e8	0.578225	0.289113	−	0.957295i	$-0.406640\pi$
$251$	1.44862e8	0.578225	0.289113	+	0.957295i	$0.406640\pi$
$252$	0	0
$253$	−3.47593e8	−1.34943
$254$	5.18421e8	1.98502
$255$	0	0
$256$	−5.29834e8	−1.97378
$257$	−5.53323e7	−0.203335	−0.101668	−	0.994818i	$-0.532418\pi$
$257$	−5.53323e7	−0.203335	−0.101668	+	0.994818i	$0.532418\pi$
$258$	0	0
$259$	−1.28609e7	−0.0459963
$260$	−1.61861e8	−0.571129
$261$	0	0
$262$	−6.99623e8	−2.40331
$263$	−1.21289e8	−0.411128	−0.205564	−	0.978644i	$-0.565903\pi$
$263$	−1.21289e8	−0.411128	−0.205564	+	0.978644i	$0.565903\pi$
$264$	0	0
$265$	4.61469e7	0.152329
$266$	4.11073e7	0.133916
$267$	0	0
$268$	2.86882e8	0.910398
$269$	5.24972e8	1.64438	0.822191	−	0.569211i	$-0.192751\pi$
$269$	5.24972e8	1.64438	0.822191	+	0.569211i	$0.192751\pi$
$270$	0	0
$271$	1.03625e8	0.316280	0.158140	−	0.987417i	$-0.449450\pi$
$271$	1.03625e8	0.316280	0.158140	+	0.987417i	$0.449450\pi$
$272$	−3.90936e8	−1.17792
$273$	0	0
$274$	−7.70062e8	−2.26151
$275$	−4.15746e8	−1.20549
$276$	0	0
$277$	1.95179e8	0.551765	0.275883	−	0.961191i	$-0.411030\pi$
$277$	1.95179e8	0.551765	0.275883	+	0.961191i	$0.411030\pi$
$278$	−2.16255e8	−0.603684
$279$	0	0
$280$	−1.45189e7	−0.0395259
$281$	−2.55753e8	−0.687621	−0.343811	−	0.939039i	$-0.611718\pi$
$281$	−2.55753e8	−0.687621	−0.343811	+	0.939039i	$0.611718\pi$
$282$	0	0
$283$	−1.42799e8	−0.374518	−0.187259	−	0.982311i	$-0.559960\pi$
$283$	−1.42799e8	−0.374518	−0.187259	+	0.982311i	$0.559960\pi$
$284$	−6.23894e8	−1.61621
$285$	0	0
$286$	−1.38317e9	−3.49618
$287$	6.24691e7	0.155983
$288$	0	0
$289$	2.46112e8	0.599777
$290$	2.86814e7	0.0690569
$291$	0	0
$292$	1.28830e9	3.02815
$293$	3.51565e8	0.816524	0.408262	−	0.912865i	$-0.366135\pi$
$293$	3.51565e8	0.816524	0.408262	+	0.912865i	$0.366135\pi$
$294$	0	0
$295$	−1.01870e7	−0.0231031
$296$	2.61784e8	0.586708
$297$	0	0
$298$	−4.06357e8	−0.889510
$299$	8.15805e8	1.76497
$300$	0	0
$301$	1.13126e8	0.239100
$302$	−5.18149e8	−1.08251
$303$	0	0
$304$	−2.67947e8	−0.547006
$305$	1.09021e8	0.220018
$306$	0	0
$307$	−6.10924e8	−1.20504	−0.602522	−	0.798102i	$-0.705837\pi$
$307$	−6.10924e8	−1.20504	−0.602522	+	0.798102i	$0.705837\pi$
$308$	−1.66733e8	−0.325157
$309$	0	0
$310$	1.61231e8	0.307384
$311$	5.80323e8	1.09398	0.546988	−	0.837140i	$-0.315774\pi$
$311$	5.80323e8	1.09398	0.546988	+	0.837140i	$0.315774\pi$
$312$	0	0
$313$	8.00376e8	1.47533	0.737664	−	0.675168i	$-0.235929\pi$
$313$	8.00376e8	1.47533	0.737664	+	0.675168i	$0.235929\pi$
$314$	1.16738e9	2.12793
$315$	0	0
$316$	1.11434e9	1.98661
$317$	6.06744e8	1.06979	0.534895	−	0.844919i	$-0.320351\pi$
$317$	6.06744e8	1.06979	0.534895	+	0.844919i	$0.320351\pi$
$318$	0	0
$319$	1.62763e8	0.280729
$320$	−1.11003e8	−0.189370
$321$	0	0
$322$	1.48085e8	0.247181
$323$	4.49930e8	0.742910
$324$	0	0
$325$	9.75763e8	1.57671
$326$	1.86435e9	2.98034
$327$	0	0
$328$	−1.27156e9	−1.98965
$329$	−9.47337e7	−0.146662
$330$	0	0
$331$	−4.95371e8	−0.750814	−0.375407	−	0.926860i	$-0.622497\pi$
$331$	−4.95371e8	−0.750814	−0.375407	+	0.926860i	$0.622497\pi$
$332$	−4.35987e7	−0.0653868
$333$	0	0
$334$	3.27665e8	0.481191
$335$	5.62335e7	0.0817219
$336$	0	0
$337$	−5.35635e8	−0.762367	−0.381183	−	0.924499i	$-0.624483\pi$
$337$	−5.35635e8	−0.762367	−0.381183	+	0.924499i	$0.624483\pi$
$338$	2.02144e9	2.84742
$339$	0	0
$340$	−3.21582e8	−0.443726
$341$	9.14962e8	1.24958
$342$	0	0
$343$	1.95792e8	0.261978
$344$	−2.30268e9	−3.04985
$345$	0	0
$346$	1.38099e9	1.79235
$347$	−1.06614e9	−1.36981	−0.684905	−	0.728633i	$-0.740156\pi$
$347$	−1.06614e9	−1.36981	−0.684905	+	0.728633i	$0.740156\pi$
$348$	0	0
$349$	9.40691e8	1.18456	0.592282	−	0.805731i	$-0.298227\pi$
$349$	9.40691e8	1.18456	0.592282	+	0.805731i	$0.298227\pi$
$350$	1.77120e8	0.220816
$351$	0	0
$352$	−8.01827e7	−0.0979899
$353$	6.36948e8	0.770713	0.385357	−	0.922768i	$-0.374078\pi$
$353$	6.36948e8	0.770713	0.385357	+	0.922768i	$0.374078\pi$
$354$	0	0
$355$	−1.22293e8	−0.145079
$356$	−2.09531e9	−2.46135
$357$	0	0
$358$	1.90819e8	0.219802
$359$	−1.13966e9	−1.30001	−0.650003	−	0.759932i	$-0.725232\pi$
$359$	−1.13966e9	−1.30001	−0.650003	+	0.759932i	$0.725232\pi$
$360$	0	0
$361$	−5.85490e8	−0.655005
$362$	1.78360e9	1.97614
$363$	0	0
$364$	3.91324e8	0.425286
$365$	2.52528e8	0.271822
$366$	0	0
$367$	−3.66527e8	−0.387057	−0.193529	−	0.981095i	$-0.561993\pi$
$367$	−3.66527e8	−0.387057	−0.193529	+	0.981095i	$0.561993\pi$
$368$	−9.65252e8	−1.00966
$369$	0	0
$370$	1.03840e8	0.106576
$371$	−1.11568e8	−0.113430
$372$	0	0
$373$	1.84626e9	1.84209	0.921047	−	0.389452i	$-0.127336\pi$
$373$	1.84626e9	1.84209	0.921047	+	0.389452i	$0.127336\pi$
$374$	−2.74805e9	−2.71628
$375$	0	0
$376$	1.92830e9	1.87076
$377$	−3.82007e8	−0.367178
$378$	0	0
$379$	−1.89436e8	−0.178742	−0.0893708	−	0.995998i	$-0.528486\pi$
$379$	−1.89436e8	−0.178742	−0.0893708	+	0.995998i	$0.528486\pi$
$380$	−2.20412e8	−0.206059
$381$	0	0
$382$	2.81415e9	2.58301
$383$	−9.23938e8	−0.840325	−0.420162	−	0.907449i	$-0.638027\pi$
$383$	−9.23938e8	−0.840325	−0.420162	+	0.907449i	$0.638027\pi$
$384$	0	0
$385$	−3.26823e7	−0.0291877
$386$	1.06650e9	0.943852
$387$	0	0
$388$	−1.67044e9	−1.45184
$389$	−1.32165e8	−0.113839	−0.0569197	−	0.998379i	$-0.518128\pi$
$389$	−1.32165e8	−0.113839	−0.0569197	+	0.998379i	$0.518128\pi$
$390$	0	0
$391$	1.62083e9	1.37125
$392$	−1.97512e9	−1.65612
$393$	0	0
$394$	−3.03457e9	−2.49954
$395$	2.18429e8	0.178328
$396$	0	0
$397$	−3.45223e8	−0.276907	−0.138453	−	0.990369i	$-0.544213\pi$
$397$	−3.45223e8	−0.276907	−0.138453	+	0.990369i	$0.544213\pi$
$398$	−3.96645e8	−0.315363
$399$	0	0
$400$	−1.15451e9	−0.901963
$401$	−1.50851e9	−1.16827	−0.584135	−	0.811656i	$-0.698566\pi$
$401$	−1.50851e9	−1.16827	−0.584135	+	0.811656i	$0.698566\pi$
$402$	0	0
$403$	−2.14743e9	−1.63437
$404$	4.50268e8	0.339732
$405$	0	0
$406$	−6.93418e7	−0.0514226
$407$	5.89279e8	0.433252
$408$	0	0
$409$	−8.44043e8	−0.610004	−0.305002	−	0.952352i	$-0.598657\pi$
$409$	−8.44043e8	−0.610004	−0.305002	+	0.952352i	$0.598657\pi$
$410$	−5.04381e8	−0.361422
$411$	0	0
$412$	−1.71157e9	−1.20574
$413$	2.46288e7	0.0172035
$414$	0	0
$415$	−8.54605e6	−0.00586945
$416$	1.88190e8	0.128165
$417$	0	0
$418$	−1.88351e9	−1.26139
$419$	1.19941e9	0.796563	0.398281	−	0.917263i	$-0.369607\pi$
$419$	1.19941e9	0.796563	0.398281	+	0.917263i	$0.369607\pi$
$420$	0	0
$421$	6.99371e8	0.456794	0.228397	−	0.973568i	$-0.426652\pi$
$421$	6.99371e8	0.456794	0.228397	+	0.973568i	$0.426652\pi$
$422$	−2.61985e9	−1.69701
$423$	0	0
$424$	2.27095e9	1.44686
$425$	1.93863e9	1.22499
$426$	0	0
$427$	−2.63575e8	−0.163835
$428$	−6.30652e9	−3.88809
$429$	0	0
$430$	−9.13390e8	−0.554009
$431$	−1.57533e9	−0.947768	−0.473884	−	0.880587i	$-0.657148\pi$
$431$	−1.57533e9	−0.947768	−0.473884	+	0.880587i	$0.657148\pi$
$432$	0	0
$433$	1.94780e9	1.15302	0.576511	−	0.817090i	$-0.304414\pi$
$433$	1.94780e9	1.15302	0.576511	+	0.817090i	$0.304414\pi$
$434$	−3.89801e8	−0.228891
$435$	0	0
$436$	−5.46471e9	−3.15765
$437$	1.11091e9	0.636788
$438$	0	0
$439$	−1.26243e9	−0.712167	−0.356084	−	0.934454i	$-0.615888\pi$
$439$	−1.26243e9	−0.712167	−0.356084	+	0.934454i	$0.615888\pi$
$440$	6.65248e8	0.372305
$441$	0	0
$442$	6.44971e9	3.55273
$443$	2.40199e9	1.31268	0.656339	−	0.754466i	$-0.272104\pi$
$443$	2.40199e9	1.31268	0.656339	+	0.754466i	$0.272104\pi$
$444$	0	0
$445$	−4.10715e8	−0.220943
$446$	3.08822e9	1.64830
$447$	0	0
$448$	2.68368e8	0.141013
$449$	−7.69955e8	−0.401424	−0.200712	−	0.979650i	$-0.564325\pi$
$449$	−7.69955e8	−0.401424	−0.200712	+	0.979650i	$0.564325\pi$
$450$	0	0
$451$	−2.86229e9	−1.46925
$452$	4.65795e9	2.37252
$453$	0	0
$454$	−4.84669e9	−2.43081
$455$	7.67059e7	0.0381759
$456$	0	0
$457$	7.07638e8	0.346820	0.173410	−	0.984850i	$-0.444521\pi$
$457$	7.07638e8	0.346820	0.173410	+	0.984850i	$0.444521\pi$
$458$	9.83046e7	0.0478129
$459$	0	0
$460$	−7.94010e8	−0.380341
$461$	−1.84947e8	−0.0879212	−0.0439606	−	0.999033i	$-0.513998\pi$
$461$	−1.84947e8	−0.0879212	−0.0439606	+	0.999033i	$0.513998\pi$
$462$	0	0
$463$	2.34876e9	1.09978	0.549890	−	0.835237i	$-0.314670\pi$
$463$	2.34876e9	1.09978	0.549890	+	0.835237i	$0.314670\pi$
$464$	4.51986e8	0.210045
$465$	0	0
$466$	−5.40704e9	−2.47519
$467$	2.14573e9	0.974912	0.487456	−	0.873147i	$-0.337925\pi$
$467$	2.14573e9	0.974912	0.487456	+	0.873147i	$0.337925\pi$
$468$	0	0
$469$	−1.35954e8	−0.0608535
$470$	7.64888e8	0.339825
$471$	0	0
$472$	−5.01318e8	−0.219440
$473$	−5.18336e9	−2.25215
$474$	0	0
$475$	1.32873e9	0.568866
$476$	7.77475e8	0.330417
$477$	0	0
$478$	−7.42659e9	−3.11023
$479$	4.02137e8	0.167186	0.0835931	−	0.996500i	$-0.473360\pi$
$479$	4.02137e8	0.167186	0.0835931	+	0.996500i	$0.473360\pi$
$480$	0	0
$481$	−1.38305e9	−0.566668
$482$	1.31529e9	0.535005
$483$	0	0
$484$	2.70843e9	1.08582
$485$	−3.27433e8	−0.130325
$486$	0	0
$487$	3.29797e9	1.29389	0.646943	−	0.762538i	$-0.276047\pi$
$487$	3.29797e9	1.29389	0.646943	+	0.762538i	$0.276047\pi$
$488$	5.36505e9	2.08980
$489$	0	0
$490$	−7.83458e8	−0.300836
$491$	−1.59461e9	−0.607953	−0.303977	−	0.952680i	$-0.598314\pi$
$491$	−1.59461e9	−0.607953	−0.303977	+	0.952680i	$0.598314\pi$
$492$	0	0
$493$	−7.58963e8	−0.285270
$494$	4.42062e9	1.64983
$495$	0	0
$496$	2.54082e9	0.934948
$497$	2.95664e8	0.108032
$498$	0	0
$499$	−4.65147e9	−1.67586	−0.837931	−	0.545776i	$-0.816235\pi$
$499$	−4.65147e9	−1.67586	−0.837931	+	0.545776i	$0.816235\pi$
$500$	−1.93027e9	−0.690594
$501$	0	0
$502$	2.82777e9	0.997656
$503$	6.98423e8	0.244698	0.122349	−	0.992487i	$-0.460957\pi$
$503$	6.98423e8	0.244698	0.122349	+	0.992487i	$0.460957\pi$
$504$	0	0
$505$	8.82600e7	0.0304961
$506$	−6.78515e9	−2.32827
$507$	0	0
$508$	6.72037e9	2.27442
$509$	1.07682e9	0.361935	0.180967	−	0.983489i	$-0.442077\pi$
$509$	1.07682e9	0.361935	0.180967	+	0.983489i	$0.442077\pi$
$510$	0	0
$511$	−6.10527e8	−0.202410
$512$	−4.98996e9	−1.64306
$513$	0	0
$514$	−1.08011e9	−0.350830
$515$	−3.35497e8	−0.108234
$516$	0	0
$517$	4.34063e9	1.38145
$518$	−2.51050e8	−0.0793609
$519$	0	0
$520$	−1.56135e9	−0.486954
$521$	3.06506e9	0.949527	0.474763	−	0.880114i	$-0.342534\pi$
$521$	3.06506e9	0.949527	0.474763	+	0.880114i	$0.342534\pi$
$522$	0	0
$523$	−5.83979e9	−1.78501	−0.892506	−	0.451036i	$-0.851055\pi$
$523$	−5.83979e9	−1.78501	−0.892506	+	0.451036i	$0.851055\pi$
$524$	−9.06932e9	−2.75369
$525$	0	0
$526$	−2.36762e9	−0.709351
$527$	−4.26647e9	−1.26979
$528$	0	0
$529$	5.97121e8	0.175375
$530$	9.00806e8	0.262824
$531$	0	0
$532$	5.32880e8	0.153440
$533$	6.71783e9	1.92169
$534$	0	0
$535$	−1.23618e9	−0.349015
$536$	2.76733e9	0.776219
$537$	0	0
$538$	1.02477e10	2.83718
$539$	−4.44601e9	−1.22295
$540$	0	0
$541$	−4.19074e9	−1.13789	−0.568945	−	0.822375i	$-0.692648\pi$
$541$	−4.19074e9	−1.13789	−0.568945	+	0.822375i	$0.692648\pi$
$542$	2.02280e9	0.545701
$543$	0	0
$544$	3.73892e8	0.0995750
$545$	−1.07117e9	−0.283447
$546$	0	0
$547$	1.83917e9	0.480470	0.240235	−	0.970715i	$-0.422775\pi$
$547$	1.83917e9	0.480470	0.240235	+	0.970715i	$0.422775\pi$
$548$	−9.98243e9	−2.59122
$549$	0	0
$550$	−8.11553e9	−2.07992
$551$	−5.20192e8	−0.132475
$552$	0	0
$553$	−5.28087e8	−0.132791
$554$	3.80997e9	0.952002
$555$	0	0
$556$	−2.80335e9	−0.691696
$557$	3.81190e9	0.934648	0.467324	−	0.884086i	$-0.345218\pi$
$557$	3.81190e9	0.934648	0.467324	+	0.884086i	$0.345218\pi$
$558$	0	0
$559$	1.21654e10	2.94568
$560$	−9.07576e7	−0.0218386
$561$	0	0
$562$	−4.99241e9	−1.18640
$563$	6.18376e9	1.46041	0.730203	−	0.683231i	$-0.239426\pi$
$563$	6.18376e9	1.46041	0.730203	+	0.683231i	$0.239426\pi$
$564$	0	0
$565$	9.13035e8	0.212970
$566$	−2.78749e9	−0.646184
$567$	0	0
$568$	−6.01823e9	−1.37800
$569$	5.12814e9	1.16699	0.583495	−	0.812117i	$-0.301685\pi$
$569$	5.12814e9	1.16699	0.583495	+	0.812117i	$0.301685\pi$
$570$	0	0
$571$	6.26225e8	0.140768	0.0703840	−	0.997520i	$-0.477578\pi$
$571$	6.26225e8	0.140768	0.0703840	+	0.997520i	$0.477578\pi$
$572$	−1.79302e10	−4.00589
$573$	0	0
$574$	1.21942e9	0.269130
$575$	4.78662e9	1.05001
$576$	0	0
$577$	−2.62762e9	−0.569440	−0.284720	−	0.958611i	$-0.591901\pi$
$577$	−2.62762e9	−0.569440	−0.284720	+	0.958611i	$0.591901\pi$
$578$	4.80419e9	1.03484
$579$	0	0
$580$	3.71801e8	0.0791247
$581$	2.06614e7	0.00437063
$582$	0	0
$583$	5.11195e9	1.06843
$584$	1.24273e10	2.58185
$585$	0	0
$586$	6.86269e9	1.40881
$587$	1.62680e9	0.331972	0.165986	−	0.986128i	$-0.446919\pi$
$587$	1.62680e9	0.331972	0.165986	+	0.986128i	$0.446919\pi$
$588$	0	0
$589$	−2.92423e9	−0.589669
$590$	−1.98855e8	−0.0398615
$591$	0	0
$592$	1.63640e9	0.324164
$593$	−2.05047e8	−0.0403797	−0.0201898	−	0.999796i	$-0.506427\pi$
$593$	−2.05047e8	−0.0403797	−0.0201898	+	0.999796i	$0.506427\pi$
$594$	0	0
$595$	1.52398e8	0.0296599
$596$	−5.26766e9	−1.01919
$597$	0	0
$598$	1.59248e10	3.04524
$599$	2.72889e9	0.518790	0.259395	−	0.965771i	$-0.416477\pi$
$599$	2.72889e9	0.518790	0.259395	+	0.965771i	$0.416477\pi$
$600$	0	0
$601$	1.75593e9	0.329949	0.164974	−	0.986298i	$-0.447246\pi$
$601$	1.75593e9	0.329949	0.164974	+	0.986298i	$0.447246\pi$
$602$	2.20827e9	0.412538
$603$	0	0
$604$	−6.71684e9	−1.24033
$605$	5.30896e8	0.0974689
$606$	0	0
$607$	3.28692e9	0.596526	0.298263	−	0.954484i	$-0.403593\pi$
$607$	3.28692e9	0.596526	0.298263	+	0.954484i	$0.403593\pi$
$608$	2.56265e8	0.0462410
$609$	0	0
$610$	2.12812e9	0.379614
$611$	−1.01875e10	−1.80686
$612$	0	0
$613$	5.79734e9	1.01652	0.508261	−	0.861203i	$-0.330289\pi$
$613$	5.79734e9	1.01652	0.508261	+	0.861203i	$0.330289\pi$
$614$	−1.19255e10	−2.07915
$615$	0	0
$616$	−1.60834e9	−0.277234
$617$	−1.42127e9	−0.243600	−0.121800	−	0.992555i	$-0.538867\pi$
$617$	−1.42127e9	−0.243600	−0.121800	+	0.992555i	$0.538867\pi$
$618$	0	0
$619$	3.85999e9	0.654138	0.327069	−	0.945001i	$-0.393939\pi$
$619$	3.85999e9	0.654138	0.327069	+	0.945001i	$0.393939\pi$
$620$	2.09006e9	0.352198
$621$	0	0
$622$	1.13281e10	1.88752
$623$	9.92968e8	0.164523
$624$	0	0
$625$	5.53294e9	0.906517
$626$	1.56236e10	2.54549
$627$	0	0
$628$	1.51329e10	2.43816
$629$	−2.74781e9	−0.440260
$630$	0	0
$631$	−5.65956e9	−0.896768	−0.448384	−	0.893841i	$-0.648000\pi$
$631$	−5.65956e9	−0.896768	−0.448384	+	0.893841i	$0.648000\pi$
$632$	1.07492e10	1.69382
$633$	0	0
$634$	1.18439e10	1.84579
$635$	1.31730e9	0.204163
$636$	0	0
$637$	1.04349e10	1.59955
$638$	3.17719e9	0.484363
$639$	0	0
$640$	−2.07418e9	−0.312763
$641$	2.57457e9	0.386101	0.193051	−	0.981189i	$-0.438162\pi$
$641$	2.57457e9	0.386101	0.193051	+	0.981189i	$0.438162\pi$
$642$	0	0
$643$	3.46249e9	0.513629	0.256815	−	0.966461i	$-0.417327\pi$
$643$	3.46249e9	0.513629	0.256815	+	0.966461i	$0.417327\pi$
$644$	1.91965e9	0.283218
$645$	0	0
$646$	8.78281e9	1.28180
$647$	−1.17598e9	−0.170701	−0.0853506	−	0.996351i	$-0.527201\pi$
$647$	−1.17598e9	−0.170701	−0.0853506	+	0.996351i	$0.527201\pi$
$648$	0	0
$649$	−1.12847e9	−0.162045
$650$	1.90473e10	2.72042
$651$	0	0
$652$	2.41679e10	3.41485
$653$	−9.71067e9	−1.36475	−0.682375	−	0.731002i	$-0.739053\pi$
$653$	−9.71067e9	−1.36475	−0.682375	+	0.731002i	$0.739053\pi$
$654$	0	0
$655$	−1.77774e9	−0.247185
$656$	−7.94847e9	−1.09931
$657$	0	0
$658$	−1.84924e9	−0.253048
$659$	9.69880e9	1.32014	0.660069	−	0.751205i	$-0.270527\pi$
$659$	9.69880e9	1.32014	0.660069	+	0.751205i	$0.270527\pi$
$660$	0	0
$661$	−1.15769e10	−1.55914	−0.779572	−	0.626313i	$-0.784563\pi$
$661$	−1.15769e10	−1.55914	−0.779572	+	0.626313i	$0.784563\pi$
$662$	−9.66984e9	−1.29544
$663$	0	0
$664$	−4.20563e8	−0.0557498
$665$	1.04453e8	0.0137736
$666$	0	0
$667$	−1.87394e9	−0.244520
$668$	4.24757e9	0.551344
$669$	0	0
$670$	1.09770e9	0.141001
$671$	1.20768e10	1.54320
$672$	0	0
$673$	−1.00199e10	−1.26710	−0.633552	−	0.773700i	$-0.718404\pi$
$673$	−1.00199e10	−1.26710	−0.633552	+	0.773700i	$0.718404\pi$
$674$	−1.04558e10	−1.31537
$675$	0	0
$676$	2.62042e10	3.26255
$677$	2.76445e8	0.0342411	0.0171206	−	0.999853i	$-0.494550\pi$
$677$	2.76445e8	0.0342411	0.0171206	+	0.999853i	$0.494550\pi$
$678$	0	0
$679$	7.91622e8	0.0970452
$680$	−3.10205e9	−0.378328
$681$	0	0
$682$	1.78604e10	2.15599
$683$	−1.25710e9	−0.150973	−0.0754864	−	0.997147i	$-0.524051\pi$
$683$	−1.25710e9	−0.150973	−0.0754864	+	0.997147i	$0.524051\pi$
$684$	0	0
$685$	−1.95672e9	−0.232601
$686$	3.82193e9	0.452011
$687$	0	0
$688$	−1.43940e10	−1.68509
$689$	−1.19978e10	−1.39745
$690$	0	0
$691$	1.23439e10	1.42325	0.711623	−	0.702562i	$-0.247960\pi$
$691$	1.23439e10	1.42325	0.711623	+	0.702562i	$0.247960\pi$
$692$	1.79020e10	2.05366
$693$	0	0
$694$	−2.08114e10	−2.36343
$695$	−5.49502e8	−0.0620902
$696$	0	0
$697$	1.33469e10	1.49302
$698$	1.83627e10	2.04382
$699$	0	0
$700$	2.29604e9	0.253009
$701$	−4.36020e9	−0.478072	−0.239036	−	0.971011i	$-0.576831\pi$
$701$	−4.36020e9	−0.478072	−0.239036	+	0.971011i	$0.576831\pi$
$702$	0	0
$703$	−1.88334e9	−0.204449
$704$	−1.22964e10	−1.32824
$705$	0	0
$706$	1.24335e10	1.32977
$707$	−2.13383e8	−0.0227086
$708$	0	0
$709$	−6.54244e9	−0.689411	−0.344705	−	0.938711i	$-0.612021\pi$
$709$	−6.54244e9	−0.689411	−0.344705	+	0.938711i	$0.612021\pi$
$710$	−2.38722e9	−0.250315
$711$	0	0
$712$	−2.02118e10	−2.09858
$713$	−1.05342e10	−1.08840
$714$	0	0
$715$	−3.51461e9	−0.359589
$716$	2.47361e9	0.251847
$717$	0	0
$718$	−2.22466e10	−2.24300
$719$	−1.24076e9	−0.124491	−0.0622453	−	0.998061i	$-0.519826\pi$
$719$	−1.24076e9	−0.124491	−0.0622453	+	0.998061i	$0.519826\pi$
$720$	0	0
$721$	8.11117e8	0.0805953
$722$	−1.14290e10	−1.13013
$723$	0	0
$724$	2.31210e10	2.26424
$725$	−2.24137e9	−0.218439
$726$	0	0
$727$	8.74540e8	0.0844130	0.0422065	−	0.999109i	$-0.486561\pi$
$727$	8.74540e8	0.0844130	0.0422065	+	0.999109i	$0.486561\pi$
$728$	3.77481e9	0.362606
$729$	0	0
$730$	4.92945e9	0.468995
$731$	2.41700e10	2.28858
$732$	0	0
$733$	−1.87673e10	−1.76010	−0.880052	−	0.474878i	$-0.842492\pi$
$733$	−1.87673e10	−1.76010	−0.880052	+	0.474878i	$0.842492\pi$
$734$	−7.15476e9	−0.667819
$735$	0	0
$736$	9.23169e8	0.0853510
$737$	6.22930e9	0.573196
$738$	0	0
$739$	1.50202e10	1.36905	0.684525	−	0.728989i	$-0.260010\pi$
$739$	1.50202e10	1.36905	0.684525	+	0.728989i	$0.260010\pi$
$740$	1.34610e9	0.122114
$741$	0	0
$742$	−2.17784e9	−0.195710
$743$	−4.97295e9	−0.444788	−0.222394	−	0.974957i	$-0.571387\pi$
$743$	−4.97295e9	−0.444788	−0.222394	+	0.974957i	$0.571387\pi$
$744$	0	0
$745$	−1.03255e9	−0.0914879
$746$	3.60397e10	3.17830
$747$	0	0
$748$	−3.56234e10	−3.11228
$749$	2.98867e9	0.259891
$750$	0	0
$751$	−1.94834e10	−1.67851	−0.839255	−	0.543737i	$-0.817009\pi$
$751$	−1.94834e10	−1.67851	−0.839255	+	0.543737i	$0.817009\pi$
$752$	1.20538e10	1.03362
$753$	0	0
$754$	−7.45692e9	−0.633519
$755$	−1.31661e9	−0.111338
$756$	0	0
$757$	5.89170e9	0.493634	0.246817	−	0.969062i	$-0.420615\pi$
$757$	5.89170e9	0.493634	0.246817	+	0.969062i	$0.420615\pi$
$758$	−3.69787e9	−0.308396
$759$	0	0
$760$	−2.12614e9	−0.175689
$761$	−1.46780e10	−1.20731	−0.603656	−	0.797245i	$-0.706290\pi$
$761$	−1.46780e10	−1.20731	−0.603656	+	0.797245i	$0.706290\pi$
$762$	0	0
$763$	2.58973e9	0.211066
$764$	3.64803e10	2.95959
$765$	0	0
$766$	−1.80356e10	−1.44988
$767$	2.64854e9	0.211945
$768$	0	0
$769$	−2.45703e10	−1.94835	−0.974177	−	0.225784i	$-0.927506\pi$
$769$	−2.45703e10	−1.94835	−0.974177	+	0.225784i	$0.927506\pi$
$770$	−6.37972e8	−0.0503598
$771$	0	0
$772$	1.38252e10	1.08146
$773$	1.40761e10	1.09611	0.548057	−	0.836441i	$-0.315368\pi$
$773$	1.40761e10	1.09611	0.548057	+	0.836441i	$0.315368\pi$
$774$	0	0
$775$	−1.25997e10	−0.972312
$776$	−1.61135e10	−1.23786
$777$	0	0
$778$	−2.57991e9	−0.196416
$779$	9.14792e9	0.693332
$780$	0	0
$781$	−1.35471e10	−1.01758
$782$	3.16392e10	2.36593
$783$	0	0
$784$	−1.23464e10	−0.915029
$785$	2.96629e9	0.218862
$786$	0	0
$787$	7.50211e9	0.548620	0.274310	−	0.961641i	$-0.411551\pi$
$787$	7.50211e9	0.548620	0.274310	+	0.961641i	$0.411551\pi$
$788$	−3.93376e10	−2.86395
$789$	0	0
$790$	4.26382e9	0.307683
$791$	−2.20741e9	−0.158586
$792$	0	0
$793$	−2.83444e10	−2.01842
$794$	−6.73889e9	−0.477768
$795$	0	0
$796$	−5.14177e9	−0.361341
$797$	−1.68153e10	−1.17652	−0.588260	−	0.808672i	$-0.700187\pi$
$797$	−1.68153e10	−1.17652	−0.588260	+	0.808672i	$0.700187\pi$
$798$	0	0
$799$	−2.02404e10	−1.40380
$800$	1.10418e9	0.0762472
$801$	0	0
$802$	−2.94467e10	−2.01570
$803$	2.79739e10	1.90655
$804$	0	0
$805$	3.76282e8	0.0254231
$806$	−4.19187e10	−2.81991
$807$	0	0
$808$	4.34340e9	0.289661
$809$	2.50538e10	1.66362	0.831810	−	0.555060i	$-0.187305\pi$
$809$	2.50538e10	1.66362	0.831810	+	0.555060i	$0.187305\pi$
$810$	0	0
$811$	−7.70837e9	−0.507446	−0.253723	−	0.967277i	$-0.581655\pi$
$811$	−7.70837e9	−0.507446	−0.253723	+	0.967277i	$0.581655\pi$
$812$	−8.98888e8	−0.0589195
$813$	0	0
$814$	1.15029e10	0.747522
$815$	4.73730e9	0.306534
$816$	0	0
$817$	1.65661e10	1.06278
$818$	−1.64760e10	−1.05249
$819$	0	0
$820$	−6.53836e9	−0.414114
$821$	1.50406e10	0.948558	0.474279	−	0.880375i	$-0.342709\pi$
$821$	1.50406e10	0.948558	0.474279	+	0.880375i	$0.342709\pi$
$822$	0	0
$823$	3.00702e10	1.88034	0.940172	−	0.340700i	$-0.110664\pi$
$823$	3.00702e10	1.88034	0.940172	+	0.340700i	$0.110664\pi$
$824$	−1.65103e10	−1.02804
$825$	0	0
$826$	4.80763e8	0.0296825
$827$	−1.24512e10	−0.765494	−0.382747	−	0.923853i	$-0.625022\pi$
$827$	−1.24512e10	−0.765494	−0.382747	+	0.923853i	$0.625022\pi$
$828$	0	0
$829$	1.67317e10	1.02000	0.509999	−	0.860175i	$-0.329646\pi$
$829$	1.67317e10	1.02000	0.509999	+	0.860175i	$0.329646\pi$
$830$	−1.66822e8	−0.0101270
$831$	0	0
$832$	2.88599e10	1.73726
$833$	2.07318e10	1.24274
$834$	0	0
$835$	8.32593e8	0.0494914
$836$	−2.44162e10	−1.44529
$837$	0	0
$838$	2.34130e10	1.37437
$839$	−1.49303e10	−0.872773	−0.436386	−	0.899759i	$-0.643742\pi$
$839$	−1.49303e10	−0.872773	−0.436386	+	0.899759i	$0.643742\pi$
$840$	0	0
$841$	−1.63724e10	−0.949131
$842$	1.36520e10	0.788140
$843$	0	0
$844$	−3.39615e10	−1.94441
$845$	5.13645e9	0.292863
$846$	0	0
$847$	−1.28353e9	−0.0725794
$848$	1.41957e10	0.799413
$849$	0	0
$850$	3.78428e10	2.11357
$851$	−6.78455e9	−0.377370
$852$	0	0
$853$	−6.25227e9	−0.344918	−0.172459	−	0.985017i	$-0.555171\pi$
$853$	−6.25227e9	−0.344918	−0.172459	+	0.985017i	$0.555171\pi$
$854$	−5.14508e9	−0.282676
$855$	0	0
$856$	−6.08342e10	−3.31505
$857$	−2.79248e10	−1.51550	−0.757751	−	0.652543i	$-0.773702\pi$
$857$	−2.79248e10	−1.51550	−0.757751	+	0.652543i	$0.773702\pi$
$858$	0	0
$859$	4.15604e9	0.223719	0.111860	−	0.993724i	$-0.464319\pi$
$859$	4.15604e9	0.223719	0.111860	+	0.993724i	$0.464319\pi$
$860$	−1.18404e10	−0.634778
$861$	0	0
$862$	−3.07511e10	−1.63525
$863$	−1.83670e10	−0.972750	−0.486375	−	0.873750i	$-0.661681\pi$
$863$	−1.83670e10	−0.972750	−0.486375	+	0.873750i	$0.661681\pi$
$864$	0	0
$865$	3.50908e9	0.184347
$866$	3.80219e10	1.98939
$867$	0	0
$868$	−5.05305e9	−0.262261
$869$	2.41966e10	1.25079
$870$	0	0
$871$	−1.46203e10	−0.749707
$872$	−5.27139e10	−2.69226
$873$	0	0
$874$	2.16854e10	1.09870
$875$	9.14756e8	0.0461612
$876$	0	0
$877$	−2.18727e10	−1.09497	−0.547487	−	0.836815i	$-0.684415\pi$
$877$	−2.18727e10	−1.09497	−0.547487	+	0.836815i	$0.684415\pi$
$878$	−2.46432e10	−1.22876
$879$	0	0
$880$	4.15845e9	0.205704
$881$	−2.12719e10	−1.04807	−0.524035	−	0.851697i	$-0.675574\pi$
$881$	−2.12719e10	−1.04807	−0.524035	+	0.851697i	$0.675574\pi$
$882$	0	0
$883$	2.46950e10	1.20711	0.603554	−	0.797322i	$-0.293751\pi$
$883$	2.46950e10	1.20711	0.603554	+	0.797322i	$0.293751\pi$
$884$	8.36086e10	4.07069
$885$	0	0
$886$	4.68878e10	2.26486
$887$	−2.50959e10	−1.20745	−0.603727	−	0.797191i	$-0.706318\pi$
$887$	−2.50959e10	−1.20745	−0.603727	+	0.797191i	$0.706318\pi$
$888$	0	0
$889$	−3.18479e9	−0.152028
$890$	−8.01731e9	−0.381209
$891$	0	0
$892$	4.00330e10	1.88861
$893$	−1.38727e10	−0.651902
$894$	0	0
$895$	4.84869e8	0.0226071
$896$	5.01466e9	0.232897
$897$	0	0
$898$	−1.50298e10	−0.692606
$899$	4.93274e9	0.226427
$900$	0	0
$901$	−2.38370e10	−1.08571
$902$	−5.58730e10	−2.53501
$903$	0	0
$904$	4.49317e10	2.02285
$905$	4.53210e9	0.203250
$906$	0	0
$907$	1.03741e10	0.461662	0.230831	−	0.972994i	$-0.425856\pi$
$907$	1.03741e10	0.461662	0.230831	+	0.972994i	$0.425856\pi$
$908$	−6.28284e10	−2.78519
$909$	0	0
$910$	1.49733e9	0.0658677
$911$	1.15863e10	0.507728	0.253864	−	0.967240i	$-0.418298\pi$
$911$	1.15863e10	0.507728	0.253864	+	0.967240i	$0.418298\pi$
$912$	0	0
$913$	−9.46693e8	−0.0411682
$914$	1.38134e10	0.598395
$915$	0	0
$916$	1.27434e9	0.0547836
$917$	4.29796e9	0.184064
$918$	0	0
$919$	−1.20996e10	−0.514240	−0.257120	−	0.966380i	$-0.582774\pi$
$919$	−1.20996e10	−0.514240	−0.257120	+	0.966380i	$0.582774\pi$
$920$	−7.65921e9	−0.324285
$921$	0	0
$922$	−3.61024e9	−0.151697
$923$	3.17953e10	1.33093
$924$	0	0
$925$	−8.11482e9	−0.337119
$926$	4.58488e10	1.89753
$927$	0	0
$928$	−4.32280e8	−0.0177561
$929$	2.16304e10	0.885134	0.442567	−	0.896735i	$-0.354068\pi$
$929$	2.16304e10	0.885134	0.442567	+	0.896735i	$0.354068\pi$
$930$	0	0
$931$	1.42095e10	0.577107
$932$	−7.00922e10	−2.83605
$933$	0	0
$934$	4.18855e10	1.68209
$935$	−6.98276e9	−0.279374
$936$	0	0
$937$	1.59669e9	0.0634063	0.0317031	−	0.999497i	$-0.489907\pi$
$937$	1.59669e9	0.0634063	0.0317031	+	0.999497i	$0.489907\pi$
$938$	−2.65387e9	−0.104995
$939$	0	0
$940$	9.91536e9	0.389369
$941$	−3.39260e10	−1.32730	−0.663650	−	0.748043i	$-0.730993\pi$
$941$	−3.39260e10	−1.32730	−0.663650	+	0.748043i	$0.730993\pi$
$942$	0	0
$943$	3.29544e10	1.27974
$944$	−3.13373e9	−0.121244
$945$	0	0
$946$	−1.01181e11	−3.88580
$947$	−2.74213e10	−1.04921	−0.524606	−	0.851345i	$-0.675787\pi$
$947$	−2.74213e10	−1.04921	−0.524606	+	0.851345i	$0.675787\pi$
$948$	0	0
$949$	−6.56553e10	−2.49366
$950$	2.59374e10	0.981507
$951$	0	0
$952$	7.49971e9	0.281719
$953$	1.02794e10	0.384720	0.192360	−	0.981324i	$-0.438386\pi$
$953$	1.02794e10	0.384720	0.192360	+	0.981324i	$0.438386\pi$
$954$	0	0
$955$	7.15074e9	0.265668
$956$	−9.62720e10	−3.56367
$957$	0	0
$958$	7.84988e9	0.288459
$959$	4.73068e9	0.173204
$960$	0	0
$961$	2.16493e8	0.00786887
$962$	−2.69976e10	−0.977715
$963$	0	0
$964$	1.70503e10	0.613004
$965$	2.70996e9	0.0970771
$966$	0	0
$967$	−2.47735e10	−0.881040	−0.440520	−	0.897743i	$-0.645206\pi$
$967$	−2.47735e10	−0.881040	−0.440520	+	0.897743i	$0.645206\pi$
$968$	2.61262e10	0.925789
$969$	0	0
$970$	−6.39163e9	−0.224859
$971$	−4.27396e10	−1.49818	−0.749088	−	0.662470i	$-0.769508\pi$
$971$	−4.27396e10	−1.49818	−0.749088	+	0.662470i	$0.769508\pi$
$972$	0	0
$973$	1.32851e9	0.0462349
$974$	6.43778e10	2.23244
$975$	0	0
$976$	3.35368e10	1.15464
$977$	−4.18609e8	−0.0143608	−0.00718039	−	0.999974i	$-0.502286\pi$
$977$	−4.18609e8	−0.0143608	−0.00718039	+	0.999974i	$0.502286\pi$
$978$	0	0
$979$	−4.54971e10	−1.54969
$980$	−1.01561e10	−0.344695
$981$	0	0
$982$	−3.11275e10	−1.04895
$983$	8.78167e9	0.294876	0.147438	−	0.989071i	$-0.452897\pi$
$983$	8.78167e9	0.294876	0.147438	+	0.989071i	$0.452897\pi$
$984$	0	0
$985$	−7.71081e9	−0.257083
$986$	−1.48153e10	−0.492198
$987$	0	0
$988$	5.73052e10	1.89036
$989$	5.96777e10	1.96167
$990$	0	0
$991$	3.19826e10	1.04389	0.521947	−	0.852978i	$-0.325206\pi$
$991$	3.19826e10	1.04389	0.521947	+	0.852978i	$0.325206\pi$
$992$	−2.43004e9	−0.0790356
$993$	0	0
$994$	5.77148e9	0.186395
$995$	−1.00787e9	−0.0324358
$996$	0	0
$997$	1.39615e10	0.446170	0.223085	−	0.974799i	$-0.428387\pi$
$997$	1.39615e10	0.446170	0.223085	+	0.974799i	$0.428387\pi$
$998$	−9.07985e10	−2.89149
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.8.a.e.1.17		18
3.2	odd	2		177.8.a.d.1.2	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.8.a.d.1.2	✓	18	3.2	odd	2
531.8.a.e.1.17		18	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:	\( N \)	\(=\)	\( 531 = 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	531.a (trivial)

\(f(q)\)	\(=\)	\(q+19.5204 q^{2} +253.046 q^{4} +49.6011 q^{5} -119.919 q^{7} +2440.94 q^{8} +O(q^{10})\)	\(q+19.5204 q^{2} +253.046 q^{4} +49.6011 q^{5} -119.919 q^{7} +2440.94 q^{8} +968.233 q^{10} +5494.59 q^{11} -12895.9 q^{13} -2340.86 q^{14} +15258.3 q^{16} -25621.3 q^{17} -17560.8 q^{19} +12551.3 q^{20} +107256. q^{22} -63260.9 q^{23} -75664.7 q^{25} -251733. q^{26} -30344.9 q^{28} +29622.4 q^{29} +166521. q^{31} -14593.0 q^{32} -500138. q^{34} -5948.10 q^{35} +107247. q^{37} -342794. q^{38} +121073. q^{40} -520929. q^{41} -943358. q^{43} +1.39038e6 q^{44} -1.23488e6 q^{46} +789984. q^{47} -809163. q^{49} -1.47701e6 q^{50} -3.26325e6 q^{52} +930361. q^{53} +272538. q^{55} -292714. q^{56} +578240. q^{58} -205379. q^{59} +2.19795e6 q^{61} +3.25055e6 q^{62} -2.23792e6 q^{64} -639650. q^{65} +1.13372e6 q^{67} -6.48336e6 q^{68} -116109. q^{70} -2.46554e6 q^{71} +5.09118e6 q^{73} +2.09351e6 q^{74} -4.44368e6 q^{76} -658903. q^{77} +4.40371e6 q^{79} +756827. q^{80} -1.01687e7 q^{82} -172296. q^{83} -1.27084e6 q^{85} -1.84147e7 q^{86} +1.34120e7 q^{88} -8.28035e6 q^{89} +1.54646e6 q^{91} -1.60079e7 q^{92} +1.54208e7 q^{94} -871035. q^{95} -6.60133e6 q^{97} -1.57952e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 24 q^{2} + 1358 q^{4} - 678 q^{5} + 3081 q^{7} - 4107 q^{8}+O(q^{10}) \) 18 * q - 24 * q^2 + 1358 * q^4 - 678 * q^5 + 3081 * q^7 - 4107 * q^8	\( 18 q - 24 q^{2} + 1358 q^{4} - 678 q^{5} + 3081 q^{7} - 4107 q^{8} + 3609 q^{10} - 15070 q^{11} + 13662 q^{13} - 20861 q^{14} + 60482 q^{16} - 71919 q^{17} + 56231 q^{19} - 143053 q^{20} + 274198 q^{22} - 150029 q^{23} + 399672 q^{25} - 182846 q^{26} + 434150 q^{28} - 591285 q^{29} + 426733 q^{31} - 1205630 q^{32} + 403548 q^{34} - 912879 q^{35} + 7703 q^{37} + 417859 q^{38} + 618020 q^{40} - 770959 q^{41} + 793050 q^{43} - 2591274 q^{44} - 4068019 q^{46} - 1410373 q^{47} + 1637427 q^{49} - 1021549 q^{50} - 3749190 q^{52} - 1037934 q^{53} + 331974 q^{55} + 391748 q^{56} + 653724 q^{58} - 3696822 q^{59} - 1374623 q^{61} - 5251718 q^{62} + 5077197 q^{64} - 3257170 q^{65} - 2436904 q^{67} - 14119909 q^{68} + 5185580 q^{70} - 14289172 q^{71} + 5482515 q^{73} - 14934154 q^{74} + 3822912 q^{76} - 23157109 q^{77} + 19786414 q^{79} - 31978143 q^{80} + 9749509 q^{82} - 30227337 q^{83} + 9946981 q^{85} - 44295864 q^{86} + 39970897 q^{88} - 31061677 q^{89} + 26377785 q^{91} - 4719698 q^{92} + 44488296 q^{94} - 15534599 q^{95} + 12084118 q^{97} - 42274744 q^{98}+O(q^{100}) \) 18 * q - 24 * q^2 + 1358 * q^4 - 678 * q^5 + 3081 * q^7 - 4107 * q^8 + 3609 * q^10 - 15070 * q^11 + 13662 * q^13 - 20861 * q^14 + 60482 * q^16 - 71919 * q^17 + 56231 * q^19 - 143053 * q^20 + 274198 * q^22 - 150029 * q^23 + 399672 * q^25 - 182846 * q^26 + 434150 * q^28 - 591285 * q^29 + 426733 * q^31 - 1205630 * q^32 + 403548 * q^34 - 912879 * q^35 + 7703 * q^37 + 417859 * q^38 + 618020 * q^40 - 770959 * q^41 + 793050 * q^43 - 2591274 * q^44 - 4068019 * q^46 - 1410373 * q^47 + 1637427 * q^49 - 1021549 * q^50 - 3749190 * q^52 - 1037934 * q^53 + 331974 * q^55 + 391748 * q^56 + 653724 * q^58 - 3696822 * q^59 - 1374623 * q^61 - 5251718 * q^62 + 5077197 * q^64 - 3257170 * q^65 - 2436904 * q^67 - 14119909 * q^68 + 5185580 * q^70 - 14289172 * q^71 + 5482515 * q^73 - 14934154 * q^74 + 3822912 * q^76 - 23157109 * q^77 + 19786414 * q^79 - 31978143 * q^80 + 9749509 * q^82 - 30227337 * q^83 + 9946981 * q^85 - 44295864 * q^86 + 39970897 * q^88 - 31061677 * q^89 + 26377785 * q^91 - 4719698 * q^92 + 44488296 * q^94 - 15534599 * q^95 + 12084118 * q^97 - 42274744 * q^98

Introduction

L-functions

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