LMFDB - Embedded newform 531.8.a.b.1.7

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:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 6 x^{15} - 1493 x^{14} + 8791 x^{13} + 890490 x^{12} - 5107725 x^{11} - 269092298 x^{10} + \cdots - 23\!\cdots\!32 $ x^16 - 6x^15 - 1493x^14 + 8791x^13 + 890490x^12 - 5107725x^11 - 269092298x^10 + 1488374176x^9 + 42885295136x^8 - 226132003872x^7 - 3353576629440x^6 + 16796366777600x^5 + 99470801612800x^4 - 494039551757568x^3 - 493048066650624x^2 + 3193975642099712*x - 2385018853548032
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{9}\cdot 3^{5} $
Twist minimal:	no (minimal twist has level 177)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-3.09726$ 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)

$f(q)$	$=$	$q-3.09726 q^{2} -118.407 q^{4} +156.435 q^{5} +11.3597 q^{7} +763.187 q^{8} +O(q^{10})$	\(q-3.09726 q^{2} -118.407 q^{4} +156.435 q^{5} +11.3597 q^{7} +763.187 q^{8} -484.520 q^{10} -4296.19 q^{11} +1741.86 q^{13} -35.1841 q^{14} +12792.3 q^{16} +4764.38 q^{17} +18278.8 q^{19} -18523.0 q^{20} +13306.4 q^{22} -82004.5 q^{23} -53653.2 q^{25} -5395.00 q^{26} -1345.07 q^{28} +157425. q^{29} +37778.8 q^{31} -137309. q^{32} -14756.5 q^{34} +1777.05 q^{35} +450442. q^{37} -56614.3 q^{38} +119389. q^{40} +437505. q^{41} -738856. q^{43} +508699. q^{44} +253989. q^{46} -869122. q^{47} -823414. q^{49} +166178. q^{50} -206248. q^{52} +991560. q^{53} -672074. q^{55} +8669.60 q^{56} -487588. q^{58} -205379. q^{59} -1.30896e6 q^{61} -117011. q^{62} -1.21213e6 q^{64} +272487. q^{65} +2.85617e6 q^{67} -564135. q^{68} -5504.01 q^{70} +4.56945e6 q^{71} +937053. q^{73} -1.39514e6 q^{74} -2.16434e6 q^{76} -48803.6 q^{77} +6.61566e6 q^{79} +2.00116e6 q^{80} -1.35507e6 q^{82} +3.65210e6 q^{83} +745314. q^{85} +2.28843e6 q^{86} -3.27880e6 q^{88} +5.71107e6 q^{89} +19787.0 q^{91} +9.70990e6 q^{92} +2.69190e6 q^{94} +2.85944e6 q^{95} -1.38687e7 q^{97} +2.55033e6 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 6 q^{2} + 974 q^{4} + 68 q^{5} - 2343 q^{7} - 819 q^{8}+O(q^{10}) $ 16 * q + 6 * q^2 + 974 * q^4 + 68 * q^5 - 2343 * q^7 - 819 * q^8	\( 16 q + 6 q^{2} + 974 q^{4} + 68 q^{5} - 2343 q^{7} - 819 q^{8} - 3479 q^{10} - 898 q^{11} - 8172 q^{13} + 13315 q^{14} + 3138 q^{16} + 44985 q^{17} - 40137 q^{19} - 130657 q^{20} + 109394 q^{22} + 2833 q^{23} + 285746 q^{25} + 129420 q^{26} + 112890 q^{28} - 144375 q^{29} - 141759 q^{31} + 36224 q^{32} - 341332 q^{34} + 78859 q^{35} - 297971 q^{37} - 329075 q^{38} - 203048 q^{40} - 659077 q^{41} - 1431608 q^{43} - 254916 q^{44} + 873113 q^{46} + 1574073 q^{47} + 1893545 q^{49} - 302533 q^{50} - 4972548 q^{52} - 587736 q^{53} - 4624036 q^{55} + 5798506 q^{56} - 6991380 q^{58} - 3286064 q^{59} - 6117131 q^{61} + 11570258 q^{62} - 19063011 q^{64} + 5335514 q^{65} - 16518710 q^{67} + 17284669 q^{68} - 39189486 q^{70} + 10882582 q^{71} - 21097441 q^{73} + 16717030 q^{74} - 40864952 q^{76} + 3404601 q^{77} - 3784458 q^{79} + 27466195 q^{80} - 24990117 q^{82} + 1951425 q^{83} - 23238675 q^{85} + 35910572 q^{86} - 27843055 q^{88} - 10499443 q^{89} + 699217 q^{91} + 20062766 q^{92} - 59358988 q^{94} + 29236333 q^{95} - 25158976 q^{97} - 2120460 q^{98}+O(q^{100}) \) 16 * q + 6 * q^2 + 974 * q^4 + 68 * q^5 - 2343 * q^7 - 819 * q^8 - 3479 * q^10 - 898 * q^11 - 8172 * q^13 + 13315 * q^14 + 3138 * q^16 + 44985 * q^17 - 40137 * q^19 - 130657 * q^20 + 109394 * q^22 + 2833 * q^23 + 285746 * q^25 + 129420 * q^26 + 112890 * q^28 - 144375 * q^29 - 141759 * q^31 + 36224 * q^32 - 341332 * q^34 + 78859 * q^35 - 297971 * q^37 - 329075 * q^38 - 203048 * q^40 - 659077 * q^41 - 1431608 * q^43 - 254916 * q^44 + 873113 * q^46 + 1574073 * q^47 + 1893545 * q^49 - 302533 * q^50 - 4972548 * q^52 - 587736 * q^53 - 4624036 * q^55 + 5798506 * q^56 - 6991380 * q^58 - 3286064 * q^59 - 6117131 * q^61 + 11570258 * q^62 - 19063011 * q^64 + 5335514 * q^65 - 16518710 * q^67 + 17284669 * q^68 - 39189486 * q^70 + 10882582 * q^71 - 21097441 * q^73 + 16717030 * q^74 - 40864952 * q^76 + 3404601 * q^77 - 3784458 * q^79 + 27466195 * q^80 - 24990117 * q^82 + 1951425 * q^83 - 23238675 * q^85 + 35910572 * q^86 - 27843055 * q^88 - 10499443 * q^89 + 699217 * q^91 + 20062766 * q^92 - 59358988 * q^94 + 29236333 * q^95 - 25158976 * q^97 - 2120460 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−3.09726	−0.273762	−0.136881	−	0.990587i	$-0.543708\pi$
$2$	−3.09726	−0.273762	−0.136881	+	0.990587i	$0.543708\pi$
$3$	0	0
$3$	0	0
$4$	−118.407	−0.925054
$5$	156.435	0.559678	0.279839	−	0.960047i	$-0.409719\pi$
$5$	156.435	0.559678	0.279839	+	0.960047i	$0.409719\pi$
$6$	0	0
$7$	11.3597	0.0125177	0.00625885	−	0.999980i	$-0.498008\pi$
$7$	11.3597	0.0125177	0.00625885	+	0.999980i	$0.498008\pi$
$8$	763.187	0.527007
$9$	0	0
$10$	−484.520	−0.153219
$11$	−4296.19	−0.973217	−0.486608	−	0.873620i	$-0.661766\pi$
$11$	−4296.19	−0.973217	−0.486608	+	0.873620i	$0.661766\pi$
$12$	0	0
$13$	1741.86	0.219893	0.109947	−	0.993938i	$-0.464932\pi$
$13$	1741.86	0.219893	0.109947	+	0.993938i	$0.464932\pi$
$14$	−35.1841	−0.00342687
$15$	0	0
$16$	12792.3	0.780780
$17$	4764.38	0.235199	0.117599	−	0.993061i	$-0.462480\pi$
$17$	4764.38	0.235199	0.117599	+	0.993061i	$0.462480\pi$
$18$	0	0
$19$	18278.8	0.611379	0.305689	−	0.952131i	$-0.401113\pi$
$19$	18278.8	0.611379	0.305689	+	0.952131i	$0.401113\pi$
$20$	−18523.0	−0.517732
$21$	0	0
$22$	13306.4	0.266430
$23$	−82004.5	−1.40537	−0.702684	−	0.711502i	$-0.748015\pi$
$23$	−82004.5	−1.40537	−0.702684	+	0.711502i	$0.748015\pi$
$24$	0	0
$25$	−53653.2	−0.686761
$26$	−5395.00	−0.0601984
$27$	0	0
$28$	−1345.07	−0.0115796
$29$	157425.	1.19862	0.599311	−	0.800517i	$-0.295441\pi$
$29$	157425.	1.19862	0.599311	+	0.800517i	$0.295441\pi$
$30$	0	0
$31$	37778.8	0.227762	0.113881	−	0.993494i	$-0.463672\pi$
$31$	37778.8	0.227762	0.113881	+	0.993494i	$0.463672\pi$
$32$	−137309.	−0.740755
$33$	0	0
$34$	−14756.5	−0.0643885
$35$	1777.05	0.00700588
$36$	0	0
$37$	450442.	1.46195	0.730975	−	0.682404i	$-0.239066\pi$
$37$	450442.	1.46195	0.730975	+	0.682404i	$0.239066\pi$
$38$	−56614.3	−0.167372
$39$	0	0
$40$	119389.	0.294954
$41$	437505.	0.991378	0.495689	−	0.868500i	$-0.334916\pi$
$41$	437505.	0.991378	0.495689	+	0.868500i	$0.334916\pi$
$42$	0	0
$43$	−738856.	−1.41716	−0.708582	−	0.705629i	$-0.750665\pi$
$43$	−738856.	−1.41716	−0.708582	+	0.705629i	$0.750665\pi$
$44$	508699.	0.900278
$45$	0	0
$46$	253989.	0.384736
$47$	−869122.	−1.22106	−0.610532	−	0.791992i	$-0.709044\pi$
$47$	−869122.	−1.22106	−0.610532	+	0.791992i	$0.709044\pi$
$48$	0	0
$49$	−823414.	−0.999843
$50$	166178.	0.188009
$51$	0	0
$52$	−206248.	−0.203413
$53$	991560.	0.914858	0.457429	−	0.889246i	$-0.348770\pi$
$53$	991560.	0.914858	0.457429	+	0.889246i	$0.348770\pi$
$54$	0	0
$55$	−672074.	−0.544688
$56$	8669.60	0.00659691
$57$	0	0
$58$	−487588.	−0.328137
$59$	−205379.	−0.130189
$59$	−205379.	−0.130189
$60$	0	0
$61$	−1.30896e6	−0.738368	−0.369184	−	0.929356i	$-0.620363\pi$
$61$	−1.30896e6	−0.738368	−0.369184	+	0.929356i	$0.620363\pi$
$62$	−117011.	−0.0623527
$63$	0	0
$64$	−1.21213e6	−0.577989
$65$	272487.	0.123069
$66$	0	0
$67$	2.85617e6	1.16017	0.580085	−	0.814556i	$-0.303019\pi$
$67$	2.85617e6	1.16017	0.580085	+	0.814556i	$0.303019\pi$
$68$	−564135.	−0.217572
$69$	0	0
$70$	−5504.01	−0.00191794
$71$	4.56945e6	1.51516	0.757582	−	0.652740i	$-0.226381\pi$
$71$	4.56945e6	1.51516	0.757582	+	0.652740i	$0.226381\pi$
$72$	0	0
$73$	937053.	0.281925	0.140963	−	0.990015i	$-0.454980\pi$
$73$	937053.	0.281925	0.140963	+	0.990015i	$0.454980\pi$
$74$	−1.39514e6	−0.400226
$75$	0	0
$76$	−2.16434e6	−0.565559
$77$	−48803.6	−0.0121824
$78$	0	0
$79$	6.61566e6	1.50966	0.754829	−	0.655922i	$-0.227720\pi$
$79$	6.61566e6	1.50966	0.754829	+	0.655922i	$0.227720\pi$
$80$	2.00116e6	0.436985
$81$	0	0
$82$	−1.35507e6	−0.271402
$83$	3.65210e6	0.701082	0.350541	−	0.936547i	$-0.385998\pi$
$83$	3.65210e6	0.701082	0.350541	+	0.936547i	$0.385998\pi$
$84$	0	0
$85$	745314.	0.131635
$86$	2.28843e6	0.387966
$87$	0	0
$88$	−3.27880e6	−0.512892
$89$	5.71107e6	0.858722	0.429361	−	0.903133i	$-0.358739\pi$
$89$	5.71107e6	0.858722	0.429361	+	0.903133i	$0.358739\pi$
$90$	0	0
$91$	19787.0	0.00275256
$92$	9.70990e6	1.30004
$93$	0	0
$94$	2.69190e6	0.334281
$95$	2.85944e6	0.342175
$96$	0	0
$97$	−1.38687e7	−1.54289	−0.771446	−	0.636295i	$-0.780466\pi$
$97$	−1.38687e7	−1.54289	−0.771446	+	0.636295i	$0.780466\pi$
$98$	2.55033e6	0.273719
$99$	0	0
$100$	6.35291e6	0.635291
$101$	−1.27095e7	−1.22745	−0.613727	−	0.789518i	$-0.710330\pi$
$101$	−1.27095e7	−1.22745	−0.613727	+	0.789518i	$0.710330\pi$
$102$	0	0
$103$	1.41595e7	1.27679	0.638394	−	0.769710i	$-0.279599\pi$
$103$	1.41595e7	1.27679	0.638394	+	0.769710i	$0.279599\pi$
$104$	1.32937e6	0.115885
$105$	0	0
$106$	−3.07112e6	−0.250453
$107$	7.44830e6	0.587779	0.293889	−	0.955839i	$-0.405050\pi$
$107$	7.44830e6	0.587779	0.293889	+	0.955839i	$0.405050\pi$
$108$	0	0
$109$	−2.45904e7	−1.81875	−0.909374	−	0.415980i	$-0.863439\pi$
$109$	−2.45904e7	−1.81875	−0.909374	+	0.415980i	$0.863439\pi$
$110$	2.08159e6	0.149115
$111$	0	0
$112$	145317.	0.00977357
$113$	−5.72325e6	−0.373137	−0.186569	−	0.982442i	$-0.559737\pi$
$113$	−5.72325e6	−0.373137	−0.186569	+	0.982442i	$0.559737\pi$
$114$	0	0
$115$	−1.28283e7	−0.786553
$116$	−1.86403e7	−1.10879
$117$	0	0
$118$	636113.	0.0356408
$119$	54122.0	0.00294415
$120$	0	0
$121$	−1.02989e6	−0.0528495
$122$	4.05420e6	0.202137
$123$	0	0
$124$	−4.47327e6	−0.210693
$125$	−2.06147e7	−0.944043
$126$	0	0
$127$	−1.29403e7	−0.560574	−0.280287	−	0.959916i	$-0.590430\pi$
$127$	−1.29403e7	−0.560574	−0.280287	+	0.959916i	$0.590430\pi$
$128$	2.13299e7	0.898986
$129$	0	0
$130$	−843965.	−0.0336917
$131$	4.66242e7	1.81201	0.906007	−	0.423263i	$-0.139115\pi$
$131$	4.66242e7	1.81201	0.906007	+	0.423263i	$0.139115\pi$
$132$	0	0
$133$	207642.	0.00765306
$134$	−8.84630e6	−0.317611
$135$	0	0
$136$	3.63611e6	0.123951
$137$	2.07610e7	0.689803	0.344902	−	0.938639i	$-0.387912\pi$
$137$	2.07610e7	0.689803	0.344902	+	0.938639i	$0.387912\pi$
$138$	0	0
$139$	−1.80615e6	−0.0570430	−0.0285215	−	0.999593i	$-0.509080\pi$
$139$	−1.80615e6	−0.0570430	−0.0285215	+	0.999593i	$0.509080\pi$
$140$	−210416.	−0.00648082
$141$	0	0
$142$	−1.41528e7	−0.414794
$143$	−7.48337e6	−0.214004
$144$	0	0
$145$	2.46268e7	0.670842
$146$	−2.90230e6	−0.0771805
$147$	0	0
$148$	−5.33354e7	−1.35238
$149$	−7.35804e7	−1.82226	−0.911130	−	0.412119i	$-0.864789\pi$
$149$	−7.35804e7	−1.82226	−0.911130	+	0.412119i	$0.864789\pi$
$150$	0	0
$151$	2.57500e6	0.0608636	0.0304318	−	0.999537i	$-0.490312\pi$
$151$	2.57500e6	0.0608636	0.0304318	+	0.999537i	$0.490312\pi$
$152$	1.39502e7	0.322201
$153$	0	0
$154$	151158.	0.00333509
$155$	5.90991e6	0.127474
$156$	0	0
$157$	−2.53317e7	−0.522414	−0.261207	−	0.965283i	$-0.584121\pi$
$157$	−2.53317e7	−0.522414	−0.261207	+	0.965283i	$0.584121\pi$
$158$	−2.04904e7	−0.413287
$159$	0	0
$160$	−2.14799e7	−0.414584
$161$	−931548.	−0.0175920
$162$	0	0
$163$	−1.12501e6	−0.0203470	−0.0101735	−	0.999948i	$-0.503238\pi$
$163$	−1.12501e6	−0.0203470	−0.0101735	+	0.999948i	$0.503238\pi$
$164$	−5.18036e7	−0.917078
$165$	0	0
$166$	−1.13115e7	−0.191930
$167$	5.17400e7	0.859645	0.429823	−	0.902913i	$-0.358576\pi$
$167$	5.17400e7	0.859645	0.429823	+	0.902913i	$0.358576\pi$
$168$	0	0
$169$	−5.97144e7	−0.951647
$170$	−2.30843e6	−0.0360368
$171$	0	0
$172$	8.74857e7	1.31095
$173$	5.17329e7	0.759636	0.379818	−	0.925061i	$-0.375987\pi$
$173$	5.17329e7	0.759636	0.379818	+	0.925061i	$0.375987\pi$
$174$	0	0
$175$	−609485.	−0.00859667
$176$	−5.49582e7	−0.759868
$177$	0	0
$178$	−1.76887e7	−0.235086
$179$	−4.40517e7	−0.574086	−0.287043	−	0.957918i	$-0.592672\pi$
$179$	−4.40517e7	−0.574086	−0.287043	+	0.957918i	$0.592672\pi$
$180$	0	0
$181$	8.76285e7	1.09843	0.549213	−	0.835683i	$-0.314928\pi$
$181$	8.76285e7	1.09843	0.549213	+	0.835683i	$0.314928\pi$
$182$	−61285.7	−0.000753545 0
$183$	0	0
$184$	−6.25848e7	−0.740638
$185$	7.04647e7	0.818221
$186$	0	0
$187$	−2.04687e7	−0.228899
$188$	1.02910e8	1.12955
$189$	0	0
$190$	−8.85644e6	−0.0936746
$191$	−1.82917e8	−1.89949	−0.949744	−	0.313028i	$-0.898656\pi$
$191$	−1.82917e8	−1.89949	−0.949744	+	0.313028i	$0.898656\pi$
$192$	0	0
$193$	−8.82731e7	−0.883848	−0.441924	−	0.897053i	$-0.645704\pi$
$193$	−8.82731e7	−0.883848	−0.441924	+	0.897053i	$0.645704\pi$
$194$	4.29551e7	0.422385
$195$	0	0
$196$	9.74979e7	0.924909
$197$	−1.70359e8	−1.58757	−0.793785	−	0.608198i	$-0.791893\pi$
$197$	−1.70359e8	−1.58757	−0.793785	+	0.608198i	$0.791893\pi$
$198$	0	0
$199$	9.62750e7	0.866020	0.433010	−	0.901389i	$-0.357451\pi$
$199$	9.62750e7	0.866020	0.433010	+	0.901389i	$0.357451\pi$
$200$	−4.09474e7	−0.361928
$201$	0	0
$202$	3.93648e7	0.336030
$203$	1.78831e6	0.0150040
$204$	0	0
$205$	6.84409e7	0.554852
$206$	−4.38558e7	−0.349536
$207$	0	0
$208$	2.22824e7	0.171688
$209$	−7.85293e7	−0.595004
$210$	0	0
$211$	−2.06758e8	−1.51521	−0.757605	−	0.652713i	$-0.773631\pi$
$211$	−2.06758e8	−1.51521	−0.757605	+	0.652713i	$0.773631\pi$
$212$	−1.17408e8	−0.846293
$213$	0	0
$214$	−2.30693e7	−0.160912
$215$	−1.15583e8	−0.793155
$216$	0	0
$217$	429156.	0.00285106
$218$	7.61629e7	0.497904
$219$	0	0
$220$	7.95782e7	0.503866
$221$	8.29888e6	0.0517186
$222$	0	0
$223$	−4.61080e7	−0.278426	−0.139213	−	0.990262i	$-0.544457\pi$
$223$	−4.61080e7	−0.278426	−0.139213	+	0.990262i	$0.544457\pi$
$224$	−1.55979e6	−0.00927255
$225$	0	0
$226$	1.77264e7	0.102151
$227$	−1.52563e8	−0.865683	−0.432841	−	0.901470i	$-0.642489\pi$
$227$	−1.52563e8	−0.865683	−0.432841	+	0.901470i	$0.642489\pi$
$228$	0	0
$229$	−2.88554e8	−1.58782	−0.793912	−	0.608033i	$-0.791959\pi$
$229$	−2.88554e8	−1.58782	−0.793912	+	0.608033i	$0.791959\pi$
$230$	3.97328e7	0.215328
$231$	0	0
$232$	1.20145e8	0.631682
$233$	−3.41845e7	−0.177045	−0.0885224	−	0.996074i	$-0.528214\pi$
$233$	−3.41845e7	−0.177045	−0.0885224	+	0.996074i	$0.528214\pi$
$234$	0	0
$235$	−1.35961e8	−0.683402
$236$	2.43183e7	0.120432
$237$	0	0
$238$	−167630.	−0.000805996 0
$239$	−1.22378e8	−0.579843	−0.289922	−	0.957050i	$-0.593629\pi$
$239$	−1.22378e8	−0.579843	−0.289922	+	0.957050i	$0.593629\pi$
$240$	0	0
$241$	−1.04747e8	−0.482040	−0.241020	−	0.970520i	$-0.577482\pi$
$241$	−1.04747e8	−0.482040	−0.241020	+	0.970520i	$0.577482\pi$
$242$	3.18983e6	0.0144682
$243$	0	0
$244$	1.54990e8	0.683030
$245$	−1.28811e8	−0.559590
$246$	0	0
$247$	3.18391e7	0.134438
$248$	2.88323e7	0.120032
$249$	0	0
$250$	6.38491e7	0.258443
$251$	−1.01531e8	−0.405268	−0.202634	−	0.979255i	$-0.564950\pi$
$251$	−1.01531e8	−0.405268	−0.202634	+	0.979255i	$0.564950\pi$
$252$	0	0
$253$	3.52307e8	1.36773
$254$	4.00797e7	0.153464
$255$	0	0
$256$	8.90886e7	0.331881
$257$	−2.43506e8	−0.894836	−0.447418	−	0.894325i	$-0.647656\pi$
$257$	−2.43506e8	−0.894836	−0.447418	+	0.894325i	$0.647656\pi$
$258$	0	0
$259$	5.11689e6	0.0183003
$260$	−3.22644e7	−0.113846
$261$	0	0
$262$	−1.44407e8	−0.496061
$263$	5.30485e8	1.79816	0.899079	−	0.437787i	$-0.144238\pi$
$263$	5.30485e8	1.79816	0.899079	+	0.437787i	$0.144238\pi$
$264$	0	0
$265$	1.55114e8	0.512026
$266$	−643123.	−0.00209512
$267$	0	0
$268$	−3.38190e8	−1.07322
$269$	1.66365e8	0.521110	0.260555	−	0.965459i	$-0.416094\pi$
$269$	1.66365e8	0.521110	0.260555	+	0.965459i	$0.416094\pi$
$270$	0	0
$271$	−5.31604e8	−1.62254	−0.811271	−	0.584670i	$-0.801224\pi$
$271$	−5.31604e8	−1.62254	−0.811271	+	0.584670i	$0.801224\pi$
$272$	6.09473e7	0.183638
$273$	0	0
$274$	−6.43022e7	−0.188842
$275$	2.30504e8	0.668367
$276$	0	0
$277$	−3.03386e8	−0.857661	−0.428831	−	0.903385i	$-0.641074\pi$
$277$	−3.03386e8	−0.857661	−0.428831	+	0.903385i	$0.641074\pi$
$278$	5.59412e6	0.0156162
$279$	0	0
$280$	1.35623e6	0.00369215
$281$	−6.31916e7	−0.169898	−0.0849489	−	0.996385i	$-0.527073\pi$
$281$	−6.31916e7	−0.169898	−0.0849489	+	0.996385i	$0.527073\pi$
$282$	0	0
$283$	−2.67112e8	−0.700552	−0.350276	−	0.936647i	$-0.613912\pi$
$283$	−2.67112e8	−0.700552	−0.350276	+	0.936647i	$0.613912\pi$
$284$	−5.41054e8	−1.40161
$285$	0	0
$286$	2.31780e7	0.0585861
$287$	4.96993e6	0.0124098
$288$	0	0
$289$	−3.87639e8	−0.944682
$290$	−7.62757e7	−0.183651
$291$	0	0
$292$	−1.10954e8	−0.260796
$293$	4.43690e8	1.03049	0.515244	−	0.857043i	$-0.327701\pi$
$293$	4.43690e8	1.03049	0.515244	+	0.857043i	$0.327701\pi$
$294$	0	0
$295$	−3.21284e7	−0.0728638
$296$	3.43771e8	0.770458
$297$	0	0
$298$	2.27898e8	0.498866
$299$	−1.42840e8	−0.309031
$300$	0	0
$301$	−8.39320e6	−0.0177396
$302$	−7.97544e6	−0.0166621
$303$	0	0
$304$	2.33828e8	0.477352
$305$	−2.04767e8	−0.413248
$306$	0	0
$307$	−4.86871e8	−0.960351	−0.480175	−	0.877173i	$-0.659427\pi$
$307$	−4.86871e8	−0.960351	−0.480175	+	0.877173i	$0.659427\pi$
$308$	5.77868e6	0.0112694
$309$	0	0
$310$	−1.83046e7	−0.0348974
$311$	5.91749e8	1.11552	0.557758	−	0.830003i	$-0.311662\pi$
$311$	5.91749e8	1.11552	0.557758	+	0.830003i	$0.311662\pi$
$312$	0	0
$313$	−6.83534e8	−1.25995	−0.629977	−	0.776614i	$-0.716936\pi$
$313$	−6.83534e8	−1.25995	−0.629977	+	0.776614i	$0.716936\pi$
$314$	7.84589e7	0.143017
$315$	0	0
$316$	−7.83340e8	−1.39651
$317$	3.17450e8	0.559716	0.279858	−	0.960041i	$-0.409713\pi$
$317$	3.17450e8	0.559716	0.279858	+	0.960041i	$0.409713\pi$
$318$	0	0
$319$	−6.76330e8	−1.16652
$320$	−1.89619e8	−0.323488
$321$	0	0
$322$	2.88525e6	0.00481602
$323$	8.70872e7	0.143796
$324$	0	0
$325$	−9.34563e7	−0.151014
$326$	3.48446e6	0.00557024
$327$	0	0
$328$	3.33898e8	0.522463
$329$	−9.87298e6	−0.0152849
$330$	0	0
$331$	1.88196e8	0.285242	0.142621	−	0.989777i	$-0.454447\pi$
$331$	1.88196e8	0.285242	0.142621	+	0.989777i	$0.454447\pi$
$332$	−4.32434e8	−0.648539
$333$	0	0
$334$	−1.60253e8	−0.235338
$335$	4.46804e8	0.649321
$336$	0	0
$337$	−7.47824e8	−1.06438	−0.532188	−	0.846626i	$-0.678630\pi$
$337$	−7.47824e8	−1.06438	−0.532188	+	0.846626i	$0.678630\pi$
$338$	1.84951e8	0.260525
$339$	0	0
$340$	−8.82504e7	−0.121770
$341$	−1.62305e8	−0.221662
$342$	0	0
$343$	−1.87090e7	−0.0250334
$344$	−5.63885e8	−0.746855
$345$	0	0
$346$	−1.60230e8	−0.207959
$347$	1.23827e8	0.159097	0.0795483	−	0.996831i	$-0.474652\pi$
$347$	1.23827e8	0.159097	0.0795483	+	0.996831i	$0.474652\pi$
$348$	0	0
$349$	−1.33403e9	−1.67987	−0.839935	−	0.542688i	$-0.817407\pi$
$349$	−1.33403e9	−1.67987	−0.839935	+	0.542688i	$0.817407\pi$
$350$	1.88774e6	0.00235344
$351$	0	0
$352$	5.89907e8	0.720915
$353$	−1.33244e9	−1.61226	−0.806132	−	0.591735i	$-0.798443\pi$
$353$	−1.33244e9	−1.61226	−0.806132	+	0.591735i	$0.798443\pi$
$354$	0	0
$355$	7.14820e8	0.848003
$356$	−6.76231e8	−0.794365
$357$	0	0
$358$	1.36440e8	0.157163
$359$	−7.56062e8	−0.862435	−0.431218	−	0.902248i	$-0.641916\pi$
$359$	−7.56062e8	−0.862435	−0.431218	+	0.902248i	$0.641916\pi$
$360$	0	0
$361$	−5.59757e8	−0.626216
$362$	−2.71409e8	−0.300707
$363$	0	0
$364$	−2.34292e6	−0.00254626
$365$	1.46588e8	0.157787
$366$	0	0
$367$	−1.10105e9	−1.16272	−0.581358	−	0.813648i	$-0.697479\pi$
$367$	−1.10105e9	−1.16272	−0.581358	+	0.813648i	$0.697479\pi$
$368$	−1.04903e9	−1.09728
$369$	0	0
$370$	−2.18248e8	−0.223998
$371$	1.12639e7	0.0114519
$372$	0	0
$373$	−1.72328e9	−1.71940	−0.859698	−	0.510803i	$-0.829348\pi$
$373$	−1.72328e9	−1.71940	−0.859698	+	0.510803i	$0.829348\pi$
$374$	6.33969e7	0.0626639
$375$	0	0
$376$	−6.63303e8	−0.643509
$377$	2.74213e8	0.263569
$378$	0	0
$379$	3.29484e8	0.310883	0.155441	−	0.987845i	$-0.450320\pi$
$379$	3.29484e8	0.310883	0.155441	+	0.987845i	$0.450320\pi$
$380$	−3.38578e8	−0.316531
$381$	0	0
$382$	5.66541e8	0.520008
$383$	6.36350e8	0.578762	0.289381	−	0.957214i	$-0.406551\pi$
$383$	6.36350e8	0.578762	0.289381	+	0.957214i	$0.406551\pi$
$384$	0	0
$385$	−7.63457e6	−0.00681824
$386$	2.73405e8	0.241964
$387$	0	0
$388$	1.64215e9	1.42726
$389$	9.45771e8	0.814634	0.407317	−	0.913287i	$-0.366464\pi$
$389$	9.45771e8	0.814634	0.407317	+	0.913287i	$0.366464\pi$
$390$	0	0
$391$	−3.90700e8	−0.330541
$392$	−6.28419e8	−0.526924
$393$	0	0
$394$	5.27647e8	0.434617
$395$	1.03492e9	0.844922
$396$	0	0
$397$	1.20371e9	0.965506	0.482753	−	0.875756i	$-0.339637\pi$
$397$	1.20371e9	0.965506	0.482753	+	0.875756i	$0.339637\pi$
$398$	−2.98189e8	−0.237083
$399$	0	0
$400$	−6.86348e8	−0.536209
$401$	−1.24141e9	−0.961409	−0.480705	−	0.876883i	$-0.659619\pi$
$401$	−1.24141e9	−0.961409	−0.480705	+	0.876883i	$0.659619\pi$
$402$	0	0
$403$	6.58053e7	0.0500834
$404$	1.50490e9	1.13546
$405$	0	0
$406$	−5.53887e6	−0.00410752
$407$	−1.93518e9	−1.42279
$408$	0	0
$409$	−1.31447e9	−0.949990	−0.474995	−	0.879988i	$-0.657550\pi$
$409$	−1.31447e9	−0.949990	−0.474995	+	0.879988i	$0.657550\pi$
$410$	−2.11980e8	−0.151897
$411$	0	0
$412$	−1.67659e9	−1.18110
$413$	−2.33305e6	−0.00162967
$414$	0	0
$415$	5.71315e8	0.392380
$416$	−2.39173e8	−0.162887
$417$	0	0
$418$	2.43226e8	0.162890
$419$	1.30216e9	0.864796	0.432398	−	0.901683i	$-0.357667\pi$
$419$	1.30216e9	0.864796	0.432398	+	0.901683i	$0.357667\pi$
$420$	0	0
$421$	2.34962e9	1.53465	0.767327	−	0.641256i	$-0.221587\pi$
$421$	2.34962e9	1.53465	0.767327	+	0.641256i	$0.221587\pi$
$422$	6.40383e8	0.414807
$423$	0	0
$424$	7.56746e8	0.482136
$425$	−2.55624e8	−0.161525
$426$	0	0
$427$	−1.48694e7	−0.00924267
$428$	−8.81930e8	−0.543727
$429$	0	0
$430$	3.57990e8	0.217136
$431$	1.37817e9	0.829147	0.414573	−	0.910016i	$-0.363931\pi$
$431$	1.37817e9	0.829147	0.414573	+	0.910016i	$0.363931\pi$
$432$	0	0
$433$	−1.93157e9	−1.14341	−0.571705	−	0.820459i	$-0.693718\pi$
$433$	−1.93157e9	−1.14341	−0.571705	+	0.820459i	$0.693718\pi$
$434$	−1.32921e6	−0.000780512 0
$435$	0	0
$436$	2.91167e9	1.68244
$437$	−1.49894e9	−0.859212
$438$	0	0
$439$	2.84522e9	1.60505	0.802526	−	0.596617i	$-0.203489\pi$
$439$	2.84522e9	1.60505	0.802526	+	0.596617i	$0.203489\pi$
$440$	−5.12918e8	−0.287054
$441$	0	0
$442$	−2.57038e7	−0.0141586
$443$	−3.69126e8	−0.201726	−0.100863	−	0.994900i	$-0.532160\pi$
$443$	−3.69126e8	−0.201726	−0.100863	+	0.994900i	$0.532160\pi$
$444$	0	0
$445$	8.93410e8	0.480608
$446$	1.42809e8	0.0762224
$447$	0	0
$448$	−1.37695e7	−0.00723510
$449$	3.21750e8	0.167748	0.0838739	−	0.996476i	$-0.473271\pi$
$449$	3.21750e8	0.167748	0.0838739	+	0.996476i	$0.473271\pi$
$450$	0	0
$451$	−1.87961e9	−0.964825
$452$	6.77673e8	0.345172
$453$	0	0
$454$	4.72528e8	0.236991
$455$	3.09538e6	0.00154054
$456$	0	0
$457$	3.67556e9	1.80143	0.900714	−	0.434413i	$-0.143044\pi$
$457$	3.67556e9	1.80143	0.900714	+	0.434413i	$0.143044\pi$
$458$	8.93727e8	0.434686
$459$	0	0
$460$	1.51897e9	0.727604
$461$	8.20164e8	0.389895	0.194947	−	0.980814i	$-0.437546\pi$
$461$	8.20164e8	0.389895	0.194947	+	0.980814i	$0.437546\pi$
$462$	0	0
$463$	−1.84327e9	−0.863089	−0.431544	−	0.902092i	$-0.642031\pi$
$463$	−1.84327e9	−0.863089	−0.431544	+	0.902092i	$0.642031\pi$
$464$	2.01383e9	0.935859
$465$	0	0
$466$	1.05878e8	0.0484682
$467$	−7.89693e8	−0.358797	−0.179399	−	0.983776i	$-0.557415\pi$
$467$	−7.89693e8	−0.358797	−0.179399	+	0.983776i	$0.557415\pi$
$468$	0	0
$469$	3.24453e7	0.0145227
$470$	4.21107e8	0.187090
$471$	0	0
$472$	−1.56743e8	−0.0686104
$473$	3.17427e9	1.37921
$474$	0	0
$475$	−9.80717e8	−0.419871
$476$	−6.40842e6	−0.00272350
$477$	0	0
$478$	3.79037e8	0.158739
$479$	−3.05692e9	−1.27089	−0.635447	−	0.772144i	$-0.719184\pi$
$479$	−3.05692e9	−1.27089	−0.635447	+	0.772144i	$0.719184\pi$
$480$	0	0
$481$	7.84606e8	0.321473
$482$	3.24430e8	0.131964
$483$	0	0
$484$	1.21946e8	0.0488887
$485$	−2.16955e9	−0.863522
$486$	0	0
$487$	2.29199e9	0.899209	0.449605	−	0.893228i	$-0.351565\pi$
$487$	2.29199e9	0.899209	0.449605	+	0.893228i	$0.351565\pi$
$488$	−9.98983e8	−0.389125
$489$	0	0
$490$	3.98960e8	0.153195
$491$	2.91988e9	1.11322	0.556608	−	0.830775i	$-0.312103\pi$
$491$	2.91988e9	1.11322	0.556608	+	0.830775i	$0.312103\pi$
$492$	0	0
$493$	7.50034e8	0.281914
$494$	−9.86142e7	−0.0368040
$495$	0	0
$496$	4.83277e8	0.177832
$497$	5.19077e7	0.0189664
$498$	0	0
$499$	2.09323e9	0.754163	0.377082	−	0.926180i	$-0.376928\pi$
$499$	2.09323e9	0.754163	0.377082	+	0.926180i	$0.376928\pi$
$500$	2.44092e9	0.873291
$501$	0	0
$502$	3.14469e8	0.110947
$503$	−2.06891e9	−0.724860	−0.362430	−	0.932011i	$-0.618053\pi$
$503$	−2.06891e9	−0.724860	−0.362430	+	0.932011i	$0.618053\pi$
$504$	0	0
$505$	−1.98821e9	−0.686979
$506$	−1.09119e9	−0.374432
$507$	0	0
$508$	1.53223e9	0.518561
$509$	3.44598e9	1.15824	0.579122	−	0.815241i	$-0.303395\pi$
$509$	3.44598e9	1.15824	0.579122	+	0.815241i	$0.303395\pi$
$510$	0	0
$511$	1.06447e7	0.00352906
$512$	−3.00615e9	−0.989843
$513$	0	0
$514$	7.54202e8	0.244972
$515$	2.21504e9	0.714589
$516$	0	0
$517$	3.73392e9	1.18836
$518$	−1.58484e7	−0.00500992
$519$	0	0
$520$	2.07959e8	0.0648583
$521$	−3.18767e9	−0.987509	−0.493755	−	0.869601i	$-0.664376\pi$
$521$	−3.18767e9	−0.987509	−0.493755	+	0.869601i	$0.664376\pi$
$522$	0	0
$523$	4.05354e9	1.23902	0.619510	−	0.784988i	$-0.287331\pi$
$523$	4.05354e9	1.23902	0.619510	+	0.784988i	$0.287331\pi$
$524$	−5.52062e9	−1.67621
$525$	0	0
$526$	−1.64305e9	−0.492267
$527$	1.79992e8	0.0535694
$528$	0	0
$529$	3.31991e9	0.975059
$530$	−4.80430e8	−0.140173
$531$	0	0
$532$	−2.45863e7	−0.00707950
$533$	7.62072e8	0.217997
$534$	0	0
$535$	1.16517e9	0.328967
$536$	2.17979e9	0.611418
$537$	0	0
$538$	−5.15277e8	−0.142660
$539$	3.53755e9	0.973064
$540$	0	0
$541$	−1.94039e9	−0.526865	−0.263432	−	0.964678i	$-0.584855\pi$
$541$	−1.94039e9	−0.526865	−0.263432	+	0.964678i	$0.584855\pi$
$542$	1.64652e9	0.444190
$543$	0	0
$544$	−6.54192e8	−0.174225
$545$	−3.84679e9	−1.01791
$546$	0	0
$547$	3.90843e9	1.02105	0.510525	−	0.859863i	$-0.329451\pi$
$547$	3.90843e9	1.02105	0.510525	+	0.859863i	$0.329451\pi$
$548$	−2.45824e9	−0.638106
$549$	0	0
$550$	−7.13933e8	−0.182974
$551$	2.87755e9	0.732812
$552$	0	0
$553$	7.51521e7	0.0188974
$554$	9.39665e8	0.234795
$555$	0	0
$556$	2.13861e8	0.0527678
$557$	1.37471e9	0.337069	0.168534	−	0.985696i	$-0.446097\pi$
$557$	1.37471e9	0.337069	0.168534	+	0.985696i	$0.446097\pi$
$558$	0	0
$559$	−1.28698e9	−0.311625
$560$	2.27326e7	0.00547005
$561$	0	0
$562$	1.95721e8	0.0465115
$563$	7.49339e9	1.76970	0.884849	−	0.465879i	$-0.154262\pi$
$563$	7.49339e9	1.76970	0.884849	+	0.465879i	$0.154262\pi$
$564$	0	0
$565$	−8.95316e8	−0.208837
$566$	8.27315e8	0.191785
$567$	0	0
$568$	3.48734e9	0.798501
$569$	−4.64326e9	−1.05665	−0.528324	−	0.849043i	$-0.677179\pi$
$569$	−4.64326e9	−1.05665	−0.528324	+	0.849043i	$0.677179\pi$
$570$	0	0
$571$	−7.07238e9	−1.58979	−0.794894	−	0.606748i	$-0.792474\pi$
$571$	−7.07238e9	−1.58979	−0.794894	+	0.606748i	$0.792474\pi$
$572$	8.86083e8	0.197965
$573$	0	0
$574$	−1.53932e7	−0.00339733
$575$	4.39980e9	0.965152
$576$	0	0
$577$	7.54152e8	0.163434	0.0817172	−	0.996656i	$-0.473960\pi$
$577$	7.54152e8	0.163434	0.0817172	+	0.996656i	$0.473960\pi$
$578$	1.20062e9	0.258618
$579$	0	0
$580$	−2.91599e9	−0.620565
$581$	4.14868e7	0.00877594
$582$	0	0
$583$	−4.25994e9	−0.890355
$584$	7.15147e8	0.148577
$585$	0	0
$586$	−1.37423e9	−0.282109
$587$	−4.19667e9	−0.856389	−0.428195	−	0.903687i	$-0.640850\pi$
$587$	−4.19667e9	−0.856389	−0.428195	+	0.903687i	$0.640850\pi$
$588$	0	0
$589$	6.90551e8	0.139249
$590$	9.95101e7	0.0199474
$591$	0	0
$592$	5.76218e9	1.14146
$593$	9.07288e8	0.178671	0.0893355	−	0.996002i	$-0.471526\pi$
$593$	9.07288e8	0.178671	0.0893355	+	0.996002i	$0.471526\pi$
$594$	0	0
$595$	8.46656e6	0.00164777
$596$	8.71244e9	1.68569
$597$	0	0
$598$	4.42414e8	0.0846009
$599$	7.00263e9	1.33127	0.665637	−	0.746276i	$-0.268160\pi$
$599$	7.00263e9	1.33127	0.665637	+	0.746276i	$0.268160\pi$
$600$	0	0
$601$	3.01689e7	0.00566890	0.00283445	−	0.999996i	$-0.499098\pi$
$601$	3.01689e7	0.00566890	0.00283445	+	0.999996i	$0.499098\pi$
$602$	2.59959e7	0.00485644
$603$	0	0
$604$	−3.04898e8	−0.0563021
$605$	−1.61110e8	−0.0295787
$606$	0	0
$607$	−1.46131e9	−0.265205	−0.132602	−	0.991169i	$-0.542333\pi$
$607$	−1.46131e9	−0.265205	−0.132602	+	0.991169i	$0.542333\pi$
$608$	−2.50985e9	−0.452882
$609$	0	0
$610$	6.34218e8	0.113132
$611$	−1.51389e9	−0.268503
$612$	0	0
$613$	−5.73275e9	−1.00520	−0.502599	−	0.864520i	$-0.667623\pi$
$613$	−5.73275e9	−1.00520	−0.502599	+	0.864520i	$0.667623\pi$
$614$	1.50797e9	0.262908
$615$	0	0
$616$	−3.72463e7	−0.00642023
$617$	−3.48905e9	−0.598010	−0.299005	−	0.954251i	$-0.596655\pi$
$617$	−3.48905e9	−0.598010	−0.299005	+	0.954251i	$0.596655\pi$
$618$	0	0
$619$	−5.02284e9	−0.851200	−0.425600	−	0.904911i	$-0.639937\pi$
$619$	−5.02284e9	−0.851200	−0.425600	+	0.904911i	$0.639937\pi$
$620$	−6.99775e8	−0.117920
$621$	0	0
$622$	−1.83280e9	−0.305386
$623$	6.48762e7	0.0107492
$624$	0	0
$625$	9.66804e8	0.158401
$626$	2.11708e9	0.344928
$627$	0	0
$628$	2.99945e9	0.483262
$629$	2.14607e9	0.343849
$630$	0	0
$631$	−7.21168e9	−1.14270	−0.571352	−	0.820705i	$-0.693581\pi$
$631$	−7.21168e9	−1.14270	−0.571352	+	0.820705i	$0.693581\pi$
$632$	5.04899e9	0.795600
$633$	0	0
$634$	−9.83226e8	−0.153229
$635$	−2.02432e9	−0.313741
$636$	0	0
$637$	−1.43427e9	−0.219859
$638$	2.09477e9	0.319348
$639$	0	0
$640$	3.33673e9	0.503143
$641$	−1.16522e10	−1.74746	−0.873728	−	0.486415i	$-0.838304\pi$
$641$	−1.16522e10	−1.74746	−0.873728	+	0.486415i	$0.838304\pi$
$642$	0	0
$643$	−1.68105e9	−0.249369	−0.124684	−	0.992196i	$-0.539792\pi$
$643$	−1.68105e9	−0.249369	−0.124684	+	0.992196i	$0.539792\pi$
$644$	1.10302e8	0.0162735
$645$	0	0
$646$	−2.69732e8	−0.0393658
$647$	−3.46626e9	−0.503148	−0.251574	−	0.967838i	$-0.580948\pi$
$647$	−3.46626e9	−0.503148	−0.251574	+	0.967838i	$0.580948\pi$
$648$	0	0
$649$	8.82348e8	0.126702
$650$	2.89459e8	0.0413419
$651$	0	0
$652$	1.33209e8	0.0188221
$653$	−8.92600e9	−1.25447	−0.627236	−	0.778830i	$-0.715814\pi$
$653$	−8.92600e9	−1.25447	−0.627236	+	0.778830i	$0.715814\pi$
$654$	0	0
$655$	7.29364e9	1.01414
$656$	5.59669e9	0.774048
$657$	0	0
$658$	3.05792e7	0.00418443
$659$	−1.71451e8	−0.0233368	−0.0116684	−	0.999932i	$-0.503714\pi$
$659$	−1.71451e8	−0.0233368	−0.0116684	+	0.999932i	$0.503714\pi$
$660$	0	0
$661$	1.10119e10	1.48305	0.741526	−	0.670924i	$-0.234102\pi$
$661$	1.10119e10	1.48305	0.741526	+	0.670924i	$0.234102\pi$
$662$	−5.82894e8	−0.0780884
$663$	0	0
$664$	2.78723e9	0.369475
$665$	3.24825e7	0.00428325
$666$	0	0
$667$	−1.29096e10	−1.68450
$668$	−6.12638e9	−0.795218
$669$	0	0
$670$	−1.38387e9	−0.177760
$671$	5.62356e9	0.718592
$672$	0	0
$673$	−5.18178e9	−0.655279	−0.327640	−	0.944803i	$-0.606253\pi$
$673$	−5.18178e9	−0.655279	−0.327640	+	0.944803i	$0.606253\pi$
$674$	2.31621e9	0.291386
$675$	0	0
$676$	7.07061e9	0.880325
$677$	8.09065e9	1.00213	0.501064	−	0.865410i	$-0.332942\pi$
$677$	8.09065e9	1.00213	0.501064	+	0.865410i	$0.332942\pi$
$678$	0	0
$679$	−1.57545e8	−0.0193135
$680$	5.68814e8	0.0693728
$681$	0	0
$682$	5.02701e8	0.0606827
$683$	−1.31475e9	−0.157896	−0.0789478	−	0.996879i	$-0.525156\pi$
$683$	−1.31475e9	−0.157896	−0.0789478	+	0.996879i	$0.525156\pi$
$684$	0	0
$685$	3.24773e9	0.386068
$686$	5.79466e7	0.00685321
$687$	0	0
$688$	−9.45166e9	−1.10649
$689$	1.72716e9	0.201171
$690$	0	0
$691$	9.51511e9	1.09709	0.548543	−	0.836123i	$-0.315183\pi$
$691$	9.51511e9	1.09709	0.548543	+	0.836123i	$0.315183\pi$
$692$	−6.12553e9	−0.702704
$693$	0	0
$694$	−3.83524e8	−0.0435546
$695$	−2.82545e8	−0.0319257
$696$	0	0
$697$	2.08444e9	0.233171
$698$	4.13183e9	0.459884
$699$	0	0
$700$	7.21673e7	0.00795238
$701$	1.01482e10	1.11270	0.556349	−	0.830949i	$-0.312202\pi$
$701$	1.01482e10	1.11270	0.556349	+	0.830949i	$0.312202\pi$
$702$	0	0
$703$	8.23354e9	0.893805
$704$	5.20755e9	0.562509
$705$	0	0
$706$	4.12692e9	0.441377
$707$	−1.44377e8	−0.0153649
$708$	0	0
$709$	1.71054e10	1.80248	0.901241	−	0.433319i	$-0.142658\pi$
$709$	1.71054e10	1.80248	0.901241	+	0.433319i	$0.142658\pi$
$710$	−2.21399e9	−0.232151
$711$	0	0
$712$	4.35862e9	0.452553
$713$	−3.09803e9	−0.320090
$714$	0	0
$715$	−1.17066e9	−0.119773
$716$	5.21602e9	0.531060
$717$	0	0
$718$	2.34172e9	0.236102
$719$	−4.37544e9	−0.439006	−0.219503	−	0.975612i	$-0.570444\pi$
$719$	−4.37544e9	−0.439006	−0.219503	+	0.975612i	$0.570444\pi$
$720$	0	0
$721$	1.60848e8	0.0159824
$722$	1.73371e9	0.171434
$723$	0	0
$724$	−1.03758e10	−1.01610
$725$	−8.44638e9	−0.823166
$726$	0	0
$727$	−1.22749e10	−1.18481	−0.592405	−	0.805641i	$-0.701821\pi$
$727$	−1.22749e10	−1.18481	−0.592405	+	0.805641i	$0.701821\pi$
$728$	1.51012e7	0.00145062
$729$	0	0
$730$	−4.54020e8	−0.0431962
$731$	−3.52019e9	−0.333315
$732$	0	0
$733$	−7.53477e9	−0.706653	−0.353327	−	0.935500i	$-0.614950\pi$
$733$	−7.53477e9	−0.706653	−0.353327	+	0.935500i	$0.614950\pi$
$734$	3.41023e9	0.318308
$735$	0	0
$736$	1.12600e10	1.04103
$737$	−1.22706e10	−1.12910
$738$	0	0
$739$	1.70023e10	1.54972	0.774858	−	0.632136i	$-0.217821\pi$
$739$	1.70023e10	1.54972	0.774858	+	0.632136i	$0.217821\pi$
$740$	−8.34351e9	−0.756899
$741$	0	0
$742$	−3.48871e7	−0.00313510
$743$	1.25008e10	1.11809	0.559045	−	0.829137i	$-0.311168\pi$
$743$	1.25008e10	1.11809	0.559045	+	0.829137i	$0.311168\pi$
$744$	0	0
$745$	−1.15105e10	−1.01988
$746$	5.33747e9	0.470705
$747$	0	0
$748$	2.42364e9	0.211744
$749$	8.46106e7	0.00735764
$750$	0	0
$751$	−6.82394e9	−0.587889	−0.293945	−	0.955822i	$-0.594968\pi$
$751$	−6.82394e9	−0.587889	−0.293945	+	0.955822i	$0.594968\pi$
$752$	−1.11181e10	−0.953382
$753$	0	0
$754$	−8.49311e8	−0.0721551
$755$	4.02819e8	0.0340640
$756$	0	0
$757$	1.76261e10	1.47680	0.738400	−	0.674363i	$-0.235582\pi$
$757$	1.76261e10	1.47680	0.738400	+	0.674363i	$0.235582\pi$
$758$	−1.02050e9	−0.0851079
$759$	0	0
$760$	2.18229e9	0.180329
$761$	8.48847e9	0.698205	0.349103	−	0.937084i	$-0.386486\pi$
$761$	8.48847e9	0.698205	0.349103	+	0.937084i	$0.386486\pi$
$762$	0	0
$763$	−2.79340e8	−0.0227665
$764$	2.16586e10	1.75713
$765$	0	0
$766$	−1.97094e9	−0.158443
$767$	−3.57741e8	−0.0286276
$768$	0	0
$769$	1.91757e10	1.52058	0.760288	−	0.649586i	$-0.225058\pi$
$769$	1.91757e10	1.52058	0.760288	+	0.649586i	$0.225058\pi$
$770$	2.36463e7	0.00186657
$771$	0	0
$772$	1.04521e10	0.817607
$773$	7.92188e8	0.0616879	0.0308440	−	0.999524i	$-0.490181\pi$
$773$	7.92188e8	0.0616879	0.0308440	+	0.999524i	$0.490181\pi$
$774$	0	0
$775$	−2.02695e9	−0.156418
$776$	−1.05844e10	−0.813114
$777$	0	0
$778$	−2.92930e9	−0.223016
$779$	7.99707e9	0.606108
$780$	0	0
$781$	−1.96312e10	−1.47458
$782$	1.21010e9	0.0904895
$783$	0	0
$784$	−1.05334e10	−0.780658
$785$	−3.96275e9	−0.292384
$786$	0	0
$787$	2.00728e10	1.46790	0.733950	−	0.679203i	$-0.237675\pi$
$787$	2.00728e10	1.46790	0.733950	+	0.679203i	$0.237675\pi$
$788$	2.01717e10	1.46859
$789$	0	0
$790$	−3.20542e9	−0.231307
$791$	−6.50146e7	−0.00467082
$792$	0	0
$793$	−2.28003e9	−0.162362
$794$	−3.72821e9	−0.264319
$795$	0	0
$796$	−1.13996e10	−0.801115
$797$	−1.65749e10	−1.15971	−0.579853	−	0.814721i	$-0.696890\pi$
$797$	−1.65749e10	−1.15971	−0.579853	+	0.814721i	$0.696890\pi$
$798$	0	0
$799$	−4.14082e9	−0.287192
$800$	7.36707e9	0.508721
$801$	0	0
$802$	3.84496e9	0.263197
$803$	−4.02576e9	−0.274374
$804$	0	0
$805$	−1.45726e8	−0.00984584
$806$	−2.03817e8	−0.0137109
$807$	0	0
$808$	−9.69977e9	−0.646877
$809$	7.11284e9	0.472305	0.236153	−	0.971716i	$-0.424113\pi$
$809$	7.11284e9	0.472305	0.236153	+	0.971716i	$0.424113\pi$
$810$	0	0
$811$	1.80923e9	0.119102	0.0595511	−	0.998225i	$-0.481033\pi$
$811$	1.80923e9	0.119102	0.0595511	+	0.998225i	$0.481033\pi$
$812$	−2.11748e8	−0.0138795
$813$	0	0
$814$	5.99378e9	0.389507
$815$	−1.75991e8	−0.0113878
$816$	0	0
$817$	−1.35054e10	−0.866424
$818$	4.07126e9	0.260071
$819$	0	0
$820$	−8.10388e9	−0.513268
$821$	−1.91315e10	−1.20655	−0.603277	−	0.797531i	$-0.706139\pi$
$821$	−1.91315e10	−1.20655	−0.603277	+	0.797531i	$0.706139\pi$
$822$	0	0
$823$	−1.55173e10	−0.970325	−0.485163	−	0.874424i	$-0.661240\pi$
$823$	−1.55173e10	−0.970325	−0.485163	+	0.874424i	$0.661240\pi$
$824$	1.08064e10	0.672876
$825$	0	0
$826$	7.22607e6	0.000446141 0
$827$	−2.67848e10	−1.64672	−0.823360	−	0.567519i	$-0.807903\pi$
$827$	−2.67848e10	−1.64672	−0.823360	+	0.567519i	$0.807903\pi$
$828$	0	0
$829$	1.50178e10	0.915515	0.457757	−	0.889077i	$-0.348653\pi$
$829$	1.50178e10	0.915515	0.457757	+	0.889077i	$0.348653\pi$
$830$	−1.76951e9	−0.107419
$831$	0	0
$832$	−2.11136e9	−0.127096
$833$	−3.92305e9	−0.235162
$834$	0	0
$835$	8.09394e9	0.481124
$836$	9.29842e9	0.550411
$837$	0	0
$838$	−4.03312e9	−0.236748
$839$	−1.78661e10	−1.04439	−0.522197	−	0.852825i	$-0.674887\pi$
$839$	−1.78661e10	−1.04439	−0.522197	+	0.852825i	$0.674887\pi$
$840$	0	0
$841$	7.53290e9	0.436693
$842$	−7.27739e9	−0.420130
$843$	0	0
$844$	2.44815e10	1.40165
$845$	−9.34141e9	−0.532616
$846$	0	0
$847$	−1.16992e7	−0.000661555 0
$848$	1.26843e10	0.714303
$849$	0	0
$850$	7.91735e8	0.0442195
$851$	−3.69382e10	−2.05458
$852$	0	0
$853$	−1.05414e10	−0.581538	−0.290769	−	0.956793i	$-0.593911\pi$
$853$	−1.05414e10	−0.581538	−0.290769	+	0.956793i	$0.593911\pi$
$854$	4.60546e7	0.00253029
$855$	0	0
$856$	5.68445e9	0.309763
$857$	1.68053e10	0.912037	0.456018	−	0.889970i	$-0.349275\pi$
$857$	1.68053e10	0.912037	0.456018	+	0.889970i	$0.349275\pi$
$858$	0	0
$859$	−3.46312e9	−0.186420	−0.0932099	−	0.995646i	$-0.529713\pi$
$859$	−3.46312e9	−0.186420	−0.0932099	+	0.995646i	$0.529713\pi$
$860$	1.36858e10	0.733712
$861$	0	0
$862$	−4.26855e9	−0.226989
$863$	−1.26642e10	−0.670716	−0.335358	−	0.942091i	$-0.608857\pi$
$863$	−1.26642e10	−0.670716	−0.335358	+	0.942091i	$0.608857\pi$
$864$	0	0
$865$	8.09282e9	0.425151
$866$	5.98257e9	0.313022
$867$	0	0
$868$	−5.08151e7	−0.00263739
$869$	−2.84222e10	−1.46922
$870$	0	0
$871$	4.97504e9	0.255113
$872$	−1.87671e10	−0.958492
$873$	0	0
$874$	4.64263e9	0.235220
$875$	−2.34177e8	−0.0118172
$876$	0	0
$877$	−1.35643e10	−0.679044	−0.339522	−	0.940598i	$-0.610265\pi$
$877$	−1.35643e10	−0.679044	−0.339522	+	0.940598i	$0.610265\pi$
$878$	−8.81238e9	−0.439403
$879$	0	0
$880$	−8.59737e9	−0.425281
$881$	−3.52328e10	−1.73593	−0.867963	−	0.496630i	$-0.834571\pi$
$881$	−3.52328e10	−1.73593	−0.867963	+	0.496630i	$0.834571\pi$
$882$	0	0
$883$	1.37573e10	0.672469	0.336234	−	0.941778i	$-0.390847\pi$
$883$	1.37573e10	0.672469	0.336234	+	0.941778i	$0.390847\pi$
$884$	−9.82645e8	−0.0478425
$885$	0	0
$886$	1.14328e9	0.0552248
$887$	2.09679e10	1.00884	0.504420	−	0.863459i	$-0.331706\pi$
$887$	2.09679e10	1.00884	0.504420	+	0.863459i	$0.331706\pi$
$888$	0	0
$889$	−1.46999e8	−0.00701709
$890$	−2.76713e9	−0.131572
$891$	0	0
$892$	5.45951e9	0.257559
$893$	−1.58865e10	−0.746532
$894$	0	0
$895$	−6.89121e9	−0.321303
$896$	2.42301e8	0.0112532
$897$	0	0
$898$	−9.96545e8	−0.0459230
$899$	5.94734e9	0.273001
$900$	0	0
$901$	4.72417e9	0.215173
$902$	5.82163e9	0.264133
$903$	0	0
$904$	−4.36792e9	−0.196646
$905$	1.37081e10	0.614764
$906$	0	0
$907$	−6.58180e9	−0.292900	−0.146450	−	0.989218i	$-0.546785\pi$
$907$	−6.58180e9	−0.292900	−0.146450	+	0.989218i	$0.546785\pi$
$908$	1.80645e10	0.800803
$909$	0	0
$910$	−9.58721e6	−0.000421743 0
$911$	−2.97498e10	−1.30368	−0.651839	−	0.758357i	$-0.726002\pi$
$911$	−2.97498e10	−1.30368	−0.651839	+	0.758357i	$0.726002\pi$
$912$	0	0
$913$	−1.56901e10	−0.682305
$914$	−1.13842e10	−0.493163
$915$	0	0
$916$	3.41668e10	1.46882
$917$	5.29638e8	0.0226822
$918$	0	0
$919$	−9.95637e9	−0.423152	−0.211576	−	0.977361i	$-0.567860\pi$
$919$	−9.95637e9	−0.423152	−0.211576	+	0.977361i	$0.567860\pi$
$920$	−9.79043e9	−0.414519
$921$	0	0
$922$	−2.54027e9	−0.106738
$923$	7.95934e9	0.333174
$924$	0	0
$925$	−2.41676e10	−1.00401
$926$	5.70909e9	0.236281
$927$	0	0
$928$	−2.16160e10	−0.887884
$929$	6.90890e9	0.282718	0.141359	−	0.989958i	$-0.454853\pi$
$929$	6.90890e9	0.282718	0.141359	+	0.989958i	$0.454853\pi$
$930$	0	0
$931$	−1.50510e10	−0.611283
$932$	4.04768e9	0.163776
$933$	0	0
$934$	2.44589e9	0.0982251
$935$	−3.20201e9	−0.128110
$936$	0	0
$937$	−6.04161e9	−0.239919	−0.119959	−	0.992779i	$-0.538276\pi$
$937$	−6.04161e9	−0.239919	−0.119959	+	0.992779i	$0.538276\pi$
$938$	−1.00492e8	−0.00397575
$939$	0	0
$940$	1.60987e10	0.632184
$941$	1.95376e10	0.764376	0.382188	−	0.924085i	$-0.375171\pi$
$941$	1.95376e10	0.764376	0.382188	+	0.924085i	$0.375171\pi$
$942$	0	0
$943$	−3.58773e10	−1.39325
$944$	−2.62727e9	−0.101649
$945$	0	0
$946$	−9.83154e9	−0.377575
$947$	2.08163e10	0.796486	0.398243	−	0.917280i	$-0.369620\pi$
$947$	2.08163e10	0.796486	0.398243	+	0.917280i	$0.369620\pi$
$948$	0	0
$949$	1.63222e9	0.0619934
$950$	3.03754e9	0.114945
$951$	0	0
$952$	4.13052e7	0.00155159
$953$	2.45125e10	0.917409	0.458705	−	0.888589i	$-0.348314\pi$
$953$	2.45125e10	0.917409	0.458705	+	0.888589i	$0.348314\pi$
$954$	0	0
$955$	−2.86145e10	−1.06310
$956$	1.44904e10	0.536387
$957$	0	0
$958$	9.46808e9	0.347923
$959$	2.35839e8	0.00863475
$960$	0	0
$961$	−2.60854e10	−0.948124
$962$	−2.43013e9	−0.0880070
$963$	0	0
$964$	1.24028e10	0.445913
$965$	−1.38090e10	−0.494670
$966$	0	0
$967$	−9.86699e9	−0.350907	−0.175454	−	0.984488i	$-0.556139\pi$
$967$	−9.86699e9	−0.350907	−0.175454	+	0.984488i	$0.556139\pi$
$968$	−7.85997e8	−0.0278521
$969$	0	0
$970$	6.71967e9	0.236400
$971$	−1.19828e10	−0.420039	−0.210020	−	0.977697i	$-0.567353\pi$
$971$	−1.19828e10	−0.420039	−0.210020	+	0.977697i	$0.567353\pi$
$972$	0	0
$973$	−2.05174e7	−0.000714047 0
$974$	−7.09889e9	−0.246169
$975$	0	0
$976$	−1.67446e10	−0.576503
$977$	−7.54352e9	−0.258787	−0.129394	−	0.991593i	$-0.541303\pi$
$977$	−7.54352e9	−0.258787	−0.129394	+	0.991593i	$0.541303\pi$
$978$	0	0
$979$	−2.45359e10	−0.835723
$980$	1.52521e10	0.517651
$981$	0	0
$982$	−9.04363e9	−0.304756
$983$	−2.03133e10	−0.682091	−0.341045	−	0.940047i	$-0.610781\pi$
$983$	−2.03133e10	−0.682091	−0.341045	+	0.940047i	$0.610781\pi$
$984$	0	0
$985$	−2.66500e10	−0.888528
$986$	−2.32305e9	−0.0771774
$987$	0	0
$988$	−3.76998e9	−0.124362
$989$	6.05895e10	1.99164
$990$	0	0
$991$	−3.47760e10	−1.13507	−0.567534	−	0.823350i	$-0.692103\pi$
$991$	−3.47760e10	−1.13507	−0.567534	+	0.823350i	$0.692103\pi$
$992$	−5.18737e9	−0.168716
$993$	0	0
$994$	−1.60772e8	−0.00519227
$995$	1.50608e10	0.484692
$996$	0	0
$997$	−1.31923e10	−0.421586	−0.210793	−	0.977531i	$-0.567605\pi$
$997$	−1.31923e10	−0.421586	−0.210793	+	0.977531i	$0.567605\pi$
$998$	−6.48329e9	−0.206461
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.8.a.b.1.7		16
3.2	odd	2		177.8.a.a.1.10	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.8.a.a.1.10	✓	16	3.2	odd	2
531.8.a.b.1.7		16	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)

\(f(q)\)	\(=\)	\(q-3.09726 q^{2} -118.407 q^{4} +156.435 q^{5} +11.3597 q^{7} +763.187 q^{8} +O(q^{10})\)	\(q-3.09726 q^{2} -118.407 q^{4} +156.435 q^{5} +11.3597 q^{7} +763.187 q^{8} -484.520 q^{10} -4296.19 q^{11} +1741.86 q^{13} -35.1841 q^{14} +12792.3 q^{16} +4764.38 q^{17} +18278.8 q^{19} -18523.0 q^{20} +13306.4 q^{22} -82004.5 q^{23} -53653.2 q^{25} -5395.00 q^{26} -1345.07 q^{28} +157425. q^{29} +37778.8 q^{31} -137309. q^{32} -14756.5 q^{34} +1777.05 q^{35} +450442. q^{37} -56614.3 q^{38} +119389. q^{40} +437505. q^{41} -738856. q^{43} +508699. q^{44} +253989. q^{46} -869122. q^{47} -823414. q^{49} +166178. q^{50} -206248. q^{52} +991560. q^{53} -672074. q^{55} +8669.60 q^{56} -487588. q^{58} -205379. q^{59} -1.30896e6 q^{61} -117011. q^{62} -1.21213e6 q^{64} +272487. q^{65} +2.85617e6 q^{67} -564135. q^{68} -5504.01 q^{70} +4.56945e6 q^{71} +937053. q^{73} -1.39514e6 q^{74} -2.16434e6 q^{76} -48803.6 q^{77} +6.61566e6 q^{79} +2.00116e6 q^{80} -1.35507e6 q^{82} +3.65210e6 q^{83} +745314. q^{85} +2.28843e6 q^{86} -3.27880e6 q^{88} +5.71107e6 q^{89} +19787.0 q^{91} +9.70990e6 q^{92} +2.69190e6 q^{94} +2.85944e6 q^{95} -1.38687e7 q^{97} +2.55033e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 6 q^{2} + 974 q^{4} + 68 q^{5} - 2343 q^{7} - 819 q^{8}+O(q^{10}) \) 16 * q + 6 * q^2 + 974 * q^4 + 68 * q^5 - 2343 * q^7 - 819 * q^8	\( 16 q + 6 q^{2} + 974 q^{4} + 68 q^{5} - 2343 q^{7} - 819 q^{8} - 3479 q^{10} - 898 q^{11} - 8172 q^{13} + 13315 q^{14} + 3138 q^{16} + 44985 q^{17} - 40137 q^{19} - 130657 q^{20} + 109394 q^{22} + 2833 q^{23} + 285746 q^{25} + 129420 q^{26} + 112890 q^{28} - 144375 q^{29} - 141759 q^{31} + 36224 q^{32} - 341332 q^{34} + 78859 q^{35} - 297971 q^{37} - 329075 q^{38} - 203048 q^{40} - 659077 q^{41} - 1431608 q^{43} - 254916 q^{44} + 873113 q^{46} + 1574073 q^{47} + 1893545 q^{49} - 302533 q^{50} - 4972548 q^{52} - 587736 q^{53} - 4624036 q^{55} + 5798506 q^{56} - 6991380 q^{58} - 3286064 q^{59} - 6117131 q^{61} + 11570258 q^{62} - 19063011 q^{64} + 5335514 q^{65} - 16518710 q^{67} + 17284669 q^{68} - 39189486 q^{70} + 10882582 q^{71} - 21097441 q^{73} + 16717030 q^{74} - 40864952 q^{76} + 3404601 q^{77} - 3784458 q^{79} + 27466195 q^{80} - 24990117 q^{82} + 1951425 q^{83} - 23238675 q^{85} + 35910572 q^{86} - 27843055 q^{88} - 10499443 q^{89} + 699217 q^{91} + 20062766 q^{92} - 59358988 q^{94} + 29236333 q^{95} - 25158976 q^{97} - 2120460 q^{98}+O(q^{100}) \) 16 * q + 6 * q^2 + 974 * q^4 + 68 * q^5 - 2343 * q^7 - 819 * q^8 - 3479 * q^10 - 898 * q^11 - 8172 * q^13 + 13315 * q^14 + 3138 * q^16 + 44985 * q^17 - 40137 * q^19 - 130657 * q^20 + 109394 * q^22 + 2833 * q^23 + 285746 * q^25 + 129420 * q^26 + 112890 * q^28 - 144375 * q^29 - 141759 * q^31 + 36224 * q^32 - 341332 * q^34 + 78859 * q^35 - 297971 * q^37 - 329075 * q^38 - 203048 * q^40 - 659077 * q^41 - 1431608 * q^43 - 254916 * q^44 + 873113 * q^46 + 1574073 * q^47 + 1893545 * q^49 - 302533 * q^50 - 4972548 * q^52 - 587736 * q^53 - 4624036 * q^55 + 5798506 * q^56 - 6991380 * q^58 - 3286064 * q^59 - 6117131 * q^61 + 11570258 * q^62 - 19063011 * q^64 + 5335514 * q^65 - 16518710 * q^67 + 17284669 * q^68 - 39189486 * q^70 + 10882582 * q^71 - 21097441 * q^73 + 16717030 * q^74 - 40864952 * q^76 + 3404601 * q^77 - 3784458 * q^79 + 27466195 * q^80 - 24990117 * q^82 + 1951425 * q^83 - 23238675 * q^85 + 35910572 * q^86 - 27843055 * q^88 - 10499443 * q^89 + 699217 * q^91 + 20062766 * q^92 - 59358988 * q^94 + 29236333 * q^95 - 25158976 * q^97 - 2120460 * q^98

Introduction

L-functions

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