LMFDB - Embedded newform 4598.2.a.bx.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4598,2,Mod(1,4598)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4598, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4598.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4598 = 2 \cdot 11^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4598.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:	$36.7152148494$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 10x^{6} + 16x^{5} + 26x^{4} - 32x^{3} - 16x^{2} + 20x - 4 $ x^8 - 2x^7 - 10x^6 + 16x^5 + 26x^4 - 32x^3 - 16x^2 + 20*x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 418)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-1.35168$ of defining polynomial
Character	$\chi$	$=$	4598.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +2.30094 q^{3} +1.00000 q^{4} +2.62639 q^{5} -2.30094 q^{6} -2.74200 q^{7} -1.00000 q^{8} +2.29431 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +2.30094 q^{3} +1.00000 q^{4} +2.62639 q^{5} -2.30094 q^{6} -2.74200 q^{7} -1.00000 q^{8} +2.29431 q^{9} -2.62639 q^{10} +2.30094 q^{12} -5.41693 q^{13} +2.74200 q^{14} +6.04315 q^{15} +1.00000 q^{16} +4.27854 q^{17} -2.29431 q^{18} +1.00000 q^{19} +2.62639 q^{20} -6.30916 q^{21} -6.21615 q^{23} -2.30094 q^{24} +1.89791 q^{25} +5.41693 q^{26} -1.62374 q^{27} -2.74200 q^{28} -5.99594 q^{29} -6.04315 q^{30} -7.27922 q^{31} -1.00000 q^{32} -4.27854 q^{34} -7.20154 q^{35} +2.29431 q^{36} -3.97289 q^{37} -1.00000 q^{38} -12.4640 q^{39} -2.62639 q^{40} -10.5357 q^{41} +6.30916 q^{42} +0.306239 q^{43} +6.02576 q^{45} +6.21615 q^{46} -4.56638 q^{47} +2.30094 q^{48} +0.518537 q^{49} -1.89791 q^{50} +9.84465 q^{51} -5.41693 q^{52} -3.72986 q^{53} +1.62374 q^{54} +2.74200 q^{56} +2.30094 q^{57} +5.99594 q^{58} +5.42719 q^{59} +6.04315 q^{60} +10.1851 q^{61} +7.27922 q^{62} -6.29100 q^{63} +1.00000 q^{64} -14.2269 q^{65} -0.0148030 q^{67} +4.27854 q^{68} -14.3030 q^{69} +7.20154 q^{70} +12.1074 q^{71} -2.29431 q^{72} +13.8664 q^{73} +3.97289 q^{74} +4.36696 q^{75} +1.00000 q^{76} +12.4640 q^{78} -11.8987 q^{79} +2.62639 q^{80} -10.6191 q^{81} +10.5357 q^{82} -1.29710 q^{83} -6.30916 q^{84} +11.2371 q^{85} -0.306239 q^{86} -13.7963 q^{87} -14.5993 q^{89} -6.02576 q^{90} +14.8532 q^{91} -6.21615 q^{92} -16.7490 q^{93} +4.56638 q^{94} +2.62639 q^{95} -2.30094 q^{96} -10.0528 q^{97} -0.518537 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} + 8 q^{4} + 2 q^{5} - 8 q^{7} - 8 q^{8} + 20 q^{9}+O(q^{10}) $ 8 * q - 8 * q^2 + 8 * q^4 + 2 * q^5 - 8 * q^7 - 8 * q^8 + 20 * q^9	$ 8 q - 8 q^{2} + 8 q^{4} + 2 q^{5} - 8 q^{7} - 8 q^{8} + 20 q^{9} - 2 q^{10} - 18 q^{13} + 8 q^{14} + 10 q^{15} + 8 q^{16} - 4 q^{17} - 20 q^{18} + 8 q^{19} + 2 q^{20} - 14 q^{21} + 12 q^{23} + 18 q^{26} - 24 q^{27} - 8 q^{28} - 14 q^{29} - 10 q^{30} - 2 q^{31} - 8 q^{32} + 4 q^{34} - 40 q^{35} + 20 q^{36} - 22 q^{37} - 8 q^{38} + 4 q^{39} - 2 q^{40} - 8 q^{41} + 14 q^{42} - 28 q^{43} - 28 q^{45} - 12 q^{46} + 6 q^{47} + 32 q^{49} + 12 q^{51} - 18 q^{52} - 24 q^{53} + 24 q^{54} + 8 q^{56} + 14 q^{58} + 46 q^{59} + 10 q^{60} + 24 q^{61} + 2 q^{62} - 30 q^{63} + 8 q^{64} + 16 q^{65} - 22 q^{67} - 4 q^{68} - 38 q^{69} + 40 q^{70} + 8 q^{71} - 20 q^{72} - 16 q^{73} + 22 q^{74} + 6 q^{75} + 8 q^{76} - 4 q^{78} - 4 q^{79} + 2 q^{80} + 28 q^{81} + 8 q^{82} - 12 q^{83} - 14 q^{84} - 48 q^{85} + 28 q^{86} - 42 q^{87} - 28 q^{89} + 28 q^{90} - 12 q^{91} + 12 q^{92} + 22 q^{93} - 6 q^{94} + 2 q^{95} - 22 q^{97} - 32 q^{98}+O(q^{100}) $ 8 * q - 8 * q^2 + 8 * q^4 + 2 * q^5 - 8 * q^7 - 8 * q^8 + 20 * q^9 - 2 * q^10 - 18 * q^13 + 8 * q^14 + 10 * q^15 + 8 * q^16 - 4 * q^17 - 20 * q^18 + 8 * q^19 + 2 * q^20 - 14 * q^21 + 12 * q^23 + 18 * q^26 - 24 * q^27 - 8 * q^28 - 14 * q^29 - 10 * q^30 - 2 * q^31 - 8 * q^32 + 4 * q^34 - 40 * q^35 + 20 * q^36 - 22 * q^37 - 8 * q^38 + 4 * q^39 - 2 * q^40 - 8 * q^41 + 14 * q^42 - 28 * q^43 - 28 * q^45 - 12 * q^46 + 6 * q^47 + 32 * q^49 + 12 * q^51 - 18 * q^52 - 24 * q^53 + 24 * q^54 + 8 * q^56 + 14 * q^58 + 46 * q^59 + 10 * q^60 + 24 * q^61 + 2 * q^62 - 30 * q^63 + 8 * q^64 + 16 * q^65 - 22 * q^67 - 4 * q^68 - 38 * q^69 + 40 * q^70 + 8 * q^71 - 20 * q^72 - 16 * q^73 + 22 * q^74 + 6 * q^75 + 8 * q^76 - 4 * q^78 - 4 * q^79 + 2 * q^80 + 28 * q^81 + 8 * q^82 - 12 * q^83 - 14 * q^84 - 48 * q^85 + 28 * q^86 - 42 * q^87 - 28 * q^89 + 28 * q^90 - 12 * q^91 + 12 * q^92 + 22 * q^93 - 6 * q^94 + 2 * q^95 - 22 * q^97 - 32 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	2.30094	1.32845	0.664224	−	0.747534i	$-0.268762\pi$
$3$	2.30094	1.32845	0.664224	+	0.747534i	$0.268762\pi$
$4$	1.00000	0.500000
$5$	2.62639	1.17456	0.587278	−	0.809385i	$-0.300200\pi$
$5$	2.62639	1.17456	0.587278	+	0.809385i	$0.300200\pi$
$6$	−2.30094	−0.939354
$7$	−2.74200	−1.03638	−0.518188	−	0.855267i	$-0.673393\pi$
$7$	−2.74200	−1.03638	−0.518188	+	0.855267i	$0.673393\pi$
$8$	−1.00000	−0.353553
$9$	2.29431	0.764772
$10$	−2.62639	−0.830536
$11$	0	0
$11$	0	0
$12$	2.30094	0.664224
$13$	−5.41693	−1.50239	−0.751193	−	0.660083i	$-0.770521\pi$
$13$	−5.41693	−1.50239	−0.751193	+	0.660083i	$0.770521\pi$
$14$	2.74200	0.732829
$15$	6.04315	1.56034
$16$	1.00000	0.250000
$17$	4.27854	1.03770	0.518849	−	0.854866i	$-0.326361\pi$
$17$	4.27854	1.03770	0.518849	+	0.854866i	$0.326361\pi$
$18$	−2.29431	−0.540775
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	2.62639	0.587278
$21$	−6.30916	−1.37677
$22$	0	0
$23$	−6.21615	−1.29616	−0.648079	−	0.761573i	$-0.724427\pi$
$23$	−6.21615	−1.29616	−0.648079	+	0.761573i	$0.724427\pi$
$24$	−2.30094	−0.469677
$25$	1.89791	0.379581
$26$	5.41693	1.06235
$27$	−1.62374	−0.312489
$28$	−2.74200	−0.518188
$29$	−5.99594	−1.11342	−0.556709	−	0.830708i	$-0.687936\pi$
$29$	−5.99594	−1.11342	−0.556709	+	0.830708i	$0.687936\pi$
$30$	−6.04315	−1.10332
$31$	−7.27922	−1.30739	−0.653693	−	0.756760i	$-0.726781\pi$
$31$	−7.27922	−1.30739	−0.653693	+	0.756760i	$0.726781\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−4.27854	−0.733763
$35$	−7.20154	−1.21728
$36$	2.29431	0.382386
$37$	−3.97289	−0.653139	−0.326570	−	0.945173i	$-0.605893\pi$
$37$	−3.97289	−0.653139	−0.326570	+	0.945173i	$0.605893\pi$
$38$	−1.00000	−0.162221
$39$	−12.4640	−1.99584
$40$	−2.62639	−0.415268
$41$	−10.5357	−1.64540	−0.822699	−	0.568477i	$-0.807533\pi$
$41$	−10.5357	−1.64540	−0.822699	+	0.568477i	$0.807533\pi$
$42$	6.30916	0.973525
$43$	0.306239	0.0467009	0.0233505	−	0.999727i	$-0.492567\pi$
$43$	0.306239	0.0467009	0.0233505	+	0.999727i	$0.492567\pi$
$44$	0	0
$45$	6.02576	0.898267
$46$	6.21615	0.916522
$47$	−4.56638	−0.666075	−0.333038	−	0.942914i	$-0.608074\pi$
$47$	−4.56638	−0.666075	−0.333038	+	0.942914i	$0.608074\pi$
$48$	2.30094	0.332112
$49$	0.518537	0.0740767
$50$	−1.89791	−0.268404
$51$	9.84465	1.37853
$52$	−5.41693	−0.751193
$53$	−3.72986	−0.512335	−0.256168	−	0.966632i	$-0.582460\pi$
$53$	−3.72986	−0.512335	−0.256168	+	0.966632i	$0.582460\pi$
$54$	1.62374	0.220963
$55$	0	0
$56$	2.74200	0.366415
$57$	2.30094	0.304767
$58$	5.99594	0.787305
$59$	5.42719	0.706560	0.353280	−	0.935518i	$-0.385066\pi$
$59$	5.42719	0.706560	0.353280	+	0.935518i	$0.385066\pi$
$60$	6.04315	0.780168
$61$	10.1851	1.30407	0.652036	−	0.758188i	$-0.273915\pi$
$61$	10.1851	1.30407	0.652036	+	0.758188i	$0.273915\pi$
$62$	7.27922	0.924462
$63$	−6.29100	−0.792591
$64$	1.00000	0.125000
$65$	−14.2269	−1.76464
$66$	0	0
$67$	−0.0148030	−0.00180848	−0.000904239	−	1.00000i	$-0.500288\pi$
$67$	−0.0148030	−0.00180848	−0.000904239		1.00000i	$0.500288\pi$
$68$	4.27854	0.518849
$69$	−14.3030	−1.72188
$70$	7.20154	0.860749
$71$	12.1074	1.43688	0.718440	−	0.695589i	$-0.244856\pi$
$71$	12.1074	1.43688	0.718440	+	0.695589i	$0.244856\pi$
$72$	−2.29431	−0.270388
$73$	13.8664	1.62294	0.811472	−	0.584392i	$-0.198667\pi$
$73$	13.8664	1.62294	0.811472	+	0.584392i	$0.198667\pi$
$74$	3.97289	0.461839
$75$	4.36696	0.504254
$76$	1.00000	0.114708
$77$	0	0
$78$	12.4640	1.41127
$79$	−11.8987	−1.33871	−0.669356	−	0.742942i	$-0.733430\pi$
$79$	−11.8987	−1.33871	−0.669356	+	0.742942i	$0.733430\pi$
$80$	2.62639	0.293639
$81$	−10.6191	−1.17990
$82$	10.5357	1.16347
$83$	−1.29710	−0.142375	−0.0711876	−	0.997463i	$-0.522679\pi$
$83$	−1.29710	−0.142375	−0.0711876	+	0.997463i	$0.522679\pi$
$84$	−6.30916	−0.688386
$85$	11.2371	1.21883
$86$	−0.306239	−0.0330226
$87$	−13.7963	−1.47912
$88$	0	0
$89$	−14.5993	−1.54753	−0.773764	−	0.633474i	$-0.781628\pi$
$89$	−14.5993	−1.54753	−0.773764	+	0.633474i	$0.781628\pi$
$90$	−6.02576	−0.635171
$91$	14.8532	1.55704
$92$	−6.21615	−0.648079
$93$	−16.7490	−1.73679
$94$	4.56638	0.470986
$95$	2.62639	0.269462
$96$	−2.30094	−0.234838
$97$	−10.0528	−1.02071	−0.510356	−	0.859963i	$-0.670486\pi$
$97$	−10.0528	−1.02071	−0.510356	+	0.859963i	$0.670486\pi$
$98$	−0.518537	−0.0523802
$99$	0	0
$100$	1.89791	0.189791
$101$	−4.03837	−0.401833	−0.200916	−	0.979608i	$-0.564392\pi$
$101$	−4.03837	−0.401833	−0.200916	+	0.979608i	$0.564392\pi$
$102$	−9.84465	−0.974765
$103$	12.3140	1.21333	0.606666	−	0.794957i	$-0.292507\pi$
$103$	12.3140	1.21333	0.606666	+	0.794957i	$0.292507\pi$
$104$	5.41693	0.531173
$105$	−16.5703	−1.61710
$106$	3.72986	0.362276
$107$	7.79569	0.753638	0.376819	−	0.926287i	$-0.377018\pi$
$107$	7.79569	0.753638	0.376819	+	0.926287i	$0.377018\pi$
$108$	−1.62374	−0.156244
$109$	14.9495	1.43191	0.715953	−	0.698148i	$-0.245992\pi$
$109$	14.9495	1.43191	0.715953	+	0.698148i	$0.245992\pi$
$110$	0	0
$111$	−9.14138	−0.867661
$112$	−2.74200	−0.259094
$113$	1.06888	0.100552	0.0502761	−	0.998735i	$-0.483990\pi$
$113$	1.06888	0.100552	0.0502761	+	0.998735i	$0.483990\pi$
$114$	−2.30094	−0.215503
$115$	−16.3260	−1.52241
$116$	−5.99594	−0.556709
$117$	−12.4281	−1.14898
$118$	−5.42719	−0.499613
$119$	−11.7317	−1.07545
$120$	−6.04315	−0.551662
$121$	0	0
$122$	−10.1851	−0.922118
$123$	−24.2420	−2.18582
$124$	−7.27922	−0.653693
$125$	−8.14730	−0.728716
$126$	6.29100	0.560447
$127$	−1.63724	−0.145281	−0.0726407	−	0.997358i	$-0.523143\pi$
$127$	−1.63724	−0.145281	−0.0726407	+	0.997358i	$0.523143\pi$
$128$	−1.00000	−0.0883883
$129$	0.704636	0.0620397
$130$	14.2269	1.24779
$131$	21.5141	1.87970	0.939849	−	0.341592i	$-0.110966\pi$
$131$	21.5141	1.87970	0.939849	+	0.341592i	$0.110966\pi$
$132$	0	0
$133$	−2.74200	−0.237761
$134$	0.0148030	0.00127879
$135$	−4.26456	−0.367035
$136$	−4.27854	−0.366882
$137$	6.40013	0.546800	0.273400	−	0.961900i	$-0.411852\pi$
$137$	6.40013	0.546800	0.273400	+	0.961900i	$0.411852\pi$
$138$	14.3030	1.21755
$139$	7.37781	0.625778	0.312889	−	0.949790i	$-0.398703\pi$
$139$	7.37781	0.625778	0.312889	+	0.949790i	$0.398703\pi$
$140$	−7.20154	−0.608641
$141$	−10.5070	−0.884846
$142$	−12.1074	−1.01603
$143$	0	0
$144$	2.29431	0.191193
$145$	−15.7477	−1.30777
$146$	−13.8664	−1.14759
$147$	1.19312	0.0984070
$148$	−3.97289	−0.326570
$149$	−6.70635	−0.549405	−0.274703	−	0.961529i	$-0.588579\pi$
$149$	−6.70635	−0.549405	−0.274703	+	0.961529i	$0.588579\pi$
$150$	−4.36696	−0.356561
$151$	−17.7575	−1.44508	−0.722542	−	0.691327i	$-0.757027\pi$
$151$	−17.7575	−1.44508	−0.722542	+	0.691327i	$0.757027\pi$
$152$	−1.00000	−0.0811107
$153$	9.81631	0.793602
$154$	0	0
$155$	−19.1180	−1.53560
$156$	−12.4640	−0.997920
$157$	1.60819	0.128348	0.0641738	−	0.997939i	$-0.479559\pi$
$157$	1.60819	0.128348	0.0641738	+	0.997939i	$0.479559\pi$
$158$	11.8987	0.946612
$159$	−8.58217	−0.680610
$160$	−2.62639	−0.207634
$161$	17.0447	1.34331
$162$	10.6191	0.834312
$163$	21.6766	1.69784	0.848922	−	0.528518i	$-0.177252\pi$
$163$	21.6766	1.69784	0.848922	+	0.528518i	$0.177252\pi$
$164$	−10.5357	−0.822699
$165$	0	0
$166$	1.29710	0.100674
$167$	−12.2081	−0.944693	−0.472346	−	0.881413i	$-0.656593\pi$
$167$	−12.2081	−0.944693	−0.472346	+	0.881413i	$0.656593\pi$
$168$	6.30916	0.486762
$169$	16.3431	1.25716
$170$	−11.2371	−0.861846
$171$	2.29431	0.175451
$172$	0.306239	0.0233505
$173$	−11.5446	−0.877720	−0.438860	−	0.898556i	$-0.644618\pi$
$173$	−11.5446	−0.877720	−0.438860	+	0.898556i	$0.644618\pi$
$174$	13.7963	1.04589
$175$	−5.20405	−0.393389
$176$	0	0
$177$	12.4876	0.938627
$178$	14.5993	1.09427
$179$	−0.875261	−0.0654201	−0.0327100	−	0.999465i	$-0.510414\pi$
$179$	−0.875261	−0.0654201	−0.0327100	+	0.999465i	$0.510414\pi$
$180$	6.02576	0.449133
$181$	5.60684	0.416753	0.208377	−	0.978049i	$-0.433182\pi$
$181$	5.60684	0.416753	0.208377	+	0.978049i	$0.433182\pi$
$182$	−14.8532	−1.10099
$183$	23.4353	1.73239
$184$	6.21615	0.458261
$185$	−10.4343	−0.767149
$186$	16.7490	1.22810
$187$	0	0
$188$	−4.56638	−0.333038
$189$	4.45228	0.323856
$190$	−2.62639	−0.190538
$191$	−9.79138	−0.708480	−0.354240	−	0.935155i	$-0.615260\pi$
$191$	−9.79138	−0.708480	−0.354240	+	0.935155i	$0.615260\pi$
$192$	2.30094	0.166056
$193$	−8.40525	−0.605023	−0.302512	−	0.953146i	$-0.597825\pi$
$193$	−8.40525	−0.605023	−0.302512	+	0.953146i	$0.597825\pi$
$194$	10.0528	0.721752
$195$	−32.7353	−2.34422
$196$	0.518537	0.0370384
$197$	3.91660	0.279046	0.139523	−	0.990219i	$-0.455443\pi$
$197$	3.91660	0.279046	0.139523	+	0.990219i	$0.455443\pi$
$198$	0	0
$199$	0.808932	0.0573437	0.0286718	−	0.999589i	$-0.490872\pi$
$199$	0.808932	0.0573437	0.0286718	+	0.999589i	$0.490872\pi$
$200$	−1.89791	−0.134202
$201$	−0.0340609	−0.00240247
$202$	4.03837	0.284139
$203$	16.4408	1.15392
$204$	9.84465	0.689263
$205$	−27.6708	−1.93261
$206$	−12.3140	−0.857955
$207$	−14.2618	−0.991265
$208$	−5.41693	−0.375596
$209$	0	0
$210$	16.5703	1.14346
$211$	−13.8172	−0.951214	−0.475607	−	0.879658i	$-0.657772\pi$
$211$	−13.8172	−0.951214	−0.475607	+	0.879658i	$0.657772\pi$
$212$	−3.72986	−0.256168
$213$	27.8583	1.90882
$214$	−7.79569	−0.532902
$215$	0.804301	0.0548529
$216$	1.62374	0.110481
$217$	19.9596	1.35494
$218$	−14.9495	−1.01251
$219$	31.9058	2.15599
$220$	0	0
$221$	−23.1765	−1.55902
$222$	9.14138	0.613529
$223$	1.06868	0.0715638	0.0357819	−	0.999360i	$-0.488608\pi$
$223$	1.06868	0.0715638	0.0357819	+	0.999360i	$0.488608\pi$
$224$	2.74200	0.183207
$225$	4.35439	0.290293
$226$	−1.06888	−0.0711011
$227$	20.1639	1.33833	0.669164	−	0.743115i	$-0.266652\pi$
$227$	20.1639	1.33833	0.669164	+	0.743115i	$0.266652\pi$
$228$	2.30094	0.152383
$229$	25.5481	1.68827	0.844133	−	0.536134i	$-0.180116\pi$
$229$	25.5481	1.68827	0.844133	+	0.536134i	$0.180116\pi$
$230$	16.3260	1.07651
$231$	0	0
$232$	5.99594	0.393653
$233$	−11.3403	−0.742928	−0.371464	−	0.928447i	$-0.621144\pi$
$233$	−11.3403	−0.742928	−0.371464	+	0.928447i	$0.621144\pi$
$234$	12.4281	0.812453
$235$	−11.9931	−0.782342
$236$	5.42719	0.353280
$237$	−27.3782	−1.77841
$238$	11.7317	0.760455
$239$	1.78866	0.115699	0.0578495	−	0.998325i	$-0.481576\pi$
$239$	1.78866	0.115699	0.0578495	+	0.998325i	$0.481576\pi$
$240$	6.04315	0.390084
$241$	19.8713	1.28002	0.640011	−	0.768366i	$-0.278930\pi$
$241$	19.8713	1.28002	0.640011	+	0.768366i	$0.278930\pi$
$242$	0	0
$243$	−19.5626	−1.25494
$244$	10.1851	0.652036
$245$	1.36188	0.0870072
$246$	24.2420	1.54561
$247$	−5.41693	−0.344671
$248$	7.27922	0.462231
$249$	−2.98454	−0.189138
$250$	8.14730	0.515280
$251$	11.0409	0.696898	0.348449	−	0.937328i	$-0.386708\pi$
$251$	11.0409	0.696898	0.348449	+	0.937328i	$0.386708\pi$
$252$	−6.29100	−0.396296
$253$	0	0
$254$	1.63724	0.102729
$255$	25.8559	1.61916
$256$	1.00000	0.0625000
$257$	−6.40353	−0.399441	−0.199721	−	0.979853i	$-0.564003\pi$
$257$	−6.40353	−0.399441	−0.199721	+	0.979853i	$0.564003\pi$
$258$	−0.704636	−0.0438687
$259$	10.8936	0.676898
$260$	−14.2269	−0.882318
$261$	−13.7566	−0.851510
$262$	−21.5141	−1.32915
$263$	−4.26381	−0.262918	−0.131459	−	0.991322i	$-0.541966\pi$
$263$	−4.26381	−0.262918	−0.131459	+	0.991322i	$0.541966\pi$
$264$	0	0
$265$	−9.79604	−0.601766
$266$	2.74200	0.168123
$267$	−33.5922	−2.05581
$268$	−0.0148030	−0.000904239 0
$269$	−2.60318	−0.158719	−0.0793595	−	0.996846i	$-0.525287\pi$
$269$	−2.60318	−0.158719	−0.0793595	+	0.996846i	$0.525287\pi$
$270$	4.26456	0.259533
$271$	−9.55649	−0.580515	−0.290258	−	0.956949i	$-0.593741\pi$
$271$	−9.55649	−0.580515	−0.290258	+	0.956949i	$0.593741\pi$
$272$	4.27854	0.259424
$273$	34.1763	2.06844
$274$	−6.40013	−0.386646
$275$	0	0
$276$	−14.3030	−0.860938
$277$	−9.48715	−0.570027	−0.285014	−	0.958523i	$-0.591998\pi$
$277$	−9.48715	−0.570027	−0.285014	+	0.958523i	$0.591998\pi$
$278$	−7.37781	−0.442492
$279$	−16.7008	−0.999852
$280$	7.20154	0.430374
$281$	9.15958	0.546415	0.273207	−	0.961955i	$-0.411915\pi$
$281$	9.15958	0.546415	0.273207	+	0.961955i	$0.411915\pi$
$282$	10.5070	0.625680
$283$	−21.1581	−1.25772	−0.628860	−	0.777519i	$-0.716478\pi$
$283$	−21.1581	−1.25772	−0.628860	+	0.777519i	$0.716478\pi$
$284$	12.1074	0.718440
$285$	6.04315	0.357965
$286$	0	0
$287$	28.8888	1.70525
$288$	−2.29431	−0.135194
$289$	1.30588	0.0768165
$290$	15.7477	0.924734
$291$	−23.1310	−1.35596
$292$	13.8664	0.811472
$293$	−22.1069	−1.29150	−0.645749	−	0.763550i	$-0.723455\pi$
$293$	−22.1069	−1.29150	−0.645749	+	0.763550i	$0.723455\pi$
$294$	−1.19312	−0.0695843
$295$	14.2539	0.829894
$296$	3.97289	0.230920
$297$	0	0
$298$	6.70635	0.388488
$299$	33.6725	1.94733
$300$	4.36696	0.252127
$301$	−0.839705	−0.0483998
$302$	17.7575	1.02183
$303$	−9.29204	−0.533814
$304$	1.00000	0.0573539
$305$	26.7501	1.53171
$306$	−9.81631	−0.561161
$307$	−14.5129	−0.828296	−0.414148	−	0.910209i	$-0.635920\pi$
$307$	−14.5129	−0.828296	−0.414148	+	0.910209i	$0.635920\pi$
$308$	0	0
$309$	28.3337	1.61185
$310$	19.1180	1.08583
$311$	−29.2145	−1.65660	−0.828300	−	0.560285i	$-0.810692\pi$
$311$	−29.2145	−1.65660	−0.828300	+	0.560285i	$0.810692\pi$
$312$	12.4640	0.705636
$313$	−27.0783	−1.53055	−0.765277	−	0.643702i	$-0.777398\pi$
$313$	−27.0783	−1.53055	−0.765277	+	0.643702i	$0.777398\pi$
$314$	−1.60819	−0.0907555
$315$	−16.5226	−0.930943
$316$	−11.8987	−0.669356
$317$	27.2677	1.53151	0.765754	−	0.643134i	$-0.222366\pi$
$317$	27.2677	1.53151	0.765754	+	0.643134i	$0.222366\pi$
$318$	8.58217	0.481264
$319$	0	0
$320$	2.62639	0.146819
$321$	17.9374	1.00117
$322$	−17.0447	−0.949862
$323$	4.27854	0.238064
$324$	−10.6191	−0.589948
$325$	−10.2808	−0.570277
$326$	−21.6766	−1.20056
$327$	34.3980	1.90221
$328$	10.5357	0.581736
$329$	12.5210	0.690305
$330$	0	0
$331$	16.1001	0.884941	0.442471	−	0.896783i	$-0.354102\pi$
$331$	16.1001	0.884941	0.442471	+	0.896783i	$0.354102\pi$
$332$	−1.29710	−0.0711876
$333$	−9.11506	−0.499502
$334$	12.2081	0.667999
$335$	−0.0388785	−0.00212416
$336$	−6.30916	−0.344193
$337$	−22.6353	−1.23303	−0.616513	−	0.787345i	$-0.711455\pi$
$337$	−22.6353	−1.23303	−0.616513	+	0.787345i	$0.711455\pi$
$338$	−16.3431	−0.888948
$339$	2.45944	0.133578
$340$	11.2371	0.609417
$341$	0	0
$342$	−2.29431	−0.124062
$343$	17.7721	0.959605
$344$	−0.306239	−0.0165113
$345$	−37.5652	−2.02244
$346$	11.5446	0.620642
$347$	18.8940	1.01428	0.507142	−	0.861863i	$-0.330702\pi$
$347$	18.8940	1.01428	0.507142	+	0.861863i	$0.330702\pi$
$348$	−13.7963	−0.739558
$349$	−16.3651	−0.876002	−0.438001	−	0.898974i	$-0.644313\pi$
$349$	−16.3651	−0.876002	−0.438001	+	0.898974i	$0.644313\pi$
$350$	5.20405	0.278168
$351$	8.79567	0.469478
$352$	0	0
$353$	−28.7651	−1.53101	−0.765506	−	0.643429i	$-0.777511\pi$
$353$	−28.7651	−1.53101	−0.765506	+	0.643429i	$0.777511\pi$
$354$	−12.4876	−0.663710
$355$	31.7986	1.68770
$356$	−14.5993	−0.773764
$357$	−26.9940	−1.42867
$358$	0.875261	0.0462590
$359$	22.4187	1.18321	0.591606	−	0.806228i	$-0.298494\pi$
$359$	22.4187	1.18321	0.591606	+	0.806228i	$0.298494\pi$
$360$	−6.02576	−0.317585
$361$	1.00000	0.0526316
$362$	−5.60684	−0.294689
$363$	0	0
$364$	14.8532	0.778519
$365$	36.4186	1.90624
$366$	−23.4353	−1.22499
$367$	21.2032	1.10680	0.553399	−	0.832916i	$-0.313330\pi$
$367$	21.2032	1.10680	0.553399	+	0.832916i	$0.313330\pi$
$368$	−6.21615	−0.324039
$369$	−24.1722	−1.25835
$370$	10.4343	0.542456
$371$	10.2272	0.530972
$372$	−16.7490	−0.868397
$373$	−34.6222	−1.79267	−0.896334	−	0.443379i	$-0.853780\pi$
$373$	−34.6222	−1.79267	−0.896334	+	0.443379i	$0.853780\pi$
$374$	0	0
$375$	−18.7464	−0.968061
$376$	4.56638	0.235493
$377$	32.4796	1.67278
$378$	−4.45228	−0.229001
$379$	18.9799	0.974933	0.487467	−	0.873142i	$-0.337921\pi$
$379$	18.9799	0.974933	0.487467	+	0.873142i	$0.337921\pi$
$380$	2.62639	0.134731
$381$	−3.76718	−0.192999
$382$	9.79138	0.500971
$383$	15.4258	0.788219	0.394110	−	0.919063i	$-0.371053\pi$
$383$	15.4258	0.788219	0.394110	+	0.919063i	$0.371053\pi$
$384$	−2.30094	−0.117419
$385$	0	0
$386$	8.40525	0.427816
$387$	0.702608	0.0357156
$388$	−10.0528	−0.510356
$389$	−18.1318	−0.919317	−0.459659	−	0.888096i	$-0.652028\pi$
$389$	−18.1318	−0.919317	−0.459659	+	0.888096i	$0.652028\pi$
$390$	32.7353	1.65762
$391$	−26.5960	−1.34502
$392$	−0.518537	−0.0261901
$393$	49.5026	2.49708
$394$	−3.91660	−0.197315
$395$	−31.2507	−1.57239
$396$	0	0
$397$	9.44526	0.474044	0.237022	−	0.971504i	$-0.423829\pi$
$397$	9.44526	0.474044	0.237022	+	0.971504i	$0.423829\pi$
$398$	−0.808932	−0.0405481
$399$	−6.30916	−0.315853
$400$	1.89791	0.0948953
$401$	26.3735	1.31703	0.658515	−	0.752568i	$-0.271185\pi$
$401$	26.3735	1.31703	0.658515	+	0.752568i	$0.271185\pi$
$402$	0.0340609	0.00169880
$403$	39.4310	1.96420
$404$	−4.03837	−0.200916
$405$	−27.8898	−1.38585
$406$	−16.4408	−0.815945
$407$	0	0
$408$	−9.84465	−0.487383
$409$	27.1570	1.34283	0.671413	−	0.741083i	$-0.265688\pi$
$409$	27.1570	1.34283	0.671413	+	0.741083i	$0.265688\pi$
$410$	27.6708	1.36656
$411$	14.7263	0.726395
$412$	12.3140	0.606666
$413$	−14.8813	−0.732262
$414$	14.2618	0.700930
$415$	−3.40668	−0.167228
$416$	5.41693	0.265587
$417$	16.9759	0.831312
$418$	0	0
$419$	16.5473	0.808387	0.404193	−	0.914674i	$-0.367552\pi$
$419$	16.5473	0.808387	0.404193	+	0.914674i	$0.367552\pi$
$420$	−16.5703	−0.808548
$421$	−37.0039	−1.80346	−0.901731	−	0.432298i	$-0.857703\pi$
$421$	−37.0039	−1.80346	−0.901731	+	0.432298i	$0.857703\pi$
$422$	13.8172	0.672610
$423$	−10.4767	−0.509395
$424$	3.72986	0.181138
$425$	8.12026	0.393891
$426$	−27.8583	−1.34974
$427$	−27.9276	−1.35151
$428$	7.79569	0.376819
$429$	0	0
$430$	−0.804301	−0.0387868
$431$	11.7109	0.564096	0.282048	−	0.959400i	$-0.408986\pi$
$431$	11.7109	0.564096	0.282048	+	0.959400i	$0.408986\pi$
$432$	−1.62374	−0.0781221
$433$	2.12441	0.102093	0.0510463	−	0.998696i	$-0.483744\pi$
$433$	2.12441	0.102093	0.0510463	+	0.998696i	$0.483744\pi$
$434$	−19.9596	−0.958091
$435$	−36.2344	−1.73730
$436$	14.9495	0.715953
$437$	−6.21615	−0.297359
$438$	−31.9058	−1.52452
$439$	−17.7590	−0.847591	−0.423795	−	0.905758i	$-0.639302\pi$
$439$	−17.7590	−0.847591	−0.423795	+	0.905758i	$0.639302\pi$
$440$	0	0
$441$	1.18969	0.0566518
$442$	23.1765	1.10239
$443$	7.42988	0.353004	0.176502	−	0.984300i	$-0.443522\pi$
$443$	7.42988	0.353004	0.176502	+	0.984300i	$0.443522\pi$
$444$	−9.14138	−0.433831
$445$	−38.3435	−1.81766
$446$	−1.06868	−0.0506033
$447$	−15.4309	−0.729856
$448$	−2.74200	−0.129547
$449$	−19.0848	−0.900668	−0.450334	−	0.892860i	$-0.648695\pi$
$449$	−19.0848	−0.900668	−0.450334	+	0.892860i	$0.648695\pi$
$450$	−4.35439	−0.205268
$451$	0	0
$452$	1.06888	0.0502761
$453$	−40.8589	−1.91972
$454$	−20.1639	−0.946341
$455$	39.0102	1.82883
$456$	−2.30094	−0.107751
$457$	2.89775	0.135551	0.0677755	−	0.997701i	$-0.478410\pi$
$457$	2.89775	0.135551	0.0677755	+	0.997701i	$0.478410\pi$
$458$	−25.5481	−1.19378
$459$	−6.94722	−0.324269
$460$	−16.3260	−0.761205
$461$	7.89943	0.367913	0.183957	−	0.982934i	$-0.441109\pi$
$461$	7.89943	0.367913	0.183957	+	0.982934i	$0.441109\pi$
$462$	0	0
$463$	20.8107	0.967155	0.483578	−	0.875301i	$-0.339337\pi$
$463$	20.8107	0.967155	0.483578	+	0.875301i	$0.339337\pi$
$464$	−5.99594	−0.278354
$465$	−43.9894	−2.03996
$466$	11.3403	0.525329
$467$	−31.5622	−1.46052	−0.730262	−	0.683168i	$-0.760602\pi$
$467$	−31.5622	−1.46052	−0.730262	+	0.683168i	$0.760602\pi$
$468$	−12.4281	−0.574491
$469$	0.0405898	0.00187426
$470$	11.9931	0.553200
$471$	3.70035	0.170503
$472$	−5.42719	−0.249807
$473$	0	0
$474$	27.3782	1.25752
$475$	1.89791	0.0870819
$476$	−11.7317	−0.537723
$477$	−8.55746	−0.391819
$478$	−1.78866	−0.0818116
$479$	−23.7066	−1.08318	−0.541591	−	0.840642i	$-0.682178\pi$
$479$	−23.7066	−1.08318	−0.541591	+	0.840642i	$0.682178\pi$
$480$	−6.04315	−0.275831
$481$	21.5209	0.981267
$482$	−19.8713	−0.905112
$483$	39.2187	1.78451
$484$	0	0
$485$	−26.4027	−1.19888
$486$	19.5626	0.887377
$487$	−24.6716	−1.11798	−0.558988	−	0.829176i	$-0.688810\pi$
$487$	−24.6716	−1.11798	−0.558988	+	0.829176i	$0.688810\pi$
$488$	−10.1851	−0.461059
$489$	49.8766	2.25550
$490$	−1.36188	−0.0615234
$491$	−34.1976	−1.54332	−0.771658	−	0.636038i	$-0.780572\pi$
$491$	−34.1976	−1.54332	−0.771658	+	0.636038i	$0.780572\pi$
$492$	−24.2420	−1.09291
$493$	−25.6538	−1.15539
$494$	5.41693	0.243719
$495$	0	0
$496$	−7.27922	−0.326847
$497$	−33.1983	−1.48915
$498$	2.98454	0.133741
$499$	23.0706	1.03278	0.516390	−	0.856353i	$-0.327275\pi$
$499$	23.0706	1.03278	0.516390	+	0.856353i	$0.327275\pi$
$500$	−8.14730	−0.364358
$501$	−28.0901	−1.25497
$502$	−11.0409	−0.492782
$503$	20.1071	0.896532	0.448266	−	0.893900i	$-0.352042\pi$
$503$	20.1071	0.896532	0.448266	+	0.893900i	$0.352042\pi$
$504$	6.29100	0.280223
$505$	−10.6063	−0.471975
$506$	0	0
$507$	37.6045	1.67007
$508$	−1.63724	−0.0726407
$509$	30.9152	1.37029	0.685145	−	0.728406i	$-0.259739\pi$
$509$	30.9152	1.37029	0.685145	+	0.728406i	$0.259739\pi$
$510$	−25.8559	−1.14492
$511$	−38.0217	−1.68198
$512$	−1.00000	−0.0441942
$513$	−1.62374	−0.0716898
$514$	6.40353	0.282448
$515$	32.3412	1.42513
$516$	0.704636	0.0310199
$517$	0	0
$518$	−10.8936	−0.478639
$519$	−26.5634	−1.16600
$520$	14.2269	0.623893
$521$	−24.5075	−1.07369	−0.536845	−	0.843681i	$-0.680384\pi$
$521$	−24.5075	−1.07369	−0.536845	+	0.843681i	$0.680384\pi$
$522$	13.7566	0.602109
$523$	−0.648533	−0.0283584	−0.0141792	−	0.999899i	$-0.504514\pi$
$523$	−0.648533	−0.0283584	−0.0141792	+	0.999899i	$0.504514\pi$
$524$	21.5141	0.939849
$525$	−11.9742	−0.522597
$526$	4.26381	0.185911
$527$	−31.1444	−1.35667
$528$	0	0
$529$	15.6406	0.680025
$530$	9.79604	0.425513
$531$	12.4517	0.540357
$532$	−2.74200	−0.118881
$533$	57.0711	2.47202
$534$	33.5922	1.45368
$535$	20.4745	0.885190
$536$	0.0148030	0.000639394 0
$537$	−2.01392	−0.0869071
$538$	2.60318	0.112231
$539$	0	0
$540$	−4.26456	−0.183518
$541$	0.839279	0.0360834	0.0180417	−	0.999837i	$-0.494257\pi$
$541$	0.839279	0.0360834	0.0180417	+	0.999837i	$0.494257\pi$
$542$	9.55649	0.410486
$543$	12.9010	0.553635
$544$	−4.27854	−0.183441
$545$	39.2633	1.68185
$546$	−34.1763	−1.46261
$547$	−35.5183	−1.51865	−0.759327	−	0.650709i	$-0.774472\pi$
$547$	−35.5183	−1.51865	−0.759327	+	0.650709i	$0.774472\pi$
$548$	6.40013	0.273400
$549$	23.3679	0.997317
$550$	0	0
$551$	−5.99594	−0.255436
$552$	14.3030	0.608775
$553$	32.6263	1.38741
$554$	9.48715	0.403070
$555$	−24.0088	−1.01912
$556$	7.37781	0.312889
$557$	−11.6586	−0.493990	−0.246995	−	0.969017i	$-0.579443\pi$
$557$	−11.6586	−0.493990	−0.246995	+	0.969017i	$0.579443\pi$
$558$	16.7008	0.707002
$559$	−1.65887	−0.0701628
$560$	−7.20154	−0.304321
$561$	0	0
$562$	−9.15958	−0.386374
$563$	−2.58856	−0.109095	−0.0545473	−	0.998511i	$-0.517372\pi$
$563$	−2.58856	−0.109095	−0.0545473	+	0.998511i	$0.517372\pi$
$564$	−10.5070	−0.442423
$565$	2.80730	0.118104
$566$	21.1581	0.889342
$567$	29.1174	1.22282
$568$	−12.1074	−0.508014
$569$	11.8494	0.496751	0.248376	−	0.968664i	$-0.420103\pi$
$569$	11.8494	0.496751	0.248376	+	0.968664i	$0.420103\pi$
$570$	−6.04315	−0.253120
$571$	7.12206	0.298049	0.149024	−	0.988834i	$-0.452387\pi$
$571$	7.12206	0.298049	0.149024	+	0.988834i	$0.452387\pi$
$572$	0	0
$573$	−22.5294	−0.941178
$574$	−28.8888	−1.20580
$575$	−11.7977	−0.491997
$576$	2.29431	0.0955964
$577$	−40.0598	−1.66771	−0.833855	−	0.551984i	$-0.813871\pi$
$577$	−40.0598	−1.66771	−0.833855	+	0.551984i	$0.813871\pi$
$578$	−1.30588	−0.0543175
$579$	−19.3400	−0.803741
$580$	−15.7477	−0.653886
$581$	3.55664	0.147554
$582$	23.1310	0.958810
$583$	0	0
$584$	−13.8664	−0.573797
$585$	−32.6411	−1.34954
$586$	22.1069	0.913227
$587$	10.7483	0.443631	0.221815	−	0.975089i	$-0.428802\pi$
$587$	10.7483	0.443631	0.221815	+	0.975089i	$0.428802\pi$
$588$	1.19312	0.0492035
$589$	−7.27922	−0.299935
$590$	−14.2539	−0.586824
$591$	9.01184	0.370698
$592$	−3.97289	−0.163285
$593$	−15.0714	−0.618909	−0.309454	−	0.950914i	$-0.600146\pi$
$593$	−15.0714	−0.618909	−0.309454	+	0.950914i	$0.600146\pi$
$594$	0	0
$595$	−30.8121	−1.26317
$596$	−6.70635	−0.274703
$597$	1.86130	0.0761781
$598$	−33.6725	−1.37697
$599$	22.2653	0.909737	0.454868	−	0.890559i	$-0.349686\pi$
$599$	22.2653	0.909737	0.454868	+	0.890559i	$0.349686\pi$
$600$	−4.36696	−0.178281
$601$	−15.6242	−0.637326	−0.318663	−	0.947868i	$-0.603234\pi$
$601$	−15.6242	−0.637326	−0.318663	+	0.947868i	$0.603234\pi$
$602$	0.839705	0.0342238
$603$	−0.0339628	−0.00138307
$604$	−17.7575	−0.722542
$605$	0	0
$606$	9.29204	0.377463
$607$	17.3993	0.706217	0.353109	−	0.935582i	$-0.385125\pi$
$607$	17.3993	0.706217	0.353109	+	0.935582i	$0.385125\pi$
$608$	−1.00000	−0.0405554
$609$	37.8293	1.53292
$610$	−26.7501	−1.08308
$611$	24.7358	1.00070
$612$	9.81631	0.396801
$613$	−45.7036	−1.84595	−0.922975	−	0.384860i	$-0.874250\pi$
$613$	−45.7036	−1.84595	−0.922975	+	0.384860i	$0.874250\pi$
$614$	14.5129	0.585694
$615$	−63.6688	−2.56737
$616$	0	0
$617$	−15.1087	−0.608252	−0.304126	−	0.952632i	$-0.598364\pi$
$617$	−15.1087	−0.608252	−0.304126	+	0.952632i	$0.598364\pi$
$618$	−28.3337	−1.13975
$619$	15.8931	0.638799	0.319399	−	0.947620i	$-0.396519\pi$
$619$	15.8931	0.638799	0.319399	+	0.947620i	$0.396519\pi$
$620$	−19.1180	−0.767799
$621$	10.0934	0.405034
$622$	29.2145	1.17139
$623$	40.0313	1.60382
$624$	−12.4640	−0.498960
$625$	−30.8875	−1.23550
$626$	27.0783	1.08226
$627$	0	0
$628$	1.60819	0.0641738
$629$	−16.9982	−0.677761
$630$	16.5226	0.658276
$631$	−20.2056	−0.804371	−0.402185	−	0.915558i	$-0.631749\pi$
$631$	−20.2056	−0.804371	−0.402185	+	0.915558i	$0.631749\pi$
$632$	11.8987	0.473306
$633$	−31.7925	−1.26364
$634$	−27.2677	−1.08294
$635$	−4.30002	−0.170641
$636$	−8.58217	−0.340305
$637$	−2.80888	−0.111292
$638$	0	0
$639$	27.7781	1.09889
$640$	−2.62639	−0.103817
$641$	33.5188	1.32391	0.661957	−	0.749542i	$-0.269726\pi$
$641$	33.5188	1.32391	0.661957	+	0.749542i	$0.269726\pi$
$642$	−17.9374	−0.707933
$643$	−38.5327	−1.51958	−0.759791	−	0.650167i	$-0.774699\pi$
$643$	−38.5327	−1.51958	−0.759791	+	0.650167i	$0.774699\pi$
$644$	17.0447	0.671654
$645$	1.85065	0.0728691
$646$	−4.27854	−0.168337
$647$	15.4199	0.606221	0.303110	−	0.952955i	$-0.401975\pi$
$647$	15.4199	0.606221	0.303110	+	0.952955i	$0.401975\pi$
$648$	10.6191	0.417156
$649$	0	0
$650$	10.2808	0.403247
$651$	45.9258	1.79997
$652$	21.6766	0.848922
$653$	−47.1190	−1.84391	−0.921955	−	0.387297i	$-0.873409\pi$
$653$	−47.1190	−1.84391	−0.921955	+	0.387297i	$0.873409\pi$
$654$	−34.3980	−1.34507
$655$	56.5044	2.20781
$656$	−10.5357	−0.411350
$657$	31.8140	1.24118
$658$	−12.5210	−0.488119
$659$	42.9260	1.67216	0.836080	−	0.548608i	$-0.184842\pi$
$659$	42.9260	1.67216	0.836080	+	0.548608i	$0.184842\pi$
$660$	0	0
$661$	−14.7231	−0.572663	−0.286331	−	0.958131i	$-0.592436\pi$
$661$	−14.7231	−0.572663	−0.286331	+	0.958131i	$0.592436\pi$
$662$	−16.1001	−0.625748
$663$	−53.3277	−2.07108
$664$	1.29710	0.0503372
$665$	−7.20154	−0.279264
$666$	9.11506	0.353202
$667$	37.2717	1.44317
$668$	−12.2081	−0.472346
$669$	2.45896	0.0950688
$670$	0.0388785	0.00150201
$671$	0	0
$672$	6.30916	0.243381
$673$	−1.95301	−0.0752829	−0.0376415	−	0.999291i	$-0.511984\pi$
$673$	−1.95301	−0.0752829	−0.0376415	+	0.999291i	$0.511984\pi$
$674$	22.6353	0.871881
$675$	−3.08170	−0.118615
$676$	16.3431	0.628581
$677$	22.6415	0.870183	0.435091	−	0.900386i	$-0.356716\pi$
$677$	22.6415	0.870183	0.435091	+	0.900386i	$0.356716\pi$
$678$	−2.45944	−0.0944541
$679$	27.5649	1.05784
$680$	−11.2371	−0.430923
$681$	46.3960	1.77790
$682$	0	0
$683$	−42.4520	−1.62438	−0.812191	−	0.583392i	$-0.801725\pi$
$683$	−42.4520	−1.62438	−0.812191	+	0.583392i	$0.801725\pi$
$684$	2.29431	0.0877253
$685$	16.8092	0.642247
$686$	−17.7721	−0.678543
$687$	58.7846	2.24277
$688$	0.306239	0.0116752
$689$	20.2044	0.769725
$690$	37.5652	1.43008
$691$	−2.80082	−0.106548	−0.0532741	−	0.998580i	$-0.516966\pi$
$691$	−2.80082	−0.106548	−0.0532741	+	0.998580i	$0.516966\pi$
$692$	−11.5446	−0.438860
$693$	0	0
$694$	−18.8940	−0.717207
$695$	19.3770	0.735011
$696$	13.7963	0.522947
$697$	−45.0773	−1.70743
$698$	16.3651	0.619427
$699$	−26.0933	−0.986940
$700$	−5.20405	−0.196695
$701$	42.0175	1.58698	0.793489	−	0.608585i	$-0.208262\pi$
$701$	42.0175	1.58698	0.793489	+	0.608585i	$0.208262\pi$
$702$	−8.79567	−0.331971
$703$	−3.97289	−0.149840
$704$	0	0
$705$	−27.5953	−1.03930
$706$	28.7651	1.08259
$707$	11.0732	0.416450
$708$	12.4876	0.469314
$709$	16.1992	0.608375	0.304187	−	0.952612i	$-0.401615\pi$
$709$	16.1992	0.608375	0.304187	+	0.952612i	$0.401615\pi$
$710$	−31.7986	−1.19338
$711$	−27.2994	−1.02381
$712$	14.5993	0.547134
$713$	45.2488	1.69458
$714$	26.9940	1.01022
$715$	0	0
$716$	−0.875261	−0.0327100
$717$	4.11561	0.153700
$718$	−22.4187	−0.836657
$719$	8.51612	0.317598	0.158799	−	0.987311i	$-0.449238\pi$
$719$	8.51612	0.317598	0.158799	+	0.987311i	$0.449238\pi$
$720$	6.02576	0.224567
$721$	−33.7648	−1.25747
$722$	−1.00000	−0.0372161
$723$	45.7226	1.70044
$724$	5.60684	0.208377
$725$	−11.3797	−0.422632
$726$	0	0
$727$	−34.0860	−1.26418	−0.632091	−	0.774895i	$-0.717803\pi$
$727$	−34.0860	−1.26418	−0.632091	+	0.774895i	$0.717803\pi$
$728$	−14.8532	−0.550496
$729$	−13.1551	−0.487226
$730$	−36.4186	−1.34791
$731$	1.31025	0.0484615
$732$	23.4353	0.866195
$733$	−19.6418	−0.725485	−0.362743	−	0.931889i	$-0.618160\pi$
$733$	−19.6418	−0.725485	−0.362743	+	0.931889i	$0.618160\pi$
$734$	−21.2032	−0.782624
$735$	3.13360	0.115585
$736$	6.21615	0.229130
$737$	0	0
$738$	24.1722	0.889791
$739$	0.422833	0.0155542	0.00777709	−	0.999970i	$-0.497524\pi$
$739$	0.422833	0.0155542	0.00777709	+	0.999970i	$0.497524\pi$
$740$	−10.4343	−0.383574
$741$	−12.4640	−0.457877
$742$	−10.2272	−0.375454
$743$	−8.69396	−0.318950	−0.159475	−	0.987202i	$-0.550980\pi$
$743$	−8.69396	−0.318950	−0.159475	+	0.987202i	$0.550980\pi$
$744$	16.7490	0.614049
$745$	−17.6135	−0.645307
$746$	34.6222	1.26761
$747$	−2.97595	−0.108884
$748$	0	0
$749$	−21.3757	−0.781053
$750$	18.7464	0.684523
$751$	16.7911	0.612715	0.306357	−	0.951917i	$-0.400890\pi$
$751$	16.7911	0.612715	0.306357	+	0.951917i	$0.400890\pi$
$752$	−4.56638	−0.166519
$753$	25.4045	0.925792
$754$	−32.4796	−1.18284
$755$	−46.6380	−1.69733
$756$	4.45228	0.161928
$757$	23.9419	0.870183	0.435091	−	0.900386i	$-0.356716\pi$
$757$	23.9419	0.870183	0.435091	+	0.900386i	$0.356716\pi$
$758$	−18.9799	−0.689382
$759$	0	0
$760$	−2.62639	−0.0952691
$761$	−7.13012	−0.258467	−0.129233	−	0.991614i	$-0.541252\pi$
$761$	−7.13012	−0.258467	−0.129233	+	0.991614i	$0.541252\pi$
$762$	3.76718	0.136471
$763$	−40.9916	−1.48399
$764$	−9.79138	−0.354240
$765$	25.7814	0.932129
$766$	−15.4258	−0.557355
$767$	−29.3987	−1.06152
$768$	2.30094	0.0830279
$769$	9.77470	0.352485	0.176242	−	0.984347i	$-0.443606\pi$
$769$	9.77470	0.352485	0.176242	+	0.984347i	$0.443606\pi$
$770$	0	0
$771$	−14.7341	−0.530637
$772$	−8.40525	−0.302512
$773$	6.37683	0.229359	0.114679	−	0.993403i	$-0.463416\pi$
$773$	6.37683	0.229359	0.114679	+	0.993403i	$0.463416\pi$
$774$	−0.702608	−0.0252547
$775$	−13.8153	−0.496259
$776$	10.0528	0.360876
$777$	25.0656	0.899224
$778$	18.1318	0.650055
$779$	−10.5357	−0.377480
$780$	−32.7353	−1.17211
$781$	0	0
$782$	26.5960	0.951073
$783$	9.73583	0.347930
$784$	0.518537	0.0185192
$785$	4.22373	0.150751
$786$	−49.5026	−1.76570
$787$	−31.8703	−1.13605	−0.568027	−	0.823010i	$-0.692293\pi$
$787$	−31.8703	−1.13605	−0.568027	+	0.823010i	$0.692293\pi$
$788$	3.91660	0.139523
$789$	−9.81075	−0.349272
$790$	31.2507	1.11185
$791$	−2.93088	−0.104210
$792$	0	0
$793$	−55.1721	−1.95922
$794$	−9.44526	−0.335200
$795$	−22.5401	−0.799414
$796$	0.808932	0.0286718
$797$	7.17570	0.254176	0.127088	−	0.991891i	$-0.459437\pi$
$797$	7.17570	0.254176	0.127088	+	0.991891i	$0.459437\pi$
$798$	6.30916	0.223342
$799$	−19.5374	−0.691185
$800$	−1.89791	−0.0671011
$801$	−33.4955	−1.18351
$802$	−26.3735	−0.931281
$803$	0	0
$804$	−0.0340609	−0.00120123
$805$	44.7659	1.57779
$806$	−39.4310	−1.38890
$807$	−5.98976	−0.210850
$808$	4.03837	0.142069
$809$	−0.228395	−0.00802993	−0.00401496	−	0.999992i	$-0.501278\pi$
$809$	−0.228395	−0.00802993	−0.00401496	+	0.999992i	$0.501278\pi$
$810$	27.8898	0.979947
$811$	−2.94450	−0.103395	−0.0516976	−	0.998663i	$-0.516463\pi$
$811$	−2.94450	−0.103395	−0.0516976	+	0.998663i	$0.516463\pi$
$812$	16.4408	0.576960
$813$	−21.9889	−0.771184
$814$	0	0
$815$	56.9312	1.99421
$816$	9.84465	0.344632
$817$	0.306239	0.0107139
$818$	−27.1570	−0.949522
$819$	34.0779	1.19078
$820$	−27.6708	−0.966306
$821$	31.6730	1.10540	0.552698	−	0.833382i	$-0.313598\pi$
$821$	31.6730	1.10540	0.552698	+	0.833382i	$0.313598\pi$
$822$	−14.7263	−0.513639
$823$	−15.9174	−0.554847	−0.277424	−	0.960748i	$-0.589481\pi$
$823$	−15.9174	−0.554847	−0.277424	+	0.960748i	$0.589481\pi$
$824$	−12.3140	−0.428977
$825$	0	0
$826$	14.8813	0.517787
$827$	−45.1869	−1.57130	−0.785650	−	0.618671i	$-0.787672\pi$
$827$	−45.1869	−1.57130	−0.785650	+	0.618671i	$0.787672\pi$
$828$	−14.2618	−0.495632
$829$	7.99671	0.277737	0.138869	−	0.990311i	$-0.455653\pi$
$829$	7.99671	0.277737	0.138869	+	0.990311i	$0.455653\pi$
$830$	3.40668	0.118248
$831$	−21.8293	−0.757251
$832$	−5.41693	−0.187798
$833$	2.21858	0.0768693
$834$	−16.9759	−0.587827
$835$	−32.0632	−1.10959
$836$	0	0
$837$	11.8195	0.408543
$838$	−16.5473	−0.571616
$839$	−48.9847	−1.69114	−0.845570	−	0.533864i	$-0.820739\pi$
$839$	−48.9847	−1.69114	−0.845570	+	0.533864i	$0.820739\pi$
$840$	16.5703	0.571729
$841$	6.95128	0.239699
$842$	37.0039	1.27524
$843$	21.0756	0.725883
$844$	−13.8172	−0.475607
$845$	42.9233	1.47661
$846$	10.4767	0.360197
$847$	0	0
$848$	−3.72986	−0.128084
$849$	−48.6835	−1.67081
$850$	−8.12026	−0.278523
$851$	24.6961	0.846572
$852$	27.8583	0.954410
$853$	−5.48357	−0.187754	−0.0938770	−	0.995584i	$-0.529926\pi$
$853$	−5.48357	−0.187754	−0.0938770	+	0.995584i	$0.529926\pi$
$854$	27.9276	0.955662
$855$	6.02576	0.206077
$856$	−7.79569	−0.266451
$857$	47.1600	1.61096	0.805478	−	0.592626i	$-0.201909\pi$
$857$	47.1600	1.61096	0.805478	+	0.592626i	$0.201909\pi$
$858$	0	0
$859$	35.3677	1.20673	0.603364	−	0.797466i	$-0.293826\pi$
$859$	35.3677	1.20673	0.603364	+	0.797466i	$0.293826\pi$
$860$	0.804301	0.0274264
$861$	66.4714	2.26534
$862$	−11.7109	−0.398876
$863$	20.0753	0.683370	0.341685	−	0.939815i	$-0.389002\pi$
$863$	20.0753	0.683370	0.341685	+	0.939815i	$0.389002\pi$
$864$	1.62374	0.0552407
$865$	−30.3206	−1.03093
$866$	−2.12441	−0.0721904
$867$	3.00475	0.102047
$868$	19.9596	0.677472
$869$	0	0
$870$	36.2344	1.22846
$871$	0.0801869	0.00271703
$872$	−14.9495	−0.506255
$873$	−23.0644	−0.780611
$874$	6.21615	0.210265
$875$	22.3399	0.755225
$876$	31.9058	1.07800
$877$	−13.6785	−0.461890	−0.230945	−	0.972967i	$-0.574182\pi$
$877$	−13.6785	−0.461890	−0.230945	+	0.972967i	$0.574182\pi$
$878$	17.7590	0.599337
$879$	−50.8666	−1.71569
$880$	0	0
$881$	−44.3568	−1.49442	−0.747209	−	0.664590i	$-0.768606\pi$
$881$	−44.3568	−1.49442	−0.747209	+	0.664590i	$0.768606\pi$
$882$	−1.18969	−0.0400589
$883$	24.4649	0.823308	0.411654	−	0.911340i	$-0.364951\pi$
$883$	24.4649	0.823308	0.411654	+	0.911340i	$0.364951\pi$
$884$	−23.1765	−0.779511
$885$	32.7973	1.10247
$886$	−7.42988	−0.249612
$887$	−9.44528	−0.317141	−0.158571	−	0.987348i	$-0.550689\pi$
$887$	−9.44528	−0.317141	−0.158571	+	0.987348i	$0.550689\pi$
$888$	9.14138	0.306765
$889$	4.48930	0.150566
$890$	38.3435	1.28528
$891$	0	0
$892$	1.06868	0.0357819
$893$	−4.56638	−0.152808
$894$	15.4309	0.516086
$895$	−2.29877	−0.0768395
$896$	2.74200	0.0916036
$897$	77.4782	2.58692
$898$	19.0848	0.636869
$899$	43.6458	1.45567
$900$	4.35439	0.145146
$901$	−15.9583	−0.531649
$902$	0	0
$903$	−1.93211	−0.0642965
$904$	−1.06888	−0.0355506
$905$	14.7257	0.489500
$906$	40.8589	1.35745
$907$	−19.0258	−0.631741	−0.315870	−	0.948802i	$-0.602296\pi$
$907$	−19.0258	−0.631741	−0.315870	+	0.948802i	$0.602296\pi$
$908$	20.1639	0.669164
$909$	−9.26529	−0.307310
$910$	−39.0102	−1.29318
$911$	−31.4552	−1.04216	−0.521079	−	0.853509i	$-0.674470\pi$
$911$	−31.4552	−1.04216	−0.521079	+	0.853509i	$0.674470\pi$
$912$	2.30094	0.0761917
$913$	0	0
$914$	−2.89775	−0.0958490
$915$	61.5503	2.03479
$916$	25.5481	0.844133
$917$	−58.9916	−1.94807
$918$	6.94722	0.229293
$919$	−1.56648	−0.0516734	−0.0258367	−	0.999666i	$-0.508225\pi$
$919$	−1.56648	−0.0516734	−0.0258367	+	0.999666i	$0.508225\pi$
$920$	16.3260	0.538253
$921$	−33.3933	−1.10035
$922$	−7.89943	−0.260154
$923$	−65.5847	−2.15875
$924$	0	0
$925$	−7.54018	−0.247919
$926$	−20.8107	−0.683882
$927$	28.2521	0.927921
$928$	5.99594	0.196826
$929$	−9.10446	−0.298708	−0.149354	−	0.988784i	$-0.547719\pi$
$929$	−9.10446	−0.298708	−0.149354	+	0.988784i	$0.547719\pi$
$930$	43.9894	1.44247
$931$	0.518537	0.0169944
$932$	−11.3403	−0.371464
$933$	−67.2206	−2.20071
$934$	31.5622	1.03275
$935$	0	0
$936$	12.4281	0.406226
$937$	−11.1056	−0.362805	−0.181402	−	0.983409i	$-0.558064\pi$
$937$	−11.1056	−0.362805	−0.181402	+	0.983409i	$0.558064\pi$
$938$	−0.0405898	−0.00132531
$939$	−62.3054	−2.03326
$940$	−11.9931	−0.391171
$941$	34.2505	1.11654	0.558268	−	0.829661i	$-0.311466\pi$
$941$	34.2505	1.11654	0.558268	+	0.829661i	$0.311466\pi$
$942$	−3.70035	−0.120564
$943$	65.4915	2.13270
$944$	5.42719	0.176640
$945$	11.6934	0.380387
$946$	0	0
$947$	38.0662	1.23699	0.618493	−	0.785791i	$-0.287744\pi$
$947$	38.0662	1.23699	0.618493	+	0.785791i	$0.287744\pi$
$948$	−27.3782	−0.889204
$949$	−75.1134	−2.43829
$950$	−1.89791	−0.0615762
$951$	62.7413	2.03453
$952$	11.7317	0.380227
$953$	−14.8133	−0.479848	−0.239924	−	0.970792i	$-0.577123\pi$
$953$	−14.8133	−0.479848	−0.239924	+	0.970792i	$0.577123\pi$
$954$	8.55746	0.277058
$955$	−25.7160	−0.832149
$956$	1.78866	0.0578495
$957$	0	0
$958$	23.7066	0.765926
$959$	−17.5491	−0.566691
$960$	6.04315	0.195042
$961$	21.9870	0.709259
$962$	−21.5209	−0.693861
$963$	17.8858	0.576361
$964$	19.8713	0.640011
$965$	−22.0754	−0.710633
$966$	−39.2187	−1.26184
$967$	−33.3782	−1.07337	−0.536684	−	0.843783i	$-0.680323\pi$
$967$	−33.3782	−1.07337	−0.536684	+	0.843783i	$0.680323\pi$
$968$	0	0
$969$	9.84465	0.316256
$970$	26.4027	0.847738
$971$	43.0686	1.38214	0.691069	−	0.722789i	$-0.257140\pi$
$971$	43.0686	1.38214	0.691069	+	0.722789i	$0.257140\pi$
$972$	−19.5626	−0.627470
$973$	−20.2299	−0.648541
$974$	24.6716	0.790528
$975$	−23.6555	−0.757583
$976$	10.1851	0.326018
$977$	−29.7229	−0.950919	−0.475459	−	0.879738i	$-0.657718\pi$
$977$	−29.7229	−0.950919	−0.475459	+	0.879738i	$0.657718\pi$
$978$	−49.8766	−1.59488
$979$	0	0
$980$	1.36188	0.0435036
$981$	34.2990	1.09508
$982$	34.1976	1.09129
$983$	21.5927	0.688699	0.344349	−	0.938842i	$-0.388100\pi$
$983$	21.5927	0.688699	0.344349	+	0.938842i	$0.388100\pi$
$984$	24.2420	0.772806
$985$	10.2865	0.327755
$986$	25.6538	0.816985
$987$	28.8100	0.917033
$988$	−5.41693	−0.172335
$989$	−1.90363	−0.0605318
$990$	0	0
$991$	−22.9486	−0.728987	−0.364494	−	0.931206i	$-0.618758\pi$
$991$	−22.9486	−0.728987	−0.364494	+	0.931206i	$0.618758\pi$
$992$	7.27922	0.231115
$993$	37.0453	1.17560
$994$	33.1983	1.05299
$995$	2.12457	0.0673534
$996$	−2.98454	−0.0945689
$997$	41.3888	1.31080	0.655399	−	0.755283i	$-0.272501\pi$
$997$	41.3888	1.31080	0.655399	+	0.755283i	$0.272501\pi$
$998$	−23.0706	−0.730286
$999$	6.45094	0.204099

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4598.2.a.bx.1.7		8
11.7	odd	10		418.2.f.f.115.1	✓	16
11.8	odd	10		418.2.f.f.229.1	yes	16
11.10	odd	2		4598.2.a.ca.1.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
418.2.f.f.115.1	✓	16	11.7	odd	10
418.2.f.f.229.1	yes	16	11.8	odd	10
4598.2.a.bx.1.7		8	1.1	even	1	trivial
4598.2.a.ca.1.7		8	11.10	odd	2

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:	\( N \)	\(=\)	\( 4598 = 2 \cdot 11^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4598.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +2.30094 q^{3} +1.00000 q^{4} +2.62639 q^{5} -2.30094 q^{6} -2.74200 q^{7} -1.00000 q^{8} +2.29431 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +2.30094 q^{3} +1.00000 q^{4} +2.62639 q^{5} -2.30094 q^{6} -2.74200 q^{7} -1.00000 q^{8} +2.29431 q^{9} -2.62639 q^{10} +2.30094 q^{12} -5.41693 q^{13} +2.74200 q^{14} +6.04315 q^{15} +1.00000 q^{16} +4.27854 q^{17} -2.29431 q^{18} +1.00000 q^{19} +2.62639 q^{20} -6.30916 q^{21} -6.21615 q^{23} -2.30094 q^{24} +1.89791 q^{25} +5.41693 q^{26} -1.62374 q^{27} -2.74200 q^{28} -5.99594 q^{29} -6.04315 q^{30} -7.27922 q^{31} -1.00000 q^{32} -4.27854 q^{34} -7.20154 q^{35} +2.29431 q^{36} -3.97289 q^{37} -1.00000 q^{38} -12.4640 q^{39} -2.62639 q^{40} -10.5357 q^{41} +6.30916 q^{42} +0.306239 q^{43} +6.02576 q^{45} +6.21615 q^{46} -4.56638 q^{47} +2.30094 q^{48} +0.518537 q^{49} -1.89791 q^{50} +9.84465 q^{51} -5.41693 q^{52} -3.72986 q^{53} +1.62374 q^{54} +2.74200 q^{56} +2.30094 q^{57} +5.99594 q^{58} +5.42719 q^{59} +6.04315 q^{60} +10.1851 q^{61} +7.27922 q^{62} -6.29100 q^{63} +1.00000 q^{64} -14.2269 q^{65} -0.0148030 q^{67} +4.27854 q^{68} -14.3030 q^{69} +7.20154 q^{70} +12.1074 q^{71} -2.29431 q^{72} +13.8664 q^{73} +3.97289 q^{74} +4.36696 q^{75} +1.00000 q^{76} +12.4640 q^{78} -11.8987 q^{79} +2.62639 q^{80} -10.6191 q^{81} +10.5357 q^{82} -1.29710 q^{83} -6.30916 q^{84} +11.2371 q^{85} -0.306239 q^{86} -13.7963 q^{87} -14.5993 q^{89} -6.02576 q^{90} +14.8532 q^{91} -6.21615 q^{92} -16.7490 q^{93} +4.56638 q^{94} +2.62639 q^{95} -2.30094 q^{96} -10.0528 q^{97} -0.518537 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} + 8 q^{4} + 2 q^{5} - 8 q^{7} - 8 q^{8} + 20 q^{9}+O(q^{10}) \) 8 * q - 8 * q^2 + 8 * q^4 + 2 * q^5 - 8 * q^7 - 8 * q^8 + 20 * q^9	\( 8 q - 8 q^{2} + 8 q^{4} + 2 q^{5} - 8 q^{7} - 8 q^{8} + 20 q^{9} - 2 q^{10} - 18 q^{13} + 8 q^{14} + 10 q^{15} + 8 q^{16} - 4 q^{17} - 20 q^{18} + 8 q^{19} + 2 q^{20} - 14 q^{21} + 12 q^{23} + 18 q^{26} - 24 q^{27} - 8 q^{28} - 14 q^{29} - 10 q^{30} - 2 q^{31} - 8 q^{32} + 4 q^{34} - 40 q^{35} + 20 q^{36} - 22 q^{37} - 8 q^{38} + 4 q^{39} - 2 q^{40} - 8 q^{41} + 14 q^{42} - 28 q^{43} - 28 q^{45} - 12 q^{46} + 6 q^{47} + 32 q^{49} + 12 q^{51} - 18 q^{52} - 24 q^{53} + 24 q^{54} + 8 q^{56} + 14 q^{58} + 46 q^{59} + 10 q^{60} + 24 q^{61} + 2 q^{62} - 30 q^{63} + 8 q^{64} + 16 q^{65} - 22 q^{67} - 4 q^{68} - 38 q^{69} + 40 q^{70} + 8 q^{71} - 20 q^{72} - 16 q^{73} + 22 q^{74} + 6 q^{75} + 8 q^{76} - 4 q^{78} - 4 q^{79} + 2 q^{80} + 28 q^{81} + 8 q^{82} - 12 q^{83} - 14 q^{84} - 48 q^{85} + 28 q^{86} - 42 q^{87} - 28 q^{89} + 28 q^{90} - 12 q^{91} + 12 q^{92} + 22 q^{93} - 6 q^{94} + 2 q^{95} - 22 q^{97} - 32 q^{98}+O(q^{100}) \) 8 * q - 8 * q^2 + 8 * q^4 + 2 * q^5 - 8 * q^7 - 8 * q^8 + 20 * q^9 - 2 * q^10 - 18 * q^13 + 8 * q^14 + 10 * q^15 + 8 * q^16 - 4 * q^17 - 20 * q^18 + 8 * q^19 + 2 * q^20 - 14 * q^21 + 12 * q^23 + 18 * q^26 - 24 * q^27 - 8 * q^28 - 14 * q^29 - 10 * q^30 - 2 * q^31 - 8 * q^32 + 4 * q^34 - 40 * q^35 + 20 * q^36 - 22 * q^37 - 8 * q^38 + 4 * q^39 - 2 * q^40 - 8 * q^41 + 14 * q^42 - 28 * q^43 - 28 * q^45 - 12 * q^46 + 6 * q^47 + 32 * q^49 + 12 * q^51 - 18 * q^52 - 24 * q^53 + 24 * q^54 + 8 * q^56 + 14 * q^58 + 46 * q^59 + 10 * q^60 + 24 * q^61 + 2 * q^62 - 30 * q^63 + 8 * q^64 + 16 * q^65 - 22 * q^67 - 4 * q^68 - 38 * q^69 + 40 * q^70 + 8 * q^71 - 20 * q^72 - 16 * q^73 + 22 * q^74 + 6 * q^75 + 8 * q^76 - 4 * q^78 - 4 * q^79 + 2 * q^80 + 28 * q^81 + 8 * q^82 - 12 * q^83 - 14 * q^84 - 48 * q^85 + 28 * q^86 - 42 * q^87 - 28 * q^89 + 28 * q^90 - 12 * q^91 + 12 * q^92 + 22 * q^93 - 6 * q^94 + 2 * q^95 - 22 * q^97 - 32 * q^98

Introduction

L-functions

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