LMFDB - Embedded newform 5239.2.a.p.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5239,2,Mod(1,5239)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5239, 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("5239.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5239 = 13^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5239.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:	$41.8336256189$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 5 x^{17} - 15 x^{16} + 105 x^{15} + 50 x^{14} - 876 x^{13} + 294 x^{12} + 3702 x^{11} + \cdots + 3 $ x^18 - 5x^17 - 15x^16 + 105x^15 + 50x^14 - 876x^13 + 294x^12 + 3702x^11 - 2615x^10 - 8398x^9 + 7365x^8 + 10062x^7 - 9022x^6 - 5896x^5 + 4145x^4 + 1407x^3 - 146x^2 - 33*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 403)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.25209$ of defining polynomial
Character	$\chi$	$=$	5239.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.25209 q^{2} -3.32589 q^{3} +3.07190 q^{4} -0.820579 q^{5} +7.49019 q^{6} -3.93743 q^{7} -2.41402 q^{8} +8.06151 q^{9} +O(q^{10})$	\(q-2.25209 q^{2} -3.32589 q^{3} +3.07190 q^{4} -0.820579 q^{5} +7.49019 q^{6} -3.93743 q^{7} -2.41402 q^{8} +8.06151 q^{9} +1.84802 q^{10} +2.18520 q^{11} -10.2168 q^{12} +8.86745 q^{14} +2.72915 q^{15} -0.707225 q^{16} -4.11685 q^{17} -18.1552 q^{18} -1.31451 q^{19} -2.52074 q^{20} +13.0955 q^{21} -4.92127 q^{22} -9.40141 q^{23} +8.02874 q^{24} -4.32665 q^{25} -16.8340 q^{27} -12.0954 q^{28} +0.345650 q^{29} -6.14629 q^{30} +1.00000 q^{31} +6.42077 q^{32} -7.26773 q^{33} +9.27151 q^{34} +3.23097 q^{35} +24.7642 q^{36} +6.03586 q^{37} +2.96038 q^{38} +1.98089 q^{40} +9.11198 q^{41} -29.4921 q^{42} +6.42673 q^{43} +6.71272 q^{44} -6.61511 q^{45} +21.1728 q^{46} -1.68889 q^{47} +2.35215 q^{48} +8.50338 q^{49} +9.74400 q^{50} +13.6922 q^{51} +2.16668 q^{53} +37.9117 q^{54} -1.79313 q^{55} +9.50503 q^{56} +4.37190 q^{57} -0.778434 q^{58} -4.38829 q^{59} +8.38368 q^{60} -5.19747 q^{61} -2.25209 q^{62} -31.7417 q^{63} -13.0457 q^{64} +16.3676 q^{66} -2.51200 q^{67} -12.6466 q^{68} +31.2680 q^{69} -7.27644 q^{70} +3.37261 q^{71} -19.4606 q^{72} -6.64010 q^{73} -13.5933 q^{74} +14.3899 q^{75} -4.03803 q^{76} -8.60408 q^{77} -7.24898 q^{79} +0.580333 q^{80} +31.8035 q^{81} -20.5210 q^{82} -1.24172 q^{83} +40.2279 q^{84} +3.37820 q^{85} -14.4736 q^{86} -1.14959 q^{87} -5.27511 q^{88} -16.0346 q^{89} +14.8978 q^{90} -28.8802 q^{92} -3.32589 q^{93} +3.80353 q^{94} +1.07866 q^{95} -21.3547 q^{96} +2.73448 q^{97} -19.1504 q^{98} +17.6160 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 5 q^{2} + 19 q^{4} + 6 q^{5} + 6 q^{6} + 4 q^{7} + 15 q^{8} + 22 q^{9}+O(q^{10}) $ 18 * q + 5 * q^2 + 19 * q^4 + 6 * q^5 + 6 * q^6 + 4 * q^7 + 15 * q^8 + 22 * q^9	\( 18 q + 5 q^{2} + 19 q^{4} + 6 q^{5} + 6 q^{6} + 4 q^{7} + 15 q^{8} + 22 q^{9} + 3 q^{10} + 17 q^{11} + 4 q^{12} + 2 q^{14} + 12 q^{15} + 17 q^{16} - 7 q^{17} - 7 q^{18} + 4 q^{19} + 15 q^{20} + 14 q^{21} + 30 q^{22} - 4 q^{23} + 48 q^{24} + 12 q^{25} - 3 q^{27} + q^{28} - 15 q^{29} - 35 q^{30} + 18 q^{31} + 35 q^{32} + 17 q^{33} - 3 q^{34} + 17 q^{35} + 35 q^{36} + 3 q^{37} + 7 q^{38} - q^{40} + 44 q^{41} - 57 q^{42} - 4 q^{43} + 32 q^{44} - 5 q^{45} + 13 q^{46} + 12 q^{47} + 89 q^{48} + 44 q^{49} + 84 q^{50} - 14 q^{51} - 14 q^{53} + 21 q^{54} - 29 q^{55} - 11 q^{56} + 16 q^{57} - 49 q^{58} + 11 q^{59} + 27 q^{60} + 4 q^{61} + 5 q^{62} + 9 q^{63} + 17 q^{64} + 26 q^{66} + 16 q^{67} - 53 q^{68} - 4 q^{69} - 22 q^{70} + 5 q^{71} + 27 q^{72} + 32 q^{73} - q^{74} + 98 q^{75} + 42 q^{76} - 11 q^{77} - 3 q^{79} + 2 q^{80} - 10 q^{81} - 22 q^{82} + 18 q^{83} - 38 q^{84} - 2 q^{85} + 42 q^{86} + 34 q^{87} + 69 q^{88} + 54 q^{89} - 16 q^{90} - 43 q^{92} - 44 q^{94} + 2 q^{95} + 85 q^{96} + 28 q^{97} + 29 q^{98} + 77 q^{99}+O(q^{100}) \) 18 * q + 5 * q^2 + 19 * q^4 + 6 * q^5 + 6 * q^6 + 4 * q^7 + 15 * q^8 + 22 * q^9 + 3 * q^10 + 17 * q^11 + 4 * q^12 + 2 * q^14 + 12 * q^15 + 17 * q^16 - 7 * q^17 - 7 * q^18 + 4 * q^19 + 15 * q^20 + 14 * q^21 + 30 * q^22 - 4 * q^23 + 48 * q^24 + 12 * q^25 - 3 * q^27 + q^28 - 15 * q^29 - 35 * q^30 + 18 * q^31 + 35 * q^32 + 17 * q^33 - 3 * q^34 + 17 * q^35 + 35 * q^36 + 3 * q^37 + 7 * q^38 - q^40 + 44 * q^41 - 57 * q^42 - 4 * q^43 + 32 * q^44 - 5 * q^45 + 13 * q^46 + 12 * q^47 + 89 * q^48 + 44 * q^49 + 84 * q^50 - 14 * q^51 - 14 * q^53 + 21 * q^54 - 29 * q^55 - 11 * q^56 + 16 * q^57 - 49 * q^58 + 11 * q^59 + 27 * q^60 + 4 * q^61 + 5 * q^62 + 9 * q^63 + 17 * q^64 + 26 * q^66 + 16 * q^67 - 53 * q^68 - 4 * q^69 - 22 * q^70 + 5 * q^71 + 27 * q^72 + 32 * q^73 - q^74 + 98 * q^75 + 42 * q^76 - 11 * q^77 - 3 * q^79 + 2 * q^80 - 10 * q^81 - 22 * q^82 + 18 * q^83 - 38 * q^84 - 2 * q^85 + 42 * q^86 + 34 * q^87 + 69 * q^88 + 54 * q^89 - 16 * q^90 - 43 * q^92 - 44 * q^94 + 2 * q^95 + 85 * q^96 + 28 * q^97 + 29 * q^98 + 77 * q^99

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 $

$2$	−2.25209	−1.59247	−0.796233	−	0.604990i	$-0.793177\pi$
$2$	−2.25209	−1.59247	−0.796233	+	0.604990i	$0.793177\pi$
$3$	−3.32589	−1.92020	−0.960100	−	0.279655i	$-0.909780\pi$
$3$	−3.32589	−1.92020	−0.960100	+	0.279655i	$0.909780\pi$
$4$	3.07190	1.53595
$5$	−0.820579	−0.366974	−0.183487	−	0.983022i	$-0.558739\pi$
$5$	−0.820579	−0.366974	−0.183487	+	0.983022i	$0.558739\pi$
$6$	7.49019	3.05786
$7$	−3.93743	−1.48821	−0.744105	−	0.668063i	$-0.767124\pi$
$7$	−3.93743	−1.48821	−0.744105	+	0.668063i	$0.767124\pi$
$8$	−2.41402	−0.853484
$9$	8.06151	2.68717
$10$	1.84802	0.584394
$11$	2.18520	0.658863	0.329431	−	0.944179i	$-0.393143\pi$
$11$	2.18520	0.658863	0.329431	+	0.944179i	$0.393143\pi$
$12$	−10.2168	−2.94933
$13$	0	0
$13$	0	0
$14$	8.86745	2.36993
$15$	2.72915	0.704664
$16$	−0.707225	−0.176806
$17$	−4.11685	−0.998483	−0.499242	−	0.866463i	$-0.666388\pi$
$17$	−4.11685	−0.998483	−0.499242	+	0.866463i	$0.666388\pi$
$18$	−18.1552	−4.27923
$19$	−1.31451	−0.301568	−0.150784	−	0.988567i	$-0.548180\pi$
$19$	−1.31451	−0.301568	−0.150784	+	0.988567i	$0.548180\pi$
$20$	−2.52074	−0.563654
$21$	13.0955	2.85766
$22$	−4.92127	−1.04922
$23$	−9.40141	−1.96033	−0.980165	−	0.198184i	$-0.936496\pi$
$23$	−9.40141	−1.96033	−0.980165	+	0.198184i	$0.936496\pi$
$24$	8.02874	1.63886
$25$	−4.32665	−0.865330
$26$	0	0
$27$	−16.8340	−3.23971
$28$	−12.0954	−2.28582
$29$	0.345650	0.0641855	0.0320928	−	0.999485i	$-0.489783\pi$
$29$	0.345650	0.0641855	0.0320928	+	0.999485i	$0.489783\pi$
$30$	−6.14629	−1.12215
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	6.42077	1.13504
$33$	−7.26773	−1.26515
$34$	9.27151	1.59005
$35$	3.23097	0.546134
$36$	24.7642	4.12736
$37$	6.03586	0.992288	0.496144	−	0.868240i	$-0.334749\pi$
$37$	6.03586	0.992288	0.496144	+	0.868240i	$0.334749\pi$
$38$	2.96038	0.480238
$39$	0	0
$40$	1.98089	0.313206
$41$	9.11198	1.42305	0.711526	−	0.702659i	$-0.248004\pi$
$41$	9.11198	1.42305	0.711526	+	0.702659i	$0.248004\pi$
$42$	−29.4921	−4.55073
$43$	6.42673	0.980067	0.490034	−	0.871703i	$-0.336984\pi$
$43$	6.42673	0.980067	0.490034	+	0.871703i	$0.336984\pi$
$44$	6.71272	1.01198
$45$	−6.61511	−0.986122
$46$	21.1728	3.12176
$47$	−1.68889	−0.246350	−0.123175	−	0.992385i	$-0.539308\pi$
$47$	−1.68889	−0.246350	−0.123175	+	0.992385i	$0.539308\pi$
$48$	2.35215	0.339503
$49$	8.50338	1.21477
$50$	9.74400	1.37801
$51$	13.6922	1.91729
$52$	0	0
$53$	2.16668	0.297617	0.148808	−	0.988866i	$-0.452456\pi$
$53$	2.16668	0.297617	0.148808	+	0.988866i	$0.452456\pi$
$54$	37.9117	5.15913
$55$	−1.79313	−0.241786
$56$	9.50503	1.27016
$57$	4.37190	0.579072
$58$	−0.778434	−0.102213
$59$	−4.38829	−0.571306	−0.285653	−	0.958333i	$-0.592211\pi$
$59$	−4.38829	−0.571306	−0.285653	+	0.958333i	$0.592211\pi$
$60$	8.38368	1.08233
$61$	−5.19747	−0.665468	−0.332734	−	0.943021i	$-0.607971\pi$
$61$	−5.19747	−0.665468	−0.332734	+	0.943021i	$0.607971\pi$
$62$	−2.25209	−0.286015
$63$	−31.7417	−3.99908
$64$	−13.0457	−1.63071
$65$	0	0
$66$	16.3676	2.01471
$67$	−2.51200	−0.306890	−0.153445	−	0.988157i	$-0.549037\pi$
$67$	−2.51200	−0.306890	−0.153445	+	0.988157i	$0.549037\pi$
$68$	−12.6466	−1.53362
$69$	31.2680	3.76423
$70$	−7.27644	−0.869701
$71$	3.37261	0.400256	0.200128	−	0.979770i	$-0.435864\pi$
$71$	3.37261	0.400256	0.200128	+	0.979770i	$0.435864\pi$
$72$	−19.4606	−2.29346
$73$	−6.64010	−0.777165	−0.388583	−	0.921414i	$-0.627035\pi$
$73$	−6.64010	−0.777165	−0.388583	+	0.921414i	$0.627035\pi$
$74$	−13.5933	−1.58019
$75$	14.3899	1.66161
$76$	−4.03803	−0.463194
$77$	−8.60408	−0.980526
$78$	0	0
$79$	−7.24898	−0.815574	−0.407787	−	0.913077i	$-0.633699\pi$
$79$	−7.24898	−0.815574	−0.407787	+	0.913077i	$0.633699\pi$
$80$	0.580333	0.0648833
$81$	31.8035	3.53372
$82$	−20.5210	−2.26616
$83$	−1.24172	−0.136296	−0.0681482	−	0.997675i	$-0.521709\pi$
$83$	−1.24172	−0.136296	−0.0681482	+	0.997675i	$0.521709\pi$
$84$	40.2279	4.38923
$85$	3.37820	0.366417
$86$	−14.4736	−1.56073
$87$	−1.14959	−0.123249
$88$	−5.27511	−0.562329
$89$	−16.0346	−1.69966	−0.849831	−	0.527056i	$-0.823296\pi$
$89$	−16.0346	−1.69966	−0.849831	+	0.527056i	$0.823296\pi$
$90$	14.8978	1.57037
$91$	0	0
$92$	−28.8802	−3.01097
$93$	−3.32589	−0.344878
$94$	3.80353	0.392304
$95$	1.07866	0.110668
$96$	−21.3547	−2.17951
$97$	2.73448	0.277645	0.138822	−	0.990317i	$-0.455668\pi$
$97$	2.73448	0.277645	0.138822	+	0.990317i	$0.455668\pi$
$98$	−19.1504	−1.93448
$99$	17.6160	1.77048
$100$	−13.2910	−1.32910
$101$	−18.8736	−1.87799	−0.938995	−	0.343932i	$-0.888241\pi$
$101$	−18.8736	−1.87799	−0.938995	+	0.343932i	$0.888241\pi$
$102$	−30.8360	−3.05322
$103$	0.456171	0.0449478	0.0224739	−	0.999747i	$-0.492846\pi$
$103$	0.456171	0.0449478	0.0224739	+	0.999747i	$0.492846\pi$
$104$	0	0
$105$	−10.7458	−1.04869
$106$	−4.87956	−0.473945
$107$	−8.54911	−0.826473	−0.413237	−	0.910624i	$-0.635602\pi$
$107$	−8.54911	−0.826473	−0.413237	+	0.910624i	$0.635602\pi$
$108$	−51.7124	−4.97603
$109$	4.56542	0.437288	0.218644	−	0.975805i	$-0.429837\pi$
$109$	4.56542	0.437288	0.218644	+	0.975805i	$0.429837\pi$
$110$	4.03829	0.385035
$111$	−20.0746	−1.90539
$112$	2.78465	0.263125
$113$	−5.43823	−0.511586	−0.255793	−	0.966732i	$-0.582336\pi$
$113$	−5.43823	−0.511586	−0.255793	+	0.966732i	$0.582336\pi$
$114$	−9.84590	−0.922153
$115$	7.71460	0.719390
$116$	1.06180	0.0985858
$117$	0	0
$118$	9.88281	0.909786
$119$	16.2098	1.48595
$120$	−6.58821	−0.601419
$121$	−6.22490	−0.565900
$122$	11.7052	1.05974
$123$	−30.3054	−2.73255
$124$	3.07190	0.275865
$125$	7.65325	0.684527
$126$	71.4851	6.36839
$127$	0.340839	0.0302446	0.0151223	−	0.999886i	$-0.495186\pi$
$127$	0.340839	0.0302446	0.0151223	+	0.999886i	$0.495186\pi$
$128$	16.5385	1.46181
$129$	−21.3746	−1.88193
$130$	0	0
$131$	6.37644	0.557112	0.278556	−	0.960420i	$-0.410144\pi$
$131$	6.37644	0.557112	0.278556	+	0.960420i	$0.410144\pi$
$132$	−22.3257	−1.94321
$133$	5.17578	0.448797
$134$	5.65725	0.488712
$135$	13.8136	1.18889
$136$	9.93815	0.852189
$137$	11.4119	0.974982	0.487491	−	0.873128i	$-0.337912\pi$
$137$	11.4119	0.974982	0.487491	+	0.873128i	$0.337912\pi$
$138$	−70.4183	−5.99441
$139$	−8.33641	−0.707085	−0.353543	−	0.935418i	$-0.615023\pi$
$139$	−8.33641	−0.707085	−0.353543	+	0.935418i	$0.615023\pi$
$140$	9.92523	0.838835
$141$	5.61705	0.473041
$142$	−7.59542	−0.637394
$143$	0	0
$144$	−5.70130	−0.475109
$145$	−0.283633	−0.0235544
$146$	14.9541	1.23761
$147$	−28.2813	−2.33260
$148$	18.5416	1.52411
$149$	−2.69139	−0.220487	−0.110243	−	0.993905i	$-0.535163\pi$
$149$	−2.69139	−0.220487	−0.110243	+	0.993905i	$0.535163\pi$
$150$	−32.4074	−2.64606
$151$	−15.1401	−1.23208	−0.616041	−	0.787714i	$-0.711265\pi$
$151$	−15.1401	−1.23208	−0.616041	+	0.787714i	$0.711265\pi$
$152$	3.17324	0.257384
$153$	−33.1881	−2.68309
$154$	19.3772	1.56146
$155$	−0.820579	−0.0659105
$156$	0	0
$157$	0.421894	0.0336708	0.0168354	−	0.999858i	$-0.494641\pi$
$157$	0.421894	0.0336708	0.0168354	+	0.999858i	$0.494641\pi$
$158$	16.3253	1.29877
$159$	−7.20614	−0.571484
$160$	−5.26874	−0.416531
$161$	37.0174	2.91738
$162$	−71.6242	−5.62733
$163$	−18.7255	−1.46670	−0.733349	−	0.679853i	$-0.762044\pi$
$163$	−18.7255	−1.46670	−0.733349	+	0.679853i	$0.762044\pi$
$164$	27.9911	2.18574
$165$	5.96374	0.464277
$166$	2.79646	0.217047
$167$	−5.43878	−0.420866	−0.210433	−	0.977608i	$-0.567487\pi$
$167$	−5.43878	−0.420866	−0.210433	+	0.977608i	$0.567487\pi$
$168$	−31.6126	−2.43897
$169$	0	0
$170$	−7.60800	−0.583507
$171$	−10.5969	−0.810366
$172$	19.7423	1.50534
$173$	7.55699	0.574547	0.287273	−	0.957849i	$-0.407251\pi$
$173$	7.55699	0.574547	0.287273	+	0.957849i	$0.407251\pi$
$174$	2.58898	0.196270
$175$	17.0359	1.28779
$176$	−1.54543	−0.116491
$177$	14.5949	1.09702
$178$	36.1113	2.70665
$179$	20.6229	1.54143	0.770716	−	0.637179i	$-0.219899\pi$
$179$	20.6229	1.54143	0.770716	+	0.637179i	$0.219899\pi$
$180$	−20.3210	−1.51463
$181$	−17.6944	−1.31521	−0.657606	−	0.753362i	$-0.728431\pi$
$181$	−17.6944	−1.31521	−0.657606	+	0.753362i	$0.728431\pi$
$182$	0	0
$183$	17.2862	1.27783
$184$	22.6952	1.67311
$185$	−4.95289	−0.364144
$186$	7.49019	0.549207
$187$	−8.99615	−0.657864
$188$	−5.18810	−0.378381
$189$	66.2828	4.82137
$190$	−2.42923	−0.176235
$191$	−24.2312	−1.75331	−0.876653	−	0.481123i	$-0.840229\pi$
$191$	−24.2312	−1.75331	−0.876653	+	0.481123i	$0.840229\pi$
$192$	43.3884	3.13129
$193$	0.173282	0.0124731	0.00623655	−	0.999981i	$-0.498015\pi$
$193$	0.173282	0.0124731	0.00623655	+	0.999981i	$0.498015\pi$
$194$	−6.15829	−0.442140
$195$	0	0
$196$	26.1216	1.86583
$197$	8.92135	0.635620	0.317810	−	0.948154i	$-0.397053\pi$
$197$	8.92135	0.635620	0.317810	+	0.948154i	$0.397053\pi$
$198$	−39.6729	−2.81943
$199$	−18.2367	−1.29277	−0.646383	−	0.763013i	$-0.723719\pi$
$199$	−18.2367	−1.29277	−0.646383	+	0.763013i	$0.723719\pi$
$200$	10.4446	0.738545
$201$	8.35462	0.589290
$202$	42.5049	2.99064
$203$	−1.36097	−0.0955216
$204$	42.0610	2.94486
$205$	−7.47710	−0.522223
$206$	−1.02734	−0.0715780
$207$	−75.7896	−5.26774
$208$	0	0
$209$	−2.87246	−0.198692
$210$	24.2006	1.67000
$211$	−15.5645	−1.07151	−0.535753	−	0.844375i	$-0.679972\pi$
$211$	−15.5645	−1.07151	−0.535753	+	0.844375i	$0.679972\pi$
$212$	6.65584	0.457125
$213$	−11.2169	−0.768571
$214$	19.2533	1.31613
$215$	−5.27364	−0.359659
$216$	40.6376	2.76504
$217$	−3.93743	−0.267290
$218$	−10.2817	−0.696367
$219$	22.0842	1.49231
$220$	−5.50832	−0.371371
$221$	0	0
$222$	45.2097	3.03428
$223$	−19.8171	−1.32705	−0.663525	−	0.748154i	$-0.730940\pi$
$223$	−19.8171	−1.32705	−0.663525	+	0.748154i	$0.730940\pi$
$224$	−25.2813	−1.68918
$225$	−34.8794	−2.32529
$226$	12.2474	0.814683
$227$	−6.83650	−0.453754	−0.226877	−	0.973923i	$-0.572852\pi$
$227$	−6.83650	−0.453754	−0.226877	+	0.973923i	$0.572852\pi$
$228$	13.4300	0.889426
$229$	−13.4953	−0.891793	−0.445896	−	0.895085i	$-0.647115\pi$
$229$	−13.4953	−0.891793	−0.445896	+	0.895085i	$0.647115\pi$
$230$	−17.3740	−1.14560
$231$	28.6162	1.88281
$232$	−0.834404	−0.0547813
$233$	−6.98898	−0.457863	−0.228932	−	0.973443i	$-0.573523\pi$
$233$	−6.98898	−0.457863	−0.228932	+	0.973443i	$0.573523\pi$
$234$	0	0
$235$	1.38587	0.0904040
$236$	−13.4804	−0.877498
$237$	24.1093	1.56607
$238$	−36.5060	−2.36633
$239$	−10.5262	−0.680884	−0.340442	−	0.940266i	$-0.610577\pi$
$239$	−10.5262	−0.680884	−0.340442	+	0.940266i	$0.610577\pi$
$240$	−1.93012	−0.124589
$241$	11.4314	0.736363	0.368182	−	0.929754i	$-0.379981\pi$
$241$	11.4314	0.736363	0.368182	+	0.929754i	$0.379981\pi$
$242$	14.0190	0.901176
$243$	−55.2726	−3.54574
$244$	−15.9661	−1.02213
$245$	−6.97769	−0.445788
$246$	68.2505	4.35149
$247$	0	0
$248$	−2.41402	−0.153290
$249$	4.12982	0.261716
$250$	−17.2358	−1.09009
$251$	−19.3227	−1.21964	−0.609820	−	0.792540i	$-0.708758\pi$
$251$	−19.3227	−1.21964	−0.609820	+	0.792540i	$0.708758\pi$
$252$	−97.5073	−6.14238
$253$	−20.5440	−1.29159
$254$	−0.767600	−0.0481635
$255$	−11.2355	−0.703595
$256$	−11.1548	−0.697174
$257$	12.5944	0.785615	0.392808	−	0.919621i	$-0.371504\pi$
$257$	12.5944	0.785615	0.392808	+	0.919621i	$0.371504\pi$
$258$	48.1374	2.99691
$259$	−23.7658	−1.47673
$260$	0	0
$261$	2.78646	0.172478
$262$	−14.3603	−0.887183
$263$	−12.2974	−0.758287	−0.379144	−	0.925338i	$-0.623781\pi$
$263$	−12.2974	−0.758287	−0.379144	+	0.925338i	$0.623781\pi$
$264$	17.5444	1.07978
$265$	−1.77793	−0.109218
$266$	−11.6563	−0.714694
$267$	53.3291	3.26369
$268$	−7.71662	−0.471367
$269$	18.3362	1.11798	0.558990	−	0.829174i	$-0.311189\pi$
$269$	18.3362	1.11798	0.558990	+	0.829174i	$0.311189\pi$
$270$	−31.1095	−1.89326
$271$	−31.1835	−1.89426	−0.947132	−	0.320843i	$-0.896034\pi$
$271$	−31.1835	−1.89426	−0.947132	+	0.320843i	$0.896034\pi$
$272$	2.91154	0.176538
$273$	0	0
$274$	−25.7005	−1.55263
$275$	−9.45460	−0.570134
$276$	96.0523	5.78167
$277$	21.5861	1.29699	0.648493	−	0.761221i	$-0.275400\pi$
$277$	21.5861	1.29699	0.648493	+	0.761221i	$0.275400\pi$
$278$	18.7743	1.12601
$279$	8.06151	0.482630
$280$	−7.79962	−0.466117
$281$	28.2168	1.68327	0.841637	−	0.540043i	$-0.181592\pi$
$281$	28.2168	1.68327	0.841637	+	0.540043i	$0.181592\pi$
$282$	−12.6501	−0.753303
$283$	11.7519	0.698578	0.349289	−	0.937015i	$-0.386423\pi$
$283$	11.7519	0.698578	0.349289	+	0.937015i	$0.386423\pi$
$284$	10.3603	0.614773
$285$	−3.58748	−0.212504
$286$	0	0
$287$	−35.8778	−2.11780
$288$	51.7611	3.05005
$289$	−0.0515371	−0.00303159
$290$	0.638766	0.0375096
$291$	−9.09457	−0.533133
$292$	−20.3977	−1.19369
$293$	−1.42068	−0.0829968	−0.0414984	−	0.999139i	$-0.513213\pi$
$293$	−1.42068	−0.0829968	−0.0414984	+	0.999139i	$0.513213\pi$
$294$	63.6919	3.71459
$295$	3.60093	0.209654
$296$	−14.5707	−0.846902
$297$	−36.7857	−2.13452
$298$	6.06124	0.351118
$299$	0	0
$300$	44.2045	2.55215
$301$	−25.3048	−1.45855
$302$	34.0968	1.96205
$303$	62.7713	3.60612
$304$	0.929651	0.0533192
$305$	4.26493	0.244209
$306$	74.7424	4.27274
$307$	−28.6352	−1.63430	−0.817149	−	0.576427i	$-0.804447\pi$
$307$	−28.6352	−1.63430	−0.817149	+	0.576427i	$0.804447\pi$
$308$	−26.4309	−1.50604
$309$	−1.51717	−0.0863089
$310$	1.84802	0.104960
$311$	11.2329	0.636961	0.318481	−	0.947929i	$-0.396827\pi$
$311$	11.2329	0.636961	0.318481	+	0.947929i	$0.396827\pi$
$312$	0	0
$313$	−1.57444	−0.0889926	−0.0444963	−	0.999010i	$-0.514168\pi$
$313$	−1.57444	−0.0889926	−0.0444963	+	0.999010i	$0.514168\pi$
$314$	−0.950143	−0.0536197
$315$	26.0465	1.46756
$316$	−22.2681	−1.25268
$317$	0.962434	0.0540557	0.0270278	−	0.999635i	$-0.491396\pi$
$317$	0.962434	0.0540557	0.0270278	+	0.999635i	$0.491396\pi$
$318$	16.2289	0.910070
$319$	0.755314	0.0422895
$320$	10.7050	0.598428
$321$	28.4334	1.58699
$322$	−83.3665	−4.64583
$323$	5.41163	0.301111
$324$	97.6971	5.42762
$325$	0	0
$326$	42.1716	2.33567
$327$	−15.1841	−0.839681
$328$	−21.9965	−1.21455
$329$	6.64989	0.366620
$330$	−13.4309	−0.739345
$331$	6.60003	0.362771	0.181385	−	0.983412i	$-0.441942\pi$
$331$	6.60003	0.362771	0.181385	+	0.983412i	$0.441942\pi$
$332$	−3.81444	−0.209345
$333$	48.6581	2.66645
$334$	12.2486	0.670215
$335$	2.06129	0.112620
$336$	−9.26143	−0.505252
$337$	−13.8035	−0.751923	−0.375962	−	0.926635i	$-0.622688\pi$
$337$	−13.8035	−0.751923	−0.375962	+	0.926635i	$0.622688\pi$
$338$	0	0
$339$	18.0869	0.982347
$340$	10.3775	0.562799
$341$	2.18520	0.118335
$342$	23.8652	1.29048
$343$	−5.91947	−0.319621
$344$	−15.5142	−0.836472
$345$	−25.6579	−1.38137
$346$	−17.0190	−0.914947
$347$	13.6232	0.731334	0.365667	−	0.930746i	$-0.380841\pi$
$347$	13.6232	0.731334	0.365667	+	0.930746i	$0.380841\pi$
$348$	−3.53143	−0.189305
$349$	−8.19265	−0.438542	−0.219271	−	0.975664i	$-0.570368\pi$
$349$	−8.19265	−0.438542	−0.219271	+	0.975664i	$0.570368\pi$
$350$	−38.3663	−2.05077
$351$	0	0
$352$	14.0307	0.747837
$353$	−10.0897	−0.537023	−0.268511	−	0.963276i	$-0.586532\pi$
$353$	−10.0897	−0.537023	−0.268511	+	0.963276i	$0.586532\pi$
$354$	−32.8691	−1.74697
$355$	−2.76749	−0.146883
$356$	−49.2566	−2.61060
$357$	−53.9120	−2.85333
$358$	−46.4447	−2.45468
$359$	3.03712	0.160293	0.0801467	−	0.996783i	$-0.474461\pi$
$359$	3.03712	0.160293	0.0801467	+	0.996783i	$0.474461\pi$
$360$	15.9690	0.841639
$361$	−17.2721	−0.909057
$362$	39.8493	2.09443
$363$	20.7033	1.08664
$364$	0	0
$365$	5.44873	0.285199
$366$	−38.9300	−2.03491
$367$	2.84240	0.148372	0.0741861	−	0.997244i	$-0.476364\pi$
$367$	2.84240	0.148372	0.0741861	+	0.997244i	$0.476364\pi$
$368$	6.64891	0.346598
$369$	73.4564	3.82399
$370$	11.1544	0.579887
$371$	−8.53117	−0.442916
$372$	−10.2168	−0.529716
$373$	−32.1489	−1.66461	−0.832305	−	0.554318i	$-0.812979\pi$
$373$	−32.1489	−1.66461	−0.832305	+	0.554318i	$0.812979\pi$
$374$	20.2601	1.04763
$375$	−25.4538	−1.31443
$376$	4.07701	0.210256
$377$	0	0
$378$	−149.275	−7.67786
$379$	30.0175	1.54189	0.770946	−	0.636900i	$-0.219784\pi$
$379$	30.0175	1.54189	0.770946	+	0.636900i	$0.219784\pi$
$380$	3.31352	0.169980
$381$	−1.13359	−0.0580757
$382$	54.5707	2.79208
$383$	36.4382	1.86190	0.930952	−	0.365142i	$-0.118980\pi$
$383$	36.4382	1.86190	0.930952	+	0.365142i	$0.118980\pi$
$384$	−55.0051	−2.80697
$385$	7.06033	0.359828
$386$	−0.390246	−0.0198630
$387$	51.8092	2.63361
$388$	8.40006	0.426448
$389$	−27.3087	−1.38461	−0.692303	−	0.721607i	$-0.743404\pi$
$389$	−27.3087	−1.38461	−0.692303	+	0.721607i	$0.743404\pi$
$390$	0	0
$391$	38.7042	1.95736
$392$	−20.5273	−1.03679
$393$	−21.2073	−1.06977
$394$	−20.0917	−1.01220
$395$	5.94836	0.299294
$396$	54.1147	2.71937
$397$	2.73344	0.137188	0.0685938	−	0.997645i	$-0.478149\pi$
$397$	2.73344	0.137188	0.0685938	+	0.997645i	$0.478149\pi$
$398$	41.0707	2.05869
$399$	−17.2141	−0.861781
$400$	3.05991	0.152996
$401$	10.2811	0.513416	0.256708	−	0.966489i	$-0.417362\pi$
$401$	10.2811	0.513416	0.256708	+	0.966489i	$0.417362\pi$
$402$	−18.8153	−0.938424
$403$	0	0
$404$	−57.9777	−2.88450
$405$	−26.0972	−1.29678
$406$	3.06503	0.152115
$407$	13.1896	0.653782
$408$	−33.0531	−1.63637
$409$	−15.9928	−0.790793	−0.395397	−	0.918511i	$-0.629393\pi$
$409$	−15.9928	−0.790793	−0.395397	+	0.918511i	$0.629393\pi$
$410$	16.8391	0.831623
$411$	−37.9546	−1.87216
$412$	1.40131	0.0690377
$413$	17.2786	0.850224
$414$	170.685	8.38870
$415$	1.01893	0.0500172
$416$	0	0
$417$	27.7260	1.35775
$418$	6.46903	0.316411
$419$	14.0519	0.686483	0.343241	−	0.939247i	$-0.388475\pi$
$419$	14.0519	0.686483	0.343241	+	0.939247i	$0.388475\pi$
$420$	−33.0102	−1.61073
$421$	15.6249	0.761510	0.380755	−	0.924676i	$-0.375664\pi$
$421$	15.6249	0.761510	0.380755	+	0.924676i	$0.375664\pi$
$422$	35.0527	1.70634
$423$	−13.6150	−0.661984
$424$	−5.23041	−0.254011
$425$	17.8122	0.864018
$426$	25.2615	1.22392
$427$	20.4647	0.990356
$428$	−26.2620	−1.26942
$429$	0	0
$430$	11.8767	0.572745
$431$	18.0364	0.868782	0.434391	−	0.900724i	$-0.356964\pi$
$431$	18.0364	0.868782	0.434391	+	0.900724i	$0.356964\pi$
$432$	11.9054	0.572800
$433$	−7.81902	−0.375758	−0.187879	−	0.982192i	$-0.560161\pi$
$433$	−7.81902	−0.375758	−0.187879	+	0.982192i	$0.560161\pi$
$434$	8.86745	0.425651
$435$	0.943330	0.0452292
$436$	14.0245	0.671653
$437$	12.3582	0.591173
$438$	−49.7356	−2.37646
$439$	−21.0358	−1.00398	−0.501992	−	0.864872i	$-0.667399\pi$
$439$	−21.0358	−1.00398	−0.501992	+	0.864872i	$0.667399\pi$
$440$	4.32864	0.206360
$441$	68.5501	3.26429
$442$	0	0
$443$	−0.554601	−0.0263499	−0.0131749	−	0.999913i	$-0.504194\pi$
$443$	−0.554601	−0.0263499	−0.0131749	+	0.999913i	$0.504194\pi$
$444$	−61.6671	−2.92659
$445$	13.1576	0.623731
$446$	44.6298	2.11328
$447$	8.95124	0.423379
$448$	51.3665	2.42684
$449$	3.51651	0.165954	0.0829771	−	0.996551i	$-0.473557\pi$
$449$	3.51651	0.165954	0.0829771	+	0.996551i	$0.473557\pi$
$450$	78.5514	3.70295
$451$	19.9115	0.937597
$452$	−16.7057	−0.785770
$453$	50.3542	2.36585
$454$	15.3964	0.722589
$455$	0	0
$456$	−10.5538	−0.494228
$457$	−18.9499	−0.886441	−0.443220	−	0.896413i	$-0.646164\pi$
$457$	−18.9499	−0.886441	−0.443220	+	0.896413i	$0.646164\pi$
$458$	30.3926	1.42015
$459$	69.3031	3.23479
$460$	23.6985	1.10495
$461$	−20.1136	−0.936782	−0.468391	−	0.883521i	$-0.655166\pi$
$461$	−20.1136	−0.936782	−0.468391	+	0.883521i	$0.655166\pi$
$462$	−64.4462	−2.99831
$463$	0.739652	0.0343746	0.0171873	−	0.999852i	$-0.494529\pi$
$463$	0.739652	0.0343746	0.0171873	+	0.999852i	$0.494529\pi$
$464$	−0.244452	−0.0113484
$465$	2.72915	0.126561
$466$	15.7398	0.729132
$467$	−0.220183	−0.0101889	−0.00509443	−	0.999987i	$-0.501622\pi$
$467$	−0.220183	−0.0101889	−0.00509443	+	0.999987i	$0.501622\pi$
$468$	0	0
$469$	9.89083	0.456716
$470$	−3.12109	−0.143965
$471$	−1.40317	−0.0646548
$472$	10.5934	0.487601
$473$	14.0437	0.645730
$474$	−54.2962	−2.49391
$475$	5.68741	0.260956
$476$	49.7950	2.28235
$477$	17.4667	0.799747
$478$	23.7060	1.08428
$479$	−28.4910	−1.30179	−0.650894	−	0.759169i	$-0.725606\pi$
$479$	−28.4910	−1.30179	−0.650894	+	0.759169i	$0.725606\pi$
$480$	17.5232	0.799823
$481$	0	0
$482$	−25.7446	−1.17263
$483$	−123.116	−5.60196
$484$	−19.1223	−0.869194
$485$	−2.24386	−0.101888
$486$	124.479	5.64648
$487$	−31.7689	−1.43958	−0.719792	−	0.694190i	$-0.755763\pi$
$487$	−31.7689	−1.43958	−0.719792	+	0.694190i	$0.755763\pi$
$488$	12.5468	0.567966
$489$	62.2790	2.81635
$490$	15.7144	0.709903
$491$	−27.5256	−1.24222	−0.621108	−	0.783725i	$-0.713317\pi$
$491$	−27.5256	−1.24222	−0.621108	+	0.783725i	$0.713317\pi$
$492$	−93.0952	−4.19706
$493$	−1.42299	−0.0640882
$494$	0	0
$495$	−14.4553	−0.649719
$496$	−0.707225	−0.0317553
$497$	−13.2794	−0.595664
$498$	−9.30071	−0.416775
$499$	−15.7883	−0.706782	−0.353391	−	0.935476i	$-0.614972\pi$
$499$	−15.7883	−0.706782	−0.353391	+	0.935476i	$0.614972\pi$
$500$	23.5100	1.05140
$501$	18.0888	0.808147
$502$	43.5165	1.94224
$503$	−26.8064	−1.19524	−0.597619	−	0.801780i	$-0.703886\pi$
$503$	−26.8064	−1.19524	−0.597619	+	0.801780i	$0.703886\pi$
$504$	76.6249	3.41315
$505$	15.4872	0.689173
$506$	46.2668	2.05681
$507$	0	0
$508$	1.04702	0.0464542
$509$	11.7571	0.521126	0.260563	−	0.965457i	$-0.416092\pi$
$509$	11.7571	0.521126	0.260563	+	0.965457i	$0.416092\pi$
$510$	25.3033	1.12045
$511$	26.1450	1.15658
$512$	−7.95543	−0.351584
$513$	22.1284	0.976993
$514$	−28.3636	−1.25107
$515$	−0.374324	−0.0164947
$516$	−65.6606	−2.89055
$517$	−3.69056	−0.162311
$518$	53.5226	2.35165
$519$	−25.1337	−1.10325
$520$	0	0
$521$	−21.6302	−0.947635	−0.473818	−	0.880623i	$-0.657124\pi$
$521$	−21.6302	−0.947635	−0.473818	+	0.880623i	$0.657124\pi$
$522$	−6.27535	−0.274665
$523$	38.1623	1.66872	0.834360	−	0.551219i	$-0.185837\pi$
$523$	38.1623	1.66872	0.834360	+	0.551219i	$0.185837\pi$
$524$	19.5878	0.855697
$525$	−56.6595	−2.47282
$526$	27.6947	1.20755
$527$	−4.11685	−0.179333
$528$	5.13992	0.223686
$529$	65.3865	2.84289
$530$	4.00406	0.173925
$531$	−35.3762	−1.53520
$532$	15.8995	0.689330
$533$	0	0
$534$	−120.102	−5.19732
$535$	7.01521	0.303294
$536$	6.06401	0.261925
$537$	−68.5896	−2.95986
$538$	−41.2949	−1.78035
$539$	18.5816	0.800366
$540$	42.4341	1.82607
$541$	−26.5071	−1.13963	−0.569815	−	0.821773i	$-0.692985\pi$
$541$	−26.5071	−1.13963	−0.569815	+	0.821773i	$0.692985\pi$
$542$	70.2281	3.01655
$543$	58.8495	2.52547
$544$	−26.4333	−1.13332
$545$	−3.74629	−0.160473
$546$	0	0
$547$	26.5611	1.13567	0.567835	−	0.823143i	$-0.307781\pi$
$547$	26.5611	1.13567	0.567835	+	0.823143i	$0.307781\pi$
$548$	35.0561	1.49752
$549$	−41.8995	−1.78823
$550$	21.2926	0.907920
$551$	−0.454359	−0.0193563
$552$	−75.4815	−3.21271
$553$	28.5424	1.21375
$554$	−48.6139	−2.06541
$555$	16.4728	0.699230
$556$	−25.6086	−1.08605
$557$	−10.5601	−0.447448	−0.223724	−	0.974653i	$-0.571821\pi$
$557$	−10.5601	−0.447448	−0.223724	+	0.974653i	$0.571821\pi$
$558$	−18.1552	−0.768573
$559$	0	0
$560$	−2.28502	−0.0965599
$561$	29.9202	1.26323
$562$	−63.5468	−2.68056
$563$	−1.41915	−0.0598102	−0.0299051	−	0.999553i	$-0.509521\pi$
$563$	−1.41915	−0.0598102	−0.0299051	+	0.999553i	$0.509521\pi$
$564$	17.2550	0.726568
$565$	4.46249	0.187739
$566$	−26.4663	−1.11246
$567$	−125.224	−5.25891
$568$	−8.14154	−0.341612
$569$	−44.8852	−1.88169	−0.940843	−	0.338843i	$-0.889964\pi$
$569$	−44.8852	−1.88169	−0.940843	+	0.338843i	$0.889964\pi$
$570$	8.07933	0.338406
$571$	−19.9875	−0.836453	−0.418226	−	0.908343i	$-0.637348\pi$
$571$	−19.9875	−0.836453	−0.418226	+	0.908343i	$0.637348\pi$
$572$	0	0
$573$	80.5901	3.36670
$574$	80.8000	3.37253
$575$	40.6766	1.69633
$576$	−105.168	−4.38200
$577$	−13.4822	−0.561272	−0.280636	−	0.959814i	$-0.590545\pi$
$577$	−13.4822	−0.561272	−0.280636	+	0.959814i	$0.590545\pi$
$578$	0.116066	0.00482771
$579$	−0.576316	−0.0239509
$580$	−0.871292	−0.0361784
$581$	4.88919	0.202838
$582$	20.4818	0.848997
$583$	4.73464	0.196089
$584$	16.0293	0.663298
$585$	0	0
$586$	3.19949	0.132170
$587$	7.96084	0.328579	0.164289	−	0.986412i	$-0.447467\pi$
$587$	7.96084	0.328579	0.164289	+	0.986412i	$0.447467\pi$
$588$	−86.8773	−3.58276
$589$	−1.31451	−0.0541633
$590$	−8.10962	−0.333868
$591$	−29.6714	−1.22052
$592$	−4.26871	−0.175443
$593$	20.3911	0.837364	0.418682	−	0.908133i	$-0.362492\pi$
$593$	20.3911	0.837364	0.418682	+	0.908133i	$0.362492\pi$
$594$	82.8447	3.39916
$595$	−13.3014	−0.545306
$596$	−8.26767	−0.338657
$597$	60.6532	2.48237
$598$	0	0
$599$	22.5529	0.921488	0.460744	−	0.887533i	$-0.347583\pi$
$599$	22.5529	0.921488	0.460744	+	0.887533i	$0.347583\pi$
$600$	−34.7376	−1.41816
$601$	−0.322888	−0.0131709	−0.00658543	−	0.999978i	$-0.502096\pi$
$601$	−0.322888	−0.0131709	−0.00658543	+	0.999978i	$0.502096\pi$
$602$	56.9887	2.32269
$603$	−20.2505	−0.824665
$604$	−46.5088	−1.89242
$605$	5.10802	0.207670
$606$	−141.366	−5.74262
$607$	−0.285177	−0.0115750	−0.00578749	−	0.999983i	$-0.501842\pi$
$607$	−0.285177	−0.0115750	−0.00578749	+	0.999983i	$0.501842\pi$
$608$	−8.44014	−0.342293
$609$	4.52644	0.183421
$610$	−9.60501	−0.388895
$611$	0	0
$612$	−101.950	−4.12110
$613$	1.73611	0.0701209	0.0350605	−	0.999385i	$-0.488838\pi$
$613$	1.73611	0.0701209	0.0350605	+	0.999385i	$0.488838\pi$
$614$	64.4890	2.60256
$615$	24.8680	1.00277
$616$	20.7704	0.836863
$617$	40.7295	1.63971	0.819854	−	0.572572i	$-0.194054\pi$
$617$	40.7295	1.63971	0.819854	+	0.572572i	$0.194054\pi$
$618$	3.41680	0.137444
$619$	−3.22138	−0.129478	−0.0647391	−	0.997902i	$-0.520622\pi$
$619$	−3.22138	−0.129478	−0.0647391	+	0.997902i	$0.520622\pi$
$620$	−2.52074	−0.101235
$621$	158.263	6.35090
$622$	−25.2976	−1.01434
$623$	63.1351	2.52945
$624$	0	0
$625$	15.3532	0.614126
$626$	3.54578	0.141718
$627$	9.55347	0.381529
$628$	1.29602	0.0517167
$629$	−24.8487	−0.990783
$630$	−58.6591	−2.33703
$631$	41.4338	1.64945	0.824726	−	0.565533i	$-0.191329\pi$
$631$	41.4338	1.64945	0.824726	+	0.565533i	$0.191329\pi$
$632$	17.4992	0.696079
$633$	51.7658	2.05751
$634$	−2.16749	−0.0860818
$635$	−0.279685	−0.0110990
$636$	−22.1366	−0.877771
$637$	0	0
$638$	−1.70103	−0.0673446
$639$	27.1884	1.07556
$640$	−13.5711	−0.536446
$641$	28.9180	1.14219	0.571097	−	0.820883i	$-0.306518\pi$
$641$	28.9180	1.14219	0.571097	+	0.820883i	$0.306518\pi$
$642$	−64.0344	−2.52724
$643$	−28.5797	−1.12708	−0.563538	−	0.826090i	$-0.690560\pi$
$643$	−28.5797	−1.12708	−0.563538	+	0.826090i	$0.690560\pi$
$644$	113.714	4.48096
$645$	17.5395	0.690618
$646$	−12.1875	−0.479509
$647$	−35.3770	−1.39081	−0.695407	−	0.718616i	$-0.744776\pi$
$647$	−35.3770	−1.39081	−0.695407	+	0.718616i	$0.744776\pi$
$648$	−76.7741	−3.01597
$649$	−9.58929	−0.376413
$650$	0	0
$651$	13.0955	0.513251
$652$	−57.5230	−2.25277
$653$	29.4978	1.15434	0.577169	−	0.816625i	$-0.304157\pi$
$653$	29.4978	1.15434	0.577169	+	0.816625i	$0.304157\pi$
$654$	34.1959	1.33716
$655$	−5.23237	−0.204446
$656$	−6.44422	−0.251605
$657$	−53.5293	−2.08838
$658$	−14.9761	−0.583831
$659$	32.7342	1.27514	0.637572	−	0.770391i	$-0.279939\pi$
$659$	32.7342	1.27514	0.637572	+	0.770391i	$0.279939\pi$
$660$	18.3200	0.713106
$661$	−28.7096	−1.11667	−0.558336	−	0.829615i	$-0.688560\pi$
$661$	−28.7096	−1.11667	−0.558336	+	0.829615i	$0.688560\pi$
$662$	−14.8639	−0.577700
$663$	0	0
$664$	2.99753	0.116327
$665$	−4.24713	−0.164697
$666$	−109.582	−4.24623
$667$	−3.24960	−0.125825
$668$	−16.7074	−0.646429
$669$	65.9093	2.54820
$670$	−4.64221	−0.179344
$671$	−11.3575	−0.438452
$672$	84.0828	3.24357
$673$	−33.3642	−1.28610	−0.643048	−	0.765826i	$-0.722330\pi$
$673$	−33.3642	−1.28610	−0.643048	+	0.765826i	$0.722330\pi$
$674$	31.0866	1.19741
$675$	72.8349	2.80342
$676$	0	0
$677$	26.0371	1.00069	0.500344	−	0.865827i	$-0.333207\pi$
$677$	26.0371	1.00069	0.500344	+	0.865827i	$0.333207\pi$
$678$	−40.7334	−1.56436
$679$	−10.7668	−0.413193
$680$	−8.15503	−0.312731
$681$	22.7374	0.871299
$682$	−4.92127	−0.188445
$683$	−34.0270	−1.30201	−0.651003	−	0.759075i	$-0.725652\pi$
$683$	−34.0270	−1.30201	−0.651003	+	0.759075i	$0.725652\pi$
$684$	−32.5527	−1.24468
$685$	−9.36434	−0.357793
$686$	13.3312	0.508986
$687$	44.8837	1.71242
$688$	−4.54514	−0.173282
$689$	0	0
$690$	57.7838	2.19979
$691$	20.6663	0.786185	0.393092	−	0.919499i	$-0.371405\pi$
$691$	20.6663	0.786185	0.393092	+	0.919499i	$0.371405\pi$
$692$	23.2143	0.882476
$693$	−69.3619	−2.63484
$694$	−30.6807	−1.16462
$695$	6.84068	0.259482
$696$	2.77513	0.105191
$697$	−37.5127	−1.42089
$698$	18.4506	0.698364
$699$	23.2445	0.879189
$700$	52.3326	1.97799
$701$	−12.0241	−0.454144	−0.227072	−	0.973878i	$-0.572915\pi$
$701$	−12.0241	−0.454144	−0.227072	+	0.973878i	$0.572915\pi$
$702$	0	0
$703$	−7.93417	−0.299243
$704$	−28.5074	−1.07441
$705$	−4.60923	−0.173594
$706$	22.7230	0.855191
$707$	74.3134	2.79484
$708$	44.8342	1.68497
$709$	−28.6343	−1.07538	−0.537692	−	0.843141i	$-0.680704\pi$
$709$	−28.6343	−1.07538	−0.537692	+	0.843141i	$0.680704\pi$
$710$	6.23264	0.233907
$711$	−58.4377	−2.19159
$712$	38.7077	1.45063
$713$	−9.40141	−0.352086
$714$	121.415	4.54383
$715$	0	0
$716$	63.3517	2.36756
$717$	35.0090	1.30743
$718$	−6.83987	−0.255262
$719$	−2.00161	−0.0746473	−0.0373237	−	0.999303i	$-0.511883\pi$
$719$	−2.00161	−0.0746473	−0.0373237	+	0.999303i	$0.511883\pi$
$720$	4.67837	0.174352
$721$	−1.79614	−0.0668918
$722$	38.8982	1.44764
$723$	−38.0196	−1.41396
$724$	−54.3554	−2.02010
$725$	−1.49551	−0.0555417
$726$	−46.6256	−1.73044
$727$	25.5153	0.946310	0.473155	−	0.880979i	$-0.343115\pi$
$727$	25.5153	0.946310	0.473155	+	0.880979i	$0.343115\pi$
$728$	0	0
$729$	88.4201	3.27482
$730$	−12.2710	−0.454170
$731$	−26.4579	−0.978581
$732$	53.1015	1.96269
$733$	10.1067	0.373298	0.186649	−	0.982427i	$-0.440237\pi$
$733$	10.1067	0.373298	0.186649	+	0.982427i	$0.440237\pi$
$734$	−6.40134	−0.236278
$735$	23.2070	0.856003
$736$	−60.3643	−2.22506
$737$	−5.48923	−0.202198
$738$	−165.430	−6.08957
$739$	52.4005	1.92758	0.963791	−	0.266658i	$-0.0859194\pi$
$739$	52.4005	1.92758	0.963791	+	0.266658i	$0.0859194\pi$
$740$	−15.2148	−0.559307
$741$	0	0
$742$	19.2130	0.705330
$743$	29.2450	1.07290	0.536448	−	0.843933i	$-0.319766\pi$
$743$	29.2450	1.07290	0.536448	+	0.843933i	$0.319766\pi$
$744$	8.02874	0.294348
$745$	2.20849	0.0809129
$746$	72.4023	2.65084
$747$	−10.0101	−0.366252
$748$	−27.6353	−1.01045
$749$	33.6615	1.22997
$750$	57.3243	2.09319
$751$	10.9998	0.401387	0.200694	−	0.979654i	$-0.435680\pi$
$751$	10.9998	0.401387	0.200694	+	0.979654i	$0.435680\pi$
$752$	1.19442	0.0435562
$753$	64.2652	2.34196
$754$	0	0
$755$	12.4236	0.452142
$756$	203.614	7.40538
$757$	−11.8324	−0.430058	−0.215029	−	0.976608i	$-0.568985\pi$
$757$	−11.8324	−0.430058	−0.215029	+	0.976608i	$0.568985\pi$
$758$	−67.6020	−2.45541
$759$	68.3269	2.48011
$760$	−2.60389	−0.0944531
$761$	11.9338	0.432601	0.216300	−	0.976327i	$-0.430601\pi$
$761$	11.9338	0.432601	0.216300	+	0.976327i	$0.430601\pi$
$762$	2.55295	0.0924836
$763$	−17.9760	−0.650777
$764$	−74.4358	−2.69299
$765$	27.2334	0.984626
$766$	−82.0620	−2.96502
$767$	0	0
$768$	37.0995	1.33871
$769$	37.7230	1.36033	0.680163	−	0.733061i	$-0.261909\pi$
$769$	37.7230	1.36033	0.680163	+	0.733061i	$0.261909\pi$
$770$	−15.9005	−0.573014
$771$	−41.8874	−1.50854
$772$	0.532305	0.0191581
$773$	31.3867	1.12890	0.564450	−	0.825467i	$-0.309088\pi$
$773$	31.3867	1.12890	0.564450	+	0.825467i	$0.309088\pi$
$774$	−116.679	−4.19394
$775$	−4.32665	−0.155418
$776$	−6.60108	−0.236965
$777$	79.0423	2.83563
$778$	61.5016	2.20494
$779$	−11.9778	−0.429148
$780$	0	0
$781$	7.36984	0.263714
$782$	−87.1653	−3.11702
$783$	−5.81867	−0.207942
$784$	−6.01380	−0.214779
$785$	−0.346197	−0.0123563
$786$	47.7607	1.70357
$787$	−38.8043	−1.38322	−0.691612	−	0.722269i	$-0.743099\pi$
$787$	−38.8043	−1.38322	−0.691612	+	0.722269i	$0.743099\pi$
$788$	27.4055	0.976280
$789$	40.8996	1.45606
$790$	−13.3962	−0.476616
$791$	21.4127	0.761347
$792$	−42.5254	−1.51107
$793$	0	0
$794$	−6.15595	−0.218467
$795$	5.91320	0.209720
$796$	−56.0214	−1.98562
$797$	31.4067	1.11248	0.556241	−	0.831021i	$-0.312243\pi$
$797$	31.4067	1.11248	0.556241	+	0.831021i	$0.312243\pi$
$798$	38.7676	1.37236
$799$	6.95291	0.245976
$800$	−27.7804	−0.982186
$801$	−129.263	−4.56728
$802$	−23.1541	−0.817598
$803$	−14.5100	−0.512045
$804$	25.6646	0.905120
$805$	−30.3757	−1.07060
$806$	0	0
$807$	−60.9843	−2.14675
$808$	45.5611	1.60283
$809$	10.4509	0.367433	0.183717	−	0.982979i	$-0.441187\pi$
$809$	10.4509	0.367433	0.183717	+	0.982979i	$0.441187\pi$
$810$	58.7733	2.06508
$811$	−20.6332	−0.724530	−0.362265	−	0.932075i	$-0.617996\pi$
$811$	−20.6332	−0.724530	−0.362265	+	0.932075i	$0.617996\pi$
$812$	−4.18077	−0.146716
$813$	103.713	3.63737
$814$	−29.7040	−1.04113
$815$	15.3658	0.538240
$816$	−9.68345	−0.338988
$817$	−8.44798	−0.295557
$818$	36.0172	1.25931
$819$	0	0
$820$	−22.9689	−0.802109
$821$	11.5665	0.403673	0.201836	−	0.979419i	$-0.435309\pi$
$821$	11.5665	0.403673	0.201836	+	0.979419i	$0.435309\pi$
$822$	85.4771	2.98136
$823$	18.8810	0.658149	0.329075	−	0.944304i	$-0.393263\pi$
$823$	18.8810	0.658149	0.329075	+	0.944304i	$0.393263\pi$
$824$	−1.10120	−0.0383623
$825$	31.4449	1.09477
$826$	−38.9129	−1.35395
$827$	7.32358	0.254666	0.127333	−	0.991860i	$-0.459358\pi$
$827$	7.32358	0.254666	0.127333	+	0.991860i	$0.459358\pi$
$828$	−232.818	−8.09099
$829$	−3.38696	−0.117634	−0.0588170	−	0.998269i	$-0.518733\pi$
$829$	−3.38696	−0.117634	−0.0588170	+	0.998269i	$0.518733\pi$
$830$	−2.29472	−0.0796508
$831$	−71.7930	−2.49047
$832$	0	0
$833$	−35.0072	−1.21293
$834$	−62.4413	−2.16217
$835$	4.46295	0.154447
$836$	−8.82392	−0.305182
$837$	−16.8340	−0.581869
$838$	−31.6462	−1.09320
$839$	−25.4159	−0.877453	−0.438726	−	0.898621i	$-0.644570\pi$
$839$	−25.4159	−0.877453	−0.438726	+	0.898621i	$0.644570\pi$
$840$	25.9407	0.895038
$841$	−28.8805	−0.995880
$842$	−35.1886	−1.21268
$843$	−93.8459	−3.23222
$844$	−47.8127	−1.64578
$845$	0	0
$846$	30.6622	1.05419
$847$	24.5101	0.842177
$848$	−1.53233	−0.0526205
$849$	−39.0854	−1.34141
$850$	−40.1146	−1.37592
$851$	−56.7456	−1.94521
$852$	−34.4573	−1.18049
$853$	25.2008	0.862861	0.431430	−	0.902146i	$-0.358009\pi$
$853$	25.2008	0.862861	0.431430	+	0.902146i	$0.358009\pi$
$854$	−46.0883	−1.57711
$855$	8.69560	0.297383
$856$	20.6377	0.705382
$857$	−11.0152	−0.376272	−0.188136	−	0.982143i	$-0.560244\pi$
$857$	−11.0152	−0.376272	−0.188136	+	0.982143i	$0.560244\pi$
$858$	0	0
$859$	40.5158	1.38238	0.691191	−	0.722672i	$-0.257086\pi$
$859$	40.5158	1.38238	0.691191	+	0.722672i	$0.257086\pi$
$860$	−16.2001	−0.552419
$861$	119.326	4.06660
$862$	−40.6195	−1.38351
$863$	35.5770	1.21105	0.605527	−	0.795824i	$-0.292962\pi$
$863$	35.5770	1.21105	0.605527	+	0.795824i	$0.292962\pi$
$864$	−108.087	−3.67720
$865$	−6.20110	−0.210844
$866$	17.6091	0.598382
$867$	0.171406	0.00582127
$868$	−12.0954	−0.410545
$869$	−15.8405	−0.537351
$870$	−2.12446	−0.0720260
$871$	0	0
$872$	−11.0210	−0.373218
$873$	22.0441	0.746078
$874$	−27.8318	−0.941424
$875$	−30.1342	−1.01872
$876$	67.8405	2.29212
$877$	−39.1136	−1.32077	−0.660387	−	0.750926i	$-0.729608\pi$
$877$	−39.1136	−1.32077	−0.660387	+	0.750926i	$0.729608\pi$
$878$	47.3745	1.59881
$879$	4.72501	0.159370
$880$	1.26815	0.0427492
$881$	−33.5468	−1.13022	−0.565110	−	0.825016i	$-0.691166\pi$
$881$	−33.5468	−1.13022	−0.565110	+	0.825016i	$0.691166\pi$
$882$	−154.381	−5.19828
$883$	7.48637	0.251936	0.125968	−	0.992034i	$-0.459796\pi$
$883$	7.48637	0.251936	0.125968	+	0.992034i	$0.459796\pi$
$884$	0	0
$885$	−11.9763	−0.402579
$886$	1.24901	0.0419613
$887$	−4.62496	−0.155291	−0.0776454	−	0.996981i	$-0.524740\pi$
$887$	−4.62496	−0.155291	−0.0776454	+	0.996981i	$0.524740\pi$
$888$	48.4603	1.62622
$889$	−1.34203	−0.0450103
$890$	−29.6321	−0.993271
$891$	69.4970	2.32824
$892$	−60.8761	−2.03828
$893$	2.22006	0.0742914
$894$	−20.1590	−0.674217
$895$	−16.9227	−0.565665
$896$	−65.1192	−2.17548
$897$	0	0
$898$	−7.91949	−0.264277
$899$	0.345650	0.0115281
$900$	−107.146	−3.57153
$901$	−8.91991	−0.297165
$902$	−44.8425	−1.49309
$903$	84.1610	2.80070
$904$	13.1280	0.436630
$905$	14.5196	0.482649
$906$	−113.402	−3.76753
$907$	−30.5803	−1.01540	−0.507701	−	0.861533i	$-0.669505\pi$
$907$	−30.5803	−1.01540	−0.507701	+	0.861533i	$0.669505\pi$
$908$	−21.0010	−0.696944
$909$	−152.149	−5.04648
$910$	0	0
$911$	8.97901	0.297488	0.148744	−	0.988876i	$-0.452477\pi$
$911$	8.97901	0.297488	0.148744	+	0.988876i	$0.452477\pi$
$912$	−3.09191	−0.102383
$913$	−2.71341	−0.0898007
$914$	42.6769	1.41163
$915$	−14.1847	−0.468931
$916$	−41.4562	−1.36975
$917$	−25.1068	−0.829100
$918$	−156.077	−5.15130
$919$	28.7341	0.947852	0.473926	−	0.880565i	$-0.342836\pi$
$919$	28.7341	0.947852	0.473926	+	0.880565i	$0.342836\pi$
$920$	−18.6232	−0.613987
$921$	95.2374	3.13818
$922$	45.2975	1.49179
$923$	0	0
$924$	87.9061	2.89190
$925$	−26.1150	−0.858657
$926$	−1.66576	−0.0547403
$927$	3.67743	0.120783
$928$	2.21934	0.0728533
$929$	−8.21797	−0.269623	−0.134811	−	0.990871i	$-0.543043\pi$
$929$	−8.21797	−0.269623	−0.134811	+	0.990871i	$0.543043\pi$
$930$	−6.14629	−0.201545
$931$	−11.1777	−0.366336
$932$	−21.4695	−0.703255
$933$	−37.3595	−1.22309
$934$	0.495872	0.0162254
$935$	7.38205	0.241419
$936$	0	0
$937$	20.4532	0.668177	0.334088	−	0.942542i	$-0.391572\pi$
$937$	20.4532	0.668177	0.334088	+	0.942542i	$0.391572\pi$
$938$	−22.2750	−0.727305
$939$	5.23641	0.170884
$940$	4.25725	0.138856
$941$	44.3624	1.44617	0.723086	−	0.690758i	$-0.242723\pi$
$941$	44.3624	1.44617	0.723086	+	0.690758i	$0.242723\pi$
$942$	3.16007	0.102961
$943$	−85.6655	−2.78965
$944$	3.10351	0.101010
$945$	−54.3903	−1.76932
$946$	−31.6277	−1.02830
$947$	36.3752	1.18204	0.591018	−	0.806658i	$-0.298726\pi$
$947$	36.3752	1.18204	0.591018	+	0.806658i	$0.298726\pi$
$948$	74.0613	2.40540
$949$	0	0
$950$	−12.8085	−0.415564
$951$	−3.20094	−0.103798
$952$	−39.1308	−1.26824
$953$	13.3934	0.433856	0.216928	−	0.976188i	$-0.430396\pi$
$953$	13.3934	0.433856	0.216928	+	0.976188i	$0.430396\pi$
$954$	−39.3367	−1.27357
$955$	19.8836	0.643418
$956$	−32.3355	−1.04580
$957$	−2.51209	−0.0812043
$958$	64.1642	2.07305
$959$	−44.9335	−1.45098
$960$	−35.6036	−1.14910
$961$	1.00000	0.0322581
$962$	0	0
$963$	−68.9187	−2.22088
$964$	35.1162	1.13102
$965$	−0.142191	−0.00457730
$966$	277.267	8.92094
$967$	−38.8876	−1.25054	−0.625271	−	0.780408i	$-0.715011\pi$
$967$	−38.8876	−1.25054	−0.625271	+	0.780408i	$0.715011\pi$
$968$	15.0270	0.482986
$969$	−17.9984	−0.578193
$970$	5.05336	0.162254
$971$	−39.2139	−1.25843	−0.629217	−	0.777230i	$-0.716624\pi$
$971$	−39.2139	−1.25843	−0.629217	+	0.777230i	$0.716624\pi$
$972$	−169.792	−5.44608
$973$	32.8241	1.05229
$974$	71.5463	2.29249
$975$	0	0
$976$	3.67578	0.117659
$977$	9.39118	0.300451	0.150225	−	0.988652i	$-0.452000\pi$
$977$	9.39118	0.300451	0.150225	+	0.988652i	$0.452000\pi$
$978$	−140.258	−4.48495
$979$	−35.0388	−1.11984
$980$	−21.4348	−0.684709
$981$	36.8042	1.17507
$982$	61.9902	1.97819
$983$	−1.17816	−0.0375776	−0.0187888	−	0.999823i	$-0.505981\pi$
$983$	−1.17816	−0.0375776	−0.0187888	+	0.999823i	$0.505981\pi$
$984$	73.1578	2.33218
$985$	−7.32066	−0.233256
$986$	3.20470	0.102058
$987$	−22.1168	−0.703985
$988$	0	0
$989$	−60.4204	−1.92126
$990$	32.5547	1.03466
$991$	57.8221	1.83678	0.918389	−	0.395678i	$-0.129490\pi$
$991$	57.8221	1.83678	0.918389	+	0.395678i	$0.129490\pi$
$992$	6.42077	0.203860
$993$	−21.9509	−0.696592
$994$	29.9065	0.948576
$995$	14.9647	0.474411
$996$	12.6864	0.401984
$997$	−44.8839	−1.42149	−0.710744	−	0.703451i	$-0.751642\pi$
$997$	−44.8839	−1.42149	−0.710744	+	0.703451i	$0.751642\pi$
$998$	35.5567	1.12553
$999$	−101.608	−3.21472

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5239.2.a.p.1.2		18
13.3	even	3		403.2.f.c.373.17	yes	36
13.9	even	3		403.2.f.c.94.17	✓	36
13.12	even	2		5239.2.a.o.1.17		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.f.c.94.17	✓	36	13.9	even	3
403.2.f.c.373.17	yes	36	13.3	even	3
5239.2.a.o.1.17		18	13.12	even	2
5239.2.a.p.1.2		18	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5239 = 13^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5239.a (trivial)

\(f(q)\)	\(=\)	\(q-2.25209 q^{2} -3.32589 q^{3} +3.07190 q^{4} -0.820579 q^{5} +7.49019 q^{6} -3.93743 q^{7} -2.41402 q^{8} +8.06151 q^{9} +O(q^{10})\)	\(q-2.25209 q^{2} -3.32589 q^{3} +3.07190 q^{4} -0.820579 q^{5} +7.49019 q^{6} -3.93743 q^{7} -2.41402 q^{8} +8.06151 q^{9} +1.84802 q^{10} +2.18520 q^{11} -10.2168 q^{12} +8.86745 q^{14} +2.72915 q^{15} -0.707225 q^{16} -4.11685 q^{17} -18.1552 q^{18} -1.31451 q^{19} -2.52074 q^{20} +13.0955 q^{21} -4.92127 q^{22} -9.40141 q^{23} +8.02874 q^{24} -4.32665 q^{25} -16.8340 q^{27} -12.0954 q^{28} +0.345650 q^{29} -6.14629 q^{30} +1.00000 q^{31} +6.42077 q^{32} -7.26773 q^{33} +9.27151 q^{34} +3.23097 q^{35} +24.7642 q^{36} +6.03586 q^{37} +2.96038 q^{38} +1.98089 q^{40} +9.11198 q^{41} -29.4921 q^{42} +6.42673 q^{43} +6.71272 q^{44} -6.61511 q^{45} +21.1728 q^{46} -1.68889 q^{47} +2.35215 q^{48} +8.50338 q^{49} +9.74400 q^{50} +13.6922 q^{51} +2.16668 q^{53} +37.9117 q^{54} -1.79313 q^{55} +9.50503 q^{56} +4.37190 q^{57} -0.778434 q^{58} -4.38829 q^{59} +8.38368 q^{60} -5.19747 q^{61} -2.25209 q^{62} -31.7417 q^{63} -13.0457 q^{64} +16.3676 q^{66} -2.51200 q^{67} -12.6466 q^{68} +31.2680 q^{69} -7.27644 q^{70} +3.37261 q^{71} -19.4606 q^{72} -6.64010 q^{73} -13.5933 q^{74} +14.3899 q^{75} -4.03803 q^{76} -8.60408 q^{77} -7.24898 q^{79} +0.580333 q^{80} +31.8035 q^{81} -20.5210 q^{82} -1.24172 q^{83} +40.2279 q^{84} +3.37820 q^{85} -14.4736 q^{86} -1.14959 q^{87} -5.27511 q^{88} -16.0346 q^{89} +14.8978 q^{90} -28.8802 q^{92} -3.32589 q^{93} +3.80353 q^{94} +1.07866 q^{95} -21.3547 q^{96} +2.73448 q^{97} -19.1504 q^{98} +17.6160 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 5 q^{2} + 19 q^{4} + 6 q^{5} + 6 q^{6} + 4 q^{7} + 15 q^{8} + 22 q^{9}+O(q^{10}) \) 18 * q + 5 * q^2 + 19 * q^4 + 6 * q^5 + 6 * q^6 + 4 * q^7 + 15 * q^8 + 22 * q^9	\( 18 q + 5 q^{2} + 19 q^{4} + 6 q^{5} + 6 q^{6} + 4 q^{7} + 15 q^{8} + 22 q^{9} + 3 q^{10} + 17 q^{11} + 4 q^{12} + 2 q^{14} + 12 q^{15} + 17 q^{16} - 7 q^{17} - 7 q^{18} + 4 q^{19} + 15 q^{20} + 14 q^{21} + 30 q^{22} - 4 q^{23} + 48 q^{24} + 12 q^{25} - 3 q^{27} + q^{28} - 15 q^{29} - 35 q^{30} + 18 q^{31} + 35 q^{32} + 17 q^{33} - 3 q^{34} + 17 q^{35} + 35 q^{36} + 3 q^{37} + 7 q^{38} - q^{40} + 44 q^{41} - 57 q^{42} - 4 q^{43} + 32 q^{44} - 5 q^{45} + 13 q^{46} + 12 q^{47} + 89 q^{48} + 44 q^{49} + 84 q^{50} - 14 q^{51} - 14 q^{53} + 21 q^{54} - 29 q^{55} - 11 q^{56} + 16 q^{57} - 49 q^{58} + 11 q^{59} + 27 q^{60} + 4 q^{61} + 5 q^{62} + 9 q^{63} + 17 q^{64} + 26 q^{66} + 16 q^{67} - 53 q^{68} - 4 q^{69} - 22 q^{70} + 5 q^{71} + 27 q^{72} + 32 q^{73} - q^{74} + 98 q^{75} + 42 q^{76} - 11 q^{77} - 3 q^{79} + 2 q^{80} - 10 q^{81} - 22 q^{82} + 18 q^{83} - 38 q^{84} - 2 q^{85} + 42 q^{86} + 34 q^{87} + 69 q^{88} + 54 q^{89} - 16 q^{90} - 43 q^{92} - 44 q^{94} + 2 q^{95} + 85 q^{96} + 28 q^{97} + 29 q^{98} + 77 q^{99}+O(q^{100}) \) 18 * q + 5 * q^2 + 19 * q^4 + 6 * q^5 + 6 * q^6 + 4 * q^7 + 15 * q^8 + 22 * q^9 + 3 * q^10 + 17 * q^11 + 4 * q^12 + 2 * q^14 + 12 * q^15 + 17 * q^16 - 7 * q^17 - 7 * q^18 + 4 * q^19 + 15 * q^20 + 14 * q^21 + 30 * q^22 - 4 * q^23 + 48 * q^24 + 12 * q^25 - 3 * q^27 + q^28 - 15 * q^29 - 35 * q^30 + 18 * q^31 + 35 * q^32 + 17 * q^33 - 3 * q^34 + 17 * q^35 + 35 * q^36 + 3 * q^37 + 7 * q^38 - q^40 + 44 * q^41 - 57 * q^42 - 4 * q^43 + 32 * q^44 - 5 * q^45 + 13 * q^46 + 12 * q^47 + 89 * q^48 + 44 * q^49 + 84 * q^50 - 14 * q^51 - 14 * q^53 + 21 * q^54 - 29 * q^55 - 11 * q^56 + 16 * q^57 - 49 * q^58 + 11 * q^59 + 27 * q^60 + 4 * q^61 + 5 * q^62 + 9 * q^63 + 17 * q^64 + 26 * q^66 + 16 * q^67 - 53 * q^68 - 4 * q^69 - 22 * q^70 + 5 * q^71 + 27 * q^72 + 32 * q^73 - q^74 + 98 * q^75 + 42 * q^76 - 11 * q^77 - 3 * q^79 + 2 * q^80 - 10 * q^81 - 22 * q^82 + 18 * q^83 - 38 * q^84 - 2 * q^85 + 42 * q^86 + 34 * q^87 + 69 * q^88 + 54 * q^89 - 16 * q^90 - 43 * q^92 - 44 * q^94 + 2 * q^95 + 85 * q^96 + 28 * q^97 + 29 * q^98 + 77 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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