LMFDB - Embedded newform 8013.2.a.b.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8013,2,Mod(1,8013)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8013, 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("8013.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8013 = 3 \cdot 2671 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8013.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:	$63.9841271397$
Analytic rank:	$0$
Dimension:	$106$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Character	$\chi$	$=$	8013.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.68470 q^{2} -1.00000 q^{3} +5.20764 q^{4} +3.29964 q^{5} +2.68470 q^{6} +2.53834 q^{7} -8.61155 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.68470 q^{2} -1.00000 q^{3} +5.20764 q^{4} +3.29964 q^{5} +2.68470 q^{6} +2.53834 q^{7} -8.61155 q^{8} +1.00000 q^{9} -8.85857 q^{10} -3.38448 q^{11} -5.20764 q^{12} -2.10791 q^{13} -6.81468 q^{14} -3.29964 q^{15} +12.7042 q^{16} +2.42812 q^{17} -2.68470 q^{18} -5.18666 q^{19} +17.1833 q^{20} -2.53834 q^{21} +9.08633 q^{22} +5.22110 q^{23} +8.61155 q^{24} +5.88765 q^{25} +5.65912 q^{26} -1.00000 q^{27} +13.2187 q^{28} +7.21769 q^{29} +8.85857 q^{30} +0.885022 q^{31} -16.8839 q^{32} +3.38448 q^{33} -6.51879 q^{34} +8.37560 q^{35} +5.20764 q^{36} -1.34140 q^{37} +13.9247 q^{38} +2.10791 q^{39} -28.4151 q^{40} +8.43898 q^{41} +6.81468 q^{42} +7.65624 q^{43} -17.6252 q^{44} +3.29964 q^{45} -14.0171 q^{46} +2.97342 q^{47} -12.7042 q^{48} -0.556854 q^{49} -15.8066 q^{50} -2.42812 q^{51} -10.9772 q^{52} -7.65239 q^{53} +2.68470 q^{54} -11.1676 q^{55} -21.8590 q^{56} +5.18666 q^{57} -19.3774 q^{58} +9.71669 q^{59} -17.1833 q^{60} +2.83907 q^{61} -2.37602 q^{62} +2.53834 q^{63} +19.9199 q^{64} -6.95536 q^{65} -9.08633 q^{66} +4.85131 q^{67} +12.6448 q^{68} -5.22110 q^{69} -22.4860 q^{70} +1.91662 q^{71} -8.61155 q^{72} -11.7804 q^{73} +3.60126 q^{74} -5.88765 q^{75} -27.0102 q^{76} -8.59095 q^{77} -5.65912 q^{78} -11.2855 q^{79} +41.9193 q^{80} +1.00000 q^{81} -22.6562 q^{82} +15.4117 q^{83} -13.2187 q^{84} +8.01194 q^{85} -20.5547 q^{86} -7.21769 q^{87} +29.1456 q^{88} -7.03493 q^{89} -8.85857 q^{90} -5.35058 q^{91} +27.1896 q^{92} -0.885022 q^{93} -7.98275 q^{94} -17.1141 q^{95} +16.8839 q^{96} -16.1263 q^{97} +1.49499 q^{98} -3.38448 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 106 q + 15 q^{2} - 106 q^{3} + 109 q^{4} + 16 q^{5} - 15 q^{6} + 35 q^{7} + 48 q^{8} + 106 q^{9}+O(q^{10}) $ 106 * q + 15 * q^2 - 106 * q^3 + 109 * q^4 + 16 * q^5 - 15 * q^6 + 35 * q^7 + 48 * q^8 + 106 * q^9	\( 106 q + 15 q^{2} - 106 q^{3} + 109 q^{4} + 16 q^{5} - 15 q^{6} + 35 q^{7} + 48 q^{8} + 106 q^{9} - 3 q^{10} + 55 q^{11} - 109 q^{12} - 8 q^{13} + 27 q^{14} - 16 q^{15} + 111 q^{16} + 28 q^{17} + 15 q^{18} + q^{19} + 54 q^{20} - 35 q^{21} + 20 q^{22} + 62 q^{23} - 48 q^{24} + 102 q^{25} + 21 q^{26} - 106 q^{27} + 79 q^{28} + 36 q^{29} + 3 q^{30} + q^{31} + 111 q^{32} - 55 q^{33} - 27 q^{34} + 72 q^{35} + 109 q^{36} + 31 q^{37} + 43 q^{38} + 8 q^{39} - 13 q^{40} + 35 q^{41} - 27 q^{42} + 98 q^{43} + 121 q^{44} + 16 q^{45} + 8 q^{46} + 75 q^{47} - 111 q^{48} + 49 q^{49} + 83 q^{50} - 28 q^{51} - 18 q^{52} + 60 q^{53} - 15 q^{54} + 14 q^{55} + 85 q^{56} - q^{57} + 65 q^{58} + 77 q^{59} - 54 q^{60} - 55 q^{61} + 83 q^{62} + 35 q^{63} + 122 q^{64} + 86 q^{65} - 20 q^{66} + 121 q^{67} + 80 q^{68} - 62 q^{69} - 11 q^{70} + 79 q^{71} + 48 q^{72} - 29 q^{73} + 91 q^{74} - 102 q^{75} - 10 q^{76} + 87 q^{77} - 21 q^{78} + 15 q^{79} + 108 q^{80} + 106 q^{81} + 21 q^{82} + 196 q^{83} - 79 q^{84} - 5 q^{85} + 65 q^{86} - 36 q^{87} + 84 q^{88} + 34 q^{89} - 3 q^{90} + 17 q^{91} + 162 q^{92} - q^{93} - 35 q^{94} + 113 q^{95} - 111 q^{96} - 63 q^{97} + 112 q^{98} + 55 q^{99}+O(q^{100}) \) 106 * q + 15 * q^2 - 106 * q^3 + 109 * q^4 + 16 * q^5 - 15 * q^6 + 35 * q^7 + 48 * q^8 + 106 * q^9 - 3 * q^10 + 55 * q^11 - 109 * q^12 - 8 * q^13 + 27 * q^14 - 16 * q^15 + 111 * q^16 + 28 * q^17 + 15 * q^18 + q^19 + 54 * q^20 - 35 * q^21 + 20 * q^22 + 62 * q^23 - 48 * q^24 + 102 * q^25 + 21 * q^26 - 106 * q^27 + 79 * q^28 + 36 * q^29 + 3 * q^30 + q^31 + 111 * q^32 - 55 * q^33 - 27 * q^34 + 72 * q^35 + 109 * q^36 + 31 * q^37 + 43 * q^38 + 8 * q^39 - 13 * q^40 + 35 * q^41 - 27 * q^42 + 98 * q^43 + 121 * q^44 + 16 * q^45 + 8 * q^46 + 75 * q^47 - 111 * q^48 + 49 * q^49 + 83 * q^50 - 28 * q^51 - 18 * q^52 + 60 * q^53 - 15 * q^54 + 14 * q^55 + 85 * q^56 - q^57 + 65 * q^58 + 77 * q^59 - 54 * q^60 - 55 * q^61 + 83 * q^62 + 35 * q^63 + 122 * q^64 + 86 * q^65 - 20 * q^66 + 121 * q^67 + 80 * q^68 - 62 * q^69 - 11 * q^70 + 79 * q^71 + 48 * q^72 - 29 * q^73 + 91 * q^74 - 102 * q^75 - 10 * q^76 + 87 * q^77 - 21 * q^78 + 15 * q^79 + 108 * q^80 + 106 * q^81 + 21 * q^82 + 196 * q^83 - 79 * q^84 - 5 * q^85 + 65 * q^86 - 36 * q^87 + 84 * q^88 + 34 * q^89 - 3 * q^90 + 17 * q^91 + 162 * q^92 - q^93 - 35 * q^94 + 113 * q^95 - 111 * q^96 - 63 * q^97 + 112 * q^98 + 55 * 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 $

$2$	−2.68470	−1.89837	−0.949186	−	0.314715i	$-0.898091\pi$
$2$	−2.68470	−1.89837	−0.949186	+	0.314715i	$0.898091\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	5.20764	2.60382
$5$	3.29964	1.47565	0.737823	−	0.674994i	$-0.235854\pi$
$5$	3.29964	1.47565	0.737823	+	0.674994i	$0.235854\pi$
$6$	2.68470	1.09603
$7$	2.53834	0.959401	0.479700	−	0.877432i	$-0.340745\pi$
$7$	2.53834	0.959401	0.479700	+	0.877432i	$0.340745\pi$
$8$	−8.61155	−3.04464
$9$	1.00000	0.333333
$10$	−8.85857	−2.80133
$11$	−3.38448	−1.02046	−0.510230	−	0.860038i	$-0.670440\pi$
$11$	−3.38448	−1.02046	−0.510230	+	0.860038i	$0.670440\pi$
$12$	−5.20764	−1.50331
$13$	−2.10791	−0.584629	−0.292315	−	0.956322i	$-0.594425\pi$
$13$	−2.10791	−0.584629	−0.292315	+	0.956322i	$0.594425\pi$
$14$	−6.81468	−1.82130
$15$	−3.29964	−0.851965
$16$	12.7042	3.17605
$17$	2.42812	0.588906	0.294453	−	0.955666i	$-0.404863\pi$
$17$	2.42812	0.588906	0.294453	+	0.955666i	$0.404863\pi$
$18$	−2.68470	−0.632791
$19$	−5.18666	−1.18990	−0.594951	−	0.803762i	$-0.702829\pi$
$19$	−5.18666	−1.18990	−0.594951	+	0.803762i	$0.702829\pi$
$20$	17.1833	3.84231
$21$	−2.53834	−0.553910
$22$	9.08633	1.93721
$23$	5.22110	1.08867	0.544337	−	0.838866i	$-0.316781\pi$
$23$	5.22110	1.08867	0.544337	+	0.838866i	$0.316781\pi$
$24$	8.61155	1.75783
$25$	5.88765	1.17753
$26$	5.65912	1.10984
$27$	−1.00000	−0.192450
$28$	13.2187	2.49810
$29$	7.21769	1.34029	0.670146	−	0.742229i	$-0.266232\pi$
$29$	7.21769	1.34029	0.670146	+	0.742229i	$0.266232\pi$
$30$	8.85857	1.61735
$31$	0.885022	0.158955	0.0794773	−	0.996837i	$-0.474675\pi$
$31$	0.885022	0.158955	0.0794773	+	0.996837i	$0.474675\pi$
$32$	−16.8839	−2.98468
$33$	3.38448	0.589163
$34$	−6.51879	−1.11796
$35$	8.37560	1.41574
$36$	5.20764	0.867939
$37$	−1.34140	−0.220525	−0.110262	−	0.993903i	$-0.535169\pi$
$37$	−1.34140	−0.220525	−0.110262	+	0.993903i	$0.535169\pi$
$38$	13.9247	2.25888
$39$	2.10791	0.337536
$40$	−28.4151	−4.49282
$41$	8.43898	1.31795	0.658973	−	0.752166i	$-0.270991\pi$
$41$	8.43898	1.31795	0.658973	+	0.752166i	$0.270991\pi$
$42$	6.81468	1.05153
$43$	7.65624	1.16757	0.583783	−	0.811910i	$-0.301572\pi$
$43$	7.65624	1.16757	0.583783	+	0.811910i	$0.301572\pi$
$44$	−17.6252	−2.65709
$45$	3.29964	0.491882
$46$	−14.0171	−2.06671
$47$	2.97342	0.433718	0.216859	−	0.976203i	$-0.430419\pi$
$47$	2.97342	0.433718	0.216859	+	0.976203i	$0.430419\pi$
$48$	−12.7042	−1.83369
$49$	−0.556854	−0.0795506
$50$	−15.8066	−2.23539
$51$	−2.42812	−0.340005
$52$	−10.9772	−1.52227
$53$	−7.65239	−1.05114	−0.525569	−	0.850751i	$-0.676147\pi$
$53$	−7.65239	−1.05114	−0.525569	+	0.850751i	$0.676147\pi$
$54$	2.68470	0.365342
$55$	−11.1676	−1.50584
$56$	−21.8590	−2.92103
$57$	5.18666	0.686990
$58$	−19.3774	−2.54437
$59$	9.71669	1.26501	0.632503	−	0.774558i	$-0.282028\pi$
$59$	9.71669	1.26501	0.632503	+	0.774558i	$0.282028\pi$
$60$	−17.1833	−2.21836
$61$	2.83907	0.363505	0.181753	−	0.983344i	$-0.441823\pi$
$61$	2.83907	0.363505	0.181753	+	0.983344i	$0.441823\pi$
$62$	−2.37602	−0.301755
$63$	2.53834	0.319800
$64$	19.9199	2.48999
$65$	−6.95536	−0.862706
$66$	−9.08633	−1.11845
$67$	4.85131	0.592682	0.296341	−	0.955082i	$-0.404233\pi$
$67$	4.85131	0.592682	0.296341	+	0.955082i	$0.404233\pi$
$68$	12.6448	1.53340
$69$	−5.22110	−0.628547
$70$	−22.4860	−2.68759
$71$	1.91662	0.227461	0.113731	−	0.993512i	$-0.463720\pi$
$71$	1.91662	0.227461	0.113731	+	0.993512i	$0.463720\pi$
$72$	−8.61155	−1.01488
$73$	−11.7804	−1.37880	−0.689398	−	0.724383i	$-0.742125\pi$
$73$	−11.7804	−1.37880	−0.689398	+	0.724383i	$0.742125\pi$
$74$	3.60126	0.418638
$75$	−5.88765	−0.679848
$76$	−27.0102	−3.09829
$77$	−8.59095	−0.979030
$78$	−5.65912	−0.640769
$79$	−11.2855	−1.26972	−0.634858	−	0.772629i	$-0.718941\pi$
$79$	−11.2855	−1.26972	−0.634858	+	0.772629i	$0.718941\pi$
$80$	41.9193	4.68672
$81$	1.00000	0.111111
$82$	−22.6562	−2.50195
$83$	15.4117	1.69166	0.845828	−	0.533456i	$-0.179107\pi$
$83$	15.4117	1.69166	0.845828	+	0.533456i	$0.179107\pi$
$84$	−13.2187	−1.44228
$85$	8.01194	0.869017
$86$	−20.5547	−2.21647
$87$	−7.21769	−0.773818
$88$	29.1456	3.10694
$89$	−7.03493	−0.745701	−0.372851	−	0.927891i	$-0.621620\pi$
$89$	−7.03493	−0.745701	−0.372851	+	0.927891i	$0.621620\pi$
$90$	−8.85857	−0.933775
$91$	−5.35058	−0.560894
$92$	27.1896	2.83471
$93$	−0.885022	−0.0917725
$94$	−7.98275	−0.823358
$95$	−17.1141	−1.75587
$96$	16.8839	1.72321
$97$	−16.1263	−1.63738	−0.818691	−	0.574234i	$-0.805300\pi$
$97$	−16.1263	−1.63738	−0.818691	+	0.574234i	$0.805300\pi$
$98$	1.49499	0.151017
$99$	−3.38448	−0.340153
$100$	30.6607	3.06607
$101$	13.9975	1.39280	0.696400	−	0.717654i	$-0.254784\pi$
$101$	13.9975	1.39280	0.696400	+	0.717654i	$0.254784\pi$
$102$	6.51879	0.645456
$103$	10.9538	1.07931	0.539655	−	0.841886i	$-0.318555\pi$
$103$	10.9538	1.07931	0.539655	+	0.841886i	$0.318555\pi$
$104$	18.1524	1.77999
$105$	−8.37560	−0.817375
$106$	20.5444	1.99545
$107$	−7.47290	−0.722433	−0.361216	−	0.932482i	$-0.617638\pi$
$107$	−7.47290	−0.722433	−0.361216	+	0.932482i	$0.617638\pi$
$108$	−5.20764	−0.501105
$109$	−6.04185	−0.578704	−0.289352	−	0.957223i	$-0.593440\pi$
$109$	−6.04185	−0.578704	−0.289352	+	0.957223i	$0.593440\pi$
$110$	29.9817	2.85864
$111$	1.34140	0.127320
$112$	32.2475	3.04710
$113$	−11.4857	−1.08049	−0.540244	−	0.841509i	$-0.681668\pi$
$113$	−11.4857	−1.08049	−0.540244	+	0.841509i	$0.681668\pi$
$114$	−13.9247	−1.30416
$115$	17.2278	1.60650
$116$	37.5871	3.48988
$117$	−2.10791	−0.194876
$118$	−26.0864	−2.40145
$119$	6.16339	0.564997
$120$	28.4151	2.59393
$121$	0.454722	0.0413384
$122$	−7.62205	−0.690068
$123$	−8.43898	−0.760917
$124$	4.60887	0.413889
$125$	2.92894	0.261972
$126$	−6.81468	−0.607100
$127$	4.53767	0.402653	0.201327	−	0.979524i	$-0.435475\pi$
$127$	4.53767	0.402653	0.201327	+	0.979524i	$0.435475\pi$
$128$	−19.7112	−1.74224
$129$	−7.65624	−0.674094
$130$	18.6731	1.63774
$131$	0.458171	0.0400306	0.0200153	−	0.999800i	$-0.493629\pi$
$131$	0.458171	0.0400306	0.0200153	+	0.999800i	$0.493629\pi$
$132$	17.6252	1.53407
$133$	−13.1655	−1.14159
$134$	−13.0243	−1.12513
$135$	−3.29964	−0.283988
$136$	−20.9099	−1.79301
$137$	11.8193	1.00979	0.504897	−	0.863179i	$-0.331530\pi$
$137$	11.8193	1.00979	0.504897	+	0.863179i	$0.331530\pi$
$138$	14.0171	1.19322
$139$	14.5972	1.23812	0.619059	−	0.785345i	$-0.287514\pi$
$139$	14.5972	1.23812	0.619059	+	0.785345i	$0.287514\pi$
$140$	43.6171	3.68632
$141$	−2.97342	−0.250407
$142$	−5.14556	−0.431806
$143$	7.13419	0.596591
$144$	12.7042	1.05868
$145$	23.8158	1.97780
$146$	31.6270	2.61747
$147$	0.556854	0.0459286
$148$	−6.98552	−0.574206
$149$	7.92429	0.649183	0.324591	−	0.945854i	$-0.394773\pi$
$149$	7.92429	0.649183	0.324591	+	0.945854i	$0.394773\pi$
$150$	15.8066	1.29060
$151$	6.62201	0.538891	0.269446	−	0.963016i	$-0.413160\pi$
$151$	6.62201	0.538891	0.269446	+	0.963016i	$0.413160\pi$
$152$	44.6652	3.62283
$153$	2.42812	0.196302
$154$	23.0642	1.85856
$155$	2.92026	0.234561
$156$	10.9772	0.878882
$157$	11.4834	0.916476	0.458238	−	0.888829i	$-0.348481\pi$
$157$	11.4834	0.916476	0.458238	+	0.888829i	$0.348481\pi$
$158$	30.2982	2.41039
$159$	7.65239	0.606874
$160$	−55.7109	−4.40433
$161$	13.2529	1.04447
$162$	−2.68470	−0.210930
$163$	−13.0785	−1.02439	−0.512193	−	0.858871i	$-0.671167\pi$
$163$	−13.0785	−1.02439	−0.512193	+	0.858871i	$0.671167\pi$
$164$	43.9471	3.43169
$165$	11.1676	0.869396
$166$	−41.3759	−3.21139
$167$	−2.66326	−0.206089	−0.103045	−	0.994677i	$-0.532858\pi$
$167$	−2.66326	−0.206089	−0.103045	+	0.994677i	$0.532858\pi$
$168$	21.8590	1.68646
$169$	−8.55671	−0.658209
$170$	−21.5097	−1.64972
$171$	−5.18666	−0.396634
$172$	39.8709	3.04013
$173$	8.02943	0.610467	0.305233	−	0.952278i	$-0.401266\pi$
$173$	8.02943	0.610467	0.305233	+	0.952278i	$0.401266\pi$
$174$	19.3774	1.46899
$175$	14.9448	1.12972
$176$	−42.9971	−3.24103
$177$	−9.71669	−0.730351
$178$	18.8867	1.41562
$179$	−3.60838	−0.269703	−0.134851	−	0.990866i	$-0.543056\pi$
$179$	−3.60838	−0.269703	−0.134851	+	0.990866i	$0.543056\pi$
$180$	17.1833	1.28077
$181$	2.40598	0.178835	0.0894174	−	0.995994i	$-0.471500\pi$
$181$	2.40598	0.178835	0.0894174	+	0.995994i	$0.471500\pi$
$182$	14.3647	1.06478
$183$	−2.83907	−0.209870
$184$	−44.9618	−3.31463
$185$	−4.42614	−0.325417
$186$	2.37602	0.174218
$187$	−8.21794	−0.600955
$188$	15.4845	1.12932
$189$	−2.53834	−0.184637
$190$	45.9464	3.33330
$191$	−7.64491	−0.553166	−0.276583	−	0.960990i	$-0.589202\pi$
$191$	−7.64491	−0.553166	−0.276583	+	0.960990i	$0.589202\pi$
$192$	−19.9199	−1.43759
$193$	10.8044	0.777719	0.388860	−	0.921297i	$-0.372869\pi$
$193$	10.8044	0.777719	0.388860	+	0.921297i	$0.372869\pi$
$194$	43.2945	3.10836
$195$	6.95536	0.498083
$196$	−2.89989	−0.207135
$197$	−22.5169	−1.60426	−0.802131	−	0.597149i	$-0.796300\pi$
$197$	−22.5169	−1.60426	−0.802131	+	0.597149i	$0.796300\pi$
$198$	9.08633	0.645738
$199$	1.13355	0.0803551	0.0401776	−	0.999193i	$-0.487208\pi$
$199$	1.13355	0.0803551	0.0401776	+	0.999193i	$0.487208\pi$
$200$	−50.7018	−3.58516
$201$	−4.85131	−0.342185
$202$	−37.5791	−2.64405
$203$	18.3209	1.28588
$204$	−12.6448	−0.885312
$205$	27.8456	1.94482
$206$	−29.4077	−2.04893
$207$	5.22110	0.362892
$208$	−26.7793	−1.85681
$209$	17.5542	1.21425
$210$	22.4860	1.55168
$211$	−9.67596	−0.666121	−0.333060	−	0.942906i	$-0.608081\pi$
$211$	−9.67596	−0.666121	−0.333060	+	0.942906i	$0.608081\pi$
$212$	−39.8509	−2.73697
$213$	−1.91662	−0.131325
$214$	20.0625	1.37145
$215$	25.2629	1.72291
$216$	8.61155	0.585942
$217$	2.24648	0.152501
$218$	16.2206	1.09860
$219$	11.7804	0.796048
$220$	−58.1567	−3.92093
$221$	−5.11827	−0.344292
$222$	−3.60126	−0.241701
$223$	29.0595	1.94597	0.972983	−	0.230877i	$-0.0741595\pi$
$223$	29.0595	1.94597	0.972983	+	0.230877i	$0.0741595\pi$
$224$	−42.8570	−2.86350
$225$	5.88765	0.392510
$226$	30.8358	2.05117
$227$	9.44869	0.627131	0.313566	−	0.949566i	$-0.398476\pi$
$227$	9.44869	0.627131	0.313566	+	0.949566i	$0.398476\pi$
$228$	27.0102	1.78880
$229$	−2.04575	−0.135187	−0.0675936	−	0.997713i	$-0.521532\pi$
$229$	−2.04575	−0.135187	−0.0675936	+	0.997713i	$0.521532\pi$
$230$	−46.2515	−3.04973
$231$	8.59095	0.565243
$232$	−62.1555	−4.08071
$233$	1.04495	0.0684567	0.0342283	−	0.999414i	$-0.489103\pi$
$233$	1.04495	0.0684567	0.0342283	+	0.999414i	$0.489103\pi$
$234$	5.65912	0.369948
$235$	9.81122	0.640014
$236$	50.6010	3.29384
$237$	11.2855	0.733071
$238$	−16.5469	−1.07257
$239$	7.23422	0.467943	0.233971	−	0.972243i	$-0.424828\pi$
$239$	7.23422	0.467943	0.233971	+	0.972243i	$0.424828\pi$
$240$	−41.9193	−2.70588
$241$	20.2812	1.30643	0.653214	−	0.757173i	$-0.273420\pi$
$241$	20.2812	1.30643	0.653214	+	0.757173i	$0.273420\pi$
$242$	−1.22079	−0.0784756
$243$	−1.00000	−0.0641500
$244$	14.7848	0.946501
$245$	−1.83742	−0.117389
$246$	22.6562	1.44450
$247$	10.9330	0.695652
$248$	−7.62141	−0.483960
$249$	−15.4117	−0.976677
$250$	−7.86333	−0.497321
$251$	−0.554658	−0.0350097	−0.0175049	−	0.999847i	$-0.505572\pi$
$251$	−0.554658	−0.0350097	−0.0175049	+	0.999847i	$0.505572\pi$
$252$	13.2187	0.832701
$253$	−17.6707	−1.11095
$254$	−12.1823	−0.764386
$255$	−8.01194	−0.501727
$256$	13.0790	0.817435
$257$	−5.07063	−0.316297	−0.158149	−	0.987415i	$-0.550552\pi$
$257$	−5.07063	−0.316297	−0.158149	+	0.987415i	$0.550552\pi$
$258$	20.5547	1.27968
$259$	−3.40492	−0.211572
$260$	−36.2210	−2.24633
$261$	7.21769	0.446764
$262$	−1.23005	−0.0759929
$263$	2.08628	0.128646	0.0643228	−	0.997929i	$-0.479511\pi$
$263$	2.08628	0.128646	0.0643228	+	0.997929i	$0.479511\pi$
$264$	−29.1456	−1.79379
$265$	−25.2502	−1.55111
$266$	35.3454	2.16717
$267$	7.03493	0.430531
$268$	25.2639	1.54324
$269$	−27.8149	−1.69590	−0.847952	−	0.530074i	$-0.822164\pi$
$269$	−27.8149	−1.69590	−0.847952	+	0.530074i	$0.822164\pi$
$270$	8.85857	0.539115
$271$	−22.2972	−1.35446	−0.677230	−	0.735772i	$-0.736820\pi$
$271$	−22.2972	−1.35446	−0.677230	+	0.735772i	$0.736820\pi$
$272$	30.8473	1.87040
$273$	5.35058	0.323832
$274$	−31.7314	−1.91697
$275$	−19.9267	−1.20162
$276$	−27.1896	−1.63662
$277$	−18.1895	−1.09290	−0.546449	−	0.837492i	$-0.684021\pi$
$277$	−18.1895	−1.09290	−0.546449	+	0.837492i	$0.684021\pi$
$278$	−39.1891	−2.35041
$279$	0.885022	0.0529849
$280$	−72.1269	−4.31041
$281$	−13.4338	−0.801392	−0.400696	−	0.916211i	$-0.631232\pi$
$281$	−13.4338	−0.801392	−0.400696	+	0.916211i	$0.631232\pi$
$282$	7.98275	0.475366
$283$	26.2636	1.56121	0.780604	−	0.625025i	$-0.214911\pi$
$283$	26.2636	1.56121	0.780604	+	0.625025i	$0.214911\pi$
$284$	9.98107	0.592268
$285$	17.1141	1.01375
$286$	−19.1532	−1.13255
$287$	21.4209	1.26444
$288$	−16.8839	−0.994894
$289$	−11.1042	−0.653189
$290$	−63.9384	−3.75459
$291$	16.1263	0.945343
$292$	−61.3482	−3.59013
$293$	26.4093	1.54284	0.771422	−	0.636323i	$-0.219546\pi$
$293$	26.4093	1.54284	0.771422	+	0.636323i	$0.219546\pi$
$294$	−1.49499	−0.0871895
$295$	32.0616	1.86670
$296$	11.5515	0.671419
$297$	3.38448	0.196388
$298$	−21.2744	−1.23239
$299$	−11.0056	−0.636471
$300$	−30.6607	−1.77020
$301$	19.4341	1.12016
$302$	−17.7781	−1.02302
$303$	−13.9975	−0.804133
$304$	−65.8924	−3.77919
$305$	9.36791	0.536405
$306$	−6.51879	−0.372654
$307$	17.8941	1.02127	0.510634	−	0.859798i	$-0.329411\pi$
$307$	17.8941	1.02127	0.510634	+	0.859798i	$0.329411\pi$
$308$	−44.7385	−2.54922
$309$	−10.9538	−0.623140
$310$	−7.84003	−0.445284
$311$	−6.01836	−0.341270	−0.170635	−	0.985334i	$-0.554582\pi$
$311$	−6.01836	−0.341270	−0.170635	+	0.985334i	$0.554582\pi$
$312$	−18.1524	−1.02768
$313$	13.0641	0.738425	0.369212	−	0.929345i	$-0.379628\pi$
$313$	13.0641	0.738425	0.369212	+	0.929345i	$0.379628\pi$
$314$	−30.8296	−1.73981
$315$	8.37560	0.471912
$316$	−58.7707	−3.30611
$317$	12.4826	0.701093	0.350547	−	0.936545i	$-0.385996\pi$
$317$	12.4826	0.701093	0.350547	+	0.936545i	$0.385996\pi$
$318$	−20.5444	−1.15207
$319$	−24.4282	−1.36771
$320$	65.7285	3.67434
$321$	7.47290	0.417097
$322$	−35.5801	−1.98280
$323$	−12.5939	−0.700741
$324$	5.20764	0.289313
$325$	−12.4106	−0.688419
$326$	35.1118	1.94466
$327$	6.04185	0.334115
$328$	−72.6727	−4.01268
$329$	7.54753	0.416109
$330$	−29.9817	−1.65044
$331$	−24.7554	−1.36068	−0.680339	−	0.732898i	$-0.738167\pi$
$331$	−24.7554	−1.36068	−0.680339	+	0.732898i	$0.738167\pi$
$332$	80.2586	4.40476
$333$	−1.34140	−0.0735083
$334$	7.15006	0.391234
$335$	16.0076	0.874589
$336$	−32.2475	−1.75925
$337$	6.73036	0.366626	0.183313	−	0.983055i	$-0.441318\pi$
$337$	6.73036	0.366626	0.183313	+	0.983055i	$0.441318\pi$
$338$	22.9722	1.24953
$339$	11.4857	0.623820
$340$	41.7233	2.26276
$341$	−2.99534	−0.162207
$342$	13.9247	0.752959
$343$	−19.1818	−1.03572
$344$	−65.9321	−3.55482
$345$	−17.2278	−0.927512
$346$	−21.5567	−1.15889
$347$	12.7871	0.686447	0.343223	−	0.939254i	$-0.388481\pi$
$347$	12.7871	0.686447	0.343223	+	0.939254i	$0.388481\pi$
$348$	−37.5871	−2.01488
$349$	−13.3883	−0.716658	−0.358329	−	0.933595i	$-0.616653\pi$
$349$	−13.3883	−0.716658	−0.358329	+	0.933595i	$0.616653\pi$
$350$	−40.1225	−2.14464
$351$	2.10791	0.112512
$352$	57.1433	3.04575
$353$	37.1769	1.97873	0.989364	−	0.145458i	$-0.0464655\pi$
$353$	37.1769	1.97873	0.989364	+	0.145458i	$0.0464655\pi$
$354$	26.0864	1.38648
$355$	6.32417	0.335652
$356$	−36.6354	−1.94167
$357$	−6.16339	−0.326201
$358$	9.68743	0.511996
$359$	30.1708	1.59235	0.796176	−	0.605065i	$-0.206853\pi$
$359$	30.1708	1.59235	0.796176	+	0.605065i	$0.206853\pi$
$360$	−28.4151	−1.49761
$361$	7.90147	0.415867
$362$	−6.45933	−0.339495
$363$	−0.454722	−0.0238667
$364$	−27.8639	−1.46046
$365$	−38.8712	−2.03461
$366$	7.62205	0.398411
$367$	36.8148	1.92172	0.960858	−	0.277042i	$-0.0893540\pi$
$367$	36.8148	1.92172	0.960858	+	0.277042i	$0.0893540\pi$
$368$	66.3299	3.45768
$369$	8.43898	0.439316
$370$	11.8829	0.617762
$371$	−19.4243	−1.00846
$372$	−4.60887	−0.238959
$373$	−21.0678	−1.09085	−0.545425	−	0.838159i	$-0.683632\pi$
$373$	−21.0678	−1.09085	−0.545425	+	0.838159i	$0.683632\pi$
$374$	22.0627	1.14084
$375$	−2.92894	−0.151250
$376$	−25.6057	−1.32052
$377$	−15.2143	−0.783574
$378$	6.81468	0.350509
$379$	27.9569	1.43605	0.718024	−	0.696018i	$-0.245047\pi$
$379$	27.9569	1.43605	0.718024	+	0.696018i	$0.245047\pi$
$380$	−89.1242	−4.57198
$381$	−4.53767	−0.232472
$382$	20.5243	1.05011
$383$	16.8839	0.862729	0.431365	−	0.902178i	$-0.358032\pi$
$383$	16.8839	0.862729	0.431365	+	0.902178i	$0.358032\pi$
$384$	19.7112	1.00588
$385$	−28.3471	−1.44470
$386$	−29.0067	−1.47640
$387$	7.65624	0.389189
$388$	−83.9801	−4.26345
$389$	−21.9794	−1.11440	−0.557200	−	0.830378i	$-0.688124\pi$
$389$	−21.9794	−1.11440	−0.557200	+	0.830378i	$0.688124\pi$
$390$	−18.6731	−0.945548
$391$	12.6775	0.641127
$392$	4.79538	0.242203
$393$	−0.458171	−0.0231117
$394$	60.4511	3.04548
$395$	−37.2381	−1.87365
$396$	−17.6252	−0.885697
$397$	−25.3942	−1.27450	−0.637250	−	0.770657i	$-0.719928\pi$
$397$	−25.3942	−1.27450	−0.637250	+	0.770657i	$0.719928\pi$
$398$	−3.04324	−0.152544
$399$	13.1655	0.659099
$400$	74.7979	3.73989
$401$	−16.8273	−0.840316	−0.420158	−	0.907451i	$-0.638025\pi$
$401$	−16.8273	−0.840316	−0.420158	+	0.907451i	$0.638025\pi$
$402$	13.0243	0.649595
$403$	−1.86555	−0.0929295
$404$	72.8937	3.62660
$405$	3.29964	0.163961
$406$	−49.1863	−2.44107
$407$	4.53995	0.225037
$408$	20.9099	1.03519
$409$	34.9308	1.72722	0.863609	−	0.504163i	$-0.168199\pi$
$409$	34.9308	1.72722	0.863609	+	0.504163i	$0.168199\pi$
$410$	−74.7572	−3.69200
$411$	−11.8193	−0.583005
$412$	57.0434	2.81032
$413$	24.6642	1.21365
$414$	−14.0171	−0.688903
$415$	50.8532	2.49628
$416$	35.5898	1.74493
$417$	−14.5972	−0.714828
$418$	−47.1277	−2.30509
$419$	21.9217	1.07094	0.535471	−	0.844553i	$-0.320134\pi$
$419$	21.9217	1.07094	0.535471	+	0.844553i	$0.320134\pi$
$420$	−43.6171	−2.12830
$421$	−21.4243	−1.04416	−0.522078	−	0.852898i	$-0.674843\pi$
$421$	−21.4243	−1.04416	−0.522078	+	0.852898i	$0.674843\pi$
$422$	25.9771	1.26455
$423$	2.97342	0.144573
$424$	65.8990	3.20034
$425$	14.2959	0.693455
$426$	5.14556	0.249303
$427$	7.20650	0.348747
$428$	−38.9161	−1.88108
$429$	−7.13419	−0.344442
$430$	−67.8233	−3.27073
$431$	−8.47571	−0.408261	−0.204130	−	0.978944i	$-0.565437\pi$
$431$	−8.47571	−0.408261	−0.204130	+	0.978944i	$0.565437\pi$
$432$	−12.7042	−0.611231
$433$	−13.2146	−0.635053	−0.317526	−	0.948249i	$-0.602852\pi$
$433$	−13.2146	−0.635053	−0.317526	+	0.948249i	$0.602852\pi$
$434$	−6.03114	−0.289504
$435$	−23.8158	−1.14188
$436$	−31.4637	−1.50684
$437$	−27.0801	−1.29542
$438$	−31.6270	−1.51120
$439$	−19.1408	−0.913541	−0.456771	−	0.889585i	$-0.650994\pi$
$439$	−19.1408	−0.913541	−0.456771	+	0.889585i	$0.650994\pi$
$440$	96.1703	4.58474
$441$	−0.556854	−0.0265169
$442$	13.7410	0.653594
$443$	35.6986	1.69609	0.848046	−	0.529922i	$-0.177779\pi$
$443$	35.6986	1.69609	0.848046	+	0.529922i	$0.177779\pi$
$444$	6.98552	0.331518
$445$	−23.2128	−1.10039
$446$	−78.0161	−3.69417
$447$	−7.92429	−0.374806
$448$	50.5634	2.38889
$449$	−9.64003	−0.454941	−0.227471	−	0.973785i	$-0.573046\pi$
$449$	−9.64003	−0.454941	−0.227471	+	0.973785i	$0.573046\pi$
$450$	−15.8066	−0.745130
$451$	−28.5616	−1.34491
$452$	−59.8136	−2.81339
$453$	−6.62201	−0.311129
$454$	−25.3669	−1.19053
$455$	−17.6550	−0.827680
$456$	−44.6652	−2.09164
$457$	12.2987	0.575309	0.287654	−	0.957734i	$-0.407125\pi$
$457$	12.2987	0.575309	0.287654	+	0.957734i	$0.407125\pi$
$458$	5.49224	0.256636
$459$	−2.42812	−0.113335
$460$	89.7160	4.18303
$461$	12.0492	0.561186	0.280593	−	0.959827i	$-0.409469\pi$
$461$	12.0492	0.561186	0.280593	+	0.959827i	$0.409469\pi$
$462$	−23.0642	−1.07304
$463$	7.25395	0.337120	0.168560	−	0.985691i	$-0.446088\pi$
$463$	7.25395	0.337120	0.168560	+	0.985691i	$0.446088\pi$
$464$	91.6950	4.25683
$465$	−2.92026	−0.135424
$466$	−2.80537	−0.129956
$467$	31.1968	1.44361	0.721807	−	0.692095i	$-0.243312\pi$
$467$	31.1968	1.44361	0.721807	+	0.692095i	$0.243312\pi$
$468$	−10.9772	−0.507423
$469$	12.3143	0.568620
$470$	−26.3402	−1.21498
$471$	−11.4834	−0.529128
$472$	−83.6758	−3.85149
$473$	−25.9124	−1.19145
$474$	−30.2982	−1.39164
$475$	−30.5373	−1.40115
$476$	32.0967	1.47115
$477$	−7.65239	−0.350379
$478$	−19.4217	−0.888330
$479$	−4.62094	−0.211136	−0.105568	−	0.994412i	$-0.533666\pi$
$479$	−4.62094	−0.211136	−0.105568	+	0.994412i	$0.533666\pi$
$480$	55.7109	2.54284
$481$	2.82755	0.128925
$482$	−54.4491	−2.48009
$483$	−13.2529	−0.603028
$484$	2.36803	0.107638
$485$	−53.2112	−2.41620
$486$	2.68470	0.121781
$487$	−7.22593	−0.327438	−0.163719	−	0.986507i	$-0.552349\pi$
$487$	−7.22593	−0.327438	−0.163719	+	0.986507i	$0.552349\pi$
$488$	−24.4488	−1.10674
$489$	13.0785	0.591429
$490$	4.93293	0.222847
$491$	−30.3370	−1.36909	−0.684544	−	0.728972i	$-0.739999\pi$
$491$	−30.3370	−1.36909	−0.684544	+	0.728972i	$0.739999\pi$
$492$	−43.9471	−1.98129
$493$	17.5254	0.789306
$494$	−29.3519	−1.32061
$495$	−11.1676	−0.501946
$496$	11.2435	0.504848
$497$	4.86503	0.218226
$498$	41.3759	1.85410
$499$	22.0817	0.988511	0.494255	−	0.869317i	$-0.335441\pi$
$499$	22.0817	0.988511	0.494255	+	0.869317i	$0.335441\pi$
$500$	15.2528	0.682128
$501$	2.66326	0.118986
$502$	1.48909	0.0664615
$503$	−10.1183	−0.451154	−0.225577	−	0.974225i	$-0.572427\pi$
$503$	−10.1183	−0.451154	−0.225577	+	0.974225i	$0.572427\pi$
$504$	−21.8590	−0.973677
$505$	46.1867	2.05528
$506$	47.4407	2.10899
$507$	8.55671	0.380017
$508$	23.6305	1.04844
$509$	−6.48186	−0.287304	−0.143652	−	0.989628i	$-0.545885\pi$
$509$	−6.48186	−0.287304	−0.143652	+	0.989628i	$0.545885\pi$
$510$	21.5097	0.952465
$511$	−29.9027	−1.32282
$512$	4.30927	0.190445
$513$	5.18666	0.228997
$514$	13.6131	0.600450
$515$	36.1436	1.59268
$516$	−39.8709	−1.75522
$517$	−10.0635	−0.442591
$518$	9.14121	0.401642
$519$	−8.02943	−0.352453
$520$	59.8964	2.62663
$521$	14.5744	0.638514	0.319257	−	0.947668i	$-0.396567\pi$
$521$	14.5744	0.638514	0.319257	+	0.947668i	$0.396567\pi$
$522$	−19.3774	−0.848124
$523$	20.2247	0.884366	0.442183	−	0.896925i	$-0.354204\pi$
$523$	20.2247	0.884366	0.442183	+	0.896925i	$0.354204\pi$
$524$	2.38599	0.104232
$525$	−14.9448	−0.652246
$526$	−5.60105	−0.244217
$527$	2.14894	0.0936094
$528$	42.9971	1.87121
$529$	4.25988	0.185212
$530$	67.7892	2.94458
$531$	9.71669	0.421669
$532$	−68.5611	−2.97250
$533$	−17.7886	−0.770510
$534$	−18.8867	−0.817308
$535$	−24.6579	−1.06605
$536$	−41.7774	−1.80451
$537$	3.60838	0.155713
$538$	74.6747	3.21946
$539$	1.88466	0.0811782
$540$	−17.1833	−0.739453
$541$	34.2024	1.47048	0.735238	−	0.677809i	$-0.237070\pi$
$541$	34.2024	1.47048	0.735238	+	0.677809i	$0.237070\pi$
$542$	59.8614	2.57127
$543$	−2.40598	−0.103250
$544$	−40.9962	−1.75770
$545$	−19.9360	−0.853962
$546$	−14.3647	−0.614754
$547$	21.1784	0.905523	0.452761	−	0.891632i	$-0.350439\pi$
$547$	21.1784	0.905523	0.452761	+	0.891632i	$0.350439\pi$
$548$	61.5508	2.62932
$549$	2.83907	0.121168
$550$	53.4972	2.28113
$551$	−37.4357	−1.59482
$552$	44.9618	1.91370
$553$	−28.6463	−1.21817
$554$	48.8333	2.07473
$555$	4.42614	0.187879
$556$	76.0169	3.22383
$557$	44.1266	1.86970	0.934852	−	0.355036i	$-0.115532\pi$
$557$	44.1266	1.86970	0.934852	+	0.355036i	$0.115532\pi$
$558$	−2.37602	−0.100585
$559$	−16.1387	−0.682593
$560$	106.405	4.49644
$561$	8.21794	0.346962
$562$	36.0657	1.52134
$563$	18.4000	0.775469	0.387735	−	0.921771i	$-0.373258\pi$
$563$	18.4000	0.775469	0.387735	+	0.921771i	$0.373258\pi$
$564$	−15.4845	−0.652014
$565$	−37.8989	−1.59442
$566$	−70.5100	−2.96376
$567$	2.53834	0.106600
$568$	−16.5051	−0.692538
$569$	−0.199485	−0.00836285	−0.00418143	−	0.999991i	$-0.501331\pi$
$569$	−0.199485	−0.00836285	−0.00418143	+	0.999991i	$0.501331\pi$
$570$	−45.9464	−1.92448
$571$	9.07627	0.379830	0.189915	−	0.981801i	$-0.439179\pi$
$571$	9.07627	0.379830	0.189915	+	0.981801i	$0.439179\pi$
$572$	37.1522	1.55341
$573$	7.64491	0.319371
$574$	−57.5089	−2.40038
$575$	30.7400	1.28195
$576$	19.9199	0.829995
$577$	−5.46074	−0.227333	−0.113667	−	0.993519i	$-0.536260\pi$
$577$	−5.46074	−0.227333	−0.113667	+	0.993519i	$0.536260\pi$
$578$	29.8115	1.24000
$579$	−10.8044	−0.449016
$580$	124.024	5.14982
$581$	39.1201	1.62297
$582$	−43.2945	−1.79461
$583$	25.8994	1.07264
$584$	101.448	4.19794
$585$	−6.95536	−0.287569
$586$	−70.9010	−2.92889
$587$	41.2300	1.70174	0.850871	−	0.525374i	$-0.176075\pi$
$587$	41.2300	1.70174	0.850871	+	0.525374i	$0.176075\pi$
$588$	2.89989	0.119590
$589$	−4.59031	−0.189140
$590$	−86.0760	−3.54369
$591$	22.5169	0.926221
$592$	−17.0414	−0.700398
$593$	−11.0736	−0.454737	−0.227368	−	0.973809i	$-0.573012\pi$
$593$	−11.0736	−0.454737	−0.227368	+	0.973809i	$0.573012\pi$
$594$	−9.08633	−0.372817
$595$	20.3370	0.833735
$596$	41.2668	1.69035
$597$	−1.13355	−0.0463931
$598$	29.5468	1.20826
$599$	−14.1544	−0.578332	−0.289166	−	0.957279i	$-0.593378\pi$
$599$	−14.1544	−0.578332	−0.289166	+	0.957279i	$0.593378\pi$
$600$	50.7018	2.06989
$601$	−34.4358	−1.40466	−0.702332	−	0.711850i	$-0.747858\pi$
$601$	−34.4358	−1.40466	−0.702332	+	0.711850i	$0.747858\pi$
$602$	−52.1748	−2.12649
$603$	4.85131	0.197561
$604$	34.4850	1.40318
$605$	1.50042	0.0610008
$606$	37.5791	1.52654
$607$	4.64546	0.188553	0.0942767	−	0.995546i	$-0.469946\pi$
$607$	4.64546	0.188553	0.0942767	+	0.995546i	$0.469946\pi$
$608$	87.5711	3.55148
$609$	−18.3209	−0.742401
$610$	−25.1501	−1.01830
$611$	−6.26770	−0.253564
$612$	12.6448	0.511135
$613$	−2.56462	−0.103584	−0.0517921	−	0.998658i	$-0.516493\pi$
$613$	−2.56462	−0.103584	−0.0517921	+	0.998658i	$0.516493\pi$
$614$	−48.0402	−1.93875
$615$	−27.8456	−1.12284
$616$	73.9814	2.98080
$617$	30.3676	1.22256	0.611278	−	0.791416i	$-0.290656\pi$
$617$	30.3676	1.22256	0.611278	+	0.791416i	$0.290656\pi$
$618$	29.4077	1.18295
$619$	23.1076	0.928773	0.464387	−	0.885633i	$-0.346275\pi$
$619$	23.1076	0.928773	0.464387	+	0.885633i	$0.346275\pi$
$620$	15.2076	0.610753
$621$	−5.22110	−0.209516
$622$	16.1575	0.647857
$623$	−17.8570	−0.715426
$624$	26.7793	1.07203
$625$	−19.7738	−0.790952
$626$	−35.0731	−1.40180
$627$	−17.5542	−0.701046
$628$	59.8014	2.38634
$629$	−3.25708	−0.129868
$630$	−22.4860	−0.895864
$631$	−1.35672	−0.0540100	−0.0270050	−	0.999635i	$-0.508597\pi$
$631$	−1.35672	−0.0540100	−0.0270050	+	0.999635i	$0.508597\pi$
$632$	97.1855	3.86583
$633$	9.67596	0.384585
$634$	−33.5121	−1.33094
$635$	14.9727	0.594174
$636$	39.8509	1.58019
$637$	1.17380	0.0465076
$638$	65.5824	2.59643
$639$	1.91662	0.0758204
$640$	−65.0399	−2.57093
$641$	−8.31578	−0.328454	−0.164227	−	0.986423i	$-0.552513\pi$
$641$	−8.31578	−0.328454	−0.164227	+	0.986423i	$0.552513\pi$
$642$	−20.0625	−0.791805
$643$	−34.1340	−1.34611	−0.673056	−	0.739592i	$-0.735019\pi$
$643$	−34.1340	−1.34611	−0.673056	+	0.739592i	$0.735019\pi$
$644$	69.0163	2.71962
$645$	−25.2629	−0.994725
$646$	33.8108	1.33027
$647$	−28.2339	−1.10999	−0.554994	−	0.831855i	$-0.687279\pi$
$647$	−28.2339	−1.10999	−0.554994	+	0.831855i	$0.687279\pi$
$648$	−8.61155	−0.338294
$649$	−32.8860	−1.29089
$650$	33.3189	1.30688
$651$	−2.24648	−0.0880466
$652$	−68.1079	−2.66731
$653$	34.8278	1.36292	0.681458	−	0.731857i	$-0.261346\pi$
$653$	34.8278	1.36292	0.681458	+	0.731857i	$0.261346\pi$
$654$	−16.2206	−0.634275
$655$	1.51180	0.0590710
$656$	107.210	4.18586
$657$	−11.7804	−0.459599
$658$	−20.2629	−0.789930
$659$	−27.4226	−1.06823	−0.534117	−	0.845411i	$-0.679356\pi$
$659$	−27.4226	−1.06823	−0.534117	+	0.845411i	$0.679356\pi$
$660$	58.1567	2.26375
$661$	−15.9984	−0.622267	−0.311133	−	0.950366i	$-0.600709\pi$
$661$	−15.9984	−0.622267	−0.311133	+	0.950366i	$0.600709\pi$
$662$	66.4608	2.58307
$663$	5.11827	0.198777
$664$	−132.719	−5.15049
$665$	−43.4414	−1.68459
$666$	3.60126	0.139546
$667$	37.6843	1.45914
$668$	−13.8693	−0.536618
$669$	−29.0595	−1.12350
$670$	−42.9757	−1.66030
$671$	−9.60877	−0.370942
$672$	42.8570	1.65324
$673$	19.7512	0.761355	0.380677	−	0.924708i	$-0.375691\pi$
$673$	19.7512	0.761355	0.380677	+	0.924708i	$0.375691\pi$
$674$	−18.0690	−0.695993
$675$	−5.88765	−0.226616
$676$	−44.5602	−1.71386
$677$	17.9764	0.690888	0.345444	−	0.938439i	$-0.387728\pi$
$677$	17.9764	0.690888	0.345444	+	0.938439i	$0.387728\pi$
$678$	−30.8358	−1.18424
$679$	−40.9341	−1.57091
$680$	−68.9952	−2.64585
$681$	−9.44869	−0.362075
$682$	8.04160	0.307929
$683$	−1.62611	−0.0622214	−0.0311107	−	0.999516i	$-0.509904\pi$
$683$	−1.62611	−0.0622214	−0.0311107	+	0.999516i	$0.509904\pi$
$684$	−27.0102	−1.03276
$685$	38.9996	1.49010
$686$	51.4975	1.96618
$687$	2.04575	0.0780504
$688$	97.2664	3.70825
$689$	16.1306	0.614525
$690$	46.2515	1.76076
$691$	25.4333	0.967528	0.483764	−	0.875198i	$-0.339269\pi$
$691$	25.4333	0.967528	0.483764	+	0.875198i	$0.339269\pi$
$692$	41.8144	1.58954
$693$	−8.59095	−0.326343
$694$	−34.3295	−1.30313
$695$	48.1655	1.82702
$696$	62.1555	2.35600
$697$	20.4909	0.776147
$698$	35.9435	1.36048
$699$	−1.04495	−0.0395235
$700$	77.8273	2.94159
$701$	−4.76449	−0.179952	−0.0899761	−	0.995944i	$-0.528679\pi$
$701$	−4.76449	−0.179952	−0.0899761	+	0.995944i	$0.528679\pi$
$702$	−5.65912	−0.213590
$703$	6.95739	0.262403
$704$	−67.4185	−2.54093
$705$	−9.81122	−0.369512
$706$	−99.8091	−3.75636
$707$	35.5303	1.33625
$708$	−50.6010	−1.90170
$709$	24.2566	0.910974	0.455487	−	0.890242i	$-0.349465\pi$
$709$	24.2566	0.910974	0.455487	+	0.890242i	$0.349465\pi$
$710$	−16.9785	−0.637193
$711$	−11.2855	−0.423239
$712$	60.5817	2.27039
$713$	4.62079	0.173050
$714$	16.5469	0.619251
$715$	23.5403	0.880357
$716$	−18.7911	−0.702257
$717$	−7.23422	−0.270167
$718$	−80.9996	−3.02288
$719$	3.82437	0.142625	0.0713125	−	0.997454i	$-0.477281\pi$
$719$	3.82437	0.142625	0.0713125	+	0.997454i	$0.477281\pi$
$720$	41.9193	1.56224
$721$	27.8044	1.03549
$722$	−21.2131	−0.789470
$723$	−20.2812	−0.754266
$724$	12.5294	0.465653
$725$	42.4953	1.57823
$726$	1.22079	0.0453079
$727$	−4.80880	−0.178349	−0.0891743	−	0.996016i	$-0.528423\pi$
$727$	−4.80880	−0.178349	−0.0891743	+	0.996016i	$0.528423\pi$
$728$	46.0768	1.70772
$729$	1.00000	0.0370370
$730$	104.358	3.86246
$731$	18.5903	0.687587
$732$	−14.7848	−0.546463
$733$	1.20437	0.0444843	0.0222422	−	0.999753i	$-0.492920\pi$
$733$	1.20437	0.0444843	0.0222422	+	0.999753i	$0.492920\pi$
$734$	−98.8368	−3.64813
$735$	1.83742	0.0677743
$736$	−88.1525	−3.24935
$737$	−16.4192	−0.604809
$738$	−22.6562	−0.833985
$739$	11.2165	0.412605	0.206303	−	0.978488i	$-0.433857\pi$
$739$	11.2165	0.412605	0.206303	+	0.978488i	$0.433857\pi$
$740$	−23.0497	−0.847325
$741$	−10.9330	−0.401635
$742$	52.1486	1.91444
$743$	24.7967	0.909704	0.454852	−	0.890567i	$-0.349692\pi$
$743$	24.7967	0.909704	0.454852	+	0.890567i	$0.349692\pi$
$744$	7.62141	0.279414
$745$	26.1473	0.957964
$746$	56.5609	2.07084
$747$	15.4117	0.563885
$748$	−42.7960	−1.56478
$749$	−18.9687	−0.693102
$750$	7.86333	0.287128
$751$	−10.6641	−0.389139	−0.194570	−	0.980889i	$-0.562331\pi$
$751$	−10.6641	−0.389139	−0.194570	+	0.980889i	$0.562331\pi$
$752$	37.7749	1.37751
$753$	0.554658	0.0202129
$754$	40.8458	1.48751
$755$	21.8503	0.795213
$756$	−13.2187	−0.480760
$757$	35.3594	1.28516	0.642580	−	0.766219i	$-0.277864\pi$
$757$	35.3594	1.28516	0.642580	+	0.766219i	$0.277864\pi$
$758$	−75.0559	−2.72615
$759$	17.6707	0.641407
$760$	147.379	5.34601
$761$	38.1446	1.38274	0.691370	−	0.722501i	$-0.257008\pi$
$761$	38.1446	1.38274	0.691370	+	0.722501i	$0.257008\pi$
$762$	12.1823	0.441318
$763$	−15.3362	−0.555209
$764$	−39.8119	−1.44034
$765$	8.01194	0.289672
$766$	−45.3284	−1.63778
$767$	−20.4819	−0.739559
$768$	−13.0790	−0.471946
$769$	−16.1504	−0.582399	−0.291200	−	0.956662i	$-0.594054\pi$
$769$	−16.1504	−0.582399	−0.291200	+	0.956662i	$0.594054\pi$
$770$	76.1035	2.74258
$771$	5.07063	0.182614
$772$	56.2655	2.02504
$773$	5.64716	0.203114	0.101557	−	0.994830i	$-0.467618\pi$
$773$	5.64716	0.203114	0.101557	+	0.994830i	$0.467618\pi$
$774$	−20.5547	−0.738825
$775$	5.21070	0.187174
$776$	138.873	4.98525
$777$	3.40492	0.122151
$778$	59.0082	2.11555
$779$	−43.7701	−1.56823
$780$	36.2210	1.29692
$781$	−6.48677	−0.232115
$782$	−34.0353	−1.21710
$783$	−7.21769	−0.257939
$784$	−7.07439	−0.252657
$785$	37.8912	1.35239
$786$	1.23005	0.0438745
$787$	27.8477	0.992662	0.496331	−	0.868133i	$-0.334680\pi$
$787$	27.8477	0.992662	0.496331	+	0.868133i	$0.334680\pi$
$788$	−117.260	−4.17720
$789$	−2.08628	−0.0742736
$790$	99.9732	3.55689
$791$	−29.1547	−1.03662
$792$	29.1456	1.03565
$793$	−5.98450	−0.212516
$794$	68.1760	2.41948
$795$	25.2502	0.895531
$796$	5.90311	0.209230
$797$	−34.0102	−1.20470	−0.602351	−	0.798231i	$-0.705769\pi$
$797$	−34.0102	−1.20470	−0.602351	+	0.798231i	$0.705769\pi$
$798$	−35.3454	−1.25121
$799$	7.21983	0.255419
$800$	−99.4066	−3.51455
$801$	−7.03493	−0.248567
$802$	45.1763	1.59523
$803$	39.8707	1.40701
$804$	−25.2639	−0.890988
$805$	43.7299	1.54128
$806$	5.00844	0.176415
$807$	27.8149	0.979130
$808$	−120.540	−4.24058
$809$	−34.4193	−1.21012	−0.605059	−	0.796181i	$-0.706851\pi$
$809$	−34.4193	−1.21012	−0.605059	+	0.796181i	$0.706851\pi$
$810$	−8.85857	−0.311258
$811$	12.0501	0.423135	0.211568	−	0.977363i	$-0.432143\pi$
$811$	12.0501	0.423135	0.211568	+	0.977363i	$0.432143\pi$
$812$	95.4087	3.34819
$813$	22.2972	0.781998
$814$	−12.1884	−0.427204
$815$	−43.1543	−1.51163
$816$	−30.8473	−1.07987
$817$	−39.7103	−1.38929
$818$	−93.7789	−3.27890
$819$	−5.35058	−0.186965
$820$	145.010	5.06396
$821$	2.33553	0.0815105	0.0407552	−	0.999169i	$-0.487024\pi$
$821$	2.33553	0.0815105	0.0407552	+	0.999169i	$0.487024\pi$
$822$	31.7314	1.10676
$823$	−49.3766	−1.72116	−0.860579	−	0.509317i	$-0.829898\pi$
$823$	−49.3766	−1.72116	−0.860579	+	0.509317i	$0.829898\pi$
$824$	−94.3292	−3.28611
$825$	19.9267	0.693757
$826$	−66.2161	−2.30395
$827$	−5.37175	−0.186794	−0.0933971	−	0.995629i	$-0.529773\pi$
$827$	−5.37175	−0.186794	−0.0933971	+	0.995629i	$0.529773\pi$
$828$	27.1896	0.944903
$829$	6.77778	0.235402	0.117701	−	0.993049i	$-0.462448\pi$
$829$	6.77778	0.235402	0.117701	+	0.993049i	$0.462448\pi$
$830$	−136.526	−4.73888
$831$	18.1895	0.630985
$832$	−41.9893	−1.45572
$833$	−1.35211	−0.0468479
$834$	39.1891	1.35701
$835$	−8.78780	−0.304114
$836$	91.4157	3.16168
$837$	−0.885022	−0.0305908
$838$	−58.8531	−2.03305
$839$	7.75515	0.267737	0.133869	−	0.990999i	$-0.457260\pi$
$839$	7.75515	0.267737	0.133869	+	0.990999i	$0.457260\pi$
$840$	72.1269	2.48862
$841$	23.0951	0.796382
$842$	57.5178	1.98220
$843$	13.4338	0.462684
$844$	−50.3889	−1.73446
$845$	−28.2341	−0.971283
$846$	−7.98275	−0.274453
$847$	1.15424	0.0396600
$848$	−97.2175	−3.33846
$849$	−26.2636	−0.901364
$850$	−38.3804	−1.31644
$851$	−7.00358	−0.240080
$852$	−9.98107	−0.341946
$853$	19.6818	0.673893	0.336947	−	0.941524i	$-0.390606\pi$
$853$	19.6818	0.673893	0.336947	+	0.941524i	$0.390606\pi$
$854$	−19.3473	−0.662052
$855$	−17.1141	−0.585291
$856$	64.3533	2.19955
$857$	−35.8215	−1.22364	−0.611819	−	0.790997i	$-0.709562\pi$
$857$	−35.8215	−1.22364	−0.611819	+	0.790997i	$0.709562\pi$
$858$	19.1532	0.653879
$859$	−5.89533	−0.201146	−0.100573	−	0.994930i	$-0.532068\pi$
$859$	−5.89533	−0.201146	−0.100573	+	0.994930i	$0.532068\pi$
$860$	131.560	4.48615
$861$	−21.4209	−0.730024
$862$	22.7548	0.775031
$863$	20.4470	0.696026	0.348013	−	0.937490i	$-0.386857\pi$
$863$	20.4470	0.696026	0.348013	+	0.937490i	$0.386857\pi$
$864$	16.8839	0.574402
$865$	26.4943	0.900832
$866$	35.4773	1.20557
$867$	11.1042	0.377119
$868$	11.6989	0.397085
$869$	38.1955	1.29569
$870$	63.9384	2.16772
$871$	−10.2261	−0.346500
$872$	52.0297	1.76195
$873$	−16.1263	−0.545794
$874$	72.7020	2.45918
$875$	7.43463	0.251336
$876$	61.3482	2.07276
$877$	−12.5166	−0.422654	−0.211327	−	0.977415i	$-0.567779\pi$
$877$	−12.5166	−0.422654	−0.211327	+	0.977415i	$0.567779\pi$
$878$	51.3874	1.73424
$879$	−26.4093	−0.890762
$880$	−141.875	−4.78261
$881$	−54.7330	−1.84400	−0.922001	−	0.387188i	$-0.873446\pi$
$881$	−54.7330	−1.84400	−0.922001	+	0.387188i	$0.873446\pi$
$882$	1.49499	0.0503389
$883$	−30.1511	−1.01466	−0.507332	−	0.861751i	$-0.669368\pi$
$883$	−30.1511	−1.01466	−0.507332	+	0.861751i	$0.669368\pi$
$884$	−26.6541	−0.896473
$885$	−32.0616	−1.07774
$886$	−95.8402	−3.21982
$887$	42.2622	1.41902	0.709512	−	0.704693i	$-0.248915\pi$
$887$	42.2622	1.41902	0.709512	+	0.704693i	$0.248915\pi$
$888$	−11.5515	−0.387644
$889$	11.5181	0.386306
$890$	62.3194	2.08895
$891$	−3.38448	−0.113384
$892$	151.331	5.06694
$893$	−15.4221	−0.516081
$894$	21.2744	0.711521
$895$	−11.9064	−0.397986
$896$	−50.0336	−1.67151
$897$	11.0056	0.367467
$898$	25.8806	0.863648
$899$	6.38781	0.213046
$900$	30.6607	1.02202
$901$	−18.5809	−0.619021
$902$	76.6794	2.55314
$903$	−19.4341	−0.646727
$904$	98.9101	3.28970
$905$	7.93886	0.263897
$906$	17.7781	0.590639
$907$	−20.5124	−0.681104	−0.340552	−	0.940226i	$-0.610614\pi$
$907$	−20.5124	−0.681104	−0.340552	+	0.940226i	$0.610614\pi$
$908$	49.2053	1.63294
$909$	13.9975	0.464267
$910$	47.3985	1.57125
$911$	−18.2896	−0.605963	−0.302981	−	0.952997i	$-0.597982\pi$
$911$	−18.2896	−0.605963	−0.302981	+	0.952997i	$0.597982\pi$
$912$	65.8924	2.18191
$913$	−52.1607	−1.72627
$914$	−33.0183	−1.09215
$915$	−9.36791	−0.309693
$916$	−10.6535	−0.352003
$917$	1.16299	0.0384054
$918$	6.51879	0.215152
$919$	54.8775	1.81024	0.905121	−	0.425153i	$-0.139780\pi$
$919$	54.8775	1.81024	0.905121	+	0.425153i	$0.139780\pi$
$920$	−148.358	−4.89121
$921$	−17.8941	−0.589629
$922$	−32.3484	−1.06534
$923$	−4.04007	−0.132980
$924$	44.7385	1.47179
$925$	−7.89770	−0.259675
$926$	−19.4747	−0.639978
$927$	10.9538	0.359770
$928$	−121.863	−4.00034
$929$	−5.26490	−0.172736	−0.0863679	−	0.996263i	$-0.527526\pi$
$929$	−5.26490	−0.172736	−0.0863679	+	0.996263i	$0.527526\pi$
$930$	7.84003	0.257085
$931$	2.88822	0.0946575
$932$	5.44170	0.178249
$933$	6.01836	0.197032
$934$	−83.7541	−2.74052
$935$	−27.1163	−0.886797
$936$	18.1524	0.593329
$937$	39.5235	1.29118	0.645588	−	0.763686i	$-0.276613\pi$
$937$	39.5235	1.29118	0.645588	+	0.763686i	$0.276613\pi$
$938$	−33.0602	−1.07945
$939$	−13.0641	−0.426330
$940$	51.0933	1.66648
$941$	48.3467	1.57606	0.788029	−	0.615638i	$-0.211102\pi$
$941$	48.3467	1.57606	0.788029	+	0.615638i	$0.211102\pi$
$942$	30.8296	1.00448
$943$	44.0607	1.43481
$944$	123.443	4.01772
$945$	−8.37560	−0.272458
$946$	69.5672	2.26182
$947$	−22.9150	−0.744637	−0.372318	−	0.928105i	$-0.621437\pi$
$947$	−22.9150	−0.744637	−0.372318	+	0.928105i	$0.621437\pi$
$948$	58.7707	1.90878
$949$	24.8321	0.806084
$950$	81.9835	2.65990
$951$	−12.4826	−0.404776
$952$	−53.0763	−1.72021
$953$	−18.7312	−0.606764	−0.303382	−	0.952869i	$-0.598116\pi$
$953$	−18.7312	−0.606764	−0.303382	+	0.952869i	$0.598116\pi$
$954$	20.5444	0.665150
$955$	−25.2255	−0.816277
$956$	37.6732	1.21844
$957$	24.4282	0.789650
$958$	12.4059	0.400815
$959$	30.0015	0.968798
$960$	−65.7285	−2.12138
$961$	−30.2167	−0.974733
$962$	−7.59114	−0.244748
$963$	−7.47290	−0.240811
$964$	105.617	3.40170
$965$	35.6507	1.14764
$966$	35.5801	1.14477
$967$	−45.3412	−1.45808	−0.729038	−	0.684474i	$-0.760032\pi$
$967$	−45.3412	−1.45808	−0.729038	+	0.684474i	$0.760032\pi$
$968$	−3.91586	−0.125861
$969$	12.5939	0.404573
$970$	142.856	4.58684
$971$	−35.9609	−1.15404	−0.577020	−	0.816730i	$-0.695785\pi$
$971$	−35.9609	−1.15404	−0.577020	+	0.816730i	$0.695785\pi$
$972$	−5.20764	−0.167035
$973$	37.0526	1.18785
$974$	19.3995	0.621599
$975$	12.4106	0.397459
$976$	36.0681	1.15451
$977$	−12.5486	−0.401465	−0.200733	−	0.979646i	$-0.564332\pi$
$977$	−12.5486	−0.401465	−0.200733	+	0.979646i	$0.564332\pi$
$978$	−35.1118	−1.12275
$979$	23.8096	0.760958
$980$	−9.56862	−0.305658
$981$	−6.04185	−0.192901
$982$	81.4457	2.59904
$983$	4.33491	0.138262	0.0691312	−	0.997608i	$-0.477977\pi$
$983$	4.33491	0.138262	0.0691312	+	0.997608i	$0.477977\pi$
$984$	72.6727	2.31672
$985$	−74.2977	−2.36732
$986$	−47.0506	−1.49840
$987$	−7.54753	−0.240241
$988$	56.9352	1.81135
$989$	39.9740	1.27110
$990$	29.9817	0.952880
$991$	−15.1065	−0.479875	−0.239937	−	0.970788i	$-0.577127\pi$
$991$	−15.1065	−0.479875	−0.239937	+	0.970788i	$0.577127\pi$
$992$	−14.9426	−0.474429
$993$	24.7554	0.785588
$994$	−13.0612	−0.414275
$995$	3.74031	0.118576
$996$	−80.2586	−2.54309
$997$	29.4703	0.933332	0.466666	−	0.884434i	$-0.345455\pi$
$997$	29.4703	0.933332	0.466666	+	0.884434i	$0.345455\pi$
$998$	−59.2827	−1.87656
$999$	1.34140	0.0424400

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8013.2.a.b.1.2	✓	106

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8013.2.a.b.1.2	✓	106	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:	\( N \)	\(=\)	\( 8013 = 3 \cdot 2671 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8013.a (trivial)

\(f(q)\)	\(=\)	\(q-2.68470 q^{2} -1.00000 q^{3} +5.20764 q^{4} +3.29964 q^{5} +2.68470 q^{6} +2.53834 q^{7} -8.61155 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.68470 q^{2} -1.00000 q^{3} +5.20764 q^{4} +3.29964 q^{5} +2.68470 q^{6} +2.53834 q^{7} -8.61155 q^{8} +1.00000 q^{9} -8.85857 q^{10} -3.38448 q^{11} -5.20764 q^{12} -2.10791 q^{13} -6.81468 q^{14} -3.29964 q^{15} +12.7042 q^{16} +2.42812 q^{17} -2.68470 q^{18} -5.18666 q^{19} +17.1833 q^{20} -2.53834 q^{21} +9.08633 q^{22} +5.22110 q^{23} +8.61155 q^{24} +5.88765 q^{25} +5.65912 q^{26} -1.00000 q^{27} +13.2187 q^{28} +7.21769 q^{29} +8.85857 q^{30} +0.885022 q^{31} -16.8839 q^{32} +3.38448 q^{33} -6.51879 q^{34} +8.37560 q^{35} +5.20764 q^{36} -1.34140 q^{37} +13.9247 q^{38} +2.10791 q^{39} -28.4151 q^{40} +8.43898 q^{41} +6.81468 q^{42} +7.65624 q^{43} -17.6252 q^{44} +3.29964 q^{45} -14.0171 q^{46} +2.97342 q^{47} -12.7042 q^{48} -0.556854 q^{49} -15.8066 q^{50} -2.42812 q^{51} -10.9772 q^{52} -7.65239 q^{53} +2.68470 q^{54} -11.1676 q^{55} -21.8590 q^{56} +5.18666 q^{57} -19.3774 q^{58} +9.71669 q^{59} -17.1833 q^{60} +2.83907 q^{61} -2.37602 q^{62} +2.53834 q^{63} +19.9199 q^{64} -6.95536 q^{65} -9.08633 q^{66} +4.85131 q^{67} +12.6448 q^{68} -5.22110 q^{69} -22.4860 q^{70} +1.91662 q^{71} -8.61155 q^{72} -11.7804 q^{73} +3.60126 q^{74} -5.88765 q^{75} -27.0102 q^{76} -8.59095 q^{77} -5.65912 q^{78} -11.2855 q^{79} +41.9193 q^{80} +1.00000 q^{81} -22.6562 q^{82} +15.4117 q^{83} -13.2187 q^{84} +8.01194 q^{85} -20.5547 q^{86} -7.21769 q^{87} +29.1456 q^{88} -7.03493 q^{89} -8.85857 q^{90} -5.35058 q^{91} +27.1896 q^{92} -0.885022 q^{93} -7.98275 q^{94} -17.1141 q^{95} +16.8839 q^{96} -16.1263 q^{97} +1.49499 q^{98} -3.38448 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 106 q + 15 q^{2} - 106 q^{3} + 109 q^{4} + 16 q^{5} - 15 q^{6} + 35 q^{7} + 48 q^{8} + 106 q^{9}+O(q^{10}) \) 106 * q + 15 * q^2 - 106 * q^3 + 109 * q^4 + 16 * q^5 - 15 * q^6 + 35 * q^7 + 48 * q^8 + 106 * q^9	\( 106 q + 15 q^{2} - 106 q^{3} + 109 q^{4} + 16 q^{5} - 15 q^{6} + 35 q^{7} + 48 q^{8} + 106 q^{9} - 3 q^{10} + 55 q^{11} - 109 q^{12} - 8 q^{13} + 27 q^{14} - 16 q^{15} + 111 q^{16} + 28 q^{17} + 15 q^{18} + q^{19} + 54 q^{20} - 35 q^{21} + 20 q^{22} + 62 q^{23} - 48 q^{24} + 102 q^{25} + 21 q^{26} - 106 q^{27} + 79 q^{28} + 36 q^{29} + 3 q^{30} + q^{31} + 111 q^{32} - 55 q^{33} - 27 q^{34} + 72 q^{35} + 109 q^{36} + 31 q^{37} + 43 q^{38} + 8 q^{39} - 13 q^{40} + 35 q^{41} - 27 q^{42} + 98 q^{43} + 121 q^{44} + 16 q^{45} + 8 q^{46} + 75 q^{47} - 111 q^{48} + 49 q^{49} + 83 q^{50} - 28 q^{51} - 18 q^{52} + 60 q^{53} - 15 q^{54} + 14 q^{55} + 85 q^{56} - q^{57} + 65 q^{58} + 77 q^{59} - 54 q^{60} - 55 q^{61} + 83 q^{62} + 35 q^{63} + 122 q^{64} + 86 q^{65} - 20 q^{66} + 121 q^{67} + 80 q^{68} - 62 q^{69} - 11 q^{70} + 79 q^{71} + 48 q^{72} - 29 q^{73} + 91 q^{74} - 102 q^{75} - 10 q^{76} + 87 q^{77} - 21 q^{78} + 15 q^{79} + 108 q^{80} + 106 q^{81} + 21 q^{82} + 196 q^{83} - 79 q^{84} - 5 q^{85} + 65 q^{86} - 36 q^{87} + 84 q^{88} + 34 q^{89} - 3 q^{90} + 17 q^{91} + 162 q^{92} - q^{93} - 35 q^{94} + 113 q^{95} - 111 q^{96} - 63 q^{97} + 112 q^{98} + 55 q^{99}+O(q^{100}) \) 106 * q + 15 * q^2 - 106 * q^3 + 109 * q^4 + 16 * q^5 - 15 * q^6 + 35 * q^7 + 48 * q^8 + 106 * q^9 - 3 * q^10 + 55 * q^11 - 109 * q^12 - 8 * q^13 + 27 * q^14 - 16 * q^15 + 111 * q^16 + 28 * q^17 + 15 * q^18 + q^19 + 54 * q^20 - 35 * q^21 + 20 * q^22 + 62 * q^23 - 48 * q^24 + 102 * q^25 + 21 * q^26 - 106 * q^27 + 79 * q^28 + 36 * q^29 + 3 * q^30 + q^31 + 111 * q^32 - 55 * q^33 - 27 * q^34 + 72 * q^35 + 109 * q^36 + 31 * q^37 + 43 * q^38 + 8 * q^39 - 13 * q^40 + 35 * q^41 - 27 * q^42 + 98 * q^43 + 121 * q^44 + 16 * q^45 + 8 * q^46 + 75 * q^47 - 111 * q^48 + 49 * q^49 + 83 * q^50 - 28 * q^51 - 18 * q^52 + 60 * q^53 - 15 * q^54 + 14 * q^55 + 85 * q^56 - q^57 + 65 * q^58 + 77 * q^59 - 54 * q^60 - 55 * q^61 + 83 * q^62 + 35 * q^63 + 122 * q^64 + 86 * q^65 - 20 * q^66 + 121 * q^67 + 80 * q^68 - 62 * q^69 - 11 * q^70 + 79 * q^71 + 48 * q^72 - 29 * q^73 + 91 * q^74 - 102 * q^75 - 10 * q^76 + 87 * q^77 - 21 * q^78 + 15 * q^79 + 108 * q^80 + 106 * q^81 + 21 * q^82 + 196 * q^83 - 79 * q^84 - 5 * q^85 + 65 * q^86 - 36 * q^87 + 84 * q^88 + 34 * q^89 - 3 * q^90 + 17 * q^91 + 162 * q^92 - q^93 - 35 * q^94 + 113 * q^95 - 111 * q^96 - 63 * q^97 + 112 * q^98 + 55 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more