LMFDB - Embedded newform 6013.2.a.e.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6013,2,Mod(1,6013)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6013, 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("6013.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6013 = 7 \cdot 859 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6013.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:	$48.0140467354$
Analytic rank:	$0$
Dimension:	$109$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Character	$\chi$	$=$	6013.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.64232 q^{2} +0.320634 q^{3} +4.98185 q^{4} +3.17576 q^{5} -0.847217 q^{6} +1.00000 q^{7} -7.87900 q^{8} -2.89719 q^{9} +O(q^{10})$	\(q-2.64232 q^{2} +0.320634 q^{3} +4.98185 q^{4} +3.17576 q^{5} -0.847217 q^{6} +1.00000 q^{7} -7.87900 q^{8} -2.89719 q^{9} -8.39136 q^{10} -2.25492 q^{11} +1.59735 q^{12} -4.14521 q^{13} -2.64232 q^{14} +1.01826 q^{15} +10.8551 q^{16} -4.34057 q^{17} +7.65531 q^{18} -3.45874 q^{19} +15.8211 q^{20} +0.320634 q^{21} +5.95822 q^{22} -0.0874563 q^{23} -2.52627 q^{24} +5.08544 q^{25} +10.9530 q^{26} -1.89084 q^{27} +4.98185 q^{28} +8.08271 q^{29} -2.69056 q^{30} -1.22874 q^{31} -12.9247 q^{32} -0.723004 q^{33} +11.4692 q^{34} +3.17576 q^{35} -14.4334 q^{36} +2.47866 q^{37} +9.13908 q^{38} -1.32909 q^{39} -25.0218 q^{40} +0.513039 q^{41} -0.847217 q^{42} +1.83803 q^{43} -11.2337 q^{44} -9.20079 q^{45} +0.231087 q^{46} +3.62406 q^{47} +3.48052 q^{48} +1.00000 q^{49} -13.4373 q^{50} -1.39173 q^{51} -20.6508 q^{52} +0.723055 q^{53} +4.99620 q^{54} -7.16108 q^{55} -7.87900 q^{56} -1.10899 q^{57} -21.3571 q^{58} -6.53732 q^{59} +5.07280 q^{60} -0.324330 q^{61} +3.24673 q^{62} -2.89719 q^{63} +12.4410 q^{64} -13.1642 q^{65} +1.91041 q^{66} +7.47223 q^{67} -21.6241 q^{68} -0.0280415 q^{69} -8.39136 q^{70} -6.11028 q^{71} +22.8270 q^{72} -1.38985 q^{73} -6.54942 q^{74} +1.63056 q^{75} -17.2309 q^{76} -2.25492 q^{77} +3.51189 q^{78} +4.31579 q^{79} +34.4733 q^{80} +8.08531 q^{81} -1.35561 q^{82} +16.0073 q^{83} +1.59735 q^{84} -13.7846 q^{85} -4.85667 q^{86} +2.59159 q^{87} +17.7665 q^{88} +0.168935 q^{89} +24.3114 q^{90} -4.14521 q^{91} -0.435694 q^{92} -0.393977 q^{93} -9.57591 q^{94} -10.9841 q^{95} -4.14410 q^{96} +4.41054 q^{97} -2.64232 q^{98} +6.53294 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 109 q + 19 q^{2} + 38 q^{3} + 111 q^{4} + 43 q^{5} + 14 q^{6} + 109 q^{7} + 48 q^{8} + 119 q^{9}+O(q^{10}) $ 109 * q + 19 * q^2 + 38 * q^3 + 111 * q^4 + 43 * q^5 + 14 * q^6 + 109 * q^7 + 48 * q^8 + 119 * q^9	\( 109 q + 19 q^{2} + 38 q^{3} + 111 q^{4} + 43 q^{5} + 14 q^{6} + 109 q^{7} + 48 q^{8} + 119 q^{9} + 15 q^{10} + 48 q^{11} + 72 q^{12} + 29 q^{13} + 19 q^{14} + 29 q^{15} + 115 q^{16} + 72 q^{17} + 33 q^{18} + 58 q^{19} + 88 q^{20} + 38 q^{21} + 4 q^{22} + 65 q^{23} + 46 q^{24} + 124 q^{25} + 49 q^{26} + 131 q^{27} + 111 q^{28} + 25 q^{29} + 2 q^{30} + 41 q^{31} + 75 q^{32} + 54 q^{33} + 23 q^{34} + 43 q^{35} + 111 q^{36} + 25 q^{37} + 54 q^{38} + 27 q^{39} + 30 q^{40} + 109 q^{41} + 14 q^{42} + 38 q^{43} + 68 q^{44} + 84 q^{45} - 9 q^{46} + 121 q^{47} + 106 q^{48} + 109 q^{49} + 14 q^{50} + 36 q^{51} + 38 q^{52} + 61 q^{53} + 31 q^{54} + 50 q^{55} + 48 q^{56} + 5 q^{57} - 20 q^{58} + 181 q^{59} + 25 q^{60} + 34 q^{61} + 75 q^{62} + 119 q^{63} + 96 q^{64} + 12 q^{65} + 19 q^{66} + 87 q^{67} + 150 q^{68} + 89 q^{69} + 15 q^{70} + 83 q^{71} + 65 q^{72} + 32 q^{73} - 19 q^{74} + 112 q^{75} + 84 q^{76} + 48 q^{77} - 34 q^{78} - 9 q^{79} + 137 q^{80} + 109 q^{81} - 19 q^{82} + 136 q^{83} + 72 q^{84} - 32 q^{85} - 24 q^{86} + 28 q^{87} - 24 q^{88} + 142 q^{89} + 19 q^{90} + 29 q^{91} + 96 q^{92} + 29 q^{93} + 9 q^{94} + 52 q^{95} + 88 q^{96} + 75 q^{97} + 19 q^{98} + 84 q^{99}+O(q^{100}) \) 109 * q + 19 * q^2 + 38 * q^3 + 111 * q^4 + 43 * q^5 + 14 * q^6 + 109 * q^7 + 48 * q^8 + 119 * q^9 + 15 * q^10 + 48 * q^11 + 72 * q^12 + 29 * q^13 + 19 * q^14 + 29 * q^15 + 115 * q^16 + 72 * q^17 + 33 * q^18 + 58 * q^19 + 88 * q^20 + 38 * q^21 + 4 * q^22 + 65 * q^23 + 46 * q^24 + 124 * q^25 + 49 * q^26 + 131 * q^27 + 111 * q^28 + 25 * q^29 + 2 * q^30 + 41 * q^31 + 75 * q^32 + 54 * q^33 + 23 * q^34 + 43 * q^35 + 111 * q^36 + 25 * q^37 + 54 * q^38 + 27 * q^39 + 30 * q^40 + 109 * q^41 + 14 * q^42 + 38 * q^43 + 68 * q^44 + 84 * q^45 - 9 * q^46 + 121 * q^47 + 106 * q^48 + 109 * q^49 + 14 * q^50 + 36 * q^51 + 38 * q^52 + 61 * q^53 + 31 * q^54 + 50 * q^55 + 48 * q^56 + 5 * q^57 - 20 * q^58 + 181 * q^59 + 25 * q^60 + 34 * q^61 + 75 * q^62 + 119 * q^63 + 96 * q^64 + 12 * q^65 + 19 * q^66 + 87 * q^67 + 150 * q^68 + 89 * q^69 + 15 * q^70 + 83 * q^71 + 65 * q^72 + 32 * q^73 - 19 * q^74 + 112 * q^75 + 84 * q^76 + 48 * q^77 - 34 * q^78 - 9 * q^79 + 137 * q^80 + 109 * q^81 - 19 * q^82 + 136 * q^83 + 72 * q^84 - 32 * q^85 - 24 * q^86 + 28 * q^87 - 24 * q^88 + 142 * q^89 + 19 * q^90 + 29 * q^91 + 96 * q^92 + 29 * q^93 + 9 * q^94 + 52 * q^95 + 88 * q^96 + 75 * q^97 + 19 * q^98 + 84 * 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 $

$2$	−2.64232	−1.86840	−0.934201	−	0.356747i	$-0.883886\pi$
$2$	−2.64232	−1.86840	−0.934201	+	0.356747i	$0.883886\pi$
$3$	0.320634	0.185118	0.0925590	−	0.995707i	$-0.470495\pi$
$3$	0.320634	0.185118	0.0925590	+	0.995707i	$0.470495\pi$
$4$	4.98185	2.49093
$5$	3.17576	1.42024	0.710121	−	0.704080i	$-0.248640\pi$
$5$	3.17576	1.42024	0.710121	+	0.704080i	$0.248640\pi$
$6$	−0.847217	−0.345875
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−7.87900	−2.78565
$9$	−2.89719	−0.965731
$10$	−8.39136	−2.65358
$11$	−2.25492	−0.679884	−0.339942	−	0.940446i	$-0.610407\pi$
$11$	−2.25492	−0.679884	−0.339942	+	0.940446i	$0.610407\pi$
$12$	1.59735	0.461115
$13$	−4.14521	−1.14967	−0.574837	−	0.818268i	$-0.694935\pi$
$13$	−4.14521	−1.14967	−0.574837	+	0.818268i	$0.694935\pi$
$14$	−2.64232	−0.706189
$15$	1.01826	0.262912
$16$	10.8551	2.71378
$17$	−4.34057	−1.05274	−0.526371	−	0.850255i	$-0.676448\pi$
$17$	−4.34057	−1.05274	−0.526371	+	0.850255i	$0.676448\pi$
$18$	7.65531	1.80437
$19$	−3.45874	−0.793489	−0.396744	−	0.917929i	$-0.629860\pi$
$19$	−3.45874	−0.793489	−0.396744	+	0.917929i	$0.629860\pi$
$20$	15.8211	3.53772
$21$	0.320634	0.0699680
$22$	5.95822	1.27030
$23$	−0.0874563	−0.0182359	−0.00911795	−	0.999958i	$-0.502902\pi$
$23$	−0.0874563	−0.0182359	−0.00911795	+	0.999958i	$0.502902\pi$
$24$	−2.52627	−0.515674
$25$	5.08544	1.01709
$26$	10.9530	2.14805
$27$	−1.89084	−0.363892
$28$	4.98185	0.941481
$29$	8.08271	1.50092	0.750461	−	0.660915i	$-0.229832\pi$
$29$	8.08271	1.50092	0.750461	+	0.660915i	$0.229832\pi$
$30$	−2.69056	−0.491226
$31$	−1.22874	−0.220689	−0.110345	−	0.993893i	$-0.535195\pi$
$31$	−1.22874	−0.220689	−0.110345	+	0.993893i	$0.535195\pi$
$32$	−12.9247	−2.28479
$33$	−0.723004	−0.125859
$34$	11.4692	1.96695
$35$	3.17576	0.536801
$36$	−14.4334	−2.40556
$37$	2.47866	0.407490	0.203745	−	0.979024i	$-0.434689\pi$
$37$	2.47866	0.407490	0.203745	+	0.979024i	$0.434689\pi$
$38$	9.13908	1.48256
$39$	−1.32909	−0.212825
$40$	−25.0218	−3.95629
$41$	0.513039	0.0801232	0.0400616	−	0.999197i	$-0.487245\pi$
$41$	0.513039	0.0801232	0.0400616	+	0.999197i	$0.487245\pi$
$42$	−0.847217	−0.130728
$43$	1.83803	0.280297	0.140149	−	0.990130i	$-0.455242\pi$
$43$	1.83803	0.280297	0.140149	+	0.990130i	$0.455242\pi$
$44$	−11.2337	−1.69354
$45$	−9.20079	−1.37157
$46$	0.231087	0.0340720
$47$	3.62406	0.528623	0.264312	−	0.964437i	$-0.414855\pi$
$47$	3.62406	0.528623	0.264312	+	0.964437i	$0.414855\pi$
$48$	3.48052	0.502370
$49$	1.00000	0.142857
$50$	−13.4373	−1.90033
$51$	−1.39173	−0.194882
$52$	−20.6508	−2.86375
$53$	0.723055	0.0993193	0.0496596	−	0.998766i	$-0.484186\pi$
$53$	0.723055	0.0993193	0.0496596	+	0.998766i	$0.484186\pi$
$54$	4.99620	0.679897
$55$	−7.16108	−0.965600
$56$	−7.87900	−1.05288
$57$	−1.10899	−0.146889
$58$	−21.3571	−2.80433
$59$	−6.53732	−0.851086	−0.425543	−	0.904938i	$-0.639917\pi$
$59$	−6.53732	−0.851086	−0.425543	+	0.904938i	$0.639917\pi$
$60$	5.07280	0.654895
$61$	−0.324330	−0.0415261	−0.0207631	−	0.999784i	$-0.506610\pi$
$61$	−0.324330	−0.0415261	−0.0207631	+	0.999784i	$0.506610\pi$
$62$	3.24673	0.412336
$63$	−2.89719	−0.365012
$64$	12.4410	1.55512
$65$	−13.1642	−1.63282
$66$	1.91041	0.235155
$67$	7.47223	0.912879	0.456439	−	0.889755i	$-0.349125\pi$
$67$	7.47223	0.912879	0.456439	+	0.889755i	$0.349125\pi$
$68$	−21.6241	−2.62230
$69$	−0.0280415	−0.00337579
$70$	−8.39136	−1.00296
$71$	−6.11028	−0.725156	−0.362578	−	0.931953i	$-0.618103\pi$
$71$	−6.11028	−0.725156	−0.362578	+	0.931953i	$0.618103\pi$
$72$	22.8270	2.69019
$73$	−1.38985	−0.162670	−0.0813349	−	0.996687i	$-0.525918\pi$
$73$	−1.38985	−0.162670	−0.0813349	+	0.996687i	$0.525918\pi$
$74$	−6.54942	−0.761355
$75$	1.63056	0.188281
$76$	−17.2309	−1.97652
$77$	−2.25492	−0.256972
$78$	3.51189	0.397643
$79$	4.31579	0.485565	0.242782	−	0.970081i	$-0.421940\pi$
$79$	4.31579	0.485565	0.242782	+	0.970081i	$0.421940\pi$
$80$	34.4733	3.85423
$81$	8.08531	0.898368
$82$	−1.35561	−0.149702
$83$	16.0073	1.75703	0.878515	−	0.477715i	$-0.158535\pi$
$83$	16.0073	1.75703	0.878515	+	0.477715i	$0.158535\pi$
$84$	1.59735	0.174285
$85$	−13.7846	−1.49515
$86$	−4.85667	−0.523708
$87$	2.59159	0.277848
$88$	17.7665	1.89392
$89$	0.168935	0.0179071	0.00895355	−	0.999960i	$-0.497150\pi$
$89$	0.168935	0.0179071	0.00895355	+	0.999960i	$0.497150\pi$
$90$	24.3114	2.56265
$91$	−4.14521	−0.434536
$92$	−0.435694	−0.0454243
$93$	−0.393977	−0.0408535
$94$	−9.57591	−0.987680
$95$	−10.9841	−1.12695
$96$	−4.14410	−0.422956
$97$	4.41054	0.447822	0.223911	−	0.974610i	$-0.428117\pi$
$97$	4.41054	0.447822	0.223911	+	0.974610i	$0.428117\pi$
$98$	−2.64232	−0.266915
$99$	6.53294	0.656585
$100$	25.3349	2.53349
$101$	14.1880	1.41176	0.705881	−	0.708331i	$-0.250552\pi$
$101$	14.1880	1.41176	0.705881	+	0.708331i	$0.250552\pi$
$102$	3.67740	0.364117
$103$	8.06600	0.794766	0.397383	−	0.917653i	$-0.369918\pi$
$103$	8.06600	0.794766	0.397383	+	0.917653i	$0.369918\pi$
$104$	32.6601	3.20259
$105$	1.01826	0.0993716
$106$	−1.91054	−0.185568
$107$	−3.05395	−0.295236	−0.147618	−	0.989044i	$-0.547161\pi$
$107$	−3.05395	−0.295236	−0.147618	+	0.989044i	$0.547161\pi$
$108$	−9.41988	−0.906429
$109$	−10.6433	−1.01945	−0.509724	−	0.860338i	$-0.670252\pi$
$109$	−10.6433	−1.01945	−0.509724	+	0.860338i	$0.670252\pi$
$110$	18.9219	1.80413
$111$	0.794744	0.0754337
$112$	10.8551	1.02571
$113$	−4.99844	−0.470214	−0.235107	−	0.971969i	$-0.575544\pi$
$113$	−4.99844	−0.470214	−0.235107	+	0.971969i	$0.575544\pi$
$114$	2.93030	0.274448
$115$	−0.277740	−0.0258994
$116$	40.2669	3.73868
$117$	12.0095	1.11028
$118$	17.2737	1.59017
$119$	−4.34057	−0.397899
$120$	−8.02283	−0.732381
$121$	−5.91533	−0.537758
$122$	0.856982	0.0775875
$123$	0.164498	0.0148323
$124$	−6.12142	−0.549720
$125$	0.271324	0.0242680
$126$	7.65531	0.681989
$127$	6.43874	0.571346	0.285673	−	0.958327i	$-0.407783\pi$
$127$	6.43874	0.571346	0.285673	+	0.958327i	$0.407783\pi$
$128$	−7.02359	−0.620803
$129$	0.589335	0.0518881
$130$	34.7840	3.05076
$131$	7.45692	0.651514	0.325757	−	0.945454i	$-0.394381\pi$
$131$	7.45692	0.651514	0.325757	+	0.945454i	$0.394381\pi$
$132$	−3.60190	−0.313505
$133$	−3.45874	−0.299910
$134$	−19.7440	−1.70562
$135$	−6.00485	−0.516815
$136$	34.1993	2.93257
$137$	18.5423	1.58418	0.792088	−	0.610407i	$-0.208994\pi$
$137$	18.5423	1.58418	0.792088	+	0.610407i	$0.208994\pi$
$138$	0.0740945	0.00630734
$139$	−12.4252	−1.05390	−0.526948	−	0.849898i	$-0.676664\pi$
$139$	−12.4252	−1.05390	−0.526948	+	0.849898i	$0.676664\pi$
$140$	15.8211	1.33713
$141$	1.16200	0.0978577
$142$	16.1453	1.35488
$143$	9.34712	0.781645
$144$	−31.4494	−2.62079
$145$	25.6687	2.13167
$146$	3.67243	0.303933
$147$	0.320634	0.0264454
$148$	12.3483	1.01503
$149$	16.0116	1.31172	0.655860	−	0.754882i	$-0.272306\pi$
$149$	16.0116	1.31172	0.655860	+	0.754882i	$0.272306\pi$
$150$	−4.30847	−0.351785
$151$	3.72561	0.303185	0.151593	−	0.988443i	$-0.451560\pi$
$151$	3.72561	0.303185	0.151593	+	0.988443i	$0.451560\pi$
$152$	27.2514	2.21038
$153$	12.5755	1.01667
$154$	5.95822	0.480127
$155$	−3.90219	−0.313432
$156$	−6.62135	−0.530132
$157$	−13.5703	−1.08303	−0.541515	−	0.840691i	$-0.682149\pi$
$157$	−13.5703	−1.08303	−0.541515	+	0.840691i	$0.682149\pi$
$158$	−11.4037	−0.907230
$159$	0.231836	0.0183858
$160$	−41.0458	−3.24495
$161$	−0.0874563	−0.00689252
$162$	−21.3640	−1.67851
$163$	7.25112	0.567952	0.283976	−	0.958831i	$-0.408346\pi$
$163$	7.25112	0.567952	0.283976	+	0.958831i	$0.408346\pi$
$164$	2.55588	0.199581
$165$	−2.29609	−0.178750
$166$	−42.2964	−3.28284
$167$	13.2017	1.02158	0.510788	−	0.859706i	$-0.329354\pi$
$167$	13.2017	1.02158	0.510788	+	0.859706i	$0.329354\pi$
$168$	−2.52627	−0.194906
$169$	4.18276	0.321751
$170$	36.4233	2.79354
$171$	10.0206	0.766297
$172$	9.15680	0.698200
$173$	12.3773	0.941030	0.470515	−	0.882392i	$-0.344068\pi$
$173$	12.3773	0.941030	0.470515	+	0.882392i	$0.344068\pi$
$174$	−6.84781	−0.519131
$175$	5.08544	0.384423
$176$	−24.4775	−1.84506
$177$	−2.09609	−0.157551
$178$	−0.446381	−0.0334577
$179$	19.9556	1.49155	0.745776	−	0.666197i	$-0.232079\pi$
$179$	19.9556	1.49155	0.745776	+	0.666197i	$0.232079\pi$
$180$	−45.8369	−3.41648
$181$	−0.979065	−0.0727734	−0.0363867	−	0.999338i	$-0.511585\pi$
$181$	−0.979065	−0.0727734	−0.0363867	+	0.999338i	$0.511585\pi$
$182$	10.9530	0.811888
$183$	−0.103991	−0.00768724
$184$	0.689068	0.0507988
$185$	7.87164	0.578734
$186$	1.04101	0.0763308
$187$	9.78763	0.715743
$188$	18.0545	1.31676
$189$	−1.89084	−0.137538
$190$	29.0235	2.10559
$191$	7.02299	0.508166	0.254083	−	0.967182i	$-0.418226\pi$
$191$	7.02299	0.508166	0.254083	+	0.967182i	$0.418226\pi$
$192$	3.98900	0.287881
$193$	16.0737	1.15701	0.578506	−	0.815678i	$-0.303636\pi$
$193$	16.0737	1.15701	0.578506	+	0.815678i	$0.303636\pi$
$194$	−11.6540	−0.836712
$195$	−4.22088	−0.302264
$196$	4.98185	0.355846
$197$	−8.03130	−0.572207	−0.286103	−	0.958199i	$-0.592360\pi$
$197$	−8.03130	−0.572207	−0.286103	+	0.958199i	$0.592360\pi$
$198$	−17.2621	−1.22677
$199$	21.6683	1.53603	0.768013	−	0.640434i	$-0.221246\pi$
$199$	21.6683	1.53603	0.768013	+	0.640434i	$0.221246\pi$
$200$	−40.0682	−2.83325
$201$	2.39585	0.168990
$202$	−37.4893	−2.63774
$203$	8.08271	0.567295
$204$	−6.93341	−0.485435
$205$	1.62929	0.113794
$206$	−21.3129	−1.48494
$207$	0.253378	0.0176110
$208$	−44.9968	−3.11997
$209$	7.79918	0.539480
$210$	−2.69056	−0.185666
$211$	−20.5592	−1.41535	−0.707677	−	0.706536i	$-0.750257\pi$
$211$	−20.5592	−1.41535	−0.707677	+	0.706536i	$0.750257\pi$
$212$	3.60215	0.247397
$213$	−1.95916	−0.134240
$214$	8.06951	0.551620
$215$	5.83715	0.398090
$216$	14.8979	1.01368
$217$	−1.22874	−0.0834126
$218$	28.1231	1.90474
$219$	−0.445633	−0.0301131
$220$	−35.6754	−2.40524
$221$	17.9926	1.21031
$222$	−2.09997	−0.140941
$223$	22.4438	1.50295	0.751475	−	0.659762i	$-0.229343\pi$
$223$	22.4438	1.50295	0.751475	+	0.659762i	$0.229343\pi$
$224$	−12.9247	−0.863569
$225$	−14.7335	−0.982233
$226$	13.2075	0.878549
$227$	10.4453	0.693277	0.346638	−	0.937999i	$-0.387323\pi$
$227$	10.4453	0.693277	0.346638	+	0.937999i	$0.387323\pi$
$228$	−5.52481	−0.365890
$229$	15.2443	1.00737	0.503686	−	0.863887i	$-0.331977\pi$
$229$	15.2443	1.00737	0.503686	+	0.863887i	$0.331977\pi$
$230$	0.733878	0.0483905
$231$	−0.723004	−0.0475702
$232$	−63.6837	−4.18104
$233$	−25.4580	−1.66781	−0.833905	−	0.551907i	$-0.813900\pi$
$233$	−25.4580	−1.66781	−0.833905	+	0.551907i	$0.813900\pi$
$234$	−31.7329	−2.07444
$235$	11.5091	0.750773
$236$	−32.5679	−2.11999
$237$	1.38379	0.0898868
$238$	11.4692	0.743436
$239$	−24.1092	−1.55950	−0.779749	−	0.626093i	$-0.784653\pi$
$239$	−24.1092	−1.55950	−0.779749	+	0.626093i	$0.784653\pi$
$240$	11.0533	0.713487
$241$	−20.0411	−1.29096	−0.645482	−	0.763776i	$-0.723343\pi$
$241$	−20.0411	−1.29096	−0.645482	+	0.763776i	$0.723343\pi$
$242$	15.6302	1.00475
$243$	8.26495	0.530197
$244$	−1.61576	−0.103439
$245$	3.17576	0.202892
$246$	−0.434655	−0.0277126
$247$	14.3372	0.912253
$248$	9.68128	0.614762
$249$	5.13248	0.325258
$250$	−0.716925	−0.0453423
$251$	21.6054	1.36372	0.681861	−	0.731481i	$-0.261171\pi$
$251$	21.6054	1.36372	0.681861	+	0.731481i	$0.261171\pi$
$252$	−14.4334	−0.909218
$253$	0.197207	0.0123983
$254$	−17.0132	−1.06750
$255$	−4.41981	−0.276779
$256$	−6.32340	−0.395212
$257$	23.0018	1.43482	0.717408	−	0.696654i	$-0.245329\pi$
$257$	23.0018	1.43482	0.717408	+	0.696654i	$0.245329\pi$
$258$	−1.55721	−0.0969478
$259$	2.47866	0.154017
$260$	−65.5820	−4.06722
$261$	−23.4172	−1.44949
$262$	−19.7036	−1.21729
$263$	−2.20155	−0.135754	−0.0678768	−	0.997694i	$-0.521622\pi$
$263$	−2.20155	−0.135754	−0.0678768	+	0.997694i	$0.521622\pi$
$264$	5.69655	0.350598
$265$	2.29625	0.141057
$266$	9.13908	0.560353
$267$	0.0541664	0.00331493
$268$	37.2256	2.27391
$269$	18.5627	1.13178	0.565892	−	0.824479i	$-0.308532\pi$
$269$	18.5627	1.13178	0.565892	+	0.824479i	$0.308532\pi$
$270$	15.8667	0.965618
$271$	−14.7983	−0.898930	−0.449465	−	0.893298i	$-0.648385\pi$
$271$	−14.7983	−0.898930	−0.449465	+	0.893298i	$0.648385\pi$
$272$	−47.1174	−2.85691
$273$	−1.32909	−0.0804405
$274$	−48.9947	−2.95988
$275$	−11.4673	−0.691501
$276$	−0.139698	−0.00840885
$277$	−12.6694	−0.761232	−0.380616	−	0.924733i	$-0.624288\pi$
$277$	−12.6694	−0.761232	−0.380616	+	0.924733i	$0.624288\pi$
$278$	32.8314	1.96910
$279$	3.55991	0.213126
$280$	−25.0218	−1.49534
$281$	−4.86512	−0.290229	−0.145114	−	0.989415i	$-0.546355\pi$
$281$	−4.86512	−0.290229	−0.145114	+	0.989415i	$0.546355\pi$
$282$	−3.07036	−0.182837
$283$	−5.17506	−0.307625	−0.153813	−	0.988100i	$-0.549155\pi$
$283$	−5.17506	−0.307625	−0.153813	+	0.988100i	$0.549155\pi$
$284$	−30.4405	−1.80631
$285$	−3.52188	−0.208618
$286$	−24.6981	−1.46043
$287$	0.513039	0.0302837
$288$	37.4454	2.20649
$289$	1.84053	0.108266
$290$	−67.8250	−3.98282
$291$	1.41417	0.0829000
$292$	−6.92403	−0.405198
$293$	−7.98020	−0.466208	−0.233104	−	0.972452i	$-0.574888\pi$
$293$	−7.98020	−0.466208	−0.233104	+	0.972452i	$0.574888\pi$
$294$	−0.847217	−0.0494107
$295$	−20.7609	−1.20875
$296$	−19.5294	−1.13512
$297$	4.26369	0.247405
$298$	−42.3077	−2.45082
$299$	0.362525	0.0209653
$300$	8.12322	0.468994
$301$	1.83803	0.105942
$302$	−9.84424	−0.566472
$303$	4.54916	0.261342
$304$	−37.5450	−2.15336
$305$	−1.02999	−0.0589772
$306$	−33.2284	−1.89954
$307$	−13.5611	−0.773973	−0.386986	−	0.922085i	$-0.626484\pi$
$307$	−13.5611	−0.773973	−0.386986	+	0.922085i	$0.626484\pi$
$308$	−11.2337	−0.640098
$309$	2.58623	0.147126
$310$	10.3108	0.585617
$311$	−9.64570	−0.546957	−0.273479	−	0.961878i	$-0.588174\pi$
$311$	−9.64570	−0.546957	−0.273479	+	0.961878i	$0.588174\pi$
$312$	10.4719	0.592857
$313$	10.3211	0.583384	0.291692	−	0.956512i	$-0.405782\pi$
$313$	10.3211	0.583384	0.291692	+	0.956512i	$0.405782\pi$
$314$	35.8572	2.02354
$315$	−9.20079	−0.518406
$316$	21.5006	1.20951
$317$	4.47127	0.251132	0.125566	−	0.992085i	$-0.459925\pi$
$317$	4.47127	0.251132	0.125566	+	0.992085i	$0.459925\pi$
$318$	−0.612585	−0.0343520
$319$	−18.2259	−1.02045
$320$	39.5095	2.20865
$321$	−0.979199	−0.0546536
$322$	0.231087	0.0128780
$323$	15.0129	0.835339
$324$	40.2798	2.23777
$325$	−21.0802	−1.16932
$326$	−19.1598	−1.06116
$327$	−3.41262	−0.188718
$328$	−4.04223	−0.223195
$329$	3.62406	0.199801
$330$	6.06699	0.333977
$331$	13.1757	0.724201	0.362100	−	0.932139i	$-0.382060\pi$
$331$	13.1757	0.724201	0.362100	+	0.932139i	$0.382060\pi$
$332$	79.7460	4.37663
$333$	−7.18117	−0.393526
$334$	−34.8831	−1.90872
$335$	23.7300	1.29651
$336$	3.48052	0.189878
$337$	7.32420	0.398975	0.199487	−	0.979900i	$-0.436072\pi$
$337$	7.32420	0.398975	0.199487	+	0.979900i	$0.436072\pi$
$338$	−11.0522	−0.601159
$339$	−1.60267	−0.0870451
$340$	−68.6728	−3.72430
$341$	2.77072	0.150043
$342$	−26.4777	−1.43175
$343$	1.00000	0.0539949
$344$	−14.4819	−0.780809
$345$	−0.0890529	−0.00479445
$346$	−32.7048	−1.75822
$347$	−27.5390	−1.47837	−0.739186	−	0.673501i	$-0.764790\pi$
$347$	−27.5390	−1.47837	−0.739186	+	0.673501i	$0.764790\pi$
$348$	12.9109	0.692098
$349$	26.4658	1.41668	0.708341	−	0.705870i	$-0.249444\pi$
$349$	26.4658	1.41668	0.708341	+	0.705870i	$0.249444\pi$
$350$	−13.4373	−0.718256
$351$	7.83793	0.418358
$352$	29.1442	1.55339
$353$	5.00160	0.266208	0.133104	−	0.991102i	$-0.457506\pi$
$353$	5.00160	0.266208	0.133104	+	0.991102i	$0.457506\pi$
$354$	5.53853	0.294369
$355$	−19.4048	−1.02990
$356$	0.841610	0.0446053
$357$	−1.39173	−0.0736583
$358$	−52.7291	−2.78682
$359$	25.4529	1.34335	0.671677	−	0.740844i	$-0.265574\pi$
$359$	25.4529	1.34335	0.671677	+	0.740844i	$0.265574\pi$
$360$	72.4930	3.82072
$361$	−7.03714	−0.370376
$362$	2.58700	0.135970
$363$	−1.89666	−0.0995486
$364$	−20.6508	−1.08240
$365$	−4.41383	−0.231030
$366$	0.274778	0.0143629
$367$	−3.59788	−0.187808	−0.0939040	−	0.995581i	$-0.529935\pi$
$367$	−3.59788	−0.187808	−0.0939040	+	0.995581i	$0.529935\pi$
$368$	−0.949350	−0.0494883
$369$	−1.48637	−0.0773775
$370$	−20.7994	−1.08131
$371$	0.723055	0.0375392
$372$	−1.96273	−0.101763
$373$	3.59247	0.186011	0.0930054	−	0.995666i	$-0.470353\pi$
$373$	3.59247	0.186011	0.0930054	+	0.995666i	$0.470353\pi$
$374$	−25.8621	−1.33730
$375$	0.0869958	0.00449244
$376$	−28.5539	−1.47256
$377$	−33.5045	−1.72557
$378$	4.99620	0.256977
$379$	1.68973	0.0867954	0.0433977	−	0.999058i	$-0.486182\pi$
$379$	1.68973	0.0867954	0.0433977	+	0.999058i	$0.486182\pi$
$380$	−54.7212	−2.80714
$381$	2.06448	0.105766
$382$	−18.5570	−0.949458
$383$	10.7477	0.549181	0.274590	−	0.961561i	$-0.411458\pi$
$383$	10.7477	0.549181	0.274590	+	0.961561i	$0.411458\pi$
$384$	−2.25200	−0.114922
$385$	−7.16108	−0.364962
$386$	−42.4719	−2.16176
$387$	−5.32514	−0.270692
$388$	21.9726	1.11549
$389$	16.3811	0.830555	0.415278	−	0.909695i	$-0.363684\pi$
$389$	16.3811	0.830555	0.415278	+	0.909695i	$0.363684\pi$
$390$	11.1529	0.564750
$391$	0.379610	0.0191977
$392$	−7.87900	−0.397950
$393$	2.39094	0.120607
$394$	21.2213	1.06911
$395$	13.7059	0.689619
$396$	32.5461	1.63551
$397$	−16.5515	−0.830695	−0.415347	−	0.909663i	$-0.636340\pi$
$397$	−16.5515	−0.830695	−0.415347	+	0.909663i	$0.636340\pi$
$398$	−57.2546	−2.86991
$399$	−1.10899	−0.0555188
$400$	55.2031	2.76015
$401$	21.4851	1.07292	0.536458	−	0.843927i	$-0.319762\pi$
$401$	21.4851	1.07292	0.536458	+	0.843927i	$0.319762\pi$
$402$	−6.33060	−0.315742
$403$	5.09340	0.253720
$404$	70.6826	3.51659
$405$	25.6770	1.27590
$406$	−21.3571	−1.05994
$407$	−5.58919	−0.277046
$408$	10.9655	0.542871
$409$	−15.9226	−0.787321	−0.393661	−	0.919256i	$-0.628791\pi$
$409$	−15.9226	−0.787321	−0.393661	+	0.919256i	$0.628791\pi$
$410$	−4.30510	−0.212614
$411$	5.94529	0.293259
$412$	40.1836	1.97970
$413$	−6.53732	−0.321680
$414$	−0.669505	−0.0329044
$415$	50.8353	2.49541
$416$	53.5757	2.62676
$417$	−3.98395	−0.195095
$418$	−20.6079	−1.00797
$419$	−2.24045	−0.109453	−0.0547266	−	0.998501i	$-0.517429\pi$
$419$	−2.24045	−0.109453	−0.0547266	+	0.998501i	$0.517429\pi$
$420$	5.07280	0.247527
$421$	5.19408	0.253144	0.126572	−	0.991957i	$-0.459603\pi$
$421$	5.19408	0.253144	0.126572	+	0.991957i	$0.459603\pi$
$422$	54.3239	2.64445
$423$	−10.4996	−0.510508
$424$	−5.69695	−0.276668
$425$	−22.0737	−1.07073
$426$	5.17673	0.250813
$427$	−0.324330	−0.0156954
$428$	−15.2143	−0.735412
$429$	2.99700	0.144697
$430$	−15.4236	−0.743792
$431$	10.7555	0.518076	0.259038	−	0.965867i	$-0.416594\pi$
$431$	10.7555	0.518076	0.259038	+	0.965867i	$0.416594\pi$
$432$	−20.5253	−0.987525
$433$	2.38804	0.114762	0.0573810	−	0.998352i	$-0.481725\pi$
$433$	2.38804	0.114762	0.0573810	+	0.998352i	$0.481725\pi$
$434$	3.24673	0.155848
$435$	8.23027	0.394611
$436$	−53.0236	−2.53937
$437$	0.302488	0.0144700
$438$	1.17751	0.0562634
$439$	13.5279	0.645650	0.322825	−	0.946459i	$-0.395368\pi$
$439$	13.5279	0.645650	0.322825	+	0.946459i	$0.395368\pi$
$440$	56.4222	2.68982
$441$	−2.89719	−0.137962
$442$	−47.5421	−2.26135
$443$	15.3217	0.727956	0.363978	−	0.931408i	$-0.381418\pi$
$443$	15.3217	0.727956	0.363978	+	0.931408i	$0.381418\pi$
$444$	3.95929	0.187900
$445$	0.536498	0.0254324
$446$	−59.3037	−2.80811
$447$	5.13386	0.242823
$448$	12.4410	0.587781
$449$	−12.1618	−0.573951	−0.286976	−	0.957938i	$-0.592650\pi$
$449$	−12.1618	−0.573951	−0.286976	+	0.957938i	$0.592650\pi$
$450$	38.9306	1.83521
$451$	−1.15686	−0.0544745
$452$	−24.9015	−1.17127
$453$	1.19456	0.0561251
$454$	−27.5997	−1.29532
$455$	−13.1642	−0.617146
$456$	8.73772	0.409181
$457$	−11.4311	−0.534724	−0.267362	−	0.963596i	$-0.586152\pi$
$457$	−11.4311	−0.534724	−0.267362	+	0.963596i	$0.586152\pi$
$458$	−40.2803	−1.88218
$459$	8.20732	0.383085
$460$	−1.38366	−0.0645135
$461$	0.207894	0.00968258	0.00484129	−	0.999988i	$-0.498459\pi$
$461$	0.207894	0.00968258	0.00484129	+	0.999988i	$0.498459\pi$
$462$	1.91041	0.0888802
$463$	13.3184	0.618958	0.309479	−	0.950906i	$-0.399845\pi$
$463$	13.3184	0.618958	0.309479	+	0.950906i	$0.399845\pi$
$464$	87.7389	4.07318
$465$	−1.25118	−0.0580219
$466$	67.2682	3.11614
$467$	39.7135	1.83772	0.918862	−	0.394580i	$-0.129110\pi$
$467$	39.7135	1.83772	0.918862	+	0.394580i	$0.129110\pi$
$468$	59.8294	2.76562
$469$	7.47223	0.345036
$470$	−30.4108	−1.40274
$471$	−4.35111	−0.200489
$472$	51.5075	2.37083
$473$	−4.14462	−0.190570
$474$	−3.65641	−0.167945
$475$	−17.5892	−0.807047
$476$	−21.6241	−0.991137
$477$	−2.09483	−0.0959157
$478$	63.7043	2.91377
$479$	−27.5611	−1.25930	−0.629649	−	0.776880i	$-0.716801\pi$
$479$	−27.5611	−1.25930	−0.629649	+	0.776880i	$0.716801\pi$
$480$	−13.1607	−0.600700
$481$	−10.2746	−0.468481
$482$	52.9551	2.41204
$483$	−0.0280415	−0.00127593
$484$	−29.4693	−1.33951
$485$	14.0068	0.636016
$486$	−21.8386	−0.990620
$487$	24.9500	1.13059	0.565296	−	0.824888i	$-0.308762\pi$
$487$	24.9500	1.13059	0.565296	+	0.824888i	$0.308762\pi$
$488$	2.55539	0.115677
$489$	2.32495	0.105138
$490$	−8.39136	−0.379083
$491$	−3.63544	−0.164065	−0.0820326	−	0.996630i	$-0.526141\pi$
$491$	−3.63544	−0.164065	−0.0820326	+	0.996630i	$0.526141\pi$
$492$	0.819503	0.0369460
$493$	−35.0836	−1.58008
$494$	−37.8834	−1.70446
$495$	20.7470	0.932510
$496$	−13.3382	−0.598902
$497$	−6.11028	−0.274083
$498$	−13.5617	−0.607713
$499$	−39.5693	−1.77137	−0.885683	−	0.464290i	$-0.846310\pi$
$499$	−39.5693	−1.77137	−0.885683	+	0.464290i	$0.846310\pi$
$500$	1.35170	0.0604497
$501$	4.23291	0.189112
$502$	−57.0884	−2.54798
$503$	2.80048	0.124867	0.0624336	−	0.998049i	$-0.480114\pi$
$503$	2.80048	0.124867	0.0624336	+	0.998049i	$0.480114\pi$
$504$	22.8270	1.01679
$505$	45.0577	2.00504
$506$	−0.521084	−0.0231650
$507$	1.34113	0.0595618
$508$	32.0768	1.42318
$509$	14.0930	0.624662	0.312331	−	0.949973i	$-0.398890\pi$
$509$	14.0930	0.624662	0.312331	+	0.949973i	$0.398890\pi$
$510$	11.6785	0.517134
$511$	−1.38985	−0.0614834
$512$	30.7556	1.35922
$513$	6.53992	0.288744
$514$	−60.7782	−2.68081
$515$	25.6156	1.12876
$516$	2.93598	0.129249
$517$	−8.17196	−0.359402
$518$	−6.54942	−0.287765
$519$	3.96859	0.174202
$520$	103.721	4.54845
$521$	37.2880	1.63362	0.816809	−	0.576908i	$-0.195741\pi$
$521$	37.2880	1.63362	0.816809	+	0.576908i	$0.195741\pi$
$522$	61.8757	2.70822
$523$	−25.1252	−1.09865	−0.549325	−	0.835609i	$-0.685115\pi$
$523$	−25.1252	−1.09865	−0.549325	+	0.835609i	$0.685115\pi$
$524$	37.1492	1.62287
$525$	1.63056	0.0711636
$526$	5.81720	0.253642
$527$	5.33345	0.232329
$528$	−7.84830	−0.341554
$529$	−22.9924	−0.999667
$530$	−6.06742	−0.263552
$531$	18.9399	0.821921
$532$	−17.2309	−0.747055
$533$	−2.12665	−0.0921156
$534$	−0.143125	−0.00619362
$535$	−9.69860	−0.419307
$536$	−58.8737	−2.54296
$537$	6.39844	0.276113
$538$	−49.0485	−2.11463
$539$	−2.25492	−0.0971263
$540$	−29.9153	−1.28735
$541$	−2.79764	−0.120280	−0.0601400	−	0.998190i	$-0.519155\pi$
$541$	−2.79764	−0.120280	−0.0601400	+	0.998190i	$0.519155\pi$
$542$	39.1017	1.67956
$543$	−0.313922	−0.0134717
$544$	56.1006	2.40529
$545$	−33.8007	−1.44786
$546$	3.51189	0.150295
$547$	1.01710	0.0434878	0.0217439	−	0.999764i	$-0.493078\pi$
$547$	1.01710	0.0434878	0.0217439	+	0.999764i	$0.493078\pi$
$548$	92.3749	3.94606
$549$	0.939646	0.0401031
$550$	30.3001	1.29200
$551$	−27.9560	−1.19096
$552$	0.220939	0.00940377
$553$	4.31579	0.183526
$554$	33.4767	1.42229
$555$	2.52391	0.107134
$556$	−61.9007	−2.62517
$557$	11.3077	0.479121	0.239560	−	0.970881i	$-0.422997\pi$
$557$	11.3077	0.479121	0.239560	+	0.970881i	$0.422997\pi$
$558$	−9.40642	−0.398206
$559$	−7.61903	−0.322251
$560$	34.4733	1.45676
$561$	3.13825	0.132497
$562$	12.8552	0.542264
$563$	−38.2528	−1.61217	−0.806083	−	0.591803i	$-0.798416\pi$
$563$	−38.2528	−1.61217	−0.806083	+	0.591803i	$0.798416\pi$
$564$	5.78889	0.243756
$565$	−15.8738	−0.667818
$566$	13.6742	0.574768
$567$	8.08531	0.339551
$568$	48.1429	2.02003
$569$	23.1702	0.971346	0.485673	−	0.874141i	$-0.338575\pi$
$569$	23.1702	0.971346	0.485673	+	0.874141i	$0.338575\pi$
$570$	9.30592	0.389782
$571$	12.5117	0.523598	0.261799	−	0.965122i	$-0.415684\pi$
$571$	12.5117	0.523598	0.261799	+	0.965122i	$0.415684\pi$
$572$	46.5659	1.94702
$573$	2.25181	0.0940706
$574$	−1.35561	−0.0565822
$575$	−0.444754	−0.0185475
$576$	−36.0439	−1.50183
$577$	18.0570	0.751721	0.375860	−	0.926676i	$-0.377347\pi$
$577$	18.0570	0.751721	0.375860	+	0.926676i	$0.377347\pi$
$578$	−4.86326	−0.202285
$579$	5.15378	0.214184
$580$	127.878	5.30984
$581$	16.0073	0.664095
$582$	−3.73668	−0.154890
$583$	−1.63043	−0.0675256
$584$	10.9506	0.453141
$585$	38.1392	1.57686
$586$	21.0862	0.871064
$587$	−28.2171	−1.16464	−0.582322	−	0.812958i	$-0.697856\pi$
$587$	−28.2171	−1.16464	−0.582322	+	0.812958i	$0.697856\pi$
$588$	1.59735	0.0658736
$589$	4.24990	0.175114
$590$	54.8570	2.25843
$591$	−2.57511	−0.105926
$592$	26.9062	1.10584
$593$	15.5471	0.638442	0.319221	−	0.947680i	$-0.396579\pi$
$593$	15.5471	0.638442	0.319221	+	0.947680i	$0.396579\pi$
$594$	−11.2660	−0.462251
$595$	−13.7846	−0.565113
$596$	79.7674	3.26740
$597$	6.94760	0.284346
$598$	−0.957906	−0.0391717
$599$	−23.3398	−0.953638	−0.476819	−	0.879002i	$-0.658210\pi$
$599$	−23.3398	−0.953638	−0.476819	+	0.879002i	$0.658210\pi$
$600$	−12.8472	−0.524485
$601$	−7.53645	−0.307418	−0.153709	−	0.988116i	$-0.549122\pi$
$601$	−7.53645	−0.307418	−0.153709	+	0.988116i	$0.549122\pi$
$602$	−4.85667	−0.197943
$603$	−21.6485	−0.881596
$604$	18.5604	0.755212
$605$	−18.7857	−0.763746
$606$	−12.0203	−0.488293
$607$	31.0866	1.26177	0.630883	−	0.775878i	$-0.282693\pi$
$607$	31.0866	1.26177	0.630883	+	0.775878i	$0.282693\pi$
$608$	44.7032	1.81295
$609$	2.59159	0.105017
$610$	2.72157	0.110193
$611$	−15.0225	−0.607744
$612$	62.6491	2.53244
$613$	−2.93536	−0.118558	−0.0592790	−	0.998241i	$-0.518880\pi$
$613$	−2.93536	−0.118558	−0.0592790	+	0.998241i	$0.518880\pi$
$614$	35.8328	1.44609
$615$	0.522405	0.0210654
$616$	17.7665	0.715833
$617$	42.2244	1.69989	0.849944	−	0.526873i	$-0.176636\pi$
$617$	42.2244	1.69989	0.849944	+	0.526873i	$0.176636\pi$
$618$	−6.83365	−0.274890
$619$	−23.9597	−0.963020	−0.481510	−	0.876440i	$-0.659912\pi$
$619$	−23.9597	−0.963020	−0.481510	+	0.876440i	$0.659912\pi$
$620$	−19.4401	−0.780735
$621$	0.165366	0.00663591
$622$	25.4870	1.02194
$623$	0.168935	0.00676825
$624$	−14.4275	−0.577562
$625$	−24.5655	−0.982621
$626$	−27.2717	−1.09000
$627$	2.50068	0.0998675
$628$	−67.6054	−2.69775
$629$	−10.7588	−0.428982
$630$	24.3114	0.968590
$631$	21.9157	0.872449	0.436224	−	0.899838i	$-0.356315\pi$
$631$	21.9157	0.872449	0.436224	+	0.899838i	$0.356315\pi$
$632$	−34.0041	−1.35261
$633$	−6.59197	−0.262007
$634$	−11.8145	−0.469215
$635$	20.4479	0.811449
$636$	1.15497	0.0457976
$637$	−4.14521	−0.164239
$638$	48.1586	1.90662
$639$	17.7027	0.700306
$640$	−22.3052	−0.881691
$641$	2.78360	0.109945	0.0549727	−	0.998488i	$-0.482493\pi$
$641$	2.78360	0.109945	0.0549727	+	0.998488i	$0.482493\pi$
$642$	2.58736	0.102115
$643$	−1.77914	−0.0701625	−0.0350812	−	0.999384i	$-0.511169\pi$
$643$	−1.77914	−0.0701625	−0.0350812	+	0.999384i	$0.511169\pi$
$644$	−0.435694	−0.0171688
$645$	1.87159	0.0736937
$646$	−39.6688	−1.56075
$647$	10.0738	0.396041	0.198021	−	0.980198i	$-0.436549\pi$
$647$	10.0738	0.396041	0.198021	+	0.980198i	$0.436549\pi$
$648$	−63.7042	−2.50254
$649$	14.7411	0.578640
$650$	55.7006	2.18476
$651$	−0.393977	−0.0154412
$652$	36.1240	1.41472
$653$	10.7235	0.419643	0.209821	−	0.977740i	$-0.432712\pi$
$653$	10.7235	0.419643	0.209821	+	0.977740i	$0.432712\pi$
$654$	9.01723	0.352601
$655$	23.6814	0.925307
$656$	5.56910	0.217437
$657$	4.02667	0.157095
$658$	−9.57591	−0.373308
$659$	16.0478	0.625135	0.312568	−	0.949896i	$-0.398811\pi$
$659$	16.0478	0.625135	0.312568	+	0.949896i	$0.398811\pi$
$660$	−11.4388	−0.445253
$661$	−46.2532	−1.79904	−0.899520	−	0.436879i	$-0.856084\pi$
$661$	−46.2532	−1.79904	−0.899520	+	0.436879i	$0.856084\pi$
$662$	−34.8143	−1.35310
$663$	5.76902	0.224050
$664$	−126.122	−4.89447
$665$	−10.9841	−0.425945
$666$	18.9749	0.735264
$667$	−0.706884	−0.0273707
$668$	65.7688	2.54467
$669$	7.19625	0.278223
$670$	−62.7022	−2.42240
$671$	0.731337	0.0282330
$672$	−4.14410	−0.159862
$673$	12.6888	0.489118	0.244559	−	0.969634i	$-0.421357\pi$
$673$	12.6888	0.489118	0.244559	+	0.969634i	$0.421357\pi$
$674$	−19.3529	−0.745445
$675$	−9.61575	−0.370110
$676$	20.8379	0.801457
$677$	35.3157	1.35729	0.678646	−	0.734466i	$-0.262567\pi$
$677$	35.3157	1.35729	0.678646	+	0.734466i	$0.262567\pi$
$678$	4.23477	0.162635
$679$	4.41054	0.169261
$680$	108.609	4.16496
$681$	3.34911	0.128338
$682$	−7.32113	−0.280341
$683$	−35.6799	−1.36525	−0.682627	−	0.730767i	$-0.739163\pi$
$683$	−35.6799	−1.36525	−0.682627	+	0.730767i	$0.739163\pi$
$684$	49.9213	1.90879
$685$	58.8858	2.24991
$686$	−2.64232	−0.100884
$687$	4.88784	0.186483
$688$	19.9521	0.760666
$689$	−2.99722	−0.114185
$690$	0.235306	0.00895795
$691$	−27.7905	−1.05720	−0.528599	−	0.848871i	$-0.677283\pi$
$691$	−27.7905	−1.05720	−0.528599	+	0.848871i	$0.677283\pi$
$692$	61.6619	2.34403
$693$	6.53294	0.248166
$694$	72.7669	2.76219
$695$	−39.4595	−1.49679
$696$	−20.4191	−0.773986
$697$	−2.22688	−0.0843491
$698$	−69.9311	−2.64693
$699$	−8.16271	−0.308742
$700$	25.3349	0.957569
$701$	−23.7986	−0.898860	−0.449430	−	0.893316i	$-0.648373\pi$
$701$	−23.7986	−0.898860	−0.449430	+	0.893316i	$0.648373\pi$
$702$	−20.7103	−0.781660
$703$	−8.57305	−0.323339
$704$	−28.0534	−1.05730
$705$	3.69022	0.138982
$706$	−13.2158	−0.497384
$707$	14.1880	0.533596
$708$	−10.4424	−0.392449
$709$	2.40624	0.0903683	0.0451842	−	0.998979i	$-0.485613\pi$
$709$	2.40624	0.0903683	0.0451842	+	0.998979i	$0.485613\pi$
$710$	51.2736	1.92426
$711$	−12.5037	−0.468925
$712$	−1.33104	−0.0498829
$713$	0.107461	0.00402446
$714$	3.67740	0.137623
$715$	29.6842	1.11013
$716$	99.4158	3.71534
$717$	−7.73024	−0.288691
$718$	−67.2548	−2.50993
$719$	−2.49024	−0.0928703	−0.0464352	−	0.998921i	$-0.514786\pi$
$719$	−2.49024	−0.0928703	−0.0464352	+	0.998921i	$0.514786\pi$
$720$	−99.8757	−3.72215
$721$	8.06600	0.300393
$722$	18.5944	0.692011
$723$	−6.42587	−0.238981
$724$	−4.87756	−0.181273
$725$	41.1041	1.52657
$726$	5.01157	0.185997
$727$	20.1483	0.747261	0.373630	−	0.927578i	$-0.378113\pi$
$727$	20.1483	0.747261	0.373630	+	0.927578i	$0.378113\pi$
$728$	32.6601	1.21046
$729$	−21.6059	−0.800219
$730$	11.6628	0.431658
$731$	−7.97810	−0.295081
$732$	−0.518068	−0.0191483
$733$	32.3164	1.19363	0.596817	−	0.802378i	$-0.296432\pi$
$733$	32.3164	1.19363	0.596817	+	0.802378i	$0.296432\pi$
$734$	9.50676	0.350901
$735$	1.01826	0.0375589
$736$	1.13035	0.0416652
$737$	−16.8493	−0.620652
$738$	3.92747	0.144572
$739$	−5.43566	−0.199954	−0.0999771	−	0.994990i	$-0.531877\pi$
$739$	−5.43566	−0.199954	−0.0999771	+	0.994990i	$0.531877\pi$
$740$	39.2153	1.44158
$741$	4.59699	0.168875
$742$	−1.91054	−0.0701382
$743$	−4.12811	−0.151446	−0.0757229	−	0.997129i	$-0.524126\pi$
$743$	−4.12811	−0.151446	−0.0757229	+	0.997129i	$0.524126\pi$
$744$	3.10415	0.113803
$745$	50.8489	1.86296
$746$	−9.49244	−0.347543
$747$	−46.3763	−1.69682
$748$	48.7605	1.78286
$749$	−3.05395	−0.111589
$750$	−0.229871	−0.00839369
$751$	29.8530	1.08935	0.544676	−	0.838647i	$-0.316653\pi$
$751$	29.8530	1.08935	0.544676	+	0.838647i	$0.316653\pi$
$752$	39.3396	1.43457
$753$	6.92743	0.252450
$754$	88.5297	3.22406
$755$	11.8316	0.430597
$756$	−9.41988	−0.342598
$757$	−4.74385	−0.172418	−0.0862090	−	0.996277i	$-0.527475\pi$
$757$	−4.74385	−0.172418	−0.0862090	+	0.996277i	$0.527475\pi$
$758$	−4.46479	−0.162169
$759$	0.0632313	0.00229515
$760$	86.5438	3.13927
$761$	−3.31318	−0.120103	−0.0600513	−	0.998195i	$-0.519126\pi$
$761$	−3.31318	−0.120103	−0.0600513	+	0.998195i	$0.519126\pi$
$762$	−5.45501	−0.197614
$763$	−10.6433	−0.385315
$764$	34.9875	1.26580
$765$	39.9366	1.44391
$766$	−28.3988	−1.02609
$767$	27.0985	0.978472
$768$	−2.02750	−0.0731609
$769$	−5.16827	−0.186373	−0.0931863	−	0.995649i	$-0.529705\pi$
$769$	−5.16827	−0.186373	−0.0931863	+	0.995649i	$0.529705\pi$
$770$	18.9219	0.681897
$771$	7.37517	0.265610
$772$	80.0768	2.88203
$773$	43.4416	1.56249	0.781243	−	0.624228i	$-0.214586\pi$
$773$	43.4416	1.56249	0.781243	+	0.624228i	$0.214586\pi$
$774$	14.0707	0.505761
$775$	−6.24870	−0.224460
$776$	−34.7506	−1.24747
$777$	0.794744	0.0285113
$778$	−43.2841	−1.55181
$779$	−1.77447	−0.0635768
$780$	−21.0278	−0.752916
$781$	13.7782	0.493022
$782$	−1.00305	−0.0358690
$783$	−15.2831	−0.546174
$784$	10.8551	0.387683
$785$	−43.0961	−1.53817
$786$	−6.31763	−0.225342
$787$	2.28367	0.0814040	0.0407020	−	0.999171i	$-0.487041\pi$
$787$	2.28367	0.0814040	0.0407020	+	0.999171i	$0.487041\pi$
$788$	−40.0107	−1.42532
$789$	−0.705892	−0.0251304
$790$	−36.2154	−1.28849
$791$	−4.99844	−0.177724
$792$	−51.4731	−1.82902
$793$	1.34441	0.0477415
$794$	43.7343	1.55207
$795$	0.736255	0.0261123
$796$	107.948	3.82613
$797$	−36.6606	−1.29859	−0.649293	−	0.760539i	$-0.724935\pi$
$797$	−36.6606	−1.29859	−0.649293	+	0.760539i	$0.724935\pi$
$798$	2.93030	0.103732
$799$	−15.7305	−0.556504
$800$	−65.7278	−2.32383
$801$	−0.489438	−0.0172935
$802$	−56.7705	−2.00464
$803$	3.13400	0.110597
$804$	11.9358	0.420942
$805$	−0.277740	−0.00978905
$806$	−13.4584	−0.474052
$807$	5.95182	0.209514
$808$	−111.787	−3.93267
$809$	−46.7473	−1.64355	−0.821774	−	0.569814i	$-0.807015\pi$
$809$	−46.7473	−1.64355	−0.821774	+	0.569814i	$0.807015\pi$
$810$	−67.8468	−2.38389
$811$	−43.6072	−1.53126	−0.765628	−	0.643284i	$-0.777571\pi$
$811$	−43.6072	−1.53126	−0.765628	+	0.643284i	$0.777571\pi$
$812$	40.2669	1.41309
$813$	−4.74482	−0.166408
$814$	14.7684	0.517633
$815$	23.0278	0.806629
$816$	−15.1074	−0.528866
$817$	−6.35727	−0.222413
$818$	42.0726	1.47103
$819$	12.0095	0.419645
$820$	8.11686	0.283453
$821$	22.3921	0.781489	0.390745	−	0.920499i	$-0.372218\pi$
$821$	22.3921	0.781489	0.390745	+	0.920499i	$0.372218\pi$
$822$	−15.7093	−0.547926
$823$	11.4757	0.400019	0.200010	−	0.979794i	$-0.435903\pi$
$823$	11.4757	0.400019	0.200010	+	0.979794i	$0.435903\pi$
$824$	−63.5520	−2.21394
$825$	−3.67679	−0.128009
$826$	17.2737	0.601028
$827$	21.5012	0.747670	0.373835	−	0.927495i	$-0.378043\pi$
$827$	21.5012	0.747670	0.373835	+	0.927495i	$0.378043\pi$
$828$	1.26229	0.0438676
$829$	−10.1045	−0.350945	−0.175472	−	0.984484i	$-0.556145\pi$
$829$	−10.1045	−0.350945	−0.175472	+	0.984484i	$0.556145\pi$
$830$	−134.323	−4.66242
$831$	−4.06225	−0.140918
$832$	−51.5704	−1.78788
$833$	−4.34057	−0.150392
$834$	10.5269	0.364516
$835$	41.9253	1.45089
$836$	38.8543	1.34380
$837$	2.32336	0.0803070
$838$	5.91999	0.204503
$839$	−18.7556	−0.647514	−0.323757	−	0.946140i	$-0.604946\pi$
$839$	−18.7556	−0.647514	−0.323757	+	0.946140i	$0.604946\pi$
$840$	−8.02283	−0.276814
$841$	36.3302	1.25277
$842$	−13.7244	−0.472975
$843$	−1.55992	−0.0537265
$844$	−102.423	−3.52554
$845$	13.2834	0.456964
$846$	27.7433	0.953834
$847$	−5.91533	−0.203253
$848$	7.84886	0.269531
$849$	−1.65930	−0.0569470
$850$	58.3257	2.00056
$851$	−0.216775	−0.00743095
$852$	−9.76025	−0.334381
$853$	36.4910	1.24943	0.624713	−	0.780854i	$-0.285216\pi$
$853$	36.4910	1.24943	0.624713	+	0.780854i	$0.285216\pi$
$854$	0.856982	0.0293253
$855$	31.8231	1.08833
$856$	24.0621	0.822424
$857$	−25.8429	−0.882777	−0.441388	−	0.897316i	$-0.645514\pi$
$857$	−25.8429	−0.882777	−0.441388	+	0.897316i	$0.645514\pi$
$858$	−7.91904	−0.270351
$859$	−1.00000	−0.0341196
$859$	−1.00000	−0.0341196
$860$	29.0798	0.991612
$861$	0.164498	0.00560606
$862$	−28.4196	−0.967974
$863$	6.51288	0.221701	0.110851	−	0.993837i	$-0.464643\pi$
$863$	6.51288	0.221701	0.110851	+	0.993837i	$0.464643\pi$
$864$	24.4386	0.831417
$865$	39.3073	1.33649
$866$	−6.30997	−0.214421
$867$	0.590135	0.0200420
$868$	−6.12142	−0.207775
$869$	−9.73177	−0.330128
$870$	−21.7470	−0.737292
$871$	−30.9740	−1.04951
$872$	83.8589	2.83982
$873$	−12.7782	−0.432476
$874$	−0.799271	−0.0270357
$875$	0.271324	0.00917244
$876$	−2.22008	−0.0750095
$877$	−23.5377	−0.794813	−0.397407	−	0.917643i	$-0.630090\pi$
$877$	−23.5377	−0.794813	−0.397407	+	0.917643i	$0.630090\pi$
$878$	−35.7449	−1.20633
$879$	−2.55872	−0.0863036
$880$	−77.7345	−2.62043
$881$	−56.5228	−1.90430	−0.952152	−	0.305626i	$-0.901134\pi$
$881$	−56.5228	−1.90430	−0.952152	+	0.305626i	$0.901134\pi$
$882$	7.65531	0.257768
$883$	7.00314	0.235674	0.117837	−	0.993033i	$-0.462404\pi$
$883$	7.00314	0.235674	0.117837	+	0.993033i	$0.462404\pi$
$884$	89.6362	3.01479
$885$	−6.65666	−0.223761
$886$	−40.4848	−1.36011
$887$	14.8868	0.499849	0.249925	−	0.968265i	$-0.419594\pi$
$887$	14.8868	0.499849	0.249925	+	0.968265i	$0.419594\pi$
$888$	−6.26179	−0.210132
$889$	6.43874	0.215948
$890$	−1.41760	−0.0475180
$891$	−18.2317	−0.610786
$892$	111.812	3.74373
$893$	−12.5347	−0.419456
$894$	−13.5653	−0.453691
$895$	63.3742	2.11836
$896$	−7.02359	−0.234642
$897$	0.116238	0.00388106
$898$	32.1354	1.07237
$899$	−9.93159	−0.331237
$900$	−73.4001	−2.44667
$901$	−3.13847	−0.104558
$902$	3.05680	0.101780
$903$	0.589335	0.0196119
$904$	39.3827	1.30985
$905$	−3.10927	−0.103356
$906$	−3.15640	−0.104864
$907$	34.0339	1.13008	0.565039	−	0.825065i	$-0.308861\pi$
$907$	34.0339	1.13008	0.565039	+	0.825065i	$0.308861\pi$
$908$	52.0368	1.72690
$909$	−41.1055	−1.36338
$910$	34.7840	1.15308
$911$	40.8576	1.35367	0.676836	−	0.736134i	$-0.263350\pi$
$911$	40.8576	1.35367	0.676836	+	0.736134i	$0.263350\pi$
$912$	−12.0382	−0.398625
$913$	−36.0952	−1.19458
$914$	30.2046	0.999079
$915$	−0.330250	−0.0109177
$916$	75.9449	2.50929
$917$	7.45692	0.246249
$918$	−21.6864	−0.715756
$919$	−31.7728	−1.04809	−0.524045	−	0.851691i	$-0.675578\pi$
$919$	−31.7728	−1.04809	−0.524045	+	0.851691i	$0.675578\pi$
$920$	2.18831	0.0721466
$921$	−4.34815	−0.143276
$922$	−0.549322	−0.0180909
$923$	25.3284	0.833694
$924$	−3.60190	−0.118494
$925$	12.6051	0.414453
$926$	−35.1914	−1.15646
$927$	−23.3688	−0.767531
$928$	−104.467	−3.42929
$929$	44.8537	1.47160	0.735802	−	0.677197i	$-0.236806\pi$
$929$	44.8537	1.47160	0.735802	+	0.677197i	$0.236806\pi$
$930$	3.30601	0.108408
$931$	−3.45874	−0.113356
$932$	−126.828	−4.15439
$933$	−3.09274	−0.101252
$934$	−104.936	−3.43360
$935$	31.0832	1.01653
$936$	−94.6226	−3.09284
$937$	−36.9287	−1.20641	−0.603204	−	0.797587i	$-0.706110\pi$
$937$	−36.9287	−1.20641	−0.603204	+	0.797587i	$0.706110\pi$
$938$	−19.7440	−0.644665
$939$	3.30930	0.107995
$940$	57.3367	1.87012
$941$	−38.0152	−1.23926	−0.619630	−	0.784894i	$-0.712717\pi$
$941$	−38.0152	−1.23926	−0.619630	+	0.784894i	$0.712717\pi$
$942$	11.4970	0.374593
$943$	−0.0448685	−0.00146112
$944$	−70.9634	−2.30966
$945$	−6.00485	−0.195338
$946$	10.9514	0.356061
$947$	5.88590	0.191266	0.0956331	−	0.995417i	$-0.469512\pi$
$947$	5.88590	0.191266	0.0956331	+	0.995417i	$0.469512\pi$
$948$	6.89383	0.223901
$949$	5.76122	0.187017
$950$	46.4762	1.50789
$951$	1.43364	0.0464890
$952$	34.1993	1.10841
$953$	−1.04219	−0.0337600	−0.0168800	−	0.999858i	$-0.505373\pi$
$953$	−1.04219	−0.0337600	−0.0168800	+	0.999858i	$0.505373\pi$
$954$	5.53521	0.179209
$955$	22.3033	0.721718
$956$	−120.109	−3.88459
$957$	−5.84383	−0.188904
$958$	72.8251	2.35287
$959$	18.5423	0.598762
$960$	12.6681	0.408861
$961$	−29.4902	−0.951296
$962$	27.1487	0.875310
$963$	8.84788	0.285119
$964$	−99.8420	−3.21569
$965$	51.0462	1.64324
$966$	0.0740945	0.00238395
$967$	28.1286	0.904553	0.452277	−	0.891878i	$-0.350612\pi$
$967$	28.1286	0.904553	0.452277	+	0.891878i	$0.350612\pi$
$968$	46.6069	1.49800
$969$	4.81364	0.154636
$970$	−37.0104	−1.18833
$971$	29.8038	0.956451	0.478225	−	0.878237i	$-0.341280\pi$
$971$	29.8038	0.956451	0.478225	+	0.878237i	$0.341280\pi$
$972$	41.1747	1.32068
$973$	−12.4252	−0.398335
$974$	−65.9259	−2.11240
$975$	−6.75902	−0.216462
$976$	−3.52064	−0.112693
$977$	5.27992	0.168920	0.0844599	−	0.996427i	$-0.473084\pi$
$977$	5.27992	0.168920	0.0844599	+	0.996427i	$0.473084\pi$
$978$	−6.14327	−0.196440
$979$	−0.380936	−0.0121748
$980$	15.8211	0.505388
$981$	30.8358	0.984513
$982$	9.60600	0.306540
$983$	−3.92318	−0.125130	−0.0625651	−	0.998041i	$-0.519928\pi$
$983$	−3.92318	−0.125130	−0.0625651	+	0.998041i	$0.519928\pi$
$984$	−1.29608	−0.0413174
$985$	−25.5055	−0.812672
$986$	92.7019	2.95223
$987$	1.16200	0.0369867
$988$	71.4257	2.27235
$989$	−0.160748	−0.00511147
$990$	−54.8203	−1.74230
$991$	−41.8517	−1.32946	−0.664732	−	0.747082i	$-0.731454\pi$
$991$	−41.8517	−1.32946	−0.664732	+	0.747082i	$0.731454\pi$
$992$	15.8812	0.504228
$993$	4.22457	0.134063
$994$	16.1453	0.512098
$995$	68.8133	2.18153
$996$	25.5693	0.810193
$997$	−18.4242	−0.583502	−0.291751	−	0.956494i	$-0.594238\pi$
$997$	−18.4242	−0.583502	−0.291751	+	0.956494i	$0.594238\pi$
$998$	104.555	3.30962
$999$	−4.68676	−0.148282

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6013.2.a.e.1.4	✓	109

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6013.2.a.e.1.4	✓	109	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:	\( N \)	\(=\)	\( 6013 = 7 \cdot 859 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6013.a (trivial)

\(f(q)\)	\(=\)	\(q-2.64232 q^{2} +0.320634 q^{3} +4.98185 q^{4} +3.17576 q^{5} -0.847217 q^{6} +1.00000 q^{7} -7.87900 q^{8} -2.89719 q^{9} +O(q^{10})\)	\(q-2.64232 q^{2} +0.320634 q^{3} +4.98185 q^{4} +3.17576 q^{5} -0.847217 q^{6} +1.00000 q^{7} -7.87900 q^{8} -2.89719 q^{9} -8.39136 q^{10} -2.25492 q^{11} +1.59735 q^{12} -4.14521 q^{13} -2.64232 q^{14} +1.01826 q^{15} +10.8551 q^{16} -4.34057 q^{17} +7.65531 q^{18} -3.45874 q^{19} +15.8211 q^{20} +0.320634 q^{21} +5.95822 q^{22} -0.0874563 q^{23} -2.52627 q^{24} +5.08544 q^{25} +10.9530 q^{26} -1.89084 q^{27} +4.98185 q^{28} +8.08271 q^{29} -2.69056 q^{30} -1.22874 q^{31} -12.9247 q^{32} -0.723004 q^{33} +11.4692 q^{34} +3.17576 q^{35} -14.4334 q^{36} +2.47866 q^{37} +9.13908 q^{38} -1.32909 q^{39} -25.0218 q^{40} +0.513039 q^{41} -0.847217 q^{42} +1.83803 q^{43} -11.2337 q^{44} -9.20079 q^{45} +0.231087 q^{46} +3.62406 q^{47} +3.48052 q^{48} +1.00000 q^{49} -13.4373 q^{50} -1.39173 q^{51} -20.6508 q^{52} +0.723055 q^{53} +4.99620 q^{54} -7.16108 q^{55} -7.87900 q^{56} -1.10899 q^{57} -21.3571 q^{58} -6.53732 q^{59} +5.07280 q^{60} -0.324330 q^{61} +3.24673 q^{62} -2.89719 q^{63} +12.4410 q^{64} -13.1642 q^{65} +1.91041 q^{66} +7.47223 q^{67} -21.6241 q^{68} -0.0280415 q^{69} -8.39136 q^{70} -6.11028 q^{71} +22.8270 q^{72} -1.38985 q^{73} -6.54942 q^{74} +1.63056 q^{75} -17.2309 q^{76} -2.25492 q^{77} +3.51189 q^{78} +4.31579 q^{79} +34.4733 q^{80} +8.08531 q^{81} -1.35561 q^{82} +16.0073 q^{83} +1.59735 q^{84} -13.7846 q^{85} -4.85667 q^{86} +2.59159 q^{87} +17.7665 q^{88} +0.168935 q^{89} +24.3114 q^{90} -4.14521 q^{91} -0.435694 q^{92} -0.393977 q^{93} -9.57591 q^{94} -10.9841 q^{95} -4.14410 q^{96} +4.41054 q^{97} -2.64232 q^{98} +6.53294 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 109 q + 19 q^{2} + 38 q^{3} + 111 q^{4} + 43 q^{5} + 14 q^{6} + 109 q^{7} + 48 q^{8} + 119 q^{9}+O(q^{10}) \) 109 * q + 19 * q^2 + 38 * q^3 + 111 * q^4 + 43 * q^5 + 14 * q^6 + 109 * q^7 + 48 * q^8 + 119 * q^9	\( 109 q + 19 q^{2} + 38 q^{3} + 111 q^{4} + 43 q^{5} + 14 q^{6} + 109 q^{7} + 48 q^{8} + 119 q^{9} + 15 q^{10} + 48 q^{11} + 72 q^{12} + 29 q^{13} + 19 q^{14} + 29 q^{15} + 115 q^{16} + 72 q^{17} + 33 q^{18} + 58 q^{19} + 88 q^{20} + 38 q^{21} + 4 q^{22} + 65 q^{23} + 46 q^{24} + 124 q^{25} + 49 q^{26} + 131 q^{27} + 111 q^{28} + 25 q^{29} + 2 q^{30} + 41 q^{31} + 75 q^{32} + 54 q^{33} + 23 q^{34} + 43 q^{35} + 111 q^{36} + 25 q^{37} + 54 q^{38} + 27 q^{39} + 30 q^{40} + 109 q^{41} + 14 q^{42} + 38 q^{43} + 68 q^{44} + 84 q^{45} - 9 q^{46} + 121 q^{47} + 106 q^{48} + 109 q^{49} + 14 q^{50} + 36 q^{51} + 38 q^{52} + 61 q^{53} + 31 q^{54} + 50 q^{55} + 48 q^{56} + 5 q^{57} - 20 q^{58} + 181 q^{59} + 25 q^{60} + 34 q^{61} + 75 q^{62} + 119 q^{63} + 96 q^{64} + 12 q^{65} + 19 q^{66} + 87 q^{67} + 150 q^{68} + 89 q^{69} + 15 q^{70} + 83 q^{71} + 65 q^{72} + 32 q^{73} - 19 q^{74} + 112 q^{75} + 84 q^{76} + 48 q^{77} - 34 q^{78} - 9 q^{79} + 137 q^{80} + 109 q^{81} - 19 q^{82} + 136 q^{83} + 72 q^{84} - 32 q^{85} - 24 q^{86} + 28 q^{87} - 24 q^{88} + 142 q^{89} + 19 q^{90} + 29 q^{91} + 96 q^{92} + 29 q^{93} + 9 q^{94} + 52 q^{95} + 88 q^{96} + 75 q^{97} + 19 q^{98} + 84 q^{99}+O(q^{100}) \) 109 * q + 19 * q^2 + 38 * q^3 + 111 * q^4 + 43 * q^5 + 14 * q^6 + 109 * q^7 + 48 * q^8 + 119 * q^9 + 15 * q^10 + 48 * q^11 + 72 * q^12 + 29 * q^13 + 19 * q^14 + 29 * q^15 + 115 * q^16 + 72 * q^17 + 33 * q^18 + 58 * q^19 + 88 * q^20 + 38 * q^21 + 4 * q^22 + 65 * q^23 + 46 * q^24 + 124 * q^25 + 49 * q^26 + 131 * q^27 + 111 * q^28 + 25 * q^29 + 2 * q^30 + 41 * q^31 + 75 * q^32 + 54 * q^33 + 23 * q^34 + 43 * q^35 + 111 * q^36 + 25 * q^37 + 54 * q^38 + 27 * q^39 + 30 * q^40 + 109 * q^41 + 14 * q^42 + 38 * q^43 + 68 * q^44 + 84 * q^45 - 9 * q^46 + 121 * q^47 + 106 * q^48 + 109 * q^49 + 14 * q^50 + 36 * q^51 + 38 * q^52 + 61 * q^53 + 31 * q^54 + 50 * q^55 + 48 * q^56 + 5 * q^57 - 20 * q^58 + 181 * q^59 + 25 * q^60 + 34 * q^61 + 75 * q^62 + 119 * q^63 + 96 * q^64 + 12 * q^65 + 19 * q^66 + 87 * q^67 + 150 * q^68 + 89 * q^69 + 15 * q^70 + 83 * q^71 + 65 * q^72 + 32 * q^73 - 19 * q^74 + 112 * q^75 + 84 * q^76 + 48 * q^77 - 34 * q^78 - 9 * q^79 + 137 * q^80 + 109 * q^81 - 19 * q^82 + 136 * q^83 + 72 * q^84 - 32 * q^85 - 24 * q^86 + 28 * q^87 - 24 * q^88 + 142 * q^89 + 19 * q^90 + 29 * q^91 + 96 * q^92 + 29 * q^93 + 9 * q^94 + 52 * q^95 + 88 * q^96 + 75 * q^97 + 19 * q^98 + 84 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more