Properties

Label 60.8.a.a.1.1
Level $60$
Weight $8$
Character 60.1
Self dual yes
Analytic conductor $18.743$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [60,8,Mod(1,60)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("60.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(60, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 60.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,-27,0,-125] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(18.7431015290\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 60.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-27.0000 q^{3} -125.000 q^{5} +1028.00 q^{7} +729.000 q^{9} +3096.00 q^{11} -13030.0 q^{13} +3375.00 q^{15} +1878.00 q^{17} -31180.0 q^{19} -27756.0 q^{21} -33288.0 q^{23} +15625.0 q^{25} -19683.0 q^{27} -213054. q^{29} -172696. q^{31} -83592.0 q^{33} -128500. q^{35} +27434.0 q^{37} +351810. q^{39} +532650. q^{41} -911908. q^{43} -91125.0 q^{45} -732648. q^{47} +233241. q^{49} -50706.0 q^{51} +409074. q^{53} -387000. q^{55} +841860. q^{57} +1.50814e6 q^{59} -302578. q^{61} +749412. q^{63} +1.62875e6 q^{65} +1.25433e6 q^{67} +898776. q^{69} +4.78128e6 q^{71} -502414. q^{73} -421875. q^{75} +3.18269e6 q^{77} -1.99137e6 q^{79} +531441. q^{81} -8.09927e6 q^{83} -234750. q^{85} +5.75246e6 q^{87} +7.48797e6 q^{89} -1.33948e7 q^{91} +4.66279e6 q^{93} +3.89750e6 q^{95} -1.71726e7 q^{97} +2.25698e6 q^{99} +O(q^{100})\)

Coefficient data

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



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −27.0000 −0.577350
\(4\) 0 0
\(5\) −125.000 −0.447214
\(6\) 0 0
\(7\) 1028.00 1.13279 0.566396 − 0.824133i \(-0.308337\pi\)
0.566396 + 0.824133i \(0.308337\pi\)
\(8\) 0 0
\(9\) 729.000 0.333333
\(10\) 0 0
\(11\) 3096.00 0.701337 0.350668 − 0.936500i \(-0.385954\pi\)
0.350668 + 0.936500i \(0.385954\pi\)
\(12\) 0 0
\(13\) −13030.0 −1.64491 −0.822456 − 0.568829i \(-0.807397\pi\)
−0.822456 + 0.568829i \(0.807397\pi\)
\(14\) 0 0
\(15\) 3375.00 0.258199
\(16\) 0 0
\(17\) 1878.00 0.0927095 0.0463548 − 0.998925i \(-0.485240\pi\)
0.0463548 + 0.998925i \(0.485240\pi\)
\(18\) 0 0
\(19\) −31180.0 −1.04289 −0.521445 − 0.853285i \(-0.674607\pi\)
−0.521445 + 0.853285i \(0.674607\pi\)
\(20\) 0 0
\(21\) −27756.0 −0.654017
\(22\) 0 0
\(23\) −33288.0 −0.570480 −0.285240 − 0.958456i \(-0.592073\pi\)
−0.285240 + 0.958456i \(0.592073\pi\)
\(24\) 0 0
\(25\) 15625.0 0.200000
\(26\) 0 0
\(27\) −19683.0 −0.192450
\(28\) 0 0
\(29\) −213054. −1.62217 −0.811086 − 0.584927i \(-0.801123\pi\)
−0.811086 + 0.584927i \(0.801123\pi\)
\(30\) 0 0
\(31\) −172696. −1.04116 −0.520579 − 0.853814i \(-0.674284\pi\)
−0.520579 + 0.853814i \(0.674284\pi\)
\(32\) 0 0
\(33\) −83592.0 −0.404917
\(34\) 0 0
\(35\) −128500. −0.506600
\(36\) 0 0
\(37\) 27434.0 0.0890396 0.0445198 − 0.999009i \(-0.485824\pi\)
0.0445198 + 0.999009i \(0.485824\pi\)
\(38\) 0 0
\(39\) 351810. 0.949690
\(40\) 0 0
\(41\) 532650. 1.20698 0.603488 − 0.797372i \(-0.293777\pi\)
0.603488 + 0.797372i \(0.293777\pi\)
\(42\) 0 0
\(43\) −911908. −1.74909 −0.874544 − 0.484947i \(-0.838839\pi\)
−0.874544 + 0.484947i \(0.838839\pi\)
\(44\) 0 0
\(45\) −91125.0 −0.149071
\(46\) 0 0
\(47\) −732648. −1.02933 −0.514663 − 0.857393i \(-0.672083\pi\)
−0.514663 + 0.857393i \(0.672083\pi\)
\(48\) 0 0
\(49\) 233241. 0.283217
\(50\) 0 0
\(51\) −50706.0 −0.0535259
\(52\) 0 0
\(53\) 409074. 0.377430 0.188715 − 0.982032i \(-0.439568\pi\)
0.188715 + 0.982032i \(0.439568\pi\)
\(54\) 0 0
\(55\) −387000. −0.313647
\(56\) 0 0
\(57\) 841860. 0.602113
\(58\) 0 0
\(59\) 1.50814e6 0.956001 0.478001 − 0.878359i \(-0.341362\pi\)
0.478001 + 0.878359i \(0.341362\pi\)
\(60\) 0 0
\(61\) −302578. −0.170680 −0.0853401 − 0.996352i \(-0.527198\pi\)
−0.0853401 + 0.996352i \(0.527198\pi\)
\(62\) 0 0
\(63\) 749412. 0.377597
\(64\) 0 0
\(65\) 1.62875e6 0.735627
\(66\) 0 0
\(67\) 1.25433e6 0.509508 0.254754 − 0.967006i \(-0.418006\pi\)
0.254754 + 0.967006i \(0.418006\pi\)
\(68\) 0 0
\(69\) 898776. 0.329367
\(70\) 0 0
\(71\) 4.78128e6 1.58540 0.792702 − 0.609609i \(-0.208674\pi\)
0.792702 + 0.609609i \(0.208674\pi\)
\(72\) 0 0
\(73\) −502414. −0.151158 −0.0755791 − 0.997140i \(-0.524081\pi\)
−0.0755791 + 0.997140i \(0.524081\pi\)
\(74\) 0 0
\(75\) −421875. −0.115470
\(76\) 0 0
\(77\) 3.18269e6 0.794468
\(78\) 0 0
\(79\) −1.99137e6 −0.454419 −0.227210 − 0.973846i \(-0.572960\pi\)
−0.227210 + 0.973846i \(0.572960\pi\)
\(80\) 0 0
\(81\) 531441. 0.111111
\(82\) 0 0
\(83\) −8.09927e6 −1.55479 −0.777396 − 0.629011i \(-0.783460\pi\)
−0.777396 + 0.629011i \(0.783460\pi\)
\(84\) 0 0
\(85\) −234750. −0.0414610
\(86\) 0 0
\(87\) 5.75246e6 0.936561
\(88\) 0 0
\(89\) 7.48797e6 1.12590 0.562949 − 0.826492i \(-0.309667\pi\)
0.562949 + 0.826492i \(0.309667\pi\)
\(90\) 0 0
\(91\) −1.33948e7 −1.86334
\(92\) 0 0
\(93\) 4.66279e6 0.601112
\(94\) 0 0
\(95\) 3.89750e6 0.466395
\(96\) 0 0
\(97\) −1.71726e7 −1.91044 −0.955222 − 0.295890i \(-0.904384\pi\)
−0.955222 + 0.295890i \(0.904384\pi\)
\(98\) 0 0
\(99\) 2.25698e6 0.233779
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 60.8.a.a.1.1 ✓ 1
3.2 odd 2 180.8.a.e.1.1 1
4.3 odd 2 240.8.a.i.1.1 1
5.2 odd 4 300.8.d.d.49.2 2
5.3 odd 4 300.8.d.d.49.1 2
5.4 even 2 300.8.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
60.8.a.a.1.1 ✓ 1 1.1 even 1 trivial
180.8.a.e.1.1 1 3.2 odd 2
240.8.a.i.1.1 1 4.3 odd 2
300.8.a.e.1.1 1 5.4 even 2
300.8.d.d.49.1 2 5.3 odd 4
300.8.d.d.49.2 2 5.2 odd 4