Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [588,8,Mod(1,588)] 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("588.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(588, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 588.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,108,0,-196] 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: \(183.682394985\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 655x^{2} - 5704x - 84 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{3}\cdot 7 \)
Twist minimal: no (minimal twist has level 84)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(30.0811\) of defining polynomial
Character \(\chi\) \(=\) 588.1

$q$-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)
 
\(f(q)\) \(=\) \(q+27.0000 q^{3} -445.586 q^{5} +729.000 q^{9} -3238.07 q^{11} +1653.08 q^{13} -12030.8 q^{15} -16919.0 q^{17} +27921.9 q^{19} +69497.4 q^{23} +120422. q^{25} +19683.0 q^{27} +69738.9 q^{29} +120774. q^{31} -87428.0 q^{33} -453767. q^{37} +44633.1 q^{39} +690187. q^{41} -752955. q^{43} -324832. q^{45} +112768. q^{47} -456813. q^{51} +678317. q^{53} +1.44284e6 q^{55} +753890. q^{57} +2.11890e6 q^{59} -1.45737e6 q^{61} -736588. q^{65} +986278. q^{67} +1.87643e6 q^{69} +3.31556e6 q^{71} +811766. q^{73} +3.25138e6 q^{75} -8.58451e6 q^{79} +531441. q^{81} -2.96018e6 q^{83} +7.53886e6 q^{85} +1.88295e6 q^{87} -8.84130e6 q^{89} +3.26091e6 q^{93} -1.24416e7 q^{95} -1.02812e7 q^{97} -2.36056e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 108 q^{3} - 196 q^{5} + 2916 q^{9} - 406 q^{11} - 1974 q^{13} - 5292 q^{15} - 7436 q^{17} + 15874 q^{19} + 6788 q^{23} - 69898 q^{25} + 78732 q^{27} - 94544 q^{29} - 55890 q^{31} - 10962 q^{33} - 93742 q^{37}+ \cdots - 295974 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(5\) −445.586 −1.59418 −0.797088 − 0.603863i \(-0.793627\pi\)
−0.797088 + 0.603863i \(0.793627\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 729.000 0.333333
\(10\) 0 0
\(11\) −3238.07 −0.733521 −0.366760 − 0.930315i \(-0.619533\pi\)
−0.366760 + 0.930315i \(0.619533\pi\)
\(12\) 0 0
\(13\) 1653.08 0.208685 0.104343 − 0.994541i \(-0.466726\pi\)
0.104343 + 0.994541i \(0.466726\pi\)
\(14\) 0 0
\(15\) −12030.8 −0.920398
\(16\) 0 0
\(17\) −16919.0 −0.835225 −0.417612 − 0.908625i \(-0.637133\pi\)
−0.417612 + 0.908625i \(0.637133\pi\)
\(18\) 0 0
\(19\) 27921.9 0.933914 0.466957 − 0.884280i \(-0.345350\pi\)
0.466957 + 0.884280i \(0.345350\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 69497.4 1.19103 0.595513 − 0.803346i \(-0.296949\pi\)
0.595513 + 0.803346i \(0.296949\pi\)
\(24\) 0 0
\(25\) 120422. 1.54140
\(26\) 0 0
\(27\) 19683.0 0.192450
\(28\) 0 0
\(29\) 69738.9 0.530985 0.265492 − 0.964113i \(-0.414466\pi\)
0.265492 + 0.964113i \(0.414466\pi\)
\(30\) 0 0
\(31\) 120774. 0.728130 0.364065 − 0.931374i \(-0.381389\pi\)
0.364065 + 0.931374i \(0.381389\pi\)
\(32\) 0 0
\(33\) −87428.0 −0.423498
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −453767. −1.47274 −0.736371 − 0.676578i \(-0.763462\pi\)
−0.736371 + 0.676578i \(0.763462\pi\)
\(38\) 0 0
\(39\) 44633.1 0.120485
\(40\) 0 0
\(41\) 690187. 1.56395 0.781976 − 0.623308i \(-0.214212\pi\)
0.781976 + 0.623308i \(0.214212\pi\)
\(42\) 0 0
\(43\) −752955. −1.44421 −0.722104 − 0.691785i \(-0.756825\pi\)
−0.722104 + 0.691785i \(0.756825\pi\)
\(44\) 0 0
\(45\) −324832. −0.531392
\(46\) 0 0
\(47\) 112768. 0.158432 0.0792159 − 0.996857i \(-0.474758\pi\)
0.0792159 + 0.996857i \(0.474758\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −456813. −0.482217
\(52\) 0 0
\(53\) 678317. 0.625846 0.312923 − 0.949779i \(-0.398692\pi\)
0.312923 + 0.949779i \(0.398692\pi\)
\(54\) 0 0
\(55\) 1.44284e6 1.16936
\(56\) 0 0
\(57\) 753890. 0.539196
\(58\) 0 0
\(59\) 2.11890e6 1.34316 0.671582 − 0.740930i \(-0.265615\pi\)
0.671582 + 0.740930i \(0.265615\pi\)
\(60\) 0 0
\(61\) −1.45737e6 −0.822085 −0.411042 − 0.911616i \(-0.634835\pi\)
−0.411042 + 0.911616i \(0.634835\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −736588. −0.332681
\(66\) 0 0
\(67\) 986278. 0.400624 0.200312 − 0.979732i \(-0.435804\pi\)
0.200312 + 0.979732i \(0.435804\pi\)
\(68\) 0 0
\(69\) 1.87643e6 0.687639
\(70\) 0 0
\(71\) 3.31556e6 1.09939 0.549697 − 0.835364i \(-0.314743\pi\)
0.549697 + 0.835364i \(0.314743\pi\)
\(72\) 0 0
\(73\) 811766. 0.244231 0.122116 − 0.992516i \(-0.461032\pi\)
0.122116 + 0.992516i \(0.461032\pi\)
\(74\) 0 0
\(75\) 3.25138e6 0.889925
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −8.58451e6 −1.95894 −0.979469 − 0.201595i \(-0.935387\pi\)
−0.979469 + 0.201595i \(0.935387\pi\)
\(80\) 0 0
\(81\) 531441. 0.111111
\(82\) 0 0
\(83\) −2.96018e6 −0.568258 −0.284129 − 0.958786i \(-0.591704\pi\)
−0.284129 + 0.958786i \(0.591704\pi\)
\(84\) 0 0
\(85\) 7.53886e6 1.33149
\(86\) 0 0
\(87\) 1.88295e6 0.306564
\(88\) 0 0
\(89\) −8.84130e6 −1.32939 −0.664693 − 0.747117i \(-0.731438\pi\)
−0.664693 + 0.747117i \(0.731438\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 3.26091e6 0.420386
\(94\) 0 0
\(95\) −1.24416e7 −1.48882
\(96\) 0 0
\(97\) −1.02812e7 −1.14378 −0.571890 − 0.820330i \(-0.693790\pi\)
−0.571890 + 0.820330i \(0.693790\pi\)
\(98\) 0 0
\(99\) −2.36056e6 −0.244507
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 588.8.a.j.1.1 4
7.2 even 3 588.8.i.o.361.4 8
7.3 odd 6 84.8.i.a.37.1 yes 8
7.4 even 3 588.8.i.o.373.4 8
7.5 odd 6 84.8.i.a.25.1 ✓ 8
7.6 odd 2 588.8.a.i.1.4 4
21.5 even 6 252.8.k.b.109.4 8
21.17 even 6 252.8.k.b.37.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.8.i.a.25.1 ✓ 8 7.5 odd 6
84.8.i.a.37.1 yes 8 7.3 odd 6
252.8.k.b.37.4 8 21.17 even 6
252.8.k.b.109.4 8 21.5 even 6
588.8.a.i.1.4 4 7.6 odd 2
588.8.a.j.1.1 4 1.1 even 1 trivial
588.8.i.o.361.4 8 7.2 even 3
588.8.i.o.373.4 8 7.4 even 3