Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [5520,2,Mod(1,5520)] 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("5520.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5520, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5520.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,-3,0,3,0,-1,0,3,0,0,0,4,0,-3,0,7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(44.0774219157\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 4x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 2760)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.254102\) of defining polynomial
Character \(\chi\) \(=\) 5520.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.00000 q^{5} -3.18953 q^{7} +1.00000 q^{9} -3.36266 q^{13} -1.00000 q^{15} +5.18953 q^{17} +3.18953 q^{21} -1.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} -1.18953 q^{29} -2.17313 q^{31} -3.18953 q^{35} +9.53579 q^{37} +3.36266 q^{39} -6.55220 q^{41} +1.01641 q^{43} +1.00000 q^{45} -6.37907 q^{47} +3.17313 q^{49} -5.18953 q^{51} -2.55220 q^{53} -8.55220 q^{59} -3.01641 q^{61} -3.18953 q^{63} -3.36266 q^{65} +7.53579 q^{67} +1.00000 q^{69} -13.9149 q^{71} +8.37907 q^{73} -1.00000 q^{75} +7.74173 q^{79} +1.00000 q^{81} +9.91486 q^{83} +5.18953 q^{85} +1.18953 q^{87} +10.3463 q^{89} +10.7253 q^{91} +2.17313 q^{93} +3.01641 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} + 3 q^{5} - q^{7} + 3 q^{9} + 4 q^{13} - 3 q^{15} + 7 q^{17} + q^{21} - 3 q^{23} + 3 q^{25} - 3 q^{27} + 5 q^{29} - q^{31} - q^{35} + 9 q^{37} - 4 q^{39} + 3 q^{41} + 3 q^{45} - 2 q^{47}+ \cdots + 6 q^{97}+O(q^{100}) \) 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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −3.18953 −1.20553 −0.602765 − 0.797919i \(-0.705934\pi\)
−0.602765 + 0.797919i \(0.705934\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −3.36266 −0.932634 −0.466317 − 0.884618i \(-0.654419\pi\)
−0.466317 + 0.884618i \(0.654419\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 5.18953 1.25865 0.629323 − 0.777143i \(-0.283332\pi\)
0.629323 + 0.777143i \(0.283332\pi\)
\(18\) 0 0
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 3.18953 0.696013
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −1.18953 −0.220891 −0.110445 − 0.993882i \(-0.535228\pi\)
−0.110445 + 0.993882i \(0.535228\pi\)
\(30\) 0 0
\(31\) −2.17313 −0.390305 −0.195153 − 0.980773i \(-0.562520\pi\)
−0.195153 + 0.980773i \(0.562520\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −3.18953 −0.539130
\(36\) 0 0
\(37\) 9.53579 1.56767 0.783837 − 0.620967i \(-0.213260\pi\)
0.783837 + 0.620967i \(0.213260\pi\)
\(38\) 0 0
\(39\) 3.36266 0.538457
\(40\) 0 0
\(41\) −6.55220 −1.02328 −0.511640 − 0.859200i \(-0.670962\pi\)
−0.511640 + 0.859200i \(0.670962\pi\)
\(42\) 0 0
\(43\) 1.01641 0.155001 0.0775003 − 0.996992i \(-0.475306\pi\)
0.0775003 + 0.996992i \(0.475306\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) −6.37907 −0.930483 −0.465241 − 0.885184i \(-0.654032\pi\)
−0.465241 + 0.885184i \(0.654032\pi\)
\(48\) 0 0
\(49\) 3.17313 0.453304
\(50\) 0 0
\(51\) −5.18953 −0.726680
\(52\) 0 0
\(53\) −2.55220 −0.350571 −0.175285 − 0.984518i \(-0.556085\pi\)
−0.175285 + 0.984518i \(0.556085\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −8.55220 −1.11340 −0.556700 − 0.830713i \(-0.687933\pi\)
−0.556700 + 0.830713i \(0.687933\pi\)
\(60\) 0 0
\(61\) −3.01641 −0.386211 −0.193106 − 0.981178i \(-0.561856\pi\)
−0.193106 + 0.981178i \(0.561856\pi\)
\(62\) 0 0
\(63\) −3.18953 −0.401844
\(64\) 0 0
\(65\) −3.36266 −0.417087
\(66\) 0 0
\(67\) 7.53579 0.920643 0.460322 − 0.887752i \(-0.347734\pi\)
0.460322 + 0.887752i \(0.347734\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −13.9149 −1.65139 −0.825695 − 0.564117i \(-0.809217\pi\)
−0.825695 + 0.564117i \(0.809217\pi\)
\(72\) 0 0
\(73\) 8.37907 0.980696 0.490348 − 0.871527i \(-0.336870\pi\)
0.490348 + 0.871527i \(0.336870\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 7.74173 0.871013 0.435506 − 0.900186i \(-0.356569\pi\)
0.435506 + 0.900186i \(0.356569\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 9.91486 1.08830 0.544148 − 0.838989i \(-0.316853\pi\)
0.544148 + 0.838989i \(0.316853\pi\)
\(84\) 0 0
\(85\) 5.18953 0.562884
\(86\) 0 0
\(87\) 1.18953 0.127531
\(88\) 0 0
\(89\) 10.3463 1.09670 0.548350 − 0.836249i \(-0.315256\pi\)
0.548350 + 0.836249i \(0.315256\pi\)
\(90\) 0 0
\(91\) 10.7253 1.12432
\(92\) 0 0
\(93\) 2.17313 0.225343
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 3.01641 0.306270 0.153135 − 0.988205i \(-0.451063\pi\)
0.153135 + 0.988205i \(0.451063\pi\)
\(98\) 0 0
\(99\) 0 0
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 5520.2.a.ca.1.1 3
4.3 odd 2 2760.2.a.t.1.3 ✓ 3
12.11 even 2 8280.2.a.bh.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2760.2.a.t.1.3 ✓ 3 4.3 odd 2
5520.2.a.ca.1.1 3 1.1 even 1 trivial
8280.2.a.bh.1.3 3 12.11 even 2