Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [9072,2,Mod(1,9072)] 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("9072.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9072, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 9072 = 2^{4} \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9072.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,0,0,-3,0,-3,0,0,0,6,0,-3,0,0,0,0] 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: \(72.4402847137\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.321.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 252)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.46050\) of defining polynomial
Character \(\chi\) \(=\) 9072.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.05408 q^{5} -1.00000 q^{7} +5.05408 q^{11} -1.00000 q^{13} +0.273346 q^{17} +5.38151 q^{19} +5.32743 q^{23} -0.780738 q^{25} -8.32743 q^{29} +10.1623 q^{31} -2.05408 q^{35} +8.16225 q^{37} -5.05408 q^{41} -4.60078 q^{43} -1.38151 q^{47} +1.00000 q^{49} +3.43560 q^{53} +10.3815 q^{55} -1.78074 q^{59} +0.780738 q^{61} -2.05408 q^{65} +8.38151 q^{67} +7.78074 q^{71} +9.38151 q^{73} -5.05408 q^{77} -12.9430 q^{79} +5.72665 q^{83} +0.561476 q^{85} -13.8171 q^{89} +1.00000 q^{91} +11.0541 q^{95} -2.21926 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{5} - 3 q^{7} + 6 q^{11} - 3 q^{13} - 3 q^{19} + 6 q^{23} + 6 q^{25} - 15 q^{29} + 3 q^{31} + 3 q^{35} - 3 q^{37} - 6 q^{41} - 3 q^{43} + 15 q^{47} + 3 q^{49} - 18 q^{53} + 12 q^{55} + 3 q^{59}+ \cdots - 15 q^{97}+O(q^{100}) \) 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
\(3\) 0 0
\(4\) 0 0
\(5\) 2.05408 0.918614 0.459307 − 0.888277i \(-0.348098\pi\)
0.459307 + 0.888277i \(0.348098\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5.05408 1.52386 0.761932 − 0.647657i \(-0.224251\pi\)
0.761932 + 0.647657i \(0.224251\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350 −0.138675 − 0.990338i \(-0.544284\pi\)
−0.138675 + 0.990338i \(0.544284\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.273346 0.0662962 0.0331481 − 0.999450i \(-0.489447\pi\)
0.0331481 + 0.999450i \(0.489447\pi\)
\(18\) 0 0
\(19\) 5.38151 1.23460 0.617302 − 0.786726i \(-0.288226\pi\)
0.617302 + 0.786726i \(0.288226\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 5.32743 1.11085 0.555423 − 0.831568i \(-0.312556\pi\)
0.555423 + 0.831568i \(0.312556\pi\)
\(24\) 0 0
\(25\) −0.780738 −0.156148
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −8.32743 −1.54637 −0.773183 − 0.634184i \(-0.781336\pi\)
−0.773183 + 0.634184i \(0.781336\pi\)
\(30\) 0 0
\(31\) 10.1623 1.82519 0.912597 − 0.408860i \(-0.134073\pi\)
0.912597 + 0.408860i \(0.134073\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.05408 −0.347204
\(36\) 0 0
\(37\) 8.16225 1.34187 0.670933 − 0.741518i \(-0.265894\pi\)
0.670933 + 0.741518i \(0.265894\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −5.05408 −0.789315 −0.394658 − 0.918828i \(-0.629137\pi\)
−0.394658 + 0.918828i \(0.629137\pi\)
\(42\) 0 0
\(43\) −4.60078 −0.701612 −0.350806 − 0.936448i \(-0.614092\pi\)
−0.350806 + 0.936448i \(0.614092\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1.38151 −0.201515 −0.100757 − 0.994911i \(-0.532127\pi\)
−0.100757 + 0.994911i \(0.532127\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 3.43560 0.471916 0.235958 − 0.971763i \(-0.424177\pi\)
0.235958 + 0.971763i \(0.424177\pi\)
\(54\) 0 0
\(55\) 10.3815 1.39984
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −1.78074 −0.231832 −0.115916 − 0.993259i \(-0.536980\pi\)
−0.115916 + 0.993259i \(0.536980\pi\)
\(60\) 0 0
\(61\) 0.780738 0.0999633 0.0499816 − 0.998750i \(-0.484084\pi\)
0.0499816 + 0.998750i \(0.484084\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −2.05408 −0.254778
\(66\) 0 0
\(67\) 8.38151 1.02396 0.511982 − 0.858996i \(-0.328911\pi\)
0.511982 + 0.858996i \(0.328911\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 7.78074 0.923404 0.461702 − 0.887035i \(-0.347239\pi\)
0.461702 + 0.887035i \(0.347239\pi\)
\(72\) 0 0
\(73\) 9.38151 1.09802 0.549012 − 0.835815i \(-0.315004\pi\)
0.549012 + 0.835815i \(0.315004\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −5.05408 −0.575966
\(78\) 0 0
\(79\) −12.9430 −1.45620 −0.728100 − 0.685471i \(-0.759596\pi\)
−0.728100 + 0.685471i \(0.759596\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 5.72665 0.628582 0.314291 − 0.949327i \(-0.398233\pi\)
0.314291 + 0.949327i \(0.398233\pi\)
\(84\) 0 0
\(85\) 0.561476 0.0609006
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −13.8171 −1.46461 −0.732306 − 0.680976i \(-0.761556\pi\)
−0.732306 + 0.680976i \(0.761556\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 11.0541 1.13413
\(96\) 0 0
\(97\) −2.21926 −0.225332 −0.112666 − 0.993633i \(-0.535939\pi\)
−0.112666 + 0.993633i \(0.535939\pi\)
\(98\) 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 9072.2.a.bt.1.3 3
3.2 odd 2 9072.2.a.bz.1.1 3
4.3 odd 2 2268.2.a.g.1.3 3
9.2 odd 6 3024.2.r.i.1009.3 6
9.4 even 3 1008.2.r.g.673.2 6
9.5 odd 6 3024.2.r.i.2017.3 6
9.7 even 3 1008.2.r.g.337.2 6
12.11 even 2 2268.2.a.j.1.1 3
36.7 odd 6 252.2.j.b.85.2 ✓ 6
36.11 even 6 756.2.j.a.253.3 6
36.23 even 6 756.2.j.a.505.3 6
36.31 odd 6 252.2.j.b.169.2 yes 6
252.11 even 6 5292.2.i.d.1549.3 6
252.23 even 6 5292.2.i.d.2125.3 6
252.31 even 6 1764.2.l.g.961.3 6
252.47 odd 6 5292.2.l.d.361.3 6
252.59 odd 6 5292.2.l.d.3313.3 6
252.67 odd 6 1764.2.l.d.961.1 6
252.79 odd 6 1764.2.l.d.949.1 6
252.83 odd 6 5292.2.j.e.1765.1 6
252.95 even 6 5292.2.l.g.3313.1 6
252.103 even 6 1764.2.i.e.1537.2 6
252.115 even 6 1764.2.i.e.373.2 6
252.131 odd 6 5292.2.i.g.2125.1 6
252.139 even 6 1764.2.j.d.1177.2 6
252.151 odd 6 1764.2.i.f.373.2 6
252.167 odd 6 5292.2.j.e.3529.1 6
252.187 even 6 1764.2.l.g.949.3 6
252.191 even 6 5292.2.l.g.361.1 6
252.223 even 6 1764.2.j.d.589.2 6
252.227 odd 6 5292.2.i.g.1549.1 6
252.247 odd 6 1764.2.i.f.1537.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
252.2.j.b.85.2 ✓ 6 36.7 odd 6
252.2.j.b.169.2 yes 6 36.31 odd 6
756.2.j.a.253.3 6 36.11 even 6
756.2.j.a.505.3 6 36.23 even 6
1008.2.r.g.337.2 6 9.7 even 3
1008.2.r.g.673.2 6 9.4 even 3
1764.2.i.e.373.2 6 252.115 even 6
1764.2.i.e.1537.2 6 252.103 even 6
1764.2.i.f.373.2 6 252.151 odd 6
1764.2.i.f.1537.2 6 252.247 odd 6
1764.2.j.d.589.2 6 252.223 even 6
1764.2.j.d.1177.2 6 252.139 even 6
1764.2.l.d.949.1 6 252.79 odd 6
1764.2.l.d.961.1 6 252.67 odd 6
1764.2.l.g.949.3 6 252.187 even 6
1764.2.l.g.961.3 6 252.31 even 6
2268.2.a.g.1.3 3 4.3 odd 2
2268.2.a.j.1.1 3 12.11 even 2
3024.2.r.i.1009.3 6 9.2 odd 6
3024.2.r.i.2017.3 6 9.5 odd 6
5292.2.i.d.1549.3 6 252.11 even 6
5292.2.i.d.2125.3 6 252.23 even 6
5292.2.i.g.1549.1 6 252.227 odd 6
5292.2.i.g.2125.1 6 252.131 odd 6
5292.2.j.e.1765.1 6 252.83 odd 6
5292.2.j.e.3529.1 6 252.167 odd 6
5292.2.l.d.361.3 6 252.47 odd 6
5292.2.l.d.3313.3 6 252.59 odd 6
5292.2.l.g.361.1 6 252.191 even 6
5292.2.l.g.3313.1 6 252.95 even 6
9072.2.a.bt.1.3 3 1.1 even 1 trivial
9072.2.a.bz.1.1 3 3.2 odd 2