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 1.2
Root \(-1.69963\) 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)
 
\(f(q)\) \(=\) \(q-1.11126 q^{5} -1.00000 q^{7} +1.88874 q^{11} -1.00000 q^{13} -5.87636 q^{17} -7.09888 q^{19} -3.98762 q^{23} -3.76509 q^{25} +0.987620 q^{29} +0.666208 q^{31} +1.11126 q^{35} -1.33379 q^{37} -1.88874 q^{41} +10.8640 q^{43} +11.0989 q^{47} +1.00000 q^{49} -12.2101 q^{53} -2.09888 q^{55} -4.76509 q^{59} +3.76509 q^{61} +1.11126 q^{65} -4.09888 q^{67} +10.7651 q^{71} -3.09888 q^{73} -1.88874 q^{77} -6.43130 q^{79} +11.8764 q^{83} +6.53018 q^{85} +14.3090 q^{89} +1.00000 q^{91} +7.88874 q^{95} +0.765092 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{5} - 3 q^{7} + 6 q^{11} - 3 q^{13} - 3 q^{19} + 6 q^{23} + 6 q^{25} - 15 q^{29} + 3 q^{31} + 3 q^{35} - 3 q^{37} - 6 q^{41} - 3 q^{43} + 15 q^{47} + 3 q^{49} - 18 q^{53} + 12 q^{55} + 3 q^{59}+ \cdots - 15 q^{97}+O(q^{100}) \) 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
\(5\) −1.11126 −0.496972 −0.248486 − 0.968635i \(-0.579933\pi\)
−0.248486 + 0.968635i \(0.579933\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.88874 0.569475 0.284738 − 0.958605i \(-0.408094\pi\)
0.284738 + 0.958605i \(0.408094\pi\)
\(12\) 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
\(17\) −5.87636 −1.42523 −0.712613 − 0.701557i \(-0.752488\pi\)
−0.712613 + 0.701557i \(0.752488\pi\)
\(18\) 0 0
\(19\) −7.09888 −1.62860 −0.814298 − 0.580447i \(-0.802878\pi\)
−0.814298 + 0.580447i \(0.802878\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.98762 −0.831476 −0.415738 − 0.909484i \(-0.636477\pi\)
−0.415738 + 0.909484i \(0.636477\pi\)
\(24\) 0 0
\(25\) −3.76509 −0.753018
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0.987620 0.183396 0.0916982 − 0.995787i \(-0.470770\pi\)
0.0916982 + 0.995787i \(0.470770\pi\)
\(30\) 0 0
\(31\) 0.666208 0.119654 0.0598272 − 0.998209i \(-0.480945\pi\)
0.0598272 + 0.998209i \(0.480945\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.11126 0.187838
\(36\) 0 0
\(37\) −1.33379 −0.219274 −0.109637 − 0.993972i \(-0.534969\pi\)
−0.109637 + 0.993972i \(0.534969\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −1.88874 −0.294971 −0.147485 − 0.989064i \(-0.547118\pi\)
−0.147485 + 0.989064i \(0.547118\pi\)
\(42\) 0 0
\(43\) 10.8640 1.65674 0.828370 − 0.560181i \(-0.189268\pi\)
0.828370 + 0.560181i \(0.189268\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 11.0989 1.61894 0.809469 − 0.587162i \(-0.199755\pi\)
0.809469 + 0.587162i \(0.199755\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −12.2101 −1.67719 −0.838596 − 0.544753i \(-0.816623\pi\)
−0.838596 + 0.544753i \(0.816623\pi\)
\(54\) 0 0
\(55\) −2.09888 −0.283014
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −4.76509 −0.620362 −0.310181 − 0.950677i \(-0.600390\pi\)
−0.310181 + 0.950677i \(0.600390\pi\)
\(60\) 0 0
\(61\) 3.76509 0.482071 0.241035 − 0.970516i \(-0.422513\pi\)
0.241035 + 0.970516i \(0.422513\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.11126 0.137835
\(66\) 0 0
\(67\) −4.09888 −0.500758 −0.250379 − 0.968148i \(-0.580555\pi\)
−0.250379 + 0.968148i \(0.580555\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 10.7651 1.27758 0.638791 − 0.769381i \(-0.279435\pi\)
0.638791 + 0.769381i \(0.279435\pi\)
\(72\) 0 0
\(73\) −3.09888 −0.362697 −0.181348 − 0.983419i \(-0.558046\pi\)
−0.181348 + 0.983419i \(0.558046\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1.88874 −0.215241
\(78\) 0 0
\(79\) −6.43130 −0.723578 −0.361789 − 0.932260i \(-0.617834\pi\)
−0.361789 + 0.932260i \(0.617834\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 11.8764 1.30360 0.651800 − 0.758391i \(-0.274014\pi\)
0.651800 + 0.758391i \(0.274014\pi\)
\(84\) 0 0
\(85\) 6.53018 0.708298
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 14.3090 1.51675 0.758377 − 0.651816i \(-0.225993\pi\)
0.758377 + 0.651816i \(0.225993\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 7.88874 0.809367
\(96\) 0 0
\(97\) 0.765092 0.0776833 0.0388417 − 0.999245i \(-0.487633\pi\)
0.0388417 + 0.999245i \(0.487633\pi\)
\(98\) 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.1.2 3
3.2 odd 2 9072.2.a.bz.1.2 3
4.3 odd 2 2268.2.a.g.1.2 3
9.2 odd 6 3024.2.r.i.1009.2 6
9.4 even 3 1008.2.r.g.673.1 6
9.5 odd 6 3024.2.r.i.2017.2 6
9.7 even 3 1008.2.r.g.337.1 6
12.11 even 2 2268.2.a.j.1.2 3
36.7 odd 6 252.2.j.b.85.3 ✓ 6
36.11 even 6 756.2.j.a.253.2 6
36.23 even 6 756.2.j.a.505.2 6
36.31 odd 6 252.2.j.b.169.3 yes 6
252.11 even 6 5292.2.i.d.1549.2 6
252.23 even 6 5292.2.i.d.2125.2 6
252.31 even 6 1764.2.l.g.961.1 6
252.47 odd 6 5292.2.l.d.361.2 6
252.59 odd 6 5292.2.l.d.3313.2 6
252.67 odd 6 1764.2.l.d.961.3 6
252.79 odd 6 1764.2.l.d.949.3 6
252.83 odd 6 5292.2.j.e.1765.2 6
252.95 even 6 5292.2.l.g.3313.2 6
252.103 even 6 1764.2.i.e.1537.3 6
252.115 even 6 1764.2.i.e.373.3 6
252.131 odd 6 5292.2.i.g.2125.2 6
252.139 even 6 1764.2.j.d.1177.1 6
252.151 odd 6 1764.2.i.f.373.1 6
252.167 odd 6 5292.2.j.e.3529.2 6
252.187 even 6 1764.2.l.g.949.1 6
252.191 even 6 5292.2.l.g.361.2 6
252.223 even 6 1764.2.j.d.589.1 6
252.227 odd 6 5292.2.i.g.1549.2 6
252.247 odd 6 1764.2.i.f.1537.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
252.2.j.b.85.3 ✓ 6 36.7 odd 6
252.2.j.b.169.3 yes 6 36.31 odd 6
756.2.j.a.253.2 6 36.11 even 6
756.2.j.a.505.2 6 36.23 even 6
1008.2.r.g.337.1 6 9.7 even 3
1008.2.r.g.673.1 6 9.4 even 3
1764.2.i.e.373.3 6 252.115 even 6
1764.2.i.e.1537.3 6 252.103 even 6
1764.2.i.f.373.1 6 252.151 odd 6
1764.2.i.f.1537.1 6 252.247 odd 6
1764.2.j.d.589.1 6 252.223 even 6
1764.2.j.d.1177.1 6 252.139 even 6
1764.2.l.d.949.3 6 252.79 odd 6
1764.2.l.d.961.3 6 252.67 odd 6
1764.2.l.g.949.1 6 252.187 even 6
1764.2.l.g.961.1 6 252.31 even 6
2268.2.a.g.1.2 3 4.3 odd 2
2268.2.a.j.1.2 3 12.11 even 2
3024.2.r.i.1009.2 6 9.2 odd 6
3024.2.r.i.2017.2 6 9.5 odd 6
5292.2.i.d.1549.2 6 252.11 even 6
5292.2.i.d.2125.2 6 252.23 even 6
5292.2.i.g.1549.2 6 252.227 odd 6
5292.2.i.g.2125.2 6 252.131 odd 6
5292.2.j.e.1765.2 6 252.83 odd 6
5292.2.j.e.3529.2 6 252.167 odd 6
5292.2.l.d.361.2 6 252.47 odd 6
5292.2.l.d.3313.2 6 252.59 odd 6
5292.2.l.g.361.2 6 252.191 even 6
5292.2.l.g.3313.2 6 252.95 even 6
9072.2.a.bt.1.2 3 1.1 even 1 trivial
9072.2.a.bz.1.2 3 3.2 odd 2