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 1.1
Root \(0.239123\) 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)
 
\(f(q)\) \(=\) \(q-3.94282 q^{5} -1.00000 q^{7} -0.942820 q^{11} -1.00000 q^{13} +5.60301 q^{17} -1.28263 q^{19} +4.66019 q^{23} +10.5458 q^{25} -7.66019 q^{29} -7.82846 q^{31} +3.94282 q^{35} -9.82846 q^{37} +0.942820 q^{41} -9.26320 q^{43} +5.28263 q^{47} +1.00000 q^{49} -9.22545 q^{53} +3.71737 q^{55} +9.54583 q^{59} -10.5458 q^{61} +3.94282 q^{65} +1.71737 q^{67} -3.54583 q^{71} +2.71737 q^{73} +0.942820 q^{77} +16.3743 q^{79} +0.396990 q^{83} -22.0917 q^{85} +5.50808 q^{89} +1.00000 q^{91} +5.05718 q^{95} -13.5458 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{5} - 3 q^{7} + 6 q^{11} - 3 q^{13} - 3 q^{19} + 6 q^{23} + 6 q^{25} - 15 q^{29} + 3 q^{31} + 3 q^{35} - 3 q^{37} - 6 q^{41} - 3 q^{43} + 15 q^{47} + 3 q^{49} - 18 q^{53} + 12 q^{55} + 3 q^{59}+ \cdots - 15 q^{97}+O(q^{100}) \) 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
\(5\) −3.94282 −1.76328 −0.881641 − 0.471920i \(-0.843561\pi\)
−0.881641 + 0.471920i \(0.843561\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −0.942820 −0.284271 −0.142135 − 0.989847i \(-0.545397\pi\)
−0.142135 + 0.989847i \(0.545397\pi\)
\(12\) 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
\(17\) 5.60301 1.35893 0.679465 − 0.733708i \(-0.262212\pi\)
0.679465 + 0.733708i \(0.262212\pi\)
\(18\) 0 0
\(19\) −1.28263 −0.294256 −0.147128 − 0.989117i \(-0.547003\pi\)
−0.147128 + 0.989117i \(0.547003\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.66019 0.971717 0.485858 − 0.874038i \(-0.338507\pi\)
0.485858 + 0.874038i \(0.338507\pi\)
\(24\) 0 0
\(25\) 10.5458 2.10917
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −7.66019 −1.42246 −0.711231 − 0.702959i \(-0.751862\pi\)
−0.711231 + 0.702959i \(0.751862\pi\)
\(30\) 0 0
\(31\) −7.82846 −1.40603 −0.703016 − 0.711174i \(-0.748164\pi\)
−0.703016 + 0.711174i \(0.748164\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 3.94282 0.666458
\(36\) 0 0
\(37\) −9.82846 −1.61579 −0.807894 − 0.589327i \(-0.799393\pi\)
−0.807894 + 0.589327i \(0.799393\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0.942820 0.147244 0.0736219 − 0.997286i \(-0.476544\pi\)
0.0736219 + 0.997286i \(0.476544\pi\)
\(42\) 0 0
\(43\) −9.26320 −1.41262 −0.706312 − 0.707900i \(-0.749643\pi\)
−0.706312 + 0.707900i \(0.749643\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 5.28263 0.770551 0.385275 − 0.922802i \(-0.374106\pi\)
0.385275 + 0.922802i \(0.374106\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −9.22545 −1.26721 −0.633607 − 0.773655i \(-0.718426\pi\)
−0.633607 + 0.773655i \(0.718426\pi\)
\(54\) 0 0
\(55\) 3.71737 0.501250
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 9.54583 1.24276 0.621381 − 0.783509i \(-0.286572\pi\)
0.621381 + 0.783509i \(0.286572\pi\)
\(60\) 0 0
\(61\) −10.5458 −1.35026 −0.675128 − 0.737701i \(-0.735911\pi\)
−0.675128 + 0.737701i \(0.735911\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 3.94282 0.489047
\(66\) 0 0
\(67\) 1.71737 0.209810 0.104905 − 0.994482i \(-0.466546\pi\)
0.104905 + 0.994482i \(0.466546\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −3.54583 −0.420813 −0.210406 − 0.977614i \(-0.567479\pi\)
−0.210406 + 0.977614i \(0.567479\pi\)
\(72\) 0 0
\(73\) 2.71737 0.318044 0.159022 − 0.987275i \(-0.449166\pi\)
0.159022 + 0.987275i \(0.449166\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.942820 0.107444
\(78\) 0 0
\(79\) 16.3743 1.84225 0.921126 − 0.389265i \(-0.127271\pi\)
0.921126 + 0.389265i \(0.127271\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0.396990 0.0435753 0.0217877 − 0.999763i \(-0.493064\pi\)
0.0217877 + 0.999763i \(0.493064\pi\)
\(84\) 0 0
\(85\) −22.0917 −2.39618
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 5.50808 0.583855 0.291928 − 0.956440i \(-0.405703\pi\)
0.291928 + 0.956440i \(0.405703\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 5.05718 0.518856
\(96\) 0 0
\(97\) −13.5458 −1.37537 −0.687685 − 0.726009i \(-0.741373\pi\)
−0.687685 + 0.726009i \(0.741373\pi\)
\(98\) 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.1.1 3
3.2 odd 2 9072.2.a.bz.1.3 3
4.3 odd 2 2268.2.a.g.1.1 3
9.2 odd 6 3024.2.r.i.1009.1 6
9.4 even 3 1008.2.r.g.673.3 6
9.5 odd 6 3024.2.r.i.2017.1 6
9.7 even 3 1008.2.r.g.337.3 6
12.11 even 2 2268.2.a.j.1.3 3
36.7 odd 6 252.2.j.b.85.1 ✓ 6
36.11 even 6 756.2.j.a.253.1 6
36.23 even 6 756.2.j.a.505.1 6
36.31 odd 6 252.2.j.b.169.1 yes 6
252.11 even 6 5292.2.i.d.1549.1 6
252.23 even 6 5292.2.i.d.2125.1 6
252.31 even 6 1764.2.l.g.961.2 6
252.47 odd 6 5292.2.l.d.361.1 6
252.59 odd 6 5292.2.l.d.3313.1 6
252.67 odd 6 1764.2.l.d.961.2 6
252.79 odd 6 1764.2.l.d.949.2 6
252.83 odd 6 5292.2.j.e.1765.3 6
252.95 even 6 5292.2.l.g.3313.3 6
252.103 even 6 1764.2.i.e.1537.1 6
252.115 even 6 1764.2.i.e.373.1 6
252.131 odd 6 5292.2.i.g.2125.3 6
252.139 even 6 1764.2.j.d.1177.3 6
252.151 odd 6 1764.2.i.f.373.3 6
252.167 odd 6 5292.2.j.e.3529.3 6
252.187 even 6 1764.2.l.g.949.2 6
252.191 even 6 5292.2.l.g.361.3 6
252.223 even 6 1764.2.j.d.589.3 6
252.227 odd 6 5292.2.i.g.1549.3 6
252.247 odd 6 1764.2.i.f.1537.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
252.2.j.b.85.1 ✓ 6 36.7 odd 6
252.2.j.b.169.1 yes 6 36.31 odd 6
756.2.j.a.253.1 6 36.11 even 6
756.2.j.a.505.1 6 36.23 even 6
1008.2.r.g.337.3 6 9.7 even 3
1008.2.r.g.673.3 6 9.4 even 3
1764.2.i.e.373.1 6 252.115 even 6
1764.2.i.e.1537.1 6 252.103 even 6
1764.2.i.f.373.3 6 252.151 odd 6
1764.2.i.f.1537.3 6 252.247 odd 6
1764.2.j.d.589.3 6 252.223 even 6
1764.2.j.d.1177.3 6 252.139 even 6
1764.2.l.d.949.2 6 252.79 odd 6
1764.2.l.d.961.2 6 252.67 odd 6
1764.2.l.g.949.2 6 252.187 even 6
1764.2.l.g.961.2 6 252.31 even 6
2268.2.a.g.1.1 3 4.3 odd 2
2268.2.a.j.1.3 3 12.11 even 2
3024.2.r.i.1009.1 6 9.2 odd 6
3024.2.r.i.2017.1 6 9.5 odd 6
5292.2.i.d.1549.1 6 252.11 even 6
5292.2.i.d.2125.1 6 252.23 even 6
5292.2.i.g.1549.3 6 252.227 odd 6
5292.2.i.g.2125.3 6 252.131 odd 6
5292.2.j.e.1765.3 6 252.83 odd 6
5292.2.j.e.3529.3 6 252.167 odd 6
5292.2.l.d.361.1 6 252.47 odd 6
5292.2.l.d.3313.1 6 252.59 odd 6
5292.2.l.g.361.3 6 252.191 even 6
5292.2.l.g.3313.3 6 252.95 even 6
9072.2.a.bt.1.1 3 1.1 even 1 trivial
9072.2.a.bz.1.3 3 3.2 odd 2