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 = [2,0,0,0,-4,0,2,0,0,0,2,0,-6,0,0,0,-14] 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: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1134)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) 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)
 
\(f(q)\) \(=\) \(q-0.267949 q^{5} +1.00000 q^{7} +6.19615 q^{11} -6.46410 q^{13} -7.00000 q^{17} -0.732051 q^{19} -4.19615 q^{23} -4.92820 q^{25} -1.53590 q^{29} -8.19615 q^{31} -0.267949 q^{35} +10.6603 q^{37} -2.53590 q^{41} +1.46410 q^{43} +4.73205 q^{47} +1.00000 q^{49} +9.46410 q^{53} -1.66025 q^{55} +4.19615 q^{59} +3.92820 q^{61} +1.73205 q^{65} +6.73205 q^{67} +6.53590 q^{71} +8.26795 q^{73} +6.19615 q^{77} +9.12436 q^{79} +16.5885 q^{83} +1.87564 q^{85} -9.92820 q^{89} -6.46410 q^{91} +0.196152 q^{95} +10.9282 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{5} + 2 q^{7} + 2 q^{11} - 6 q^{13} - 14 q^{17} + 2 q^{19} + 2 q^{23} + 4 q^{25} - 10 q^{29} - 6 q^{31} - 4 q^{35} + 4 q^{37} - 12 q^{41} - 4 q^{43} + 6 q^{47} + 2 q^{49} + 12 q^{53} + 14 q^{55}+ \cdots + 8 q^{97}+O(q^{100}) \) 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
\(5\) −0.267949 −0.119831 −0.0599153 − 0.998203i \(-0.519083\pi\)
−0.0599153 + 0.998203i \(0.519083\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 6.19615 1.86821 0.934105 − 0.356998i \(-0.116200\pi\)
0.934105 + 0.356998i \(0.116200\pi\)
\(12\) 0 0
\(13\) −6.46410 −1.79282 −0.896410 − 0.443227i \(-0.853834\pi\)
−0.896410 + 0.443227i \(0.853834\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −7.00000 −1.69775 −0.848875 − 0.528594i \(-0.822719\pi\)
−0.848875 + 0.528594i \(0.822719\pi\)
\(18\) 0 0
\(19\) −0.732051 −0.167944 −0.0839720 − 0.996468i \(-0.526761\pi\)
−0.0839720 + 0.996468i \(0.526761\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.19615 −0.874958 −0.437479 − 0.899229i \(-0.644129\pi\)
−0.437479 + 0.899229i \(0.644129\pi\)
\(24\) 0 0
\(25\) −4.92820 −0.985641
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −1.53590 −0.285209 −0.142605 − 0.989780i \(-0.545548\pi\)
−0.142605 + 0.989780i \(0.545548\pi\)
\(30\) 0 0
\(31\) −8.19615 −1.47207 −0.736036 − 0.676942i \(-0.763305\pi\)
−0.736036 + 0.676942i \(0.763305\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −0.267949 −0.0452917
\(36\) 0 0
\(37\) 10.6603 1.75253 0.876267 − 0.481825i \(-0.160026\pi\)
0.876267 + 0.481825i \(0.160026\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2.53590 −0.396041 −0.198020 − 0.980198i \(-0.563451\pi\)
−0.198020 + 0.980198i \(0.563451\pi\)
\(42\) 0 0
\(43\) 1.46410 0.223273 0.111637 − 0.993749i \(-0.464391\pi\)
0.111637 + 0.993749i \(0.464391\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.73205 0.690241 0.345120 − 0.938558i \(-0.387838\pi\)
0.345120 + 0.938558i \(0.387838\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 9.46410 1.29999 0.649997 − 0.759937i \(-0.274770\pi\)
0.649997 + 0.759937i \(0.274770\pi\)
\(54\) 0 0
\(55\) −1.66025 −0.223869
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.19615 0.546293 0.273146 − 0.961973i \(-0.411936\pi\)
0.273146 + 0.961973i \(0.411936\pi\)
\(60\) 0 0
\(61\) 3.92820 0.502955 0.251477 − 0.967863i \(-0.419084\pi\)
0.251477 + 0.967863i \(0.419084\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.73205 0.214834
\(66\) 0 0
\(67\) 6.73205 0.822451 0.411225 − 0.911534i \(-0.365101\pi\)
0.411225 + 0.911534i \(0.365101\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 6.53590 0.775668 0.387834 − 0.921729i \(-0.373223\pi\)
0.387834 + 0.921729i \(0.373223\pi\)
\(72\) 0 0
\(73\) 8.26795 0.967690 0.483845 − 0.875154i \(-0.339240\pi\)
0.483845 + 0.875154i \(0.339240\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 6.19615 0.706117
\(78\) 0 0
\(79\) 9.12436 1.02657 0.513285 − 0.858218i \(-0.328428\pi\)
0.513285 + 0.858218i \(0.328428\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 16.5885 1.82082 0.910410 − 0.413707i \(-0.135766\pi\)
0.910410 + 0.413707i \(0.135766\pi\)
\(84\) 0 0
\(85\) 1.87564 0.203442
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −9.92820 −1.05239 −0.526194 − 0.850365i \(-0.676381\pi\)
−0.526194 + 0.850365i \(0.676381\pi\)
\(90\) 0 0
\(91\) −6.46410 −0.677622
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0.196152 0.0201248
\(96\) 0 0
\(97\) 10.9282 1.10959 0.554795 − 0.831987i \(-0.312797\pi\)
0.554795 + 0.831987i \(0.312797\pi\)
\(98\) 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.y.1.2 2
3.2 odd 2 9072.2.a.bp.1.1 2
4.3 odd 2 1134.2.a.m.1.2 yes 2
12.11 even 2 1134.2.a.l.1.1 ✓ 2
28.27 even 2 7938.2.a.bt.1.1 2
36.7 odd 6 1134.2.f.r.757.1 4
36.11 even 6 1134.2.f.s.757.2 4
36.23 even 6 1134.2.f.s.379.2 4
36.31 odd 6 1134.2.f.r.379.1 4
84.83 odd 2 7938.2.a.bg.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1134.2.a.l.1.1 ✓ 2 12.11 even 2
1134.2.a.m.1.2 yes 2 4.3 odd 2
1134.2.f.r.379.1 4 36.31 odd 6
1134.2.f.r.757.1 4 36.7 odd 6
1134.2.f.s.379.2 4 36.23 even 6
1134.2.f.s.757.2 4 36.11 even 6
7938.2.a.bg.1.2 2 84.83 odd 2
7938.2.a.bt.1.1 2 28.27 even 2
9072.2.a.y.1.2 2 1.1 even 1 trivial
9072.2.a.bp.1.1 2 3.2 odd 2