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 = [3,0,0,0,0,0,-3,0,0,0,3,0,-6,0,0,0,-6] 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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.837.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 6x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 4536)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.52892\) 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)
 
\(f(q)\) \(=\) \(q+0.133492 q^{5} -1.00000 q^{7} -3.92434 q^{11} -2.13349 q^{13} +2.92434 q^{17} +5.05784 q^{19} +6.79085 q^{23} -4.98218 q^{25} -7.19133 q^{29} +5.05784 q^{31} -0.133492 q^{35} +0.0578359 q^{37} +2.00000 q^{41} -10.7152 q^{43} +12.9065 q^{47} +1.00000 q^{49} -13.2492 q^{53} -0.523868 q^{55} +2.94216 q^{59} +7.98218 q^{61} -0.284804 q^{65} -2.19133 q^{67} +6.86651 q^{71} -8.92434 q^{73} +3.92434 q^{77} -0.0756560 q^{79} -9.05784 q^{83} +0.390376 q^{85} +4.65736 q^{89} +2.13349 q^{91} +0.675180 q^{95} +19.5817 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{7} + 3 q^{11} - 6 q^{13} - 6 q^{17} + 6 q^{23} + 15 q^{25} - 6 q^{29} - 15 q^{37} + 6 q^{41} - 3 q^{43} - 6 q^{47} + 3 q^{49} - 9 q^{53} + 12 q^{55} + 24 q^{59} - 6 q^{61} - 30 q^{65} + 9 q^{67}+ \cdots + 30 q^{97}+O(q^{100}) \) 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
\(5\) 0.133492 0.0596994 0.0298497 − 0.999554i \(-0.490497\pi\)
0.0298497 + 0.999554i \(0.490497\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −3.92434 −1.18323 −0.591617 − 0.806219i \(-0.701510\pi\)
−0.591617 + 0.806219i \(0.701510\pi\)
\(12\) 0 0
\(13\) −2.13349 −0.591724 −0.295862 − 0.955231i \(-0.595607\pi\)
−0.295862 + 0.955231i \(0.595607\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.92434 0.709258 0.354629 − 0.935007i \(-0.384607\pi\)
0.354629 + 0.935007i \(0.384607\pi\)
\(18\) 0 0
\(19\) 5.05784 1.16035 0.580174 − 0.814493i \(-0.302985\pi\)
0.580174 + 0.814493i \(0.302985\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 6.79085 1.41599 0.707995 − 0.706217i \(-0.249600\pi\)
0.707995 + 0.706217i \(0.249600\pi\)
\(24\) 0 0
\(25\) −4.98218 −0.996436
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −7.19133 −1.33540 −0.667698 − 0.744432i \(-0.732720\pi\)
−0.667698 + 0.744432i \(0.732720\pi\)
\(30\) 0 0
\(31\) 5.05784 0.908414 0.454207 − 0.890896i \(-0.349923\pi\)
0.454207 + 0.890896i \(0.349923\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −0.133492 −0.0225643
\(36\) 0 0
\(37\) 0.0578359 0.00950817 0.00475408 − 0.999989i \(-0.498487\pi\)
0.00475408 + 0.999989i \(0.498487\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 − 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) −10.7152 −1.63405 −0.817026 − 0.576601i \(-0.804379\pi\)
−0.817026 + 0.576601i \(0.804379\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 12.9065 1.88261 0.941305 − 0.337557i \(-0.109601\pi\)
0.941305 + 0.337557i \(0.109601\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −13.2492 −1.81991 −0.909956 − 0.414704i \(-0.863885\pi\)
−0.909956 + 0.414704i \(0.863885\pi\)
\(54\) 0 0
\(55\) −0.523868 −0.0706384
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 2.94216 0.383037 0.191519 − 0.981489i \(-0.438659\pi\)
0.191519 + 0.981489i \(0.438659\pi\)
\(60\) 0 0
\(61\) 7.98218 1.02201 0.511007 − 0.859577i \(-0.329273\pi\)
0.511007 + 0.859577i \(0.329273\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −0.284804 −0.0353256
\(66\) 0 0
\(67\) −2.19133 −0.267713 −0.133857 − 0.991001i \(-0.542736\pi\)
−0.133857 + 0.991001i \(0.542736\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 6.86651 0.814905 0.407452 − 0.913226i \(-0.366417\pi\)
0.407452 + 0.913226i \(0.366417\pi\)
\(72\) 0 0
\(73\) −8.92434 −1.04452 −0.522258 − 0.852788i \(-0.674910\pi\)
−0.522258 + 0.852788i \(0.674910\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.92434 0.447221
\(78\) 0 0
\(79\) −0.0756560 −0.00851197 −0.00425598 − 0.999991i \(-0.501355\pi\)
−0.00425598 + 0.999991i \(0.501355\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −9.05784 −0.994227 −0.497113 − 0.867686i \(-0.665607\pi\)
−0.497113 + 0.867686i \(0.665607\pi\)
\(84\) 0 0
\(85\) 0.390376 0.0423423
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 4.65736 0.493679 0.246840 − 0.969056i \(-0.420608\pi\)
0.246840 + 0.969056i \(0.420608\pi\)
\(90\) 0 0
\(91\) 2.13349 0.223651
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0.675180 0.0692720
\(96\) 0 0
\(97\) 19.5817 1.98822 0.994110 − 0.108372i \(-0.0345638\pi\)
0.994110 + 0.108372i \(0.0345638\pi\)
\(98\) 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.bx.1.2 3
3.2 odd 2 9072.2.a.bw.1.2 3
4.3 odd 2 4536.2.a.t.1.2 ✓ 3
12.11 even 2 4536.2.a.u.1.2 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4536.2.a.t.1.2 ✓ 3 4.3 odd 2
4536.2.a.u.1.2 yes 3 12.11 even 2
9072.2.a.bw.1.2 3 3.2 odd 2
9072.2.a.bx.1.2 3 1.1 even 1 trivial