↑

Properties

Label 9152.2.a.cv
Level $9152$
Weight $2$
Character orbit 9152.a
Self dual yes
Analytic conductor $73.079$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [9152,2,Mod(1,9152)] 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("9152.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9152, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 9152 = 2^{6} \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9152.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,-1,0,1,0,-4,0,9,0,8,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(73.0790879299\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} - 16x^{6} + 16x^{5} + 68x^{4} - 70x^{3} - 89x^{2} + 91x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 4576)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} - \beta_{6} q^{5} + ( - \beta_{3} - 1) q^{7} + (\beta_{2} + 1) q^{9} + q^{11} - q^{13} + (\beta_{7} + \beta_{3} - \beta_{2} - 1) q^{15} + ( - \beta_{7} - \beta_{6} - \beta_{4} + \cdots - 1) q^{17}+ \cdots + (\beta_{2} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - q^{3} + q^{5} - 4 q^{7} + 9 q^{9} + 8 q^{11} - 8 q^{13} - 13 q^{15} - 8 q^{17} - 2 q^{19} + 12 q^{21} - 17 q^{23} + 23 q^{25} + 5 q^{27} - 23 q^{31} - q^{33} + 2 q^{35} + q^{37} + q^{39} - 14 q^{41}+ \cdots + 9 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} - 16x^{6} + 16x^{5} + 68x^{4} - 70x^{3} - 89x^{2} + 91x + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + 2\nu^{6} - 14\nu^{5} - 26\nu^{4} + 42\nu^{3} + 56\nu^{2} - 37\nu - 12 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} + 14\nu^{5} - 42\nu^{3} + 2\nu^{2} + 31\nu + 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{7} - 2\nu^{6} + 46\nu^{5} + 26\nu^{4} - 178\nu^{3} - 52\nu^{2} + 215\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -2\nu^{7} - \nu^{6} + 31\nu^{5} + 14\nu^{4} - 122\nu^{3} - 36\nu^{2} + 143\nu + 14 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -13\nu^{7} - 6\nu^{6} + 198\nu^{5} + 82\nu^{4} - 746\nu^{3} - 192\nu^{2} + 821\nu + 64 ) / 4 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + 7\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{7} + 2\beta_{6} - 2\beta_{5} + \beta_{4} - \beta_{3} + 11\beta_{2} + 2\beta _1 + 28 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 13\beta_{7} - 13\beta_{6} - 12\beta_{5} - 14\beta_{4} + \beta_{3} + 62\beta _1 - 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -13\beta_{7} + 26\beta_{6} - 26\beta_{5} + 14\beta_{4} - 11\beta_{3} + 114\beta_{2} + 29\beta _1 + 253 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 140\beta_{7} - 140\beta_{6} - 126\beta_{5} - 156\beta_{4} + 14\beta_{3} + 2\beta_{2} + 605\beta _1 - 74 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
3.21103
1.63161
1.48238
1.35792
−0.0818190
−1.59352
−1.83233
−3.17526
0 −3.21103 0 2.06531 0 −2.48057 0 7.31070 0
1.2 0 −1.63161 0 1.18797 0 3.62258 0 −0.337857 0
1.3 0 −1.48238 0 3.10373 0 −2.05291 0 −0.802538 0
1.4 0 −1.35792 0 −3.75642 0 −4.54826 0 −1.15606 0
1.5 0 0.0818190 0 −2.12622 0 1.15548 0 −2.99331 0
1.6 0 1.59352 0 4.06972 0 −1.51716 0 −0.460695 0
1.7 0 1.83233 0 −0.00914290 0 2.03468 0 0.357448 0
1.8 0 3.17526 0 −3.53494 0 −0.213848 0 7.08231 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(11\) \( -1 \)
\(13\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9152.2.a.cv 8
4.b odd 2 1 9152.2.a.cw 8
8.b even 2 1 4576.2.a.u yes 8
8.d odd 2 1 4576.2.a.t ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4576.2.a.t ✓ 8 8.d odd 2 1
4576.2.a.u yes 8 8.b even 2 1
9152.2.a.cv 8 1.a even 1 1 trivial
9152.2.a.cw 8 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9152))\):

\( T_{3}^{8} + T_{3}^{7} - 16T_{3}^{6} - 16T_{3}^{5} + 68T_{3}^{4} + 70T_{3}^{3} - 89T_{3}^{2} - 91T_{3} + 8 \) Copy content Toggle raw display
\( T_{5}^{8} - T_{5}^{7} - 31T_{5}^{6} + 31T_{5}^{5} + 290T_{5}^{4} - 310T_{5}^{3} - 768T_{5}^{2} + 868T_{5} + 8 \) Copy content Toggle raw display
\( T_{7}^{8} + 4T_{7}^{7} - 18T_{7}^{6} - 72T_{7}^{5} + 56T_{7}^{4} + 304T_{7}^{3} + 21T_{7}^{2} - 308T_{7} - 64 \) Copy content Toggle raw display
\( T_{17}^{8} + 8T_{17}^{7} - 61T_{17}^{6} - 538T_{17}^{5} + 622T_{17}^{4} + 7328T_{17}^{3} - 3564T_{17}^{2} - 20760T_{17} + 2176 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + T^{7} - 16 T^{6} + \cdots + 8 \) Copy content Toggle raw display
$5$ \( T^{8} - T^{7} - 31 T^{6} + \cdots + 8 \) Copy content Toggle raw display
$7$ \( T^{8} + 4 T^{7} + \cdots - 64 \) Copy content Toggle raw display
$11$ \( (T - 1)^{8} \) Copy content Toggle raw display
$13$ \( (T + 1)^{8} \) Copy content Toggle raw display
$17$ \( T^{8} + 8 T^{7} + \cdots + 2176 \) Copy content Toggle raw display
$19$ \( T^{8} + 2 T^{7} + \cdots + 8192 \) Copy content Toggle raw display
$23$ \( T^{8} + 17 T^{7} + \cdots + 13824 \) Copy content Toggle raw display
$29$ \( T^{8} - 124 T^{6} + \cdots - 45056 \) Copy content Toggle raw display
$31$ \( T^{8} + 23 T^{7} + \cdots + 389888 \) Copy content Toggle raw display
$37$ \( T^{8} - T^{7} + \cdots + 6842752 \) Copy content Toggle raw display
$41$ \( T^{8} + 14 T^{7} + \cdots - 53504 \) Copy content Toggle raw display
$43$ \( T^{8} + 10 T^{7} + \cdots - 689344 \) Copy content Toggle raw display
$47$ \( T^{8} + 28 T^{7} + \cdots + 1904704 \) Copy content Toggle raw display
$53$ \( T^{8} - 14 T^{7} + \cdots - 26512 \) Copy content Toggle raw display
$59$ \( T^{8} - 31 T^{7} + \cdots + 615168 \) Copy content Toggle raw display
$61$ \( T^{8} - 162 T^{6} + \cdots - 2048 \) Copy content Toggle raw display
$67$ \( T^{8} + 21 T^{7} + \cdots - 10550272 \) Copy content Toggle raw display
$71$ \( T^{8} + 61 T^{7} + \cdots - 466576 \) Copy content Toggle raw display
$73$ \( T^{8} - 371 T^{6} + \cdots + 4283392 \) Copy content Toggle raw display
$79$ \( T^{8} - 300 T^{6} + \cdots + 516352 \) Copy content Toggle raw display
$83$ \( T^{8} - 22 T^{7} + \cdots + 1408 \) Copy content Toggle raw display
$89$ \( T^{8} - 9 T^{7} + \cdots - 34816 \) Copy content Toggle raw display
$97$ \( T^{8} + 11 T^{7} + \cdots + 512 \) Copy content Toggle raw display
show more
show less