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Properties

Label 9216.2.a.bs
Level $9216$
Weight $2$
Character orbit 9216.a
Self dual yes
Analytic conductor $73.590$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9216,2,Mod(1,9216)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9216, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9216.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9216 = 2^{10} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9216.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(73.5901305028\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.8.3288334336.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 12x^{6} - 8x^{5} + 24x^{4} + 8x^{3} - 16x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 4608)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
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.

\(f(q)\) \(=\) \( q - \beta_{4} q^{5} + \beta_{7} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{5} + \beta_{7} q^{7} - \beta_{2} q^{11} + ( - \beta_{5} + 2) q^{13} + ( - \beta_{6} - \beta_{4}) q^{17} + (2 \beta_{7} + \beta_1) q^{19} + (\beta_{3} - \beta_{2}) q^{23} + ( - 2 \beta_{5} + 1) q^{25} + ( - 2 \beta_{6} + \beta_{4}) q^{29} + (\beta_{7} + 2 \beta_1) q^{31} + (2 \beta_{3} + \beta_{2}) q^{35} + (\beta_{5} + 2) q^{37} + (\beta_{6} - 3 \beta_{4}) q^{41} + (2 \beta_{7} - \beta_1) q^{43} + ( - \beta_{3} - 3 \beta_{2}) q^{47} + (2 \beta_{5} + 1) q^{49} + (2 \beta_{6} - \beta_{4}) q^{53} + ( - 2 \beta_{7} - 2 \beta_1) q^{55} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{59} + ( - \beta_{5} + 6) q^{61} + (\beta_{6} - 3 \beta_{4}) q^{65} - 2 \beta_1 q^{67} - 4 \beta_{3} q^{71} + ( - 6 \beta_{5} + 2) q^{73} + ( - 2 \beta_{6} + 2 \beta_{4}) q^{77} - \beta_{7} q^{79} + ( - 4 \beta_{3} - \beta_{2}) q^{83} + (2 \beta_{5} + 8) q^{85} - 4 \beta_{4} q^{89} + (2 \beta_{7} + \beta_1) q^{91} + (3 \beta_{3} + 5 \beta_{2}) q^{95} + (8 \beta_{5} - 4) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 16 q^{13} + 8 q^{25} + 16 q^{37} + 8 q^{49} + 48 q^{61} + 16 q^{73} + 64 q^{85} - 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 12x^{6} - 8x^{5} + 24x^{4} + 8x^{3} - 16x^{2} + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( -2\nu^{6} + 2\nu^{5} + 18\nu^{4} + 4\nu^{3} - 18\nu^{2} - 8\nu + 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -2\nu^{7} + 22\nu^{5} + 20\nu^{4} - 32\nu^{3} - 30\nu^{2} + 12\nu + 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{7} + 2\nu^{6} - 22\nu^{5} - 40\nu^{4} + 12\nu^{3} + 42\nu^{2} - 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -2\nu^{7} - \nu^{6} + 24\nu^{5} + 27\nu^{4} - 38\nu^{3} - 32\nu^{2} + 16\nu + 6 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 2\nu^{7} + 2\nu^{6} - 24\nu^{5} - 37\nu^{4} + 28\nu^{3} + 38\nu^{2} - 12\nu - 6 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -4\nu^{7} - \nu^{6} + 44\nu^{5} + 47\nu^{4} - 50\nu^{3} - 42\nu^{2} + 16\nu + 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 4\nu^{7} + 3\nu^{6} - 46\nu^{5} - 66\nu^{4} + 48\nu^{3} + 68\nu^{2} - 14\nu - 14 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -2\beta_{5} - 2\beta_{4} + \beta_{3} + \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{7} + \beta_{6} - 2\beta_{5} - \beta_{4} + \beta_{2} + \beta _1 + 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 6\beta_{7} + 4\beta_{6} - 16\beta_{5} - 16\beta_{4} + 7\beta_{3} + 11\beta_{2} + 6\beta _1 + 12 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 10\beta_{7} + 5\beta_{6} - 13\beta_{5} - 9\beta_{4} + 2\beta_{3} + 8\beta_{2} + 6\beta _1 + 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 50\beta_{7} + 29\beta_{6} - 94\beta_{5} - 83\beta_{4} + 32\beta_{3} + 63\beta_{2} + 40\beta _1 + 110 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 218\beta_{7} + 114\beta_{6} - 322\beta_{5} - 248\beta_{4} + 73\beta_{3} + 207\beta_{2} + 144\beta _1 + 504 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 672\beta_{7} + 372\beta_{6} - 1142\beta_{5} - 956\beta_{4} + 339\beta_{3} + 752\beta_{2} + 497\beta _1 + 1512 ) / 2 \) 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.

comment: embeddings in the coefficient field
 
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
0.528036
−2.08509
−1.05636
−2.13875
−0.357857
0.724535
0.886177
3.49930
0 0 0 −2.97127 0 −2.27411 0 0 0
1.2 0 0 0 −2.97127 0 2.27411 0 0 0
1.3 0 0 0 −1.78089 0 −3.29066 0 0 0
1.4 0 0 0 −1.78089 0 3.29066 0 0 0
1.5 0 0 0 1.78089 0 −3.29066 0 0 0
1.6 0 0 0 1.78089 0 3.29066 0 0 0
1.7 0 0 0 2.97127 0 −2.27411 0 0 0
1.8 0 0 0 2.97127 0 2.27411 0 0 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
\(2\) \(-1\)
\(3\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9216.2.a.bs 8
3.b odd 2 1 inner 9216.2.a.bs 8
4.b odd 2 1 inner 9216.2.a.bs 8
8.b even 2 1 9216.2.a.br 8
8.d odd 2 1 9216.2.a.br 8
12.b even 2 1 inner 9216.2.a.bs 8
24.f even 2 1 9216.2.a.br 8
24.h odd 2 1 9216.2.a.br 8
32.g even 8 2 4608.2.k.bk ✓ 16
32.g even 8 2 4608.2.k.bl yes 16
32.h odd 8 2 4608.2.k.bk ✓ 16
32.h odd 8 2 4608.2.k.bl yes 16
96.o even 8 2 4608.2.k.bk ✓ 16
96.o even 8 2 4608.2.k.bl yes 16
96.p odd 8 2 4608.2.k.bk ✓ 16
96.p odd 8 2 4608.2.k.bl yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4608.2.k.bk ✓ 16 32.g even 8 2
4608.2.k.bk ✓ 16 32.h odd 8 2
4608.2.k.bk ✓ 16 96.o even 8 2
4608.2.k.bk ✓ 16 96.p odd 8 2
4608.2.k.bl yes 16 32.g even 8 2
4608.2.k.bl yes 16 32.h odd 8 2
4608.2.k.bl yes 16 96.o even 8 2
4608.2.k.bl yes 16 96.p odd 8 2
9216.2.a.br 8 8.b even 2 1
9216.2.a.br 8 8.d odd 2 1
9216.2.a.br 8 24.f even 2 1
9216.2.a.br 8 24.h odd 2 1
9216.2.a.bs 8 1.a even 1 1 trivial
9216.2.a.bs 8 3.b odd 2 1 inner
9216.2.a.bs 8 4.b odd 2 1 inner
9216.2.a.bs 8 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9216))\):

\( T_{5}^{4} - 12T_{5}^{2} + 28 \) Copy content Toggle raw display
\( T_{7}^{4} - 16T_{7}^{2} + 56 \) Copy content Toggle raw display
\( T_{11}^{4} - 16T_{11}^{2} + 32 \) Copy content Toggle raw display
\( T_{13}^{2} - 4T_{13} + 2 \) Copy content Toggle raw display
\( T_{17}^{4} - 40T_{17}^{2} + 112 \) Copy content Toggle raw display
\( T_{19}^{4} - 64T_{19}^{2} + 224 \) Copy content Toggle raw display
\( T_{67}^{4} - 128T_{67}^{2} + 3584 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} - 12 T^{2} + 28)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} - 16 T^{2} + 56)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 16 T^{2} + 32)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 4 T + 2)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} - 40 T^{2} + 112)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 64 T^{2} + 224)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 32 T^{2} + 128)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 76 T^{2} + 1372)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 112 T^{2} + 2744)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 4 T + 2)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} - 104 T^{2} + 112)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 128 T^{2} + 224)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 160 T^{2} + 128)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 76 T^{2} + 1372)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 128 T^{2} + 2048)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 12 T + 34)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} - 128 T^{2} + 3584)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 256 T^{2} + 8192)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 4 T - 68)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} - 16 T^{2} + 56)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 272 T^{2} + 16928)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 192 T^{2} + 7168)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 8 T - 112)^{4} \) Copy content Toggle raw display
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