Properties

Label 9216.2.a.x
Level $9216$
Weight $2$
Character orbit 9216.a
Self dual yes
Analytic conductor $73.590$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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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: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.4352.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 6x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 48)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} - 1) q^{5} + ( - \beta_{2} - 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} - 1) q^{5} + ( - \beta_{2} - 1) q^{7} + (\beta_{3} + \beta_{2} - \beta_1) q^{11} + ( - \beta_{3} - \beta_{2} + 2) q^{13} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{17} + (\beta_{3} - \beta_{2} - \beta_1) q^{19} - 2 \beta_1 q^{23} + (2 \beta_{3} + 2 \beta_1 + 1) q^{25} + (\beta_{3} + 2 \beta_1 - 3) q^{29} + (\beta_{2} - 3) q^{31} + (\beta_{3} + \beta_{2} + 3 \beta_1) q^{35} + ( - \beta_{3} + \beta_{2} + 2 \beta_1 + 4) q^{37} + ( - \beta_{3} + \beta_{2} - 3 \beta_1) q^{41} + (3 \beta_{3} + \beta_{2} + \beta_1) q^{43} + 2 \beta_1 q^{47} + (2 \beta_{2} - 4 \beta_1 + 1) q^{49} + ( - \beta_{3} - 2 \beta_1 - 5) q^{53} + ( - 4 \beta_1 - 4) q^{55} - 4 \beta_1 q^{59} + (\beta_{3} - \beta_{2} + 2 \beta_1 + 4) q^{61} + ( - \beta_{3} + \beta_{2} + 5 \beta_1 + 2) q^{65} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 4) q^{67} + ( - 2 \beta_{2} - 2) q^{71} + (2 \beta_{2} + 4 \beta_1 - 2) q^{73} + ( - 2 \beta_{2} + 2 \beta_1 - 6) q^{77} + (4 \beta_{3} + \beta_{2} + 4 \beta_1 - 3) q^{79} + (3 \beta_{3} - \beta_{2} + 5 \beta_1) q^{83} + ( - 2 \beta_{2} - 2 \beta_1 + 6) q^{85} + (2 \beta_{3} + 2 \beta_{2} - 6 \beta_1 + 2) q^{89} + (\beta_{3} - \beta_{2} - \beta_1 + 4) q^{91} + (2 \beta_{2} + 2 \beta_1 - 6) q^{95} + ( - 2 \beta_{3} - 2 \beta_{2} - 6 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{5} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{5} - 4 q^{7} + 8 q^{13} + 4 q^{25} - 12 q^{29} - 12 q^{31} + 16 q^{37} + 4 q^{49} - 20 q^{53} - 16 q^{55} + 16 q^{61} + 8 q^{65} + 16 q^{67} - 8 q^{71} - 8 q^{73} - 24 q^{77} - 12 q^{79} + 24 q^{85} + 8 q^{89} + 16 q^{91} - 24 q^{95}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu^{3} - \nu^{2} - 4\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{2} + 2\nu + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{3} + 2\nu^{2} + 4\nu - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{3} + 2\beta_{2} + 4\beta _1 + 3 \) Copy content Toggle raw display

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
2.68554
−1.74912
0.334904
−1.27133
0 0 0 −3.79793 0 −2.15894 0 0 0
1.2 0 0 0 −2.47363 0 2.55765 0 0 0
1.3 0 0 0 0.473626 0 −4.55765 0 0 0
1.4 0 0 0 1.79793 0 0.158942 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)

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 9216.2.a.x 4
3.b odd 2 1 3072.2.a.n 4
4.b odd 2 1 9216.2.a.y 4
8.b even 2 1 9216.2.a.bn 4
8.d odd 2 1 9216.2.a.bo 4
12.b even 2 1 3072.2.a.t 4
24.f even 2 1 3072.2.a.i 4
24.h odd 2 1 3072.2.a.o 4
32.g even 8 2 576.2.k.b 8
32.g even 8 2 1152.2.k.f 8
32.h odd 8 2 144.2.k.b 8
32.h odd 8 2 1152.2.k.c 8
48.i odd 4 2 3072.2.d.i 8
48.k even 4 2 3072.2.d.f 8
96.o even 8 2 48.2.j.a 8
96.o even 8 2 384.2.j.b 8
96.p odd 8 2 192.2.j.a 8
96.p odd 8 2 384.2.j.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.2.j.a 8 96.o even 8 2
144.2.k.b 8 32.h odd 8 2
192.2.j.a 8 96.p odd 8 2
384.2.j.a 8 96.p odd 8 2
384.2.j.b 8 96.o even 8 2
576.2.k.b 8 32.g even 8 2
1152.2.k.c 8 32.h odd 8 2
1152.2.k.f 8 32.g even 8 2
3072.2.a.i 4 24.f even 2 1
3072.2.a.n 4 3.b odd 2 1
3072.2.a.o 4 24.h odd 2 1
3072.2.a.t 4 12.b even 2 1
3072.2.d.f 8 48.k even 4 2
3072.2.d.i 8 48.i odd 4 2
9216.2.a.x 4 1.a even 1 1 trivial
9216.2.a.y 4 4.b odd 2 1
9216.2.a.bn 4 8.b even 2 1
9216.2.a.bo 4 8.d odd 2 1

Hecke kernels

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

\( T_{5}^{4} + 4T_{5}^{3} - 4T_{5}^{2} - 16T_{5} + 8 \) Copy content Toggle raw display
\( T_{7}^{4} + 4T_{7}^{3} - 8T_{7}^{2} - 24T_{7} + 4 \) Copy content Toggle raw display
\( T_{11}^{4} - 24T_{11}^{2} + 32T_{11} + 32 \) Copy content Toggle raw display
\( T_{13}^{4} - 8T_{13}^{3} + 4T_{13}^{2} + 48T_{13} + 4 \) Copy content Toggle raw display
\( T_{17}^{4} - 32T_{17}^{2} + 64T_{17} + 16 \) Copy content Toggle raw display
\( T_{19}^{4} - 32T_{19}^{2} - 64T_{19} + 16 \) Copy content Toggle raw display
\( T_{67}^{4} - 16T_{67}^{3} + 256T_{67} + 256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 4 T^{3} - 4 T^{2} - 16 T + 8 \) Copy content Toggle raw display
$7$ \( T^{4} + 4 T^{3} - 8 T^{2} - 24 T + 4 \) Copy content Toggle raw display
$11$ \( T^{4} - 24 T^{2} + 32 T + 32 \) Copy content Toggle raw display
$13$ \( T^{4} - 8 T^{3} + 4 T^{2} + 48 T + 4 \) Copy content Toggle raw display
$17$ \( T^{4} - 32 T^{2} + 64 T + 16 \) Copy content Toggle raw display
$19$ \( T^{4} - 32 T^{2} - 64 T + 16 \) Copy content Toggle raw display
$23$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 12 T^{3} + 28 T^{2} + \cdots - 248 \) Copy content Toggle raw display
$31$ \( T^{4} + 12 T^{3} + 40 T^{2} + 24 T - 28 \) Copy content Toggle raw display
$37$ \( T^{4} - 16 T^{3} + 52 T^{2} + \cdots - 1052 \) Copy content Toggle raw display
$41$ \( T^{4} - 64 T^{2} - 192 T - 112 \) Copy content Toggle raw display
$43$ \( T^{4} - 96 T^{2} - 256 T - 112 \) Copy content Toggle raw display
$47$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 20 T^{3} + 124 T^{2} + \cdots + 136 \) Copy content Toggle raw display
$59$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} - 16 T^{3} + 52 T^{2} + \cdots - 1052 \) Copy content Toggle raw display
$67$ \( T^{4} - 16 T^{3} + 256 T + 256 \) Copy content Toggle raw display
$71$ \( T^{4} + 8 T^{3} - 32 T^{2} - 192 T + 64 \) Copy content Toggle raw display
$73$ \( T^{4} + 8 T^{3} - 96 T^{2} + 64 T + 64 \) Copy content Toggle raw display
$79$ \( T^{4} + 12 T^{3} - 168 T^{2} + \cdots - 10108 \) Copy content Toggle raw display
$83$ \( T^{4} - 216 T^{2} + 160 T + 32 \) Copy content Toggle raw display
$89$ \( T^{4} - 8 T^{3} - 200 T^{2} + \cdots - 1904 \) Copy content Toggle raw display
$97$ \( T^{4} - 224 T^{2} + 768 T + 512 \) Copy content Toggle raw display
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