Properties

Label 9408.2.a.do
Level $9408$
Weight $2$
Character orbit 9408.a
Self dual yes
Analytic conductor $75.123$
Analytic rank $1$
Dimension $2$
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] = [9408,2,Mod(1,9408)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9408, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("9408.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9408 = 2^{6} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9408.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: \(75.1232582216\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
comment: defining polynomial
 
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: \( 2 \)
Twist minimal: no (minimal twist has level 672)
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 \(\beta = 2\sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + \beta q^{5} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} + \beta q^{5} + q^{9} + ( - \beta + 2) q^{11} + 2 q^{13} - \beta q^{15} + (\beta - 4) q^{17} - 2 \beta q^{19} + (\beta - 2) q^{23} + 7 q^{25} - q^{27} + (2 \beta - 2) q^{29} + ( - 2 \beta - 4) q^{31} + (\beta - 2) q^{33} + 2 q^{37} - 2 q^{39} + ( - \beta - 8) q^{41} - 8 q^{43} + \beta q^{45} + (2 \beta + 4) q^{47} + ( - \beta + 4) q^{51} + 2 q^{53} + (2 \beta - 12) q^{55} + 2 \beta q^{57} + ( - 2 \beta + 8) q^{59} + ( - 2 \beta - 2) q^{61} + 2 \beta q^{65} + ( - 2 \beta + 4) q^{67} + ( - \beta + 2) q^{69} + ( - \beta - 6) q^{71} + ( - 2 \beta - 6) q^{73} - 7 q^{75} + (2 \beta + 4) q^{79} + q^{81} + 4 q^{83} + ( - 4 \beta + 12) q^{85} + ( - 2 \beta + 2) q^{87} - \beta q^{89} + (2 \beta + 4) q^{93} - 24 q^{95} + (2 \beta + 2) q^{97} + ( - \beta + 2) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 2 q^{9} + 4 q^{11} + 4 q^{13} - 8 q^{17} - 4 q^{23} + 14 q^{25} - 2 q^{27} - 4 q^{29} - 8 q^{31} - 4 q^{33} + 4 q^{37} - 4 q^{39} - 16 q^{41} - 16 q^{43} + 8 q^{47} + 8 q^{51} + 4 q^{53} - 24 q^{55} + 16 q^{59} - 4 q^{61} + 8 q^{67} + 4 q^{69} - 12 q^{71} - 12 q^{73} - 14 q^{75} + 8 q^{79} + 2 q^{81} + 8 q^{83} + 24 q^{85} + 4 q^{87} + 8 q^{93} - 48 q^{95} + 4 q^{97} + 4 q^{99}+O(q^{100}) \) 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
−1.73205
1.73205
0 −1.00000 0 −3.46410 0 0 0 1.00000 0
1.2 0 −1.00000 0 3.46410 0 0 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(7\) \(-1\)

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 9408.2.a.do 2
4.b odd 2 1 9408.2.a.dx 2
7.b odd 2 1 1344.2.a.v 2
8.b even 2 1 4704.2.a.bn 2
8.d odd 2 1 4704.2.a.bm 2
21.c even 2 1 4032.2.a.bs 2
28.d even 2 1 1344.2.a.u 2
56.e even 2 1 672.2.a.j yes 2
56.h odd 2 1 672.2.a.i 2
84.h odd 2 1 4032.2.a.br 2
112.j even 4 2 5376.2.c.bn 4
112.l odd 4 2 5376.2.c.bh 4
168.e odd 2 1 2016.2.a.s 2
168.i even 2 1 2016.2.a.t 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.2.a.i 2 56.h odd 2 1
672.2.a.j yes 2 56.e even 2 1
1344.2.a.u 2 28.d even 2 1
1344.2.a.v 2 7.b odd 2 1
2016.2.a.s 2 168.e odd 2 1
2016.2.a.t 2 168.i even 2 1
4032.2.a.br 2 84.h odd 2 1
4032.2.a.bs 2 21.c even 2 1
4704.2.a.bm 2 8.d odd 2 1
4704.2.a.bn 2 8.b even 2 1
5376.2.c.bh 4 112.l odd 4 2
5376.2.c.bn 4 112.j even 4 2
9408.2.a.do 2 1.a even 1 1 trivial
9408.2.a.dx 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9408))\):

\( T_{5}^{2} - 12 \) Copy content Toggle raw display
\( T_{11}^{2} - 4T_{11} - 8 \) Copy content Toggle raw display
\( T_{13} - 2 \) Copy content Toggle raw display
\( T_{17}^{2} + 8T_{17} + 4 \) Copy content Toggle raw display
\( T_{19}^{2} - 48 \) Copy content Toggle raw display
\( T_{31}^{2} + 8T_{31} - 32 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 12 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 4T - 8 \) Copy content Toggle raw display
$13$ \( (T - 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 8T + 4 \) Copy content Toggle raw display
$19$ \( T^{2} - 48 \) Copy content Toggle raw display
$23$ \( T^{2} + 4T - 8 \) Copy content Toggle raw display
$29$ \( T^{2} + 4T - 44 \) Copy content Toggle raw display
$31$ \( T^{2} + 8T - 32 \) Copy content Toggle raw display
$37$ \( (T - 2)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 16T + 52 \) Copy content Toggle raw display
$43$ \( (T + 8)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 8T - 32 \) Copy content Toggle raw display
$53$ \( (T - 2)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 16T + 16 \) Copy content Toggle raw display
$61$ \( T^{2} + 4T - 44 \) Copy content Toggle raw display
$67$ \( T^{2} - 8T - 32 \) Copy content Toggle raw display
$71$ \( T^{2} + 12T + 24 \) Copy content Toggle raw display
$73$ \( T^{2} + 12T - 12 \) Copy content Toggle raw display
$79$ \( T^{2} - 8T - 32 \) Copy content Toggle raw display
$83$ \( (T - 4)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 12 \) Copy content Toggle raw display
$97$ \( T^{2} - 4T - 44 \) Copy content Toggle raw display
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