Properties

Label 9240.2.a.cj
Level $9240$
Weight $2$
Character orbit 9240.a
Self dual yes
Analytic conductor $73.782$
Analytic rank $0$
Dimension $5$
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] = [9240,2,Mod(1,9240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9240, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("9240.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9240 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9240.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: \(73.7817714677\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.9229792.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 12x^{3} + 6x^{2} + 24x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} - q^{5} + q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} - q^{5} + q^{7} + q^{9} + q^{11} + ( - \beta_{2} - 1) q^{13} + q^{15} - \beta_{4} q^{17} + ( - \beta_{2} + \beta_1 - 1) q^{19} - q^{21} + (\beta_{3} - \beta_1 + 1) q^{23} + q^{25} - q^{27} + (\beta_{3} - \beta_{2} - \beta_1) q^{29} + ( - \beta_{4} + \beta_1) q^{31} - q^{33} - q^{35} + ( - \beta_{2} + 3) q^{37} + (\beta_{2} + 1) q^{39} + ( - \beta_{3} - 1) q^{41} + (\beta_{3} - \beta_{2} + \beta_1 + 2) q^{43} - q^{45} + ( - \beta_{3} + 1) q^{47} + q^{49} + \beta_{4} q^{51} + (\beta_{2} + \beta_1 - 1) q^{53} - q^{55} + (\beta_{2} - \beta_1 + 1) q^{57} + ( - \beta_{3} + \beta_{2} - \beta_1 - 2) q^{59} + ( - \beta_{2} - \beta_1 + 1) q^{61} + q^{63} + (\beta_{2} + 1) q^{65} + ( - \beta_{3} + \beta_{2}) q^{67} + ( - \beta_{3} + \beta_1 - 1) q^{69} + (\beta_{3} - 2 \beta_1 + 3) q^{71} + (\beta_{4} - \beta_{3} + \beta_1 - 1) q^{73} - q^{75} + q^{77} + (\beta_{4} + \beta_1) q^{79} + q^{81} + (2 \beta_{4} - \beta_{2} + \beta_1 - 1) q^{83} + \beta_{4} q^{85} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{87} + ( - \beta_{4} + \beta_{3} - 1) q^{89} + ( - \beta_{2} - 1) q^{91} + (\beta_{4} - \beta_1) q^{93} + (\beta_{2} - \beta_1 + 1) q^{95} + ( - \beta_{3} + \beta_{2} + \beta_1 + 4) q^{97} + q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 5 q^{3} - 5 q^{5} + 5 q^{7} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 5 q^{3} - 5 q^{5} + 5 q^{7} + 5 q^{9} + 5 q^{11} - 4 q^{13} + 5 q^{15} - 2 q^{19} - 5 q^{21} + 2 q^{23} + 5 q^{25} - 5 q^{27} - 2 q^{29} + 2 q^{31} - 5 q^{33} - 5 q^{35} + 16 q^{37} + 4 q^{39} - 4 q^{41} + 12 q^{43} - 5 q^{45} + 6 q^{47} + 5 q^{49} - 4 q^{53} - 5 q^{55} + 2 q^{57} - 12 q^{59} + 4 q^{61} + 5 q^{63} + 4 q^{65} - 2 q^{69} + 10 q^{71} - 2 q^{73} - 5 q^{75} + 5 q^{77} + 2 q^{79} + 5 q^{81} - 2 q^{83} + 2 q^{87} - 6 q^{89} - 4 q^{91} - 2 q^{93} + 2 q^{95} + 22 q^{97} + 5 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( \nu^{3} - \nu^{2} - 10\nu + 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{4} - \nu^{3} - 10\nu^{2} + 10 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} - 11\nu^{2} - 2\nu + 12 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{4} - \beta_{3} - \beta _1 + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{4} - \beta_{3} + 4\beta_{2} - \beta _1 + 21 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 11\beta_{4} - 11\beta_{3} + 4\beta_{2} - 3\beta _1 + 15 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 21\beta_{4} - 13\beta_{3} + 44\beta_{2} - 13\beta _1 + 185 ) / 4 \) 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
−2.81601
3.44700
−1.57176
1.61637
0.324405
0 −1.00000 0 −1.00000 0 1.00000 0 1.00000 0
1.2 0 −1.00000 0 −1.00000 0 1.00000 0 1.00000 0
1.3 0 −1.00000 0 −1.00000 0 1.00000 0 1.00000 0
1.4 0 −1.00000 0 −1.00000 0 1.00000 0 1.00000 0
1.5 0 −1.00000 0 −1.00000 0 1.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(1\)
\(7\) \(-1\)
\(11\) \(-1\)

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 9240.2.a.cj 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9240.2.a.cj 5 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9240))\):

\( T_{13}^{5} + 4T_{13}^{4} - 38T_{13}^{3} - 76T_{13}^{2} + 376T_{13} + 16 \) Copy content Toggle raw display
\( T_{17}^{5} - 75T_{17}^{3} + 86T_{17}^{2} + 1204T_{17} - 2600 \) Copy content Toggle raw display
\( T_{19}^{5} + 2T_{19}^{4} - 63T_{19}^{3} - 144T_{19}^{2} + 896T_{19} + 2048 \) Copy content Toggle raw display
\( T_{23}^{5} - 2T_{23}^{4} - 79T_{23}^{3} + 124T_{23}^{2} + 936T_{23} + 416 \) Copy content Toggle raw display
\( T_{37}^{5} - 16T_{37}^{4} + 58T_{37}^{3} + 124T_{37}^{2} - 584T_{37} - 272 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \) Copy content Toggle raw display
$3$ \( (T + 1)^{5} \) Copy content Toggle raw display
$5$ \( (T + 1)^{5} \) Copy content Toggle raw display
$7$ \( (T - 1)^{5} \) Copy content Toggle raw display
$11$ \( (T - 1)^{5} \) Copy content Toggle raw display
$13$ \( T^{5} + 4 T^{4} - 38 T^{3} - 76 T^{2} + \cdots + 16 \) Copy content Toggle raw display
$17$ \( T^{5} - 75 T^{3} + 86 T^{2} + \cdots - 2600 \) Copy content Toggle raw display
$19$ \( T^{5} + 2 T^{4} - 63 T^{3} + \cdots + 2048 \) Copy content Toggle raw display
$23$ \( T^{5} - 2 T^{4} - 79 T^{3} + 124 T^{2} + \cdots + 416 \) Copy content Toggle raw display
$29$ \( T^{5} + 2 T^{4} - 103 T^{3} + \cdots - 1688 \) Copy content Toggle raw display
$31$ \( T^{5} - 2 T^{4} - 102 T^{3} + \cdots - 2176 \) Copy content Toggle raw display
$37$ \( T^{5} - 16 T^{4} + 58 T^{3} + \cdots - 272 \) Copy content Toggle raw display
$41$ \( T^{5} + 4 T^{4} - 70 T^{3} + \cdots - 1264 \) Copy content Toggle raw display
$43$ \( T^{5} - 12 T^{4} - 79 T^{3} + \cdots - 17536 \) Copy content Toggle raw display
$47$ \( T^{5} - 6 T^{4} - 62 T^{3} + \cdots - 3328 \) Copy content Toggle raw display
$53$ \( T^{5} + 4 T^{4} - 91 T^{3} + \cdots - 2824 \) Copy content Toggle raw display
$59$ \( T^{5} + 12 T^{4} - 79 T^{3} + \cdots + 17536 \) Copy content Toggle raw display
$61$ \( T^{5} - 4 T^{4} - 91 T^{3} + \cdots + 2824 \) Copy content Toggle raw display
$67$ \( T^{5} - 84 T^{3} + 144 T^{2} + \cdots - 640 \) Copy content Toggle raw display
$71$ \( T^{5} - 10 T^{4} - 118 T^{3} + \cdots + 512 \) Copy content Toggle raw display
$73$ \( T^{5} + 2 T^{4} - 148 T^{3} - 152 T^{2} + \cdots + 64 \) Copy content Toggle raw display
$79$ \( T^{5} - 2 T^{4} - 118 T^{3} + \cdots - 4096 \) Copy content Toggle raw display
$83$ \( T^{5} + 2 T^{4} - 359 T^{3} + \cdots - 21184 \) Copy content Toggle raw display
$89$ \( T^{5} + 6 T^{4} - 123 T^{3} + \cdots - 104 \) Copy content Toggle raw display
$97$ \( T^{5} - 22 T^{4} + 89 T^{3} + \cdots + 2888 \) Copy content Toggle raw display
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