Properties

Label 3040.2.a.s
Level $3040$
Weight $2$
Character orbit 3040.a
Self dual yes
Analytic conductor $24.275$
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] = [3040,2,Mod(1,3040)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3040, 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("3040.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3040 = 2^{5} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3040.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: \(24.2745222145\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.78292.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 10x^{2} + 8x + 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - q^{5} + ( - \beta_{2} - 1) q^{7} + (\beta_{2} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - q^{5} + ( - \beta_{2} - 1) q^{7} + (\beta_{2} + 2) q^{9} + ( - \beta_{3} - \beta_1 - 1) q^{11} + (\beta_{3} + 1) q^{13} - \beta_1 q^{15} + (\beta_{3} + \beta_{2} + \beta_1 - 2) q^{17} + q^{19} + ( - \beta_{3} + \beta_{2} - 3 \beta_1) q^{21} + ( - \beta_{2} - 5) q^{23} + q^{25} + (\beta_{3} - \beta_{2} + \beta_1) q^{27} + (\beta_{3} - \beta_{2} - \beta_1) q^{29} + ( - 2 \beta_{3} - 2 \beta_1) q^{31} + ( - 2 \beta_{3} - 2 \beta_{2} + \cdots - 2) q^{33}+ \cdots + ( - 3 \beta_{3} - 2 \beta_{2} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{3} - 4 q^{5} - 5 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{3} - 4 q^{5} - 5 q^{7} + 9 q^{9} - 6 q^{11} + 5 q^{13} - q^{15} - 5 q^{17} + 4 q^{19} - 3 q^{21} - 21 q^{23} + 4 q^{25} + q^{27} - q^{29} - 4 q^{31} - 14 q^{33} + 5 q^{35} + 10 q^{37} - 7 q^{39} - 2 q^{41} + 6 q^{43} - 9 q^{45} - 24 q^{47} + 5 q^{49} + 13 q^{51} - 5 q^{53} + 6 q^{55} + q^{57} - 11 q^{59} - 6 q^{61} - 38 q^{63} - 5 q^{65} + 19 q^{67} - 7 q^{69} - 2 q^{71} + 5 q^{73} + q^{75} + 8 q^{77} + 4 q^{79} - 16 q^{81} - 10 q^{83} + 5 q^{85} - 31 q^{87} + 20 q^{89} - 5 q^{91} - 26 q^{93} - 4 q^{95} - 6 q^{97} - 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 10x^{2} + 8x + 18 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + \nu^{2} - 7\nu - 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - \beta_{2} + 7\beta_1 \) 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
−2.78678
−1.09502
2.19719
2.68461
0 −2.78678 0 −1.00000 0 −3.76616 0 4.76616 0
1.2 0 −1.09502 0 −1.00000 0 2.80094 0 −1.80094 0
1.3 0 2.19719 0 −1.00000 0 −0.827631 0 1.82763 0
1.4 0 2.68461 0 −1.00000 0 −3.20715 0 4.20715 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(5\) \( +1 \)
\(19\) \( -1 \)

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 3040.2.a.s yes 4
4.b odd 2 1 3040.2.a.q 4
8.b even 2 1 6080.2.a.ce 4
8.d odd 2 1 6080.2.a.cg 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3040.2.a.q 4 4.b odd 2 1
3040.2.a.s yes 4 1.a even 1 1 trivial
6080.2.a.ce 4 8.b even 2 1
6080.2.a.cg 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3040))\):

\( T_{3}^{4} - T_{3}^{3} - 10T_{3}^{2} + 8T_{3} + 18 \) Copy content Toggle raw display
\( T_{7}^{4} + 5T_{7}^{3} - 4T_{7}^{2} - 40T_{7} - 28 \) Copy content Toggle raw display
\( T_{11}^{4} + 6T_{11}^{3} - 8T_{11}^{2} - 28T_{11} + 32 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - T^{3} + \cdots + 18 \) Copy content Toggle raw display
$5$ \( (T + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 5 T^{3} + \cdots - 28 \) Copy content Toggle raw display
$11$ \( T^{4} + 6 T^{3} + \cdots + 32 \) Copy content Toggle raw display
$13$ \( T^{4} - 5 T^{3} + \cdots - 86 \) Copy content Toggle raw display
$17$ \( T^{4} + 5 T^{3} + \cdots - 168 \) Copy content Toggle raw display
$19$ \( (T - 1)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + 21 T^{3} + \cdots + 324 \) Copy content Toggle raw display
$29$ \( T^{4} + T^{3} + \cdots + 72 \) Copy content Toggle raw display
$31$ \( T^{4} + 4 T^{3} + \cdots + 752 \) Copy content Toggle raw display
$37$ \( T^{4} - 10 T^{3} + \cdots - 44 \) Copy content Toggle raw display
$41$ \( T^{4} + 2 T^{3} + \cdots + 3456 \) Copy content Toggle raw display
$43$ \( T^{4} - 6 T^{3} + \cdots + 1048 \) Copy content Toggle raw display
$47$ \( T^{4} + 24 T^{3} + \cdots - 1376 \) Copy content Toggle raw display
$53$ \( T^{4} + 5 T^{3} + \cdots + 726 \) Copy content Toggle raw display
$59$ \( T^{4} + 11 T^{3} + \cdots + 2312 \) Copy content Toggle raw display
$61$ \( T^{4} + 6 T^{3} + \cdots + 32 \) Copy content Toggle raw display
$67$ \( T^{4} - 19 T^{3} + \cdots - 1778 \) Copy content Toggle raw display
$71$ \( T^{4} + 2 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$73$ \( T^{4} - 5 T^{3} + \cdots - 872 \) Copy content Toggle raw display
$79$ \( T^{4} - 4 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$83$ \( T^{4} + 10 T^{3} + \cdots + 648 \) Copy content Toggle raw display
$89$ \( T^{4} - 20 T^{3} + \cdots + 432 \) Copy content Toggle raw display
$97$ \( T^{4} + 6 T^{3} + \cdots + 3108 \) Copy content Toggle raw display
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