Properties

Label 3040.2.a.u
Level $3040$
Weight $2$
Character orbit 3040.a
Self dual yes
Analytic conductor $24.275$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

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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: \(5\)
Coefficient field: 5.5.387268.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 7x^{3} + 4x^{2} + 12x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 1) q^{3} - q^{5} + (\beta_{4} + 1) q^{7} + (\beta_{4} - \beta_{3} - \beta_{2} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 1) q^{3} - q^{5} + (\beta_{4} + 1) q^{7} + (\beta_{4} - \beta_{3} - \beta_{2} + 2) q^{9} + ( - \beta_{4} - \beta_{2} + \beta_1) q^{11} + ( - \beta_{4} + \beta_{3} - 1) q^{13} + (\beta_1 + 1) q^{15} + (\beta_{3} + 2 \beta_{2} + \beta_1 - 3) q^{17} + q^{19} + ( - 2 \beta_{4} + \beta_{2} - \beta_1 - 3) q^{21} + ( - \beta_{4} + 2 \beta_{2} + 2 \beta_1 + 1) q^{23} + q^{25} + ( - 2 \beta_{4} + \beta_{3} + 4 \beta_{2} + \cdots - 5) q^{27}+ \cdots + (\beta_{4} - \beta_{2} + 3 \beta_1 + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 4 q^{3} - 5 q^{5} + 4 q^{7} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 4 q^{3} - 5 q^{5} + 4 q^{7} + 7 q^{9} - 2 q^{11} - 4 q^{13} + 4 q^{15} - 12 q^{17} + 5 q^{19} - 10 q^{21} + 8 q^{23} + 5 q^{25} - 16 q^{27} - 6 q^{29} + 10 q^{31} - 18 q^{33} - 4 q^{35} - 6 q^{37} + 18 q^{39} - 8 q^{41} - 12 q^{43} - 7 q^{45} + 16 q^{47} + 7 q^{49} + 14 q^{51} - 18 q^{53} + 2 q^{55} - 4 q^{57} - 8 q^{59} + 2 q^{61} + 36 q^{63} + 4 q^{65} - 10 q^{67} - 22 q^{69} - 18 q^{71} - 28 q^{73} - 4 q^{75} - 28 q^{77} + 14 q^{79} + 25 q^{81} - 8 q^{83} + 12 q^{85} + 24 q^{87} - 30 q^{89} - 28 q^{91} - 24 q^{93} - 5 q^{95} - 18 q^{97} + 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - x^{4} - 7x^{3} + 4x^{2} + 12x - 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - \nu^{2} - 3\nu + 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{3} + \nu^{2} + 5\nu - 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 2\nu^{3} - 3\nu^{2} + 6\nu - 1 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + \beta_{2} + 2\beta _1 + 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} + 3\beta_{2} + \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{4} + 5\beta_{3} + 9\beta_{2} + 10\beta _1 + 24 ) / 2 \) 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
−1.88930
−1.50372
2.42170
1.81079
0.160536
0 −3.45876 0 −1.00000 0 4.18444 0 8.96304 0
1.2 0 −1.76491 0 −1.00000 0 −3.89255 0 0.114895 0
1.3 0 −1.44292 0 −1.00000 0 2.92540 0 −0.917992 0
1.4 0 0.531822 0 −1.00000 0 −0.0955795 0 −2.71717 0
1.5 0 2.13476 0 −1.00000 0 0.878290 0 1.55722 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
\(2\) \( -1 \)
\(5\) \( +1 \)
\(19\) \( -1 \)

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.u 5
4.b odd 2 1 3040.2.a.x yes 5
8.b even 2 1 6080.2.a.cl 5
8.d odd 2 1 6080.2.a.ci 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3040.2.a.u 5 1.a even 1 1 trivial
3040.2.a.x yes 5 4.b odd 2 1
6080.2.a.ci 5 8.d odd 2 1
6080.2.a.cl 5 8.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3040))\):

\( T_{3}^{5} + 4T_{3}^{4} - 3T_{3}^{3} - 20T_{3}^{2} - 8T_{3} + 10 \) Copy content Toggle raw display
\( T_{7}^{5} - 4T_{7}^{4} - 13T_{7}^{3} + 60T_{7}^{2} - 36T_{7} - 4 \) Copy content Toggle raw display
\( T_{11}^{5} + 2T_{11}^{4} - 32T_{11}^{3} - 44T_{11}^{2} + 208T_{11} - 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \) Copy content Toggle raw display
$3$ \( T^{5} + 4 T^{4} + \cdots + 10 \) Copy content Toggle raw display
$5$ \( (T + 1)^{5} \) Copy content Toggle raw display
$7$ \( T^{5} - 4 T^{4} + \cdots - 4 \) Copy content Toggle raw display
$11$ \( T^{5} + 2 T^{4} + \cdots - 64 \) Copy content Toggle raw display
$13$ \( T^{5} + 4 T^{4} + \cdots + 2 \) Copy content Toggle raw display
$17$ \( T^{5} + 12 T^{4} + \cdots - 1192 \) Copy content Toggle raw display
$19$ \( (T - 1)^{5} \) Copy content Toggle raw display
$23$ \( T^{5} - 8 T^{4} + \cdots + 772 \) Copy content Toggle raw display
$29$ \( T^{5} + 6 T^{4} + \cdots + 5480 \) Copy content Toggle raw display
$31$ \( T^{5} - 10 T^{4} + \cdots - 1888 \) Copy content Toggle raw display
$37$ \( T^{5} + 6 T^{4} + \cdots + 2216 \) Copy content Toggle raw display
$41$ \( T^{5} + 8 T^{4} + \cdots + 320 \) Copy content Toggle raw display
$43$ \( T^{5} + 12 T^{4} + \cdots - 1328 \) Copy content Toggle raw display
$47$ \( T^{5} - 16 T^{4} + \cdots + 6592 \) Copy content Toggle raw display
$53$ \( T^{5} + 18 T^{4} + \cdots + 28486 \) Copy content Toggle raw display
$59$ \( T^{5} + 8 T^{4} + \cdots + 392 \) Copy content Toggle raw display
$61$ \( T^{5} - 2 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$67$ \( T^{5} + 10 T^{4} + \cdots - 50 \) Copy content Toggle raw display
$71$ \( T^{5} + 18 T^{4} + \cdots + 22016 \) Copy content Toggle raw display
$73$ \( T^{5} + 28 T^{4} + \cdots - 5480 \) Copy content Toggle raw display
$79$ \( T^{5} - 14 T^{4} + \cdots - 3296 \) Copy content Toggle raw display
$83$ \( T^{5} + 8 T^{4} + \cdots - 5840 \) Copy content Toggle raw display
$89$ \( T^{5} + 30 T^{4} + \cdots - 736 \) Copy content Toggle raw display
$97$ \( T^{5} + 18 T^{4} + \cdots + 100552 \) Copy content Toggle raw display
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