Properties

Label 3040.2.a.w
Level $3040$
Weight $2$
Character orbit 3040.a
Self dual yes
Analytic conductor $24.275$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.2363492.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 11x^{3} - 6x^{2} + 14x + 10 \) 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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + q^{5} - \beta_{2} q^{7} + ( - \beta_{3} + 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + q^{5} - \beta_{2} q^{7} + ( - \beta_{3} + 4) q^{9} + ( - \beta_{4} + \beta_{3} + 2) q^{11} + ( - \beta_{4} - \beta_{2} + \beta_1 + 1) q^{13} - \beta_1 q^{15} + ( - \beta_{4} + 1) q^{17} + q^{19} + (3 \beta_{4} - \beta_{3} - \beta_{2} + 2) q^{21} - \beta_{2} q^{23} + q^{25} + (\beta_{4} - 2 \beta_{3} - 2 \beta_1 + 1) q^{27} + ( - \beta_{4} - \beta_{3} + \beta_{2} + 2) q^{29} + ( - 2 \beta_{4} + 2 \beta_{3} - 2) q^{31} + (2 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{33} - \beta_{2} q^{35} + ( - 2 \beta_{4} + \beta_{3} + \beta_1 + 1) q^{37} + (4 \beta_{4} + \beta_{2} - 2 \beta_1 - 4) q^{39} + (2 \beta_{3} + 2) q^{41} + ( - \beta_{4} - \beta_{3}) q^{43} + ( - \beta_{3} + 4) q^{45} + ( - \beta_{4} + \beta_{3} + 2 \beta_{2} + \cdots - 6) q^{47}+ \cdots + ( - 5 \beta_{4} + \beta_{3} + 2 \beta_{2} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 2 q^{3} + 5 q^{5} - 2 q^{7} + 19 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 2 q^{3} + 5 q^{5} - 2 q^{7} + 19 q^{9} + 10 q^{11} + 2 q^{15} + 4 q^{17} + 5 q^{19} + 10 q^{21} - 2 q^{23} + 5 q^{25} + 8 q^{27} + 10 q^{29} - 10 q^{31} + 10 q^{33} - 2 q^{35} + 2 q^{37} - 10 q^{39} + 12 q^{41} - 2 q^{43} + 19 q^{45} - 22 q^{47} + 19 q^{49} + 14 q^{51} - 18 q^{53} + 10 q^{55} + 2 q^{57} + 4 q^{59} + 6 q^{61} - 10 q^{63} + 20 q^{67} + 10 q^{69} - 2 q^{71} - 12 q^{73} + 2 q^{75} - 4 q^{77} - 14 q^{79} + 13 q^{81} - 14 q^{83} + 4 q^{85} + 12 q^{87} + 22 q^{89} + 40 q^{91} + 8 q^{93} + 5 q^{95} - 10 q^{97} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - x^{4} - 11x^{3} - 6x^{2} + 14x + 10 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{4} - 3\nu^{3} - 6\nu^{2} + 7\nu + 6 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{4} + 2\nu^{3} + 9\nu^{2} - 4\nu - 10 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{4} - 2\nu^{3} - 9\nu^{2} + 2\nu + 11 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( 2\nu^{4} - 5\nu^{3} - 14\nu^{2} + 8\nu + 13 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{3} - \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{4} - 3\beta_{3} - \beta_{2} - 2\beta _1 + 9 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{4} - 6\beta_{3} - 4\beta_{2} - 4\beta _1 + 11 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 30\beta_{4} - 47\beta_{3} - 23\beta_{2} - 34\beta _1 + 101 ) / 2 \) 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
1.19018
−1.82337
−0.778695
−1.50919
3.92107
0 −2.78093 0 1.00000 0 0.646760 0 4.73359 0
1.2 0 −2.52807 0 1.00000 0 −4.03789 0 3.39116 0
1.3 0 1.30486 0 1.00000 0 2.73996 0 −1.29735 0
1.4 0 2.73033 0 1.00000 0 −4.47310 0 4.45473 0
1.5 0 3.27382 0 1.00000 0 3.12428 0 7.71786 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.w yes 5
4.b odd 2 1 3040.2.a.v 5
8.b even 2 1 6080.2.a.cj 5
8.d odd 2 1 6080.2.a.ck 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3040.2.a.v 5 4.b odd 2 1
3040.2.a.w yes 5 1.a even 1 1 trivial
6080.2.a.cj 5 8.b even 2 1
6080.2.a.ck 5 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3040))\):

\( T_{3}^{5} - 2T_{3}^{4} - 15T_{3}^{3} + 26T_{3}^{2} + 56T_{3} - 82 \) Copy content Toggle raw display
\( T_{7}^{5} + 2T_{7}^{4} - 25T_{7}^{3} - 18T_{7}^{2} + 176T_{7} - 100 \) Copy content Toggle raw display
\( T_{11}^{5} - 10T_{11}^{4} + 12T_{11}^{3} + 100T_{11}^{2} - 112T_{11} - 320 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \) Copy content Toggle raw display
$3$ \( T^{5} - 2 T^{4} + \cdots - 82 \) Copy content Toggle raw display
$5$ \( (T - 1)^{5} \) Copy content Toggle raw display
$7$ \( T^{5} + 2 T^{4} + \cdots - 100 \) Copy content Toggle raw display
$11$ \( T^{5} - 10 T^{4} + \cdots - 320 \) Copy content Toggle raw display
$13$ \( T^{5} - 35 T^{3} + \cdots - 34 \) Copy content Toggle raw display
$17$ \( T^{5} - 4 T^{4} + \cdots - 8 \) Copy content Toggle raw display
$19$ \( (T - 1)^{5} \) Copy content Toggle raw display
$23$ \( T^{5} + 2 T^{4} + \cdots - 100 \) Copy content Toggle raw display
$29$ \( T^{5} - 10 T^{4} + \cdots + 712 \) Copy content Toggle raw display
$31$ \( T^{5} + 10 T^{4} + \cdots + 32 \) Copy content Toggle raw display
$37$ \( T^{5} - 2 T^{4} + \cdots - 2056 \) Copy content Toggle raw display
$41$ \( T^{5} - 12 T^{4} + \cdots + 128 \) Copy content Toggle raw display
$43$ \( T^{5} + 2 T^{4} + \cdots - 1008 \) Copy content Toggle raw display
$47$ \( T^{5} + 22 T^{4} + \cdots - 43840 \) Copy content Toggle raw display
$53$ \( T^{5} + 18 T^{4} + \cdots + 2 \) Copy content Toggle raw display
$59$ \( T^{5} - 4 T^{4} + \cdots + 30200 \) Copy content Toggle raw display
$61$ \( T^{5} - 6 T^{4} + \cdots - 2816 \) Copy content Toggle raw display
$67$ \( T^{5} - 20 T^{4} + \cdots - 81334 \) Copy content Toggle raw display
$71$ \( T^{5} + 2 T^{4} + \cdots - 128 \) Copy content Toggle raw display
$73$ \( T^{5} + 12 T^{4} + \cdots - 136 \) Copy content Toggle raw display
$79$ \( T^{5} + 14 T^{4} + \cdots + 215776 \) Copy content Toggle raw display
$83$ \( T^{5} + 14 T^{4} + \cdots + 417232 \) Copy content Toggle raw display
$89$ \( T^{5} - 22 T^{4} + \cdots + 7264 \) Copy content Toggle raw display
$97$ \( T^{5} + 10 T^{4} + \cdots + 3160 \) Copy content Toggle raw display
show more
show less