↑

Properties

Label 3240.2.a.s
Level $3240$
Weight $2$
Character orbit 3240.a
Self dual yes
Analytic conductor $25.872$
Analytic rank $1$
Dimension $4$
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] = [3240,2,Mod(1,3240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3240, 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("3240.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3240 = 2^{3} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3240.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: \(25.8715302549\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.29268.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 9x^{2} + 5x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 360)
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.

\(f(q)\) \(=\) \( q - q^{5} - \beta_{2} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{5} - \beta_{2} q^{7} + (\beta_{3} + \beta_{2}) q^{11} + (\beta_1 + 1) q^{13} + ( - \beta_{3} + \beta_{2} - 2) q^{17} + ( - \beta_{3} - \beta_{2} - 2 \beta_1) q^{19} + (\beta_{2} - 2) q^{23} + q^{25} + ( - \beta_{3} - \beta_{2} - \beta_1 - 2) q^{29} + (2 \beta_{2} + \beta_1 - 1) q^{31} + \beta_{2} q^{35} + (2 \beta_{2} + \beta_1 + 1) q^{37} + (2 \beta_{3} + \beta_1 - 2) q^{41} + (\beta_{3} - \beta_{2} - 2) q^{43} + ( - 3 \beta_{2} - \beta_1 - 1) q^{47} + ( - \beta_{3} + \beta_{2} - 2 \beta_1) q^{49} + ( - 2 \beta_{3} - 4) q^{53} + ( - \beta_{3} - \beta_{2}) q^{55} + ( - \beta_{3} - \beta_{2} + \beta_1 - 3) q^{59} + (\beta_{3} + \beta_{2} + 5) q^{61} + ( - \beta_1 - 1) q^{65} + (\beta_{3} + 2 \beta_1 - 2) q^{67} + ( - \beta_1 - 3) q^{71} + (\beta_{3} - \beta_{2} - \beta_1 + 3) q^{73} + (2 \beta_{3} + 3 \beta_1 - 7) q^{77} + (4 \beta_{3} + 2 \beta_1 - 4) q^{79} + (\beta_{2} + 2 \beta_1 - 6) q^{83} + (\beta_{3} - \beta_{2} + 2) q^{85} + (3 \beta_{3} + \beta_{2} + \beta_1 - 4) q^{89} + (2 \beta_{2} - \beta_1 + 3) q^{91} + (\beta_{3} + \beta_{2} + 2 \beta_1) q^{95} + ( - \beta_{3} + 3 \beta_{2} - \beta_1 - 1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{5} - q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{5} - q^{7} - q^{11} + 4 q^{13} - 5 q^{17} + q^{19} - 7 q^{23} + 4 q^{25} - 7 q^{29} - 2 q^{31} + q^{35} + 6 q^{37} - 12 q^{41} - 11 q^{43} - 7 q^{47} + 3 q^{49} - 12 q^{53} + q^{55} - 11 q^{59} + 19 q^{61} - 4 q^{65} - 10 q^{67} - 12 q^{71} + 9 q^{73} - 32 q^{77} - 24 q^{79} - 23 q^{83} + 5 q^{85} - 21 q^{89} + 14 q^{91} - q^{95} + q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 9x^{2} + 5x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} + \nu - 5 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 2\nu^{2} - 5\nu + 8 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} - 2\nu^{2} + 7\nu + 10 ) / 2 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} + 2\beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{3} - \beta_{2} + \beta _1 + 14 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 3\beta_{2} + 4\beta _1 + 3 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
−1.25548
2.85121
1.83719
−2.43292
0 0 0 −1.00000 0 −4.57301 0 0 0
1.2 0 0 0 −1.00000 0 −0.331895 0 0 0
1.3 0 0 0 −1.00000 0 0.867736 0 0 0
1.4 0 0 0 −1.00000 0 3.03717 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 3240.2.a.s 4
3.b odd 2 1 3240.2.a.u 4
4.b odd 2 1 6480.2.a.bz 4
9.c even 3 2 1080.2.q.e 8
9.d odd 6 2 360.2.q.e ✓ 8
12.b even 2 1 6480.2.a.cb 4
36.f odd 6 2 2160.2.q.l 8
36.h even 6 2 720.2.q.l 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.q.e ✓ 8 9.d odd 6 2
720.2.q.l 8 36.h even 6 2
1080.2.q.e 8 9.c even 3 2
2160.2.q.l 8 36.f odd 6 2
3240.2.a.s 4 1.a even 1 1 trivial
3240.2.a.u 4 3.b odd 2 1
6480.2.a.bz 4 4.b odd 2 1
6480.2.a.cb 4 12.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3240))\):

\( T_{7}^{4} + T_{7}^{3} - 15T_{7}^{2} + 7T_{7} + 4 \) Copy content Toggle raw display
\( T_{11}^{4} + T_{11}^{3} - 42T_{11}^{2} - 20T_{11} + 436 \) Copy content Toggle raw display
\( T_{17}^{4} + 5T_{17}^{3} - 30T_{17}^{2} - 40T_{17} + 172 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + T^{3} - 15 T^{2} + \cdots + 4 \) Copy content Toggle raw display
$11$ \( T^{4} + T^{3} + \cdots + 436 \) Copy content Toggle raw display
$13$ \( T^{4} - 4 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$17$ \( T^{4} + 5 T^{3} + \cdots + 172 \) Copy content Toggle raw display
$19$ \( T^{4} - T^{3} + \cdots + 1348 \) Copy content Toggle raw display
$23$ \( T^{4} + 7 T^{3} + \cdots - 62 \) Copy content Toggle raw display
$29$ \( T^{4} + 7 T^{3} + \cdots - 206 \) Copy content Toggle raw display
$31$ \( T^{4} + 2 T^{3} + \cdots + 424 \) Copy content Toggle raw display
$37$ \( T^{4} - 6 T^{3} + \cdots + 144 \) Copy content Toggle raw display
$41$ \( T^{4} + 12 T^{3} + \cdots - 2367 \) Copy content Toggle raw display
$43$ \( T^{4} + 11 T^{3} + \cdots - 212 \) Copy content Toggle raw display
$47$ \( T^{4} + 7 T^{3} + \cdots + 1072 \) Copy content Toggle raw display
$53$ \( T^{4} + 12 T^{3} + \cdots + 144 \) Copy content Toggle raw display
$59$ \( T^{4} + 11 T^{3} + \cdots + 472 \) Copy content Toggle raw display
$61$ \( T^{4} - 19 T^{3} + \cdots - 14 \) Copy content Toggle raw display
$67$ \( T^{4} + 10 T^{3} + \cdots + 1453 \) Copy content Toggle raw display
$71$ \( T^{4} + 12 T^{3} + \cdots - 72 \) Copy content Toggle raw display
$73$ \( T^{4} - 9 T^{3} + \cdots - 1152 \) Copy content Toggle raw display
$79$ \( T^{4} + 24 T^{3} + \cdots - 37872 \) Copy content Toggle raw display
$83$ \( T^{4} + 23 T^{3} + \cdots - 5276 \) Copy content Toggle raw display
$89$ \( T^{4} + 21 T^{3} + \cdots - 7506 \) Copy content Toggle raw display
$97$ \( T^{4} - T^{3} + \cdots - 1208 \) Copy content Toggle raw display
show more
show less