Properties

Label 4680.2.a.bc
Level $4680$
Weight $2$
Character orbit 4680.a
Self dual yes
Analytic conductor $37.370$
Analytic rank $1$
Dimension $2$
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] = [4680,2,Mod(1,4680)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4680, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("4680.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4680 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4680.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: \(37.3699881460\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 520)
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 \(\beta = \sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{5} + ( - \beta - 3) q^{11} - q^{13} - 2 q^{17} + (\beta + 3) q^{19} + ( - \beta + 1) q^{23} + q^{25} + (2 \beta - 4) q^{29} + (3 \beta + 1) q^{31} + (2 \beta - 4) q^{37} + ( - 4 \beta + 2) q^{41} + (\beta - 1) q^{43} + (4 \beta - 4) q^{47} - 7 q^{49} + ( - 4 \beta + 2) q^{53} + ( - \beta - 3) q^{55} + ( - 3 \beta - 1) q^{59} + 2 \beta q^{61} - q^{65} + ( - 2 \beta + 6) q^{67} + (\beta + 3) q^{71} + ( - 2 \beta - 12) q^{73} + ( - 2 \beta + 2) q^{79} - 4 q^{83} - 2 q^{85} - 10 q^{89} + (\beta + 3) q^{95} - 10 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{5} - 6 q^{11} - 2 q^{13} - 4 q^{17} + 6 q^{19} + 2 q^{23} + 2 q^{25} - 8 q^{29} + 2 q^{31} - 8 q^{37} + 4 q^{41} - 2 q^{43} - 8 q^{47} - 14 q^{49} + 4 q^{53} - 6 q^{55} - 2 q^{59} - 2 q^{65} + 12 q^{67} + 6 q^{71} - 24 q^{73} + 4 q^{79} - 8 q^{83} - 4 q^{85} - 20 q^{89} + 6 q^{95} - 20 q^{97}+O(q^{100}) \) 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
1.61803
−0.618034
0 0 0 1.00000 0 0 0 0 0
1.2 0 0 0 1.00000 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(5\) \(-1\)
\(13\) \(1\)

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 4680.2.a.bc 2
3.b odd 2 1 520.2.a.g 2
4.b odd 2 1 9360.2.a.cs 2
12.b even 2 1 1040.2.a.i 2
15.d odd 2 1 2600.2.a.o 2
15.e even 4 2 2600.2.d.j 4
24.f even 2 1 4160.2.a.bm 2
24.h odd 2 1 4160.2.a.x 2
39.d odd 2 1 6760.2.a.t 2
60.h even 2 1 5200.2.a.bz 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
520.2.a.g 2 3.b odd 2 1
1040.2.a.i 2 12.b even 2 1
2600.2.a.o 2 15.d odd 2 1
2600.2.d.j 4 15.e even 4 2
4160.2.a.x 2 24.h odd 2 1
4160.2.a.bm 2 24.f even 2 1
4680.2.a.bc 2 1.a even 1 1 trivial
5200.2.a.bz 2 60.h even 2 1
6760.2.a.t 2 39.d odd 2 1
9360.2.a.cs 2 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4680))\):

\( T_{7} \) Copy content Toggle raw display
\( T_{11}^{2} + 6T_{11} + 4 \) Copy content Toggle raw display
\( T_{17} + 2 \) Copy content Toggle raw display
\( T_{19}^{2} - 6T_{19} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 6T + 4 \) Copy content Toggle raw display
$13$ \( (T + 1)^{2} \) Copy content Toggle raw display
$17$ \( (T + 2)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 6T + 4 \) Copy content Toggle raw display
$23$ \( T^{2} - 2T - 4 \) Copy content Toggle raw display
$29$ \( T^{2} + 8T - 4 \) Copy content Toggle raw display
$31$ \( T^{2} - 2T - 44 \) Copy content Toggle raw display
$37$ \( T^{2} + 8T - 4 \) Copy content Toggle raw display
$41$ \( T^{2} - 4T - 76 \) Copy content Toggle raw display
$43$ \( T^{2} + 2T - 4 \) Copy content Toggle raw display
$47$ \( T^{2} + 8T - 64 \) Copy content Toggle raw display
$53$ \( T^{2} - 4T - 76 \) Copy content Toggle raw display
$59$ \( T^{2} + 2T - 44 \) Copy content Toggle raw display
$61$ \( T^{2} - 20 \) Copy content Toggle raw display
$67$ \( T^{2} - 12T + 16 \) Copy content Toggle raw display
$71$ \( T^{2} - 6T + 4 \) Copy content Toggle raw display
$73$ \( T^{2} + 24T + 124 \) Copy content Toggle raw display
$79$ \( T^{2} - 4T - 16 \) Copy content Toggle raw display
$83$ \( (T + 4)^{2} \) Copy content Toggle raw display
$89$ \( (T + 10)^{2} \) Copy content Toggle raw display
$97$ \( (T + 10)^{2} \) Copy content Toggle raw display
show more
show less