Properties

Label 4680.2.a.z
Level $4680$
Weight $2$
Character orbit 4680.a
Self dual yes
Analytic conductor $37.370$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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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: \(\Q(\sqrt{6}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
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 \(\beta = \sqrt{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{5} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{5} + 2 q^{7} + ( - \beta + 2) q^{11} - q^{13} + (2 \beta - 2) q^{17} + ( - \beta - 2) q^{19} + (\beta - 4) q^{23} + q^{25} - 4 q^{29} + (\beta - 2) q^{31} - 2 q^{35} - 2 \beta q^{37} + (2 \beta - 6) q^{41} + ( - 3 \beta + 4) q^{43} + 2 q^{47} - 3 q^{49} + ( - 2 \beta - 6) q^{53} + (\beta - 2) q^{55} + (3 \beta - 2) q^{59} + (4 \beta + 4) q^{61} + q^{65} + ( - 2 \beta - 10) q^{67} + (\beta + 6) q^{71} + 6 \beta q^{73} + ( - 2 \beta + 4) q^{77} - 2 \beta q^{79} - 2 q^{83} + ( - 2 \beta + 2) q^{85} - 6 q^{89} - 2 q^{91} + (\beta + 2) q^{95} + ( - 4 \beta - 2) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{5} + 4 q^{7} + 4 q^{11} - 2 q^{13} - 4 q^{17} - 4 q^{19} - 8 q^{23} + 2 q^{25} - 8 q^{29} - 4 q^{31} - 4 q^{35} - 12 q^{41} + 8 q^{43} + 4 q^{47} - 6 q^{49} - 12 q^{53} - 4 q^{55} - 4 q^{59} + 8 q^{61} + 2 q^{65} - 20 q^{67} + 12 q^{71} + 8 q^{77} - 4 q^{83} + 4 q^{85} - 12 q^{89} - 4 q^{91} + 4 q^{95} - 4 q^{97}+O(q^{100}) \) 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
2.44949
−2.44949
0 0 0 −1.00000 0 2.00000 0 0 0
1.2 0 0 0 −1.00000 0 2.00000 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.z 2
3.b odd 2 1 520.2.a.f 2
4.b odd 2 1 9360.2.a.cc 2
12.b even 2 1 1040.2.a.k 2
15.d odd 2 1 2600.2.a.q 2
15.e even 4 2 2600.2.d.h 4
24.f even 2 1 4160.2.a.ba 2
24.h odd 2 1 4160.2.a.bg 2
39.d odd 2 1 6760.2.a.s 2
60.h even 2 1 5200.2.a.bv 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
520.2.a.f 2 3.b odd 2 1
1040.2.a.k 2 12.b even 2 1
2600.2.a.q 2 15.d odd 2 1
2600.2.d.h 4 15.e even 4 2
4160.2.a.ba 2 24.f even 2 1
4160.2.a.bg 2 24.h odd 2 1
4680.2.a.z 2 1.a even 1 1 trivial
5200.2.a.bv 2 60.h even 2 1
6760.2.a.s 2 39.d odd 2 1
9360.2.a.cc 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4680))\):

\( T_{7} - 2 \) Copy content Toggle raw display
\( T_{11}^{2} - 4T_{11} - 2 \) Copy content Toggle raw display
\( T_{17}^{2} + 4T_{17} - 20 \) Copy content Toggle raw display
\( T_{19}^{2} + 4T_{19} - 2 \) 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
$7$ \( (T - 2)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 4T - 2 \) Copy content Toggle raw display
$13$ \( (T + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 4T - 20 \) Copy content Toggle raw display
$19$ \( T^{2} + 4T - 2 \) Copy content Toggle raw display
$23$ \( T^{2} + 8T + 10 \) Copy content Toggle raw display
$29$ \( (T + 4)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 4T - 2 \) Copy content Toggle raw display
$37$ \( T^{2} - 24 \) Copy content Toggle raw display
$41$ \( T^{2} + 12T + 12 \) Copy content Toggle raw display
$43$ \( T^{2} - 8T - 38 \) Copy content Toggle raw display
$47$ \( (T - 2)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 12T + 12 \) Copy content Toggle raw display
$59$ \( T^{2} + 4T - 50 \) Copy content Toggle raw display
$61$ \( T^{2} - 8T - 80 \) Copy content Toggle raw display
$67$ \( T^{2} + 20T + 76 \) Copy content Toggle raw display
$71$ \( T^{2} - 12T + 30 \) Copy content Toggle raw display
$73$ \( T^{2} - 216 \) Copy content Toggle raw display
$79$ \( T^{2} - 24 \) Copy content Toggle raw display
$83$ \( (T + 2)^{2} \) Copy content Toggle raw display
$89$ \( (T + 6)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 4T - 92 \) Copy content Toggle raw display
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