Properties

Label 3840.2.a.bk
Level $3840$
Weight $2$
Character orbit 3840.a
Self dual yes
Analytic conductor $30.663$
Analytic rank $0$
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] = [3840,2,Mod(1,3840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3840, 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("3840.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3840 = 2^{8} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3840.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: \(30.6625543762\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1920)
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 \(\beta = 2\sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} - q^{5} + (\beta + 2) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} - q^{5} + (\beta + 2) q^{7} + q^{9} + (\beta - 2) q^{11} - 2 q^{13} - q^{15} + 2 \beta q^{17} + ( - \beta - 4) q^{19} + (\beta + 2) q^{21} + ( - \beta + 4) q^{23} + q^{25} + q^{27} + 6 q^{29} + 4 q^{31} + (\beta - 2) q^{33} + ( - \beta - 2) q^{35} + (2 \beta + 2) q^{37} - 2 q^{39} + ( - 2 \beta + 2) q^{41} + ( - 2 \beta - 4) q^{43} - q^{45} + ( - \beta + 8) q^{47} + (4 \beta + 5) q^{49} + 2 \beta q^{51} + ( - 4 \beta + 2) q^{53} + ( - \beta + 2) q^{55} + ( - \beta - 4) q^{57} + (3 \beta + 6) q^{59} + 4 q^{61} + (\beta + 2) q^{63} + 2 q^{65} + 4 q^{67} + ( - \beta + 4) q^{69} - 2 \beta q^{71} + ( - 4 \beta - 2) q^{73} + q^{75} + 4 q^{77} + 4 q^{79} + q^{81} + ( - 2 \beta + 8) q^{83} - 2 \beta q^{85} + 6 q^{87} + ( - 2 \beta + 6) q^{89} + ( - 2 \beta - 4) q^{91} + 4 q^{93} + (\beta + 4) q^{95} - 14 q^{97} + (\beta - 2) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 2 q^{5} + 4 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - 2 q^{5} + 4 q^{7} + 2 q^{9} - 4 q^{11} - 4 q^{13} - 2 q^{15} - 8 q^{19} + 4 q^{21} + 8 q^{23} + 2 q^{25} + 2 q^{27} + 12 q^{29} + 8 q^{31} - 4 q^{33} - 4 q^{35} + 4 q^{37} - 4 q^{39} + 4 q^{41} - 8 q^{43} - 2 q^{45} + 16 q^{47} + 10 q^{49} + 4 q^{53} + 4 q^{55} - 8 q^{57} + 12 q^{59} + 8 q^{61} + 4 q^{63} + 4 q^{65} + 8 q^{67} + 8 q^{69} - 4 q^{73} + 2 q^{75} + 8 q^{77} + 8 q^{79} + 2 q^{81} + 16 q^{83} + 12 q^{87} + 12 q^{89} - 8 q^{91} + 8 q^{93} + 8 q^{95} - 28 q^{97} - 4 q^{99}+O(q^{100}) \) 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
−1.41421
1.41421
0 1.00000 0 −1.00000 0 −0.828427 0 1.00000 0
1.2 0 1.00000 0 −1.00000 0 4.82843 0 1.00000 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 3840.2.a.bk 2
4.b odd 2 1 3840.2.a.bc 2
8.b even 2 1 3840.2.a.bh 2
8.d odd 2 1 3840.2.a.bl 2
16.e even 4 2 1920.2.k.i 4
16.f odd 4 2 1920.2.k.l yes 4
48.i odd 4 2 5760.2.k.n 4
48.k even 4 2 5760.2.k.w 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1920.2.k.i 4 16.e even 4 2
1920.2.k.l yes 4 16.f odd 4 2
3840.2.a.bc 2 4.b odd 2 1
3840.2.a.bh 2 8.b even 2 1
3840.2.a.bk 2 1.a even 1 1 trivial
3840.2.a.bl 2 8.d odd 2 1
5760.2.k.n 4 48.i odd 4 2
5760.2.k.w 4 48.k even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3840))\):

\( T_{7}^{2} - 4T_{7} - 4 \) Copy content Toggle raw display
\( T_{11}^{2} + 4T_{11} - 4 \) Copy content Toggle raw display
\( T_{13} + 2 \) Copy content Toggle raw display
\( T_{17}^{2} - 32 \) Copy content Toggle raw display
\( T_{19}^{2} + 8T_{19} + 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T - 1)^{2} \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 4T - 4 \) Copy content Toggle raw display
$11$ \( T^{2} + 4T - 4 \) Copy content Toggle raw display
$13$ \( (T + 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 32 \) Copy content Toggle raw display
$19$ \( T^{2} + 8T + 8 \) Copy content Toggle raw display
$23$ \( T^{2} - 8T + 8 \) Copy content Toggle raw display
$29$ \( (T - 6)^{2} \) Copy content Toggle raw display
$31$ \( (T - 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 4T - 28 \) Copy content Toggle raw display
$41$ \( T^{2} - 4T - 28 \) Copy content Toggle raw display
$43$ \( T^{2} + 8T - 16 \) Copy content Toggle raw display
$47$ \( T^{2} - 16T + 56 \) Copy content Toggle raw display
$53$ \( T^{2} - 4T - 124 \) Copy content Toggle raw display
$59$ \( T^{2} - 12T - 36 \) Copy content Toggle raw display
$61$ \( (T - 4)^{2} \) Copy content Toggle raw display
$67$ \( (T - 4)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 32 \) Copy content Toggle raw display
$73$ \( T^{2} + 4T - 124 \) Copy content Toggle raw display
$79$ \( (T - 4)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 16T + 32 \) Copy content Toggle raw display
$89$ \( T^{2} - 12T + 4 \) Copy content Toggle raw display
$97$ \( (T + 14)^{2} \) Copy content Toggle raw display
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