Properties

Label 3840.2.a.bf
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: \(\Q(\sqrt{3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 960)
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 \(\beta = 2\sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + q^{5} - 2 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} + q^{5} - 2 q^{7} + q^{9} + ( - \beta - 2) q^{11} + (\beta + 2) q^{13} - q^{15} + \beta q^{17} + 2 \beta q^{19} + 2 q^{21} - 4 q^{23} + q^{25} - q^{27} + ( - 2 \beta - 2) q^{29} + ( - \beta + 4) q^{31} + (\beta + 2) q^{33} - 2 q^{35} + (\beta + 6) q^{37} + ( - \beta - 2) q^{39} + ( - 2 \beta + 2) q^{41} - 2 \beta q^{43} + q^{45} - 4 q^{47} - 3 q^{49} - \beta q^{51} + ( - 2 \beta + 6) q^{53} + ( - \beta - 2) q^{55} - 2 \beta q^{57} + (\beta + 6) q^{59} + 4 q^{61} - 2 q^{63} + (\beta + 2) q^{65} + 2 \beta q^{67} + 4 q^{69} + 2 \beta q^{71} - 10 q^{73} - q^{75} + (2 \beta + 4) q^{77} + (3 \beta + 4) q^{79} + q^{81} + 8 q^{83} + \beta q^{85} + (2 \beta + 2) q^{87} + ( - 2 \beta + 6) q^{89} + ( - 2 \beta - 4) q^{91} + (\beta - 4) q^{93} + 2 \beta q^{95} + ( - 2 \beta - 6) q^{97} + ( - \beta - 2) q^{99} +O(q^{100}) \) 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 \( 2 q - 2 q^{3} + 2 q^{5} - 4 q^{7} + 2 q^{9} - 4 q^{11} + 4 q^{13} - 2 q^{15} + 4 q^{21} - 8 q^{23} + 2 q^{25} - 2 q^{27} - 4 q^{29} + 8 q^{31} + 4 q^{33} - 4 q^{35} + 12 q^{37} - 4 q^{39} + 4 q^{41} + 2 q^{45} - 8 q^{47} - 6 q^{49} + 12 q^{53} - 4 q^{55} + 12 q^{59} + 8 q^{61} - 4 q^{63} + 4 q^{65} + 8 q^{69} - 20 q^{73} - 2 q^{75} + 8 q^{77} + 8 q^{79} + 2 q^{81} + 16 q^{83} + 4 q^{87} + 12 q^{89} - 8 q^{91} - 8 q^{93} - 12 q^{97} - 4 q^{99}+O(q^{100}) \) 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
1.73205
−1.73205
0 −1.00000 0 1.00000 0 −2.00000 0 1.00000 0
1.2 0 −1.00000 0 1.00000 0 −2.00000 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.bf 2
4.b odd 2 1 3840.2.a.bn 2
8.b even 2 1 3840.2.a.bi 2
8.d odd 2 1 3840.2.a.be 2
16.e even 4 2 960.2.k.f yes 4
16.f odd 4 2 960.2.k.e 4
48.i odd 4 2 2880.2.k.k 4
48.k even 4 2 2880.2.k.f 4
80.i odd 4 2 4800.2.d.n 4
80.j even 4 2 4800.2.d.r 4
80.k odd 4 2 4800.2.k.o 4
80.q even 4 2 4800.2.k.i 4
80.s even 4 2 4800.2.d.i 4
80.t odd 4 2 4800.2.d.m 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
960.2.k.e 4 16.f odd 4 2
960.2.k.f yes 4 16.e even 4 2
2880.2.k.f 4 48.k even 4 2
2880.2.k.k 4 48.i odd 4 2
3840.2.a.be 2 8.d odd 2 1
3840.2.a.bf 2 1.a even 1 1 trivial
3840.2.a.bi 2 8.b even 2 1
3840.2.a.bn 2 4.b odd 2 1
4800.2.d.i 4 80.s even 4 2
4800.2.d.m 4 80.t odd 4 2
4800.2.d.n 4 80.i odd 4 2
4800.2.d.r 4 80.j even 4 2
4800.2.k.i 4 80.q even 4 2
4800.2.k.o 4 80.k odd 4 2

Hecke kernels

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

\( T_{7} + 2 \) Copy content Toggle raw display
\( T_{11}^{2} + 4T_{11} - 8 \) Copy content Toggle raw display
\( T_{13}^{2} - 4T_{13} - 8 \) Copy content Toggle raw display
\( T_{17}^{2} - 12 \) Copy content Toggle raw display
\( T_{19}^{2} - 48 \) 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
$7$ \( (T + 2)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 4T - 8 \) Copy content Toggle raw display
$13$ \( T^{2} - 4T - 8 \) Copy content Toggle raw display
$17$ \( T^{2} - 12 \) Copy content Toggle raw display
$19$ \( T^{2} - 48 \) Copy content Toggle raw display
$23$ \( (T + 4)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 4T - 44 \) Copy content Toggle raw display
$31$ \( T^{2} - 8T + 4 \) Copy content Toggle raw display
$37$ \( T^{2} - 12T + 24 \) Copy content Toggle raw display
$41$ \( T^{2} - 4T - 44 \) Copy content Toggle raw display
$43$ \( T^{2} - 48 \) Copy content Toggle raw display
$47$ \( (T + 4)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 12T - 12 \) Copy content Toggle raw display
$59$ \( T^{2} - 12T + 24 \) Copy content Toggle raw display
$61$ \( (T - 4)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 48 \) Copy content Toggle raw display
$71$ \( T^{2} - 48 \) Copy content Toggle raw display
$73$ \( (T + 10)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 8T - 92 \) Copy content Toggle raw display
$83$ \( (T - 8)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 12T - 12 \) Copy content Toggle raw display
$97$ \( T^{2} + 12T - 12 \) Copy content Toggle raw display
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