Properties

Label 3840.2.a.br
Level $3840$
Weight $2$
Character orbit 3840.a
Self dual yes
Analytic conductor $30.663$
Analytic rank $0$
Dimension $3$
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: \(3\)
Coefficient field: 3.3.316.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 120)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + q^{5} + (\beta_1 + 1) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + q^{5} + (\beta_1 + 1) q^{7} + q^{9} + (\beta_{2} - \beta_1 + 1) q^{11} - \beta_{2} q^{13} + q^{15} + ( - \beta_{2} + 2) q^{17} + ( - \beta_1 + 1) q^{19} + (\beta_1 + 1) q^{21} + (\beta_1 - 1) q^{23} + q^{25} + q^{27} + 2 q^{29} + (\beta_{2} + 2) q^{31} + (\beta_{2} - \beta_1 + 1) q^{33} + (\beta_1 + 1) q^{35} + (\beta_{2} - 2 \beta_1 - 2) q^{37} - \beta_{2} q^{39} + (2 \beta_1 + 4) q^{41} + (2 \beta_1 + 2) q^{43} + q^{45} + ( - 2 \beta_{2} + \beta_1 - 1) q^{47} + (2 \beta_{2} + 5) q^{49} + ( - \beta_{2} + 2) q^{51} + 2 q^{53} + (\beta_{2} - \beta_1 + 1) q^{55} + ( - \beta_1 + 1) q^{57} + ( - \beta_{2} + \beta_1 + 3) q^{59} + (2 \beta_{2} - 2 \beta_1 + 2) q^{61} + (\beta_1 + 1) q^{63} - \beta_{2} q^{65} - 4 q^{67} + (\beta_1 - 1) q^{69} + (2 \beta_{2} - 2 \beta_1 - 2) q^{71} + 6 q^{73} + q^{75} + (4 \beta_1 - 4) q^{77} + (\beta_{2} - 6) q^{79} + q^{81} + (2 \beta_1 + 6) q^{83} + ( - \beta_{2} + 2) q^{85} + 2 q^{87} + ( - 2 \beta_1 + 4) q^{89} + ( - 2 \beta_{2} - 2 \beta_1 - 6) q^{91} + (\beta_{2} + 2) q^{93} + ( - \beta_1 + 1) q^{95} + ( - 2 \beta_{2} + 6) q^{97} + (\beta_{2} - \beta_1 + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} + 3 q^{5} + 2 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{3} + 3 q^{5} + 2 q^{7} + 3 q^{9} + 4 q^{11} + 3 q^{15} + 6 q^{17} + 4 q^{19} + 2 q^{21} - 4 q^{23} + 3 q^{25} + 3 q^{27} + 6 q^{29} + 6 q^{31} + 4 q^{33} + 2 q^{35} - 4 q^{37} + 10 q^{41} + 4 q^{43} + 3 q^{45} - 4 q^{47} + 15 q^{49} + 6 q^{51} + 6 q^{53} + 4 q^{55} + 4 q^{57} + 8 q^{59} + 8 q^{61} + 2 q^{63} - 12 q^{67} - 4 q^{69} - 4 q^{71} + 18 q^{73} + 3 q^{75} - 16 q^{77} - 18 q^{79} + 3 q^{81} + 16 q^{83} + 6 q^{85} + 6 q^{87} + 14 q^{89} - 16 q^{91} + 6 q^{93} + 4 q^{95} + 18 q^{97} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( 2\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\nu^{2} - 6 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} + 6 ) / 2 \) 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
−1.81361
0.470683
2.34292
0 1.00000 0 1.00000 0 −3.62721 0 1.00000 0
1.2 0 1.00000 0 1.00000 0 0.941367 0 1.00000 0
1.3 0 1.00000 0 1.00000 0 4.68585 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.br 3
4.b odd 2 1 3840.2.a.bp 3
8.b even 2 1 3840.2.a.bo 3
8.d odd 2 1 3840.2.a.bq 3
16.e even 4 2 480.2.k.b 6
16.f odd 4 2 120.2.k.b 6
48.i odd 4 2 1440.2.k.f 6
48.k even 4 2 360.2.k.f 6
80.i odd 4 2 2400.2.d.e 6
80.j even 4 2 600.2.d.e 6
80.k odd 4 2 600.2.k.c 6
80.q even 4 2 2400.2.k.c 6
80.s even 4 2 600.2.d.f 6
80.t odd 4 2 2400.2.d.f 6
240.t even 4 2 1800.2.k.p 6
240.z odd 4 2 1800.2.d.r 6
240.bb even 4 2 7200.2.d.r 6
240.bd odd 4 2 1800.2.d.q 6
240.bf even 4 2 7200.2.d.q 6
240.bm odd 4 2 7200.2.k.p 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.k.b 6 16.f odd 4 2
360.2.k.f 6 48.k even 4 2
480.2.k.b 6 16.e even 4 2
600.2.d.e 6 80.j even 4 2
600.2.d.f 6 80.s even 4 2
600.2.k.c 6 80.k odd 4 2
1440.2.k.f 6 48.i odd 4 2
1800.2.d.q 6 240.bd odd 4 2
1800.2.d.r 6 240.z odd 4 2
1800.2.k.p 6 240.t even 4 2
2400.2.d.e 6 80.i odd 4 2
2400.2.d.f 6 80.t odd 4 2
2400.2.k.c 6 80.q even 4 2
3840.2.a.bo 3 8.b even 2 1
3840.2.a.bp 3 4.b odd 2 1
3840.2.a.bq 3 8.d odd 2 1
3840.2.a.br 3 1.a even 1 1 trivial
7200.2.d.q 6 240.bf even 4 2
7200.2.d.r 6 240.bb even 4 2
7200.2.k.p 6 240.bm odd 4 2

Hecke kernels

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

\( T_{7}^{3} - 2T_{7}^{2} - 16T_{7} + 16 \) Copy content Toggle raw display
\( T_{11}^{3} - 4T_{11}^{2} - 24T_{11} + 64 \) Copy content Toggle raw display
\( T_{13}^{3} - 28T_{13} - 16 \) Copy content Toggle raw display
\( T_{17}^{3} - 6T_{17}^{2} - 16T_{17} + 32 \) Copy content Toggle raw display
\( T_{19}^{3} - 4T_{19}^{2} - 12T_{19} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( (T - 1)^{3} \) Copy content Toggle raw display
$5$ \( (T - 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 2 T^{2} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( T^{3} - 4 T^{2} + \cdots + 64 \) Copy content Toggle raw display
$13$ \( T^{3} - 28T - 16 \) Copy content Toggle raw display
$17$ \( T^{3} - 6 T^{2} + \cdots + 32 \) Copy content Toggle raw display
$19$ \( T^{3} - 4 T^{2} + \cdots + 16 \) Copy content Toggle raw display
$23$ \( T^{3} + 4 T^{2} + \cdots - 16 \) Copy content Toggle raw display
$29$ \( (T - 2)^{3} \) Copy content Toggle raw display
$31$ \( T^{3} - 6 T^{2} + \cdots + 64 \) Copy content Toggle raw display
$37$ \( T^{3} + 4 T^{2} + \cdots - 256 \) Copy content Toggle raw display
$41$ \( T^{3} - 10 T^{2} + \cdots + 232 \) Copy content Toggle raw display
$43$ \( T^{3} - 4 T^{2} + \cdots + 128 \) Copy content Toggle raw display
$47$ \( T^{3} + 4 T^{2} + \cdots - 496 \) Copy content Toggle raw display
$53$ \( (T - 2)^{3} \) Copy content Toggle raw display
$59$ \( T^{3} - 8 T^{2} + \cdots + 32 \) Copy content Toggle raw display
$61$ \( T^{3} - 8 T^{2} + \cdots + 512 \) Copy content Toggle raw display
$67$ \( (T + 4)^{3} \) Copy content Toggle raw display
$71$ \( T^{3} + 4 T^{2} + \cdots + 64 \) Copy content Toggle raw display
$73$ \( (T - 6)^{3} \) Copy content Toggle raw display
$79$ \( T^{3} + 18 T^{2} + \cdots + 64 \) Copy content Toggle raw display
$83$ \( T^{3} - 16 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$89$ \( T^{3} - 14 T^{2} + \cdots + 184 \) Copy content Toggle raw display
$97$ \( T^{3} - 18 T^{2} + \cdots + 328 \) Copy content Toggle raw display
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