Properties

Label 3744.2.m.e
Level $3744$
Weight $2$
Character orbit 3744.m
Analytic conductor $29.896$
Analytic rank $0$
Dimension $4$
CM discriminant -312
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3744,2,Mod(1585,3744)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3744, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3744.1585");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3744 = 2^{5} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3744.m (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(29.8959905168\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{13})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - x^{2} + 2x + 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{41}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 936)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{13} - 2 \beta_{2} q^{19} - 5 q^{25} - \beta_{3} q^{29} + 2 \beta_{2} q^{37} - \beta_1 q^{41} - 2 \beta_1 q^{47} + 7 q^{49} - \beta_{3} q^{53} - 2 \beta_{2} q^{67} + 4 \beta_1 q^{71} - 2 q^{79} + 5 \beta_1 q^{89}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 20 q^{25} + 28 q^{49} - 8 q^{79}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - x^{2} + 2x + 27 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 4\nu^{3} - 6\nu^{2} + 14\nu - 6 ) / 21 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -4\nu^{3} + 6\nu^{2} + 28\nu - 15 ) / 21 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{2} - 2\nu - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + \beta_{2} + \beta _1 + 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{3} - 4\beta_{2} + 17\beta _1 + 8 ) / 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3744\mathbb{Z}\right)^\times\).

\(n\) \(703\) \(2017\) \(2081\) \(2341\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1585.1
−1.30278 1.41421i
−1.30278 + 1.41421i
2.30278 + 1.41421i
2.30278 1.41421i
0 0 0 0 0 0 0 0 0
1585.2 0 0 0 0 0 0 0 0 0
1585.3 0 0 0 0 0 0 0 0 0
1585.4 0 0 0 0 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
312.b odd 2 1 CM by \(\Q(\sqrt{-78}) \)
3.b odd 2 1 inner
8.b even 2 1 inner
13.b even 2 1 inner
24.h odd 2 1 inner
39.d odd 2 1 inner
104.e even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3744.2.m.e 4
3.b odd 2 1 inner 3744.2.m.e 4
4.b odd 2 1 936.2.m.e 4
8.b even 2 1 inner 3744.2.m.e 4
8.d odd 2 1 936.2.m.e 4
12.b even 2 1 936.2.m.e 4
13.b even 2 1 inner 3744.2.m.e 4
24.f even 2 1 936.2.m.e 4
24.h odd 2 1 inner 3744.2.m.e 4
39.d odd 2 1 inner 3744.2.m.e 4
52.b odd 2 1 936.2.m.e 4
104.e even 2 1 inner 3744.2.m.e 4
104.h odd 2 1 936.2.m.e 4
156.h even 2 1 936.2.m.e 4
312.b odd 2 1 CM 3744.2.m.e 4
312.h even 2 1 936.2.m.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
936.2.m.e 4 4.b odd 2 1
936.2.m.e 4 8.d odd 2 1
936.2.m.e 4 12.b even 2 1
936.2.m.e 4 24.f even 2 1
936.2.m.e 4 52.b odd 2 1
936.2.m.e 4 104.h odd 2 1
936.2.m.e 4 156.h even 2 1
936.2.m.e 4 312.h even 2 1
3744.2.m.e 4 1.a even 1 1 trivial
3744.2.m.e 4 3.b odd 2 1 inner
3744.2.m.e 4 8.b even 2 1 inner
3744.2.m.e 4 13.b even 2 1 inner
3744.2.m.e 4 24.h odd 2 1 inner
3744.2.m.e 4 39.d odd 2 1 inner
3744.2.m.e 4 104.e even 2 1 inner
3744.2.m.e 4 312.b odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} \) acting on \(S_{2}^{\mathrm{new}}(3744, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} - 13)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} - 52)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + 104)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} - 52)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} + 32)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 104)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} - 52)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 128)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( (T + 2)^{4} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} + 200)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
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