Properties

Label 2736.3.o.a
Level $2736$
Weight $3$
Character orbit 2736.o
Self dual yes
Analytic conductor $74.551$
Analytic rank $0$
Dimension $1$
CM discriminant -19
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2736,3,Mod(721,2736)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2736, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2736.721");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2736.o (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: yes
Analytic conductor: \(74.5506003290\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 19)
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)
 
\(f(q)\) \(=\) \( q + 9 q^{5} + 5 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 9 q^{5} + 5 q^{7} + 3 q^{11} - 15 q^{17} + 19 q^{19} - 30 q^{23} + 56 q^{25} + 45 q^{35} + 85 q^{43} + 75 q^{47} - 24 q^{49} + 27 q^{55} + 103 q^{61} - 25 q^{73} + 15 q^{77} + 90 q^{83} - 135 q^{85} + 171 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times\).

\(n\) \(1009\) \(1217\) \(1711\) \(2053\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
721.1
0
0 0 0 9.00000 0 5.00000 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
19.b odd 2 1 CM by \(\Q(\sqrt{-19}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.3.o.a 1
3.b odd 2 1 304.3.e.a 1
4.b odd 2 1 171.3.c.a 1
12.b even 2 1 19.3.b.a 1
19.b odd 2 1 CM 2736.3.o.a 1
24.f even 2 1 1216.3.e.a 1
24.h odd 2 1 1216.3.e.b 1
57.d even 2 1 304.3.e.a 1
60.h even 2 1 475.3.c.a 1
60.l odd 4 2 475.3.d.a 2
76.d even 2 1 171.3.c.a 1
228.b odd 2 1 19.3.b.a 1
228.m even 6 2 361.3.d.a 2
228.n odd 6 2 361.3.d.a 2
228.u odd 18 6 361.3.f.a 6
228.v even 18 6 361.3.f.a 6
456.l odd 2 1 1216.3.e.a 1
456.p even 2 1 1216.3.e.b 1
1140.p odd 2 1 475.3.c.a 1
1140.w even 4 2 475.3.d.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.3.b.a 1 12.b even 2 1
19.3.b.a 1 228.b odd 2 1
171.3.c.a 1 4.b odd 2 1
171.3.c.a 1 76.d even 2 1
304.3.e.a 1 3.b odd 2 1
304.3.e.a 1 57.d even 2 1
361.3.d.a 2 228.m even 6 2
361.3.d.a 2 228.n odd 6 2
361.3.f.a 6 228.u odd 18 6
361.3.f.a 6 228.v even 18 6
475.3.c.a 1 60.h even 2 1
475.3.c.a 1 1140.p odd 2 1
475.3.d.a 2 60.l odd 4 2
475.3.d.a 2 1140.w even 4 2
1216.3.e.a 1 24.f even 2 1
1216.3.e.a 1 456.l odd 2 1
1216.3.e.b 1 24.h odd 2 1
1216.3.e.b 1 456.p even 2 1
2736.3.o.a 1 1.a even 1 1 trivial
2736.3.o.a 1 19.b odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(2736, [\chi])\):

\( T_{5} - 9 \) Copy content Toggle raw display
\( T_{7} - 5 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 9 \) Copy content Toggle raw display
$7$ \( T - 5 \) Copy content Toggle raw display
$11$ \( T - 3 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T + 15 \) Copy content Toggle raw display
$19$ \( T - 19 \) Copy content Toggle raw display
$23$ \( T + 30 \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T - 85 \) Copy content Toggle raw display
$47$ \( T - 75 \) Copy content Toggle raw display
$53$ \( T \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T - 103 \) Copy content Toggle raw display
$67$ \( T \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T + 25 \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T - 90 \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T \) Copy content Toggle raw display
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