Properties

Label 2736.2.k.g
Level $2736$
Weight $2$
Character orbit 2736.k
Analytic conductor $21.847$
Analytic rank $0$
Dimension $2$
CM discriminant -19
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2736,2,Mod(2431,2736)]
 
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([1, 0, 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("2736.2431");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2736.k (of order \(2\), degree \(1\), 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: \(21.8470699930\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-19}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 304)
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 \(\beta = \sqrt{-19}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{5} + \beta q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{5} + \beta q^{7} - \beta q^{11} - 7 q^{17} - \beta q^{19} - 2 \beta q^{23} - 4 q^{25} + \beta q^{35} - 3 \beta q^{43} - \beta q^{47} - 12 q^{49} - \beta q^{55} + 15 q^{61} + 11 q^{73} + 19 q^{77} + 2 \beta q^{83} - 7 q^{85} - \beta q^{95} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{5} - 14 q^{17} - 8 q^{25} - 24 q^{49} + 30 q^{61} + 22 q^{73} + 38 q^{77} - 14 q^{85}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2431.1
0.500000 2.17945i
0.500000 + 2.17945i
0 0 0 1.00000 0 4.35890i 0 0 0
2431.2 0 0 0 1.00000 0 4.35890i 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 \(\Q(\sqrt{-19}) \)
4.b odd 2 1 inner
76.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.2.k.g 2
3.b odd 2 1 304.2.h.a 2
4.b odd 2 1 inner 2736.2.k.g 2
12.b even 2 1 304.2.h.a 2
19.b odd 2 1 CM 2736.2.k.g 2
24.f even 2 1 1216.2.h.a 2
24.h odd 2 1 1216.2.h.a 2
57.d even 2 1 304.2.h.a 2
76.d even 2 1 inner 2736.2.k.g 2
228.b odd 2 1 304.2.h.a 2
456.l odd 2 1 1216.2.h.a 2
456.p even 2 1 1216.2.h.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
304.2.h.a 2 3.b odd 2 1
304.2.h.a 2 12.b even 2 1
304.2.h.a 2 57.d even 2 1
304.2.h.a 2 228.b odd 2 1
1216.2.h.a 2 24.f even 2 1
1216.2.h.a 2 24.h odd 2 1
1216.2.h.a 2 456.l odd 2 1
1216.2.h.a 2 456.p even 2 1
2736.2.k.g 2 1.a even 1 1 trivial
2736.2.k.g 2 4.b odd 2 1 inner
2736.2.k.g 2 19.b odd 2 1 CM
2736.2.k.g 2 76.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2736, [\chi])\):

\( T_{5} - 1 \) Copy content Toggle raw display
\( T_{7}^{2} + 19 \) Copy content Toggle raw display
\( T_{11}^{2} + 19 \) Copy content Toggle raw display
\( T_{31} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 19 \) Copy content Toggle raw display
$11$ \( T^{2} + 19 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T + 7)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 19 \) Copy content Toggle raw display
$23$ \( T^{2} + 76 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 171 \) Copy content Toggle raw display
$47$ \( T^{2} + 19 \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T - 15)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( (T - 11)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 76 \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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