Properties

Label 2736.2.f.c
Level $2736$
Weight $2$
Character orbit 2736.f
Analytic conductor $21.847$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2736,2,Mod(1025,2736)]
 
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, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2736.1025");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2736.f (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: \(21.8470699930\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1368)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{5} - 2 q^{7} + 3 \beta q^{11} - 2 \beta q^{13} - 5 \beta q^{17} + ( - 3 \beta + 1) q^{19} + \beta q^{23} + 3 q^{25} - 10 q^{29} + 2 \beta q^{31} - 2 \beta q^{35} - 4 \beta q^{37} + 10 q^{41} + \cdots - 6 \beta q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{7} + 2 q^{19} + 6 q^{25} - 20 q^{29} + 20 q^{41} - 24 q^{43} - 6 q^{49} + 20 q^{53} - 12 q^{55} + 24 q^{59} + 16 q^{61} + 8 q^{65} + 16 q^{71} + 12 q^{73} + 20 q^{85} + 12 q^{89} + 12 q^{95}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1025.1
1.41421i
1.41421i
0 0 0 1.41421i 0 −2.00000 0 0 0
1025.2 0 0 0 1.41421i 0 −2.00000 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
57.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.f.c 2
3.b odd 2 1 2736.2.f.d 2
4.b odd 2 1 1368.2.f.a 2
12.b even 2 1 1368.2.f.b yes 2
19.b odd 2 1 2736.2.f.d 2
57.d even 2 1 inner 2736.2.f.c 2
76.d even 2 1 1368.2.f.b yes 2
228.b odd 2 1 1368.2.f.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1368.2.f.a 2 4.b odd 2 1
1368.2.f.a 2 228.b odd 2 1
1368.2.f.b yes 2 12.b even 2 1
1368.2.f.b yes 2 76.d even 2 1
2736.2.f.c 2 1.a even 1 1 trivial
2736.2.f.c 2 57.d even 2 1 inner
2736.2.f.d 2 3.b odd 2 1
2736.2.f.d 2 19.b odd 2 1

Hecke kernels

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

\( T_{5}^{2} + 2 \) Copy content Toggle raw display
\( T_{7} + 2 \) Copy content Toggle raw display
\( T_{11}^{2} + 18 \) Copy content Toggle raw display
\( T_{29} + 10 \) 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
$5$ \( T^{2} + 2 \) Copy content Toggle raw display
$7$ \( (T + 2)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 18 \) Copy content Toggle raw display
$13$ \( T^{2} + 8 \) Copy content Toggle raw display
$17$ \( T^{2} + 50 \) Copy content Toggle raw display
$19$ \( T^{2} - 2T + 19 \) Copy content Toggle raw display
$23$ \( T^{2} + 2 \) Copy content Toggle raw display
$29$ \( (T + 10)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 8 \) Copy content Toggle raw display
$37$ \( T^{2} + 32 \) Copy content Toggle raw display
$41$ \( (T - 10)^{2} \) Copy content Toggle raw display
$43$ \( (T + 12)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 2 \) Copy content Toggle raw display
$53$ \( (T - 10)^{2} \) Copy content Toggle raw display
$59$ \( (T - 12)^{2} \) Copy content Toggle raw display
$61$ \( (T - 8)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 200 \) Copy content Toggle raw display
$71$ \( (T - 8)^{2} \) Copy content Toggle raw display
$73$ \( (T - 6)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 128 \) Copy content Toggle raw display
$83$ \( T^{2} + 18 \) Copy content Toggle raw display
$89$ \( (T - 6)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 72 \) Copy content Toggle raw display
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