Properties

Label 2736.1.bk.a
Level $2736$
Weight $1$
Character orbit 2736.bk
Analytic conductor $1.365$
Analytic rank $0$
Dimension $2$
Projective image $D_{6}$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2736.bk (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.36544187456\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.2.225194688.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{7} +O(q^{10})\) \( q + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{7} -\zeta_{6} q^{13} - q^{19} + \zeta_{6} q^{25} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{31} - q^{37} + ( -1 - \zeta_{6} ) q^{43} + ( -1 - \zeta_{6} + \zeta_{6}^{2} ) q^{49} + \zeta_{6} q^{61} + ( 1 - \zeta_{6}^{2} ) q^{67} + \zeta_{6}^{2} q^{73} + ( 1 + \zeta_{6} ) q^{79} + ( -1 + \zeta_{6}^{2} ) q^{91} -2 \zeta_{6}^{2} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q - q^{13} - 2q^{19} + q^{25} - 2q^{37} - 3q^{43} - 4q^{49} + q^{61} + 3q^{67} - q^{73} + 3q^{79} - 3q^{91} + 2q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(1009\) \(1217\) \(1711\) \(2053\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\) \(-1\) \(1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
847.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 0 0 1.73205i 0 0 0
2287.1 0 0 0 0 0 1.73205i 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
76.g odd 6 1 inner
228.m even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.1.bk.a 2
3.b odd 2 1 CM 2736.1.bk.a 2
4.b odd 2 1 2736.1.bk.b yes 2
12.b even 2 1 2736.1.bk.b yes 2
19.c even 3 1 2736.1.bk.b yes 2
57.h odd 6 1 2736.1.bk.b yes 2
76.g odd 6 1 inner 2736.1.bk.a 2
228.m even 6 1 inner 2736.1.bk.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2736.1.bk.a 2 1.a even 1 1 trivial
2736.1.bk.a 2 3.b odd 2 1 CM
2736.1.bk.a 2 76.g odd 6 1 inner
2736.1.bk.a 2 228.m even 6 1 inner
2736.1.bk.b yes 2 4.b odd 2 1
2736.1.bk.b yes 2 12.b even 2 1
2736.1.bk.b yes 2 19.c even 3 1
2736.1.bk.b yes 2 57.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{43}^{2} + 3 T_{43} + 3 \) acting on \(S_{1}^{\mathrm{new}}(2736, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 3 + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 1 + T + T^{2} \)
$17$ \( T^{2} \)
$19$ \( ( 1 + T )^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( 3 + T^{2} \)
$37$ \( ( 1 + T )^{2} \)
$41$ \( T^{2} \)
$43$ \( 3 + 3 T + T^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( 1 - T + T^{2} \)
$67$ \( 3 - 3 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 1 + T + T^{2} \)
$79$ \( 3 - 3 T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( 4 - 2 T + T^{2} \)
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