Properties

Label 2736.2.k.c
Level $2736$
Weight $2$
Character orbit 2736.k
Analytic conductor $21.847$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

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

Self dual: no
Analytic conductor: \(21.8470699930\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -2 + 4 \zeta_{6} ) q^{7} +O(q^{10})\) \( q + ( -2 + 4 \zeta_{6} ) q^{7} + ( 4 - 8 \zeta_{6} ) q^{13} + ( -3 - 2 \zeta_{6} ) q^{19} -5 q^{25} + 4 q^{31} + ( 4 - 8 \zeta_{6} ) q^{37} + ( 6 - 12 \zeta_{6} ) q^{43} -5 q^{49} -14 q^{61} + 16 q^{67} + 10 q^{73} -4 q^{79} + 24 q^{91} + ( 8 - 16 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q - 8q^{19} - 10q^{25} + 8q^{31} - 10q^{49} - 28q^{61} + 32q^{67} + 20q^{73} - 8q^{79} + 48q^{91} + O(q^{100}) \)

Character values

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

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

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.

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 0.866025i
0.500000 + 0.866025i
0 0 0 0 0 3.46410i 0 0 0
2431.2 0 0 0 0 0 3.46410i 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.d even 2 1 inner
228.b odd 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.c 2
3.b odd 2 1 CM 2736.2.k.c 2
4.b odd 2 1 2736.2.k.d yes 2
12.b even 2 1 2736.2.k.d yes 2
19.b odd 2 1 2736.2.k.d yes 2
57.d even 2 1 2736.2.k.d yes 2
76.d even 2 1 inner 2736.2.k.c 2
228.b odd 2 1 inner 2736.2.k.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2736.2.k.c 2 1.a even 1 1 trivial
2736.2.k.c 2 3.b odd 2 1 CM
2736.2.k.c 2 76.d even 2 1 inner
2736.2.k.c 2 228.b odd 2 1 inner
2736.2.k.d yes 2 4.b odd 2 1
2736.2.k.d yes 2 12.b even 2 1
2736.2.k.d yes 2 19.b odd 2 1
2736.2.k.d yes 2 57.d even 2 1

Hecke kernels

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

\( T_{5} \)
\( T_{7}^{2} + 12 \)
\( T_{11} \)
\( T_{31} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 12 + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 48 + T^{2} \)
$17$ \( T^{2} \)
$19$ \( 19 + 8 T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( ( -4 + T )^{2} \)
$37$ \( 48 + T^{2} \)
$41$ \( T^{2} \)
$43$ \( 108 + T^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( ( 14 + T )^{2} \)
$67$ \( ( -16 + T )^{2} \)
$71$ \( T^{2} \)
$73$ \( ( -10 + T )^{2} \)
$79$ \( ( 4 + T )^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( 192 + T^{2} \)
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