Properties

Label 2736.1.o.c
Level $2736$
Weight $1$
Character orbit 2736.o
Self dual yes
Analytic conductor $1.365$
Analytic rank $0$
Dimension $2$
Projective image $D_{6}$
CM discriminant -19
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.o (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(1.36544187456\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 684)
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.0.155952.1

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{5} - q^{7} +O(q^{10})\) \( q -\beta q^{5} - q^{7} -\beta q^{11} + \beta q^{17} + q^{19} + 2 q^{25} + \beta q^{35} + q^{43} + \beta q^{47} + 3 q^{55} + q^{61} - q^{73} + \beta q^{77} -3 q^{85} -\beta q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{7} + O(q^{10}) \) \( 2q - 2q^{7} + 2q^{19} + 4q^{25} + 2q^{43} + 6q^{55} + 2q^{61} - 2q^{73} - 6q^{85} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
721.1
1.73205
−1.73205
0 0 0 −1.73205 0 −1.00000 0 0 0
721.2 0 0 0 1.73205 0 −1.00000 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 \(\Q(\sqrt{-19}) \)
3.b odd 2 1 inner
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 2736.1.o.c 2
3.b odd 2 1 inner 2736.1.o.c 2
4.b odd 2 1 684.1.h.b 2
12.b even 2 1 684.1.h.b 2
19.b odd 2 1 CM 2736.1.o.c 2
57.d even 2 1 inner 2736.1.o.c 2
76.d even 2 1 684.1.h.b 2
228.b odd 2 1 684.1.h.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
684.1.h.b 2 4.b odd 2 1
684.1.h.b 2 12.b even 2 1
684.1.h.b 2 76.d even 2 1
684.1.h.b 2 228.b odd 2 1
2736.1.o.c 2 1.a even 1 1 trivial
2736.1.o.c 2 3.b odd 2 1 inner
2736.1.o.c 2 19.b odd 2 1 CM
2736.1.o.c 2 57.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 3 \) acting on \(S_{1}^{\mathrm{new}}(2736, [\chi])\).

Hecke characteristic polynomials

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