Properties

Label 2736.1.o.a
Level $2736$
Weight $1$
Character orbit 2736.o
Self dual yes
Analytic conductor $1.365$
Analytic rank $0$
Dimension $1$
Projective image $D_{2}$
CM/RM discs -3, -19, 57
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: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 171)
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(\sqrt{-3}, \sqrt{-19})\)
Artin image: $D_4$
Artin field: Galois closure of 4.0.8208.1

$q$-expansion

\(f(q)\) \(=\) \( q + 2 q^{7} + O(q^{10}) \) \( q + 2 q^{7} + q^{19} - q^{25} - 2 q^{43} + 3 q^{49} - 2 q^{61} + 2 q^{73} + O(q^{100}) \)

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
0
0 0 0 0 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
19.b odd 2 1 CM by \(\Q(\sqrt{-19}) \)
57.d even 2 1 RM by \(\Q(\sqrt{57}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.1.o.a 1
3.b odd 2 1 CM 2736.1.o.a 1
4.b odd 2 1 171.1.c.a 1
12.b even 2 1 171.1.c.a 1
19.b odd 2 1 CM 2736.1.o.a 1
36.f odd 6 2 1539.1.o.b 2
36.h even 6 2 1539.1.o.b 2
57.d even 2 1 RM 2736.1.o.a 1
76.d even 2 1 171.1.c.a 1
76.f even 6 2 3249.1.p.a 2
76.g odd 6 2 3249.1.p.a 2
76.k even 18 6 3249.1.ba.c 6
76.l odd 18 6 3249.1.ba.c 6
228.b odd 2 1 171.1.c.a 1
228.m even 6 2 3249.1.p.a 2
228.n odd 6 2 3249.1.p.a 2
228.u odd 18 6 3249.1.ba.c 6
228.v even 18 6 3249.1.ba.c 6
684.w even 6 2 1539.1.o.b 2
684.bh odd 6 2 1539.1.o.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
171.1.c.a 1 4.b odd 2 1
171.1.c.a 1 12.b even 2 1
171.1.c.a 1 76.d even 2 1
171.1.c.a 1 228.b odd 2 1
1539.1.o.b 2 36.f odd 6 2
1539.1.o.b 2 36.h even 6 2
1539.1.o.b 2 684.w even 6 2
1539.1.o.b 2 684.bh odd 6 2
2736.1.o.a 1 1.a even 1 1 trivial
2736.1.o.a 1 3.b odd 2 1 CM
2736.1.o.a 1 19.b odd 2 1 CM
2736.1.o.a 1 57.d even 2 1 RM
3249.1.p.a 2 76.f even 6 2
3249.1.p.a 2 76.g odd 6 2
3249.1.p.a 2 228.m even 6 2
3249.1.p.a 2 228.n odd 6 2
3249.1.ba.c 6 76.k even 18 6
3249.1.ba.c 6 76.l odd 18 6
3249.1.ba.c 6 228.u odd 18 6
3249.1.ba.c 6 228.v even 18 6

Hecke kernels

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

Hecke characteristic polynomials

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