Properties

Label 2736.1.b.b
Level $2736$
Weight $1$
Character orbit 2736.b
Analytic conductor $1.365$
Analytic rank $0$
Dimension $8$
Projective image $D_{12}$
CM discriminant -19
Inner twists $8$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(1.36544187456\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Projective image: \(D_{12}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( -\zeta_{24}^{5} - \zeta_{24}^{7} ) q^{5} -\zeta_{24}^{6} q^{7} +O(q^{10})\) \( q + ( -\zeta_{24}^{5} - \zeta_{24}^{7} ) q^{5} -\zeta_{24}^{6} q^{7} + ( \zeta_{24} - \zeta_{24}^{11} ) q^{11} + ( -\zeta_{24} - \zeta_{24}^{11} ) q^{17} + \zeta_{24}^{6} q^{19} + ( \zeta_{24}^{3} - \zeta_{24}^{9} ) q^{23} + ( -1 - \zeta_{24}^{2} + \zeta_{24}^{10} ) q^{25} + ( -\zeta_{24} + \zeta_{24}^{11} ) q^{35} + ( \zeta_{24}^{4} + \zeta_{24}^{8} ) q^{43} + ( -\zeta_{24}^{5} + \zeta_{24}^{7} ) q^{47} + ( -\zeta_{24}^{4} - 2 \zeta_{24}^{6} - \zeta_{24}^{8} ) q^{55} + ( -\zeta_{24}^{2} + \zeta_{24}^{10} ) q^{61} - q^{73} + ( -\zeta_{24}^{5} - \zeta_{24}^{7} ) q^{77} + ( -\zeta_{24}^{3} + \zeta_{24}^{9} ) q^{83} + ( -\zeta_{24}^{4} + \zeta_{24}^{8} ) q^{85} + ( \zeta_{24} - \zeta_{24}^{11} ) q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q - 8q^{25} - 8q^{73} - 8q^{85} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2735.1
0.965926 + 0.258819i
−0.965926 + 0.258819i
0.258819 + 0.965926i
−0.258819 + 0.965926i
−0.258819 0.965926i
0.258819 0.965926i
−0.965926 0.258819i
0.965926 0.258819i
0 0 0 1.93185i 0 1.00000i 0 0 0
2735.2 0 0 0 1.93185i 0 1.00000i 0 0 0
2735.3 0 0 0 0.517638i 0 1.00000i 0 0 0
2735.4 0 0 0 0.517638i 0 1.00000i 0 0 0
2735.5 0 0 0 0.517638i 0 1.00000i 0 0 0
2735.6 0 0 0 0.517638i 0 1.00000i 0 0 0
2735.7 0 0 0 1.93185i 0 1.00000i 0 0 0
2735.8 0 0 0 1.93185i 0 1.00000i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2735.8
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
4.b odd 2 1 inner
12.b even 2 1 inner
57.d even 2 1 inner
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 2736.1.b.b 8
3.b odd 2 1 inner 2736.1.b.b 8
4.b odd 2 1 inner 2736.1.b.b 8
12.b even 2 1 inner 2736.1.b.b 8
19.b odd 2 1 CM 2736.1.b.b 8
57.d even 2 1 inner 2736.1.b.b 8
76.d even 2 1 inner 2736.1.b.b 8
228.b odd 2 1 inner 2736.1.b.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2736.1.b.b 8 1.a even 1 1 trivial
2736.1.b.b 8 3.b odd 2 1 inner
2736.1.b.b 8 4.b odd 2 1 inner
2736.1.b.b 8 12.b even 2 1 inner
2736.1.b.b 8 19.b odd 2 1 CM
2736.1.b.b 8 57.d even 2 1 inner
2736.1.b.b 8 76.d even 2 1 inner
2736.1.b.b 8 228.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 4 T_{5}^{2} + 1 \) acting on \(S_{1}^{\mathrm{new}}(2736, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( ( 1 + 4 T^{2} + T^{4} )^{2} \)
$7$ \( ( 1 + T^{2} )^{4} \)
$11$ \( ( 1 - 4 T^{2} + T^{4} )^{2} \)
$13$ \( T^{8} \)
$17$ \( ( 1 + 4 T^{2} + T^{4} )^{2} \)
$19$ \( ( 1 + T^{2} )^{4} \)
$23$ \( ( -2 + T^{2} )^{4} \)
$29$ \( T^{8} \)
$31$ \( T^{8} \)
$37$ \( T^{8} \)
$41$ \( T^{8} \)
$43$ \( ( 3 + T^{2} )^{4} \)
$47$ \( ( 1 - 4 T^{2} + T^{4} )^{2} \)
$53$ \( T^{8} \)
$59$ \( T^{8} \)
$61$ \( ( -3 + T^{2} )^{4} \)
$67$ \( T^{8} \)
$71$ \( T^{8} \)
$73$ \( ( 1 + T )^{8} \)
$79$ \( T^{8} \)
$83$ \( ( -2 + T^{2} )^{4} \)
$89$ \( T^{8} \)
$97$ \( T^{8} \)
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