Properties

Label 3024.1.cd.a
Level $3024$
Weight $1$
Character orbit 3024.cd
Analytic conductor $1.509$
Analytic rank $0$
Dimension $2$
Projective image $D_{6}$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3024.cd (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.50917259820\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{6}\)
Projective field Galois closure of 6.0.112021056.1

$q$-expansion

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

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3024\mathbb{Z}\right)^\times\).

\(n\) \(757\) \(785\) \(1135\) \(2593\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(\zeta_{6}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1999.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0 0 0 0 −1.00000 0 0 0
2431.1 0 0 0 0 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
28.g odd 6 1 inner
84.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3024.1.cd.a 2
3.b odd 2 1 CM 3024.1.cd.a 2
4.b odd 2 1 3024.1.cd.d yes 2
7.c even 3 1 3024.1.cd.d yes 2
12.b even 2 1 3024.1.cd.d yes 2
21.h odd 6 1 3024.1.cd.d yes 2
28.g odd 6 1 inner 3024.1.cd.a 2
84.n even 6 1 inner 3024.1.cd.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3024.1.cd.a 2 1.a even 1 1 trivial
3024.1.cd.a 2 3.b odd 2 1 CM
3024.1.cd.a 2 28.g odd 6 1 inner
3024.1.cd.a 2 84.n even 6 1 inner
3024.1.cd.d yes 2 4.b odd 2 1
3024.1.cd.d yes 2 7.c even 3 1
3024.1.cd.d yes 2 12.b even 2 1
3024.1.cd.d yes 2 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3024, [\chi])\):

\( T_{13} - 1 \)
\( T_{19} \)
\( T_{31}^{2} - 3 T_{31} + 3 \)

Hecke characteristic polynomials

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