Properties

Label 3024.1.dc.c
Level $3024$
Weight $1$
Character orbit 3024.dc
Analytic conductor $1.509$
Analytic rank $0$
Dimension $4$
Projective image $S_{4}$
CM/RM no
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.dc (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.50917259820\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} - 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1512)
Projective image \(S_{4}\)
Projective field Galois closure of 4.2.21168.3

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{5} + \beta_{2} q^{7} +O(q^{10})\) \( q + \beta_{1} q^{5} + \beta_{2} q^{7} + ( \beta_{1} - \beta_{3} ) q^{17} + ( 1 - \beta_{2} ) q^{19} -\beta_{1} q^{23} + \beta_{2} q^{25} -\beta_{3} q^{29} + \beta_{2} q^{31} + \beta_{3} q^{35} + \beta_{3} q^{41} + q^{43} -\beta_{1} q^{47} + ( -1 + \beta_{2} ) q^{49} + ( -\beta_{1} + \beta_{3} ) q^{53} + ( -1 + \beta_{2} ) q^{61} -\beta_{2} q^{73} + ( -2 + 2 \beta_{2} ) q^{79} + 2 q^{85} -\beta_{1} q^{89} + ( \beta_{1} - \beta_{3} ) q^{95} + q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{7} + O(q^{10}) \) \( 4q + 2q^{7} + 2q^{19} + 2q^{25} + 2q^{31} + 4q^{43} - 2q^{49} - 2q^{61} - 2q^{73} - 4q^{79} + 8q^{85} + 4q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{2}\)
\(\nu^{3}\)\(=\)\(2 \beta_{3}\)

Character values

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

\(n\) \(757\) \(785\) \(1135\) \(2593\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1 + \beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2321.1
−1.22474 + 0.707107i
1.22474 0.707107i
−1.22474 0.707107i
1.22474 + 0.707107i
0 0 0 −1.22474 + 0.707107i 0 0.500000 0.866025i 0 0 0
2321.2 0 0 0 1.22474 0.707107i 0 0.500000 0.866025i 0 0 0
2753.1 0 0 0 −1.22474 0.707107i 0 0.500000 + 0.866025i 0 0 0
2753.2 0 0 0 1.22474 + 0.707107i 0 0.500000 + 0.866025i 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 inner
7.c even 3 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3024.1.dc.c 4
3.b odd 2 1 inner 3024.1.dc.c 4
4.b odd 2 1 1512.1.cu.a 4
7.c even 3 1 inner 3024.1.dc.c 4
12.b even 2 1 1512.1.cu.a 4
21.h odd 6 1 inner 3024.1.dc.c 4
28.g odd 6 1 1512.1.cu.a 4
84.n even 6 1 1512.1.cu.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1512.1.cu.a 4 4.b odd 2 1
1512.1.cu.a 4 12.b even 2 1
1512.1.cu.a 4 28.g odd 6 1
1512.1.cu.a 4 84.n even 6 1
3024.1.dc.c 4 1.a even 1 1 trivial
3024.1.dc.c 4 3.b odd 2 1 inner
3024.1.dc.c 4 7.c even 3 1 inner
3024.1.dc.c 4 21.h odd 6 1 inner

Hecke kernels

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

\( T_{5}^{4} - 2 T_{5}^{2} + 4 \)
\( T_{13} \)

Hecke characteristic polynomials

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