Properties

Label 3024.2.b.r
Level 3024
Weight 2
Character orbit 3024.b
Analytic conductor 24.147
Analytic rank 0
Dimension 4
CM no
Inner twists 4

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(24.1467615712\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{7})\)
Defining polynomial: \(x^{4} + 7 x^{2} + 49\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + 2 \beta_{2} q^{5} + \beta_{1} q^{7} +O(q^{10})\) \( q + 2 \beta_{2} q^{5} + \beta_{1} q^{7} -\beta_{3} q^{13} + \beta_{2} q^{17} + 2 \beta_{1} q^{19} -\beta_{3} q^{23} -7 q^{25} -3 \beta_{1} q^{29} + \beta_{1} q^{31} -2 \beta_{3} q^{35} + 4 q^{37} -4 \beta_{2} q^{41} + \beta_{2} q^{43} -6 q^{47} + 7 q^{49} + 3 \beta_{1} q^{53} -3 q^{59} -2 \beta_{3} q^{61} -6 \beta_{1} q^{65} -7 \beta_{2} q^{67} -\beta_{3} q^{71} + 2 \beta_{3} q^{73} + 4 \beta_{2} q^{79} + 12 q^{83} -6 q^{85} -3 \beta_{2} q^{89} + 7 \beta_{2} q^{91} -4 \beta_{3} q^{95} + 2 \beta_{3} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 28q^{25} + 16q^{37} - 24q^{47} + 28q^{49} - 12q^{59} + 48q^{83} - 24q^{85} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 7 x^{2} + 49\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} \)\(/7\)
\(\beta_{2}\)\(=\)\((\)\( 2 \nu^{2} + 7 \)\()/7\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} + 14 \nu \)\()/7\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(7 \beta_{2} - 7\)\()/2\)
\(\nu^{3}\)\(=\)\(7 \beta_{1}\)

Character values

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

\(n\) \(757\) \(785\) \(1135\) \(2593\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1567.1
1.32288 2.29129i
−1.32288 + 2.29129i
1.32288 + 2.29129i
−1.32288 2.29129i
0 0 0 3.46410i 0 −2.64575 0 0 0
1567.2 0 0 0 3.46410i 0 2.64575 0 0 0
1567.3 0 0 0 3.46410i 0 −2.64575 0 0 0
1567.4 0 0 0 3.46410i 0 2.64575 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
12.b even 2 1 inner
21.c even 2 1 inner
28.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3024.2.b.r 4
3.b odd 2 1 3024.2.b.s yes 4
4.b odd 2 1 3024.2.b.s yes 4
7.b odd 2 1 3024.2.b.s yes 4
12.b even 2 1 inner 3024.2.b.r 4
21.c even 2 1 inner 3024.2.b.r 4
28.d even 2 1 inner 3024.2.b.r 4
84.h odd 2 1 3024.2.b.s yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3024.2.b.r 4 1.a even 1 1 trivial
3024.2.b.r 4 12.b even 2 1 inner
3024.2.b.r 4 21.c even 2 1 inner
3024.2.b.r 4 28.d even 2 1 inner
3024.2.b.s yes 4 3.b odd 2 1
3024.2.b.s yes 4 4.b odd 2 1
3024.2.b.s yes 4 7.b odd 2 1
3024.2.b.s yes 4 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3024, [\chi])\):

\( T_{5}^{2} + 12 \)
\( T_{11} \)
\( T_{13}^{2} + 21 \)
\( T_{17}^{2} + 3 \)
\( T_{19}^{2} - 28 \)
\( T_{29}^{2} - 63 \)
\( T_{47} + 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( ( 1 + 2 T^{2} + 25 T^{4} )^{2} \)
$7$ \( ( 1 - 7 T^{2} )^{2} \)
$11$ \( ( 1 - 11 T^{2} )^{4} \)
$13$ \( ( 1 - 5 T^{2} + 169 T^{4} )^{2} \)
$17$ \( ( 1 - 31 T^{2} + 289 T^{4} )^{2} \)
$19$ \( ( 1 + 10 T^{2} + 361 T^{4} )^{2} \)
$23$ \( ( 1 - 25 T^{2} + 529 T^{4} )^{2} \)
$29$ \( ( 1 - 5 T^{2} + 841 T^{4} )^{2} \)
$31$ \( ( 1 + 55 T^{2} + 961 T^{4} )^{2} \)
$37$ \( ( 1 - 4 T + 37 T^{2} )^{4} \)
$41$ \( ( 1 - 34 T^{2} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 - 13 T + 43 T^{2} )^{2}( 1 + 13 T + 43 T^{2} )^{2} \)
$47$ \( ( 1 + 6 T + 47 T^{2} )^{4} \)
$53$ \( ( 1 + 43 T^{2} + 2809 T^{4} )^{2} \)
$59$ \( ( 1 + 3 T + 59 T^{2} )^{4} \)
$61$ \( ( 1 - 38 T^{2} + 3721 T^{4} )^{2} \)
$67$ \( ( 1 - 11 T + 67 T^{2} )^{2}( 1 + 11 T + 67 T^{2} )^{2} \)
$71$ \( ( 1 - 121 T^{2} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 - 62 T^{2} + 5329 T^{4} )^{2} \)
$79$ \( ( 1 - 110 T^{2} + 6241 T^{4} )^{2} \)
$83$ \( ( 1 - 12 T + 83 T^{2} )^{4} \)
$89$ \( ( 1 - 151 T^{2} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 - 110 T^{2} + 9409 T^{4} )^{2} \)
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