Properties

Label 3024.2.b.q
Level 3024
Weight 2
Character orbit 3024.b
Analytic conductor 24.147
Analytic rank 0
Dimension 4
CM discriminant -84
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: \(\Q(\sqrt{-6}, \sqrt{7})\)
Defining polynomial: \(x^{4} + 24 x^{2} + 81\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{5} -\beta_{2} q^{7} +O(q^{10})\) \( q + \beta_{1} q^{5} -\beta_{2} q^{7} -\beta_{1} q^{11} + ( \beta_{1} - \beta_{3} ) q^{17} + ( -6 + \beta_{2} ) q^{19} + \beta_{3} q^{23} + ( -7 + 3 \beta_{2} ) q^{25} + ( -3 + 2 \beta_{2} ) q^{31} + ( 2 \beta_{1} - \beta_{3} ) q^{35} + ( 4 + 3 \beta_{2} ) q^{37} -\beta_{3} q^{41} + 7 q^{49} + ( 12 - 3 \beta_{2} ) q^{55} + ( \beta_{1} - 2 \beta_{3} ) q^{71} + ( -2 \beta_{1} + \beta_{3} ) q^{77} + ( -9 + 9 \beta_{2} ) q^{85} + ( 4 \beta_{1} + \beta_{3} ) q^{89} + ( -8 \beta_{1} + \beta_{3} ) q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 24q^{19} - 28q^{25} - 12q^{31} + 16q^{37} + 28q^{49} + 48q^{55} - 36q^{85} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 24 x^{2} + 81\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{2} + 12 \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} + 18 \nu \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(3 \beta_{2} - 12\)
\(\nu^{3}\)\(=\)\(3 \beta_{3} - 18 \beta_{1}\)

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
4.46512i
2.01563i
2.01563i
4.46512i
0 0 0 4.46512i 0 2.64575 0 0 0
1567.2 0 0 0 2.01563i 0 −2.64575 0 0 0
1567.3 0 0 0 2.01563i 0 −2.64575 0 0 0
1567.4 0 0 0 4.46512i 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
84.h odd 2 1 CM by \(\Q(\sqrt{-21}) \)
3.b odd 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.q 4
3.b odd 2 1 inner 3024.2.b.q 4
4.b odd 2 1 3024.2.b.t yes 4
7.b odd 2 1 3024.2.b.t yes 4
12.b even 2 1 3024.2.b.t yes 4
21.c even 2 1 3024.2.b.t yes 4
28.d even 2 1 inner 3024.2.b.q 4
84.h odd 2 1 CM 3024.2.b.q 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3024.2.b.q 4 1.a even 1 1 trivial
3024.2.b.q 4 3.b odd 2 1 inner
3024.2.b.q 4 28.d even 2 1 inner
3024.2.b.q 4 84.h odd 2 1 CM
3024.2.b.t yes 4 4.b odd 2 1
3024.2.b.t yes 4 7.b odd 2 1
3024.2.b.t yes 4 12.b even 2 1
3024.2.b.t yes 4 21.c even 2 1

Hecke kernels

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

\( T_{5}^{4} + 24 T_{5}^{2} + 81 \)
\( T_{11}^{4} + 24 T_{11}^{2} + 81 \)
\( T_{13} \)
\( T_{17}^{2} + 54 \)
\( T_{19}^{2} + 12 T_{19} + 29 \)
\( T_{29} \)
\( T_{47} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 + 4 T^{2} - 9 T^{4} + 100 T^{6} + 625 T^{8} \)
$7$ \( ( 1 - 7 T^{2} )^{2} \)
$11$ \( 1 - 20 T^{2} + 279 T^{4} - 2420 T^{6} + 14641 T^{8} \)
$13$ \( ( 1 - 13 T^{2} )^{4} \)
$17$ \( ( 1 + 20 T^{2} + 289 T^{4} )^{2} \)
$19$ \( ( 1 + 12 T + 67 T^{2} + 228 T^{3} + 361 T^{4} )^{2} \)
$23$ \( 1 + 4 T^{2} - 513 T^{4} + 2116 T^{6} + 279841 T^{8} \)
$29$ \( ( 1 + 29 T^{2} )^{4} \)
$31$ \( ( 1 + 6 T + 43 T^{2} + 186 T^{3} + 961 T^{4} )^{2} \)
$37$ \( ( 1 - 8 T + 27 T^{2} - 296 T^{3} + 1369 T^{4} )^{2} \)
$41$ \( 1 - 68 T^{2} + 2943 T^{4} - 114308 T^{6} + 2825761 T^{8} \)
$43$ \( ( 1 - 43 T^{2} )^{4} \)
$47$ \( ( 1 + 47 T^{2} )^{4} \)
$53$ \( ( 1 + 53 T^{2} )^{4} \)
$59$ \( ( 1 + 59 T^{2} )^{4} \)
$61$ \( ( 1 - 61 T^{2} )^{4} \)
$67$ \( ( 1 - 67 T^{2} )^{4} \)
$71$ \( 1 + 100 T^{2} + 4959 T^{4} + 504100 T^{6} + 25411681 T^{8} \)
$73$ \( ( 1 - 73 T^{2} )^{4} \)
$79$ \( ( 1 - 79 T^{2} )^{4} \)
$83$ \( ( 1 + 83 T^{2} )^{4} \)
$89$ \( 1 + 172 T^{2} + 21663 T^{4} + 1362412 T^{6} + 62742241 T^{8} \)
$97$ \( ( 1 - 97 T^{2} )^{4} \)
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