Properties

Label 3024.2.b.u
Level $3024$
Weight $2$
Character orbit 3024.b
Analytic conductor $24.147$
Analytic rank $0$
Dimension $8$
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: \(8\)
Coefficient field: 8.0.4161798144.13
Defining polynomial: \(x^{8} + 6 x^{6} + 29 x^{4} + 42 x^{2} + 49\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{3} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{5} + ( -1 - \beta_{5} ) q^{7} +O(q^{10})\) \( q + \beta_{1} q^{5} + ( -1 - \beta_{5} ) q^{7} + \beta_{2} q^{11} + ( \beta_{3} + \beta_{5} ) q^{13} + ( -\beta_{1} + \beta_{2} ) q^{17} + ( 1 + \beta_{4} ) q^{19} + ( -2 \beta_{1} + \beta_{2} ) q^{23} + ( -4 - \beta_{4} ) q^{25} + \beta_{6} q^{29} + ( -1 + \beta_{4} ) q^{31} + ( -\beta_{1} + \beta_{6} ) q^{35} + q^{37} + ( \beta_{1} - 2 \beta_{2} ) q^{41} + ( 2 \beta_{3} + \beta_{5} ) q^{43} + \beta_{7} q^{47} + ( -5 + 2 \beta_{5} ) q^{49} + ( \beta_{6} + \beta_{7} ) q^{53} + ( -3 + 2 \beta_{4} ) q^{55} -\beta_{6} q^{59} -\beta_{3} q^{61} + ( -2 \beta_{6} - \beta_{7} ) q^{65} + ( \beta_{3} + 3 \beta_{5} ) q^{67} + \beta_{1} q^{71} + ( -2 \beta_{3} + 3 \beta_{5} ) q^{73} + ( -\beta_{2} - \beta_{7} ) q^{77} + ( 2 \beta_{3} - \beta_{5} ) q^{79} -\beta_{7} q^{83} + ( 6 + 3 \beta_{4} ) q^{85} + ( -3 \beta_{1} + 2 \beta_{2} ) q^{89} + ( 6 - \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{91} + ( 4 \beta_{1} - 3 \beta_{2} ) q^{95} + ( \beta_{3} + 4 \beta_{5} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{7} + O(q^{10}) \) \( 8q - 8q^{7} + 8q^{19} - 32q^{25} - 8q^{31} + 8q^{37} - 40q^{49} - 24q^{55} + 48q^{85} + 48q^{91} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 6 x^{6} + 29 x^{4} + 42 x^{2} + 49\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -6 \nu^{7} - 29 \nu^{5} - 174 \nu^{3} - 455 \nu \)\()/203\)
\(\beta_{2}\)\(=\)\((\)\( -10 \nu^{7} - 116 \nu^{5} - 493 \nu^{3} - 1232 \nu \)\()/203\)
\(\beta_{3}\)\(=\)\((\)\( 24 \nu^{6} + 116 \nu^{4} + 696 \nu^{2} + 602 \)\()/203\)
\(\beta_{4}\)\(=\)\((\)\( 3 \nu^{6} - 135 \)\()/29\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{6} - 6 \nu^{4} - 22 \nu^{2} - 21 \)\()/7\)
\(\beta_{6}\)\(=\)\((\)\( -54 \nu^{7} - 261 \nu^{5} - 957 \nu^{3} - 441 \nu \)\()/203\)
\(\beta_{7}\)\(=\)\((\)\( 90 \nu^{7} + 435 \nu^{5} + 2001 \nu^{3} + 735 \nu \)\()/203\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{7} - \beta_{6} - 6 \beta_{1}\)\()/12\)
\(\nu^{2}\)\(=\)\((\)\(6 \beta_{5} - 2 \beta_{4} + 9 \beta_{3} - 18\)\()/12\)
\(\nu^{3}\)\(=\)\((\)\(3 \beta_{7} + 5 \beta_{6}\)\()/6\)
\(\nu^{4}\)\(=\)\((\)\(-12 \beta_{5} - 4 \beta_{4} - 11 \beta_{3} - 22\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(-11 \beta_{7} - 23 \beta_{6} - 36 \beta_{2} + 102 \beta_{1}\)\()/12\)
\(\nu^{6}\)\(=\)\((\)\(29 \beta_{4} + 135\)\()/3\)
\(\nu^{7}\)\(=\)\((\)\(-45 \beta_{7} - 103 \beta_{6} + 174 \beta_{2} - 444 \beta_{1}\)\()/12\)

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
1.05050 + 1.81952i
−1.05050 + 1.81952i
−0.629640 + 1.09057i
0.629640 + 1.09057i
0.629640 1.09057i
−0.629640 1.09057i
−1.05050 1.81952i
1.05050 1.81952i
0 0 0 3.63904i 0 −1.00000 2.44949i 0 0 0
1567.2 0 0 0 3.63904i 0 −1.00000 + 2.44949i 0 0 0
1567.3 0 0 0 2.18114i 0 −1.00000 2.44949i 0 0 0
1567.4 0 0 0 2.18114i 0 −1.00000 + 2.44949i 0 0 0
1567.5 0 0 0 2.18114i 0 −1.00000 2.44949i 0 0 0
1567.6 0 0 0 2.18114i 0 −1.00000 + 2.44949i 0 0 0
1567.7 0 0 0 3.63904i 0 −1.00000 2.44949i 0 0 0
1567.8 0 0 0 3.63904i 0 −1.00000 + 2.44949i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1567.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
28.d even 2 1 inner
84.h odd 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.u 8
3.b odd 2 1 inner 3024.2.b.u 8
4.b odd 2 1 3024.2.b.v yes 8
7.b odd 2 1 3024.2.b.v yes 8
12.b even 2 1 3024.2.b.v yes 8
21.c even 2 1 3024.2.b.v yes 8
28.d even 2 1 inner 3024.2.b.u 8
84.h odd 2 1 inner 3024.2.b.u 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3024.2.b.u 8 1.a even 1 1 trivial
3024.2.b.u 8 3.b odd 2 1 inner
3024.2.b.u 8 28.d even 2 1 inner
3024.2.b.u 8 84.h odd 2 1 inner
3024.2.b.v yes 8 4.b odd 2 1
3024.2.b.v yes 8 7.b odd 2 1
3024.2.b.v yes 8 12.b even 2 1
3024.2.b.v yes 8 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 \(S_{2}^{\mathrm{new}}(3024, [\chi])\):

\( T_{5}^{4} + 18 T_{5}^{2} + 63 \)
\( T_{11}^{4} + 30 T_{11}^{2} + 63 \)
\( T_{13}^{4} + 36 T_{13}^{2} + 36 \)
\( T_{17}^{4} + 36 T_{17}^{2} + 252 \)
\( T_{19}^{2} - 2 T_{19} - 17 \)
\( T_{29}^{4} - 108 T_{29}^{2} + 2268 \)
\( T_{47}^{4} - 180 T_{47}^{2} + 2268 \)