Properties

Label 3024.2.b.f
Level $3024$
Weight $2$
Character orbit 3024.b
Analytic conductor $24.147$
Analytic rank $0$
Dimension $2$
CM discriminant -3
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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -2 + 3 \zeta_{6} ) q^{7} +O(q^{10})\) \( q + ( -2 + 3 \zeta_{6} ) q^{7} + ( 3 - 6 \zeta_{6} ) q^{13} -7 q^{19} + 5 q^{25} + 4 q^{31} -11 q^{37} + ( 6 - 12 \zeta_{6} ) q^{43} + ( -5 - 3 \zeta_{6} ) q^{49} + ( 9 - 18 \zeta_{6} ) q^{61} + ( -9 + 18 \zeta_{6} ) q^{67} + ( 9 - 18 \zeta_{6} ) q^{73} + ( 3 - 6 \zeta_{6} ) q^{79} + ( 12 + 3 \zeta_{6} ) q^{91} + ( 3 - 6 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{7} + O(q^{10}) \) \( 2q - q^{7} - 14q^{19} + 10q^{25} + 8q^{31} - 22q^{37} - 13q^{49} + 27q^{91} + O(q^{100}) \)

Character values

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

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

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
0.500000 0.866025i
0.500000 + 0.866025i
0 0 0 0 0 −0.500000 2.59808i 0 0 0
1567.2 0 0 0 0 0 −0.500000 + 2.59808i 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.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.f 2
3.b odd 2 1 CM 3024.2.b.f 2
4.b odd 2 1 3024.2.b.k yes 2
7.b odd 2 1 3024.2.b.k yes 2
12.b even 2 1 3024.2.b.k yes 2
21.c even 2 1 3024.2.b.k yes 2
28.d even 2 1 inner 3024.2.b.f 2
84.h odd 2 1 inner 3024.2.b.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3024.2.b.f 2 1.a even 1 1 trivial
3024.2.b.f 2 3.b odd 2 1 CM
3024.2.b.f 2 28.d even 2 1 inner
3024.2.b.f 2 84.h odd 2 1 inner
3024.2.b.k yes 2 4.b odd 2 1
3024.2.b.k yes 2 7.b odd 2 1
3024.2.b.k yes 2 12.b even 2 1
3024.2.b.k yes 2 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 \(S_{2}^{\mathrm{new}}(3024, [\chi])\):

\( T_{5} \)
\( T_{11} \)
\( T_{13}^{2} + 27 \)
\( T_{17} \)
\( T_{19} + 7 \)
\( T_{29} \)
\( T_{47} \)