Properties

Label 6048.2.a.ba
Level 6048
Weight 2
Character orbit 6048.a
Self dual Yes
Analytic conductor 48.294
Analytic rank 1
Dimension 2
CM No
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 6048 = 2^{5} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6048.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.2935231425\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta ) q^{5} + q^{7} +O(q^{10})\) \( q + ( -1 + \beta ) q^{5} + q^{7} + ( -1 + \beta ) q^{11} -2 \beta q^{13} + 2 \beta q^{17} + ( 1 - 2 \beta ) q^{19} + ( -1 + 3 \beta ) q^{23} + ( -2 - 2 \beta ) q^{25} -2 \beta q^{29} + ( -1 - 4 \beta ) q^{31} + ( -1 + \beta ) q^{35} + ( 3 + 2 \beta ) q^{37} + ( -5 - 5 \beta ) q^{41} + ( 2 - 2 \beta ) q^{43} + ( -6 - 4 \beta ) q^{47} + q^{49} + 4 q^{53} + ( 3 - 2 \beta ) q^{55} + ( -2 - 4 \beta ) q^{59} + 8 \beta q^{61} + ( -4 + 2 \beta ) q^{65} + ( 4 + 6 \beta ) q^{67} + ( -7 - \beta ) q^{71} + ( -6 - 2 \beta ) q^{73} + ( -1 + \beta ) q^{77} + ( -2 + 2 \beta ) q^{79} + ( -8 + 2 \beta ) q^{83} + ( 4 - 2 \beta ) q^{85} + ( 3 + 3 \beta ) q^{89} -2 \beta q^{91} + ( -5 + 3 \beta ) q^{95} + ( -4 + 4 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{5} + 2q^{7} + O(q^{10}) \) \( 2q - 2q^{5} + 2q^{7} - 2q^{11} + 2q^{19} - 2q^{23} - 4q^{25} - 2q^{31} - 2q^{35} + 6q^{37} - 10q^{41} + 4q^{43} - 12q^{47} + 2q^{49} + 8q^{53} + 6q^{55} - 4q^{59} - 8q^{65} + 8q^{67} - 14q^{71} - 12q^{73} - 2q^{77} - 4q^{79} - 16q^{83} + 8q^{85} + 6q^{89} - 10q^{95} - 8q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.41421
1.41421
0 0 0 −2.41421 0 1.00000 0 0 0
1.2 0 0 0 0.414214 0 1.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(7\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6048))\):

\( T_{5}^{2} + 2 T_{5} - 1 \)
\( T_{11}^{2} + 2 T_{11} - 1 \)
\( T_{13}^{2} - 8 \)
\( T_{17}^{2} - 8 \)