Properties

Label 6048.2.a.bc
Level 6048
Weight 2
Character orbit 6048.a
Self dual Yes
Analytic conductor 48.294
Analytic rank 0
Dimension 2
CM No
Inner twists 1

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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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( -\beta q^{5} \) \(- q^{7}\) \(+O(q^{10})\) \( q\) \( -\beta q^{5} \) \(- q^{7}\) \( -2 \beta q^{11} \) \( + ( -5 + \beta ) q^{13} \) \( + ( 1 - 2 \beta ) q^{17} \) \( + ( -2 + 2 \beta ) q^{19} \) \( + ( -1 - 3 \beta ) q^{23} \) \( + ( -1 + \beta ) q^{25} \) \( + ( -7 - \beta ) q^{29} \) \( + ( -1 + \beta ) q^{31} \) \( + \beta q^{35} \) \( + ( 2 - \beta ) q^{37} \) \( + ( 10 - \beta ) q^{41} \) \( + ( 3 - 4 \beta ) q^{43} \) \( + ( 2 - \beta ) q^{47} \) \(+ q^{49}\) \( + ( -1 + 3 \beta ) q^{53} \) \( + ( 8 + 2 \beta ) q^{55} \) \( + ( 3 + 2 \beta ) q^{59} \) \( + ( 2 - 4 \beta ) q^{61} \) \( + ( -4 + 4 \beta ) q^{65} \) \( + ( 11 - 3 \beta ) q^{67} \) \( + ( 1 - \beta ) q^{71} \) \( + ( -2 - 2 \beta ) q^{73} \) \( + 2 \beta q^{77} \) \( + ( 6 + \beta ) q^{79} \) \( -\beta q^{83} \) \( + ( 8 + \beta ) q^{85} \) \( + ( 9 - 3 \beta ) q^{89} \) \( + ( 5 - \beta ) q^{91} \) \( -8 q^{95} \) \( + ( -12 + 4 \beta ) q^{97} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut q^{5} \) \(\mathstrut -\mathstrut 2q^{7} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut q^{5} \) \(\mathstrut -\mathstrut 2q^{7} \) \(\mathstrut -\mathstrut 2q^{11} \) \(\mathstrut -\mathstrut 9q^{13} \) \(\mathstrut -\mathstrut 2q^{19} \) \(\mathstrut -\mathstrut 5q^{23} \) \(\mathstrut -\mathstrut q^{25} \) \(\mathstrut -\mathstrut 15q^{29} \) \(\mathstrut -\mathstrut q^{31} \) \(\mathstrut +\mathstrut q^{35} \) \(\mathstrut +\mathstrut 3q^{37} \) \(\mathstrut +\mathstrut 19q^{41} \) \(\mathstrut +\mathstrut 2q^{43} \) \(\mathstrut +\mathstrut 3q^{47} \) \(\mathstrut +\mathstrut 2q^{49} \) \(\mathstrut +\mathstrut q^{53} \) \(\mathstrut +\mathstrut 18q^{55} \) \(\mathstrut +\mathstrut 8q^{59} \) \(\mathstrut -\mathstrut 4q^{65} \) \(\mathstrut +\mathstrut 19q^{67} \) \(\mathstrut +\mathstrut q^{71} \) \(\mathstrut -\mathstrut 6q^{73} \) \(\mathstrut +\mathstrut 2q^{77} \) \(\mathstrut +\mathstrut 13q^{79} \) \(\mathstrut -\mathstrut q^{83} \) \(\mathstrut +\mathstrut 17q^{85} \) \(\mathstrut +\mathstrut 15q^{89} \) \(\mathstrut +\mathstrut 9q^{91} \) \(\mathstrut -\mathstrut 16q^{95} \) \(\mathstrut -\mathstrut 20q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
2.56155
−1.56155
0 0 0 −2.56155 0 −1.00000 0 0 0
1.2 0 0 0 1.56155 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6048))\):

\(T_{5}^{2} \) \(\mathstrut +\mathstrut T_{5} \) \(\mathstrut -\mathstrut 4 \)
\(T_{11}^{2} \) \(\mathstrut +\mathstrut 2 T_{11} \) \(\mathstrut -\mathstrut 16 \)
\(T_{13}^{2} \) \(\mathstrut +\mathstrut 9 T_{13} \) \(\mathstrut +\mathstrut 16 \)
\(T_{17}^{2} \) \(\mathstrut -\mathstrut 17 \)