Properties

Label 6012.2.a.a
Level 6012
Weight 2
Character orbit 6012.a
Self dual Yes
Analytic conductor 48.006
Analytic rank 0
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6012 = 2^{2} \cdot 3^{2} \cdot 167 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6012.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.0060616952\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{13})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + 3 q^{5} \) \( + ( 2 - \beta ) q^{7} \) \(+O(q^{10})\) \( q\) \( + 3 q^{5} \) \( + ( 2 - \beta ) q^{7} \) \( + ( 5 - \beta ) q^{13} \) \( -\beta q^{17} \) \( + 2 q^{19} \) \( + \beta q^{23} \) \( + 4 q^{25} \) \( + ( -3 - 2 \beta ) q^{29} \) \( + ( 2 + 2 \beta ) q^{31} \) \( + ( 6 - 3 \beta ) q^{35} \) \( + ( 5 - 2 \beta ) q^{37} \) \( + ( -3 + 4 \beta ) q^{41} \) \( + ( 5 + 2 \beta ) q^{43} \) \( + ( 3 - 2 \beta ) q^{47} \) \( -3 \beta q^{49} \) \( + 2 \beta q^{53} \) \( + ( 6 - 6 \beta ) q^{59} \) \( + ( -7 + 6 \beta ) q^{61} \) \( + ( 15 - 3 \beta ) q^{65} \) \( + ( -1 + 4 \beta ) q^{67} \) \( + ( 3 - 3 \beta ) q^{71} \) \( + ( -10 + 3 \beta ) q^{73} \) \( + ( 11 - 2 \beta ) q^{79} \) \( + ( -3 + 2 \beta ) q^{83} \) \( -3 \beta q^{85} \) \( + 6 \beta q^{89} \) \( + ( 13 - 6 \beta ) q^{91} \) \( + 6 q^{95} \) \( + ( -7 + 5 \beta ) q^{97} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut 6q^{5} \) \(\mathstrut +\mathstrut 3q^{7} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 6q^{5} \) \(\mathstrut +\mathstrut 3q^{7} \) \(\mathstrut +\mathstrut 9q^{13} \) \(\mathstrut -\mathstrut q^{17} \) \(\mathstrut +\mathstrut 4q^{19} \) \(\mathstrut +\mathstrut q^{23} \) \(\mathstrut +\mathstrut 8q^{25} \) \(\mathstrut -\mathstrut 8q^{29} \) \(\mathstrut +\mathstrut 6q^{31} \) \(\mathstrut +\mathstrut 9q^{35} \) \(\mathstrut +\mathstrut 8q^{37} \) \(\mathstrut -\mathstrut 2q^{41} \) \(\mathstrut +\mathstrut 12q^{43} \) \(\mathstrut +\mathstrut 4q^{47} \) \(\mathstrut -\mathstrut 3q^{49} \) \(\mathstrut +\mathstrut 2q^{53} \) \(\mathstrut +\mathstrut 6q^{59} \) \(\mathstrut -\mathstrut 8q^{61} \) \(\mathstrut +\mathstrut 27q^{65} \) \(\mathstrut +\mathstrut 2q^{67} \) \(\mathstrut +\mathstrut 3q^{71} \) \(\mathstrut -\mathstrut 17q^{73} \) \(\mathstrut +\mathstrut 20q^{79} \) \(\mathstrut -\mathstrut 4q^{83} \) \(\mathstrut -\mathstrut 3q^{85} \) \(\mathstrut +\mathstrut 6q^{89} \) \(\mathstrut +\mathstrut 20q^{91} \) \(\mathstrut +\mathstrut 12q^{95} \) \(\mathstrut -\mathstrut 9q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
2.30278
−1.30278
0 0 0 3.00000 0 −0.302776 0 0 0
1.2 0 0 0 3.00000 0 3.30278 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
\(2\) \(-1\)
\(3\) \(-1\)
\(167\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{5} \) \(\mathstrut -\mathstrut 3 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6012))\).