Properties

Label 6027.2.a.n
Level 6027
Weight 2
Character orbit 6027.a
Self dual Yes
Analytic conductor 48.126
Analytic rank 0
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6027 = 3 \cdot 7^{2} \cdot 41 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6027.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.1258372982\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \(+ q^{3}\) \( -2 q^{4} \) \( -2 \beta q^{5} \) \(+ q^{9}\) \(+O(q^{10})\) \( q\) \(+ q^{3}\) \( -2 q^{4} \) \( -2 \beta q^{5} \) \(+ q^{9}\) \( + ( 4 - \beta ) q^{11} \) \( -2 q^{12} \) \( + ( -2 + 2 \beta ) q^{13} \) \( -2 \beta q^{15} \) \( + 4 q^{16} \) \( -5 q^{17} \) \( + ( -4 - 2 \beta ) q^{19} \) \( + 4 \beta q^{20} \) \( + ( 2 - 2 \beta ) q^{23} \) \( + 7 q^{25} \) \(+ q^{27}\) \( + 3 \beta q^{29} \) \( + ( -4 + \beta ) q^{31} \) \( + ( 4 - \beta ) q^{33} \) \( -2 q^{36} \) \(- q^{37}\) \( + ( -2 + 2 \beta ) q^{39} \) \(+ q^{41}\) \( + ( 3 - 4 \beta ) q^{43} \) \( + ( -8 + 2 \beta ) q^{44} \) \( -2 \beta q^{45} \) \( + ( 5 - 4 \beta ) q^{47} \) \( + 4 q^{48} \) \( -5 q^{51} \) \( + ( 4 - 4 \beta ) q^{52} \) \( + ( 4 - 4 \beta ) q^{53} \) \( + ( 6 - 8 \beta ) q^{55} \) \( + ( -4 - 2 \beta ) q^{57} \) \( -4 \beta q^{59} \) \( + 4 \beta q^{60} \) \( -\beta q^{61} \) \( -8 q^{64} \) \( + ( -12 + 4 \beta ) q^{65} \) \( + ( 4 - 2 \beta ) q^{67} \) \( + 10 q^{68} \) \( + ( 2 - 2 \beta ) q^{69} \) \( + ( 8 - 3 \beta ) q^{71} \) \( + ( 4 - 3 \beta ) q^{73} \) \( + 7 q^{75} \) \( + ( 8 + 4 \beta ) q^{76} \) \( + ( -6 - 4 \beta ) q^{79} \) \( -8 \beta q^{80} \) \(+ q^{81}\) \( -10 q^{83} \) \( + 10 \beta q^{85} \) \( + 3 \beta q^{87} \) \( + ( -2 + 4 \beta ) q^{89} \) \( + ( -4 + 4 \beta ) q^{92} \) \( + ( -4 + \beta ) q^{93} \) \( + ( 12 + 8 \beta ) q^{95} \) \( + ( 8 + 6 \beta ) q^{97} \) \( + ( 4 - \beta ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut 2q^{3} \) \(\mathstrut -\mathstrut 4q^{4} \) \(\mathstrut +\mathstrut 2q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 2q^{3} \) \(\mathstrut -\mathstrut 4q^{4} \) \(\mathstrut +\mathstrut 2q^{9} \) \(\mathstrut +\mathstrut 8q^{11} \) \(\mathstrut -\mathstrut 4q^{12} \) \(\mathstrut -\mathstrut 4q^{13} \) \(\mathstrut +\mathstrut 8q^{16} \) \(\mathstrut -\mathstrut 10q^{17} \) \(\mathstrut -\mathstrut 8q^{19} \) \(\mathstrut +\mathstrut 4q^{23} \) \(\mathstrut +\mathstrut 14q^{25} \) \(\mathstrut +\mathstrut 2q^{27} \) \(\mathstrut -\mathstrut 8q^{31} \) \(\mathstrut +\mathstrut 8q^{33} \) \(\mathstrut -\mathstrut 4q^{36} \) \(\mathstrut -\mathstrut 2q^{37} \) \(\mathstrut -\mathstrut 4q^{39} \) \(\mathstrut +\mathstrut 2q^{41} \) \(\mathstrut +\mathstrut 6q^{43} \) \(\mathstrut -\mathstrut 16q^{44} \) \(\mathstrut +\mathstrut 10q^{47} \) \(\mathstrut +\mathstrut 8q^{48} \) \(\mathstrut -\mathstrut 10q^{51} \) \(\mathstrut +\mathstrut 8q^{52} \) \(\mathstrut +\mathstrut 8q^{53} \) \(\mathstrut +\mathstrut 12q^{55} \) \(\mathstrut -\mathstrut 8q^{57} \) \(\mathstrut -\mathstrut 16q^{64} \) \(\mathstrut -\mathstrut 24q^{65} \) \(\mathstrut +\mathstrut 8q^{67} \) \(\mathstrut +\mathstrut 20q^{68} \) \(\mathstrut +\mathstrut 4q^{69} \) \(\mathstrut +\mathstrut 16q^{71} \) \(\mathstrut +\mathstrut 8q^{73} \) \(\mathstrut +\mathstrut 14q^{75} \) \(\mathstrut +\mathstrut 16q^{76} \) \(\mathstrut -\mathstrut 12q^{79} \) \(\mathstrut +\mathstrut 2q^{81} \) \(\mathstrut -\mathstrut 20q^{83} \) \(\mathstrut -\mathstrut 4q^{89} \) \(\mathstrut -\mathstrut 8q^{92} \) \(\mathstrut -\mathstrut 8q^{93} \) \(\mathstrut +\mathstrut 24q^{95} \) \(\mathstrut +\mathstrut 16q^{97} \) \(\mathstrut +\mathstrut 8q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
1.73205
−1.73205
0 1.00000 −2.00000 −3.46410 0 0 0 1.00000 0
1.2 0 1.00000 −2.00000 3.46410 0 0 0 1.00000 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
\(3\) \(-1\)
\(7\) \(-1\)
\(41\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6027))\):

\(T_{2} \)
\(T_{5}^{2} \) \(\mathstrut -\mathstrut 12 \)
\(T_{13}^{2} \) \(\mathstrut +\mathstrut 4 T_{13} \) \(\mathstrut -\mathstrut 8 \)