Properties

Label 6027.2.a.p
Level 6027
Weight 2
Character orbit 6027.a
Self dual Yes
Analytic conductor 48.126
Analytic rank 1
Dimension 3
CM No
Inner twists 1

Related objects

Downloads

Learn more about

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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.1620.1
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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \(- q^{3}\) \( -2 q^{4} \) \( -\beta_{1} q^{5} \) \(+ q^{9}\) \(+O(q^{10})\) \( q\) \(- q^{3}\) \( -2 q^{4} \) \( -\beta_{1} q^{5} \) \(+ q^{9}\) \( + ( -\beta_{1} + \beta_{2} ) q^{11} \) \( + 2 q^{12} \) \( + ( -1 - \beta_{1} - \beta_{2} ) q^{13} \) \( + \beta_{1} q^{15} \) \( + 4 q^{16} \) \( -2 q^{17} \) \( + ( -3 - \beta_{1} ) q^{19} \) \( + 2 \beta_{1} q^{20} \) \( + ( 2 + \beta_{1} ) q^{23} \) \( + ( 3 + 2 \beta_{1} + \beta_{2} ) q^{25} \) \(- q^{27}\) \( + 2 \beta_{1} q^{29} \) \( + ( 1 + 2 \beta_{1} - \beta_{2} ) q^{31} \) \( + ( \beta_{1} - \beta_{2} ) q^{33} \) \( -2 q^{36} \) \(- q^{37}\) \( + ( 1 + \beta_{1} + \beta_{2} ) q^{39} \) \(- q^{41}\) \( + ( -7 - \beta_{2} ) q^{43} \) \( + ( 2 \beta_{1} - 2 \beta_{2} ) q^{44} \) \( -\beta_{1} q^{45} \) \( + ( -\beta_{1} - 3 \beta_{2} ) q^{47} \) \( -4 q^{48} \) \( + 2 q^{51} \) \( + ( 2 + 2 \beta_{1} + 2 \beta_{2} ) q^{52} \) \( + ( -2 - 2 \beta_{1} - 2 \beta_{2} ) q^{53} \) \( + ( 10 + 2 \beta_{1} + 3 \beta_{2} ) q^{55} \) \( + ( 3 + \beta_{1} ) q^{57} \) \( + ( 6 + 2 \beta_{1} + 2 \beta_{2} ) q^{59} \) \( -2 \beta_{1} q^{60} \) \( + ( 2 + 2 \beta_{1} + 3 \beta_{2} ) q^{61} \) \( -8 q^{64} \) \( + ( 6 + 3 \beta_{1} - \beta_{2} ) q^{65} \) \( + ( 1 - \beta_{1} - \beta_{2} ) q^{67} \) \( + 4 q^{68} \) \( + ( -2 - \beta_{1} ) q^{69} \) \( + ( 2 \beta_{1} + 2 \beta_{2} ) q^{71} \) \( + ( -7 - \beta_{2} ) q^{73} \) \( + ( -3 - 2 \beta_{1} - \beta_{2} ) q^{75} \) \( + ( 6 + 2 \beta_{1} ) q^{76} \) \( + ( 7 - \beta_{1} + 2 \beta_{2} ) q^{79} \) \( -4 \beta_{1} q^{80} \) \(+ q^{81}\) \( + ( -2 - \beta_{1} - 4 \beta_{2} ) q^{83} \) \( + 2 \beta_{1} q^{85} \) \( -2 \beta_{1} q^{87} \) \( + ( 4 + \beta_{1} + 3 \beta_{2} ) q^{89} \) \( + ( -4 - 2 \beta_{1} ) q^{92} \) \( + ( -1 - 2 \beta_{1} + \beta_{2} ) q^{93} \) \( + ( 8 + 5 \beta_{1} + \beta_{2} ) q^{95} \) \( + ( 6 - 4 \beta_{1} ) q^{97} \) \( + ( -\beta_{1} + \beta_{2} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(3q \) \(\mathstrut -\mathstrut 3q^{3} \) \(\mathstrut -\mathstrut 6q^{4} \) \(\mathstrut +\mathstrut 3q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(3q \) \(\mathstrut -\mathstrut 3q^{3} \) \(\mathstrut -\mathstrut 6q^{4} \) \(\mathstrut +\mathstrut 3q^{9} \) \(\mathstrut +\mathstrut 6q^{12} \) \(\mathstrut -\mathstrut 3q^{13} \) \(\mathstrut +\mathstrut 12q^{16} \) \(\mathstrut -\mathstrut 6q^{17} \) \(\mathstrut -\mathstrut 9q^{19} \) \(\mathstrut +\mathstrut 6q^{23} \) \(\mathstrut +\mathstrut 9q^{25} \) \(\mathstrut -\mathstrut 3q^{27} \) \(\mathstrut +\mathstrut 3q^{31} \) \(\mathstrut -\mathstrut 6q^{36} \) \(\mathstrut -\mathstrut 3q^{37} \) \(\mathstrut +\mathstrut 3q^{39} \) \(\mathstrut -\mathstrut 3q^{41} \) \(\mathstrut -\mathstrut 21q^{43} \) \(\mathstrut -\mathstrut 12q^{48} \) \(\mathstrut +\mathstrut 6q^{51} \) \(\mathstrut +\mathstrut 6q^{52} \) \(\mathstrut -\mathstrut 6q^{53} \) \(\mathstrut +\mathstrut 30q^{55} \) \(\mathstrut +\mathstrut 9q^{57} \) \(\mathstrut +\mathstrut 18q^{59} \) \(\mathstrut +\mathstrut 6q^{61} \) \(\mathstrut -\mathstrut 24q^{64} \) \(\mathstrut +\mathstrut 18q^{65} \) \(\mathstrut +\mathstrut 3q^{67} \) \(\mathstrut +\mathstrut 12q^{68} \) \(\mathstrut -\mathstrut 6q^{69} \) \(\mathstrut -\mathstrut 21q^{73} \) \(\mathstrut -\mathstrut 9q^{75} \) \(\mathstrut +\mathstrut 18q^{76} \) \(\mathstrut +\mathstrut 21q^{79} \) \(\mathstrut +\mathstrut 3q^{81} \) \(\mathstrut -\mathstrut 6q^{83} \) \(\mathstrut +\mathstrut 12q^{89} \) \(\mathstrut -\mathstrut 12q^{92} \) \(\mathstrut -\mathstrut 3q^{93} \) \(\mathstrut +\mathstrut 24q^{95} \) \(\mathstrut +\mathstrut 18q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3}\mathstrut -\mathstrut \) \(12\) \(x\mathstrut -\mathstrut \) \(14\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 2 \nu - 8 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(2\) \(\beta_{1}\mathstrut +\mathstrut \) \(8\)

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
3.94338
−1.39091
−2.55247
0 −1.00000 −2.00000 −3.94338 0 0 0 1.00000 0
1.2 0 −1.00000 −2.00000 1.39091 0 0 0 1.00000 0
1.3 0 −1.00000 −2.00000 2.55247 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6027))\):

\(T_{2} \)
\(T_{5}^{3} \) \(\mathstrut -\mathstrut 12 T_{5} \) \(\mathstrut +\mathstrut 14 \)
\(T_{13}^{3} \) \(\mathstrut +\mathstrut 3 T_{13}^{2} \) \(\mathstrut -\mathstrut 15 T_{13} \) \(\mathstrut -\mathstrut 35 \)