Properties

Label 6039.2.a.b
Level 6039
Weight 2
Character orbit 6039.a
Self dual Yes
Analytic conductor 48.222
Analytic rank 1
Dimension 6
CM No
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 6039 = 3^{2} \cdot 11 \cdot 61 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6039.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.2216577807\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.2661761.1
Coefficient ring: \(\Z[a_1, a_2]\)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( -\beta_{3} q^{2} \) \( -\beta_{5} q^{4} \) \( + ( -\beta_{1} + \beta_{3} - \beta_{5} ) q^{5} \) \( + ( -1 + \beta_{1} ) q^{7} \) \( + ( 1 + \beta_{1} - \beta_{4} ) q^{8} \) \(+O(q^{10})\) \( q\) \( -\beta_{3} q^{2} \) \( -\beta_{5} q^{4} \) \( + ( -\beta_{1} + \beta_{3} - \beta_{5} ) q^{5} \) \( + ( -1 + \beta_{1} ) q^{7} \) \( + ( 1 + \beta_{1} - \beta_{4} ) q^{8} \) \( + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} ) q^{10} \) \(+ q^{11}\) \( + ( -1 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{13} \) \( + ( -\beta_{2} - \beta_{4} ) q^{14} \) \( + ( -1 + \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{16} \) \( + ( 1 + \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{5} ) q^{17} \) \( + ( -\beta_{3} - \beta_{4} + \beta_{5} ) q^{19} \) \( + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} ) q^{20} \) \( -\beta_{3} q^{22} \) \( + ( 1 + \beta_{4} + 2 \beta_{5} ) q^{23} \) \( + ( -2 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{25} \) \( + ( -2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} ) q^{26} \) \( + ( 1 + \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{28} \) \( + \beta_{4} q^{29} \) \( + ( -2 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{31} \) \( + ( -1 - \beta_{1} - \beta_{2} + 3 \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{32} \) \( + ( -3 - 5 \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{34} \) \( + ( -1 + \beta_{1} - \beta_{3} + \beta_{5} ) q^{35} \) \( + ( -2 + \beta_{1} - \beta_{2} - 2 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} ) q^{37} \) \( + ( 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{38} \) \( + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{40} \) \( + ( 1 + \beta_{1} - 2 \beta_{2} + 2 \beta_{4} + \beta_{5} ) q^{41} \) \( + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} + 3 \beta_{4} - \beta_{5} ) q^{43} \) \( -\beta_{5} q^{44} \) \( + ( -1 - 3 \beta_{1} + 3 \beta_{3} + 2 \beta_{4} ) q^{46} \) \( + ( -2 - 2 \beta_{1} - \beta_{2} + \beta_{4} - 4 \beta_{5} ) q^{47} \) \( + ( -4 - \beta_{1} + \beta_{2} ) q^{49} \) \( + ( -2 + 4 \beta_{1} + 2 \beta_{2} - \beta_{4} + 4 \beta_{5} ) q^{50} \) \( + ( 1 - \beta_{1} + \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{52} \) \( + ( 3 + \beta_{1} - \beta_{2} - 4 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} ) q^{53} \) \( + ( -\beta_{1} + \beta_{3} - \beta_{5} ) q^{55} \) \( + ( \beta_{2} + 2 \beta_{4} - \beta_{5} ) q^{56} \) \( + ( 1 - \beta_{1} ) q^{58} \) \( + ( 2 + \beta_{1} - \beta_{2} - 3 \beta_{3} - 2 \beta_{4} + 3 \beta_{5} ) q^{59} \) \(- q^{61}\) \( + ( 2 \beta_{1} + \beta_{2} - \beta_{4} + 2 \beta_{5} ) q^{62} \) \( + ( -1 - \beta_{1} + 3 \beta_{2} + 3 \beta_{3} + \beta_{4} ) q^{64} \) \( + ( 2 - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{65} \) \( + ( -1 - 2 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} ) q^{67} \) \( + ( -3 - \beta_{1} + \beta_{2} + 3 \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{68} \) \( + ( 1 - \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{5} ) q^{70} \) \( + ( 3 - \beta_{3} - 4 \beta_{4} ) q^{71} \) \( + ( -1 + 5 \beta_{1} + 3 \beta_{2} - 3 \beta_{4} + 5 \beta_{5} ) q^{73} \) \( + ( -2 + 4 \beta_{1} - \beta_{2} + 4 \beta_{3} - \beta_{5} ) q^{74} \) \( + ( -2 + \beta_{4} ) q^{76} \) \( + ( -1 + \beta_{1} ) q^{77} \) \( + ( 2 + 4 \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} + 5 \beta_{5} ) q^{79} \) \( + ( -4 + 4 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{80} \) \( + ( 1 + \beta_{1} - \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{82} \) \( + ( -2 - 6 \beta_{1} - 4 \beta_{2} + 3 \beta_{3} + \beta_{4} ) q^{83} \) \( + ( 1 + \beta_{1} - 3 \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{85} \) \( + ( 2 + \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{86} \) \( + ( 1 + \beta_{1} - \beta_{4} ) q^{88} \) \( + ( 4 + \beta_{1} - 4 \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{89} \) \( + ( 2 \beta_{1} - \beta_{2} - 2 \beta_{4} + 2 \beta_{5} ) q^{91} \) \( + ( -6 - 2 \beta_{1} + 3 \beta_{2} + 4 \beta_{3} + \beta_{4} - \beta_{5} ) q^{92} \) \( + ( 5 + 5 \beta_{1} + 2 \beta_{2} - 5 \beta_{3} - 3 \beta_{4} + \beta_{5} ) q^{94} \) \( + ( 2 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{95} \) \( + ( -4 - \beta_{1} - 4 \beta_{2} + \beta_{4} ) q^{97} \) \( + ( -2 \beta_{1} + \beta_{2} + 6 \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(6q \) \(\mathstrut +\mathstrut 2q^{4} \) \(\mathstrut +\mathstrut q^{5} \) \(\mathstrut -\mathstrut 5q^{7} \) \(\mathstrut +\mathstrut 6q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(6q \) \(\mathstrut +\mathstrut 2q^{4} \) \(\mathstrut +\mathstrut q^{5} \) \(\mathstrut -\mathstrut 5q^{7} \) \(\mathstrut +\mathstrut 6q^{8} \) \(\mathstrut -\mathstrut 7q^{10} \) \(\mathstrut +\mathstrut 6q^{11} \) \(\mathstrut -\mathstrut 4q^{13} \) \(\mathstrut -\mathstrut q^{14} \) \(\mathstrut -\mathstrut 10q^{16} \) \(\mathstrut +\mathstrut 5q^{17} \) \(\mathstrut -\mathstrut 3q^{19} \) \(\mathstrut +\mathstrut 6q^{20} \) \(\mathstrut +\mathstrut 3q^{23} \) \(\mathstrut -\mathstrut 11q^{25} \) \(\mathstrut -\mathstrut q^{26} \) \(\mathstrut +\mathstrut 4q^{28} \) \(\mathstrut +\mathstrut q^{29} \) \(\mathstrut -\mathstrut 10q^{31} \) \(\mathstrut -\mathstrut 3q^{32} \) \(\mathstrut -\mathstrut 19q^{34} \) \(\mathstrut -\mathstrut 7q^{35} \) \(\mathstrut -\mathstrut 19q^{37} \) \(\mathstrut +\mathstrut 3q^{38} \) \(\mathstrut +\mathstrut 5q^{40} \) \(\mathstrut +\mathstrut 7q^{41} \) \(\mathstrut -\mathstrut 2q^{43} \) \(\mathstrut +\mathstrut 2q^{44} \) \(\mathstrut -\mathstrut 7q^{46} \) \(\mathstrut -\mathstrut 5q^{47} \) \(\mathstrut -\mathstrut 25q^{49} \) \(\mathstrut -\mathstrut 17q^{50} \) \(\mathstrut +\mathstrut 2q^{52} \) \(\mathstrut +\mathstrut 9q^{53} \) \(\mathstrut +\mathstrut q^{55} \) \(\mathstrut +\mathstrut 4q^{56} \) \(\mathstrut +\mathstrut 5q^{58} \) \(\mathstrut +\mathstrut 5q^{59} \) \(\mathstrut -\mathstrut 6q^{61} \) \(\mathstrut -\mathstrut 3q^{62} \) \(\mathstrut -\mathstrut 6q^{64} \) \(\mathstrut +\mathstrut 11q^{65} \) \(\mathstrut -\mathstrut 14q^{67} \) \(\mathstrut -\mathstrut 18q^{68} \) \(\mathstrut +\mathstrut 7q^{70} \) \(\mathstrut +\mathstrut 14q^{71} \) \(\mathstrut -\mathstrut 14q^{73} \) \(\mathstrut -\mathstrut 6q^{74} \) \(\mathstrut -\mathstrut 11q^{76} \) \(\mathstrut -\mathstrut 5q^{77} \) \(\mathstrut +\mathstrut 5q^{79} \) \(\mathstrut -\mathstrut 26q^{80} \) \(\mathstrut +\mathstrut q^{82} \) \(\mathstrut -\mathstrut 17q^{83} \) \(\mathstrut +\mathstrut 2q^{85} \) \(\mathstrut +\mathstrut 7q^{86} \) \(\mathstrut +\mathstrut 6q^{88} \) \(\mathstrut +\mathstrut 25q^{89} \) \(\mathstrut -\mathstrut 4q^{91} \) \(\mathstrut -\mathstrut 35q^{92} \) \(\mathstrut +\mathstrut 30q^{94} \) \(\mathstrut -\mathstrut 3q^{95} \) \(\mathstrut -\mathstrut 24q^{97} \) \(\mathstrut +\mathstrut 2q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6}\mathstrut -\mathstrut \) \(x^{5}\mathstrut -\mathstrut \) \(6\) \(x^{4}\mathstrut +\mathstrut \) \(3\) \(x^{3}\mathstrut +\mathstrut \) \(9\) \(x^{2}\mathstrut -\mathstrut \) \(x\mathstrut -\mathstrut \) \(1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 2 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 3 \nu + 1 \)
\(\beta_{4}\)\(=\)\( \nu^{4} - 2 \nu^{3} - 3 \nu^{2} + 5 \nu + 1 \)
\(\beta_{5}\)\(=\)\( \nu^{5} - \nu^{4} - 5 \nu^{3} + 2 \nu^{2} + 5 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(2\)
\(\nu^{3}\)\(=\)\(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(4\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(\beta_{4}\mathstrut +\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(5\) \(\beta_{2}\mathstrut +\mathstrut \) \(6\) \(\beta_{1}\mathstrut +\mathstrut \) \(7\)
\(\nu^{5}\)\(=\)\(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(7\) \(\beta_{3}\mathstrut +\mathstrut \) \(8\) \(\beta_{2}\mathstrut +\mathstrut \) \(19\) \(\beta_{1}\mathstrut +\mathstrut \) \(8\)

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
−0.303283
2.36588
−1.34697
0.387870
−1.68584
1.58234
−1.78997 0 1.20400 3.29725 0 −1.30328 1.42482 0 −5.90199
1.2 −1.54773 0 0.395474 −0.422675 0 1.36588 2.48338 0 0.654188
1.3 −0.782747 0 −1.38731 0.742408 0 −2.34697 2.65140 0 −0.581118
1.4 0.255699 0 −1.93462 −2.57819 0 −0.612130 −1.00608 0 −0.659240
1.5 1.57580 0 0.483130 0.593176 0 −2.68584 −2.39028 0 0.934724
1.6 2.28896 0 3.23932 −0.631975 0 0.582341 2.83675 0 −1.44656
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(11\) \(-1\)
\(61\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{6} \) \(\mathstrut -\mathstrut 7 T_{2}^{4} \) \(\mathstrut -\mathstrut 2 T_{2}^{3} \) \(\mathstrut +\mathstrut 12 T_{2}^{2} \) \(\mathstrut +\mathstrut 5 T_{2} \) \(\mathstrut -\mathstrut 2 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6039))\).