Properties

Label 6039.2.a.a
Level $6039$
Weight $2$
Character orbit 6039.a
Self dual yes
Analytic conductor $48.222$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

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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: \(5\)
Coefficient field: 5.5.24217.1
Defining polynomial: \(x^{5} - 5 x^{3} - x^{2} + 3 x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 671)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + ( -2 \beta_{1} - \beta_{2} + \beta_{4} ) q^{4} + \beta_{2} q^{5} + ( -1 + \beta_{1} + 2 \beta_{2} - \beta_{4} ) q^{7} + ( 2 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{8} +O(q^{10})\) \( q -\beta_{1} q^{2} + ( -2 \beta_{1} - \beta_{2} + \beta_{4} ) q^{4} + \beta_{2} q^{5} + ( -1 + \beta_{1} + 2 \beta_{2} - \beta_{4} ) q^{7} + ( 2 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{8} + ( 1 - \beta_{3} ) q^{10} - q^{11} + ( -2 + \beta_{1} - \beta_{4} ) q^{13} + ( 1 + 3 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{14} + ( -2 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{16} + ( 1 - \beta_{2} ) q^{17} + ( -3 + 2 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{19} + ( -\beta_{1} - \beta_{2} ) q^{20} + \beta_{1} q^{22} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} ) q^{23} + ( -3 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{25} + ( -1 + 4 \beta_{1} + \beta_{2} - \beta_{4} ) q^{26} + ( -2 + 3 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} ) q^{28} + ( 3 - \beta_{1} - 4 \beta_{2} + 3 \beta_{3} + \beta_{4} ) q^{29} + ( -1 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{31} + ( -1 - \beta_{3} - 2 \beta_{4} ) q^{32} + ( -1 - \beta_{1} + \beta_{3} ) q^{34} + ( 3 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{35} + ( -3 \beta_{1} - 3 \beta_{2} + \beta_{3} + 3 \beta_{4} ) q^{37} + ( -1 + 7 \beta_{1} + 3 \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{38} + ( -1 - 2 \beta_{1} - \beta_{2} + 3 \beta_{3} + \beta_{4} ) q^{40} + ( 1 + \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{41} + ( 2 - 3 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} ) q^{43} + ( 2 \beta_{1} + \beta_{2} - \beta_{4} ) q^{44} + ( -1 + 2 \beta_{1} - 2 \beta_{3} - \beta_{4} ) q^{46} + ( 1 + 3 \beta_{1} + \beta_{2} - \beta_{4} ) q^{47} + ( \beta_{1} - \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{49} + ( -1 + 5 \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} ) q^{50} + ( -2 + 7 \beta_{1} + 4 \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{52} + ( -1 + 3 \beta_{1} + \beta_{2} - 5 \beta_{3} - 3 \beta_{4} ) q^{53} -\beta_{2} q^{55} + ( -6 + 2 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} - \beta_{4} ) q^{56} + ( -3 - 5 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} + \beta_{4} ) q^{58} + ( 2 - 2 \beta_{1} + 3 \beta_{3} ) q^{59} + q^{61} + ( -5 + 3 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{62} + ( 2 + 5 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} ) q^{64} + ( -1 - 2 \beta_{2} ) q^{65} + ( -1 + 4 \beta_{1} - 4 \beta_{4} ) q^{67} + ( -\beta_{1} + \beta_{4} ) q^{68} + ( -3 + \beta_{1} + 4 \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{70} + ( -1 - \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{71} + ( -2 - 2 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} + \beta_{4} ) q^{73} + ( -6 \beta_{1} - 4 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} ) q^{74} + ( -3 + 11 \beta_{1} + 6 \beta_{2} - \beta_{3} - 3 \beta_{4} ) q^{76} + ( 1 - \beta_{1} - 2 \beta_{2} + \beta_{4} ) q^{77} + ( -5 - 2 \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{79} + ( 2 - \beta_{1} - 3 \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{80} + ( 1 + \beta_{1} + 3 \beta_{2} - \beta_{3} - \beta_{4} ) q^{82} + ( 1 - 4 \beta_{1} - 4 \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{83} + ( -2 - \beta_{1} + \beta_{3} ) q^{85} + ( -1 - 8 \beta_{1} - 5 \beta_{2} + 4 \beta_{3} + 3 \beta_{4} ) q^{86} + ( -2 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{88} + ( 4 - \beta_{1} - 5 \beta_{2} - 2 \beta_{3} ) q^{89} + ( 2 - 4 \beta_{1} - 5 \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{91} + ( -3 + 3 \beta_{1} - 2 \beta_{3} - 4 \beta_{4} ) q^{92} + ( -4 + 5 \beta_{1} + 3 \beta_{2} - \beta_{3} - 3 \beta_{4} ) q^{94} + ( -1 + \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{95} + ( -1 + 6 \beta_{1} + 4 \beta_{2} - 7 \beta_{3} - 2 \beta_{4} ) q^{97} + ( -4 + 2 \beta_{1} + 4 \beta_{2} + \beta_{3} - \beta_{4} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q + 2q^{2} + 2q^{5} - q^{7} + 6q^{8} + O(q^{10}) \) \( 5q + 2q^{2} + 2q^{5} - q^{7} + 6q^{8} + 5q^{10} - 5q^{11} - 10q^{13} + 3q^{14} + 2q^{16} + 3q^{17} - 13q^{19} - 2q^{22} - 15q^{25} - 9q^{26} - 12q^{28} + 7q^{29} - 13q^{31} - q^{32} - 3q^{34} + 13q^{35} - 6q^{37} - 9q^{38} - 5q^{40} + 9q^{41} + 2q^{43} - 7q^{46} + 3q^{47} - 6q^{49} - 9q^{50} - 12q^{52} - 3q^{53} - 2q^{55} - 28q^{56} - 15q^{58} + 14q^{59} + 5q^{61} - 31q^{62} + 6q^{64} - 9q^{65} - 5q^{67} - 5q^{70} - 3q^{71} - 4q^{73} - 2q^{74} - 19q^{76} + q^{77} - 27q^{79} + 2q^{80} + 11q^{82} + 3q^{83} - 8q^{85} - 5q^{86} - 6q^{88} + 12q^{89} + 4q^{91} - 13q^{92} - 18q^{94} - 7q^{95} - 5q^{97} - 14q^{98} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{4} - 5 \nu^{2} + 2 \)
\(\beta_{2}\)\(=\)\( -\nu^{4} + 5 \nu^{2} + \nu - 2 \)
\(\beta_{3}\)\(=\)\( -\nu^{4} + \nu^{3} + 5 \nu^{2} - 3 \nu - 3 \)
\(\beta_{4}\)\(=\)\( 2 \nu^{4} - \nu^{3} - 9 \nu^{2} + 3 \nu + 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{2} + \beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4} + \beta_{3} - \beta_{1} + 2\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 3 \beta_{2} + 4 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(5 \beta_{4} + 5 \beta_{3} - 4 \beta_{1} + 8\)

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
−0.369680
2.17442
−0.722813
0.878095
−1.96003
−1.33536 0 −0.216816 −1.70504 0 −3.82358 2.96025 0 2.27684
1.2 −0.714533 0 −1.48944 1.45989 0 1.23480 2.49332 0 −1.04314
1.3 0.339328 0 −1.88486 −0.383484 0 0.840700 −1.31824 0 −0.130127
1.4 1.26073 0 −0.410549 2.13883 0 2.81011 −3.03906 0 2.69649
1.5 2.44983 0 4.00166 0.489803 0 −2.06203 4.90374 0 1.19993
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6039.2.a.a 5
3.b odd 2 1 671.2.a.a 5
33.d even 2 1 7381.2.a.g 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
671.2.a.a 5 3.b odd 2 1
6039.2.a.a 5 1.a even 1 1 trivial
7381.2.a.g 5 33.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{5} - 2 T_{2}^{4} - 3 T_{2}^{3} + 4 T_{2}^{2} + 2 T_{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6039))\).