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 $1$

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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: \(6\)
Coefficient field: 6.6.2661761.1
Defining polynomial: \(x^{6} - x^{5} - 6 x^{4} + 3 x^{3} + 9 x^{2} - x - 1\)
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 \(1,\beta_1,\ldots,\beta_{5}\) 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})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 2q^{4} + q^{5} - 5q^{7} + 6q^{8} + O(q^{10}) \) \( 6q + 2q^{4} + q^{5} - 5q^{7} + 6q^{8} - 7q^{10} + 6q^{11} - 4q^{13} - q^{14} - 10q^{16} + 5q^{17} - 3q^{19} + 6q^{20} + 3q^{23} - 11q^{25} - q^{26} + 4q^{28} + q^{29} - 10q^{31} - 3q^{32} - 19q^{34} - 7q^{35} - 19q^{37} + 3q^{38} + 5q^{40} + 7q^{41} - 2q^{43} + 2q^{44} - 7q^{46} - 5q^{47} - 25q^{49} - 17q^{50} + 2q^{52} + 9q^{53} + q^{55} + 4q^{56} + 5q^{58} + 5q^{59} - 6q^{61} - 3q^{62} - 6q^{64} + 11q^{65} - 14q^{67} - 18q^{68} + 7q^{70} + 14q^{71} - 14q^{73} - 6q^{74} - 11q^{76} - 5q^{77} + 5q^{79} - 26q^{80} + q^{82} - 17q^{83} + 2q^{85} + 7q^{86} + 6q^{88} + 25q^{89} - 4q^{91} - 35q^{92} + 30q^{94} - 3q^{95} - 24q^{97} + 2q^{98} + O(q^{100}) \)

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

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

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
−0.303283
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:

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.b 6
3.b odd 2 1 671.2.a.b 6
33.d even 2 1 7381.2.a.h 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
671.2.a.b 6 3.b odd 2 1
6039.2.a.b 6 1.a even 1 1 trivial
7381.2.a.h 6 33.d even 2 1

Hecke kernels

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