Properties

Label 539.2.a.i
Level $539$
Weight $2$
Character orbit 539.a
Self dual yes
Analytic conductor $4.304$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 539 = 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 539.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.30393666895\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.257.1
Defining polynomial: \(x^{3} - x^{2} - 4 x + 3\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 77)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + \beta_{1} q^{3} + ( 1 + \beta_{1} - \beta_{2} ) q^{4} + ( 1 - \beta_{1} + \beta_{2} ) q^{5} + ( \beta_{1} + \beta_{2} ) q^{6} + ( -3 + \beta_{2} ) q^{8} + \beta_{2} q^{9} +O(q^{10})\) \( q + \beta_{2} q^{2} + \beta_{1} q^{3} + ( 1 + \beta_{1} - \beta_{2} ) q^{4} + ( 1 - \beta_{1} + \beta_{2} ) q^{5} + ( \beta_{1} + \beta_{2} ) q^{6} + ( -3 + \beta_{2} ) q^{8} + \beta_{2} q^{9} + ( 3 - \beta_{2} ) q^{10} - q^{11} + 3 q^{12} + ( 4 - \beta_{1} ) q^{13} + ( -3 + 2 \beta_{1} ) q^{15} + ( 1 - \beta_{1} - 2 \beta_{2} ) q^{16} + ( 1 + \beta_{2} ) q^{17} + ( 3 + \beta_{1} - \beta_{2} ) q^{18} + ( 3 + 2 \beta_{1} - \beta_{2} ) q^{19} + ( -5 + \beta_{1} + 2 \beta_{2} ) q^{20} -\beta_{2} q^{22} + ( 4 - \beta_{2} ) q^{23} + ( -2 \beta_{1} + \beta_{2} ) q^{24} + ( 2 - 3 \beta_{1} ) q^{25} + ( -\beta_{1} + 3 \beta_{2} ) q^{26} + ( -2 \beta_{1} + \beta_{2} ) q^{27} + ( -2 - 3 \beta_{1} - \beta_{2} ) q^{29} + ( 2 \beta_{1} - \beta_{2} ) q^{30} + ( 2 - 3 \beta_{1} - \beta_{2} ) q^{31} -3 \beta_{1} q^{32} -\beta_{1} q^{33} + ( 3 + \beta_{1} ) q^{34} + ( -3 + 3 \beta_{2} ) q^{36} + ( -2 + 2 \beta_{1} + 2 \beta_{2} ) q^{37} + ( -3 + \beta_{1} + 6 \beta_{2} ) q^{38} + ( -3 + 4 \beta_{1} - \beta_{2} ) q^{39} + ( 3 \beta_{1} - 4 \beta_{2} ) q^{40} + ( 2 - \beta_{1} - 4 \beta_{2} ) q^{41} + ( 1 - \beta_{1} - 2 \beta_{2} ) q^{43} + ( -1 - \beta_{1} + \beta_{2} ) q^{44} + ( 3 - \beta_{2} ) q^{45} + ( -3 - \beta_{1} + 5 \beta_{2} ) q^{46} + ( 1 + \beta_{2} ) q^{47} + ( -3 - \beta_{1} - 3 \beta_{2} ) q^{48} + ( -3 \beta_{1} - \beta_{2} ) q^{50} + ( 2 \beta_{1} + \beta_{2} ) q^{51} + ( 1 + 4 \beta_{1} - 4 \beta_{2} ) q^{52} + ( 6 - \beta_{1} + 2 \beta_{2} ) q^{53} + ( 3 - \beta_{1} - 3 \beta_{2} ) q^{54} + ( -1 + \beta_{1} - \beta_{2} ) q^{55} + ( 6 + 2 \beta_{1} + \beta_{2} ) q^{57} + ( -3 - 4 \beta_{1} - 4 \beta_{2} ) q^{58} + ( -3 + \beta_{1} - 6 \beta_{2} ) q^{59} + ( 3 - 3 \beta_{1} + 3 \beta_{2} ) q^{60} + ( 8 - 2 \beta_{2} ) q^{61} + ( -3 - 4 \beta_{1} ) q^{62} + ( -2 - \beta_{1} + \beta_{2} ) q^{64} + ( 7 - 6 \beta_{1} + 4 \beta_{2} ) q^{65} + ( -\beta_{1} - \beta_{2} ) q^{66} + ( -6 + 2 \beta_{1} ) q^{67} + ( -2 + \beta_{1} + 2 \beta_{2} ) q^{68} + ( 3 \beta_{1} - \beta_{2} ) q^{69} + ( 1 + 4 \beta_{1} - 3 \beta_{2} ) q^{71} + ( 3 + \beta_{1} - 4 \beta_{2} ) q^{72} + ( 5 + 5 \beta_{1} ) q^{73} + ( 6 + 4 \beta_{1} - 2 \beta_{2} ) q^{74} + ( -9 + 2 \beta_{1} - 3 \beta_{2} ) q^{75} + ( 12 + 3 \beta_{1} - 6 \beta_{2} ) q^{76} + ( -3 + 3 \beta_{1} + 2 \beta_{2} ) q^{78} + ( 3 \beta_{1} - \beta_{2} ) q^{79} + ( -2 - 3 \beta_{1} + 3 \beta_{2} ) q^{80} + ( -6 + \beta_{1} - 4 \beta_{2} ) q^{81} + ( -12 - 5 \beta_{1} + 5 \beta_{2} ) q^{82} + ( 3 + 2 \beta_{1} + \beta_{2} ) q^{83} + ( 4 - \beta_{1} ) q^{85} + ( -6 - 3 \beta_{1} + 2 \beta_{2} ) q^{86} + ( -9 - 3 \beta_{1} - 4 \beta_{2} ) q^{87} + ( 3 - \beta_{2} ) q^{88} + ( -\beta_{1} + \beta_{2} ) q^{89} + ( -3 - \beta_{1} + 4 \beta_{2} ) q^{90} + ( 7 + 4 \beta_{1} - 7 \beta_{2} ) q^{92} + ( -9 + \beta_{1} - 4 \beta_{2} ) q^{93} + ( 3 + \beta_{1} ) q^{94} + ( -6 + \beta_{1} + 4 \beta_{2} ) q^{95} + ( -9 - 3 \beta_{2} ) q^{96} + ( -4 + 3 \beta_{1} ) q^{97} -\beta_{2} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + q^{3} + 4q^{4} + 2q^{5} + q^{6} - 9q^{8} + O(q^{10}) \) \( 3q + q^{3} + 4q^{4} + 2q^{5} + q^{6} - 9q^{8} + 9q^{10} - 3q^{11} + 9q^{12} + 11q^{13} - 7q^{15} + 2q^{16} + 3q^{17} + 10q^{18} + 11q^{19} - 14q^{20} + 12q^{23} - 2q^{24} + 3q^{25} - q^{26} - 2q^{27} - 9q^{29} + 2q^{30} + 3q^{31} - 3q^{32} - q^{33} + 10q^{34} - 9q^{36} - 4q^{37} - 8q^{38} - 5q^{39} + 3q^{40} + 5q^{41} + 2q^{43} - 4q^{44} + 9q^{45} - 10q^{46} + 3q^{47} - 10q^{48} - 3q^{50} + 2q^{51} + 7q^{52} + 17q^{53} + 8q^{54} - 2q^{55} + 20q^{57} - 13q^{58} - 8q^{59} + 6q^{60} + 24q^{61} - 13q^{62} - 7q^{64} + 15q^{65} - q^{66} - 16q^{67} - 5q^{68} + 3q^{69} + 7q^{71} + 10q^{72} + 20q^{73} + 22q^{74} - 25q^{75} + 39q^{76} - 6q^{78} + 3q^{79} - 9q^{80} - 17q^{81} - 41q^{82} + 11q^{83} + 11q^{85} - 21q^{86} - 30q^{87} + 9q^{88} - q^{89} - 10q^{90} + 25q^{92} - 26q^{93} + 10q^{94} - 17q^{95} - 27q^{96} - 9q^{97} + O(q^{100}) \)

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

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

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
0.713538
−1.91223
2.19869
−2.49086 0.713538 4.20440 −2.20440 −1.77733 0 −5.49086 −2.49086 5.49086
1.2 0.656620 −1.91223 −1.56885 3.56885 −1.25561 0 −2.34338 0.656620 2.34338
1.3 1.83424 2.19869 1.36445 0.635552 4.03293 0 −1.16576 1.83424 1.16576
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(-1\)
\(11\) \(1\)

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 539.2.a.i 3
3.b odd 2 1 4851.2.a.bn 3
4.b odd 2 1 8624.2.a.ck 3
7.b odd 2 1 539.2.a.h 3
7.c even 3 2 539.2.e.l 6
7.d odd 6 2 77.2.e.b 6
11.b odd 2 1 5929.2.a.w 3
21.c even 2 1 4851.2.a.bo 3
21.g even 6 2 693.2.i.g 6
28.d even 2 1 8624.2.a.cl 3
28.f even 6 2 1232.2.q.k 6
77.b even 2 1 5929.2.a.v 3
77.i even 6 2 847.2.e.d 6
77.n even 30 8 847.2.n.d 24
77.p odd 30 8 847.2.n.e 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.e.b 6 7.d odd 6 2
539.2.a.h 3 7.b odd 2 1
539.2.a.i 3 1.a even 1 1 trivial
539.2.e.l 6 7.c even 3 2
693.2.i.g 6 21.g even 6 2
847.2.e.d 6 77.i even 6 2
847.2.n.d 24 77.n even 30 8
847.2.n.e 24 77.p odd 30 8
1232.2.q.k 6 28.f even 6 2
4851.2.a.bn 3 3.b odd 2 1
4851.2.a.bo 3 21.c even 2 1
5929.2.a.v 3 77.b even 2 1
5929.2.a.w 3 11.b odd 2 1
8624.2.a.ck 3 4.b odd 2 1
8624.2.a.cl 3 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(539))\):

\( T_{2}^{3} - 5 T_{2} + 3 \)
\( T_{3}^{3} - T_{3}^{2} - 4 T_{3} + 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 3 - 5 T + T^{3} \)
$3$ \( 3 - 4 T - T^{2} + T^{3} \)
$5$ \( 5 - 7 T - 2 T^{2} + T^{3} \)
$7$ \( T^{3} \)
$11$ \( ( 1 + T )^{3} \)
$13$ \( -35 + 36 T - 11 T^{2} + T^{3} \)
$17$ \( 7 - 2 T - 3 T^{2} + T^{3} \)
$19$ \( 57 + 20 T - 11 T^{2} + T^{3} \)
$23$ \( -47 + 43 T - 12 T^{2} + T^{3} \)
$29$ \( -53 - 20 T + 9 T^{2} + T^{3} \)
$31$ \( 107 - 44 T - 3 T^{2} + T^{3} \)
$37$ \( -152 - 36 T + 4 T^{2} + T^{3} \)
$41$ \( 109 - 80 T - 5 T^{2} + T^{3} \)
$43$ \( 41 - 25 T - 2 T^{2} + T^{3} \)
$47$ \( 7 - 2 T - 3 T^{2} + T^{3} \)
$53$ \( -21 + 74 T - 17 T^{2} + T^{3} \)
$59$ \( -1323 - 157 T + 8 T^{2} + T^{3} \)
$61$ \( -376 + 172 T - 24 T^{2} + T^{3} \)
$67$ \( 72 + 68 T + 16 T^{2} + T^{3} \)
$71$ \( 419 - 86 T - 7 T^{2} + T^{3} \)
$73$ \( 625 + 25 T - 20 T^{2} + T^{3} \)
$79$ \( 141 - 38 T - 3 T^{2} + T^{3} \)
$83$ \( 3 + 16 T - 11 T^{2} + T^{3} \)
$89$ \( -3 - 8 T + T^{2} + T^{3} \)
$97$ \( -47 - 12 T + 9 T^{2} + T^{3} \)
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