Properties

Label 5929.2.a.w
Level $5929$
Weight $2$
Character orbit 5929.a
Self dual yes
Analytic conductor $47.343$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5929 = 7^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5929.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(47.3433033584\)
Analytic rank: \(1\)
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} + 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} + ( 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} + ( 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} + ( 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} + ( -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} + ( -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} + ( -\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} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + q^{3} + 4q^{4} + 2q^{5} - q^{6} + 9q^{8} + O(q^{10}) \) \( 3q + q^{3} + 4q^{4} + 2q^{5} - q^{6} + 9q^{8} - 9q^{10} + 9q^{12} - 11q^{13} - 7q^{15} + 2q^{16} - 3q^{17} - 10q^{18} - 11q^{19} - 14q^{20} + 12q^{23} + 2q^{24} + 3q^{25} - q^{26} - 2q^{27} + 9q^{29} - 2q^{30} + 3q^{31} + 3q^{32} + 10q^{34} - 9q^{36} - 4q^{37} - 8q^{38} + 5q^{39} - 3q^{40} - 5q^{41} - 2q^{43} + 9q^{45} + 10q^{46} + 3q^{47} - 10q^{48} + 3q^{50} - 2q^{51} - 7q^{52} + 17q^{53} - 8q^{54} - 20q^{57} - 13q^{58} - 8q^{59} + 6q^{60} - 24q^{61} + 13q^{62} - 7q^{64} - 15q^{65} - 16q^{67} + 5q^{68} + 3q^{69} + 7q^{71} - 10q^{72} - 20q^{73} - 22q^{74} - 25q^{75} - 39q^{76} - 6q^{78} - 3q^{79} - 9q^{80} - 17q^{81} - 41q^{82} - 11q^{83} - 11q^{85} - 21q^{86} + 30q^{87} - q^{89} + 10q^{90} + 25q^{92} - 26q^{93} - 10q^{94} + 17q^{95} + 27q^{96} - 9q^{97} + O(q^{100}) \)

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
2.19869
−1.91223
0.713538
−1.83424 2.19869 1.36445 0.635552 −4.03293 0 1.16576 1.83424 −1.16576
1.2 −0.656620 −1.91223 −1.56885 3.56885 1.25561 0 2.34338 0.656620 −2.34338
1.3 2.49086 0.713538 4.20440 −2.20440 1.77733 0 5.49086 −2.49086 −5.49086
\(n\): 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 5929.2.a.w 3
7.b odd 2 1 5929.2.a.v 3
7.d odd 6 2 847.2.e.d 6
11.b odd 2 1 539.2.a.i 3
33.d even 2 1 4851.2.a.bn 3
44.c even 2 1 8624.2.a.ck 3
77.b even 2 1 539.2.a.h 3
77.h odd 6 2 539.2.e.l 6
77.i even 6 2 77.2.e.b 6
77.n even 30 8 847.2.n.e 24
77.p odd 30 8 847.2.n.d 24
231.h odd 2 1 4851.2.a.bo 3
231.k odd 6 2 693.2.i.g 6
308.g odd 2 1 8624.2.a.cl 3
308.m odd 6 2 1232.2.q.k 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.e.b 6 77.i even 6 2
539.2.a.h 3 77.b even 2 1
539.2.a.i 3 11.b odd 2 1
539.2.e.l 6 77.h odd 6 2
693.2.i.g 6 231.k odd 6 2
847.2.e.d 6 7.d odd 6 2
847.2.n.d 24 77.p odd 30 8
847.2.n.e 24 77.n even 30 8
1232.2.q.k 6 308.m odd 6 2
4851.2.a.bn 3 33.d even 2 1
4851.2.a.bo 3 231.h odd 2 1
5929.2.a.v 3 7.b odd 2 1
5929.2.a.w 3 1.a even 1 1 trivial
8624.2.a.ck 3 44.c even 2 1
8624.2.a.cl 3 308.g odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5929))\):

\( T_{2}^{3} - 5 T_{2} - 3 \)
\( T_{3}^{3} - T_{3}^{2} - 4 T_{3} + 3 \)
\( T_{5}^{3} - 2 T_{5}^{2} - 7 T_{5} + 5 \)

Hecke characteristic polynomials

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