Properties

Label 5733.2.a.x
Level $5733$
Weight $2$
Character orbit 5733.a
Self dual yes
Analytic conductor $45.778$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5733 = 3^{2} \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5733.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(45.7782354788\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
Defining polynomial: \(x^{3} - x^{2} - 4 x + 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + ( 1 - \beta_{1} ) q^{5} + ( -1 - \beta_{2} ) q^{8} +O(q^{10})\) \( q -\beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + ( 1 - \beta_{1} ) q^{5} + ( -1 - \beta_{2} ) q^{8} + ( 3 - \beta_{1} + \beta_{2} ) q^{10} + ( -1 + \beta_{1} - \beta_{2} ) q^{11} - q^{13} + ( -1 + 2 \beta_{1} - \beta_{2} ) q^{16} + ( 1 + \beta_{1} + \beta_{2} ) q^{17} + ( 1 + \beta_{1} ) q^{19} -2 \beta_{1} q^{20} + ( -2 + 2 \beta_{1} ) q^{22} + ( -4 + 2 \beta_{1} + \beta_{2} ) q^{23} + ( -1 - 2 \beta_{1} + \beta_{2} ) q^{25} + \beta_{1} q^{26} + ( -8 - \beta_{2} ) q^{29} + ( 1 + \beta_{1} - 2 \beta_{2} ) q^{31} + ( -3 + 2 \beta_{1} + \beta_{2} ) q^{32} + ( -4 - 2 \beta_{1} - 2 \beta_{2} ) q^{34} + ( -1 + 3 \beta_{1} + \beta_{2} ) q^{37} + ( -3 - \beta_{1} - \beta_{2} ) q^{38} + 2 \beta_{1} q^{40} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{41} + ( 4 - 2 \beta_{1} - 3 \beta_{2} ) q^{43} -4 q^{44} + ( -7 + 3 \beta_{1} - 3 \beta_{2} ) q^{46} + ( -3 + \beta_{1} - 4 \beta_{2} ) q^{47} + ( 5 + \beta_{2} ) q^{50} + ( -1 - \beta_{2} ) q^{52} + ( -2 - 2 \beta_{1} + 3 \beta_{2} ) q^{53} + ( -3 + 3 \beta_{1} - \beta_{2} ) q^{55} + ( 1 + 9 \beta_{1} + \beta_{2} ) q^{58} + ( -2 + 2 \beta_{1} + 4 \beta_{2} ) q^{59} + 2 q^{61} + ( -1 + \beta_{1} + \beta_{2} ) q^{62} + ( -5 - 2 \beta_{1} - \beta_{2} ) q^{64} + ( -1 + \beta_{1} ) q^{65} + ( -2 - 6 \beta_{1} + 4 \beta_{2} ) q^{67} + ( 6 + 4 \beta_{1} + 2 \beta_{2} ) q^{68} + ( 3 - 3 \beta_{1} + \beta_{2} ) q^{71} + ( 3 + \beta_{1} + 4 \beta_{2} ) q^{73} + ( -10 - 4 \beta_{2} ) q^{74} + ( 2 + 2 \beta_{1} + 2 \beta_{2} ) q^{76} + ( -6 + 4 \beta_{1} - \beta_{2} ) q^{79} + ( -6 + 4 \beta_{1} - 2 \beta_{2} ) q^{80} + ( -4 + 2 \beta_{1} ) q^{82} + ( -1 - 9 \beta_{1} + 4 \beta_{2} ) q^{83} + ( -3 - \beta_{1} - \beta_{2} ) q^{85} + ( 9 - \beta_{1} + 5 \beta_{2} ) q^{86} + 4 q^{88} + ( -1 + 5 \beta_{1} - 2 \beta_{2} ) q^{89} + ( 2 + 6 \beta_{1} - 2 \beta_{2} ) q^{92} + ( 1 + 7 \beta_{1} + 3 \beta_{2} ) q^{94} + ( -2 - \beta_{2} ) q^{95} + ( 3 + \beta_{1} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - q^{2} + 3q^{4} + 2q^{5} - 3q^{8} + O(q^{10}) \) \( 3q - q^{2} + 3q^{4} + 2q^{5} - 3q^{8} + 8q^{10} - 2q^{11} - 3q^{13} - q^{16} + 4q^{17} + 4q^{19} - 2q^{20} - 4q^{22} - 10q^{23} - 5q^{25} + q^{26} - 24q^{29} + 4q^{31} - 7q^{32} - 14q^{34} - 10q^{38} + 2q^{40} + 2q^{41} + 10q^{43} - 12q^{44} - 18q^{46} - 8q^{47} + 15q^{50} - 3q^{52} - 8q^{53} - 6q^{55} + 12q^{58} - 4q^{59} + 6q^{61} - 2q^{62} - 17q^{64} - 2q^{65} - 12q^{67} + 22q^{68} + 6q^{71} + 10q^{73} - 30q^{74} + 8q^{76} - 14q^{79} - 14q^{80} - 10q^{82} - 12q^{83} - 10q^{85} + 26q^{86} + 12q^{88} + 2q^{89} + 12q^{92} + 10q^{94} - 6q^{95} + 10q^{97} + O(q^{100}) \)

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

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

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
2.34292
0.470683
−1.81361
−2.34292 0 3.48929 −1.34292 0 0 −3.48929 0 3.14637
1.2 −0.470683 0 −1.77846 0.529317 0 0 1.77846 0 −0.249141
1.3 1.81361 0 1.28917 2.81361 0 0 −1.28917 0 5.10278
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(-1\)
\(13\) \(1\)

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 5733.2.a.x 3
3.b odd 2 1 637.2.a.j 3
7.b odd 2 1 819.2.a.i 3
21.c even 2 1 91.2.a.d 3
21.g even 6 2 637.2.e.j 6
21.h odd 6 2 637.2.e.i 6
39.d odd 2 1 8281.2.a.bg 3
84.h odd 2 1 1456.2.a.t 3
105.g even 2 1 2275.2.a.m 3
168.e odd 2 1 5824.2.a.bs 3
168.i even 2 1 5824.2.a.by 3
273.g even 2 1 1183.2.a.i 3
273.o odd 4 2 1183.2.c.f 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.a.d 3 21.c even 2 1
637.2.a.j 3 3.b odd 2 1
637.2.e.i 6 21.h odd 6 2
637.2.e.j 6 21.g even 6 2
819.2.a.i 3 7.b odd 2 1
1183.2.a.i 3 273.g even 2 1
1183.2.c.f 6 273.o odd 4 2
1456.2.a.t 3 84.h odd 2 1
2275.2.a.m 3 105.g even 2 1
5733.2.a.x 3 1.a even 1 1 trivial
5824.2.a.bs 3 168.e odd 2 1
5824.2.a.by 3 168.i even 2 1
8281.2.a.bg 3 39.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5733))\):

\( T_{2}^{3} + T_{2}^{2} - 4 T_{2} - 2 \)
\( T_{5}^{3} - 2 T_{5}^{2} - 3 T_{5} + 2 \)
\( T_{11}^{3} + 2 T_{11}^{2} - 6 T_{11} - 8 \)
\( T_{17}^{3} - 4 T_{17}^{2} - 10 T_{17} - 4 \)
\( T_{19}^{3} - 4 T_{19}^{2} + T_{19} + 4 \)
\( T_{31}^{3} - 4 T_{31}^{2} - 19 T_{31} - 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -2 - 4 T + T^{2} + T^{3} \)
$3$ \( T^{3} \)
$5$ \( 2 - 3 T - 2 T^{2} + T^{3} \)
$7$ \( T^{3} \)
$11$ \( -8 - 6 T + 2 T^{2} + T^{3} \)
$13$ \( ( 1 + T )^{3} \)
$17$ \( -4 - 10 T - 4 T^{2} + T^{3} \)
$19$ \( 4 + T - 4 T^{2} + T^{3} \)
$23$ \( -136 + T + 10 T^{2} + T^{3} \)
$29$ \( 454 + 185 T + 24 T^{2} + T^{3} \)
$31$ \( -16 - 19 T - 4 T^{2} + T^{3} \)
$37$ \( -124 - 58 T + T^{3} \)
$41$ \( -8 - 28 T - 2 T^{2} + T^{3} \)
$43$ \( 628 - 71 T - 10 T^{2} + T^{3} \)
$47$ \( -544 - 79 T + 8 T^{2} + T^{3} \)
$53$ \( 22 - 35 T + 8 T^{2} + T^{3} \)
$59$ \( -688 - 156 T + 4 T^{2} + T^{3} \)
$61$ \( ( -2 + T )^{3} \)
$67$ \( -976 - 124 T + 12 T^{2} + T^{3} \)
$71$ \( -16 - 22 T - 6 T^{2} + T^{3} \)
$73$ \( 274 - 99 T - 10 T^{2} + T^{3} \)
$79$ \( -16 + 5 T + 14 T^{2} + T^{3} \)
$83$ \( -3268 - 271 T + 12 T^{2} + T^{3} \)
$89$ \( 422 - 95 T - 2 T^{2} + T^{3} \)
$97$ \( -22 + 29 T - 10 T^{2} + T^{3} \)
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