Properties

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

Related objects

Downloads

Learn more

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: 3.3.404.1
Defining polynomial: \(x^{3} - x^{2} - 5 x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 637)
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.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{1} ) q^{2} + ( 2 - \beta_{1} + \beta_{2} ) q^{4} + ( -2 + \beta_{1} ) q^{5} + ( 2 - 2 \beta_{1} + 2 \beta_{2} ) q^{8} +O(q^{10})\) \( q + ( 1 - \beta_{1} ) q^{2} + ( 2 - \beta_{1} + \beta_{2} ) q^{4} + ( -2 + \beta_{1} ) q^{5} + ( 2 - 2 \beta_{1} + 2 \beta_{2} ) q^{8} + ( -5 + 2 \beta_{1} - \beta_{2} ) q^{10} + ( 1 + \beta_{1} ) q^{11} - q^{13} + ( 2 - 4 \beta_{1} + 2 \beta_{2} ) q^{16} + ( -1 - \beta_{2} ) q^{17} + ( 2 + 2 \beta_{1} - \beta_{2} ) q^{19} + ( -6 + 5 \beta_{1} - 3 \beta_{2} ) q^{20} + ( -2 - \beta_{1} - \beta_{2} ) q^{22} + ( -1 + 3 \beta_{1} - \beta_{2} ) q^{23} + ( 2 - 3 \beta_{1} + \beta_{2} ) q^{25} + ( -1 + \beta_{1} ) q^{26} + ( 3 - 2 \beta_{2} ) q^{29} + ( -\beta_{1} - 2 \beta_{2} ) q^{31} + ( 8 - 2 \beta_{1} + 2 \beta_{2} ) q^{32} + ( 3 \beta_{1} - \beta_{2} ) q^{34} + ( -3 + \beta_{1} - 2 \beta_{2} ) q^{37} + ( -3 - 3 \beta_{2} ) q^{38} + ( -8 + 8 \beta_{1} - 6 \beta_{2} ) q^{40} + ( -2 - 4 \beta_{1} + 4 \beta_{2} ) q^{41} + ( 3 + 2 \beta_{1} - 2 \beta_{2} ) q^{43} + 2 \beta_{1} q^{44} + ( -9 + 3 \beta_{1} - 4 \beta_{2} ) q^{46} + ( -6 + \beta_{2} ) q^{47} + ( 10 - 4 \beta_{1} + 4 \beta_{2} ) q^{50} + ( -2 + \beta_{1} - \beta_{2} ) q^{52} + ( -5 + \beta_{1} + \beta_{2} ) q^{53} + ( 1 + \beta_{2} ) q^{55} + ( 5 + \beta_{1} - 2 \beta_{2} ) q^{58} + ( -8 + \beta_{1} + \beta_{2} ) q^{59} + ( -8 - 2 \beta_{1} + 2 \beta_{2} ) q^{61} + ( 5 + 4 \beta_{1} - \beta_{2} ) q^{62} + ( 8 - 4 \beta_{1} ) q^{64} + ( 2 - \beta_{1} ) q^{65} + ( -4 + 2 \beta_{1} - 4 \beta_{2} ) q^{67} + ( -6 + 2 \beta_{1} - 2 \beta_{2} ) q^{68} + ( -1 - 3 \beta_{1} + 2 \beta_{2} ) q^{71} + ( 4 - 6 \beta_{1} - \beta_{2} ) q^{73} + ( -4 + 7 \beta_{1} - 3 \beta_{2} ) q^{74} + ( -4 + 5 \beta_{1} - \beta_{2} ) q^{76} + ( -1 + 2 \beta_{1} + 2 \beta_{2} ) q^{79} + ( -14 + 10 \beta_{1} - 8 \beta_{2} ) q^{80} + ( 6 - 6 \beta_{1} + 8 \beta_{2} ) q^{82} + ( -8 + 3 \beta_{1} - 2 \beta_{2} ) q^{83} + ( 1 - 3 \beta_{1} + 2 \beta_{2} ) q^{85} + ( -1 + \beta_{1} - 4 \beta_{2} ) q^{86} + ( -2 + 2 \beta_{1} ) q^{88} + ( 4 - 4 \beta_{1} + 3 \beta_{2} ) q^{89} + ( -12 + 11 \beta_{1} - 5 \beta_{2} ) q^{92} + ( -7 + 4 \beta_{1} + \beta_{2} ) q^{94} + ( 1 - 2 \beta_{1} + 4 \beta_{2} ) q^{95} + ( -2 + 4 \beta_{1} - \beta_{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{2} + 6 q^{4} - 5 q^{5} + 6 q^{8} + O(q^{10}) \) \( 3 q + 2 q^{2} + 6 q^{4} - 5 q^{5} + 6 q^{8} - 14 q^{10} + 4 q^{11} - 3 q^{13} + 4 q^{16} - 4 q^{17} + 7 q^{19} - 16 q^{20} - 8 q^{22} - q^{23} + 4 q^{25} - 2 q^{26} + 7 q^{29} - 3 q^{31} + 24 q^{32} + 2 q^{34} - 10 q^{37} - 12 q^{38} - 22 q^{40} - 6 q^{41} + 9 q^{43} + 2 q^{44} - 28 q^{46} - 17 q^{47} + 30 q^{50} - 6 q^{52} - 13 q^{53} + 4 q^{55} + 14 q^{58} - 22 q^{59} - 24 q^{61} + 18 q^{62} + 20 q^{64} + 5 q^{65} - 14 q^{67} - 18 q^{68} - 4 q^{71} + 5 q^{73} - 8 q^{74} - 8 q^{76} + q^{79} - 40 q^{80} + 20 q^{82} - 23 q^{83} + 2 q^{85} - 6 q^{86} - 4 q^{88} + 11 q^{89} - 30 q^{92} - 16 q^{94} + 5 q^{95} - 3 q^{97} + O(q^{100}) \)

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

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

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
2.86620
−0.210756
−1.65544
−1.86620 0 1.48270 0.866198 0 0 0.965392 0 −1.61650
1.2 1.21076 0 −0.534070 −2.21076 0 0 −3.06814 0 −2.67669
1.3 2.65544 0 5.05137 −3.65544 0 0 8.10275 0 −9.70682
\(n\): 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.bd 3
3.b odd 2 1 637.2.a.i yes 3
7.b odd 2 1 5733.2.a.be 3
21.c even 2 1 637.2.a.h 3
21.g even 6 2 637.2.e.l 6
21.h odd 6 2 637.2.e.k 6
39.d odd 2 1 8281.2.a.bk 3
273.g even 2 1 8281.2.a.bh 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.a.h 3 21.c even 2 1
637.2.a.i yes 3 3.b odd 2 1
637.2.e.k 6 21.h odd 6 2
637.2.e.l 6 21.g even 6 2
5733.2.a.bd 3 1.a even 1 1 trivial
5733.2.a.be 3 7.b odd 2 1
8281.2.a.bh 3 273.g even 2 1
8281.2.a.bk 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5733))\):

\( T_{2}^{3} - 2 T_{2}^{2} - 4 T_{2} + 6 \)
\( T_{5}^{3} + 5 T_{5}^{2} + 3 T_{5} - 7 \)
\( T_{11}^{3} - 4 T_{11}^{2} + 2 \)
\( T_{17}^{3} + 4 T_{17}^{2} - 2 T_{17} - 14 \)
\( T_{19}^{3} - 7 T_{19}^{2} - 3 T_{19} + 63 \)
\( T_{31}^{3} + 3 T_{31}^{2} - 41 T_{31} - 49 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 6 - 4 T - 2 T^{2} + T^{3} \)
$3$ \( T^{3} \)
$5$ \( -7 + 3 T + 5 T^{2} + T^{3} \)
$7$ \( T^{3} \)
$11$ \( 2 - 4 T^{2} + T^{3} \)
$13$ \( ( 1 + T )^{3} \)
$17$ \( -14 - 2 T + 4 T^{2} + T^{3} \)
$19$ \( 63 - 3 T - 7 T^{2} + T^{3} \)
$23$ \( 43 - 41 T + T^{2} + T^{3} \)
$29$ \( 3 - 13 T - 7 T^{2} + T^{3} \)
$31$ \( -49 - 41 T + 3 T^{2} + T^{3} \)
$37$ \( -82 + 8 T + 10 T^{2} + T^{3} \)
$41$ \( -504 - 116 T + 6 T^{2} + T^{3} \)
$43$ \( 101 - 5 T - 9 T^{2} + T^{3} \)
$47$ \( 147 + 89 T + 17 T^{2} + T^{3} \)
$53$ \( -9 + 39 T + 13 T^{2} + T^{3} \)
$59$ \( 252 + 144 T + 22 T^{2} + T^{3} \)
$61$ \( 224 + 160 T + 24 T^{2} + T^{3} \)
$67$ \( -648 - 36 T + 14 T^{2} + T^{3} \)
$71$ \( -194 - 44 T + 4 T^{2} + T^{3} \)
$73$ \( 1561 - 219 T - 5 T^{2} + T^{3} \)
$79$ \( -99 - 69 T - T^{2} + T^{3} \)
$83$ \( 203 + 127 T + 23 T^{2} + T^{3} \)
$89$ \( -21 - 55 T - 11 T^{2} + T^{3} \)
$97$ \( -7 - 71 T + 3 T^{2} + T^{3} \)
show more
show less