Properties

Label 5733.2.a.bm
Level $5733$
Weight $2$
Character orbit 5733.a
Self dual yes
Analytic conductor $45.778$
Analytic rank $1$
Dimension $5$
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: \(5\)
Coefficient field: 5.5.746052.1
Defining polynomial: \(x^{5} - x^{4} - 7 x^{3} + 8 x + 2\)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{1} ) q^{2} + ( 2 - \beta_{1} + \beta_{3} - \beta_{4} ) q^{4} -\beta_{2} q^{5} + ( -2 + 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{8} +O(q^{10})\) \( q + ( -1 + \beta_{1} ) q^{2} + ( 2 - \beta_{1} + \beta_{3} - \beta_{4} ) q^{4} -\beta_{2} q^{5} + ( -2 + 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{8} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{10} + ( -2 + \beta_{3} ) q^{11} - q^{13} + ( 2 - 3 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{16} + ( -1 + 2 \beta_{1} + \beta_{2} + \beta_{4} ) q^{17} + ( -2 - 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} ) q^{19} + ( 1 - \beta_{1} + 3 \beta_{3} ) q^{20} + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} ) q^{22} + ( -2 + \beta_{4} ) q^{23} + ( 1 - \beta_{2} - 2 \beta_{3} + 2 \beta_{4} ) q^{25} + ( 1 - \beta_{1} ) q^{26} + ( 1 - 2 \beta_{1} - \beta_{4} ) q^{29} + ( 2 + 2 \beta_{2} + \beta_{4} ) q^{31} + ( -5 + \beta_{1} + \beta_{2} - 4 \beta_{3} ) q^{32} + ( 5 - 3 \beta_{1} - \beta_{2} + 2 \beta_{3} - 3 \beta_{4} ) q^{34} + ( 2 \beta_{1} - 2 \beta_{3} - \beta_{4} ) q^{37} + ( -2 - \beta_{1} - \beta_{3} + 4 \beta_{4} ) q^{38} + ( -6 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{40} + ( 4 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} - 3 \beta_{4} ) q^{41} + ( -2 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{43} + ( -1 + \beta_{1} - 2 \beta_{2} + 3 \beta_{4} ) q^{44} + ( 1 - 3 \beta_{1} - \beta_{3} - \beta_{4} ) q^{46} + ( -\beta_{3} + 4 \beta_{4} ) q^{47} + ( -2 - 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{50} + ( -2 + \beta_{1} - \beta_{3} + \beta_{4} ) q^{52} + ( -3 - 4 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} ) q^{53} + ( 2 - 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} ) q^{55} + ( -6 + 2 \beta_{1} - \beta_{3} + 3 \beta_{4} ) q^{58} + ( 2 - \beta_{3} ) q^{59} + ( 3 + 4 \beta_{3} - 2 \beta_{4} ) q^{61} + ( -5 - \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} ) q^{62} + ( 3 - 4 \beta_{1} - \beta_{2} + 4 \beta_{3} + 3 \beta_{4} ) q^{64} + \beta_{2} q^{65} + ( 4 + 2 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} + \beta_{4} ) q^{67} + ( -8 + 7 \beta_{1} + \beta_{2} - 3 \beta_{3} + 2 \beta_{4} ) q^{68} + ( -2 - 4 \beta_{1} + \beta_{3} - \beta_{4} ) q^{71} + ( -\beta_{2} + 2 \beta_{3} ) q^{73} + ( 7 - \beta_{1} - 2 \beta_{2} + 5 \beta_{3} + \beta_{4} ) q^{74} + ( -1 - 3 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{76} + ( 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} ) q^{79} + ( 11 - 5 \beta_{1} - 3 \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{80} + ( -8 + 8 \beta_{1} + \beta_{2} + 3 \beta_{4} ) q^{82} + ( 2 + 2 \beta_{1} + 3 \beta_{2} + \beta_{4} ) q^{83} + ( -4 + \beta_{2} - 4 \beta_{4} ) q^{85} + ( -6 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} + 3 \beta_{4} ) q^{86} + ( -1 - 2 \beta_{3} - 2 \beta_{4} ) q^{88} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} ) q^{89} + ( -5 + \beta_{1} - \beta_{2} - \beta_{3} + 3 \beta_{4} ) q^{92} + ( -4 - 5 \beta_{1} - \beta_{2} - 3 \beta_{3} - 3 \beta_{4} ) q^{94} + ( 2 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + 6 \beta_{4} ) q^{95} + ( 2 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q - 4q^{2} + 8q^{4} + 2q^{5} - 9q^{8} + O(q^{10}) \) \( 5q - 4q^{2} + 8q^{4} + 2q^{5} - 9q^{8} + 5q^{10} - 11q^{11} - 5q^{13} + 10q^{16} - 5q^{17} - 9q^{19} + q^{20} + 8q^{22} - 10q^{23} + 9q^{25} + 4q^{26} + 3q^{29} + 6q^{31} - 22q^{32} + 22q^{34} + 4q^{37} - 10q^{38} - 28q^{40} + 14q^{41} + 2q^{43} + 3q^{46} + q^{47} - 9q^{50} - 8q^{52} - 17q^{53} - 27q^{58} + 11q^{59} + 11q^{61} - 23q^{62} + 9q^{64} - 2q^{65} + 13q^{67} - 32q^{68} - 15q^{71} + 33q^{74} - 8q^{76} + 2q^{79} + 55q^{80} - 34q^{82} + 6q^{83} - 22q^{85} - 28q^{86} - 3q^{88} - 4q^{89} - 21q^{92} - 20q^{94} + 12q^{95} + 12q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - x^{4} - 7 x^{3} + 8 x + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{3} - 2 \nu^{2} - 4 \nu + 2 \)
\(\beta_{3}\)\(=\)\( \nu^{4} - 2 \nu^{3} - 4 \nu^{2} + 3 \nu + 1 \)
\(\beta_{4}\)\(=\)\( \nu^{4} - 2 \nu^{3} - 5 \nu^{2} + 4 \nu + 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{4} + \beta_{3} + \beta_{1} + 3\)
\(\nu^{3}\)\(=\)\(-2 \beta_{4} + 2 \beta_{3} + \beta_{2} + 6 \beta_{1} + 4\)
\(\nu^{4}\)\(=\)\(-8 \beta_{4} + 9 \beta_{3} + 2 \beta_{2} + 13 \beta_{1} + 19\)

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
−1.72525
−1.21332
−0.265608
1.19566
3.00852
−2.72525 0 5.42699 2.18716 0 0 −9.33940 0 −5.96057
1.2 −2.21332 0 2.89879 −2.12280 0 0 −1.98932 0 4.69843
1.3 −1.26561 0 −0.398235 −2.90260 0 0 3.03523 0 3.67356
1.4 0.195656 0 −1.96172 3.93251 0 0 −0.775135 0 0.769420
1.5 2.00852 0 2.03417 0.905722 0 0 0.0686323 0 1.81916
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.bm 5
3.b odd 2 1 637.2.a.k 5
7.b odd 2 1 5733.2.a.bl 5
7.d odd 6 2 819.2.j.h 10
21.c even 2 1 637.2.a.l 5
21.g even 6 2 91.2.e.c 10
21.h odd 6 2 637.2.e.m 10
39.d odd 2 1 8281.2.a.bx 5
84.j odd 6 2 1456.2.r.p 10
273.g even 2 1 8281.2.a.bw 5
273.ba even 6 2 1183.2.e.f 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.e.c 10 21.g even 6 2
637.2.a.k 5 3.b odd 2 1
637.2.a.l 5 21.c even 2 1
637.2.e.m 10 21.h odd 6 2
819.2.j.h 10 7.d odd 6 2
1183.2.e.f 10 273.ba even 6 2
1456.2.r.p 10 84.j odd 6 2
5733.2.a.bl 5 7.b odd 2 1
5733.2.a.bm 5 1.a even 1 1 trivial
8281.2.a.bw 5 273.g even 2 1
8281.2.a.bx 5 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5733))\):

\( T_{2}^{5} + 4 T_{2}^{4} - T_{2}^{3} - 17 T_{2}^{2} - 12 T_{2} + 3 \)
\( T_{5}^{5} - 2 T_{5}^{4} - 15 T_{5}^{3} + 20 T_{5}^{2} + 48 T_{5} - 48 \)
\( T_{11}^{5} + 11 T_{11}^{4} + 36 T_{11}^{3} + 22 T_{11}^{2} - 45 T_{11} - 33 \)
\( T_{17}^{5} + 5 T_{17}^{4} - 22 T_{17}^{3} - 106 T_{17}^{2} + 93 T_{17} + 429 \)
\( T_{19}^{5} + 9 T_{19}^{4} - 14 T_{19}^{3} - 176 T_{19}^{2} + 173 T_{19} + 223 \)
\( T_{31}^{5} - 6 T_{31}^{4} - 61 T_{31}^{3} + 102 T_{31}^{2} + 508 T_{31} + 356 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 3 - 12 T - 17 T^{2} - T^{3} + 4 T^{4} + T^{5} \)
$3$ \( T^{5} \)
$5$ \( -48 + 48 T + 20 T^{2} - 15 T^{3} - 2 T^{4} + T^{5} \)
$7$ \( T^{5} \)
$11$ \( -33 - 45 T + 22 T^{2} + 36 T^{3} + 11 T^{4} + T^{5} \)
$13$ \( ( 1 + T )^{5} \)
$17$ \( 429 + 93 T - 106 T^{2} - 22 T^{3} + 5 T^{4} + T^{5} \)
$19$ \( 223 + 173 T - 176 T^{2} - 14 T^{3} + 9 T^{4} + T^{5} \)
$23$ \( -12 - 12 T + 26 T^{2} + 31 T^{3} + 10 T^{4} + T^{5} \)
$29$ \( 108 + 144 T + 19 T^{2} - 25 T^{3} - 3 T^{4} + T^{5} \)
$31$ \( 356 + 508 T + 102 T^{2} - 61 T^{3} - 6 T^{4} + T^{5} \)
$37$ \( -7036 + 660 T + 678 T^{2} - 111 T^{3} - 4 T^{4} + T^{5} \)
$41$ \( -1584 - 2544 T + 940 T^{2} - 28 T^{3} - 14 T^{4} + T^{5} \)
$43$ \( 64 - 288 T + 308 T^{2} - 72 T^{3} - 2 T^{4} + T^{5} \)
$47$ \( -5169 + 2811 T + 26 T^{2} - 124 T^{3} - T^{4} + T^{5} \)
$53$ \( -19959 - 12759 T - 2426 T^{2} - 74 T^{3} + 17 T^{4} + T^{5} \)
$59$ \( 33 - 45 T - 22 T^{2} + 36 T^{3} - 11 T^{4} + T^{5} \)
$61$ \( 8461 + 5881 T + 766 T^{2} - 122 T^{3} - 11 T^{4} + T^{5} \)
$67$ \( -22699 - 591 T + 2160 T^{2} - 162 T^{3} - 13 T^{4} + T^{5} \)
$71$ \( 6336 - 456 T - 853 T^{2} - 25 T^{3} + 15 T^{4} + T^{5} \)
$73$ \( 712 + 700 T - 42 T^{2} - 75 T^{3} + T^{5} \)
$79$ \( -1000 + 1500 T - 190 T^{2} - 137 T^{3} - 2 T^{4} + T^{5} \)
$83$ \( -7488 + 2688 T + 308 T^{2} - 124 T^{3} - 6 T^{4} + T^{5} \)
$89$ \( 7692 + 2148 T - 694 T^{2} - 155 T^{3} + 4 T^{4} + T^{5} \)
$97$ \( 2384 - 2240 T + 612 T^{2} - 16 T^{3} - 12 T^{4} + T^{5} \)
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