Properties

Label 1143.2.a.f
Level $1143$
Weight $2$
Character orbit 1143.a
Self dual yes
Analytic conductor $9.127$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1143 = 3^{2} \cdot 127 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1143.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.12690095103\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{7})\)
Defining polynomial: \(x^{4} - 5 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{1} + \beta_{3} ) q^{2} + q^{4} -\beta_{1} q^{5} + ( -1 + \beta_{2} ) q^{7} + ( -\beta_{1} - \beta_{3} ) q^{8} +O(q^{10})\) \( q + ( \beta_{1} + \beta_{3} ) q^{2} + q^{4} -\beta_{1} q^{5} + ( -1 + \beta_{2} ) q^{7} + ( -\beta_{1} - \beta_{3} ) q^{8} + ( -2 - \beta_{2} ) q^{10} -\beta_{3} q^{11} + ( -4 - \beta_{2} ) q^{13} -3 \beta_{3} q^{14} -5 q^{16} -2 \beta_{3} q^{17} + ( -2 - 2 \beta_{2} ) q^{19} -\beta_{1} q^{20} + ( -1 + \beta_{2} ) q^{22} + ( -2 + \beta_{2} ) q^{25} + ( -5 \beta_{1} - 2 \beta_{3} ) q^{26} + ( -1 + \beta_{2} ) q^{28} + ( 2 \beta_{1} - \beta_{3} ) q^{29} + ( -2 - 2 \beta_{2} ) q^{31} + ( -3 \beta_{1} - 3 \beta_{3} ) q^{32} + ( -2 + 2 \beta_{2} ) q^{34} + ( -\beta_{1} - \beta_{3} ) q^{35} + \beta_{2} q^{37} + ( -4 \beta_{1} + 2 \beta_{3} ) q^{38} + ( 2 + \beta_{2} ) q^{40} + ( -2 \beta_{1} + 4 \beta_{3} ) q^{41} + 3 \beta_{2} q^{43} -\beta_{3} q^{44} + \beta_{1} q^{47} + ( -1 - 3 \beta_{2} ) q^{49} + ( -\beta_{1} - 4 \beta_{3} ) q^{50} + ( -4 - \beta_{2} ) q^{52} + ( 2 \beta_{1} - \beta_{3} ) q^{53} - q^{55} + 3 \beta_{3} q^{56} + ( 3 + 3 \beta_{2} ) q^{58} + 4 \beta_{3} q^{59} + ( -1 - \beta_{2} ) q^{61} + ( -4 \beta_{1} + 2 \beta_{3} ) q^{62} + q^{64} + ( 6 \beta_{1} + \beta_{3} ) q^{65} + ( -4 - \beta_{2} ) q^{67} -2 \beta_{3} q^{68} -3 q^{70} + ( -4 \beta_{1} - \beta_{3} ) q^{71} + ( -5 + 3 \beta_{2} ) q^{73} + ( \beta_{1} - 2 \beta_{3} ) q^{74} + ( -2 - 2 \beta_{2} ) q^{76} + ( \beta_{1} + 4 \beta_{3} ) q^{77} -2 \beta_{2} q^{79} + 5 \beta_{1} q^{80} -6 \beta_{2} q^{82} + ( 8 \beta_{1} + 2 \beta_{3} ) q^{83} -2 q^{85} + ( 3 \beta_{1} - 6 \beta_{3} ) q^{86} + ( 1 - \beta_{2} ) q^{88} + ( -\beta_{1} - 4 \beta_{3} ) q^{89} + ( -1 - 2 \beta_{2} ) q^{91} + ( 2 + \beta_{2} ) q^{94} + ( 6 \beta_{1} + 2 \beta_{3} ) q^{95} -8 q^{97} + ( -4 \beta_{1} + 5 \beta_{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{4} - 6q^{7} + O(q^{10}) \) \( 4q + 4q^{4} - 6q^{7} - 6q^{10} - 14q^{13} - 20q^{16} - 4q^{19} - 6q^{22} - 10q^{25} - 6q^{28} - 4q^{31} - 12q^{34} - 2q^{37} + 6q^{40} - 6q^{43} + 2q^{49} - 14q^{52} - 4q^{55} + 6q^{58} - 2q^{61} + 4q^{64} - 14q^{67} - 12q^{70} - 26q^{73} - 4q^{76} + 4q^{79} + 12q^{82} - 8q^{85} + 6q^{88} + 6q^{94} - 32q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 5 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 5 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 5 \beta_{1}\)

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
0.456850
−2.18890
2.18890
−0.456850
−1.73205 0 1.00000 −0.456850 0 −3.79129 1.73205 0 0.791288
1.2 −1.73205 0 1.00000 2.18890 0 0.791288 1.73205 0 −3.79129
1.3 1.73205 0 1.00000 −2.18890 0 0.791288 −1.73205 0 −3.79129
1.4 1.73205 0 1.00000 0.456850 0 −3.79129 −1.73205 0 0.791288
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(127\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1143.2.a.f 4
3.b odd 2 1 inner 1143.2.a.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1143.2.a.f 4 1.a even 1 1 trivial
1143.2.a.f 4 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1143))\):

\( T_{2}^{2} - 3 \)
\( T_{5}^{4} - 5 T_{5}^{2} + 1 \)
\( T_{7}^{2} + 3 T_{7} - 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -3 + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( 1 - 5 T^{2} + T^{4} \)
$7$ \( ( -3 + 3 T + T^{2} )^{2} \)
$11$ \( 1 - 5 T^{2} + T^{4} \)
$13$ \( ( 7 + 7 T + T^{2} )^{2} \)
$17$ \( 16 - 20 T^{2} + T^{4} \)
$19$ \( ( -20 + 2 T + T^{2} )^{2} \)
$23$ \( T^{4} \)
$29$ \( 225 - 33 T^{2} + T^{4} \)
$31$ \( ( -20 + 2 T + T^{2} )^{2} \)
$37$ \( ( -5 + T + T^{2} )^{2} \)
$41$ \( 3600 - 132 T^{2} + T^{4} \)
$43$ \( ( -45 + 3 T + T^{2} )^{2} \)
$47$ \( 1 - 5 T^{2} + T^{4} \)
$53$ \( 225 - 33 T^{2} + T^{4} \)
$59$ \( 256 - 80 T^{2} + T^{4} \)
$61$ \( ( -5 + T + T^{2} )^{2} \)
$67$ \( ( 7 + 7 T + T^{2} )^{2} \)
$71$ \( 9 - 69 T^{2} + T^{4} \)
$73$ \( ( -5 + 13 T + T^{2} )^{2} \)
$79$ \( ( -20 - 2 T + T^{2} )^{2} \)
$83$ \( 144 - 276 T^{2} + T^{4} \)
$89$ \( 9 - 69 T^{2} + T^{4} \)
$97$ \( ( 8 + T )^{4} \)
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