Properties

Label 1288.2.a.l
Level $1288$
Weight $2$
Character orbit 1288.a
Self dual yes
Analytic conductor $10.285$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1288 = 2^{3} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1288.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(10.2847317803\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \(x^{3} - x^{2} - 3 x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{2} q^{3} + ( 2 - 2 \beta_{1} - \beta_{2} ) q^{5} - q^{7} + ( -\beta_{1} - \beta_{2} ) q^{9} +O(q^{10})\) \( q -\beta_{2} q^{3} + ( 2 - 2 \beta_{1} - \beta_{2} ) q^{5} - q^{7} + ( -\beta_{1} - \beta_{2} ) q^{9} + ( -1 - \beta_{1} + \beta_{2} ) q^{11} -2 \beta_{1} q^{13} + ( 1 + \beta_{1} - 3 \beta_{2} ) q^{15} + ( 4 + 2 \beta_{1} - \beta_{2} ) q^{17} + \beta_{2} q^{21} + q^{23} + ( 6 - \beta_{1} - \beta_{2} ) q^{25} + ( 2 + 2 \beta_{2} ) q^{27} + ( 3 + \beta_{1} + \beta_{2} ) q^{29} + ( 2 + 4 \beta_{1} - \beta_{2} ) q^{31} + ( -4 + 2 \beta_{1} + 2 \beta_{2} ) q^{33} + ( -2 + 2 \beta_{1} + \beta_{2} ) q^{35} + ( 8 - 2 \beta_{1} ) q^{37} + ( -2 + 2 \beta_{1} ) q^{39} + ( 2 - 4 \beta_{1} - 4 \beta_{2} ) q^{41} + ( -2 - 2 \beta_{1} - 4 \beta_{2} ) q^{43} + ( 4 + 2 \beta_{1} - \beta_{2} ) q^{45} + ( 2 \beta_{1} - \beta_{2} ) q^{47} + q^{49} + ( 5 - 3 \beta_{1} - 5 \beta_{2} ) q^{51} + ( -6 + 4 \beta_{1} + 4 \beta_{2} ) q^{53} + ( 2 \beta_{1} + 6 \beta_{2} ) q^{55} + ( -6 - 2 \beta_{1} + 3 \beta_{2} ) q^{59} + ( 4 \beta_{1} + 7 \beta_{2} ) q^{61} + ( \beta_{1} + \beta_{2} ) q^{63} + ( 6 + 2 \beta_{1} + 4 \beta_{2} ) q^{65} + ( -1 - 5 \beta_{1} + \beta_{2} ) q^{67} -\beta_{2} q^{69} + 4 \beta_{2} q^{71} + ( 2 + 6 \beta_{2} ) q^{73} + ( 2 - 7 \beta_{2} ) q^{75} + ( 1 + \beta_{1} - \beta_{2} ) q^{77} + ( 5 - 3 \beta_{1} - \beta_{2} ) q^{79} + ( -6 + 5 \beta_{1} + 3 \beta_{2} ) q^{81} + ( -2 - 2 \beta_{1} ) q^{83} + ( 3 - 9 \beta_{1} - 11 \beta_{2} ) q^{85} + ( -2 - 2 \beta_{2} ) q^{87} + ( 6 + 5 \beta_{2} ) q^{89} + 2 \beta_{1} q^{91} + ( 7 - 5 \beta_{1} - 3 \beta_{2} ) q^{93} + ( 2 + 2 \beta_{1} + 9 \beta_{2} ) q^{97} + ( -1 + 3 \beta_{1} + 3 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 4q^{5} - 3q^{7} - q^{9} + O(q^{10}) \) \( 3q + 4q^{5} - 3q^{7} - q^{9} - 4q^{11} - 2q^{13} + 4q^{15} + 14q^{17} + 3q^{23} + 17q^{25} + 6q^{27} + 10q^{29} + 10q^{31} - 10q^{33} - 4q^{35} + 22q^{37} - 4q^{39} + 2q^{41} - 8q^{43} + 14q^{45} + 2q^{47} + 3q^{49} + 12q^{51} - 14q^{53} + 2q^{55} - 20q^{59} + 4q^{61} + q^{63} + 20q^{65} - 8q^{67} + 6q^{73} + 6q^{75} + 4q^{77} + 12q^{79} - 13q^{81} - 8q^{83} - 6q^{87} + 18q^{89} + 2q^{91} + 16q^{93} + 8q^{97} + O(q^{100}) \)

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

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

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
−1.48119
2.17009
0.311108
0 −1.67513 0 3.28726 0 −1.00000 0 −0.193937 0
1.2 0 −0.539189 0 −2.87936 0 −1.00000 0 −2.70928 0
1.3 0 2.21432 0 3.59210 0 −1.00000 0 1.90321 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(7\) \(1\)
\(23\) \(-1\)

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 1288.2.a.l 3
4.b odd 2 1 2576.2.a.y 3
7.b odd 2 1 9016.2.a.z 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1288.2.a.l 3 1.a even 1 1 trivial
2576.2.a.y 3 4.b odd 2 1
9016.2.a.z 3 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1288))\):

\( T_{3}^{3} - 4 T_{3} - 2 \)
\( T_{5}^{3} - 4 T_{5}^{2} - 8 T_{5} + 34 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( -2 - 4 T + T^{3} \)
$5$ \( 34 - 8 T - 4 T^{2} + T^{3} \)
$7$ \( ( 1 + T )^{3} \)
$11$ \( -20 - 4 T + 4 T^{2} + T^{3} \)
$13$ \( -8 - 12 T + 2 T^{2} + T^{3} \)
$17$ \( 34 + 44 T - 14 T^{2} + T^{3} \)
$19$ \( T^{3} \)
$23$ \( ( -1 + T )^{3} \)
$29$ \( -20 + 28 T - 10 T^{2} + T^{3} \)
$31$ \( 310 - 32 T - 10 T^{2} + T^{3} \)
$37$ \( -296 + 148 T - 22 T^{2} + T^{3} \)
$41$ \( 104 - 84 T - 2 T^{2} + T^{3} \)
$43$ \( -304 - 40 T + 8 T^{2} + T^{3} \)
$47$ \( 50 - 20 T - 2 T^{2} + T^{3} \)
$53$ \( -344 - 20 T + 14 T^{2} + T^{3} \)
$59$ \( -230 + 72 T + 20 T^{2} + T^{3} \)
$61$ \( 1030 - 188 T - 4 T^{2} + T^{3} \)
$67$ \( -436 - 76 T + 8 T^{2} + T^{3} \)
$71$ \( 128 - 64 T + T^{3} \)
$73$ \( 712 - 132 T - 6 T^{2} + T^{3} \)
$79$ \( 100 + 20 T - 12 T^{2} + T^{3} \)
$83$ \( -16 + 8 T + 8 T^{2} + T^{3} \)
$89$ \( 634 + 8 T - 18 T^{2} + T^{3} \)
$97$ \( 2734 - 280 T - 8 T^{2} + T^{3} \)
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