Properties

Label 1288.2.a.n
Level $1288$
Weight $2$
Character orbit 1288.a
Self dual yes
Analytic conductor $10.285$
Analytic rank $1$
Dimension $4$
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: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.8468.1
Defining polynomial: \(x^{4} - x^{3} - 5 x^{2} + 3 x + 4\)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{1} ) q^{3} + ( -\beta_{2} + \beta_{3} ) q^{5} + q^{7} + ( 1 - 2 \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( -1 + \beta_{1} ) q^{3} + ( -\beta_{2} + \beta_{3} ) q^{5} + q^{7} + ( 1 - 2 \beta_{1} + \beta_{2} ) q^{9} + ( -3 + \beta_{1} - \beta_{3} ) q^{11} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{13} + ( -1 - \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{15} + ( -2 \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{17} + ( -2 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{19} + ( -1 + \beta_{1} ) q^{21} - q^{23} + ( \beta_{1} - 4 \beta_{2} + \beta_{3} ) q^{25} + ( -3 + \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{27} + ( -2 - 2 \beta_{1} + 3 \beta_{2} ) q^{29} + ( 1 + \beta_{1} + 2 \beta_{2} ) q^{31} + ( 6 - 4 \beta_{1} ) q^{33} + ( -\beta_{2} + \beta_{3} ) q^{35} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{37} + ( -3 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{39} + ( 1 + \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{41} + ( -2 + 2 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{43} + ( 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{45} + ( -1 - \beta_{1} ) q^{47} + q^{49} + ( -3 + 5 \beta_{1} - 6 \beta_{2} + 3 \beta_{3} ) q^{51} + ( -6 + 4 \beta_{1} ) q^{53} + ( -4 - 2 \beta_{1} + 6 \beta_{2} - 2 \beta_{3} ) q^{55} + ( -2 + 2 \beta_{1} - 6 \beta_{2} + 2 \beta_{3} ) q^{57} + ( -4 - \beta_{2} - \beta_{3} ) q^{59} + ( 2 + 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{61} + ( 1 - 2 \beta_{1} + \beta_{2} ) q^{63} + 2 \beta_{3} q^{65} + ( -3 - 3 \beta_{1} + 4 \beta_{2} - \beta_{3} ) q^{67} + ( 1 - \beta_{1} ) q^{69} + ( -7 + 3 \beta_{1} - \beta_{2} + \beta_{3} ) q^{71} + ( 3 + 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{73} + ( -1 - 5 \beta_{1} + 6 \beta_{2} - 4 \beta_{3} ) q^{75} + ( -3 + \beta_{1} - \beta_{3} ) q^{77} + ( 3 - \beta_{1} + \beta_{3} ) q^{79} + ( -\beta_{1} + 2 \beta_{2} - 3 \beta_{3} ) q^{81} + ( -10 + 2 \beta_{1} - 6 \beta_{2} + 4 \beta_{3} ) q^{83} + ( -7 + \beta_{1} + 6 \beta_{2} - 5 \beta_{3} ) q^{85} + ( -1 + 3 \beta_{1} - 5 \beta_{2} + 3 \beta_{3} ) q^{87} + ( 2 - 4 \beta_{1} - \beta_{2} + \beta_{3} ) q^{89} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{91} + ( 4 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{93} + ( -8 + 8 \beta_{2} - 4 \beta_{3} ) q^{95} + ( 2 - 2 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{97} + ( -9 + 7 \beta_{1} - 4 \beta_{2} + 3 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 3q^{3} + 4q^{7} + q^{9} + O(q^{10}) \) \( 4q - 3q^{3} + 4q^{7} + q^{9} - 10q^{11} - 3q^{13} - 6q^{15} - 4q^{17} - 10q^{19} - 3q^{21} - 4q^{23} + 4q^{25} - 9q^{27} - 13q^{29} + 3q^{31} + 20q^{33} - 11q^{39} + q^{41} - 8q^{43} + 4q^{45} - 5q^{47} + 4q^{49} - 4q^{51} - 20q^{53} - 22q^{55} - 2q^{57} - 14q^{59} + 8q^{61} + q^{63} - 2q^{65} - 18q^{67} + 3q^{69} - 25q^{71} + 15q^{73} - 11q^{75} - 10q^{77} + 10q^{79} - 36q^{83} - 28q^{85} + q^{87} + 4q^{89} - 3q^{91} + 17q^{93} - 36q^{95} + 6q^{97} - 28q^{99} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 4 \nu - 1 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 4 \beta_{1} + 1\)

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
−1.89122
−0.704624
1.31743
2.27841
0 −2.89122 0 −0.776183 0 1.00000 0 5.35915 0
1.2 0 −1.70462 0 3.97216 0 1.00000 0 −0.0942558 0
1.3 0 0.317431 0 −2.71878 0 1.00000 0 −2.89924 0
1.4 0 1.27841 0 −0.477194 0 1.00000 0 −1.36566 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.n 4
4.b odd 2 1 2576.2.a.ba 4
7.b odd 2 1 9016.2.a.bf 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1288.2.a.n 4 1.a even 1 1 trivial
2576.2.a.ba 4 4.b odd 2 1
9016.2.a.bf 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1288))\):

\( T_{3}^{4} + 3 T_{3}^{3} - 2 T_{3}^{2} - 6 T_{3} + 2 \)
\( T_{5}^{4} - 12 T_{5}^{2} - 14 T_{5} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 2 - 6 T - 2 T^{2} + 3 T^{3} + T^{4} \)
$5$ \( -4 - 14 T - 12 T^{2} + T^{4} \)
$7$ \( ( -1 + T )^{4} \)
$11$ \( -136 - 52 T + 20 T^{2} + 10 T^{3} + T^{4} \)
$13$ \( -8 + 28 T - 26 T^{2} + 3 T^{3} + T^{4} \)
$17$ \( 32 - 94 T - 40 T^{2} + 4 T^{3} + T^{4} \)
$19$ \( 128 - 160 T - 8 T^{2} + 10 T^{3} + T^{4} \)
$23$ \( ( 1 + T )^{4} \)
$29$ \( 4 - 240 T + 10 T^{2} + 13 T^{3} + T^{4} \)
$31$ \( 2 + 2 T - 36 T^{2} - 3 T^{3} + T^{4} \)
$37$ \( 16 + 144 T - 64 T^{2} + T^{4} \)
$41$ \( 232 + 148 T - 118 T^{2} - T^{3} + T^{4} \)
$43$ \( 3712 - 464 T - 120 T^{2} + 8 T^{3} + T^{4} \)
$47$ \( -2 - 6 T + 4 T^{2} + 5 T^{3} + T^{4} \)
$53$ \( -272 - 336 T + 64 T^{2} + 20 T^{3} + T^{4} \)
$59$ \( -128 + 14 T + 52 T^{2} + 14 T^{3} + T^{4} \)
$61$ \( -4 - 18 T - 24 T^{2} - 8 T^{3} + T^{4} \)
$67$ \( -1432 - 532 T + 36 T^{2} + 18 T^{3} + T^{4} \)
$71$ \( 256 + 512 T + 192 T^{2} + 25 T^{3} + T^{4} \)
$73$ \( 584 + 380 T + 6 T^{2} - 15 T^{3} + T^{4} \)
$79$ \( -136 + 52 T + 20 T^{2} - 10 T^{3} + T^{4} \)
$83$ \( -31552 - 3120 T + 248 T^{2} + 36 T^{3} + T^{4} \)
$89$ \( 3512 + 250 T - 116 T^{2} - 4 T^{3} + T^{4} \)
$97$ \( 584 + 438 T - 80 T^{2} - 6 T^{3} + T^{4} \)
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