Properties

Label 1288.2.a.k
Level $1288$
Weight $2$
Character orbit 1288.a
Self dual yes
Analytic conductor $10.285$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} + ( -2 - \beta ) q^{5} - q^{7} - q^{9} +O(q^{10})\) \( q + \beta q^{3} + ( -2 - \beta ) q^{5} - q^{7} - q^{9} + 2 q^{11} + 2 \beta q^{13} + ( -2 - 2 \beta ) q^{15} + ( 4 + \beta ) q^{17} + 2 q^{19} -\beta q^{21} - q^{23} + ( 1 + 4 \beta ) q^{25} -4 \beta q^{27} + 6 \beta q^{29} + ( 2 - \beta ) q^{31} + 2 \beta q^{33} + ( 2 + \beta ) q^{35} + ( 2 - 6 \beta ) q^{37} + 4 q^{39} + 6 q^{41} + 8 q^{43} + ( 2 + \beta ) q^{45} + ( 6 + \beta ) q^{47} + q^{49} + ( 2 + 4 \beta ) q^{51} + ( 2 + 2 \beta ) q^{53} + ( -4 - 2 \beta ) q^{55} + 2 \beta q^{57} + ( 8 - 5 \beta ) q^{59} + ( -6 - 3 \beta ) q^{61} + q^{63} + ( -4 - 4 \beta ) q^{65} + ( 6 + 4 \beta ) q^{67} -\beta q^{69} + ( -4 + 6 \beta ) q^{71} + ( 2 + 6 \beta ) q^{73} + ( 8 + \beta ) q^{75} -2 q^{77} + ( 2 - 6 \beta ) q^{79} -5 q^{81} + ( 2 - 2 \beta ) q^{83} + ( -10 - 6 \beta ) q^{85} + 12 q^{87} + ( 8 + 5 \beta ) q^{89} -2 \beta q^{91} + ( -2 + 2 \beta ) q^{93} + ( -4 - 2 \beta ) q^{95} -11 \beta q^{97} -2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{5} - 2q^{7} - 2q^{9} + O(q^{10}) \) \( 2q - 4q^{5} - 2q^{7} - 2q^{9} + 4q^{11} - 4q^{15} + 8q^{17} + 4q^{19} - 2q^{23} + 2q^{25} + 4q^{31} + 4q^{35} + 4q^{37} + 8q^{39} + 12q^{41} + 16q^{43} + 4q^{45} + 12q^{47} + 2q^{49} + 4q^{51} + 4q^{53} - 8q^{55} + 16q^{59} - 12q^{61} + 2q^{63} - 8q^{65} + 12q^{67} - 8q^{71} + 4q^{73} + 16q^{75} - 4q^{77} + 4q^{79} - 10q^{81} + 4q^{83} - 20q^{85} + 24q^{87} + 16q^{89} - 4q^{93} - 8q^{95} - 4q^{99} + O(q^{100}) \)

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
−1.41421
1.41421
0 −1.41421 0 −0.585786 0 −1.00000 0 −1.00000 0
1.2 0 1.41421 0 −3.41421 0 −1.00000 0 −1.00000 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.k 2
4.b odd 2 1 2576.2.a.q 2
7.b odd 2 1 9016.2.a.v 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1288.2.a.k 2 1.a even 1 1 trivial
2576.2.a.q 2 4.b odd 2 1
9016.2.a.v 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1288))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -2 + T^{2} \)
$5$ \( 2 + 4 T + T^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( ( -2 + T )^{2} \)
$13$ \( -8 + T^{2} \)
$17$ \( 14 - 8 T + T^{2} \)
$19$ \( ( -2 + T )^{2} \)
$23$ \( ( 1 + T )^{2} \)
$29$ \( -72 + T^{2} \)
$31$ \( 2 - 4 T + T^{2} \)
$37$ \( -68 - 4 T + T^{2} \)
$41$ \( ( -6 + T )^{2} \)
$43$ \( ( -8 + T )^{2} \)
$47$ \( 34 - 12 T + T^{2} \)
$53$ \( -4 - 4 T + T^{2} \)
$59$ \( 14 - 16 T + T^{2} \)
$61$ \( 18 + 12 T + T^{2} \)
$67$ \( 4 - 12 T + T^{2} \)
$71$ \( -56 + 8 T + T^{2} \)
$73$ \( -68 - 4 T + T^{2} \)
$79$ \( -68 - 4 T + T^{2} \)
$83$ \( -4 - 4 T + T^{2} \)
$89$ \( 14 - 16 T + T^{2} \)
$97$ \( -242 + T^{2} \)
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