Properties

Label 1088.2.a.v
Level $1088$
Weight $2$
Character orbit 1088.a
Self dual yes
Analytic conductor $8.688$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1088 = 2^{6} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1088.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.68772373992\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 544)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{3} + (\beta_{2} + \beta_1) q^{5} + (\beta_{2} - 1) q^{7} + (\beta_{2} - \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{3} + (\beta_{2} + \beta_1) q^{5} + (\beta_{2} - 1) q^{7} + (\beta_{2} - \beta_1 + 1) q^{9} + (\beta_1 + 3) q^{11} + ( - \beta_{2} + \beta_1 + 2) q^{13} + ( - 2 \beta_1 - 2) q^{15} + q^{17} + ( - 2 \beta_1 + 2) q^{19} + (\beta_{2} - \beta_1) q^{21} + ( - 3 \beta_{2} - 1) q^{23} + (4 \beta_1 + 3) q^{25} + (2 \beta_{2} + 2) q^{27} + ( - \beta_{2} - \beta_1 + 4) q^{29} + (\beta_{2} - 2 \beta_1 - 3) q^{31} + ( - \beta_{2} - 3 \beta_1) q^{33} + ( - 2 \beta_{2} + 6) q^{35} + ( - \beta_{2} - \beta_1 + 4) q^{37} + ( - 2 \beta_{2} - 2) q^{39} + ( - 2 \beta_{2} + 2 \beta_1 + 2) q^{41} + ( - 2 \beta_{2} - 2) q^{43} + ( - \beta_{2} - \beta_1 + 4) q^{45} + (2 \beta_{2} + 2 \beta_1) q^{47} + ( - 3 \beta_{2} - \beta_1 + 1) q^{49} + ( - \beta_1 + 1) q^{51} + (2 \beta_{2} - 2 \beta_1 + 2) q^{53} + (4 \beta_{2} + 6 \beta_1 + 2) q^{55} + (2 \beta_{2} - 2 \beta_1 + 8) q^{57} + (2 \beta_{2} + 4 \beta_1 - 2) q^{59} + ( - \beta_{2} - 5 \beta_1 + 8) q^{61} + ( - \beta_{2} - 2 \beta_1 + 7) q^{63} + (4 \beta_{2} + 4 \beta_1 - 4) q^{65} + (4 \beta_1 - 4) q^{67} + ( - 3 \beta_{2} + 7 \beta_1 - 4) q^{69} + (\beta_{2} + 2 \beta_1 - 11) q^{71} + ( - 4 \beta_1 + 6) q^{73} + ( - 4 \beta_{2} - 3 \beta_1 - 9) q^{75} + (3 \beta_{2} + \beta_1 - 4) q^{77} + ( - 3 \beta_{2} + 4 \beta_1 + 3) q^{79} + ( - \beta_{2} - 3 \beta_1 + 1) q^{81} + (2 \beta_{2} + 2) q^{83} + (\beta_{2} + \beta_1) q^{85} + ( - 2 \beta_1 + 6) q^{87} + (3 \beta_{2} + \beta_1 + 2) q^{89} + (4 \beta_{2} + 2 \beta_1 - 10) q^{91} + (3 \beta_{2} + \beta_1 + 4) q^{93} + ( - 4 \beta_1 - 4) q^{95} + ( - 4 \beta_1 - 2) q^{97} + (2 \beta_{2} - \beta_1 - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{3} + 2 q^{5} - 2 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 2 q^{3} + 2 q^{5} - 2 q^{7} + 3 q^{9} + 10 q^{11} + 6 q^{13} - 8 q^{15} + 3 q^{17} + 4 q^{19} - 6 q^{23} + 13 q^{25} + 8 q^{27} + 10 q^{29} - 10 q^{31} - 4 q^{33} + 16 q^{35} + 10 q^{37} - 8 q^{39} + 6 q^{41} - 8 q^{43} + 10 q^{45} + 4 q^{47} - q^{49} + 2 q^{51} + 6 q^{53} + 16 q^{55} + 24 q^{57} + 18 q^{61} + 18 q^{63} - 4 q^{65} - 8 q^{67} - 8 q^{69} - 30 q^{71} + 14 q^{73} - 34 q^{75} - 8 q^{77} + 10 q^{79} - q^{81} + 8 q^{83} + 2 q^{85} + 16 q^{87} + 10 q^{89} - 24 q^{91} + 16 q^{93} - 16 q^{95} - 10 q^{97} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{2} + 2\nu + 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta _1 + 2 \) Copy content Toggle raw display

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
2.17009
−1.48119
0.311108
0 −1.70928 0 4.34017 0 0.630898 0 −0.0783777 0
1.2 0 0.806063 0 −2.96239 0 −4.15633 0 −2.35026 0
1.3 0 2.90321 0 0.622216 0 1.52543 0 5.42864 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(17\) \(-1\)

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 1088.2.a.v 3
3.b odd 2 1 9792.2.a.dc 3
4.b odd 2 1 1088.2.a.u 3
8.b even 2 1 544.2.a.i 3
8.d odd 2 1 544.2.a.j yes 3
12.b even 2 1 9792.2.a.dd 3
24.f even 2 1 4896.2.a.bf 3
24.h odd 2 1 4896.2.a.be 3
136.e odd 2 1 9248.2.a.t 3
136.h even 2 1 9248.2.a.u 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
544.2.a.i 3 8.b even 2 1
544.2.a.j yes 3 8.d odd 2 1
1088.2.a.u 3 4.b odd 2 1
1088.2.a.v 3 1.a even 1 1 trivial
4896.2.a.be 3 24.h odd 2 1
4896.2.a.bf 3 24.f even 2 1
9248.2.a.t 3 136.e odd 2 1
9248.2.a.u 3 136.h even 2 1
9792.2.a.dc 3 3.b odd 2 1
9792.2.a.dd 3 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1088))\):

\( T_{3}^{3} - 2T_{3}^{2} - 4T_{3} + 4 \) Copy content Toggle raw display
\( T_{5}^{3} - 2T_{5}^{2} - 12T_{5} + 8 \) Copy content Toggle raw display
\( T_{7}^{3} + 2T_{7}^{2} - 8T_{7} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} - 2 T^{2} - 4 T + 4 \) Copy content Toggle raw display
$5$ \( T^{3} - 2 T^{2} - 12 T + 8 \) Copy content Toggle raw display
$7$ \( T^{3} + 2 T^{2} - 8 T + 4 \) Copy content Toggle raw display
$11$ \( T^{3} - 10 T^{2} + 28 T - 20 \) Copy content Toggle raw display
$13$ \( T^{3} - 6 T^{2} - 4 T + 40 \) Copy content Toggle raw display
$17$ \( (T - 1)^{3} \) Copy content Toggle raw display
$19$ \( T^{3} - 4 T^{2} - 16 T + 32 \) Copy content Toggle raw display
$23$ \( T^{3} + 6 T^{2} - 72 T - 428 \) Copy content Toggle raw display
$29$ \( T^{3} - 10 T^{2} + 20 T + 8 \) Copy content Toggle raw display
$31$ \( T^{3} + 10T^{2} - 148 \) Copy content Toggle raw display
$37$ \( T^{3} - 10 T^{2} + 20 T + 8 \) Copy content Toggle raw display
$41$ \( T^{3} - 6 T^{2} - 52 T + 248 \) Copy content Toggle raw display
$43$ \( T^{3} + 8 T^{2} - 16 T - 160 \) Copy content Toggle raw display
$47$ \( T^{3} - 4 T^{2} - 48 T + 64 \) Copy content Toggle raw display
$53$ \( T^{3} - 6 T^{2} - 52 T - 8 \) Copy content Toggle raw display
$59$ \( T^{3} - 112T - 416 \) Copy content Toggle raw display
$61$ \( T^{3} - 18 T^{2} - 28 T + 1096 \) Copy content Toggle raw display
$67$ \( T^{3} + 8 T^{2} - 64 T - 256 \) Copy content Toggle raw display
$71$ \( T^{3} + 30 T^{2} + 272 T + 668 \) Copy content Toggle raw display
$73$ \( T^{3} - 14 T^{2} - 20 T + 344 \) Copy content Toggle raw display
$79$ \( T^{3} - 10 T^{2} - 152 T + 1444 \) Copy content Toggle raw display
$83$ \( T^{3} - 8 T^{2} - 16 T + 160 \) Copy content Toggle raw display
$89$ \( T^{3} - 10 T^{2} - 52 T + 536 \) Copy content Toggle raw display
$97$ \( T^{3} + 10 T^{2} - 52 T - 200 \) Copy content Toggle raw display
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