Properties

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

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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: \(2\)
Coefficient field: \(\Q(\sqrt{10}) \)
Defining polynomial: \( x^{2} - 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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 \(\beta = \sqrt{10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} + 2 q^{5} - \beta q^{7} + 7 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{3} + 2 q^{5} - \beta q^{7} + 7 q^{9} + \beta q^{11} + 4 q^{13} + 2 \beta q^{15} - q^{17} - 2 \beta q^{19} - 10 q^{21} + \beta q^{23} - q^{25} + 4 \beta q^{27} - 10 q^{29} - \beta q^{31} + 10 q^{33} - 2 \beta q^{35} + 2 q^{37} + 4 \beta q^{39} + 10 q^{41} - 2 \beta q^{43} + 14 q^{45} + 3 q^{49} - \beta q^{51} + 6 q^{53} + 2 \beta q^{55} - 20 q^{57} + 2 \beta q^{59} + 10 q^{61} - 7 \beta q^{63} + 8 q^{65} - 4 \beta q^{67} + 10 q^{69} + \beta q^{71} - 6 q^{73} - \beta q^{75} - 10 q^{77} - 3 \beta q^{79} + 19 q^{81} - 2 \beta q^{83} - 2 q^{85} - 10 \beta q^{87} - 4 \beta q^{91} - 10 q^{93} - 4 \beta q^{95} - 2 q^{97} + 7 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{5} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{5} + 14 q^{9} + 8 q^{13} - 2 q^{17} - 20 q^{21} - 2 q^{25} - 20 q^{29} + 20 q^{33} + 4 q^{37} + 20 q^{41} + 28 q^{45} + 6 q^{49} + 12 q^{53} - 40 q^{57} + 20 q^{61} + 16 q^{65} + 20 q^{69} - 12 q^{73} - 20 q^{77} + 38 q^{81} - 4 q^{85} - 20 q^{93} - 4 q^{97}+O(q^{100}) \) 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
−3.16228
3.16228
0 −3.16228 0 2.00000 0 3.16228 0 7.00000 0
1.2 0 3.16228 0 2.00000 0 −3.16228 0 7.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1088.2.a.r 2
3.b odd 2 1 9792.2.a.ch 2
4.b odd 2 1 inner 1088.2.a.r 2
8.b even 2 1 544.2.a.h 2
8.d odd 2 1 544.2.a.h 2
12.b even 2 1 9792.2.a.ch 2
24.f even 2 1 4896.2.a.y 2
24.h odd 2 1 4896.2.a.y 2
136.e odd 2 1 9248.2.a.q 2
136.h even 2 1 9248.2.a.q 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
544.2.a.h 2 8.b even 2 1
544.2.a.h 2 8.d odd 2 1
1088.2.a.r 2 1.a even 1 1 trivial
1088.2.a.r 2 4.b odd 2 1 inner
4896.2.a.y 2 24.f even 2 1
4896.2.a.y 2 24.h odd 2 1
9248.2.a.q 2 136.e odd 2 1
9248.2.a.q 2 136.h even 2 1
9792.2.a.ch 2 3.b odd 2 1
9792.2.a.ch 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1088))\):

\( T_{3}^{2} - 10 \) Copy content Toggle raw display
\( T_{5} - 2 \) Copy content Toggle raw display
\( T_{7}^{2} - 10 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 10 \) Copy content Toggle raw display
$5$ \( (T - 2)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 10 \) Copy content Toggle raw display
$11$ \( T^{2} - 10 \) Copy content Toggle raw display
$13$ \( (T - 4)^{2} \) Copy content Toggle raw display
$17$ \( (T + 1)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 40 \) Copy content Toggle raw display
$23$ \( T^{2} - 10 \) Copy content Toggle raw display
$29$ \( (T + 10)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 10 \) Copy content Toggle raw display
$37$ \( (T - 2)^{2} \) Copy content Toggle raw display
$41$ \( (T - 10)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 40 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( (T - 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 40 \) Copy content Toggle raw display
$61$ \( (T - 10)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 160 \) Copy content Toggle raw display
$71$ \( T^{2} - 10 \) Copy content Toggle raw display
$73$ \( (T + 6)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 90 \) Copy content Toggle raw display
$83$ \( T^{2} - 40 \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( (T + 2)^{2} \) Copy content Toggle raw display
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