Properties

Label 3584.2.a.a
Level $3584$
Weight $2$
Character orbit 3584.a
Self dual yes
Analytic conductor $28.618$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3584 = 2^{9} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3584.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.6183840844\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + \beta q^{3} - q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{3} - q^{7} - q^{9} - \beta q^{11} - \beta q^{19} - \beta q^{21} + 4 q^{23} - 5 q^{25} - 4 \beta q^{27} + 2 \beta q^{29} + 4 q^{31} - 2 q^{33} + 2 \beta q^{37} - 6 q^{41} - 3 \beta q^{43} - 4 q^{47} + q^{49} - 6 \beta q^{53} - 2 q^{57} - 5 \beta q^{59} - 4 \beta q^{61} + q^{63} + \beta q^{67} + 4 \beta q^{69} - 8 q^{71} - 8 q^{73} - 5 \beta q^{75} + \beta q^{77} - 4 q^{79} - 5 q^{81} + 5 \beta q^{83} + 4 q^{87} - 8 q^{89} + 4 \beta q^{93} - 8 q^{97} + \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{7} - 2 q^{9} + 8 q^{23} - 10 q^{25} + 8 q^{31} - 4 q^{33} - 12 q^{41} - 8 q^{47} + 2 q^{49} - 4 q^{57} + 2 q^{63} - 16 q^{71} - 16 q^{73} - 8 q^{79} - 10 q^{81} + 8 q^{87} - 16 q^{89} - 16 q^{97}+O(q^{100}) \) 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
−1.41421
1.41421
0 −1.41421 0 0 0 −1.00000 0 −1.00000 0
1.2 0 1.41421 0 0 0 −1.00000 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3584.2.a.a 2
4.b odd 2 1 3584.2.a.b yes 2
8.b even 2 1 inner 3584.2.a.a 2
8.d odd 2 1 3584.2.a.b yes 2
16.e even 4 2 3584.2.b.b 2
16.f odd 4 2 3584.2.b.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3584.2.a.a 2 1.a even 1 1 trivial
3584.2.a.a 2 8.b even 2 1 inner
3584.2.a.b yes 2 4.b odd 2 1
3584.2.a.b yes 2 8.d odd 2 1
3584.2.b.a 2 16.f odd 4 2
3584.2.b.b 2 16.e even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3584))\):

\( T_{3}^{2} - 2 \) Copy content Toggle raw display
\( T_{5} \) Copy content Toggle raw display
\( T_{23} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 2 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 2 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 2 \) Copy content Toggle raw display
$23$ \( (T - 4)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 8 \) Copy content Toggle raw display
$31$ \( (T - 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 8 \) Copy content Toggle raw display
$41$ \( (T + 6)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 18 \) Copy content Toggle raw display
$47$ \( (T + 4)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 72 \) Copy content Toggle raw display
$59$ \( T^{2} - 50 \) Copy content Toggle raw display
$61$ \( T^{2} - 32 \) Copy content Toggle raw display
$67$ \( T^{2} - 2 \) Copy content Toggle raw display
$71$ \( (T + 8)^{2} \) Copy content Toggle raw display
$73$ \( (T + 8)^{2} \) Copy content Toggle raw display
$79$ \( (T + 4)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 50 \) Copy content Toggle raw display
$89$ \( (T + 8)^{2} \) Copy content Toggle raw display
$97$ \( (T + 8)^{2} \) Copy content Toggle raw display
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