Properties

Label 7623.2.a.ck
Level $7623$
Weight $2$
Character orbit 7623.a
Self dual yes
Analytic conductor $60.870$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7623.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(60.8699614608\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{7})\)
Defining polynomial: \( x^{4} - 8x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{3} q^{5} + q^{7} - 2 \beta_1 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{3} q^{5} + q^{7} - 2 \beta_1 q^{8} - \beta_{2} q^{10} + ( - \beta_{2} - 2) q^{13} + \beta_1 q^{14} - 4 q^{16} + (\beta_{3} + 2 \beta_1) q^{17} + (\beta_{2} + 2) q^{19} + (2 \beta_{3} - \beta_1) q^{23} + 2 q^{25} + ( - 2 \beta_{3} - 2 \beta_1) q^{26} + (2 \beta_{3} - \beta_1) q^{29} + (2 \beta_{2} - 2) q^{31} + (\beta_{2} + 4) q^{34} - \beta_{3} q^{35} + (2 \beta_{2} - 4) q^{37} + (2 \beta_{3} + 2 \beta_1) q^{38} + 2 \beta_{2} q^{40} - 4 \beta_1 q^{41} - q^{43} + (2 \beta_{2} - 2) q^{46} + (\beta_{3} + 2 \beta_1) q^{47} + q^{49} + 2 \beta_1 q^{50} + ( - 4 \beta_{3} - 2 \beta_1) q^{53} - 2 \beta_1 q^{56} + (2 \beta_{2} - 2) q^{58} + ( - \beta_{3} - 4 \beta_1) q^{59} + (\beta_{2} - 2) q^{61} + (4 \beta_{3} - 2 \beta_1) q^{62} + 8 q^{64} + (2 \beta_{3} + 7 \beta_1) q^{65} + ( - 2 \beta_{2} + 1) q^{67} - \beta_{2} q^{70} + \beta_1 q^{71} + ( - \beta_{2} - 12) q^{73} + (4 \beta_{3} - 4 \beta_1) q^{74} + ( - 2 \beta_{2} - 6) q^{79} + 4 \beta_{3} q^{80} - 8 q^{82} + (\beta_{3} + 6 \beta_1) q^{83} + ( - 2 \beta_{2} - 7) q^{85} - \beta_1 q^{86} + (\beta_{3} - 6 \beta_1) q^{89} + ( - \beta_{2} - 2) q^{91} + (\beta_{2} + 4) q^{94} + ( - 2 \beta_{3} - 7 \beta_1) q^{95} + ( - 3 \beta_{2} + 4) q^{97} + \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{7} - 8 q^{13} - 16 q^{16} + 8 q^{19} + 8 q^{25} - 8 q^{31} + 16 q^{34} - 16 q^{37} - 4 q^{43} - 8 q^{46} + 4 q^{49} - 8 q^{58} - 8 q^{61} + 32 q^{64} + 4 q^{67} - 48 q^{73} - 24 q^{79} - 32 q^{82} - 28 q^{85} - 8 q^{91} + 16 q^{94} + 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 8x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} - 5\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 11\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5\beta_{2} + 11\beta_1 ) / 2 \) 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
−2.57794
1.16372
2.57794
−1.16372
−1.41421 0 0 −2.64575 0 1.00000 2.82843 0 3.74166
1.2 −1.41421 0 0 2.64575 0 1.00000 2.82843 0 −3.74166
1.3 1.41421 0 0 −2.64575 0 1.00000 −2.82843 0 −3.74166
1.4 1.41421 0 0 2.64575 0 1.00000 −2.82843 0 3.74166
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(7\) \(-1\)
\(11\) \(-1\)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7623.2.a.ck yes 4
3.b odd 2 1 inner 7623.2.a.ck yes 4
11.b odd 2 1 7623.2.a.cj 4
33.d even 2 1 7623.2.a.cj 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7623.2.a.cj 4 11.b odd 2 1
7623.2.a.cj 4 33.d even 2 1
7623.2.a.ck yes 4 1.a even 1 1 trivial
7623.2.a.ck yes 4 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7623))\):

\( T_{2}^{2} - 2 \) Copy content Toggle raw display
\( T_{5}^{2} - 7 \) Copy content Toggle raw display
\( T_{13}^{2} + 4T_{13} - 10 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} - 7)^{2} \) Copy content Toggle raw display
$7$ \( (T - 1)^{4} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 4 T - 10)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 30T^{2} + 1 \) Copy content Toggle raw display
$19$ \( (T^{2} - 4 T - 10)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 60T^{2} + 676 \) Copy content Toggle raw display
$29$ \( T^{4} - 60T^{2} + 676 \) Copy content Toggle raw display
$31$ \( (T^{2} + 4 T - 52)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 8 T - 40)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$43$ \( (T + 1)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} - 30T^{2} + 1 \) Copy content Toggle raw display
$53$ \( T^{4} - 240 T^{2} + 10816 \) Copy content Toggle raw display
$59$ \( T^{4} - 78T^{2} + 625 \) Copy content Toggle raw display
$61$ \( (T^{2} + 4 T - 10)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 2 T - 55)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 24 T + 130)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 12 T - 20)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 158T^{2} + 4225 \) Copy content Toggle raw display
$89$ \( T^{4} - 158T^{2} + 4225 \) Copy content Toggle raw display
$97$ \( (T^{2} - 8 T - 110)^{2} \) Copy content Toggle raw display
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