Properties

Label 4704.2.a.br
Level $4704$
Weight $2$
Character orbit 4704.a
Self dual yes
Analytic conductor $37.562$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4704 = 2^{5} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4704.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(37.5616291108\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
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 \(\beta = 2\sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

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

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 1.00000 0 −2.82843 0 0 0 1.00000 0
1.2 0 1.00000 0 2.82843 0 0 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(7\) \(-1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4704.2.a.br yes 2
4.b odd 2 1 4704.2.a.bi 2
7.b odd 2 1 4704.2.a.bi 2
8.b even 2 1 9408.2.a.dk 2
8.d odd 2 1 9408.2.a.eb 2
28.d even 2 1 inner 4704.2.a.br yes 2
56.e even 2 1 9408.2.a.dk 2
56.h odd 2 1 9408.2.a.eb 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4704.2.a.bi 2 4.b odd 2 1
4704.2.a.bi 2 7.b odd 2 1
4704.2.a.br yes 2 1.a even 1 1 trivial
4704.2.a.br yes 2 28.d even 2 1 inner
9408.2.a.dk 2 8.b even 2 1
9408.2.a.dk 2 56.e even 2 1
9408.2.a.eb 2 8.d odd 2 1
9408.2.a.eb 2 56.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4704))\):

\( T_{5}^{2} - 8 \)
\( T_{11}^{2} - 8 \)
\( T_{13} \)
\( T_{19} - 4 \)
\( T_{31} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( -8 + T^{2} \)
$7$ \( T^{2} \)
$11$ \( -8 + T^{2} \)
$13$ \( T^{2} \)
$17$ \( -8 + T^{2} \)
$19$ \( ( -4 + T )^{2} \)
$23$ \( -72 + T^{2} \)
$29$ \( ( -2 + T )^{2} \)
$31$ \( T^{2} \)
$37$ \( ( 6 + T )^{2} \)
$41$ \( -72 + T^{2} \)
$43$ \( -128 + T^{2} \)
$47$ \( ( -8 + T )^{2} \)
$53$ \( ( -6 + T )^{2} \)
$59$ \( ( -12 + T )^{2} \)
$61$ \( -32 + T^{2} \)
$67$ \( -32 + T^{2} \)
$71$ \( -8 + T^{2} \)
$73$ \( -32 + T^{2} \)
$79$ \( -32 + T^{2} \)
$83$ \( ( 4 + T )^{2} \)
$89$ \( -8 + T^{2} \)
$97$ \( -288 + T^{2} \)
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