Properties

Label 3528.2.a.bh
Level $3528$
Weight $2$
Character orbit 3528.a
Self dual yes
Analytic conductor $28.171$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3528.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.1712218331\)
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: \( 1 \)
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.

\(f(q)\) \(=\) \( q + \beta q^{5} +O(q^{10})\) \( q + \beta q^{5} + \beta q^{13} -\beta q^{17} + 4 q^{23} -3 q^{25} + 6 q^{29} + 4 \beta q^{31} -4 q^{37} -5 \beta q^{41} + 4 q^{43} + 8 \beta q^{47} + 4 q^{53} -4 \beta q^{59} + 3 \beta q^{61} + 2 q^{65} + 4 q^{67} + 12 q^{71} -5 \beta q^{73} + 8 q^{79} -4 \beta q^{83} -2 q^{85} + 9 \beta q^{89} + 3 \beta q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q + 8q^{23} - 6q^{25} + 12q^{29} - 8q^{37} + 8q^{43} + 8q^{53} + 4q^{65} + 8q^{67} + 24q^{71} + 16q^{79} - 4q^{85} + O(q^{100}) \)

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 0 0 −1.41421 0 0 0 0 0
1.2 0 0 0 1.41421 0 0 0 0 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
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3528.2.a.bh yes 2
3.b odd 2 1 3528.2.a.bg 2
4.b odd 2 1 7056.2.a.cn 2
7.b odd 2 1 inner 3528.2.a.bh yes 2
7.c even 3 2 3528.2.s.bg 4
7.d odd 6 2 3528.2.s.bg 4
12.b even 2 1 7056.2.a.cp 2
21.c even 2 1 3528.2.a.bg 2
21.g even 6 2 3528.2.s.bh 4
21.h odd 6 2 3528.2.s.bh 4
28.d even 2 1 7056.2.a.cn 2
84.h odd 2 1 7056.2.a.cp 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3528.2.a.bg 2 3.b odd 2 1
3528.2.a.bg 2 21.c even 2 1
3528.2.a.bh yes 2 1.a even 1 1 trivial
3528.2.a.bh yes 2 7.b odd 2 1 inner
3528.2.s.bg 4 7.c even 3 2
3528.2.s.bg 4 7.d odd 6 2
3528.2.s.bh 4 21.g even 6 2
3528.2.s.bh 4 21.h odd 6 2
7056.2.a.cn 2 4.b odd 2 1
7056.2.a.cn 2 28.d even 2 1
7056.2.a.cp 2 12.b even 2 1
7056.2.a.cp 2 84.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3528))\):

\( T_{5}^{2} - 2 \)
\( T_{11} \)
\( T_{13}^{2} - 2 \)
\( T_{23} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 + 8 T^{2} + 25 T^{4} \)
$7$ 1
$11$ \( ( 1 + 11 T^{2} )^{2} \)
$13$ \( 1 + 24 T^{2} + 169 T^{4} \)
$17$ \( 1 + 32 T^{2} + 289 T^{4} \)
$19$ \( ( 1 + 19 T^{2} )^{2} \)
$23$ \( ( 1 - 4 T + 23 T^{2} )^{2} \)
$29$ \( ( 1 - 6 T + 29 T^{2} )^{2} \)
$31$ \( 1 + 30 T^{2} + 961 T^{4} \)
$37$ \( ( 1 + 4 T + 37 T^{2} )^{2} \)
$41$ \( 1 + 32 T^{2} + 1681 T^{4} \)
$43$ \( ( 1 - 4 T + 43 T^{2} )^{2} \)
$47$ \( 1 - 34 T^{2} + 2209 T^{4} \)
$53$ \( ( 1 - 4 T + 53 T^{2} )^{2} \)
$59$ \( 1 + 86 T^{2} + 3481 T^{4} \)
$61$ \( 1 + 104 T^{2} + 3721 T^{4} \)
$67$ \( ( 1 - 4 T + 67 T^{2} )^{2} \)
$71$ \( ( 1 - 12 T + 71 T^{2} )^{2} \)
$73$ \( 1 + 96 T^{2} + 5329 T^{4} \)
$79$ \( ( 1 - 8 T + 79 T^{2} )^{2} \)
$83$ \( 1 + 134 T^{2} + 6889 T^{4} \)
$89$ \( 1 + 16 T^{2} + 7921 T^{4} \)
$97$ \( 1 + 176 T^{2} + 9409 T^{4} \)
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