Properties

Label 3528.2.a.g
Level $3528$
Weight $2$
Character orbit 3528.a
Self dual yes
Analytic conductor $28.171$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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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: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 504)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2q^{5} + O(q^{10}) \) \( q - 2q^{5} + 6q^{11} + 6q^{13} + 2q^{17} - 4q^{19} + 2q^{23} - q^{25} + 8q^{29} - 4q^{31} - 6q^{37} - 10q^{41} - 4q^{43} + 4q^{47} - 4q^{53} - 12q^{55} + 12q^{59} + 2q^{61} - 12q^{65} + 12q^{67} + 6q^{71} + 2q^{73} - 8q^{79} - 4q^{85} + 14q^{89} + 8q^{95} + 2q^{97} + O(q^{100}) \)

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
0
0 0 0 −2.00000 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

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3528.2.a.g 1
3.b odd 2 1 3528.2.a.t 1
4.b odd 2 1 7056.2.a.j 1
7.b odd 2 1 504.2.a.g yes 1
7.c even 3 2 3528.2.s.s 2
7.d odd 6 2 3528.2.s.c 2
12.b even 2 1 7056.2.a.bv 1
21.c even 2 1 504.2.a.d 1
21.g even 6 2 3528.2.s.z 2
21.h odd 6 2 3528.2.s.k 2
28.d even 2 1 1008.2.a.i 1
56.e even 2 1 4032.2.a.i 1
56.h odd 2 1 4032.2.a.j 1
84.h odd 2 1 1008.2.a.c 1
168.e odd 2 1 4032.2.a.ba 1
168.i even 2 1 4032.2.a.bl 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.a.d 1 21.c even 2 1
504.2.a.g yes 1 7.b odd 2 1
1008.2.a.c 1 84.h odd 2 1
1008.2.a.i 1 28.d even 2 1
3528.2.a.g 1 1.a even 1 1 trivial
3528.2.a.t 1 3.b odd 2 1
3528.2.s.c 2 7.d odd 6 2
3528.2.s.k 2 21.h odd 6 2
3528.2.s.s 2 7.c even 3 2
3528.2.s.z 2 21.g even 6 2
4032.2.a.i 1 56.e even 2 1
4032.2.a.j 1 56.h odd 2 1
4032.2.a.ba 1 168.e odd 2 1
4032.2.a.bl 1 168.i even 2 1
7056.2.a.j 1 4.b odd 2 1
7056.2.a.bv 1 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3528))\):

\( T_{5} + 2 \)
\( T_{11} - 6 \)
\( T_{13} - 6 \)
\( T_{23} - 2 \)