Properties

Label 3525.2.a.f
Level $3525$
Weight $2$
Character orbit 3525.a
Self dual yes
Analytic conductor $28.147$
Analytic rank $2$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3525 = 3 \cdot 5^{2} \cdot 47 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3525.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.1472667125\)
Analytic rank: \(2\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 705)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{3} - 2q^{4} - 2q^{7} + q^{9} + O(q^{10}) \) \( q - q^{3} - 2q^{4} - 2q^{7} + q^{9} - 6q^{11} + 2q^{12} - 5q^{13} + 4q^{16} - 6q^{17} + 2q^{19} + 2q^{21} - 9q^{23} - q^{27} + 4q^{28} - 6q^{29} + 2q^{31} + 6q^{33} - 2q^{36} + 4q^{37} + 5q^{39} - 11q^{43} + 12q^{44} + q^{47} - 4q^{48} - 3q^{49} + 6q^{51} + 10q^{52} - 6q^{53} - 2q^{57} + 9q^{59} - q^{61} - 2q^{63} - 8q^{64} + 4q^{67} + 12q^{68} + 9q^{69} - 15q^{71} - 11q^{73} - 4q^{76} + 12q^{77} + 11q^{79} + q^{81} - 4q^{84} + 6q^{87} - 3q^{89} + 10q^{91} + 18q^{92} - 2q^{93} - 2q^{97} - 6q^{99} + O(q^{100}) \)

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 −1.00000 −2.00000 0 0 −2.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(1\)
\(47\) \(-1\)

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 3525.2.a.f 1
5.b even 2 1 705.2.a.d 1
15.d odd 2 1 2115.2.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
705.2.a.d 1 5.b even 2 1
2115.2.a.f 1 15.d odd 2 1
3525.2.a.f 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3525))\):

\( T_{2} \)
\( T_{7} + 2 \)
\( T_{11} + 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 1 + T \)
$5$ \( T \)
$7$ \( 2 + T \)
$11$ \( 6 + T \)
$13$ \( 5 + T \)
$17$ \( 6 + T \)
$19$ \( -2 + T \)
$23$ \( 9 + T \)
$29$ \( 6 + T \)
$31$ \( -2 + T \)
$37$ \( -4 + T \)
$41$ \( T \)
$43$ \( 11 + T \)
$47$ \( -1 + T \)
$53$ \( 6 + T \)
$59$ \( -9 + T \)
$61$ \( 1 + T \)
$67$ \( -4 + T \)
$71$ \( 15 + T \)
$73$ \( 11 + T \)
$79$ \( -11 + T \)
$83$ \( T \)
$89$ \( 3 + T \)
$97$ \( 2 + T \)
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