Properties

Label 3525.2.a.l
Level $3525$
Weight $2$
Character orbit 3525.a
Self dual yes
Analytic conductor $28.147$
Analytic rank $0$
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: \(0\)
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

\(f(q)\) \(=\) \( q + q^{2} + q^{3} - q^{4} + q^{6} + 5q^{7} - 3q^{8} + q^{9} + O(q^{10}) \) \( q + q^{2} + q^{3} - q^{4} + q^{6} + 5q^{7} - 3q^{8} + q^{9} + 6q^{11} - q^{12} - 3q^{13} + 5q^{14} - q^{16} + 3q^{17} + q^{18} - q^{19} + 5q^{21} + 6q^{22} + 5q^{23} - 3q^{24} - 3q^{26} + q^{27} - 5q^{28} - 7q^{29} + 5q^{32} + 6q^{33} + 3q^{34} - q^{36} - q^{38} - 3q^{39} - 5q^{41} + 5q^{42} + 6q^{43} - 6q^{44} + 5q^{46} - q^{47} - q^{48} + 18q^{49} + 3q^{51} + 3q^{52} - 5q^{53} + q^{54} - 15q^{56} - q^{57} - 7q^{58} - 9q^{59} - 7q^{61} + 5q^{63} + 7q^{64} + 6q^{66} + 8q^{67} - 3q^{68} + 5q^{69} - 3q^{71} - 3q^{72} + 10q^{73} + q^{76} + 30q^{77} - 3q^{78} - 10q^{79} + q^{81} - 5q^{82} + 8q^{83} - 5q^{84} + 6q^{86} - 7q^{87} - 18q^{88} + 14q^{89} - 15q^{91} - 5q^{92} - q^{94} + 5q^{96} - 12q^{97} + 18q^{98} + 6q^{99} + O(q^{100}) \)

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
1.00000 1.00000 −1.00000 0 1.00000 5.00000 −3.00000 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.l 1
5.b even 2 1 705.2.a.a 1
15.d odd 2 1 2115.2.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
705.2.a.a 1 5.b even 2 1
2115.2.a.g 1 15.d odd 2 1
3525.2.a.l 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3525))\):

\( T_{2} - 1 \)
\( T_{7} - 5 \)
\( T_{11} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( -1 + T \)
$5$ \( T \)
$7$ \( -5 + T \)
$11$ \( -6 + T \)
$13$ \( 3 + T \)
$17$ \( -3 + T \)
$19$ \( 1 + T \)
$23$ \( -5 + T \)
$29$ \( 7 + T \)
$31$ \( T \)
$37$ \( T \)
$41$ \( 5 + T \)
$43$ \( -6 + T \)
$47$ \( 1 + T \)
$53$ \( 5 + T \)
$59$ \( 9 + T \)
$61$ \( 7 + T \)
$67$ \( -8 + T \)
$71$ \( 3 + T \)
$73$ \( -10 + T \)
$79$ \( 10 + T \)
$83$ \( -8 + T \)
$89$ \( -14 + T \)
$97$ \( 12 + T \)
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