Properties

Label 3525.2.a.a
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

\(f(q)\) \(=\) \( q - 2q^{2} - q^{3} + 2q^{4} + 2q^{6} - 4q^{7} + q^{9} + O(q^{10}) \) \( q - 2q^{2} - q^{3} + 2q^{4} + 2q^{6} - 4q^{7} + q^{9} - 2q^{12} - 5q^{13} + 8q^{14} - 4q^{16} - 6q^{17} - 2q^{18} - 2q^{19} + 4q^{21} - q^{23} + 10q^{26} - q^{27} - 8q^{28} - 6q^{29} - 8q^{31} + 8q^{32} + 12q^{34} + 2q^{36} - 2q^{37} + 4q^{38} + 5q^{39} - 2q^{41} - 8q^{42} + 5q^{43} + 2q^{46} - q^{47} + 4q^{48} + 9q^{49} + 6q^{51} - 10q^{52} - 12q^{53} + 2q^{54} + 2q^{57} + 12q^{58} - 3q^{59} - q^{61} + 16q^{62} - 4q^{63} - 8q^{64} - 8q^{67} - 12q^{68} + q^{69} + q^{71} + 13q^{73} + 4q^{74} - 4q^{76} - 10q^{78} - q^{79} + q^{81} + 4q^{82} - 12q^{83} + 8q^{84} - 10q^{86} + 6q^{87} - 15q^{89} + 20q^{91} - 2q^{92} + 8q^{93} + 2q^{94} - 8q^{96} + 8q^{97} - 18q^{98} + O(q^{100}) \)

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
−2.00000 −1.00000 2.00000 0 2.00000 −4.00000 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.a 1
5.b even 2 1 3525.2.a.o 1
5.c odd 4 2 705.2.c.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
705.2.c.a 2 5.c odd 4 2
3525.2.a.a 1 1.a even 1 1 trivial
3525.2.a.o 1 5.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3525))\):

\( T_{2} + 2 \)
\( T_{7} + 4 \)
\( T_{11} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2 + T \)
$3$ \( 1 + T \)
$5$ \( T \)
$7$ \( 4 + T \)
$11$ \( T \)
$13$ \( 5 + T \)
$17$ \( 6 + T \)
$19$ \( 2 + T \)
$23$ \( 1 + T \)
$29$ \( 6 + T \)
$31$ \( 8 + T \)
$37$ \( 2 + T \)
$41$ \( 2 + T \)
$43$ \( -5 + T \)
$47$ \( 1 + T \)
$53$ \( 12 + T \)
$59$ \( 3 + T \)
$61$ \( 1 + T \)
$67$ \( 8 + T \)
$71$ \( -1 + T \)
$73$ \( -13 + T \)
$79$ \( 1 + T \)
$83$ \( 12 + T \)
$89$ \( 15 + T \)
$97$ \( -8 + T \)
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