Properties

Label 525.2.a.b
Level $525$
Weight $2$
Character orbit 525.a
Self dual yes
Analytic conductor $4.192$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 525.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.19214610612\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 105)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} + q^{3} - q^{4} - q^{6} - q^{7} + 3q^{8} + q^{9} + O(q^{10}) \) \( q - q^{2} + q^{3} - q^{4} - q^{6} - q^{7} + 3q^{8} + q^{9} - 6q^{11} - q^{12} - 2q^{13} + q^{14} - q^{16} + 4q^{17} - q^{18} - 6q^{19} - q^{21} + 6q^{22} + 3q^{24} + 2q^{26} + q^{27} + q^{28} - 2q^{29} - 10q^{31} - 5q^{32} - 6q^{33} - 4q^{34} - q^{36} - 4q^{37} + 6q^{38} - 2q^{39} + 2q^{41} + q^{42} - 4q^{43} + 6q^{44} - q^{48} + q^{49} + 4q^{51} + 2q^{52} + 6q^{53} - q^{54} - 3q^{56} - 6q^{57} + 2q^{58} - 8q^{59} - 2q^{61} + 10q^{62} - q^{63} + 7q^{64} + 6q^{66} - 16q^{67} - 4q^{68} + 10q^{71} + 3q^{72} - 6q^{73} + 4q^{74} + 6q^{76} + 6q^{77} + 2q^{78} + 4q^{79} + q^{81} - 2q^{82} + 8q^{83} + q^{84} + 4q^{86} - 2q^{87} - 18q^{88} + 6q^{89} + 2q^{91} - 10q^{93} - 5q^{96} - 2q^{97} - q^{98} - 6q^{99} + O(q^{100}) \)

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 −1.00000 3.00000 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(-1\)
\(7\) \(1\)

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 525.2.a.b 1
3.b odd 2 1 1575.2.a.i 1
4.b odd 2 1 8400.2.a.bj 1
5.b even 2 1 525.2.a.c 1
5.c odd 4 2 105.2.d.a 2
7.b odd 2 1 3675.2.a.d 1
15.d odd 2 1 1575.2.a.e 1
15.e even 4 2 315.2.d.c 2
20.d odd 2 1 8400.2.a.ch 1
20.e even 4 2 1680.2.t.f 2
35.c odd 2 1 3675.2.a.l 1
35.f even 4 2 735.2.d.a 2
35.k even 12 4 735.2.q.b 4
35.l odd 12 4 735.2.q.a 4
60.l odd 4 2 5040.2.t.e 2
105.k odd 4 2 2205.2.d.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.d.a 2 5.c odd 4 2
315.2.d.c 2 15.e even 4 2
525.2.a.b 1 1.a even 1 1 trivial
525.2.a.c 1 5.b even 2 1
735.2.d.a 2 35.f even 4 2
735.2.q.a 4 35.l odd 12 4
735.2.q.b 4 35.k even 12 4
1575.2.a.e 1 15.d odd 2 1
1575.2.a.i 1 3.b odd 2 1
1680.2.t.f 2 20.e even 4 2
2205.2.d.f 2 105.k odd 4 2
3675.2.a.d 1 7.b odd 2 1
3675.2.a.l 1 35.c odd 2 1
5040.2.t.e 2 60.l odd 4 2
8400.2.a.bj 1 4.b odd 2 1
8400.2.a.ch 1 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(525))\):

\( T_{2} + 1 \)
\( T_{11} + 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( -1 + T \)
$5$ \( T \)
$7$ \( 1 + T \)
$11$ \( 6 + T \)
$13$ \( 2 + T \)
$17$ \( -4 + T \)
$19$ \( 6 + T \)
$23$ \( T \)
$29$ \( 2 + T \)
$31$ \( 10 + T \)
$37$ \( 4 + T \)
$41$ \( -2 + T \)
$43$ \( 4 + T \)
$47$ \( T \)
$53$ \( -6 + T \)
$59$ \( 8 + T \)
$61$ \( 2 + T \)
$67$ \( 16 + T \)
$71$ \( -10 + T \)
$73$ \( 6 + T \)
$79$ \( -4 + T \)
$83$ \( -8 + T \)
$89$ \( -6 + T \)
$97$ \( 2 + T \)
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