Properties

Label 525.4.a.f
Level $525$
Weight $4$
Character orbit 525.a
Self dual yes
Analytic conductor $30.976$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 525.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(30.9760027530\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 2q^{2} - 3q^{3} - 4q^{4} - 6q^{6} + 7q^{7} - 24q^{8} + 9q^{9} + O(q^{10}) \) \( q + 2q^{2} - 3q^{3} - 4q^{4} - 6q^{6} + 7q^{7} - 24q^{8} + 9q^{9} - 21q^{11} + 12q^{12} - 24q^{13} + 14q^{14} - 16q^{16} + 22q^{17} + 18q^{18} + 16q^{19} - 21q^{21} - 42q^{22} + 25q^{23} + 72q^{24} - 48q^{26} - 27q^{27} - 28q^{28} + 167q^{29} + 10q^{31} + 160q^{32} + 63q^{33} + 44q^{34} - 36q^{36} + 133q^{37} + 32q^{38} + 72q^{39} - 168q^{41} - 42q^{42} + 97q^{43} + 84q^{44} + 50q^{46} + 400q^{47} + 48q^{48} + 49q^{49} - 66q^{51} + 96q^{52} + 182q^{53} - 54q^{54} - 168q^{56} - 48q^{57} + 334q^{58} + 488q^{59} + 28q^{61} + 20q^{62} + 63q^{63} + 448q^{64} + 126q^{66} + 967q^{67} - 88q^{68} - 75q^{69} - 285q^{71} - 216q^{72} + 838q^{73} + 266q^{74} - 64q^{76} - 147q^{77} + 144q^{78} - 469q^{79} + 81q^{81} - 336q^{82} + 406q^{83} + 84q^{84} + 194q^{86} - 501q^{87} + 504q^{88} + 324q^{89} - 168q^{91} - 100q^{92} - 30q^{93} + 800q^{94} - 480q^{96} + 114q^{97} + 98q^{98} - 189q^{99} + O(q^{100}) \)

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
2.00000 −3.00000 −4.00000 0 −6.00000 7.00000 −24.0000 9.00000 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 525.4.a.f yes 1
3.b odd 2 1 1575.4.a.d 1
5.b even 2 1 525.4.a.d 1
5.c odd 4 2 525.4.d.e 2
15.d odd 2 1 1575.4.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.4.a.d 1 5.b even 2 1
525.4.a.f yes 1 1.a even 1 1 trivial
525.4.d.e 2 5.c odd 4 2
1575.4.a.d 1 3.b odd 2 1
1575.4.a.h 1 15.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(525))\):

\( T_{2} - 2 \)
\( T_{11} + 21 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -2 + T \)
$3$ \( 3 + T \)
$5$ \( T \)
$7$ \( -7 + T \)
$11$ \( 21 + T \)
$13$ \( 24 + T \)
$17$ \( -22 + T \)
$19$ \( -16 + T \)
$23$ \( -25 + T \)
$29$ \( -167 + T \)
$31$ \( -10 + T \)
$37$ \( -133 + T \)
$41$ \( 168 + T \)
$43$ \( -97 + T \)
$47$ \( -400 + T \)
$53$ \( -182 + T \)
$59$ \( -488 + T \)
$61$ \( -28 + T \)
$67$ \( -967 + T \)
$71$ \( 285 + T \)
$73$ \( -838 + T \)
$79$ \( 469 + T \)
$83$ \( -406 + T \)
$89$ \( -324 + T \)
$97$ \( -114 + T \)
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