Properties

Label 525.4.a.d
Level $525$
Weight $4$
Character orbit 525.a
Self dual yes
Analytic conductor $30.976$
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: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 525.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(30.9760027530\)
Analytic rank: \(1\)
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.d 1
3.b odd 2 1 1575.4.a.h 1
5.b even 2 1 525.4.a.f yes 1
5.c odd 4 2 525.4.d.e 2
15.d odd 2 1 1575.4.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.4.a.d 1 1.a even 1 1 trivial
525.4.a.f yes 1 5.b even 2 1
525.4.d.e 2 5.c odd 4 2
1575.4.a.d 1 15.d odd 2 1
1575.4.a.h 1 3.b 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

$p$ $F_p(T)$
$2$ \( 1 + 2 T + 8 T^{2} \)
$3$ \( 1 - 3 T \)
$5$ 1
$7$ \( 1 + 7 T \)
$11$ \( 1 + 21 T + 1331 T^{2} \)
$13$ \( 1 - 24 T + 2197 T^{2} \)
$17$ \( 1 + 22 T + 4913 T^{2} \)
$19$ \( 1 - 16 T + 6859 T^{2} \)
$23$ \( 1 + 25 T + 12167 T^{2} \)
$29$ \( 1 - 167 T + 24389 T^{2} \)
$31$ \( 1 - 10 T + 29791 T^{2} \)
$37$ \( 1 + 133 T + 50653 T^{2} \)
$41$ \( 1 + 168 T + 68921 T^{2} \)
$43$ \( 1 + 97 T + 79507 T^{2} \)
$47$ \( 1 + 400 T + 103823 T^{2} \)
$53$ \( 1 + 182 T + 148877 T^{2} \)
$59$ \( 1 - 488 T + 205379 T^{2} \)
$61$ \( 1 - 28 T + 226981 T^{2} \)
$67$ \( 1 + 967 T + 300763 T^{2} \)
$71$ \( 1 + 285 T + 357911 T^{2} \)
$73$ \( 1 + 838 T + 389017 T^{2} \)
$79$ \( 1 + 469 T + 493039 T^{2} \)
$83$ \( 1 + 406 T + 571787 T^{2} \)
$89$ \( 1 - 324 T + 704969 T^{2} \)
$97$ \( 1 + 114 T + 912673 T^{2} \)
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