Properties

Label 525.4.a.h
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

\(f(q)\) \(=\) \( q + 3q^{2} + 3q^{3} + q^{4} + 9q^{6} - 7q^{7} - 21q^{8} + 9q^{9} + O(q^{10}) \) \( q + 3q^{2} + 3q^{3} + q^{4} + 9q^{6} - 7q^{7} - 21q^{8} + 9q^{9} - 6q^{11} + 3q^{12} - 41q^{13} - 21q^{14} - 71q^{16} - 27q^{17} + 27q^{18} - 4q^{19} - 21q^{21} - 18q^{22} - 75q^{23} - 63q^{24} - 123q^{26} + 27q^{27} - 7q^{28} - 123q^{29} - 205q^{31} - 45q^{32} - 18q^{33} - 81q^{34} + 9q^{36} + 262q^{37} - 12q^{38} - 123q^{39} + 57q^{41} - 63q^{42} - 407q^{43} - 6q^{44} - 225q^{46} + 60q^{47} - 213q^{48} + 49q^{49} - 81q^{51} - 41q^{52} - 327q^{53} + 81q^{54} + 147q^{56} - 12q^{57} - 369q^{58} + 33q^{59} - 427q^{61} - 615q^{62} - 63q^{63} + 433q^{64} - 54q^{66} + 628q^{67} - 27q^{68} - 225q^{69} + 300q^{71} - 189q^{72} - 98q^{73} + 786q^{74} - 4q^{76} + 42q^{77} - 369q^{78} + 686q^{79} + 81q^{81} + 171q^{82} - 1401q^{83} - 21q^{84} - 1221q^{86} - 369q^{87} + 126q^{88} + 714q^{89} + 287q^{91} - 75q^{92} - 615q^{93} + 180q^{94} - 135q^{96} - 494q^{97} + 147q^{98} - 54q^{99} + O(q^{100}) \)

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
3.00000 3.00000 1.00000 0 9.00000 −7.00000 −21.0000 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.h yes 1
3.b odd 2 1 1575.4.a.a 1
5.b even 2 1 525.4.a.c 1
5.c odd 4 2 525.4.d.d 2
15.d odd 2 1 1575.4.a.j 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.4.a.c 1 5.b even 2 1
525.4.a.h yes 1 1.a even 1 1 trivial
525.4.d.d 2 5.c odd 4 2
1575.4.a.a 1 3.b odd 2 1
1575.4.a.j 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(525))\):

\( T_{2} - 3 \)
\( T_{11} + 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 3 T + 8 T^{2} \)
$3$ \( 1 - 3 T \)
$5$ 1
$7$ \( 1 + 7 T \)
$11$ \( 1 + 6 T + 1331 T^{2} \)
$13$ \( 1 + 41 T + 2197 T^{2} \)
$17$ \( 1 + 27 T + 4913 T^{2} \)
$19$ \( 1 + 4 T + 6859 T^{2} \)
$23$ \( 1 + 75 T + 12167 T^{2} \)
$29$ \( 1 + 123 T + 24389 T^{2} \)
$31$ \( 1 + 205 T + 29791 T^{2} \)
$37$ \( 1 - 262 T + 50653 T^{2} \)
$41$ \( 1 - 57 T + 68921 T^{2} \)
$43$ \( 1 + 407 T + 79507 T^{2} \)
$47$ \( 1 - 60 T + 103823 T^{2} \)
$53$ \( 1 + 327 T + 148877 T^{2} \)
$59$ \( 1 - 33 T + 205379 T^{2} \)
$61$ \( 1 + 427 T + 226981 T^{2} \)
$67$ \( 1 - 628 T + 300763 T^{2} \)
$71$ \( 1 - 300 T + 357911 T^{2} \)
$73$ \( 1 + 98 T + 389017 T^{2} \)
$79$ \( 1 - 686 T + 493039 T^{2} \)
$83$ \( 1 + 1401 T + 571787 T^{2} \)
$89$ \( 1 - 714 T + 704969 T^{2} \)
$97$ \( 1 + 494 T + 912673 T^{2} \)
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