Properties

Label 525.6.a.c
Level $525$
Weight $6$
Character orbit 525.a
Self dual yes
Analytic conductor $84.202$
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: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 525.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(84.2015054018\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} + 9q^{3} - 31q^{4} - 9q^{6} + 49q^{7} + 63q^{8} + 81q^{9} + O(q^{10}) \) \( q - q^{2} + 9q^{3} - 31q^{4} - 9q^{6} + 49q^{7} + 63q^{8} + 81q^{9} - 340q^{11} - 279q^{12} - 454q^{13} - 49q^{14} + 929q^{16} + 798q^{17} - 81q^{18} + 892q^{19} + 441q^{21} + 340q^{22} + 3192q^{23} + 567q^{24} + 454q^{26} + 729q^{27} - 1519q^{28} - 8242q^{29} - 2496q^{31} - 2945q^{32} - 3060q^{33} - 798q^{34} - 2511q^{36} - 9798q^{37} - 892q^{38} - 4086q^{39} + 19834q^{41} - 441q^{42} + 17236q^{43} + 10540q^{44} - 3192q^{46} - 8928q^{47} + 8361q^{48} + 2401q^{49} + 7182q^{51} + 14074q^{52} - 150q^{53} - 729q^{54} + 3087q^{56} + 8028q^{57} + 8242q^{58} - 42396q^{59} + 14758q^{61} + 2496q^{62} + 3969q^{63} - 26783q^{64} + 3060q^{66} + 1676q^{67} - 24738q^{68} + 28728q^{69} + 14568q^{71} + 5103q^{72} - 78378q^{73} + 9798q^{74} - 27652q^{76} - 16660q^{77} + 4086q^{78} - 2272q^{79} + 6561q^{81} - 19834q^{82} + 37764q^{83} - 13671q^{84} - 17236q^{86} - 74178q^{87} - 21420q^{88} - 117286q^{89} - 22246q^{91} - 98952q^{92} - 22464q^{93} + 8928q^{94} - 26505q^{96} - 10002q^{97} - 2401q^{98} - 27540q^{99} + O(q^{100}) \)

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 9.00000 −31.0000 0 −9.00000 49.0000 63.0000 81.0000 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 525.6.a.c 1
5.b even 2 1 21.6.a.b 1
5.c odd 4 2 525.6.d.d 2
15.d odd 2 1 63.6.a.c 1
20.d odd 2 1 336.6.a.l 1
35.c odd 2 1 147.6.a.e 1
35.i odd 6 2 147.6.e.e 2
35.j even 6 2 147.6.e.f 2
60.h even 2 1 1008.6.a.t 1
105.g even 2 1 441.6.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.a.b 1 5.b even 2 1
63.6.a.c 1 15.d odd 2 1
147.6.a.e 1 35.c odd 2 1
147.6.e.e 2 35.i odd 6 2
147.6.e.f 2 35.j even 6 2
336.6.a.l 1 20.d odd 2 1
441.6.a.d 1 105.g even 2 1
525.6.a.c 1 1.a even 1 1 trivial
525.6.d.d 2 5.c odd 4 2
1008.6.a.t 1 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 1 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(525))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( -9 + T \)
$5$ \( T \)
$7$ \( -49 + T \)
$11$ \( 340 + T \)
$13$ \( 454 + T \)
$17$ \( -798 + T \)
$19$ \( -892 + T \)
$23$ \( -3192 + T \)
$29$ \( 8242 + T \)
$31$ \( 2496 + T \)
$37$ \( 9798 + T \)
$41$ \( -19834 + T \)
$43$ \( -17236 + T \)
$47$ \( 8928 + T \)
$53$ \( 150 + T \)
$59$ \( 42396 + T \)
$61$ \( -14758 + T \)
$67$ \( -1676 + T \)
$71$ \( -14568 + T \)
$73$ \( 78378 + T \)
$79$ \( 2272 + T \)
$83$ \( -37764 + T \)
$89$ \( 117286 + T \)
$97$ \( 10002 + T \)
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