Properties

Label 525.4.a.b
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: no (minimal twist has level 21)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 4q^{2} + 3q^{3} + 8q^{4} - 12q^{6} + 7q^{7} + 9q^{9} + O(q^{10}) \) \( q - 4q^{2} + 3q^{3} + 8q^{4} - 12q^{6} + 7q^{7} + 9q^{9} + 62q^{11} + 24q^{12} + 62q^{13} - 28q^{14} - 64q^{16} - 84q^{17} - 36q^{18} + 100q^{19} + 21q^{21} - 248q^{22} + 42q^{23} - 248q^{26} + 27q^{27} + 56q^{28} - 10q^{29} - 48q^{31} + 256q^{32} + 186q^{33} + 336q^{34} + 72q^{36} + 246q^{37} - 400q^{38} + 186q^{39} - 248q^{41} - 84q^{42} - 68q^{43} + 496q^{44} - 168q^{46} - 324q^{47} - 192q^{48} + 49q^{49} - 252q^{51} + 496q^{52} - 258q^{53} - 108q^{54} + 300q^{57} + 40q^{58} + 120q^{59} + 622q^{61} + 192q^{62} + 63q^{63} - 512q^{64} - 744q^{66} - 904q^{67} - 672q^{68} + 126q^{69} - 678q^{71} + 642q^{73} - 984q^{74} + 800q^{76} + 434q^{77} - 744q^{78} + 740q^{79} + 81q^{81} + 992q^{82} - 468q^{83} + 168q^{84} + 272q^{86} - 30q^{87} + 200q^{89} + 434q^{91} + 336q^{92} - 144q^{93} + 1296q^{94} + 768q^{96} + 1266q^{97} - 196q^{98} + 558q^{99} + O(q^{100}) \)

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
−4.00000 3.00000 8.00000 0 −12.0000 7.00000 0 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.b 1
3.b odd 2 1 1575.4.a.k 1
5.b even 2 1 21.4.a.b 1
5.c odd 4 2 525.4.d.b 2
15.d odd 2 1 63.4.a.a 1
20.d odd 2 1 336.4.a.h 1
35.c odd 2 1 147.4.a.g 1
35.i odd 6 2 147.4.e.b 2
35.j even 6 2 147.4.e.c 2
40.e odd 2 1 1344.4.a.i 1
40.f even 2 1 1344.4.a.w 1
60.h even 2 1 1008.4.a.m 1
105.g even 2 1 441.4.a.b 1
105.o odd 6 2 441.4.e.m 2
105.p even 6 2 441.4.e.n 2
140.c even 2 1 2352.4.a.l 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.a.b 1 5.b even 2 1
63.4.a.a 1 15.d odd 2 1
147.4.a.g 1 35.c odd 2 1
147.4.e.b 2 35.i odd 6 2
147.4.e.c 2 35.j even 6 2
336.4.a.h 1 20.d odd 2 1
441.4.a.b 1 105.g even 2 1
441.4.e.m 2 105.o odd 6 2
441.4.e.n 2 105.p even 6 2
525.4.a.b 1 1.a even 1 1 trivial
525.4.d.b 2 5.c odd 4 2
1008.4.a.m 1 60.h even 2 1
1344.4.a.i 1 40.e odd 2 1
1344.4.a.w 1 40.f even 2 1
1575.4.a.k 1 3.b odd 2 1
2352.4.a.l 1 140.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(525))\):

\( T_{2} + 4 \)
\( T_{11} - 62 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 + T \)
$3$ \( -3 + T \)
$5$ \( T \)
$7$ \( -7 + T \)
$11$ \( -62 + T \)
$13$ \( -62 + T \)
$17$ \( 84 + T \)
$19$ \( -100 + T \)
$23$ \( -42 + T \)
$29$ \( 10 + T \)
$31$ \( 48 + T \)
$37$ \( -246 + T \)
$41$ \( 248 + T \)
$43$ \( 68 + T \)
$47$ \( 324 + T \)
$53$ \( 258 + T \)
$59$ \( -120 + T \)
$61$ \( -622 + T \)
$67$ \( 904 + T \)
$71$ \( 678 + T \)
$73$ \( -642 + T \)
$79$ \( -740 + T \)
$83$ \( 468 + T \)
$89$ \( -200 + T \)
$97$ \( -1266 + T \)
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