Properties

Label 53.2.a.a
Level 53
Weight 2
Character orbit 53.a
Self dual yes
Analytic conductor 0.423
Analytic rank 1
Dimension 1
CM no
Inner twists 1

Related objects

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Newspace parameters

Level: \( N \) \(=\) \( 53 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 53.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.423207130713\)
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

\(f(q)\) \(=\) \( q - q^{2} - 3q^{3} - q^{4} + 3q^{6} - 4q^{7} + 3q^{8} + 6q^{9} + O(q^{10}) \) \( q - q^{2} - 3q^{3} - q^{4} + 3q^{6} - 4q^{7} + 3q^{8} + 6q^{9} + 3q^{12} - 3q^{13} + 4q^{14} - q^{16} - 3q^{17} - 6q^{18} - 5q^{19} + 12q^{21} + 7q^{23} - 9q^{24} - 5q^{25} + 3q^{26} - 9q^{27} + 4q^{28} - 7q^{29} + 4q^{31} - 5q^{32} + 3q^{34} - 6q^{36} + 5q^{37} + 5q^{38} + 9q^{39} + 6q^{41} - 12q^{42} - 2q^{43} - 7q^{46} - 2q^{47} + 3q^{48} + 9q^{49} + 5q^{50} + 9q^{51} + 3q^{52} - q^{53} + 9q^{54} - 12q^{56} + 15q^{57} + 7q^{58} - 2q^{59} - 8q^{61} - 4q^{62} - 24q^{63} + 7q^{64} - 12q^{67} + 3q^{68} - 21q^{69} + q^{71} + 18q^{72} - 4q^{73} - 5q^{74} + 15q^{75} + 5q^{76} - 9q^{78} - q^{79} + 9q^{81} - 6q^{82} - q^{83} - 12q^{84} + 2q^{86} + 21q^{87} - 14q^{89} + 12q^{91} - 7q^{92} - 12q^{93} + 2q^{94} + 15q^{96} + q^{97} - 9q^{98} + O(q^{100}) \)

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 −3.00000 −1.00000 0 3.00000 −4.00000 3.00000 6.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 53.2.a.a 1
3.b odd 2 1 477.2.a.a 1
4.b odd 2 1 848.2.a.g 1
5.b even 2 1 1325.2.a.e 1
5.c odd 4 2 1325.2.b.c 2
7.b odd 2 1 2597.2.a.a 1
8.b even 2 1 3392.2.a.s 1
8.d odd 2 1 3392.2.a.a 1
11.b odd 2 1 6413.2.a.h 1
12.b even 2 1 7632.2.a.j 1
13.b even 2 1 8957.2.a.b 1
53.b even 2 1 2809.2.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
53.2.a.a 1 1.a even 1 1 trivial
477.2.a.a 1 3.b odd 2 1
848.2.a.g 1 4.b odd 2 1
1325.2.a.e 1 5.b even 2 1
1325.2.b.c 2 5.c odd 4 2
2597.2.a.a 1 7.b odd 2 1
2809.2.a.a 1 53.b even 2 1
3392.2.a.a 1 8.d odd 2 1
3392.2.a.s 1 8.b even 2 1
6413.2.a.h 1 11.b odd 2 1
7632.2.a.j 1 12.b even 2 1
8957.2.a.b 1 13.b even 2 1

Atkin-Lehner signs

\( p \) Sign
\(53\) \(1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(53))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + 2 T^{2} \)
$3$ \( 1 + 3 T + 3 T^{2} \)
$5$ \( 1 + 5 T^{2} \)
$7$ \( 1 + 4 T + 7 T^{2} \)
$11$ \( 1 + 11 T^{2} \)
$13$ \( 1 + 3 T + 13 T^{2} \)
$17$ \( 1 + 3 T + 17 T^{2} \)
$19$ \( 1 + 5 T + 19 T^{2} \)
$23$ \( 1 - 7 T + 23 T^{2} \)
$29$ \( 1 + 7 T + 29 T^{2} \)
$31$ \( 1 - 4 T + 31 T^{2} \)
$37$ \( 1 - 5 T + 37 T^{2} \)
$41$ \( 1 - 6 T + 41 T^{2} \)
$43$ \( 1 + 2 T + 43 T^{2} \)
$47$ \( 1 + 2 T + 47 T^{2} \)
$53$ \( 1 + T \)
$59$ \( 1 + 2 T + 59 T^{2} \)
$61$ \( 1 + 8 T + 61 T^{2} \)
$67$ \( 1 + 12 T + 67 T^{2} \)
$71$ \( 1 - T + 71 T^{2} \)
$73$ \( 1 + 4 T + 73 T^{2} \)
$79$ \( 1 + T + 79 T^{2} \)
$83$ \( 1 + T + 83 T^{2} \)
$89$ \( 1 + 14 T + 89 T^{2} \)
$97$ \( 1 - T + 97 T^{2} \)
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