Properties

Label 50.2.a.a
Level $50$
Weight $2$
Character orbit 50.a
Self dual yes
Analytic conductor $0.399$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 50.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.399252010106\)
Analytic rank: \(0\)
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

\(f(q)\) \(=\) \( q - q^{2} + q^{3} + q^{4} - q^{6} + 2q^{7} - q^{8} - 2q^{9} + O(q^{10}) \) \( q - q^{2} + q^{3} + q^{4} - q^{6} + 2q^{7} - q^{8} - 2q^{9} - 3q^{11} + q^{12} - 4q^{13} - 2q^{14} + q^{16} - 3q^{17} + 2q^{18} + 5q^{19} + 2q^{21} + 3q^{22} + 6q^{23} - q^{24} + 4q^{26} - 5q^{27} + 2q^{28} + 2q^{31} - q^{32} - 3q^{33} + 3q^{34} - 2q^{36} + 2q^{37} - 5q^{38} - 4q^{39} - 3q^{41} - 2q^{42} - 4q^{43} - 3q^{44} - 6q^{46} + 12q^{47} + q^{48} - 3q^{49} - 3q^{51} - 4q^{52} + 6q^{53} + 5q^{54} - 2q^{56} + 5q^{57} + 2q^{61} - 2q^{62} - 4q^{63} + q^{64} + 3q^{66} - 13q^{67} - 3q^{68} + 6q^{69} + 12q^{71} + 2q^{72} + 11q^{73} - 2q^{74} + 5q^{76} - 6q^{77} + 4q^{78} - 10q^{79} + q^{81} + 3q^{82} - 9q^{83} + 2q^{84} + 4q^{86} + 3q^{88} + 15q^{89} - 8q^{91} + 6q^{92} + 2q^{93} - 12q^{94} - q^{96} + 2q^{97} + 3q^{98} + 6q^{99} + O(q^{100}) \)

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 1.00000 1.00000 0 −1.00000 2.00000 −1.00000 −2.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(-1\)

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 50.2.a.a 1
3.b odd 2 1 450.2.a.g 1
4.b odd 2 1 400.2.a.d 1
5.b even 2 1 50.2.a.b yes 1
5.c odd 4 2 50.2.b.a 2
7.b odd 2 1 2450.2.a.g 1
8.b even 2 1 1600.2.a.j 1
8.d odd 2 1 1600.2.a.p 1
11.b odd 2 1 6050.2.a.bi 1
12.b even 2 1 3600.2.a.l 1
13.b even 2 1 8450.2.a.v 1
15.d odd 2 1 450.2.a.c 1
15.e even 4 2 450.2.c.c 2
20.d odd 2 1 400.2.a.f 1
20.e even 4 2 400.2.c.c 2
35.c odd 2 1 2450.2.a.bd 1
35.f even 4 2 2450.2.c.m 2
40.e odd 2 1 1600.2.a.i 1
40.f even 2 1 1600.2.a.q 1
40.i odd 4 2 1600.2.c.i 2
40.k even 4 2 1600.2.c.h 2
55.d odd 2 1 6050.2.a.h 1
60.h even 2 1 3600.2.a.bc 1
60.l odd 4 2 3600.2.f.f 2
65.d even 2 1 8450.2.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.2.a.a 1 1.a even 1 1 trivial
50.2.a.b yes 1 5.b even 2 1
50.2.b.a 2 5.c odd 4 2
400.2.a.d 1 4.b odd 2 1
400.2.a.f 1 20.d odd 2 1
400.2.c.c 2 20.e even 4 2
450.2.a.c 1 15.d odd 2 1
450.2.a.g 1 3.b odd 2 1
450.2.c.c 2 15.e even 4 2
1600.2.a.i 1 40.e odd 2 1
1600.2.a.j 1 8.b even 2 1
1600.2.a.p 1 8.d odd 2 1
1600.2.a.q 1 40.f even 2 1
1600.2.c.h 2 40.k even 4 2
1600.2.c.i 2 40.i odd 4 2
2450.2.a.g 1 7.b odd 2 1
2450.2.a.bd 1 35.c odd 2 1
2450.2.c.m 2 35.f even 4 2
3600.2.a.l 1 12.b even 2 1
3600.2.a.bc 1 60.h even 2 1
3600.2.f.f 2 60.l odd 4 2
6050.2.a.h 1 55.d odd 2 1
6050.2.a.bi 1 11.b odd 2 1
8450.2.a.d 1 65.d even 2 1
8450.2.a.v 1 13.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(50))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( -1 + T \)
$5$ \( T \)
$7$ \( -2 + T \)
$11$ \( 3 + T \)
$13$ \( 4 + T \)
$17$ \( 3 + T \)
$19$ \( -5 + T \)
$23$ \( -6 + T \)
$29$ \( T \)
$31$ \( -2 + T \)
$37$ \( -2 + T \)
$41$ \( 3 + T \)
$43$ \( 4 + T \)
$47$ \( -12 + T \)
$53$ \( -6 + T \)
$59$ \( T \)
$61$ \( -2 + T \)
$67$ \( 13 + T \)
$71$ \( -12 + T \)
$73$ \( -11 + T \)
$79$ \( 10 + T \)
$83$ \( 9 + T \)
$89$ \( -15 + T \)
$97$ \( -2 + T \)
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