Properties

Label 475.2.a.b
Level $475$
Weight $2$
Character orbit 475.a
Self dual yes
Analytic conductor $3.793$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 475.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.79289409601\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 19)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 2q^{3} - 2q^{4} + q^{7} + q^{9} + O(q^{10}) \) \( q + 2q^{3} - 2q^{4} + q^{7} + q^{9} + 3q^{11} - 4q^{12} + 4q^{13} + 4q^{16} + 3q^{17} + q^{19} + 2q^{21} - 4q^{27} - 2q^{28} + 6q^{29} - 4q^{31} + 6q^{33} - 2q^{36} - 2q^{37} + 8q^{39} - 6q^{41} + q^{43} - 6q^{44} + 3q^{47} + 8q^{48} - 6q^{49} + 6q^{51} - 8q^{52} - 12q^{53} + 2q^{57} - 6q^{59} - q^{61} + q^{63} - 8q^{64} + 4q^{67} - 6q^{68} + 6q^{71} + 7q^{73} - 2q^{76} + 3q^{77} + 8q^{79} - 11q^{81} - 12q^{83} - 4q^{84} + 12q^{87} + 12q^{89} + 4q^{91} - 8q^{93} - 8q^{97} + 3q^{99} + O(q^{100}) \)

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

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(19\) \(-1\)

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 475.2.a.b 1
3.b odd 2 1 4275.2.a.i 1
4.b odd 2 1 7600.2.a.c 1
5.b even 2 1 19.2.a.a 1
5.c odd 4 2 475.2.b.a 2
15.d odd 2 1 171.2.a.b 1
19.b odd 2 1 9025.2.a.d 1
20.d odd 2 1 304.2.a.f 1
35.c odd 2 1 931.2.a.a 1
35.i odd 6 2 931.2.f.b 2
35.j even 6 2 931.2.f.c 2
40.e odd 2 1 1216.2.a.b 1
40.f even 2 1 1216.2.a.o 1
55.d odd 2 1 2299.2.a.b 1
60.h even 2 1 2736.2.a.c 1
65.d even 2 1 3211.2.a.a 1
85.c even 2 1 5491.2.a.b 1
95.d odd 2 1 361.2.a.b 1
95.h odd 6 2 361.2.c.a 2
95.i even 6 2 361.2.c.c 2
95.o odd 18 6 361.2.e.e 6
95.p even 18 6 361.2.e.d 6
105.g even 2 1 8379.2.a.j 1
285.b even 2 1 3249.2.a.d 1
380.d even 2 1 5776.2.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.2.a.a 1 5.b even 2 1
171.2.a.b 1 15.d odd 2 1
304.2.a.f 1 20.d odd 2 1
361.2.a.b 1 95.d odd 2 1
361.2.c.a 2 95.h odd 6 2
361.2.c.c 2 95.i even 6 2
361.2.e.d 6 95.p even 18 6
361.2.e.e 6 95.o odd 18 6
475.2.a.b 1 1.a even 1 1 trivial
475.2.b.a 2 5.c odd 4 2
931.2.a.a 1 35.c odd 2 1
931.2.f.b 2 35.i odd 6 2
931.2.f.c 2 35.j even 6 2
1216.2.a.b 1 40.e odd 2 1
1216.2.a.o 1 40.f even 2 1
2299.2.a.b 1 55.d odd 2 1
2736.2.a.c 1 60.h even 2 1
3211.2.a.a 1 65.d even 2 1
3249.2.a.d 1 285.b even 2 1
4275.2.a.i 1 3.b odd 2 1
5491.2.a.b 1 85.c even 2 1
5776.2.a.c 1 380.d even 2 1
7600.2.a.c 1 4.b odd 2 1
8379.2.a.j 1 105.g even 2 1
9025.2.a.d 1 19.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(475))\).

Hecke characteristic polynomials

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