Properties

Label 475.2.a.a
Level $475$
Weight $2$
Character orbit 475.a
Self dual yes
Analytic conductor $3.793$
Analytic rank $1$
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: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 95)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} - q^{4} + 2 q^{7} + 3 q^{8} - 3 q^{9} + O(q^{10}) \) \( q - q^{2} - q^{4} + 2 q^{7} + 3 q^{8} - 3 q^{9} - 4 q^{11} - 2 q^{13} - 2 q^{14} - q^{16} + 4 q^{17} + 3 q^{18} + q^{19} + 4 q^{22} - 6 q^{23} + 2 q^{26} - 2 q^{28} - 6 q^{29} - 4 q^{31} - 5 q^{32} - 4 q^{34} + 3 q^{36} - 10 q^{37} - q^{38} - 10 q^{41} + 2 q^{43} + 4 q^{44} + 6 q^{46} - 6 q^{47} - 3 q^{49} + 2 q^{52} + 10 q^{53} + 6 q^{56} + 6 q^{58} + 2 q^{61} + 4 q^{62} - 6 q^{63} + 7 q^{64} + 8 q^{67} - 4 q^{68} + 4 q^{71} - 9 q^{72} + 4 q^{73} + 10 q^{74} - q^{76} - 8 q^{77} + 4 q^{79} + 9 q^{81} + 10 q^{82} - 18 q^{83} - 2 q^{86} - 12 q^{88} - 2 q^{89} - 4 q^{91} + 6 q^{92} + 6 q^{94} + 6 q^{97} + 3 q^{98} + 12 q^{99} + O(q^{100}) \)

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 0 −1.00000 0 0 2.00000 3.00000 −3.00000 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.a 1
3.b odd 2 1 4275.2.a.p 1
4.b odd 2 1 7600.2.a.i 1
5.b even 2 1 475.2.a.c 1
5.c odd 4 2 95.2.b.a 2
15.d odd 2 1 4275.2.a.e 1
15.e even 4 2 855.2.c.b 2
19.b odd 2 1 9025.2.a.h 1
20.d odd 2 1 7600.2.a.l 1
20.e even 4 2 1520.2.d.b 2
95.d odd 2 1 9025.2.a.c 1
95.g even 4 2 1805.2.b.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.b.a 2 5.c odd 4 2
475.2.a.a 1 1.a even 1 1 trivial
475.2.a.c 1 5.b even 2 1
855.2.c.b 2 15.e even 4 2
1520.2.d.b 2 20.e even 4 2
1805.2.b.c 2 95.g even 4 2
4275.2.a.e 1 15.d odd 2 1
4275.2.a.p 1 3.b odd 2 1
7600.2.a.i 1 4.b odd 2 1
7600.2.a.l 1 20.d odd 2 1
9025.2.a.c 1 95.d odd 2 1
9025.2.a.h 1 19.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(475))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( T \)
$5$ \( T \)
$7$ \( -2 + T \)
$11$ \( 4 + T \)
$13$ \( 2 + T \)
$17$ \( -4 + T \)
$19$ \( -1 + T \)
$23$ \( 6 + T \)
$29$ \( 6 + T \)
$31$ \( 4 + T \)
$37$ \( 10 + T \)
$41$ \( 10 + T \)
$43$ \( -2 + T \)
$47$ \( 6 + T \)
$53$ \( -10 + T \)
$59$ \( T \)
$61$ \( -2 + T \)
$67$ \( -8 + T \)
$71$ \( -4 + T \)
$73$ \( -4 + T \)
$79$ \( -4 + T \)
$83$ \( 18 + T \)
$89$ \( 2 + T \)
$97$ \( -6 + T \)
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