Properties

Label 475.1.c.b
Level $475$
Weight $1$
Character orbit 475.c
Analytic conductor $0.237$
Analytic rank $0$
Dimension $2$
Projective image $D_{4}$
CM discriminant -95
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 475.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.237055881001\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \(x^{2} + 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 95)
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.475.1
Artin image: $SD_{16}$
Artin field: Galois closure of 8.2.107171875.1

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{2} -\beta q^{3} - q^{4} -2 q^{6} - q^{9} +O(q^{10})\) \( q -\beta q^{2} -\beta q^{3} - q^{4} -2 q^{6} - q^{9} + \beta q^{12} + \beta q^{13} - q^{16} + \beta q^{18} + q^{19} + 2 q^{26} + \beta q^{32} + q^{36} + \beta q^{37} -\beta q^{38} + 2 q^{39} + \beta q^{48} - q^{49} -\beta q^{52} -\beta q^{53} -\beta q^{57} + q^{64} -\beta q^{67} + 2 q^{74} - q^{76} -2 \beta q^{78} - q^{81} + 2 q^{96} + \beta q^{97} + \beta q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} - 4q^{6} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{4} - 4q^{6} - 2q^{9} - 2q^{16} + 2q^{19} + 4q^{26} + 2q^{36} + 4q^{39} - 2q^{49} + 2q^{64} + 4q^{74} - 2q^{76} - 2q^{81} + 4q^{96} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/475\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
151.1
1.41421i
1.41421i
1.41421i 1.41421i −1.00000 0 −2.00000 0 0 −1.00000 0
151.2 1.41421i 1.41421i −1.00000 0 −2.00000 0 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
95.d odd 2 1 CM by \(\Q(\sqrt{-95}) \)
5.b even 2 1 inner
19.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.1.c.b 2
5.b even 2 1 inner 475.1.c.b 2
5.c odd 4 2 95.1.d.b 2
15.e even 4 2 855.1.g.c 2
19.b odd 2 1 inner 475.1.c.b 2
20.e even 4 2 1520.1.m.b 2
95.d odd 2 1 CM 475.1.c.b 2
95.g even 4 2 95.1.d.b 2
95.l even 12 4 1805.1.h.b 4
95.m odd 12 4 1805.1.h.b 4
95.q odd 36 12 1805.1.o.b 12
95.r even 36 12 1805.1.o.b 12
285.j odd 4 2 855.1.g.c 2
380.j odd 4 2 1520.1.m.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.1.d.b 2 5.c odd 4 2
95.1.d.b 2 95.g even 4 2
475.1.c.b 2 1.a even 1 1 trivial
475.1.c.b 2 5.b even 2 1 inner
475.1.c.b 2 19.b odd 2 1 inner
475.1.c.b 2 95.d odd 2 1 CM
855.1.g.c 2 15.e even 4 2
855.1.g.c 2 285.j odd 4 2
1520.1.m.b 2 20.e even 4 2
1520.1.m.b 2 380.j odd 4 2
1805.1.h.b 4 95.l even 12 4
1805.1.h.b 4 95.m odd 12 4
1805.1.o.b 12 95.q odd 36 12
1805.1.o.b 12 95.r even 36 12

Hecke kernels

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

Hecke characteristic polynomials

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