Properties

Label 475.1.m.a
Level $475$
Weight $1$
Character orbit 475.m
Analytic conductor $0.237$
Analytic rank $0$
Dimension $4$
Projective image $D_{10}$
CM discriminant -19
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 475.m (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.237055881001\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Defining polynomial: \(x^{4} - x^{3} + x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{10}\)
Projective field: Galois closure of 10.0.1889113616943359375.4

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + \zeta_{10}^{3} q^{4} -\zeta_{10} q^{5} + ( \zeta_{10} + \zeta_{10}^{4} ) q^{7} + \zeta_{10} q^{9} +O(q^{10})\) \( q + \zeta_{10}^{3} q^{4} -\zeta_{10} q^{5} + ( \zeta_{10} + \zeta_{10}^{4} ) q^{7} + \zeta_{10} q^{9} + ( -1 + \zeta_{10}^{3} ) q^{11} -\zeta_{10} q^{16} + ( 1 - \zeta_{10}^{4} ) q^{17} + \zeta_{10}^{2} q^{19} -\zeta_{10}^{4} q^{20} + ( -1 - \zeta_{10}^{3} ) q^{23} + \zeta_{10}^{2} q^{25} + ( -\zeta_{10}^{2} + \zeta_{10}^{4} ) q^{28} + ( 1 - \zeta_{10}^{2} ) q^{35} + \zeta_{10}^{4} q^{36} + ( -\zeta_{10}^{2} - \zeta_{10}^{3} ) q^{43} + ( -\zeta_{10} - \zeta_{10}^{3} ) q^{44} -\zeta_{10}^{2} q^{45} + ( -1 + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{49} + ( \zeta_{10} - \zeta_{10}^{4} ) q^{55} + ( \zeta_{10} - \zeta_{10}^{2} ) q^{61} + ( -1 + \zeta_{10}^{2} ) q^{63} -\zeta_{10}^{4} q^{64} + ( \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{68} - q^{76} + ( -\zeta_{10} - \zeta_{10}^{2} ) q^{77} + \zeta_{10}^{2} q^{80} + \zeta_{10}^{2} q^{81} + ( 1 - \zeta_{10}^{4} ) q^{83} + ( -1 - \zeta_{10} ) q^{85} + ( \zeta_{10} - \zeta_{10}^{3} ) q^{92} -\zeta_{10}^{3} q^{95} + ( -\zeta_{10} + \zeta_{10}^{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{4} - q^{5} + q^{9} + O(q^{10}) \) \( 4 q + q^{4} - q^{5} + q^{9} - 3 q^{11} - q^{16} + 5 q^{17} - q^{19} + q^{20} - 5 q^{23} - q^{25} + 5 q^{35} - q^{36} - 2 q^{44} + q^{45} - 6 q^{49} + 2 q^{55} + 2 q^{61} - 5 q^{63} + q^{64} - 4 q^{76} - q^{80} - q^{81} + 5 q^{83} - 5 q^{85} - q^{95} - 2 q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(77\) \(401\)
\(\chi(n)\) \(\zeta_{10}^{3}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
94.1
−0.309017 + 0.951057i
0.809017 + 0.587785i
0.809017 0.587785i
−0.309017 0.951057i
0 0 0.809017 0.587785i 0.309017 0.951057i 0 1.90211i 0 −0.309017 + 0.951057i 0
189.1 0 0 −0.309017 + 0.951057i −0.809017 0.587785i 0 1.17557i 0 0.809017 + 0.587785i 0
284.1 0 0 −0.309017 0.951057i −0.809017 + 0.587785i 0 1.17557i 0 0.809017 0.587785i 0
379.1 0 0 0.809017 + 0.587785i 0.309017 + 0.951057i 0 1.90211i 0 −0.309017 0.951057i 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
19.b odd 2 1 CM by \(\Q(\sqrt{-19}) \)
25.e even 10 1 inner
475.m odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.1.m.a 4
5.b even 2 1 2375.1.m.a 4
5.c odd 4 2 2375.1.o.b 8
19.b odd 2 1 CM 475.1.m.a 4
25.d even 5 1 2375.1.m.a 4
25.e even 10 1 inner 475.1.m.a 4
25.f odd 20 2 2375.1.o.b 8
95.d odd 2 1 2375.1.m.a 4
95.g even 4 2 2375.1.o.b 8
475.m odd 10 1 inner 475.1.m.a 4
475.o odd 10 1 2375.1.m.a 4
475.v even 20 2 2375.1.o.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
475.1.m.a 4 1.a even 1 1 trivial
475.1.m.a 4 19.b odd 2 1 CM
475.1.m.a 4 25.e even 10 1 inner
475.1.m.a 4 475.m odd 10 1 inner
2375.1.m.a 4 5.b even 2 1
2375.1.m.a 4 25.d even 5 1
2375.1.m.a 4 95.d odd 2 1
2375.1.m.a 4 475.o odd 10 1
2375.1.o.b 8 5.c odd 4 2
2375.1.o.b 8 25.f odd 20 2
2375.1.o.b 8 95.g even 4 2
2375.1.o.b 8 475.v even 20 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(475, [\chi])\).

Hecke characteristic polynomials

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