Properties

Label 471.1.d.b
Level $471$
Weight $1$
Character orbit 471.d
Self dual yes
Analytic conductor $0.235$
Analytic rank $0$
Dimension $2$
Projective image $D_{4}$
CM discriminant -471
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 471 = 3 \cdot 157 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 471.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.235059620950\)
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: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.73947.1
Artin image: $D_8$
Artin field: Galois closure of 8.0.313461333.1

$q$-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.

\(f(q)\) \(=\) \( q -\beta q^{2} + q^{3} + q^{4} + \beta q^{5} -\beta q^{6} + q^{9} +O(q^{10})\) \( q -\beta q^{2} + q^{3} + q^{4} + \beta q^{5} -\beta q^{6} + q^{9} -2 q^{10} + q^{12} -2 q^{13} + \beta q^{15} - q^{16} -\beta q^{18} + \beta q^{20} -\beta q^{23} + q^{25} + 2 \beta q^{26} + q^{27} -\beta q^{29} -2 q^{30} + \beta q^{32} + q^{36} -2 q^{39} + \beta q^{41} + \beta q^{45} + 2 q^{46} - q^{48} + q^{49} -\beta q^{50} -2 q^{52} + \beta q^{53} -\beta q^{54} + 2 q^{58} -\beta q^{59} + \beta q^{60} - q^{64} -2 \beta q^{65} -2 q^{67} -\beta q^{69} + q^{75} + 2 \beta q^{78} -\beta q^{80} + q^{81} -2 q^{82} -\beta q^{83} -\beta q^{87} -2 q^{90} -\beta q^{92} + \beta q^{96} -\beta q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{4} + 2 q^{9} + O(q^{10}) \) \( 2 q + 2 q^{3} + 2 q^{4} + 2 q^{9} - 4 q^{10} + 2 q^{12} - 4 q^{13} - 2 q^{16} + 2 q^{25} + 2 q^{27} - 4 q^{30} + 2 q^{36} - 4 q^{39} + 4 q^{46} - 2 q^{48} + 2 q^{49} - 4 q^{52} + 4 q^{58} - 2 q^{64} - 4 q^{67} + 2 q^{75} + 2 q^{81} - 4 q^{82} - 4 q^{90} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/471\mathbb{Z}\right)^\times\).

\(n\) \(158\) \(319\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
470.1
1.41421
−1.41421
−1.41421 1.00000 1.00000 1.41421 −1.41421 0 0 1.00000 −2.00000
470.2 1.41421 1.00000 1.00000 −1.41421 1.41421 0 0 1.00000 −2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
471.d odd 2 1 CM by \(\Q(\sqrt{-471}) \)
3.b odd 2 1 inner
157.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 471.1.d.b 2
3.b odd 2 1 inner 471.1.d.b 2
157.b even 2 1 inner 471.1.d.b 2
471.d odd 2 1 CM 471.1.d.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
471.1.d.b 2 1.a even 1 1 trivial
471.1.d.b 2 3.b odd 2 1 inner
471.1.d.b 2 157.b even 2 1 inner
471.1.d.b 2 471.d odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 2 \) acting on \(S_{1}^{\mathrm{new}}(471, [\chi])\).

Hecke characteristic polynomials

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