Properties

Label 639.1.d.a
Level $639$
Weight $1$
Character orbit 639.d
Self dual yes
Analytic conductor $0.319$
Analytic rank $0$
Dimension $3$
Projective image $D_{7}$
CM discriminant -71
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 639 = 3^{2} \cdot 71 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 639.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.318902543072\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Defining polynomial: \(x^{3} - x^{2} - 2 x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 71)
Projective image: \(D_{7}\)
Projective field: Galois closure of 7.1.357911.1
Artin image: $D_{14}$
Artin field: Galois closure of 14.0.280155320935227.1

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} -\beta_{2} q^{5} + ( 1 + \beta_{2} ) q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} -\beta_{2} q^{5} + ( 1 + \beta_{2} ) q^{8} + ( -1 - \beta_{2} ) q^{10} + \beta_{1} q^{16} -\beta_{1} q^{19} + ( -1 - \beta_{1} ) q^{20} + ( \beta_{1} - \beta_{2} ) q^{25} + ( 1 - \beta_{1} + \beta_{2} ) q^{29} + q^{32} -\beta_{1} q^{37} + ( -2 - \beta_{2} ) q^{38} + ( -1 - \beta_{1} ) q^{40} + \beta_{2} q^{43} + q^{49} + q^{50} + ( -1 + \beta_{1} ) q^{58} - q^{71} + \beta_{2} q^{73} + ( -2 - \beta_{2} ) q^{74} + ( -1 - \beta_{1} - \beta_{2} ) q^{76} + \beta_{2} q^{79} + ( -1 - \beta_{2} ) q^{80} + \beta_{1} q^{83} + ( 1 + \beta_{2} ) q^{86} + ( 1 - \beta_{1} + \beta_{2} ) q^{89} + ( 1 + \beta_{2} ) q^{95} + \beta_{1} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} + 2 q^{4} + q^{5} + 2 q^{8} + O(q^{10}) \) \( 3 q + q^{2} + 2 q^{4} + q^{5} + 2 q^{8} - 2 q^{10} + q^{16} - q^{19} - 4 q^{20} + 2 q^{25} + q^{29} + 3 q^{32} - q^{37} - 5 q^{38} - 4 q^{40} - q^{43} + 3 q^{49} + 3 q^{50} - 2 q^{58} - 3 q^{71} - q^{73} - 5 q^{74} - 3 q^{76} - q^{79} - 2 q^{80} + q^{83} + 2 q^{86} + q^{89} + 2 q^{95} + q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/639\mathbb{Z}\right)^\times\).

\(n\) \(433\) \(569\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
496.1
−1.24698
0.445042
1.80194
−1.24698 0 0.554958 0.445042 0 0 0.554958 0 −0.554958
496.2 0.445042 0 −0.801938 1.80194 0 0 −0.801938 0 0.801938
496.3 1.80194 0 2.24698 −1.24698 0 0 2.24698 0 −2.24698
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
71.b odd 2 1 CM by \(\Q(\sqrt{-71}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 639.1.d.a 3
3.b odd 2 1 71.1.b.a 3
12.b even 2 1 1136.1.h.a 3
15.d odd 2 1 1775.1.d.b 3
15.e even 4 2 1775.1.c.a 6
21.c even 2 1 3479.1.d.e 3
21.g even 6 2 3479.1.g.d 6
21.h odd 6 2 3479.1.g.e 6
71.b odd 2 1 CM 639.1.d.a 3
213.b even 2 1 71.1.b.a 3
852.d odd 2 1 1136.1.h.a 3
1065.h even 2 1 1775.1.d.b 3
1065.l odd 4 2 1775.1.c.a 6
1491.e odd 2 1 3479.1.d.e 3
1491.n odd 6 2 3479.1.g.d 6
1491.p even 6 2 3479.1.g.e 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
71.1.b.a 3 3.b odd 2 1
71.1.b.a 3 213.b even 2 1
639.1.d.a 3 1.a even 1 1 trivial
639.1.d.a 3 71.b odd 2 1 CM
1136.1.h.a 3 12.b even 2 1
1136.1.h.a 3 852.d odd 2 1
1775.1.c.a 6 15.e even 4 2
1775.1.c.a 6 1065.l odd 4 2
1775.1.d.b 3 15.d odd 2 1
1775.1.d.b 3 1065.h even 2 1
3479.1.d.e 3 21.c even 2 1
3479.1.d.e 3 1491.e odd 2 1
3479.1.g.d 6 21.g even 6 2
3479.1.g.d 6 1491.n odd 6 2
3479.1.g.e 6 21.h odd 6 2
3479.1.g.e 6 1491.p even 6 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(639, [\chi])\).

Hecke characteristic polynomials

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