# 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 $2$

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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

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