# Properties

 Label 179.1.b.a Level $179$ Weight $1$ Character orbit 179.b Self dual yes Analytic conductor $0.089$ Analytic rank $0$ Dimension $2$ Projective image $D_{5}$ CM discriminant -179 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$179$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 179.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.0893326372613$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{5}$$ Projective field: Galois closure of 5.1.32041.1 Artin image: $D_5$ Artin field: Galois closure of 5.1.32041.1

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta ) q^{3} + q^{4} -\beta q^{5} + ( 1 - \beta ) q^{9} +O(q^{10})$$ $$q + ( -1 + \beta ) q^{3} + q^{4} -\beta q^{5} + ( 1 - \beta ) q^{9} + ( -1 + \beta ) q^{12} -\beta q^{13} - q^{15} + q^{16} + ( -1 + \beta ) q^{17} + ( -1 + \beta ) q^{19} -\beta q^{20} + \beta q^{25} - q^{27} + ( -1 + \beta ) q^{29} -\beta q^{31} + ( 1 - \beta ) q^{36} - q^{39} -\beta q^{43} + q^{45} + 2 q^{47} + ( -1 + \beta ) q^{48} + q^{49} + ( 2 - \beta ) q^{51} -\beta q^{52} + ( 2 - \beta ) q^{57} + ( -1 + \beta ) q^{59} - q^{60} + ( -1 + \beta ) q^{61} + q^{64} + ( 1 + \beta ) q^{65} + ( -1 + \beta ) q^{67} + ( -1 + \beta ) q^{68} + q^{75} + ( -1 + \beta ) q^{76} -\beta q^{80} -\beta q^{83} - q^{85} + ( 2 - \beta ) q^{87} -\beta q^{89} - q^{93} - q^{95} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{3} + 2q^{4} - q^{5} + q^{9} + O(q^{10})$$ $$2q - q^{3} + 2q^{4} - q^{5} + q^{9} - q^{12} - q^{13} - 2q^{15} + 2q^{16} - q^{17} - q^{19} - q^{20} + q^{25} - 2q^{27} - q^{29} - q^{31} + q^{36} - 2q^{39} - q^{43} + 2q^{45} + 4q^{47} - q^{48} + 2q^{49} + 3q^{51} - q^{52} + 3q^{57} - q^{59} - 2q^{60} - q^{61} + 2q^{64} + 3q^{65} - q^{67} - q^{68} + 2q^{75} - q^{76} - q^{80} - q^{83} - 2q^{85} + 3q^{87} - q^{89} - 2q^{93} - 2q^{95} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/179\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
178.1
 −0.618034 1.61803
0 −1.61803 1.00000 0.618034 0 0 0 1.61803 0
178.2 0 0.618034 1.00000 −1.61803 0 0 0 −0.618034 0
 $$n$$: 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
179.b odd 2 1 CM by $$\Q(\sqrt{-179})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 179.1.b.a 2
3.b odd 2 1 1611.1.c.a 2
4.b odd 2 1 2864.1.h.b 2
179.b odd 2 1 CM 179.1.b.a 2
537.d even 2 1 1611.1.c.a 2
716.c even 2 1 2864.1.h.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
179.1.b.a 2 1.a even 1 1 trivial
179.1.b.a 2 179.b odd 2 1 CM
1611.1.c.a 2 3.b odd 2 1
1611.1.c.a 2 537.d even 2 1
2864.1.h.b 2 4.b odd 2 1
2864.1.h.b 2 716.c even 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(179, [\chi])$$.

## Hecke characteristic polynomials

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