# Properties

 Label 79.1.b.a Level $79$ Weight $1$ Character orbit 79.b Self dual yes Analytic conductor $0.039$ Analytic rank $0$ Dimension $2$ Projective image $D_{5}$ CM discriminant -79 Inner twists $2$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$79$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 79.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.0394261359980$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{5}$$ Projective field: Galois closure of 5.1.6241.1 Artin image: $D_5$ Artin field: Galois closure of 5.1.6241.1

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta ) q^{2} + ( 1 - \beta ) q^{4} -\beta q^{5} - q^{8} + q^{9} +O(q^{10})$$ $$q + ( -1 + \beta ) q^{2} + ( 1 - \beta ) q^{4} -\beta q^{5} - q^{8} + q^{9} - q^{10} + ( -1 + \beta ) q^{11} + ( -1 + \beta ) q^{13} + ( -1 + \beta ) q^{18} -\beta q^{19} + q^{20} + ( 2 - \beta ) q^{22} -\beta q^{23} + \beta q^{25} + ( 2 - \beta ) q^{26} + ( -1 + \beta ) q^{31} + q^{32} + ( 1 - \beta ) q^{36} - q^{38} + \beta q^{40} + ( -2 + \beta ) q^{44} -\beta q^{45} - q^{46} + q^{49} + q^{50} + ( -2 + \beta ) q^{52} - q^{55} + ( 2 - \beta ) q^{62} + ( -1 + \beta ) q^{64} - q^{65} -\beta q^{67} - q^{72} -\beta q^{73} + q^{76} + q^{79} + q^{81} + 2 q^{83} + ( 1 - \beta ) q^{88} + ( -1 + \beta ) q^{89} - q^{90} + q^{92} + ( 1 + \beta ) q^{95} -\beta q^{97} + ( -1 + \beta ) q^{98} + ( -1 + \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} + q^{4} - q^{5} - 2q^{8} + 2q^{9} + O(q^{10})$$ $$2q - q^{2} + q^{4} - q^{5} - 2q^{8} + 2q^{9} - 2q^{10} - q^{11} - q^{13} - q^{18} - q^{19} + 2q^{20} + 3q^{22} - q^{23} + q^{25} + 3q^{26} - q^{31} + 2q^{32} + q^{36} - 2q^{38} + q^{40} - 3q^{44} - q^{45} - 2q^{46} + 2q^{49} + 2q^{50} - 3q^{52} - 2q^{55} + 3q^{62} - q^{64} - 2q^{65} - q^{67} - 2q^{72} - q^{73} + 2q^{76} + 2q^{79} + 2q^{81} + 4q^{83} + q^{88} - q^{89} - 2q^{90} + 2q^{92} + 3q^{95} - q^{97} - q^{98} - q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/79\mathbb{Z}\right)^\times$$.

 $$n$$ $$3$$ $$\chi(n)$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
78.1
 −0.618034 1.61803
−1.61803 0 1.61803 0.618034 0 0 −1.00000 1.00000 −1.00000
78.2 0.618034 0 −0.618034 −1.61803 0 0 −1.00000 1.00000 −1.00000
 $$n$$: 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
79.b odd 2 1 CM by $$\Q(\sqrt{-79})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 79.1.b.a 2
3.b odd 2 1 711.1.d.b 2
4.b odd 2 1 1264.1.e.a 2
5.b even 2 1 1975.1.d.c 2
5.c odd 4 2 1975.1.c.a 4
7.b odd 2 1 3871.1.c.c 2
7.c even 3 2 3871.1.m.c 4
7.d odd 6 2 3871.1.m.b 4
79.b odd 2 1 CM 79.1.b.a 2
237.b even 2 1 711.1.d.b 2
316.d even 2 1 1264.1.e.a 2
395.c odd 2 1 1975.1.d.c 2
395.f even 4 2 1975.1.c.a 4
553.d even 2 1 3871.1.c.c 2
553.l even 6 2 3871.1.m.b 4
553.m odd 6 2 3871.1.m.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
79.1.b.a 2 1.a even 1 1 trivial
79.1.b.a 2 79.b odd 2 1 CM
711.1.d.b 2 3.b odd 2 1
711.1.d.b 2 237.b even 2 1
1264.1.e.a 2 4.b odd 2 1
1264.1.e.a 2 316.d even 2 1
1975.1.c.a 4 5.c odd 4 2
1975.1.c.a 4 395.f even 4 2
1975.1.d.c 2 5.b even 2 1
1975.1.d.c 2 395.c odd 2 1
3871.1.c.c 2 7.b odd 2 1
3871.1.c.c 2 553.d even 2 1
3871.1.m.b 4 7.d odd 6 2
3871.1.m.b 4 553.l even 6 2
3871.1.m.c 4 7.c even 3 2
3871.1.m.c 4 553.m odd 6 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(79, [\chi])$$.

## Hecke characteristic polynomials

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