# Properties

 Label 47.1.b.a Level 47 Weight 1 Character orbit 47.b Self dual yes Analytic conductor 0.023 Analytic rank 0 Dimension 2 Projective image $$D_{5}$$ CM discriminant -47 Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$47$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 47.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.0234560555938$$ 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 5.1.2209.1 Artin image $D_5$ Artin field Galois closure of 5.1.2209.1

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta ) q^{2} -\beta q^{3} + ( 1 - \beta ) q^{4} - q^{6} + ( -1 + \beta ) q^{7} - q^{8} + \beta q^{9} +O(q^{10})$$ $$q + ( -1 + \beta ) q^{2} -\beta q^{3} + ( 1 - \beta ) q^{4} - q^{6} + ( -1 + \beta ) q^{7} - q^{8} + \beta q^{9} + q^{12} + ( 2 - \beta ) q^{14} -\beta q^{17} + q^{18} - q^{21} + \beta q^{24} + q^{25} - q^{27} + ( -2 + \beta ) q^{28} + q^{32} - q^{34} - q^{36} -\beta q^{37} + ( 1 - \beta ) q^{42} + q^{47} + ( 1 - \beta ) q^{49} + ( -1 + \beta ) q^{50} + ( 1 + \beta ) q^{51} + ( -1 + \beta ) q^{53} + ( 1 - \beta ) q^{54} + ( 1 - \beta ) q^{56} + ( -1 + \beta ) q^{59} + ( -1 + \beta ) q^{61} + q^{63} + ( -1 + \beta ) q^{64} + q^{68} -\beta q^{71} -\beta q^{72} - q^{74} -\beta q^{75} -\beta q^{79} + 2 q^{83} + ( -1 + \beta ) q^{84} + ( -1 + \beta ) q^{89} + ( -1 + \beta ) q^{94} -\beta q^{96} + ( -1 + \beta ) q^{97} + ( -2 + \beta ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} - q^{3} + q^{4} - 2q^{6} - q^{7} - 2q^{8} + q^{9} + O(q^{10})$$ $$2q - q^{2} - q^{3} + q^{4} - 2q^{6} - q^{7} - 2q^{8} + q^{9} + 2q^{12} + 3q^{14} - q^{17} + 2q^{18} - 2q^{21} + q^{24} + 2q^{25} - 2q^{27} - 3q^{28} + 2q^{32} - 2q^{34} - 2q^{36} - q^{37} + q^{42} + 2q^{47} + q^{49} - q^{50} + 3q^{51} - q^{53} + q^{54} + q^{56} - q^{59} - q^{61} + 2q^{63} - q^{64} + 2q^{68} - q^{71} - q^{72} - 2q^{74} - q^{75} - q^{79} + 4q^{83} - q^{84} - q^{89} - q^{94} - q^{96} - q^{97} - 3q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/47\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$\chi(n)$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
46.1
 −0.618034 1.61803
−1.61803 0.618034 1.61803 0 −1.00000 −1.61803 −1.00000 −0.618034 0
46.2 0.618034 −1.61803 −0.618034 0 −1.00000 0.618034 −1.00000 1.61803 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
47.b odd 2 1 CM by $$\Q(\sqrt{-47})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 47.1.b.a 2
3.b odd 2 1 423.1.d.a 2
4.b odd 2 1 752.1.g.a 2
5.b even 2 1 1175.1.d.c 2
5.c odd 4 2 1175.1.b.b 4
7.b odd 2 1 2303.1.d.c 2
7.c even 3 2 2303.1.f.c 4
7.d odd 6 2 2303.1.f.b 4
8.b even 2 1 3008.1.g.b 2
8.d odd 2 1 3008.1.g.a 2
9.c even 3 2 3807.1.f.b 4
9.d odd 6 2 3807.1.f.a 4
47.b odd 2 1 CM 47.1.b.a 2
47.c even 23 22 2209.1.d.a 44
47.d odd 46 22 2209.1.d.a 44
141.c even 2 1 423.1.d.a 2
188.b even 2 1 752.1.g.a 2
235.b odd 2 1 1175.1.d.c 2
235.e even 4 2 1175.1.b.b 4
329.c even 2 1 2303.1.d.c 2
329.f odd 6 2 2303.1.f.c 4
329.g even 6 2 2303.1.f.b 4
376.e odd 2 1 3008.1.g.b 2
376.h even 2 1 3008.1.g.a 2
423.f odd 6 2 3807.1.f.b 4
423.g even 6 2 3807.1.f.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
47.1.b.a 2 1.a even 1 1 trivial
47.1.b.a 2 47.b odd 2 1 CM
423.1.d.a 2 3.b odd 2 1
423.1.d.a 2 141.c even 2 1
752.1.g.a 2 4.b odd 2 1
752.1.g.a 2 188.b even 2 1
1175.1.b.b 4 5.c odd 4 2
1175.1.b.b 4 235.e even 4 2
1175.1.d.c 2 5.b even 2 1
1175.1.d.c 2 235.b odd 2 1
2209.1.d.a 44 47.c even 23 22
2209.1.d.a 44 47.d odd 46 22
2303.1.d.c 2 7.b odd 2 1
2303.1.d.c 2 329.c even 2 1
2303.1.f.b 4 7.d odd 6 2
2303.1.f.b 4 329.g even 6 2
2303.1.f.c 4 7.c even 3 2
2303.1.f.c 4 329.f odd 6 2
3008.1.g.a 2 8.d odd 2 1
3008.1.g.a 2 376.h even 2 1
3008.1.g.b 2 8.b even 2 1
3008.1.g.b 2 376.e odd 2 1
3807.1.f.a 4 9.d odd 6 2
3807.1.f.a 4 423.g even 6 2
3807.1.f.b 4 9.c even 3 2
3807.1.f.b 4 423.f odd 6 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(47, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2} + T^{3} + T^{4}$$
$3$ $$1 + T + T^{2} + T^{3} + T^{4}$$
$5$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$7$ $$1 + T + T^{2} + T^{3} + T^{4}$$
$11$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$13$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$17$ $$1 + T + T^{2} + T^{3} + T^{4}$$
$19$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$23$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$29$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$31$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$37$ $$1 + T + T^{2} + T^{3} + T^{4}$$
$41$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$43$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$47$ $$( 1 - T )^{2}$$
$53$ $$1 + T + T^{2} + T^{3} + T^{4}$$
$59$ $$1 + T + T^{2} + T^{3} + T^{4}$$
$61$ $$1 + T + T^{2} + T^{3} + T^{4}$$
$67$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$71$ $$1 + T + T^{2} + T^{3} + T^{4}$$
$73$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$79$ $$1 + T + T^{2} + T^{3} + T^{4}$$
$83$ $$( 1 - T )^{4}$$
$89$ $$1 + T + T^{2} + T^{3} + T^{4}$$
$97$ $$1 + T + T^{2} + T^{3} + T^{4}$$