Properties

 Label 131.1.b.a Level $131$ Weight $1$ Character orbit 131.b Self dual yes Analytic conductor $0.065$ Analytic rank $0$ Dimension $2$ Projective image $D_{5}$ CM discriminant -131 Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$131$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 131.b (of order $$2$$, degree $$1$$, minimal)

Newform invariants

 Self dual: yes Analytic conductor: $$0.0653775166549$$ 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 5.1.17161.1 Artin image: $D_5$ Artin field: Galois closure of 5.1.17161.1

$q$-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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta ) q^{3} + q^{4} -\beta q^{5} -\beta q^{7} + ( 1 - \beta ) q^{9} +O(q^{10})$$ $$q + ( -1 + \beta ) q^{3} + q^{4} -\beta q^{5} -\beta q^{7} + ( 1 - \beta ) q^{9} + ( -1 + \beta ) q^{11} + ( -1 + \beta ) q^{12} + ( -1 + \beta ) q^{13} - q^{15} + q^{16} -\beta q^{20} - q^{21} + \beta q^{25} - q^{27} -\beta q^{28} + ( 2 - \beta ) q^{33} + ( 1 + \beta ) q^{35} + ( 1 - \beta ) q^{36} + ( 2 - \beta ) q^{39} + ( -1 + \beta ) q^{41} -\beta q^{43} + ( -1 + \beta ) q^{44} + q^{45} + ( -1 + \beta ) q^{48} + \beta q^{49} + ( -1 + \beta ) q^{52} + 2 q^{53} - q^{55} -\beta q^{59} - q^{60} -\beta q^{61} + q^{63} + q^{64} - q^{65} + q^{75} - q^{77} -\beta q^{80} - q^{84} + 2 q^{89} - q^{91} + ( -2 + \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{3} + 2q^{4} - q^{5} - q^{7} + q^{9} + O(q^{10})$$ $$2q - q^{3} + 2q^{4} - q^{5} - q^{7} + q^{9} - q^{11} - q^{12} - q^{13} - 2q^{15} + 2q^{16} - q^{20} - 2q^{21} + q^{25} - 2q^{27} - q^{28} + 3q^{33} + 3q^{35} + q^{36} + 3q^{39} - q^{41} - q^{43} - q^{44} + 2q^{45} - q^{48} + q^{49} - q^{52} + 4q^{53} - 2q^{55} - q^{59} - 2q^{60} - q^{61} + 2q^{63} + 2q^{64} - 2q^{65} + 2q^{75} - 2q^{77} - q^{80} - 2q^{84} + 4q^{89} - 2q^{91} - 3q^{99} + O(q^{100})$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/131\mathbb{Z}\right)^\times$$.

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

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
130.1
 −0.618034 1.61803
0 −1.61803 1.00000 0.618034 0 0.618034 0 1.61803 0
130.2 0 0.618034 1.00000 −1.61803 0 −1.61803 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
131.b odd 2 1 CM by $$\Q(\sqrt{-131})$$

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 131.1.b.a 2
3.b odd 2 1 1179.1.c.a 2
4.b odd 2 1 2096.1.h.b 2
5.b even 2 1 3275.1.c.d 2
5.c odd 4 2 3275.1.d.a 4
131.b odd 2 1 CM 131.1.b.a 2
393.d even 2 1 1179.1.c.a 2
524.b even 2 1 2096.1.h.b 2
655.d odd 2 1 3275.1.c.d 2
655.e even 4 2 3275.1.d.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
131.1.b.a 2 1.a even 1 1 trivial
131.1.b.a 2 131.b odd 2 1 CM
1179.1.c.a 2 3.b odd 2 1
1179.1.c.a 2 393.d even 2 1
2096.1.h.b 2 4.b odd 2 1
2096.1.h.b 2 524.b even 2 1
3275.1.c.d 2 5.b even 2 1
3275.1.c.d 2 655.d odd 2 1
3275.1.d.a 4 5.c odd 4 2
3275.1.d.a 4 655.e even 4 2

Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(131, [\chi])$$.

Hecke characteristic polynomials

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