# Properties

 Label 531.2.a.b Level $531$ Weight $2$ Character orbit 531.a Self dual yes Analytic conductor $4.240$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$531 = 3^{2} \cdot 59$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 531.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$4.24005634733$$ Analytic rank: $$1$$ 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: no (minimal twist has level 177) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + ( -1 + \beta ) q^{4} + ( 1 - 2 \beta ) q^{5} + ( -3 - \beta ) q^{7} + ( 1 - 2 \beta ) q^{8} +O(q^{10})$$ $$q + \beta q^{2} + ( -1 + \beta ) q^{4} + ( 1 - 2 \beta ) q^{5} + ( -3 - \beta ) q^{7} + ( 1 - 2 \beta ) q^{8} + ( -2 - \beta ) q^{10} + ( -1 + 2 \beta ) q^{11} + ( -5 + 2 \beta ) q^{13} + ( -1 - 4 \beta ) q^{14} -3 \beta q^{16} + 3 \beta q^{17} -5 \beta q^{19} + ( -3 + \beta ) q^{20} + ( 2 + \beta ) q^{22} + ( 4 - \beta ) q^{23} + ( 2 - 3 \beta ) q^{26} + ( 2 - 3 \beta ) q^{28} + ( -7 - \beta ) q^{29} + ( -5 + 9 \beta ) q^{31} + ( -5 + \beta ) q^{32} + ( 3 + 3 \beta ) q^{34} + ( -1 + 7 \beta ) q^{35} + ( -2 - 3 \beta ) q^{37} + ( -5 - 5 \beta ) q^{38} + 5 q^{40} + ( -5 + 5 \beta ) q^{41} + ( -5 + 6 \beta ) q^{43} + ( 3 - \beta ) q^{44} + ( -1 + 3 \beta ) q^{46} + ( 9 - 3 \beta ) q^{47} + ( 3 + 7 \beta ) q^{49} + ( 7 - 5 \beta ) q^{52} + ( -5 + 2 \beta ) q^{53} -5 q^{55} + ( -1 + 7 \beta ) q^{56} + ( -1 - 8 \beta ) q^{58} + q^{59} + ( -5 - 3 \beta ) q^{61} + ( 9 + 4 \beta ) q^{62} + ( 1 + 2 \beta ) q^{64} + ( -9 + 8 \beta ) q^{65} + ( 7 - 6 \beta ) q^{67} + 3 q^{68} + ( 7 + 6 \beta ) q^{70} + ( 3 - 8 \beta ) q^{71} + ( -1 - 3 \beta ) q^{73} + ( -3 - 5 \beta ) q^{74} -5 q^{76} + ( 1 - 7 \beta ) q^{77} -3 q^{79} + ( 6 + 3 \beta ) q^{80} + 5 q^{82} + ( 1 - \beta ) q^{83} + ( -6 - 3 \beta ) q^{85} + ( 6 + \beta ) q^{86} -5 q^{88} + ( 7 - 11 \beta ) q^{89} + ( 13 - 3 \beta ) q^{91} + ( -5 + 4 \beta ) q^{92} + ( -3 + 6 \beta ) q^{94} + ( 10 + 5 \beta ) q^{95} + 3 q^{97} + ( 7 + 10 \beta ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{2} - q^{4} - 7q^{7} + O(q^{10})$$ $$2q + q^{2} - q^{4} - 7q^{7} - 5q^{10} - 8q^{13} - 6q^{14} - 3q^{16} + 3q^{17} - 5q^{19} - 5q^{20} + 5q^{22} + 7q^{23} + q^{26} + q^{28} - 15q^{29} - q^{31} - 9q^{32} + 9q^{34} + 5q^{35} - 7q^{37} - 15q^{38} + 10q^{40} - 5q^{41} - 4q^{43} + 5q^{44} + q^{46} + 15q^{47} + 13q^{49} + 9q^{52} - 8q^{53} - 10q^{55} + 5q^{56} - 10q^{58} + 2q^{59} - 13q^{61} + 22q^{62} + 4q^{64} - 10q^{65} + 8q^{67} + 6q^{68} + 20q^{70} - 2q^{71} - 5q^{73} - 11q^{74} - 10q^{76} - 5q^{77} - 6q^{79} + 15q^{80} + 10q^{82} + q^{83} - 15q^{85} + 13q^{86} - 10q^{88} + 3q^{89} + 23q^{91} - 6q^{92} + 25q^{95} + 6q^{97} + 24q^{98} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −0.618034 1.61803
−0.618034 0 −1.61803 2.23607 0 −2.38197 2.23607 0 −1.38197
1.2 1.61803 0 0.618034 −2.23607 0 −4.61803 −2.23607 0 −3.61803
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$59$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 531.2.a.b 2
3.b odd 2 1 177.2.a.b 2
4.b odd 2 1 8496.2.a.bb 2
12.b even 2 1 2832.2.a.o 2
15.d odd 2 1 4425.2.a.t 2
21.c even 2 1 8673.2.a.k 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.2.a.b 2 3.b odd 2 1
531.2.a.b 2 1.a even 1 1 trivial
2832.2.a.o 2 12.b even 2 1
4425.2.a.t 2 15.d odd 2 1
8496.2.a.bb 2 4.b odd 2 1
8673.2.a.k 2 21.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - T_{2} - 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(531))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 - T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$-5 + T^{2}$$
$7$ $$11 + 7 T + T^{2}$$
$11$ $$-5 + T^{2}$$
$13$ $$11 + 8 T + T^{2}$$
$17$ $$-9 - 3 T + T^{2}$$
$19$ $$-25 + 5 T + T^{2}$$
$23$ $$11 - 7 T + T^{2}$$
$29$ $$55 + 15 T + T^{2}$$
$31$ $$-101 + T + T^{2}$$
$37$ $$1 + 7 T + T^{2}$$
$41$ $$-25 + 5 T + T^{2}$$
$43$ $$-41 + 4 T + T^{2}$$
$47$ $$45 - 15 T + T^{2}$$
$53$ $$11 + 8 T + T^{2}$$
$59$ $$( -1 + T )^{2}$$
$61$ $$31 + 13 T + T^{2}$$
$67$ $$-29 - 8 T + T^{2}$$
$71$ $$-79 + 2 T + T^{2}$$
$73$ $$-5 + 5 T + T^{2}$$
$79$ $$( 3 + T )^{2}$$
$83$ $$-1 - T + T^{2}$$
$89$ $$-149 - 3 T + T^{2}$$
$97$ $$( -3 + T )^{2}$$