# Properties

 Label 531.2.a.d Level $531$ Weight $2$ Character orbit 531.a Self dual yes Analytic conductor $4.240$ Analytic rank $0$ Dimension $3$ 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: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.229.1 Defining polynomial: $$x^{3} - 4 x - 1$$ 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + ( 1 - \beta_{1} + \beta_{2} ) q^{5} + ( 3 + \beta_{1} ) q^{7} - q^{8} +O(q^{10})$$ $$q -\beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + ( 1 - \beta_{1} + \beta_{2} ) q^{5} + ( 3 + \beta_{1} ) q^{7} - q^{8} + ( 2 - 2 \beta_{1} + \beta_{2} ) q^{10} + ( 1 + \beta_{1} + \beta_{2} ) q^{11} + ( 1 - \beta_{1} - \beta_{2} ) q^{13} + ( -3 - 3 \beta_{1} - \beta_{2} ) q^{14} + ( -2 + \beta_{1} - 2 \beta_{2} ) q^{16} + ( -2 + 2 \beta_{1} - 3 \beta_{2} ) q^{17} + ( 2 - \beta_{2} ) q^{19} + ( 3 - \beta_{1} ) q^{20} + ( -4 - 2 \beta_{1} - \beta_{2} ) q^{22} + ( 2 \beta_{1} + \beta_{2} ) q^{23} + ( -3 \beta_{1} + \beta_{2} ) q^{25} + ( 4 + \beta_{2} ) q^{26} + ( 4 + 2 \beta_{1} + 3 \beta_{2} ) q^{28} + ( 3 - \beta_{1} - 2 \beta_{2} ) q^{29} + ( 5 - \beta_{1} + 2 \beta_{2} ) q^{31} + ( 1 + 4 \beta_{1} - \beta_{2} ) q^{32} + ( -3 + 5 \beta_{1} - 2 \beta_{2} ) q^{34} + ( 1 - \beta_{1} + 2 \beta_{2} ) q^{35} + ( -2 - 2 \beta_{1} - \beta_{2} ) q^{37} + ( 1 - \beta_{1} ) q^{38} + ( -1 + \beta_{1} - \beta_{2} ) q^{40} + ( 1 - 3 \beta_{1} + 2 \beta_{2} ) q^{41} + ( 1 + 5 \beta_{1} - 3 \beta_{2} ) q^{43} + ( 5 + 3 \beta_{1} ) q^{44} + ( -7 - \beta_{1} - 2 \beta_{2} ) q^{46} + ( -5 + \beta_{1} - 4 \beta_{2} ) q^{47} + ( 5 + 6 \beta_{1} + \beta_{2} ) q^{49} + ( 8 - \beta_{1} + 3 \beta_{2} ) q^{50} + ( -3 - 3 \beta_{1} + 2 \beta_{2} ) q^{52} + ( -1 + 5 \beta_{1} - \beta_{2} ) q^{53} + ( 1 + \beta_{1} - \beta_{2} ) q^{55} + ( -3 - \beta_{1} ) q^{56} + ( 5 - \beta_{1} + \beta_{2} ) q^{58} - q^{59} + ( 1 + \beta_{1} + 4 \beta_{2} ) q^{61} + ( 1 - 7 \beta_{1} + \beta_{2} ) q^{62} + ( -7 - 2 \beta_{1} ) q^{64} + ( 1 - 3 \beta_{1} + 3 \beta_{2} ) q^{65} + ( 5 + \beta_{1} + 5 \beta_{2} ) q^{67} + ( -9 + \beta_{1} + \beta_{2} ) q^{68} + ( 1 - 3 \beta_{1} + \beta_{2} ) q^{70} + ( -9 + 3 \beta_{1} - \beta_{2} ) q^{71} + ( 1 - 3 \beta_{1} - 4 \beta_{2} ) q^{73} + ( 7 + 3 \beta_{1} + 2 \beta_{2} ) q^{74} + ( -1 - \beta_{1} + 3 \beta_{2} ) q^{76} + ( 7 + 5 \beta_{1} + 4 \beta_{2} ) q^{77} + ( 1 - 3 \beta_{1} + \beta_{2} ) q^{79} + ( -8 + 4 \beta_{1} - \beta_{2} ) q^{80} + ( 7 - 3 \beta_{1} + 3 \beta_{2} ) q^{82} + ( -1 + 5 \beta_{1} - 6 \beta_{2} ) q^{83} + ( -12 + 6 \beta_{1} - \beta_{2} ) q^{85} + ( -12 + 2 \beta_{1} - 5 \beta_{2} ) q^{86} + ( -1 - \beta_{1} - \beta_{2} ) q^{88} + ( 9 - 3 \beta_{1} + 4 \beta_{2} ) q^{89} + ( -1 - 3 \beta_{1} - 4 \beta_{2} ) q^{91} + ( 5 + 5 \beta_{1} - \beta_{2} ) q^{92} + ( 1 + 9 \beta_{1} - \beta_{2} ) q^{94} + ( -2 \beta_{1} + 3 \beta_{2} ) q^{95} + ( 5 - 5 \beta_{1} + \beta_{2} ) q^{97} + ( -19 - 6 \beta_{1} - 6 \beta_{2} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + 2q^{4} + 2q^{5} + 9q^{7} - 3q^{8} + O(q^{10})$$ $$3q + 2q^{4} + 2q^{5} + 9q^{7} - 3q^{8} + 5q^{10} + 2q^{11} + 4q^{13} - 8q^{14} - 4q^{16} - 3q^{17} + 7q^{19} + 9q^{20} - 11q^{22} - q^{23} - q^{25} + 11q^{26} + 9q^{28} + 11q^{29} + 13q^{31} + 4q^{32} - 7q^{34} + q^{35} - 5q^{37} + 3q^{38} - 2q^{40} + q^{41} + 6q^{43} + 15q^{44} - 19q^{46} - 11q^{47} + 14q^{49} + 21q^{50} - 11q^{52} - 2q^{53} + 4q^{55} - 9q^{56} + 14q^{58} - 3q^{59} - q^{61} + 2q^{62} - 21q^{64} + 10q^{67} - 28q^{68} + 2q^{70} - 26q^{71} + 7q^{73} + 19q^{74} - 6q^{76} + 17q^{77} + 2q^{79} - 23q^{80} + 18q^{82} + 3q^{83} - 35q^{85} - 31q^{86} - 2q^{88} + 23q^{89} + q^{91} + 16q^{92} + 4q^{94} - 3q^{95} + 14q^{97} - 51q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 4 x - 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$

## 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
 2.11491 −0.254102 −1.86081
−2.11491 0 2.47283 0.357926 0 5.11491 −1.00000 0 −0.756981
1.2 0.254102 0 −1.93543 −1.68133 0 2.74590 −1.00000 0 −0.427229
1.3 1.86081 0 1.46260 3.32340 0 1.13919 −1.00000 0 6.18421
 $$n$$: 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.d 3
3.b odd 2 1 177.2.a.d 3
4.b odd 2 1 8496.2.a.bl 3
12.b even 2 1 2832.2.a.t 3
15.d odd 2 1 4425.2.a.w 3
21.c even 2 1 8673.2.a.s 3

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} - 4 T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(531))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 4 T + T^{3}$$
$3$ $$T^{3}$$
$5$ $$2 - 5 T - 2 T^{2} + T^{3}$$
$7$ $$-16 + 23 T - 9 T^{2} + T^{3}$$
$11$ $$-4 - 11 T - 2 T^{2} + T^{3}$$
$13$ $$26 - 7 T - 4 T^{2} + T^{3}$$
$17$ $$-98 - 43 T + 3 T^{2} + T^{3}$$
$19$ $$-4 + 11 T - 7 T^{2} + T^{3}$$
$23$ $$-64 - 27 T + T^{2} + T^{3}$$
$29$ $$74 + 9 T - 11 T^{2} + T^{3}$$
$31$ $$28 + 37 T - 13 T^{2} + T^{3}$$
$37$ $$14 - 19 T + 5 T^{2} + T^{3}$$
$41$ $$-74 - 39 T - T^{2} + T^{3}$$
$43$ $$592 - 91 T - 6 T^{2} + T^{3}$$
$47$ $$-496 - 37 T + 11 T^{2} + T^{3}$$
$53$ $$58 - 89 T + 2 T^{2} + T^{3}$$
$59$ $$( 1 + T )^{3}$$
$61$ $$98 - 101 T + T^{2} + T^{3}$$
$67$ $$784 - 119 T - 10 T^{2} + T^{3}$$
$71$ $$424 + 193 T + 26 T^{2} + T^{3}$$
$73$ $$718 - 141 T - 7 T^{2} + T^{3}$$
$79$ $$-32 - 31 T - 2 T^{2} + T^{3}$$
$83$ $$148 - 199 T - 3 T^{2} + T^{3}$$
$89$ $$278 + 91 T - 23 T^{2} + T^{3}$$
$97$ $$202 - 25 T - 14 T^{2} + T^{3}$$