# Properties

 Label 8673.2.a.s Level $8673$ Weight $2$ Character orbit 8673.a Self dual yes Analytic conductor $69.254$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8673 = 3 \cdot 7^{2} \cdot 59$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8673.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$69.2542536731$$ Analytic rank: $$1$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + q^{3} + ( 1 + \beta_{2} ) q^{4} + ( 1 - \beta_{1} + \beta_{2} ) q^{5} + \beta_{1} q^{6} + q^{8} + q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + q^{3} + ( 1 + \beta_{2} ) q^{4} + ( 1 - \beta_{1} + \beta_{2} ) q^{5} + \beta_{1} q^{6} + q^{8} + q^{9} + ( -2 + 2 \beta_{1} - \beta_{2} ) q^{10} + ( -1 - \beta_{1} - \beta_{2} ) q^{11} + ( 1 + \beta_{2} ) q^{12} + ( -1 + \beta_{1} + \beta_{2} ) q^{13} + ( 1 - \beta_{1} + \beta_{2} ) q^{15} + ( -2 + \beta_{1} - 2 \beta_{2} ) q^{16} + ( -2 + 2 \beta_{1} - 3 \beta_{2} ) q^{17} + \beta_{1} q^{18} + ( -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} + q^{24} + ( -3 \beta_{1} + \beta_{2} ) q^{25} + ( 4 + \beta_{2} ) q^{26} + q^{27} + ( -3 + \beta_{1} + 2 \beta_{2} ) q^{29} + ( -2 + 2 \beta_{1} - \beta_{2} ) q^{30} + ( -5 + \beta_{1} - 2 \beta_{2} ) q^{31} + ( -1 - 4 \beta_{1} + \beta_{2} ) q^{32} + ( -1 - \beta_{1} - \beta_{2} ) q^{33} + ( 3 - 5 \beta_{1} + 2 \beta_{2} ) q^{34} + ( 1 + \beta_{2} ) q^{36} + ( -2 - 2 \beta_{1} - \beta_{2} ) q^{37} + ( 1 - \beta_{1} ) q^{38} + ( -1 + \beta_{1} + \beta_{2} ) q^{39} + ( 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} + ( 1 - \beta_{1} + \beta_{2} ) q^{45} + ( -7 - \beta_{1} - 2 \beta_{2} ) q^{46} + ( -5 + \beta_{1} - 4 \beta_{2} ) q^{47} + ( -2 + \beta_{1} - 2 \beta_{2} ) q^{48} + ( -8 + \beta_{1} - 3 \beta_{2} ) q^{50} + ( -2 + 2 \beta_{1} - 3 \beta_{2} ) q^{51} + ( 3 + 3 \beta_{1} - 2 \beta_{2} ) q^{52} + ( 1 - 5 \beta_{1} + \beta_{2} ) q^{53} + \beta_{1} q^{54} + ( -1 - \beta_{1} + \beta_{2} ) q^{55} + ( -2 + \beta_{2} ) q^{57} + ( 5 - \beta_{1} + \beta_{2} ) q^{58} - q^{59} + ( 3 - \beta_{1} ) q^{60} + ( -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} + ( -4 - 2 \beta_{1} - \beta_{2} ) q^{66} + ( 5 + \beta_{1} + 5 \beta_{2} ) q^{67} + ( -9 + \beta_{1} + \beta_{2} ) q^{68} + ( -2 \beta_{1} - \beta_{2} ) q^{69} + ( 9 - 3 \beta_{1} + \beta_{2} ) q^{71} + q^{72} + ( -1 + 3 \beta_{1} + 4 \beta_{2} ) q^{73} + ( -7 - 3 \beta_{1} - 2 \beta_{2} ) q^{74} + ( -3 \beta_{1} + \beta_{2} ) q^{75} + ( 1 + \beta_{1} - 3 \beta_{2} ) q^{76} + ( 4 + \beta_{2} ) q^{78} + ( 1 - 3 \beta_{1} + \beta_{2} ) q^{79} + ( -8 + 4 \beta_{1} - \beta_{2} ) q^{80} + q^{81} + ( -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} + ( -3 + \beta_{1} + 2 \beta_{2} ) q^{87} + ( -1 - \beta_{1} - \beta_{2} ) q^{88} + ( 9 - 3 \beta_{1} + 4 \beta_{2} ) q^{89} + ( -2 + 2 \beta_{1} - \beta_{2} ) q^{90} + ( -5 - 5 \beta_{1} + \beta_{2} ) q^{92} + ( -5 + \beta_{1} - 2 \beta_{2} ) q^{93} + ( -1 - 9 \beta_{1} + \beta_{2} ) q^{94} + ( 2 \beta_{1} - 3 \beta_{2} ) q^{95} + ( -1 - 4 \beta_{1} + \beta_{2} ) q^{96} + ( -5 + 5 \beta_{1} - \beta_{2} ) q^{97} + ( -1 - \beta_{1} - \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + 3q^{3} + 2q^{4} + 2q^{5} + 3q^{8} + 3q^{9} + O(q^{10})$$ $$3q + 3q^{3} + 2q^{4} + 2q^{5} + 3q^{8} + 3q^{9} - 5q^{10} - 2q^{11} + 2q^{12} - 4q^{13} + 2q^{15} - 4q^{16} - 3q^{17} - 7q^{19} + 9q^{20} - 11q^{22} + q^{23} + 3q^{24} - q^{25} + 11q^{26} + 3q^{27} - 11q^{29} - 5q^{30} - 13q^{31} - 4q^{32} - 2q^{33} + 7q^{34} + 2q^{36} - 5q^{37} + 3q^{38} - 4q^{39} + 2q^{40} + q^{41} + 6q^{43} - 15q^{44} + 2q^{45} - 19q^{46} - 11q^{47} - 4q^{48} - 21q^{50} - 3q^{51} + 11q^{52} + 2q^{53} - 4q^{55} - 7q^{57} + 14q^{58} - 3q^{59} + 9q^{60} + q^{61} + 2q^{62} - 21q^{64} - 11q^{66} + 10q^{67} - 28q^{68} + q^{69} + 26q^{71} + 3q^{72} - 7q^{73} - 19q^{74} - q^{75} + 6q^{76} + 11q^{78} + 2q^{79} - 23q^{80} + 3q^{81} - 18q^{82} + 3q^{83} - 35q^{85} + 31q^{86} - 11q^{87} - 2q^{88} + 23q^{89} - 5q^{90} - 16q^{92} - 13q^{93} - 4q^{94} + 3q^{95} - 4q^{96} - 14q^{97} - 2q^{99} + O(q^{100})$$

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
 −1.86081 −0.254102 2.11491
−1.86081 1.00000 1.46260 3.32340 −1.86081 0 1.00000 1.00000 −6.18421
1.2 −0.254102 1.00000 −1.93543 −1.68133 −0.254102 0 1.00000 1.00000 0.427229
1.3 2.11491 1.00000 2.47283 0.357926 2.11491 0 1.00000 1.00000 0.756981
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$7$$ $$-1$$
$$59$$ $$1$$

## 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 8673.2.a.s 3
7.b odd 2 1 177.2.a.d 3
21.c even 2 1 531.2.a.d 3
28.d even 2 1 2832.2.a.t 3
35.c odd 2 1 4425.2.a.w 3
84.h odd 2 1 8496.2.a.bl 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.2.a.d 3 7.b odd 2 1
531.2.a.d 3 21.c even 2 1
2832.2.a.t 3 28.d even 2 1
4425.2.a.w 3 35.c odd 2 1
8496.2.a.bl 3 84.h odd 2 1
8673.2.a.s 3 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(8673))$$:

 $$T_{2}^{3} - 4 T_{2} - 1$$ $$T_{5}^{3} - 2 T_{5}^{2} - 5 T_{5} + 2$$ $$T_{11}^{3} + 2 T_{11}^{2} - 11 T_{11} + 4$$

## Hecke characteristic polynomials

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