# Properties

 Label 8673.2.a.k Level $8673$ Weight $2$ Character orbit 8673.a Self dual yes Analytic conductor $69.254$ Analytic rank $0$ Dimension $2$ 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: $$0$$ 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: 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.

 $$f(q)$$ $$=$$ $$q -\beta q^{2} + q^{3} + ( -1 + \beta ) q^{4} + ( 1 - 2 \beta ) q^{5} -\beta q^{6} + ( -1 + 2 \beta ) q^{8} + q^{9} +O(q^{10})$$ $$q -\beta q^{2} + q^{3} + ( -1 + \beta ) q^{4} + ( 1 - 2 \beta ) q^{5} -\beta q^{6} + ( -1 + 2 \beta ) q^{8} + q^{9} + ( 2 + \beta ) q^{10} + ( 1 - 2 \beta ) q^{11} + ( -1 + \beta ) q^{12} + ( 5 - 2 \beta ) q^{13} + ( 1 - 2 \beta ) q^{15} -3 \beta q^{16} + 3 \beta q^{17} -\beta q^{18} + 5 \beta q^{19} + ( -3 + \beta ) q^{20} + ( 2 + \beta ) q^{22} + ( -4 + \beta ) q^{23} + ( -1 + 2 \beta ) q^{24} + ( 2 - 3 \beta ) q^{26} + q^{27} + ( 7 + \beta ) q^{29} + ( 2 + \beta ) q^{30} + ( 5 - 9 \beta ) q^{31} + ( 5 - \beta ) q^{32} + ( 1 - 2 \beta ) q^{33} + ( -3 - 3 \beta ) q^{34} + ( -1 + \beta ) q^{36} + ( -2 - 3 \beta ) q^{37} + ( -5 - 5 \beta ) q^{38} + ( 5 - 2 \beta ) q^{39} -5 q^{40} + ( -5 + 5 \beta ) q^{41} + ( -5 + 6 \beta ) q^{43} + ( -3 + \beta ) q^{44} + ( 1 - 2 \beta ) q^{45} + ( -1 + 3 \beta ) q^{46} + ( 9 - 3 \beta ) q^{47} -3 \beta q^{48} + 3 \beta q^{51} + ( -7 + 5 \beta ) q^{52} + ( 5 - 2 \beta ) q^{53} -\beta q^{54} + 5 q^{55} + 5 \beta q^{57} + ( -1 - 8 \beta ) q^{58} + q^{59} + ( -3 + \beta ) q^{60} + ( 5 + 3 \beta ) q^{61} + ( 9 + 4 \beta ) q^{62} + ( 1 + 2 \beta ) q^{64} + ( 9 - 8 \beta ) q^{65} + ( 2 + \beta ) q^{66} + ( 7 - 6 \beta ) q^{67} + 3 q^{68} + ( -4 + \beta ) q^{69} + ( -3 + 8 \beta ) q^{71} + ( -1 + 2 \beta ) q^{72} + ( 1 + 3 \beta ) q^{73} + ( 3 + 5 \beta ) q^{74} + 5 q^{76} + ( 2 - 3 \beta ) q^{78} -3 q^{79} + ( 6 + 3 \beta ) q^{80} + q^{81} -5 q^{82} + ( 1 - \beta ) q^{83} + ( -6 - 3 \beta ) q^{85} + ( -6 - \beta ) q^{86} + ( 7 + \beta ) q^{87} -5 q^{88} + ( 7 - 11 \beta ) q^{89} + ( 2 + \beta ) q^{90} + ( 5 - 4 \beta ) q^{92} + ( 5 - 9 \beta ) q^{93} + ( 3 - 6 \beta ) q^{94} + ( -10 - 5 \beta ) q^{95} + ( 5 - \beta ) q^{96} -3 q^{97} + ( 1 - 2 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} + 2q^{3} - q^{4} - q^{6} + 2q^{9} + O(q^{10})$$ $$2q - q^{2} + 2q^{3} - q^{4} - q^{6} + 2q^{9} + 5q^{10} - q^{12} + 8q^{13} - 3q^{16} + 3q^{17} - q^{18} + 5q^{19} - 5q^{20} + 5q^{22} - 7q^{23} + q^{26} + 2q^{27} + 15q^{29} + 5q^{30} + q^{31} + 9q^{32} - 9q^{34} - q^{36} - 7q^{37} - 15q^{38} + 8q^{39} - 10q^{40} - 5q^{41} - 4q^{43} - 5q^{44} + q^{46} + 15q^{47} - 3q^{48} + 3q^{51} - 9q^{52} + 8q^{53} - q^{54} + 10q^{55} + 5q^{57} - 10q^{58} + 2q^{59} - 5q^{60} + 13q^{61} + 22q^{62} + 4q^{64} + 10q^{65} + 5q^{66} + 8q^{67} + 6q^{68} - 7q^{69} + 2q^{71} + 5q^{73} + 11q^{74} + 10q^{76} + q^{78} - 6q^{79} + 15q^{80} + 2q^{81} - 10q^{82} + q^{83} - 15q^{85} - 13q^{86} + 15q^{87} - 10q^{88} + 3q^{89} + 5q^{90} + 6q^{92} + q^{93} - 25q^{95} + 9q^{96} - 6q^{97} + O(q^{100})$$

## 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
 1.61803 −0.618034
−1.61803 1.00000 0.618034 −2.23607 −1.61803 0 2.23607 1.00000 3.61803
1.2 0.618034 1.00000 −1.61803 2.23607 0.618034 0 −2.23607 1.00000 1.38197
 $$n$$: 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.k 2
7.b odd 2 1 177.2.a.b 2
21.c even 2 1 531.2.a.b 2
28.d even 2 1 2832.2.a.o 2
35.c odd 2 1 4425.2.a.t 2
84.h odd 2 1 8496.2.a.bb 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.2.a.b 2 7.b odd 2 1
531.2.a.b 2 21.c even 2 1
2832.2.a.o 2 28.d even 2 1
4425.2.a.t 2 35.c odd 2 1
8496.2.a.bb 2 84.h odd 2 1
8673.2.a.k 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8673))$$:

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

## Hecke characteristic polynomials

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