# Properties

 Label 693.2.a.f Level $693$ Weight $2$ Character orbit 693.a Self dual yes Analytic conductor $5.534$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$693 = 3^{2} \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 693.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$5.53363286007$$ 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 231) 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.

 $$f(q)$$ $$=$$ $$q -\beta q^{2} + ( -1 + \beta ) q^{4} - q^{5} + q^{7} + ( -1 + 2 \beta ) q^{8} +O(q^{10})$$ $$q -\beta q^{2} + ( -1 + \beta ) q^{4} - q^{5} + q^{7} + ( -1 + 2 \beta ) q^{8} + \beta q^{10} - q^{11} + ( 1 - 4 \beta ) q^{13} -\beta q^{14} -3 \beta q^{16} + ( -4 + 2 \beta ) q^{17} + ( -3 + 6 \beta ) q^{19} + ( 1 - \beta ) q^{20} + \beta q^{22} + ( -2 + 6 \beta ) q^{23} -4 q^{25} + ( 4 + 3 \beta ) q^{26} + ( -1 + \beta ) q^{28} -5 q^{29} + ( -4 + 2 \beta ) q^{31} + ( 5 - \beta ) q^{32} + ( -2 + 2 \beta ) q^{34} - q^{35} -7 q^{37} + ( -6 - 3 \beta ) q^{38} + ( 1 - 2 \beta ) q^{40} -4 \beta q^{41} + ( 2 - 6 \beta ) q^{43} + ( 1 - \beta ) q^{44} + ( -6 - 4 \beta ) q^{46} + ( 1 + 2 \beta ) q^{47} + q^{49} + 4 \beta q^{50} + ( -5 + \beta ) q^{52} + ( 6 - 10 \beta ) q^{53} + q^{55} + ( -1 + 2 \beta ) q^{56} + 5 \beta q^{58} + ( 5 - 10 \beta ) q^{59} + 2 q^{61} + ( -2 + 2 \beta ) q^{62} + ( 1 + 2 \beta ) q^{64} + ( -1 + 4 \beta ) q^{65} + ( -11 - 2 \beta ) q^{67} + ( 6 - 4 \beta ) q^{68} + \beta q^{70} -4 \beta q^{71} + ( 7 + 4 \beta ) q^{73} + 7 \beta q^{74} + ( 9 - 3 \beta ) q^{76} - q^{77} + ( -12 + 4 \beta ) q^{79} + 3 \beta q^{80} + ( 4 + 4 \beta ) q^{82} + ( -8 - 2 \beta ) q^{83} + ( 4 - 2 \beta ) q^{85} + ( 6 + 4 \beta ) q^{86} + ( 1 - 2 \beta ) q^{88} + ( 2 - 4 \beta ) q^{89} + ( 1 - 4 \beta ) q^{91} + ( 8 - 2 \beta ) q^{92} + ( -2 - 3 \beta ) q^{94} + ( 3 - 6 \beta ) q^{95} + ( 6 - 6 \beta ) q^{97} -\beta q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - q^{4} - 2 q^{5} + 2 q^{7} + O(q^{10})$$ $$2 q - q^{2} - q^{4} - 2 q^{5} + 2 q^{7} + q^{10} - 2 q^{11} - 2 q^{13} - q^{14} - 3 q^{16} - 6 q^{17} + q^{20} + q^{22} + 2 q^{23} - 8 q^{25} + 11 q^{26} - q^{28} - 10 q^{29} - 6 q^{31} + 9 q^{32} - 2 q^{34} - 2 q^{35} - 14 q^{37} - 15 q^{38} - 4 q^{41} - 2 q^{43} + q^{44} - 16 q^{46} + 4 q^{47} + 2 q^{49} + 4 q^{50} - 9 q^{52} + 2 q^{53} + 2 q^{55} + 5 q^{58} + 4 q^{61} - 2 q^{62} + 4 q^{64} + 2 q^{65} - 24 q^{67} + 8 q^{68} + q^{70} - 4 q^{71} + 18 q^{73} + 7 q^{74} + 15 q^{76} - 2 q^{77} - 20 q^{79} + 3 q^{80} + 12 q^{82} - 18 q^{83} + 6 q^{85} + 16 q^{86} - 2 q^{91} + 14 q^{92} - 7 q^{94} + 6 q^{97} - q^{98} + O(q^{100})$$

## 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
 1.61803 −0.618034
−1.61803 0 0.618034 −1.00000 0 1.00000 2.23607 0 1.61803
1.2 0.618034 0 −1.61803 −1.00000 0 1.00000 −2.23607 0 −0.618034
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$7$$ $$-1$$
$$11$$ $$1$$

## 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 693.2.a.f 2
3.b odd 2 1 231.2.a.c 2
7.b odd 2 1 4851.2.a.w 2
11.b odd 2 1 7623.2.a.bm 2
12.b even 2 1 3696.2.a.be 2
15.d odd 2 1 5775.2.a.be 2
21.c even 2 1 1617.2.a.p 2
33.d even 2 1 2541.2.a.t 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.a.c 2 3.b odd 2 1
693.2.a.f 2 1.a even 1 1 trivial
1617.2.a.p 2 21.c even 2 1
2541.2.a.t 2 33.d even 2 1
3696.2.a.be 2 12.b even 2 1
4851.2.a.w 2 7.b odd 2 1
5775.2.a.be 2 15.d odd 2 1
7623.2.a.bm 2 11.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(693))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$( 1 + T )^{2}$$
$7$ $$( -1 + T )^{2}$$
$11$ $$( 1 + T )^{2}$$
$13$ $$-19 + 2 T + T^{2}$$
$17$ $$4 + 6 T + T^{2}$$
$19$ $$-45 + T^{2}$$
$23$ $$-44 - 2 T + T^{2}$$
$29$ $$( 5 + T )^{2}$$
$31$ $$4 + 6 T + T^{2}$$
$37$ $$( 7 + T )^{2}$$
$41$ $$-16 + 4 T + T^{2}$$
$43$ $$-44 + 2 T + T^{2}$$
$47$ $$-1 - 4 T + T^{2}$$
$53$ $$-124 - 2 T + T^{2}$$
$59$ $$-125 + T^{2}$$
$61$ $$( -2 + T )^{2}$$
$67$ $$139 + 24 T + T^{2}$$
$71$ $$-16 + 4 T + T^{2}$$
$73$ $$61 - 18 T + T^{2}$$
$79$ $$80 + 20 T + T^{2}$$
$83$ $$76 + 18 T + T^{2}$$
$89$ $$-20 + T^{2}$$
$97$ $$-36 - 6 T + T^{2}$$