Properties

 Label 693.2.a.h Level $693$ Weight $2$ Character orbit 693.a Self dual yes Analytic conductor $5.534$ Analytic rank $0$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 77) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.

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

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
−2.23607 0 3.00000 2.00000 0 1.00000 −2.23607 0 −4.47214
1.2 2.23607 0 3.00000 2.00000 0 1.00000 2.23607 0 4.47214
 $$n$$: 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.h 2
3.b odd 2 1 77.2.a.d 2
7.b odd 2 1 4851.2.a.y 2
11.b odd 2 1 7623.2.a.bl 2
12.b even 2 1 1232.2.a.m 2
15.d odd 2 1 1925.2.a.r 2
15.e even 4 2 1925.2.b.h 4
21.c even 2 1 539.2.a.f 2
21.g even 6 2 539.2.e.j 4
21.h odd 6 2 539.2.e.i 4
24.f even 2 1 4928.2.a.bv 2
24.h odd 2 1 4928.2.a.bm 2
33.d even 2 1 847.2.a.f 2
33.f even 10 2 847.2.f.b 4
33.f even 10 2 847.2.f.m 4
33.h odd 10 2 847.2.f.a 4
33.h odd 10 2 847.2.f.n 4
84.h odd 2 1 8624.2.a.ce 2
231.h odd 2 1 5929.2.a.m 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.a.d 2 3.b odd 2 1
539.2.a.f 2 21.c even 2 1
539.2.e.i 4 21.h odd 6 2
539.2.e.j 4 21.g even 6 2
693.2.a.h 2 1.a even 1 1 trivial
847.2.a.f 2 33.d even 2 1
847.2.f.a 4 33.h odd 10 2
847.2.f.b 4 33.f even 10 2
847.2.f.m 4 33.f even 10 2
847.2.f.n 4 33.h odd 10 2
1232.2.a.m 2 12.b even 2 1
1925.2.a.r 2 15.d odd 2 1
1925.2.b.h 4 15.e even 4 2
4851.2.a.y 2 7.b odd 2 1
4928.2.a.bm 2 24.h odd 2 1
4928.2.a.bv 2 24.f even 2 1
5929.2.a.m 2 231.h odd 2 1
7623.2.a.bl 2 11.b odd 2 1
8624.2.a.ce 2 84.h odd 2 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-5 + T^{2}$$
$3$ $$T^{2}$$
$5$ $$( -2 + T )^{2}$$
$7$ $$( -1 + T )^{2}$$
$11$ $$( -1 + T )^{2}$$
$13$ $$-4 - 2 T + T^{2}$$
$17$ $$-4 - 2 T + T^{2}$$
$19$ $$-16 - 4 T + T^{2}$$
$23$ $$-16 - 4 T + T^{2}$$
$29$ $$-4 + 8 T + T^{2}$$
$31$ $$20 + 10 T + T^{2}$$
$37$ $$-4 + 8 T + T^{2}$$
$41$ $$76 - 18 T + T^{2}$$
$43$ $$( -8 + T )^{2}$$
$47$ $$20 + 10 T + T^{2}$$
$53$ $$-4 + 8 T + T^{2}$$
$59$ $$-4 + 2 T + T^{2}$$
$61$ $$20 + 10 T + T^{2}$$
$67$ $$80 - 20 T + T^{2}$$
$71$ $$16 - 12 T + T^{2}$$
$73$ $$4 + 6 T + T^{2}$$
$79$ $$-80 + T^{2}$$
$83$ $$-176 + 4 T + T^{2}$$
$89$ $$( 2 + T )^{2}$$
$97$ $$-164 - 8 T + T^{2}$$