# Properties

 Label 5733.2.a.w Level $5733$ Weight $2$ Character orbit 5733.a Self dual yes Analytic conductor $45.778$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5733 = 3^{2} \cdot 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5733.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$45.7782354788$$ 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 91) 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.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta ) q^{2} + 3 \beta q^{4} + ( -1 + 2 \beta ) q^{5} + ( 1 + 4 \beta ) q^{8} +O(q^{10})$$ $$q + ( 1 + \beta ) q^{2} + 3 \beta q^{4} + ( -1 + 2 \beta ) q^{5} + ( 1 + 4 \beta ) q^{8} + ( 1 + 3 \beta ) q^{10} + 3 q^{11} + q^{13} + ( 5 + 3 \beta ) q^{16} + ( -5 + 4 \beta ) q^{17} -3 q^{19} + ( 6 + 3 \beta ) q^{20} + ( 3 + 3 \beta ) q^{22} + ( 5 + 2 \beta ) q^{23} + ( 1 + \beta ) q^{26} + ( 2 - 4 \beta ) q^{29} -5 q^{31} + ( 6 + 3 \beta ) q^{32} + ( -1 + 3 \beta ) q^{34} + ( -5 + 6 \beta ) q^{37} + ( -3 - 3 \beta ) q^{38} + ( 7 + 6 \beta ) q^{40} + ( 2 - 4 \beta ) q^{41} -8 q^{43} + 9 \beta q^{44} + ( 7 + 9 \beta ) q^{46} + ( -1 - 4 \beta ) q^{47} + 3 \beta q^{52} + ( 1 + 4 \beta ) q^{53} + ( -3 + 6 \beta ) q^{55} + ( -2 - 6 \beta ) q^{58} + ( 5 - 4 \beta ) q^{59} -3 q^{61} + ( -5 - 5 \beta ) q^{62} + ( -1 + 6 \beta ) q^{64} + ( -1 + 2 \beta ) q^{65} -3 q^{67} + ( 12 - 3 \beta ) q^{68} + ( -4 + 8 \beta ) q^{71} + ( -7 + 6 \beta ) q^{73} + ( 1 + 7 \beta ) q^{74} -9 \beta q^{76} + ( 7 - 6 \beta ) q^{79} + ( 1 + 13 \beta ) q^{80} + ( -2 - 6 \beta ) q^{82} + ( 13 - 6 \beta ) q^{85} + ( -8 - 8 \beta ) q^{86} + ( 3 + 12 \beta ) q^{88} + ( -1 + 2 \beta ) q^{89} + ( 6 + 21 \beta ) q^{92} + ( -5 - 9 \beta ) q^{94} + ( 3 - 6 \beta ) q^{95} + ( 10 - 12 \beta ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 3q^{2} + 3q^{4} + 6q^{8} + O(q^{10})$$ $$2q + 3q^{2} + 3q^{4} + 6q^{8} + 5q^{10} + 6q^{11} + 2q^{13} + 13q^{16} - 6q^{17} - 6q^{19} + 15q^{20} + 9q^{22} + 12q^{23} + 3q^{26} - 10q^{31} + 15q^{32} + q^{34} - 4q^{37} - 9q^{38} + 20q^{40} - 16q^{43} + 9q^{44} + 23q^{46} - 6q^{47} + 3q^{52} + 6q^{53} - 10q^{58} + 6q^{59} - 6q^{61} - 15q^{62} + 4q^{64} - 6q^{67} + 21q^{68} - 8q^{73} + 9q^{74} - 9q^{76} + 8q^{79} + 15q^{80} - 10q^{82} + 20q^{85} - 24q^{86} + 18q^{88} + 33q^{92} - 19q^{94} + 8q^{97} + O(q^{100})$$

## 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
 −0.618034 1.61803
0.381966 0 −1.85410 −2.23607 0 0 −1.47214 0 −0.854102
1.2 2.61803 0 4.85410 2.23607 0 0 7.47214 0 5.85410
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$7$$ $$-1$$
$$13$$ $$-1$$

## 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 5733.2.a.w 2
3.b odd 2 1 637.2.a.e 2
7.b odd 2 1 5733.2.a.v 2
7.d odd 6 2 819.2.j.c 4
21.c even 2 1 637.2.a.f 2
21.g even 6 2 91.2.e.b 4
21.h odd 6 2 637.2.e.h 4
39.d odd 2 1 8281.2.a.ba 2
84.j odd 6 2 1456.2.r.j 4
273.g even 2 1 8281.2.a.z 2
273.ba even 6 2 1183.2.e.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.e.b 4 21.g even 6 2
637.2.a.e 2 3.b odd 2 1
637.2.a.f 2 21.c even 2 1
637.2.e.h 4 21.h odd 6 2
819.2.j.c 4 7.d odd 6 2
1183.2.e.d 4 273.ba even 6 2
1456.2.r.j 4 84.j odd 6 2
5733.2.a.v 2 7.b odd 2 1
5733.2.a.w 2 1.a even 1 1 trivial
8281.2.a.z 2 273.g even 2 1
8281.2.a.ba 2 39.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5733))$$:

 $$T_{2}^{2} - 3 T_{2} + 1$$ $$T_{5}^{2} - 5$$ $$T_{11} - 3$$ $$T_{17}^{2} + 6 T_{17} - 11$$ $$T_{19} + 3$$ $$T_{31} + 5$$

## Hecke characteristic polynomials

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