Properties

 Label 5733.2.a.s 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: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ 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 $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

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

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
 −1.41421 1.41421
−1.41421 0 0 4.41421 0 0 2.82843 0 −6.24264
1.2 1.41421 0 0 1.58579 0 0 −2.82843 0 2.24264
 $$n$$: 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.s 2
3.b odd 2 1 637.2.a.g 2
7.b odd 2 1 819.2.a.h 2
21.c even 2 1 91.2.a.c 2
21.g even 6 2 637.2.e.f 4
21.h odd 6 2 637.2.e.g 4
39.d odd 2 1 8281.2.a.v 2
84.h odd 2 1 1456.2.a.q 2
105.g even 2 1 2275.2.a.j 2
168.e odd 2 1 5824.2.a.bk 2
168.i even 2 1 5824.2.a.bl 2
273.g even 2 1 1183.2.a.d 2
273.o odd 4 2 1183.2.c.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.a.c 2 21.c even 2 1
637.2.a.g 2 3.b odd 2 1
637.2.e.f 4 21.g even 6 2
637.2.e.g 4 21.h odd 6 2
819.2.a.h 2 7.b odd 2 1
1183.2.a.d 2 273.g even 2 1
1183.2.c.d 4 273.o odd 4 2
1456.2.a.q 2 84.h odd 2 1
2275.2.a.j 2 105.g even 2 1
5733.2.a.s 2 1.a even 1 1 trivial
5824.2.a.bk 2 168.e odd 2 1
5824.2.a.bl 2 168.i even 2 1
8281.2.a.v 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5733))$$:

 $$T_{2}^{2} - 2$$ $$T_{5}^{2} - 6 T_{5} + 7$$ $$T_{11}^{2} - 18$$ $$T_{17}^{2} - 2$$ $$T_{19}^{2} - 6 T_{19} - 9$$ $$T_{31}^{2} - 2 T_{31} - 17$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-2 + T^{2}$$
$3$ $$T^{2}$$
$5$ $$7 - 6 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$-18 + T^{2}$$
$13$ $$( -1 + T )^{2}$$
$17$ $$-2 + T^{2}$$
$19$ $$-9 - 6 T + T^{2}$$
$23$ $$1 - 6 T + T^{2}$$
$29$ $$1 + 6 T + T^{2}$$
$31$ $$-17 - 2 T + T^{2}$$
$37$ $$-14 + 4 T + T^{2}$$
$41$ $$28 - 12 T + T^{2}$$
$43$ $$( 5 + T )^{2}$$
$47$ $$7 - 6 T + T^{2}$$
$53$ $$1 - 6 T + T^{2}$$
$59$ $$4 - 12 T + T^{2}$$
$61$ $$( 6 + T )^{2}$$
$67$ $$-36 + 12 T + T^{2}$$
$71$ $$-14 - 12 T + T^{2}$$
$73$ $$7 - 10 T + T^{2}$$
$79$ $$-23 - 14 T + T^{2}$$
$83$ $$63 - 18 T + T^{2}$$
$89$ $$7 - 6 T + T^{2}$$
$97$ $$-161 - 2 T + T^{2}$$