Properties

 Label 5733.2.a.x Level $5733$ Weight $2$ Character orbit 5733.a Self dual yes Analytic conductor $45.778$ Analytic rank $1$ Dimension $3$ 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: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.316.1 Defining polynomial: $$x^{3} - x^{2} - 4 x + 2$$ 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + ( 1 - \beta_{1} ) q^{5} + ( -1 - \beta_{2} ) q^{8} +O(q^{10})$$ $$q -\beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + ( 1 - \beta_{1} ) q^{5} + ( -1 - \beta_{2} ) q^{8} + ( 3 - \beta_{1} + \beta_{2} ) q^{10} + ( -1 + \beta_{1} - \beta_{2} ) q^{11} - q^{13} + ( -1 + 2 \beta_{1} - \beta_{2} ) q^{16} + ( 1 + \beta_{1} + \beta_{2} ) q^{17} + ( 1 + \beta_{1} ) q^{19} -2 \beta_{1} q^{20} + ( -2 + 2 \beta_{1} ) q^{22} + ( -4 + 2 \beta_{1} + \beta_{2} ) q^{23} + ( -1 - 2 \beta_{1} + \beta_{2} ) q^{25} + \beta_{1} q^{26} + ( -8 - \beta_{2} ) q^{29} + ( 1 + \beta_{1} - 2 \beta_{2} ) q^{31} + ( -3 + 2 \beta_{1} + \beta_{2} ) q^{32} + ( -4 - 2 \beta_{1} - 2 \beta_{2} ) q^{34} + ( -1 + 3 \beta_{1} + \beta_{2} ) q^{37} + ( -3 - \beta_{1} - \beta_{2} ) q^{38} + 2 \beta_{1} q^{40} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{41} + ( 4 - 2 \beta_{1} - 3 \beta_{2} ) q^{43} -4 q^{44} + ( -7 + 3 \beta_{1} - 3 \beta_{2} ) q^{46} + ( -3 + \beta_{1} - 4 \beta_{2} ) q^{47} + ( 5 + \beta_{2} ) q^{50} + ( -1 - \beta_{2} ) q^{52} + ( -2 - 2 \beta_{1} + 3 \beta_{2} ) q^{53} + ( -3 + 3 \beta_{1} - \beta_{2} ) q^{55} + ( 1 + 9 \beta_{1} + \beta_{2} ) q^{58} + ( -2 + 2 \beta_{1} + 4 \beta_{2} ) q^{59} + 2 q^{61} + ( -1 + \beta_{1} + \beta_{2} ) q^{62} + ( -5 - 2 \beta_{1} - \beta_{2} ) q^{64} + ( -1 + \beta_{1} ) q^{65} + ( -2 - 6 \beta_{1} + 4 \beta_{2} ) q^{67} + ( 6 + 4 \beta_{1} + 2 \beta_{2} ) q^{68} + ( 3 - 3 \beta_{1} + \beta_{2} ) q^{71} + ( 3 + \beta_{1} + 4 \beta_{2} ) q^{73} + ( -10 - 4 \beta_{2} ) q^{74} + ( 2 + 2 \beta_{1} + 2 \beta_{2} ) q^{76} + ( -6 + 4 \beta_{1} - \beta_{2} ) q^{79} + ( -6 + 4 \beta_{1} - 2 \beta_{2} ) q^{80} + ( -4 + 2 \beta_{1} ) q^{82} + ( -1 - 9 \beta_{1} + 4 \beta_{2} ) q^{83} + ( -3 - \beta_{1} - \beta_{2} ) q^{85} + ( 9 - \beta_{1} + 5 \beta_{2} ) q^{86} + 4 q^{88} + ( -1 + 5 \beta_{1} - 2 \beta_{2} ) q^{89} + ( 2 + 6 \beta_{1} - 2 \beta_{2} ) q^{92} + ( 1 + 7 \beta_{1} + 3 \beta_{2} ) q^{94} + ( -2 - \beta_{2} ) q^{95} + ( 3 + \beta_{1} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - q^{2} + 3q^{4} + 2q^{5} - 3q^{8} + O(q^{10})$$ $$3q - q^{2} + 3q^{4} + 2q^{5} - 3q^{8} + 8q^{10} - 2q^{11} - 3q^{13} - q^{16} + 4q^{17} + 4q^{19} - 2q^{20} - 4q^{22} - 10q^{23} - 5q^{25} + q^{26} - 24q^{29} + 4q^{31} - 7q^{32} - 14q^{34} - 10q^{38} + 2q^{40} + 2q^{41} + 10q^{43} - 12q^{44} - 18q^{46} - 8q^{47} + 15q^{50} - 3q^{52} - 8q^{53} - 6q^{55} + 12q^{58} - 4q^{59} + 6q^{61} - 2q^{62} - 17q^{64} - 2q^{65} - 12q^{67} + 22q^{68} + 6q^{71} + 10q^{73} - 30q^{74} + 8q^{76} - 14q^{79} - 14q^{80} - 10q^{82} - 12q^{83} - 10q^{85} + 26q^{86} + 12q^{88} + 2q^{89} + 12q^{92} + 10q^{94} - 6q^{95} + 10q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 4 x + 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$

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
 2.34292 0.470683 −1.81361
−2.34292 0 3.48929 −1.34292 0 0 −3.48929 0 3.14637
1.2 −0.470683 0 −1.77846 0.529317 0 0 1.77846 0 −0.249141
1.3 1.81361 0 1.28917 2.81361 0 0 −1.28917 0 5.10278
 $$n$$: 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.x 3
3.b odd 2 1 637.2.a.j 3
7.b odd 2 1 819.2.a.i 3
21.c even 2 1 91.2.a.d 3
21.g even 6 2 637.2.e.j 6
21.h odd 6 2 637.2.e.i 6
39.d odd 2 1 8281.2.a.bg 3
84.h odd 2 1 1456.2.a.t 3
105.g even 2 1 2275.2.a.m 3
168.e odd 2 1 5824.2.a.bs 3
168.i even 2 1 5824.2.a.by 3
273.g even 2 1 1183.2.a.i 3
273.o odd 4 2 1183.2.c.f 6

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

 $$T_{2}^{3} + T_{2}^{2} - 4 T_{2} - 2$$ $$T_{5}^{3} - 2 T_{5}^{2} - 3 T_{5} + 2$$ $$T_{11}^{3} + 2 T_{11}^{2} - 6 T_{11} - 8$$ $$T_{17}^{3} - 4 T_{17}^{2} - 10 T_{17} - 4$$ $$T_{19}^{3} - 4 T_{19}^{2} + T_{19} + 4$$ $$T_{31}^{3} - 4 T_{31}^{2} - 19 T_{31} - 16$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-2 - 4 T + T^{2} + T^{3}$$
$3$ $$T^{3}$$
$5$ $$2 - 3 T - 2 T^{2} + T^{3}$$
$7$ $$T^{3}$$
$11$ $$-8 - 6 T + 2 T^{2} + T^{3}$$
$13$ $$( 1 + T )^{3}$$
$17$ $$-4 - 10 T - 4 T^{2} + T^{3}$$
$19$ $$4 + T - 4 T^{2} + T^{3}$$
$23$ $$-136 + T + 10 T^{2} + T^{3}$$
$29$ $$454 + 185 T + 24 T^{2} + T^{3}$$
$31$ $$-16 - 19 T - 4 T^{2} + T^{3}$$
$37$ $$-124 - 58 T + T^{3}$$
$41$ $$-8 - 28 T - 2 T^{2} + T^{3}$$
$43$ $$628 - 71 T - 10 T^{2} + T^{3}$$
$47$ $$-544 - 79 T + 8 T^{2} + T^{3}$$
$53$ $$22 - 35 T + 8 T^{2} + T^{3}$$
$59$ $$-688 - 156 T + 4 T^{2} + T^{3}$$
$61$ $$( -2 + T )^{3}$$
$67$ $$-976 - 124 T + 12 T^{2} + T^{3}$$
$71$ $$-16 - 22 T - 6 T^{2} + T^{3}$$
$73$ $$274 - 99 T - 10 T^{2} + T^{3}$$
$79$ $$-16 + 5 T + 14 T^{2} + T^{3}$$
$83$ $$-3268 - 271 T + 12 T^{2} + T^{3}$$
$89$ $$422 - 95 T - 2 T^{2} + T^{3}$$
$97$ $$-22 + 29 T - 10 T^{2} + T^{3}$$