Properties

 Label 5733.2.a.u 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 39) 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.

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

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
−0.414214 0 −1.82843 −2.82843 0 0 1.58579 0 1.17157
1.2 2.41421 0 3.82843 2.82843 0 0 4.41421 0 6.82843
 $$n$$: 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.u 2
3.b odd 2 1 1911.2.a.h 2
7.b odd 2 1 117.2.a.c 2
21.c even 2 1 39.2.a.b 2
28.d even 2 1 1872.2.a.w 2
35.c odd 2 1 2925.2.a.v 2
35.f even 4 2 2925.2.c.u 4
56.e even 2 1 7488.2.a.co 2
56.h odd 2 1 7488.2.a.cl 2
63.l odd 6 2 1053.2.e.e 4
63.o even 6 2 1053.2.e.m 4
84.h odd 2 1 624.2.a.k 2
91.b odd 2 1 1521.2.a.f 2
91.i even 4 2 1521.2.b.j 4
105.g even 2 1 975.2.a.l 2
105.k odd 4 2 975.2.c.h 4
168.e odd 2 1 2496.2.a.bi 2
168.i even 2 1 2496.2.a.bf 2
231.h odd 2 1 4719.2.a.p 2
273.g even 2 1 507.2.a.h 2
273.o odd 4 2 507.2.b.e 4
273.u even 6 2 507.2.e.d 4
273.bn even 6 2 507.2.e.h 4
273.ca odd 12 4 507.2.j.f 8
1092.d odd 2 1 8112.2.a.bm 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.a.b 2 21.c even 2 1
117.2.a.c 2 7.b odd 2 1
507.2.a.h 2 273.g even 2 1
507.2.b.e 4 273.o odd 4 2
507.2.e.d 4 273.u even 6 2
507.2.e.h 4 273.bn even 6 2
507.2.j.f 8 273.ca odd 12 4
624.2.a.k 2 84.h odd 2 1
975.2.a.l 2 105.g even 2 1
975.2.c.h 4 105.k odd 4 2
1053.2.e.e 4 63.l odd 6 2
1053.2.e.m 4 63.o even 6 2
1521.2.a.f 2 91.b odd 2 1
1521.2.b.j 4 91.i even 4 2
1872.2.a.w 2 28.d even 2 1
1911.2.a.h 2 3.b odd 2 1
2496.2.a.bf 2 168.i even 2 1
2496.2.a.bi 2 168.e odd 2 1
2925.2.a.v 2 35.c odd 2 1
2925.2.c.u 4 35.f even 4 2
4719.2.a.p 2 231.h odd 2 1
5733.2.a.u 2 1.a even 1 1 trivial
7488.2.a.cl 2 56.h odd 2 1
7488.2.a.co 2 56.e even 2 1
8112.2.a.bm 2 1092.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5733))$$:

 $$T_{2}^{2} - 2 T_{2} - 1$$ $$T_{5}^{2} - 8$$ $$T_{11} - 2$$ $$T_{17}^{2} - 4 T_{17} - 28$$ $$T_{19}^{2} - 8$$ $$T_{31}^{2} - 8 T_{31} + 8$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 - 2 T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$-8 + T^{2}$$
$7$ $$T^{2}$$
$11$ $$( -2 + T )^{2}$$
$13$ $$( -1 + T )^{2}$$
$17$ $$-28 - 4 T + T^{2}$$
$19$ $$-8 + T^{2}$$
$23$ $$( -4 + T )^{2}$$
$29$ $$( 2 + T )^{2}$$
$31$ $$8 - 8 T + T^{2}$$
$37$ $$-28 + 4 T + T^{2}$$
$41$ $$56 - 16 T + T^{2}$$
$43$ $$-16 - 8 T + T^{2}$$
$47$ $$4 + 12 T + T^{2}$$
$53$ $$( -2 + T )^{2}$$
$59$ $$-28 - 4 T + T^{2}$$
$61$ $$-124 + 4 T + T^{2}$$
$67$ $$8 - 8 T + T^{2}$$
$71$ $$( 2 + T )^{2}$$
$73$ $$4 + 12 T + T^{2}$$
$79$ $$-128 + T^{2}$$
$83$ $$-28 + 4 T + T^{2}$$
$89$ $$136 - 24 T + T^{2}$$
$97$ $$-28 - 4 T + T^{2}$$