# 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

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5733,2,Mod(1,5733)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5733, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("5733.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$45.7782354788$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ 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

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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.

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

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5733))$$:

 $$T_{2}^{2} - 3T_{2} + 1$$ T2^2 - 3*T2 + 1 $$T_{5}^{2} - 5$$ T5^2 - 5 $$T_{11} - 3$$ T11 - 3 $$T_{17}^{2} + 6T_{17} - 11$$ T17^2 + 6*T17 - 11 $$T_{19} + 3$$ T19 + 3 $$T_{31} + 5$$ T31 + 5

## Hecke characteristic polynomials

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