# Properties

 Label 5733.2.a.be Level $5733$ Weight $2$ Character orbit 5733.a Self dual yes Analytic conductor $45.778$ Analytic rank $0$ Dimension $3$ 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: $$3$$ Coefficient field: 3.3.404.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 5x - 1$$ x^3 - x^2 - 5*x - 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 637) 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 + 1) q^{2} + (\beta_{2} - \beta_1 + 2) q^{4} + ( - \beta_1 + 2) q^{5} + (2 \beta_{2} - 2 \beta_1 + 2) q^{8}+O(q^{10})$$ q + (-b1 + 1) * q^2 + (b2 - b1 + 2) * q^4 + (-b1 + 2) * q^5 + (2*b2 - 2*b1 + 2) * q^8 $$q + ( - \beta_1 + 1) q^{2} + (\beta_{2} - \beta_1 + 2) q^{4} + ( - \beta_1 + 2) q^{5} + (2 \beta_{2} - 2 \beta_1 + 2) q^{8} + (\beta_{2} - 2 \beta_1 + 5) q^{10} + (\beta_1 + 1) q^{11} + q^{13} + (2 \beta_{2} - 4 \beta_1 + 2) q^{16} + (\beta_{2} + 1) q^{17} + (\beta_{2} - 2 \beta_1 - 2) q^{19} + (3 \beta_{2} - 5 \beta_1 + 6) q^{20} + ( - \beta_{2} - \beta_1 - 2) q^{22} + ( - \beta_{2} + 3 \beta_1 - 1) q^{23} + (\beta_{2} - 3 \beta_1 + 2) q^{25} + ( - \beta_1 + 1) q^{26} + ( - 2 \beta_{2} + 3) q^{29} + (2 \beta_{2} + \beta_1) q^{31} + (2 \beta_{2} - 2 \beta_1 + 8) q^{32} + (\beta_{2} - 3 \beta_1) q^{34} + ( - 2 \beta_{2} + \beta_1 - 3) q^{37} + (3 \beta_{2} + 3) q^{38} + (6 \beta_{2} - 8 \beta_1 + 8) q^{40} + ( - 4 \beta_{2} + 4 \beta_1 + 2) q^{41} + ( - 2 \beta_{2} + 2 \beta_1 + 3) q^{43} + 2 \beta_1 q^{44} + ( - 4 \beta_{2} + 3 \beta_1 - 9) q^{46} + ( - \beta_{2} + 6) q^{47} + (4 \beta_{2} - 4 \beta_1 + 10) q^{50} + (\beta_{2} - \beta_1 + 2) q^{52} + (\beta_{2} + \beta_1 - 5) q^{53} + ( - \beta_{2} - 1) q^{55} + ( - 2 \beta_{2} + \beta_1 + 5) q^{58} + ( - \beta_{2} - \beta_1 + 8) q^{59} + ( - 2 \beta_{2} + 2 \beta_1 + 8) q^{61} + (\beta_{2} - 4 \beta_1 - 5) q^{62} + ( - 4 \beta_1 + 8) q^{64} + ( - \beta_1 + 2) q^{65} + ( - 4 \beta_{2} + 2 \beta_1 - 4) q^{67} + (2 \beta_{2} - 2 \beta_1 + 6) q^{68} + (2 \beta_{2} - 3 \beta_1 - 1) q^{71} + (\beta_{2} + 6 \beta_1 - 4) q^{73} + ( - 3 \beta_{2} + 7 \beta_1 - 4) q^{74} + (\beta_{2} - 5 \beta_1 + 4) q^{76} + (2 \beta_{2} + 2 \beta_1 - 1) q^{79} + (8 \beta_{2} - 10 \beta_1 + 14) q^{80} + ( - 8 \beta_{2} + 6 \beta_1 - 6) q^{82} + (2 \beta_{2} - 3 \beta_1 + 8) q^{83} + (2 \beta_{2} - 3 \beta_1 + 1) q^{85} + ( - 4 \beta_{2} + \beta_1 - 1) q^{86} + (2 \beta_1 - 2) q^{88} + ( - 3 \beta_{2} + 4 \beta_1 - 4) q^{89} + ( - 5 \beta_{2} + 11 \beta_1 - 12) q^{92} + ( - \beta_{2} - 4 \beta_1 + 7) q^{94} + (4 \beta_{2} - 2 \beta_1 + 1) q^{95} + (\beta_{2} - 4 \beta_1 + 2) q^{97}+O(q^{100})$$ q + (-b1 + 1) * q^2 + (b2 - b1 + 2) * q^4 + (-b1 + 2) * q^5 + (2*b2 - 2*b1 + 2) * q^8 + (b2 - 2*b1 + 5) * q^10 + (b1 + 1) * q^11 + q^13 + (2*b2 - 4*b1 + 2) * q^16 + (b2 + 1) * q^17 + (b2 - 2*b1 - 2) * q^19 + (3*b2 - 5*b1 + 6) * q^20 + (-b2 - b1 - 2) * q^22 + (-b2 + 3*b1 - 1) * q^23 + (b2 - 3*b1 + 2) * q^25 + (-b1 + 1) * q^26 + (-2*b2 + 3) * q^29 + (2*b2 + b1) * q^31 + (2*b2 - 2*b1 + 8) * q^32 + (b2 - 3*b1) * q^34 + (-2*b2 + b1 - 3) * q^37 + (3*b2 + 3) * q^38 + (6*b2 - 8*b1 + 8) * q^40 + (-4*b2 + 4*b1 + 2) * q^41 + (-2*b2 + 2*b1 + 3) * q^43 + 2*b1 * q^44 + (-4*b2 + 3*b1 - 9) * q^46 + (-b2 + 6) * q^47 + (4*b2 - 4*b1 + 10) * q^50 + (b2 - b1 + 2) * q^52 + (b2 + b1 - 5) * q^53 + (-b2 - 1) * q^55 + (-2*b2 + b1 + 5) * q^58 + (-b2 - b1 + 8) * q^59 + (-2*b2 + 2*b1 + 8) * q^61 + (b2 - 4*b1 - 5) * q^62 + (-4*b1 + 8) * q^64 + (-b1 + 2) * q^65 + (-4*b2 + 2*b1 - 4) * q^67 + (2*b2 - 2*b1 + 6) * q^68 + (2*b2 - 3*b1 - 1) * q^71 + (b2 + 6*b1 - 4) * q^73 + (-3*b2 + 7*b1 - 4) * q^74 + (b2 - 5*b1 + 4) * q^76 + (2*b2 + 2*b1 - 1) * q^79 + (8*b2 - 10*b1 + 14) * q^80 + (-8*b2 + 6*b1 - 6) * q^82 + (2*b2 - 3*b1 + 8) * q^83 + (2*b2 - 3*b1 + 1) * q^85 + (-4*b2 + b1 - 1) * q^86 + (2*b1 - 2) * q^88 + (-3*b2 + 4*b1 - 4) * q^89 + (-5*b2 + 11*b1 - 12) * q^92 + (-b2 - 4*b1 + 7) * q^94 + (4*b2 - 2*b1 + 1) * q^95 + (b2 - 4*b1 + 2) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 2 q^{2} + 6 q^{4} + 5 q^{5} + 6 q^{8}+O(q^{10})$$ 3 * q + 2 * q^2 + 6 * q^4 + 5 * q^5 + 6 * q^8 $$3 q + 2 q^{2} + 6 q^{4} + 5 q^{5} + 6 q^{8} + 14 q^{10} + 4 q^{11} + 3 q^{13} + 4 q^{16} + 4 q^{17} - 7 q^{19} + 16 q^{20} - 8 q^{22} - q^{23} + 4 q^{25} + 2 q^{26} + 7 q^{29} + 3 q^{31} + 24 q^{32} - 2 q^{34} - 10 q^{37} + 12 q^{38} + 22 q^{40} + 6 q^{41} + 9 q^{43} + 2 q^{44} - 28 q^{46} + 17 q^{47} + 30 q^{50} + 6 q^{52} - 13 q^{53} - 4 q^{55} + 14 q^{58} + 22 q^{59} + 24 q^{61} - 18 q^{62} + 20 q^{64} + 5 q^{65} - 14 q^{67} + 18 q^{68} - 4 q^{71} - 5 q^{73} - 8 q^{74} + 8 q^{76} + q^{79} + 40 q^{80} - 20 q^{82} + 23 q^{83} + 2 q^{85} - 6 q^{86} - 4 q^{88} - 11 q^{89} - 30 q^{92} + 16 q^{94} + 5 q^{95} + 3 q^{97}+O(q^{100})$$ 3 * q + 2 * q^2 + 6 * q^4 + 5 * q^5 + 6 * q^8 + 14 * q^10 + 4 * q^11 + 3 * q^13 + 4 * q^16 + 4 * q^17 - 7 * q^19 + 16 * q^20 - 8 * q^22 - q^23 + 4 * q^25 + 2 * q^26 + 7 * q^29 + 3 * q^31 + 24 * q^32 - 2 * q^34 - 10 * q^37 + 12 * q^38 + 22 * q^40 + 6 * q^41 + 9 * q^43 + 2 * q^44 - 28 * q^46 + 17 * q^47 + 30 * q^50 + 6 * q^52 - 13 * q^53 - 4 * q^55 + 14 * q^58 + 22 * q^59 + 24 * q^61 - 18 * q^62 + 20 * q^64 + 5 * q^65 - 14 * q^67 + 18 * q^68 - 4 * q^71 - 5 * q^73 - 8 * q^74 + 8 * q^76 + q^79 + 40 * q^80 - 20 * q^82 + 23 * q^83 + 2 * q^85 - 6 * q^86 - 4 * q^88 - 11 * q^89 - 30 * q^92 + 16 * q^94 + 5 * q^95 + 3 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 5x - 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$ v^2 - v - 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 3$$ b2 + b1 + 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.

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
 2.86620 −0.210756 −1.65544
−1.86620 0 1.48270 −0.866198 0 0 0.965392 0 1.61650
1.2 1.21076 0 −0.534070 2.21076 0 0 −3.06814 0 2.67669
1.3 2.65544 0 5.05137 3.65544 0 0 8.10275 0 9.70682
 $$n$$: 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.be 3
3.b odd 2 1 637.2.a.h 3
7.b odd 2 1 5733.2.a.bd 3
21.c even 2 1 637.2.a.i yes 3
21.g even 6 2 637.2.e.k 6
21.h odd 6 2 637.2.e.l 6
39.d odd 2 1 8281.2.a.bh 3
273.g even 2 1 8281.2.a.bk 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.a.h 3 3.b odd 2 1
637.2.a.i yes 3 21.c even 2 1
637.2.e.k 6 21.g even 6 2
637.2.e.l 6 21.h odd 6 2
5733.2.a.bd 3 7.b odd 2 1
5733.2.a.be 3 1.a even 1 1 trivial
8281.2.a.bh 3 39.d odd 2 1
8281.2.a.bk 3 273.g even 2 1

## Hecke kernels

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

 $$T_{2}^{3} - 2T_{2}^{2} - 4T_{2} + 6$$ T2^3 - 2*T2^2 - 4*T2 + 6 $$T_{5}^{3} - 5T_{5}^{2} + 3T_{5} + 7$$ T5^3 - 5*T5^2 + 3*T5 + 7 $$T_{11}^{3} - 4T_{11}^{2} + 2$$ T11^3 - 4*T11^2 + 2 $$T_{17}^{3} - 4T_{17}^{2} - 2T_{17} + 14$$ T17^3 - 4*T17^2 - 2*T17 + 14 $$T_{19}^{3} + 7T_{19}^{2} - 3T_{19} - 63$$ T19^3 + 7*T19^2 - 3*T19 - 63 $$T_{31}^{3} - 3T_{31}^{2} - 41T_{31} + 49$$ T31^3 - 3*T31^2 - 41*T31 + 49

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} - 2 T^{2} - 4 T + 6$$
$3$ $$T^{3}$$
$5$ $$T^{3} - 5 T^{2} + 3 T + 7$$
$7$ $$T^{3}$$
$11$ $$T^{3} - 4T^{2} + 2$$
$13$ $$(T - 1)^{3}$$
$17$ $$T^{3} - 4 T^{2} - 2 T + 14$$
$19$ $$T^{3} + 7 T^{2} - 3 T - 63$$
$23$ $$T^{3} + T^{2} - 41 T + 43$$
$29$ $$T^{3} - 7 T^{2} - 13 T + 3$$
$31$ $$T^{3} - 3 T^{2} - 41 T + 49$$
$37$ $$T^{3} + 10 T^{2} + 8 T - 82$$
$41$ $$T^{3} - 6 T^{2} - 116 T + 504$$
$43$ $$T^{3} - 9 T^{2} - 5 T + 101$$
$47$ $$T^{3} - 17 T^{2} + 89 T - 147$$
$53$ $$T^{3} + 13 T^{2} + 39 T - 9$$
$59$ $$T^{3} - 22 T^{2} + 144 T - 252$$
$61$ $$T^{3} - 24 T^{2} + 160 T - 224$$
$67$ $$T^{3} + 14 T^{2} - 36 T - 648$$
$71$ $$T^{3} + 4 T^{2} - 44 T - 194$$
$73$ $$T^{3} + 5 T^{2} - 219 T - 1561$$
$79$ $$T^{3} - T^{2} - 69 T - 99$$
$83$ $$T^{3} - 23 T^{2} + 127 T - 203$$
$89$ $$T^{3} + 11 T^{2} - 55 T + 21$$
$97$ $$T^{3} - 3 T^{2} - 71 T + 7$$