# Properties

 Label 5733.2.a.br Level $5733$ Weight $2$ Character orbit 5733.a Self dual yes Analytic conductor $45.778$ Analytic rank $1$ Dimension $6$ 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: $$1$$ Dimension: $$6$$ Coefficient field: 6.6.4507648.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 2x^{5} - 5x^{4} + 8x^{3} + 7x^{2} - 6x - 1$$ x^6 - 2*x^5 - 5*x^4 + 8*x^3 + 7*x^2 - 6*x - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + (\beta_{4} - \beta_{3} - \beta_1 + 1) q^{4} + ( - \beta_{4} - 1) q^{5} + ( - \beta_{5} - \beta_{3} - \beta_1) q^{8}+O(q^{10})$$ q + b2 * q^2 + (b4 - b3 - b1 + 1) * q^4 + (-b4 - 1) * q^5 + (-b5 - b3 - b1) * q^8 $$q + \beta_{2} q^{2} + (\beta_{4} - \beta_{3} - \beta_1 + 1) q^{4} + ( - \beta_{4} - 1) q^{5} + ( - \beta_{5} - \beta_{3} - \beta_1) q^{8} + ( - 2 \beta_{2} - \beta_1 + 1) q^{10} + (2 \beta_{5} + \beta_{4}) q^{11} + q^{13} + ( - \beta_{4} - \beta_{3} + \beta_{2}) q^{16} + (\beta_{5} + \beta_{4} + \beta_{3} + \cdots - 3) q^{17}+ \cdots + ( - 2 \beta_{5} - \beta_{4} - 5 \beta_{3} + \cdots - 4) q^{97}+O(q^{100})$$ q + b2 * q^2 + (b4 - b3 - b1 + 1) * q^4 + (-b4 - 1) * q^5 + (-b5 - b3 - b1) * q^8 + (-2*b2 - b1 + 1) * q^10 + (2*b5 + b4) * q^11 + q^13 + (-b4 - b3 + b2) * q^16 + (b5 + b4 + b3 + b2 + 2*b1 - 3) * q^17 + (-b5 - b4 + b3 - b2) * q^19 + (b3 + b2 + b1 - 3) * q^20 + (-2*b5 - 2*b4 + 4*b3 - b2 + b1 - 3) * q^22 + (-2*b5 - b4 + b3 + b1) * q^23 + (b4 + b3 + b1 - 1) * q^25 + b2 * q^26 + (2*b4 - 2*b3 + 1) * q^29 + (b5 - 3*b4 - 2*b2 + b1 + 1) * q^31 + (b5 + b4 + b3 - b2 - b1 + 4) * q^32 + (3*b3 - 3*b2 + 3*b1 - 1) * q^34 + (2*b5 + b4 - 2*b3 + 2*b2 + 2*b1) * q^37 + (2*b5 - b3 + b1 - 1) * q^38 + (b5 + b4 + b2 + 3*b1) * q^40 + (-b5 - 2*b3 - 2*b2 + 3*b1) * q^41 + (-2*b5 + 2*b4 - 4*b1 + 1) * q^43 + (2*b5 - b4 - 2*b3 - 3*b2 + 4*b1) * q^44 + (3*b5 + 2*b4 - 3*b3 + b2 + b1 + 2) * q^46 + (-3*b5 - b4 + 3*b3 - 3*b2 - 6) * q^47 + (b5 + b3 + 3*b1 - 2) * q^50 + (b4 - b3 - b1 + 1) * q^52 + (b4 - b3 + 4*b2 + b1 + 2) * q^53 + (-2*b5 + 3*b3 - 3*b1 - 1) * q^55 + (-2*b5 + 3*b2 - 2) * q^58 + (-2*b5 - 2*b4 + 3*b3 - b2 + b1 - 5) * q^59 + (3*b5 - b4 - 2*b3 + b2 - 3*b1 + 2) * q^61 + (-b5 - 3*b4 + 5*b3 - 3*b2 - 5) * q^62 + (4*b3 + 2*b2 + 2*b1 - 4) * q^64 + (-b4 - 1) * q^65 + (3*b5 + b4 + b3 + b2 - b1 + 4) * q^67 + (b5 - 5*b4 + 4*b3 - 3*b2 + 5*b1 - 6) * q^68 + (-3*b5 - 5*b4 + 5*b3 - 4*b2 - b1 - 2) * q^71 + (-5*b5 - b4 + b3 - b2 - 2*b1 - 2) * q^73 + (-4*b5 + 4*b3 - b2 - b1 + 1) * q^74 + (-b5 + 3*b3 - b2 - 3) * q^76 + (3*b4 + 3*b3 - 3*b2 - 2*b1 - 3) * q^79 + (-b5 + 2*b3 - 2*b2 + b1 + 4) * q^80 + (-b5 - b4 + 3*b3 + b2 + 3*b1 - 8) * q^82 + (2*b5 - b4 - 2*b1 - 7) * q^83 + (-2*b5 + b4 - 2*b3 - 4*b2 - 6*b1) * q^85 + (2*b5 + 2*b4 - 8*b3 + 5*b2 - 2*b1 + 4) * q^86 + (-b4 + 3*b3 - b2 + 2*b1 - 8) * q^88 + (2*b5 - 3*b4 + b3 - b2 + b1 - 4) * q^89 + (-2*b5 + 4*b3 + b2 - 3*b1 - 3) * q^92 + (6*b5 - 3*b3 - 4*b2 + 5*b1 - 5) * q^94 + (2*b5 - 2*b3 + 2*b2 + 2*b1 + 1) * q^95 + (-2*b5 - b4 - 5*b3 + 3*b2 + 3*b1 - 4) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 4 q^{4} - 6 q^{5}+O(q^{10})$$ 6 * q + 4 * q^4 - 6 * q^5 $$6 q + 4 q^{4} - 6 q^{5} + 4 q^{10} - 4 q^{11} + 6 q^{13} - 16 q^{17} + 2 q^{19} - 16 q^{20} - 12 q^{22} + 6 q^{23} - 4 q^{25} + 6 q^{29} + 6 q^{31} + 20 q^{32} - 8 q^{38} + 4 q^{40} + 8 q^{41} + 2 q^{43} + 4 q^{44} + 8 q^{46} - 30 q^{47} - 8 q^{50} + 4 q^{52} + 14 q^{53} - 8 q^{55} - 8 q^{58} - 24 q^{59} - 28 q^{62} - 20 q^{64} - 6 q^{65} + 16 q^{67} - 28 q^{68} - 8 q^{71} - 6 q^{73} + 12 q^{74} - 16 q^{76} - 22 q^{79} + 28 q^{80} - 40 q^{82} - 50 q^{83} - 8 q^{85} + 16 q^{86} - 44 q^{88} - 26 q^{89} - 20 q^{92} - 32 q^{94} + 6 q^{95} - 14 q^{97}+O(q^{100})$$ 6 * q + 4 * q^4 - 6 * q^5 + 4 * q^10 - 4 * q^11 + 6 * q^13 - 16 * q^17 + 2 * q^19 - 16 * q^20 - 12 * q^22 + 6 * q^23 - 4 * q^25 + 6 * q^29 + 6 * q^31 + 20 * q^32 - 8 * q^38 + 4 * q^40 + 8 * q^41 + 2 * q^43 + 4 * q^44 + 8 * q^46 - 30 * q^47 - 8 * q^50 + 4 * q^52 + 14 * q^53 - 8 * q^55 - 8 * q^58 - 24 * q^59 - 28 * q^62 - 20 * q^64 - 6 * q^65 + 16 * q^67 - 28 * q^68 - 8 * q^71 - 6 * q^73 + 12 * q^74 - 16 * q^76 - 22 * q^79 + 28 * q^80 - 40 * q^82 - 50 * q^83 - 8 * q^85 + 16 * q^86 - 44 * q^88 - 26 * q^89 - 20 * q^92 - 32 * q^94 + 6 * q^95 - 14 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2x^{5} - 5x^{4} + 8x^{3} + 7x^{2} - 6x - 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 2$$ v^2 - v - 2 $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 3\nu + 1$$ v^3 - v^2 - 3*v + 1 $$\beta_{4}$$ $$=$$ $$\nu^{4} - \nu^{3} - 4\nu^{2} + 2\nu + 2$$ v^4 - v^3 - 4*v^2 + 2*v + 2 $$\beta_{5}$$ $$=$$ $$\nu^{5} - 2\nu^{4} - 4\nu^{3} + 6\nu^{2} + 4\nu - 2$$ v^5 - 2*v^4 - 4*v^3 + 6*v^2 + 4*v - 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 2$$ b2 + b1 + 2 $$\nu^{3}$$ $$=$$ $$\beta_{3} + \beta_{2} + 4\beta _1 + 1$$ b3 + b2 + 4*b1 + 1 $$\nu^{4}$$ $$=$$ $$\beta_{4} + \beta_{3} + 5\beta_{2} + 6\beta _1 + 7$$ b4 + b3 + 5*b2 + 6*b1 + 7 $$\nu^{5}$$ $$=$$ $$\beta_{5} + 2\beta_{4} + 6\beta_{3} + 8\beta_{2} + 18\beta _1 + 8$$ b5 + 2*b4 + 6*b3 + 8*b2 + 18*b1 + 8

## 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
 0.758419 −0.146243 1.90903 −1.20475 2.35100 −1.66745
−2.18322 0 2.76645 −2.11065 0 0 −1.67333 0 4.60802
1.2 −1.83237 0 1.35758 −2.62555 0 0 1.17715 0 4.81098
1.3 −0.264627 0 −1.92997 1.43515 0 0 1.03998 0 −0.379780
1.4 0.656184 0 −1.56942 1.35996 0 0 −2.34220 0 0.892385
1.5 1.17619 0 −0.616586 −3.14862 0 0 −3.07759 0 −3.70337
1.6 2.44785 0 3.99195 −0.910286 0 0 4.87599 0 −2.22824
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 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.br 6
3.b odd 2 1 637.2.a.n yes 6
7.b odd 2 1 5733.2.a.bu 6
21.c even 2 1 637.2.a.m 6
21.g even 6 2 637.2.e.o 12
21.h odd 6 2 637.2.e.n 12
39.d odd 2 1 8281.2.a.cd 6
273.g even 2 1 8281.2.a.cc 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.a.m 6 21.c even 2 1
637.2.a.n yes 6 3.b odd 2 1
637.2.e.n 12 21.h odd 6 2
637.2.e.o 12 21.g even 6 2
5733.2.a.br 6 1.a even 1 1 trivial
5733.2.a.bu 6 7.b odd 2 1
8281.2.a.cc 6 273.g even 2 1
8281.2.a.cd 6 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5733))$$:

 $$T_{2}^{6} - 8T_{2}^{4} + 14T_{2}^{2} - 4T_{2} - 2$$ T2^6 - 8*T2^4 + 14*T2^2 - 4*T2 - 2 $$T_{5}^{6} + 6T_{5}^{5} + 5T_{5}^{4} - 24T_{5}^{3} - 31T_{5}^{2} + 26T_{5} + 31$$ T5^6 + 6*T5^5 + 5*T5^4 - 24*T5^3 - 31*T5^2 + 26*T5 + 31 $$T_{11}^{6} + 4T_{11}^{5} - 38T_{11}^{4} - 156T_{11}^{3} + 186T_{11}^{2} + 692T_{11} - 562$$ T11^6 + 4*T11^5 - 38*T11^4 - 156*T11^3 + 186*T11^2 + 692*T11 - 562 $$T_{17}^{6} + 16T_{17}^{5} + 56T_{17}^{4} - 324T_{17}^{3} - 2792T_{17}^{2} - 6792T_{17} - 5294$$ T17^6 + 16*T17^5 + 56*T17^4 - 324*T17^3 - 2792*T17^2 - 6792*T17 - 5294 $$T_{19}^{6} - 2T_{19}^{5} - 17T_{19}^{4} + 16T_{19}^{3} + 83T_{19}^{2} - 6T_{19} - 73$$ T19^6 - 2*T19^5 - 17*T19^4 + 16*T19^3 + 83*T19^2 - 6*T19 - 73 $$T_{31}^{6} - 6T_{31}^{5} - 115T_{31}^{4} + 508T_{31}^{3} + 4181T_{31}^{2} - 10046T_{31} - 44249$$ T31^6 - 6*T31^5 - 115*T31^4 + 508*T31^3 + 4181*T31^2 - 10046*T31 - 44249

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} - 8 T^{4} + \cdots - 2$$
$3$ $$T^{6}$$
$5$ $$T^{6} + 6 T^{5} + \cdots + 31$$
$7$ $$T^{6}$$
$11$ $$T^{6} + 4 T^{5} + \cdots - 562$$
$13$ $$(T - 1)^{6}$$
$17$ $$T^{6} + 16 T^{5} + \cdots - 5294$$
$19$ $$T^{6} - 2 T^{5} + \cdots - 73$$
$23$ $$T^{6} - 6 T^{5} + \cdots + 529$$
$29$ $$T^{6} - 6 T^{5} + \cdots + 529$$
$31$ $$T^{6} - 6 T^{5} + \cdots - 44249$$
$37$ $$T^{6} - 82 T^{4} + \cdots + 254$$
$41$ $$T^{6} - 8 T^{5} + \cdots + 28784$$
$43$ $$T^{6} - 2 T^{5} + \cdots + 35153$$
$47$ $$T^{6} + 30 T^{5} + \cdots - 135617$$
$53$ $$T^{6} - 14 T^{5} + \cdots - 1319$$
$59$ $$T^{6} + 24 T^{5} + \cdots + 1532$$
$61$ $$T^{6} - 246 T^{4} + \cdots - 216584$$
$67$ $$T^{6} - 16 T^{5} + \cdots + 6112$$
$71$ $$T^{6} + 8 T^{5} + \cdots - 1206162$$
$73$ $$T^{6} + 6 T^{5} + \cdots - 142657$$
$79$ $$T^{6} + 22 T^{5} + \cdots + 7913$$
$83$ $$T^{6} + 50 T^{5} + \cdots - 167041$$
$89$ $$T^{6} + 26 T^{5} + \cdots + 9959$$
$97$ $$T^{6} + 14 T^{5} + \cdots + 217287$$