# Properties

 Label 1323.2.a.u Level $1323$ Weight $2$ Character orbit 1323.a Self dual yes Analytic conductor $10.564$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1323,2,Mod(1,1323)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1323, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1323.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1323 = 3^{3} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1323.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: $$10.5642081874$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 6$$ x^2 - 6 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 189) 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 $$\beta = \sqrt{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + 4 q^{4} - \beta q^{5} + 2 \beta q^{8} +O(q^{10})$$ q + b * q^2 + 4 * q^4 - b * q^5 + 2*b * q^8 $$q + \beta q^{2} + 4 q^{4} - \beta q^{5} + 2 \beta q^{8} - 6 q^{10} + 2 \beta q^{11} + 4 q^{13} + 4 q^{16} + \beta q^{17} + q^{19} - 4 \beta q^{20} + 12 q^{22} + \beta q^{23} + q^{25} + 4 \beta q^{26} - 3 \beta q^{29} + 7 q^{31} + 6 q^{34} + 8 q^{37} + \beta q^{38} - 12 q^{40} - 3 \beta q^{41} - q^{43} + 8 \beta q^{44} + 6 q^{46} - \beta q^{47} + \beta q^{50} + 16 q^{52} - \beta q^{53} - 12 q^{55} - 18 q^{58} + 4 \beta q^{59} - 5 q^{61} + 7 \beta q^{62} - 8 q^{64} - 4 \beta q^{65} + 2 q^{67} + 4 \beta q^{68} + q^{73} + 8 \beta q^{74} + 4 q^{76} - 4 q^{79} - 4 \beta q^{80} - 18 q^{82} - 6 \beta q^{83} - 6 q^{85} - \beta q^{86} + 24 q^{88} - \beta q^{89} + 4 \beta q^{92} - 6 q^{94} - \beta q^{95} + q^{97} +O(q^{100})$$ q + b * q^2 + 4 * q^4 - b * q^5 + 2*b * q^8 - 6 * q^10 + 2*b * q^11 + 4 * q^13 + 4 * q^16 + b * q^17 + q^19 - 4*b * q^20 + 12 * q^22 + b * q^23 + q^25 + 4*b * q^26 - 3*b * q^29 + 7 * q^31 + 6 * q^34 + 8 * q^37 + b * q^38 - 12 * q^40 - 3*b * q^41 - q^43 + 8*b * q^44 + 6 * q^46 - b * q^47 + b * q^50 + 16 * q^52 - b * q^53 - 12 * q^55 - 18 * q^58 + 4*b * q^59 - 5 * q^61 + 7*b * q^62 - 8 * q^64 - 4*b * q^65 + 2 * q^67 + 4*b * q^68 + q^73 + 8*b * q^74 + 4 * q^76 - 4 * q^79 - 4*b * q^80 - 18 * q^82 - 6*b * q^83 - 6 * q^85 - b * q^86 + 24 * q^88 - b * q^89 + 4*b * q^92 - 6 * q^94 - b * q^95 + q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 8 q^{4}+O(q^{10})$$ 2 * q + 8 * q^4 $$2 q + 8 q^{4} - 12 q^{10} + 8 q^{13} + 8 q^{16} + 2 q^{19} + 24 q^{22} + 2 q^{25} + 14 q^{31} + 12 q^{34} + 16 q^{37} - 24 q^{40} - 2 q^{43} + 12 q^{46} + 32 q^{52} - 24 q^{55} - 36 q^{58} - 10 q^{61} - 16 q^{64} + 4 q^{67} + 2 q^{73} + 8 q^{76} - 8 q^{79} - 36 q^{82} - 12 q^{85} + 48 q^{88} - 12 q^{94} + 2 q^{97}+O(q^{100})$$ 2 * q + 8 * q^4 - 12 * q^10 + 8 * q^13 + 8 * q^16 + 2 * q^19 + 24 * q^22 + 2 * q^25 + 14 * q^31 + 12 * q^34 + 16 * q^37 - 24 * q^40 - 2 * q^43 + 12 * q^46 + 32 * q^52 - 24 * q^55 - 36 * q^58 - 10 * q^61 - 16 * q^64 + 4 * q^67 + 2 * q^73 + 8 * q^76 - 8 * q^79 - 36 * q^82 - 12 * q^85 + 48 * q^88 - 12 * q^94 + 2 * 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
 −2.44949 2.44949
−2.44949 0 4.00000 2.44949 0 0 −4.89898 0 −6.00000
1.2 2.44949 0 4.00000 −2.44949 0 0 4.89898 0 −6.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$7$$ $$-1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1323.2.a.u 2
3.b odd 2 1 inner 1323.2.a.u 2
7.b odd 2 1 1323.2.a.v 2
7.d odd 6 2 189.2.e.d 4
21.c even 2 1 1323.2.a.v 2
21.g even 6 2 189.2.e.d 4
63.i even 6 2 567.2.h.g 4
63.k odd 6 2 567.2.g.g 4
63.s even 6 2 567.2.g.g 4
63.t odd 6 2 567.2.h.g 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.e.d 4 7.d odd 6 2
189.2.e.d 4 21.g even 6 2
567.2.g.g 4 63.k odd 6 2
567.2.g.g 4 63.s even 6 2
567.2.h.g 4 63.i even 6 2
567.2.h.g 4 63.t odd 6 2
1323.2.a.u 2 1.a even 1 1 trivial
1323.2.a.u 2 3.b odd 2 1 inner
1323.2.a.v 2 7.b odd 2 1
1323.2.a.v 2 21.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1323))$$:

 $$T_{2}^{2} - 6$$ T2^2 - 6 $$T_{5}^{2} - 6$$ T5^2 - 6 $$T_{13} - 4$$ T13 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 6$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 6$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 24$$
$13$ $$(T - 4)^{2}$$
$17$ $$T^{2} - 6$$
$19$ $$(T - 1)^{2}$$
$23$ $$T^{2} - 6$$
$29$ $$T^{2} - 54$$
$31$ $$(T - 7)^{2}$$
$37$ $$(T - 8)^{2}$$
$41$ $$T^{2} - 54$$
$43$ $$(T + 1)^{2}$$
$47$ $$T^{2} - 6$$
$53$ $$T^{2} - 6$$
$59$ $$T^{2} - 96$$
$61$ $$(T + 5)^{2}$$
$67$ $$(T - 2)^{2}$$
$71$ $$T^{2}$$
$73$ $$(T - 1)^{2}$$
$79$ $$(T + 4)^{2}$$
$83$ $$T^{2} - 216$$
$89$ $$T^{2} - 6$$
$97$ $$(T - 1)^{2}$$