# Properties

 Label 1323.2.h.a Level $1323$ Weight $2$ Character orbit 1323.h 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(226,1323)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1323, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2, 2]))

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

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

chi := DirichletCharacter("1323.226");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1323 = 3^{3} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1323.h (of order $$3$$, degree $$2$$, not minimal)

## 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: no Analytic conductor: $$10.5642081874$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 63) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{4} + ( - \zeta_{6} + 1) q^{5} + 3 q^{8}+O(q^{10})$$ q - q^2 - q^4 + (-z + 1) * q^5 + 3 * q^8 $$q - q^{2} - q^{4} + ( - \zeta_{6} + 1) q^{5} + 3 q^{8} + (\zeta_{6} - 1) q^{10} + 5 \zeta_{6} q^{11} - 5 \zeta_{6} q^{13} - q^{16} + (3 \zeta_{6} - 3) q^{17} + \zeta_{6} q^{19} + (\zeta_{6} - 1) q^{20} - 5 \zeta_{6} q^{22} + ( - 3 \zeta_{6} + 3) q^{23} + 4 \zeta_{6} q^{25} + 5 \zeta_{6} q^{26} + (\zeta_{6} - 1) q^{29} - 5 q^{32} + ( - 3 \zeta_{6} + 3) q^{34} - 3 \zeta_{6} q^{37} - \zeta_{6} q^{38} + ( - 3 \zeta_{6} + 3) q^{40} + 5 \zeta_{6} q^{41} + ( - \zeta_{6} + 1) q^{43} - 5 \zeta_{6} q^{44} + (3 \zeta_{6} - 3) q^{46} - 4 \zeta_{6} q^{50} + 5 \zeta_{6} q^{52} + (9 \zeta_{6} - 9) q^{53} + 5 q^{55} + ( - \zeta_{6} + 1) q^{58} + 14 q^{61} + 7 q^{64} - 5 q^{65} + 4 q^{67} + ( - 3 \zeta_{6} + 3) q^{68} + 12 q^{71} + ( - 3 \zeta_{6} + 3) q^{73} + 3 \zeta_{6} q^{74} - \zeta_{6} q^{76} + 8 q^{79} + (\zeta_{6} - 1) q^{80} - 5 \zeta_{6} q^{82} + ( - 9 \zeta_{6} + 9) q^{83} + 3 \zeta_{6} q^{85} + (\zeta_{6} - 1) q^{86} + 15 \zeta_{6} q^{88} + 13 \zeta_{6} q^{89} + (3 \zeta_{6} - 3) q^{92} + q^{95} + (9 \zeta_{6} - 9) q^{97} +O(q^{100})$$ q - q^2 - q^4 + (-z + 1) * q^5 + 3 * q^8 + (z - 1) * q^10 + 5*z * q^11 - 5*z * q^13 - q^16 + (3*z - 3) * q^17 + z * q^19 + (z - 1) * q^20 - 5*z * q^22 + (-3*z + 3) * q^23 + 4*z * q^25 + 5*z * q^26 + (z - 1) * q^29 - 5 * q^32 + (-3*z + 3) * q^34 - 3*z * q^37 - z * q^38 + (-3*z + 3) * q^40 + 5*z * q^41 + (-z + 1) * q^43 - 5*z * q^44 + (3*z - 3) * q^46 - 4*z * q^50 + 5*z * q^52 + (9*z - 9) * q^53 + 5 * q^55 + (-z + 1) * q^58 + 14 * q^61 + 7 * q^64 - 5 * q^65 + 4 * q^67 + (-3*z + 3) * q^68 + 12 * q^71 + (-3*z + 3) * q^73 + 3*z * q^74 - z * q^76 + 8 * q^79 + (z - 1) * q^80 - 5*z * q^82 + (-9*z + 9) * q^83 + 3*z * q^85 + (z - 1) * q^86 + 15*z * q^88 + 13*z * q^89 + (3*z - 3) * q^92 + q^95 + (9*z - 9) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 2 q^{4} + q^{5} + 6 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 - 2 * q^4 + q^5 + 6 * q^8 $$2 q - 2 q^{2} - 2 q^{4} + q^{5} + 6 q^{8} - q^{10} + 5 q^{11} - 5 q^{13} - 2 q^{16} - 3 q^{17} + q^{19} - q^{20} - 5 q^{22} + 3 q^{23} + 4 q^{25} + 5 q^{26} - q^{29} - 10 q^{32} + 3 q^{34} - 3 q^{37} - q^{38} + 3 q^{40} + 5 q^{41} + q^{43} - 5 q^{44} - 3 q^{46} - 4 q^{50} + 5 q^{52} - 9 q^{53} + 10 q^{55} + q^{58} + 28 q^{61} + 14 q^{64} - 10 q^{65} + 8 q^{67} + 3 q^{68} + 24 q^{71} + 3 q^{73} + 3 q^{74} - q^{76} + 16 q^{79} - q^{80} - 5 q^{82} + 9 q^{83} + 3 q^{85} - q^{86} + 15 q^{88} + 13 q^{89} - 3 q^{92} + 2 q^{95} - 9 q^{97}+O(q^{100})$$ 2 * q - 2 * q^2 - 2 * q^4 + q^5 + 6 * q^8 - q^10 + 5 * q^11 - 5 * q^13 - 2 * q^16 - 3 * q^17 + q^19 - q^20 - 5 * q^22 + 3 * q^23 + 4 * q^25 + 5 * q^26 - q^29 - 10 * q^32 + 3 * q^34 - 3 * q^37 - q^38 + 3 * q^40 + 5 * q^41 + q^43 - 5 * q^44 - 3 * q^46 - 4 * q^50 + 5 * q^52 - 9 * q^53 + 10 * q^55 + q^58 + 28 * q^61 + 14 * q^64 - 10 * q^65 + 8 * q^67 + 3 * q^68 + 24 * q^71 + 3 * q^73 + 3 * q^74 - q^76 + 16 * q^79 - q^80 - 5 * q^82 + 9 * q^83 + 3 * q^85 - q^86 + 15 * q^88 + 13 * q^89 - 3 * q^92 + 2 * q^95 - 9 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1323\mathbb{Z}\right)^\times$$.

 $$n$$ $$785$$ $$1081$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$-\zeta_{6}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
226.1
 0.5 − 0.866025i 0.5 + 0.866025i
−1.00000 0 −1.00000 0.500000 + 0.866025i 0 0 3.00000 0 −0.500000 0.866025i
802.1 −1.00000 0 −1.00000 0.500000 0.866025i 0 0 3.00000 0 −0.500000 + 0.866025i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.h even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1323.2.h.a 2
3.b odd 2 1 441.2.h.a 2
7.b odd 2 1 189.2.h.a 2
7.c even 3 1 1323.2.f.b 2
7.c even 3 1 1323.2.g.a 2
7.d odd 6 1 189.2.g.a 2
7.d odd 6 1 1323.2.f.a 2
9.c even 3 1 1323.2.g.a 2
9.d odd 6 1 441.2.g.a 2
21.c even 2 1 63.2.h.a yes 2
21.g even 6 1 63.2.g.a 2
21.g even 6 1 441.2.f.b 2
21.h odd 6 1 441.2.f.a 2
21.h odd 6 1 441.2.g.a 2
28.d even 2 1 3024.2.q.b 2
28.f even 6 1 3024.2.t.d 2
63.g even 3 1 1323.2.f.b 2
63.h even 3 1 inner 1323.2.h.a 2
63.h even 3 1 3969.2.a.a 1
63.i even 6 1 63.2.h.a yes 2
63.i even 6 1 3969.2.a.d 1
63.j odd 6 1 441.2.h.a 2
63.j odd 6 1 3969.2.a.f 1
63.k odd 6 1 567.2.e.b 2
63.k odd 6 1 1323.2.f.a 2
63.l odd 6 1 189.2.g.a 2
63.l odd 6 1 567.2.e.b 2
63.n odd 6 1 441.2.f.a 2
63.o even 6 1 63.2.g.a 2
63.o even 6 1 567.2.e.a 2
63.s even 6 1 441.2.f.b 2
63.s even 6 1 567.2.e.a 2
63.t odd 6 1 189.2.h.a 2
63.t odd 6 1 3969.2.a.c 1
84.h odd 2 1 1008.2.q.c 2
84.j odd 6 1 1008.2.t.d 2
252.r odd 6 1 1008.2.q.c 2
252.s odd 6 1 1008.2.t.d 2
252.bi even 6 1 3024.2.t.d 2
252.bj even 6 1 3024.2.q.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.g.a 2 21.g even 6 1
63.2.g.a 2 63.o even 6 1
63.2.h.a yes 2 21.c even 2 1
63.2.h.a yes 2 63.i even 6 1
189.2.g.a 2 7.d odd 6 1
189.2.g.a 2 63.l odd 6 1
189.2.h.a 2 7.b odd 2 1
189.2.h.a 2 63.t odd 6 1
441.2.f.a 2 21.h odd 6 1
441.2.f.a 2 63.n odd 6 1
441.2.f.b 2 21.g even 6 1
441.2.f.b 2 63.s even 6 1
441.2.g.a 2 9.d odd 6 1
441.2.g.a 2 21.h odd 6 1
441.2.h.a 2 3.b odd 2 1
441.2.h.a 2 63.j odd 6 1
567.2.e.a 2 63.o even 6 1
567.2.e.a 2 63.s even 6 1
567.2.e.b 2 63.k odd 6 1
567.2.e.b 2 63.l odd 6 1
1008.2.q.c 2 84.h odd 2 1
1008.2.q.c 2 252.r odd 6 1
1008.2.t.d 2 84.j odd 6 1
1008.2.t.d 2 252.s odd 6 1
1323.2.f.a 2 7.d odd 6 1
1323.2.f.a 2 63.k odd 6 1
1323.2.f.b 2 7.c even 3 1
1323.2.f.b 2 63.g even 3 1
1323.2.g.a 2 7.c even 3 1
1323.2.g.a 2 9.c even 3 1
1323.2.h.a 2 1.a even 1 1 trivial
1323.2.h.a 2 63.h even 3 1 inner
3024.2.q.b 2 28.d even 2 1
3024.2.q.b 2 252.bj even 6 1
3024.2.t.d 2 28.f even 6 1
3024.2.t.d 2 252.bi even 6 1
3969.2.a.a 1 63.h even 3 1
3969.2.a.c 1 63.t odd 6 1
3969.2.a.d 1 63.i even 6 1
3969.2.a.f 1 63.j odd 6 1

## Hecke kernels

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

 $$T_{2} + 1$$ T2 + 1 $$T_{5}^{2} - T_{5} + 1$$ T5^2 - T5 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - T + 1$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 5T + 25$$
$13$ $$T^{2} + 5T + 25$$
$17$ $$T^{2} + 3T + 9$$
$19$ $$T^{2} - T + 1$$
$23$ $$T^{2} - 3T + 9$$
$29$ $$T^{2} + T + 1$$
$31$ $$T^{2}$$
$37$ $$T^{2} + 3T + 9$$
$41$ $$T^{2} - 5T + 25$$
$43$ $$T^{2} - T + 1$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 9T + 81$$
$59$ $$T^{2}$$
$61$ $$(T - 14)^{2}$$
$67$ $$(T - 4)^{2}$$
$71$ $$(T - 12)^{2}$$
$73$ $$T^{2} - 3T + 9$$
$79$ $$(T - 8)^{2}$$
$83$ $$T^{2} - 9T + 81$$
$89$ $$T^{2} - 13T + 169$$
$97$ $$T^{2} + 9T + 81$$