# Properties

 Label 1323.2.f.a Level $1323$ Weight $2$ Character orbit 1323.f 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(442,1323)]

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, 0]))

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

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

chi := DirichletCharacter("1323.442");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1323 = 3^{3} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1323.f (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: $$\Z[a_1, a_2]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{6} + 1) q^{2} + \zeta_{6} q^{4} - \zeta_{6} q^{5} + 3 q^{8} +O(q^{10})$$ q + (-z + 1) * q^2 + z * q^4 - z * q^5 + 3 * q^8 $$q + ( - \zeta_{6} + 1) q^{2} + \zeta_{6} q^{4} - \zeta_{6} q^{5} + 3 q^{8} - q^{10} + ( - 5 \zeta_{6} + 5) q^{11} + 5 \zeta_{6} q^{13} + ( - \zeta_{6} + 1) q^{16} - 3 q^{17} + q^{19} + ( - \zeta_{6} + 1) q^{20} - 5 \zeta_{6} q^{22} + 3 \zeta_{6} q^{23} + ( - 4 \zeta_{6} + 4) q^{25} + 5 q^{26} + (\zeta_{6} - 1) q^{29} + 5 \zeta_{6} q^{32} + (3 \zeta_{6} - 3) q^{34} + 3 q^{37} + ( - \zeta_{6} + 1) q^{38} - 3 \zeta_{6} q^{40} - 5 \zeta_{6} q^{41} + ( - \zeta_{6} + 1) q^{43} + 5 q^{44} + 3 q^{46} - 4 \zeta_{6} q^{50} + (5 \zeta_{6} - 5) q^{52} + 9 q^{53} - 5 q^{55} + \zeta_{6} q^{58} + ( - 14 \zeta_{6} + 14) q^{61} + 7 q^{64} + ( - 5 \zeta_{6} + 5) q^{65} - 4 \zeta_{6} q^{67} - 3 \zeta_{6} q^{68} + 12 q^{71} + 3 q^{73} + ( - 3 \zeta_{6} + 3) q^{74} + \zeta_{6} q^{76} + (8 \zeta_{6} - 8) q^{79} - q^{80} - 5 q^{82} + (9 \zeta_{6} - 9) q^{83} + 3 \zeta_{6} q^{85} - \zeta_{6} q^{86} + ( - 15 \zeta_{6} + 15) q^{88} + 13 q^{89} + (3 \zeta_{6} - 3) q^{92} - \zeta_{6} q^{95} + ( - 9 \zeta_{6} + 9) q^{97} +O(q^{100})$$ q + (-z + 1) * q^2 + z * q^4 - z * q^5 + 3 * q^8 - q^10 + (-5*z + 5) * q^11 + 5*z * q^13 + (-z + 1) * q^16 - 3 * q^17 + q^19 + (-z + 1) * q^20 - 5*z * q^22 + 3*z * q^23 + (-4*z + 4) * q^25 + 5 * q^26 + (z - 1) * q^29 + 5*z * q^32 + (3*z - 3) * q^34 + 3 * q^37 + (-z + 1) * q^38 - 3*z * q^40 - 5*z * q^41 + (-z + 1) * q^43 + 5 * q^44 + 3 * q^46 - 4*z * q^50 + (5*z - 5) * q^52 + 9 * q^53 - 5 * q^55 + z * q^58 + (-14*z + 14) * q^61 + 7 * q^64 + (-5*z + 5) * q^65 - 4*z * q^67 - 3*z * q^68 + 12 * q^71 + 3 * q^73 + (-3*z + 3) * q^74 + z * q^76 + (8*z - 8) * q^79 - q^80 - 5 * q^82 + (9*z - 9) * q^83 + 3*z * q^85 - z * q^86 + (-15*z + 15) * q^88 + 13 * q^89 + (3*z - 3) * q^92 - z * q^95 + (-9*z + 9) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} + q^{4} - q^{5} + 6 q^{8}+O(q^{10})$$ 2 * q + q^2 + q^4 - q^5 + 6 * q^8 $$2 q + q^{2} + q^{4} - q^{5} + 6 q^{8} - 2 q^{10} + 5 q^{11} + 5 q^{13} + q^{16} - 6 q^{17} + 2 q^{19} + q^{20} - 5 q^{22} + 3 q^{23} + 4 q^{25} + 10 q^{26} - q^{29} + 5 q^{32} - 3 q^{34} + 6 q^{37} + q^{38} - 3 q^{40} - 5 q^{41} + q^{43} + 10 q^{44} + 6 q^{46} - 4 q^{50} - 5 q^{52} + 18 q^{53} - 10 q^{55} + q^{58} + 14 q^{61} + 14 q^{64} + 5 q^{65} - 4 q^{67} - 3 q^{68} + 24 q^{71} + 6 q^{73} + 3 q^{74} + q^{76} - 8 q^{79} - 2 q^{80} - 10 q^{82} - 9 q^{83} + 3 q^{85} - q^{86} + 15 q^{88} + 26 q^{89} - 3 q^{92} - q^{95} + 9 q^{97}+O(q^{100})$$ 2 * q + q^2 + q^4 - q^5 + 6 * q^8 - 2 * q^10 + 5 * q^11 + 5 * q^13 + q^16 - 6 * q^17 + 2 * q^19 + q^20 - 5 * q^22 + 3 * q^23 + 4 * q^25 + 10 * q^26 - q^29 + 5 * q^32 - 3 * q^34 + 6 * q^37 + q^38 - 3 * q^40 - 5 * q^41 + q^43 + 10 * q^44 + 6 * q^46 - 4 * q^50 - 5 * q^52 + 18 * q^53 - 10 * q^55 + q^58 + 14 * q^61 + 14 * q^64 + 5 * q^65 - 4 * q^67 - 3 * q^68 + 24 * q^71 + 6 * q^73 + 3 * q^74 + q^76 - 8 * q^79 - 2 * q^80 - 10 * q^82 - 9 * q^83 + 3 * q^85 - q^86 + 15 * q^88 + 26 * q^89 - 3 * q^92 - q^95 + 9 * q^97

## Character values

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

 $$n$$ $$785$$ $$1081$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
442.1
 0.5 − 0.866025i 0.5 + 0.866025i
0.500000 + 0.866025i 0 0.500000 0.866025i −0.500000 + 0.866025i 0 0 3.00000 0 −1.00000
883.1 0.500000 0.866025i 0 0.500000 + 0.866025i −0.500000 0.866025i 0 0 3.00000 0 −1.00000
 $$n$$: 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
9.c 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.f.a 2
3.b odd 2 1 441.2.f.b 2
7.b odd 2 1 1323.2.f.b 2
7.c even 3 1 189.2.g.a 2
7.c even 3 1 189.2.h.a 2
7.d odd 6 1 1323.2.g.a 2
7.d odd 6 1 1323.2.h.a 2
9.c even 3 1 inner 1323.2.f.a 2
9.c even 3 1 3969.2.a.c 1
9.d odd 6 1 441.2.f.b 2
9.d odd 6 1 3969.2.a.d 1
21.c even 2 1 441.2.f.a 2
21.g even 6 1 441.2.g.a 2
21.g even 6 1 441.2.h.a 2
21.h odd 6 1 63.2.g.a 2
21.h odd 6 1 63.2.h.a yes 2
28.g odd 6 1 3024.2.q.b 2
28.g odd 6 1 3024.2.t.d 2
63.g even 3 1 189.2.h.a 2
63.g even 3 1 567.2.e.b 2
63.h even 3 1 189.2.g.a 2
63.h even 3 1 567.2.e.b 2
63.i even 6 1 441.2.g.a 2
63.j odd 6 1 63.2.g.a 2
63.j odd 6 1 567.2.e.a 2
63.k odd 6 1 1323.2.h.a 2
63.l odd 6 1 1323.2.f.b 2
63.l odd 6 1 3969.2.a.a 1
63.n odd 6 1 63.2.h.a yes 2
63.n odd 6 1 567.2.e.a 2
63.o even 6 1 441.2.f.a 2
63.o even 6 1 3969.2.a.f 1
63.s even 6 1 441.2.h.a 2
63.t odd 6 1 1323.2.g.a 2
84.n even 6 1 1008.2.q.c 2
84.n even 6 1 1008.2.t.d 2
252.o even 6 1 1008.2.q.c 2
252.u odd 6 1 3024.2.t.d 2
252.bb even 6 1 1008.2.t.d 2
252.bl odd 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.h odd 6 1
63.2.g.a 2 63.j odd 6 1
63.2.h.a yes 2 21.h odd 6 1
63.2.h.a yes 2 63.n odd 6 1
189.2.g.a 2 7.c even 3 1
189.2.g.a 2 63.h even 3 1
189.2.h.a 2 7.c even 3 1
189.2.h.a 2 63.g even 3 1
441.2.f.a 2 21.c even 2 1
441.2.f.a 2 63.o even 6 1
441.2.f.b 2 3.b odd 2 1
441.2.f.b 2 9.d odd 6 1
441.2.g.a 2 21.g even 6 1
441.2.g.a 2 63.i even 6 1
441.2.h.a 2 21.g even 6 1
441.2.h.a 2 63.s even 6 1
567.2.e.a 2 63.j odd 6 1
567.2.e.a 2 63.n odd 6 1
567.2.e.b 2 63.g even 3 1
567.2.e.b 2 63.h even 3 1
1008.2.q.c 2 84.n even 6 1
1008.2.q.c 2 252.o even 6 1
1008.2.t.d 2 84.n even 6 1
1008.2.t.d 2 252.bb even 6 1
1323.2.f.a 2 1.a even 1 1 trivial
1323.2.f.a 2 9.c even 3 1 inner
1323.2.f.b 2 7.b odd 2 1
1323.2.f.b 2 63.l odd 6 1
1323.2.g.a 2 7.d odd 6 1
1323.2.g.a 2 63.t odd 6 1
1323.2.h.a 2 7.d odd 6 1
1323.2.h.a 2 63.k odd 6 1
3024.2.q.b 2 28.g odd 6 1
3024.2.q.b 2 252.bl odd 6 1
3024.2.t.d 2 28.g odd 6 1
3024.2.t.d 2 252.u odd 6 1
3969.2.a.a 1 63.l odd 6 1
3969.2.a.c 1 9.c even 3 1
3969.2.a.d 1 9.d odd 6 1
3969.2.a.f 1 63.o even 6 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2} + T + 1$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 5T + 25$$
$13$ $$T^{2} - 5T + 25$$
$17$ $$(T + 3)^{2}$$
$19$ $$(T - 1)^{2}$$
$23$ $$T^{2} - 3T + 9$$
$29$ $$T^{2} + T + 1$$
$31$ $$T^{2}$$
$37$ $$(T - 3)^{2}$$
$41$ $$T^{2} + 5T + 25$$
$43$ $$T^{2} - T + 1$$
$47$ $$T^{2}$$
$53$ $$(T - 9)^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} - 14T + 196$$
$67$ $$T^{2} + 4T + 16$$
$71$ $$(T - 12)^{2}$$
$73$ $$(T - 3)^{2}$$
$79$ $$T^{2} + 8T + 64$$
$83$ $$T^{2} + 9T + 81$$
$89$ $$(T - 13)^{2}$$
$97$ $$T^{2} - 9T + 81$$