# Properties

 Label 2268.1.m.a Level $2268$ Weight $1$ Character orbit 2268.m Analytic conductor $1.132$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -3 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2268,1,Mod(53,2268)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2268, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 5, 4]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("2268.53");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2268 = 2^{2} \cdot 3^{4} \cdot 7$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2268.m (of order $$6$$, 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: $$1.13187944865$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 84) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.588.1 Artin image: $C_3\times S_3$ Artin field: Galois closure of 6.0.15431472.1

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + \zeta_{6}^{2} q^{7}+O(q^{10})$$ q + z^2 * q^7 $$q + \zeta_{6}^{2} q^{7} - \zeta_{6}^{2} q^{13} + \zeta_{6} q^{19} + q^{25} + \zeta_{6} q^{31} + \zeta_{6} q^{37} + \zeta_{6} q^{43} - \zeta_{6} q^{49} + \zeta_{6}^{2} q^{61} + \zeta_{6} q^{67} - \zeta_{6}^{2} q^{73} - \zeta_{6}^{2} q^{79} + \zeta_{6} q^{91} - \zeta_{6} q^{97} +O(q^{100})$$ q + z^2 * q^7 - z^2 * q^13 + z * q^19 + q^25 + z * q^31 + z * q^37 + z * q^43 - z * q^49 + z^2 * q^61 + z * q^67 - z^2 * q^73 - z^2 * q^79 + z * q^91 - z * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{7}+O(q^{10})$$ 2 * q - q^7 $$2 q - q^{7} + q^{13} + q^{19} + 2 q^{25} + q^{31} + q^{37} + q^{43} - q^{49} - 2 q^{61} + q^{67} + q^{73} + q^{79} + q^{91} - 2 q^{97}+O(q^{100})$$ 2 * q - q^7 + q^13 + q^19 + 2 * q^25 + q^31 + q^37 + q^43 - q^49 - 2 * q^61 + q^67 + q^73 + q^79 + q^91 - 2 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2268\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$1135$$ $$1541$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$ $$-\zeta_{6}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
53.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 0 0 0 −0.500000 + 0.866025i 0 0 0
1241.1 0 0 0 0 0 −0.500000 0.866025i 0 0 0
 $$n$$: 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
63.g even 3 1 inner
63.n odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2268.1.m.a 2
3.b odd 2 1 CM 2268.1.m.a 2
7.c even 3 1 2268.1.bh.b 2
9.c even 3 1 84.1.p.a 2
9.c even 3 1 2268.1.bh.b 2
9.d odd 6 1 84.1.p.a 2
9.d odd 6 1 2268.1.bh.b 2
21.h odd 6 1 2268.1.bh.b 2
36.f odd 6 1 336.1.bn.a 2
36.h even 6 1 336.1.bn.a 2
45.h odd 6 1 2100.1.bn.c 2
45.j even 6 1 2100.1.bn.c 2
45.k odd 12 2 2100.1.bh.a 4
45.l even 12 2 2100.1.bh.a 4
63.g even 3 1 588.1.c.b 1
63.g even 3 1 inner 2268.1.m.a 2
63.h even 3 1 84.1.p.a 2
63.i even 6 1 588.1.p.a 2
63.j odd 6 1 84.1.p.a 2
63.k odd 6 1 588.1.c.a 1
63.l odd 6 1 588.1.p.a 2
63.n odd 6 1 588.1.c.b 1
63.n odd 6 1 inner 2268.1.m.a 2
63.o even 6 1 588.1.p.a 2
63.s even 6 1 588.1.c.a 1
63.t odd 6 1 588.1.p.a 2
72.j odd 6 1 1344.1.bn.b 2
72.l even 6 1 1344.1.bn.a 2
72.n even 6 1 1344.1.bn.b 2
72.p odd 6 1 1344.1.bn.a 2
252.n even 6 1 2352.1.d.b 1
252.o even 6 1 2352.1.d.a 1
252.r odd 6 1 2352.1.bn.a 2
252.s odd 6 1 2352.1.bn.a 2
252.u odd 6 1 336.1.bn.a 2
252.bb even 6 1 336.1.bn.a 2
252.bi even 6 1 2352.1.bn.a 2
252.bj even 6 1 2352.1.bn.a 2
252.bl odd 6 1 2352.1.d.a 1
252.bn odd 6 1 2352.1.d.b 1
315.r even 6 1 2100.1.bn.c 2
315.br odd 6 1 2100.1.bn.c 2
315.bt odd 12 2 2100.1.bh.a 4
315.bv even 12 2 2100.1.bh.a 4
504.bi odd 6 1 1344.1.bn.b 2
504.bt even 6 1 1344.1.bn.a 2
504.ce odd 6 1 1344.1.bn.a 2
504.cq even 6 1 1344.1.bn.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.1.p.a 2 9.c even 3 1
84.1.p.a 2 9.d odd 6 1
84.1.p.a 2 63.h even 3 1
84.1.p.a 2 63.j odd 6 1
336.1.bn.a 2 36.f odd 6 1
336.1.bn.a 2 36.h even 6 1
336.1.bn.a 2 252.u odd 6 1
336.1.bn.a 2 252.bb even 6 1
588.1.c.a 1 63.k odd 6 1
588.1.c.a 1 63.s even 6 1
588.1.c.b 1 63.g even 3 1
588.1.c.b 1 63.n odd 6 1
588.1.p.a 2 63.i even 6 1
588.1.p.a 2 63.l odd 6 1
588.1.p.a 2 63.o even 6 1
588.1.p.a 2 63.t odd 6 1
1344.1.bn.a 2 72.l even 6 1
1344.1.bn.a 2 72.p odd 6 1
1344.1.bn.a 2 504.bt even 6 1
1344.1.bn.a 2 504.ce odd 6 1
1344.1.bn.b 2 72.j odd 6 1
1344.1.bn.b 2 72.n even 6 1
1344.1.bn.b 2 504.bi odd 6 1
1344.1.bn.b 2 504.cq even 6 1
2100.1.bh.a 4 45.k odd 12 2
2100.1.bh.a 4 45.l even 12 2
2100.1.bh.a 4 315.bt odd 12 2
2100.1.bh.a 4 315.bv even 12 2
2100.1.bn.c 2 45.h odd 6 1
2100.1.bn.c 2 45.j even 6 1
2100.1.bn.c 2 315.r even 6 1
2100.1.bn.c 2 315.br odd 6 1
2268.1.m.a 2 1.a even 1 1 trivial
2268.1.m.a 2 3.b odd 2 1 CM
2268.1.m.a 2 63.g even 3 1 inner
2268.1.m.a 2 63.n odd 6 1 inner
2268.1.bh.b 2 7.c even 3 1
2268.1.bh.b 2 9.c even 3 1
2268.1.bh.b 2 9.d odd 6 1
2268.1.bh.b 2 21.h odd 6 1
2352.1.d.a 1 252.o even 6 1
2352.1.d.a 1 252.bl odd 6 1
2352.1.d.b 1 252.n even 6 1
2352.1.d.b 1 252.bn odd 6 1
2352.1.bn.a 2 252.r odd 6 1
2352.1.bn.a 2 252.s odd 6 1
2352.1.bn.a 2 252.bi even 6 1
2352.1.bn.a 2 252.bj even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{13}^{2} - T_{13} + 1$$ acting on $$S_{1}^{\mathrm{new}}(2268, [\chi])$$.

## Hecke characteristic polynomials

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