# Properties

 Label 2548.1.bi.a Level $2548$ Weight $1$ Character orbit 2548.bi Analytic conductor $1.272$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -4 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2548,1,Mod(471,2548)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2548, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 2, 2]))

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

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

chi := DirichletCharacter("2548.471");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2548 = 2^{2} \cdot 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2548.bi (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: $$1.27161765219$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 52) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.676.1 Artin image: $C_6\times S_3$ Artin field: Galois closure of $$\mathbb{Q}[x]/(x^{12} - \cdots)$$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + \zeta_{6}^{2} q^{5} + q^{8} - \zeta_{6} q^{9} +O(q^{10})$$ q + q^2 + q^4 + z^2 * q^5 + q^8 - z * q^9 $$q + q^{2} + q^{4} + \zeta_{6}^{2} q^{5} + q^{8} - \zeta_{6} q^{9} + \zeta_{6}^{2} q^{10} + \zeta_{6} q^{13} + q^{16} + q^{17} - \zeta_{6} q^{18} + \zeta_{6}^{2} q^{20} + \zeta_{6} q^{26} + \zeta_{6} q^{29} + q^{32} + q^{34} - \zeta_{6} q^{36} - q^{37} + \zeta_{6}^{2} q^{40} - \zeta_{6} q^{41} + q^{45} + \zeta_{6} q^{52} + \zeta_{6} q^{53} + \zeta_{6} q^{58} - \zeta_{6} q^{61} + q^{64} - q^{65} + q^{68} - \zeta_{6} q^{72} - \zeta_{6} q^{73} - q^{74} + \zeta_{6}^{2} q^{80} + \zeta_{6}^{2} q^{81} - \zeta_{6} q^{82} + \zeta_{6}^{2} q^{85} - q^{89} + q^{90} - \zeta_{6}^{2} q^{97} +O(q^{100})$$ q + q^2 + q^4 + z^2 * q^5 + q^8 - z * q^9 + z^2 * q^10 + z * q^13 + q^16 + q^17 - z * q^18 + z^2 * q^20 + z * q^26 + z * q^29 + q^32 + q^34 - z * q^36 - q^37 + z^2 * q^40 - z * q^41 + q^45 + z * q^52 + z * q^53 + z * q^58 - z * q^61 + q^64 - q^65 + q^68 - z * q^72 - z * q^73 - q^74 + z^2 * q^80 + z^2 * q^81 - z * q^82 + z^2 * q^85 - q^89 + q^90 - z^2 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{4} - q^{5} + 2 q^{8} - q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^4 - q^5 + 2 * q^8 - q^9 $$2 q + 2 q^{2} + 2 q^{4} - q^{5} + 2 q^{8} - q^{9} - q^{10} + q^{13} + 2 q^{16} + 2 q^{17} - q^{18} - q^{20} + q^{26} + q^{29} + 2 q^{32} + 2 q^{34} - q^{36} - 2 q^{37} - q^{40} - q^{41} + 2 q^{45} + q^{52} + q^{53} + q^{58} - q^{61} + 2 q^{64} - 2 q^{65} + 2 q^{68} - q^{72} - q^{73} - 2 q^{74} - q^{80} - q^{81} - q^{82} - q^{85} - 4 q^{89} + 2 q^{90} + 2 q^{97}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^4 - q^5 + 2 * q^8 - q^9 - q^10 + q^13 + 2 * q^16 + 2 * q^17 - q^18 - q^20 + q^26 + q^29 + 2 * q^32 + 2 * q^34 - q^36 - 2 * q^37 - q^40 - q^41 + 2 * q^45 + q^52 + q^53 + q^58 - q^61 + 2 * q^64 - 2 * q^65 + 2 * q^68 - q^72 - q^73 - 2 * q^74 - q^80 - q^81 - q^82 - q^85 - 4 * q^89 + 2 * q^90 + 2 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2548\mathbb{Z}\right)^\times$$.

 $$n$$ $$197$$ $$885$$ $$1275$$ $$\chi(n)$$ $$\zeta_{6}^{2}$$ $$\zeta_{6}^{2}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
471.1
 0.5 + 0.866025i 0.5 − 0.866025i
1.00000 0 1.00000 −0.500000 + 0.866025i 0 0 1.00000 −0.500000 0.866025i −0.500000 + 0.866025i
1439.1 1.00000 0 1.00000 −0.500000 0.866025i 0 0 1.00000 −0.500000 + 0.866025i −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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
91.h even 3 1 inner
364.bi odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2548.1.bi.a 2
4.b odd 2 1 CM 2548.1.bi.a 2
7.b odd 2 1 2548.1.bi.b 2
7.c even 3 1 2548.1.q.a 2
7.c even 3 1 2548.1.bn.a 2
7.d odd 6 1 52.1.j.a 2
7.d odd 6 1 2548.1.q.b 2
13.c even 3 1 2548.1.q.a 2
21.g even 6 1 468.1.br.a 2
28.d even 2 1 2548.1.bi.b 2
28.f even 6 1 52.1.j.a 2
28.f even 6 1 2548.1.q.b 2
28.g odd 6 1 2548.1.q.a 2
28.g odd 6 1 2548.1.bn.a 2
35.i odd 6 1 1300.1.bc.a 2
35.k even 12 2 1300.1.w.a 4
52.j odd 6 1 2548.1.q.a 2
56.j odd 6 1 832.1.bb.a 2
56.m even 6 1 832.1.bb.a 2
84.j odd 6 1 468.1.br.a 2
91.g even 3 1 2548.1.bn.a 2
91.h even 3 1 inner 2548.1.bi.a 2
91.l odd 6 1 676.1.c.a 1
91.m odd 6 1 52.1.j.a 2
91.n odd 6 1 2548.1.q.b 2
91.p odd 6 1 676.1.j.a 2
91.s odd 6 1 676.1.j.a 2
91.v odd 6 1 676.1.c.b 1
91.v odd 6 1 2548.1.bi.b 2
91.w even 12 2 676.1.i.a 4
91.ba even 12 2 676.1.b.a 2
91.bb even 12 2 676.1.i.a 4
112.v even 12 2 3328.1.v.b 4
112.x odd 12 2 3328.1.v.b 4
140.s even 6 1 1300.1.bc.a 2
140.x odd 12 2 1300.1.w.a 4
273.bf even 6 1 468.1.br.a 2
364.q odd 6 1 2548.1.bn.a 2
364.v even 6 1 2548.1.q.b 2
364.w even 6 1 676.1.c.a 1
364.x even 6 1 676.1.j.a 2
364.ba even 6 1 676.1.c.b 1
364.ba even 6 1 2548.1.bi.b 2
364.bi odd 6 1 inner 2548.1.bi.a 2
364.bp even 6 1 676.1.j.a 2
364.br even 6 1 52.1.j.a 2
364.bw odd 12 2 676.1.i.a 4
364.bz odd 12 2 676.1.b.a 2
364.cg odd 12 2 676.1.i.a 4
455.bw odd 6 1 1300.1.bc.a 2
455.db even 12 2 1300.1.w.a 4
728.bf even 6 1 832.1.bb.a 2
728.de odd 6 1 832.1.bb.a 2
1092.cc odd 6 1 468.1.br.a 2
1456.fj odd 12 2 3328.1.v.b 4
1456.fm even 12 2 3328.1.v.b 4
1820.bt even 6 1 1300.1.bc.a 2
1820.er odd 12 2 1300.1.w.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
52.1.j.a 2 7.d odd 6 1
52.1.j.a 2 28.f even 6 1
52.1.j.a 2 91.m odd 6 1
52.1.j.a 2 364.br even 6 1
468.1.br.a 2 21.g even 6 1
468.1.br.a 2 84.j odd 6 1
468.1.br.a 2 273.bf even 6 1
468.1.br.a 2 1092.cc odd 6 1
676.1.b.a 2 91.ba even 12 2
676.1.b.a 2 364.bz odd 12 2
676.1.c.a 1 91.l odd 6 1
676.1.c.a 1 364.w even 6 1
676.1.c.b 1 91.v odd 6 1
676.1.c.b 1 364.ba even 6 1
676.1.i.a 4 91.w even 12 2
676.1.i.a 4 91.bb even 12 2
676.1.i.a 4 364.bw odd 12 2
676.1.i.a 4 364.cg odd 12 2
676.1.j.a 2 91.p odd 6 1
676.1.j.a 2 91.s odd 6 1
676.1.j.a 2 364.x even 6 1
676.1.j.a 2 364.bp even 6 1
832.1.bb.a 2 56.j odd 6 1
832.1.bb.a 2 56.m even 6 1
832.1.bb.a 2 728.bf even 6 1
832.1.bb.a 2 728.de odd 6 1
1300.1.w.a 4 35.k even 12 2
1300.1.w.a 4 140.x odd 12 2
1300.1.w.a 4 455.db even 12 2
1300.1.w.a 4 1820.er odd 12 2
1300.1.bc.a 2 35.i odd 6 1
1300.1.bc.a 2 140.s even 6 1
1300.1.bc.a 2 455.bw odd 6 1
1300.1.bc.a 2 1820.bt even 6 1
2548.1.q.a 2 7.c even 3 1
2548.1.q.a 2 13.c even 3 1
2548.1.q.a 2 28.g odd 6 1
2548.1.q.a 2 52.j odd 6 1
2548.1.q.b 2 7.d odd 6 1
2548.1.q.b 2 28.f even 6 1
2548.1.q.b 2 91.n odd 6 1
2548.1.q.b 2 364.v even 6 1
2548.1.bi.a 2 1.a even 1 1 trivial
2548.1.bi.a 2 4.b odd 2 1 CM
2548.1.bi.a 2 91.h even 3 1 inner
2548.1.bi.a 2 364.bi odd 6 1 inner
2548.1.bi.b 2 7.b odd 2 1
2548.1.bi.b 2 28.d even 2 1
2548.1.bi.b 2 91.v odd 6 1
2548.1.bi.b 2 364.ba even 6 1
2548.1.bn.a 2 7.c even 3 1
2548.1.bn.a 2 28.g odd 6 1
2548.1.bn.a 2 91.g even 3 1
2548.1.bn.a 2 364.q odd 6 1
3328.1.v.b 4 112.v even 12 2
3328.1.v.b 4 112.x odd 12 2
3328.1.v.b 4 1456.fj odd 12 2
3328.1.v.b 4 1456.fm even 12 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + T_{5} + 1$$ acting on $$S_{1}^{\mathrm{new}}(2548, [\chi])$$.

## Hecke characteristic polynomials

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