# Properties

 Label 2496.1.cb.b Level $2496$ Weight $1$ Character orbit 2496.cb Analytic conductor $1.246$ Analytic rank $0$ Dimension $2$ Projective image $D_{6}$ CM discriminant -3 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2496,1,Mod(257,2496)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2496, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 3, 5]))

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

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

chi := DirichletCharacter("2496.257");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2496 = 2^{6} \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2496.cb (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: $$1.24566627153$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 156) Projective image: $$D_{6}$$ Projective field: Galois closure of 6.0.160398576.1

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \zeta_{6}^{2} q^{3} + ( - \zeta_{6}^{2} + 1) q^{7} - \zeta_{6} q^{9} +O(q^{10})$$ q - z^2 * q^3 + (-z^2 + 1) * q^7 - z * q^9 $$q - \zeta_{6}^{2} q^{3} + ( - \zeta_{6}^{2} + 1) q^{7} - \zeta_{6} q^{9} - \zeta_{6} q^{13} + ( - \zeta_{6}^{2} - \zeta_{6}) q^{21} - q^{25} - q^{27} + (\zeta_{6}^{2} + \zeta_{6}) q^{31} - q^{39} + \zeta_{6} q^{43} + ( - \zeta_{6}^{2} - \zeta_{6} + 1) q^{49} + \zeta_{6} q^{61} + ( - \zeta_{6} - 1) q^{63} + (\zeta_{6} + 1) q^{67} + ( - \zeta_{6}^{2} - \zeta_{6}) q^{73} + \zeta_{6}^{2} q^{75} - q^{79} + \zeta_{6}^{2} q^{81} + ( - \zeta_{6} - 1) q^{91} + (\zeta_{6} + 1) q^{93} + (\zeta_{6}^{2} - 1) q^{97} +O(q^{100})$$ q - z^2 * q^3 + (-z^2 + 1) * q^7 - z * q^9 - z * q^13 + (-z^2 - z) * q^21 - q^25 - q^27 + (z^2 + z) * q^31 - q^39 + z * q^43 + (-z^2 - z + 1) * q^49 + z * q^61 + (-z - 1) * q^63 + (z + 1) * q^67 + (-z^2 - z) * q^73 + z^2 * q^75 - q^79 + z^2 * q^81 + (-z - 1) * q^91 + (z + 1) * q^93 + (z^2 - 1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} + 3 q^{7} - q^{9}+O(q^{10})$$ 2 * q + q^3 + 3 * q^7 - q^9 $$2 q + q^{3} + 3 q^{7} - q^{9} - q^{13} - 2 q^{25} - 2 q^{27} - 2 q^{39} + q^{43} + 2 q^{49} + q^{61} - 3 q^{63} + 3 q^{67} - q^{75} - 2 q^{79} - q^{81} - 3 q^{91} + 3 q^{93} - 3 q^{97}+O(q^{100})$$ 2 * q + q^3 + 3 * q^7 - q^9 - q^13 - 2 * q^25 - 2 * q^27 - 2 * q^39 + q^43 + 2 * q^49 + q^61 - 3 * q^63 + 3 * q^67 - q^75 - 2 * q^79 - q^81 - 3 * q^91 + 3 * q^93 - 3 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2496\mathbb{Z}\right)^\times$$.

 $$n$$ $$703$$ $$769$$ $$833$$ $$1093$$ $$\chi(n)$$ $$1$$ $$\zeta_{6}$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
257.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0.500000 + 0.866025i 0 0 0 1.50000 + 0.866025i 0 −0.500000 + 0.866025i 0
641.1 0 0.500000 0.866025i 0 0 0 1.50000 0.866025i 0 −0.500000 0.866025i 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 $$\Q(\sqrt{-3})$$
13.e even 6 1 inner
39.h odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2496.1.cb.b 2
3.b odd 2 1 CM 2496.1.cb.b 2
4.b odd 2 1 2496.1.cb.a 2
8.b even 2 1 624.1.cb.a 2
8.d odd 2 1 156.1.s.a 2
12.b even 2 1 2496.1.cb.a 2
13.e even 6 1 inner 2496.1.cb.b 2
24.f even 2 1 156.1.s.a 2
24.h odd 2 1 624.1.cb.a 2
39.h odd 6 1 inner 2496.1.cb.b 2
40.e odd 2 1 3900.1.ca.b 2
40.k even 4 2 3900.1.br.b 4
52.i odd 6 1 2496.1.cb.a 2
104.h odd 2 1 2028.1.s.a 2
104.m even 4 2 2028.1.o.b 4
104.n odd 6 1 2028.1.g.a 2
104.n odd 6 1 2028.1.s.a 2
104.p odd 6 1 156.1.s.a 2
104.p odd 6 1 2028.1.g.a 2
104.s even 6 1 624.1.cb.a 2
104.u even 12 2 2028.1.d.c 2
104.u even 12 2 2028.1.o.b 4
120.m even 2 1 3900.1.ca.b 2
120.q odd 4 2 3900.1.br.b 4
156.r even 6 1 2496.1.cb.a 2
312.h even 2 1 2028.1.s.a 2
312.w odd 4 2 2028.1.o.b 4
312.ba even 6 1 156.1.s.a 2
312.ba even 6 1 2028.1.g.a 2
312.bg odd 6 1 624.1.cb.a 2
312.bn even 6 1 2028.1.g.a 2
312.bn even 6 1 2028.1.s.a 2
312.bq odd 12 2 2028.1.d.c 2
312.bq odd 12 2 2028.1.o.b 4
520.cd odd 6 1 3900.1.ca.b 2
520.cs even 12 2 3900.1.br.b 4
1560.dv even 6 1 3900.1.ca.b 2
1560.es odd 12 2 3900.1.br.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.1.s.a 2 8.d odd 2 1
156.1.s.a 2 24.f even 2 1
156.1.s.a 2 104.p odd 6 1
156.1.s.a 2 312.ba even 6 1
624.1.cb.a 2 8.b even 2 1
624.1.cb.a 2 24.h odd 2 1
624.1.cb.a 2 104.s even 6 1
624.1.cb.a 2 312.bg odd 6 1
2028.1.d.c 2 104.u even 12 2
2028.1.d.c 2 312.bq odd 12 2
2028.1.g.a 2 104.n odd 6 1
2028.1.g.a 2 104.p odd 6 1
2028.1.g.a 2 312.ba even 6 1
2028.1.g.a 2 312.bn even 6 1
2028.1.o.b 4 104.m even 4 2
2028.1.o.b 4 104.u even 12 2
2028.1.o.b 4 312.w odd 4 2
2028.1.o.b 4 312.bq odd 12 2
2028.1.s.a 2 104.h odd 2 1
2028.1.s.a 2 104.n odd 6 1
2028.1.s.a 2 312.h even 2 1
2028.1.s.a 2 312.bn even 6 1
2496.1.cb.a 2 4.b odd 2 1
2496.1.cb.a 2 12.b even 2 1
2496.1.cb.a 2 52.i odd 6 1
2496.1.cb.a 2 156.r even 6 1
2496.1.cb.b 2 1.a even 1 1 trivial
2496.1.cb.b 2 3.b odd 2 1 CM
2496.1.cb.b 2 13.e even 6 1 inner
2496.1.cb.b 2 39.h odd 6 1 inner
3900.1.br.b 4 40.k even 4 2
3900.1.br.b 4 120.q odd 4 2
3900.1.br.b 4 520.cs even 12 2
3900.1.br.b 4 1560.es odd 12 2
3900.1.ca.b 2 40.e odd 2 1
3900.1.ca.b 2 120.m even 2 1
3900.1.ca.b 2 520.cd odd 6 1
3900.1.ca.b 2 1560.dv even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{2} - 3T_{7} + 3$$ acting on $$S_{1}^{\mathrm{new}}(2496, [\chi])$$.

## Hecke characteristic polynomials

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