# Properties

 Label 2496.1.l.c Level $2496$ Weight $1$ Character orbit 2496.l Analytic conductor $1.246$ Analytic rank $0$ Dimension $4$ Projective image $D_{4}$ RM discriminant 156 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2496,1,Mod(1793,2496)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2496, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))

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

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

chi := DirichletCharacter("2496.1793");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2496 = 2^{6} \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2496.l (of order $$2$$, degree $$1$$, 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: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 1248) Projective image: $$D_{4}$$ Projective field: Galois closure of 4.0.8112.1

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \zeta_{8}^{2} q^{3} + (\zeta_{8}^{3} - \zeta_{8}) q^{5} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{7} - q^{9}+O(q^{10})$$ q + z^2 * q^3 + (z^3 - z) * q^5 + (-z^3 - z) * q^7 - q^9 $$q + \zeta_{8}^{2} q^{3} + (\zeta_{8}^{3} - \zeta_{8}) q^{5} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{7} - q^{9} + q^{13} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{15} + (\zeta_{8}^{3} + \zeta_{8}) q^{19} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{21} + \zeta_{8}^{2} q^{23} + q^{25} - \zeta_{8}^{2} q^{27} + (\zeta_{8}^{3} + \zeta_{8}) q^{31} + \zeta_{8}^{2} q^{35} + \zeta_{8}^{2} q^{39} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{41} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{45} - q^{49} + (\zeta_{8}^{3} - \zeta_{8}) q^{57} + (\zeta_{8}^{3} + \zeta_{8}) q^{63} + (\zeta_{8}^{3} - \zeta_{8}) q^{65} + (\zeta_{8}^{3} + \zeta_{8}) q^{67} - 2 q^{69} + \zeta_{8}^{2} q^{75} + q^{81} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{89} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{91} + (\zeta_{8}^{3} - \zeta_{8}) q^{93} - \zeta_{8}^{2} q^{95} +O(q^{100})$$ q + z^2 * q^3 + (z^3 - z) * q^5 + (-z^3 - z) * q^7 - q^9 + q^13 + (-z^3 - z) * q^15 + (z^3 + z) * q^19 + (-z^3 + z) * q^21 + z^2 * q^23 + q^25 - z^2 * q^27 + (z^3 + z) * q^31 + z^2 * q^35 + z^2 * q^39 + (-z^3 + z) * q^41 + (-z^3 + z) * q^45 - q^49 + (z^3 - z) * q^57 + (z^3 + z) * q^63 + (z^3 - z) * q^65 + (z^3 + z) * q^67 - 2 * q^69 + z^2 * q^75 + q^81 + (-z^3 + z) * q^89 + (-z^3 - z) * q^91 + (z^3 - z) * q^93 - z^2 * q^95 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^9 $$4 q - 4 q^{9} + 4 q^{13} + 4 q^{25} - 4 q^{49} - 8 q^{69} + 4 q^{81}+O(q^{100})$$ 4 * q - 4 * q^9 + 4 * q^13 + 4 * q^25 - 4 * q^49 - 8 * q^69 + 4 * q^81

## Character values

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

 $$n$$ $$703$$ $$769$$ $$833$$ $$1093$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1793.1
 0.707107 − 0.707107i −0.707107 + 0.707107i 0.707107 + 0.707107i −0.707107 − 0.707107i
0 1.00000i 0 −1.41421 0 1.41421i 0 −1.00000 0
1793.2 0 1.00000i 0 1.41421 0 1.41421i 0 −1.00000 0
1793.3 0 1.00000i 0 −1.41421 0 1.41421i 0 −1.00000 0
1793.4 0 1.00000i 0 1.41421 0 1.41421i 0 −1.00000 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
156.h even 2 1 RM by $$\Q(\sqrt{39})$$
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner
13.b even 2 1 inner
39.d odd 2 1 inner
52.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2496.1.l.c 4
3.b odd 2 1 inner 2496.1.l.c 4
4.b odd 2 1 inner 2496.1.l.c 4
8.b even 2 1 1248.1.l.a 4
8.d odd 2 1 1248.1.l.a 4
12.b even 2 1 inner 2496.1.l.c 4
13.b even 2 1 inner 2496.1.l.c 4
24.f even 2 1 1248.1.l.a 4
24.h odd 2 1 1248.1.l.a 4
39.d odd 2 1 inner 2496.1.l.c 4
52.b odd 2 1 inner 2496.1.l.c 4
104.e even 2 1 1248.1.l.a 4
104.h odd 2 1 1248.1.l.a 4
156.h even 2 1 RM 2496.1.l.c 4
312.b odd 2 1 1248.1.l.a 4
312.h even 2 1 1248.1.l.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1248.1.l.a 4 8.b even 2 1
1248.1.l.a 4 8.d odd 2 1
1248.1.l.a 4 24.f even 2 1
1248.1.l.a 4 24.h odd 2 1
1248.1.l.a 4 104.e even 2 1
1248.1.l.a 4 104.h odd 2 1
1248.1.l.a 4 312.b odd 2 1
1248.1.l.a 4 312.h even 2 1
2496.1.l.c 4 1.a even 1 1 trivial
2496.1.l.c 4 3.b odd 2 1 inner
2496.1.l.c 4 4.b odd 2 1 inner
2496.1.l.c 4 12.b even 2 1 inner
2496.1.l.c 4 13.b even 2 1 inner
2496.1.l.c 4 39.d odd 2 1 inner
2496.1.l.c 4 52.b odd 2 1 inner
2496.1.l.c 4 156.h even 2 1 RM

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(2496, [\chi])$$:

 $$T_{5}^{2} - 2$$ T5^2 - 2 $$T_{43}$$ T43

## Hecke characteristic polynomials

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