# Properties

 Label 48.4.c.a Level $48$ Weight $4$ Character orbit 48.c Analytic conductor $2.832$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,4,Mod(47,48)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(48, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("48.47");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$48 = 2^{4} \cdot 3$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 48.c (of order $$2$$, degree $$1$$, 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: $$2.83209168028$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2\cdot 3$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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 $$\beta = 3\sqrt{-3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{3} - 6 \beta q^{7} - 27 q^{9} +O(q^{10})$$ q - b * q^3 - 6*b * q^7 - 27 * q^9 $$q - \beta q^{3} - 6 \beta q^{7} - 27 q^{9} + 70 q^{13} + 30 \beta q^{19} - 162 q^{21} + 125 q^{25} + 27 \beta q^{27} - 30 \beta q^{31} + 110 q^{37} - 70 \beta q^{39} - 42 \beta q^{43} - 629 q^{49} + 810 q^{57} + 182 q^{61} + 162 \beta q^{63} + 126 \beta q^{67} - 1190 q^{73} - 125 \beta q^{75} + 210 \beta q^{79} + 729 q^{81} - 420 \beta q^{91} - 810 q^{93} + 1330 q^{97} +O(q^{100})$$ q - b * q^3 - 6*b * q^7 - 27 * q^9 + 70 * q^13 + 30*b * q^19 - 162 * q^21 + 125 * q^25 + 27*b * q^27 - 30*b * q^31 + 110 * q^37 - 70*b * q^39 - 42*b * q^43 - 629 * q^49 + 810 * q^57 + 182 * q^61 + 162*b * q^63 + 126*b * q^67 - 1190 * q^73 - 125*b * q^75 + 210*b * q^79 + 729 * q^81 - 420*b * q^91 - 810 * q^93 + 1330 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 54 q^{9}+O(q^{10})$$ 2 * q - 54 * q^9 $$2 q - 54 q^{9} + 140 q^{13} - 324 q^{21} + 250 q^{25} + 220 q^{37} - 1258 q^{49} + 1620 q^{57} + 364 q^{61} - 2380 q^{73} + 1458 q^{81} - 1620 q^{93} + 2660 q^{97}+O(q^{100})$$ 2 * q - 54 * q^9 + 140 * q^13 - 324 * q^21 + 250 * q^25 + 220 * q^37 - 1258 * q^49 + 1620 * q^57 + 364 * q^61 - 2380 * q^73 + 1458 * q^81 - 1620 * q^93 + 2660 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/48\mathbb{Z}\right)^\times$$.

 $$n$$ $$17$$ $$31$$ $$37$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
47.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 5.19615i 0 0 0 31.1769i 0 −27.0000 0
47.2 0 5.19615i 0 0 0 31.1769i 0 −27.0000 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 $$\Q(\sqrt{-3})$$
4.b odd 2 1 inner
12.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 48.4.c.a 2
3.b odd 2 1 CM 48.4.c.a 2
4.b odd 2 1 inner 48.4.c.a 2
8.b even 2 1 192.4.c.a 2
8.d odd 2 1 192.4.c.a 2
12.b even 2 1 inner 48.4.c.a 2
16.e even 4 2 768.4.f.a 4
16.f odd 4 2 768.4.f.a 4
24.f even 2 1 192.4.c.a 2
24.h odd 2 1 192.4.c.a 2
48.i odd 4 2 768.4.f.a 4
48.k even 4 2 768.4.f.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.4.c.a 2 1.a even 1 1 trivial
48.4.c.a 2 3.b odd 2 1 CM
48.4.c.a 2 4.b odd 2 1 inner
48.4.c.a 2 12.b even 2 1 inner
192.4.c.a 2 8.b even 2 1
192.4.c.a 2 8.d odd 2 1
192.4.c.a 2 24.f even 2 1
192.4.c.a 2 24.h odd 2 1
768.4.f.a 4 16.e even 4 2
768.4.f.a 4 16.f odd 4 2
768.4.f.a 4 48.i odd 4 2
768.4.f.a 4 48.k even 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}$$ acting on $$S_{4}^{\mathrm{new}}(48, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 27$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 972$$
$11$ $$T^{2}$$
$13$ $$(T - 70)^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2} + 24300$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2} + 24300$$
$37$ $$(T - 110)^{2}$$
$41$ $$T^{2}$$
$43$ $$T^{2} + 47628$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$(T - 182)^{2}$$
$67$ $$T^{2} + 428652$$
$71$ $$T^{2}$$
$73$ $$(T + 1190)^{2}$$
$79$ $$T^{2} + 1190700$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$(T - 1330)^{2}$$