# Properties

 Label 48.2.c.a Level $48$ Weight $2$ Character orbit 48.c Analytic conductor $0.383$ 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] = [48,2,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, 2, names="a")

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

chi := DirichletCharacter("48.47");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$48 = 2^{4} \cdot 3$$ Weight: $$k$$ $$=$$ $$2$$ 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: $$0.383281929702$$ 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: $$2$$ 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 $$\beta = \sqrt{-3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{3} + 2 \beta q^{7} - 3 q^{9} +O(q^{10})$$ q - b * q^3 + 2*b * q^7 - 3 * q^9 $$q - \beta q^{3} + 2 \beta q^{7} - 3 q^{9} - 2 q^{13} - 2 \beta q^{19} + 6 q^{21} + 5 q^{25} + 3 \beta q^{27} - 6 \beta q^{31} - 10 q^{37} + 2 \beta q^{39} + 6 \beta q^{43} - 5 q^{49} - 6 q^{57} + 14 q^{61} - 6 \beta q^{63} - 2 \beta q^{67} + 10 q^{73} - 5 \beta q^{75} + 10 \beta q^{79} + 9 q^{81} - 4 \beta q^{91} - 18 q^{93} - 14 q^{97} +O(q^{100})$$ q - b * q^3 + 2*b * q^7 - 3 * q^9 - 2 * q^13 - 2*b * q^19 + 6 * q^21 + 5 * q^25 + 3*b * q^27 - 6*b * q^31 - 10 * q^37 + 2*b * q^39 + 6*b * q^43 - 5 * q^49 - 6 * q^57 + 14 * q^61 - 6*b * q^63 - 2*b * q^67 + 10 * q^73 - 5*b * q^75 + 10*b * q^79 + 9 * q^81 - 4*b * q^91 - 18 * q^93 - 14 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 6 q^{9}+O(q^{10})$$ 2 * q - 6 * q^9 $$2 q - 6 q^{9} - 4 q^{13} + 12 q^{21} + 10 q^{25} - 20 q^{37} - 10 q^{49} - 12 q^{57} + 28 q^{61} + 20 q^{73} + 18 q^{81} - 36 q^{93} - 28 q^{97}+O(q^{100})$$ 2 * q - 6 * q^9 - 4 * q^13 + 12 * q^21 + 10 * q^25 - 20 * q^37 - 10 * q^49 - 12 * q^57 + 28 * q^61 + 20 * q^73 + 18 * q^81 - 36 * q^93 - 28 * 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 1.73205i 0 0 0 3.46410i 0 −3.00000 0
47.2 0 1.73205i 0 0 0 3.46410i 0 −3.00000 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
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 48.2.c.a 2
3.b odd 2 1 CM 48.2.c.a 2
4.b odd 2 1 inner 48.2.c.a 2
5.b even 2 1 1200.2.h.e 2
5.c odd 4 2 1200.2.o.i 4
7.b odd 2 1 2352.2.h.c 2
8.b even 2 1 192.2.c.a 2
8.d odd 2 1 192.2.c.a 2
9.c even 3 1 1296.2.s.b 2
9.c even 3 1 1296.2.s.e 2
9.d odd 6 1 1296.2.s.b 2
9.d odd 6 1 1296.2.s.e 2
12.b even 2 1 inner 48.2.c.a 2
15.d odd 2 1 1200.2.h.e 2
15.e even 4 2 1200.2.o.i 4
16.e even 4 2 768.2.f.d 4
16.f odd 4 2 768.2.f.d 4
20.d odd 2 1 1200.2.h.e 2
20.e even 4 2 1200.2.o.i 4
21.c even 2 1 2352.2.h.c 2
24.f even 2 1 192.2.c.a 2
24.h odd 2 1 192.2.c.a 2
28.d even 2 1 2352.2.h.c 2
36.f odd 6 1 1296.2.s.b 2
36.f odd 6 1 1296.2.s.e 2
36.h even 6 1 1296.2.s.b 2
36.h even 6 1 1296.2.s.e 2
48.i odd 4 2 768.2.f.d 4
48.k even 4 2 768.2.f.d 4
60.h even 2 1 1200.2.h.e 2
60.l odd 4 2 1200.2.o.i 4
84.h odd 2 1 2352.2.h.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.2.c.a 2 1.a even 1 1 trivial
48.2.c.a 2 3.b odd 2 1 CM
48.2.c.a 2 4.b odd 2 1 inner
48.2.c.a 2 12.b even 2 1 inner
192.2.c.a 2 8.b even 2 1
192.2.c.a 2 8.d odd 2 1
192.2.c.a 2 24.f even 2 1
192.2.c.a 2 24.h odd 2 1
768.2.f.d 4 16.e even 4 2
768.2.f.d 4 16.f odd 4 2
768.2.f.d 4 48.i odd 4 2
768.2.f.d 4 48.k even 4 2
1200.2.h.e 2 5.b even 2 1
1200.2.h.e 2 15.d odd 2 1
1200.2.h.e 2 20.d odd 2 1
1200.2.h.e 2 60.h even 2 1
1200.2.o.i 4 5.c odd 4 2
1200.2.o.i 4 15.e even 4 2
1200.2.o.i 4 20.e even 4 2
1200.2.o.i 4 60.l odd 4 2
1296.2.s.b 2 9.c even 3 1
1296.2.s.b 2 9.d odd 6 1
1296.2.s.b 2 36.f odd 6 1
1296.2.s.b 2 36.h even 6 1
1296.2.s.e 2 9.c even 3 1
1296.2.s.e 2 9.d odd 6 1
1296.2.s.e 2 36.f odd 6 1
1296.2.s.e 2 36.h even 6 1
2352.2.h.c 2 7.b odd 2 1
2352.2.h.c 2 21.c even 2 1
2352.2.h.c 2 28.d even 2 1
2352.2.h.c 2 84.h odd 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(48, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 3$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 12$$
$11$ $$T^{2}$$
$13$ $$(T + 2)^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2} + 12$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2} + 108$$
$37$ $$(T + 10)^{2}$$
$41$ $$T^{2}$$
$43$ $$T^{2} + 108$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$(T - 14)^{2}$$
$67$ $$T^{2} + 12$$
$71$ $$T^{2}$$
$73$ $$(T - 10)^{2}$$
$79$ $$T^{2} + 300$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$(T + 14)^{2}$$