# Properties

 Label 48.2.a.a Level $48$ Weight $2$ Character orbit 48.a Self dual yes Analytic conductor $0.383$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,2,Mod(1,48)]

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([0, 0, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("48.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$48 = 2^{4} \cdot 3$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 48.a (trivial)

## 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: yes Analytic conductor: $$0.383281929702$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 24) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + q^{3} - 2 q^{5} + q^{9}+O(q^{10})$$ q + q^3 - 2 * q^5 + q^9 $$q + q^{3} - 2 q^{5} + q^{9} - 4 q^{11} - 2 q^{13} - 2 q^{15} + 2 q^{17} + 4 q^{19} + 8 q^{23} - q^{25} + q^{27} + 6 q^{29} - 8 q^{31} - 4 q^{33} + 6 q^{37} - 2 q^{39} - 6 q^{41} - 4 q^{43} - 2 q^{45} - 7 q^{49} + 2 q^{51} - 2 q^{53} + 8 q^{55} + 4 q^{57} - 4 q^{59} - 2 q^{61} + 4 q^{65} + 4 q^{67} + 8 q^{69} - 8 q^{71} + 10 q^{73} - q^{75} + 8 q^{79} + q^{81} + 4 q^{83} - 4 q^{85} + 6 q^{87} - 6 q^{89} - 8 q^{93} - 8 q^{95} + 2 q^{97} - 4 q^{99}+O(q^{100})$$ q + q^3 - 2 * q^5 + q^9 - 4 * q^11 - 2 * q^13 - 2 * q^15 + 2 * q^17 + 4 * q^19 + 8 * q^23 - q^25 + q^27 + 6 * q^29 - 8 * q^31 - 4 * q^33 + 6 * q^37 - 2 * q^39 - 6 * q^41 - 4 * q^43 - 2 * q^45 - 7 * q^49 + 2 * q^51 - 2 * q^53 + 8 * q^55 + 4 * q^57 - 4 * q^59 - 2 * q^61 + 4 * q^65 + 4 * q^67 + 8 * q^69 - 8 * q^71 + 10 * q^73 - q^75 + 8 * q^79 + q^81 + 4 * q^83 - 4 * q^85 + 6 * q^87 - 6 * q^89 - 8 * q^93 - 8 * q^95 + 2 * q^97 - 4 * q^99

## Expression as an eta quotient

$$f(z) = \dfrac{\eta(4z)^{4}\eta(12z)^{4}}{\eta(2z)\eta(6z)\eta(8z)\eta(24z)}=q\prod_{n=1}^\infty(1 - q^{2n})^{-1}(1 - q^{4n})^{4}(1 - q^{6n})^{-1}(1 - q^{8n})^{-1}(1 - q^{12n})^{4}(1 - q^{24n})^{-1}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 1.00000 0 −2.00000 0 0 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 48.2.a.a 1
3.b odd 2 1 144.2.a.b 1
4.b odd 2 1 24.2.a.a 1
5.b even 2 1 1200.2.a.d 1
5.c odd 4 2 1200.2.f.b 2
7.b odd 2 1 2352.2.a.i 1
7.c even 3 2 2352.2.q.l 2
7.d odd 6 2 2352.2.q.r 2
8.b even 2 1 192.2.a.b 1
8.d odd 2 1 192.2.a.d 1
9.c even 3 2 1296.2.i.m 2
9.d odd 6 2 1296.2.i.e 2
11.b odd 2 1 5808.2.a.s 1
12.b even 2 1 72.2.a.a 1
13.b even 2 1 8112.2.a.be 1
15.d odd 2 1 3600.2.a.v 1
15.e even 4 2 3600.2.f.r 2
16.e even 4 2 768.2.d.d 2
16.f odd 4 2 768.2.d.e 2
20.d odd 2 1 600.2.a.h 1
20.e even 4 2 600.2.f.e 2
21.c even 2 1 7056.2.a.q 1
24.f even 2 1 576.2.a.d 1
24.h odd 2 1 576.2.a.b 1
28.d even 2 1 1176.2.a.i 1
28.f even 6 2 1176.2.q.a 2
28.g odd 6 2 1176.2.q.i 2
36.f odd 6 2 648.2.i.g 2
36.h even 6 2 648.2.i.b 2
40.e odd 2 1 4800.2.a.q 1
40.f even 2 1 4800.2.a.cc 1
40.i odd 4 2 4800.2.f.bg 2
40.k even 4 2 4800.2.f.d 2
44.c even 2 1 2904.2.a.c 1
48.i odd 4 2 2304.2.d.k 2
48.k even 4 2 2304.2.d.i 2
52.b odd 2 1 4056.2.a.i 1
52.f even 4 2 4056.2.c.e 2
56.e even 2 1 9408.2.a.h 1
56.h odd 2 1 9408.2.a.cc 1
60.h even 2 1 1800.2.a.m 1
60.l odd 4 2 1800.2.f.c 2
68.d odd 2 1 6936.2.a.p 1
76.d even 2 1 8664.2.a.j 1
84.h odd 2 1 3528.2.a.d 1
84.j odd 6 2 3528.2.s.y 2
84.n even 6 2 3528.2.s.j 2
132.d odd 2 1 8712.2.a.u 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.2.a.a 1 4.b odd 2 1
48.2.a.a 1 1.a even 1 1 trivial
72.2.a.a 1 12.b even 2 1
144.2.a.b 1 3.b odd 2 1
192.2.a.b 1 8.b even 2 1
192.2.a.d 1 8.d odd 2 1
576.2.a.b 1 24.h odd 2 1
576.2.a.d 1 24.f even 2 1
600.2.a.h 1 20.d odd 2 1
600.2.f.e 2 20.e even 4 2
648.2.i.b 2 36.h even 6 2
648.2.i.g 2 36.f odd 6 2
768.2.d.d 2 16.e even 4 2
768.2.d.e 2 16.f odd 4 2
1176.2.a.i 1 28.d even 2 1
1176.2.q.a 2 28.f even 6 2
1176.2.q.i 2 28.g odd 6 2
1200.2.a.d 1 5.b even 2 1
1200.2.f.b 2 5.c odd 4 2
1296.2.i.e 2 9.d odd 6 2
1296.2.i.m 2 9.c even 3 2
1800.2.a.m 1 60.h even 2 1
1800.2.f.c 2 60.l odd 4 2
2304.2.d.i 2 48.k even 4 2
2304.2.d.k 2 48.i odd 4 2
2352.2.a.i 1 7.b odd 2 1
2352.2.q.l 2 7.c even 3 2
2352.2.q.r 2 7.d odd 6 2
2904.2.a.c 1 44.c even 2 1
3528.2.a.d 1 84.h odd 2 1
3528.2.s.j 2 84.n even 6 2
3528.2.s.y 2 84.j odd 6 2
3600.2.a.v 1 15.d odd 2 1
3600.2.f.r 2 15.e even 4 2
4056.2.a.i 1 52.b odd 2 1
4056.2.c.e 2 52.f even 4 2
4800.2.a.q 1 40.e odd 2 1
4800.2.a.cc 1 40.f even 2 1
4800.2.f.d 2 40.k even 4 2
4800.2.f.bg 2 40.i odd 4 2
5808.2.a.s 1 11.b odd 2 1
6936.2.a.p 1 68.d odd 2 1
7056.2.a.q 1 21.c even 2 1
8112.2.a.be 1 13.b even 2 1
8664.2.a.j 1 76.d even 2 1
8712.2.a.u 1 132.d odd 2 1
9408.2.a.h 1 56.e even 2 1
9408.2.a.cc 1 56.h odd 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(\Gamma_0(48))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 1$$
$5$ $$T + 2$$
$7$ $$T$$
$11$ $$T + 4$$
$13$ $$T + 2$$
$17$ $$T - 2$$
$19$ $$T - 4$$
$23$ $$T - 8$$
$29$ $$T - 6$$
$31$ $$T + 8$$
$37$ $$T - 6$$
$41$ $$T + 6$$
$43$ $$T + 4$$
$47$ $$T$$
$53$ $$T + 2$$
$59$ $$T + 4$$
$61$ $$T + 2$$
$67$ $$T - 4$$
$71$ $$T + 8$$
$73$ $$T - 10$$
$79$ $$T - 8$$
$83$ $$T - 4$$
$89$ $$T + 6$$
$97$ $$T - 2$$