# Properties

 Label 48.3.g.a Level $48$ Weight $3$ Character orbit 48.g Analytic conductor $1.308$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,3,Mod(31,48)]

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, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("48.31");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$48 = 2^{4} \cdot 3$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 48.g (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: $$1.30790526893$$ 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: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \beta q^{3} + 6 q^{5} - 4 \beta q^{7} - 3 q^{9} +O(q^{10})$$ q - b * q^3 + 6 * q^5 - 4*b * q^7 - 3 * q^9 $$q - \beta q^{3} + 6 q^{5} - 4 \beta q^{7} - 3 q^{9} + 12 \beta q^{11} - 14 q^{13} - 6 \beta q^{15} - 6 q^{17} + 4 \beta q^{19} - 12 q^{21} + 11 q^{25} + 3 \beta q^{27} + 30 q^{29} - 12 \beta q^{31} + 36 q^{33} - 24 \beta q^{35} + 26 q^{37} + 14 \beta q^{39} - 54 q^{41} + 12 \beta q^{43} - 18 q^{45} - 24 \beta q^{47} + q^{49} + 6 \beta q^{51} - 18 q^{53} + 72 \beta q^{55} + 12 q^{57} - 12 \beta q^{59} - 70 q^{61} + 12 \beta q^{63} - 84 q^{65} - 68 \beta q^{67} + 48 \beta q^{71} + 82 q^{73} - 11 \beta q^{75} + 144 q^{77} - 44 \beta q^{79} + 9 q^{81} - 12 \beta q^{83} - 36 q^{85} - 30 \beta q^{87} + 114 q^{89} + 56 \beta q^{91} - 36 q^{93} + 24 \beta q^{95} + 34 q^{97} - 36 \beta q^{99} +O(q^{100})$$ q - b * q^3 + 6 * q^5 - 4*b * q^7 - 3 * q^9 + 12*b * q^11 - 14 * q^13 - 6*b * q^15 - 6 * q^17 + 4*b * q^19 - 12 * q^21 + 11 * q^25 + 3*b * q^27 + 30 * q^29 - 12*b * q^31 + 36 * q^33 - 24*b * q^35 + 26 * q^37 + 14*b * q^39 - 54 * q^41 + 12*b * q^43 - 18 * q^45 - 24*b * q^47 + q^49 + 6*b * q^51 - 18 * q^53 + 72*b * q^55 + 12 * q^57 - 12*b * q^59 - 70 * q^61 + 12*b * q^63 - 84 * q^65 - 68*b * q^67 + 48*b * q^71 + 82 * q^73 - 11*b * q^75 + 144 * q^77 - 44*b * q^79 + 9 * q^81 - 12*b * q^83 - 36 * q^85 - 30*b * q^87 + 114 * q^89 + 56*b * q^91 - 36 * q^93 + 24*b * q^95 + 34 * q^97 - 36*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 12 q^{5} - 6 q^{9}+O(q^{10})$$ 2 * q + 12 * q^5 - 6 * q^9 $$2 q + 12 q^{5} - 6 q^{9} - 28 q^{13} - 12 q^{17} - 24 q^{21} + 22 q^{25} + 60 q^{29} + 72 q^{33} + 52 q^{37} - 108 q^{41} - 36 q^{45} + 2 q^{49} - 36 q^{53} + 24 q^{57} - 140 q^{61} - 168 q^{65} + 164 q^{73} + 288 q^{77} + 18 q^{81} - 72 q^{85} + 228 q^{89} - 72 q^{93} + 68 q^{97}+O(q^{100})$$ 2 * q + 12 * q^5 - 6 * q^9 - 28 * q^13 - 12 * q^17 - 24 * q^21 + 22 * q^25 + 60 * q^29 + 72 * q^33 + 52 * q^37 - 108 * q^41 - 36 * q^45 + 2 * q^49 - 36 * q^53 + 24 * q^57 - 140 * q^61 - 168 * q^65 + 164 * q^73 + 288 * q^77 + 18 * q^81 - 72 * q^85 + 228 * q^89 - 72 * q^93 + 68 * 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
31.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 1.73205i 0 6.00000 0 6.92820i 0 −3.00000 0
31.2 0 1.73205i 0 6.00000 0 6.92820i 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
4.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 48.3.g.a 2
3.b odd 2 1 144.3.g.b 2
4.b odd 2 1 inner 48.3.g.a 2
5.b even 2 1 1200.3.e.h 2
5.c odd 4 2 1200.3.j.a 4
7.b odd 2 1 2352.3.m.a 2
8.b even 2 1 192.3.g.a 2
8.d odd 2 1 192.3.g.a 2
9.c even 3 1 1296.3.o.a 2
9.c even 3 1 1296.3.o.c 2
9.d odd 6 1 1296.3.o.n 2
9.d odd 6 1 1296.3.o.p 2
12.b even 2 1 144.3.g.b 2
15.d odd 2 1 3600.3.e.t 2
15.e even 4 2 3600.3.j.i 4
16.e even 4 2 768.3.b.b 4
16.f odd 4 2 768.3.b.b 4
20.d odd 2 1 1200.3.e.h 2
20.e even 4 2 1200.3.j.a 4
24.f even 2 1 576.3.g.i 2
24.h odd 2 1 576.3.g.i 2
28.d even 2 1 2352.3.m.a 2
36.f odd 6 1 1296.3.o.a 2
36.f odd 6 1 1296.3.o.c 2
36.h even 6 1 1296.3.o.n 2
36.h even 6 1 1296.3.o.p 2
48.i odd 4 2 2304.3.b.n 4
48.k even 4 2 2304.3.b.n 4
60.h even 2 1 3600.3.e.t 2
60.l odd 4 2 3600.3.j.i 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.3.g.a 2 1.a even 1 1 trivial
48.3.g.a 2 4.b odd 2 1 inner
144.3.g.b 2 3.b odd 2 1
144.3.g.b 2 12.b even 2 1
192.3.g.a 2 8.b even 2 1
192.3.g.a 2 8.d odd 2 1
576.3.g.i 2 24.f even 2 1
576.3.g.i 2 24.h odd 2 1
768.3.b.b 4 16.e even 4 2
768.3.b.b 4 16.f odd 4 2
1200.3.e.h 2 5.b even 2 1
1200.3.e.h 2 20.d odd 2 1
1200.3.j.a 4 5.c odd 4 2
1200.3.j.a 4 20.e even 4 2
1296.3.o.a 2 9.c even 3 1
1296.3.o.a 2 36.f odd 6 1
1296.3.o.c 2 9.c even 3 1
1296.3.o.c 2 36.f odd 6 1
1296.3.o.n 2 9.d odd 6 1
1296.3.o.n 2 36.h even 6 1
1296.3.o.p 2 9.d odd 6 1
1296.3.o.p 2 36.h even 6 1
2304.3.b.n 4 48.i odd 4 2
2304.3.b.n 4 48.k even 4 2
2352.3.m.a 2 7.b odd 2 1
2352.3.m.a 2 28.d even 2 1
3600.3.e.t 2 15.d odd 2 1
3600.3.e.t 2 60.h even 2 1
3600.3.j.i 4 15.e even 4 2
3600.3.j.i 4 60.l odd 4 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(48, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 3$$
$5$ $$(T - 6)^{2}$$
$7$ $$T^{2} + 48$$
$11$ $$T^{2} + 432$$
$13$ $$(T + 14)^{2}$$
$17$ $$(T + 6)^{2}$$
$19$ $$T^{2} + 48$$
$23$ $$T^{2}$$
$29$ $$(T - 30)^{2}$$
$31$ $$T^{2} + 432$$
$37$ $$(T - 26)^{2}$$
$41$ $$(T + 54)^{2}$$
$43$ $$T^{2} + 432$$
$47$ $$T^{2} + 1728$$
$53$ $$(T + 18)^{2}$$
$59$ $$T^{2} + 432$$
$61$ $$(T + 70)^{2}$$
$67$ $$T^{2} + 13872$$
$71$ $$T^{2} + 6912$$
$73$ $$(T - 82)^{2}$$
$79$ $$T^{2} + 5808$$
$83$ $$T^{2} + 432$$
$89$ $$(T - 114)^{2}$$
$97$ $$(T - 34)^{2}$$