Properties

 Label 48.4.a.a Level $48$ Weight $4$ Character orbit 48.a Self dual yes Analytic conductor $2.832$ Analytic rank $1$ 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] = [48,4,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, 4, names="a")

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

chi := DirichletCharacter("48.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$48 = 2^{4} \cdot 3$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$2.83209168028$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 12) 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)

 $$f(q)$$ $$=$$ $$q - 3 q^{3} - 18 q^{5} - 8 q^{7} + 9 q^{9}+O(q^{10})$$ q - 3 * q^3 - 18 * q^5 - 8 * q^7 + 9 * q^9 $$q - 3 q^{3} - 18 q^{5} - 8 q^{7} + 9 q^{9} - 36 q^{11} - 10 q^{13} + 54 q^{15} + 18 q^{17} + 100 q^{19} + 24 q^{21} - 72 q^{23} + 199 q^{25} - 27 q^{27} - 234 q^{29} + 16 q^{31} + 108 q^{33} + 144 q^{35} - 226 q^{37} + 30 q^{39} + 90 q^{41} - 452 q^{43} - 162 q^{45} - 432 q^{47} - 279 q^{49} - 54 q^{51} + 414 q^{53} + 648 q^{55} - 300 q^{57} + 684 q^{59} + 422 q^{61} - 72 q^{63} + 180 q^{65} - 332 q^{67} + 216 q^{69} + 360 q^{71} + 26 q^{73} - 597 q^{75} + 288 q^{77} - 512 q^{79} + 81 q^{81} + 1188 q^{83} - 324 q^{85} + 702 q^{87} - 630 q^{89} + 80 q^{91} - 48 q^{93} - 1800 q^{95} - 1054 q^{97} - 324 q^{99}+O(q^{100})$$ q - 3 * q^3 - 18 * q^5 - 8 * q^7 + 9 * q^9 - 36 * q^11 - 10 * q^13 + 54 * q^15 + 18 * q^17 + 100 * q^19 + 24 * q^21 - 72 * q^23 + 199 * q^25 - 27 * q^27 - 234 * q^29 + 16 * q^31 + 108 * q^33 + 144 * q^35 - 226 * q^37 + 30 * q^39 + 90 * q^41 - 452 * q^43 - 162 * q^45 - 432 * q^47 - 279 * q^49 - 54 * q^51 + 414 * q^53 + 648 * q^55 - 300 * q^57 + 684 * q^59 + 422 * q^61 - 72 * q^63 + 180 * q^65 - 332 * q^67 + 216 * q^69 + 360 * q^71 + 26 * q^73 - 597 * q^75 + 288 * q^77 - 512 * q^79 + 81 * q^81 + 1188 * q^83 - 324 * q^85 + 702 * q^87 - 630 * q^89 + 80 * q^91 - 48 * q^93 - 1800 * q^95 - 1054 * q^97 - 324 * q^99

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 −3.00000 0 −18.0000 0 −8.00000 0 9.00000 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 48.4.a.a 1
3.b odd 2 1 144.4.a.g 1
4.b odd 2 1 12.4.a.a 1
5.b even 2 1 1200.4.a.be 1
5.c odd 4 2 1200.4.f.d 2
7.b odd 2 1 2352.4.a.bk 1
8.b even 2 1 192.4.a.l 1
8.d odd 2 1 192.4.a.f 1
12.b even 2 1 36.4.a.a 1
16.e even 4 2 768.4.d.j 2
16.f odd 4 2 768.4.d.g 2
20.d odd 2 1 300.4.a.b 1
20.e even 4 2 300.4.d.e 2
24.f even 2 1 576.4.a.b 1
24.h odd 2 1 576.4.a.a 1
28.d even 2 1 588.4.a.c 1
28.f even 6 2 588.4.i.e 2
28.g odd 6 2 588.4.i.d 2
36.f odd 6 2 324.4.e.h 2
36.h even 6 2 324.4.e.a 2
44.c even 2 1 1452.4.a.d 1
52.b odd 2 1 2028.4.a.c 1
52.f even 4 2 2028.4.b.c 2
60.h even 2 1 900.4.a.g 1
60.l odd 4 2 900.4.d.c 2
84.h odd 2 1 1764.4.a.b 1
84.j odd 6 2 1764.4.k.o 2
84.n even 6 2 1764.4.k.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.4.a.a 1 4.b odd 2 1
36.4.a.a 1 12.b even 2 1
48.4.a.a 1 1.a even 1 1 trivial
144.4.a.g 1 3.b odd 2 1
192.4.a.f 1 8.d odd 2 1
192.4.a.l 1 8.b even 2 1
300.4.a.b 1 20.d odd 2 1
300.4.d.e 2 20.e even 4 2
324.4.e.a 2 36.h even 6 2
324.4.e.h 2 36.f odd 6 2
576.4.a.a 1 24.h odd 2 1
576.4.a.b 1 24.f even 2 1
588.4.a.c 1 28.d even 2 1
588.4.i.d 2 28.g odd 6 2
588.4.i.e 2 28.f even 6 2
768.4.d.g 2 16.f odd 4 2
768.4.d.j 2 16.e even 4 2
900.4.a.g 1 60.h even 2 1
900.4.d.c 2 60.l odd 4 2
1200.4.a.be 1 5.b even 2 1
1200.4.f.d 2 5.c odd 4 2
1452.4.a.d 1 44.c even 2 1
1764.4.a.b 1 84.h odd 2 1
1764.4.k.b 2 84.n even 6 2
1764.4.k.o 2 84.j odd 6 2
2028.4.a.c 1 52.b odd 2 1
2028.4.b.c 2 52.f even 4 2
2352.4.a.bk 1 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} + 18$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(48))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 3$$
$5$ $$T + 18$$
$7$ $$T + 8$$
$11$ $$T + 36$$
$13$ $$T + 10$$
$17$ $$T - 18$$
$19$ $$T - 100$$
$23$ $$T + 72$$
$29$ $$T + 234$$
$31$ $$T - 16$$
$37$ $$T + 226$$
$41$ $$T - 90$$
$43$ $$T + 452$$
$47$ $$T + 432$$
$53$ $$T - 414$$
$59$ $$T - 684$$
$61$ $$T - 422$$
$67$ $$T + 332$$
$71$ $$T - 360$$
$73$ $$T - 26$$
$79$ $$T + 512$$
$83$ $$T - 1188$$
$89$ $$T + 630$$
$97$ $$T + 1054$$