# Properties

 Label 24.2.a.a Level $24$ Weight $2$ Character orbit 24.a Self dual yes Analytic conductor $0.192$ 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] = [24,2,Mod(1,24)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(24, 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("24.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$24 = 2^{3} \cdot 3$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 24.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: $$0.191640964851$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes 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

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

## 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 24.2.a.a 1
3.b odd 2 1 72.2.a.a 1
4.b odd 2 1 48.2.a.a 1
5.b even 2 1 600.2.a.h 1
5.c odd 4 2 600.2.f.e 2
7.b odd 2 1 1176.2.a.i 1
7.c even 3 2 1176.2.q.i 2
7.d odd 6 2 1176.2.q.a 2
8.b even 2 1 192.2.a.d 1
8.d odd 2 1 192.2.a.b 1
9.c even 3 2 648.2.i.g 2
9.d odd 6 2 648.2.i.b 2
11.b odd 2 1 2904.2.a.c 1
12.b even 2 1 144.2.a.b 1
13.b even 2 1 4056.2.a.i 1
13.d odd 4 2 4056.2.c.e 2
15.d odd 2 1 1800.2.a.m 1
15.e even 4 2 1800.2.f.c 2
16.e even 4 2 768.2.d.e 2
16.f odd 4 2 768.2.d.d 2
17.b even 2 1 6936.2.a.p 1
19.b odd 2 1 8664.2.a.j 1
20.d odd 2 1 1200.2.a.d 1
20.e even 4 2 1200.2.f.b 2
21.c even 2 1 3528.2.a.d 1
21.g even 6 2 3528.2.s.y 2
21.h odd 6 2 3528.2.s.j 2
24.f even 2 1 576.2.a.b 1
24.h odd 2 1 576.2.a.d 1
28.d even 2 1 2352.2.a.i 1
28.f even 6 2 2352.2.q.r 2
28.g odd 6 2 2352.2.q.l 2
33.d even 2 1 8712.2.a.u 1
36.f odd 6 2 1296.2.i.m 2
36.h even 6 2 1296.2.i.e 2
40.e odd 2 1 4800.2.a.cc 1
40.f even 2 1 4800.2.a.q 1
40.i odd 4 2 4800.2.f.d 2
40.k even 4 2 4800.2.f.bg 2
44.c even 2 1 5808.2.a.s 1
48.i odd 4 2 2304.2.d.i 2
48.k even 4 2 2304.2.d.k 2
52.b odd 2 1 8112.2.a.be 1
56.e even 2 1 9408.2.a.cc 1
56.h odd 2 1 9408.2.a.h 1
60.h even 2 1 3600.2.a.v 1
60.l odd 4 2 3600.2.f.r 2
84.h odd 2 1 7056.2.a.q 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.2.a.a 1 1.a even 1 1 trivial
48.2.a.a 1 4.b odd 2 1
72.2.a.a 1 3.b odd 2 1
144.2.a.b 1 12.b even 2 1
192.2.a.b 1 8.d odd 2 1
192.2.a.d 1 8.b even 2 1
576.2.a.b 1 24.f even 2 1
576.2.a.d 1 24.h odd 2 1
600.2.a.h 1 5.b even 2 1
600.2.f.e 2 5.c odd 4 2
648.2.i.b 2 9.d odd 6 2
648.2.i.g 2 9.c even 3 2
768.2.d.d 2 16.f odd 4 2
768.2.d.e 2 16.e even 4 2
1176.2.a.i 1 7.b odd 2 1
1176.2.q.a 2 7.d odd 6 2
1176.2.q.i 2 7.c even 3 2
1200.2.a.d 1 20.d odd 2 1
1200.2.f.b 2 20.e even 4 2
1296.2.i.e 2 36.h even 6 2
1296.2.i.m 2 36.f odd 6 2
1800.2.a.m 1 15.d odd 2 1
1800.2.f.c 2 15.e even 4 2
2304.2.d.i 2 48.i odd 4 2
2304.2.d.k 2 48.k even 4 2
2352.2.a.i 1 28.d even 2 1
2352.2.q.l 2 28.g odd 6 2
2352.2.q.r 2 28.f even 6 2
2904.2.a.c 1 11.b odd 2 1
3528.2.a.d 1 21.c even 2 1
3528.2.s.j 2 21.h odd 6 2
3528.2.s.y 2 21.g even 6 2
3600.2.a.v 1 60.h even 2 1
3600.2.f.r 2 60.l odd 4 2
4056.2.a.i 1 13.b even 2 1
4056.2.c.e 2 13.d odd 4 2
4800.2.a.q 1 40.f even 2 1
4800.2.a.cc 1 40.e odd 2 1
4800.2.f.d 2 40.i odd 4 2
4800.2.f.bg 2 40.k even 4 2
5808.2.a.s 1 44.c even 2 1
6936.2.a.p 1 17.b even 2 1
7056.2.a.q 1 84.h odd 2 1
8112.2.a.be 1 52.b odd 2 1
8664.2.a.j 1 19.b odd 2 1
8712.2.a.u 1 33.d even 2 1
9408.2.a.h 1 56.h odd 2 1
9408.2.a.cc 1 56.e even 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(\Gamma_0(24))$$.

## Hecke characteristic polynomials

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