Properties

 Label 1224.2.a.d Level $1224$ Weight $2$ Character orbit 1224.a Self dual yes Analytic conductor $9.774$ 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] = [1224,2,Mod(1,1224)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1224, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1224.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1224 = 2^{3} \cdot 3^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1224.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: $$9.77368920740$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 136) 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)

 $$f(q)$$ $$=$$ $$q+O(q^{10})$$ q $$q - 2 q^{11} - 6 q^{13} + q^{17} + 4 q^{19} - 4 q^{23} - 5 q^{25} - 8 q^{31} - 4 q^{37} - 6 q^{41} + 8 q^{43} + 8 q^{47} - 7 q^{49} - 10 q^{53} + 12 q^{61} + 8 q^{67} - 12 q^{71} + 2 q^{73} - 4 q^{79} - 16 q^{83} - 10 q^{89} - 18 q^{97}+O(q^{100})$$ q - 2 * q^11 - 6 * q^13 + q^17 + 4 * q^19 - 4 * q^23 - 5 * q^25 - 8 * q^31 - 4 * q^37 - 6 * q^41 + 8 * q^43 + 8 * q^47 - 7 * q^49 - 10 * q^53 + 12 * q^61 + 8 * q^67 - 12 * q^71 + 2 * q^73 - 4 * q^79 - 16 * q^83 - 10 * q^89 - 18 * q^97

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 0 0 0 0 0 0 0 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
$$2$$ $$1$$
$$3$$ $$-1$$
$$17$$ $$-1$$

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 1224.2.a.d 1
3.b odd 2 1 136.2.a.b 1
4.b odd 2 1 2448.2.a.j 1
8.b even 2 1 9792.2.a.be 1
8.d odd 2 1 9792.2.a.bd 1
12.b even 2 1 272.2.a.a 1
15.d odd 2 1 3400.2.a.b 1
15.e even 4 2 3400.2.e.c 2
21.c even 2 1 6664.2.a.b 1
24.f even 2 1 1088.2.a.m 1
24.h odd 2 1 1088.2.a.c 1
51.c odd 2 1 2312.2.a.a 1
51.f odd 4 2 2312.2.b.b 2
60.h even 2 1 6800.2.a.w 1
204.h even 2 1 4624.2.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.2.a.b 1 3.b odd 2 1
272.2.a.a 1 12.b even 2 1
1088.2.a.c 1 24.h odd 2 1
1088.2.a.m 1 24.f even 2 1
1224.2.a.d 1 1.a even 1 1 trivial
2312.2.a.a 1 51.c odd 2 1
2312.2.b.b 2 51.f odd 4 2
2448.2.a.j 1 4.b odd 2 1
3400.2.a.b 1 15.d odd 2 1
3400.2.e.c 2 15.e even 4 2
4624.2.a.f 1 204.h even 2 1
6664.2.a.b 1 21.c even 2 1
6800.2.a.w 1 60.h even 2 1
9792.2.a.bd 1 8.d odd 2 1
9792.2.a.be 1 8.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1224))$$:

 $$T_{5}$$ T5 $$T_{7}$$ T7

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T$$
$11$ $$T + 2$$
$13$ $$T + 6$$
$17$ $$T - 1$$
$19$ $$T - 4$$
$23$ $$T + 4$$
$29$ $$T$$
$31$ $$T + 8$$
$37$ $$T + 4$$
$41$ $$T + 6$$
$43$ $$T - 8$$
$47$ $$T - 8$$
$53$ $$T + 10$$
$59$ $$T$$
$61$ $$T - 12$$
$67$ $$T - 8$$
$71$ $$T + 12$$
$73$ $$T - 2$$
$79$ $$T + 4$$
$83$ $$T + 16$$
$89$ $$T + 10$$
$97$ $$T + 18$$