# Properties

 Label 144.4.a.d Level $144$ Weight $4$ Character orbit 144.a Self dual yes Analytic conductor $8.496$ Analytic rank $1$ Dimension $1$ CM discriminant -3 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [144,4,Mod(1,144)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(144, 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("144.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$144 = 2^{4} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 144.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: $$8.49627504083$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 9) Fricke sign: $$-1$$ Sato-Tate group: $N(\mathrm{U}(1))$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - 20 q^{7}+O(q^{10})$$ q - 20 * q^7 $$q - 20 q^{7} - 70 q^{13} - 56 q^{19} - 125 q^{25} - 308 q^{31} + 110 q^{37} + 520 q^{43} + 57 q^{49} + 182 q^{61} + 880 q^{67} + 1190 q^{73} - 884 q^{79} + 1400 q^{91} - 1330 q^{97}+O(q^{100})$$ q - 20 * q^7 - 70 * q^13 - 56 * q^19 - 125 * q^25 - 308 * q^31 + 110 * q^37 + 520 * q^43 + 57 * q^49 + 182 * q^61 + 880 * q^67 + 1190 * q^73 - 884 * q^79 + 1400 * q^91 - 1330 * 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 −20.0000 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
$$2$$ $$-1$$
$$3$$ $$+1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 144.4.a.d 1
3.b odd 2 1 CM 144.4.a.d 1
4.b odd 2 1 9.4.a.a 1
8.b even 2 1 576.4.a.l 1
8.d odd 2 1 576.4.a.m 1
12.b even 2 1 9.4.a.a 1
20.d odd 2 1 225.4.a.d 1
20.e even 4 2 225.4.b.g 2
24.f even 2 1 576.4.a.m 1
24.h odd 2 1 576.4.a.l 1
28.d even 2 1 441.4.a.f 1
28.f even 6 2 441.4.e.j 2
28.g odd 6 2 441.4.e.i 2
36.f odd 6 2 81.4.c.b 2
36.h even 6 2 81.4.c.b 2
44.c even 2 1 1089.4.a.g 1
52.b odd 2 1 1521.4.a.g 1
60.h even 2 1 225.4.a.d 1
60.l odd 4 2 225.4.b.g 2
84.h odd 2 1 441.4.a.f 1
84.j odd 6 2 441.4.e.j 2
84.n even 6 2 441.4.e.i 2
132.d odd 2 1 1089.4.a.g 1
156.h even 2 1 1521.4.a.g 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.4.a.a 1 4.b odd 2 1
9.4.a.a 1 12.b even 2 1
81.4.c.b 2 36.f odd 6 2
81.4.c.b 2 36.h even 6 2
144.4.a.d 1 1.a even 1 1 trivial
144.4.a.d 1 3.b odd 2 1 CM
225.4.a.d 1 20.d odd 2 1
225.4.a.d 1 60.h even 2 1
225.4.b.g 2 20.e even 4 2
225.4.b.g 2 60.l odd 4 2
441.4.a.f 1 28.d even 2 1
441.4.a.f 1 84.h odd 2 1
441.4.e.i 2 28.g odd 6 2
441.4.e.i 2 84.n even 6 2
441.4.e.j 2 28.f even 6 2
441.4.e.j 2 84.j odd 6 2
576.4.a.l 1 8.b even 2 1
576.4.a.l 1 24.h odd 2 1
576.4.a.m 1 8.d odd 2 1
576.4.a.m 1 24.f even 2 1
1089.4.a.g 1 44.c even 2 1
1089.4.a.g 1 132.d odd 2 1
1521.4.a.g 1 52.b odd 2 1
1521.4.a.g 1 156.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(144))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T + 20$$
$11$ $$T$$
$13$ $$T + 70$$
$17$ $$T$$
$19$ $$T + 56$$
$23$ $$T$$
$29$ $$T$$
$31$ $$T + 308$$
$37$ $$T - 110$$
$41$ $$T$$
$43$ $$T - 520$$
$47$ $$T$$
$53$ $$T$$
$59$ $$T$$
$61$ $$T - 182$$
$67$ $$T - 880$$
$71$ $$T$$
$73$ $$T - 1190$$
$79$ $$T + 884$$
$83$ $$T$$
$89$ $$T$$
$97$ $$T + 1330$$