# Properties

 Label 144.4.a.a Level $144$ Weight $4$ Character orbit 144.a Self dual yes Analytic conductor $8.496$ 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] = [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: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 72) 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)

 $$f(q)$$ $$=$$ $$q - 16 q^{5} + 12 q^{7}+O(q^{10})$$ q - 16 * q^5 + 12 * q^7 $$q - 16 q^{5} + 12 q^{7} + 64 q^{11} + 58 q^{13} - 32 q^{17} + 136 q^{19} - 128 q^{23} + 131 q^{25} + 144 q^{29} - 20 q^{31} - 192 q^{35} - 18 q^{37} + 288 q^{41} + 200 q^{43} + 384 q^{47} - 199 q^{49} - 496 q^{53} - 1024 q^{55} - 128 q^{59} - 458 q^{61} - 928 q^{65} + 496 q^{67} + 512 q^{71} - 602 q^{73} + 768 q^{77} - 1108 q^{79} + 704 q^{83} + 512 q^{85} + 960 q^{89} + 696 q^{91} - 2176 q^{95} + 206 q^{97}+O(q^{100})$$ q - 16 * q^5 + 12 * q^7 + 64 * q^11 + 58 * q^13 - 32 * q^17 + 136 * q^19 - 128 * q^23 + 131 * q^25 + 144 * q^29 - 20 * q^31 - 192 * q^35 - 18 * q^37 + 288 * q^41 + 200 * q^43 + 384 * q^47 - 199 * q^49 - 496 * q^53 - 1024 * q^55 - 128 * q^59 - 458 * q^61 - 928 * q^65 + 496 * q^67 + 512 * q^71 - 602 * q^73 + 768 * q^77 - 1108 * q^79 + 704 * q^83 + 512 * q^85 + 960 * q^89 + 696 * q^91 - 2176 * q^95 + 206 * 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 −16.0000 0 12.0000 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

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 144.4.a.a 1
3.b odd 2 1 144.4.a.f 1
4.b odd 2 1 72.4.a.a 1
8.b even 2 1 576.4.a.x 1
8.d odd 2 1 576.4.a.w 1
12.b even 2 1 72.4.a.d yes 1
20.d odd 2 1 1800.4.a.z 1
20.e even 4 2 1800.4.f.b 2
24.f even 2 1 576.4.a.c 1
24.h odd 2 1 576.4.a.d 1
36.f odd 6 2 648.4.i.l 2
36.h even 6 2 648.4.i.a 2
60.h even 2 1 1800.4.a.ba 1
60.l odd 4 2 1800.4.f.x 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.4.a.a 1 4.b odd 2 1
72.4.a.d yes 1 12.b even 2 1
144.4.a.a 1 1.a even 1 1 trivial
144.4.a.f 1 3.b odd 2 1
576.4.a.c 1 24.f even 2 1
576.4.a.d 1 24.h odd 2 1
576.4.a.w 1 8.d odd 2 1
576.4.a.x 1 8.b even 2 1
648.4.i.a 2 36.h even 6 2
648.4.i.l 2 36.f odd 6 2
1800.4.a.z 1 20.d odd 2 1
1800.4.a.ba 1 60.h even 2 1
1800.4.f.b 2 20.e even 4 2
1800.4.f.x 2 60.l odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} + 16$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(144))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 16$$
$7$ $$T - 12$$
$11$ $$T - 64$$
$13$ $$T - 58$$
$17$ $$T + 32$$
$19$ $$T - 136$$
$23$ $$T + 128$$
$29$ $$T - 144$$
$31$ $$T + 20$$
$37$ $$T + 18$$
$41$ $$T - 288$$
$43$ $$T - 200$$
$47$ $$T - 384$$
$53$ $$T + 496$$
$59$ $$T + 128$$
$61$ $$T + 458$$
$67$ $$T - 496$$
$71$ $$T - 512$$
$73$ $$T + 602$$
$79$ $$T + 1108$$
$83$ $$T - 704$$
$89$ $$T - 960$$
$97$ $$T - 206$$