# Properties

 Label 72.4.a.a Level $72$ Weight $4$ Character orbit 72.a Self dual yes Analytic conductor $4.248$ 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] = [72,4,Mod(1,72)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(72, 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("72.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$72 = 2^{3} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 72.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: $$4.24813752041$$ Analytic rank: $$1$$ 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)

 $$f(q)$$ $$=$$ $$q - 16 q^{5} - 12 q^{7}+O(q^{10})$$ 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 72.4.a.a 1
3.b odd 2 1 72.4.a.d yes 1
4.b odd 2 1 144.4.a.a 1
5.b even 2 1 1800.4.a.z 1
5.c odd 4 2 1800.4.f.b 2
8.b even 2 1 576.4.a.w 1
8.d odd 2 1 576.4.a.x 1
9.c even 3 2 648.4.i.l 2
9.d odd 6 2 648.4.i.a 2
12.b even 2 1 144.4.a.f 1
15.d odd 2 1 1800.4.a.ba 1
15.e even 4 2 1800.4.f.x 2
24.f even 2 1 576.4.a.d 1
24.h odd 2 1 576.4.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.4.a.a 1 1.a even 1 1 trivial
72.4.a.d yes 1 3.b odd 2 1
144.4.a.a 1 4.b odd 2 1
144.4.a.f 1 12.b even 2 1
576.4.a.c 1 24.h odd 2 1
576.4.a.d 1 24.f even 2 1
576.4.a.w 1 8.b even 2 1
576.4.a.x 1 8.d odd 2 1
648.4.i.a 2 9.d odd 6 2
648.4.i.l 2 9.c even 3 2
1800.4.a.z 1 5.b even 2 1
1800.4.a.ba 1 15.d odd 2 1
1800.4.f.b 2 5.c odd 4 2
1800.4.f.x 2 15.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} + 16$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(72))$$.

## Hecke characteristic polynomials

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