# Properties

 Label 72.4.a.b 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: no (minimal twist has level 24) 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)

 $$f(q)$$ $$=$$ $$q - 14 q^{5} - 24 q^{7}+O(q^{10})$$ q - 14 * q^5 - 24 * q^7 $$q - 14 q^{5} - 24 q^{7} + 28 q^{11} - 74 q^{13} - 82 q^{17} + 92 q^{19} - 8 q^{23} + 71 q^{25} + 138 q^{29} + 80 q^{31} + 336 q^{35} + 30 q^{37} - 282 q^{41} + 4 q^{43} - 240 q^{47} + 233 q^{49} + 130 q^{53} - 392 q^{55} - 596 q^{59} - 218 q^{61} + 1036 q^{65} - 436 q^{67} - 856 q^{71} - 998 q^{73} - 672 q^{77} - 32 q^{79} + 1508 q^{83} + 1148 q^{85} + 246 q^{89} + 1776 q^{91} - 1288 q^{95} + 866 q^{97}+O(q^{100})$$ q - 14 * q^5 - 24 * q^7 + 28 * q^11 - 74 * q^13 - 82 * q^17 + 92 * q^19 - 8 * q^23 + 71 * q^25 + 138 * q^29 + 80 * q^31 + 336 * q^35 + 30 * q^37 - 282 * q^41 + 4 * q^43 - 240 * q^47 + 233 * q^49 + 130 * q^53 - 392 * q^55 - 596 * q^59 - 218 * q^61 + 1036 * q^65 - 436 * q^67 - 856 * q^71 - 998 * q^73 - 672 * q^77 - 32 * q^79 + 1508 * q^83 + 1148 * q^85 + 246 * q^89 + 1776 * q^91 - 1288 * q^95 + 866 * 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 −14.0000 0 −24.0000 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.b 1
3.b odd 2 1 24.4.a.a 1
4.b odd 2 1 144.4.a.b 1
5.b even 2 1 1800.4.a.bg 1
5.c odd 4 2 1800.4.f.q 2
8.b even 2 1 576.4.a.u 1
8.d odd 2 1 576.4.a.v 1
9.c even 3 2 648.4.i.k 2
9.d odd 6 2 648.4.i.b 2
12.b even 2 1 48.4.a.b 1
15.d odd 2 1 600.4.a.h 1
15.e even 4 2 600.4.f.b 2
21.c even 2 1 1176.4.a.a 1
24.f even 2 1 192.4.a.g 1
24.h odd 2 1 192.4.a.a 1
48.i odd 4 2 768.4.d.o 2
48.k even 4 2 768.4.d.b 2
60.h even 2 1 1200.4.a.u 1
60.l odd 4 2 1200.4.f.p 2
84.h odd 2 1 2352.4.a.w 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.4.a.a 1 3.b odd 2 1
48.4.a.b 1 12.b even 2 1
72.4.a.b 1 1.a even 1 1 trivial
144.4.a.b 1 4.b odd 2 1
192.4.a.a 1 24.h odd 2 1
192.4.a.g 1 24.f even 2 1
576.4.a.u 1 8.b even 2 1
576.4.a.v 1 8.d odd 2 1
600.4.a.h 1 15.d odd 2 1
600.4.f.b 2 15.e even 4 2
648.4.i.b 2 9.d odd 6 2
648.4.i.k 2 9.c even 3 2
768.4.d.b 2 48.k even 4 2
768.4.d.o 2 48.i odd 4 2
1176.4.a.a 1 21.c even 2 1
1200.4.a.u 1 60.h even 2 1
1200.4.f.p 2 60.l odd 4 2
1800.4.a.bg 1 5.b even 2 1
1800.4.f.q 2 5.c odd 4 2
2352.4.a.w 1 84.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} + 14$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(72))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 14$$
$7$ $$T + 24$$
$11$ $$T - 28$$
$13$ $$T + 74$$
$17$ $$T + 82$$
$19$ $$T - 92$$
$23$ $$T + 8$$
$29$ $$T - 138$$
$31$ $$T - 80$$
$37$ $$T - 30$$
$41$ $$T + 282$$
$43$ $$T - 4$$
$47$ $$T + 240$$
$53$ $$T - 130$$
$59$ $$T + 596$$
$61$ $$T + 218$$
$67$ $$T + 436$$
$71$ $$T + 856$$
$73$ $$T + 998$$
$79$ $$T + 32$$
$83$ $$T - 1508$$
$89$ $$T - 246$$
$97$ $$T - 866$$