# Properties

 Label 144.16.a.f Level $144$ Weight $16$ Character orbit 144.a Self dual yes Analytic conductor $205.479$ 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] = [144,16,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, 16, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("144.1");

S:= CuspForms(chi, 16);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$144 = 2^{4} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$16$$ 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: $$205.478647344$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1) 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)

 $$f(q)$$ $$=$$ $$q - 52110 q^{5} - 2822456 q^{7}+O(q^{10})$$ q - 52110 * q^5 - 2822456 * q^7 $$q - 52110 q^{5} - 2822456 q^{7} + 20586852 q^{11} - 190073338 q^{13} - 1646527986 q^{17} - 1563257180 q^{19} + 9451116072 q^{23} - 27802126025 q^{25} + 36902568330 q^{29} - 71588483552 q^{31} + 147078182160 q^{35} - 1033652081554 q^{37} - 1641974018202 q^{41} + 492403109308 q^{43} - 3410684952624 q^{47} + 3218696361993 q^{49} - 6797151655902 q^{53} - 1072780857720 q^{55} + 9858856815540 q^{59} + 4931842626902 q^{61} + 9904721643180 q^{65} + 28837826625364 q^{67} + 125050114914552 q^{71} - 82171455513478 q^{73} - 58105483948512 q^{77} + 25413078694480 q^{79} - 281736730890468 q^{83} + 85800573350460 q^{85} - 715618564776810 q^{89} + 536473633278128 q^{91} + 81461331649800 q^{95} + 612786136081826 q^{97}+O(q^{100})$$ q - 52110 * q^5 - 2822456 * q^7 + 20586852 * q^11 - 190073338 * q^13 - 1646527986 * q^17 - 1563257180 * q^19 + 9451116072 * q^23 - 27802126025 * q^25 + 36902568330 * q^29 - 71588483552 * q^31 + 147078182160 * q^35 - 1033652081554 * q^37 - 1641974018202 * q^41 + 492403109308 * q^43 - 3410684952624 * q^47 + 3218696361993 * q^49 - 6797151655902 * q^53 - 1072780857720 * q^55 + 9858856815540 * q^59 + 4931842626902 * q^61 + 9904721643180 * q^65 + 28837826625364 * q^67 + 125050114914552 * q^71 - 82171455513478 * q^73 - 58105483948512 * q^77 + 25413078694480 * q^79 - 281736730890468 * q^83 + 85800573350460 * q^85 - 715618564776810 * q^89 + 536473633278128 * q^91 + 81461331649800 * q^95 + 612786136081826 * 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 −52110.0 0 −2.82246e6 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 144.16.a.f 1
3.b odd 2 1 16.16.a.d 1
4.b odd 2 1 9.16.a.a 1
12.b even 2 1 1.16.a.a 1
24.f even 2 1 64.16.a.i 1
24.h odd 2 1 64.16.a.c 1
60.h even 2 1 25.16.a.a 1
60.l odd 4 2 25.16.b.a 2
84.h odd 2 1 49.16.a.a 1
84.j odd 6 2 49.16.c.b 2
84.n even 6 2 49.16.c.c 2
132.d odd 2 1 121.16.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.16.a.a 1 12.b even 2 1
9.16.a.a 1 4.b odd 2 1
16.16.a.d 1 3.b odd 2 1
25.16.a.a 1 60.h even 2 1
25.16.b.a 2 60.l odd 4 2
49.16.a.a 1 84.h odd 2 1
49.16.c.b 2 84.j odd 6 2
49.16.c.c 2 84.n even 6 2
64.16.a.c 1 24.h odd 2 1
64.16.a.i 1 24.f even 2 1
121.16.a.a 1 132.d odd 2 1
144.16.a.f 1 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} + 52110$$ acting on $$S_{16}^{\mathrm{new}}(\Gamma_0(144))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 52110$$
$7$ $$T + 2822456$$
$11$ $$T - 20586852$$
$13$ $$T + 190073338$$
$17$ $$T + 1646527986$$
$19$ $$T + 1563257180$$
$23$ $$T - 9451116072$$
$29$ $$T - 36902568330$$
$31$ $$T + 71588483552$$
$37$ $$T + 1033652081554$$
$41$ $$T + 1641974018202$$
$43$ $$T - 492403109308$$
$47$ $$T + 3410684952624$$
$53$ $$T + 6797151655902$$
$59$ $$T - 9858856815540$$
$61$ $$T - 4931842626902$$
$67$ $$T - 28837826625364$$
$71$ $$T - 125050114914552$$
$73$ $$T + 82171455513478$$
$79$ $$T - 25413078694480$$
$83$ $$T + 281736730890468$$
$89$ $$T + 715618564776810$$
$97$ $$T - 612786136081826$$