# Properties

 Label 144.8.a.g Level $144$ Weight $8$ Character orbit 144.a Self dual yes Analytic conductor $44.983$ 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] = [144,8,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, 8, names="a")

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

chi := DirichletCharacter("144.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$144 = 2^{4} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$8$$ 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: $$44.9834436697$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 8) 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)

 $$f(q)$$ $$=$$ $$q + 82 q^{5} + 456 q^{7}+O(q^{10})$$ q + 82 * q^5 + 456 * q^7 $$q + 82 q^{5} + 456 q^{7} - 2524 q^{11} - 10778 q^{13} + 11150 q^{17} - 4124 q^{19} + 81704 q^{23} - 71401 q^{25} - 99798 q^{29} + 40480 q^{31} + 37392 q^{35} - 419442 q^{37} - 141402 q^{41} + 690428 q^{43} - 682032 q^{47} - 615607 q^{49} - 1813118 q^{53} - 206968 q^{55} - 966028 q^{59} + 1887670 q^{61} - 883796 q^{65} - 2965868 q^{67} - 2548232 q^{71} - 1680326 q^{73} - 1150944 q^{77} - 4038064 q^{79} - 5385764 q^{83} + 914300 q^{85} + 6473046 q^{89} - 4914768 q^{91} - 338168 q^{95} - 6065758 q^{97}+O(q^{100})$$ q + 82 * q^5 + 456 * q^7 - 2524 * q^11 - 10778 * q^13 + 11150 * q^17 - 4124 * q^19 + 81704 * q^23 - 71401 * q^25 - 99798 * q^29 + 40480 * q^31 + 37392 * q^35 - 419442 * q^37 - 141402 * q^41 + 690428 * q^43 - 682032 * q^47 - 615607 * q^49 - 1813118 * q^53 - 206968 * q^55 - 966028 * q^59 + 1887670 * q^61 - 883796 * q^65 - 2965868 * q^67 - 2548232 * q^71 - 1680326 * q^73 - 1150944 * q^77 - 4038064 * q^79 - 5385764 * q^83 + 914300 * q^85 + 6473046 * q^89 - 4914768 * q^91 - 338168 * q^95 - 6065758 * 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 82.0000 0 456.000 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 144.8.a.g 1
3.b odd 2 1 16.8.a.c 1
4.b odd 2 1 72.8.a.d 1
8.b even 2 1 576.8.a.k 1
8.d odd 2 1 576.8.a.j 1
12.b even 2 1 8.8.a.a 1
15.d odd 2 1 400.8.a.b 1
15.e even 4 2 400.8.c.b 2
24.f even 2 1 64.8.a.g 1
24.h odd 2 1 64.8.a.a 1
48.i odd 4 2 256.8.b.c 2
48.k even 4 2 256.8.b.e 2
60.h even 2 1 200.8.a.i 1
60.l odd 4 2 200.8.c.a 2
84.h odd 2 1 392.8.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.8.a.a 1 12.b even 2 1
16.8.a.c 1 3.b odd 2 1
64.8.a.a 1 24.h odd 2 1
64.8.a.g 1 24.f even 2 1
72.8.a.d 1 4.b odd 2 1
144.8.a.g 1 1.a even 1 1 trivial
200.8.a.i 1 60.h even 2 1
200.8.c.a 2 60.l odd 4 2
256.8.b.c 2 48.i odd 4 2
256.8.b.e 2 48.k even 4 2
392.8.a.d 1 84.h odd 2 1
400.8.a.b 1 15.d odd 2 1
400.8.c.b 2 15.e even 4 2
576.8.a.j 1 8.d odd 2 1
576.8.a.k 1 8.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} - 82$$ acting on $$S_{8}^{\mathrm{new}}(\Gamma_0(144))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T - 82$$
$7$ $$T - 456$$
$11$ $$T + 2524$$
$13$ $$T + 10778$$
$17$ $$T - 11150$$
$19$ $$T + 4124$$
$23$ $$T - 81704$$
$29$ $$T + 99798$$
$31$ $$T - 40480$$
$37$ $$T + 419442$$
$41$ $$T + 141402$$
$43$ $$T - 690428$$
$47$ $$T + 682032$$
$53$ $$T + 1813118$$
$59$ $$T + 966028$$
$61$ $$T - 1887670$$
$67$ $$T + 2965868$$
$71$ $$T + 2548232$$
$73$ $$T + 1680326$$
$79$ $$T + 4038064$$
$83$ $$T + 5385764$$
$89$ $$T - 6473046$$
$97$ $$T + 6065758$$