# Properties

 Label 288.4.a.h Level $288$ Weight $4$ Character orbit 288.a Self dual yes Analytic conductor $16.993$ 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] = [288,4,Mod(1,288)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(288, 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("288.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$288 = 2^{5} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 288.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: $$16.9925500817$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 32) 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)

 $$f(q)$$ $$=$$ $$q + 10 q^{5} - 16 q^{7}+O(q^{10})$$ q + 10 * q^5 - 16 * q^7 $$q + 10 q^{5} - 16 q^{7} - 40 q^{11} - 50 q^{13} + 30 q^{17} - 40 q^{19} + 48 q^{23} - 25 q^{25} + 34 q^{29} - 320 q^{31} - 160 q^{35} + 310 q^{37} - 410 q^{41} - 152 q^{43} - 416 q^{47} - 87 q^{49} + 410 q^{53} - 400 q^{55} - 200 q^{59} + 30 q^{61} - 500 q^{65} - 776 q^{67} + 400 q^{71} - 630 q^{73} + 640 q^{77} + 1120 q^{79} + 552 q^{83} + 300 q^{85} + 326 q^{89} + 800 q^{91} - 400 q^{95} - 110 q^{97}+O(q^{100})$$ q + 10 * q^5 - 16 * q^7 - 40 * q^11 - 50 * q^13 + 30 * q^17 - 40 * q^19 + 48 * q^23 - 25 * q^25 + 34 * q^29 - 320 * q^31 - 160 * q^35 + 310 * q^37 - 410 * q^41 - 152 * q^43 - 416 * q^47 - 87 * q^49 + 410 * q^53 - 400 * q^55 - 200 * q^59 + 30 * q^61 - 500 * q^65 - 776 * q^67 + 400 * q^71 - 630 * q^73 + 640 * q^77 + 1120 * q^79 + 552 * q^83 + 300 * q^85 + 326 * q^89 + 800 * q^91 - 400 * q^95 - 110 * 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 10.0000 0 −16.0000 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 288.4.a.h 1
3.b odd 2 1 32.4.a.a 1
4.b odd 2 1 288.4.a.i 1
8.b even 2 1 576.4.a.g 1
8.d odd 2 1 576.4.a.h 1
12.b even 2 1 32.4.a.c yes 1
15.d odd 2 1 800.4.a.k 1
15.e even 4 2 800.4.c.b 2
21.c even 2 1 1568.4.a.o 1
24.f even 2 1 64.4.a.a 1
24.h odd 2 1 64.4.a.e 1
48.i odd 4 2 256.4.b.e 2
48.k even 4 2 256.4.b.c 2
60.h even 2 1 800.4.a.a 1
60.l odd 4 2 800.4.c.a 2
84.h odd 2 1 1568.4.a.c 1
120.i odd 2 1 1600.4.a.e 1
120.m even 2 1 1600.4.a.bw 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.4.a.a 1 3.b odd 2 1
32.4.a.c yes 1 12.b even 2 1
64.4.a.a 1 24.f even 2 1
64.4.a.e 1 24.h odd 2 1
256.4.b.c 2 48.k even 4 2
256.4.b.e 2 48.i odd 4 2
288.4.a.h 1 1.a even 1 1 trivial
288.4.a.i 1 4.b odd 2 1
576.4.a.g 1 8.b even 2 1
576.4.a.h 1 8.d odd 2 1
800.4.a.a 1 60.h even 2 1
800.4.a.k 1 15.d odd 2 1
800.4.c.a 2 60.l odd 4 2
800.4.c.b 2 15.e even 4 2
1568.4.a.c 1 84.h odd 2 1
1568.4.a.o 1 21.c even 2 1
1600.4.a.e 1 120.i odd 2 1
1600.4.a.bw 1 120.m even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(288))$$:

 $$T_{5} - 10$$ T5 - 10 $$T_{7} + 16$$ T7 + 16 $$T_{11} + 40$$ T11 + 40

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T - 10$$
$7$ $$T + 16$$
$11$ $$T + 40$$
$13$ $$T + 50$$
$17$ $$T - 30$$
$19$ $$T + 40$$
$23$ $$T - 48$$
$29$ $$T - 34$$
$31$ $$T + 320$$
$37$ $$T - 310$$
$41$ $$T + 410$$
$43$ $$T + 152$$
$47$ $$T + 416$$
$53$ $$T - 410$$
$59$ $$T + 200$$
$61$ $$T - 30$$
$67$ $$T + 776$$
$71$ $$T - 400$$
$73$ $$T + 630$$
$79$ $$T - 1120$$
$83$ $$T - 552$$
$89$ $$T - 326$$
$97$ $$T + 110$$