# Properties

 Label 288.2.a.d Level $288$ Weight $2$ Character orbit 288.a Self dual yes Analytic conductor $2.300$ Analytic rank $0$ Dimension $1$ CM discriminant -4 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [288,2,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, 2, names="a")

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

chi := DirichletCharacter("288.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$288 = 2^{5} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ 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: $$2.29969157821$$ Analytic rank: $$0$$ 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: $N(\mathrm{U}(1))$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + 2 q^{5}+O(q^{10})$$ q + 2 * q^5 $$q + 2 q^{5} + 6 q^{13} - 2 q^{17} - q^{25} + 10 q^{29} - 2 q^{37} - 10 q^{41} - 7 q^{49} - 14 q^{53} - 10 q^{61} + 12 q^{65} - 6 q^{73} - 4 q^{85} - 10 q^{89} + 18 q^{97}+O(q^{100})$$ q + 2 * q^5 + 6 * q^13 - 2 * q^17 - q^25 + 10 * q^29 - 2 * q^37 - 10 * q^41 - 7 * q^49 - 14 * q^53 - 10 * q^61 + 12 * q^65 - 6 * q^73 - 4 * q^85 - 10 * q^89 + 18 * 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 2.00000 0 0 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 288.2.a.d 1
3.b odd 2 1 32.2.a.a 1
4.b odd 2 1 CM 288.2.a.d 1
5.b even 2 1 7200.2.a.v 1
5.c odd 4 2 7200.2.f.m 2
8.b even 2 1 576.2.a.c 1
8.d odd 2 1 576.2.a.c 1
9.c even 3 2 2592.2.i.e 2
9.d odd 6 2 2592.2.i.t 2
12.b even 2 1 32.2.a.a 1
15.d odd 2 1 800.2.a.d 1
15.e even 4 2 800.2.c.e 2
16.e even 4 2 2304.2.d.j 2
16.f odd 4 2 2304.2.d.j 2
20.d odd 2 1 7200.2.a.v 1
20.e even 4 2 7200.2.f.m 2
21.c even 2 1 1568.2.a.e 1
21.g even 6 2 1568.2.i.f 2
21.h odd 6 2 1568.2.i.g 2
24.f even 2 1 64.2.a.a 1
24.h odd 2 1 64.2.a.a 1
33.d even 2 1 3872.2.a.f 1
36.f odd 6 2 2592.2.i.e 2
36.h even 6 2 2592.2.i.t 2
39.d odd 2 1 5408.2.a.g 1
48.i odd 4 2 256.2.b.b 2
48.k even 4 2 256.2.b.b 2
51.c odd 2 1 9248.2.a.f 1
60.h even 2 1 800.2.a.d 1
60.l odd 4 2 800.2.c.e 2
84.h odd 2 1 1568.2.a.e 1
84.j odd 6 2 1568.2.i.f 2
84.n even 6 2 1568.2.i.g 2
96.o even 8 4 1024.2.e.j 4
96.p odd 8 4 1024.2.e.j 4
120.i odd 2 1 1600.2.a.n 1
120.m even 2 1 1600.2.a.n 1
120.q odd 4 2 1600.2.c.l 2
120.w even 4 2 1600.2.c.l 2
132.d odd 2 1 3872.2.a.f 1
156.h even 2 1 5408.2.a.g 1
168.e odd 2 1 3136.2.a.m 1
168.i even 2 1 3136.2.a.m 1
204.h even 2 1 9248.2.a.f 1
264.m even 2 1 7744.2.a.v 1
264.p odd 2 1 7744.2.a.v 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.2.a.a 1 3.b odd 2 1
32.2.a.a 1 12.b even 2 1
64.2.a.a 1 24.f even 2 1
64.2.a.a 1 24.h odd 2 1
256.2.b.b 2 48.i odd 4 2
256.2.b.b 2 48.k even 4 2
288.2.a.d 1 1.a even 1 1 trivial
288.2.a.d 1 4.b odd 2 1 CM
576.2.a.c 1 8.b even 2 1
576.2.a.c 1 8.d odd 2 1
800.2.a.d 1 15.d odd 2 1
800.2.a.d 1 60.h even 2 1
800.2.c.e 2 15.e even 4 2
800.2.c.e 2 60.l odd 4 2
1024.2.e.j 4 96.o even 8 4
1024.2.e.j 4 96.p odd 8 4
1568.2.a.e 1 21.c even 2 1
1568.2.a.e 1 84.h odd 2 1
1568.2.i.f 2 21.g even 6 2
1568.2.i.f 2 84.j odd 6 2
1568.2.i.g 2 21.h odd 6 2
1568.2.i.g 2 84.n even 6 2
1600.2.a.n 1 120.i odd 2 1
1600.2.a.n 1 120.m even 2 1
1600.2.c.l 2 120.q odd 4 2
1600.2.c.l 2 120.w even 4 2
2304.2.d.j 2 16.e even 4 2
2304.2.d.j 2 16.f odd 4 2
2592.2.i.e 2 9.c even 3 2
2592.2.i.e 2 36.f odd 6 2
2592.2.i.t 2 9.d odd 6 2
2592.2.i.t 2 36.h even 6 2
3136.2.a.m 1 168.e odd 2 1
3136.2.a.m 1 168.i even 2 1
3872.2.a.f 1 33.d even 2 1
3872.2.a.f 1 132.d odd 2 1
5408.2.a.g 1 39.d odd 2 1
5408.2.a.g 1 156.h even 2 1
7200.2.a.v 1 5.b even 2 1
7200.2.a.v 1 20.d odd 2 1
7200.2.f.m 2 5.c odd 4 2
7200.2.f.m 2 20.e even 4 2
7744.2.a.v 1 264.m even 2 1
7744.2.a.v 1 264.p odd 2 1
9248.2.a.f 1 51.c odd 2 1
9248.2.a.f 1 204.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(288))$$:

 $$T_{5} - 2$$ T5 - 2 $$T_{7}$$ T7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T - 2$$
$7$ $$T$$
$11$ $$T$$
$13$ $$T - 6$$
$17$ $$T + 2$$
$19$ $$T$$
$23$ $$T$$
$29$ $$T - 10$$
$31$ $$T$$
$37$ $$T + 2$$
$41$ $$T + 10$$
$43$ $$T$$
$47$ $$T$$
$53$ $$T + 14$$
$59$ $$T$$
$61$ $$T + 10$$
$67$ $$T$$
$71$ $$T$$
$73$ $$T + 6$$
$79$ $$T$$
$83$ $$T$$
$89$ $$T + 10$$
$97$ $$T - 18$$