# Properties

 Label 288.2.a.b Level $288$ Weight $2$ Character orbit 288.a Self dual yes Analytic conductor $2.300$ 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] = [288,2,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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 96) 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)

 $$f(q)$$ $$=$$ $$q - 2 q^{5} - 4 q^{7}+O(q^{10})$$ q - 2 * q^5 - 4 * q^7 $$q - 2 q^{5} - 4 q^{7} - 4 q^{11} - 2 q^{13} + 6 q^{17} - 4 q^{19} - q^{25} - 2 q^{29} + 4 q^{31} + 8 q^{35} - 2 q^{37} - 2 q^{41} + 4 q^{43} - 8 q^{47} + 9 q^{49} - 10 q^{53} + 8 q^{55} + 4 q^{59} + 6 q^{61} + 4 q^{65} + 4 q^{67} + 16 q^{71} - 6 q^{73} + 16 q^{77} + 4 q^{79} - 12 q^{83} - 12 q^{85} - 10 q^{89} + 8 q^{91} + 8 q^{95} - 14 q^{97}+O(q^{100})$$ q - 2 * q^5 - 4 * q^7 - 4 * q^11 - 2 * q^13 + 6 * q^17 - 4 * q^19 - q^25 - 2 * q^29 + 4 * q^31 + 8 * q^35 - 2 * q^37 - 2 * q^41 + 4 * q^43 - 8 * q^47 + 9 * q^49 - 10 * q^53 + 8 * q^55 + 4 * q^59 + 6 * q^61 + 4 * q^65 + 4 * q^67 + 16 * q^71 - 6 * q^73 + 16 * q^77 + 4 * q^79 - 12 * q^83 - 12 * q^85 - 10 * q^89 + 8 * q^91 + 8 * q^95 - 14 * 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 −4.00000 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 288.2.a.b 1
3.b odd 2 1 96.2.a.b yes 1
4.b odd 2 1 288.2.a.c 1
5.b even 2 1 7200.2.a.bx 1
5.c odd 4 2 7200.2.f.f 2
8.b even 2 1 576.2.a.g 1
8.d odd 2 1 576.2.a.h 1
9.c even 3 2 2592.2.i.w 2
9.d odd 6 2 2592.2.i.h 2
12.b even 2 1 96.2.a.a 1
15.d odd 2 1 2400.2.a.q 1
15.e even 4 2 2400.2.f.r 2
16.e even 4 2 2304.2.d.s 2
16.f odd 4 2 2304.2.d.c 2
20.d odd 2 1 7200.2.a.e 1
20.e even 4 2 7200.2.f.x 2
21.c even 2 1 4704.2.a.e 1
24.f even 2 1 192.2.a.c 1
24.h odd 2 1 192.2.a.a 1
36.f odd 6 2 2592.2.i.q 2
36.h even 6 2 2592.2.i.b 2
48.i odd 4 2 768.2.d.h 2
48.k even 4 2 768.2.d.a 2
60.h even 2 1 2400.2.a.r 1
60.l odd 4 2 2400.2.f.a 2
84.h odd 2 1 4704.2.a.t 1
120.i odd 2 1 4800.2.a.co 1
120.m even 2 1 4800.2.a.f 1
120.q odd 4 2 4800.2.f.bh 2
120.w even 4 2 4800.2.f.e 2
168.e odd 2 1 9408.2.a.bj 1
168.i even 2 1 9408.2.a.ct 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.2.a.a 1 12.b even 2 1
96.2.a.b yes 1 3.b odd 2 1
192.2.a.a 1 24.h odd 2 1
192.2.a.c 1 24.f even 2 1
288.2.a.b 1 1.a even 1 1 trivial
288.2.a.c 1 4.b odd 2 1
576.2.a.g 1 8.b even 2 1
576.2.a.h 1 8.d odd 2 1
768.2.d.a 2 48.k even 4 2
768.2.d.h 2 48.i odd 4 2
2304.2.d.c 2 16.f odd 4 2
2304.2.d.s 2 16.e even 4 2
2400.2.a.q 1 15.d odd 2 1
2400.2.a.r 1 60.h even 2 1
2400.2.f.a 2 60.l odd 4 2
2400.2.f.r 2 15.e even 4 2
2592.2.i.b 2 36.h even 6 2
2592.2.i.h 2 9.d odd 6 2
2592.2.i.q 2 36.f odd 6 2
2592.2.i.w 2 9.c even 3 2
4704.2.a.e 1 21.c even 2 1
4704.2.a.t 1 84.h odd 2 1
4800.2.a.f 1 120.m even 2 1
4800.2.a.co 1 120.i odd 2 1
4800.2.f.e 2 120.w even 4 2
4800.2.f.bh 2 120.q odd 4 2
7200.2.a.e 1 20.d odd 2 1
7200.2.a.bx 1 5.b even 2 1
7200.2.f.f 2 5.c odd 4 2
7200.2.f.x 2 20.e even 4 2
9408.2.a.bj 1 168.e odd 2 1
9408.2.a.ct 1 168.i 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 $$T_{7} + 4$$ T7 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 2$$
$7$ $$T + 4$$
$11$ $$T + 4$$
$13$ $$T + 2$$
$17$ $$T - 6$$
$19$ $$T + 4$$
$23$ $$T$$
$29$ $$T + 2$$
$31$ $$T - 4$$
$37$ $$T + 2$$
$41$ $$T + 2$$
$43$ $$T - 4$$
$47$ $$T + 8$$
$53$ $$T + 10$$
$59$ $$T - 4$$
$61$ $$T - 6$$
$67$ $$T - 4$$
$71$ $$T - 16$$
$73$ $$T + 6$$
$79$ $$T - 4$$
$83$ $$T + 12$$
$89$ $$T + 10$$
$97$ $$T + 14$$