# Properties

 Label 288.1.t.a Level $288$ Weight $1$ Character orbit 288.t Analytic conductor $0.144$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -8 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [288,1,Mod(79,288)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(288, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 3, 4]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("288.79");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$288 = 2^{5} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 288.t (of order $$6$$, degree $$2$$, not minimal)

## 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: no Analytic conductor: $$0.143730723638$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 72) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.648.1 Artin image: $C_6\times S_3$ Artin field: Galois closure of 12.0.110075314176.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q - \zeta_{6}^{2} q^{3} - \zeta_{6} q^{9} +O(q^{10})$$ q - z^2 * q^3 - z * q^9 $$q - \zeta_{6}^{2} q^{3} - \zeta_{6} q^{9} + \zeta_{6}^{2} q^{11} - q^{17} + q^{19} + \zeta_{6}^{2} q^{25} - q^{27} + \zeta_{6} q^{33} + \zeta_{6} q^{41} + \zeta_{6}^{2} q^{43} - \zeta_{6} q^{49} + \zeta_{6}^{2} q^{51} - \zeta_{6}^{2} q^{57} - \zeta_{6} q^{59} - \zeta_{6} q^{67} - q^{73} + \zeta_{6} q^{75} + \zeta_{6}^{2} q^{81} - \zeta_{6}^{2} q^{83} + q^{89} - \zeta_{6}^{2} q^{97} + q^{99} +O(q^{100})$$ q - z^2 * q^3 - z * q^9 + z^2 * q^11 - q^17 + q^19 + z^2 * q^25 - q^27 + z * q^33 + z * q^41 + z^2 * q^43 - z * q^49 + z^2 * q^51 - z^2 * q^57 - z * q^59 - z * q^67 - q^73 + z * q^75 + z^2 * q^81 - z^2 * q^83 + q^89 - z^2 * q^97 + q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} - q^{9}+O(q^{10})$$ 2 * q + q^3 - q^9 $$2 q + q^{3} - q^{9} - q^{11} - 2 q^{17} + 2 q^{19} - q^{25} - 2 q^{27} + q^{33} + q^{41} - q^{43} - q^{49} - q^{51} + q^{57} - q^{59} - q^{67} - 2 q^{73} + q^{75} - q^{81} + 2 q^{83} + 4 q^{89} + q^{97} + 2 q^{99}+O(q^{100})$$ 2 * q + q^3 - q^9 - q^11 - 2 * q^17 + 2 * q^19 - q^25 - 2 * q^27 + q^33 + q^41 - q^43 - q^49 - q^51 + q^57 - q^59 - q^67 - 2 * q^73 + q^75 - q^81 + 2 * q^83 + 4 * q^89 + q^97 + 2 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/288\mathbb{Z}\right)^\times$$.

 $$n$$ $$37$$ $$65$$ $$127$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{6}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
79.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0.500000 0.866025i 0 0 0 0 0 −0.500000 0.866025i 0
175.1 0 0.500000 + 0.866025i 0 0 0 0 0 −0.500000 + 0.866025i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
9.c even 3 1 inner
72.p odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 288.1.t.a 2
3.b odd 2 1 864.1.t.a 2
4.b odd 2 1 72.1.p.a 2
8.b even 2 1 72.1.p.a 2
8.d odd 2 1 CM 288.1.t.a 2
9.c even 3 1 inner 288.1.t.a 2
9.c even 3 1 2592.1.b.b 1
9.d odd 6 1 864.1.t.a 2
9.d odd 6 1 2592.1.b.a 1
12.b even 2 1 216.1.p.a 2
16.e even 4 2 2304.1.o.c 4
16.f odd 4 2 2304.1.o.c 4
20.d odd 2 1 1800.1.bk.d 2
20.e even 4 2 1800.1.ba.b 4
24.f even 2 1 864.1.t.a 2
24.h odd 2 1 216.1.p.a 2
28.d even 2 1 3528.1.cg.a 2
28.f even 6 1 3528.1.ba.a 2
28.f even 6 1 3528.1.ce.b 2
28.g odd 6 1 3528.1.ba.b 2
28.g odd 6 1 3528.1.ce.a 2
36.f odd 6 1 72.1.p.a 2
36.f odd 6 1 648.1.b.b 1
36.h even 6 1 216.1.p.a 2
36.h even 6 1 648.1.b.a 1
40.f even 2 1 1800.1.bk.d 2
40.i odd 4 2 1800.1.ba.b 4
56.h odd 2 1 3528.1.cg.a 2
56.j odd 6 1 3528.1.ba.a 2
56.j odd 6 1 3528.1.ce.b 2
56.p even 6 1 3528.1.ba.b 2
56.p even 6 1 3528.1.ce.a 2
72.j odd 6 1 216.1.p.a 2
72.j odd 6 1 648.1.b.a 1
72.l even 6 1 864.1.t.a 2
72.l even 6 1 2592.1.b.a 1
72.n even 6 1 72.1.p.a 2
72.n even 6 1 648.1.b.b 1
72.p odd 6 1 inner 288.1.t.a 2
72.p odd 6 1 2592.1.b.b 1
144.v odd 12 2 2304.1.o.c 4
144.x even 12 2 2304.1.o.c 4
180.p odd 6 1 1800.1.bk.d 2
180.x even 12 2 1800.1.ba.b 4
252.n even 6 1 3528.1.ce.b 2
252.u odd 6 1 3528.1.ba.b 2
252.bi even 6 1 3528.1.cg.a 2
252.bj even 6 1 3528.1.ba.a 2
252.bl odd 6 1 3528.1.ce.a 2
360.bk even 6 1 1800.1.bk.d 2
360.bu odd 12 2 1800.1.ba.b 4
504.w even 6 1 3528.1.ce.a 2
504.bn odd 6 1 3528.1.cg.a 2
504.bp odd 6 1 3528.1.ba.a 2
504.cq even 6 1 3528.1.ba.b 2
504.cw odd 6 1 3528.1.ce.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.1.p.a 2 4.b odd 2 1
72.1.p.a 2 8.b even 2 1
72.1.p.a 2 36.f odd 6 1
72.1.p.a 2 72.n even 6 1
216.1.p.a 2 12.b even 2 1
216.1.p.a 2 24.h odd 2 1
216.1.p.a 2 36.h even 6 1
216.1.p.a 2 72.j odd 6 1
288.1.t.a 2 1.a even 1 1 trivial
288.1.t.a 2 8.d odd 2 1 CM
288.1.t.a 2 9.c even 3 1 inner
288.1.t.a 2 72.p odd 6 1 inner
648.1.b.a 1 36.h even 6 1
648.1.b.a 1 72.j odd 6 1
648.1.b.b 1 36.f odd 6 1
648.1.b.b 1 72.n even 6 1
864.1.t.a 2 3.b odd 2 1
864.1.t.a 2 9.d odd 6 1
864.1.t.a 2 24.f even 2 1
864.1.t.a 2 72.l even 6 1
1800.1.ba.b 4 20.e even 4 2
1800.1.ba.b 4 40.i odd 4 2
1800.1.ba.b 4 180.x even 12 2
1800.1.ba.b 4 360.bu odd 12 2
1800.1.bk.d 2 20.d odd 2 1
1800.1.bk.d 2 40.f even 2 1
1800.1.bk.d 2 180.p odd 6 1
1800.1.bk.d 2 360.bk even 6 1
2304.1.o.c 4 16.e even 4 2
2304.1.o.c 4 16.f odd 4 2
2304.1.o.c 4 144.v odd 12 2
2304.1.o.c 4 144.x even 12 2
2592.1.b.a 1 9.d odd 6 1
2592.1.b.a 1 72.l even 6 1
2592.1.b.b 1 9.c even 3 1
2592.1.b.b 1 72.p odd 6 1
3528.1.ba.a 2 28.f even 6 1
3528.1.ba.a 2 56.j odd 6 1
3528.1.ba.a 2 252.bj even 6 1
3528.1.ba.a 2 504.bp odd 6 1
3528.1.ba.b 2 28.g odd 6 1
3528.1.ba.b 2 56.p even 6 1
3528.1.ba.b 2 252.u odd 6 1
3528.1.ba.b 2 504.cq even 6 1
3528.1.ce.a 2 28.g odd 6 1
3528.1.ce.a 2 56.p even 6 1
3528.1.ce.a 2 252.bl odd 6 1
3528.1.ce.a 2 504.w even 6 1
3528.1.ce.b 2 28.f even 6 1
3528.1.ce.b 2 56.j odd 6 1
3528.1.ce.b 2 252.n even 6 1
3528.1.ce.b 2 504.cw odd 6 1
3528.1.cg.a 2 28.d even 2 1
3528.1.cg.a 2 56.h odd 2 1
3528.1.cg.a 2 252.bi even 6 1
3528.1.cg.a 2 504.bn odd 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(288, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - T + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2} + T + 1$$
$13$ $$T^{2}$$
$17$ $$(T + 1)^{2}$$
$19$ $$(T - 1)^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2}$$
$41$ $$T^{2} - T + 1$$
$43$ $$T^{2} + T + 1$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2} + T + 1$$
$61$ $$T^{2}$$
$67$ $$T^{2} + T + 1$$
$71$ $$T^{2}$$
$73$ $$(T + 1)^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2} - 2T + 4$$
$89$ $$(T - 2)^{2}$$
$97$ $$T^{2} - T + 1$$