Properties

 Label 588.3.p.c Level $588$ Weight $3$ Character orbit 588.p Analytic conductor $16.022$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [588,3,Mod(557,588)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(588, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 4]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("588.557");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$588 = 2^{2} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 588.p (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: $$16.0218395444$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ 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: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 12) Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 \zeta_{6} q^{3} + (9 \zeta_{6} - 9) q^{9}+O(q^{10})$$ q + 3*z * q^3 + (9*z - 9) * q^9 $$q + 3 \zeta_{6} q^{3} + (9 \zeta_{6} - 9) q^{9} - 22 q^{13} + (26 \zeta_{6} - 26) q^{19} - 25 \zeta_{6} q^{25} - 27 q^{27} + 46 \zeta_{6} q^{31} + (26 \zeta_{6} - 26) q^{37} - 66 \zeta_{6} q^{39} - 22 q^{43} - 78 q^{57} + (74 \zeta_{6} - 74) q^{61} - 122 \zeta_{6} q^{67} + 46 \zeta_{6} q^{73} + ( - 75 \zeta_{6} + 75) q^{75} + ( - 142 \zeta_{6} + 142) q^{79} - 81 \zeta_{6} q^{81} + (138 \zeta_{6} - 138) q^{93} + 2 q^{97} +O(q^{100})$$ q + 3*z * q^3 + (9*z - 9) * q^9 - 22 * q^13 + (26*z - 26) * q^19 - 25*z * q^25 - 27 * q^27 + 46*z * q^31 + (26*z - 26) * q^37 - 66*z * q^39 - 22 * q^43 - 78 * q^57 + (74*z - 74) * q^61 - 122*z * q^67 + 46*z * q^73 + (-75*z + 75) * q^75 + (-142*z + 142) * q^79 - 81*z * q^81 + (138*z - 138) * q^93 + 2 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{3} - 9 q^{9}+O(q^{10})$$ 2 * q + 3 * q^3 - 9 * q^9 $$2 q + 3 q^{3} - 9 q^{9} - 44 q^{13} - 26 q^{19} - 25 q^{25} - 54 q^{27} + 46 q^{31} - 26 q^{37} - 66 q^{39} - 44 q^{43} - 156 q^{57} - 74 q^{61} - 122 q^{67} + 46 q^{73} + 75 q^{75} + 142 q^{79} - 81 q^{81} - 138 q^{93} + 4 q^{97}+O(q^{100})$$ 2 * q + 3 * q^3 - 9 * q^9 - 44 * q^13 - 26 * q^19 - 25 * q^25 - 54 * q^27 + 46 * q^31 - 26 * q^37 - 66 * q^39 - 44 * q^43 - 156 * q^57 - 74 * q^61 - 122 * q^67 + 46 * q^73 + 75 * q^75 + 142 * q^79 - 81 * q^81 - 138 * q^93 + 4 * q^97

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/588\mathbb{Z}\right)^\times$$.

 $$n$$ $$197$$ $$295$$ $$493$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1 + \zeta_{6}$$

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
557.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 1.50000 2.59808i 0 0 0 0 0 −4.50000 7.79423i 0
569.1 0 1.50000 + 2.59808i 0 0 0 0 0 −4.50000 + 7.79423i 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
7.c even 3 1 inner
21.h odd 6 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 588.3.p.c 2
3.b odd 2 1 CM 588.3.p.c 2
7.b odd 2 1 588.3.p.b 2
7.c even 3 1 12.3.c.a 1
7.c even 3 1 inner 588.3.p.c 2
7.d odd 6 1 588.3.c.c 1
7.d odd 6 1 588.3.p.b 2
21.c even 2 1 588.3.p.b 2
21.g even 6 1 588.3.c.c 1
21.g even 6 1 588.3.p.b 2
21.h odd 6 1 12.3.c.a 1
21.h odd 6 1 inner 588.3.p.c 2
28.g odd 6 1 48.3.e.a 1
35.j even 6 1 300.3.g.b 1
35.l odd 12 2 300.3.b.a 2
56.k odd 6 1 192.3.e.a 1
56.p even 6 1 192.3.e.b 1
63.g even 3 1 324.3.g.b 2
63.h even 3 1 324.3.g.b 2
63.j odd 6 1 324.3.g.b 2
63.n odd 6 1 324.3.g.b 2
77.h odd 6 1 1452.3.e.b 1
84.n even 6 1 48.3.e.a 1
105.o odd 6 1 300.3.g.b 1
105.x even 12 2 300.3.b.a 2
112.u odd 12 2 768.3.h.b 2
112.w even 12 2 768.3.h.a 2
140.p odd 6 1 1200.3.l.b 1
140.w even 12 2 1200.3.c.c 2
168.s odd 6 1 192.3.e.b 1
168.v even 6 1 192.3.e.a 1
231.l even 6 1 1452.3.e.b 1
252.o even 6 1 1296.3.q.b 2
252.u odd 6 1 1296.3.q.b 2
252.bb even 6 1 1296.3.q.b 2
252.bl odd 6 1 1296.3.q.b 2
336.bt odd 12 2 768.3.h.a 2
336.bu even 12 2 768.3.h.b 2
420.ba even 6 1 1200.3.l.b 1
420.bp odd 12 2 1200.3.c.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.3.c.a 1 7.c even 3 1
12.3.c.a 1 21.h odd 6 1
48.3.e.a 1 28.g odd 6 1
48.3.e.a 1 84.n even 6 1
192.3.e.a 1 56.k odd 6 1
192.3.e.a 1 168.v even 6 1
192.3.e.b 1 56.p even 6 1
192.3.e.b 1 168.s odd 6 1
300.3.b.a 2 35.l odd 12 2
300.3.b.a 2 105.x even 12 2
300.3.g.b 1 35.j even 6 1
300.3.g.b 1 105.o odd 6 1
324.3.g.b 2 63.g even 3 1
324.3.g.b 2 63.h even 3 1
324.3.g.b 2 63.j odd 6 1
324.3.g.b 2 63.n odd 6 1
588.3.c.c 1 7.d odd 6 1
588.3.c.c 1 21.g even 6 1
588.3.p.b 2 7.b odd 2 1
588.3.p.b 2 7.d odd 6 1
588.3.p.b 2 21.c even 2 1
588.3.p.b 2 21.g even 6 1
588.3.p.c 2 1.a even 1 1 trivial
588.3.p.c 2 3.b odd 2 1 CM
588.3.p.c 2 7.c even 3 1 inner
588.3.p.c 2 21.h odd 6 1 inner
768.3.h.a 2 112.w even 12 2
768.3.h.a 2 336.bt odd 12 2
768.3.h.b 2 112.u odd 12 2
768.3.h.b 2 336.bu even 12 2
1200.3.c.c 2 140.w even 12 2
1200.3.c.c 2 420.bp odd 12 2
1200.3.l.b 1 140.p odd 6 1
1200.3.l.b 1 420.ba even 6 1
1296.3.q.b 2 252.o even 6 1
1296.3.q.b 2 252.u odd 6 1
1296.3.q.b 2 252.bb even 6 1
1296.3.q.b 2 252.bl odd 6 1
1452.3.e.b 1 77.h odd 6 1
1452.3.e.b 1 231.l even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(588, [\chi])$$:

 $$T_{5}$$ T5 $$T_{13} + 22$$ T13 + 22

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 3T + 9$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$(T + 22)^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2} + 26T + 676$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2} - 46T + 2116$$
$37$ $$T^{2} + 26T + 676$$
$41$ $$T^{2}$$
$43$ $$(T + 22)^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} + 74T + 5476$$
$67$ $$T^{2} + 122T + 14884$$
$71$ $$T^{2}$$
$73$ $$T^{2} - 46T + 2116$$
$79$ $$T^{2} - 142T + 20164$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$(T - 2)^{2}$$