# Properties

 Label 588.2.i.g Level $588$ Weight $2$ Character orbit 588.i Analytic conductor $4.695$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [588,2,Mod(361,588)]

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, 0, 4]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("588.361");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$588 = 2^{2} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 588.i (of order $$3$$, 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: $$4.69520363885$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{6} + 1) q^{3} + 2 \zeta_{6} q^{5} - \zeta_{6} q^{9} +O(q^{10})$$ q + (-z + 1) * q^3 + 2*z * q^5 - z * q^9 $$q + ( - \zeta_{6} + 1) q^{3} + 2 \zeta_{6} q^{5} - \zeta_{6} q^{9} + (2 \zeta_{6} - 2) q^{11} + 4 q^{13} + 2 q^{15} + ( - 6 \zeta_{6} + 6) q^{17} + 8 \zeta_{6} q^{19} + 6 \zeta_{6} q^{23} + ( - \zeta_{6} + 1) q^{25} - q^{27} - 10 q^{29} + ( - 4 \zeta_{6} + 4) q^{31} + 2 \zeta_{6} q^{33} - 6 \zeta_{6} q^{37} + ( - 4 \zeta_{6} + 4) q^{39} + 6 q^{41} + 4 q^{43} + ( - 2 \zeta_{6} + 2) q^{45} + 8 \zeta_{6} q^{47} - 6 \zeta_{6} q^{51} + (2 \zeta_{6} - 2) q^{53} - 4 q^{55} + 8 q^{57} + (4 \zeta_{6} - 4) q^{59} - 8 \zeta_{6} q^{61} + 8 \zeta_{6} q^{65} + ( - 8 \zeta_{6} + 8) q^{67} + 6 q^{69} - 10 q^{71} + ( - 4 \zeta_{6} + 4) q^{73} - \zeta_{6} q^{75} - 4 \zeta_{6} q^{79} + (\zeta_{6} - 1) q^{81} - 12 q^{83} + 12 q^{85} + (10 \zeta_{6} - 10) q^{87} - 14 \zeta_{6} q^{89} - 4 \zeta_{6} q^{93} + (16 \zeta_{6} - 16) q^{95} - 4 q^{97} + 2 q^{99} +O(q^{100})$$ q + (-z + 1) * q^3 + 2*z * q^5 - z * q^9 + (2*z - 2) * q^11 + 4 * q^13 + 2 * q^15 + (-6*z + 6) * q^17 + 8*z * q^19 + 6*z * q^23 + (-z + 1) * q^25 - q^27 - 10 * q^29 + (-4*z + 4) * q^31 + 2*z * q^33 - 6*z * q^37 + (-4*z + 4) * q^39 + 6 * q^41 + 4 * q^43 + (-2*z + 2) * q^45 + 8*z * q^47 - 6*z * q^51 + (2*z - 2) * q^53 - 4 * q^55 + 8 * q^57 + (4*z - 4) * q^59 - 8*z * q^61 + 8*z * q^65 + (-8*z + 8) * q^67 + 6 * q^69 - 10 * q^71 + (-4*z + 4) * q^73 - z * q^75 - 4*z * q^79 + (z - 1) * q^81 - 12 * q^83 + 12 * q^85 + (10*z - 10) * q^87 - 14*z * q^89 - 4*z * q^93 + (16*z - 16) * q^95 - 4 * q^97 + 2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} + 2 q^{5} - q^{9}+O(q^{10})$$ 2 * q + q^3 + 2 * q^5 - q^9 $$2 q + q^{3} + 2 q^{5} - q^{9} - 2 q^{11} + 8 q^{13} + 4 q^{15} + 6 q^{17} + 8 q^{19} + 6 q^{23} + q^{25} - 2 q^{27} - 20 q^{29} + 4 q^{31} + 2 q^{33} - 6 q^{37} + 4 q^{39} + 12 q^{41} + 8 q^{43} + 2 q^{45} + 8 q^{47} - 6 q^{51} - 2 q^{53} - 8 q^{55} + 16 q^{57} - 4 q^{59} - 8 q^{61} + 8 q^{65} + 8 q^{67} + 12 q^{69} - 20 q^{71} + 4 q^{73} - q^{75} - 4 q^{79} - q^{81} - 24 q^{83} + 24 q^{85} - 10 q^{87} - 14 q^{89} - 4 q^{93} - 16 q^{95} - 8 q^{97} + 4 q^{99}+O(q^{100})$$ 2 * q + q^3 + 2 * q^5 - q^9 - 2 * q^11 + 8 * q^13 + 4 * q^15 + 6 * q^17 + 8 * q^19 + 6 * q^23 + q^25 - 2 * q^27 - 20 * q^29 + 4 * q^31 + 2 * q^33 - 6 * q^37 + 4 * q^39 + 12 * q^41 + 8 * q^43 + 2 * q^45 + 8 * q^47 - 6 * q^51 - 2 * q^53 - 8 * q^55 + 16 * q^57 - 4 * q^59 - 8 * q^61 + 8 * q^65 + 8 * q^67 + 12 * q^69 - 20 * q^71 + 4 * q^73 - q^75 - 4 * q^79 - q^81 - 24 * q^83 + 24 * q^85 - 10 * q^87 - 14 * q^89 - 4 * q^93 - 16 * q^95 - 8 * q^97 + 4 * q^99

## Character values

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

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
361.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0.500000 0.866025i 0 1.00000 + 1.73205i 0 0 0 −0.500000 0.866025i 0
373.1 0 0.500000 + 0.866025i 0 1.00000 1.73205i 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 588.2.i.g 2
3.b odd 2 1 1764.2.k.c 2
4.b odd 2 1 2352.2.q.k 2
7.b odd 2 1 588.2.i.a 2
7.c even 3 1 588.2.a.b 1
7.c even 3 1 inner 588.2.i.g 2
7.d odd 6 1 588.2.a.e yes 1
7.d odd 6 1 588.2.i.a 2
21.c even 2 1 1764.2.k.i 2
21.g even 6 1 1764.2.a.b 1
21.g even 6 1 1764.2.k.i 2
21.h odd 6 1 1764.2.a.i 1
21.h odd 6 1 1764.2.k.c 2
28.d even 2 1 2352.2.q.p 2
28.f even 6 1 2352.2.a.j 1
28.f even 6 1 2352.2.q.p 2
28.g odd 6 1 2352.2.a.p 1
28.g odd 6 1 2352.2.q.k 2
56.j odd 6 1 9408.2.a.l 1
56.k odd 6 1 9408.2.a.bf 1
56.m even 6 1 9408.2.a.ca 1
56.p even 6 1 9408.2.a.cu 1
84.j odd 6 1 7056.2.a.n 1
84.n even 6 1 7056.2.a.bu 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
588.2.a.b 1 7.c even 3 1
588.2.a.e yes 1 7.d odd 6 1
588.2.i.a 2 7.b odd 2 1
588.2.i.a 2 7.d odd 6 1
588.2.i.g 2 1.a even 1 1 trivial
588.2.i.g 2 7.c even 3 1 inner
1764.2.a.b 1 21.g even 6 1
1764.2.a.i 1 21.h odd 6 1
1764.2.k.c 2 3.b odd 2 1
1764.2.k.c 2 21.h odd 6 1
1764.2.k.i 2 21.c even 2 1
1764.2.k.i 2 21.g even 6 1
2352.2.a.j 1 28.f even 6 1
2352.2.a.p 1 28.g odd 6 1
2352.2.q.k 2 4.b odd 2 1
2352.2.q.k 2 28.g odd 6 1
2352.2.q.p 2 28.d even 2 1
2352.2.q.p 2 28.f even 6 1
7056.2.a.n 1 84.j odd 6 1
7056.2.a.bu 1 84.n even 6 1
9408.2.a.l 1 56.j odd 6 1
9408.2.a.bf 1 56.k odd 6 1
9408.2.a.ca 1 56.m even 6 1
9408.2.a.cu 1 56.p even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(588, [\chi])$$:

 $$T_{5}^{2} - 2T_{5} + 4$$ T5^2 - 2*T5 + 4 $$T_{13} - 4$$ T13 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - T + 1$$
$5$ $$T^{2} - 2T + 4$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 2T + 4$$
$13$ $$(T - 4)^{2}$$
$17$ $$T^{2} - 6T + 36$$
$19$ $$T^{2} - 8T + 64$$
$23$ $$T^{2} - 6T + 36$$
$29$ $$(T + 10)^{2}$$
$31$ $$T^{2} - 4T + 16$$
$37$ $$T^{2} + 6T + 36$$
$41$ $$(T - 6)^{2}$$
$43$ $$(T - 4)^{2}$$
$47$ $$T^{2} - 8T + 64$$
$53$ $$T^{2} + 2T + 4$$
$59$ $$T^{2} + 4T + 16$$
$61$ $$T^{2} + 8T + 64$$
$67$ $$T^{2} - 8T + 64$$
$71$ $$(T + 10)^{2}$$
$73$ $$T^{2} - 4T + 16$$
$79$ $$T^{2} + 4T + 16$$
$83$ $$(T + 12)^{2}$$
$89$ $$T^{2} + 14T + 196$$
$97$ $$(T + 4)^{2}$$