# Properties

 Label 588.6.i.e Level $588$ Weight $6$ Character orbit 588.i Analytic conductor $94.306$ 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] = [588,6,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, 6, names="a")

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

chi := DirichletCharacter("588.361");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$588 = 2^{2} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ 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: $$94.3056860500$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 84) 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.

 $$f(q)$$ $$=$$ $$q + ( - 9 \zeta_{6} + 9) q^{3} - 34 \zeta_{6} q^{5} - 81 \zeta_{6} q^{9} +O(q^{10})$$ q + (-9*z + 9) * q^3 - 34*z * q^5 - 81*z * q^9 $$q + ( - 9 \zeta_{6} + 9) q^{3} - 34 \zeta_{6} q^{5} - 81 \zeta_{6} q^{9} + ( - 332 \zeta_{6} + 332) q^{11} + 1026 q^{13} - 306 q^{15} + ( - 922 \zeta_{6} + 922) q^{17} + 452 \zeta_{6} q^{19} + 3776 \zeta_{6} q^{23} + ( - 1969 \zeta_{6} + 1969) q^{25} - 729 q^{27} + 1166 q^{29} + (9792 \zeta_{6} - 9792) q^{31} - 2988 \zeta_{6} q^{33} - 2390 \zeta_{6} q^{37} + ( - 9234 \zeta_{6} + 9234) q^{39} + 7230 q^{41} + 4652 q^{43} + (2754 \zeta_{6} - 2754) q^{45} + 24672 \zeta_{6} q^{47} - 8298 \zeta_{6} q^{51} + (1110 \zeta_{6} - 1110) q^{53} - 11288 q^{55} + 4068 q^{57} + ( - 46892 \zeta_{6} + 46892) q^{59} - 9762 \zeta_{6} q^{61} - 34884 \zeta_{6} q^{65} + ( - 26252 \zeta_{6} + 26252) q^{67} + 33984 q^{69} + 65440 q^{71} + (5606 \zeta_{6} - 5606) q^{73} - 17721 \zeta_{6} q^{75} + 9840 \zeta_{6} q^{79} + (6561 \zeta_{6} - 6561) q^{81} - 61108 q^{83} - 31348 q^{85} + ( - 10494 \zeta_{6} + 10494) q^{87} - 62958 \zeta_{6} q^{89} + 88128 \zeta_{6} q^{93} + ( - 15368 \zeta_{6} + 15368) q^{95} + 37838 q^{97} - 26892 q^{99} +O(q^{100})$$ q + (-9*z + 9) * q^3 - 34*z * q^5 - 81*z * q^9 + (-332*z + 332) * q^11 + 1026 * q^13 - 306 * q^15 + (-922*z + 922) * q^17 + 452*z * q^19 + 3776*z * q^23 + (-1969*z + 1969) * q^25 - 729 * q^27 + 1166 * q^29 + (9792*z - 9792) * q^31 - 2988*z * q^33 - 2390*z * q^37 + (-9234*z + 9234) * q^39 + 7230 * q^41 + 4652 * q^43 + (2754*z - 2754) * q^45 + 24672*z * q^47 - 8298*z * q^51 + (1110*z - 1110) * q^53 - 11288 * q^55 + 4068 * q^57 + (-46892*z + 46892) * q^59 - 9762*z * q^61 - 34884*z * q^65 + (-26252*z + 26252) * q^67 + 33984 * q^69 + 65440 * q^71 + (5606*z - 5606) * q^73 - 17721*z * q^75 + 9840*z * q^79 + (6561*z - 6561) * q^81 - 61108 * q^83 - 31348 * q^85 + (-10494*z + 10494) * q^87 - 62958*z * q^89 + 88128*z * q^93 + (-15368*z + 15368) * q^95 + 37838 * q^97 - 26892 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 9 q^{3} - 34 q^{5} - 81 q^{9}+O(q^{10})$$ 2 * q + 9 * q^3 - 34 * q^5 - 81 * q^9 $$2 q + 9 q^{3} - 34 q^{5} - 81 q^{9} + 332 q^{11} + 2052 q^{13} - 612 q^{15} + 922 q^{17} + 452 q^{19} + 3776 q^{23} + 1969 q^{25} - 1458 q^{27} + 2332 q^{29} - 9792 q^{31} - 2988 q^{33} - 2390 q^{37} + 9234 q^{39} + 14460 q^{41} + 9304 q^{43} - 2754 q^{45} + 24672 q^{47} - 8298 q^{51} - 1110 q^{53} - 22576 q^{55} + 8136 q^{57} + 46892 q^{59} - 9762 q^{61} - 34884 q^{65} + 26252 q^{67} + 67968 q^{69} + 130880 q^{71} - 5606 q^{73} - 17721 q^{75} + 9840 q^{79} - 6561 q^{81} - 122216 q^{83} - 62696 q^{85} + 10494 q^{87} - 62958 q^{89} + 88128 q^{93} + 15368 q^{95} + 75676 q^{97} - 53784 q^{99}+O(q^{100})$$ 2 * q + 9 * q^3 - 34 * q^5 - 81 * q^9 + 332 * q^11 + 2052 * q^13 - 612 * q^15 + 922 * q^17 + 452 * q^19 + 3776 * q^23 + 1969 * q^25 - 1458 * q^27 + 2332 * q^29 - 9792 * q^31 - 2988 * q^33 - 2390 * q^37 + 9234 * q^39 + 14460 * q^41 + 9304 * q^43 - 2754 * q^45 + 24672 * q^47 - 8298 * q^51 - 1110 * q^53 - 22576 * q^55 + 8136 * q^57 + 46892 * q^59 - 9762 * q^61 - 34884 * q^65 + 26252 * q^67 + 67968 * q^69 + 130880 * q^71 - 5606 * q^73 - 17721 * q^75 + 9840 * q^79 - 6561 * q^81 - 122216 * q^83 - 62696 * q^85 + 10494 * q^87 - 62958 * q^89 + 88128 * q^93 + 15368 * q^95 + 75676 * q^97 - 53784 * 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 4.50000 7.79423i 0 −17.0000 29.4449i 0 0 0 −40.5000 70.1481i 0
373.1 0 4.50000 + 7.79423i 0 −17.0000 + 29.4449i 0 0 0 −40.5000 + 70.1481i 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 588.6.i.e 2
7.b odd 2 1 588.6.i.c 2
7.c even 3 1 588.6.a.b 1
7.c even 3 1 inner 588.6.i.e 2
7.d odd 6 1 84.6.a.b 1
7.d odd 6 1 588.6.i.c 2
21.g even 6 1 252.6.a.c 1
28.f even 6 1 336.6.a.e 1
84.j odd 6 1 1008.6.a.u 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.6.a.b 1 7.d odd 6 1
252.6.a.c 1 21.g even 6 1
336.6.a.e 1 28.f even 6 1
588.6.a.b 1 7.c even 3 1
588.6.i.c 2 7.b odd 2 1
588.6.i.c 2 7.d odd 6 1
588.6.i.e 2 1.a even 1 1 trivial
588.6.i.e 2 7.c even 3 1 inner
1008.6.a.u 1 84.j odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 34T_{5} + 1156$$ acting on $$S_{6}^{\mathrm{new}}(588, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 9T + 81$$
$5$ $$T^{2} + 34T + 1156$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 332T + 110224$$
$13$ $$(T - 1026)^{2}$$
$17$ $$T^{2} - 922T + 850084$$
$19$ $$T^{2} - 452T + 204304$$
$23$ $$T^{2} - 3776 T + 14258176$$
$29$ $$(T - 1166)^{2}$$
$31$ $$T^{2} + 9792 T + 95883264$$
$37$ $$T^{2} + 2390 T + 5712100$$
$41$ $$(T - 7230)^{2}$$
$43$ $$(T - 4652)^{2}$$
$47$ $$T^{2} - 24672 T + 608707584$$
$53$ $$T^{2} + 1110 T + 1232100$$
$59$ $$T^{2} + \cdots + 2198859664$$
$61$ $$T^{2} + 9762 T + 95296644$$
$67$ $$T^{2} - 26252 T + 689167504$$
$71$ $$(T - 65440)^{2}$$
$73$ $$T^{2} + 5606 T + 31427236$$
$79$ $$T^{2} - 9840 T + 96825600$$
$83$ $$(T + 61108)^{2}$$
$89$ $$T^{2} + \cdots + 3963709764$$
$97$ $$(T - 37838)^{2}$$