# Properties

 Label 588.6.i.b 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: $$\Z[a_1, \ldots, a_{19}]$$ 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.

 $$f(q)$$ $$=$$ $$q + (9 \zeta_{6} - 9) q^{3} + 6 \zeta_{6} q^{5} - 81 \zeta_{6} q^{9} +O(q^{10})$$ q + (9*z - 9) * q^3 + 6*z * q^5 - 81*z * q^9 $$q + (9 \zeta_{6} - 9) q^{3} + 6 \zeta_{6} q^{5} - 81 \zeta_{6} q^{9} + ( - 108 \zeta_{6} + 108) q^{11} + 346 q^{13} - 54 q^{15} + (1398 \zeta_{6} - 1398) q^{17} - 1012 \zeta_{6} q^{19} + 1536 \zeta_{6} q^{23} + ( - 3089 \zeta_{6} + 3089) q^{25} + 729 q^{27} - 3762 q^{29} + (736 \zeta_{6} - 736) q^{31} + 972 \zeta_{6} q^{33} - 2054 \zeta_{6} q^{37} + (3114 \zeta_{6} - 3114) q^{39} + 15534 q^{41} + 11036 q^{43} + ( - 486 \zeta_{6} + 486) q^{45} + 4560 \zeta_{6} q^{47} - 12582 \zeta_{6} q^{51} + ( - 7962 \zeta_{6} + 7962) q^{53} + 648 q^{55} + 9108 q^{57} + (7020 \zeta_{6} - 7020) q^{59} + 26870 \zeta_{6} q^{61} + 2076 \zeta_{6} q^{65} + (52148 \zeta_{6} - 52148) q^{67} - 13824 q^{69} - 2544 q^{71} + (9766 \zeta_{6} - 9766) q^{73} + 27801 \zeta_{6} q^{75} - 68672 \zeta_{6} q^{79} + (6561 \zeta_{6} - 6561) q^{81} + 61668 q^{83} - 8388 q^{85} + ( - 33858 \zeta_{6} + 33858) q^{87} - 41454 \zeta_{6} q^{89} - 6624 \zeta_{6} q^{93} + ( - 6072 \zeta_{6} + 6072) q^{95} + 111262 q^{97} - 8748 q^{99} +O(q^{100})$$ q + (9*z - 9) * q^3 + 6*z * q^5 - 81*z * q^9 + (-108*z + 108) * q^11 + 346 * q^13 - 54 * q^15 + (1398*z - 1398) * q^17 - 1012*z * q^19 + 1536*z * q^23 + (-3089*z + 3089) * q^25 + 729 * q^27 - 3762 * q^29 + (736*z - 736) * q^31 + 972*z * q^33 - 2054*z * q^37 + (3114*z - 3114) * q^39 + 15534 * q^41 + 11036 * q^43 + (-486*z + 486) * q^45 + 4560*z * q^47 - 12582*z * q^51 + (-7962*z + 7962) * q^53 + 648 * q^55 + 9108 * q^57 + (7020*z - 7020) * q^59 + 26870*z * q^61 + 2076*z * q^65 + (52148*z - 52148) * q^67 - 13824 * q^69 - 2544 * q^71 + (9766*z - 9766) * q^73 + 27801*z * q^75 - 68672*z * q^79 + (6561*z - 6561) * q^81 + 61668 * q^83 - 8388 * q^85 + (-33858*z + 33858) * q^87 - 41454*z * q^89 - 6624*z * q^93 + (-6072*z + 6072) * q^95 + 111262 * q^97 - 8748 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 9 q^{3} + 6 q^{5} - 81 q^{9}+O(q^{10})$$ 2 * q - 9 * q^3 + 6 * q^5 - 81 * q^9 $$2 q - 9 q^{3} + 6 q^{5} - 81 q^{9} + 108 q^{11} + 692 q^{13} - 108 q^{15} - 1398 q^{17} - 1012 q^{19} + 1536 q^{23} + 3089 q^{25} + 1458 q^{27} - 7524 q^{29} - 736 q^{31} + 972 q^{33} - 2054 q^{37} - 3114 q^{39} + 31068 q^{41} + 22072 q^{43} + 486 q^{45} + 4560 q^{47} - 12582 q^{51} + 7962 q^{53} + 1296 q^{55} + 18216 q^{57} - 7020 q^{59} + 26870 q^{61} + 2076 q^{65} - 52148 q^{67} - 27648 q^{69} - 5088 q^{71} - 9766 q^{73} + 27801 q^{75} - 68672 q^{79} - 6561 q^{81} + 123336 q^{83} - 16776 q^{85} + 33858 q^{87} - 41454 q^{89} - 6624 q^{93} + 6072 q^{95} + 222524 q^{97} - 17496 q^{99}+O(q^{100})$$ 2 * q - 9 * q^3 + 6 * q^5 - 81 * q^9 + 108 * q^11 + 692 * q^13 - 108 * q^15 - 1398 * q^17 - 1012 * q^19 + 1536 * q^23 + 3089 * q^25 + 1458 * q^27 - 7524 * q^29 - 736 * q^31 + 972 * q^33 - 2054 * q^37 - 3114 * q^39 + 31068 * q^41 + 22072 * q^43 + 486 * q^45 + 4560 * q^47 - 12582 * q^51 + 7962 * q^53 + 1296 * q^55 + 18216 * q^57 - 7020 * q^59 + 26870 * q^61 + 2076 * q^65 - 52148 * q^67 - 27648 * q^69 - 5088 * q^71 - 9766 * q^73 + 27801 * q^75 - 68672 * q^79 - 6561 * q^81 + 123336 * q^83 - 16776 * q^85 + 33858 * q^87 - 41454 * q^89 - 6624 * q^93 + 6072 * q^95 + 222524 * q^97 - 17496 * 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 3.00000 + 5.19615i 0 0 0 −40.5000 70.1481i 0
373.1 0 −4.50000 7.79423i 0 3.00000 5.19615i 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.b 2
7.b odd 2 1 588.6.i.f 2
7.c even 3 1 588.6.a.e 1
7.c even 3 1 inner 588.6.i.b 2
7.d odd 6 1 84.6.a.a 1
7.d odd 6 1 588.6.i.f 2
21.g even 6 1 252.6.a.b 1
28.f even 6 1 336.6.a.n 1
84.j odd 6 1 1008.6.a.o 1

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 6T_{5} + 36$$ acting on $$S_{6}^{\mathrm{new}}(588, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 9T + 81$$
$5$ $$T^{2} - 6T + 36$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 108T + 11664$$
$13$ $$(T - 346)^{2}$$
$17$ $$T^{2} + 1398 T + 1954404$$
$19$ $$T^{2} + 1012 T + 1024144$$
$23$ $$T^{2} - 1536 T + 2359296$$
$29$ $$(T + 3762)^{2}$$
$31$ $$T^{2} + 736T + 541696$$
$37$ $$T^{2} + 2054 T + 4218916$$
$41$ $$(T - 15534)^{2}$$
$43$ $$(T - 11036)^{2}$$
$47$ $$T^{2} - 4560 T + 20793600$$
$53$ $$T^{2} - 7962 T + 63393444$$
$59$ $$T^{2} + 7020 T + 49280400$$
$61$ $$T^{2} - 26870 T + 721996900$$
$67$ $$T^{2} + \cdots + 2719413904$$
$71$ $$(T + 2544)^{2}$$
$73$ $$T^{2} + 9766 T + 95374756$$
$79$ $$T^{2} + \cdots + 4715843584$$
$83$ $$(T - 61668)^{2}$$
$89$ $$T^{2} + \cdots + 1718434116$$
$97$ $$(T - 111262)^{2}$$