# Properties

 Label 588.6.a.e Level $588$ Weight $6$ Character orbit 588.a Self dual yes Analytic conductor $94.306$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [588,6,Mod(1,588)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(588, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("588.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$588 = 2^{2} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 588.a (trivial)

## 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: yes Analytic conductor: $$94.3056860500$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 84) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + 9 q^{3} - 6 q^{5} + 81 q^{9}+O(q^{10})$$ q + 9 * q^3 - 6 * q^5 + 81 * q^9 $$q + 9 q^{3} - 6 q^{5} + 81 q^{9} - 108 q^{11} + 346 q^{13} - 54 q^{15} + 1398 q^{17} + 1012 q^{19} - 1536 q^{23} - 3089 q^{25} + 729 q^{27} - 3762 q^{29} + 736 q^{31} - 972 q^{33} + 2054 q^{37} + 3114 q^{39} + 15534 q^{41} + 11036 q^{43} - 486 q^{45} - 4560 q^{47} + 12582 q^{51} - 7962 q^{53} + 648 q^{55} + 9108 q^{57} + 7020 q^{59} - 26870 q^{61} - 2076 q^{65} + 52148 q^{67} - 13824 q^{69} - 2544 q^{71} + 9766 q^{73} - 27801 q^{75} + 68672 q^{79} + 6561 q^{81} + 61668 q^{83} - 8388 q^{85} - 33858 q^{87} + 41454 q^{89} + 6624 q^{93} - 6072 q^{95} + 111262 q^{97} - 8748 q^{99}+O(q^{100})$$ q + 9 * q^3 - 6 * q^5 + 81 * q^9 - 108 * q^11 + 346 * q^13 - 54 * q^15 + 1398 * q^17 + 1012 * q^19 - 1536 * q^23 - 3089 * q^25 + 729 * q^27 - 3762 * q^29 + 736 * q^31 - 972 * q^33 + 2054 * q^37 + 3114 * q^39 + 15534 * q^41 + 11036 * q^43 - 486 * q^45 - 4560 * q^47 + 12582 * q^51 - 7962 * q^53 + 648 * q^55 + 9108 * q^57 + 7020 * q^59 - 26870 * q^61 - 2076 * q^65 + 52148 * q^67 - 13824 * q^69 - 2544 * q^71 + 9766 * q^73 - 27801 * q^75 + 68672 * q^79 + 6561 * q^81 + 61668 * q^83 - 8388 * q^85 - 33858 * q^87 + 41454 * q^89 + 6624 * q^93 - 6072 * q^95 + 111262 * q^97 - 8748 * q^99

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 9.00000 0 −6.00000 0 0 0 81.0000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$7$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} + 6$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(588))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 9$$
$5$ $$T + 6$$
$7$ $$T$$
$11$ $$T + 108$$
$13$ $$T - 346$$
$17$ $$T - 1398$$
$19$ $$T - 1012$$
$23$ $$T + 1536$$
$29$ $$T + 3762$$
$31$ $$T - 736$$
$37$ $$T - 2054$$
$41$ $$T - 15534$$
$43$ $$T - 11036$$
$47$ $$T + 4560$$
$53$ $$T + 7962$$
$59$ $$T - 7020$$
$61$ $$T + 26870$$
$67$ $$T - 52148$$
$71$ $$T + 2544$$
$73$ $$T - 9766$$
$79$ $$T - 68672$$
$83$ $$T - 61668$$
$89$ $$T - 41454$$
$97$ $$T - 111262$$