# Properties

 Label 588.6.a.b Level $588$ Weight $6$ Character orbit 588.a Self dual yes Analytic conductor $94.306$ Analytic rank $1$ 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: $$1$$ 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)

 $$f(q)$$ $$=$$ $$q - 9 q^{3} + 34 q^{5} + 81 q^{9}+O(q^{10})$$ q - 9 * q^3 + 34 * q^5 + 81 * q^9 $$q - 9 q^{3} + 34 q^{5} + 81 q^{9} - 332 q^{11} + 1026 q^{13} - 306 q^{15} - 922 q^{17} - 452 q^{19} - 3776 q^{23} - 1969 q^{25} - 729 q^{27} + 1166 q^{29} + 9792 q^{31} + 2988 q^{33} + 2390 q^{37} - 9234 q^{39} + 7230 q^{41} + 4652 q^{43} + 2754 q^{45} - 24672 q^{47} + 8298 q^{51} + 1110 q^{53} - 11288 q^{55} + 4068 q^{57} - 46892 q^{59} + 9762 q^{61} + 34884 q^{65} - 26252 q^{67} + 33984 q^{69} + 65440 q^{71} + 5606 q^{73} + 17721 q^{75} - 9840 q^{79} + 6561 q^{81} - 61108 q^{83} - 31348 q^{85} - 10494 q^{87} + 62958 q^{89} - 88128 q^{93} - 15368 q^{95} + 37838 q^{97} - 26892 q^{99}+O(q^{100})$$ q - 9 * q^3 + 34 * q^5 + 81 * q^9 - 332 * q^11 + 1026 * q^13 - 306 * q^15 - 922 * q^17 - 452 * q^19 - 3776 * q^23 - 1969 * q^25 - 729 * q^27 + 1166 * q^29 + 9792 * q^31 + 2988 * q^33 + 2390 * q^37 - 9234 * q^39 + 7230 * q^41 + 4652 * q^43 + 2754 * q^45 - 24672 * q^47 + 8298 * q^51 + 1110 * q^53 - 11288 * q^55 + 4068 * q^57 - 46892 * q^59 + 9762 * q^61 + 34884 * q^65 - 26252 * q^67 + 33984 * q^69 + 65440 * q^71 + 5606 * q^73 + 17721 * q^75 - 9840 * q^79 + 6561 * q^81 - 61108 * q^83 - 31348 * q^85 - 10494 * q^87 + 62958 * q^89 - 88128 * q^93 - 15368 * q^95 + 37838 * q^97 - 26892 * 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 34.0000 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.b 1
7.b odd 2 1 84.6.a.b 1
7.c even 3 2 588.6.i.e 2
7.d odd 6 2 588.6.i.c 2
21.c even 2 1 252.6.a.c 1
28.d even 2 1 336.6.a.e 1
84.h odd 2 1 1008.6.a.u 1

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} - 34$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(588))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 9$$
$5$ $$T - 34$$
$7$ $$T$$
$11$ $$T + 332$$
$13$ $$T - 1026$$
$17$ $$T + 922$$
$19$ $$T + 452$$
$23$ $$T + 3776$$
$29$ $$T - 1166$$
$31$ $$T - 9792$$
$37$ $$T - 2390$$
$41$ $$T - 7230$$
$43$ $$T - 4652$$
$47$ $$T + 24672$$
$53$ $$T - 1110$$
$59$ $$T + 46892$$
$61$ $$T - 9762$$
$67$ $$T + 26252$$
$71$ $$T - 65440$$
$73$ $$T - 5606$$
$79$ $$T + 9840$$
$83$ $$T + 61108$$
$89$ $$T - 62958$$
$97$ $$T - 37838$$