# Properties

 Label 588.6.a.k Level $588$ Weight $6$ Character orbit 588.a Self dual yes Analytic conductor $94.306$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Learn more

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: $$2$$ Coefficient field: $$\Q(\sqrt{5569})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1392$$ x^2 - x - 1392 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{5569}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 9 q^{3} + (\beta + 3) q^{5} + 81 q^{9}+O(q^{10})$$ q + 9 * q^3 + (b + 3) * q^5 + 81 * q^9 $$q + 9 q^{3} + (\beta + 3) q^{5} + 81 q^{9} + (7 \beta - 45) q^{11} + (6 \beta - 384) q^{13} + (9 \beta + 27) q^{15} + ( - \beta - 963) q^{17} + (24 \beta - 1124) q^{19} + (\beta + 3177) q^{23} + (6 \beta + 2453) q^{25} + 729 q^{27} + ( - 40 \beta + 5286) q^{29} + ( - 72 \beta + 1656) q^{31} + (63 \beta - 405) q^{33} + (150 \beta + 1052) q^{37} + (54 \beta - 3456) q^{39} + (33 \beta - 633) q^{41} + ( - 240 \beta - 2884) q^{43} + (81 \beta + 243) q^{45} + (162 \beta - 7806) q^{47} + ( - 9 \beta - 8667) q^{51} + (234 \beta + 8256) q^{53} + ( - 24 \beta + 38848) q^{55} + (216 \beta - 10116) q^{57} + ( - 50 \beta + 6570) q^{59} + ( - 264 \beta + 2898) q^{61} + ( - 366 \beta + 32262) q^{65} + ( - 462 \beta - 28058) q^{67} + (9 \beta + 28593) q^{69} + (895 \beta + 5511) q^{71} + (174 \beta + 42692) q^{73} + (54 \beta + 22077) q^{75} + (78 \beta - 9810) q^{79} + 6561 q^{81} + (320 \beta + 22212) q^{83} + ( - 966 \beta - 8458) q^{85} + ( - 360 \beta + 47574) q^{87} + ( - 399 \beta - 105609) q^{89} + ( - 648 \beta + 14904) q^{93} + ( - 1052 \beta + 130284) q^{95} + ( - 1614 \beta - 22432) q^{97} + (567 \beta - 3645) q^{99}+O(q^{100})$$ q + 9 * q^3 + (b + 3) * q^5 + 81 * q^9 + (7*b - 45) * q^11 + (6*b - 384) * q^13 + (9*b + 27) * q^15 + (-b - 963) * q^17 + (24*b - 1124) * q^19 + (b + 3177) * q^23 + (6*b + 2453) * q^25 + 729 * q^27 + (-40*b + 5286) * q^29 + (-72*b + 1656) * q^31 + (63*b - 405) * q^33 + (150*b + 1052) * q^37 + (54*b - 3456) * q^39 + (33*b - 633) * q^41 + (-240*b - 2884) * q^43 + (81*b + 243) * q^45 + (162*b - 7806) * q^47 + (-9*b - 8667) * q^51 + (234*b + 8256) * q^53 + (-24*b + 38848) * q^55 + (216*b - 10116) * q^57 + (-50*b + 6570) * q^59 + (-264*b + 2898) * q^61 + (-366*b + 32262) * q^65 + (-462*b - 28058) * q^67 + (9*b + 28593) * q^69 + (895*b + 5511) * q^71 + (174*b + 42692) * q^73 + (54*b + 22077) * q^75 + (78*b - 9810) * q^79 + 6561 * q^81 + (320*b + 22212) * q^83 + (-966*b - 8458) * q^85 + (-360*b + 47574) * q^87 + (-399*b - 105609) * q^89 + (-648*b + 14904) * q^93 + (-1052*b + 130284) * q^95 + (-1614*b - 22432) * q^97 + (567*b - 3645) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 18 q^{3} + 6 q^{5} + 162 q^{9}+O(q^{10})$$ 2 * q + 18 * q^3 + 6 * q^5 + 162 * q^9 $$2 q + 18 q^{3} + 6 q^{5} + 162 q^{9} - 90 q^{11} - 768 q^{13} + 54 q^{15} - 1926 q^{17} - 2248 q^{19} + 6354 q^{23} + 4906 q^{25} + 1458 q^{27} + 10572 q^{29} + 3312 q^{31} - 810 q^{33} + 2104 q^{37} - 6912 q^{39} - 1266 q^{41} - 5768 q^{43} + 486 q^{45} - 15612 q^{47} - 17334 q^{51} + 16512 q^{53} + 77696 q^{55} - 20232 q^{57} + 13140 q^{59} + 5796 q^{61} + 64524 q^{65} - 56116 q^{67} + 57186 q^{69} + 11022 q^{71} + 85384 q^{73} + 44154 q^{75} - 19620 q^{79} + 13122 q^{81} + 44424 q^{83} - 16916 q^{85} + 95148 q^{87} - 211218 q^{89} + 29808 q^{93} + 260568 q^{95} - 44864 q^{97} - 7290 q^{99}+O(q^{100})$$ 2 * q + 18 * q^3 + 6 * q^5 + 162 * q^9 - 90 * q^11 - 768 * q^13 + 54 * q^15 - 1926 * q^17 - 2248 * q^19 + 6354 * q^23 + 4906 * q^25 + 1458 * q^27 + 10572 * q^29 + 3312 * q^31 - 810 * q^33 + 2104 * q^37 - 6912 * q^39 - 1266 * q^41 - 5768 * q^43 + 486 * q^45 - 15612 * q^47 - 17334 * q^51 + 16512 * q^53 + 77696 * q^55 - 20232 * q^57 + 13140 * q^59 + 5796 * q^61 + 64524 * q^65 - 56116 * q^67 + 57186 * q^69 + 11022 * q^71 + 85384 * q^73 + 44154 * q^75 - 19620 * q^79 + 13122 * q^81 + 44424 * q^83 - 16916 * q^85 + 95148 * q^87 - 211218 * q^89 + 29808 * q^93 + 260568 * q^95 - 44864 * q^97 - 7290 * 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
 −36.8129 37.8129
0 9.00000 0 −71.6257 0 0 0 81.0000 0
1.2 0 9.00000 0 77.6257 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.k 2
7.b odd 2 1 84.6.a.c 2
7.c even 3 2 588.6.i.i 4
7.d odd 6 2 588.6.i.l 4
21.c even 2 1 252.6.a.h 2
28.d even 2 1 336.6.a.x 2
84.h odd 2 1 1008.6.a.bo 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.6.a.c 2 7.b odd 2 1
252.6.a.h 2 21.c even 2 1
336.6.a.x 2 28.d even 2 1
588.6.a.k 2 1.a even 1 1 trivial
588.6.i.i 4 7.c even 3 2
588.6.i.l 4 7.d odd 6 2
1008.6.a.bo 2 84.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 6T_{5} - 5560$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(588))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T - 9)^{2}$$
$5$ $$T^{2} - 6T - 5560$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 90T - 270856$$
$13$ $$T^{2} + 768T - 53028$$
$17$ $$T^{2} + 1926 T + 921800$$
$19$ $$T^{2} + 2248 T - 1944368$$
$23$ $$T^{2} - 6354 T + 10087760$$
$29$ $$T^{2} - 10572 T + 19031396$$
$31$ $$T^{2} - 3312 T - 26127360$$
$37$ $$T^{2} - 2104 T - 124195796$$
$41$ $$T^{2} + 1266 T - 5663952$$
$43$ $$T^{2} + 5768 T - 312456944$$
$47$ $$T^{2} + 15612 T - 85219200$$
$53$ $$T^{2} - 16512 T - 236774628$$
$59$ $$T^{2} - 13140 T + 29242400$$
$61$ $$T^{2} - 5796 T - 379738620$$
$67$ $$T^{2} + 56116 T - 401418272$$
$71$ $$T^{2} + \cdots - 4430537104$$
$73$ $$T^{2} + \cdots + 1653999820$$
$79$ $$T^{2} + 19620 T + 62354304$$
$83$ $$T^{2} - 44424 T - 76892656$$
$89$ $$T^{2} + \cdots + 10266670512$$
$97$ $$T^{2} + \cdots - 14004028100$$
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