# Properties

 Label 588.6.i.l Level $588$ Weight $6$ Character orbit 588.i Analytic conductor $94.306$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{5569})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} + 1393x^{2} + 1392x + 1937664$$ x^4 - x^3 + 1393*x^2 + 1392*x + 1937664 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - 9 \beta_1 + 9) q^{3} + (\beta_{2} + 3 \beta_1) q^{5} - 81 \beta_1 q^{9}+O(q^{10})$$ q + (-9*b1 + 9) * q^3 + (b2 + 3*b1) * q^5 - 81*b1 * q^9 $$q + ( - 9 \beta_1 + 9) q^{3} + (\beta_{2} + 3 \beta_1) q^{5} - 81 \beta_1 q^{9} + (7 \beta_{3} + 7 \beta_{2} - 45 \beta_1 + 45) q^{11} + (6 \beta_{3} + 384) q^{13} + ( - 9 \beta_{3} + 27) q^{15} + (\beta_{3} + \beta_{2} + 963 \beta_1 - 963) q^{17} + (24 \beta_{2} - 1124 \beta_1) q^{19} + ( - \beta_{2} - 3177 \beta_1) q^{23} + (6 \beta_{3} + 6 \beta_{2} + \cdots - 2453) q^{25}+ \cdots + ( - 567 \beta_{3} - 3645) q^{99}+O(q^{100})$$ q + (-9*b1 + 9) * q^3 + (b2 + 3*b1) * q^5 - 81*b1 * q^9 + (7*b3 + 7*b2 - 45*b1 + 45) * q^11 + (6*b3 + 384) * q^13 + (-9*b3 + 27) * q^15 + (b3 + b2 + 963*b1 - 963) * q^17 + (24*b2 - 1124*b1) * q^19 + (-b2 - 3177*b1) * q^23 + (6*b3 + 6*b2 + 2453*b1 - 2453) * q^25 - 729 * q^27 + (40*b3 + 5286) * q^29 + (72*b3 + 72*b2 - 1656*b1 + 1656) * q^31 + (63*b2 - 405*b1) * q^33 + (-150*b2 - 1052*b1) * q^37 + (54*b3 + 54*b2 - 3456*b1 + 3456) * q^39 + (33*b3 + 633) * q^41 + (240*b3 - 2884) * q^43 + (-81*b3 - 81*b2 - 243*b1 + 243) * q^45 + (162*b2 - 7806*b1) * q^47 + (9*b2 + 8667*b1) * q^51 + (234*b3 + 234*b2 + 8256*b1 - 8256) * q^53 + (-24*b3 - 38848) * q^55 + (-216*b3 - 10116) * q^57 + (50*b3 + 50*b2 - 6570*b1 + 6570) * q^59 + (-264*b2 + 2898*b1) * q^61 + (366*b2 - 32262*b1) * q^65 + (-462*b3 - 462*b2 - 28058*b1 + 28058) * q^67 + (9*b3 - 28593) * q^69 + (-895*b3 + 5511) * q^71 + (-174*b3 - 174*b2 - 42692*b1 + 42692) * q^73 + (54*b2 + 22077*b1) * q^75 + (-78*b2 + 9810*b1) * q^79 + (6561*b1 - 6561) * q^81 + (320*b3 - 22212) * q^83 + (966*b3 - 8458) * q^85 + (360*b3 + 360*b2 - 47574*b1 + 47574) * q^87 + (-399*b2 - 105609*b1) * q^89 + (648*b2 - 14904*b1) * q^93 + (-1052*b3 - 1052*b2 + 130284*b1 - 130284) * q^95 + (-1614*b3 + 22432) * q^97 + (-567*b3 - 3645) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 18 q^{3} + 6 q^{5} - 162 q^{9}+O(q^{10})$$ 4 * q + 18 * q^3 + 6 * q^5 - 162 * q^9 $$4 q + 18 q^{3} + 6 q^{5} - 162 q^{9} + 90 q^{11} + 1536 q^{13} + 108 q^{15} - 1926 q^{17} - 2248 q^{19} - 6354 q^{23} - 4906 q^{25} - 2916 q^{27} + 21144 q^{29} + 3312 q^{31} - 810 q^{33} - 2104 q^{37} + 6912 q^{39} + 2532 q^{41} - 11536 q^{43} + 486 q^{45} - 15612 q^{47} + 17334 q^{51} - 16512 q^{53} - 155392 q^{55} - 40464 q^{57} + 13140 q^{59} + 5796 q^{61} - 64524 q^{65} + 56116 q^{67} - 114372 q^{69} + 22044 q^{71} + 85384 q^{73} + 44154 q^{75} + 19620 q^{79} - 13122 q^{81} - 88848 q^{83} - 33832 q^{85} + 95148 q^{87} - 211218 q^{89} - 29808 q^{93} - 260568 q^{95} + 89728 q^{97} - 14580 q^{99}+O(q^{100})$$ 4 * q + 18 * q^3 + 6 * q^5 - 162 * q^9 + 90 * q^11 + 1536 * q^13 + 108 * q^15 - 1926 * q^17 - 2248 * q^19 - 6354 * q^23 - 4906 * q^25 - 2916 * q^27 + 21144 * q^29 + 3312 * q^31 - 810 * q^33 - 2104 * q^37 + 6912 * q^39 + 2532 * q^41 - 11536 * q^43 + 486 * q^45 - 15612 * q^47 + 17334 * q^51 - 16512 * q^53 - 155392 * q^55 - 40464 * q^57 + 13140 * q^59 + 5796 * q^61 - 64524 * q^65 + 56116 * q^67 - 114372 * q^69 + 22044 * q^71 + 85384 * q^73 + 44154 * q^75 + 19620 * q^79 - 13122 * q^81 - 88848 * q^83 - 33832 * q^85 + 95148 * q^87 - 211218 * q^89 - 29808 * q^93 - 260568 * q^95 + 89728 * q^97 - 14580 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 1393x^{2} + 1392x + 1937664$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{3} + 1393\nu^{2} - 1393\nu + 1937664 ) / 1939056$$ (-v^3 + 1393*v^2 - 1393*v + 1937664) / 1939056 $$\beta_{2}$$ $$=$$ $$( \nu^{3} - 1393\nu^{2} + 3879505\nu - 1937664 ) / 1939056$$ (v^3 - 1393*v^2 + 3879505*v - 1937664) / 1939056 $$\beta_{3}$$ $$=$$ $$( 2\nu^{3} + 4177 ) / 1393$$ (2*v^3 + 4177) / 1393
 $$\nu$$ $$=$$ $$( \beta_{2} + \beta_1 ) / 2$$ (b2 + b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + \beta_{2} + 2785\beta _1 - 2785 ) / 2$$ (b3 + b2 + 2785*b1 - 2785) / 2 $$\nu^{3}$$ $$=$$ $$( 1393\beta_{3} - 4177 ) / 2$$ (1393*b3 - 4177) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/588\mathbb{Z}\right)^\times$$.

 $$n$$ $$197$$ $$295$$ $$493$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\beta_{1}$$

## 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
 −18.4064 − 31.8809i 18.9064 + 32.7469i −18.4064 + 31.8809i 18.9064 − 32.7469i
0 4.50000 7.79423i 0 −35.8129 62.0297i 0 0 0 −40.5000 70.1481i 0
361.2 0 4.50000 7.79423i 0 38.8129 + 67.2259i 0 0 0 −40.5000 70.1481i 0
373.1 0 4.50000 + 7.79423i 0 −35.8129 + 62.0297i 0 0 0 −40.5000 + 70.1481i 0
373.2 0 4.50000 + 7.79423i 0 38.8129 67.2259i 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.l 4
7.b odd 2 1 588.6.i.i 4
7.c even 3 1 84.6.a.c 2
7.c even 3 1 inner 588.6.i.l 4
7.d odd 6 1 588.6.a.k 2
7.d odd 6 1 588.6.i.i 4
21.h odd 6 1 252.6.a.h 2
28.g odd 6 1 336.6.a.x 2
84.n even 6 1 1008.6.a.bo 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} - 6T_{5}^{3} + 5596T_{5}^{2} + 33360T_{5} + 30913600$$ acting on $$S_{6}^{\mathrm{new}}(588, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} - 9 T + 81)^{2}$$
$5$ $$T^{4} - 6 T^{3} + \cdots + 30913600$$
$7$ $$T^{4}$$
$11$ $$T^{4} + \cdots + 73362972736$$
$13$ $$(T^{2} - 768 T - 53028)^{2}$$
$17$ $$T^{4} + \cdots + 849715240000$$
$19$ $$T^{4} + \cdots + 3780566919424$$
$23$ $$T^{4} + \cdots + 101762901817600$$
$29$ $$(T^{2} - 10572 T + 19031396)^{2}$$
$31$ $$T^{4} + \cdots + 682638940569600$$
$37$ $$T^{4} + \cdots + 15\!\cdots\!16$$
$41$ $$(T^{2} - 1266 T - 5663952)^{2}$$
$43$ $$(T^{2} + 5768 T - 312456944)^{2}$$
$47$ $$T^{4} + \cdots + 72\!\cdots\!00$$
$53$ $$T^{4} + \cdots + 56\!\cdots\!84$$
$59$ $$T^{4} + \cdots + 855117957760000$$
$61$ $$T^{4} + \cdots + 14\!\cdots\!00$$
$67$ $$T^{4} + \cdots + 16\!\cdots\!84$$
$71$ $$(T^{2} - 11022 T - 4430537104)^{2}$$
$73$ $$T^{4} + \cdots + 27\!\cdots\!00$$
$79$ $$T^{4} + \cdots + 38\!\cdots\!16$$
$83$ $$(T^{2} + 44424 T - 76892656)^{2}$$
$89$ $$T^{4} + \cdots + 10\!\cdots\!44$$
$97$ $$(T^{2} - 44864 T - 14004028100)^{2}$$