# Properties

 Label 588.6.i.n 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: $$\Q(\sqrt{-3}, \sqrt{505})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} + 127x^{2} + 126x + 15876$$ x^4 - x^3 + 127*x^2 + 126*x + 15876 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{2}\cdot 3^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - 9 \beta_1 + 9) q^{3} + (\beta_{2} + 39 \beta_1) q^{5} - 81 \beta_1 q^{9}+O(q^{10})$$ q + (-9*b1 + 9) * q^3 + (b2 + 39*b1) * q^5 - 81*b1 * q^9 $$q + ( - 9 \beta_1 + 9) q^{3} + (\beta_{2} + 39 \beta_1) q^{5} - 81 \beta_1 q^{9} + ( - 5 \beta_{3} - 5 \beta_{2} + \cdots - 87) q^{11}+ \cdots + (405 \beta_{3} + 7047) q^{99}+O(q^{100})$$ q + (-9*b1 + 9) * q^3 + (b2 + 39*b1) * q^5 - 81*b1 * q^9 + (-5*b3 - 5*b2 + 87*b1 - 87) * q^11 + (-6*b3 - 104) * q^13 + (-9*b3 + 351) * q^15 + (-19*b3 - 19*b2 - 741*b1 + 741) * q^17 + (-36*b2 + 176*b1) * q^19 + (-21*b2 - 1677*b1) * q^23 + (78*b3 + 78*b2 + 2941*b1 - 2941) * q^25 - 729 * q^27 + (100*b3 + 138) * q^29 + (36*b3 + 36*b2 - 3260*b1 + 3260) * q^31 + (-45*b2 + 783*b1) * q^33 + (-30*b2 - 6932*b1) * q^37 + (-54*b3 - 54*b2 + 936*b1 - 936) * q^39 + (-11*b3 + 6465) * q^41 + (-120*b3 + 6356) * q^43 + (-81*b3 - 81*b2 - 3159*b1 + 3159) * q^45 + (-58*b2 - 14058*b1) * q^47 + (-171*b2 - 6669*b1) * q^51 + (114*b3 + 114*b2 - 23496*b1 + 23496) * q^53 + (-108*b3 + 19332) * q^55 + (324*b3 + 1584) * q^57 + (-90*b3 - 90*b2 + 32778*b1 - 32778) * q^59 + (-300*b2 - 6574*b1) * q^61 + (130*b2 + 23214*b1) * q^65 + (-102*b3 - 102*b2 - 37618*b1 + 37618) * q^67 + (189*b3 - 15093) * q^69 + (-187*b3 - 33021) * q^71 + (-126*b3 - 126*b2 - 30248*b1 + 30248) * q^73 + (702*b2 + 26469*b1) * q^75 + (738*b2 + 17458*b1) * q^79 + (6561*b1 - 6561) * q^81 + (80*b3 + 41244) * q^83 + (-1482*b3 + 115254) * q^85 + (900*b3 + 900*b2 - 1242*b1 + 1242) * q^87 + (1261*b2 + 21255*b1) * q^89 + (324*b2 - 29340*b1) * q^93 + (-1228*b3 - 1228*b2 - 156756*b1 + 156756) * q^95 + (-126*b3 - 106628) * q^97 + (405*b3 + 7047) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 18 q^{3} + 78 q^{5} - 162 q^{9}+O(q^{10})$$ 4 * q + 18 * q^3 + 78 * q^5 - 162 * q^9 $$4 q + 18 q^{3} + 78 q^{5} - 162 q^{9} - 174 q^{11} - 416 q^{13} + 1404 q^{15} + 1482 q^{17} + 352 q^{19} - 3354 q^{23} - 5882 q^{25} - 2916 q^{27} + 552 q^{29} + 6520 q^{31} + 1566 q^{33} - 13864 q^{37} - 1872 q^{39} + 25860 q^{41} + 25424 q^{43} + 6318 q^{45} - 28116 q^{47} - 13338 q^{51} + 46992 q^{53} + 77328 q^{55} + 6336 q^{57} - 65556 q^{59} - 13148 q^{61} + 46428 q^{65} + 75236 q^{67} - 60372 q^{69} - 132084 q^{71} + 60496 q^{73} + 52938 q^{75} + 34916 q^{79} - 13122 q^{81} + 164976 q^{83} + 461016 q^{85} + 2484 q^{87} + 42510 q^{89} - 58680 q^{93} + 313512 q^{95} - 426512 q^{97} + 28188 q^{99}+O(q^{100})$$ 4 * q + 18 * q^3 + 78 * q^5 - 162 * q^9 - 174 * q^11 - 416 * q^13 + 1404 * q^15 + 1482 * q^17 + 352 * q^19 - 3354 * q^23 - 5882 * q^25 - 2916 * q^27 + 552 * q^29 + 6520 * q^31 + 1566 * q^33 - 13864 * q^37 - 1872 * q^39 + 25860 * q^41 + 25424 * q^43 + 6318 * q^45 - 28116 * q^47 - 13338 * q^51 + 46992 * q^53 + 77328 * q^55 + 6336 * q^57 - 65556 * q^59 - 13148 * q^61 + 46428 * q^65 + 75236 * q^67 - 60372 * q^69 - 132084 * q^71 + 60496 * q^73 + 52938 * q^75 + 34916 * q^79 - 13122 * q^81 + 164976 * q^83 + 461016 * q^85 + 2484 * q^87 + 42510 * q^89 - 58680 * q^93 + 313512 * q^95 - 426512 * q^97 + 28188 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 127x^{2} + 126x + 15876$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{3} + 127\nu^{2} - 127\nu + 15876 ) / 16002$$ (-v^3 + 127*v^2 - 127*v + 15876) / 16002 $$\beta_{2}$$ $$=$$ $$( \nu^{3} - 127\nu^{2} + 32131\nu - 15876 ) / 5334$$ (v^3 - 127*v^2 + 32131*v - 15876) / 5334 $$\beta_{3}$$ $$=$$ $$( 6\nu^{3} + 1137 ) / 127$$ (6*v^3 + 1137) / 127
 $$\nu$$ $$=$$ $$( \beta_{2} + 3\beta_1 ) / 6$$ (b2 + 3*b1) / 6 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + \beta_{2} + 759\beta _1 - 759 ) / 6$$ (b3 + b2 + 759*b1 - 759) / 6 $$\nu^{3}$$ $$=$$ $$( 127\beta_{3} - 1137 ) / 6$$ (127*b3 - 1137) / 6

## 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
 −5.36805 − 9.29774i 5.86805 + 10.1638i −5.36805 + 9.29774i 5.86805 − 10.1638i
0 4.50000 7.79423i 0 −14.2083 24.6095i 0 0 0 −40.5000 70.1481i 0
361.2 0 4.50000 7.79423i 0 53.2083 + 92.1595i 0 0 0 −40.5000 70.1481i 0
373.1 0 4.50000 + 7.79423i 0 −14.2083 + 24.6095i 0 0 0 −40.5000 + 70.1481i 0
373.2 0 4.50000 + 7.79423i 0 53.2083 92.1595i 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.n 4
7.b odd 2 1 588.6.i.h 4
7.c even 3 1 588.6.a.g 2
7.c even 3 1 inner 588.6.i.n 4
7.d odd 6 1 84.6.a.d 2
7.d odd 6 1 588.6.i.h 4
21.g even 6 1 252.6.a.e 2
28.f even 6 1 336.6.a.u 2
84.j odd 6 1 1008.6.a.bf 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} - 78T_{5}^{3} + 9108T_{5}^{2} + 235872T_{5} + 9144576$$ acting on $$S_{6}^{\mathrm{new}}(588, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} - 9 T + 81)^{2}$$
$5$ $$T^{4} - 78 T^{3} + \cdots + 9144576$$
$7$ $$T^{4}$$
$11$ $$T^{4} + \cdots + 11247875136$$
$13$ $$(T^{2} + 208 T - 152804)^{2}$$
$17$ $$T^{4} + \cdots + 1191730288896$$
$19$ $$T^{4} + \cdots + 34331912110336$$
$23$ $$T^{4} + \cdots + 652838144256$$
$29$ $$(T^{2} - 276 T - 45430956)^{2}$$
$31$ $$T^{4} + \cdots + 22441821798400$$
$37$ $$T^{4} + \cdots + 19\!\cdots\!76$$
$41$ $$(T^{2} - 12930 T + 41246280)^{2}$$
$43$ $$(T^{2} - 12712 T - 25049264)^{2}$$
$47$ $$T^{4} + \cdots + 33\!\cdots\!56$$
$53$ $$T^{4} + \cdots + 24\!\cdots\!16$$
$59$ $$T^{4} + \cdots + 10\!\cdots\!56$$
$61$ $$T^{4} + \cdots + 13\!\cdots\!76$$
$67$ $$T^{4} + \cdots + 18\!\cdots\!36$$
$71$ $$(T^{2} + 66042 T + 931452336)^{2}$$
$73$ $$T^{4} + \cdots + 71\!\cdots\!56$$
$79$ $$T^{4} + \cdots + 47\!\cdots\!56$$
$83$ $$(T^{2} - 82488 T + 1671979536)^{2}$$
$89$ $$T^{4} + \cdots + 45\!\cdots\!00$$
$97$ $$(T^{2} + 213256 T + 11297373964)^{2}$$