# Properties

 Label 432.4.i.d Level $432$ Weight $4$ Character orbit 432.i Analytic conductor $25.489$ Analytic rank $0$ Dimension $6$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [432,4,Mod(145,432)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(432, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("432.145");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$432 = 2^{4} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 432.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: $$25.4888251225$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.6831243.2 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} + 13x^{4} + 49x^{2} + 48$$ x^6 + 13*x^4 + 49*x^2 + 48 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{2}\cdot 3^{5}$$ Twist minimal: no (minimal twist has level 36) 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{5} + 2 \beta_1 - 2) q^{5} + (\beta_{4} + \beta_{2} + 2 \beta_1) q^{7}+O(q^{10})$$ q + (b5 + 2*b1 - 2) * q^5 + (b4 + b2 + 2*b1) * q^7 $$q + (\beta_{5} + 2 \beta_1 - 2) q^{5} + (\beta_{4} + \beta_{2} + 2 \beta_1) q^{7} + ( - 2 \beta_{4} + 3 \beta_{2} + 17 \beta_1) q^{11} + (5 \beta_{5} - 4 \beta_{3} - 4 \beta_{2} + \cdots + 4) q^{13}+ \cdots + (70 \beta_{4} - 56 \beta_{2} + 31 \beta_1) q^{97}+O(q^{100})$$ q + (b5 + 2*b1 - 2) * q^5 + (b4 + b2 + 2*b1) * q^7 + (-2*b4 + 3*b2 + 17*b1) * q^11 + (5*b5 - 4*b3 - 4*b2 - 4*b1 + 4) * q^13 + (b5 - b4 + 6*b3 + 37) * q^17 + (-7*b5 + 7*b4 + 2*b3 - 5) * q^19 + (-5*b5 + 3*b3 + 3*b2 - 70*b1 + 70) * q^23 + (3*b4 - 6*b2 - b1) * q^25 + (5*b4 - 152*b1) * q^29 + (15*b5 - 3*b3 - 3*b2 + 16*b1 - 16) * q^31 + (-b5 + b4 + 15*b3 - 184) * q^35 + (-10*b5 + 10*b4 + 8*b3 - 16) * q^37 + (-14*b5 + 12*b3 + 12*b2 + 299*b1 - 299) * q^41 + (-27*b2 - 43*b1) * q^43 + (39*b4 - 21*b2 + 174*b1) * q^47 + (13*b5 + 22*b3 + 22*b2 + 75*b1 - 75) * q^49 + (-22*b5 + 22*b4 + 368) * q^53 + (33*b5 - 33*b4 + 15*b3 + 36) * q^55 + (34*b5 + 15*b3 + 15*b2 - 151*b1 + 151) * q^59 + (-3*b4 - 12*b2 - 134*b1) * q^61 + (37*b4 + 6*b2 - 370*b1) * q^65 + (24*b5 - 3*b3 - 3*b2 - 71*b1 + 71) * q^67 + (-52*b5 + 52*b4 + 12*b3 + 20) * q^71 + (21*b5 - 21*b4 + 30*b3 + 125) * q^73 + (65*b5 + 12*b3 + 12*b2 + 376*b1 - 376) * q^77 + (23*b4 + 5*b2 - 184*b1) * q^79 + (-15*b4 + 33*b2 - 204*b1) * q^83 + (30*b5 - 60*b3 - 60*b2 - 396*b1 + 396) * q^85 + (-44*b5 + 44*b4 + 60*b3 + 154) * q^89 + (-75*b5 + 75*b4 + 33*b3 + 44) * q^91 + (-44*b5 + 24*b3 + 24*b2 + 728*b1 - 728) * q^95 + (70*b4 - 56*b2 + 31*b1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 6 q^{5} + 6 q^{7}+O(q^{10})$$ 6 * q - 6 * q^5 + 6 * q^7 $$6 q - 6 q^{5} + 6 q^{7} + 51 q^{11} + 12 q^{13} + 222 q^{17} - 30 q^{19} + 210 q^{23} - 3 q^{25} - 456 q^{29} - 48 q^{31} - 1104 q^{35} - 96 q^{37} - 897 q^{41} - 129 q^{43} + 522 q^{47} - 225 q^{49} + 2208 q^{53} + 216 q^{55} + 453 q^{59} - 402 q^{61} - 1110 q^{65} + 213 q^{67} + 120 q^{71} + 750 q^{73} - 1128 q^{77} - 552 q^{79} - 612 q^{83} + 1188 q^{85} + 924 q^{89} + 264 q^{91} - 2184 q^{95} + 93 q^{97}+O(q^{100})$$ 6 * q - 6 * q^5 + 6 * q^7 + 51 * q^11 + 12 * q^13 + 222 * q^17 - 30 * q^19 + 210 * q^23 - 3 * q^25 - 456 * q^29 - 48 * q^31 - 1104 * q^35 - 96 * q^37 - 897 * q^41 - 129 * q^43 + 522 * q^47 - 225 * q^49 + 2208 * q^53 + 216 * q^55 + 453 * q^59 - 402 * q^61 - 1110 * q^65 + 213 * q^67 + 120 * q^71 + 750 * q^73 - 1128 * q^77 - 552 * q^79 - 612 * q^83 + 1188 * q^85 + 924 * q^89 + 264 * q^91 - 2184 * q^95 + 93 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 13x^{4} + 49x^{2} + 48$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{5} + 9\nu^{3} + 17\nu + 4 ) / 8$$ (v^5 + 9*v^3 + 17*v + 4) / 8 $$\beta_{2}$$ $$=$$ $$( \nu^{5} + 21\nu^{3} + 12\nu^{2} + 77\nu + 52 ) / 4$$ (v^5 + 21*v^3 + 12*v^2 + 77*v + 52) / 4 $$\beta_{3}$$ $$=$$ $$-6\nu^{2} - 26$$ -6*v^2 - 26 $$\beta_{4}$$ $$=$$ $$( -\nu^{5} + 6\nu^{4} - 3\nu^{3} + 54\nu^{2} + 31\nu + 92 ) / 4$$ (-v^5 + 6*v^4 - 3*v^3 + 54*v^2 + 31*v + 92) / 4 $$\beta_{5}$$ $$=$$ $$( -\nu^{5} - 6\nu^{4} - 3\nu^{3} - 54\nu^{2} + 31\nu - 92 ) / 4$$ (-v^5 - 6*v^4 - 3*v^3 - 54*v^2 + 31*v - 92) / 4
 $$\nu$$ $$=$$ $$( 2\beta_{5} + 2\beta_{4} - \beta_{3} - 2\beta_{2} + 12\beta _1 - 6 ) / 18$$ (2*b5 + 2*b4 - b3 - 2*b2 + 12*b1 - 6) / 18 $$\nu^{2}$$ $$=$$ $$( -\beta_{3} - 26 ) / 6$$ (-b3 - 26) / 6 $$\nu^{3}$$ $$=$$ $$( -5\beta_{5} - 5\beta_{4} + 4\beta_{3} + 8\beta_{2} - 36\beta _1 + 18 ) / 9$$ (-5*b5 - 5*b4 + 4*b3 + 8*b2 - 36*b1 + 18) / 9 $$\nu^{4}$$ $$=$$ $$( -2\beta_{5} + 2\beta_{4} + 9\beta_{3} + 142 ) / 6$$ (-2*b5 + 2*b4 + 9*b3 + 142) / 6 $$\nu^{5}$$ $$=$$ $$( 56\beta_{5} + 56\beta_{4} - 55\beta_{3} - 110\beta_{2} + 588\beta _1 - 294 ) / 18$$ (56*b5 + 56*b4 - 55*b3 - 110*b2 + 588*b1 - 294) / 18

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/432\mathbb{Z}\right)^\times$$.

 $$n$$ $$271$$ $$325$$ $$353$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1 + \beta_{1}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
145.1
 1.23396i 2.63162i − 2.13353i − 1.23396i − 2.63162i 2.13353i
0 0 0 −6.92194 + 11.9892i 0 15.3540 + 26.5939i 0 0 0
145.2 0 0 0 −2.44901 + 4.24182i 0 −5.32725 9.22708i 0 0 0
145.3 0 0 0 6.37096 11.0348i 0 −7.02674 12.1707i 0 0 0
289.1 0 0 0 −6.92194 11.9892i 0 15.3540 26.5939i 0 0 0
289.2 0 0 0 −2.44901 4.24182i 0 −5.32725 + 9.22708i 0 0 0
289.3 0 0 0 6.37096 + 11.0348i 0 −7.02674 + 12.1707i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 145.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 432.4.i.d 6
3.b odd 2 1 144.4.i.d 6
4.b odd 2 1 108.4.e.a 6
9.c even 3 1 inner 432.4.i.d 6
9.c even 3 1 1296.4.a.w 3
9.d odd 6 1 144.4.i.d 6
9.d odd 6 1 1296.4.a.v 3
12.b even 2 1 36.4.e.a 6
36.f odd 6 1 108.4.e.a 6
36.f odd 6 1 324.4.a.d 3
36.h even 6 1 36.4.e.a 6
36.h even 6 1 324.4.a.c 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.4.e.a 6 12.b even 2 1
36.4.e.a 6 36.h even 6 1
108.4.e.a 6 4.b odd 2 1
108.4.e.a 6 36.f odd 6 1
144.4.i.d 6 3.b odd 2 1
144.4.i.d 6 9.d odd 6 1
324.4.a.c 3 36.h even 6 1
324.4.a.d 3 36.f odd 6 1
432.4.i.d 6 1.a even 1 1 trivial
432.4.i.d 6 9.c even 3 1 inner
1296.4.a.v 3 9.d odd 6 1
1296.4.a.w 3 9.c even 3 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{6} + 6T_{5}^{5} + 207T_{5}^{4} + 702T_{5}^{3} + 34425T_{5}^{2} + 147744T_{5} + 746496$$ acting on $$S_{4}^{\mathrm{new}}(432, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6}$$
$5$ $$T^{6} + 6 T^{5} + \cdots + 746496$$
$7$ $$T^{6} - 6 T^{5} + \cdots + 21141604$$
$11$ $$T^{6} + \cdots + 4386545361$$
$13$ $$T^{6} + \cdots + 4155865156$$
$17$ $$(T^{3} - 111 T^{2} + \cdots + 577476)^{2}$$
$19$ $$(T^{3} + 15 T^{2} + \cdots + 216368)^{2}$$
$23$ $$T^{6} - 210 T^{5} + \cdots + 5391684$$
$29$ $$T^{6} + \cdots + 8292430196964$$
$31$ $$T^{6} + \cdots + 9331729724944$$
$37$ $$(T^{3} + 48 T^{2} + \cdots - 682352)^{2}$$
$41$ $$T^{6} + \cdots + 139158426750849$$
$43$ $$T^{6} + \cdots + 2031049672201$$
$47$ $$T^{6} + \cdots + 41\!\cdots\!24$$
$53$ $$(T^{3} - 1104 T^{2} + \cdots - 11853648)^{2}$$
$59$ $$T^{6} + \cdots + 93\!\cdots\!69$$
$61$ $$T^{6} + \cdots + 1463461189696$$
$67$ $$T^{6} + \cdots + 9579414973969$$
$71$ $$(T^{3} - 60 T^{2} + \cdots + 113211648)^{2}$$
$73$ $$(T^{3} - 375 T^{2} + \cdots + 158369284)^{2}$$
$79$ $$T^{6} + \cdots + 318578661907984$$
$83$ $$T^{6} + \cdots + 12101965948944$$
$89$ $$(T^{3} - 462 T^{2} + \cdots - 170122248)^{2}$$
$97$ $$T^{6} + \cdots + 74\!\cdots\!29$$