# Properties

 Label 432.2.u.a Level $432$ Weight $2$ Character orbit 432.u Analytic conductor $3.450$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [432,2,Mod(49,432)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(432, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([0, 0, 14]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("432.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$432 = 2^{4} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 432.u (of order $$9$$, degree $$6$$, 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: $$3.44953736732$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\Q(\zeta_{18})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - x^{3} + 1$$ x^6 - x^3 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 54) Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

## $q$-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 primitive root of unity $$\zeta_{18}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\zeta_{18}^{5} - 2 \zeta_{18}^{2}) q^{3} + (\zeta_{18}^{5} + \zeta_{18}^{4} - \zeta_{18}^{3} - \zeta_{18}^{2}) q^{5} + ( - \zeta_{18}^{5} + 2 \zeta_{18}^{4} - \zeta_{18}^{3}) q^{7} + 3 \zeta_{18} q^{9}+O(q^{10})$$ q + (z^5 - 2*z^2) * q^3 + (z^5 + z^4 - z^3 - z^2) * q^5 + (-z^5 + 2*z^4 - z^3) * q^7 + 3*z * q^9 $$q + (\zeta_{18}^{5} - 2 \zeta_{18}^{2}) q^{3} + (\zeta_{18}^{5} + \zeta_{18}^{4} - \zeta_{18}^{3} - \zeta_{18}^{2}) q^{5} + ( - \zeta_{18}^{5} + 2 \zeta_{18}^{4} - \zeta_{18}^{3}) q^{7} + 3 \zeta_{18} q^{9} + ( - 3 \zeta_{18}^{5} + 3 \zeta_{18}^{3} + 2 \zeta_{18}^{2} + \zeta_{18} - 1) q^{11} + ( - 2 \zeta_{18}^{4} - 3 \zeta_{18}^{2} - 2) q^{13} + (\zeta_{18}^{5} - \zeta_{18}^{4} - 2 \zeta_{18}^{3} + \zeta_{18}^{2} + 2 \zeta_{18} + 1) q^{15} + ( - 4 \zeta_{18}^{5} - 4 \zeta_{18}^{4} + 2 \zeta_{18}^{3} + \zeta_{18}^{2} + 3 \zeta_{18} - 2) q^{17} + (3 \zeta_{18}^{5} - 4 \zeta_{18}^{4} - 3 \zeta_{18}^{3} - 4 \zeta_{18}^{2} + 3 \zeta_{18}) q^{19} + (\zeta_{18}^{5} + 2 \zeta_{18}^{4} - 4 \zeta_{18}^{3} + \zeta_{18}^{2} - \zeta_{18} + 2) q^{21} + ( - \zeta_{18}^{5} - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{3} + 2) q^{23} + (\zeta_{18}^{5} + 2 \zeta_{18}^{4} - \zeta_{18}^{3} + \zeta_{18}^{2} - 2 \zeta_{18} - 1) q^{25} + ( - 3 \zeta_{18}^{3} - 3) q^{27} + ( - 3 \zeta_{18}^{5} - \zeta_{18}^{4} - \zeta_{18}^{3} + \zeta_{18}^{2} + \zeta_{18} + 3) q^{29} + ( - 4 \zeta_{18}^{5} - 3 \zeta_{18} + 3) q^{31} + ( - 4 \zeta_{18}^{5} + 4 \zeta_{18}^{4} - \zeta_{18}^{3} - \zeta_{18}^{2} - 5 \zeta_{18} - 1) q^{33} + (3 \zeta_{18}^{5} - 2 \zeta_{18}^{4} - \zeta_{18}^{3} - 2 \zeta_{18}^{2} + 3 \zeta_{18}) q^{35} + (3 \zeta_{18}^{5} + 3 \zeta_{18}^{4} - 5 \zeta_{18}^{3} - \zeta_{18}^{2} - 2 \zeta_{18} + 5) q^{37} + ( - 2 \zeta_{18}^{5} + 3 \zeta_{18}^{4} + 4 \zeta_{18}^{3} + 4 \zeta_{18}^{2} + 3 \zeta_{18} - 2) q^{39} + ( - 3 \zeta_{18}^{4} + 5 \zeta_{18}^{3} + \zeta_{18}^{2} + 5 \zeta_{18} - 3) q^{41} + (2 \zeta_{18}^{5} - 2 \zeta_{18}^{3} + 2 \zeta_{18}^{2} - 4 \zeta_{18} + 4) q^{43} + (3 \zeta_{18}^{5} - 3 \zeta_{18}^{4} - 3) q^{45} + ( - 2 \zeta_{18}^{5} + 3 \zeta_{18}^{4} - \zeta_{18}^{3} + \zeta_{18}^{2} - 1) q^{47} + ( - \zeta_{18}^{5} - 4 \zeta_{18}^{4} + \zeta_{18}^{3} + \zeta_{18}^{2} + 3 \zeta_{18} + 3) q^{49} + ( - 4 \zeta_{18}^{5} + 7 \zeta_{18}^{4} + 5 \zeta_{18}^{3} + 2 \zeta_{18}^{2} - 5 \zeta_{18} - 7) q^{51} + (2 \zeta_{18}^{5} + \zeta_{18}^{4} - 3 \zeta_{18}^{2} - 3 \zeta_{18} - 2) q^{53} + (\zeta_{18}^{5} + 4 \zeta_{18}^{4} - 5 \zeta_{18}^{2} - 5 \zeta_{18} + 3) q^{55} + (3 \zeta_{18}^{5} - 2 \zeta_{18}^{4} + 5 \zeta_{18}^{3} + 3 \zeta_{18}^{2} + 7 \zeta_{18} - 7) q^{57} + (\zeta_{18}^{5} - 3 \zeta_{18}^{4} + 4 \zeta_{18}^{3} - \zeta_{18}^{2} - \zeta_{18} - 1) q^{59} + (\zeta_{18}^{5} + 2 \zeta_{18}^{4} - 4 \zeta_{18}^{3} - 5 \zeta_{18}^{2} + 5) q^{61} + (6 \zeta_{18}^{5} - 3 \zeta_{18}^{4} - 3 \zeta_{18}^{3} + 3) q^{63} + ( - \zeta_{18}^{5} + \zeta_{18}^{3} + 4 \zeta_{18}^{2} + \zeta_{18} + 3) q^{65} + (3 \zeta_{18}^{3} + \zeta_{18}^{2} + 3 \zeta_{18}) q^{67} + (4 \zeta_{18}^{5} + 2 \zeta_{18}^{4} + 4 \zeta_{18}^{3} - 2 \zeta_{18}^{2} - \zeta_{18} - 2) q^{69} + ( - 5 \zeta_{18}^{5} - 5 \zeta_{18}^{4} + 4 \zeta_{18}^{3} - \zeta_{18}^{2} + 6 \zeta_{18} - 4) q^{71} + ( - 5 \zeta_{18}^{5} - 2 \zeta_{18}^{4} + \zeta_{18}^{3} - 2 \zeta_{18}^{2} - 5 \zeta_{18}) q^{73} + ( - 3 \zeta_{18}^{4} - 2 \zeta_{18}^{3} + 3 \zeta_{18}^{2} + 4) q^{75} + (\zeta_{18}^{5} + \zeta_{18}^{4} + \zeta_{18}^{3} - 7 \zeta_{18} + 6) q^{77} + (8 \zeta_{18}^{5} + 5 \zeta_{18}^{3} - 5 \zeta_{18}^{2} - 8) q^{79} + 9 \zeta_{18}^{2} q^{81} + ( - 2 \zeta_{18}^{5} + 2 \zeta_{18}^{3} - 2 \zeta_{18}^{2} + 2) q^{83} + (5 \zeta_{18}^{4} + 5 \zeta_{18}^{3} - 7 \zeta_{18} + 2) q^{85} + (4 \zeta_{18}^{5} + 5 \zeta_{18}^{4} + \zeta_{18}^{3} - 5 \zeta_{18}^{2} - 4 \zeta_{18} - 2) q^{87} + (\zeta_{18}^{5} + 3 \zeta_{18}^{4} - 5 \zeta_{18}^{3} + 3 \zeta_{18}^{2} + \zeta_{18}) q^{89} + (\zeta_{18}^{5} + \zeta_{18}^{4} - 4 \zeta_{18}^{3} + 4 \zeta_{18}^{2} - 5 \zeta_{18} + 4) q^{91} + (3 \zeta_{18}^{5} + 8 \zeta_{18}^{4} + 3 \zeta_{18}^{3} - 6 \zeta_{18}^{2} - 4 \zeta_{18} + 3) q^{93} + ( - 5 \zeta_{18}^{4} + 3 \zeta_{18}^{3} + 10 \zeta_{18}^{2} + 3 \zeta_{18} - 5) q^{95} + ( - 2 \zeta_{18}^{5} + 2 \zeta_{18}^{3} - \zeta_{18}^{2} + 6 \zeta_{18} - 3) q^{97} + (9 \zeta_{18}^{4} - 3 \zeta_{18}^{3} + 3 \zeta_{18}^{2} - 3 \zeta_{18} + 9) q^{99}+O(q^{100})$$ q + (z^5 - 2*z^2) * q^3 + (z^5 + z^4 - z^3 - z^2) * q^5 + (-z^5 + 2*z^4 - z^3) * q^7 + 3*z * q^9 + (-3*z^5 + 3*z^3 + 2*z^2 + z - 1) * q^11 + (-2*z^4 - 3*z^2 - 2) * q^13 + (z^5 - z^4 - 2*z^3 + z^2 + 2*z + 1) * q^15 + (-4*z^5 - 4*z^4 + 2*z^3 + z^2 + 3*z - 2) * q^17 + (3*z^5 - 4*z^4 - 3*z^3 - 4*z^2 + 3*z) * q^19 + (z^5 + 2*z^4 - 4*z^3 + z^2 - z + 2) * q^21 + (-z^5 - 2*z^4 - 2*z^3 + 2) * q^23 + (z^5 + 2*z^4 - z^3 + z^2 - 2*z - 1) * q^25 + (-3*z^3 - 3) * q^27 + (-3*z^5 - z^4 - z^3 + z^2 + z + 3) * q^29 + (-4*z^5 - 3*z + 3) * q^31 + (-4*z^5 + 4*z^4 - z^3 - z^2 - 5*z - 1) * q^33 + (3*z^5 - 2*z^4 - z^3 - 2*z^2 + 3*z) * q^35 + (3*z^5 + 3*z^4 - 5*z^3 - z^2 - 2*z + 5) * q^37 + (-2*z^5 + 3*z^4 + 4*z^3 + 4*z^2 + 3*z - 2) * q^39 + (-3*z^4 + 5*z^3 + z^2 + 5*z - 3) * q^41 + (2*z^5 - 2*z^3 + 2*z^2 - 4*z + 4) * q^43 + (3*z^5 - 3*z^4 - 3) * q^45 + (-2*z^5 + 3*z^4 - z^3 + z^2 - 1) * q^47 + (-z^5 - 4*z^4 + z^3 + z^2 + 3*z + 3) * q^49 + (-4*z^5 + 7*z^4 + 5*z^3 + 2*z^2 - 5*z - 7) * q^51 + (2*z^5 + z^4 - 3*z^2 - 3*z - 2) * q^53 + (z^5 + 4*z^4 - 5*z^2 - 5*z + 3) * q^55 + (3*z^5 - 2*z^4 + 5*z^3 + 3*z^2 + 7*z - 7) * q^57 + (z^5 - 3*z^4 + 4*z^3 - z^2 - z - 1) * q^59 + (z^5 + 2*z^4 - 4*z^3 - 5*z^2 + 5) * q^61 + (6*z^5 - 3*z^4 - 3*z^3 + 3) * q^63 + (-z^5 + z^3 + 4*z^2 + z + 3) * q^65 + (3*z^3 + z^2 + 3*z) * q^67 + (4*z^5 + 2*z^4 + 4*z^3 - 2*z^2 - z - 2) * q^69 + (-5*z^5 - 5*z^4 + 4*z^3 - z^2 + 6*z - 4) * q^71 + (-5*z^5 - 2*z^4 + z^3 - 2*z^2 - 5*z) * q^73 + (-3*z^4 - 2*z^3 + 3*z^2 + 4) * q^75 + (z^5 + z^4 + z^3 - 7*z + 6) * q^77 + (8*z^5 + 5*z^3 - 5*z^2 - 8) * q^79 + 9*z^2 * q^81 + (-2*z^5 + 2*z^3 - 2*z^2 + 2) * q^83 + (5*z^4 + 5*z^3 - 7*z + 2) * q^85 + (4*z^5 + 5*z^4 + z^3 - 5*z^2 - 4*z - 2) * q^87 + (z^5 + 3*z^4 - 5*z^3 + 3*z^2 + z) * q^89 + (z^5 + z^4 - 4*z^3 + 4*z^2 - 5*z + 4) * q^91 + (3*z^5 + 8*z^4 + 3*z^3 - 6*z^2 - 4*z + 3) * q^93 + (-5*z^4 + 3*z^3 + 10*z^2 + 3*z - 5) * q^95 + (-2*z^5 + 2*z^3 - z^2 + 6*z - 3) * q^97 + (9*z^4 - 3*z^3 + 3*z^2 - 3*z + 9) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 3 q^{5} - 3 q^{7}+O(q^{10})$$ 6 * q - 3 * q^5 - 3 * q^7 $$6 q - 3 q^{5} - 3 q^{7} + 3 q^{11} - 12 q^{13} - 6 q^{17} - 9 q^{19} + 6 q^{23} - 9 q^{25} - 27 q^{27} + 15 q^{29} + 18 q^{31} - 9 q^{33} - 3 q^{35} + 15 q^{37} - 3 q^{41} + 18 q^{43} - 18 q^{45} - 9 q^{47} + 21 q^{49} - 27 q^{51} - 12 q^{53} + 18 q^{55} - 27 q^{57} + 6 q^{59} + 18 q^{61} + 9 q^{63} + 21 q^{65} + 9 q^{67} - 12 q^{71} + 3 q^{73} + 18 q^{75} + 39 q^{77} - 33 q^{79} + 18 q^{83} + 27 q^{85} - 9 q^{87} - 15 q^{89} + 12 q^{91} + 27 q^{93} - 21 q^{95} - 12 q^{97} + 45 q^{99}+O(q^{100})$$ 6 * q - 3 * q^5 - 3 * q^7 + 3 * q^11 - 12 * q^13 - 6 * q^17 - 9 * q^19 + 6 * q^23 - 9 * q^25 - 27 * q^27 + 15 * q^29 + 18 * q^31 - 9 * q^33 - 3 * q^35 + 15 * q^37 - 3 * q^41 + 18 * q^43 - 18 * q^45 - 9 * q^47 + 21 * q^49 - 27 * q^51 - 12 * q^53 + 18 * q^55 - 27 * q^57 + 6 * q^59 + 18 * q^61 + 9 * q^63 + 21 * q^65 + 9 * q^67 - 12 * q^71 + 3 * q^73 + 18 * q^75 + 39 * q^77 - 33 * q^79 + 18 * q^83 + 27 * q^85 - 9 * q^87 - 15 * q^89 + 12 * q^91 + 27 * q^93 - 21 * q^95 - 12 * q^97 + 45 * q^99

## Character values

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

 $$n$$ $$271$$ $$325$$ $$353$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{18}^{5}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 0.939693 + 0.342020i 0.939693 − 0.342020i −0.173648 − 0.984808i −0.766044 − 0.642788i −0.766044 + 0.642788i −0.173648 + 0.984808i
0 −1.70574 0.300767i 0 −1.26604 + 0.460802i 0 0.0209445 + 0.118782i 0 2.81908 + 1.02606i 0
97.1 0 −1.70574 + 0.300767i 0 −1.26604 0.460802i 0 0.0209445 0.118782i 0 2.81908 1.02606i 0
193.1 0 1.11334 1.32683i 0 0.439693 2.49362i 0 1.79813 1.50881i 0 −0.520945 2.95442i 0
241.1 0 0.592396 1.62760i 0 −0.673648 + 0.565258i 0 −3.31908 + 1.20805i 0 −2.29813 1.92836i 0
337.1 0 0.592396 + 1.62760i 0 −0.673648 0.565258i 0 −3.31908 1.20805i 0 −2.29813 + 1.92836i 0
385.1 0 1.11334 + 1.32683i 0 0.439693 + 2.49362i 0 1.79813 + 1.50881i 0 −0.520945 + 2.95442i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 49.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
27.e even 9 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 432.2.u.a 6
4.b odd 2 1 54.2.e.a 6
12.b even 2 1 162.2.e.a 6
27.e even 9 1 inner 432.2.u.a 6
36.f odd 6 1 486.2.e.b 6
36.f odd 6 1 486.2.e.d 6
36.h even 6 1 486.2.e.a 6
36.h even 6 1 486.2.e.c 6
108.j odd 18 1 54.2.e.a 6
108.j odd 18 1 486.2.e.b 6
108.j odd 18 1 486.2.e.d 6
108.j odd 18 1 1458.2.a.a 3
108.j odd 18 2 1458.2.c.d 6
108.l even 18 1 162.2.e.a 6
108.l even 18 1 486.2.e.a 6
108.l even 18 1 486.2.e.c 6
108.l even 18 1 1458.2.a.d 3
108.l even 18 2 1458.2.c.a 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.2.e.a 6 4.b odd 2 1
54.2.e.a 6 108.j odd 18 1
162.2.e.a 6 12.b even 2 1
162.2.e.a 6 108.l even 18 1
432.2.u.a 6 1.a even 1 1 trivial
432.2.u.a 6 27.e even 9 1 inner
486.2.e.a 6 36.h even 6 1
486.2.e.a 6 108.l even 18 1
486.2.e.b 6 36.f odd 6 1
486.2.e.b 6 108.j odd 18 1
486.2.e.c 6 36.h even 6 1
486.2.e.c 6 108.l even 18 1
486.2.e.d 6 36.f odd 6 1
486.2.e.d 6 108.j odd 18 1
1458.2.a.a 3 108.j odd 18 1
1458.2.a.d 3 108.l even 18 1
1458.2.c.a 6 108.l even 18 2
1458.2.c.d 6 108.j odd 18 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{6} + 3T_{5}^{5} + 9T_{5}^{4} + 24T_{5}^{3} + 36T_{5}^{2} + 27T_{5} + 9$$ acting on $$S_{2}^{\mathrm{new}}(432, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6} + 9T^{3} + 27$$
$5$ $$T^{6} + 3 T^{5} + 9 T^{4} + 24 T^{3} + \cdots + 9$$
$7$ $$T^{6} + 3 T^{5} - 6 T^{4} - 8 T^{3} + \cdots + 1$$
$11$ $$T^{6} - 3 T^{5} + 9 T^{4} - 24 T^{3} + \cdots + 3249$$
$13$ $$T^{6} + 12 T^{5} + 78 T^{4} + \cdots + 289$$
$17$ $$T^{6} + 6 T^{5} + 63 T^{4} + \cdots + 25281$$
$19$ $$T^{6} + 9 T^{5} + 93 T^{4} + \cdots + 32041$$
$23$ $$T^{6} - 6 T^{5} + 36 T^{4} - 111 T^{3} + \cdots + 9$$
$29$ $$T^{6} - 15 T^{5} + 99 T^{4} - 300 T^{3} + \cdots + 9$$
$31$ $$T^{6} - 18 T^{5} + 171 T^{4} + \cdots + 5041$$
$37$ $$T^{6} - 15 T^{5} + 171 T^{4} + \cdots + 289$$
$41$ $$T^{6} + 3 T^{5} + 36 T^{4} + \cdots + 47961$$
$43$ $$T^{6} - 18 T^{5} + 144 T^{4} + \cdots + 64$$
$47$ $$T^{6} + 9 T^{5} + 9 T^{4} - 72 T^{3} + \cdots + 81$$
$53$ $$(T^{3} + 6 T^{2} - 9 T + 3)^{2}$$
$59$ $$T^{6} - 6 T^{5} + 36 T^{4} + \cdots + 3249$$
$61$ $$T^{6} - 18 T^{5} + 153 T^{4} + \cdots + 2809$$
$67$ $$T^{6} - 9 T^{5} + 45 T^{4} - 80 T^{3} + \cdots + 1$$
$71$ $$T^{6} + 12 T^{5} + 189 T^{4} + \cdots + 106929$$
$73$ $$T^{6} - 3 T^{5} + 123 T^{4} + \cdots + 72361$$
$79$ $$T^{6} + 33 T^{5} + 510 T^{4} + \cdots + 466489$$
$83$ $$T^{6} - 18 T^{5} + 144 T^{4} + \cdots + 5184$$
$89$ $$T^{6} + 15 T^{5} + 189 T^{4} + \cdots + 25281$$
$97$ $$T^{6} + 12 T^{5} + 51 T^{4} + \cdots + 16129$$