# Properties

 Label 432.9.e.g Level $432$ Weight $9$ Character orbit 432.e Analytic conductor $175.988$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [432,9,Mod(161,432)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(432, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("432.161");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$432 = 2^{4} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 432.e (of order $$2$$, degree $$1$$, 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: $$175.987559546$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 2$$ x^2 + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{5}\cdot 3$$ Twist minimal: no (minimal twist has level 54) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$\beta = 96\sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 5 \beta q^{5} + 2065 q^{7}+O(q^{10})$$ q + 5*b * q^5 + 2065 * q^7 $$q + 5 \beta q^{5} + 2065 q^{7} - 49 \beta q^{11} + 8063 q^{13} - 159 \beta q^{17} + 226609 q^{19} + 2713 \beta q^{23} - 70175 q^{25} - 6902 \beta q^{29} - 826370 q^{31} + 10325 \beta q^{35} + 1344575 q^{37} - 38242 \beta q^{41} + 6147742 q^{43} - 43537 \beta q^{47} - 1500576 q^{49} - 5658 \beta q^{53} + 4515840 q^{55} - 3491 \beta q^{59} - 14985697 q^{61} + 40315 \beta q^{65} + 10023697 q^{67} - 335028 \beta q^{71} - 23261569 q^{73} - 101185 \beta q^{77} - 14267183 q^{79} - 266578 \beta q^{83} + 14653440 q^{85} + 847701 \beta q^{89} + 16650095 q^{91} + 1133045 \beta q^{95} - 40571617 q^{97} +O(q^{100})$$ q + 5*b * q^5 + 2065 * q^7 - 49*b * q^11 + 8063 * q^13 - 159*b * q^17 + 226609 * q^19 + 2713*b * q^23 - 70175 * q^25 - 6902*b * q^29 - 826370 * q^31 + 10325*b * q^35 + 1344575 * q^37 - 38242*b * q^41 + 6147742 * q^43 - 43537*b * q^47 - 1500576 * q^49 - 5658*b * q^53 + 4515840 * q^55 - 3491*b * q^59 - 14985697 * q^61 + 40315*b * q^65 + 10023697 * q^67 - 335028*b * q^71 - 23261569 * q^73 - 101185*b * q^77 - 14267183 * q^79 - 266578*b * q^83 + 14653440 * q^85 + 847701*b * q^89 + 16650095 * q^91 + 1133045*b * q^95 - 40571617 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4130 q^{7}+O(q^{10})$$ 2 * q + 4130 * q^7 $$2 q + 4130 q^{7} + 16126 q^{13} + 453218 q^{19} - 140350 q^{25} - 1652740 q^{31} + 2689150 q^{37} + 12295484 q^{43} - 3001152 q^{49} + 9031680 q^{55} - 29971394 q^{61} + 20047394 q^{67} - 46523138 q^{73} - 28534366 q^{79} + 29306880 q^{85} + 33300190 q^{91} - 81143234 q^{97}+O(q^{100})$$ 2 * q + 4130 * q^7 + 16126 * q^13 + 453218 * q^19 - 140350 * q^25 - 1652740 * q^31 + 2689150 * q^37 + 12295484 * q^43 - 3001152 * q^49 + 9031680 * q^55 - 29971394 * q^61 + 20047394 * q^67 - 46523138 * q^73 - 28534366 * q^79 + 29306880 * q^85 + 33300190 * q^91 - 81143234 * q^97

## Character values

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

 $$n$$ $$271$$ $$325$$ $$353$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
161.1
 − 1.41421i 1.41421i
0 0 0 678.823i 0 2065.00 0 0 0
161.2 0 0 0 678.823i 0 2065.00 0 0 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
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 432.9.e.g 2
3.b odd 2 1 inner 432.9.e.g 2
4.b odd 2 1 54.9.b.a 2
12.b even 2 1 54.9.b.a 2
36.f odd 6 2 162.9.d.c 4
36.h even 6 2 162.9.d.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.9.b.a 2 4.b odd 2 1
54.9.b.a 2 12.b even 2 1
162.9.d.c 4 36.f odd 6 2
162.9.d.c 4 36.h even 6 2
432.9.e.g 2 1.a even 1 1 trivial
432.9.e.g 2 3.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{9}^{\mathrm{new}}(432, [\chi])$$:

 $$T_{5}^{2} + 460800$$ T5^2 + 460800 $$T_{7} - 2065$$ T7 - 2065

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 460800$$
$7$ $$(T - 2065)^{2}$$
$11$ $$T^{2} + 44255232$$
$13$ $$(T - 8063)^{2}$$
$17$ $$T^{2} + 465979392$$
$19$ $$(T - 226609)^{2}$$
$23$ $$T^{2} + 135666321408$$
$29$ $$T^{2} + 878056316928$$
$31$ $$(T + 826370)^{2}$$
$37$ $$(T - 1344575)^{2}$$
$41$ $$T^{2} + 26955888795648$$
$43$ $$(T - 6147742)^{2}$$
$47$ $$T^{2} + 34937309841408$$
$53$ $$T^{2} + 590062952448$$
$59$ $$T^{2} + 224632276992$$
$61$ $$(T + 14985697)^{2}$$
$67$ $$(T - 10023697)^{2}$$
$71$ $$T^{2} + 20\!\cdots\!88$$
$73$ $$(T + 23261569)^{2}$$
$79$ $$(T + 14267183)^{2}$$
$83$ $$T^{2} + 13\!\cdots\!88$$
$89$ $$T^{2} + 13\!\cdots\!32$$
$97$ $$(T + 40571617)^{2}$$