# Properties

 Label 144.9.g.f Level $144$ Weight $9$ Character orbit 144.g Analytic conductor $58.663$ 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] = [144,9,Mod(127,144)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(144, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("144.127");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$144 = 2^{4} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 144.g (of order $$2$$, degree $$1$$, 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: $$58.6625198488$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 48) 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 $$\beta = 4\sqrt{-3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 90 q^{5} + 133 \beta q^{7}+O(q^{10})$$ q + 90 * q^5 + 133*b * q^7 $$q + 90 q^{5} + 133 \beta q^{7} - 747 \beta q^{11} - 16358 q^{13} + 42678 q^{17} - 16519 \beta q^{19} + 36180 \beta q^{23} - 382525 q^{25} + 1270530 q^{29} - 74037 \beta q^{31} + 11970 \beta q^{35} - 2262142 q^{37} + 872694 q^{41} - 239901 \beta q^{43} - 115038 \beta q^{47} + 4915729 q^{49} - 1061694 q^{53} - 67230 \beta q^{55} - 3373497 \beta q^{59} + 15301010 q^{61} - 1472220 q^{65} + 1325051 \beta q^{67} - 3295368 \beta q^{71} + 18916354 q^{73} + 4768848 q^{77} - 7817221 \beta q^{79} - 9412101 \beta q^{83} + 3841020 q^{85} + 89813214 q^{89} - 2175614 \beta q^{91} - 1486710 \beta q^{95} - 75778238 q^{97} +O(q^{100})$$ q + 90 * q^5 + 133*b * q^7 - 747*b * q^11 - 16358 * q^13 + 42678 * q^17 - 16519*b * q^19 + 36180*b * q^23 - 382525 * q^25 + 1270530 * q^29 - 74037*b * q^31 + 11970*b * q^35 - 2262142 * q^37 + 872694 * q^41 - 239901*b * q^43 - 115038*b * q^47 + 4915729 * q^49 - 1061694 * q^53 - 67230*b * q^55 - 3373497*b * q^59 + 15301010 * q^61 - 1472220 * q^65 + 1325051*b * q^67 - 3295368*b * q^71 + 18916354 * q^73 + 4768848 * q^77 - 7817221*b * q^79 - 9412101*b * q^83 + 3841020 * q^85 + 89813214 * q^89 - 2175614*b * q^91 - 1486710*b * q^95 - 75778238 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 180 q^{5}+O(q^{10})$$ 2 * q + 180 * q^5 $$2 q + 180 q^{5} - 32716 q^{13} + 85356 q^{17} - 765050 q^{25} + 2541060 q^{29} - 4524284 q^{37} + 1745388 q^{41} + 9831458 q^{49} - 2123388 q^{53} + 30602020 q^{61} - 2944440 q^{65} + 37832708 q^{73} + 9537696 q^{77} + 7682040 q^{85} + 179626428 q^{89} - 151556476 q^{97}+O(q^{100})$$ 2 * q + 180 * q^5 - 32716 * q^13 + 85356 * q^17 - 765050 * q^25 + 2541060 * q^29 - 4524284 * q^37 + 1745388 * q^41 + 9831458 * q^49 - 2123388 * q^53 + 30602020 * q^61 - 2944440 * q^65 + 37832708 * q^73 + 9537696 * q^77 + 7682040 * q^85 + 179626428 * q^89 - 151556476 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/144\mathbb{Z}\right)^\times$$.

 $$n$$ $$37$$ $$65$$ $$127$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
127.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 0 90.0000 0 921.451i 0 0 0
127.2 0 0 0 90.0000 0 921.451i 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
4.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 144.9.g.f 2
3.b odd 2 1 48.9.g.a 2
4.b odd 2 1 inner 144.9.g.f 2
12.b even 2 1 48.9.g.a 2
24.f even 2 1 192.9.g.b 2
24.h odd 2 1 192.9.g.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.9.g.a 2 3.b odd 2 1
48.9.g.a 2 12.b even 2 1
144.9.g.f 2 1.a even 1 1 trivial
144.9.g.f 2 4.b odd 2 1 inner
192.9.g.b 2 24.f even 2 1
192.9.g.b 2 24.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} - 90$$ acting on $$S_{9}^{\mathrm{new}}(144, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$(T - 90)^{2}$$
$7$ $$T^{2} + 849072$$
$11$ $$T^{2} + 26784432$$
$13$ $$(T + 16358)^{2}$$
$17$ $$(T - 42678)^{2}$$
$19$ $$T^{2} + 13098113328$$
$23$ $$T^{2} + 62831635200$$
$29$ $$(T - 1270530)^{2}$$
$31$ $$T^{2} + 263110913712$$
$37$ $$(T + 2262142)^{2}$$
$41$ $$(T - 872694)^{2}$$
$43$ $$T^{2} + 2762519510448$$
$47$ $$T^{2} + 635219589312$$
$53$ $$(T + 1061694)^{2}$$
$59$ $$T^{2} + \cdots + 546263136432432$$
$61$ $$(T - 15301010)^{2}$$
$67$ $$T^{2} + 84276487324848$$
$71$ $$T^{2} + \cdots + 521253612260352$$
$73$ $$(T - 18916354)^{2}$$
$79$ $$T^{2} + 29\!\cdots\!68$$
$83$ $$T^{2} + 42\!\cdots\!48$$
$89$ $$(T - 89813214)^{2}$$
$97$ $$(T + 75778238)^{2}$$