# Properties

 Label 144.9.g.c 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: $$2^{3}\cdot 3^{2}$$ 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 $$\beta = 36\sqrt{-3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 726 q^{5} + 49 \beta q^{7}+O(q^{10})$$ q - 726 * q^5 + 49*b * q^7 $$q - 726 q^{5} + 49 \beta q^{7} + 213 \beta q^{11} + 39034 q^{13} + 65814 q^{17} + 2089 \beta q^{19} - 8052 \beta q^{23} + 136451 q^{25} - 202062 q^{29} + 19175 \beta q^{31} - 35574 \beta q^{35} - 1876030 q^{37} - 3091050 q^{41} + 36307 \beta q^{43} + 101946 \beta q^{47} - 3570287 q^{49} + 1066482 q^{53} - 154638 \beta q^{55} - 92433 \beta q^{59} + 17154194 q^{61} - 28338684 q^{65} - 439869 \beta q^{67} + 638832 \beta q^{71} - 53286014 q^{73} - 40579056 q^{77} + 292999 \beta q^{79} - 124917 \beta q^{83} - 47780964 q^{85} - 86667234 q^{89} + 1912666 \beta q^{91} - 1516614 \beta q^{95} - 73901822 q^{97} +O(q^{100})$$ q - 726 * q^5 + 49*b * q^7 + 213*b * q^11 + 39034 * q^13 + 65814 * q^17 + 2089*b * q^19 - 8052*b * q^23 + 136451 * q^25 - 202062 * q^29 + 19175*b * q^31 - 35574*b * q^35 - 1876030 * q^37 - 3091050 * q^41 + 36307*b * q^43 + 101946*b * q^47 - 3570287 * q^49 + 1066482 * q^53 - 154638*b * q^55 - 92433*b * q^59 + 17154194 * q^61 - 28338684 * q^65 - 439869*b * q^67 + 638832*b * q^71 - 53286014 * q^73 - 40579056 * q^77 + 292999*b * q^79 - 124917*b * q^83 - 47780964 * q^85 - 86667234 * q^89 + 1912666*b * q^91 - 1516614*b * q^95 - 73901822 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 1452 q^{5}+O(q^{10})$$ 2 * q - 1452 * q^5 $$2 q - 1452 q^{5} + 78068 q^{13} + 131628 q^{17} + 272902 q^{25} - 404124 q^{29} - 3752060 q^{37} - 6182100 q^{41} - 7140574 q^{49} + 2132964 q^{53} + 34308388 q^{61} - 56677368 q^{65} - 106572028 q^{73} - 81158112 q^{77} - 95561928 q^{85} - 173334468 q^{89} - 147803644 q^{97}+O(q^{100})$$ 2 * q - 1452 * q^5 + 78068 * q^13 + 131628 * q^17 + 272902 * q^25 - 404124 * q^29 - 3752060 * q^37 - 6182100 * q^41 - 7140574 * q^49 + 2132964 * q^53 + 34308388 * q^61 - 56677368 * q^65 - 106572028 * q^73 - 81158112 * q^77 - 95561928 * q^85 - 173334468 * q^89 - 147803644 * 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 −726.000 0 3055.34i 0 0 0
127.2 0 0 0 −726.000 0 3055.34i 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.c 2
3.b odd 2 1 48.9.g.b 2
4.b odd 2 1 inner 144.9.g.c 2
12.b even 2 1 48.9.g.b 2
24.f even 2 1 192.9.g.a 2
24.h odd 2 1 192.9.g.a 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} + 726$$ acting on $$S_{9}^{\mathrm{new}}(144, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$(T + 726)^{2}$$
$7$ $$T^{2} + 9335088$$
$11$ $$T^{2} + 176394672$$
$13$ $$(T - 39034)^{2}$$
$17$ $$(T - 65814)^{2}$$
$19$ $$T^{2} + 16966924848$$
$23$ $$T^{2} + 252077329152$$
$29$ $$(T + 202062)^{2}$$
$31$ $$T^{2} + 1429542270000$$
$37$ $$(T + 1876030)^{2}$$
$41$ $$(T + 3091050)^{2}$$
$43$ $$T^{2} + 5125154792112$$
$47$ $$T^{2} + 40407933129408$$
$53$ $$(T - 1066482)^{2}$$
$59$ $$T^{2} + 33218525693232$$
$61$ $$(T - 17154194)^{2}$$
$67$ $$T^{2} + \cdots + 752268658081968$$
$71$ $$T^{2} + 15\!\cdots\!12$$
$73$ $$(T + 53286014)^{2}$$
$79$ $$T^{2} + \cdots + 333778633635888$$
$83$ $$T^{2} + 60669350784432$$
$89$ $$(T + 86667234)^{2}$$
$97$ $$(T + 73901822)^{2}$$