# Properties

 Label 144.2.i.b Level $144$ Weight $2$ Character orbit 144.i Analytic conductor $1.150$ 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] = [144,2,Mod(49,144)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(144, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("144.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$144 = 2^{4} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 144.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: $$1.14984578911$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 72) 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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (2 \zeta_{6} - 1) q^{3} + \zeta_{6} q^{5} + (3 \zeta_{6} - 3) q^{7} - 3 q^{9}+O(q^{10})$$ q + (2*z - 1) * q^3 + z * q^5 + (3*z - 3) * q^7 - 3 * q^9 $$q + (2 \zeta_{6} - 1) q^{3} + \zeta_{6} q^{5} + (3 \zeta_{6} - 3) q^{7} - 3 q^{9} + ( - 5 \zeta_{6} + 5) q^{11} + 5 \zeta_{6} q^{13} + (\zeta_{6} - 2) q^{15} - 2 q^{17} + 4 q^{19} + ( - 3 \zeta_{6} - 3) q^{21} - \zeta_{6} q^{23} + ( - 4 \zeta_{6} + 4) q^{25} + ( - 6 \zeta_{6} + 3) q^{27} + ( - 9 \zeta_{6} + 9) q^{29} - \zeta_{6} q^{31} + (5 \zeta_{6} + 5) q^{33} - 3 q^{35} - 6 q^{37} + (5 \zeta_{6} - 10) q^{39} - 3 \zeta_{6} q^{41} + ( - \zeta_{6} + 1) q^{43} - 3 \zeta_{6} q^{45} + (3 \zeta_{6} - 3) q^{47} - 2 \zeta_{6} q^{49} + ( - 4 \zeta_{6} + 2) q^{51} + 2 q^{53} + 5 q^{55} + (8 \zeta_{6} - 4) q^{57} + 11 \zeta_{6} q^{59} + (7 \zeta_{6} - 7) q^{61} + ( - 9 \zeta_{6} + 9) q^{63} + (5 \zeta_{6} - 5) q^{65} - \zeta_{6} q^{67} + ( - \zeta_{6} + 2) q^{69} - 4 q^{71} - 2 q^{73} + (4 \zeta_{6} + 4) q^{75} + 15 \zeta_{6} q^{77} + ( - \zeta_{6} + 1) q^{79} + 9 q^{81} + ( - \zeta_{6} + 1) q^{83} - 2 \zeta_{6} q^{85} + (9 \zeta_{6} + 9) q^{87} - 18 q^{89} - 15 q^{91} + ( - \zeta_{6} + 2) q^{93} + 4 \zeta_{6} q^{95} + ( - 13 \zeta_{6} + 13) q^{97} + (15 \zeta_{6} - 15) q^{99} +O(q^{100})$$ q + (2*z - 1) * q^3 + z * q^5 + (3*z - 3) * q^7 - 3 * q^9 + (-5*z + 5) * q^11 + 5*z * q^13 + (z - 2) * q^15 - 2 * q^17 + 4 * q^19 + (-3*z - 3) * q^21 - z * q^23 + (-4*z + 4) * q^25 + (-6*z + 3) * q^27 + (-9*z + 9) * q^29 - z * q^31 + (5*z + 5) * q^33 - 3 * q^35 - 6 * q^37 + (5*z - 10) * q^39 - 3*z * q^41 + (-z + 1) * q^43 - 3*z * q^45 + (3*z - 3) * q^47 - 2*z * q^49 + (-4*z + 2) * q^51 + 2 * q^53 + 5 * q^55 + (8*z - 4) * q^57 + 11*z * q^59 + (7*z - 7) * q^61 + (-9*z + 9) * q^63 + (5*z - 5) * q^65 - z * q^67 + (-z + 2) * q^69 - 4 * q^71 - 2 * q^73 + (4*z + 4) * q^75 + 15*z * q^77 + (-z + 1) * q^79 + 9 * q^81 + (-z + 1) * q^83 - 2*z * q^85 + (9*z + 9) * q^87 - 18 * q^89 - 15 * q^91 + (-z + 2) * q^93 + 4*z * q^95 + (-13*z + 13) * q^97 + (15*z - 15) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{5} - 3 q^{7} - 6 q^{9}+O(q^{10})$$ 2 * q + q^5 - 3 * q^7 - 6 * q^9 $$2 q + q^{5} - 3 q^{7} - 6 q^{9} + 5 q^{11} + 5 q^{13} - 3 q^{15} - 4 q^{17} + 8 q^{19} - 9 q^{21} - q^{23} + 4 q^{25} + 9 q^{29} - q^{31} + 15 q^{33} - 6 q^{35} - 12 q^{37} - 15 q^{39} - 3 q^{41} + q^{43} - 3 q^{45} - 3 q^{47} - 2 q^{49} + 4 q^{53} + 10 q^{55} + 11 q^{59} - 7 q^{61} + 9 q^{63} - 5 q^{65} - q^{67} + 3 q^{69} - 8 q^{71} - 4 q^{73} + 12 q^{75} + 15 q^{77} + q^{79} + 18 q^{81} + q^{83} - 2 q^{85} + 27 q^{87} - 36 q^{89} - 30 q^{91} + 3 q^{93} + 4 q^{95} + 13 q^{97} - 15 q^{99}+O(q^{100})$$ 2 * q + q^5 - 3 * q^7 - 6 * q^9 + 5 * q^11 + 5 * q^13 - 3 * q^15 - 4 * q^17 + 8 * q^19 - 9 * q^21 - q^23 + 4 * q^25 + 9 * q^29 - q^31 + 15 * q^33 - 6 * q^35 - 12 * q^37 - 15 * q^39 - 3 * q^41 + q^43 - 3 * q^45 - 3 * q^47 - 2 * q^49 + 4 * q^53 + 10 * q^55 + 11 * q^59 - 7 * q^61 + 9 * q^63 - 5 * q^65 - q^67 + 3 * q^69 - 8 * q^71 - 4 * q^73 + 12 * q^75 + 15 * q^77 + q^79 + 18 * q^81 + q^83 - 2 * q^85 + 27 * q^87 - 36 * q^89 - 30 * q^91 + 3 * q^93 + 4 * q^95 + 13 * q^97 - 15 * q^99

## Character values

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

 $$n$$ $$37$$ $$65$$ $$127$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 1.73205i 0 0.500000 0.866025i 0 −1.50000 2.59808i 0 −3.00000 0
97.1 0 1.73205i 0 0.500000 + 0.866025i 0 −1.50000 + 2.59808i 0 −3.00000 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
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 144.2.i.b 2
3.b odd 2 1 432.2.i.a 2
4.b odd 2 1 72.2.i.a 2
8.b even 2 1 576.2.i.c 2
8.d odd 2 1 576.2.i.d 2
9.c even 3 1 inner 144.2.i.b 2
9.c even 3 1 1296.2.a.e 1
9.d odd 6 1 432.2.i.a 2
9.d odd 6 1 1296.2.a.i 1
12.b even 2 1 216.2.i.a 2
24.f even 2 1 1728.2.i.h 2
24.h odd 2 1 1728.2.i.g 2
36.f odd 6 1 72.2.i.a 2
36.f odd 6 1 648.2.a.a 1
36.h even 6 1 216.2.i.a 2
36.h even 6 1 648.2.a.c 1
72.j odd 6 1 1728.2.i.g 2
72.j odd 6 1 5184.2.a.n 1
72.l even 6 1 1728.2.i.h 2
72.l even 6 1 5184.2.a.i 1
72.n even 6 1 576.2.i.c 2
72.n even 6 1 5184.2.a.x 1
72.p odd 6 1 576.2.i.d 2
72.p odd 6 1 5184.2.a.s 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.2.i.a 2 4.b odd 2 1
72.2.i.a 2 36.f odd 6 1
144.2.i.b 2 1.a even 1 1 trivial
144.2.i.b 2 9.c even 3 1 inner
216.2.i.a 2 12.b even 2 1
216.2.i.a 2 36.h even 6 1
432.2.i.a 2 3.b odd 2 1
432.2.i.a 2 9.d odd 6 1
576.2.i.c 2 8.b even 2 1
576.2.i.c 2 72.n even 6 1
576.2.i.d 2 8.d odd 2 1
576.2.i.d 2 72.p odd 6 1
648.2.a.a 1 36.f odd 6 1
648.2.a.c 1 36.h even 6 1
1296.2.a.e 1 9.c even 3 1
1296.2.a.i 1 9.d odd 6 1
1728.2.i.g 2 24.h odd 2 1
1728.2.i.g 2 72.j odd 6 1
1728.2.i.h 2 24.f even 2 1
1728.2.i.h 2 72.l even 6 1
5184.2.a.i 1 72.l even 6 1
5184.2.a.n 1 72.j odd 6 1
5184.2.a.s 1 72.p odd 6 1
5184.2.a.x 1 72.n even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - T_{5} + 1$$ acting on $$S_{2}^{\mathrm{new}}(144, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 3$$
$5$ $$T^{2} - T + 1$$
$7$ $$T^{2} + 3T + 9$$
$11$ $$T^{2} - 5T + 25$$
$13$ $$T^{2} - 5T + 25$$
$17$ $$(T + 2)^{2}$$
$19$ $$(T - 4)^{2}$$
$23$ $$T^{2} + T + 1$$
$29$ $$T^{2} - 9T + 81$$
$31$ $$T^{2} + T + 1$$
$37$ $$(T + 6)^{2}$$
$41$ $$T^{2} + 3T + 9$$
$43$ $$T^{2} - T + 1$$
$47$ $$T^{2} + 3T + 9$$
$53$ $$(T - 2)^{2}$$
$59$ $$T^{2} - 11T + 121$$
$61$ $$T^{2} + 7T + 49$$
$67$ $$T^{2} + T + 1$$
$71$ $$(T + 4)^{2}$$
$73$ $$(T + 2)^{2}$$
$79$ $$T^{2} - T + 1$$
$83$ $$T^{2} - T + 1$$
$89$ $$(T + 18)^{2}$$
$97$ $$T^{2} - 13T + 169$$