# Properties

 Label 360.2.q.a Level $360$ Weight $2$ Character orbit 360.q Analytic conductor $2.875$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [360,2,Mod(121,360)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(360, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("360.121");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$360 = 2^{3} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 360.q (of order $$3$$, degree $$2$$, 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: $$2.87461447277$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{6} - 1) q^{3} + \zeta_{6} q^{5} + 3 \zeta_{6} q^{9}+O(q^{10})$$ q + (-z - 1) * q^3 + z * q^5 + 3*z * q^9 $$q + ( - \zeta_{6} - 1) q^{3} + \zeta_{6} q^{5} + 3 \zeta_{6} q^{9} + ( - 5 \zeta_{6} + 5) q^{11} + ( - 2 \zeta_{6} + 1) q^{15} + 3 q^{17} + 5 q^{19} - 6 \zeta_{6} q^{23} + (\zeta_{6} - 1) q^{25} + ( - 6 \zeta_{6} + 3) q^{27} + ( - 10 \zeta_{6} + 10) q^{29} + 2 \zeta_{6} q^{31} + (5 \zeta_{6} - 10) q^{33} + 4 q^{37} + 3 \zeta_{6} q^{41} + (3 \zeta_{6} - 3) q^{43} + (3 \zeta_{6} - 3) q^{45} + (4 \zeta_{6} - 4) q^{47} + 7 \zeta_{6} q^{49} + ( - 3 \zeta_{6} - 3) q^{51} - 6 q^{53} + 5 q^{55} + ( - 5 \zeta_{6} - 5) q^{57} + 3 \zeta_{6} q^{59} + (2 \zeta_{6} - 2) q^{61} + 11 \zeta_{6} q^{67} + (12 \zeta_{6} - 6) q^{69} - 14 q^{71} - 15 q^{73} + ( - \zeta_{6} + 2) q^{75} + (10 \zeta_{6} - 10) q^{79} + (9 \zeta_{6} - 9) q^{81} + ( - 12 \zeta_{6} + 12) q^{83} + 3 \zeta_{6} q^{85} + (10 \zeta_{6} - 20) q^{87} + 14 q^{89} + ( - 4 \zeta_{6} + 2) q^{93} + 5 \zeta_{6} q^{95} + ( - 13 \zeta_{6} + 13) q^{97} + 15 q^{99} +O(q^{100})$$ q + (-z - 1) * q^3 + z * q^5 + 3*z * q^9 + (-5*z + 5) * q^11 + (-2*z + 1) * q^15 + 3 * q^17 + 5 * q^19 - 6*z * q^23 + (z - 1) * q^25 + (-6*z + 3) * q^27 + (-10*z + 10) * q^29 + 2*z * q^31 + (5*z - 10) * q^33 + 4 * q^37 + 3*z * q^41 + (3*z - 3) * q^43 + (3*z - 3) * q^45 + (4*z - 4) * q^47 + 7*z * q^49 + (-3*z - 3) * q^51 - 6 * q^53 + 5 * q^55 + (-5*z - 5) * q^57 + 3*z * q^59 + (2*z - 2) * q^61 + 11*z * q^67 + (12*z - 6) * q^69 - 14 * q^71 - 15 * q^73 + (-z + 2) * q^75 + (10*z - 10) * q^79 + (9*z - 9) * q^81 + (-12*z + 12) * q^83 + 3*z * q^85 + (10*z - 20) * q^87 + 14 * q^89 + (-4*z + 2) * q^93 + 5*z * q^95 + (-13*z + 13) * q^97 + 15 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{3} + q^{5} + 3 q^{9}+O(q^{10})$$ 2 * q - 3 * q^3 + q^5 + 3 * q^9 $$2 q - 3 q^{3} + q^{5} + 3 q^{9} + 5 q^{11} + 6 q^{17} + 10 q^{19} - 6 q^{23} - q^{25} + 10 q^{29} + 2 q^{31} - 15 q^{33} + 8 q^{37} + 3 q^{41} - 3 q^{43} - 3 q^{45} - 4 q^{47} + 7 q^{49} - 9 q^{51} - 12 q^{53} + 10 q^{55} - 15 q^{57} + 3 q^{59} - 2 q^{61} + 11 q^{67} - 28 q^{71} - 30 q^{73} + 3 q^{75} - 10 q^{79} - 9 q^{81} + 12 q^{83} + 3 q^{85} - 30 q^{87} + 28 q^{89} + 5 q^{95} + 13 q^{97} + 30 q^{99}+O(q^{100})$$ 2 * q - 3 * q^3 + q^5 + 3 * q^9 + 5 * q^11 + 6 * q^17 + 10 * q^19 - 6 * q^23 - q^25 + 10 * q^29 + 2 * q^31 - 15 * q^33 + 8 * q^37 + 3 * q^41 - 3 * q^43 - 3 * q^45 - 4 * q^47 + 7 * q^49 - 9 * q^51 - 12 * q^53 + 10 * q^55 - 15 * q^57 + 3 * q^59 - 2 * q^61 + 11 * q^67 - 28 * q^71 - 30 * q^73 + 3 * q^75 - 10 * q^79 - 9 * q^81 + 12 * q^83 + 3 * q^85 - 30 * q^87 + 28 * q^89 + 5 * q^95 + 13 * q^97 + 30 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/360\mathbb{Z}\right)^\times$$.

 $$n$$ $$181$$ $$217$$ $$271$$ $$281$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-\zeta_{6}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
121.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 −1.50000 + 0.866025i 0 0.500000 0.866025i 0 0 0 1.50000 2.59808i 0
241.1 0 −1.50000 0.866025i 0 0.500000 + 0.866025i 0 0 0 1.50000 + 2.59808i 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 360.2.q.a 2
3.b odd 2 1 1080.2.q.a 2
4.b odd 2 1 720.2.q.e 2
9.c even 3 1 inner 360.2.q.a 2
9.c even 3 1 3240.2.a.b 1
9.d odd 6 1 1080.2.q.a 2
9.d odd 6 1 3240.2.a.f 1
12.b even 2 1 2160.2.q.c 2
36.f odd 6 1 720.2.q.e 2
36.f odd 6 1 6480.2.a.e 1
36.h even 6 1 2160.2.q.c 2
36.h even 6 1 6480.2.a.q 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.q.a 2 1.a even 1 1 trivial
360.2.q.a 2 9.c even 3 1 inner
720.2.q.e 2 4.b odd 2 1
720.2.q.e 2 36.f odd 6 1
1080.2.q.a 2 3.b odd 2 1
1080.2.q.a 2 9.d odd 6 1
2160.2.q.c 2 12.b even 2 1
2160.2.q.c 2 36.h even 6 1
3240.2.a.b 1 9.c even 3 1
3240.2.a.f 1 9.d odd 6 1
6480.2.a.e 1 36.f odd 6 1
6480.2.a.q 1 36.h even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}$$ acting on $$S_{2}^{\mathrm{new}}(360, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 3T + 3$$
$5$ $$T^{2} - T + 1$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 5T + 25$$
$13$ $$T^{2}$$
$17$ $$(T - 3)^{2}$$
$19$ $$(T - 5)^{2}$$
$23$ $$T^{2} + 6T + 36$$
$29$ $$T^{2} - 10T + 100$$
$31$ $$T^{2} - 2T + 4$$
$37$ $$(T - 4)^{2}$$
$41$ $$T^{2} - 3T + 9$$
$43$ $$T^{2} + 3T + 9$$
$47$ $$T^{2} + 4T + 16$$
$53$ $$(T + 6)^{2}$$
$59$ $$T^{2} - 3T + 9$$
$61$ $$T^{2} + 2T + 4$$
$67$ $$T^{2} - 11T + 121$$
$71$ $$(T + 14)^{2}$$
$73$ $$(T + 15)^{2}$$
$79$ $$T^{2} + 10T + 100$$
$83$ $$T^{2} - 12T + 144$$
$89$ $$(T - 14)^{2}$$
$97$ $$T^{2} - 13T + 169$$