# Properties

 Label 180.2.i.a Level $180$ Weight $2$ Character orbit 180.i Analytic conductor $1.437$ 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] = [180,2,Mod(61,180)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(180, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4, 0]))

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

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

chi := DirichletCharacter("180.61");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$180 = 2^{2} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 180.i (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: $$1.43730723638$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{6} + 2) q^{3} - \zeta_{6} q^{5} + ( - \zeta_{6} + 1) q^{7} + ( - 3 \zeta_{6} + 3) q^{9} +O(q^{10})$$ q + (-z + 2) * q^3 - z * q^5 + (-z + 1) * q^7 + (-3*z + 3) * q^9 $$q + ( - \zeta_{6} + 2) q^{3} - \zeta_{6} q^{5} + ( - \zeta_{6} + 1) q^{7} + ( - 3 \zeta_{6} + 3) q^{9} + 4 \zeta_{6} q^{13} + ( - \zeta_{6} - 1) q^{15} - 6 q^{17} + 2 q^{19} + ( - 2 \zeta_{6} + 1) q^{21} + 3 \zeta_{6} q^{23} + (\zeta_{6} - 1) q^{25} + ( - 6 \zeta_{6} + 3) q^{27} + (3 \zeta_{6} - 3) q^{29} + 10 \zeta_{6} q^{31} - q^{35} - 10 q^{37} + (4 \zeta_{6} + 4) q^{39} - 9 \zeta_{6} q^{41} + ( - 4 \zeta_{6} + 4) q^{43} - 3 q^{45} + (9 \zeta_{6} - 9) q^{47} + 6 \zeta_{6} q^{49} + (6 \zeta_{6} - 12) q^{51} - 6 q^{53} + ( - 2 \zeta_{6} + 4) q^{57} + 6 \zeta_{6} q^{59} + ( - \zeta_{6} + 1) q^{61} - 3 \zeta_{6} q^{63} + ( - 4 \zeta_{6} + 4) q^{65} - 11 \zeta_{6} q^{67} + (3 \zeta_{6} + 3) q^{69} + 12 q^{71} - 4 q^{73} + (2 \zeta_{6} - 1) q^{75} + ( - 10 \zeta_{6} + 10) q^{79} - 9 \zeta_{6} q^{81} + ( - 9 \zeta_{6} + 9) q^{83} + 6 \zeta_{6} q^{85} + (6 \zeta_{6} - 3) q^{87} + 9 q^{89} + 4 q^{91} + (10 \zeta_{6} + 10) q^{93} - 2 \zeta_{6} q^{95} + ( - 10 \zeta_{6} + 10) q^{97} +O(q^{100})$$ q + (-z + 2) * q^3 - z * q^5 + (-z + 1) * q^7 + (-3*z + 3) * q^9 + 4*z * q^13 + (-z - 1) * q^15 - 6 * q^17 + 2 * q^19 + (-2*z + 1) * q^21 + 3*z * q^23 + (z - 1) * q^25 + (-6*z + 3) * q^27 + (3*z - 3) * q^29 + 10*z * q^31 - q^35 - 10 * q^37 + (4*z + 4) * q^39 - 9*z * q^41 + (-4*z + 4) * q^43 - 3 * q^45 + (9*z - 9) * q^47 + 6*z * q^49 + (6*z - 12) * q^51 - 6 * q^53 + (-2*z + 4) * q^57 + 6*z * q^59 + (-z + 1) * q^61 - 3*z * q^63 + (-4*z + 4) * q^65 - 11*z * q^67 + (3*z + 3) * q^69 + 12 * q^71 - 4 * q^73 + (2*z - 1) * q^75 + (-10*z + 10) * q^79 - 9*z * q^81 + (-9*z + 9) * q^83 + 6*z * q^85 + (6*z - 3) * q^87 + 9 * q^89 + 4 * q^91 + (10*z + 10) * q^93 - 2*z * q^95 + (-10*z + 10) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{3} - q^{5} + q^{7} + 3 q^{9}+O(q^{10})$$ 2 * q + 3 * q^3 - q^5 + q^7 + 3 * q^9 $$2 q + 3 q^{3} - q^{5} + q^{7} + 3 q^{9} + 4 q^{13} - 3 q^{15} - 12 q^{17} + 4 q^{19} + 3 q^{23} - q^{25} - 3 q^{29} + 10 q^{31} - 2 q^{35} - 20 q^{37} + 12 q^{39} - 9 q^{41} + 4 q^{43} - 6 q^{45} - 9 q^{47} + 6 q^{49} - 18 q^{51} - 12 q^{53} + 6 q^{57} + 6 q^{59} + q^{61} - 3 q^{63} + 4 q^{65} - 11 q^{67} + 9 q^{69} + 24 q^{71} - 8 q^{73} + 10 q^{79} - 9 q^{81} + 9 q^{83} + 6 q^{85} + 18 q^{89} + 8 q^{91} + 30 q^{93} - 2 q^{95} + 10 q^{97}+O(q^{100})$$ 2 * q + 3 * q^3 - q^5 + q^7 + 3 * q^9 + 4 * q^13 - 3 * q^15 - 12 * q^17 + 4 * q^19 + 3 * q^23 - q^25 - 3 * q^29 + 10 * q^31 - 2 * q^35 - 20 * q^37 + 12 * q^39 - 9 * q^41 + 4 * q^43 - 6 * q^45 - 9 * q^47 + 6 * q^49 - 18 * q^51 - 12 * q^53 + 6 * q^57 + 6 * q^59 + q^61 - 3 * q^63 + 4 * q^65 - 11 * q^67 + 9 * q^69 + 24 * q^71 - 8 * q^73 + 10 * q^79 - 9 * q^81 + 9 * q^83 + 6 * q^85 + 18 * q^89 + 8 * q^91 + 30 * q^93 - 2 * q^95 + 10 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/180\mathbb{Z}\right)^\times$$.

 $$n$$ $$37$$ $$91$$ $$101$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
61.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 1.50000 0.866025i 0 −0.500000 0.866025i 0 0.500000 0.866025i 0 1.50000 2.59808i 0
121.1 0 1.50000 + 0.866025i 0 −0.500000 + 0.866025i 0 0.500000 + 0.866025i 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 180.2.i.a 2
3.b odd 2 1 540.2.i.a 2
4.b odd 2 1 720.2.q.a 2
5.b even 2 1 900.2.i.a 2
5.c odd 4 2 900.2.s.a 4
9.c even 3 1 inner 180.2.i.a 2
9.c even 3 1 1620.2.a.e 1
9.d odd 6 1 540.2.i.a 2
9.d odd 6 1 1620.2.a.b 1
12.b even 2 1 2160.2.q.e 2
15.d odd 2 1 2700.2.i.a 2
15.e even 4 2 2700.2.s.a 4
36.f odd 6 1 720.2.q.a 2
36.f odd 6 1 6480.2.a.t 1
36.h even 6 1 2160.2.q.e 2
36.h even 6 1 6480.2.a.h 1
45.h odd 6 1 2700.2.i.a 2
45.h odd 6 1 8100.2.a.h 1
45.j even 6 1 900.2.i.a 2
45.j even 6 1 8100.2.a.i 1
45.k odd 12 2 900.2.s.a 4
45.k odd 12 2 8100.2.d.e 2
45.l even 12 2 2700.2.s.a 4
45.l even 12 2 8100.2.d.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.2.i.a 2 1.a even 1 1 trivial
180.2.i.a 2 9.c even 3 1 inner
540.2.i.a 2 3.b odd 2 1
540.2.i.a 2 9.d odd 6 1
720.2.q.a 2 4.b odd 2 1
720.2.q.a 2 36.f odd 6 1
900.2.i.a 2 5.b even 2 1
900.2.i.a 2 45.j even 6 1
900.2.s.a 4 5.c odd 4 2
900.2.s.a 4 45.k odd 12 2
1620.2.a.b 1 9.d odd 6 1
1620.2.a.e 1 9.c even 3 1
2160.2.q.e 2 12.b even 2 1
2160.2.q.e 2 36.h even 6 1
2700.2.i.a 2 15.d odd 2 1
2700.2.i.a 2 45.h odd 6 1
2700.2.s.a 4 15.e even 4 2
2700.2.s.a 4 45.l even 12 2
6480.2.a.h 1 36.h even 6 1
6480.2.a.t 1 36.f odd 6 1
8100.2.a.h 1 45.h odd 6 1
8100.2.a.i 1 45.j even 6 1
8100.2.d.e 2 45.k odd 12 2
8100.2.d.f 2 45.l even 12 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{2} - T_{7} + 1$$ acting on $$S_{2}^{\mathrm{new}}(180, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 3T + 3$$
$5$ $$T^{2} + T + 1$$
$7$ $$T^{2} - T + 1$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 4T + 16$$
$17$ $$(T + 6)^{2}$$
$19$ $$(T - 2)^{2}$$
$23$ $$T^{2} - 3T + 9$$
$29$ $$T^{2} + 3T + 9$$
$31$ $$T^{2} - 10T + 100$$
$37$ $$(T + 10)^{2}$$
$41$ $$T^{2} + 9T + 81$$
$43$ $$T^{2} - 4T + 16$$
$47$ $$T^{2} + 9T + 81$$
$53$ $$(T + 6)^{2}$$
$59$ $$T^{2} - 6T + 36$$
$61$ $$T^{2} - T + 1$$
$67$ $$T^{2} + 11T + 121$$
$71$ $$(T - 12)^{2}$$
$73$ $$(T + 4)^{2}$$
$79$ $$T^{2} - 10T + 100$$
$83$ $$T^{2} - 9T + 81$$
$89$ $$(T - 9)^{2}$$
$97$ $$T^{2} - 10T + 100$$