# Properties

 Label 180.1.p.a Level $180$ Weight $1$ Character orbit 180.p Analytic conductor $0.090$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -20 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [180,1,Mod(79,180)]

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([3, 4, 3]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("180.79");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$180 = 2^{2} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 180.p (of order $$6$$, 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: $$0.0898317022739$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.1620.1 Artin image: $C_3\times S_3$ Artin field: Galois closure of 6.0.648000.1

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + \zeta_{6}^{2} q^{2} - \zeta_{6} q^{3} - \zeta_{6} q^{4} - \zeta_{6} q^{5} + q^{6} - \zeta_{6}^{2} q^{7} + q^{8} + \zeta_{6}^{2} q^{9} +O(q^{10})$$ q + z^2 * q^2 - z * q^3 - z * q^4 - z * q^5 + q^6 - z^2 * q^7 + q^8 + z^2 * q^9 $$q + \zeta_{6}^{2} q^{2} - \zeta_{6} q^{3} - \zeta_{6} q^{4} - \zeta_{6} q^{5} + q^{6} - \zeta_{6}^{2} q^{7} + q^{8} + \zeta_{6}^{2} q^{9} + q^{10} + \zeta_{6}^{2} q^{12} + \zeta_{6} q^{14} + \zeta_{6}^{2} q^{15} + \zeta_{6}^{2} q^{16} - \zeta_{6} q^{18} + \zeta_{6}^{2} q^{20} - q^{21} + \zeta_{6} q^{23} - \zeta_{6} q^{24} + \zeta_{6}^{2} q^{25} + q^{27} - q^{28} - \zeta_{6}^{2} q^{29} - \zeta_{6} q^{30} - \zeta_{6} q^{32} - q^{35} + q^{36} - \zeta_{6} q^{40} + \zeta_{6} q^{41} - \zeta_{6}^{2} q^{42} + \zeta_{6}^{2} q^{43} + q^{45} - q^{46} - \zeta_{6}^{2} q^{47} + q^{48} - \zeta_{6} q^{50} + \zeta_{6}^{2} q^{54} - \zeta_{6}^{2} q^{56} + \zeta_{6} q^{58} + q^{60} - \zeta_{6}^{2} q^{61} + \zeta_{6} q^{63} + q^{64} + \zeta_{6} q^{67} - \zeta_{6}^{2} q^{69} - \zeta_{6}^{2} q^{70} + \zeta_{6}^{2} q^{72} + q^{75} + q^{80} - \zeta_{6} q^{81} - q^{82} - \zeta_{6}^{2} q^{83} + \zeta_{6} q^{84} - 2 \zeta_{6} q^{86} - q^{87} - q^{89} + \zeta_{6}^{2} q^{90} - \zeta_{6}^{2} q^{92} + \zeta_{6} q^{94} + \zeta_{6}^{2} q^{96} +O(q^{100})$$ q + z^2 * q^2 - z * q^3 - z * q^4 - z * q^5 + q^6 - z^2 * q^7 + q^8 + z^2 * q^9 + q^10 + z^2 * q^12 + z * q^14 + z^2 * q^15 + z^2 * q^16 - z * q^18 + z^2 * q^20 - q^21 + z * q^23 - z * q^24 + z^2 * q^25 + q^27 - q^28 - z^2 * q^29 - z * q^30 - z * q^32 - q^35 + q^36 - z * q^40 + z * q^41 - z^2 * q^42 + z^2 * q^43 + q^45 - q^46 - z^2 * q^47 + q^48 - z * q^50 + z^2 * q^54 - z^2 * q^56 + z * q^58 + q^60 - z^2 * q^61 + z * q^63 + q^64 + z * q^67 - z^2 * q^69 - z^2 * q^70 + z^2 * q^72 + q^75 + q^80 - z * q^81 - q^82 - z^2 * q^83 + z * q^84 - 2*z * q^86 - q^87 - q^89 + z^2 * q^90 - z^2 * q^92 + z * q^94 + z^2 * q^96 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - q^{3} - q^{4} - q^{5} + 2 q^{6} + q^{7} + 2 q^{8} - q^{9}+O(q^{10})$$ 2 * q - q^2 - q^3 - q^4 - q^5 + 2 * q^6 + q^7 + 2 * q^8 - q^9 $$2 q - q^{2} - q^{3} - q^{4} - q^{5} + 2 q^{6} + q^{7} + 2 q^{8} - q^{9} + 2 q^{10} - q^{12} + q^{14} - q^{15} - q^{16} - q^{18} - q^{20} - 2 q^{21} + q^{23} - q^{24} - q^{25} + 2 q^{27} - 2 q^{28} + q^{29} - q^{30} - q^{32} - 2 q^{35} + 2 q^{36} - q^{40} + q^{41} + q^{42} - 2 q^{43} + 2 q^{45} - 2 q^{46} + q^{47} + 2 q^{48} - q^{50} - q^{54} + q^{56} + q^{58} + 2 q^{60} + q^{61} + q^{63} + 2 q^{64} + q^{67} + q^{69} + q^{70} - q^{72} + 2 q^{75} + 2 q^{80} - q^{81} - 2 q^{82} + q^{83} + q^{84} - 2 q^{86} - 2 q^{87} - 2 q^{89} - q^{90} + q^{92} + q^{94} - q^{96}+O(q^{100})$$ 2 * q - q^2 - q^3 - q^4 - q^5 + 2 * q^6 + q^7 + 2 * q^8 - q^9 + 2 * q^10 - q^12 + q^14 - q^15 - q^16 - q^18 - q^20 - 2 * q^21 + q^23 - q^24 - q^25 + 2 * q^27 - 2 * q^28 + q^29 - q^30 - q^32 - 2 * q^35 + 2 * q^36 - q^40 + q^41 + q^42 - 2 * q^43 + 2 * q^45 - 2 * q^46 + q^47 + 2 * q^48 - q^50 - q^54 + q^56 + q^58 + 2 * q^60 + q^61 + q^63 + 2 * q^64 + q^67 + q^69 + q^70 - q^72 + 2 * q^75 + 2 * q^80 - q^81 - 2 * q^82 + q^83 + q^84 - 2 * q^86 - 2 * q^87 - 2 * q^89 - q^90 + q^92 + q^94 - q^96

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
79.1
 0.5 + 0.866025i 0.5 − 0.866025i
−0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i −0.500000 0.866025i 1.00000 0.500000 0.866025i 1.00000 −0.500000 + 0.866025i 1.00000
139.1 −0.500000 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 1.00000 0.500000 + 0.866025i 1.00000 −0.500000 0.866025i 1.00000
 $$n$$: 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
20.d odd 2 1 CM by $$\Q(\sqrt{-5})$$
9.c even 3 1 inner
180.p odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 180.1.p.a 2
3.b odd 2 1 540.1.p.b 2
4.b odd 2 1 180.1.p.b yes 2
5.b even 2 1 180.1.p.b yes 2
5.c odd 4 2 900.1.t.a 4
8.b even 2 1 2880.1.bu.b 2
8.d odd 2 1 2880.1.bu.a 2
9.c even 3 1 inner 180.1.p.a 2
9.c even 3 1 1620.1.f.d 1
9.d odd 6 1 540.1.p.b 2
9.d odd 6 1 1620.1.f.a 1
12.b even 2 1 540.1.p.a 2
15.d odd 2 1 540.1.p.a 2
15.e even 4 2 2700.1.t.a 4
20.d odd 2 1 CM 180.1.p.a 2
20.e even 4 2 900.1.t.a 4
36.f odd 6 1 180.1.p.b yes 2
36.f odd 6 1 1620.1.f.b 1
36.h even 6 1 540.1.p.a 2
36.h even 6 1 1620.1.f.c 1
40.e odd 2 1 2880.1.bu.b 2
40.f even 2 1 2880.1.bu.a 2
45.h odd 6 1 540.1.p.a 2
45.h odd 6 1 1620.1.f.c 1
45.j even 6 1 180.1.p.b yes 2
45.j even 6 1 1620.1.f.b 1
45.k odd 12 2 900.1.t.a 4
45.l even 12 2 2700.1.t.a 4
60.h even 2 1 540.1.p.b 2
60.l odd 4 2 2700.1.t.a 4
72.n even 6 1 2880.1.bu.b 2
72.p odd 6 1 2880.1.bu.a 2
180.n even 6 1 540.1.p.b 2
180.n even 6 1 1620.1.f.a 1
180.p odd 6 1 inner 180.1.p.a 2
180.p odd 6 1 1620.1.f.d 1
180.v odd 12 2 2700.1.t.a 4
180.x even 12 2 900.1.t.a 4
360.z odd 6 1 2880.1.bu.b 2
360.bk even 6 1 2880.1.bu.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.1.p.a 2 1.a even 1 1 trivial
180.1.p.a 2 9.c even 3 1 inner
180.1.p.a 2 20.d odd 2 1 CM
180.1.p.a 2 180.p odd 6 1 inner
180.1.p.b yes 2 4.b odd 2 1
180.1.p.b yes 2 5.b even 2 1
180.1.p.b yes 2 36.f odd 6 1
180.1.p.b yes 2 45.j even 6 1
540.1.p.a 2 12.b even 2 1
540.1.p.a 2 15.d odd 2 1
540.1.p.a 2 36.h even 6 1
540.1.p.a 2 45.h odd 6 1
540.1.p.b 2 3.b odd 2 1
540.1.p.b 2 9.d odd 6 1
540.1.p.b 2 60.h even 2 1
540.1.p.b 2 180.n even 6 1
900.1.t.a 4 5.c odd 4 2
900.1.t.a 4 20.e even 4 2
900.1.t.a 4 45.k odd 12 2
900.1.t.a 4 180.x even 12 2
1620.1.f.a 1 9.d odd 6 1
1620.1.f.a 1 180.n even 6 1
1620.1.f.b 1 36.f odd 6 1
1620.1.f.b 1 45.j even 6 1
1620.1.f.c 1 36.h even 6 1
1620.1.f.c 1 45.h odd 6 1
1620.1.f.d 1 9.c even 3 1
1620.1.f.d 1 180.p odd 6 1
2700.1.t.a 4 15.e even 4 2
2700.1.t.a 4 45.l even 12 2
2700.1.t.a 4 60.l odd 4 2
2700.1.t.a 4 180.v odd 12 2
2880.1.bu.a 2 8.d odd 2 1
2880.1.bu.a 2 40.f even 2 1
2880.1.bu.a 2 72.p odd 6 1
2880.1.bu.a 2 360.bk even 6 1
2880.1.bu.b 2 8.b even 2 1
2880.1.bu.b 2 40.e odd 2 1
2880.1.bu.b 2 72.n even 6 1
2880.1.bu.b 2 360.z odd 6 1

## Hecke kernels

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

## Hecke characteristic polynomials

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