# Properties

 Label 180.2.i.b Level $180$ Weight $2$ Character orbit 180.i Analytic conductor $1.437$ Analytic rank $0$ Dimension $6$ 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: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.954288.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27$$ x^6 - x^5 - 2*x^4 + 3*x^3 - 6*x^2 - 9*x + 27 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3$$ 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 basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{3} + (\beta_{2} + 1) q^{5} + (\beta_{5} - \beta_{4} - \beta_{3} + \beta_{2}) q^{7} + (\beta_{3} + \beta_{2} + 1) q^{9}+O(q^{10})$$ q - b1 * q^3 + (b2 + 1) * q^5 + (b5 - b4 - b3 + b2) * q^7 + (b3 + b2 + 1) * q^9 $$q - \beta_1 q^{3} + (\beta_{2} + 1) q^{5} + (\beta_{5} - \beta_{4} - \beta_{3} + \beta_{2}) q^{7} + (\beta_{3} + \beta_{2} + 1) q^{9} + (2 \beta_{4} - \beta_{3} - 2 \beta_1 + 1) q^{11} + (\beta_{5} + 2 \beta_{4} - \beta_{3} - 3 \beta_{2} - 2) q^{13} - \beta_{4} q^{15} + ( - \beta_{5} + \beta_{2} - 2 \beta_1 + 1) q^{17} + (\beta_{5} - \beta_{2} + 2 \beta_1 + 1) q^{19} + (\beta_{5} - 2 \beta_{4} + \beta_{3} - \beta_{2} + 2 \beta_1 - 5) q^{21} + ( - \beta_{5} + \beta_{2} + \beta_1 + 1) q^{23} + \beta_{2} q^{25} + ( - \beta_{5} - \beta_{4} + 5 \beta_{2} - \beta_1 + 3) q^{27} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - \beta_{2}) q^{29} + ( - \beta_{5} - 2 \beta_{4} + \beta_{3} - \beta_{2} - 2) q^{31} + ( - \beta_{5} + 2 \beta_{3} - 5 \beta_{2} - 1) q^{33} + ( - \beta_{4} - \beta_{3} + \beta_1 - 1) q^{35} + (\beta_{5} - 2 \beta_{4} - 2 \beta_{3} - \beta_{2} + 4 \beta_1 + 3) q^{37} + ( - 2 \beta_{5} + 2 \beta_{4} + \beta_{3} - 4 \beta_{2} - 5) q^{39} + ( - \beta_{5} + 2 \beta_{4} - \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 1) q^{41} + ( - 2 \beta_{4} + \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 1) q^{43} + (\beta_{5} + \beta_{2}) q^{45} + (\beta_{5} - 3 \beta_{4} + 5 \beta_{2} + 2 \beta_1 - 1) q^{47} + (\beta_{5} + 2 \beta_{4} - \beta_{3} - 7 \beta_{2} - 6) q^{49} + (\beta_{5} + \beta_{3} - \beta_{2} + 4) q^{51} + (2 \beta_{4} + 2 \beta_{3} - 2 \beta_1 - 2) q^{53} + ( - \beta_{5} + \beta_{2} - 2 \beta_1 + 1) q^{55} + ( - \beta_{5} - \beta_{3} + \beta_{2} - 2 \beta_1 - 4) q^{57} + (2 \beta_{5} - 2 \beta_{2} - 2 \beta_1 - 2) q^{59} + ( - 2 \beta_{5} + 4 \beta_{4} + \beta_{3} + 7 \beta_{2} - 2 \beta_1 + 1) q^{61} + ( - \beta_{3} + 8 \beta_{2} + 3 \beta_1 - 1) q^{63} + (2 \beta_{4} - \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 1) q^{65} + (2 \beta_{5} + 2 \beta_{4} - \beta_{3} - 4 \beta_{2} - \beta_1 - 3) q^{67} + (\beta_{5} - 2 \beta_{3} - 4 \beta_{2} + 1) q^{69} + (\beta_{5} - 2 \beta_{4} - 2 \beta_{3} - \beta_{2} + 4 \beta_1 + 7) q^{71} + 8 q^{73} + ( - \beta_{4} + \beta_1) q^{75} + ( - \beta_{5} - 6 \beta_{4} + 3 \beta_{3} + \beta_{2} - 2 \beta_1 - 2) q^{77} + 2 \beta_{2} q^{79} + (2 \beta_{5} - 4 \beta_{4} - \beta_{2} + 2 \beta_1 + 3) q^{81} + (\beta_{5} - \beta_{4} - \beta_{3} - 7 \beta_{2}) q^{83} + ( - \beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_{2}) q^{85} + ( - 2 \beta_{5} + 3 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 10) q^{87} - 3 q^{89} + ( - 3 \beta_{5} + 4 \beta_{4} + 4 \beta_{3} + 3 \beta_{2} - 10 \beta_1 + 3) q^{91} + (2 \beta_{5} + 2 \beta_{4} - \beta_{3} + 4 \beta_{2} + 5) q^{93} + (\beta_{5} + 2 \beta_{4} - \beta_{3} + \beta_{2} + 2) q^{95} + (2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 6 \beta_{2}) q^{97} + ( - \beta_{5} + 6 \beta_{4} - \beta_{3} + 7 \beta_{2} - 6 \beta_1 + 8) q^{99}+O(q^{100})$$ q - b1 * q^3 + (b2 + 1) * q^5 + (b5 - b4 - b3 + b2) * q^7 + (b3 + b2 + 1) * q^9 + (2*b4 - b3 - 2*b1 + 1) * q^11 + (b5 + 2*b4 - b3 - 3*b2 - 2) * q^13 - b4 * q^15 + (-b5 + b2 - 2*b1 + 1) * q^17 + (b5 - b2 + 2*b1 + 1) * q^19 + (b5 - 2*b4 + b3 - b2 + 2*b1 - 5) * q^21 + (-b5 + b2 + b1 + 1) * q^23 + b2 * q^25 + (-b5 - b4 + 5*b2 - b1 + 3) * q^27 + (-2*b5 + 2*b4 + 2*b3 - b2) * q^29 + (-b5 - 2*b4 + b3 - b2 - 2) * q^31 + (-b5 + 2*b3 - 5*b2 - 1) * q^33 + (-b4 - b3 + b1 - 1) * q^35 + (b5 - 2*b4 - 2*b3 - b2 + 4*b1 + 3) * q^37 + (-2*b5 + 2*b4 + b3 - 4*b2 - 5) * q^39 + (-b5 + 2*b4 - b3 - 2*b2 + 2*b1 - 1) * q^41 + (-2*b4 + b3 + 2*b2 + 2*b1 - 1) * q^43 + (b5 + b2) * q^45 + (b5 - 3*b4 + 5*b2 + 2*b1 - 1) * q^47 + (b5 + 2*b4 - b3 - 7*b2 - 6) * q^49 + (b5 + b3 - b2 + 4) * q^51 + (2*b4 + 2*b3 - 2*b1 - 2) * q^53 + (-b5 + b2 - 2*b1 + 1) * q^55 + (-b5 - b3 + b2 - 2*b1 - 4) * q^57 + (2*b5 - 2*b2 - 2*b1 - 2) * q^59 + (-2*b5 + 4*b4 + b3 + 7*b2 - 2*b1 + 1) * q^61 + (-b3 + 8*b2 + 3*b1 - 1) * q^63 + (2*b4 - b3 - 2*b2 - 2*b1 + 1) * q^65 + (2*b5 + 2*b4 - b3 - 4*b2 - b1 - 3) * q^67 + (b5 - 2*b3 - 4*b2 + 1) * q^69 + (b5 - 2*b4 - 2*b3 - b2 + 4*b1 + 7) * q^71 + 8 * q^73 + (-b4 + b1) * q^75 + (-b5 - 6*b4 + 3*b3 + b2 - 2*b1 - 2) * q^77 + 2*b2 * q^79 + (2*b5 - 4*b4 - b2 + 2*b1 + 3) * q^81 + (b5 - b4 - b3 - 7*b2) * q^83 + (-b5 - 2*b4 + b3 + b2) * q^85 + (-2*b5 + 3*b4 - 2*b3 + 2*b2 - 3*b1 + 10) * q^87 - 3 * q^89 + (-3*b5 + 4*b4 + 4*b3 + 3*b2 - 10*b1 + 3) * q^91 + (2*b5 + 2*b4 - b3 + 4*b2 + 5) * q^93 + (b5 + 2*b4 - b3 + b2 + 2) * q^95 + (2*b5 - 2*b4 - 2*b3 + 6*b2) * q^97 + (-b5 + 6*b4 - b3 + 7*b2 - 6*b1 + 8) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - q^{3} + 3 q^{5} - 3 q^{7} + 5 q^{9}+O(q^{10})$$ 6 * q - q^3 + 3 * q^5 - 3 * q^7 + 5 * q^9 $$6 q - q^{3} + 3 q^{5} - 3 q^{7} + 5 q^{9} - 6 q^{13} + q^{15} + 12 q^{19} - 20 q^{21} + 3 q^{23} - 3 q^{25} + 2 q^{27} + 3 q^{29} - 6 q^{31} + 12 q^{33} - 6 q^{35} + 24 q^{37} - 20 q^{39} - 3 q^{41} - 6 q^{43} - 2 q^{45} - 15 q^{47} - 18 q^{49} + 30 q^{51} - 12 q^{53} - 32 q^{57} - 6 q^{59} - 21 q^{61} - 29 q^{63} + 6 q^{65} - 9 q^{67} + 15 q^{69} + 48 q^{71} + 48 q^{73} + 2 q^{75} - 6 q^{77} - 6 q^{79} + 29 q^{81} + 21 q^{83} + 42 q^{87} - 18 q^{89} + 16 q^{93} + 6 q^{95} - 18 q^{97} + 12 q^{99}+O(q^{100})$$ 6 * q - q^3 + 3 * q^5 - 3 * q^7 + 5 * q^9 - 6 * q^13 + q^15 + 12 * q^19 - 20 * q^21 + 3 * q^23 - 3 * q^25 + 2 * q^27 + 3 * q^29 - 6 * q^31 + 12 * q^33 - 6 * q^35 + 24 * q^37 - 20 * q^39 - 3 * q^41 - 6 * q^43 - 2 * q^45 - 15 * q^47 - 18 * q^49 + 30 * q^51 - 12 * q^53 - 32 * q^57 - 6 * q^59 - 21 * q^61 - 29 * q^63 + 6 * q^65 - 9 * q^67 + 15 * q^69 + 48 * q^71 + 48 * q^73 + 2 * q^75 - 6 * q^77 - 6 * q^79 + 29 * q^81 + 21 * q^83 + 42 * q^87 - 18 * q^89 + 16 * q^93 + 6 * q^95 - 18 * q^97 + 12 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -2\nu^{5} - \nu^{4} - 2\nu^{3} + 12\nu + 9 ) / 27$$ (-2*v^5 - v^4 - 2*v^3 + 12*v + 9) / 27 $$\beta_{3}$$ $$=$$ $$( 2\nu^{5} + \nu^{4} + 2\nu^{3} + 27\nu^{2} - 12\nu - 36 ) / 27$$ (2*v^5 + v^4 + 2*v^3 + 27*v^2 - 12*v - 36) / 27 $$\beta_{4}$$ $$=$$ $$( -\nu^{5} - 2\nu^{4} + 2\nu^{3} + 6\nu + 18 ) / 9$$ (-v^5 - 2*v^4 + 2*v^3 + 6*v + 18) / 9 $$\beta_{5}$$ $$=$$ $$( -7\nu^{5} + \nu^{4} + 11\nu^{3} + 15\nu + 72 ) / 27$$ (-7*v^5 + v^4 + 11*v^3 + 15*v + 72) / 27
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2} + 1$$ b3 + b2 + 1 $$\nu^{3}$$ $$=$$ $$\beta_{5} + \beta_{4} - 5\beta_{2} + \beta _1 - 3$$ b5 + b4 - 5*b2 + b1 - 3 $$\nu^{4}$$ $$=$$ $$2\beta_{5} - 4\beta_{4} - \beta_{2} + 2\beta _1 + 3$$ 2*b5 - 4*b4 - b2 + 2*b1 + 3 $$\nu^{5}$$ $$=$$ $$-2\beta_{5} + \beta_{4} - 8\beta_{2} + 4\beta _1 + 6$$ -2*b5 + b4 - 8*b2 + 4*b1 + 6

## Character values

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

 $$n$$ $$37$$ $$91$$ $$101$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1 - \beta_{2}$$

## 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
 1.71903 − 0.211943i 0.403374 + 1.68443i −1.62241 − 0.606458i 1.71903 + 0.211943i 0.403374 − 1.68443i −1.62241 + 0.606458i
0 −1.71903 + 0.211943i 0 0.500000 + 0.866025i 0 −1.36710 + 2.36788i 0 2.91016 0.728674i 0
61.2 0 −0.403374 1.68443i 0 0.500000 + 0.866025i 0 1.91751 3.32123i 0 −2.67458 + 1.35891i 0
61.3 0 1.62241 + 0.606458i 0 0.500000 + 0.866025i 0 −2.05042 + 3.55142i 0 2.26442 + 1.96784i 0
121.1 0 −1.71903 0.211943i 0 0.500000 0.866025i 0 −1.36710 2.36788i 0 2.91016 + 0.728674i 0
121.2 0 −0.403374 + 1.68443i 0 0.500000 0.866025i 0 1.91751 + 3.32123i 0 −2.67458 1.35891i 0
121.3 0 1.62241 0.606458i 0 0.500000 0.866025i 0 −2.05042 3.55142i 0 2.26442 1.96784i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 61.3 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.b 6
3.b odd 2 1 540.2.i.b 6
4.b odd 2 1 720.2.q.k 6
5.b even 2 1 900.2.i.c 6
5.c odd 4 2 900.2.s.c 12
9.c even 3 1 inner 180.2.i.b 6
9.c even 3 1 1620.2.a.i 3
9.d odd 6 1 540.2.i.b 6
9.d odd 6 1 1620.2.a.j 3
12.b even 2 1 2160.2.q.i 6
15.d odd 2 1 2700.2.i.c 6
15.e even 4 2 2700.2.s.c 12
36.f odd 6 1 720.2.q.k 6
36.f odd 6 1 6480.2.a.bt 3
36.h even 6 1 2160.2.q.i 6
36.h even 6 1 6480.2.a.bw 3
45.h odd 6 1 2700.2.i.c 6
45.h odd 6 1 8100.2.a.u 3
45.j even 6 1 900.2.i.c 6
45.j even 6 1 8100.2.a.v 3
45.k odd 12 2 900.2.s.c 12
45.k odd 12 2 8100.2.d.p 6
45.l even 12 2 2700.2.s.c 12
45.l even 12 2 8100.2.d.o 6

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{6} + 3T_{7}^{5} + 24T_{7}^{4} + 41T_{7}^{3} + 354T_{7}^{2} + 645T_{7} + 1849$$ acting on $$S_{2}^{\mathrm{new}}(180, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6} + T^{5} - 2 T^{4} - 3 T^{3} + \cdots + 27$$
$5$ $$(T^{2} - T + 1)^{3}$$
$7$ $$T^{6} + 3 T^{5} + 24 T^{4} + \cdots + 1849$$
$11$ $$T^{6} + 24 T^{4} - 72 T^{3} + \cdots + 1296$$
$13$ $$T^{6} + 6 T^{5} + 48 T^{4} + \cdots + 5776$$
$17$ $$(T^{3} - 24 T + 36)^{2}$$
$19$ $$(T^{3} - 6 T^{2} - 12 T + 4)^{2}$$
$23$ $$T^{6} - 3 T^{5} + 24 T^{4} + 63 T^{3} + \cdots + 81$$
$29$ $$T^{6} - 3 T^{5} + 78 T^{4} + \cdots + 77841$$
$31$ $$T^{6} + 6 T^{5} + 48 T^{4} - 64 T^{3} + \cdots + 16$$
$37$ $$(T^{3} - 12 T^{2} - 36 T + 436)^{2}$$
$41$ $$T^{6} + 3 T^{5} + 90 T^{4} + \cdots + 6561$$
$43$ $$T^{6} + 6 T^{5} + 48 T^{4} + \cdots + 5776$$
$47$ $$T^{6} + 15 T^{5} + 186 T^{4} + \cdots + 729$$
$53$ $$(T^{3} + 6 T^{2} - 60 T + 72)^{2}$$
$59$ $$T^{6} + 6 T^{5} + 96 T^{4} + \cdots + 5184$$
$61$ $$T^{6} + 21 T^{5} + 378 T^{4} + \cdots + 167281$$
$67$ $$T^{6} + 9 T^{5} + 102 T^{4} + \cdots + 22801$$
$71$ $$(T^{3} - 24 T^{2} + 108 T + 324)^{2}$$
$73$ $$(T - 8)^{6}$$
$79$ $$(T^{2} + 2 T + 4)^{3}$$
$83$ $$T^{6} - 21 T^{5} + 312 T^{4} + \cdots + 59049$$
$89$ $$(T + 3)^{6}$$
$97$ $$T^{6} + 18 T^{5} + 288 T^{4} + \cdots + 179776$$