# Properties

 Label 180.3.b.a Level $180$ Weight $3$ Character orbit 180.b Analytic conductor $4.905$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [180,3,Mod(89,180)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(180, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("180.89");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$180 = 2^{2} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 180.b (of order $$2$$, degree $$1$$, 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: $$4.90464475849$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{23})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 24x^{2} + 121$$ x^4 + 24*x^2 + 121 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{3} - \beta_1) q^{5} - \beta_{2} q^{7}+O(q^{10})$$ q + (b3 - b1) * q^5 - b2 * q^7 $$q + (\beta_{3} - \beta_1) q^{5} - \beta_{2} q^{7} + 7 \beta_1 q^{11} - 3 \beta_{2} q^{13} + 4 \beta_{3} q^{17} + 12 q^{19} + 2 \beta_{3} q^{23} + ( - 2 \beta_{2} + 21) q^{25} - 6 \beta_1 q^{29} - 38 q^{31} + ( - 2 \beta_{3} - 23 \beta_1) q^{35} - \beta_{2} q^{37} + 49 \beta_1 q^{41} + 10 \beta_{2} q^{43} - 16 \beta_{3} q^{47} + 3 q^{49} + (7 \beta_{2} + 14) q^{55} - 59 \beta_1 q^{59} - 70 q^{61} + ( - 6 \beta_{3} - 69 \beta_1) q^{65} + 16 \beta_{2} q^{67} + 84 \beta_1 q^{71} - 2 \beta_{2} q^{73} + 14 \beta_{3} q^{77} + 30 q^{79} - 28 \beta_{3} q^{83} + ( - 4 \beta_{2} + 92) q^{85} + 23 \beta_1 q^{89} - 138 q^{91} + (12 \beta_{3} - 12 \beta_1) q^{95} - 14 \beta_{2} q^{97}+O(q^{100})$$ q + (b3 - b1) * q^5 - b2 * q^7 + 7*b1 * q^11 - 3*b2 * q^13 + 4*b3 * q^17 + 12 * q^19 + 2*b3 * q^23 + (-2*b2 + 21) * q^25 - 6*b1 * q^29 - 38 * q^31 + (-2*b3 - 23*b1) * q^35 - b2 * q^37 + 49*b1 * q^41 + 10*b2 * q^43 - 16*b3 * q^47 + 3 * q^49 + (7*b2 + 14) * q^55 - 59*b1 * q^59 - 70 * q^61 + (-6*b3 - 69*b1) * q^65 + 16*b2 * q^67 + 84*b1 * q^71 - 2*b2 * q^73 + 14*b3 * q^77 + 30 * q^79 - 28*b3 * q^83 + (-4*b2 + 92) * q^85 + 23*b1 * q^89 - 138 * q^91 + (12*b3 - 12*b1) * q^95 - 14*b2 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q + 48 q^{19} + 84 q^{25} - 152 q^{31} + 12 q^{49} + 56 q^{55} - 280 q^{61} + 120 q^{79} + 368 q^{85} - 552 q^{91}+O(q^{100})$$ 4 * q + 48 * q^19 + 84 * q^25 - 152 * q^31 + 12 * q^49 + 56 * q^55 - 280 * q^61 + 120 * q^79 + 368 * q^85 - 552 * q^91

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 24x^{2} + 121$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + 13\nu ) / 11$$ (v^3 + 13*v) / 11 $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 35\nu ) / 11$$ (v^3 + 35*v) / 11 $$\beta_{3}$$ $$=$$ $$\nu^{2} + 12$$ v^2 + 12
 $$\nu$$ $$=$$ $$( \beta_{2} - \beta_1 ) / 2$$ (b2 - b1) / 2 $$\nu^{2}$$ $$=$$ $$\beta_{3} - 12$$ b3 - 12 $$\nu^{3}$$ $$=$$ $$( -13\beta_{2} + 35\beta_1 ) / 2$$ (-13*b2 + 35*b1) / 2

## Character values

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

 $$n$$ $$37$$ $$91$$ $$101$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
89.1
 − 4.09827i 4.09827i 2.68406i − 2.68406i
0 0 0 −4.79583 1.41421i 0 6.78233i 0 0 0
89.2 0 0 0 −4.79583 + 1.41421i 0 6.78233i 0 0 0
89.3 0 0 0 4.79583 1.41421i 0 6.78233i 0 0 0
89.4 0 0 0 4.79583 + 1.41421i 0 6.78233i 0 0 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
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 180.3.b.a 4
3.b odd 2 1 inner 180.3.b.a 4
4.b odd 2 1 720.3.c.b 4
5.b even 2 1 inner 180.3.b.a 4
5.c odd 4 2 900.3.g.c 4
8.b even 2 1 2880.3.c.c 4
8.d odd 2 1 2880.3.c.f 4
9.c even 3 2 1620.3.t.c 8
9.d odd 6 2 1620.3.t.c 8
12.b even 2 1 720.3.c.b 4
15.d odd 2 1 inner 180.3.b.a 4
15.e even 4 2 900.3.g.c 4
20.d odd 2 1 720.3.c.b 4
20.e even 4 2 3600.3.l.r 4
24.f even 2 1 2880.3.c.f 4
24.h odd 2 1 2880.3.c.c 4
40.e odd 2 1 2880.3.c.f 4
40.f even 2 1 2880.3.c.c 4
45.h odd 6 2 1620.3.t.c 8
45.j even 6 2 1620.3.t.c 8
60.h even 2 1 720.3.c.b 4
60.l odd 4 2 3600.3.l.r 4
120.i odd 2 1 2880.3.c.c 4
120.m even 2 1 2880.3.c.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.3.b.a 4 1.a even 1 1 trivial
180.3.b.a 4 3.b odd 2 1 inner
180.3.b.a 4 5.b even 2 1 inner
180.3.b.a 4 15.d odd 2 1 inner
720.3.c.b 4 4.b odd 2 1
720.3.c.b 4 12.b even 2 1
720.3.c.b 4 20.d odd 2 1
720.3.c.b 4 60.h even 2 1
900.3.g.c 4 5.c odd 4 2
900.3.g.c 4 15.e even 4 2
1620.3.t.c 8 9.c even 3 2
1620.3.t.c 8 9.d odd 6 2
1620.3.t.c 8 45.h odd 6 2
1620.3.t.c 8 45.j even 6 2
2880.3.c.c 4 8.b even 2 1
2880.3.c.c 4 24.h odd 2 1
2880.3.c.c 4 40.f even 2 1
2880.3.c.c 4 120.i odd 2 1
2880.3.c.f 4 8.d odd 2 1
2880.3.c.f 4 24.f even 2 1
2880.3.c.f 4 40.e odd 2 1
2880.3.c.f 4 120.m even 2 1
3600.3.l.r 4 20.e even 4 2
3600.3.l.r 4 60.l odd 4 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(180, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4} - 42T^{2} + 625$$
$7$ $$(T^{2} + 46)^{2}$$
$11$ $$(T^{2} + 98)^{2}$$
$13$ $$(T^{2} + 414)^{2}$$
$17$ $$(T^{2} - 368)^{2}$$
$19$ $$(T - 12)^{4}$$
$23$ $$(T^{2} - 92)^{2}$$
$29$ $$(T^{2} + 72)^{2}$$
$31$ $$(T + 38)^{4}$$
$37$ $$(T^{2} + 46)^{2}$$
$41$ $$(T^{2} + 4802)^{2}$$
$43$ $$(T^{2} + 4600)^{2}$$
$47$ $$(T^{2} - 5888)^{2}$$
$53$ $$T^{4}$$
$59$ $$(T^{2} + 6962)^{2}$$
$61$ $$(T + 70)^{4}$$
$67$ $$(T^{2} + 11776)^{2}$$
$71$ $$(T^{2} + 14112)^{2}$$
$73$ $$(T^{2} + 184)^{2}$$
$79$ $$(T - 30)^{4}$$
$83$ $$(T^{2} - 18032)^{2}$$
$89$ $$(T^{2} + 1058)^{2}$$
$97$ $$(T^{2} + 9016)^{2}$$