# Properties

 Label 180.3.l.a Level $180$ Weight $3$ Character orbit 180.l Analytic conductor $4.905$ 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] = [180,3,Mod(37,180)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(180, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("180.37");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$180 = 2^{2} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 180.l (of order $$4$$, 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: $$4.90464475849$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 20) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - 4 i + 3) q^{5} + ( - 7 i - 7) q^{7}+O(q^{10})$$ q + (-4*i + 3) * q^5 + (-7*i - 7) * q^7 $$q + ( - 4 i + 3) q^{5} + ( - 7 i - 7) q^{7} - 10 q^{11} + ( - 9 i + 9) q^{13} + ( - i - 1) q^{17} - 8 i q^{19} + ( - 23 i + 23) q^{23} + ( - 24 i - 7) q^{25} + 8 i q^{29} - 14 q^{31} + (7 i - 49) q^{35} + (33 i + 33) q^{37} + 14 q^{41} + (15 i - 15) q^{43} + (39 i + 39) q^{47} + 49 i q^{49} + ( - 7 i + 7) q^{53} + (40 i - 30) q^{55} + 56 i q^{59} + 42 q^{61} + ( - 63 i - 9) q^{65} + ( - 7 i - 7) q^{67} - 98 q^{71} + ( - 49 i + 49) q^{73} + (70 i + 70) q^{77} + 96 i q^{79} + ( - 63 i + 63) q^{83} + (i - 7) q^{85} + 112 i q^{89} - 126 q^{91} + ( - 24 i - 32) q^{95} + (33 i + 33) q^{97} +O(q^{100})$$ q + (-4*i + 3) * q^5 + (-7*i - 7) * q^7 - 10 * q^11 + (-9*i + 9) * q^13 + (-i - 1) * q^17 - 8*i * q^19 + (-23*i + 23) * q^23 + (-24*i - 7) * q^25 + 8*i * q^29 - 14 * q^31 + (7*i - 49) * q^35 + (33*i + 33) * q^37 + 14 * q^41 + (15*i - 15) * q^43 + (39*i + 39) * q^47 + 49*i * q^49 + (-7*i + 7) * q^53 + (40*i - 30) * q^55 + 56*i * q^59 + 42 * q^61 + (-63*i - 9) * q^65 + (-7*i - 7) * q^67 - 98 * q^71 + (-49*i + 49) * q^73 + (70*i + 70) * q^77 + 96*i * q^79 + (-63*i + 63) * q^83 + (i - 7) * q^85 + 112*i * q^89 - 126 * q^91 + (-24*i - 32) * q^95 + (33*i + 33) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{5} - 14 q^{7}+O(q^{10})$$ 2 * q + 6 * q^5 - 14 * q^7 $$2 q + 6 q^{5} - 14 q^{7} - 20 q^{11} + 18 q^{13} - 2 q^{17} + 46 q^{23} - 14 q^{25} - 28 q^{31} - 98 q^{35} + 66 q^{37} + 28 q^{41} - 30 q^{43} + 78 q^{47} + 14 q^{53} - 60 q^{55} + 84 q^{61} - 18 q^{65} - 14 q^{67} - 196 q^{71} + 98 q^{73} + 140 q^{77} + 126 q^{83} - 14 q^{85} - 252 q^{91} - 64 q^{95} + 66 q^{97}+O(q^{100})$$ 2 * q + 6 * q^5 - 14 * q^7 - 20 * q^11 + 18 * q^13 - 2 * q^17 + 46 * q^23 - 14 * q^25 - 28 * q^31 - 98 * q^35 + 66 * q^37 + 28 * q^41 - 30 * q^43 + 78 * q^47 + 14 * q^53 - 60 * q^55 + 84 * q^61 - 18 * q^65 - 14 * q^67 - 196 * q^71 + 98 * q^73 + 140 * q^77 + 126 * q^83 - 14 * q^85 - 252 * q^91 - 64 * q^95 + 66 * q^97

## Character values

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

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1
 1.00000i − 1.00000i
0 0 0 3.00000 4.00000i 0 −7.00000 7.00000i 0 0 0
73.1 0 0 0 3.00000 + 4.00000i 0 −7.00000 + 7.00000i 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
5.c odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 180.3.l.a 2
3.b odd 2 1 20.3.f.a 2
4.b odd 2 1 720.3.bh.e 2
5.b even 2 1 900.3.l.a 2
5.c odd 4 1 inner 180.3.l.a 2
5.c odd 4 1 900.3.l.a 2
12.b even 2 1 80.3.p.a 2
15.d odd 2 1 100.3.f.a 2
15.e even 4 1 20.3.f.a 2
15.e even 4 1 100.3.f.a 2
20.e even 4 1 720.3.bh.e 2
21.c even 2 1 980.3.l.a 2
24.f even 2 1 320.3.p.g 2
24.h odd 2 1 320.3.p.c 2
60.h even 2 1 400.3.p.d 2
60.l odd 4 1 80.3.p.a 2
60.l odd 4 1 400.3.p.d 2
105.k odd 4 1 980.3.l.a 2
120.q odd 4 1 320.3.p.g 2
120.w even 4 1 320.3.p.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.3.f.a 2 3.b odd 2 1
20.3.f.a 2 15.e even 4 1
80.3.p.a 2 12.b even 2 1
80.3.p.a 2 60.l odd 4 1
100.3.f.a 2 15.d odd 2 1
100.3.f.a 2 15.e even 4 1
180.3.l.a 2 1.a even 1 1 trivial
180.3.l.a 2 5.c odd 4 1 inner
320.3.p.c 2 24.h odd 2 1
320.3.p.c 2 120.w even 4 1
320.3.p.g 2 24.f even 2 1
320.3.p.g 2 120.q odd 4 1
400.3.p.d 2 60.h even 2 1
400.3.p.d 2 60.l odd 4 1
720.3.bh.e 2 4.b odd 2 1
720.3.bh.e 2 20.e even 4 1
900.3.l.a 2 5.b even 2 1
900.3.l.a 2 5.c odd 4 1
980.3.l.a 2 21.c even 2 1
980.3.l.a 2 105.k odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{2} + 14T_{7} + 98$$ acting on $$S_{3}^{\mathrm{new}}(180, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 6T + 25$$
$7$ $$T^{2} + 14T + 98$$
$11$ $$(T + 10)^{2}$$
$13$ $$T^{2} - 18T + 162$$
$17$ $$T^{2} + 2T + 2$$
$19$ $$T^{2} + 64$$
$23$ $$T^{2} - 46T + 1058$$
$29$ $$T^{2} + 64$$
$31$ $$(T + 14)^{2}$$
$37$ $$T^{2} - 66T + 2178$$
$41$ $$(T - 14)^{2}$$
$43$ $$T^{2} + 30T + 450$$
$47$ $$T^{2} - 78T + 3042$$
$53$ $$T^{2} - 14T + 98$$
$59$ $$T^{2} + 3136$$
$61$ $$(T - 42)^{2}$$
$67$ $$T^{2} + 14T + 98$$
$71$ $$(T + 98)^{2}$$
$73$ $$T^{2} - 98T + 4802$$
$79$ $$T^{2} + 9216$$
$83$ $$T^{2} - 126T + 7938$$
$89$ $$T^{2} + 12544$$
$97$ $$T^{2} - 66T + 2178$$