# Properties

 Label 180.3.f.a Level $180$ Weight $3$ Character orbit 180.f Self dual yes Analytic conductor $4.905$ Analytic rank $0$ Dimension $1$ CM discriminant -20 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [180,3,Mod(19,180)]

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([1, 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("180.19");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$180 = 2^{2} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 180.f (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: yes Analytic conductor: $$4.90464475849$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 20) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - 2 q^{2} + 4 q^{4} + 5 q^{5} + 4 q^{7} - 8 q^{8}+O(q^{10})$$ q - 2 * q^2 + 4 * q^4 + 5 * q^5 + 4 * q^7 - 8 * q^8 $$q - 2 q^{2} + 4 q^{4} + 5 q^{5} + 4 q^{7} - 8 q^{8} - 10 q^{10} - 8 q^{14} + 16 q^{16} + 20 q^{20} + 44 q^{23} + 25 q^{25} + 16 q^{28} + 22 q^{29} - 32 q^{32} + 20 q^{35} - 40 q^{40} - 62 q^{41} + 76 q^{43} - 88 q^{46} - 4 q^{47} - 33 q^{49} - 50 q^{50} - 32 q^{56} - 44 q^{58} - 58 q^{61} + 64 q^{64} - 116 q^{67} - 40 q^{70} + 80 q^{80} + 124 q^{82} - 76 q^{83} - 152 q^{86} + 142 q^{89} + 176 q^{92} + 8 q^{94} + 66 q^{98}+O(q^{100})$$ q - 2 * q^2 + 4 * q^4 + 5 * q^5 + 4 * q^7 - 8 * q^8 - 10 * q^10 - 8 * q^14 + 16 * q^16 + 20 * q^20 + 44 * q^23 + 25 * q^25 + 16 * q^28 + 22 * q^29 - 32 * q^32 + 20 * q^35 - 40 * q^40 - 62 * q^41 + 76 * q^43 - 88 * q^46 - 4 * q^47 - 33 * q^49 - 50 * q^50 - 32 * q^56 - 44 * q^58 - 58 * q^61 + 64 * q^64 - 116 * q^67 - 40 * q^70 + 80 * q^80 + 124 * q^82 - 76 * q^83 - 152 * q^86 + 142 * q^89 + 176 * q^92 + 8 * q^94 + 66 * q^98

## Character values

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

 $$n$$ $$37$$ $$91$$ $$101$$ $$\chi(n)$$ $$1$$ $$1$$ $$0$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 0
−2.00000 0 4.00000 5.00000 0 4.00000 −8.00000 0 −10.0000
 $$n$$: 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 $$\Q(\sqrt{-5})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 180.3.f.a 1
3.b odd 2 1 20.3.d.b yes 1
4.b odd 2 1 180.3.f.b 1
5.b even 2 1 180.3.f.b 1
5.c odd 4 2 900.3.c.h 2
12.b even 2 1 20.3.d.a 1
15.d odd 2 1 20.3.d.a 1
15.e even 4 2 100.3.b.c 2
20.d odd 2 1 CM 180.3.f.a 1
20.e even 4 2 900.3.c.h 2
24.f even 2 1 320.3.h.a 1
24.h odd 2 1 320.3.h.b 1
48.i odd 4 2 1280.3.e.b 2
48.k even 4 2 1280.3.e.c 2
60.h even 2 1 20.3.d.b yes 1
60.l odd 4 2 100.3.b.c 2
120.i odd 2 1 320.3.h.a 1
120.m even 2 1 320.3.h.b 1
120.q odd 4 2 1600.3.b.f 2
120.w even 4 2 1600.3.b.f 2
240.t even 4 2 1280.3.e.b 2
240.bm odd 4 2 1280.3.e.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.3.d.a 1 12.b even 2 1
20.3.d.a 1 15.d odd 2 1
20.3.d.b yes 1 3.b odd 2 1
20.3.d.b yes 1 60.h even 2 1
100.3.b.c 2 15.e even 4 2
100.3.b.c 2 60.l odd 4 2
180.3.f.a 1 1.a even 1 1 trivial
180.3.f.a 1 20.d odd 2 1 CM
180.3.f.b 1 4.b odd 2 1
180.3.f.b 1 5.b even 2 1
320.3.h.a 1 24.f even 2 1
320.3.h.a 1 120.i odd 2 1
320.3.h.b 1 24.h odd 2 1
320.3.h.b 1 120.m even 2 1
900.3.c.h 2 5.c odd 4 2
900.3.c.h 2 20.e even 4 2
1280.3.e.b 2 48.i odd 4 2
1280.3.e.b 2 240.t even 4 2
1280.3.e.c 2 48.k even 4 2
1280.3.e.c 2 240.bm odd 4 2
1600.3.b.f 2 120.q odd 4 2
1600.3.b.f 2 120.w even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(180, [\chi])$$:

 $$T_{7} - 4$$ T7 - 4 $$T_{13}$$ T13 $$T_{23} - 44$$ T23 - 44

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 2$$
$3$ $$T$$
$5$ $$T - 5$$
$7$ $$T - 4$$
$11$ $$T$$
$13$ $$T$$
$17$ $$T$$
$19$ $$T$$
$23$ $$T - 44$$
$29$ $$T - 22$$
$31$ $$T$$
$37$ $$T$$
$41$ $$T + 62$$
$43$ $$T - 76$$
$47$ $$T + 4$$
$53$ $$T$$
$59$ $$T$$
$61$ $$T + 58$$
$67$ $$T + 116$$
$71$ $$T$$
$73$ $$T$$
$79$ $$T$$
$83$ $$T + 76$$
$89$ $$T - 142$$
$97$ $$T$$