# Properties

 Label 180.2.k.a Level $180$ Weight $2$ Character orbit 180.k Analytic conductor $1.437$ Analytic rank $0$ Dimension $2$ CM discriminant -4 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [180,2,Mod(127,180)]

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([2, 0, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("180.127");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$180 = 2^{2} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 180.k (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: $$1.43730723638$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - i - 1) q^{2} + 2 i q^{4} + ( - 2 i + 1) q^{5} + ( - 2 i + 2) q^{8}+O(q^{10})$$ q + (-i - 1) * q^2 + 2*i * q^4 + (-2*i + 1) * q^5 + (-2*i + 2) * q^8 $$q + ( - i - 1) q^{2} + 2 i q^{4} + ( - 2 i + 1) q^{5} + ( - 2 i + 2) q^{8} + (i - 3) q^{10} + ( - 5 i + 5) q^{13} - 4 q^{16} + ( - 3 i - 3) q^{17} + (2 i + 4) q^{20} + ( - 4 i - 3) q^{25} - 10 q^{26} + 10 i q^{29} + (4 i + 4) q^{32} + 6 i q^{34} + (5 i + 5) q^{37} + ( - 6 i - 2) q^{40} + 10 q^{41} + 7 i q^{49} + (7 i - 1) q^{50} + (10 i + 10) q^{52} + (9 i - 9) q^{53} + ( - 10 i + 10) q^{58} - 12 q^{61} - 8 i q^{64} + ( - 15 i - 5) q^{65} + ( - 6 i + 6) q^{68} + (5 i - 5) q^{73} - 10 i q^{74} + (8 i - 4) q^{80} + ( - 10 i - 10) q^{82} + (3 i - 9) q^{85} - 10 i q^{89} + ( - 5 i - 5) q^{97} + ( - 7 i + 7) q^{98} +O(q^{100})$$ q + (-i - 1) * q^2 + 2*i * q^4 + (-2*i + 1) * q^5 + (-2*i + 2) * q^8 + (i - 3) * q^10 + (-5*i + 5) * q^13 - 4 * q^16 + (-3*i - 3) * q^17 + (2*i + 4) * q^20 + (-4*i - 3) * q^25 - 10 * q^26 + 10*i * q^29 + (4*i + 4) * q^32 + 6*i * q^34 + (5*i + 5) * q^37 + (-6*i - 2) * q^40 + 10 * q^41 + 7*i * q^49 + (7*i - 1) * q^50 + (10*i + 10) * q^52 + (9*i - 9) * q^53 + (-10*i + 10) * q^58 - 12 * q^61 - 8*i * q^64 + (-15*i - 5) * q^65 + (-6*i + 6) * q^68 + (5*i - 5) * q^73 - 10*i * q^74 + (8*i - 4) * q^80 + (-10*i - 10) * q^82 + (3*i - 9) * q^85 - 10*i * q^89 + (-5*i - 5) * q^97 + (-7*i + 7) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{5} + 4 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^5 + 4 * q^8 $$2 q - 2 q^{2} + 2 q^{5} + 4 q^{8} - 6 q^{10} + 10 q^{13} - 8 q^{16} - 6 q^{17} + 8 q^{20} - 6 q^{25} - 20 q^{26} + 8 q^{32} + 10 q^{37} - 4 q^{40} + 20 q^{41} - 2 q^{50} + 20 q^{52} - 18 q^{53} + 20 q^{58} - 24 q^{61} - 10 q^{65} + 12 q^{68} - 10 q^{73} - 8 q^{80} - 20 q^{82} - 18 q^{85} - 10 q^{97} + 14 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^5 + 4 * q^8 - 6 * q^10 + 10 * q^13 - 8 * q^16 - 6 * q^17 + 8 * q^20 - 6 * q^25 - 20 * q^26 + 8 * q^32 + 10 * q^37 - 4 * q^40 + 20 * q^41 - 2 * q^50 + 20 * q^52 - 18 * q^53 + 20 * q^58 - 24 * q^61 - 10 * q^65 + 12 * q^68 - 10 * q^73 - 8 * q^80 - 20 * q^82 - 18 * q^85 - 10 * q^97 + 14 * q^98

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
127.1
 1.00000i − 1.00000i
−1.00000 1.00000i 0 2.00000i 1.00000 2.00000i 0 0 2.00000 2.00000i 0 −3.00000 + 1.00000i
163.1 −1.00000 + 1.00000i 0 2.00000i 1.00000 + 2.00000i 0 0 2.00000 + 2.00000i 0 −3.00000 1.00000i
 $$n$$: 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
5.c odd 4 1 inner
20.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 180.2.k.a 2
3.b odd 2 1 180.2.k.b yes 2
4.b odd 2 1 CM 180.2.k.a 2
5.b even 2 1 900.2.k.d 2
5.c odd 4 1 inner 180.2.k.a 2
5.c odd 4 1 900.2.k.d 2
12.b even 2 1 180.2.k.b yes 2
15.d odd 2 1 900.2.k.b 2
15.e even 4 1 180.2.k.b yes 2
15.e even 4 1 900.2.k.b 2
20.d odd 2 1 900.2.k.d 2
20.e even 4 1 inner 180.2.k.a 2
20.e even 4 1 900.2.k.d 2
60.h even 2 1 900.2.k.b 2
60.l odd 4 1 180.2.k.b yes 2
60.l odd 4 1 900.2.k.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.2.k.a 2 1.a even 1 1 trivial
180.2.k.a 2 4.b odd 2 1 CM
180.2.k.a 2 5.c odd 4 1 inner
180.2.k.a 2 20.e even 4 1 inner
180.2.k.b yes 2 3.b odd 2 1
180.2.k.b yes 2 12.b even 2 1
180.2.k.b yes 2 15.e even 4 1
180.2.k.b yes 2 60.l odd 4 1
900.2.k.b 2 15.d odd 2 1
900.2.k.b 2 15.e even 4 1
900.2.k.b 2 60.h even 2 1
900.2.k.b 2 60.l odd 4 1
900.2.k.d 2 5.b even 2 1
900.2.k.d 2 5.c odd 4 1
900.2.k.d 2 20.d odd 2 1
900.2.k.d 2 20.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(180, [\chi])$$:

 $$T_{7}$$ T7 $$T_{13}^{2} - 10T_{13} + 50$$ T13^2 - 10*T13 + 50 $$T_{17}^{2} + 6T_{17} + 18$$ T17^2 + 6*T17 + 18

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T + 2$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 2T + 5$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 10T + 50$$
$17$ $$T^{2} + 6T + 18$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2} + 100$$
$31$ $$T^{2}$$
$37$ $$T^{2} - 10T + 50$$
$41$ $$(T - 10)^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 18T + 162$$
$59$ $$T^{2}$$
$61$ $$(T + 12)^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 10T + 50$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2} + 100$$
$97$ $$T^{2} + 10T + 50$$