# Properties

 Label 180.2.w.a Level $180$ Weight $2$ Character orbit 180.w Analytic conductor $1.437$ Analytic rank $0$ Dimension $24$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [180,2,Mod(77,180)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(180, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([0, 10, 3]))

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

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

chi := DirichletCharacter("180.77");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$180 = 2^{2} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 180.w (of order $$12$$, degree $$4$$, 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: $$24$$ Relative dimension: $$6$$ over $$\Q(\zeta_{12})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$24 q - 2 q^{3}+O(q^{10})$$ 24 * q - 2 * q^3 $$\operatorname{Tr}(f)(q) =$$ $$24 q - 2 q^{3} + 12 q^{11} + 10 q^{15} + 4 q^{21} - 24 q^{23} + 6 q^{25} - 26 q^{27} - 26 q^{33} + 12 q^{37} - 36 q^{41} - 10 q^{45} - 42 q^{47} - 76 q^{51} + 12 q^{55} - 14 q^{57} - 12 q^{61} + 34 q^{63} - 24 q^{65} + 6 q^{67} + 26 q^{75} + 96 q^{77} + 56 q^{81} + 60 q^{83} - 24 q^{85} + 88 q^{87} - 24 q^{91} + 52 q^{93} + 60 q^{95} - 18 q^{97}+O(q^{100})$$ 24 * q - 2 * q^3 + 12 * q^11 + 10 * q^15 + 4 * q^21 - 24 * q^23 + 6 * q^25 - 26 * q^27 - 26 * q^33 + 12 * q^37 - 36 * q^41 - 10 * q^45 - 42 * q^47 - 76 * q^51 + 12 * q^55 - 14 * q^57 - 12 * q^61 + 34 * q^63 - 24 * q^65 + 6 * q^67 + 26 * q^75 + 96 * q^77 + 56 * q^81 + 60 * q^83 - 24 * q^85 + 88 * q^87 - 24 * q^91 + 52 * q^93 + 60 * q^95 - 18 * q^97

## 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
77.1 0 −1.69800 0.341771i 0 1.88305 + 1.20587i 0 −0.165071 + 0.616053i 0 2.76639 + 1.16065i 0
77.2 0 −1.60329 + 0.655337i 0 −1.86002 1.24110i 0 0.901926 3.36603i 0 2.14107 2.10139i 0
77.3 0 0.438358 1.67566i 0 1.00297 1.99851i 0 −0.290218 + 1.08311i 0 −2.61569 1.46908i 0
77.4 0 0.483525 + 1.66319i 0 −2.23512 0.0652184i 0 −1.23944 + 4.62566i 0 −2.53241 + 1.60839i 0
77.5 0 1.04282 + 1.38294i 0 1.58055 1.58172i 0 1.04520 3.90075i 0 −0.825048 + 2.88432i 0
77.6 0 1.70261 0.318010i 0 0.494584 + 2.18068i 0 −0.252396 + 0.941955i 0 2.79774 1.08289i 0
113.1 0 −1.66319 + 0.483525i 0 −1.17404 1.90306i 0 4.62566 + 1.23944i 0 2.53241 1.60839i 0
113.2 0 −1.38294 + 1.04282i 0 −0.579537 + 2.15966i 0 −3.90075 1.04520i 0 0.825048 2.88432i 0
113.3 0 −0.655337 1.60329i 0 −2.00484 0.990270i 0 −3.36603 0.901926i 0 −2.14107 + 2.10139i 0
113.4 0 0.318010 + 1.70261i 0 2.13582 0.662020i 0 0.941955 + 0.252396i 0 −2.79774 + 1.08289i 0
113.5 0 0.341771 1.69800i 0 1.98584 + 1.02783i 0 0.616053 + 0.165071i 0 −2.76639 1.16065i 0
113.6 0 1.67566 + 0.438358i 0 −1.22927 + 1.86786i 0 1.08311 + 0.290218i 0 2.61569 + 1.46908i 0
137.1 0 −1.66319 0.483525i 0 −1.17404 + 1.90306i 0 4.62566 1.23944i 0 2.53241 + 1.60839i 0
137.2 0 −1.38294 1.04282i 0 −0.579537 2.15966i 0 −3.90075 + 1.04520i 0 0.825048 + 2.88432i 0
137.3 0 −0.655337 + 1.60329i 0 −2.00484 + 0.990270i 0 −3.36603 + 0.901926i 0 −2.14107 2.10139i 0
137.4 0 0.318010 1.70261i 0 2.13582 + 0.662020i 0 0.941955 0.252396i 0 −2.79774 1.08289i 0
137.5 0 0.341771 + 1.69800i 0 1.98584 1.02783i 0 0.616053 0.165071i 0 −2.76639 + 1.16065i 0
137.6 0 1.67566 0.438358i 0 −1.22927 1.86786i 0 1.08311 0.290218i 0 2.61569 1.46908i 0
173.1 0 −1.69800 + 0.341771i 0 1.88305 1.20587i 0 −0.165071 0.616053i 0 2.76639 1.16065i 0
173.2 0 −1.60329 0.655337i 0 −1.86002 + 1.24110i 0 0.901926 + 3.36603i 0 2.14107 + 2.10139i 0
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 77.6 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
9.d odd 6 1 inner
45.l even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 180.2.w.a 24
3.b odd 2 1 540.2.x.a 24
4.b odd 2 1 720.2.cu.d 24
5.b even 2 1 900.2.be.e 24
5.c odd 4 1 inner 180.2.w.a 24
5.c odd 4 1 900.2.be.e 24
9.c even 3 1 540.2.x.a 24
9.c even 3 1 1620.2.j.b 24
9.d odd 6 1 inner 180.2.w.a 24
9.d odd 6 1 1620.2.j.b 24
15.d odd 2 1 2700.2.bf.e 24
15.e even 4 1 540.2.x.a 24
15.e even 4 1 2700.2.bf.e 24
20.e even 4 1 720.2.cu.d 24
36.h even 6 1 720.2.cu.d 24
45.h odd 6 1 900.2.be.e 24
45.j even 6 1 2700.2.bf.e 24
45.k odd 12 1 540.2.x.a 24
45.k odd 12 1 1620.2.j.b 24
45.k odd 12 1 2700.2.bf.e 24
45.l even 12 1 inner 180.2.w.a 24
45.l even 12 1 900.2.be.e 24
45.l even 12 1 1620.2.j.b 24
180.v odd 12 1 720.2.cu.d 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.2.w.a 24 1.a even 1 1 trivial
180.2.w.a 24 5.c odd 4 1 inner
180.2.w.a 24 9.d odd 6 1 inner
180.2.w.a 24 45.l even 12 1 inner
540.2.x.a 24 3.b odd 2 1
540.2.x.a 24 9.c even 3 1
540.2.x.a 24 15.e even 4 1
540.2.x.a 24 45.k odd 12 1
720.2.cu.d 24 4.b odd 2 1
720.2.cu.d 24 20.e even 4 1
720.2.cu.d 24 36.h even 6 1
720.2.cu.d 24 180.v odd 12 1
900.2.be.e 24 5.b even 2 1
900.2.be.e 24 5.c odd 4 1
900.2.be.e 24 45.h odd 6 1
900.2.be.e 24 45.l even 12 1
1620.2.j.b 24 9.c even 3 1
1620.2.j.b 24 9.d odd 6 1
1620.2.j.b 24 45.k odd 12 1
1620.2.j.b 24 45.l even 12 1
2700.2.bf.e 24 15.d odd 2 1
2700.2.bf.e 24 15.e even 4 1
2700.2.bf.e 24 45.j even 6 1
2700.2.bf.e 24 45.k odd 12 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(180, [\chi])$$.