# Properties

 Label 180.2.n.d Level $180$ Weight $2$ Character orbit 180.n Analytic conductor $1.437$ Analytic rank $0$ Dimension $48$ CM no Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [180,2,Mod(59,180)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(180, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 5, 3]))

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

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

chi := DirichletCharacter("180.59");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$180 = 2^{2} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 180.n (of order $$6$$, 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: $$48$$ Relative dimension: $$24$$ over $$\Q(\zeta_{6})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$48 q - 2 q^{4} - 18 q^{5} - 12 q^{9}+O(q^{10})$$ 48 * q - 2 * q^4 - 18 * q^5 - 12 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$48 q - 2 q^{4} - 18 q^{5} - 12 q^{9} - 16 q^{10} - 42 q^{14} + 30 q^{16} - 12 q^{21} - 6 q^{24} - 26 q^{25} - 12 q^{30} - 4 q^{34} + 96 q^{36} + 10 q^{40} + 96 q^{41} - 66 q^{45} + 4 q^{46} + 32 q^{49} - 90 q^{50} - 30 q^{54} + 6 q^{56} - 42 q^{60} + 8 q^{61} - 20 q^{64} - 6 q^{65} + 36 q^{66} - 96 q^{69} + 4 q^{70} + 72 q^{74} + 36 q^{81} - 6 q^{84} - 4 q^{85} + 108 q^{86} - 6 q^{90} - 62 q^{94} + 54 q^{96}+O(q^{100})$$ 48 * q - 2 * q^4 - 18 * q^5 - 12 * q^9 - 16 * q^10 - 42 * q^14 + 30 * q^16 - 12 * q^21 - 6 * q^24 - 26 * q^25 - 12 * q^30 - 4 * q^34 + 96 * q^36 + 10 * q^40 + 96 * q^41 - 66 * q^45 + 4 * q^46 + 32 * q^49 - 90 * q^50 - 30 * q^54 + 6 * q^56 - 42 * q^60 + 8 * q^61 - 20 * q^64 - 6 * q^65 + 36 * q^66 - 96 * q^69 + 4 * q^70 + 72 * q^74 + 36 * q^81 - 6 * q^84 - 4 * q^85 + 108 * q^86 - 6 * q^90 - 62 * q^94 + 54 * q^96

## 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
59.1 −1.41350 + 0.0449209i 1.62879 + 0.589100i 1.99596 0.126991i −1.91817 + 1.14918i −2.32876 0.759526i 1.44473 + 2.50234i −2.81559 + 0.269163i 2.30592 + 1.91904i 2.65972 1.71053i
59.2 −1.38574 + 0.282343i −0.959839 1.44177i 1.84056 0.782509i 0.520093 2.17474i 1.73716 + 1.72692i 0.550457 + 0.953419i −2.32961 + 1.60403i −1.15742 + 2.76774i −0.106691 + 3.16048i
59.3 −1.38444 0.288666i 0.112475 + 1.72840i 1.83334 + 0.799280i 2.23134 0.145382i 0.343213 2.42533i 0.573658 + 0.993605i −2.30743 1.63578i −2.97470 + 0.388803i −3.13112 0.442837i
59.4 −1.25828 0.645549i 0.424796 1.67915i 1.16653 + 1.62456i −1.98302 + 1.03326i −1.61849 + 1.83861i −1.60868 2.78632i −0.419091 2.79721i −2.63910 1.42659i 3.16221 0.0199985i
59.5 −1.24631 0.668359i −1.66946 + 0.461412i 1.10659 + 1.66597i −1.69235 1.46149i 2.38906 + 0.540736i 0.667441 + 1.15604i −0.265693 2.81592i 2.57420 1.54062i 1.13240 + 2.95257i
59.6 −1.00727 0.992675i 1.30213 1.14213i 0.0291929 + 1.99979i 1.64183 1.51803i −2.44536 0.142161i 1.65751 + 2.87089i 1.95573 2.04331i 0.391092 2.97440i −3.16067 0.100737i
59.7 −0.937387 + 1.05892i −0.959839 1.44177i −0.242609 1.98523i −1.62334 + 1.53778i 2.42646 + 0.335110i 0.550457 + 0.953419i 2.32961 + 1.60403i −1.15742 + 2.76774i −0.106691 3.16048i
59.8 −0.745653 + 1.20167i 1.62879 + 0.589100i −0.888004 1.79205i 0.0361315 2.23578i −1.92241 + 1.51800i 1.44473 + 2.50234i 2.81559 + 0.269163i 2.30592 + 1.91904i 2.65972 + 1.71053i
59.9 −0.442228 + 1.34329i 0.112475 + 1.72840i −1.60887 1.18808i 0.989764 + 2.00509i −2.37148 0.613258i 0.573658 + 0.993605i 2.30743 1.63578i −2.97470 + 0.388803i −3.13112 + 0.442837i
59.10 −0.356046 1.36866i −1.30213 + 1.14213i −1.74646 + 0.974612i 1.64183 1.51803i 2.02680 + 1.37553i −1.65751 2.87089i 1.95573 + 2.04331i 0.391092 2.97440i −2.66223 1.70662i
59.11 −0.0700780 + 1.41248i 0.424796 1.67915i −1.99018 0.197967i −0.0966770 2.23398i 2.34199 + 0.717686i −1.60868 2.78632i 0.419091 2.79721i −2.63910 1.42659i 3.16221 + 0.0199985i
59.12 −0.0443404 + 1.41352i −1.66946 + 0.461412i −1.99607 0.125352i −2.11186 0.734875i −0.578190 2.38027i 0.667441 + 1.15604i 0.265693 2.81592i 2.57420 1.54062i 1.13240 2.95257i
59.13 0.0443404 1.41352i 1.66946 0.461412i −1.99607 0.125352i −1.69235 1.46149i −0.578190 2.38027i −0.667441 1.15604i −0.265693 + 2.81592i 2.57420 1.54062i −2.14088 + 2.32737i
59.14 0.0700780 1.41248i −0.424796 + 1.67915i −1.99018 0.197967i −1.98302 + 1.03326i 2.34199 + 0.717686i 1.60868 + 2.78632i −0.419091 + 2.79721i −2.63910 1.42659i 1.32049 + 2.87338i
59.15 0.356046 + 1.36866i 1.30213 1.14213i −1.74646 + 0.974612i −0.493735 + 2.18088i 2.02680 + 1.37553i 1.65751 + 2.87089i −1.95573 2.04331i 0.391092 2.97440i −3.16067 + 0.100737i
59.16 0.442228 1.34329i −0.112475 1.72840i −1.60887 1.18808i 2.23134 0.145382i −2.37148 0.613258i −0.573658 0.993605i −2.30743 + 1.63578i −2.97470 + 0.388803i 0.791468 3.06163i
59.17 0.745653 1.20167i −1.62879 0.589100i −0.888004 1.79205i −1.91817 + 1.14918i −1.92241 + 1.51800i −1.44473 2.50234i −2.81559 0.269163i 2.30592 + 1.91904i −0.0493612 + 3.16189i
59.18 0.937387 1.05892i 0.959839 + 1.44177i −0.242609 1.98523i 0.520093 2.17474i 2.42646 + 0.335110i −0.550457 0.953419i −2.32961 1.60403i −1.15742 + 2.76774i −1.81534 2.58931i
59.19 1.00727 + 0.992675i −1.30213 + 1.14213i 0.0291929 + 1.99979i −0.493735 + 2.18088i −2.44536 0.142161i −1.65751 2.87089i −1.95573 + 2.04331i 0.391092 2.97440i −2.66223 + 1.70662i
59.20 1.24631 + 0.668359i 1.66946 0.461412i 1.10659 + 1.66597i −2.11186 0.734875i 2.38906 + 0.540736i −0.667441 1.15604i 0.265693 + 2.81592i 2.57420 1.54062i −2.14088 2.32737i
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 59.24 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 inner
5.b even 2 1 inner
9.d odd 6 1 inner
20.d odd 2 1 inner
36.h even 6 1 inner
45.h odd 6 1 inner
180.n even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 180.2.n.d 48
3.b odd 2 1 540.2.n.d 48
4.b odd 2 1 inner 180.2.n.d 48
5.b even 2 1 inner 180.2.n.d 48
5.c odd 4 2 900.2.r.g 48
9.c even 3 1 540.2.n.d 48
9.d odd 6 1 inner 180.2.n.d 48
12.b even 2 1 540.2.n.d 48
15.d odd 2 1 540.2.n.d 48
20.d odd 2 1 inner 180.2.n.d 48
20.e even 4 2 900.2.r.g 48
36.f odd 6 1 540.2.n.d 48
36.h even 6 1 inner 180.2.n.d 48
45.h odd 6 1 inner 180.2.n.d 48
45.j even 6 1 540.2.n.d 48
45.l even 12 2 900.2.r.g 48
60.h even 2 1 540.2.n.d 48
180.n even 6 1 inner 180.2.n.d 48
180.p odd 6 1 540.2.n.d 48
180.v odd 12 2 900.2.r.g 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.2.n.d 48 1.a even 1 1 trivial
180.2.n.d 48 4.b odd 2 1 inner
180.2.n.d 48 5.b even 2 1 inner
180.2.n.d 48 9.d odd 6 1 inner
180.2.n.d 48 20.d odd 2 1 inner
180.2.n.d 48 36.h even 6 1 inner
180.2.n.d 48 45.h odd 6 1 inner
180.2.n.d 48 180.n even 6 1 inner
540.2.n.d 48 3.b odd 2 1
540.2.n.d 48 9.c even 3 1
540.2.n.d 48 12.b even 2 1
540.2.n.d 48 15.d odd 2 1
540.2.n.d 48 36.f odd 6 1
540.2.n.d 48 45.j even 6 1
540.2.n.d 48 60.h even 2 1
540.2.n.d 48 180.p odd 6 1
900.2.r.g 48 5.c odd 4 2
900.2.r.g 48 20.e even 4 2
900.2.r.g 48 45.l even 12 2
900.2.r.g 48 180.v odd 12 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{24} + 34 T_{7}^{22} + 730 T_{7}^{20} + 9700 T_{7}^{18} + 94189 T_{7}^{16} + 620404 T_{7}^{14} + 2963314 T_{7}^{12} + 8702080 T_{7}^{10} + 18514873 T_{7}^{8} + 26555616 T_{7}^{6} + 27788076 T_{7}^{4} + \cdots + 7290000$$ acting on $$S_{2}^{\mathrm{new}}(180, [\chi])$$.