Properties

 Label 546.2.bg.a Level $546$ Weight $2$ Character orbit 546.bg Analytic conductor $4.360$ Analytic rank $0$ Dimension $36$ Inner twists $4$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [546,2,Mod(311,546)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(546, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 1, 3]))

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

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

chi := DirichletCharacter("546.311");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$546 = 2 \cdot 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 546.bg (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: $$4.35983195036$$ Analytic rank: $$0$$ Dimension: $$36$$ Relative dimension: $$18$$ over $$\Q(\zeta_{6})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The algebraic $$q$$-expansion of this newform has not been computed, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$36 q - 18 q^{2} - 18 q^{4} + 36 q^{8} + 4 q^{9}+O(q^{10})$$ 36 * q - 18 * q^2 - 18 * q^4 + 36 * q^8 + 4 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$36 q - 18 q^{2} - 18 q^{4} + 36 q^{8} + 4 q^{9} - 14 q^{15} - 18 q^{16} + 4 q^{18} + 23 q^{21} + 14 q^{25} - 6 q^{26} + 7 q^{30} - 18 q^{32} + 24 q^{33} - 8 q^{36} - 10 q^{39} - 16 q^{42} - 16 q^{43} - 9 q^{45} + 72 q^{47} + 12 q^{49} - 28 q^{50} - 3 q^{51} + 6 q^{52} + 9 q^{54} - 8 q^{57} + 24 q^{59} + 7 q^{60} - 36 q^{61} - 39 q^{63} + 36 q^{64} + 18 q^{65} - 24 q^{66} - 72 q^{71} + 4 q^{72} + 54 q^{75} + 20 q^{78} + 20 q^{79} - 20 q^{81} - 24 q^{82} - 7 q^{84} + 8 q^{86} - 24 q^{87} - 72 q^{89} - 2 q^{91} - 14 q^{93} - 72 q^{94} - 12 q^{98} + 72 q^{99}+O(q^{100})$$ 36 * q - 18 * q^2 - 18 * q^4 + 36 * q^8 + 4 * q^9 - 14 * q^15 - 18 * q^16 + 4 * q^18 + 23 * q^21 + 14 * q^25 - 6 * q^26 + 7 * q^30 - 18 * q^32 + 24 * q^33 - 8 * q^36 - 10 * q^39 - 16 * q^42 - 16 * q^43 - 9 * q^45 + 72 * q^47 + 12 * q^49 - 28 * q^50 - 3 * q^51 + 6 * q^52 + 9 * q^54 - 8 * q^57 + 24 * q^59 + 7 * q^60 - 36 * q^61 - 39 * q^63 + 36 * q^64 + 18 * q^65 - 24 * q^66 - 72 * q^71 + 4 * q^72 + 54 * q^75 + 20 * q^78 + 20 * q^79 - 20 * q^81 - 24 * q^82 - 7 * q^84 + 8 * q^86 - 24 * q^87 - 72 * q^89 - 2 * q^91 - 14 * q^93 - 72 * q^94 - 12 * q^98 + 72 * q^99

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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
311.1 −0.500000 0.866025i −1.71076 0.270722i −0.500000 + 0.866025i 2.20332 1.27209i 0.620929 + 1.61693i 2.32714 1.25873i 1.00000 2.85342 + 0.926282i −2.20332 1.27209i
311.2 −0.500000 0.866025i −1.63622 0.568150i −0.500000 + 0.866025i −1.08726 + 0.627730i 0.326076 + 1.70108i 1.14208 2.38656i 1.00000 2.35441 + 1.85923i 1.08726 + 0.627730i
311.3 −0.500000 0.866025i −1.62062 + 0.611211i −0.500000 + 0.866025i 0.0759308 0.0438386i 1.33964 + 1.09790i −2.62099 0.361112i 1.00000 2.25284 1.98109i −0.0759308 0.0438386i
311.4 −0.500000 0.866025i −1.34147 + 1.09565i −0.500000 + 0.866025i −2.44950 + 1.41422i 1.61960 + 0.613923i −2.60728 + 0.449563i 1.00000 0.599095 2.93957i 2.44950 + 1.41422i
311.5 −0.500000 0.866025i −1.31014 1.13293i −0.500000 + 0.866025i −1.08726 + 0.627730i −0.326076 + 1.70108i −1.14208 + 2.38656i 1.00000 0.432936 + 2.96860i 1.08726 + 0.627730i
311.6 −0.500000 0.866025i −1.08983 1.34620i −0.500000 + 0.866025i 2.20332 1.27209i −0.620929 + 1.61693i −2.32714 + 1.25873i 1.00000 −0.624526 + 2.93427i −2.20332 1.27209i
311.7 −0.500000 0.866025i −0.924091 + 1.46494i −0.500000 + 0.866025i 3.23425 1.86729i 1.73072 + 0.0678152i −0.481958 2.60148i 1.00000 −1.29211 2.70748i −3.23425 1.86729i
311.8 −0.500000 0.866025i −0.418670 + 1.68069i −0.500000 + 0.866025i 0.0916492 0.0529137i 1.66485 0.477766i 2.29863 + 1.31008i 1.00000 −2.64943 1.40731i −0.0916492 0.0529137i
311.9 −0.500000 0.866025i −0.280988 1.70911i −0.500000 + 0.866025i 0.0759308 0.0438386i −1.33964 + 1.09790i 2.62099 + 0.361112i 1.00000 −2.84209 + 0.960476i −0.0759308 0.0438386i
311.10 −0.500000 0.866025i 0.278126 1.70957i −0.500000 + 0.866025i −2.44950 + 1.41422i −1.61960 + 0.613923i 2.60728 0.449563i 1.00000 −2.84529 0.950954i 2.44950 + 1.41422i
311.11 −0.500000 0.866025i 0.658943 + 1.60181i −0.500000 + 0.866025i −1.00187 + 0.578432i 1.05774 1.37157i −0.296298 2.62911i 1.00000 −2.13159 + 2.11100i 1.00187 + 0.578432i
311.12 −0.500000 0.866025i 0.806632 1.53276i −0.500000 + 0.866025i 3.23425 1.86729i −1.73072 + 0.0678152i 0.481958 + 2.60148i 1.00000 −1.69869 2.47274i −3.23425 1.86729i
311.13 −0.500000 0.866025i 0.909476 + 1.47406i −0.500000 + 0.866025i −3.26421 + 1.88459i 0.821836 1.52466i −0.385108 + 2.61757i 1.00000 −1.34571 + 2.68124i 3.26421 + 1.88459i
311.14 −0.500000 0.866025i 1.24618 1.20292i −0.500000 + 0.866025i 0.0916492 0.0529137i −1.66485 0.477766i −2.29863 1.31008i 1.00000 0.105953 2.99813i −0.0916492 0.0529137i
311.15 −0.500000 0.866025i 1.40752 + 1.00940i −0.500000 + 0.866025i 2.19770 1.26884i 0.170404 1.72365i −2.61925 0.373536i 1.00000 0.962231 + 2.84150i −2.19770 1.26884i
311.16 −0.500000 0.866025i 1.57792 + 0.714250i −0.500000 + 0.866025i 2.19770 1.26884i −0.170404 1.72365i 2.61925 + 0.373536i 1.00000 1.97969 + 2.25407i −2.19770 1.26884i
311.17 −0.500000 0.866025i 1.71668 0.230243i −0.500000 + 0.866025i −1.00187 + 0.578432i −1.05774 1.37157i 0.296298 + 2.62911i 1.00000 2.89398 0.790508i 1.00187 + 0.578432i
311.18 −0.500000 0.866025i 1.73131 + 0.0505991i −0.500000 + 0.866025i −3.26421 + 1.88459i −0.821836 1.52466i 0.385108 2.61757i 1.00000 2.99488 + 0.175206i 3.26421 + 1.88459i
467.1 −0.500000 + 0.866025i −1.71076 + 0.270722i −0.500000 0.866025i 2.20332 + 1.27209i 0.620929 1.61693i 2.32714 + 1.25873i 1.00000 2.85342 0.926282i −2.20332 + 1.27209i
467.2 −0.500000 + 0.866025i −1.63622 + 0.568150i −0.500000 0.866025i −1.08726 0.627730i 0.326076 1.70108i 1.14208 + 2.38656i 1.00000 2.35441 1.85923i 1.08726 0.627730i
See all 36 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 311.18 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
39.d odd 2 1 inner
273.ba even 6 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 546.2.bg.a 36
3.b odd 2 1 546.2.bg.b yes 36
7.d odd 6 1 inner 546.2.bg.a 36
13.b even 2 1 546.2.bg.b yes 36
21.g even 6 1 546.2.bg.b yes 36
39.d odd 2 1 inner 546.2.bg.a 36
91.s odd 6 1 546.2.bg.b yes 36
273.ba even 6 1 inner 546.2.bg.a 36

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.bg.a 36 1.a even 1 1 trivial
546.2.bg.a 36 7.d odd 6 1 inner
546.2.bg.a 36 39.d odd 2 1 inner
546.2.bg.a 36 273.ba even 6 1 inner
546.2.bg.b yes 36 3.b odd 2 1
546.2.bg.b yes 36 13.b even 2 1
546.2.bg.b yes 36 21.g even 6 1
546.2.bg.b yes 36 91.s odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{18} - 26 T_{5}^{16} + 475 T_{5}^{14} - 3 T_{5}^{13} - 4279 T_{5}^{12} - 111 T_{5}^{11} + \cdots + 12$$ acting on $$S_{2}^{\mathrm{new}}(546, [\chi])$$.