Properties

Label 546.2.bg.b
Level $546$
Weight $2$
Character orbit 546.bg
Analytic conductor $4.360$
Analytic rank $0$
Dimension $36$
CM no
Inner twists $4$

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Newspace parameters

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

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

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

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.

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 467.18
Significant digits:
Format:

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{5}^{18} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(546, [\chi])\).