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Properties

Label 546.2.u.a
Level $546$
Weight $2$
Character orbit 546.u
Analytic conductor $4.360$
Analytic rank $0$
Dimension $76$
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.u (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.35983195036\)
Analytic rank: \(0\)
Dimension: \(76\)
Relative dimension: \(38\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 76q + 38q^{4} + 10q^{7} - 8q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 76q + 38q^{4} + 10q^{7} - 8q^{9} + 2q^{13} + 12q^{15} - 38q^{16} - 8q^{21} - 42q^{25} + 20q^{28} + 20q^{30} + 12q^{31} - 4q^{36} - 6q^{37} + 12q^{39} + 8q^{42} - 2q^{43} - 4q^{46} + 2q^{49} + 10q^{51} - 14q^{52} + 18q^{54} + 24q^{55} + 8q^{57} + 16q^{58} - 12q^{60} + 32q^{63} - 76q^{64} - 24q^{66} + 96q^{67} - 30q^{69} - 54q^{73} - 12q^{75} + 18q^{76} - 12q^{78} - 60q^{79} + 8q^{81} + 8q^{84} + 8q^{85} - 24q^{87} + 48q^{91} + 16q^{93} - 66q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
185.1 −0.866025 − 0.500000i −1.67595 + 0.437249i 0.500000 + 0.866025i 1.40279 + 2.42970i 1.67004 + 0.459307i 1.42569 − 2.22877i − 1.00000i 2.61763 − 1.46562i − 2.80557i
185.2 −0.866025 − 0.500000i −1.59603 + 0.672816i 0.500000 + 0.866025i 1.07511 + 1.86215i 1.71861 + 0.215340i 0.447408 + 2.60765i − 1.00000i 2.09464 − 2.14767i − 2.15023i
185.3 −0.866025 − 0.500000i −1.56696 − 0.737995i 0.500000 + 0.866025i −1.41378 − 2.44875i 0.988029 + 1.42260i −2.45899 + 0.976416i − 1.00000i 1.91073 + 2.31282i 2.82757i
185.4 −0.866025 − 0.500000i −1.51699 − 0.835899i 0.500000 + 0.866025i −0.699535 − 1.21163i 0.895807 + 1.48241i 1.97618 + 1.75918i − 1.00000i 1.60255 + 2.53611i 1.39907i
185.5 −0.866025 − 0.500000i −1.38893 + 1.03484i 0.500000 + 0.866025i −1.75377 − 3.03762i 1.72026 − 0.201730i 1.37341 − 2.26136i − 1.00000i 0.858232 − 2.87462i 3.50754i
185.6 −0.866025 − 0.500000i −1.18075 − 1.26722i 0.500000 + 0.866025i 0.386492 + 0.669424i 0.388948 + 1.68782i 0.662023 − 2.56159i − 1.00000i −0.211678 + 2.99252i − 0.772984i
185.7 −0.866025 − 0.500000i −0.685609 + 1.59058i 0.500000 + 0.866025i −0.587127 − 1.01693i 1.38904 − 1.03468i −1.73688 + 1.99580i − 1.00000i −2.05988 − 2.18103i 1.17425i
185.8 −0.866025 − 0.500000i −0.266485 + 1.71143i 0.500000 + 0.866025i 0.279399 + 0.483934i 1.08650 − 1.34890i 0.253211 − 2.63361i − 1.00000i −2.85797 − 0.912140i − 0.558798i
185.9 −0.866025 − 0.500000i −0.219501 − 1.71809i 0.500000 + 0.866025i 0.356048 + 0.616693i −0.668950 + 1.59766i −2.23074 − 1.42260i − 1.00000i −2.90364 + 0.754242i − 0.712096i
185.10 −0.866025 − 0.500000i −0.0864870 − 1.72989i 0.500000 + 0.866025i −0.886718 − 1.53584i −0.790045 + 1.54137i 2.61154 + 0.424103i − 1.00000i −2.98504 + 0.299226i 1.77344i
185.11 −0.866025 − 0.500000i 0.323515 + 1.70157i 0.500000 + 0.866025i 1.99252 + 3.45115i 0.570612 − 1.63536i −2.44529 − 1.01022i − 1.00000i −2.79068 + 1.10097i − 3.98504i
185.12 −0.866025 − 0.500000i 0.696855 + 1.58568i 0.500000 + 0.866025i −1.42622 − 2.47029i 0.189348 − 1.72167i 2.54641 + 0.718201i − 1.00000i −2.02879 + 2.20998i 2.85245i
185.13 −0.866025 − 0.500000i 0.804592 − 1.53383i 0.500000 + 0.866025i 2.14723 + 3.71911i −1.46371 + 0.926039i 2.58099 − 0.581819i − 1.00000i −1.70526 − 2.46821i − 4.29446i
185.14 −0.866025 − 0.500000i 0.896713 − 1.48186i 0.500000 + 0.866025i −0.277881 − 0.481304i −1.51751 + 0.834971i −0.887101 + 2.49260i − 1.00000i −1.39181 − 2.65760i 0.555761i
185.15 −0.866025 − 0.500000i 1.11365 − 1.32657i 0.500000 + 0.866025i −2.12998 − 3.68924i −1.62773 + 0.592023i −1.07162 − 2.41901i − 1.00000i −0.519589 − 2.95466i 4.25996i
185.16 −0.866025 − 0.500000i 1.30626 + 1.13741i 0.500000 + 0.866025i −0.604752 − 1.04746i −0.562548 − 1.63815i −2.45540 + 0.985399i − 1.00000i 0.412611 + 2.97149i 1.20950i
185.17 −0.866025 − 0.500000i 1.62717 + 0.593575i 0.500000 + 0.866025i 1.76347 + 3.05441i −1.11238 − 1.32763i 1.45770 + 2.20797i − 1.00000i 2.29534 + 1.93169i − 3.52693i
185.18 −0.866025 − 0.500000i 1.69393 + 0.361382i 0.500000 + 0.866025i 0.0272832 + 0.0472559i −1.28630 − 1.15993i 2.23165 − 1.42118i − 1.00000i 2.73881 + 1.22431i − 0.0545664i
185.19 −0.866025 − 0.500000i 1.72102 − 0.195177i 0.500000 + 0.866025i 0.349437 + 0.605242i −1.58803 − 0.691481i −1.78017 − 1.95729i − 1.00000i 2.92381 − 0.671806i − 0.698873i
185.20 0.866025 + 0.500000i −1.72102 − 0.195177i 0.500000 + 0.866025i −0.349437 − 0.605242i −1.39286 − 1.02954i −1.78017 − 1.95729i 1.00000i 2.92381 + 0.671806i − 0.698873i
See all 76 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 425.38
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
91.m odd 6 1 inner
273.bf 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.u.a ✓ 76
3.b odd 2 1 inner 546.2.u.a ✓ 76
7.d odd 6 1 546.2.bb.a yes 76
13.c even 3 1 546.2.bb.a yes 76
21.g even 6 1 546.2.bb.a yes 76
39.i odd 6 1 546.2.bb.a yes 76
91.m odd 6 1 inner 546.2.u.a ✓ 76
273.bf even 6 1 inner 546.2.u.a ✓ 76
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.u.a ✓ 76 1.a even 1 1 trivial
546.2.u.a ✓ 76 3.b odd 2 1 inner
546.2.u.a ✓ 76 91.m odd 6 1 inner
546.2.u.a ✓ 76 273.bf even 6 1 inner
546.2.bb.a yes 76 7.d odd 6 1
546.2.bb.a yes 76 13.c even 3 1
546.2.bb.a yes 76 21.g even 6 1
546.2.bb.a yes 76 39.i odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(546, [\chi])\).