Properties

Label 546.2.bu.b
Level $546$
Weight $2$
Character orbit 546.bu
Analytic conductor $4.360$
Analytic rank $0$
Dimension $56$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 546.bu (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.35983195036\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(14\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 56q + 8q^{6} - 24q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 56q + 8q^{6} - 24q^{9} + 24q^{10} + 8q^{11} + 24q^{13} + 4q^{15} + 28q^{16} - 4q^{17} - 8q^{18} + 8q^{19} + 4q^{21} - 8q^{23} - 8q^{24} - 4q^{26} - 24q^{27} - 8q^{30} - 8q^{31} + 8q^{33} - 24q^{34} + 24q^{35} - 12q^{36} - 8q^{37} - 20q^{39} - 28q^{41} + 8q^{44} + 72q^{45} - 20q^{46} + 64q^{50} + 16q^{54} + 8q^{55} - 28q^{56} + 4q^{58} - 8q^{59} + 20q^{60} + 8q^{61} - 32q^{62} - 16q^{63} - 24q^{65} + 32q^{66} - 16q^{69} - 112q^{71} + 8q^{73} + 48q^{74} + 40q^{75} + 8q^{76} + 16q^{79} + 12q^{81} - 4q^{83} - 4q^{84} + 32q^{85} - 16q^{86} - 144q^{87} + 88q^{89} - 8q^{90} - 8q^{91} + 52q^{93} - 8q^{94} - 48q^{95} + 4q^{96} - 64q^{97} + 56q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
71.1 −0.965926 + 0.258819i −1.67137 + 0.454432i 0.866025 0.500000i −0.359298 0.359298i 1.49681 0.871531i −0.258819 + 0.965926i −0.707107 + 0.707107i 2.58698 1.51905i 0.440048 + 0.254062i
71.2 −0.965926 + 0.258819i −1.38004 1.04666i 0.866025 0.500000i −1.68505 1.68505i 1.60391 + 0.653816i −0.258819 + 0.965926i −0.707107 + 0.707107i 0.809005 + 2.88886i 2.06376 + 1.19151i
71.3 −0.965926 + 0.258819i −0.211319 1.71911i 0.866025 0.500000i 0.0940280 + 0.0940280i 0.649057 + 1.60584i −0.258819 + 0.965926i −0.707107 + 0.707107i −2.91069 + 0.726562i −0.115160 0.0664879i
71.4 −0.965926 + 0.258819i −0.186842 + 1.72194i 0.866025 0.500000i −2.04268 2.04268i −0.265197 1.71163i −0.258819 + 0.965926i −0.707107 + 0.707107i −2.93018 0.643461i 2.50176 + 1.44439i
71.5 −0.965926 + 0.258819i 0.458354 + 1.67030i 0.866025 0.500000i 1.24412 + 1.24412i −0.875042 1.49476i −0.258819 + 0.965926i −0.707107 + 0.707107i −2.57982 + 1.53118i −1.52373 0.879726i
71.6 −0.965926 + 0.258819i 1.13690 1.30670i 0.866025 0.500000i 1.58301 + 1.58301i −0.759962 + 1.55642i −0.258819 + 0.965926i −0.707107 + 0.707107i −0.414918 2.97117i −1.93879 1.11936i
71.7 −0.965926 + 0.258819i 1.59550 + 0.674078i 0.866025 0.500000i −1.28362 1.28362i −1.71560 0.238164i −0.258819 + 0.965926i −0.707107 + 0.707107i 2.09124 + 2.15098i 1.57211 + 0.907660i
71.8 0.965926 0.258819i −1.69149 + 0.372632i 0.866025 0.500000i 1.11112 + 1.11112i −1.53741 + 0.797725i 0.258819 0.965926i 0.707107 0.707107i 2.72229 1.26061i 1.36084 + 0.785680i
71.9 0.965926 0.258819i −0.884905 + 1.48894i 0.866025 0.500000i −1.36895 1.36895i −0.469386 + 1.66724i 0.258819 0.965926i 0.707107 0.707107i −1.43389 2.63514i −1.67662 0.967996i
71.10 0.965926 0.258819i −0.625096 1.61532i 0.866025 0.500000i 0.828570 + 0.828570i −1.02187 1.39849i 0.258819 0.965926i 0.707107 0.707107i −2.21851 + 2.01946i 1.01479 + 0.585888i
71.11 0.965926 0.258819i 0.162199 1.72444i 0.866025 0.500000i −2.61115 2.61115i −0.289646 1.70766i 0.258819 0.965926i 0.707107 0.707107i −2.94738 0.559404i −3.19799 1.84636i
71.12 0.965926 0.258819i 0.505064 + 1.65678i 0.866025 0.500000i 2.95685 + 2.95685i 0.916660 + 1.46960i 0.258819 0.965926i 0.707107 0.707107i −2.48982 + 1.67356i 3.62139 + 2.09081i
71.13 0.965926 0.258819i 1.18291 1.26520i 0.866025 0.500000i 2.02216 + 2.02216i 0.815146 1.52825i 0.258819 0.965926i 0.707107 0.707107i −0.201451 2.99323i 2.47663 + 1.42988i
71.14 0.965926 0.258819i 1.61014 + 0.638319i 0.866025 0.500000i −0.489109 0.489109i 1.72048 + 0.199834i 0.258819 0.965926i 0.707107 0.707107i 2.18510 + 2.05556i −0.599034 0.345852i
197.1 −0.258819 + 0.965926i −1.64483 + 0.542708i −0.866025 0.500000i 1.01262 + 1.01262i −0.0985019 1.72925i −0.965926 + 0.258819i 0.707107 0.707107i 2.41094 1.78532i −1.24020 + 0.716030i
197.2 −0.258819 + 0.965926i −1.34996 1.08518i −0.866025 0.500000i 0.429580 + 0.429580i 1.39760 1.02310i −0.965926 + 0.258819i 0.707107 0.707107i 0.644780 + 2.92989i −0.526126 + 0.303759i
197.3 −0.258819 + 0.965926i −1.12111 + 1.32027i −0.866025 0.500000i −0.381662 0.381662i −0.985117 1.42462i −0.965926 + 0.258819i 0.707107 0.707107i −0.486221 2.96034i 0.467439 0.269876i
197.4 −0.258819 + 0.965926i −0.413071 1.68207i −0.866025 0.500000i −2.27299 2.27299i 1.73167 + 0.0363565i −0.965926 + 0.258819i 0.707107 0.707107i −2.65874 + 1.38963i 2.78383 1.60724i
197.5 −0.258819 + 0.965926i 0.984178 1.42527i −0.866025 0.500000i −1.05727 1.05727i 1.12198 + 1.31953i −0.965926 + 0.258819i 0.707107 0.707107i −1.06279 2.80544i 1.29489 0.747605i
197.6 −0.258819 + 0.965926i 1.01587 + 1.40286i −0.866025 0.500000i −1.99319 1.99319i −1.61798 + 0.618168i −0.965926 + 0.258819i 0.707107 0.707107i −0.936020 + 2.85024i 2.44115 1.40940i
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 449.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
39.k even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 546.2.bu.b yes 56
3.b odd 2 1 546.2.bu.a 56
13.f odd 12 1 546.2.bu.a 56
39.k even 12 1 inner 546.2.bu.b yes 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.bu.a 56 3.b odd 2 1
546.2.bu.a 56 13.f odd 12 1
546.2.bu.b yes 56 1.a even 1 1 trivial
546.2.bu.b yes 56 39.k even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(80\!\cdots\!08\)\( T_{5}^{32} + \)\(85\!\cdots\!04\)\( T_{5}^{31} + \)\(53\!\cdots\!20\)\( T_{5}^{30} + \)\(16\!\cdots\!88\)\( T_{5}^{29} + \)\(14\!\cdots\!64\)\( T_{5}^{28} + \)\(16\!\cdots\!12\)\( T_{5}^{27} + \)\(10\!\cdots\!60\)\( T_{5}^{26} + \)\(17\!\cdots\!96\)\( T_{5}^{25} + \)\(14\!\cdots\!21\)\( T_{5}^{24} + \)\(17\!\cdots\!08\)\( T_{5}^{23} + \)\(11\!\cdots\!64\)\( T_{5}^{22} + \)\(10\!\cdots\!24\)\( T_{5}^{21} + \)\(78\!\cdots\!28\)\( T_{5}^{20} + \)\(94\!\cdots\!20\)\( T_{5}^{19} + \)\(59\!\cdots\!24\)\( T_{5}^{18} + \)\(31\!\cdots\!88\)\( T_{5}^{17} + \)\(19\!\cdots\!76\)\( T_{5}^{16} + \)\(24\!\cdots\!28\)\( T_{5}^{15} + \)\(15\!\cdots\!32\)\( T_{5}^{14} + \)\(50\!\cdots\!60\)\( T_{5}^{13} + \)\(18\!\cdots\!96\)\( T_{5}^{12} + \)\(22\!\cdots\!76\)\( T_{5}^{11} + \)\(14\!\cdots\!80\)\( T_{5}^{10} + \)\(32\!\cdots\!56\)\( T_{5}^{9} + \)\(22\!\cdots\!16\)\( T_{5}^{8} + \)\(25\!\cdots\!60\)\( T_{5}^{7} + \)\(17\!\cdots\!72\)\( T_{5}^{6} + \)\(36\!\cdots\!64\)\( T_{5}^{5} - 406256678912 T_{5}^{4} - \)\(20\!\cdots\!92\)\( T_{5}^{3} + \)\(56\!\cdots\!08\)\( T_{5}^{2} - 474052952064 T_{5} + 19932921856 \)">\(T_{5}^{56} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(546, [\chi])\).