Properties

Label 546.2.bu.a
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.

\(\operatorname{Tr}(f)(q) = \) \( 56q - 8q^{6} + 24q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 56q - 8q^{6} + 24q^{9} + 24q^{10} - 8q^{11} + 24q^{13} - 4q^{15} + 28q^{16} + 4q^{17} + 8q^{19} + 4q^{21} + 8q^{23} + 8q^{24} + 4q^{26} - 24q^{27} - 16q^{30} - 8q^{31} - 32q^{33} - 24q^{34} - 24q^{35} - 12q^{36} - 8q^{37} - 60q^{39} + 28q^{41} - 8q^{44} - 40q^{45} - 20q^{46} - 64q^{50} + 8q^{54} + 8q^{55} + 28q^{56} + 40q^{57} + 4q^{58} + 8q^{59} + 4q^{60} + 8q^{61} + 32q^{62} + 24q^{65} + 32q^{66} - 56q^{69} + 112q^{71} + 16q^{72} + 8q^{73} - 48q^{74} - 40q^{75} + 8q^{76} - 8q^{78} + 16q^{79} + 12q^{81} + 4q^{83} - 4q^{84} + 32q^{85} + 16q^{86} + 144q^{87} - 88q^{89} + 8q^{90} - 8q^{91} - 76q^{93} - 8q^{94} + 48q^{95} - 4q^{96} - 64q^{97} - 40q^{99} + O(q^{100}) \)

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 −1.68734 + 0.390990i 0.866025 0.500000i −2.95685 2.95685i 1.52865 0.814384i 0.258819 0.965926i −0.707107 + 0.707107i 2.69425 1.31947i 3.62139 + 2.09081i
71.2 −0.965926 + 0.258819i −1.35787 1.07526i 0.866025 0.500000i 0.489109 + 0.489109i 1.58990 + 0.687181i 0.258819 0.965926i −0.707107 + 0.707107i 0.687622 + 2.92013i −0.599034 0.345852i
71.3 −0.965926 + 0.258819i −0.847008 + 1.51082i 0.866025 0.500000i 1.36895 + 1.36895i 0.427118 1.67856i 0.258819 0.965926i −0.707107 + 0.707107i −1.56515 2.55935i −1.67662 0.967996i
71.4 −0.965926 + 0.258819i 0.504239 1.65703i 0.866025 0.500000i −2.02216 2.02216i −0.0581866 + 1.73107i 0.258819 0.965926i −0.707107 + 0.707107i −2.49149 1.67108i 2.47663 + 1.42988i
71.5 −0.965926 + 0.258819i 0.523037 + 1.65119i 0.866025 0.500000i −1.11112 1.11112i −0.932575 1.45956i 0.258819 0.965926i −0.707107 + 0.707107i −2.45286 + 1.72727i 1.36084 + 0.785680i
71.6 −0.965926 + 0.258819i 1.41231 1.00269i 0.866025 0.500000i 2.61115 + 2.61115i −1.10467 + 1.33405i 0.258819 0.965926i −0.707107 + 0.707107i 0.989233 2.83221i −3.19799 1.84636i
71.7 −0.965926 + 0.258819i 1.71146 0.266310i 0.866025 0.500000i −0.828570 0.828570i −1.58421 + 0.700193i 0.258819 0.965926i −0.707107 + 0.707107i 2.85816 0.911557i 1.01479 + 0.585888i
71.8 0.965926 0.258819i −1.67570 + 0.438205i 0.866025 0.500000i −1.24412 1.24412i −1.50519 + 0.856977i −0.258819 + 0.965926i 0.707107 0.707107i 2.61595 1.46860i −1.52373 0.879726i
71.9 0.965926 0.258819i −1.39783 + 1.02278i 0.866025 0.500000i 2.04268 + 2.04268i −1.08548 + 1.34972i −0.258819 + 0.965926i 0.707107 0.707107i 0.907836 2.85934i 2.50176 + 1.44439i
71.10 0.965926 0.258819i −1.38152 1.04470i 0.866025 0.500000i 1.28362 + 1.28362i −1.60483 0.651543i −0.258819 + 0.965926i 0.707107 0.707107i 0.817188 + 2.88656i 1.57211 + 0.907660i
71.11 0.965926 0.258819i 0.442137 + 1.67467i 0.866025 0.500000i 0.359298 + 0.359298i 0.860508 + 1.50317i −0.258819 + 0.965926i 0.707107 0.707107i −2.60903 + 1.48087i 0.440048 + 0.254062i
71.12 0.965926 0.258819i 0.563184 1.63793i 0.866025 0.500000i −1.58301 1.58301i 0.120065 1.72788i −0.258819 + 0.965926i 0.707107 0.707107i −2.36565 1.84491i −1.93879 1.11936i
71.13 0.965926 0.258819i 1.59445 0.676548i 0.866025 0.500000i −0.0940280 0.0940280i 1.36502 1.06617i −0.258819 + 0.965926i 0.707107 0.707107i 2.08457 2.15745i −0.115160 0.0664879i
71.14 0.965926 0.258819i 1.59645 + 0.671817i 0.866025 0.500000i 1.68505 + 1.68505i 1.71593 + 0.235733i −0.258819 + 0.965926i 0.707107 0.707107i 2.09732 + 2.14505i 2.06376 + 1.19151i
197.1 −0.258819 + 0.965926i −1.73063 0.0700467i −0.866025 0.500000i 2.88717 + 2.88717i 0.515581 1.65353i 0.965926 0.258819i 0.707107 0.707107i 2.99019 + 0.242450i −3.53605 + 2.04154i
197.2 −0.258819 + 0.965926i −1.60917 0.640767i −0.866025 0.500000i −2.11813 2.11813i 1.03542 1.38849i 0.965926 0.258819i 0.707107 0.707107i 2.17884 + 2.06220i 2.59417 1.49774i
197.3 −0.258819 + 0.965926i −0.703834 + 1.58260i −0.866025 0.500000i −2.95010 2.95010i −1.34651 1.08946i 0.965926 0.258819i 0.707107 0.707107i −2.00924 2.22777i 3.61312 2.08604i
197.4 −0.258819 + 0.965926i 0.684122 + 1.59122i −0.866025 0.500000i 0.803802 + 0.803802i −1.71406 + 0.248973i 0.965926 0.258819i 0.707107 0.707107i −2.06396 + 2.17717i −0.984452 + 0.568374i
197.5 −0.258819 + 0.965926i 0.957713 1.44319i −0.866025 0.500000i −0.0544192 0.0544192i 1.14614 + 1.29860i 0.965926 0.258819i 0.707107 0.707107i −1.16557 2.76432i 0.0666496 0.0384802i
197.6 −0.258819 + 0.965926i 1.63803 + 0.562898i −0.866025 0.500000i 1.20435 + 1.20435i −0.967672 + 1.43653i 0.965926 0.258819i 0.707107 0.707107i 2.36629 + 1.84409i −1.47502 + 0.851606i
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.a 56
3.b odd 2 1 546.2.bu.b yes 56
13.f odd 12 1 546.2.bu.b yes 56
39.k even 12 1 inner 546.2.bu.a 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.bu.a 56 1.a even 1 1 trivial
546.2.bu.a 56 39.k even 12 1 inner
546.2.bu.b yes 56 3.b odd 2 1
546.2.bu.b yes 56 13.f odd 12 1

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 \(S_{2}^{\mathrm{new}}(546, [\chi])\).