Properties

Label 546.2.by.b
Level $546$
Weight $2$
Character orbit 546.by
Analytic conductor $4.360$
Analytic rank $0$
Dimension $40$
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.by (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.35983195036\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(10\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 40q - 4q^{7} - 40q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 40q - 4q^{7} - 40q^{9} - 4q^{11} + 20q^{12} + 4q^{14} + 20q^{16} + 8q^{17} + 8q^{19} - 8q^{21} - 4q^{22} + 24q^{23} + 24q^{25} - 8q^{26} + 4q^{28} - 12q^{29} + 24q^{31} - 4q^{33} + 8q^{34} + 28q^{35} - 8q^{37} - 8q^{38} - 16q^{39} - 20q^{41} - 12q^{42} - 24q^{43} + 8q^{44} - 4q^{46} - 16q^{47} + 4q^{49} - 16q^{50} - 24q^{51} - 4q^{52} - 4q^{53} - 24q^{55} + 12q^{56} + 8q^{57} + 24q^{58} - 12q^{59} - 32q^{62} + 4q^{63} - 4q^{65} - 24q^{67} + 24q^{68} + 8q^{69} + 52q^{70} - 28q^{71} + 108q^{73} + 20q^{74} - 36q^{75} - 4q^{76} + 12q^{77} + 4q^{78} + 40q^{81} - 48q^{82} - 60q^{83} - 8q^{84} - 4q^{85} - 20q^{86} - 36q^{87} - 60q^{89} - 40q^{91} - 16q^{92} - 48q^{95} + 48q^{97} - 8q^{98} + 4q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −0.965926 0.258819i 1.00000i 0.866025 + 0.500000i −4.03777 + 1.08192i −0.258819 + 0.965926i −0.0127227 + 2.64572i −0.707107 0.707107i −1.00000 4.18021
19.2 −0.965926 0.258819i 1.00000i 0.866025 + 0.500000i −1.37297 + 0.367885i −0.258819 + 0.965926i −0.220772 2.63652i −0.707107 0.707107i −1.00000 1.42140
19.3 −0.965926 0.258819i 1.00000i 0.866025 + 0.500000i 0.195603 0.0524115i −0.258819 + 0.965926i −2.35070 + 1.21417i −0.707107 0.707107i −1.00000 −0.202503
19.4 −0.965926 0.258819i 1.00000i 0.866025 + 0.500000i 1.30611 0.349971i −0.258819 + 0.965926i 2.64352 0.108591i −0.707107 0.707107i −1.00000 −1.35218
19.5 −0.965926 0.258819i 1.00000i 0.866025 + 0.500000i 3.90903 1.04742i −0.258819 + 0.965926i −1.71472 2.01488i −0.707107 0.707107i −1.00000 −4.04692
19.6 0.965926 + 0.258819i 1.00000i 0.866025 + 0.500000i −3.15172 + 0.844500i 0.258819 0.965926i −2.64535 0.0462630i 0.707107 + 0.707107i −1.00000 −3.26290
19.7 0.965926 + 0.258819i 1.00000i 0.866025 + 0.500000i −1.66136 + 0.445159i 0.258819 0.965926i 2.48285 + 0.914026i 0.707107 + 0.707107i −1.00000 −1.71996
19.8 0.965926 + 0.258819i 1.00000i 0.866025 + 0.500000i −1.45998 + 0.391201i 0.258819 0.965926i 0.541277 2.58979i 0.707107 + 0.707107i −1.00000 −1.51148
19.9 0.965926 + 0.258819i 1.00000i 0.866025 + 0.500000i 2.82421 0.756745i 0.258819 0.965926i −2.36822 1.17963i 0.707107 + 0.707107i −1.00000 2.92384
19.10 0.965926 + 0.258819i 1.00000i 0.866025 + 0.500000i 3.44884 0.924115i 0.258819 0.965926i 2.64483 + 0.0697092i 0.707107 + 0.707107i −1.00000 3.57050
115.1 −0.965926 + 0.258819i 1.00000i 0.866025 0.500000i −4.03777 1.08192i −0.258819 0.965926i −0.0127227 2.64572i −0.707107 + 0.707107i −1.00000 4.18021
115.2 −0.965926 + 0.258819i 1.00000i 0.866025 0.500000i −1.37297 0.367885i −0.258819 0.965926i −0.220772 + 2.63652i −0.707107 + 0.707107i −1.00000 1.42140
115.3 −0.965926 + 0.258819i 1.00000i 0.866025 0.500000i 0.195603 + 0.0524115i −0.258819 0.965926i −2.35070 1.21417i −0.707107 + 0.707107i −1.00000 −0.202503
115.4 −0.965926 + 0.258819i 1.00000i 0.866025 0.500000i 1.30611 + 0.349971i −0.258819 0.965926i 2.64352 + 0.108591i −0.707107 + 0.707107i −1.00000 −1.35218
115.5 −0.965926 + 0.258819i 1.00000i 0.866025 0.500000i 3.90903 + 1.04742i −0.258819 0.965926i −1.71472 + 2.01488i −0.707107 + 0.707107i −1.00000 −4.04692
115.6 0.965926 0.258819i 1.00000i 0.866025 0.500000i −3.15172 0.844500i 0.258819 + 0.965926i −2.64535 + 0.0462630i 0.707107 0.707107i −1.00000 −3.26290
115.7 0.965926 0.258819i 1.00000i 0.866025 0.500000i −1.66136 0.445159i 0.258819 + 0.965926i 2.48285 0.914026i 0.707107 0.707107i −1.00000 −1.71996
115.8 0.965926 0.258819i 1.00000i 0.866025 0.500000i −1.45998 0.391201i 0.258819 + 0.965926i 0.541277 + 2.58979i 0.707107 0.707107i −1.00000 −1.51148
115.9 0.965926 0.258819i 1.00000i 0.866025 0.500000i 2.82421 + 0.756745i 0.258819 + 0.965926i −2.36822 + 1.17963i 0.707107 0.707107i −1.00000 2.92384
115.10 0.965926 0.258819i 1.00000i 0.866025 0.500000i 3.44884 + 0.924115i 0.258819 + 0.965926i 2.64483 0.0697092i 0.707107 0.707107i −1.00000 3.57050
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 535.10
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.w 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.by.b 40
7.d odd 6 1 546.2.cg.b yes 40
13.f odd 12 1 546.2.cg.b yes 40
91.w even 12 1 inner 546.2.by.b 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.by.b 40 1.a even 1 1 trivial
546.2.by.b 40 91.w even 12 1 inner
546.2.cg.b yes 40 7.d odd 6 1
546.2.cg.b yes 40 13.f odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(11\!\cdots\!91\)\( T_{5}^{16} + 994272484676 T_{5}^{15} - 784664655000 T_{5}^{14} - \)\(31\!\cdots\!28\)\( T_{5}^{13} - \)\(37\!\cdots\!44\)\( T_{5}^{12} - \)\(14\!\cdots\!36\)\( T_{5}^{11} + \)\(37\!\cdots\!82\)\( T_{5}^{10} + \)\(73\!\cdots\!92\)\( T_{5}^{9} + \)\(73\!\cdots\!21\)\( T_{5}^{8} + \)\(60\!\cdots\!04\)\( T_{5}^{7} + \)\(31\!\cdots\!80\)\( T_{5}^{6} + \)\(16\!\cdots\!24\)\( T_{5}^{5} + \)\(10\!\cdots\!88\)\( T_{5}^{4} - 390810811392 T_{5}^{3} + 322066181376 T_{5}^{2} - 127675450368 T_{5} + 14841086976 \)">\(T_{5}^{40} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(546, [\chi])\).