Properties

Label 546.2.bx.b
Level $546$
Weight $2$
Character orbit 546.bx
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.bx (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.

\(\operatorname{Tr}(f)(q) = \) \( 40q + 8q^{7} + 20q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 40q + 8q^{7} + 20q^{9} + 8q^{11} - 40q^{12} - 8q^{14} + 20q^{16} - 16q^{17} - 16q^{19} + 4q^{21} + 8q^{22} - 32q^{26} - 8q^{28} + 8q^{33} - 16q^{34} + 40q^{35} + 40q^{37} + 16q^{38} - 16q^{39} + 4q^{41} - 12q^{42} + 24q^{43} + 8q^{44} - 4q^{46} - 16q^{47} - 20q^{49} - 16q^{50} + 8q^{52} + 32q^{53} + 72q^{55} + 12q^{56} + 8q^{57} - 36q^{58} - 84q^{59} + 48q^{61} + 4q^{62} + 4q^{63} - 16q^{65} - 12q^{68} + 8q^{69} - 20q^{70} - 40q^{71} + 48q^{73} + 8q^{74} - 36q^{75} - 16q^{76} + 24q^{77} - 8q^{78} - 48q^{79} - 20q^{81} - 36q^{83} - 8q^{84} + 8q^{85} - 8q^{86} - 12q^{89} + 32q^{91} - 16q^{92} - 24q^{93} - 96q^{95} - 8q^{98} - 8q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1 −0.258819 0.965926i 0.866025 + 0.500000i −0.866025 + 0.500000i −2.52974 2.52974i 0.258819 0.965926i 2.60750 + 0.448249i 0.707107 + 0.707107i 0.500000 + 0.866025i −1.78880 + 3.09829i
97.2 −0.258819 0.965926i 0.866025 + 0.500000i −0.866025 + 0.500000i −0.413783 0.413783i 0.258819 0.965926i 1.23233 + 2.34123i 0.707107 + 0.707107i 0.500000 + 0.866025i −0.292588 + 0.506778i
97.3 −0.258819 0.965926i 0.866025 + 0.500000i −0.866025 + 0.500000i −0.301627 0.301627i 0.258819 0.965926i −2.51185 0.831019i 0.707107 + 0.707107i 0.500000 + 0.866025i −0.213282 + 0.369416i
97.4 −0.258819 0.965926i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.201763 + 0.201763i 0.258819 0.965926i −0.864369 2.50057i 0.707107 + 0.707107i 0.500000 + 0.866025i 0.142668 0.247108i
97.5 −0.258819 0.965926i 0.866025 + 0.500000i −0.866025 + 0.500000i 3.04339 + 3.04339i 0.258819 0.965926i 2.10952 1.59685i 0.707107 + 0.707107i 0.500000 + 0.866025i 2.15200 3.72738i
97.6 0.258819 + 0.965926i 0.866025 + 0.500000i −0.866025 + 0.500000i −2.46294 2.46294i −0.258819 + 0.965926i 0.0731721 + 2.64474i −0.707107 0.707107i 0.500000 + 0.866025i 1.74156 3.01647i
97.7 0.258819 + 0.965926i 0.866025 + 0.500000i −0.866025 + 0.500000i −1.65224 1.65224i −0.258819 + 0.965926i 0.918006 2.48138i −0.707107 0.707107i 0.500000 + 0.866025i 1.16831 2.02357i
97.8 0.258819 + 0.965926i 0.866025 + 0.500000i −0.866025 + 0.500000i −0.0374438 0.0374438i −0.258819 + 0.965926i 2.56811 + 0.636228i −0.707107 0.707107i 0.500000 + 0.866025i 0.0264768 0.0458592i
97.9 0.258819 + 0.965926i 0.866025 + 0.500000i −0.866025 + 0.500000i 1.52143 + 1.52143i −0.258819 + 0.965926i −2.62909 0.296473i −0.707107 0.707107i 0.500000 + 0.866025i −1.07582 + 1.86337i
97.10 0.258819 + 0.965926i 0.866025 + 0.500000i −0.866025 + 0.500000i 2.63119 + 2.63119i −0.258819 + 0.965926i 0.228714 + 2.63585i −0.707107 0.707107i 0.500000 + 0.866025i −1.86053 + 3.22253i
223.1 −0.965926 0.258819i −0.866025 + 0.500000i 0.866025 + 0.500000i −2.53071 2.53071i 0.965926 0.258819i 0.289566 + 2.62986i −0.707107 0.707107i 0.500000 0.866025i 1.78948 + 3.09948i
223.2 −0.965926 0.258819i −0.866025 + 0.500000i 0.866025 + 0.500000i −1.23174 1.23174i 0.965926 0.258819i −2.50370 0.855258i −0.707107 0.707107i 0.500000 0.866025i 0.870974 + 1.50857i
223.3 −0.965926 0.258819i −0.866025 + 0.500000i 0.866025 + 0.500000i 0.781970 + 0.781970i 0.965926 0.258819i −2.22139 1.43716i −0.707107 0.707107i 0.500000 0.866025i −0.552937 0.957714i
223.4 −0.965926 0.258819i −0.866025 + 0.500000i 0.866025 + 0.500000i 1.20611 + 1.20611i 0.965926 0.258819i 2.09456 1.61642i −0.707107 0.707107i 0.500000 0.866025i −0.852845 1.47717i
223.5 −0.965926 0.258819i −0.866025 + 0.500000i 0.866025 + 0.500000i 1.77438 + 1.77438i 0.965926 0.258819i 1.76783 + 1.96844i −0.707107 0.707107i 0.500000 0.866025i −1.25468 2.17316i
223.6 0.965926 + 0.258819i −0.866025 + 0.500000i 0.866025 + 0.500000i −2.50346 2.50346i −0.965926 + 0.258819i −0.691933 + 2.55367i 0.707107 + 0.707107i 0.500000 0.866025i −1.77021 3.06610i
223.7 0.965926 + 0.258819i −0.866025 + 0.500000i 0.866025 + 0.500000i −1.14071 1.14071i −0.965926 + 0.258819i 2.54651 0.717822i 0.707107 + 0.707107i 0.500000 0.866025i −0.806606 1.39708i
223.8 0.965926 + 0.258819i −0.866025 + 0.500000i 0.866025 + 0.500000i −1.10131 1.10131i −0.965926 + 0.258819i −2.04402 1.67988i 0.707107 + 0.707107i 0.500000 0.866025i −0.778741 1.34882i
223.9 0.965926 + 0.258819i −0.866025 + 0.500000i 0.866025 + 0.500000i 1.73144 + 1.73144i −0.965926 + 0.258819i 0.130508 + 2.64253i 0.707107 + 0.707107i 0.500000 0.866025i 1.22431 + 2.12057i
223.10 0.965926 + 0.258819i −0.866025 + 0.500000i 0.866025 + 0.500000i 3.01404 + 3.01404i −0.965926 + 0.258819i 0.900010 2.48797i 0.707107 + 0.707107i 0.500000 0.866025i 2.13125 + 3.69143i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 475.10
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.bc 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.bx.b yes 40
7.b odd 2 1 546.2.bx.a 40
13.f odd 12 1 546.2.bx.a 40
91.bc even 12 1 inner 546.2.bx.b yes 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.bx.a 40 7.b odd 2 1
546.2.bx.a 40 13.f odd 12 1
546.2.bx.b yes 40 1.a even 1 1 trivial
546.2.bx.b yes 40 91.bc even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(10\!\cdots\!48\)\( T_{5}^{16} + \)\(12\!\cdots\!32\)\( T_{5}^{15} + 847065499392 T_{5}^{14} + 427715229824 T_{5}^{13} + \)\(52\!\cdots\!88\)\( T_{5}^{12} + \)\(69\!\cdots\!12\)\( T_{5}^{11} + \)\(45\!\cdots\!72\)\( T_{5}^{10} + 257856619648 T_{5}^{9} + \)\(65\!\cdots\!96\)\( T_{5}^{8} + \)\(76\!\cdots\!00\)\( T_{5}^{7} + \)\(45\!\cdots\!60\)\( T_{5}^{6} + 778894921728 T_{5}^{5} + 53270120448 T_{5}^{4} + 25885495296 T_{5}^{3} + 40155512832 T_{5}^{2} + 2938208256 T_{5} + 107495424 \)">\(T_{5}^{40} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(546, [\chi])\).