Properties

Label 546.2.bx.a
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.

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

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 −3.04339 3.04339i −0.258819 + 0.965926i −1.02848 2.43767i 0.707107 + 0.707107i 0.500000 + 0.866025i −2.15200 + 3.72738i
97.2 −0.258819 0.965926i −0.866025 0.500000i −0.866025 + 0.500000i −0.201763 0.201763i −0.258819 + 0.965926i 1.99885 1.73338i 0.707107 + 0.707107i 0.500000 + 0.866025i −0.142668 + 0.247108i
97.3 −0.258819 0.965926i −0.866025 0.500000i −0.866025 + 0.500000i 0.301627 + 0.301627i −0.258819 + 0.965926i 2.59084 + 0.536243i 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 0.413783 + 0.413783i −0.258819 + 0.965926i −2.23784 + 1.41140i 0.707107 + 0.707107i 0.500000 + 0.866025i 0.292588 0.506778i
97.5 −0.258819 0.965926i −0.866025 0.500000i −0.866025 + 0.500000i 2.52974 + 2.52974i −0.258819 + 0.965926i −2.48229 0.915557i 0.707107 + 0.707107i 0.500000 + 0.866025i 1.78880 3.09829i
97.6 0.258819 + 0.965926i −0.866025 0.500000i −0.866025 + 0.500000i −2.63119 2.63119i 0.258819 0.965926i −1.51600 + 2.16835i −0.707107 0.707107i 0.500000 + 0.866025i 1.86053 3.22253i
97.7 0.258819 + 0.965926i −0.866025 0.500000i −0.866025 + 0.500000i −1.52143 1.52143i 0.258819 0.965926i 2.42509 + 1.05779i −0.707107 0.707107i 0.500000 + 0.866025i 1.07582 1.86337i
97.8 0.258819 + 0.965926i −0.866025 0.500000i −0.866025 + 0.500000i 0.0374438 + 0.0374438i 0.258819 0.965926i −2.54217 0.733067i −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 1.65224 + 1.65224i 0.258819 0.965926i 0.445675 2.60794i −0.707107 0.707107i 0.500000 + 0.866025i −1.16831 + 2.02357i
97.10 0.258819 + 0.965926i −0.866025 0.500000i −0.866025 + 0.500000i 2.46294 + 2.46294i 0.258819 0.965926i −1.38574 + 2.25383i −0.707107 0.707107i 0.500000 + 0.866025i −1.74156 + 3.01647i
223.1 −0.965926 0.258819i 0.866025 0.500000i 0.866025 + 0.500000i −1.77438 1.77438i −0.965926 + 0.258819i 0.546766 2.58864i −0.707107 0.707107i 0.500000 0.866025i 1.25468 + 2.17316i
223.2 −0.965926 0.258819i 0.866025 0.500000i 0.866025 + 0.500000i −1.20611 1.20611i −0.965926 + 0.258819i 2.62215 + 0.352575i −0.707107 0.707107i 0.500000 0.866025i 0.852845 + 1.47717i
223.3 −0.965926 0.258819i 0.866025 0.500000i 0.866025 + 0.500000i −0.781970 0.781970i −0.965926 + 0.258819i −1.20520 + 2.35531i −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 1.23174 + 1.23174i −0.965926 + 0.258819i −1.74064 + 1.99253i −0.707107 0.707107i 0.500000 0.866025i −0.870974 1.50857i
223.5 −0.965926 0.258819i 0.866025 0.500000i 0.866025 + 0.500000i 2.53071 + 2.53071i −0.965926 + 0.258819i −1.06416 2.42231i −0.707107 0.707107i 0.500000 0.866025i −1.78948 3.09948i
223.6 0.965926 + 0.258819i 0.866025 0.500000i 0.866025 + 0.500000i −3.01404 3.01404i 0.965926 0.258819i 2.02342 + 1.70464i 0.707107 + 0.707107i 0.500000 0.866025i −2.13125 3.69143i
223.7 0.965926 + 0.258819i 0.866025 0.500000i 0.866025 + 0.500000i −1.73144 1.73144i 0.965926 0.258819i −1.20824 2.35375i 0.707107 + 0.707107i 0.500000 0.866025i −1.22431 2.12057i
223.8 0.965926 + 0.258819i 0.866025 0.500000i 0.866025 + 0.500000i 1.10131 + 1.10131i 0.965926 0.258819i −0.930231 + 2.47683i 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 1.14071 + 1.14071i 0.965926 0.258819i 2.56426 0.651604i 0.707107 + 0.707107i 0.500000 0.866025i 0.806606 + 1.39708i
223.10 0.965926 + 0.258819i 0.866025 0.500000i 0.866025 + 0.500000i 2.50346 + 2.50346i 0.965926 0.258819i −1.87607 1.86558i 0.707107 + 0.707107i 0.500000 0.866025i 1.77021 + 3.06610i
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.a 40
7.b odd 2 1 546.2.bx.b yes 40
13.f odd 12 1 546.2.bx.b yes 40
91.bc even 12 1 inner 546.2.bx.a 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.bx.a 40 1.a even 1 1 trivial
546.2.bx.a 40 91.bc even 12 1 inner
546.2.bx.b yes 40 7.b odd 2 1
546.2.bx.b yes 40 13.f odd 12 1

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