Properties

Label 378.2.u.c
Level $378$
Weight $2$
Character orbit 378.u
Analytic conductor $3.018$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 378.u (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.01834519640\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(4\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 3 q^{3} + 3 q^{5} + 6 q^{6} + 12 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 3 q^{3} + 3 q^{5} + 6 q^{6} + 12 q^{8} - 3 q^{9} + 3 q^{10} - 9 q^{13} - 6 q^{18} + 15 q^{19} - 6 q^{20} - 3 q^{21} + 6 q^{23} + 3 q^{24} + 33 q^{25} + 18 q^{26} - 18 q^{27} - 24 q^{28} - 30 q^{29} + 15 q^{30} - 6 q^{33} - 3 q^{34} + 3 q^{35} - 12 q^{36} - 3 q^{37} + 15 q^{38} - 18 q^{39} - 3 q^{40} + 27 q^{41} + 18 q^{43} - 3 q^{44} + 12 q^{45} + 15 q^{46} + 3 q^{48} + 3 q^{50} + 24 q^{51} - 9 q^{52} + 6 q^{53} + 45 q^{54} - 66 q^{55} - 6 q^{57} + 3 q^{58} - 30 q^{59} - 30 q^{60} - 57 q^{61} - 18 q^{62} + 12 q^{63} - 12 q^{64} + 24 q^{65} - 54 q^{66} + 39 q^{67} + 3 q^{68} + 36 q^{69} + 3 q^{70} - 24 q^{71} - 6 q^{72} + 36 q^{73} - 12 q^{74} + 15 q^{75} - 15 q^{76} - 9 q^{77} + 18 q^{78} - 33 q^{79} + 6 q^{80} + 9 q^{81} - 15 q^{83} + 3 q^{84} + 36 q^{85} - 18 q^{86} - 84 q^{87} - 9 q^{88} - 3 q^{89} - 30 q^{90} - 9 q^{91} - 30 q^{92} - 45 q^{93} + 45 q^{94} + 63 q^{95} - 30 q^{97} + 12 q^{98} - 57 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1 −0.766044 0.642788i −1.71357 + 0.252371i 0.173648 + 0.984808i 3.65119 + 1.32893i 1.47489 + 0.908132i −0.173648 + 0.984808i 0.500000 0.866025i 2.87262 0.864908i −1.94276 3.36496i
43.2 −0.766044 0.642788i −0.762464 + 1.55520i 0.173648 + 0.984808i −1.51130 0.550067i 1.58375 0.701250i −0.173648 + 0.984808i 0.500000 0.866025i −1.83730 2.37157i 0.804144 + 1.39282i
43.3 −0.766044 0.642788i 0.217603 1.71833i 0.173648 + 0.984808i −3.75298 1.36597i −1.27121 + 1.17644i −0.173648 + 0.984808i 0.500000 0.866025i −2.90530 0.747827i 1.99692 + 3.45877i
43.4 −0.766044 0.642788i 0.992383 + 1.41957i 0.173648 + 0.984808i 0.999746 + 0.363878i 0.152272 1.72534i −0.173648 + 0.984808i 0.500000 0.866025i −1.03035 + 2.81751i −0.531954 0.921371i
85.1 0.939693 + 0.342020i −1.33338 1.10549i 0.766044 + 0.642788i −0.412508 + 2.33945i −0.874866 1.49486i −0.766044 + 0.642788i 0.500000 + 0.866025i 0.555794 + 2.94807i −1.18777 + 2.05728i
85.2 0.939693 + 0.342020i −0.0350640 + 1.73170i 0.766044 + 0.642788i −0.00462235 + 0.0262146i −0.625224 + 1.61527i −0.766044 + 0.642788i 0.500000 + 0.866025i −2.99754 0.121440i −0.0133095 + 0.0230528i
85.3 0.939693 + 0.342020i 0.0792099 1.73024i 0.766044 + 0.642788i 0.488347 2.76955i 0.666209 1.59880i −0.766044 + 0.642788i 0.500000 + 0.866025i −2.98745 0.274104i 1.40614 2.43551i
85.4 0.939693 + 0.342020i 1.72892 0.104015i 0.766044 + 0.642788i −0.163613 + 0.927896i 1.66023 + 0.493585i −0.766044 + 0.642788i 0.500000 + 0.866025i 2.97836 0.359668i −0.471105 + 0.815978i
169.1 0.939693 0.342020i −1.33338 + 1.10549i 0.766044 0.642788i −0.412508 2.33945i −0.874866 + 1.49486i −0.766044 0.642788i 0.500000 0.866025i 0.555794 2.94807i −1.18777 2.05728i
169.2 0.939693 0.342020i −0.0350640 1.73170i 0.766044 0.642788i −0.00462235 0.0262146i −0.625224 1.61527i −0.766044 0.642788i 0.500000 0.866025i −2.99754 + 0.121440i −0.0133095 0.0230528i
169.3 0.939693 0.342020i 0.0792099 + 1.73024i 0.766044 0.642788i 0.488347 + 2.76955i 0.666209 + 1.59880i −0.766044 0.642788i 0.500000 0.866025i −2.98745 + 0.274104i 1.40614 + 2.43551i
169.4 0.939693 0.342020i 1.72892 + 0.104015i 0.766044 0.642788i −0.163613 0.927896i 1.66023 0.493585i −0.766044 0.642788i 0.500000 0.866025i 2.97836 + 0.359668i −0.471105 0.815978i
211.1 −0.766044 + 0.642788i −1.71357 0.252371i 0.173648 0.984808i 3.65119 1.32893i 1.47489 0.908132i −0.173648 0.984808i 0.500000 + 0.866025i 2.87262 + 0.864908i −1.94276 + 3.36496i
211.2 −0.766044 + 0.642788i −0.762464 1.55520i 0.173648 0.984808i −1.51130 + 0.550067i 1.58375 + 0.701250i −0.173648 0.984808i 0.500000 + 0.866025i −1.83730 + 2.37157i 0.804144 1.39282i
211.3 −0.766044 + 0.642788i 0.217603 + 1.71833i 0.173648 0.984808i −3.75298 + 1.36597i −1.27121 1.17644i −0.173648 0.984808i 0.500000 + 0.866025i −2.90530 + 0.747827i 1.99692 3.45877i
211.4 −0.766044 + 0.642788i 0.992383 1.41957i 0.173648 0.984808i 0.999746 0.363878i 0.152272 + 1.72534i −0.173648 0.984808i 0.500000 + 0.866025i −1.03035 2.81751i −0.531954 + 0.921371i
295.1 −0.173648 + 0.984808i −1.63475 + 0.572349i −0.939693 0.342020i 0.789233 0.662245i −0.279782 1.70930i 0.939693 0.342020i 0.500000 0.866025i 2.34483 1.87130i 0.515135 + 0.892241i
295.2 −0.173648 + 0.984808i −1.49215 0.879480i −0.939693 0.342020i −2.11377 + 1.77366i 1.12523 1.31676i 0.939693 0.342020i 0.500000 0.866025i 1.45303 + 2.62463i −1.37966 2.38965i
295.3 −0.173648 + 0.984808i 1.13330 + 1.30982i −0.939693 0.342020i 2.14766 1.80210i −1.48672 + 0.888630i 0.939693 0.342020i 0.500000 0.866025i −0.431280 + 2.96884i 1.40179 + 2.42797i
295.4 −0.173648 + 0.984808i 1.31996 1.12148i −0.939693 0.342020i 1.38261 1.16015i 0.875229 + 1.49465i 0.939693 0.342020i 0.500000 0.866025i 0.484585 2.96060i 0.902434 + 1.56306i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 337.4
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
27.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 378.2.u.c 24
27.e even 9 1 inner 378.2.u.c 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.u.c 24 1.a even 1 1 trivial
378.2.u.c 24 27.e even 9 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{24} - 3 T_{5}^{23} - 12 T_{5}^{22} + 53 T_{5}^{21} + 24 T_{5}^{20} - 408 T_{5}^{19} + 2017 T_{5}^{18} - 3447 T_{5}^{17} + 3699 T_{5}^{16} - 6721 T_{5}^{15} + 105204 T_{5}^{14} - 250746 T_{5}^{13} + 709801 T_{5}^{12} + \cdots + 4096 \) acting on \(S_{2}^{\mathrm{new}}(378, [\chi])\). Copy content Toggle raw display