Properties

Label 378.2.u.e
Level $378$
Weight $2$
Character orbit 378.u
Analytic conductor $3.018$
Analytic rank $0$
Dimension $36$
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: \(36\)
Relative dimension: \(6\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 3 q^{3} + 3 q^{5} + 6 q^{6} + 18 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q + 3 q^{3} + 3 q^{5} + 6 q^{6} + 18 q^{8} + 3 q^{9} - 3 q^{10} + 9 q^{13} - 6 q^{15} + 6 q^{18} - 15 q^{19} - 6 q^{20} + 3 q^{21} - 3 q^{24} - 3 q^{25} - 18 q^{26} - 30 q^{27} + 36 q^{28} - 6 q^{29} - 9 q^{30} + 60 q^{33} + 9 q^{34} + 3 q^{35} - 27 q^{37} + 15 q^{38} + 6 q^{39} - 3 q^{40} - 9 q^{41} - 6 q^{43} + 9 q^{44} + 42 q^{45} - 9 q^{46} + 36 q^{47} + 3 q^{48} - 15 q^{50} - 36 q^{51} + 9 q^{52} - 42 q^{53} - 27 q^{54} + 30 q^{55} - 18 q^{57} - 21 q^{58} + 12 q^{59} + 24 q^{60} + 3 q^{61} + 18 q^{62} + 12 q^{63} - 18 q^{64} - 84 q^{65} + 18 q^{66} - 69 q^{67} + 9 q^{68} - 48 q^{69} - 3 q^{70} + 12 q^{71} + 6 q^{72} - 12 q^{73} + 9 q^{75} - 15 q^{76} + 9 q^{77} - 48 q^{78} - 51 q^{79} - 6 q^{80} - 69 q^{81} + 15 q^{83} + 3 q^{84} - 12 q^{85} + 6 q^{86} + 84 q^{87} - 9 q^{88} - 3 q^{89} + 6 q^{90} - 9 q^{91} + 18 q^{92} - 21 q^{93} + 9 q^{94} - 75 q^{95} - 42 q^{97} + 18 q^{98} + 39 q^{99}+O(q^{100}) \) 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 −1.73154 0.0419083i 0.173648 + 0.984808i −0.354259 0.128940i 1.29950 + 1.14512i 0.173648 0.984808i 0.500000 0.866025i 2.99649 + 0.145132i 0.188497 + 0.326487i
43.2 −0.766044 0.642788i −1.15570 1.29010i 0.173648 + 0.984808i −2.06967 0.753297i 0.0560567 + 1.73114i 0.173648 0.984808i 0.500000 0.866025i −0.328718 + 2.98194i 1.10125 + 1.90741i
43.3 −0.766044 0.642788i −0.910669 + 1.47332i 0.173648 + 0.984808i 1.95320 + 0.710905i 1.64465 0.543265i 0.173648 0.984808i 0.500000 0.866025i −1.34137 2.68342i −1.03927 1.80007i
43.4 −0.766044 0.642788i 0.391603 1.68720i 0.173648 + 0.984808i 2.08466 + 0.758756i −1.38450 + 1.04075i 0.173648 0.984808i 0.500000 0.866025i −2.69329 1.32143i −1.10923 1.92124i
43.5 −0.766044 0.642788i 1.44903 + 0.948848i 0.173648 + 0.984808i −3.88962 1.41571i −0.500114 1.65828i 0.173648 0.984808i 0.500000 0.866025i 1.19938 + 2.74982i 2.06962 + 3.58470i
43.6 −0.766044 0.642788i 1.69123 + 0.373800i 0.173648 + 0.984808i 3.54173 + 1.28909i −1.05529 1.37345i 0.173648 0.984808i 0.500000 0.866025i 2.72055 + 1.26437i −1.88452 3.26408i
85.1 0.939693 + 0.342020i −1.62983 0.586228i 0.766044 + 0.642788i 0.184482 1.04625i −1.33103 1.10831i 0.766044 0.642788i 0.500000 + 0.866025i 2.31267 + 1.91090i 0.531195 0.920056i
85.2 0.939693 + 0.342020i −1.33644 + 1.10178i 0.766044 + 0.642788i −0.619299 + 3.51222i −1.63268 + 0.578248i 0.766044 0.642788i 0.500000 + 0.866025i 0.572147 2.94494i −1.78320 + 3.08859i
85.3 0.939693 + 0.342020i 0.873169 1.49585i 0.766044 + 0.642788i −0.605889 + 3.43617i 1.33212 1.10700i 0.766044 0.642788i 0.500000 + 0.866025i −1.47515 2.61227i −1.74459 + 3.02172i
85.4 0.939693 + 0.342020i 0.927705 + 1.46266i 0.766044 + 0.642788i 0.587615 3.33253i 0.371500 + 1.69174i 0.766044 0.642788i 0.500000 + 0.866025i −1.27873 + 2.71383i 1.69197 2.93058i
85.5 0.939693 + 0.342020i 1.28430 + 1.16214i 0.766044 + 0.642788i −0.345781 + 1.96102i 0.809368 + 1.53131i 0.766044 0.642788i 0.500000 + 0.866025i 0.298838 + 2.98508i −0.995637 + 1.72449i
85.6 0.939693 + 0.342020i 1.32079 1.12050i 0.766044 + 0.642788i 0.359180 2.03701i 1.62437 0.601188i 0.766044 0.642788i 0.500000 + 0.866025i 0.488967 2.95988i 1.03422 1.79132i
169.1 0.939693 0.342020i −1.62983 + 0.586228i 0.766044 0.642788i 0.184482 + 1.04625i −1.33103 + 1.10831i 0.766044 + 0.642788i 0.500000 0.866025i 2.31267 1.91090i 0.531195 + 0.920056i
169.2 0.939693 0.342020i −1.33644 1.10178i 0.766044 0.642788i −0.619299 3.51222i −1.63268 0.578248i 0.766044 + 0.642788i 0.500000 0.866025i 0.572147 + 2.94494i −1.78320 3.08859i
169.3 0.939693 0.342020i 0.873169 + 1.49585i 0.766044 0.642788i −0.605889 3.43617i 1.33212 + 1.10700i 0.766044 + 0.642788i 0.500000 0.866025i −1.47515 + 2.61227i −1.74459 3.02172i
169.4 0.939693 0.342020i 0.927705 1.46266i 0.766044 0.642788i 0.587615 + 3.33253i 0.371500 1.69174i 0.766044 + 0.642788i 0.500000 0.866025i −1.27873 2.71383i 1.69197 + 2.93058i
169.5 0.939693 0.342020i 1.28430 1.16214i 0.766044 0.642788i −0.345781 1.96102i 0.809368 1.53131i 0.766044 + 0.642788i 0.500000 0.866025i 0.298838 2.98508i −0.995637 1.72449i
169.6 0.939693 0.342020i 1.32079 + 1.12050i 0.766044 0.642788i 0.359180 + 2.03701i 1.62437 + 0.601188i 0.766044 + 0.642788i 0.500000 0.866025i 0.488967 + 2.95988i 1.03422 + 1.79132i
211.1 −0.766044 + 0.642788i −1.73154 + 0.0419083i 0.173648 0.984808i −0.354259 + 0.128940i 1.29950 1.14512i 0.173648 + 0.984808i 0.500000 + 0.866025i 2.99649 0.145132i 0.188497 0.326487i
211.2 −0.766044 + 0.642788i −1.15570 + 1.29010i 0.173648 0.984808i −2.06967 + 0.753297i 0.0560567 1.73114i 0.173648 + 0.984808i 0.500000 + 0.866025i −0.328718 2.98194i 1.10125 1.90741i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 337.6
Significant digits:
Format:

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.e 36
27.e even 9 1 inner 378.2.u.e 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.u.e 36 1.a even 1 1 trivial
378.2.u.e 36 27.e even 9 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{36} - 3 T_{5}^{35} + 6 T_{5}^{34} - 15 T_{5}^{33} - 9 T_{5}^{32} + 297 T_{5}^{31} + 2244 T_{5}^{30} - 11889 T_{5}^{29} + 44739 T_{5}^{28} - 143496 T_{5}^{27} + 493074 T_{5}^{26} - 1076814 T_{5}^{25} + \cdots + 7766544384 \) acting on \(S_{2}^{\mathrm{new}}(378, [\chi])\). Copy content Toggle raw display