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 \(\Q(\zeta_{9})\)
Coefficient ring index: multiple of None
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.

\(\operatorname{Tr}(f)(q) = \) \( 36q + 3q^{3} + 3q^{5} + 6q^{6} + 18q^{8} + 3q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 36q + 3q^{3} + 3q^{5} + 6q^{6} + 18q^{8} + 3q^{9} - 3q^{10} + 9q^{13} - 6q^{15} + 6q^{18} - 15q^{19} - 6q^{20} + 3q^{21} - 3q^{24} - 3q^{25} - 18q^{26} - 30q^{27} + 36q^{28} - 6q^{29} - 9q^{30} + 60q^{33} + 9q^{34} + 3q^{35} - 27q^{37} + 15q^{38} + 6q^{39} - 3q^{40} - 9q^{41} - 6q^{43} + 9q^{44} + 42q^{45} - 9q^{46} + 36q^{47} + 3q^{48} - 15q^{50} - 36q^{51} + 9q^{52} - 42q^{53} - 27q^{54} + 30q^{55} - 18q^{57} - 21q^{58} + 12q^{59} + 24q^{60} + 3q^{61} + 18q^{62} + 12q^{63} - 18q^{64} - 84q^{65} + 18q^{66} - 69q^{67} + 9q^{68} - 48q^{69} - 3q^{70} + 12q^{71} + 6q^{72} - 12q^{73} + 9q^{75} - 15q^{76} + 9q^{77} - 48q^{78} - 51q^{79} - 6q^{80} - 69q^{81} + 15q^{83} + 3q^{84} - 12q^{85} + 6q^{86} + 84q^{87} - 9q^{88} - 3q^{89} + 6q^{90} - 9q^{91} + 18q^{92} - 21q^{93} + 9q^{94} - 75q^{95} - 42q^{97} + 18q^{98} + 39q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
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 \(T_{5}^{36} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(378, [\chi])\).

Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database