Properties

Label 378.2.w.b
Level 378
Weight 2
Character orbit 378.w
Analytic conductor 3.018
Analytic rank 0
Dimension 72
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.w (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.01834519640\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(12\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 72q + 3q^{6} + 3q^{7} + 36q^{8} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 72q + 3q^{6} + 3q^{7} + 36q^{8} - 12q^{10} - 6q^{11} + 12q^{13} - 3q^{14} - 6q^{15} + 24q^{17} - 36q^{19} + 18q^{21} + 6q^{22} - 6q^{23} + 30q^{25} - 18q^{26} + 6q^{27} + 3q^{29} + 15q^{30} - 9q^{31} - 18q^{33} - 9q^{34} + 9q^{35} - 3q^{36} + 51q^{39} - 6q^{41} - 27q^{42} - 24q^{43} + 18q^{45} + 27q^{47} + 6q^{48} - 69q^{49} + 6q^{50} + 6q^{51} - 6q^{52} - 15q^{53} + 45q^{54} - 72q^{55} + 6q^{56} + 57q^{57} - 3q^{58} + 15q^{59} - 33q^{60} - 18q^{61} - 24q^{62} - 12q^{63} - 36q^{64} - 90q^{65} - 36q^{66} - 66q^{67} - 18q^{68} - 39q^{69} - 12q^{70} + 12q^{71} + 30q^{73} + 9q^{74} - 21q^{75} - 87q^{77} + 6q^{78} - 45q^{79} - 6q^{80} - 24q^{81} + 33q^{82} + 18q^{83} + 6q^{84} + 51q^{85} - 12q^{86} - 18q^{87} - 12q^{88} + 72q^{89} - 69q^{90} - 30q^{91} + 12q^{92} - 48q^{93} + 21q^{95} + 48q^{97} + 6q^{98} + 90q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1 0.939693 0.342020i −1.63077 + 0.583588i 0.766044 0.642788i 2.72763 + 0.992778i −1.33283 + 1.10615i −1.62553 + 2.08750i 0.500000 0.866025i 2.31885 1.90340i 2.90269
25.2 0.939693 0.342020i −1.37994 1.04679i 0.766044 0.642788i 0.829059 + 0.301753i −1.65474 0.511691i 1.89032 + 1.85113i 0.500000 0.866025i 0.808471 + 2.88901i 0.882266
25.3 0.939693 0.342020i −1.28916 1.15675i 0.766044 0.642788i −2.54535 0.926430i −1.60704 0.646075i −2.64478 0.0716849i 0.500000 0.866025i 0.323845 + 2.98247i −2.70870
25.4 0.939693 0.342020i −1.25782 + 1.19075i 0.766044 0.642788i −1.38782 0.505123i −0.774703 + 1.54914i 0.493387 2.59934i 0.500000 0.866025i 0.164223 2.99550i −1.47688
25.5 0.939693 0.342020i −0.902405 + 1.47840i 0.766044 0.642788i −3.70320 1.34785i −0.342341 + 1.69788i −0.207321 + 2.63762i 0.500000 0.866025i −1.37133 2.66823i −3.94086
25.6 0.939693 0.342020i −0.214796 + 1.71868i 0.766044 0.642788i 3.87748 + 1.41129i 0.385982 + 1.68850i 2.09775 1.61228i 0.500000 0.866025i −2.90773 0.738330i 4.12633
25.7 0.939693 0.342020i −0.0864462 1.72989i 0.766044 0.642788i 1.47620 + 0.537294i −0.672891 1.59600i 0.815021 2.51709i 0.500000 0.866025i −2.98505 + 0.299085i 1.57094
25.8 0.939693 0.342020i 1.05262 + 1.37550i 0.766044 0.642788i −0.324117 0.117969i 1.45959 + 0.932530i 0.0472752 + 2.64533i 0.500000 0.866025i −0.783993 + 2.89575i −0.344919
25.9 0.939693 0.342020i 1.10671 1.33237i 0.766044 0.642788i −2.28151 0.830401i 0.584268 1.63053i −2.63284 0.261069i 0.500000 0.866025i −0.550401 2.94908i −2.42793
25.10 0.939693 0.342020i 1.47894 0.901527i 0.766044 0.642788i 1.05304 + 0.383275i 1.08140 1.35298i 1.66470 + 2.05639i 0.500000 0.866025i 1.37450 2.66660i 1.12062
25.11 0.939693 0.342020i 1.56589 + 0.740254i 0.766044 0.642788i 1.77013 + 0.644273i 1.72464 + 0.160044i −2.34950 1.21648i 0.500000 0.866025i 1.90405 + 2.31832i 1.88373
25.12 0.939693 0.342020i 1.73083 + 0.0649612i 0.766044 0.642788i −3.37095 1.22693i 1.64867 0.530936i 2.18547 1.49122i 0.500000 0.866025i 2.99156 + 0.224874i −3.58729
121.1 0.939693 + 0.342020i −1.63077 0.583588i 0.766044 + 0.642788i 2.72763 0.992778i −1.33283 1.10615i −1.62553 2.08750i 0.500000 + 0.866025i 2.31885 + 1.90340i 2.90269
121.2 0.939693 + 0.342020i −1.37994 + 1.04679i 0.766044 + 0.642788i 0.829059 0.301753i −1.65474 + 0.511691i 1.89032 1.85113i 0.500000 + 0.866025i 0.808471 2.88901i 0.882266
121.3 0.939693 + 0.342020i −1.28916 + 1.15675i 0.766044 + 0.642788i −2.54535 + 0.926430i −1.60704 + 0.646075i −2.64478 + 0.0716849i 0.500000 + 0.866025i 0.323845 2.98247i −2.70870
121.4 0.939693 + 0.342020i −1.25782 1.19075i 0.766044 + 0.642788i −1.38782 + 0.505123i −0.774703 1.54914i 0.493387 + 2.59934i 0.500000 + 0.866025i 0.164223 + 2.99550i −1.47688
121.5 0.939693 + 0.342020i −0.902405 1.47840i 0.766044 + 0.642788i −3.70320 + 1.34785i −0.342341 1.69788i −0.207321 2.63762i 0.500000 + 0.866025i −1.37133 + 2.66823i −3.94086
121.6 0.939693 + 0.342020i −0.214796 1.71868i 0.766044 + 0.642788i 3.87748 1.41129i 0.385982 1.68850i 2.09775 + 1.61228i 0.500000 + 0.866025i −2.90773 + 0.738330i 4.12633
121.7 0.939693 + 0.342020i −0.0864462 + 1.72989i 0.766044 + 0.642788i 1.47620 0.537294i −0.672891 + 1.59600i 0.815021 + 2.51709i 0.500000 + 0.866025i −2.98505 0.299085i 1.57094
121.8 0.939693 + 0.342020i 1.05262 1.37550i 0.766044 + 0.642788i −0.324117 + 0.117969i 1.45959 0.932530i 0.0472752 2.64533i 0.500000 + 0.866025i −0.783993 2.89575i −0.344919
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 373.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
189.w 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.w.b yes 72
7.c even 3 1 378.2.v.a 72
27.e even 9 1 378.2.v.a 72
189.w even 9 1 inner 378.2.w.b yes 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.v.a 72 7.c even 3 1
378.2.v.a 72 27.e even 9 1
378.2.w.b yes 72 1.a even 1 1 trivial
378.2.w.b yes 72 189.w even 9 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{5}^{72} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(378, [\chi])\).

Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database