↑

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 $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [378,2,Mod(25,378)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(378, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([10, 12]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("378.25");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
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

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.01834519640\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(12\) 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.

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

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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 25.12
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 \( T_{5}^{72} - 15 T_{5}^{70} + 41 T_{5}^{69} + 246 T_{5}^{68} - 531 T_{5}^{67} + 4291 T_{5}^{66} + \cdots + 2678994081 \) acting on \(S_{2}^{\mathrm{new}}(378, [\chi])\). Copy content Toggle raw display