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Properties

Label 378.2.z.a
Level $378$
Weight $2$
Character orbit 378.z
Analytic conductor $3.018$
Analytic rank $0$
Dimension $144$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [378,2,Mod(41,378)]
 
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([17, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("378.41");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 378.z (of order \(18\), 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: \(144\)
Relative dimension: \(24\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 144 q - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 144 q - 12 q^{9} + 12 q^{11} + 6 q^{14} + 12 q^{15} + 24 q^{21} - 60 q^{23} + 12 q^{29} - 72 q^{30} + 54 q^{35} - 12 q^{36} + 48 q^{39} + 24 q^{42} + 18 q^{49} - 60 q^{50} - 36 q^{51} + 6 q^{56} - 60 q^{57} - 36 q^{60} - 78 q^{63} + 72 q^{64} - 120 q^{65} + 36 q^{70} - 72 q^{71} - 24 q^{72} - 36 q^{74} + 66 q^{77} - 60 q^{78} - 72 q^{79} + 12 q^{84} - 72 q^{85} - 48 q^{86} - 18 q^{91} + 12 q^{92} + 24 q^{93} - 120 q^{95} + 36 q^{98} + 144 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
41.1 −0.984808 + 0.173648i −1.71625 + 0.233448i 0.939693 − 0.342020i −0.0848896 − 0.0712308i 1.64964 − 0.527925i −1.74235 − 1.99104i −0.866025 + 0.500000i 2.89100 − 0.801309i 0.0959690 + 0.0554077i
41.2 −0.984808 + 0.173648i −1.56441 − 0.743380i 0.939693 − 0.342020i −0.941136 − 0.789707i 1.66973 + 0.460429i 2.62387 + 0.339554i −0.866025 + 0.500000i 1.89477 + 2.32590i 1.06397 + 0.614283i
41.3 −0.984808 + 0.173648i −1.34465 + 1.09174i 0.939693 − 0.342020i −2.60134 − 2.18278i 1.13465 − 1.30865i −0.615646 + 2.57313i −0.866025 + 0.500000i 0.616190 − 2.93604i 2.94085 + 1.69790i
41.4 −0.984808 + 0.173648i −0.826027 − 1.52239i 0.939693 − 0.342020i 1.99843 + 1.67688i 1.07784 + 1.35583i 0.187552 + 2.63910i −0.866025 + 0.500000i −1.63536 + 2.51507i −2.25926 − 1.30438i
41.5 −0.984808 + 0.173648i −0.754516 + 1.55907i 0.939693 − 0.342020i 0.486827 + 0.408497i 0.472323 − 1.66641i −0.719601 − 2.54601i −0.866025 + 0.500000i −1.86141 − 2.35269i −0.550366 − 0.317754i
41.6 −0.984808 + 0.173648i −0.0283469 − 1.73182i 0.939693 − 0.342020i −2.71537 − 2.27847i 0.328643 + 1.70059i 0.173696 − 2.64004i −0.866025 + 0.500000i −2.99839 + 0.0981834i 3.06977 + 1.77233i
41.7 −0.984808 + 0.173648i 0.0283469 + 1.73182i 0.939693 − 0.342020i 2.71537 + 2.27847i −0.328643 − 1.70059i −1.56393 + 2.13404i −0.866025 + 0.500000i −2.99839 + 0.0981834i −3.06977 − 1.77233i
41.8 −0.984808 + 0.173648i 0.754516 − 1.55907i 0.939693 − 0.342020i −0.486827 − 0.408497i −0.472323 + 1.66641i −2.18779 + 1.48781i −0.866025 + 0.500000i −1.86141 − 2.35269i 0.550366 + 0.317754i
41.9 −0.984808 + 0.173648i 0.826027 + 1.52239i 0.939693 − 0.342020i −1.99843 − 1.67688i −1.07784 − 1.35583i 1.84005 − 1.90111i −0.866025 + 0.500000i −1.63536 + 2.51507i 2.25926 + 1.30438i
41.10 −0.984808 + 0.173648i 1.34465 − 1.09174i 0.939693 − 0.342020i 2.60134 + 2.18278i −1.13465 + 1.30865i 1.18236 − 2.36686i −0.866025 + 0.500000i 0.616190 − 2.93604i −2.94085 − 1.69790i
41.11 −0.984808 + 0.173648i 1.56441 + 0.743380i 0.939693 − 0.342020i 0.941136 + 0.789707i −1.66973 − 0.460429i 2.22826 + 1.42648i −0.866025 + 0.500000i 1.89477 + 2.32590i −1.06397 − 0.614283i
41.12 −0.984808 + 0.173648i 1.71625 − 0.233448i 0.939693 − 0.342020i 0.0848896 + 0.0712308i −1.64964 + 0.527925i −2.61453 + 0.405264i −0.866025 + 0.500000i 2.89100 − 0.801309i −0.0959690 − 0.0554077i
41.13 0.984808 − 0.173648i −1.64560 − 0.540384i 0.939693 − 0.342020i 0.881233 + 0.739442i −1.71443 − 0.246420i −0.336437 + 2.62427i 0.866025 − 0.500000i 2.41597 + 1.77851i 0.996248 + 0.575184i
41.14 0.984808 − 0.173648i −1.47964 + 0.900370i 0.939693 − 0.342020i −1.23395 − 1.03541i −1.30081 + 1.14363i 1.54951 − 2.14453i 0.866025 − 0.500000i 1.37867 − 2.66445i −1.39500 − 0.805406i
41.15 0.984808 − 0.173648i −1.41919 + 0.992922i 0.939693 − 0.342020i 3.28093 + 2.75302i −1.22521 + 1.22428i −2.27542 − 1.34999i 0.866025 − 0.500000i 1.02821 − 2.81829i 3.70914 + 2.14147i
41.16 0.984808 − 0.173648i −1.33409 − 1.10463i 0.939693 − 0.342020i −1.22837 − 1.03073i −1.50564 − 0.856190i −2.38077 − 1.15409i 0.866025 − 0.500000i 0.559569 + 2.94735i −1.38870 − 0.801764i
41.17 0.984808 − 0.173648i −0.419652 − 1.68044i 0.939693 − 0.342020i 1.95067 + 1.63681i −0.705083 − 1.58204i 2.43278 − 1.04000i 0.866025 − 0.500000i −2.64778 + 1.41040i 2.20527 + 1.27321i
41.18 0.984808 − 0.173648i −0.0750633 − 1.73042i 0.939693 − 0.342020i −2.14704 − 1.80158i −0.374408 − 1.69110i 2.53928 + 0.743006i 0.866025 − 0.500000i −2.98873 + 0.259783i −2.42726 − 1.40138i
41.19 0.984808 − 0.173648i 0.0750633 + 1.73042i 0.939693 − 0.342020i 2.14704 + 1.80158i 0.374408 + 1.69110i 2.42280 + 1.06304i 0.866025 − 0.500000i −2.98873 + 0.259783i 2.42726 + 1.40138i
41.20 0.984808 − 0.173648i 0.419652 + 1.68044i 0.939693 − 0.342020i −1.95067 − 1.63681i 0.705083 + 1.58204i 1.19511 + 2.36045i 0.866025 − 0.500000i −2.64778 + 1.41040i −2.20527 − 1.27321i
See next 80 embeddings (of 144 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 41.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
27.f odd 18 1 inner
189.be even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 378.2.z.a ✓ 144
7.b odd 2 1 inner 378.2.z.a ✓ 144
27.f odd 18 1 inner 378.2.z.a ✓ 144
189.be even 18 1 inner 378.2.z.a ✓ 144
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.z.a ✓ 144 1.a even 1 1 trivial
378.2.z.a ✓ 144 7.b odd 2 1 inner
378.2.z.a ✓ 144 27.f odd 18 1 inner
378.2.z.a ✓ 144 189.be even 18 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(378, [\chi])\).