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Properties

Label 351.3.k.a
Level $351$
Weight $3$
Character orbit 351.k
Analytic conductor $9.564$
Analytic rank $0$
Dimension $52$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [351,3,Mod(152,351)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(351, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("351.152");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 351 = 3^{3} \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 351.k (of order \(6\), degree \(2\), not 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: \(9.56405727905\)
Analytic rank: \(0\)
Dimension: \(52\)
Relative dimension: \(26\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 117)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 52 q - 98 q^{4} + 6 q^{5} - q^{7} - 6 q^{10} + 4 q^{13} + 6 q^{14} + 166 q^{16} + 5 q^{19} - 21 q^{20} + 30 q^{22} + 75 q^{23} + 88 q^{25} + 132 q^{26} - 22 q^{28} + 14 q^{31} - 6 q^{34} - 228 q^{35} - 13 q^{37}+ \cdots + 405 q^{98}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
152.1 − 3.86461i 0 −10.9352 −0.468040 − 0.270223i 0 −0.276958 + 0.479705i 26.8020i 0 −1.04431 + 1.80879i
152.2 − 3.53953i 0 −8.52830 6.00755 + 3.46846i 0 −2.39062 + 4.14067i 16.0281i 0 12.2767 − 21.2639i
152.3 − 3.27217i 0 −6.70711 3.48512 + 2.01214i 0 1.40029 − 2.42538i 8.85812i 0 6.58406 − 11.4039i
152.4 − 3.09261i 0 −5.56427 −6.70477 − 3.87100i 0 6.32403 − 10.9535i 4.83767i 0 −11.9715 + 20.7353i
152.5 − 2.75995i 0 −3.61733 −7.95372 − 4.59208i 0 −2.71987 + 4.71096i − 1.05616i 0 −12.6739 + 21.9519i
152.6 − 2.57904i 0 −2.65146 0.817198 + 0.471810i 0 −2.52350 + 4.37084i − 3.47795i 0 1.21682 − 2.10759i
152.7 − 2.44466i 0 −1.97636 −2.73246 − 1.57759i 0 −3.82542 + 6.62581i − 4.94710i 0 −3.85666 + 6.67994i
152.8 − 1.86232i 0 0.531747 −0.895843 − 0.517215i 0 2.55841 − 4.43129i − 8.43958i 0 −0.963222 + 1.66835i
152.9 − 1.68712i 0 1.15363 3.08450 + 1.78084i 0 6.39840 − 11.0823i − 8.69479i 0 3.00448 − 5.20392i
152.10 − 1.67065i 0 1.20892 8.08978 + 4.67064i 0 −4.17857 + 7.23749i − 8.70230i 0 7.80302 − 13.5152i
152.11 − 0.630376i 0 3.60263 −1.96357 − 1.13367i 0 −4.09731 + 7.09675i − 4.79251i 0 −0.714636 + 1.23779i
152.12 − 0.487810i 0 3.76204 6.04718 + 3.49134i 0 3.63606 − 6.29784i − 3.78640i 0 1.70311 − 2.94987i
152.13 − 0.434891i 0 3.81087 −3.11857 − 1.80051i 0 3.94503 − 6.83299i − 3.39687i 0 −0.783024 + 1.35624i
152.14 − 0.376029i 0 3.85860 −0.994641 − 0.574256i 0 −2.57212 + 4.45503i − 2.95506i 0 −0.215937 + 0.374014i
152.15 0.306140i 0 3.90628 −5.23834 − 3.02436i 0 0.307870 − 0.533246i 2.42043i 0 0.925877 − 1.60367i
152.16 1.13140i 0 2.71993 7.06649 + 4.07984i 0 −3.97102 + 6.87801i 7.60294i 0 −4.61594 + 7.99504i
152.17 1.15571i 0 2.66432 3.81803 + 2.20434i 0 2.07846 − 3.60000i 7.70206i 0 −2.54759 + 4.41256i
152.18 1.28020i 0 2.36108 −2.95232 − 1.70452i 0 −5.94919 + 10.3043i 8.14347i 0 2.18213 − 3.77956i
152.19 1.86111i 0 0.536256 4.76252 + 2.74964i 0 2.83533 − 4.91094i 8.44249i 0 −5.11740 + 8.86359i
152.20 1.89941i 0 0.392234 −5.82785 − 3.36471i 0 2.17329 − 3.76424i 8.34266i 0 6.39097 − 11.0695i
See all 52 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 152.26
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
117.k odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 351.3.k.a 52
3.b odd 2 1 117.3.k.a ✓ 52
9.c even 3 1 117.3.u.a yes 52
9.d odd 6 1 351.3.u.a 52
13.c even 3 1 351.3.u.a 52
39.i odd 6 1 117.3.u.a yes 52
117.h even 3 1 117.3.k.a ✓ 52
117.k odd 6 1 inner 351.3.k.a 52
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.3.k.a ✓ 52 3.b odd 2 1
117.3.k.a ✓ 52 117.h even 3 1
117.3.u.a yes 52 9.c even 3 1
117.3.u.a yes 52 39.i odd 6 1
351.3.k.a 52 1.a even 1 1 trivial
351.3.k.a 52 117.k odd 6 1 inner
351.3.u.a 52 9.d odd 6 1
351.3.u.a 52 13.c even 3 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(351, [\chi])\).