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Properties

Label 351.3.u.a
Level $351$
Weight $3$
Character orbit 351.u
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(35,351)]
 
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([1, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("351.35");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 351 = 3^{3} \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 351.u (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.

\(\operatorname{Tr}(f)(q) = \) \( 52 q + 3 q^{2} + 49 q^{4} + 6 q^{5} + 2 q^{7} - 6 q^{10} - 33 q^{11} + 4 q^{13} + 6 q^{14} - 83 q^{16} + 5 q^{19} - 15 q^{22} + 88 q^{25} - 132 q^{26} - 22 q^{28} + 30 q^{29} + 14 q^{31} + 63 q^{32} - 6 q^{34}+ \cdots - 405 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
35.1 −3.21405 + 1.85563i 0 4.88675 − 8.46411i 6.34683 − 3.66434i 0 5.81323 21.4270i 0 −13.5994 + 23.5548i
35.2 −3.13273 + 1.80868i 0 4.54266 − 7.86812i 1.92042 − 1.10875i 0 −9.66874 18.3955i 0 −4.01077 + 6.94686i
35.3 −2.95186 + 1.70426i 0 3.80900 − 6.59737i −6.08619 + 3.51386i 0 −9.14903 12.3320i 0 11.9771 − 20.7449i
35.4 −2.92701 + 1.68991i 0 3.71158 − 6.42864i −0.0255066 + 0.0147263i 0 11.8046 11.5696i 0 0.0497721 − 0.0862077i
35.5 −2.32292 + 1.34114i 0 1.59730 − 2.76660i 1.20758 − 0.697196i 0 0.0654956 − 2.16032i 0 −1.87007 + 3.23906i
35.6 −2.19669 + 1.26826i 0 1.21696 − 2.10784i −4.69139 + 2.70857i 0 0.439627 − 3.97239i 0 6.87035 − 11.8998i
35.7 −1.64494 + 0.949706i 0 −0.196117 + 0.339685i −5.82785 + 3.36471i 0 −4.34657 − 8.34266i 0 6.39097 − 11.0695i
35.8 −1.61177 + 0.930557i 0 −0.268128 + 0.464411i 4.76252 − 2.74964i 0 −5.67067 − 8.44249i 0 −5.11740 + 8.86359i
35.9 −1.10869 + 0.640101i 0 −1.18054 + 2.04476i −2.95232 + 1.70452i 0 11.8984 − 8.14347i 0 2.18213 − 3.77956i
35.10 −1.00088 + 0.577857i 0 −1.33216 + 2.30737i 3.81803 − 2.20434i 0 −4.15692 − 7.70206i 0 −2.54759 + 4.41256i
35.11 −0.979822 + 0.565700i 0 −1.35997 + 2.35553i 7.06649 − 4.07984i 0 7.94204 − 7.60294i 0 −4.61594 + 7.99504i
35.12 −0.265125 + 0.153070i 0 −1.95314 + 3.38294i −5.23834 + 3.02436i 0 −0.615739 − 2.42043i 0 0.925877 − 1.60367i
35.13 0.325651 − 0.188014i 0 −1.92930 + 3.34165i −0.994641 + 0.574256i 0 5.14423 2.95506i 0 −0.215937 + 0.374014i
35.14 0.376626 − 0.217445i 0 −1.90544 + 3.30031i −3.11857 + 1.80051i 0 −7.89006 3.39687i 0 −0.783024 + 1.35624i
35.15 0.422456 − 0.243905i 0 −1.88102 + 3.25802i 6.04718 − 3.49134i 0 −7.27212 3.78640i 0 1.70311 − 2.94987i
35.16 0.545922 − 0.315188i 0 −1.80131 + 3.11997i −1.96357 + 1.13367i 0 8.19462 4.79251i 0 −0.714636 + 1.23779i
35.17 1.44683 − 0.835327i 0 −0.604458 + 1.04695i 8.08978 − 4.67064i 0 8.35713 8.70230i 0 7.80302 − 13.5152i
35.18 1.46109 − 0.843560i 0 −0.576814 + 0.999071i 3.08450 − 1.78084i 0 −12.7968 8.69479i 0 3.00448 − 5.20392i
35.19 1.61282 − 0.931162i 0 −0.265873 + 0.460506i −0.895843 + 0.517215i 0 −5.11682 8.43958i 0 −0.963222 + 1.66835i
35.20 2.11714 − 1.22233i 0 0.988182 − 1.71158i −2.73246 + 1.57759i 0 7.65083 4.94710i 0 −3.85666 + 6.67994i
See all 52 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 35.26
Significant digits:
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Inner twists

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

Hecke kernels

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