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Properties

Label 351.2.f.a
Level $351$
Weight $2$
Character orbit 351.f
Analytic conductor $2.803$
Analytic rank $0$
Dimension $24$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [351,2,Mod(100,351)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("351.100"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(351, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 351 = 3^{3} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 351.f (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.80274911095\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{3})\)
Twist minimal: no (minimal twist has level 117)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - q^{2} - 9 q^{4} + 2 q^{5} - 6 q^{7} + 18 q^{8} + 3 q^{11} - 2 q^{14} - 3 q^{16} - 6 q^{17} - 3 q^{19} - 22 q^{20} + 9 q^{22} + 34 q^{23} - 6 q^{25} + 12 q^{26} - 12 q^{29} - 6 q^{31} - 19 q^{32}+ \cdots + 61 q^{98}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
100.1 −1.32814 − 2.30041i 0 −2.52792 + 4.37849i −0.324360 − 0.561808i 0 1.54792 8.11720 0 −0.861592 + 1.49232i
100.2 −1.14137 − 1.97690i 0 −1.60543 + 2.78069i 0.461458 + 0.799268i 0 0.908370 2.76408 0 1.05338 − 1.82452i
100.3 −1.00395 − 1.73890i 0 −1.01584 + 1.75949i 1.37329 + 2.37860i 0 −2.23810 0.0636126 0 2.75743 − 4.77600i
100.4 −0.602131 − 1.04292i 0 0.274876 − 0.476100i −1.89177 − 3.27665i 0 0.300456 −3.07057 0 −2.27819 + 3.94594i
100.5 −0.433689 − 0.751171i 0 0.623828 − 1.08050i 0.0324057 + 0.0561283i 0 −3.92419 −2.81694 0 0.0281080 − 0.0486844i
100.6 −0.0816825 − 0.141478i 0 0.986656 − 1.70894i 1.55806 + 2.69863i 0 −0.136429 −0.649100 0 0.254532 − 0.440862i
100.7 0.108182 + 0.187377i 0 0.976593 − 1.69151i 0.702153 + 1.21616i 0 3.31452 0.855329 0 −0.151921 + 0.263135i
100.8 0.348782 + 0.604108i 0 0.756702 − 1.31065i −1.44568 − 2.50399i 0 −3.17282 2.45082 0 1.00846 − 1.74670i
100.9 0.567922 + 0.983670i 0 0.354929 − 0.614756i 0.0587384 + 0.101738i 0 0.849445 3.07798 0 −0.0667177 + 0.115558i
100.10 0.900808 + 1.56024i 0 −0.622909 + 1.07891i −1.73153 − 2.99909i 0 3.24477 1.35875 0 3.11954 − 5.40321i
100.11 1.02543 + 1.77610i 0 −1.10302 + 1.91049i 0.737604 + 1.27757i 0 1.16435 −0.422561 0 −1.51272 + 2.62012i
100.12 1.13984 + 1.97426i 0 −1.59846 + 2.76861i 1.46964 + 2.54549i 0 −4.85829 −2.72859 0 −3.35030 + 5.80289i
172.1 −1.32814 + 2.30041i 0 −2.52792 − 4.37849i −0.324360 + 0.561808i 0 1.54792 8.11720 0 −0.861592 − 1.49232i
172.2 −1.14137 + 1.97690i 0 −1.60543 − 2.78069i 0.461458 − 0.799268i 0 0.908370 2.76408 0 1.05338 + 1.82452i
172.3 −1.00395 + 1.73890i 0 −1.01584 − 1.75949i 1.37329 − 2.37860i 0 −2.23810 0.0636126 0 2.75743 + 4.77600i
172.4 −0.602131 + 1.04292i 0 0.274876 + 0.476100i −1.89177 + 3.27665i 0 0.300456 −3.07057 0 −2.27819 − 3.94594i
172.5 −0.433689 + 0.751171i 0 0.623828 + 1.08050i 0.0324057 − 0.0561283i 0 −3.92419 −2.81694 0 0.0281080 + 0.0486844i
172.6 −0.0816825 + 0.141478i 0 0.986656 + 1.70894i 1.55806 − 2.69863i 0 −0.136429 −0.649100 0 0.254532 + 0.440862i
172.7 0.108182 − 0.187377i 0 0.976593 + 1.69151i 0.702153 − 1.21616i 0 3.31452 0.855329 0 −0.151921 − 0.263135i
172.8 0.348782 − 0.604108i 0 0.756702 + 1.31065i −1.44568 + 2.50399i 0 −3.17282 2.45082 0 1.00846 + 1.74670i
See all 24 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 100.12
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
117.f even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 351.2.f.a 24
3.b odd 2 1 117.2.f.a ✓ 24
9.c even 3 1 351.2.h.a 24
9.d odd 6 1 117.2.h.a yes 24
13.c even 3 1 351.2.h.a 24
39.i odd 6 1 117.2.h.a yes 24
117.f even 3 1 inner 351.2.f.a 24
117.u odd 6 1 117.2.f.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.2.f.a ✓ 24 3.b odd 2 1
117.2.f.a ✓ 24 117.u odd 6 1
117.2.h.a yes 24 9.d odd 6 1
117.2.h.a yes 24 39.i odd 6 1
351.2.f.a 24 1.a even 1 1 trivial
351.2.f.a 24 117.f even 3 1 inner
351.2.h.a 24 9.c even 3 1
351.2.h.a 24 13.c even 3 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(351, [\chi])\).