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Properties

Label 351.2.l.b
Level $351$
Weight $2$
Character orbit 351.l
Analytic conductor $2.803$
Analytic rank $0$
Dimension $22$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [351,2,Mod(127,351)] 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("351.127"); 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, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 351 = 3^{3} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 351.l (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [22] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.80274911095\)
Analytic rank: \(0\)
Dimension: \(22\)
Relative dimension: \(11\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 22 q - 20 q^{4} + 3 q^{5} - 6 q^{7} - 7 q^{10} + 9 q^{14} + 24 q^{16} - 9 q^{17} - 6 q^{19} + 24 q^{20} + 26 q^{22} - 6 q^{23} + 4 q^{25} + 12 q^{26} + 3 q^{28} - 48 q^{29} - 27 q^{31} + 15 q^{34} + 27 q^{35}+ \cdots - 117 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1 − 2.59035i 0 −4.70993 −1.18696 + 0.685292i 0 −3.20825 + 1.85228i 7.01967i 0 1.77515 + 3.07465i
127.2 − 2.31664i 0 −3.36682 1.09505 − 0.632228i 0 3.61589 − 2.08764i 3.16642i 0 −1.46464 − 2.53684i
127.3 − 1.40574i 0 0.0239006 2.61504 − 1.50979i 0 −2.76663 + 1.59731i − 2.84507i 0 −2.12237 − 3.67605i
127.4 − 1.05773i 0 0.881215 2.71101 − 1.56520i 0 0.784891 − 0.453157i − 3.04754i 0 −1.65555 − 2.86750i
127.5 − 0.968164i 0 1.06266 −3.54737 + 2.04808i 0 −3.06763 + 1.77110i − 2.96516i 0 1.98287 + 3.43444i
127.6 0.391710i 0 1.84656 1.60580 − 0.927107i 0 −0.0712374 + 0.0411289i 1.50674i 0 0.363157 + 0.629006i
127.7 0.571953i 0 1.67287 −0.796103 + 0.459630i 0 −1.67386 + 0.966405i 2.10071i 0 −0.262887 − 0.455333i
127.8 0.782672i 0 1.38742 −0.0536139 + 0.0309540i 0 3.25250 − 1.87783i 2.65124i 0 −0.0242268 − 0.0419621i
127.9 1.93463i 0 −1.74278 2.26677 − 1.30872i 0 2.01692 − 1.16447i 0.497616i 0 2.53189 + 4.38536i
127.10 2.00617i 0 −2.02472 −0.778411 + 0.449416i 0 −2.10878 + 1.21751i − 0.0495935i 0 −0.901605 − 1.56163i
127.11 2.65149i 0 −5.03038 −2.43120 + 1.40366i 0 0.226187 − 0.130589i − 8.03502i 0 −3.72178 − 6.44631i
199.1 − 2.65149i 0 −5.03038 −2.43120 − 1.40366i 0 0.226187 + 0.130589i 8.03502i 0 −3.72178 + 6.44631i
199.2 − 2.00617i 0 −2.02472 −0.778411 − 0.449416i 0 −2.10878 − 1.21751i 0.0495935i 0 −0.901605 + 1.56163i
199.3 − 1.93463i 0 −1.74278 2.26677 + 1.30872i 0 2.01692 + 1.16447i − 0.497616i 0 2.53189 − 4.38536i
199.4 − 0.782672i 0 1.38742 −0.0536139 − 0.0309540i 0 3.25250 + 1.87783i − 2.65124i 0 −0.0242268 + 0.0419621i
199.5 − 0.571953i 0 1.67287 −0.796103 − 0.459630i 0 −1.67386 − 0.966405i − 2.10071i 0 −0.262887 + 0.455333i
199.6 − 0.391710i 0 1.84656 1.60580 + 0.927107i 0 −0.0712374 − 0.0411289i − 1.50674i 0 0.363157 − 0.629006i
199.7 0.968164i 0 1.06266 −3.54737 − 2.04808i 0 −3.06763 − 1.77110i 2.96516i 0 1.98287 − 3.43444i
199.8 1.05773i 0 0.881215 2.71101 + 1.56520i 0 0.784891 + 0.453157i 3.04754i 0 −1.65555 + 2.86750i
199.9 1.40574i 0 0.0239006 2.61504 + 1.50979i 0 −2.76663 − 1.59731i 2.84507i 0 −2.12237 + 3.67605i
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 127.11
Significant digits:
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Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 351.2.l.b 22
3.b odd 2 1 117.2.l.b ✓ 22
9.c even 3 1 351.2.r.b 22
9.d odd 6 1 117.2.r.b yes 22
13.e even 6 1 351.2.r.b 22
39.h odd 6 1 117.2.r.b yes 22
117.l even 6 1 inner 351.2.l.b 22
117.v odd 6 1 117.2.l.b ✓ 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.2.l.b ✓ 22 3.b odd 2 1
117.2.l.b ✓ 22 117.v odd 6 1
117.2.r.b yes 22 9.d odd 6 1
117.2.r.b yes 22 39.h odd 6 1
351.2.l.b 22 1.a even 1 1 trivial
351.2.l.b 22 117.l even 6 1 inner
351.2.r.b 22 9.c even 3 1
351.2.r.b 22 13.e even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{22} + 32 T_{2}^{20} + 432 T_{2}^{18} + 3212 T_{2}^{16} + 14428 T_{2}^{14} + 40524 T_{2}^{12} + \cdots + 243 \) acting on \(S_{2}^{\mathrm{new}}(351, [\chi])\). Copy content Toggle raw display