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Properties

Label 351.2.r.b
Level $351$
Weight $2$
Character orbit 351.r
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(10,351)] 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("351.10"); 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([2, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 351 = 3^{3} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 351.r (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.

\(\operatorname{Tr}(f)(q) = \) \( 22 q + 10 q^{4} - 3 q^{5} - 7 q^{10} - 3 q^{11} + 3 q^{13} + 9 q^{14} - 12 q^{16} - 9 q^{17} - 6 q^{19} - 13 q^{22} + 12 q^{23} + 4 q^{25} + 12 q^{26} + 3 q^{28} + 24 q^{29} + 27 q^{31} - 15 q^{34} + 27 q^{35}+ \cdots - 117 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1 −2.29626 − 1.32574i 0 2.51519 + 4.35644i 2.43120 + 1.40366i 0 0.261179i − 8.03502i 0 −3.72178 − 6.44631i
10.2 −1.73739 − 1.00309i 0 1.01236 + 1.75346i 0.778411 + 0.449416i 0 − 2.43501i − 0.0495935i 0 −0.901605 − 1.56163i
10.3 −1.67544 − 0.967314i 0 0.871392 + 1.50930i −2.26677 − 1.30872i 0 2.32894i 0.497616i 0 2.53189 + 4.38536i
10.4 −0.677814 − 0.391336i 0 −0.693712 − 1.20154i 0.0536139 + 0.0309540i 0 3.75567i 2.65124i 0 −0.0242268 − 0.0419621i
10.5 −0.495326 − 0.285977i 0 −0.836435 − 1.44875i 0.796103 + 0.459630i 0 − 1.93281i 2.10071i 0 −0.262887 − 0.455333i
10.6 −0.339230 − 0.195855i 0 −0.923282 − 1.59917i −1.60580 − 0.927107i 0 − 0.0822579i 1.50674i 0 0.363157 + 0.629006i
10.7 0.838455 + 0.484082i 0 −0.531329 − 0.920289i 3.54737 + 2.04808i 0 − 3.54220i − 2.96516i 0 1.98287 + 3.43444i
10.8 0.916018 + 0.528863i 0 −0.440607 − 0.763154i −2.71101 − 1.56520i 0 0.906314i − 3.04754i 0 −1.65555 − 2.86750i
10.9 1.21740 + 0.702869i 0 −0.0119503 − 0.0206986i −2.61504 − 1.50979i 0 − 3.19463i − 2.84507i 0 −2.12237 − 3.67605i
10.10 2.00627 + 1.15832i 0 1.68341 + 2.91575i −1.09505 − 0.632228i 0 4.17527i 3.16642i 0 −1.46464 − 2.53684i
10.11 2.24331 + 1.29518i 0 2.35496 + 4.07892i 1.18696 + 0.685292i 0 − 3.70457i 7.01967i 0 1.77515 + 3.07465i
316.1 −2.29626 + 1.32574i 0 2.51519 − 4.35644i 2.43120 − 1.40366i 0 − 0.261179i 8.03502i 0 −3.72178 + 6.44631i
316.2 −1.73739 + 1.00309i 0 1.01236 − 1.75346i 0.778411 − 0.449416i 0 2.43501i 0.0495935i 0 −0.901605 + 1.56163i
316.3 −1.67544 + 0.967314i 0 0.871392 − 1.50930i −2.26677 + 1.30872i 0 − 2.32894i − 0.497616i 0 2.53189 − 4.38536i
316.4 −0.677814 + 0.391336i 0 −0.693712 + 1.20154i 0.0536139 − 0.0309540i 0 − 3.75567i − 2.65124i 0 −0.0242268 + 0.0419621i
316.5 −0.495326 + 0.285977i 0 −0.836435 + 1.44875i 0.796103 − 0.459630i 0 1.93281i − 2.10071i 0 −0.262887 + 0.455333i
316.6 −0.339230 + 0.195855i 0 −0.923282 + 1.59917i −1.60580 + 0.927107i 0 0.0822579i − 1.50674i 0 0.363157 − 0.629006i
316.7 0.838455 − 0.484082i 0 −0.531329 + 0.920289i 3.54737 − 2.04808i 0 3.54220i 2.96516i 0 1.98287 − 3.43444i
316.8 0.916018 − 0.528863i 0 −0.440607 + 0.763154i −2.71101 + 1.56520i 0 − 0.906314i 3.04754i 0 −1.65555 + 2.86750i
316.9 1.21740 − 0.702869i 0 −0.0119503 + 0.0206986i −2.61504 + 1.50979i 0 3.19463i 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 10.11
Significant digits:
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Inner twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{22} - 16 T_{2}^{20} + 168 T_{2}^{18} - 1012 T_{2}^{16} + 4402 T_{2}^{14} - 11910 T_{2}^{12} + \cdots + 243 \) acting on \(S_{2}^{\mathrm{new}}(351, [\chi])\). Copy content Toggle raw display