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Properties

Label 1664.2.n.a
Level $1664$
Weight $2$
Character orbit 1664.n
Analytic conductor $13.287$
Analytic rank $0$
Dimension $48$
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] = [1664,2,Mod(417,1664)] 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("1664.417"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1664, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 3, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1664 = 2^{7} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1664.n (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [48,0,0,0,0,0,0,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(13.2871068963\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 208)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 8 q^{11} + 16 q^{15} - 16 q^{19} + 16 q^{29} - 24 q^{31} + 24 q^{35} + 16 q^{37} - 8 q^{43} + 40 q^{47} - 48 q^{49} - 24 q^{51} - 16 q^{53} - 32 q^{61} - 40 q^{63} + 16 q^{67} - 32 q^{69} - 40 q^{75}+ \cdots - 40 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
417.1 0 −2.17257 − 2.17257i 0 2.25200 − 2.25200i 0 2.10748i 0 6.44015i 0
417.2 0 −2.05439 − 2.05439i 0 −0.347700 + 0.347700i 0 − 5.08331i 0 5.44107i 0
417.3 0 −1.99491 − 1.99491i 0 0.414832 − 0.414832i 0 0.607342i 0 4.95936i 0
417.4 0 −1.84134 − 1.84134i 0 −0.917928 + 0.917928i 0 2.65087i 0 3.78103i 0
417.5 0 −1.56862 − 1.56862i 0 −2.41649 + 2.41649i 0 3.76757i 0 1.92113i 0
417.6 0 −1.56058 − 1.56058i 0 −3.08656 + 3.08656i 0 − 2.75937i 0 1.87081i 0
417.7 0 −1.13990 − 1.13990i 0 −0.631774 + 0.631774i 0 − 0.0885755i 0 − 0.401269i 0
417.8 0 −1.08262 − 1.08262i 0 2.20245 − 2.20245i 0 4.01772i 0 − 0.655887i 0
417.9 0 −0.390961 − 0.390961i 0 2.58479 − 2.58479i 0 − 2.97780i 0 − 2.69430i 0
417.10 0 −0.326558 − 0.326558i 0 0.499927 − 0.499927i 0 0.517149i 0 − 2.78672i 0
417.11 0 −0.316792 − 0.316792i 0 2.69572 − 2.69572i 0 − 1.94658i 0 − 2.79929i 0
417.12 0 −0.152350 − 0.152350i 0 −1.59716 + 1.59716i 0 − 3.10196i 0 − 2.95358i 0
417.13 0 −0.140536 − 0.140536i 0 −1.99922 + 1.99922i 0 0.679168i 0 − 2.96050i 0
417.14 0 0.474808 + 0.474808i 0 −0.481789 + 0.481789i 0 4.68072i 0 − 2.54911i 0
417.15 0 0.559046 + 0.559046i 0 −0.101861 + 0.101861i 0 3.07699i 0 − 2.37494i 0
417.16 0 0.718176 + 0.718176i 0 −2.02033 + 2.02033i 0 − 0.407753i 0 − 1.96845i 0
417.17 0 0.902776 + 0.902776i 0 1.29480 − 1.29480i 0 3.20594i 0 − 1.36999i 0
417.18 0 1.23013 + 1.23013i 0 0.845708 − 0.845708i 0 − 3.63419i 0 0.0264265i 0
417.19 0 1.33118 + 1.33118i 0 −1.78745 + 1.78745i 0 − 4.71876i 0 0.544086i 0
417.20 0 1.41610 + 1.41610i 0 1.67490 − 1.67490i 0 − 1.84335i 0 1.01065i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 417.24
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1664.2.n.a 48
4.b odd 2 1 1664.2.n.b 48
8.b even 2 1 832.2.n.a 48
8.d odd 2 1 208.2.n.a ✓ 48
16.e even 4 1 832.2.n.a 48
16.e even 4 1 inner 1664.2.n.a 48
16.f odd 4 1 208.2.n.a ✓ 48
16.f odd 4 1 1664.2.n.b 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
208.2.n.a ✓ 48 8.d odd 2 1
208.2.n.a ✓ 48 16.f odd 4 1
832.2.n.a 48 8.b even 2 1
832.2.n.a 48 16.e even 4 1
1664.2.n.a 48 1.a even 1 1 trivial
1664.2.n.a 48 16.e even 4 1 inner
1664.2.n.b 48 4.b odd 2 1
1664.2.n.b 48 16.f odd 4 1