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Properties

Label 765.2.be.d
Level $765$
Weight $2$
Character orbit 765.be
Analytic conductor $6.109$
Analytic rank $0$
Dimension $48$
Inner twists $4$

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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] = [765,2,Mod(406,765)] 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("765.406"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(765, base_ring=CyclotomicField(8)) chi = DirichletCharacter(H, H._module([0, 0, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 765 = 3^{2} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 765.be (of order \(8\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [48] 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: \(6.10855575463\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(12\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 48 q^{16} + 32 q^{19} + 80 q^{22} - 48 q^{28} + 8 q^{34} - 48 q^{37} - 16 q^{40} + 16 q^{43} - 32 q^{46} - 48 q^{49} - 96 q^{52} + 64 q^{58} + 112 q^{61} + 32 q^{67} + 80 q^{76} + 48 q^{79} + 176 q^{82}+ \cdots - 32 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
406.1 −1.86027 − 1.86027i 0 4.92123i −0.923880 + 0.382683i 0 1.64664 + 0.682062i 5.43427 − 5.43427i 0 2.43056 + 1.00677i
406.2 −1.84029 − 1.84029i 0 4.77335i 0.923880 − 0.382683i 0 −2.71531 − 1.12472i 5.10378 − 5.10378i 0 −2.40446 − 0.995959i
406.3 −1.19537 − 1.19537i 0 0.857805i −0.923880 + 0.382683i 0 −3.10354 − 1.28553i −1.36534 + 1.36534i 0 1.56182 + 0.646928i
406.4 −1.03636 − 1.03636i 0 0.148087i 0.923880 − 0.382683i 0 0.191998 + 0.0795280i −1.91925 + 1.91925i 0 −1.35407 − 0.560874i
406.5 −0.562729 − 0.562729i 0 − 1.36667i −0.923880 + 0.382683i 0 4.16587 + 1.72556i −1.89452 + 1.89452i 0 0.735241 + 0.304547i
406.6 −0.200519 − 0.200519i 0 − 1.91958i 0.923880 − 0.382683i 0 −0.185664 − 0.0769047i −0.785951 + 0.785951i 0 −0.261991 − 0.108520i
406.7 0.200519 + 0.200519i 0 − 1.91958i −0.923880 + 0.382683i 0 −0.185664 − 0.0769047i 0.785951 − 0.785951i 0 −0.261991 − 0.108520i
406.8 0.562729 + 0.562729i 0 − 1.36667i 0.923880 − 0.382683i 0 4.16587 + 1.72556i 1.89452 − 1.89452i 0 0.735241 + 0.304547i
406.9 1.03636 + 1.03636i 0 0.148087i −0.923880 + 0.382683i 0 0.191998 + 0.0795280i 1.91925 − 1.91925i 0 −1.35407 − 0.560874i
406.10 1.19537 + 1.19537i 0 0.857805i 0.923880 − 0.382683i 0 −3.10354 − 1.28553i 1.36534 − 1.36534i 0 1.56182 + 0.646928i
406.11 1.84029 + 1.84029i 0 4.77335i −0.923880 + 0.382683i 0 −2.71531 − 1.12472i −5.10378 + 5.10378i 0 −2.40446 − 0.995959i
406.12 1.86027 + 1.86027i 0 4.92123i 0.923880 − 0.382683i 0 1.64664 + 0.682062i −5.43427 + 5.43427i 0 2.43056 + 1.00677i
451.1 −1.72972 + 1.72972i 0 − 3.98383i −0.382683 + 0.923880i 0 −1.92605 − 4.64990i 3.43147 + 3.43147i 0 −0.936116 − 2.25998i
451.2 −1.42794 + 1.42794i 0 − 2.07804i 0.382683 − 0.923880i 0 0.0474368 + 0.114522i 0.111436 + 0.111436i 0 0.772797 + 1.86570i
451.3 −1.41983 + 1.41983i 0 − 2.03181i −0.382683 + 0.923880i 0 0.941276 + 2.27244i 0.0451599 + 0.0451599i 0 −0.768404 − 1.85509i
451.4 −1.01031 + 1.01031i 0 − 0.0414414i 0.382683 − 0.923880i 0 0.371080 + 0.895867i −1.97875 − 1.97875i 0 0.546774 + 1.32003i
451.5 −0.452810 + 0.452810i 0 1.58993i −0.382683 + 0.923880i 0 −1.12434 − 2.71439i −1.62555 − 1.62555i 0 −0.245059 − 0.591625i
451.6 −0.142462 + 0.142462i 0 1.95941i −0.382683 + 0.923880i 0 1.69059 + 4.08146i −0.564064 − 0.564064i 0 −0.0770997 − 0.186135i
451.7 0.142462 − 0.142462i 0 1.95941i 0.382683 − 0.923880i 0 1.69059 + 4.08146i 0.564064 + 0.564064i 0 −0.0770997 − 0.186135i
451.8 0.452810 − 0.452810i 0 1.58993i 0.382683 − 0.923880i 0 −1.12434 − 2.71439i 1.62555 + 1.62555i 0 −0.245059 − 0.591625i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 406.12
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
17.d even 8 1 inner
51.g odd 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 765.2.be.d ✓ 48
3.b odd 2 1 inner 765.2.be.d ✓ 48
17.d even 8 1 inner 765.2.be.d ✓ 48
51.g odd 8 1 inner 765.2.be.d ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
765.2.be.d ✓ 48 1.a even 1 1 trivial
765.2.be.d ✓ 48 3.b odd 2 1 inner
765.2.be.d ✓ 48 17.d even 8 1 inner
765.2.be.d ✓ 48 51.g odd 8 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{48} + 180 T_{2}^{44} + 13036 T_{2}^{40} + 489816 T_{2}^{36} + 10336296 T_{2}^{32} + 124976100 T_{2}^{28} + \cdots + 2401 \) acting on \(S_{2}^{\mathrm{new}}(765, [\chi])\). Copy content Toggle raw display