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Properties

Label 765.2.be.c
Level $765$
Weight $2$
Character orbit 765.be
Analytic conductor $6.109$
Analytic rank $0$
Dimension $32$
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] = [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 = [32] 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: \(32\)
Relative dimension: \(8\) over \(\Q(\zeta_{8})\)
Twist minimal: no (minimal twist has level 255)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 8 q^{7} - 56 q^{16} - 8 q^{17} - 16 q^{19} + 8 q^{22} + 72 q^{26} - 24 q^{28} - 32 q^{29} + 8 q^{34} - 16 q^{37} - 24 q^{40} - 32 q^{41} - 16 q^{43} - 16 q^{44} + 8 q^{46} + 8 q^{49} + 8 q^{50} + 128 q^{52}+ \cdots - 24 q^{97}+O(q^{100}) \) 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 −1.81449 − 1.81449i 0 4.58477i 0.923880 − 0.382683i 0 2.79106 + 1.15609i 4.69005 − 4.69005i 0 −2.37075 − 0.981997i
406.2 −1.37042 − 1.37042i 0 1.75608i −0.923880 + 0.382683i 0 0.108602 + 0.0449843i −0.334278 + 0.334278i 0 1.79053 + 0.741663i
406.3 −0.470313 − 0.470313i 0 − 1.55761i 0.923880 − 0.382683i 0 −1.36091 − 0.563709i −1.67319 + 1.67319i 0 −0.614493 − 0.254531i
406.4 −0.245139 − 0.245139i 0 − 1.87981i −0.923880 + 0.382683i 0 0.860352 + 0.356370i −0.951095 + 0.951095i 0 0.320290 + 0.132669i
406.5 1.06986 + 1.06986i 0 0.289222i −0.923880 + 0.382683i 0 −4.38000 − 1.81426i 1.83030 − 1.83030i 0 −1.39785 − 0.579007i
406.6 1.07263 + 1.07263i 0 0.301074i 0.923880 − 0.382683i 0 −1.95554 − 0.810011i 1.82232 − 1.82232i 0 1.40146 + 0.580504i
406.7 1.25280 + 1.25280i 0 1.13900i −0.923880 + 0.382683i 0 3.81159 + 1.57881i 1.07866 − 1.07866i 0 −1.63686 − 0.678009i
406.8 1.91928 + 1.91928i 0 5.36729i 0.923880 − 0.382683i 0 3.53907 + 1.46593i −6.46277 + 6.46277i 0 2.50766 + 1.03871i
451.1 −1.78971 + 1.78971i 0 − 4.40611i −0.382683 + 0.923880i 0 0.211600 + 0.510848i 4.30624 + 4.30624i 0 −0.968583 − 2.33837i
451.2 −1.73834 + 1.73834i 0 − 4.04366i 0.382683 − 0.923880i 0 −0.920704 − 2.22278i 3.55258 + 3.55258i 0 0.940784 + 2.27125i
451.3 −0.876678 + 0.876678i 0 0.462871i 0.382683 − 0.923880i 0 0.999316 + 2.41256i −2.15915 − 2.15915i 0 0.474455 + 1.14544i
451.4 −0.689027 + 0.689027i 0 1.05048i −0.382683 + 0.923880i 0 1.50637 + 3.63669i −2.10187 − 2.10187i 0 −0.372899 − 0.900257i
451.5 0.0443119 − 0.0443119i 0 1.99607i 0.382683 − 0.923880i 0 −0.295776 − 0.714067i 0.177073 + 0.177073i 0 −0.0239814 − 0.0578963i
451.6 0.392111 − 0.392111i 0 1.69250i −0.382683 + 0.923880i 0 −2.00796 − 4.84764i 1.44787 + 1.44787i 0 0.212209 + 0.512318i
451.7 1.37952 − 1.37952i 0 − 1.80613i −0.382683 + 0.923880i 0 1.12408 + 2.71378i 0.267441 + 0.267441i 0 0.746589 + 1.80243i
451.8 1.86360 − 1.86360i 0 − 4.94602i 0.382683 − 0.923880i 0 −0.0311391 − 0.0751765i −5.49020 − 5.49020i 0 −1.00857 − 2.43491i
586.1 −1.81449 + 1.81449i 0 − 4.58477i 0.923880 + 0.382683i 0 2.79106 − 1.15609i 4.69005 + 4.69005i 0 −2.37075 + 0.981997i
586.2 −1.37042 + 1.37042i 0 − 1.75608i −0.923880 − 0.382683i 0 0.108602 − 0.0449843i −0.334278 − 0.334278i 0 1.79053 − 0.741663i
586.3 −0.470313 + 0.470313i 0 1.55761i 0.923880 + 0.382683i 0 −1.36091 + 0.563709i −1.67319 − 1.67319i 0 −0.614493 + 0.254531i
586.4 −0.245139 + 0.245139i 0 1.87981i −0.923880 − 0.382683i 0 0.860352 − 0.356370i −0.951095 − 0.951095i 0 0.320290 − 0.132669i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 406.8
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.d even 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.c 32
3.b odd 2 1 255.2.w.b ✓ 32
17.d even 8 1 inner 765.2.be.c 32
51.g odd 8 1 255.2.w.b ✓ 32
51.i even 16 1 4335.2.a.bl 16
51.i even 16 1 4335.2.a.bm 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
255.2.w.b ✓ 32 3.b odd 2 1
255.2.w.b ✓ 32 51.g odd 8 1
765.2.be.c 32 1.a even 1 1 trivial
765.2.be.c 32 17.d even 8 1 inner
4335.2.a.bl 16 51.i even 16 1
4335.2.a.bm 16 51.i even 16 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{32} + 138 T_{2}^{28} - 24 T_{2}^{25} + 6837 T_{2}^{24} - 320 T_{2}^{23} - 296 T_{2}^{21} + \cdots + 289 \) acting on \(S_{2}^{\mathrm{new}}(765, [\chi])\). Copy content Toggle raw display