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Properties

Label 790.2.j.a
Level $790$
Weight $2$
Character orbit 790.j
Analytic conductor $6.308$
Analytic rank $0$
Dimension $80$
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] = [790,2,Mod(339,790)] 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("790.339"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(790, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 790 = 2 \cdot 5 \cdot 79 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 790.j (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.30818175968\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(40\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 80 q + 40 q^{4} + 4 q^{6} + 36 q^{9} - 4 q^{10} - 8 q^{11} + 16 q^{14} - 8 q^{15} - 40 q^{16} - 8 q^{19} - 72 q^{21} - 4 q^{24} + 6 q^{25} - 8 q^{29} - 10 q^{30} - 16 q^{31} - 2 q^{35} - 36 q^{36} + 24 q^{39}+ \cdots + 52 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
339.1 −0.866025 + 0.500000i −2.91982 + 1.68576i 0.500000 − 0.866025i −1.75498 − 1.38566i 1.68576 − 2.91982i 3.40032 + 1.96318i 1.00000i 4.18358 − 7.24618i 2.21269 + 0.322524i
339.2 −0.866025 + 0.500000i −2.47336 + 1.42800i 0.500000 − 0.866025i −1.77519 + 1.35967i 1.42800 − 2.47336i −2.33335 − 1.34716i 1.00000i 2.57835 − 4.46583i 0.857523 − 2.06510i
339.3 −0.866025 + 0.500000i −2.23260 + 1.28899i 0.500000 − 0.866025i 2.06429 − 0.859474i 1.28899 − 2.23260i −3.56828 − 2.06015i 1.00000i 1.82301 − 3.15755i −1.35799 + 1.77647i
339.4 −0.866025 + 0.500000i −2.16589 + 1.25048i 0.500000 − 0.866025i 1.46933 + 1.68555i 1.25048 − 2.16589i 0.153833 + 0.0888155i 1.00000i 1.62739 − 2.81872i −2.11525 − 0.725065i
339.5 −0.866025 + 0.500000i −1.88067 + 1.08581i 0.500000 − 0.866025i 1.95450 − 1.08624i 1.08581 − 1.88067i 1.92927 + 1.11387i 1.00000i 0.857950 − 1.48601i −1.14953 + 1.91796i
339.6 −0.866025 + 0.500000i −1.52028 + 0.877733i 0.500000 − 0.866025i −0.599051 − 2.15433i 0.877733 − 1.52028i −0.132461 − 0.0764766i 1.00000i 0.0408318 − 0.0707228i 1.59596 + 1.56618i
339.7 −0.866025 + 0.500000i −1.26805 + 0.732111i 0.500000 − 0.866025i 1.35871 + 1.77592i 0.732111 − 1.26805i 3.77541 + 2.17973i 1.00000i −0.428026 + 0.741363i −2.06464 − 0.858635i
339.8 −0.866025 + 0.500000i −0.841332 + 0.485743i 0.500000 − 0.866025i −2.13918 − 0.651098i 0.485743 − 0.841332i −1.50793 − 0.870602i 1.00000i −1.02811 + 1.78073i 2.17813 − 0.505720i
339.9 −0.866025 + 0.500000i −0.465382 + 0.268688i 0.500000 − 0.866025i 0.812378 + 2.08328i 0.268688 − 0.465382i −2.40043 − 1.38589i 1.00000i −1.35561 + 2.34799i −1.74518 − 1.39798i
339.10 −0.866025 + 0.500000i 0.0143725 − 0.00829795i 0.500000 − 0.866025i 0.785660 − 2.09350i −0.00829795 + 0.0143725i 3.24789 + 1.87517i 1.00000i −1.49986 + 2.59784i 0.366348 + 2.20585i
339.11 −0.866025 + 0.500000i 0.202116 − 0.116692i 0.500000 − 0.866025i −2.02581 + 0.946629i −0.116692 + 0.202116i 1.96198 + 1.13275i 1.00000i −1.47277 + 2.55091i 1.28109 − 1.83271i
339.12 −0.866025 + 0.500000i 0.512034 − 0.295623i 0.500000 − 0.866025i 0.479545 + 2.18404i −0.295623 + 0.512034i 0.189884 + 0.109629i 1.00000i −1.32521 + 2.29534i −1.50732 − 1.65166i
339.13 −0.866025 + 0.500000i 0.666203 − 0.384632i 0.500000 − 0.866025i 0.687020 − 2.12791i −0.384632 + 0.666203i −4.07878 − 2.35488i 1.00000i −1.20412 + 2.08559i 0.468979 + 2.18633i
339.14 −0.866025 + 0.500000i 0.795047 − 0.459021i 0.500000 − 0.866025i 2.06922 + 0.847547i −0.459021 + 0.795047i −0.0946389 − 0.0546398i 1.00000i −1.07860 + 1.86819i −2.21577 + 0.300611i
339.15 −0.866025 + 0.500000i 1.37490 − 0.793797i 0.500000 − 0.866025i −2.10708 − 0.748488i −0.793797 + 1.37490i −3.21473 − 1.85603i 1.00000i −0.239773 + 0.415299i 2.19902 − 0.405328i
339.16 −0.866025 + 0.500000i 1.70082 − 0.981971i 0.500000 − 0.866025i −0.925617 + 2.03549i −0.981971 + 1.70082i 1.62485 + 0.938105i 1.00000i 0.428534 − 0.742242i −0.216139 − 2.22560i
339.17 −0.866025 + 0.500000i 1.91183 − 1.10379i 0.500000 − 0.866025i −1.55503 − 1.60682i −1.10379 + 1.91183i 1.84254 + 1.06379i 1.00000i 0.936725 − 1.62246i 2.15011 + 0.614032i
339.18 −0.866025 + 0.500000i 1.95468 − 1.12854i 0.500000 − 0.866025i 2.19721 + 0.415036i −1.12854 + 1.95468i −2.21518 − 1.27893i 1.00000i 1.04718 − 1.81377i −2.11036 + 0.739175i
339.19 −0.866025 + 0.500000i 2.18581 − 1.26198i 0.500000 − 0.866025i 1.44862 − 1.70338i −1.26198 + 2.18581i 1.64823 + 0.951608i 1.00000i 1.68519 − 2.91883i −0.402846 + 2.19948i
339.20 −0.866025 + 0.500000i 2.71753 − 1.56897i 0.500000 − 0.866025i −1.57854 + 1.58374i −1.56897 + 2.71753i −3.69254 − 2.13189i 1.00000i 3.42333 − 5.92938i 0.575184 − 2.16082i
See all 80 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 339.40
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
79.c even 3 1 inner
395.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 790.2.j.a ✓ 80
5.b even 2 1 inner 790.2.j.a ✓ 80
79.c even 3 1 inner 790.2.j.a ✓ 80
395.j even 6 1 inner 790.2.j.a ✓ 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
790.2.j.a ✓ 80 1.a even 1 1 trivial
790.2.j.a ✓ 80 5.b even 2 1 inner
790.2.j.a ✓ 80 79.c even 3 1 inner
790.2.j.a ✓ 80 395.j even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(790, [\chi])\).