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Properties

Label 820.2.x.a
Level $820$
Weight $2$
Character orbit 820.x
Analytic conductor $6.548$
Analytic rank $0$
Dimension $84$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [820,2,Mod(273,820)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(820, base_ring=CyclotomicField(8)) chi = DirichletCharacter(H, H._module([0, 6, 1])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("820.273"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 820 = 2^{2} \cdot 5 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 820.x (of order \(8\), degree \(4\), 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: \(6.54773296574\)
Analytic rank: \(0\)
Dimension: \(84\)
Relative dimension: \(21\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 84 q - 8 q^{9} + 20 q^{15} - 12 q^{17} - 8 q^{21} + 12 q^{27} - 28 q^{29} + 20 q^{35} + 24 q^{37} + 16 q^{39} + 20 q^{45} - 4 q^{47} + 24 q^{49} + 28 q^{53} + 16 q^{55} - 8 q^{57} + 4 q^{61} + 72 q^{63}+ \cdots - 48 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
273.1 0 −2.81169 − 1.16464i 0 2.15389 − 0.600634i 0 −0.269021 + 0.649473i 0 4.42790 + 4.42790i 0
273.2 0 −2.58129 − 1.06921i 0 −1.41563 − 1.73090i 0 0.277392 − 0.669683i 0 3.39855 + 3.39855i 0
273.3 0 −2.53901 − 1.05169i 0 −0.771074 + 2.09892i 0 0.595144 − 1.43680i 0 3.21921 + 3.21921i 0
273.4 0 −2.14326 − 0.887768i 0 −1.64975 + 1.50941i 0 0.810001 − 1.95552i 0 1.68412 + 1.68412i 0
273.5 0 −1.88141 − 0.779304i 0 0.312265 − 2.21416i 0 −0.913439 + 2.20524i 0 0.811052 + 0.811052i 0
273.6 0 −1.30072 − 0.538776i 0 1.59126 + 1.57096i 0 −0.188362 + 0.454747i 0 −0.719728 − 0.719728i 0
273.7 0 −1.03141 − 0.427224i 0 0.661175 − 2.13608i 0 1.80577 − 4.35952i 0 −1.24003 − 1.24003i 0
273.8 0 −0.967083 − 0.400579i 0 −2.12327 + 0.701240i 0 −1.65654 + 3.99924i 0 −1.34653 − 1.34653i 0
273.9 0 −0.741636 − 0.307196i 0 2.17900 − 0.501977i 0 −1.54758 + 3.73618i 0 −1.66567 − 1.66567i 0
273.10 0 −0.637865 − 0.264212i 0 −2.14189 − 0.642112i 0 0.168176 − 0.406013i 0 −1.78426 − 1.78426i 0
273.11 0 −0.283526 − 0.117440i 0 0.607107 + 2.15207i 0 −0.664030 + 1.60311i 0 −2.05473 − 2.05473i 0
273.12 0 0.455346 + 0.188611i 0 0.155576 + 2.23065i 0 1.56570 − 3.77995i 0 −1.94955 − 1.94955i 0
273.13 0 0.677221 + 0.280514i 0 2.21931 + 0.273212i 0 1.12911 − 2.72591i 0 −1.74138 − 1.74138i 0
273.14 0 1.00868 + 0.417807i 0 −1.23676 − 1.86291i 0 −0.290359 + 0.700988i 0 −1.27846 − 1.27846i 0
273.15 0 1.17715 + 0.487592i 0 2.02779 − 0.942363i 0 −0.0522617 + 0.126171i 0 −0.973380 − 0.973380i 0
273.16 0 1.62905 + 0.674773i 0 −1.92538 + 1.13706i 0 −1.09593 + 2.64580i 0 0.0771547 + 0.0771547i 0
273.17 0 1.70683 + 0.706990i 0 −1.61561 − 1.54590i 0 0.839794 − 2.02744i 0 0.292099 + 0.292099i 0
273.18 0 2.21506 + 0.917510i 0 0.0611172 − 2.23523i 0 −2.00677 + 4.84476i 0 1.94336 + 1.94336i 0
273.19 0 2.42535 + 1.00461i 0 −1.21756 + 1.87551i 0 1.20713 − 2.91427i 0 2.75177 + 2.75177i 0
273.20 0 2.52347 + 1.04526i 0 1.32280 + 1.80283i 0 −0.949785 + 2.29298i 0 3.15402 + 3.15402i 0
See all 84 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 273.21
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
205.o even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 820.2.x.a ✓ 84
5.c odd 4 1 820.2.y.a yes 84
41.e odd 8 1 820.2.y.a yes 84
205.o even 8 1 inner 820.2.x.a ✓ 84
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
820.2.x.a ✓ 84 1.a even 1 1 trivial
820.2.x.a ✓ 84 205.o even 8 1 inner
820.2.y.a yes 84 5.c odd 4 1
820.2.y.a yes 84 41.e odd 8 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(820, [\chi])\).