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Properties

Label 820.2.y.a
Level $820$
Weight $2$
Character orbit 820.y
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(137,820)] 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, 2, 5])) 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("820.137"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 820 = 2^{2} \cdot 5 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 820.y (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.

\(\operatorname{Tr}(f)(q) = \) \( 84 q + 8 q^{9} + 4 q^{13} + 4 q^{15} - 16 q^{17} - 8 q^{21} - 12 q^{27} + 28 q^{29} + 40 q^{33} - 20 q^{35} + 24 q^{37} - 16 q^{39} - 20 q^{45} + 28 q^{47} - 24 q^{49} - 32 q^{53} + 16 q^{55} - 8 q^{57}+ \cdots + 48 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
137.1 0 −2.96515 − 1.22820i 0 −1.89572 + 1.18586i 0 1.18435 − 2.85928i 0 5.16230 + 5.16230i 0
137.2 0 −2.90256 − 1.20228i 0 −0.575127 − 2.16084i 0 0.543853 − 1.31298i 0 4.85807 + 4.85807i 0
137.3 0 −2.17457 − 0.900738i 0 −1.55478 + 1.60706i 0 −1.37999 + 3.33158i 0 1.79612 + 1.79612i 0
137.4 0 −2.13627 − 0.884872i 0 1.66644 − 1.49097i 0 −0.602718 + 1.45509i 0 1.65933 + 1.65933i 0
137.5 0 −2.04447 − 0.846849i 0 2.12876 + 0.684393i 0 1.81325 − 4.37757i 0 1.34140 + 1.34140i 0
137.6 0 −1.70434 − 0.705960i 0 −1.80257 − 1.32316i 0 −0.486096 + 1.17354i 0 0.285071 + 0.285071i 0
137.7 0 −1.41402 − 0.585707i 0 1.93611 + 1.11869i 0 −0.549988 + 1.32779i 0 −0.464917 − 0.464917i 0
137.8 0 −0.764781 − 0.316783i 0 −0.0345804 + 2.23580i 0 0.341602 − 0.824700i 0 −1.63678 − 1.63678i 0
137.9 0 −0.629944 − 0.260931i 0 −1.47443 − 1.68109i 0 1.11716 − 2.69706i 0 −1.79258 − 1.79258i 0
137.10 0 −0.172330 − 0.0713814i 0 −0.358390 + 2.20716i 0 0.396806 − 0.957974i 0 −2.09672 − 2.09672i 0
137.11 0 −0.0282677 − 0.0117088i 0 2.08219 + 0.815171i 0 −1.57883 + 3.81164i 0 −2.12066 − 2.12066i 0
137.12 0 0.124058 + 0.0513864i 0 −0.0384730 − 2.23574i 0 −1.97802 + 4.77536i 0 −2.10857 − 2.10857i 0
137.13 0 0.374577 + 0.155155i 0 1.08943 − 1.95273i 0 0.751690 − 1.81474i 0 −2.00509 − 2.00509i 0
137.14 0 0.960443 + 0.397829i 0 −2.14104 + 0.644933i 0 0.619523 − 1.49566i 0 −1.35714 − 1.35714i 0
137.15 0 1.03876 + 0.430268i 0 −2.16617 − 0.554719i 0 −0.831762 + 2.00805i 0 −1.22743 − 1.22743i 0
137.16 0 2.05568 + 0.851491i 0 0.997785 + 2.00111i 0 −1.62885 + 3.93239i 0 1.37946 + 1.37946i 0
137.17 0 2.07085 + 0.857774i 0 2.22915 − 0.175717i 0 0.0957430 − 0.231144i 0 1.43132 + 1.43132i 0
137.18 0 2.30222 + 0.953612i 0 0.939983 + 2.02890i 0 1.81993 − 4.39370i 0 2.26953 + 2.26953i 0
137.19 0 2.47497 + 1.02516i 0 −1.56727 + 1.59488i 0 −0.661333 + 1.59660i 0 2.95318 + 2.95318i 0
137.20 0 2.55440 + 1.05807i 0 −1.22454 − 1.87097i 0 1.67533 − 4.04462i 0 3.28415 + 3.28415i 0
See all 84 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 137.21
Significant digits:
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Inner twists

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

Hecke kernels

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