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Properties

Label 820.2.bq.a
Level $820$
Weight $2$
Character orbit 820.bq
Analytic conductor $6.548$
Analytic rank $0$
Dimension $176$
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] = [820,2,Mod(49,820)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(820, base_ring=CyclotomicField(20)) chi = DirichletCharacter(H, H._module([0, 10, 19])) 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("820.49"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 820 = 2^{2} \cdot 5 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 820.bq (of order \(20\), degree \(8\), 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: \(176\)
Relative dimension: \(22\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 176 q + 4 q^{11} - 10 q^{15} - 4 q^{19} + 12 q^{25} + 8 q^{29} - 8 q^{31} - 6 q^{35} + 40 q^{39} + 28 q^{41} - 4 q^{45} + 20 q^{49} - 32 q^{51} - 50 q^{55} + 12 q^{59} + 40 q^{61} - 10 q^{65} - 28 q^{69}+ \cdots + 4 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1 0 −2.28615 + 2.28615i 0 −0.249857 + 2.22206i 0 0.324162 − 2.04668i 0 − 7.45299i 0
49.2 0 −2.22470 + 2.22470i 0 −0.804275 − 2.08642i 0 −0.555695 + 3.50852i 0 − 6.89860i 0
49.3 0 −2.10870 + 2.10870i 0 2.22522 + 0.220000i 0 0.00805431 − 0.0508529i 0 − 5.89321i 0
49.4 0 −1.52630 + 1.52630i 0 2.01925 + 0.960543i 0 −0.663644 + 4.19008i 0 − 1.65918i 0
49.5 0 −1.42551 + 1.42551i 0 0.0240788 − 2.23594i 0 0.459361 − 2.90029i 0 − 1.06416i 0
49.6 0 −1.37884 + 1.37884i 0 −2.23133 + 0.145551i 0 −0.0463388 + 0.292571i 0 − 0.802406i 0
49.7 0 −1.13142 + 1.13142i 0 −1.70024 + 1.45230i 0 −0.770520 + 4.86487i 0 0.439760i 0
49.8 0 −1.05623 + 1.05623i 0 2.03080 − 0.935877i 0 0.269596 − 1.70217i 0 0.768736i 0
49.9 0 −0.412193 + 0.412193i 0 −1.18491 + 1.89631i 0 0.252713 − 1.59557i 0 2.66019i 0
49.10 0 −0.341338 + 0.341338i 0 −0.760852 − 2.10264i 0 0.0854014 − 0.539203i 0 2.76698i 0
49.11 0 −0.142650 + 0.142650i 0 1.77418 + 1.36098i 0 0.670976 − 4.23638i 0 2.95930i 0
49.12 0 0.142650 − 0.142650i 0 0.635377 − 2.14390i 0 −0.670976 + 4.23638i 0 2.95930i 0
49.13 0 0.341338 − 0.341338i 0 0.620360 + 2.14829i 0 −0.0854014 + 0.539203i 0 2.76698i 0
49.14 0 0.412193 − 0.412193i 0 −2.07323 − 0.837678i 0 −0.252713 + 1.59557i 0 2.66019i 0
49.15 0 1.05623 − 1.05623i 0 2.19304 − 0.436532i 0 −0.269596 + 1.70217i 0 0.768736i 0
49.16 0 1.13142 − 1.13142i 0 −2.22917 − 0.175559i 0 0.770520 − 4.86487i 0 0.439760i 0
49.17 0 1.37884 − 1.37884i 0 −1.89073 + 1.19379i 0 0.0463388 − 0.292571i 0 − 0.802406i 0
49.18 0 1.42551 − 1.42551i 0 1.33373 + 1.79476i 0 −0.459361 + 2.90029i 0 − 1.06416i 0
49.19 0 1.52630 − 1.52630i 0 1.06901 − 1.96398i 0 0.663644 − 4.19008i 0 − 1.65918i 0
49.20 0 2.10870 − 2.10870i 0 1.67093 − 1.48593i 0 −0.00805431 + 0.0508529i 0 − 5.89321i 0
See next 80 embeddings (of 176 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
41.g even 20 1 inner
205.s even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 820.2.bq.a ✓ 176
5.b even 2 1 inner 820.2.bq.a ✓ 176
41.g even 20 1 inner 820.2.bq.a ✓ 176
205.s even 20 1 inner 820.2.bq.a ✓ 176
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
820.2.bq.a ✓ 176 1.a even 1 1 trivial
820.2.bq.a ✓ 176 5.b even 2 1 inner
820.2.bq.a ✓ 176 41.g even 20 1 inner
820.2.bq.a ✓ 176 205.s even 20 1 inner

Hecke kernels

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