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Properties

Label 160.8.f.a
Level $160$
Weight $8$
Character orbit 160.f
Analytic conductor $49.982$
Analytic rank $0$
Dimension $40$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [160,8,Mod(49,160)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(160, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 8, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("160.49"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Level: \( N \) \(=\) \( 160 = 2^{5} \cdot 5 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 160.f (of order \(2\), degree \(1\), not 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: \(49.9816040775\)
Analytic rank: \(0\)
Dimension: \(40\)
Twist minimal: no (minimal twist has level 40)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 26240 q^{9} + 4376 q^{15} - 8064 q^{25} + 625520 q^{31} + 957856 q^{39} - 441288 q^{41} - 3294176 q^{49} - 2251752 q^{55} - 2463240 q^{65} - 3847152 q^{71} - 9729840 q^{79} + 9702688 q^{81} + 15759840 q^{89}+ \cdots + 2170824 q^{95}+O(q^{100}) \) 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 −85.1361 0 −163.129 − 226.967i 0 − 1132.18i 0 5061.16 0
49.2 0 −85.1361 0 −163.129 + 226.967i 0 1132.18i 0 5061.16 0
49.3 0 −80.4449 0 276.446 + 41.2614i 0 − 281.883i 0 4284.38 0
49.4 0 −80.4449 0 276.446 − 41.2614i 0 281.883i 0 4284.38 0
49.5 0 −70.5054 0 −35.7394 + 277.214i 0 − 1255.84i 0 2784.01 0
49.6 0 −70.5054 0 −35.7394 − 277.214i 0 1255.84i 0 2784.01 0
49.7 0 −54.6083 0 96.2712 + 262.406i 0 75.2701i 0 795.063 0
49.8 0 −54.6083 0 96.2712 − 262.406i 0 − 75.2701i 0 795.063 0
49.9 0 −50.9148 0 −244.226 − 135.936i 0 − 534.587i 0 405.322 0
49.10 0 −50.9148 0 −244.226 + 135.936i 0 534.587i 0 405.322 0
49.11 0 −46.1976 0 −271.075 + 68.1421i 0 − 1088.21i 0 −52.7861 0
49.12 0 −46.1976 0 −271.075 − 68.1421i 0 1088.21i 0 −52.7861 0
49.13 0 −35.0518 0 246.912 − 130.994i 0 1440.38i 0 −958.369 0
49.14 0 −35.0518 0 246.912 + 130.994i 0 − 1440.38i 0 −958.369 0
49.15 0 −21.4733 0 162.054 − 227.735i 0 − 1461.90i 0 −1725.90 0
49.16 0 −21.4733 0 162.054 + 227.735i 0 1461.90i 0 −1725.90 0
49.17 0 −18.1977 0 −5.66487 − 279.451i 0 − 646.347i 0 −1855.84 0
49.18 0 −18.1977 0 −5.66487 + 279.451i 0 646.347i 0 −1855.84 0
49.19 0 −3.15519 0 235.904 − 149.914i 0 − 123.632i 0 −2177.04 0
49.20 0 −3.15519 0 235.904 + 149.914i 0 123.632i 0 −2177.04 0
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
8.b even 2 1 inner
40.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 160.8.f.a 40
4.b odd 2 1 40.8.f.a ✓ 40
5.b even 2 1 inner 160.8.f.a 40
8.b even 2 1 inner 160.8.f.a 40
8.d odd 2 1 40.8.f.a ✓ 40
20.d odd 2 1 40.8.f.a ✓ 40
40.e odd 2 1 40.8.f.a ✓ 40
40.f even 2 1 inner 160.8.f.a 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.8.f.a ✓ 40 4.b odd 2 1
40.8.f.a ✓ 40 8.d odd 2 1
40.8.f.a ✓ 40 20.d odd 2 1
40.8.f.a ✓ 40 40.e odd 2 1
160.8.f.a 40 1.a even 1 1 trivial
160.8.f.a 40 5.b even 2 1 inner
160.8.f.a 40 8.b even 2 1 inner
160.8.f.a 40 40.f even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(160, [\chi])\).