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Properties

Label 160.8.d.a
Level $160$
Weight $8$
Character orbit 160.d
Analytic conductor $49.982$
Analytic rank $0$
Dimension $28$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [160,8,Mod(81,160)] 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, 0])) 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("160.81"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Level: \( N \) \(=\) \( 160 = 2^{5} \cdot 5 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 160.d (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: \(28\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q - 1372 q^{7} - 20412 q^{9} + 13500 q^{15} - 2588 q^{23} - 437500 q^{25} - 268024 q^{31} - 99016 q^{33} + 283944 q^{39} - 601208 q^{41} + 2076460 q^{47} + 4316268 q^{49} - 1331000 q^{55} + 3788536 q^{57}+ \cdots + 15198608 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
81.1 0 − 91.4325i 0 125.000i 0 386.271 0 −6172.90 0
81.2 0 − 81.3925i 0 − 125.000i 0 −1696.29 0 −4437.73 0
81.3 0 − 69.9529i 0 − 125.000i 0 814.432 0 −2706.41 0
81.4 0 − 69.7326i 0 125.000i 0 −1358.60 0 −2675.64 0
81.5 0 − 63.3835i 0 125.000i 0 633.994 0 −1830.47 0
81.6 0 − 58.0742i 0 − 125.000i 0 772.054 0 −1185.62 0
81.7 0 − 53.5442i 0 125.000i 0 −30.3611 0 −679.986 0
81.8 0 − 47.3200i 0 − 125.000i 0 −608.992 0 −52.1788 0
81.9 0 − 42.7235i 0 125.000i 0 −1344.00 0 361.702 0
81.10 0 − 28.9351i 0 − 125.000i 0 −724.987 0 1349.76 0
81.11 0 − 21.9337i 0 125.000i 0 1766.19 0 1705.91 0
81.12 0 − 15.3942i 0 125.000i 0 206.895 0 1950.02 0
81.13 0 − 13.4707i 0 − 125.000i 0 1011.70 0 2005.54 0
81.14 0 − 4.99890i 0 − 125.000i 0 −514.310 0 2162.01 0
81.15 0 4.99890i 0 125.000i 0 −514.310 0 2162.01 0
81.16 0 13.4707i 0 125.000i 0 1011.70 0 2005.54 0
81.17 0 15.3942i 0 − 125.000i 0 206.895 0 1950.02 0
81.18 0 21.9337i 0 − 125.000i 0 1766.19 0 1705.91 0
81.19 0 28.9351i 0 125.000i 0 −724.987 0 1349.76 0
81.20 0 42.7235i 0 − 125.000i 0 −1344.00 0 361.702 0
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 81.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b 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.d.a 28
4.b odd 2 1 40.8.d.a ✓ 28
8.b even 2 1 inner 160.8.d.a 28
8.d odd 2 1 40.8.d.a ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.8.d.a ✓ 28 4.b odd 2 1
40.8.d.a ✓ 28 8.d odd 2 1
160.8.d.a 28 1.a even 1 1 trivial
160.8.d.a 28 8.b even 2 1 inner

Hecke kernels

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