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Properties

Label 167.6.a.b
Level $167$
Weight $6$
Character orbit 167.a
Self dual yes
Analytic conductor $26.784$
Analytic rank $0$
Dimension $40$
CM no
Inner twists $1$

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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] = [167,6,Mod(1,167)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("167.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(167, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 167 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 167.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [40] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(26.7840979088\)
Analytic rank: \(0\)
Dimension: \(40\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 11 q^{2} + 35 q^{3} + 709 q^{4} + 124 q^{5} + 184 q^{6} + 587 q^{7} + 756 q^{8} + 4129 q^{9} + 333 q^{10} + 816 q^{11} + 2241 q^{12} + 2035 q^{13} + 915 q^{14} + 1761 q^{15} + 14773 q^{16} + 4109 q^{17}+ \cdots - 6728 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −11.2161 26.5611 93.7999 63.6485 −297.911 31.3109 −693.151 462.492 −713.886
1.2 −10.6519 6.57642 81.4632 −70.5555 −70.0515 42.6624 −526.878 −199.751 751.551
1.3 −10.1017 11.0437 70.0434 5.30089 −111.560 −246.773 −384.301 −121.037 −53.5477
1.4 −10.0038 −16.9341 68.0759 −23.0069 169.405 118.948 −360.896 43.7628 230.156
1.5 −8.26625 −27.7320 36.3309 14.1520 229.239 −115.182 −35.8000 526.062 −116.984
1.6 −7.98406 −23.8588 31.7452 −62.2413 190.490 254.626 2.03424 326.243 496.938
1.7 −7.95950 −3.29979 31.3536 −23.1848 26.2647 −233.198 5.14531 −232.111 184.540
1.8 −7.74828 13.6410 28.0358 75.3330 −105.694 −96.9164 30.7160 −56.9243 −583.700
1.9 −7.12709 22.2975 18.7955 32.1068 −158.916 209.360 94.1099 254.178 −228.828
1.10 −6.87020 −29.2215 15.1996 77.8306 200.758 130.544 115.422 610.898 −534.712
1.11 −5.82157 11.3603 1.89063 −76.7638 −66.1347 213.626 175.284 −113.944 446.886
1.12 −5.65600 −5.19192 −0.00970892 24.3090 29.3655 97.9079 181.047 −216.044 −137.492
1.13 −5.11586 −18.6467 −5.82794 29.5338 95.3937 −4.65753 193.523 104.698 −151.091
1.14 −4.54190 3.89135 −11.3711 −67.3664 −17.6741 −93.3366 196.987 −227.857 305.972
1.15 −3.87757 22.5061 −16.9645 −22.5126 −87.2689 −158.702 189.863 263.525 87.2939
1.16 −2.94140 −12.5547 −23.3481 95.1293 36.9285 234.082 162.801 −85.3786 −279.814
1.17 −2.11225 27.1341 −27.5384 54.9221 −57.3140 161.040 125.760 493.261 −116.009
1.18 0.0725453 −2.86754 −31.9947 17.5002 −0.208027 −202.665 −4.64252 −234.777 1.26956
1.19 0.124291 27.9267 −31.9846 −108.219 3.47103 161.854 −7.95269 536.899 −13.4507
1.20 0.494765 6.35086 −31.7552 −106.759 3.14218 −142.460 −31.5438 −202.667 −52.8206
See all 40 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.40
Significant digits:
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Atkin-Lehner signs

\( p \) Sign
\(167\) \( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 167.6.a.b ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
167.6.a.b ✓ 40 1.a even 1 1 trivial