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Properties

Label 167.8.a.b
Level $167$
Weight $8$
Character orbit 167.a
Self dual yes
Analytic conductor $52.168$
Analytic rank $0$
Dimension $54$
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] = [167,8,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, 8); 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, 8, names="a")
 
Level: \( N \) \(=\) \( 167 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 167.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [54] 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: \(52.1682992558\)
Analytic rank: \(0\)
Dimension: \(54\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 54 q + 23 q^{2} + 107 q^{3} + 3749 q^{4} + 874 q^{5} + 304 q^{6} + 4115 q^{7} + 3180 q^{8} + 47983 q^{9} + 12917 q^{10} + 4176 q^{11} + 16761 q^{12} + 33055 q^{13} + 25523 q^{14} + 39501 q^{15} + 270069 q^{16}+ \cdots + 5876500 q^{99}+O(q^{100}) \) 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 −22.4792 −38.6381 377.315 −134.378 868.554 137.604 −5604.41 −694.099 3020.72
1.2 −21.7348 −6.85239 344.402 329.376 148.936 −1066.73 −4703.46 −2140.04 −7158.93
1.3 −20.7391 73.4499 302.109 −309.236 −1523.28 1261.56 −3610.86 3207.89 6413.26
1.4 −20.5788 17.1687 295.486 337.493 −353.312 1169.15 −3446.66 −1892.23 −6945.19
1.5 −19.2359 −82.6989 242.018 −508.012 1590.79 23.3749 −2193.24 4652.11 9772.04
1.6 −18.7404 33.7877 223.202 381.091 −633.193 −530.841 −1784.12 −1045.39 −7141.78
1.7 −18.7280 −53.0958 222.738 224.805 994.378 −1278.59 −1774.26 632.165 −4210.14
1.8 −18.3365 4.57531 208.228 −218.090 −83.8952 −235.839 −1471.10 −2166.07 3999.01
1.9 −16.0731 81.9673 130.344 531.454 −1317.47 307.881 −37.6828 4531.63 −8542.12
1.10 −15.3958 17.3195 109.029 −527.451 −266.646 1592.87 292.069 −1887.04 8120.50
1.11 −14.0432 84.0779 69.2123 −149.547 −1180.72 836.086 825.570 4882.09 2100.12
1.12 −13.3133 −7.63745 49.2445 −19.2064 101.680 86.9555 1048.50 −2128.67 255.701
1.13 −12.5288 69.4815 28.9699 −533.654 −870.517 −1021.45 1240.72 2640.68 6686.02
1.14 −11.8391 6.78124 12.1652 360.958 −80.2841 −1232.19 1371.38 −2141.01 −4273.43
1.15 −11.7811 −76.4459 10.7939 −315.455 900.615 70.8199 1380.81 3656.97 3716.40
1.16 −11.5226 31.5861 4.76995 −119.243 −363.954 −1026.77 1419.93 −1189.32 1373.99
1.17 −11.3037 −72.7325 −0.226769 −111.352 822.145 1113.89 1449.43 3103.01 1258.69
1.18 −10.0033 −68.4415 −27.9333 45.0772 684.643 −451.275 1559.85 2497.23 −450.922
1.19 −8.14257 77.9612 −61.6985 −21.6690 −634.805 −132.163 1544.63 3890.96 176.442
1.20 −6.03451 −1.04614 −91.5847 250.600 6.31292 1526.80 1325.09 −2185.91 −1512.25
See all 54 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.54
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 167.8.a.b ✓ 54
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
167.8.a.b ✓ 54 1.a even 1 1 trivial