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Properties

Label 538.8.a.b
Level $538$
Weight $8$
Character orbit 538.a
Self dual yes
Analytic conductor $168.063$
Analytic rank $0$
Dimension $37$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [538,8,Mod(1,538)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(538, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("538.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 538 = 2 \cdot 269 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 538.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(168.063143710\)
Analytic rank: \(0\)
Dimension: \(37\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 37 q - 296 q^{2} + 80 q^{3} + 2368 q^{4} + 624 q^{5} - 640 q^{6} + 453 q^{7} - 18944 q^{8} + 26707 q^{9} - 4992 q^{10} + 12312 q^{11} + 5120 q^{12} - 2673 q^{13} - 3624 q^{14} + 38182 q^{15} + 151552 q^{16}+ \cdots - 31108161 q^{99}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −8.00000 −86.3900 64.0000 −4.83256 691.120 −1234.41 −512.000 5276.23 38.6605
1.2 −8.00000 −84.1077 64.0000 439.841 672.862 1264.47 −512.000 4887.11 −3518.73
1.3 −8.00000 −76.7774 64.0000 −436.364 614.219 −815.525 −512.000 3707.77 3490.91
1.4 −8.00000 −67.8298 64.0000 289.818 542.639 −87.4120 −512.000 2413.89 −2318.55
1.5 −8.00000 −66.6431 64.0000 −38.8553 533.145 952.805 −512.000 2254.30 310.843
1.6 −8.00000 −65.8915 64.0000 −67.1672 527.132 1448.62 −512.000 2154.68 537.337
1.7 −8.00000 −63.0323 64.0000 −242.543 504.259 −1045.70 −512.000 1786.08 1940.34
1.8 −8.00000 −53.5974 64.0000 422.327 428.779 1358.76 −512.000 685.680 −3378.61
1.9 −8.00000 −52.3112 64.0000 −409.466 418.490 440.021 −512.000 549.463 3275.73
1.10 −8.00000 −48.7222 64.0000 165.782 389.778 −1320.62 −512.000 186.857 −1326.26
1.11 −8.00000 −47.2845 64.0000 259.043 378.276 −285.965 −512.000 48.8225 −2072.34
1.12 −8.00000 −37.8586 64.0000 −407.058 302.869 521.075 −512.000 −753.728 3256.47
1.13 −8.00000 −25.6095 64.0000 −255.788 204.876 −1147.04 −512.000 −1531.16 2046.31
1.14 −8.00000 −24.3629 64.0000 −116.636 194.903 1267.01 −512.000 −1593.45 933.090
1.15 −8.00000 −18.4868 64.0000 117.318 147.895 −512.012 −512.000 −1845.24 −938.543
1.16 −8.00000 −2.40564 64.0000 307.971 19.2451 1505.23 −512.000 −2181.21 −2463.77
1.17 −8.00000 1.29627 64.0000 −505.522 −10.3702 316.327 −512.000 −2185.32 4044.18
1.18 −8.00000 1.56345 64.0000 206.112 −12.5076 −1184.30 −512.000 −2184.56 −1648.90
1.19 −8.00000 2.39663 64.0000 −162.235 −19.1730 677.883 −512.000 −2181.26 1297.88
1.20 −8.00000 3.68486 64.0000 −24.6615 −29.4789 748.149 −512.000 −2173.42 197.292
See all 37 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.37
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(269\) \( +1 \)

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 538.8.a.b ✓ 37
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
538.8.a.b ✓ 37 1.a even 1 1 trivial