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Properties

Label 538.8.a.a
Level $538$
Weight $8$
Character orbit 538.a
Self dual yes
Analytic conductor $168.063$
Analytic rank $1$
Dimension $35$
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: \(1\)
Dimension: \(35\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 35 q + 280 q^{2} - 66 q^{3} + 2240 q^{4} - 1126 q^{5} - 528 q^{6} - 3196 q^{7} + 17920 q^{8} + 16893 q^{9} - 9008 q^{10} - 16053 q^{11} - 4224 q^{12} - 16615 q^{13} - 25568 q^{14} - 42338 q^{15} + 143360 q^{16}+ \cdots - 70193806 q^{99}+O(q^{100}) \) 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 −87.4696 64.0000 −14.9114 −699.757 696.948 512.000 5463.92 −119.291
1.2 8.00000 −86.5070 64.0000 265.568 −692.056 824.433 512.000 5296.46 2124.54
1.3 8.00000 −79.2150 64.0000 −296.391 −633.720 −1053.73 512.000 4088.02 −2371.13
1.4 8.00000 −70.6144 64.0000 −122.340 −564.915 328.106 512.000 2799.39 −978.720
1.5 8.00000 −64.4195 64.0000 25.5214 −515.356 1552.39 512.000 1962.87 204.172
1.6 8.00000 −62.5719 64.0000 −49.0557 −500.575 −1323.63 512.000 1728.24 −392.445
1.7 8.00000 −59.0455 64.0000 86.3281 −472.364 −1159.96 512.000 1299.37 690.625
1.8 8.00000 −51.8557 64.0000 377.145 −414.846 −341.085 512.000 502.015 3017.16
1.9 8.00000 −44.4394 64.0000 −497.631 −355.516 −1574.96 512.000 −212.136 −3981.05
1.10 8.00000 −43.6477 64.0000 419.845 −349.181 158.174 512.000 −281.880 3358.76
1.11 8.00000 −42.5865 64.0000 −551.172 −340.692 −307.941 512.000 −373.393 −4409.38
1.12 8.00000 −32.2126 64.0000 173.271 −257.701 363.058 512.000 −1149.35 1386.16
1.13 8.00000 −26.4233 64.0000 −307.454 −211.386 −534.463 512.000 −1488.81 −2459.63
1.14 8.00000 −20.1389 64.0000 116.446 −161.111 −206.552 512.000 −1781.42 931.569
1.15 8.00000 −19.0568 64.0000 319.121 −152.454 −1291.59 512.000 −1823.84 2552.97
1.16 8.00000 −15.0242 64.0000 −254.360 −120.194 −377.492 512.000 −1961.27 −2034.88
1.17 8.00000 −14.6799 64.0000 −134.756 −117.439 1219.17 512.000 −1971.50 −1078.04
1.18 8.00000 −4.92513 64.0000 −427.609 −39.4010 1017.60 512.000 −2162.74 −3420.88
1.19 8.00000 0.525252 64.0000 201.246 4.20202 768.758 512.000 −2186.72 1609.97
1.20 8.00000 9.38981 64.0000 −35.2592 75.1185 1733.62 512.000 −2098.83 −282.073
See all 35 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.35
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.a ✓ 35
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
538.8.a.a ✓ 35 1.a even 1 1 trivial