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Properties

Label 538.8.a.c
Level $538$
Weight $8$
Character orbit 538.a
Self dual yes
Analytic conductor $168.063$
Analytic rank $1$
Dimension $40$
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: \(40\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 320 q^{2} - 109 q^{3} + 2560 q^{4} - 751 q^{5} + 872 q^{6} - 233 q^{7} - 20480 q^{8} + 29623 q^{9} + 6008 q^{10} - 12977 q^{11} - 6976 q^{12} - 4870 q^{13} + 1864 q^{14} - 9068 q^{15} + 163840 q^{16}+ \cdots + 9909340 q^{99}+O(q^{100}) \) 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 −88.0215 64.0000 −492.867 704.172 1216.54 −512.000 5560.79 3942.93
1.2 −8.00000 −86.9213 64.0000 −78.3854 695.370 51.4362 −512.000 5368.31 627.083
1.3 −8.00000 −84.1165 64.0000 317.613 672.932 132.828 −512.000 4888.58 −2540.91
1.4 −8.00000 −82.6709 64.0000 −228.141 661.368 −449.827 −512.000 4647.48 1825.13
1.5 −8.00000 −78.7674 64.0000 294.812 630.140 89.8790 −512.000 4017.31 −2358.49
1.6 −8.00000 −72.1201 64.0000 −232.395 576.961 1331.71 −512.000 3014.30 1859.16
1.7 −8.00000 −66.3983 64.0000 433.449 531.186 −1762.13 −512.000 2221.73 −3467.59
1.8 −8.00000 −61.6300 64.0000 68.6323 493.040 353.255 −512.000 1611.25 −549.058
1.9 −8.00000 −57.3727 64.0000 −154.183 458.982 −1126.39 −512.000 1104.63 1233.46
1.10 −8.00000 −54.3544 64.0000 −133.374 434.836 −588.564 −512.000 767.405 1066.99
1.11 −8.00000 −38.6224 64.0000 27.7308 308.980 1156.53 −512.000 −695.307 −221.846
1.12 −8.00000 −33.0172 64.0000 26.6415 264.137 631.905 −512.000 −1096.87 −213.132
1.13 −8.00000 −29.8214 64.0000 183.000 238.571 −278.933 −512.000 −1297.69 −1464.00
1.14 −8.00000 −29.1710 64.0000 −496.219 233.368 −305.036 −512.000 −1336.05 3969.75
1.15 −8.00000 −27.3504 64.0000 −426.693 218.803 −434.276 −512.000 −1438.96 3413.55
1.16 −8.00000 −25.6095 64.0000 548.719 204.876 −21.1122 −512.000 −1531.15 −4389.75
1.17 −8.00000 −25.5264 64.0000 439.995 204.211 −363.962 −512.000 −1535.40 −3519.96
1.18 −8.00000 −19.6652 64.0000 −242.251 157.322 −1441.33 −512.000 −1800.28 1938.01
1.19 −8.00000 −15.6593 64.0000 356.182 125.275 −732.429 −512.000 −1941.78 −2849.46
1.20 −8.00000 −14.7346 64.0000 −379.054 117.877 1370.51 −512.000 −1969.89 3032.43
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.40
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.c ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
538.8.a.c ✓ 40 1.a even 1 1 trivial