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Properties

Label 538.6.a.d
Level $538$
Weight $6$
Character orbit 538.a
Self dual yes
Analytic conductor $86.286$
Analytic rank $0$
Dimension $32$
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] = [538,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("538.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 538 = 2 \cdot 269 \)
Weight: \( k \) \(=\) \( 6 \)
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: \(86.2864950594\)
Analytic rank: \(0\)
Dimension: \(32\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 128 q^{2} + 32 q^{3} + 512 q^{4} + 214 q^{5} + 128 q^{6} + 428 q^{7} + 2048 q^{8} + 3196 q^{9} + 856 q^{10} + 1357 q^{11} + 512 q^{12} + 1747 q^{13} + 1712 q^{14} + 2728 q^{15} + 8192 q^{16} + 3766 q^{17}+ \cdots + 338392 q^{99}+O(q^{100}) \) 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 4.00000 −29.6073 16.0000 24.4163 −118.429 80.4758 64.0000 633.594 97.6651
1.2 4.00000 −29.3470 16.0000 25.2305 −117.388 −166.437 64.0000 618.245 100.922
1.3 4.00000 −28.4796 16.0000 −79.3785 −113.918 32.3272 64.0000 568.087 −317.514
1.4 4.00000 −22.8734 16.0000 5.07497 −91.4936 −147.634 64.0000 280.192 20.2999
1.5 4.00000 −21.8767 16.0000 −3.38430 −87.5069 186.867 64.0000 235.591 −13.5372
1.6 4.00000 −18.1958 16.0000 98.2549 −72.7833 −6.69779 64.0000 88.0879 393.020
1.7 4.00000 −17.4506 16.0000 −51.4365 −69.8026 194.240 64.0000 61.5252 −205.746
1.8 4.00000 −16.3894 16.0000 −28.8885 −65.5576 −95.9166 64.0000 25.6125 −115.554
1.9 4.00000 −14.9754 16.0000 98.0560 −59.9018 −30.1626 64.0000 −18.7360 392.224
1.10 4.00000 −8.12007 16.0000 20.9594 −32.4803 −59.5157 64.0000 −177.064 83.8375
1.11 4.00000 −7.35365 16.0000 78.1821 −29.4146 209.235 64.0000 −188.924 312.729
1.12 4.00000 −6.54989 16.0000 −39.7210 −26.1996 −213.201 64.0000 −200.099 −158.884
1.13 4.00000 −5.97697 16.0000 88.0506 −23.9079 −8.25317 64.0000 −207.276 352.202
1.14 4.00000 −5.34471 16.0000 −20.8011 −21.3788 58.6538 64.0000 −214.434 −83.2043
1.15 4.00000 −4.57567 16.0000 −25.2862 −18.3027 −163.843 64.0000 −222.063 −101.145
1.16 4.00000 −4.54690 16.0000 −104.024 −18.1876 85.1535 64.0000 −222.326 −416.096
1.17 4.00000 −2.16251 16.0000 −46.7687 −8.65004 167.747 64.0000 −238.324 −187.075
1.18 4.00000 4.48488 16.0000 9.63518 17.9395 −200.095 64.0000 −222.886 38.5407
1.19 4.00000 7.78577 16.0000 −101.708 31.1431 −122.487 64.0000 −182.382 −406.832
1.20 4.00000 9.23058 16.0000 70.0627 36.9223 186.451 64.0000 −157.796 280.251
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.32
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 538.6.a.d ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
538.6.a.d ✓ 32 1.a even 1 1 trivial