↑

Properties

Label 538.6.a.c
Level $538$
Weight $6$
Character orbit 538.a
Self dual yes
Analytic conductor $86.286$
Analytic rank $1$
Dimension $30$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(1\)
Dimension: \(30\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 30 q - 120 q^{2} - 30 q^{3} + 480 q^{4} - 136 q^{5} + 120 q^{6} - 123 q^{7} - 1920 q^{8} + 2670 q^{9} + 544 q^{10} - 1058 q^{11} - 480 q^{12} - 371 q^{13} + 492 q^{14} - 1364 q^{15} + 7680 q^{16} - 1918 q^{17}+ \cdots - 78063 q^{99}+O(q^{100}) \) 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 −30.9251 16.0000 80.4208 123.700 −129.956 −64.0000 713.363 −321.683
1.2 −4.00000 −28.8260 16.0000 −7.34973 115.304 103.117 −64.0000 587.939 29.3989
1.3 −4.00000 −27.1105 16.0000 10.2995 108.442 251.758 −64.0000 491.979 −41.1981
1.4 −4.00000 −25.4076 16.0000 −105.902 101.630 −235.883 −64.0000 402.544 423.608
1.5 −4.00000 −23.0229 16.0000 −7.03191 92.0915 −152.036 −64.0000 287.053 28.1276
1.6 −4.00000 −20.8118 16.0000 45.7716 83.2472 168.664 −64.0000 190.131 −183.086
1.7 −4.00000 −18.8532 16.0000 −69.9105 75.4127 251.245 −64.0000 112.442 279.642
1.8 −4.00000 −17.8255 16.0000 7.99471 71.3018 −253.891 −64.0000 74.7468 −31.9789
1.9 −4.00000 −15.8265 16.0000 27.2096 63.3061 12.3272 −64.0000 7.47886 −108.838
1.10 −4.00000 −12.2410 16.0000 −56.0956 48.9638 66.3991 −64.0000 −93.1590 224.383
1.11 −4.00000 −8.28314 16.0000 36.5236 33.1326 79.5135 −64.0000 −174.390 −146.095
1.12 −4.00000 −7.47143 16.0000 −71.6988 29.8857 −164.816 −64.0000 −187.178 286.795
1.13 −4.00000 −7.00462 16.0000 30.1411 28.0185 −141.413 −64.0000 −193.935 −120.565
1.14 −4.00000 −5.37473 16.0000 −58.0252 21.4989 −22.6689 −64.0000 −214.112 232.101
1.15 −4.00000 −4.33734 16.0000 105.731 17.3493 −161.969 −64.0000 −224.188 −422.923
1.16 −4.00000 0.296724 16.0000 −103.439 −1.18689 −37.7073 −64.0000 −242.912 413.755
1.17 −4.00000 2.68265 16.0000 25.8421 −10.7306 96.3372 −64.0000 −235.803 −103.369
1.18 −4.00000 3.94626 16.0000 66.0339 −15.7851 122.020 −64.0000 −227.427 −264.136
1.19 −4.00000 8.15361 16.0000 55.7791 −32.6145 −53.6485 −64.0000 −176.519 −223.116
1.20 −4.00000 9.07209 16.0000 62.0192 −36.2884 −113.415 −64.0000 −160.697 −248.077
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.30
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.c ✓ 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
538.6.a.c ✓ 30 1.a even 1 1 trivial