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Properties

Label 538.6.a.b
Level $538$
Weight $6$
Character orbit 538.a
Self dual yes
Analytic conductor $86.286$
Analytic rank $0$
Dimension $27$
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: \(27\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 27 q - 108 q^{2} + 33 q^{3} + 432 q^{4} + 139 q^{5} - 132 q^{6} - 25 q^{7} - 1728 q^{8} + 2346 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 27 q - 108 q^{2} + 33 q^{3} + 432 q^{4} + 139 q^{5} - 132 q^{6} - 25 q^{7} - 1728 q^{8} + 2346 q^{9} - 556 q^{10} + 1241 q^{11} + 528 q^{12} - 202 q^{13} + 100 q^{14} + 1786 q^{15} + 6912 q^{16} + 1550 q^{17} - 9384 q^{18} - 66 q^{19} + 2224 q^{20} - 1217 q^{21} - 4964 q^{22} + 10568 q^{23} - 2112 q^{24} + 15880 q^{25} + 808 q^{26} + 8256 q^{27} - 400 q^{28} - 2122 q^{29} - 7144 q^{30} - 1621 q^{31} - 27648 q^{32} + 18369 q^{33} - 6200 q^{34} + 23123 q^{35} + 37536 q^{36} + 8542 q^{37} + 264 q^{38} + 589 q^{39} - 8896 q^{40} + 10859 q^{41} + 4868 q^{42} + 23759 q^{43} + 19856 q^{44} + 28006 q^{45} - 42272 q^{46} + 38457 q^{47} + 8448 q^{48} + 57376 q^{49} - 63520 q^{50} + 58237 q^{51} - 3232 q^{52} + 73586 q^{53} - 33024 q^{54} + 25923 q^{55} + 1600 q^{56} + 52534 q^{57} + 8488 q^{58} + 68927 q^{59} + 28576 q^{60} - 76016 q^{61} + 6484 q^{62} - 20738 q^{63} + 110592 q^{64} + 74867 q^{65} - 73476 q^{66} + 94787 q^{67} + 24800 q^{68} - 90601 q^{69} - 92492 q^{70} + 15523 q^{71} - 150144 q^{72} - 96279 q^{73} - 34168 q^{74} - 123786 q^{75} - 1056 q^{76} - 199866 q^{77} - 2356 q^{78} - 246217 q^{79} + 35584 q^{80} - 54473 q^{81} - 43436 q^{82} - 56658 q^{83} - 19472 q^{84} - 120541 q^{85} - 95036 q^{86} - 352954 q^{87} - 79424 q^{88} - 83070 q^{89} - 112024 q^{90} - 132348 q^{91} + 169088 q^{92} + 40727 q^{93} - 153828 q^{94} - 72090 q^{95} - 33792 q^{96} - 36273 q^{97} - 229504 q^{98} + 335378 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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 −29.2728 16.0000 −50.9628 117.091 −3.17793 −64.0000 613.894 203.851
1.2 −4.00000 −24.3343 16.0000 −84.4416 97.3371 −27.4793 −64.0000 349.157 337.766
1.3 −4.00000 −22.7912 16.0000 54.5062 91.1647 54.5641 −64.0000 276.437 −218.025
1.4 −4.00000 −21.2938 16.0000 −39.3815 85.1754 −2.28910 −64.0000 210.428 157.526
1.5 −4.00000 −19.0757 16.0000 93.2221 76.3028 72.1299 −64.0000 120.882 −372.888
1.6 −4.00000 −19.0431 16.0000 92.3025 76.1725 −74.0523 −64.0000 119.641 −369.210
1.7 −4.00000 −18.8542 16.0000 33.4033 75.4169 −183.453 −64.0000 112.482 −133.613
1.8 −4.00000 −15.3559 16.0000 15.2647 61.4236 −104.139 −64.0000 −7.19597 −61.0589
1.9 −4.00000 −14.5248 16.0000 −87.9895 58.0992 116.671 −64.0000 −32.0301 351.958
1.10 −4.00000 −8.18103 16.0000 −4.49984 32.7241 129.673 −64.0000 −176.071 17.9994
1.11 −4.00000 −5.64678 16.0000 86.8943 22.5871 216.055 −64.0000 −211.114 −347.577
1.12 −4.00000 0.0232818 16.0000 79.7413 −0.0931271 169.626 −64.0000 −242.999 −318.965
1.13 −4.00000 0.615828 16.0000 −16.0468 −2.46331 103.596 −64.0000 −242.621 64.1872
1.14 −4.00000 2.64462 16.0000 −42.4071 −10.5785 −172.158 −64.0000 −236.006 169.629
1.15 −4.00000 3.44941 16.0000 −20.1864 −13.7977 −171.258 −64.0000 −231.102 80.7457
1.16 −4.00000 5.43763 16.0000 −50.9250 −21.7505 53.3146 −64.0000 −213.432 203.700
1.17 −4.00000 9.69825 16.0000 −72.2025 −38.7930 −54.9958 −64.0000 −148.944 288.810
1.18 −4.00000 10.6837 16.0000 72.6773 −42.7347 −163.422 −64.0000 −128.859 −290.709
1.19 −4.00000 15.4253 16.0000 −30.3539 −61.7011 145.934 −64.0000 −5.06117 121.416
1.20 −4.00000 16.4424 16.0000 9.98004 −65.7694 −46.2862 −64.0000 27.3512 −39.9202
See all 27 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.27
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.b ✓ 27
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
538.6.a.b ✓ 27 1.a even 1 1 trivial