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Properties

Label 538.2.d.a
Level $538$
Weight $2$
Character orbit 538.d
Analytic conductor $4.296$
Analytic rank $0$
Dimension $726$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [538,2,Mod(5,538)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(538, base_ring=CyclotomicField(134))
 
chi = DirichletCharacter(H, H._module([104]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("538.5");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 538 = 2 \cdot 269 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 538.d (of order \(67\), degree \(66\), minimal)

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: no
Analytic conductor: \(4.29595162874\)
Analytic rank: \(0\)
Dimension: \(726\)
Relative dimension: \(11\) over \(\Q(\zeta_{67})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{67}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 726 q + 11 q^{2} + q^{3} - 11 q^{4} + q^{5} - q^{6} + 4 q^{7} + 11 q^{8} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 726 q + 11 q^{2} + q^{3} - 11 q^{4} + q^{5} - q^{6} + 4 q^{7} + 11 q^{8} - 10 q^{9} - q^{10} + 5 q^{11} + q^{12} - 7 q^{13} - 4 q^{14} - 2 q^{15} - 11 q^{16} + 4 q^{17} + 10 q^{18} + 5 q^{19} + q^{20} - 191 q^{21} - 5 q^{22} + 6 q^{23} - q^{24} - 8 q^{25} + 7 q^{26} - 185 q^{27} + 4 q^{28} + 7 q^{29} - 65 q^{30} + 4 q^{31} + 11 q^{32} + 18 q^{33} - 4 q^{34} + 22 q^{35} - 10 q^{36} + 216 q^{37} - 5 q^{38} - 4 q^{39} - q^{40} - 122 q^{41} - 10 q^{42} + 5 q^{43} - 62 q^{44} + 9 q^{45} - 6 q^{46} + 211 q^{47} + q^{48} + 9 q^{49} + 8 q^{50} + 18 q^{51} - 7 q^{52} - 141 q^{53} - 16 q^{54} - 28 q^{55} - 4 q^{56} + 4 q^{57} - 7 q^{58} + q^{59} + 132 q^{60} - 60 q^{61} - 4 q^{62} - 11 q^{64} + 24 q^{65} - 18 q^{66} + 15 q^{67} + 4 q^{68} - 8 q^{69} - 22 q^{70} + 8 q^{71} + 10 q^{72} + 65 q^{73} - 15 q^{74} - 15 q^{75} + 5 q^{76} - 2 q^{77} + 4 q^{78} - 61 q^{79} + q^{80} - 315 q^{81} - 280 q^{82} + 198 q^{83} - 258 q^{84} - 126 q^{85} - 5 q^{86} + 6 q^{87} - 5 q^{88} - 36 q^{89} - 9 q^{90} + 28 q^{91} + 6 q^{92} + 152 q^{93} - 10 q^{94} + 6 q^{95} - q^{96} - 24 q^{97} - 9 q^{98} + 25 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 0.762685 − 0.646770i −0.776184 + 2.46734i 0.163377 − 0.986564i −0.704737 − 3.29023i 1.00382 + 2.38382i −2.42768 + 2.48527i −0.513474 − 0.858105i −3.02562 − 2.11270i −2.66552 − 2.05361i
5.2 0.762685 − 0.646770i −0.747651 + 2.37664i 0.163377 − 0.986564i 0.634546 + 2.96253i 0.966918 + 2.29619i −0.839181 + 0.859090i −0.513474 − 0.858105i −2.62974 − 1.83627i 2.40003 + 1.84907i
5.3 0.762685 − 0.646770i −0.591238 + 1.87944i 0.163377 − 0.986564i −0.0746505 − 0.348523i 0.764634 + 1.81581i 2.45258 − 2.51077i −0.513474 − 0.858105i −0.723021 − 0.504862i −0.282349 − 0.217532i
5.4 0.762685 − 0.646770i −0.456170 + 1.45008i 0.163377 − 0.986564i −0.177575 − 0.829052i 0.589953 + 1.40099i −0.732047 + 0.749414i −0.513474 − 0.858105i 0.565057 + 0.394562i −0.671640 − 0.517455i
5.5 0.762685 − 0.646770i −0.0562281 + 0.178739i 0.163377 − 0.986564i 0.190837 + 0.890968i 0.0727184 + 0.172688i −3.49785 + 3.58084i −0.513474 − 0.858105i 2.43091 + 1.69743i 0.721800 + 0.556100i
5.6 0.762685 − 0.646770i 0.104324 − 0.331627i 0.163377 − 0.986564i −0.661637 − 3.08901i −0.134920 − 0.320401i 2.51990 − 2.57969i −0.513474 − 0.858105i 2.36060 + 1.64833i −2.50250 − 1.92801i
5.7 0.762685 − 0.646770i 0.282253 − 0.897228i 0.163377 − 0.986564i 0.214342 + 1.00070i −0.365030 − 0.866855i 1.50863 − 1.54442i −0.513474 − 0.858105i 1.73434 + 1.21104i 0.810701 + 0.624593i
5.8 0.762685 − 0.646770i 0.283548 − 0.901345i 0.163377 − 0.986564i 0.686321 + 3.20425i −0.366705 − 0.870832i 0.576604 − 0.590284i −0.513474 − 0.858105i 1.72767 + 1.20638i 2.59586 + 1.99994i
5.9 0.762685 − 0.646770i 0.591541 − 1.88040i 0.163377 − 0.986564i −0.904268 − 4.22179i −0.765025 − 1.81674i −1.51720 + 1.55319i −0.513474 − 0.858105i −0.726277 − 0.507136i −3.42020 − 2.63504i
5.10 0.762685 − 0.646770i 0.690820 − 2.19599i 0.163377 − 0.986564i 0.00920101 + 0.0429571i −0.893420 − 2.12165i −1.35729 + 1.38949i −0.513474 − 0.858105i −1.88543 − 1.31653i 0.0348008 + 0.0268118i
5.11 0.762685 − 0.646770i 0.933025 − 2.96591i 0.163377 − 0.986564i 0.0415518 + 0.193994i −1.20666 − 2.86551i 2.29470 − 2.34914i −0.513474 − 0.858105i −5.46640 − 3.81701i 0.157161 + 0.121082i
21.1 −0.0702762 − 0.997528i −3.00595 − 0.570494i −0.990123 + 0.140205i −0.140608 − 0.262135i −0.357837 + 3.03861i 2.00498 + 0.478869i 0.209440 + 0.977821i 5.91885 + 2.33061i −0.251605 + 0.158682i
21.2 −0.0702762 − 0.997528i −2.08992 − 0.396644i −0.990123 + 0.140205i 0.942818 + 1.75769i −0.248791 + 2.11263i −3.39297 − 0.810374i 0.209440 + 0.977821i 1.41906 + 0.558771i 1.68708 − 1.06401i
21.3 −0.0702762 − 0.997528i −1.83107 − 0.347517i −0.990123 + 0.140205i −1.56002 − 2.90832i −0.217977 + 1.85097i −1.26988 − 0.303298i 0.209440 + 0.977821i 0.440666 + 0.173517i −2.79150 + 1.76054i
21.4 −0.0702762 − 0.997528i −1.29336 − 0.245465i −0.990123 + 0.140205i −0.311985 − 0.581631i −0.153965 + 1.30741i 3.46621 + 0.827867i 0.209440 + 0.977821i −1.17887 − 0.464191i −0.558268 + 0.352088i
21.5 −0.0702762 − 0.997528i −0.948653 − 0.180043i −0.990123 + 0.140205i 1.76758 + 3.29528i −0.112931 + 0.958960i −2.20438 − 0.526493i 0.209440 + 0.977821i −1.92387 − 0.757543i 3.16291 − 1.99479i
21.6 −0.0702762 − 0.997528i 0.673422 + 0.127808i −0.990123 + 0.140205i −0.370999 − 0.691651i 0.0801662 − 0.680739i −1.80627 − 0.431408i 0.209440 + 0.977821i −2.35423 − 0.927003i −0.663869 + 0.418689i
21.7 −0.0702762 − 0.997528i 0.964062 + 0.182968i −0.990123 + 0.140205i 0.876703 + 1.63443i 0.114765 − 0.974537i 2.63572 + 0.629513i 0.209440 + 0.977821i −1.89546 − 0.746355i 1.56878 − 0.989397i
21.8 −0.0702762 − 0.997528i 1.91835 + 0.364080i −0.990123 + 0.140205i −1.94187 − 3.62021i 0.228366 − 1.93919i 4.90486 + 1.17147i 0.209440 + 0.977821i 0.756111 + 0.297726i −3.47479 + 2.19148i
21.9 −0.0702762 − 0.997528i 2.04464 + 0.388049i −0.990123 + 0.140205i −0.320692 − 0.597863i 0.243400 − 2.06686i −4.26515 − 1.01868i 0.209440 + 0.977821i 1.23858 + 0.487703i −0.573848 + 0.361914i
See next 80 embeddings (of 726 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.11
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
269.d even 67 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 538.2.d.a ✓ 726
269.d even 67 1 inner 538.2.d.a ✓ 726
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
538.2.d.a ✓ 726 1.a even 1 1 trivial
538.2.d.a ✓ 726 269.d even 67 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{726} - T_{3}^{725} + 22 T_{3}^{724} + 38 T_{3}^{723} + 333 T_{3}^{722} + 805 T_{3}^{721} + \cdots + 91\!\cdots\!81 \) acting on \(S_{2}^{\mathrm{new}}(538, [\chi])\). Copy content Toggle raw display