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Properties

Label 538.3.c.b
Level $538$
Weight $3$
Character orbit 538.c
Analytic conductor $14.659$
Analytic rank $0$
Dimension $46$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [538,3,Mod(187,538)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(538, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("538.187");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 538 = 2 \cdot 269 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 538.c (of order \(4\), degree \(2\), 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: \(14.6594382226\)
Analytic rank: \(0\)
Dimension: \(46\)
Relative dimension: \(23\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 46 q - 46 q^{2} - 6 q^{3} - 32 q^{5} + 4 q^{7} + 92 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 46 q - 46 q^{2} - 6 q^{3} - 32 q^{5} + 4 q^{7} + 92 q^{8} + 32 q^{10} + 12 q^{12} - 8 q^{14} - 10 q^{15} - 184 q^{16} - 50 q^{17} + 118 q^{18} + 10 q^{19} + 16 q^{21} + 8 q^{22} - 124 q^{23} - 24 q^{24} + 178 q^{25} + 4 q^{26} + 90 q^{27} + 8 q^{28} + 40 q^{29} + 40 q^{31} + 184 q^{32} - 64 q^{33} - 14 q^{35} - 236 q^{36} - 32 q^{37} - 20 q^{38} - 54 q^{39} - 64 q^{40} - 132 q^{41} - 16 q^{42} - 16 q^{44} + 124 q^{46} - 60 q^{47} + 24 q^{48} - 178 q^{50} - 8 q^{52} - 188 q^{53} - 180 q^{54} - 40 q^{57} - 80 q^{58} + 190 q^{59} + 20 q^{60} - 80 q^{62} - 266 q^{63} + 128 q^{66} - 164 q^{67} + 100 q^{68} + 170 q^{69} + 28 q^{70} - 92 q^{71} + 236 q^{72} + 32 q^{74} - 190 q^{75} + 20 q^{76} + 20 q^{77} + 108 q^{78} + 128 q^{80} - 298 q^{81} + 132 q^{82} + 190 q^{83} + 298 q^{85} + 228 q^{86} - 92 q^{87} + 16 q^{88} - 528 q^{90} + 2 q^{91} + 232 q^{93} + 60 q^{94} - 280 q^{95} - 226 q^{98} + 508 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
187.1 −1.00000 − 1.00000i −4.16653 − 4.16653i 2.00000i −6.39863 8.33306i −6.86857 + 6.86857i 2.00000 − 2.00000i 25.7199i 6.39863 + 6.39863i
187.2 −1.00000 − 1.00000i −3.73758 − 3.73758i 2.00000i 3.56643 7.47516i 0.891381 − 0.891381i 2.00000 − 2.00000i 18.9390i −3.56643 − 3.56643i
187.3 −1.00000 − 1.00000i −3.24321 − 3.24321i 2.00000i 4.65083 6.48641i 4.88017 − 4.88017i 2.00000 − 2.00000i 12.0368i −4.65083 − 4.65083i
187.4 −1.00000 − 1.00000i −2.93814 − 2.93814i 2.00000i −0.402162 5.87628i −0.366019 + 0.366019i 2.00000 − 2.00000i 8.26535i 0.402162 + 0.402162i
187.5 −1.00000 − 1.00000i −2.38876 − 2.38876i 2.00000i −4.67342 4.77753i 4.02389 − 4.02389i 2.00000 − 2.00000i 2.41237i 4.67342 + 4.67342i
187.6 −1.00000 − 1.00000i −2.33317 − 2.33317i 2.00000i −8.37238 4.66635i 7.86410 − 7.86410i 2.00000 − 2.00000i 1.88739i 8.37238 + 8.37238i
187.7 −1.00000 − 1.00000i −2.21999 − 2.21999i 2.00000i −7.23380 4.43998i −6.31833 + 6.31833i 2.00000 − 2.00000i 0.856692i 7.23380 + 7.23380i
187.8 −1.00000 − 1.00000i −1.96983 − 1.96983i 2.00000i 5.21476 3.93966i −6.17471 + 6.17471i 2.00000 − 2.00000i − 1.23954i −5.21476 − 5.21476i
187.9 −1.00000 − 1.00000i −1.00320 − 1.00320i 2.00000i 7.12333 2.00640i 8.68530 − 8.68530i 2.00000 − 2.00000i − 6.98717i −7.12333 − 7.12333i
187.10 −1.00000 − 1.00000i −0.937502 − 0.937502i 2.00000i 2.85894 1.87500i −4.26196 + 4.26196i 2.00000 − 2.00000i − 7.24218i −2.85894 − 2.85894i
187.11 −1.00000 − 1.00000i −0.629399 − 0.629399i 2.00000i −4.51452 1.25880i −1.06915 + 1.06915i 2.00000 − 2.00000i − 8.20771i 4.51452 + 4.51452i
187.12 −1.00000 − 1.00000i 0.0373023 + 0.0373023i 2.00000i 8.88487 − 0.0746046i −3.71313 + 3.71313i 2.00000 − 2.00000i − 8.99722i −8.88487 − 8.88487i
187.13 −1.00000 − 1.00000i 0.227895 + 0.227895i 2.00000i −3.63179 − 0.455790i 2.58294 − 2.58294i 2.00000 − 2.00000i − 8.89613i 3.63179 + 3.63179i
187.14 −1.00000 − 1.00000i 0.800377 + 0.800377i 2.00000i −1.60802 − 1.60075i 2.83686 − 2.83686i 2.00000 − 2.00000i − 7.71879i 1.60802 + 1.60802i
187.15 −1.00000 − 1.00000i 0.944590 + 0.944590i 2.00000i −6.79358 − 1.88918i −6.27837 + 6.27837i 2.00000 − 2.00000i − 7.21550i 6.79358 + 6.79358i
187.16 −1.00000 − 1.00000i 1.31676 + 1.31676i 2.00000i 0.898633 − 2.63352i 7.62068 − 7.62068i 2.00000 − 2.00000i − 5.53228i −0.898633 − 0.898633i
187.17 −1.00000 − 1.00000i 1.35493 + 1.35493i 2.00000i 4.13408 − 2.70986i 1.20624 − 1.20624i 2.00000 − 2.00000i − 5.32832i −4.13408 − 4.13408i
187.18 −1.00000 − 1.00000i 2.36640 + 2.36640i 2.00000i 6.44002 − 4.73279i −3.37537 + 3.37537i 2.00000 − 2.00000i 2.19967i −6.44002 − 6.44002i
187.19 −1.00000 − 1.00000i 2.50903 + 2.50903i 2.00000i −9.33859 − 5.01806i 7.56973 − 7.56973i 2.00000 − 2.00000i 3.59045i 9.33859 + 9.33859i
187.20 −1.00000 − 1.00000i 2.52768 + 2.52768i 2.00000i 0.166601 − 5.05536i −8.52444 + 8.52444i 2.00000 − 2.00000i 3.77835i −0.166601 − 0.166601i
See all 46 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 187.23
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
269.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 538.3.c.b ✓ 46
269.c odd 4 1 inner 538.3.c.b ✓ 46
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
538.3.c.b ✓ 46 1.a even 1 1 trivial
538.3.c.b ✓ 46 269.c odd 4 1 inner