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Properties

Label 138.8.d.a
Level $138$
Weight $8$
Character orbit 138.d
Analytic conductor $43.109$
Analytic rank $0$
Dimension $56$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [138,8,Mod(137,138)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(138, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("138.137");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 138 = 2 \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 138.d (of order \(2\), degree \(1\), 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: \(43.1091335168\)
Analytic rank: \(0\)
Dimension: \(56\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 56 q - 52 q^{3} - 3584 q^{4} - 928 q^{6} - 2808 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q - 52 q^{3} - 3584 q^{4} - 928 q^{6} - 2808 q^{9} + 3328 q^{12} + 9976 q^{13} + 229376 q^{16} - 55168 q^{18} + 59392 q^{24} + 603984 q^{25} + 56564 q^{27} + 867760 q^{31} + 179712 q^{36} - 2338104 q^{39} - 1175936 q^{46} - 212992 q^{48} - 13409192 q^{49} - 638464 q^{52} + 2037088 q^{54} - 2209792 q^{55} + 1705664 q^{58} - 14680064 q^{64} - 8006564 q^{69} - 8819136 q^{70} + 3530752 q^{72} + 1955896 q^{73} - 2119732 q^{75} + 2325632 q^{78} - 8860384 q^{81} - 13670400 q^{82} + 22448712 q^{85} - 16147112 q^{87} - 20431184 q^{93} - 9890368 q^{94} - 3801088 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
137.1 − 8.00000i 20.7097 − 41.9298i −64.0000 −516.371 −335.438 − 165.678i 1728.96i 512.000i −1329.21 − 1736.71i 4130.97i
137.2 − 8.00000i 20.7097 − 41.9298i −64.0000 516.371 −335.438 − 165.678i − 1728.96i 512.000i −1329.21 − 1736.71i − 4130.97i
137.3 8.00000i 20.7097 + 41.9298i −64.0000 −516.371 −335.438 + 165.678i − 1728.96i − 512.000i −1329.21 + 1736.71i − 4130.97i
137.4 8.00000i 20.7097 + 41.9298i −64.0000 516.371 −335.438 + 165.678i 1728.96i − 512.000i −1329.21 + 1736.71i 4130.97i
137.5 − 8.00000i −16.3648 − 43.8086i −64.0000 −411.725 −350.469 + 130.918i − 1155.46i 512.000i −1651.39 + 1433.84i 3293.80i
137.6 − 8.00000i −16.3648 − 43.8086i −64.0000 411.725 −350.469 + 130.918i 1155.46i 512.000i −1651.39 + 1433.84i − 3293.80i
137.7 8.00000i −16.3648 + 43.8086i −64.0000 −411.725 −350.469 − 130.918i 1155.46i − 512.000i −1651.39 − 1433.84i − 3293.80i
137.8 8.00000i −16.3648 + 43.8086i −64.0000 411.725 −350.469 − 130.918i − 1155.46i − 512.000i −1651.39 − 1433.84i 3293.80i
137.9 − 8.00000i −11.8782 + 45.2317i −64.0000 −275.717 361.854 + 95.0253i − 1253.68i 512.000i −1904.82 − 1074.54i 2205.73i
137.10 − 8.00000i −11.8782 + 45.2317i −64.0000 275.717 361.854 + 95.0253i 1253.68i 512.000i −1904.82 − 1074.54i − 2205.73i
137.11 8.00000i −11.8782 − 45.2317i −64.0000 −275.717 361.854 − 95.0253i 1253.68i − 512.000i −1904.82 + 1074.54i − 2205.73i
137.12 8.00000i −11.8782 − 45.2317i −64.0000 275.717 361.854 − 95.0253i − 1253.68i − 512.000i −1904.82 + 1074.54i 2205.73i
137.13 − 8.00000i 24.3475 − 39.9274i −64.0000 −163.600 −319.419 − 194.780i − 1065.32i 512.000i −1001.40 − 1944.27i 1308.80i
137.14 − 8.00000i 24.3475 − 39.9274i −64.0000 163.600 −319.419 − 194.780i 1065.32i 512.000i −1001.40 − 1944.27i − 1308.80i
137.15 8.00000i 24.3475 + 39.9274i −64.0000 −163.600 −319.419 + 194.780i 1065.32i − 512.000i −1001.40 + 1944.27i − 1308.80i
137.16 8.00000i 24.3475 + 39.9274i −64.0000 163.600 −319.419 + 194.780i − 1065.32i − 512.000i −1001.40 + 1944.27i 1308.80i
137.17 − 8.00000i −38.9650 − 25.8598i −64.0000 −269.338 −206.879 + 311.720i 774.680i 512.000i 849.539 + 2015.25i 2154.70i
137.18 − 8.00000i −38.9650 − 25.8598i −64.0000 269.338 −206.879 + 311.720i − 774.680i 512.000i 849.539 + 2015.25i − 2154.70i
137.19 8.00000i −38.9650 + 25.8598i −64.0000 −269.338 −206.879 − 311.720i − 774.680i − 512.000i 849.539 − 2015.25i − 2154.70i
137.20 8.00000i −38.9650 + 25.8598i −64.0000 269.338 −206.879 − 311.720i 774.680i − 512.000i 849.539 − 2015.25i 2154.70i
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 137.56
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
23.b odd 2 1 inner
69.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 138.8.d.a ✓ 56
3.b odd 2 1 inner 138.8.d.a ✓ 56
23.b odd 2 1 inner 138.8.d.a ✓ 56
69.c even 2 1 inner 138.8.d.a ✓ 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.8.d.a ✓ 56 1.a even 1 1 trivial
138.8.d.a ✓ 56 3.b odd 2 1 inner
138.8.d.a ✓ 56 23.b odd 2 1 inner
138.8.d.a ✓ 56 69.c even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(138, [\chi])\).