↑

Properties

Label 129.4.d.b
Level $129$
Weight $4$
Character orbit 129.d
Analytic conductor $7.611$
Analytic rank $0$
Dimension $40$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [129,4,Mod(128,129)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(129, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("129.128");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 129 = 3 \cdot 43 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 129.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: \(7.61124639074\)
Analytic rank: \(0\)
Dimension: \(40\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 172 q^{4} - 6 q^{6} + 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q + 172 q^{4} - 6 q^{6} + 54 q^{9} - 108 q^{10} - 172 q^{13} - 56 q^{15} + 100 q^{16} + 104 q^{21} - 70 q^{24} + 1192 q^{25} + 572 q^{31} - 854 q^{36} + 60 q^{40} - 860 q^{43} + 76 q^{49} - 184 q^{52} - 2030 q^{54} - 1320 q^{57} - 2100 q^{58} + 1088 q^{60} + 28 q^{64} - 1584 q^{66} + 1784 q^{67} - 436 q^{78} + 2876 q^{79} - 3026 q^{81} + 3100 q^{84} - 2166 q^{87} - 4306 q^{90} + 2786 q^{96} + 3392 q^{97} + 7076 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
128.1 −5.27650 −2.88180 − 4.32380i 19.8414 7.18332 15.2058 + 22.8145i 7.72553i −62.4811 −10.3905 + 24.9206i −37.9028
128.2 −5.27650 −2.88180 + 4.32380i 19.8414 7.18332 15.2058 − 22.8145i − 7.72553i −62.4811 −10.3905 − 24.9206i −37.9028
128.3 −5.04106 3.97147 − 3.35074i 17.4122 −5.66794 −20.0204 + 16.8913i − 20.7753i −47.4476 4.54509 − 26.6147i 28.5724
128.4 −5.04106 3.97147 + 3.35074i 17.4122 −5.66794 −20.0204 − 16.8913i 20.7753i −47.4476 4.54509 + 26.6147i 28.5724
128.5 −4.06771 4.44323 − 2.69401i 8.54624 18.3154 −18.0738 + 10.9584i 30.6303i −2.22195 12.4846 − 23.9402i −74.5016
128.6 −4.06771 4.44323 + 2.69401i 8.54624 18.3154 −18.0738 − 10.9584i − 30.6303i −2.22195 12.4846 + 23.9402i −74.5016
128.7 −3.98576 −0.915028 − 5.11495i 7.88630 −20.6038 3.64708 + 20.3870i 17.3737i 0.453195 −25.3254 + 9.36065i 82.1220
128.8 −3.98576 −0.915028 + 5.11495i 7.88630 −20.6038 3.64708 − 20.3870i − 17.3737i 0.453195 −25.3254 − 9.36065i 82.1220
128.9 −3.92435 −5.14192 − 0.748758i 7.40050 −2.99210 20.1787 + 2.93839i − 22.8501i 2.35265 25.8787 + 7.70011i 11.7420
128.10 −3.92435 −5.14192 + 0.748758i 7.40050 −2.99210 20.1787 − 2.93839i 22.8501i 2.35265 25.8787 − 7.70011i 11.7420
128.11 −3.06801 −2.08321 − 4.76027i 1.41267 15.0080 6.39132 + 14.6046i − 12.2345i 20.2100 −18.3204 + 19.8333i −46.0447
128.12 −3.06801 −2.08321 + 4.76027i 1.41267 15.0080 6.39132 − 14.6046i 12.2345i 20.2100 −18.3204 − 19.8333i −46.0447
128.13 −2.44225 4.99892 − 1.41802i −2.03543 −7.59046 −12.2086 + 3.46316i 15.7306i 24.5090 22.9784 − 14.1772i 18.5378
128.14 −2.44225 4.99892 + 1.41802i −2.03543 −7.59046 −12.2086 − 3.46316i − 15.7306i 24.5090 22.9784 + 14.1772i 18.5378
128.15 −2.28958 1.52984 − 4.96584i −2.75784 0.329325 −3.50269 + 11.3697i − 11.9797i 24.6309 −22.3192 − 15.1939i −0.754014
128.16 −2.28958 1.52984 + 4.96584i −2.75784 0.329325 −3.50269 − 11.3697i 11.9797i 24.6309 −22.3192 + 15.1939i −0.754014
128.17 −1.00191 −4.59953 − 2.41750i −6.99618 15.6870 4.60830 + 2.42211i 19.9392i 15.0248 15.3113 + 22.2388i −15.7169
128.18 −1.00191 −4.59953 + 2.41750i −6.99618 15.6870 4.60830 − 2.42211i − 19.9392i 15.0248 15.3113 − 22.2388i −15.7169
128.19 −0.538613 −4.22240 − 3.02842i −7.70990 −12.8958 2.27424 + 1.63115i − 14.8086i 8.46156 8.65732 + 25.5744i 6.94587
128.20 −0.538613 −4.22240 + 3.02842i −7.70990 −12.8958 2.27424 − 1.63115i 14.8086i 8.46156 8.65732 − 25.5744i 6.94587
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 128.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
43.b odd 2 1 inner
129.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 129.4.d.b ✓ 40
3.b odd 2 1 inner 129.4.d.b ✓ 40
43.b odd 2 1 inner 129.4.d.b ✓ 40
129.d even 2 1 inner 129.4.d.b ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
129.4.d.b ✓ 40 1.a even 1 1 trivial
129.4.d.b ✓ 40 3.b odd 2 1 inner
129.4.d.b ✓ 40 43.b odd 2 1 inner
129.4.d.b ✓ 40 129.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} - 123 T_{2}^{18} + 6396 T_{2}^{16} - 183544 T_{2}^{14} + 3181233 T_{2}^{12} + \cdots + 245472000 \) acting on \(S_{4}^{\mathrm{new}}(129, [\chi])\). Copy content Toggle raw display