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Properties

Label 129.3.c.a
Level $129$
Weight $3$
Character orbit 129.c
Analytic conductor $3.515$
Analytic rank $0$
Dimension $28$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [129,3,Mod(44,129)]
 
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, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("129.44");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 129 = 3 \cdot 43 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 129.c (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: \(3.51499541025\)
Analytic rank: \(0\)
Dimension: \(28\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 2 q^{3} - 60 q^{4} + 6 q^{6} + 8 q^{7} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q + 2 q^{3} - 60 q^{4} + 6 q^{6} + 8 q^{7} - 6 q^{9} + 20 q^{10} - 36 q^{12} - 32 q^{13} + 16 q^{15} + 100 q^{16} + 8 q^{18} - 32 q^{19} + 36 q^{21} + 36 q^{22} + 62 q^{24} - 96 q^{25} + 2 q^{27} - 4 q^{28} - 144 q^{30} + 84 q^{31} - 74 q^{33} - 68 q^{34} + 114 q^{36} + 28 q^{37} + 22 q^{39} - 52 q^{40} - 92 q^{42} - 16 q^{45} - 344 q^{46} + 268 q^{48} + 68 q^{49} - 54 q^{51} + 368 q^{52} + 2 q^{54} + 276 q^{55} - 156 q^{57} + 236 q^{58} - 104 q^{60} - 256 q^{61} + 92 q^{63} - 180 q^{64} - 16 q^{66} - 184 q^{67} + 74 q^{69} - 96 q^{70} - 96 q^{72} + 124 q^{73} - 74 q^{75} + 80 q^{76} + 524 q^{78} - 356 q^{79} + 202 q^{81} + 140 q^{82} - 452 q^{84} + 164 q^{85} + 110 q^{87} - 184 q^{88} - 370 q^{90} - 404 q^{91} - 202 q^{93} - 164 q^{94} - 414 q^{96} + 364 q^{97} + 256 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
44.1 − 3.78815i −0.282858 − 2.98664i −10.3501 5.74959i −11.3138 + 1.07151i −1.22483 24.0551i −8.83998 + 1.68959i 21.7803
44.2 − 3.73174i 2.99075 − 0.235406i −9.92585 − 7.45539i −0.878475 − 11.1607i 6.69854 22.1137i 8.88917 − 1.40808i −27.8215
44.3 − 3.56985i −0.207085 + 2.99284i −8.74382 0.952129i 10.6840 + 0.739261i −12.5757 16.9347i −8.91423 − 1.23954i 3.39896
44.4 − 2.94964i 2.16268 + 2.07914i −4.70038 6.42379i 6.13271 − 6.37913i 7.91609 2.06587i 0.354368 + 8.99302i 18.9479
44.5 − 2.81281i −2.16960 − 2.07191i −3.91191 − 5.53402i −5.82790 + 6.10269i −1.24220 − 0.247783i 0.414369 + 8.99046i −15.5662
44.6 − 2.70752i −2.99110 + 0.230860i −3.33067 8.84636i 0.625059 + 8.09848i 3.15451 − 1.81222i 8.89341 − 1.38105i 23.9517
44.7 − 2.64778i −1.24322 + 2.73028i −3.01072 − 3.99011i 7.22916 + 3.29176i 9.79888 − 2.61939i −5.90882 − 6.78865i −10.5649
44.8 − 2.48144i 2.39820 − 1.80239i −2.15753 0.428619i −4.47252 − 5.95099i −8.03706 − 4.57197i 2.50277 − 8.64501i 1.06359
44.9 − 1.63973i 0.867947 − 2.87170i 1.31127 1.12612i −4.70883 − 1.42320i 11.1361 − 8.70907i −7.49334 − 4.98497i 1.84653
44.10 − 1.57306i −2.76779 + 1.15731i 1.52550 − 1.96996i 1.82051 + 4.35388i −9.96110 − 8.69191i 6.32128 − 6.40636i −3.09885
44.11 − 1.17273i 1.87332 + 2.34322i 2.62471 − 8.70068i 2.74796 − 2.19689i −2.60390 − 7.76897i −1.98137 + 8.77919i −10.2035
44.12 − 1.08495i 2.92747 + 0.655684i 2.82288 3.41513i 0.711385 − 3.17616i −0.682946 − 7.40249i 8.14016 + 3.83899i 3.70525
44.13 − 0.302356i −2.51175 − 1.64045i 3.90858 3.37326i −0.496001 + 0.759443i 5.75689 − 2.39120i 3.61782 + 8.24084i 1.01992
44.14 − 0.248951i −0.0469577 + 2.99963i 3.93802 6.18926i 0.746762 + 0.0116902i −4.13327 − 1.97618i −8.99559 − 0.281712i 1.54082
44.15 0.248951i −0.0469577 − 2.99963i 3.93802 − 6.18926i 0.746762 − 0.0116902i −4.13327 1.97618i −8.99559 + 0.281712i 1.54082
44.16 0.302356i −2.51175 + 1.64045i 3.90858 − 3.37326i −0.496001 − 0.759443i 5.75689 2.39120i 3.61782 − 8.24084i 1.01992
44.17 1.08495i 2.92747 − 0.655684i 2.82288 − 3.41513i 0.711385 + 3.17616i −0.682946 7.40249i 8.14016 − 3.83899i 3.70525
44.18 1.17273i 1.87332 − 2.34322i 2.62471 8.70068i 2.74796 + 2.19689i −2.60390 7.76897i −1.98137 − 8.77919i −10.2035
44.19 1.57306i −2.76779 − 1.15731i 1.52550 1.96996i 1.82051 − 4.35388i −9.96110 8.69191i 6.32128 + 6.40636i −3.09885
44.20 1.63973i 0.867947 + 2.87170i 1.31127 − 1.12612i −4.70883 + 1.42320i 11.1361 8.70907i −7.49334 + 4.98497i 1.84653
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 44.28
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 129.3.c.a ✓ 28
3.b odd 2 1 inner 129.3.c.a ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
129.3.c.a ✓ 28 1.a even 1 1 trivial
129.3.c.a ✓ 28 3.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(129, [\chi])\).