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Properties

Label 129.2.j.a
Level $129$
Weight $2$
Character orbit 129.j
Analytic conductor $1.030$
Analytic rank $0$
Dimension $72$
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] = [129,2,Mod(2,129)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(129, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([7, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("129.2");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 129 = 3 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 129.j (of order \(14\), degree \(6\), 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: \(1.03007018607\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(12\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q - 7 q^{3} - 18 q^{4} - 14 q^{6} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q - 7 q^{3} - 18 q^{4} - 14 q^{6} - 9 q^{9} - 10 q^{10} + 7 q^{12} - 12 q^{13} + q^{15} - 10 q^{16} - 7 q^{18} - 56 q^{19} + 18 q^{21} - 56 q^{22} + 2 q^{24} - 14 q^{25} - 7 q^{27} + 42 q^{28} - 42 q^{30} + 54 q^{31} + 14 q^{33} + 28 q^{34} + 6 q^{36} - 7 q^{39} + 86 q^{40} - 4 q^{43} - 21 q^{45} - 42 q^{46} + 35 q^{48} + 24 q^{49} - 28 q^{51} + 76 q^{52} - 47 q^{54} - 56 q^{55} - 19 q^{57} - 42 q^{58} - 81 q^{60} - 14 q^{61} + 7 q^{63} - 42 q^{64} + 55 q^{66} - 34 q^{67} + 63 q^{69} + 154 q^{70} - 14 q^{72} - 42 q^{73} + 7 q^{75} - 70 q^{76} + 82 q^{78} - 84 q^{79} - 5 q^{81} - 14 q^{82} + 144 q^{84} + 78 q^{87} + 56 q^{88} + 148 q^{90} + 56 q^{91} - 70 q^{94} - 64 q^{96} + 18 q^{97} + 26 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −2.36898 + 1.14084i −0.405974 + 1.68380i 3.06358 − 3.84161i 0.602399 − 2.63928i −0.959206 − 4.45205i − 3.07810i −1.70472 + 7.46888i −2.67037 − 1.36716i 1.58393 + 6.93966i
2.2 −1.79909 + 0.866396i 1.60410 + 0.653342i 1.23910 − 1.55378i −0.490759 + 2.15015i −3.45198 + 0.214367i 0.451945i 0.00561411 − 0.0245970i 2.14629 + 2.09606i −0.979966 − 4.29351i
2.3 −1.44652 + 0.696607i 0.716198 − 1.57704i 0.360178 − 0.451649i 0.245120 − 1.07394i 0.0625841 + 2.78013i − 0.328256i 0.508139 − 2.22630i −1.97412 − 2.25895i 0.393544 + 1.72423i
2.4 −0.754176 + 0.363192i 0.0167182 + 1.73197i −0.810106 + 1.01584i 0.252915 − 1.10809i −0.641646 − 1.30014i 2.80770i 0.614550 − 2.69252i −2.99944 + 0.0579108i 0.211709 + 0.927556i
2.5 −0.614866 + 0.296104i −1.73140 − 0.0475088i −0.956597 + 1.19953i 0.450842 − 1.97527i 1.07865 − 0.483462i − 4.09686i 0.536711 − 2.35148i 2.99549 + 0.164513i 0.307677 + 1.34802i
2.6 −0.296007 + 0.142549i −0.942683 − 1.45305i −1.17968 + 1.47927i −0.788427 + 3.45432i 0.486171 + 0.295733i 0.343870i 0.284539 − 1.24665i −1.22270 + 2.73953i −0.259032 − 1.13489i
2.7 0.296007 − 0.142549i 1.72379 − 0.168942i −1.17968 + 1.47927i 0.788427 − 3.45432i 0.486171 − 0.295733i 0.343870i −0.284539 + 1.24665i 2.94292 − 0.582442i −0.259032 − 1.13489i
2.8 0.614866 − 0.296104i 1.11665 + 1.32404i −0.956597 + 1.19953i −0.450842 + 1.97527i 1.07865 + 0.483462i − 4.09686i −0.536711 + 2.35148i −0.506170 + 2.95699i 0.307677 + 1.34802i
2.9 0.754176 − 0.363192i −1.36453 + 1.06679i −0.810106 + 1.01584i −0.252915 + 1.10809i −0.641646 + 1.30014i 2.80770i −0.614550 + 2.69252i 0.723897 − 2.91135i 0.211709 + 0.927556i
2.10 1.44652 − 0.696607i 0.786439 − 1.54322i 0.360178 − 0.451649i −0.245120 + 1.07394i 0.0625841 − 2.78013i − 0.328256i −0.508139 + 2.22630i −1.76303 − 2.42729i 0.393544 + 1.72423i
2.11 1.79909 − 0.866396i −1.51094 − 0.846785i 1.23910 − 1.55378i 0.490759 − 2.15015i −3.45198 − 0.214367i 0.451945i −0.00561411 + 0.0245970i 1.56591 + 2.55889i −0.979966 − 4.29351i
2.12 2.36898 − 1.14084i −1.06333 + 1.36724i 3.06358 − 3.84161i −0.602399 + 2.63928i −0.959206 + 4.45205i − 3.07810i 1.70472 − 7.46888i −0.738668 − 2.90764i 1.58393 + 6.93966i
8.1 −0.554094 + 2.42764i −1.29040 + 1.15536i −3.78449 − 1.82252i −0.610568 + 0.765628i −2.08981 − 3.77281i − 0.100351i 3.41631 − 4.28391i 0.330265 − 2.98177i −1.52036 − 1.90647i
8.2 −0.474726 + 2.07991i 1.40123 + 1.01811i −2.29873 − 1.10701i 0.645467 − 0.809389i −2.78278 + 2.43111i 1.72701i 0.733441 − 0.919706i 0.926901 + 2.85322i 1.37704 + 1.72675i
8.3 −0.413464 + 1.81150i −0.661436 − 1.60078i −1.30865 − 0.630215i −1.73105 + 2.17067i 3.17330 − 0.536329i 3.00347i −0.634282 + 0.795364i −2.12500 + 2.11763i −3.21645 − 4.03330i
8.4 −0.365638 + 1.60196i 1.08035 − 1.35382i −0.630658 − 0.303709i 1.43751 − 1.80258i 1.77376 + 2.22569i − 2.64471i −1.33187 + 1.67011i −0.665683 − 2.92521i 2.36207 + 2.96194i
8.5 −0.178779 + 0.783281i −1.72901 + 0.102631i 1.22037 + 0.587699i 2.53156 − 3.17448i 0.228721 − 1.37265i 3.17594i −1.68036 + 2.10711i 2.97893 − 0.354901i 2.03392 + 2.55046i
8.6 −0.111728 + 0.489514i −0.00531643 + 1.73204i 1.57480 + 0.758382i −0.242334 + 0.303878i −0.847264 − 0.196121i − 2.03404i −1.17330 + 1.47127i −2.99994 − 0.0184166i −0.121677 − 0.152578i
8.7 0.111728 − 0.489514i −0.756295 − 1.55821i 1.57480 + 0.758382i 0.242334 − 0.303878i −0.847264 + 0.196121i − 2.03404i 1.17330 − 1.47127i −1.85604 + 2.35693i −0.121677 − 0.152578i
8.8 0.178779 − 0.783281i −1.60231 + 0.657721i 1.22037 + 0.587699i −2.53156 + 3.17448i 0.228721 + 1.37265i 3.17594i 1.68036 − 2.10711i 2.13481 − 2.10775i 2.03392 + 2.55046i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
43.f odd 14 1 inner
129.j even 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 129.2.j.a ✓ 72
3.b odd 2 1 inner 129.2.j.a ✓ 72
43.f odd 14 1 inner 129.2.j.a ✓ 72
129.j even 14 1 inner 129.2.j.a ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
129.2.j.a ✓ 72 1.a even 1 1 trivial
129.2.j.a ✓ 72 3.b odd 2 1 inner
129.2.j.a ✓ 72 43.f odd 14 1 inner
129.2.j.a ✓ 72 129.j even 14 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(129, [\chi])\).