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Properties

Label 819.2.et.c
Level $819$
Weight $2$
Character orbit 819.et
Analytic conductor $6.540$
Analytic rank $0$
Dimension $36$
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] = [819,2,Mod(136,819)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(819, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 2, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("819.136");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 819.et (of order \(12\), degree \(4\), 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: \(6.53974792554\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(9\) over \(\Q(\zeta_{12})\)
Twist minimal: no (minimal twist has level 273)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q + 6 q^{7} + 8 q^{11} - 42 q^{14} - 24 q^{16} + 8 q^{17} - 18 q^{19} - 14 q^{20} + 4 q^{22} + 24 q^{25} + 50 q^{26} + 34 q^{28} - 8 q^{29} + 6 q^{31} + 50 q^{32} - 24 q^{34} - 14 q^{35} - 14 q^{37} + 8 q^{38} - 30 q^{40} - 34 q^{41} + 30 q^{43} - 28 q^{44} - 32 q^{46} + 10 q^{47} + 6 q^{49} + 20 q^{50} + 4 q^{52} + 8 q^{53} - 30 q^{55} + 92 q^{56} + 72 q^{58} + 70 q^{59} - 60 q^{61} + 48 q^{62} + 44 q^{65} - 46 q^{67} + 80 q^{70} - 42 q^{71} - 56 q^{73} - 40 q^{74} + 12 q^{76} - 24 q^{77} - 170 q^{80} + 24 q^{82} + 60 q^{83} + 2 q^{85} - 12 q^{86} + 84 q^{88} - 64 q^{89} - 86 q^{91} + 100 q^{92} - 66 q^{94} + 36 q^{97} + 22 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
136.1 −1.74432 + 1.74432i 0 − 4.08534i −0.130892 + 0.488495i 0 1.09376 + 2.40909i 3.63751 + 3.63751i 0 −0.623777 − 1.08041i
136.2 −1.12005 + 1.12005i 0 − 0.509024i 0.973479 − 3.63307i 0 2.28455 − 1.33448i −1.66997 − 1.66997i 0 2.97888 + 5.15957i
136.3 −1.09987 + 1.09987i 0 − 0.419447i −0.745735 + 2.78312i 0 −1.80794 + 1.93167i −1.73841 − 1.73841i 0 −2.24087 − 3.88130i
136.4 −0.556084 + 0.556084i 0 1.38154i −0.542987 + 2.02645i 0 −0.405927 − 2.61443i −1.88042 − 1.88042i 0 −0.824932 − 1.42882i
136.5 −0.374685 + 0.374685i 0 1.71922i 0.545981 − 2.03763i 0 −2.03549 + 1.69021i −1.39354 − 1.39354i 0 0.558898 + 0.968040i
136.6 0.411775 − 0.411775i 0 1.66088i 0.180309 − 0.672922i 0 2.60113 + 0.483875i 1.50746 + 1.50746i 0 −0.202846 − 0.351339i
136.7 1.20543 − 1.20543i 0 − 0.906108i −0.363968 + 1.35835i 0 0.864271 − 2.50061i 1.31861 + 1.31861i 0 1.19865 + 2.07613i
136.8 1.48267 − 1.48267i 0 − 2.39661i −0.507149 + 1.89270i 0 −0.313052 + 2.62717i −0.588043 − 0.588043i 0 2.05432 + 3.55819i
136.9 1.79515 − 1.79515i 0 − 4.44512i 0.590961 − 2.20549i 0 −2.51335 − 0.826467i −4.38935 − 4.38935i 0 −2.89833 − 5.02005i
145.1 −1.92842 + 1.92842i 0 − 5.43762i 3.41458 − 0.914933i 0 2.45097 − 0.996354i 6.62917 + 6.62917i 0 −4.82036 + 8.34912i
145.2 −1.55654 + 1.55654i 0 − 2.84566i −3.12520 + 0.837395i 0 1.93981 + 1.79920i 1.31631 + 1.31631i 0 3.56107 − 6.16796i
145.3 −0.698661 + 0.698661i 0 1.02375i 0.912136 − 0.244406i 0 2.61412 − 0.407922i −2.11257 − 2.11257i 0 −0.466517 + 0.808031i
145.4 −0.430820 + 0.430820i 0 1.62879i −1.97745 + 0.529856i 0 −1.23433 + 2.34018i −1.56335 − 1.56335i 0 0.623652 − 1.08020i
145.5 −0.111217 + 0.111217i 0 1.97526i 2.13060 − 0.570893i 0 −0.399954 − 2.61535i −0.442117 − 0.442117i 0 −0.173466 + 0.300452i
145.6 0.745928 − 0.745928i 0 0.887184i −3.80456 + 1.01943i 0 0.148943 − 2.64156i 2.15363 + 2.15363i 0 −2.07751 + 3.59835i
145.7 0.926196 − 0.926196i 0 0.284323i −0.409991 + 0.109857i 0 −2.25606 + 1.38209i 2.11573 + 2.11573i 0 −0.277983 + 0.481481i
145.8 1.30773 − 1.30773i 0 − 1.42031i 0.0744995 − 0.0199621i 0 2.55141 + 0.700217i 0.758075 + 0.758075i 0 0.0713202 − 0.123530i
145.9 1.74581 − 1.74581i 0 − 4.09571i 2.78539 − 0.746344i 0 −2.58285 + 0.573466i −3.65872 − 3.65872i 0 3.55979 − 6.16574i
271.1 −1.74432 − 1.74432i 0 4.08534i −0.130892 − 0.488495i 0 1.09376 − 2.40909i 3.63751 − 3.63751i 0 −0.623777 + 1.08041i
271.2 −1.12005 − 1.12005i 0 0.509024i 0.973479 + 3.63307i 0 2.28455 + 1.33448i −1.66997 + 1.66997i 0 2.97888 − 5.15957i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 136.9
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.ba even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 819.2.et.c 36
3.b odd 2 1 273.2.bt.a ✓ 36
7.d odd 6 1 819.2.gh.c 36
13.f odd 12 1 819.2.gh.c 36
21.g even 6 1 273.2.cg.a yes 36
39.k even 12 1 273.2.cg.a yes 36
91.ba even 12 1 inner 819.2.et.c 36
273.bs odd 12 1 273.2.bt.a ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.bt.a ✓ 36 3.b odd 2 1
273.2.bt.a ✓ 36 273.bs odd 12 1
273.2.cg.a yes 36 21.g even 6 1
273.2.cg.a yes 36 39.k even 12 1
819.2.et.c 36 1.a even 1 1 trivial
819.2.et.c 36 91.ba even 12 1 inner
819.2.gh.c 36 7.d odd 6 1
819.2.gh.c 36 13.f odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{36} + 126 T_{2}^{32} - 10 T_{2}^{31} + 88 T_{2}^{29} + 5749 T_{2}^{28} - 616 T_{2}^{27} + \cdots + 2304 \) acting on \(S_{2}^{\mathrm{new}}(819, [\chi])\). Copy content Toggle raw display