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Properties

Label 819.2.fd.a
Level $819$
Weight $2$
Character orbit 819.fd
Analytic conductor $6.540$
Analytic rank $0$
Dimension $152$
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] = [819,2,Mod(422,819)]
 
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([6, 4, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("819.422");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 819.fd (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: \(152\)
Relative dimension: \(38\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 152 q + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 152 q + 8 q^{7} + 88 q^{16} + 4 q^{19} + 16 q^{28} - 16 q^{31} + 48 q^{34} - 8 q^{37} - 40 q^{40} + 72 q^{43} - 72 q^{46} - 28 q^{49} - 64 q^{52} + 16 q^{55} + 16 q^{58} + 64 q^{61} - 124 q^{67} - 40 q^{70} - 4 q^{73} - 72 q^{76} - 16 q^{79} - 88 q^{85} - 4 q^{91} + 16 q^{94} - 140 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
422.1 −0.721923 + 2.69425i 0 −5.00577 − 2.89009i 1.44439 − 0.387023i 0 −0.839225 + 2.50912i 7.45574 − 7.45574i 0 4.17095i
422.2 −0.693595 + 2.58853i 0 −4.48737 − 2.59078i 0.0521985 − 0.0139865i 0 −1.31721 − 2.29455i 6.02887 − 6.02887i 0 0.144818i
422.3 −0.644817 + 2.40649i 0 −3.64336 − 2.10349i −3.93701 + 1.05492i 0 2.58382 + 0.569112i 3.88800 − 3.88800i 0 − 10.1546i
422.4 −0.644802 + 2.40643i 0 −3.64310 − 2.10335i −1.24398 + 0.333323i 0 2.39684 + 1.12035i 3.88738 − 3.88738i 0 − 3.20848i
422.5 −0.587370 + 2.19209i 0 −2.72822 − 1.57514i 2.94636 − 0.789474i 0 1.12741 + 2.39352i 1.84588 − 1.84588i 0 6.92240i
422.6 −0.537980 + 2.00777i 0 −2.00966 − 1.16028i 0.843889 − 0.226119i 0 −2.61706 − 0.388576i 0.471147 − 0.471147i 0 1.81598i
422.7 −0.532229 + 1.98631i 0 −1.93010 − 1.11434i 0.860714 − 0.230628i 0 1.29321 − 2.30816i 0.332527 − 0.332527i 0 1.83239i
422.8 −0.522127 + 1.94861i 0 −1.79240 − 1.03484i −3.42992 + 0.919045i 0 −2.47541 + 0.934007i 0.0993975 − 0.0993975i 0 − 7.16342i
422.9 −0.430612 + 1.60706i 0 −0.665180 − 0.384042i 2.15596 − 0.577687i 0 −2.54858 + 0.710441i −1.44929 + 1.44929i 0 3.71352i
422.10 −0.425317 + 1.58730i 0 −0.606589 − 0.350214i 4.18572 − 1.12156i 0 1.60346 + 2.10450i −1.51009 + 1.51009i 0 7.12103i
422.11 −0.379577 + 1.41660i 0 −0.130631 − 0.0754196i −1.98134 + 0.530898i 0 0.146963 − 2.64167i −1.91762 + 1.91762i 0 − 3.00829i
422.12 −0.338049 + 1.26162i 0 0.254651 + 0.147023i −1.58236 + 0.423993i 0 −0.251590 + 2.63376i −2.11871 + 2.11871i 0 − 2.13967i
422.13 −0.253967 + 0.947818i 0 0.898190 + 0.518570i −2.38765 + 0.639769i 0 2.63901 − 0.188759i −2.10732 + 2.10732i 0 − 2.42554i
422.14 −0.253597 + 0.946437i 0 0.900618 + 0.519972i −1.09172 + 0.292525i 0 −1.58656 − 2.11727i −2.10620 + 2.10620i 0 − 1.10743i
422.15 −0.224727 + 0.838694i 0 1.07915 + 0.623045i 2.23020 − 0.597581i 0 2.48018 − 0.921252i −1.99299 + 1.99299i 0 2.00475i
422.16 −0.180001 + 0.671774i 0 1.31317 + 0.758159i −1.93835 + 0.519380i 0 2.19826 + 1.47229i −1.72923 + 1.72923i 0 − 1.39562i
422.17 −0.117431 + 0.438260i 0 1.55377 + 0.897069i 3.41884 − 0.916074i 0 −2.33620 − 1.24184i −1.21727 + 1.21727i 0 1.60591i
422.18 −0.100788 + 0.376148i 0 1.60072 + 0.924177i 3.31187 − 0.887413i 0 0.147031 − 2.64166i −1.05968 + 1.05968i 0 1.33519i
422.19 −0.0284252 + 0.106084i 0 1.72160 + 0.993969i −0.562114 + 0.150618i 0 −0.778314 + 2.52868i −0.309700 + 0.309700i 0 − 0.0639128i
422.20 0.0284252 − 0.106084i 0 1.72160 + 0.993969i 0.562114 − 0.150618i 0 −0.778314 + 2.52868i 0.309700 − 0.309700i 0 − 0.0639128i
See next 80 embeddings (of 152 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 422.38
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
91.bd odd 12 1 inner
273.bw 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.fd.a yes 152
3.b odd 2 1 inner 819.2.fd.a yes 152
7.c even 3 1 819.2.ez.a ✓ 152
13.f odd 12 1 819.2.ez.a ✓ 152
21.h odd 6 1 819.2.ez.a ✓ 152
39.k even 12 1 819.2.ez.a ✓ 152
91.bd odd 12 1 inner 819.2.fd.a yes 152
273.bw even 12 1 inner 819.2.fd.a yes 152
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
819.2.ez.a ✓ 152 7.c even 3 1
819.2.ez.a ✓ 152 13.f odd 12 1
819.2.ez.a ✓ 152 21.h odd 6 1
819.2.ez.a ✓ 152 39.k even 12 1
819.2.fd.a yes 152 1.a even 1 1 trivial
819.2.fd.a yes 152 3.b odd 2 1 inner
819.2.fd.a yes 152 91.bd odd 12 1 inner
819.2.fd.a yes 152 273.bw even 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(819, [\chi])\).