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Properties

Label 819.2.s.g
Level $819$
Weight $2$
Character orbit 819.s
Analytic conductor $6.540$
Analytic rank $0$
Dimension $36$
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(289,819)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(819, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("819.289");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 819.s (of order \(3\), degree \(2\), 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: \(18\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 44 q^{4} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q + 44 q^{4} + 4 q^{7} + 8 q^{10} + 20 q^{16} + 4 q^{19} - 10 q^{22} - 22 q^{25} + 16 q^{28} - 18 q^{31} + 8 q^{34} - 20 q^{37} + 14 q^{40} + 20 q^{43} + 8 q^{46} - 12 q^{49} + 10 q^{52} + 42 q^{55} - 12 q^{58} + 14 q^{61} - 12 q^{64} + 12 q^{67} - 24 q^{70} + 6 q^{73} + 14 q^{76} - 24 q^{79} + 38 q^{82} + 28 q^{85} - 12 q^{88} + 92 q^{91} + 28 q^{94} + 28 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1 −2.68495 0 5.20897 −0.261327 + 0.452632i 0 0.345095 + 2.62315i −8.61592 0 0.701650 − 1.21529i
289.2 −2.37301 0 3.63116 0.794473 − 1.37607i 0 2.03905 − 1.68590i −3.87075 0 −1.88529 + 3.26542i
289.3 −2.20299 0 2.85316 −1.98729 + 3.44208i 0 1.55404 − 2.14125i −1.87950 0 4.37797 − 7.58287i
289.4 −2.11630 0 2.47873 1.28400 − 2.22396i 0 −2.36737 + 1.18134i −1.01313 0 −2.71733 + 4.70656i
289.5 −1.85815 0 1.45273 −1.41690 + 2.45414i 0 −2.59533 − 0.514050i 1.01691 0 2.63281 − 4.56016i
289.6 −1.14867 0 −0.680564 0.912102 − 1.57981i 0 0.248536 − 2.63405i 3.07908 0 −1.04770 + 1.81467i
289.7 −1.09380 0 −0.803600 0.179458 − 0.310830i 0 2.59911 + 0.494573i 3.06658 0 −0.196291 + 0.339986i
289.8 −0.883334 0 −1.21972 −0.803460 + 1.39163i 0 0.969010 + 2.46191i 2.84409 0 0.709724 − 1.22928i
289.9 −0.281329 0 −1.92085 2.04579 − 3.54342i 0 −1.79215 + 1.94633i 1.10305 0 −0.575541 + 0.996867i
289.10 0.281329 0 −1.92085 −2.04579 + 3.54342i 0 −1.79215 + 1.94633i −1.10305 0 −0.575541 + 0.996867i
289.11 0.883334 0 −1.21972 0.803460 − 1.39163i 0 0.969010 + 2.46191i −2.84409 0 0.709724 − 1.22928i
289.12 1.09380 0 −0.803600 −0.179458 + 0.310830i 0 2.59911 + 0.494573i −3.06658 0 −0.196291 + 0.339986i
289.13 1.14867 0 −0.680564 −0.912102 + 1.57981i 0 0.248536 − 2.63405i −3.07908 0 −1.04770 + 1.81467i
289.14 1.85815 0 1.45273 1.41690 − 2.45414i 0 −2.59533 − 0.514050i −1.01691 0 2.63281 − 4.56016i
289.15 2.11630 0 2.47873 −1.28400 + 2.22396i 0 −2.36737 + 1.18134i 1.01313 0 −2.71733 + 4.70656i
289.16 2.20299 0 2.85316 1.98729 − 3.44208i 0 1.55404 − 2.14125i 1.87950 0 4.37797 − 7.58287i
289.17 2.37301 0 3.63116 −0.794473 + 1.37607i 0 2.03905 − 1.68590i 3.87075 0 −1.88529 + 3.26542i
289.18 2.68495 0 5.20897 0.261327 − 0.452632i 0 0.345095 + 2.62315i 8.61592 0 0.701650 − 1.21529i
802.1 −2.68495 0 5.20897 −0.261327 − 0.452632i 0 0.345095 − 2.62315i −8.61592 0 0.701650 + 1.21529i
802.2 −2.37301 0 3.63116 0.794473 + 1.37607i 0 2.03905 + 1.68590i −3.87075 0 −1.88529 − 3.26542i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
91.h even 3 1 inner
273.s odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 819.2.s.g yes 36
3.b odd 2 1 inner 819.2.s.g yes 36
7.c even 3 1 819.2.n.g ✓ 36
13.c even 3 1 819.2.n.g ✓ 36
21.h odd 6 1 819.2.n.g ✓ 36
39.i odd 6 1 819.2.n.g ✓ 36
91.h even 3 1 inner 819.2.s.g yes 36
273.s odd 6 1 inner 819.2.s.g yes 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
819.2.n.g ✓ 36 7.c even 3 1
819.2.n.g ✓ 36 13.c even 3 1
819.2.n.g ✓ 36 21.h odd 6 1
819.2.n.g ✓ 36 39.i odd 6 1
819.2.s.g yes 36 1.a even 1 1 trivial
819.2.s.g yes 36 3.b odd 2 1 inner
819.2.s.g yes 36 91.h even 3 1 inner
819.2.s.g yes 36 273.s odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(819, [\chi])\):

\( T_{2}^{18} - 29 T_{2}^{16} + 349 T_{2}^{14} - 2259 T_{2}^{12} + 8520 T_{2}^{10} - 18979 T_{2}^{8} + \cdots - 297 \) Copy content Toggle raw display
\( T_{11}^{36} + 106 T_{11}^{34} + 6917 T_{11}^{32} + 288288 T_{11}^{30} + 8833317 T_{11}^{28} + \cdots + 2370515963904 \) Copy content Toggle raw display