Properties

Label 819.2.et.d
Level $819$
Weight $2$
Character orbit 819.et
Analytic conductor $6.540$
Analytic rank $0$
Dimension $40$
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: \(40\)
Relative dimension: \(10\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 2 q^{7} + 8 q^{11} + 18 q^{14} - 64 q^{16} - 8 q^{17} - 14 q^{19} + 14 q^{20} + 4 q^{22} - 24 q^{25} + 10 q^{26} - 2 q^{28} - 8 q^{29} - 8 q^{31} - 10 q^{32} + 24 q^{34} + 22 q^{35} + 12 q^{37} - 8 q^{38} - 30 q^{40} - 2 q^{41} - 66 q^{43} - 28 q^{44} + 40 q^{46} - 10 q^{47} + 38 q^{49} + 20 q^{50} + 40 q^{52} + 8 q^{53} + 42 q^{55} - 20 q^{56} - 48 q^{58} + 26 q^{59} - 12 q^{61} + 24 q^{62} + 44 q^{65} + 46 q^{67} + 32 q^{70} + 6 q^{71} + 10 q^{73} - 40 q^{74} + 64 q^{76} + 24 q^{77} - 34 q^{80} + 24 q^{82} - 12 q^{83} + 2 q^{85} - 12 q^{86} - 84 q^{88} + 16 q^{89} + 26 q^{91} - 236 q^{92} + 30 q^{94} + 62 q^{97} + 14 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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 −1.93326 + 1.93326i 0 5.47495i 0.212233 0.792066i 0 2.21517 1.44673i 6.71797 + 6.71797i 0 1.12096 + 1.94157i
136.2 −1.57115 + 1.57115i 0 2.93701i −0.922649 + 3.44337i 0 −2.64488 + 0.0678683i 1.47218 + 1.47218i 0 −3.96043 6.85967i
136.3 −1.09526 + 1.09526i 0 0.399185i −0.128983 + 0.481371i 0 2.51963 + 0.807124i −1.75331 1.75331i 0 −0.385956 0.668496i
136.4 −1.07562 + 1.07562i 0 0.313900i 0.962166 3.59085i 0 −2.21486 1.44721i −1.81360 1.81360i 0 2.82746 + 4.89730i
136.5 0.184060 0.184060i 0 1.93224i −0.900201 + 3.35960i 0 0.939961 2.47315i 0.723769 + 0.723769i 0 0.452676 + 0.784058i
136.6 0.326747 0.326747i 0 1.78647i 0.562238 2.09830i 0 2.25993 1.37576i 1.23722 + 1.23722i 0 −0.501904 0.869322i
136.7 0.783932 0.783932i 0 0.770902i 0.994915 3.71307i 0 0.0389254 + 2.64546i 2.17220 + 2.17220i 0 −2.13085 3.69074i
136.8 1.03264 1.03264i 0 0.132693i −0.456824 + 1.70489i 0 −2.58707 + 0.554121i 1.92826 + 1.92826i 0 1.28880 + 2.23227i
136.9 1.59737 1.59737i 0 3.10321i −0.609466 + 2.27456i 0 1.40839 + 2.23974i −1.76223 1.76223i 0 2.65977 + 4.60686i
136.10 1.75053 1.75053i 0 4.12868i 0.286571 1.06950i 0 1.02891 2.43749i −3.72630 3.72630i 0 −1.37053 2.37383i
145.1 −1.85908 + 1.85908i 0 4.91234i −0.502992 + 0.134776i 0 −1.89546 + 1.84587i 5.41427 + 5.41427i 0 0.684542 1.18566i
145.2 −1.55234 + 1.55234i 0 2.81950i 0.926472 0.248247i 0 0.619609 2.57218i 1.27214 + 1.27214i 0 −1.05283 + 1.82356i
145.3 −0.884731 + 0.884731i 0 0.434503i −3.68041 + 0.986163i 0 −2.53212 0.767050i −2.15388 2.15388i 0 2.38368 4.12866i
145.4 −0.837153 + 0.837153i 0 0.598351i 1.58368 0.424345i 0 2.46703 + 0.955901i −2.17522 2.17522i 0 −0.970539 + 1.68102i
145.5 −0.465913 + 0.465913i 0 1.56585i 3.81958 1.02345i 0 −2.61148 + 0.424440i −1.66138 1.66138i 0 −1.30275 + 2.25643i
145.6 0.240784 0.240784i 0 1.88405i −2.06336 + 0.552877i 0 1.35855 2.27032i 0.935214 + 0.935214i 0 −0.363700 + 0.629948i
145.7 0.507981 0.507981i 0 1.48391i −1.10096 + 0.295002i 0 0.718657 + 2.54628i 1.76976 + 1.76976i 0 −0.409414 + 0.709125i
145.8 1.22934 1.22934i 0 1.02253i 1.95691 0.524353i 0 −1.82011 1.92021i 1.20163 + 1.20163i 0 1.76110 3.05031i
145.9 1.65256 1.65256i 0 3.46192i −2.69306 + 0.721604i 0 −2.54381 0.727358i −2.41591 2.41591i 0 −3.25795 + 5.64294i
145.10 1.96855 1.96855i 0 5.75037i 1.75415 0.470023i 0 2.27503 + 1.35065i −7.38279 7.38279i 0 2.52787 4.37840i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 136.10
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.d 40
3.b odd 2 1 273.2.bt.b 40
7.d odd 6 1 819.2.gh.d 40
13.f odd 12 1 819.2.gh.d 40
21.g even 6 1 273.2.cg.b yes 40
39.k even 12 1 273.2.cg.b yes 40
91.ba even 12 1 inner 819.2.et.d 40
273.bs odd 12 1 273.2.bt.b 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.bt.b 40 3.b odd 2 1
273.2.bt.b 40 273.bs odd 12 1
273.2.cg.b yes 40 21.g even 6 1
273.2.cg.b yes 40 39.k even 12 1
819.2.et.d 40 1.a even 1 1 trivial
819.2.et.d 40 91.ba even 12 1 inner
819.2.gh.d 40 7.d odd 6 1
819.2.gh.d 40 13.f odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{40} + 168 T_{2}^{36} + 2 T_{2}^{35} - 8 T_{2}^{33} + 10786 T_{2}^{32} + 308 T_{2}^{31} + 2 T_{2}^{30} - 1192 T_{2}^{29} + 335772 T_{2}^{28} + 18394 T_{2}^{27} + 312 T_{2}^{26} - 78396 T_{2}^{25} + 5319379 T_{2}^{24} + \cdots + 59049 \) acting on \(S_{2}^{\mathrm{new}}(819, [\chi])\). Copy content Toggle raw display