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Properties

Label 819.2.fn.f
Level $819$
Weight $2$
Character orbit 819.fn
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(73,819)]
 
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, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("819.73");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 819.fn (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
\(\operatorname{Tr}(f)(q) = \) \( 36 q - 6 q^{7} - 4 q^{11} + 36 q^{14} + 12 q^{16} - 4 q^{17} - 18 q^{19} - 44 q^{20} - 8 q^{22} + 12 q^{23} - 48 q^{25} + 32 q^{26} + 4 q^{28} + 16 q^{29} - 6 q^{31} - 76 q^{32} - 48 q^{34} - 8 q^{35} - 8 q^{37} - 16 q^{38} + 60 q^{40} + 32 q^{41} - 4 q^{44} + 28 q^{46} - 14 q^{47} + 6 q^{49} + 68 q^{50} - 62 q^{52} + 8 q^{53} + 8 q^{56} + 36 q^{58} - 26 q^{59} + 36 q^{61} - 48 q^{62} + 8 q^{65} - 40 q^{67} - 36 q^{68} - 64 q^{70} + 36 q^{71} - 8 q^{73} - 40 q^{74} - 60 q^{76} - 60 q^{77} - 26 q^{80} - 24 q^{83} + 44 q^{85} - 48 q^{86} + 168 q^{88} - 10 q^{89} + 4 q^{91} + 40 q^{92} + 36 q^{97} - 38 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
73.1 −0.703128 + 2.62411i 0 −4.65951 − 2.69017i 1.03350 + 0.276925i 0 −1.53729 − 2.15331i 6.49357 − 6.49357i 0 −1.45336 + 2.51730i
73.2 −0.542736 + 2.02552i 0 −2.07611 − 1.19864i −0.821926 − 0.220234i 0 2.59841 − 0.498246i 0.589092 − 0.589092i 0 0.892178 − 1.54530i
73.3 −0.345245 + 1.28847i 0 0.191081 + 0.110321i −1.14557 − 0.306956i 0 −2.38403 − 1.14734i −2.09457 + 2.09457i 0 0.791009 − 1.37007i
73.4 −0.0704795 + 0.263033i 0 1.66783 + 0.962923i 2.42785 + 0.650540i 0 1.21724 + 2.34911i −0.755936 + 0.755936i 0 −0.342227 + 0.592755i
73.5 −0.0455765 + 0.170094i 0 1.70520 + 0.984495i 0.317778 + 0.0851485i 0 1.64846 − 2.06944i −0.494209 + 0.494209i 0 −0.0289665 + 0.0501714i
73.6 0.164102 − 0.612438i 0 1.38390 + 0.798995i −2.57523 − 0.690032i 0 −1.84289 + 1.89836i 1.61311 − 1.61311i 0 −0.845203 + 1.46393i
73.7 0.458633 − 1.71164i 0 −0.987324 − 0.570032i −2.18934 − 0.586631i 0 −0.537086 − 2.59066i 1.07751 − 1.07751i 0 −2.00820 + 3.47831i
73.8 0.517606 − 1.93173i 0 −1.73162 − 0.999754i 3.58432 + 0.960415i 0 −2.59127 + 0.534135i 0.000695828 0 0.000695828i 0 3.71053 − 6.42683i
73.9 0.566824 − 2.11542i 0 −2.42164 − 1.39814i −0.631372 − 0.169176i 0 1.92845 + 1.81137i −1.23310 + 1.23310i 0 −0.715753 + 1.23972i
460.1 −0.703128 − 2.62411i 0 −4.65951 + 2.69017i 1.03350 − 0.276925i 0 −1.53729 + 2.15331i 6.49357 + 6.49357i 0 −1.45336 − 2.51730i
460.2 −0.542736 − 2.02552i 0 −2.07611 + 1.19864i −0.821926 + 0.220234i 0 2.59841 + 0.498246i 0.589092 + 0.589092i 0 0.892178 + 1.54530i
460.3 −0.345245 − 1.28847i 0 0.191081 − 0.110321i −1.14557 + 0.306956i 0 −2.38403 + 1.14734i −2.09457 − 2.09457i 0 0.791009 + 1.37007i
460.4 −0.0704795 − 0.263033i 0 1.66783 − 0.962923i 2.42785 − 0.650540i 0 1.21724 − 2.34911i −0.755936 − 0.755936i 0 −0.342227 − 0.592755i
460.5 −0.0455765 − 0.170094i 0 1.70520 − 0.984495i 0.317778 − 0.0851485i 0 1.64846 + 2.06944i −0.494209 − 0.494209i 0 −0.0289665 − 0.0501714i
460.6 0.164102 + 0.612438i 0 1.38390 − 0.798995i −2.57523 + 0.690032i 0 −1.84289 − 1.89836i 1.61311 + 1.61311i 0 −0.845203 − 1.46393i
460.7 0.458633 + 1.71164i 0 −0.987324 + 0.570032i −2.18934 + 0.586631i 0 −0.537086 + 2.59066i 1.07751 + 1.07751i 0 −2.00820 − 3.47831i
460.8 0.517606 + 1.93173i 0 −1.73162 + 0.999754i 3.58432 − 0.960415i 0 −2.59127 − 0.534135i 0.000695828 0 0.000695828i 0 3.71053 + 6.42683i
460.9 0.566824 + 2.11542i 0 −2.42164 + 1.39814i −0.631372 + 0.169176i 0 1.92845 − 1.81137i −1.23310 − 1.23310i 0 −0.715753 − 1.23972i
577.1 −2.52924 − 0.677708i 0 4.20573 + 2.42818i −1.05058 + 3.92082i 0 −2.39518 + 1.12389i −5.28864 − 5.28864i 0 5.31435 − 9.20472i
577.2 −2.20302 − 0.590298i 0 2.77280 + 1.60088i 0.460631 − 1.71910i 0 2.30189 + 1.30433i −1.93810 − 1.93810i 0 −2.02956 + 3.51530i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 73.9
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.bb 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.fn.f 36
3.b odd 2 1 273.2.bz.a ✓ 36
7.d odd 6 1 819.2.fn.g 36
13.d odd 4 1 819.2.fn.g 36
21.g even 6 1 273.2.bz.b yes 36
39.f even 4 1 273.2.bz.b yes 36
91.bb even 12 1 inner 819.2.fn.f 36
273.cb odd 12 1 273.2.bz.a ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.bz.a ✓ 36 3.b odd 2 1
273.2.bz.a ✓ 36 273.cb odd 12 1
273.2.bz.b yes 36 21.g even 6 1
273.2.bz.b yes 36 39.f even 4 1
819.2.fn.f 36 1.a even 1 1 trivial
819.2.fn.f 36 91.bb even 12 1 inner
819.2.fn.g 36 7.d odd 6 1
819.2.fn.g 36 13.d odd 4 1

Hecke kernels

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

\( T_{2}^{36} - 63 T_{2}^{32} + 44 T_{2}^{31} - 284 T_{2}^{29} + 3022 T_{2}^{28} - 2452 T_{2}^{27} + \cdots + 144 \) Copy content Toggle raw display
\( T_{19}^{36} + 18 T_{19}^{35} + 177 T_{19}^{34} + 1148 T_{19}^{33} + 825 T_{19}^{32} + \cdots + 85\!\cdots\!89 \) Copy content Toggle raw display