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Properties

Label 819.2.t.a
Level $819$
Weight $2$
Character orbit 819.t
Analytic conductor $6.540$
Analytic rank $0$
Dimension $84$
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(22,819)]
 
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([2, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("819.22");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 819.t (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: \(84\)
Relative dimension: \(42\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 84 q - 8 q^{2} - q^{3} + 84 q^{4} + 8 q^{5} + 3 q^{6} + 42 q^{7} - 18 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 84 q - 8 q^{2} - q^{3} + 84 q^{4} + 8 q^{5} + 3 q^{6} + 42 q^{7} - 18 q^{8} - q^{9} - 20 q^{11} + 4 q^{12} + 6 q^{13} - 4 q^{14} + 9 q^{15} + 84 q^{16} + q^{17} + 14 q^{18} + 20 q^{20} + q^{21} - 14 q^{23} - 21 q^{24} - 36 q^{25} - 5 q^{26} - 40 q^{27} + 42 q^{28} + 26 q^{29} + 14 q^{30} - 24 q^{32} - 4 q^{33} - 8 q^{35} - 63 q^{36} - 28 q^{38} + 17 q^{39} + 24 q^{41} - 3 q^{42} - 96 q^{44} - 13 q^{45} + 6 q^{46} + 13 q^{47} - 26 q^{48} - 42 q^{49} + 62 q^{50} - 21 q^{51} + 33 q^{52} + 16 q^{53} - 61 q^{54} - 6 q^{55} - 9 q^{56} - 35 q^{57} - 36 q^{58} - 80 q^{59} - 26 q^{60} - 6 q^{61} - 9 q^{62} - 2 q^{63} + 102 q^{64} + 29 q^{65} + 46 q^{66} + 17 q^{68} + 66 q^{69} + 20 q^{71} + 19 q^{72} - 24 q^{73} - 27 q^{74} + 24 q^{75} - 18 q^{76} - 10 q^{77} - 125 q^{78} - 39 q^{79} + 60 q^{80} + 39 q^{81} + 15 q^{82} + 55 q^{83} + 8 q^{84} - 18 q^{85} - 14 q^{86} + q^{87} + 24 q^{88} + 51 q^{89} - 33 q^{90} + 3 q^{91} + 6 q^{92} + 9 q^{93} + 27 q^{94} + 12 q^{95} - 41 q^{96} - 21 q^{97} + 4 q^{98} - 39 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
22.1 −2.70539 −0.420748 + 1.68017i 5.31911 −1.33316 + 2.30910i 1.13828 − 4.54551i 0.500000 − 0.866025i −8.97947 −2.64594 − 1.41386i 3.60671 − 6.24700i
22.2 −2.69529 −0.124924 − 1.72754i 5.26458 0.704821 − 1.22079i 0.336705 + 4.65622i 0.500000 − 0.866025i −8.79899 −2.96879 + 0.431621i −1.89970 + 3.29037i
22.3 −2.69420 1.29698 + 1.14797i 5.25869 1.70057 − 2.94547i −3.49433 − 3.09286i 0.500000 − 0.866025i −8.77956 0.364330 + 2.97780i −4.58167 + 7.93568i
22.4 −2.51998 −1.66917 + 0.462449i 4.35028 0.509951 − 0.883262i 4.20628 − 1.16536i 0.500000 − 0.866025i −5.92264 2.57228 − 1.54381i −1.28506 + 2.22580i
22.5 −2.44291 1.38524 − 1.03976i 3.96783 −1.57836 + 2.73379i −3.38403 + 2.54005i 0.500000 − 0.866025i −4.80725 0.837788 − 2.88064i 3.85579 − 6.67843i
22.6 −2.23718 −0.752034 + 1.56027i 3.00499 2.09038 − 3.62065i 1.68244 − 3.49061i 0.500000 − 0.866025i −2.24836 −1.86889 − 2.34675i −4.67658 + 8.10007i
22.7 −2.00395 1.12689 + 1.31534i 2.01583 −1.12710 + 1.95219i −2.25824 − 2.63588i 0.500000 − 0.866025i −0.0317315 −0.460237 + 2.96449i 2.25865 − 3.91210i
22.8 −1.99732 −0.106157 − 1.72879i 1.98928 −1.03962 + 1.80068i 0.212028 + 3.45295i 0.500000 − 0.866025i 0.0214090 −2.97746 + 0.367046i 2.07646 − 3.59654i
22.9 −1.93402 −1.71377 − 0.250962i 1.74042 0.506059 − 0.876520i 3.31446 + 0.485365i 0.500000 − 0.866025i 0.502034 2.87404 + 0.860184i −0.978727 + 1.69520i
22.10 −1.84191 1.71465 − 0.244913i 1.39263 0.524500 − 0.908460i −3.15823 + 0.451107i 0.500000 − 0.866025i 1.11872 2.88004 − 0.839878i −0.966081 + 1.67330i
22.11 −1.78451 0.664855 − 1.59936i 1.18447 2.03038 − 3.51672i −1.18644 + 2.85408i 0.500000 − 0.866025i 1.45531 −2.11594 − 2.12669i −3.62323 + 6.27562i
22.12 −1.60939 −1.19370 − 1.25502i 0.590132 0.721228 − 1.24920i 1.92112 + 2.01982i 0.500000 − 0.866025i 2.26903 −0.150174 + 2.99624i −1.16074 + 2.01045i
22.13 −1.51995 −1.64128 + 0.553350i 0.310251 −2.12790 + 3.68564i 2.49467 − 0.841065i 0.500000 − 0.866025i 2.56834 2.38761 − 1.81641i 3.23431 − 5.60199i
22.14 −1.49200 0.229954 + 1.71672i 0.226067 −0.0587897 + 0.101827i −0.343092 − 2.56135i 0.500000 − 0.866025i 2.64671 −2.89424 + 0.789533i 0.0877143 − 0.151926i
22.15 −1.03400 1.70818 − 0.286564i −0.930835 −1.81140 + 3.13743i −1.76627 + 0.296309i 0.500000 − 0.866025i 3.03050 2.83576 − 0.979007i 1.87299 − 3.24412i
22.16 −0.779632 −0.642027 + 1.60866i −1.39217 −0.550261 + 0.953080i 0.500545 − 1.25417i 0.500000 − 0.866025i 2.64465 −2.17560 − 2.06561i 0.429001 − 0.743051i
22.17 −0.752771 1.18587 − 1.26242i −1.43334 0.392595 − 0.679995i −0.892690 + 0.950315i 0.500000 − 0.866025i 2.58452 −0.187418 − 2.99414i −0.295535 + 0.511881i
22.18 −0.641494 1.46821 + 0.918884i −1.58849 0.506882 − 0.877945i −0.941850 − 0.589459i 0.500000 − 0.866025i 2.30199 1.31130 + 2.69824i −0.325162 + 0.563197i
22.19 −0.561479 −1.03988 + 1.38516i −1.68474 1.57371 − 2.72574i 0.583868 − 0.777737i 0.500000 − 0.866025i 2.06891 −0.837320 − 2.88078i −0.883604 + 1.53045i
22.20 −0.444793 −1.11744 − 1.32338i −1.80216 −0.630994 + 1.09291i 0.497031 + 0.588628i 0.500000 − 0.866025i 1.69117 −0.502643 + 2.95759i 0.280662 − 0.486120i
See all 84 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 22.42
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
117.h even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 819.2.t.a yes 84
9.c even 3 1 819.2.i.b ✓ 84
13.c even 3 1 819.2.i.b ✓ 84
117.h even 3 1 inner 819.2.t.a yes 84
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
819.2.i.b ✓ 84 9.c even 3 1
819.2.i.b ✓ 84 13.c even 3 1
819.2.t.a yes 84 1.a even 1 1 trivial
819.2.t.a yes 84 117.h even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{42} + 4 T_{2}^{41} - 55 T_{2}^{40} - 233 T_{2}^{39} + 1367 T_{2}^{38} + 6219 T_{2}^{37} + \cdots + 3481 \) acting on \(S_{2}^{\mathrm{new}}(819, [\chi])\). Copy content Toggle raw display