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Properties

Label 819.2.r.a
Level $819$
Weight $2$
Character orbit 819.r
Analytic conductor $6.540$
Analytic rank $0$
Dimension $96$
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(625,819)]
 
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, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("819.625");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 819.r (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: \(96\)
Relative dimension: \(48\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 96 q - 12 q^{2} + 96 q^{4} - 4 q^{5} + 6 q^{6} - 36 q^{8} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q - 12 q^{2} + 96 q^{4} - 4 q^{5} + 6 q^{6} - 36 q^{8} - 8 q^{9} + 20 q^{11} - 15 q^{12} + 48 q^{13} + 7 q^{14} + 96 q^{16} - 3 q^{17} + 5 q^{18} - 12 q^{20} + 29 q^{23} - 2 q^{24} - 48 q^{25} - 6 q^{26} + 30 q^{28} + 10 q^{29} + 39 q^{30} - 18 q^{31} - 134 q^{32} - 16 q^{35} - 48 q^{36} - 6 q^{37} - 15 q^{38} - 12 q^{41} - 85 q^{42} + 29 q^{44} - 13 q^{45} - 6 q^{46} + 30 q^{47} - 86 q^{48} + 6 q^{49} + 2 q^{50} + 5 q^{51} + 48 q^{52} + 50 q^{53} + 19 q^{54} - 36 q^{55} + 22 q^{56} + 55 q^{57} + 6 q^{58} + 70 q^{59} - 35 q^{60} + 12 q^{61} - 38 q^{62} + 70 q^{63} + 96 q^{64} - 8 q^{65} - 71 q^{66} + 9 q^{68} + 50 q^{69} - 58 q^{71} + 6 q^{72} + 80 q^{74} - 34 q^{75} + q^{77} + 3 q^{78} - 12 q^{79} - 60 q^{80} - 24 q^{81} - 30 q^{82} - 34 q^{83} + 76 q^{84} + 27 q^{85} + 50 q^{86} + 20 q^{87} - 29 q^{89} - 43 q^{90} + 63 q^{92} + 25 q^{93} + 24 q^{94} - 132 q^{95} - 4 q^{96} - 64 q^{98} - 67 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
625.1 −2.78522 −0.690190 + 1.58860i 5.75746 1.45529 + 2.52064i 1.92233 − 4.42459i 2.47006 − 0.948040i −10.4654 −2.04727 − 2.19287i −4.05331 − 7.02054i
625.2 −2.75587 −0.963077 − 1.43961i 5.59483 0.613842 + 1.06321i 2.65412 + 3.96739i −2.17042 − 1.51303i −9.90689 −1.14497 + 2.77291i −1.69167 − 2.93006i
625.3 −2.67922 0.579745 − 1.63214i 5.17825 −1.73238 − 3.00057i −1.55327 + 4.37288i 1.08153 + 2.41460i −8.51524 −2.32779 − 1.89246i 4.64143 + 8.03920i
625.4 −2.59957 −1.54771 + 0.777565i 4.75774 −1.45751 − 2.52448i 4.02336 − 2.02133i −2.64333 + 0.113059i −7.16893 1.79079 − 2.40688i 3.78889 + 6.56255i
625.5 −2.46639 0.707761 + 1.58085i 4.08307 −1.91149 − 3.31081i −1.74561 − 3.89898i 1.80004 − 1.93904i −5.13766 −1.99815 + 2.23772i 4.71449 + 8.16573i
625.6 −2.42808 1.70038 − 0.329690i 3.89559 0.264941 + 0.458891i −4.12867 + 0.800514i 2.47047 + 0.946995i −4.60265 2.78261 − 1.12120i −0.643298 − 1.11423i
625.7 −2.37795 1.51464 + 0.840167i 3.65465 1.43947 + 2.49323i −3.60173 − 1.99788i −2.33187 + 1.24996i −3.93469 1.58824 + 2.54509i −3.42298 − 5.92878i
625.8 −2.22795 −1.73125 − 0.0527506i 2.96376 −0.422424 − 0.731659i 3.85713 + 0.117526i 0.416788 − 2.61272i −2.14722 2.99443 + 0.182649i 0.941139 + 1.63010i
625.9 −2.04534 1.19446 − 1.25430i 2.18342 0.217406 + 0.376558i −2.44307 + 2.56547i −2.05888 + 1.66163i −0.375155 −0.146538 − 2.99642i −0.444669 − 0.770189i
625.10 −1.87780 −1.15128 − 1.29405i 1.52614 −1.11868 − 1.93760i 2.16188 + 2.42996i 0.593535 + 2.57832i 0.889814 −0.349105 + 2.97962i 2.10065 + 3.63844i
625.11 −1.84050 −1.06656 + 1.36472i 1.38746 1.64389 + 2.84730i 1.96300 − 2.51177i −2.44063 + 1.02142i 1.12739 −0.724920 − 2.91110i −3.02558 − 5.24046i
625.12 −1.77324 −0.913516 − 1.47156i 1.14437 0.150523 + 0.260714i 1.61988 + 2.60942i 0.981264 − 2.45706i 1.51724 −1.33098 + 2.68859i −0.266913 − 0.462307i
625.13 −1.61549 1.12424 + 1.31761i 0.609802 0.228286 + 0.395404i −1.81619 − 2.12858i −0.446242 − 2.60785i 2.24585 −0.472174 + 2.96261i −0.368794 − 0.638770i
625.14 −1.60093 0.594456 + 1.62684i 0.562973 0.333868 + 0.578276i −0.951682 − 2.60446i 2.20181 + 1.46698i 2.30058 −2.29324 + 1.93417i −0.534498 − 0.925778i
625.15 −1.45176 −1.63260 + 0.578465i 0.107598 1.10800 + 1.91911i 2.37014 − 0.839790i 2.64572 − 0.0131658i 2.74731 2.33076 − 1.88880i −1.60854 − 2.78608i
625.16 −1.20600 0.466465 − 1.66806i −0.545575 2.14803 + 3.72051i −0.562555 + 2.01167i 1.59573 + 2.11037i 3.06995 −2.56482 − 1.55618i −2.59052 − 4.48691i
625.17 −1.10850 1.42635 − 0.982615i −0.771223 −1.89749 − 3.28654i −1.58111 + 1.08923i −2.31362 − 1.28341i 3.07191 1.06894 − 2.80310i 2.10337 + 3.64314i
625.18 −0.954989 0.117850 + 1.72804i −1.08800 −0.940589 − 1.62915i −0.112545 − 1.65026i −1.25162 + 2.33098i 2.94900 −2.97222 + 0.407297i 0.898252 + 1.55582i
625.19 −0.919929 −1.24150 + 1.20776i −1.15373 −1.17586 − 2.03665i 1.14209 − 1.11105i −0.707790 − 2.54932i 2.90121 0.0826321 − 2.99886i 1.08171 + 1.87357i
625.20 −0.792014 −0.244204 − 1.71475i −1.37271 −0.183236 − 0.317374i 0.193413 + 1.35810i −2.39652 + 1.12102i 2.67124 −2.88073 + 0.837499i 0.145125 + 0.251364i
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 625.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.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.r.a yes 96
7.c even 3 1 819.2.q.b ✓ 96
9.c even 3 1 819.2.q.b ✓ 96
63.h even 3 1 inner 819.2.r.a yes 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
819.2.q.b ✓ 96 7.c even 3 1
819.2.q.b ✓ 96 9.c even 3 1
819.2.r.a yes 96 1.a even 1 1 trivial
819.2.r.a yes 96 63.h even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{48} + 6 T_{2}^{47} - 54 T_{2}^{46} - 382 T_{2}^{45} + 1254 T_{2}^{44} + 11255 T_{2}^{43} + \cdots - 709 \) acting on \(S_{2}^{\mathrm{new}}(819, [\chi])\). Copy content Toggle raw display