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Properties

Label 819.2.fw.a
Level $819$
Weight $2$
Character orbit 819.fw
Analytic conductor $6.540$
Analytic rank $0$
Dimension $56$
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(71,819)]
 
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([6, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("819.71");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 819.fw (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: \(56\)
Relative dimension: \(14\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 56 q - 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q - 8 q^{5} + 12 q^{10} + 24 q^{11} + 20 q^{13} + 44 q^{16} + 8 q^{19} + 12 q^{20} + 16 q^{22} + 16 q^{23} + 44 q^{26} - 24 q^{29} - 16 q^{31} - 36 q^{34} - 24 q^{35} - 12 q^{37} - 32 q^{38} - 32 q^{40} - 112 q^{44} - 72 q^{46} + 24 q^{47} + 144 q^{50} - 12 q^{56} + 36 q^{58} + 20 q^{59} + 12 q^{61} + 28 q^{62} + 120 q^{65} + 16 q^{67} - 168 q^{68} - 48 q^{70} + 8 q^{71} + 4 q^{73} - 144 q^{74} - 16 q^{76} - 8 q^{77} - 64 q^{79} - 80 q^{80} - 120 q^{82} - 40 q^{83} + 16 q^{85} + 176 q^{86} - 72 q^{88} + 32 q^{89} - 16 q^{91} - 72 q^{95} - 12 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
71.1 −2.61147 + 0.699741i 0 4.59808 − 2.65470i −0.739467 − 0.739467i 0 −0.258819 + 0.965926i −6.32668 + 6.32668i 0 2.44853 + 1.41366i
71.2 −2.38628 + 0.639403i 0 3.55346 − 2.05159i 1.48012 + 1.48012i 0 0.258819 − 0.965926i −3.67401 + 3.67401i 0 −4.47838 − 2.58560i
71.3 −1.51743 + 0.406595i 0 0.405235 − 0.233963i −1.57186 − 1.57186i 0 0.258819 − 0.965926i 1.70189 − 1.70189i 0 3.02431 + 1.74609i
71.4 −1.50575 + 0.403464i 0 0.372440 − 0.215029i −0.180744 − 0.180744i 0 −0.258819 + 0.965926i 1.73052 − 1.73052i 0 0.345079 + 0.199231i
71.5 −1.47035 + 0.393978i 0 0.274651 − 0.158570i 1.62939 + 1.62939i 0 −0.258819 + 0.965926i 1.81138 − 1.81138i 0 −3.03772 − 1.75383i
71.6 −0.762040 + 0.204188i 0 −1.19304 + 0.688801i 0.488957 + 0.488957i 0 0.258819 − 0.965926i 1.88420 − 1.88420i 0 −0.472444 − 0.272766i
71.7 0.277208 − 0.0742778i 0 −1.66072 + 0.958819i −0.539776 − 0.539776i 0 0.258819 − 0.965926i −0.795009 + 0.795009i 0 −0.189724 − 0.109537i
71.8 0.292140 − 0.0782786i 0 −1.65283 + 0.954263i 0.228742 + 0.228742i 0 −0.258819 + 0.965926i −0.835882 + 0.835882i 0 0.0847301 + 0.0489189i
71.9 0.503692 − 0.134964i 0 −1.49656 + 0.864040i −2.14124 − 2.14124i 0 −0.258819 + 0.965926i −1.37465 + 1.37465i 0 −1.36751 − 0.789535i
71.10 0.827962 − 0.221852i 0 −1.09575 + 0.632630i −2.57700 − 2.57700i 0 0.258819 − 0.965926i −1.97911 + 1.97911i 0 −2.70537 − 1.56194i
71.11 1.32687 − 0.355535i 0 −0.0978615 + 0.0565004i 2.51306 + 2.51306i 0 −0.258819 + 0.965926i −2.05244 + 2.05244i 0 4.22799 + 2.44103i
71.12 2.19775 − 0.588884i 0 2.75125 − 1.58844i 1.49126 + 1.49126i 0 0.258819 − 0.965926i 1.89343 − 1.89343i 0 4.15559 + 2.39923i
71.13 2.32877 − 0.623991i 0 3.30174 − 1.90626i −2.92830 − 2.92830i 0 0.258819 − 0.965926i 3.08993 − 3.08993i 0 −8.64656 − 4.99209i
71.14 2.49893 − 0.669587i 0 4.06426 − 2.34650i 0.846855 + 0.846855i 0 −0.258819 + 0.965926i 4.92643 − 4.92643i 0 2.68327 + 1.54919i
197.1 −0.705642 + 2.63349i 0 −4.70529 − 2.71660i −0.00714653 − 0.00714653i 0 −0.965926 + 0.258819i 6.61870 − 6.61870i 0 0.0238632 − 0.0137774i
197.2 −0.601811 + 2.24599i 0 −2.95024 − 1.70332i −0.246737 − 0.246737i 0 0.965926 − 0.258819i 2.31278 − 2.31278i 0 0.702659 − 0.405680i
197.3 −0.512565 + 1.91292i 0 −1.66448 − 0.960987i −0.613475 − 0.613475i 0 −0.965926 + 0.258819i −0.109264 + 0.109264i 0 1.48797 − 0.859082i
197.4 −0.436764 + 1.63002i 0 −0.734167 − 0.423871i −1.89544 − 1.89544i 0 0.965926 − 0.258819i −1.37494 + 1.37494i 0 3.91748 − 2.26176i
197.5 −0.246210 + 0.918869i 0 0.948350 + 0.547530i 2.13201 + 2.13201i 0 0.965926 − 0.258819i −2.08192 + 2.08192i 0 −2.48396 + 1.43411i
197.6 −0.157338 + 0.587192i 0 1.41201 + 0.815225i −3.03236 − 3.03236i 0 −0.965926 + 0.258819i −1.56056 + 1.56056i 0 2.25768 − 1.30347i
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 71.14
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
39.k 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.fw.a ✓ 56
3.b odd 2 1 819.2.fw.b yes 56
13.f odd 12 1 819.2.fw.b yes 56
39.k even 12 1 inner 819.2.fw.a ✓ 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
819.2.fw.a ✓ 56 1.a even 1 1 trivial
819.2.fw.a ✓ 56 39.k even 12 1 inner
819.2.fw.b yes 56 3.b odd 2 1
819.2.fw.b yes 56 13.f odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{56} - 109 T_{2}^{52} - 16 T_{2}^{49} + 8003 T_{2}^{48} - 176 T_{2}^{47} - 228 T_{2}^{46} + \cdots + 36481 \) acting on \(S_{2}^{\mathrm{new}}(819, [\chi])\). Copy content Toggle raw display