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Properties

Label 799.2.j.b
Level $799$
Weight $2$
Character orbit 799.j
Analytic conductor $6.380$
Analytic rank $0$
Dimension $480$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [799,2,Mod(46,799)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(799, base_ring=CyclotomicField(16))
 
chi = DirichletCharacter(H, H._module([13, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("799.46");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 799 = 17 \cdot 47 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 799.j (of order \(16\), degree \(8\), 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: \(6.38004712150\)
Analytic rank: \(0\)
Dimension: \(480\)
Relative dimension: \(60\) over \(\Q(\zeta_{16})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 480 q - 16 q^{2} - 16 q^{3} - 16 q^{4} - 16 q^{6} - 16 q^{7} - 16 q^{8} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 480 q - 16 q^{2} - 16 q^{3} - 16 q^{4} - 16 q^{6} - 16 q^{7} - 16 q^{8} - 16 q^{9} - 16 q^{12} - 48 q^{14} - 32 q^{17} - 32 q^{18} - 16 q^{21} + 16 q^{24} - 16 q^{27} - 112 q^{28} - 64 q^{32} + 112 q^{34} + 80 q^{36} - 80 q^{37} - 16 q^{42} + 32 q^{47} + 192 q^{48} - 16 q^{49} + 48 q^{51} + 48 q^{53} + 416 q^{54} - 80 q^{55} - 336 q^{56} + 48 q^{59} + 48 q^{61} - 16 q^{63} + 112 q^{64} + 16 q^{65} - 16 q^{68} + 48 q^{71} - 80 q^{72} - 16 q^{74} - 80 q^{75} - 80 q^{79} + 80 q^{81} - 80 q^{83} - 16 q^{89} + 152 q^{94} + 144 q^{95} - 96 q^{96} - 16 q^{97} - 16 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
46.1 −1.02117 − 2.46533i 0.575417 − 0.384481i −3.62083 + 3.62083i −0.458609 + 2.30558i −1.53547 − 1.02597i −2.77842 + 0.552662i 7.69336 + 3.18669i −0.964772 + 2.32916i 6.15233 − 1.22377i
46.2 −1.02117 − 2.46533i 0.575417 − 0.384481i −3.62083 + 3.62083i 0.458609 − 2.30558i −1.53547 − 1.02597i −2.77842 + 0.552662i 7.69336 + 3.18669i −0.964772 + 2.32916i −6.15233 + 1.22377i
46.3 −0.948475 − 2.28982i 0.665635 − 0.444763i −2.92946 + 2.92946i −0.258302 + 1.29857i −1.64977 − 1.10234i 3.07331 − 0.611319i 4.90682 + 2.03247i −0.902795 + 2.17954i 3.21849 − 0.640197i
46.4 −0.948475 − 2.28982i 0.665635 − 0.444763i −2.92946 + 2.92946i 0.258302 − 1.29857i −1.64977 − 1.10234i 3.07331 − 0.611319i 4.90682 + 2.03247i −0.902795 + 2.17954i −3.21849 + 0.640197i
46.5 −0.905955 − 2.18717i −1.03257 + 0.689939i −2.54874 + 2.54874i −0.822504 + 4.13501i 2.44447 + 1.63334i 2.85258 − 0.567413i 3.50924 + 1.45358i −0.557873 + 1.34682i 9.78911 − 1.94718i
46.6 −0.905955 − 2.18717i −1.03257 + 0.689939i −2.54874 + 2.54874i 0.822504 − 4.13501i 2.44447 + 1.63334i 2.85258 − 0.567413i 3.50924 + 1.45358i −0.557873 + 1.34682i −9.78911 + 1.94718i
46.7 −0.864719 − 2.08762i −1.49319 + 0.997721i −2.19619 + 2.19619i −0.611441 + 3.07392i 3.37405 + 2.25447i −4.81110 + 0.956987i 2.30867 + 0.956281i 0.0861332 − 0.207944i 6.94589 − 1.38162i
46.8 −0.864719 − 2.08762i −1.49319 + 0.997721i −2.19619 + 2.19619i 0.611441 − 3.07392i 3.37405 + 2.25447i −4.81110 + 0.956987i 2.30867 + 0.956281i 0.0861332 − 0.207944i −6.94589 + 1.38162i
46.9 −0.824202 − 1.98980i 1.72982 − 1.15583i −1.86578 + 1.86578i −0.437520 + 2.19956i −3.72559 − 2.48936i −2.66517 + 0.530135i 1.27070 + 0.526340i 0.508288 − 1.22712i 4.73729 − 0.942305i
46.10 −0.824202 − 1.98980i 1.72982 − 1.15583i −1.86578 + 1.86578i 0.437520 − 2.19956i −3.72559 − 2.48936i −2.66517 + 0.530135i 1.27070 + 0.526340i 0.508288 − 1.22712i −4.73729 + 0.942305i
46.11 −0.762142 − 1.83997i −2.59929 + 1.73679i −1.39043 + 1.39043i −0.277008 + 1.39261i 5.17667 + 3.45894i −1.04401 + 0.207667i −0.0618905 − 0.0256359i 2.59182 − 6.25720i 2.77349 − 0.551682i
46.12 −0.762142 − 1.83997i −2.59929 + 1.73679i −1.39043 + 1.39043i 0.277008 − 1.39261i 5.17667 + 3.45894i −1.04401 + 0.207667i −0.0618905 − 0.0256359i 2.59182 − 6.25720i −2.77349 + 0.551682i
46.13 −0.722833 − 1.74507i 2.08604 − 1.39385i −1.10858 + 1.10858i −0.658684 + 3.31143i −3.94022 − 2.63277i 2.30206 − 0.457908i −0.754282 − 0.312434i 1.26070 − 3.04361i 6.25480 − 1.24416i
46.14 −0.722833 − 1.74507i 2.08604 − 1.39385i −1.10858 + 1.10858i 0.658684 − 3.31143i −3.94022 − 2.63277i 2.30206 − 0.457908i −0.754282 − 0.312434i 1.26070 − 3.04361i −6.25480 + 1.24416i
46.15 −0.668212 − 1.61321i −0.415463 + 0.277604i −0.741717 + 0.741717i −0.242078 + 1.21701i 0.725450 + 0.484730i 0.919661 − 0.182932i −1.53425 − 0.635506i −1.05250 + 2.54097i 2.12505 − 0.422699i
46.16 −0.668212 − 1.61321i −0.415463 + 0.277604i −0.741717 + 0.741717i 0.242078 − 1.21701i 0.725450 + 0.484730i 0.919661 − 0.182932i −1.53425 − 0.635506i −1.05250 + 2.54097i −2.12505 + 0.422699i
46.17 −0.553997 − 1.33747i 0.196793 − 0.131493i −0.0676924 + 0.0676924i −0.166961 + 0.839371i −0.284890 − 0.190358i −2.35899 + 0.469232i −2.54690 − 1.05496i −1.12661 + 2.71988i 1.21513 − 0.241704i
46.18 −0.553997 − 1.33747i 0.196793 − 0.131493i −0.0676924 + 0.0676924i 0.166961 − 0.839371i −0.284890 − 0.190358i −2.35899 + 0.469232i −2.54690 − 1.05496i −1.12661 + 2.71988i −1.21513 + 0.241704i
46.19 −0.421289 − 1.01708i −1.87064 + 1.24992i 0.557242 − 0.557242i −0.610358 + 3.06847i 2.05935 + 1.37601i −0.880713 + 0.175185i −2.83569 − 1.17458i 0.788935 − 1.90466i 3.37803 − 0.671932i
46.20 −0.421289 − 1.01708i −1.87064 + 1.24992i 0.557242 − 0.557242i 0.610358 − 3.06847i 2.05935 + 1.37601i −0.880713 + 0.175185i −2.83569 − 1.17458i 0.788935 − 1.90466i −3.37803 + 0.671932i
See next 80 embeddings (of 480 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 46.60
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.e odd 16 1 inner
47.b odd 2 1 inner
799.j even 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 799.2.j.b ✓ 480
17.e odd 16 1 inner 799.2.j.b ✓ 480
47.b odd 2 1 inner 799.2.j.b ✓ 480
799.j even 16 1 inner 799.2.j.b ✓ 480
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
799.2.j.b ✓ 480 1.a even 1 1 trivial
799.2.j.b ✓ 480 17.e odd 16 1 inner
799.2.j.b ✓ 480 47.b odd 2 1 inner
799.2.j.b ✓ 480 799.j even 16 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{240} + 8 T_{2}^{239} + 36 T_{2}^{238} + 120 T_{2}^{237} + 328 T_{2}^{236} + 768 T_{2}^{235} + \cdots + 2670475637281 \) acting on \(S_{2}^{\mathrm{new}}(799, [\chi])\). Copy content Toggle raw display