↑

Properties

Label 799.2.f.b
Level $799$
Weight $2$
Character orbit 799.f
Analytic conductor $6.380$
Analytic rank $0$
Dimension $80$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [799,2,Mod(565,799)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(799, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([3, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("799.565");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 799 = 17 \cdot 47 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 799.f (of order \(4\), 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: \(6.38004712150\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(40\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 80 q - 2 q^{3} - 92 q^{4} - 4 q^{5} - 2 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 80 q - 2 q^{3} - 92 q^{4} - 4 q^{5} - 2 q^{6} + 6 q^{10} + 2 q^{11} - 4 q^{12} - 44 q^{13} - 12 q^{14} + 132 q^{16} - 16 q^{17} + 44 q^{18} + 34 q^{20} - 36 q^{21} - 14 q^{22} + 6 q^{23} + 46 q^{24} - 8 q^{27} - 26 q^{28} - 4 q^{29} - 20 q^{30} - 16 q^{31} + 80 q^{33} - 42 q^{34} + 24 q^{35} - 24 q^{38} + 8 q^{39} - 2 q^{40} + 30 q^{41} + 2 q^{44} + 34 q^{45} - 72 q^{46} - 80 q^{47} - 92 q^{48} + 152 q^{50} - 12 q^{51} + 116 q^{52} + 50 q^{54} - 60 q^{55} + 26 q^{56} - 6 q^{57} + 30 q^{58} - 28 q^{61} - 54 q^{62} - 78 q^{63} - 172 q^{64} + 10 q^{65} + 100 q^{67} - 54 q^{68} + 44 q^{69} + 30 q^{71} - 100 q^{72} + 22 q^{73} + 4 q^{74} + 100 q^{75} - 20 q^{78} - 46 q^{79} - 186 q^{80} - 188 q^{81} - 52 q^{82} + 260 q^{84} - 18 q^{85} + 72 q^{86} + 160 q^{88} - 44 q^{89} + 70 q^{90} - 16 q^{91} + 56 q^{92} - 6 q^{95} - 102 q^{96} - 68 q^{97} - 56 q^{98} + 56 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
565.1 − 2.78023i 0.620384 − 0.620384i −5.72969 −1.26381 + 1.26381i −1.72481 − 1.72481i −0.521996 − 0.521996i 10.3694i 2.23025i 3.51368 + 3.51368i
565.2 − 2.72927i −1.45496 + 1.45496i −5.44893 −1.96613 + 1.96613i 3.97099 + 3.97099i 3.20093 + 3.20093i 9.41307i − 1.23382i 5.36612 + 5.36612i
565.3 − 2.55807i 0.954138 − 0.954138i −4.54373 1.58740 − 1.58740i −2.44075 − 2.44075i −1.30946 − 1.30946i 6.50703i 1.17924i −4.06068 − 4.06068i
565.4 − 2.42641i −2.17173 + 2.17173i −3.88747 −0.238155 + 0.238155i 5.26952 + 5.26952i −2.02978 − 2.02978i 4.57977i − 6.43285i 0.577861 + 0.577861i
565.5 − 2.42030i 2.37114 − 2.37114i −3.85785 −1.48270 + 1.48270i −5.73886 − 5.73886i −2.56036 − 2.56036i 4.49655i − 8.24460i 3.58858 + 3.58858i
565.6 − 2.10932i 0.00872145 − 0.00872145i −2.44922 −0.377685 + 0.377685i −0.0183963 − 0.0183963i 1.51798 + 1.51798i 0.947537i 2.99985i 0.796656 + 0.796656i
565.7 − 2.02001i 1.13033 − 1.13033i −2.08042 1.22760 − 1.22760i −2.28328 − 2.28328i 3.21383 + 3.21383i 0.162454i 0.444693i −2.47976 − 2.47976i
565.8 − 1.84024i 1.67288 − 1.67288i −1.38650 0.891936 − 0.891936i −3.07850 − 3.07850i −1.20572 − 1.20572i − 1.12899i − 2.59703i −1.64138 − 1.64138i
565.9 − 1.57449i −2.06637 + 2.06637i −0.479006 1.92463 − 1.92463i 3.25347 + 3.25347i −1.51903 − 1.51903i − 2.39478i − 5.53977i −3.03031 − 3.03031i
565.10 − 1.55750i −1.00137 + 1.00137i −0.425797 −1.26123 + 1.26123i 1.55963 + 1.55963i 0.163533 + 0.163533i − 2.45182i 0.994528i 1.96437 + 1.96437i
565.11 − 1.55341i 0.112910 − 0.112910i −0.413087 3.04466 − 3.04466i −0.175396 − 0.175396i 1.42045 + 1.42045i − 2.46513i 2.97450i −4.72961 − 4.72961i
565.12 − 1.10133i −0.619953 + 0.619953i 0.787069 2.03051 − 2.03051i 0.682774 + 0.682774i −0.210484 − 0.210484i − 3.06949i 2.23132i −2.23626 − 2.23626i
565.13 − 1.01279i −0.223142 + 0.223142i 0.974264 −1.13698 + 1.13698i 0.225995 + 0.225995i −1.36995 − 1.36995i − 3.01229i 2.90042i 1.15152 + 1.15152i
565.14 − 0.979531i −1.20179 + 1.20179i 1.04052 −3.10638 + 3.10638i 1.17719 + 1.17719i −1.90650 − 1.90650i − 2.97828i 0.111382i 3.04280 + 3.04280i
565.15 − 0.802137i 0.247870 − 0.247870i 1.35658 −1.12140 + 1.12140i −0.198825 − 0.198825i 1.73932 + 1.73932i − 2.69243i 2.87712i 0.899517 + 0.899517i
565.16 − 0.777016i 2.25605 − 2.25605i 1.39625 −2.47429 + 2.47429i −1.75299 − 1.75299i 2.19055 + 2.19055i − 2.63894i − 7.17957i 1.92256 + 1.92256i
565.17 − 0.336601i −2.01642 + 2.01642i 1.88670 −0.290865 + 0.290865i 0.678730 + 0.678730i −3.56716 − 3.56716i − 1.30827i − 5.13193i 0.0979054 + 0.0979054i
565.18 − 0.196067i 1.25736 − 1.25736i 1.96156 0.994590 − 0.994590i −0.246526 − 0.246526i 0.921169 + 0.921169i − 0.776730i − 0.161894i −0.195006 − 0.195006i
565.19 − 0.192949i −1.20502 + 1.20502i 1.96277 2.21976 − 2.21976i 0.232508 + 0.232508i 3.34948 + 3.34948i − 0.764613i 0.0958317i −0.428300 − 0.428300i
565.20 − 0.127103i 1.81028 − 1.81028i 1.98384 0.718121 − 0.718121i −0.230091 − 0.230091i −3.54335 − 3.54335i − 0.506357i − 3.55421i −0.0912750 − 0.0912750i
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 565.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.c even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 799.2.f.b ✓ 80
17.c even 4 1 inner 799.2.f.b ✓ 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
799.2.f.b ✓ 80 1.a even 1 1 trivial
799.2.f.b ✓ 80 17.c even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{80} + 126 T_{2}^{78} + 7607 T_{2}^{76} + 293024 T_{2}^{74} + 8090897 T_{2}^{72} + 170566064 T_{2}^{70} + \cdots + 23409 \) acting on \(S_{2}^{\mathrm{new}}(799, [\chi])\). Copy content Toggle raw display