Properties

Label 799.2.b.c
Level $799$
Weight $2$
Character orbit 799.b
Analytic conductor $6.380$
Analytic rank $0$
Dimension $40$
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] = [799,2,Mod(424,799)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(799, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("799.424");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 799 = 17 \cdot 47 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 799.b (of order \(2\), degree \(1\), 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: \(40\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 6 q^{2} + 46 q^{4} - 18 q^{8} - 46 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 6 q^{2} + 46 q^{4} - 18 q^{8} - 46 q^{9} + 14 q^{13} - 18 q^{15} + 58 q^{16} + 2 q^{17} - 10 q^{18} - 22 q^{19} + 24 q^{21} - 58 q^{25} - 6 q^{26} + 34 q^{30} - 92 q^{32} + 50 q^{33} + 8 q^{34} - 18 q^{35} - 68 q^{36} + 6 q^{38} + 36 q^{42} - 26 q^{43} + 40 q^{47} - 78 q^{49} + 50 q^{50} + 20 q^{51} - 24 q^{52} - 22 q^{53} + 22 q^{55} + 18 q^{59} - 74 q^{60} + 94 q^{64} - 96 q^{66} + 52 q^{67} + 71 q^{68} - 26 q^{69} + 2 q^{70} + 16 q^{72} - 2 q^{76} - 60 q^{77} + 96 q^{81} + 10 q^{83} + 138 q^{84} + 38 q^{85} - 152 q^{86} - 10 q^{87} - 6 q^{89} + 14 q^{93} - 6 q^{94} + 60 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
424.1 −2.73159 2.93228i 5.46157 0.487885i 8.00977i 0.803779i −9.45559 −5.59824 1.33270i
424.2 −2.73159 2.93228i 5.46157 0.487885i 8.00977i 0.803779i −9.45559 −5.59824 1.33270i
424.3 −2.67224 1.20994i 5.14087 1.97515i 3.23326i 5.12847i −8.39315 1.53604 5.27807i
424.4 −2.67224 1.20994i 5.14087 1.97515i 3.23326i 5.12847i −8.39315 1.53604 5.27807i
424.5 −2.60183 0.179412i 4.76951 3.93490i 0.466798i 2.38196i −7.20579 2.96781 10.2379i
424.6 −2.60183 0.179412i 4.76951 3.93490i 0.466798i 2.38196i −7.20579 2.96781 10.2379i
424.7 −2.12048 2.93288i 2.49642 4.40553i 6.21910i 0.798156i −1.05264 −5.60179 9.34181i
424.8 −2.12048 2.93288i 2.49642 4.40553i 6.21910i 0.798156i −1.05264 −5.60179 9.34181i
424.9 −1.85975 0.458272i 1.45866 2.56209i 0.852271i 0.0216061i 1.00675 2.78999 4.76485i
424.10 −1.85975 0.458272i 1.45866 2.56209i 0.852271i 0.0216061i 1.00675 2.78999 4.76485i
424.11 −1.50908 1.00039i 0.277334 3.40854i 1.50967i 4.30933i 2.59965 1.99922 5.14378i
424.12 −1.50908 1.00039i 0.277334 3.40854i 1.50967i 4.30933i 2.59965 1.99922 5.14378i
424.13 −1.19759 2.00820i −0.565770 1.69879i 2.40500i 3.98040i 3.07275 −1.03285 2.03446i
424.14 −1.19759 2.00820i −0.565770 1.69879i 2.40500i 3.98040i 3.07275 −1.03285 2.03446i
424.15 −1.13957 2.92459i −0.701377 0.283746i 3.33278i 3.87426i 3.07841 −5.55325 0.323349i
424.16 −1.13957 2.92459i −0.701377 0.283746i 3.33278i 3.87426i 3.07841 −5.55325 0.323349i
424.17 −0.850575 3.02328i −1.27652 0.477702i 2.57153i 3.15138i 2.78693 −6.14023 0.406322i
424.18 −0.850575 3.02328i −1.27652 0.477702i 2.57153i 3.15138i 2.78693 −6.14023 0.406322i
424.19 −0.262611 0.974571i −1.93104 1.23574i 0.255933i 1.43778i 1.03233 2.05021 0.324520i
424.20 −0.262611 0.974571i −1.93104 1.23574i 0.255933i 1.43778i 1.03233 2.05021 0.324520i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 424.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 799.2.b.c 40
17.b even 2 1 inner 799.2.b.c 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
799.2.b.c 40 1.a even 1 1 trivial
799.2.b.c 40 17.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} + 3 T_{2}^{19} - 27 T_{2}^{18} - 83 T_{2}^{17} + 297 T_{2}^{16} + 950 T_{2}^{15} - 1701 T_{2}^{14} - 5829 T_{2}^{13} + 5329 T_{2}^{12} + 20800 T_{2}^{11} - 8418 T_{2}^{10} - 43846 T_{2}^{9} + 3462 T_{2}^{8} + 52825 T_{2}^{7} + \cdots + 17 \) acting on \(S_{2}^{\mathrm{new}}(799, [\chi])\). Copy content Toggle raw display