Properties

Label 799.1.e.b
Level $799$
Weight $1$
Character orbit 799.e
Analytic conductor $0.399$
Analytic rank $0$
Dimension $8$
Projective image $D_{20}$
CM discriminant -47
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [799,1,Mod(140,799)]
 
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, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("799.140");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 799 = 17 \cdot 47 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 799.e (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: \(0.398752945094\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{20})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{20}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{20} - \cdots)\)

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + (\zeta_{20}^{7} + \zeta_{20}^{3}) q^{2} + (\zeta_{20}^{4} - \zeta_{20}) q^{3} + (\zeta_{20}^{6} - \zeta_{20}^{4} - 1) q^{4} + ( - \zeta_{20}^{8} + \zeta_{20}^{7} - \zeta_{20}^{4} - \zeta_{20}) q^{6} + (\zeta_{20}^{3} + \zeta_{20}^{2}) q^{7} + (\zeta_{20}^{9} - \zeta_{20}^{7} - \zeta_{20}^{3} + \zeta_{20}) q^{8} + (\zeta_{20}^{8} - \zeta_{20}^{5} + \zeta_{20}^{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{20}^{7} + \zeta_{20}^{3}) q^{2} + (\zeta_{20}^{4} - \zeta_{20}) q^{3} + (\zeta_{20}^{6} - \zeta_{20}^{4} - 1) q^{4} + ( - \zeta_{20}^{8} + \zeta_{20}^{7} - \zeta_{20}^{4} - \zeta_{20}) q^{6} + (\zeta_{20}^{3} + \zeta_{20}^{2}) q^{7} + (\zeta_{20}^{9} - \zeta_{20}^{7} - \zeta_{20}^{3} + \zeta_{20}) q^{8} + (\zeta_{20}^{8} - \zeta_{20}^{5} + \zeta_{20}^{2}) q^{9} + ( - \zeta_{20}^{8} - \zeta_{20}^{7} + \zeta_{20}^{5} - \zeta_{20}^{4} + \zeta_{20} - 1) q^{12} + (\zeta_{20}^{9} + \zeta_{20}^{6} + \zeta_{20}^{5} - 1) q^{14} + (\zeta_{20}^{8} - \zeta_{20}^{6} + \zeta_{20}^{4} - \zeta_{20}^{2} + 1) q^{16} - \zeta_{20}^{6} q^{17} + (\zeta_{20}^{9} - \zeta_{20}^{8} - \zeta_{20}^{5} + \zeta_{20}^{2} - \zeta_{20}) q^{18} + (\zeta_{20}^{7} + \zeta_{20}^{6} - \zeta_{20}^{4} - \zeta_{20}^{3}) q^{21} + (\zeta_{20}^{8} - \zeta_{20}^{7} + \zeta_{20}^{5} + \zeta_{20}^{4} - \zeta_{20}^{3} - \zeta_{20}^{2} + \zeta_{20} + 1) q^{24} - \zeta_{20}^{5} q^{25} + ( - \zeta_{20}^{9} + \zeta_{20}^{6} - \zeta_{20}^{3} - \zeta_{20}^{2}) q^{27} + (\zeta_{20}^{9} + \zeta_{20}^{8} - \zeta_{20}^{7} - \zeta_{20}^{6} - \zeta_{20}^{3} - \zeta_{20}^{2}) q^{28} + ( - \zeta_{20}^{9} + \zeta_{20}^{7} - 2 \zeta_{20}^{5} + \zeta_{20}^{3} - \zeta_{20}) q^{32} + ( - \zeta_{20}^{9} + \zeta_{20}^{3}) q^{34} + (\zeta_{20}^{9} - \zeta_{20}^{6} + \zeta_{20}^{5} - \zeta_{20}^{4} + \zeta_{20}^{2} + \zeta_{20}) q^{36} + ( - \zeta_{20}^{4} + \zeta_{20}) q^{37} + (\zeta_{20}^{9} - \zeta_{20}^{7} - \zeta_{20}^{6} - \zeta_{20}^{4} - \zeta_{20}^{3} + \zeta_{20}) q^{42} - q^{47} + ( - \zeta_{20}^{9} + \zeta_{20}^{8} + \zeta_{20}^{7} - \zeta_{20}^{6} - \zeta_{20}^{5} + \zeta_{20}^{4} + \zeta_{20}^{3} - \zeta_{20}^{2} - \zeta_{20} + 1) q^{48} + (\zeta_{20}^{6} + \zeta_{20}^{5} + \zeta_{20}^{4}) q^{49} + ( - \zeta_{20}^{8} + \zeta_{20}^{2}) q^{50} + (\zeta_{20}^{7} + 1) q^{51} + ( - \zeta_{20}^{8} - \zeta_{20}^{2}) q^{53} + (\zeta_{20}^{6} - \zeta_{20}^{5} - \zeta_{20}^{3} + \zeta_{20}^{2} + 1) q^{54} + ( - \zeta_{20}^{9} - \zeta_{20}^{6} - \zeta_{20}^{5} + \zeta_{20}^{4} + \zeta_{20}^{3} - \zeta_{20}^{2} - \zeta_{20} + 1) q^{56} + (\zeta_{20}^{7} + \zeta_{20}^{3}) q^{59} + (\zeta_{20}^{8} - \zeta_{20}^{7}) q^{61} + ( - \zeta_{20}^{8} - \zeta_{20}^{7} + \zeta_{20}^{5} + \zeta_{20}^{4} - \zeta_{20} - 1) q^{63} + ( - 2 \zeta_{20}^{8} + \zeta_{20}^{6} - \zeta_{20}^{4} + 2 \zeta_{20}^{2} - 1) q^{64} + (\zeta_{20}^{6} + \zeta_{20}^{2} - 1) q^{68} + ( - \zeta_{20}^{9} - \zeta_{20}^{6}) q^{71} + (\zeta_{20}^{8} - \zeta_{20}^{7} - \zeta_{20}^{6} + \zeta_{20}^{5} + \zeta_{20}^{4} + \zeta_{20}^{3} - \zeta_{20}^{2}) q^{72} + (\zeta_{20}^{8} - \zeta_{20}^{7} + \zeta_{20}^{4} + \zeta_{20}) q^{74} + ( - \zeta_{20}^{9} + \zeta_{20}^{6}) q^{75} + (\zeta_{20}^{4} + \zeta_{20}) q^{79} + ( - \zeta_{20}^{7} - \zeta_{20}^{6} + \zeta_{20}^{4} - \zeta_{20}^{3} + 1) q^{81} + ( - \zeta_{20}^{9} + \zeta_{20}^{8} - \zeta_{20}^{6} + \zeta_{20}^{4} - \zeta_{20}^{2} + \zeta_{20} + 2) q^{84} + (\zeta_{20}^{8} - \zeta_{20}^{2}) q^{89} + ( - \zeta_{20}^{7} - \zeta_{20}^{3}) q^{94} + ( - 2 \zeta_{20}^{9} - \zeta_{20}^{8} + \zeta_{20}^{7} + 2 \zeta_{20}^{6} - \zeta_{20}^{5} - \zeta_{20}^{4} + \zeta_{20}^{3} + \zeta_{20}^{2} + \cdots - 1) q^{96} + \cdots + (\zeta_{20}^{9} + \zeta_{20}^{8} + \zeta_{20}^{7} - \zeta_{20}^{3} - \zeta_{20}^{2} - \zeta_{20}) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{3} - 4 q^{4} + 4 q^{6} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{3} - 4 q^{4} + 4 q^{6} + 2 q^{7} - 4 q^{12} - 6 q^{14} - 2 q^{17} + 4 q^{18} + 4 q^{21} + 2 q^{24} - 6 q^{28} + 2 q^{37} - 8 q^{47} + 4 q^{50} + 8 q^{51} + 10 q^{54} + 2 q^{56} - 2 q^{61} - 8 q^{63} + 4 q^{64} - 4 q^{68} - 2 q^{71} - 8 q^{72} - 4 q^{74} + 2 q^{75} - 2 q^{79} - 12 q^{81} + 8 q^{84} - 4 q^{89} + 2 q^{96} + 2 q^{97} - 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/799\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(377\)
\(\chi(n)\) \(-1\) \(-\zeta_{20}^{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
140.1
0.587785 0.809017i
−0.587785 0.809017i
0.951057 + 0.309017i
−0.951057 + 0.309017i
0.951057 0.309017i
−0.951057 0.309017i
0.587785 + 0.809017i
−0.587785 + 0.809017i
0.618034i −1.39680 + 1.39680i 0.618034 0 0.863271 + 0.863271i −1.26007 1.26007i 1.00000i 2.90211i 0
140.2 0.618034i −0.221232 + 0.221232i 0.618034 0 0.136729 + 0.136729i 0.642040 + 0.642040i 1.00000i 0.902113i 0
140.3 1.61803i −0.642040 + 0.642040i −1.61803 0 −1.03884 1.03884i 1.39680 + 1.39680i 1.00000i 0.175571i 0
140.4 1.61803i 1.26007 1.26007i −1.61803 0 2.03884 + 2.03884i 0.221232 + 0.221232i 1.00000i 2.17557i 0
234.1 1.61803i −0.642040 0.642040i −1.61803 0 −1.03884 + 1.03884i 1.39680 1.39680i 1.00000i 0.175571i 0
234.2 1.61803i 1.26007 + 1.26007i −1.61803 0 2.03884 2.03884i 0.221232 0.221232i 1.00000i 2.17557i 0
234.3 0.618034i −1.39680 1.39680i 0.618034 0 0.863271 0.863271i −1.26007 + 1.26007i 1.00000i 2.90211i 0
234.4 0.618034i −0.221232 0.221232i 0.618034 0 0.136729 0.136729i 0.642040 0.642040i 1.00000i 0.902113i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 140.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
47.b odd 2 1 CM by \(\Q(\sqrt{-47}) \)
17.c even 4 1 inner
799.e odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 799.1.e.b 8
17.c even 4 1 inner 799.1.e.b 8
47.b odd 2 1 CM 799.1.e.b 8
799.e odd 4 1 inner 799.1.e.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
799.1.e.b 8 1.a even 1 1 trivial
799.1.e.b 8 17.c even 4 1 inner
799.1.e.b 8 47.b odd 2 1 CM
799.1.e.b 8 799.e odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 3T_{2}^{2} + 1 \) acting on \(S_{1}^{\mathrm{new}}(799, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 3 T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} + 2 T^{7} + 2 T^{6} + 11 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 2 T^{7} + 2 T^{6} + 11 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( (T^{4} + T^{3} + T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} - 2 T^{7} + 2 T^{6} + 11 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( (T + 1)^{8} \) Copy content Toggle raw display
$53$ \( (T^{4} + 5 T^{2} + 5)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 3 T^{2} + 1)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} + 2 T^{7} + 2 T^{6} + 11 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( T^{8} + 2 T^{7} + 2 T^{6} + 11 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( T^{8} + 2 T^{7} + 2 T^{6} + 11 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( (T^{2} + T - 1)^{4} \) Copy content Toggle raw display
$97$ \( T^{8} - 2 T^{7} + 2 T^{6} + 11 T^{4} + \cdots + 1 \) Copy content Toggle raw display
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