Properties

Label 799.1.c.d
Level $799$
Weight $1$
Character orbit 799.c
Analytic conductor $0.399$
Analytic rank $0$
Dimension $4$
Projective image $D_{10}$
CM discriminant -47
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [799,1,Mod(798,799)]
 
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, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("799.798");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 799 = 17 \cdot 47 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 799.c (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: \(0.398752945094\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{10}\)
Projective field: Galois closure of 10.2.6928449225617.1

$q$-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.

\(f(q)\) \(=\) \( q + (\zeta_{10}^{3} - \zeta_{10}^{2}) q^{2} + (\zeta_{10}^{4} + \zeta_{10}) q^{3} + (\zeta_{10}^{4} - \zeta_{10} + 1) q^{4} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \cdots + \zeta_{10}) q^{6}+ \cdots + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{10}^{3} - \zeta_{10}^{2}) q^{2} + (\zeta_{10}^{4} + \zeta_{10}) q^{3} + (\zeta_{10}^{4} - \zeta_{10} + 1) q^{4} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \cdots + \zeta_{10}) q^{6}+ \cdots + ( - \zeta_{10}^{4} - \zeta_{10}^{2} + \zeta_{10}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} + 2 q^{4} + 4 q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} + 2 q^{4} + 4 q^{8} - 6 q^{9} - q^{17} - 8 q^{18} - 4 q^{25} - 4 q^{32} + 2 q^{34} - 8 q^{36} + 10 q^{42} + 4 q^{47} - 6 q^{49} - 2 q^{50} + 5 q^{51} - 2 q^{53} + 2 q^{59} - 2 q^{64} + 2 q^{68} - 6 q^{72} + 4 q^{81} + 8 q^{83} + 10 q^{84} - 2 q^{89} + 2 q^{94} + 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(52\) \(377\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
798.1
0.809017 0.587785i
0.809017 + 0.587785i
−0.309017 0.951057i
−0.309017 + 0.951057i
−0.618034 1.17557i −0.618034 0 0.726543i 1.90211i 1.00000 −0.381966 0
798.2 −0.618034 1.17557i −0.618034 0 0.726543i 1.90211i 1.00000 −0.381966 0
798.3 1.61803 1.90211i 1.61803 0 3.07768i 1.17557i 1.00000 −2.61803 0
798.4 1.61803 1.90211i 1.61803 0 3.07768i 1.17557i 1.00000 −2.61803 0
\(n\): e.g. 2-40 or 990-1000
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.b even 2 1 inner
799.c odd 2 1 inner

Twists

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

Hecke kernels

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

Hecke characteristic polynomials

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