Properties

Label 799.1.c.c
Level $799$
Weight $1$
Character orbit 799.c
Self dual yes
Analytic conductor $0.399$
Analytic rank $0$
Dimension $4$
Projective image $D_{8}$
CM discriminant -799
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: yes
Analytic conductor: \(0.398752945094\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{16})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 4x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{8}\)
Projective field: Galois closure of 8.2.8671400783.1

$q$-expansion

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

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + q^{4} - \beta_1 q^{5} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + q^{4} - \beta_1 q^{5} + q^{9} + (\beta_{3} + \beta_1) q^{10} - \beta_{3} q^{11} - q^{16} - q^{17} - \beta_{2} q^{18} - \beta_1 q^{20} + ( - \beta_{3} + \beta_1) q^{22} + \beta_1 q^{23} + (\beta_{2} + 1) q^{25} + \beta_{3} q^{29} + \beta_{3} q^{31} + \beta_{2} q^{32} + \beta_{2} q^{34} + q^{36} + \beta_1 q^{41} - \beta_{3} q^{44} - \beta_1 q^{45} + ( - \beta_{3} - \beta_1) q^{46} - q^{47} + q^{49} + ( - \beta_{2} - 2) q^{50} + \beta_{2} q^{55} + (\beta_{3} - \beta_1) q^{58} + \beta_{2} q^{59} + (\beta_{3} - \beta_1) q^{62} - q^{64} - q^{68} - \beta_{3} q^{73} + \beta_1 q^{80} + q^{81} + ( - \beta_{3} - \beta_1) q^{82} + \beta_1 q^{85} + (\beta_{3} + \beta_1) q^{90} + \beta_1 q^{92} + \beta_{2} q^{94} - \beta_{2} q^{98} - \beta_{3} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{4} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{4} + 4 q^{9} - 4 q^{16} - 4 q^{17} + 4 q^{25} + 4 q^{36} - 4 q^{47} + 4 q^{49} - 8 q^{50} - 4 q^{64} - 4 q^{68} + 4 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{16} + \zeta_{16}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 3\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 3\beta_1 \) 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
1.84776
−1.84776
0.765367
−0.765367
−1.41421 0 1.00000 −1.84776 0 0 0 1.00000 2.61313
798.2 −1.41421 0 1.00000 1.84776 0 0 0 1.00000 −2.61313
798.3 1.41421 0 1.00000 −0.765367 0 0 0 1.00000 −1.08239
798.4 1.41421 0 1.00000 0.765367 0 0 0 1.00000 1.08239
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
799.c odd 2 1 CM by \(\Q(\sqrt{-799}) \)
17.b even 2 1 inner
47.b 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.c 4
17.b even 2 1 inner 799.1.c.c 4
47.b odd 2 1 inner 799.1.c.c 4
799.c odd 2 1 CM 799.1.c.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
799.1.c.c 4 1.a even 1 1 trivial
799.1.c.c 4 17.b even 2 1 inner
799.1.c.c 4 47.b odd 2 1 inner
799.1.c.c 4 799.c odd 2 1 CM

Hecke kernels

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

Hecke characteristic polynomials

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