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 $4$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 799 = 17 \cdot 47 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 799.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.398752945094\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{16})^+\)
Defining polynomial: \(x^{4} - 4 x^{2} + 2\)
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

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.

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

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.

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 \(S_{1}^{\mathrm{new}}(799, [\chi])\).

Hecke characteristic polynomials

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