Properties

Label 799.1.c.b
Level $799$
Weight $1$
Character orbit 799.c
Self dual yes
Analytic conductor $0.399$
Analytic rank $0$
Dimension $2$
Projective image $D_{4}$
CM discriminant -799
Inner twists $4$

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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: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.13583.1
Artin image: $D_8$
Artin field: Galois closure of 8.2.8671400783.1

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{4} -\beta q^{5} + q^{9} +O(q^{10})\) \( q - q^{4} -\beta q^{5} + q^{9} + \beta q^{11} + q^{16} + q^{17} + \beta q^{20} -\beta q^{23} + q^{25} + \beta q^{29} + \beta q^{31} - q^{36} -\beta q^{41} -\beta q^{44} -\beta q^{45} + q^{47} + q^{49} -2 q^{53} -2 q^{55} - q^{64} - q^{68} + \beta q^{73} -\beta q^{80} + q^{81} -2 q^{83} -\beta q^{85} -2 q^{89} + \beta q^{92} + \beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{4} + 2q^{9} + 2q^{16} + 2q^{17} + 2q^{25} - 2q^{36} + 2q^{47} + 2q^{49} - 4q^{53} - 4q^{55} - 2q^{64} - 2q^{68} + 2q^{81} - 4q^{83} - 4q^{89} + O(q^{100}) \)

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
1.41421
−1.41421
0 0 −1.00000 −1.41421 0 0 0 1.00000 0
798.2 0 0 −1.00000 1.41421 0 0 0 1.00000 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
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.b 2
17.b even 2 1 inner 799.1.c.b 2
47.b odd 2 1 inner 799.1.c.b 2
799.c odd 2 1 CM 799.1.c.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
799.1.c.b 2 1.a even 1 1 trivial
799.1.c.b 2 17.b even 2 1 inner
799.1.c.b 2 47.b odd 2 1 inner
799.1.c.b 2 799.c odd 2 1 CM

Hecke kernels

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

Hecke characteristic polynomials

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