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 $4$

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: no
Analytic conductor: \(0.398752945094\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Defining polynomial: \(x^{4} - x^{3} + x^{2} - x + 1\)
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

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

\(f(q)\) \(=\) \( q + ( -\zeta_{10}^{2} + \zeta_{10}^{3} ) q^{2} + ( \zeta_{10} + \zeta_{10}^{4} ) q^{3} + ( 1 - \zeta_{10} + \zeta_{10}^{4} ) q^{4} + ( \zeta_{10} - \zeta_{10}^{2} - \zeta_{10}^{3} + \zeta_{10}^{4} ) q^{6} + ( \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{7} + ( \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} - \zeta_{10}^{4} ) q^{8} + ( -1 + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{9} +O(q^{10})\) \( q + ( -\zeta_{10}^{2} + \zeta_{10}^{3} ) q^{2} + ( \zeta_{10} + \zeta_{10}^{4} ) q^{3} + ( 1 - \zeta_{10} + \zeta_{10}^{4} ) q^{4} + ( \zeta_{10} - \zeta_{10}^{2} - \zeta_{10}^{3} + \zeta_{10}^{4} ) q^{6} + ( \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{7} + ( \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} - \zeta_{10}^{4} ) q^{8} + ( -1 + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{9} + ( \zeta_{10} - \zeta_{10}^{2} - \zeta_{10}^{3} + \zeta_{10}^{4} ) q^{12} + ( -\zeta_{10} - \zeta_{10}^{4} ) q^{14} + ( 1 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} + \zeta_{10}^{4} ) q^{16} -\zeta_{10} q^{17} + ( -2 + \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} - \zeta_{10}^{4} ) q^{18} + ( -\zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} + \zeta_{10}^{4} ) q^{21} + ( \zeta_{10} + \zeta_{10}^{4} ) q^{24} - q^{25} + ( -\zeta_{10} + \zeta_{10}^{2} + \zeta_{10}^{3} - \zeta_{10}^{4} ) q^{27} + ( -\zeta_{10} - \zeta_{10}^{4} ) q^{28} + ( -2 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} - \zeta_{10}^{4} ) q^{32} + ( \zeta_{10}^{3} - \zeta_{10}^{4} ) q^{34} + ( -1 + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{36} + ( \zeta_{10} + \zeta_{10}^{4} ) q^{37} + ( 2 - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{42} + q^{47} + ( -1 - \zeta_{10} + \zeta_{10}^{4} ) q^{49} + ( \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{50} + ( 1 - \zeta_{10}^{2} ) q^{51} + ( \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{53} + ( -2 \zeta_{10} + \zeta_{10}^{2} + \zeta_{10}^{3} - 2 \zeta_{10}^{4} ) q^{54} + ( \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{56} + ( -\zeta_{10}^{2} + \zeta_{10}^{3} ) q^{59} + ( -\zeta_{10}^{2} - \zeta_{10}^{3} ) q^{61} + ( \zeta_{10} - \zeta_{10}^{2} - \zeta_{10}^{3} + \zeta_{10}^{4} ) q^{63} + ( 1 - \zeta_{10} + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} + \zeta_{10}^{4} ) q^{64} + ( 1 - \zeta_{10} + \zeta_{10}^{2} ) q^{68} + ( -\zeta_{10} - \zeta_{10}^{4} ) q^{71} + ( -2 + \zeta_{10} - \zeta_{10}^{4} ) q^{72} + ( \zeta_{10} - \zeta_{10}^{2} - \zeta_{10}^{3} + \zeta_{10}^{4} ) q^{74} + ( -\zeta_{10} - \zeta_{10}^{4} ) q^{75} + ( \zeta_{10} + \zeta_{10}^{4} ) q^{79} + ( 1 - \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} + \zeta_{10}^{4} ) q^{81} + 2 q^{83} + ( 2 - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{84} + ( \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{89} + ( -\zeta_{10}^{2} + \zeta_{10}^{3} ) q^{94} + ( -\zeta_{10} - \zeta_{10}^{4} ) q^{96} + ( \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{97} + ( \zeta_{10} - \zeta_{10}^{4} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{2} + 2q^{4} + 4q^{8} - 6q^{9} + O(q^{10}) \) \( 4q + 2q^{2} + 2q^{4} + 4q^{8} - 6q^{9} - q^{17} - 8q^{18} - 4q^{25} - 4q^{32} + 2q^{34} - 8q^{36} + 10q^{42} + 4q^{47} - 6q^{49} - 2q^{50} + 5q^{51} - 2q^{53} + 2q^{59} - 2q^{64} + 2q^{68} - 6q^{72} + 4q^{81} + 8q^{83} + 10q^{84} - 2q^{89} + 2q^{94} + 2q^{98} + O(q^{100}) \)

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

Hecke characteristic polynomials

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