Properties

Label 799.1.e.b
Level $799$
Weight $1$
Character orbit 799.e
Analytic conductor $0.399$
Analytic rank $0$
Dimension $8$
Projective image $D_{20}$
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.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.398752945094\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{20})\)
Defining polynomial: \(x^{8} - x^{6} + x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{20}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{20} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( \zeta_{20}^{3} + \zeta_{20}^{7} ) q^{2} + ( -\zeta_{20} + \zeta_{20}^{4} ) q^{3} + ( -1 - \zeta_{20}^{4} + \zeta_{20}^{6} ) q^{4} + ( -\zeta_{20} - \zeta_{20}^{4} + \zeta_{20}^{7} - \zeta_{20}^{8} ) q^{6} + ( \zeta_{20}^{2} + \zeta_{20}^{3} ) q^{7} + ( \zeta_{20} - \zeta_{20}^{3} - \zeta_{20}^{7} + \zeta_{20}^{9} ) q^{8} + ( \zeta_{20}^{2} - \zeta_{20}^{5} + \zeta_{20}^{8} ) q^{9} +O(q^{10})\) \( q + ( \zeta_{20}^{3} + \zeta_{20}^{7} ) q^{2} + ( -\zeta_{20} + \zeta_{20}^{4} ) q^{3} + ( -1 - \zeta_{20}^{4} + \zeta_{20}^{6} ) q^{4} + ( -\zeta_{20} - \zeta_{20}^{4} + \zeta_{20}^{7} - \zeta_{20}^{8} ) q^{6} + ( \zeta_{20}^{2} + \zeta_{20}^{3} ) q^{7} + ( \zeta_{20} - \zeta_{20}^{3} - \zeta_{20}^{7} + \zeta_{20}^{9} ) q^{8} + ( \zeta_{20}^{2} - \zeta_{20}^{5} + \zeta_{20}^{8} ) q^{9} + ( -1 + \zeta_{20} - \zeta_{20}^{4} + \zeta_{20}^{5} - \zeta_{20}^{7} - \zeta_{20}^{8} ) q^{12} + ( -1 + \zeta_{20}^{5} + \zeta_{20}^{6} + \zeta_{20}^{9} ) q^{14} + ( 1 - \zeta_{20}^{2} + \zeta_{20}^{4} - \zeta_{20}^{6} + \zeta_{20}^{8} ) q^{16} -\zeta_{20}^{6} q^{17} + ( -\zeta_{20} + \zeta_{20}^{2} - \zeta_{20}^{8} + \zeta_{20}^{9} ) q^{18} + ( -\zeta_{20}^{3} - \zeta_{20}^{4} + \zeta_{20}^{6} + \zeta_{20}^{7} ) q^{21} + ( 1 + \zeta_{20} - \zeta_{20}^{2} - \zeta_{20}^{3} + \zeta_{20}^{4} + \zeta_{20}^{5} - \zeta_{20}^{7} + \zeta_{20}^{8} ) q^{24} -\zeta_{20}^{5} q^{25} + ( -\zeta_{20}^{2} - \zeta_{20}^{3} + \zeta_{20}^{6} - \zeta_{20}^{9} ) q^{27} + ( -\zeta_{20}^{2} - \zeta_{20}^{3} - \zeta_{20}^{6} - \zeta_{20}^{7} + \zeta_{20}^{8} + \zeta_{20}^{9} ) q^{28} + ( -\zeta_{20} + \zeta_{20}^{3} - 2 \zeta_{20}^{5} + \zeta_{20}^{7} - \zeta_{20}^{9} ) q^{32} + ( \zeta_{20}^{3} - \zeta_{20}^{9} ) q^{34} + ( \zeta_{20} - \zeta_{20}^{4} + \zeta_{20}^{5} - \zeta_{20}^{6} + \zeta_{20}^{9} ) q^{36} + ( \zeta_{20} - \zeta_{20}^{4} ) q^{37} + ( \zeta_{20} - \zeta_{20}^{3} - \zeta_{20}^{4} - \zeta_{20}^{6} - \zeta_{20}^{7} + \zeta_{20}^{9} ) q^{42} - q^{47} + ( 1 - \zeta_{20} - \zeta_{20}^{2} + \zeta_{20}^{3} + \zeta_{20}^{4} - \zeta_{20}^{5} - \zeta_{20}^{6} + \zeta_{20}^{7} + \zeta_{20}^{8} - \zeta_{20}^{9} ) q^{48} + ( \zeta_{20}^{4} + \zeta_{20}^{5} + \zeta_{20}^{6} ) q^{49} + ( \zeta_{20}^{2} - \zeta_{20}^{8} ) q^{50} + ( 1 + \zeta_{20}^{7} ) q^{51} + ( -\zeta_{20}^{2} - \zeta_{20}^{8} ) q^{53} + ( 1 + \zeta_{20}^{2} - \zeta_{20}^{3} - \zeta_{20}^{5} ) q^{54} + ( 1 - \zeta_{20} - \zeta_{20}^{2} + \zeta_{20}^{3} + \zeta_{20}^{4} - \zeta_{20}^{5} - \zeta_{20}^{6} - \zeta_{20}^{9} ) q^{56} + ( \zeta_{20}^{3} + \zeta_{20}^{7} ) q^{59} + ( -\zeta_{20}^{7} + \zeta_{20}^{8} ) q^{61} + ( -1 - \zeta_{20} + \zeta_{20}^{4} + \zeta_{20}^{5} - \zeta_{20}^{7} - \zeta_{20}^{8} ) q^{63} + ( -1 + 2 \zeta_{20}^{2} - \zeta_{20}^{4} + \zeta_{20}^{6} - 2 \zeta_{20}^{8} ) q^{64} + ( -1 + \zeta_{20}^{2} + \zeta_{20}^{6} ) q^{68} + ( -\zeta_{20}^{6} - \zeta_{20}^{9} ) q^{71} + ( -\zeta_{20}^{2} + \zeta_{20}^{3} + \zeta_{20}^{4} - \zeta_{20}^{6} - \zeta_{20}^{7} + \zeta_{20}^{8} ) q^{72} + ( \zeta_{20} + \zeta_{20}^{4} - \zeta_{20}^{7} + \zeta_{20}^{8} ) q^{74} + ( \zeta_{20}^{6} - \zeta_{20}^{9} ) q^{75} + ( \zeta_{20} + \zeta_{20}^{4} ) q^{79} + ( -1 + \zeta_{20}^{3} + \zeta_{20}^{4} - \zeta_{20}^{6} - \zeta_{20}^{7} ) q^{81} + ( 2 + \zeta_{20} - \zeta_{20}^{2} + \zeta_{20}^{4} - \zeta_{20}^{6} + \zeta_{20}^{8} - \zeta_{20}^{9} ) q^{84} + ( -\zeta_{20}^{2} + \zeta_{20}^{8} ) q^{89} + ( -\zeta_{20}^{3} - \zeta_{20}^{7} ) q^{94} + ( -1 - \zeta_{20} + \zeta_{20}^{2} + \zeta_{20}^{3} - \zeta_{20}^{4} - \zeta_{20}^{5} + 2 \zeta_{20}^{6} + \zeta_{20}^{7} - \zeta_{20}^{8} - 2 \zeta_{20}^{9} ) q^{96} + ( \zeta_{20}^{2} - \zeta_{20}^{3} ) q^{97} + ( -\zeta_{20} - \zeta_{20}^{2} - \zeta_{20}^{3} + \zeta_{20}^{7} + \zeta_{20}^{8} + \zeta_{20}^{9} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 2q^{3} - 4q^{4} + 4q^{6} + 2q^{7} + O(q^{10}) \) \( 8q - 2q^{3} - 4q^{4} + 4q^{6} + 2q^{7} - 4q^{12} - 6q^{14} - 2q^{17} + 4q^{18} + 4q^{21} + 2q^{24} - 6q^{28} + 2q^{37} - 8q^{47} + 4q^{50} + 8q^{51} + 10q^{54} + 2q^{56} - 2q^{61} - 8q^{63} + 4q^{64} - 4q^{68} - 2q^{71} - 8q^{72} - 4q^{74} + 2q^{75} - 2q^{79} - 12q^{81} + 8q^{84} - 4q^{89} + 2q^{96} + 2q^{97} - 4q^{98} + O(q^{100}) \)

Character values

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

\(n\) \(52\) \(377\)
\(\chi(n)\) \(-1\) \(-\zeta_{20}^{5}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
140.1
0.587785 0.809017i
−0.587785 0.809017i
0.951057 + 0.309017i
−0.951057 + 0.309017i
0.951057 0.309017i
−0.951057 0.309017i
0.587785 + 0.809017i
−0.587785 + 0.809017i
0.618034i −1.39680 + 1.39680i 0.618034 0 0.863271 + 0.863271i −1.26007 1.26007i 1.00000i 2.90211i 0
140.2 0.618034i −0.221232 + 0.221232i 0.618034 0 0.136729 + 0.136729i 0.642040 + 0.642040i 1.00000i 0.902113i 0
140.3 1.61803i −0.642040 + 0.642040i −1.61803 0 −1.03884 1.03884i 1.39680 + 1.39680i 1.00000i 0.175571i 0
140.4 1.61803i 1.26007 1.26007i −1.61803 0 2.03884 + 2.03884i 0.221232 + 0.221232i 1.00000i 2.17557i 0
234.1 1.61803i −0.642040 0.642040i −1.61803 0 −1.03884 + 1.03884i 1.39680 1.39680i 1.00000i 0.175571i 0
234.2 1.61803i 1.26007 + 1.26007i −1.61803 0 2.03884 2.03884i 0.221232 0.221232i 1.00000i 2.17557i 0
234.3 0.618034i −1.39680 1.39680i 0.618034 0 0.863271 0.863271i −1.26007 + 1.26007i 1.00000i 2.90211i 0
234.4 0.618034i −0.221232 0.221232i 0.618034 0 0.136729 0.136729i 0.642040 0.642040i 1.00000i 0.902113i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 234.4
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.c even 4 1 inner
799.e odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 799.1.e.b 8
17.c even 4 1 inner 799.1.e.b 8
47.b odd 2 1 CM 799.1.e.b 8
799.e odd 4 1 inner 799.1.e.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
799.1.e.b 8 1.a even 1 1 trivial
799.1.e.b 8 17.c even 4 1 inner
799.1.e.b 8 47.b odd 2 1 CM
799.1.e.b 8 799.e odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 3 T_{2}^{2} + 1 \) acting on \(S_{1}^{\mathrm{new}}(799, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 3 T^{2} + T^{4} )^{2} \)
$3$ \( 1 + 6 T + 18 T^{2} + 20 T^{3} + 11 T^{4} + 2 T^{6} + 2 T^{7} + T^{8} \)
$5$ \( T^{8} \)
$7$ \( 1 - 6 T + 18 T^{2} - 20 T^{3} + 11 T^{4} + 2 T^{6} - 2 T^{7} + T^{8} \)
$11$ \( T^{8} \)
$13$ \( T^{8} \)
$17$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
$19$ \( T^{8} \)
$23$ \( T^{8} \)
$29$ \( T^{8} \)
$31$ \( T^{8} \)
$37$ \( 1 - 6 T + 18 T^{2} - 20 T^{3} + 11 T^{4} + 2 T^{6} - 2 T^{7} + T^{8} \)
$41$ \( T^{8} \)
$43$ \( T^{8} \)
$47$ \( ( 1 + T )^{8} \)
$53$ \( ( 5 + 5 T^{2} + T^{4} )^{2} \)
$59$ \( ( 1 + 3 T^{2} + T^{4} )^{2} \)
$61$ \( 1 + 6 T + 18 T^{2} + 20 T^{3} + 11 T^{4} + 2 T^{6} + 2 T^{7} + T^{8} \)
$67$ \( T^{8} \)
$71$ \( 1 + 6 T + 18 T^{2} + 20 T^{3} + 11 T^{4} + 2 T^{6} + 2 T^{7} + T^{8} \)
$73$ \( T^{8} \)
$79$ \( 1 + 6 T + 18 T^{2} + 20 T^{3} + 11 T^{4} + 2 T^{6} + 2 T^{7} + T^{8} \)
$83$ \( T^{8} \)
$89$ \( ( -1 + T + T^{2} )^{4} \)
$97$ \( 1 - 6 T + 18 T^{2} - 20 T^{3} + 11 T^{4} + 2 T^{6} - 2 T^{7} + T^{8} \)
show more
show less