Properties

Label 799.1.h.a
Level $799$
Weight $1$
Character orbit 799.h
Analytic conductor $0.399$
Analytic rank $0$
Dimension $4$
Projective image $D_{8}$
CM discriminant -47
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 799 = 17 \cdot 47 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 799.h (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.398752945094\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{8}\)
Projective field: Galois closure of 8.2.42602592046879.2

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( 1 - \zeta_{8} ) q^{3} + \zeta_{8}^{2} q^{4} + ( \zeta_{8} - \zeta_{8}^{2} ) q^{7} + ( 1 - \zeta_{8} + \zeta_{8}^{2} ) q^{9} +O(q^{10})\) \( q + ( 1 - \zeta_{8} ) q^{3} + \zeta_{8}^{2} q^{4} + ( \zeta_{8} - \zeta_{8}^{2} ) q^{7} + ( 1 - \zeta_{8} + \zeta_{8}^{2} ) q^{9} + ( \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{12} - q^{16} - q^{17} + ( \zeta_{8} - 2 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{21} + \zeta_{8} q^{25} + ( 1 - \zeta_{8} + \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{27} + ( 1 + \zeta_{8}^{3} ) q^{28} + ( -1 + \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{36} + ( -\zeta_{8}^{2} - \zeta_{8}^{3} ) q^{37} -\zeta_{8}^{2} q^{47} + ( -1 + \zeta_{8} ) q^{48} + ( -1 + \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{49} + ( -1 + \zeta_{8} ) q^{51} + ( -1 + \zeta_{8}^{2} ) q^{53} -2 \zeta_{8} q^{59} + ( -1 + \zeta_{8}^{3} ) q^{61} + ( 1 + \zeta_{8} - 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{63} -\zeta_{8}^{2} q^{64} -\zeta_{8}^{2} q^{68} + ( 1 - \zeta_{8} ) q^{71} + ( \zeta_{8} - \zeta_{8}^{2} ) q^{75} + ( 1 + \zeta_{8}^{3} ) q^{79} + ( -\zeta_{8} + \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{81} + ( -1 + \zeta_{8}^{2} ) q^{83} + ( 2 - \zeta_{8} + \zeta_{8}^{3} ) q^{84} + ( -\zeta_{8}^{2} + \zeta_{8}^{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{3} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{3} + 4q^{9} - 4q^{16} - 4q^{17} + 4q^{27} + 4q^{28} - 4q^{36} - 4q^{48} - 4q^{49} - 4q^{51} - 4q^{53} - 4q^{61} + 4q^{63} + 4q^{71} + 4q^{79} - 4q^{83} + 8q^{84} + O(q^{100}) \)

Character values

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

\(n\) \(52\) \(377\)
\(\chi(n)\) \(-1\) \(-\zeta_{8}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
93.1
0.707107 + 0.707107i
−0.707107 0.707107i
−0.707107 + 0.707107i
0.707107 0.707107i
0 0.292893 0.707107i 1.00000i 0 0 0.707107 0.292893i 0 0.292893 + 0.292893i 0
281.1 0 1.70711 + 0.707107i 1.00000i 0 0 −0.707107 1.70711i 0 1.70711 + 1.70711i 0
563.1 0 1.70711 0.707107i 1.00000i 0 0 −0.707107 + 1.70711i 0 1.70711 1.70711i 0
610.1 0 0.292893 + 0.707107i 1.00000i 0 0 0.707107 + 0.292893i 0 0.292893 0.292893i 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.d even 8 1 inner
799.h odd 8 1 inner

Twists

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

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

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 2 - 4 T + 6 T^{2} - 4 T^{3} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( 2 - 4 T + 2 T^{2} + T^{4} \)
$11$ \( T^{4} \)
$13$ \( T^{4} \)
$17$ \( ( 1 + T )^{4} \)
$19$ \( T^{4} \)
$23$ \( T^{4} \)
$29$ \( T^{4} \)
$31$ \( T^{4} \)
$37$ \( 2 + 4 T + 2 T^{2} + T^{4} \)
$41$ \( T^{4} \)
$43$ \( T^{4} \)
$47$ \( ( 1 + T^{2} )^{2} \)
$53$ \( ( 2 + 2 T + T^{2} )^{2} \)
$59$ \( 16 + T^{4} \)
$61$ \( 2 + 4 T + 6 T^{2} + 4 T^{3} + T^{4} \)
$67$ \( T^{4} \)
$71$ \( 2 - 4 T + 6 T^{2} - 4 T^{3} + T^{4} \)
$73$ \( T^{4} \)
$79$ \( 2 - 4 T + 6 T^{2} - 4 T^{3} + T^{4} \)
$83$ \( ( 2 + 2 T + T^{2} )^{2} \)
$89$ \( T^{4} \)
$97$ \( 2 + 4 T + 2 T^{2} + T^{4} \)
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