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: \( x^{4} + 1 \) Copy content Toggle raw display
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.

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

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 \(S_{1}^{\mathrm{new}}(799, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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