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 $4$

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

\(f(q)\) \(=\) \( q + (\zeta_{10}^{3} - \zeta_{10}^{2}) q^{2} + (\zeta_{10}^{4} + \zeta_{10}) q^{3} + (\zeta_{10}^{4} - \zeta_{10} + 1) q^{4} + (\zeta_{10}^{4} - \zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10}) q^{6} + (\zeta_{10}^{3} + \zeta_{10}^{2}) q^{7} + ( - \zeta_{10}^{4} + \zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10}) q^{8} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{10}^{3} - \zeta_{10}^{2}) q^{2} + (\zeta_{10}^{4} + \zeta_{10}) q^{3} + (\zeta_{10}^{4} - \zeta_{10} + 1) q^{4} + (\zeta_{10}^{4} - \zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10}) q^{6} + (\zeta_{10}^{3} + \zeta_{10}^{2}) q^{7} + ( - \zeta_{10}^{4} + \zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10}) q^{8} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - 1) q^{9} + (\zeta_{10}^{4} - \zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10}) q^{12} + ( - \zeta_{10}^{4} - \zeta_{10}) q^{14} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{16} - \zeta_{10} q^{17} + ( - \zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2} + \zeta_{10} - 2) q^{18} + (\zeta_{10}^{4} + \zeta_{10}^{3} - \zeta_{10}^{2} - \zeta_{10}) q^{21} + (\zeta_{10}^{4} + \zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10}) q^{24} - q^{25} + ( - \zeta_{10}^{4} + \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10}) q^{27} + ( - \zeta_{10}^{4} - \zeta_{10}^{2} - \zeta_{10}) q^{28} + ( - \zeta_{10}^{4} + \zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 2) q^{32} + ( - \zeta_{10}^{4} + \zeta_{10}^{3}) q^{34} + ( - 2 \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} - 1) q^{36} + (\zeta_{10}^{4} + \zeta_{10}) q^{37} + (\zeta_{10}^{3} - \zeta_{10}^{2} + 2) q^{42} + q^{47} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10}) q^{48} + (\zeta_{10}^{4} - \zeta_{10} - 1) q^{49} + ( - \zeta_{10}^{3} + \zeta_{10}^{2}) q^{50} + ( - \zeta_{10}^{2} + 1) q^{51} + ( - \zeta_{10}^{3} + \zeta_{10}^{2}) q^{53} + ( - 2 \zeta_{10}^{4} + \zeta_{10}^{3} + \zeta_{10}^{2} - 2 \zeta_{10}) q^{54} + (\zeta_{10}^{3} + \zeta_{10}^{2}) q^{56} + (\zeta_{10}^{3} - \zeta_{10}^{2}) q^{59} + ( - \zeta_{10}^{3} - \zeta_{10}^{2}) q^{61} + (\zeta_{10}^{4} - \zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10}) q^{63} + (\zeta_{10}^{4} - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - \zeta_{10} + 1) q^{64} + (\zeta_{10}^{2} - \zeta_{10} + 1) q^{68} + ( - \zeta_{10}^{4} - \zeta_{10}) q^{71} + ( - \zeta_{10}^{4} - \zeta_{10}^{2} + 2 \zeta_{10} - 2) q^{72} + (\zeta_{10}^{4} - \zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10}) q^{74} + ( - \zeta_{10}^{4} - \zeta_{10}) q^{75} + (\zeta_{10}^{4} + \zeta_{10}) q^{79} + (\zeta_{10}^{4} + \zeta_{10}^{3} - \zeta_{10}^{2} - \zeta_{10} + 1) q^{81} + q^{83} + (\zeta_{10}^{3} - \zeta_{10}^{2} + 2) q^{84} + ( - \zeta_{10}^{3} + \zeta_{10}^{2}) q^{89} + (\zeta_{10}^{3} - \zeta_{10}^{2}) q^{94} + ( - \zeta_{10}^{4} + \zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10}) q^{96} + (\zeta_{10}^{3} + \zeta_{10}^{2}) q^{97} + ( - \zeta_{10}^{4} - \zeta_{10}^{2} + \zeta_{10}) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} + 2 q^{4} + 4 q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} + 2 q^{4} + 4 q^{8} - 6 q^{9} - q^{17} - 8 q^{18} - 4 q^{25} - 4 q^{32} + 2 q^{34} - 8 q^{36} + 10 q^{42} + 4 q^{47} - 6 q^{49} - 2 q^{50} + 5 q^{51} - 2 q^{53} + 2 q^{59} - 2 q^{64} + 2 q^{68} - 6 q^{72} + 4 q^{81} + 8 q^{83} + 10 q^{84} - 2 q^{89} + 2 q^{94} + 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(52\) \(377\)
\(\chi(n)\) \(-1\) \(-1\)

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

Hecke characteristic polynomials

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