Properties

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

Newform invariants

Self dual: no
Analytic conductor: \(0.398752945094\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.230911.1

$q$-expansion

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

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

Character values

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

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

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
1.00000i
1.00000i
2.00000i 1.00000 1.00000i −3.00000 0 −2.00000 2.00000i −1.00000 1.00000i 4.00000i 1.00000i 0
234.1 2.00000i 1.00000 + 1.00000i −3.00000 0 −2.00000 + 2.00000i −1.00000 + 1.00000i 4.00000i 1.00000i 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.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.a 2
17.c even 4 1 inner 799.1.e.a 2
47.b odd 2 1 CM 799.1.e.a 2
799.e odd 4 1 inner 799.1.e.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
799.1.e.a 2 1.a even 1 1 trivial
799.1.e.a 2 17.c even 4 1 inner
799.1.e.a 2 47.b odd 2 1 CM
799.1.e.a 2 799.e odd 4 1 inner

Hecke kernels

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

Hecke characteristic polynomials

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