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: \(x^{2} + 1\)
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.

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

Character values

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

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

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 \(S_{1}^{\mathrm{new}}(799, [\chi])\).

Hecke characteristic polynomials

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