Properties

Label 1792.1.c.d
Level $1792$
Weight $1$
Character orbit 1792.c
Analytic conductor $0.894$
Analytic rank $0$
Dimension $2$
Projective image $D_{4}$
CM discriminant -56
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1792 = 2^{8} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1792.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.894324502638\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \(x^{2} + 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 896)
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.0.1568.1
Artin image: $SD_{16}$
Artin field: Galois closure of 8.2.5754585088.2

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} -\beta q^{5} + q^{7} - q^{9} +O(q^{10})\) \( q + \beta q^{3} -\beta q^{5} + q^{7} - q^{9} -\beta q^{13} + 2 q^{15} -\beta q^{19} + \beta q^{21} - q^{25} -\beta q^{35} + 2 q^{39} + \beta q^{45} + q^{49} + 2 q^{57} + \beta q^{59} + \beta q^{61} - q^{63} -2 q^{65} -2 q^{71} -\beta q^{75} + 2 q^{79} - q^{81} + \beta q^{83} -\beta q^{91} -2 q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{7} - 2q^{9} + O(q^{10}) \) \( 2q + 2q^{7} - 2q^{9} + 4q^{15} - 2q^{25} + 4q^{39} + 2q^{49} + 4q^{57} - 2q^{63} - 4q^{65} - 4q^{71} + 4q^{79} - 2q^{81} - 4q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1792\mathbb{Z}\right)^\times\).

\(n\) \(1023\) \(1025\) \(1541\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
769.1
1.41421i
1.41421i
0 1.41421i 0 1.41421i 0 1.00000 0 −1.00000 0
769.2 0 1.41421i 0 1.41421i 0 1.00000 0 −1.00000 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
56.h odd 2 1 CM by \(\Q(\sqrt{-14}) \)
7.b odd 2 1 inner
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1792.1.c.d 2
4.b odd 2 1 1792.1.c.c 2
7.b odd 2 1 inner 1792.1.c.d 2
8.b even 2 1 inner 1792.1.c.d 2
8.d odd 2 1 1792.1.c.c 2
16.e even 4 2 896.1.h.a 2
16.f odd 4 2 896.1.h.b yes 2
28.d even 2 1 1792.1.c.c 2
56.e even 2 1 1792.1.c.c 2
56.h odd 2 1 CM 1792.1.c.d 2
112.j even 4 2 896.1.h.b yes 2
112.l odd 4 2 896.1.h.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
896.1.h.a 2 16.e even 4 2
896.1.h.a 2 112.l odd 4 2
896.1.h.b yes 2 16.f odd 4 2
896.1.h.b yes 2 112.j even 4 2
1792.1.c.c 2 4.b odd 2 1
1792.1.c.c 2 8.d odd 2 1
1792.1.c.c 2 28.d even 2 1
1792.1.c.c 2 56.e even 2 1
1792.1.c.d 2 1.a even 1 1 trivial
1792.1.c.d 2 7.b odd 2 1 inner
1792.1.c.d 2 8.b even 2 1 inner
1792.1.c.d 2 56.h odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1792, [\chi])\):

\( T_{3}^{2} + 2 \)
\( T_{23} \)
\( T_{71} + 2 \)

Hecke characteristic polynomials

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