# 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 $4$

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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

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