# Properties

 Label 1792.2.f.g Level $1792$ Weight $2$ Character orbit 1792.f Analytic conductor $14.309$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$14.3091920422$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 896) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{3} -\beta_{3} q^{5} + ( 2 + \beta_{2} ) q^{7} - q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{3} -\beta_{3} q^{5} + ( 2 + \beta_{2} ) q^{7} - q^{9} + 2 \beta_{3} q^{11} -\beta_{3} q^{13} -2 \beta_{2} q^{15} -4 \beta_{2} q^{17} -3 \beta_{1} q^{19} + ( -2 \beta_{1} + \beta_{3} ) q^{21} -4 \beta_{2} q^{23} - q^{25} + 4 \beta_{1} q^{27} -4 \beta_{1} q^{29} + 4 q^{31} + 4 \beta_{2} q^{33} + ( -3 \beta_{1} - 2 \beta_{3} ) q^{35} -2 \beta_{2} q^{39} + 2 \beta_{3} q^{43} + \beta_{3} q^{45} + 12 q^{47} + ( 1 + 4 \beta_{2} ) q^{49} -4 \beta_{3} q^{51} -4 \beta_{1} q^{53} + 12 q^{55} + 6 q^{57} + 7 \beta_{1} q^{59} -3 \beta_{3} q^{61} + ( -2 - \beta_{2} ) q^{63} -6 q^{65} + 2 \beta_{3} q^{67} -4 \beta_{3} q^{69} -2 \beta_{2} q^{71} + 4 \beta_{2} q^{73} + \beta_{1} q^{75} + ( 6 \beta_{1} + 4 \beta_{3} ) q^{77} -6 \beta_{2} q^{79} -5 q^{81} -\beta_{1} q^{83} + 12 \beta_{1} q^{85} + 8 q^{87} -4 \beta_{2} q^{89} + ( -3 \beta_{1} - 2 \beta_{3} ) q^{91} -4 \beta_{1} q^{93} -6 \beta_{2} q^{95} -4 \beta_{2} q^{97} -2 \beta_{3} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 8q^{7} - 4q^{9} + O(q^{10})$$ $$4q + 8q^{7} - 4q^{9} - 4q^{25} + 16q^{31} + 48q^{47} + 4q^{49} + 48q^{55} + 24q^{57} - 8q^{63} - 24q^{65} - 20q^{81} + 32q^{87} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 2 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{3}$$$$/2$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 1$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} + 4 \nu$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 1$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1791.1
 −0.707107 + 1.22474i −0.707107 − 1.22474i 0.707107 + 1.22474i 0.707107 − 1.22474i
0 −1.41421 0 2.44949i 0 2.00000 1.73205i 0 −1.00000 0
1791.2 0 −1.41421 0 2.44949i 0 2.00000 + 1.73205i 0 −1.00000 0
1791.3 0 1.41421 0 2.44949i 0 2.00000 + 1.73205i 0 −1.00000 0
1791.4 0 1.41421 0 2.44949i 0 2.00000 1.73205i 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
8.b even 2 1 inner
28.d even 2 1 inner
56.e even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1792.2.f.g 4
4.b odd 2 1 1792.2.f.c 4
7.b odd 2 1 1792.2.f.c 4
8.b even 2 1 inner 1792.2.f.g 4
8.d odd 2 1 1792.2.f.c 4
16.e even 4 2 896.2.e.c 4
16.f odd 4 2 896.2.e.d yes 4
28.d even 2 1 inner 1792.2.f.g 4
56.e even 2 1 inner 1792.2.f.g 4
56.h odd 2 1 1792.2.f.c 4
112.j even 4 2 896.2.e.c 4
112.l odd 4 2 896.2.e.d yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
896.2.e.c 4 16.e even 4 2
896.2.e.c 4 112.j even 4 2
896.2.e.d yes 4 16.f odd 4 2
896.2.e.d yes 4 112.l odd 4 2
1792.2.f.c 4 4.b odd 2 1
1792.2.f.c 4 7.b odd 2 1
1792.2.f.c 4 8.d odd 2 1
1792.2.f.c 4 56.h odd 2 1
1792.2.f.g 4 1.a even 1 1 trivial
1792.2.f.g 4 8.b even 2 1 inner
1792.2.f.g 4 28.d even 2 1 inner
1792.2.f.g 4 56.e even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1792, [\chi])$$:

 $$T_{3}^{2} - 2$$ $$T_{5}^{2} + 6$$ $$T_{29}^{2} - 32$$ $$T_{31} - 4$$ $$T_{37}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( -2 + T^{2} )^{2}$$
$5$ $$( 6 + T^{2} )^{2}$$
$7$ $$( 7 - 4 T + T^{2} )^{2}$$
$11$ $$( 24 + T^{2} )^{2}$$
$13$ $$( 6 + T^{2} )^{2}$$
$17$ $$( 48 + T^{2} )^{2}$$
$19$ $$( -18 + T^{2} )^{2}$$
$23$ $$( 48 + T^{2} )^{2}$$
$29$ $$( -32 + T^{2} )^{2}$$
$31$ $$( -4 + T )^{4}$$
$37$ $$T^{4}$$
$41$ $$T^{4}$$
$43$ $$( 24 + T^{2} )^{2}$$
$47$ $$( -12 + T )^{4}$$
$53$ $$( -32 + T^{2} )^{2}$$
$59$ $$( -98 + T^{2} )^{2}$$
$61$ $$( 54 + T^{2} )^{2}$$
$67$ $$( 24 + T^{2} )^{2}$$
$71$ $$( 12 + T^{2} )^{2}$$
$73$ $$( 48 + T^{2} )^{2}$$
$79$ $$( 108 + T^{2} )^{2}$$
$83$ $$( -2 + T^{2} )^{2}$$
$89$ $$( 48 + T^{2} )^{2}$$
$97$ $$( 48 + T^{2} )^{2}$$