# Properties

 Label 1792.2.e.b Level $1792$ Weight $2$ Character orbit 1792.e Analytic conductor $14.309$ Analytic rank $0$ Dimension $4$ CM discriminant -7 Inner twists $8$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$14.3091920422$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{7})$$ Defining polynomial: $$x^{4} - 3 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{29}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 28) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{3} q^{7} + 3 q^{9} +O(q^{10})$$ $$q -\beta_{3} q^{7} + 3 q^{9} -\beta_{2} q^{11} -2 \beta_{3} q^{23} -5 q^{25} -\beta_{1} q^{29} -3 \beta_{1} q^{37} -\beta_{2} q^{43} -7 q^{49} + 5 \beta_{1} q^{53} -3 \beta_{3} q^{63} -3 \beta_{2} q^{67} -2 \beta_{3} q^{71} + 7 \beta_{1} q^{77} -6 \beta_{3} q^{79} + 9 q^{81} -3 \beta_{2} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 12q^{9} + O(q^{10})$$ $$4q + 12q^{9} - 20q^{25} - 28q^{49} + 36q^{81} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 3 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{3} - \nu$$ $$\beta_{2}$$ $$=$$ $$-\nu^{3} + 5 \nu$$ $$\beta_{3}$$ $$=$$ $$2 \nu^{2} - 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + \beta_{1}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + 3$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{2} + 5 \beta_{1}$$$$)/4$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
895.1
 1.32288 + 0.500000i −1.32288 − 0.500000i 1.32288 − 0.500000i −1.32288 + 0.500000i
0 0 0 0 0 2.64575i 0 3.00000 0
895.2 0 0 0 0 0 2.64575i 0 3.00000 0
895.3 0 0 0 0 0 2.64575i 0 3.00000 0
895.4 0 0 0 0 0 2.64575i 0 3.00000 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
7.b odd 2 1 CM by $$\Q(\sqrt{-7})$$
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
28.d even 2 1 inner
56.e even 2 1 inner
56.h odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1792.2.e.b 4
4.b odd 2 1 inner 1792.2.e.b 4
7.b odd 2 1 CM 1792.2.e.b 4
8.b even 2 1 inner 1792.2.e.b 4
8.d odd 2 1 inner 1792.2.e.b 4
16.e even 4 1 28.2.d.a 2
16.e even 4 1 448.2.f.b 2
16.f odd 4 1 28.2.d.a 2
16.f odd 4 1 448.2.f.b 2
28.d even 2 1 inner 1792.2.e.b 4
48.i odd 4 1 252.2.b.a 2
48.i odd 4 1 4032.2.b.e 2
48.k even 4 1 252.2.b.a 2
48.k even 4 1 4032.2.b.e 2
56.e even 2 1 inner 1792.2.e.b 4
56.h odd 2 1 inner 1792.2.e.b 4
80.i odd 4 1 700.2.c.d 4
80.j even 4 1 700.2.c.d 4
80.k odd 4 1 700.2.g.a 2
80.q even 4 1 700.2.g.a 2
80.s even 4 1 700.2.c.d 4
80.t odd 4 1 700.2.c.d 4
112.j even 4 1 28.2.d.a 2
112.j even 4 1 448.2.f.b 2
112.l odd 4 1 28.2.d.a 2
112.l odd 4 1 448.2.f.b 2
112.u odd 12 2 196.2.f.b 4
112.v even 12 2 196.2.f.b 4
112.w even 12 2 196.2.f.b 4
112.x odd 12 2 196.2.f.b 4
336.v odd 4 1 252.2.b.a 2
336.v odd 4 1 4032.2.b.e 2
336.y even 4 1 252.2.b.a 2
336.y even 4 1 4032.2.b.e 2
560.r even 4 1 700.2.c.d 4
560.u odd 4 1 700.2.c.d 4
560.be even 4 1 700.2.g.a 2
560.bf odd 4 1 700.2.g.a 2
560.bm odd 4 1 700.2.c.d 4
560.bn even 4 1 700.2.c.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.2.d.a 2 16.e even 4 1
28.2.d.a 2 16.f odd 4 1
28.2.d.a 2 112.j even 4 1
28.2.d.a 2 112.l odd 4 1
196.2.f.b 4 112.u odd 12 2
196.2.f.b 4 112.v even 12 2
196.2.f.b 4 112.w even 12 2
196.2.f.b 4 112.x odd 12 2
252.2.b.a 2 48.i odd 4 1
252.2.b.a 2 48.k even 4 1
252.2.b.a 2 336.v odd 4 1
252.2.b.a 2 336.y even 4 1
448.2.f.b 2 16.e even 4 1
448.2.f.b 2 16.f odd 4 1
448.2.f.b 2 112.j even 4 1
448.2.f.b 2 112.l odd 4 1
700.2.c.d 4 80.i odd 4 1
700.2.c.d 4 80.j even 4 1
700.2.c.d 4 80.s even 4 1
700.2.c.d 4 80.t odd 4 1
700.2.c.d 4 560.r even 4 1
700.2.c.d 4 560.u odd 4 1
700.2.c.d 4 560.bm odd 4 1
700.2.c.d 4 560.bn even 4 1
700.2.g.a 2 80.k odd 4 1
700.2.g.a 2 80.q even 4 1
700.2.g.a 2 560.be even 4 1
700.2.g.a 2 560.bf odd 4 1
1792.2.e.b 4 1.a even 1 1 trivial
1792.2.e.b 4 4.b odd 2 1 inner
1792.2.e.b 4 7.b odd 2 1 CM
1792.2.e.b 4 8.b even 2 1 inner
1792.2.e.b 4 8.d odd 2 1 inner
1792.2.e.b 4 28.d even 2 1 inner
1792.2.e.b 4 56.e even 2 1 inner
1792.2.e.b 4 56.h odd 2 1 inner
4032.2.b.e 2 48.i odd 4 1
4032.2.b.e 2 48.k even 4 1
4032.2.b.e 2 336.v odd 4 1
4032.2.b.e 2 336.y even 4 1

## Hecke kernels

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

 $$T_{3}$$ $$T_{11}^{2} - 28$$ $$T_{31}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$( 7 + T^{2} )^{2}$$
$11$ $$( -28 + T^{2} )^{2}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$( 28 + T^{2} )^{2}$$
$29$ $$( 4 + T^{2} )^{2}$$
$31$ $$T^{4}$$
$37$ $$( 36 + T^{2} )^{2}$$
$41$ $$T^{4}$$
$43$ $$( -28 + T^{2} )^{2}$$
$47$ $$T^{4}$$
$53$ $$( 100 + T^{2} )^{2}$$
$59$ $$T^{4}$$
$61$ $$T^{4}$$
$67$ $$( -252 + T^{2} )^{2}$$
$71$ $$( 28 + T^{2} )^{2}$$
$73$ $$T^{4}$$
$79$ $$( 252 + T^{2} )^{2}$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$T^{4}$$