# Properties

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

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$14.3091920422$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 56) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{5} - q^{7} + 3 q^{9} +O(q^{10})$$ q - b * q^5 - q^7 + 3 * q^9 $$q - \beta q^{5} - q^{7} + 3 q^{9} - 2 \beta q^{11} + \beta q^{13} - 6 q^{17} - 4 \beta q^{19} + q^{25} + 3 \beta q^{29} - 8 q^{31} + \beta q^{35} + \beta q^{37} - 2 q^{41} - 2 \beta q^{43} - 3 \beta q^{45} + 8 q^{47} + q^{49} - 3 \beta q^{53} - 8 q^{55} - 3 \beta q^{61} - 3 q^{63} + 4 q^{65} + 2 \beta q^{67} - 8 q^{71} - 10 q^{73} + 2 \beta q^{77} - 16 q^{79} + 9 q^{81} - 4 \beta q^{83} + 6 \beta q^{85} + 6 q^{89} - \beta q^{91} - 16 q^{95} - 6 q^{97} - 6 \beta q^{99} +O(q^{100})$$ q - b * q^5 - q^7 + 3 * q^9 - 2*b * q^11 + b * q^13 - 6 * q^17 - 4*b * q^19 + q^25 + 3*b * q^29 - 8 * q^31 + b * q^35 + b * q^37 - 2 * q^41 - 2*b * q^43 - 3*b * q^45 + 8 * q^47 + q^49 - 3*b * q^53 - 8 * q^55 - 3*b * q^61 - 3 * q^63 + 4 * q^65 + 2*b * q^67 - 8 * q^71 - 10 * q^73 + 2*b * q^77 - 16 * q^79 + 9 * q^81 - 4*b * q^83 + 6*b * q^85 + 6 * q^89 - b * q^91 - 16 * q^95 - 6 * q^97 - 6*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{7} + 6 q^{9}+O(q^{10})$$ 2 * q - 2 * q^7 + 6 * q^9 $$2 q - 2 q^{7} + 6 q^{9} - 12 q^{17} + 2 q^{25} - 16 q^{31} - 4 q^{41} + 16 q^{47} + 2 q^{49} - 16 q^{55} - 6 q^{63} + 8 q^{65} - 16 q^{71} - 20 q^{73} - 32 q^{79} + 18 q^{81} + 12 q^{89} - 32 q^{95} - 12 q^{97}+O(q^{100})$$ 2 * q - 2 * q^7 + 6 * q^9 - 12 * q^17 + 2 * q^25 - 16 * q^31 - 4 * q^41 + 16 * q^47 + 2 * q^49 - 16 * q^55 - 6 * q^63 + 8 * q^65 - 16 * q^71 - 20 * q^73 - 32 * q^79 + 18 * q^81 + 12 * q^89 - 32 * q^95 - 12 * q^97

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
897.1
 1.00000i − 1.00000i
0 0 0 2.00000i 0 −1.00000 0 3.00000 0
897.2 0 0 0 2.00000i 0 −1.00000 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
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 1792.2.b.d 2
4.b odd 2 1 1792.2.b.i 2
8.b even 2 1 inner 1792.2.b.d 2
8.d odd 2 1 1792.2.b.i 2
16.e even 4 1 112.2.a.b 1
16.e even 4 1 448.2.a.e 1
16.f odd 4 1 56.2.a.a 1
16.f odd 4 1 448.2.a.d 1
48.i odd 4 1 1008.2.a.d 1
48.i odd 4 1 4032.2.a.bk 1
48.k even 4 1 504.2.a.c 1
48.k even 4 1 4032.2.a.bb 1
80.i odd 4 1 2800.2.g.p 2
80.j even 4 1 1400.2.g.g 2
80.k odd 4 1 1400.2.a.g 1
80.q even 4 1 2800.2.a.p 1
80.s even 4 1 1400.2.g.g 2
80.t odd 4 1 2800.2.g.p 2
112.j even 4 1 392.2.a.d 1
112.j even 4 1 3136.2.a.q 1
112.l odd 4 1 784.2.a.e 1
112.l odd 4 1 3136.2.a.p 1
112.u odd 12 2 392.2.i.c 2
112.v even 12 2 392.2.i.d 2
112.w even 12 2 784.2.i.e 2
112.x odd 12 2 784.2.i.g 2
176.i even 4 1 6776.2.a.g 1
208.o odd 4 1 9464.2.a.c 1
336.v odd 4 1 3528.2.a.x 1
336.y even 4 1 7056.2.a.bo 1
336.br odd 12 2 3528.2.s.e 2
336.bu even 12 2 3528.2.s.t 2
560.be even 4 1 9800.2.a.u 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.a.a 1 16.f odd 4 1
112.2.a.b 1 16.e even 4 1
392.2.a.d 1 112.j even 4 1
392.2.i.c 2 112.u odd 12 2
392.2.i.d 2 112.v even 12 2
448.2.a.d 1 16.f odd 4 1
448.2.a.e 1 16.e even 4 1
504.2.a.c 1 48.k even 4 1
784.2.a.e 1 112.l odd 4 1
784.2.i.e 2 112.w even 12 2
784.2.i.g 2 112.x odd 12 2
1008.2.a.d 1 48.i odd 4 1
1400.2.a.g 1 80.k odd 4 1
1400.2.g.g 2 80.j even 4 1
1400.2.g.g 2 80.s even 4 1
1792.2.b.d 2 1.a even 1 1 trivial
1792.2.b.d 2 8.b even 2 1 inner
1792.2.b.i 2 4.b odd 2 1
1792.2.b.i 2 8.d odd 2 1
2800.2.a.p 1 80.q even 4 1
2800.2.g.p 2 80.i odd 4 1
2800.2.g.p 2 80.t odd 4 1
3136.2.a.p 1 112.l odd 4 1
3136.2.a.q 1 112.j even 4 1
3528.2.a.x 1 336.v odd 4 1
3528.2.s.e 2 336.br odd 12 2
3528.2.s.t 2 336.bu even 12 2
4032.2.a.bb 1 48.k even 4 1
4032.2.a.bk 1 48.i odd 4 1
6776.2.a.g 1 176.i even 4 1
7056.2.a.bo 1 336.y even 4 1
9464.2.a.c 1 208.o odd 4 1
9800.2.a.u 1 560.be even 4 1

## Hecke kernels

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

 $$T_{3}$$ T3 $$T_{5}^{2} + 4$$ T5^2 + 4 $$T_{11}^{2} + 16$$ T11^2 + 16 $$T_{23}$$ T23 $$T_{31} + 8$$ T31 + 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 4$$
$7$ $$(T + 1)^{2}$$
$11$ $$T^{2} + 16$$
$13$ $$T^{2} + 4$$
$17$ $$(T + 6)^{2}$$
$19$ $$T^{2} + 64$$
$23$ $$T^{2}$$
$29$ $$T^{2} + 36$$
$31$ $$(T + 8)^{2}$$
$37$ $$T^{2} + 4$$
$41$ $$(T + 2)^{2}$$
$43$ $$T^{2} + 16$$
$47$ $$(T - 8)^{2}$$
$53$ $$T^{2} + 36$$
$59$ $$T^{2}$$
$61$ $$T^{2} + 36$$
$67$ $$T^{2} + 16$$
$71$ $$(T + 8)^{2}$$
$73$ $$(T + 10)^{2}$$
$79$ $$(T + 16)^{2}$$
$83$ $$T^{2} + 64$$
$89$ $$(T - 6)^{2}$$
$97$ $$(T + 6)^{2}$$