# Properties

 Label 1120.2.w.a Level $1120$ Weight $2$ Character orbit 1120.w Analytic conductor $8.943$ Analytic rank $0$ Dimension $8$ CM discriminant -56 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1120 = 2^{5} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1120.w (of order $$4$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.94324502638$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: 8.0.40282095616.8 Defining polynomial: $$x^{8} - 4 x^{6} + 8 x^{4} - 36 x^{2} + 81$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 280) Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{5} - \beta_{6} ) q^{3} + ( \beta_{1} - \beta_{5} - \beta_{6} ) q^{5} -\beta_{2} q^{7} + ( -\beta_{2} + \beta_{3} + \beta_{7} ) q^{9} +O(q^{10})$$ $$q + ( -\beta_{5} - \beta_{6} ) q^{3} + ( \beta_{1} - \beta_{5} - \beta_{6} ) q^{5} -\beta_{2} q^{7} + ( -\beta_{2} + \beta_{3} + \beta_{7} ) q^{9} + ( 2 \beta_{1} - 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{13} + ( -2 - \beta_{2} + 3 \beta_{3} ) q^{15} + ( -2 \beta_{1} + \beta_{4} + 2 \beta_{5} - \beta_{6} ) q^{19} + ( 2 \beta_{1} + \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{21} + ( 3 + 3 \beta_{3} + 2 \beta_{7} ) q^{23} + ( -3 + 3 \beta_{3} - \beta_{7} ) q^{25} + ( \beta_{1} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{27} + ( \beta_{1} - 2 \beta_{4} + 2 \beta_{5} ) q^{35} + ( 3 \beta_{2} - 8 \beta_{3} - 3 \beta_{7} ) q^{39} + ( 2 \beta_{1} + \beta_{4} + 4 \beta_{5} ) q^{45} -7 \beta_{3} q^{49} + ( -1 - 2 \beta_{2} + \beta_{3} ) q^{57} + ( 7 \beta_{1} + 2 \beta_{4} - 7 \beta_{5} - 2 \beta_{6} ) q^{59} + ( -\beta_{1} - 6 \beta_{4} - \beta_{5} - 6 \beta_{6} ) q^{61} + ( -7 - 7 \beta_{3} - \beta_{7} ) q^{63} + ( 4 + 5 \beta_{2} + \beta_{3} - 2 \beta_{7} ) q^{65} + ( 5 \beta_{1} + 7 \beta_{4} - 5 \beta_{5} - 7 \beta_{6} ) q^{69} + ( 4 + 3 \beta_{2} + 3 \beta_{7} ) q^{71} + ( 2 \beta_{1} + \beta_{4} + 4 \beta_{5} + 5 \beta_{6} ) q^{75} + ( \beta_{2} + 12 \beta_{3} - \beta_{7} ) q^{79} + ( -3 + \beta_{2} + \beta_{7} ) q^{81} + ( 3 \beta_{1} - 3 \beta_{4} + 4 \beta_{5} + 4 \beta_{6} ) q^{83} + ( -4 \beta_{1} - 9 \beta_{4} - 4 \beta_{5} - 9 \beta_{6} ) q^{91} + ( 4 - 4 \beta_{2} - 5 \beta_{3} - \beta_{7} ) q^{95} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q - 16q^{15} + 24q^{23} - 24q^{25} - 8q^{57} - 56q^{63} + 32q^{65} + 32q^{71} - 24q^{81} + 32q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4 x^{6} + 8 x^{4} - 36 x^{2} + 81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$2 \nu^{6} + \nu^{4} + 43 \nu^{2} - 81$$$$)/45$$ $$\beta_{3}$$ $$=$$ $$($$$$-2 \nu^{6} - \nu^{4} + 2 \nu^{2} + 36$$$$)/45$$ $$\beta_{4}$$ $$=$$ $$($$$$4 \nu^{7} + 2 \nu^{5} + 41 \nu^{3} - 162 \nu$$$$)/135$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{7} + 4 \nu^{5} - 8 \nu^{3} + 36 \nu$$$$)/27$$ $$\beta_{6}$$ $$=$$ $$($$$$-2 \nu^{7} - \nu^{5} + 2 \nu^{3} + 36 \nu$$$$)/45$$ $$\beta_{7}$$ $$=$$ $$($$$$-7 \nu^{6} + 19 \nu^{4} - 38 \nu^{2} + 171$$$$)/45$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2} + 1$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{6} + 3 \beta_{4} + 2 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{7} - 5 \beta_{3} + 2 \beta_{2}$$ $$\nu^{5}$$ $$=$$ $$-\beta_{6} + 6 \beta_{5} + 6 \beta_{4}$$ $$\nu^{6}$$ $$=$$ $$-\beta_{7} - 19 \beta_{3} + 19$$ $$\nu^{7}$$ $$=$$ $$-20 \beta_{6} - 3 \beta_{5} + 20 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1120\mathbb{Z}\right)^\times$$.

 $$n$$ $$351$$ $$421$$ $$801$$ $$897$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$ $$-\beta_{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
433.1
 1.71331 − 0.254137i −1.03179 − 1.39119i 1.03179 + 1.39119i −1.71331 + 0.254137i 1.71331 + 0.254137i −1.03179 + 1.39119i 1.03179 − 1.39119i −1.71331 − 0.254137i
0 −1.96744 1.96744i 0 −0.254137 2.22158i 0 −1.87083 + 1.87083i 0 4.74166i 0
433.2 0 −0.359404 0.359404i 0 −1.39119 1.75060i 0 1.87083 1.87083i 0 2.74166i 0
433.3 0 0.359404 + 0.359404i 0 1.39119 + 1.75060i 0 1.87083 1.87083i 0 2.74166i 0
433.4 0 1.96744 + 1.96744i 0 0.254137 + 2.22158i 0 −1.87083 + 1.87083i 0 4.74166i 0
657.1 0 −1.96744 + 1.96744i 0 −0.254137 + 2.22158i 0 −1.87083 1.87083i 0 4.74166i 0
657.2 0 −0.359404 + 0.359404i 0 −1.39119 + 1.75060i 0 1.87083 + 1.87083i 0 2.74166i 0
657.3 0 0.359404 0.359404i 0 1.39119 1.75060i 0 1.87083 + 1.87083i 0 2.74166i 0
657.4 0 1.96744 1.96744i 0 0.254137 2.22158i 0 −1.87083 1.87083i 0 4.74166i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 657.4 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 $$\Q(\sqrt{-14})$$
5.c odd 4 1 inner
7.b odd 2 1 inner
8.b even 2 1 inner
35.f even 4 1 inner
40.i odd 4 1 inner
280.s even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1120.2.w.a 8
4.b odd 2 1 280.2.s.a 8
5.c odd 4 1 inner 1120.2.w.a 8
7.b odd 2 1 inner 1120.2.w.a 8
8.b even 2 1 inner 1120.2.w.a 8
8.d odd 2 1 280.2.s.a 8
20.e even 4 1 280.2.s.a 8
28.d even 2 1 280.2.s.a 8
35.f even 4 1 inner 1120.2.w.a 8
40.i odd 4 1 inner 1120.2.w.a 8
40.k even 4 1 280.2.s.a 8
56.e even 2 1 280.2.s.a 8
56.h odd 2 1 CM 1120.2.w.a 8
140.j odd 4 1 280.2.s.a 8
280.s even 4 1 inner 1120.2.w.a 8
280.y odd 4 1 280.2.s.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.s.a 8 4.b odd 2 1
280.2.s.a 8 8.d odd 2 1
280.2.s.a 8 20.e even 4 1
280.2.s.a 8 28.d even 2 1
280.2.s.a 8 40.k even 4 1
280.2.s.a 8 56.e even 2 1
280.2.s.a 8 140.j odd 4 1
280.2.s.a 8 280.y odd 4 1
1120.2.w.a 8 1.a even 1 1 trivial
1120.2.w.a 8 5.c odd 4 1 inner
1120.2.w.a 8 7.b odd 2 1 inner
1120.2.w.a 8 8.b even 2 1 inner
1120.2.w.a 8 35.f even 4 1 inner
1120.2.w.a 8 40.i odd 4 1 inner
1120.2.w.a 8 56.h odd 2 1 CM
1120.2.w.a 8 280.s even 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} + 60 T_{3}^{4} + 4$$ acting on $$S_{2}^{\mathrm{new}}(1120, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$4 + 60 T^{4} + T^{8}$$
$5$ $$625 + 300 T^{2} + 72 T^{4} + 12 T^{6} + T^{8}$$
$7$ $$( 49 + T^{4} )^{2}$$
$11$ $$T^{8}$$
$13$ $$3694084 + 3900 T^{4} + T^{8}$$
$17$ $$T^{8}$$
$19$ $$( 338 + 64 T^{2} + T^{4} )^{2}$$
$23$ $$( 100 + 120 T + 72 T^{2} - 12 T^{3} + T^{4} )^{2}$$
$29$ $$T^{8}$$
$31$ $$T^{8}$$
$37$ $$T^{8}$$
$41$ $$T^{8}$$
$43$ $$T^{8}$$
$47$ $$T^{8}$$
$53$ $$T^{8}$$
$59$ $$( 25538 + 344 T^{2} + T^{4} )^{2}$$
$61$ $$( 8978 - 256 T^{2} + T^{4} )^{2}$$
$67$ $$T^{8}$$
$71$ $$( -110 - 8 T + T^{2} )^{4}$$
$73$ $$T^{8}$$
$79$ $$( 16900 + 316 T^{2} + T^{4} )^{2}$$
$83$ $$416241604 + 94620 T^{4} + T^{8}$$
$89$ $$T^{8}$$
$97$ $$T^{8}$$