# Properties

 Label 2240.2.e.e Level $2240$ Weight $2$ Character orbit 2240.e Analytic conductor $17.886$ Analytic rank $0$ Dimension $8$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2240 = 2^{6} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2240.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$17.8864900528$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.12745506816.5 Defining polynomial: $$x^{8} - 8 x^{6} + 55 x^{4} - 72 x^{2} + 81$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 560) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\beta_{5} q^{3} + \beta_{4} q^{5} -\beta_{1} q^{7} + q^{9} +O(q^{10})$$ $$q -\beta_{5} q^{3} + \beta_{4} q^{5} -\beta_{1} q^{7} + q^{9} + 2 \beta_{3} q^{11} + ( \beta_{4} - \beta_{6} ) q^{13} + ( \beta_{1} + \beta_{3} ) q^{15} + ( -2 \beta_{4} + 2 \beta_{6} ) q^{17} + ( -2 \beta_{2} - \beta_{5} ) q^{19} + ( \beta_{4} + \beta_{6} ) q^{21} + ( -2 - \beta_{7} ) q^{25} -4 \beta_{5} q^{27} + 6 q^{29} + ( -2 \beta_{4} + 2 \beta_{6} ) q^{33} + ( \beta_{2} - 3 \beta_{5} ) q^{35} -2 \beta_{7} q^{37} + 2 \beta_{3} q^{39} + ( -2 \beta_{4} - 2 \beta_{6} ) q^{41} -2 \beta_{1} q^{43} + \beta_{4} q^{45} -2 \beta_{5} q^{47} + 7 q^{49} -4 \beta_{3} q^{51} + 2 \beta_{7} q^{53} + ( -2 \beta_{2} - 4 \beta_{5} ) q^{55} + 2 \beta_{7} q^{57} + ( -2 \beta_{2} - \beta_{5} ) q^{59} + ( -3 \beta_{4} - 3 \beta_{6} ) q^{61} -\beta_{1} q^{63} + ( 3 - \beta_{7} ) q^{65} + 2 \beta_{1} q^{67} + ( 4 \beta_{4} - 4 \beta_{6} ) q^{73} + ( -2 \beta_{2} + \beta_{5} ) q^{75} -2 \beta_{7} q^{77} -4 \beta_{3} q^{79} -5 q^{81} -7 \beta_{5} q^{83} + ( -6 + 2 \beta_{7} ) q^{85} -6 \beta_{5} q^{87} + ( 2 \beta_{4} + 2 \beta_{6} ) q^{89} + ( 2 \beta_{2} + \beta_{5} ) q^{91} + ( 3 \beta_{1} - 7 \beta_{3} ) q^{95} + ( 6 \beta_{4} - 6 \beta_{6} ) q^{97} + 2 \beta_{3} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 8q^{9} + O(q^{10})$$ $$8q + 8q^{9} - 16q^{25} + 48q^{29} + 56q^{49} + 24q^{65} - 40q^{81} - 48q^{85} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 8 x^{6} + 55 x^{4} - 72 x^{2} + 81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{6} - 148$$$$)/55$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{7} + 11 \nu^{5} - 88 \nu^{3} + 336 \nu$$$$)/99$$ $$\beta_{3}$$ $$=$$ $$($$$$16 \nu^{6} - 110 \nu^{4} + 880 \nu^{2} - 657$$$$)/495$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{7} - 5 \nu^{5} + 40 \nu^{3} + 48 \nu$$$$)/45$$ $$\beta_{5}$$ $$=$$ $$($$$$-8 \nu^{7} + 55 \nu^{5} - 341 \nu^{3} + 81 \nu$$$$)/297$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{7} - 8 \nu^{5} + 55 \nu^{3} - 45 \nu$$$$)/27$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{6} + 8 \nu^{4} - 46 \nu^{2} + 36$$$$)/9$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{6} + \beta_{5} + \beta_{2}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{7} + 4 \beta_{3} + \beta_{1} + 4$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$5 \beta_{6} + 11 \beta_{5} + 5 \beta_{4}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$8 \beta_{7} + 23 \beta_{3} - 8 \beta_{1} - 23$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-24 \beta_{6} + 24 \beta_{5} + 55 \beta_{4} - 31 \beta_{2}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$-55 \beta_{1} - 148$$ $$\nu^{7}$$ $$=$$ $$($$$$-368 \beta_{6} - 368 \beta_{5} + 165 \beta_{4} - 203 \beta_{2}$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times$$.

 $$n$$ $$897$$ $$1471$$ $$1541$$ $$1921$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2239.1
 −1.00781 − 0.581861i 2.23256 + 1.28897i 1.00781 − 0.581861i −2.23256 + 1.28897i 2.23256 − 1.28897i −1.00781 + 0.581861i −2.23256 − 1.28897i 1.00781 + 0.581861i
0 1.41421i 0 −1.22474 1.87083i 0 2.64575 0 1.00000 0
2239.2 0 1.41421i 0 −1.22474 + 1.87083i 0 −2.64575 0 1.00000 0
2239.3 0 1.41421i 0 1.22474 1.87083i 0 2.64575 0 1.00000 0
2239.4 0 1.41421i 0 1.22474 + 1.87083i 0 −2.64575 0 1.00000 0
2239.5 0 1.41421i 0 −1.22474 1.87083i 0 −2.64575 0 1.00000 0
2239.6 0 1.41421i 0 −1.22474 + 1.87083i 0 2.64575 0 1.00000 0
2239.7 0 1.41421i 0 1.22474 1.87083i 0 −2.64575 0 1.00000 0
2239.8 0 1.41421i 0 1.22474 + 1.87083i 0 2.64575 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2239.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
7.b odd 2 1 inner
20.d odd 2 1 inner
28.d even 2 1 inner
35.c odd 2 1 inner
140.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2240.2.e.e 8
4.b odd 2 1 inner 2240.2.e.e 8
5.b even 2 1 inner 2240.2.e.e 8
7.b odd 2 1 inner 2240.2.e.e 8
8.b even 2 1 560.2.e.d 8
8.d odd 2 1 560.2.e.d 8
20.d odd 2 1 inner 2240.2.e.e 8
28.d even 2 1 inner 2240.2.e.e 8
35.c odd 2 1 inner 2240.2.e.e 8
40.e odd 2 1 560.2.e.d 8
40.f even 2 1 560.2.e.d 8
40.i odd 4 2 2800.2.k.k 8
40.k even 4 2 2800.2.k.k 8
56.e even 2 1 560.2.e.d 8
56.h odd 2 1 560.2.e.d 8
140.c even 2 1 inner 2240.2.e.e 8
280.c odd 2 1 560.2.e.d 8
280.n even 2 1 560.2.e.d 8
280.s even 4 2 2800.2.k.k 8
280.y odd 4 2 2800.2.k.k 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
560.2.e.d 8 8.b even 2 1
560.2.e.d 8 8.d odd 2 1
560.2.e.d 8 40.e odd 2 1
560.2.e.d 8 40.f even 2 1
560.2.e.d 8 56.e even 2 1
560.2.e.d 8 56.h odd 2 1
560.2.e.d 8 280.c odd 2 1
560.2.e.d 8 280.n even 2 1
2240.2.e.e 8 1.a even 1 1 trivial
2240.2.e.e 8 4.b odd 2 1 inner
2240.2.e.e 8 5.b even 2 1 inner
2240.2.e.e 8 7.b odd 2 1 inner
2240.2.e.e 8 20.d odd 2 1 inner
2240.2.e.e 8 28.d even 2 1 inner
2240.2.e.e 8 35.c odd 2 1 inner
2240.2.e.e 8 140.c even 2 1 inner
2800.2.k.k 8 40.i odd 4 2
2800.2.k.k 8 40.k even 4 2
2800.2.k.k 8 280.s even 4 2
2800.2.k.k 8 280.y odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2240, [\chi])$$:

 $$T_{3}^{2} + 2$$ $$T_{11}^{2} + 12$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( 2 + T^{2} )^{4}$$
$5$ $$( 25 + 4 T^{2} + T^{4} )^{2}$$
$7$ $$( -7 + T^{2} )^{4}$$
$11$ $$( 12 + T^{2} )^{4}$$
$13$ $$( -6 + T^{2} )^{4}$$
$17$ $$( -24 + T^{2} )^{4}$$
$19$ $$( -42 + T^{2} )^{4}$$
$23$ $$T^{8}$$
$29$ $$( -6 + T )^{8}$$
$31$ $$T^{8}$$
$37$ $$( 84 + T^{2} )^{4}$$
$41$ $$( 56 + T^{2} )^{4}$$
$43$ $$( -28 + T^{2} )^{4}$$
$47$ $$( 8 + T^{2} )^{4}$$
$53$ $$( 84 + T^{2} )^{4}$$
$59$ $$( -42 + T^{2} )^{4}$$
$61$ $$( 126 + T^{2} )^{4}$$
$67$ $$( -28 + T^{2} )^{4}$$
$71$ $$T^{8}$$
$73$ $$( -96 + T^{2} )^{4}$$
$79$ $$( 48 + T^{2} )^{4}$$
$83$ $$( 98 + T^{2} )^{4}$$
$89$ $$( 56 + T^{2} )^{4}$$
$97$ $$( -216 + T^{2} )^{4}$$