# Properties

 Label 2240.2.k.c Level $2240$ Weight $2$ Character orbit 2240.k Analytic conductor $17.886$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$17.8864900528$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.303595776.1 Defining polynomial: $$x^{8} + 5 x^{6} + 16 x^{4} + 45 x^{2} + 81$$ 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.

 $$f(q)$$ $$=$$ $$q -\beta_{6} q^{3} + \beta_{4} q^{5} + ( -\beta_{1} + \beta_{5} + \beta_{6} ) q^{7} -\beta_{7} q^{9} +O(q^{10})$$ $$q -\beta_{6} q^{3} + \beta_{4} q^{5} + ( -\beta_{1} + \beta_{5} + \beta_{6} ) q^{7} -\beta_{7} q^{9} -\beta_{1} q^{11} + ( \beta_{2} + 3 \beta_{4} ) q^{13} -\beta_{1} q^{15} + ( -\beta_{2} - \beta_{4} ) q^{17} + 2 \beta_{3} q^{19} + ( -3 + 2 \beta_{4} + \beta_{7} ) q^{21} + ( -2 \beta_{1} - 2 \beta_{5} ) q^{23} - q^{25} + ( -2 \beta_{3} + \beta_{6} ) q^{27} + ( 3 + \beta_{7} ) q^{29} + ( -4 \beta_{3} + 2 \beta_{6} ) q^{31} + ( -\beta_{2} + 3 \beta_{4} ) q^{33} + ( \beta_{1} - \beta_{3} + \beta_{6} ) q^{35} + ( -4 + 2 \beta_{7} ) q^{37} + ( -\beta_{1} + 2 \beta_{5} ) q^{39} + ( -2 \beta_{2} + 2 \beta_{4} ) q^{41} + 2 \beta_{5} q^{43} -\beta_{2} q^{45} + ( -6 \beta_{3} + \beta_{6} ) q^{47} + ( -1 - 4 \beta_{4} - 2 \beta_{7} ) q^{49} + ( -\beta_{1} - 2 \beta_{5} ) q^{51} + ( -2 - 4 \beta_{7} ) q^{53} + \beta_{6} q^{55} + ( -2 + 2 \beta_{7} ) q^{57} + ( -2 \beta_{3} + 4 \beta_{6} ) q^{59} + ( 2 \beta_{2} - 4 \beta_{4} ) q^{61} + ( \beta_{1} + 2 \beta_{3} - 3 \beta_{5} + 2 \beta_{6} ) q^{63} + ( -3 - \beta_{7} ) q^{65} + ( -4 \beta_{1} + 2 \beta_{5} ) q^{67} + ( -4 \beta_{2} + 8 \beta_{4} ) q^{69} + ( 4 \beta_{1} - 2 \beta_{5} ) q^{71} + ( -2 \beta_{2} + 2 \beta_{4} ) q^{73} + \beta_{6} q^{75} + ( -2 + \beta_{2} - 3 \beta_{4} ) q^{77} + ( -9 \beta_{1} + 4 \beta_{5} ) q^{79} + ( -1 + 2 \beta_{7} ) q^{81} -6 \beta_{3} q^{83} + ( 1 + \beta_{7} ) q^{85} + ( 2 \beta_{3} - \beta_{6} ) q^{87} + ( 4 \beta_{2} + 8 \beta_{4} ) q^{89} + ( \beta_{1} - 6 \beta_{3} - 2 \beta_{5} + 4 \beta_{6} ) q^{91} + ( -2 - 2 \beta_{7} ) q^{93} + 2 \beta_{5} q^{95} + ( 3 \beta_{2} - 5 \beta_{4} ) q^{97} + ( -2 \beta_{1} - 2 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 4q^{9} + O(q^{10})$$ $$8q + 4q^{9} - 28q^{21} - 8q^{25} + 20q^{29} - 40q^{37} - 24q^{57} - 20q^{65} - 16q^{77} - 16q^{81} + 4q^{85} - 8q^{93} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 5 x^{6} + 16 x^{4} + 45 x^{2} + 81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{6} + 4 \nu^{4} + 20 \nu^{2} + 27$$$$)/36$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{7} + 8 \nu^{5} + 40 \nu^{3} + 165 \nu$$$$)/72$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{7} + 4 \nu^{5} + 2 \nu^{3} - 9 \nu$$$$)/54$$ $$\beta_{4}$$ $$=$$ $$($$$$5 \nu^{7} + 16 \nu^{5} + 8 \nu^{3} + 81 \nu$$$$)/216$$ $$\beta_{5}$$ $$=$$ $$($$$$5 \nu^{6} + 16 \nu^{4} + 80 \nu^{2} + 153$$$$)/72$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{7} + 5 \nu^{5} + 16 \nu^{3} + 18 \nu$$$$)/27$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{6} + 5 \nu^{4} + 7 \nu^{2} + 18$$$$)/9$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{6} + \beta_{4} + \beta_{2}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{7} + 2 \beta_{5} + \beta_{1} - 3$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{6} - 4 \beta_{4} - \beta_{3}$$ $$\nu^{4}$$ $$=$$ $$($$$$5 \beta_{7} - 6 \beta_{5} + 5 \beta_{1} - 1$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-\beta_{6} + 15 \beta_{4} + 16 \beta_{3} - \beta_{2}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$8 \beta_{5} - 16 \beta_{1} - 5$$ $$\nu^{7}$$ $$=$$ $$($$$$13 \beta_{6} + 35 \beta_{4} - 48 \beta_{3} - 13 \beta_{2}$$$$)/2$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1791.1
 −1.26217 + 1.18614i −1.26217 − 1.18614i −0.396143 − 1.68614i −0.396143 + 1.68614i 0.396143 − 1.68614i 0.396143 + 1.68614i 1.26217 + 1.18614i 1.26217 − 1.18614i
0 −2.52434 0 1.00000i 0 2.52434 + 0.792287i 0 3.37228 0
1791.2 0 −2.52434 0 1.00000i 0 2.52434 0.792287i 0 3.37228 0
1791.3 0 −0.792287 0 1.00000i 0 0.792287 + 2.52434i 0 −2.37228 0
1791.4 0 −0.792287 0 1.00000i 0 0.792287 2.52434i 0 −2.37228 0
1791.5 0 0.792287 0 1.00000i 0 −0.792287 2.52434i 0 −2.37228 0
1791.6 0 0.792287 0 1.00000i 0 −0.792287 + 2.52434i 0 −2.37228 0
1791.7 0 2.52434 0 1.00000i 0 −2.52434 0.792287i 0 3.37228 0
1791.8 0 2.52434 0 1.00000i 0 −2.52434 + 0.792287i 0 3.37228 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1791.8 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
7.b odd 2 1 inner
28.d 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.k.c 8
4.b odd 2 1 inner 2240.2.k.c 8
7.b odd 2 1 inner 2240.2.k.c 8
8.b even 2 1 560.2.k.a 8
8.d odd 2 1 560.2.k.a 8
24.f even 2 1 5040.2.d.e 8
24.h odd 2 1 5040.2.d.e 8
28.d even 2 1 inner 2240.2.k.c 8
40.e odd 2 1 2800.2.k.l 8
40.f even 2 1 2800.2.k.l 8
40.i odd 4 1 2800.2.e.i 8
40.i odd 4 1 2800.2.e.j 8
40.k even 4 1 2800.2.e.i 8
40.k even 4 1 2800.2.e.j 8
56.e even 2 1 560.2.k.a 8
56.h odd 2 1 560.2.k.a 8
168.e odd 2 1 5040.2.d.e 8
168.i even 2 1 5040.2.d.e 8
280.c odd 2 1 2800.2.k.l 8
280.n even 2 1 2800.2.k.l 8
280.s even 4 1 2800.2.e.i 8
280.s even 4 1 2800.2.e.j 8
280.y odd 4 1 2800.2.e.i 8
280.y odd 4 1 2800.2.e.j 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
560.2.k.a 8 8.b even 2 1
560.2.k.a 8 8.d odd 2 1
560.2.k.a 8 56.e even 2 1
560.2.k.a 8 56.h odd 2 1
2240.2.k.c 8 1.a even 1 1 trivial
2240.2.k.c 8 4.b odd 2 1 inner
2240.2.k.c 8 7.b odd 2 1 inner
2240.2.k.c 8 28.d even 2 1 inner
2800.2.e.i 8 40.i odd 4 1
2800.2.e.i 8 40.k even 4 1
2800.2.e.i 8 280.s even 4 1
2800.2.e.i 8 280.y odd 4 1
2800.2.e.j 8 40.i odd 4 1
2800.2.e.j 8 40.k even 4 1
2800.2.e.j 8 280.s even 4 1
2800.2.e.j 8 280.y odd 4 1
2800.2.k.l 8 40.e odd 2 1
2800.2.k.l 8 40.f even 2 1
2800.2.k.l 8 280.c odd 2 1
2800.2.k.l 8 280.n even 2 1
5040.2.d.e 8 24.f even 2 1
5040.2.d.e 8 24.h odd 2 1
5040.2.d.e 8 168.e odd 2 1
5040.2.d.e 8 168.i even 2 1

## Hecke kernels

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

 $$T_{3}^{4} - 7 T_{3}^{2} + 4$$ $$T_{19}^{2} - 12$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( 4 - 7 T^{2} + T^{4} )^{2}$$
$5$ $$( 1 + T^{2} )^{4}$$
$7$ $$2401 - 34 T^{4} + T^{8}$$
$11$ $$( 4 + 7 T^{2} + T^{4} )^{2}$$
$13$ $$( 4 + 29 T^{2} + T^{4} )^{2}$$
$17$ $$( 64 + 17 T^{2} + T^{4} )^{2}$$
$19$ $$( -12 + T^{2} )^{4}$$
$23$ $$( 256 + 76 T^{2} + T^{4} )^{2}$$
$29$ $$( -2 - 5 T + T^{2} )^{4}$$
$31$ $$( 256 - 76 T^{2} + T^{4} )^{2}$$
$37$ $$( -8 + 10 T + T^{2} )^{4}$$
$41$ $$( 576 + 84 T^{2} + T^{4} )^{2}$$
$43$ $$( 12 + T^{2} )^{4}$$
$47$ $$( 7744 - 187 T^{2} + T^{4} )^{2}$$
$53$ $$( -132 + T^{2} )^{4}$$
$59$ $$( -44 + T^{2} )^{4}$$
$61$ $$( 64 + 116 T^{2} + T^{4} )^{2}$$
$67$ $$( 44 + T^{2} )^{4}$$
$71$ $$( 44 + T^{2} )^{4}$$
$73$ $$( 576 + 84 T^{2} + T^{4} )^{2}$$
$79$ $$( 49284 + 447 T^{2} + T^{4} )^{2}$$
$83$ $$( -108 + T^{2} )^{4}$$
$89$ $$( 9216 + 336 T^{2} + T^{4} )^{2}$$
$97$ $$( 1024 + 233 T^{2} + T^{4} )^{2}$$