# Properties

 Label 2240.2.k.e 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: 8.0.342102016.5 Defining polynomial: $$x^{8} + x^{6} + 4 x^{4} + 4 x^{2} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 140) 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.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{2} - \beta_{5} ) q^{3} + \beta_{1} q^{5} + ( -\beta_{2} - \beta_{3} - \beta_{5} ) q^{7} + ( 3 - \beta_{1} + \beta_{6} - \beta_{7} ) q^{9} +O(q^{10})$$ $$q + ( -\beta_{2} - \beta_{5} ) q^{3} + \beta_{1} q^{5} + ( -\beta_{2} - \beta_{3} - \beta_{5} ) q^{7} + ( 3 - \beta_{1} + \beta_{6} - \beta_{7} ) q^{9} + ( \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{11} + 2 \beta_{1} q^{13} + ( -\beta_{3} - \beta_{4} ) q^{15} + ( 3 \beta_{1} - \beta_{6} - \beta_{7} ) q^{17} + ( \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{19} + ( 5 + 2 \beta_{1} + \beta_{6} ) q^{21} + ( -2 \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{23} - q^{25} + ( -2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{27} + 2 q^{29} + ( 2 \beta_{3} - 2 \beta_{4} ) q^{31} + ( 6 \beta_{1} + 2 \beta_{6} + 2 \beta_{7} ) q^{33} + ( -\beta_{3} - \beta_{4} + \beta_{5} ) q^{35} -2 q^{37} + ( -2 \beta_{3} - 2 \beta_{4} ) q^{39} + ( -3 \beta_{1} + \beta_{6} + \beta_{7} ) q^{41} + ( -2 \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{43} + ( 2 \beta_{1} + \beta_{6} + \beta_{7} ) q^{45} + ( -3 \beta_{3} + 3 \beta_{4} ) q^{47} + ( 4 + 5 \beta_{1} + \beta_{7} ) q^{49} + ( -2 \beta_{2} + 2 \beta_{5} ) q^{51} -2 q^{53} + ( \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{55} + ( -8 + 2 \beta_{1} - 2 \beta_{6} + 2 \beta_{7} ) q^{57} + ( -\beta_{2} - 3 \beta_{3} + 3 \beta_{4} - \beta_{5} ) q^{59} + 2 \beta_{1} q^{61} + ( -2 \beta_{2} - 5 \beta_{4} - 4 \beta_{5} ) q^{63} -2 q^{65} + ( -4 \beta_{2} - \beta_{3} - \beta_{4} + 4 \beta_{5} ) q^{67} + ( -7 \beta_{1} - 3 \beta_{6} - 3 \beta_{7} ) q^{69} + ( -\beta_{2} - 3 \beta_{3} - 3 \beta_{4} + \beta_{5} ) q^{71} + ( -3 \beta_{1} - 3 \beta_{6} - 3 \beta_{7} ) q^{73} + ( \beta_{2} + \beta_{5} ) q^{75} + ( -4 + 3 \beta_{1} + \beta_{6} + 3 \beta_{7} ) q^{77} + ( \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{79} + ( 7 - \beta_{1} + \beta_{6} - \beta_{7} ) q^{81} + ( -\beta_{2} - \beta_{5} ) q^{83} + ( -2 - \beta_{1} + \beta_{6} - \beta_{7} ) q^{85} + ( -2 \beta_{2} - 2 \beta_{5} ) q^{87} -12 \beta_{1} q^{89} + ( -2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{91} + ( 4 - 2 \beta_{1} + 2 \beta_{6} - 2 \beta_{7} ) q^{93} + ( -\beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{95} + ( -3 \beta_{1} + \beta_{6} + \beta_{7} ) q^{97} + ( \beta_{2} - 9 \beta_{3} - 9 \beta_{4} - \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 16q^{9} + O(q^{10})$$ $$8q + 16q^{9} + 36q^{21} - 8q^{25} + 16q^{29} - 16q^{37} + 36q^{49} - 16q^{53} - 48q^{57} - 16q^{65} - 24q^{77} + 48q^{81} - 24q^{85} + 16q^{93} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{7} + \nu^{5} + 2 \nu^{3}$$$$)/16$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{7} + \nu^{6} - 3 \nu^{5} - 5 \nu^{4} - 2 \nu^{3} + 2 \nu^{2} - 8$$$$)/16$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{7} + \nu^{6} - \nu^{5} + 3 \nu^{4} + 6 \nu^{3} + 10 \nu^{2} - 8 \nu + 8$$$$)/16$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{7} + \nu^{6} + \nu^{5} + 3 \nu^{4} - 6 \nu^{3} + 10 \nu^{2} + 8 \nu + 8$$$$)/16$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{7} - \nu^{6} - 3 \nu^{5} + 5 \nu^{4} - 2 \nu^{3} - 2 \nu^{2} + 8$$$$)/16$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{7} - 4 \nu^{6} + 3 \nu^{5} - 4 \nu^{4} + 10 \nu^{3} + 24 \nu - 16$$$$)/16$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{7} + 2 \nu^{6} + \nu^{5} + 2 \nu^{4} + 4 \nu^{3} + 12 \nu + 8$$$$)/8$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - \beta_{1}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{7} + \beta_{6} - \beta_{5} + 3 \beta_{4} + 3 \beta_{3} + \beta_{2} - \beta_{1}$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{7} + \beta_{6} + \beta_{5} - 3 \beta_{4} + 3 \beta_{3} + \beta_{2} + 7 \beta_{1}$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$\beta_{7} - \beta_{6} + 5 \beta_{5} + \beta_{4} + \beta_{3} - 5 \beta_{2} + \beta_{1} - 8$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$-\beta_{7} - \beta_{6} - 9 \beta_{5} + 3 \beta_{4} - 3 \beta_{3} - 9 \beta_{2} + 9 \beta_{1}$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$7 \beta_{7} - 7 \beta_{6} - 5 \beta_{5} - \beta_{4} - \beta_{3} + 5 \beta_{2} + 7 \beta_{1} - 8$$$$)/4$$ $$\nu^{7}$$ $$=$$ $$($$$$\beta_{7} + \beta_{6} - 7 \beta_{5} - 3 \beta_{4} + 3 \beta_{3} - 7 \beta_{2} - 41 \beta_{1}$$$$)/4$$

## 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
 1.17915 − 0.780776i 1.17915 + 0.780776i −0.599676 + 1.28078i −0.599676 − 1.28078i 0.599676 + 1.28078i 0.599676 − 1.28078i −1.17915 − 0.780776i −1.17915 + 0.780776i
0 −3.02045 0 1.00000i 0 −2.17238 + 1.51022i 0 6.12311 0
1791.2 0 −3.02045 0 1.00000i 0 −2.17238 1.51022i 0 6.12311 0
1791.3 0 −0.936426 0 1.00000i 0 −2.60399 + 0.468213i 0 −2.12311 0
1791.4 0 −0.936426 0 1.00000i 0 −2.60399 0.468213i 0 −2.12311 0
1791.5 0 0.936426 0 1.00000i 0 2.60399 0.468213i 0 −2.12311 0
1791.6 0 0.936426 0 1.00000i 0 2.60399 + 0.468213i 0 −2.12311 0
1791.7 0 3.02045 0 1.00000i 0 2.17238 1.51022i 0 6.12311 0
1791.8 0 3.02045 0 1.00000i 0 2.17238 + 1.51022i 0 6.12311 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.e 8
4.b odd 2 1 inner 2240.2.k.e 8
7.b odd 2 1 inner 2240.2.k.e 8
8.b even 2 1 140.2.g.c 8
8.d odd 2 1 140.2.g.c 8
24.f even 2 1 1260.2.c.c 8
24.h odd 2 1 1260.2.c.c 8
28.d even 2 1 inner 2240.2.k.e 8
40.e odd 2 1 700.2.g.j 8
40.f even 2 1 700.2.g.j 8
40.i odd 4 1 700.2.c.i 8
40.i odd 4 1 700.2.c.j 8
40.k even 4 1 700.2.c.i 8
40.k even 4 1 700.2.c.j 8
56.e even 2 1 140.2.g.c 8
56.h odd 2 1 140.2.g.c 8
56.j odd 6 2 980.2.o.e 16
56.k odd 6 2 980.2.o.e 16
56.m even 6 2 980.2.o.e 16
56.p even 6 2 980.2.o.e 16
168.e odd 2 1 1260.2.c.c 8
168.i even 2 1 1260.2.c.c 8
280.c odd 2 1 700.2.g.j 8
280.n even 2 1 700.2.g.j 8
280.s even 4 1 700.2.c.i 8
280.s even 4 1 700.2.c.j 8
280.y odd 4 1 700.2.c.i 8
280.y odd 4 1 700.2.c.j 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.g.c 8 8.b even 2 1
140.2.g.c 8 8.d odd 2 1
140.2.g.c 8 56.e even 2 1
140.2.g.c 8 56.h odd 2 1
700.2.c.i 8 40.i odd 4 1
700.2.c.i 8 40.k even 4 1
700.2.c.i 8 280.s even 4 1
700.2.c.i 8 280.y odd 4 1
700.2.c.j 8 40.i odd 4 1
700.2.c.j 8 40.k even 4 1
700.2.c.j 8 280.s even 4 1
700.2.c.j 8 280.y odd 4 1
700.2.g.j 8 40.e odd 2 1
700.2.g.j 8 40.f even 2 1
700.2.g.j 8 280.c odd 2 1
700.2.g.j 8 280.n even 2 1
980.2.o.e 16 56.j odd 6 2
980.2.o.e 16 56.k odd 6 2
980.2.o.e 16 56.m even 6 2
980.2.o.e 16 56.p even 6 2
1260.2.c.c 8 24.f even 2 1
1260.2.c.c 8 24.h odd 2 1
1260.2.c.c 8 168.e odd 2 1
1260.2.c.c 8 168.i even 2 1
2240.2.k.e 8 1.a even 1 1 trivial
2240.2.k.e 8 4.b odd 2 1 inner
2240.2.k.e 8 7.b odd 2 1 inner
2240.2.k.e 8 28.d even 2 1 inner

## Hecke kernels

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

 $$T_{3}^{4} - 10 T_{3}^{2} + 8$$ $$T_{19}^{4} - 28 T_{19}^{2} + 128$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( 8 - 10 T^{2} + T^{4} )^{2}$$
$5$ $$( 1 + T^{2} )^{4}$$
$7$ $$2401 - 882 T^{2} + 162 T^{4} - 18 T^{6} + T^{8}$$
$11$ $$( 128 + 28 T^{2} + T^{4} )^{2}$$
$13$ $$( 4 + T^{2} )^{4}$$
$17$ $$( 64 + 52 T^{2} + T^{4} )^{2}$$
$19$ $$( 128 - 28 T^{2} + T^{4} )^{2}$$
$23$ $$( 1352 + 74 T^{2} + T^{4} )^{2}$$
$29$ $$( -2 + T )^{8}$$
$31$ $$( 512 - 56 T^{2} + T^{4} )^{2}$$
$37$ $$( 2 + T )^{8}$$
$41$ $$( 64 + 52 T^{2} + T^{4} )^{2}$$
$43$ $$( 8 + 58 T^{2} + T^{4} )^{2}$$
$47$ $$( 2592 - 126 T^{2} + T^{4} )^{2}$$
$53$ $$( 2 + T )^{8}$$
$59$ $$( 512 - 124 T^{2} + T^{4} )^{2}$$
$61$ $$( 4 + T^{2} )^{4}$$
$67$ $$( 2888 + 218 T^{2} + T^{4} )^{2}$$
$71$ $$( 2048 + 92 T^{2} + T^{4} )^{2}$$
$73$ $$( 20736 + 324 T^{2} + T^{4} )^{2}$$
$79$ $$( 32 + 20 T^{2} + T^{4} )^{2}$$
$83$ $$( 8 - 10 T^{2} + T^{4} )^{2}$$
$89$ $$( 144 + T^{2} )^{4}$$
$97$ $$( 64 + 52 T^{2} + T^{4} )^{2}$$