# Properties

 Label 2240.2.e.d Level $2240$ Weight $2$ Character orbit 2240.e Analytic conductor $17.886$ Analytic rank $0$ Dimension $8$ CM discriminant -35 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: 8.0.121550625.1 Defining polynomial: $$x^{8} - x^{7} - 4 x^{6} - 9 x^{5} + 23 x^{4} + 18 x^{3} - 16 x^{2} + 8 x + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{8}$$ Twist minimal: no (minimal twist has level 560) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{6} q^{3} + \beta_{2} q^{5} + ( \beta_{1} - \beta_{6} ) q^{7} + ( -3 + \beta_{5} ) q^{9} +O(q^{10})$$ $$q + \beta_{6} q^{3} + \beta_{2} q^{5} + ( \beta_{1} - \beta_{6} ) q^{7} + ( -3 + \beta_{5} ) q^{9} + ( \beta_{4} + \beta_{7} ) q^{11} + ( -\beta_{2} + \beta_{3} ) q^{13} + ( 3 \beta_{4} - \beta_{7} ) q^{15} + ( \beta_{2} + 3 \beta_{3} ) q^{17} + ( 4 - \beta_{5} ) q^{21} + 5 q^{25} + ( 2 \beta_{1} - 5 \beta_{6} ) q^{27} + ( -4 - \beta_{5} ) q^{29} + ( -\beta_{2} - \beta_{3} ) q^{33} + ( -\beta_{4} + 2 \beta_{7} ) q^{35} + ( -7 \beta_{4} + 3 \beta_{7} ) q^{39} + 5 \beta_{3} q^{45} + ( -4 \beta_{1} - \beta_{6} ) q^{47} -7 q^{49} + ( -9 \beta_{4} + 5 \beta_{7} ) q^{51} + ( 4 \beta_{1} - \beta_{6} ) q^{55} + ( \beta_{1} + 6 \beta_{6} ) q^{63} + ( -8 + \beta_{5} ) q^{65} + ( -2 \beta_{4} + 4 \beta_{7} ) q^{71} -6 \beta_{2} q^{73} + 5 \beta_{6} q^{75} + ( -5 \beta_{2} - 3 \beta_{3} ) q^{77} + ( -\beta_{4} + 3 \beta_{7} ) q^{79} + ( 17 - 2 \beta_{5} ) q^{81} + ( -6 \beta_{1} + 6 \beta_{6} ) q^{83} + ( -4 + 3 \beta_{5} ) q^{85} + ( -2 \beta_{1} + \beta_{6} ) q^{87} + ( 5 \beta_{4} - 3 \beta_{7} ) q^{91} + ( 5 \beta_{2} + 7 \beta_{3} ) q^{97} + ( 4 \beta_{4} + 2 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 20q^{9} + O(q^{10})$$ $$8q - 20q^{9} + 28q^{21} + 40q^{25} - 36q^{29} - 56q^{49} - 60q^{65} + 128q^{81} - 20q^{85} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{7} - 4 x^{6} - 9 x^{5} + 23 x^{4} + 18 x^{3} - 16 x^{2} + 8 x + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-7 \nu^{7} - 93 \nu^{6} + 244 \nu^{5} + 547 \nu^{4} + 659 \nu^{3} - 3622 \nu^{2} - 1884 \nu + 1240$$$$)/1224$$ $$\beta_{2}$$ $$=$$ $$($$$$25 \nu^{7} - 3 \nu^{6} - 70 \nu^{5} - 205 \nu^{4} - 95 \nu^{3} + 40 \nu^{2} - 120 \nu + 2624$$$$)/1224$$ $$\beta_{3}$$ $$=$$ $$($$$$-89 \nu^{7} + 129 \nu^{6} + 392 \nu^{5} + 485 \nu^{4} - 2375 \nu^{3} - 1550 \nu^{2} + 3324 \nu - 1720$$$$)/1224$$ $$\beta_{4}$$ $$=$$ $$($$$$41 \nu^{7} - 111 \nu^{6} - 74 \nu^{5} - 173 \nu^{4} + 1517 \nu^{3} - 1036 \nu^{2} - 768 \nu + 1072$$$$)/408$$ $$\beta_{5}$$ $$=$$ $$($$$$-65 \nu^{7} + 69 \nu^{6} + 284 \nu^{5} + 533 \nu^{4} - 1691 \nu^{3} - 1226 \nu^{2} + 2556 \nu + 32$$$$)/408$$ $$\beta_{6}$$ $$=$$ $$($$$$313 \nu^{7} - 621 \nu^{6} - 652 \nu^{5} - 2077 \nu^{4} + 9235 \nu^{3} - 3110 \nu^{2} - 3420 \nu + 5696$$$$)/1224$$ $$\beta_{7}$$ $$=$$ $$($$$$-107 \nu^{7} + 123 \nu^{6} + 320 \nu^{5} + 959 \nu^{4} - 2429 \nu^{3} - 722 \nu^{2} + 228 \nu - 1096$$$$)/408$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{7} + 3 \beta_{6} + \beta_{5} - 5 \beta_{4} - 3 \beta_{3} - 3 \beta_{2} + 4$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{7} - 2 \beta_{6} + \beta_{4} - 5 \beta_{2} + 2 \beta_{1} + 10$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-9 \beta_{7} - 9 \beta_{6} + 9 \beta_{5} + \beta_{4} - 21 \beta_{3} - 9 \beta_{2} + 12 \beta_{1} - 8$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$5 \beta_{7} + 33 \beta_{6} + 5 \beta_{5} - 63 \beta_{4} - 11 \beta_{3} - 33 \beta_{2} + 22 \beta_{1} + 58$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$-40 \beta_{7} - 45 \beta_{6} + 20 \beta_{4} - 36 \beta_{2} + 45 \beta_{1} + 81$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-91 \beta_{7} - 17 \beta_{6} + 91 \beta_{5} - 143 \beta_{4} - 203 \beta_{3} - 17 \beta_{2} + 186 \beta_{1} - 234$$$$)/4$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2239.1
 −1.44918 + 1.77086i −0.862555 − 0.141174i 2.25820 − 0.369600i 0.553538 + 0.676408i 2.25820 + 0.369600i 0.553538 − 0.676408i −1.44918 − 1.77086i −0.862555 + 0.141174i
0 3.25937i 0 −2.23607 0 2.64575i 0 −7.62348 0
2239.2 0 3.25937i 0 2.23607 0 2.64575i 0 −7.62348 0
2239.3 0 0.613616i 0 −2.23607 0 2.64575i 0 2.62348 0
2239.4 0 0.613616i 0 2.23607 0 2.64575i 0 2.62348 0
2239.5 0 0.613616i 0 −2.23607 0 2.64575i 0 2.62348 0
2239.6 0 0.613616i 0 2.23607 0 2.64575i 0 2.62348 0
2239.7 0 3.25937i 0 −2.23607 0 2.64575i 0 −7.62348 0
2239.8 0 3.25937i 0 2.23607 0 2.64575i 0 −7.62348 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
35.c odd 2 1 CM by $$\Q(\sqrt{-35})$$
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
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.d 8
4.b odd 2 1 inner 2240.2.e.d 8
5.b even 2 1 inner 2240.2.e.d 8
7.b odd 2 1 inner 2240.2.e.d 8
8.b even 2 1 560.2.e.c 8
8.d odd 2 1 560.2.e.c 8
20.d odd 2 1 inner 2240.2.e.d 8
28.d even 2 1 inner 2240.2.e.d 8
35.c odd 2 1 CM 2240.2.e.d 8
40.e odd 2 1 560.2.e.c 8
40.f even 2 1 560.2.e.c 8
40.i odd 4 2 2800.2.k.p 8
40.k even 4 2 2800.2.k.p 8
56.e even 2 1 560.2.e.c 8
56.h odd 2 1 560.2.e.c 8
140.c even 2 1 inner 2240.2.e.d 8
280.c odd 2 1 560.2.e.c 8
280.n even 2 1 560.2.e.c 8
280.s even 4 2 2800.2.k.p 8
280.y odd 4 2 2800.2.k.p 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
560.2.e.c 8 8.b even 2 1
560.2.e.c 8 8.d odd 2 1
560.2.e.c 8 40.e odd 2 1
560.2.e.c 8 40.f even 2 1
560.2.e.c 8 56.e even 2 1
560.2.e.c 8 56.h odd 2 1
560.2.e.c 8 280.c odd 2 1
560.2.e.c 8 280.n even 2 1
2240.2.e.d 8 1.a even 1 1 trivial
2240.2.e.d 8 4.b odd 2 1 inner
2240.2.e.d 8 5.b even 2 1 inner
2240.2.e.d 8 7.b odd 2 1 inner
2240.2.e.d 8 20.d odd 2 1 inner
2240.2.e.d 8 28.d even 2 1 inner
2240.2.e.d 8 35.c odd 2 1 CM
2240.2.e.d 8 140.c even 2 1 inner
2800.2.k.p 8 40.i odd 4 2
2800.2.k.p 8 40.k even 4 2
2800.2.k.p 8 280.s even 4 2
2800.2.k.p 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 $$S_{2}^{\mathrm{new}}(2240, [\chi])$$:

 $$T_{3}^{4} + 11 T_{3}^{2} + 4$$ $$T_{11}^{4} + 31 T_{11}^{2} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( 4 + 11 T^{2} + T^{4} )^{2}$$
$5$ $$( -5 + T^{2} )^{4}$$
$7$ $$( 7 + T^{2} )^{4}$$
$11$ $$( 4 + 31 T^{2} + T^{4} )^{2}$$
$13$ $$( 36 - 33 T^{2} + T^{4} )^{2}$$
$17$ $$( 2116 - 97 T^{2} + T^{4} )^{2}$$
$19$ $$T^{8}$$
$23$ $$T^{8}$$
$29$ $$( -6 + 9 T + T^{2} )^{4}$$
$31$ $$T^{8}$$
$37$ $$T^{8}$$
$41$ $$T^{8}$$
$43$ $$T^{8}$$
$47$ $$( 6084 + 219 T^{2} + T^{4} )^{2}$$
$53$ $$T^{8}$$
$59$ $$T^{8}$$
$61$ $$T^{8}$$
$67$ $$T^{8}$$
$71$ $$( 140 + T^{2} )^{4}$$
$73$ $$( -180 + T^{2} )^{4}$$
$79$ $$( 6084 + 159 T^{2} + T^{4} )^{2}$$
$83$ $$( 252 + T^{2} )^{4}$$
$89$ $$T^{8}$$
$97$ $$( 60516 - 537 T^{2} + T^{4} )^{2}$$