# Properties

 Label 2240.2.n.g Level $2240$ Weight $2$ Character orbit 2240.n 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.n (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$17.8864900528$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.1731891456.1 Defining polynomial: $$x^{8} - 9 x^{6} + 65 x^{4} - 144 x^{2} + 256$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + ( -2 + \beta_{5} ) q^{5} + ( -2 - \beta_{2} ) q^{7} + ( 1 + \beta_{1} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + ( -2 + \beta_{5} ) q^{5} + ( -2 - \beta_{2} ) q^{7} + ( 1 + \beta_{1} ) q^{9} + ( -2 \beta_{3} + \beta_{7} ) q^{11} + ( \beta_{4} + \beta_{5} ) q^{13} + ( -2 \beta_{1} - \beta_{4} + \beta_{5} ) q^{15} + \beta_{7} q^{17} + ( 2 \beta_{4} + 2 \beta_{5} ) q^{19} + ( -2 \beta_{1} + \beta_{6} ) q^{21} + ( -4 + 2 \beta_{1} ) q^{23} + ( 3 - 4 \beta_{5} ) q^{25} + ( 4 - \beta_{1} ) q^{27} + \beta_{6} q^{29} + 2 \beta_{7} q^{31} + ( 4 \beta_{3} - \beta_{7} ) q^{33} + ( 4 + 2 \beta_{2} + \beta_{3} - 2 \beta_{5} ) q^{35} + 2 \beta_{7} q^{37} + ( -\beta_{4} - 3 \beta_{5} ) q^{39} + 2 \beta_{6} q^{41} + ( -4 \beta_{2} - 2 \beta_{6} ) q^{43} + ( -2 - 2 \beta_{1} - \beta_{4} + 2 \beta_{5} ) q^{45} + ( -2 \beta_{2} - \beta_{6} ) q^{47} + ( 1 + 4 \beta_{2} ) q^{49} + ( 4 \beta_{3} + \beta_{7} ) q^{51} + ( -2 \beta_{2} + 4 \beta_{3} - \beta_{6} - 2 \beta_{7} ) q^{55} + ( -2 \beta_{4} - 6 \beta_{5} ) q^{57} + 4 \beta_{5} q^{59} + ( -4 + 2 \beta_{1} ) q^{61} + ( -2 - 2 \beta_{1} - \beta_{2} + \beta_{6} ) q^{63} + ( -2 + \beta_{1} - 2 \beta_{4} - 2 \beta_{5} ) q^{65} -4 \beta_{2} q^{67} + ( 8 - 2 \beta_{1} ) q^{69} + ( 4 \beta_{4} - 8 \beta_{5} ) q^{71} + 4 \beta_{3} q^{73} + ( 3 \beta_{1} + 4 \beta_{4} - 4 \beta_{5} ) q^{75} + ( 4 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} - 2 \beta_{7} ) q^{77} + ( -3 \beta_{4} + 3 \beta_{5} ) q^{79} -7 q^{81} -4 q^{83} + ( -\beta_{6} - 2 \beta_{7} ) q^{85} + ( -4 \beta_{2} + \beta_{6} ) q^{87} -2 \beta_{6} q^{89} + ( 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - \beta_{7} ) q^{91} + ( 8 \beta_{3} + 2 \beta_{7} ) q^{93} + ( -4 + 2 \beta_{1} - 4 \beta_{4} - 4 \beta_{5} ) q^{95} + ( 8 \beta_{3} - \beta_{7} ) q^{97} + 2 \beta_{3} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 4q^{3} - 16q^{5} - 16q^{7} + 12q^{9} + O(q^{10})$$ $$8q + 4q^{3} - 16q^{5} - 16q^{7} + 12q^{9} - 8q^{15} - 8q^{21} - 24q^{23} + 24q^{25} + 28q^{27} + 32q^{35} - 24q^{45} + 8q^{49} - 24q^{61} - 24q^{63} - 12q^{65} + 56q^{69} + 12q^{75} - 56q^{81} - 32q^{83} - 24q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 9 x^{6} + 65 x^{4} - 144 x^{2} + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{6} + 181$$$$)/65$$ $$\beta_{2}$$ $$=$$ $$($$$$9 \nu^{6} - 65 \nu^{4} + 585 \nu^{2} - 776$$$$)/520$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{7} + 25 \nu^{5} - 145 \nu^{3} + 544 \nu$$$$)/320$$ $$\beta_{4}$$ $$=$$ $$($$$$-9 \nu^{7} + 65 \nu^{5} - 585 \nu^{3} + 256 \nu$$$$)/1040$$ $$\beta_{5}$$ $$=$$ $$($$$$9 \nu^{7} - 65 \nu^{5} + 377 \nu^{3} - 256 \nu$$$$)/832$$ $$\beta_{6}$$ $$=$$ $$($$$$-37 \nu^{6} + 325 \nu^{4} - 1885 \nu^{2} + 2728$$$$)/520$$ $$\beta_{7}$$ $$=$$ $$($$$$-49 \nu^{7} + 585 \nu^{5} - 4225 \nu^{3} + 16416 \nu$$$$)/4160$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} - \beta_{4} - \beta_{3}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{6} + 5 \beta_{2} - \beta_{1} + 5$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$-4 \beta_{5} - 5 \beta_{4}$$ $$\nu^{4}$$ $$=$$ $$($$$$9 \beta_{6} + 29 \beta_{2} + 9 \beta_{1} - 29$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-29 \beta_{7} - 36 \beta_{5} - 29 \beta_{4} + 65 \beta_{3}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$65 \beta_{1} - 181$$ $$\nu^{7}$$ $$=$$ $$($$$$-181 \beta_{7} + 260 \beta_{5} + 181 \beta_{4} + 441 \beta_{3}$$$$)/2$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1119.1
 2.21837 + 1.28078i −2.21837 + 1.28078i −2.21837 − 1.28078i 2.21837 − 1.28078i −1.35234 − 0.780776i 1.35234 − 0.780776i 1.35234 + 0.780776i −1.35234 + 0.780776i
0 −1.56155 0 −2.00000 1.00000i 0 −2.00000 1.73205i 0 −0.561553 0
1119.2 0 −1.56155 0 −2.00000 1.00000i 0 −2.00000 + 1.73205i 0 −0.561553 0
1119.3 0 −1.56155 0 −2.00000 + 1.00000i 0 −2.00000 1.73205i 0 −0.561553 0
1119.4 0 −1.56155 0 −2.00000 + 1.00000i 0 −2.00000 + 1.73205i 0 −0.561553 0
1119.5 0 2.56155 0 −2.00000 1.00000i 0 −2.00000 1.73205i 0 3.56155 0
1119.6 0 2.56155 0 −2.00000 1.00000i 0 −2.00000 + 1.73205i 0 3.56155 0
1119.7 0 2.56155 0 −2.00000 + 1.00000i 0 −2.00000 1.73205i 0 3.56155 0
1119.8 0 2.56155 0 −2.00000 + 1.00000i 0 −2.00000 + 1.73205i 0 3.56155 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1119.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 inner
56.h odd 2 1 inner
280.n 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.n.g yes 8
4.b odd 2 1 2240.2.n.a 8
5.b even 2 1 2240.2.n.a 8
7.b odd 2 1 2240.2.n.b yes 8
8.b even 2 1 2240.2.n.b yes 8
8.d odd 2 1 2240.2.n.h yes 8
20.d odd 2 1 inner 2240.2.n.g yes 8
28.d even 2 1 2240.2.n.h yes 8
35.c odd 2 1 2240.2.n.h yes 8
40.e odd 2 1 2240.2.n.b yes 8
40.f even 2 1 2240.2.n.h yes 8
56.e even 2 1 2240.2.n.a 8
56.h odd 2 1 inner 2240.2.n.g yes 8
140.c even 2 1 2240.2.n.b yes 8
280.c odd 2 1 2240.2.n.a 8
280.n even 2 1 inner 2240.2.n.g yes 8

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

## Hecke kernels

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

 $$T_{3}^{2} - T_{3} - 4$$ $$T_{11}^{4} - 39 T_{11}^{2} + 36$$ $$T_{17}^{4} - 27 T_{17}^{2} + 144$$ $$T_{23}^{2} + 6 T_{23} - 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( -4 - T + T^{2} )^{4}$$
$5$ $$( 5 + 4 T + T^{2} )^{4}$$
$7$ $$( 7 + 4 T + T^{2} )^{4}$$
$11$ $$( 36 - 39 T^{2} + T^{4} )^{2}$$
$13$ $$( 4 + 13 T^{2} + T^{4} )^{2}$$
$17$ $$( 144 - 27 T^{2} + T^{4} )^{2}$$
$19$ $$( 64 + 52 T^{2} + T^{4} )^{2}$$
$23$ $$( -8 + 6 T + T^{2} )^{4}$$
$29$ $$( 144 + 27 T^{2} + T^{4} )^{2}$$
$31$ $$( 2304 - 108 T^{2} + T^{4} )^{2}$$
$37$ $$( 2304 - 108 T^{2} + T^{4} )^{2}$$
$41$ $$( 2304 + 108 T^{2} + T^{4} )^{2}$$
$43$ $$( 576 + 156 T^{2} + T^{4} )^{2}$$
$47$ $$( 36 + 39 T^{2} + T^{4} )^{2}$$
$53$ $$T^{8}$$
$59$ $$( 16 + T^{2} )^{4}$$
$61$ $$( -8 + 6 T + T^{2} )^{4}$$
$67$ $$( 48 + T^{2} )^{4}$$
$71$ $$( 1024 + 208 T^{2} + T^{4} )^{2}$$
$73$ $$( -48 + T^{2} )^{4}$$
$79$ $$( 1296 + 81 T^{2} + T^{4} )^{2}$$
$83$ $$( 4 + T )^{8}$$
$89$ $$( 2304 + 108 T^{2} + T^{4} )^{2}$$
$97$ $$( 24336 - 363 T^{2} + T^{4} )^{2}$$