# Properties

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

## Newform invariants

 Self dual: no Analytic conductor: $$17.8864900528$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{10}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{24}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 - \zeta_{24} - \zeta_{24}^{3} + 2 \zeta_{24}^{4} + \zeta_{24}^{5} ) q^{5} + ( -\zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} - \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{7} -3 q^{9} +O(q^{10})$$ $$q + ( -1 - \zeta_{24} - \zeta_{24}^{3} + 2 \zeta_{24}^{4} + \zeta_{24}^{5} ) q^{5} + ( -\zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} - \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{7} -3 q^{9} + ( 4 \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{11} + ( 2 - 4 \zeta_{24}^{4} ) q^{13} + 4 q^{17} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{19} + ( 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{23} + ( -1 - 2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{25} + ( 4 - 8 \zeta_{24}^{4} ) q^{29} + ( -4 \zeta_{24} - 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} + 8 \zeta_{24}^{7} ) q^{31} + ( 2 \zeta_{24} + 6 \zeta_{24}^{2} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 3 \zeta_{24}^{6} ) q^{35} + ( -4 \zeta_{24} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} ) q^{37} + ( 4 \zeta_{24} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} + 8 \zeta_{24}^{7} ) q^{41} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{43} + ( 3 + 3 \zeta_{24} + 3 \zeta_{24}^{3} - 6 \zeta_{24}^{4} - 3 \zeta_{24}^{5} ) q^{45} + 10 \zeta_{24}^{6} q^{47} + ( 5 + 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{49} + ( 8 \zeta_{24} + 8 \zeta_{24}^{3} - 8 \zeta_{24}^{5} ) q^{53} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 6 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{55} + ( -6 \zeta_{24} + 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} ) q^{59} + ( 10 \zeta_{24} + 10 \zeta_{24}^{3} - 10 \zeta_{24}^{5} ) q^{61} + ( 3 \zeta_{24} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} + 3 \zeta_{24}^{6} - 6 \zeta_{24}^{7} ) q^{63} + ( 6 + 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{65} + ( 10 \zeta_{24} - 10 \zeta_{24}^{3} - 10 \zeta_{24}^{5} ) q^{67} + 4 \zeta_{24}^{6} q^{71} + 12 q^{73} + ( 2 - 6 \zeta_{24} - 6 \zeta_{24}^{3} - 4 \zeta_{24}^{4} + 6 \zeta_{24}^{5} ) q^{77} + 12 \zeta_{24}^{6} q^{79} + 9 q^{81} + ( 16 \zeta_{24}^{2} - 8 \zeta_{24}^{6} ) q^{83} + ( -4 - 4 \zeta_{24} - 4 \zeta_{24}^{3} + 8 \zeta_{24}^{4} + 4 \zeta_{24}^{5} ) q^{85} + ( 4 \zeta_{24} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} + 8 \zeta_{24}^{7} ) q^{89} + ( -6 \zeta_{24} - 4 \zeta_{24}^{2} + 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} + 2 \zeta_{24}^{6} ) q^{91} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 4 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{95} + 4 q^{97} + ( -12 \zeta_{24}^{2} + 6 \zeta_{24}^{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 24q^{9} + O(q^{10})$$ $$8q - 24q^{9} + 32q^{17} - 8q^{25} + 40q^{49} + 48q^{65} + 96q^{73} + 72q^{81} + 32q^{97} + O(q^{100})$$

## 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
 0.965926 − 0.258819i −0.258819 − 0.965926i 0.965926 + 0.258819i −0.258819 + 0.965926i 0.258819 + 0.965926i −0.965926 + 0.258819i 0.258819 − 0.965926i −0.965926 − 0.258819i
0 0 0 −1.41421 1.73205i 0 −2.44949 + 1.00000i 0 −3.00000 0
1119.2 0 0 0 −1.41421 1.73205i 0 2.44949 1.00000i 0 −3.00000 0
1119.3 0 0 0 −1.41421 + 1.73205i 0 −2.44949 1.00000i 0 −3.00000 0
1119.4 0 0 0 −1.41421 + 1.73205i 0 2.44949 + 1.00000i 0 −3.00000 0
1119.5 0 0 0 1.41421 1.73205i 0 −2.44949 1.00000i 0 −3.00000 0
1119.6 0 0 0 1.41421 1.73205i 0 2.44949 + 1.00000i 0 −3.00000 0
1119.7 0 0 0 1.41421 + 1.73205i 0 −2.44949 + 1.00000i 0 −3.00000 0
1119.8 0 0 0 1.41421 + 1.73205i 0 2.44949 1.00000i 0 −3.00000 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
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
35.c odd 2 1 inner
140.c even 2 1 inner
280.c 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.e yes 8
4.b odd 2 1 inner 2240.2.n.e yes 8
5.b even 2 1 2240.2.n.c 8
7.b odd 2 1 2240.2.n.c 8
8.b even 2 1 inner 2240.2.n.e yes 8
8.d odd 2 1 inner 2240.2.n.e yes 8
20.d odd 2 1 2240.2.n.c 8
28.d even 2 1 2240.2.n.c 8
35.c odd 2 1 inner 2240.2.n.e yes 8
40.e odd 2 1 2240.2.n.c 8
40.f even 2 1 2240.2.n.c 8
56.e even 2 1 2240.2.n.c 8
56.h odd 2 1 2240.2.n.c 8
140.c even 2 1 inner 2240.2.n.e yes 8
280.c odd 2 1 inner 2240.2.n.e yes 8
280.n even 2 1 inner 2240.2.n.e yes 8

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

 $$T_{3}$$ $$T_{11}^{2} - 12$$ $$T_{17} - 4$$ $$T_{23}^{2} - 24$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$( 25 + 2 T^{2} + T^{4} )^{2}$$
$7$ $$( 49 - 10 T^{2} + T^{4} )^{2}$$
$11$ $$( -12 + T^{2} )^{4}$$
$13$ $$( 12 + T^{2} )^{4}$$
$17$ $$( -4 + T )^{8}$$
$19$ $$( 8 + T^{2} )^{4}$$
$23$ $$( -24 + T^{2} )^{4}$$
$29$ $$( 48 + T^{2} )^{4}$$
$31$ $$( -96 + T^{2} )^{4}$$
$37$ $$( -32 + T^{2} )^{4}$$
$41$ $$( 96 + T^{2} )^{4}$$
$43$ $$( 8 + T^{2} )^{4}$$
$47$ $$( 100 + T^{2} )^{4}$$
$53$ $$( -128 + T^{2} )^{4}$$
$59$ $$( 72 + T^{2} )^{4}$$
$61$ $$( -200 + T^{2} )^{4}$$
$67$ $$( 200 + T^{2} )^{4}$$
$71$ $$( 16 + T^{2} )^{4}$$
$73$ $$( -12 + T )^{8}$$
$79$ $$( 144 + T^{2} )^{4}$$
$83$ $$( -192 + T^{2} )^{4}$$
$89$ $$( 96 + T^{2} )^{4}$$
$97$ $$( -4 + T )^{8}$$