# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2240 = 2^{6} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2240.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$17.8864900528$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ 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 $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - 2 \zeta_{12}^{2} ) q^{3} -\zeta_{12}^{3} q^{5} + q^{7} +O(q^{10})$$ $$q + ( 1 - 2 \zeta_{12}^{2} ) q^{3} -\zeta_{12}^{3} q^{5} + q^{7} + \zeta_{12}^{3} q^{11} -\zeta_{12}^{3} q^{13} + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{15} + ( -4 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{17} + 2 \zeta_{12}^{3} q^{19} + ( 1 - 2 \zeta_{12}^{2} ) q^{21} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{23} - q^{25} + ( 3 - 6 \zeta_{12}^{2} ) q^{27} + ( 3 - 6 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{29} + 4 q^{31} + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{33} -\zeta_{12}^{3} q^{35} + ( 2 - 4 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{37} + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{39} + ( -4 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{41} + ( 2 - 4 \zeta_{12}^{2} ) q^{43} + ( 3 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{47} + q^{49} + ( -4 + 8 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{51} + ( 2 - 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{53} + q^{55} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{57} + ( 2 - 4 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{59} + ( 2 - 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{61} - q^{65} + ( -2 + 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{67} + 6 \zeta_{12}^{3} q^{69} + ( 4 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{71} + 4 q^{73} + ( -1 + 2 \zeta_{12}^{2} ) q^{75} + \zeta_{12}^{3} q^{77} + ( 4 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{79} -9 q^{81} + ( -2 + 4 \zeta_{12}^{2} + 12 \zeta_{12}^{3} ) q^{83} + ( -1 + 2 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{85} + ( -9 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{87} + ( 6 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{89} -\zeta_{12}^{3} q^{91} + ( 4 - 8 \zeta_{12}^{2} ) q^{93} + 2 q^{95} + ( -16 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{7} + O(q^{10})$$ $$4q + 4q^{7} - 16q^{17} - 4q^{25} + 16q^{31} - 16q^{41} + 12q^{47} + 4q^{49} + 4q^{55} - 4q^{65} + 16q^{71} + 16q^{73} + 16q^{79} - 36q^{81} - 36q^{87} + 24q^{89} + 8q^{95} - 64q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1121.1
 0.866025 + 0.500000i −0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
0 1.73205i 0 1.00000i 0 1.00000 0 0 0
1121.2 0 1.73205i 0 1.00000i 0 1.00000 0 0 0
1121.3 0 1.73205i 0 1.00000i 0 1.00000 0 0 0
1121.4 0 1.73205i 0 1.00000i 0 1.00000 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b 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.b.d yes 4
4.b odd 2 1 2240.2.b.c 4
8.b even 2 1 inner 2240.2.b.d yes 4
8.d odd 2 1 2240.2.b.c 4
16.e even 4 1 8960.2.a.y 2
16.e even 4 1 8960.2.a.ba 2
16.f odd 4 1 8960.2.a.z 2
16.f odd 4 1 8960.2.a.bb 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2240.2.b.c 4 4.b odd 2 1
2240.2.b.c 4 8.d odd 2 1
2240.2.b.d yes 4 1.a even 1 1 trivial
2240.2.b.d yes 4 8.b even 2 1 inner
8960.2.a.y 2 16.e even 4 1
8960.2.a.z 2 16.f odd 4 1
8960.2.a.ba 2 16.e even 4 1
8960.2.a.bb 2 16.f odd 4 1

## Hecke kernels

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

 $$T_{3}^{2} + 3$$ $$T_{23}^{2} - 12$$ $$T_{31} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 3 + T^{2} )^{2}$$
$5$ $$( 1 + T^{2} )^{2}$$
$7$ $$( -1 + T )^{4}$$
$11$ $$( 1 + T^{2} )^{2}$$
$13$ $$( 1 + T^{2} )^{2}$$
$17$ $$( 13 + 8 T + T^{2} )^{2}$$
$19$ $$( 4 + T^{2} )^{2}$$
$23$ $$( -12 + T^{2} )^{2}$$
$29$ $$121 + 86 T^{2} + T^{4}$$
$31$ $$( -4 + T )^{4}$$
$37$ $$16 + 56 T^{2} + T^{4}$$
$41$ $$( -32 + 8 T + T^{2} )^{2}$$
$43$ $$( 12 + T^{2} )^{2}$$
$47$ $$( -39 - 6 T + T^{2} )^{2}$$
$53$ $$64 + 32 T^{2} + T^{4}$$
$59$ $$576 + 96 T^{2} + T^{4}$$
$61$ $$64 + 32 T^{2} + T^{4}$$
$67$ $$64 + 32 T^{2} + T^{4}$$
$71$ $$( 4 - 8 T + T^{2} )^{2}$$
$73$ $$( -4 + T )^{4}$$
$79$ $$( 13 - 8 T + T^{2} )^{2}$$
$83$ $$17424 + 312 T^{2} + T^{4}$$
$89$ $$( 24 - 12 T + T^{2} )^{2}$$
$97$ $$( 253 + 32 T + T^{2} )^{2}$$