# Properties

 Label 2240.2.h.d Level $2240$ Weight $2$ Character orbit 2240.h Analytic conductor $17.886$ Analytic rank $0$ Dimension $4$ 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.h (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: $$\Z[a_1, \ldots, a_{7}]$$ 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{12}^{3} q^{3} + q^{5} + ( 2 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{7} + 2 q^{9} +O(q^{10})$$ $$q + \zeta_{12}^{3} q^{3} + q^{5} + ( 2 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{7} + 2 q^{9} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{11} + q^{13} + \zeta_{12}^{3} q^{15} + ( 1 - 2 \zeta_{12}^{2} ) q^{17} + 2 \zeta_{12}^{3} q^{19} + ( 1 + 2 \zeta_{12}^{2} ) q^{21} + q^{25} + 5 \zeta_{12}^{3} q^{27} + ( 1 - 2 \zeta_{12}^{2} ) q^{29} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{31} + ( -3 + 6 \zeta_{12}^{2} ) q^{33} + ( 2 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{35} + ( 4 - 8 \zeta_{12}^{2} ) q^{37} + \zeta_{12}^{3} q^{39} + ( 6 - 12 \zeta_{12}^{2} ) q^{41} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{43} + 2 q^{45} + ( -14 \zeta_{12} + 7 \zeta_{12}^{3} ) q^{47} + ( 3 - 8 \zeta_{12}^{2} ) q^{49} + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{51} + ( 2 - 4 \zeta_{12}^{2} ) q^{53} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{55} -2 q^{57} + 6 \zeta_{12}^{3} q^{59} + 8 q^{61} + ( 4 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{63} + q^{65} + ( -12 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{67} + ( -4 + 8 \zeta_{12}^{2} ) q^{73} + \zeta_{12}^{3} q^{75} + ( 15 - 12 \zeta_{12}^{2} ) q^{77} -\zeta_{12}^{3} q^{79} + q^{81} + 12 \zeta_{12}^{3} q^{83} + ( 1 - 2 \zeta_{12}^{2} ) q^{85} + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{87} + ( -4 + 8 \zeta_{12}^{2} ) q^{89} + ( 2 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{91} + ( 2 - 4 \zeta_{12}^{2} ) q^{93} + 2 \zeta_{12}^{3} q^{95} + ( -5 + 10 \zeta_{12}^{2} ) q^{97} + ( 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{5} + 8q^{9} + O(q^{10})$$ $$4q + 4q^{5} + 8q^{9} + 4q^{13} + 8q^{21} + 4q^{25} + 8q^{45} - 4q^{49} - 8q^{57} + 32q^{61} + 4q^{65} + 36q^{77} + 4q^{81} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
671.1
 −0.866025 − 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i
0 1.00000i 0 1.00000 0 −1.73205 + 2.00000i 0 2.00000 0
671.2 0 1.00000i 0 1.00000 0 1.73205 + 2.00000i 0 2.00000 0
671.3 0 1.00000i 0 1.00000 0 −1.73205 2.00000i 0 2.00000 0
671.4 0 1.00000i 0 1.00000 0 1.73205 2.00000i 0 2.00000 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
4.b odd 2 1 inner
56.e even 2 1 inner
56.h odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2240.2.h.d yes 4
4.b odd 2 1 inner 2240.2.h.d yes 4
7.b odd 2 1 2240.2.h.b 4
8.b even 2 1 2240.2.h.b 4
8.d odd 2 1 2240.2.h.b 4
28.d even 2 1 2240.2.h.b 4
56.e even 2 1 inner 2240.2.h.d yes 4
56.h odd 2 1 inner 2240.2.h.d yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2240.2.h.b 4 7.b odd 2 1
2240.2.h.b 4 8.b even 2 1
2240.2.h.b 4 8.d odd 2 1
2240.2.h.b 4 28.d even 2 1
2240.2.h.d yes 4 1.a even 1 1 trivial
2240.2.h.d yes 4 4.b odd 2 1 inner
2240.2.h.d yes 4 56.e even 2 1 inner
2240.2.h.d yes 4 56.h odd 2 1 inner

## Hecke kernels

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

 $$T_{3}^{2} + 1$$ $$T_{13} - 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 1 + T^{2} )^{2}$$
$5$ $$( -1 + T )^{4}$$
$7$ $$49 + 2 T^{2} + T^{4}$$
$11$ $$( -27 + T^{2} )^{2}$$
$13$ $$( -1 + T )^{4}$$
$17$ $$( 3 + T^{2} )^{2}$$
$19$ $$( 4 + T^{2} )^{2}$$
$23$ $$T^{4}$$
$29$ $$( 3 + T^{2} )^{2}$$
$31$ $$( -12 + T^{2} )^{2}$$
$37$ $$( 48 + T^{2} )^{2}$$
$41$ $$( 108 + T^{2} )^{2}$$
$43$ $$( -12 + T^{2} )^{2}$$
$47$ $$( -147 + T^{2} )^{2}$$
$53$ $$( 12 + T^{2} )^{2}$$
$59$ $$( 36 + T^{2} )^{2}$$
$61$ $$( -8 + T )^{4}$$
$67$ $$( -108 + T^{2} )^{2}$$
$71$ $$T^{4}$$
$73$ $$( 48 + T^{2} )^{2}$$
$79$ $$( 1 + T^{2} )^{2}$$
$83$ $$( 144 + T^{2} )^{2}$$
$89$ $$( 48 + T^{2} )^{2}$$
$97$ $$( 75 + T^{2} )^{2}$$