# Properties

 Label 2240.2.h.a 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: $$\Q(i, \sqrt{6})$$ Defining polynomial: $$x^{4} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} - q^{5} + ( \beta_{1} + \beta_{2} ) q^{7} - q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} - q^{5} + ( \beta_{1} + \beta_{2} ) q^{7} - q^{9} + 2 q^{13} -\beta_{1} q^{15} + \beta_{3} q^{17} -\beta_{1} q^{19} + ( -2 + \beta_{3} ) q^{21} -3 \beta_{1} q^{23} + q^{25} + 2 \beta_{1} q^{27} + 2 \beta_{3} q^{29} + ( 2 \beta_{1} + 4 \beta_{2} ) q^{31} + ( -\beta_{1} - \beta_{2} ) q^{35} -\beta_{3} q^{37} + 2 \beta_{1} q^{39} + ( -\beta_{1} - 2 \beta_{2} ) q^{43} + q^{45} + ( \beta_{1} + 2 \beta_{2} ) q^{47} + ( 5 + \beta_{3} ) q^{49} + ( -2 \beta_{1} - 4 \beta_{2} ) q^{51} + \beta_{3} q^{53} + 4 q^{57} + 3 \beta_{1} q^{59} + 10 q^{61} + ( -\beta_{1} - \beta_{2} ) q^{63} -2 q^{65} + ( -3 \beta_{1} - 6 \beta_{2} ) q^{67} + 12 q^{69} + 3 \beta_{1} q^{71} -\beta_{3} q^{73} + \beta_{1} q^{75} -5 \beta_{1} q^{79} -11 q^{81} -3 \beta_{1} q^{83} -\beta_{3} q^{85} + ( -4 \beta_{1} - 8 \beta_{2} ) q^{87} + 2 \beta_{3} q^{89} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{91} + 4 \beta_{3} q^{93} + \beta_{1} q^{95} + \beta_{3} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{5} - 4q^{9} + O(q^{10})$$ $$4q - 4q^{5} - 4q^{9} + 8q^{13} - 8q^{21} + 4q^{25} + 4q^{45} + 20q^{49} + 16q^{57} + 40q^{61} - 8q^{65} + 48q^{69} - 44q^{81} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu^{2}$$$$/3$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} - \nu^{2} + 3 \nu$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$($$$$2 \nu^{3} + 6 \nu$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + 2 \beta_{2} + \beta_{1}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$3 \beta_{1}$$$$/2$$ $$\nu^{3}$$ $$=$$ $$($$$$3 \beta_{3} - 6 \beta_{2} - 3 \beta_{1}$$$$)/4$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
671.1
 −1.22474 + 1.22474i 1.22474 − 1.22474i −1.22474 − 1.22474i 1.22474 + 1.22474i
0 2.00000i 0 −1.00000 0 −2.44949 1.00000i 0 −1.00000 0
671.2 0 2.00000i 0 −1.00000 0 2.44949 1.00000i 0 −1.00000 0
671.3 0 2.00000i 0 −1.00000 0 −2.44949 + 1.00000i 0 −1.00000 0
671.4 0 2.00000i 0 −1.00000 0 2.44949 + 1.00000i 0 −1.00000 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.a 4
4.b odd 2 1 inner 2240.2.h.a 4
7.b odd 2 1 2240.2.h.c yes 4
8.b even 2 1 2240.2.h.c yes 4
8.d odd 2 1 2240.2.h.c yes 4
28.d even 2 1 2240.2.h.c yes 4
56.e even 2 1 inner 2240.2.h.a 4
56.h odd 2 1 inner 2240.2.h.a 4

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

## Hecke kernels

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

 $$T_{3}^{2} + 4$$ $$T_{13} - 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 4 + T^{2} )^{2}$$
$5$ $$( 1 + T )^{4}$$
$7$ $$49 - 10 T^{2} + T^{4}$$
$11$ $$T^{4}$$
$13$ $$( -2 + T )^{4}$$
$17$ $$( 24 + T^{2} )^{2}$$
$19$ $$( 4 + T^{2} )^{2}$$
$23$ $$( 36 + T^{2} )^{2}$$
$29$ $$( 96 + T^{2} )^{2}$$
$31$ $$( -96 + T^{2} )^{2}$$
$37$ $$( 24 + T^{2} )^{2}$$
$41$ $$T^{4}$$
$43$ $$( -24 + T^{2} )^{2}$$
$47$ $$( -24 + T^{2} )^{2}$$
$53$ $$( 24 + T^{2} )^{2}$$
$59$ $$( 36 + T^{2} )^{2}$$
$61$ $$( -10 + T )^{4}$$
$67$ $$( -216 + T^{2} )^{2}$$
$71$ $$( 36 + T^{2} )^{2}$$
$73$ $$( 24 + T^{2} )^{2}$$
$79$ $$( 100 + T^{2} )^{2}$$
$83$ $$( 36 + T^{2} )^{2}$$
$89$ $$( 96 + T^{2} )^{2}$$
$97$ $$( 24 + T^{2} )^{2}$$