# Properties

 Label 2240.2.b.e Level $2240$ Weight $2$ Character orbit 2240.b Analytic conductor $17.886$ Analytic rank $0$ Dimension $6$ 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: $$6$$ Coefficient field: 6.0.3534400.1 Defining polynomial: $$x^{6} - 2 x^{5} - 3 x^{4} + 16 x^{3} + x^{2} - 12 x + 40$$ 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 basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{5} q^{3} + \beta_{4} q^{5} - q^{7} + ( -2 - \beta_{1} - \beta_{3} ) q^{9} +O(q^{10})$$ $$q -\beta_{5} q^{3} + \beta_{4} q^{5} - q^{7} + ( -2 - \beta_{1} - \beta_{3} ) q^{9} + ( -\beta_{2} - \beta_{4} - \beta_{5} ) q^{11} + ( \beta_{2} + \beta_{4} - \beta_{5} ) q^{13} -\beta_{1} q^{15} + ( -\beta_{1} + 2 \beta_{3} ) q^{17} -2 \beta_{5} q^{19} + \beta_{5} q^{21} + ( 2 + 2 \beta_{1} + 2 \beta_{3} ) q^{23} - q^{25} + ( -4 \beta_{4} + \beta_{5} ) q^{27} + ( -2 \beta_{2} + 2 \beta_{4} - \beta_{5} ) q^{29} + ( 2 - 2 \beta_{3} ) q^{31} + ( -4 - \beta_{1} ) q^{33} -\beta_{4} q^{35} + ( -2 \beta_{2} + 2 \beta_{4} + 2 \beta_{5} ) q^{37} + ( -6 - \beta_{1} - 2 \beta_{3} ) q^{39} + ( -6 - 2 \beta_{1} ) q^{41} -2 \beta_{4} q^{43} + ( \beta_{2} - 2 \beta_{4} + \beta_{5} ) q^{45} + ( -3 - \beta_{1} + \beta_{3} ) q^{47} + q^{49} + ( 3 \beta_{2} - 7 \beta_{4} - \beta_{5} ) q^{51} + ( 2 \beta_{2} - 6 \beta_{4} ) q^{53} + ( 1 - \beta_{1} - \beta_{3} ) q^{55} + ( -10 - 2 \beta_{1} - 2 \beta_{3} ) q^{57} + ( 2 \beta_{2} - 6 \beta_{4} ) q^{59} + ( 2 \beta_{2} - 8 \beta_{4} + 2 \beta_{5} ) q^{61} + ( 2 + \beta_{1} + \beta_{3} ) q^{63} + ( -1 - \beta_{1} + \beta_{3} ) q^{65} + ( -6 \beta_{4} - 2 \beta_{5} ) q^{67} + ( 8 \beta_{4} - 6 \beta_{5} ) q^{69} + ( 8 + 2 \beta_{1} ) q^{71} + ( -8 - 2 \beta_{1} - 2 \beta_{3} ) q^{73} + \beta_{5} q^{75} + ( \beta_{2} + \beta_{4} + \beta_{5} ) q^{77} + ( -2 + 5 \beta_{1} + 2 \beta_{3} ) q^{79} + ( -1 + 2 \beta_{1} - 2 \beta_{3} ) q^{81} -2 \beta_{5} q^{83} + ( -2 \beta_{2} + \beta_{5} ) q^{85} + ( -3 - 5 \beta_{1} + \beta_{3} ) q^{87} + ( -8 + 2 \beta_{1} + 2 \beta_{3} ) q^{89} + ( -\beta_{2} - \beta_{4} + \beta_{5} ) q^{91} + ( -2 \beta_{2} + 2 \beta_{4} ) q^{93} -2 \beta_{1} q^{95} + ( 4 + \beta_{1} - 2 \beta_{3} ) q^{97} + ( -2 \beta_{2} - 8 \beta_{4} + 2 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 6q^{7} - 10q^{9} + O(q^{10})$$ $$6q - 6q^{7} - 10q^{9} - 4q^{17} + 8q^{23} - 6q^{25} + 16q^{31} - 24q^{33} - 32q^{39} - 36q^{41} - 20q^{47} + 6q^{49} + 8q^{55} - 56q^{57} + 10q^{63} - 8q^{65} + 48q^{71} - 44q^{73} - 16q^{79} - 2q^{81} - 20q^{87} - 52q^{89} + 28q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2 x^{5} - 3 x^{4} + 16 x^{3} + x^{2} - 12 x + 40$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-11 \nu^{5} - 30 \nu^{4} + 53 \nu^{3} + 115 \nu^{2} - 236 \nu - 660$$$$)/445$$ $$\beta_{2}$$ $$=$$ $$($$$$-27 \nu^{5} + 250 \nu^{4} - 679 \nu^{3} + 80 \nu^{2} + 2293 \nu - 2510$$$$)/890$$ $$\beta_{3}$$ $$=$$ $$($$$$-17 \nu^{5} + 75 \nu^{4} + \nu^{3} - 510 \nu^{2} + 768 \nu + 315$$$$)/445$$ $$\beta_{4}$$ $$=$$ $$($$$$-9 \nu^{5} + 24 \nu^{4} + 11 \nu^{3} - 92 \nu^{2} - 7 \nu - 6$$$$)/178$$ $$\beta_{5}$$ $$=$$ $$($$$$-28 \nu^{5} + 45 \nu^{4} + 54 \nu^{3} - 395 \nu^{2} - 358 \nu + 100$$$$)/445$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{5} + \beta_{3} + \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-3 \beta_{5} + 4 \beta_{4} - \beta_{3} + \beta_{1} + 3$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-8 \beta_{5} + 11 \beta_{4} + \beta_{3} - 3 \beta_{2} - 7$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-13 \beta_{5} + 30 \beta_{4} - 7 \beta_{3} - 2 \beta_{2} - 15 \beta_{1} - 19$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-13 \beta_{5} + 13 \beta_{4} - 8 \beta_{3} - 9 \beta_{2} - 51 \beta_{1} - 92$$$$)/2$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1121.1
 2.19082 − 1.44755i 0.627553 − 1.14620i −1.81837 − 0.301352i −1.81837 + 0.301352i 0.627553 + 1.14620i 2.19082 + 1.44755i
0 2.89511i 0 1.00000i 0 −1.00000 0 −5.38164 0
1121.2 0 2.29240i 0 1.00000i 0 −1.00000 0 −2.25511 0
1121.3 0 0.602705i 0 1.00000i 0 −1.00000 0 2.63675 0
1121.4 0 0.602705i 0 1.00000i 0 −1.00000 0 2.63675 0
1121.5 0 2.29240i 0 1.00000i 0 −1.00000 0 −2.25511 0
1121.6 0 2.89511i 0 1.00000i 0 −1.00000 0 −5.38164 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1121.6 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.e 6
4.b odd 2 1 2240.2.b.f yes 6
8.b even 2 1 inner 2240.2.b.e 6
8.d odd 2 1 2240.2.b.f yes 6
16.e even 4 1 8960.2.a.bk 3
16.e even 4 1 8960.2.a.bq 3
16.f odd 4 1 8960.2.a.bh 3
16.f odd 4 1 8960.2.a.bn 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2240.2.b.e 6 1.a even 1 1 trivial
2240.2.b.e 6 8.b even 2 1 inner
2240.2.b.f yes 6 4.b odd 2 1
2240.2.b.f yes 6 8.d odd 2 1
8960.2.a.bh 3 16.f odd 4 1
8960.2.a.bk 3 16.e even 4 1
8960.2.a.bn 3 16.f odd 4 1
8960.2.a.bq 3 16.e even 4 1

## Hecke kernels

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

 $$T_{3}^{6} + 14 T_{3}^{4} + 49 T_{3}^{2} + 16$$ $$T_{23}^{3} - 4 T_{23}^{2} - 60 T_{23} + 160$$ $$T_{31}^{3} - 8 T_{31}^{2} - 24 T_{31} + 32$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$16 + 49 T^{2} + 14 T^{4} + T^{6}$$
$5$ $$( 1 + T^{2} )^{3}$$
$7$ $$( 1 + T )^{6}$$
$11$ $$100 + 201 T^{2} + 38 T^{4} + T^{6}$$
$13$ $$3364 + 689 T^{2} + 46 T^{4} + T^{6}$$
$17$ $$( -106 - 55 T + 2 T^{2} + T^{3} )^{2}$$
$19$ $$1024 + 784 T^{2} + 56 T^{4} + T^{6}$$
$23$ $$( 160 - 60 T - 4 T^{2} + T^{3} )^{2}$$
$29$ $$35344 + 3353 T^{2} + 102 T^{4} + T^{6}$$
$31$ $$( 32 - 24 T - 8 T^{2} + T^{3} )^{2}$$
$37$ $$25600 + 4496 T^{2} + 168 T^{4} + T^{6}$$
$41$ $$( 80 + 80 T + 18 T^{2} + T^{3} )^{2}$$
$43$ $$( 4 + T^{2} )^{3}$$
$47$ $$( -64 + 13 T + 10 T^{2} + T^{3} )^{2}$$
$53$ $$25600 + 6720 T^{2} + 176 T^{4} + T^{6}$$
$59$ $$25600 + 6720 T^{2} + 176 T^{4} + T^{6}$$
$61$ $$25600 + 16256 T^{2} + 292 T^{4} + T^{6}$$
$67$ $$256 + 5824 T^{2} + 164 T^{4} + T^{6}$$
$71$ $$( -320 + 164 T - 24 T^{2} + T^{3} )^{2}$$
$73$ $$( -160 + 96 T + 22 T^{2} + T^{3} )^{2}$$
$79$ $$( -1472 - 179 T + 8 T^{2} + T^{3} )^{2}$$
$83$ $$1024 + 784 T^{2} + 56 T^{4} + T^{6}$$
$89$ $$( 160 + 160 T + 26 T^{2} + T^{3} )^{2}$$
$97$ $$( 230 + 9 T - 14 T^{2} + T^{3} )^{2}$$