# Properties

 Label 2240.2.k.d Level $2240$ Weight $2$ Character orbit 2240.k Analytic conductor $17.886$ Analytic rank $0$ Dimension $8$ 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.k (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$17.8864900528$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.116319195136.7 Defining polynomial: $$x^{8} + 77 x^{4} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 560) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{7} q^{3} -\beta_{3} q^{5} + ( -\beta_{4} + \beta_{6} ) q^{7} + ( 1 + \beta_{2} ) q^{9} +O(q^{10})$$ $$q + \beta_{7} q^{3} -\beta_{3} q^{5} + ( -\beta_{4} + \beta_{6} ) q^{7} + ( 1 + \beta_{2} ) q^{9} + ( \beta_{1} + \beta_{4} + \beta_{6} ) q^{11} + \beta_{5} q^{13} -\beta_{6} q^{15} + \beta_{5} q^{17} + ( -2 \beta_{1} + 2 \beta_{4} + 2 \beta_{7} ) q^{19} + ( 1 + 3 \beta_{3} - \beta_{5} ) q^{21} + ( -\beta_{1} - \beta_{4} + 2 \beta_{6} ) q^{23} - q^{25} + ( -\beta_{1} + \beta_{4} + 3 \beta_{7} ) q^{27} + \beta_{2} q^{29} -2 \beta_{7} q^{31} + ( 6 \beta_{3} - \beta_{5} ) q^{33} + ( -\beta_{1} + \beta_{7} ) q^{35} + ( 2 + 2 \beta_{2} ) q^{37} + ( \beta_{1} + \beta_{4} - 5 \beta_{6} ) q^{39} -6 \beta_{3} q^{41} + ( 3 \beta_{1} + 3 \beta_{4} ) q^{43} + ( -\beta_{3} + \beta_{5} ) q^{45} + ( -2 \beta_{1} + 2 \beta_{4} + \beta_{7} ) q^{47} + ( -2 - \beta_{2} - 3 \beta_{3} - \beta_{5} ) q^{49} + ( \beta_{1} + \beta_{4} - 5 \beta_{6} ) q^{51} + 6 q^{53} + ( \beta_{1} - \beta_{4} + \beta_{7} ) q^{55} + ( 4 + 2 \beta_{2} ) q^{57} + ( -2 \beta_{1} + 2 \beta_{4} - 2 \beta_{7} ) q^{59} + ( -6 \beta_{3} - 2 \beta_{5} ) q^{61} + ( -\beta_{1} + 2 \beta_{4} + 5 \beta_{6} + \beta_{7} ) q^{63} -\beta_{2} q^{65} + ( \beta_{1} + \beta_{4} + 4 \beta_{6} ) q^{67} + ( 6 \beta_{3} - 2 \beta_{5} ) q^{69} + ( 2 \beta_{1} + 2 \beta_{4} + 2 \beta_{6} ) q^{71} + 6 \beta_{3} q^{73} -\beta_{7} q^{75} + ( -2 \beta_{2} + 6 \beta_{3} + \beta_{5} ) q^{77} + ( 3 \beta_{1} + 3 \beta_{4} - 3 \beta_{6} ) q^{79} + 7 q^{81} + ( 3 \beta_{1} - 3 \beta_{4} ) q^{83} -\beta_{2} q^{85} + ( -\beta_{1} + \beta_{4} + 5 \beta_{7} ) q^{87} + 12 \beta_{3} q^{89} + ( 3 \beta_{1} + \beta_{4} - \beta_{6} + 4 \beta_{7} ) q^{91} + ( -8 - 2 \beta_{2} ) q^{93} + ( 2 \beta_{1} + 2 \beta_{4} - 2 \beta_{6} ) q^{95} + ( -12 \beta_{3} + \beta_{5} ) q^{97} + ( -4 \beta_{1} - 4 \beta_{4} + 8 \beta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 12q^{9} + O(q^{10})$$ $$8q + 12q^{9} + 8q^{21} - 8q^{25} + 4q^{29} + 24q^{37} - 20q^{49} + 48q^{53} + 40q^{57} - 4q^{65} - 8q^{77} + 56q^{81} - 4q^{85} - 72q^{93} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 77 x^{4} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{4} + 43$$$$)/9$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{6} + 79 \nu^{2}$$$$)/18$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{7} + 79 \nu^{3}$$$$)/18$$ $$\beta_{5}$$ $$=$$ $$($$$$-5 \nu^{6} - 377 \nu^{2}$$$$)/18$$ $$\beta_{6}$$ $$=$$ $$($$$$9 \nu^{7} + 2 \nu^{5} + 693 \nu^{3} + 158 \nu$$$$)/36$$ $$\beta_{7}$$ $$=$$ $$($$$$-9 \nu^{7} + 2 \nu^{5} - 693 \nu^{3} + 158 \nu$$$$)/36$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{5} + 5 \beta_{3}$$ $$\nu^{3}$$ $$=$$ $$\beta_{7} - \beta_{6} + 9 \beta_{4}$$ $$\nu^{4}$$ $$=$$ $$9 \beta_{2} - 43$$ $$\nu^{5}$$ $$=$$ $$9 \beta_{7} + 9 \beta_{6} - 79 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-79 \beta_{5} - 377 \beta_{3}$$ $$\nu^{7}$$ $$=$$ $$-79 \beta_{7} + 79 \beta_{6} - 693 \beta_{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1791.1
 −0.337637 − 0.337637i −0.337637 + 0.337637i −2.09428 − 2.09428i −2.09428 + 2.09428i 2.09428 + 2.09428i 2.09428 − 2.09428i 0.337637 + 0.337637i 0.337637 − 0.337637i
0 −2.96176 0 1.00000i 0 −0.337637 2.62412i 0 5.77200 0
1791.2 0 −2.96176 0 1.00000i 0 −0.337637 + 2.62412i 0 5.77200 0
1791.3 0 −0.477491 0 1.00000i 0 −2.09428 + 1.61679i 0 −2.77200 0
1791.4 0 −0.477491 0 1.00000i 0 −2.09428 1.61679i 0 −2.77200 0
1791.5 0 0.477491 0 1.00000i 0 2.09428 1.61679i 0 −2.77200 0
1791.6 0 0.477491 0 1.00000i 0 2.09428 + 1.61679i 0 −2.77200 0
1791.7 0 2.96176 0 1.00000i 0 0.337637 + 2.62412i 0 5.77200 0
1791.8 0 2.96176 0 1.00000i 0 0.337637 2.62412i 0 5.77200 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1791.8 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
7.b odd 2 1 inner
28.d 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.k.d 8
4.b odd 2 1 inner 2240.2.k.d 8
7.b odd 2 1 inner 2240.2.k.d 8
8.b even 2 1 560.2.k.b 8
8.d odd 2 1 560.2.k.b 8
24.f even 2 1 5040.2.d.d 8
24.h odd 2 1 5040.2.d.d 8
28.d even 2 1 inner 2240.2.k.d 8
40.e odd 2 1 2800.2.k.m 8
40.f even 2 1 2800.2.k.m 8
40.i odd 4 1 2800.2.e.g 8
40.i odd 4 1 2800.2.e.h 8
40.k even 4 1 2800.2.e.g 8
40.k even 4 1 2800.2.e.h 8
56.e even 2 1 560.2.k.b 8
56.h odd 2 1 560.2.k.b 8
168.e odd 2 1 5040.2.d.d 8
168.i even 2 1 5040.2.d.d 8
280.c odd 2 1 2800.2.k.m 8
280.n even 2 1 2800.2.k.m 8
280.s even 4 1 2800.2.e.g 8
280.s even 4 1 2800.2.e.h 8
280.y odd 4 1 2800.2.e.g 8
280.y odd 4 1 2800.2.e.h 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
560.2.k.b 8 8.b even 2 1
560.2.k.b 8 8.d odd 2 1
560.2.k.b 8 56.e even 2 1
560.2.k.b 8 56.h odd 2 1
2240.2.k.d 8 1.a even 1 1 trivial
2240.2.k.d 8 4.b odd 2 1 inner
2240.2.k.d 8 7.b odd 2 1 inner
2240.2.k.d 8 28.d even 2 1 inner
2800.2.e.g 8 40.i odd 4 1
2800.2.e.g 8 40.k even 4 1
2800.2.e.g 8 280.s even 4 1
2800.2.e.g 8 280.y odd 4 1
2800.2.e.h 8 40.i odd 4 1
2800.2.e.h 8 40.k even 4 1
2800.2.e.h 8 280.s even 4 1
2800.2.e.h 8 280.y odd 4 1
2800.2.k.m 8 40.e odd 2 1
2800.2.k.m 8 40.f even 2 1
2800.2.k.m 8 280.c odd 2 1
2800.2.k.m 8 280.n even 2 1
5040.2.d.d 8 24.f even 2 1
5040.2.d.d 8 24.h odd 2 1
5040.2.d.d 8 168.e odd 2 1
5040.2.d.d 8 168.i even 2 1

## Hecke kernels

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

 $$T_{3}^{4} - 9 T_{3}^{2} + 2$$ $$T_{19}^{4} - 76 T_{19}^{2} + 1152$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( 2 - 9 T^{2} + T^{4} )^{2}$$
$5$ $$( 1 + T^{2} )^{4}$$
$7$ $$2401 + 490 T^{2} + 50 T^{4} + 10 T^{6} + T^{8}$$
$11$ $$( 288 + 35 T^{2} + T^{4} )^{2}$$
$13$ $$( 324 + 37 T^{2} + T^{4} )^{2}$$
$17$ $$( 324 + 37 T^{2} + T^{4} )^{2}$$
$19$ $$( 1152 - 76 T^{2} + T^{4} )^{2}$$
$23$ $$( 288 + 38 T^{2} + T^{4} )^{2}$$
$29$ $$( -18 - T + T^{2} )^{4}$$
$31$ $$( 32 - 36 T^{2} + T^{4} )^{2}$$
$37$ $$( -64 - 6 T + T^{2} )^{4}$$
$41$ $$( 36 + T^{2} )^{4}$$
$43$ $$( 648 + 162 T^{2} + T^{4} )^{2}$$
$47$ $$( 162 - 65 T^{2} + T^{4} )^{2}$$
$53$ $$( -6 + T )^{8}$$
$59$ $$( 4608 - 140 T^{2} + T^{4} )^{2}$$
$61$ $$( 2304 + 196 T^{2} + T^{4} )^{2}$$
$67$ $$( 5832 + 194 T^{2} + T^{4} )^{2}$$
$71$ $$( 4608 + 140 T^{2} + T^{4} )^{2}$$
$73$ $$( 36 + T^{2} )^{4}$$
$79$ $$( 5832 + 171 T^{2} + T^{4} )^{2}$$
$83$ $$( 648 - 162 T^{2} + T^{4} )^{2}$$
$89$ $$( 144 + T^{2} )^{4}$$
$97$ $$( 19044 + 349 T^{2} + T^{4} )^{2}$$