# Properties

 Label 2240.2.g.j Level $2240$ Weight $2$ Character orbit 2240.g 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.g (of order $$2$$, degree $$1$$, not 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: $$1$$ Twist minimal: no (minimal twist has level 70) 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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{1} + \beta_{3} ) q^{3} + ( -1 - \beta_{2} - \beta_{3} ) q^{5} -\beta_{2} q^{7} -3 q^{9} +O(q^{10})$$ $$q + ( \beta_{1} + \beta_{3} ) q^{3} + ( -1 - \beta_{2} - \beta_{3} ) q^{5} -\beta_{2} q^{7} -3 q^{9} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{11} + ( -\beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{13} + ( 3 + 3 \beta_{2} - 2 \beta_{3} ) q^{15} + 2 \beta_{2} q^{17} + ( 4 + \beta_{1} - \beta_{3} ) q^{19} + ( \beta_{1} - \beta_{3} ) q^{21} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{23} + ( -2 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{25} + ( 2 - 2 \beta_{1} + 2 \beta_{3} ) q^{29} + ( 4 - 2 \beta_{1} + 2 \beta_{3} ) q^{31} -12 \beta_{2} q^{33} + ( -1 - \beta_{1} + \beta_{2} ) q^{35} -2 \beta_{2} q^{37} + ( 6 - 2 \beta_{1} + 2 \beta_{3} ) q^{39} + ( -6 - 2 \beta_{1} + 2 \beta_{3} ) q^{41} + ( 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{43} + ( 3 + 3 \beta_{2} + 3 \beta_{3} ) q^{45} + ( -2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{47} - q^{49} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{51} + ( 2 \beta_{1} - 6 \beta_{2} + 2 \beta_{3} ) q^{53} + ( -6 + 4 \beta_{1} + 6 \beta_{2} ) q^{55} + ( 4 \beta_{1} + 6 \beta_{2} + 4 \beta_{3} ) q^{57} + ( -4 - \beta_{1} + \beta_{3} ) q^{59} + ( -6 - \beta_{1} + \beta_{3} ) q^{61} + 3 \beta_{2} q^{63} + ( -1 + 2 \beta_{1} - 5 \beta_{2} + 2 \beta_{3} ) q^{65} + 8 \beta_{2} q^{67} + ( -12 - 2 \beta_{1} + 2 \beta_{3} ) q^{69} + ( -6 - 2 \beta_{1} + 2 \beta_{3} ) q^{71} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{73} + ( \beta_{1} - 12 \beta_{2} - \beta_{3} ) q^{75} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{77} + ( -2 + 2 \beta_{1} - 2 \beta_{3} ) q^{79} -9 q^{81} + ( \beta_{1} + \beta_{3} ) q^{83} + ( 2 + 2 \beta_{1} - 2 \beta_{2} ) q^{85} + ( 2 \beta_{1} - 12 \beta_{2} + 2 \beta_{3} ) q^{87} + 10 q^{89} + ( 2 - \beta_{1} + \beta_{3} ) q^{91} + ( 4 \beta_{1} - 12 \beta_{2} + 4 \beta_{3} ) q^{93} + ( -1 - 2 \beta_{1} - 7 \beta_{2} - 4 \beta_{3} ) q^{95} + ( -4 \beta_{1} + 6 \beta_{2} - 4 \beta_{3} ) q^{97} + ( 6 \beta_{1} - 6 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{5} - 12q^{9} + O(q^{10})$$ $$4q - 4q^{5} - 12q^{9} + 12q^{15} + 16q^{19} + 8q^{29} + 16q^{31} - 4q^{35} + 24q^{39} - 24q^{41} + 12q^{45} - 4q^{49} - 24q^{55} - 16q^{59} - 24q^{61} - 4q^{65} - 48q^{69} - 24q^{71} - 8q^{79} - 36q^{81} + 8q^{85} + 40q^{89} + 8q^{91} - 4q^{95} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/3$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$3 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$3 \beta_{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 −1.22474 − 1.22474i 1.22474 − 1.22474i −1.22474 + 1.22474i 1.22474 + 1.22474i
0 2.44949i 0 −2.22474 + 0.224745i 0 1.00000i 0 −3.00000 0
449.2 0 2.44949i 0 0.224745 + 2.22474i 0 1.00000i 0 −3.00000 0
449.3 0 2.44949i 0 −2.22474 0.224745i 0 1.00000i 0 −3.00000 0
449.4 0 2.44949i 0 0.224745 2.22474i 0 1.00000i 0 −3.00000 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
5.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.g.j 4
4.b odd 2 1 2240.2.g.i 4
5.b even 2 1 inner 2240.2.g.j 4
8.b even 2 1 70.2.c.a 4
8.d odd 2 1 560.2.g.e 4
20.d odd 2 1 2240.2.g.i 4
24.f even 2 1 5040.2.t.t 4
24.h odd 2 1 630.2.g.g 4
40.e odd 2 1 560.2.g.e 4
40.f even 2 1 70.2.c.a 4
40.i odd 4 1 350.2.a.g 2
40.i odd 4 1 350.2.a.h 2
40.k even 4 1 2800.2.a.bl 2
40.k even 4 1 2800.2.a.bm 2
56.h odd 2 1 490.2.c.e 4
56.j odd 6 2 490.2.i.f 8
56.p even 6 2 490.2.i.c 8
120.i odd 2 1 630.2.g.g 4
120.m even 2 1 5040.2.t.t 4
120.w even 4 1 3150.2.a.bs 2
120.w even 4 1 3150.2.a.bt 2
280.c odd 2 1 490.2.c.e 4
280.s even 4 1 2450.2.a.bl 2
280.s even 4 1 2450.2.a.bq 2
280.bf even 6 2 490.2.i.c 8
280.bk odd 6 2 490.2.i.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.c.a 4 8.b even 2 1
70.2.c.a 4 40.f even 2 1
350.2.a.g 2 40.i odd 4 1
350.2.a.h 2 40.i odd 4 1
490.2.c.e 4 56.h odd 2 1
490.2.c.e 4 280.c odd 2 1
490.2.i.c 8 56.p even 6 2
490.2.i.c 8 280.bf even 6 2
490.2.i.f 8 56.j odd 6 2
490.2.i.f 8 280.bk odd 6 2
560.2.g.e 4 8.d odd 2 1
560.2.g.e 4 40.e odd 2 1
630.2.g.g 4 24.h odd 2 1
630.2.g.g 4 120.i odd 2 1
2240.2.g.i 4 4.b odd 2 1
2240.2.g.i 4 20.d odd 2 1
2240.2.g.j 4 1.a even 1 1 trivial
2240.2.g.j 4 5.b even 2 1 inner
2450.2.a.bl 2 280.s even 4 1
2450.2.a.bq 2 280.s even 4 1
2800.2.a.bl 2 40.k even 4 1
2800.2.a.bm 2 40.k even 4 1
3150.2.a.bs 2 120.w even 4 1
3150.2.a.bt 2 120.w even 4 1
5040.2.t.t 4 24.f even 2 1
5040.2.t.t 4 120.m even 2 1

## Hecke kernels

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

 $$T_{3}^{2} + 6$$ $$T_{11}^{2} - 24$$ $$T_{19}^{2} - 8 T_{19} + 10$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 6 + T^{2} )^{2}$$
$5$ $$25 + 20 T + 8 T^{2} + 4 T^{3} + T^{4}$$
$7$ $$( 1 + T^{2} )^{2}$$
$11$ $$( -24 + T^{2} )^{2}$$
$13$ $$4 + 20 T^{2} + T^{4}$$
$17$ $$( 4 + T^{2} )^{2}$$
$19$ $$( 10 - 8 T + T^{2} )^{2}$$
$23$ $$400 + 56 T^{2} + T^{4}$$
$29$ $$( -20 - 4 T + T^{2} )^{2}$$
$31$ $$( -8 - 8 T + T^{2} )^{2}$$
$37$ $$( 4 + T^{2} )^{2}$$
$41$ $$( 12 + 12 T + T^{2} )^{2}$$
$43$ $$64 + 80 T^{2} + T^{4}$$
$47$ $$64 + 80 T^{2} + T^{4}$$
$53$ $$144 + 120 T^{2} + T^{4}$$
$59$ $$( 10 + 8 T + T^{2} )^{2}$$
$61$ $$( 30 + 12 T + T^{2} )^{2}$$
$67$ $$( 64 + T^{2} )^{2}$$
$71$ $$( 12 + 12 T + T^{2} )^{2}$$
$73$ $$400 + 56 T^{2} + T^{4}$$
$79$ $$( -20 + 4 T + T^{2} )^{2}$$
$83$ $$( 6 + T^{2} )^{2}$$
$89$ $$( -10 + T )^{4}$$
$97$ $$3600 + 264 T^{2} + T^{4}$$