# Properties

 Label 2240.2.n.f Level $2240$ Weight $2$ Character orbit 2240.n Analytic conductor $17.886$ Analytic rank $0$ Dimension $8$ CM discriminant -35 Inner twists $16$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2240 = 2^{6} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2240.n (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$17.8864900528$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.384160000.1 Defining polynomial: $$x^{8} - 9 x^{6} + 37 x^{4} - 36 x^{2} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{8}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + \beta_{6} q^{5} -\beta_{3} q^{7} + 2 q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + \beta_{6} q^{5} -\beta_{3} q^{7} + 2 q^{9} -\beta_{5} q^{11} -3 \beta_{6} q^{13} -5 \beta_{4} q^{15} + 3 \beta_{2} q^{17} -\beta_{7} q^{21} -5 q^{25} -\beta_{1} q^{27} -\beta_{7} q^{29} + 5 \beta_{2} q^{33} + \beta_{5} q^{35} + 15 \beta_{4} q^{39} + 2 \beta_{6} q^{45} + 3 \beta_{3} q^{47} -7 q^{49} -3 \beta_{5} q^{51} -5 \beta_{3} q^{55} -2 \beta_{3} q^{63} + 15 q^{65} -12 \beta_{4} q^{71} -4 \beta_{2} q^{73} -5 \beta_{1} q^{75} + 7 \beta_{6} q^{77} + \beta_{4} q^{79} -11 q^{81} + 4 \beta_{1} q^{83} -3 \beta_{7} q^{85} -5 \beta_{3} q^{87} -3 \beta_{5} q^{91} -7 \beta_{2} q^{97} -2 \beta_{5} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 16q^{9} + O(q^{10})$$ $$8q + 16q^{9} - 40q^{25} - 56q^{49} + 120q^{65} - 88q^{81} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 9 x^{6} + 37 x^{4} - 36 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{6} + 5 \nu^{4} - 5 \nu^{2} - 38$$$$)/18$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{7} + 13 \nu^{5} - 65 \nu^{3} + 104 \nu$$$$)/24$$ $$\beta_{3}$$ $$=$$ $$($$$$2 \nu^{6} - 16 \nu^{4} + 64 \nu^{2} - 35$$$$)/9$$ $$\beta_{4}$$ $$=$$ $$($$$$7 \nu^{7} - 59 \nu^{5} + 215 \nu^{3} - 88 \nu$$$$)/72$$ $$\beta_{5}$$ $$=$$ $$($$$$-11 \nu^{7} + 103 \nu^{5} - 451 \nu^{3} + 704 \nu$$$$)/72$$ $$\beta_{6}$$ $$=$$ $$($$$$5 \nu^{7} - 41 \nu^{5} + 157 \nu^{3} - 64 \nu$$$$)/24$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{6} + 9 \nu^{4} - 33 \nu^{2} + 18$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{6} + \beta_{5} - \beta_{4} - \beta_{2}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{7} + 3 \beta_{3} + 3 \beta_{1} + 9$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$5 \beta_{6} + \beta_{5} - 10 \beta_{4} - 2 \beta_{2}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$9 \beta_{7} + 21 \beta_{3} + 3 \beta_{1} + 7$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$55 \beta_{6} - 5 \beta_{5} - 121 \beta_{4} + 11 \beta_{2}$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$20 \beta_{7} + 45 \beta_{3} - 36 \beta_{1} - 81$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$169 \beta_{6} - 91 \beta_{5} - 377 \beta_{4} + 203 \beta_{2}$$$$)/4$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1119.1
 −0.817582 − 0.309017i 0.817582 − 0.309017i 0.817582 + 0.309017i −0.817582 + 0.309017i −2.14046 − 0.809017i 2.14046 − 0.809017i 2.14046 + 0.809017i −2.14046 + 0.809017i
0 −2.23607 0 2.23607i 0 2.64575i 0 2.00000 0
1119.2 0 −2.23607 0 2.23607i 0 2.64575i 0 2.00000 0
1119.3 0 −2.23607 0 2.23607i 0 2.64575i 0 2.00000 0
1119.4 0 −2.23607 0 2.23607i 0 2.64575i 0 2.00000 0
1119.5 0 2.23607 0 2.23607i 0 2.64575i 0 2.00000 0
1119.6 0 2.23607 0 2.23607i 0 2.64575i 0 2.00000 0
1119.7 0 2.23607 0 2.23607i 0 2.64575i 0 2.00000 0
1119.8 0 2.23607 0 2.23607i 0 2.64575i 0 2.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1119.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
35.c odd 2 1 CM by $$\Q(\sqrt{-35})$$
4.b odd 2 1 inner
5.b even 2 1 inner
7.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
20.d odd 2 1 inner
28.d even 2 1 inner
40.e odd 2 1 inner
40.f even 2 1 inner
56.e even 2 1 inner
56.h odd 2 1 inner
140.c even 2 1 inner
280.c odd 2 1 inner
280.n 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.n.f 8
4.b odd 2 1 inner 2240.2.n.f 8
5.b even 2 1 inner 2240.2.n.f 8
7.b odd 2 1 inner 2240.2.n.f 8
8.b even 2 1 inner 2240.2.n.f 8
8.d odd 2 1 inner 2240.2.n.f 8
20.d odd 2 1 inner 2240.2.n.f 8
28.d even 2 1 inner 2240.2.n.f 8
35.c odd 2 1 CM 2240.2.n.f 8
40.e odd 2 1 inner 2240.2.n.f 8
40.f even 2 1 inner 2240.2.n.f 8
56.e even 2 1 inner 2240.2.n.f 8
56.h odd 2 1 inner 2240.2.n.f 8
140.c even 2 1 inner 2240.2.n.f 8
280.c odd 2 1 inner 2240.2.n.f 8
280.n even 2 1 inner 2240.2.n.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2240.2.n.f 8 1.a even 1 1 trivial
2240.2.n.f 8 4.b odd 2 1 inner
2240.2.n.f 8 5.b even 2 1 inner
2240.2.n.f 8 7.b odd 2 1 inner
2240.2.n.f 8 8.b even 2 1 inner
2240.2.n.f 8 8.d odd 2 1 inner
2240.2.n.f 8 20.d odd 2 1 inner
2240.2.n.f 8 28.d even 2 1 inner
2240.2.n.f 8 35.c odd 2 1 CM
2240.2.n.f 8 40.e odd 2 1 inner
2240.2.n.f 8 40.f even 2 1 inner
2240.2.n.f 8 56.e even 2 1 inner
2240.2.n.f 8 56.h odd 2 1 inner
2240.2.n.f 8 140.c even 2 1 inner
2240.2.n.f 8 280.c odd 2 1 inner
2240.2.n.f 8 280.n even 2 1 inner

## Hecke kernels

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

 $$T_{3}^{2} - 5$$ $$T_{11}^{2} - 35$$ $$T_{17}^{2} - 63$$ $$T_{23}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( -5 + T^{2} )^{4}$$
$5$ $$( 5 + T^{2} )^{4}$$
$7$ $$( 7 + T^{2} )^{4}$$
$11$ $$( -35 + T^{2} )^{4}$$
$13$ $$( 45 + T^{2} )^{4}$$
$17$ $$( -63 + T^{2} )^{4}$$
$19$ $$T^{8}$$
$23$ $$T^{8}$$
$29$ $$( 35 + T^{2} )^{4}$$
$31$ $$T^{8}$$
$37$ $$T^{8}$$
$41$ $$T^{8}$$
$43$ $$T^{8}$$
$47$ $$( 63 + T^{2} )^{4}$$
$53$ $$T^{8}$$
$59$ $$T^{8}$$
$61$ $$T^{8}$$
$67$ $$T^{8}$$
$71$ $$( 144 + T^{2} )^{4}$$
$73$ $$( -112 + T^{2} )^{4}$$
$79$ $$( 1 + T^{2} )^{4}$$
$83$ $$( -80 + T^{2} )^{4}$$
$89$ $$T^{8}$$
$97$ $$( -343 + T^{2} )^{4}$$