# Properties

 Label 280.2.n.a Level $280$ Weight $2$ Character orbit 280.n Analytic conductor $2.236$ Analytic rank $0$ Dimension $4$ CM discriminant -40 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$280 = 2^{3} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 280.n (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.23581125660$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-5})$$ Defining polynomial: $$x^{4} + 4 x^{2} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + 2 q^{4} + \beta_{3} q^{5} + ( -\beta_{1} + \beta_{3} ) q^{7} -2 \beta_{1} q^{8} -3 q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{2} + 2 q^{4} + \beta_{3} q^{5} + ( -\beta_{1} + \beta_{3} ) q^{7} -2 \beta_{1} q^{8} -3 q^{9} + \beta_{2} q^{10} -2 q^{11} -2 \beta_{3} q^{13} + ( 2 + \beta_{2} ) q^{14} + 4 q^{16} + 3 \beta_{1} q^{18} -2 \beta_{2} q^{19} + 2 \beta_{3} q^{20} + 2 \beta_{1} q^{22} -6 \beta_{1} q^{23} -5 q^{25} -2 \beta_{2} q^{26} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{28} -4 \beta_{1} q^{32} + ( -5 + \beta_{2} ) q^{35} -6 q^{36} + 8 \beta_{1} q^{37} -4 \beta_{3} q^{38} + 2 \beta_{2} q^{40} -4 \beta_{2} q^{41} -4 q^{44} -3 \beta_{3} q^{45} + 12 q^{46} + 6 \beta_{3} q^{47} + ( -3 + 2 \beta_{2} ) q^{49} + 5 \beta_{1} q^{50} -4 \beta_{3} q^{52} + 4 \beta_{1} q^{53} -2 \beta_{3} q^{55} + ( 4 + 2 \beta_{2} ) q^{56} + 2 \beta_{2} q^{59} + ( 3 \beta_{1} - 3 \beta_{3} ) q^{63} + 8 q^{64} + 10 q^{65} + ( 5 \beta_{1} + 2 \beta_{3} ) q^{70} + 6 \beta_{1} q^{72} -16 q^{74} -4 \beta_{2} q^{76} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{77} + 4 \beta_{3} q^{80} + 9 q^{81} -8 \beta_{3} q^{82} + 4 \beta_{1} q^{88} + 4 \beta_{2} q^{89} -3 \beta_{2} q^{90} + ( 10 - 2 \beta_{2} ) q^{91} -12 \beta_{1} q^{92} + 6 \beta_{2} q^{94} -10 \beta_{1} q^{95} + ( 3 \beta_{1} + 4 \beta_{3} ) q^{98} + 6 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 8q^{4} - 12q^{9} + O(q^{10})$$ $$4q + 8q^{4} - 12q^{9} - 8q^{11} + 8q^{14} + 16q^{16} - 20q^{25} - 20q^{35} - 24q^{36} - 16q^{44} + 48q^{46} - 12q^{49} + 16q^{56} + 32q^{64} + 40q^{65} - 64q^{74} + 36q^{81} + 40q^{91} + 24q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 4 x^{2} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + \nu$$$$)/3$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 7 \nu$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} + 2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} - 2$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{2} + 7 \beta_{1}$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/280\mathbb{Z}\right)^\times$$.

 $$n$$ $$57$$ $$71$$ $$141$$ $$241$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
139.1
 −0.707107 + 1.58114i −0.707107 − 1.58114i 0.707107 − 1.58114i 0.707107 + 1.58114i
−1.41421 0 2.00000 2.23607i 0 −1.41421 2.23607i −2.82843 −3.00000 3.16228i
139.2 −1.41421 0 2.00000 2.23607i 0 −1.41421 + 2.23607i −2.82843 −3.00000 3.16228i
139.3 1.41421 0 2.00000 2.23607i 0 1.41421 2.23607i 2.82843 −3.00000 3.16228i
139.4 1.41421 0 2.00000 2.23607i 0 1.41421 + 2.23607i 2.82843 −3.00000 3.16228i
 $$n$$: 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
40.e odd 2 1 CM by $$\Q(\sqrt{-10})$$
5.b even 2 1 inner
7.b odd 2 1 inner
8.d odd 2 1 inner
35.c odd 2 1 inner
56.e even 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 280.2.n.a 4
4.b odd 2 1 1120.2.n.a 4
5.b even 2 1 inner 280.2.n.a 4
7.b odd 2 1 inner 280.2.n.a 4
8.b even 2 1 1120.2.n.a 4
8.d odd 2 1 inner 280.2.n.a 4
20.d odd 2 1 1120.2.n.a 4
28.d even 2 1 1120.2.n.a 4
35.c odd 2 1 inner 280.2.n.a 4
40.e odd 2 1 CM 280.2.n.a 4
40.f even 2 1 1120.2.n.a 4
56.e even 2 1 inner 280.2.n.a 4
56.h odd 2 1 1120.2.n.a 4
140.c even 2 1 1120.2.n.a 4
280.c odd 2 1 1120.2.n.a 4
280.n even 2 1 inner 280.2.n.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.n.a 4 1.a even 1 1 trivial
280.2.n.a 4 5.b even 2 1 inner
280.2.n.a 4 7.b odd 2 1 inner
280.2.n.a 4 8.d odd 2 1 inner
280.2.n.a 4 35.c odd 2 1 inner
280.2.n.a 4 40.e odd 2 1 CM
280.2.n.a 4 56.e even 2 1 inner
280.2.n.a 4 280.n even 2 1 inner
1120.2.n.a 4 4.b odd 2 1
1120.2.n.a 4 8.b even 2 1
1120.2.n.a 4 20.d odd 2 1
1120.2.n.a 4 28.d even 2 1
1120.2.n.a 4 40.f even 2 1
1120.2.n.a 4 56.h odd 2 1
1120.2.n.a 4 140.c even 2 1
1120.2.n.a 4 280.c odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}$$ acting on $$S_{2}^{\mathrm{new}}(280, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -2 + T^{2} )^{2}$$
$3$ $$T^{4}$$
$5$ $$( 5 + T^{2} )^{2}$$
$7$ $$49 + 6 T^{2} + T^{4}$$
$11$ $$( 2 + T )^{4}$$
$13$ $$( 20 + T^{2} )^{2}$$
$17$ $$T^{4}$$
$19$ $$( 40 + T^{2} )^{2}$$
$23$ $$( -72 + T^{2} )^{2}$$
$29$ $$T^{4}$$
$31$ $$T^{4}$$
$37$ $$( -128 + T^{2} )^{2}$$
$41$ $$( 160 + T^{2} )^{2}$$
$43$ $$T^{4}$$
$47$ $$( 180 + T^{2} )^{2}$$
$53$ $$( -32 + T^{2} )^{2}$$
$59$ $$( 40 + T^{2} )^{2}$$
$61$ $$T^{4}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$T^{4}$$
$79$ $$T^{4}$$
$83$ $$T^{4}$$
$89$ $$( 160 + T^{2} )^{2}$$
$97$ $$T^{4}$$