# Properties

 Label 280.2.q.b Level $280$ Weight $2$ Character orbit 280.q Analytic conductor $2.236$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$2.23581125660$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{6} ) q^{3} + \zeta_{6} q^{5} + ( -1 + 3 \zeta_{6} ) q^{7} + 2 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 1 - \zeta_{6} ) q^{3} + \zeta_{6} q^{5} + ( -1 + 3 \zeta_{6} ) q^{7} + 2 \zeta_{6} q^{9} + ( 2 - 2 \zeta_{6} ) q^{11} + 4 q^{13} + q^{15} -6 \zeta_{6} q^{19} + ( 2 + \zeta_{6} ) q^{21} -3 \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{25} + 5 q^{27} -3 q^{29} -2 \zeta_{6} q^{33} + ( -3 + 2 \zeta_{6} ) q^{35} + 12 \zeta_{6} q^{37} + ( 4 - 4 \zeta_{6} ) q^{39} -7 q^{41} -9 q^{43} + ( -2 + 2 \zeta_{6} ) q^{45} + ( -8 + 3 \zeta_{6} ) q^{49} + ( 6 - 6 \zeta_{6} ) q^{53} + 2 q^{55} -6 q^{57} + ( 10 - 10 \zeta_{6} ) q^{59} -5 \zeta_{6} q^{61} + ( -6 + 4 \zeta_{6} ) q^{63} + 4 \zeta_{6} q^{65} + ( -11 + 11 \zeta_{6} ) q^{67} -3 q^{69} -10 q^{71} + ( 8 - 8 \zeta_{6} ) q^{73} + \zeta_{6} q^{75} + ( 4 + 2 \zeta_{6} ) q^{77} -6 \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{81} -3 q^{83} + ( -3 + 3 \zeta_{6} ) q^{87} -17 \zeta_{6} q^{89} + ( -4 + 12 \zeta_{6} ) q^{91} + ( 6 - 6 \zeta_{6} ) q^{95} -2 q^{97} + 4 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{3} + q^{5} + q^{7} + 2q^{9} + O(q^{10})$$ $$2q + q^{3} + q^{5} + q^{7} + 2q^{9} + 2q^{11} + 8q^{13} + 2q^{15} - 6q^{19} + 5q^{21} - 3q^{23} - q^{25} + 10q^{27} - 6q^{29} - 2q^{33} - 4q^{35} + 12q^{37} + 4q^{39} - 14q^{41} - 18q^{43} - 2q^{45} - 13q^{49} + 6q^{53} + 4q^{55} - 12q^{57} + 10q^{59} - 5q^{61} - 8q^{63} + 4q^{65} - 11q^{67} - 6q^{69} - 20q^{71} + 8q^{73} + q^{75} + 10q^{77} - 6q^{79} - q^{81} - 6q^{83} - 3q^{87} - 17q^{89} + 4q^{91} + 6q^{95} - 4q^{97} + 8q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$57$$ $$71$$ $$141$$ $$241$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
81.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0.500000 0.866025i 0 0.500000 + 0.866025i 0 0.500000 + 2.59808i 0 1.00000 + 1.73205i 0
121.1 0 0.500000 + 0.866025i 0 0.500000 0.866025i 0 0.500000 2.59808i 0 1.00000 1.73205i 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 280.2.q.b 2
3.b odd 2 1 2520.2.bi.d 2
4.b odd 2 1 560.2.q.e 2
5.b even 2 1 1400.2.q.c 2
5.c odd 4 2 1400.2.bh.c 4
7.b odd 2 1 1960.2.q.d 2
7.c even 3 1 inner 280.2.q.b 2
7.c even 3 1 1960.2.a.c 1
7.d odd 6 1 1960.2.a.l 1
7.d odd 6 1 1960.2.q.d 2
21.h odd 6 1 2520.2.bi.d 2
28.f even 6 1 3920.2.a.q 1
28.g odd 6 1 560.2.q.e 2
28.g odd 6 1 3920.2.a.v 1
35.i odd 6 1 9800.2.a.o 1
35.j even 6 1 1400.2.q.c 2
35.j even 6 1 9800.2.a.z 1
35.l odd 12 2 1400.2.bh.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.q.b 2 1.a even 1 1 trivial
280.2.q.b 2 7.c even 3 1 inner
560.2.q.e 2 4.b odd 2 1
560.2.q.e 2 28.g odd 6 1
1400.2.q.c 2 5.b even 2 1
1400.2.q.c 2 35.j even 6 1
1400.2.bh.c 4 5.c odd 4 2
1400.2.bh.c 4 35.l odd 12 2
1960.2.a.c 1 7.c even 3 1
1960.2.a.l 1 7.d odd 6 1
1960.2.q.d 2 7.b odd 2 1
1960.2.q.d 2 7.d odd 6 1
2520.2.bi.d 2 3.b odd 2 1
2520.2.bi.d 2 21.h odd 6 1
3920.2.a.q 1 28.f even 6 1
3920.2.a.v 1 28.g odd 6 1
9800.2.a.o 1 35.i odd 6 1
9800.2.a.z 1 35.j even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - T_{3} + 1$$ acting on $$S_{2}^{\mathrm{new}}(280, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$1 - T + T^{2}$$
$5$ $$1 - T + T^{2}$$
$7$ $$7 - T + T^{2}$$
$11$ $$4 - 2 T + T^{2}$$
$13$ $$( -4 + T )^{2}$$
$17$ $$T^{2}$$
$19$ $$36 + 6 T + T^{2}$$
$23$ $$9 + 3 T + T^{2}$$
$29$ $$( 3 + T )^{2}$$
$31$ $$T^{2}$$
$37$ $$144 - 12 T + T^{2}$$
$41$ $$( 7 + T )^{2}$$
$43$ $$( 9 + T )^{2}$$
$47$ $$T^{2}$$
$53$ $$36 - 6 T + T^{2}$$
$59$ $$100 - 10 T + T^{2}$$
$61$ $$25 + 5 T + T^{2}$$
$67$ $$121 + 11 T + T^{2}$$
$71$ $$( 10 + T )^{2}$$
$73$ $$64 - 8 T + T^{2}$$
$79$ $$36 + 6 T + T^{2}$$
$83$ $$( 3 + T )^{2}$$
$89$ $$289 + 17 T + T^{2}$$
$97$ $$( 2 + T )^{2}$$