# Properties

 Label 280.2.q.c 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: $$\Z[a_1, \ldots, a_{5}]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( 2 - 2 \zeta_{6} ) q^{3} + \zeta_{6} q^{5} + ( 3 - 2 \zeta_{6} ) q^{7} -\zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 2 - 2 \zeta_{6} ) q^{3} + \zeta_{6} q^{5} + ( 3 - 2 \zeta_{6} ) q^{7} -\zeta_{6} q^{9} + ( 1 - \zeta_{6} ) q^{11} -3 q^{13} + 2 q^{15} + ( 2 - 2 \zeta_{6} ) q^{17} + 5 \zeta_{6} q^{19} + ( 2 - 6 \zeta_{6} ) q^{21} -7 \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{25} + 4 q^{27} -6 q^{29} + ( -4 + 4 \zeta_{6} ) q^{31} -2 \zeta_{6} q^{33} + ( 2 + \zeta_{6} ) q^{35} + 5 \zeta_{6} q^{37} + ( -6 + 6 \zeta_{6} ) q^{39} -5 q^{41} + 6 q^{43} + ( 1 - \zeta_{6} ) q^{45} + 9 \zeta_{6} q^{47} + ( 5 - 8 \zeta_{6} ) q^{49} -4 \zeta_{6} q^{51} + ( -11 + 11 \zeta_{6} ) q^{53} + q^{55} + 10 q^{57} + ( -8 + 8 \zeta_{6} ) q^{59} + 12 \zeta_{6} q^{61} + ( -2 - \zeta_{6} ) q^{63} -3 \zeta_{6} q^{65} + ( 4 - 4 \zeta_{6} ) q^{67} -14 q^{69} -4 q^{71} + ( -12 + 12 \zeta_{6} ) q^{73} + 2 \zeta_{6} q^{75} + ( 1 - 3 \zeta_{6} ) q^{77} -14 \zeta_{6} q^{79} + ( 11 - 11 \zeta_{6} ) q^{81} -4 q^{83} + 2 q^{85} + ( -12 + 12 \zeta_{6} ) q^{87} -6 \zeta_{6} q^{89} + ( -9 + 6 \zeta_{6} ) q^{91} + 8 \zeta_{6} q^{93} + ( -5 + 5 \zeta_{6} ) q^{95} + 6 q^{97} - q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{3} + q^{5} + 4q^{7} - q^{9} + O(q^{10})$$ $$2q + 2q^{3} + q^{5} + 4q^{7} - q^{9} + q^{11} - 6q^{13} + 4q^{15} + 2q^{17} + 5q^{19} - 2q^{21} - 7q^{23} - q^{25} + 8q^{27} - 12q^{29} - 4q^{31} - 2q^{33} + 5q^{35} + 5q^{37} - 6q^{39} - 10q^{41} + 12q^{43} + q^{45} + 9q^{47} + 2q^{49} - 4q^{51} - 11q^{53} + 2q^{55} + 20q^{57} - 8q^{59} + 12q^{61} - 5q^{63} - 3q^{65} + 4q^{67} - 28q^{69} - 8q^{71} - 12q^{73} + 2q^{75} - q^{77} - 14q^{79} + 11q^{81} - 8q^{83} + 4q^{85} - 12q^{87} - 6q^{89} - 12q^{91} + 8q^{93} - 5q^{95} + 12q^{97} - 2q^{99} + O(q^{100})$$

## 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 1.00000 1.73205i 0 0.500000 + 0.866025i 0 2.00000 1.73205i 0 −0.500000 0.866025i 0
121.1 0 1.00000 + 1.73205i 0 0.500000 0.866025i 0 2.00000 + 1.73205i 0 −0.500000 + 0.866025i 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.c 2
3.b odd 2 1 2520.2.bi.e 2
4.b odd 2 1 560.2.q.c 2
5.b even 2 1 1400.2.q.a 2
5.c odd 4 2 1400.2.bh.e 4
7.b odd 2 1 1960.2.q.c 2
7.c even 3 1 inner 280.2.q.c 2
7.c even 3 1 1960.2.a.a 1
7.d odd 6 1 1960.2.a.m 1
7.d odd 6 1 1960.2.q.c 2
21.h odd 6 1 2520.2.bi.e 2
28.f even 6 1 3920.2.a.i 1
28.g odd 6 1 560.2.q.c 2
28.g odd 6 1 3920.2.a.bf 1
35.i odd 6 1 9800.2.a.g 1
35.j even 6 1 1400.2.q.a 2
35.j even 6 1 9800.2.a.bi 1
35.l odd 12 2 1400.2.bh.e 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 2 T_{3} + 4$$ acting on $$S_{2}^{\mathrm{new}}(280, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$4 - 2 T + T^{2}$$
$5$ $$1 - T + T^{2}$$
$7$ $$7 - 4 T + T^{2}$$
$11$ $$1 - T + T^{2}$$
$13$ $$( 3 + T )^{2}$$
$17$ $$4 - 2 T + T^{2}$$
$19$ $$25 - 5 T + T^{2}$$
$23$ $$49 + 7 T + T^{2}$$
$29$ $$( 6 + T )^{2}$$
$31$ $$16 + 4 T + T^{2}$$
$37$ $$25 - 5 T + T^{2}$$
$41$ $$( 5 + T )^{2}$$
$43$ $$( -6 + T )^{2}$$
$47$ $$81 - 9 T + T^{2}$$
$53$ $$121 + 11 T + T^{2}$$
$59$ $$64 + 8 T + T^{2}$$
$61$ $$144 - 12 T + T^{2}$$
$67$ $$16 - 4 T + T^{2}$$
$71$ $$( 4 + T )^{2}$$
$73$ $$144 + 12 T + T^{2}$$
$79$ $$196 + 14 T + T^{2}$$
$83$ $$( 4 + T )^{2}$$
$89$ $$36 + 6 T + T^{2}$$
$97$ $$( -6 + T )^{2}$$