Properties

 Label 280.1.bi.b Level $280$ Weight $1$ Character orbit 280.bi Analytic conductor $0.140$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -40 Inner twists $4$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$280 = 2^{3} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 280.bi (of order $$6$$, degree $$2$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$0.139738203537$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.1960.1 Artin image: $C_6\times S_3$ Artin field: Galois closure of $$\mathbb{Q}[x]/(x^{12} - \cdots)$$

$q$-expansion

The $$q$$-expansion and trace form are shown below.

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

Character values

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

 $$n$$ $$57$$ $$71$$ $$141$$ $$241$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$ $$\zeta_{6}^{2}$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
179.1
 0.5 − 0.866025i 0.5 + 0.866025i
0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0.500000 + 0.866025i 0 −1.00000 −1.00000 −0.500000 0.866025i −0.500000 + 0.866025i
219.1 0.500000 0.866025i 0 −0.500000 0.866025i 0.500000 0.866025i 0 −1.00000 −1.00000 −0.500000 + 0.866025i −0.500000 0.866025i
 $$n$$: 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 $$\Q(\sqrt{-10})$$
7.c even 3 1 inner
280.bi odd 6 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 280.1.bi.b yes 2
3.b odd 2 1 2520.1.ef.a 2
4.b odd 2 1 1120.1.by.b 2
5.b even 2 1 280.1.bi.a 2
5.c odd 4 2 1400.1.ba.a 4
7.b odd 2 1 1960.1.bi.b 2
7.c even 3 1 inner 280.1.bi.b yes 2
7.c even 3 1 1960.1.i.a 1
7.d odd 6 1 1960.1.i.b 1
7.d odd 6 1 1960.1.bi.b 2
8.b even 2 1 1120.1.by.a 2
8.d odd 2 1 280.1.bi.a 2
15.d odd 2 1 2520.1.ef.b 2
20.d odd 2 1 1120.1.by.a 2
21.h odd 6 1 2520.1.ef.a 2
24.f even 2 1 2520.1.ef.b 2
28.g odd 6 1 1120.1.by.b 2
35.c odd 2 1 1960.1.bi.a 2
35.i odd 6 1 1960.1.i.c 1
35.i odd 6 1 1960.1.bi.a 2
35.j even 6 1 280.1.bi.a 2
35.j even 6 1 1960.1.i.d 1
35.l odd 12 2 1400.1.ba.a 4
40.e odd 2 1 CM 280.1.bi.b yes 2
40.f even 2 1 1120.1.by.b 2
40.k even 4 2 1400.1.ba.a 4
56.e even 2 1 1960.1.bi.a 2
56.k odd 6 1 280.1.bi.a 2
56.k odd 6 1 1960.1.i.d 1
56.m even 6 1 1960.1.i.c 1
56.m even 6 1 1960.1.bi.a 2
56.p even 6 1 1120.1.by.a 2
105.o odd 6 1 2520.1.ef.b 2
120.m even 2 1 2520.1.ef.a 2
140.p odd 6 1 1120.1.by.a 2
168.v even 6 1 2520.1.ef.b 2
280.n even 2 1 1960.1.bi.b 2
280.ba even 6 1 1960.1.i.b 1
280.ba even 6 1 1960.1.bi.b 2
280.bf even 6 1 1120.1.by.b 2
280.bi odd 6 1 inner 280.1.bi.b yes 2
280.bi odd 6 1 1960.1.i.a 1
280.br even 12 2 1400.1.ba.a 4
840.cv even 6 1 2520.1.ef.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.1.bi.a 2 5.b even 2 1
280.1.bi.a 2 8.d odd 2 1
280.1.bi.a 2 35.j even 6 1
280.1.bi.a 2 56.k odd 6 1
280.1.bi.b yes 2 1.a even 1 1 trivial
280.1.bi.b yes 2 7.c even 3 1 inner
280.1.bi.b yes 2 40.e odd 2 1 CM
280.1.bi.b yes 2 280.bi odd 6 1 inner
1120.1.by.a 2 8.b even 2 1
1120.1.by.a 2 20.d odd 2 1
1120.1.by.a 2 56.p even 6 1
1120.1.by.a 2 140.p odd 6 1
1120.1.by.b 2 4.b odd 2 1
1120.1.by.b 2 28.g odd 6 1
1120.1.by.b 2 40.f even 2 1
1120.1.by.b 2 280.bf even 6 1
1400.1.ba.a 4 5.c odd 4 2
1400.1.ba.a 4 35.l odd 12 2
1400.1.ba.a 4 40.k even 4 2
1400.1.ba.a 4 280.br even 12 2
1960.1.i.a 1 7.c even 3 1
1960.1.i.a 1 280.bi odd 6 1
1960.1.i.b 1 7.d odd 6 1
1960.1.i.b 1 280.ba even 6 1
1960.1.i.c 1 35.i odd 6 1
1960.1.i.c 1 56.m even 6 1
1960.1.i.d 1 35.j even 6 1
1960.1.i.d 1 56.k odd 6 1
1960.1.bi.a 2 35.c odd 2 1
1960.1.bi.a 2 35.i odd 6 1
1960.1.bi.a 2 56.e even 2 1
1960.1.bi.a 2 56.m even 6 1
1960.1.bi.b 2 7.b odd 2 1
1960.1.bi.b 2 7.d odd 6 1
1960.1.bi.b 2 280.n even 2 1
1960.1.bi.b 2 280.ba even 6 1
2520.1.ef.a 2 3.b odd 2 1
2520.1.ef.a 2 21.h odd 6 1
2520.1.ef.a 2 120.m even 2 1
2520.1.ef.a 2 840.cv even 6 1
2520.1.ef.b 2 15.d odd 2 1
2520.1.ef.b 2 24.f even 2 1
2520.1.ef.b 2 105.o odd 6 1
2520.1.ef.b 2 168.v even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{13} - 1$$ acting on $$S_{1}^{\mathrm{new}}(280, [\chi])$$.

Hecke characteristic polynomials

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