# Properties

 Label 980.1.n.b Level $980$ Weight $1$ Character orbit 980.n Analytic conductor $0.489$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -35 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$980 = 2^{2} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 980.n (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.489083712380$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 140) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.140.1 Artin image: $C_3\times S_3$ Artin field: Galois closure of 6.0.33614000.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{3} + \zeta_{6}^{2} q^{5} +O(q^{10})$$ $$q + \zeta_{6} q^{3} + \zeta_{6}^{2} q^{5} + \zeta_{6} q^{11} - q^{13} - q^{15} + \zeta_{6} q^{17} -\zeta_{6} q^{25} + q^{27} - q^{29} + \zeta_{6}^{2} q^{33} -\zeta_{6} q^{39} -\zeta_{6}^{2} q^{47} + \zeta_{6}^{2} q^{51} - q^{55} -\zeta_{6}^{2} q^{65} + 2 q^{71} -2 \zeta_{6} q^{73} -\zeta_{6}^{2} q^{75} -\zeta_{6}^{2} q^{79} + \zeta_{6} q^{81} + 2 q^{83} - q^{85} -\zeta_{6} q^{87} - q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{3} - q^{5} + O(q^{10})$$ $$2q + q^{3} - q^{5} + q^{11} - 2q^{13} - 2q^{15} + q^{17} - q^{25} + 2q^{27} - 2q^{29} - q^{33} - q^{39} + q^{47} - q^{51} - 2q^{55} + q^{65} + 4q^{71} - 2q^{73} + q^{75} + q^{79} + q^{81} + 4q^{83} - 2q^{85} - q^{87} - 2q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/980\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$197$$ $$491$$ $$\chi(n)$$ $$-\zeta_{6}^{2}$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
129.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0.500000 0.866025i 0 −0.500000 0.866025i 0 0 0 0 0
509.1 0 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0 0 0 0 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
35.c odd 2 1 CM by $$\Q(\sqrt{-35})$$
7.c even 3 1 inner
35.i odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 980.1.n.b 2
4.b odd 2 1 3920.1.br.a 2
5.b even 2 1 980.1.n.a 2
7.b odd 2 1 980.1.n.a 2
7.c even 3 1 140.1.h.a 1
7.c even 3 1 inner 980.1.n.b 2
7.d odd 6 1 140.1.h.b yes 1
7.d odd 6 1 980.1.n.a 2
20.d odd 2 1 3920.1.br.b 2
21.g even 6 1 1260.1.p.b 1
21.h odd 6 1 1260.1.p.a 1
28.d even 2 1 3920.1.br.b 2
28.f even 6 1 560.1.p.a 1
28.f even 6 1 3920.1.br.b 2
28.g odd 6 1 560.1.p.b 1
28.g odd 6 1 3920.1.br.a 2
35.c odd 2 1 CM 980.1.n.b 2
35.i odd 6 1 140.1.h.a 1
35.i odd 6 1 inner 980.1.n.b 2
35.j even 6 1 140.1.h.b yes 1
35.j even 6 1 980.1.n.a 2
35.k even 12 2 700.1.d.a 2
35.l odd 12 2 700.1.d.a 2
56.j odd 6 1 2240.1.p.b 1
56.k odd 6 1 2240.1.p.a 1
56.m even 6 1 2240.1.p.d 1
56.p even 6 1 2240.1.p.c 1
105.o odd 6 1 1260.1.p.b 1
105.p even 6 1 1260.1.p.a 1
140.c even 2 1 3920.1.br.a 2
140.p odd 6 1 560.1.p.a 1
140.p odd 6 1 3920.1.br.b 2
140.s even 6 1 560.1.p.b 1
140.s even 6 1 3920.1.br.a 2
140.w even 12 2 2800.1.f.c 2
140.x odd 12 2 2800.1.f.c 2
280.ba even 6 1 2240.1.p.a 1
280.bf even 6 1 2240.1.p.b 1
280.bi odd 6 1 2240.1.p.d 1
280.bk odd 6 1 2240.1.p.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.1.h.a 1 7.c even 3 1
140.1.h.a 1 35.i odd 6 1
140.1.h.b yes 1 7.d odd 6 1
140.1.h.b yes 1 35.j even 6 1
560.1.p.a 1 28.f even 6 1
560.1.p.a 1 140.p odd 6 1
560.1.p.b 1 28.g odd 6 1
560.1.p.b 1 140.s even 6 1
700.1.d.a 2 35.k even 12 2
700.1.d.a 2 35.l odd 12 2
980.1.n.a 2 5.b even 2 1
980.1.n.a 2 7.b odd 2 1
980.1.n.a 2 7.d odd 6 1
980.1.n.a 2 35.j even 6 1
980.1.n.b 2 1.a even 1 1 trivial
980.1.n.b 2 7.c even 3 1 inner
980.1.n.b 2 35.c odd 2 1 CM
980.1.n.b 2 35.i odd 6 1 inner
1260.1.p.a 1 21.h odd 6 1
1260.1.p.a 1 105.p even 6 1
1260.1.p.b 1 21.g even 6 1
1260.1.p.b 1 105.o odd 6 1
2240.1.p.a 1 56.k odd 6 1
2240.1.p.a 1 280.ba even 6 1
2240.1.p.b 1 56.j odd 6 1
2240.1.p.b 1 280.bf even 6 1
2240.1.p.c 1 56.p even 6 1
2240.1.p.c 1 280.bk odd 6 1
2240.1.p.d 1 56.m even 6 1
2240.1.p.d 1 280.bi odd 6 1
2800.1.f.c 2 140.w even 12 2
2800.1.f.c 2 140.x odd 12 2
3920.1.br.a 2 4.b odd 2 1
3920.1.br.a 2 28.g odd 6 1
3920.1.br.a 2 140.c even 2 1
3920.1.br.a 2 140.s even 6 1
3920.1.br.b 2 20.d odd 2 1
3920.1.br.b 2 28.d even 2 1
3920.1.br.b 2 28.f even 6 1
3920.1.br.b 2 140.p odd 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 $$S_{1}^{\mathrm{new}}(980, [\chi])$$.

## Hecke characteristic polynomials

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