# Properties

 Label 980.2.c.a Level $980$ Weight $2$ Character orbit 980.c Analytic conductor $7.825$ Analytic rank $0$ Dimension $8$ CM discriminant -4 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$980 = 2^{2} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 980.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.82533939809$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\Q(\zeta_{16})$$ Defining polynomial: $$x^{8} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$7^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{16}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{16}^{2} - \zeta_{16}^{6} ) q^{2} -2 q^{4} + ( -2 \zeta_{16} + \zeta_{16}^{5} ) q^{5} + ( 2 \zeta_{16}^{2} + 2 \zeta_{16}^{6} ) q^{8} + 3 q^{9} +O(q^{10})$$ $$q + ( -\zeta_{16}^{2} - \zeta_{16}^{6} ) q^{2} -2 q^{4} + ( -2 \zeta_{16} + \zeta_{16}^{5} ) q^{5} + ( 2 \zeta_{16}^{2} + 2 \zeta_{16}^{6} ) q^{8} + 3 q^{9} + ( 3 \zeta_{16}^{3} + \zeta_{16}^{7} ) q^{10} + ( 3 \zeta_{16} - 2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} - 3 \zeta_{16}^{7} ) q^{13} + 4 q^{16} + ( -\zeta_{16} - 4 \zeta_{16}^{3} + 4 \zeta_{16}^{5} + \zeta_{16}^{7} ) q^{17} + ( -3 \zeta_{16}^{2} - 3 \zeta_{16}^{6} ) q^{18} + ( 4 \zeta_{16} - 2 \zeta_{16}^{5} ) q^{20} + ( 3 \zeta_{16}^{2} - 4 \zeta_{16}^{6} ) q^{25} + ( -5 \zeta_{16} - \zeta_{16}^{3} - \zeta_{16}^{5} - 5 \zeta_{16}^{7} ) q^{26} + ( 7 \zeta_{16}^{2} - 7 \zeta_{16}^{6} ) q^{29} + ( -4 \zeta_{16}^{2} - 4 \zeta_{16}^{6} ) q^{32} + ( -3 \zeta_{16} + 5 \zeta_{16}^{3} + 5 \zeta_{16}^{5} - 3 \zeta_{16}^{7} ) q^{34} -6 q^{36} + ( -5 \zeta_{16}^{2} - 5 \zeta_{16}^{6} ) q^{37} + ( -6 \zeta_{16}^{3} - 2 \zeta_{16}^{7} ) q^{40} + ( -4 \zeta_{16} - 5 \zeta_{16}^{3} - 5 \zeta_{16}^{5} - 4 \zeta_{16}^{7} ) q^{41} + ( -6 \zeta_{16} + 3 \zeta_{16}^{5} ) q^{45} + ( -1 - 7 \zeta_{16}^{4} ) q^{50} + ( -6 \zeta_{16} + 4 \zeta_{16}^{3} - 4 \zeta_{16}^{5} + 6 \zeta_{16}^{7} ) q^{52} -14 \zeta_{16}^{4} q^{53} -14 \zeta_{16}^{4} q^{58} + ( -5 \zeta_{16} + 6 \zeta_{16}^{3} + 6 \zeta_{16}^{5} - 5 \zeta_{16}^{7} ) q^{61} -8 q^{64} + ( -4 - 8 \zeta_{16}^{2} + 7 \zeta_{16}^{4} - \zeta_{16}^{6} ) q^{65} + ( 2 \zeta_{16} + 8 \zeta_{16}^{3} - 8 \zeta_{16}^{5} - 2 \zeta_{16}^{7} ) q^{68} + ( 6 \zeta_{16}^{2} + 6 \zeta_{16}^{6} ) q^{72} + ( 8 \zeta_{16} - 3 \zeta_{16}^{3} + 3 \zeta_{16}^{5} - 8 \zeta_{16}^{7} ) q^{73} -10 q^{74} + ( -8 \zeta_{16} + 4 \zeta_{16}^{5} ) q^{80} + 9 q^{81} + ( -9 \zeta_{16} - \zeta_{16}^{3} + \zeta_{16}^{5} + 9 \zeta_{16}^{7} ) q^{82} + ( 6 - 2 \zeta_{16}^{2} + 7 \zeta_{16}^{4} - 9 \zeta_{16}^{6} ) q^{85} + ( 5 \zeta_{16} + 8 \zeta_{16}^{3} + 8 \zeta_{16}^{5} + 5 \zeta_{16}^{7} ) q^{89} + ( 9 \zeta_{16}^{3} + 3 \zeta_{16}^{7} ) q^{90} + ( 4 \zeta_{16} + 9 \zeta_{16}^{3} - 9 \zeta_{16}^{5} - 4 \zeta_{16}^{7} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 16q^{4} + 24q^{9} + O(q^{10})$$ $$8q - 16q^{4} + 24q^{9} + 32q^{16} - 48q^{36} - 8q^{50} - 64q^{64} - 32q^{65} - 80q^{74} + 72q^{81} + 48q^{85} + O(q^{100})$$

## Character values

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

 $$n$$ $$101$$ $$197$$ $$491$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
979.1
 0.923880 + 0.382683i −0.382683 − 0.923880i 0.382683 + 0.923880i −0.923880 − 0.382683i 0.923880 − 0.382683i −0.382683 + 0.923880i 0.382683 − 0.923880i −0.923880 + 0.382683i
1.41421i 0 −2.00000 −2.23044 + 0.158513i 0 0 2.82843i 3.00000 0.224171 + 3.15432i
979.2 1.41421i 0 −2.00000 −0.158513 + 2.23044i 0 0 2.82843i 3.00000 3.15432 + 0.224171i
979.3 1.41421i 0 −2.00000 0.158513 2.23044i 0 0 2.82843i 3.00000 −3.15432 0.224171i
979.4 1.41421i 0 −2.00000 2.23044 0.158513i 0 0 2.82843i 3.00000 −0.224171 3.15432i
979.5 1.41421i 0 −2.00000 −2.23044 0.158513i 0 0 2.82843i 3.00000 0.224171 3.15432i
979.6 1.41421i 0 −2.00000 −0.158513 2.23044i 0 0 2.82843i 3.00000 3.15432 0.224171i
979.7 1.41421i 0 −2.00000 0.158513 + 2.23044i 0 0 2.82843i 3.00000 −3.15432 + 0.224171i
979.8 1.41421i 0 −2.00000 2.23044 + 0.158513i 0 0 2.82843i 3.00000 −0.224171 + 3.15432i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 979.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
5.b even 2 1 inner
7.b odd 2 1 inner
20.d odd 2 1 inner
28.d even 2 1 inner
35.c odd 2 1 inner
140.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 980.2.c.a 8
4.b odd 2 1 CM 980.2.c.a 8
5.b even 2 1 inner 980.2.c.a 8
7.b odd 2 1 inner 980.2.c.a 8
7.c even 3 2 980.2.s.c 16
7.d odd 6 2 980.2.s.c 16
20.d odd 2 1 inner 980.2.c.a 8
28.d even 2 1 inner 980.2.c.a 8
28.f even 6 2 980.2.s.c 16
28.g odd 6 2 980.2.s.c 16
35.c odd 2 1 inner 980.2.c.a 8
35.i odd 6 2 980.2.s.c 16
35.j even 6 2 980.2.s.c 16
140.c even 2 1 inner 980.2.c.a 8
140.p odd 6 2 980.2.s.c 16
140.s even 6 2 980.2.s.c 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
980.2.c.a 8 1.a even 1 1 trivial
980.2.c.a 8 4.b odd 2 1 CM
980.2.c.a 8 5.b even 2 1 inner
980.2.c.a 8 7.b odd 2 1 inner
980.2.c.a 8 20.d odd 2 1 inner
980.2.c.a 8 28.d even 2 1 inner
980.2.c.a 8 35.c odd 2 1 inner
980.2.c.a 8 140.c even 2 1 inner
980.2.s.c 16 7.c even 3 2
980.2.s.c 16 7.d odd 6 2
980.2.s.c 16 28.f even 6 2
980.2.s.c 16 28.g odd 6 2
980.2.s.c 16 35.i odd 6 2
980.2.s.c 16 35.j even 6 2
980.2.s.c 16 140.p odd 6 2
980.2.s.c 16 140.s even 6 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}$$ acting on $$S_{2}^{\mathrm{new}}(980, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 2 + T^{2} )^{4}$$
$3$ $$T^{8}$$
$5$ $$625 - 48 T^{4} + T^{8}$$
$7$ $$T^{8}$$
$11$ $$T^{8}$$
$13$ $$( 578 - 52 T^{2} + T^{4} )^{2}$$
$17$ $$( 1058 - 68 T^{2} + T^{4} )^{2}$$
$19$ $$T^{8}$$
$23$ $$T^{8}$$
$29$ $$( -98 + T^{2} )^{4}$$
$31$ $$T^{8}$$
$37$ $$( 50 + T^{2} )^{4}$$
$41$ $$( 1922 + 164 T^{2} + T^{4} )^{2}$$
$43$ $$T^{8}$$
$47$ $$T^{8}$$
$53$ $$( 196 + T^{2} )^{4}$$
$59$ $$T^{8}$$
$61$ $$( 10082 + 244 T^{2} + T^{4} )^{2}$$
$67$ $$T^{8}$$
$71$ $$T^{8}$$
$73$ $$( 21218 - 292 T^{2} + T^{4} )^{2}$$
$79$ $$T^{8}$$
$83$ $$T^{8}$$
$89$ $$( 3362 + 356 T^{2} + T^{4} )^{2}$$
$97$ $$( 37538 - 388 T^{2} + T^{4} )^{2}$$