# Properties

 Label 980.1.y.a Level $980$ Weight $1$ Character orbit 980.y Analytic conductor $0.489$ Analytic rank $0$ Dimension $16$ Projective image $D_{8}$ CM discriminant -4 Inner twists $16$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$980 = 2^{2} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 980.y (of order $$12$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.489083712380$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$4$$ over $$\Q(\zeta_{12})$$ Coefficient field: $$\Q(\zeta_{48})$$ Defining polynomial: $$x^{16} - x^{8} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{8}$$ Projective field: Galois closure of 8.0.823543000000.2

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\zeta_{48}^{2} q^{2} + \zeta_{48}^{4} q^{4} + \zeta_{48}^{5} q^{5} -\zeta_{48}^{6} q^{8} + \zeta_{48}^{20} q^{9} +O(q^{10})$$ $$q -\zeta_{48}^{2} q^{2} + \zeta_{48}^{4} q^{4} + \zeta_{48}^{5} q^{5} -\zeta_{48}^{6} q^{8} + \zeta_{48}^{20} q^{9} -\zeta_{48}^{7} q^{10} + ( -\zeta_{48}^{15} - \zeta_{48}^{21} ) q^{13} + \zeta_{48}^{8} q^{16} + ( \zeta_{48}^{7} - \zeta_{48}^{13} ) q^{17} -\zeta_{48}^{22} q^{18} + \zeta_{48}^{9} q^{20} + \zeta_{48}^{10} q^{25} + ( \zeta_{48}^{17} + \zeta_{48}^{23} ) q^{26} + ( \zeta_{48}^{6} + \zeta_{48}^{18} ) q^{29} -\zeta_{48}^{10} q^{32} + ( -\zeta_{48}^{9} + \zeta_{48}^{15} ) q^{34} - q^{36} -\zeta_{48}^{11} q^{40} + ( \zeta_{48}^{3} + \zeta_{48}^{21} ) q^{41} -\zeta_{48} q^{45} -\zeta_{48}^{12} q^{50} + ( \zeta_{48} - \zeta_{48}^{19} ) q^{52} + ( -\zeta_{48}^{4} + \zeta_{48}^{16} ) q^{53} + ( -\zeta_{48}^{8} - \zeta_{48}^{20} ) q^{58} + ( -\zeta_{48}^{17} - \zeta_{48}^{23} ) q^{61} + \zeta_{48}^{12} q^{64} + ( \zeta_{48}^{2} - \zeta_{48}^{20} ) q^{65} + ( \zeta_{48}^{11} - \zeta_{48}^{17} ) q^{68} + \zeta_{48}^{2} q^{72} + ( \zeta_{48} + \zeta_{48}^{19} ) q^{73} + \zeta_{48}^{13} q^{80} -\zeta_{48}^{16} q^{81} + ( -\zeta_{48}^{5} - \zeta_{48}^{23} ) q^{82} + ( \zeta_{48}^{12} - \zeta_{48}^{18} ) q^{85} + ( -\zeta_{48}^{5} - \zeta_{48}^{11} ) q^{89} + \zeta_{48}^{3} q^{90} + ( -\zeta_{48}^{3} - \zeta_{48}^{9} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + O(q^{10})$$ $$16q + 8q^{16} - 16q^{36} - 8q^{53} - 8q^{58} + 8q^{81} + O(q^{100})$$

## Character values

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

 $$n$$ $$101$$ $$197$$ $$491$$ $$\chi(n)$$ $$\zeta_{48}^{8}$$ $$-\zeta_{48}^{12}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
227.1
 0.793353 − 0.608761i −0.793353 + 0.608761i 0.608761 + 0.793353i −0.608761 − 0.793353i −0.991445 − 0.130526i 0.991445 + 0.130526i −0.130526 + 0.991445i 0.130526 − 0.991445i −0.991445 + 0.130526i 0.991445 − 0.130526i −0.130526 − 0.991445i 0.130526 + 0.991445i 0.793353 + 0.608761i −0.793353 − 0.608761i 0.608761 − 0.793353i −0.608761 + 0.793353i
−0.258819 + 0.965926i 0 −0.866025 0.500000i −0.991445 + 0.130526i 0 0 0.707107 0.707107i 0.866025 0.500000i 0.130526 0.991445i
227.2 −0.258819 + 0.965926i 0 −0.866025 0.500000i 0.991445 0.130526i 0 0 0.707107 0.707107i 0.866025 0.500000i −0.130526 + 0.991445i
227.3 0.258819 0.965926i 0 −0.866025 0.500000i −0.130526 0.991445i 0 0 −0.707107 + 0.707107i 0.866025 0.500000i −0.991445 0.130526i
227.4 0.258819 0.965926i 0 −0.866025 0.500000i 0.130526 + 0.991445i 0 0 −0.707107 + 0.707107i 0.866025 0.500000i 0.991445 + 0.130526i
423.1 −0.965926 0.258819i 0 0.866025 + 0.500000i −0.793353 0.608761i 0 0 −0.707107 0.707107i −0.866025 + 0.500000i 0.608761 + 0.793353i
423.2 −0.965926 0.258819i 0 0.866025 + 0.500000i 0.793353 + 0.608761i 0 0 −0.707107 0.707107i −0.866025 + 0.500000i −0.608761 0.793353i
423.3 0.965926 + 0.258819i 0 0.866025 + 0.500000i −0.608761 + 0.793353i 0 0 0.707107 + 0.707107i −0.866025 + 0.500000i −0.793353 + 0.608761i
423.4 0.965926 + 0.258819i 0 0.866025 + 0.500000i 0.608761 0.793353i 0 0 0.707107 + 0.707107i −0.866025 + 0.500000i 0.793353 0.608761i
607.1 −0.965926 + 0.258819i 0 0.866025 0.500000i −0.793353 + 0.608761i 0 0 −0.707107 + 0.707107i −0.866025 0.500000i 0.608761 0.793353i
607.2 −0.965926 + 0.258819i 0 0.866025 0.500000i 0.793353 0.608761i 0 0 −0.707107 + 0.707107i −0.866025 0.500000i −0.608761 + 0.793353i
607.3 0.965926 0.258819i 0 0.866025 0.500000i −0.608761 0.793353i 0 0 0.707107 0.707107i −0.866025 0.500000i −0.793353 0.608761i
607.4 0.965926 0.258819i 0 0.866025 0.500000i 0.608761 + 0.793353i 0 0 0.707107 0.707107i −0.866025 0.500000i 0.793353 + 0.608761i
803.1 −0.258819 0.965926i 0 −0.866025 + 0.500000i −0.991445 0.130526i 0 0 0.707107 + 0.707107i 0.866025 + 0.500000i 0.130526 + 0.991445i
803.2 −0.258819 0.965926i 0 −0.866025 + 0.500000i 0.991445 + 0.130526i 0 0 0.707107 + 0.707107i 0.866025 + 0.500000i −0.130526 0.991445i
803.3 0.258819 + 0.965926i 0 −0.866025 + 0.500000i −0.130526 + 0.991445i 0 0 −0.707107 0.707107i 0.866025 + 0.500000i −0.991445 + 0.130526i
803.4 0.258819 + 0.965926i 0 −0.866025 + 0.500000i 0.130526 0.991445i 0 0 −0.707107 0.707107i 0.866025 + 0.500000i 0.991445 0.130526i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 803.4 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.c odd 4 1 inner
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
20.e even 4 1 inner
28.d even 2 1 inner
28.f even 6 1 inner
28.g odd 6 1 inner
35.f even 4 1 inner
35.k even 12 1 inner
35.l odd 12 1 inner
140.j odd 4 1 inner
140.w even 12 1 inner
140.x odd 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 980.1.y.a 16
4.b odd 2 1 CM 980.1.y.a 16
5.c odd 4 1 inner 980.1.y.a 16
7.b odd 2 1 inner 980.1.y.a 16
7.c even 3 1 980.1.j.a 8
7.c even 3 1 inner 980.1.y.a 16
7.d odd 6 1 980.1.j.a 8
7.d odd 6 1 inner 980.1.y.a 16
20.e even 4 1 inner 980.1.y.a 16
28.d even 2 1 inner 980.1.y.a 16
28.f even 6 1 980.1.j.a 8
28.f even 6 1 inner 980.1.y.a 16
28.g odd 6 1 980.1.j.a 8
28.g odd 6 1 inner 980.1.y.a 16
35.f even 4 1 inner 980.1.y.a 16
35.k even 12 1 980.1.j.a 8
35.k even 12 1 inner 980.1.y.a 16
35.l odd 12 1 980.1.j.a 8
35.l odd 12 1 inner 980.1.y.a 16
140.j odd 4 1 inner 980.1.y.a 16
140.w even 12 1 980.1.j.a 8
140.w even 12 1 inner 980.1.y.a 16
140.x odd 12 1 980.1.j.a 8
140.x odd 12 1 inner 980.1.y.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
980.1.j.a 8 7.c even 3 1
980.1.j.a 8 7.d odd 6 1
980.1.j.a 8 28.f even 6 1
980.1.j.a 8 28.g odd 6 1
980.1.j.a 8 35.k even 12 1
980.1.j.a 8 35.l odd 12 1
980.1.j.a 8 140.w even 12 1
980.1.j.a 8 140.x odd 12 1
980.1.y.a 16 1.a even 1 1 trivial
980.1.y.a 16 4.b odd 2 1 CM
980.1.y.a 16 5.c odd 4 1 inner
980.1.y.a 16 7.b odd 2 1 inner
980.1.y.a 16 7.c even 3 1 inner
980.1.y.a 16 7.d odd 6 1 inner
980.1.y.a 16 20.e even 4 1 inner
980.1.y.a 16 28.d even 2 1 inner
980.1.y.a 16 28.f even 6 1 inner
980.1.y.a 16 28.g odd 6 1 inner
980.1.y.a 16 35.f even 4 1 inner
980.1.y.a 16 35.k even 12 1 inner
980.1.y.a 16 35.l odd 12 1 inner
980.1.y.a 16 140.j odd 4 1 inner
980.1.y.a 16 140.w even 12 1 inner
980.1.y.a 16 140.x odd 12 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(980, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T^{4} + T^{8} )^{2}$$
$3$ $$T^{16}$$
$5$ $$1 - T^{8} + T^{16}$$
$7$ $$T^{16}$$
$11$ $$T^{16}$$
$13$ $$( 4 + 12 T^{4} + T^{8} )^{2}$$
$17$ $$16 - 48 T^{4} + 140 T^{8} - 12 T^{12} + T^{16}$$
$19$ $$T^{16}$$
$23$ $$T^{16}$$
$29$ $$( 2 + T^{2} )^{8}$$
$31$ $$T^{16}$$
$37$ $$T^{16}$$
$41$ $$( 2 + 4 T^{2} + T^{4} )^{4}$$
$43$ $$T^{16}$$
$47$ $$T^{16}$$
$53$ $$( 4 + 4 T + 2 T^{2} + 2 T^{3} + T^{4} )^{4}$$
$59$ $$T^{16}$$
$61$ $$( 4 - 8 T^{2} + 14 T^{4} - 4 T^{6} + T^{8} )^{2}$$
$67$ $$T^{16}$$
$71$ $$T^{16}$$
$73$ $$16 - 48 T^{4} + 140 T^{8} - 12 T^{12} + T^{16}$$
$79$ $$T^{16}$$
$83$ $$T^{16}$$
$89$ $$( 4 + 8 T^{2} + 14 T^{4} + 4 T^{6} + T^{8} )^{2}$$
$97$ $$( 4 + 12 T^{4} + T^{8} )^{2}$$
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