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:

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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