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

Downloads

Learn more

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:

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

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