# Properties

 Label 980.2.q.f Level $980$ Weight $2$ Character orbit 980.q Analytic conductor $7.825$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$7.82533939809$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 140) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 \zeta_{12} q^{3} + ( -\zeta_{12} + 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{5} + 6 \zeta_{12}^{2} q^{9} +O(q^{10})$$ $$q + 3 \zeta_{12} q^{3} + ( -\zeta_{12} + 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{5} + 6 \zeta_{12}^{2} q^{9} + ( -3 + 3 \zeta_{12}^{2} ) q^{11} -\zeta_{12}^{3} q^{13} + ( -3 + 6 \zeta_{12}^{3} ) q^{15} + 5 \zeta_{12} q^{17} -8 \zeta_{12}^{2} q^{19} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{23} + ( -3 - 4 \zeta_{12} + 3 \zeta_{12}^{2} ) q^{25} + 9 \zeta_{12}^{3} q^{27} + q^{29} + ( 2 - 2 \zeta_{12}^{2} ) q^{31} + ( -9 \zeta_{12} + 9 \zeta_{12}^{3} ) q^{33} + ( 10 \zeta_{12} - 10 \zeta_{12}^{3} ) q^{37} + ( 3 - 3 \zeta_{12}^{2} ) q^{39} -6 q^{41} + 4 \zeta_{12}^{3} q^{43} + ( -12 - 6 \zeta_{12} + 12 \zeta_{12}^{2} ) q^{45} + ( 11 \zeta_{12} - 11 \zeta_{12}^{3} ) q^{47} + 15 \zeta_{12}^{2} q^{51} + 6 \zeta_{12} q^{53} + ( -6 - 3 \zeta_{12}^{3} ) q^{55} -24 \zeta_{12}^{3} q^{57} + ( -10 + 10 \zeta_{12}^{2} ) q^{59} + ( 2 \zeta_{12} + \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{65} + 10 \zeta_{12} q^{67} -6 q^{69} -10 \zeta_{12} q^{73} + ( -9 \zeta_{12} - 12 \zeta_{12}^{2} + 9 \zeta_{12}^{3} ) q^{75} -7 \zeta_{12}^{2} q^{79} + ( -9 + 9 \zeta_{12}^{2} ) q^{81} -12 \zeta_{12}^{3} q^{83} + ( -5 + 10 \zeta_{12}^{3} ) q^{85} + 3 \zeta_{12} q^{87} + 8 \zeta_{12}^{2} q^{89} + ( 6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{93} + ( 16 + 8 \zeta_{12} - 16 \zeta_{12}^{2} ) q^{95} + 3 \zeta_{12}^{3} q^{97} -18 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{5} + 12 q^{9} + O(q^{10})$$ $$4 q + 4 q^{5} + 12 q^{9} - 6 q^{11} - 12 q^{15} - 16 q^{19} - 6 q^{25} + 4 q^{29} + 4 q^{31} + 6 q^{39} - 24 q^{41} - 24 q^{45} + 30 q^{51} - 24 q^{55} - 20 q^{59} + 2 q^{65} - 24 q^{69} - 24 q^{75} - 14 q^{79} - 18 q^{81} - 20 q^{85} + 16 q^{89} + 32 q^{95} - 72 q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$101$$ $$197$$ $$491$$ $$\chi(n)$$ $$-\zeta_{12}^{2}$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
569.1
 −0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i
0 −2.59808 + 1.50000i 0 1.86603 1.23205i 0 0 0 3.00000 5.19615i 0
569.2 0 2.59808 1.50000i 0 0.133975 2.23205i 0 0 0 3.00000 5.19615i 0
949.1 0 −2.59808 1.50000i 0 1.86603 + 1.23205i 0 0 0 3.00000 + 5.19615i 0
949.2 0 2.59808 + 1.50000i 0 0.133975 + 2.23205i 0 0 0 3.00000 + 5.19615i 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
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 980.2.q.f 4
5.b even 2 1 inner 980.2.q.f 4
7.b odd 2 1 980.2.q.c 4
7.c even 3 1 140.2.e.a 2
7.c even 3 1 inner 980.2.q.f 4
7.d odd 6 1 980.2.e.b 2
7.d odd 6 1 980.2.q.c 4
21.h odd 6 1 1260.2.k.c 2
28.g odd 6 1 560.2.g.a 2
35.c odd 2 1 980.2.q.c 4
35.i odd 6 1 980.2.e.b 2
35.i odd 6 1 980.2.q.c 4
35.j even 6 1 140.2.e.a 2
35.j even 6 1 inner 980.2.q.f 4
35.k even 12 1 4900.2.a.b 1
35.k even 12 1 4900.2.a.w 1
35.l odd 12 1 700.2.a.a 1
35.l odd 12 1 700.2.a.j 1
56.k odd 6 1 2240.2.g.f 2
56.p even 6 1 2240.2.g.e 2
84.n even 6 1 5040.2.t.s 2
105.o odd 6 1 1260.2.k.c 2
105.x even 12 1 6300.2.a.c 1
105.x even 12 1 6300.2.a.t 1
140.p odd 6 1 560.2.g.a 2
140.w even 12 1 2800.2.a.a 1
140.w even 12 1 2800.2.a.bf 1
280.bf even 6 1 2240.2.g.e 2
280.bi odd 6 1 2240.2.g.f 2
420.ba even 6 1 5040.2.t.s 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.e.a 2 7.c even 3 1
140.2.e.a 2 35.j even 6 1
560.2.g.a 2 28.g odd 6 1
560.2.g.a 2 140.p odd 6 1
700.2.a.a 1 35.l odd 12 1
700.2.a.j 1 35.l odd 12 1
980.2.e.b 2 7.d odd 6 1
980.2.e.b 2 35.i odd 6 1
980.2.q.c 4 7.b odd 2 1
980.2.q.c 4 7.d odd 6 1
980.2.q.c 4 35.c odd 2 1
980.2.q.c 4 35.i odd 6 1
980.2.q.f 4 1.a even 1 1 trivial
980.2.q.f 4 5.b even 2 1 inner
980.2.q.f 4 7.c even 3 1 inner
980.2.q.f 4 35.j even 6 1 inner
1260.2.k.c 2 21.h odd 6 1
1260.2.k.c 2 105.o odd 6 1
2240.2.g.e 2 56.p even 6 1
2240.2.g.e 2 280.bf even 6 1
2240.2.g.f 2 56.k odd 6 1
2240.2.g.f 2 280.bi odd 6 1
2800.2.a.a 1 140.w even 12 1
2800.2.a.bf 1 140.w even 12 1
4900.2.a.b 1 35.k even 12 1
4900.2.a.w 1 35.k even 12 1
5040.2.t.s 2 84.n even 6 1
5040.2.t.s 2 420.ba even 6 1
6300.2.a.c 1 105.x even 12 1
6300.2.a.t 1 105.x even 12 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(980, [\chi])$$:

 $$T_{3}^{4} - 9 T_{3}^{2} + 81$$ $$T_{11}^{2} + 3 T_{11} + 9$$ $$T_{19}^{2} + 8 T_{19} + 64$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$81 - 9 T^{2} + T^{4}$$
$5$ $$25 - 20 T + 11 T^{2} - 4 T^{3} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$( 9 + 3 T + T^{2} )^{2}$$
$13$ $$( 1 + T^{2} )^{2}$$
$17$ $$625 - 25 T^{2} + T^{4}$$
$19$ $$( 64 + 8 T + T^{2} )^{2}$$
$23$ $$16 - 4 T^{2} + T^{4}$$
$29$ $$( -1 + T )^{4}$$
$31$ $$( 4 - 2 T + T^{2} )^{2}$$
$37$ $$10000 - 100 T^{2} + T^{4}$$
$41$ $$( 6 + T )^{4}$$
$43$ $$( 16 + T^{2} )^{2}$$
$47$ $$14641 - 121 T^{2} + T^{4}$$
$53$ $$1296 - 36 T^{2} + T^{4}$$
$59$ $$( 100 + 10 T + T^{2} )^{2}$$
$61$ $$T^{4}$$
$67$ $$10000 - 100 T^{2} + T^{4}$$
$71$ $$T^{4}$$
$73$ $$10000 - 100 T^{2} + T^{4}$$
$79$ $$( 49 + 7 T + T^{2} )^{2}$$
$83$ $$( 144 + T^{2} )^{2}$$
$89$ $$( 64 - 8 T + T^{2} )^{2}$$
$97$ $$( 9 + T^{2} )^{2}$$