# Properties

 Label 980.2.q.d 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: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q + ( 2 \zeta_{12} - \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{5} -3 \zeta_{12}^{2} q^{9} +O(q^{10})$$ $$q + ( 2 \zeta_{12} - \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{5} -3 \zeta_{12}^{2} q^{9} -4 \zeta_{12}^{3} q^{13} -4 \zeta_{12} q^{17} + 4 \zeta_{12}^{2} q^{19} + ( -8 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{23} + ( 3 - 4 \zeta_{12} - 3 \zeta_{12}^{2} ) q^{25} -2 q^{29} + ( 8 - 8 \zeta_{12}^{2} ) q^{31} + ( -8 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{37} + 6 q^{41} -8 \zeta_{12}^{3} q^{43} + ( -3 - 6 \zeta_{12} + 3 \zeta_{12}^{2} ) q^{45} + ( 8 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{47} + ( -4 + 4 \zeta_{12}^{2} ) q^{59} + 6 \zeta_{12}^{2} q^{61} + ( -4 \zeta_{12} - 8 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{65} -8 \zeta_{12} q^{67} + 12 q^{71} -4 \zeta_{12} q^{73} -4 \zeta_{12}^{2} q^{79} + ( -9 + 9 \zeta_{12}^{2} ) q^{81} + ( -8 + 4 \zeta_{12}^{3} ) q^{85} -10 \zeta_{12}^{2} q^{89} + ( 4 + 8 \zeta_{12} - 4 \zeta_{12}^{2} ) q^{95} -12 \zeta_{12}^{3} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{5} - 6q^{9} + O(q^{10})$$ $$4q - 2q^{5} - 6q^{9} + 8q^{19} + 6q^{25} - 8q^{29} + 16q^{31} + 24q^{41} - 6q^{45} - 8q^{59} + 12q^{61} - 16q^{65} + 48q^{71} - 8q^{79} - 18q^{81} - 32q^{85} - 20q^{89} + 8q^{95} + O(q^{100})$$

## 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 0 0 −2.23205 0.133975i 0 0 0 −1.50000 + 2.59808i 0
569.2 0 0 0 1.23205 + 1.86603i 0 0 0 −1.50000 + 2.59808i 0
949.1 0 0 0 −2.23205 + 0.133975i 0 0 0 −1.50000 2.59808i 0
949.2 0 0 0 1.23205 1.86603i 0 0 0 −1.50000 2.59808i 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.d 4
5.b even 2 1 inner 980.2.q.d 4
7.b odd 2 1 980.2.q.e 4
7.c even 3 1 140.2.e.b 2
7.c even 3 1 inner 980.2.q.d 4
7.d odd 6 1 980.2.e.a 2
7.d odd 6 1 980.2.q.e 4
21.h odd 6 1 1260.2.k.b 2
28.g odd 6 1 560.2.g.c 2
35.c odd 2 1 980.2.q.e 4
35.i odd 6 1 980.2.e.a 2
35.i odd 6 1 980.2.q.e 4
35.j even 6 1 140.2.e.b 2
35.j even 6 1 inner 980.2.q.d 4
35.k even 12 1 4900.2.a.l 1
35.k even 12 1 4900.2.a.m 1
35.l odd 12 1 700.2.a.f 1
35.l odd 12 1 700.2.a.h 1
56.k odd 6 1 2240.2.g.c 2
56.p even 6 1 2240.2.g.d 2
84.n even 6 1 5040.2.t.g 2
105.o odd 6 1 1260.2.k.b 2
105.x even 12 1 6300.2.a.g 1
105.x even 12 1 6300.2.a.y 1
140.p odd 6 1 560.2.g.c 2
140.w even 12 1 2800.2.a.o 1
140.w even 12 1 2800.2.a.s 1
280.bf even 6 1 2240.2.g.d 2
280.bi odd 6 1 2240.2.g.c 2
420.ba even 6 1 5040.2.t.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.e.b 2 7.c even 3 1
140.2.e.b 2 35.j even 6 1
560.2.g.c 2 28.g odd 6 1
560.2.g.c 2 140.p odd 6 1
700.2.a.f 1 35.l odd 12 1
700.2.a.h 1 35.l odd 12 1
980.2.e.a 2 7.d odd 6 1
980.2.e.a 2 35.i odd 6 1
980.2.q.d 4 1.a even 1 1 trivial
980.2.q.d 4 5.b even 2 1 inner
980.2.q.d 4 7.c even 3 1 inner
980.2.q.d 4 35.j even 6 1 inner
980.2.q.e 4 7.b odd 2 1
980.2.q.e 4 7.d odd 6 1
980.2.q.e 4 35.c odd 2 1
980.2.q.e 4 35.i odd 6 1
1260.2.k.b 2 21.h odd 6 1
1260.2.k.b 2 105.o odd 6 1
2240.2.g.c 2 56.k odd 6 1
2240.2.g.c 2 280.bi odd 6 1
2240.2.g.d 2 56.p even 6 1
2240.2.g.d 2 280.bf even 6 1
2800.2.a.o 1 140.w even 12 1
2800.2.a.s 1 140.w even 12 1
4900.2.a.l 1 35.k even 12 1
4900.2.a.m 1 35.k even 12 1
5040.2.t.g 2 84.n even 6 1
5040.2.t.g 2 420.ba even 6 1
6300.2.a.g 1 105.x even 12 1
6300.2.a.y 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 $$S_{2}^{\mathrm{new}}(980, [\chi])$$:

 $$T_{3}$$ $$T_{11}$$ $$T_{19}^{2} - 4 T_{19} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$25 + 10 T - T^{2} + 2 T^{3} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$T^{4}$$
$13$ $$( 16 + T^{2} )^{2}$$
$17$ $$256 - 16 T^{2} + T^{4}$$
$19$ $$( 16 - 4 T + T^{2} )^{2}$$
$23$ $$4096 - 64 T^{2} + T^{4}$$
$29$ $$( 2 + T )^{4}$$
$31$ $$( 64 - 8 T + T^{2} )^{2}$$
$37$ $$4096 - 64 T^{2} + T^{4}$$
$41$ $$( -6 + T )^{4}$$
$43$ $$( 64 + T^{2} )^{2}$$
$47$ $$4096 - 64 T^{2} + T^{4}$$
$53$ $$T^{4}$$
$59$ $$( 16 + 4 T + T^{2} )^{2}$$
$61$ $$( 36 - 6 T + T^{2} )^{2}$$
$67$ $$4096 - 64 T^{2} + T^{4}$$
$71$ $$( -12 + T )^{4}$$
$73$ $$256 - 16 T^{2} + T^{4}$$
$79$ $$( 16 + 4 T + T^{2} )^{2}$$
$83$ $$T^{4}$$
$89$ $$( 100 + 10 T + T^{2} )^{2}$$
$97$ $$( 144 + T^{2} )^{2}$$