# Properties

 Label 980.2.k.a Level $980$ Weight $2$ Character orbit 980.k Analytic conductor $7.825$ Analytic rank $0$ Dimension $2$ CM discriminant -4 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$980 = 2^{2} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 980.k (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.82533939809$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 20) Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 - i ) q^{2} + 2 i q^{4} + ( 2 - i ) q^{5} + ( 2 - 2 i ) q^{8} -3 i q^{9} +O(q^{10})$$ $$q + ( -1 - i ) q^{2} + 2 i q^{4} + ( 2 - i ) q^{5} + ( 2 - 2 i ) q^{8} -3 i q^{9} + ( -3 - i ) q^{10} + ( 1 - i ) q^{13} -4 q^{16} + ( -3 - 3 i ) q^{17} + ( -3 + 3 i ) q^{18} + ( 2 + 4 i ) q^{20} + ( 3 - 4 i ) q^{25} -2 q^{26} + 4 i q^{29} + ( 4 + 4 i ) q^{32} + 6 i q^{34} + 6 q^{36} + ( -7 - 7 i ) q^{37} + ( 2 - 6 i ) q^{40} + 8 q^{41} + ( -3 - 6 i ) q^{45} + ( -7 + i ) q^{50} + ( 2 + 2 i ) q^{52} + ( 9 - 9 i ) q^{53} + ( 4 - 4 i ) q^{58} -12 q^{61} -8 i q^{64} + ( 1 - 3 i ) q^{65} + ( 6 - 6 i ) q^{68} + ( -6 - 6 i ) q^{72} + ( 11 - 11 i ) q^{73} + 14 i q^{74} + ( -8 + 4 i ) q^{80} -9 q^{81} + ( -8 - 8 i ) q^{82} + ( -9 - 3 i ) q^{85} + 16 i q^{89} + ( -3 + 9 i ) q^{90} + ( -13 - 13 i ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 4 q^{5} + 4 q^{8} + O(q^{10})$$ $$2 q - 2 q^{2} + 4 q^{5} + 4 q^{8} - 6 q^{10} + 2 q^{13} - 8 q^{16} - 6 q^{17} - 6 q^{18} + 4 q^{20} + 6 q^{25} - 4 q^{26} + 8 q^{32} + 12 q^{36} - 14 q^{37} + 4 q^{40} + 16 q^{41} - 6 q^{45} - 14 q^{50} + 4 q^{52} + 18 q^{53} + 8 q^{58} - 24 q^{61} + 2 q^{65} + 12 q^{68} - 12 q^{72} + 22 q^{73} - 16 q^{80} - 18 q^{81} - 16 q^{82} - 18 q^{85} - 6 q^{90} - 26 q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$101$$ $$197$$ $$491$$ $$\chi(n)$$ $$1$$ $$i$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
687.1
 1.00000i − 1.00000i
−1.00000 1.00000i 0 2.00000i 2.00000 1.00000i 0 0 2.00000 2.00000i 3.00000i −3.00000 1.00000i
883.1 −1.00000 + 1.00000i 0 2.00000i 2.00000 + 1.00000i 0 0 2.00000 + 2.00000i 3.00000i −3.00000 + 1.00000i
 $$n$$: 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
5.c odd 4 1 inner
20.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 980.2.k.a 2
4.b odd 2 1 CM 980.2.k.a 2
5.c odd 4 1 inner 980.2.k.a 2
7.b odd 2 1 20.2.e.a 2
7.c even 3 2 980.2.x.c 4
7.d odd 6 2 980.2.x.d 4
20.e even 4 1 inner 980.2.k.a 2
21.c even 2 1 180.2.k.c 2
28.d even 2 1 20.2.e.a 2
28.f even 6 2 980.2.x.d 4
28.g odd 6 2 980.2.x.c 4
35.c odd 2 1 100.2.e.b 2
35.f even 4 1 20.2.e.a 2
35.f even 4 1 100.2.e.b 2
35.k even 12 2 980.2.x.d 4
35.l odd 12 2 980.2.x.c 4
56.e even 2 1 320.2.n.e 2
56.h odd 2 1 320.2.n.e 2
84.h odd 2 1 180.2.k.c 2
105.g even 2 1 900.2.k.c 2
105.k odd 4 1 180.2.k.c 2
105.k odd 4 1 900.2.k.c 2
112.j even 4 1 1280.2.o.g 2
112.j even 4 1 1280.2.o.j 2
112.l odd 4 1 1280.2.o.g 2
112.l odd 4 1 1280.2.o.j 2
140.c even 2 1 100.2.e.b 2
140.j odd 4 1 20.2.e.a 2
140.j odd 4 1 100.2.e.b 2
140.w even 12 2 980.2.x.c 4
140.x odd 12 2 980.2.x.d 4
280.c odd 2 1 1600.2.n.h 2
280.n even 2 1 1600.2.n.h 2
280.s even 4 1 320.2.n.e 2
280.s even 4 1 1600.2.n.h 2
280.y odd 4 1 320.2.n.e 2
280.y odd 4 1 1600.2.n.h 2
420.o odd 2 1 900.2.k.c 2
420.w even 4 1 180.2.k.c 2
420.w even 4 1 900.2.k.c 2
560.r even 4 1 1280.2.o.j 2
560.u odd 4 1 1280.2.o.g 2
560.bm odd 4 1 1280.2.o.j 2
560.bn even 4 1 1280.2.o.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.2.e.a 2 7.b odd 2 1
20.2.e.a 2 28.d even 2 1
20.2.e.a 2 35.f even 4 1
20.2.e.a 2 140.j odd 4 1
100.2.e.b 2 35.c odd 2 1
100.2.e.b 2 35.f even 4 1
100.2.e.b 2 140.c even 2 1
100.2.e.b 2 140.j odd 4 1
180.2.k.c 2 21.c even 2 1
180.2.k.c 2 84.h odd 2 1
180.2.k.c 2 105.k odd 4 1
180.2.k.c 2 420.w even 4 1
320.2.n.e 2 56.e even 2 1
320.2.n.e 2 56.h odd 2 1
320.2.n.e 2 280.s even 4 1
320.2.n.e 2 280.y odd 4 1
900.2.k.c 2 105.g even 2 1
900.2.k.c 2 105.k odd 4 1
900.2.k.c 2 420.o odd 2 1
900.2.k.c 2 420.w even 4 1
980.2.k.a 2 1.a even 1 1 trivial
980.2.k.a 2 4.b odd 2 1 CM
980.2.k.a 2 5.c odd 4 1 inner
980.2.k.a 2 20.e even 4 1 inner
980.2.x.c 4 7.c even 3 2
980.2.x.c 4 28.g odd 6 2
980.2.x.c 4 35.l odd 12 2
980.2.x.c 4 140.w even 12 2
980.2.x.d 4 7.d odd 6 2
980.2.x.d 4 28.f even 6 2
980.2.x.d 4 35.k even 12 2
980.2.x.d 4 140.x odd 12 2
1280.2.o.g 2 112.j even 4 1
1280.2.o.g 2 112.l odd 4 1
1280.2.o.g 2 560.u odd 4 1
1280.2.o.g 2 560.bn even 4 1
1280.2.o.j 2 112.j even 4 1
1280.2.o.j 2 112.l odd 4 1
1280.2.o.j 2 560.r even 4 1
1280.2.o.j 2 560.bm odd 4 1
1600.2.n.h 2 280.c odd 2 1
1600.2.n.h 2 280.n even 2 1
1600.2.n.h 2 280.s even 4 1
1600.2.n.h 2 280.y odd 4 1

## Hecke kernels

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

 $$T_{3}$$ $$T_{13}^{2} - 2 T_{13} + 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$2 + 2 T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$5 - 4 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$2 - 2 T + T^{2}$$
$17$ $$18 + 6 T + T^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$16 + T^{2}$$
$31$ $$T^{2}$$
$37$ $$98 + 14 T + T^{2}$$
$41$ $$( -8 + T )^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2}$$
$53$ $$162 - 18 T + T^{2}$$
$59$ $$T^{2}$$
$61$ $$( 12 + T )^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$242 - 22 T + T^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$256 + T^{2}$$
$97$ $$338 + 26 T + T^{2}$$