# Properties

 Label 1980.2.a.f Level $1980$ Weight $2$ Character orbit 1980.a Self dual yes Analytic conductor $15.810$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$1980 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1980.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$15.8103796002$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 660) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{5} + 2 q^{7} + O(q^{10})$$ $$q + q^{5} + 2 q^{7} - q^{11} + 2 q^{13} + 2 q^{19} + q^{25} + 8 q^{31} + 2 q^{35} + 2 q^{37} + 2 q^{43} - 3 q^{49} - 6 q^{53} - q^{55} + 12 q^{59} + 2 q^{61} + 2 q^{65} - 4 q^{67} + 2 q^{73} - 2 q^{77} - 10 q^{79} + 12 q^{83} + 6 q^{89} + 4 q^{91} + 2 q^{95} + 14 q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 0 0 1.00000 0 2.00000 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$-1$$
$$11$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1980.2.a.f 1
3.b odd 2 1 660.2.a.d 1
4.b odd 2 1 7920.2.a.y 1
5.b even 2 1 9900.2.a.e 1
5.c odd 4 2 9900.2.c.d 2
12.b even 2 1 2640.2.a.b 1
15.d odd 2 1 3300.2.a.b 1
15.e even 4 2 3300.2.c.i 2
33.d even 2 1 7260.2.a.l 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
660.2.a.d 1 3.b odd 2 1
1980.2.a.f 1 1.a even 1 1 trivial
2640.2.a.b 1 12.b even 2 1
3300.2.a.b 1 15.d odd 2 1
3300.2.c.i 2 15.e even 4 2
7260.2.a.l 1 33.d even 2 1
7920.2.a.y 1 4.b odd 2 1
9900.2.a.e 1 5.b even 2 1
9900.2.c.d 2 5.c odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1980))$$:

 $$T_{7} - 2$$ $$T_{13} - 2$$ $$T_{17}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$-1 + T$$
$7$ $$-2 + T$$
$11$ $$1 + T$$
$13$ $$-2 + T$$
$17$ $$T$$
$19$ $$-2 + T$$
$23$ $$T$$
$29$ $$T$$
$31$ $$-8 + T$$
$37$ $$-2 + T$$
$41$ $$T$$
$43$ $$-2 + T$$
$47$ $$T$$
$53$ $$6 + T$$
$59$ $$-12 + T$$
$61$ $$-2 + T$$
$67$ $$4 + T$$
$71$ $$T$$
$73$ $$-2 + T$$
$79$ $$10 + T$$
$83$ $$-12 + T$$
$89$ $$-6 + T$$
$97$ $$-14 + T$$
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