# Properties

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

# Related objects

## 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 220) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{5} - 4 q^{7}+O(q^{10})$$ q - q^5 - 4 * q^7 $$q - q^{5} - 4 q^{7} + q^{11} - 4 q^{13} - 4 q^{19} + 6 q^{23} + q^{25} + 6 q^{29} + 8 q^{31} + 4 q^{35} + 2 q^{37} - 6 q^{41} + 8 q^{43} - 6 q^{47} + 9 q^{49} + 6 q^{53} - q^{55} + 12 q^{59} + 2 q^{61} + 4 q^{65} - 10 q^{67} + 12 q^{71} - 16 q^{73} - 4 q^{77} + 8 q^{79} - 6 q^{89} + 16 q^{91} + 4 q^{95} + 14 q^{97}+O(q^{100})$$ q - q^5 - 4 * q^7 + q^11 - 4 * q^13 - 4 * q^19 + 6 * q^23 + q^25 + 6 * q^29 + 8 * q^31 + 4 * q^35 + 2 * q^37 - 6 * q^41 + 8 * q^43 - 6 * q^47 + 9 * q^49 + 6 * q^53 - q^55 + 12 * q^59 + 2 * q^61 + 4 * q^65 - 10 * q^67 + 12 * q^71 - 16 * q^73 - 4 * q^77 + 8 * q^79 - 6 * q^89 + 16 * q^91 + 4 * q^95 + 14 * q^97

## 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 −4.00000 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.a 1
3.b odd 2 1 220.2.a.a 1
4.b odd 2 1 7920.2.a.o 1
5.b even 2 1 9900.2.a.bd 1
5.c odd 4 2 9900.2.c.m 2
12.b even 2 1 880.2.a.j 1
15.d odd 2 1 1100.2.a.e 1
15.e even 4 2 1100.2.b.a 2
24.f even 2 1 3520.2.a.d 1
24.h odd 2 1 3520.2.a.bd 1
33.d even 2 1 2420.2.a.b 1
60.h even 2 1 4400.2.a.e 1
60.l odd 4 2 4400.2.b.f 2
132.d odd 2 1 9680.2.a.bb 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
220.2.a.a 1 3.b odd 2 1
880.2.a.j 1 12.b even 2 1
1100.2.a.e 1 15.d odd 2 1
1100.2.b.a 2 15.e even 4 2
1980.2.a.a 1 1.a even 1 1 trivial
2420.2.a.b 1 33.d even 2 1
3520.2.a.d 1 24.f even 2 1
3520.2.a.bd 1 24.h odd 2 1
4400.2.a.e 1 60.h even 2 1
4400.2.b.f 2 60.l odd 4 2
7920.2.a.o 1 4.b odd 2 1
9680.2.a.bb 1 132.d odd 2 1
9900.2.a.bd 1 5.b even 2 1
9900.2.c.m 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1980))$$:

 $$T_{7} + 4$$ T7 + 4 $$T_{13} + 4$$ T13 + 4 $$T_{17}$$ T17

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 1$$
$7$ $$T + 4$$
$11$ $$T - 1$$
$13$ $$T + 4$$
$17$ $$T$$
$19$ $$T + 4$$
$23$ $$T - 6$$
$29$ $$T - 6$$
$31$ $$T - 8$$
$37$ $$T - 2$$
$41$ $$T + 6$$
$43$ $$T - 8$$
$47$ $$T + 6$$
$53$ $$T - 6$$
$59$ $$T - 12$$
$61$ $$T - 2$$
$67$ $$T + 10$$
$71$ $$T - 12$$
$73$ $$T + 16$$
$79$ $$T - 8$$
$83$ $$T$$
$89$ $$T + 6$$
$97$ $$T - 14$$