# Properties

 Label 1530.2.a.b Level $1530$ Weight $2$ Character orbit 1530.a Self dual yes Analytic conductor $12.217$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1530 = 2 \cdot 3^{2} \cdot 5 \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1530.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$12.2171115093$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 510) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} - q^{5} - q^{8} + O(q^{10})$$ $$q - q^{2} + q^{4} - q^{5} - q^{8} + q^{10} - 4q^{11} - 2q^{13} + q^{16} - q^{17} + 4q^{19} - q^{20} + 4q^{22} + q^{25} + 2q^{26} + 2q^{29} + 8q^{31} - q^{32} + q^{34} + 6q^{37} - 4q^{38} + q^{40} + 6q^{41} - 4q^{43} - 4q^{44} - 7q^{49} - q^{50} - 2q^{52} + 10q^{53} + 4q^{55} - 2q^{58} + 4q^{59} - 2q^{61} - 8q^{62} + q^{64} + 2q^{65} + 4q^{67} - q^{68} - 6q^{73} - 6q^{74} + 4q^{76} + 8q^{79} - q^{80} - 6q^{82} + 12q^{83} + q^{85} + 4q^{86} + 4q^{88} + 6q^{89} - 4q^{95} - 14q^{97} + 7q^{98} + O(q^{100})$$

## 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
−1.00000 0 1.00000 −1.00000 0 0 −1.00000 0 1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$
$$5$$ $$1$$
$$17$$ $$1$$

## 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 1530.2.a.b 1
3.b odd 2 1 510.2.a.e 1
5.b even 2 1 7650.2.a.bx 1
12.b even 2 1 4080.2.a.ba 1
15.d odd 2 1 2550.2.a.l 1
15.e even 4 2 2550.2.d.k 2
51.c odd 2 1 8670.2.a.v 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
510.2.a.e 1 3.b odd 2 1
1530.2.a.b 1 1.a even 1 1 trivial
2550.2.a.l 1 15.d odd 2 1
2550.2.d.k 2 15.e even 4 2
4080.2.a.ba 1 12.b even 2 1
7650.2.a.bx 1 5.b even 2 1
8670.2.a.v 1 51.c odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1530))$$:

 $$T_{7}$$ $$T_{11} + 4$$ $$T_{13} + 2$$

## Hecke characteristic polynomials

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