# Properties

 Label 4080.2.a.ba Level $4080$ Weight $2$ Character orbit 4080.a Self dual yes Analytic conductor $32.579$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4080 = 2^{4} \cdot 3 \cdot 5 \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4080.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$32.5789640247$$ Analytic rank: $$1$$ 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

 $$f(q)$$ $$=$$ $$q + q^{3} + q^{5} + q^{9} + O(q^{10})$$ $$q + q^{3} + q^{5} + q^{9} - 4q^{11} - 2q^{13} + q^{15} + q^{17} - 4q^{19} + q^{25} + q^{27} - 2q^{29} - 8q^{31} - 4q^{33} + 6q^{37} - 2q^{39} - 6q^{41} + 4q^{43} + q^{45} - 7q^{49} + q^{51} - 10q^{53} - 4q^{55} - 4q^{57} + 4q^{59} - 2q^{61} - 2q^{65} - 4q^{67} - 6q^{73} + q^{75} - 8q^{79} + q^{81} + 12q^{83} + q^{85} - 2q^{87} - 6q^{89} - 8q^{93} - 4q^{95} - 14q^{97} - 4q^{99} + O(q^{100})$$

## 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 1.00000 0 1.00000 0 0 0 1.00000 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$-1$$
$$17$$ $$-1$$

## 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 4080.2.a.ba 1
4.b odd 2 1 510.2.a.e 1
12.b even 2 1 1530.2.a.b 1
20.d odd 2 1 2550.2.a.l 1
20.e even 4 2 2550.2.d.k 2
60.h even 2 1 7650.2.a.bx 1
68.d 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 4.b odd 2 1
1530.2.a.b 1 12.b even 2 1
2550.2.a.l 1 20.d odd 2 1
2550.2.d.k 2 20.e even 4 2
4080.2.a.ba 1 1.a even 1 1 trivial
7650.2.a.bx 1 60.h even 2 1
8670.2.a.v 1 68.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4080))$$:

 $$T_{7}$$ $$T_{11} + 4$$ $$T_{13} + 2$$ $$T_{19} + 4$$

## Hecke characteristic polynomials

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