# Properties

 Label 5712.2.a.bb Level $5712$ Weight $2$ Character orbit 5712.a Self dual yes Analytic conductor $45.611$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5712 = 2^{4} \cdot 3 \cdot 7 \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5712.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$45.6105496346$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 714) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{3} + 3q^{5} + q^{7} + q^{9} + O(q^{10})$$ $$q + q^{3} + 3q^{5} + q^{7} + q^{9} - q^{11} + q^{13} + 3q^{15} - q^{17} - 6q^{19} + q^{21} + 2q^{23} + 4q^{25} + q^{27} - 2q^{29} - q^{33} + 3q^{35} + 5q^{37} + q^{39} + 4q^{41} + 9q^{43} + 3q^{45} + q^{49} - q^{51} + 11q^{53} - 3q^{55} - 6q^{57} - 4q^{59} + 6q^{61} + q^{63} + 3q^{65} + 11q^{67} + 2q^{69} + 12q^{71} - 5q^{73} + 4q^{75} - q^{77} + 15q^{79} + q^{81} - q^{83} - 3q^{85} - 2q^{87} + 9q^{89} + q^{91} - 18q^{95} - 9q^{97} - q^{99} + O(q^{100})$$

## 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 3.00000 0 1.00000 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$7$$ $$-1$$
$$17$$ $$1$$

## 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 5712.2.a.bb 1
4.b odd 2 1 714.2.a.g 1
12.b even 2 1 2142.2.a.a 1
28.d even 2 1 4998.2.a.bj 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
714.2.a.g 1 4.b odd 2 1
2142.2.a.a 1 12.b even 2 1
4998.2.a.bj 1 28.d even 2 1
5712.2.a.bb 1 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5712))$$:

 $$T_{5} - 3$$ $$T_{11} + 1$$ $$T_{13} - 1$$ $$T_{19} + 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$-1 + T$$
$5$ $$-3 + T$$
$7$ $$-1 + T$$
$11$ $$1 + T$$
$13$ $$-1 + T$$
$17$ $$1 + T$$
$19$ $$6 + T$$
$23$ $$-2 + T$$
$29$ $$2 + T$$
$31$ $$T$$
$37$ $$-5 + T$$
$41$ $$-4 + T$$
$43$ $$-9 + T$$
$47$ $$T$$
$53$ $$-11 + T$$
$59$ $$4 + T$$
$61$ $$-6 + T$$
$67$ $$-11 + T$$
$71$ $$-12 + T$$
$73$ $$5 + T$$
$79$ $$-15 + T$$
$83$ $$1 + T$$
$89$ $$-9 + T$$
$97$ $$9 + T$$