# Properties

 Label 5712.2.a.q Level $5712$ Weight $2$ Character orbit 5712.a Self dual yes Analytic conductor $45.611$ Analytic rank $1$ 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: $$1$$ 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

 $$f(q)$$ $$=$$ $$q + q^{3} - 2q^{5} + q^{7} + q^{9} + O(q^{10})$$ $$q + q^{3} - 2q^{5} + q^{7} + q^{9} - 6q^{13} - 2q^{15} + q^{17} + q^{21} + 8q^{23} - q^{25} + q^{27} - 6q^{29} + 8q^{31} - 2q^{35} + 10q^{37} - 6q^{39} - 6q^{41} - 12q^{43} - 2q^{45} + q^{49} + q^{51} - 10q^{53} + 8q^{59} + 6q^{61} + q^{63} + 12q^{65} - 12q^{67} + 8q^{69} - 6q^{73} - q^{75} + 8q^{79} + q^{81} - 16q^{83} - 2q^{85} - 6q^{87} + 2q^{89} - 6q^{91} + 8q^{93} + 2q^{97} + O(q^{100})$$

## 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 1.00000 0 −2.00000 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.q 1
4.b odd 2 1 714.2.a.e 1
12.b even 2 1 2142.2.a.g 1
28.d even 2 1 4998.2.a.bp 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
714.2.a.e 1 4.b odd 2 1
2142.2.a.g 1 12.b even 2 1
4998.2.a.bp 1 28.d even 2 1
5712.2.a.q 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5712))$$:

 $$T_{5} + 2$$ $$T_{11}$$ $$T_{13} + 6$$ $$T_{19}$$

## Hecke characteristic polynomials

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