# Properties

 Label 5712.2.a.o 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

 $$f(q)$$ $$=$$ $$q + q^{3} - 2q^{5} - q^{7} + q^{9} + O(q^{10})$$ $$q + q^{3} - 2q^{5} - q^{7} + q^{9} - 4q^{11} - 2q^{13} - 2q^{15} + q^{17} - 4q^{19} - q^{21} - 8q^{23} - q^{25} + q^{27} + 6q^{29} - 4q^{33} + 2q^{35} - 2q^{37} - 2q^{39} + 10q^{41} + 4q^{43} - 2q^{45} + q^{49} + q^{51} + 6q^{53} + 8q^{55} - 4q^{57} + 4q^{59} + 6q^{61} - q^{63} + 4q^{65} + 12q^{67} - 8q^{69} + 8q^{71} - 6q^{73} - q^{75} + 4q^{77} + q^{81} + 12q^{83} - 2q^{85} + 6q^{87} - 6q^{89} + 2q^{91} + 8q^{95} + 2q^{97} - 4q^{99} + O(q^{100})$$

## 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 −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.o 1
4.b odd 2 1 714.2.a.f 1
12.b even 2 1 2142.2.a.h 1
28.d even 2 1 4998.2.a.bq 1

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

 $$T_{5} + 2$$ $$T_{11} + 4$$ $$T_{13} + 2$$ $$T_{19} + 4$$

## Hecke characteristic polynomials

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