# Properties

 Label 714.2.a.f Level $714$ Weight $2$ Character orbit 714.a Self dual yes Analytic conductor $5.701$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$714 = 2 \cdot 3 \cdot 7 \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 714.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$5.70131870432$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} + q^{4} - 2q^{5} - q^{6} + q^{7} + q^{8} + q^{9} + O(q^{10})$$ $$q + q^{2} - q^{3} + q^{4} - 2q^{5} - q^{6} + q^{7} + q^{8} + q^{9} - 2q^{10} + 4q^{11} - q^{12} - 2q^{13} + q^{14} + 2q^{15} + q^{16} + q^{17} + q^{18} + 4q^{19} - 2q^{20} - q^{21} + 4q^{22} + 8q^{23} - q^{24} - q^{25} - 2q^{26} - q^{27} + q^{28} + 6q^{29} + 2q^{30} + q^{32} - 4q^{33} + q^{34} - 2q^{35} + q^{36} - 2q^{37} + 4q^{38} + 2q^{39} - 2q^{40} + 10q^{41} - q^{42} - 4q^{43} + 4q^{44} - 2q^{45} + 8q^{46} - q^{48} + q^{49} - q^{50} - q^{51} - 2q^{52} + 6q^{53} - q^{54} - 8q^{55} + q^{56} - 4q^{57} + 6q^{58} - 4q^{59} + 2q^{60} + 6q^{61} + q^{63} + q^{64} + 4q^{65} - 4q^{66} - 12q^{67} + q^{68} - 8q^{69} - 2q^{70} - 8q^{71} + q^{72} - 6q^{73} - 2q^{74} + q^{75} + 4q^{76} + 4q^{77} + 2q^{78} - 2q^{80} + q^{81} + 10q^{82} - 12q^{83} - q^{84} - 2q^{85} - 4q^{86} - 6q^{87} + 4q^{88} - 6q^{89} - 2q^{90} - 2q^{91} + 8q^{92} - 8q^{95} - q^{96} + 2q^{97} + q^{98} + 4q^{99} + O(q^{100})$$

## 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
1.00000 −1.00000 1.00000 −2.00000 −1.00000 1.00000 1.00000 1.00000 −2.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$
$$7$$ $$-1$$
$$17$$ $$-1$$

## 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 714.2.a.f 1
3.b odd 2 1 2142.2.a.h 1
4.b odd 2 1 5712.2.a.o 1
7.b odd 2 1 4998.2.a.bq 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
714.2.a.f 1 1.a even 1 1 trivial
2142.2.a.h 1 3.b odd 2 1
4998.2.a.bq 1 7.b odd 2 1
5712.2.a.o 1 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(714))$$:

 $$T_{5} + 2$$ $$T_{11} - 4$$

## Hecke characteristic polynomials

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