# Properties

 Label 690.2.a.a Level $690$ Weight $2$ Character orbit 690.a Self dual yes Analytic conductor $5.510$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$690 = 2 \cdot 3 \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 690.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$5.50967773947$$ Analytic rank: $$1$$ 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

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} - 2q^{7} - q^{8} + q^{9} + O(q^{10})$$ $$q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} - 2q^{7} - q^{8} + q^{9} + q^{10} + 6q^{11} - q^{12} - 2q^{13} + 2q^{14} + q^{15} + q^{16} - q^{18} - 4q^{19} - q^{20} + 2q^{21} - 6q^{22} - q^{23} + q^{24} + q^{25} + 2q^{26} - q^{27} - 2q^{28} - 2q^{29} - q^{30} - 8q^{31} - q^{32} - 6q^{33} + 2q^{35} + q^{36} - 4q^{37} + 4q^{38} + 2q^{39} + q^{40} + 2q^{41} - 2q^{42} - 8q^{43} + 6q^{44} - q^{45} + q^{46} - q^{48} - 3q^{49} - q^{50} - 2q^{52} - 2q^{53} + q^{54} - 6q^{55} + 2q^{56} + 4q^{57} + 2q^{58} - 4q^{59} + q^{60} + 8q^{62} - 2q^{63} + q^{64} + 2q^{65} + 6q^{66} + q^{69} - 2q^{70} - 8q^{71} - q^{72} + 6q^{73} + 4q^{74} - q^{75} - 4q^{76} - 12q^{77} - 2q^{78} - 14q^{79} - q^{80} + q^{81} - 2q^{82} - 6q^{83} + 2q^{84} + 8q^{86} + 2q^{87} - 6q^{88} - 16q^{89} + q^{90} + 4q^{91} - q^{92} + 8q^{93} + 4q^{95} + q^{96} + 2q^{97} + 3q^{98} + 6q^{99} + O(q^{100})$$

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

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$1$$
$$5$$ $$1$$
$$23$$ $$1$$

## 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 690.2.a.a 1
3.b odd 2 1 2070.2.a.q 1
4.b odd 2 1 5520.2.a.y 1
5.b even 2 1 3450.2.a.ba 1
5.c odd 4 2 3450.2.d.u 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.2.a.a 1 1.a even 1 1 trivial
2070.2.a.q 1 3.b odd 2 1
3450.2.a.ba 1 5.b even 2 1
3450.2.d.u 2 5.c odd 4 2
5520.2.a.y 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(690))$$:

 $$T_{7} + 2$$ $$T_{11} - 6$$ $$T_{17}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T$$
$3$ $$1 + T$$
$5$ $$1 + T$$
$7$ $$2 + T$$
$11$ $$-6 + T$$
$13$ $$2 + T$$
$17$ $$T$$
$19$ $$4 + T$$
$23$ $$1 + T$$
$29$ $$2 + T$$
$31$ $$8 + T$$
$37$ $$4 + T$$
$41$ $$-2 + T$$
$43$ $$8 + T$$
$47$ $$T$$
$53$ $$2 + T$$
$59$ $$4 + T$$
$61$ $$T$$
$67$ $$T$$
$71$ $$8 + T$$
$73$ $$-6 + T$$
$79$ $$14 + T$$
$83$ $$6 + T$$
$89$ $$16 + T$$
$97$ $$-2 + T$$