# Properties

 Label 690.2.a.k Level $690$ Weight $2$ Character orbit 690.a Self dual yes Analytic conductor $5.510$ Analytic rank $0$ 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: $$0$$ 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

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} + q^{8} + q^{9} + O(q^{10})$$ $$q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} + q^{8} + q^{9} + q^{10} + 4q^{11} + q^{12} - 2q^{13} + q^{15} + q^{16} - 6q^{17} + q^{18} + 4q^{19} + q^{20} + 4q^{22} - q^{23} + q^{24} + q^{25} - 2q^{26} + q^{27} - 2q^{29} + q^{30} + q^{32} + 4q^{33} - 6q^{34} + q^{36} - 2q^{37} + 4q^{38} - 2q^{39} + q^{40} + 10q^{41} - 4q^{43} + 4q^{44} + q^{45} - q^{46} + q^{48} - 7q^{49} + q^{50} - 6q^{51} - 2q^{52} + 6q^{53} + q^{54} + 4q^{55} + 4q^{57} - 2q^{58} - 4q^{59} + q^{60} - 10q^{61} + q^{64} - 2q^{65} + 4q^{66} - 12q^{67} - 6q^{68} - q^{69} - 8q^{71} + q^{72} + 10q^{73} - 2q^{74} + q^{75} + 4q^{76} - 2q^{78} - 8q^{79} + q^{80} + q^{81} + 10q^{82} - 4q^{83} - 6q^{85} - 4q^{86} - 2q^{87} + 4q^{88} + 18q^{89} + q^{90} - q^{92} + 4q^{95} + q^{96} + 2q^{97} - 7q^{98} + 4q^{99} + O(q^{100})$$

## 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 1.00000 1.00000 0 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.k 1
3.b odd 2 1 2070.2.a.b 1
4.b odd 2 1 5520.2.a.i 1
5.b even 2 1 3450.2.a.d 1
5.c odd 4 2 3450.2.d.t 2

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

 $$T_{7}$$ $$T_{11} - 4$$ $$T_{17} + 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + T$$
$3$ $$-1 + T$$
$5$ $$-1 + T$$
$7$ $$T$$
$11$ $$-4 + T$$
$13$ $$2 + T$$
$17$ $$6 + T$$
$19$ $$-4 + T$$
$23$ $$1 + T$$
$29$ $$2 + T$$
$31$ $$T$$
$37$ $$2 + T$$
$41$ $$-10 + T$$
$43$ $$4 + T$$
$47$ $$T$$
$53$ $$-6 + T$$
$59$ $$4 + T$$
$61$ $$10 + T$$
$67$ $$12 + T$$
$71$ $$8 + T$$
$73$ $$-10 + T$$
$79$ $$8 + T$$
$83$ $$4 + T$$
$89$ $$-18 + T$$
$97$ $$-2 + T$$