# Properties

 Label 390.2.a.f Level $390$ Weight $2$ Character orbit 390.a Self dual yes Analytic conductor $3.114$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$390 = 2 \cdot 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 390.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$3.11416567883$$ 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

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + q^{8} + q^{9} + O(q^{10})$$ $$q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + q^{8} + q^{9} + q^{10} + 4q^{11} - q^{12} + q^{13} - q^{15} + q^{16} - 6q^{17} + q^{18} + 4q^{19} + q^{20} + 4q^{22} + 8q^{23} - q^{24} + q^{25} + q^{26} - q^{27} + 6q^{29} - q^{30} - 8q^{31} + q^{32} - 4q^{33} - 6q^{34} + q^{36} - 10q^{37} + 4q^{38} - q^{39} + q^{40} - 6q^{41} + 4q^{43} + 4q^{44} + q^{45} + 8q^{46} - q^{48} - 7q^{49} + q^{50} + 6q^{51} + q^{52} - 10q^{53} - q^{54} + 4q^{55} - 4q^{57} + 6q^{58} + 4q^{59} - q^{60} - 2q^{61} - 8q^{62} + q^{64} + q^{65} - 4q^{66} - 12q^{67} - 6q^{68} - 8q^{69} + 16q^{71} + q^{72} + 2q^{73} - 10q^{74} - q^{75} + 4q^{76} - q^{78} - 16q^{79} + q^{80} + q^{81} - 6q^{82} - 12q^{83} - 6q^{85} + 4q^{86} - 6q^{87} + 4q^{88} + 10q^{89} + q^{90} + 8q^{92} + 8q^{93} + 4q^{95} - q^{96} - 6q^{97} - 7q^{98} + 4q^{99} + O(q^{100})$$

## 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
$$2$$ $$-1$$
$$3$$ $$1$$
$$5$$ $$-1$$
$$13$$ $$-1$$

## 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 390.2.a.f 1
3.b odd 2 1 1170.2.a.a 1
4.b odd 2 1 3120.2.a.w 1
5.b even 2 1 1950.2.a.k 1
5.c odd 4 2 1950.2.e.g 2
12.b even 2 1 9360.2.a.p 1
13.b even 2 1 5070.2.a.a 1
13.d odd 4 2 5070.2.b.d 2
15.d odd 2 1 5850.2.a.bo 1
15.e even 4 2 5850.2.e.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.a.f 1 1.a even 1 1 trivial
1170.2.a.a 1 3.b odd 2 1
1950.2.a.k 1 5.b even 2 1
1950.2.e.g 2 5.c odd 4 2
3120.2.a.w 1 4.b odd 2 1
5070.2.a.a 1 13.b even 2 1
5070.2.b.d 2 13.d odd 4 2
5850.2.a.bo 1 15.d odd 2 1
5850.2.e.e 2 15.e even 4 2
9360.2.a.p 1 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(390))$$:

 $$T_{7}$$ $$T_{11} - 4$$ $$T_{31} + 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + T$$
$3$ $$1 + T$$
$5$ $$-1 + T$$
$7$ $$T$$
$11$ $$-4 + T$$
$13$ $$-1 + T$$
$17$ $$6 + T$$
$19$ $$-4 + T$$
$23$ $$-8 + T$$
$29$ $$-6 + T$$
$31$ $$8 + T$$
$37$ $$10 + T$$
$41$ $$6 + T$$
$43$ $$-4 + T$$
$47$ $$T$$
$53$ $$10 + T$$
$59$ $$-4 + T$$
$61$ $$2 + T$$
$67$ $$12 + T$$
$71$ $$-16 + T$$
$73$ $$-2 + T$$
$79$ $$16 + T$$
$83$ $$12 + T$$
$89$ $$-10 + T$$
$97$ $$6 + T$$