# Properties

 Label 390.2.a.g 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

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + 2q^{7} + q^{8} + q^{9} + O(q^{10})$$ $$q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + 2q^{7} + q^{8} + q^{9} - q^{10} + q^{12} + q^{13} + 2q^{14} - q^{15} + q^{16} + q^{18} + 2q^{19} - q^{20} + 2q^{21} - 6q^{23} + q^{24} + q^{25} + q^{26} + q^{27} + 2q^{28} - q^{30} - 4q^{31} + q^{32} - 2q^{35} + q^{36} + 2q^{37} + 2q^{38} + q^{39} - q^{40} - 6q^{41} + 2q^{42} - 4q^{43} - q^{45} - 6q^{46} + q^{48} - 3q^{49} + q^{50} + q^{52} - 6q^{53} + q^{54} + 2q^{56} + 2q^{57} - q^{60} - 10q^{61} - 4q^{62} + 2q^{63} + q^{64} - q^{65} + 8q^{67} - 6q^{69} - 2q^{70} + q^{72} + 8q^{73} + 2q^{74} + q^{75} + 2q^{76} + q^{78} + 8q^{79} - q^{80} + q^{81} - 6q^{82} - 12q^{83} + 2q^{84} - 4q^{86} + 6q^{89} - q^{90} + 2q^{91} - 6q^{92} - 4q^{93} - 2q^{95} + q^{96} + 8q^{97} - 3q^{98} + O(q^{100})$$

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

## 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.g 1
3.b odd 2 1 1170.2.a.g 1
4.b odd 2 1 3120.2.a.b 1
5.b even 2 1 1950.2.a.b 1
5.c odd 4 2 1950.2.e.k 2
12.b even 2 1 9360.2.a.bg 1
13.b even 2 1 5070.2.a.k 1
13.d odd 4 2 5070.2.b.n 2
15.d odd 2 1 5850.2.a.bk 1
15.e even 4 2 5850.2.e.r 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.a.g 1 1.a even 1 1 trivial
1170.2.a.g 1 3.b odd 2 1
1950.2.a.b 1 5.b even 2 1
1950.2.e.k 2 5.c odd 4 2
3120.2.a.b 1 4.b odd 2 1
5070.2.a.k 1 13.b even 2 1
5070.2.b.n 2 13.d odd 4 2
5850.2.a.bk 1 15.d odd 2 1
5850.2.e.r 2 15.e even 4 2
9360.2.a.bg 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(390))$$:

 $$T_{7} - 2$$ $$T_{11}$$ $$T_{31} + 4$$

## Hecke characteristic polynomials

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