# Properties

 Label 390.2.a.c 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

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} + 4q^{7} - q^{8} + q^{9} + O(q^{10})$$ $$q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} + 4q^{7} - q^{8} + q^{9} + q^{10} + q^{12} - q^{13} - 4q^{14} - q^{15} + q^{16} - 2q^{17} - q^{18} + 4q^{19} - q^{20} + 4q^{21} + 8q^{23} - q^{24} + q^{25} + q^{26} + q^{27} + 4q^{28} + 2q^{29} + q^{30} - 8q^{31} - q^{32} + 2q^{34} - 4q^{35} + q^{36} + 2q^{37} - 4q^{38} - q^{39} + q^{40} - 6q^{41} - 4q^{42} + 12q^{43} - q^{45} - 8q^{46} + q^{48} + 9q^{49} - q^{50} - 2q^{51} - q^{52} + 10q^{53} - q^{54} - 4q^{56} + 4q^{57} - 2q^{58} - q^{60} - 10q^{61} + 8q^{62} + 4q^{63} + q^{64} + q^{65} - 4q^{67} - 2q^{68} + 8q^{69} + 4q^{70} - 16q^{71} - q^{72} - 6q^{73} - 2q^{74} + q^{75} + 4q^{76} + q^{78} - 8q^{79} - q^{80} + q^{81} + 6q^{82} - 4q^{83} + 4q^{84} + 2q^{85} - 12q^{86} + 2q^{87} - 14q^{89} + q^{90} - 4q^{91} + 8q^{92} - 8q^{93} - 4q^{95} - q^{96} - 6q^{97} - 9q^{98} + O(q^{100})$$

## 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 4.00000 −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.c 1
3.b odd 2 1 1170.2.a.n 1
4.b odd 2 1 3120.2.a.a 1
5.b even 2 1 1950.2.a.n 1
5.c odd 4 2 1950.2.e.e 2
12.b even 2 1 9360.2.a.bc 1
13.b even 2 1 5070.2.a.u 1
13.d odd 4 2 5070.2.b.i 2
15.d odd 2 1 5850.2.a.c 1
15.e even 4 2 5850.2.e.m 2

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T$$
$3$ $$-1 + T$$
$5$ $$1 + T$$
$7$ $$-4 + T$$
$11$ $$T$$
$13$ $$1 + T$$
$17$ $$2 + T$$
$19$ $$-4 + T$$
$23$ $$-8 + T$$
$29$ $$-2 + T$$
$31$ $$8 + T$$
$37$ $$-2 + T$$
$41$ $$6 + T$$
$43$ $$-12 + T$$
$47$ $$T$$
$53$ $$-10 + T$$
$59$ $$T$$
$61$ $$10 + T$$
$67$ $$4 + T$$
$71$ $$16 + T$$
$73$ $$6 + T$$
$79$ $$8 + T$$
$83$ $$4 + T$$
$89$ $$14 + T$$
$97$ $$6 + T$$