# Properties

 Label 390.2.a.e 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

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + 2 q^{7} + q^{8} + q^{9}+O(q^{10})$$ q + q^2 - q^3 + q^4 - q^5 - q^6 + 2 * q^7 + q^8 + q^9 $$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + 2 q^{7} + q^{8} + q^{9} - q^{10} + 4 q^{11} - q^{12} - q^{13} + 2 q^{14} + q^{15} + q^{16} + 8 q^{17} + q^{18} - 6 q^{19} - q^{20} - 2 q^{21} + 4 q^{22} + 6 q^{23} - q^{24} + q^{25} - q^{26} - q^{27} + 2 q^{28} - 4 q^{29} + q^{30} + q^{32} - 4 q^{33} + 8 q^{34} - 2 q^{35} + q^{36} - 2 q^{37} - 6 q^{38} + q^{39} - q^{40} - 2 q^{41} - 2 q^{42} - 4 q^{43} + 4 q^{44} - q^{45} + 6 q^{46} - q^{48} - 3 q^{49} + q^{50} - 8 q^{51} - q^{52} - 10 q^{53} - q^{54} - 4 q^{55} + 2 q^{56} + 6 q^{57} - 4 q^{58} + 4 q^{59} + q^{60} - 10 q^{61} + 2 q^{63} + q^{64} + q^{65} - 4 q^{66} + 12 q^{67} + 8 q^{68} - 6 q^{69} - 2 q^{70} - 8 q^{71} + q^{72} - 8 q^{73} - 2 q^{74} - q^{75} - 6 q^{76} + 8 q^{77} + q^{78} + 8 q^{79} - q^{80} + q^{81} - 2 q^{82} + 12 q^{83} - 2 q^{84} - 8 q^{85} - 4 q^{86} + 4 q^{87} + 4 q^{88} - 14 q^{89} - q^{90} - 2 q^{91} + 6 q^{92} + 6 q^{95} - q^{96} - 16 q^{97} - 3 q^{98} + 4 q^{99}+O(q^{100})$$ q + q^2 - q^3 + q^4 - q^5 - q^6 + 2 * q^7 + q^8 + q^9 - q^10 + 4 * q^11 - q^12 - q^13 + 2 * q^14 + q^15 + q^16 + 8 * q^17 + q^18 - 6 * q^19 - q^20 - 2 * q^21 + 4 * q^22 + 6 * q^23 - q^24 + q^25 - q^26 - q^27 + 2 * q^28 - 4 * q^29 + q^30 + q^32 - 4 * q^33 + 8 * q^34 - 2 * q^35 + q^36 - 2 * q^37 - 6 * q^38 + q^39 - q^40 - 2 * q^41 - 2 * q^42 - 4 * q^43 + 4 * q^44 - q^45 + 6 * q^46 - q^48 - 3 * q^49 + q^50 - 8 * q^51 - q^52 - 10 * q^53 - q^54 - 4 * q^55 + 2 * q^56 + 6 * q^57 - 4 * q^58 + 4 * q^59 + q^60 - 10 * q^61 + 2 * q^63 + q^64 + q^65 - 4 * q^66 + 12 * q^67 + 8 * q^68 - 6 * q^69 - 2 * q^70 - 8 * q^71 + q^72 - 8 * q^73 - 2 * q^74 - q^75 - 6 * q^76 + 8 * q^77 + q^78 + 8 * q^79 - q^80 + q^81 - 2 * q^82 + 12 * q^83 - 2 * q^84 - 8 * q^85 - 4 * q^86 + 4 * q^87 + 4 * q^88 - 14 * q^89 - q^90 - 2 * q^91 + 6 * q^92 + 6 * q^95 - q^96 - 16 * q^97 - 3 * q^98 + 4 * q^99

## 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.e 1
3.b odd 2 1 1170.2.a.e 1
4.b odd 2 1 3120.2.a.o 1
5.b even 2 1 1950.2.a.h 1
5.c odd 4 2 1950.2.e.f 2
12.b even 2 1 9360.2.a.bh 1
13.b even 2 1 5070.2.a.e 1
13.d odd 4 2 5070.2.b.e 2
15.d odd 2 1 5850.2.a.bi 1
15.e even 4 2 5850.2.e.i 2

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

 $$T_{7} - 2$$ T7 - 2 $$T_{11} - 4$$ T11 - 4 $$T_{31}$$ T31

## Hecke characteristic polynomials

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