# Properties

 Label 2730.2.a.o Level 2730 Weight 2 Character orbit 2730.a Self dual yes Analytic conductor 21.799 Analytic rank 0 Dimension 1 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2730 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2730.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$21.7991597518$$ 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

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} + q^{7} - q^{8} + q^{9} + O(q^{10})$$ $$q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} + q^{7} - q^{8} + q^{9} + q^{10} + q^{12} + q^{13} - q^{14} - q^{15} + q^{16} - 6q^{17} - q^{18} - 4q^{19} - q^{20} + q^{21} - q^{24} + q^{25} - q^{26} + q^{27} + q^{28} + 6q^{29} + q^{30} + 8q^{31} - q^{32} + 6q^{34} - q^{35} + q^{36} + 2q^{37} + 4q^{38} + q^{39} + q^{40} + 6q^{41} - q^{42} + 8q^{43} - q^{45} + 12q^{47} + q^{48} + q^{49} - q^{50} - 6q^{51} + q^{52} - 6q^{53} - q^{54} - q^{56} - 4q^{57} - 6q^{58} - 12q^{59} - q^{60} + 14q^{61} - 8q^{62} + q^{63} + q^{64} - q^{65} - 4q^{67} - 6q^{68} + q^{70} - 12q^{71} - q^{72} + 2q^{73} - 2q^{74} + q^{75} - 4q^{76} - q^{78} + 8q^{79} - q^{80} + q^{81} - 6q^{82} + q^{84} + 6q^{85} - 8q^{86} + 6q^{87} + 6q^{89} + q^{90} + q^{91} + 8q^{93} - 12q^{94} + 4q^{95} - q^{96} + 2q^{97} - q^{98} + O(q^{100})$$

## 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 1.00000 −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
$$2$$ $$1$$
$$3$$ $$-1$$
$$5$$ $$1$$
$$7$$ $$-1$$
$$13$$ $$-1$$

## 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 2730.2.a.o 1
3.b odd 2 1 8190.2.a.bx 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2730.2.a.o 1 1.a even 1 1 trivial
8190.2.a.bx 1 3.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2730))$$:

 $$T_{11}$$ $$T_{17} + 6$$ $$T_{19} + 4$$ $$T_{23}$$ $$T_{29} - 6$$

## Hecke characteristic polynomials

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