# Properties

 Label 270.2.a.d Level $270$ Weight $2$ Character orbit 270.a Self dual yes Analytic conductor $2.156$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$270 = 2 \cdot 3^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 270.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$2.15596085457$$ 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

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + q^{5} + 2 q^{7} + q^{8} + O(q^{10})$$ $$q + q^{2} + q^{4} + q^{5} + 2 q^{7} + q^{8} + q^{10} - 3 q^{11} - q^{13} + 2 q^{14} + q^{16} - 3 q^{17} + 8 q^{19} + q^{20} - 3 q^{22} + 3 q^{23} + q^{25} - q^{26} + 2 q^{28} - 9 q^{29} - 7 q^{31} + q^{32} - 3 q^{34} + 2 q^{35} + 2 q^{37} + 8 q^{38} + q^{40} - 12 q^{41} - 7 q^{43} - 3 q^{44} + 3 q^{46} - 3 q^{47} - 3 q^{49} + q^{50} - q^{52} + 12 q^{53} - 3 q^{55} + 2 q^{56} - 9 q^{58} + 12 q^{59} - 10 q^{61} - 7 q^{62} + q^{64} - q^{65} - 4 q^{67} - 3 q^{68} + 2 q^{70} + 2 q^{73} + 2 q^{74} + 8 q^{76} - 6 q^{77} - q^{79} + q^{80} - 12 q^{82} + 18 q^{83} - 3 q^{85} - 7 q^{86} - 3 q^{88} - 2 q^{91} + 3 q^{92} - 3 q^{94} + 8 q^{95} + 14 q^{97} - 3 q^{98} + O(q^{100})$$

## 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 0 1.00000 1.00000 0 2.00000 1.00000 0 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$-1$$

## 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 270.2.a.d yes 1
3.b odd 2 1 270.2.a.a 1
4.b odd 2 1 2160.2.a.p 1
5.b even 2 1 1350.2.a.c 1
5.c odd 4 2 1350.2.c.a 2
8.b even 2 1 8640.2.a.z 1
8.d odd 2 1 8640.2.a.f 1
9.c even 3 2 810.2.e.a 2
9.d odd 6 2 810.2.e.k 2
12.b even 2 1 2160.2.a.a 1
15.d odd 2 1 1350.2.a.p 1
15.e even 4 2 1350.2.c.l 2
24.f even 2 1 8640.2.a.bo 1
24.h odd 2 1 8640.2.a.by 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.2.a.a 1 3.b odd 2 1
270.2.a.d yes 1 1.a even 1 1 trivial
810.2.e.a 2 9.c even 3 2
810.2.e.k 2 9.d odd 6 2
1350.2.a.c 1 5.b even 2 1
1350.2.a.p 1 15.d odd 2 1
1350.2.c.a 2 5.c odd 4 2
1350.2.c.l 2 15.e even 4 2
2160.2.a.a 1 12.b even 2 1
2160.2.a.p 1 4.b odd 2 1
8640.2.a.f 1 8.d odd 2 1
8640.2.a.z 1 8.b even 2 1
8640.2.a.bo 1 24.f even 2 1
8640.2.a.by 1 24.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(270))$$:

 $$T_{11} + 3$$ $$T_{13} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + T$$
$3$ $$T$$
$5$ $$-1 + T$$
$7$ $$-2 + T$$
$11$ $$3 + T$$
$13$ $$1 + T$$
$17$ $$3 + T$$
$19$ $$-8 + T$$
$23$ $$-3 + T$$
$29$ $$9 + T$$
$31$ $$7 + T$$
$37$ $$-2 + T$$
$41$ $$12 + T$$
$43$ $$7 + T$$
$47$ $$3 + T$$
$53$ $$-12 + T$$
$59$ $$-12 + T$$
$61$ $$10 + T$$
$67$ $$4 + T$$
$71$ $$T$$
$73$ $$-2 + T$$
$79$ $$1 + T$$
$83$ $$-18 + T$$
$89$ $$T$$
$97$ $$-14 + T$$