# Properties

 Label 330.2.a.a Level $330$ Weight $2$ Character orbit 330.a Self dual yes Analytic conductor $2.635$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$330 = 2 \cdot 3 \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 330.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$2.63506326670$$ 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

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} - q^{8} + q^{9} + O(q^{10})$$ $$q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} - q^{8} + q^{9} + q^{10} + q^{11} - q^{12} + 2q^{13} + q^{15} + q^{16} - 2q^{17} - q^{18} + 8q^{19} - q^{20} - q^{22} + 4q^{23} + q^{24} + q^{25} - 2q^{26} - q^{27} + 2q^{29} - q^{30} + 8q^{31} - q^{32} - q^{33} + 2q^{34} + q^{36} - 2q^{37} - 8q^{38} - 2q^{39} + q^{40} + 6q^{41} + 8q^{43} + q^{44} - q^{45} - 4q^{46} - 4q^{47} - q^{48} - 7q^{49} - q^{50} + 2q^{51} + 2q^{52} + 2q^{53} + q^{54} - q^{55} - 8q^{57} - 2q^{58} + 4q^{59} + q^{60} - 6q^{61} - 8q^{62} + q^{64} - 2q^{65} + q^{66} - 12q^{67} - 2q^{68} - 4q^{69} - 12q^{71} - q^{72} + 2q^{73} + 2q^{74} - q^{75} + 8q^{76} + 2q^{78} - q^{80} + q^{81} - 6q^{82} + 4q^{83} + 2q^{85} - 8q^{86} - 2q^{87} - q^{88} - 6q^{89} + q^{90} + 4q^{92} - 8q^{93} + 4q^{94} - 8q^{95} + q^{96} - 14q^{97} + 7q^{98} + q^{99} + O(q^{100})$$

## 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 0 −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
$$2$$ $$1$$
$$3$$ $$1$$
$$5$$ $$1$$
$$11$$ $$-1$$

## 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 330.2.a.a 1
3.b odd 2 1 990.2.a.j 1
4.b odd 2 1 2640.2.a.n 1
5.b even 2 1 1650.2.a.r 1
5.c odd 4 2 1650.2.c.l 2
11.b odd 2 1 3630.2.a.n 1
12.b even 2 1 7920.2.a.bb 1
15.d odd 2 1 4950.2.a.k 1
15.e even 4 2 4950.2.c.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
330.2.a.a 1 1.a even 1 1 trivial
990.2.a.j 1 3.b odd 2 1
1650.2.a.r 1 5.b even 2 1
1650.2.c.l 2 5.c odd 4 2
2640.2.a.n 1 4.b odd 2 1
3630.2.a.n 1 11.b odd 2 1
4950.2.a.k 1 15.d odd 2 1
4950.2.c.g 2 15.e even 4 2
7920.2.a.bb 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(330))$$:

 $$T_{7}$$ $$T_{13} - 2$$

## Hecke characteristic polynomials

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