# Properties

 Label 330.2.a.e 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

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} + q^{8} + q^{9} + O(q^{10})$$ $$q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} + q^{8} + q^{9} + q^{10} - q^{11} + q^{12} - 2q^{13} + q^{15} + q^{16} + 2q^{17} + q^{18} - 4q^{19} + q^{20} - q^{22} + q^{24} + q^{25} - 2q^{26} + q^{27} - 2q^{29} + q^{30} + q^{32} - q^{33} + 2q^{34} + q^{36} - 2q^{37} - 4q^{38} - 2q^{39} + q^{40} + 2q^{41} - 12q^{43} - q^{44} + q^{45} + 8q^{47} + q^{48} - 7q^{49} + q^{50} + 2q^{51} - 2q^{52} + 6q^{53} + q^{54} - q^{55} - 4q^{57} - 2q^{58} - 12q^{59} + q^{60} + 6q^{61} + q^{64} - 2q^{65} - q^{66} + 4q^{67} + 2q^{68} + q^{72} - 6q^{73} - 2q^{74} + q^{75} - 4q^{76} - 2q^{78} - 16q^{79} + q^{80} + q^{81} + 2q^{82} + 4q^{83} + 2q^{85} - 12q^{86} - 2q^{87} - q^{88} + 10q^{89} + q^{90} + 8q^{94} - 4q^{95} + q^{96} + 2q^{97} - 7q^{98} - q^{99} + O(q^{100})$$

## 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.e 1
3.b odd 2 1 990.2.a.c 1
4.b odd 2 1 2640.2.a.h 1
5.b even 2 1 1650.2.a.b 1
5.c odd 4 2 1650.2.c.i 2
11.b odd 2 1 3630.2.a.k 1
12.b even 2 1 7920.2.a.g 1
15.d odd 2 1 4950.2.a.bk 1
15.e even 4 2 4950.2.c.v 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
330.2.a.e 1 1.a even 1 1 trivial
990.2.a.c 1 3.b odd 2 1
1650.2.a.b 1 5.b even 2 1
1650.2.c.i 2 5.c odd 4 2
2640.2.a.h 1 4.b odd 2 1
3630.2.a.k 1 11.b odd 2 1
4950.2.a.bk 1 15.d odd 2 1
4950.2.c.v 2 15.e even 4 2
7920.2.a.g 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

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