# Properties

 Label 3366.2.a.k Level $3366$ Weight $2$ Character orbit 3366.a Self dual yes Analytic conductor $26.878$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$3366 = 2 \cdot 3^{2} \cdot 11 \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3366.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$26.8776453204$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1122) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + 2q^{5} - q^{8} + O(q^{10})$$ $$q - q^{2} + q^{4} + 2q^{5} - q^{8} - 2q^{10} + q^{11} - 2q^{13} + q^{16} - q^{17} - 4q^{19} + 2q^{20} - q^{22} - q^{25} + 2q^{26} + 2q^{29} - 8q^{31} - q^{32} + q^{34} - 10q^{37} + 4q^{38} - 2q^{40} + 6q^{41} + 4q^{43} + q^{44} - 8q^{47} - 7q^{49} + q^{50} - 2q^{52} - 6q^{53} + 2q^{55} - 2q^{58} + 12q^{59} - 10q^{61} + 8q^{62} + q^{64} - 4q^{65} + 4q^{67} - q^{68} - 16q^{71} + 2q^{73} + 10q^{74} - 4q^{76} - 8q^{79} + 2q^{80} - 6q^{82} + 12q^{83} - 2q^{85} - 4q^{86} - q^{88} - 10q^{89} + 8q^{94} - 8q^{95} + 2q^{97} + 7q^{98} + O(q^{100})$$

## 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 2.00000 0 0 −1.00000 0 −2.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$
$$11$$ $$-1$$
$$17$$ $$1$$

## 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 3366.2.a.k 1
3.b odd 2 1 1122.2.a.e 1
12.b even 2 1 8976.2.a.w 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1122.2.a.e 1 3.b odd 2 1
3366.2.a.k 1 1.a even 1 1 trivial
8976.2.a.w 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3366))$$:

 $$T_{5} - 2$$ $$T_{7}$$ $$T_{13} + 2$$ $$T_{19} + 4$$ $$T_{23}$$

## Hecke characteristic polynomials

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