# Properties

 Label 3366.2.a.c 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

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} - 2q^{5} - 2q^{7} - q^{8} + O(q^{10})$$ $$q - q^{2} + q^{4} - 2q^{5} - 2q^{7} - q^{8} + 2q^{10} + q^{11} + 4q^{13} + 2q^{14} + q^{16} - q^{17} + 2q^{19} - 2q^{20} - q^{22} - 2q^{23} - q^{25} - 4q^{26} - 2q^{28} - 2q^{29} - 4q^{31} - q^{32} + q^{34} + 4q^{35} + 6q^{37} - 2q^{38} + 2q^{40} - 6q^{41} + 2q^{43} + q^{44} + 2q^{46} - 3q^{49} + q^{50} + 4q^{52} + 12q^{53} - 2q^{55} + 2q^{56} + 2q^{58} + 14q^{59} + 6q^{61} + 4q^{62} + q^{64} - 8q^{65} - 4q^{67} - q^{68} - 4q^{70} + 2q^{71} - 8q^{73} - 6q^{74} + 2q^{76} - 2q^{77} - 2q^{79} - 2q^{80} + 6q^{82} + 12q^{83} + 2q^{85} - 2q^{86} - q^{88} - 6q^{89} - 8q^{91} - 2q^{92} - 4q^{95} - 6q^{97} + 3q^{98} + O(q^{100})$$

## 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 −2.00000 −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.c 1
3.b odd 2 1 1122.2.a.l 1
12.b even 2 1 8976.2.a.r 1

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

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

## Hecke characteristic polynomials

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