# Properties

 Label 3381.2.a.j Level $3381$ Weight $2$ Character orbit 3381.a Self dual yes Analytic conductor $26.997$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3381 = 3 \cdot 7^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3381.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$26.9974209234$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 483) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{3} - 2q^{4} + q^{9} + O(q^{10})$$ $$q + q^{3} - 2q^{4} + q^{9} + 6q^{11} - 2q^{12} + 5q^{13} + 4q^{16} - 6q^{17} - q^{19} - q^{23} - 5q^{25} + q^{27} + 6q^{29} + 5q^{31} + 6q^{33} - 2q^{36} - 7q^{37} + 5q^{39} - q^{43} - 12q^{44} + 6q^{47} + 4q^{48} - 6q^{51} - 10q^{52} + 12q^{53} - q^{57} - 6q^{59} + 14q^{61} - 8q^{64} + 5q^{67} + 12q^{68} - q^{69} - 6q^{71} - 7q^{73} - 5q^{75} + 2q^{76} + 5q^{79} + q^{81} - 12q^{83} + 6q^{87} - 6q^{89} + 2q^{92} + 5q^{93} - 10q^{97} + 6q^{99} + O(q^{100})$$

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

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$7$$ $$1$$
$$23$$ $$1$$

## 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 3381.2.a.j 1
7.b odd 2 1 3381.2.a.g 1
7.c even 3 2 483.2.i.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.2.i.b 2 7.c even 3 2
3381.2.a.g 1 7.b odd 2 1
3381.2.a.j 1 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3381))$$:

 $$T_{2}$$ $$T_{5}$$ $$T_{11} - 6$$ $$T_{13} - 5$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$-1 + T$$
$5$ $$T$$
$7$ $$T$$
$11$ $$-6 + T$$
$13$ $$-5 + T$$
$17$ $$6 + T$$
$19$ $$1 + T$$
$23$ $$1 + T$$
$29$ $$-6 + T$$
$31$ $$-5 + T$$
$37$ $$7 + T$$
$41$ $$T$$
$43$ $$1 + T$$
$47$ $$-6 + T$$
$53$ $$-12 + T$$
$59$ $$6 + T$$
$61$ $$-14 + T$$
$67$ $$-5 + T$$
$71$ $$6 + T$$
$73$ $$7 + T$$
$79$ $$-5 + T$$
$83$ $$12 + T$$
$89$ $$6 + T$$
$97$ $$10 + T$$