# Properties

 Label 3381.2.a.v Level $3381$ Weight $2$ Character orbit 3381.a Self dual yes Analytic conductor $26.997$ Analytic rank $0$ Dimension $3$ 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: $$3$$ Coefficient field: 3.3.837.1 Defining polynomial: $$x^{3} - 6 x - 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 483) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} - q^{3} + ( 2 + \beta_{2} ) q^{4} + ( -1 + \beta_{1} ) q^{5} -\beta_{1} q^{6} + ( 1 + 2 \beta_{1} ) q^{8} + q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} - q^{3} + ( 2 + \beta_{2} ) q^{4} + ( -1 + \beta_{1} ) q^{5} -\beta_{1} q^{6} + ( 1 + 2 \beta_{1} ) q^{8} + q^{9} + ( 4 - \beta_{1} + \beta_{2} ) q^{10} + ( 2 - \beta_{1} + \beta_{2} ) q^{11} + ( -2 - \beta_{2} ) q^{12} + ( -3 + \beta_{2} ) q^{13} + ( 1 - \beta_{1} ) q^{15} + ( 4 + \beta_{1} ) q^{16} + ( -2 + \beta_{1} + \beta_{2} ) q^{17} + \beta_{1} q^{18} + ( 2 - \beta_{1} + \beta_{2} ) q^{19} + ( -1 + 4 \beta_{1} - \beta_{2} ) q^{20} + ( -3 + 4 \beta_{1} - \beta_{2} ) q^{22} - q^{23} + ( -1 - 2 \beta_{1} ) q^{24} + ( -2 \beta_{1} + \beta_{2} ) q^{25} + ( 1 - \beta_{1} ) q^{26} - q^{27} + ( 2 + 3 \beta_{1} - \beta_{2} ) q^{29} + ( -4 + \beta_{1} - \beta_{2} ) q^{30} + ( 6 - \beta_{1} + \beta_{2} ) q^{31} + ( 2 + \beta_{2} ) q^{32} + ( -2 + \beta_{1} - \beta_{2} ) q^{33} + ( 5 + \beta_{2} ) q^{34} + ( 2 + \beta_{2} ) q^{36} + ( 2 - 3 \beta_{1} + \beta_{2} ) q^{37} + ( -3 + 4 \beta_{1} - \beta_{2} ) q^{38} + ( 3 - \beta_{2} ) q^{39} + ( 7 - \beta_{1} + 2 \beta_{2} ) q^{40} + ( 2 - \beta_{1} + 3 \beta_{2} ) q^{41} + ( -3 - 2 \beta_{1} + \beta_{2} ) q^{43} + ( 11 - 3 \beta_{1} + 2 \beta_{2} ) q^{44} + ( -1 + \beta_{1} ) q^{45} -\beta_{1} q^{46} + ( -6 - 2 \beta_{2} ) q^{47} + ( -4 - \beta_{1} ) q^{48} + ( -7 + 2 \beta_{1} - 2 \beta_{2} ) q^{50} + ( 2 - \beta_{1} - \beta_{2} ) q^{51} + ( 2 + \beta_{1} - 3 \beta_{2} ) q^{52} + ( 1 - \beta_{1} ) q^{53} -\beta_{1} q^{54} + ( -5 + 5 \beta_{1} - 2 \beta_{2} ) q^{55} + ( -2 + \beta_{1} - \beta_{2} ) q^{57} + ( 11 + 3 \beta_{2} ) q^{58} + ( -1 + \beta_{1} + 2 \beta_{2} ) q^{59} + ( 1 - 4 \beta_{1} + \beta_{2} ) q^{60} + ( 1 + 2 \beta_{1} - \beta_{2} ) q^{61} + ( -3 + 8 \beta_{1} - \beta_{2} ) q^{62} + ( -7 + 2 \beta_{1} ) q^{64} + ( 4 - \beta_{1} - \beta_{2} ) q^{65} + ( 3 - 4 \beta_{1} + \beta_{2} ) q^{66} + ( -7 + 3 \beta_{1} + 2 \beta_{2} ) q^{67} + ( 5 + 5 \beta_{1} - 2 \beta_{2} ) q^{68} + q^{69} + ( 3 + 2 \beta_{1} + 3 \beta_{2} ) q^{71} + ( 1 + 2 \beta_{1} ) q^{72} + ( -4 - 3 \beta_{1} - \beta_{2} ) q^{73} + ( -11 + 4 \beta_{1} - 3 \beta_{2} ) q^{74} + ( 2 \beta_{1} - \beta_{2} ) q^{75} + ( 11 - 3 \beta_{1} + 2 \beta_{2} ) q^{76} + ( -1 + \beta_{1} ) q^{78} + ( 4 + 3 \beta_{1} - \beta_{2} ) q^{79} + ( 3 \beta_{1} + \beta_{2} ) q^{80} + q^{81} + ( -1 + 8 \beta_{1} - \beta_{2} ) q^{82} + ( -4 + \beta_{1} + \beta_{2} ) q^{83} + ( 7 - \beta_{1} ) q^{85} + ( -7 - \beta_{1} - 2 \beta_{2} ) q^{86} + ( -2 - 3 \beta_{1} + \beta_{2} ) q^{87} + ( -4 + 7 \beta_{1} - \beta_{2} ) q^{88} + ( -5 - \beta_{2} ) q^{89} + ( 4 - \beta_{1} + \beta_{2} ) q^{90} + ( -2 - \beta_{2} ) q^{92} + ( -6 + \beta_{1} - \beta_{2} ) q^{93} + ( -2 - 10 \beta_{1} ) q^{94} + ( -5 + 5 \beta_{1} - 2 \beta_{2} ) q^{95} + ( -2 - \beta_{2} ) q^{96} + ( 4 - 3 \beta_{1} - \beta_{2} ) q^{97} + ( 2 - \beta_{1} + \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - 3q^{3} + 6q^{4} - 3q^{5} + 3q^{8} + 3q^{9} + O(q^{10})$$ $$3q - 3q^{3} + 6q^{4} - 3q^{5} + 3q^{8} + 3q^{9} + 12q^{10} + 6q^{11} - 6q^{12} - 9q^{13} + 3q^{15} + 12q^{16} - 6q^{17} + 6q^{19} - 3q^{20} - 9q^{22} - 3q^{23} - 3q^{24} + 3q^{26} - 3q^{27} + 6q^{29} - 12q^{30} + 18q^{31} + 6q^{32} - 6q^{33} + 15q^{34} + 6q^{36} + 6q^{37} - 9q^{38} + 9q^{39} + 21q^{40} + 6q^{41} - 9q^{43} + 33q^{44} - 3q^{45} - 18q^{47} - 12q^{48} - 21q^{50} + 6q^{51} + 6q^{52} + 3q^{53} - 15q^{55} - 6q^{57} + 33q^{58} - 3q^{59} + 3q^{60} + 3q^{61} - 9q^{62} - 21q^{64} + 12q^{65} + 9q^{66} - 21q^{67} + 15q^{68} + 3q^{69} + 9q^{71} + 3q^{72} - 12q^{73} - 33q^{74} + 33q^{76} - 3q^{78} + 12q^{79} + 3q^{81} - 3q^{82} - 12q^{83} + 21q^{85} - 21q^{86} - 6q^{87} - 12q^{88} - 15q^{89} + 12q^{90} - 6q^{92} - 18q^{93} - 6q^{94} - 15q^{95} - 6q^{96} + 12q^{97} + 6q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 6 x - 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 4$$

## 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
 −2.36147 −0.167449 2.52892
−2.36147 −1.00000 3.57653 −3.36147 2.36147 0 −3.72294 1.00000 7.93800
1.2 −0.167449 −1.00000 −1.97196 −1.16745 0.167449 0 0.665102 1.00000 0.195488
1.3 2.52892 −1.00000 4.39543 1.52892 −2.52892 0 6.05784 1.00000 3.86651
 $$n$$: 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.v 3
7.b odd 2 1 483.2.a.h 3
21.c even 2 1 1449.2.a.l 3
28.d even 2 1 7728.2.a.bt 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.2.a.h 3 7.b odd 2 1
1449.2.a.l 3 21.c even 2 1
3381.2.a.v 3 1.a even 1 1 trivial
7728.2.a.bt 3 28.d even 2 1

## Hecke kernels

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

 $$T_{2}^{3} - 6 T_{2} - 1$$ $$T_{5}^{3} + 3 T_{5}^{2} - 3 T_{5} - 6$$ $$T_{11}^{3} - 6 T_{11}^{2} - 3 T_{11} + 20$$ $$T_{13}^{3} + 9 T_{13}^{2} + 15 T_{13} + 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 - 6 T + T^{3}$$
$3$ $$( 1 + T )^{3}$$
$5$ $$-6 - 3 T + 3 T^{2} + T^{3}$$
$7$ $$T^{3}$$
$11$ $$20 - 3 T - 6 T^{2} + T^{3}$$
$13$ $$6 + 15 T + 9 T^{2} + T^{3}$$
$17$ $$-50 - 9 T + 6 T^{2} + T^{3}$$
$19$ $$20 - 3 T - 6 T^{2} + T^{3}$$
$23$ $$( 1 + T )^{3}$$
$29$ $$262 - 45 T - 6 T^{2} + T^{3}$$
$31$ $$-128 + 93 T - 18 T^{2} + T^{3}$$
$37$ $$-50 - 45 T - 6 T^{2} + T^{3}$$
$41$ $$590 - 93 T - 6 T^{2} + T^{3}$$
$43$ $$-124 - 3 T + 9 T^{2} + T^{3}$$
$47$ $$-192 + 60 T + 18 T^{2} + T^{3}$$
$53$ $$6 - 3 T - 3 T^{2} + T^{3}$$
$59$ $$-12 - 57 T + 3 T^{2} + T^{3}$$
$61$ $$90 - 27 T - 3 T^{2} + T^{3}$$
$67$ $$-908 + 27 T + 21 T^{2} + T^{3}$$
$71$ $$424 - 123 T - 9 T^{2} + T^{3}$$
$73$ $$10 - 27 T + 12 T^{2} + T^{3}$$
$79$ $$320 - 9 T - 12 T^{2} + T^{3}$$
$83$ $$-36 + 27 T + 12 T^{2} + T^{3}$$
$89$ $$50 + 63 T + 15 T^{2} + T^{3}$$
$97$ $$482 - 27 T - 12 T^{2} + T^{3}$$