# Properties

 Label 3381.2.a.w Level $3381$ Weight $2$ Character orbit 3381.a Self dual yes Analytic conductor $26.997$ Analytic rank $1$ Dimension $4$ 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: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.24197.1 Defining polynomial: $$x^{4} - 6 x^{2} - x + 2$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + q^{3} + ( 1 + \beta_{2} ) q^{4} + ( -1 + \beta_{3} ) q^{5} -\beta_{1} q^{6} + ( -1 - \beta_{1} - \beta_{3} ) q^{8} + q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{2} + q^{3} + ( 1 + \beta_{2} ) q^{4} + ( -1 + \beta_{3} ) q^{5} -\beta_{1} q^{6} + ( -1 - \beta_{1} - \beta_{3} ) q^{8} + q^{9} + ( -1 + \beta_{1} - \beta_{2} ) q^{10} + ( -1 - \beta_{2} + \beta_{3} ) q^{11} + ( 1 + \beta_{2} ) q^{12} + ( -2 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{13} + ( -1 + \beta_{3} ) q^{15} + ( 2 + \beta_{1} ) q^{16} + ( -3 - \beta_{1} + \beta_{2} ) q^{17} -\beta_{1} q^{18} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{19} + ( 3 \beta_{1} - \beta_{2} - \beta_{3} ) q^{20} + ( 3 \beta_{1} - \beta_{2} + \beta_{3} ) q^{22} - q^{23} + ( -1 - \beta_{1} - \beta_{3} ) q^{24} + ( 1 + \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{25} + ( -1 + 4 \beta_{1} + \beta_{3} ) q^{26} + q^{27} + ( 1 + \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{29} + ( -1 + \beta_{1} - \beta_{2} ) q^{30} + ( -1 - \beta_{1} + \beta_{2} ) q^{31} + ( -1 - \beta_{2} + 2 \beta_{3} ) q^{32} + ( -1 - \beta_{2} + \beta_{3} ) q^{33} + ( 2 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{34} + ( 1 + \beta_{2} ) q^{36} + ( 5 + 3 \beta_{1} - \beta_{2} ) q^{37} + ( -6 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{38} + ( -2 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{39} + ( -5 + \beta_{3} ) q^{40} + ( -1 - \beta_{2} - \beta_{3} ) q^{41} + ( 2 + 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{43} + ( -7 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{44} + ( -1 + \beta_{3} ) q^{45} + \beta_{1} q^{46} + ( -2 + \beta_{1} - \beta_{3} ) q^{47} + ( 2 + \beta_{1} ) q^{48} + ( 1 + \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{50} + ( -3 - \beta_{1} + \beta_{2} ) q^{51} + ( -9 - \beta_{1} - 3 \beta_{2} + 2 \beta_{3} ) q^{52} + ( -1 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{53} -\beta_{1} q^{54} + ( 5 - 2 \beta_{1} - \beta_{3} ) q^{55} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{57} + ( -2 - 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{58} + ( 1 - 7 \beta_{1} + 2 \beta_{3} ) q^{59} + ( 3 \beta_{1} - \beta_{2} - \beta_{3} ) q^{60} + ( -6 + 2 \beta_{1} + \beta_{2} ) q^{61} + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{62} + ( -5 + \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{64} + ( -3 - 5 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{65} + ( 3 \beta_{1} - \beta_{2} + \beta_{3} ) q^{66} + ( 1 + 2 \beta_{1} + 3 \beta_{3} ) q^{67} + ( 3 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{68} - q^{69} + ( -4 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{71} + ( -1 - \beta_{1} - \beta_{3} ) q^{72} + ( -5 + \beta_{1} - \beta_{2} - 4 \beta_{3} ) q^{73} + ( -8 - 3 \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{74} + ( 1 + \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{75} + ( 7 + 4 \beta_{1} + 3 \beta_{3} ) q^{76} + ( -1 + 4 \beta_{1} + \beta_{3} ) q^{78} + ( -3 - \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{79} + ( -1 - \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{80} + q^{81} + ( 2 + 3 \beta_{1} + \beta_{2} + \beta_{3} ) q^{82} + ( -1 - \beta_{1} - 3 \beta_{2} + 4 \beta_{3} ) q^{83} + ( 3 + 4 \beta_{1} - 2 \beta_{2} - 5 \beta_{3} ) q^{85} + ( -9 - 4 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{86} + ( 1 + \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{87} + ( -3 + 5 \beta_{1} + \beta_{2} ) q^{88} + ( \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{89} + ( -1 + \beta_{1} - \beta_{2} ) q^{90} + ( -1 - \beta_{2} ) q^{92} + ( -1 - \beta_{1} + \beta_{2} ) q^{93} + ( -2 + 2 \beta_{1} ) q^{94} + ( -1 + 2 \beta_{2} - \beta_{3} ) q^{95} + ( -1 - \beta_{2} + 2 \beta_{3} ) q^{96} + ( 1 - 3 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{97} + ( -1 - \beta_{2} + \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{3} + 4q^{4} - 5q^{5} - 3q^{8} + 4q^{9} + O(q^{10})$$ $$4q + 4q^{3} + 4q^{4} - 5q^{5} - 3q^{8} + 4q^{9} - 4q^{10} - 5q^{11} + 4q^{12} - 7q^{13} - 5q^{15} + 8q^{16} - 12q^{17} - 3q^{19} + q^{20} - q^{22} - 4q^{23} - 3q^{24} + 7q^{25} - 5q^{26} + 4q^{27} + 6q^{29} - 4q^{30} - 4q^{31} - 6q^{32} - 5q^{33} + 9q^{34} + 4q^{36} + 20q^{37} - 23q^{38} - 7q^{39} - 21q^{40} - 3q^{41} + 9q^{43} - 27q^{44} - 5q^{45} - 7q^{47} + 8q^{48} + 3q^{50} - 12q^{51} - 38q^{52} - 6q^{53} + 21q^{55} - 3q^{57} - 7q^{58} + 2q^{59} + q^{60} - 24q^{61} + 9q^{62} - 21q^{64} - 14q^{65} - q^{66} + q^{67} + 13q^{68} - 4q^{69} - 17q^{71} - 3q^{72} - 16q^{73} - 33q^{74} + 7q^{75} + 25q^{76} - 5q^{78} - 10q^{79} - 6q^{80} + 4q^{81} + 7q^{82} - 8q^{83} + 17q^{85} - 35q^{86} + 6q^{87} - 12q^{88} + 3q^{89} - 4q^{90} - 4q^{92} - 4q^{93} - 8q^{94} - 3q^{95} - 6q^{96} + 2q^{97} - 5q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 6 x^{2} - x + 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - 5 \nu - 1$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + 5 \beta_{1} + 1$$

## 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
 2.46506 0.509552 −0.700017 −2.27460
−2.46506 1.00000 4.07653 0.653724 −2.46506 0 −5.11879 1.00000 −1.61147
1.2 −0.509552 1.00000 −1.74036 −4.41546 −0.509552 0 1.90591 1.00000 2.24991
1.3 0.700017 1.00000 −1.50998 1.15706 0.700017 0 −2.45704 1.00000 0.809960
1.4 2.27460 1.00000 3.17380 −2.39532 2.27460 0 2.66992 1.00000 −5.44840
 $$n$$: 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.w 4
7.b odd 2 1 483.2.a.i 4
21.c even 2 1 1449.2.a.p 4
28.d even 2 1 7728.2.a.cd 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.2.a.i 4 7.b odd 2 1
1449.2.a.p 4 21.c even 2 1
3381.2.a.w 4 1.a even 1 1 trivial
7728.2.a.cd 4 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3381))$$:

 $$T_{2}^{4} - 6 T_{2}^{2} + T_{2} + 2$$ $$T_{5}^{4} + 5 T_{5}^{3} - T_{5}^{2} - 14 T_{5} + 8$$ $$T_{11}^{4} + 5 T_{11}^{3} - 9 T_{11}^{2} - 65 T_{11} - 68$$ $$T_{13}^{4} + 7 T_{13}^{3} - 11 T_{13}^{2} - 152 T_{13} - 236$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$2 + T - 6 T^{2} + T^{4}$$
$3$ $$( -1 + T )^{4}$$
$5$ $$8 - 14 T - T^{2} + 5 T^{3} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$-68 - 65 T - 9 T^{2} + 5 T^{3} + T^{4}$$
$13$ $$-236 - 152 T - 11 T^{2} + 7 T^{3} + T^{4}$$
$17$ $$-104 - 10 T + 37 T^{2} + 12 T^{3} + T^{4}$$
$19$ $$52 - 51 T - 37 T^{2} + 3 T^{3} + T^{4}$$
$23$ $$( 1 + T )^{4}$$
$29$ $$-436 + 264 T - 31 T^{2} - 6 T^{3} + T^{4}$$
$31$ $$-16 - 46 T - 11 T^{2} + 4 T^{3} + T^{4}$$
$37$ $$-1868 + 280 T + 91 T^{2} - 20 T^{3} + T^{4}$$
$41$ $$-34 - 71 T - 27 T^{2} + 3 T^{3} + T^{4}$$
$43$ $$272 + 184 T - 39 T^{2} - 9 T^{3} + T^{4}$$
$47$ $$-32 - 28 T + 6 T^{2} + 7 T^{3} + T^{4}$$
$53$ $$-202 - 249 T - 68 T^{2} + 6 T^{3} + T^{4}$$
$59$ $$17564 + 263 T - 278 T^{2} - 2 T^{3} + T^{4}$$
$61$ $$-1286 + 191 T + 172 T^{2} + 24 T^{3} + T^{4}$$
$67$ $$4208 - 24 T - 141 T^{2} - T^{3} + T^{4}$$
$71$ $$64 + 128 T + 79 T^{2} + 17 T^{3} + T^{4}$$
$73$ $$-6656 - 2270 T - 95 T^{2} + 16 T^{3} + T^{4}$$
$79$ $$128 - 214 T - 17 T^{2} + 10 T^{3} + T^{4}$$
$83$ $$-7256 - 2450 T - 195 T^{2} + 8 T^{3} + T^{4}$$
$89$ $$-92 - 380 T - 113 T^{2} - 3 T^{3} + T^{4}$$
$97$ $$4 + 356 T - 87 T^{2} - 2 T^{3} + T^{4}$$