# Properties

 Label 3381.2.a.n Level $3381$ Weight $2$ Character orbit 3381.a Self dual yes Analytic conductor $26.997$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 1.61803 −0.618034
−2.61803 −1.00000 4.85410 3.61803 2.61803 0 −7.47214 1.00000 −9.47214
1.2 −0.381966 −1.00000 −1.85410 1.38197 0.381966 0 1.47214 1.00000 −0.527864
 $$n$$: 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.n 2
7.b odd 2 1 483.2.a.c 2
21.c even 2 1 1449.2.a.k 2
28.d even 2 1 7728.2.a.v 2

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

 $$T_{2}^{2} + 3 T_{2} + 1$$ $$T_{5}^{2} - 5 T_{5} + 5$$ $$T_{11}^{2} + 2 T_{11} - 19$$ $$T_{13}^{2} - 7 T_{13} + 11$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 3 T + T^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$5 - 5 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$-19 + 2 T + T^{2}$$
$13$ $$11 - 7 T + T^{2}$$
$17$ $$( -1 + T )^{2}$$
$19$ $$( -3 + T )^{2}$$
$23$ $$( 1 + T )^{2}$$
$29$ $$-11 + 6 T + T^{2}$$
$31$ $$31 - 12 T + T^{2}$$
$37$ $$-11 + 6 T + T^{2}$$
$41$ $$-41 + 4 T + T^{2}$$
$43$ $$-25 - 5 T + T^{2}$$
$47$ $$-20 - 10 T + T^{2}$$
$53$ $$19 + 9 T + T^{2}$$
$59$ $$79 - 19 T + T^{2}$$
$61$ $$-149 + 3 T + T^{2}$$
$67$ $$45 - 15 T + T^{2}$$
$71$ $$59 + 19 T + T^{2}$$
$73$ $$-29 - 8 T + T^{2}$$
$79$ $$5 - 10 T + T^{2}$$
$83$ $$( -3 + T )^{2}$$
$89$ $$31 + 13 T + T^{2}$$
$97$ $$( 5 + T )^{2}$$