# Properties

 Label 3381.2.a.r Level $3381$ Weight $2$ Character orbit 3381.a Self dual yes Analytic conductor $26.997$ Analytic rank $1$ 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: $$1$$ 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.

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

## 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
−1.61803 1.00000 0.618034 0.618034 −1.61803 0 2.23607 1.00000 −1.00000
1.2 0.618034 1.00000 −1.61803 −1.61803 0.618034 0 −2.23607 1.00000 −1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

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

## 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.r 2
7.b odd 2 1 483.2.a.d 2
21.c even 2 1 1449.2.a.h 2
28.d even 2 1 7728.2.a.bn 2

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + T + T^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$-1 + T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$-5 + T^{2}$$
$13$ $$11 - 7 T + T^{2}$$
$17$ $$-45 + T^{2}$$
$19$ $$-19 + 2 T + T^{2}$$
$23$ $$( 1 + T )^{2}$$
$29$ $$31 + 12 T + T^{2}$$
$31$ $$-45 + T^{2}$$
$37$ $$( 11 + T )^{2}$$
$41$ $$-11 + 6 T + T^{2}$$
$43$ $$-1 + T + T^{2}$$
$47$ $$20 - 10 T + T^{2}$$
$53$ $$25 + 15 T + T^{2}$$
$59$ $$109 + 21 T + T^{2}$$
$61$ $$-9 - 3 T + T^{2}$$
$67$ $$-31 - T + T^{2}$$
$71$ $$29 - 11 T + T^{2}$$
$73$ $$-9 - 12 T + T^{2}$$
$79$ $$5 + 10 T + T^{2}$$
$83$ $$-121 + 4 T + T^{2}$$
$89$ $$109 + 21 T + T^{2}$$
$97$ $$-171 - 6 T + T^{2}$$