# Properties

 Label 3381.2.a.p 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: $$\Q(\sqrt{13})$$ Defining polynomial: $$x^{2} - x - 3$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{13})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta q^{2} - q^{3} + ( 1 + \beta ) q^{4} + ( 3 - \beta ) q^{5} + \beta q^{6} -3 q^{8} + q^{9} +O(q^{10})$$ $$q -\beta q^{2} - q^{3} + ( 1 + \beta ) q^{4} + ( 3 - \beta ) q^{5} + \beta q^{6} -3 q^{8} + q^{9} + ( 3 - 2 \beta ) q^{10} -5 q^{11} + ( -1 - \beta ) q^{12} -\beta q^{13} + ( -3 + \beta ) q^{15} + ( -2 + \beta ) q^{16} + ( 1 + 2 \beta ) q^{17} -\beta q^{18} + ( -3 + 2 \beta ) q^{19} + \beta q^{20} + 5 \beta q^{22} + q^{23} + 3 q^{24} + ( 7 - 5 \beta ) q^{25} + ( 3 + \beta ) q^{26} - q^{27} + ( -3 + 4 \beta ) q^{29} + ( -3 + 2 \beta ) q^{30} -3 q^{31} + ( 3 + \beta ) q^{32} + 5 q^{33} + ( -6 - 3 \beta ) q^{34} + ( 1 + \beta ) q^{36} -9 q^{37} + ( -6 + \beta ) q^{38} + \beta q^{39} + ( -9 + 3 \beta ) q^{40} + ( 3 + 4 \beta ) q^{41} + ( -6 + 5 \beta ) q^{43} + ( -5 - 5 \beta ) q^{44} + ( 3 - \beta ) q^{45} -\beta q^{46} + ( 4 + 2 \beta ) q^{47} + ( 2 - \beta ) q^{48} + ( 15 - 2 \beta ) q^{50} + ( -1 - 2 \beta ) q^{51} + ( -3 - 2 \beta ) q^{52} + ( -1 - 5 \beta ) q^{53} + \beta q^{54} + ( -15 + 5 \beta ) q^{55} + ( 3 - 2 \beta ) q^{57} + ( -12 - \beta ) q^{58} + ( 3 - 3 \beta ) q^{59} -\beta q^{60} + ( 8 - 3 \beta ) q^{61} + 3 \beta q^{62} + ( 1 - 6 \beta ) q^{64} + ( 3 - 2 \beta ) q^{65} -5 \beta q^{66} + ( -5 - 3 \beta ) q^{67} + ( 7 + 5 \beta ) q^{68} - q^{69} + ( -6 + 3 \beta ) q^{71} -3 q^{72} + ( -7 + 4 \beta ) q^{73} + 9 \beta q^{74} + ( -7 + 5 \beta ) q^{75} + ( 3 + \beta ) q^{76} + ( -3 - \beta ) q^{78} - q^{79} + ( -9 + 4 \beta ) q^{80} + q^{81} + ( -12 - 7 \beta ) q^{82} + ( -1 - 2 \beta ) q^{83} + ( -3 + 3 \beta ) q^{85} + ( -15 + \beta ) q^{86} + ( 3 - 4 \beta ) q^{87} + 15 q^{88} + ( -4 - 3 \beta ) q^{89} + ( 3 - 2 \beta ) q^{90} + ( 1 + \beta ) q^{92} + 3 q^{93} + ( -6 - 6 \beta ) q^{94} + ( -15 + 7 \beta ) q^{95} + ( -3 - \beta ) q^{96} + ( 13 + 2 \beta ) q^{97} -5 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - 2 q^{3} + 3 q^{4} + 5 q^{5} + q^{6} - 6 q^{8} + 2 q^{9} + O(q^{10})$$ $$2 q - q^{2} - 2 q^{3} + 3 q^{4} + 5 q^{5} + q^{6} - 6 q^{8} + 2 q^{9} + 4 q^{10} - 10 q^{11} - 3 q^{12} - q^{13} - 5 q^{15} - 3 q^{16} + 4 q^{17} - q^{18} - 4 q^{19} + q^{20} + 5 q^{22} + 2 q^{23} + 6 q^{24} + 9 q^{25} + 7 q^{26} - 2 q^{27} - 2 q^{29} - 4 q^{30} - 6 q^{31} + 7 q^{32} + 10 q^{33} - 15 q^{34} + 3 q^{36} - 18 q^{37} - 11 q^{38} + q^{39} - 15 q^{40} + 10 q^{41} - 7 q^{43} - 15 q^{44} + 5 q^{45} - q^{46} + 10 q^{47} + 3 q^{48} + 28 q^{50} - 4 q^{51} - 8 q^{52} - 7 q^{53} + q^{54} - 25 q^{55} + 4 q^{57} - 25 q^{58} + 3 q^{59} - q^{60} + 13 q^{61} + 3 q^{62} - 4 q^{64} + 4 q^{65} - 5 q^{66} - 13 q^{67} + 19 q^{68} - 2 q^{69} - 9 q^{71} - 6 q^{72} - 10 q^{73} + 9 q^{74} - 9 q^{75} + 7 q^{76} - 7 q^{78} - 2 q^{79} - 14 q^{80} + 2 q^{81} - 31 q^{82} - 4 q^{83} - 3 q^{85} - 29 q^{86} + 2 q^{87} + 30 q^{88} - 11 q^{89} + 4 q^{90} + 3 q^{92} + 6 q^{93} - 18 q^{94} - 23 q^{95} - 7 q^{96} + 28 q^{97} - 10 q^{99} + O(q^{100})$$

## 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
 2.30278 −1.30278
−2.30278 −1.00000 3.30278 0.697224 2.30278 0 −3.00000 1.00000 −1.60555
1.2 1.30278 −1.00000 −0.302776 4.30278 −1.30278 0 −3.00000 1.00000 5.60555
 $$n$$: 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.p 2
7.b odd 2 1 483.2.a.f 2
21.c even 2 1 1449.2.a.j 2
28.d even 2 1 7728.2.a.x 2

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-3 + T + T^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$3 - 5 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$( 5 + T )^{2}$$
$13$ $$-3 + T + T^{2}$$
$17$ $$-9 - 4 T + T^{2}$$
$19$ $$-9 + 4 T + T^{2}$$
$23$ $$( -1 + T )^{2}$$
$29$ $$-51 + 2 T + T^{2}$$
$31$ $$( 3 + T )^{2}$$
$37$ $$( 9 + T )^{2}$$
$41$ $$-27 - 10 T + T^{2}$$
$43$ $$-69 + 7 T + T^{2}$$
$47$ $$12 - 10 T + T^{2}$$
$53$ $$-69 + 7 T + T^{2}$$
$59$ $$-27 - 3 T + T^{2}$$
$61$ $$13 - 13 T + T^{2}$$
$67$ $$13 + 13 T + T^{2}$$
$71$ $$-9 + 9 T + T^{2}$$
$73$ $$-27 + 10 T + T^{2}$$
$79$ $$( 1 + T )^{2}$$
$83$ $$-9 + 4 T + T^{2}$$
$89$ $$1 + 11 T + T^{2}$$
$97$ $$183 - 28 T + T^{2}$$