# Properties

 Label 3381.2.a.q Level $3381$ Weight $2$ Character orbit 3381.a Self dual yes Analytic conductor $26.997$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

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## 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: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 2.56155 −1.56155
−2.56155 −1.00000 4.56155 2.56155 2.56155 0 −6.56155 1.00000 −6.56155
1.2 1.56155 −1.00000 0.438447 −1.56155 −1.56155 0 −2.43845 1.00000 −2.43845
 $$n$$: 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.q 2
7.b odd 2 1 3381.2.a.s 2
7.d odd 6 2 483.2.i.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.2.i.e 4 7.d odd 6 2
3381.2.a.q 2 1.a even 1 1 trivial
3381.2.a.s 2 7.b odd 2 1

## Hecke kernels

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

 $$T_{2}^{2} + T_{2} - 4$$ $$T_{5}^{2} - T_{5} - 4$$ $$T_{11} + 2$$ $$T_{13}^{2} - 4 T_{13} - 13$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-4 + T + T^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$-4 - T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$( 2 + T )^{2}$$
$13$ $$-13 - 4 T + T^{2}$$
$17$ $$26 - 11 T + T^{2}$$
$19$ $$8 + 7 T + T^{2}$$
$23$ $$( -1 + T )^{2}$$
$29$ $$8 + 10 T + T^{2}$$
$31$ $$-38 - T + T^{2}$$
$37$ $$26 - 11 T + T^{2}$$
$41$ $$-64 - 4 T + T^{2}$$
$43$ $$4 + 13 T + T^{2}$$
$47$ $$-38 + T + T^{2}$$
$53$ $$-100 - 5 T + T^{2}$$
$59$ $$8 + 10 T + T^{2}$$
$61$ $$( 6 + T )^{2}$$
$67$ $$83 + 20 T + T^{2}$$
$71$ $$-94 - 7 T + T^{2}$$
$73$ $$83 + 20 T + T^{2}$$
$79$ $$52 + 15 T + T^{2}$$
$83$ $$64 + 18 T + T^{2}$$
$89$ $$( -10 + T )^{2}$$
$97$ $$-144 + 6 T + T^{2}$$
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