# Properties

 Label 3549.2.a.p Level $3549$ Weight $2$ Character orbit 3549.a Self dual yes Analytic conductor $28.339$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3549 = 3 \cdot 7 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3549.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$28.3389076774$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{14})^+$$ Defining polynomial: $$x^{3} - x^{2} - 2 x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + q^{3} + \beta_{2} q^{4} - q^{5} + \beta_{1} q^{6} + q^{7} + ( 1 - 2 \beta_{1} + \beta_{2} ) q^{8} + q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + q^{3} + \beta_{2} q^{4} - q^{5} + \beta_{1} q^{6} + q^{7} + ( 1 - 2 \beta_{1} + \beta_{2} ) q^{8} + q^{9} -\beta_{1} q^{10} -\beta_{1} q^{11} + \beta_{2} q^{12} + \beta_{1} q^{14} - q^{15} + ( -3 + \beta_{1} - 3 \beta_{2} ) q^{16} + ( -3 + \beta_{1} - \beta_{2} ) q^{17} + \beta_{1} q^{18} + ( -2 - 3 \beta_{2} ) q^{19} -\beta_{2} q^{20} + q^{21} + ( -2 - \beta_{2} ) q^{22} + ( -4 - \beta_{1} - \beta_{2} ) q^{23} + ( 1 - 2 \beta_{1} + \beta_{2} ) q^{24} -4 q^{25} + q^{27} + \beta_{2} q^{28} + ( 1 - 3 \beta_{1} + \beta_{2} ) q^{29} -\beta_{1} q^{30} + ( 1 - \beta_{1} + \beta_{2} ) q^{31} + ( -3 + \beta_{1} - 4 \beta_{2} ) q^{32} -\beta_{1} q^{33} + ( 1 - 3 \beta_{1} ) q^{34} - q^{35} + \beta_{2} q^{36} + ( -1 + 4 \beta_{2} ) q^{37} + ( -3 - 2 \beta_{1} - 3 \beta_{2} ) q^{38} + ( -1 + 2 \beta_{1} - \beta_{2} ) q^{40} + ( 3 + 2 \beta_{1} + 3 \beta_{2} ) q^{41} + \beta_{1} q^{42} -8 q^{43} + ( -1 - \beta_{2} ) q^{44} - q^{45} + ( -3 - 4 \beta_{1} - 2 \beta_{2} ) q^{46} + ( 4 - 3 \beta_{1} - 3 \beta_{2} ) q^{47} + ( -3 + \beta_{1} - 3 \beta_{2} ) q^{48} + q^{49} -4 \beta_{1} q^{50} + ( -3 + \beta_{1} - \beta_{2} ) q^{51} + ( -3 + 8 \beta_{1} - 2 \beta_{2} ) q^{53} + \beta_{1} q^{54} + \beta_{1} q^{55} + ( 1 - 2 \beta_{1} + \beta_{2} ) q^{56} + ( -2 - 3 \beta_{2} ) q^{57} + ( -5 + \beta_{1} - 2 \beta_{2} ) q^{58} + ( 5 + 4 \beta_{1} - 2 \beta_{2} ) q^{59} -\beta_{2} q^{60} + ( -8 + \beta_{1} - 3 \beta_{2} ) q^{61} + ( -1 + \beta_{1} ) q^{62} + q^{63} + ( 4 - 5 \beta_{1} + 3 \beta_{2} ) q^{64} + ( -2 - \beta_{2} ) q^{66} + ( 6 - 3 \beta_{1} + 3 \beta_{2} ) q^{67} + ( -\beta_{1} - \beta_{2} ) q^{68} + ( -4 - \beta_{1} - \beta_{2} ) q^{69} -\beta_{1} q^{70} + ( -3 + 3 \beta_{1} ) q^{71} + ( 1 - 2 \beta_{1} + \beta_{2} ) q^{72} + ( 1 - 3 \beta_{1} + 3 \beta_{2} ) q^{73} + ( 4 - \beta_{1} + 4 \beta_{2} ) q^{74} -4 q^{75} + ( -3 - 3 \beta_{1} + \beta_{2} ) q^{76} -\beta_{1} q^{77} + ( -6 + 7 \beta_{1} - 2 \beta_{2} ) q^{79} + ( 3 - \beta_{1} + 3 \beta_{2} ) q^{80} + q^{81} + ( 7 + 3 \beta_{1} + 5 \beta_{2} ) q^{82} + ( 1 - 6 \beta_{1} - \beta_{2} ) q^{83} + \beta_{2} q^{84} + ( 3 - \beta_{1} + \beta_{2} ) q^{85} -8 \beta_{1} q^{86} + ( 1 - 3 \beta_{1} + \beta_{2} ) q^{87} + ( 3 - \beta_{1} + \beta_{2} ) q^{88} + ( -3 - 4 \beta_{1} + 6 \beta_{2} ) q^{89} -\beta_{1} q^{90} + ( -2 - \beta_{1} - 4 \beta_{2} ) q^{92} + ( 1 - \beta_{1} + \beta_{2} ) q^{93} + ( -9 + 4 \beta_{1} - 6 \beta_{2} ) q^{94} + ( 2 + 3 \beta_{2} ) q^{95} + ( -3 + \beta_{1} - 4 \beta_{2} ) q^{96} + ( -5 + 3 \beta_{1} + 4 \beta_{2} ) q^{97} + \beta_{1} q^{98} -\beta_{1} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + q^{2} + 3q^{3} - q^{4} - 3q^{5} + q^{6} + 3q^{7} + 3q^{9} + O(q^{10})$$ $$3q + q^{2} + 3q^{3} - q^{4} - 3q^{5} + q^{6} + 3q^{7} + 3q^{9} - q^{10} - q^{11} - q^{12} + q^{14} - 3q^{15} - 5q^{16} - 7q^{17} + q^{18} - 3q^{19} + q^{20} + 3q^{21} - 5q^{22} - 12q^{23} - 12q^{25} + 3q^{27} - q^{28} - q^{29} - q^{30} + q^{31} - 4q^{32} - q^{33} - 3q^{35} - q^{36} - 7q^{37} - 8q^{38} + 8q^{41} + q^{42} - 24q^{43} - 2q^{44} - 3q^{45} - 11q^{46} + 12q^{47} - 5q^{48} + 3q^{49} - 4q^{50} - 7q^{51} + q^{53} + q^{54} + q^{55} - 3q^{57} - 12q^{58} + 21q^{59} + q^{60} - 20q^{61} - 2q^{62} + 3q^{63} + 4q^{64} - 5q^{66} + 12q^{67} - 12q^{69} - q^{70} - 6q^{71} - 3q^{73} + 7q^{74} - 12q^{75} - 13q^{76} - q^{77} - 9q^{79} + 5q^{80} + 3q^{81} + 19q^{82} - 2q^{83} - q^{84} + 7q^{85} - 8q^{86} - q^{87} + 7q^{88} - 19q^{89} - q^{90} - 3q^{92} + q^{93} - 17q^{94} + 3q^{95} - 4q^{96} - 16q^{97} + q^{98} - q^{99} + O(q^{100})$$

## 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
 −1.24698 0.445042 1.80194
−1.24698 1.00000 −0.445042 −1.00000 −1.24698 1.00000 3.04892 1.00000 1.24698
1.2 0.445042 1.00000 −1.80194 −1.00000 0.445042 1.00000 −1.69202 1.00000 −0.445042
1.3 1.80194 1.00000 1.24698 −1.00000 1.80194 1.00000 −1.35690 1.00000 −1.80194
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$7$$ $$-1$$
$$13$$ $$1$$

## 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 3549.2.a.p yes 3
13.b even 2 1 3549.2.a.j 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3549.2.a.j 3 13.b even 2 1
3549.2.a.p yes 3 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3549))$$:

 $$T_{2}^{3} - T_{2}^{2} - 2 T_{2} + 1$$ $$T_{5} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 2 T - T^{2} + T^{3}$$
$3$ $$( -1 + T )^{3}$$
$5$ $$( 1 + T )^{3}$$
$7$ $$( -1 + T )^{3}$$
$11$ $$-1 - 2 T + T^{2} + T^{3}$$
$13$ $$T^{3}$$
$17$ $$7 + 14 T + 7 T^{2} + T^{3}$$
$19$ $$-13 - 18 T + 3 T^{2} + T^{3}$$
$23$ $$43 + 41 T + 12 T^{2} + T^{3}$$
$29$ $$-29 - 16 T + T^{2} + T^{3}$$
$31$ $$1 - 2 T - T^{2} + T^{3}$$
$37$ $$-91 - 21 T + 7 T^{2} + T^{3}$$
$41$ $$-13 - 23 T - 8 T^{2} + T^{3}$$
$43$ $$( 8 + T )^{3}$$
$47$ $$377 - 15 T - 12 T^{2} + T^{3}$$
$53$ $$449 - 121 T - T^{2} + T^{3}$$
$59$ $$-91 + 119 T - 21 T^{2} + T^{3}$$
$61$ $$169 + 117 T + 20 T^{2} + T^{3}$$
$67$ $$27 + 27 T - 12 T^{2} + T^{3}$$
$71$ $$-27 - 9 T + 6 T^{2} + T^{3}$$
$73$ $$-13 - 18 T + 3 T^{2} + T^{3}$$
$79$ $$41 - 64 T + 9 T^{2} + T^{3}$$
$83$ $$13 - 99 T + 2 T^{2} + T^{3}$$
$89$ $$29 + 55 T + 19 T^{2} + T^{3}$$
$97$ $$-617 - T + 16 T^{2} + T^{3}$$