# Properties

 Label 3549.2.a.n 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: $$\Q(\zeta_{18})^+$$ Defining polynomial: $$x^{3} - 3 x - 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 273) 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.

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

## 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
 −1.53209 −0.347296 1.87939
−1.53209 1.00000 0.347296 −0.120615 −1.53209 1.00000 2.53209 1.00000 0.184793
1.2 −0.347296 1.00000 −1.87939 −3.53209 −0.347296 1.00000 1.34730 1.00000 1.22668
1.3 1.87939 1.00000 1.53209 −2.34730 1.87939 1.00000 −0.879385 1.00000 −4.41147
 $$n$$: 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.n 3
13.b even 2 1 3549.2.a.o 3
13.e even 6 2 273.2.k.a 6
39.h odd 6 2 819.2.o.g 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.k.a 6 13.e even 6 2
819.2.o.g 6 39.h odd 6 2
3549.2.a.n 3 1.a even 1 1 trivial
3549.2.a.o 3 13.b even 2 1

## Hecke kernels

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

 $$T_{2}^{3} - 3 T_{2} - 1$$ $$T_{5}^{3} + 6 T_{5}^{2} + 9 T_{5} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 - 3 T + T^{3}$$
$3$ $$( -1 + T )^{3}$$
$5$ $$1 + 9 T + 6 T^{2} + T^{3}$$
$7$ $$( -1 + T )^{3}$$
$11$ $$-71 - 9 T + 6 T^{2} + T^{3}$$
$13$ $$T^{3}$$
$17$ $$159 - 27 T - 6 T^{2} + T^{3}$$
$19$ $$-17 + 15 T + 9 T^{2} + T^{3}$$
$23$ $$57 - 36 T - 3 T^{2} + T^{3}$$
$29$ $$323 - 48 T - 9 T^{2} + T^{3}$$
$31$ $$-27 + 9 T^{2} + T^{3}$$
$37$ $$-327 - 9 T + 12 T^{2} + T^{3}$$
$41$ $$-856 - 132 T + 6 T^{2} + T^{3}$$
$43$ $$73 - 48 T - 9 T^{2} + T^{3}$$
$47$ $$-3 - 18 T + 3 T^{2} + T^{3}$$
$53$ $$867 - 153 T - 3 T^{2} + T^{3}$$
$59$ $$-1293 - 72 T + 15 T^{2} + T^{3}$$
$61$ $$-127 + 39 T + 15 T^{2} + T^{3}$$
$67$ $$-1583 - 174 T + 9 T^{2} + T^{3}$$
$71$ $$111 + 9 T - 12 T^{2} + T^{3}$$
$73$ $$807 + 279 T + 30 T^{2} + T^{3}$$
$79$ $$271 + 138 T + 21 T^{2} + T^{3}$$
$83$ $$111 + 153 T + 24 T^{2} + T^{3}$$
$89$ $$1 + 15 T - 9 T^{2} + T^{3}$$
$97$ $$-57 - 126 T + 3 T^{2} + T^{3}$$