Properties

 Label 3549.2.a.o Level $3549$ Weight $2$ Character orbit 3549.a Self dual yes Analytic conductor $28.339$ Analytic rank $0$ 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: $$0$$ 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
 1.87939 −0.347296 −1.53209
−1.87939 1.00000 1.53209 2.34730 −1.87939 −1.00000 0.879385 1.00000 −4.41147
1.2 0.347296 1.00000 −1.87939 3.53209 0.347296 −1.00000 −1.34730 1.00000 1.22668
1.3 1.53209 1.00000 0.347296 0.120615 1.53209 −1.00000 −2.53209 1.00000 0.184793
 $$n$$: 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.o 3
13.b even 2 1 3549.2.a.n 3
13.c even 3 2 273.2.k.a 6
39.i 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.c even 3 2
819.2.o.g 6 39.i odd 6 2
3549.2.a.n 3 13.b even 2 1
3549.2.a.o 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3549))$$:

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

Hecke characteristic polynomials

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