Properties

 Label 3549.2.a.t 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: 3.3.316.1 Defining polynomial: $$x^{3} - x^{2} - 4 x + 2$$ 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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{1} ) q^{2} - q^{3} + ( 2 - 2 \beta_{1} + \beta_{2} ) q^{4} + ( 1 + \beta_{2} ) q^{5} + ( -1 + \beta_{1} ) q^{6} + q^{7} + ( 5 - 3 \beta_{1} + 2 \beta_{2} ) q^{8} + q^{9} +O(q^{10})$$ $$q + ( 1 - \beta_{1} ) q^{2} - q^{3} + ( 2 - 2 \beta_{1} + \beta_{2} ) q^{4} + ( 1 + \beta_{2} ) q^{5} + ( -1 + \beta_{1} ) q^{6} + q^{7} + ( 5 - 3 \beta_{1} + 2 \beta_{2} ) q^{8} + q^{9} -2 \beta_{1} q^{10} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{11} + ( -2 + 2 \beta_{1} - \beta_{2} ) q^{12} + ( 1 - \beta_{1} ) q^{14} + ( -1 - \beta_{2} ) q^{15} + ( 8 - 6 \beta_{1} + \beta_{2} ) q^{16} + ( -2 - 2 \beta_{1} ) q^{17} + ( 1 - \beta_{1} ) q^{18} + ( 3 - 2 \beta_{1} - \beta_{2} ) q^{19} + ( 4 - 2 \beta_{1} ) q^{20} - q^{21} + ( -4 + 4 \beta_{1} - 2 \beta_{2} ) q^{22} + ( -3 + \beta_{2} ) q^{23} + ( -5 + 3 \beta_{1} - 2 \beta_{2} ) q^{24} + ( 2 \beta_{1} + \beta_{2} ) q^{25} - q^{27} + ( 2 - 2 \beta_{1} + \beta_{2} ) q^{28} + ( -1 + 2 \beta_{1} + \beta_{2} ) q^{29} + 2 \beta_{1} q^{30} + ( 3 - 2 \beta_{1} + 3 \beta_{2} ) q^{31} + ( 15 - 9 \beta_{1} + 2 \beta_{2} ) q^{32} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{33} + ( 4 + 2 \beta_{2} ) q^{34} + ( 1 + \beta_{2} ) q^{35} + ( 2 - 2 \beta_{1} + \beta_{2} ) q^{36} + ( -4 + 2 \beta_{2} ) q^{37} + ( 10 - 4 \beta_{1} + 2 \beta_{2} ) q^{38} + ( 10 - 2 \beta_{1} + 2 \beta_{2} ) q^{40} + ( 4 - 2 \beta_{1} ) q^{41} + ( -1 + \beta_{1} ) q^{42} + ( -1 + 2 \beta_{1} - \beta_{2} ) q^{43} + ( -14 + 6 \beta_{1} ) q^{44} + ( 1 + \beta_{2} ) q^{45} + ( -4 + 2 \beta_{1} ) q^{46} + ( 5 + 2 \beta_{1} - \beta_{2} ) q^{47} + ( -8 + 6 \beta_{1} - \beta_{2} ) q^{48} + q^{49} + ( -7 + \beta_{1} - 2 \beta_{2} ) q^{50} + ( 2 + 2 \beta_{1} ) q^{51} + ( -1 - 2 \beta_{1} - 3 \beta_{2} ) q^{53} + ( -1 + \beta_{1} ) q^{54} + ( -6 + 2 \beta_{2} ) q^{55} + ( 5 - 3 \beta_{1} + 2 \beta_{2} ) q^{56} + ( -3 + 2 \beta_{1} + \beta_{2} ) q^{57} + ( -8 + 2 \beta_{1} - 2 \beta_{2} ) q^{58} + ( 4 + 2 \beta_{2} ) q^{59} + ( -4 + 2 \beta_{1} ) q^{60} + ( 2 + 4 \beta_{1} ) q^{61} + ( 6 - 8 \beta_{1} + 2 \beta_{2} ) q^{62} + q^{63} + ( 24 - 14 \beta_{1} + 7 \beta_{2} ) q^{64} + ( 4 - 4 \beta_{1} + 2 \beta_{2} ) q^{66} + ( -2 + 4 \beta_{1} + 2 \beta_{2} ) q^{67} + ( 6 - 2 \beta_{1} ) q^{68} + ( 3 - \beta_{2} ) q^{69} -2 \beta_{1} q^{70} + ( 2 - 2 \beta_{1} + 4 \beta_{2} ) q^{71} + ( 5 - 3 \beta_{1} + 2 \beta_{2} ) q^{72} + ( 1 + 2 \beta_{1} - 5 \beta_{2} ) q^{73} + ( -6 + 2 \beta_{1} ) q^{74} + ( -2 \beta_{1} - \beta_{2} ) q^{75} + ( 14 - 12 \beta_{1} + 6 \beta_{2} ) q^{76} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{77} + ( -5 + 2 \beta_{1} - \beta_{2} ) q^{79} + ( 6 - 10 \beta_{1} + 2 \beta_{2} ) q^{80} + q^{81} + ( 10 - 6 \beta_{1} + 2 \beta_{2} ) q^{82} + ( 1 - 2 \beta_{1} - \beta_{2} ) q^{83} + ( -2 + 2 \beta_{1} - \beta_{2} ) q^{84} + ( -4 - 4 \beta_{1} - 4 \beta_{2} ) q^{85} + ( -6 + 4 \beta_{1} - 2 \beta_{2} ) q^{86} + ( 1 - 2 \beta_{1} - \beta_{2} ) q^{87} + ( -24 + 12 \beta_{1} - 2 \beta_{2} ) q^{88} + ( 3 + 4 \beta_{1} - \beta_{2} ) q^{89} -2 \beta_{1} q^{90} + ( -4 + 6 \beta_{1} - 4 \beta_{2} ) q^{92} + ( -3 + 2 \beta_{1} - 3 \beta_{2} ) q^{93} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{94} + ( -3 - 6 \beta_{1} + \beta_{2} ) q^{95} + ( -15 + 9 \beta_{1} - 2 \beta_{2} ) q^{96} + ( 1 + 6 \beta_{1} - 5 \beta_{2} ) q^{97} + ( 1 - \beta_{1} ) q^{98} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + 2q^{2} - 3q^{3} + 4q^{4} + 3q^{5} - 2q^{6} + 3q^{7} + 12q^{8} + 3q^{9} + O(q^{10})$$ $$3q + 2q^{2} - 3q^{3} + 4q^{4} + 3q^{5} - 2q^{6} + 3q^{7} + 12q^{8} + 3q^{9} - 2q^{10} + 2q^{11} - 4q^{12} + 2q^{14} - 3q^{15} + 18q^{16} - 8q^{17} + 2q^{18} + 7q^{19} + 10q^{20} - 3q^{21} - 8q^{22} - 9q^{23} - 12q^{24} + 2q^{25} - 3q^{27} + 4q^{28} - q^{29} + 2q^{30} + 7q^{31} + 36q^{32} - 2q^{33} + 12q^{34} + 3q^{35} + 4q^{36} - 12q^{37} + 26q^{38} + 28q^{40} + 10q^{41} - 2q^{42} - q^{43} - 36q^{44} + 3q^{45} - 10q^{46} + 17q^{47} - 18q^{48} + 3q^{49} - 20q^{50} + 8q^{51} - 5q^{53} - 2q^{54} - 18q^{55} + 12q^{56} - 7q^{57} - 22q^{58} + 12q^{59} - 10q^{60} + 10q^{61} + 10q^{62} + 3q^{63} + 58q^{64} + 8q^{66} - 2q^{67} + 16q^{68} + 9q^{69} - 2q^{70} + 4q^{71} + 12q^{72} + 5q^{73} - 16q^{74} - 2q^{75} + 30q^{76} + 2q^{77} - 13q^{79} + 8q^{80} + 3q^{81} + 24q^{82} + q^{83} - 4q^{84} - 16q^{85} - 14q^{86} + q^{87} - 60q^{88} + 13q^{89} - 2q^{90} - 6q^{92} - 7q^{93} - 2q^{94} - 15q^{95} - 36q^{96} + 9q^{97} + 2q^{98} + 2q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 4 x + 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$

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
 2.34292 0.470683 −1.81361
−1.34292 −1.00000 −0.196558 3.48929 1.34292 1.00000 2.94981 1.00000 −4.68585
1.2 0.529317 −1.00000 −1.71982 −1.77846 −0.529317 1.00000 −1.96896 1.00000 −0.941367
1.3 2.81361 −1.00000 5.91638 1.28917 −2.81361 1.00000 11.0192 1.00000 3.62721
 $$n$$: 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.t 3
13.b even 2 1 273.2.a.d 3
39.d odd 2 1 819.2.a.j 3
52.b odd 2 1 4368.2.a.bq 3
65.d even 2 1 6825.2.a.bd 3
91.b odd 2 1 1911.2.a.n 3
273.g even 2 1 5733.2.a.bc 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.a.d 3 13.b even 2 1
819.2.a.j 3 39.d odd 2 1
1911.2.a.n 3 91.b odd 2 1
3549.2.a.t 3 1.a even 1 1 trivial
4368.2.a.bq 3 52.b odd 2 1
5733.2.a.bc 3 273.g even 2 1
6825.2.a.bd 3 65.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3549))$$:

 $$T_{2}^{3} - 2 T_{2}^{2} - 3 T_{2} + 2$$ $$T_{5}^{3} - 3 T_{5}^{2} - 4 T_{5} + 8$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$2 - 3 T - 2 T^{2} + T^{3}$$
$3$ $$( 1 + T )^{3}$$
$5$ $$8 - 4 T - 3 T^{2} + T^{3}$$
$7$ $$( -1 + T )^{3}$$
$11$ $$-8 - 28 T - 2 T^{2} + T^{3}$$
$13$ $$T^{3}$$
$17$ $$-32 + 4 T + 8 T^{2} + T^{3}$$
$19$ $$128 - 16 T - 7 T^{2} + T^{3}$$
$23$ $$8 + 20 T + 9 T^{2} + T^{3}$$
$29$ $$-76 - 32 T + T^{2} + T^{3}$$
$31$ $$272 - 40 T - 7 T^{2} + T^{3}$$
$37$ $$-32 + 20 T + 12 T^{2} + T^{3}$$
$41$ $$16 + 16 T - 10 T^{2} + T^{3}$$
$43$ $$16 - 16 T + T^{2} + T^{3}$$
$47$ $$-68 + 80 T - 17 T^{2} + T^{3}$$
$53$ $$148 - 96 T + 5 T^{2} + T^{3}$$
$59$ $$64 + 20 T - 12 T^{2} + T^{3}$$
$61$ $$232 - 36 T - 10 T^{2} + T^{3}$$
$67$ $$-608 - 128 T + 2 T^{2} + T^{3}$$
$71$ $$496 - 92 T - 4 T^{2} + T^{3}$$
$73$ $$-436 - 144 T - 5 T^{2} + T^{3}$$
$79$ $$32 + 40 T + 13 T^{2} + T^{3}$$
$83$ $$76 - 32 T - T^{2} + T^{3}$$
$89$ $$344 - 4 T - 13 T^{2} + T^{3}$$
$97$ $$524 - 184 T - 9 T^{2} + T^{3}$$