# Properties

 Label 3549.2.a.l 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: 3.3.473.1 Defining polynomial: $$x^{3} - 5 x - 1$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 5 x - 1$$:

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

## 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
 −2.12842 −0.201640 2.33006
−2.12842 −1.00000 2.53017 2.65859 2.12842 −1.00000 −1.12842 1.00000 −5.65859
1.2 −0.201640 −1.00000 −1.95934 −3.75770 0.201640 −1.00000 0.798360 1.00000 0.757702
1.3 2.33006 −1.00000 3.42917 −0.900885 −2.33006 −1.00000 3.33006 1.00000 −2.09911
 $$n$$: 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.l 3
13.b even 2 1 3549.2.a.m 3
13.e even 6 2 273.2.k.b 6
39.h odd 6 2 819.2.o.f 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.k.b 6 13.e even 6 2
819.2.o.f 6 39.h odd 6 2
3549.2.a.l 3 1.a even 1 1 trivial
3549.2.a.m 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3549))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 - 5 T + T^{3}$$
$3$ $$( 1 + T )^{3}$$
$5$ $$-9 - 9 T + 2 T^{2} + T^{3}$$
$7$ $$( 1 + T )^{3}$$
$11$ $$-11 + 11 T + 8 T^{2} + T^{3}$$
$13$ $$T^{3}$$
$17$ $$-43 + 43 T - 12 T^{2} + T^{3}$$
$19$ $$-27 - 17 T + 3 T^{2} + T^{3}$$
$23$ $$-15 + 8 T + 7 T^{2} + T^{3}$$
$29$ $$-13 + 22 T - 9 T^{2} + T^{3}$$
$31$ $$-169 - 36 T + 5 T^{2} + T^{3}$$
$37$ $$-127 - 89 T + T^{3}$$
$41$ $$-216 - 68 T + 6 T^{2} + T^{3}$$
$43$ $$-15 + 8 T + 7 T^{2} + T^{3}$$
$47$ $$-405 - 44 T + 9 T^{2} + T^{3}$$
$53$ $$3 + 23 T + 13 T^{2} + T^{3}$$
$59$ $$-1635 - 158 T + 11 T^{2} + T^{3}$$
$61$ $$275 + 55 T - 19 T^{2} + T^{3}$$
$67$ $$1 + 102 T + 21 T^{2} + T^{3}$$
$71$ $$219 + 127 T + 22 T^{2} + T^{3}$$
$73$ $$-71 + 57 T - 14 T^{2} + T^{3}$$
$79$ $$163 - 62 T + T^{2} + T^{3}$$
$83$ $$365 + 279 T + 32 T^{2} + T^{3}$$
$89$ $$143 - 77 T - 3 T^{2} + T^{3}$$
$97$ $$373 + 54 T - 21 T^{2} + T^{3}$$