# Properties

 Label 3549.2.a.q 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: 3.3.148.1 Defining polynomial: $$x^{3} - x^{2} - 3 x + 1$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 3 x + 1$$:

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

## 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
 −1.48119 0.311108 2.17009
−1.48119 1.00000 0.193937 4.15633 −1.48119 −1.00000 2.67513 1.00000 −6.15633
1.2 0.311108 1.00000 −1.90321 −1.52543 0.311108 −1.00000 −1.21432 1.00000 −0.474572
1.3 2.17009 1.00000 2.70928 −0.630898 2.17009 −1.00000 1.53919 1.00000 −1.36910
 $$n$$: 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.q 3
13.b even 2 1 3549.2.a.k 3
13.d odd 4 2 273.2.c.b 6
39.f even 4 2 819.2.c.c 6
52.f even 4 2 4368.2.h.o 6
91.i even 4 2 1911.2.c.h 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.c.b 6 13.d odd 4 2
819.2.c.c 6 39.f even 4 2
1911.2.c.h 6 91.i even 4 2
3549.2.a.k 3 13.b even 2 1
3549.2.a.q 3 1.a even 1 1 trivial
4368.2.h.o 6 52.f even 4 2

## Hecke kernels

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

 $$T_{2}^{3} - T_{2}^{2} - 3 T_{2} + 1$$ $$T_{5}^{3} - 2 T_{5}^{2} - 8 T_{5} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 3 T - T^{2} + T^{3}$$
$3$ $$( -1 + T )^{3}$$
$5$ $$-4 - 8 T - 2 T^{2} + T^{3}$$
$7$ $$( 1 + T )^{3}$$
$11$ $$20 + 28 T + 10 T^{2} + T^{3}$$
$13$ $$T^{3}$$
$17$ $$-16 - 16 T + T^{3}$$
$19$ $$-32 + 8 T^{2} + T^{3}$$
$23$ $$-8 - 20 T + 2 T^{2} + T^{3}$$
$29$ $$-40 - 52 T + 2 T^{2} + T^{3}$$
$31$ $$-320 - 16 T + 12 T^{2} + T^{3}$$
$37$ $$-32 - 16 T + 4 T^{2} + T^{3}$$
$41$ $$20 + 32 T + 14 T^{2} + T^{3}$$
$43$ $$128 + 32 T - 16 T^{2} + T^{3}$$
$47$ $$-52 - 28 T + T^{3}$$
$53$ $$( 6 + T )^{3}$$
$59$ $$4 + 12 T + 8 T^{2} + T^{3}$$
$61$ $$2392 - 172 T - 14 T^{2} + T^{3}$$
$67$ $$32 + 64 T + 16 T^{2} + T^{3}$$
$71$ $$740 - 148 T - 6 T^{2} + T^{3}$$
$73$ $$-208 - 8 T + 16 T^{2} + T^{3}$$
$79$ $$80 + 72 T + 16 T^{2} + T^{3}$$
$83$ $$-1252 - 172 T + 8 T^{2} + T^{3}$$
$89$ $$76 - 88 T + 6 T^{2} + T^{3}$$
$97$ $$16 - 16 T + T^{3}$$