# Properties

 Label 3549.2.a.r 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: $$\Q(\zeta_{14})^+$$ Defining polynomial: $$x^{3} - x^{2} - 2 x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

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

## 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
 1.80194 0.445042 −1.24698
−0.801938 −1.00000 −1.35690 0 0.801938 −1.00000 2.69202 1.00000 0
1.2 0.554958 −1.00000 −1.69202 0 −0.554958 −1.00000 −2.04892 1.00000 0
1.3 2.24698 −1.00000 3.04892 0 −2.24698 −1.00000 2.35690 1.00000 0
 $$n$$: 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.r yes 3
13.b even 2 1 3549.2.a.g 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3549.2.a.g 3 13.b even 2 1
3549.2.a.r yes 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3549))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T - 2 T^{2} + T^{3}$$
$3$ $$( 1 + T )^{3}$$
$5$ $$T^{3}$$
$7$ $$( 1 + T )^{3}$$
$11$ $$1 - 2 T - T^{2} + T^{3}$$
$13$ $$T^{3}$$
$17$ $$8 - 8 T - 2 T^{2} + T^{3}$$
$19$ $$-104 - 16 T + 6 T^{2} + T^{3}$$
$23$ $$-13 - 58 T - T^{2} + T^{3}$$
$29$ $$-43 + 6 T + 9 T^{2} + T^{3}$$
$31$ $$8 + 24 T + 10 T^{2} + T^{3}$$
$37$ $$91 - 14 T - 7 T^{2} + T^{3}$$
$41$ $$-232 + 124 T - 20 T^{2} + T^{3}$$
$43$ $$-29 + 40 T - 13 T^{2} + T^{3}$$
$47$ $$56 - 84 T + T^{3}$$
$53$ $$-377 - 92 T + 5 T^{2} + T^{3}$$
$59$ $$64 - 32 T - 4 T^{2} + T^{3}$$
$61$ $$-104 - 116 T - 4 T^{2} + T^{3}$$
$67$ $$127 + 104 T + 19 T^{2} + T^{3}$$
$71$ $$-7 + 14 T - 7 T^{2} + T^{3}$$
$73$ $$728 - 56 T - 14 T^{2} + T^{3}$$
$79$ $$-43 - 64 T + 9 T^{2} + T^{3}$$
$83$ $$232 - 64 T - 2 T^{2} + T^{3}$$
$89$ $$-56 - 84 T + T^{3}$$
$97$ $$-344 - 44 T + 8 T^{2} + T^{3}$$