# Properties

 Label 3549.2.a.i 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: 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.

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

## 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 −1.24698 0.445042
−2.24698 1.00000 3.04892 1.69202 −2.24698 1.00000 −2.35690 1.00000 −3.80194
1.2 −0.554958 1.00000 −1.69202 1.35690 −0.554958 1.00000 2.04892 1.00000 −0.753020
1.3 0.801938 1.00000 −1.35690 −3.04892 0.801938 1.00000 −2.69202 1.00000 −2.44504
 $$n$$: 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.i 3
13.b even 2 1 3549.2.a.u 3
13.c even 3 2 273.2.k.c 6
39.i odd 6 2 819.2.o.e 6

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

 $$T_{2}^{3} + 2 T_{2}^{2} - T_{2} - 1$$ $$T_{5}^{3} - 7 T_{5} + 7$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 - T + 2 T^{2} + T^{3}$$
$3$ $$( -1 + T )^{3}$$
$5$ $$7 - 7 T + T^{3}$$
$7$ $$( -1 + T )^{3}$$
$11$ $$13 - 15 T + 2 T^{2} + T^{3}$$
$13$ $$T^{3}$$
$17$ $$-1 - T + 2 T^{2} + T^{3}$$
$19$ $$43 + 47 T + 13 T^{2} + T^{3}$$
$23$ $$139 - 46 T - 3 T^{2} + T^{3}$$
$29$ $$169 - 72 T - T^{2} + T^{3}$$
$31$ $$-211 - 4 T + 11 T^{2} + T^{3}$$
$37$ $$-533 - 109 T + 4 T^{2} + T^{3}$$
$41$ $$8 - 36 T - 2 T^{2} + T^{3}$$
$43$ $$-211 - 4 T + 11 T^{2} + T^{3}$$
$47$ $$127 + 104 T + 19 T^{2} + T^{3}$$
$53$ $$41 - 29 T - 5 T^{2} + T^{3}$$
$59$ $$-211 - 4 T + 11 T^{2} + T^{3}$$
$61$ $$679 - 105 T - 7 T^{2} + T^{3}$$
$67$ $$-29 + 26 T + 15 T^{2} + T^{3}$$
$71$ $$-533 + 17 T + 18 T^{2} + T^{3}$$
$73$ $$-377 - 79 T + 6 T^{2} + T^{3}$$
$79$ $$27 - 18 T - 3 T^{2} + T^{3}$$
$83$ $$13 - 99 T + 2 T^{2} + T^{3}$$
$89$ $$127 + 87 T + 17 T^{2} + T^{3}$$
$97$ $$-13 - 30 T + 13 T^{2} + T^{3}$$