Properties

 Label 3549.2.a.u 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: $$\Q(\zeta_{14})^+$$ 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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{1} ) q^{2} + q^{3} + ( 1 - 2 \beta_{1} + \beta_{2} ) q^{4} + ( \beta_{1} + \beta_{2} ) q^{5} + ( 1 - \beta_{1} ) q^{6} - q^{7} + ( 2 - \beta_{1} + 2 \beta_{2} ) q^{8} + q^{9} +O(q^{10})$$ $$q + ( 1 - \beta_{1} ) q^{2} + q^{3} + ( 1 - 2 \beta_{1} + \beta_{2} ) q^{4} + ( \beta_{1} + \beta_{2} ) q^{5} + ( 1 - \beta_{1} ) q^{6} - q^{7} + ( 2 - \beta_{1} + 2 \beta_{2} ) q^{8} + q^{9} + ( -3 + \beta_{1} - \beta_{2} ) q^{10} + ( 2 - 3 \beta_{1} + \beta_{2} ) q^{11} + ( 1 - 2 \beta_{1} + \beta_{2} ) q^{12} + ( -1 + \beta_{1} ) q^{14} + ( \beta_{1} + \beta_{2} ) q^{15} + ( \beta_{1} - \beta_{2} ) q^{16} + ( -1 + \beta_{1} ) q^{17} + ( 1 - \beta_{1} ) q^{18} + ( 5 - 2 \beta_{1} ) q^{19} + ( -4 + 2 \beta_{1} - 3 \beta_{2} ) q^{20} - q^{21} + ( 7 - 5 \beta_{1} + 3 \beta_{2} ) q^{22} + ( 4 - 4 \beta_{1} + 5 \beta_{2} ) q^{23} + ( 2 - \beta_{1} + 2 \beta_{2} ) q^{24} + ( \beta_{1} + 2 \beta_{2} ) q^{25} + q^{27} + ( -1 + 2 \beta_{1} - \beta_{2} ) q^{28} + ( 4 - 5 \beta_{1} + 6 \beta_{2} ) q^{29} + ( -3 + \beta_{1} - \beta_{2} ) q^{30} + ( 1 + 5 \beta_{1} - 3 \beta_{2} ) q^{31} + ( -5 + 3 \beta_{1} - 5 \beta_{2} ) q^{32} + ( 2 - 3 \beta_{1} + \beta_{2} ) q^{33} + ( -3 + 2 \beta_{1} - \beta_{2} ) q^{34} + ( -\beta_{1} - \beta_{2} ) q^{35} + ( 1 - 2 \beta_{1} + \beta_{2} ) q^{36} + ( 2 - 5 \beta_{1} - 3 \beta_{2} ) q^{37} + ( 9 - 7 \beta_{1} + 2 \beta_{2} ) q^{38} + ( 1 + 4 \beta_{1} ) q^{40} + ( -2 + 4 \beta_{1} ) q^{41} + ( -1 + \beta_{1} ) q^{42} + ( -6 + 2 \beta_{1} - 5 \beta_{2} ) q^{43} + ( 10 - 6 \beta_{1} + 3 \beta_{2} ) q^{44} + ( \beta_{1} + \beta_{2} ) q^{45} + ( 7 - 8 \beta_{1} + 4 \beta_{2} ) q^{46} + ( 5 + 3 \beta_{1} - \beta_{2} ) q^{47} + ( \beta_{1} - \beta_{2} ) q^{48} + q^{49} + ( -4 + \beta_{1} - \beta_{2} ) q^{50} + ( -1 + \beta_{1} ) q^{51} + ( -1 + 4 \beta_{1} - 4 \beta_{2} ) q^{53} + ( 1 - \beta_{1} ) q^{54} + ( -7 + 3 \beta_{1} - 4 \beta_{2} ) q^{55} + ( -2 + \beta_{1} - 2 \beta_{2} ) q^{56} + ( 5 - 2 \beta_{1} ) q^{57} + ( 8 - 9 \beta_{1} + 5 \beta_{2} ) q^{58} + ( 6 - 2 \beta_{1} + 5 \beta_{2} ) q^{59} + ( -4 + 2 \beta_{1} - 3 \beta_{2} ) q^{60} + ( 1 - 2 \beta_{1} - 6 \beta_{2} ) q^{61} + ( -6 + 4 \beta_{1} - 5 \beta_{2} ) q^{62} - q^{63} + ( -6 + 6 \beta_{1} - \beta_{2} ) q^{64} + ( 7 - 5 \beta_{1} + 3 \beta_{2} ) q^{66} + ( 6 + \beta_{1} + 4 \beta_{2} ) q^{67} + ( -4 + 3 \beta_{1} - 2 \beta_{2} ) q^{68} + ( 4 - 4 \beta_{1} + 5 \beta_{2} ) q^{69} + ( 3 - \beta_{1} + \beta_{2} ) q^{70} + ( 9 - 2 \beta_{1} + 7 \beta_{2} ) q^{71} + ( 2 - \beta_{1} + 2 \beta_{2} ) q^{72} + ( 5 - 2 \beta_{1} + 7 \beta_{2} ) q^{73} + ( 15 - 7 \beta_{1} + 5 \beta_{2} ) q^{74} + ( \beta_{1} + 2 \beta_{2} ) q^{75} + ( 11 - 12 \beta_{1} + 7 \beta_{2} ) q^{76} + ( -2 + 3 \beta_{1} - \beta_{2} ) q^{77} + ( 3 - 3 \beta_{1} + 3 \beta_{2} ) q^{79} + ( 1 - \beta_{1} + 2 \beta_{2} ) q^{80} + q^{81} + ( -10 + 6 \beta_{1} - 4 \beta_{2} ) q^{82} + ( -2 + \beta_{1} - 7 \beta_{2} ) q^{83} + ( -1 + 2 \beta_{1} - \beta_{2} ) q^{84} + ( 3 - \beta_{1} + \beta_{2} ) q^{85} + ( -5 + 8 \beta_{1} - 2 \beta_{2} ) q^{86} + ( 4 - 5 \beta_{1} + 6 \beta_{2} ) q^{87} + ( 5 - 6 \beta_{1} ) q^{88} + ( 5 - 2 \beta_{2} ) q^{89} + ( -3 + \beta_{1} - \beta_{2} ) q^{90} + ( 11 - 7 \beta_{1} - 2 \beta_{2} ) q^{92} + ( 1 + 5 \beta_{1} - 3 \beta_{2} ) q^{93} + ( -2 \beta_{1} - 3 \beta_{2} ) q^{94} + ( -6 + 5 \beta_{1} + \beta_{2} ) q^{95} + ( -5 + 3 \beta_{1} - 5 \beta_{2} ) q^{96} + ( 1 + 3 \beta_{1} - 7 \beta_{2} ) q^{97} + ( 1 - \beta_{1} ) q^{98} + ( 2 - 3 \beta_{1} + \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 0.445042 −1.24698
−0.801938 1.00000 −1.35690 3.04892 −0.801938 −1.00000 2.69202 1.00000 −2.44504
1.2 0.554958 1.00000 −1.69202 −1.35690 0.554958 −1.00000 −2.04892 1.00000 −0.753020
1.3 2.24698 1.00000 3.04892 −1.69202 2.24698 −1.00000 2.35690 1.00000 −3.80194
 $$n$$: 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.u 3
13.b even 2 1 3549.2.a.i 3
13.e even 6 2 273.2.k.c 6
39.h 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.e even 6 2
819.2.o.e 6 39.h odd 6 2
3549.2.a.i 3 13.b even 2 1
3549.2.a.u 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3549))$$:

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

Hecke characteristic polynomials

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