# Properties

 Label 3549.2.a.x Level $3549$ Weight $2$ Character orbit 3549.a Self dual yes Analytic conductor $28.339$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: 4.4.64436.1 Defining polynomial: $$x^{4} - x^{3} - 7 x^{2} + 7 x + 2$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} - q^{3} + ( 2 + \beta_{2} ) q^{4} + ( 1 + \beta_{2} ) q^{5} -\beta_{1} q^{6} - q^{7} + ( -1 + 2 \beta_{1} + \beta_{3} ) q^{8} + q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} - q^{3} + ( 2 + \beta_{2} ) q^{4} + ( 1 + \beta_{2} ) q^{5} -\beta_{1} q^{6} - q^{7} + ( -1 + 2 \beta_{1} + \beta_{3} ) q^{8} + q^{9} + ( -1 + 3 \beta_{1} + \beta_{3} ) q^{10} + ( 1 - \beta_{1} + \beta_{3} ) q^{11} + ( -2 - \beta_{2} ) q^{12} -\beta_{1} q^{14} + ( -1 - \beta_{2} ) q^{15} + ( 5 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{16} + ( -2 - 2 \beta_{1} ) q^{17} + \beta_{1} q^{18} + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{19} + ( 11 - \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{20} + q^{21} + ( -3 + \beta_{1} + \beta_{3} ) q^{22} + ( 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{23} + ( 1 - 2 \beta_{1} - \beta_{3} ) q^{24} + ( 5 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{25} - q^{27} + ( -2 - \beta_{2} ) q^{28} + ( -2 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{29} + ( 1 - 3 \beta_{1} - \beta_{3} ) q^{30} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{31} + ( -2 + 3 \beta_{1} ) q^{32} + ( -1 + \beta_{1} - \beta_{3} ) q^{33} + ( -8 - 2 \beta_{1} - 2 \beta_{2} ) q^{34} + ( -1 - \beta_{2} ) q^{35} + ( 2 + \beta_{2} ) q^{36} + ( -2 + 2 \beta_{1} - 2 \beta_{3} ) q^{37} + ( -6 + 4 \beta_{1} - 2 \beta_{2} ) q^{38} + ( -3 + 9 \beta_{1} + \beta_{3} ) q^{40} + ( 7 + \beta_{1} + \beta_{3} ) q^{41} + \beta_{1} q^{42} + ( -4 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{43} + ( 3 - \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{44} + ( 1 + \beta_{2} ) q^{45} + ( 10 + 2 \beta_{1} + 2 \beta_{2} ) q^{46} + ( 1 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{47} + ( -5 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{48} + q^{49} + ( -4 + 7 \beta_{1} + 2 \beta_{3} ) q^{50} + ( 2 + 2 \beta_{1} ) q^{51} + ( -2 + \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{53} -\beta_{1} q^{54} + ( 2 + 2 \beta_{2} - 2 \beta_{3} ) q^{55} + ( 1 - 2 \beta_{1} - \beta_{3} ) q^{56} + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{57} + ( 4 + 2 \beta_{2} + 2 \beta_{3} ) q^{58} + ( 7 - \beta_{1} - \beta_{3} ) q^{59} + ( -11 + \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{60} + ( -2 + 2 \beta_{3} ) q^{61} + ( 4 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{62} - q^{63} + ( 2 + \beta_{2} - 2 \beta_{3} ) q^{64} + ( 3 - \beta_{1} - \beta_{3} ) q^{66} + ( 6 - 4 \beta_{1} - 2 \beta_{2} ) q^{67} + ( -2 - 8 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{68} + ( -3 \beta_{1} - \beta_{2} + \beta_{3} ) q^{69} + ( 1 - 3 \beta_{1} - \beta_{3} ) q^{70} + ( 1 - \beta_{1} + \beta_{3} ) q^{71} + ( -1 + 2 \beta_{1} + \beta_{3} ) q^{72} + ( -8 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{73} + ( 6 - 2 \beta_{1} - 2 \beta_{3} ) q^{74} + ( -5 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{75} + ( 14 - 8 \beta_{1} + 2 \beta_{2} ) q^{76} + ( -1 + \beta_{1} - \beta_{3} ) q^{77} + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{79} + ( 15 - \beta_{1} + 6 \beta_{2} - \beta_{3} ) q^{80} + q^{81} + ( 5 + 7 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{82} + ( 1 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{83} + ( 2 + \beta_{2} ) q^{84} + ( -6 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{85} + ( 4 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{86} + ( 2 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{87} + ( -1 + 5 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{88} + ( 1 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{89} + ( -1 + 3 \beta_{1} + \beta_{3} ) q^{90} + ( 6 + 8 \beta_{1} + 4 \beta_{3} ) q^{92} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{93} + ( -11 + 3 \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{94} + ( 12 - 7 \beta_{1} + \beta_{2} + \beta_{3} ) q^{95} + ( 2 - 3 \beta_{1} ) q^{96} + ( -10 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{97} + \beta_{1} q^{98} + ( 1 - \beta_{1} + \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + q^{2} - 4q^{3} + 7q^{4} + 3q^{5} - q^{6} - 4q^{7} - 3q^{8} + 4q^{9} + O(q^{10})$$ $$4q + q^{2} - 4q^{3} + 7q^{4} + 3q^{5} - q^{6} - 4q^{7} - 3q^{8} + 4q^{9} - 2q^{10} + 2q^{11} - 7q^{12} - q^{14} - 3q^{15} + 17q^{16} - 10q^{17} + q^{18} + 7q^{19} + 40q^{20} + 4q^{21} - 12q^{22} + 3q^{23} + 3q^{24} + 17q^{25} - 4q^{27} - 7q^{28} - 9q^{29} + 2q^{30} - q^{31} - 5q^{32} - 2q^{33} - 32q^{34} - 3q^{35} + 7q^{36} - 4q^{37} - 18q^{38} - 4q^{40} + 28q^{41} + q^{42} - 17q^{43} + 10q^{44} + 3q^{45} + 40q^{46} + 3q^{47} - 17q^{48} + 4q^{49} - 11q^{50} + 10q^{51} - 5q^{53} - q^{54} + 8q^{55} + 3q^{56} - 7q^{57} + 12q^{58} + 28q^{59} - 40q^{60} - 10q^{61} + 14q^{62} - 4q^{63} + 9q^{64} + 12q^{66} + 22q^{67} - 12q^{68} - 3q^{69} + 2q^{70} + 2q^{71} - 3q^{72} - 31q^{73} + 24q^{74} - 17q^{75} + 46q^{76} - 2q^{77} - q^{79} + 54q^{80} + 4q^{81} + 24q^{82} + 3q^{83} + 7q^{84} - 2q^{85} + 10q^{86} + 9q^{87} + 4q^{88} + 5q^{89} - 2q^{90} + 28q^{92} + q^{93} - 36q^{94} + 39q^{95} + 5q^{96} - 43q^{97} + q^{98} + 2q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 7 x^{2} + 7 x + 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 4$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - 6 \nu + 1$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 4$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + 6 \beta_{1} - 1$$

## 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
 −2.60520 −0.233455 1.29051 2.54814
−2.60520 −1.00000 4.78706 3.78706 2.60520 −1.00000 −7.26084 1.00000 −9.86604
1.2 −0.233455 −1.00000 −1.94550 −2.94550 0.233455 −1.00000 0.921097 1.00000 0.687642
1.3 1.29051 −1.00000 −0.334573 −1.33457 −1.29051 −1.00000 −3.01280 1.00000 −1.72229
1.4 2.54814 −1.00000 4.49301 3.49301 −2.54814 −1.00000 6.35254 1.00000 8.90068
 $$n$$: 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.x 4
13.b even 2 1 3549.2.a.v 4
13.d odd 4 2 273.2.c.c 8
39.f even 4 2 819.2.c.d 8
52.f even 4 2 4368.2.h.q 8
91.i even 4 2 1911.2.c.l 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.c.c 8 13.d odd 4 2
819.2.c.d 8 39.f even 4 2
1911.2.c.l 8 91.i even 4 2
3549.2.a.v 4 13.b even 2 1
3549.2.a.x 4 1.a even 1 1 trivial
4368.2.h.q 8 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3549))$$:

 $$T_{2}^{4} - T_{2}^{3} - 7 T_{2}^{2} + 7 T_{2} + 2$$ $$T_{5}^{4} - 3 T_{5}^{3} - 14 T_{5}^{2} + 28 T_{5} + 52$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$2 + 7 T - 7 T^{2} - T^{3} + T^{4}$$
$3$ $$( 1 + T )^{4}$$
$5$ $$52 + 28 T - 14 T^{2} - 3 T^{3} + T^{4}$$
$7$ $$( 1 + T )^{4}$$
$11$ $$-32 + 60 T - 20 T^{2} - 2 T^{3} + T^{4}$$
$13$ $$T^{4}$$
$17$ $$-160 - 112 T + 8 T^{2} + 10 T^{3} + T^{4}$$
$19$ $$32 + 96 T - 24 T^{2} - 7 T^{3} + T^{4}$$
$23$ $$1352 + 140 T - 78 T^{2} - 3 T^{3} + T^{4}$$
$29$ $$-440 - 244 T - 14 T^{2} + 9 T^{3} + T^{4}$$
$31$ $$-64 - 112 T - 44 T^{2} + T^{3} + T^{4}$$
$37$ $$-512 - 480 T - 80 T^{2} + 4 T^{3} + T^{4}$$
$41$ $$1336 - 1020 T + 268 T^{2} - 28 T^{3} + T^{4}$$
$43$ $$-896 - 160 T + 64 T^{2} + 17 T^{3} + T^{4}$$
$47$ $$2596 + 56 T - 116 T^{2} - 3 T^{3} + T^{4}$$
$53$ $$712 - 356 T - 142 T^{2} + 5 T^{3} + T^{4}$$
$59$ $$1168 - 996 T + 268 T^{2} - 28 T^{3} + T^{4}$$
$61$ $$320 - 136 T - 28 T^{2} + 10 T^{3} + T^{4}$$
$67$ $$-8128 + 1888 T + 16 T^{2} - 22 T^{3} + T^{4}$$
$71$ $$-32 + 60 T - 20 T^{2} - 2 T^{3} + T^{4}$$
$73$ $$2288 + 1448 T + 328 T^{2} + 31 T^{3} + T^{4}$$
$79$ $$-80 - 104 T - 32 T^{2} + T^{3} + T^{4}$$
$83$ $$2596 + 56 T - 116 T^{2} - 3 T^{3} + T^{4}$$
$89$ $$-388 + 684 T - 114 T^{2} - 5 T^{3} + T^{4}$$
$97$ $$9104 + 4080 T + 648 T^{2} + 43 T^{3} + T^{4}$$