# Properties

 Label 3549.2.a.y Level $3549$ Weight $2$ Character orbit 3549.a Self dual yes Analytic conductor $28.339$ Analytic rank $1$ Dimension $6$ 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: $$6$$ Coefficient field: 6.6.121819537.1 Defining polynomial: $$x^{6} - 3 x^{5} - 25 x^{4} + 55 x^{3} + 224 x^{2} - 252 x - 728$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 3 x^{5} - 25 x^{4} + 55 x^{3} + 224 x^{2} - 252 x - 728$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{2} - \nu - 10$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} - \nu^{2} - 10 \nu$$$$)/2$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{4} - 2 \nu^{3} - 17 \nu^{2} + 18 \nu + 76$$$$)/4$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{5} - 3 \nu^{4} - 17 \nu^{3} + 39 \nu^{2} + 72 \nu - 92$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2} + \beta_{1} + 10$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3} + 2 \beta_{2} + 11 \beta_{1} + 10$$ $$\nu^{4}$$ $$=$$ $$4 \beta_{4} + 4 \beta_{3} + 38 \beta_{2} + 21 \beta_{1} + 114$$ $$\nu^{5}$$ $$=$$ $$4 \beta_{5} + 12 \beta_{4} + 46 \beta_{3} + 70 \beta_{2} + 139 \beta_{1} + 214$$

## 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
 −2.07801 3.07801 −2.55940 3.55940 −3.06987 4.06987
−1.80194 −1.00000 1.24698 −2.07801 1.80194 −1.00000 1.35690 1.00000 3.74444
1.2 −1.80194 −1.00000 1.24698 3.07801 1.80194 −1.00000 1.35690 1.00000 −5.54638
1.3 −0.445042 −1.00000 −1.80194 −2.55940 0.445042 −1.00000 1.69202 1.00000 1.13904
1.4 −0.445042 −1.00000 −1.80194 3.55940 0.445042 −1.00000 1.69202 1.00000 −1.58408
1.5 1.24698 −1.00000 −0.445042 −3.06987 −1.24698 −1.00000 −3.04892 1.00000 −3.82806
1.6 1.24698 −1.00000 −0.445042 4.06987 −1.24698 −1.00000 −3.04892 1.00000 5.07504
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 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.y 6
13.b even 2 1 3549.2.a.z yes 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3549.2.a.y 6 1.a even 1 1 trivial
3549.2.a.z yes 6 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3549))$$:

 $$T_{2}^{3} + T_{2}^{2} - 2 T_{2} - 1$$ $$T_{5}^{6} - 3 T_{5}^{5} - 25 T_{5}^{4} + 55 T_{5}^{3} + 224 T_{5}^{2} - 252 T_{5} - 728$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 - 2 T + T^{2} + T^{3} )^{2}$$
$3$ $$( 1 + T )^{6}$$
$5$ $$-728 - 252 T + 224 T^{2} + 55 T^{3} - 25 T^{4} - 3 T^{5} + T^{6}$$
$7$ $$( 1 + T )^{6}$$
$11$ $$-281 - 434 T + 426 T^{2} + 42 T^{3} - 47 T^{4} + T^{6}$$
$13$ $$T^{6}$$
$17$ $$-8 - 192 T - 886 T^{2} - 451 T^{3} - 34 T^{4} + 9 T^{5} + T^{6}$$
$19$ $$64 - 256 T - 484 T^{2} + 231 T^{3} + 6 T^{4} - 11 T^{5} + T^{6}$$
$23$ $$2353 + 1439 T - 323 T^{2} - 259 T^{3} - T^{4} + 11 T^{5} + T^{6}$$
$29$ $$12649 + 10696 T + 322 T^{2} - 734 T^{3} - 67 T^{4} + 10 T^{5} + T^{6}$$
$31$ $$4472 - 4676 T + 132 T^{2} + 665 T^{3} - 96 T^{4} - 7 T^{5} + T^{6}$$
$37$ $$7267 - 2053 T - 3208 T^{2} - 483 T^{3} + 80 T^{4} + 20 T^{5} + T^{6}$$
$41$ $$-1856 + 1776 T + 356 T^{2} - 431 T^{3} - 45 T^{4} + 10 T^{5} + T^{6}$$
$43$ $$-108352 + 21792 T + 5988 T^{2} - 903 T^{3} - 128 T^{4} + 9 T^{5} + T^{6}$$
$47$ $$-302184 + 156456 T - 24054 T^{2} - 263 T^{3} + 401 T^{4} - 36 T^{5} + T^{6}$$
$53$ $$-39403 + 16569 T + 2744 T^{2} - 909 T^{3} - 78 T^{4} + 12 T^{5} + T^{6}$$
$59$ $$498568 - 82236 T - 32312 T^{2} - 69 T^{3} + 495 T^{4} + 41 T^{5} + T^{6}$$
$61$ $$-7384 - 20204 T - 11518 T^{2} - 1429 T^{3} + 93 T^{4} + 24 T^{5} + T^{6}$$
$67$ $$-39733 - 27105 T + 75 T^{2} + 1683 T^{3} - 89 T^{4} - 15 T^{5} + T^{6}$$
$71$ $$16457 - 5425 T - 5509 T^{2} + 2421 T^{3} - 149 T^{4} - 13 T^{5} + T^{6}$$
$73$ $$58024 - 1432 T - 6574 T^{2} + 651 T^{3} + 146 T^{4} - 25 T^{5} + T^{6}$$
$79$ $$-91141 + 14004 T + 8802 T^{2} - 1480 T^{3} - 125 T^{4} + 16 T^{5} + T^{6}$$
$83$ $$-218792 + 22512 T + 13342 T^{2} - 519 T^{3} - 225 T^{4} + 2 T^{5} + T^{6}$$
$89$ $$-51064 + 48572 T + 27840 T^{2} + 2043 T^{3} - 265 T^{4} - 15 T^{5} + T^{6}$$
$97$ $$11752 - 10848 T - 1058 T^{2} + 1089 T^{3} - 87 T^{4} - 10 T^{5} + T^{6}$$