# Properties

 Label 546.2.l.i Level $546$ Weight $2$ Character orbit 546.l Analytic conductor $4.360$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$546 = 2 \cdot 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 546.l (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.35983195036$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{17})$$ Defining polynomial: $$x^{4} - x^{3} + 5 x^{2} + 4 x + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{2} ) q^{2} + ( -1 + \beta_{2} ) q^{3} -\beta_{2} q^{4} + ( -1 + \beta_{3} ) q^{5} -\beta_{2} q^{6} + \beta_{2} q^{7} + q^{8} -\beta_{2} q^{9} +O(q^{10})$$ $$q + ( -1 + \beta_{2} ) q^{2} + ( -1 + \beta_{2} ) q^{3} -\beta_{2} q^{4} + ( -1 + \beta_{3} ) q^{5} -\beta_{2} q^{6} + \beta_{2} q^{7} + q^{8} -\beta_{2} q^{9} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{10} + ( \beta_{1} + \beta_{3} ) q^{11} + q^{12} + ( -1 - 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{13} - q^{14} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{15} + ( -1 + \beta_{2} ) q^{16} + 5 \beta_{2} q^{17} + q^{18} -3 \beta_{1} q^{19} + ( \beta_{1} + \beta_{2} ) q^{20} - q^{21} -\beta_{1} q^{22} + ( 4 - 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{23} + ( -1 + \beta_{2} ) q^{24} -3 \beta_{3} q^{25} + ( \beta_{1} - \beta_{2} - \beta_{3} ) q^{26} + q^{27} + ( 1 - \beta_{2} ) q^{28} + ( 1 - \beta_{2} ) q^{29} + ( \beta_{1} + \beta_{2} ) q^{30} -2 \beta_{3} q^{31} -\beta_{2} q^{32} -\beta_{1} q^{33} -5 q^{34} + ( -\beta_{1} - \beta_{2} ) q^{35} + ( -1 + \beta_{2} ) q^{36} + ( -3 - 5 \beta_{1} + 3 \beta_{2} - 5 \beta_{3} ) q^{37} -3 \beta_{3} q^{38} + ( \beta_{1} - \beta_{2} - \beta_{3} ) q^{39} + ( -1 + \beta_{3} ) q^{40} + ( 1 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{41} + ( 1 - \beta_{2} ) q^{42} + ( -2 \beta_{1} + 8 \beta_{2} ) q^{43} -\beta_{3} q^{44} + ( \beta_{1} + \beta_{2} ) q^{45} + ( 2 \beta_{1} + 4 \beta_{2} ) q^{46} + 3 \beta_{3} q^{47} -\beta_{2} q^{48} + ( -1 + \beta_{2} ) q^{49} + ( 3 \beta_{1} + 3 \beta_{3} ) q^{50} -5 q^{51} + ( 1 + \beta_{1} + 2 \beta_{3} ) q^{52} + ( -9 - 2 \beta_{3} ) q^{53} + ( -1 + \beta_{2} ) q^{54} + ( 4 - 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{55} + \beta_{2} q^{56} -3 \beta_{3} q^{57} + \beta_{2} q^{58} + ( -2 \beta_{1} - 8 \beta_{2} ) q^{59} + ( -1 + \beta_{3} ) q^{60} + \beta_{2} q^{61} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{62} + ( 1 - \beta_{2} ) q^{63} + q^{64} + ( -3 + 3 \beta_{1} + 7 \beta_{2} + \beta_{3} ) q^{65} -\beta_{3} q^{66} + ( 4 + 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{67} + ( 5 - 5 \beta_{2} ) q^{68} + ( 2 \beta_{1} + 4 \beta_{2} ) q^{69} + ( 1 - \beta_{3} ) q^{70} + 4 \beta_{2} q^{71} -\beta_{2} q^{72} + ( -5 - \beta_{3} ) q^{73} + ( 5 \beta_{1} - 3 \beta_{2} ) q^{74} + ( 3 \beta_{1} + 3 \beta_{3} ) q^{75} + ( 3 \beta_{1} + 3 \beta_{3} ) q^{76} + \beta_{3} q^{77} + ( 1 + \beta_{1} + 2 \beta_{3} ) q^{78} + ( 4 + \beta_{3} ) q^{79} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{80} + ( -1 + \beta_{2} ) q^{81} + ( -2 \beta_{1} + \beta_{2} ) q^{82} + 4 \beta_{3} q^{83} + \beta_{2} q^{84} + ( -5 \beta_{1} - 5 \beta_{2} ) q^{85} + ( -8 - 2 \beta_{3} ) q^{86} + \beta_{2} q^{87} + ( \beta_{1} + \beta_{3} ) q^{88} + ( 2 - 3 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} ) q^{89} + ( -1 + \beta_{3} ) q^{90} + ( -1 - \beta_{1} - 2 \beta_{3} ) q^{91} + ( -4 + 2 \beta_{3} ) q^{92} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{93} + ( -3 \beta_{1} - 3 \beta_{3} ) q^{94} + ( 6 \beta_{1} + 12 \beta_{2} ) q^{95} + q^{96} + ( -4 \beta_{1} - 2 \beta_{2} ) q^{97} -\beta_{2} q^{98} -\beta_{3} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{2} - 2q^{3} - 2q^{4} - 6q^{5} - 2q^{6} + 2q^{7} + 4q^{8} - 2q^{9} + O(q^{10})$$ $$4q - 2q^{2} - 2q^{3} - 2q^{4} - 6q^{5} - 2q^{6} + 2q^{7} + 4q^{8} - 2q^{9} + 3q^{10} - q^{11} + 4q^{12} - 2q^{13} - 4q^{14} + 3q^{15} - 2q^{16} + 10q^{17} + 4q^{18} - 3q^{19} + 3q^{20} - 4q^{21} - q^{22} + 10q^{23} - 2q^{24} + 6q^{25} + q^{26} + 4q^{27} + 2q^{28} + 2q^{29} + 3q^{30} + 4q^{31} - 2q^{32} - q^{33} - 20q^{34} - 3q^{35} - 2q^{36} - q^{37} + 6q^{38} + q^{39} - 6q^{40} + 2q^{42} + 14q^{43} + 2q^{44} + 3q^{45} + 10q^{46} - 6q^{47} - 2q^{48} - 2q^{49} - 3q^{50} - 20q^{51} + q^{52} - 32q^{53} - 2q^{54} + 10q^{55} + 2q^{56} + 6q^{57} + 2q^{58} - 18q^{59} - 6q^{60} + 2q^{61} - 2q^{62} + 2q^{63} + 4q^{64} + 3q^{65} + 2q^{66} + 6q^{67} + 10q^{68} + 10q^{69} + 6q^{70} + 8q^{71} - 2q^{72} - 18q^{73} - q^{74} - 3q^{75} - 3q^{76} - 2q^{77} + q^{78} + 14q^{79} + 3q^{80} - 2q^{81} - 8q^{83} + 2q^{84} - 15q^{85} - 28q^{86} + 2q^{87} - q^{88} + 7q^{89} - 6q^{90} - q^{91} - 20q^{92} - 2q^{93} + 3q^{94} + 30q^{95} + 4q^{96} - 8q^{97} - 2q^{98} + 2q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 5 x^{2} + 4 x + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} + 5 \nu^{2} - 5 \nu + 16$$$$)/20$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} + 4$$$$)/5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 4 \beta_{2} + \beta_{1} - 4$$ $$\nu^{3}$$ $$=$$ $$5 \beta_{3} - 4$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/546\mathbb{Z}\right)^\times$$.

 $$n$$ $$157$$ $$365$$ $$379$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
211.1
 1.28078 − 2.21837i −0.780776 + 1.35234i 1.28078 + 2.21837i −0.780776 − 1.35234i
−0.500000 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i −3.56155 −0.500000 + 0.866025i 0.500000 0.866025i 1.00000 −0.500000 + 0.866025i 1.78078 + 3.08440i
211.2 −0.500000 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i 0.561553 −0.500000 + 0.866025i 0.500000 0.866025i 1.00000 −0.500000 + 0.866025i −0.280776 0.486319i
295.1 −0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i −3.56155 −0.500000 0.866025i 0.500000 + 0.866025i 1.00000 −0.500000 0.866025i 1.78078 3.08440i
295.2 −0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i 0.561553 −0.500000 0.866025i 0.500000 + 0.866025i 1.00000 −0.500000 0.866025i −0.280776 + 0.486319i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 546.2.l.i 4
3.b odd 2 1 1638.2.r.z 4
13.c even 3 1 inner 546.2.l.i 4
13.c even 3 1 7098.2.a.bw 2
13.e even 6 1 7098.2.a.bq 2
39.i odd 6 1 1638.2.r.z 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.l.i 4 1.a even 1 1 trivial
546.2.l.i 4 13.c even 3 1 inner
1638.2.r.z 4 3.b odd 2 1
1638.2.r.z 4 39.i odd 6 1
7098.2.a.bq 2 13.e even 6 1
7098.2.a.bw 2 13.c even 3 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(546, [\chi])$$:

 $$T_{5}^{2} + 3 T_{5} - 2$$ $$T_{11}^{4} + T_{11}^{3} + 5 T_{11}^{2} - 4 T_{11} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T + T^{2} )^{2}$$
$3$ $$( 1 + T + T^{2} )^{2}$$
$5$ $$( -2 + 3 T + T^{2} )^{2}$$
$7$ $$( 1 - T + T^{2} )^{2}$$
$11$ $$16 - 4 T + 5 T^{2} + T^{3} + T^{4}$$
$13$ $$( 13 + T + T^{2} )^{2}$$
$17$ $$( 25 - 5 T + T^{2} )^{2}$$
$19$ $$1296 - 108 T + 45 T^{2} + 3 T^{3} + T^{4}$$
$23$ $$64 - 80 T + 92 T^{2} - 10 T^{3} + T^{4}$$
$29$ $$( 1 - T + T^{2} )^{2}$$
$31$ $$( -16 - 2 T + T^{2} )^{2}$$
$37$ $$11236 - 106 T + 107 T^{2} + T^{3} + T^{4}$$
$41$ $$289 + 17 T^{2} + T^{4}$$
$43$ $$1024 - 448 T + 164 T^{2} - 14 T^{3} + T^{4}$$
$47$ $$( -36 + 3 T + T^{2} )^{2}$$
$53$ $$( 47 + 16 T + T^{2} )^{2}$$
$59$ $$4096 + 1152 T + 260 T^{2} + 18 T^{3} + T^{4}$$
$61$ $$( 1 - T + T^{2} )^{2}$$
$67$ $$64 + 48 T + 44 T^{2} - 6 T^{3} + T^{4}$$
$71$ $$( 16 - 4 T + T^{2} )^{2}$$
$73$ $$( 16 + 9 T + T^{2} )^{2}$$
$79$ $$( 8 - 7 T + T^{2} )^{2}$$
$83$ $$( -64 + 4 T + T^{2} )^{2}$$
$89$ $$676 + 182 T + 75 T^{2} - 7 T^{3} + T^{4}$$
$97$ $$2704 - 416 T + 116 T^{2} + 8 T^{3} + T^{4}$$