# Properties

 Label 546.2.l.k 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: $$\Q(\sqrt{-3}, \sqrt{-43})$$ Defining polynomial: $$x^{4} - x^{3} - 10 x^{2} - 11 x + 121$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 10 x^{2} - 11 x + 121$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 10 \nu^{2} - 10 \nu - 11$$$$)/110$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + 10 \nu + 11$$$$)/10$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 11 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$-10 \beta_{3} + 10 \beta_{1} + 11$$

## Character values

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

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

## 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
 −2.58945 − 2.07237i 3.08945 + 1.20635i −2.58945 + 2.07237i 3.08945 − 1.20635i
−0.500000 0.866025i 0.500000 + 0.866025i −0.500000 + 0.866025i −2.00000 0.500000 0.866025i 0.500000 0.866025i 1.00000 −0.500000 + 0.866025i 1.00000 + 1.73205i
211.2 −0.500000 0.866025i 0.500000 + 0.866025i −0.500000 + 0.866025i −2.00000 0.500000 0.866025i 0.500000 0.866025i 1.00000 −0.500000 + 0.866025i 1.00000 + 1.73205i
295.1 −0.500000 + 0.866025i 0.500000 0.866025i −0.500000 0.866025i −2.00000 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 −0.500000 0.866025i 1.00000 1.73205i
295.2 −0.500000 + 0.866025i 0.500000 0.866025i −0.500000 0.866025i −2.00000 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 −0.500000 0.866025i 1.00000 1.73205i
 $$n$$: 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.k 4
3.b odd 2 1 1638.2.r.ba 4
13.c even 3 1 inner 546.2.l.k 4
13.c even 3 1 7098.2.a.br 2
13.e even 6 1 7098.2.a.bk 2
39.i odd 6 1 1638.2.r.ba 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.l.k 4 1.a even 1 1 trivial
546.2.l.k 4 13.c even 3 1 inner
1638.2.r.ba 4 3.b odd 2 1
1638.2.r.ba 4 39.i odd 6 1
7098.2.a.bk 2 13.e even 6 1
7098.2.a.br 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 $$S_{2}^{\mathrm{new}}(546, [\chi])$$:

 $$T_{5} + 2$$ $$T_{11}^{4} + T_{11}^{3} + 33 T_{11}^{2} - 32 T_{11} + 1024$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T + T^{2} )^{2}$$
$3$ $$( 1 - T + T^{2} )^{2}$$
$5$ $$( 2 + T )^{4}$$
$7$ $$( 1 - T + T^{2} )^{2}$$
$11$ $$1024 - 32 T + 33 T^{2} + T^{3} + T^{4}$$
$13$ $$( 13 + 3 T + T^{2} )^{2}$$
$17$ $$( 25 + 5 T + T^{2} )^{2}$$
$19$ $$676 + 130 T + 51 T^{2} - 5 T^{3} + T^{4}$$
$23$ $$676 + 130 T + 51 T^{2} - 5 T^{3} + T^{4}$$
$29$ $$676 - 130 T + 51 T^{2} + 5 T^{3} + T^{4}$$
$31$ $$( -30 + 3 T + T^{2} )^{2}$$
$37$ $$( 4 + 2 T + T^{2} )^{2}$$
$41$ $$900 + 90 T + 39 T^{2} - 3 T^{3} + T^{4}$$
$43$ $$100 + 130 T + 159 T^{2} + 13 T^{3} + T^{4}$$
$47$ $$( -30 + 3 T + T^{2} )^{2}$$
$53$ $$( 3 + T )^{4}$$
$59$ $$100 - 130 T + 159 T^{2} - 13 T^{3} + T^{4}$$
$61$ $$( 9 - 3 T + T^{2} )^{2}$$
$67$ $$3364 - 1102 T + 303 T^{2} - 19 T^{3} + T^{4}$$
$71$ $$400 - 140 T + 69 T^{2} + 7 T^{3} + T^{4}$$
$73$ $$( -4 + T )^{4}$$
$79$ $$( 24 - 15 T + T^{2} )^{2}$$
$83$ $$( -32 - T + T^{2} )^{2}$$
$89$ $$( 81 + 9 T + T^{2} )^{2}$$
$97$ $$16384 - 256 T + 132 T^{2} + 2 T^{3} + T^{4}$$