# Properties

 Label 546.2.l.d Level $546$ Weight $2$ Character orbit 546.l Analytic conductor $4.360$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ 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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$157$$ $$365$$ $$379$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{6}$$

## 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
 0.5 − 0.866025i 0.5 + 0.866025i
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
 $$n$$: 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.d 2
3.b odd 2 1 1638.2.r.k 2
13.c even 3 1 inner 546.2.l.d 2
13.c even 3 1 7098.2.a.h 1
13.e even 6 1 7098.2.a.bf 1
39.i odd 6 1 1638.2.r.k 2

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

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2}$$
$3$ $$1 + T + T^{2}$$
$5$ $$( 2 + T )^{2}$$
$7$ $$1 - T + T^{2}$$
$11$ $$16 - 4 T + T^{2}$$
$13$ $$13 + 7 T + T^{2}$$
$17$ $$49 + 7 T + T^{2}$$
$19$ $$4 + 2 T + T^{2}$$
$23$ $$1 - T + T^{2}$$
$29$ $$4 - 2 T + T^{2}$$
$31$ $$( 9 + T )^{2}$$
$37$ $$4 - 2 T + T^{2}$$
$41$ $$4 + 2 T + T^{2}$$
$43$ $$25 - 5 T + T^{2}$$
$47$ $$( -6 + T )^{2}$$
$53$ $$( -3 + T )^{2}$$
$59$ $$225 + 15 T + T^{2}$$
$61$ $$49 - 7 T + T^{2}$$
$67$ $$25 - 5 T + T^{2}$$
$71$ $$1 + T + T^{2}$$
$73$ $$( -12 + T )^{2}$$
$79$ $$( 4 + T )^{2}$$
$83$ $$( 1 + T )^{2}$$
$89$ $$9 + 3 T + T^{2}$$
$97$ $$256 - 16 T + T^{2}$$