Properties

 Label 546.2.l.g 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.

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

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.g 2
3.b odd 2 1 1638.2.r.d 2
13.c even 3 1 inner 546.2.l.g 2
13.c even 3 1 7098.2.a.e 1
13.e even 6 1 7098.2.a.r 1
39.i odd 6 1 1638.2.r.d 2

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

 $$T_{5} - 2$$ $$T_{11}^{2} - T_{11} + 1$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2}$$
$3$ $$1 - T + T^{2}$$
$5$ $$( -2 + T )^{2}$$
$7$ $$1 - T + T^{2}$$
$11$ $$1 - T + T^{2}$$
$13$ $$13 - 5 T + T^{2}$$
$17$ $$1 - T + T^{2}$$
$19$ $$1 + T + T^{2}$$
$23$ $$36 - 6 T + T^{2}$$
$29$ $$81 + 9 T + T^{2}$$
$31$ $$( -2 + T )^{2}$$
$37$ $$4 - 2 T + T^{2}$$
$41$ $$25 + 5 T + T^{2}$$
$43$ $$4 - 2 T + T^{2}$$
$47$ $$( 7 + T )^{2}$$
$53$ $$( 1 + T )^{2}$$
$59$ $$100 + 10 T + T^{2}$$
$61$ $$121 - 11 T + T^{2}$$
$67$ $$4 + 2 T + T^{2}$$
$71$ $$64 - 8 T + T^{2}$$
$73$ $$( -8 + T )^{2}$$
$79$ $$( -11 + T )^{2}$$
$83$ $$( 12 + T )^{2}$$
$89$ $$121 + 11 T + T^{2}$$
$97$ $$196 - 14 T + T^{2}$$