# Properties

 Label 546.2.l.b 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.

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

## 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 3.00000 0.500000 0.866025i 0.500000 0.866025i 1.00000 −0.500000 + 0.866025i −1.50000 2.59808i
295.1 −0.500000 + 0.866025i 0.500000 0.866025i −0.500000 0.866025i 3.00000 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 −0.500000 0.866025i −1.50000 + 2.59808i
 $$n$$: 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.b 2
3.b odd 2 1 1638.2.r.o 2
13.c even 3 1 inner 546.2.l.b 2
13.c even 3 1 7098.2.a.v 1
13.e even 6 1 7098.2.a.a 1
39.i odd 6 1 1638.2.r.o 2

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ $$1 - T + T^{2}$$
$5$ $$( -3 + T )^{2}$$
$7$ $$1 - T + T^{2}$$
$11$ $$16 + 4 T + T^{2}$$
$13$ $$13 - 7 T + T^{2}$$
$17$ $$25 - 5 T + T^{2}$$
$19$ $$16 + 4 T + T^{2}$$
$23$ $$16 - 4 T + T^{2}$$
$29$ $$81 - 9 T + T^{2}$$
$31$ $$T^{2}$$
$37$ $$49 + 7 T + T^{2}$$
$41$ $$49 + 7 T + T^{2}$$
$43$ $$64 + 8 T + T^{2}$$
$47$ $$( -12 + T )^{2}$$
$53$ $$( -7 + T )^{2}$$
$59$ $$64 - 8 T + T^{2}$$
$61$ $$49 + 7 T + T^{2}$$
$67$ $$144 + 12 T + T^{2}$$
$71$ $$144 + 12 T + T^{2}$$
$73$ $$( 1 + T )^{2}$$
$79$ $$( 16 + T )^{2}$$
$83$ $$( 8 + T )^{2}$$
$89$ $$36 - 6 T + T^{2}$$
$97$ $$324 + 18 T + T^{2}$$