# Properties

 Label 546.2.l.a Level $546$ Weight $2$ Character orbit 546.l Analytic conductor $4.360$ Analytic rank $1$ 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: $$1$$ 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.

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

## 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 0 −0.500000 + 0.866025i −0.500000 + 0.866025i 1.00000 −0.500000 + 0.866025i 0
295.1 −0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i 0 −0.500000 0.866025i −0.500000 0.866025i 1.00000 −0.500000 0.866025i 0
 $$n$$: 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.a 2
3.b odd 2 1 1638.2.r.r 2
13.c even 3 1 inner 546.2.l.a 2
13.c even 3 1 7098.2.a.bb 1
13.e even 6 1 7098.2.a.l 1
39.i odd 6 1 1638.2.r.r 2

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

 $$T_{5}$$ $$T_{11}^{2} + 3 T_{11} + 9$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ $$1 + T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$1 + T + T^{2}$$
$11$ $$9 + 3 T + T^{2}$$
$13$ $$13 + 7 T + T^{2}$$
$17$ $$9 + 3 T + T^{2}$$
$19$ $$25 + 5 T + T^{2}$$
$23$ $$36 + 6 T + T^{2}$$
$29$ $$9 - 3 T + T^{2}$$
$31$ $$( 4 + T )^{2}$$
$37$ $$16 - 4 T + T^{2}$$
$41$ $$9 - 3 T + T^{2}$$
$43$ $$64 + 8 T + T^{2}$$
$47$ $$( -9 + T )^{2}$$
$53$ $$( 9 + T )^{2}$$
$59$ $$36 - 6 T + T^{2}$$
$61$ $$25 + 5 T + T^{2}$$
$67$ $$196 + 14 T + T^{2}$$
$71$ $$36 - 6 T + T^{2}$$
$73$ $$( 4 + T )^{2}$$
$79$ $$( 1 + T )^{2}$$
$83$ $$( 6 + T )^{2}$$
$89$ $$81 - 9 T + T^{2}$$
$97$ $$64 + 8 T + T^{2}$$