# Properties

 Label 546.2.l.e 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.

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

## 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.e 2
3.b odd 2 1 1638.2.r.f 2
13.c even 3 1 inner 546.2.l.e 2
13.c even 3 1 7098.2.a.m 1
13.e even 6 1 7098.2.a.ba 1
39.i odd 6 1 1638.2.r.f 2

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2}$$
$3$ $$1 + T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$1 + T + T^{2}$$
$11$ $$9 - 3 T + T^{2}$$
$13$ $$13 - 5 T + T^{2}$$
$17$ $$9 - 3 T + T^{2}$$
$19$ $$25 + 5 T + T^{2}$$
$23$ $$36 - 6 T + T^{2}$$
$29$ $$81 - 9 T + T^{2}$$
$31$ $$( -8 + T )^{2}$$
$37$ $$64 + 8 T + T^{2}$$
$41$ $$9 + 3 T + T^{2}$$
$43$ $$64 + 8 T + T^{2}$$
$47$ $$( -3 + T )^{2}$$
$53$ $$( 3 + T )^{2}$$
$59$ $$36 - 6 T + T^{2}$$
$61$ $$49 - 7 T + T^{2}$$
$67$ $$4 + 2 T + T^{2}$$
$71$ $$36 - 6 T + T^{2}$$
$73$ $$( 16 + T )^{2}$$
$79$ $$( 13 + T )^{2}$$
$83$ $$( -18 + T )^{2}$$
$89$ $$225 - 15 T + T^{2}$$
$97$ $$256 - 16 T + T^{2}$$