# Properties

 Label 546.2.i.e Level $546$ Weight $2$ Character orbit 546.i 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.i (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.

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

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
79.1
 0.5 − 0.866025i 0.5 + 0.866025i
0.500000 0.866025i −0.500000 0.866025i −0.500000 0.866025i 1.00000 1.73205i −1.00000 −2.50000 + 0.866025i −1.00000 −0.500000 + 0.866025i −1.00000 1.73205i
235.1 0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 1.00000 + 1.73205i −1.00000 −2.50000 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
7.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.i.e 2
3.b odd 2 1 1638.2.j.a 2
7.c even 3 1 inner 546.2.i.e 2
7.c even 3 1 3822.2.a.k 1
7.d odd 6 1 3822.2.a.i 1
21.h odd 6 1 1638.2.j.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.i.e 2 1.a even 1 1 trivial
546.2.i.e 2 7.c even 3 1 inner
1638.2.j.a 2 3.b odd 2 1
1638.2.j.a 2 21.h odd 6 1
3822.2.a.i 1 7.d odd 6 1
3822.2.a.k 1 7.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 $$S_{2}^{\mathrm{new}}(546, [\chi])$$:

 $$T_{5}^{2} - 2 T_{5} + 4$$ $$T_{17}^{2} + 5 T_{17} + 25$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2}$$
$3$ $$1 + T + T^{2}$$
$5$ $$4 - 2 T + T^{2}$$
$7$ $$7 + 5 T + T^{2}$$
$11$ $$9 + 3 T + T^{2}$$
$13$ $$( 1 + T )^{2}$$
$17$ $$25 + 5 T + T^{2}$$
$19$ $$1 + T + T^{2}$$
$23$ $$T^{2}$$
$29$ $$( 1 + T )^{2}$$
$31$ $$16 + 4 T + T^{2}$$
$37$ $$4 - 2 T + T^{2}$$
$41$ $$( -10 + T )^{2}$$
$43$ $$( 10 + T )^{2}$$
$47$ $$1 - T + T^{2}$$
$53$ $$9 - 3 T + T^{2}$$
$59$ $$9 + 3 T + T^{2}$$
$61$ $$25 - 5 T + T^{2}$$
$67$ $$25 + 5 T + T^{2}$$
$71$ $$( 1 + T )^{2}$$
$73$ $$144 + 12 T + T^{2}$$
$79$ $$36 - 6 T + T^{2}$$
$83$ $$( -16 + T )^{2}$$
$89$ $$196 - 14 T + T^{2}$$
$97$ $$( -4 + T )^{2}$$