# Properties

 Label 546.2.k.a Level $546$ Weight $2$ Character orbit 546.k 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.k (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.

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

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
373.1
 0.5 + 0.866025i 0.5 − 0.866025i
−0.500000 0.866025i −1.00000 −0.500000 + 0.866025i 0 0.500000 + 0.866025i −2.50000 0.866025i 1.00000 1.00000 0
445.1 −0.500000 + 0.866025i −1.00000 −0.500000 0.866025i 0 0.500000 0.866025i −2.50000 + 0.866025i 1.00000 1.00000 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
91.g 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.k.a yes 2
3.b odd 2 1 1638.2.p.d 2
7.c even 3 1 546.2.j.a 2
13.c even 3 1 546.2.j.a 2
21.h odd 6 1 1638.2.m.a 2
39.i odd 6 1 1638.2.m.a 2
91.g even 3 1 inner 546.2.k.a yes 2
273.bm odd 6 1 1638.2.p.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.j.a 2 7.c even 3 1
546.2.j.a 2 13.c even 3 1
546.2.k.a yes 2 1.a even 1 1 trivial
546.2.k.a yes 2 91.g even 3 1 inner
1638.2.m.a 2 21.h odd 6 1
1638.2.m.a 2 39.i odd 6 1
1638.2.p.d 2 3.b odd 2 1
1638.2.p.d 2 273.bm odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}$$ acting on $$S_{2}^{\mathrm{new}}(546, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$T^{2}$$
$7$ $$7 + 5 T + T^{2}$$
$11$ $$( 2 + T )^{2}$$
$13$ $$13 - 7 T + T^{2}$$
$17$ $$16 + 4 T + T^{2}$$
$19$ $$( -1 + T )^{2}$$
$23$ $$16 - 4 T + T^{2}$$
$29$ $$36 + 6 T + T^{2}$$
$31$ $$T^{2}$$
$37$ $$121 - 11 T + T^{2}$$
$41$ $$4 - 2 T + T^{2}$$
$43$ $$1 - T + T^{2}$$
$47$ $$T^{2}$$
$53$ $$16 + 4 T + T^{2}$$
$59$ $$16 + 4 T + T^{2}$$
$61$ $$( -1 + T )^{2}$$
$67$ $$( -12 + T )^{2}$$
$71$ $$36 + 6 T + T^{2}$$
$73$ $$49 - 7 T + T^{2}$$
$79$ $$64 + 8 T + T^{2}$$
$83$ $$( 14 + T )^{2}$$
$89$ $$36 + 6 T + T^{2}$$
$97$ $$81 + 9 T + T^{2}$$