# Properties

 Label 546.2.bi.c Level $546$ Weight $2$ Character orbit 546.bi 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.bi (of order $$6$$, 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + q^{2} + ( -2 + \zeta_{6} ) q^{3} + q^{4} + ( -4 + 2 \zeta_{6} ) q^{5} + ( -2 + \zeta_{6} ) q^{6} + ( -2 + 3 \zeta_{6} ) q^{7} + q^{8} + ( 3 - 3 \zeta_{6} ) q^{9} +O(q^{10})$$ $$q + q^{2} + ( -2 + \zeta_{6} ) q^{3} + q^{4} + ( -4 + 2 \zeta_{6} ) q^{5} + ( -2 + \zeta_{6} ) q^{6} + ( -2 + 3 \zeta_{6} ) q^{7} + q^{8} + ( 3 - 3 \zeta_{6} ) q^{9} + ( -4 + 2 \zeta_{6} ) q^{10} -6 \zeta_{6} q^{11} + ( -2 + \zeta_{6} ) q^{12} + ( -1 - 3 \zeta_{6} ) q^{13} + ( -2 + 3 \zeta_{6} ) q^{14} + ( 6 - 6 \zeta_{6} ) q^{15} + q^{16} + ( 3 - 3 \zeta_{6} ) q^{18} + ( 5 - 5 \zeta_{6} ) q^{19} + ( -4 + 2 \zeta_{6} ) q^{20} + ( 1 - 5 \zeta_{6} ) q^{21} -6 \zeta_{6} q^{22} + ( -2 + 4 \zeta_{6} ) q^{23} + ( -2 + \zeta_{6} ) q^{24} + ( 7 - 7 \zeta_{6} ) q^{25} + ( -1 - 3 \zeta_{6} ) q^{26} + ( -3 + 6 \zeta_{6} ) q^{27} + ( -2 + 3 \zeta_{6} ) q^{28} + ( -4 - 4 \zeta_{6} ) q^{29} + ( 6 - 6 \zeta_{6} ) q^{30} + ( -8 + 8 \zeta_{6} ) q^{31} + q^{32} + ( 6 + 6 \zeta_{6} ) q^{33} + ( 2 - 10 \zeta_{6} ) q^{35} + ( 3 - 3 \zeta_{6} ) q^{36} + ( -5 + 10 \zeta_{6} ) q^{37} + ( 5 - 5 \zeta_{6} ) q^{38} + ( 5 + 2 \zeta_{6} ) q^{39} + ( -4 + 2 \zeta_{6} ) q^{40} + ( -6 - 6 \zeta_{6} ) q^{41} + ( 1 - 5 \zeta_{6} ) q^{42} -\zeta_{6} q^{43} -6 \zeta_{6} q^{44} + ( -6 + 12 \zeta_{6} ) q^{45} + ( -2 + 4 \zeta_{6} ) q^{46} + ( -8 + 4 \zeta_{6} ) q^{47} + ( -2 + \zeta_{6} ) q^{48} + ( -5 - 3 \zeta_{6} ) q^{49} + ( 7 - 7 \zeta_{6} ) q^{50} + ( -1 - 3 \zeta_{6} ) q^{52} + ( -2 - 2 \zeta_{6} ) q^{53} + ( -3 + 6 \zeta_{6} ) q^{54} + ( 12 + 12 \zeta_{6} ) q^{55} + ( -2 + 3 \zeta_{6} ) q^{56} + ( -5 + 10 \zeta_{6} ) q^{57} + ( -4 - 4 \zeta_{6} ) q^{58} + ( 6 - 6 \zeta_{6} ) q^{60} + ( 3 + 3 \zeta_{6} ) q^{61} + ( -8 + 8 \zeta_{6} ) q^{62} + ( 3 + 6 \zeta_{6} ) q^{63} + q^{64} + ( 10 + 4 \zeta_{6} ) q^{65} + ( 6 + 6 \zeta_{6} ) q^{66} + ( -4 + 2 \zeta_{6} ) q^{67} -6 \zeta_{6} q^{69} + ( 2 - 10 \zeta_{6} ) q^{70} + ( 3 - 3 \zeta_{6} ) q^{72} + ( 7 - 7 \zeta_{6} ) q^{73} + ( -5 + 10 \zeta_{6} ) q^{74} + ( -7 + 14 \zeta_{6} ) q^{75} + ( 5 - 5 \zeta_{6} ) q^{76} + ( 18 - 6 \zeta_{6} ) q^{77} + ( 5 + 2 \zeta_{6} ) q^{78} + 8 \zeta_{6} q^{79} + ( -4 + 2 \zeta_{6} ) q^{80} -9 \zeta_{6} q^{81} + ( -6 - 6 \zeta_{6} ) q^{82} + ( -2 + 4 \zeta_{6} ) q^{83} + ( 1 - 5 \zeta_{6} ) q^{84} -\zeta_{6} q^{86} + 12 q^{87} -6 \zeta_{6} q^{88} + ( -2 + 4 \zeta_{6} ) q^{89} + ( -6 + 12 \zeta_{6} ) q^{90} + ( 11 - 6 \zeta_{6} ) q^{91} + ( -2 + 4 \zeta_{6} ) q^{92} + ( 8 - 16 \zeta_{6} ) q^{93} + ( -8 + 4 \zeta_{6} ) q^{94} + ( -10 + 20 \zeta_{6} ) q^{95} + ( -2 + \zeta_{6} ) q^{96} -7 \zeta_{6} q^{97} + ( -5 - 3 \zeta_{6} ) q^{98} -18 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} - 3q^{3} + 2q^{4} - 6q^{5} - 3q^{6} - q^{7} + 2q^{8} + 3q^{9} + O(q^{10})$$ $$2q + 2q^{2} - 3q^{3} + 2q^{4} - 6q^{5} - 3q^{6} - q^{7} + 2q^{8} + 3q^{9} - 6q^{10} - 6q^{11} - 3q^{12} - 5q^{13} - q^{14} + 6q^{15} + 2q^{16} + 3q^{18} + 5q^{19} - 6q^{20} - 3q^{21} - 6q^{22} - 3q^{24} + 7q^{25} - 5q^{26} - q^{28} - 12q^{29} + 6q^{30} - 8q^{31} + 2q^{32} + 18q^{33} - 6q^{35} + 3q^{36} + 5q^{38} + 12q^{39} - 6q^{40} - 18q^{41} - 3q^{42} - q^{43} - 6q^{44} - 12q^{47} - 3q^{48} - 13q^{49} + 7q^{50} - 5q^{52} - 6q^{53} + 36q^{55} - q^{56} - 12q^{58} + 6q^{60} + 9q^{61} - 8q^{62} + 12q^{63} + 2q^{64} + 24q^{65} + 18q^{66} - 6q^{67} - 6q^{69} - 6q^{70} + 3q^{72} + 7q^{73} + 5q^{76} + 30q^{77} + 12q^{78} + 8q^{79} - 6q^{80} - 9q^{81} - 18q^{82} - 3q^{84} - q^{86} + 24q^{87} - 6q^{88} + 16q^{91} - 12q^{94} - 3q^{96} - 7q^{97} - 13q^{98} - 36q^{99} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1
 0.5 + 0.866025i 0.5 − 0.866025i
1.00000 −1.50000 + 0.866025i 1.00000 −3.00000 + 1.73205i −1.50000 + 0.866025i −0.500000 + 2.59808i 1.00000 1.50000 2.59808i −3.00000 + 1.73205i
257.1 1.00000 −1.50000 0.866025i 1.00000 −3.00000 1.73205i −1.50000 0.866025i −0.500000 2.59808i 1.00000 1.50000 + 2.59808i −3.00000 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
273.br even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 546.2.bi.c yes 2
3.b odd 2 1 546.2.bi.a 2
7.d odd 6 1 546.2.bn.b yes 2
13.e even 6 1 546.2.bn.c yes 2
21.g even 6 1 546.2.bn.c yes 2
39.h odd 6 1 546.2.bn.b yes 2
91.l odd 6 1 546.2.bi.a 2
273.br even 6 1 inner 546.2.bi.c yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.bi.a 2 3.b odd 2 1
546.2.bi.a 2 91.l odd 6 1
546.2.bi.c yes 2 1.a even 1 1 trivial
546.2.bi.c yes 2 273.br even 6 1 inner
546.2.bn.b yes 2 7.d odd 6 1
546.2.bn.b yes 2 39.h odd 6 1
546.2.bn.c yes 2 13.e even 6 1
546.2.bn.c yes 2 21.g even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 6 T_{5} + 12$$ acting on $$S_{2}^{\mathrm{new}}(546, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{2}$$
$3$ $$3 + 3 T + T^{2}$$
$5$ $$12 + 6 T + T^{2}$$
$7$ $$7 + T + T^{2}$$
$11$ $$36 + 6 T + T^{2}$$
$13$ $$13 + 5 T + T^{2}$$
$17$ $$T^{2}$$
$19$ $$25 - 5 T + T^{2}$$
$23$ $$12 + T^{2}$$
$29$ $$48 + 12 T + T^{2}$$
$31$ $$64 + 8 T + T^{2}$$
$37$ $$75 + T^{2}$$
$41$ $$108 + 18 T + T^{2}$$
$43$ $$1 + T + T^{2}$$
$47$ $$48 + 12 T + T^{2}$$
$53$ $$12 + 6 T + T^{2}$$
$59$ $$T^{2}$$
$61$ $$27 - 9 T + T^{2}$$
$67$ $$12 + 6 T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$49 - 7 T + T^{2}$$
$79$ $$64 - 8 T + T^{2}$$
$83$ $$12 + T^{2}$$
$89$ $$12 + T^{2}$$
$97$ $$49 + 7 T + T^{2}$$