# Properties

 Label 546.2.bi.d 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.

 $$f(q)$$ $$=$$ $$q + q^{2} + ( -1 - \zeta_{6} ) q^{3} + q^{4} + ( -2 + \zeta_{6} ) q^{5} + ( -1 - \zeta_{6} ) q^{6} + ( -1 - 2 \zeta_{6} ) q^{7} + q^{8} + 3 \zeta_{6} q^{9} +O(q^{10})$$ $$q + q^{2} + ( -1 - \zeta_{6} ) q^{3} + q^{4} + ( -2 + \zeta_{6} ) q^{5} + ( -1 - \zeta_{6} ) q^{6} + ( -1 - 2 \zeta_{6} ) q^{7} + q^{8} + 3 \zeta_{6} q^{9} + ( -2 + \zeta_{6} ) q^{10} -3 \zeta_{6} q^{11} + ( -1 - \zeta_{6} ) q^{12} + ( -3 + 4 \zeta_{6} ) q^{13} + ( -1 - 2 \zeta_{6} ) q^{14} + 3 q^{15} + q^{16} -6 q^{17} + 3 \zeta_{6} q^{18} + ( -7 + 7 \zeta_{6} ) q^{19} + ( -2 + \zeta_{6} ) q^{20} + ( -1 + 5 \zeta_{6} ) q^{21} -3 \zeta_{6} q^{22} + ( 2 - 4 \zeta_{6} ) q^{23} + ( -1 - \zeta_{6} ) q^{24} + ( -2 + 2 \zeta_{6} ) q^{25} + ( -3 + 4 \zeta_{6} ) q^{26} + ( 3 - 6 \zeta_{6} ) q^{27} + ( -1 - 2 \zeta_{6} ) q^{28} + ( -5 - 5 \zeta_{6} ) q^{29} + 3 q^{30} + ( 1 - \zeta_{6} ) q^{31} + q^{32} + ( -3 + 6 \zeta_{6} ) q^{33} -6 q^{34} + ( 4 + \zeta_{6} ) q^{35} + 3 \zeta_{6} q^{36} + ( -7 + 7 \zeta_{6} ) q^{38} + ( 7 - 5 \zeta_{6} ) q^{39} + ( -2 + \zeta_{6} ) q^{40} + ( 3 + 3 \zeta_{6} ) q^{41} + ( -1 + 5 \zeta_{6} ) q^{42} -\zeta_{6} q^{43} -3 \zeta_{6} q^{44} + ( -3 - 3 \zeta_{6} ) q^{45} + ( 2 - 4 \zeta_{6} ) q^{46} + ( 2 - \zeta_{6} ) q^{47} + ( -1 - \zeta_{6} ) q^{48} + ( -3 + 8 \zeta_{6} ) q^{49} + ( -2 + 2 \zeta_{6} ) q^{50} + ( 6 + 6 \zeta_{6} ) q^{51} + ( -3 + 4 \zeta_{6} ) q^{52} + ( -1 - \zeta_{6} ) q^{53} + ( 3 - 6 \zeta_{6} ) q^{54} + ( 3 + 3 \zeta_{6} ) q^{55} + ( -1 - 2 \zeta_{6} ) q^{56} + ( 14 - 7 \zeta_{6} ) q^{57} + ( -5 - 5 \zeta_{6} ) q^{58} + ( 6 - 12 \zeta_{6} ) q^{59} + 3 q^{60} + ( 1 + \zeta_{6} ) q^{61} + ( 1 - \zeta_{6} ) q^{62} + ( 6 - 9 \zeta_{6} ) q^{63} + q^{64} + ( 2 - 7 \zeta_{6} ) q^{65} + ( -3 + 6 \zeta_{6} ) q^{66} + ( -2 + \zeta_{6} ) q^{67} -6 q^{68} + ( -6 + 6 \zeta_{6} ) q^{69} + ( 4 + \zeta_{6} ) q^{70} + 9 \zeta_{6} q^{71} + 3 \zeta_{6} q^{72} + ( 13 - 13 \zeta_{6} ) q^{73} + ( 4 - 2 \zeta_{6} ) q^{75} + ( -7 + 7 \zeta_{6} ) q^{76} + ( -6 + 9 \zeta_{6} ) q^{77} + ( 7 - 5 \zeta_{6} ) q^{78} -\zeta_{6} q^{79} + ( -2 + \zeta_{6} ) q^{80} + ( -9 + 9 \zeta_{6} ) q^{81} + ( 3 + 3 \zeta_{6} ) q^{82} + ( 2 - 4 \zeta_{6} ) q^{83} + ( -1 + 5 \zeta_{6} ) q^{84} + ( 12 - 6 \zeta_{6} ) q^{85} -\zeta_{6} q^{86} + 15 \zeta_{6} q^{87} -3 \zeta_{6} q^{88} + ( -4 + 8 \zeta_{6} ) q^{89} + ( -3 - 3 \zeta_{6} ) q^{90} + ( 11 - 6 \zeta_{6} ) q^{91} + ( 2 - 4 \zeta_{6} ) q^{92} + ( -2 + \zeta_{6} ) q^{93} + ( 2 - \zeta_{6} ) q^{94} + ( 7 - 14 \zeta_{6} ) q^{95} + ( -1 - \zeta_{6} ) q^{96} -19 \zeta_{6} q^{97} + ( -3 + 8 \zeta_{6} ) q^{98} + ( 9 - 9 \zeta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} - 3q^{3} + 2q^{4} - 3q^{5} - 3q^{6} - 4q^{7} + 2q^{8} + 3q^{9} + O(q^{10})$$ $$2q + 2q^{2} - 3q^{3} + 2q^{4} - 3q^{5} - 3q^{6} - 4q^{7} + 2q^{8} + 3q^{9} - 3q^{10} - 3q^{11} - 3q^{12} - 2q^{13} - 4q^{14} + 6q^{15} + 2q^{16} - 12q^{17} + 3q^{18} - 7q^{19} - 3q^{20} + 3q^{21} - 3q^{22} - 3q^{24} - 2q^{25} - 2q^{26} - 4q^{28} - 15q^{29} + 6q^{30} + q^{31} + 2q^{32} - 12q^{34} + 9q^{35} + 3q^{36} - 7q^{38} + 9q^{39} - 3q^{40} + 9q^{41} + 3q^{42} - q^{43} - 3q^{44} - 9q^{45} + 3q^{47} - 3q^{48} + 2q^{49} - 2q^{50} + 18q^{51} - 2q^{52} - 3q^{53} + 9q^{55} - 4q^{56} + 21q^{57} - 15q^{58} + 6q^{60} + 3q^{61} + q^{62} + 3q^{63} + 2q^{64} - 3q^{65} - 3q^{67} - 12q^{68} - 6q^{69} + 9q^{70} + 9q^{71} + 3q^{72} + 13q^{73} + 6q^{75} - 7q^{76} - 3q^{77} + 9q^{78} - q^{79} - 3q^{80} - 9q^{81} + 9q^{82} + 3q^{84} + 18q^{85} - q^{86} + 15q^{87} - 3q^{88} - 9q^{90} + 16q^{91} - 3q^{93} + 3q^{94} - 3q^{96} - 19q^{97} + 2q^{98} + 9q^{99} + O(q^{100})$$

## 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 −1.50000 + 0.866025i −1.50000 0.866025i −2.00000 1.73205i 1.00000 1.50000 + 2.59808i −1.50000 + 0.866025i
257.1 1.00000 −1.50000 + 0.866025i 1.00000 −1.50000 0.866025i −1.50000 + 0.866025i −2.00000 + 1.73205i 1.00000 1.50000 2.59808i −1.50000 0.866025i
 $$n$$: 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.d yes 2
3.b odd 2 1 546.2.bi.b 2
7.d odd 6 1 546.2.bn.a yes 2
13.e even 6 1 546.2.bn.d yes 2
21.g even 6 1 546.2.bn.d yes 2
39.h odd 6 1 546.2.bn.a yes 2
91.l odd 6 1 546.2.bi.b 2
273.br even 6 1 inner 546.2.bi.d yes 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 3 T_{5} + 3$$ acting on $$S_{2}^{\mathrm{new}}(546, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{2}$$
$3$ $$3 + 3 T + T^{2}$$
$5$ $$3 + 3 T + T^{2}$$
$7$ $$7 + 4 T + T^{2}$$
$11$ $$9 + 3 T + T^{2}$$
$13$ $$13 + 2 T + T^{2}$$
$17$ $$( 6 + T )^{2}$$
$19$ $$49 + 7 T + T^{2}$$
$23$ $$12 + T^{2}$$
$29$ $$75 + 15 T + T^{2}$$
$31$ $$1 - T + T^{2}$$
$37$ $$T^{2}$$
$41$ $$27 - 9 T + T^{2}$$
$43$ $$1 + T + T^{2}$$
$47$ $$3 - 3 T + T^{2}$$
$53$ $$3 + 3 T + T^{2}$$
$59$ $$108 + T^{2}$$
$61$ $$3 - 3 T + T^{2}$$
$67$ $$3 + 3 T + T^{2}$$
$71$ $$81 - 9 T + T^{2}$$
$73$ $$169 - 13 T + T^{2}$$
$79$ $$1 + T + T^{2}$$
$83$ $$12 + T^{2}$$
$89$ $$48 + T^{2}$$
$97$ $$361 + 19 T + T^{2}$$