Properties

 Label 546.2.q.d Level $546$ Weight $2$ Character orbit 546.q 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.q (of order $$6$$, 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: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{6} ) q^{2} + ( 1 + \zeta_{6} ) q^{3} -\zeta_{6} q^{4} + ( 2 - 4 \zeta_{6} ) q^{5} + ( 2 - \zeta_{6} ) q^{6} + ( 2 + \zeta_{6} ) q^{7} - q^{8} + 3 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 1 - \zeta_{6} ) q^{2} + ( 1 + \zeta_{6} ) q^{3} -\zeta_{6} q^{4} + ( 2 - 4 \zeta_{6} ) q^{5} + ( 2 - \zeta_{6} ) q^{6} + ( 2 + \zeta_{6} ) q^{7} - q^{8} + 3 \zeta_{6} q^{9} + ( -2 - 2 \zeta_{6} ) q^{10} + ( -3 + 3 \zeta_{6} ) q^{11} + ( 1 - 2 \zeta_{6} ) q^{12} + ( 4 - \zeta_{6} ) q^{13} + ( 3 - 2 \zeta_{6} ) q^{14} + ( 6 - 6 \zeta_{6} ) q^{15} + ( -1 + \zeta_{6} ) q^{16} -3 \zeta_{6} q^{17} + 3 q^{18} -7 \zeta_{6} q^{19} + ( -4 + 2 \zeta_{6} ) q^{20} + ( 1 + 4 \zeta_{6} ) q^{21} + 3 \zeta_{6} q^{22} + ( 4 + 4 \zeta_{6} ) q^{23} + ( -1 - \zeta_{6} ) q^{24} -7 q^{25} + ( 3 - 4 \zeta_{6} ) q^{26} + ( -3 + 6 \zeta_{6} ) q^{27} + ( 1 - 3 \zeta_{6} ) q^{28} + ( -1 - \zeta_{6} ) q^{29} -6 \zeta_{6} q^{30} -4 q^{31} + \zeta_{6} q^{32} + ( -6 + 3 \zeta_{6} ) q^{33} -3 q^{34} + ( 8 - 10 \zeta_{6} ) q^{35} + ( 3 - 3 \zeta_{6} ) q^{36} + ( -4 - 4 \zeta_{6} ) q^{37} -7 q^{38} + ( 5 + 2 \zeta_{6} ) q^{39} + ( -2 + 4 \zeta_{6} ) q^{40} + ( -3 - 3 \zeta_{6} ) q^{41} + ( 5 - \zeta_{6} ) q^{42} + 8 \zeta_{6} q^{43} + 3 q^{44} + ( 12 - 6 \zeta_{6} ) q^{45} + ( 8 - 4 \zeta_{6} ) q^{46} + ( -5 + 10 \zeta_{6} ) q^{47} + ( -2 + \zeta_{6} ) q^{48} + ( 3 + 5 \zeta_{6} ) q^{49} + ( -7 + 7 \zeta_{6} ) q^{50} + ( 3 - 6 \zeta_{6} ) q^{51} + ( -1 - 3 \zeta_{6} ) q^{52} + ( -5 + 10 \zeta_{6} ) q^{53} + ( 3 + 3 \zeta_{6} ) q^{54} + ( 6 + 6 \zeta_{6} ) q^{55} + ( -2 - \zeta_{6} ) q^{56} + ( 7 - 14 \zeta_{6} ) q^{57} + ( -2 + \zeta_{6} ) q^{58} + ( 12 - 6 \zeta_{6} ) q^{59} -6 q^{60} + ( 6 - 3 \zeta_{6} ) q^{61} + ( -4 + 4 \zeta_{6} ) q^{62} + ( -3 + 9 \zeta_{6} ) q^{63} + q^{64} + ( 4 - 14 \zeta_{6} ) q^{65} + ( -3 + 6 \zeta_{6} ) q^{66} + ( -3 + 3 \zeta_{6} ) q^{68} + 12 \zeta_{6} q^{69} + ( -2 - 8 \zeta_{6} ) q^{70} + 6 \zeta_{6} q^{71} -3 \zeta_{6} q^{72} -4 q^{73} + ( -8 + 4 \zeta_{6} ) q^{74} + ( -7 - 7 \zeta_{6} ) q^{75} + ( -7 + 7 \zeta_{6} ) q^{76} + ( -9 + 6 \zeta_{6} ) q^{77} + ( 7 - 5 \zeta_{6} ) q^{78} -11 q^{79} + ( 2 + 2 \zeta_{6} ) q^{80} + ( -9 + 9 \zeta_{6} ) q^{81} + ( -6 + 3 \zeta_{6} ) q^{82} + ( -8 + 16 \zeta_{6} ) q^{83} + ( 4 - 5 \zeta_{6} ) q^{84} + ( -12 + 6 \zeta_{6} ) q^{85} + 8 q^{86} -3 \zeta_{6} q^{87} + ( 3 - 3 \zeta_{6} ) q^{88} + ( -5 - 5 \zeta_{6} ) q^{89} + ( 6 - 12 \zeta_{6} ) q^{90} + ( 9 + \zeta_{6} ) q^{91} + ( 4 - 8 \zeta_{6} ) q^{92} + ( -4 - 4 \zeta_{6} ) q^{93} + ( 5 + 5 \zeta_{6} ) q^{94} + ( -28 + 14 \zeta_{6} ) q^{95} + ( -1 + 2 \zeta_{6} ) q^{96} + 2 \zeta_{6} q^{97} + ( 8 - 3 \zeta_{6} ) q^{98} -9 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{2} + 3q^{3} - q^{4} + 3q^{6} + 5q^{7} - 2q^{8} + 3q^{9} + O(q^{10})$$ $$2q + q^{2} + 3q^{3} - q^{4} + 3q^{6} + 5q^{7} - 2q^{8} + 3q^{9} - 6q^{10} - 3q^{11} + 7q^{13} + 4q^{14} + 6q^{15} - q^{16} - 3q^{17} + 6q^{18} - 7q^{19} - 6q^{20} + 6q^{21} + 3q^{22} + 12q^{23} - 3q^{24} - 14q^{25} + 2q^{26} - q^{28} - 3q^{29} - 6q^{30} - 8q^{31} + q^{32} - 9q^{33} - 6q^{34} + 6q^{35} + 3q^{36} - 12q^{37} - 14q^{38} + 12q^{39} - 9q^{41} + 9q^{42} + 8q^{43} + 6q^{44} + 18q^{45} + 12q^{46} - 3q^{48} + 11q^{49} - 7q^{50} - 5q^{52} + 9q^{54} + 18q^{55} - 5q^{56} - 3q^{58} + 18q^{59} - 12q^{60} + 9q^{61} - 4q^{62} + 3q^{63} + 2q^{64} - 6q^{65} - 3q^{68} + 12q^{69} - 12q^{70} + 6q^{71} - 3q^{72} - 8q^{73} - 12q^{74} - 21q^{75} - 7q^{76} - 12q^{77} + 9q^{78} - 22q^{79} + 6q^{80} - 9q^{81} - 9q^{82} + 3q^{84} - 18q^{85} + 16q^{86} - 3q^{87} + 3q^{88} - 15q^{89} + 19q^{91} - 12q^{93} + 15q^{94} - 42q^{95} + 2q^{97} + 13q^{98} - 18q^{99} + O(q^{100})$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
251.1
 0.5 + 0.866025i 0.5 − 0.866025i
0.500000 0.866025i 1.50000 + 0.866025i −0.500000 0.866025i 3.46410i 1.50000 0.866025i 2.50000 + 0.866025i −1.00000 1.50000 + 2.59808i −3.00000 1.73205i
335.1 0.500000 + 0.866025i 1.50000 0.866025i −0.500000 + 0.866025i 3.46410i 1.50000 + 0.866025i 2.50000 0.866025i −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.u 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.q.d yes 2
3.b odd 2 1 546.2.q.b yes 2
7.b odd 2 1 546.2.q.c yes 2
13.e even 6 1 546.2.q.a 2
21.c even 2 1 546.2.q.a 2
39.h odd 6 1 546.2.q.c yes 2
91.t odd 6 1 546.2.q.b yes 2
273.u even 6 1 inner 546.2.q.d yes 2

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(546, [\chi])$$:

 $$T_{5}^{2} + 12$$ $$T_{11}^{2} + 3 T_{11} + 9$$ $$T_{17}^{2} + 3 T_{17} + 9$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2}$$
$3$ $$3 - 3 T + T^{2}$$
$5$ $$12 + T^{2}$$
$7$ $$7 - 5 T + T^{2}$$
$11$ $$9 + 3 T + T^{2}$$
$13$ $$13 - 7 T + T^{2}$$
$17$ $$9 + 3 T + T^{2}$$
$19$ $$49 + 7 T + T^{2}$$
$23$ $$48 - 12 T + T^{2}$$
$29$ $$3 + 3 T + T^{2}$$
$31$ $$( 4 + T )^{2}$$
$37$ $$48 + 12 T + T^{2}$$
$41$ $$27 + 9 T + T^{2}$$
$43$ $$64 - 8 T + T^{2}$$
$47$ $$75 + T^{2}$$
$53$ $$75 + T^{2}$$
$59$ $$108 - 18 T + T^{2}$$
$61$ $$27 - 9 T + T^{2}$$
$67$ $$T^{2}$$
$71$ $$36 - 6 T + T^{2}$$
$73$ $$( 4 + T )^{2}$$
$79$ $$( 11 + T )^{2}$$
$83$ $$192 + T^{2}$$
$89$ $$75 + 15 T + T^{2}$$
$97$ $$4 - 2 T + T^{2}$$