# Properties

 Label 546.2.q.h Level $546$ Weight $2$ Character orbit 546.q Analytic conductor $4.360$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-11})$$ Defining polynomial: $$x^{4} - x^{3} - 2 x^{2} - 3 x + 9$$ 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + ( 1 - \beta_{1} + \beta_{3} ) q^{3} + ( -1 + \beta_{2} ) q^{4} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{5} + ( \beta_{2} + \beta_{3} ) q^{6} + ( 3 - \beta_{2} ) q^{7} - q^{8} + ( -2 - \beta_{1} + \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{2} + ( 1 - \beta_{1} + \beta_{3} ) q^{3} + ( -1 + \beta_{2} ) q^{4} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{5} + ( \beta_{2} + \beta_{3} ) q^{6} + ( 3 - \beta_{2} ) q^{7} - q^{8} + ( -2 - \beta_{1} + \beta_{3} ) q^{9} + ( 1 + \beta_{3} ) q^{10} + ( 1 - 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{11} + ( -1 + \beta_{1} + \beta_{2} ) q^{12} + ( -3 - \beta_{2} ) q^{13} + ( 1 + 2 \beta_{2} ) q^{14} + ( -2 - \beta_{1} - \beta_{2} ) q^{15} -\beta_{2} q^{16} + ( -1 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{17} + ( -2 \beta_{2} + \beta_{3} ) q^{18} + ( -1 + \beta_{1} - 2 \beta_{3} ) q^{19} + ( \beta_{1} + \beta_{2} ) q^{20} + ( 3 - 3 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{21} + ( -1 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{22} + ( 3 - 2 \beta_{2} - \beta_{3} ) q^{23} + ( -1 + \beta_{1} - \beta_{3} ) q^{24} + ( 2 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{25} + ( 1 - 4 \beta_{2} ) q^{26} + ( -5 + 2 \beta_{1} - 2 \beta_{3} ) q^{27} + ( -2 + 3 \beta_{2} ) q^{28} + ( 1 - \beta_{2} - \beta_{3} ) q^{29} + ( 1 - \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{30} + ( -2 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{31} + ( 1 - \beta_{2} ) q^{32} + ( -5 - \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{33} + ( -2 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{34} + ( 2 - 3 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} ) q^{35} + ( 2 + \beta_{1} - 2 \beta_{2} ) q^{36} + ( -2 + 4 \beta_{2} + 6 \beta_{3} ) q^{37} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{38} + ( -3 + 3 \beta_{1} - \beta_{2} - 4 \beta_{3} ) q^{39} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{40} + ( 9 - 5 \beta_{2} - \beta_{3} ) q^{41} + ( 1 - \beta_{1} + 2 \beta_{2} + 3 \beta_{3} ) q^{42} + ( 4 - 4 \beta_{2} ) q^{43} + ( -2 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{44} + ( -5 + 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} ) q^{45} + ( 2 - \beta_{1} + \beta_{2} ) q^{46} + ( 2 - \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{47} + ( -\beta_{2} - \beta_{3} ) q^{48} + ( 8 - 5 \beta_{2} ) q^{49} + ( 1 - 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{50} + ( -4 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{51} + ( 4 - 3 \beta_{2} ) q^{52} + ( -6 + 5 \beta_{1} + 7 \beta_{2} - 5 \beta_{3} ) q^{53} + ( -5 \beta_{2} - 2 \beta_{3} ) q^{54} + ( -4 + 2 \beta_{2} ) q^{55} + ( -3 + \beta_{2} ) q^{56} + ( 2 + \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{57} + ( 1 - \beta_{1} ) q^{58} + ( 4 + \beta_{1} + 5 \beta_{2} ) q^{59} + ( 3 - 2 \beta_{2} + \beta_{3} ) q^{60} + ( -3 - 3 \beta_{2} ) q^{61} + ( -2 + 4 \beta_{1} - 2 \beta_{3} ) q^{62} + ( -6 - 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{63} + q^{64} + ( -4 + 3 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} ) q^{65} + ( -4 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{66} + ( 2 - 4 \beta_{2} - 6 \beta_{3} ) q^{67} + ( -1 + 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{68} + ( 3 - 3 \beta_{1} + \beta_{2} + \beta_{3} ) q^{69} + ( 3 - \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{70} + ( -6 - 3 \beta_{1} + 9 \beta_{2} + 6 \beta_{3} ) q^{71} + ( 2 + \beta_{1} - \beta_{3} ) q^{72} + ( 6 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{73} + ( -4 + 6 \beta_{1} + 2 \beta_{2} ) q^{74} + ( -1 - 2 \beta_{1} + 5 \beta_{2} + \beta_{3} ) q^{75} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{76} + ( 4 - 5 \beta_{1} + \beta_{2} + \beta_{3} ) q^{77} + ( 1 - \beta_{1} - 4 \beta_{2} - 3 \beta_{3} ) q^{78} + ( 12 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{79} + ( -1 - \beta_{3} ) q^{80} + ( 1 + 5 \beta_{1} - 5 \beta_{3} ) q^{81} + ( 5 - \beta_{1} + 4 \beta_{2} ) q^{82} + ( -2 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{83} + ( -2 + 2 \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{84} + ( -2 - 2 \beta_{2} ) q^{85} + 4 q^{86} + ( 1 - \beta_{1} + 2 \beta_{2} ) q^{87} + ( -1 + 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{88} + ( -1 - 3 \beta_{2} - 7 \beta_{3} ) q^{89} + ( -2 - \beta_{1} - 3 \beta_{2} - 2 \beta_{3} ) q^{90} + ( -10 + \beta_{2} ) q^{91} + ( -1 - \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{92} + ( 4 + 2 \beta_{1} - 10 \beta_{2} ) q^{93} + ( 3 - \beta_{2} + \beta_{3} ) q^{94} + ( 2 + 2 \beta_{1} + 4 \beta_{2} ) q^{95} + ( 1 - \beta_{1} - \beta_{2} ) q^{96} + ( -6 + 2 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{97} + ( 5 + 3 \beta_{2} ) q^{98} + ( -8 + 5 \beta_{1} + \beta_{2} - \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{2} + 2q^{3} - 2q^{4} + q^{6} + 10q^{7} - 4q^{8} - 10q^{9} + O(q^{10})$$ $$4q + 2q^{2} + 2q^{3} - 2q^{4} + q^{6} + 10q^{7} - 4q^{8} - 10q^{9} + 3q^{10} + 3q^{11} - q^{12} - 14q^{13} + 8q^{14} - 11q^{15} - 2q^{16} - 3q^{17} - 5q^{18} - q^{19} + 3q^{20} + 5q^{21} - 3q^{22} + 9q^{23} - 2q^{24} + 6q^{25} - 4q^{26} - 16q^{27} - 2q^{28} + 3q^{29} - 4q^{30} - 4q^{31} + 2q^{32} - 15q^{33} - 6q^{34} - 3q^{35} + 5q^{36} - 6q^{37} - 2q^{38} - 7q^{39} + 27q^{41} + 4q^{42} + 8q^{43} - 6q^{44} - 11q^{45} + 9q^{46} - q^{48} + 22q^{49} + 3q^{50} - 18q^{51} + 10q^{52} - 8q^{54} - 12q^{55} - 10q^{56} + 16q^{57} + 3q^{58} + 27q^{59} + 7q^{60} - 18q^{61} - 2q^{62} - 25q^{63} + 4q^{64} - 3q^{65} - 18q^{66} + 6q^{67} - 3q^{68} + 10q^{69} + 6q^{70} - 15q^{71} + 10q^{72} + 20q^{73} - 6q^{74} + 3q^{75} - q^{76} + 12q^{77} - 2q^{78} + 50q^{79} - 3q^{80} + 14q^{81} + 27q^{82} - q^{84} - 12q^{85} + 16q^{86} + 7q^{87} - 3q^{88} - 3q^{89} - 13q^{90} - 38q^{91} - 2q^{93} + 9q^{94} + 18q^{95} + q^{96} - 10q^{97} + 26q^{98} - 24q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 2 x^{2} - 3 x + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 2 \nu^{2} - 2 \nu - 3$$$$)/6$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + 2 \nu + 3$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 3 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$-2 \beta_{3} + 2 \beta_{1} + 3$$

## Character values

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

 $$n$$ $$157$$ $$365$$ $$379$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1 - \beta_{2}$$

## 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
 −1.18614 + 1.26217i 1.68614 − 0.396143i 1.68614 + 0.396143i −1.18614 − 1.26217i
0.500000 0.866025i 0.500000 1.65831i −0.500000 0.866025i 0.792287i −1.18614 1.26217i 2.50000 + 0.866025i −1.00000 −2.50000 1.65831i −0.686141 0.396143i
251.2 0.500000 0.866025i 0.500000 + 1.65831i −0.500000 0.866025i 2.52434i 1.68614 + 0.396143i 2.50000 + 0.866025i −1.00000 −2.50000 + 1.65831i 2.18614 + 1.26217i
335.1 0.500000 + 0.866025i 0.500000 1.65831i −0.500000 + 0.866025i 2.52434i 1.68614 0.396143i 2.50000 0.866025i −1.00000 −2.50000 1.65831i 2.18614 1.26217i
335.2 0.500000 + 0.866025i 0.500000 + 1.65831i −0.500000 + 0.866025i 0.792287i −1.18614 + 1.26217i 2.50000 0.866025i −1.00000 −2.50000 + 1.65831i −0.686141 + 0.396143i
 $$n$$: 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.h yes 4
3.b odd 2 1 546.2.q.f yes 4
7.b odd 2 1 546.2.q.g yes 4
13.e even 6 1 546.2.q.e 4
21.c even 2 1 546.2.q.e 4
39.h odd 6 1 546.2.q.g yes 4
91.t odd 6 1 546.2.q.f yes 4
273.u even 6 1 inner 546.2.q.h yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.q.e 4 13.e even 6 1
546.2.q.e 4 21.c even 2 1
546.2.q.f yes 4 3.b odd 2 1
546.2.q.f yes 4 91.t odd 6 1
546.2.q.g yes 4 7.b odd 2 1
546.2.q.g yes 4 39.h odd 6 1
546.2.q.h yes 4 1.a even 1 1 trivial
546.2.q.h yes 4 273.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 $$S_{2}^{\mathrm{new}}(546, [\chi])$$:

 $$T_{5}^{4} + 7 T_{5}^{2} + 4$$ $$T_{11}^{4} - 3 T_{11}^{3} + 15 T_{11}^{2} + 18 T_{11} + 36$$ $$T_{17}^{4} + 3 T_{17}^{3} + 15 T_{17}^{2} - 18 T_{17} + 36$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T + T^{2} )^{2}$$
$3$ $$( 3 - T + T^{2} )^{2}$$
$5$ $$4 + 7 T^{2} + T^{4}$$
$7$ $$( 7 - 5 T + T^{2} )^{2}$$
$11$ $$36 + 18 T + 15 T^{2} - 3 T^{3} + T^{4}$$
$13$ $$( 13 + 7 T + T^{2} )^{2}$$
$17$ $$36 - 18 T + 15 T^{2} + 3 T^{3} + T^{4}$$
$19$ $$64 - 8 T + 9 T^{2} + T^{3} + T^{4}$$
$23$ $$16 - 36 T + 31 T^{2} - 9 T^{3} + T^{4}$$
$29$ $$4 + 6 T + T^{2} - 3 T^{3} + T^{4}$$
$31$ $$( -32 + 2 T + T^{2} )^{2}$$
$37$ $$9216 - 576 T - 84 T^{2} + 6 T^{3} + T^{4}$$
$41$ $$3364 - 1566 T + 301 T^{2} - 27 T^{3} + T^{4}$$
$43$ $$( 16 - 4 T + T^{2} )^{2}$$
$47$ $$16 + 19 T^{2} + T^{4}$$
$53$ $$1024 + 211 T^{2} + T^{4}$$
$59$ $$3364 - 1566 T + 301 T^{2} - 27 T^{3} + T^{4}$$
$61$ $$( 27 + 9 T + T^{2} )^{2}$$
$67$ $$9216 + 576 T - 84 T^{2} - 6 T^{3} + T^{4}$$
$71$ $$324 - 270 T + 243 T^{2} + 15 T^{3} + T^{4}$$
$73$ $$( -8 - 10 T + T^{2} )^{2}$$
$79$ $$( 148 - 25 T + T^{2} )^{2}$$
$83$ $$64 + 28 T^{2} + T^{4}$$
$89$ $$17956 - 402 T - 131 T^{2} + 3 T^{3} + T^{4}$$
$97$ $$64 - 80 T + 108 T^{2} + 10 T^{3} + T^{4}$$