# Properties

 Label 546.2.bq.a Level $546$ Weight $2$ Character orbit 546.bq Analytic conductor $4.360$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$546 = 2 \cdot 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 546.bq (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: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ 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 $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$157$$ $$365$$ $$379$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-\zeta_{12}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
419.1
 −0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i
−0.866025 + 0.500000i 0.866025 + 1.50000i 0.500000 0.866025i 3.46410 −1.50000 0.866025i 0.500000 + 2.59808i 1.00000i −1.50000 + 2.59808i −3.00000 + 1.73205i
419.2 0.866025 0.500000i −0.866025 1.50000i 0.500000 0.866025i −3.46410 −1.50000 0.866025i 0.500000 + 2.59808i 1.00000i −1.50000 + 2.59808i −3.00000 + 1.73205i
503.1 −0.866025 0.500000i 0.866025 1.50000i 0.500000 + 0.866025i 3.46410 −1.50000 + 0.866025i 0.500000 2.59808i 1.00000i −1.50000 2.59808i −3.00000 1.73205i
503.2 0.866025 + 0.500000i −0.866025 + 1.50000i 0.500000 + 0.866025i −3.46410 −1.50000 + 0.866025i 0.500000 2.59808i 1.00000i −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
3.b odd 2 1 inner
91.n odd 6 1 inner
273.bn 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.bq.a 4
3.b odd 2 1 inner 546.2.bq.a 4
7.b odd 2 1 546.2.bq.b yes 4
13.c even 3 1 546.2.bq.b yes 4
21.c even 2 1 546.2.bq.b yes 4
39.i odd 6 1 546.2.bq.b yes 4
91.n odd 6 1 inner 546.2.bq.a 4
273.bn even 6 1 inner 546.2.bq.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.bq.a 4 1.a even 1 1 trivial
546.2.bq.a 4 3.b odd 2 1 inner
546.2.bq.a 4 91.n odd 6 1 inner
546.2.bq.a 4 273.bn even 6 1 inner
546.2.bq.b yes 4 7.b odd 2 1
546.2.bq.b yes 4 13.c even 3 1
546.2.bq.b yes 4 21.c even 2 1
546.2.bq.b yes 4 39.i odd 6 1

## Hecke kernels

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

 $$T_{5}^{2} - 12$$ $$T_{61}^{2} + 21 T_{61} + 147$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4}$$
$3$ $$9 + 3 T^{2} + T^{4}$$
$5$ $$( -12 + T^{2} )^{2}$$
$7$ $$( 7 - T + T^{2} )^{2}$$
$11$ $$T^{4}$$
$13$ $$( 13 + 7 T + T^{2} )^{2}$$
$17$ $$9 + 3 T^{2} + T^{4}$$
$19$ $$T^{4}$$
$23$ $$81 - 9 T^{2} + T^{4}$$
$29$ $$1296 - 36 T^{2} + T^{4}$$
$31$ $$( 3 + T^{2} )^{2}$$
$37$ $$( 16 + 4 T + T^{2} )^{2}$$
$41$ $$2304 + 48 T^{2} + T^{4}$$
$43$ $$( 1 + T + T^{2} )^{2}$$
$47$ $$( -12 + T^{2} )^{2}$$
$53$ $$( 81 + T^{2} )^{2}$$
$59$ $$9 + 3 T^{2} + T^{4}$$
$61$ $$( 147 + 21 T + T^{2} )^{2}$$
$67$ $$( 49 - 7 T + T^{2} )^{2}$$
$71$ $$50625 - 225 T^{2} + T^{4}$$
$73$ $$( 192 + T^{2} )^{2}$$
$79$ $$( 10 + T )^{4}$$
$83$ $$( -147 + T^{2} )^{2}$$
$89$ $$59049 + 243 T^{2} + T^{4}$$
$97$ $$( 300 - 30 T + T^{2} )^{2}$$