# Properties

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

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## Newspace parameters

 Level: $$N$$ $$=$$ $$546 = 2 \cdot 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 546.e (of order $$2$$, degree $$1$$, 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

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

## Character values

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

 $$n$$ $$157$$ $$365$$ $$379$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
545.1
 0.5 − 0.866025i 0.5 + 0.866025i
−1.00000 1.50000 0.866025i 1.00000 1.73205i −1.50000 + 0.866025i 2.50000 0.866025i −1.00000 1.50000 2.59808i 1.73205i
545.2 −1.00000 1.50000 + 0.866025i 1.00000 1.73205i −1.50000 0.866025i 2.50000 + 0.866025i −1.00000 1.50000 + 2.59808i 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.g even 2 1 inner

## Twists

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

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

## Hecke kernels

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

 $$T_{5}^{2} + 3$$ $$T_{17} - 3$$ $$T_{19} + 2$$ $$T_{71} + 9$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$3 - 3 T + T^{2}$$
$5$ $$3 + T^{2}$$
$7$ $$7 - 5 T + T^{2}$$
$11$ $$T^{2}$$
$13$ $$13 + 2 T + T^{2}$$
$17$ $$( -3 + T )^{2}$$
$19$ $$( 2 + T )^{2}$$
$23$ $$12 + T^{2}$$
$29$ $$48 + T^{2}$$
$31$ $$( -8 + T )^{2}$$
$37$ $$3 + T^{2}$$
$41$ $$T^{2}$$
$43$ $$( 11 + T )^{2}$$
$47$ $$147 + T^{2}$$
$53$ $$12 + T^{2}$$
$59$ $$108 + T^{2}$$
$61$ $$48 + T^{2}$$
$67$ $$192 + T^{2}$$
$71$ $$( 9 + T )^{2}$$
$73$ $$( -2 + T )^{2}$$
$79$ $$( -10 + T )^{2}$$
$83$ $$12 + T^{2}$$
$89$ $$12 + T^{2}$$
$97$ $$( 8 + T )^{2}$$
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