# Properties

 Label 546.2.j.a Level $546$ Weight $2$ Character orbit 546.j 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.j (of order $$3$$, 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
289.1
 0.5 − 0.866025i 0.5 + 0.866025i
1.00000 0.500000 0.866025i 1.00000 0 0.500000 0.866025i 0.500000 + 2.59808i 1.00000 −0.500000 0.866025i 0
529.1 1.00000 0.500000 + 0.866025i 1.00000 0 0.500000 + 0.866025i 0.500000 2.59808i 1.00000 −0.500000 + 0.866025i 0
 $$n$$: 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
91.h even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 546.2.j.a 2
3.b odd 2 1 1638.2.m.a 2
7.c even 3 1 546.2.k.a yes 2
13.c even 3 1 546.2.k.a yes 2
21.h odd 6 1 1638.2.p.d 2
39.i odd 6 1 1638.2.p.d 2
91.h even 3 1 inner 546.2.j.a 2
273.s odd 6 1 1638.2.m.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.j.a 2 1.a even 1 1 trivial
546.2.j.a 2 91.h even 3 1 inner
546.2.k.a yes 2 7.c even 3 1
546.2.k.a yes 2 13.c even 3 1
1638.2.m.a 2 3.b odd 2 1
1638.2.m.a 2 273.s odd 6 1
1638.2.p.d 2 21.h odd 6 1
1638.2.p.d 2 39.i odd 6 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{2}$$
$3$ $$1 - T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$7 - T + T^{2}$$
$11$ $$4 - 2 T + T^{2}$$
$13$ $$13 - 7 T + T^{2}$$
$17$ $$( -4 + T )^{2}$$
$19$ $$1 + T + T^{2}$$
$23$ $$( 4 + T )^{2}$$
$29$ $$36 + 6 T + T^{2}$$
$31$ $$T^{2}$$
$37$ $$( 11 + T )^{2}$$
$41$ $$4 - 2 T + T^{2}$$
$43$ $$1 - T + T^{2}$$
$47$ $$T^{2}$$
$53$ $$16 + 4 T + T^{2}$$
$59$ $$( -4 + T )^{2}$$
$61$ $$1 + T + T^{2}$$
$67$ $$144 + 12 T + T^{2}$$
$71$ $$36 + 6 T + T^{2}$$
$73$ $$49 - 7 T + T^{2}$$
$79$ $$64 + 8 T + T^{2}$$
$83$ $$( 14 + T )^{2}$$
$89$ $$( -6 + T )^{2}$$
$97$ $$81 + 9 T + T^{2}$$