# Properties

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

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$546 = 2 \cdot 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 546.i (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: $$\Z[a_1, a_2]$$ 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.

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

## Character values

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

 $$n$$ $$157$$ $$365$$ $$379$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
79.1
 0.5 − 0.866025i 0.5 + 0.866025i
−0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i 0 −1.00000 −2.50000 + 0.866025i 1.00000 −0.500000 + 0.866025i 0
235.1 −0.500000 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i 0 −1.00000 −2.50000 0.866025i 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
7.c 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.i.d 2
3.b odd 2 1 1638.2.j.i 2
7.c even 3 1 inner 546.2.i.d 2
7.c even 3 1 3822.2.a.u 1
7.d odd 6 1 3822.2.a.bf 1
21.h odd 6 1 1638.2.j.i 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.i.d 2 1.a even 1 1 trivial
546.2.i.d 2 7.c even 3 1 inner
1638.2.j.i 2 3.b odd 2 1
1638.2.j.i 2 21.h odd 6 1
3822.2.a.u 1 7.c even 3 1
3822.2.a.bf 1 7.d odd 6 1

## Hecke kernels

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

 $$T_{5}$$ $$T_{17}^{2} + 7 T_{17} + 49$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ $$1 - T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$7 + 5 T + T^{2}$$
$11$ $$25 - 5 T + T^{2}$$
$13$ $$( 1 + T )^{2}$$
$17$ $$49 + 7 T + T^{2}$$
$19$ $$49 + 7 T + T^{2}$$
$23$ $$4 + 2 T + T^{2}$$
$29$ $$( 9 + T )^{2}$$
$31$ $$T^{2}$$
$37$ $$16 + 4 T + T^{2}$$
$41$ $$( -4 + T )^{2}$$
$43$ $$( -2 + T )^{2}$$
$47$ $$9 - 3 T + T^{2}$$
$53$ $$1 + T + T^{2}$$
$59$ $$49 + 7 T + T^{2}$$
$61$ $$169 + 13 T + T^{2}$$
$67$ $$9 + 3 T + T^{2}$$
$71$ $$( -9 + T )^{2}$$
$73$ $$100 - 10 T + T^{2}$$
$79$ $$196 + 14 T + T^{2}$$
$83$ $$( -16 + T )^{2}$$
$89$ $$144 - 12 T + T^{2}$$
$97$ $$( -6 + T )^{2}$$
show more
show less