# Properties

 Label 547.3.b.a Level 547 Weight 3 Character orbit 547.b Self dual yes Analytic conductor 14.905 Analytic rank 0 Dimension 3 CM discriminant -547 Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$547$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 547.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$14.9046704605$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.14769.1 Defining polynomial: $$x^{3} - 33 x - 56$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 4 q^{4} + 9 q^{9} +O(q^{10})$$ $$q + 4 q^{4} + 9 q^{9} + ( 3 \beta_{1} + 2 \beta_{2} ) q^{11} + ( -4 \beta_{1} - \beta_{2} ) q^{13} + 16 q^{16} + ( \beta_{1} - 5 \beta_{2} ) q^{19} + 25 q^{25} + ( -3 \beta_{1} + 7 \beta_{2} ) q^{29} + 36 q^{36} + ( 12 \beta_{1} + 8 \beta_{2} ) q^{44} + ( 12 \beta_{1} - 5 \beta_{2} ) q^{47} + 49 q^{49} + ( -16 \beta_{1} - 4 \beta_{2} ) q^{52} + ( -9 \beta_{1} - 14 \beta_{2} ) q^{53} + 64 q^{64} + ( 4 \beta_{1} + 19 \beta_{2} ) q^{67} + ( -17 \beta_{1} + 10 \beta_{2} ) q^{73} + ( 4 \beta_{1} - 20 \beta_{2} ) q^{76} + 81 q^{81} + ( 29 \beta_{1} - \beta_{2} ) q^{97} + ( 27 \beta_{1} + 18 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + 12q^{4} + 27q^{9} + O(q^{10})$$ $$3q + 12q^{4} + 27q^{9} + 48q^{16} + 75q^{25} + 108q^{36} + 147q^{49} + 192q^{64} + 243q^{81} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 33 x - 56$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{2} - 3 \nu - 22$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2} + 3 \beta_{1} + 22$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/547\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
546.1
 −1.90719 −4.54841 6.45559
0 0 4.00000 0 0 0 0 9.00000 0
546.2 0 0 4.00000 0 0 0 0 9.00000 0
546.3 0 0 4.00000 0 0 0 0 9.00000 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
547.b odd 2 1 CM by $$\Q(\sqrt{-547})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 547.3.b.a 3
547.b odd 2 1 CM 547.3.b.a 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
547.3.b.a 3 1.a even 1 1 trivial
547.3.b.a 3 547.b odd 2 1 CM

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{3}^{\mathrm{new}}(547, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 2 T )^{3}( 1 + 2 T )^{3}$$
$3$ $$( 1 - 3 T )^{3}( 1 + 3 T )^{3}$$
$5$ $$( 1 - 5 T )^{3}( 1 + 5 T )^{3}$$
$7$ $$( 1 - 7 T )^{3}( 1 + 7 T )^{3}$$
$11$ $$1 - 474 T^{3} + 1771561 T^{6}$$
$13$ $$1 + 4358 T^{3} + 4826809 T^{6}$$
$17$ $$( 1 - 17 T )^{3}( 1 + 17 T )^{3}$$
$19$ $$1 + 5974 T^{3} + 47045881 T^{6}$$
$23$ $$( 1 - 23 T )^{3}( 1 + 23 T )^{3}$$
$29$ $$1 - 40026 T^{3} + 594823321 T^{6}$$
$31$ $$( 1 - 31 T )^{3}( 1 + 31 T )^{3}$$
$37$ $$( 1 - 37 T )^{3}( 1 + 37 T )^{3}$$
$41$ $$( 1 - 41 T )^{3}( 1 + 41 T )^{3}$$
$43$ $$( 1 - 43 T )^{3}( 1 + 43 T )^{3}$$
$47$ $$1 + 57102 T^{3} + 10779215329 T^{6}$$
$53$ $$1 - 289002 T^{3} + 22164361129 T^{6}$$
$59$ $$( 1 - 59 T )^{3}( 1 + 59 T )^{3}$$
$61$ $$( 1 - 61 T )^{3}( 1 + 61 T )^{3}$$
$67$ $$1 + 363382 T^{3} + 90458382169 T^{6}$$
$71$ $$( 1 - 71 T )^{3}( 1 + 71 T )^{3}$$
$73$ $$1 - 462962 T^{3} + 151334226289 T^{6}$$
$79$ $$( 1 - 79 T )^{3}( 1 + 79 T )^{3}$$
$83$ $$( 1 - 83 T )^{3}( 1 + 83 T )^{3}$$
$89$ $$( 1 - 89 T )^{3}( 1 + 89 T )^{3}$$
$97$ $$1 - 1265218 T^{3} + 832972004929 T^{6}$$