# Properties

 Label 546.2.g.b Level $546$ Weight $2$ Character orbit 546.g Analytic conductor $4.360$ Analytic rank $0$ Dimension $4$ 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.g (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.35983195036$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{2} + ( \beta_{2} + \beta_{3} ) q^{3} - q^{4} + ( -2 - \beta_{1} + \beta_{3} ) q^{5} + ( 1 + \beta_{1} ) q^{6} + ( 2 + \beta_{2} - \beta_{3} ) q^{7} + \beta_{2} q^{8} + ( -1 - \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{2} + ( \beta_{2} + \beta_{3} ) q^{3} - q^{4} + ( -2 - \beta_{1} + \beta_{3} ) q^{5} + ( 1 + \beta_{1} ) q^{6} + ( 2 + \beta_{2} - \beta_{3} ) q^{7} + \beta_{2} q^{8} + ( -1 - \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{9} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{10} + ( -\beta_{2} - \beta_{3} ) q^{12} -\beta_{2} q^{13} + ( 1 - \beta_{1} - 2 \beta_{2} ) q^{14} + ( 2 - \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{15} + q^{16} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{17} + ( -2 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{18} -6 \beta_{2} q^{19} + ( 2 + \beta_{1} - \beta_{3} ) q^{20} + ( -1 - \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{21} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{23} + ( -1 - \beta_{1} ) q^{24} + ( 3 + 2 \beta_{1} - 2 \beta_{3} ) q^{25} - q^{26} + ( 4 + \beta_{1} - 3 \beta_{2} ) q^{27} + ( -2 - \beta_{2} + \beta_{3} ) q^{28} + ( -3 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} ) q^{29} + ( -4 - \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{30} + ( -\beta_{1} - 6 \beta_{2} - \beta_{3} ) q^{31} -\beta_{2} q^{32} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{34} + ( -6 - 3 \beta_{1} + \beta_{3} ) q^{35} + ( 1 + \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{36} + ( -8 - \beta_{1} + \beta_{3} ) q^{37} -6 q^{38} + ( 1 + \beta_{1} ) q^{39} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{40} + ( 4 + \beta_{1} - \beta_{3} ) q^{41} + ( 4 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{42} + ( -8 - 2 \beta_{1} + 2 \beta_{3} ) q^{43} + ( 6 + 3 \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{45} + ( -2 - \beta_{1} + \beta_{3} ) q^{46} + ( 4 + 2 \beta_{1} - 2 \beta_{3} ) q^{47} + ( \beta_{2} + \beta_{3} ) q^{48} + ( 3 + 3 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} ) q^{49} + ( -2 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} ) q^{50} + ( 4 - 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{51} + \beta_{2} q^{52} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{53} + ( -3 - 4 \beta_{2} - \beta_{3} ) q^{54} + ( -1 + \beta_{1} + 2 \beta_{2} ) q^{56} + ( 6 + 6 \beta_{1} ) q^{57} + ( -2 - 3 \beta_{1} + 3 \beta_{3} ) q^{58} + ( 2 + 2 \beta_{1} - 2 \beta_{3} ) q^{59} + ( -2 + \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{60} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{61} + ( -6 - \beta_{1} + \beta_{3} ) q^{62} + ( -2 - 5 \beta_{1} - 3 \beta_{2} ) q^{63} - q^{64} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{65} + ( 2 + 4 \beta_{1} - 4 \beta_{3} ) q^{67} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{68} + ( 4 + \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{69} + ( \beta_{1} + 6 \beta_{2} + 3 \beta_{3} ) q^{70} + ( 2 \beta_{1} + 12 \beta_{2} + 2 \beta_{3} ) q^{71} + ( 2 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{72} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{73} + ( \beta_{1} + 8 \beta_{2} + \beta_{3} ) q^{74} + ( -4 + 2 \beta_{1} + 7 \beta_{2} + \beta_{3} ) q^{75} + 6 \beta_{2} q^{76} + ( -\beta_{2} - \beta_{3} ) q^{78} + ( -4 - 4 \beta_{1} + 4 \beta_{3} ) q^{79} + ( -2 - \beta_{1} + \beta_{3} ) q^{80} + ( 1 + 4 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{81} + ( -\beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{82} -14 q^{83} + ( 1 + \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{84} + 8 q^{85} + ( 2 \beta_{1} + 8 \beta_{2} + 2 \beta_{3} ) q^{86} + ( 8 - \beta_{1} + 6 \beta_{2} - 3 \beta_{3} ) q^{87} + ( -5 \beta_{1} + 5 \beta_{3} ) q^{89} + ( 4 + \beta_{1} - 6 \beta_{2} - 3 \beta_{3} ) q^{90} + ( 1 - \beta_{1} - 2 \beta_{2} ) q^{91} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{92} + ( 8 + 5 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{93} + ( -2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{94} + ( 6 \beta_{1} + 12 \beta_{2} + 6 \beta_{3} ) q^{95} + ( 1 + \beta_{1} ) q^{96} + ( -\beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{97} + ( 2 - 3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{3} - 4q^{4} - 4q^{5} + 2q^{6} + 6q^{7} + O(q^{10})$$ $$4q + 2q^{3} - 4q^{4} - 4q^{5} + 2q^{6} + 6q^{7} - 2q^{12} + 6q^{14} + 8q^{15} + 4q^{16} + 8q^{17} - 8q^{18} + 4q^{20} - 2q^{24} + 4q^{25} - 4q^{26} + 14q^{27} - 6q^{28} - 12q^{30} - 16q^{35} - 28q^{37} - 24q^{38} + 2q^{39} + 12q^{41} + 16q^{42} - 24q^{43} + 20q^{45} - 4q^{46} + 8q^{47} + 2q^{48} + 24q^{51} - 14q^{54} - 6q^{56} + 12q^{57} + 4q^{58} - 8q^{60} - 20q^{62} + 2q^{63} - 4q^{64} - 8q^{67} - 8q^{68} + 12q^{69} + 4q^{70} + 8q^{72} - 18q^{75} - 2q^{78} - 4q^{80} + 4q^{81} - 56q^{83} + 32q^{85} + 28q^{87} + 20q^{89} + 8q^{90} + 6q^{91} + 20q^{93} + 2q^{96} + 8q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3 x^{2} + 1$$:

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
209.1
 0.618034i − 1.61803i − 0.618034i 1.61803i
1.00000i −0.618034 + 1.61803i −1.00000 −3.23607 1.61803 + 0.618034i 2.61803 + 0.381966i 1.00000i −2.23607 2.00000i 3.23607i
209.2 1.00000i 1.61803 0.618034i −1.00000 1.23607 −0.618034 1.61803i 0.381966 + 2.61803i 1.00000i 2.23607 2.00000i 1.23607i
209.3 1.00000i −0.618034 1.61803i −1.00000 −3.23607 1.61803 0.618034i 2.61803 0.381966i 1.00000i −2.23607 + 2.00000i 3.23607i
209.4 1.00000i 1.61803 + 0.618034i −1.00000 1.23607 −0.618034 + 1.61803i 0.381966 2.61803i 1.00000i 2.23607 + 2.00000i 1.23607i
 $$n$$: 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
21.c 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.g.b yes 4
3.b odd 2 1 546.2.g.a 4
7.b odd 2 1 546.2.g.a 4
21.c even 2 1 inner 546.2.g.b yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.g.a 4 3.b odd 2 1
546.2.g.a 4 7.b odd 2 1
546.2.g.b yes 4 1.a even 1 1 trivial
546.2.g.b yes 4 21.c even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 2 T_{5} - 4$$ acting on $$S_{2}^{\mathrm{new}}(546, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{2}$$
$3$ $$9 - 6 T + 2 T^{2} - 2 T^{3} + T^{4}$$
$5$ $$( -4 + 2 T + T^{2} )^{2}$$
$7$ $$49 - 42 T + 18 T^{2} - 6 T^{3} + T^{4}$$
$11$ $$T^{4}$$
$13$ $$( 1 + T^{2} )^{2}$$
$17$ $$( -16 - 4 T + T^{2} )^{2}$$
$19$ $$( 36 + T^{2} )^{2}$$
$23$ $$16 + 12 T^{2} + T^{4}$$
$29$ $$1936 + 92 T^{2} + T^{4}$$
$31$ $$400 + 60 T^{2} + T^{4}$$
$37$ $$( 44 + 14 T + T^{2} )^{2}$$
$41$ $$( 4 - 6 T + T^{2} )^{2}$$
$43$ $$( 16 + 12 T + T^{2} )^{2}$$
$47$ $$( -16 - 4 T + T^{2} )^{2}$$
$53$ $$16 + 12 T^{2} + T^{4}$$
$59$ $$( -20 + T^{2} )^{2}$$
$61$ $$( 20 + T^{2} )^{2}$$
$67$ $$( -76 + 4 T + T^{2} )^{2}$$
$71$ $$6400 + 240 T^{2} + T^{4}$$
$73$ $$16 + 12 T^{2} + T^{4}$$
$79$ $$( -80 + T^{2} )^{2}$$
$83$ $$( 14 + T )^{4}$$
$89$ $$( -100 - 10 T + T^{2} )^{2}$$
$97$ $$16 + 28 T^{2} + T^{4}$$