# Properties

 Label 273.2.bl.b Level $273$ Weight $2$ Character orbit 273.bl Analytic conductor $2.180$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$273 = 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 273.bl (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.17991597518$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-7})$$ Defining polynomial: $$x^{4} - x^{3} - x^{2} - 2 x + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{3} ) q^{2} + q^{3} + ( \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{4} + ( -1 - \beta_{2} ) q^{5} + ( 1 - \beta_{3} ) q^{6} + ( -1 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{7} + ( 1 - 2 \beta_{2} ) q^{8} + q^{9} +O(q^{10})$$ $$q + ( 1 - \beta_{3} ) q^{2} + q^{3} + ( \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{4} + ( -1 - \beta_{2} ) q^{5} + ( 1 - \beta_{3} ) q^{6} + ( -1 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{7} + ( 1 - 2 \beta_{2} ) q^{8} + q^{9} + ( -2 + \beta_{1} + \beta_{3} ) q^{10} + ( -2 + 4 \beta_{2} ) q^{11} + ( \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{12} + ( -3 + 4 \beta_{2} ) q^{13} + ( -1 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{14} + ( -1 - \beta_{2} ) q^{15} + ( 1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{16} + ( -1 + \beta_{2} ) q^{17} + ( 1 - \beta_{3} ) q^{18} + ( -2 + 4 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{19} + ( -1 - \beta_{2} + 3 \beta_{3} ) q^{20} + ( -1 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{21} + ( 2 - 4 \beta_{1} + 2 \beta_{3} ) q^{22} + ( 2 - 4 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{23} + ( 1 - 2 \beta_{2} ) q^{24} -2 \beta_{2} q^{25} + ( 1 - 4 \beta_{1} + 3 \beta_{3} ) q^{26} + q^{27} + ( -4 + \beta_{1} - 3 \beta_{2} ) q^{28} + ( 7 - 7 \beta_{2} ) q^{29} + ( -2 + \beta_{1} + \beta_{3} ) q^{30} + ( -4 + \beta_{2} + 2 \beta_{3} ) q^{31} + ( 4 - \beta_{1} + 3 \beta_{2} ) q^{32} + ( -2 + 4 \beta_{2} ) q^{33} + ( -\beta_{1} + \beta_{3} ) q^{34} + ( 1 - 2 \beta_{1} - 3 \beta_{2} + 4 \beta_{3} ) q^{35} + ( \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{36} + ( -2 - \beta_{2} + 4 \beta_{3} ) q^{37} + ( -2 + 4 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{38} + ( -3 + 4 \beta_{2} ) q^{39} + ( -3 + 3 \beta_{2} ) q^{40} + ( -1 + 4 \beta_{1} + 3 \beta_{2} ) q^{41} + ( -1 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{42} + ( -2 + 4 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} ) q^{43} + ( 4 - 6 \beta_{1} - 2 \beta_{2} ) q^{44} + ( -1 - \beta_{2} ) q^{45} + ( 9 - 7 \beta_{1} + 2 \beta_{2} ) q^{46} + ( 1 + 2 \beta_{1} + 3 \beta_{2} ) q^{47} + ( 1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{48} -7 q^{49} + ( -2 + 2 \beta_{1} ) q^{50} + ( -1 + \beta_{2} ) q^{51} + ( 4 - 7 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} ) q^{52} + ( 4 - 8 \beta_{1} - 9 \beta_{2} + 4 \beta_{3} ) q^{53} + ( 1 - \beta_{3} ) q^{54} + ( 6 - 6 \beta_{2} ) q^{55} + ( -1 + 2 \beta_{1} + 2 \beta_{3} ) q^{56} + ( -2 + 4 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{57} + ( 7 \beta_{1} - 7 \beta_{3} ) q^{58} + ( -3 - 2 \beta_{1} - 5 \beta_{2} ) q^{59} + ( -1 - \beta_{2} + 3 \beta_{3} ) q^{60} + ( 2 + 4 \beta_{1} + 4 \beta_{3} ) q^{61} + ( -5 - 3 \beta_{1} + 2 \beta_{2} + 6 \beta_{3} ) q^{62} + ( -1 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{63} + ( 9 - 2 \beta_{1} - 2 \beta_{3} ) q^{64} + ( 7 - 5 \beta_{2} ) q^{65} + ( 2 - 4 \beta_{1} + 2 \beta_{3} ) q^{66} + ( -2 + 8 \beta_{1} + 4 \beta_{2} - 8 \beta_{3} ) q^{67} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{68} + ( 2 - 4 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{69} + ( -2 - 3 \beta_{1} + 4 \beta_{2} + 3 \beta_{3} ) q^{70} + ( 4 - 5 \beta_{2} + 6 \beta_{3} ) q^{71} + ( 1 - 2 \beta_{2} ) q^{72} + ( 10 - 5 \beta_{2} ) q^{73} + ( -7 - 3 \beta_{1} + 4 \beta_{2} + 6 \beta_{3} ) q^{74} -2 \beta_{2} q^{75} + ( -8 + 2 \beta_{1} - 6 \beta_{2} ) q^{76} + ( 2 - 4 \beta_{1} - 4 \beta_{3} ) q^{77} + ( 1 - 4 \beta_{1} + 3 \beta_{3} ) q^{78} + ( 7 - 2 \beta_{1} - 9 \beta_{2} + 4 \beta_{3} ) q^{79} + ( -2 + 3 \beta_{1} + 4 \beta_{2} - 3 \beta_{3} ) q^{80} + q^{81} + ( -6 + \beta_{1} + \beta_{3} ) q^{82} + ( 2 - 4 \beta_{2} ) q^{83} + ( -4 + \beta_{1} - 3 \beta_{2} ) q^{84} + ( 2 - \beta_{2} ) q^{85} + ( -5 + 3 \beta_{1} - 2 \beta_{2} ) q^{86} + ( 7 - 7 \beta_{2} ) q^{87} + 6 q^{88} + ( -18 + 9 \beta_{2} ) q^{89} + ( -2 + \beta_{1} + \beta_{3} ) q^{90} + ( 3 - 6 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{91} + ( 15 - 5 \beta_{1} - 5 \beta_{3} ) q^{92} + ( -4 + \beta_{2} + 2 \beta_{3} ) q^{93} + ( -\beta_{1} - \beta_{3} ) q^{94} + ( 2 - 4 \beta_{1} - 6 \beta_{2} + 8 \beta_{3} ) q^{95} + ( 4 - \beta_{1} + 3 \beta_{2} ) q^{96} + ( 6 - \beta_{2} - 4 \beta_{3} ) q^{97} + ( -7 + 7 \beta_{3} ) q^{98} + ( -2 + 4 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 3q^{2} + 4q^{3} + q^{4} - 6q^{5} + 3q^{6} + 4q^{9} + O(q^{10})$$ $$4q + 3q^{2} + 4q^{3} + q^{4} - 6q^{5} + 3q^{6} + 4q^{9} - 6q^{10} + q^{12} - 4q^{13} - 7q^{14} - 6q^{15} - q^{16} - 2q^{17} + 3q^{18} - 3q^{20} + 6q^{22} + 8q^{23} - 4q^{25} + 3q^{26} + 4q^{27} - 21q^{28} + 14q^{29} - 6q^{30} - 12q^{31} + 21q^{32} + q^{36} - 6q^{37} - 14q^{38} - 4q^{39} - 6q^{40} + 6q^{41} - 7q^{42} + 6q^{44} - 6q^{45} + 33q^{46} + 12q^{47} - q^{48} - 28q^{49} - 6q^{50} - 2q^{51} + 5q^{52} - 6q^{53} + 3q^{54} + 12q^{55} - 24q^{59} - 3q^{60} + 16q^{61} - 13q^{62} + 32q^{64} + 18q^{65} + 6q^{66} + q^{68} + 8q^{69} + 12q^{71} + 30q^{73} - 17q^{74} - 4q^{75} - 42q^{76} + 3q^{78} + 12q^{79} + 4q^{81} - 22q^{82} - 21q^{84} + 6q^{85} - 21q^{86} + 14q^{87} + 24q^{88} - 54q^{89} - 6q^{90} + 50q^{92} - 12q^{93} - 2q^{94} + 21q^{96} + 18q^{97} - 21q^{98} + O(q^{100})$$

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

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/273\mathbb{Z}\right)^\times$$.

 $$n$$ $$92$$ $$106$$ $$157$$ $$\chi(n)$$ $$1$$ $$\beta_{2}$$ $$-1 + \beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
88.1
 1.39564 − 0.228425i −0.895644 + 1.09445i 1.39564 + 0.228425i −0.895644 − 1.09445i
−0.395644 0.228425i 1.00000 −0.895644 1.55130i −1.50000 + 0.866025i −0.395644 0.228425i 2.64575i 1.73205i 1.00000 0.791288
88.2 1.89564 + 1.09445i 1.00000 1.39564 + 2.41733i −1.50000 + 0.866025i 1.89564 + 1.09445i 2.64575i 1.73205i 1.00000 −3.79129
121.1 −0.395644 + 0.228425i 1.00000 −0.895644 + 1.55130i −1.50000 0.866025i −0.395644 + 0.228425i 2.64575i 1.73205i 1.00000 0.791288
121.2 1.89564 1.09445i 1.00000 1.39564 2.41733i −1.50000 0.866025i 1.89564 1.09445i 2.64575i 1.73205i 1.00000 −3.79129
 $$n$$: 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.u even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.2.bl.b yes 4
3.b odd 2 1 819.2.do.d 4
7.c even 3 1 273.2.t.b 4
13.e even 6 1 273.2.t.b 4
21.h odd 6 1 819.2.bm.d 4
39.h odd 6 1 819.2.bm.d 4
91.u even 6 1 inner 273.2.bl.b yes 4
273.x odd 6 1 819.2.do.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.t.b 4 7.c even 3 1
273.2.t.b 4 13.e even 6 1
273.2.bl.b yes 4 1.a even 1 1 trivial
273.2.bl.b yes 4 91.u even 6 1 inner
819.2.bm.d 4 21.h odd 6 1
819.2.bm.d 4 39.h odd 6 1
819.2.do.d 4 3.b odd 2 1
819.2.do.d 4 273.x odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 3 T_{2}^{3} + 2 T_{2}^{2} + 3 T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(273, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 3 T + 2 T^{2} - 3 T^{3} + T^{4}$$
$3$ $$( -1 + T )^{4}$$
$5$ $$( 3 + 3 T + T^{2} )^{2}$$
$7$ $$( 7 + T^{2} )^{2}$$
$11$ $$( 12 + T^{2} )^{2}$$
$13$ $$( 13 + 2 T + T^{2} )^{2}$$
$17$ $$( 1 + T + T^{2} )^{2}$$
$19$ $$( 28 + T^{2} )^{2}$$
$23$ $$25 + 40 T + 69 T^{2} - 8 T^{3} + T^{4}$$
$29$ $$( 49 - 7 T + T^{2} )^{2}$$
$31$ $$25 + 60 T + 53 T^{2} + 12 T^{3} + T^{4}$$
$37$ $$625 - 150 T - 13 T^{2} + 6 T^{3} + T^{4}$$
$41$ $$625 + 150 T - 13 T^{2} - 6 T^{3} + T^{4}$$
$43$ $$441 + 21 T^{2} + T^{4}$$
$47$ $$25 - 60 T + 53 T^{2} - 12 T^{3} + T^{4}$$
$53$ $$5625 - 450 T + 111 T^{2} + 6 T^{3} + T^{4}$$
$59$ $$1681 + 984 T + 233 T^{2} + 24 T^{3} + T^{4}$$
$61$ $$( -68 - 8 T + T^{2} )^{2}$$
$67$ $$10000 + 248 T^{2} + T^{4}$$
$71$ $$2601 + 612 T - 3 T^{2} - 12 T^{3} + T^{4}$$
$73$ $$( 75 - 15 T + T^{2} )^{2}$$
$79$ $$225 - 180 T + 129 T^{2} - 12 T^{3} + T^{4}$$
$83$ $$( 12 + T^{2} )^{2}$$
$89$ $$( 243 + 27 T + T^{2} )^{2}$$
$97$ $$1 + 18 T + 107 T^{2} - 18 T^{3} + T^{4}$$