# Properties

 Label 273.2.p.d Level $273$ Weight $2$ Character orbit 273.p 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.p (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.17991597518$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(i, \sqrt{6})$$ Defining polynomial: $$x^{4} + 9$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/3$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$3 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$3 \beta_{3}$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
34.1
 −1.22474 − 1.22474i 1.22474 + 1.22474i −1.22474 + 1.22474i 1.22474 − 1.22474i
−1.22474 1.22474i 1.00000i 1.00000i 2.00000 2.00000i −1.22474 + 1.22474i 2.44949 1.00000i −1.22474 + 1.22474i −1.00000 −4.89898
34.2 1.22474 + 1.22474i 1.00000i 1.00000i 2.00000 2.00000i 1.22474 1.22474i −2.44949 1.00000i 1.22474 1.22474i −1.00000 4.89898
265.1 −1.22474 + 1.22474i 1.00000i 1.00000i 2.00000 + 2.00000i −1.22474 1.22474i 2.44949 + 1.00000i −1.22474 1.22474i −1.00000 −4.89898
265.2 1.22474 1.22474i 1.00000i 1.00000i 2.00000 + 2.00000i 1.22474 + 1.22474i −2.44949 + 1.00000i 1.22474 + 1.22474i −1.00000 4.89898
 $$n$$: 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.i even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.2.p.d yes 4
3.b odd 2 1 819.2.y.a 4
7.b odd 2 1 273.2.p.a 4
13.d odd 4 1 273.2.p.a 4
21.c even 2 1 819.2.y.d 4
39.f even 4 1 819.2.y.d 4
91.i even 4 1 inner 273.2.p.d yes 4
273.o odd 4 1 819.2.y.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.p.a 4 7.b odd 2 1
273.2.p.a 4 13.d odd 4 1
273.2.p.d yes 4 1.a even 1 1 trivial
273.2.p.d yes 4 91.i even 4 1 inner
819.2.y.a 4 3.b odd 2 1
819.2.y.a 4 273.o odd 4 1
819.2.y.d 4 21.c even 2 1
819.2.y.d 4 39.f even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(273, [\chi])$$:

 $$T_{2}^{4} + 9$$ $$T_{5}^{2} - 4 T_{5} + 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$9 + T^{4}$$
$3$ $$( 1 + T^{2} )^{2}$$
$5$ $$( 8 - 4 T + T^{2} )^{2}$$
$7$ $$49 - 10 T^{2} + T^{4}$$
$11$ $$16 + 32 T + 32 T^{2} - 8 T^{3} + T^{4}$$
$13$ $$( 13 - 4 T + T^{2} )^{2}$$
$17$ $$( 2 + T )^{4}$$
$19$ $$36 + 72 T + 72 T^{2} + 12 T^{3} + T^{4}$$
$23$ $$64 + 80 T^{2} + T^{4}$$
$29$ $$( -20 + 4 T + T^{2} )^{2}$$
$31$ $$36 + 72 T + 72 T^{2} + 12 T^{3} + T^{4}$$
$37$ $$900 + 360 T + 72 T^{2} - 12 T^{3} + T^{4}$$
$41$ $$( 32 + 8 T + T^{2} )^{2}$$
$43$ $$400 + 56 T^{2} + T^{4}$$
$47$ $$400 - 320 T + 128 T^{2} - 16 T^{3} + T^{4}$$
$53$ $$( -92 + 4 T + T^{2} )^{2}$$
$59$ $$16 - 32 T + 32 T^{2} + 8 T^{3} + T^{4}$$
$61$ $$( 100 + T^{2} )^{2}$$
$67$ $$11236 + 424 T + 8 T^{2} - 4 T^{3} + T^{4}$$
$71$ $$144 + T^{4}$$
$73$ $$900 + 360 T + 72 T^{2} - 12 T^{3} + T^{4}$$
$79$ $$( -20 - 4 T + T^{2} )^{2}$$
$83$ $$16 - 32 T + 32 T^{2} + 8 T^{3} + T^{4}$$
$89$ $$( 8 - 4 T + T^{2} )^{2}$$
$97$ $$900 + 360 T + 72 T^{2} - 12 T^{3} + T^{4}$$