# Properties

 Label 273.2.p.c 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: $$\Q(i, \sqrt{10})$$ Defining polynomial: $$x^{4} + 25$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{3} -2 \beta_{2} q^{4} + ( 1 - \beta_{2} + \beta_{3} ) q^{5} + ( -1 - \beta_{1} + \beta_{2} ) q^{7} - q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{3} -2 \beta_{2} q^{4} + ( 1 - \beta_{2} + \beta_{3} ) q^{5} + ( -1 - \beta_{1} + \beta_{2} ) q^{7} - q^{9} + ( 1 - \beta_{2} - 2 \beta_{3} ) q^{11} -2 q^{12} + ( -2 - 2 \beta_{2} + \beta_{3} ) q^{13} + ( -1 + \beta_{1} - \beta_{2} ) q^{15} -4 q^{16} + ( 2 - \beta_{1} + \beta_{3} ) q^{17} + ( -2 + 2 \beta_{2} - \beta_{3} ) q^{19} + ( -2 + 2 \beta_{1} - 2 \beta_{2} ) q^{20} + ( 1 + \beta_{2} + \beta_{3} ) q^{21} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{23} + ( 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{25} + \beta_{2} q^{27} + ( 2 + 2 \beta_{2} + 2 \beta_{3} ) q^{28} + ( 5 - \beta_{1} + \beta_{3} ) q^{29} + ( 4 - 4 \beta_{2} - \beta_{3} ) q^{31} + ( -1 - 2 \beta_{1} - \beta_{2} ) q^{33} + ( 5 - 2 \beta_{1} + 2 \beta_{2} ) q^{35} + 2 \beta_{2} q^{36} + ( -2 + 2 \beta_{2} - 4 \beta_{3} ) q^{37} + ( -2 + \beta_{1} + 2 \beta_{2} ) q^{39} + ( 4 - 4 \beta_{2} - 2 \beta_{3} ) q^{41} + ( 2 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{43} + ( -2 - 4 \beta_{1} - 2 \beta_{2} ) q^{44} + ( -1 + \beta_{2} - \beta_{3} ) q^{45} + ( 5 + \beta_{1} + 5 \beta_{2} ) q^{47} + 4 \beta_{2} q^{48} + ( 2 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} ) q^{49} + ( \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{51} + ( -4 + 2 \beta_{1} + 4 \beta_{2} ) q^{52} + ( -1 - \beta_{1} + \beta_{3} ) q^{53} + ( -\beta_{1} + 8 \beta_{2} - \beta_{3} ) q^{55} + ( 2 - \beta_{1} + 2 \beta_{2} ) q^{57} + ( 2 + 4 \beta_{1} + 2 \beta_{2} ) q^{59} + ( -2 + 2 \beta_{2} - 2 \beta_{3} ) q^{60} + ( -\beta_{1} - \beta_{3} ) q^{61} + ( 1 + \beta_{1} - \beta_{2} ) q^{63} + 8 \beta_{2} q^{64} + ( -4 + 3 \beta_{1} - 5 \beta_{2} - \beta_{3} ) q^{65} -4 \beta_{1} q^{67} + ( 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{68} + ( -1 - \beta_{1} + \beta_{3} ) q^{69} + ( 6 + 6 \beta_{2} ) q^{71} + ( -4 - \beta_{1} - 4 \beta_{2} ) q^{73} + ( -2 + 2 \beta_{1} - 2 \beta_{3} ) q^{75} + ( 4 - 2 \beta_{1} + 4 \beta_{2} ) q^{76} + ( -10 + \beta_{1} + 2 \beta_{2} + 3 \beta_{3} ) q^{77} + ( -5 - 2 \beta_{1} + 2 \beta_{3} ) q^{79} + ( -4 + 4 \beta_{2} - 4 \beta_{3} ) q^{80} + q^{81} + ( 5 - 5 \beta_{2} - \beta_{3} ) q^{83} + ( 2 + 2 \beta_{1} - 2 \beta_{2} ) q^{84} + ( 7 - 7 \beta_{2} + 4 \beta_{3} ) q^{85} + ( \beta_{1} - 5 \beta_{2} + \beta_{3} ) q^{87} + ( 7 - \beta_{1} + 7 \beta_{2} ) q^{89} + ( 9 + \beta_{1} + \beta_{3} ) q^{91} + ( -2 - 2 \beta_{1} + 2 \beta_{3} ) q^{92} + ( -4 - \beta_{1} - 4 \beta_{2} ) q^{93} + ( -3 \beta_{1} + 9 \beta_{2} - 3 \beta_{3} ) q^{95} + ( 2 - 2 \beta_{2} + \beta_{3} ) q^{97} + ( -1 + \beta_{2} + 2 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{5} - 4q^{7} - 4q^{9} + O(q^{10})$$ $$4q + 4q^{5} - 4q^{7} - 4q^{9} + 4q^{11} - 8q^{12} - 8q^{13} - 4q^{15} - 16q^{16} + 8q^{17} - 8q^{19} - 8q^{20} + 4q^{21} + 8q^{28} + 20q^{29} + 16q^{31} - 4q^{33} + 20q^{35} - 8q^{37} - 8q^{39} + 16q^{41} - 8q^{44} - 4q^{45} + 20q^{47} - 16q^{52} - 4q^{53} + 8q^{57} + 8q^{59} - 8q^{60} + 4q^{63} - 16q^{65} - 4q^{69} + 24q^{71} - 16q^{73} - 8q^{75} + 16q^{76} - 40q^{77} - 20q^{79} - 16q^{80} + 4q^{81} + 20q^{83} + 8q^{84} + 28q^{85} + 28q^{89} + 36q^{91} - 8q^{92} - 16q^{93} + 8q^{97} - 4q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 25$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/5$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$5 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$5 \beta_{3}$$

## 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
 1.58114 + 1.58114i −1.58114 − 1.58114i 1.58114 − 1.58114i −1.58114 + 1.58114i
0 1.00000i 2.00000i −0.581139 + 0.581139i 0 −2.58114 0.581139i 0 −1.00000 0
34.2 0 1.00000i 2.00000i 2.58114 2.58114i 0 0.581139 + 2.58114i 0 −1.00000 0
265.1 0 1.00000i 2.00000i −0.581139 0.581139i 0 −2.58114 + 0.581139i 0 −1.00000 0
265.2 0 1.00000i 2.00000i 2.58114 + 2.58114i 0 0.581139 2.58114i 0 −1.00000 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
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.c yes 4
3.b odd 2 1 819.2.y.b 4
7.b odd 2 1 273.2.p.b 4
13.d odd 4 1 273.2.p.b 4
21.c even 2 1 819.2.y.c 4
39.f even 4 1 819.2.y.c 4
91.i even 4 1 inner 273.2.p.c yes 4
273.o odd 4 1 819.2.y.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.p.b 4 7.b odd 2 1
273.2.p.b 4 13.d odd 4 1
273.2.p.c yes 4 1.a even 1 1 trivial
273.2.p.c yes 4 91.i even 4 1 inner
819.2.y.b 4 3.b odd 2 1
819.2.y.b 4 273.o odd 4 1
819.2.y.c 4 21.c even 2 1
819.2.y.c 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 $$S_{2}^{\mathrm{new}}(273, [\chi])$$:

 $$T_{2}$$ $$T_{5}^{4} - 4 T_{5}^{3} + 8 T_{5}^{2} + 12 T_{5} + 9$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 1 + T^{2} )^{2}$$
$5$ $$9 + 12 T + 8 T^{2} - 4 T^{3} + T^{4}$$
$7$ $$49 + 28 T + 8 T^{2} + 4 T^{3} + T^{4}$$
$11$ $$324 + 72 T + 8 T^{2} - 4 T^{3} + T^{4}$$
$13$ $$169 + 104 T + 32 T^{2} + 8 T^{3} + T^{4}$$
$17$ $$( -6 - 4 T + T^{2} )^{2}$$
$19$ $$9 + 24 T + 32 T^{2} + 8 T^{3} + T^{4}$$
$23$ $$81 + 22 T^{2} + T^{4}$$
$29$ $$( 15 - 10 T + T^{2} )^{2}$$
$31$ $$729 - 432 T + 128 T^{2} - 16 T^{3} + T^{4}$$
$37$ $$5184 - 576 T + 32 T^{2} + 8 T^{3} + T^{4}$$
$41$ $$144 - 192 T + 128 T^{2} - 16 T^{3} + T^{4}$$
$43$ $$1521 + 82 T^{2} + T^{4}$$
$47$ $$2025 - 900 T + 200 T^{2} - 20 T^{3} + T^{4}$$
$53$ $$( -9 + 2 T + T^{2} )^{2}$$
$59$ $$5184 + 576 T + 32 T^{2} - 8 T^{3} + T^{4}$$
$61$ $$( 10 + T^{2} )^{2}$$
$67$ $$6400 + T^{4}$$
$71$ $$( 72 - 12 T + T^{2} )^{2}$$
$73$ $$729 + 432 T + 128 T^{2} + 16 T^{3} + T^{4}$$
$79$ $$( -15 + 10 T + T^{2} )^{2}$$
$83$ $$2025 - 900 T + 200 T^{2} - 20 T^{3} + T^{4}$$
$89$ $$8649 - 2604 T + 392 T^{2} - 28 T^{3} + T^{4}$$
$97$ $$9 - 24 T + 32 T^{2} - 8 T^{3} + T^{4}$$