Properties

 Label 273.2.ba.b Level $273$ Weight $2$ Character orbit 273.ba Analytic conductor $2.180$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

Related objects

Newspace parameters

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

Newform invariants

 Self dual: no Analytic conductor: $$2.17991597518$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 + \zeta_{6} ) q^{3} + ( 2 - 2 \zeta_{6} ) q^{4} + ( 2 - 3 \zeta_{6} ) q^{7} + 3 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 1 + \zeta_{6} ) q^{3} + ( 2 - 2 \zeta_{6} ) q^{4} + ( 2 - 3 \zeta_{6} ) q^{7} + 3 \zeta_{6} q^{9} + ( 4 - 2 \zeta_{6} ) q^{12} + ( -3 - \zeta_{6} ) q^{13} -4 \zeta_{6} q^{16} + 7 \zeta_{6} q^{19} + ( 5 - 4 \zeta_{6} ) q^{21} + ( -5 + 5 \zeta_{6} ) q^{25} + ( -3 + 6 \zeta_{6} ) q^{27} + ( -2 - 4 \zeta_{6} ) q^{28} + ( -7 + 7 \zeta_{6} ) q^{31} + 6 q^{36} + ( 14 - 7 \zeta_{6} ) q^{37} + ( -2 - 5 \zeta_{6} ) q^{39} -5 q^{43} + ( 4 - 8 \zeta_{6} ) q^{48} + ( -5 - 3 \zeta_{6} ) q^{49} + ( -8 + 6 \zeta_{6} ) q^{52} + ( -7 + 14 \zeta_{6} ) q^{57} + ( -8 + 4 \zeta_{6} ) q^{61} + ( 9 - 3 \zeta_{6} ) q^{63} -8 q^{64} + ( -7 - 7 \zeta_{6} ) q^{67} + ( -7 + 7 \zeta_{6} ) q^{73} + ( -10 + 5 \zeta_{6} ) q^{75} + 14 q^{76} -13 \zeta_{6} q^{79} + ( -9 + 9 \zeta_{6} ) q^{81} + ( 2 - 10 \zeta_{6} ) q^{84} + ( -9 + 10 \zeta_{6} ) q^{91} + ( -14 + 7 \zeta_{6} ) q^{93} + 14 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 3q^{3} + 2q^{4} + q^{7} + 3q^{9} + O(q^{10})$$ $$2q + 3q^{3} + 2q^{4} + q^{7} + 3q^{9} + 6q^{12} - 7q^{13} - 4q^{16} + 7q^{19} + 6q^{21} - 5q^{25} - 8q^{28} - 7q^{31} + 12q^{36} + 21q^{37} - 9q^{39} - 10q^{43} - 13q^{49} - 10q^{52} - 12q^{61} + 15q^{63} - 16q^{64} - 21q^{67} - 7q^{73} - 15q^{75} + 28q^{76} - 13q^{79} - 9q^{81} - 6q^{84} - 8q^{91} - 21q^{93} + 28q^{97} + O(q^{100})$$

Character values

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

 $$n$$ $$92$$ $$106$$ $$157$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$\zeta_{6}$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
38.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 1.50000 + 0.866025i 1.00000 1.73205i 0 0 0.500000 2.59808i 0 1.50000 + 2.59808i 0
194.1 0 1.50000 0.866025i 1.00000 + 1.73205i 0 0 0.500000 + 2.59808i 0 1.50000 2.59808i 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
91.s odd 6 1 inner
273.ba 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.ba.b yes 2
3.b odd 2 1 CM 273.2.ba.b yes 2
7.d odd 6 1 273.2.ba.a 2
13.b even 2 1 273.2.ba.a 2
21.g even 6 1 273.2.ba.a 2
39.d odd 2 1 273.2.ba.a 2
91.s odd 6 1 inner 273.2.ba.b yes 2
273.ba even 6 1 inner 273.2.ba.b yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.ba.a 2 7.d odd 6 1
273.2.ba.a 2 13.b even 2 1
273.2.ba.a 2 21.g even 6 1
273.2.ba.a 2 39.d odd 2 1
273.2.ba.b yes 2 1.a even 1 1 trivial
273.2.ba.b yes 2 3.b odd 2 1 CM
273.2.ba.b yes 2 91.s odd 6 1 inner
273.2.ba.b yes 2 273.ba even 6 1 inner

Hecke kernels

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

 $$T_{2}$$ $$T_{19}^{2} - 7 T_{19} + 49$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$3 - 3 T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$7 - T + T^{2}$$
$11$ $$T^{2}$$
$13$ $$13 + 7 T + T^{2}$$
$17$ $$T^{2}$$
$19$ $$49 - 7 T + T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$49 + 7 T + T^{2}$$
$37$ $$147 - 21 T + T^{2}$$
$41$ $$T^{2}$$
$43$ $$( 5 + T )^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$48 + 12 T + T^{2}$$
$67$ $$147 + 21 T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$49 + 7 T + T^{2}$$
$79$ $$169 + 13 T + T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$( -14 + T )^{2}$$