# Properties

 Label 273.2.cd.a Level $273$ Weight $2$ Character orbit 273.cd Analytic conductor $2.180$ Analytic rank $0$ Dimension $4$ 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.cd (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.17991597518$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{12}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{12}^{3} - \zeta_{12}) q^{3} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{4} + ( - 2 \zeta_{12}^{3} + 3 \zeta_{12}) q^{7} + (3 \zeta_{12}^{2} - 3) q^{9}+O(q^{10})$$ q + (-z^3 - z) * q^3 + (-2*z^3 + 2*z) * q^4 + (-2*z^3 + 3*z) * q^7 + (3*z^2 - 3) * q^9 $$q + ( - \zeta_{12}^{3} - \zeta_{12}) q^{3} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{4} + ( - 2 \zeta_{12}^{3} + 3 \zeta_{12}) q^{7} + (3 \zeta_{12}^{2} - 3) q^{9} + ( - 2 \zeta_{12}^{2} - 2) q^{12} + ( - 4 \zeta_{12}^{3} + \zeta_{12}) q^{13} + ( - 4 \zeta_{12}^{2} + 4) q^{16} + (3 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 5 \zeta_{12} - 5) q^{19} + ( - 4 \zeta_{12}^{2} - 1) q^{21} + ( - 5 \zeta_{12}^{3} + 5 \zeta_{12}) q^{25} + ( - 3 \zeta_{12}^{3} + 6 \zeta_{12}) q^{27} + ( - 4 \zeta_{12}^{2} + 6) q^{28} + (\zeta_{12}^{3} + \zeta_{12}^{2} - 6 \zeta_{12} + 5) q^{31} + 6 \zeta_{12}^{3} q^{36} + (7 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - 4 \zeta_{12} + 4) q^{37} + (2 \zeta_{12}^{2} - 7) q^{39} + (14 \zeta_{12}^{2} - 7) q^{43} + (4 \zeta_{12}^{3} - 8 \zeta_{12}) q^{48} + ( - 3 \zeta_{12}^{2} + 8) q^{49} + ( - 8 \zeta_{12}^{2} + 2) q^{52} + (\zeta_{12}^{3} + 7 \zeta_{12}^{2} + 7 \zeta_{12} + 1) q^{57} + (8 \zeta_{12}^{3} - 4 \zeta_{12}) q^{61} + (9 \zeta_{12}^{3} - 3 \zeta_{12}) q^{63} - 8 \zeta_{12}^{3} q^{64} + (9 \zeta_{12}^{3} + 9 \zeta_{12}^{2} - 2 \zeta_{12} - 7) q^{67} + (8 \zeta_{12}^{3} - 8 \zeta_{12}^{2} + \zeta_{12} + 9) q^{73} + ( - 5 \zeta_{12}^{2} - 5) q^{75} + (10 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 6 \zeta_{12} - 10) q^{76} + ( - 14 \zeta_{12}^{3} + 7 \zeta_{12}) q^{79} - 9 \zeta_{12}^{2} q^{81} + (2 \zeta_{12}^{3} - 10 \zeta_{12}) q^{84} + ( - 11 \zeta_{12}^{2} + 6) q^{91} + ( - 7 \zeta_{12}^{3} + 11 \zeta_{12}^{2} - 4 \zeta_{12} - 4) q^{93} + (3 \zeta_{12}^{3} + 8 \zeta_{12}^{2} + 8 \zeta_{12} + 3) q^{97} +O(q^{100})$$ q + (-z^3 - z) * q^3 + (-2*z^3 + 2*z) * q^4 + (-2*z^3 + 3*z) * q^7 + (3*z^2 - 3) * q^9 + (-2*z^2 - 2) * q^12 + (-4*z^3 + z) * q^13 + (-4*z^2 + 4) * q^16 + (3*z^3 + 2*z^2 - 5*z - 5) * q^19 + (-4*z^2 - 1) * q^21 + (-5*z^3 + 5*z) * q^25 + (-3*z^3 + 6*z) * q^27 + (-4*z^2 + 6) * q^28 + (z^3 + z^2 - 6*z + 5) * q^31 + 6*z^3 * q^36 + (7*z^3 + 3*z^2 - 4*z + 4) * q^37 + (2*z^2 - 7) * q^39 + (14*z^2 - 7) * q^43 + (4*z^3 - 8*z) * q^48 + (-3*z^2 + 8) * q^49 + (-8*z^2 + 2) * q^52 + (z^3 + 7*z^2 + 7*z + 1) * q^57 + (8*z^3 - 4*z) * q^61 + (9*z^3 - 3*z) * q^63 - 8*z^3 * q^64 + (9*z^3 + 9*z^2 - 2*z - 7) * q^67 + (8*z^3 - 8*z^2 + z + 9) * q^73 + (-5*z^2 - 5) * q^75 + (10*z^3 + 6*z^2 - 6*z - 10) * q^76 + (-14*z^3 + 7*z) * q^79 - 9*z^2 * q^81 + (2*z^3 - 10*z) * q^84 + (-11*z^2 + 6) * q^91 + (-7*z^3 + 11*z^2 - 4*z - 4) * q^93 + (3*z^3 + 8*z^2 + 8*z + 3) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 6 q^{9}+O(q^{10})$$ 4 * q - 6 * q^9 $$4 q - 6 q^{9} - 12 q^{12} + 8 q^{16} - 16 q^{19} - 12 q^{21} + 16 q^{28} + 22 q^{31} + 22 q^{37} - 24 q^{39} + 26 q^{49} - 8 q^{52} + 18 q^{57} - 10 q^{67} + 20 q^{73} - 30 q^{75} - 28 q^{76} - 18 q^{81} + 2 q^{91} + 6 q^{93} + 28 q^{97}+O(q^{100})$$ 4 * q - 6 * q^9 - 12 * q^12 + 8 * q^16 - 16 * q^19 - 12 * q^21 + 16 * q^28 + 22 * q^31 + 22 * q^37 - 24 * q^39 + 26 * q^49 - 8 * q^52 + 18 * q^57 - 10 * q^67 + 20 * q^73 - 30 * q^75 - 28 * q^76 - 18 * q^81 + 2 * q^91 + 6 * q^93 + 28 * q^97

## Character values

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

 $$n$$ $$92$$ $$106$$ $$157$$ $$\chi(n)$$ $$-1$$ $$\zeta_{12}^{3}$$ $$-1 + \zeta_{12}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
44.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
0 0.866025 + 1.50000i −1.73205 + 1.00000i 0 0 −2.59808 + 0.500000i 0 −1.50000 + 2.59808i 0
86.1 0 −0.866025 1.50000i 1.73205 1.00000i 0 0 2.59808 0.500000i 0 −1.50000 + 2.59808i 0
200.1 0 −0.866025 + 1.50000i 1.73205 + 1.00000i 0 0 2.59808 + 0.500000i 0 −1.50000 2.59808i 0
242.1 0 0.866025 1.50000i −1.73205 1.00000i 0 0 −2.59808 0.500000i 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.z odd 12 1 inner
273.cd even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.2.cd.a 4
3.b odd 2 1 CM 273.2.cd.a 4
7.c even 3 1 273.2.cd.b yes 4
13.d odd 4 1 273.2.cd.b yes 4
21.h odd 6 1 273.2.cd.b yes 4
39.f even 4 1 273.2.cd.b yes 4
91.z odd 12 1 inner 273.2.cd.a 4
273.cd even 12 1 inner 273.2.cd.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.cd.a 4 1.a even 1 1 trivial
273.2.cd.a 4 3.b odd 2 1 CM
273.2.cd.a 4 91.z odd 12 1 inner
273.2.cd.a 4 273.cd even 12 1 inner
273.2.cd.b yes 4 7.c even 3 1
273.2.cd.b yes 4 13.d odd 4 1
273.2.cd.b yes 4 21.h odd 6 1
273.2.cd.b yes 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 $$S_{2}^{\mathrm{new}}(273, [\chi])$$:

 $$T_{2}$$ T2 $$T_{5}$$ T5 $$T_{19}^{4} + 16T_{19}^{3} + 65T_{19}^{2} - 22T_{19} + 121$$ T19^4 + 16*T19^3 + 65*T19^2 - 22*T19 + 121

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 3T^{2} + 9$$
$5$ $$T^{4}$$
$7$ $$T^{4} - 13T^{2} + 49$$
$11$ $$T^{4}$$
$13$ $$T^{4} + 23T^{2} + 169$$
$17$ $$T^{4}$$
$19$ $$T^{4} + 16 T^{3} + 65 T^{2} + \cdots + 121$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$T^{4} - 22 T^{3} + 137 T^{2} + \cdots + 169$$
$37$ $$T^{4} - 22 T^{3} + 221 T^{2} + \cdots + 5329$$
$41$ $$T^{4}$$
$43$ $$(T^{2} + 147)^{2}$$
$47$ $$T^{4}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$T^{4} + 48T^{2} + 2304$$
$67$ $$T^{4} + 10 T^{3} + 281 T^{2} + \cdots + 169$$
$71$ $$T^{4}$$
$73$ $$T^{4} - 20 T^{3} + 389 T^{2} + \cdots + 9409$$
$79$ $$T^{4} + 147 T^{2} + 21609$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$T^{4} - 28 T^{3} + 392 T^{2} - 56 T + 4$$