# Properties

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

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## 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: $$x^{4} - x^{2} + 1$$ 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.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{12} + \zeta_{12}^{3} ) q^{3} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{4} + ( -1 + 3 \zeta_{12}^{2} ) q^{7} + ( -3 + 3 \zeta_{12}^{2} ) q^{9} +O(q^{10})$$ $$q + ( \zeta_{12} + \zeta_{12}^{3} ) q^{3} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{4} + ( -1 + 3 \zeta_{12}^{2} ) q^{7} + ( -3 + 3 \zeta_{12}^{2} ) q^{9} + ( 2 + 2 \zeta_{12}^{2} ) q^{12} + ( -\zeta_{12} - 3 \zeta_{12}^{3} ) q^{13} + ( 4 - 4 \zeta_{12}^{2} ) q^{16} + ( -2 - 2 \zeta_{12} + 5 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{19} + ( -4 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{21} + ( 5 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{25} + ( -6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{27} + ( 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{28} + ( -5 - \zeta_{12} + 6 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{31} + 6 \zeta_{12}^{3} q^{36} + ( -3 + 3 \zeta_{12} - 4 \zeta_{12}^{2} - 7 \zeta_{12}^{3} ) q^{37} + ( 7 - 5 \zeta_{12}^{2} ) q^{39} + ( 7 - 14 \zeta_{12}^{2} ) q^{43} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{48} + ( -8 + 3 \zeta_{12}^{2} ) q^{49} + ( -2 - 6 \zeta_{12}^{2} ) q^{52} + ( 8 - 7 \zeta_{12} - 7 \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{57} + ( 4 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{61} + ( -6 - 3 \zeta_{12}^{2} ) q^{63} -8 \zeta_{12}^{3} q^{64} + ( 7 - 9 \zeta_{12} + 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{67} + ( -9 - 8 \zeta_{12} + \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{73} + ( 5 + 5 \zeta_{12}^{2} ) q^{75} + ( -4 + 6 \zeta_{12} - 6 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{76} + ( -7 \zeta_{12} + 14 \zeta_{12}^{3} ) q^{79} -9 \zeta_{12}^{2} q^{81} + ( -8 + 10 \zeta_{12}^{2} ) q^{84} + ( 10 \zeta_{12} - 9 \zeta_{12}^{3} ) q^{91} + ( -11 - 11 \zeta_{12} + 4 \zeta_{12}^{2} + 7 \zeta_{12}^{3} ) q^{93} + ( 11 - 8 \zeta_{12} - 8 \zeta_{12}^{2} + 11 \zeta_{12}^{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{7} - 6q^{9} + O(q^{10})$$ $$4q + 2q^{7} - 6q^{9} + 12q^{12} + 8q^{16} + 2q^{19} - 8q^{31} - 20q^{37} + 18q^{39} - 26q^{49} - 20q^{52} + 18q^{57} - 30q^{63} + 32q^{67} - 34q^{73} + 30q^{75} - 28q^{76} - 18q^{81} - 12q^{84} - 36q^{93} + 28q^{97} + O(q^{100})$$

## Character values

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

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

## 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 0.500000 + 2.59808i 0 −1.50000 + 2.59808i 0
86.1 0 0.866025 + 1.50000i 1.73205 1.00000i 0 0 0.500000 + 2.59808i 0 −1.50000 + 2.59808i 0
200.1 0 0.866025 1.50000i 1.73205 + 1.00000i 0 0 0.500000 2.59808i 0 −1.50000 2.59808i 0
242.1 0 −0.866025 + 1.50000i −1.73205 1.00000i 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.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.b yes 4
3.b odd 2 1 CM 273.2.cd.b yes 4
7.c even 3 1 273.2.cd.a 4
13.d odd 4 1 273.2.cd.a 4
21.h odd 6 1 273.2.cd.a 4
39.f even 4 1 273.2.cd.a 4
91.z odd 12 1 inner 273.2.cd.b yes 4
273.cd even 12 1 inner 273.2.cd.b yes 4

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

## Hecke kernels

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

 $$T_{2}$$ $$T_{5}$$ $$T_{19}^{4} - 2 T_{19}^{3} + 65 T_{19}^{2} + 176 T_{19} + 121$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$9 + 3 T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$( 7 - T + T^{2} )^{2}$$
$11$ $$T^{4}$$
$13$ $$169 + 23 T^{2} + T^{4}$$
$17$ $$T^{4}$$
$19$ $$121 + 176 T + 65 T^{2} - 2 T^{3} + T^{4}$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$169 + 286 T + 137 T^{2} + 8 T^{3} + T^{4}$$
$37$ $$5329 + 1606 T + 221 T^{2} + 20 T^{3} + T^{4}$$
$41$ $$T^{4}$$
$43$ $$( 147 + T^{2} )^{2}$$
$47$ $$T^{4}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$2304 + 48 T^{2} + T^{4}$$
$67$ $$169 - 130 T + 281 T^{2} - 32 T^{3} + T^{4}$$
$71$ $$T^{4}$$
$73$ $$9409 + 1940 T + 389 T^{2} + 34 T^{3} + T^{4}$$
$79$ $$21609 + 147 T^{2} + T^{4}$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$4 - 56 T + 392 T^{2} - 28 T^{3} + T^{4}$$
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