# Properties

 Label 273.2.n.a Level $273$ Weight $2$ Character orbit 273.n Analytic conductor $2.180$ Analytic rank $1$ 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.n (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.17991597518$$ Analytic rank: $$1$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ 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 primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$92$$ $$106$$ $$157$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{8}^{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
8.1
 −0.707107 + 0.707107i 0.707107 − 0.707107i −0.707107 − 0.707107i 0.707107 + 0.707107i
−1.00000 + 1.00000i −1.70711 0.292893i 0 −2.70711 + 2.70711i 2.00000 1.41421i −0.707107 + 0.707107i −2.00000 2.00000i 2.82843 + 1.00000i 5.41421i
8.2 −1.00000 + 1.00000i −0.292893 1.70711i 0 −1.29289 + 1.29289i 2.00000 + 1.41421i 0.707107 0.707107i −2.00000 2.00000i −2.82843 + 1.00000i 2.58579i
239.1 −1.00000 1.00000i −1.70711 + 0.292893i 0 −2.70711 2.70711i 2.00000 + 1.41421i −0.707107 0.707107i −2.00000 + 2.00000i 2.82843 1.00000i 5.41421i
239.2 −1.00000 1.00000i −0.292893 + 1.70711i 0 −1.29289 1.29289i 2.00000 1.41421i 0.707107 + 0.707107i −2.00000 + 2.00000i −2.82843 1.00000i 2.58579i
 $$n$$: 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
39.f 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.n.a 4
3.b odd 2 1 273.2.n.b yes 4
13.d odd 4 1 273.2.n.b yes 4
39.f even 4 1 inner 273.2.n.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.n.a 4 1.a even 1 1 trivial
273.2.n.a 4 39.f even 4 1 inner
273.2.n.b yes 4 3.b odd 2 1
273.2.n.b yes 4 13.d odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 2 T_{2} + 2$$ acting on $$S_{2}^{\mathrm{new}}(273, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 2 + 2 T + T^{2} )^{2}$$
$3$ $$9 + 12 T + 8 T^{2} + 4 T^{3} + T^{4}$$
$5$ $$49 + 56 T + 32 T^{2} + 8 T^{3} + T^{4}$$
$7$ $$1 + T^{4}$$
$11$ $$16 + T^{4}$$
$13$ $$169 + 24 T^{2} + T^{4}$$
$17$ $$( 4 + T )^{4}$$
$19$ $$529 + 368 T + 128 T^{2} + 16 T^{3} + T^{4}$$
$23$ $$( -7 + 2 T + T^{2} )^{2}$$
$29$ $$961 + 66 T^{2} + T^{4}$$
$31$ $$1681 - 328 T + 32 T^{2} + 8 T^{3} + T^{4}$$
$37$ $$2116 + 920 T + 200 T^{2} + 20 T^{3} + T^{4}$$
$41$ $$16 + 64 T + 128 T^{2} - 16 T^{3} + T^{4}$$
$43$ $$961 + 66 T^{2} + T^{4}$$
$47$ $$529 - 368 T + 128 T^{2} - 16 T^{3} + T^{4}$$
$53$ $$16129 + 258 T^{2} + T^{4}$$
$59$ $$16 + T^{4}$$
$61$ $$( -82 + 8 T + T^{2} )^{2}$$
$67$ $$8464 - 736 T + 32 T^{2} + 8 T^{3} + T^{4}$$
$71$ $$3844 + 1736 T + 392 T^{2} + 28 T^{3} + T^{4}$$
$73$ $$1 - 8 T + 32 T^{2} + 8 T^{3} + T^{4}$$
$79$ $$( 1 - 6 T + T^{2} )^{2}$$
$83$ $$1 + T^{4}$$
$89$ $$14161 + 3808 T + 512 T^{2} + 32 T^{3} + T^{4}$$
$97$ $$16129 + 4064 T + 512 T^{2} + 32 T^{3} + T^{4}$$