# Properties

 Label 273.2.by.a Level $273$ Weight $2$ Character orbit 273.by 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.by (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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{2} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{3} + ( -1 - \zeta_{12}^{2} ) q^{4} + ( -\zeta_{12} - \zeta_{12}^{2} ) q^{5} + ( \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{6} + ( 1 - 3 \zeta_{12}^{2} ) q^{7} + ( -\zeta_{12} - \zeta_{12}^{2} ) q^{8} + ( 1 - \zeta_{12}^{2} ) q^{9} +O(q^{10})$$ $$q + ( 1 - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{2} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{3} + ( -1 - \zeta_{12}^{2} ) q^{4} + ( -\zeta_{12} - \zeta_{12}^{2} ) q^{5} + ( \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{6} + ( 1 - 3 \zeta_{12}^{2} ) q^{7} + ( -\zeta_{12} - \zeta_{12}^{2} ) q^{8} + ( 1 - \zeta_{12}^{2} ) q^{9} -\zeta_{12}^{2} q^{10} + ( -2 + 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{11} + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{12} + ( -2 \zeta_{12} - 3 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{13} + ( -2 + 3 \zeta_{12} - \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{14} + ( -1 - \zeta_{12} ) q^{15} + ( 2 \zeta_{12} - \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{16} + ( 3 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{17} + ( \zeta_{12} - \zeta_{12}^{2} ) q^{18} + ( 2 + 2 \zeta_{12} - 2 \zeta_{12}^{2} ) q^{19} + ( -1 + \zeta_{12} + 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{20} + ( -2 \zeta_{12} - \zeta_{12}^{3} ) q^{21} + ( -2 + 2 \zeta_{12}^{2} ) q^{22} + ( 6 + \zeta_{12} - 3 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{23} + ( -1 - \zeta_{12} ) q^{24} + ( -1 + 2 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{25} + ( -3 + 3 \zeta_{12} - 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{26} -\zeta_{12}^{3} q^{27} + ( -4 + 5 \zeta_{12}^{2} ) q^{28} + 7 \zeta_{12}^{2} q^{29} -\zeta_{12} q^{30} + ( 6 - 2 \zeta_{12} - 2 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{31} + ( -5 + 5 \zeta_{12} + 4 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{32} + ( 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{33} + ( 3 - 3 \zeta_{12} + 3 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{34} + ( -3 - \zeta_{12} + 2 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{35} + ( -2 + \zeta_{12}^{2} ) q^{36} + ( 5 - 5 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{37} + ( -2 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{38} + ( -2 - 3 \zeta_{12} + 2 \zeta_{12}^{2} ) q^{39} + ( -1 + 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{40} + ( -7 + 3 \zeta_{12} + 4 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{41} + ( 3 - 3 \zeta_{12} - 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{42} -8 \zeta_{12} q^{43} + ( 4 + 2 \zeta_{12} - 2 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{44} + ( -1 - \zeta_{12} + \zeta_{12}^{3} ) q^{45} + ( 3 + 3 \zeta_{12} - 5 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{46} + ( 4 - 4 \zeta_{12} + 4 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{47} + ( 2 - \zeta_{12} + 2 \zeta_{12}^{2} ) q^{48} + ( -8 + 3 \zeta_{12}^{2} ) q^{49} + ( 4 - 5 \zeta_{12} + \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{50} + ( 3 - 6 \zeta_{12}^{2} ) q^{51} + ( -3 + 4 \zeta_{12} + 6 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{52} + ( 6 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{53} + ( 1 - \zeta_{12} ) q^{54} + ( 4 + 4 \zeta_{12} - 2 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{55} + ( -3 - \zeta_{12} + 2 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{56} + ( 2 - 2 \zeta_{12}^{3} ) q^{57} + ( 7 - 7 \zeta_{12} + 7 \zeta_{12}^{3} ) q^{58} + ( -3 + 3 \zeta_{12} + \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{59} + ( 1 + \zeta_{12} + \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{60} + ( -2 - 3 \zeta_{12} - 2 \zeta_{12}^{2} ) q^{61} + ( 6 \zeta_{12} - 8 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{62} + ( -2 - \zeta_{12}^{2} ) q^{63} + ( -1 + 2 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{64} + ( -1 + 2 \zeta_{12} + 3 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{65} + 2 \zeta_{12}^{3} q^{66} + ( -2 - 7 \zeta_{12} - 5 \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{67} + ( -9 \zeta_{12} + 9 \zeta_{12}^{3} ) q^{68} + ( 1 + 3 \zeta_{12} - \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{69} + ( -3 + 2 \zeta_{12}^{2} ) q^{70} + ( -4 + 4 \zeta_{12} + 4 \zeta_{12}^{2} ) q^{71} + ( -1 - \zeta_{12} + \zeta_{12}^{3} ) q^{72} + ( -5 - 9 \zeta_{12} + 9 \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{73} + ( 5 - 5 \zeta_{12}^{2} ) q^{74} + ( \zeta_{12} - 3 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{75} + ( -4 - 2 \zeta_{12} + 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{76} + ( 4 + 6 \zeta_{12} + 2 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{77} + ( 3 - 5 \zeta_{12} - \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{78} + ( 7 + 10 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{79} + ( 1 + 2 \zeta_{12} - 3 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{80} -\zeta_{12}^{2} q^{81} + ( -10 + 3 \zeta_{12} + 10 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{82} + ( 4 - 8 \zeta_{12} - 8 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{83} + ( \zeta_{12} + 4 \zeta_{12}^{3} ) q^{84} + ( -6 - 6 \zeta_{12} + 3 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{85} + ( 8 - 8 \zeta_{12} - 8 \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{86} + 7 \zeta_{12} q^{87} + ( 4 + 4 \zeta_{12} - 2 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{88} + ( 6 + 5 \zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{89} - q^{90} + ( -9 + 4 \zeta_{12} + 6 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{91} + ( -9 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{92} + ( -2 + 4 \zeta_{12} + 6 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{93} + ( 16 - 12 \zeta_{12} - 8 \zeta_{12}^{2} + 12 \zeta_{12}^{3} ) q^{94} + ( -2 - 2 \zeta_{12} - 2 \zeta_{12}^{2} ) q^{95} + ( 5 - \zeta_{12} - \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{96} + ( 3 + 3 \zeta_{12} - \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{97} + ( -5 - 3 \zeta_{12} + 8 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{98} + ( 2 \zeta_{12} + 2 \zeta_{12}^{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{2} - 6q^{4} - 2q^{5} + 2q^{6} - 2q^{7} - 2q^{8} + 2q^{9} + O(q^{10})$$ $$4q + 2q^{2} - 6q^{4} - 2q^{5} + 2q^{6} - 2q^{7} - 2q^{8} + 2q^{9} - 2q^{10} - 4q^{11} - 6q^{13} - 10q^{14} - 4q^{15} - 2q^{16} - 2q^{18} + 4q^{19} - 4q^{22} + 18q^{23} - 4q^{24} - 16q^{26} - 6q^{28} + 14q^{29} + 20q^{31} - 12q^{32} + 4q^{33} + 18q^{34} - 8q^{35} - 6q^{36} + 10q^{37} - 8q^{38} - 4q^{39} - 20q^{41} + 8q^{42} + 12q^{44} - 4q^{45} + 2q^{46} + 24q^{47} + 12q^{48} - 26q^{49} + 18q^{50} + 24q^{53} + 4q^{54} + 12q^{55} - 8q^{56} + 8q^{57} + 28q^{58} - 10q^{59} + 6q^{60} - 12q^{61} - 16q^{62} - 10q^{63} + 2q^{65} - 18q^{67} + 2q^{69} - 8q^{70} - 8q^{71} - 4q^{72} - 2q^{73} + 10q^{74} - 6q^{75} - 12q^{76} + 20q^{77} + 10q^{78} + 28q^{79} - 2q^{80} - 2q^{81} - 20q^{82} - 18q^{85} + 16q^{86} + 12q^{88} + 22q^{89} - 4q^{90} - 24q^{91} - 36q^{92} + 4q^{93} + 48q^{94} - 12q^{95} + 18q^{96} + 10q^{97} - 4q^{98} + 4q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$92$$ $$106$$ $$157$$ $$\chi(n)$$ $$1$$ $$\zeta_{12}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
76.1
 −0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i 0.866025 + 0.500000i
0.500000 1.86603i −0.866025 + 0.500000i −1.50000 0.866025i 0.366025 0.366025i 0.500000 + 1.86603i −0.500000 2.59808i 0.366025 0.366025i 0.500000 0.866025i −0.500000 0.866025i
97.1 0.500000 + 1.86603i −0.866025 0.500000i −1.50000 + 0.866025i 0.366025 + 0.366025i 0.500000 1.86603i −0.500000 + 2.59808i 0.366025 + 0.366025i 0.500000 + 0.866025i −0.500000 + 0.866025i
202.1 0.500000 0.133975i 0.866025 + 0.500000i −1.50000 + 0.866025i −1.36603 + 1.36603i 0.500000 + 0.133975i −0.500000 + 2.59808i −1.36603 + 1.36603i 0.500000 + 0.866025i −0.500000 + 0.866025i
223.1 0.500000 + 0.133975i 0.866025 0.500000i −1.50000 0.866025i −1.36603 1.36603i 0.500000 0.133975i −0.500000 2.59808i −1.36603 1.36603i 0.500000 0.866025i −0.500000 0.866025i
 $$n$$: 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.bc 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.by.a 4
3.b odd 2 1 819.2.fm.b 4
7.b odd 2 1 273.2.by.b yes 4
13.f odd 12 1 273.2.by.b yes 4
21.c even 2 1 819.2.fm.a 4
39.k even 12 1 819.2.fm.a 4
91.bc even 12 1 inner 273.2.by.a 4
273.ca odd 12 1 819.2.fm.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.by.a 4 1.a even 1 1 trivial
273.2.by.a 4 91.bc even 12 1 inner
273.2.by.b yes 4 7.b odd 2 1
273.2.by.b yes 4 13.f odd 12 1
819.2.fm.a 4 21.c even 2 1
819.2.fm.a 4 39.k even 12 1
819.2.fm.b 4 3.b odd 2 1
819.2.fm.b 4 273.ca odd 12 1

## Hecke kernels

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

 $$T_{2}^{4} - 2 T_{2}^{3} + 5 T_{2}^{2} - 4 T_{2} + 1$$ $$T_{5}^{4} + 2 T_{5}^{3} + 2 T_{5}^{2} - 2 T_{5} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 4 T + 5 T^{2} - 2 T^{3} + T^{4}$$
$3$ $$1 - T^{2} + T^{4}$$
$5$ $$1 - 2 T + 2 T^{2} + 2 T^{3} + T^{4}$$
$7$ $$( 7 + T + T^{2} )^{2}$$
$11$ $$16 + 32 T + 20 T^{2} + 4 T^{3} + T^{4}$$
$13$ $$169 + 78 T + 23 T^{2} + 6 T^{3} + T^{4}$$
$17$ $$729 + 27 T^{2} + T^{4}$$
$19$ $$64 - 32 T + 8 T^{2} - 4 T^{3} + T^{4}$$
$23$ $$676 - 468 T + 134 T^{2} - 18 T^{3} + T^{4}$$
$29$ $$( 49 - 7 T + T^{2} )^{2}$$
$31$ $$1936 - 880 T + 200 T^{2} - 20 T^{3} + T^{4}$$
$37$ $$625 - 500 T + 125 T^{2} - 10 T^{3} + T^{4}$$
$41$ $$5329 + 1606 T + 221 T^{2} + 20 T^{3} + T^{4}$$
$43$ $$4096 - 64 T^{2} + T^{4}$$
$47$ $$2304 - 1152 T + 288 T^{2} - 24 T^{3} + T^{4}$$
$53$ $$( 33 - 12 T + T^{2} )^{2}$$
$59$ $$4 - 4 T + 26 T^{2} + 10 T^{3} + T^{4}$$
$61$ $$9 + 36 T + 51 T^{2} + 12 T^{3} + T^{4}$$
$67$ $$4356 + 396 T + 90 T^{2} + 18 T^{3} + T^{4}$$
$71$ $$1024 + 256 T + 32 T^{2} + 8 T^{3} + T^{4}$$
$73$ $$14641 - 242 T + 2 T^{2} + 2 T^{3} + T^{4}$$
$79$ $$( -26 - 14 T + T^{2} )^{2}$$
$83$ $$9216 + T^{4}$$
$89$ $$2116 - 644 T + 170 T^{2} - 22 T^{3} + T^{4}$$
$97$ $$4 + 4 T + 26 T^{2} - 10 T^{3} + T^{4}$$