# Properties

 Label 144.2.s.e Level $144$ Weight $2$ Character orbit 144.s Analytic conductor $1.150$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$144 = 2^{4} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 144.s (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.14984578911$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{3} + ( 1 + \zeta_{12}^{2} ) q^{5} + ( 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{7} + 3 q^{9} +O(q^{10})$$ $$q + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{3} + ( 1 + \zeta_{12}^{2} ) q^{5} + ( 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{7} + 3 q^{9} + ( 3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{11} + ( -1 + \zeta_{12}^{2} ) q^{13} -3 \zeta_{12} q^{15} + ( 2 - 4 \zeta_{12}^{2} ) q^{17} -6 \zeta_{12}^{3} q^{19} + ( -6 + 3 \zeta_{12}^{2} ) q^{21} + ( -3 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{23} -2 \zeta_{12}^{2} q^{25} + ( -6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{27} + ( -10 + 5 \zeta_{12}^{2} ) q^{29} + 3 \zeta_{12} q^{31} -9 \zeta_{12}^{2} q^{33} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{35} -4 q^{37} + ( \zeta_{12} - 2 \zeta_{12}^{3} ) q^{39} + ( 3 + 3 \zeta_{12}^{2} ) q^{41} + ( 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{43} + ( 3 + 3 \zeta_{12}^{2} ) q^{45} + ( -3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{47} + ( 2 - 2 \zeta_{12}^{2} ) q^{49} + 6 \zeta_{12}^{3} q^{51} + ( 6 - 12 \zeta_{12}^{2} ) q^{53} + 9 \zeta_{12}^{3} q^{55} + ( -6 + 12 \zeta_{12}^{2} ) q^{57} + ( 3 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{59} + 7 \zeta_{12}^{2} q^{61} + ( 9 \zeta_{12} - 9 \zeta_{12}^{3} ) q^{63} + ( -2 + \zeta_{12}^{2} ) q^{65} -9 \zeta_{12} q^{67} + ( 9 - 9 \zeta_{12}^{2} ) q^{69} + ( -12 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{71} + 4 q^{73} + ( 2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{75} + ( 9 + 9 \zeta_{12}^{2} ) q^{77} + ( -15 \zeta_{12} + 15 \zeta_{12}^{3} ) q^{79} + 9 q^{81} + ( -3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{83} + ( 6 - 6 \zeta_{12}^{2} ) q^{85} + ( 15 \zeta_{12} - 15 \zeta_{12}^{3} ) q^{87} + ( -2 + 4 \zeta_{12}^{2} ) q^{89} + 3 \zeta_{12}^{3} q^{91} + ( -3 - 3 \zeta_{12}^{2} ) q^{93} + ( 6 \zeta_{12} - 12 \zeta_{12}^{3} ) q^{95} -\zeta_{12}^{2} q^{97} + ( 9 \zeta_{12} + 9 \zeta_{12}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 6q^{5} + 12q^{9} + O(q^{10})$$ $$4q + 6q^{5} + 12q^{9} - 2q^{13} - 18q^{21} - 4q^{25} - 30q^{29} - 18q^{33} - 16q^{37} + 18q^{41} + 18q^{45} + 4q^{49} + 14q^{61} - 6q^{65} + 18q^{69} + 16q^{73} + 54q^{77} + 36q^{81} + 12q^{85} - 18q^{93} - 2q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/144\mathbb{Z}\right)^\times$$.

 $$n$$ $$37$$ $$65$$ $$127$$ $$\chi(n)$$ $$1$$ $$1 - \zeta_{12}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
47.1
 0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i −0.866025 − 0.500000i
0 −1.73205 0 1.50000 0.866025i 0 2.59808 + 1.50000i 0 3.00000 0
47.2 0 1.73205 0 1.50000 0.866025i 0 −2.59808 1.50000i 0 3.00000 0
95.1 0 −1.73205 0 1.50000 + 0.866025i 0 2.59808 1.50000i 0 3.00000 0
95.2 0 1.73205 0 1.50000 + 0.866025i 0 −2.59808 + 1.50000i 0 3.00000 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
4.b odd 2 1 inner
9.d odd 6 1 inner
36.h even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 144.2.s.e 4
3.b odd 2 1 432.2.s.e 4
4.b odd 2 1 inner 144.2.s.e 4
8.b even 2 1 576.2.s.e 4
8.d odd 2 1 576.2.s.e 4
9.c even 3 1 432.2.s.e 4
9.c even 3 1 1296.2.c.f 4
9.d odd 6 1 inner 144.2.s.e 4
9.d odd 6 1 1296.2.c.f 4
12.b even 2 1 432.2.s.e 4
24.f even 2 1 1728.2.s.e 4
24.h odd 2 1 1728.2.s.e 4
36.f odd 6 1 432.2.s.e 4
36.f odd 6 1 1296.2.c.f 4
36.h even 6 1 inner 144.2.s.e 4
36.h even 6 1 1296.2.c.f 4
72.j odd 6 1 576.2.s.e 4
72.j odd 6 1 5184.2.c.f 4
72.l even 6 1 576.2.s.e 4
72.l even 6 1 5184.2.c.f 4
72.n even 6 1 1728.2.s.e 4
72.n even 6 1 5184.2.c.f 4
72.p odd 6 1 1728.2.s.e 4
72.p odd 6 1 5184.2.c.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.2.s.e 4 1.a even 1 1 trivial
144.2.s.e 4 4.b odd 2 1 inner
144.2.s.e 4 9.d odd 6 1 inner
144.2.s.e 4 36.h even 6 1 inner
432.2.s.e 4 3.b odd 2 1
432.2.s.e 4 9.c even 3 1
432.2.s.e 4 12.b even 2 1
432.2.s.e 4 36.f odd 6 1
576.2.s.e 4 8.b even 2 1
576.2.s.e 4 8.d odd 2 1
576.2.s.e 4 72.j odd 6 1
576.2.s.e 4 72.l even 6 1
1296.2.c.f 4 9.c even 3 1
1296.2.c.f 4 9.d odd 6 1
1296.2.c.f 4 36.f odd 6 1
1296.2.c.f 4 36.h even 6 1
1728.2.s.e 4 24.f even 2 1
1728.2.s.e 4 24.h odd 2 1
1728.2.s.e 4 72.n even 6 1
1728.2.s.e 4 72.p odd 6 1
5184.2.c.f 4 72.j odd 6 1
5184.2.c.f 4 72.l even 6 1
5184.2.c.f 4 72.n even 6 1
5184.2.c.f 4 72.p odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(144, [\chi])$$:

 $$T_{5}^{2} - 3 T_{5} + 3$$ $$T_{7}^{4} - 9 T_{7}^{2} + 81$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( -3 + T^{2} )^{2}$$
$5$ $$( 3 - 3 T + T^{2} )^{2}$$
$7$ $$81 - 9 T^{2} + T^{4}$$
$11$ $$729 + 27 T^{2} + T^{4}$$
$13$ $$( 1 + T + T^{2} )^{2}$$
$17$ $$( 12 + T^{2} )^{2}$$
$19$ $$( 36 + T^{2} )^{2}$$
$23$ $$729 + 27 T^{2} + T^{4}$$
$29$ $$( 75 + 15 T + T^{2} )^{2}$$
$31$ $$81 - 9 T^{2} + T^{4}$$
$37$ $$( 4 + T )^{4}$$
$41$ $$( 27 - 9 T + T^{2} )^{2}$$
$43$ $$81 - 9 T^{2} + T^{4}$$
$47$ $$729 + 27 T^{2} + T^{4}$$
$53$ $$( 108 + T^{2} )^{2}$$
$59$ $$729 + 27 T^{2} + T^{4}$$
$61$ $$( 49 - 7 T + T^{2} )^{2}$$
$67$ $$6561 - 81 T^{2} + T^{4}$$
$71$ $$( -108 + T^{2} )^{2}$$
$73$ $$( -4 + T )^{4}$$
$79$ $$50625 - 225 T^{2} + T^{4}$$
$83$ $$729 + 27 T^{2} + T^{4}$$
$89$ $$( 12 + T^{2} )^{2}$$
$97$ $$( 1 + T + T^{2} )^{2}$$