# Properties

 Label 144.3.q.a Level $144$ Weight $3$ Character orbit 144.q Analytic conductor $3.924$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$144 = 2^{4} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 144.q (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.92371580679$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 9) 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 $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 3 - 3 \zeta_{6} ) q^{3} + ( 4 - 2 \zeta_{6} ) q^{5} + ( 2 - 2 \zeta_{6} ) q^{7} -9 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 3 - 3 \zeta_{6} ) q^{3} + ( 4 - 2 \zeta_{6} ) q^{5} + ( 2 - 2 \zeta_{6} ) q^{7} -9 \zeta_{6} q^{9} + ( 1 + \zeta_{6} ) q^{11} + 4 \zeta_{6} q^{13} + ( 6 - 12 \zeta_{6} ) q^{15} + ( 9 - 18 \zeta_{6} ) q^{17} -11 q^{19} -6 \zeta_{6} q^{21} + ( 32 - 16 \zeta_{6} ) q^{23} + ( -13 + 13 \zeta_{6} ) q^{25} -27 q^{27} + ( 26 + 26 \zeta_{6} ) q^{29} + 32 \zeta_{6} q^{31} + ( 6 - 3 \zeta_{6} ) q^{33} + ( 4 - 8 \zeta_{6} ) q^{35} -34 q^{37} + 12 q^{39} + ( -14 + 7 \zeta_{6} ) q^{41} + ( -61 + 61 \zeta_{6} ) q^{43} + ( -18 - 18 \zeta_{6} ) q^{45} + ( 28 + 28 \zeta_{6} ) q^{47} + 45 \zeta_{6} q^{49} + ( -27 - 27 \zeta_{6} ) q^{51} + 6 q^{55} + ( -33 + 33 \zeta_{6} ) q^{57} + ( -58 + 29 \zeta_{6} ) q^{59} + ( -56 + 56 \zeta_{6} ) q^{61} -18 q^{63} + ( 8 + 8 \zeta_{6} ) q^{65} -31 \zeta_{6} q^{67} + ( 48 - 96 \zeta_{6} ) q^{69} + ( 18 - 36 \zeta_{6} ) q^{71} + 65 q^{73} + 39 \zeta_{6} q^{75} + ( 4 - 2 \zeta_{6} ) q^{77} + ( 38 - 38 \zeta_{6} ) q^{79} + ( -81 + 81 \zeta_{6} ) q^{81} + ( 28 + 28 \zeta_{6} ) q^{83} -54 \zeta_{6} q^{85} + ( 156 - 78 \zeta_{6} ) q^{87} + ( 72 - 144 \zeta_{6} ) q^{89} + 8 q^{91} + 96 q^{93} + ( -44 + 22 \zeta_{6} ) q^{95} + ( 115 - 115 \zeta_{6} ) q^{97} + ( 9 - 18 \zeta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 3q^{3} + 6q^{5} + 2q^{7} - 9q^{9} + O(q^{10})$$ $$2q + 3q^{3} + 6q^{5} + 2q^{7} - 9q^{9} + 3q^{11} + 4q^{13} - 22q^{19} - 6q^{21} + 48q^{23} - 13q^{25} - 54q^{27} + 78q^{29} + 32q^{31} + 9q^{33} - 68q^{37} + 24q^{39} - 21q^{41} - 61q^{43} - 54q^{45} + 84q^{47} + 45q^{49} - 81q^{51} + 12q^{55} - 33q^{57} - 87q^{59} - 56q^{61} - 36q^{63} + 24q^{65} - 31q^{67} + 130q^{73} + 39q^{75} + 6q^{77} + 38q^{79} - 81q^{81} + 84q^{83} - 54q^{85} + 234q^{87} + 16q^{91} + 192q^{93} - 66q^{95} + 115q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$37$$ $$65$$ $$127$$ $$\chi(n)$$ $$1$$ $$\zeta_{6}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
65.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 1.50000 2.59808i 0 3.00000 1.73205i 0 1.00000 1.73205i 0 −4.50000 7.79423i 0
113.1 0 1.50000 + 2.59808i 0 3.00000 + 1.73205i 0 1.00000 + 1.73205i 0 −4.50000 + 7.79423i 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
9.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 144.3.q.a 2
3.b odd 2 1 432.3.q.a 2
4.b odd 2 1 9.3.d.a 2
8.b even 2 1 576.3.q.a 2
8.d odd 2 1 576.3.q.b 2
9.c even 3 1 432.3.q.a 2
9.c even 3 1 1296.3.e.a 2
9.d odd 6 1 inner 144.3.q.a 2
9.d odd 6 1 1296.3.e.a 2
12.b even 2 1 27.3.d.a 2
20.d odd 2 1 225.3.j.a 2
20.e even 4 2 225.3.i.a 4
24.f even 2 1 1728.3.q.a 2
24.h odd 2 1 1728.3.q.b 2
28.d even 2 1 441.3.r.a 2
28.f even 6 1 441.3.j.b 2
28.f even 6 1 441.3.n.a 2
28.g odd 6 1 441.3.j.a 2
28.g odd 6 1 441.3.n.b 2
36.f odd 6 1 27.3.d.a 2
36.f odd 6 1 81.3.b.a 2
36.h even 6 1 9.3.d.a 2
36.h even 6 1 81.3.b.a 2
60.h even 2 1 675.3.j.a 2
60.l odd 4 2 675.3.i.a 4
72.j odd 6 1 576.3.q.a 2
72.l even 6 1 576.3.q.b 2
72.n even 6 1 1728.3.q.b 2
72.p odd 6 1 1728.3.q.a 2
180.n even 6 1 225.3.j.a 2
180.p odd 6 1 675.3.j.a 2
180.v odd 12 2 225.3.i.a 4
180.x even 12 2 675.3.i.a 4
252.o even 6 1 441.3.j.a 2
252.r odd 6 1 441.3.n.a 2
252.s odd 6 1 441.3.r.a 2
252.bb even 6 1 441.3.n.b 2
252.bn odd 6 1 441.3.j.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.3.d.a 2 4.b odd 2 1
9.3.d.a 2 36.h even 6 1
27.3.d.a 2 12.b even 2 1
27.3.d.a 2 36.f odd 6 1
81.3.b.a 2 36.f odd 6 1
81.3.b.a 2 36.h even 6 1
144.3.q.a 2 1.a even 1 1 trivial
144.3.q.a 2 9.d odd 6 1 inner
225.3.i.a 4 20.e even 4 2
225.3.i.a 4 180.v odd 12 2
225.3.j.a 2 20.d odd 2 1
225.3.j.a 2 180.n even 6 1
432.3.q.a 2 3.b odd 2 1
432.3.q.a 2 9.c even 3 1
441.3.j.a 2 28.g odd 6 1
441.3.j.a 2 252.o even 6 1
441.3.j.b 2 28.f even 6 1
441.3.j.b 2 252.bn odd 6 1
441.3.n.a 2 28.f even 6 1
441.3.n.a 2 252.r odd 6 1
441.3.n.b 2 28.g odd 6 1
441.3.n.b 2 252.bb even 6 1
441.3.r.a 2 28.d even 2 1
441.3.r.a 2 252.s odd 6 1
576.3.q.a 2 8.b even 2 1
576.3.q.a 2 72.j odd 6 1
576.3.q.b 2 8.d odd 2 1
576.3.q.b 2 72.l even 6 1
675.3.i.a 4 60.l odd 4 2
675.3.i.a 4 180.x even 12 2
675.3.j.a 2 60.h even 2 1
675.3.j.a 2 180.p odd 6 1
1296.3.e.a 2 9.c even 3 1
1296.3.e.a 2 9.d odd 6 1
1728.3.q.a 2 24.f even 2 1
1728.3.q.a 2 72.p odd 6 1
1728.3.q.b 2 24.h odd 2 1
1728.3.q.b 2 72.n even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 6 T_{5} + 12$$ acting on $$S_{3}^{\mathrm{new}}(144, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$9 - 3 T + T^{2}$$
$5$ $$12 - 6 T + T^{2}$$
$7$ $$4 - 2 T + T^{2}$$
$11$ $$3 - 3 T + T^{2}$$
$13$ $$16 - 4 T + T^{2}$$
$17$ $$243 + T^{2}$$
$19$ $$( 11 + T )^{2}$$
$23$ $$768 - 48 T + T^{2}$$
$29$ $$2028 - 78 T + T^{2}$$
$31$ $$1024 - 32 T + T^{2}$$
$37$ $$( 34 + T )^{2}$$
$41$ $$147 + 21 T + T^{2}$$
$43$ $$3721 + 61 T + T^{2}$$
$47$ $$2352 - 84 T + T^{2}$$
$53$ $$T^{2}$$
$59$ $$2523 + 87 T + T^{2}$$
$61$ $$3136 + 56 T + T^{2}$$
$67$ $$961 + 31 T + T^{2}$$
$71$ $$972 + T^{2}$$
$73$ $$( -65 + T )^{2}$$
$79$ $$1444 - 38 T + T^{2}$$
$83$ $$2352 - 84 T + T^{2}$$
$89$ $$15552 + T^{2}$$
$97$ $$13225 - 115 T + T^{2}$$