# Properties

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

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$1.14984578911$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 18) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 1 + \zeta_{6} ) q^{3} + ( 2 - 2 \zeta_{6} ) q^{7} + 3 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 1 + \zeta_{6} ) q^{3} + ( 2 - 2 \zeta_{6} ) q^{7} + 3 \zeta_{6} q^{9} + ( -3 + 3 \zeta_{6} ) q^{11} -2 \zeta_{6} q^{13} -3 q^{17} + q^{19} + ( 4 - 2 \zeta_{6} ) q^{21} -6 \zeta_{6} q^{23} + ( 5 - 5 \zeta_{6} ) q^{25} + ( -3 + 6 \zeta_{6} ) q^{27} + ( -6 + 6 \zeta_{6} ) q^{29} -4 \zeta_{6} q^{31} + ( -6 + 3 \zeta_{6} ) q^{33} -4 q^{37} + ( 2 - 4 \zeta_{6} ) q^{39} -9 \zeta_{6} q^{41} + ( -1 + \zeta_{6} ) q^{43} + ( -6 + 6 \zeta_{6} ) q^{47} + 3 \zeta_{6} q^{49} + ( -3 - 3 \zeta_{6} ) q^{51} + 12 q^{53} + ( 1 + \zeta_{6} ) q^{57} + 3 \zeta_{6} q^{59} + ( -8 + 8 \zeta_{6} ) q^{61} + 6 q^{63} + 5 \zeta_{6} q^{67} + ( 6 - 12 \zeta_{6} ) q^{69} + 12 q^{71} + 11 q^{73} + ( 10 - 5 \zeta_{6} ) q^{75} + 6 \zeta_{6} q^{77} + ( -4 + 4 \zeta_{6} ) q^{79} + ( -9 + 9 \zeta_{6} ) q^{81} + ( 12 - 12 \zeta_{6} ) q^{83} + ( -12 + 6 \zeta_{6} ) q^{87} + 6 q^{89} -4 q^{91} + ( 4 - 8 \zeta_{6} ) q^{93} + ( -5 + 5 \zeta_{6} ) q^{97} -9 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 3q^{3} + 2q^{7} + 3q^{9} + O(q^{10})$$ $$2q + 3q^{3} + 2q^{7} + 3q^{9} - 3q^{11} - 2q^{13} - 6q^{17} + 2q^{19} + 6q^{21} - 6q^{23} + 5q^{25} - 6q^{29} - 4q^{31} - 9q^{33} - 8q^{37} - 9q^{41} - q^{43} - 6q^{47} + 3q^{49} - 9q^{51} + 24q^{53} + 3q^{57} + 3q^{59} - 8q^{61} + 12q^{63} + 5q^{67} + 24q^{71} + 22q^{73} + 15q^{75} + 6q^{77} - 4q^{79} - 9q^{81} + 12q^{83} - 18q^{87} + 12q^{89} - 8q^{91} - 5q^{97} - 18q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 1.50000 0.866025i 0 0 0 1.00000 + 1.73205i 0 1.50000 2.59808i 0
97.1 0 1.50000 + 0.866025i 0 0 0 1.00000 1.73205i 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
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 144.2.i.c 2
3.b odd 2 1 432.2.i.b 2
4.b odd 2 1 18.2.c.a 2
8.b even 2 1 576.2.i.a 2
8.d odd 2 1 576.2.i.g 2
9.c even 3 1 inner 144.2.i.c 2
9.c even 3 1 1296.2.a.g 1
9.d odd 6 1 432.2.i.b 2
9.d odd 6 1 1296.2.a.f 1
12.b even 2 1 54.2.c.a 2
20.d odd 2 1 450.2.e.i 2
20.e even 4 2 450.2.j.e 4
24.f even 2 1 1728.2.i.e 2
24.h odd 2 1 1728.2.i.f 2
28.d even 2 1 882.2.f.d 2
28.f even 6 1 882.2.e.g 2
28.f even 6 1 882.2.h.b 2
28.g odd 6 1 882.2.e.i 2
28.g odd 6 1 882.2.h.c 2
36.f odd 6 1 18.2.c.a 2
36.f odd 6 1 162.2.a.c 1
36.h even 6 1 54.2.c.a 2
36.h even 6 1 162.2.a.b 1
60.h even 2 1 1350.2.e.c 2
60.l odd 4 2 1350.2.j.a 4
72.j odd 6 1 1728.2.i.f 2
72.j odd 6 1 5184.2.a.p 1
72.l even 6 1 1728.2.i.e 2
72.l even 6 1 5184.2.a.q 1
72.n even 6 1 576.2.i.a 2
72.n even 6 1 5184.2.a.o 1
72.p odd 6 1 576.2.i.g 2
72.p odd 6 1 5184.2.a.r 1
84.h odd 2 1 2646.2.f.g 2
84.j odd 6 1 2646.2.e.c 2
84.j odd 6 1 2646.2.h.i 2
84.n even 6 1 2646.2.e.b 2
84.n even 6 1 2646.2.h.h 2
180.n even 6 1 1350.2.e.c 2
180.n even 6 1 4050.2.a.v 1
180.p odd 6 1 450.2.e.i 2
180.p odd 6 1 4050.2.a.c 1
180.v odd 12 2 1350.2.j.a 4
180.v odd 12 2 4050.2.c.r 2
180.x even 12 2 450.2.j.e 4
180.x even 12 2 4050.2.c.c 2
252.n even 6 1 882.2.e.g 2
252.o even 6 1 2646.2.e.b 2
252.r odd 6 1 2646.2.h.i 2
252.s odd 6 1 2646.2.f.g 2
252.s odd 6 1 7938.2.a.i 1
252.u odd 6 1 882.2.h.c 2
252.bb even 6 1 2646.2.h.h 2
252.bi even 6 1 882.2.f.d 2
252.bi even 6 1 7938.2.a.x 1
252.bj even 6 1 882.2.h.b 2
252.bl odd 6 1 882.2.e.i 2
252.bn odd 6 1 2646.2.e.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.2.c.a 2 4.b odd 2 1
18.2.c.a 2 36.f odd 6 1
54.2.c.a 2 12.b even 2 1
54.2.c.a 2 36.h even 6 1
144.2.i.c 2 1.a even 1 1 trivial
144.2.i.c 2 9.c even 3 1 inner
162.2.a.b 1 36.h even 6 1
162.2.a.c 1 36.f odd 6 1
432.2.i.b 2 3.b odd 2 1
432.2.i.b 2 9.d odd 6 1
450.2.e.i 2 20.d odd 2 1
450.2.e.i 2 180.p odd 6 1
450.2.j.e 4 20.e even 4 2
450.2.j.e 4 180.x even 12 2
576.2.i.a 2 8.b even 2 1
576.2.i.a 2 72.n even 6 1
576.2.i.g 2 8.d odd 2 1
576.2.i.g 2 72.p odd 6 1
882.2.e.g 2 28.f even 6 1
882.2.e.g 2 252.n even 6 1
882.2.e.i 2 28.g odd 6 1
882.2.e.i 2 252.bl odd 6 1
882.2.f.d 2 28.d even 2 1
882.2.f.d 2 252.bi even 6 1
882.2.h.b 2 28.f even 6 1
882.2.h.b 2 252.bj even 6 1
882.2.h.c 2 28.g odd 6 1
882.2.h.c 2 252.u odd 6 1
1296.2.a.f 1 9.d odd 6 1
1296.2.a.g 1 9.c even 3 1
1350.2.e.c 2 60.h even 2 1
1350.2.e.c 2 180.n even 6 1
1350.2.j.a 4 60.l odd 4 2
1350.2.j.a 4 180.v odd 12 2
1728.2.i.e 2 24.f even 2 1
1728.2.i.e 2 72.l even 6 1
1728.2.i.f 2 24.h odd 2 1
1728.2.i.f 2 72.j odd 6 1
2646.2.e.b 2 84.n even 6 1
2646.2.e.b 2 252.o even 6 1
2646.2.e.c 2 84.j odd 6 1
2646.2.e.c 2 252.bn odd 6 1
2646.2.f.g 2 84.h odd 2 1
2646.2.f.g 2 252.s odd 6 1
2646.2.h.h 2 84.n even 6 1
2646.2.h.h 2 252.bb even 6 1
2646.2.h.i 2 84.j odd 6 1
2646.2.h.i 2 252.r odd 6 1
4050.2.a.c 1 180.p odd 6 1
4050.2.a.v 1 180.n even 6 1
4050.2.c.c 2 180.x even 12 2
4050.2.c.r 2 180.v odd 12 2
5184.2.a.o 1 72.n even 6 1
5184.2.a.p 1 72.j odd 6 1
5184.2.a.q 1 72.l even 6 1
5184.2.a.r 1 72.p odd 6 1
7938.2.a.i 1 252.s odd 6 1
7938.2.a.x 1 252.bi even 6 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$3 - 3 T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$4 - 2 T + T^{2}$$
$11$ $$9 + 3 T + T^{2}$$
$13$ $$4 + 2 T + T^{2}$$
$17$ $$( 3 + T )^{2}$$
$19$ $$( -1 + T )^{2}$$
$23$ $$36 + 6 T + T^{2}$$
$29$ $$36 + 6 T + T^{2}$$
$31$ $$16 + 4 T + T^{2}$$
$37$ $$( 4 + T )^{2}$$
$41$ $$81 + 9 T + T^{2}$$
$43$ $$1 + T + T^{2}$$
$47$ $$36 + 6 T + T^{2}$$
$53$ $$( -12 + T )^{2}$$
$59$ $$9 - 3 T + T^{2}$$
$61$ $$64 + 8 T + T^{2}$$
$67$ $$25 - 5 T + T^{2}$$
$71$ $$( -12 + T )^{2}$$
$73$ $$( -11 + T )^{2}$$
$79$ $$16 + 4 T + T^{2}$$
$83$ $$144 - 12 T + T^{2}$$
$89$ $$( -6 + T )^{2}$$
$97$ $$25 + 5 T + T^{2}$$