# Properties

 Label 144.2.i.b 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 72) 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + 2 \zeta_{6} ) q^{3} + \zeta_{6} q^{5} + ( -3 + 3 \zeta_{6} ) q^{7} -3 q^{9} +O(q^{10})$$ $$q + ( -1 + 2 \zeta_{6} ) q^{3} + \zeta_{6} q^{5} + ( -3 + 3 \zeta_{6} ) q^{7} -3 q^{9} + ( 5 - 5 \zeta_{6} ) q^{11} + 5 \zeta_{6} q^{13} + ( -2 + \zeta_{6} ) q^{15} -2 q^{17} + 4 q^{19} + ( -3 - 3 \zeta_{6} ) q^{21} -\zeta_{6} q^{23} + ( 4 - 4 \zeta_{6} ) q^{25} + ( 3 - 6 \zeta_{6} ) q^{27} + ( 9 - 9 \zeta_{6} ) q^{29} -\zeta_{6} q^{31} + ( 5 + 5 \zeta_{6} ) q^{33} -3 q^{35} -6 q^{37} + ( -10 + 5 \zeta_{6} ) q^{39} -3 \zeta_{6} q^{41} + ( 1 - \zeta_{6} ) q^{43} -3 \zeta_{6} q^{45} + ( -3 + 3 \zeta_{6} ) q^{47} -2 \zeta_{6} q^{49} + ( 2 - 4 \zeta_{6} ) q^{51} + 2 q^{53} + 5 q^{55} + ( -4 + 8 \zeta_{6} ) q^{57} + 11 \zeta_{6} q^{59} + ( -7 + 7 \zeta_{6} ) q^{61} + ( 9 - 9 \zeta_{6} ) q^{63} + ( -5 + 5 \zeta_{6} ) q^{65} -\zeta_{6} q^{67} + ( 2 - \zeta_{6} ) q^{69} -4 q^{71} -2 q^{73} + ( 4 + 4 \zeta_{6} ) q^{75} + 15 \zeta_{6} q^{77} + ( 1 - \zeta_{6} ) q^{79} + 9 q^{81} + ( 1 - \zeta_{6} ) q^{83} -2 \zeta_{6} q^{85} + ( 9 + 9 \zeta_{6} ) q^{87} -18 q^{89} -15 q^{91} + ( 2 - \zeta_{6} ) q^{93} + 4 \zeta_{6} q^{95} + ( 13 - 13 \zeta_{6} ) q^{97} + ( -15 + 15 \zeta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{5} - 3q^{7} - 6q^{9} + O(q^{10})$$ $$2q + q^{5} - 3q^{7} - 6q^{9} + 5q^{11} + 5q^{13} - 3q^{15} - 4q^{17} + 8q^{19} - 9q^{21} - q^{23} + 4q^{25} + 9q^{29} - q^{31} + 15q^{33} - 6q^{35} - 12q^{37} - 15q^{39} - 3q^{41} + q^{43} - 3q^{45} - 3q^{47} - 2q^{49} + 4q^{53} + 10q^{55} + 11q^{59} - 7q^{61} + 9q^{63} - 5q^{65} - q^{67} + 3q^{69} - 8q^{71} - 4q^{73} + 12q^{75} + 15q^{77} + q^{79} + 18q^{81} + q^{83} - 2q^{85} + 27q^{87} - 36q^{89} - 30q^{91} + 3q^{93} + 4q^{95} + 13q^{97} - 15q^{99} + O(q^{100})$$

## 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 1.73205i 0 0.500000 0.866025i 0 −1.50000 2.59808i 0 −3.00000 0
97.1 0 1.73205i 0 0.500000 + 0.866025i 0 −1.50000 + 2.59808i 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
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.b 2
3.b odd 2 1 432.2.i.a 2
4.b odd 2 1 72.2.i.a 2
8.b even 2 1 576.2.i.c 2
8.d odd 2 1 576.2.i.d 2
9.c even 3 1 inner 144.2.i.b 2
9.c even 3 1 1296.2.a.e 1
9.d odd 6 1 432.2.i.a 2
9.d odd 6 1 1296.2.a.i 1
12.b even 2 1 216.2.i.a 2
24.f even 2 1 1728.2.i.h 2
24.h odd 2 1 1728.2.i.g 2
36.f odd 6 1 72.2.i.a 2
36.f odd 6 1 648.2.a.a 1
36.h even 6 1 216.2.i.a 2
36.h even 6 1 648.2.a.c 1
72.j odd 6 1 1728.2.i.g 2
72.j odd 6 1 5184.2.a.n 1
72.l even 6 1 1728.2.i.h 2
72.l even 6 1 5184.2.a.i 1
72.n even 6 1 576.2.i.c 2
72.n even 6 1 5184.2.a.x 1
72.p odd 6 1 576.2.i.d 2
72.p odd 6 1 5184.2.a.s 1

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - T_{5} + 1$$ acting on $$S_{2}^{\mathrm{new}}(144, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + 3 T^{2}$$
$5$ $$1 - T - 4 T^{2} - 5 T^{3} + 25 T^{4}$$
$7$ $$1 + 3 T + 2 T^{2} + 21 T^{3} + 49 T^{4}$$
$11$ $$1 - 5 T + 14 T^{2} - 55 T^{3} + 121 T^{4}$$
$13$ $$( 1 - 7 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} )$$
$17$ $$( 1 + 2 T + 17 T^{2} )^{2}$$
$19$ $$( 1 - 4 T + 19 T^{2} )^{2}$$
$23$ $$1 + T - 22 T^{2} + 23 T^{3} + 529 T^{4}$$
$29$ $$1 - 9 T + 52 T^{2} - 261 T^{3} + 841 T^{4}$$
$31$ $$1 + T - 30 T^{2} + 31 T^{3} + 961 T^{4}$$
$37$ $$( 1 + 6 T + 37 T^{2} )^{2}$$
$41$ $$1 + 3 T - 32 T^{2} + 123 T^{3} + 1681 T^{4}$$
$43$ $$1 - T - 42 T^{2} - 43 T^{3} + 1849 T^{4}$$
$47$ $$1 + 3 T - 38 T^{2} + 141 T^{3} + 2209 T^{4}$$
$53$ $$( 1 - 2 T + 53 T^{2} )^{2}$$
$59$ $$1 - 11 T + 62 T^{2} - 649 T^{3} + 3481 T^{4}$$
$61$ $$1 + 7 T - 12 T^{2} + 427 T^{3} + 3721 T^{4}$$
$67$ $$1 + T - 66 T^{2} + 67 T^{3} + 4489 T^{4}$$
$71$ $$( 1 + 4 T + 71 T^{2} )^{2}$$
$73$ $$( 1 + 2 T + 73 T^{2} )^{2}$$
$79$ $$1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4}$$
$83$ $$1 - T - 82 T^{2} - 83 T^{3} + 6889 T^{4}$$
$89$ $$( 1 + 18 T + 89 T^{2} )^{2}$$
$97$ $$1 - 13 T + 72 T^{2} - 1261 T^{3} + 9409 T^{4}$$