# Properties

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

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## Newspace parameters

 Level: $$N$$ $$=$$ $$144 = 2^{4} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 144.g (of order $$2$$, degree $$1$$, 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 48) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -6 q^{5} + ( 4 - 8 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q -6 q^{5} + ( 4 - 8 \zeta_{6} ) q^{7} + ( 12 - 24 \zeta_{6} ) q^{11} -14 q^{13} + 6 q^{17} + ( -4 + 8 \zeta_{6} ) q^{19} + 11 q^{25} -30 q^{29} + ( 12 - 24 \zeta_{6} ) q^{31} + ( -24 + 48 \zeta_{6} ) q^{35} + 26 q^{37} + 54 q^{41} + ( -12 + 24 \zeta_{6} ) q^{43} + ( -24 + 48 \zeta_{6} ) q^{47} + q^{49} + 18 q^{53} + ( -72 + 144 \zeta_{6} ) q^{55} + ( -12 + 24 \zeta_{6} ) q^{59} -70 q^{61} + 84 q^{65} + ( 68 - 136 \zeta_{6} ) q^{67} + ( 48 - 96 \zeta_{6} ) q^{71} + 82 q^{73} -144 q^{77} + ( 44 - 88 \zeta_{6} ) q^{79} + ( -12 + 24 \zeta_{6} ) q^{83} -36 q^{85} -114 q^{89} + ( -56 + 112 \zeta_{6} ) q^{91} + ( 24 - 48 \zeta_{6} ) q^{95} + 34 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 12q^{5} + O(q^{10})$$ $$2q - 12q^{5} - 28q^{13} + 12q^{17} + 22q^{25} - 60q^{29} + 52q^{37} + 108q^{41} + 2q^{49} + 36q^{53} - 140q^{61} + 168q^{65} + 164q^{73} - 288q^{77} - 72q^{85} - 228q^{89} + 68q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$37$$ $$65$$ $$127$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
127.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 0 −6.00000 0 6.92820i 0 0 0
127.2 0 0 0 −6.00000 0 6.92820i 0 0 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 144.3.g.b 2
3.b odd 2 1 48.3.g.a 2
4.b odd 2 1 inner 144.3.g.b 2
5.b even 2 1 3600.3.e.t 2
5.c odd 4 2 3600.3.j.i 4
8.b even 2 1 576.3.g.i 2
8.d odd 2 1 576.3.g.i 2
9.c even 3 1 1296.3.o.n 2
9.c even 3 1 1296.3.o.p 2
9.d odd 6 1 1296.3.o.a 2
9.d odd 6 1 1296.3.o.c 2
12.b even 2 1 48.3.g.a 2
15.d odd 2 1 1200.3.e.h 2
15.e even 4 2 1200.3.j.a 4
16.e even 4 2 2304.3.b.n 4
16.f odd 4 2 2304.3.b.n 4
20.d odd 2 1 3600.3.e.t 2
20.e even 4 2 3600.3.j.i 4
21.c even 2 1 2352.3.m.a 2
24.f even 2 1 192.3.g.a 2
24.h odd 2 1 192.3.g.a 2
36.f odd 6 1 1296.3.o.n 2
36.f odd 6 1 1296.3.o.p 2
36.h even 6 1 1296.3.o.a 2
36.h even 6 1 1296.3.o.c 2
48.i odd 4 2 768.3.b.b 4
48.k even 4 2 768.3.b.b 4
60.h even 2 1 1200.3.e.h 2
60.l odd 4 2 1200.3.j.a 4
84.h odd 2 1 2352.3.m.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.3.g.a 2 3.b odd 2 1
48.3.g.a 2 12.b even 2 1
144.3.g.b 2 1.a even 1 1 trivial
144.3.g.b 2 4.b odd 2 1 inner
192.3.g.a 2 24.f even 2 1
192.3.g.a 2 24.h odd 2 1
576.3.g.i 2 8.b even 2 1
576.3.g.i 2 8.d odd 2 1
768.3.b.b 4 48.i odd 4 2
768.3.b.b 4 48.k even 4 2
1200.3.e.h 2 15.d odd 2 1
1200.3.e.h 2 60.h even 2 1
1200.3.j.a 4 15.e even 4 2
1200.3.j.a 4 60.l odd 4 2
1296.3.o.a 2 9.d odd 6 1
1296.3.o.a 2 36.h even 6 1
1296.3.o.c 2 9.d odd 6 1
1296.3.o.c 2 36.h even 6 1
1296.3.o.n 2 9.c even 3 1
1296.3.o.n 2 36.f odd 6 1
1296.3.o.p 2 9.c even 3 1
1296.3.o.p 2 36.f odd 6 1
2304.3.b.n 4 16.e even 4 2
2304.3.b.n 4 16.f odd 4 2
2352.3.m.a 2 21.c even 2 1
2352.3.m.a 2 84.h odd 2 1
3600.3.e.t 2 5.b even 2 1
3600.3.e.t 2 20.d odd 2 1
3600.3.j.i 4 5.c odd 4 2
3600.3.j.i 4 20.e even 4 2

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$( 6 + T )^{2}$$
$7$ $$48 + T^{2}$$
$11$ $$432 + T^{2}$$
$13$ $$( 14 + T )^{2}$$
$17$ $$( -6 + T )^{2}$$
$19$ $$48 + T^{2}$$
$23$ $$T^{2}$$
$29$ $$( 30 + T )^{2}$$
$31$ $$432 + T^{2}$$
$37$ $$( -26 + T )^{2}$$
$41$ $$( -54 + T )^{2}$$
$43$ $$432 + T^{2}$$
$47$ $$1728 + T^{2}$$
$53$ $$( -18 + T )^{2}$$
$59$ $$432 + T^{2}$$
$61$ $$( 70 + T )^{2}$$
$67$ $$13872 + T^{2}$$
$71$ $$6912 + T^{2}$$
$73$ $$( -82 + T )^{2}$$
$79$ $$5808 + T^{2}$$
$83$ $$432 + T^{2}$$
$89$ $$( 114 + T )^{2}$$
$97$ $$( -34 + T )^{2}$$
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