# Properties

 Label 144.2.a.a Level $144$ Weight $2$ Character orbit 144.a Self dual yes Analytic conductor $1.150$ Analytic rank $0$ Dimension $1$ CM discriminant -3 Inner twists $2$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$144 = 2^{4} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 144.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$1.14984578911$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 36) Fricke sign: $$-1$$ Sato-Tate group: $N(\mathrm{U}(1))$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 4 q^{7} + O(q^{10})$$ $$q + 4 q^{7} + 2 q^{13} - 8 q^{19} - 5 q^{25} + 4 q^{31} - 10 q^{37} - 8 q^{43} + 9 q^{49} + 14 q^{61} + 16 q^{67} - 10 q^{73} + 4 q^{79} + 8 q^{91} + 14 q^{97} + O(q^{100})$$

## Expression as an eta quotient

$$f(z) = \dfrac{\eta(12z)^{12}}{\eta(6z)^{4}\eta(24z)^{4}}=q\prod_{n=1}^\infty(1 - q^{6n})^{-4}(1 - q^{12n})^{12}(1 - q^{24n})^{-4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 0 0 0 0 4.00000 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 144.2.a.a 1
3.b odd 2 1 CM 144.2.a.a 1
4.b odd 2 1 36.2.a.a 1
5.b even 2 1 3600.2.a.e 1
5.c odd 4 2 3600.2.f.m 2
7.b odd 2 1 7056.2.a.bb 1
8.b even 2 1 576.2.a.f 1
8.d odd 2 1 576.2.a.e 1
9.c even 3 2 1296.2.i.h 2
9.d odd 6 2 1296.2.i.h 2
12.b even 2 1 36.2.a.a 1
15.d odd 2 1 3600.2.a.e 1
15.e even 4 2 3600.2.f.m 2
16.e even 4 2 2304.2.d.a 2
16.f odd 4 2 2304.2.d.q 2
20.d odd 2 1 900.2.a.g 1
20.e even 4 2 900.2.d.b 2
21.c even 2 1 7056.2.a.bb 1
24.f even 2 1 576.2.a.e 1
24.h odd 2 1 576.2.a.f 1
28.d even 2 1 1764.2.a.e 1
28.f even 6 2 1764.2.k.g 2
28.g odd 6 2 1764.2.k.h 2
36.f odd 6 2 324.2.e.c 2
36.h even 6 2 324.2.e.c 2
44.c even 2 1 4356.2.a.g 1
48.i odd 4 2 2304.2.d.a 2
48.k even 4 2 2304.2.d.q 2
52.b odd 2 1 6084.2.a.i 1
52.f even 4 2 6084.2.b.f 2
60.h even 2 1 900.2.a.g 1
60.l odd 4 2 900.2.d.b 2
84.h odd 2 1 1764.2.a.e 1
84.j odd 6 2 1764.2.k.g 2
84.n even 6 2 1764.2.k.h 2
132.d odd 2 1 4356.2.a.g 1
156.h even 2 1 6084.2.a.i 1
156.l odd 4 2 6084.2.b.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.2.a.a 1 4.b odd 2 1
36.2.a.a 1 12.b even 2 1
144.2.a.a 1 1.a even 1 1 trivial
144.2.a.a 1 3.b odd 2 1 CM
324.2.e.c 2 36.f odd 6 2
324.2.e.c 2 36.h even 6 2
576.2.a.e 1 8.d odd 2 1
576.2.a.e 1 24.f even 2 1
576.2.a.f 1 8.b even 2 1
576.2.a.f 1 24.h odd 2 1
900.2.a.g 1 20.d odd 2 1
900.2.a.g 1 60.h even 2 1
900.2.d.b 2 20.e even 4 2
900.2.d.b 2 60.l odd 4 2
1296.2.i.h 2 9.c even 3 2
1296.2.i.h 2 9.d odd 6 2
1764.2.a.e 1 28.d even 2 1
1764.2.a.e 1 84.h odd 2 1
1764.2.k.g 2 28.f even 6 2
1764.2.k.g 2 84.j odd 6 2
1764.2.k.h 2 28.g odd 6 2
1764.2.k.h 2 84.n even 6 2
2304.2.d.a 2 16.e even 4 2
2304.2.d.a 2 48.i odd 4 2
2304.2.d.q 2 16.f odd 4 2
2304.2.d.q 2 48.k even 4 2
3600.2.a.e 1 5.b even 2 1
3600.2.a.e 1 15.d odd 2 1
3600.2.f.m 2 5.c odd 4 2
3600.2.f.m 2 15.e even 4 2
4356.2.a.g 1 44.c even 2 1
4356.2.a.g 1 132.d odd 2 1
6084.2.a.i 1 52.b odd 2 1
6084.2.a.i 1 156.h even 2 1
6084.2.b.f 2 52.f even 4 2
6084.2.b.f 2 156.l odd 4 2
7056.2.a.bb 1 7.b odd 2 1
7056.2.a.bb 1 21.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(144))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$-4 + T$$
$11$ $$T$$
$13$ $$-2 + T$$
$17$ $$T$$
$19$ $$8 + T$$
$23$ $$T$$
$29$ $$T$$
$31$ $$-4 + T$$
$37$ $$10 + T$$
$41$ $$T$$
$43$ $$8 + T$$
$47$ $$T$$
$53$ $$T$$
$59$ $$T$$
$61$ $$-14 + T$$
$67$ $$-16 + T$$
$71$ $$T$$
$73$ $$10 + T$$
$79$ $$-4 + T$$
$83$ $$T$$
$89$ $$T$$
$97$ $$-14 + T$$
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