# Properties

 Label 144.16.a.j Level $144$ Weight $16$ Character orbit 144.a Self dual yes Analytic conductor $205.479$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$144 = 2^{4} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$16$$ Character orbit: $$[\chi]$$ $$=$$ 144.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$205.478647344$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 6) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 114810 q^{5} + 3034528 q^{7} + O(q^{10})$$ $$q + 114810 q^{5} + 3034528 q^{7} - 103451700 q^{11} - 104365834 q^{13} - 997689762 q^{17} - 4934015444 q^{19} + 8324920200 q^{23} - 17336242025 q^{25} - 104128242846 q^{29} + 296696681512 q^{31} + 348394159680 q^{35} - 178337455666 q^{37} + 1790882416086 q^{41} + 2863459422772 q^{43} + 4332907521600 q^{47} + 4460798672841 q^{49} - 9732317104422 q^{53} - 11877289677000 q^{55} - 13514837176500 q^{59} + 5352663511190 q^{61} - 11982241401540 q^{65} + 53233909720108 q^{67} - 20229661643400 q^{71} + 26264166466106 q^{73} - 313927080297600 q^{77} + 339031361615128 q^{79} + 131684771045076 q^{83} - 114544761575220 q^{85} + 39352148322678 q^{89} - 316701045516352 q^{91} - 566474313125640 q^{95} + 1128750908801474 q^{97} + O(q^{100})$$

## 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 114810. 0 3.03453e6 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

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 144.16.a.j 1
3.b odd 2 1 48.16.a.d 1
4.b odd 2 1 18.16.a.b 1
12.b even 2 1 6.16.a.b 1
60.h even 2 1 150.16.a.f 1
60.l odd 4 2 150.16.c.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.16.a.b 1 12.b even 2 1
18.16.a.b 1 4.b odd 2 1
48.16.a.d 1 3.b odd 2 1
144.16.a.j 1 1.a even 1 1 trivial
150.16.a.f 1 60.h even 2 1
150.16.c.a 2 60.l odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} - 114810$$ acting on $$S_{16}^{\mathrm{new}}(\Gamma_0(144))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$-114810 + T$$
$7$ $$-3034528 + T$$
$11$ $$103451700 + T$$
$13$ $$104365834 + T$$
$17$ $$997689762 + T$$
$19$ $$4934015444 + T$$
$23$ $$-8324920200 + T$$
$29$ $$104128242846 + T$$
$31$ $$-296696681512 + T$$
$37$ $$178337455666 + T$$
$41$ $$-1790882416086 + T$$
$43$ $$-2863459422772 + T$$
$47$ $$-4332907521600 + T$$
$53$ $$9732317104422 + T$$
$59$ $$13514837176500 + T$$
$61$ $$-5352663511190 + T$$
$67$ $$-53233909720108 + T$$
$71$ $$20229661643400 + T$$
$73$ $$-26264166466106 + T$$
$79$ $$-339031361615128 + T$$
$83$ $$-131684771045076 + T$$
$89$ $$-39352148322678 + T$$
$97$ $$-1128750908801474 + T$$
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