Properties

 Label 144.4.a.f Level $144$ Weight $4$ Character orbit 144.a Self dual yes Analytic conductor $8.496$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

Related objects

Newspace parameters

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

Newform invariants

 Self dual: yes Analytic conductor: $$8.49627504083$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 72) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

 $$f(q)$$ $$=$$ $$q + 16q^{5} + 12q^{7} + O(q^{10})$$ $$q + 16q^{5} + 12q^{7} - 64q^{11} + 58q^{13} + 32q^{17} + 136q^{19} + 128q^{23} + 131q^{25} - 144q^{29} - 20q^{31} + 192q^{35} - 18q^{37} - 288q^{41} + 200q^{43} - 384q^{47} - 199q^{49} + 496q^{53} - 1024q^{55} + 128q^{59} - 458q^{61} + 928q^{65} + 496q^{67} - 512q^{71} - 602q^{73} - 768q^{77} - 1108q^{79} - 704q^{83} + 512q^{85} - 960q^{89} + 696q^{91} + 2176q^{95} + 206q^{97} + O(q^{100})$$

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 16.0000 0 12.0000 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 144.4.a.f 1
3.b odd 2 1 144.4.a.a 1
4.b odd 2 1 72.4.a.d yes 1
8.b even 2 1 576.4.a.d 1
8.d odd 2 1 576.4.a.c 1
12.b even 2 1 72.4.a.a 1
20.d odd 2 1 1800.4.a.ba 1
20.e even 4 2 1800.4.f.x 2
24.f even 2 1 576.4.a.w 1
24.h odd 2 1 576.4.a.x 1
36.f odd 6 2 648.4.i.a 2
36.h even 6 2 648.4.i.l 2
60.h even 2 1 1800.4.a.z 1
60.l odd 4 2 1800.4.f.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.4.a.a 1 12.b even 2 1
72.4.a.d yes 1 4.b odd 2 1
144.4.a.a 1 3.b odd 2 1
144.4.a.f 1 1.a even 1 1 trivial
576.4.a.c 1 8.d odd 2 1
576.4.a.d 1 8.b even 2 1
576.4.a.w 1 24.f even 2 1
576.4.a.x 1 24.h odd 2 1
648.4.i.a 2 36.f odd 6 2
648.4.i.l 2 36.h even 6 2
1800.4.a.z 1 60.h even 2 1
1800.4.a.ba 1 20.d odd 2 1
1800.4.f.b 2 60.l odd 4 2
1800.4.f.x 2 20.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} - 16$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(144))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$-16 + T$$
$7$ $$-12 + T$$
$11$ $$64 + T$$
$13$ $$-58 + T$$
$17$ $$-32 + T$$
$19$ $$-136 + T$$
$23$ $$-128 + T$$
$29$ $$144 + T$$
$31$ $$20 + T$$
$37$ $$18 + T$$
$41$ $$288 + T$$
$43$ $$-200 + T$$
$47$ $$384 + T$$
$53$ $$-496 + T$$
$59$ $$-128 + T$$
$61$ $$458 + T$$
$67$ $$-496 + T$$
$71$ $$512 + T$$
$73$ $$602 + T$$
$79$ $$1108 + T$$
$83$ $$704 + T$$
$89$ $$960 + T$$
$97$ $$-206 + T$$