# Properties

 Label 144.8.a.g Level $144$ Weight $8$ Character orbit 144.a Self dual yes Analytic conductor $44.983$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$144 = 2^{4} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 144.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$44.9834436697$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 8) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 82q^{5} + 456q^{7} + O(q^{10})$$ $$q + 82q^{5} + 456q^{7} - 2524q^{11} - 10778q^{13} + 11150q^{17} - 4124q^{19} + 81704q^{23} - 71401q^{25} - 99798q^{29} + 40480q^{31} + 37392q^{35} - 419442q^{37} - 141402q^{41} + 690428q^{43} - 682032q^{47} - 615607q^{49} - 1813118q^{53} - 206968q^{55} - 966028q^{59} + 1887670q^{61} - 883796q^{65} - 2965868q^{67} - 2548232q^{71} - 1680326q^{73} - 1150944q^{77} - 4038064q^{79} - 5385764q^{83} + 914300q^{85} + 6473046q^{89} - 4914768q^{91} - 338168q^{95} - 6065758q^{97} + O(q^{100})$$

## 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 82.0000 0 456.000 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 144.8.a.g 1
3.b odd 2 1 16.8.a.c 1
4.b odd 2 1 72.8.a.d 1
8.b even 2 1 576.8.a.k 1
8.d odd 2 1 576.8.a.j 1
12.b even 2 1 8.8.a.a 1
15.d odd 2 1 400.8.a.b 1
15.e even 4 2 400.8.c.b 2
24.f even 2 1 64.8.a.g 1
24.h odd 2 1 64.8.a.a 1
48.i odd 4 2 256.8.b.c 2
48.k even 4 2 256.8.b.e 2
60.h even 2 1 200.8.a.i 1
60.l odd 4 2 200.8.c.a 2
84.h odd 2 1 392.8.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.8.a.a 1 12.b even 2 1
16.8.a.c 1 3.b odd 2 1
64.8.a.a 1 24.h odd 2 1
64.8.a.g 1 24.f even 2 1
72.8.a.d 1 4.b odd 2 1
144.8.a.g 1 1.a even 1 1 trivial
200.8.a.i 1 60.h even 2 1
200.8.c.a 2 60.l odd 4 2
256.8.b.c 2 48.i odd 4 2
256.8.b.e 2 48.k even 4 2
392.8.a.d 1 84.h odd 2 1
400.8.a.b 1 15.d odd 2 1
400.8.c.b 2 15.e even 4 2
576.8.a.j 1 8.d odd 2 1
576.8.a.k 1 8.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} - 82$$ acting on $$S_{8}^{\mathrm{new}}(\Gamma_0(144))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$-82 + T$$
$7$ $$-456 + T$$
$11$ $$2524 + T$$
$13$ $$10778 + T$$
$17$ $$-11150 + T$$
$19$ $$4124 + T$$
$23$ $$-81704 + T$$
$29$ $$99798 + T$$
$31$ $$-40480 + T$$
$37$ $$419442 + T$$
$41$ $$141402 + T$$
$43$ $$-690428 + T$$
$47$ $$682032 + T$$
$53$ $$1813118 + T$$
$59$ $$966028 + T$$
$61$ $$-1887670 + T$$
$67$ $$2965868 + T$$
$71$ $$2548232 + T$$
$73$ $$1680326 + T$$
$79$ $$4038064 + T$$
$83$ $$5385764 + T$$
$89$ $$-6473046 + T$$
$97$ $$6065758 + T$$