Properties

 Label 648.4.a.a Level $648$ Weight $4$ Character orbit 648.a Self dual yes Analytic conductor $38.233$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$648 = 2^{3} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 648.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$38.2332376837$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

 $$f(q)$$ $$=$$ $$q - 5q^{5} + 36q^{7} + O(q^{10})$$ $$q - 5q^{5} + 36q^{7} + 64q^{11} - 65q^{13} + 59q^{17} - 28q^{19} + 160q^{23} - 100q^{25} - 57q^{29} + 164q^{31} - 180q^{35} - 321q^{37} - 246q^{41} - 8q^{43} + 84q^{47} + 953q^{49} + 478q^{53} - 320q^{55} - 32q^{59} + 415q^{61} + 325q^{65} - 220q^{67} + 884q^{71} - 77q^{73} + 2304q^{77} - 80q^{79} + 1268q^{83} - 295q^{85} + 123q^{89} - 2340q^{91} + 140q^{95} + 1346q^{97} + O(q^{100})$$

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 −5.00000 0 36.0000 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 648.4.a.a 1
3.b odd 2 1 648.4.a.b yes 1
4.b odd 2 1 1296.4.a.c 1
9.c even 3 2 648.4.i.j 2
9.d odd 6 2 648.4.i.c 2
12.b even 2 1 1296.4.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
648.4.a.a 1 1.a even 1 1 trivial
648.4.a.b yes 1 3.b odd 2 1
648.4.i.c 2 9.d odd 6 2
648.4.i.j 2 9.c even 3 2
1296.4.a.c 1 4.b odd 2 1
1296.4.a.f 1 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} + 5$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(648))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$5 + T$$
$7$ $$-36 + T$$
$11$ $$-64 + T$$
$13$ $$65 + T$$
$17$ $$-59 + T$$
$19$ $$28 + T$$
$23$ $$-160 + T$$
$29$ $$57 + T$$
$31$ $$-164 + T$$
$37$ $$321 + T$$
$41$ $$246 + T$$
$43$ $$8 + T$$
$47$ $$-84 + T$$
$53$ $$-478 + T$$
$59$ $$32 + T$$
$61$ $$-415 + T$$
$67$ $$220 + T$$
$71$ $$-884 + T$$
$73$ $$77 + T$$
$79$ $$80 + T$$
$83$ $$-1268 + T$$
$89$ $$-123 + T$$
$97$ $$-1346 + T$$