Properties

 Label 552.2.a.a Level $552$ Weight $2$ Character orbit 552.a Self dual yes Analytic conductor $4.408$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$552 = 2^{3} \cdot 3 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 552.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$4.40774219157$$ Analytic rank: $$1$$ 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

 $$f(q)$$ $$=$$ $$q - q^{3} - 2q^{5} + 2q^{7} + q^{9} + O(q^{10})$$ $$q - q^{3} - 2q^{5} + 2q^{7} + q^{9} - 2q^{11} - 2q^{13} + 2q^{15} - 4q^{17} - 2q^{21} - q^{23} - q^{25} - q^{27} - 10q^{29} + 2q^{33} - 4q^{35} - 4q^{37} + 2q^{39} - 6q^{41} - 4q^{43} - 2q^{45} + 8q^{47} - 3q^{49} + 4q^{51} - 6q^{53} + 4q^{55} + 4q^{59} + 8q^{61} + 2q^{63} + 4q^{65} - 4q^{67} + q^{69} + 8q^{71} + 6q^{73} + q^{75} - 4q^{77} + 6q^{79} + q^{81} - 6q^{83} + 8q^{85} + 10q^{87} - 4q^{89} - 4q^{91} + 10q^{97} - 2q^{99} + O(q^{100})$$

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 −1.00000 0 −2.00000 0 2.00000 0 1.00000 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
$$2$$ $$1$$
$$3$$ $$1$$
$$23$$ $$1$$

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 552.2.a.a 1
3.b odd 2 1 1656.2.a.h 1
4.b odd 2 1 1104.2.a.d 1
8.b even 2 1 4416.2.a.ba 1
8.d odd 2 1 4416.2.a.l 1
12.b even 2 1 3312.2.a.m 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
552.2.a.a 1 1.a even 1 1 trivial
1104.2.a.d 1 4.b odd 2 1
1656.2.a.h 1 3.b odd 2 1
3312.2.a.m 1 12.b even 2 1
4416.2.a.l 1 8.d odd 2 1
4416.2.a.ba 1 8.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(552))$$:

 $$T_{5} + 2$$ $$T_{7} - 2$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$1 + T$$
$5$ $$2 + T$$
$7$ $$-2 + T$$
$11$ $$2 + T$$
$13$ $$2 + T$$
$17$ $$4 + T$$
$19$ $$T$$
$23$ $$1 + T$$
$29$ $$10 + T$$
$31$ $$T$$
$37$ $$4 + T$$
$41$ $$6 + T$$
$43$ $$4 + T$$
$47$ $$-8 + T$$
$53$ $$6 + T$$
$59$ $$-4 + T$$
$61$ $$-8 + T$$
$67$ $$4 + T$$
$71$ $$-8 + T$$
$73$ $$-6 + T$$
$79$ $$-6 + T$$
$83$ $$6 + T$$
$89$ $$4 + T$$
$97$ $$-10 + T$$