# Properties

 Label 552.2.a.f Level $552$ Weight $2$ Character orbit 552.a Self dual yes Analytic conductor $4.408$ Analytic rank $0$ Dimension $2$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{3} + ( 1 + \beta ) q^{5} + 2 q^{7} + q^{9} +O(q^{10})$$ $$q + q^{3} + ( 1 + \beta ) q^{5} + 2 q^{7} + q^{9} + ( 1 + \beta ) q^{11} -2 \beta q^{13} + ( 1 + \beta ) q^{15} + ( -2 - 2 \beta ) q^{17} + ( 1 - \beta ) q^{19} + 2 q^{21} + q^{23} + ( 1 + 2 \beta ) q^{25} + q^{27} + ( -4 - 2 \beta ) q^{29} + ( 6 - 2 \beta ) q^{31} + ( 1 + \beta ) q^{33} + ( 2 + 2 \beta ) q^{35} + ( 1 + 3 \beta ) q^{37} -2 \beta q^{39} -2 q^{41} + ( 3 + \beta ) q^{43} + ( 1 + \beta ) q^{45} -4 \beta q^{47} -3 q^{49} + ( -2 - 2 \beta ) q^{51} + ( 1 + 5 \beta ) q^{53} + ( 6 + 2 \beta ) q^{55} + ( 1 - \beta ) q^{57} -4 q^{59} + ( -5 - 3 \beta ) q^{61} + 2 q^{63} + ( -10 - 2 \beta ) q^{65} + ( 1 + 3 \beta ) q^{67} + q^{69} + 4 \beta q^{71} + 2 \beta q^{73} + ( 1 + 2 \beta ) q^{75} + ( 2 + 2 \beta ) q^{77} + 14 q^{79} + q^{81} + ( -9 - \beta ) q^{83} + ( -12 - 4 \beta ) q^{85} + ( -4 - 2 \beta ) q^{87} + ( -8 - 4 \beta ) q^{89} -4 \beta q^{91} + ( 6 - 2 \beta ) q^{93} -4 q^{95} + 2 \beta q^{97} + ( 1 + \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{3} + 2q^{5} + 4q^{7} + 2q^{9} + O(q^{10})$$ $$2q + 2q^{3} + 2q^{5} + 4q^{7} + 2q^{9} + 2q^{11} + 2q^{15} - 4q^{17} + 2q^{19} + 4q^{21} + 2q^{23} + 2q^{25} + 2q^{27} - 8q^{29} + 12q^{31} + 2q^{33} + 4q^{35} + 2q^{37} - 4q^{41} + 6q^{43} + 2q^{45} - 6q^{49} - 4q^{51} + 2q^{53} + 12q^{55} + 2q^{57} - 8q^{59} - 10q^{61} + 4q^{63} - 20q^{65} + 2q^{67} + 2q^{69} + 2q^{75} + 4q^{77} + 28q^{79} + 2q^{81} - 18q^{83} - 24q^{85} - 8q^{87} - 16q^{89} + 12q^{93} - 8q^{95} + 2q^{99} + O(q^{100})$$

## 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
 −0.618034 1.61803
0 1.00000 0 −1.23607 0 2.00000 0 1.00000 0
1.2 0 1.00000 0 3.23607 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.f 2
3.b odd 2 1 1656.2.a.l 2
4.b odd 2 1 1104.2.a.k 2
8.b even 2 1 4416.2.a.bd 2
8.d odd 2 1 4416.2.a.bj 2
12.b even 2 1 3312.2.a.w 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
552.2.a.f 2 1.a even 1 1 trivial
1104.2.a.k 2 4.b odd 2 1
1656.2.a.l 2 3.b odd 2 1
3312.2.a.w 2 12.b even 2 1
4416.2.a.bd 2 8.b even 2 1
4416.2.a.bj 2 8.d odd 2 1

## Hecke kernels

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

 $$T_{5}^{2} - 2 T_{5} - 4$$ $$T_{7} - 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$-4 - 2 T + T^{2}$$
$7$ $$( -2 + T )^{2}$$
$11$ $$-4 - 2 T + T^{2}$$
$13$ $$-20 + T^{2}$$
$17$ $$-16 + 4 T + T^{2}$$
$19$ $$-4 - 2 T + T^{2}$$
$23$ $$( -1 + T )^{2}$$
$29$ $$-4 + 8 T + T^{2}$$
$31$ $$16 - 12 T + T^{2}$$
$37$ $$-44 - 2 T + T^{2}$$
$41$ $$( 2 + T )^{2}$$
$43$ $$4 - 6 T + T^{2}$$
$47$ $$-80 + T^{2}$$
$53$ $$-124 - 2 T + T^{2}$$
$59$ $$( 4 + T )^{2}$$
$61$ $$-20 + 10 T + T^{2}$$
$67$ $$-44 - 2 T + T^{2}$$
$71$ $$-80 + T^{2}$$
$73$ $$-20 + T^{2}$$
$79$ $$( -14 + T )^{2}$$
$83$ $$76 + 18 T + T^{2}$$
$89$ $$-16 + 16 T + T^{2}$$
$97$ $$-20 + T^{2}$$