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

Downloads

Learn more

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:

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

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