Properties

Label 572.2.a.b
Level $572$
Weight $2$
Character orbit 572.a
Self dual yes
Analytic conductor $4.567$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 572 = 2^{2} \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 572.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.56744299562\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \(x^{2} - x - 3\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{13})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 - \beta ) q^{3} + ( -2 + \beta ) q^{7} + ( 1 + 3 \beta ) q^{9} +O(q^{10})\) \( q + ( -1 - \beta ) q^{3} + ( -2 + \beta ) q^{7} + ( 1 + 3 \beta ) q^{9} - q^{11} + q^{13} + ( -2 + 2 \beta ) q^{17} + ( -3 - \beta ) q^{19} - q^{21} + ( -4 + \beta ) q^{23} -5 q^{25} + ( -7 - 4 \beta ) q^{27} + ( 4 - 4 \beta ) q^{29} + 2 \beta q^{31} + ( 1 + \beta ) q^{33} -4 \beta q^{37} + ( -1 - \beta ) q^{39} + ( 4 - \beta ) q^{41} + ( -2 - 2 \beta ) q^{43} -6 q^{47} -3 \beta q^{49} + ( -4 - 2 \beta ) q^{51} + ( -7 + \beta ) q^{53} + ( 6 + 5 \beta ) q^{57} + ( 10 + 2 \beta ) q^{59} + ( -4 + 6 \beta ) q^{61} + ( 7 - 2 \beta ) q^{63} + ( -6 + 2 \beta ) q^{67} + ( 1 + 2 \beta ) q^{69} + ( -8 + 2 \beta ) q^{71} + ( 1 - 5 \beta ) q^{73} + ( 5 + 5 \beta ) q^{75} + ( 2 - \beta ) q^{77} + ( -6 - 4 \beta ) q^{79} + ( 16 + 6 \beta ) q^{81} + ( 2 - 5 \beta ) q^{83} + ( 8 + 4 \beta ) q^{87} + ( -2 + \beta ) q^{91} + ( -6 - 4 \beta ) q^{93} + ( 12 - 4 \beta ) q^{97} + ( -1 - 3 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 3q^{3} - 3q^{7} + 5q^{9} + O(q^{10}) \) \( 2q - 3q^{3} - 3q^{7} + 5q^{9} - 2q^{11} + 2q^{13} - 2q^{17} - 7q^{19} - 2q^{21} - 7q^{23} - 10q^{25} - 18q^{27} + 4q^{29} + 2q^{31} + 3q^{33} - 4q^{37} - 3q^{39} + 7q^{41} - 6q^{43} - 12q^{47} - 3q^{49} - 10q^{51} - 13q^{53} + 17q^{57} + 22q^{59} - 2q^{61} + 12q^{63} - 10q^{67} + 4q^{69} - 14q^{71} - 3q^{73} + 15q^{75} + 3q^{77} - 16q^{79} + 38q^{81} - q^{83} + 20q^{87} - 3q^{91} - 16q^{93} + 20q^{97} - 5q^{99} + O(q^{100}) \)

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
2.30278
−1.30278
0 −3.30278 0 0 0 0.302776 0 7.90833 0
1.2 0 0.302776 0 0 0 −3.30278 0 −2.90833 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(11\) \(1\)
\(13\) \(-1\)

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 572.2.a.b 2
3.b odd 2 1 5148.2.a.h 2
4.b odd 2 1 2288.2.a.r 2
8.b even 2 1 9152.2.a.bq 2
8.d odd 2 1 9152.2.a.bg 2
11.b odd 2 1 6292.2.a.k 2
13.b even 2 1 7436.2.a.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
572.2.a.b 2 1.a even 1 1 trivial
2288.2.a.r 2 4.b odd 2 1
5148.2.a.h 2 3.b odd 2 1
6292.2.a.k 2 11.b odd 2 1
7436.2.a.e 2 13.b even 2 1
9152.2.a.bg 2 8.d odd 2 1
9152.2.a.bq 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(572))\):

\( T_{3}^{2} + 3 T_{3} - 1 \)
\( T_{5} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -1 + 3 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( -1 + 3 T + T^{2} \)
$11$ \( ( 1 + T )^{2} \)
$13$ \( ( -1 + T )^{2} \)
$17$ \( -12 + 2 T + T^{2} \)
$19$ \( 9 + 7 T + T^{2} \)
$23$ \( 9 + 7 T + T^{2} \)
$29$ \( -48 - 4 T + T^{2} \)
$31$ \( -12 - 2 T + T^{2} \)
$37$ \( -48 + 4 T + T^{2} \)
$41$ \( 9 - 7 T + T^{2} \)
$43$ \( -4 + 6 T + T^{2} \)
$47$ \( ( 6 + T )^{2} \)
$53$ \( 39 + 13 T + T^{2} \)
$59$ \( 108 - 22 T + T^{2} \)
$61$ \( -116 + 2 T + T^{2} \)
$67$ \( 12 + 10 T + T^{2} \)
$71$ \( 36 + 14 T + T^{2} \)
$73$ \( -79 + 3 T + T^{2} \)
$79$ \( 12 + 16 T + T^{2} \)
$83$ \( -81 + T + T^{2} \)
$89$ \( T^{2} \)
$97$ \( 48 - 20 T + T^{2} \)
show more
show less