Properties

Label 572.2.f.b
Level $572$
Weight $2$
Character orbit 572.f
Analytic conductor $4.567$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 572 = 2^{2} \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 572.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.56744299562\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{21})\)
Defining polynomial: \(x^{4} + 11 x^{2} + 25\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{3} ) q^{3} + ( \beta_{1} - \beta_{2} ) q^{7} + ( 3 - \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( 1 - \beta_{3} ) q^{3} + ( \beta_{1} - \beta_{2} ) q^{7} + ( 3 - \beta_{3} ) q^{9} -\beta_{2} q^{11} + ( 2 - 3 \beta_{2} ) q^{13} + ( -4 + 2 \beta_{3} ) q^{17} + ( -\beta_{1} + 4 \beta_{2} ) q^{19} + ( 2 \beta_{1} - 5 \beta_{2} ) q^{21} + ( 2 - \beta_{3} ) q^{23} + 5 q^{25} + 5 q^{27} + ( 2 + 2 \beta_{3} ) q^{29} + ( -2 \beta_{1} - 4 \beta_{2} ) q^{31} + \beta_{1} q^{33} + ( -4 \beta_{1} - 2 \beta_{2} ) q^{37} + ( 2 + 3 \beta_{1} - 2 \beta_{3} ) q^{39} + ( -3 \beta_{1} - 3 \beta_{2} ) q^{41} + 2 q^{43} + 6 \beta_{2} q^{47} + ( -2 + 3 \beta_{3} ) q^{49} + ( -14 + 4 \beta_{3} ) q^{51} + ( -5 + \beta_{3} ) q^{53} + ( -5 \beta_{1} + 5 \beta_{2} ) q^{57} -6 \beta_{2} q^{59} + 10 q^{61} + ( 4 \beta_{1} - 7 \beta_{2} ) q^{63} + ( -4 \beta_{1} + 4 \beta_{2} ) q^{67} + ( 7 - 2 \beta_{3} ) q^{69} + ( 6 \beta_{1} + 6 \beta_{2} ) q^{71} + ( 5 \beta_{1} + 4 \beta_{2} ) q^{73} + ( 5 - 5 \beta_{3} ) q^{75} + ( -2 + \beta_{3} ) q^{77} -8 q^{79} + ( -4 - 2 \beta_{3} ) q^{81} + ( -3 \beta_{1} + 3 \beta_{2} ) q^{83} + ( -8 - 2 \beta_{3} ) q^{87} -6 \beta_{2} q^{89} + ( -6 + 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} ) q^{91} + ( 2 \beta_{1} + 10 \beta_{2} ) q^{93} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{97} + ( \beta_{1} - 2 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{3} + 10q^{9} + O(q^{10}) \) \( 4q + 2q^{3} + 10q^{9} + 8q^{13} - 12q^{17} + 6q^{23} + 20q^{25} + 20q^{27} + 12q^{29} + 4q^{39} + 8q^{43} - 2q^{49} - 48q^{51} - 18q^{53} + 40q^{61} + 24q^{69} + 10q^{75} - 6q^{77} - 32q^{79} - 20q^{81} - 36q^{87} - 18q^{91} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 11 x^{2} + 25\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 6 \nu \)\()/5\)
\(\beta_{3}\)\(=\)\( \nu^{2} + 6 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} - 6\)
\(\nu^{3}\)\(=\)\(5 \beta_{2} - 6 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/572\mathbb{Z}\right)^\times\).

\(n\) \(287\) \(353\) \(365\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
441.1
1.79129i
1.79129i
2.79129i
2.79129i
0 −1.79129 0 0 0 0.791288i 0 0.208712 0
441.2 0 −1.79129 0 0 0 0.791288i 0 0.208712 0
441.3 0 2.79129 0 0 0 3.79129i 0 4.79129 0
441.4 0 2.79129 0 0 0 3.79129i 0 4.79129 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 572.2.f.b 4
3.b odd 2 1 5148.2.e.b 4
4.b odd 2 1 2288.2.j.f 4
13.b even 2 1 inner 572.2.f.b 4
13.d odd 4 1 7436.2.a.i 2
13.d odd 4 1 7436.2.a.j 2
39.d odd 2 1 5148.2.e.b 4
52.b odd 2 1 2288.2.j.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
572.2.f.b 4 1.a even 1 1 trivial
572.2.f.b 4 13.b even 2 1 inner
2288.2.j.f 4 4.b odd 2 1
2288.2.j.f 4 52.b odd 2 1
5148.2.e.b 4 3.b odd 2 1
5148.2.e.b 4 39.d odd 2 1
7436.2.a.i 2 13.d odd 4 1
7436.2.a.j 2 13.d odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - T_{3} - 5 \) acting on \(S_{2}^{\mathrm{new}}(572, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( -5 - T + T^{2} )^{2} \)
$5$ \( T^{4} \)
$7$ \( 9 + 15 T^{2} + T^{4} \)
$11$ \( ( 1 + T^{2} )^{2} \)
$13$ \( ( 13 - 4 T + T^{2} )^{2} \)
$17$ \( ( -12 + 6 T + T^{2} )^{2} \)
$19$ \( 225 + 51 T^{2} + T^{4} \)
$23$ \( ( -3 - 3 T + T^{2} )^{2} \)
$29$ \( ( -12 - 6 T + T^{2} )^{2} \)
$31$ \( 144 + 60 T^{2} + T^{4} \)
$37$ \( ( 84 + T^{2} )^{2} \)
$41$ \( 2025 + 99 T^{2} + T^{4} \)
$43$ \( ( -2 + T )^{4} \)
$47$ \( ( 36 + T^{2} )^{2} \)
$53$ \( ( 15 + 9 T + T^{2} )^{2} \)
$59$ \( ( 36 + T^{2} )^{2} \)
$61$ \( ( -10 + T )^{4} \)
$67$ \( 2304 + 240 T^{2} + T^{4} \)
$71$ \( 32400 + 396 T^{2} + T^{4} \)
$73$ \( 16641 + 267 T^{2} + T^{4} \)
$79$ \( ( 8 + T )^{4} \)
$83$ \( 729 + 135 T^{2} + T^{4} \)
$89$ \( ( 36 + T^{2} )^{2} \)
$97$ \( 144 + 60 T^{2} + T^{4} \)
show more
show less