Properties

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

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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: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 4 i q^{7} -3 q^{9} +O(q^{10})\) \( q + 4 i q^{7} -3 q^{9} -i q^{11} + ( -3 + 2 i ) q^{13} -2 q^{17} + 4 i q^{19} -4 q^{23} + 5 q^{25} -6 q^{29} -4 i q^{31} + 8 i q^{37} + 12 i q^{41} + 12 q^{43} -4 i q^{47} -9 q^{49} + 6 q^{53} + 4 i q^{59} -10 q^{61} -12 i q^{63} + 4 i q^{67} -4 i q^{71} + 4 i q^{73} + 4 q^{77} -8 q^{79} + 9 q^{81} -12 i q^{83} -16 i q^{89} + ( -8 - 12 i ) q^{91} + 8 i q^{97} + 3 i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 6q^{9} + O(q^{10}) \) \( 2q - 6q^{9} - 6q^{13} - 4q^{17} - 8q^{23} + 10q^{25} - 12q^{29} + 24q^{43} - 18q^{49} + 12q^{53} - 20q^{61} + 8q^{77} - 16q^{79} + 18q^{81} - 16q^{91} + O(q^{100}) \)

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
1.00000i
1.00000i
0 0 0 0 0 4.00000i 0 −3.00000 0
441.2 0 0 0 0 0 4.00000i 0 −3.00000 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.a 2
3.b odd 2 1 5148.2.e.a 2
4.b odd 2 1 2288.2.j.b 2
13.b even 2 1 inner 572.2.f.a 2
13.d odd 4 1 7436.2.a.a 1
13.d odd 4 1 7436.2.a.b 1
39.d odd 2 1 5148.2.e.a 2
52.b odd 2 1 2288.2.j.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
572.2.f.a 2 1.a even 1 1 trivial
572.2.f.a 2 13.b even 2 1 inner
2288.2.j.b 2 4.b odd 2 1
2288.2.j.b 2 52.b odd 2 1
5148.2.e.a 2 3.b odd 2 1
5148.2.e.a 2 39.d odd 2 1
7436.2.a.a 1 13.d odd 4 1
7436.2.a.b 1 13.d odd 4 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 16 + T^{2} \)
$11$ \( 1 + T^{2} \)
$13$ \( 13 + 6 T + T^{2} \)
$17$ \( ( 2 + T )^{2} \)
$19$ \( 16 + T^{2} \)
$23$ \( ( 4 + T )^{2} \)
$29$ \( ( 6 + T )^{2} \)
$31$ \( 16 + T^{2} \)
$37$ \( 64 + T^{2} \)
$41$ \( 144 + T^{2} \)
$43$ \( ( -12 + T )^{2} \)
$47$ \( 16 + T^{2} \)
$53$ \( ( -6 + T )^{2} \)
$59$ \( 16 + T^{2} \)
$61$ \( ( 10 + T )^{2} \)
$67$ \( 16 + T^{2} \)
$71$ \( 16 + T^{2} \)
$73$ \( 16 + T^{2} \)
$79$ \( ( 8 + T )^{2} \)
$83$ \( 144 + T^{2} \)
$89$ \( 256 + T^{2} \)
$97$ \( 64 + T^{2} \)
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