Properties

Label 572.2.b.a
Level $572$
Weight $2$
Character orbit 572.b
Analytic conductor $4.567$
Analytic rank $0$
Dimension $4$
CM discriminant -52
Inner twists $8$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(4.56744299562\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-13})\)
Defining polynomial: \(x^{4} - 12 x^{2} + 49\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{2} q^{2} -2 q^{4} + \beta_{2} q^{7} + 2 \beta_{2} q^{8} + 3 q^{9} +O(q^{10})\) \( q -\beta_{2} q^{2} -2 q^{4} + \beta_{2} q^{7} + 2 \beta_{2} q^{8} + 3 q^{9} + ( \beta_{1} - 2 \beta_{2} ) q^{11} + \beta_{3} q^{13} + 2 q^{14} + 4 q^{16} + 2 \beta_{3} q^{17} -3 \beta_{2} q^{18} -5 \beta_{2} q^{19} + ( -3 - \beta_{3} ) q^{22} + 5 q^{25} + ( 2 \beta_{1} - \beta_{2} ) q^{26} -2 \beta_{2} q^{28} -2 \beta_{3} q^{29} + ( 2 \beta_{1} - \beta_{2} ) q^{31} -4 \beta_{2} q^{32} + ( 4 \beta_{1} - 2 \beta_{2} ) q^{34} -6 q^{36} -10 q^{38} + ( -2 \beta_{1} + 4 \beta_{2} ) q^{44} + ( -2 \beta_{1} + \beta_{2} ) q^{47} + 5 q^{49} -5 \beta_{2} q^{50} -2 \beta_{3} q^{52} -2 q^{53} -4 q^{56} + ( -4 \beta_{1} + 2 \beta_{2} ) q^{58} + ( 6 \beta_{1} - 3 \beta_{2} ) q^{59} + 4 \beta_{3} q^{61} -2 \beta_{3} q^{62} + 3 \beta_{2} q^{63} -8 q^{64} + ( -2 \beta_{1} + \beta_{2} ) q^{67} -4 \beta_{3} q^{68} + ( -6 \beta_{1} + 3 \beta_{2} ) q^{71} + 6 \beta_{2} q^{72} + 10 \beta_{2} q^{76} + ( 3 + \beta_{3} ) q^{77} + 9 q^{81} + 7 \beta_{2} q^{83} + ( 6 + 2 \beta_{3} ) q^{88} + ( -2 \beta_{1} + \beta_{2} ) q^{91} + 2 \beta_{3} q^{94} -5 \beta_{2} q^{98} + ( 3 \beta_{1} - 6 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 8q^{4} + 12q^{9} + O(q^{10}) \) \( 4q - 8q^{4} + 12q^{9} + 8q^{14} + 16q^{16} - 12q^{22} + 20q^{25} - 24q^{36} - 40q^{38} + 20q^{49} - 8q^{53} - 16q^{56} - 32q^{64} + 12q^{77} + 36q^{81} + 24q^{88} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 12 x^{2} + 49\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} - 5 \nu \)\()/7\)
\(\beta_{3}\)\(=\)\( \nu^{2} - 6 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 6\)
\(\nu^{3}\)\(=\)\(7 \beta_{2} + 5 \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
571.1
−2.54951 + 0.707107i
2.54951 + 0.707107i
−2.54951 0.707107i
2.54951 0.707107i
1.41421i 0 −2.00000 0 0 1.41421i 2.82843i 3.00000 0
571.2 1.41421i 0 −2.00000 0 0 1.41421i 2.82843i 3.00000 0
571.3 1.41421i 0 −2.00000 0 0 1.41421i 2.82843i 3.00000 0
571.4 1.41421i 0 −2.00000 0 0 1.41421i 2.82843i 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
52.b odd 2 1 CM by \(\Q(\sqrt{-13}) \)
4.b odd 2 1 inner
11.b odd 2 1 inner
13.b even 2 1 inner
44.c even 2 1 inner
143.d odd 2 1 inner
572.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.b.a 4
4.b odd 2 1 inner 572.2.b.a 4
11.b odd 2 1 inner 572.2.b.a 4
13.b even 2 1 inner 572.2.b.a 4
44.c even 2 1 inner 572.2.b.a 4
52.b odd 2 1 CM 572.2.b.a 4
143.d odd 2 1 inner 572.2.b.a 4
572.b even 2 1 inner 572.2.b.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
572.2.b.a 4 1.a even 1 1 trivial
572.2.b.a 4 4.b odd 2 1 inner
572.2.b.a 4 11.b odd 2 1 inner
572.2.b.a 4 13.b even 2 1 inner
572.2.b.a 4 44.c even 2 1 inner
572.2.b.a 4 52.b odd 2 1 CM
572.2.b.a 4 143.d odd 2 1 inner
572.2.b.a 4 572.b even 2 1 inner

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

$p$ $F_p(T)$
$2$ \( ( 2 + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( T^{4} \)
$7$ \( ( 2 + T^{2} )^{2} \)
$11$ \( 121 - 4 T^{2} + T^{4} \)
$13$ \( ( 13 + T^{2} )^{2} \)
$17$ \( ( 52 + T^{2} )^{2} \)
$19$ \( ( 50 + T^{2} )^{2} \)
$23$ \( T^{4} \)
$29$ \( ( 52 + T^{2} )^{2} \)
$31$ \( ( -26 + T^{2} )^{2} \)
$37$ \( T^{4} \)
$41$ \( T^{4} \)
$43$ \( T^{4} \)
$47$ \( ( -26 + T^{2} )^{2} \)
$53$ \( ( 2 + T )^{4} \)
$59$ \( ( -234 + T^{2} )^{2} \)
$61$ \( ( 208 + T^{2} )^{2} \)
$67$ \( ( -26 + T^{2} )^{2} \)
$71$ \( ( -234 + T^{2} )^{2} \)
$73$ \( T^{4} \)
$79$ \( T^{4} \)
$83$ \( ( 98 + T^{2} )^{2} \)
$89$ \( T^{4} \)
$97$ \( T^{4} \)
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