Properties

Label 572.2.i.a
Level $572$
Weight $2$
Character orbit 572.i
Analytic conductor $4.567$
Analytic rank $0$
Dimension $4$
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.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.56744299562\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\zeta_{12} + \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{3} - q^{5} + ( -1 + \zeta_{12} + \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{7} + ( -1 + 2 \zeta_{12} + \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{9} +O(q^{10})\) \( q + ( -\zeta_{12} + \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{3} - q^{5} + ( -1 + \zeta_{12} + \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{7} + ( -1 + 2 \zeta_{12} + \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{9} + \zeta_{12}^{2} q^{11} + ( -3 \zeta_{12} - 2 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{13} + ( \zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{15} + ( -2 + 3 \zeta_{12} + 2 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{17} + ( 1 + \zeta_{12} - \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{19} + ( -4 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{21} + ( -2 \zeta_{12} - 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{23} -4 q^{25} -4 q^{27} + ( -\zeta_{12} + 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{29} + 6 q^{31} + ( -1 + \zeta_{12} + \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{33} + ( 1 - \zeta_{12} - \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{35} -7 \zeta_{12}^{2} q^{37} + ( 5 - 5 \zeta_{12} + \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{39} + ( -\zeta_{12} - 4 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{41} + ( 4 + 4 \zeta_{12} - 4 \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{43} + ( 1 - 2 \zeta_{12} - \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{45} + ( 5 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{47} + ( 2 \zeta_{12} + 3 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{49} + ( -11 + 10 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{51} + ( -1 - 12 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{53} -\zeta_{12}^{2} q^{55} -2 q^{57} + ( 7 - \zeta_{12} - 7 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{59} + ( -10 - \zeta_{12} + 10 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{61} + ( 3 \zeta_{12} - 7 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{63} + ( 3 \zeta_{12} + 2 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{65} + ( 5 \zeta_{12} + 7 \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{67} + ( -2 - 2 \zeta_{12} + 2 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{69} + ( -6 + 6 \zeta_{12}^{2} ) q^{71} + ( 2 + 10 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{73} + ( 4 \zeta_{12} - 4 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{75} + ( -1 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{77} + ( 11 - 6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{79} + ( -2 \zeta_{12} - \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{81} + ( 3 - 6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{83} + ( 2 - 3 \zeta_{12} - 2 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{85} + ( -5 + 3 \zeta_{12} + 5 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{87} + ( -8 \zeta_{12} + 4 \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{89} + ( -1 - 4 \zeta_{12} + 6 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{91} + ( -6 \zeta_{12} + 6 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{93} + ( -1 - \zeta_{12} + \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{95} + ( 16 - 16 \zeta_{12}^{2} ) q^{97} + ( -1 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{3} - 4q^{5} - 2q^{7} - 2q^{9} + O(q^{10}) \) \( 4q + 2q^{3} - 4q^{5} - 2q^{7} - 2q^{9} + 2q^{11} - 4q^{13} - 2q^{15} - 4q^{17} + 2q^{19} - 16q^{21} - 8q^{23} - 16q^{25} - 16q^{27} + 4q^{29} + 24q^{31} - 2q^{33} + 2q^{35} - 14q^{37} + 22q^{39} - 8q^{41} + 8q^{43} + 2q^{45} + 20q^{47} + 6q^{49} - 44q^{51} - 4q^{53} - 2q^{55} - 8q^{57} + 14q^{59} - 20q^{61} - 14q^{63} + 4q^{65} + 14q^{67} - 4q^{69} - 12q^{71} + 8q^{73} - 8q^{75} - 4q^{77} + 44q^{79} - 2q^{81} + 12q^{83} + 4q^{85} - 10q^{87} + 8q^{89} + 8q^{91} + 12q^{93} - 2q^{95} + 32q^{97} - 4q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(287\) \(353\) \(365\)
\(\chi(n)\) \(1\) \(-1 + \zeta_{12}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
133.1
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
0 −0.366025 0.633975i 0 −1.00000 0 0.366025 0.633975i 0 1.23205 2.13397i 0
133.2 0 1.36603 + 2.36603i 0 −1.00000 0 −1.36603 + 2.36603i 0 −2.23205 + 3.86603i 0
529.1 0 −0.366025 + 0.633975i 0 −1.00000 0 0.366025 + 0.633975i 0 1.23205 + 2.13397i 0
529.2 0 1.36603 2.36603i 0 −1.00000 0 −1.36603 2.36603i 0 −2.23205 3.86603i 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.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 572.2.i.a 4
13.c even 3 1 inner 572.2.i.a 4
13.c even 3 1 7436.2.a.f 2
13.e even 6 1 7436.2.a.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
572.2.i.a 4 1.a even 1 1 trivial
572.2.i.a 4 13.c even 3 1 inner
7436.2.a.f 2 13.c even 3 1
7436.2.a.g 2 13.e even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 2 T_{3}^{3} + 6 T_{3}^{2} + 4 T_{3} + 4 \) acting on \(S_{2}^{\mathrm{new}}(572, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 4 + 4 T + 6 T^{2} - 2 T^{3} + T^{4} \)
$5$ \( ( 1 + T )^{4} \)
$7$ \( 4 - 4 T + 6 T^{2} + 2 T^{3} + T^{4} \)
$11$ \( ( 1 - T + T^{2} )^{2} \)
$13$ \( 169 + 52 T + 3 T^{2} + 4 T^{3} + T^{4} \)
$17$ \( 529 - 92 T + 39 T^{2} + 4 T^{3} + T^{4} \)
$19$ \( 4 + 4 T + 6 T^{2} - 2 T^{3} + T^{4} \)
$23$ \( 16 + 32 T + 60 T^{2} + 8 T^{3} + T^{4} \)
$29$ \( 1 - 4 T + 15 T^{2} - 4 T^{3} + T^{4} \)
$31$ \( ( -6 + T )^{4} \)
$37$ \( ( 49 + 7 T + T^{2} )^{2} \)
$41$ \( 169 + 104 T + 51 T^{2} + 8 T^{3} + T^{4} \)
$43$ \( 1024 + 256 T + 96 T^{2} - 8 T^{3} + T^{4} \)
$47$ \( ( 22 - 10 T + T^{2} )^{2} \)
$53$ \( ( -107 + 2 T + T^{2} )^{2} \)
$59$ \( 2116 - 644 T + 150 T^{2} - 14 T^{3} + T^{4} \)
$61$ \( 9409 + 1940 T + 303 T^{2} + 20 T^{3} + T^{4} \)
$67$ \( 676 + 364 T + 222 T^{2} - 14 T^{3} + T^{4} \)
$71$ \( ( 36 + 6 T + T^{2} )^{2} \)
$73$ \( ( -71 - 4 T + T^{2} )^{2} \)
$79$ \( ( 94 - 22 T + T^{2} )^{2} \)
$83$ \( ( -18 - 6 T + T^{2} )^{2} \)
$89$ \( 30976 + 1408 T + 240 T^{2} - 8 T^{3} + T^{4} \)
$97$ \( ( 256 - 16 T + T^{2} )^{2} \)
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