Properties

Label 572.2.i.b
Level $572$
Weight $2$
Character orbit 572.i
Analytic conductor $4.567$
Analytic rank $0$
Dimension $6$
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: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.309123.1
Defining polynomial: \(x^{6} - 3 x^{5} + 10 x^{4} - 15 x^{3} + 19 x^{2} - 12 x + 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{1} - \beta_{5} ) q^{3} + ( 1 + \beta_{1} - \beta_{3} ) q^{5} + ( -1 - \beta_{2} + \beta_{4} + \beta_{5} ) q^{7} + ( \beta_{2} + \beta_{5} ) q^{9} +O(q^{10})\) \( q + ( \beta_{1} - \beta_{5} ) q^{3} + ( 1 + \beta_{1} - \beta_{3} ) q^{5} + ( -1 - \beta_{2} + \beta_{4} + \beta_{5} ) q^{7} + ( \beta_{2} + \beta_{5} ) q^{9} + \beta_{4} q^{11} + ( 1 + 3 \beta_{4} ) q^{13} + ( -\beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{15} + ( 2 - 2 \beta_{2} - 2 \beta_{4} - \beta_{5} ) q^{17} + ( -1 + \beta_{4} + \beta_{5} ) q^{19} + ( 4 - \beta_{1} + \beta_{3} ) q^{21} + ( -3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 3 \beta_{5} ) q^{23} + ( 1 - 3 \beta_{3} ) q^{25} + ( 2 + \beta_{1} + \beta_{3} ) q^{27} + ( 3 \beta_{1} + \beta_{2} + \beta_{3} - 3 \beta_{5} ) q^{29} + ( 1 - 4 \beta_{1} ) q^{31} -\beta_{5} q^{33} + ( -2 - 5 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} ) q^{35} + ( \beta_{1} - 3 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{37} + ( \beta_{1} - 4 \beta_{5} ) q^{39} + ( -\beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{41} + ( -4 + 5 \beta_{2} + 4 \beta_{4} + \beta_{5} ) q^{43} + ( 5 + 2 \beta_{2} - 5 \beta_{4} - \beta_{5} ) q^{45} + ( 1 - 2 \beta_{1} - \beta_{3} ) q^{47} + ( 3 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} ) q^{49} + ( -1 + 5 \beta_{1} - \beta_{3} ) q^{51} + ( 3 - 3 \beta_{1} + \beta_{3} ) q^{53} + ( \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{55} + ( 3 - 2 \beta_{1} + \beta_{3} ) q^{57} + ( -6 - \beta_{2} + 6 \beta_{4} + 3 \beta_{5} ) q^{59} + ( -1 + \beta_{2} + \beta_{4} + 4 \beta_{5} ) q^{61} + ( 3 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} + \beta_{4} - 3 \beta_{5} ) q^{63} + ( 1 + 4 \beta_{1} - 3 \beta_{2} - 4 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} ) q^{65} + ( -2 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} ) q^{67} + ( 7 - 3 \beta_{2} - 7 \beta_{4} - 5 \beta_{5} ) q^{69} + ( 4 + 3 \beta_{2} - 4 \beta_{4} + 4 \beta_{5} ) q^{71} + ( 2 + 4 \beta_{3} ) q^{73} + ( -2 \beta_{1} - 3 \beta_{4} + 2 \beta_{5} ) q^{75} + ( -1 + \beta_{1} + \beta_{3} ) q^{77} + ( -8 + 2 \beta_{1} - 6 \beta_{3} ) q^{79} + ( -\beta_{1} + 4 \beta_{2} + 4 \beta_{3} + 4 \beta_{4} + \beta_{5} ) q^{81} + ( -3 \beta_{1} + \beta_{3} ) q^{83} + ( -6 - 3 \beta_{2} + 6 \beta_{4} + 3 \beta_{5} ) q^{85} + ( -10 + 3 \beta_{2} + 10 \beta_{4} + 2 \beta_{5} ) q^{87} + ( -3 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} - 9 \beta_{4} + 3 \beta_{5} ) q^{89} + ( -4 + 3 \beta_{1} - \beta_{2} + 3 \beta_{3} + \beta_{4} + \beta_{5} ) q^{91} + ( 5 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} - 12 \beta_{4} - 5 \beta_{5} ) q^{93} + ( 1 - 2 \beta_{2} - \beta_{4} - 2 \beta_{5} ) q^{95} + ( -1 + \beta_{4} + 4 \beta_{5} ) q^{97} + ( \beta_{1} - \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - q^{3} + 6q^{5} - 5q^{7} + O(q^{10}) \) \( 6q - q^{3} + 6q^{5} - 5q^{7} + 3q^{11} + 15q^{13} + 6q^{15} + 5q^{17} - 4q^{19} + 24q^{21} + q^{23} + 12q^{25} + 8q^{27} - 4q^{29} + 14q^{31} + q^{33} - 9q^{35} + 8q^{37} + 2q^{39} + 5q^{41} - 8q^{43} + 18q^{45} + 12q^{47} - 12q^{49} - 14q^{51} + 22q^{53} + 3q^{55} + 20q^{57} - 22q^{59} - 6q^{61} + 4q^{63} + 15q^{65} - 7q^{67} + 23q^{69} + 11q^{71} + 4q^{73} - 7q^{75} - 10q^{77} - 40q^{79} + 9q^{81} + 4q^{83} - 24q^{85} - 29q^{87} - 27q^{89} - 35q^{91} - 37q^{93} + 3q^{95} - 7q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 3 x^{5} + 10 x^{4} - 15 x^{3} + 19 x^{2} - 12 x + 3\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} - \nu + 2 \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{5} + \nu^{4} - 8 \nu^{3} + 5 \nu^{2} - 18 \nu + 6 \)\()/3\)
\(\beta_{3}\)\(=\)\( \nu^{4} - 2 \nu^{3} + 6 \nu^{2} - 5 \nu + 3 \)
\(\beta_{4}\)\(=\)\((\)\( -2 \nu^{5} + 5 \nu^{4} - 16 \nu^{3} + 19 \nu^{2} - 21 \nu + 9 \)\()/3\)
\(\beta_{5}\)\(=\)\((\)\( 2 \nu^{5} - 5 \nu^{4} + 19 \nu^{3} - 22 \nu^{2} + 30 \nu - 9 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-2 \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} + \beta_{1} + 2\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(-2 \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} + 4 \beta_{1} - 4\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(7 \beta_{5} + 5 \beta_{4} + 2 \beta_{3} + 4 \beta_{2} + \beta_{1} - 10\)\()/3\)
\(\nu^{4}\)\(=\)\((\)\(16 \beta_{5} + 11 \beta_{4} + 8 \beta_{3} + 10 \beta_{2} - 17 \beta_{1} + 5\)\()/3\)
\(\nu^{5}\)\(=\)\((\)\(-14 \beta_{5} - 16 \beta_{4} + 5 \beta_{3} - 5 \beta_{2} - 23 \beta_{1} + 47\)\()/3\)

Character values

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

\(n\) \(287\) \(353\) \(365\)
\(\chi(n)\) \(1\) \(-1 + \beta_{4}\) \(1\)

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
0.500000 2.05195i
0.500000 + 1.41036i
0.500000 0.224437i
0.500000 + 2.05195i
0.500000 1.41036i
0.500000 + 0.224437i
0 −1.23025 2.13086i 0 −2.05408 0 −1.43346 + 2.48283i 0 −1.52704 + 2.64491i 0
133.2 0 −0.119562 0.207087i 0 3.94282 0 −2.21053 + 3.82876i 0 1.47141 2.54856i 0
133.3 0 0.849814 + 1.47192i 0 1.11126 0 1.14400 1.98146i 0 0.0556321 0.0963576i 0
529.1 0 −1.23025 + 2.13086i 0 −2.05408 0 −1.43346 2.48283i 0 −1.52704 2.64491i 0
529.2 0 −0.119562 + 0.207087i 0 3.94282 0 −2.21053 3.82876i 0 1.47141 + 2.54856i 0
529.3 0 0.849814 1.47192i 0 1.11126 0 1.14400 + 1.98146i 0 0.0556321 + 0.0963576i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 529.3
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.b 6
13.c even 3 1 inner 572.2.i.b 6
13.c even 3 1 7436.2.a.n 3
13.e even 6 1 7436.2.a.m 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
572.2.i.b 6 1.a even 1 1 trivial
572.2.i.b 6 13.c even 3 1 inner
7436.2.a.m 3 13.e even 6 1
7436.2.a.n 3 13.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} + T_{3}^{5} + 5 T_{3}^{4} - 2 T_{3}^{3} + 17 T_{3}^{2} + 4 T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(572, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( 1 + 4 T + 17 T^{2} - 2 T^{3} + 5 T^{4} + T^{5} + T^{6} \)
$5$ \( ( 9 - 6 T - 3 T^{2} + T^{3} )^{2} \)
$7$ \( 841 + 116 T + 161 T^{2} + 38 T^{3} + 29 T^{4} + 5 T^{5} + T^{6} \)
$11$ \( ( 1 - T + T^{2} )^{3} \)
$13$ \( ( 13 - 5 T + T^{2} )^{3} \)
$17$ \( 6561 - 1458 T + 729 T^{2} - 72 T^{3} + 43 T^{4} - 5 T^{5} + T^{6} \)
$19$ \( 9 - 3 T + 13 T^{2} + 10 T^{3} + 15 T^{4} + 4 T^{5} + T^{6} \)
$23$ \( 8649 + 5022 T + 2823 T^{2} + 240 T^{3} + 55 T^{4} - T^{5} + T^{6} \)
$29$ \( 31329 + 7965 T + 2733 T^{2} + 174 T^{3} + 61 T^{4} + 4 T^{5} + T^{6} \)
$31$ \( ( 123 - 53 T - 7 T^{2} + T^{3} )^{2} \)
$37$ \( 729 + 945 T + 1009 T^{2} + 334 T^{3} + 99 T^{4} - 8 T^{5} + T^{6} \)
$41$ \( 6561 - 1458 T + 729 T^{2} - 72 T^{3} + 43 T^{4} - 5 T^{5} + T^{6} \)
$43$ \( 1270129 + 149891 T + 26705 T^{2} + 1190 T^{3} + 197 T^{4} + 8 T^{5} + T^{6} \)
$47$ \( ( 99 - 15 T - 6 T^{2} + T^{3} )^{2} \)
$53$ \( ( 3 - 11 T^{2} + T^{3} )^{2} \)
$59$ \( 7569 - 9657 T + 14235 T^{2} + 2616 T^{3} + 373 T^{4} + 22 T^{5} + T^{6} \)
$61$ \( 3481 - 3363 T + 2895 T^{2} - 460 T^{3} + 93 T^{4} + 6 T^{5} + T^{6} \)
$67$ \( 3969 + 4284 T + 5065 T^{2} - 350 T^{3} + 117 T^{4} + 7 T^{5} + T^{6} \)
$71$ \( 335241 - 38214 T + 10725 T^{2} - 432 T^{3} + 187 T^{4} - 11 T^{5} + T^{6} \)
$73$ \( ( 392 - 100 T - 2 T^{2} + T^{3} )^{2} \)
$79$ \( ( -2472 - 92 T + 20 T^{2} + T^{3} )^{2} \)
$83$ \( ( -69 - 39 T - 2 T^{2} + T^{3} )^{2} \)
$89$ \( 729 - 4374 T + 26973 T^{2} + 4428 T^{3} + 567 T^{4} + 27 T^{5} + T^{6} \)
$97$ \( 15129 + 6519 T + 3670 T^{2} - 125 T^{3} + 102 T^{4} + 7 T^{5} + T^{6} \)
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