Properties

Label 547.3.b.a
Level 547
Weight 3
Character orbit 547.b
Self dual yes
Analytic conductor 14.905
Analytic rank 0
Dimension 3
CM discriminant -547
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 547 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 547.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(14.9046704605\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.14769.1
Defining polynomial: \(x^{3} - 33 x - 56\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 4 q^{4} + 9 q^{9} +O(q^{10})\) \( q + 4 q^{4} + 9 q^{9} + ( 3 \beta_{1} + 2 \beta_{2} ) q^{11} + ( -4 \beta_{1} - \beta_{2} ) q^{13} + 16 q^{16} + ( \beta_{1} - 5 \beta_{2} ) q^{19} + 25 q^{25} + ( -3 \beta_{1} + 7 \beta_{2} ) q^{29} + 36 q^{36} + ( 12 \beta_{1} + 8 \beta_{2} ) q^{44} + ( 12 \beta_{1} - 5 \beta_{2} ) q^{47} + 49 q^{49} + ( -16 \beta_{1} - 4 \beta_{2} ) q^{52} + ( -9 \beta_{1} - 14 \beta_{2} ) q^{53} + 64 q^{64} + ( 4 \beta_{1} + 19 \beta_{2} ) q^{67} + ( -17 \beta_{1} + 10 \beta_{2} ) q^{73} + ( 4 \beta_{1} - 20 \beta_{2} ) q^{76} + 81 q^{81} + ( 29 \beta_{1} - \beta_{2} ) q^{97} + ( 27 \beta_{1} + 18 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 12q^{4} + 27q^{9} + O(q^{10}) \) \( 3q + 12q^{4} + 27q^{9} + 48q^{16} + 75q^{25} + 108q^{36} + 147q^{49} + 192q^{64} + 243q^{81} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - 33 x - 56\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{2} - 3 \nu - 22 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{2} + 3 \beta_{1} + 22\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/547\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
546.1
−1.90719
−4.54841
6.45559
0 0 4.00000 0 0 0 0 9.00000 0
546.2 0 0 4.00000 0 0 0 0 9.00000 0
546.3 0 0 4.00000 0 0 0 0 9.00000 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
547.b odd 2 1 CM by \(\Q(\sqrt{-547}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 547.3.b.a 3
547.b odd 2 1 CM 547.3.b.a 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
547.3.b.a 3 1.a even 1 1 trivial
547.3.b.a 3 547.b odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{3}^{\mathrm{new}}(547, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 2 T )^{3}( 1 + 2 T )^{3} \)
$3$ \( ( 1 - 3 T )^{3}( 1 + 3 T )^{3} \)
$5$ \( ( 1 - 5 T )^{3}( 1 + 5 T )^{3} \)
$7$ \( ( 1 - 7 T )^{3}( 1 + 7 T )^{3} \)
$11$ \( 1 - 474 T^{3} + 1771561 T^{6} \)
$13$ \( 1 + 4358 T^{3} + 4826809 T^{6} \)
$17$ \( ( 1 - 17 T )^{3}( 1 + 17 T )^{3} \)
$19$ \( 1 + 5974 T^{3} + 47045881 T^{6} \)
$23$ \( ( 1 - 23 T )^{3}( 1 + 23 T )^{3} \)
$29$ \( 1 - 40026 T^{3} + 594823321 T^{6} \)
$31$ \( ( 1 - 31 T )^{3}( 1 + 31 T )^{3} \)
$37$ \( ( 1 - 37 T )^{3}( 1 + 37 T )^{3} \)
$41$ \( ( 1 - 41 T )^{3}( 1 + 41 T )^{3} \)
$43$ \( ( 1 - 43 T )^{3}( 1 + 43 T )^{3} \)
$47$ \( 1 + 57102 T^{3} + 10779215329 T^{6} \)
$53$ \( 1 - 289002 T^{3} + 22164361129 T^{6} \)
$59$ \( ( 1 - 59 T )^{3}( 1 + 59 T )^{3} \)
$61$ \( ( 1 - 61 T )^{3}( 1 + 61 T )^{3} \)
$67$ \( 1 + 363382 T^{3} + 90458382169 T^{6} \)
$71$ \( ( 1 - 71 T )^{3}( 1 + 71 T )^{3} \)
$73$ \( 1 - 462962 T^{3} + 151334226289 T^{6} \)
$79$ \( ( 1 - 79 T )^{3}( 1 + 79 T )^{3} \)
$83$ \( ( 1 - 83 T )^{3}( 1 + 83 T )^{3} \)
$89$ \( ( 1 - 89 T )^{3}( 1 + 89 T )^{3} \)
$97$ \( 1 - 1265218 T^{3} + 832972004929 T^{6} \)
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