Properties

Label 3744.1.dx.a
Level $3744$
Weight $1$
Character orbit 3744.dx
Analytic conductor $1.868$
Analytic rank $0$
Dimension $8$
Projective image $D_{12}$
CM discriminant -4
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 3744 = 2^{5} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3744.dx (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.86849940730\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{12}\)
Projective field Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + ( -\zeta_{24} + \zeta_{24}^{11} ) q^{5} +O(q^{10})\) \( q + ( -\zeta_{24} + \zeta_{24}^{11} ) q^{5} -\zeta_{24}^{2} q^{13} + ( \zeta_{24}^{3} - \zeta_{24}^{5} ) q^{17} + ( 1 + \zeta_{24}^{2} - \zeta_{24}^{10} ) q^{25} + ( -\zeta_{24}^{7} + \zeta_{24}^{9} ) q^{29} + ( 1 + \zeta_{24}^{4} ) q^{37} + ( \zeta_{24}^{7} + \zeta_{24}^{9} ) q^{41} + \zeta_{24}^{8} q^{49} + ( \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{53} + \zeta_{24}^{4} q^{61} + ( \zeta_{24} + \zeta_{24}^{3} ) q^{65} + \zeta_{24}^{6} q^{73} + ( -\zeta_{24}^{2} + \zeta_{24}^{6} ) q^{85} + ( -\zeta_{24}^{5} - \zeta_{24}^{11} ) q^{89} + 2 \zeta_{24}^{10} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 8q^{25} + 12q^{37} - 4q^{49} + 4q^{61} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3744\mathbb{Z}\right)^\times\).

\(n\) \(703\) \(2017\) \(2081\) \(2341\)
\(\chi(n)\) \(1\) \(\zeta_{24}^{4}\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1889.1
0.965926 + 0.258819i
0.258819 0.965926i
−0.258819 + 0.965926i
−0.965926 0.258819i
0.965926 0.258819i
0.258819 + 0.965926i
−0.258819 0.965926i
−0.965926 + 0.258819i
0 0 0 −1.93185 0 0 0 0 0
1889.2 0 0 0 −0.517638 0 0 0 0 0
1889.3 0 0 0 0.517638 0 0 0 0 0
1889.4 0 0 0 1.93185 0 0 0 0 0
2753.1 0 0 0 −1.93185 0 0 0 0 0
2753.2 0 0 0 −0.517638 0 0 0 0 0
2753.3 0 0 0 0.517638 0 0 0 0 0
2753.4 0 0 0 1.93185 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2753.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
3.b odd 2 1 inner
12.b even 2 1 inner
13.e even 6 1 inner
39.h odd 6 1 inner
52.i odd 6 1 inner
156.r even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3744.1.dx.a 8
3.b odd 2 1 inner 3744.1.dx.a 8
4.b odd 2 1 CM 3744.1.dx.a 8
12.b even 2 1 inner 3744.1.dx.a 8
13.e even 6 1 inner 3744.1.dx.a 8
39.h odd 6 1 inner 3744.1.dx.a 8
52.i odd 6 1 inner 3744.1.dx.a 8
156.r even 6 1 inner 3744.1.dx.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3744.1.dx.a 8 1.a even 1 1 trivial
3744.1.dx.a 8 3.b odd 2 1 inner
3744.1.dx.a 8 4.b odd 2 1 CM
3744.1.dx.a 8 12.b even 2 1 inner
3744.1.dx.a 8 13.e even 6 1 inner
3744.1.dx.a 8 39.h odd 6 1 inner
3744.1.dx.a 8 52.i odd 6 1 inner
3744.1.dx.a 8 156.r even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(3744, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( ( 1 - 4 T^{2} + T^{4} )^{2} \)
$7$ \( T^{8} \)
$11$ \( T^{8} \)
$13$ \( ( 1 - T^{2} + T^{4} )^{2} \)
$17$ \( 1 - 4 T^{2} + 15 T^{4} - 4 T^{6} + T^{8} \)
$19$ \( T^{8} \)
$23$ \( T^{8} \)
$29$ \( 1 - 4 T^{2} + 15 T^{4} - 4 T^{6} + T^{8} \)
$31$ \( T^{8} \)
$37$ \( ( 3 - 3 T + T^{2} )^{4} \)
$41$ \( 1 + 4 T^{2} + 15 T^{4} + 4 T^{6} + T^{8} \)
$43$ \( T^{8} \)
$47$ \( T^{8} \)
$53$ \( ( 1 + 4 T^{2} + T^{4} )^{2} \)
$59$ \( T^{8} \)
$61$ \( ( 1 - T + T^{2} )^{4} \)
$67$ \( T^{8} \)
$71$ \( T^{8} \)
$73$ \( ( 1 + T^{2} )^{4} \)
$79$ \( T^{8} \)
$83$ \( T^{8} \)
$89$ \( ( 4 + 2 T^{2} + T^{4} )^{2} \)
$97$ \( ( 16 - 4 T^{2} + T^{4} )^{2} \)
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