Properties

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

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Newspace parameters

Level: \( N \) \(=\) \( 3744 = 2^{5} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3744.gs (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.86849940730\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
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.

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

Character values

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

\(n\) \(703\) \(2017\) \(2081\) \(2341\)
\(\chi(n)\) \(1\) \(\zeta_{12}^{5}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
865.1
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
0 0 0 1.36603 + 1.36603i 0 0 0 0 0
1441.1 0 0 0 −0.366025 0.366025i 0 0 0 0 0
2017.1 0 0 0 1.36603 1.36603i 0 0 0 0 0
2593.1 0 0 0 −0.366025 + 0.366025i 0 0 0 0 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
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
13.f odd 12 1 inner
52.l even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3744.1.gs.b yes 4
3.b odd 2 1 3744.1.gs.a 4
4.b odd 2 1 CM 3744.1.gs.b yes 4
12.b even 2 1 3744.1.gs.a 4
13.f odd 12 1 inner 3744.1.gs.b yes 4
39.k even 12 1 3744.1.gs.a 4
52.l even 12 1 inner 3744.1.gs.b yes 4
156.v odd 12 1 3744.1.gs.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3744.1.gs.a 4 3.b odd 2 1
3744.1.gs.a 4 12.b even 2 1
3744.1.gs.a 4 39.k even 12 1
3744.1.gs.a 4 156.v odd 12 1
3744.1.gs.b yes 4 1.a even 1 1 trivial
3744.1.gs.b yes 4 4.b odd 2 1 CM
3744.1.gs.b yes 4 13.f odd 12 1 inner
3744.1.gs.b yes 4 52.l even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3744, [\chi])\):

\( T_{5}^{4} - 2 T_{5}^{3} + 2 T_{5}^{2} + 2 T_{5} + 1 \)
\( T_{17}^{2} + 3 T_{17} + 3 \)

Hecke characteristic polynomials

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