Properties

Label 3744.1.l.a
Level $3744$
Weight $1$
Character orbit 3744.l
Analytic conductor $1.868$
Analytic rank $0$
Dimension $4$
Projective image $D_{4}$
CM discriminant -52
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.l (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.86849940730\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Projective image \(D_{4}\)
Projective field Galois closure of 4.2.292032.4

$q$-expansion

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

\(f(q)\) \(=\) \( q +O(q^{10})\) \( q + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{11} - q^{13} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{17} + 2 \zeta_{8}^{2} q^{19} - q^{25} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{29} + 2 \zeta_{8}^{2} q^{31} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{47} + q^{49} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{53} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{59} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{71} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{83} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 4q^{13} - 4q^{25} + 4q^{49} + O(q^{100}) \)

Character values

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

\(n\) \(703\) \(2017\) \(2081\) \(2341\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3041.1
0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 + 0.707107i
−0.707107 0.707107i
0 0 0 0 0 0 0 0 0
3041.2 0 0 0 0 0 0 0 0 0
3041.3 0 0 0 0 0 0 0 0 0
3041.4 0 0 0 0 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
52.b odd 2 1 CM by \(\Q(\sqrt{-13}) \)
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner
13.b even 2 1 inner
39.d odd 2 1 inner
156.h even 2 1 inner

Twists

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

Hecke kernels

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

\( T_{5} \)
\( T_{7} \)

Hecke characteristic polynomials

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