Properties

Label 3744.1.dy.a
Level $3744$
Weight $1$
Character orbit 3744.dy
Analytic conductor $1.868$
Analytic rank $0$
Dimension $4$
Projective image $A_{4}$
CM/RM no
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.dy (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.86849940730\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 416)
Projective image \(A_{4}\)
Projective field Galois closure of 4.0.10816.1

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{12} q^{7} +O(q^{10})\) \( q + \zeta_{12} q^{7} -\zeta_{12}^{5} q^{11} + q^{13} -\zeta_{12}^{4} q^{17} + \zeta_{12} q^{19} + \zeta_{12}^{5} q^{23} - q^{25} -\zeta_{12}^{2} q^{29} -\zeta_{12}^{2} q^{37} + \zeta_{12}^{2} q^{41} -\zeta_{12} q^{43} + 2 \zeta_{12}^{3} q^{47} -\zeta_{12} q^{59} -\zeta_{12}^{4} q^{61} -\zeta_{12}^{5} q^{67} + \zeta_{12} q^{71} + q^{77} + 2 \zeta_{12}^{3} q^{79} -\zeta_{12}^{2} q^{89} + \zeta_{12} q^{91} -\zeta_{12}^{4} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q + 4q^{13} + 2q^{17} - 4q^{25} - 2q^{29} - 2q^{37} + 2q^{41} + 2q^{61} + 4q^{77} - 2q^{89} + 2q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(703\) \(2017\) \(2081\) \(2341\)
\(\chi(n)\) \(-1\) \(\zeta_{12}^{4}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
991.1
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
0 0 0 0 0 −0.866025 0.500000i 0 0 0
991.2 0 0 0 0 0 0.866025 + 0.500000i 0 0 0
1855.1 0 0 0 0 0 −0.866025 + 0.500000i 0 0 0
1855.2 0 0 0 0 0 0.866025 0.500000i 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 inner
13.c even 3 1 inner
52.j odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3744.1.dy.a 4
3.b odd 2 1 416.1.bb.a 4
4.b odd 2 1 inner 3744.1.dy.a 4
12.b even 2 1 416.1.bb.a 4
13.c even 3 1 inner 3744.1.dy.a 4
24.f even 2 1 832.1.bb.b 4
24.h odd 2 1 832.1.bb.b 4
39.i odd 6 1 416.1.bb.a 4
48.i odd 4 1 3328.1.v.a 4
48.i odd 4 1 3328.1.v.c 4
48.k even 4 1 3328.1.v.a 4
48.k even 4 1 3328.1.v.c 4
52.j odd 6 1 inner 3744.1.dy.a 4
156.p even 6 1 416.1.bb.a 4
312.bh odd 6 1 832.1.bb.b 4
312.bn even 6 1 832.1.bb.b 4
624.cl even 12 1 3328.1.v.a 4
624.cl even 12 1 3328.1.v.c 4
624.cw odd 12 1 3328.1.v.a 4
624.cw odd 12 1 3328.1.v.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
416.1.bb.a 4 3.b odd 2 1
416.1.bb.a 4 12.b even 2 1
416.1.bb.a 4 39.i odd 6 1
416.1.bb.a 4 156.p even 6 1
832.1.bb.b 4 24.f even 2 1
832.1.bb.b 4 24.h odd 2 1
832.1.bb.b 4 312.bh odd 6 1
832.1.bb.b 4 312.bn even 6 1
3328.1.v.a 4 48.i odd 4 1
3328.1.v.a 4 48.k even 4 1
3328.1.v.a 4 624.cl even 12 1
3328.1.v.a 4 624.cw odd 12 1
3328.1.v.c 4 48.i odd 4 1
3328.1.v.c 4 48.k even 4 1
3328.1.v.c 4 624.cl even 12 1
3328.1.v.c 4 624.cw odd 12 1
3744.1.dy.a 4 1.a even 1 1 trivial
3744.1.dy.a 4 4.b odd 2 1 inner
3744.1.dy.a 4 13.c even 3 1 inner
3744.1.dy.a 4 52.j odd 6 1 inner

Hecke kernels

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

Hecke characteristic polynomials

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