Properties

Label 3744.1.ck.a
Level $3744$
Weight $1$
Character orbit 3744.ck
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.ck (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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(A_{4}\)
Projective field Galois closure of 4.0.876096.1

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{12}^{5} q^{3} -\zeta_{12}^{4} q^{5} -\zeta_{12}^{4} q^{9} +O(q^{10})\) \( q + \zeta_{12}^{5} q^{3} -\zeta_{12}^{4} q^{5} -\zeta_{12}^{4} q^{9} + \zeta_{12}^{3} q^{11} + \zeta_{12}^{2} q^{13} + \zeta_{12}^{3} q^{15} -\zeta_{12}^{2} q^{17} -\zeta_{12}^{5} q^{19} + \zeta_{12}^{3} q^{27} + q^{29} -\zeta_{12} q^{31} -\zeta_{12}^{2} q^{33} + \zeta_{12}^{4} q^{37} -\zeta_{12} q^{39} + 2 \zeta_{12} q^{43} -\zeta_{12}^{2} q^{45} + \zeta_{12}^{5} q^{47} -\zeta_{12}^{2} q^{49} + \zeta_{12} q^{51} + \zeta_{12} q^{55} + \zeta_{12}^{4} q^{57} -\zeta_{12}^{3} q^{59} + q^{65} -\zeta_{12}^{5} q^{71} + 2 q^{73} -\zeta_{12}^{5} q^{79} -\zeta_{12}^{2} q^{81} -\zeta_{12}^{5} q^{83} - q^{85} + \zeta_{12}^{5} q^{87} + \zeta_{12}^{4} q^{89} + q^{93} -\zeta_{12}^{3} q^{95} + \zeta_{12} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{5} + 2q^{9} + O(q^{10}) \) \( 4q + 2q^{5} + 2q^{9} + 2q^{13} - 2q^{17} + 4q^{29} - 2q^{33} - 2q^{37} - 2q^{45} - 2q^{49} - 2q^{57} + 4q^{65} + 8q^{73} - 2q^{81} - 4q^{85} - 2q^{89} + 4q^{93} + O(q^{100}) \)

Character values

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

\(n\) \(703\) \(2017\) \(2081\) \(2341\)
\(\chi(n)\) \(-1\) \(-\zeta_{12}^{2}\) \(\zeta_{12}^{4}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
607.1
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
0 −0.866025 + 0.500000i 0 0.500000 0.866025i 0 0 0 0.500000 0.866025i 0
607.2 0 0.866025 0.500000i 0 0.500000 0.866025i 0 0 0 0.500000 0.866025i 0
2239.1 0 −0.866025 0.500000i 0 0.500000 + 0.866025i 0 0 0 0.500000 + 0.866025i 0
2239.2 0 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0 0 0 0.500000 + 0.866025i 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
117.h even 3 1 inner
468.y 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.ck.a yes 4
4.b odd 2 1 inner 3744.1.ck.a yes 4
9.c even 3 1 3744.1.cc.a 4
13.c even 3 1 3744.1.cc.a 4
36.f odd 6 1 3744.1.cc.a 4
52.j odd 6 1 3744.1.cc.a 4
117.h even 3 1 inner 3744.1.ck.a yes 4
468.y odd 6 1 inner 3744.1.ck.a yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3744.1.cc.a 4 9.c even 3 1
3744.1.cc.a 4 13.c even 3 1
3744.1.cc.a 4 36.f odd 6 1
3744.1.cc.a 4 52.j odd 6 1
3744.1.ck.a yes 4 1.a even 1 1 trivial
3744.1.ck.a yes 4 4.b odd 2 1 inner
3744.1.ck.a yes 4 117.h even 3 1 inner
3744.1.ck.a yes 4 468.y odd 6 1 inner

Hecke kernels

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

Hecke characteristic polynomials

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