Properties

Label 3969.1.k.e
Level $3969$
Weight $1$
Character orbit 3969.k
Analytic conductor $1.981$
Analytic rank $0$
Dimension $4$
Projective image $D_{6}$
CM discriminant -7
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 3969 = 3^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3969.k (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.98078903514\)
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: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 567)
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.2.6751269.1

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( \zeta_{12} + \zeta_{12}^{3} ) q^{2} + ( -1 + \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{4} + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{8} +O(q^{10})\) \( q + ( \zeta_{12} + \zeta_{12}^{3} ) q^{2} + ( -1 + \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{4} + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{8} + ( \zeta_{12} - \zeta_{12}^{5} ) q^{11} -\zeta_{12}^{2} q^{16} + ( 1 + 2 \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{22} + q^{25} + \zeta_{12}^{4} q^{37} + \zeta_{12}^{4} q^{43} + ( 2 \zeta_{12}^{3} + 2 \zeta_{12}^{5} ) q^{44} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{50} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{53} - q^{64} -\zeta_{12}^{4} q^{67} + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{71} + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{74} + \zeta_{12}^{2} q^{79} + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{86} + ( -2 - \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{88} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + O(q^{10}) \) \( 4 q - 4 q^{4} - 2 q^{16} + 6 q^{22} + 4 q^{25} - 2 q^{37} - 2 q^{43} - 4 q^{64} + 2 q^{67} + 2 q^{79} - 12 q^{88} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3969\mathbb{Z}\right)^\times\).

\(n\) \(2108\) \(3727\)
\(\chi(n)\) \(-\zeta_{12}^{2}\) \(-\zeta_{12}^{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
460.1
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 1.50000i 0 −1.00000 + 1.73205i 0 0 0 1.73205 0 0
460.2 0.866025 + 1.50000i 0 −1.00000 + 1.73205i 0 0 0 −1.73205 0 0
1648.1 −0.866025 + 1.50000i 0 −1.00000 1.73205i 0 0 0 1.73205 0 0
1648.2 0.866025 1.50000i 0 −1.00000 1.73205i 0 0 0 −1.73205 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
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
3.b odd 2 1 inner
21.c even 2 1 inner
63.g even 3 1 inner
63.k odd 6 1 inner
63.n odd 6 1 inner
63.s even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3969.1.k.e 4
3.b odd 2 1 inner 3969.1.k.e 4
7.b odd 2 1 CM 3969.1.k.e 4
7.c even 3 1 567.1.l.d 4
7.c even 3 1 3969.1.t.e 4
7.d odd 6 1 567.1.l.d 4
7.d odd 6 1 3969.1.t.e 4
9.c even 3 1 3969.1.m.c 4
9.c even 3 1 3969.1.t.e 4
9.d odd 6 1 3969.1.m.c 4
9.d odd 6 1 3969.1.t.e 4
21.c even 2 1 inner 3969.1.k.e 4
21.g even 6 1 567.1.l.d 4
21.g even 6 1 3969.1.t.e 4
21.h odd 6 1 567.1.l.d 4
21.h odd 6 1 3969.1.t.e 4
63.g even 3 1 567.1.d.c 2
63.g even 3 1 inner 3969.1.k.e 4
63.h even 3 1 567.1.l.d 4
63.h even 3 1 3969.1.m.c 4
63.i even 6 1 567.1.l.d 4
63.i even 6 1 3969.1.m.c 4
63.j odd 6 1 567.1.l.d 4
63.j odd 6 1 3969.1.m.c 4
63.k odd 6 1 567.1.d.c 2
63.k odd 6 1 inner 3969.1.k.e 4
63.l odd 6 1 3969.1.m.c 4
63.l odd 6 1 3969.1.t.e 4
63.n odd 6 1 567.1.d.c 2
63.n odd 6 1 inner 3969.1.k.e 4
63.o even 6 1 3969.1.m.c 4
63.o even 6 1 3969.1.t.e 4
63.s even 6 1 567.1.d.c 2
63.s even 6 1 inner 3969.1.k.e 4
63.t odd 6 1 567.1.l.d 4
63.t odd 6 1 3969.1.m.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
567.1.d.c 2 63.g even 3 1
567.1.d.c 2 63.k odd 6 1
567.1.d.c 2 63.n odd 6 1
567.1.d.c 2 63.s even 6 1
567.1.l.d 4 7.c even 3 1
567.1.l.d 4 7.d odd 6 1
567.1.l.d 4 21.g even 6 1
567.1.l.d 4 21.h odd 6 1
567.1.l.d 4 63.h even 3 1
567.1.l.d 4 63.i even 6 1
567.1.l.d 4 63.j odd 6 1
567.1.l.d 4 63.t odd 6 1
3969.1.k.e 4 1.a even 1 1 trivial
3969.1.k.e 4 3.b odd 2 1 inner
3969.1.k.e 4 7.b odd 2 1 CM
3969.1.k.e 4 21.c even 2 1 inner
3969.1.k.e 4 63.g even 3 1 inner
3969.1.k.e 4 63.k odd 6 1 inner
3969.1.k.e 4 63.n odd 6 1 inner
3969.1.k.e 4 63.s even 6 1 inner
3969.1.m.c 4 9.c even 3 1
3969.1.m.c 4 9.d odd 6 1
3969.1.m.c 4 63.h even 3 1
3969.1.m.c 4 63.i even 6 1
3969.1.m.c 4 63.j odd 6 1
3969.1.m.c 4 63.l odd 6 1
3969.1.m.c 4 63.o even 6 1
3969.1.m.c 4 63.t odd 6 1
3969.1.t.e 4 7.c even 3 1
3969.1.t.e 4 7.d odd 6 1
3969.1.t.e 4 9.c even 3 1
3969.1.t.e 4 9.d odd 6 1
3969.1.t.e 4 21.g even 6 1
3969.1.t.e 4 21.h odd 6 1
3969.1.t.e 4 63.l odd 6 1
3969.1.t.e 4 63.o even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3969, [\chi])\):

\( T_{2}^{4} + 3 T_{2}^{2} + 9 \)
\( T_{13} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 9 + 3 T^{2} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( -3 + T^{2} )^{2} \)
$13$ \( T^{4} \)
$17$ \( T^{4} \)
$19$ \( T^{4} \)
$23$ \( T^{4} \)
$29$ \( T^{4} \)
$31$ \( T^{4} \)
$37$ \( ( 1 + T + T^{2} )^{2} \)
$41$ \( T^{4} \)
$43$ \( ( 1 + T + T^{2} )^{2} \)
$47$ \( T^{4} \)
$53$ \( 9 + 3 T^{2} + T^{4} \)
$59$ \( T^{4} \)
$61$ \( T^{4} \)
$67$ \( ( 1 - T + T^{2} )^{2} \)
$71$ \( ( -3 + T^{2} )^{2} \)
$73$ \( T^{4} \)
$79$ \( ( 1 - T + T^{2} )^{2} \)
$83$ \( T^{4} \)
$89$ \( T^{4} \)
$97$ \( T^{4} \)
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