Properties

Label 3969.1.t.c
Level $3969$
Weight $1$
Character orbit 3969.t
Analytic conductor $1.981$
Analytic rank $0$
Dimension $2$
Projective image $D_{2}$
CM/RM discs -3, -7, 21
Inner twists $8$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(1.98078903514\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 63)
Projective image \(D_{2}\)
Projective field Galois closure of \(\Q(\sqrt{-3}, \sqrt{-7})\)
Artin image $C_3\times D_4$
Artin field Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q - q^{4} +O(q^{10})\) \( q - q^{4} + q^{16} + \zeta_{6}^{2} q^{25} -2 \zeta_{6}^{2} q^{37} + 2 \zeta_{6} q^{43} - q^{64} + 2 q^{67} + 2 q^{79} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + O(q^{10}) \) \( 2q - 2q^{4} + 2q^{16} - q^{25} + 2q^{37} + 2q^{43} - 2q^{64} + 4q^{67} + 4q^{79} + O(q^{100}) \)

Character values

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

\(n\) \(2108\) \(3727\)
\(\chi(n)\) \(\zeta_{6}^{2}\) \(-\zeta_{6}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2971.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0 −1.00000 0 0 0 0 0 0
3106.1 0 0 −1.00000 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
21.c even 2 1 RM by \(\Q(\sqrt{21}) \)
63.h even 3 1 inner
63.i even 6 1 inner
63.j odd 6 1 inner
63.t odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3969.1.t.c 2
3.b odd 2 1 CM 3969.1.t.c 2
7.b odd 2 1 CM 3969.1.t.c 2
7.c even 3 1 567.1.l.b 2
7.c even 3 1 3969.1.k.b 2
7.d odd 6 1 567.1.l.b 2
7.d odd 6 1 3969.1.k.b 2
9.c even 3 1 441.1.m.a 2
9.c even 3 1 3969.1.k.b 2
9.d odd 6 1 441.1.m.a 2
9.d odd 6 1 3969.1.k.b 2
21.c even 2 1 RM 3969.1.t.c 2
21.g even 6 1 567.1.l.b 2
21.g even 6 1 3969.1.k.b 2
21.h odd 6 1 567.1.l.b 2
21.h odd 6 1 3969.1.k.b 2
63.g even 3 1 441.1.m.a 2
63.g even 3 1 567.1.l.b 2
63.h even 3 1 63.1.d.a 1
63.h even 3 1 inner 3969.1.t.c 2
63.i even 6 1 63.1.d.a 1
63.i even 6 1 inner 3969.1.t.c 2
63.j odd 6 1 63.1.d.a 1
63.j odd 6 1 inner 3969.1.t.c 2
63.k odd 6 1 441.1.m.a 2
63.k odd 6 1 567.1.l.b 2
63.l odd 6 1 441.1.m.a 2
63.l odd 6 1 3969.1.k.b 2
63.n odd 6 1 441.1.m.a 2
63.n odd 6 1 567.1.l.b 2
63.o even 6 1 441.1.m.a 2
63.o even 6 1 3969.1.k.b 2
63.s even 6 1 441.1.m.a 2
63.s even 6 1 567.1.l.b 2
63.t odd 6 1 63.1.d.a 1
63.t odd 6 1 inner 3969.1.t.c 2
252.r odd 6 1 1008.1.f.a 1
252.u odd 6 1 1008.1.f.a 1
252.bb even 6 1 1008.1.f.a 1
252.bj even 6 1 1008.1.f.a 1
315.q odd 6 1 1575.1.h.b 1
315.r even 6 1 1575.1.h.b 1
315.bq even 6 1 1575.1.h.b 1
315.br odd 6 1 1575.1.h.b 1
315.bs even 12 2 1575.1.e.b 2
315.bt odd 12 2 1575.1.e.b 2
315.bu odd 12 2 1575.1.e.b 2
315.bv even 12 2 1575.1.e.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.1.d.a 1 63.h even 3 1
63.1.d.a 1 63.i even 6 1
63.1.d.a 1 63.j odd 6 1
63.1.d.a 1 63.t odd 6 1
441.1.m.a 2 9.c even 3 1
441.1.m.a 2 9.d odd 6 1
441.1.m.a 2 63.g even 3 1
441.1.m.a 2 63.k odd 6 1
441.1.m.a 2 63.l odd 6 1
441.1.m.a 2 63.n odd 6 1
441.1.m.a 2 63.o even 6 1
441.1.m.a 2 63.s even 6 1
567.1.l.b 2 7.c even 3 1
567.1.l.b 2 7.d odd 6 1
567.1.l.b 2 21.g even 6 1
567.1.l.b 2 21.h odd 6 1
567.1.l.b 2 63.g even 3 1
567.1.l.b 2 63.k odd 6 1
567.1.l.b 2 63.n odd 6 1
567.1.l.b 2 63.s even 6 1
1008.1.f.a 1 252.r odd 6 1
1008.1.f.a 1 252.u odd 6 1
1008.1.f.a 1 252.bb even 6 1
1008.1.f.a 1 252.bj even 6 1
1575.1.e.b 2 315.bs even 12 2
1575.1.e.b 2 315.bt odd 12 2
1575.1.e.b 2 315.bu odd 12 2
1575.1.e.b 2 315.bv even 12 2
1575.1.h.b 1 315.q odd 6 1
1575.1.h.b 1 315.r even 6 1
1575.1.h.b 1 315.bq even 6 1
1575.1.h.b 1 315.br odd 6 1
3969.1.k.b 2 7.c even 3 1
3969.1.k.b 2 7.d odd 6 1
3969.1.k.b 2 9.c even 3 1
3969.1.k.b 2 9.d odd 6 1
3969.1.k.b 2 21.g even 6 1
3969.1.k.b 2 21.h odd 6 1
3969.1.k.b 2 63.l odd 6 1
3969.1.k.b 2 63.o even 6 1
3969.1.t.c 2 1.a even 1 1 trivial
3969.1.t.c 2 3.b odd 2 1 CM
3969.1.t.c 2 7.b odd 2 1 CM
3969.1.t.c 2 21.c even 2 1 RM
3969.1.t.c 2 63.h even 3 1 inner
3969.1.t.c 2 63.i even 6 1 inner
3969.1.t.c 2 63.j odd 6 1 inner
3969.1.t.c 2 63.t odd 6 1 inner

Hecke kernels

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

\( T_{2} \)
\( T_{13} \)

Hecke characteristic polynomials

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