# Properties

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

# Related objects

## 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{25}]$$ 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.

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

## Character values

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

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

## 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
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
−1.73205 0 2.00000 0 0 0 −1.73205 0 0
2971.2 1.73205 0 2.00000 0 0 0 1.73205 0 0
3106.1 −1.73205 0 2.00000 0 0 0 −1.73205 0 0
3106.2 1.73205 0 2.00000 0 0 0 1.73205 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.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.e 4
3.b odd 2 1 inner 3969.1.t.e 4
7.b odd 2 1 CM 3969.1.t.e 4
7.c even 3 1 567.1.l.d 4
7.c even 3 1 3969.1.k.e 4
7.d odd 6 1 567.1.l.d 4
7.d odd 6 1 3969.1.k.e 4
9.c even 3 1 3969.1.k.e 4
9.c even 3 1 3969.1.m.c 4
9.d odd 6 1 3969.1.k.e 4
9.d odd 6 1 3969.1.m.c 4
21.c even 2 1 inner 3969.1.t.e 4
21.g even 6 1 567.1.l.d 4
21.g even 6 1 3969.1.k.e 4
21.h odd 6 1 567.1.l.d 4
21.h odd 6 1 3969.1.k.e 4
63.g even 3 1 567.1.l.d 4
63.g even 3 1 3969.1.m.c 4
63.h even 3 1 567.1.d.c 2
63.h even 3 1 inner 3969.1.t.e 4
63.i even 6 1 567.1.d.c 2
63.i even 6 1 inner 3969.1.t.e 4
63.j odd 6 1 567.1.d.c 2
63.j odd 6 1 inner 3969.1.t.e 4
63.k odd 6 1 567.1.l.d 4
63.k odd 6 1 3969.1.m.c 4
63.l odd 6 1 3969.1.k.e 4
63.l odd 6 1 3969.1.m.c 4
63.n odd 6 1 567.1.l.d 4
63.n odd 6 1 3969.1.m.c 4
63.o even 6 1 3969.1.k.e 4
63.o even 6 1 3969.1.m.c 4
63.s even 6 1 567.1.l.d 4
63.s even 6 1 3969.1.m.c 4
63.t odd 6 1 567.1.d.c 2
63.t odd 6 1 inner 3969.1.t.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
567.1.d.c 2 63.h even 3 1
567.1.d.c 2 63.i even 6 1
567.1.d.c 2 63.j odd 6 1
567.1.d.c 2 63.t odd 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.g even 3 1
567.1.l.d 4 63.k odd 6 1
567.1.l.d 4 63.n odd 6 1
567.1.l.d 4 63.s even 6 1
3969.1.k.e 4 7.c even 3 1
3969.1.k.e 4 7.d odd 6 1
3969.1.k.e 4 9.c even 3 1
3969.1.k.e 4 9.d odd 6 1
3969.1.k.e 4 21.g even 6 1
3969.1.k.e 4 21.h odd 6 1
3969.1.k.e 4 63.l odd 6 1
3969.1.k.e 4 63.o even 6 1
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.g even 3 1
3969.1.m.c 4 63.k odd 6 1
3969.1.m.c 4 63.l odd 6 1
3969.1.m.c 4 63.n odd 6 1
3969.1.m.c 4 63.o even 6 1
3969.1.m.c 4 63.s even 6 1
3969.1.t.e 4 1.a even 1 1 trivial
3969.1.t.e 4 3.b odd 2 1 inner
3969.1.t.e 4 7.b odd 2 1 CM
3969.1.t.e 4 21.c even 2 1 inner
3969.1.t.e 4 63.h even 3 1 inner
3969.1.t.e 4 63.i even 6 1 inner
3969.1.t.e 4 63.j odd 6 1 inner
3969.1.t.e 4 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 $$S_{1}^{\mathrm{new}}(3969, [\chi])$$:

 $$T_{2}^{2} - 3$$ $$T_{13}$$

## Hecke characteristic polynomials

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