# Properties

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

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3969,1,Mod(1079,3969)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3969, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 1, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3969.1079");

S:= CuspForms(chi, 1);

N := Newforms(S);

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

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$1.98078903514$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{2} + 4$$ x^4 - 2*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 441) Projective image: $$D_{4}$$ Projective field: Galois closure of 4.0.189.1 Artin image: $C_3\times \SD_{16}$ Artin field: Galois closure of $$\mathbb{Q}[x]/(x^{24} - \cdots)$$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{3} + \beta_1) q^{2} + ( - \beta_{2} + 1) q^{4}+O(q^{10})$$ q + (-b3 + b1) * q^2 + (-b2 + 1) * q^4 $$q + ( - \beta_{3} + \beta_1) q^{2} + ( - \beta_{2} + 1) q^{4} + ( - \beta_{3} + \beta_1) q^{11} + \beta_{2} q^{16} + ( - 2 \beta_{2} + 2) q^{22} + \beta_1 q^{23} - \beta_{2} q^{25} + (\beta_{3} - \beta_1) q^{29} + \beta_1 q^{32} - \beta_{3} q^{44} + 2 q^{46} - \beta_1 q^{50} - \beta_{3} q^{53} + (2 \beta_{2} - 2) q^{58} + q^{64} + ( - 2 \beta_{2} + 2) q^{67} + \beta_{3} q^{71} - 2 \beta_{2} q^{79} + ( - \beta_{3} + \beta_1) q^{92}+O(q^{100})$$ q + (-b3 + b1) * q^2 + (-b2 + 1) * q^4 + (-b3 + b1) * q^11 + b2 * q^16 + (-2*b2 + 2) * q^22 + b1 * q^23 - b2 * q^25 + (b3 - b1) * q^29 + b1 * q^32 - b3 * q^44 + 2 * q^46 - b1 * q^50 - b3 * q^53 + (2*b2 - 2) * q^58 + q^64 + (-2*b2 + 2) * q^67 + b3 * q^71 - 2*b2 * q^79 + (-b3 + b1) * q^92 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{4}+O(q^{10})$$ 4 * q + 2 * q^4 $$4 q + 2 q^{4} + 2 q^{16} + 4 q^{22} - 2 q^{25} + 8 q^{46} - 4 q^{58} + 4 q^{64} + 4 q^{67} - 4 q^{79}+O(q^{100})$$ 4 * q + 2 * q^4 + 2 * q^16 + 4 * q^22 - 2 * q^25 + 8 * q^46 - 4 * q^58 + 4 * q^64 + 4 * q^67 - 4 * q^79

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{2} + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 2$$ (v^3) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$2\beta_{2}$$ 2*b2 $$\nu^{3}$$ $$=$$ $$2\beta_{3}$$ 2*b3

## Character values

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

 $$n$$ $$2108$$ $$3727$$ $$\chi(n)$$ $$1 - \beta_{2}$$ $$1$$

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1079.1
 −1.22474 + 0.707107i 1.22474 − 0.707107i −1.22474 − 0.707107i 1.22474 + 0.707107i
−1.22474 0.707107i 0 0.500000 + 0.866025i 0 0 0 0 0 0
1079.2 1.22474 + 0.707107i 0 0.500000 + 0.866025i 0 0 0 0 0 0
2402.1 −1.22474 + 0.707107i 0 0.500000 0.866025i 0 0 0 0 0 0
2402.2 1.22474 0.707107i 0 0.500000 0.866025i 0 0 0 0 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
9.c even 3 1 inner
9.d odd 6 1 inner
21.c even 2 1 inner
63.l odd 6 1 inner
63.o 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.r.c 4
3.b odd 2 1 inner 3969.1.r.c 4
7.b odd 2 1 CM 3969.1.r.c 4
7.c even 3 1 3969.1.j.b 4
7.c even 3 1 3969.1.n.b 4
7.d odd 6 1 3969.1.j.b 4
7.d odd 6 1 3969.1.n.b 4
9.c even 3 1 441.1.b.a 2
9.c even 3 1 inner 3969.1.r.c 4
9.d odd 6 1 441.1.b.a 2
9.d odd 6 1 inner 3969.1.r.c 4
21.c even 2 1 inner 3969.1.r.c 4
21.g even 6 1 3969.1.j.b 4
21.g even 6 1 3969.1.n.b 4
21.h odd 6 1 3969.1.j.b 4
21.h odd 6 1 3969.1.n.b 4
63.g even 3 1 441.1.q.a 4
63.g even 3 1 3969.1.j.b 4
63.h even 3 1 441.1.q.a 4
63.h even 3 1 3969.1.n.b 4
63.i even 6 1 441.1.q.a 4
63.i even 6 1 3969.1.n.b 4
63.j odd 6 1 441.1.q.a 4
63.j odd 6 1 3969.1.n.b 4
63.k odd 6 1 441.1.q.a 4
63.k odd 6 1 3969.1.j.b 4
63.l odd 6 1 441.1.b.a 2
63.l odd 6 1 inner 3969.1.r.c 4
63.n odd 6 1 441.1.q.a 4
63.n odd 6 1 3969.1.j.b 4
63.o even 6 1 441.1.b.a 2
63.o even 6 1 inner 3969.1.r.c 4
63.s even 6 1 441.1.q.a 4
63.s even 6 1 3969.1.j.b 4
63.t odd 6 1 441.1.q.a 4
63.t odd 6 1 3969.1.n.b 4

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

## Hecke kernels

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

 $$T_{2}^{4} - 2T_{2}^{2} + 4$$ T2^4 - 2*T2^2 + 4 $$T_{13}$$ T13

## Hecke characteristic polynomials

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