# Properties

 Label 3969.2.a.f Level $3969$ Weight $2$ Character orbit 3969.a Self dual yes Analytic conductor $31.693$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3969,2,Mod(1,3969)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3969, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("3969.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3969 = 3^{4} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3969.a (trivial)

## 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: yes Analytic conductor: $$31.6926245622$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 63) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{4} + q^{5} - 3 q^{8}+O(q^{10})$$ q + q^2 - q^4 + q^5 - 3 * q^8 $$q + q^{2} - q^{4} + q^{5} - 3 q^{8} + q^{10} + 5 q^{11} + 5 q^{13} - q^{16} - 3 q^{17} - q^{19} - q^{20} + 5 q^{22} + 3 q^{23} - 4 q^{25} + 5 q^{26} - q^{29} + 5 q^{32} - 3 q^{34} + 3 q^{37} - q^{38} - 3 q^{40} + 5 q^{41} - q^{43} - 5 q^{44} + 3 q^{46} - 4 q^{50} - 5 q^{52} - 9 q^{53} + 5 q^{55} - q^{58} + 14 q^{61} + 7 q^{64} + 5 q^{65} + 4 q^{67} + 3 q^{68} - 12 q^{71} - 3 q^{73} + 3 q^{74} + q^{76} + 8 q^{79} - q^{80} + 5 q^{82} + 9 q^{83} - 3 q^{85} - q^{86} - 15 q^{88} + 13 q^{89} - 3 q^{92} - q^{95} + 9 q^{97}+O(q^{100})$$ q + q^2 - q^4 + q^5 - 3 * q^8 + q^10 + 5 * q^11 + 5 * q^13 - q^16 - 3 * q^17 - q^19 - q^20 + 5 * q^22 + 3 * q^23 - 4 * q^25 + 5 * q^26 - q^29 + 5 * q^32 - 3 * q^34 + 3 * q^37 - q^38 - 3 * q^40 + 5 * q^41 - q^43 - 5 * q^44 + 3 * q^46 - 4 * q^50 - 5 * q^52 - 9 * q^53 + 5 * q^55 - q^58 + 14 * q^61 + 7 * q^64 + 5 * q^65 + 4 * q^67 + 3 * q^68 - 12 * q^71 - 3 * q^73 + 3 * q^74 + q^76 + 8 * q^79 - q^80 + 5 * q^82 + 9 * q^83 - 3 * q^85 - q^86 - 15 * q^88 + 13 * q^89 - 3 * q^92 - q^95 + 9 * q^97

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
1.00000 0 −1.00000 1.00000 0 0 −3.00000 0 1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$7$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3969.2.a.f 1
3.b odd 2 1 3969.2.a.a 1
7.b odd 2 1 3969.2.a.d 1
7.d odd 6 2 567.2.e.a 2
9.c even 3 2 441.2.f.a 2
9.d odd 6 2 1323.2.f.b 2
21.c even 2 1 3969.2.a.c 1
21.g even 6 2 567.2.e.b 2
63.g even 3 2 441.2.g.a 2
63.h even 3 2 441.2.h.a 2
63.i even 6 2 189.2.h.a 2
63.j odd 6 2 1323.2.h.a 2
63.k odd 6 2 63.2.g.a 2
63.l odd 6 2 441.2.f.b 2
63.n odd 6 2 1323.2.g.a 2
63.o even 6 2 1323.2.f.a 2
63.s even 6 2 189.2.g.a 2
63.t odd 6 2 63.2.h.a yes 2
252.n even 6 2 1008.2.t.d 2
252.r odd 6 2 3024.2.q.b 2
252.bj even 6 2 1008.2.q.c 2
252.bn odd 6 2 3024.2.t.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.g.a 2 63.k odd 6 2
63.2.h.a yes 2 63.t odd 6 2
189.2.g.a 2 63.s even 6 2
189.2.h.a 2 63.i even 6 2
441.2.f.a 2 9.c even 3 2
441.2.f.b 2 63.l odd 6 2
441.2.g.a 2 63.g even 3 2
441.2.h.a 2 63.h even 3 2
567.2.e.a 2 7.d odd 6 2
567.2.e.b 2 21.g even 6 2
1008.2.q.c 2 252.bj even 6 2
1008.2.t.d 2 252.n even 6 2
1323.2.f.a 2 63.o even 6 2
1323.2.f.b 2 9.d odd 6 2
1323.2.g.a 2 63.n odd 6 2
1323.2.h.a 2 63.j odd 6 2
3024.2.q.b 2 252.r odd 6 2
3024.2.t.d 2 252.bn odd 6 2
3969.2.a.a 1 3.b odd 2 1
3969.2.a.c 1 21.c even 2 1
3969.2.a.d 1 7.b odd 2 1
3969.2.a.f 1 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3969))$$:

 $$T_{2} - 1$$ T2 - 1 $$T_{5} - 1$$ T5 - 1 $$T_{11} - 5$$ T11 - 5 $$T_{13} - 5$$ T13 - 5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T$$
$5$ $$T - 1$$
$7$ $$T$$
$11$ $$T - 5$$
$13$ $$T - 5$$
$17$ $$T + 3$$
$19$ $$T + 1$$
$23$ $$T - 3$$
$29$ $$T + 1$$
$31$ $$T$$
$37$ $$T - 3$$
$41$ $$T - 5$$
$43$ $$T + 1$$
$47$ $$T$$
$53$ $$T + 9$$
$59$ $$T$$
$61$ $$T - 14$$
$67$ $$T - 4$$
$71$ $$T + 12$$
$73$ $$T + 3$$
$79$ $$T - 8$$
$83$ $$T - 9$$
$89$ $$T - 13$$
$97$ $$T - 9$$