# Properties

 Label 3969.2.a.m Level $3969$ Weight $2$ Character orbit 3969.a Self dual yes Analytic conductor $31.693$ Analytic rank $0$ Dimension $3$ 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: $$3$$ Coefficient field: 3.3.321.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 4x + 1$$ x^3 - x^2 - 4*x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ 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)

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

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + (\beta_{2} + \beta_1 + 1) q^{4} + (\beta_{2} + 2) q^{5} + ( - \beta_{2} - \beta_1 - 2) q^{8}+O(q^{10})$$ q - b1 * q^2 + (b2 + b1 + 1) * q^4 + (b2 + 2) * q^5 + (-b2 - b1 - 2) * q^8 $$q - \beta_1 q^{2} + (\beta_{2} + \beta_1 + 1) q^{4} + (\beta_{2} + 2) q^{5} + ( - \beta_{2} - \beta_1 - 2) q^{8} + ( - 3 \beta_1 + 1) q^{10} + (\beta_{2} + 2 \beta_1 - 1) q^{11} - q^{13} + ( - \beta_{2} + 2 \beta_1) q^{16} + ( - \beta_{2} - \beta_1 + 4) q^{17} + (\beta_{2} + \beta_1 + 1) q^{19} + (\beta_{2} + 2 \beta_1 + 5) q^{20} + ( - 2 \beta_{2} - 2 \beta_1 - 5) q^{22} + (2 \beta_{2} - \beta_1 + 1) q^{23} + (2 \beta_{2} - \beta_1 + 3) q^{25} + \beta_1 q^{26} + \beta_1 q^{29} + (\beta_{2} - 2 \beta_1 + 2) q^{31} + (\beta_1 - 3) q^{32} + (\beta_{2} - 2 \beta_1 + 2) q^{34} + 3 \beta_{2} q^{37} + ( - \beta_{2} - 3 \beta_1 - 2) q^{38} + ( - 2 \beta_{2} - 2 \beta_1 - 7) q^{40} + ( - \beta_{2} + 7) q^{41} + ( - \beta_{2} - 4 \beta_1) q^{43} + (5 \beta_1 + 6) q^{44} + (\beta_{2} - 2 \beta_1 + 5) q^{46} + (3 \beta_{2} + 3 \beta_1 + 3) q^{47} + (\beta_{2} - 4 \beta_1 + 5) q^{50} + ( - \beta_{2} - \beta_1 - 1) q^{52} + (2 \beta_{2} - \beta_1 - 5) q^{53} + ( - \beta_{2} + 5 \beta_1) q^{55} + ( - \beta_{2} - \beta_1 - 3) q^{58} + (2 \beta_{2} - \beta_1 + 4) q^{59} + ( - 2 \beta_{2} + \beta_1 + 1) q^{61} + (2 \beta_{2} - \beta_1 + 7) q^{62} + (\beta_{2} - 2 \beta_1 - 3) q^{64} + ( - \beta_{2} - 2) q^{65} + ( - \beta_{2} - 7 \beta_1 + 2) q^{67} + (4 \beta_{2} + \beta_1 - 1) q^{68} + ( - 2 \beta_{2} + \beta_1 + 2) q^{71} + (5 \beta_{2} - \beta_1 + 1) q^{73} + ( - 3 \beta_1 + 3) q^{74} + (\beta_{2} + 4 \beta_1 + 6) q^{76} + ( - \beta_{2} + 5 \beta_1 + 3) q^{79} + (7 \beta_1 - 6) q^{80} + ( - 6 \beta_1 - 1) q^{82} + ( - \beta_{2} - \beta_1 + 4) q^{83} + (4 \beta_{2} - 2 \beta_1 + 5) q^{85} + (4 \beta_{2} + 5 \beta_1 + 11) q^{86} + ( - \beta_{2} - 7 \beta_1 - 5) q^{88} + (\beta_{2} + 6 \beta_1 - 1) q^{89} + ( - 2 \beta_{2} - 2 \beta_1 + 5) q^{92} + ( - 3 \beta_{2} - 9 \beta_1 - 6) q^{94} + (\beta_{2} + 2 \beta_1 + 5) q^{95} + (2 \beta_{2} + 5 \beta_1 - 2) q^{97}+O(q^{100})$$ q - b1 * q^2 + (b2 + b1 + 1) * q^4 + (b2 + 2) * q^5 + (-b2 - b1 - 2) * q^8 + (-3*b1 + 1) * q^10 + (b2 + 2*b1 - 1) * q^11 - q^13 + (-b2 + 2*b1) * q^16 + (-b2 - b1 + 4) * q^17 + (b2 + b1 + 1) * q^19 + (b2 + 2*b1 + 5) * q^20 + (-2*b2 - 2*b1 - 5) * q^22 + (2*b2 - b1 + 1) * q^23 + (2*b2 - b1 + 3) * q^25 + b1 * q^26 + b1 * q^29 + (b2 - 2*b1 + 2) * q^31 + (b1 - 3) * q^32 + (b2 - 2*b1 + 2) * q^34 + 3*b2 * q^37 + (-b2 - 3*b1 - 2) * q^38 + (-2*b2 - 2*b1 - 7) * q^40 + (-b2 + 7) * q^41 + (-b2 - 4*b1) * q^43 + (5*b1 + 6) * q^44 + (b2 - 2*b1 + 5) * q^46 + (3*b2 + 3*b1 + 3) * q^47 + (b2 - 4*b1 + 5) * q^50 + (-b2 - b1 - 1) * q^52 + (2*b2 - b1 - 5) * q^53 + (-b2 + 5*b1) * q^55 + (-b2 - b1 - 3) * q^58 + (2*b2 - b1 + 4) * q^59 + (-2*b2 + b1 + 1) * q^61 + (2*b2 - b1 + 7) * q^62 + (b2 - 2*b1 - 3) * q^64 + (-b2 - 2) * q^65 + (-b2 - 7*b1 + 2) * q^67 + (4*b2 + b1 - 1) * q^68 + (-2*b2 + b1 + 2) * q^71 + (5*b2 - b1 + 1) * q^73 + (-3*b1 + 3) * q^74 + (b2 + 4*b1 + 6) * q^76 + (-b2 + 5*b1 + 3) * q^79 + (7*b1 - 6) * q^80 + (-6*b1 - 1) * q^82 + (-b2 - b1 + 4) * q^83 + (4*b2 - 2*b1 + 5) * q^85 + (4*b2 + 5*b1 + 11) * q^86 + (-b2 - 7*b1 - 5) * q^88 + (b2 + 6*b1 - 1) * q^89 + (-2*b2 - 2*b1 + 5) * q^92 + (-3*b2 - 9*b1 - 6) * q^94 + (b2 + 2*b1 + 5) * q^95 + (2*b2 + 5*b1 - 2) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - q^{2} + 3 q^{4} + 5 q^{5} - 6 q^{8}+O(q^{10})$$ 3 * q - q^2 + 3 * q^4 + 5 * q^5 - 6 * q^8 $$3 q - q^{2} + 3 q^{4} + 5 q^{5} - 6 q^{8} - 2 q^{11} - 3 q^{13} + 3 q^{16} + 12 q^{17} + 3 q^{19} + 16 q^{20} - 15 q^{22} + 6 q^{25} + q^{26} + q^{29} + 3 q^{31} - 8 q^{32} + 3 q^{34} - 3 q^{37} - 8 q^{38} - 21 q^{40} + 22 q^{41} - 3 q^{43} + 23 q^{44} + 12 q^{46} + 9 q^{47} + 10 q^{50} - 3 q^{52} - 18 q^{53} + 6 q^{55} - 9 q^{58} + 9 q^{59} + 6 q^{61} + 18 q^{62} - 12 q^{64} - 5 q^{65} - 6 q^{68} + 9 q^{71} - 3 q^{73} + 6 q^{74} + 21 q^{76} + 15 q^{79} - 11 q^{80} - 9 q^{82} + 12 q^{83} + 9 q^{85} + 34 q^{86} - 21 q^{88} + 2 q^{89} + 15 q^{92} - 24 q^{94} + 16 q^{95} - 3 q^{97}+O(q^{100})$$ 3 * q - q^2 + 3 * q^4 + 5 * q^5 - 6 * q^8 - 2 * q^11 - 3 * q^13 + 3 * q^16 + 12 * q^17 + 3 * q^19 + 16 * q^20 - 15 * q^22 + 6 * q^25 + q^26 + q^29 + 3 * q^31 - 8 * q^32 + 3 * q^34 - 3 * q^37 - 8 * q^38 - 21 * q^40 + 22 * q^41 - 3 * q^43 + 23 * q^44 + 12 * q^46 + 9 * q^47 + 10 * q^50 - 3 * q^52 - 18 * q^53 + 6 * q^55 - 9 * q^58 + 9 * q^59 + 6 * q^61 + 18 * q^62 - 12 * q^64 - 5 * q^65 - 6 * q^68 + 9 * q^71 - 3 * q^73 + 6 * q^74 + 21 * q^76 + 15 * q^79 - 11 * q^80 - 9 * q^82 + 12 * q^83 + 9 * q^85 + 34 * q^86 - 21 * q^88 + 2 * q^89 + 15 * q^92 - 24 * q^94 + 16 * q^95 - 3 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 4x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$ v^2 - v - 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 3$$ b2 + b1 + 3

## 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
 2.46050 0.239123 −1.69963
−2.46050 0 4.05408 2.59358 0 0 −5.05408 0 −6.38151
1.2 −0.239123 0 −1.94282 −1.18194 0 0 0.942820 0 0.282630
1.3 1.69963 0 0.888736 3.58836 0 0 −1.88874 0 6.09888
 $$n$$: 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.m 3
3.b odd 2 1 3969.2.a.p 3
7.b odd 2 1 567.2.a.d 3
9.c even 3 2 441.2.f.d 6
9.d odd 6 2 1323.2.f.c 6
21.c even 2 1 567.2.a.g 3
28.d even 2 1 9072.2.a.bq 3
63.g even 3 2 441.2.g.d 6
63.h even 3 2 441.2.h.b 6
63.i even 6 2 1323.2.h.d 6
63.j odd 6 2 1323.2.h.e 6
63.k odd 6 2 441.2.g.e 6
63.l odd 6 2 63.2.f.b 6
63.n odd 6 2 1323.2.g.b 6
63.o even 6 2 189.2.f.a 6
63.s even 6 2 1323.2.g.c 6
63.t odd 6 2 441.2.h.c 6
84.h odd 2 1 9072.2.a.cd 3
252.s odd 6 2 3024.2.r.g 6
252.bi even 6 2 1008.2.r.k 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.f.b 6 63.l odd 6 2
189.2.f.a 6 63.o even 6 2
441.2.f.d 6 9.c even 3 2
441.2.g.d 6 63.g even 3 2
441.2.g.e 6 63.k odd 6 2
441.2.h.b 6 63.h even 3 2
441.2.h.c 6 63.t odd 6 2
567.2.a.d 3 7.b odd 2 1
567.2.a.g 3 21.c even 2 1
1008.2.r.k 6 252.bi even 6 2
1323.2.f.c 6 9.d odd 6 2
1323.2.g.b 6 63.n odd 6 2
1323.2.g.c 6 63.s even 6 2
1323.2.h.d 6 63.i even 6 2
1323.2.h.e 6 63.j odd 6 2
3024.2.r.g 6 252.s odd 6 2
3969.2.a.m 3 1.a even 1 1 trivial
3969.2.a.p 3 3.b odd 2 1
9072.2.a.bq 3 28.d even 2 1
9072.2.a.cd 3 84.h odd 2 1

## Hecke kernels

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

 $$T_{2}^{3} + T_{2}^{2} - 4T_{2} - 1$$ T2^3 + T2^2 - 4*T2 - 1 $$T_{5}^{3} - 5T_{5}^{2} + 2T_{5} + 11$$ T5^3 - 5*T5^2 + 2*T5 + 11 $$T_{11}^{3} + 2T_{11}^{2} - 19T_{11} - 47$$ T11^3 + 2*T11^2 - 19*T11 - 47 $$T_{13} + 1$$ T13 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} + T^{2} - 4T - 1$$
$3$ $$T^{3}$$
$5$ $$T^{3} - 5 T^{2} + 2 T + 11$$
$7$ $$T^{3}$$
$11$ $$T^{3} + 2 T^{2} - 19 T - 47$$
$13$ $$(T + 1)^{3}$$
$17$ $$T^{3} - 12 T^{2} + 39 T - 27$$
$19$ $$T^{3} - 3 T^{2} - 6 T + 7$$
$23$ $$T^{3} - 33T - 9$$
$29$ $$T^{3} - T^{2} - 4T + 1$$
$31$ $$T^{3} - 3 T^{2} - 24 T - 27$$
$37$ $$T^{3} + 3 T^{2} - 54 T + 81$$
$41$ $$T^{3} - 22 T^{2} + 155 T - 353$$
$43$ $$T^{3} + 3 T^{2} - 66 T + 121$$
$47$ $$T^{3} - 9 T^{2} - 54 T + 189$$
$53$ $$T^{3} + 18 T^{2} + 75 T + 9$$
$59$ $$T^{3} - 9 T^{2} - 6 T + 63$$
$61$ $$T^{3} - 6 T^{2} - 21 T + 67$$
$67$ $$T^{3} - 207T + 683$$
$71$ $$T^{3} - 9 T^{2} - 6 T + 81$$
$73$ $$T^{3} + 3 T^{2} - 168 T + 243$$
$79$ $$T^{3} - 15 T^{2} - 48 T + 769$$
$83$ $$T^{3} - 12 T^{2} + 39 T - 27$$
$89$ $$T^{3} - 2 T^{2} - 151 T - 379$$
$97$ $$T^{3} + 3 T^{2} - 114 T - 603$$