Properties

 Label 3969.2.a.q 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: $$\Q(\zeta_{18})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - 3x - 1$$ x^3 - 3*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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 + 1) q^{2} + (\beta_{2} - 2 \beta_1 + 1) q^{4} + (\beta_{2} - \beta_1 - 1) q^{5} + (3 \beta_{2} - 2 \beta_1 + 2) q^{8}+O(q^{10})$$ q + (-b1 + 1) * q^2 + (b2 - 2*b1 + 1) * q^4 + (b2 - b1 - 1) * q^5 + (3*b2 - 2*b1 + 2) * q^8 $$q + ( - \beta_1 + 1) q^{2} + (\beta_{2} - 2 \beta_1 + 1) q^{4} + (\beta_{2} - \beta_1 - 1) q^{5} + (3 \beta_{2} - 2 \beta_1 + 2) q^{8} + (2 \beta_{2} - \beta_1) q^{10} + (\beta_{2} - \beta_1 + 2) q^{11} + (4 \beta_{2} - 2 \beta_1 + 1) q^{13} + (3 \beta_{2} - 3 \beta_1 + 1) q^{16} + (\beta_{2} - 2) q^{17} + ( - \beta_{2} + 2 \beta_1 + 1) q^{19} + (\beta_{2} - \beta_1 + 2) q^{20} + (2 \beta_{2} - 4 \beta_1 + 3) q^{22} + (2 \beta_{2} + \beta_1 + 4) q^{23} + ( - 2 \beta_{2} + \beta_1 - 2) q^{25} + (6 \beta_{2} - 7 \beta_1 + 1) q^{26} + ( - 4 \beta_{2} + 5 \beta_1 + 3) q^{29} + ( - 3 \beta_{2} - 3 \beta_1 + 1) q^{31} - 3 \beta_1 q^{32} + (\beta_{2} + \beta_1 - 3) q^{34} + (3 \beta_{2} + 3 \beta_1 - 1) q^{37} + ( - 3 \beta_{2} + 2 \beta_1 - 2) q^{38} + ( - 2 \beta_{2} - 2 \beta_1 + 3) q^{40} + ( - \beta_{2} - \beta_1) q^{41} + ( - \beta_{2} - \beta_1 - 1) q^{43} + (4 \beta_{2} - 7 \beta_1 + 5) q^{44} + (\beta_{2} - 5 \beta_1) q^{46} + ( - 3 \beta_{2} - 2 \beta_1 - 1) q^{47} + ( - 3 \beta_{2} + 5 \beta_1 - 2) q^{50} + (5 \beta_{2} - 10 \beta_1 + 7) q^{52} + (2 \beta_{2} - 3 \beta_1 + 2) q^{53} + (\beta_{2} - 2 \beta_1) q^{55} + ( - 9 \beta_{2} + 6 \beta_1 - 3) q^{58} + (5 \beta_1 + 1) q^{59} + ( - 3 \beta_1 - 2) q^{61} + ( - \beta_1 + 10) q^{62} + ( - 3 \beta_{2} + 3 \beta_1 + 4) q^{64} + ( - 5 \beta_{2} - \beta_1 + 5) q^{65} + (3 \beta_{2} - 4) q^{67} + ( - 2 \beta_{2} + 3 \beta_1 - 2) q^{68} + ( - 6 \beta_{2} + 3 \beta_1 + 3) q^{71} + ( - \beta_{2} - 4 \beta_1 + 7) q^{73} + (\beta_1 - 10) q^{74} + ( - 3 \beta_{2} + 3 \beta_1 - 5) q^{76} + (3 \beta_{2} - 7) q^{79} + ( - 2 \beta_{2} - \beta_1 + 5) q^{80} + 3 q^{82} + (\beta_{2} + 4 \beta_1 + 6) q^{83} + ( - 4 \beta_{2} + 2 \beta_1 + 3) q^{85} + (\beta_1 + 2) q^{86} + (7 \beta_{2} - 8 \beta_1 + 9) q^{88} + ( - 7 \beta_{2} + 3 \beta_1 - 4) q^{89} + (2 \beta_{2} - 8 \beta_1 + 1) q^{92} + ( - \beta_{2} + 2 \beta_1 + 6) q^{94} + (\beta_{2} - \beta_1 - 4) q^{95} + ( - 8 \beta_{2} + 7 \beta_1 + 1) q^{97}+O(q^{100})$$ q + (-b1 + 1) * q^2 + (b2 - 2*b1 + 1) * q^4 + (b2 - b1 - 1) * q^5 + (3*b2 - 2*b1 + 2) * q^8 + (2*b2 - b1) * q^10 + (b2 - b1 + 2) * q^11 + (4*b2 - 2*b1 + 1) * q^13 + (3*b2 - 3*b1 + 1) * q^16 + (b2 - 2) * q^17 + (-b2 + 2*b1 + 1) * q^19 + (b2 - b1 + 2) * q^20 + (2*b2 - 4*b1 + 3) * q^22 + (2*b2 + b1 + 4) * q^23 + (-2*b2 + b1 - 2) * q^25 + (6*b2 - 7*b1 + 1) * q^26 + (-4*b2 + 5*b1 + 3) * q^29 + (-3*b2 - 3*b1 + 1) * q^31 - 3*b1 * q^32 + (b2 + b1 - 3) * q^34 + (3*b2 + 3*b1 - 1) * q^37 + (-3*b2 + 2*b1 - 2) * q^38 + (-2*b2 - 2*b1 + 3) * q^40 + (-b2 - b1) * q^41 + (-b2 - b1 - 1) * q^43 + (4*b2 - 7*b1 + 5) * q^44 + (b2 - 5*b1) * q^46 + (-3*b2 - 2*b1 - 1) * q^47 + (-3*b2 + 5*b1 - 2) * q^50 + (5*b2 - 10*b1 + 7) * q^52 + (2*b2 - 3*b1 + 2) * q^53 + (b2 - 2*b1) * q^55 + (-9*b2 + 6*b1 - 3) * q^58 + (5*b1 + 1) * q^59 + (-3*b1 - 2) * q^61 + (-b1 + 10) * q^62 + (-3*b2 + 3*b1 + 4) * q^64 + (-5*b2 - b1 + 5) * q^65 + (3*b2 - 4) * q^67 + (-2*b2 + 3*b1 - 2) * q^68 + (-6*b2 + 3*b1 + 3) * q^71 + (-b2 - 4*b1 + 7) * q^73 + (b1 - 10) * q^74 + (-3*b2 + 3*b1 - 5) * q^76 + (3*b2 - 7) * q^79 + (-2*b2 - b1 + 5) * q^80 + 3 * q^82 + (b2 + 4*b1 + 6) * q^83 + (-4*b2 + 2*b1 + 3) * q^85 + (b1 + 2) * q^86 + (7*b2 - 8*b1 + 9) * q^88 + (-7*b2 + 3*b1 - 4) * q^89 + (2*b2 - 8*b1 + 1) * q^92 + (-b2 + 2*b1 + 6) * q^94 + (b2 - b1 - 4) * q^95 + (-8*b2 + 7*b1 + 1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{2} + 3 q^{4} - 3 q^{5} + 6 q^{8}+O(q^{10})$$ 3 * q + 3 * q^2 + 3 * q^4 - 3 * q^5 + 6 * q^8 $$3 q + 3 q^{2} + 3 q^{4} - 3 q^{5} + 6 q^{8} + 6 q^{11} + 3 q^{13} + 3 q^{16} - 6 q^{17} + 3 q^{19} + 6 q^{20} + 9 q^{22} + 12 q^{23} - 6 q^{25} + 3 q^{26} + 9 q^{29} + 3 q^{31} - 9 q^{34} - 3 q^{37} - 6 q^{38} + 9 q^{40} - 3 q^{43} + 15 q^{44} - 3 q^{47} - 6 q^{50} + 21 q^{52} + 6 q^{53} - 9 q^{58} + 3 q^{59} - 6 q^{61} + 30 q^{62} + 12 q^{64} + 15 q^{65} - 12 q^{67} - 6 q^{68} + 9 q^{71} + 21 q^{73} - 30 q^{74} - 15 q^{76} - 21 q^{79} + 15 q^{80} + 9 q^{82} + 18 q^{83} + 9 q^{85} + 6 q^{86} + 27 q^{88} - 12 q^{89} + 3 q^{92} + 18 q^{94} - 12 q^{95} + 3 q^{97}+O(q^{100})$$ 3 * q + 3 * q^2 + 3 * q^4 - 3 * q^5 + 6 * q^8 + 6 * q^11 + 3 * q^13 + 3 * q^16 - 6 * q^17 + 3 * q^19 + 6 * q^20 + 9 * q^22 + 12 * q^23 - 6 * q^25 + 3 * q^26 + 9 * q^29 + 3 * q^31 - 9 * q^34 - 3 * q^37 - 6 * q^38 + 9 * q^40 - 3 * q^43 + 15 * q^44 - 3 * q^47 - 6 * q^50 + 21 * q^52 + 6 * q^53 - 9 * q^58 + 3 * q^59 - 6 * q^61 + 30 * q^62 + 12 * q^64 + 15 * q^65 - 12 * q^67 - 6 * q^68 + 9 * q^71 + 21 * q^73 - 30 * q^74 - 15 * q^76 - 21 * q^79 + 15 * q^80 + 9 * q^82 + 18 * q^83 + 9 * q^85 + 6 * q^86 + 27 * q^88 - 12 * q^89 + 3 * q^92 + 18 * q^94 - 12 * q^95 + 3 * q^97

Basis of coefficient ring in terms of $$\nu = \zeta_{18} + \zeta_{18}^{-1}$$:

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

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
 1.87939 −0.347296 −1.53209
−0.879385 0 −1.22668 −1.34730 0 0 2.83750 0 1.18479
1.2 1.34730 0 −0.184793 −2.53209 0 0 −2.94356 0 −3.41147
1.3 2.53209 0 4.41147 0.879385 0 0 6.10607 0 2.22668
 $$n$$: 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.q 3
3.b odd 2 1 3969.2.a.l 3
7.b odd 2 1 567.2.a.h 3
9.c even 3 2 441.2.f.c 6
9.d odd 6 2 1323.2.f.d 6
21.c even 2 1 567.2.a.c 3
28.d even 2 1 9072.2.a.ca 3
63.g even 3 2 441.2.g.b 6
63.h even 3 2 441.2.h.e 6
63.i even 6 2 1323.2.h.c 6
63.j odd 6 2 1323.2.h.b 6
63.k odd 6 2 441.2.g.c 6
63.l odd 6 2 63.2.f.a 6
63.n odd 6 2 1323.2.g.e 6
63.o even 6 2 189.2.f.b 6
63.s even 6 2 1323.2.g.d 6
63.t odd 6 2 441.2.h.d 6
84.h odd 2 1 9072.2.a.bs 3
252.s odd 6 2 3024.2.r.k 6
252.bi even 6 2 1008.2.r.h 6

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

Hecke kernels

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

 $$T_{2}^{3} - 3T_{2}^{2} + 3$$ T2^3 - 3*T2^2 + 3 $$T_{5}^{3} + 3T_{5}^{2} - 3$$ T5^3 + 3*T5^2 - 3 $$T_{11}^{3} - 6T_{11}^{2} + 9T_{11} - 3$$ T11^3 - 6*T11^2 + 9*T11 - 3 $$T_{13}^{3} - 3T_{13}^{2} - 33T_{13} + 107$$ T13^3 - 3*T13^2 - 33*T13 + 107

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} - 3T^{2} + 3$$
$3$ $$T^{3}$$
$5$ $$T^{3} + 3T^{2} - 3$$
$7$ $$T^{3}$$
$11$ $$T^{3} - 6 T^{2} + \cdots - 3$$
$13$ $$T^{3} - 3 T^{2} + \cdots + 107$$
$17$ $$T^{3} + 6 T^{2} + \cdots + 3$$
$19$ $$T^{3} - 3 T^{2} + \cdots + 17$$
$23$ $$T^{3} - 12 T^{2} + \cdots + 3$$
$29$ $$T^{3} - 9 T^{2} + \cdots + 333$$
$31$ $$T^{3} - 3 T^{2} + \cdots + 323$$
$37$ $$T^{3} + 3 T^{2} + \cdots - 323$$
$41$ $$T^{3} - 9T + 9$$
$43$ $$T^{3} + 3 T^{2} + \cdots + 1$$
$47$ $$T^{3} + 3 T^{2} + \cdots + 51$$
$53$ $$T^{3} - 6 T^{2} + \cdots - 3$$
$59$ $$T^{3} - 3 T^{2} + \cdots - 51$$
$61$ $$T^{3} + 6 T^{2} + \cdots - 19$$
$67$ $$T^{3} + 12 T^{2} + \cdots - 17$$
$71$ $$T^{3} - 9 T^{2} + \cdots - 27$$
$73$ $$T^{3} - 21 T^{2} + \cdots + 269$$
$79$ $$T^{3} + 21 T^{2} + \cdots + 181$$
$83$ $$T^{3} - 18 T^{2} + \cdots - 9$$
$89$ $$T^{3} + 12 T^{2} + \cdots - 813$$
$97$ $$T^{3} - 3 T^{2} + \cdots + 323$$