# Properties

 Label 3971.2.a.e Level $3971$ Weight $2$ Character orbit 3971.a Self dual yes Analytic conductor $31.709$ Analytic rank $2$ 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] = [3971,2,Mod(1,3971)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3971, 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("3971.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3971 = 11 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3971.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: $$31.7085946427$$ Analytic rank: $$2$$ 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 209) 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.

 $$f(q)$$ $$=$$ $$q + (\beta_1 - 1) q^{2} - 2 q^{3} + (\beta_{2} - 2 \beta_1 + 1) q^{4} + ( - \beta_{2} + \beta_1 - 2) q^{5} + ( - 2 \beta_1 + 2) q^{6} + (\beta_1 - 2) q^{7} + ( - 3 \beta_{2} + 2 \beta_1 - 2) q^{8} + q^{9}+O(q^{10})$$ q + (b1 - 1) * q^2 - 2 * q^3 + (b2 - 2*b1 + 1) * q^4 + (-b2 + b1 - 2) * q^5 + (-2*b1 + 2) * q^6 + (b1 - 2) * q^7 + (-3*b2 + 2*b1 - 2) * q^8 + q^9 $$q + (\beta_1 - 1) q^{2} - 2 q^{3} + (\beta_{2} - 2 \beta_1 + 1) q^{4} + ( - \beta_{2} + \beta_1 - 2) q^{5} + ( - 2 \beta_1 + 2) q^{6} + (\beta_1 - 2) q^{7} + ( - 3 \beta_{2} + 2 \beta_1 - 2) q^{8} + q^{9} + (2 \beta_{2} - 4 \beta_1 + 3) q^{10} + q^{11} + ( - 2 \beta_{2} + 4 \beta_1 - 2) q^{12} + (2 \beta_{2} - 3 \beta_1 - 2) q^{13} + (\beta_{2} - 3 \beta_1 + 4) q^{14} + (2 \beta_{2} - 2 \beta_1 + 4) q^{15} + (3 \beta_{2} - 3 \beta_1 + 1) q^{16} + ( - \beta_1 - 5) q^{17} + (\beta_1 - 1) q^{18} + ( - 4 \beta_{2} + 7 \beta_1 - 5) q^{20} + ( - 2 \beta_1 + 4) q^{21} + (\beta_1 - 1) q^{22} + ( - 2 \beta_{2} + \beta_1 - 3) q^{23} + (6 \beta_{2} - 4 \beta_1 + 4) q^{24} + (4 \beta_{2} - 5 \beta_1 + 1) q^{25} + ( - 5 \beta_{2} + 3 \beta_1 - 2) q^{26} + 4 q^{27} + ( - 4 \beta_{2} + 6 \beta_1 - 5) q^{28} + ( - 2 \beta_1 - 4) q^{29} + ( - 4 \beta_{2} + 8 \beta_1 - 6) q^{30} + ( - \beta_{2} - 3 \beta_1 + 1) q^{31} + 3 \beta_1 q^{32} - 2 q^{33} + ( - \beta_{2} - 4 \beta_1 + 3) q^{34} + (3 \beta_{2} - 5 \beta_1 + 5) q^{35} + (\beta_{2} - 2 \beta_1 + 1) q^{36} + (\beta_{2} - 3) q^{37} + ( - 4 \beta_{2} + 6 \beta_1 + 4) q^{39} + (7 \beta_{2} - 8 \beta_1 + 9) q^{40} + ( - \beta_{2} + 2 \beta_1) q^{41} + ( - 2 \beta_{2} + 6 \beta_1 - 8) q^{42} + (\beta_{2} - 3 \beta_1 - 6) q^{43} + (\beta_{2} - 2 \beta_1 + 1) q^{44} + ( - \beta_{2} + \beta_1 - 2) q^{45} + (3 \beta_{2} - 6 \beta_1 + 3) q^{46} + (3 \beta_{2} + 3 \beta_1 - 3) q^{47} + ( - 6 \beta_{2} + 6 \beta_1 - 2) q^{48} + (\beta_{2} - 4 \beta_1 - 1) q^{49} + ( - 9 \beta_{2} + 10 \beta_1 - 7) q^{50} + (2 \beta_1 + 10) q^{51} + (4 \beta_{2} - 4 \beta_1 + 7) q^{52} + ( - 6 \beta_{2} - 3) q^{53} + (4 \beta_1 - 4) q^{54} + ( - \beta_{2} + \beta_1 - 2) q^{55} + (8 \beta_{2} - 9 \beta_1 + 5) q^{56} + ( - 2 \beta_{2} - 2 \beta_1) q^{58} + ( - 4 \beta_{2} + 6 \beta_1 - 1) q^{59} + (8 \beta_{2} - 14 \beta_1 + 10) q^{60} + (3 \beta_{2} - 4 \beta_1 - 6) q^{61} + ( - 2 \beta_{2} + 3 \beta_1 - 8) q^{62} + (\beta_1 - 2) q^{63} + ( - 3 \beta_{2} + 3 \beta_1 + 4) q^{64} + ( - 3 \beta_{2} + 7 \beta_1 - 1) q^{65} + ( - 2 \beta_1 + 2) q^{66} + ( - \beta_{2} + 6 \beta_1 - 5) q^{67} + ( - 3 \beta_{2} + 8 \beta_1 - 2) q^{68} + (4 \beta_{2} - 2 \beta_1 + 6) q^{69} + ( - 8 \beta_{2} + 13 \beta_1 - 12) q^{70} + (7 \beta_{2} - 7 \beta_1 + 2) q^{71} + ( - 3 \beta_{2} + 2 \beta_1 - 2) q^{72} + (\beta_{2} + 2 \beta_1 - 11) q^{73} + ( - \beta_{2} - 2 \beta_1 + 4) q^{74} + ( - 8 \beta_{2} + 10 \beta_1 - 2) q^{75} + (\beta_1 - 2) q^{77} + (10 \beta_{2} - 6 \beta_1 + 4) q^{78} + ( - 3 \beta_{2} + 3 \beta_1 - 4) q^{79} + ( - 7 \beta_{2} + 10 \beta_1 - 8) q^{80} - 11 q^{81} + (3 \beta_{2} - 3 \beta_1 + 3) q^{82} + (7 \beta_{2} - 3 \beta_1 - 2) q^{83} + (8 \beta_{2} - 12 \beta_1 + 10) q^{84} + (4 \beta_{2} - 2 \beta_1 + 9) q^{85} + ( - 4 \beta_{2} - 2 \beta_1 + 1) q^{86} + (4 \beta_1 + 8) q^{87} + ( - 3 \beta_{2} + 2 \beta_1 - 2) q^{88} + ( - 5 \beta_{2} - \beta_1 - 1) q^{89} + (2 \beta_{2} - 4 \beta_1 + 3) q^{90} + ( - 7 \beta_{2} + 6 \beta_1) q^{91} + ( - 5 \beta_{2} + 10 \beta_1 - 6) q^{92} + (2 \beta_{2} + 6 \beta_1 - 2) q^{93} + ( - 3 \beta_1 + 12) q^{94} - 6 \beta_1 q^{96} + (6 \beta_{2} - 3 \beta_1 + 2) q^{97} + ( - 5 \beta_{2} + 4 \beta_1 - 6) q^{98} + q^{99}+O(q^{100})$$ q + (b1 - 1) * q^2 - 2 * q^3 + (b2 - 2*b1 + 1) * q^4 + (-b2 + b1 - 2) * q^5 + (-2*b1 + 2) * q^6 + (b1 - 2) * q^7 + (-3*b2 + 2*b1 - 2) * q^8 + q^9 + (2*b2 - 4*b1 + 3) * q^10 + q^11 + (-2*b2 + 4*b1 - 2) * q^12 + (2*b2 - 3*b1 - 2) * q^13 + (b2 - 3*b1 + 4) * q^14 + (2*b2 - 2*b1 + 4) * q^15 + (3*b2 - 3*b1 + 1) * q^16 + (-b1 - 5) * q^17 + (b1 - 1) * q^18 + (-4*b2 + 7*b1 - 5) * q^20 + (-2*b1 + 4) * q^21 + (b1 - 1) * q^22 + (-2*b2 + b1 - 3) * q^23 + (6*b2 - 4*b1 + 4) * q^24 + (4*b2 - 5*b1 + 1) * q^25 + (-5*b2 + 3*b1 - 2) * q^26 + 4 * q^27 + (-4*b2 + 6*b1 - 5) * q^28 + (-2*b1 - 4) * q^29 + (-4*b2 + 8*b1 - 6) * q^30 + (-b2 - 3*b1 + 1) * q^31 + 3*b1 * q^32 - 2 * q^33 + (-b2 - 4*b1 + 3) * q^34 + (3*b2 - 5*b1 + 5) * q^35 + (b2 - 2*b1 + 1) * q^36 + (b2 - 3) * q^37 + (-4*b2 + 6*b1 + 4) * q^39 + (7*b2 - 8*b1 + 9) * q^40 + (-b2 + 2*b1) * q^41 + (-2*b2 + 6*b1 - 8) * q^42 + (b2 - 3*b1 - 6) * q^43 + (b2 - 2*b1 + 1) * q^44 + (-b2 + b1 - 2) * q^45 + (3*b2 - 6*b1 + 3) * q^46 + (3*b2 + 3*b1 - 3) * q^47 + (-6*b2 + 6*b1 - 2) * q^48 + (b2 - 4*b1 - 1) * q^49 + (-9*b2 + 10*b1 - 7) * q^50 + (2*b1 + 10) * q^51 + (4*b2 - 4*b1 + 7) * q^52 + (-6*b2 - 3) * q^53 + (4*b1 - 4) * q^54 + (-b2 + b1 - 2) * q^55 + (8*b2 - 9*b1 + 5) * q^56 + (-2*b2 - 2*b1) * q^58 + (-4*b2 + 6*b1 - 1) * q^59 + (8*b2 - 14*b1 + 10) * q^60 + (3*b2 - 4*b1 - 6) * q^61 + (-2*b2 + 3*b1 - 8) * q^62 + (b1 - 2) * q^63 + (-3*b2 + 3*b1 + 4) * q^64 + (-3*b2 + 7*b1 - 1) * q^65 + (-2*b1 + 2) * q^66 + (-b2 + 6*b1 - 5) * q^67 + (-3*b2 + 8*b1 - 2) * q^68 + (4*b2 - 2*b1 + 6) * q^69 + (-8*b2 + 13*b1 - 12) * q^70 + (7*b2 - 7*b1 + 2) * q^71 + (-3*b2 + 2*b1 - 2) * q^72 + (b2 + 2*b1 - 11) * q^73 + (-b2 - 2*b1 + 4) * q^74 + (-8*b2 + 10*b1 - 2) * q^75 + (b1 - 2) * q^77 + (10*b2 - 6*b1 + 4) * q^78 + (-3*b2 + 3*b1 - 4) * q^79 + (-7*b2 + 10*b1 - 8) * q^80 - 11 * q^81 + (3*b2 - 3*b1 + 3) * q^82 + (7*b2 - 3*b1 - 2) * q^83 + (8*b2 - 12*b1 + 10) * q^84 + (4*b2 - 2*b1 + 9) * q^85 + (-4*b2 - 2*b1 + 1) * q^86 + (4*b1 + 8) * q^87 + (-3*b2 + 2*b1 - 2) * q^88 + (-5*b2 - b1 - 1) * q^89 + (2*b2 - 4*b1 + 3) * q^90 + (-7*b2 + 6*b1) * q^91 + (-5*b2 + 10*b1 - 6) * q^92 + (2*b2 + 6*b1 - 2) * q^93 + (-3*b1 + 12) * q^94 - 6*b1 * q^96 + (6*b2 - 3*b1 + 2) * q^97 + (-5*b2 + 4*b1 - 6) * q^98 + q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{2} - 6 q^{3} + 3 q^{4} - 6 q^{5} + 6 q^{6} - 6 q^{7} - 6 q^{8} + 3 q^{9}+O(q^{10})$$ 3 * q - 3 * q^2 - 6 * q^3 + 3 * q^4 - 6 * q^5 + 6 * q^6 - 6 * q^7 - 6 * q^8 + 3 * q^9 $$3 q - 3 q^{2} - 6 q^{3} + 3 q^{4} - 6 q^{5} + 6 q^{6} - 6 q^{7} - 6 q^{8} + 3 q^{9} + 9 q^{10} + 3 q^{11} - 6 q^{12} - 6 q^{13} + 12 q^{14} + 12 q^{15} + 3 q^{16} - 15 q^{17} - 3 q^{18} - 15 q^{20} + 12 q^{21} - 3 q^{22} - 9 q^{23} + 12 q^{24} + 3 q^{25} - 6 q^{26} + 12 q^{27} - 15 q^{28} - 12 q^{29} - 18 q^{30} + 3 q^{31} - 6 q^{33} + 9 q^{34} + 15 q^{35} + 3 q^{36} - 9 q^{37} + 12 q^{39} + 27 q^{40} - 24 q^{42} - 18 q^{43} + 3 q^{44} - 6 q^{45} + 9 q^{46} - 9 q^{47} - 6 q^{48} - 3 q^{49} - 21 q^{50} + 30 q^{51} + 21 q^{52} - 9 q^{53} - 12 q^{54} - 6 q^{55} + 15 q^{56} - 3 q^{59} + 30 q^{60} - 18 q^{61} - 24 q^{62} - 6 q^{63} + 12 q^{64} - 3 q^{65} + 6 q^{66} - 15 q^{67} - 6 q^{68} + 18 q^{69} - 36 q^{70} + 6 q^{71} - 6 q^{72} - 33 q^{73} + 12 q^{74} - 6 q^{75} - 6 q^{77} + 12 q^{78} - 12 q^{79} - 24 q^{80} - 33 q^{81} + 9 q^{82} - 6 q^{83} + 30 q^{84} + 27 q^{85} + 3 q^{86} + 24 q^{87} - 6 q^{88} - 3 q^{89} + 9 q^{90} - 18 q^{92} - 6 q^{93} + 36 q^{94} + 6 q^{97} - 18 q^{98} + 3 q^{99}+O(q^{100})$$ 3 * q - 3 * q^2 - 6 * q^3 + 3 * q^4 - 6 * q^5 + 6 * q^6 - 6 * q^7 - 6 * q^8 + 3 * q^9 + 9 * q^10 + 3 * q^11 - 6 * q^12 - 6 * q^13 + 12 * q^14 + 12 * q^15 + 3 * q^16 - 15 * q^17 - 3 * q^18 - 15 * q^20 + 12 * q^21 - 3 * q^22 - 9 * q^23 + 12 * q^24 + 3 * q^25 - 6 * q^26 + 12 * q^27 - 15 * q^28 - 12 * q^29 - 18 * q^30 + 3 * q^31 - 6 * q^33 + 9 * q^34 + 15 * q^35 + 3 * q^36 - 9 * q^37 + 12 * q^39 + 27 * q^40 - 24 * q^42 - 18 * q^43 + 3 * q^44 - 6 * q^45 + 9 * q^46 - 9 * q^47 - 6 * q^48 - 3 * q^49 - 21 * q^50 + 30 * q^51 + 21 * q^52 - 9 * q^53 - 12 * q^54 - 6 * q^55 + 15 * q^56 - 3 * q^59 + 30 * q^60 - 18 * q^61 - 24 * q^62 - 6 * q^63 + 12 * q^64 - 3 * q^65 + 6 * q^66 - 15 * q^67 - 6 * q^68 + 18 * q^69 - 36 * q^70 + 6 * q^71 - 6 * q^72 - 33 * q^73 + 12 * q^74 - 6 * q^75 - 6 * q^77 + 12 * q^78 - 12 * q^79 - 24 * q^80 - 33 * q^81 + 9 * q^82 - 6 * q^83 + 30 * q^84 + 27 * q^85 + 3 * q^86 + 24 * q^87 - 6 * q^88 - 3 * q^89 + 9 * q^90 - 18 * q^92 - 6 * q^93 + 36 * q^94 + 6 * q^97 - 18 * q^98 + 3 * q^99

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
 −1.53209 −0.347296 1.87939
−2.53209 −2.00000 4.41147 −3.87939 5.06418 −3.53209 −6.10607 1.00000 9.82295
1.2 −1.34730 −2.00000 −0.184793 −0.467911 2.69459 −2.34730 2.94356 1.00000 0.630415
1.3 0.879385 −2.00000 −1.22668 −1.65270 −1.75877 −0.120615 −2.83750 1.00000 −1.45336
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$11$$ $$-1$$
$$19$$ $$1$$

## 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 3971.2.a.e 3
19.b odd 2 1 3971.2.a.f 3
19.e even 9 2 209.2.j.a 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
209.2.j.a 6 19.e even 9 2
3971.2.a.e 3 1.a even 1 1 trivial
3971.2.a.f 3 19.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3971))$$:

 $$T_{2}^{3} + 3T_{2}^{2} - 3$$ T2^3 + 3*T2^2 - 3 $$T_{3} + 2$$ T3 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} + 3T^{2} - 3$$
$3$ $$(T + 2)^{3}$$
$5$ $$T^{3} + 6 T^{2} + 9 T + 3$$
$7$ $$T^{3} + 6 T^{2} + 9 T + 1$$
$11$ $$(T - 1)^{3}$$
$13$ $$T^{3} + 6 T^{2} - 9 T - 71$$
$17$ $$T^{3} + 15 T^{2} + 72 T + 111$$
$19$ $$T^{3}$$
$23$ $$T^{3} + 9 T^{2} + 18 T - 9$$
$29$ $$T^{3} + 12 T^{2} + 36 T + 24$$
$31$ $$T^{3} - 3 T^{2} - 36 T + 127$$
$37$ $$T^{3} + 9 T^{2} + 24 T + 19$$
$41$ $$T^{3} - 9T + 9$$
$43$ $$T^{3} + 18 T^{2} + 87 T + 73$$
$47$ $$T^{3} + 9 T^{2} - 54 T - 459$$
$53$ $$T^{3} + 9 T^{2} - 81 T - 513$$
$59$ $$T^{3} + 3 T^{2} - 81 T + 213$$
$61$ $$T^{3} + 18 T^{2} + 69 T - 107$$
$67$ $$T^{3} + 15 T^{2} - 18 T - 359$$
$71$ $$T^{3} - 6 T^{2} - 135 T - 57$$
$73$ $$T^{3} + 33 T^{2} + 342 T + 1063$$
$79$ $$T^{3} + 12 T^{2} + 21 T - 17$$
$83$ $$T^{3} + 6 T^{2} - 99 T + 219$$
$89$ $$T^{3} + 3 T^{2} - 90 T - 111$$
$97$ $$T^{3} - 6 T^{2} - 69 T + 397$$