# Properties

 Label 2166.2.a.r Level $2166$ Weight $2$ Character orbit 2166.a Self dual yes Analytic conductor $17.296$ 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] = [2166,2,Mod(1,2166)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2166, 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("2166.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2166 = 2 \cdot 3 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2166.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: $$17.2955970778$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 114) 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.

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

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
1.00000 −1.00000 1.00000 −3.53209 −1.00000 3.71688 1.00000 1.00000 −3.53209
1.2 1.00000 −1.00000 1.00000 −2.34730 −1.00000 3.57398 1.00000 1.00000 −2.34730
1.3 1.00000 −1.00000 1.00000 −0.120615 −1.00000 −4.29086 1.00000 1.00000 −0.120615
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$
$$19$$ $$1$$

## 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 2166.2.a.r 3
3.b odd 2 1 6498.2.a.bp 3
19.b odd 2 1 2166.2.a.p 3
19.e even 9 2 114.2.i.c 6
57.d even 2 1 6498.2.a.bu 3
57.l odd 18 2 342.2.u.b 6
76.l odd 18 2 912.2.bo.d 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.2.i.c 6 19.e even 9 2
342.2.u.b 6 57.l odd 18 2
912.2.bo.d 6 76.l odd 18 2
2166.2.a.p 3 19.b odd 2 1
2166.2.a.r 3 1.a even 1 1 trivial
6498.2.a.bp 3 3.b odd 2 1
6498.2.a.bu 3 57.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2166))$$:

 $$T_{5}^{3} + 6T_{5}^{2} + 9T_{5} + 1$$ T5^3 + 6*T5^2 + 9*T5 + 1 $$T_{7}^{3} - 3T_{7}^{2} - 18T_{7} + 57$$ T7^3 - 3*T7^2 - 18*T7 + 57 $$T_{13}^{3} - 6T_{13}^{2} + 3T_{13} + 1$$ T13^3 - 6*T13^2 + 3*T13 + 1 $$T_{29}^{3} - 6T_{29}^{2} + 9T_{29} - 1$$ T29^3 - 6*T29^2 + 9*T29 - 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{3}$$
$3$ $$(T + 1)^{3}$$
$5$ $$T^{3} + 6 T^{2} + 9 T + 1$$
$7$ $$T^{3} - 3 T^{2} - 18 T + 57$$
$11$ $$T^{3} - 21T + 37$$
$13$ $$T^{3} - 6 T^{2} + 3 T + 1$$
$17$ $$T^{3} + 6 T^{2} + 9 T + 3$$
$19$ $$T^{3}$$
$23$ $$T^{3} - 6 T^{2} - 69 T + 397$$
$29$ $$T^{3} - 6 T^{2} + 9 T - 1$$
$31$ $$T^{3} - 15 T^{2} + 72 T - 109$$
$37$ $$T^{3} + 3 T^{2} - 45 T + 17$$
$41$ $$T^{3} - 9 T^{2} + 6 T + 53$$
$43$ $$T^{3} - 9 T^{2} - 30 T + 251$$
$47$ $$T^{3} + 9 T^{2} - 12 T - 71$$
$53$ $$T^{3} - 27 T^{2} + 231 T - 629$$
$59$ $$T^{3} - 18 T^{2} + 45 T - 9$$
$61$ $$T^{3} - 3 T^{2} - 18 T - 17$$
$67$ $$T^{3} - 12 T^{2} - 225 T + 2649$$
$71$ $$T^{3} - 21 T^{2} + 90 T + 163$$
$73$ $$T^{3} - 15 T^{2} + 54 T - 57$$
$79$ $$T^{3} - 15 T^{2} + 12 T + 181$$
$83$ $$T^{3} + 3 T^{2} - 36 T + 51$$
$89$ $$T^{3} - 3 T^{2} - 144 T + 489$$
$97$ $$T^{3} - 12 T^{2} - 123 T + 467$$