# Properties

 Label 143.2.e.a Level $143$ Weight $2$ Character orbit 143.e Analytic conductor $1.142$ Analytic rank $0$ Dimension $6$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [143,2,Mod(100,143)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(143, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4]))

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

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

chi := DirichletCharacter("143.100");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$143 = 11 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 143.e (of order $$3$$, degree $$2$$, minimal)

## 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: no Analytic conductor: $$1.14186074890$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.1714608.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 3x^{5} + 12x^{4} - 19x^{3} + 30x^{2} - 21x + 7$$ x^6 - 3*x^5 + 12*x^4 - 19*x^3 + 30*x^2 - 21*x + 7 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{4} - 1) q^{2} + (\beta_{5} - \beta_{3}) q^{3} - \beta_{4} q^{4} + (\beta_{4} + \beta_{3} + 2 \beta_{2} + \cdots + 2) q^{5}+ \cdots + ( - \beta_{5} - \beta_{2} - 2 \beta_1) q^{9}+O(q^{10})$$ q + (-b4 - 1) * q^2 + (b5 - b3) * q^3 - b4 * q^4 + (b4 + b3 + 2*b2 + b1 + 2) * q^5 - b5 * q^6 + (-b4 - b2 - 2*b1) * q^7 - 3 * q^8 + (-b5 - b2 - 2*b1) * q^9 $$q + ( - \beta_{4} - 1) q^{2} + (\beta_{5} - \beta_{3}) q^{3} - \beta_{4} q^{4} + (\beta_{4} + \beta_{3} + 2 \beta_{2} + \cdots + 2) q^{5}+ \cdots + (\beta_{4} + \beta_{3} + 2 \beta_{2} + \cdots + 2) q^{99}+O(q^{100})$$ q + (-b4 - 1) * q^2 + (b5 - b3) * q^3 - b4 * q^4 + (b4 + b3 + 2*b2 + b1 + 2) * q^5 - b5 * q^6 + (-b4 - b2 - 2*b1) * q^7 - 3 * q^8 + (-b5 - b2 - 2*b1) * q^9 + (b5 - b4 - b3 - b2 + b1 - 2) * q^10 + (-b4 - 1) * q^11 - b3 * q^12 + (-b5 + 2*b4 + b1 + 2) * q^13 + (b4 + 2*b2 + b1 + 1) * q^14 + (3*b5 - 2*b4 - 3*b3 - 2) * q^15 + (b4 + 1) * q^16 + (-2*b5 - 2*b4 - b2 - 2*b1) * q^17 + (b4 + b3 + 2*b2 + b1 + 2) * q^18 + (2*b5 + 3*b4 + b2 + 2*b1) * q^19 + (b5 + b2 + 2*b1) * q^20 + (-b4 + b3 - 2*b2 - b1 - 3) * q^21 + b4 * q^22 + (-2*b5 + 2*b3) * q^23 + (-3*b5 + 3*b3) * q^24 + (2*b3 + 4) * q^25 + (-3*b4 + b3 - b2 - b1 - 1) * q^26 + 2 * q^27 + (b2 - b1 + 1) * q^28 + (b5 + b4 - b3 - 2*b2 + 2*b1 - 1) * q^29 + (-3*b5 + 2*b4) * q^30 + (-2*b4 + 2*b3 - 4*b2 - 2*b1) * q^31 + 5*b4 * q^32 - b5 * q^33 + (b4 + 2*b3 + 2*b2 + b1) * q^34 + (2*b5 + 7*b4 + b2 + 2*b1) * q^35 + (-b5 + b4 + b3 + b2 - b1 + 2) * q^36 + (3*b5 - b4 - 3*b3 + b2 - b1) * q^37 + (-b4 - 2*b3 - 2*b2 - b1 + 1) * q^38 + (2*b5 + b4 + 3*b2 + 2*b1 + 6) * q^39 + (-3*b4 - 3*b3 - 6*b2 - 3*b1 - 6) * q^40 + (-b5 - b4 + b3 - 2*b2 + 2*b1 - 3) * q^41 + (b5 + 2*b4 - b3 + b2 - b1 + 3) * q^42 + (2*b5 - 2*b4) * q^43 - q^44 + (-2*b5 + 9*b4) * q^45 + 2*b5 * q^46 + (-2*b4 - 3*b3 - 4*b2 - 2*b1 - 4) * q^47 + b5 * q^48 + (-3*b5 - b4 + 3*b3 + b2 - b1) * q^49 + (2*b5 - 4*b4 - 2*b3 - 4) * q^50 + (b4 + 2*b3 + 2*b2 + b1 + 7) * q^51 + (-b5 - b4 + b3 - b2 + 1) * q^52 + (2*b4 + 4*b2 + 2*b1 + 3) * q^53 + (-2*b4 - 2) * q^54 + (b5 - b4 - b3 - b2 + b1 - 2) * q^55 + (3*b4 + 3*b2 + 6*b1) * q^56 + (-b4 - b3 - 2*b2 - b1 - 7) * q^57 + (-b5 - 3*b4 - 2*b2 - 4*b1) * q^58 + (-3*b5 - 8*b4 - 2*b2 - 4*b1) * q^59 + (-3*b3 - 2) * q^60 + (3*b5 + 3*b4) * q^61 + (2*b5 - 2*b4 - 2*b3 + 2*b2 - 2*b1) * q^62 + (-2*b5 - 6*b4 + 2*b3 + b2 - b1 - 5) * q^63 + 7 * q^64 + (-6*b5 + 3*b3 + 3*b2 - 2*b1 + 3) * q^65 + b3 * q^66 + (-3*b5 - 4*b4 + 3*b3 + 2*b2 - 2*b1 - 2) * q^67 + (-2*b5 - b4 + 2*b3 + b2 - b1) * q^68 + (2*b5 - 6*b4 + 2*b2 + 4*b1) * q^69 + (-b4 - 2*b3 - 2*b2 - b1 + 5) * q^70 + 6*b4 * q^71 + (3*b5 + 3*b2 + 6*b1) * q^72 + (5*b4 + 4*b3 + 10*b2 + 5*b1 + 8) * q^73 + (-3*b5 + 2*b4 + b2 + 2*b1) * q^74 + (6*b5 - 8*b4 - 6*b3 - 2*b2 + 2*b1 - 10) * q^75 + (2*b5 + 2*b4 - 2*b3 - b2 + b1 + 1) * q^76 + (b4 + 2*b2 + b1 + 1) * q^77 + (-5*b4 - 2*b3 - 2*b2 + b1 - 7) * q^78 + (-b4 - 4*b3 - 2*b2 - b1 - 5) * q^79 + (-b5 + b4 + b3 + b2 - b1 + 2) * q^80 + (-b5 + 3*b4 + b3 + 3*b2 - 3*b1 + 6) * q^81 + (b5 - b4 - 2*b2 - 4*b1) * q^82 + (-b4 - 4*b3 - 2*b2 - b1 - 1) * q^83 + (b5 + b4 - b2 - 2*b1) * q^84 + (-3*b5 + 11*b4 + 2*b2 + 4*b1) * q^85 + (-2*b3 - 2) * q^86 + (-2*b5 + b4 + b2 + 2*b1) * q^87 + (3*b4 + 3) * q^88 + (3*b5 - 6*b4 - 3*b3 - 3*b2 + 3*b1 - 9) * q^89 + (2*b3 + 9) * q^90 + (2*b5 + 2*b4 - 3*b3 - 4*b2 - 4*b1 + 6) * q^91 + 2*b3 * q^92 + (2*b5 - 12*b4 - 2*b3 - 4*b2 + 4*b1 - 16) * q^93 + (-3*b5 + 2*b4 + 3*b3 + 2*b2 - 2*b1 + 4) * q^94 + (2*b5 - 11*b4 - 3*b2 - 6*b1) * q^95 + 5*b3 * q^96 + (b5 + 5*b4 - b2 - 2*b1) * q^97 + (3*b5 + 2*b4 + b2 + 2*b1) * q^98 + (b4 + b3 + 2*b2 + b1 + 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 3 q^{2} + 3 q^{4} + 6 q^{5} - 18 q^{8} - 3 q^{9}+O(q^{10})$$ 6 * q - 3 * q^2 + 3 * q^4 + 6 * q^5 - 18 * q^8 - 3 * q^9 $$6 q - 3 q^{2} + 3 q^{4} + 6 q^{5} - 18 q^{8} - 3 q^{9} - 3 q^{10} - 3 q^{11} + 9 q^{13} - 6 q^{15} + 3 q^{16} + 3 q^{17} + 6 q^{18} - 6 q^{19} + 3 q^{20} - 12 q^{21} - 3 q^{22} + 24 q^{25} + 3 q^{26} + 12 q^{27} + 3 q^{29} - 6 q^{30} + 12 q^{31} - 15 q^{32} - 6 q^{34} - 18 q^{35} + 3 q^{36} - 3 q^{37} + 12 q^{38} + 30 q^{39} - 18 q^{40} - 3 q^{41} + 6 q^{42} + 6 q^{43} - 6 q^{44} - 27 q^{45} - 12 q^{47} - 3 q^{49} - 12 q^{50} + 36 q^{51} + 12 q^{52} + 6 q^{53} - 6 q^{54} - 3 q^{55} - 36 q^{57} + 3 q^{58} + 18 q^{59} - 12 q^{60} - 9 q^{61} - 6 q^{62} - 18 q^{63} + 42 q^{64} + 3 q^{65} - 12 q^{67} - 3 q^{68} + 24 q^{69} + 36 q^{70} - 18 q^{71} + 9 q^{72} + 18 q^{73} - 3 q^{74} - 24 q^{75} + 6 q^{76} - 18 q^{78} - 24 q^{79} + 3 q^{80} + 9 q^{81} - 3 q^{82} - 6 q^{84} - 27 q^{85} - 12 q^{86} + 9 q^{88} - 18 q^{89} + 54 q^{90} + 30 q^{91} - 36 q^{93} + 6 q^{94} + 24 q^{95} - 18 q^{97} - 3 q^{98} + 6 q^{99}+O(q^{100})$$ 6 * q - 3 * q^2 + 3 * q^4 + 6 * q^5 - 18 * q^8 - 3 * q^9 - 3 * q^10 - 3 * q^11 + 9 * q^13 - 6 * q^15 + 3 * q^16 + 3 * q^17 + 6 * q^18 - 6 * q^19 + 3 * q^20 - 12 * q^21 - 3 * q^22 + 24 * q^25 + 3 * q^26 + 12 * q^27 + 3 * q^29 - 6 * q^30 + 12 * q^31 - 15 * q^32 - 6 * q^34 - 18 * q^35 + 3 * q^36 - 3 * q^37 + 12 * q^38 + 30 * q^39 - 18 * q^40 - 3 * q^41 + 6 * q^42 + 6 * q^43 - 6 * q^44 - 27 * q^45 - 12 * q^47 - 3 * q^49 - 12 * q^50 + 36 * q^51 + 12 * q^52 + 6 * q^53 - 6 * q^54 - 3 * q^55 - 36 * q^57 + 3 * q^58 + 18 * q^59 - 12 * q^60 - 9 * q^61 - 6 * q^62 - 18 * q^63 + 42 * q^64 + 3 * q^65 - 12 * q^67 - 3 * q^68 + 24 * q^69 + 36 * q^70 - 18 * q^71 + 9 * q^72 + 18 * q^73 - 3 * q^74 - 24 * q^75 + 6 * q^76 - 18 * q^78 - 24 * q^79 + 3 * q^80 + 9 * q^81 - 3 * q^82 - 6 * q^84 - 27 * q^85 - 12 * q^86 + 9 * q^88 - 18 * q^89 + 54 * q^90 + 30 * q^91 - 36 * q^93 + 6 * q^94 + 24 * q^95 - 18 * q^97 - 3 * q^98 + 6 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 3x^{5} + 12x^{4} - 19x^{3} + 30x^{2} - 21x + 7$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{5} + \nu^{4} + 2\nu^{3} + 10\nu^{2} - 7\nu ) / 7$$ (v^5 + v^4 + 2*v^3 + 10*v^2 - 7*v) / 7 $$\beta_{3}$$ $$=$$ $$\nu^{2} - \nu + 3$$ v^2 - v + 3 $$\beta_{4}$$ $$=$$ $$( -2\nu^{5} + 5\nu^{4} - 18\nu^{3} + 22\nu^{2} - 28\nu + 7 ) / 7$$ (-2*v^5 + 5*v^4 - 18*v^3 + 22*v^2 - 28*v + 7) / 7 $$\beta_{5}$$ $$=$$ $$( 2\nu^{5} - 5\nu^{4} + 25\nu^{3} - 29\nu^{2} + 56\nu - 14 ) / 7$$ (2*v^5 - 5*v^4 + 25*v^3 - 29*v^2 + 56*v - 14) / 7
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta _1 - 3$$ b3 + b1 - 3 $$\nu^{3}$$ $$=$$ $$\beta_{5} + \beta_{4} + \beta_{3} - 3\beta _1 - 2$$ b5 + b4 + b3 - 3*b1 - 2 $$\nu^{4}$$ $$=$$ $$2\beta_{5} + 3\beta_{4} - 4\beta_{3} + 2\beta_{2} - 6\beta _1 + 13$$ 2*b5 + 3*b4 - 4*b3 + 2*b2 - 6*b1 + 13 $$\nu^{5}$$ $$=$$ $$-4\beta_{5} - 5\beta_{4} - 8\beta_{3} + 5\beta_{2} + 9\beta _1 + 21$$ -4*b5 - 5*b4 - 8*b3 + 5*b2 + 9*b1 + 21

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/143\mathbb{Z}\right)^\times$$.

 $$n$$ $$67$$ $$79$$ $$\chi(n)$$ $$\beta_{4}$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
100.1
 0.5 + 0.385124i 0.5 − 1.75780i 0.5 + 2.23871i 0.5 − 0.385124i 0.5 + 1.75780i 0.5 − 2.23871i
−0.500000 + 0.866025i −1.30084 + 2.25312i 0.500000 + 0.866025i 3.76873 −1.30084 2.25312i −0.0835276 0.144674i −3.00000 −1.88437 3.26382i −1.88437 + 3.26382i
100.2 −0.500000 + 0.866025i 0.169938 0.294342i 0.500000 + 0.866025i −2.88448 0.169938 + 0.294342i 1.77230 + 3.06972i −3.00000 1.44224 + 2.49804i 1.44224 2.49804i
100.3 −0.500000 + 0.866025i 1.13090 1.95878i 0.500000 + 0.866025i 2.11575 1.13090 + 1.95878i −1.68878 2.92505i −3.00000 −1.05787 1.83229i −1.05787 + 1.83229i
133.1 −0.500000 0.866025i −1.30084 2.25312i 0.500000 0.866025i 3.76873 −1.30084 + 2.25312i −0.0835276 + 0.144674i −3.00000 −1.88437 + 3.26382i −1.88437 3.26382i
133.2 −0.500000 0.866025i 0.169938 + 0.294342i 0.500000 0.866025i −2.88448 0.169938 0.294342i 1.77230 3.06972i −3.00000 1.44224 2.49804i 1.44224 + 2.49804i
133.3 −0.500000 0.866025i 1.13090 + 1.95878i 0.500000 0.866025i 2.11575 1.13090 1.95878i −1.68878 + 2.92505i −3.00000 −1.05787 + 1.83229i −1.05787 1.83229i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 100.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 143.2.e.a 6
13.c even 3 1 inner 143.2.e.a 6
13.c even 3 1 1859.2.a.h 3
13.e even 6 1 1859.2.a.e 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
143.2.e.a 6 1.a even 1 1 trivial
143.2.e.a 6 13.c even 3 1 inner
1859.2.a.e 3 13.e even 6 1
1859.2.a.h 3 13.c even 3 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(143, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + T + 1)^{3}$$
$3$ $$T^{6} + 6 T^{4} + \cdots + 4$$
$5$ $$(T^{3} - 3 T^{2} - 9 T + 23)^{2}$$
$7$ $$T^{6} + 12 T^{4} + \cdots + 4$$
$11$ $$(T^{2} + T + 1)^{3}$$
$13$ $$T^{6} - 9 T^{5} + \cdots + 2197$$
$17$ $$T^{6} - 3 T^{5} + \cdots + 2401$$
$19$ $$T^{6} + 6 T^{5} + \cdots + 4356$$
$23$ $$T^{6} + 24 T^{4} + \cdots + 256$$
$29$ $$T^{6} - 3 T^{5} + \cdots + 3249$$
$31$ $$(T^{3} - 6 T^{2} + \cdots + 536)^{2}$$
$37$ $$T^{6} + 3 T^{5} + \cdots + 77841$$
$41$ $$T^{6} + 3 T^{5} + \cdots + 5329$$
$43$ $$T^{6} - 6 T^{5} + \cdots + 576$$
$47$ $$(T^{3} + 6 T^{2} + \cdots - 122)^{2}$$
$53$ $$(T^{3} - 3 T^{2} - 45 T + 63)^{2}$$
$59$ $$T^{6} - 18 T^{5} + \cdots + 33124$$
$61$ $$T^{6} + 9 T^{5} + \cdots + 35721$$
$67$ $$T^{6} + 12 T^{5} + \cdots + 40804$$
$71$ $$(T^{2} + 6 T + 36)^{3}$$
$73$ $$(T^{3} - 9 T^{2} + \cdots + 2483)^{2}$$
$79$ $$(T^{3} + 12 T^{2} + \cdots + 22)^{2}$$
$83$ $$(T^{3} - 84 T + 294)^{2}$$
$89$ $$T^{6} + 18 T^{5} + \cdots + 571536$$
$97$ $$T^{6} + 18 T^{5} + \cdots + 13456$$