# Properties

 Label 143.2.e.b 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: 6.0.3518667.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - x^{5} + 7x^{4} - 8x^{3} + 43x^{2} - 42x + 49$$ x^6 - x^5 + 7*x^4 - 8*x^3 + 43*x^2 - 42*x + 49 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} + 7x^{4} - 8x^{3} + 43x^{2} - 42x + 49$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -\nu^{5} + 7\nu^{4} - 49\nu^{3} + 43\nu^{2} - 42\nu + 294 ) / 259$$ (-v^5 + 7*v^4 - 49*v^3 + 43*v^2 - 42*v + 294) / 259 $$\beta_{3}$$ $$=$$ $$( -6\nu^{5} + 5\nu^{4} - 35\nu^{3} - \nu^{2} - 215\nu - 49 ) / 259$$ (-6*v^5 + 5*v^4 - 35*v^3 - v^2 - 215*v - 49) / 259 $$\beta_{4}$$ $$=$$ $$( -5\nu^{5} + 35\nu^{4} + 14\nu^{3} + 215\nu^{2} - 210\nu + 952 ) / 259$$ (-5*v^5 + 35*v^4 + 14*v^3 + 215*v^2 - 210*v + 952) / 259 $$\beta_{5}$$ $$=$$ $$( 18\nu^{5} + 22\nu^{4} + 105\nu^{3} + 3\nu^{2} + 608\nu + 147 ) / 259$$ (18*v^5 + 22*v^4 + 105*v^3 + 3*v^2 + 608*v + 147) / 259
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{5} + \beta_{4} - 4\beta_{3} + \beta_{2} - 5$$ -b5 + b4 - 4*b3 + b2 - 5 $$\nu^{3}$$ $$=$$ $$\beta_{4} - 5\beta_{2} + 2$$ b4 - 5*b2 + 2 $$\nu^{4}$$ $$=$$ $$7\beta_{5} + 21\beta_{3} + \beta_1$$ 7*b5 + 21*b3 + b1 $$\nu^{5}$$ $$=$$ $$6\beta_{5} - 6\beta_{4} - 25\beta_{3} + 29\beta_{2} - 35\beta _1 - 19$$ 6*b5 - 6*b4 - 25*b3 + 29*b2 - 35*b1 - 19

## Character values

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

 $$n$$ $$67$$ $$79$$ $$\chi(n)$$ $$-1 - \beta_{3}$$ $$1$$

## 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
 −1.25351 + 2.17114i 0.610938 − 1.05818i 1.14257 − 1.97899i −1.25351 − 2.17114i 0.610938 + 1.05818i 1.14257 + 1.97899i
−1.25351 + 2.17114i −1.14257 + 1.97899i −2.14257 3.71104i −1.22188 −2.86445 4.96137i −0.389062 0.673875i 5.72889 −1.11094 1.92420i 1.53163 2.65287i
100.2 0.610938 1.05818i 1.25351 2.17114i 0.253509 + 0.439091i −2.28514 −1.53163 2.65287i 0.142571 + 0.246941i 3.06327 −1.64257 2.84502i −1.39608 + 2.41808i
100.3 1.14257 1.97899i −0.610938 + 1.05818i −1.61094 2.79023i 2.50702 1.39608 + 2.41808i −2.25351 3.90319i −2.79216 0.753509 + 1.30512i 2.86445 4.96137i
133.1 −1.25351 2.17114i −1.14257 1.97899i −2.14257 + 3.71104i −1.22188 −2.86445 + 4.96137i −0.389062 + 0.673875i 5.72889 −1.11094 + 1.92420i 1.53163 + 2.65287i
133.2 0.610938 + 1.05818i 1.25351 + 2.17114i 0.253509 0.439091i −2.28514 −1.53163 + 2.65287i 0.142571 0.246941i 3.06327 −1.64257 + 2.84502i −1.39608 2.41808i
133.3 1.14257 + 1.97899i −0.610938 1.05818i −1.61094 + 2.79023i 2.50702 1.39608 2.41808i −2.25351 + 3.90319i −2.79216 0.753509 1.30512i 2.86445 + 4.96137i
 $$n$$: 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.b 6
13.c even 3 1 inner 143.2.e.b 6
13.c even 3 1 1859.2.a.f 3
13.e even 6 1 1859.2.a.g 3

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} - T_{2}^{5} + 7T_{2}^{4} - 8T_{2}^{3} + 43T_{2}^{2} - 42T_{2} + 49$$ acting on $$S_{2}^{\mathrm{new}}(143, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} - T^{5} + \cdots + 49$$
$3$ $$T^{6} + T^{5} + \cdots + 49$$
$5$ $$(T^{3} + T^{2} - 6 T - 7)^{2}$$
$7$ $$T^{6} + 5 T^{5} + \cdots + 1$$
$11$ $$(T^{2} + T + 1)^{3}$$
$13$ $$(T^{2} + 5 T + 13)^{3}$$
$17$ $$T^{6} - 7 T^{5} + \cdots + 49$$
$19$ $$T^{6} + 2 T^{5} + \cdots + 5929$$
$23$ $$T^{6} + 3 T^{5} + \cdots + 1$$
$29$ $$T^{6} - 4 T^{5} + \cdots + 2401$$
$31$ $$(T^{3} - T^{2} - 25 T - 31)^{2}$$
$37$ $$T^{6} + 12 T^{5} + \cdots + 961$$
$41$ $$T^{6} + T^{5} + \cdots + 961$$
$43$ $$T^{6} - 4 T^{5} + \cdots + 337561$$
$47$ $$(T^{3} + 8 T^{2} + \cdots - 259)^{2}$$
$53$ $$(T^{3} + 9 T^{2} + 8 T - 49)^{2}$$
$59$ $$T^{6} + 30 T^{5} + \cdots + 625681$$
$61$ $$T^{6} - 18 T^{5} + \cdots + 49$$
$67$ $$T^{6} + 15 T^{5} + \cdots + 121$$
$71$ $$T^{6} - 11 T^{5} + \cdots + 961$$
$73$ $$(T + 2)^{6}$$
$79$ $$(T^{3} - 12 T^{2} + \cdots + 88)^{2}$$
$83$ $$(T^{3} + 14 T^{2} + \cdots - 341)^{2}$$
$89$ $$T^{6} - 17 T^{5} + \cdots + 22801$$
$97$ $$T^{6} + 11 T^{5} + \cdots + 49$$