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 $2$

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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: \( x^{6} - 3x^{5} + 12x^{4} - 19x^{3} + 30x^{2} - 21x + 7 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + ( - \beta_{4} - 1) q^{2} + (\beta_{5} - \beta_{3}) q^{3} - \beta_{4} q^{4} + (\beta_{4} + \beta_{3} + 2 \beta_{2} + \cdots + 2) q^{5}+ \cdots + ( - \beta_{5} - \beta_{2} - 2 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} - 1) q^{2} + (\beta_{5} - \beta_{3}) q^{3} - \beta_{4} q^{4} + (\beta_{4} + \beta_{3} + 2 \beta_{2} + \cdots + 2) q^{5}+ \cdots + (\beta_{4} + \beta_{3} + 2 \beta_{2} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{2} + 3 q^{4} + 6 q^{5} - 18 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 3 q^{2} + 3 q^{4} + 6 q^{5} - 18 q^{8} - 3 q^{9} - 3 q^{10} - 3 q^{11} + 9 q^{13} - 6 q^{15} + 3 q^{16} + 3 q^{17} + 6 q^{18} - 6 q^{19} + 3 q^{20} - 12 q^{21} - 3 q^{22} + 24 q^{25} + 3 q^{26} + 12 q^{27} + 3 q^{29} - 6 q^{30} + 12 q^{31} - 15 q^{32} - 6 q^{34} - 18 q^{35} + 3 q^{36} - 3 q^{37} + 12 q^{38} + 30 q^{39} - 18 q^{40} - 3 q^{41} + 6 q^{42} + 6 q^{43} - 6 q^{44} - 27 q^{45} - 12 q^{47} - 3 q^{49} - 12 q^{50} + 36 q^{51} + 12 q^{52} + 6 q^{53} - 6 q^{54} - 3 q^{55} - 36 q^{57} + 3 q^{58} + 18 q^{59} - 12 q^{60} - 9 q^{61} - 6 q^{62} - 18 q^{63} + 42 q^{64} + 3 q^{65} - 12 q^{67} - 3 q^{68} + 24 q^{69} + 36 q^{70} - 18 q^{71} + 9 q^{72} + 18 q^{73} - 3 q^{74} - 24 q^{75} + 6 q^{76} - 18 q^{78} - 24 q^{79} + 3 q^{80} + 9 q^{81} - 3 q^{82} - 6 q^{84} - 27 q^{85} - 12 q^{86} + 9 q^{88} - 18 q^{89} + 54 q^{90} + 30 q^{91} - 36 q^{93} + 6 q^{94} + 24 q^{95} - 18 q^{97} - 3 q^{98} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} + 12x^{4} - 19x^{3} + 30x^{2} - 21x + 7 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} + \nu^{4} + 2\nu^{3} + 10\nu^{2} - 7\nu ) / 7 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - \nu + 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{5} + 5\nu^{4} - 18\nu^{3} + 22\nu^{2} - 28\nu + 7 ) / 7 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{5} - 5\nu^{4} + 25\nu^{3} - 29\nu^{2} + 56\nu - 14 ) / 7 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{4} + \beta_{3} - 3\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{5} + 3\beta_{4} - 4\beta_{3} + 2\beta_{2} - 6\beta _1 + 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -4\beta_{5} - 5\beta_{4} - 8\beta_{3} + 5\beta_{2} + 9\beta _1 + 21 \) Copy content Toggle raw display

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
0.500000 + 0.385124i
0.500000 1.75780i
0.500000 + 2.23871i
0.500000 0.385124i
0.500000 + 1.75780i
0.500000 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:

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 \(S_{2}^{\mathrm{new}}(143, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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