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 $2$

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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: 6.0.3518667.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 7x^{4} - 8x^{3} + 43x^{2} - 42x + 49 \) Copy content Toggle raw display
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 7x^{4} - 8x^{3} + 43x^{2} - 42x + 49 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + 7\nu^{4} - 49\nu^{3} + 43\nu^{2} - 42\nu + 294 ) / 259 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -6\nu^{5} + 5\nu^{4} - 35\nu^{3} - \nu^{2} - 215\nu - 49 ) / 259 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -5\nu^{5} + 35\nu^{4} + 14\nu^{3} + 215\nu^{2} - 210\nu + 952 ) / 259 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 18\nu^{5} + 22\nu^{4} + 105\nu^{3} + 3\nu^{2} + 608\nu + 147 ) / 259 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} + \beta_{4} - 4\beta_{3} + \beta_{2} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} - 5\beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7\beta_{5} + 21\beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 6\beta_{5} - 6\beta_{4} - 25\beta_{3} + 29\beta_{2} - 35\beta _1 - 19 \) Copy content Toggle raw display

Character values

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

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

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:

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

Hecke characteristic polynomials

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