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Properties

Label 140.1.p.b
Level $140$
Weight $1$
Character orbit 140.p
Analytic conductor $0.070$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -20
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [140,1,Mod(39,140)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(140, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 4]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("140.39");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 140 = 2^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 140.p (of order \(6\), 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: \(0.0698691017686\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.980.1
Artin image: $C_6\times S_3$
Artin field: Galois closure of 12.0.153664000000.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + \zeta_{6} q^{2} + \zeta_{6}^{2} q^{3} + \zeta_{6}^{2} q^{4} - \zeta_{6} q^{5} - q^{6} - \zeta_{6}^{2} q^{7} - q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{6} q^{2} + \zeta_{6}^{2} q^{3} + \zeta_{6}^{2} q^{4} - \zeta_{6} q^{5} - q^{6} - \zeta_{6}^{2} q^{7} - q^{8} - \zeta_{6}^{2} q^{10} - \zeta_{6} q^{12} + q^{14} + q^{15} - \zeta_{6} q^{16} + q^{20} + \zeta_{6} q^{21} - \zeta_{6} q^{23} - \zeta_{6}^{2} q^{24} + \zeta_{6}^{2} q^{25} - q^{27} + \zeta_{6} q^{28} - q^{29} + \zeta_{6} q^{30} - \zeta_{6}^{2} q^{32} - q^{35} + \zeta_{6} q^{40} - q^{41} + \zeta_{6}^{2} q^{42} + q^{43} - \zeta_{6}^{2} q^{46} + \zeta_{6} q^{47} + q^{48} - \zeta_{6} q^{49} - q^{50} - \zeta_{6} q^{54} + \zeta_{6}^{2} q^{56} - \zeta_{6} q^{58} + \zeta_{6}^{2} q^{60} + \zeta_{6} q^{61} + q^{64} + \zeta_{6}^{2} q^{67} + q^{69} - \zeta_{6} q^{70} - \zeta_{6} q^{75} + \zeta_{6}^{2} q^{80} - \zeta_{6}^{2} q^{81} - \zeta_{6} q^{82} + q^{83} - q^{84} + \zeta_{6} q^{86} - \zeta_{6}^{2} q^{87} + \zeta_{6} q^{89} + q^{92} + 2 \zeta_{6}^{2} q^{94} + \zeta_{6} q^{96} - \zeta_{6}^{2} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - q^{3} - q^{4} - q^{5} - 2 q^{6} + q^{7} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - q^{3} - q^{4} - q^{5} - 2 q^{6} + q^{7} - 2 q^{8} + q^{10} - q^{12} + 2 q^{14} + 2 q^{15} - q^{16} + 2 q^{20} + q^{21} - q^{23} + q^{24} - q^{25} - 2 q^{27} + q^{28} - 2 q^{29} + q^{30} + q^{32} - 2 q^{35} + q^{40} - 2 q^{41} - q^{42} + 2 q^{43} + q^{46} + 2 q^{47} + 2 q^{48} - q^{49} - 2 q^{50} - q^{54} - q^{56} - q^{58} - q^{60} + q^{61} + 2 q^{64} - q^{67} + 2 q^{69} - q^{70} - q^{75} - q^{80} + q^{81} - q^{82} + 2 q^{83} - 2 q^{84} + q^{86} + q^{87} + q^{89} + 2 q^{92} - 2 q^{94} + q^{96} + q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/140\mathbb{Z}\right)^\times\).

\(n\) \(57\) \(71\) \(101\)
\(\chi(n)\) \(-1\) \(-1\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
39.1
0.500000 + 0.866025i
0.500000 − 0.866025i
0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 − 0.866025i −1.00000 0.500000 − 0.866025i −1.00000 0 0.500000 − 0.866025i
79.1 0.500000 − 0.866025i −0.500000 − 0.866025i −0.500000 − 0.866025i −0.500000 + 0.866025i −1.00000 0.500000 + 0.866025i −1.00000 0 0.500000 + 0.866025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
7.c even 3 1 inner
140.p odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 140.1.p.b yes 2
3.b odd 2 1 1260.1.ci.a 2
4.b odd 2 1 140.1.p.a ✓ 2
5.b even 2 1 140.1.p.a ✓ 2
5.c odd 4 2 700.1.u.a 4
7.b odd 2 1 980.1.p.b 2
7.c even 3 1 inner 140.1.p.b yes 2
7.c even 3 1 980.1.f.b 1
7.d odd 6 1 980.1.f.a 1
7.d odd 6 1 980.1.p.b 2
8.b even 2 1 2240.1.bt.b 2
8.d odd 2 1 2240.1.bt.a 2
12.b even 2 1 1260.1.ci.b 2
15.d odd 2 1 1260.1.ci.b 2
20.d odd 2 1 CM 140.1.p.b yes 2
20.e even 4 2 700.1.u.a 4
21.h odd 6 1 1260.1.ci.a 2
28.d even 2 1 980.1.p.a 2
28.f even 6 1 980.1.f.d 1
28.f even 6 1 980.1.p.a 2
28.g odd 6 1 140.1.p.a ✓ 2
28.g odd 6 1 980.1.f.c 1
35.c odd 2 1 980.1.p.a 2
35.i odd 6 1 980.1.f.d 1
35.i odd 6 1 980.1.p.a 2
35.j even 6 1 140.1.p.a ✓ 2
35.j even 6 1 980.1.f.c 1
35.l odd 12 2 700.1.u.a 4
40.e odd 2 1 2240.1.bt.b 2
40.f even 2 1 2240.1.bt.a 2
56.k odd 6 1 2240.1.bt.a 2
56.p even 6 1 2240.1.bt.b 2
60.h even 2 1 1260.1.ci.a 2
84.n even 6 1 1260.1.ci.b 2
105.o odd 6 1 1260.1.ci.b 2
140.c even 2 1 980.1.p.b 2
140.p odd 6 1 inner 140.1.p.b yes 2
140.p odd 6 1 980.1.f.b 1
140.s even 6 1 980.1.f.a 1
140.s even 6 1 980.1.p.b 2
140.w even 12 2 700.1.u.a 4
280.bf even 6 1 2240.1.bt.a 2
280.bi odd 6 1 2240.1.bt.b 2
420.ba even 6 1 1260.1.ci.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.1.p.a ✓ 2 4.b odd 2 1
140.1.p.a ✓ 2 5.b even 2 1
140.1.p.a ✓ 2 28.g odd 6 1
140.1.p.a ✓ 2 35.j even 6 1
140.1.p.b yes 2 1.a even 1 1 trivial
140.1.p.b yes 2 7.c even 3 1 inner
140.1.p.b yes 2 20.d odd 2 1 CM
140.1.p.b yes 2 140.p odd 6 1 inner
700.1.u.a 4 5.c odd 4 2
700.1.u.a 4 20.e even 4 2
700.1.u.a 4 35.l odd 12 2
700.1.u.a 4 140.w even 12 2
980.1.f.a 1 7.d odd 6 1
980.1.f.a 1 140.s even 6 1
980.1.f.b 1 7.c even 3 1
980.1.f.b 1 140.p odd 6 1
980.1.f.c 1 28.g odd 6 1
980.1.f.c 1 35.j even 6 1
980.1.f.d 1 28.f even 6 1
980.1.f.d 1 35.i odd 6 1
980.1.p.a 2 28.d even 2 1
980.1.p.a 2 28.f even 6 1
980.1.p.a 2 35.c odd 2 1
980.1.p.a 2 35.i odd 6 1
980.1.p.b 2 7.b odd 2 1
980.1.p.b 2 7.d odd 6 1
980.1.p.b 2 140.c even 2 1
980.1.p.b 2 140.s even 6 1
1260.1.ci.a 2 3.b odd 2 1
1260.1.ci.a 2 21.h odd 6 1
1260.1.ci.a 2 60.h even 2 1
1260.1.ci.a 2 420.ba even 6 1
1260.1.ci.b 2 12.b even 2 1
1260.1.ci.b 2 15.d odd 2 1
1260.1.ci.b 2 84.n even 6 1
1260.1.ci.b 2 105.o odd 6 1
2240.1.bt.a 2 8.d odd 2 1
2240.1.bt.a 2 40.f even 2 1
2240.1.bt.a 2 56.k odd 6 1
2240.1.bt.a 2 280.bf even 6 1
2240.1.bt.b 2 8.b even 2 1
2240.1.bt.b 2 40.e odd 2 1
2240.1.bt.b 2 56.p even 6 1
2240.1.bt.b 2 280.bi odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + T_{3} + 1 \) acting on \(S_{1}^{\mathrm{new}}(140, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$7$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$29$ \( (T + 1)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( (T + 1)^{2} \) Copy content Toggle raw display
$43$ \( (T - 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$67$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( (T - 1)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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