Properties

Label 980.1.p.b
Level $980$
Weight $1$
Character orbit 980.p
Analytic conductor $0.489$
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] = [980,1,Mod(79,980)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(980, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("980.79");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 980.p (of order \(6\), degree \(2\), not 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: \(0.489083712380\)
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: no (minimal twist has level 140)
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 \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

$q$-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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/980\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(197\) \(491\)
\(\chi(n)\) \(\zeta_{6}^{2}\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
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 −1.00000 0 −0.500000 0.866025i
459.1 0.500000 + 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 0 −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 980.1.p.b 2
4.b odd 2 1 980.1.p.a 2
5.b even 2 1 980.1.p.a 2
7.b odd 2 1 140.1.p.b yes 2
7.c even 3 1 980.1.f.a 1
7.c even 3 1 inner 980.1.p.b 2
7.d odd 6 1 140.1.p.b yes 2
7.d odd 6 1 980.1.f.b 1
20.d odd 2 1 CM 980.1.p.b 2
21.c even 2 1 1260.1.ci.a 2
21.g even 6 1 1260.1.ci.a 2
28.d even 2 1 140.1.p.a 2
28.f even 6 1 140.1.p.a 2
28.f even 6 1 980.1.f.c 1
28.g odd 6 1 980.1.f.d 1
28.g odd 6 1 980.1.p.a 2
35.c odd 2 1 140.1.p.a 2
35.f even 4 2 700.1.u.a 4
35.i odd 6 1 140.1.p.a 2
35.i odd 6 1 980.1.f.c 1
35.j even 6 1 980.1.f.d 1
35.j even 6 1 980.1.p.a 2
35.k even 12 2 700.1.u.a 4
56.e even 2 1 2240.1.bt.a 2
56.h odd 2 1 2240.1.bt.b 2
56.j odd 6 1 2240.1.bt.b 2
56.m even 6 1 2240.1.bt.a 2
84.h odd 2 1 1260.1.ci.b 2
84.j odd 6 1 1260.1.ci.b 2
105.g even 2 1 1260.1.ci.b 2
105.p even 6 1 1260.1.ci.b 2
140.c even 2 1 140.1.p.b yes 2
140.j odd 4 2 700.1.u.a 4
140.p odd 6 1 980.1.f.a 1
140.p odd 6 1 inner 980.1.p.b 2
140.s even 6 1 140.1.p.b yes 2
140.s even 6 1 980.1.f.b 1
140.x odd 12 2 700.1.u.a 4
280.c odd 2 1 2240.1.bt.a 2
280.n even 2 1 2240.1.bt.b 2
280.ba even 6 1 2240.1.bt.b 2
280.bk odd 6 1 2240.1.bt.a 2
420.o odd 2 1 1260.1.ci.a 2
420.be odd 6 1 1260.1.ci.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.1.p.a 2 28.d even 2 1
140.1.p.a 2 28.f even 6 1
140.1.p.a 2 35.c odd 2 1
140.1.p.a 2 35.i odd 6 1
140.1.p.b yes 2 7.b odd 2 1
140.1.p.b yes 2 7.d odd 6 1
140.1.p.b yes 2 140.c even 2 1
140.1.p.b yes 2 140.s even 6 1
700.1.u.a 4 35.f even 4 2
700.1.u.a 4 35.k even 12 2
700.1.u.a 4 140.j odd 4 2
700.1.u.a 4 140.x odd 12 2
980.1.f.a 1 7.c even 3 1
980.1.f.a 1 140.p odd 6 1
980.1.f.b 1 7.d odd 6 1
980.1.f.b 1 140.s even 6 1
980.1.f.c 1 28.f even 6 1
980.1.f.c 1 35.i odd 6 1
980.1.f.d 1 28.g odd 6 1
980.1.f.d 1 35.j even 6 1
980.1.p.a 2 4.b odd 2 1
980.1.p.a 2 5.b even 2 1
980.1.p.a 2 28.g odd 6 1
980.1.p.a 2 35.j even 6 1
980.1.p.b 2 1.a even 1 1 trivial
980.1.p.b 2 7.c even 3 1 inner
980.1.p.b 2 20.d odd 2 1 CM
980.1.p.b 2 140.p odd 6 1 inner
1260.1.ci.a 2 21.c even 2 1
1260.1.ci.a 2 21.g even 6 1
1260.1.ci.a 2 420.o odd 2 1
1260.1.ci.a 2 420.be odd 6 1
1260.1.ci.b 2 84.h odd 2 1
1260.1.ci.b 2 84.j odd 6 1
1260.1.ci.b 2 105.g even 2 1
1260.1.ci.b 2 105.p even 6 1
2240.1.bt.a 2 56.e even 2 1
2240.1.bt.a 2 56.m even 6 1
2240.1.bt.a 2 280.c odd 2 1
2240.1.bt.a 2 280.bk odd 6 1
2240.1.bt.b 2 56.h odd 2 1
2240.1.bt.b 2 56.j odd 6 1
2240.1.bt.b 2 280.n even 2 1
2240.1.bt.b 2 280.ba even 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 \(S_{1}^{\mathrm{new}}(980, [\chi])\). 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
$7$ \( T^{2} \) 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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