Properties

Label 980.2.i.f
Level $980$
Weight $2$
Character orbit 980.i
Analytic conductor $7.825$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [980,2,Mod(361,980)]
 
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([0, 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("980.361");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 980.i (of order \(3\), 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: \(7.82533939809\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 140)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{6} + 1) q^{3} - \zeta_{6} q^{5} + 2 \zeta_{6} q^{9} + (6 \zeta_{6} - 6) q^{11} - 2 q^{13} - q^{15} + (6 \zeta_{6} - 6) q^{17} + 8 \zeta_{6} q^{19} - 3 \zeta_{6} q^{23} + (\zeta_{6} - 1) q^{25} + \cdots - 12 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} - q^{5} + 2 q^{9} - 6 q^{11} - 4 q^{13} - 2 q^{15} - 6 q^{17} + 8 q^{19} - 3 q^{23} - q^{25} + 10 q^{27} + 6 q^{29} + 2 q^{31} + 6 q^{33} - 8 q^{37} - 2 q^{39} + 6 q^{41} + 10 q^{43} + 2 q^{45}+ \cdots - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0.500000 0.866025i 0 −0.500000 0.866025i 0 0 0 1.00000 + 1.73205i 0
961.1 0 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0 0 0 1.00000 1.73205i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 980.2.i.f 2
7.b odd 2 1 140.2.i.a 2
7.c even 3 1 980.2.a.e 1
7.c even 3 1 inner 980.2.i.f 2
7.d odd 6 1 140.2.i.a 2
7.d odd 6 1 980.2.a.g 1
21.c even 2 1 1260.2.s.c 2
21.g even 6 1 1260.2.s.c 2
21.g even 6 1 8820.2.a.p 1
21.h odd 6 1 8820.2.a.a 1
28.d even 2 1 560.2.q.f 2
28.f even 6 1 560.2.q.f 2
28.f even 6 1 3920.2.a.k 1
28.g odd 6 1 3920.2.a.w 1
35.c odd 2 1 700.2.i.b 2
35.f even 4 2 700.2.r.a 4
35.i odd 6 1 700.2.i.b 2
35.i odd 6 1 4900.2.a.i 1
35.j even 6 1 4900.2.a.q 1
35.k even 12 2 700.2.r.a 4
35.k even 12 2 4900.2.e.m 2
35.l odd 12 2 4900.2.e.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.i.a 2 7.b odd 2 1
140.2.i.a 2 7.d odd 6 1
560.2.q.f 2 28.d even 2 1
560.2.q.f 2 28.f even 6 1
700.2.i.b 2 35.c odd 2 1
700.2.i.b 2 35.i odd 6 1
700.2.r.a 4 35.f even 4 2
700.2.r.a 4 35.k even 12 2
980.2.a.e 1 7.c even 3 1
980.2.a.g 1 7.d odd 6 1
980.2.i.f 2 1.a even 1 1 trivial
980.2.i.f 2 7.c even 3 1 inner
1260.2.s.c 2 21.c even 2 1
1260.2.s.c 2 21.g even 6 1
3920.2.a.k 1 28.f even 6 1
3920.2.a.w 1 28.g odd 6 1
4900.2.a.i 1 35.i odd 6 1
4900.2.a.q 1 35.j even 6 1
4900.2.e.m 2 35.k even 12 2
4900.2.e.n 2 35.l odd 12 2
8820.2.a.a 1 21.h odd 6 1
8820.2.a.p 1 21.g even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(980, [\chi])\):

\( T_{3}^{2} - T_{3} + 1 \) Copy content Toggle raw display
\( T_{11}^{2} + 6T_{11} + 36 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) 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
$11$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$13$ \( (T + 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$19$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$23$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$29$ \( (T - 3)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$37$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$41$ \( (T - 3)^{2} \) Copy content Toggle raw display
$43$ \( (T - 5)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$67$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$79$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$83$ \( (T + 3)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$97$ \( (T - 10)^{2} \) Copy content Toggle raw display
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