Properties

Label 980.2.i.i
Level $980$
Weight $2$
Character orbit 980.i
Analytic conductor $7.825$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 20)
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.

\(f(q)\) \(=\) \( q + ( - 2 \zeta_{6} + 2) q^{3} + \zeta_{6} q^{5} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \zeta_{6} + 2) q^{3} + \zeta_{6} q^{5} - \zeta_{6} q^{9} + 2 q^{13} + 2 q^{15} + ( - 6 \zeta_{6} + 6) q^{17} + 4 \zeta_{6} q^{19} - 6 \zeta_{6} q^{23} + (\zeta_{6} - 1) q^{25} + 4 q^{27} + 6 q^{29} + ( - 4 \zeta_{6} + 4) q^{31} - 2 \zeta_{6} q^{37} + ( - 4 \zeta_{6} + 4) q^{39} + 6 q^{41} - 10 q^{43} + ( - \zeta_{6} + 1) q^{45} + 6 \zeta_{6} q^{47} - 12 \zeta_{6} q^{51} + ( - 6 \zeta_{6} + 6) q^{53} + 8 q^{57} + (12 \zeta_{6} - 12) q^{59} - 2 \zeta_{6} q^{61} + 2 \zeta_{6} q^{65} + (2 \zeta_{6} - 2) q^{67} - 12 q^{69} - 12 q^{71} + (2 \zeta_{6} - 2) q^{73} + 2 \zeta_{6} q^{75} - 8 \zeta_{6} q^{79} + ( - 11 \zeta_{6} + 11) q^{81} + 6 q^{83} + 6 q^{85} + ( - 12 \zeta_{6} + 12) q^{87} + 6 \zeta_{6} q^{89} - 8 \zeta_{6} q^{93} + (4 \zeta_{6} - 4) q^{95} + 2 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + q^{5} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + q^{5} - q^{9} + 4 q^{13} + 4 q^{15} + 6 q^{17} + 4 q^{19} - 6 q^{23} - q^{25} + 8 q^{27} + 12 q^{29} + 4 q^{31} - 2 q^{37} + 4 q^{39} + 12 q^{41} - 20 q^{43} + q^{45} + 6 q^{47} - 12 q^{51} + 6 q^{53} + 16 q^{57} - 12 q^{59} - 2 q^{61} + 2 q^{65} - 2 q^{67} - 24 q^{69} - 24 q^{71} - 2 q^{73} + 2 q^{75} - 8 q^{79} + 11 q^{81} + 12 q^{83} + 12 q^{85} + 12 q^{87} + 6 q^{89} - 8 q^{93} - 4 q^{95} + 4 q^{97}+O(q^{100}) \) 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 1.00000 1.73205i 0 0.500000 + 0.866025i 0 0 0 −0.500000 0.866025i 0
961.1 0 1.00000 + 1.73205i 0 0.500000 0.866025i 0 0 0 −0.500000 + 0.866025i 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.i 2
7.b odd 2 1 980.2.i.c 2
7.c even 3 1 20.2.a.a 1
7.c even 3 1 inner 980.2.i.i 2
7.d odd 6 1 980.2.a.h 1
7.d odd 6 1 980.2.i.c 2
21.g even 6 1 8820.2.a.g 1
21.h odd 6 1 180.2.a.a 1
28.f even 6 1 3920.2.a.h 1
28.g odd 6 1 80.2.a.b 1
35.i odd 6 1 4900.2.a.e 1
35.j even 6 1 100.2.a.a 1
35.k even 12 2 4900.2.e.f 2
35.l odd 12 2 100.2.c.a 2
56.k odd 6 1 320.2.a.a 1
56.p even 6 1 320.2.a.f 1
63.g even 3 1 1620.2.i.h 2
63.h even 3 1 1620.2.i.h 2
63.j odd 6 1 1620.2.i.b 2
63.n odd 6 1 1620.2.i.b 2
77.h odd 6 1 2420.2.a.a 1
84.n even 6 1 720.2.a.h 1
91.r even 6 1 3380.2.a.c 1
91.z odd 12 2 3380.2.f.b 2
105.o odd 6 1 900.2.a.b 1
105.x even 12 2 900.2.d.c 2
112.u odd 12 2 1280.2.d.g 2
112.w even 12 2 1280.2.d.c 2
119.j even 6 1 5780.2.a.f 1
119.n even 12 2 5780.2.c.a 2
133.r odd 6 1 7220.2.a.f 1
140.p odd 6 1 400.2.a.c 1
140.w even 12 2 400.2.c.b 2
168.s odd 6 1 2880.2.a.m 1
168.v even 6 1 2880.2.a.f 1
280.bf even 6 1 1600.2.a.c 1
280.bi odd 6 1 1600.2.a.w 1
280.br even 12 2 1600.2.c.e 2
280.bt odd 12 2 1600.2.c.d 2
308.n even 6 1 9680.2.a.ba 1
420.ba even 6 1 3600.2.a.be 1
420.bp odd 12 2 3600.2.f.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.2.a.a 1 7.c even 3 1
80.2.a.b 1 28.g odd 6 1
100.2.a.a 1 35.j even 6 1
100.2.c.a 2 35.l odd 12 2
180.2.a.a 1 21.h odd 6 1
320.2.a.a 1 56.k odd 6 1
320.2.a.f 1 56.p even 6 1
400.2.a.c 1 140.p odd 6 1
400.2.c.b 2 140.w even 12 2
720.2.a.h 1 84.n even 6 1
900.2.a.b 1 105.o odd 6 1
900.2.d.c 2 105.x even 12 2
980.2.a.h 1 7.d odd 6 1
980.2.i.c 2 7.b odd 2 1
980.2.i.c 2 7.d odd 6 1
980.2.i.i 2 1.a even 1 1 trivial
980.2.i.i 2 7.c even 3 1 inner
1280.2.d.c 2 112.w even 12 2
1280.2.d.g 2 112.u odd 12 2
1600.2.a.c 1 280.bf even 6 1
1600.2.a.w 1 280.bi odd 6 1
1600.2.c.d 2 280.bt odd 12 2
1600.2.c.e 2 280.br even 12 2
1620.2.i.b 2 63.j odd 6 1
1620.2.i.b 2 63.n odd 6 1
1620.2.i.h 2 63.g even 3 1
1620.2.i.h 2 63.h even 3 1
2420.2.a.a 1 77.h odd 6 1
2880.2.a.f 1 168.v even 6 1
2880.2.a.m 1 168.s odd 6 1
3380.2.a.c 1 91.r even 6 1
3380.2.f.b 2 91.z odd 12 2
3600.2.a.be 1 420.ba even 6 1
3600.2.f.j 2 420.bp odd 12 2
3920.2.a.h 1 28.f even 6 1
4900.2.a.e 1 35.i odd 6 1
4900.2.e.f 2 35.k even 12 2
5780.2.a.f 1 119.j even 6 1
5780.2.c.a 2 119.n even 12 2
7220.2.a.f 1 133.r odd 6 1
8820.2.a.g 1 21.g even 6 1
9680.2.a.ba 1 308.n even 6 1

Hecke kernels

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

\( T_{3}^{2} - 2T_{3} + 4 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display

Hecke characteristic polynomials

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