Properties

Label 1960.2.q.a
Level $1960$
Weight $2$
Character orbit 1960.q
Analytic conductor $15.651$
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] = [1960,2,Mod(361,1960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1960, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 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("1960.361");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1960 = 2^{3} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1960.q (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: \(15.6506787962\)
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 280)
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.

\(f(q)\) \(=\) \( q + (3 \zeta_{6} - 3) q^{3} + \zeta_{6} q^{5} - 6 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (3 \zeta_{6} - 3) q^{3} + \zeta_{6} q^{5} - 6 \zeta_{6} q^{9} + ( - 5 \zeta_{6} + 5) q^{11} + 5 q^{13} - 3 q^{15} + (7 \zeta_{6} - 7) q^{17} - 2 \zeta_{6} q^{19} + 2 \zeta_{6} q^{23} + (\zeta_{6} - 1) q^{25} + 9 q^{27} + 7 q^{29} + ( - 4 \zeta_{6} + 4) q^{31} + 15 \zeta_{6} q^{33} + 6 \zeta_{6} q^{37} + (15 \zeta_{6} - 15) q^{39} + 12 q^{41} - 2 q^{43} + ( - 6 \zeta_{6} + 6) q^{45} + \zeta_{6} q^{47} - 21 \zeta_{6} q^{51} + 5 q^{55} + 6 q^{57} + (4 \zeta_{6} - 4) q^{59} + 4 \zeta_{6} q^{61} + 5 \zeta_{6} q^{65} + (8 \zeta_{6} - 8) q^{67} - 6 q^{69} + ( - 6 \zeta_{6} + 6) q^{73} - 3 \zeta_{6} q^{75} + 3 \zeta_{6} q^{79} + (9 \zeta_{6} - 9) q^{81} + 4 q^{83} - 7 q^{85} + (21 \zeta_{6} - 21) q^{87} + 12 \zeta_{6} q^{93} + ( - 2 \zeta_{6} + 2) q^{95} - 13 q^{97} - 30 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{3} + q^{5} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{3} + q^{5} - 6 q^{9} + 5 q^{11} + 10 q^{13} - 6 q^{15} - 7 q^{17} - 2 q^{19} + 2 q^{23} - q^{25} + 18 q^{27} + 14 q^{29} + 4 q^{31} + 15 q^{33} + 6 q^{37} - 15 q^{39} + 24 q^{41} - 4 q^{43} + 6 q^{45} + q^{47} - 21 q^{51} + 10 q^{55} + 12 q^{57} - 4 q^{59} + 4 q^{61} + 5 q^{65} - 8 q^{67} - 12 q^{69} + 6 q^{73} - 3 q^{75} + 3 q^{79} - 9 q^{81} + 8 q^{83} - 14 q^{85} - 21 q^{87} + 12 q^{93} + 2 q^{95} - 26 q^{97} - 60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1960\mathbb{Z}\right)^\times\).

\(n\) \(981\) \(1081\) \(1177\) \(1471\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\) \(1\) \(1\)

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 −1.50000 + 2.59808i 0 0.500000 + 0.866025i 0 0 0 −3.00000 5.19615i 0
961.1 0 −1.50000 2.59808i 0 0.500000 0.866025i 0 0 0 −3.00000 + 5.19615i 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 1960.2.q.a 2
7.b odd 2 1 1960.2.q.o 2
7.c even 3 1 1960.2.a.o 1
7.c even 3 1 inner 1960.2.q.a 2
7.d odd 6 1 280.2.a.a 1
7.d odd 6 1 1960.2.q.o 2
21.g even 6 1 2520.2.a.i 1
28.f even 6 1 560.2.a.f 1
28.g odd 6 1 3920.2.a.c 1
35.i odd 6 1 1400.2.a.n 1
35.j even 6 1 9800.2.a.a 1
35.k even 12 2 1400.2.g.a 2
56.j odd 6 1 2240.2.a.z 1
56.m even 6 1 2240.2.a.a 1
84.j odd 6 1 5040.2.a.a 1
140.s even 6 1 2800.2.a.c 1
140.x odd 12 2 2800.2.g.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.a.a 1 7.d odd 6 1
560.2.a.f 1 28.f even 6 1
1400.2.a.n 1 35.i odd 6 1
1400.2.g.a 2 35.k even 12 2
1960.2.a.o 1 7.c even 3 1
1960.2.q.a 2 1.a even 1 1 trivial
1960.2.q.a 2 7.c even 3 1 inner
1960.2.q.o 2 7.b odd 2 1
1960.2.q.o 2 7.d odd 6 1
2240.2.a.a 1 56.m even 6 1
2240.2.a.z 1 56.j odd 6 1
2520.2.a.i 1 21.g even 6 1
2800.2.a.c 1 140.s even 6 1
2800.2.g.b 2 140.x odd 12 2
3920.2.a.c 1 28.g odd 6 1
5040.2.a.a 1 84.j odd 6 1
9800.2.a.a 1 35.j even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1960, [\chi])\):

\( T_{3}^{2} + 3T_{3} + 9 \) Copy content Toggle raw display
\( T_{11}^{2} - 5T_{11} + 25 \) Copy content Toggle raw display
\( T_{13} - 5 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$5$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$13$ \( (T - 5)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$19$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$23$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$29$ \( (T - 7)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$37$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$41$ \( (T - 12)^{2} \) Copy content Toggle raw display
$43$ \( (T + 2)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$61$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$67$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$79$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$83$ \( (T - 4)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( (T + 13)^{2} \) Copy content Toggle raw display
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