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Properties

Label 1960.2.q.u
Level $1960$
Weight $2$
Character orbit 1960.q
Analytic conductor $15.651$
Analytic rank $0$
Dimension $4$
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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{17})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 5x^{2} + 4x + 16 \) 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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + ( - \beta_{2} + 1) q^{5} + (\beta_{3} + \beta_{2} + \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + ( - \beta_{2} + 1) q^{5} + (\beta_{3} + \beta_{2} + \beta_1 - 1) q^{9} + \beta_1 q^{11} + ( - 3 \beta_{3} - 2) q^{13} - \beta_{3} q^{15} + (6 \beta_{2} - \beta_1) q^{17} + ( - 2 \beta_{3} + 4 \beta_{2} + \cdots - 4) q^{19}+ \cdots + (2 \beta_{3} - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{3} + 2 q^{5} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{3} + 2 q^{5} - 3 q^{9} + q^{11} - 2 q^{13} + 2 q^{15} + 11 q^{17} - 6 q^{19} + 2 q^{23} - 2 q^{25} - 14 q^{27} + 10 q^{29} - 4 q^{31} - 9 q^{33} + 25 q^{39} - 12 q^{41} - 12 q^{43} + 3 q^{45} + 9 q^{47} + 3 q^{51} + 18 q^{53} + 2 q^{55} + 28 q^{57} - 8 q^{59} - 22 q^{61} - q^{65} + 12 q^{67} + 36 q^{69} + 8 q^{73} + q^{75} + 11 q^{79} + 14 q^{81} - 48 q^{83} + 22 q^{85} + 11 q^{87} + 2 q^{89} + 36 q^{93} + 6 q^{95} - 30 q^{97} - 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 5x^{2} + 4x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 5\nu^{2} - 5\nu + 16 ) / 20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 4 ) / 5 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 4\beta_{2} + \beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 5\beta_{3} - 4 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(981\) \(1081\) \(1177\) \(1471\)
\(\chi(n)\) \(1\) \(-1 + \beta_{2}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
−0.780776 + 1.35234i
1.28078 − 2.21837i
−0.780776 − 1.35234i
1.28078 + 2.21837i
0 −0.780776 + 1.35234i 0 0.500000 + 0.866025i 0 0 0 0.280776 + 0.486319i 0
361.2 0 1.28078 − 2.21837i 0 0.500000 + 0.866025i 0 0 0 −1.78078 − 3.08440i 0
961.1 0 −0.780776 − 1.35234i 0 0.500000 − 0.866025i 0 0 0 0.280776 − 0.486319i 0
961.2 0 1.28078 + 2.21837i 0 0.500000 − 0.866025i 0 0 0 −1.78078 + 3.08440i 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.u 4
7.b odd 2 1 1960.2.q.s 4
7.c even 3 1 1960.2.a.r 2
7.c even 3 1 inner 1960.2.q.u 4
7.d odd 6 1 280.2.a.d ✓ 2
7.d odd 6 1 1960.2.q.s 4
21.g even 6 1 2520.2.a.w 2
28.f even 6 1 560.2.a.g 2
28.g odd 6 1 3920.2.a.bu 2
35.i odd 6 1 1400.2.a.p 2
35.j even 6 1 9800.2.a.by 2
35.k even 12 2 1400.2.g.k 4
56.j odd 6 1 2240.2.a.be 2
56.m even 6 1 2240.2.a.bi 2
84.j odd 6 1 5040.2.a.bq 2
140.s even 6 1 2800.2.a.bn 2
140.x odd 12 2 2800.2.g.u 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.a.d ✓ 2 7.d odd 6 1
560.2.a.g 2 28.f even 6 1
1400.2.a.p 2 35.i odd 6 1
1400.2.g.k 4 35.k even 12 2
1960.2.a.r 2 7.c even 3 1
1960.2.q.s 4 7.b odd 2 1
1960.2.q.s 4 7.d odd 6 1
1960.2.q.u 4 1.a even 1 1 trivial
1960.2.q.u 4 7.c even 3 1 inner
2240.2.a.be 2 56.j odd 6 1
2240.2.a.bi 2 56.m even 6 1
2520.2.a.w 2 21.g even 6 1
2800.2.a.bn 2 140.s even 6 1
2800.2.g.u 4 140.x odd 12 2
3920.2.a.bu 2 28.g odd 6 1
5040.2.a.bq 2 84.j odd 6 1
9800.2.a.by 2 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 \(S_{2}^{\mathrm{new}}(1960, [\chi])\):

\( T_{3}^{4} - T_{3}^{3} + 5T_{3}^{2} + 4T_{3} + 16 \) Copy content Toggle raw display
\( T_{11}^{4} - T_{11}^{3} + 5T_{11}^{2} + 4T_{11} + 16 \) Copy content Toggle raw display
\( T_{13}^{2} + T_{13} - 38 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - T^{3} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - T^{3} + \cdots + 16 \) Copy content Toggle raw display
$13$ \( (T^{2} + T - 38)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 11 T^{3} + \cdots + 676 \) Copy content Toggle raw display
$19$ \( T^{4} + 6 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$23$ \( T^{4} - 2 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$29$ \( (T^{2} - 5 T + 2)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 4 T^{3} + \cdots + 4096 \) Copy content Toggle raw display
$37$ \( T^{4} + 68T^{2} + 4624 \) Copy content Toggle raw display
$41$ \( (T^{2} + 6 T - 8)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 6 T - 8)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 9 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$53$ \( T^{4} - 18 T^{3} + \cdots + 4096 \) Copy content Toggle raw display
$59$ \( (T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} + 22 T^{3} + \cdots + 10816 \) Copy content Toggle raw display
$67$ \( T^{4} - 12 T^{3} + \cdots + 1024 \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} - 8 T^{3} + \cdots + 2704 \) Copy content Toggle raw display
$79$ \( T^{4} - 11 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$83$ \( (T + 12)^{4} \) Copy content Toggle raw display
$89$ \( T^{4} - 2 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$97$ \( (T^{2} + 15 T + 18)^{2} \) Copy content Toggle raw display
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