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Properties

Label 960.1.u.a
Level $960$
Weight $1$
Character orbit 960.u
Analytic conductor $0.479$
Analytic rank $0$
Dimension $4$
Projective image $D_{4}$
CM discriminant -24
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [960,1,Mod(287,960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(960, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 2, 2, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("960.287");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 960.u (of order \(4\), degree \(2\), 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: \(0.479102412128\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.0.9000.1

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{8}^{3} q^{3} - \zeta_{8}^{3} q^{5} + ( - \zeta_{8}^{2} - 1) q^{7} - \zeta_{8}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{8}^{3} q^{3} - \zeta_{8}^{3} q^{5} + ( - \zeta_{8}^{2} - 1) q^{7} - \zeta_{8}^{2} q^{9} + (\zeta_{8}^{3} + \zeta_{8}) q^{11} - \zeta_{8}^{2} q^{15} + (\zeta_{8}^{3} - \zeta_{8}) q^{21} - \zeta_{8}^{2} q^{25} - \zeta_{8} q^{27} + (\zeta_{8}^{3} + \zeta_{8}) q^{29} + (\zeta_{8}^{2} + 1) q^{33} + (\zeta_{8}^{3} - \zeta_{8}) q^{35} - \zeta_{8} q^{45} + \zeta_{8}^{2} q^{49} + \zeta_{8} q^{53} + (\zeta_{8}^{2} + 1) q^{55} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{59} + (\zeta_{8}^{2} - 1) q^{63} + ( - \zeta_{8}^{2} - 1) q^{73} - \zeta_{8} q^{75} + ( - 2 \zeta_{8}^{3} + \zeta_{8}) q^{77} - q^{81} + (\zeta_{8}^{2} + 1) q^{87} + (\zeta_{8}^{2} - 1) q^{97} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{7} + 4 q^{33} + 4 q^{55} - 4 q^{63} - 4 q^{73} - 4 q^{81} + 4 q^{87} - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/960\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(577\) \(641\) \(901\)
\(\chi(n)\) \(-1\) \(-\zeta_{8}^{2}\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
287.1
−0.707107 + 0.707107i
0.707107 − 0.707107i
−0.707107 − 0.707107i
0.707107 + 0.707107i
0 −0.707107 − 0.707107i 0 −0.707107 − 0.707107i 0 −1.00000 + 1.00000i 0 1.00000i 0
287.2 0 0.707107 + 0.707107i 0 0.707107 + 0.707107i 0 −1.00000 + 1.00000i 0 1.00000i 0
863.1 0 −0.707107 + 0.707107i 0 −0.707107 + 0.707107i 0 −1.00000 − 1.00000i 0 − 1.00000i 0
863.2 0 0.707107 − 0.707107i 0 0.707107 − 0.707107i 0 −1.00000 − 1.00000i 0 − 1.00000i 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
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)
3.b odd 2 1 inner
8.b even 2 1 inner
20.e even 4 1 inner
40.k even 4 1 inner
60.l odd 4 1 inner
120.q odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 960.1.u.a ✓ 4
3.b odd 2 1 inner 960.1.u.a ✓ 4
4.b odd 2 1 960.1.u.b yes 4
5.c odd 4 1 960.1.u.b yes 4
8.b even 2 1 inner 960.1.u.a ✓ 4
8.d odd 2 1 960.1.u.b yes 4
12.b even 2 1 960.1.u.b yes 4
15.e even 4 1 960.1.u.b yes 4
16.e even 4 2 3840.1.bj.c 4
16.f odd 4 2 3840.1.bj.b 4
20.e even 4 1 inner 960.1.u.a ✓ 4
24.f even 2 1 960.1.u.b yes 4
24.h odd 2 1 CM 960.1.u.a ✓ 4
40.i odd 4 1 960.1.u.b yes 4
40.k even 4 1 inner 960.1.u.a ✓ 4
48.i odd 4 2 3840.1.bj.c 4
48.k even 4 2 3840.1.bj.b 4
60.l odd 4 1 inner 960.1.u.a ✓ 4
80.i odd 4 1 3840.1.bj.b 4
80.j even 4 1 3840.1.bj.c 4
80.s even 4 1 3840.1.bj.c 4
80.t odd 4 1 3840.1.bj.b 4
120.q odd 4 1 inner 960.1.u.a ✓ 4
120.w even 4 1 960.1.u.b yes 4
240.z odd 4 1 3840.1.bj.c 4
240.bb even 4 1 3840.1.bj.b 4
240.bd odd 4 1 3840.1.bj.c 4
240.bf even 4 1 3840.1.bj.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
960.1.u.a ✓ 4 1.a even 1 1 trivial
960.1.u.a ✓ 4 3.b odd 2 1 inner
960.1.u.a ✓ 4 8.b even 2 1 inner
960.1.u.a ✓ 4 20.e even 4 1 inner
960.1.u.a ✓ 4 24.h odd 2 1 CM
960.1.u.a ✓ 4 40.k even 4 1 inner
960.1.u.a ✓ 4 60.l odd 4 1 inner
960.1.u.a ✓ 4 120.q odd 4 1 inner
960.1.u.b yes 4 4.b odd 2 1
960.1.u.b yes 4 5.c odd 4 1
960.1.u.b yes 4 8.d odd 2 1
960.1.u.b yes 4 12.b even 2 1
960.1.u.b yes 4 15.e even 4 1
960.1.u.b yes 4 24.f even 2 1
960.1.u.b yes 4 40.i odd 4 1
960.1.u.b yes 4 120.w even 4 1
3840.1.bj.b 4 16.f odd 4 2
3840.1.bj.b 4 48.k even 4 2
3840.1.bj.b 4 80.i odd 4 1
3840.1.bj.b 4 80.t odd 4 1
3840.1.bj.b 4 240.bb even 4 1
3840.1.bj.b 4 240.bf even 4 1
3840.1.bj.c 4 16.e even 4 2
3840.1.bj.c 4 48.i odd 4 2
3840.1.bj.c 4 80.j even 4 1
3840.1.bj.c 4 80.s even 4 1
3840.1.bj.c 4 240.z odd 4 1
3840.1.bj.c 4 240.bd odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} + 2T_{7} + 2 \) acting on \(S_{1}^{\mathrm{new}}(960, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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