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Properties

Label 3960.1.bp.a
Level $3960$
Weight $1$
Character orbit 3960.bp
Analytic conductor $1.976$
Analytic rank $0$
Dimension $8$
Projective image $D_{8}$
CM discriminant -264
Inner twists $8$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3960,1,Mod(307,3960)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3960.307"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3960, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 2, 0, 1, 2])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 3960 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3960.bp (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-8,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.97629745003\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{16})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{8}\)
Projective field: Galois closure of 8.2.9487368000000.2

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q - \zeta_{16}^{6} q^{2} - \zeta_{16}^{4} q^{4} - \zeta_{16}^{7} q^{5} + (\zeta_{16}^{3} + \zeta_{16}) q^{7} - \zeta_{16}^{2} q^{8} - \zeta_{16}^{5} q^{10} - \zeta_{16}^{4} q^{11} + ( - \zeta_{16}^{3} + \zeta_{16}) q^{13} + \cdots + (\zeta_{16}^{4} + \zeta_{16}^{2} + 1) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{16} - 8 q^{17} + 8 q^{35} - 8 q^{44} + 8 q^{65} - 8 q^{67} + 8 q^{68} + 8 q^{70} + 8 q^{82} + 8 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3960\mathbb{Z}\right)^\times\).

\(n\) \(991\) \(1981\) \(2377\) \(2521\) \(3521\)
\(\chi(n)\) \(-1\) \(-1\) \(\zeta_{16}^{4}\) \(-1\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1
−0.382683 + 0.923880i
0.382683 − 0.923880i
−0.923880 − 0.382683i
0.923880 + 0.382683i
−0.382683 − 0.923880i
0.382683 + 0.923880i
−0.923880 + 0.382683i
0.923880 − 0.382683i
−0.707107 + 0.707107i 0 − 1.00000i −0.382683 − 0.923880i 0 0.541196 + 0.541196i 0.707107 + 0.707107i 0 0.923880 + 0.382683i
307.2 −0.707107 + 0.707107i 0 − 1.00000i 0.382683 + 0.923880i 0 −0.541196 − 0.541196i 0.707107 + 0.707107i 0 −0.923880 − 0.382683i
307.3 0.707107 − 0.707107i 0 − 1.00000i −0.923880 + 0.382683i 0 −1.30656 − 1.30656i −0.707107 − 0.707107i 0 −0.382683 + 0.923880i
307.4 0.707107 − 0.707107i 0 − 1.00000i 0.923880 − 0.382683i 0 1.30656 + 1.30656i −0.707107 − 0.707107i 0 0.382683 − 0.923880i
2683.1 −0.707107 − 0.707107i 0 1.00000i −0.382683 + 0.923880i 0 0.541196 − 0.541196i 0.707107 − 0.707107i 0 0.923880 − 0.382683i
2683.2 −0.707107 − 0.707107i 0 1.00000i 0.382683 − 0.923880i 0 −0.541196 + 0.541196i 0.707107 − 0.707107i 0 −0.923880 + 0.382683i
2683.3 0.707107 + 0.707107i 0 1.00000i −0.923880 − 0.382683i 0 −1.30656 + 1.30656i −0.707107 + 0.707107i 0 −0.382683 − 0.923880i
2683.4 0.707107 + 0.707107i 0 1.00000i 0.923880 + 0.382683i 0 1.30656 − 1.30656i −0.707107 + 0.707107i 0 0.382683 + 0.923880i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 307.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
264.p odd 2 1 CM by \(\Q(\sqrt{-66}) \)
8.d odd 2 1 inner
15.e even 4 1 inner
33.d even 2 1 inner
55.e even 4 1 inner
120.q odd 4 1 inner
440.w odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3960.1.bp.a ✓ 8
3.b odd 2 1 3960.1.bp.b yes 8
5.c odd 4 1 3960.1.bp.b yes 8
8.d odd 2 1 inner 3960.1.bp.a ✓ 8
11.b odd 2 1 3960.1.bp.b yes 8
15.e even 4 1 inner 3960.1.bp.a ✓ 8
24.f even 2 1 3960.1.bp.b yes 8
33.d even 2 1 inner 3960.1.bp.a ✓ 8
40.k even 4 1 3960.1.bp.b yes 8
55.e even 4 1 inner 3960.1.bp.a ✓ 8
88.g even 2 1 3960.1.bp.b yes 8
120.q odd 4 1 inner 3960.1.bp.a ✓ 8
165.l odd 4 1 3960.1.bp.b yes 8
264.p odd 2 1 CM 3960.1.bp.a ✓ 8
440.w odd 4 1 inner 3960.1.bp.a ✓ 8
1320.bt even 4 1 3960.1.bp.b yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3960.1.bp.a ✓ 8 1.a even 1 1 trivial
3960.1.bp.a ✓ 8 8.d odd 2 1 inner
3960.1.bp.a ✓ 8 15.e even 4 1 inner
3960.1.bp.a ✓ 8 33.d even 2 1 inner
3960.1.bp.a ✓ 8 55.e even 4 1 inner
3960.1.bp.a ✓ 8 120.q odd 4 1 inner
3960.1.bp.a ✓ 8 264.p odd 2 1 CM
3960.1.bp.a ✓ 8 440.w odd 4 1 inner
3960.1.bp.b yes 8 3.b odd 2 1
3960.1.bp.b yes 8 5.c odd 4 1
3960.1.bp.b yes 8 11.b odd 2 1
3960.1.bp.b yes 8 24.f even 2 1
3960.1.bp.b yes 8 40.k even 4 1
3960.1.bp.b yes 8 88.g even 2 1
3960.1.bp.b yes 8 165.l odd 4 1
3960.1.bp.b yes 8 1320.bt even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{17}^{2} + 2T_{17} + 2 \) acting on \(S_{1}^{\mathrm{new}}(3960, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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