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Properties

Label 3960.1.eg.a
Level $3960$
Weight $1$
Character orbit 3960.eg
Analytic conductor $1.976$
Analytic rank $0$
Dimension $16$
Projective image $D_{20}$
CM discriminant -40
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3960,1,Mod(899,3960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3960, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 5, 5, 5, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3960.899");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3960 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3960.eg (of order \(10\), degree \(4\), 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: \(1.97629745003\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{40})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{12} + x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{20}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{20} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{40}^{16} q^{2} - \zeta_{40}^{12} q^{4} + \zeta_{40}^{14} q^{5} + ( - \zeta_{40}^{15} + \zeta_{40}^{9}) q^{7} + \zeta_{40}^{8} q^{8} - \zeta_{40}^{10} q^{10} + \zeta_{40}^{17} q^{11} + ( - \zeta_{40}^{17} + \zeta_{40}^{15}) q^{13} + \cdots + ( - \zeta_{40}^{14} + \zeta_{40}^{6} - 1) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{2} - 4 q^{4} - 4 q^{8} - 4 q^{16} + 4 q^{25} + 16 q^{32} + 4 q^{49} + 4 q^{50} - 4 q^{64} + 4 q^{77} - 8 q^{91} + 20 q^{94} + 8 q^{95} - 16 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(991\) \(1981\) \(2377\) \(2521\) \(3521\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(\zeta_{40}^{12}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
899.1
−0.987688 − 0.156434i
0.987688 + 0.156434i
−0.156434 + 0.987688i
0.156434 − 0.987688i
0.453990 + 0.891007i
−0.453990 − 0.891007i
0.891007 − 0.453990i
−0.891007 + 0.453990i
−0.987688 + 0.156434i
0.987688 − 0.156434i
−0.156434 − 0.987688i
0.156434 + 0.987688i
0.453990 − 0.891007i
−0.453990 + 0.891007i
0.891007 + 0.453990i
−0.891007 − 0.453990i
−0.809017 + 0.587785i 0 0.309017 − 0.951057i −0.587785 + 0.809017i 0 −0.863541 − 0.280582i 0.309017 + 0.951057i 0 − 1.00000i
899.2 −0.809017 + 0.587785i 0 0.309017 − 0.951057i −0.587785 + 0.809017i 0 0.863541 + 0.280582i 0.309017 + 0.951057i 0 − 1.00000i
899.3 −0.809017 + 0.587785i 0 0.309017 − 0.951057i 0.587785 − 0.809017i 0 −1.69480 − 0.550672i 0.309017 + 0.951057i 0 1.00000i
899.4 −0.809017 + 0.587785i 0 0.309017 − 0.951057i 0.587785 − 0.809017i 0 1.69480 + 0.550672i 0.309017 + 0.951057i 0 1.00000i
1619.1 0.309017 − 0.951057i 0 −0.809017 − 0.587785i −0.951057 + 0.309017i 0 −0.183900 + 0.253116i −0.809017 + 0.587785i 0 1.00000i
1619.2 0.309017 − 0.951057i 0 −0.809017 − 0.587785i −0.951057 + 0.309017i 0 0.183900 − 0.253116i −0.809017 + 0.587785i 0 1.00000i
1619.3 0.309017 − 0.951057i 0 −0.809017 − 0.587785i 0.951057 − 0.309017i 0 −1.16110 + 1.59811i −0.809017 + 0.587785i 0 − 1.00000i
1619.4 0.309017 − 0.951057i 0 −0.809017 − 0.587785i 0.951057 − 0.309017i 0 1.16110 − 1.59811i −0.809017 + 0.587785i 0 − 1.00000i
2339.1 −0.809017 − 0.587785i 0 0.309017 + 0.951057i −0.587785 − 0.809017i 0 −0.863541 + 0.280582i 0.309017 − 0.951057i 0 1.00000i
2339.2 −0.809017 − 0.587785i 0 0.309017 + 0.951057i −0.587785 − 0.809017i 0 0.863541 − 0.280582i 0.309017 − 0.951057i 0 1.00000i
2339.3 −0.809017 − 0.587785i 0 0.309017 + 0.951057i 0.587785 + 0.809017i 0 −1.69480 + 0.550672i 0.309017 − 0.951057i 0 − 1.00000i
2339.4 −0.809017 − 0.587785i 0 0.309017 + 0.951057i 0.587785 + 0.809017i 0 1.69480 − 0.550672i 0.309017 − 0.951057i 0 − 1.00000i
3779.1 0.309017 + 0.951057i 0 −0.809017 + 0.587785i −0.951057 − 0.309017i 0 −0.183900 − 0.253116i −0.809017 − 0.587785i 0 − 1.00000i
3779.2 0.309017 + 0.951057i 0 −0.809017 + 0.587785i −0.951057 − 0.309017i 0 0.183900 + 0.253116i −0.809017 − 0.587785i 0 − 1.00000i
3779.3 0.309017 + 0.951057i 0 −0.809017 + 0.587785i 0.951057 + 0.309017i 0 −1.16110 − 1.59811i −0.809017 − 0.587785i 0 1.00000i
3779.4 0.309017 + 0.951057i 0 −0.809017 + 0.587785i 0.951057 + 0.309017i 0 1.16110 + 1.59811i −0.809017 − 0.587785i 0 1.00000i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 899.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
40.e odd 2 1 CM by \(\Q(\sqrt{-10}) \)
15.d odd 2 1 inner
24.f even 2 1 inner
33.f even 10 1 inner
55.h odd 10 1 inner
88.k even 10 1 inner
1320.da odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3960.1.eg.a ✓ 16
3.b odd 2 1 3960.1.eg.b yes 16
5.b even 2 1 3960.1.eg.b yes 16
8.d odd 2 1 3960.1.eg.b yes 16
11.d odd 10 1 3960.1.eg.b yes 16
15.d odd 2 1 inner 3960.1.eg.a ✓ 16
24.f even 2 1 inner 3960.1.eg.a ✓ 16
33.f even 10 1 inner 3960.1.eg.a ✓ 16
40.e odd 2 1 CM 3960.1.eg.a ✓ 16
55.h odd 10 1 inner 3960.1.eg.a ✓ 16
88.k even 10 1 inner 3960.1.eg.a ✓ 16
120.m even 2 1 3960.1.eg.b yes 16
165.r even 10 1 3960.1.eg.b yes 16
264.r odd 10 1 3960.1.eg.b yes 16
440.bm even 10 1 3960.1.eg.b yes 16
1320.da odd 10 1 inner 3960.1.eg.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3960.1.eg.a ✓ 16 1.a even 1 1 trivial
3960.1.eg.a ✓ 16 15.d odd 2 1 inner
3960.1.eg.a ✓ 16 24.f even 2 1 inner
3960.1.eg.a ✓ 16 33.f even 10 1 inner
3960.1.eg.a ✓ 16 40.e odd 2 1 CM
3960.1.eg.a ✓ 16 55.h odd 10 1 inner
3960.1.eg.a ✓ 16 88.k even 10 1 inner
3960.1.eg.a ✓ 16 1320.da odd 10 1 inner
3960.1.eg.b yes 16 3.b odd 2 1
3960.1.eg.b yes 16 5.b even 2 1
3960.1.eg.b yes 16 8.d odd 2 1
3960.1.eg.b yes 16 11.d odd 10 1
3960.1.eg.b yes 16 120.m even 2 1
3960.1.eg.b yes 16 165.r even 10 1
3960.1.eg.b yes 16 264.r odd 10 1
3960.1.eg.b yes 16 440.bm even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{47}^{4} + 5T_{47} + 5 \) acting on \(S_{1}^{\mathrm{new}}(3960, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} - T^{6} + T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} - 4 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{16} - T^{12} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{16} - 4 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{16} \) Copy content Toggle raw display
$19$ \( (T^{8} - 4 T^{6} + 6 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 3 T^{2} + 1)^{4} \) Copy content Toggle raw display
$29$ \( T^{16} \) Copy content Toggle raw display
$31$ \( T^{16} \) Copy content Toggle raw display
$37$ \( T^{16} + 4 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{16} + 4 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( T^{16} \) Copy content Toggle raw display
$47$ \( (T^{4} + 5 T + 5)^{4} \) Copy content Toggle raw display
$53$ \( (T^{4} - 5 T + 5)^{4} \) Copy content Toggle raw display
$59$ \( T^{16} - 4 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$61$ \( T^{16} \) Copy content Toggle raw display
$67$ \( T^{16} \) Copy content Toggle raw display
$71$ \( T^{16} \) Copy content Toggle raw display
$73$ \( T^{16} \) Copy content Toggle raw display
$79$ \( T^{16} \) Copy content Toggle raw display
$83$ \( T^{16} \) Copy content Toggle raw display
$89$ \( (T^{8} + 8 T^{6} + 19 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} \) Copy content Toggle raw display
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