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Properties

Label 3960.1.dh.d
Level $3960$
Weight $1$
Character orbit 3960.dh
Analytic conductor $1.976$
Analytic rank $0$
Dimension $8$
Projective image $D_{10}$
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(379,3960)]
 
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, 0, 5, 4]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3960.379");
 
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.dh (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: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{20})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{10}\)
Projective field: Galois closure of 10.2.1066786981539840000.1

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{20}^{2} q^{2} + \zeta_{20}^{4} q^{4} + \zeta_{20}^{8} q^{5} + (\zeta_{20}^{5} + \zeta_{20}^{3}) q^{7} + \zeta_{20}^{6} q^{8} - q^{10} + \zeta_{20}^{9} q^{11} + ( - \zeta_{20}^{9} + \zeta_{20}^{5}) q^{13}+ \cdots + (\zeta_{20}^{8} - \zeta_{20}^{2} - 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{2} - 2 q^{4} - 2 q^{5} + 2 q^{8} - 8 q^{10} - 2 q^{16} + 4 q^{19} - 2 q^{20} - 4 q^{23} - 2 q^{25} - 8 q^{32} + 6 q^{38} + 2 q^{40} + 4 q^{46} + 4 q^{47} - 8 q^{49} + 2 q^{50} + 4 q^{53} - 2 q^{64}+ \cdots - 12 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(991\) \(1981\) \(2377\) \(2521\) \(3521\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(\zeta_{20}^{4}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
379.1
0.587785 − 0.809017i
−0.587785 + 0.809017i
−0.951057 − 0.309017i
0.951057 + 0.309017i
0.587785 + 0.809017i
−0.587785 − 0.809017i
−0.951057 + 0.309017i
0.951057 − 0.309017i
−0.309017 − 0.951057i 0 −0.809017 + 0.587785i 0.309017 − 0.951057i 0 −0.951057 + 0.690983i 0.809017 + 0.587785i 0 −1.00000
379.2 −0.309017 − 0.951057i 0 −0.809017 + 0.587785i 0.309017 − 0.951057i 0 0.951057 − 0.690983i 0.809017 + 0.587785i 0 −1.00000
1819.1 0.809017 + 0.587785i 0 0.309017 + 0.951057i −0.809017 + 0.587785i 0 −0.587785 − 1.80902i −0.309017 + 0.951057i 0 −1.00000
1819.2 0.809017 + 0.587785i 0 0.309017 + 0.951057i −0.809017 + 0.587785i 0 0.587785 + 1.80902i −0.309017 + 0.951057i 0 −1.00000
2539.1 −0.309017 + 0.951057i 0 −0.809017 − 0.587785i 0.309017 + 0.951057i 0 −0.951057 − 0.690983i 0.809017 − 0.587785i 0 −1.00000
2539.2 −0.309017 + 0.951057i 0 −0.809017 − 0.587785i 0.309017 + 0.951057i 0 0.951057 + 0.690983i 0.809017 − 0.587785i 0 −1.00000
3259.1 0.809017 − 0.587785i 0 0.309017 − 0.951057i −0.809017 − 0.587785i 0 −0.587785 + 1.80902i −0.309017 − 0.951057i 0 −1.00000
3259.2 0.809017 − 0.587785i 0 0.309017 − 0.951057i −0.809017 − 0.587785i 0 0.587785 − 1.80902i −0.309017 − 0.951057i 0 −1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 379.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
40.e odd 2 1 CM by \(\Q(\sqrt{-10}) \)
11.c even 5 1 inner
15.d odd 2 1 inner
24.f even 2 1 inner
165.o odd 10 1 inner
264.w even 10 1 inner
440.bh 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.dh.d yes 8
3.b odd 2 1 3960.1.dh.c ✓ 8
5.b even 2 1 3960.1.dh.c ✓ 8
8.d odd 2 1 3960.1.dh.c ✓ 8
11.c even 5 1 inner 3960.1.dh.d yes 8
15.d odd 2 1 inner 3960.1.dh.d yes 8
24.f even 2 1 inner 3960.1.dh.d yes 8
33.h odd 10 1 3960.1.dh.c ✓ 8
40.e odd 2 1 CM 3960.1.dh.d yes 8
55.j even 10 1 3960.1.dh.c ✓ 8
88.l odd 10 1 3960.1.dh.c ✓ 8
120.m even 2 1 3960.1.dh.c ✓ 8
165.o odd 10 1 inner 3960.1.dh.d yes 8
264.w even 10 1 inner 3960.1.dh.d yes 8
440.bh odd 10 1 inner 3960.1.dh.d yes 8
1320.bx even 10 1 3960.1.dh.c ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3960.1.dh.c ✓ 8 3.b odd 2 1
3960.1.dh.c ✓ 8 5.b even 2 1
3960.1.dh.c ✓ 8 8.d odd 2 1
3960.1.dh.c ✓ 8 33.h odd 10 1
3960.1.dh.c ✓ 8 55.j even 10 1
3960.1.dh.c ✓ 8 88.l odd 10 1
3960.1.dh.c ✓ 8 120.m even 2 1
3960.1.dh.c ✓ 8 1320.bx even 10 1
3960.1.dh.d yes 8 1.a even 1 1 trivial
3960.1.dh.d yes 8 11.c even 5 1 inner
3960.1.dh.d yes 8 15.d odd 2 1 inner
3960.1.dh.d yes 8 24.f even 2 1 inner
3960.1.dh.d yes 8 40.e odd 2 1 CM
3960.1.dh.d yes 8 165.o odd 10 1 inner
3960.1.dh.d yes 8 264.w even 10 1 inner
3960.1.dh.d yes 8 440.bh odd 10 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3960, [\chi])\):

\( T_{7}^{8} + 5T_{7}^{6} + 10T_{7}^{4} + 25 \) Copy content Toggle raw display
\( T_{23}^{2} + T_{23} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + 5 T^{6} + \cdots + 25 \) Copy content Toggle raw display
$11$ \( T^{8} - T^{6} + T^{4} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{8} + 5 T^{6} + \cdots + 25 \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( (T^{4} - 2 T^{3} + 4 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + T - 1)^{4} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} + 10 T^{4} + \cdots + 25 \) Copy content Toggle raw display
$41$ \( T^{8} + 5 T^{6} + \cdots + 25 \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( (T^{4} - 2 T^{3} + 4 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 2 T^{3} + 4 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 10 T^{4} + \cdots + 25 \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( (T^{4} - 5 T^{2} + 5)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
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