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Properties

Label 1960.1.bu.a
Level $1960$
Weight $1$
Character orbit 1960.bu
Analytic conductor $0.978$
Analytic rank $0$
Dimension $16$
Projective image $D_{8}$
CM discriminant -56
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1960,1,Mod(373,1960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1960, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 6, 9, 4]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1960.373");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1960 = 2^{3} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1960.bu (of order \(12\), degree \(4\), 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: \(0.978167424761\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\Q(\zeta_{48})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{8} + 1 \) 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 8.2.21952000000.3

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{48}^{2} q^{2} + ( - \zeta_{48}^{19} - \zeta_{48}) q^{3} + \zeta_{48}^{4} q^{4} - \zeta_{48}^{17} q^{5} + (\zeta_{48}^{21} + \zeta_{48}^{3}) q^{6} - \zeta_{48}^{6} q^{8} + (\zeta_{48}^{20} + \cdots + \zeta_{48}^{2}) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{48}^{2} q^{2} + ( - \zeta_{48}^{19} - \zeta_{48}) q^{3} + \zeta_{48}^{4} q^{4} - \zeta_{48}^{17} q^{5} + (\zeta_{48}^{21} + \zeta_{48}^{3}) q^{6} - \zeta_{48}^{6} q^{8} + (\zeta_{48}^{20} + \cdots + \zeta_{48}^{2}) q^{9} + \cdots + (\zeta_{48}^{11} - \zeta_{48}^{5}) q^{96} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{16} - 8 q^{18} - 8 q^{23} - 16 q^{36} - 16 q^{57} + 8 q^{60} + 8 q^{65} - 8 q^{72} - 16 q^{78} + 8 q^{81} + 16 q^{92}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(981\) \(1081\) \(1177\) \(1471\)
\(\chi(n)\) \(-1\) \(-\zeta_{48}^{8}\) \(-\zeta_{48}^{12}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
373.1
0.793353 + 0.608761i
−0.793353 − 0.608761i
0.608761 − 0.793353i
−0.608761 + 0.793353i
0.793353 − 0.608761i
−0.793353 + 0.608761i
0.608761 + 0.793353i
−0.608761 − 0.793353i
0.991445 − 0.130526i
−0.991445 + 0.130526i
−0.130526 − 0.991445i
0.130526 + 0.991445i
0.991445 + 0.130526i
−0.991445 − 0.130526i
−0.130526 + 0.991445i
0.130526 − 0.991445i
−0.258819 − 0.965926i −1.78480 − 0.478235i −0.866025 + 0.500000i −0.130526 + 0.991445i 1.84776i 0 0.707107 + 0.707107i 2.09077 + 1.20711i 0.991445 − 0.130526i
373.2 −0.258819 − 0.965926i 1.78480 + 0.478235i −0.866025 + 0.500000i 0.130526 − 0.991445i − 1.84776i 0 0.707107 + 0.707107i 2.09077 + 1.20711i −0.991445 + 0.130526i
373.3 0.258819 + 0.965926i −0.739288 − 0.198092i −0.866025 + 0.500000i 0.991445 + 0.130526i − 0.765367i 0 −0.707107 − 0.707107i −0.358719 − 0.207107i 0.130526 + 0.991445i
373.4 0.258819 + 0.965926i 0.739288 + 0.198092i −0.866025 + 0.500000i −0.991445 − 0.130526i 0.765367i 0 −0.707107 − 0.707107i −0.358719 − 0.207107i −0.130526 − 0.991445i
557.1 −0.258819 + 0.965926i −1.78480 + 0.478235i −0.866025 − 0.500000i −0.130526 − 0.991445i − 1.84776i 0 0.707107 − 0.707107i 2.09077 − 1.20711i 0.991445 + 0.130526i
557.2 −0.258819 + 0.965926i 1.78480 − 0.478235i −0.866025 − 0.500000i 0.130526 + 0.991445i 1.84776i 0 0.707107 − 0.707107i 2.09077 − 1.20711i −0.991445 − 0.130526i
557.3 0.258819 − 0.965926i −0.739288 + 0.198092i −0.866025 − 0.500000i 0.991445 − 0.130526i 0.765367i 0 −0.707107 + 0.707107i −0.358719 + 0.207107i 0.130526 − 0.991445i
557.4 0.258819 − 0.965926i 0.739288 − 0.198092i −0.866025 − 0.500000i −0.991445 + 0.130526i − 0.765367i 0 −0.707107 + 0.707107i −0.358719 + 0.207107i −0.130526 + 0.991445i
1157.1 −0.965926 + 0.258819i −0.198092 + 0.739288i 0.866025 − 0.500000i 0.608761 + 0.793353i − 0.765367i 0 −0.707107 + 0.707107i 0.358719 + 0.207107i −0.793353 − 0.608761i
1157.2 −0.965926 + 0.258819i 0.198092 − 0.739288i 0.866025 − 0.500000i −0.608761 − 0.793353i 0.765367i 0 −0.707107 + 0.707107i 0.358719 + 0.207107i 0.793353 + 0.608761i
1157.3 0.965926 − 0.258819i −0.478235 + 1.78480i 0.866025 − 0.500000i 0.793353 − 0.608761i 1.84776i 0 0.707107 − 0.707107i −2.09077 − 1.20711i 0.608761 − 0.793353i
1157.4 0.965926 − 0.258819i 0.478235 − 1.78480i 0.866025 − 0.500000i −0.793353 + 0.608761i − 1.84776i 0 0.707107 − 0.707107i −2.09077 − 1.20711i −0.608761 + 0.793353i
1733.1 −0.965926 − 0.258819i −0.198092 − 0.739288i 0.866025 + 0.500000i 0.608761 − 0.793353i 0.765367i 0 −0.707107 − 0.707107i 0.358719 − 0.207107i −0.793353 + 0.608761i
1733.2 −0.965926 − 0.258819i 0.198092 + 0.739288i 0.866025 + 0.500000i −0.608761 + 0.793353i − 0.765367i 0 −0.707107 − 0.707107i 0.358719 − 0.207107i 0.793353 − 0.608761i
1733.3 0.965926 + 0.258819i −0.478235 − 1.78480i 0.866025 + 0.500000i 0.793353 + 0.608761i − 1.84776i 0 0.707107 + 0.707107i −2.09077 + 1.20711i 0.608761 + 0.793353i
1733.4 0.965926 + 0.258819i 0.478235 + 1.78480i 0.866025 + 0.500000i −0.793353 − 0.608761i 1.84776i 0 0.707107 + 0.707107i −2.09077 + 1.20711i −0.608761 − 0.793353i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 373.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
56.h odd 2 1 CM by \(\Q(\sqrt{-14}) \)
5.c odd 4 1 inner
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
8.b even 2 1 inner
35.f even 4 1 inner
35.k even 12 1 inner
35.l odd 12 1 inner
40.i odd 4 1 inner
56.j odd 6 1 inner
56.p even 6 1 inner
280.s even 4 1 inner
280.bt odd 12 1 inner
280.bv even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1960.1.bu.a 16
5.c odd 4 1 inner 1960.1.bu.a 16
7.b odd 2 1 inner 1960.1.bu.a 16
7.c even 3 1 1960.1.u.a ✓ 8
7.c even 3 1 inner 1960.1.bu.a 16
7.d odd 6 1 1960.1.u.a ✓ 8
7.d odd 6 1 inner 1960.1.bu.a 16
8.b even 2 1 inner 1960.1.bu.a 16
35.f even 4 1 inner 1960.1.bu.a 16
35.k even 12 1 1960.1.u.a ✓ 8
35.k even 12 1 inner 1960.1.bu.a 16
35.l odd 12 1 1960.1.u.a ✓ 8
35.l odd 12 1 inner 1960.1.bu.a 16
40.i odd 4 1 inner 1960.1.bu.a 16
56.h odd 2 1 CM 1960.1.bu.a 16
56.j odd 6 1 1960.1.u.a ✓ 8
56.j odd 6 1 inner 1960.1.bu.a 16
56.p even 6 1 1960.1.u.a ✓ 8
56.p even 6 1 inner 1960.1.bu.a 16
280.s even 4 1 inner 1960.1.bu.a 16
280.bt odd 12 1 1960.1.u.a ✓ 8
280.bt odd 12 1 inner 1960.1.bu.a 16
280.bv even 12 1 1960.1.u.a ✓ 8
280.bv even 12 1 inner 1960.1.bu.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1960.1.u.a ✓ 8 7.c even 3 1
1960.1.u.a ✓ 8 7.d odd 6 1
1960.1.u.a ✓ 8 35.k even 12 1
1960.1.u.a ✓ 8 35.l odd 12 1
1960.1.u.a ✓ 8 56.j odd 6 1
1960.1.u.a ✓ 8 56.p even 6 1
1960.1.u.a ✓ 8 280.bt odd 12 1
1960.1.u.a ✓ 8 280.bv even 12 1
1960.1.bu.a 16 1.a even 1 1 trivial
1960.1.bu.a 16 5.c odd 4 1 inner
1960.1.bu.a 16 7.b odd 2 1 inner
1960.1.bu.a 16 7.c even 3 1 inner
1960.1.bu.a 16 7.d odd 6 1 inner
1960.1.bu.a 16 8.b even 2 1 inner
1960.1.bu.a 16 35.f even 4 1 inner
1960.1.bu.a 16 35.k even 12 1 inner
1960.1.bu.a 16 35.l odd 12 1 inner
1960.1.bu.a 16 40.i odd 4 1 inner
1960.1.bu.a 16 56.h odd 2 1 CM
1960.1.bu.a 16 56.j odd 6 1 inner
1960.1.bu.a 16 56.p even 6 1 inner
1960.1.bu.a 16 280.s even 4 1 inner
1960.1.bu.a 16 280.bt odd 12 1 inner
1960.1.bu.a 16 280.bv even 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1960, [\chi])\).

Hecke characteristic polynomials

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