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Properties

Label 1760.1.bw.a
Level $1760$
Weight $1$
Character orbit 1760.bw
Analytic conductor $0.878$
Analytic rank $0$
Dimension $16$
Projective image $D_{16}$
CM discriminant -55
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] = [1760,1,Mod(109,1760)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1760, base_ring=CyclotomicField(8)) chi = DirichletCharacter(H, H._module([0, 7, 4, 4])) N = Newforms(chi, 1, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1760.109"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1760 = 2^{5} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1760.bw (of order \(8\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.878354422234\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{8})\)
Coefficient field: \(\Q(\zeta_{32})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{16}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{16} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{32}^{3} q^{2} + \zeta_{32}^{6} q^{4} + \zeta_{32}^{10} q^{5} + ( - \zeta_{32}^{13} + \zeta_{32}^{11}) q^{7} - \zeta_{32}^{9} q^{8} - \zeta_{32}^{12} q^{9} - \zeta_{32}^{13} q^{10} - \zeta_{32}^{2} q^{11} + \cdots + \zeta_{32}^{14} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{14} - 16 q^{20} + 16 q^{59} + 16 q^{71}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1760\mathbb{Z}\right)^\times\).

\(n\) \(321\) \(991\) \(1057\) \(1541\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(-\zeta_{32}^{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
−0.555570 + 0.831470i
−0.831470 − 0.555570i
0.831470 + 0.555570i
0.555570 − 0.831470i
−0.555570 − 0.831470i
−0.831470 + 0.555570i
0.831470 − 0.555570i
0.555570 + 0.831470i
0.980785 − 0.195090i
−0.195090 − 0.980785i
0.195090 + 0.980785i
−0.980785 + 0.195090i
0.980785 + 0.195090i
−0.195090 + 0.980785i
0.195090 − 0.980785i
−0.980785 − 0.195090i
−0.980785 − 0.195090i 0 0.923880 + 0.382683i −0.923880 + 0.382683i 0 1.17588 − 1.17588i −0.831470 − 0.555570i −0.707107 − 0.707107i 0.980785 − 0.195090i
109.2 −0.195090 + 0.980785i 0 −0.923880 − 0.382683i 0.923880 − 0.382683i 0 −0.785695 + 0.785695i 0.555570 − 0.831470i −0.707107 − 0.707107i 0.195090 + 0.980785i
109.3 0.195090 − 0.980785i 0 −0.923880 − 0.382683i 0.923880 − 0.382683i 0 0.785695 − 0.785695i −0.555570 + 0.831470i −0.707107 − 0.707107i −0.195090 − 0.980785i
109.4 0.980785 + 0.195090i 0 0.923880 + 0.382683i −0.923880 + 0.382683i 0 −1.17588 + 1.17588i 0.831470 + 0.555570i −0.707107 − 0.707107i −0.980785 + 0.195090i
549.1 −0.980785 + 0.195090i 0 0.923880 − 0.382683i −0.923880 − 0.382683i 0 1.17588 + 1.17588i −0.831470 + 0.555570i −0.707107 + 0.707107i 0.980785 + 0.195090i
549.2 −0.195090 − 0.980785i 0 −0.923880 + 0.382683i 0.923880 + 0.382683i 0 −0.785695 − 0.785695i 0.555570 + 0.831470i −0.707107 + 0.707107i 0.195090 − 0.980785i
549.3 0.195090 + 0.980785i 0 −0.923880 + 0.382683i 0.923880 + 0.382683i 0 0.785695 + 0.785695i −0.555570 − 0.831470i −0.707107 + 0.707107i −0.195090 + 0.980785i
549.4 0.980785 − 0.195090i 0 0.923880 − 0.382683i −0.923880 − 0.382683i 0 −1.17588 − 1.17588i 0.831470 − 0.555570i −0.707107 + 0.707107i −0.980785 − 0.195090i
989.1 −0.831470 + 0.555570i 0 0.382683 − 0.923880i −0.382683 − 0.923880i 0 0.275899 − 0.275899i 0.195090 + 0.980785i 0.707107 + 0.707107i 0.831470 + 0.555570i
989.2 −0.555570 − 0.831470i 0 −0.382683 + 0.923880i 0.382683 + 0.923880i 0 1.38704 − 1.38704i 0.980785 − 0.195090i 0.707107 + 0.707107i 0.555570 − 0.831470i
989.3 0.555570 + 0.831470i 0 −0.382683 + 0.923880i 0.382683 + 0.923880i 0 −1.38704 + 1.38704i −0.980785 + 0.195090i 0.707107 + 0.707107i −0.555570 + 0.831470i
989.4 0.831470 − 0.555570i 0 0.382683 − 0.923880i −0.382683 − 0.923880i 0 −0.275899 + 0.275899i −0.195090 − 0.980785i 0.707107 + 0.707107i −0.831470 − 0.555570i
1429.1 −0.831470 − 0.555570i 0 0.382683 + 0.923880i −0.382683 + 0.923880i 0 0.275899 + 0.275899i 0.195090 − 0.980785i 0.707107 − 0.707107i 0.831470 − 0.555570i
1429.2 −0.555570 + 0.831470i 0 −0.382683 − 0.923880i 0.382683 − 0.923880i 0 1.38704 + 1.38704i 0.980785 + 0.195090i 0.707107 − 0.707107i 0.555570 + 0.831470i
1429.3 0.555570 − 0.831470i 0 −0.382683 − 0.923880i 0.382683 − 0.923880i 0 −1.38704 − 1.38704i −0.980785 − 0.195090i 0.707107 − 0.707107i −0.555570 − 0.831470i
1429.4 0.831470 + 0.555570i 0 0.382683 + 0.923880i −0.382683 + 0.923880i 0 −0.275899 − 0.275899i −0.195090 + 0.980785i 0.707107 − 0.707107i −0.831470 + 0.555570i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 109.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
55.d odd 2 1 CM by \(\Q(\sqrt{-55}) \)
5.b even 2 1 inner
11.b odd 2 1 inner
32.g even 8 1 inner
160.z even 8 1 inner
352.o odd 8 1 inner
1760.bw odd 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1760.1.bw.a ✓ 16
5.b even 2 1 inner 1760.1.bw.a ✓ 16
11.b odd 2 1 inner 1760.1.bw.a ✓ 16
32.g even 8 1 inner 1760.1.bw.a ✓ 16
55.d odd 2 1 CM 1760.1.bw.a ✓ 16
160.z even 8 1 inner 1760.1.bw.a ✓ 16
352.o odd 8 1 inner 1760.1.bw.a ✓ 16
1760.bw odd 8 1 inner 1760.1.bw.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1760.1.bw.a ✓ 16 1.a even 1 1 trivial
1760.1.bw.a ✓ 16 5.b even 2 1 inner
1760.1.bw.a ✓ 16 11.b odd 2 1 inner
1760.1.bw.a ✓ 16 32.g even 8 1 inner
1760.1.bw.a ✓ 16 55.d odd 2 1 CM
1760.1.bw.a ✓ 16 160.z even 8 1 inner
1760.1.bw.a ✓ 16 352.o odd 8 1 inner
1760.1.bw.a ✓ 16 1760.bw odd 8 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1760, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 1 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} + 24 T^{12} + \cdots + 4 \) Copy content Toggle raw display
$11$ \( (T^{8} + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + 16 T^{10} + \cdots + 4 \) Copy content Toggle raw display
$17$ \( (T^{8} + 8 T^{6} + 20 T^{4} + \cdots + 2)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} \) Copy content Toggle raw display
$23$ \( T^{16} \) Copy content Toggle raw display
$29$ \( T^{16} \) Copy content Toggle raw display
$31$ \( (T^{4} - 4 T^{2} + 2)^{4} \) Copy content Toggle raw display
$37$ \( T^{16} \) Copy content Toggle raw display
$41$ \( T^{16} \) Copy content Toggle raw display
$43$ \( T^{16} - 16 T^{10} + \cdots + 4 \) Copy content Toggle raw display
$47$ \( T^{16} \) Copy content Toggle raw display
$53$ \( T^{16} \) Copy content Toggle raw display
$59$ \( (T^{4} - 4 T^{3} + 6 T^{2} + \cdots + 2)^{4} \) Copy content Toggle raw display
$61$ \( T^{16} \) Copy content Toggle raw display
$67$ \( T^{16} \) Copy content Toggle raw display
$71$ \( (T^{2} - 2 T + 2)^{8} \) Copy content Toggle raw display
$73$ \( T^{16} + 24 T^{12} + \cdots + 4 \) Copy content Toggle raw display
$79$ \( T^{16} \) Copy content Toggle raw display
$83$ \( T^{16} + 16 T^{10} + \cdots + 4 \) Copy content Toggle raw display
$89$ \( (T^{8} + 12 T^{4} + 4)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} \) Copy content Toggle raw display
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