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Properties

Label 880.1.x.a
Level $880$
Weight $1$
Character orbit 880.x
Analytic conductor $0.439$
Analytic rank $0$
Dimension $8$
Projective image $D_{8}$
CM discriminant -55
Inner twists $8$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [880,1,Mod(109,880)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(880, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 3, 2, 2])) 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("880.109"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Level: \( N \) \(=\) \( 880 = 2^{4} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 880.x (of order \(4\), degree \(2\), 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: \(0.439177211117\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{8}\)
Projective field: Galois closure of 8.0.7676100608000.1

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{16}^{5} q^{2} - \zeta_{16}^{2} q^{4} + \zeta_{16}^{6} q^{5} + ( - \zeta_{16}^{5} - \zeta_{16}^{3}) q^{7} - \zeta_{16}^{7} q^{8} + \zeta_{16}^{4} q^{9} - \zeta_{16}^{3} q^{10} - \zeta_{16}^{6} q^{11} + \cdots + \zeta_{16}^{2} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{14} + 8 q^{20} - 8 q^{44} - 8 q^{49} + 8 q^{59} - 8 q^{70} - 8 q^{81} - 8 q^{86} - 8 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/880\mathbb{Z}\right)^\times\).

\(n\) \(111\) \(177\) \(321\) \(661\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-\zeta_{16}^{4}\)

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
−0.382683 + 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
0.382683 + 0.923880i
−0.923880 − 0.382683i 0 0.707107 + 0.707107i 0.707107 − 0.707107i 0 0.765367i −0.382683 − 0.923880i 1.00000i −0.923880 + 0.382683i
109.2 −0.382683 + 0.923880i 0 −0.707107 − 0.707107i −0.707107 + 0.707107i 0 − 1.84776i 0.923880 − 0.382683i 1.00000i −0.382683 − 0.923880i
109.3 0.382683 − 0.923880i 0 −0.707107 − 0.707107i −0.707107 + 0.707107i 0 1.84776i −0.923880 + 0.382683i 1.00000i 0.382683 + 0.923880i
109.4 0.923880 + 0.382683i 0 0.707107 + 0.707107i 0.707107 − 0.707107i 0 − 0.765367i 0.382683 + 0.923880i 1.00000i 0.923880 − 0.382683i
549.1 −0.923880 + 0.382683i 0 0.707107 − 0.707107i 0.707107 + 0.707107i 0 − 0.765367i −0.382683 + 0.923880i − 1.00000i −0.923880 − 0.382683i
549.2 −0.382683 − 0.923880i 0 −0.707107 + 0.707107i −0.707107 − 0.707107i 0 1.84776i 0.923880 + 0.382683i − 1.00000i −0.382683 + 0.923880i
549.3 0.382683 + 0.923880i 0 −0.707107 + 0.707107i −0.707107 − 0.707107i 0 − 1.84776i −0.923880 − 0.382683i − 1.00000i 0.382683 − 0.923880i
549.4 0.923880 − 0.382683i 0 0.707107 − 0.707107i 0.707107 + 0.707107i 0 0.765367i 0.382683 − 0.923880i − 1.00000i 0.923880 + 0.382683i
\(n\): 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
16.e even 4 1 inner
80.q even 4 1 inner
176.l odd 4 1 inner
880.x odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 880.1.x.a ✓ 8
4.b odd 2 1 3520.1.x.a 8
5.b even 2 1 inner 880.1.x.a ✓ 8
11.b odd 2 1 inner 880.1.x.a ✓ 8
16.e even 4 1 inner 880.1.x.a ✓ 8
16.f odd 4 1 3520.1.x.a 8
20.d odd 2 1 3520.1.x.a 8
44.c even 2 1 3520.1.x.a 8
55.d odd 2 1 CM 880.1.x.a ✓ 8
80.k odd 4 1 3520.1.x.a 8
80.q even 4 1 inner 880.1.x.a ✓ 8
176.i even 4 1 3520.1.x.a 8
176.l odd 4 1 inner 880.1.x.a ✓ 8
220.g even 2 1 3520.1.x.a 8
880.x odd 4 1 inner 880.1.x.a ✓ 8
880.bi even 4 1 3520.1.x.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
880.1.x.a ✓ 8 1.a even 1 1 trivial
880.1.x.a ✓ 8 5.b even 2 1 inner
880.1.x.a ✓ 8 11.b odd 2 1 inner
880.1.x.a ✓ 8 16.e even 4 1 inner
880.1.x.a ✓ 8 55.d odd 2 1 CM
880.1.x.a ✓ 8 80.q even 4 1 inner
880.1.x.a ✓ 8 176.l odd 4 1 inner
880.1.x.a ✓ 8 880.x odd 4 1 inner
3520.1.x.a 8 4.b odd 2 1
3520.1.x.a 8 16.f odd 4 1
3520.1.x.a 8 20.d odd 2 1
3520.1.x.a 8 44.c even 2 1
3520.1.x.a 8 80.k odd 4 1
3520.1.x.a 8 176.i even 4 1
3520.1.x.a 8 220.g even 2 1
3520.1.x.a 8 880.bi even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(880, [\chi])\).

Hecke characteristic polynomials

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