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Properties

Label 2856.1.dg.b
Level $2856$
Weight $1$
Character orbit 2856.dg
Analytic conductor $1.425$
Analytic rank $0$
Dimension $8$
Projective image $D_{8}$
CM discriminant -168
Inner twists $8$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2856,1,Mod(83,2856)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2856, base_ring=CyclotomicField(8)) chi = DirichletCharacter(H, H._module([4, 4, 4, 4, 3])) 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("2856.83"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Level: \( N \) \(=\) \( 2856 = 2^{3} \cdot 3 \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2856.dg (of order \(8\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-8,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(18)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.42532967608\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{8})\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{8}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{8} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{16}^{6} q^{2} + \zeta_{16}^{5} q^{3} - \zeta_{16}^{4} q^{4} - \zeta_{16}^{3} q^{6} - \zeta_{16}^{7} q^{7} + \zeta_{16}^{2} q^{8} - \zeta_{16}^{2} q^{9} + \zeta_{16} q^{12} + ( - \zeta_{16}^{7} - \zeta_{16}) q^{13} + \cdots + \zeta_{16}^{4} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{16} + 8 q^{18} - 8 q^{29} - 8 q^{43} + 8 q^{46} - 8 q^{50} + 8 q^{53} + 8 q^{58} - 8 q^{71} + 8 q^{84} - 8 q^{91} - 8 q^{92}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2856\mathbb{Z}\right)^\times\).

\(n\) \(409\) \(953\) \(1429\) \(2143\) \(2689\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(-1\) \(-\zeta_{16}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
83.1
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
0.382683 + 0.923880i
−0.707107 − 0.707107i −0.382683 − 0.923880i 1.00000i 0 −0.382683 + 0.923880i 0.923880 + 0.382683i 0.707107 − 0.707107i −0.707107 + 0.707107i 0
83.2 −0.707107 − 0.707107i 0.382683 + 0.923880i 1.00000i 0 0.382683 − 0.923880i −0.923880 − 0.382683i 0.707107 − 0.707107i −0.707107 + 0.707107i 0
587.1 0.707107 − 0.707107i −0.923880 − 0.382683i − 1.00000i 0 −0.923880 + 0.382683i −0.382683 − 0.923880i −0.707107 − 0.707107i 0.707107 + 0.707107i 0
587.2 0.707107 − 0.707107i 0.923880 + 0.382683i − 1.00000i 0 0.923880 − 0.382683i 0.382683 + 0.923880i −0.707107 − 0.707107i 0.707107 + 0.707107i 0
2099.1 −0.707107 + 0.707107i −0.382683 + 0.923880i − 1.00000i 0 −0.382683 − 0.923880i 0.923880 − 0.382683i 0.707107 + 0.707107i −0.707107 − 0.707107i 0
2099.2 −0.707107 + 0.707107i 0.382683 − 0.923880i − 1.00000i 0 0.382683 + 0.923880i −0.923880 + 0.382683i 0.707107 + 0.707107i −0.707107 − 0.707107i 0
2603.1 0.707107 + 0.707107i −0.923880 + 0.382683i 1.00000i 0 −0.923880 − 0.382683i −0.382683 + 0.923880i −0.707107 + 0.707107i 0.707107 − 0.707107i 0
2603.2 0.707107 + 0.707107i 0.923880 − 0.382683i 1.00000i 0 0.923880 + 0.382683i 0.382683 − 0.923880i −0.707107 + 0.707107i 0.707107 − 0.707107i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 83.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
168.e odd 2 1 CM by \(\Q(\sqrt{-42}) \)
7.b odd 2 1 inner
17.d even 8 1 inner
24.f even 2 1 inner
119.l odd 8 1 inner
408.bd even 8 1 inner
2856.dg odd 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2856.1.dg.b yes 8
3.b odd 2 1 2856.1.dg.a ✓ 8
7.b odd 2 1 inner 2856.1.dg.b yes 8
8.d odd 2 1 2856.1.dg.a ✓ 8
17.d even 8 1 inner 2856.1.dg.b yes 8
21.c even 2 1 2856.1.dg.a ✓ 8
24.f even 2 1 inner 2856.1.dg.b yes 8
51.g odd 8 1 2856.1.dg.a ✓ 8
56.e even 2 1 2856.1.dg.a ✓ 8
119.l odd 8 1 inner 2856.1.dg.b yes 8
136.p odd 8 1 2856.1.dg.a ✓ 8
168.e odd 2 1 CM 2856.1.dg.b yes 8
357.w even 8 1 2856.1.dg.a ✓ 8
408.bd even 8 1 inner 2856.1.dg.b yes 8
952.bo even 8 1 2856.1.dg.a ✓ 8
2856.dg odd 8 1 inner 2856.1.dg.b yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2856.1.dg.a ✓ 8 3.b odd 2 1
2856.1.dg.a ✓ 8 8.d odd 2 1
2856.1.dg.a ✓ 8 21.c even 2 1
2856.1.dg.a ✓ 8 51.g odd 8 1
2856.1.dg.a ✓ 8 56.e even 2 1
2856.1.dg.a ✓ 8 136.p odd 8 1
2856.1.dg.a ✓ 8 357.w even 8 1
2856.1.dg.a ✓ 8 952.bo even 8 1
2856.1.dg.b yes 8 1.a even 1 1 trivial
2856.1.dg.b yes 8 7.b odd 2 1 inner
2856.1.dg.b yes 8 17.d even 8 1 inner
2856.1.dg.b yes 8 24.f even 2 1 inner
2856.1.dg.b yes 8 119.l odd 8 1 inner
2856.1.dg.b yes 8 168.e odd 2 1 CM
2856.1.dg.b yes 8 408.bd even 8 1 inner
2856.1.dg.b yes 8 2856.dg odd 8 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{23}^{4} + 2T_{23}^{2} - 4T_{23} + 2 \) acting on \(S_{1}^{\mathrm{new}}(2856, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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