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Properties

Label 3060.1.ds.c
Level $3060$
Weight $1$
Character orbit 3060.ds
Analytic conductor $1.527$
Analytic rank $0$
Dimension $8$
Projective image $D_{16}$
CM discriminant -4
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3060,1,Mod(343,3060)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3060, base_ring=CyclotomicField(16)) chi = DirichletCharacter(H, H._module([8, 0, 12, 1])) 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("3060.343"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Level: \( N \) \(=\) \( 3060 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3060.ds (of order \(16\), degree \(8\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(26)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.52713893866\)
Analytic rank: \(0\)
Dimension: \(8\)
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: \(\Z[a_1, a_2]\)
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.

\(f(q)\) \(=\) \( q - \zeta_{16}^{5} q^{2} - \zeta_{16}^{2} q^{4} + \zeta_{16}^{3} q^{5} + \zeta_{16}^{7} q^{8} + q^{10} + (\zeta_{16}^{5} + \zeta_{16}^{3}) q^{13} + \zeta_{16}^{4} q^{16} + \zeta_{16}^{6} q^{17} - \zeta_{16}^{5} q^{20} + \cdots - \zeta_{16}^{4} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{10} + 8 q^{26} - 8 q^{29} - 8 q^{53} - 8 q^{65} + 8 q^{68} + 8 q^{73}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3060\mathbb{Z}\right)^\times\).

\(n\) \(1261\) \(1361\) \(1531\) \(1837\)
\(\chi(n)\) \(-\zeta_{16}\) \(1\) \(-1\) \(-\zeta_{16}^{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
343.1
−0.923880 − 0.382683i
0.923880 − 0.382683i
0.382683 + 0.923880i
−0.382683 − 0.923880i
−0.382683 + 0.923880i
0.382683 − 0.923880i
−0.923880 + 0.382683i
0.923880 + 0.382683i
−0.382683 + 0.923880i 0 −0.707107 − 0.707107i −0.382683 − 0.923880i 0 0 0.923880 − 0.382683i 0 1.00000
487.1 0.382683 + 0.923880i 0 −0.707107 + 0.707107i 0.382683 − 0.923880i 0 0 −0.923880 − 0.382683i 0 1.00000
1027.1 −0.923880 + 0.382683i 0 0.707107 − 0.707107i −0.923880 − 0.382683i 0 0 −0.382683 + 0.923880i 0 1.00000
1387.1 0.923880 − 0.382683i 0 0.707107 − 0.707107i 0.923880 + 0.382683i 0 0 0.382683 − 0.923880i 0 1.00000
1423.1 0.923880 + 0.382683i 0 0.707107 + 0.707107i 0.923880 − 0.382683i 0 0 0.382683 + 0.923880i 0 1.00000
1603.1 −0.923880 − 0.382683i 0 0.707107 + 0.707107i −0.923880 + 0.382683i 0 0 −0.382683 − 0.923880i 0 1.00000
1927.1 −0.382683 − 0.923880i 0 −0.707107 + 0.707107i −0.382683 + 0.923880i 0 0 0.923880 + 0.382683i 0 1.00000
2683.1 0.382683 − 0.923880i 0 −0.707107 − 0.707107i 0.382683 + 0.923880i 0 0 −0.923880 + 0.382683i 0 1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 343.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
85.o even 16 1 inner
340.bc odd 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3060.1.ds.c yes 8
3.b odd 2 1 3060.1.ds.b ✓ 8
4.b odd 2 1 CM 3060.1.ds.c yes 8
5.c odd 4 1 3060.1.eg.c yes 8
12.b even 2 1 3060.1.ds.b ✓ 8
15.e even 4 1 3060.1.eg.b yes 8
17.e odd 16 1 3060.1.eg.c yes 8
20.e even 4 1 3060.1.eg.c yes 8
51.i even 16 1 3060.1.eg.b yes 8
60.l odd 4 1 3060.1.eg.b yes 8
68.i even 16 1 3060.1.eg.c yes 8
85.o even 16 1 inner 3060.1.ds.c yes 8
204.t odd 16 1 3060.1.eg.b yes 8
255.bc odd 16 1 3060.1.ds.b ✓ 8
340.bc odd 16 1 inner 3060.1.ds.c yes 8
1020.ch even 16 1 3060.1.ds.b ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3060.1.ds.b ✓ 8 3.b odd 2 1
3060.1.ds.b ✓ 8 12.b even 2 1
3060.1.ds.b ✓ 8 255.bc odd 16 1
3060.1.ds.b ✓ 8 1020.ch even 16 1
3060.1.ds.c yes 8 1.a even 1 1 trivial
3060.1.ds.c yes 8 4.b odd 2 1 CM
3060.1.ds.c yes 8 85.o even 16 1 inner
3060.1.ds.c yes 8 340.bc odd 16 1 inner
3060.1.eg.b yes 8 15.e even 4 1
3060.1.eg.b yes 8 51.i even 16 1
3060.1.eg.b yes 8 60.l odd 4 1
3060.1.eg.b yes 8 204.t odd 16 1
3060.1.eg.c yes 8 5.c odd 4 1
3060.1.eg.c yes 8 17.e odd 16 1
3060.1.eg.c yes 8 20.e even 4 1
3060.1.eg.c yes 8 68.i even 16 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{29}^{8} + 8T_{29}^{7} + 28T_{29}^{6} + 56T_{29}^{5} + 70T_{29}^{4} + 56T_{29}^{3} + 28T_{29}^{2} + 8T_{29} + 2 \) acting on \(S_{1}^{\mathrm{new}}(3060, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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