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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3552,1,Mod(1553,3552)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3552.1553"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3552, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1, 1])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 3552 = 2^{5} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3552.i (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,0,0,0,-8,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.77267892487\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 888)
Projective image: \(D_{8}\)
Projective field: Galois closure of 8.2.207267213312.3

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{16}^{4} q^{3} + ( - \zeta_{16}^{5} - \zeta_{16}^{3}) q^{5} + ( - \zeta_{16}^{6} + \zeta_{16}^{2}) q^{7} - q^{9} + (\zeta_{16}^{7} - \zeta_{16}) q^{15} + (\zeta_{16}^{7} - \zeta_{16}) q^{17} + \cdots + ( - \zeta_{16}^{5} + \zeta_{16}^{3}) q^{89} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{9} - 8 q^{25} + 8 q^{49} + 8 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3552\mathbb{Z}\right)^\times\).

\(n\) \(223\) \(2369\) \(3073\) \(3109\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1553.1
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
0.923880 − 0.382683i
0 − 1.00000i 0 − 1.84776i 0 1.41421 0 −1.00000 0
1553.2 0 − 1.00000i 0 − 0.765367i 0 −1.41421 0 −1.00000 0
1553.3 0 − 1.00000i 0 0.765367i 0 −1.41421 0 −1.00000 0
1553.4 0 − 1.00000i 0 1.84776i 0 1.41421 0 −1.00000 0
1553.5 0 1.00000i 0 − 1.84776i 0 1.41421 0 −1.00000 0
1553.6 0 1.00000i 0 − 0.765367i 0 −1.41421 0 −1.00000 0
1553.7 0 1.00000i 0 0.765367i 0 −1.41421 0 −1.00000 0
1553.8 0 1.00000i 0 1.84776i 0 1.41421 0 −1.00000 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1553.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
111.d odd 2 1 CM by \(\Q(\sqrt{-111}) \)
3.b odd 2 1 inner
8.b even 2 1 inner
24.h odd 2 1 inner
37.b even 2 1 inner
296.e even 2 1 inner
888.i odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3552.1.i.e 8
3.b odd 2 1 inner 3552.1.i.e 8
4.b odd 2 1 888.1.i.e ✓ 8
8.b even 2 1 inner 3552.1.i.e 8
8.d odd 2 1 888.1.i.e ✓ 8
12.b even 2 1 888.1.i.e ✓ 8
24.f even 2 1 888.1.i.e ✓ 8
24.h odd 2 1 inner 3552.1.i.e 8
37.b even 2 1 inner 3552.1.i.e 8
111.d odd 2 1 CM 3552.1.i.e 8
148.b odd 2 1 888.1.i.e ✓ 8
296.e even 2 1 inner 3552.1.i.e 8
296.h odd 2 1 888.1.i.e ✓ 8
444.g even 2 1 888.1.i.e ✓ 8
888.c even 2 1 888.1.i.e ✓ 8
888.i odd 2 1 inner 3552.1.i.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
888.1.i.e ✓ 8 4.b odd 2 1
888.1.i.e ✓ 8 8.d odd 2 1
888.1.i.e ✓ 8 12.b even 2 1
888.1.i.e ✓ 8 24.f even 2 1
888.1.i.e ✓ 8 148.b odd 2 1
888.1.i.e ✓ 8 296.h odd 2 1
888.1.i.e ✓ 8 444.g even 2 1
888.1.i.e ✓ 8 888.c even 2 1
3552.1.i.e 8 1.a even 1 1 trivial
3552.1.i.e 8 3.b odd 2 1 inner
3552.1.i.e 8 8.b even 2 1 inner
3552.1.i.e 8 24.h odd 2 1 inner
3552.1.i.e 8 37.b even 2 1 inner
3552.1.i.e 8 111.d odd 2 1 CM
3552.1.i.e 8 296.e even 2 1 inner
3552.1.i.e 8 888.i odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3552, [\chi])\):

\( T_{5}^{4} + 4T_{5}^{2} + 2 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display

Hecke characteristic polynomials

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