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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2304,1,Mod(127,2304)] 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("2304.127"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2304, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 0])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2304.b (of order \(2\), degree \(1\), not 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: \(1.14984578911\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 144)
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(\zeta_{12})\)
Artin image: $D_4:C_2$
Artin field: Galois closure of 8.0.3057647616.1

$q$-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.

\(f(q)\) \(=\) \( q - 2 i q^{13} + q^{25} - 2 i q^{37} + q^{49} + 2 i q^{61} + 2 q^{73} - 2 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{25} + 2 q^{49} + 4 q^{73} - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times\).

\(n\) \(1279\) \(1793\) \(2053\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
1.00000i
− 1.00000i
0 0 0 0 0 0 0 0 0
127.2 0 0 0 0 0 0 0 0 0
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
12.b even 2 1 RM by \(\Q(\sqrt{3}) \)
8.b even 2 1 inner
8.d odd 2 1 inner
24.f even 2 1 inner
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2304.1.b.a 2
3.b odd 2 1 CM 2304.1.b.a 2
4.b odd 2 1 CM 2304.1.b.a 2
8.b even 2 1 inner 2304.1.b.a 2
8.d odd 2 1 inner 2304.1.b.a 2
12.b even 2 1 RM 2304.1.b.a 2
16.e even 4 1 144.1.g.a ✓ 1
16.e even 4 1 576.1.g.a 1
16.f odd 4 1 144.1.g.a ✓ 1
16.f odd 4 1 576.1.g.a 1
24.f even 2 1 inner 2304.1.b.a 2
24.h odd 2 1 inner 2304.1.b.a 2
48.i odd 4 1 144.1.g.a ✓ 1
48.i odd 4 1 576.1.g.a 1
48.k even 4 1 144.1.g.a ✓ 1
48.k even 4 1 576.1.g.a 1
80.i odd 4 1 3600.1.j.a 2
80.j even 4 1 3600.1.j.a 2
80.k odd 4 1 3600.1.e.b 1
80.q even 4 1 3600.1.e.b 1
80.s even 4 1 3600.1.j.a 2
80.t odd 4 1 3600.1.j.a 2
144.u even 12 2 1296.1.o.b 2
144.v odd 12 2 1296.1.o.b 2
144.w odd 12 2 1296.1.o.b 2
144.x even 12 2 1296.1.o.b 2
240.t even 4 1 3600.1.e.b 1
240.z odd 4 1 3600.1.j.a 2
240.bb even 4 1 3600.1.j.a 2
240.bd odd 4 1 3600.1.j.a 2
240.bf even 4 1 3600.1.j.a 2
240.bm odd 4 1 3600.1.e.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.1.g.a ✓ 1 16.e even 4 1
144.1.g.a ✓ 1 16.f odd 4 1
144.1.g.a ✓ 1 48.i odd 4 1
144.1.g.a ✓ 1 48.k even 4 1
576.1.g.a 1 16.e even 4 1
576.1.g.a 1 16.f odd 4 1
576.1.g.a 1 48.i odd 4 1
576.1.g.a 1 48.k even 4 1
1296.1.o.b 2 144.u even 12 2
1296.1.o.b 2 144.v odd 12 2
1296.1.o.b 2 144.w odd 12 2
1296.1.o.b 2 144.x even 12 2
2304.1.b.a 2 1.a even 1 1 trivial
2304.1.b.a 2 3.b odd 2 1 CM
2304.1.b.a 2 4.b odd 2 1 CM
2304.1.b.a 2 8.b even 2 1 inner
2304.1.b.a 2 8.d odd 2 1 inner
2304.1.b.a 2 12.b even 2 1 RM
2304.1.b.a 2 24.f even 2 1 inner
2304.1.b.a 2 24.h odd 2 1 inner
3600.1.e.b 1 80.k odd 4 1
3600.1.e.b 1 80.q even 4 1
3600.1.e.b 1 240.t even 4 1
3600.1.e.b 1 240.bm odd 4 1
3600.1.j.a 2 80.i odd 4 1
3600.1.j.a 2 80.j even 4 1
3600.1.j.a 2 80.s even 4 1
3600.1.j.a 2 80.t odd 4 1
3600.1.j.a 2 240.z odd 4 1
3600.1.j.a 2 240.bb even 4 1
3600.1.j.a 2 240.bd odd 4 1
3600.1.j.a 2 240.bf even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(2304, [\chi])\).

Hecke characteristic polynomials

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