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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3040,1,Mod(1329,3040)] 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("3040.1329"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3040, 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: \( N \) \(=\) \( 3040 = 2^{5} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3040.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: \(1.51715763840\)
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 760)
Projective image: \(D_{8}\)
Projective field: Galois closure of 8.0.66724352000.2

$q$-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.

\(f(q)\) \(=\) \( q + ( - \zeta_{16}^{5} - \zeta_{16}^{3}) q^{3} + \zeta_{16}^{4} q^{5} + (\zeta_{16}^{6} - \zeta_{16}^{2} - 1) q^{9} + (\zeta_{16}^{6} + \zeta_{16}^{2}) q^{11} + (\zeta_{16}^{7} + \zeta_{16}) q^{13} + ( - \zeta_{16}^{7} + \zeta_{16}) q^{15}+ \cdots + ( - \zeta_{16}^{6} + \cdots - \zeta_{16}^{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{9} - 8 q^{25} + 8 q^{49} + 8 q^{81} - 8 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3040\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(1217\) \(1921\) \(2661\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1329.1
−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 − 1.84776i 0 − 1.00000i 0 0 0 −2.41421 0
1329.2 0 − 1.84776i 0 1.00000i 0 0 0 −2.41421 0
1329.3 0 − 0.765367i 0 − 1.00000i 0 0 0 0.414214 0
1329.4 0 − 0.765367i 0 1.00000i 0 0 0 0.414214 0
1329.5 0 0.765367i 0 − 1.00000i 0 0 0 0.414214 0
1329.6 0 0.765367i 0 1.00000i 0 0 0 0.414214 0
1329.7 0 1.84776i 0 − 1.00000i 0 0 0 −2.41421 0
1329.8 0 1.84776i 0 1.00000i 0 0 0 −2.41421 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1329.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
95.d odd 2 1 CM by \(\Q(\sqrt{-95}) \)
5.b even 2 1 inner
8.b even 2 1 inner
19.b odd 2 1 inner
40.f even 2 1 inner
152.g odd 2 1 inner
760.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3040.1.b.a 8
4.b odd 2 1 760.1.b.a ✓ 8
5.b even 2 1 inner 3040.1.b.a 8
8.b even 2 1 inner 3040.1.b.a 8
8.d odd 2 1 760.1.b.a ✓ 8
19.b odd 2 1 inner 3040.1.b.a 8
20.d odd 2 1 760.1.b.a ✓ 8
20.e even 4 2 3800.1.o.g 8
40.e odd 2 1 760.1.b.a ✓ 8
40.f even 2 1 inner 3040.1.b.a 8
40.k even 4 2 3800.1.o.g 8
76.d even 2 1 760.1.b.a ✓ 8
95.d odd 2 1 CM 3040.1.b.a 8
152.b even 2 1 760.1.b.a ✓ 8
152.g odd 2 1 inner 3040.1.b.a 8
380.d even 2 1 760.1.b.a ✓ 8
380.j odd 4 2 3800.1.o.g 8
760.b odd 2 1 inner 3040.1.b.a 8
760.p even 2 1 760.1.b.a ✓ 8
760.y odd 4 2 3800.1.o.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.1.b.a ✓ 8 4.b odd 2 1
760.1.b.a ✓ 8 8.d odd 2 1
760.1.b.a ✓ 8 20.d odd 2 1
760.1.b.a ✓ 8 40.e odd 2 1
760.1.b.a ✓ 8 76.d even 2 1
760.1.b.a ✓ 8 152.b even 2 1
760.1.b.a ✓ 8 380.d even 2 1
760.1.b.a ✓ 8 760.p even 2 1
3040.1.b.a 8 1.a even 1 1 trivial
3040.1.b.a 8 5.b even 2 1 inner
3040.1.b.a 8 8.b even 2 1 inner
3040.1.b.a 8 19.b odd 2 1 inner
3040.1.b.a 8 40.f even 2 1 inner
3040.1.b.a 8 95.d odd 2 1 CM
3040.1.b.a 8 152.g odd 2 1 inner
3040.1.b.a 8 760.b odd 2 1 inner
3800.1.o.g 8 20.e even 4 2
3800.1.o.g 8 40.k even 4 2
3800.1.o.g 8 380.j odd 4 2
3800.1.o.g 8 760.y odd 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(3040, [\chi])\).

Hecke characteristic polynomials

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