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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3040,1,Mod(719,3040)] 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("3040.719"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3040, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([9, 9, 9, 4])) 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.dv (of order \(18\), degree \(6\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.51715763840\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 760)
Projective image: \(D_{9}\)
Projective field: Galois closure of 9.1.43477921384960000.1

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{18}^{7} q^{5} + ( - \zeta_{18}^{8} - \zeta_{18}^{4}) q^{7} + \zeta_{18}^{8} q^{9} + ( - \zeta_{18}^{4} - \zeta_{18}^{2}) q^{11} + \zeta_{18}^{5} q^{13} + \zeta_{18} q^{19} + (\zeta_{18}^{7} - \zeta_{18}^{6}) q^{23} + \cdots + (\zeta_{18}^{3} + \zeta_{18}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{23} + 3 q^{35} - 3 q^{41} - 3 q^{45} - 3 q^{49} + 6 q^{53} - 6 q^{55} + 3 q^{63} + 3 q^{65} - 6 q^{77} + 6 q^{89} + 6 q^{91} + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(191\) \(1217\) \(1921\) \(2661\)
\(\chi(n)\) \(-1\) \(-1\) \(\zeta_{18}^{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
719.1
−0.766044 − 0.642788i
0.939693 − 0.342020i
0.939693 + 0.342020i
−0.173648 − 0.984808i
−0.173648 + 0.984808i
−0.766044 + 0.642788i
0 0 0 0.173648 − 0.984808i 0 0.173648 + 0.300767i 0 0.766044 − 0.642788i 0
879.1 0 0 0 0.766044 + 0.642788i 0 0.766044 + 1.32683i 0 −0.939693 − 0.342020i 0
1999.1 0 0 0 0.766044 − 0.642788i 0 0.766044 − 1.32683i 0 −0.939693 + 0.342020i 0
2479.1 0 0 0 −0.939693 − 0.342020i 0 −0.939693 + 1.62760i 0 0.173648 − 0.984808i 0
2639.1 0 0 0 −0.939693 + 0.342020i 0 −0.939693 − 1.62760i 0 0.173648 + 0.984808i 0
2799.1 0 0 0 0.173648 + 0.984808i 0 0.173648 − 0.300767i 0 0.766044 + 0.642788i 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 719.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
40.e odd 2 1 CM by \(\Q(\sqrt{-10}) \)
19.e even 9 1 inner
760.bz odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3040.1.dv.b 6
4.b odd 2 1 760.1.bz.a ✓ 6
5.b even 2 1 3040.1.dv.a 6
8.b even 2 1 760.1.bz.b yes 6
8.d odd 2 1 3040.1.dv.a 6
19.e even 9 1 inner 3040.1.dv.b 6
20.d odd 2 1 760.1.bz.b yes 6
20.e even 4 2 3800.1.cv.f 12
40.e odd 2 1 CM 3040.1.dv.b 6
40.f even 2 1 760.1.bz.a ✓ 6
40.i odd 4 2 3800.1.cv.f 12
76.l odd 18 1 760.1.bz.a ✓ 6
95.p even 18 1 3040.1.dv.a 6
152.t even 18 1 760.1.bz.b yes 6
152.u odd 18 1 3040.1.dv.a 6
380.ba odd 18 1 760.1.bz.b yes 6
380.bj even 36 2 3800.1.cv.f 12
760.bz odd 18 1 inner 3040.1.dv.b 6
760.cj even 18 1 760.1.bz.a ✓ 6
760.cq odd 36 2 3800.1.cv.f 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.1.bz.a ✓ 6 4.b odd 2 1
760.1.bz.a ✓ 6 40.f even 2 1
760.1.bz.a ✓ 6 76.l odd 18 1
760.1.bz.a ✓ 6 760.cj even 18 1
760.1.bz.b yes 6 8.b even 2 1
760.1.bz.b yes 6 20.d odd 2 1
760.1.bz.b yes 6 152.t even 18 1
760.1.bz.b yes 6 380.ba odd 18 1
3040.1.dv.a 6 5.b even 2 1
3040.1.dv.a 6 8.d odd 2 1
3040.1.dv.a 6 95.p even 18 1
3040.1.dv.a 6 152.u odd 18 1
3040.1.dv.b 6 1.a even 1 1 trivial
3040.1.dv.b 6 19.e even 9 1 inner
3040.1.dv.b 6 40.e odd 2 1 CM
3040.1.dv.b 6 760.bz odd 18 1 inner
3800.1.cv.f 12 20.e even 4 2
3800.1.cv.f 12 40.i odd 4 2
3800.1.cv.f 12 380.bj even 36 2
3800.1.cv.f 12 760.cq odd 36 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{6} + 3T_{7}^{4} - 2T_{7}^{3} + 9T_{7}^{2} - 3T_{7} + 1 \) acting on \(S_{1}^{\mathrm{new}}(3040, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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