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Properties

Label 1860.2.bj.a
Level $1860$
Weight $2$
Character orbit 1860.bj
Analytic conductor $14.852$
Analytic rank $0$
Dimension $64$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1860,2,Mod(769,1860)] 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("1860.769"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1860, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 3, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1860 = 2^{2} \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1860.bj (of order \(6\), degree \(2\), 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: \(14.8521747760\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(32\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 64 q + 32 q^{9} - 8 q^{15} - 4 q^{19} - 4 q^{21} - 32 q^{29} - 20 q^{31} - 44 q^{35} - 8 q^{41} + 32 q^{49} - 20 q^{55} - 20 q^{59} + 40 q^{61} - 14 q^{65} + 52 q^{71} + 8 q^{75} + 12 q^{79} - 32 q^{81}+ \cdots + 84 q^{95}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
769.1 0 −0.866025 + 0.500000i 0 −2.02460 + 0.949213i 0 3.85461 − 2.22546i 0 0.500000 − 0.866025i 0
769.2 0 −0.866025 + 0.500000i 0 1.88891 − 1.19666i 0 3.39795 − 1.96181i 0 0.500000 − 0.866025i 0
769.3 0 −0.866025 + 0.500000i 0 −0.547706 − 2.16795i 0 3.24249 − 1.87205i 0 0.500000 − 0.866025i 0
769.4 0 −0.866025 + 0.500000i 0 2.12630 + 0.691996i 0 3.01933 − 1.74321i 0 0.500000 − 0.866025i 0
769.5 0 −0.866025 + 0.500000i 0 1.05226 − 1.97300i 0 −2.21158 + 1.27686i 0 0.500000 − 0.866025i 0
769.6 0 −0.866025 + 0.500000i 0 −0.703449 + 2.12254i 0 −1.24288 + 0.717578i 0 0.500000 − 0.866025i 0
769.7 0 −0.866025 + 0.500000i 0 −0.798308 + 2.08871i 0 −1.21248 + 0.700028i 0 0.500000 − 0.866025i 0
769.8 0 −0.866025 + 0.500000i 0 −2.20295 − 0.383429i 0 0.335467 − 0.193682i 0 0.500000 − 0.866025i 0
769.9 0 −0.866025 + 0.500000i 0 −1.03024 − 1.98459i 0 0.252191 − 0.145603i 0 0.500000 − 0.866025i 0
769.10 0 −0.866025 + 0.500000i 0 −0.609333 + 2.15144i 0 0.746338 − 0.430899i 0 0.500000 − 0.866025i 0
769.11 0 −0.866025 + 0.500000i 0 −0.151759 − 2.23091i 0 0.905619 − 0.522860i 0 0.500000 − 0.866025i 0
769.12 0 −0.866025 + 0.500000i 0 1.14140 + 1.92281i 0 2.31757 − 1.33805i 0 0.500000 − 0.866025i 0
769.13 0 −0.866025 + 0.500000i 0 2.02390 − 0.950691i 0 −2.34936 + 1.35640i 0 0.500000 − 0.866025i 0
769.14 0 −0.866025 + 0.500000i 0 1.84520 + 1.26302i 0 −2.73160 + 1.57709i 0 0.500000 − 0.866025i 0
769.15 0 −0.866025 + 0.500000i 0 −2.18690 − 0.466344i 0 −2.83235 + 1.63526i 0 0.500000 − 0.866025i 0
769.16 0 −0.866025 + 0.500000i 0 1.90931 + 1.16385i 0 −3.75926 + 2.17041i 0 0.500000 − 0.866025i 0
769.17 0 0.866025 − 0.500000i 0 −1.96258 − 1.07158i 0 3.75926 − 2.17041i 0 0.500000 − 0.866025i 0
769.18 0 0.866025 − 0.500000i 0 1.49731 + 1.66074i 0 2.83235 − 1.63526i 0 0.500000 − 0.866025i 0
769.19 0 0.866025 − 0.500000i 0 −2.01641 − 0.966481i 0 2.73160 − 1.57709i 0 0.500000 − 0.866025i 0
769.20 0 0.866025 − 0.500000i 0 −0.188629 − 2.22810i 0 2.34936 − 1.35640i 0 0.500000 − 0.866025i 0
See all 64 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 769.32
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
31.c even 3 1 inner
155.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1860.2.bj.a ✓ 64
5.b even 2 1 inner 1860.2.bj.a ✓ 64
31.c even 3 1 inner 1860.2.bj.a ✓ 64
155.j even 6 1 inner 1860.2.bj.a ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1860.2.bj.a ✓ 64 1.a even 1 1 trivial
1860.2.bj.a ✓ 64 5.b even 2 1 inner
1860.2.bj.a ✓ 64 31.c even 3 1 inner
1860.2.bj.a ✓ 64 155.j even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(1860, [\chi])\).