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Properties

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

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1860,2,Mod(433,1860)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1860, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 3, 2])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1860.433"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1860 = 2^{2} \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1860.s (of order \(4\), 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 \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 64 q - 8 q^{7} + 16 q^{25} + 8 q^{31} - 8 q^{33} + 24 q^{35} + 16 q^{41} + 56 q^{47} + 8 q^{63} - 32 q^{67} - 16 q^{71} - 64 q^{81} - 32 q^{87} - 32 q^{95} + 64 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
433.1 0 −0.707107 + 0.707107i 0 1.97917 − 1.04062i 0 −0.798115 + 0.798115i 0 − 1.00000i 0
433.2 0 −0.707107 + 0.707107i 0 1.61302 − 1.54861i 0 −2.80826 + 2.80826i 0 − 1.00000i 0
433.3 0 −0.707107 + 0.707107i 0 −0.143549 + 2.23146i 0 −1.60544 + 1.60544i 0 − 1.00000i 0
433.4 0 −0.707107 + 0.707107i 0 −0.198698 − 2.22722i 0 0.548034 − 0.548034i 0 − 1.00000i 0
433.5 0 −0.707107 + 0.707107i 0 1.82341 − 1.29429i 0 2.05794 − 2.05794i 0 − 1.00000i 0
433.6 0 −0.707107 + 0.707107i 0 −0.193900 − 2.22765i 0 −2.45774 + 2.45774i 0 − 1.00000i 0
433.7 0 −0.707107 + 0.707107i 0 −1.64250 + 1.51730i 0 0.674956 − 0.674956i 0 − 1.00000i 0
433.8 0 −0.707107 + 0.707107i 0 −2.23589 + 0.0281135i 0 −0.339712 + 0.339712i 0 − 1.00000i 0
433.9 0 −0.707107 + 0.707107i 0 1.94432 + 1.10436i 0 −1.30181 + 1.30181i 0 − 1.00000i 0
433.10 0 −0.707107 + 0.707107i 0 1.31391 + 1.80932i 0 2.30228 − 2.30228i 0 − 1.00000i 0
433.11 0 −0.707107 + 0.707107i 0 −2.16858 + 0.545223i 0 −3.45556 + 3.45556i 0 − 1.00000i 0
433.12 0 −0.707107 + 0.707107i 0 −1.28778 + 1.82801i 0 −0.110456 + 0.110456i 0 − 1.00000i 0
433.13 0 −0.707107 + 0.707107i 0 −1.72803 − 1.41912i 0 −0.575654 + 0.575654i 0 − 1.00000i 0
433.14 0 −0.707107 + 0.707107i 0 −2.03335 − 0.930324i 0 3.55204 − 3.55204i 0 − 1.00000i 0
433.15 0 −0.707107 + 0.707107i 0 0.773223 + 2.09812i 0 −0.100270 + 0.100270i 0 − 1.00000i 0
433.16 0 −0.707107 + 0.707107i 0 2.18523 − 0.474092i 0 2.41778 − 2.41778i 0 − 1.00000i 0
433.17 0 0.707107 − 0.707107i 0 0.773223 + 2.09812i 0 −0.100270 + 0.100270i 0 − 1.00000i 0
433.18 0 0.707107 − 0.707107i 0 −0.193900 − 2.22765i 0 −2.45774 + 2.45774i 0 − 1.00000i 0
433.19 0 0.707107 − 0.707107i 0 1.97917 − 1.04062i 0 −0.798115 + 0.798115i 0 − 1.00000i 0
433.20 0 0.707107 − 0.707107i 0 −1.72803 − 1.41912i 0 −0.575654 + 0.575654i 0 − 1.00000i 0
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 433.32
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
31.b odd 2 1 inner
155.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1860.2.s.a ✓ 64
5.c odd 4 1 inner 1860.2.s.a ✓ 64
31.b odd 2 1 inner 1860.2.s.a ✓ 64
155.f even 4 1 inner 1860.2.s.a ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1860.2.s.a ✓ 64 1.a even 1 1 trivial
1860.2.s.a ✓ 64 5.c odd 4 1 inner
1860.2.s.a ✓ 64 31.b odd 2 1 inner
1860.2.s.a ✓ 64 155.f even 4 1 inner

Hecke kernels

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