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Properties

Label 1860.3.k.a
Level $1860$
Weight $3$
Character orbit 1860.k
Analytic conductor $50.681$
Analytic rank $0$
Dimension $44$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1860,3,Mod(61,1860)] 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("1860.61"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1860, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 1860 = 2^{2} \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1860.k (of order \(2\), degree \(1\), 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: \(50.6813291710\)
Analytic rank: \(0\)
Dimension: \(44\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 44 q - 16 q^{7} - 132 q^{9} - 32 q^{19} + 220 q^{25} - 60 q^{31} - 24 q^{33} - 40 q^{35} - 24 q^{39} - 80 q^{41} + 96 q^{47} + 252 q^{49} - 168 q^{51} - 248 q^{59} + 48 q^{63} + 208 q^{67} + 120 q^{71}+ \cdots + 96 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
61.1 0 − 1.73205i 0 2.23607 0 −0.925426 0 −3.00000 0
61.2 0 1.73205i 0 2.23607 0 −0.925426 0 −3.00000 0
61.3 0 − 1.73205i 0 −2.23607 0 7.53425 0 −3.00000 0
61.4 0 1.73205i 0 −2.23607 0 7.53425 0 −3.00000 0
61.5 0 − 1.73205i 0 −2.23607 0 13.2955 0 −3.00000 0
61.6 0 1.73205i 0 −2.23607 0 13.2955 0 −3.00000 0
61.7 0 − 1.73205i 0 2.23607 0 −13.0283 0 −3.00000 0
61.8 0 1.73205i 0 2.23607 0 −13.0283 0 −3.00000 0
61.9 0 − 1.73205i 0 2.23607 0 −8.13053 0 −3.00000 0
61.10 0 1.73205i 0 2.23607 0 −8.13053 0 −3.00000 0
61.11 0 − 1.73205i 0 −2.23607 0 −8.44549 0 −3.00000 0
61.12 0 1.73205i 0 −2.23607 0 −8.44549 0 −3.00000 0
61.13 0 − 1.73205i 0 −2.23607 0 −6.24009 0 −3.00000 0
61.14 0 1.73205i 0 −2.23607 0 −6.24009 0 −3.00000 0
61.15 0 − 1.73205i 0 2.23607 0 4.75757 0 −3.00000 0
61.16 0 1.73205i 0 2.23607 0 4.75757 0 −3.00000 0
61.17 0 − 1.73205i 0 2.23607 0 5.45471 0 −3.00000 0
61.18 0 1.73205i 0 2.23607 0 5.45471 0 −3.00000 0
61.19 0 − 1.73205i 0 −2.23607 0 8.11271 0 −3.00000 0
61.20 0 1.73205i 0 −2.23607 0 8.11271 0 −3.00000 0
See all 44 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 61.44
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1860.3.k.a ✓ 44
31.b odd 2 1 inner 1860.3.k.a ✓ 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1860.3.k.a ✓ 44 1.a even 1 1 trivial
1860.3.k.a ✓ 44 31.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(1860, [\chi])\).