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Properties

Label 1710.4.f.a
Level $1710$
Weight $4$
Character orbit 1710.f
Analytic conductor $100.893$
Analytic rank $0$
Dimension $40$
Inner twists $2$

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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] = [1710,4,Mod(341,1710)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1710, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 4, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1710.341"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1710 = 2 \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1710.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [40,-80] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(100.893266110\)
Analytic rank: \(0\)
Dimension: \(40\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 80 q^{2} + 160 q^{4} - 56 q^{7} - 320 q^{8} + 112 q^{14} + 640 q^{16} - 76 q^{19} - 1000 q^{25} - 224 q^{28} - 120 q^{29} - 1280 q^{32} + 152 q^{38} - 312 q^{41} + 56 q^{43} + 2112 q^{49} + 2000 q^{50}+ \cdots - 4224 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
341.1 −2.00000 0 4.00000 − 5.00000i 0 −27.0790 −8.00000 0 10.0000i
341.2 −2.00000 0 4.00000 5.00000i 0 −27.0790 −8.00000 0 − 10.0000i
341.3 −2.00000 0 4.00000 − 5.00000i 0 32.6046 −8.00000 0 10.0000i
341.4 −2.00000 0 4.00000 5.00000i 0 32.6046 −8.00000 0 − 10.0000i
341.5 −2.00000 0 4.00000 − 5.00000i 0 17.5238 −8.00000 0 10.0000i
341.6 −2.00000 0 4.00000 5.00000i 0 17.5238 −8.00000 0 − 10.0000i
341.7 −2.00000 0 4.00000 − 5.00000i 0 22.4759 −8.00000 0 10.0000i
341.8 −2.00000 0 4.00000 5.00000i 0 22.4759 −8.00000 0 − 10.0000i
341.9 −2.00000 0 4.00000 − 5.00000i 0 −6.26670 −8.00000 0 10.0000i
341.10 −2.00000 0 4.00000 5.00000i 0 −6.26670 −8.00000 0 − 10.0000i
341.11 −2.00000 0 4.00000 − 5.00000i 0 26.6820 −8.00000 0 10.0000i
341.12 −2.00000 0 4.00000 5.00000i 0 26.6820 −8.00000 0 − 10.0000i
341.13 −2.00000 0 4.00000 − 5.00000i 0 −20.6657 −8.00000 0 10.0000i
341.14 −2.00000 0 4.00000 5.00000i 0 −20.6657 −8.00000 0 − 10.0000i
341.15 −2.00000 0 4.00000 − 5.00000i 0 10.8134 −8.00000 0 10.0000i
341.16 −2.00000 0 4.00000 5.00000i 0 10.8134 −8.00000 0 − 10.0000i
341.17 −2.00000 0 4.00000 − 5.00000i 0 5.35183 −8.00000 0 10.0000i
341.18 −2.00000 0 4.00000 5.00000i 0 5.35183 −8.00000 0 − 10.0000i
341.19 −2.00000 0 4.00000 − 5.00000i 0 −5.26733 −8.00000 0 10.0000i
341.20 −2.00000 0 4.00000 5.00000i 0 −5.26733 −8.00000 0 − 10.0000i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 341.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
57.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1710.4.f.a ✓ 40
3.b odd 2 1 1710.4.f.b yes 40
19.b odd 2 1 1710.4.f.b yes 40
57.d even 2 1 inner 1710.4.f.a ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1710.4.f.a ✓ 40 1.a even 1 1 trivial
1710.4.f.a ✓ 40 57.d even 2 1 inner
1710.4.f.b yes 40 3.b odd 2 1
1710.4.f.b yes 40 19.b odd 2 1