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Properties

Label 1760.2.f.d
Level $1760$
Weight $2$
Character orbit 1760.f
Analytic conductor $14.054$
Analytic rank $0$
Dimension $24$
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] = [1760,2,Mod(351,1760)] 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("1760.351"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1760, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1760 = 2^{5} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1760.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,0,0,0,-24,0,0,0,-32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,24,0,0, 0,0,0,0,0,32] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(33)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(14.0536707557\)
Analytic rank: \(0\)
Dimension: \(24\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 24 q^{5} - 32 q^{9} + 24 q^{25} + 32 q^{33} + 24 q^{37} + 32 q^{45} + 16 q^{49} + 8 q^{53} - 16 q^{69} + 24 q^{77} + 24 q^{81} - 24 q^{89} + 8 q^{93} + 48 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
351.1 0 − 2.97875i 0 −1.00000 0 −4.81963 0 −5.87292 0
351.2 0 2.97875i 0 −1.00000 0 −4.81963 0 −5.87292 0
351.3 0 − 0.664406i 0 −1.00000 0 −3.22449 0 2.55856 0
351.4 0 0.664406i 0 −1.00000 0 −3.22449 0 2.55856 0
351.5 0 − 2.43031i 0 −1.00000 0 2.50020 0 −2.90640 0
351.6 0 2.43031i 0 −1.00000 0 2.50020 0 −2.90640 0
351.7 0 − 2.84742i 0 −1.00000 0 −0.280035 0 −5.10782 0
351.8 0 2.84742i 0 −1.00000 0 −0.280035 0 −5.10782 0
351.9 0 − 1.59277i 0 −1.00000 0 −1.92830 0 0.463073 0
351.10 0 1.59277i 0 −1.00000 0 −1.92830 0 0.463073 0
351.11 0 − 0.366738i 0 −1.00000 0 −1.52516 0 2.86550 0
351.12 0 0.366738i 0 −1.00000 0 −1.52516 0 2.86550 0
351.13 0 − 0.366738i 0 −1.00000 0 1.52516 0 2.86550 0
351.14 0 0.366738i 0 −1.00000 0 1.52516 0 2.86550 0
351.15 0 − 1.59277i 0 −1.00000 0 1.92830 0 0.463073 0
351.16 0 1.59277i 0 −1.00000 0 1.92830 0 0.463073 0
351.17 0 − 2.84742i 0 −1.00000 0 0.280035 0 −5.10782 0
351.18 0 2.84742i 0 −1.00000 0 0.280035 0 −5.10782 0
351.19 0 − 2.43031i 0 −1.00000 0 −2.50020 0 −2.90640 0
351.20 0 2.43031i 0 −1.00000 0 −2.50020 0 −2.90640 0
See all 24 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 351.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
11.b odd 2 1 inner
44.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1760.2.f.d ✓ 24
4.b odd 2 1 inner 1760.2.f.d ✓ 24
8.b even 2 1 3520.2.f.m 24
8.d odd 2 1 3520.2.f.m 24
11.b odd 2 1 inner 1760.2.f.d ✓ 24
44.c even 2 1 inner 1760.2.f.d ✓ 24
88.b odd 2 1 3520.2.f.m 24
88.g even 2 1 3520.2.f.m 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1760.2.f.d ✓ 24 1.a even 1 1 trivial
1760.2.f.d ✓ 24 4.b odd 2 1 inner
1760.2.f.d ✓ 24 11.b odd 2 1 inner
1760.2.f.d ✓ 24 44.c even 2 1 inner
3520.2.f.m 24 8.b even 2 1
3520.2.f.m 24 8.d odd 2 1
3520.2.f.m 24 88.b odd 2 1
3520.2.f.m 24 88.g even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1760, [\chi])\):

\( T_{3}^{12} + 26T_{3}^{10} + 245T_{3}^{8} + 996T_{3}^{6} + 1588T_{3}^{4} + 672T_{3}^{2} + 64 \) Copy content Toggle raw display
\( T_{7}^{12} - 46T_{7}^{10} + 705T_{7}^{8} - 4640T_{7}^{6} + 13392T_{7}^{4} - 14080T_{7}^{2} + 1024 \) Copy content Toggle raw display