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Properties

Label 87.2.i.a
Level $87$
Weight $2$
Character orbit 87.i
Analytic conductor $0.695$
Analytic rank $0$
Dimension $24$
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] = [87,2,Mod(4,87)] 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("87.4"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(87, base_ring=CyclotomicField(14)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 87 = 3 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 87.i (of order \(14\), degree \(6\), 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: \(0.694698497585\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(4\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 2 q^{4} - 4 q^{5} - 2 q^{6} + 8 q^{7} + 4 q^{9} - 28 q^{11} - 10 q^{13} - 14 q^{15} - 22 q^{16} - 20 q^{20} + 4 q^{22} + 18 q^{23} - 18 q^{25} + 28 q^{26} + 8 q^{28} + 28 q^{29} - 40 q^{30} + 28 q^{31}+ \cdots - 42 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −1.39841 − 0.319179i 0.433884 + 0.900969i 0.0517506 + 0.0249218i −0.831893 + 3.64476i −0.319179 − 1.39841i 0.406749 − 0.195880i 2.17847 + 1.73727i −0.623490 + 0.781831i 2.32666 − 4.83137i
4.2 −0.0665871 − 0.0151981i −0.433884 − 0.900969i −1.79773 − 0.865743i 0.452835 − 1.98400i 0.0151981 + 0.0665871i 3.73941 − 1.80080i 0.213346 + 0.170138i −0.623490 + 0.781831i −0.0603060 + 0.125227i
4.3 1.26950 + 0.289755i 0.433884 + 0.900969i −0.274272 − 0.132082i −0.0725658 + 0.317931i 0.289755 + 1.26950i 0.994220 − 0.478791i −2.34603 − 1.87090i −0.623490 + 0.781831i −0.184244 + 0.382587i
4.4 1.88753 + 0.430815i −0.433884 − 0.900969i 1.57521 + 0.758583i −0.103334 + 0.452737i −0.430815 − 1.88753i −2.33844 + 1.12613i −0.380909 − 0.303765i −0.623490 + 0.781831i −0.390092 + 0.810035i
13.1 −0.583108 + 1.21084i 0.781831 + 0.623490i 0.120869 + 0.151565i −0.242958 − 0.117003i −1.21084 + 0.583108i 0.650721 − 0.815979i −2.87447 + 0.656078i 0.222521 + 0.974928i 0.283342 − 0.225958i
13.2 0.105380 − 0.218824i −0.781831 − 0.623490i 1.21020 + 1.51754i 0.879217 + 0.423409i −0.218824 + 0.105380i 2.00910 − 2.51933i 0.933178 − 0.212992i 0.222521 + 0.974928i 0.185304 − 0.147775i
13.3 0.827672 − 1.71868i 0.781831 + 0.623490i −1.02184 − 1.28134i −1.54672 − 0.744859i 1.71868 − 0.827672i −0.774211 + 0.970830i 0.671559 − 0.153279i 0.222521 + 0.974928i −2.56035 + 2.04181i
13.4 1.00695 − 2.09096i −0.781831 − 0.623490i −2.11117 − 2.64732i 1.71239 + 0.824646i −2.09096 + 1.00695i −2.13259 + 2.67418i −3.13608 + 0.715791i 0.222521 + 0.974928i 3.44860 − 2.75017i
22.1 −1.39841 + 0.319179i 0.433884 − 0.900969i 0.0517506 − 0.0249218i −0.831893 − 3.64476i −0.319179 + 1.39841i 0.406749 + 0.195880i 2.17847 − 1.73727i −0.623490 − 0.781831i 2.32666 + 4.83137i
22.2 −0.0665871 + 0.0151981i −0.433884 + 0.900969i −1.79773 + 0.865743i 0.452835 + 1.98400i 0.0151981 − 0.0665871i 3.73941 + 1.80080i 0.213346 − 0.170138i −0.623490 − 0.781831i −0.0603060 − 0.125227i
22.3 1.26950 − 0.289755i 0.433884 − 0.900969i −0.274272 + 0.132082i −0.0725658 − 0.317931i 0.289755 − 1.26950i 0.994220 + 0.478791i −2.34603 + 1.87090i −0.623490 − 0.781831i −0.184244 − 0.382587i
22.4 1.88753 − 0.430815i −0.433884 + 0.900969i 1.57521 − 0.758583i −0.103334 − 0.452737i −0.430815 + 1.88753i −2.33844 − 1.12613i −0.380909 + 0.303765i −0.623490 − 0.781831i −0.390092 − 0.810035i
34.1 −2.01867 + 1.60984i −0.974928 + 0.222521i 1.03842 − 4.54960i −1.92590 − 2.41500i 1.60984 − 2.01867i 0.0378611 + 0.165880i 2.98734 + 6.20327i 0.900969 − 0.433884i 7.77550 + 1.77471i
34.2 −1.76451 + 1.40715i 0.974928 − 0.222521i 0.688387 − 3.01602i 1.56067 + 1.95702i −1.40715 + 1.76451i 0.193639 + 0.848388i 1.07087 + 2.22368i 0.900969 − 0.433884i −5.50765 − 1.25709i
34.3 −0.287620 + 0.229369i −0.974928 + 0.222521i −0.414927 + 1.81791i 0.561619 + 0.704248i 0.229369 − 0.287620i 0.684660 + 2.99969i −0.616866 − 1.28093i 0.900969 − 0.433884i −0.323065 − 0.0737376i
34.4 1.02189 − 0.814927i 0.974928 − 0.222521i −0.0648970 + 0.284332i −2.44337 − 3.06390i 0.814927 − 1.02189i 0.528882 + 2.31718i 1.29960 + 2.69865i 0.900969 − 0.433884i −4.99370 − 1.13978i
64.1 −2.01867 − 1.60984i −0.974928 − 0.222521i 1.03842 + 4.54960i −1.92590 + 2.41500i 1.60984 + 2.01867i 0.0378611 − 0.165880i 2.98734 − 6.20327i 0.900969 + 0.433884i 7.77550 − 1.77471i
64.2 −1.76451 − 1.40715i 0.974928 + 0.222521i 0.688387 + 3.01602i 1.56067 − 1.95702i −1.40715 − 1.76451i 0.193639 − 0.848388i 1.07087 − 2.22368i 0.900969 + 0.433884i −5.50765 + 1.25709i
64.3 −0.287620 − 0.229369i −0.974928 − 0.222521i −0.414927 − 1.81791i 0.561619 − 0.704248i 0.229369 + 0.287620i 0.684660 − 2.99969i −0.616866 + 1.28093i 0.900969 + 0.433884i −0.323065 + 0.0737376i
64.4 1.02189 + 0.814927i 0.974928 + 0.222521i −0.0648970 − 0.284332i −2.44337 + 3.06390i 0.814927 + 1.02189i 0.528882 − 2.31718i 1.29960 − 2.69865i 0.900969 + 0.433884i −4.99370 + 1.13978i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.4
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.e even 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 87.2.i.a ✓ 24
3.b odd 2 1 261.2.o.b 24
29.e even 14 1 inner 87.2.i.a ✓ 24
29.f odd 28 1 2523.2.a.s 12
29.f odd 28 1 2523.2.a.v 12
87.h odd 14 1 261.2.o.b 24
87.k even 28 1 7569.2.a.bn 12
87.k even 28 1 7569.2.a.bt 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
87.2.i.a ✓ 24 1.a even 1 1 trivial
87.2.i.a ✓ 24 29.e even 14 1 inner
261.2.o.b 24 3.b odd 2 1
261.2.o.b 24 87.h odd 14 1
2523.2.a.s 12 29.f odd 28 1
2523.2.a.v 12 29.f odd 28 1
7569.2.a.bn 12 87.k even 28 1
7569.2.a.bt 12 87.k even 28 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(87, [\chi])\).