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Properties

Label 186.2.p.a
Level $186$
Weight $2$
Character orbit 186.p
Analytic conductor $1.485$
Analytic rank $0$
Dimension $80$
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] = [186,2,Mod(11,186)] 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("186.11"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(186, base_ring=CyclotomicField(30)) chi = DirichletCharacter(H, H._module([15, 23])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 186 = 2 \cdot 3 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 186.p (of order \(30\), degree \(8\), 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: \(1.48521747760\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(10\) over \(\Q(\zeta_{30})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 80 q + 20 q^{4} + 8 q^{7} + 4 q^{9} - 4 q^{10} - 10 q^{15} - 20 q^{16} - 8 q^{18} - 4 q^{19} + 30 q^{21} - 2 q^{22} + 12 q^{25} - 38 q^{28} - 86 q^{31} - 28 q^{34} - 4 q^{36} - 144 q^{37} - 16 q^{39} - 6 q^{40}+ \cdots + 150 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −0.951057 + 0.309017i −1.55075 − 0.771484i 0.809017 − 0.587785i −1.31329 + 0.758228i 1.71325 + 0.254518i 0.0465366 + 0.442766i −0.587785 + 0.809017i 1.80963 + 2.39275i 1.01471 − 1.12695i
11.2 −0.951057 + 0.309017i −0.878235 + 1.49288i 0.809017 − 0.587785i −2.08872 + 1.20592i 0.373925 − 1.69121i 0.0316894 + 0.301504i −0.587785 + 0.809017i −1.45741 − 2.62221i 1.61384 − 1.79235i
11.3 −0.951057 + 0.309017i −0.127863 − 1.72732i 0.809017 − 0.587785i 1.72504 − 0.995952i 0.655377 + 1.60327i −0.0669056 − 0.636564i −0.587785 + 0.809017i −2.96730 + 0.441720i −1.33284 + 1.48027i
11.4 −0.951057 + 0.309017i 1.70155 + 0.323596i 0.809017 − 0.587785i 1.52111 − 0.878214i −1.71827 + 0.218051i −0.525722 − 5.00191i −0.587785 + 0.809017i 2.79057 + 1.10123i −1.17528 + 1.30528i
11.5 −0.951057 + 0.309017i 1.72891 − 0.104282i 0.809017 − 0.587785i −1.24540 + 0.719033i −1.61207 + 0.633440i 0.425612 + 4.04943i −0.587785 + 0.809017i 2.97825 − 0.360587i 0.962254 − 1.06869i
11.6 0.951057 − 0.309017i −1.57651 + 0.717375i 0.809017 − 0.587785i 2.08872 − 1.20592i −1.27767 + 1.16943i 0.0316894 + 0.301504i 0.587785 − 0.809017i 1.97075 − 2.26189i 1.61384 − 1.79235i
11.7 0.951057 − 0.309017i −0.143963 − 1.72606i 0.809017 − 0.587785i −1.52111 + 0.878214i −0.670298 − 1.59709i −0.525722 − 5.00191i 0.587785 − 0.809017i −2.95855 + 0.496976i −1.17528 + 1.30528i
11.8 0.951057 − 0.309017i 0.284431 − 1.70854i 0.809017 − 0.587785i 1.24540 − 0.719033i −0.257457 − 1.71281i 0.425612 + 4.04943i 0.587785 − 0.809017i −2.83820 − 0.971921i 0.962254 − 1.06869i
11.9 0.951057 − 0.309017i 0.605161 + 1.62289i 0.809017 − 0.587785i 1.31329 − 0.758228i 1.07704 + 1.35646i 0.0465366 + 0.442766i 0.587785 − 0.809017i −2.26756 + 1.96422i 1.01471 − 1.12695i
11.10 0.951057 − 0.309017i 1.70450 + 0.307717i 0.809017 − 0.587785i −1.72504 + 0.995952i 1.71616 − 0.234063i −0.0669056 − 0.636564i 0.587785 − 0.809017i 2.81062 + 1.04900i −1.33284 + 1.48027i
17.1 −0.951057 − 0.309017i −1.55075 + 0.771484i 0.809017 + 0.587785i −1.31329 − 0.758228i 1.71325 − 0.254518i 0.0465366 − 0.442766i −0.587785 − 0.809017i 1.80963 − 2.39275i 1.01471 + 1.12695i
17.2 −0.951057 − 0.309017i −0.878235 − 1.49288i 0.809017 + 0.587785i −2.08872 − 1.20592i 0.373925 + 1.69121i 0.0316894 − 0.301504i −0.587785 − 0.809017i −1.45741 + 2.62221i 1.61384 + 1.79235i
17.3 −0.951057 − 0.309017i −0.127863 + 1.72732i 0.809017 + 0.587785i 1.72504 + 0.995952i 0.655377 − 1.60327i −0.0669056 + 0.636564i −0.587785 − 0.809017i −2.96730 − 0.441720i −1.33284 − 1.48027i
17.4 −0.951057 − 0.309017i 1.70155 − 0.323596i 0.809017 + 0.587785i 1.52111 + 0.878214i −1.71827 − 0.218051i −0.525722 + 5.00191i −0.587785 − 0.809017i 2.79057 − 1.10123i −1.17528 − 1.30528i
17.5 −0.951057 − 0.309017i 1.72891 + 0.104282i 0.809017 + 0.587785i −1.24540 − 0.719033i −1.61207 − 0.633440i 0.425612 − 4.04943i −0.587785 − 0.809017i 2.97825 + 0.360587i 0.962254 + 1.06869i
17.6 0.951057 + 0.309017i −1.57651 − 0.717375i 0.809017 + 0.587785i 2.08872 + 1.20592i −1.27767 − 1.16943i 0.0316894 − 0.301504i 0.587785 + 0.809017i 1.97075 + 2.26189i 1.61384 + 1.79235i
17.7 0.951057 + 0.309017i −0.143963 + 1.72606i 0.809017 + 0.587785i −1.52111 − 0.878214i −0.670298 + 1.59709i −0.525722 + 5.00191i 0.587785 + 0.809017i −2.95855 − 0.496976i −1.17528 − 1.30528i
17.8 0.951057 + 0.309017i 0.284431 + 1.70854i 0.809017 + 0.587785i 1.24540 + 0.719033i −0.257457 + 1.71281i 0.425612 − 4.04943i 0.587785 + 0.809017i −2.83820 + 0.971921i 0.962254 + 1.06869i
17.9 0.951057 + 0.309017i 0.605161 − 1.62289i 0.809017 + 0.587785i 1.31329 + 0.758228i 1.07704 − 1.35646i 0.0465366 − 0.442766i 0.587785 + 0.809017i −2.26756 − 1.96422i 1.01471 + 1.12695i
17.10 0.951057 + 0.309017i 1.70450 − 0.307717i 0.809017 + 0.587785i −1.72504 − 0.995952i 1.71616 + 0.234063i −0.0669056 + 0.636564i 0.587785 + 0.809017i 2.81062 − 1.04900i −1.33284 − 1.48027i
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.10
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
31.h odd 30 1 inner
93.p even 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 186.2.p.a ✓ 80
3.b odd 2 1 inner 186.2.p.a ✓ 80
31.h odd 30 1 inner 186.2.p.a ✓ 80
93.p even 30 1 inner 186.2.p.a ✓ 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
186.2.p.a ✓ 80 1.a even 1 1 trivial
186.2.p.a ✓ 80 3.b odd 2 1 inner
186.2.p.a ✓ 80 31.h odd 30 1 inner
186.2.p.a ✓ 80 93.p even 30 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(186, [\chi])\).