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Properties

Label 185.2.n.a
Level $185$
Weight $2$
Character orbit 185.n
Analytic conductor $1.477$
Analytic rank $0$
Dimension $36$
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] = [185,2,Mod(84,185)] 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("185.84"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(185, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 185 = 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 185.n (of order \(6\), degree \(2\), 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: \(1.47723243739\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(18\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 16 q^{4} - 2 q^{5} - 16 q^{6} + 14 q^{9} - 12 q^{10} + 12 q^{11} - 8 q^{14} - 10 q^{15} - 16 q^{16} - 8 q^{19} + 22 q^{20} - 26 q^{21} - 42 q^{24} + 12 q^{26} - 16 q^{29} + 18 q^{34} - 16 q^{35}+ \cdots + 64 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
84.1 −2.33663 − 1.34906i 2.35586 − 1.36016i 2.63990 + 4.57244i 0.261363 − 2.22074i −7.33972 −1.64348 + 0.948866i − 8.84926i 2.20006 − 3.81062i −3.60661 + 4.83646i
84.2 −1.99875 − 1.15398i −1.81123 + 1.04571i 1.66334 + 2.88098i −0.0902553 − 2.23425i 4.82692 3.28655 − 1.89749i − 3.06190i 0.687027 − 1.18997i −2.39787 + 4.56985i
84.3 −1.85494 − 1.07095i 0.858786 − 0.495820i 1.29387 + 2.24104i −1.59352 + 1.56866i −2.12399 −1.20076 + 0.693262i − 1.25887i −1.00832 + 1.74647i 4.63584 − 1.20318i
84.4 −1.64422 − 0.949293i 1.16436 − 0.672243i 0.802314 + 1.38965i 1.62372 + 1.53738i −2.55262 2.34261 − 1.35251i 0.750648i −0.596178 + 1.03261i −1.21033 − 4.06918i
84.5 −1.58456 − 0.914845i −1.14999 + 0.663947i 0.673882 + 1.16720i 2.12799 − 0.686767i 2.42963 −2.49908 + 1.44285i 1.19339i −0.618348 + 1.07101i −4.00021 − 0.858562i
84.6 −0.843193 − 0.486818i −1.64972 + 0.952467i −0.526017 − 0.911089i −1.79344 + 1.33551i 1.85471 3.89368 − 2.24802i 2.97157i 0.314386 − 0.544533i 2.16236 − 0.253015i
84.7 −0.704704 − 0.406861i −0.142682 + 0.0823772i −0.668928 − 1.15862i −1.50396 − 1.65472i 0.134064 −2.76887 + 1.59861i 2.71609i −1.48643 + 2.57457i 0.386605 + 1.77799i
84.8 −0.398514 − 0.230082i 2.88437 − 1.66529i −0.894124 − 1.54867i −2.18772 + 0.462462i −1.53262 −0.0944407 + 0.0545254i 1.74322i 4.04641 − 7.00858i 0.978243 + 0.319058i
84.9 −0.153805 − 0.0887996i −1.48057 + 0.854808i −0.984229 − 1.70474i 0.854987 + 2.06616i 0.303627 −1.21910 + 0.703845i 0.704795i −0.0386059 + 0.0668674i 0.0519722 − 0.393709i
84.10 0.153805 + 0.0887996i 1.48057 − 0.854808i −0.984229 − 1.70474i 1.36185 + 1.77352i 0.303627 1.21910 − 0.703845i − 0.704795i −0.0386059 + 0.0668674i 0.0519722 + 0.393709i
84.11 0.398514 + 0.230082i −2.88437 + 1.66529i −0.894124 − 1.54867i 1.49437 − 1.66339i −1.53262 0.0944407 − 0.0545254i − 1.74322i 4.04641 − 7.00858i 0.978243 − 0.319058i
84.12 0.704704 + 0.406861i 0.142682 − 0.0823772i −0.668928 − 1.15862i −0.681051 − 2.12983i 0.134064 2.76887 − 1.59861i − 2.71609i −1.48643 + 2.57457i 0.386605 − 1.77799i
84.13 0.843193 + 0.486818i 1.64972 − 0.952467i −0.526017 − 0.911089i 2.05330 − 0.885408i 1.85471 −3.89368 + 2.24802i − 2.97157i 0.314386 − 0.544533i 2.16236 + 0.253015i
84.14 1.58456 + 0.914845i 1.14999 − 0.663947i 0.673882 + 1.16720i −1.65875 + 1.49951i 2.42963 2.49908 − 1.44285i − 1.19339i −0.618348 + 1.07101i −4.00021 + 0.858562i
84.15 1.64422 + 0.949293i −1.16436 + 0.672243i 0.802314 + 1.38965i 0.519553 + 2.17487i −2.55262 −2.34261 + 1.35251i − 0.750648i −0.596178 + 1.03261i −1.21033 + 4.06918i
84.16 1.85494 + 1.07095i −0.858786 + 0.495820i 1.29387 + 2.24104i 2.15526 − 0.595704i −2.12399 1.20076 − 0.693262i 1.25887i −1.00832 + 1.74647i 4.63584 + 1.20318i
84.17 1.99875 + 1.15398i 1.81123 − 1.04571i 1.66334 + 2.88098i −1.88979 − 1.19529i 4.82692 −3.28655 + 1.89749i 3.06190i 0.687027 − 1.18997i −2.39787 − 4.56985i
84.18 2.33663 + 1.34906i −2.35586 + 1.36016i 2.63990 + 4.57244i −2.05390 − 0.884024i −7.33972 1.64348 − 0.948866i 8.84926i 2.20006 − 3.81062i −3.60661 − 4.83646i
174.1 −2.33663 + 1.34906i 2.35586 + 1.36016i 2.63990 − 4.57244i 0.261363 + 2.22074i −7.33972 −1.64348 − 0.948866i 8.84926i 2.20006 + 3.81062i −3.60661 − 4.83646i
174.2 −1.99875 + 1.15398i −1.81123 − 1.04571i 1.66334 − 2.88098i −0.0902553 + 2.23425i 4.82692 3.28655 + 1.89749i 3.06190i 0.687027 + 1.18997i −2.39787 − 4.56985i
See all 36 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 84.18
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
37.c even 3 1 inner
185.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 185.2.n.a ✓ 36
5.b even 2 1 inner 185.2.n.a ✓ 36
5.c odd 4 2 925.2.e.f 36
37.c even 3 1 inner 185.2.n.a ✓ 36
185.n even 6 1 inner 185.2.n.a ✓ 36
185.s odd 12 2 925.2.e.f 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
185.2.n.a ✓ 36 1.a even 1 1 trivial
185.2.n.a ✓ 36 5.b even 2 1 inner
185.2.n.a ✓ 36 37.c even 3 1 inner
185.2.n.a ✓ 36 185.n even 6 1 inner
925.2.e.f 36 5.c odd 4 2
925.2.e.f 36 185.s odd 12 2

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(185, [\chi])\).