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Properties

Label 171.3.j.a
Level $171$
Weight $3$
Character orbit 171.j
Analytic conductor $4.659$
Analytic rank $0$
Dimension $76$
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] = [171,3,Mod(68,171)] 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("171.68"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(171, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([5, 4])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 171 = 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 171.j (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: \(4.65941252056\)
Analytic rank: \(0\)
Dimension: \(76\)
Relative dimension: \(38\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 76 q - 5 q^{3} - 146 q^{4} - 3 q^{5} + 23 q^{6} - q^{7} - 13 q^{9} + 6 q^{10} - 24 q^{11} + 15 q^{12} + 8 q^{13} - 3 q^{14} - 14 q^{15} + 262 q^{16} - 135 q^{17} + 30 q^{18} + 4 q^{19} + 69 q^{20} - 40 q^{21}+ \cdots - 64 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
68.1 − 3.82545i 2.13284 − 2.10974i −10.6341 −5.58931 + 3.22699i −8.07069 − 8.15907i −1.96728 − 3.40743i 25.3782i 0.0980157 − 8.99947i 12.3447 + 21.3816i
68.2 − 3.59401i −1.31663 + 2.69564i −8.91687 8.32337 − 4.80550i 9.68815 + 4.73198i −2.79484 − 4.84080i 17.6713i −5.53297 − 7.09833i −17.2710 − 29.9142i
68.3 − 3.52844i 1.16233 + 2.76568i −8.44990 −2.10325 + 1.21431i 9.75854 − 4.10122i 6.90575 + 11.9611i 15.7012i −6.29798 + 6.42927i 4.28463 + 7.42120i
68.4 − 3.47060i −2.99457 + 0.180454i −8.04505 −2.80877 + 1.62164i 0.626282 + 10.3929i 1.88424 + 3.26360i 14.0387i 8.93487 − 1.08076i 5.62807 + 9.74810i
68.5 − 3.32549i −1.77023 − 2.42204i −7.05886 2.83609 − 1.63742i −8.05448 + 5.88686i −4.41382 − 7.64495i 10.1722i −2.73260 + 8.57513i −5.44521 − 9.43138i
68.6 − 3.10239i 2.99536 + 0.166742i −5.62485 3.72606 − 2.15124i 0.517300 − 9.29280i 0.122000 + 0.211310i 5.04093i 8.94439 + 0.998906i −6.67400 − 11.5597i
68.7 − 2.85275i 0.843129 + 2.87909i −4.13816 −5.53158 + 3.19366i 8.21330 − 2.40523i −5.89881 − 10.2170i 0.394147i −7.57827 + 4.85488i 9.11071 + 15.7802i
68.8 − 2.40847i 1.11760 − 2.78406i −1.80072 2.42738 − 1.40145i −6.70532 − 2.69169i 1.31674 + 2.28066i − 5.29689i −6.50196 − 6.22290i −3.37535 − 5.84627i
68.9 − 2.24727i −1.89263 + 2.32765i −1.05020 −1.86611 + 1.07740i 5.23085 + 4.25324i 0.850966 + 1.47392i − 6.62898i −1.83590 − 8.81076i 2.42120 + 4.19365i
68.10 − 2.14545i −0.852190 − 2.87642i −0.602949 −8.38130 + 4.83894i −6.17120 + 1.82833i 3.39934 + 5.88784i − 7.28820i −7.54755 + 4.90250i 10.3817 + 17.9816i
68.11 − 1.84565i −2.85716 − 0.914684i 0.593585 7.52822 − 4.34642i −1.68819 + 5.27331i 6.64307 + 11.5061i − 8.47814i 7.32670 + 5.22680i −8.02196 − 13.8944i
68.12 − 1.74490i −2.89076 + 0.802171i 0.955312 −1.19670 + 0.690913i 1.39971 + 5.04411i −2.54525 − 4.40851i − 8.64654i 7.71304 − 4.63778i 1.20558 + 2.08812i
68.13 − 1.51035i 2.85016 + 0.936275i 1.71884 −5.52694 + 3.19098i 1.41410 − 4.30473i 3.52050 + 6.09769i − 8.63745i 7.24678 + 5.33706i 4.81950 + 8.34762i
68.14 − 1.43562i 2.12101 + 2.12163i 1.93899 4.29193 − 2.47795i 3.04586 − 3.04497i −4.02177 − 6.96591i − 8.52615i −0.00260476 + 9.00000i −3.55740 − 6.16159i
68.15 − 1.01670i −1.89164 − 2.32845i 2.96632 0.598131 − 0.345331i −2.36734 + 1.92323i −3.52922 − 6.11279i − 7.08266i −1.84340 + 8.80919i −0.351098 − 0.608119i
68.16 − 0.893947i 0.637080 + 2.93157i 3.20086 4.25671 − 2.45761i 2.62067 − 0.569516i 3.03123 + 5.25024i − 6.43719i −8.18826 + 3.73530i −2.19697 − 3.80527i
68.17 − 0.704384i 2.27899 − 1.95095i 3.50384 −3.99773 + 2.30809i −1.37422 − 1.60528i −5.58841 − 9.67942i − 5.28559i 1.38760 − 8.89239i 1.62578 + 2.81594i
68.18 − 0.236363i 2.74241 − 1.21623i 3.94413 1.55464 − 0.897570i −0.287471 − 0.648204i 4.33853 + 7.51455i − 1.87770i 6.04159 − 6.67077i −0.212153 − 0.367459i
68.19 0.0424081i −0.913329 + 2.85759i 3.99820 −6.00310 + 3.46589i −0.121185 − 0.0387325i 1.72622 + 2.98990i 0.339188i −7.33166 − 5.21984i −0.146982 − 0.254580i
68.20 0.220265i −2.45385 + 1.72587i 3.95148 4.96938 − 2.86907i −0.380150 − 0.540498i −3.79245 − 6.56872i 1.75144i 3.04274 − 8.47005i 0.631957 + 1.09458i
See all 76 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 68.38
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
171.j odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 171.3.j.a ✓ 76
3.b odd 2 1 513.3.j.a 76
9.c even 3 1 513.3.n.a 76
9.d odd 6 1 171.3.n.a yes 76
19.c even 3 1 171.3.n.a yes 76
57.h odd 6 1 513.3.n.a 76
171.h even 3 1 513.3.j.a 76
171.j odd 6 1 inner 171.3.j.a ✓ 76
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
171.3.j.a ✓ 76 1.a even 1 1 trivial
171.3.j.a ✓ 76 171.j odd 6 1 inner
171.3.n.a yes 76 9.d odd 6 1
171.3.n.a yes 76 19.c even 3 1
513.3.j.a 76 3.b odd 2 1
513.3.j.a 76 171.h even 3 1
513.3.n.a 76 9.c even 3 1
513.3.n.a 76 57.h odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(171, [\chi])\).