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Properties

Label 738.2.s.b
Level $738$
Weight $2$
Character orbit 738.s
Analytic conductor $5.893$
Analytic rank $0$
Dimension $176$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [738,2,Mod(133,738)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(738, base_ring=CyclotomicField(30)) chi = DirichletCharacter(H, H._module([20, 6])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("738.133"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 738 = 2 \cdot 3^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 738.s (of order \(15\), degree \(8\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [176] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.89295966917\)
Analytic rank: \(0\)
Dimension: \(176\)
Relative dimension: \(22\) over \(\Q(\zeta_{15})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 176 q + 22 q^{2} + 4 q^{3} + 22 q^{4} + 2 q^{5} - 7 q^{6} - 2 q^{7} - 44 q^{8} + 4 q^{9} - 4 q^{10} + 6 q^{11} - 7 q^{12} + 8 q^{14} + 3 q^{15} + 22 q^{16} + 20 q^{17} - 7 q^{18} - 12 q^{19} + 2 q^{20}+ \cdots + 23 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
133.1 0.913545 − 0.406737i −1.72248 − 0.181787i 0.669131 − 0.743145i 2.73264 − 3.03490i −1.64751 + 0.534527i 0.158369 + 1.50678i 0.309017 − 0.951057i 2.93391 + 0.626249i 1.26198 − 3.88398i
133.2 0.913545 − 0.406737i −1.71217 + 0.261665i 0.669131 − 0.743145i 0.213251 − 0.236839i −1.45772 + 0.935446i 0.502149 + 4.77763i 0.309017 − 0.951057i 2.86306 − 0.896031i 0.0984834 − 0.303101i
133.3 0.913545 − 0.406737i −1.65077 − 0.524358i 0.669131 − 0.743145i −2.30338 + 2.55816i −1.72133 + 0.192405i 0.0386506 + 0.367736i 0.309017 − 0.951057i 2.45010 + 1.73119i −1.06374 + 3.27386i
133.4 0.913545 − 0.406737i −1.53082 + 0.810303i 0.669131 − 0.743145i −1.72294 + 1.91352i −1.06889 + 1.36289i −0.359442 − 3.41987i 0.309017 − 0.951057i 1.68682 − 2.48086i −0.795687 + 2.44887i
133.5 0.913545 − 0.406737i −1.43831 + 0.965015i 0.669131 − 0.743145i 0.148397 − 0.164812i −0.921458 + 1.46660i −0.00595688 − 0.0566760i 0.309017 − 0.951057i 1.13749 − 2.77599i 0.0685326 − 0.210922i
133.6 0.913545 − 0.406737i −1.25797 − 1.19060i 0.669131 − 0.743145i 1.47464 − 1.63776i −1.63347 − 0.576002i −0.463983 − 4.41450i 0.309017 − 0.951057i 0.164964 + 2.99546i 0.681018 − 2.09596i
133.7 0.913545 − 0.406737i −0.845909 − 1.51144i 0.669131 − 0.743145i −0.564842 + 0.627320i −1.38753 − 1.03670i 0.369679 + 3.51726i 0.309017 − 0.951057i −1.56888 + 2.55707i −0.260854 + 0.802827i
133.8 0.913545 − 0.406737i −0.502062 + 1.65769i 0.669131 − 0.743145i 2.54079 − 2.82183i 0.215587 + 1.71858i −0.0377180 − 0.358863i 0.309017 − 0.951057i −2.49587 − 1.66452i 1.17338 − 3.61130i
133.9 0.913545 − 0.406737i −0.448012 + 1.67311i 0.669131 − 0.743145i −1.68705 + 1.87366i 0.271235 + 1.71068i −0.484875 − 4.61328i 0.309017 − 0.951057i −2.59857 − 1.49914i −0.779113 + 2.39786i
133.10 0.913545 − 0.406737i −0.367738 − 1.69256i 0.669131 − 0.743145i 0.215709 − 0.239569i −1.02437 − 1.39666i −0.160088 − 1.52313i 0.309017 − 0.951057i −2.72954 + 1.24484i 0.0996184 − 0.306594i
133.11 0.913545 − 0.406737i −0.276192 − 1.70989i 0.669131 − 0.743145i −2.70137 + 3.00017i −0.947788 − 1.44972i 0.115353 + 1.09751i 0.309017 − 0.951057i −2.84744 + 0.944516i −1.24754 + 3.83954i
133.12 0.913545 − 0.406737i −0.0560254 + 1.73114i 0.669131 − 0.743145i −0.392878 + 0.436335i 0.652938 + 1.60427i 0.182529 + 1.73665i 0.309017 − 0.951057i −2.99372 − 0.193976i −0.181438 + 0.558410i
133.13 0.913545 − 0.406737i 0.158109 − 1.72482i 0.669131 − 0.743145i 2.01177 − 2.23430i −0.557108 − 1.64001i 0.292492 + 2.78287i 0.309017 − 0.951057i −2.95000 − 0.545419i 0.929075 − 2.85940i
133.14 0.913545 − 0.406737i 0.820657 + 1.52529i 0.669131 − 0.743145i 1.47789 − 1.64136i 1.37010 + 1.05963i −0.362341 − 3.44744i 0.309017 − 0.951057i −1.65304 + 2.50349i 0.682517 − 2.10057i
133.15 0.913545 − 0.406737i 1.00247 − 1.41247i 0.669131 − 0.743145i −0.814759 + 0.904882i 0.341298 − 1.69809i −0.197266 − 1.87686i 0.309017 − 0.951057i −0.990118 − 2.83190i −0.376271 + 1.15804i
133.16 0.913545 − 0.406737i 1.22126 + 1.22822i 0.669131 − 0.743145i −1.05959 + 1.17680i 1.61524 + 0.625302i 0.308707 + 2.93715i 0.309017 − 0.951057i −0.0170417 + 2.99995i −0.489340 + 1.50603i
133.17 0.913545 − 0.406737i 1.26060 − 1.18781i 0.669131 − 0.743145i 1.42871 − 1.58674i 0.668486 − 1.59785i −0.268411 − 2.55376i 0.309017 − 0.951057i 0.178208 − 2.99470i 0.659804 − 2.03067i
133.18 0.913545 − 0.406737i 1.27255 − 1.17500i 0.669131 − 0.743145i −0.294230 + 0.326775i 0.684619 − 1.59100i 0.427454 + 4.06695i 0.309017 − 0.951057i 0.238770 − 2.99048i −0.135881 + 0.418198i
133.19 0.913545 − 0.406737i 1.45990 + 0.932030i 0.669131 − 0.743145i 1.79990 − 1.99900i 1.71278 + 0.257655i 0.158953 + 1.51234i 0.309017 − 0.951057i 1.26264 + 2.72135i 0.831229 − 2.55826i
133.20 0.913545 − 0.406737i 1.69660 − 0.348657i 0.669131 − 0.743145i 2.08937 − 2.32048i 1.40811 − 1.00858i 0.331646 + 3.15540i 0.309017 − 0.951057i 2.75688 − 1.18306i 0.964908 − 2.96968i
See next 80 embeddings (of 176 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 133.22
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
41.d even 5 1 inner
369.s even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 738.2.s.b ✓ 176
9.c even 3 1 inner 738.2.s.b ✓ 176
41.d even 5 1 inner 738.2.s.b ✓ 176
369.s even 15 1 inner 738.2.s.b ✓ 176
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
738.2.s.b ✓ 176 1.a even 1 1 trivial
738.2.s.b ✓ 176 9.c even 3 1 inner
738.2.s.b ✓ 176 41.d even 5 1 inner
738.2.s.b ✓ 176 369.s even 15 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{176} - 2 T_{5}^{175} - 65 T_{5}^{174} + 150 T_{5}^{173} + 1714 T_{5}^{172} + \cdots + 17\!\cdots\!00 \) acting on \(S_{2}^{\mathrm{new}}(738, [\chi])\). Copy content Toggle raw display