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Properties

Label 738.2.i.b
Level $738$
Weight $2$
Character orbit 738.i
Analytic conductor $5.893$
Analytic rank $0$
Dimension $36$
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(409,738)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(738, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 3])) 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("738.409"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 738 = 2 \cdot 3^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 738.i (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [36] 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: \(36\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 18 q^{2} - 18 q^{4} + 4 q^{5} + 36 q^{8} - 14 q^{9} - 8 q^{10} - 18 q^{16} + 4 q^{18} + 4 q^{20} - 16 q^{21} - 4 q^{23} - 26 q^{25} + 8 q^{31} - 18 q^{32} - 6 q^{33} + 10 q^{36} - 60 q^{37} - 64 q^{39}+ \cdots - 76 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
409.1 −0.500000 − 0.866025i −1.68332 + 0.407978i −0.500000 + 0.866025i 1.84171 − 3.18994i 1.19498 + 1.25381i 3.70331 − 2.13810i 1.00000 2.66711 − 1.37351i −3.68343
409.2 −0.500000 − 0.866025i −1.57206 + 0.727065i −0.500000 + 0.866025i 0.714167 − 1.23697i 1.41569 + 0.997912i −2.06322 + 1.19120i 1.00000 1.94275 − 2.28598i −1.42833
409.3 −0.500000 − 0.866025i −1.42570 − 0.983551i −0.500000 + 0.866025i 1.13494 − 1.96578i −0.138929 + 1.72647i −4.05054 + 2.33858i 1.00000 1.06526 + 2.80450i −2.26989
409.4 −0.500000 − 0.866025i −1.35012 + 1.08497i −0.500000 + 0.866025i −1.58544 + 2.74607i 1.61468 + 0.626753i 1.86289 − 1.07554i 1.00000 0.645660 − 2.92970i 3.17089
409.5 −0.500000 − 0.866025i −1.23882 − 1.21051i −0.500000 + 0.866025i 0.290704 − 0.503513i −0.428919 + 1.67810i 1.28705 − 0.743077i 1.00000 0.0693495 + 2.99920i −0.581407
409.6 −0.500000 − 0.866025i −0.793227 − 1.53974i −0.500000 + 0.866025i −1.18467 + 2.05191i −0.936838 + 1.45682i 4.22182 − 2.43747i 1.00000 −1.74158 + 2.44272i 2.36934
409.7 −0.500000 − 0.866025i −0.528977 + 1.64930i −0.500000 + 0.866025i 1.55925 − 2.70071i 1.69282 − 0.366541i −0.0882889 + 0.0509736i 1.00000 −2.44037 − 1.74488i −3.11851
409.8 −0.500000 − 0.866025i −0.359981 + 1.69423i −0.500000 + 0.866025i −0.0761831 + 0.131953i 1.64724 − 0.535362i 1.94692 − 1.12405i 1.00000 −2.74083 − 1.21978i 0.152366
409.9 −0.500000 − 0.866025i −0.127771 − 1.72733i −0.500000 + 0.866025i −1.69449 + 2.93494i −1.43203 + 0.974319i −0.610625 + 0.352544i 1.00000 −2.96735 + 0.441407i 3.38898
409.10 −0.500000 − 0.866025i 0.127771 + 1.72733i −0.500000 + 0.866025i −1.69449 + 2.93494i 1.43203 − 0.974319i 0.610625 − 0.352544i 1.00000 −2.96735 + 0.441407i 3.38898
409.11 −0.500000 − 0.866025i 0.359981 − 1.69423i −0.500000 + 0.866025i −0.0761831 + 0.131953i −1.64724 + 0.535362i −1.94692 + 1.12405i 1.00000 −2.74083 − 1.21978i 0.152366
409.12 −0.500000 − 0.866025i 0.528977 − 1.64930i −0.500000 + 0.866025i 1.55925 − 2.70071i −1.69282 + 0.366541i 0.0882889 − 0.0509736i 1.00000 −2.44037 − 1.74488i −3.11851
409.13 −0.500000 − 0.866025i 0.793227 + 1.53974i −0.500000 + 0.866025i −1.18467 + 2.05191i 0.936838 − 1.45682i −4.22182 + 2.43747i 1.00000 −1.74158 + 2.44272i 2.36934
409.14 −0.500000 − 0.866025i 1.23882 + 1.21051i −0.500000 + 0.866025i 0.290704 − 0.503513i 0.428919 − 1.67810i −1.28705 + 0.743077i 1.00000 0.0693495 + 2.99920i −0.581407
409.15 −0.500000 − 0.866025i 1.35012 − 1.08497i −0.500000 + 0.866025i −1.58544 + 2.74607i −1.61468 − 0.626753i −1.86289 + 1.07554i 1.00000 0.645660 − 2.92970i 3.17089
409.16 −0.500000 − 0.866025i 1.42570 + 0.983551i −0.500000 + 0.866025i 1.13494 − 1.96578i 0.138929 − 1.72647i 4.05054 − 2.33858i 1.00000 1.06526 + 2.80450i −2.26989
409.17 −0.500000 − 0.866025i 1.57206 − 0.727065i −0.500000 + 0.866025i 0.714167 − 1.23697i −1.41569 − 0.997912i 2.06322 − 1.19120i 1.00000 1.94275 − 2.28598i −1.42833
409.18 −0.500000 − 0.866025i 1.68332 − 0.407978i −0.500000 + 0.866025i 1.84171 − 3.18994i −1.19498 − 1.25381i −3.70331 + 2.13810i 1.00000 2.66711 − 1.37351i −3.68343
655.1 −0.500000 + 0.866025i −1.68332 − 0.407978i −0.500000 − 0.866025i 1.84171 + 3.18994i 1.19498 − 1.25381i 3.70331 + 2.13810i 1.00000 2.66711 + 1.37351i −3.68343
655.2 −0.500000 + 0.866025i −1.57206 − 0.727065i −0.500000 − 0.866025i 0.714167 + 1.23697i 1.41569 − 0.997912i −2.06322 − 1.19120i 1.00000 1.94275 + 2.28598i −1.42833
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 409.18
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.b even 2 1 inner
369.i even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 738.2.i.b ✓ 36
3.b odd 2 1 2214.2.i.b 36
9.c even 3 1 inner 738.2.i.b ✓ 36
9.d odd 6 1 2214.2.i.b 36
41.b even 2 1 inner 738.2.i.b ✓ 36
123.b odd 2 1 2214.2.i.b 36
369.i even 6 1 inner 738.2.i.b ✓ 36
369.k odd 6 1 2214.2.i.b 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
738.2.i.b ✓ 36 1.a even 1 1 trivial
738.2.i.b ✓ 36 9.c even 3 1 inner
738.2.i.b ✓ 36 41.b even 2 1 inner
738.2.i.b ✓ 36 369.i even 6 1 inner
2214.2.i.b 36 3.b odd 2 1
2214.2.i.b 36 9.d odd 6 1
2214.2.i.b 36 123.b odd 2 1
2214.2.i.b 36 369.k odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{18} - 2 T_{5}^{17} + 31 T_{5}^{16} - 52 T_{5}^{15} + 608 T_{5}^{14} - 979 T_{5}^{13} + \cdots + 7056 \) acting on \(S_{2}^{\mathrm{new}}(738, [\chi])\). Copy content Toggle raw display