Properties

Label 137.4.f.a
Level $137$
Weight $4$
Character orbit 137.f
Analytic conductor $8.083$
Analytic rank $0$
Dimension $544$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [137,4,Mod(4,137)] 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("137.4"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(137, base_ring=CyclotomicField(34)) chi = DirichletCharacter(H, H._module([5])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 137 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 137.f (of order \(34\), degree \(16\), 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: \(8.08326167079\)
Analytic rank: \(0\)
Dimension: \(544\)
Relative dimension: \(34\) over \(\Q(\zeta_{34})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{34}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 544 q - 13 q^{2} - 17 q^{3} - 231 q^{4} - 17 q^{5} - 17 q^{6} + 29 q^{7} + 67 q^{8} + 293 q^{9} - 13 q^{11} - 221 q^{12} - 17 q^{13} - 125 q^{14} - 193 q^{15} - 1579 q^{16} - 51 q^{17} - 1023 q^{18} - 5 q^{19}+ \cdots + 3681 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −0.500113 − 5.39707i 8.07542 + 4.02108i −21.0145 + 3.92829i 8.48016 + 0.785803i 17.6635 − 45.5946i 19.4087 + 12.0174i 19.8445 + 69.7460i 32.7722 + 43.3974i − 46.1610i
4.2 −0.499552 − 5.39102i −1.61193 − 0.802643i −20.9498 + 3.91620i −13.5238 − 1.25316i −3.52183 + 9.09089i 10.0975 + 6.25209i 19.7247 + 69.3251i −14.3171 − 18.9589i 73.5329i
4.3 −0.452325 − 4.88137i −8.07038 − 4.01857i −15.7594 + 2.94594i 9.17291 + 0.849995i −15.9657 + 41.2122i 24.8176 + 15.3664i 10.7760 + 37.8737i 32.7110 + 43.3163i − 45.1608i
4.4 −0.438082 − 4.72765i 1.95644 + 0.974190i −14.2950 + 2.67220i −1.11331 − 0.103163i 3.74855 − 9.67613i −10.4237 − 6.45411i 8.50102 + 29.8780i −13.3925 − 17.7346i 5.30852i
4.5 −0.435184 − 4.69638i 0.148622 + 0.0740051i −14.0028 + 2.61758i 20.9005 + 1.93671i 0.282878 − 0.730193i −12.2029 − 7.55572i 8.06113 + 28.3319i −16.2545 − 21.5245i − 98.9995i
4.6 −0.434369 − 4.68759i −6.36160 − 3.16770i −13.9211 + 2.60230i −0.896581 − 0.0830804i −12.0856 + 31.1965i −29.1399 − 18.0427i 7.93885 + 27.9022i 14.1645 + 18.7568i 4.23889i
4.7 −0.383947 − 4.14345i 8.14274 + 4.05460i −9.15699 + 1.71174i −18.8775 − 1.74926i 13.6737 − 35.2958i −29.6623 − 18.3661i 1.49816 + 5.26550i 33.5932 + 44.4847i 78.8895i
4.8 −0.345749 − 3.73123i 4.15266 + 2.06778i −5.93872 + 1.11014i −0.160741 − 0.0148948i 6.27957 − 16.2094i 10.5300 + 6.51988i −2.00830 − 7.05843i −3.30227 − 4.37291i 0.604909i
4.9 −0.300444 − 3.24230i 1.12272 + 0.559047i −2.55848 + 0.478263i −13.7838 − 1.27725i 1.47529 − 3.80815i 11.1749 + 6.91919i −4.80945 − 16.9035i −15.3232 − 20.2912i 45.0748i
4.10 −0.297463 − 3.21013i −3.19996 − 1.59339i −2.35268 + 0.439793i 10.7812 + 0.999026i −4.16312 + 10.7463i 20.0668 + 12.4248i −4.94644 − 17.3849i −8.57030 − 11.3489i − 34.9063i
4.11 −0.282866 − 3.05261i −7.20079 − 3.58557i −1.37461 + 0.256959i −19.8321 − 1.83772i −8.90847 + 22.9954i 0.761574 + 0.471546i −5.53849 − 19.4658i 22.7240 + 30.0914i 61.0596i
4.12 −0.227026 − 2.45000i −4.79900 − 2.38962i 1.91285 − 0.357573i 4.08364 + 0.378405i −4.76505 + 12.3000i −8.60938 − 5.33070i −6.69708 − 23.5378i 1.04897 + 1.38905i − 10.0908i
4.13 −0.204591 − 2.20788i 5.97094 + 2.97317i 3.03089 − 0.566572i 16.3136 + 1.51168i 5.34282 − 13.7914i −20.2519 − 12.5394i −6.72545 − 23.6375i 10.5412 + 13.9588i − 36.3278i
4.14 −0.204190 − 2.20356i 7.55886 + 3.76386i 3.04982 − 0.570110i 5.16412 + 0.478526i 6.75044 − 17.4249i 14.2409 + 8.81761i −6.72393 − 23.6322i 26.6985 + 35.3545i − 11.4771i
4.15 −0.0745552 − 0.804579i 0.913439 + 0.454838i 7.22200 − 1.35003i −6.18805 − 0.573408i 0.297852 − 0.768844i −19.7351 − 12.2195i −3.39365 − 11.9275i −15.6436 − 20.7155i 5.02153i
4.16 −0.0502622 − 0.542415i 6.16434 + 3.06948i 7.57210 − 1.41547i −14.9668 − 1.38688i 1.35510 − 3.49791i 16.9221 + 10.4777i −2.34096 − 8.22763i 12.3063 + 16.2962i 8.18794i
4.17 −0.00891234 − 0.0961795i −8.89434 − 4.42886i 7.85461 − 1.46828i 13.2220 + 1.22520i −0.346696 + 0.894924i −2.34663 − 1.45297i −0.422690 − 1.48560i 43.2234 + 57.2371i − 1.28261i
4.18 −0.00188101 − 0.0202993i 0.538681 + 0.268231i 7.86338 − 1.46992i 16.2169 + 1.50272i 0.00443165 − 0.0114394i 13.6307 + 8.43978i −0.0892613 − 0.313721i −16.0529 − 21.2575i − 0.332019i
4.19 0.00834223 + 0.0900270i −4.21490 − 2.09877i 7.85575 − 1.46849i −8.28063 − 0.767313i 0.153784 − 0.396963i 27.2219 + 16.8551i 0.395679 + 1.39067i −2.91057 − 3.85422i − 0.751881i
4.20 0.0668061 + 0.720953i −5.98279 − 2.97908i 7.34848 − 1.37367i −10.0589 − 0.932091i 1.74809 − 4.51233i −5.41908 − 3.35535i 3.06642 + 10.7773i 10.6478 + 14.0999i − 7.31423i
See next 80 embeddings (of 544 total)
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 4.34
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
137.f even 34 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 137.4.f.a ✓ 544
137.f even 34 1 inner 137.4.f.a ✓ 544
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
137.4.f.a ✓ 544 1.a even 1 1 trivial
137.4.f.a ✓ 544 137.f even 34 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(137, [\chi])\).