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Properties

Label 27.4.e.a
Level $27$
Weight $4$
Character orbit 27.e
Analytic conductor $1.593$
Analytic rank $0$
Dimension $48$
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] = [27,4,Mod(4,27)] 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("27.4"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(27, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([2])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 27 = 3^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 27.e (of order \(9\), degree \(6\), 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: \(1.59305157015\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(8\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 6 q^{2} - 6 q^{3} - 6 q^{4} + 6 q^{5} - 18 q^{6} - 6 q^{7} - 75 q^{8} - 54 q^{9} - 3 q^{10} + 57 q^{11} + 147 q^{12} - 6 q^{13} + 51 q^{14} - 36 q^{15} + 18 q^{16} - 207 q^{17} - 639 q^{18} - 3 q^{19}+ \cdots - 1242 q^{99}+O(q^{100}) \) 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 −4.97861 − 1.81207i −3.67671 − 3.67176i 15.3746 + 12.9008i −1.82513 + 10.3508i 11.6515 + 24.9427i −5.92244 + 4.96952i −31.9745 − 55.3815i 0.0364268 + 27.0000i 27.8430 − 48.2255i
4.2 −4.05233 − 1.47493i 4.37853 + 2.79794i 8.11761 + 6.81148i 3.22691 − 18.3007i −13.6165 − 17.7962i 14.4087 − 12.0903i −5.59920 − 9.69809i 11.3431 + 24.5017i −40.0687 + 69.4011i
4.3 −2.12171 − 0.772239i −0.789571 + 5.13581i −2.22306 − 1.86537i −3.33920 + 18.9376i 5.64131 − 10.2870i 0.500915 − 0.420318i 12.3077 + 21.3175i −25.7532 − 8.11018i 21.7091 − 37.6013i
4.4 −0.982993 − 0.357780i −5.19600 − 0.0395600i −5.29009 − 4.43891i 2.68017 − 15.2000i 5.09348 + 1.89791i −18.0349 + 15.1331i 7.79628 + 13.5036i 26.9969 + 0.411107i −8.07284 + 13.9826i
4.5 −0.932125 − 0.339266i 1.23404 − 5.04749i −5.37460 − 4.50982i 0.0250883 − 0.142283i −2.86272 + 4.28623i 19.4381 − 16.3105i 7.44756 + 12.8996i −23.9543 − 12.4576i −0.0716572 + 0.124114i
4.6 1.82340 + 0.663664i 5.12138 + 0.878355i −3.24401 − 2.72205i −0.470388 + 2.66770i 8.75540 + 5.00047i −9.12942 + 7.66050i −11.8703 − 20.5600i 25.4570 + 8.99678i −2.62817 + 4.55212i
4.7 3.53135 + 1.28531i −2.46998 + 4.57157i 4.69008 + 3.93544i 1.06026 − 6.01302i −14.5982 + 12.9691i 13.2194 − 11.0924i −3.52788 − 6.11047i −14.7984 − 22.5833i 11.4727 − 19.8713i
4.8 4.06758 + 1.48048i −1.65472 − 4.92564i 8.22505 + 6.90164i −0.745703 + 4.22909i 0.561612 − 22.4852i −15.6540 + 13.1353i 5.92381 + 10.2603i −21.5238 + 16.3011i −9.29428 + 16.0982i
7.1 −4.97861 + 1.81207i −3.67671 + 3.67176i 15.3746 − 12.9008i −1.82513 − 10.3508i 11.6515 − 24.9427i −5.92244 − 4.96952i −31.9745 + 55.3815i 0.0364268 − 27.0000i 27.8430 + 48.2255i
7.2 −4.05233 + 1.47493i 4.37853 − 2.79794i 8.11761 − 6.81148i 3.22691 + 18.3007i −13.6165 + 17.7962i 14.4087 + 12.0903i −5.59920 + 9.69809i 11.3431 − 24.5017i −40.0687 − 69.4011i
7.3 −2.12171 + 0.772239i −0.789571 − 5.13581i −2.22306 + 1.86537i −3.33920 − 18.9376i 5.64131 + 10.2870i 0.500915 + 0.420318i 12.3077 − 21.3175i −25.7532 + 8.11018i 21.7091 + 37.6013i
7.4 −0.982993 + 0.357780i −5.19600 + 0.0395600i −5.29009 + 4.43891i 2.68017 + 15.2000i 5.09348 − 1.89791i −18.0349 − 15.1331i 7.79628 − 13.5036i 26.9969 − 0.411107i −8.07284 − 13.9826i
7.5 −0.932125 + 0.339266i 1.23404 + 5.04749i −5.37460 + 4.50982i 0.0250883 + 0.142283i −2.86272 − 4.28623i 19.4381 + 16.3105i 7.44756 − 12.8996i −23.9543 + 12.4576i −0.0716572 − 0.124114i
7.6 1.82340 − 0.663664i 5.12138 − 0.878355i −3.24401 + 2.72205i −0.470388 − 2.66770i 8.75540 − 5.00047i −9.12942 − 7.66050i −11.8703 + 20.5600i 25.4570 − 8.99678i −2.62817 − 4.55212i
7.7 3.53135 − 1.28531i −2.46998 − 4.57157i 4.69008 − 3.93544i 1.06026 + 6.01302i −14.5982 − 12.9691i 13.2194 + 11.0924i −3.52788 + 6.11047i −14.7984 + 22.5833i 11.4727 + 19.8713i
7.8 4.06758 − 1.48048i −1.65472 + 4.92564i 8.22505 − 6.90164i −0.745703 − 4.22909i 0.561612 + 22.4852i −15.6540 − 13.1353i 5.92381 − 10.2603i −21.5238 − 16.3011i −9.29428 − 16.0982i
13.1 −0.874714 − 4.96075i 2.71299 − 4.43167i −16.3263 + 5.94230i 11.9326 + 10.0126i −24.3575 − 9.58200i −5.29732 − 1.92807i 23.6100 + 40.8938i −12.2794 − 24.0461i 39.2325 − 67.9527i
13.2 −0.592608 − 3.36085i −5.15968 − 0.614572i −3.42657 + 1.24717i −8.41296 − 7.05931i 0.992184 + 17.7051i 4.14024 + 1.50692i −7.42861 − 12.8667i 26.2446 + 6.34199i −18.7397 + 32.4581i
13.3 −0.404735 − 2.29536i 4.52897 + 2.54724i 2.41266 − 0.878135i −4.78113 − 4.01185i 4.01380 − 11.4266i 2.53990 + 0.924448i −12.3152 − 21.3306i 14.0232 + 23.0727i −7.27356 + 12.5982i
13.4 −0.0518956 − 0.294314i −2.82128 + 4.36353i 7.43361 − 2.70561i 15.3604 + 12.8889i 1.43066 + 0.603897i −20.1945 − 7.35018i −2.37749 − 4.11794i −11.0807 − 24.6215i 2.99626 − 5.18968i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.8
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
27.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 27.4.e.a ✓ 48
3.b odd 2 1 81.4.e.a 48
9.c even 3 1 243.4.e.c 48
9.c even 3 1 243.4.e.d 48
9.d odd 6 1 243.4.e.a 48
9.d odd 6 1 243.4.e.b 48
27.e even 9 1 inner 27.4.e.a ✓ 48
27.e even 9 1 243.4.e.c 48
27.e even 9 1 243.4.e.d 48
27.e even 9 1 729.4.a.d 24
27.f odd 18 1 81.4.e.a 48
27.f odd 18 1 243.4.e.a 48
27.f odd 18 1 243.4.e.b 48
27.f odd 18 1 729.4.a.c 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.4.e.a ✓ 48 1.a even 1 1 trivial
27.4.e.a ✓ 48 27.e even 9 1 inner
81.4.e.a 48 3.b odd 2 1
81.4.e.a 48 27.f odd 18 1
243.4.e.a 48 9.d odd 6 1
243.4.e.a 48 27.f odd 18 1
243.4.e.b 48 9.d odd 6 1
243.4.e.b 48 27.f odd 18 1
243.4.e.c 48 9.c even 3 1
243.4.e.c 48 27.e even 9 1
243.4.e.d 48 9.c even 3 1
243.4.e.d 48 27.e even 9 1
729.4.a.c 24 27.f odd 18 1
729.4.a.d 24 27.e even 9 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(27, [\chi])\).