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Properties

Label 27.3.f.a
Level $27$
Weight $3$
Character orbit 27.f
Analytic conductor $0.736$
Analytic rank $0$
Dimension $30$
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] = [27,3,Mod(2,27)] 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("27.2"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(27, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 27 = 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 27.f (of order \(18\), 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: \(0.735696713773\)
Analytic rank: \(0\)
Dimension: \(30\)
Relative dimension: \(5\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 30 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 15 q^{5} - 18 q^{6} - 6 q^{7} - 9 q^{8} - 3 q^{10} - 6 q^{11} - 15 q^{12} - 6 q^{13} - 15 q^{14} - 9 q^{15} - 18 q^{16} - 9 q^{17} + 63 q^{18} - 3 q^{19} + 213 q^{20}+ \cdots + 513 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −3.46291 − 0.610605i 1.47633 + 2.61160i 7.86014 + 2.86086i 3.16671 + 3.77394i −3.51774 − 9.94519i −3.18911 + 1.16074i −13.2912 − 7.67367i −4.64090 + 7.71116i −8.66165 − 15.0024i
2.2 −2.31604 − 0.408381i −1.62484 − 2.52189i 1.43851 + 0.523575i −3.71692 − 4.42965i 2.73330 + 6.50435i 4.57693 − 1.66587i 5.02894 + 2.90346i −3.71981 + 8.19530i 6.79956 + 11.7772i
2.3 0.115908 + 0.0204377i 2.32380 − 1.89736i −3.74575 − 1.36334i 3.98394 + 4.74788i 0.308125 − 0.172426i −7.49258 + 2.72708i −0.814011 − 0.469969i 1.80008 − 8.81815i 0.364735 + 0.631740i
2.4 1.14332 + 0.201599i 1.10490 + 2.78912i −2.49222 − 0.907094i −3.46013 − 4.12362i 0.700975 + 3.41162i 9.89907 − 3.60297i −6.68824 − 3.86146i −6.55839 + 6.16340i −3.12473 − 5.41219i
2.5 2.58003 + 0.454929i −2.92175 − 0.680712i 2.69082 + 0.979377i 0.519123 + 0.618667i −7.22853 − 3.08544i −5.56035 + 2.02380i −2.57852 − 1.48871i 8.07326 + 3.97774i 1.05790 + 1.83234i
5.1 −2.17948 − 2.59740i −2.99958 − 0.0504108i −1.30178 + 7.38274i −1.19547 − 3.28452i 6.40658 + 7.90098i −1.88718 − 10.7027i 10.2675 − 5.92795i 8.99492 + 0.302422i −5.92572 + 10.2637i
5.2 −1.24712 − 1.48626i 2.80515 + 1.06355i 0.0409354 − 0.232156i −1.47839 − 4.06185i −1.91764 − 5.49555i 1.54363 + 8.75434i −7.11704 + 4.10903i 6.73771 + 5.96685i −4.19322 + 7.26288i
5.3 −0.374063 − 0.445791i −0.428198 − 2.96928i 0.635786 − 3.60572i 2.62195 + 7.20376i −1.16351 + 1.30159i 0.231638 + 1.31369i −3.86112 + 2.22922i −8.63329 + 2.54288i 2.23059 − 3.86350i
5.4 0.837612 + 0.998227i −0.987122 + 2.83295i 0.399729 − 2.26698i 0.149473 + 0.410673i −3.65475 + 1.38754i −1.05651 − 5.99176i 7.11182 − 4.10601i −7.05118 − 5.59293i −0.284745 + 0.493192i
5.5 2.13670 + 2.54642i −2.03568 − 2.20363i −1.22417 + 6.94260i −2.35247 − 6.46335i 1.26173 − 9.89219i 1.10811 + 6.28443i −8.77937 + 5.06877i −0.712002 + 8.97179i 11.4319 − 19.8006i
11.1 −2.17948 + 2.59740i −2.99958 + 0.0504108i −1.30178 − 7.38274i −1.19547 + 3.28452i 6.40658 − 7.90098i −1.88718 + 10.7027i 10.2675 + 5.92795i 8.99492 − 0.302422i −5.92572 − 10.2637i
11.2 −1.24712 + 1.48626i 2.80515 − 1.06355i 0.0409354 + 0.232156i −1.47839 + 4.06185i −1.91764 + 5.49555i 1.54363 − 8.75434i −7.11704 − 4.10903i 6.73771 − 5.96685i −4.19322 − 7.26288i
11.3 −0.374063 + 0.445791i −0.428198 + 2.96928i 0.635786 + 3.60572i 2.62195 − 7.20376i −1.16351 − 1.30159i 0.231638 − 1.31369i −3.86112 − 2.22922i −8.63329 − 2.54288i 2.23059 + 3.86350i
11.4 0.837612 − 0.998227i −0.987122 − 2.83295i 0.399729 + 2.26698i 0.149473 − 0.410673i −3.65475 − 1.38754i −1.05651 + 5.99176i 7.11182 + 4.10601i −7.05118 + 5.59293i −0.284745 − 0.493192i
11.5 2.13670 − 2.54642i −2.03568 + 2.20363i −1.22417 − 6.94260i −2.35247 + 6.46335i 1.26173 + 9.89219i 1.10811 − 6.28443i −8.77937 − 5.06877i −0.712002 − 8.97179i 11.4319 + 19.8006i
14.1 −3.46291 + 0.610605i 1.47633 − 2.61160i 7.86014 − 2.86086i 3.16671 − 3.77394i −3.51774 + 9.94519i −3.18911 − 1.16074i −13.2912 + 7.67367i −4.64090 − 7.71116i −8.66165 + 15.0024i
14.2 −2.31604 + 0.408381i −1.62484 + 2.52189i 1.43851 − 0.523575i −3.71692 + 4.42965i 2.73330 − 6.50435i 4.57693 + 1.66587i 5.02894 − 2.90346i −3.71981 − 8.19530i 6.79956 − 11.7772i
14.3 0.115908 − 0.0204377i 2.32380 + 1.89736i −3.74575 + 1.36334i 3.98394 − 4.74788i 0.308125 + 0.172426i −7.49258 − 2.72708i −0.814011 + 0.469969i 1.80008 + 8.81815i 0.364735 − 0.631740i
14.4 1.14332 − 0.201599i 1.10490 − 2.78912i −2.49222 + 0.907094i −3.46013 + 4.12362i 0.700975 − 3.41162i 9.89907 + 3.60297i −6.68824 + 3.86146i −6.55839 − 6.16340i −3.12473 + 5.41219i
14.5 2.58003 − 0.454929i −2.92175 + 0.680712i 2.69082 − 0.979377i 0.519123 − 0.618667i −7.22853 + 3.08544i −5.56035 − 2.02380i −2.57852 + 1.48871i 8.07326 − 3.97774i 1.05790 − 1.83234i
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.5
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
27.f odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 27.3.f.a ✓ 30
3.b odd 2 1 81.3.f.a 30
4.b odd 2 1 432.3.bc.a 30
9.c even 3 1 243.3.f.c 30
9.c even 3 1 243.3.f.d 30
9.d odd 6 1 243.3.f.a 30
9.d odd 6 1 243.3.f.b 30
27.e even 9 1 81.3.f.a 30
27.e even 9 1 243.3.f.a 30
27.e even 9 1 243.3.f.b 30
27.e even 9 1 729.3.b.a 30
27.f odd 18 1 inner 27.3.f.a ✓ 30
27.f odd 18 1 243.3.f.c 30
27.f odd 18 1 243.3.f.d 30
27.f odd 18 1 729.3.b.a 30
108.l even 18 1 432.3.bc.a 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.3.f.a ✓ 30 1.a even 1 1 trivial
27.3.f.a ✓ 30 27.f odd 18 1 inner
81.3.f.a 30 3.b odd 2 1
81.3.f.a 30 27.e even 9 1
243.3.f.a 30 9.d odd 6 1
243.3.f.a 30 27.e even 9 1
243.3.f.b 30 9.d odd 6 1
243.3.f.b 30 27.e even 9 1
243.3.f.c 30 9.c even 3 1
243.3.f.c 30 27.f odd 18 1
243.3.f.d 30 9.c even 3 1
243.3.f.d 30 27.f odd 18 1
432.3.bc.a 30 4.b odd 2 1
432.3.bc.a 30 108.l even 18 1
729.3.b.a 30 27.e even 9 1
729.3.b.a 30 27.f odd 18 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(27, [\chi])\).