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Properties

Label 243.3.f.a
Level $243$
Weight $3$
Character orbit 243.f
Analytic conductor $6.621$
Analytic rank $0$
Dimension $30$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [243,3,Mod(26,243)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(243, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("243.26");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 243 = 3^{5} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 243.f (of order \(18\), degree \(6\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.62127042396\)
Analytic rank: \(0\)
Dimension: \(30\)
Relative dimension: \(5\) over \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 27)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 30 q - 3 q^{2} + 3 q^{4} - 21 q^{5} + 3 q^{7} + 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q - 3 q^{2} + 3 q^{4} - 21 q^{5} + 3 q^{7} + 9 q^{8} - 3 q^{10} - 57 q^{11} + 3 q^{13} + 114 q^{14} + 27 q^{16} + 9 q^{17} - 3 q^{19} + 183 q^{20} + 75 q^{22} - 48 q^{23} + 21 q^{25} - 12 q^{28} + 78 q^{29} - 87 q^{31} - 243 q^{32} - 153 q^{34} + 252 q^{35} - 3 q^{37} - 321 q^{38} - 168 q^{40} + 357 q^{41} - 87 q^{43} - 639 q^{44} - 3 q^{46} + 51 q^{47} - 69 q^{49} - 168 q^{50} - 36 q^{52} - 12 q^{55} + 177 q^{56} + 138 q^{58} - 48 q^{59} + 147 q^{61} + 900 q^{62} - 51 q^{64} - 624 q^{65} + 12 q^{67} + 477 q^{68} - 6 q^{70} - 315 q^{71} - 66 q^{73} + 480 q^{74} - 57 q^{76} - 453 q^{77} - 15 q^{79} - 12 q^{82} + 591 q^{83} + 243 q^{85} - 669 q^{86} + 591 q^{88} - 72 q^{89} + 96 q^{91} - 564 q^{92} + 957 q^{94} + 606 q^{95} + 696 q^{97} - 882 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1 −3.70189 − 0.652743i 0 9.51913 + 3.46468i −0.343682 − 0.409585i 0 −2.61708 + 0.952538i −19.9557 − 11.5214i 0 1.00492 + 1.74057i
26.2 −1.51019 − 0.266287i 0 −1.54901 − 0.563792i −0.0324054 − 0.0386192i 0 −9.26350 + 3.37164i 7.50132 + 4.33089i 0 0.0386545 + 0.0669515i
26.3 −0.449062 − 0.0791817i 0 −3.56338 − 1.29697i −1.57741 − 1.87988i 0 4.06554 − 1.47973i 3.07708 + 1.77655i 0 0.559502 + 0.969085i
26.4 2.39954 + 0.423104i 0 1.82001 + 0.662430i −5.80974 − 6.92378i 0 1.62667 − 0.592058i −4.35357 − 2.51353i 0 −11.0112 − 19.0720i
26.5 2.58795 + 0.456325i 0 2.73048 + 0.993814i 4.01814 + 4.78863i 0 7.62807 − 2.77639i −2.49037 − 1.43782i 0 8.21356 + 14.2263i
53.1 −1.13691 + 3.12364i 0 −5.40039 − 4.53146i −6.77366 − 1.19438i 0 4.88842 − 4.10187i 8.77937 − 5.06877i 0 11.4319 − 19.8006i
53.2 −0.445684 + 1.22451i 0 1.76340 + 1.47966i 0.430389 + 0.0758893i 0 −4.66076 + 3.91084i −7.11182 + 4.10601i 0 −0.284745 + 0.493192i
53.3 0.199035 − 0.546844i 0 2.80475 + 2.35347i 7.54962 + 1.33120i 0 1.02187 − 0.857448i 3.86112 − 2.22922i 0 2.23059 − 3.86350i
53.4 0.663577 − 1.82316i 0 0.180586 + 0.151529i −4.25686 − 0.750600i 0 6.80967 − 5.71399i 7.11704 − 4.10903i 0 −4.19322 + 7.26288i
53.5 1.15968 − 3.18619i 0 −5.74275 − 4.81874i −3.44221 − 0.606955i 0 −8.32523 + 6.98570i −10.2675 + 5.92795i 0 −5.92572 + 10.2637i
107.1 −2.26025 + 2.69367i 0 −1.45250 − 8.23751i −1.68497 + 4.62942i 0 0.589325 − 3.34223i 13.2912 + 7.67367i 0 −8.66165 − 15.0024i
107.2 −1.51169 + 1.80156i 0 −0.265826 − 1.50758i 1.97773 − 5.43377i 0 −0.845783 + 4.79667i −5.02894 − 2.90346i 0 6.79956 + 11.7772i
107.3 0.0756536 − 0.0901605i 0 0.692187 + 3.92559i −2.11981 + 5.82413i 0 1.38457 − 7.85231i 0.814011 + 0.469969i 0 0.364735 + 0.631740i
107.4 0.746252 − 0.889349i 0 0.460544 + 2.61187i 1.84110 − 5.05837i 0 −1.82927 + 10.3743i 6.68824 + 3.86146i 0 −3.12473 − 5.41219i
107.5 1.68399 − 2.00691i 0 −0.497243 − 2.82001i −0.276220 + 0.758907i 0 1.02751 − 5.82730i 2.57852 + 1.48871i 0 1.05790 + 1.83234i
134.1 −2.26025 − 2.69367i 0 −1.45250 + 8.23751i −1.68497 − 4.62942i 0 0.589325 + 3.34223i 13.2912 − 7.67367i 0 −8.66165 + 15.0024i
134.2 −1.51169 − 1.80156i 0 −0.265826 + 1.50758i 1.97773 + 5.43377i 0 −0.845783 − 4.79667i −5.02894 + 2.90346i 0 6.79956 − 11.7772i
134.3 0.0756536 + 0.0901605i 0 0.692187 − 3.92559i −2.11981 − 5.82413i 0 1.38457 + 7.85231i 0.814011 − 0.469969i 0 0.364735 − 0.631740i
134.4 0.746252 + 0.889349i 0 0.460544 − 2.61187i 1.84110 + 5.05837i 0 −1.82927 − 10.3743i 6.68824 − 3.86146i 0 −3.12473 + 5.41219i
134.5 1.68399 + 2.00691i 0 −0.497243 + 2.82001i −0.276220 − 0.758907i 0 1.02751 + 5.82730i 2.57852 − 1.48871i 0 1.05790 − 1.83234i
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 26.5
Significant digits:
Format:

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 243.3.f.a 30
3.b odd 2 1 243.3.f.d 30
9.c even 3 1 81.3.f.a 30
9.c even 3 1 243.3.f.b 30
9.d odd 6 1 27.3.f.a ✓ 30
9.d odd 6 1 243.3.f.c 30
27.e even 9 1 27.3.f.a ✓ 30
27.e even 9 1 243.3.f.c 30
27.e even 9 1 243.3.f.d 30
27.e even 9 1 729.3.b.a 30
27.f odd 18 1 81.3.f.a 30
27.f odd 18 1 inner 243.3.f.a 30
27.f odd 18 1 243.3.f.b 30
27.f odd 18 1 729.3.b.a 30
36.h even 6 1 432.3.bc.a 30
108.j odd 18 1 432.3.bc.a 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.3.f.a ✓ 30 9.d odd 6 1
27.3.f.a ✓ 30 27.e even 9 1
81.3.f.a 30 9.c even 3 1
81.3.f.a 30 27.f odd 18 1
243.3.f.a 30 1.a even 1 1 trivial
243.3.f.a 30 27.f odd 18 1 inner
243.3.f.b 30 9.c even 3 1
243.3.f.b 30 27.f odd 18 1
243.3.f.c 30 9.d odd 6 1
243.3.f.c 30 27.e even 9 1
243.3.f.d 30 3.b odd 2 1
243.3.f.d 30 27.e even 9 1
432.3.bc.a 30 36.h even 6 1
432.3.bc.a 30 108.j odd 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 can be constructed as the kernel of the linear operator \( T_{2}^{30} + 3 T_{2}^{29} + 3 T_{2}^{28} - 3 T_{2}^{27} - 36 T_{2}^{26} - 36 T_{2}^{25} - 1371 T_{2}^{24} + \cdots + 682587 \) acting on \(S_{3}^{\mathrm{new}}(243, [\chi])\). Copy content Toggle raw display