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Properties

Label 729.6.a.e
Level $729$
Weight $6$
Character orbit 729.a
Self dual yes
Analytic conductor $116.920$
Analytic rank $0$
Dimension $42$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [729,6,Mod(1,729)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(729, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("729.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 729 = 3^{6} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 729.a (trivial)

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: yes
Analytic conductor: \(116.919804644\)
Analytic rank: \(0\)
Dimension: \(42\)
Twist minimal: no (minimal twist has level 27)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 42 q + 12 q^{2} + 624 q^{4} + 150 q^{5} + 573 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 42 q + 12 q^{2} + 624 q^{4} + 150 q^{5} + 573 q^{8} + 3 q^{10} + 1452 q^{11} + 2256 q^{14} + 8448 q^{16} + 3465 q^{17} + 3 q^{19} + 4128 q^{20} + 96 q^{22} + 5019 q^{23} + 18750 q^{25} + 3903 q^{26} - 6 q^{28} + 13008 q^{29} + 24273 q^{32} + 35868 q^{35} + 3 q^{37} + 51801 q^{38} + 96 q^{40} + 55833 q^{41} + 110757 q^{44} + 3 q^{46} + 90129 q^{47} + 57624 q^{49} + 145362 q^{50} + 3072 q^{52} + 103203 q^{53} - 6 q^{55} + 227154 q^{56} - 192 q^{58} + 176856 q^{59} - 31851 q^{61} + 246066 q^{62} + 86019 q^{64} + 167160 q^{65} - 801 q^{67} + 374589 q^{68} + 9375 q^{70} + 279531 q^{71} + 27012 q^{73} + 413970 q^{74} + 96 q^{76} + 185190 q^{77} + 462057 q^{80} - 6 q^{82} + 295536 q^{83} + 319803 q^{86} + 3072 q^{88} + 154827 q^{89} + 91002 q^{91} + 330558 q^{92} + 96 q^{94} + 353244 q^{95} + 463410 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −10.9511 0 87.9259 27.4634 0 −157.899 −612.448 0 −300.754
1.2 −10.3172 0 74.4455 −13.8184 0 −109.502 −437.920 0 142.567
1.3 −9.58877 0 59.9445 6.44812 0 −132.129 −267.953 0 −61.8295
1.4 −9.35933 0 55.5970 63.9114 0 207.073 −220.852 0 −598.168
1.5 −8.67982 0 43.3393 −26.8958 0 155.058 −98.4235 0 233.451
1.6 −8.44164 0 39.2612 25.1134 0 −71.3510 −61.2966 0 −211.998
1.7 −8.14030 0 34.2644 87.0225 0 76.4591 −18.4332 0 −708.389
1.8 −8.08006 0 33.2873 −19.8788 0 75.2303 −10.4014 0 160.622
1.9 −7.76978 0 28.3695 −87.5892 0 −118.132 28.2079 0 680.549
1.10 −6.94684 0 16.2587 −77.1463 0 −12.3071 109.353 0 535.923
1.11 −5.91459 0 2.98238 22.5002 0 90.8299 171.627 0 −133.079
1.12 −5.08001 0 −6.19352 16.0034 0 −124.463 194.023 0 −81.2975
1.13 −4.58821 0 −10.9483 −95.1620 0 −70.9631 197.056 0 436.623
1.14 −4.14873 0 −14.7880 97.3845 0 216.100 194.111 0 −404.022
1.15 −3.20822 0 −21.7073 −12.9868 0 46.9455 172.305 0 41.6645
1.16 −2.33600 0 −26.5431 72.9161 0 −190.269 136.757 0 −170.332
1.17 −1.73962 0 −28.9737 91.0786 0 −76.4078 106.071 0 −158.443
1.18 −1.57653 0 −29.5145 −59.1354 0 65.7043 96.9798 0 93.2289
1.19 −1.44191 0 −29.9209 −31.6424 0 −172.321 89.2843 0 45.6255
1.20 −1.21452 0 −30.5249 −59.2856 0 188.415 75.9377 0 72.0035
See all 42 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.42
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 729.6.a.e 42
3.b odd 2 1 729.6.a.c 42
27.e even 9 2 81.6.e.a 84
27.f odd 18 2 27.6.e.a ✓ 84
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.6.e.a ✓ 84 27.f odd 18 2
81.6.e.a 84 27.e even 9 2
729.6.a.c 42 3.b odd 2 1
729.6.a.e 42 1.a even 1 1 trivial