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Properties

Label 729.6.a.c
Level $729$
Weight $6$
Character orbit 729.a
Self dual yes
Analytic conductor $116.920$
Analytic rank $1$
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: \(1\)
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 −10.9004 0 86.8198 73.1107 0 98.2458 −597.560 0 −796.939
1.2 −10.6834 0 82.1353 −40.5848 0 95.0301 −535.616 0 433.585
1.3 −10.2979 0 74.0464 −99.2344 0 −83.9362 −432.988 0 1021.90
1.4 −10.2912 0 73.9087 −72.1108 0 169.987 −431.291 0 742.106
1.5 −9.54668 0 59.1391 46.8536 0 92.4653 −259.089 0 −447.296
1.6 −9.01543 0 49.2780 101.294 0 −187.870 −155.768 0 −913.207
1.7 −8.52657 0 40.7024 −19.0301 0 −8.80284 −74.2014 0 162.261
1.8 −8.18865 0 35.0540 −47.1718 0 −227.158 −25.0084 0 386.274
1.9 −7.10946 0 18.5445 −46.5276 0 213.671 95.6616 0 330.786
1.10 −6.98644 0 16.8104 77.5082 0 −88.8035 106.121 0 −541.507
1.11 −6.28480 0 7.49875 48.8607 0 −157.508 153.986 0 −307.080
1.12 −5.70404 0 0.536059 −83.3058 0 171.266 179.472 0 475.179
1.13 −5.27048 0 −4.22203 0.133479 0 20.0016 190.908 0 −0.703501
1.14 −4.94484 0 −7.54855 55.6596 0 79.2288 195.561 0 −275.228
1.15 −4.69277 0 −9.97793 55.0515 0 24.1983 196.993 0 −258.344
1.16 −4.34767 0 −13.0977 −30.4326 0 −195.355 196.070 0 132.311
1.17 −3.46279 0 −20.0091 −30.9121 0 −203.370 180.097 0 107.042
1.18 −2.97026 0 −23.1775 −89.6268 0 76.9105 163.892 0 266.215
1.19 −2.88095 0 −23.7001 16.0197 0 243.284 160.469 0 −46.1520
1.20 −0.239374 0 −31.9427 42.7526 0 10.4066 15.3062 0 −10.2339
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.c 42
3.b odd 2 1 729.6.a.e 42
27.e even 9 2 27.6.e.a ✓ 84
27.f odd 18 2 81.6.e.a 84
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.6.e.a ✓ 84 27.e even 9 2
81.6.e.a 84 27.f odd 18 2
729.6.a.c 42 1.a even 1 1 trivial
729.6.a.e 42 3.b odd 2 1