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Properties

Label 729.4.a.c
Level $729$
Weight $4$
Character orbit 729.a
Self dual yes
Analytic conductor $43.012$
Analytic rank $1$
Dimension $24$
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] = [729,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("729.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 729 = 3^{6} \)
Weight: \( k \) \(=\) \( 4 \)
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: \(43.0123923942\)
Analytic rank: \(1\)
Dimension: \(24\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 6 q^{2} + 84 q^{4} - 30 q^{5} - 75 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 6 q^{2} + 84 q^{4} - 30 q^{5} - 75 q^{8} + 3 q^{10} - 132 q^{11} - 192 q^{14} + 240 q^{16} - 207 q^{17} + 3 q^{19} - 552 q^{20} + 24 q^{22} - 588 q^{23} + 300 q^{25} - 957 q^{26} - 6 q^{28} - 834 q^{29} - 1431 q^{32} - 1257 q^{35} + 3 q^{37} - 1587 q^{38} + 24 q^{40} - 1344 q^{41} - 2211 q^{44} + 3 q^{46} - 1716 q^{47} + 294 q^{49} - 1932 q^{50} + 192 q^{52} - 1368 q^{53} - 6 q^{55} - 2958 q^{56} - 48 q^{58} - 2388 q^{59} + 540 q^{61} - 2118 q^{62} + 195 q^{64} - 2175 q^{65} + 378 q^{67} - 2979 q^{68} + 375 q^{70} - 3105 q^{71} + 219 q^{73} - 2514 q^{74} + 24 q^{76} - 2937 q^{77} - 4935 q^{80} - 6 q^{82} - 3669 q^{83} - 3837 q^{86} + 192 q^{88} - 5202 q^{89} - 267 q^{91} - 5250 q^{92} + 24 q^{94} - 6972 q^{95} - 4392 q^{98}+O(q^{100}) \) 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 −5.30748 0 20.1693 −5.51457 0 24.6230 −64.5884 0 29.2684
1.2 −5.29813 0 20.0701 10.5105 0 −7.73120 −63.9491 0 −55.6861
1.3 −5.21012 0 19.1454 −14.0463 0 −10.3001 −58.0686 0 73.1831
1.4 −4.31240 0 10.5968 −18.5830 0 18.8092 −11.1984 0 80.1375
1.5 −3.68865 0 5.60613 1.61561 0 33.9652 8.83016 0 −5.95943
1.6 −3.18906 0 2.17013 14.3358 0 −32.2016 18.5918 0 −45.7179
1.7 −3.18772 0 2.16159 7.17014 0 4.85729 18.6113 0 −22.8564
1.8 −2.95196 0 0.714070 −17.0434 0 −12.8302 21.5078 0 50.3115
1.9 −2.25787 0 −2.90200 19.2297 0 0.653898 24.6154 0 −43.4182
1.10 −1.04608 0 −6.90572 −15.4345 0 −23.5429 15.5926 0 16.1457
1.11 −0.991947 0 −7.01604 −0.144478 0 25.3747 14.8951 0 0.143314
1.12 −0.841078 0 −7.29259 10.8166 0 −26.6917 12.8623 0 −9.09757
1.13 −0.644468 0 −7.58466 0.514271 0 −9.60308 10.0438 0 −0.331431
1.14 0.298855 0 −7.91069 −20.0516 0 21.4905 −4.75498 0 −5.99252
1.15 1.15359 0 −6.66923 −7.67988 0 12.9720 −16.9223 0 −8.85943
1.16 1.82173 0 −4.68131 13.5279 0 22.8583 −23.1019 0 24.6442
1.17 1.94042 0 −4.23475 2.70886 0 −11.9176 −23.7406 0 5.25633
1.18 2.33077 0 −2.56749 6.24132 0 −2.70290 −24.6304 0 14.5471
1.19 3.41269 0 3.64648 10.9823 0 −4.40595 −14.8572 0 37.4794
1.20 3.75799 0 6.12246 −6.10578 0 17.2567 −7.05576 0 −22.9454
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.24
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 729.4.a.c 24
3.b odd 2 1 729.4.a.d 24
27.e even 9 2 81.4.e.a 48
27.e even 9 2 243.4.e.a 48
27.e even 9 2 243.4.e.b 48
27.f odd 18 2 27.4.e.a ✓ 48
27.f odd 18 2 243.4.e.c 48
27.f odd 18 2 243.4.e.d 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.4.e.a ✓ 48 27.f odd 18 2
81.4.e.a 48 27.e even 9 2
243.4.e.a 48 27.e even 9 2
243.4.e.b 48 27.e even 9 2
243.4.e.c 48 27.f odd 18 2
243.4.e.d 48 27.f odd 18 2
729.4.a.c 24 1.a even 1 1 trivial
729.4.a.d 24 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} + 6 T_{2}^{23} - 120 T_{2}^{22} - 735 T_{2}^{21} + 6102 T_{2}^{20} + 38547 T_{2}^{19} + \cdots - 1375087104 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(729))\). Copy content Toggle raw display