Properties

Label 729.2.e.g
Level $729$
Weight $2$
Character orbit 729.e
Analytic conductor $5.821$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [729,2,Mod(82,729)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(729, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("729.82");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 729 = 3^{6} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 729.e (of order \(9\), 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: \(5.82109430735\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 243)
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{18}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{18}^{5} + \zeta_{18}^{4} + \cdots + 1) q^{2}+ \cdots + (3 \zeta_{18}^{5} + \cdots + 3 \zeta_{18}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{18}^{5} + \zeta_{18}^{4} + \cdots + 1) q^{2}+ \cdots + ( - 3 \zeta_{18}^{3} + 3 \zeta_{18}^{2} + \cdots + 3) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{2} - 9 q^{4} + 6 q^{5} + 9 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 3 q^{2} - 9 q^{4} + 6 q^{5} + 9 q^{7} + 6 q^{8} + 3 q^{11} + 9 q^{13} + 6 q^{14} + 9 q^{16} + 9 q^{17} + 3 q^{19} + 6 q^{20} - 9 q^{22} + 15 q^{23} + 18 q^{25} + 24 q^{26} - 24 q^{28} - 15 q^{29} + 9 q^{31} - 18 q^{32} - 9 q^{34} - 6 q^{35} + 3 q^{37} - 3 q^{38} + 27 q^{40} - 3 q^{41} + 9 q^{43} - 15 q^{44} - 9 q^{46} - 15 q^{47} + 9 q^{49} - 15 q^{50} - 9 q^{52} - 36 q^{53} + 18 q^{55} + 3 q^{56} - 36 q^{58} + 6 q^{59} + 18 q^{61} - 12 q^{62} - 12 q^{64} + 21 q^{65} - 9 q^{67} - 18 q^{70} - 9 q^{71} - 6 q^{73} + 15 q^{74} + 9 q^{76} - 3 q^{77} + 9 q^{79} + 6 q^{80} + 36 q^{82} + 21 q^{83} + 9 q^{85} + 15 q^{86} + 9 q^{88} + 6 q^{91} - 48 q^{92} - 9 q^{94} - 42 q^{95} + 36 q^{97} + 9 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/729\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-\zeta_{18}^{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
82.1
−0.766044 + 0.642788i
−0.173648 + 0.984808i
0.939693 + 0.342020i
0.939693 0.342020i
−0.173648 0.984808i
−0.766044 0.642788i
−0.439693 2.49362i 0 −4.14543 + 1.50881i −0.358441 0.300767i 0 3.03209 + 1.10359i 3.05303 + 5.28801i 0 −0.592396 + 1.02606i
163.1 1.26604 0.460802i 0 −0.141559 + 0.118782i −0.286989 1.62760i 0 1.84730 + 1.55007i −1.47178 + 2.54920i 0 −1.11334 1.92836i
325.1 0.673648 0.565258i 0 −0.213011 + 1.20805i 3.64543 1.32683i 0 −0.379385 2.15160i 1.41875 + 2.45734i 0 1.70574 2.95442i
406.1 0.673648 + 0.565258i 0 −0.213011 1.20805i 3.64543 + 1.32683i 0 −0.379385 + 2.15160i 1.41875 2.45734i 0 1.70574 + 2.95442i
568.1 1.26604 + 0.460802i 0 −0.141559 0.118782i −0.286989 + 1.62760i 0 1.84730 1.55007i −1.47178 2.54920i 0 −1.11334 + 1.92836i
649.1 −0.439693 + 2.49362i 0 −4.14543 1.50881i −0.358441 + 0.300767i 0 3.03209 1.10359i 3.05303 5.28801i 0 −0.592396 1.02606i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 82.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
27.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 729.2.e.g 6
3.b odd 2 1 729.2.e.b 6
9.c even 3 1 729.2.e.a 6
9.c even 3 1 729.2.e.h 6
9.d odd 6 1 729.2.e.c 6
9.d odd 6 1 729.2.e.i 6
27.e even 9 1 243.2.a.e 3
27.e even 9 2 243.2.c.f 6
27.e even 9 1 729.2.e.a 6
27.e even 9 1 inner 729.2.e.g 6
27.e even 9 1 729.2.e.h 6
27.f odd 18 1 243.2.a.f yes 3
27.f odd 18 2 243.2.c.e 6
27.f odd 18 1 729.2.e.b 6
27.f odd 18 1 729.2.e.c 6
27.f odd 18 1 729.2.e.i 6
108.j odd 18 1 3888.2.a.bd 3
108.l even 18 1 3888.2.a.bk 3
135.n odd 18 1 6075.2.a.bq 3
135.p even 18 1 6075.2.a.bv 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
243.2.a.e 3 27.e even 9 1
243.2.a.f yes 3 27.f odd 18 1
243.2.c.e 6 27.f odd 18 2
243.2.c.f 6 27.e even 9 2
729.2.e.a 6 9.c even 3 1
729.2.e.a 6 27.e even 9 1
729.2.e.b 6 3.b odd 2 1
729.2.e.b 6 27.f odd 18 1
729.2.e.c 6 9.d odd 6 1
729.2.e.c 6 27.f odd 18 1
729.2.e.g 6 1.a even 1 1 trivial
729.2.e.g 6 27.e even 9 1 inner
729.2.e.h 6 9.c even 3 1
729.2.e.h 6 27.e even 9 1
729.2.e.i 6 9.d odd 6 1
729.2.e.i 6 27.f odd 18 1
3888.2.a.bd 3 108.j odd 18 1
3888.2.a.bk 3 108.l even 18 1
6075.2.a.bq 3 135.n odd 18 1
6075.2.a.bv 3 135.p even 18 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(729, [\chi])\):

\( T_{2}^{6} - 3T_{2}^{5} + 9T_{2}^{4} - 24T_{2}^{3} + 36T_{2}^{2} - 27T_{2} + 9 \) Copy content Toggle raw display
\( T_{5}^{6} - 6T_{5}^{5} + 9T_{5}^{4} - 3T_{5}^{3} + 36T_{5}^{2} + 27T_{5} + 9 \) Copy content Toggle raw display
\( T_{7}^{6} - 9T_{7}^{5} + 36T_{7}^{4} - 91T_{7}^{3} + 189T_{7}^{2} - 306T_{7} + 289 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - 3 T^{5} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} - 6 T^{5} + \cdots + 9 \) Copy content Toggle raw display
$7$ \( T^{6} - 9 T^{5} + \cdots + 289 \) Copy content Toggle raw display
$11$ \( T^{6} - 3 T^{5} + \cdots + 9 \) Copy content Toggle raw display
$13$ \( T^{6} - 9 T^{5} + \cdots + 289 \) Copy content Toggle raw display
$17$ \( (T^{2} - 3 T + 9)^{3} \) Copy content Toggle raw display
$19$ \( T^{6} - 3 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$23$ \( T^{6} - 15 T^{5} + \cdots + 2601 \) Copy content Toggle raw display
$29$ \( T^{6} + 15 T^{5} + \cdots + 3249 \) Copy content Toggle raw display
$31$ \( T^{6} - 9 T^{5} + \cdots + 361 \) Copy content Toggle raw display
$37$ \( T^{6} - 3 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{6} + 3 T^{5} + \cdots + 47961 \) Copy content Toggle raw display
$43$ \( T^{6} - 9 T^{5} + \cdots + 361 \) Copy content Toggle raw display
$47$ \( T^{6} + 15 T^{5} + \cdots + 71289 \) Copy content Toggle raw display
$53$ \( (T^{3} + 18 T^{2} + \cdots + 81)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} - 6 T^{5} + \cdots + 103041 \) Copy content Toggle raw display
$61$ \( T^{6} - 18 T^{5} + \cdots + 2809 \) Copy content Toggle raw display
$67$ \( T^{6} + 9 T^{5} + \cdots + 11881 \) Copy content Toggle raw display
$71$ \( T^{6} + 9 T^{5} + \cdots + 998001 \) Copy content Toggle raw display
$73$ \( T^{6} + 6 T^{5} + \cdots + 157609 \) Copy content Toggle raw display
$79$ \( T^{6} - 9 T^{5} + \cdots + 2809 \) Copy content Toggle raw display
$83$ \( T^{6} - 21 T^{5} + \cdots + 2601 \) Copy content Toggle raw display
$89$ \( T^{6} + 189 T^{4} + \cdots + 998001 \) Copy content Toggle raw display
$97$ \( T^{6} - 36 T^{5} + \cdots + 361 \) Copy content Toggle raw display
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