Properties

Label 729.2.e.j
Level $729$
Weight $2$
Character orbit 729.e
Analytic conductor $5.821$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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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: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{9})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 18x^{10} + 105x^{8} + 266x^{6} + 306x^{4} + 132x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
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 basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{11} - \beta_{7} - \beta_{6} - 1) q^{2} + ( - \beta_{11} + \beta_{7} + \beta_{3} + \cdots + 1) q^{4}+ \cdots + (\beta_{11} - \beta_{10} - \beta_{9} + \cdots - 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{11} - \beta_{7} - \beta_{6} - 1) q^{2} + ( - \beta_{11} + \beta_{7} + \beta_{3} + \cdots + 1) q^{4}+ \cdots + (4 \beta_{11} + 2 \beta_{10} + \cdots + 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{2} + 6 q^{4} + 6 q^{5} - 3 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 6 q^{2} + 6 q^{4} + 6 q^{5} - 3 q^{7} + 6 q^{8} - 6 q^{10} + 12 q^{11} - 3 q^{13} + 15 q^{14} - 36 q^{16} - 9 q^{17} - 12 q^{19} + 42 q^{20} + 6 q^{22} + 6 q^{23} + 6 q^{25} - 48 q^{26} + 6 q^{28} + 12 q^{29} + 6 q^{31} + 54 q^{32} - 9 q^{34} + 30 q^{35} - 3 q^{37} + 42 q^{38} - 57 q^{40} + 24 q^{41} + 6 q^{43} - 33 q^{44} + 3 q^{46} + 21 q^{47} + 33 q^{49} + 21 q^{50} + 45 q^{52} + 18 q^{53} + 30 q^{55} + 3 q^{56} + 33 q^{58} + 15 q^{59} + 33 q^{61} - 30 q^{62} - 6 q^{64} - 6 q^{65} + 42 q^{67} - 18 q^{68} + 24 q^{70} - 12 q^{73} - 3 q^{74} - 87 q^{76} - 57 q^{77} - 48 q^{79} + 42 q^{80} - 42 q^{82} + 12 q^{83} - 36 q^{85} - 30 q^{86} + 30 q^{88} - 9 q^{89} - 18 q^{91} - 48 q^{92} + 33 q^{94} + 30 q^{95} - 3 q^{97} + 18 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 18x^{10} + 105x^{8} + 266x^{6} + 306x^{4} + 132x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{11} - 16\nu^{9} + \nu^{8} - 72\nu^{7} + 15\nu^{6} - 106\nu^{5} + 58\nu^{4} - 21\nu^{3} + 63\nu^{2} + 31\nu + 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2 \nu^{10} + \nu^{9} + 33 \nu^{8} + 15 \nu^{7} + 160 \nu^{6} + 58 \nu^{5} + 285 \nu^{4} + 63 \nu^{3} + \cdots + 1 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 2 \nu^{10} + 3 \nu^{9} - 33 \nu^{8} + 47 \nu^{7} - 160 \nu^{6} + 204 \nu^{5} - 285 \nu^{4} + \cdots + 11 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{10} + 34\nu^{8} + 175\nu^{6} + 343\nu^{4} + 222\nu^{2} - 3 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -5\nu^{10} - 82\nu^{8} + \nu^{7} - 394\nu^{6} + 15\nu^{5} - 702\nu^{4} + 58\nu^{3} - 411\nu^{2} + 59\nu + 3 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -5\nu^{10} - 82\nu^{8} - \nu^{7} - 394\nu^{6} - 15\nu^{5} - 702\nu^{4} - 58\nu^{3} - 411\nu^{2} - 59\nu + 3 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{11} + 5 \nu^{10} + 16 \nu^{9} + 83 \nu^{8} + 71 \nu^{7} + 409 \nu^{6} + 91 \nu^{5} + 760 \nu^{4} + \cdots + 5 ) / 8 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 3 \nu^{11} + 4 \nu^{10} - 49 \nu^{9} + 66 \nu^{8} - 232 \nu^{7} + 321 \nu^{6} - 389 \nu^{5} + \cdots + 7 ) / 4 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -3\nu^{11} - 50\nu^{9} - 249\nu^{7} - 477\nu^{5} - 335\nu^{3} - 42\nu - 2 ) / 4 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 6 \nu^{11} + 6 \nu^{10} + 99 \nu^{9} + 99 \nu^{8} + 481 \nu^{7} + 482 \nu^{6} + 866 \nu^{5} + 881 \nu^{4} + \cdots + 13 ) / 8 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 3 \nu^{11} + 4 \nu^{10} + 49 \nu^{9} + 66 \nu^{8} + 232 \nu^{7} + 321 \nu^{6} + 389 \nu^{5} + 583 \nu^{4} + \cdots + 11 ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( - 2 \beta_{11} + \beta_{10} - 2 \beta_{9} + \beta_{8} + \beta_{7} + 3 \beta_{6} - 2 \beta_{5} + \cdots + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} - \beta_{10} + \beta_{7} - \beta_{6} + 2\beta_{5} - \beta_{4} + \beta_{2} + \beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 11 \beta_{11} - \beta_{10} + 20 \beta_{9} - 10 \beta_{8} - 7 \beta_{7} - 27 \beta_{6} + 20 \beta_{5} + \cdots - 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 9 \beta_{11} + 12 \beta_{10} + 3 \beta_{8} - 12 \beta_{7} + 16 \beta_{6} - 20 \beta_{5} + 10 \beta_{4} + \cdots + 30 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 92 \beta_{11} - 11 \beta_{10} - 200 \beta_{9} + 103 \beta_{8} + 79 \beta_{7} + 261 \beta_{6} + \cdots - 8 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 81 \beta_{11} - 118 \beta_{10} - 37 \beta_{8} + 126 \beta_{7} - 170 \beta_{6} + 192 \beta_{5} + \cdots - 269 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 860 \beta_{11} + 164 \beta_{10} + 1958 \beta_{9} - 1024 \beta_{8} - 838 \beta_{7} - 2538 \beta_{6} + \cdots + 119 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 756 \beta_{11} + 1137 \beta_{10} + 381 \beta_{8} - 1253 \beta_{7} + 1685 \beta_{6} - 1842 \beta_{5} + \cdots + 2539 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 8237 \beta_{11} - 1763 \beta_{10} - 18998 \beta_{9} + 10000 \beta_{8} + 8413 \beta_{7} + 24597 \beta_{6} + \cdots - 1262 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 7197 \beta_{11} - 10951 \beta_{10} - 3754 \beta_{8} + 12223 \beta_{7} - 16403 \beta_{6} + 17722 \beta_{5} + \cdots - 24325 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 79331 \beta_{11} + 17618 \beta_{10} + 183710 \beta_{9} - 96949 \beta_{8} - 82456 \beta_{7} + \cdots + 12524 ) / 3 \) Copy content Toggle raw display

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(\beta_{7}\)

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
3.10658i
1.13697i
1.91182i
1.22778i
1.37340i
0.0878222i
1.37340i
0.0878222i
1.91182i
1.22778i
3.10658i
1.13697i
−0.369007 2.09274i 0 −2.36403 + 0.860436i −1.58643 1.33117i 0 −4.55626 1.65834i 0.547989 + 0.949144i 0 −2.20040 + 3.81121i
82.2 0.135052 + 0.765917i 0 1.31100 0.477163i 1.82039 + 1.52749i 0 2.35052 + 0.855521i 1.32025 + 2.28674i 0 −0.924081 + 1.60056i
163.1 −2.30957 + 0.840614i 0 3.09538 2.59733i 0.534882 + 3.03347i 0 −2.03666 1.70896i −2.50784 + 4.34371i 0 −3.78532 6.55636i
163.2 1.48321 0.539846i 0 0.376403 0.315840i 0.291470 + 1.65301i 0 2.12905 + 1.78649i −1.19062 + 2.06222i 0 1.32468 + 2.29442i
325.1 −2.07220 + 1.73878i 0 0.923353 5.23659i −1.57153 + 0.571989i 0 0.0869267 + 0.492986i 4.48686 + 7.77147i 0 2.26195 3.91782i
325.2 0.132507 0.111187i 0 −0.342101 + 1.94015i 3.51122 1.27798i 0 0.526414 + 2.98544i 0.343364 + 0.594724i 0 0.323168 0.559743i
406.1 −2.07220 1.73878i 0 0.923353 + 5.23659i −1.57153 0.571989i 0 0.0869267 0.492986i 4.48686 7.77147i 0 2.26195 + 3.91782i
406.2 0.132507 + 0.111187i 0 −0.342101 1.94015i 3.51122 + 1.27798i 0 0.526414 2.98544i 0.343364 0.594724i 0 0.323168 + 0.559743i
568.1 −2.30957 0.840614i 0 3.09538 + 2.59733i 0.534882 3.03347i 0 −2.03666 + 1.70896i −2.50784 4.34371i 0 −3.78532 + 6.55636i
568.2 1.48321 + 0.539846i 0 0.376403 + 0.315840i 0.291470 1.65301i 0 2.12905 1.78649i −1.19062 2.06222i 0 1.32468 2.29442i
649.1 −0.369007 + 2.09274i 0 −2.36403 0.860436i −1.58643 + 1.33117i 0 −4.55626 + 1.65834i 0.547989 0.949144i 0 −2.20040 3.81121i
649.2 0.135052 0.765917i 0 1.31100 + 0.477163i 1.82039 1.52749i 0 2.35052 0.855521i 1.32025 2.28674i 0 −0.924081 1.60056i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 82.2
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.j 12
3.b odd 2 1 729.2.e.u 12
9.c even 3 1 729.2.e.s 12
9.c even 3 1 729.2.e.t 12
9.d odd 6 1 729.2.e.k 12
9.d odd 6 1 729.2.e.l 12
27.e even 9 1 729.2.a.b 6
27.e even 9 2 729.2.c.d 12
27.e even 9 1 inner 729.2.e.j 12
27.e even 9 1 729.2.e.s 12
27.e even 9 1 729.2.e.t 12
27.f odd 18 1 729.2.a.e yes 6
27.f odd 18 2 729.2.c.a 12
27.f odd 18 1 729.2.e.k 12
27.f odd 18 1 729.2.e.l 12
27.f odd 18 1 729.2.e.u 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
729.2.a.b 6 27.e even 9 1
729.2.a.e yes 6 27.f odd 18 1
729.2.c.a 12 27.f odd 18 2
729.2.c.d 12 27.e even 9 2
729.2.e.j 12 1.a even 1 1 trivial
729.2.e.j 12 27.e even 9 1 inner
729.2.e.k 12 9.d odd 6 1
729.2.e.k 12 27.f odd 18 1
729.2.e.l 12 9.d odd 6 1
729.2.e.l 12 27.f odd 18 1
729.2.e.s 12 9.c even 3 1
729.2.e.s 12 27.e even 9 1
729.2.e.t 12 9.c even 3 1
729.2.e.t 12 27.e even 9 1
729.2.e.u 12 3.b odd 2 1
729.2.e.u 12 27.f odd 18 1

Hecke kernels

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

\( T_{2}^{12} + 6 T_{2}^{11} + 15 T_{2}^{10} + 12 T_{2}^{9} - 18 T_{2}^{8} - 72 T_{2}^{7} - 69 T_{2}^{6} + \cdots + 9 \) Copy content Toggle raw display
\( T_{5}^{12} - 6 T_{5}^{11} + 15 T_{5}^{10} - 21 T_{5}^{9} - 9 T_{5}^{8} + 135 T_{5}^{7} + 363 T_{5}^{6} + \cdots + 25281 \) Copy content Toggle raw display
\( T_{7}^{12} + 3 T_{7}^{11} - 12 T_{7}^{10} - 2 T_{7}^{9} + 162 T_{7}^{8} - 522 T_{7}^{7} + 921 T_{7}^{6} + \cdots + 18496 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 6 T^{11} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} - 6 T^{11} + \cdots + 25281 \) Copy content Toggle raw display
$7$ \( T^{12} + 3 T^{11} + \cdots + 18496 \) Copy content Toggle raw display
$11$ \( T^{12} - 12 T^{11} + \cdots + 788544 \) Copy content Toggle raw display
$13$ \( T^{12} + 3 T^{11} + \cdots + 7921 \) Copy content Toggle raw display
$17$ \( T^{12} + 9 T^{11} + \cdots + 210681 \) Copy content Toggle raw display
$19$ \( T^{12} + 12 T^{11} + \cdots + 87616 \) Copy content Toggle raw display
$23$ \( T^{12} - 6 T^{11} + \cdots + 207936 \) Copy content Toggle raw display
$29$ \( T^{12} - 12 T^{11} + \cdots + 97594641 \) Copy content Toggle raw display
$31$ \( T^{12} - 6 T^{11} + \cdots + 1032256 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 259499881 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 534025881 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 300814336 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 314565696 \) Copy content Toggle raw display
$53$ \( (T^{6} - 9 T^{5} + \cdots + 1944)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} - 15 T^{11} + \cdots + 166464 \) Copy content Toggle raw display
$61$ \( T^{12} - 33 T^{11} + \cdots + 2483776 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 298874944 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 862949376 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 3317875201 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 8681766976 \) Copy content Toggle raw display
$83$ \( T^{12} - 12 T^{11} + \cdots + 18870336 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 15365337849 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 333099001 \) Copy content Toggle raw display
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