Properties

Label 729.2.e.k
Level $729$
Weight $2$
Character orbit 729.e
Analytic conductor $5.821$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 729 = 3^{6} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 729.e (of order \(9\), degree \(6\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.82109430735\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{9})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
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

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.

\(f(q)\) \(=\) \( q + (\beta_{10} - \beta_{7} - \beta_{5}) q^{2} + (\beta_{11} - \beta_{10} - \beta_{8} - \beta_{6} + 2 \beta_{5} + \beta_{4} - 1) q^{4} + (\beta_{11} - \beta_{10} + \beta_{9} + \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} - 1) q^{5} + ( - \beta_{11} + \beta_{8} + 2 \beta_{7} + 3 \beta_{6} - \beta_{4} - \beta_{3} - \beta_{2} + 1) q^{7} + ( - \beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} + 2 \beta_{7} + \beta_{6} - \beta_{5} + \beta_{3} - 3 \beta_1 + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{10} - \beta_{7} - \beta_{5}) q^{2} + (\beta_{11} - \beta_{10} - \beta_{8} - \beta_{6} + 2 \beta_{5} + \beta_{4} - 1) q^{4} + (\beta_{11} - \beta_{10} + \beta_{9} + \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} - 1) q^{5} + ( - \beta_{11} + \beta_{8} + 2 \beta_{7} + 3 \beta_{6} - \beta_{4} - \beta_{3} - \beta_{2} + 1) q^{7} + ( - \beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} + 2 \beta_{7} + \beta_{6} - \beta_{5} + \beta_{3} - 3 \beta_1 + 1) q^{8} + ( - \beta_{10} + \beta_{8} + 2 \beta_{7} + 2 \beta_{6} + \beta_{4} + \beta_{3} - 2 \beta_1) q^{10} + (\beta_{8} - 2 \beta_{7} + \beta_{6} - 2 \beta_{5} + \beta_{4} + \beta_{3} - 2 \beta_1 + 2) q^{11} + (\beta_{11} - \beta_{10} + 2 \beta_{9} + \beta_{8} - \beta_{7} - \beta_{5} + \beta_{3} - \beta_{2} - \beta_1) q^{13} + ( - \beta_{11} + \beta_{10} + \beta_{8} - 3 \beta_{7} + 3 \beta_{6} - 5 \beta_{5} - \beta_{3} + \cdots + 3) q^{14}+ \cdots + ( - 4 \beta_{11} - 2 \beta_{10} - 5 \beta_{9} + 2 \beta_{8} + 8 \beta_{7} + 8 \beta_{6} + \cdots - 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 3 q^{2} - 3 q^{4} - 6 q^{5} + 6 q^{7} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 3 q^{2} - 3 q^{4} - 6 q^{5} + 6 q^{7} - 6 q^{8} - 6 q^{10} + 15 q^{11} - 3 q^{13} + 21 q^{14} + 9 q^{16} + 9 q^{17} - 12 q^{19} + 3 q^{20} + 33 q^{22} - 15 q^{23} - 12 q^{25} + 48 q^{26} + 6 q^{28} + 6 q^{29} - 12 q^{31} + 27 q^{32} + 27 q^{34} - 30 q^{35} - 3 q^{37} + 39 q^{38} + 24 q^{40} + 39 q^{41} + 24 q^{43} + 33 q^{44} + 3 q^{46} + 42 q^{47} - 30 q^{49} + 15 q^{50} - 45 q^{52} - 18 q^{53} + 30 q^{55} - 12 q^{56} - 30 q^{58} - 15 q^{59} - 3 q^{61} + 30 q^{62} - 6 q^{64} + 6 q^{65} - 3 q^{67} - 36 q^{68} - 75 q^{70} - 12 q^{73} - 60 q^{74} + 30 q^{76} - 33 q^{77} + 33 q^{79} - 42 q^{80} - 42 q^{82} + 33 q^{83} - 18 q^{85} + 30 q^{86} - 42 q^{88} + 9 q^{89} - 18 q^{91} - 33 q^{92} - 66 q^{94} - 12 q^{95} + 15 q^{97} - 18 q^{98}+O(q^{100}) \) 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
\(\beta_{2}\)\(=\) \( ( 2 \nu^{10} + \nu^{9} + 33 \nu^{8} + 15 \nu^{7} + 160 \nu^{6} + 58 \nu^{5} + 285 \nu^{4} + 63 \nu^{3} + 163 \nu^{2} + 16 \nu + 1 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 2 \nu^{10} + 3 \nu^{9} - 33 \nu^{8} + 47 \nu^{7} - 160 \nu^{6} + 204 \nu^{5} - 285 \nu^{4} + 301 \nu^{3} - 159 \nu^{2} + 126 \nu + 11 ) / 8 \) 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
\(\beta_{7}\)\(=\) \( ( \nu^{11} + 5 \nu^{10} + 16 \nu^{9} + 83 \nu^{8} + 71 \nu^{7} + 409 \nu^{6} + 91 \nu^{5} + 760 \nu^{4} - 37 \nu^{3} + 474 \nu^{2} - 90 \nu + 5 ) / 8 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 3 \nu^{11} + 4 \nu^{10} - 49 \nu^{9} + 66 \nu^{8} - 232 \nu^{7} + 321 \nu^{6} - 389 \nu^{5} + 583 \nu^{4} - 158 \nu^{3} + 356 \nu^{2} + 78 \nu + 7 ) / 4 \) 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
\(\beta_{10}\)\(=\) \( ( 6 \nu^{11} + 6 \nu^{10} + 99 \nu^{9} + 99 \nu^{8} + 481 \nu^{7} + 482 \nu^{6} + 866 \nu^{5} + 881 \nu^{4} + 495 \nu^{3} + 549 \nu^{2} - 22 \nu + 13 ) / 8 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 3 \nu^{11} + 4 \nu^{10} + 49 \nu^{9} + 66 \nu^{8} + 232 \nu^{7} + 321 \nu^{6} + 389 \nu^{5} + 583 \nu^{4} + 158 \nu^{3} + 356 \nu^{2} - 78 \nu + 11 ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( - 2 \beta_{11} + \beta_{10} - 2 \beta_{9} + \beta_{8} + \beta_{7} + 3 \beta_{6} - 2 \beta_{5} - \beta_{4} - 2 \beta_{3} + \beta_{2} + \beta _1 + 1 ) / 3 \) 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
\(\nu^{3}\)\(=\) \( ( 11 \beta_{11} - \beta_{10} + 20 \beta_{9} - 10 \beta_{8} - 7 \beta_{7} - 27 \beta_{6} + 20 \beta_{5} + 7 \beta_{4} + 14 \beta_{3} - \beta_{2} - 7 \beta _1 - 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 9 \beta_{11} + 12 \beta_{10} + 3 \beta_{8} - 12 \beta_{7} + 16 \beta_{6} - 20 \beta_{5} + 10 \beta_{4} - 12 \beta_{2} - 12 \beta _1 + 30 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 92 \beta_{11} - 11 \beta_{10} - 200 \beta_{9} + 103 \beta_{8} + 79 \beta_{7} + 261 \beta_{6} - 182 \beta_{5} - 61 \beta_{4} - 122 \beta_{3} - 11 \beta_{2} + 43 \beta _1 - 8 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 81 \beta_{11} - 118 \beta_{10} - 37 \beta_{8} + 126 \beta_{7} - 170 \beta_{6} + 192 \beta_{5} - 100 \beta_{4} + 118 \beta_{2} + 126 \beta _1 - 269 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 860 \beta_{11} + 164 \beta_{10} + 1958 \beta_{9} - 1024 \beta_{8} - 838 \beta_{7} - 2538 \beta_{6} + 1700 \beta_{5} + 568 \beta_{4} + 1136 \beta_{3} + 164 \beta_{2} - 298 \beta _1 + 119 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 756 \beta_{11} + 1137 \beta_{10} + 381 \beta_{8} - 1253 \beta_{7} + 1685 \beta_{6} - 1842 \beta_{5} + 983 \beta_{4} - 1137 \beta_{2} - 1253 \beta _1 + 2539 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 8237 \beta_{11} - 1763 \beta_{10} - 18998 \beta_{9} + 10000 \beta_{8} + 8413 \beta_{7} + 24597 \beta_{6} - 16184 \beta_{5} - 5407 \beta_{4} - 10814 \beta_{3} - 1763 \beta_{2} + 2401 \beta _1 - 1262 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 7197 \beta_{11} - 10951 \beta_{10} - 3754 \beta_{8} + 12223 \beta_{7} - 16403 \beta_{6} + 17722 \beta_{5} - 9563 \beta_{4} + 10951 \beta_{2} + 12223 \beta _1 - 24325 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 79331 \beta_{11} + 17618 \beta_{10} + 183710 \beta_{9} - 96949 \beta_{8} - 82456 \beta_{7} - 237822 \beta_{6} + 155366 \beta_{5} + 51904 \beta_{4} + 103808 \beta_{3} + 17618 \beta_{2} + \cdots + 12524 ) / 3 \) Copy content Toggle raw display

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(\beta_{5}\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
82.1
1.91182i
1.22778i
1.37340i
0.0878222i
1.13697i
3.10658i
1.13697i
3.10658i
1.37340i
0.0878222i
1.91182i
1.22778i
−0.426791 2.42045i 0 −3.79704 + 1.38201i −2.35962 1.97995i 0 2.49833 + 0.909318i 2.50784 + 4.34371i 0 −3.78532 + 6.55636i
82.2 0.274087 + 1.55442i 0 −0.461727 + 0.168055i −1.28581 1.07892i 0 −2.61167 0.950570i 1.19062 + 2.06222i 0 1.32468 2.29442i
163.1 −2.54193 + 0.925187i 0 4.07335 3.41794i −0.290407 1.64698i 0 0.383475 + 0.321774i −4.48686 + 7.77147i 0 2.26195 + 3.91782i
163.2 0.162544 0.0591613i 0 −1.50917 + 1.26634i 0.648847 + 3.67980i 0 2.32226 + 1.94861i −0.343364 + 0.594724i 0 0.323168 + 0.559743i
325.1 −0.595778 + 0.499917i 0 −0.242262 + 1.37394i 2.23304 0.812759i 0 −0.434359 2.46337i −1.32025 2.28674i 0 −0.924081 + 1.60056i
325.2 1.62787 1.36594i 0 0.436855 2.47753i −1.94605 + 0.708303i 0 0.841963 + 4.77501i −0.547989 0.949144i 0 −2.20040 + 3.81121i
406.1 −0.595778 0.499917i 0 −0.242262 1.37394i 2.23304 + 0.812759i 0 −0.434359 + 2.46337i −1.32025 + 2.28674i 0 −0.924081 1.60056i
406.2 1.62787 + 1.36594i 0 0.436855 + 2.47753i −1.94605 0.708303i 0 0.841963 4.77501i −0.547989 + 0.949144i 0 −2.20040 3.81121i
568.1 −2.54193 0.925187i 0 4.07335 + 3.41794i −0.290407 + 1.64698i 0 0.383475 0.321774i −4.48686 7.77147i 0 2.26195 3.91782i
568.2 0.162544 + 0.0591613i 0 −1.50917 1.26634i 0.648847 3.67980i 0 2.32226 1.94861i −0.343364 0.594724i 0 0.323168 0.559743i
649.1 −0.426791 + 2.42045i 0 −3.79704 1.38201i −2.35962 + 1.97995i 0 2.49833 0.909318i 2.50784 4.34371i 0 −3.78532 6.55636i
649.2 0.274087 1.55442i 0 −0.461727 0.168055i −1.28581 + 1.07892i 0 −2.61167 + 0.950570i 1.19062 2.06222i 0 1.32468 + 2.29442i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 649.2
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.k 12
3.b odd 2 1 729.2.e.t 12
9.c even 3 1 729.2.e.l 12
9.c even 3 1 729.2.e.u 12
9.d odd 6 1 729.2.e.j 12
9.d odd 6 1 729.2.e.s 12
27.e even 9 1 729.2.a.e yes 6
27.e even 9 2 729.2.c.a 12
27.e even 9 1 inner 729.2.e.k 12
27.e even 9 1 729.2.e.l 12
27.e even 9 1 729.2.e.u 12
27.f odd 18 1 729.2.a.b 6
27.f odd 18 2 729.2.c.d 12
27.f odd 18 1 729.2.e.j 12
27.f odd 18 1 729.2.e.s 12
27.f odd 18 1 729.2.e.t 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
729.2.a.b 6 27.f odd 18 1
729.2.a.e yes 6 27.e even 9 1
729.2.c.a 12 27.e even 9 2
729.2.c.d 12 27.f odd 18 2
729.2.e.j 12 9.d odd 6 1
729.2.e.j 12 27.f odd 18 1
729.2.e.k 12 1.a even 1 1 trivial
729.2.e.k 12 27.e even 9 1 inner
729.2.e.l 12 9.c even 3 1
729.2.e.l 12 27.e even 9 1
729.2.e.s 12 9.d odd 6 1
729.2.e.s 12 27.f odd 18 1
729.2.e.t 12 3.b odd 2 1
729.2.e.t 12 27.f odd 18 1
729.2.e.u 12 9.c even 3 1
729.2.e.u 12 27.e even 9 1

Hecke kernels

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

\( T_{2}^{12} + 3 T_{2}^{11} + 6 T_{2}^{10} + 15 T_{2}^{9} + 18 T_{2}^{8} + 45 T_{2}^{7} + 228 T_{2}^{6} + 126 T_{2}^{5} + 495 T_{2}^{4} + 387 T_{2}^{3} + 135 T_{2}^{2} - 81 T_{2} + 9 \) Copy content Toggle raw display
\( T_{5}^{12} + 6 T_{5}^{11} + 24 T_{5}^{10} + 75 T_{5}^{9} + 162 T_{5}^{8} - 81 T_{5}^{7} - 1203 T_{5}^{6} - 1854 T_{5}^{5} + 2547 T_{5}^{4} + 15624 T_{5}^{3} + 34290 T_{5}^{2} + 41499 T_{5} + 25281 \) Copy content Toggle raw display
\( T_{7}^{12} - 6 T_{7}^{11} + 33 T_{7}^{10} - 83 T_{7}^{9} - 18 T_{7}^{8} + 621 T_{7}^{7} - 861 T_{7}^{6} - 198 T_{7}^{5} + 5832 T_{7}^{4} - 38936 T_{7}^{3} + 101712 T_{7}^{2} - 65280 T_{7} + 18496 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 3 T^{11} + 6 T^{10} + 15 T^{9} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 6 T^{11} + 24 T^{10} + \cdots + 25281 \) Copy content Toggle raw display
$7$ \( T^{12} - 6 T^{11} + 33 T^{10} + \cdots + 18496 \) Copy content Toggle raw display
$11$ \( T^{12} - 15 T^{11} + 96 T^{10} + \cdots + 788544 \) Copy content Toggle raw display
$13$ \( T^{12} + 3 T^{11} + 51 T^{10} + \cdots + 7921 \) Copy content Toggle raw display
$17$ \( T^{12} - 9 T^{11} + 81 T^{10} + \cdots + 210681 \) Copy content Toggle raw display
$19$ \( T^{12} + 12 T^{11} + 129 T^{10} + \cdots + 87616 \) Copy content Toggle raw display
$23$ \( T^{12} + 15 T^{11} + 132 T^{10} + \cdots + 207936 \) Copy content Toggle raw display
$29$ \( T^{12} - 6 T^{11} + 33 T^{10} + \cdots + 97594641 \) Copy content Toggle raw display
$31$ \( T^{12} + 12 T^{11} + 24 T^{10} + \cdots + 1032256 \) Copy content Toggle raw display
$37$ \( T^{12} + 3 T^{11} + 138 T^{10} + \cdots + 259499881 \) Copy content Toggle raw display
$41$ \( T^{12} - 39 T^{11} + \cdots + 534025881 \) Copy content Toggle raw display
$43$ \( T^{12} - 24 T^{11} + \cdots + 300814336 \) Copy content Toggle raw display
$47$ \( T^{12} - 42 T^{11} + \cdots + 314565696 \) Copy content Toggle raw display
$53$ \( (T^{6} + 9 T^{5} - 81 T^{4} - 729 T^{3} + \cdots + 1944)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + 15 T^{11} + 222 T^{10} + \cdots + 166464 \) Copy content Toggle raw display
$61$ \( T^{12} + 3 T^{11} - 156 T^{10} + \cdots + 2483776 \) Copy content Toggle raw display
$67$ \( T^{12} + 3 T^{11} - 84 T^{10} + \cdots + 298874944 \) Copy content Toggle raw display
$71$ \( T^{12} + 180 T^{10} + \cdots + 862949376 \) Copy content Toggle raw display
$73$ \( T^{12} + 12 T^{11} + \cdots + 3317875201 \) Copy content Toggle raw display
$79$ \( T^{12} - 33 T^{11} + \cdots + 8681766976 \) Copy content Toggle raw display
$83$ \( T^{12} - 33 T^{11} + 510 T^{10} + \cdots + 18870336 \) Copy content Toggle raw display
$89$ \( T^{12} - 9 T^{11} + \cdots + 15365337849 \) Copy content Toggle raw display
$97$ \( T^{12} - 15 T^{11} + \cdots + 333099001 \) Copy content Toggle raw display
show more
show less