Properties

Label 729.2.c.d
Level $729$
Weight $2$
Character orbit 729.c
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.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.82109430735\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{9} - \beta_{6} + \beta_{5} - \beta_{3} + \beta_{2}) q^{2} + ( - \beta_{11} + \beta_{10} - \beta_{7} - \beta_{3} - 1) q^{4} + (\beta_{10} - \beta_{7} + \beta_{6} - \beta_{2} + \beta_1 - 1) q^{5} + ( - \beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} + \beta_{3} - \beta_{2}) q^{7} + (\beta_{10} - 3 \beta_{9} - \beta_{5} - 3 \beta_1 - 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{9} - \beta_{6} + \beta_{5} - \beta_{3} + \beta_{2}) q^{2} + ( - \beta_{11} + \beta_{10} - \beta_{7} - \beta_{3} - 1) q^{4} + (\beta_{10} - \beta_{7} + \beta_{6} - \beta_{2} + \beta_1 - 1) q^{5} + ( - \beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} + \beta_{3} - \beta_{2}) q^{7} + (\beta_{10} - 3 \beta_{9} - \beta_{5} - 3 \beta_1 - 1) q^{8} + ( - \beta_{9} - \beta_{5} + \beta_{4} - 3 \beta_1 + 2) q^{10} + (\beta_{11} - \beta_{9} + \beta_{7} + \beta_{6} - \beta_{4} + \beta_{2} - 1) q^{11} + (\beta_{11} - \beta_{8} - \beta_{3} + \beta_{2} - \beta_1 - 1) q^{13} + (\beta_{11} + 3 \beta_{8} + \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 3) q^{14} + (\beta_{11} - \beta_{9} + \beta_{8} + 2 \beta_{7} + \beta_{6} - \beta_{4} - 3 \beta_{2} - 1) q^{16} + (\beta_{10} + \beta_{9} - \beta_{5} - \beta_{4} + 1) q^{17} + ( - \beta_{10} + \beta_{9} - \beta_{4} + \beta_1 + 2) q^{19} + (\beta_{11} + 3 \beta_{9} + 3 \beta_{8} + \beta_{7} - 3 \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} + \cdots - 1) q^{20}+ \cdots + ( - 4 \beta_{10} - 4 \beta_{9} - 2 \beta_{5} - 2 \beta_{4} + 6 \beta_1 - 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 3 q^{2} - 9 q^{4} - 3 q^{5} - 6 q^{7} - 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 3 q^{2} - 9 q^{4} - 3 q^{5} - 6 q^{7} - 12 q^{8} + 12 q^{10} - 6 q^{11} - 6 q^{13} + 24 q^{14} - 15 q^{16} + 18 q^{17} + 24 q^{19} - 21 q^{20} - 3 q^{22} - 12 q^{23} - 9 q^{25} - 48 q^{26} + 6 q^{28} + 21 q^{29} - 15 q^{31} - 60 q^{35} + 6 q^{37} + 15 q^{38} - 3 q^{40} - 12 q^{41} - 6 q^{43} + 66 q^{44} - 6 q^{46} - 15 q^{47} - 12 q^{49} - 24 q^{50} - 3 q^{52} + 18 q^{53} + 30 q^{55} + 12 q^{56} + 15 q^{58} + 6 q^{59} - 24 q^{61} + 60 q^{62} + 12 q^{64} - 15 q^{65} - 15 q^{67} + 36 q^{68} + 15 q^{70} + 24 q^{73} + 24 q^{74} - 9 q^{76} + 15 q^{77} - 24 q^{79} + 42 q^{80} - 42 q^{82} - 6 q^{83} + 18 q^{85} - 30 q^{86} + 21 q^{88} + 18 q^{89} + 36 q^{91} + 6 q^{92} + 6 q^{94} - 33 q^{95} + 21 q^{97} - 36 q^{98}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( ( \nu^{8} + 15\nu^{6} + 58\nu^{4} + 63\nu^{2} + 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{11} + 16\nu^{9} + \nu^{8} + 70\nu^{7} + 15\nu^{6} + 76\nu^{5} + 58\nu^{4} - 95\nu^{3} + 63\nu^{2} - 149\nu + 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{11} - 2 \nu^{10} + 19 \nu^{9} - 34 \nu^{8} + 119 \nu^{7} - 175 \nu^{6} + 310 \nu^{5} - 343 \nu^{4} + 322 \nu^{3} - 222 \nu^{2} + 95 \nu + 3 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{10} + 33\nu^{8} + 160\nu^{6} + 285\nu^{4} + 163\nu^{2} + 1 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2\nu^{10} - 34\nu^{8} - 175\nu^{6} - 343\nu^{4} - 222\nu^{2} + 3 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - \nu^{11} - 5 \nu^{10} - 16 \nu^{9} - 83 \nu^{8} - 73 \nu^{7} - 409 \nu^{6} - 121 \nu^{5} - 760 \nu^{4} - 79 \nu^{3} - 474 \nu^{2} - 28 \nu - 5 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 6 \nu^{10} + 3 \nu^{9} - 99 \nu^{8} + 49 \nu^{7} - 482 \nu^{6} + 234 \nu^{5} - 881 \nu^{4} + 417 \nu^{3} - 549 \nu^{2} + 256 \nu - 17 ) / 8 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3\nu^{11} + 50\nu^{9} + 249\nu^{7} + 477\nu^{5} + 335\nu^{3} + 42\nu - 2 ) / 4 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -5\nu^{10} - 83\nu^{8} - 409\nu^{6} - 760\nu^{4} - 474\nu^{2} - 5 ) / 4 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -6\nu^{10} - 99\nu^{8} - 482\nu^{6} - 881\nu^{4} - 549\nu^{2} - 13 ) / 4 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 16 \nu^{11} + 2 \nu^{10} + 265 \nu^{9} + 33 \nu^{8} + 1301 \nu^{7} + 160 \nu^{6} + 2416 \nu^{5} + 285 \nu^{4} + 1555 \nu^{3} + 163 \nu^{2} + 86 \nu + 5 ) / 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{10} + 2\beta_{7} + \beta_{5} - 2\beta_{3} + 2\beta_{2} - \beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{4} + \beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 6 \beta_{11} + 4 \beta_{10} + 4 \beta_{9} - 18 \beta_{8} - 8 \beta_{7} - 8 \beta_{6} - 7 \beta_{5} - 3 \beta_{4} + 14 \beta_{3} - 10 \beta_{2} + 5 \beta _1 - 16 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -3\beta_{10} + 4\beta_{9} - 10\beta_{5} - 9\beta_{4} - 8\beta _1 + 21 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 76 \beta_{11} - 27 \beta_{10} - 38 \beta_{9} + 222 \beta_{8} + 54 \beta_{7} + 76 \beta_{6} + 61 \beta_{5} + 38 \beta_{4} - 122 \beta_{3} + 82 \beta_{2} - 41 \beta _1 + 176 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 37\beta_{10} - 52\beta_{9} + 100\beta_{5} + 81\beta_{4} + 74\beta _1 - 188 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 792 \beta_{11} + 232 \beta_{10} + 342 \beta_{9} - 2286 \beta_{8} - 464 \beta_{7} - 684 \beta_{6} - 568 \beta_{5} - 396 \beta_{4} + 1136 \beta_{3} - 776 \beta_{2} + 388 \beta _1 - 1771 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -381\beta_{10} + 548\beta_{9} - 983\beta_{5} - 756\beta_{4} - 705\beta _1 + 1783 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 7842 \beta_{11} - 2158 \beta_{10} - 3178 \beta_{9} + 22524 \beta_{8} + 4316 \beta_{7} + 6356 \beta_{6} + 5407 \beta_{5} + 3921 \beta_{4} - 10814 \beta_{3} + 7498 \beta_{2} - 3749 \beta _1 + 17341 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 3754\beta_{10} - 5452\beta_{9} + 9563\beta_{5} + 7197\beta_{4} + 6771\beta _1 - 17128 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 76378 \beta_{11} + 20571 \beta_{10} + 30176 \beta_{9} - 218946 \beta_{8} - 41142 \beta_{7} - 60352 \beta_{6} - 51904 \beta_{5} - 38189 \beta_{4} + 103808 \beta_{3} - 72508 \beta_{2} + \cdots - 168233 ) / 3 \) Copy content Toggle raw display

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(-1 - \beta_{8}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
244.1
1.22778i
3.10658i
1.37340i
1.91182i
1.13697i
0.0878222i
1.22778i
3.10658i
1.37340i
1.91182i
1.13697i
0.0878222i
−1.22889 2.12851i 0 −2.02036 + 3.49937i −1.54013 + 2.66759i 0 1.32933 + 2.30247i 5.01568 0 7.57064
244.2 −0.388866 0.673536i 0 0.697566 1.20822i −1.18817 + 2.05798i 0 1.25069 + 2.16626i −2.64050 0 1.84816
244.3 −0.0864880 0.149802i 0 0.985040 1.70614i 1.86828 3.23596i 0 −1.51575 2.62535i −0.686728 0 −0.646335
244.4 0.789202 + 1.36694i 0 −0.245680 + 0.425530i −0.839254 + 1.45363i 0 −1.38964 2.40693i 2.38124 0 −2.64936
244.5 1.06251 + 1.84033i 0 −1.25787 + 2.17870i 1.03547 1.79349i 0 −2.42434 4.19907i −1.09598 0 4.40081
244.6 1.35253 + 2.34265i 0 −2.65869 + 4.60498i −0.836192 + 1.44833i 0 −0.250296 0.433525i −8.97372 0 −4.52391
487.1 −1.22889 + 2.12851i 0 −2.02036 3.49937i −1.54013 2.66759i 0 1.32933 2.30247i 5.01568 0 7.57064
487.2 −0.388866 + 0.673536i 0 0.697566 + 1.20822i −1.18817 2.05798i 0 1.25069 2.16626i −2.64050 0 1.84816
487.3 −0.0864880 + 0.149802i 0 0.985040 + 1.70614i 1.86828 + 3.23596i 0 −1.51575 + 2.62535i −0.686728 0 −0.646335
487.4 0.789202 1.36694i 0 −0.245680 0.425530i −0.839254 1.45363i 0 −1.38964 + 2.40693i 2.38124 0 −2.64936
487.5 1.06251 1.84033i 0 −1.25787 2.17870i 1.03547 + 1.79349i 0 −2.42434 + 4.19907i −1.09598 0 4.40081
487.6 1.35253 2.34265i 0 −2.65869 4.60498i −0.836192 1.44833i 0 −0.250296 + 0.433525i −8.97372 0 −4.52391
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 487.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 729.2.c.d 12
3.b odd 2 1 729.2.c.a 12
9.c even 3 1 729.2.a.b 6
9.c even 3 1 inner 729.2.c.d 12
9.d odd 6 1 729.2.a.e yes 6
9.d odd 6 1 729.2.c.a 12
27.e even 9 2 729.2.e.j 12
27.e even 9 2 729.2.e.s 12
27.e even 9 2 729.2.e.t 12
27.f odd 18 2 729.2.e.k 12
27.f odd 18 2 729.2.e.l 12
27.f odd 18 2 729.2.e.u 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
729.2.a.b 6 9.c even 3 1
729.2.a.e yes 6 9.d odd 6 1
729.2.c.a 12 3.b odd 2 1
729.2.c.a 12 9.d odd 6 1
729.2.c.d 12 1.a even 1 1 trivial
729.2.c.d 12 9.c even 3 1 inner
729.2.e.j 12 27.e even 9 2
729.2.e.k 12 27.f odd 18 2
729.2.e.l 12 27.f odd 18 2
729.2.e.s 12 27.e even 9 2
729.2.e.t 12 27.e even 9 2
729.2.e.u 12 27.f odd 18 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} - 3 T_{2}^{11} + 15 T_{2}^{10} - 24 T_{2}^{9} + 99 T_{2}^{8} - 144 T_{2}^{7} + 381 T_{2}^{6} - 207 T_{2}^{5} + 360 T_{2}^{4} + 126 T_{2}^{3} + 324 T_{2}^{2} + 54 T_{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(729, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 3 T^{11} + 15 T^{10} - 24 T^{9} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 3 T^{11} + 24 T^{10} + \cdots + 25281 \) Copy content Toggle raw display
$7$ \( T^{12} + 6 T^{11} + 45 T^{10} + \cdots + 18496 \) Copy content Toggle raw display
$11$ \( T^{12} + 6 T^{11} + 51 T^{10} + \cdots + 788544 \) Copy content Toggle raw display
$13$ \( T^{12} + 6 T^{11} + 54 T^{10} + \cdots + 7921 \) Copy content Toggle raw display
$17$ \( (T^{6} - 9 T^{5} + 135 T^{3} - 108 T^{2} + \cdots + 459)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} - 12 T^{5} + 15 T^{4} + 155 T^{3} + \cdots - 296)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + 12 T^{11} + 123 T^{10} + \cdots + 207936 \) Copy content Toggle raw display
$29$ \( T^{12} - 21 T^{11} + 330 T^{10} + \cdots + 97594641 \) Copy content Toggle raw display
$31$ \( T^{12} + 15 T^{11} + 180 T^{10} + \cdots + 1032256 \) Copy content Toggle raw display
$37$ \( (T^{6} - 3 T^{5} - 129 T^{4} + 461 T^{3} + \cdots - 16109)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + 12 T^{11} + \cdots + 534025881 \) Copy content Toggle raw display
$43$ \( T^{12} + 6 T^{11} + 144 T^{10} + \cdots + 300814336 \) Copy content Toggle raw display
$47$ \( T^{12} + 15 T^{11} + \cdots + 314565696 \) 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
$59$ \( T^{12} - 6 T^{11} + 141 T^{10} + \cdots + 166464 \) Copy content Toggle raw display
$61$ \( T^{12} + 24 T^{11} + 459 T^{10} + \cdots + 2483776 \) Copy content Toggle raw display
$67$ \( T^{12} + 15 T^{11} + \cdots + 298874944 \) Copy content Toggle raw display
$71$ \( (T^{6} - 180 T^{4} + 864 T^{3} + \cdots + 29376)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} - 12 T^{5} - 156 T^{4} + \cdots + 57601)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + 24 T^{11} + \cdots + 8681766976 \) Copy content Toggle raw display
$83$ \( T^{12} + 6 T^{11} + 123 T^{10} + \cdots + 18870336 \) Copy content Toggle raw display
$89$ \( (T^{6} - 9 T^{5} - 225 T^{4} + \cdots - 123957)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} - 21 T^{11} + \cdots + 333099001 \) Copy content Toggle raw display
show more
show less