Properties

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

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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: 12.0.1952986685049.1
Defining polynomial: \( x^{12} - 6 x^{11} + 27 x^{10} - 80 x^{9} + 186 x^{8} - 330 x^{7} + 463 x^{6} - 504 x^{5} + 420 x^{4} - 258 x^{3} + 108 x^{2} - 27 x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3^{3} \)
Twist minimal: no (minimal twist has level 27)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{10} + \beta_{8} + \beta_{6}) q^{2} + (\beta_{11} + \beta_{10} - \beta_{6} + \beta_{4} - \beta_{3} + \beta_{2} - 1) q^{4} + ( - \beta_{9} + \beta_{7} + \beta_{5} + \beta_1 - 1) q^{5} + (\beta_{11} - \beta_{10} + \beta_{8} - \beta_{7}) q^{7} + ( - \beta_{5} - \beta_{4} - 2 \beta_{2} + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{10} + \beta_{8} + \beta_{6}) q^{2} + (\beta_{11} + \beta_{10} - \beta_{6} + \beta_{4} - \beta_{3} + \beta_{2} - 1) q^{4} + ( - \beta_{9} + \beta_{7} + \beta_{5} + \beta_1 - 1) q^{5} + (\beta_{11} - \beta_{10} + \beta_{8} - \beta_{7}) q^{7} + ( - \beta_{5} - \beta_{4} - 2 \beta_{2} + 1) q^{8} + (\beta_{5} + \beta_{3} + \beta_{2} - \beta_1 + 1) q^{10} + (2 \beta_{9} + \beta_{7}) q^{11} + (\beta_{9} - \beta_{7} - 2 \beta_{6} - \beta_{5} - 2 \beta_{3} - \beta_1 - 1) q^{13} + (\beta_{11} - 2 \beta_{10} - \beta_{9} + \beta_{8} + \beta_{5} + \beta_{4} - \beta_{2} - 1) q^{14} + (\beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} + \beta_{6}) q^{16} + (\beta_{4} - \beta_{3} - \beta_{2} + 1) q^{17} + ( - \beta_{4} - \beta_{3} + \beta_{2} - \beta_1) q^{19} + ( - \beta_{11} - \beta_{10} + \beta_{9} + 2 \beta_{8}) q^{20} + (2 \beta_{10} - \beta_{8} - \beta_{7} - 2 \beta_{6} - 2 \beta_{5} - 2 \beta_{3} + \beta_{2} - \beta_1 - 2) q^{22} + ( - 2 \beta_{10} - 2 \beta_{9} + 2 \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{3} + \cdots - 3) q^{23}+ \cdots + (4 \beta_{4} - 3 \beta_{3} + 2 \beta_{2} - \beta_1 - 9) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 3 q^{2} - 3 q^{4} - 6 q^{5} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 3 q^{2} - 3 q^{4} - 6 q^{5} + 12 q^{8} + 6 q^{10} - 12 q^{11} - 6 q^{14} + 3 q^{16} + 18 q^{17} + 6 q^{19} - 6 q^{20} - 6 q^{22} - 15 q^{23} + 6 q^{25} + 30 q^{26} - 12 q^{28} - 12 q^{29} + 24 q^{35} + 6 q^{37} + 3 q^{38} - 6 q^{40} - 15 q^{41} + 6 q^{44} + 6 q^{46} - 21 q^{47} + 12 q^{49} - 3 q^{50} - 12 q^{52} + 18 q^{53} - 12 q^{55} + 6 q^{56} + 12 q^{58} - 24 q^{59} + 9 q^{61} - 24 q^{62} - 24 q^{64} + 6 q^{65} + 9 q^{67} + 9 q^{68} - 15 q^{70} + 54 q^{71} - 12 q^{73} + 12 q^{74} - 6 q^{76} + 12 q^{77} - 42 q^{80} - 12 q^{82} - 12 q^{83} + 21 q^{86} - 12 q^{88} + 18 q^{89} - 12 q^{91} - 6 q^{92} - 6 q^{94} - 12 q^{95} - 90 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} + 27 x^{10} - 80 x^{9} + 186 x^{8} - 330 x^{7} + 463 x^{6} - 504 x^{5} + 420 x^{4} - 258 x^{3} + 108 x^{2} - 27 x + 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} - \nu + 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{8} - 4\nu^{7} + 16\nu^{6} - 34\nu^{5} + 62\nu^{4} - 72\nu^{3} + 64\nu^{2} - 33\nu + 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{10} - 5 \nu^{9} + 22 \nu^{8} - 58 \nu^{7} + 127 \nu^{6} - 199 \nu^{5} + 249 \nu^{4} - 224 \nu^{3} + 145 \nu^{2} - 58 \nu + 9 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( - 2 \nu^{10} + 10 \nu^{9} - 43 \nu^{8} + 112 \nu^{7} - 239 \nu^{6} + 367 \nu^{5} - 448 \nu^{4} + 395 \nu^{3} - 256 \nu^{2} + 104 \nu - 20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( - 2 \nu^{10} + 10 \nu^{9} - 44 \nu^{8} + 116 \nu^{7} - 255 \nu^{6} + 401 \nu^{5} - 509 \nu^{4} + 465 \nu^{3} - 314 \nu^{2} + 132 \nu - 25 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( 11 \nu^{11} - 61 \nu^{10} + 270 \nu^{9} - 761 \nu^{8} + 1715 \nu^{7} - 2888 \nu^{6} + 3852 \nu^{5} - 3892 \nu^{4} + 2948 \nu^{3} - 1562 \nu^{2} + 500 \nu - 71 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( - 14 \nu^{11} + 78 \nu^{10} - 344 \nu^{9} + 970 \nu^{8} - 2177 \nu^{7} + 3659 \nu^{6} - 4853 \nu^{5} + 4884 \nu^{4} - 3671 \nu^{3} + 1934 \nu^{2} - 613 \nu + 85 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( - 20 \nu^{11} + 111 \nu^{10} - 490 \nu^{9} + 1379 \nu^{8} - 3097 \nu^{7} + 5198 \nu^{6} - 6900 \nu^{5} + 6936 \nu^{4} - 5222 \nu^{3} + 2750 \nu^{2} - 876 \nu + 124 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( 42 \nu^{11} - 231 \nu^{10} + 1019 \nu^{9} - 2853 \nu^{8} + 6396 \nu^{7} - 10689 \nu^{6} + 14157 \nu^{5} - 14172 \nu^{4} + 10648 \nu^{3} - 5589 \nu^{2} + 1785 \nu - 257 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( 68 \nu^{11} - 374 \nu^{10} + 1649 \nu^{9} - 4616 \nu^{8} + 10342 \nu^{7} - 17277 \nu^{6} + 22863 \nu^{5} - 22873 \nu^{4} + 17165 \nu^{3} - 9002 \nu^{2} + 2871 \nu - 412 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( - 98 \nu^{11} + 540 \nu^{10} - 2381 \nu^{9} + 6671 \nu^{8} - 14949 \nu^{7} + 24987 \nu^{6} - 33072 \nu^{5} + 33097 \nu^{4} - 24840 \nu^{3} + 13028 \nu^{2} - 4153 \nu + 595 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 2\beta_{11} + 2\beta_{10} + \beta_{9} - 2\beta_{8} - 2\beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{11} + 2\beta_{10} + \beta_{9} - 2\beta_{8} - 2\beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + 3\beta _1 - 5 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 4 \beta_{11} - 5 \beta_{10} + \beta_{9} + 9 \beta_{8} - 3 \beta_{7} + 4 \beta_{6} + 3 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 3 \beta _1 - 5 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 10 \beta_{11} - 12 \beta_{10} + \beta_{9} + 20 \beta_{8} - 6 \beta_{7} + 10 \beta_{6} + 10 \beta_{5} - 8 \beta_{4} + 5 \beta_{3} + 7 \beta_{2} - 12 \beta _1 + 16 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 7 \beta_{11} + 14 \beta_{10} - 12 \beta_{9} - 25 \beta_{8} + 14 \beta_{7} - 6 \beta_{6} + 2 \beta_{5} - 4 \beta_{4} - 3 \beta_{3} + 2 \beta_{2} - 23 \beta _1 + 30 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 47 \beta_{11} + 73 \beta_{10} - 38 \beta_{9} - 126 \beta_{8} + 57 \beta_{7} - 44 \beta_{6} - 48 \beta_{5} + 34 \beta_{4} - 28 \beta_{3} - 37 \beta_{2} + 42 \beta _1 - 56 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 8 \beta_{11} - 6 \beta_{10} + 28 \beta_{9} + 2 \beta_{8} - 15 \beta_{7} - 14 \beta_{6} - 80 \beta_{5} + 67 \beta_{4} - 28 \beta_{3} - 65 \beta_{2} + 150 \beta _1 - 191 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 212 \beta_{11} - 394 \beta_{10} + 291 \beta_{9} + 644 \beta_{8} - 340 \beta_{7} + 174 \beta_{6} + 143 \beta_{5} - 91 \beta_{4} + 99 \beta_{3} + 113 \beta_{2} - 86 \beta _1 + 129 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 226 \beta_{11} - 359 \beta_{10} + 199 \beta_{9} + 624 \beta_{8} - 288 \beta_{7} + 214 \beta_{6} + 600 \beta_{5} - 455 \beta_{4} + 287 \beta_{3} + 488 \beta_{2} - 846 \beta _1 + 1108 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 842 \beta_{11} + 1734 \beta_{10} - 1457 \beta_{9} - 2695 \beta_{8} + 1539 \beta_{7} - 596 \beta_{6} - 23 \beta_{5} - 44 \beta_{4} - 115 \beta_{3} - 14 \beta_{2} - 324 \beta _1 + 358 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 1918 \beta_{11} + 3656 \beta_{10} - 2790 \beta_{9} - 5875 \beta_{8} + 3164 \beta_{7} - 1515 \beta_{6} - 3154 \beta_{5} + 2312 \beta_{4} - 1632 \beta_{3} - 2578 \beta_{2} + 4090 \beta _1 - 5451 ) / 3 \) Copy content Toggle raw display

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(-1 - \beta_{9}\)

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
0.500000 + 1.00210i
0.500000 + 2.22827i
0.500000 1.27297i
0.500000 0.258654i
0.500000 1.68614i
0.500000 0.0126039i
0.500000 1.00210i
0.500000 2.22827i
0.500000 + 1.27297i
0.500000 + 0.258654i
0.500000 + 1.68614i
0.500000 + 0.0126039i
−1.20081 2.07986i 0 −1.88389 + 3.26300i 0.0465417 0.0806126i 0 0.289931 + 0.502175i 4.24555 0 −0.223551
244.2 −1.05831 1.83305i 0 −1.24005 + 2.14782i −1.34155 + 2.32363i 0 −0.486166 0.842065i 1.01617 0 5.67911
244.3 −0.400763 0.694143i 0 0.678777 1.17568i −1.37492 + 2.38143i 0 −1.18842 2.05840i −2.69117 0 2.20407
244.4 −0.207733 0.359804i 0 0.913694 1.58256i 1.10759 1.91841i 0 0.659815 + 1.14283i −1.59015 0 −0.920335
244.5 0.527162 + 0.913072i 0 0.444200 0.769376i −0.872894 + 1.51190i 0 −1.22962 2.12977i 3.04531 0 −1.84063
244.6 0.840456 + 1.45571i 0 −0.412733 + 0.714874i −0.564772 + 0.978214i 0 1.95446 + 3.38523i 1.97429 0 −1.89866
487.1 −1.20081 + 2.07986i 0 −1.88389 3.26300i 0.0465417 + 0.0806126i 0 0.289931 0.502175i 4.24555 0 −0.223551
487.2 −1.05831 + 1.83305i 0 −1.24005 2.14782i −1.34155 2.32363i 0 −0.486166 + 0.842065i 1.01617 0 5.67911
487.3 −0.400763 + 0.694143i 0 0.678777 + 1.17568i −1.37492 2.38143i 0 −1.18842 + 2.05840i −2.69117 0 2.20407
487.4 −0.207733 + 0.359804i 0 0.913694 + 1.58256i 1.10759 + 1.91841i 0 0.659815 1.14283i −1.59015 0 −0.920335
487.5 0.527162 0.913072i 0 0.444200 + 0.769376i −0.872894 1.51190i 0 −1.22962 + 2.12977i 3.04531 0 −1.84063
487.6 0.840456 1.45571i 0 −0.412733 0.714874i −0.564772 0.978214i 0 1.95446 3.38523i 1.97429 0 −1.89866
\(n\): 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.b 12
3.b odd 2 1 729.2.c.e 12
9.c even 3 1 729.2.a.d 6
9.c even 3 1 inner 729.2.c.b 12
9.d odd 6 1 729.2.a.a 6
9.d odd 6 1 729.2.c.e 12
27.e even 9 2 81.2.e.a 12
27.e even 9 2 243.2.e.a 12
27.e even 9 2 243.2.e.b 12
27.f odd 18 2 27.2.e.a 12
27.f odd 18 2 243.2.e.c 12
27.f odd 18 2 243.2.e.d 12
108.l even 18 2 432.2.u.c 12
135.n odd 18 2 675.2.l.c 12
135.q even 36 4 675.2.u.b 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.2.e.a 12 27.f odd 18 2
81.2.e.a 12 27.e even 9 2
243.2.e.a 12 27.e even 9 2
243.2.e.b 12 27.e even 9 2
243.2.e.c 12 27.f odd 18 2
243.2.e.d 12 27.f odd 18 2
432.2.u.c 12 108.l even 18 2
675.2.l.c 12 135.n odd 18 2
675.2.u.b 24 135.q even 36 4
729.2.a.a 6 9.d odd 6 1
729.2.a.d 6 9.c even 3 1
729.2.c.b 12 1.a even 1 1 trivial
729.2.c.b 12 9.c even 3 1 inner
729.2.c.e 12 3.b odd 2 1
729.2.c.e 12 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 3 T_{2}^{11} + 12 T_{2}^{10} + 15 T_{2}^{9} + 45 T_{2}^{8} + 45 T_{2}^{7} + 123 T_{2}^{6} + 63 T_{2}^{5} + 117 T_{2}^{4} + 72 T_{2}^{3} + 81 T_{2}^{2} + 27 T_{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(729, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 3 T^{11} + 12 T^{10} + 15 T^{9} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 6 T^{11} + 30 T^{10} + 84 T^{9} + \cdots + 9 \) Copy content Toggle raw display
$7$ \( T^{12} + 15 T^{10} + 22 T^{9} + \cdots + 289 \) Copy content Toggle raw display
$11$ \( T^{12} + 12 T^{11} + 93 T^{10} + 420 T^{9} + \cdots + 9 \) Copy content Toggle raw display
$13$ \( T^{12} + 24 T^{10} + 4 T^{9} + 486 T^{8} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( (T^{6} - 9 T^{5} + 9 T^{4} + 54 T^{3} + \cdots + 27)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} - 3 T^{5} - 30 T^{4} + 38 T^{3} + \cdots + 19)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + 15 T^{11} + 165 T^{10} + \cdots + 106929 \) Copy content Toggle raw display
$29$ \( T^{12} + 12 T^{11} + 147 T^{10} + \cdots + 45369 \) Copy content Toggle raw display
$31$ \( T^{12} + 51 T^{10} + 382 T^{9} + \cdots + 26569 \) Copy content Toggle raw display
$37$ \( (T^{6} - 3 T^{5} - 57 T^{4} + 254 T^{3} + \cdots + 4933)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + 15 T^{11} + 219 T^{10} + \cdots + 11229201 \) Copy content Toggle raw display
$43$ \( T^{12} + 96 T^{10} + 346 T^{9} + \cdots + 3308761 \) Copy content Toggle raw display
$47$ \( T^{12} + 21 T^{11} + 336 T^{10} + \cdots + 42732369 \) Copy content Toggle raw display
$53$ \( (T^{6} - 9 T^{5} - 108 T^{4} + 513 T^{3} + \cdots - 12393)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + 24 T^{11} + \cdots + 176384961 \) Copy content Toggle raw display
$61$ \( T^{12} - 9 T^{11} + 240 T^{10} + \cdots + 273670849 \) Copy content Toggle raw display
$67$ \( T^{12} - 9 T^{11} + 195 T^{10} + \cdots + 8288641 \) Copy content Toggle raw display
$71$ \( (T^{6} - 27 T^{5} + 225 T^{4} - 486 T^{3} + \cdots + 27)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 6 T^{5} - 174 T^{4} - 250 T^{3} + \cdots - 431)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + 177 T^{10} - 140 T^{9} + \cdots + 3508129 \) Copy content Toggle raw display
$83$ \( T^{12} + 12 T^{11} + \cdots + 6951057129 \) Copy content Toggle raw display
$89$ \( (T^{6} - 9 T^{5} - 180 T^{4} + 1026 T^{3} + \cdots + 32589)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + 204 T^{10} + \cdots + 66765241 \) Copy content Toggle raw display
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