Newspace parameters
Level: | \( N \) | \(=\) | \( 1728 = 2^{6} \cdot 3^{3} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1728.r (of order \(6\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(13.7981494693\) |
Analytic rank: | \(0\) |
Dimension: | \(12\) |
Relative dimension: | \(6\) over \(\Q(\zeta_{6})\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{12} - 16x^{8} - 24x^{7} + 96x^{5} + 304x^{4} + 384x^{3} + 288x^{2} + 144x + 36 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
Coefficient ring index: | \( 2^{6}\cdot 3^{4} \) |
Twist minimal: | no (minimal twist has level 576) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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.
Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 16x^{8} - 24x^{7} + 96x^{5} + 304x^{4} + 384x^{3} + 288x^{2} + 144x + 36 \) :
\(\beta_{1}\) | \(=\) | \( ( - 275 \nu^{11} + 300 \nu^{10} - 650 \nu^{9} + 1148 \nu^{8} + 1747 \nu^{7} + 3300 \nu^{6} + 5062 \nu^{5} - 27262 \nu^{4} - 51252 \nu^{3} - 44964 \nu^{2} - 26082 \nu - 96192 ) / 51972 \) |
\(\beta_{2}\) | \(=\) | \( ( 839 \nu^{11} + 7708 \nu^{10} - 10784 \nu^{9} + 8976 \nu^{8} - 22648 \nu^{7} - 131336 \nu^{6} - 31862 \nu^{5} + 209952 \nu^{4} + 921960 \nu^{3} + 1654212 \nu^{2} + \cdots + 275832 ) / 103944 \) |
\(\beta_{3}\) | \(=\) | \( ( 962 \nu^{11} - 2392 \nu^{10} + 2887 \nu^{9} - 2220 \nu^{8} - 13990 \nu^{7} + 14442 \nu^{6} + 11380 \nu^{5} + 57708 \nu^{4} + 98118 \nu^{3} - 132080 \nu^{2} + 49284 \nu + 15984 ) / 51972 \) |
\(\beta_{4}\) | \(=\) | \( ( - 118 \nu^{11} + 90 \nu^{10} - 53 \nu^{9} + 32 \nu^{8} + 1866 \nu^{7} + 1416 \nu^{6} - 1364 \nu^{5} - 10408 \nu^{4} - 27758 \nu^{3} - 22776 \nu^{2} - 12468 \nu - 5172 ) / 1704 \) |
\(\beta_{5}\) | \(=\) | \( ( 2793 \nu^{11} - 3602 \nu^{10} + 4065 \nu^{9} - 2808 \nu^{8} - 44568 \nu^{7} - 7530 \nu^{6} + 17238 \nu^{5} + 219348 \nu^{4} + 562962 \nu^{3} + 348980 \nu^{2} + \cdots + 133704 ) / 34648 \) |
\(\beta_{6}\) | \(=\) | \( ( 6049 \nu^{11} - 4740 \nu^{10} + 3454 \nu^{9} - 2632 \nu^{8} - 93803 \nu^{7} - 72588 \nu^{6} + 65542 \nu^{5} + 530594 \nu^{4} + 1432764 \nu^{3} + 1177668 \nu^{2} + \cdots + 268416 ) / 51972 \) |
\(\beta_{7}\) | \(=\) | \( ( 1688 \nu^{11} - 1054 \nu^{10} + 840 \nu^{9} - 684 \nu^{8} - 26378 \nu^{7} - 24587 \nu^{6} + 12648 \nu^{5} + 151968 \nu^{4} + 420176 \nu^{3} + 400856 \nu^{2} + 282372 \nu + 105120 ) / 12993 \) |
\(\beta_{8}\) | \(=\) | \( ( - 15676 \nu^{11} + 10866 \nu^{10} - 681 \nu^{9} - 11340 \nu^{8} + 267718 \nu^{7} + 188112 \nu^{6} - 234336 \nu^{5} - 1340808 \nu^{4} - 3664246 \nu^{3} + \cdots - 1037028 ) / 103944 \) |
\(\beta_{9}\) | \(=\) | \( ( - 19630 \nu^{11} + 7150 \nu^{10} - 3611 \nu^{9} + 1848 \nu^{8} + 317744 \nu^{7} + 348166 \nu^{6} - 103124 \nu^{5} - 1841148 \nu^{4} - 5346606 \nu^{3} + \cdots - 1369008 ) / 103944 \) |
\(\beta_{10}\) | \(=\) | \( ( 284 \nu^{11} - 222 \nu^{10} + 159 \nu^{9} - 192 \nu^{8} - 4304 \nu^{7} - 3408 \nu^{6} + 2664 \nu^{5} + 26820 \nu^{4} + 65498 \nu^{3} + 53952 \nu^{2} + 29640 \nu + 4860 ) / 1464 \) |
\(\beta_{11}\) | \(=\) | \( ( - 6916 \nu^{11} + 3388 \nu^{10} - 2205 \nu^{9} + 1476 \nu^{8} + 110866 \nu^{7} + 108978 \nu^{6} - 40656 \nu^{5} - 637476 \nu^{4} - 1820914 \nu^{3} - 1805344 \nu^{2} + \cdots - 462528 ) / 34648 \) |
\(\nu\) | \(=\) | \( ( -2\beta_{11} + \beta_{9} - \beta_{6} - \beta_{5} - \beta_{2} + \beta_1 ) / 6 \) |
\(\nu^{2}\) | \(=\) | \( ( -2\beta_{11} + 2\beta_{9} + 2\beta_{5} - 9\beta_{3} - 2\beta_{2} ) / 6 \) |
\(\nu^{3}\) | \(=\) | \( ( - 4 \beta_{11} + 8 \beta_{9} + 3 \beta_{7} + 8 \beta_{6} + 4 \beta_{5} + 12 \beta_{4} - 6 \beta_{3} + 4 \beta_{2} + 4 \beta _1 + 6 ) / 6 \) |
\(\nu^{4}\) | \(=\) | \( ( 4\beta_{10} - 2\beta_{8} + 9\beta_{6} + 30\beta_{4} + 9\beta _1 + 28 ) / 3 \) |
\(\nu^{5}\) | \(=\) | \( ( - 22 \beta_{11} + 3 \beta_{10} - 22 \beta_{9} - 42 \beta_{7} + 19 \beta_{6} - 38 \beta_{5} + 39 \beta_{4} + 21 \beta_{3} - 38 \beta_{2} + 38 \beta _1 + 78 ) / 6 \) |
\(\nu^{6}\) | \(=\) | \( ( -56\beta_{11} - 16\beta_{9} - 81\beta_{7} - 40\beta_{5} - 56\beta_{2} ) / 3 \) |
\(\nu^{7}\) | \(=\) | \( ( - 94 \beta_{11} + 35 \beta_{9} - 12 \beta_{8} - 60 \beta_{7} + 47 \beta_{6} - 35 \beta_{5} + 108 \beta_{4} - 60 \beta_{3} - 59 \beta_{2} - 47 \beta _1 - 120 ) / 3 \) |
\(\nu^{8}\) | \(=\) | \( ( 35\beta_{10} - 70\beta_{8} + 249\beta_{6} + 669\beta_{4} - 70 ) / 3 \) |
\(\nu^{9}\) | \(=\) | \( ( 164 \beta_{11} + 72 \beta_{10} - 472 \beta_{9} - 72 \beta_{8} - 321 \beta_{7} + 472 \beta_{6} - 308 \beta_{5} + 1140 \beta_{4} + 642 \beta_{3} - 164 \beta_{2} + 236 \beta _1 + 498 ) / 3 \) |
\(\nu^{10}\) | \(=\) | \( ( -328\beta_{11} - 1442\beta_{9} - 1908\beta_{7} - 1442\beta_{5} + 1908\beta_{3} - 1114\beta_{2} ) / 3 \) |
\(\nu^{11}\) | \(=\) | \( ( - 1586 \beta_{11} - 393 \beta_{10} - 1586 \beta_{9} - 3342 \beta_{7} - 1193 \beta_{6} - 2386 \beta_{5} - 2949 \beta_{4} + 1671 \beta_{3} - 2386 \beta_{2} - 2386 \beta _1 - 5898 ) / 3 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times\).
\(n\) | \(325\) | \(703\) | \(1217\) |
\(\chi(n)\) | \(-1\) | \(1\) | \(\beta_{4}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
289.1 |
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0 | 0 | 0 | 1.50000 | − | 0.866025i | 0 | −2.17731 | + | 3.77121i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||
289.2 | 0 | 0 | 0 | 1.50000 | − | 0.866025i | 0 | −1.80664 | + | 3.12920i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||
289.3 | 0 | 0 | 0 | 1.50000 | − | 0.866025i | 0 | −0.495361 | + | 0.857990i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||
289.4 | 0 | 0 | 0 | 1.50000 | − | 0.866025i | 0 | 0.495361 | − | 0.857990i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||
289.5 | 0 | 0 | 0 | 1.50000 | − | 0.866025i | 0 | 1.80664 | − | 3.12920i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||
289.6 | 0 | 0 | 0 | 1.50000 | − | 0.866025i | 0 | 2.17731 | − | 3.77121i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||
1441.1 | 0 | 0 | 0 | 1.50000 | + | 0.866025i | 0 | −2.17731 | − | 3.77121i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||
1441.2 | 0 | 0 | 0 | 1.50000 | + | 0.866025i | 0 | −1.80664 | − | 3.12920i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||
1441.3 | 0 | 0 | 0 | 1.50000 | + | 0.866025i | 0 | −0.495361 | − | 0.857990i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||
1441.4 | 0 | 0 | 0 | 1.50000 | + | 0.866025i | 0 | 0.495361 | + | 0.857990i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||
1441.5 | 0 | 0 | 0 | 1.50000 | + | 0.866025i | 0 | 1.80664 | + | 3.12920i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||
1441.6 | 0 | 0 | 0 | 1.50000 | + | 0.866025i | 0 | 2.17731 | + | 3.77121i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
72.n | even | 6 | 1 | inner |
72.p | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1728.2.r.f | 12 | |
3.b | odd | 2 | 1 | 576.2.r.e | ✓ | 12 | |
4.b | odd | 2 | 1 | inner | 1728.2.r.f | 12 | |
8.b | even | 2 | 1 | 1728.2.r.e | 12 | ||
8.d | odd | 2 | 1 | 1728.2.r.e | 12 | ||
9.c | even | 3 | 1 | 1728.2.r.e | 12 | ||
9.c | even | 3 | 1 | 5184.2.d.r | 12 | ||
9.d | odd | 6 | 1 | 576.2.r.f | yes | 12 | |
9.d | odd | 6 | 1 | 5184.2.d.q | 12 | ||
12.b | even | 2 | 1 | 576.2.r.e | ✓ | 12 | |
24.f | even | 2 | 1 | 576.2.r.f | yes | 12 | |
24.h | odd | 2 | 1 | 576.2.r.f | yes | 12 | |
36.f | odd | 6 | 1 | 1728.2.r.e | 12 | ||
36.f | odd | 6 | 1 | 5184.2.d.r | 12 | ||
36.h | even | 6 | 1 | 576.2.r.f | yes | 12 | |
36.h | even | 6 | 1 | 5184.2.d.q | 12 | ||
72.j | odd | 6 | 1 | 576.2.r.e | ✓ | 12 | |
72.j | odd | 6 | 1 | 5184.2.d.q | 12 | ||
72.l | even | 6 | 1 | 576.2.r.e | ✓ | 12 | |
72.l | even | 6 | 1 | 5184.2.d.q | 12 | ||
72.n | even | 6 | 1 | inner | 1728.2.r.f | 12 | |
72.n | even | 6 | 1 | 5184.2.d.r | 12 | ||
72.p | odd | 6 | 1 | inner | 1728.2.r.f | 12 | |
72.p | odd | 6 | 1 | 5184.2.d.r | 12 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
576.2.r.e | ✓ | 12 | 3.b | odd | 2 | 1 | |
576.2.r.e | ✓ | 12 | 12.b | even | 2 | 1 | |
576.2.r.e | ✓ | 12 | 72.j | odd | 6 | 1 | |
576.2.r.e | ✓ | 12 | 72.l | even | 6 | 1 | |
576.2.r.f | yes | 12 | 9.d | odd | 6 | 1 | |
576.2.r.f | yes | 12 | 24.f | even | 2 | 1 | |
576.2.r.f | yes | 12 | 24.h | odd | 2 | 1 | |
576.2.r.f | yes | 12 | 36.h | even | 6 | 1 | |
1728.2.r.e | 12 | 8.b | even | 2 | 1 | ||
1728.2.r.e | 12 | 8.d | odd | 2 | 1 | ||
1728.2.r.e | 12 | 9.c | even | 3 | 1 | ||
1728.2.r.e | 12 | 36.f | odd | 6 | 1 | ||
1728.2.r.f | 12 | 1.a | even | 1 | 1 | trivial | |
1728.2.r.f | 12 | 4.b | odd | 2 | 1 | inner | |
1728.2.r.f | 12 | 72.n | even | 6 | 1 | inner | |
1728.2.r.f | 12 | 72.p | odd | 6 | 1 | inner | |
5184.2.d.q | 12 | 9.d | odd | 6 | 1 | ||
5184.2.d.q | 12 | 36.h | even | 6 | 1 | ||
5184.2.d.q | 12 | 72.j | odd | 6 | 1 | ||
5184.2.d.q | 12 | 72.l | even | 6 | 1 | ||
5184.2.d.r | 12 | 9.c | even | 3 | 1 | ||
5184.2.d.r | 12 | 36.f | odd | 6 | 1 | ||
5184.2.d.r | 12 | 72.n | even | 6 | 1 | ||
5184.2.d.r | 12 | 72.p | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} - 3T_{5} + 3 \)
acting on \(S_{2}^{\mathrm{new}}(1728, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{12} \)
$3$
\( T^{12} \)
$5$
\( (T^{2} - 3 T + 3)^{6} \)
$7$
\( T^{12} + 33 T^{10} + 810 T^{8} + \cdots + 59049 \)
$11$
\( T^{12} - 39 T^{10} + 1350 T^{8} + \cdots + 6561 \)
$13$
\( (T^{6} + 3 T^{5} - 24 T^{4} - 81 T^{3} + \cdots + 243)^{2} \)
$17$
\( (T^{3} - 24 T - 36)^{4} \)
$19$
\( (T^{2} + 4)^{6} \)
$23$
\( T^{12} + 117 T^{10} + \cdots + 2492305929 \)
$29$
\( (T^{6} + 9 T^{5} - 36 T^{4} - 567 T^{3} + \cdots + 93987)^{2} \)
$31$
\( T^{12} + 117 T^{10} + \cdots + 95004009 \)
$37$
\( (T^{6} + 60 T^{4} + 1008 T^{2} + \cdots + 3888)^{2} \)
$41$
\( (T^{6} + 9 T^{5} + 78 T^{4} + 45 T^{3} + \cdots + 81)^{2} \)
$43$
\( T^{12} - 123 T^{10} + \cdots + 47458321 \)
$47$
\( T^{12} + 297 T^{10} + \cdots + 514609673769 \)
$53$
\( (T^{6} + 180 T^{4} + 1728 T^{2} + \cdots + 432)^{2} \)
$59$
\( T^{12} - 147 T^{10} + 20898 T^{8} + \cdots + 531441 \)
$61$
\( (T^{6} - 9 T^{5} - 36 T^{4} + 567 T^{3} + \cdots + 4563)^{2} \)
$67$
\( T^{12} - 231 T^{10} + \cdots + 352275361 \)
$71$
\( (T^{6} - 288 T^{4} + 19872 T^{2} + \cdots - 124848)^{2} \)
$73$
\( (T^{3} - 84 T - 164)^{4} \)
$79$
\( T^{12} + 93 T^{10} + 7758 T^{8} + \cdots + 4782969 \)
$83$
\( T^{12} - 171 T^{10} + \cdots + 43046721 \)
$89$
\( (T^{3} - 12 T^{2} - 36 T + 108)^{4} \)
$97$
\( (T^{6} + 3 T^{5} + 222 T^{4} + \cdots + 1408969)^{2} \)
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