Newspace parameters
Level: | \( N \) | \(=\) | \( 432 = 2^{4} \cdot 3^{3} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 432.u (of order \(9\), degree \(6\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(3.44953736732\) |
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
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Defining polynomial: | \( x^{12} - 6 x^{11} + 33 x^{10} - 110 x^{9} + 318 x^{8} - 678 x^{7} + 1225 x^{6} - 1698 x^{5} + 1905 x^{4} - 1584 x^{3} + 936 x^{2} - 342 x + 57 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 3 \) |
Twist minimal: | no (minimal twist has level 54) |
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.
Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} + 33 x^{10} - 110 x^{9} + 318 x^{8} - 678 x^{7} + 1225 x^{6} - 1698 x^{5} + 1905 x^{4} - 1584 x^{3} + 936 x^{2} - 342 x + 57 \) :
\(\beta_{1}\) | \(=\) | \( ( 6 \nu^{11} - 33 \nu^{10} + 127 \nu^{9} - 324 \nu^{8} + 438 \nu^{7} - 252 \nu^{6} - 1278 \nu^{5} + 3234 \nu^{4} - 5701 \nu^{3} + 5358 \nu^{2} - 3477 \nu + 1060 ) / 218 \) |
\(\beta_{2}\) | \(=\) | \( ( 36 \nu^{11} - 89 \nu^{10} + 544 \nu^{9} - 745 \nu^{8} + 2301 \nu^{7} - 1512 \nu^{6} + 3777 \nu^{5} - 2069 \nu^{4} + 5579 \nu^{3} - 6002 \nu^{2} + 5080 \nu - 1706 ) / 218 \) |
\(\beta_{3}\) | \(=\) | \( ( - 27 \nu^{11} + 94 \nu^{10} - 408 \nu^{9} + 586 \nu^{8} - 445 \nu^{7} - 2572 \nu^{6} + 9021 \nu^{5} - 18150 \nu^{4} + 24401 \nu^{3} - 20623 \nu^{2} + 10469 \nu - 2263 ) / 218 \) |
\(\beta_{4}\) | \(=\) | \( ( 26 \nu^{11} - 34 \nu^{10} + 187 \nu^{9} + 449 \nu^{8} - 1590 \nu^{7} + 6865 \nu^{6} - 12623 \nu^{5} + 20118 \nu^{4} - 19981 \nu^{3} + 12972 \nu^{2} - 4712 \nu + 524 ) / 218 \) |
\(\beta_{5}\) | \(=\) | \( ( - 2 \nu^{11} + 120 \nu^{10} - 551 \nu^{9} + 2615 \nu^{8} - 6686 \nu^{7} + 15780 \nu^{6} - 23990 \nu^{5} + 31404 \nu^{4} - 25822 \nu^{3} + 15545 \nu^{2} - 4618 \nu + 555 ) / 218 \) |
\(\beta_{6}\) | \(=\) | \( ( - 27 \nu^{11} + 203 \nu^{10} - 953 \nu^{9} + 3311 \nu^{8} - 8075 \nu^{7} + 16285 \nu^{6} - 23134 \nu^{5} + 26758 \nu^{4} - 19635 \nu^{3} + 9570 \nu^{2} - 1957 \nu - 83 ) / 218 \) |
\(\beta_{7}\) | \(=\) | \( ( \nu^{10} - 5 \nu^{9} + 25 \nu^{8} - 70 \nu^{7} + 173 \nu^{6} - 295 \nu^{5} + 412 \nu^{4} - 404 \nu^{3} + 279 \nu^{2} - 116 \nu + 26 ) / 2 \) |
\(\beta_{8}\) | \(=\) | \( ( 2 \nu^{11} + 98 \nu^{10} - 539 \nu^{9} + 2726 \nu^{8} - 8138 \nu^{7} + 20190 \nu^{6} - 36614 \nu^{5} + 52308 \nu^{4} - 55710 \nu^{3} + 41571 \nu^{2} - 19689 \nu + 4350 ) / 218 \) |
\(\beta_{9}\) | \(=\) | \( ( 26 \nu^{11} - 252 \nu^{10} + 1277 \nu^{9} - 4892 \nu^{8} + 13234 \nu^{7} - 28887 \nu^{6} + 47327 \nu^{5} - 60760 \nu^{4} + 56973 \nu^{3} - 36296 \nu^{2} + 13927 \nu - 2201 ) / 218 \) |
\(\beta_{10}\) | \(=\) | \( ( 36 \nu^{11} - 307 \nu^{10} + 1634 \nu^{9} - 6086 \nu^{8} + 17125 \nu^{7} - 37373 \nu^{6} + 64054 \nu^{5} - 84255 \nu^{4} + 84604 \nu^{3} - 58431 \nu^{2} + 25899 \nu - 5194 ) / 218 \) |
\(\beta_{11}\) | \(=\) | \( ( 91 \nu^{11} - 664 \nu^{10} + 3325 \nu^{9} - 11563 \nu^{8} + 30405 \nu^{7} - 62791 \nu^{6} + 99754 \nu^{5} - 123498 \nu^{4} + 112914 \nu^{3} - 71337 \nu^{2} + 28743 \nu - 5469 ) / 218 \) |
\(\nu\) | \(=\) | \( ( 2 \beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} + 2 \beta_{6} - \beta_{5} + \beta_{4} + 2 \beta_{3} - \beta_{2} - \beta _1 + 2 ) / 3 \) |
\(\nu^{2}\) | \(=\) | \( ( 2 \beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} + 4 \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + 5 \beta_{3} - \beta_{2} - \beta _1 - 7 ) / 3 \) |
\(\nu^{3}\) | \(=\) | \( ( - 7 \beta_{11} + 2 \beta_{10} + 5 \beta_{9} - 5 \beta_{8} + \beta_{7} - 13 \beta_{6} + 5 \beta_{5} - 2 \beta_{4} - 4 \beta_{3} + 2 \beta_{2} - 4 \beta _1 - 10 ) / 3 \) |
\(\nu^{4}\) | \(=\) | \( ( - 16 \beta_{11} + 11 \beta_{10} + 8 \beta_{9} - 8 \beta_{8} - 20 \beta_{7} - 7 \beta_{6} + 14 \beta_{5} - 2 \beta_{4} - 31 \beta_{3} - \beta_{2} - 7 \beta _1 + 32 ) / 3 \) |
\(\nu^{5}\) | \(=\) | \( ( 23 \beta_{11} + 14 \beta_{10} - 34 \beta_{9} + 25 \beta_{8} - 26 \beta_{7} + 65 \beta_{6} - 10 \beta_{5} + 4 \beta_{4} - 10 \beta_{3} - 16 \beta_{2} + 29 \beta _1 + 77 ) / 3 \) |
\(\nu^{6}\) | \(=\) | \( ( 110 \beta_{11} - 49 \beta_{10} - 88 \beta_{9} + 67 \beta_{8} + 85 \beta_{7} + 101 \beta_{6} - 94 \beta_{5} - 17 \beta_{4} + 161 \beta_{3} + 17 \beta_{2} + 104 \beta _1 - 121 ) / 3 \) |
\(\nu^{7}\) | \(=\) | \( ( - 25 \beta_{11} - 187 \beta_{10} + 158 \beta_{9} - 101 \beta_{8} + 229 \beta_{7} - 271 \beta_{6} - 46 \beta_{5} - 56 \beta_{4} + 212 \beta_{3} + 149 \beta_{2} - 82 \beta _1 - 535 ) / 3 \) |
\(\nu^{8}\) | \(=\) | \( ( - 652 \beta_{11} + 41 \beta_{10} + 761 \beta_{9} - 509 \beta_{8} - 263 \beta_{7} - 826 \beta_{6} + 515 \beta_{5} + 151 \beta_{4} - 724 \beta_{3} - 19 \beta_{2} - 829 \beta _1 + 257 ) / 3 \) |
\(\nu^{9}\) | \(=\) | \( ( - 478 \beta_{11} + 1304 \beta_{10} - 268 \beta_{9} + 160 \beta_{8} - 1571 \beta_{7} + 821 \beta_{6} + 812 \beta_{5} + 640 \beta_{4} - 1951 \beta_{3} - 1108 \beta_{2} - 406 \beta _1 + 3359 ) / 3 \) |
\(\nu^{10}\) | \(=\) | \( ( 3353 \beta_{11} + 1451 \beta_{10} - 5224 \beta_{9} + 3352 \beta_{8} + 25 \beta_{7} + 5630 \beta_{6} - 2308 \beta_{5} - 521 \beta_{4} + 2375 \beta_{3} - 913 \beta_{2} + 4949 \beta _1 + 1451 ) / 3 \) |
\(\nu^{11}\) | \(=\) | \( ( 6041 \beta_{11} - 6547 \beta_{10} - 3685 \beta_{9} + 2323 \beta_{8} + 9241 \beta_{7} + 326 \beta_{6} - 7273 \beta_{5} - 5168 \beta_{4} + 14021 \beta_{3} + 6587 \beta_{2} + 7973 \beta _1 - 18736 ) / 3 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/432\mathbb{Z}\right)^\times\).
\(n\) | \(271\) | \(325\) | \(353\) |
\(\chi(n)\) | \(1\) | \(1\) | \(\beta_{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
49.1 |
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0 | −0.247510 | − | 1.71428i | 0 | −1.96209 | + | 0.714144i | 0 | 0.696712 | + | 3.95125i | 0 | −2.87748 | + | 0.848600i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
49.2 | 0 | 1.36085 | + | 1.07149i | 0 | 0.696050 | − | 0.253341i | 0 | −0.717657 | − | 4.07003i | 0 | 0.703829 | + | 2.91627i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
97.1 | 0 | −0.247510 | + | 1.71428i | 0 | −1.96209 | − | 0.714144i | 0 | 0.696712 | − | 3.95125i | 0 | −2.87748 | − | 0.848600i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
97.2 | 0 | 1.36085 | − | 1.07149i | 0 | 0.696050 | + | 0.253341i | 0 | −0.717657 | + | 4.07003i | 0 | 0.703829 | − | 2.91627i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
193.1 | 0 | −0.552775 | − | 1.64147i | 0 | −0.177398 | + | 1.00607i | 0 | −2.04289 | + | 1.71418i | 0 | −2.38888 | + | 1.81473i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
193.2 | 0 | 1.14517 | + | 1.29945i | 0 | 0.617090 | − | 3.49969i | 0 | 0.244752 | − | 0.205371i | 0 | −0.377165 | + | 2.97620i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
241.1 | 0 | −1.56529 | + | 0.741539i | 0 | −3.10057 | + | 2.60168i | 0 | −0.144365 | + | 0.0525446i | 0 | 1.90024 | − | 2.32144i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
241.2 | 0 | −0.140451 | − | 1.72635i | 0 | 2.42692 | − | 2.03643i | 0 | 3.46344 | − | 1.26059i | 0 | −2.96055 | + | 0.484935i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
337.1 | 0 | −1.56529 | − | 0.741539i | 0 | −3.10057 | − | 2.60168i | 0 | −0.144365 | − | 0.0525446i | 0 | 1.90024 | + | 2.32144i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
337.2 | 0 | −0.140451 | + | 1.72635i | 0 | 2.42692 | + | 2.03643i | 0 | 3.46344 | + | 1.26059i | 0 | −2.96055 | − | 0.484935i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
385.1 | 0 | −0.552775 | + | 1.64147i | 0 | −0.177398 | − | 1.00607i | 0 | −2.04289 | − | 1.71418i | 0 | −2.38888 | − | 1.81473i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
385.2 | 0 | 1.14517 | − | 1.29945i | 0 | 0.617090 | + | 3.49969i | 0 | 0.244752 | + | 0.205371i | 0 | −0.377165 | − | 2.97620i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
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 | 432.2.u.b | 12 | |
4.b | odd | 2 | 1 | 54.2.e.b | ✓ | 12 | |
12.b | even | 2 | 1 | 162.2.e.b | 12 | ||
27.e | even | 9 | 1 | inner | 432.2.u.b | 12 | |
36.f | odd | 6 | 1 | 486.2.e.f | 12 | ||
36.f | odd | 6 | 1 | 486.2.e.h | 12 | ||
36.h | even | 6 | 1 | 486.2.e.e | 12 | ||
36.h | even | 6 | 1 | 486.2.e.g | 12 | ||
108.j | odd | 18 | 1 | 54.2.e.b | ✓ | 12 | |
108.j | odd | 18 | 1 | 486.2.e.f | 12 | ||
108.j | odd | 18 | 1 | 486.2.e.h | 12 | ||
108.j | odd | 18 | 1 | 1458.2.a.g | 6 | ||
108.j | odd | 18 | 2 | 1458.2.c.f | 12 | ||
108.l | even | 18 | 1 | 162.2.e.b | 12 | ||
108.l | even | 18 | 1 | 486.2.e.e | 12 | ||
108.l | even | 18 | 1 | 486.2.e.g | 12 | ||
108.l | even | 18 | 1 | 1458.2.a.f | 6 | ||
108.l | even | 18 | 2 | 1458.2.c.g | 12 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
54.2.e.b | ✓ | 12 | 4.b | odd | 2 | 1 | |
54.2.e.b | ✓ | 12 | 108.j | odd | 18 | 1 | |
162.2.e.b | 12 | 12.b | even | 2 | 1 | ||
162.2.e.b | 12 | 108.l | even | 18 | 1 | ||
432.2.u.b | 12 | 1.a | even | 1 | 1 | trivial | |
432.2.u.b | 12 | 27.e | even | 9 | 1 | inner | |
486.2.e.e | 12 | 36.h | even | 6 | 1 | ||
486.2.e.e | 12 | 108.l | even | 18 | 1 | ||
486.2.e.f | 12 | 36.f | odd | 6 | 1 | ||
486.2.e.f | 12 | 108.j | odd | 18 | 1 | ||
486.2.e.g | 12 | 36.h | even | 6 | 1 | ||
486.2.e.g | 12 | 108.l | even | 18 | 1 | ||
486.2.e.h | 12 | 36.f | odd | 6 | 1 | ||
486.2.e.h | 12 | 108.j | odd | 18 | 1 | ||
1458.2.a.f | 6 | 108.l | even | 18 | 1 | ||
1458.2.a.g | 6 | 108.j | odd | 18 | 1 | ||
1458.2.c.f | 12 | 108.j | odd | 18 | 2 | ||
1458.2.c.g | 12 | 108.l | even | 18 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{12} + 3 T_{5}^{11} + 9 T_{5}^{10} + 24 T_{5}^{9} + 162 T_{5}^{8} - 27 T_{5}^{7} + 1053 T_{5}^{6} + 5184 T_{5}^{5} + 3564 T_{5}^{4} - 3672 T_{5}^{3} + 2592 T_{5}^{2} - 7776 T_{5} + 5184 \)
acting on \(S_{2}^{\mathrm{new}}(432, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{12} \)
$3$
\( T^{12} + 6 T^{10} + 18 T^{8} + 18 T^{7} + \cdots + 729 \)
$5$
\( T^{12} + 3 T^{11} + 9 T^{10} + 24 T^{9} + \cdots + 5184 \)
$7$
\( T^{12} - 3 T^{11} + 24 T^{10} - 88 T^{9} + \cdots + 64 \)
$11$
\( T^{12} - 12 T^{11} + 90 T^{10} - 537 T^{9} + \cdots + 81 \)
$13$
\( T^{12} - 12 T^{11} + 48 T^{10} + \cdots + 23104 \)
$17$
\( T^{12} + 6 T^{11} + 54 T^{10} + \cdots + 110889 \)
$19$
\( T^{12} - 9 T^{11} + 78 T^{10} + \cdots + 94249 \)
$23$
\( T^{12} + 30 T^{11} + 414 T^{10} + \cdots + 5184 \)
$29$
\( T^{12} - 15 T^{11} + 81 T^{10} + \cdots + 5184 \)
$31$
\( T^{12} + 81 T^{10} - 421 T^{9} + \cdots + 4032064 \)
$37$
\( T^{12} + 15 T^{11} + \cdots + 142659136 \)
$41$
\( T^{12} + 12 T^{11} + 117 T^{10} + \cdots + 2653641 \)
$43$
\( T^{12} + 9 T^{11} + 36 T^{10} + \cdots + 49674304 \)
$47$
\( T^{12} - 9 T^{11} + 99 T^{10} + \cdots + 419904 \)
$53$
\( (T^{6} + 6 T^{5} - 63 T^{4} + 3 T^{3} + \cdots - 72)^{2} \)
$59$
\( T^{12} + 12 T^{11} + 9 T^{10} + \cdots + 82464561 \)
$61$
\( T^{12} + 36 T^{11} + 531 T^{10} + \cdots + 1000000 \)
$67$
\( T^{12} + 36 T^{11} + \cdots + 249393368449 \)
$71$
\( T^{12} + 12 T^{11} + \cdots + 488586816 \)
$73$
\( T^{12} + 21 T^{11} + 390 T^{10} + \cdots + 72361 \)
$79$
\( T^{12} + 39 T^{11} + \cdots + 591851584 \)
$83$
\( T^{12} + 18 T^{11} + \cdots + 13756474944 \)
$89$
\( T^{12} - 12 T^{11} + \cdots + 126899100441 \)
$97$
\( T^{12} - 39 T^{11} + \cdots + 373532435929 \)
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