Newspace parameters
Level: | \( N \) | \(=\) | \( 162 = 2 \cdot 3^{4} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 162.e (of order \(9\), degree \(6\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.29357651274\) |
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: |
\( x^{12} - 6 x^{11} + 33 x^{10} - 110 x^{9} + 318 x^{8} - 678 x^{7} + 1225 x^{6} - 1698 x^{5} + 1905 x^{4} - 1584 x^{3} + 936 x^{2} - 342 x + 57 \)
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Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 3^{3} \) |
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
\( x^{12} - 6 x^{11} + 33 x^{10} - 110 x^{9} + 318 x^{8} - 678 x^{7} + 1225 x^{6} - 1698 x^{5} + 1905 x^{4} - 1584 x^{3} + 936 x^{2} - 342 x + 57 \)
:
\(\beta_{1}\) | \(=\) |
\( ( \nu^{6} - 3\nu^{5} + 14\nu^{4} - 23\nu^{3} + 41\nu^{2} - 30\nu + 11 ) / 2 \)
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\(\beta_{2}\) | \(=\) |
\( ( 6 \nu^{11} - 33 \nu^{10} + 127 \nu^{9} - 324 \nu^{8} + 438 \nu^{7} - 252 \nu^{6} - 1278 \nu^{5} + 3234 \nu^{4} - 5701 \nu^{3} + 5358 \nu^{2} - 3477 \nu + 1060 ) / 218 \)
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\(\beta_{3}\) | \(=\) |
\( ( 2 \nu^{11} - 11 \nu^{10} + 115 \nu^{9} - 435 \nu^{8} + 1781 \nu^{7} - 4226 \nu^{6} + 9493 \nu^{5} - 13637 \nu^{4} + 16775 \nu^{3} - 12275 \nu^{2} + 5163 \nu - 882 ) / 218 \)
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\(\beta_{4}\) | \(=\) |
\( ( - 27 \nu^{11} + 94 \nu^{10} - 408 \nu^{9} + 586 \nu^{8} - 445 \nu^{7} - 2572 \nu^{6} + 9021 \nu^{5} - 18150 \nu^{4} + 24401 \nu^{3} - 20623 \nu^{2} + 10469 \nu - 2263 ) / 218 \)
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\(\beta_{5}\) | \(=\) |
\( ( - 26 \nu^{11} + 34 \nu^{10} - 187 \nu^{9} - 449 \nu^{8} + 1590 \nu^{7} - 6865 \nu^{6} + 12623 \nu^{5} - 20118 \nu^{4} + 19981 \nu^{3} - 12972 \nu^{2} + 4712 \nu - 524 ) / 218 \)
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\(\beta_{6}\) | \(=\) |
\( ( - 27 \nu^{11} + 203 \nu^{10} - 953 \nu^{9} + 3311 \nu^{8} - 8075 \nu^{7} + 16285 \nu^{6} - 23134 \nu^{5} + 26758 \nu^{4} - 19635 \nu^{3} + 9570 \nu^{2} - 1957 \nu - 83 ) / 218 \)
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\(\beta_{7}\) | \(=\) |
\( ( - 39 \nu^{11} - 58 \nu^{10} + 210 \nu^{9} - 3126 \nu^{8} + 8816 \nu^{7} - 26048 \nu^{6} + 44604 \nu^{5} - 65711 \nu^{4} + 63925 \nu^{3} - 42784 \nu^{2} + 16878 \nu - 3184 ) / 218 \)
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\(\beta_{8}\) | \(=\) |
\( ( - 26 \nu^{11} + 252 \nu^{10} - 1277 \nu^{9} + 4892 \nu^{8} - 13234 \nu^{7} + 28887 \nu^{6} - 47327 \nu^{5} + 60760 \nu^{4} - 56973 \nu^{3} + 36296 \nu^{2} - 13927 \nu + 2201 ) / 218 \)
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\(\beta_{9}\) | \(=\) |
\( ( - 19 \nu^{11} + 268 \nu^{10} - 1365 \nu^{9} + 5713 \nu^{8} - 15666 \nu^{7} + 35787 \nu^{6} - 59173 \nu^{5} + 78267 \nu^{4} - 73416 \nu^{3} + 47888 \nu^{2} - 18038 \nu + 3256 ) / 218 \)
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\(\beta_{10}\) | \(=\) |
\( ( 106 \nu^{11} - 583 \nu^{10} + 3043 \nu^{9} - 9321 \nu^{8} + 24960 \nu^{7} - 47943 \nu^{6} + 77593 \nu^{5} - 93068 \nu^{4} + 88252 \nu^{3} - 58487 \nu^{2} + 25228 \nu - 5544 ) / 218 \)
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\(\beta_{11}\) | \(=\) |
\( ( - 106 \nu^{11} + 583 \nu^{10} - 3043 \nu^{9} + 9321 \nu^{8} - 24960 \nu^{7} + 47943 \nu^{6} - 77593 \nu^{5} + 92850 \nu^{4} - 87816 \nu^{3} + 56961 \nu^{2} - 23920 \nu + 4454 ) / 218 \)
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\(\nu\) | \(=\) |
\( ( -\beta_{8} + \beta_{6} - \beta_{5} + \beta_{4} - 2\beta_{3} + \beta_{2} + \beta_1 ) / 3 \)
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\(\nu^{2}\) | \(=\) |
\( ( -\beta_{11} - \beta_{10} + \beta_{9} - 2\beta_{8} - \beta_{7} + \beta_{5} + 2\beta_{4} - 2\beta_{3} + 2\beta_{2} - 9 ) / 3 \)
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\(\nu^{3}\) | \(=\) |
\( ( - \beta_{11} - 2 \beta_{10} + 2 \beta_{9} - \beta_{7} - 6 \beta_{6} + 4 \beta_{5} - 3 \beta_{4} + 7 \beta_{3} - 8 \beta_{2} - 5 \beta _1 - 4 ) / 3 \)
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\(\nu^{4}\) | \(=\) |
\( ( 2 \beta_{11} - 3 \beta_{9} + 8 \beta_{8} + 5 \beta_{7} - 6 \beta_{6} - 5 \beta_{5} - 14 \beta_{4} + 16 \beta_{3} - 24 \beta_{2} - 4 \beta _1 + 40 ) / 3 \)
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\(\nu^{5}\) | \(=\) |
\( ( 10 \beta_{10} - 13 \beta_{9} + 21 \beta_{8} + 12 \beta_{7} + 28 \beta_{6} - 16 \beta_{5} + 3 \beta_{4} - 23 \beta_{3} + 29 \beta_{2} + 24 \beta _1 + 49 ) / 3 \)
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\(\nu^{6}\) | \(=\) |
\( ( - 10 \beta_{11} + 25 \beta_{10} + 8 \beta_{9} + 3 \beta_{8} - 16 \beta_{7} + 60 \beta_{6} + 43 \beta_{5} + 84 \beta_{4} - 110 \beta_{3} + 187 \beta_{2} + 49 \beta _1 - 169 ) / 3 \)
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\(\nu^{7}\) | \(=\) |
\( ( 23 \beta_{11} - 9 \beta_{10} + 87 \beta_{9} - 142 \beta_{8} - 88 \beta_{7} - 105 \beta_{6} + 121 \beta_{5} + 70 \beta_{4} + 25 \beta_{3} + 12 \beta_{2} - 79 \beta _1 - 395 ) / 3 \)
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\(\nu^{8}\) | \(=\) |
\( ( 108 \beta_{11} - 188 \beta_{10} + 32 \beta_{9} - 297 \beta_{8} + 6 \beta_{7} - 431 \beta_{6} - 265 \beta_{5} - 429 \beta_{4} + 652 \beta_{3} - 1153 \beta_{2} - 363 \beta _1 + 607 ) / 3 \)
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\(\nu^{9}\) | \(=\) |
\( ( - 133 \beta_{11} - 263 \beta_{10} - 511 \beta_{9} + 483 \beta_{8} + 533 \beta_{7} + 261 \beta_{6} - 1040 \beta_{5} - 891 \beta_{4} + 478 \beta_{3} - 1265 \beta_{2} + 49 \beta _1 + 2735 ) / 3 \)
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\(\nu^{10}\) | \(=\) |
\( ( - 976 \beta_{11} + 849 \beta_{10} - 717 \beta_{9} + 2720 \beta_{8} + 476 \beta_{7} + 2778 \beta_{6} + 1021 \beta_{5} + 1633 \beta_{4} - 3353 \beta_{3} + 5760 \beta_{2} + 2135 \beta _1 - 1121 ) / 3 \)
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\(\nu^{11}\) | \(=\) |
\( ( 45 \beta_{11} + 2914 \beta_{10} + 2441 \beta_{9} + 558 \beta_{8} - 2784 \beta_{7} + 742 \beta_{6} + 7907 \beta_{5} + 7221 \beta_{4} - 6041 \beta_{3} + 13586 \beta_{2} + 2025 \beta _1 - 16958 ) / 3 \)
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Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/162\mathbb{Z}\right)^\times\).
\(n\) | \(83\) |
\(\chi(n)\) | \(-\beta_{6}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
19.1 |
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0.939693 | − | 0.342020i | 0 | 0.766044 | − | 0.642788i | −0.617090 | − | 3.49969i | 0 | −0.244752 | − | 0.205371i | 0.500000 | − | 0.866025i | 0 | −1.77684 | − | 3.07758i | ||||||||||||||||||||||||||||||||||||||||||
19.2 | 0.939693 | − | 0.342020i | 0 | 0.766044 | − | 0.642788i | 0.177398 | + | 1.00607i | 0 | 2.04289 | + | 1.71418i | 0.500000 | − | 0.866025i | 0 | 0.510796 | + | 0.884725i | |||||||||||||||||||||||||||||||||||||||||||
37.1 | −0.766044 | + | 0.642788i | 0 | 0.173648 | − | 0.984808i | −0.696050 | + | 0.253341i | 0 | 0.717657 | + | 4.07003i | 0.500000 | + | 0.866025i | 0 | 0.370360 | − | 0.641483i | |||||||||||||||||||||||||||||||||||||||||||
37.2 | −0.766044 | + | 0.642788i | 0 | 0.173648 | − | 0.984808i | 1.96209 | − | 0.714144i | 0 | −0.696712 | − | 3.95125i | 0.500000 | + | 0.866025i | 0 | −1.04401 | + | 1.80828i | |||||||||||||||||||||||||||||||||||||||||||
73.1 | −0.173648 | + | 0.984808i | 0 | −0.939693 | − | 0.342020i | −2.42692 | + | 2.03643i | 0 | −3.46344 | + | 1.26059i | 0.500000 | − | 0.866025i | 0 | −1.58406 | − | 2.74367i | |||||||||||||||||||||||||||||||||||||||||||
73.2 | −0.173648 | + | 0.984808i | 0 | −0.939693 | − | 0.342020i | 3.10057 | − | 2.60168i | 0 | 0.144365 | − | 0.0525446i | 0.500000 | − | 0.866025i | 0 | 2.02375 | + | 3.50524i | |||||||||||||||||||||||||||||||||||||||||||
91.1 | −0.173648 | − | 0.984808i | 0 | −0.939693 | + | 0.342020i | −2.42692 | − | 2.03643i | 0 | −3.46344 | − | 1.26059i | 0.500000 | + | 0.866025i | 0 | −1.58406 | + | 2.74367i | |||||||||||||||||||||||||||||||||||||||||||
91.2 | −0.173648 | − | 0.984808i | 0 | −0.939693 | + | 0.342020i | 3.10057 | + | 2.60168i | 0 | 0.144365 | + | 0.0525446i | 0.500000 | + | 0.866025i | 0 | 2.02375 | − | 3.50524i | |||||||||||||||||||||||||||||||||||||||||||
127.1 | −0.766044 | − | 0.642788i | 0 | 0.173648 | + | 0.984808i | −0.696050 | − | 0.253341i | 0 | 0.717657 | − | 4.07003i | 0.500000 | − | 0.866025i | 0 | 0.370360 | + | 0.641483i | |||||||||||||||||||||||||||||||||||||||||||
127.2 | −0.766044 | − | 0.642788i | 0 | 0.173648 | + | 0.984808i | 1.96209 | + | 0.714144i | 0 | −0.696712 | + | 3.95125i | 0.500000 | − | 0.866025i | 0 | −1.04401 | − | 1.80828i | |||||||||||||||||||||||||||||||||||||||||||
145.1 | 0.939693 | + | 0.342020i | 0 | 0.766044 | + | 0.642788i | −0.617090 | + | 3.49969i | 0 | −0.244752 | + | 0.205371i | 0.500000 | + | 0.866025i | 0 | −1.77684 | + | 3.07758i | |||||||||||||||||||||||||||||||||||||||||||
145.2 | 0.939693 | + | 0.342020i | 0 | 0.766044 | + | 0.642788i | 0.177398 | − | 1.00607i | 0 | 2.04289 | − | 1.71418i | 0.500000 | + | 0.866025i | 0 | 0.510796 | − | 0.884725i | |||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
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1.a | even | 1 | 1 | trivial |
27.e | even | 9 | 1 | inner |
Twists
By twisting character orbit | |||||||
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Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 162.2.e.b | 12 | |
3.b | odd | 2 | 1 | 54.2.e.b | ✓ | 12 | |
9.c | even | 3 | 1 | 486.2.e.e | 12 | ||
9.c | even | 3 | 1 | 486.2.e.g | 12 | ||
9.d | odd | 6 | 1 | 486.2.e.f | 12 | ||
9.d | odd | 6 | 1 | 486.2.e.h | 12 | ||
12.b | even | 2 | 1 | 432.2.u.b | 12 | ||
27.e | even | 9 | 1 | inner | 162.2.e.b | 12 | |
27.e | even | 9 | 1 | 486.2.e.e | 12 | ||
27.e | even | 9 | 1 | 486.2.e.g | 12 | ||
27.e | even | 9 | 1 | 1458.2.a.f | 6 | ||
27.e | even | 9 | 2 | 1458.2.c.g | 12 | ||
27.f | odd | 18 | 1 | 54.2.e.b | ✓ | 12 | |
27.f | odd | 18 | 1 | 486.2.e.f | 12 | ||
27.f | odd | 18 | 1 | 486.2.e.h | 12 | ||
27.f | odd | 18 | 1 | 1458.2.a.g | 6 | ||
27.f | odd | 18 | 2 | 1458.2.c.f | 12 | ||
108.l | even | 18 | 1 | 432.2.u.b | 12 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
54.2.e.b | ✓ | 12 | 3.b | odd | 2 | 1 | |
54.2.e.b | ✓ | 12 | 27.f | odd | 18 | 1 | |
162.2.e.b | 12 | 1.a | even | 1 | 1 | trivial | |
162.2.e.b | 12 | 27.e | even | 9 | 1 | inner | |
432.2.u.b | 12 | 12.b | even | 2 | 1 | ||
432.2.u.b | 12 | 108.l | even | 18 | 1 | ||
486.2.e.e | 12 | 9.c | even | 3 | 1 | ||
486.2.e.e | 12 | 27.e | even | 9 | 1 | ||
486.2.e.f | 12 | 9.d | odd | 6 | 1 | ||
486.2.e.f | 12 | 27.f | odd | 18 | 1 | ||
486.2.e.g | 12 | 9.c | even | 3 | 1 | ||
486.2.e.g | 12 | 27.e | even | 9 | 1 | ||
486.2.e.h | 12 | 9.d | odd | 6 | 1 | ||
486.2.e.h | 12 | 27.f | odd | 18 | 1 | ||
1458.2.a.f | 6 | 27.e | even | 9 | 1 | ||
1458.2.a.g | 6 | 27.f | odd | 18 | 1 | ||
1458.2.c.f | 12 | 27.f | odd | 18 | 2 | ||
1458.2.c.g | 12 | 27.e | even | 9 | 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 \(S_{2}^{\mathrm{new}}(162, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{6} - T^{3} + 1)^{2} \)
$3$
\( T^{12} \)
$5$
\( T^{12} - 3 T^{11} + 9 T^{10} - 24 T^{9} + \cdots + 5184 \)
$7$
\( T^{12} + 3 T^{11} + 24 T^{10} + 88 T^{9} + \cdots + 64 \)
$11$
\( T^{12} - 12 T^{11} + 90 T^{10} - 537 T^{9} + \cdots + 81 \)
$13$
\( T^{12} - 12 T^{11} + 48 T^{10} + \cdots + 23104 \)
$17$
\( T^{12} - 6 T^{11} + 54 T^{10} + \cdots + 110889 \)
$19$
\( T^{12} + 9 T^{11} + 78 T^{10} + \cdots + 94249 \)
$23$
\( T^{12} + 30 T^{11} + 414 T^{10} + \cdots + 5184 \)
$29$
\( T^{12} + 15 T^{11} + 81 T^{10} + \cdots + 5184 \)
$31$
\( T^{12} + 81 T^{10} + 421 T^{9} + \cdots + 4032064 \)
$37$
\( T^{12} + 15 T^{11} + \cdots + 142659136 \)
$41$
\( T^{12} - 12 T^{11} + 117 T^{10} + \cdots + 2653641 \)
$43$
\( T^{12} - 9 T^{11} + 36 T^{10} + \cdots + 49674304 \)
$47$
\( T^{12} - 9 T^{11} + 99 T^{10} + \cdots + 419904 \)
$53$
\( (T^{6} - 6 T^{5} - 63 T^{4} - 3 T^{3} + \cdots - 72)^{2} \)
$59$
\( T^{12} + 12 T^{11} + 9 T^{10} + \cdots + 82464561 \)
$61$
\( T^{12} + 36 T^{11} + 531 T^{10} + \cdots + 1000000 \)
$67$
\( T^{12} - 36 T^{11} + \cdots + 249393368449 \)
$71$
\( T^{12} + 12 T^{11} + \cdots + 488586816 \)
$73$
\( T^{12} + 21 T^{11} + 390 T^{10} + \cdots + 72361 \)
$79$
\( T^{12} - 39 T^{11} + \cdots + 591851584 \)
$83$
\( T^{12} + 18 T^{11} + \cdots + 13756474944 \)
$89$
\( T^{12} + 12 T^{11} + \cdots + 126899100441 \)
$97$
\( T^{12} - 39 T^{11} + \cdots + 373532435929 \)
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