Newspace parameters
Level: | \( N \) | \(=\) | \( 1512 = 2^{3} \cdot 3^{3} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1512.k (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(12.0733807856\) |
Analytic rank: | \(0\) |
Dimension: | \(16\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{16} - 24x^{14} + 230x^{12} - 1052x^{10} + 2139x^{8} - 1244x^{6} + 1134x^{4} - 104x^{2} + 169 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
Coefficient ring index: | \( 2^{20} \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) 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 \( x^{16} - 24x^{14} + 230x^{12} - 1052x^{10} + 2139x^{8} - 1244x^{6} + 1134x^{4} - 104x^{2} + 169 \) :
\(\beta_{1}\) | \(=\) | \( ( - 305389 \nu^{14} + 7316240 \nu^{12} - 69689225 \nu^{10} + 319235369 \nu^{8} - 695756145 \nu^{6} + 747498217 \nu^{4} - 1154216584 \nu^{2} + \cdots + 237296969 ) / 192947092 \) |
\(\beta_{2}\) | \(=\) | \( ( - 2694 \nu^{14} + 61469 \nu^{12} - 552649 \nu^{10} + 2285409 \nu^{8} - 3789335 \nu^{6} + 779859 \nu^{4} - 572101 \nu^{2} - 10061233 ) / 1663337 \) |
\(\beta_{3}\) | \(=\) | \( ( 2694 \nu^{14} - 61469 \nu^{12} + 552649 \nu^{10} - 2285409 \nu^{8} + 3789335 \nu^{6} - 779859 \nu^{4} + 3898775 \nu^{2} + 81211 ) / 1663337 \) |
\(\beta_{4}\) | \(=\) | \( ( - 11520 \nu^{14} + 293628 \nu^{12} - 3040296 \nu^{10} + 15624254 \nu^{8} - 39217560 \nu^{6} + 38849348 \nu^{4} - 16270592 \nu^{2} + 6370193 ) / 3710521 \) |
\(\beta_{5}\) | \(=\) | \( ( - 1139534 \nu^{14} + 28712659 \nu^{12} - 295900503 \nu^{10} + 1529685268 \nu^{8} - 3939691239 \nu^{6} + 4125515812 \nu^{4} + \cdots + 691999581 ) / 192947092 \) |
\(\beta_{6}\) | \(=\) | \( ( - 1429545 \nu^{14} + 33944011 \nu^{12} - 320576374 \nu^{10} + 1427099575 \nu^{8} - 2709051014 \nu^{6} + 1088455103 \nu^{4} - 1464953121 \nu^{2} + \cdots - 44292248 ) / 192947092 \) |
\(\beta_{7}\) | \(=\) | \( ( - 458582 \nu^{15} + 16643678 \nu^{13} - 243643736 \nu^{11} + 1832626360 \nu^{9} - 7303260208 \nu^{7} + 13830825760 \nu^{5} + \cdots + 2564873142 \nu ) / 627078049 \) |
\(\beta_{8}\) | \(=\) | \( ( 88700 \nu^{14} - 2103563 \nu^{12} + 19736021 \nu^{10} - 86408618 \nu^{8} + 154953525 \nu^{6} - 27036722 \nu^{4} + 23842785 \nu^{2} + 15626169 ) / 6653348 \) |
\(\beta_{9}\) | \(=\) | \( ( 4302859 \nu^{15} - 113264303 \nu^{13} + 1228806350 \nu^{11} - 6828867199 \nu^{9} + 19868045926 \nu^{7} - 27619863991 \nu^{5} + \cdots - 10467201680 \nu ) / 2508312196 \) |
\(\beta_{10}\) | \(=\) | \( ( - 403473 \nu^{15} + 10071623 \nu^{13} - 101747496 \nu^{11} + 504132793 \nu^{9} - 1172875960 \nu^{7} + 854627793 \nu^{5} + 21762817 \nu^{3} + \cdots + 225114682 \nu ) / 86493524 \) |
\(\beta_{11}\) | \(=\) | \( ( 3300067 \nu^{15} - 75367115 \nu^{13} + 666756659 \nu^{11} - 2588222763 \nu^{9} + 3036067465 \nu^{7} + 3953470823 \nu^{5} - 407547934 \nu^{3} + \cdots + 1131065416 \nu ) / 627078049 \) |
\(\beta_{12}\) | \(=\) | \( ( 6600134 \nu^{15} - 150734230 \nu^{13} + 1333513318 \nu^{11} - 5176445526 \nu^{9} + 6072134930 \nu^{7} + 7906941646 \nu^{5} + \cdots + 4770443028 \nu ) / 627078049 \) |
\(\beta_{13}\) | \(=\) | \( ( - 248063 \nu^{15} + 6021151 \nu^{13} - 58426783 \nu^{11} + 271352435 \nu^{9} - 561779405 \nu^{7} + 316469269 \nu^{5} - 175583178 \nu^{3} + \cdots - 8003320 \nu ) / 21623381 \) |
\(\beta_{14}\) | \(=\) | \( ( - 10755599 \nu^{15} + 252302706 \nu^{13} - 2332684002 \nu^{11} + 9941641297 \nu^{9} - 16587964242 \nu^{7} + 138651525 \nu^{5} + \cdots + 1098391515 \nu ) / 627078049 \) |
\(\beta_{15}\) | \(=\) | \( ( - 1723809 \nu^{15} + 41505147 \nu^{13} - 399318104 \nu^{11} + 1835444329 \nu^{9} - 3743153672 \nu^{7} + 2039706321 \nu^{5} + \cdots - 140493626 \nu ) / 86493524 \) |
\(\nu\) | \(=\) | \( ( \beta_{12} - 2\beta_{11} ) / 4 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta_{3} + \beta_{2} + 6 ) / 2 \) |
\(\nu^{3}\) | \(=\) | \( ( 3\beta_{15} - \beta_{14} + 7\beta_{12} - 10\beta_{11} - 3\beta_{10} + 4\beta_{9} - \beta_{7} ) / 4 \) |
\(\nu^{4}\) | \(=\) | \( ( \beta_{8} + 4\beta_{6} + \beta_{5} - 4\beta_{4} + 13\beta_{3} + 5\beta_{2} + 2\beta _1 + 29 ) / 2 \) |
\(\nu^{5}\) | \(=\) | \( ( 30\beta_{15} - 2\beta_{14} - 5\beta_{13} + 51\beta_{12} - 43\beta_{11} - 30\beta_{10} + 24\beta_{9} - 20\beta_{7} ) / 4 \) |
\(\nu^{6}\) | \(=\) | \( ( 13\beta_{8} + 78\beta_{6} + 35\beta_{5} - 113\beta_{4} + 231\beta_{3} + 35\beta_{2} + 36\beta _1 + 212 ) / 4 \) |
\(\nu^{7}\) | \(=\) | \( ( 245 \beta_{15} + 7 \beta_{14} - 77 \beta_{13} + 314 \beta_{12} - 99 \beta_{11} - 217 \beta_{10} + 96 \beta_{9} - 217 \beta_{7} ) / 4 \) |
\(\nu^{8}\) | \(=\) | \( ( 20\beta_{8} + 266\beta_{6} + 194\beta_{5} - 559\beta_{4} + 864\beta_{3} + 2\beta_{2} + 178\beta _1 + 29 ) / 2 \) |
\(\nu^{9}\) | \(=\) | \( ( 1728 \beta_{15} + 88 \beta_{14} - 771 \beta_{13} + 1556 \beta_{12} + 629 \beta_{11} - 1344 \beta_{10} - 128 \beta_{9} - 1868 \beta_{7} ) / 4 \) |
\(\nu^{10}\) | \(=\) | \( ( - 319 \beta_{8} + 2600 \beta_{6} + 3285 \beta_{5} - 8921 \beta_{4} + 11245 \beta_{3} - 1473 \beta_{2} + 3398 \beta _1 - 8758 ) / 4 \) |
\(\nu^{11}\) | \(=\) | \( ( 10560 \beta_{15} + 286 \beta_{14} - 5995 \beta_{13} + 4951 \beta_{12} + 12093 \beta_{11} - 7172 \beta_{10} - 7564 \beta_{9} - 14166 \beta_{7} ) / 4 \) |
\(\nu^{12}\) | \(=\) | \( ( - 3605 \beta_{8} + 2573 \beta_{6} + 11182 \beta_{5} - 29241 \beta_{4} + 31239 \beta_{3} - 9280 \beta_{2} + 14687 \beta _1 - 56567 ) / 2 \) |
\(\nu^{13}\) | \(=\) | \( ( 53248 \beta_{15} - 2884 \beta_{14} - 36868 \beta_{13} - 10003 \beta_{12} + 123074 \beta_{11} - 30784 \beta_{10} - 98032 \beta_{9} - 97052 \beta_{7} ) / 4 \) |
\(\nu^{14}\) | \(=\) | \( ( - 41422 \beta_{8} - 36017 \beta_{6} + 58447 \beta_{5} - 148066 \beta_{4} + 133427 \beta_{3} - 83574 \beta_{2} + 112843 \beta _1 - 517223 ) / 2 \) |
\(\nu^{15}\) | \(=\) | \( ( 176793 \beta_{15} - 58799 \beta_{14} - 158604 \beta_{13} - 357861 \beta_{12} + 1003586 \beta_{11} - 72873 \beta_{10} - 927572 \beta_{9} - 595917 \beta_{7} ) / 4 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1512\mathbb{Z}\right)^\times\).
\(n\) | \(757\) | \(785\) | \(1081\) | \(1135\) |
\(\chi(n)\) | \(1\) | \(-1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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377.1 |
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0 | 0 | 0 | −2.86833 | 0 | 2.43500 | − | 1.03478i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
377.2 | 0 | 0 | 0 | −2.86833 | 0 | 2.43500 | + | 1.03478i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
377.3 | 0 | 0 | 0 | −2.70790 | 0 | −0.946562 | − | 2.47063i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
377.4 | 0 | 0 | 0 | −2.70790 | 0 | −0.946562 | + | 2.47063i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
377.5 | 0 | 0 | 0 | −1.10598 | 0 | −2.64465 | − | 0.0763047i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
377.6 | 0 | 0 | 0 | −1.10598 | 0 | −2.64465 | + | 0.0763047i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
377.7 | 0 | 0 | 0 | −0.465643 | 0 | 0.656211 | − | 2.56308i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
377.8 | 0 | 0 | 0 | −0.465643 | 0 | 0.656211 | + | 2.56308i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
377.9 | 0 | 0 | 0 | 0.465643 | 0 | 0.656211 | − | 2.56308i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
377.10 | 0 | 0 | 0 | 0.465643 | 0 | 0.656211 | + | 2.56308i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
377.11 | 0 | 0 | 0 | 1.10598 | 0 | −2.64465 | − | 0.0763047i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
377.12 | 0 | 0 | 0 | 1.10598 | 0 | −2.64465 | + | 0.0763047i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
377.13 | 0 | 0 | 0 | 2.70790 | 0 | −0.946562 | − | 2.47063i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
377.14 | 0 | 0 | 0 | 2.70790 | 0 | −0.946562 | + | 2.47063i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
377.15 | 0 | 0 | 0 | 2.86833 | 0 | 2.43500 | − | 1.03478i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
377.16 | 0 | 0 | 0 | 2.86833 | 0 | 2.43500 | + | 1.03478i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
7.b | odd | 2 | 1 | inner |
21.c | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1512.2.k.a | ✓ | 16 |
3.b | odd | 2 | 1 | inner | 1512.2.k.a | ✓ | 16 |
4.b | odd | 2 | 1 | 3024.2.k.k | 16 | ||
7.b | odd | 2 | 1 | inner | 1512.2.k.a | ✓ | 16 |
12.b | even | 2 | 1 | 3024.2.k.k | 16 | ||
21.c | even | 2 | 1 | inner | 1512.2.k.a | ✓ | 16 |
28.d | even | 2 | 1 | 3024.2.k.k | 16 | ||
84.h | odd | 2 | 1 | 3024.2.k.k | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1512.2.k.a | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
1512.2.k.a | ✓ | 16 | 3.b | odd | 2 | 1 | inner |
1512.2.k.a | ✓ | 16 | 7.b | odd | 2 | 1 | inner |
1512.2.k.a | ✓ | 16 | 21.c | even | 2 | 1 | inner |
3024.2.k.k | 16 | 4.b | odd | 2 | 1 | ||
3024.2.k.k | 16 | 12.b | even | 2 | 1 | ||
3024.2.k.k | 16 | 28.d | even | 2 | 1 | ||
3024.2.k.k | 16 | 84.h | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} - 17T_{5}^{6} + 83T_{5}^{4} - 91T_{5}^{2} + 16 \)
acting on \(S_{2}^{\mathrm{new}}(1512, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{16} \)
$3$
\( T^{16} \)
$5$
\( (T^{8} - 17 T^{6} + 83 T^{4} - 91 T^{2} + \cdots + 16)^{2} \)
$7$
\( (T^{8} + T^{7} + 5 T^{5} - 34 T^{4} + \cdots + 2401)^{2} \)
$11$
\( (T^{8} + 51 T^{6} + 779 T^{4} + 3801 T^{2} + \cdots + 784)^{2} \)
$13$
\( (T^{8} + 55 T^{6} + 956 T^{4} + 5264 T^{2} + \cdots + 3136)^{2} \)
$17$
\( (T^{8} - 64 T^{6} + 1088 T^{4} + \cdots + 4096)^{2} \)
$19$
\( (T^{8} + 52 T^{6} + 734 T^{4} + 1700 T^{2} + \cdots + 841)^{2} \)
$23$
\( (T^{8} + 163 T^{6} + 8379 T^{4} + \cdots + 559504)^{2} \)
$29$
\( (T^{8} + 60 T^{6} + 944 T^{4} + 3648 T^{2} + \cdots + 4096)^{2} \)
$31$
\( (T^{8} + 61 T^{6} + 791 T^{4} + 3431 T^{2} + \cdots + 3844)^{2} \)
$37$
\( (T^{4} + 2 T^{3} - 96 T^{2} - 322 T - 161)^{4} \)
$41$
\( (T^{8} - 193 T^{6} + 11699 T^{4} + \cdots + 1827904)^{2} \)
$43$
\( (T^{4} - 2 T^{3} - 132 T^{2} + 376 T + 64)^{4} \)
$47$
\( (T^{8} - 356 T^{6} + 42608 T^{4} + \cdots + 30647296)^{2} \)
$53$
\( (T^{8} + 240 T^{6} + 15104 T^{4} + \cdots + 1048576)^{2} \)
$59$
\( (T^{8} - 324 T^{6} + 27504 T^{4} + \cdots + 4064256)^{2} \)
$61$
\( (T^{8} + 171 T^{6} + 8496 T^{4} + \cdots + 331776)^{2} \)
$67$
\( (T^{4} + 7 T^{3} - 154 T^{2} - 1372 T - 2744)^{4} \)
$71$
\( (T^{8} + 227 T^{6} + 18747 T^{4} + \cdots + 8479744)^{2} \)
$73$
\( (T^{8} + 263 T^{6} + 23708 T^{4} + \cdots + 10291264)^{2} \)
$79$
\( (T^{4} + 11 T^{3} - 70 T^{2} - 92 T + 184)^{4} \)
$83$
\( (T^{8} - 532 T^{6} + 85040 T^{4} + \cdots + 51380224)^{2} \)
$89$
\( (T^{8} - 361 T^{6} + 28979 T^{4} + \cdots + 8737936)^{2} \)
$97$
\( (T^{8} + 491 T^{6} + 87296 T^{4} + \cdots + 177209344)^{2} \)
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