Newspace parameters
Level: | \( N \) | \(=\) | \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1008.ca (of order \(6\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(8.04892052375\) |
Analytic rank: | \(0\) |
Dimension: | \(16\) |
Relative dimension: | \(8\) over \(\Q(\zeta_{6})\) |
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: | \( x^{16} - 8 x^{15} + 23 x^{14} - 8 x^{13} - 131 x^{12} + 380 x^{11} - 289 x^{10} - 880 x^{9} + 2785 x^{8} - 2640 x^{7} - 2601 x^{6} + 10260 x^{5} - 10611 x^{4} - 1944 x^{3} + \cdots + 6561 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
Coefficient ring index: | \( 3^{4} \) |
Twist minimal: | no (minimal twist has level 126) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) 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 \( x^{16} - 8 x^{15} + 23 x^{14} - 8 x^{13} - 131 x^{12} + 380 x^{11} - 289 x^{10} - 880 x^{9} + 2785 x^{8} - 2640 x^{7} - 2601 x^{6} + 10260 x^{5} - 10611 x^{4} - 1944 x^{3} + \cdots + 6561 \) :
\(\beta_{1}\) | \(=\) | \( ( - 154 \nu^{15} + 1325 \nu^{14} - 3608 \nu^{13} + 224 \nu^{12} + 22478 \nu^{11} - 55022 \nu^{10} + 23518 \nu^{9} + 159688 \nu^{8} - 382978 \nu^{7} + 226785 \nu^{6} + \cdots + 1285227 ) / 47385 \) |
\(\beta_{2}\) | \(=\) | \( ( 1342 \nu^{15} - 9134 \nu^{14} + 18833 \nu^{13} + 17821 \nu^{12} - 164858 \nu^{11} + 301448 \nu^{10} + 55817 \nu^{9} - 1253167 \nu^{8} + 2222275 \nu^{7} - 414219 \nu^{6} + \cdots - 6445089 ) / 142155 \) |
\(\beta_{3}\) | \(=\) | \( ( 2846 \nu^{15} - 22369 \nu^{14} + 55246 \nu^{13} + 17972 \nu^{12} - 402586 \nu^{11} + 878875 \nu^{10} - 166676 \nu^{9} - 3023495 \nu^{8} + 6386423 \nu^{7} + \cdots - 20783061 ) / 142155 \) |
\(\beta_{4}\) | \(=\) | \( ( 2782 \nu^{15} - 15918 \nu^{14} + 26947 \nu^{13} + 42629 \nu^{12} - 270897 \nu^{11} + 425335 \nu^{10} + 220488 \nu^{9} - 2001225 \nu^{8} + 3082271 \nu^{7} + \cdots - 7405182 ) / 47385 \) |
\(\beta_{5}\) | \(=\) | \( ( - 5666 \nu^{15} + 35256 \nu^{14} - 67793 \nu^{13} - 74626 \nu^{12} + 609708 \nu^{11} - 1074116 \nu^{10} - 260022 \nu^{9} + 4521984 \nu^{8} - 7792711 \nu^{7} + \cdots + 21264930 ) / 47385 \) |
\(\beta_{6}\) | \(=\) | \( ( - 432 \nu^{15} + 2816 \nu^{14} - 5740 \nu^{13} - 5150 \nu^{12} + 49010 \nu^{11} - 90874 \nu^{10} - 11735 \nu^{9} + 363706 \nu^{8} - 657698 \nu^{7} + 149366 \nu^{6} + \cdots + 1853118 ) / 3645 \) |
\(\beta_{7}\) | \(=\) | \( ( 4120 \nu^{15} - 25571 \nu^{14} + 48788 \nu^{13} + 55006 \nu^{12} - 441224 \nu^{11} + 771188 \nu^{10} + 200900 \nu^{9} - 3269857 \nu^{8} + 5585140 \nu^{7} + \cdots - 14963454 ) / 28431 \) |
\(\beta_{8}\) | \(=\) | \( ( 20555 \nu^{15} - 129232 \nu^{14} + 250807 \nu^{13} + 267434 \nu^{12} - 2233957 \nu^{11} + 3963841 \nu^{10} + 897553 \nu^{9} - 16558319 \nu^{8} + 28688468 \nu^{7} + \cdots - 77795964 ) / 142155 \) |
\(\beta_{9}\) | \(=\) | \( ( - 21860 \nu^{15} + 133486 \nu^{14} - 249901 \nu^{13} - 298607 \nu^{12} + 2298061 \nu^{11} - 3952333 \nu^{10} - 1174129 \nu^{9} + 17020292 \nu^{8} + \cdots + 75541167 ) / 142155 \) |
\(\beta_{10}\) | \(=\) | \( ( 2015 \nu^{15} - 12538 \nu^{14} + 24088 \nu^{13} + 26576 \nu^{12} - 216643 \nu^{11} + 381184 \nu^{10} + 93322 \nu^{9} - 1605386 \nu^{8} + 2760692 \nu^{7} - 423483 \nu^{6} + \cdots - 7453296 ) / 10935 \) |
\(\beta_{11}\) | \(=\) | \( ( 29357 \nu^{15} - 190006 \nu^{14} + 382390 \nu^{13} + 360905 \nu^{12} - 3304405 \nu^{11} + 6053149 \nu^{10} + 947140 \nu^{9} - 24541961 \nu^{8} + 43845563 \nu^{7} + \cdots - 122535423 ) / 142155 \) |
\(\beta_{12}\) | \(=\) | \( ( - 11745 \nu^{15} + 73063 \nu^{14} - 139508 \nu^{13} - 157621 \nu^{12} + 1262908 \nu^{11} - 2204834 \nu^{10} - 583027 \nu^{9} + 9367136 \nu^{8} - 15968257 \nu^{7} + \cdots + 42448941 ) / 47385 \) |
\(\beta_{13}\) | \(=\) | \( ( - 45758 \nu^{15} + 290392 \nu^{14} - 571483 \nu^{13} - 583571 \nu^{12} + 5039353 \nu^{11} - 9054625 \nu^{10} - 1803427 \nu^{9} + 37411250 \nu^{8} + \cdots + 181431333 ) / 142155 \) |
\(\beta_{14}\) | \(=\) | \( ( - 17267 \nu^{15} + 108242 \nu^{14} - 209241 \nu^{13} - 227007 \nu^{12} + 1874096 \nu^{11} - 3311892 \nu^{10} - 782399 \nu^{9} + 13905028 \nu^{8} + \cdots + 65023155 ) / 47385 \) |
\(\beta_{15}\) | \(=\) | \( ( - 56402 \nu^{15} + 348301 \nu^{14} - 661000 \nu^{13} - 757115 \nu^{12} + 6004885 \nu^{11} - 10449439 \nu^{10} - 2822275 \nu^{9} + 44476256 \nu^{8} + \cdots + 200862828 ) / 142155 \) |
\(\nu\) | \(=\) | \( ( -\beta_{14} - \beta_{10} + \beta_{9} - \beta_{7} + 2\beta_{4} + \beta_{3} - 2\beta_{2} + 1 ) / 3 \) |
\(\nu^{2}\) | \(=\) | \( ( 2 \beta_{15} - \beta_{12} - \beta_{10} - \beta_{9} + 2 \beta_{8} - \beta_{7} - 2 \beta_{6} - \beta_{5} + 2 \beta_{4} - \beta_{3} - 2 \beta_{2} + \beta _1 + 2 ) / 3 \) |
\(\nu^{3}\) | \(=\) | \( ( 2 \beta_{15} + 2 \beta_{14} - 2 \beta_{13} - 4 \beta_{12} + \beta_{11} - 2 \beta_{10} - \beta_{9} - 3 \beta_{8} - \beta_{6} - 2 \beta_{5} + \beta_{4} - 2 \beta_{3} - 5 \beta_{2} - 3 ) / 3 \) |
\(\nu^{4}\) | \(=\) | \( ( 3 \beta_{15} + 10 \beta_{14} - 4 \beta_{13} - 6 \beta_{12} + 5 \beta_{11} + 4 \beta_{10} - 3 \beta_{9} - \beta_{8} - \beta_{7} + 2 \beta_{6} - 5 \beta_{5} - 2 \beta_{4} - 8 \beta_{3} + 3 \beta_{2} - 2 \beta _1 + 11 ) / 3 \) |
\(\nu^{5}\) | \(=\) | \( ( - 3 \beta_{15} - 2 \beta_{14} - 6 \beta_{13} - 6 \beta_{12} + 3 \beta_{11} - 8 \beta_{10} - 4 \beta_{9} - 3 \beta_{8} - 20 \beta_{7} + 15 \beta_{6} - 3 \beta_{5} - 5 \beta_{4} + 2 \beta_{3} - 13 \beta_{2} - 9 \beta _1 - 7 ) / 3 \) |
\(\nu^{6}\) | \(=\) | \( ( - 7 \beta_{15} - 3 \beta_{14} + 4 \beta_{13} - 4 \beta_{12} - 2 \beta_{11} - 10 \beta_{10} + 7 \beta_{9} + 33 \beta_{8} - 44 \beta_{7} + 8 \beta_{6} - 20 \beta_{5} - \beta_{4} + 6 \beta_{3} - 12 \beta_{2} + 9 \beta _1 - 31 ) / 3 \) |
\(\nu^{7}\) | \(=\) | \( ( - 6 \beta_{15} - 35 \beta_{14} + \beta_{13} - 18 \beta_{12} - 38 \beta_{11} - 59 \beta_{10} + \beta_{9} + 22 \beta_{8} - 36 \beta_{7} + \beta_{6} - 34 \beta_{5} - 35 \beta_{4} + 21 \beta_{3} - 68 \beta_{2} + 11 \beta _1 - 85 ) / 3 \) |
\(\nu^{8}\) | \(=\) | \( ( - 10 \beta_{15} + 35 \beta_{14} - 18 \beta_{13} - 40 \beta_{12} - 48 \beta_{11} + 7 \beta_{10} + 84 \beta_{9} + 38 \beta_{8} + 13 \beta_{7} - 17 \beta_{6} - 70 \beta_{5} - 41 \beta_{4} - 21 \beta_{3} - 22 \beta_{2} + 34 \beta _1 - 36 ) / 3 \) |
\(\nu^{9}\) | \(=\) | \( ( 32 \beta_{14} - 126 \beta_{13} - 72 \beta_{12} - 90 \beta_{11} - 46 \beta_{10} - 20 \beta_{9} - 12 \beta_{8} + 20 \beta_{7} + 87 \beta_{6} + 48 \beta_{5} - 124 \beta_{4} + 10 \beta_{3} - 98 \beta_{2} - 42 \beta _1 + 64 ) / 3 \) |
\(\nu^{10}\) | \(=\) | \( ( - 90 \beta_{15} + 171 \beta_{14} - 167 \beta_{13} - 36 \beta_{12} - 5 \beta_{11} + 153 \beta_{10} + 143 \beta_{9} + 229 \beta_{8} - 220 \beta_{7} + 199 \beta_{6} + 146 \beta_{5} - 15 \beta_{4} + 40 \beta_{3} - 85 \beta_{2} + 170 \beta _1 - 225 ) / 3 \) |
\(\nu^{11}\) | \(=\) | \( ( 100 \beta_{15} + 83 \beta_{14} - 141 \beta_{13} - 113 \beta_{12} - 90 \beta_{11} - 27 \beta_{10} - 412 \beta_{9} + 403 \beta_{8} - 42 \beta_{7} + 335 \beta_{6} + 286 \beta_{5} - 360 \beta_{4} + 272 \beta_{3} - 489 \beta_{2} + 383 \beta _1 - 325 ) / 3 \) |
\(\nu^{12}\) | \(=\) | \( ( 251 \beta_{15} + 97 \beta_{14} + 298 \beta_{13} - 319 \beta_{12} - 128 \beta_{11} + 867 \beta_{10} + 501 \beta_{9} + 471 \beta_{8} + 395 \beta_{7} + 236 \beta_{6} - 284 \beta_{5} - 510 \beta_{4} + 263 \beta_{3} - 847 \beta_{2} + \cdots - 845 ) / 3 \) |
\(\nu^{13}\) | \(=\) | \( ( 1440 \beta_{15} - 170 \beta_{14} + 597 \beta_{13} - 1254 \beta_{12} - 999 \beta_{11} + 1288 \beta_{10} - 721 \beta_{9} - 87 \beta_{8} + 2173 \beta_{7} + 198 \beta_{6} - 549 \beta_{5} - 1706 \beta_{4} + 926 \beta_{3} + \cdots + 1901 ) / 3 \) |
\(\nu^{14}\) | \(=\) | \( ( 835 \beta_{15} - 918 \beta_{14} + 882 \beta_{13} - 1877 \beta_{12} - 1842 \beta_{11} + 2872 \beta_{10} + 1198 \beta_{9} - 389 \beta_{8} - 1067 \beta_{7} + 101 \beta_{6} - 881 \beta_{5} - 872 \beta_{4} + 2104 \beta_{3} + \cdots - 1013 ) / 3 \) |
\(\nu^{15}\) | \(=\) | \( ( 1012 \beta_{15} + 838 \beta_{14} + 431 \beta_{13} - 2534 \beta_{12} - 3259 \beta_{11} + 1145 \beta_{10} - 5435 \beta_{9} + 951 \beta_{8} - 2874 \beta_{7} - 1079 \beta_{6} - 214 \beta_{5} - 1879 \beta_{4} + 4997 \beta_{3} + \cdots + 561 ) / 3 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).
\(n\) | \(127\) | \(577\) | \(757\) | \(785\) |
\(\chi(n)\) | \(1\) | \(-\beta_{7}\) | \(1\) | \(-\beta_{7}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
257.1 |
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0 | −1.64774 | − | 0.533822i | 0 | −0.450129 | − | 0.779646i | 0 | −1.57151 | − | 2.12847i | 0 | 2.43007 | + | 1.75919i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
257.2 | 0 | −1.52765 | + | 0.816261i | 0 | −1.82207 | − | 3.15592i | 0 | 1.58246 | − | 2.12034i | 0 | 1.66744 | − | 2.49392i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
257.3 | 0 | −1.38631 | − | 1.03834i | 0 | 0.714925 | + | 1.23829i | 0 | −0.327442 | + | 2.62541i | 0 | 0.843698 | + | 2.87892i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
257.4 | 0 | 0.290993 | + | 1.70743i | 0 | −0.0338034 | − | 0.0585493i | 0 | −1.19767 | + | 2.35915i | 0 | −2.83065 | + | 0.993700i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
257.5 | 0 | 0.734581 | − | 1.56856i | 0 | 0.483662 | + | 0.837727i | 0 | 2.16249 | − | 1.52435i | 0 | −1.92078 | − | 2.30447i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
257.6 | 0 | 0.890915 | − | 1.48535i | 0 | 1.14095 | + | 1.97618i | 0 | −1.42337 | − | 2.23025i | 0 | −1.41254 | − | 2.64665i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
257.7 | 0 | 1.08509 | + | 1.35003i | 0 | 1.77612 | + | 3.07634i | 0 | −2.63804 | + | 0.201867i | 0 | −0.645160 | + | 2.92981i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
257.8 | 0 | 1.56012 | + | 0.752355i | 0 | −1.80966 | − | 3.13442i | 0 | 2.41308 | + | 1.08492i | 0 | 1.86792 | + | 2.34752i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
353.1 | 0 | −1.64774 | + | 0.533822i | 0 | −0.450129 | + | 0.779646i | 0 | −1.57151 | + | 2.12847i | 0 | 2.43007 | − | 1.75919i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
353.2 | 0 | −1.52765 | − | 0.816261i | 0 | −1.82207 | + | 3.15592i | 0 | 1.58246 | + | 2.12034i | 0 | 1.66744 | + | 2.49392i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
353.3 | 0 | −1.38631 | + | 1.03834i | 0 | 0.714925 | − | 1.23829i | 0 | −0.327442 | − | 2.62541i | 0 | 0.843698 | − | 2.87892i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
353.4 | 0 | 0.290993 | − | 1.70743i | 0 | −0.0338034 | + | 0.0585493i | 0 | −1.19767 | − | 2.35915i | 0 | −2.83065 | − | 0.993700i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
353.5 | 0 | 0.734581 | + | 1.56856i | 0 | 0.483662 | − | 0.837727i | 0 | 2.16249 | + | 1.52435i | 0 | −1.92078 | + | 2.30447i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
353.6 | 0 | 0.890915 | + | 1.48535i | 0 | 1.14095 | − | 1.97618i | 0 | −1.42337 | + | 2.23025i | 0 | −1.41254 | + | 2.64665i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
353.7 | 0 | 1.08509 | − | 1.35003i | 0 | 1.77612 | − | 3.07634i | 0 | −2.63804 | − | 0.201867i | 0 | −0.645160 | − | 2.92981i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
353.8 | 0 | 1.56012 | − | 0.752355i | 0 | −1.80966 | + | 3.13442i | 0 | 2.41308 | − | 1.08492i | 0 | 1.86792 | − | 2.34752i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
63.i | even | 6 | 1 | inner |
Twists
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{16} + 24 T_{5}^{14} - 24 T_{5}^{13} + 423 T_{5}^{12} - 450 T_{5}^{11} + 3582 T_{5}^{10} - 5814 T_{5}^{9} + 22536 T_{5}^{8} - 25002 T_{5}^{7} + 42201 T_{5}^{6} - 19494 T_{5}^{5} + 32724 T_{5}^{4} - 11826 T_{5}^{3} + \cdots + 81 \)
acting on \(S_{2}^{\mathrm{new}}(1008, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{16} \)
$3$
\( T^{16} + 12 T^{13} + 9 T^{12} + \cdots + 6561 \)
$5$
\( T^{16} + 24 T^{14} - 24 T^{13} + 423 T^{12} + \cdots + 81 \)
$7$
\( T^{16} + 2 T^{15} + 6 T^{14} + \cdots + 5764801 \)
$11$
\( T^{16} + 12 T^{15} + 18 T^{14} + \cdots + 61732449 \)
$13$
\( T^{16} - 6 T^{15} - 57 T^{14} + \cdots + 390971529 \)
$17$
\( T^{16} + 18 T^{15} + 231 T^{14} + \cdots + 56070144 \)
$19$
\( T^{16} - 72 T^{14} + 4167 T^{12} + \cdots + 9199089 \)
$23$
\( T^{16} - 6 T^{15} - 54 T^{14} + \cdots + 187388721 \)
$29$
\( T^{16} - 6 T^{15} - 36 T^{14} + \cdots + 1108809 \)
$31$
\( T^{16} + 204 T^{14} + \cdots + 65610000 \)
$37$
\( T^{16} + 2 T^{15} + \cdots + 32746159681 \)
$41$
\( T^{16} + 6 T^{15} + 105 T^{14} - 210 T^{13} + \cdots + 81 \)
$43$
\( T^{16} - 2 T^{15} + \cdots + 2999643361 \)
$47$
\( (T^{8} - 18 T^{7} + 3 T^{6} + 1650 T^{5} + \cdots + 766944)^{2} \)
$53$
\( T^{16} + 36 T^{15} + \cdots + 36759242529 \)
$59$
\( (T^{8} + 30 T^{7} + 228 T^{6} + \cdots + 465300)^{2} \)
$61$
\( T^{16} + 504 T^{14} + \cdots + 547560000 \)
$67$
\( (T^{8} - 14 T^{7} - 101 T^{6} + \cdots + 51028)^{2} \)
$71$
\( T^{16} + 486 T^{14} + \cdots + 65610000 \)
$73$
\( T^{16} - 150 T^{14} + \cdots + 71115489 \)
$79$
\( (T^{8} + 16 T^{7} - 149 T^{6} + \cdots - 985100)^{2} \)
$83$
\( T^{16} + 177 T^{14} + \cdots + 953512641 \)
$89$
\( T^{16} + \cdots + 131145120363321 \)
$97$
\( T^{16} - 6 T^{15} + \cdots + 9120206721024 \)
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