Newspace parameters
Level: | \( N \) | \(=\) | \( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 1680.l (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(45.7766844125\) |
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: | \( x^{16} + 46x^{14} + 823x^{12} + 7252x^{10} + 32831x^{8} + 71486x^{6} + 60809x^{4} + 15680x^{2} + 576 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
Coefficient ring index: | \( 2^{11}\cdot 3^{2} \) |
Twist minimal: | no (minimal twist has level 105) |
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 \( x^{16} + 46x^{14} + 823x^{12} + 7252x^{10} + 32831x^{8} + 71486x^{6} + 60809x^{4} + 15680x^{2} + 576 \) :
\(\beta_{1}\) | \(=\) | \( 2\nu \) |
\(\beta_{2}\) | \(=\) | \( ( 19 \nu^{15} + 346 \nu^{14} + 314 \nu^{13} + 11114 \nu^{12} - 6979 \nu^{11} + 96196 \nu^{10} - 199188 \nu^{9} - 172404 \nu^{8} - 1667083 \nu^{7} - 5356966 \nu^{6} + \cdots - 1400928 ) / 468672 \) |
\(\beta_{3}\) | \(=\) | \( ( 355 \nu^{15} + 362 \nu^{14} + 14860 \nu^{13} + 13434 \nu^{12} + 235239 \nu^{11} + 173056 \nu^{10} + 1782656 \nu^{9} + 883956 \nu^{8} + 6908345 \nu^{7} + 1385434 \nu^{6} + \cdots + 480768 ) / 468672 \) |
\(\beta_{4}\) | \(=\) | \( ( - 355 \nu^{15} - 454 \nu^{14} - 14860 \nu^{13} - 17010 \nu^{12} - 235239 \nu^{11} - 222000 \nu^{10} - 1782656 \nu^{9} - 1143564 \nu^{8} - 6908345 \nu^{7} + \cdots + 357696 ) / 468672 \) |
\(\beta_{5}\) | \(=\) | \( ( 9 \nu^{15} - 536 \nu^{14} - 1136 \nu^{13} - 19136 \nu^{12} - 51355 \nu^{11} - 226568 \nu^{10} - 720276 \nu^{9} - 882084 \nu^{8} - 4021041 \nu^{7} + 703220 \nu^{6} + \cdots + 1054080 ) / 234336 \) |
\(\beta_{6}\) | \(=\) | \( ( 187 \nu^{14} + 7587 \nu^{12} + 114130 \nu^{10} + 791734 \nu^{8} + 2620631 \nu^{6} + 3968375 \nu^{4} + 2637324 \nu^{2} + 780032 ) / 78112 \) |
\(\beta_{7}\) | \(=\) | \( ( - 187 \nu^{14} - 7587 \nu^{12} - 114130 \nu^{10} - 791734 \nu^{8} - 2620631 \nu^{6} - 3968375 \nu^{4} - 2559212 \nu^{2} - 311360 ) / 78112 \) |
\(\beta_{8}\) | \(=\) | \( ( 321 \nu^{15} + 14812 \nu^{13} + 265971 \nu^{11} + 2352364 \nu^{9} + 10668555 \nu^{7} + 23042624 \nu^{5} + 18482577 \nu^{3} + 3269960 \nu ) / 234336 \) |
\(\beta_{9}\) | \(=\) | \( ( - 365 \nu^{15} + 174 \nu^{14} - 16310 \nu^{13} + 5702 \nu^{12} - 279615 \nu^{11} + 53512 \nu^{10} - 2303744 \nu^{9} - 1872 \nu^{8} - 9262303 \nu^{7} - 2088654 \nu^{6} + \cdots - 1769184 ) / 234336 \) |
\(\beta_{10}\) | \(=\) | \( ( 48 \nu^{15} - 1040 \nu^{14} + 4519 \nu^{13} - 43396 \nu^{12} + 130499 \nu^{11} - 680212 \nu^{10} + 1650778 \nu^{9} - 5011884 \nu^{8} + 9843186 \nu^{7} - 17945092 \nu^{6} + \cdots - 455520 ) / 234336 \) |
\(\beta_{11}\) | \(=\) | \( ( - 1141 \nu^{15} - 346 \nu^{14} - 45836 \nu^{13} - 11114 \nu^{12} - 677801 \nu^{11} - 96196 \nu^{10} - 4551216 \nu^{9} + 172404 \nu^{8} - 14056703 \nu^{7} + \cdots + 1400928 ) / 468672 \) |
\(\beta_{12}\) | \(=\) | \( ( 695 \nu^{15} - 354 \nu^{14} + 32427 \nu^{13} - 12274 \nu^{12} + 589430 \nu^{11} - 134626 \nu^{10} + 5278382 \nu^{9} - 355776 \nu^{8} + 24214099 \nu^{7} + 1985766 \nu^{6} + \cdots + 225744 ) / 234336 \) |
\(\beta_{13}\) | \(=\) | \( ( 695 \nu^{15} - 354 \nu^{14} + 29986 \nu^{13} - 12274 \nu^{12} + 494231 \nu^{11} - 134626 \nu^{10} + 3935832 \nu^{9} - 355776 \nu^{8} + 15909817 \nu^{7} + 1985766 \nu^{6} + \cdots + 225744 ) / 234336 \) |
\(\beta_{14}\) | \(=\) | \( ( - 1823 \nu^{15} - 956 \nu^{14} - 86142 \nu^{13} - 40980 \nu^{12} - 1587409 \nu^{11} - 664816 \nu^{10} - 14396428 \nu^{9} - 5104704 \nu^{8} - 66456433 \nu^{7} + \cdots + 1631712 ) / 468672 \) |
\(\beta_{15}\) | \(=\) | \( ( 511 \nu^{15} + 87 \nu^{14} + 22834 \nu^{13} + 2851 \nu^{12} + 393902 \nu^{11} + 26756 \nu^{10} + 3318976 \nu^{9} - 936 \nu^{8} + 14253143 \nu^{7} - 1044327 \nu^{6} + \cdots - 884592 ) / 117168 \) |
\(\nu\) | \(=\) | \( ( \beta_1 ) / 2 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{7} + \beta_{6} - 6 \) |
\(\nu^{3}\) | \(=\) | \( ( 2\beta_{13} + 2\beta_{11} - 2\beta_{8} + 2\beta_{3} + 4\beta_{2} - 9\beta _1 + 2 ) / 2 \) |
\(\nu^{4}\) | \(=\) | \( \beta_{15} + \beta_{13} - \beta_{12} - \beta_{11} + 2 \beta_{10} - 2 \beta_{8} - 18 \beta_{7} - 12 \beta_{6} + \beta_{5} + 4 \beta_{4} - 2 \beta_{3} + \beta_{2} + 2 \beta _1 + 60 \) |
\(\nu^{5}\) | \(=\) | \( ( 4 \beta_{15} + 4 \beta_{14} - 44 \beta_{13} + 4 \beta_{12} - 32 \beta_{11} - 4 \beta_{10} - 4 \beta_{9} + 44 \beta_{8} - 4 \beta_{6} + 4 \beta_{5} - 32 \beta_{3} - 68 \beta_{2} + 101 \beta _1 - 36 ) / 2 \) |
\(\nu^{6}\) | \(=\) | \( - 22 \beta_{15} - 16 \beta_{13} + 18 \beta_{12} + 20 \beta_{11} - 36 \beta_{10} - 2 \beta_{9} + 38 \beta_{8} + 261 \beta_{7} + 149 \beta_{6} - 20 \beta_{5} - 88 \beta_{4} + 34 \beta_{3} - 18 \beta_{2} - 44 \beta _1 - 692 \) |
\(\nu^{7}\) | \(=\) | \( ( - 68 \beta_{15} - 96 \beta_{14} + 706 \beta_{13} - 92 \beta_{12} + 450 \beta_{11} + 96 \beta_{10} + 84 \beta_{9} - 782 \beta_{8} + 96 \beta_{6} - 80 \beta_{5} + 454 \beta_{3} + 968 \beta_{2} - 1249 \beta _1 + 534 ) / 2 \) |
\(\nu^{8}\) | \(=\) | \( 343 \beta_{15} - 12 \beta_{14} + 217 \beta_{13} - 271 \beta_{12} - 313 \beta_{11} + 530 \beta_{10} + 42 \beta_{9} - 572 \beta_{8} - 3598 \beta_{7} - 1932 \beta_{6} + 313 \beta_{5} + 1468 \beta_{4} - 416 \beta_{3} + 271 \beta_{2} + \cdots + 8626 \) |
\(\nu^{9}\) | \(=\) | \( ( 760 \beta_{15} + 1708 \beta_{14} - 10316 \beta_{13} + 1720 \beta_{12} - 6232 \beta_{11} - 1708 \beta_{10} - 1288 \beta_{9} + 12560 \beta_{8} - 1708 \beta_{6} + 1180 \beta_{5} - 6236 \beta_{3} - 13120 \beta_{2} + \cdots - 7416 ) / 2 \) |
\(\nu^{10}\) | \(=\) | \( - 4724 \beta_{15} + 480 \beta_{14} - 2808 \beta_{13} + 3924 \beta_{12} + 4560 \beta_{11} - 7368 \beta_{10} - 636 \beta_{9} + 8012 \beta_{8} + 48769 \beta_{7} + 25729 \beta_{6} - 4568 \beta_{5} - 22320 \beta_{4} + \cdots - 112226 \) |
\(\nu^{11}\) | \(=\) | \( ( - 5656 \beta_{15} - 27344 \beta_{14} + 145458 \beta_{13} - 29912 \beta_{12} + 86626 \beta_{11} + 27344 \beta_{10} + 17624 \beta_{9} - 191530 \beta_{8} + 27344 \beta_{6} - 15376 \beta_{5} + \cdots + 100170 ) / 2 \) |
\(\nu^{12}\) | \(=\) | \( 61357 \beta_{15} - 12128 \beta_{14} + 35657 \beta_{13} - 56229 \beta_{12} - 64673 \beta_{11} + 100330 \beta_{10} + 8444 \beta_{9} - 109142 \beta_{8} - 657138 \beta_{7} - 347836 \beta_{6} + \cdots + 1494128 \) |
\(\nu^{13}\) | \(=\) | \( ( - 2244 \beta_{15} + 417444 \beta_{14} - 2020764 \beta_{13} + 497580 \beta_{12} - 1209904 \beta_{11} - 417444 \beta_{10} - 228540 \beta_{9} + 2841732 \beta_{8} - 417444 \beta_{6} + \cdots - 1336524 ) / 2 \) |
\(\nu^{14}\) | \(=\) | \( - 771362 \beta_{15} + 249912 \beta_{14} - 448648 \beta_{13} + 802782 \beta_{12} + 907004 \beta_{11} - 1355652 \beta_{10} - 104222 \beta_{9} + 1469922 \beta_{8} + 8840469 \beta_{7} + \cdots - 20142648 \) |
\(\nu^{15}\) | \(=\) | \( ( 1193348 \beta_{15} - 6218992 \beta_{14} + 27899826 \beta_{13} - 8009844 \beta_{12} + 16948834 \beta_{11} + 6218992 \beta_{10} + 2877004 \beta_{9} - 41525334 \beta_{8} + \cdots + 17741342 ) / 2 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times\).
\(n\) | \(241\) | \(337\) | \(421\) | \(1121\) | \(1471\) |
\(\chi(n)\) | \(1\) | \(1\) | \(1\) | \(-1\) | \(1\) |
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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1121.1 |
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0 | −2.93192 | − | 0.635503i | 0 | − | 2.23607i | 0 | 2.64575 | 0 | 8.19227 | + | 3.72648i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1121.2 | 0 | −2.93192 | + | 0.635503i | 0 | 2.23607i | 0 | 2.64575 | 0 | 8.19227 | − | 3.72648i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1121.3 | 0 | −2.91626 | − | 0.703870i | 0 | − | 2.23607i | 0 | −2.64575 | 0 | 8.00914 | + | 4.10533i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1121.4 | 0 | −2.91626 | + | 0.703870i | 0 | 2.23607i | 0 | −2.64575 | 0 | 8.00914 | − | 4.10533i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1121.5 | 0 | −1.91543 | − | 2.30892i | 0 | 2.23607i | 0 | 2.64575 | 0 | −1.66223 | + | 8.84517i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1121.6 | 0 | −1.91543 | + | 2.30892i | 0 | − | 2.23607i | 0 | 2.64575 | 0 | −1.66223 | − | 8.84517i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1121.7 | 0 | −1.47033 | − | 2.61498i | 0 | − | 2.23607i | 0 | 2.64575 | 0 | −4.67625 | + | 7.68978i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1121.8 | 0 | −1.47033 | + | 2.61498i | 0 | 2.23607i | 0 | 2.64575 | 0 | −4.67625 | − | 7.68978i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1121.9 | 0 | −0.926467 | − | 2.85336i | 0 | − | 2.23607i | 0 | −2.64575 | 0 | −7.28332 | + | 5.28709i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1121.10 | 0 | −0.926467 | + | 2.85336i | 0 | 2.23607i | 0 | −2.64575 | 0 | −7.28332 | − | 5.28709i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1121.11 | 0 | 0.217001 | − | 2.99214i | 0 | 2.23607i | 0 | −2.64575 | 0 | −8.90582 | − | 1.29859i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1121.12 | 0 | 0.217001 | + | 2.99214i | 0 | − | 2.23607i | 0 | −2.64575 | 0 | −8.90582 | + | 1.29859i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1121.13 | 0 | 2.94860 | − | 0.552947i | 0 | − | 2.23607i | 0 | −2.64575 | 0 | 8.38850 | − | 3.26084i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1121.14 | 0 | 2.94860 | + | 0.552947i | 0 | 2.23607i | 0 | −2.64575 | 0 | 8.38850 | + | 3.26084i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1121.15 | 0 | 2.99481 | − | 0.176471i | 0 | − | 2.23607i | 0 | 2.64575 | 0 | 8.93772 | − | 1.05699i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1121.16 | 0 | 2.99481 | + | 0.176471i | 0 | 2.23607i | 0 | 2.64575 | 0 | 8.93772 | + | 1.05699i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1680.3.l.a | 16 | |
3.b | odd | 2 | 1 | inner | 1680.3.l.a | 16 | |
4.b | odd | 2 | 1 | 105.3.c.a | ✓ | 16 | |
12.b | even | 2 | 1 | 105.3.c.a | ✓ | 16 | |
20.d | odd | 2 | 1 | 525.3.c.b | 16 | ||
20.e | even | 4 | 2 | 525.3.f.b | 32 | ||
60.h | even | 2 | 1 | 525.3.c.b | 16 | ||
60.l | odd | 4 | 2 | 525.3.f.b | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
105.3.c.a | ✓ | 16 | 4.b | odd | 2 | 1 | |
105.3.c.a | ✓ | 16 | 12.b | even | 2 | 1 | |
525.3.c.b | 16 | 20.d | odd | 2 | 1 | ||
525.3.c.b | 16 | 60.h | even | 2 | 1 | ||
525.3.f.b | 32 | 20.e | even | 4 | 2 | ||
525.3.f.b | 32 | 60.l | odd | 4 | 2 | ||
1680.3.l.a | 16 | 1.a | even | 1 | 1 | trivial | |
1680.3.l.a | 16 | 3.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{11}^{16} + 1302 T_{11}^{14} + 695913 T_{11}^{12} + 201895536 T_{11}^{10} + 35000873568 T_{11}^{8} + 3726118349952 T_{11}^{6} + 238332652195072 T_{11}^{4} + \cdots + 12\!\cdots\!76 \)
acting on \(S_{3}^{\mathrm{new}}(1680, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{16} \)
$3$
\( T^{16} + 8 T^{15} + 21 T^{14} + \cdots + 43046721 \)
$5$
\( (T^{2} + 5)^{8} \)
$7$
\( (T^{2} - 7)^{8} \)
$11$
\( T^{16} + 1302 T^{14} + \cdots + 12\!\cdots\!76 \)
$13$
\( (T^{8} - 607 T^{6} + 1452 T^{5} + \cdots + 187600704)^{2} \)
$17$
\( T^{16} + 2706 T^{14} + \cdots + 60\!\cdots\!16 \)
$19$
\( (T^{8} - 8 T^{7} - 2028 T^{6} + \cdots + 18126732544)^{2} \)
$23$
\( T^{16} + 3984 T^{14} + \cdots + 13\!\cdots\!56 \)
$29$
\( T^{16} + 8078 T^{14} + \cdots + 53\!\cdots\!16 \)
$31$
\( (T^{8} - 36 T^{7} - 2100 T^{6} + \cdots - 949999104)^{2} \)
$37$
\( (T^{8} + 20 T^{7} + \cdots - 722645680896)^{2} \)
$41$
\( T^{16} + 9352 T^{14} + \cdots + 59\!\cdots\!56 \)
$43$
\( (T^{8} + 140 T^{7} + \cdots + 104284349696)^{2} \)
$47$
\( T^{16} + 18850 T^{14} + \cdots + 11\!\cdots\!56 \)
$53$
\( T^{16} + 18536 T^{14} + \cdots + 12\!\cdots\!16 \)
$59$
\( T^{16} + 40896 T^{14} + \cdots + 26\!\cdots\!16 \)
$61$
\( (T^{8} + 28 T^{7} + \cdots - 58902532217856)^{2} \)
$67$
\( (T^{8} - 60 T^{7} + \cdots + 49453479094784)^{2} \)
$71$
\( T^{16} + 63416 T^{14} + \cdots + 10\!\cdots\!76 \)
$73$
\( (T^{8} + 104 T^{7} + \cdots - 1772600244224)^{2} \)
$79$
\( (T^{8} - 102 T^{7} + \cdots + 110155963598144)^{2} \)
$83$
\( T^{16} + 67984 T^{14} + \cdots + 77\!\cdots\!16 \)
$89$
\( T^{16} + 93704 T^{14} + \cdots + 54\!\cdots\!16 \)
$97$
\( (T^{8} - 364 T^{7} + \cdots - 99213562086336)^{2} \)
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