Newspace parameters
Level: | \( N \) | \(=\) | \( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 840.f (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(6.70743376979\) |
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: | \( x^{16} - 2 x^{15} + x^{14} - 4 x^{13} + 10 x^{12} - 32 x^{11} + 71 x^{10} - 70 x^{9} + 74 x^{8} - 210 x^{7} + 639 x^{6} - 864 x^{5} + 810 x^{4} - 972 x^{3} + 729 x^{2} - 4374 x + 6561 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
Coefficient ring index: | \( 2^{16} \) |
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 \( x^{16} - 2 x^{15} + x^{14} - 4 x^{13} + 10 x^{12} - 32 x^{11} + 71 x^{10} - 70 x^{9} + 74 x^{8} - 210 x^{7} + 639 x^{6} - 864 x^{5} + 810 x^{4} - 972 x^{3} + 729 x^{2} - 4374 x + 6561 \) :
\(\beta_{1}\) | \(=\) | \( \nu^{2} \) |
\(\beta_{2}\) | \(=\) | \( ( 13 \nu^{15} + 55 \nu^{14} - 50 \nu^{13} + 74 \nu^{12} - 176 \nu^{11} + 160 \nu^{10} - 517 \nu^{9} + 1025 \nu^{8} - 2863 \nu^{7} + 627 \nu^{6} - 1296 \nu^{5} + 8640 \nu^{4} + 5346 \nu^{3} + \cdots - 37179 ) / 46656 \) |
\(\beta_{3}\) | \(=\) | \( ( 17 \nu^{15} - 17 \nu^{14} + 22 \nu^{13} + 114 \nu^{12} - 48 \nu^{11} - 152 \nu^{10} + 135 \nu^{9} + 497 \nu^{8} - 1483 \nu^{7} + 763 \nu^{6} - 1296 \nu^{5} - 1944 \nu^{4} - 4806 \nu^{3} + \cdots + 9477 ) / 46656 \) |
\(\beta_{4}\) | \(=\) | \( ( 7 \nu^{15} - 19 \nu^{14} + 14 \nu^{13} - 30 \nu^{12} + 84 \nu^{11} - 220 \nu^{10} + 333 \nu^{9} - 473 \nu^{8} + 67 \nu^{7} - 799 \nu^{6} + 1092 \nu^{5} - 3372 \nu^{4} + 3474 \nu^{3} + 4158 \nu^{2} + \cdots + 8019 ) / 15552 \) |
\(\beta_{5}\) | \(=\) | \( ( - 17 \nu^{15} - 5 \nu^{14} - 182 \nu^{13} + 218 \nu^{12} - 392 \nu^{11} + 1072 \nu^{10} - 1687 \nu^{9} + 2741 \nu^{8} - 4333 \nu^{7} + 12159 \nu^{6} - 12744 \nu^{5} + 18576 \nu^{4} + \cdots + 111537 ) / 34992 \) |
\(\beta_{6}\) | \(=\) | \( ( 19 \nu^{15} + 45 \nu^{14} - 126 \nu^{13} + 10 \nu^{12} - 184 \nu^{11} - 276 \nu^{10} - 683 \nu^{9} + 2127 \nu^{8} - 1977 \nu^{7} + 1393 \nu^{6} + 4104 \nu^{5} + 9756 \nu^{4} - 10098 \nu^{3} + \cdots - 102789 ) / 23328 \) |
\(\beta_{7}\) | \(=\) | \( ( 2 \nu^{15} - \nu^{14} - 4 \nu^{13} - 5 \nu^{12} + 8 \nu^{11} - 34 \nu^{10} + 46 \nu^{9} + 73 \nu^{8} - 62 \nu^{7} - 198 \nu^{6} + 648 \nu^{5} + 189 \nu^{4} - 972 \nu^{3} + 486 \nu^{2} - 1458 \nu - 6561 ) / 2187 \) |
\(\beta_{8}\) | \(=\) | \( ( 71 \nu^{15} + 5 \nu^{14} + 182 \nu^{13} - 218 \nu^{12} + 284 \nu^{11} - 1936 \nu^{10} + 1957 \nu^{9} - 3605 \nu^{8} + 5251 \nu^{7} - 7839 \nu^{6} + 24300 \nu^{5} - 9936 \nu^{4} + \cdots - 181521 ) / 69984 \) |
\(\beta_{9}\) | \(=\) | \( ( 75 \nu^{15} - 47 \nu^{14} + 22 \nu^{13} - 62 \nu^{12} + 68 \nu^{11} - 1892 \nu^{10} + 1849 \nu^{9} - 421 \nu^{8} - 1129 \nu^{7} - 3835 \nu^{6} + 29940 \nu^{5} - 16596 \nu^{4} + 25866 \nu^{3} + \cdots - 280665 ) / 46656 \) |
\(\beta_{10}\) | \(=\) | \( ( - 131 \nu^{15} + 19 \nu^{14} - 140 \nu^{13} + 704 \nu^{12} - 914 \nu^{11} + 2446 \nu^{10} - 1291 \nu^{9} + 5975 \nu^{8} - 9199 \nu^{7} + 19959 \nu^{6} - 36666 \nu^{5} - 1242 \nu^{4} + \cdots + 194643 ) / 69984 \) |
\(\beta_{11}\) | \(=\) | \( ( 15 \nu^{15} - 13 \nu^{14} + 24 \nu^{13} - 72 \nu^{12} - 34 \nu^{11} - 362 \nu^{10} + 651 \nu^{9} - 577 \nu^{8} + 483 \nu^{7} - 1305 \nu^{6} + 4982 \nu^{5} - 3570 \nu^{4} + 12096 \nu^{3} + \cdots - 60021 ) / 7776 \) |
\(\beta_{12}\) | \(=\) | \( ( 6 \nu^{15} - 4 \nu^{14} - \nu^{13} - 16 \nu^{12} + 28 \nu^{11} - 184 \nu^{10} + 215 \nu^{9} - 68 \nu^{8} + 217 \nu^{7} - 884 \nu^{6} + 2748 \nu^{5} - 2520 \nu^{4} + 1917 \nu^{3} - 1296 \nu^{2} + \cdots - 26244 ) / 2916 \) |
\(\beta_{13}\) | \(=\) | \( ( 57 \nu^{15} + 7 \nu^{14} + 70 \nu^{13} - 302 \nu^{12} + 440 \nu^{11} - 2048 \nu^{10} + 1951 \nu^{9} - 2839 \nu^{8} + 5717 \nu^{7} - 17941 \nu^{6} + 32376 \nu^{5} - 16992 \nu^{4} + \cdots - 305451 ) / 23328 \) |
\(\beta_{14}\) | \(=\) | \( ( 35 \nu^{15} - 13 \nu^{14} + 11 \nu^{13} - 209 \nu^{12} + 185 \nu^{11} - 775 \nu^{10} + 940 \nu^{9} - 824 \nu^{8} + 2371 \nu^{7} - 6489 \nu^{6} + 13437 \nu^{5} - 3699 \nu^{4} + 20817 \nu^{3} + \cdots - 131220 ) / 11664 \) |
\(\beta_{15}\) | \(=\) | \( ( 427 \nu^{15} - 491 \nu^{14} - 2 \nu^{13} - 1102 \nu^{12} + 2332 \nu^{11} - 9980 \nu^{10} + 15569 \nu^{9} - 10273 \nu^{8} + 7943 \nu^{7} - 38103 \nu^{6} + 156204 \nu^{5} + \cdots - 1047573 ) / 139968 \) |
\(\nu\) | \(=\) | \( ( \beta_{15} - \beta_{12} - \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} ) / 4 \) |
\(\nu^{2}\) | \(=\) | \( \beta_1 \) |
\(\nu^{3}\) | \(=\) | \( ( 4 \beta_{14} - 2 \beta_{12} - 2 \beta_{11} + 4 \beta_{10} + 2 \beta_{9} + 2 \beta_{8} - 4 \beta_{7} + \beta_{5} + 4 \beta_{4} + 2 \beta_{2} + 2 \beta _1 + 2 ) / 4 \) |
\(\nu^{4}\) | \(=\) | \( \beta_{13} - 2\beta_{12} + \beta_{10} + 2\beta_{8} + 2\beta_{7} + \beta_{5} + \beta_{4} - \beta_{2} - 1 \) |
\(\nu^{5}\) | \(=\) | \( ( - \beta_{15} - \beta_{12} - 2 \beta_{11} - 2 \beta_{10} + 14 \beta_{9} - 3 \beta_{8} + 3 \beta_{7} - 7 \beta_{6} + 2 \beta_{5} - 13 \beta_{4} - 13 \beta_{3} + \beta_{2} + 2 \beta _1 + 32 ) / 4 \) |
\(\nu^{6}\) | \(=\) | \( 4 \beta_{15} - 3 \beta_{13} - 2 \beta_{12} - \beta_{11} + \beta_{9} + 2 \beta_{8} - 4 \beta_{7} + 4 \beta_{6} - 4 \beta_{5} - 4 \beta_{4} - 3 \beta_{3} - 4 \beta_{2} + 3 \beta _1 - 6 \) |
\(\nu^{7}\) | \(=\) | \( ( - 5 \beta_{15} + 8 \beta_{14} + 21 \beta_{12} - 2 \beta_{11} + 6 \beta_{10} - 34 \beta_{9} + 41 \beta_{8} - \beta_{7} - 5 \beta_{6} + 23 \beta_{5} - 25 \beta_{4} + 29 \beta_{3} - 25 \beta_{2} + 66 \beta_1 ) / 4 \) |
\(\nu^{8}\) | \(=\) | \( - 4 \beta_{15} + 16 \beta_{14} - 7 \beta_{13} - 2 \beta_{12} - 8 \beta_{11} + 13 \beta_{10} + 8 \beta_{9} + 6 \beta_{8} - 3 \beta_{7} + 4 \beta_{6} - 7 \beta_{5} + \beta_{4} + 12 \beta_{3} - 9 \beta_{2} + 3 \beta _1 + 27 \) |
\(\nu^{9}\) | \(=\) | \( ( 72 \beta_{15} - 20 \beta_{14} - 30 \beta_{12} + 24 \beta_{11} + 58 \beta_{10} - 108 \beta_{9} + 74 \beta_{8} + 62 \beta_{7} + 16 \beta_{6} - 35 \beta_{5} - 58 \beta_{4} - 6 \beta_{3} + 16 \beta_{2} + 40 \beta _1 + 22 ) / 4 \) |
\(\nu^{10}\) | \(=\) | \( 8 \beta_{15} + 16 \beta_{14} - 16 \beta_{13} + 16 \beta_{12} - 5 \beta_{11} - 13 \beta_{10} + 5 \beta_{9} - 28 \beta_{8} - 10 \beta_{7} - 40 \beta_{6} + 3 \beta_{5} - 69 \beta_{4} - 3 \beta_{3} + 53 \beta_{2} + 39 \beta _1 - 8 \) |
\(\nu^{11}\) | \(=\) | \( ( 165 \beta_{15} + 136 \beta_{14} - 128 \beta_{13} - 123 \beta_{12} - 280 \beta_{11} + 112 \beta_{10} + 88 \beta_{9} + 179 \beta_{8} - 85 \beta_{7} + 67 \beta_{6} - 200 \beta_{5} + 131 \beta_{4} - 189 \beta_{3} + 61 \beta_{2} + 216 \beta _1 - 800 ) / 4 \) |
\(\nu^{12}\) | \(=\) | \( - 20 \beta_{15} - 16 \beta_{14} + 80 \beta_{13} + 32 \beta_{12} - 35 \beta_{11} + 65 \beta_{10} - 61 \beta_{9} + 168 \beta_{8} + 57 \beta_{7} - 100 \beta_{6} + 157 \beta_{5} + 5 \beta_{4} + 135 \beta_{3} + 67 \beta_{2} + 57 \beta _1 - 123 \) |
\(\nu^{13}\) | \(=\) | \( ( - 447 \beta_{15} + 88 \beta_{14} - 384 \beta_{13} - 161 \beta_{12} - 56 \beta_{11} - 224 \beta_{10} + 1016 \beta_{9} + 71 \beta_{8} + 479 \beta_{7} - 215 \beta_{6} - 689 \beta_{5} - 505 \beta_{4} + 313 \beta_{3} - 473 \beta_{2} + \cdots + 160 ) / 4 \) |
\(\nu^{14}\) | \(=\) | \( 184 \beta_{15} - 256 \beta_{14} - 160 \beta_{13} + 208 \beta_{12} + 39 \beta_{11} - 169 \beta_{10} - 327 \beta_{9} + 180 \beta_{8} + 51 \beta_{7} + 160 \beta_{6} - 377 \beta_{5} - 449 \beta_{4} - 383 \beta_{3} + 369 \beta_{2} + \cdots + 208 \) |
\(\nu^{15}\) | \(=\) | \( ( 568 \beta_{15} + 364 \beta_{14} - 1408 \beta_{13} + 2214 \beta_{12} + 230 \beta_{11} - 2300 \beta_{10} - 2150 \beta_{9} - 494 \beta_{8} - 1340 \beta_{7} - 624 \beta_{6} + 129 \beta_{5} - 3196 \beta_{4} + 4800 \beta_{3} + \cdots - 558 ) / 4 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/840\mathbb{Z}\right)^\times\).
\(n\) | \(241\) | \(281\) | \(337\) | \(421\) | \(631\) |
\(\chi(n)\) | \(-1\) | \(-1\) | \(1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
41.1 |
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0 | −1.69668 | − | 0.348228i | 0 | 1.00000 | 0 | −2.63166 | − | 0.272689i | 0 | 2.75748 | + | 1.18166i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
41.2 | 0 | −1.69668 | + | 0.348228i | 0 | 1.00000 | 0 | −2.63166 | + | 0.272689i | 0 | 2.75748 | − | 1.18166i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
41.3 | 0 | −1.31688 | − | 1.12510i | 0 | 1.00000 | 0 | 2.64497 | + | 0.0644212i | 0 | 0.468322 | + | 2.96322i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
41.4 | 0 | −1.31688 | + | 1.12510i | 0 | 1.00000 | 0 | 2.64497 | − | 0.0644212i | 0 | 0.468322 | − | 2.96322i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
41.5 | 0 | −0.869033 | − | 1.49826i | 0 | 1.00000 | 0 | −0.807952 | + | 2.51937i | 0 | −1.48956 | + | 2.60407i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
41.6 | 0 | −0.869033 | + | 1.49826i | 0 | 1.00000 | 0 | −0.807952 | − | 2.51937i | 0 | −1.48956 | − | 2.60407i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
41.7 | 0 | −0.462633 | − | 1.66912i | 0 | 1.00000 | 0 | 1.62879 | − | 2.08496i | 0 | −2.57194 | + | 1.54438i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
41.8 | 0 | −0.462633 | + | 1.66912i | 0 | 1.00000 | 0 | 1.62879 | + | 2.08496i | 0 | −2.57194 | − | 1.54438i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
41.9 | 0 | 0.227581 | − | 1.71703i | 0 | 1.00000 | 0 | −1.22074 | + | 2.34729i | 0 | −2.89641 | − | 0.781528i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
41.10 | 0 | 0.227581 | + | 1.71703i | 0 | 1.00000 | 0 | −1.22074 | − | 2.34729i | 0 | −2.89641 | + | 0.781528i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
41.11 | 0 | 1.08593 | − | 1.34935i | 0 | 1.00000 | 0 | 2.53123 | + | 0.769995i | 0 | −0.641511 | − | 2.93061i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
41.12 | 0 | 1.08593 | + | 1.34935i | 0 | 1.00000 | 0 | 2.53123 | − | 0.769995i | 0 | −0.641511 | + | 2.93061i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
41.13 | 0 | 1.30235 | − | 1.14188i | 0 | 1.00000 | 0 | −1.35345 | − | 2.27336i | 0 | 0.392236 | − | 2.97425i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
41.14 | 0 | 1.30235 | + | 1.14188i | 0 | 1.00000 | 0 | −1.35345 | + | 2.27336i | 0 | 0.392236 | + | 2.97425i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
41.15 | 0 | 1.72936 | − | 0.0964469i | 0 | 1.00000 | 0 | 0.208829 | + | 2.63750i | 0 | 2.98140 | − | 0.333584i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
41.16 | 0 | 1.72936 | + | 0.0964469i | 0 | 1.00000 | 0 | 0.208829 | − | 2.63750i | 0 | 2.98140 | + | 0.333584i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
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 | 840.2.f.b | yes | 16 |
3.b | odd | 2 | 1 | 840.2.f.a | ✓ | 16 | |
4.b | odd | 2 | 1 | 1680.2.f.l | 16 | ||
7.b | odd | 2 | 1 | 840.2.f.a | ✓ | 16 | |
12.b | even | 2 | 1 | 1680.2.f.k | 16 | ||
21.c | even | 2 | 1 | inner | 840.2.f.b | yes | 16 |
28.d | even | 2 | 1 | 1680.2.f.k | 16 | ||
84.h | odd | 2 | 1 | 1680.2.f.l | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
840.2.f.a | ✓ | 16 | 3.b | odd | 2 | 1 | |
840.2.f.a | ✓ | 16 | 7.b | odd | 2 | 1 | |
840.2.f.b | yes | 16 | 1.a | even | 1 | 1 | trivial |
840.2.f.b | yes | 16 | 21.c | even | 2 | 1 | inner |
1680.2.f.k | 16 | 12.b | even | 2 | 1 | ||
1680.2.f.k | 16 | 28.d | even | 2 | 1 | ||
1680.2.f.l | 16 | 4.b | odd | 2 | 1 | ||
1680.2.f.l | 16 | 84.h | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{17}^{8} - 79T_{17}^{6} + 98T_{17}^{5} + 1612T_{17}^{4} - 4048T_{17}^{3} - 1232T_{17}^{2} + 4640T_{17} + 1856 \)
acting on \(S_{2}^{\mathrm{new}}(840, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{16} \)
$3$
\( T^{16} + T^{14} - 2 T^{13} + 2 T^{12} + \cdots + 6561 \)
$5$
\( (T - 1)^{16} \)
$7$
\( T^{16} - 2 T^{15} + 4 T^{14} + \cdots + 5764801 \)
$11$
\( T^{16} + 78 T^{14} + 2169 T^{12} + \cdots + 4096 \)
$13$
\( T^{16} + 146 T^{14} + \cdots + 256000000 \)
$17$
\( (T^{8} - 79 T^{6} + 98 T^{5} + 1612 T^{4} + \cdots + 1856)^{2} \)
$19$
\( T^{16} + 112 T^{14} + 4672 T^{12} + \cdots + 65536 \)
$23$
\( T^{16} + 192 T^{14} + \cdots + 16777216 \)
$29$
\( T^{16} + 242 T^{14} + \cdots + 2316304384 \)
$31$
\( T^{16} + 260 T^{14} + \cdots + 89718784 \)
$37$
\( (T^{8} - 6 T^{7} - 212 T^{6} + \cdots + 1573888)^{2} \)
$41$
\( (T^{8} - 16 T^{7} - 20 T^{6} + 1328 T^{5} + \cdots - 80896)^{2} \)
$43$
\( (T^{8} - 16 T^{7} + 4 T^{6} + 672 T^{5} + \cdots + 8192)^{2} \)
$47$
\( (T^{8} - 2 T^{7} - 199 T^{6} + \cdots + 696832)^{2} \)
$53$
\( T^{16} + 512 T^{14} + \cdots + 2083195715584 \)
$59$
\( (T^{8} + 12 T^{7} - 144 T^{6} + \cdots - 495616)^{2} \)
$61$
\( T^{16} + 444 T^{14} + \cdots + 1073741824 \)
$67$
\( (T^{8} - 316 T^{6} - 192 T^{5} + \cdots + 6326272)^{2} \)
$71$
\( T^{16} + 580 T^{14} + \cdots + 83534872576 \)
$73$
\( T^{16} + 684 T^{14} + \cdots + 3761489575936 \)
$79$
\( (T^{8} + 2 T^{7} - 439 T^{6} + \cdots + 352256)^{2} \)
$83$
\( (T^{8} - 10 T^{7} - 188 T^{6} + \cdots + 1067008)^{2} \)
$89$
\( (T^{8} + 12 T^{7} - 232 T^{6} + \cdots - 2134016)^{2} \)
$97$
\( T^{16} + \cdots + 895060875034624 \)
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