Newspace parameters
Level: | \( N \) | \(=\) | \( 650 = 2 \cdot 5^{2} \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 650.t (of order \(12\), degree \(4\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(5.19027613138\) |
Analytic rank: | \(0\) |
Dimension: | \(16\) |
Relative dimension: | \(4\) over \(\Q(\zeta_{12})\) |
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: | \( x^{16} + 28x^{14} + 294x^{12} + 1516x^{10} + 4147x^{8} + 6012x^{6} + 4338x^{4} + 1296x^{2} + 81 \) |
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 3^{2} \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 \( x^{16} + 28x^{14} + 294x^{12} + 1516x^{10} + 4147x^{8} + 6012x^{6} + 4338x^{4} + 1296x^{2} + 81 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( ( 4 \nu^{15} - 8 \nu^{14} + 145 \nu^{13} - 221 \nu^{12} + 2010 \nu^{11} - 2295 \nu^{10} + 13435 \nu^{9} - 11759 \nu^{8} + 45196 \nu^{7} - 31121 \nu^{6} + 72339 \nu^{5} - 36969 \nu^{4} + \cdots + 756 ) / 4968 \) |
\(\beta_{3}\) | \(=\) | \( ( 53 \nu^{15} - 138 \nu^{14} + 1490 \nu^{13} - 3657 \nu^{12} + 15489 \nu^{11} - 35397 \nu^{10} + 75911 \nu^{9} - 163254 \nu^{8} + 178775 \nu^{7} - 382053 \nu^{6} + 168459 \nu^{5} + \cdots - 9315 ) / 14904 \) |
\(\beta_{4}\) | \(=\) | \( ( 37\nu^{14} + 979\nu^{12} + 9381\nu^{10} + 41836\nu^{8} + 89761\nu^{6} + 83688\nu^{4} + 26658\nu^{2} + 4887 ) / 2484 \) |
\(\beta_{5}\) | \(=\) | \( ( 89 \nu^{15} - 132 \nu^{14} + 2381 \nu^{13} - 3750 \nu^{12} + 23229 \nu^{11} - 39834 \nu^{10} + 106781 \nu^{9} - 204270 \nu^{8} + 243575 \nu^{7} - 532230 \nu^{6} + 265785 \nu^{5} + \cdots - 35964 ) / 14904 \) |
\(\beta_{6}\) | \(=\) | \( ( 44 \nu^{15} - 37 \nu^{14} + 1250 \nu^{13} - 979 \nu^{12} + 13278 \nu^{11} - 9381 \nu^{10} + 68090 \nu^{9} - 41836 \nu^{8} + 177410 \nu^{7} - 89761 \nu^{6} + 221994 \nu^{5} + \cdots - 2403 ) / 4968 \) |
\(\beta_{7}\) | \(=\) | \( ( 89 \nu^{15} + 132 \nu^{14} + 2381 \nu^{13} + 3750 \nu^{12} + 23229 \nu^{11} + 39834 \nu^{10} + 106781 \nu^{9} + 204270 \nu^{8} + 243575 \nu^{7} + 532230 \nu^{6} + 265785 \nu^{5} + \cdots + 35964 ) / 14904 \) |
\(\beta_{8}\) | \(=\) | \( ( 29 \nu^{15} - 81 \nu^{14} + 758 \nu^{13} - 2229 \nu^{12} + 7086 \nu^{11} - 22659 \nu^{10} + 29870 \nu^{9} - 109926 \nu^{8} + 54086 \nu^{7} - 267171 \nu^{6} + 15876 \nu^{5} + \cdots - 6939 ) / 4968 \) |
\(\beta_{9}\) | \(=\) | \( ( 61 \nu^{15} - 77 \nu^{14} + 1711 \nu^{13} - 2015 \nu^{12} + 17991 \nu^{11} - 18924 \nu^{10} + 92638 \nu^{9} - 81380 \nu^{8} + 250675 \nu^{7} - 162704 \nu^{6} + 349914 \nu^{5} + \cdots + 7587 ) / 4968 \) |
\(\beta_{10}\) | \(=\) | \( ( - 29 \nu^{15} - 81 \nu^{14} - 758 \nu^{13} - 2229 \nu^{12} - 7086 \nu^{11} - 22659 \nu^{10} - 29870 \nu^{9} - 109926 \nu^{8} - 54086 \nu^{7} - 267171 \nu^{6} - 15876 \nu^{5} + \cdots - 6939 ) / 4968 \) |
\(\beta_{11}\) | \(=\) | \( ( - 73 \nu^{15} - 38 \nu^{14} - 1939 \nu^{13} - 998 \nu^{12} - 18639 \nu^{11} - 9504 \nu^{10} - 83056 \nu^{9} - 42659 \nu^{8} - 176779 \nu^{7} - 96230 \nu^{6} - 161004 \nu^{5} + \cdots - 3861 ) / 4968 \) |
\(\beta_{12}\) | \(=\) | \( ( -2\nu^{14} - 53\nu^{12} - 510\nu^{10} - 2303\nu^{8} - 5132\nu^{6} - 5331\nu^{4} - 2142\nu^{2} + 36\nu - 162 ) / 72 \) |
\(\beta_{13}\) | \(=\) | \( ( - 19 \nu^{15} - 23 \nu^{14} - 522 \nu^{13} - 598 \nu^{12} - 5327 \nu^{11} - 5543 \nu^{10} - 26401 \nu^{9} - 23207 \nu^{8} - 68493 \nu^{7} - 43861 \nu^{6} - 92525 \nu^{5} - 29555 \nu^{4} + \cdots + 2277 ) / 1656 \) |
\(\beta_{14}\) | \(=\) | \( ( - 77 \nu^{15} - 46 \nu^{14} - 2084 \nu^{13} - 1219 \nu^{12} - 20649 \nu^{11} - 11799 \nu^{10} - 96491 \nu^{9} - 54418 \nu^{8} - 221975 \nu^{7} - 127351 \nu^{6} - 233343 \nu^{5} + \cdots - 3105 ) / 4968 \) |
\(\beta_{15}\) | \(=\) | \( ( 2\nu^{15} + 54\nu^{13} + 535\nu^{11} + 2522\nu^{9} + 5991\nu^{7} + 6892\nu^{5} + 3345\nu^{3} + 450\nu + 36 ) / 72 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{10} + \beta_{8} + \beta_{7} - \beta_{5} + \beta_{4} - 4 \) |
\(\nu^{3}\) | \(=\) | \( - 2 \beta_{15} + 2 \beta_{14} - \beta_{13} - \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} + 2 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} + \beta_{4} - \beta_{3} - 2 \beta_{2} - 5 \beta_1 \) |
\(\nu^{4}\) | \(=\) | \( 2 \beta_{14} + 2 \beta_{13} - 4 \beta_{12} + 2 \beta_{11} - 9 \beta_{10} + 2 \beta_{9} - 9 \beta_{8} - 6 \beta_{7} + 14 \beta_{5} - 7 \beta_{4} + 4 \beta_{3} - 4 \beta_{2} + 6 \beta _1 + 28 \) |
\(\nu^{5}\) | \(=\) | \( 38 \beta_{15} - 24 \beta_{14} + 21 \beta_{13} + 15 \beta_{11} + 17 \beta_{10} - 21 \beta_{9} - 17 \beta_{8} - 30 \beta_{7} - 24 \beta_{6} - 30 \beta_{5} - 12 \beta_{4} + 9 \beta_{3} + 36 \beta_{2} + 37 \beta _1 - 7 \) |
\(\nu^{6}\) | \(=\) | \( - 22 \beta_{14} - 27 \beta_{13} + 62 \beta_{12} - 40 \beta_{11} + 88 \beta_{10} - 27 \beta_{9} + 88 \beta_{8} + 41 \beta_{7} - 165 \beta_{5} + 67 \beta_{4} - 62 \beta_{3} + 67 \beta_{2} - 93 \beta _1 - 259 \) |
\(\nu^{7}\) | \(=\) | \( - 524 \beta_{15} + 265 \beta_{14} - 293 \beta_{13} - 182 \beta_{11} - 222 \beta_{10} + 293 \beta_{9} + 222 \beta_{8} + 397 \beta_{7} + 268 \beta_{6} + 397 \beta_{5} + 134 \beta_{4} - 83 \beta_{3} - 475 \beta_{2} - 342 \beta _1 + 128 \) |
\(\nu^{8}\) | \(=\) | \( 234 \beta_{14} + 316 \beta_{13} - 792 \beta_{12} + 546 \beta_{11} - 923 \beta_{10} + 316 \beta_{9} - 923 \beta_{8} - 328 \beta_{7} + 1888 \beta_{5} - 739 \beta_{4} + 780 \beta_{3} - 862 \beta_{2} + 1176 \beta _1 + 2713 \) |
\(\nu^{9}\) | \(=\) | \( 6506 \beta_{15} - 2942 \beta_{14} + 3637 \beta_{13} + 2101 \beta_{11} + 2695 \beta_{10} - 3637 \beta_{9} - 2695 \beta_{8} - 4904 \beta_{7} - 3008 \beta_{6} - 4904 \beta_{5} - 1504 \beta_{4} + 841 \beta_{3} + \cdots - 1749 \) |
\(\nu^{10}\) | \(=\) | \( - 2594 \beta_{14} - 3605 \beta_{13} + 9514 \beta_{12} - 6680 \beta_{11} + 10101 \beta_{10} - 3605 \beta_{9} + 10101 \beta_{8} + 2994 \beta_{7} - 21542 \beta_{5} + 8458 \beta_{4} - 9274 \beta_{3} + \cdots - 29935 \) |
\(\nu^{11}\) | \(=\) | \( - 77366 \beta_{15} + 33081 \beta_{14} - 43200 \beta_{13} - 23991 \beta_{11} - 31763 \beta_{10} + 43200 \beta_{9} + 31763 \beta_{8} + 58377 \beta_{7} + 34050 \beta_{6} + 58377 \beta_{5} + \cdots + 21658 \) |
\(\nu^{12}\) | \(=\) | \( 29416 \beta_{14} + 41016 \beta_{13} - 111416 \beta_{12} + 78628 \beta_{11} - 113122 \beta_{10} + 41016 \beta_{9} - 113122 \beta_{8} - 30038 \beta_{7} + 246126 \beta_{5} - 97402 \beta_{4} + \cdots + 337525 \) |
\(\nu^{13}\) | \(=\) | \( 902396 \beta_{15} - 375304 \beta_{14} + 503534 \beta_{13} + 273782 \beta_{11} + 369198 \beta_{10} - 503534 \beta_{9} - 369198 \beta_{8} - 681940 \beta_{7} - 387580 \beta_{6} - 681940 \beta_{5} + \cdots - 257408 \) |
\(\nu^{14}\) | \(=\) | \( - 336384 \beta_{14} - 467572 \beta_{13} + 1289976 \beta_{12} - 911652 \beta_{11} + 1281923 \beta_{10} - 467572 \beta_{9} + 1281923 \beta_{8} + 319945 \beta_{7} - 2816017 \beta_{5} + \cdots - 3840844 \) |
\(\nu^{15}\) | \(=\) | \( - 10431278 \beta_{15} + 4279454 \beta_{14} - 5818687 \beta_{13} - 3128719 \beta_{11} - 4262047 \beta_{10} + 5818687 \beta_{9} + 4262047 \beta_{8} + 7891298 \beta_{7} + \cdots + 3002670 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/650\mathbb{Z}\right)^\times\).
\(n\) | \(27\) | \(301\) |
\(\chi(n)\) | \(\beta_{5} + \beta_{7}\) | \(-\beta_{5}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
7.1 |
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−0.866025 | − | 0.500000i | −0.333429 | − | 1.24438i | 0.500000 | + | 0.866025i | 0 | −0.333429 | + | 1.24438i | −1.27944 | − | 2.21606i | − | 1.00000i | 1.16078 | − | 0.670177i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
7.2 | −0.866025 | − | 0.500000i | −0.272763 | − | 1.01796i | 0.500000 | + | 0.866025i | 0 | −0.272763 | + | 1.01796i | −0.303090 | − | 0.524968i | − | 1.00000i | 1.63622 | − | 0.944675i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
7.3 | −0.866025 | − | 0.500000i | 0.0746104 | + | 0.278450i | 0.500000 | + | 0.866025i | 0 | 0.0746104 | − | 0.278450i | 2.39698 | + | 4.15170i | − | 1.00000i | 2.52611 | − | 1.45845i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
7.4 | −0.866025 | − | 0.500000i | 0.531582 | + | 1.98389i | 0.500000 | + | 0.866025i | 0 | 0.531582 | − | 1.98389i | −1.54650 | − | 2.67861i | − | 1.00000i | −1.05516 | + | 0.609199i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
93.1 | −0.866025 | + | 0.500000i | −0.333429 | + | 1.24438i | 0.500000 | − | 0.866025i | 0 | −0.333429 | − | 1.24438i | −1.27944 | + | 2.21606i | 1.00000i | 1.16078 | + | 0.670177i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
93.2 | −0.866025 | + | 0.500000i | −0.272763 | + | 1.01796i | 0.500000 | − | 0.866025i | 0 | −0.272763 | − | 1.01796i | −0.303090 | + | 0.524968i | 1.00000i | 1.63622 | + | 0.944675i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
93.3 | −0.866025 | + | 0.500000i | 0.0746104 | − | 0.278450i | 0.500000 | − | 0.866025i | 0 | 0.0746104 | + | 0.278450i | 2.39698 | − | 4.15170i | 1.00000i | 2.52611 | + | 1.45845i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
93.4 | −0.866025 | + | 0.500000i | 0.531582 | − | 1.98389i | 0.500000 | − | 0.866025i | 0 | 0.531582 | + | 1.98389i | −1.54650 | + | 2.67861i | 1.00000i | −1.05516 | − | 0.609199i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
557.1 | 0.866025 | − | 0.500000i | −3.27048 | − | 0.876322i | 0.500000 | − | 0.866025i | 0 | −3.27048 | + | 0.876322i | 2.08773 | − | 3.61605i | − | 1.00000i | 7.33001 | + | 4.23198i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
557.2 | 0.866025 | − | 0.500000i | −0.752511 | − | 0.201635i | 0.500000 | − | 0.866025i | 0 | −0.752511 | + | 0.201635i | −1.21153 | + | 2.09844i | − | 1.00000i | −2.07246 | − | 1.19654i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
557.3 | 0.866025 | − | 0.500000i | 1.71844 | + | 0.460454i | 0.500000 | − | 0.866025i | 0 | 1.71844 | − | 0.460454i | 0.386889 | − | 0.670111i | − | 1.00000i | 0.142931 | + | 0.0825211i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
557.4 | 0.866025 | − | 0.500000i | 2.30455 | + | 0.617503i | 0.500000 | − | 0.866025i | 0 | 2.30455 | − | 0.617503i | 1.46897 | − | 2.54433i | − | 1.00000i | 2.33157 | + | 1.34613i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
643.1 | 0.866025 | + | 0.500000i | −3.27048 | + | 0.876322i | 0.500000 | + | 0.866025i | 0 | −3.27048 | − | 0.876322i | 2.08773 | + | 3.61605i | 1.00000i | 7.33001 | − | 4.23198i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
643.2 | 0.866025 | + | 0.500000i | −0.752511 | + | 0.201635i | 0.500000 | + | 0.866025i | 0 | −0.752511 | − | 0.201635i | −1.21153 | − | 2.09844i | 1.00000i | −2.07246 | + | 1.19654i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
643.3 | 0.866025 | + | 0.500000i | 1.71844 | − | 0.460454i | 0.500000 | + | 0.866025i | 0 | 1.71844 | + | 0.460454i | 0.386889 | + | 0.670111i | 1.00000i | 0.142931 | − | 0.0825211i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
643.4 | 0.866025 | + | 0.500000i | 2.30455 | − | 0.617503i | 0.500000 | + | 0.866025i | 0 | 2.30455 | + | 0.617503i | 1.46897 | + | 2.54433i | 1.00000i | 2.33157 | − | 1.34613i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
65.t | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 650.2.t.h | yes | 16 |
5.b | even | 2 | 1 | 650.2.t.f | ✓ | 16 | |
5.c | odd | 4 | 1 | 650.2.w.f | yes | 16 | |
5.c | odd | 4 | 1 | 650.2.w.h | yes | 16 | |
13.f | odd | 12 | 1 | 650.2.w.f | yes | 16 | |
65.o | even | 12 | 1 | 650.2.t.f | ✓ | 16 | |
65.s | odd | 12 | 1 | 650.2.w.h | yes | 16 | |
65.t | even | 12 | 1 | inner | 650.2.t.h | yes | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
650.2.t.f | ✓ | 16 | 5.b | even | 2 | 1 | |
650.2.t.f | ✓ | 16 | 65.o | even | 12 | 1 | |
650.2.t.h | yes | 16 | 1.a | even | 1 | 1 | trivial |
650.2.t.h | yes | 16 | 65.t | even | 12 | 1 | inner |
650.2.w.f | yes | 16 | 5.c | odd | 4 | 1 | |
650.2.w.f | yes | 16 | 13.f | odd | 12 | 1 | |
650.2.w.h | yes | 16 | 5.c | odd | 4 | 1 | |
650.2.w.h | yes | 16 | 65.s | odd | 12 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{16} - 12 T_{3}^{14} + 12 T_{3}^{13} + 23 T_{3}^{12} - 120 T_{3}^{11} + 372 T_{3}^{10} - 312 T_{3}^{9} + 304 T_{3}^{8} - 828 T_{3}^{7} - 450 T_{3}^{6} + 540 T_{3}^{5} + 1035 T_{3}^{4} + 1620 T_{3}^{3} + 810 T_{3}^{2} + 81 \)
acting on \(S_{2}^{\mathrm{new}}(650, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{4} - T^{2} + 1)^{4} \)
$3$
\( T^{16} - 12 T^{14} + 12 T^{13} + 23 T^{12} + \cdots + 81 \)
$5$
\( T^{16} \)
$7$
\( T^{16} - 4 T^{15} + 44 T^{14} + \cdots + 279841 \)
$11$
\( T^{16} + 4 T^{15} + 20 T^{14} + 244 T^{13} + \cdots + 81 \)
$13$
\( T^{16} + 12 T^{15} + \cdots + 815730721 \)
$17$
\( T^{16} + 8 T^{15} + 8 T^{14} + \cdots + 9199089 \)
$19$
\( T^{16} + 16 T^{15} + 158 T^{14} + \cdots + 81 \)
$23$
\( T^{16} + 4 T^{15} - 40 T^{14} + \cdots + 11881809 \)
$29$
\( T^{16} + 36 T^{15} + \cdots + 7001170929 \)
$31$
\( T^{16} + 8 T^{15} + 32 T^{14} + \cdots + 20736 \)
$37$
\( T^{16} + 20 T^{15} + 380 T^{14} + \cdots + 187489 \)
$41$
\( T^{16} - 32 T^{15} + \cdots + 5749430625 \)
$43$
\( T^{16} + 126 T^{14} + \cdots + 45270647361 \)
$47$
\( (T^{8} - 16 T^{7} - 96 T^{6} + \cdots - 2124351)^{2} \)
$53$
\( T^{16} + 44 T^{15} + \cdots + 178024081041 \)
$59$
\( T^{16} - 24 T^{15} + 198 T^{14} + \cdots + 3470769 \)
$61$
\( T^{16} + 20 T^{15} + 382 T^{14} + \cdots + 42849 \)
$67$
\( T^{16} - 178 T^{14} + \cdots + 36051756129 \)
$71$
\( T^{16} - 16 T^{15} + \cdots + 10\!\cdots\!64 \)
$73$
\( T^{16} + 608 T^{14} + \cdots + 8100000000 \)
$79$
\( T^{16} + 340 T^{14} + \cdots + 211086032481 \)
$83$
\( (T^{8} + 40 T^{7} + 522 T^{6} + \cdots - 419175)^{2} \)
$89$
\( T^{16} + \cdots + 709755295760289 \)
$97$
\( T^{16} - 96 T^{15} + \cdots + 45\!\cdots\!89 \)
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