Properties

Label 2016.2.p
Level 2016
Weight 2
Character orbit p
Rep. character \(\chi_{2016}(559,\cdot)\)
Character field \(\Q\)
Dimension 38
Newform subspaces 7
Sturm bound 768
Trace bound 43

Related objects

Downloads

Learn more about

Defining parameters

Level: \( N \) = \( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 2016.p (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 56 \)
Character field: \(\Q\)
Newform subspaces: \( 7 \)
Sturm bound: \(768\)
Trace bound: \(43\)
Distinguishing \(T_p\): \(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(2016, [\chi])\).

Total New Old
Modular forms 416 42 374
Cusp forms 352 38 314
Eisenstein series 64 4 60

Trace form

\( 38q + O(q^{10}) \) \( 38q - 8q^{11} + 26q^{25} - 10q^{49} + 24q^{65} - 16q^{67} + 24q^{91} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(2016, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
2016.2.p.a \(2\) \(16.098\) \(\Q(\sqrt{-7}) \) \(\Q(\sqrt{-7}) \) \(0\) \(0\) \(0\) \(0\) \(q-\beta q^{7}-4q^{11}-2\beta q^{23}-5q^{25}+\cdots\)
2016.2.p.b \(4\) \(16.098\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) \(q-2\zeta_{12}q^{5}+(\zeta_{12}-\zeta_{12}^{3})q^{7}-2q^{11}+\cdots\)
2016.2.p.c \(4\) \(16.098\) \(\Q(\sqrt{-2}, \sqrt{7})\) \(\Q(\sqrt{-42}) \) \(0\) \(0\) \(0\) \(0\) \(q+\beta _{3}q^{7}-2\beta _{3}q^{13}+\beta _{2}q^{17}+\beta _{1}q^{23}+\cdots\)
2016.2.p.d \(4\) \(16.098\) \(\Q(i, \sqrt{7})\) \(\Q(\sqrt{-7}) \) \(0\) \(0\) \(0\) \(0\) \(q-\beta _{3}q^{7}-\beta _{2}q^{11}+4\beta _{1}q^{23}-5q^{25}+\cdots\)
2016.2.p.e \(4\) \(16.098\) \(\Q(i, \sqrt{6})\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{3}q^{5}+(\beta _{1}-\beta _{3})q^{7}+2q^{11}+\beta _{3}q^{13}+\cdots\)
2016.2.p.f \(4\) \(16.098\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) \(q+2\zeta_{12}q^{5}+(\zeta_{12}-\zeta_{12}^{3})q^{7}+2q^{11}+\cdots\)
2016.2.p.g \(16\) \(16.098\) \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{3}q^{5}+\beta _{6}q^{7}-\beta _{4}q^{11}+(\beta _{6}-\beta _{8}+\cdots)q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(2016, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(2016, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(168, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(224, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(504, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(672, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ (\( ( 1 + 5 T^{2} )^{2} \))(\( ( 1 - 2 T^{2} + 25 T^{4} )^{2} \))(\( ( 1 + 5 T^{2} )^{4} \))(\( ( 1 + 5 T^{2} )^{4} \))(\( ( 1 + 4 T^{2} + 25 T^{4} )^{2} \))(\( ( 1 - 2 T^{2} + 25 T^{4} )^{2} \))(\( ( 1 + 16 T^{2} + 140 T^{4} + 832 T^{6} + 4326 T^{8} + 20800 T^{10} + 87500 T^{12} + 250000 T^{14} + 390625 T^{16} )^{2} \))
$7$ (\( 1 + 7 T^{2} \))(\( 1 + 2 T^{2} + 49 T^{4} \))(\( ( 1 - 7 T^{2} )^{2} \))(\( ( 1 + 7 T^{2} )^{2} \))(\( 1 - 10 T^{2} + 49 T^{4} \))(\( 1 + 2 T^{2} + 49 T^{4} \))(\( 1 + 4 T^{2} + 68 T^{4} + 572 T^{6} + 3382 T^{8} + 28028 T^{10} + 163268 T^{12} + 470596 T^{14} + 5764801 T^{16} \))
$11$ (\( ( 1 + 4 T + 11 T^{2} )^{2} \))(\( ( 1 + 2 T + 11 T^{2} )^{4} \))(\( ( 1 + 11 T^{2} )^{4} \))(\( ( 1 - 6 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 - 2 T + 11 T^{2} )^{4} \))(\( ( 1 - 2 T + 11 T^{2} )^{4} \))(\( ( 1 + 2 T + 20 T^{2} + 6 T^{3} + 182 T^{4} + 66 T^{5} + 2420 T^{6} + 2662 T^{7} + 14641 T^{8} )^{4} \))
$13$ (\( ( 1 + 13 T^{2} )^{2} \))(\( ( 1 + 14 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 - 2 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 + 13 T^{2} )^{4} \))(\( ( 1 + 20 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 + 14 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 + 36 T^{2} + 692 T^{4} + 12604 T^{6} + 195894 T^{8} + 2130076 T^{10} + 19764212 T^{12} + 173765124 T^{14} + 815730721 T^{16} )^{2} \))
$17$ (\( ( 1 - 17 T^{2} )^{2} \))(\( ( 1 - 22 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 + 22 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 - 17 T^{2} )^{4} \))(\( ( 1 - 10 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 - 22 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 - 72 T^{2} + 2652 T^{4} - 66024 T^{6} + 1253414 T^{8} - 19080936 T^{10} + 221497692 T^{12} - 1737904968 T^{14} + 6975757441 T^{16} )^{2} \))
$19$ (\( ( 1 - 19 T^{2} )^{2} \))(\( ( 1 + 10 T^{2} + 361 T^{4} )^{2} \))(\( ( 1 - 19 T^{2} )^{4} \))(\( ( 1 - 19 T^{2} )^{4} \))(\( ( 1 - 32 T^{2} + 361 T^{4} )^{2} \))(\( ( 1 + 10 T^{2} + 361 T^{4} )^{2} \))(\( ( 1 - 56 T^{2} + 1532 T^{4} - 36808 T^{6} + 804262 T^{8} - 13287688 T^{10} + 199651772 T^{12} - 2634569336 T^{14} + 16983563041 T^{16} )^{2} \))
$23$ (\( ( 1 - 8 T + 23 T^{2} )( 1 + 8 T + 23 T^{2} ) \))(\( ( 1 - 42 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 - 38 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 + 18 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 - 30 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 - 42 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 - 88 T^{2} + 3548 T^{4} - 96952 T^{6} + 2287782 T^{8} - 51287608 T^{10} + 992875868 T^{12} - 13027158232 T^{14} + 78310985281 T^{16} )^{2} \))
$29$ (\( ( 1 - 2 T + 29 T^{2} )( 1 + 2 T + 29 T^{2} ) \))(\( ( 1 + 6 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 26 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 54 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 10 T + 29 T^{2} )^{2}( 1 + 10 T + 29 T^{2} )^{2} \))(\( ( 1 + 6 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 164 T^{2} + 12980 T^{4} - 650812 T^{6} + 22500790 T^{8} - 547332892 T^{10} + 9180507380 T^{12} - 97551024644 T^{14} + 500246412961 T^{16} )^{2} \))
$31$ (\( ( 1 + 31 T^{2} )^{2} \))(\( ( 1 + 50 T^{2} + 961 T^{4} )^{2} \))(\( ( 1 + 34 T^{2} + 961 T^{4} )^{2} \))(\( ( 1 + 31 T^{2} )^{4} \))(\( ( 1 + 38 T^{2} + 961 T^{4} )^{2} \))(\( ( 1 + 50 T^{2} + 961 T^{4} )^{2} \))(\( ( 1 + 68 T^{2} + 5252 T^{4} + 204988 T^{6} + 8264182 T^{8} + 196993468 T^{10} + 4850332292 T^{12} + 60350250308 T^{14} + 852891037441 T^{16} )^{2} \))
$37$ (\( ( 1 - 6 T + 37 T^{2} )( 1 + 6 T + 37 T^{2} ) \))(\( ( 1 - 58 T^{2} + 1369 T^{4} )^{2} \))(\( ( 1 - 37 T^{2} )^{4} \))(\( ( 1 - 6 T + 37 T^{2} )^{2}( 1 + 6 T + 37 T^{2} )^{2} \))(\( ( 1 - 10 T^{2} + 1369 T^{4} )^{2} \))(\( ( 1 - 58 T^{2} + 1369 T^{4} )^{2} \))(\( ( 1 - 232 T^{2} + 25148 T^{4} - 1670040 T^{6} + 74486502 T^{8} - 2286284760 T^{10} + 47131400828 T^{12} - 595248526888 T^{14} + 3512479453921 T^{16} )^{2} \))
$41$ (\( ( 1 - 41 T^{2} )^{2} \))(\( ( 1 + 26 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 - 26 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 - 41 T^{2} )^{4} \))(\( ( 1 - 41 T^{2} )^{4} \))(\( ( 1 + 26 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 - 120 T^{2} + 10204 T^{4} - 627512 T^{6} + 28679654 T^{8} - 1054847672 T^{10} + 28834065244 T^{12} - 570012508920 T^{14} + 7984925229121 T^{16} )^{2} \))
$43$ (\( ( 1 + 12 T + 43 T^{2} )^{2} \))(\( ( 1 + 6 T + 43 T^{2} )^{4} \))(\( ( 1 + 2 T + 43 T^{2} )^{4} \))(\( ( 1 - 12 T + 43 T^{2} )^{4} \))(\( ( 1 - 6 T + 43 T^{2} )^{4} \))(\( ( 1 + 6 T + 43 T^{2} )^{4} \))(\( ( 1 - 2 T + 120 T^{2} - 250 T^{3} + 6782 T^{4} - 10750 T^{5} + 221880 T^{6} - 159014 T^{7} + 3418801 T^{8} )^{4} \))
$47$ (\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 + 46 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 + 47 T^{2} )^{4} \))(\( ( 1 + 47 T^{2} )^{4} \))(\( ( 1 + 70 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 + 46 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 + 120 T^{2} + 10396 T^{4} + 591944 T^{6} + 31424966 T^{8} + 1307604296 T^{10} + 50729163676 T^{12} + 1293505839480 T^{14} + 23811286661761 T^{16} )^{2} \))
$53$ (\( ( 1 - 10 T + 53 T^{2} )( 1 + 10 T + 53 T^{2} ) \))(\( ( 1 - 14 T + 53 T^{2} )^{2}( 1 + 14 T + 53 T^{2} )^{2} \))(\( ( 1 + 22 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 - 6 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 - 14 T + 53 T^{2} )^{2}( 1 + 14 T + 53 T^{2} )^{2} \))(\( ( 1 - 14 T + 53 T^{2} )^{2}( 1 + 14 T + 53 T^{2} )^{2} \))(\( ( 1 - 260 T^{2} + 33908 T^{4} - 2877468 T^{6} + 176791542 T^{8} - 8082807612 T^{10} + 267550429748 T^{12} - 5762733893540 T^{14} + 62259690411361 T^{16} )^{2} \))
$59$ (\( ( 1 - 59 T^{2} )^{2} \))(\( ( 1 - 70 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 + 106 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 - 59 T^{2} )^{4} \))(\( ( 1 - 112 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 - 70 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 - 264 T^{2} + 34076 T^{4} - 3023352 T^{6} + 203740198 T^{8} - 10524288312 T^{10} + 412911193436 T^{12} - 11135660881224 T^{14} + 146830437604321 T^{16} )^{2} \))
$61$ (\( ( 1 + 61 T^{2} )^{2} \))(\( ( 1 + 14 T^{2} + 3721 T^{4} )^{2} \))(\( ( 1 + 94 T^{2} + 3721 T^{4} )^{2} \))(\( ( 1 + 61 T^{2} )^{4} \))(\( ( 1 + 68 T^{2} + 3721 T^{4} )^{2} \))(\( ( 1 + 14 T^{2} + 3721 T^{4} )^{2} \))(\( ( 1 + 324 T^{2} + 46708 T^{4} + 4167580 T^{6} + 280236854 T^{8} + 15507565180 T^{10} + 646711541428 T^{12} + 16692601292964 T^{14} + 191707312997281 T^{16} )^{2} \))
$67$ (\( ( 1 + 4 T + 67 T^{2} )^{2} \))(\( ( 1 + 10 T + 67 T^{2} )^{4} \))(\( ( 1 - 10 T + 67 T^{2} )^{4} \))(\( ( 1 + 4 T + 67 T^{2} )^{4} \))(\( ( 1 - 2 T + 67 T^{2} )^{4} \))(\( ( 1 + 10 T + 67 T^{2} )^{4} \))(\( ( 1 - 10 T + 208 T^{2} - 1122 T^{3} + 16622 T^{4} - 75174 T^{5} + 933712 T^{6} - 3007630 T^{7} + 20151121 T^{8} )^{4} \))
$71$ (\( ( 1 - 16 T + 71 T^{2} )( 1 + 16 T + 71 T^{2} ) \))(\( ( 1 - 138 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 + 58 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 + 114 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 42 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 138 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 456 T^{2} + 94300 T^{4} - 11817416 T^{6} + 1002933542 T^{8} - 59571594056 T^{10} + 2396321518300 T^{12} - 58413729467976 T^{14} + 645753531245761 T^{16} )^{2} \))
$73$ (\( ( 1 - 73 T^{2} )^{2} \))(\( ( 1 - 73 T^{2} )^{4} \))(\( ( 1 - 73 T^{2} )^{4} \))(\( ( 1 - 73 T^{2} )^{4} \))(\( ( 1 + 70 T^{2} + 5329 T^{4} )^{2} \))(\( ( 1 - 73 T^{2} )^{4} \))(\( ( 1 - 328 T^{2} + 54492 T^{4} - 6043256 T^{6} + 500200454 T^{8} - 32204511224 T^{10} + 1547476948572 T^{12} - 49637626222792 T^{14} + 806460091894081 T^{16} )^{2} \))
$79$ (\( ( 1 - 8 T + 79 T^{2} )( 1 + 8 T + 79 T^{2} ) \))(\( ( 1 - 14 T^{2} + 6241 T^{4} )^{2} \))(\( ( 1 - 79 T^{2} )^{4} \))(\( ( 1 - 8 T + 79 T^{2} )^{2}( 1 + 8 T + 79 T^{2} )^{2} \))(\( ( 1 - 122 T^{2} + 6241 T^{4} )^{2} \))(\( ( 1 - 14 T^{2} + 6241 T^{4} )^{2} \))(\( ( 1 - 444 T^{2} + 95492 T^{4} - 12990660 T^{6} + 1221053302 T^{8} - 81074709060 T^{10} + 3719421134852 T^{12} - 107930830251324 T^{14} + 1517108809906561 T^{16} )^{2} \))
$83$ (\( ( 1 - 83 T^{2} )^{2} \))(\( ( 1 - 118 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 + 58 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 - 83 T^{2} )^{4} \))(\( ( 1 - 160 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 - 118 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 - 120 T^{2} + 20284 T^{4} - 1531784 T^{6} + 164981030 T^{8} - 10552459976 T^{10} + 962644583164 T^{12} - 39232844804280 T^{14} + 2252292232139041 T^{16} )^{2} \))
$89$ (\( ( 1 - 89 T^{2} )^{2} \))(\( ( 1 - 70 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 - 122 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 - 89 T^{2} )^{4} \))(\( ( 1 + 38 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 - 70 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 - 312 T^{2} + 51484 T^{4} - 5696504 T^{6} + 530357222 T^{8} - 45122008184 T^{10} + 3230221535644 T^{12} - 155058162779832 T^{14} + 3936588805702081 T^{16} )^{2} \))
$97$ (\( ( 1 - 97 T^{2} )^{2} \))(\( ( 1 - 14 T + 97 T^{2} )^{2}( 1 + 14 T + 97 T^{2} )^{2} \))(\( ( 1 - 97 T^{2} )^{4} \))(\( ( 1 - 97 T^{2} )^{4} \))(\( ( 1 - 170 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 - 14 T + 97 T^{2} )^{2}( 1 + 14 T + 97 T^{2} )^{2} \))(\( ( 1 - 232 T^{2} + 17244 T^{4} + 409768 T^{6} - 157308730 T^{8} + 3855507112 T^{10} + 1526598921564 T^{12} - 193249505143528 T^{14} + 7837433594376961 T^{16} )^{2} \))
show more
show less