Properties

Label 2016.2.cq
Level 2016
Weight 2
Character orbit cq
Rep. character \(\chi_{2016}(863,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 64
Newform subspaces 5
Sturm bound 768
Trace bound 11

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Defining parameters

Level: \( N \) = \( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 2016.cq (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 84 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 5 \)
Sturm bound: \(768\)
Trace bound: \(11\)
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 832 64 768
Cusp forms 704 64 640
Eisenstein series 128 0 128

Trace form

\( 64q + O(q^{10}) \) \( 64q - 32q^{13} + 48q^{25} - 16q^{37} + 32q^{49} + 32q^{61} + 48q^{73} - 64q^{85} + 160q^{97} + O(q^{100}) \)

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.cq.a \(8\) \(16.098\) \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(0\) \(q+(3\zeta_{24}-3\zeta_{24}^{3}+\zeta_{24}^{5})q^{5}+(3\zeta_{24}+\cdots)q^{7}+\cdots\)
2016.2.cq.b \(8\) \(16.098\) \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(0\) \(q+(2\zeta_{24}-2\zeta_{24}^{3}-\zeta_{24}^{5})q^{5}+(\zeta_{24}+\cdots)q^{7}+\cdots\)
2016.2.cq.c \(8\) \(16.098\) \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(0\) \(q+(2\zeta_{24}-2\zeta_{24}^{3}-\zeta_{24}^{5})q^{5}+(-\zeta_{24}+\cdots)q^{7}+\cdots\)
2016.2.cq.d \(8\) \(16.098\) \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(0\) \(q+(3\zeta_{24}-3\zeta_{24}^{3}+\zeta_{24}^{5})q^{5}+(-3\zeta_{24}+\cdots)q^{7}+\cdots\)
2016.2.cq.e \(32\) \(16.098\) None \(0\) \(0\) \(0\) \(0\)

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

\( S_{2}^{\mathrm{old}}(2016, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(252, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(336, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(672, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1008, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( \))(\( \))(\( \))(\( \))
$3$ (\( \))(\( \))(\( \))(\( \))
$5$ (\( ( 1 - 2 T^{2} + 25 T^{4} )^{2}( 1 + 2 T^{2} - 21 T^{4} + 50 T^{6} + 625 T^{8} ) \))(\( 1 + 8 T^{2} + 30 T^{4} - 128 T^{6} - 1069 T^{8} - 3200 T^{10} + 18750 T^{12} + 125000 T^{14} + 390625 T^{16} \))(\( 1 + 8 T^{2} + 30 T^{4} - 128 T^{6} - 1069 T^{8} - 3200 T^{10} + 18750 T^{12} + 125000 T^{14} + 390625 T^{16} \))(\( ( 1 - 2 T^{2} + 25 T^{4} )^{2}( 1 + 2 T^{2} - 21 T^{4} + 50 T^{6} + 625 T^{8} ) \))
$7$ (\( ( 1 - 13 T^{2} + 49 T^{4} )^{2} \))(\( 1 + 10 T^{2} + 51 T^{4} + 490 T^{6} + 2401 T^{8} \))(\( 1 + 10 T^{2} + 51 T^{4} + 490 T^{6} + 2401 T^{8} \))(\( ( 1 - 13 T^{2} + 49 T^{4} )^{2} \))
$11$ (\( ( 1 + 6 T + 7 T^{2} + 42 T^{3} + 324 T^{4} + 462 T^{5} + 847 T^{6} + 7986 T^{7} + 14641 T^{8} )^{2} \))(\( ( 1 + 4 T + 8 T^{2} - 56 T^{3} - 233 T^{4} - 616 T^{5} + 968 T^{6} + 5324 T^{7} + 14641 T^{8} )^{2} \))(\( ( 1 - 4 T + 8 T^{2} + 56 T^{3} - 233 T^{4} + 616 T^{5} + 968 T^{6} - 5324 T^{7} + 14641 T^{8} )^{2} \))(\( ( 1 - 6 T + 7 T^{2} - 42 T^{3} + 324 T^{4} - 462 T^{5} + 847 T^{6} - 7986 T^{7} + 14641 T^{8} )^{2} \))
$13$ (\( ( 1 + 8 T^{2} + 169 T^{4} )^{4} \))(\( ( 1 + 2 T + 19 T^{2} + 26 T^{3} + 169 T^{4} )^{4} \))(\( ( 1 + 2 T + 19 T^{2} + 26 T^{3} + 169 T^{4} )^{4} \))(\( ( 1 + 8 T^{2} + 169 T^{4} )^{4} \))
$17$ (\( ( 1 + 32 T^{2} + 735 T^{4} + 9248 T^{6} + 83521 T^{8} )^{2} \))(\( 1 + 44 T^{2} + 1002 T^{4} + 15664 T^{6} + 229331 T^{8} + 4526896 T^{10} + 83688042 T^{12} + 1062053036 T^{14} + 6975757441 T^{16} \))(\( 1 + 44 T^{2} + 1002 T^{4} + 15664 T^{6} + 229331 T^{8} + 4526896 T^{10} + 83688042 T^{12} + 1062053036 T^{14} + 6975757441 T^{16} \))(\( ( 1 + 32 T^{2} + 735 T^{4} + 9248 T^{6} + 83521 T^{8} )^{2} \))
$19$ (\( 1 + 32 T^{2} + 334 T^{4} - 1024 T^{6} - 23885 T^{8} - 369664 T^{10} + 43527214 T^{12} + 1505468192 T^{14} + 16983563041 T^{16} \))(\( 1 + 38 T^{2} + 433 T^{4} + 10982 T^{6} + 370516 T^{8} + 3964502 T^{10} + 56428993 T^{12} + 1787743478 T^{14} + 16983563041 T^{16} \))(\( 1 + 38 T^{2} + 433 T^{4} + 10982 T^{6} + 370516 T^{8} + 3964502 T^{10} + 56428993 T^{12} + 1787743478 T^{14} + 16983563041 T^{16} \))(\( 1 + 32 T^{2} + 334 T^{4} - 1024 T^{6} - 23885 T^{8} - 369664 T^{10} + 43527214 T^{12} + 1505468192 T^{14} + 16983563041 T^{16} \))
$23$ (\( ( 1 - 12 T + 64 T^{2} - 408 T^{3} + 2559 T^{4} - 9384 T^{5} + 33856 T^{6} - 146004 T^{7} + 279841 T^{8} )^{2} \))(\( ( 1 + 2 T - 19 T^{2} + 46 T^{3} + 529 T^{4} )^{4} \))(\( ( 1 - 2 T - 19 T^{2} - 46 T^{3} + 529 T^{4} )^{4} \))(\( ( 1 + 12 T + 64 T^{2} + 408 T^{3} + 2559 T^{4} + 9384 T^{5} + 33856 T^{6} + 146004 T^{7} + 279841 T^{8} )^{2} \))
$29$ (\( ( 1 - 62 T^{2} + 1995 T^{4} - 52142 T^{6} + 707281 T^{8} )^{2} \))(\( ( 1 - 20 T^{2} - 266 T^{4} - 16820 T^{6} + 707281 T^{8} )^{2} \))(\( ( 1 - 20 T^{2} - 266 T^{4} - 16820 T^{6} + 707281 T^{8} )^{2} \))(\( ( 1 - 62 T^{2} + 1995 T^{4} - 52142 T^{6} + 707281 T^{8} )^{2} \))
$31$ (\( ( 1 + 37 T^{2} + 408 T^{4} + 35557 T^{6} + 923521 T^{8} )^{2} \))(\( 1 + 118 T^{2} + 8529 T^{4} + 409814 T^{6} + 14801540 T^{8} + 393831254 T^{10} + 7876710609 T^{12} + 104725434358 T^{14} + 852891037441 T^{16} \))(\( 1 + 118 T^{2} + 8529 T^{4} + 409814 T^{6} + 14801540 T^{8} + 393831254 T^{10} + 7876710609 T^{12} + 104725434358 T^{14} + 852891037441 T^{16} \))(\( ( 1 + 37 T^{2} + 408 T^{4} + 35557 T^{6} + 923521 T^{8} )^{2} \))
$37$ (\( ( 1 + 4 T - 44 T^{2} - 56 T^{3} + 1639 T^{4} - 2072 T^{5} - 60236 T^{6} + 202612 T^{7} + 1874161 T^{8} )^{2} \))(\( ( 1 + 10 T + 33 T^{2} - 70 T^{3} - 580 T^{4} - 2590 T^{5} + 45177 T^{6} + 506530 T^{7} + 1874161 T^{8} )^{2} \))(\( ( 1 + 10 T + 33 T^{2} - 70 T^{3} - 580 T^{4} - 2590 T^{5} + 45177 T^{6} + 506530 T^{7} + 1874161 T^{8} )^{2} \))(\( ( 1 + 4 T - 44 T^{2} - 56 T^{3} + 1639 T^{4} - 2072 T^{5} - 60236 T^{6} + 202612 T^{7} + 1874161 T^{8} )^{2} \))
$41$ (\( ( 1 - 46 T^{2} + 1681 T^{4} )^{4} \))(\( ( 1 - 96 T^{2} + 4514 T^{4} - 161376 T^{6} + 2825761 T^{8} )^{2} \))(\( ( 1 - 96 T^{2} + 4514 T^{4} - 161376 T^{6} + 2825761 T^{8} )^{2} \))(\( ( 1 - 46 T^{2} + 1681 T^{4} )^{4} \))
$43$ (\( ( 1 - 64 T^{2} + 2130 T^{4} - 118336 T^{6} + 3418801 T^{8} )^{2} \))(\( ( 1 - 134 T^{2} + 8115 T^{4} - 247766 T^{6} + 3418801 T^{8} )^{2} \))(\( ( 1 - 134 T^{2} + 8115 T^{4} - 247766 T^{6} + 3418801 T^{8} )^{2} \))(\( ( 1 - 64 T^{2} + 2130 T^{4} - 118336 T^{6} + 3418801 T^{8} )^{2} \))
$47$ (\( ( 1 + 12 T + 16 T^{2} + 408 T^{3} + 7407 T^{4} + 19176 T^{5} + 35344 T^{6} + 1245876 T^{7} + 4879681 T^{8} )^{2} \))(\( ( 1 + 16 T + 100 T^{2} + 992 T^{3} + 10095 T^{4} + 46624 T^{5} + 220900 T^{6} + 1661168 T^{7} + 4879681 T^{8} )^{2} \))(\( ( 1 - 16 T + 100 T^{2} - 992 T^{3} + 10095 T^{4} - 46624 T^{5} + 220900 T^{6} - 1661168 T^{7} + 4879681 T^{8} )^{2} \))(\( ( 1 - 12 T + 16 T^{2} - 408 T^{3} + 7407 T^{4} - 19176 T^{5} + 35344 T^{6} - 1245876 T^{7} + 4879681 T^{8} )^{2} \))
$53$ (\( ( 1 + 94 T^{2} + 2809 T^{4} )^{2}( 1 - 94 T^{2} + 6027 T^{4} - 264046 T^{6} + 7890481 T^{8} ) \))(\( 1 - 4 T^{2} - 2406 T^{4} + 12784 T^{6} - 2069341 T^{8} + 35910256 T^{10} - 18984497286 T^{12} - 88657444516 T^{14} + 62259690411361 T^{16} \))(\( 1 - 4 T^{2} - 2406 T^{4} + 12784 T^{6} - 2069341 T^{8} + 35910256 T^{10} - 18984497286 T^{12} - 88657444516 T^{14} + 62259690411361 T^{16} \))(\( ( 1 + 94 T^{2} + 2809 T^{4} )^{2}( 1 - 94 T^{2} + 6027 T^{4} - 264046 T^{6} + 7890481 T^{8} ) \))
$59$ (\( ( 1 + 6 T - 41 T^{2} - 246 T^{3} + 324 T^{4} - 14514 T^{5} - 142721 T^{6} + 1232274 T^{7} + 12117361 T^{8} )^{2} \))(\( ( 1 - 2 T - 55 T^{2} - 118 T^{3} + 3481 T^{4} )^{4} \))(\( ( 1 + 2 T - 55 T^{2} + 118 T^{3} + 3481 T^{4} )^{4} \))(\( ( 1 - 6 T - 41 T^{2} + 246 T^{3} + 324 T^{4} + 14514 T^{5} - 142721 T^{6} - 1232274 T^{7} + 12117361 T^{8} )^{2} \))
$61$ (\( ( 1 + 16 T + 88 T^{2} + 736 T^{3} + 8887 T^{4} + 44896 T^{5} + 327448 T^{6} + 3631696 T^{7} + 13845841 T^{8} )^{2} \))(\( ( 1 - 8 T + 3 T^{2} - 488 T^{3} + 3721 T^{4} )^{4} \))(\( ( 1 - 8 T + 3 T^{2} - 488 T^{3} + 3721 T^{4} )^{4} \))(\( ( 1 + 16 T + 88 T^{2} + 736 T^{3} + 8887 T^{4} + 44896 T^{5} + 327448 T^{6} + 3631696 T^{7} + 13845841 T^{8} )^{2} \))
$67$ (\( 1 + 92 T^{2} + 1978 T^{4} - 229264 T^{6} - 16077821 T^{8} - 1029166096 T^{10} + 39858917338 T^{12} + 8322171159548 T^{14} + 406067677556641 T^{16} \))(\( 1 + 166 T^{2} + 11889 T^{4} + 1110374 T^{6} + 101604884 T^{8} + 4984468886 T^{10} + 239576677569 T^{12} + 15016091440054 T^{14} + 406067677556641 T^{16} \))(\( 1 + 166 T^{2} + 11889 T^{4} + 1110374 T^{6} + 101604884 T^{8} + 4984468886 T^{10} + 239576677569 T^{12} + 15016091440054 T^{14} + 406067677556641 T^{16} \))(\( 1 + 92 T^{2} + 1978 T^{4} - 229264 T^{6} - 16077821 T^{8} - 1029166096 T^{10} + 39858917338 T^{12} + 8322171159548 T^{14} + 406067677556641 T^{16} \))
$71$ (\( ( 1 + 12 T + 160 T^{2} + 852 T^{3} + 5041 T^{4} )^{4} \))(\( ( 1 + 140 T^{2} + 5041 T^{4} )^{4} \))(\( ( 1 + 140 T^{2} + 5041 T^{4} )^{4} \))(\( ( 1 - 12 T + 160 T^{2} - 852 T^{3} + 5041 T^{4} )^{4} \))
$73$ (\( ( 1 - 8 T - 26 T^{2} + 448 T^{3} - 1901 T^{4} + 32704 T^{5} - 138554 T^{6} - 3112136 T^{7} + 28398241 T^{8} )^{2} \))(\( ( 1 + 18 T + 129 T^{2} + 882 T^{3} + 9044 T^{4} + 64386 T^{5} + 687441 T^{6} + 7002306 T^{7} + 28398241 T^{8} )^{2} \))(\( ( 1 + 18 T + 129 T^{2} + 882 T^{3} + 9044 T^{4} + 64386 T^{5} + 687441 T^{6} + 7002306 T^{7} + 28398241 T^{8} )^{2} \))(\( ( 1 - 8 T - 26 T^{2} + 448 T^{3} - 1901 T^{4} + 32704 T^{5} - 138554 T^{6} - 3112136 T^{7} + 28398241 T^{8} )^{2} \))
$79$ (\( 1 + 170 T^{2} + 9481 T^{4} + 1179290 T^{6} + 157782580 T^{8} + 7359948890 T^{10} + 369285717961 T^{12} + 41324867438570 T^{14} + 1517108809906561 T^{16} \))(\( 1 + 182 T^{2} + 15889 T^{4} + 865046 T^{6} + 51368260 T^{8} + 5398752086 T^{10} + 618877837009 T^{12} + 44241916904822 T^{14} + 1517108809906561 T^{16} \))(\( 1 + 182 T^{2} + 15889 T^{4} + 865046 T^{6} + 51368260 T^{8} + 5398752086 T^{10} + 618877837009 T^{12} + 44241916904822 T^{14} + 1517108809906561 T^{16} \))(\( 1 + 170 T^{2} + 9481 T^{4} + 1179290 T^{6} + 157782580 T^{8} + 7359948890 T^{10} + 369285717961 T^{12} + 41324867438570 T^{14} + 1517108809906561 T^{16} \))
$83$ (\( ( 1 + 18 T + 229 T^{2} + 1494 T^{3} + 6889 T^{4} )^{4} \))(\( ( 1 - 8 T + 84 T^{2} - 664 T^{3} + 6889 T^{4} )^{4} \))(\( ( 1 + 8 T + 84 T^{2} + 664 T^{3} + 6889 T^{4} )^{4} \))(\( ( 1 - 18 T + 229 T^{2} - 1494 T^{3} + 6889 T^{4} )^{4} \))
$89$ (\( 1 + 220 T^{2} + 25066 T^{4} + 1648240 T^{6} + 98356915 T^{8} + 13055709040 T^{10} + 1572697012906 T^{12} + 109335884011420 T^{14} + 3936588805702081 T^{16} \))(\( 1 + 164 T^{2} + 12522 T^{4} - 240752 T^{6} - 69173581 T^{8} - 1906996592 T^{10} + 785658341802 T^{12} + 81504931717604 T^{14} + 3936588805702081 T^{16} \))(\( 1 + 164 T^{2} + 12522 T^{4} - 240752 T^{6} - 69173581 T^{8} - 1906996592 T^{10} + 785658341802 T^{12} + 81504931717604 T^{14} + 3936588805702081 T^{16} \))(\( 1 + 220 T^{2} + 25066 T^{4} + 1648240 T^{6} + 98356915 T^{8} + 13055709040 T^{10} + 1572697012906 T^{12} + 109335884011420 T^{14} + 3936588805702081 T^{16} \))
$97$ (\( ( 1 - 15 T + 97 T^{2} )^{8} \))(\( ( 1 + 186 T^{2} + 9409 T^{4} )^{4} \))(\( ( 1 + 186 T^{2} + 9409 T^{4} )^{4} \))(\( ( 1 - 15 T + 97 T^{2} )^{8} \))
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