Properties

Label 224.2.i
Level 224
Weight 2
Character orbit i
Rep. character \(\chi_{224}(65,\cdot)\)
Character field \(\Q(\zeta_{3})\)
Dimension 16
Newform subspaces 4
Sturm bound 64
Trace bound 5

Related objects

Downloads

Learn more about

Defining parameters

Level: \( N \) = \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 224.i (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 4 \)
Sturm bound: \(64\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(224, [\chi])\).

Total New Old
Modular forms 80 16 64
Cusp forms 48 16 32
Eisenstein series 32 0 32

Trace form

\( 16q - 8q^{9} + O(q^{10}) \) \( 16q - 8q^{9} - 16q^{13} + 24q^{21} - 16q^{25} + 16q^{29} - 8q^{33} + 8q^{37} + 16q^{41} - 8q^{45} - 16q^{49} - 8q^{53} - 48q^{57} - 24q^{61} + 8q^{65} + 24q^{73} - 48q^{77} + 32q^{81} - 80q^{85} + 40q^{89} + 8q^{93} - 16q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(224, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
224.2.i.a \(4\) \(1.789\) \(\Q(\sqrt{2}, \sqrt{-3})\) None \(0\) \(-2\) \(-2\) \(-4\) \(q+(-1+\beta _{1}-\beta _{2})q^{3}+(-2\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots\)
224.2.i.b \(4\) \(1.789\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(-2\) \(0\) \(q+(-\zeta_{12}-\zeta_{12}^{3})q^{3}+(-1+\zeta_{12}^{2}+\cdots)q^{5}+\cdots\)
224.2.i.c \(4\) \(1.789\) \(\Q(\sqrt{-3}, \sqrt{7})\) None \(0\) \(0\) \(6\) \(0\) \(q+\beta _{1}q^{3}-3\beta _{2}q^{5}-\beta _{3}q^{7}+4\beta _{2}q^{9}+\cdots\)
224.2.i.d \(4\) \(1.789\) \(\Q(\sqrt{2}, \sqrt{-3})\) None \(0\) \(2\) \(-2\) \(4\) \(q+(1+\beta _{1}+\beta _{2})q^{3}+(2\beta _{1}+\beta _{2}+2\beta _{3})q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(224, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(224, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(112, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 + 2 T - T^{2} - 2 T^{3} + 4 T^{4} - 6 T^{5} - 9 T^{6} + 54 T^{7} + 81 T^{8} \))(\( ( 1 - 3 T^{2} )^{2}( 1 + 3 T^{2} + 9 T^{4} ) \))(\( 1 + T^{2} - 8 T^{4} + 9 T^{6} + 81 T^{8} \))(\( 1 - 2 T - T^{2} + 2 T^{3} + 4 T^{4} + 6 T^{5} - 9 T^{6} - 54 T^{7} + 81 T^{8} \))
$5$ (\( 1 + 2 T + T^{2} - 14 T^{3} - 36 T^{4} - 70 T^{5} + 25 T^{6} + 250 T^{7} + 625 T^{8} \))(\( ( 1 + T - 4 T^{2} + 5 T^{3} + 25 T^{4} )^{2} \))(\( ( 1 - 3 T + 4 T^{2} - 15 T^{3} + 25 T^{4} )^{2} \))(\( 1 + 2 T + T^{2} - 14 T^{3} - 36 T^{4} - 70 T^{5} + 25 T^{6} + 250 T^{7} + 625 T^{8} \))
$7$ (\( ( 1 + 2 T + 7 T^{2} )^{2} \))(\( 1 + 2 T^{2} + 49 T^{4} \))(\( ( 1 - 7 T^{2} )^{2} \))(\( ( 1 - 2 T + 7 T^{2} )^{2} \))
$11$ (\( 1 - 2 T - 17 T^{2} + 2 T^{3} + 276 T^{4} + 22 T^{5} - 2057 T^{6} - 2662 T^{7} + 14641 T^{8} \))(\( 1 + 5 T^{2} - 96 T^{4} + 605 T^{6} + 14641 T^{8} \))(\( 1 - 15 T^{2} + 104 T^{4} - 1815 T^{6} + 14641 T^{8} \))(\( 1 + 2 T - 17 T^{2} - 2 T^{3} + 276 T^{4} - 22 T^{5} - 2057 T^{6} + 2662 T^{7} + 14641 T^{8} \))
$13$ (\( ( 1 + 18 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 + 13 T^{2} )^{4} \))(\( ( 1 + 4 T + 13 T^{2} )^{4} \))(\( ( 1 + 18 T^{2} + 169 T^{4} )^{2} \))
$17$ (\( 1 - 6 T + T^{2} - 6 T^{3} + 324 T^{4} - 102 T^{5} + 289 T^{6} - 29478 T^{7} + 83521 T^{8} \))(\( ( 1 + 5 T + 8 T^{2} + 85 T^{3} + 289 T^{4} )^{2} \))(\( ( 1 + T - 16 T^{2} + 17 T^{3} + 289 T^{4} )^{2} \))(\( 1 - 6 T + T^{2} - 6 T^{3} + 324 T^{4} - 102 T^{5} + 289 T^{6} - 29478 T^{7} + 83521 T^{8} \))
$19$ (\( 1 + 10 T + 39 T^{2} + 230 T^{3} + 1460 T^{4} + 4370 T^{5} + 14079 T^{6} + 68590 T^{7} + 130321 T^{8} \))(\( 1 - 35 T^{2} + 864 T^{4} - 12635 T^{6} + 130321 T^{8} \))(\( 1 + 25 T^{2} + 264 T^{4} + 9025 T^{6} + 130321 T^{8} \))(\( 1 - 10 T + 39 T^{2} - 230 T^{3} + 1460 T^{4} - 4370 T^{5} + 14079 T^{6} - 68590 T^{7} + 130321 T^{8} \))
$23$ (\( 1 - 2 T - 25 T^{2} + 34 T^{3} + 220 T^{4} + 782 T^{5} - 13225 T^{6} - 24334 T^{7} + 279841 T^{8} \))(\( 1 - 43 T^{2} + 1320 T^{4} - 22747 T^{6} + 279841 T^{8} \))(\( 1 - 39 T^{2} + 992 T^{4} - 20631 T^{6} + 279841 T^{8} \))(\( 1 + 2 T - 25 T^{2} - 34 T^{3} + 220 T^{4} - 782 T^{5} - 13225 T^{6} + 24334 T^{7} + 279841 T^{8} \))
$29$ (\( ( 1 + 50 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 8 T + 29 T^{2} )^{4} \))(\( ( 1 + 4 T + 29 T^{2} )^{4} \))(\( ( 1 + 50 T^{2} + 841 T^{4} )^{2} \))
$31$ (\( 1 + 14 T + 87 T^{2} + 658 T^{3} + 4844 T^{4} + 20398 T^{5} + 83607 T^{6} + 417074 T^{7} + 923521 T^{8} \))(\( ( 1 - 46 T^{2} + 961 T^{4} )( 1 + 59 T^{2} + 961 T^{4} ) \))(\( 1 - 55 T^{2} + 2064 T^{4} - 52855 T^{6} + 923521 T^{8} \))(\( 1 - 14 T + 87 T^{2} - 658 T^{3} + 4844 T^{4} - 20398 T^{5} + 83607 T^{6} - 417074 T^{7} + 923521 T^{8} \))
$37$ (\( 1 + 6 T - 15 T^{2} - 138 T^{3} - 100 T^{4} - 5106 T^{5} - 20535 T^{6} + 303918 T^{7} + 1874161 T^{8} \))(\( ( 1 - 5 T - 12 T^{2} - 185 T^{3} + 1369 T^{4} )^{2} \))(\( ( 1 - 5 T - 12 T^{2} - 185 T^{3} + 1369 T^{4} )^{2} \))(\( 1 + 6 T - 15 T^{2} - 138 T^{3} - 100 T^{4} - 5106 T^{5} - 20535 T^{6} + 303918 T^{7} + 1874161 T^{8} \))
$41$ (\( ( 1 + 8 T + 90 T^{2} + 328 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 - 4 T + 41 T^{2} )^{4} \))(\( ( 1 - 8 T + 41 T^{2} )^{4} \))(\( ( 1 + 8 T + 90 T^{2} + 328 T^{3} + 1681 T^{4} )^{2} \))
$43$ (\( ( 1 - 8 T + 70 T^{2} - 344 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 + 38 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 26 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 + 8 T + 70 T^{2} + 344 T^{3} + 1849 T^{4} )^{2} \))
$47$ (\( 1 + 18 T + 151 T^{2} + 1422 T^{3} + 12492 T^{4} + 66834 T^{5} + 333559 T^{6} + 1868814 T^{7} + 4879681 T^{8} \))(\( 1 - 19 T^{2} - 1848 T^{4} - 41971 T^{6} + 4879681 T^{8} \))(\( 1 - 87 T^{2} + 5360 T^{4} - 192183 T^{6} + 4879681 T^{8} \))(\( 1 - 18 T + 151 T^{2} - 1422 T^{3} + 12492 T^{4} - 66834 T^{5} + 333559 T^{6} - 1868814 T^{7} + 4879681 T^{8} \))
$53$ (\( ( 1 - T - 52 T^{2} - 53 T^{3} + 2809 T^{4} )^{2} \))(\( ( 1 - T - 52 T^{2} - 53 T^{3} + 2809 T^{4} )^{2} \))(\( ( 1 + 7 T - 4 T^{2} + 371 T^{3} + 2809 T^{4} )^{2} \))(\( ( 1 - T - 52 T^{2} - 53 T^{3} + 2809 T^{4} )^{2} \))
$59$ (\( 1 + 2 T - 17 T^{2} - 194 T^{3} - 3276 T^{4} - 11446 T^{5} - 59177 T^{6} + 410758 T^{7} + 12117361 T^{8} \))(\( 1 - 115 T^{2} + 9744 T^{4} - 400315 T^{6} + 12117361 T^{8} \))(\( 1 - 111 T^{2} + 8840 T^{4} - 386391 T^{6} + 12117361 T^{8} \))(\( 1 - 2 T - 17 T^{2} + 194 T^{3} - 3276 T^{4} + 11446 T^{5} - 59177 T^{6} - 410758 T^{7} + 12117361 T^{8} \))
$61$ (\( 1 + 6 T - 63 T^{2} - 138 T^{3} + 3884 T^{4} - 8418 T^{5} - 234423 T^{6} + 1361886 T^{7} + 13845841 T^{8} \))(\( ( 1 + 11 T + 60 T^{2} + 671 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 - 5 T - 36 T^{2} - 305 T^{3} + 3721 T^{4} )^{2} \))(\( 1 + 6 T - 63 T^{2} - 138 T^{3} + 3884 T^{4} - 8418 T^{5} - 234423 T^{6} + 1361886 T^{7} + 13845841 T^{8} \))
$67$ (\( 1 - 14 T + 31 T^{2} - 434 T^{3} + 9604 T^{4} - 29078 T^{5} + 139159 T^{6} - 4210682 T^{7} + 20151121 T^{8} \))(\( ( 1 - 109 T^{2} + 4489 T^{4} )( 1 + 122 T^{2} + 4489 T^{4} ) \))(\( 1 - 127 T^{2} + 11640 T^{4} - 570103 T^{6} + 20151121 T^{8} \))(\( 1 + 14 T + 31 T^{2} + 434 T^{3} + 9604 T^{4} + 29078 T^{5} + 139159 T^{6} + 4210682 T^{7} + 20151121 T^{8} \))
$71$ (\( ( 1 - 16 T + 174 T^{2} - 1136 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 - 50 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 + 71 T^{2} )^{4} \))(\( ( 1 + 16 T + 174 T^{2} + 1136 T^{3} + 5041 T^{4} )^{2} \))
$73$ (\( 1 - 18 T + 129 T^{2} - 882 T^{3} + 9044 T^{4} - 64386 T^{5} + 687441 T^{6} - 7002306 T^{7} + 28398241 T^{8} \))(\( ( 1 + 15 T + 152 T^{2} + 1095 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 - 9 T + 8 T^{2} - 657 T^{3} + 5329 T^{4} )^{2} \))(\( 1 - 18 T + 129 T^{2} - 882 T^{3} + 9044 T^{4} - 64386 T^{5} + 687441 T^{6} - 7002306 T^{7} + 28398241 T^{8} \))
$79$ (\( 1 - 2 T - 105 T^{2} + 98 T^{3} + 5324 T^{4} + 7742 T^{5} - 655305 T^{6} - 986078 T^{7} + 38950081 T^{8} \))(\( 1 - 155 T^{2} + 17784 T^{4} - 967355 T^{6} + 38950081 T^{8} \))(\( 1 - 151 T^{2} + 16560 T^{4} - 942391 T^{6} + 38950081 T^{8} \))(\( 1 + 2 T - 105 T^{2} - 98 T^{3} + 5324 T^{4} - 7742 T^{5} - 655305 T^{6} + 986078 T^{7} + 38950081 T^{8} \))
$83$ (\( ( 1 - 8 T + 54 T^{2} - 664 T^{3} + 6889 T^{4} )^{2} \))(\( ( 1 + 118 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 + 54 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 + 8 T + 54 T^{2} + 664 T^{3} + 6889 T^{4} )^{2} \))
$89$ (\( ( 1 - 9 T - 8 T^{2} - 801 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 + 7 T - 40 T^{2} + 623 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 - 9 T - 8 T^{2} - 801 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 - 9 T - 8 T^{2} - 801 T^{3} + 7921 T^{4} )^{2} \))
$97$ (\( ( 1 + 8 T + 202 T^{2} + 776 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 - 12 T + 97 T^{2} )^{4} \))(\( ( 1 + 8 T + 97 T^{2} )^{4} \))(\( ( 1 + 8 T + 202 T^{2} + 776 T^{3} + 9409 T^{4} )^{2} \))
show more
show less