Properties

Label 112.2.w
Level 112
Weight 2
Character orbit w
Rep. character \(\chi_{112}(37,\cdot)\)
Character field \(\Q(\zeta_{12})\)
Dimension 56
Newform subspaces 3
Sturm bound 32
Trace bound 2

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Defining parameters

Level: \( N \) = \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 112.w (of order \(12\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 112 \)
Character field: \(\Q(\zeta_{12})\)
Newform subspaces: \( 3 \)
Sturm bound: \(32\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(112, [\chi])\).

Total New Old
Modular forms 72 72 0
Cusp forms 56 56 0
Eisenstein series 16 16 0

Trace form

\( 56q - 2q^{2} - 2q^{3} - 4q^{4} - 2q^{5} - 8q^{6} - 20q^{8} + O(q^{10}) \) \( 56q - 2q^{2} - 2q^{3} - 4q^{4} - 2q^{5} - 8q^{6} - 20q^{8} + 4q^{10} - 6q^{11} - 2q^{12} - 8q^{13} - 24q^{14} - 16q^{15} + 8q^{16} - 4q^{17} + 10q^{18} - 2q^{19} + 8q^{20} - 10q^{21} - 36q^{22} + 18q^{24} - 2q^{26} - 20q^{27} - 14q^{28} + 8q^{29} + 14q^{30} + 20q^{31} + 8q^{32} - 4q^{33} - 32q^{34} - 2q^{35} - 64q^{36} - 10q^{37} + 18q^{38} + 50q^{40} + 30q^{42} - 16q^{43} - 14q^{44} - 28q^{45} - 44q^{47} + 40q^{48} - 8q^{49} + 24q^{50} + 30q^{51} + 4q^{52} + 6q^{53} + 82q^{54} + 72q^{56} - 14q^{58} - 18q^{59} + 22q^{60} - 2q^{61} + 28q^{62} - 36q^{63} + 80q^{64} - 4q^{65} + 2q^{66} + 30q^{67} + 24q^{68} - 20q^{69} + 76q^{70} - 24q^{72} - 38q^{74} - 24q^{75} - 68q^{76} - 18q^{77} + 132q^{78} - 4q^{79} - 52q^{80} - 8q^{81} - 18q^{82} + 32q^{83} + 68q^{84} + 12q^{85} - 44q^{86} - 34q^{88} - 20q^{90} + 32q^{91} + 68q^{92} + 10q^{93} - 54q^{94} + 20q^{95} - 128q^{96} - 16q^{97} - 28q^{98} + 8q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(112, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
112.2.w.a \(4\) \(0.894\) \(\Q(\zeta_{12})\) None \(-2\) \(-4\) \(-6\) \(0\) \(q+(\zeta_{12}-\zeta_{12}^{2}-\zeta_{12}^{3})q^{2}+(-1-\zeta_{12}+\cdots)q^{3}+\cdots\)
112.2.w.b \(4\) \(0.894\) \(\Q(\zeta_{12})\) None \(4\) \(2\) \(0\) \(0\) \(q+(1+\zeta_{12}^{3})q^{2}+(1-\zeta_{12}^{2}+\zeta_{12}^{3})q^{3}+\cdots\)
112.2.w.c \(48\) \(0.894\) None \(-4\) \(0\) \(4\) \(0\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4} \))(\( ( 1 - 2 T + 2 T^{2} )^{2} \))
$3$ (\( 1 + 4 T + 5 T^{2} - 4 T^{3} - 20 T^{4} - 12 T^{5} + 45 T^{6} + 108 T^{7} + 81 T^{8} \))(\( 1 - 2 T + 5 T^{2} - 10 T^{3} + 16 T^{4} - 30 T^{5} + 45 T^{6} - 54 T^{7} + 81 T^{8} \))
$5$ (\( ( 1 + 4 T + 5 T^{2} )^{2}( 1 - 2 T - T^{2} - 10 T^{3} + 25 T^{4} ) \))(\( ( 1 + 2 T + 5 T^{2} )^{2}( 1 - 4 T + 11 T^{2} - 20 T^{3} + 25 T^{4} ) \))
$7$ (\( 1 + 2 T^{2} + 49 T^{4} \))(\( 1 + 2 T^{2} + 49 T^{4} \))
$11$ (\( 1 - 8 T + 41 T^{2} - 152 T^{3} + 532 T^{4} - 1672 T^{5} + 4961 T^{6} - 10648 T^{7} + 14641 T^{8} \))(\( 1 + 10 T + 41 T^{2} + 82 T^{3} + 136 T^{4} + 902 T^{5} + 4961 T^{6} + 13310 T^{7} + 14641 T^{8} \))
$13$ (\( 1 - 8 T + 32 T^{2} - 120 T^{3} + 446 T^{4} - 1560 T^{5} + 5408 T^{6} - 17576 T^{7} + 28561 T^{8} \))(\( 1 - 8 T + 32 T^{2} - 120 T^{3} + 446 T^{4} - 1560 T^{5} + 5408 T^{6} - 17576 T^{7} + 28561 T^{8} \))
$17$ (\( 1 + 6 T + 5 T^{2} - 18 T^{3} + 60 T^{4} - 306 T^{5} + 1445 T^{6} + 29478 T^{7} + 83521 T^{8} \))(\( 1 + 6 T + 5 T^{2} - 18 T^{3} + 60 T^{4} - 306 T^{5} + 1445 T^{6} + 29478 T^{7} + 83521 T^{8} \))
$19$ (\( 1 + 8 T + 41 T^{2} + 168 T^{3} + 644 T^{4} + 3192 T^{5} + 14801 T^{6} + 54872 T^{7} + 130321 T^{8} \))(\( 1 - 10 T + 41 T^{2} - 66 T^{3} - 40 T^{4} - 1254 T^{5} + 14801 T^{6} - 68590 T^{7} + 130321 T^{8} \))
$23$ (\( 1 + 12 T + 105 T^{2} + 684 T^{3} + 3824 T^{4} + 15732 T^{5} + 55545 T^{6} + 146004 T^{7} + 279841 T^{8} \))(\( 1 - 12 T + 105 T^{2} - 684 T^{3} + 3824 T^{4} - 15732 T^{5} + 55545 T^{6} - 146004 T^{7} + 279841 T^{8} \))
$29$ (\( 1 + 8 T + 32 T^{2} + 248 T^{3} + 1918 T^{4} + 7192 T^{5} + 26912 T^{6} + 195112 T^{7} + 707281 T^{8} \))(\( 1 + 8 T + 32 T^{2} + 248 T^{3} + 1918 T^{4} + 7192 T^{5} + 26912 T^{6} + 195112 T^{7} + 707281 T^{8} \))
$31$ (\( 1 + 4 T - 47 T^{2} + 4 T^{3} + 2512 T^{4} + 124 T^{5} - 45167 T^{6} + 119164 T^{7} + 923521 T^{8} \))(\( 1 + 4 T - 47 T^{2} + 4 T^{3} + 2512 T^{4} + 124 T^{5} - 45167 T^{6} + 119164 T^{7} + 923521 T^{8} \))
$37$ (\( ( 1 - 12 T + 37 T^{2} )^{2}( 1 + 2 T - 33 T^{2} + 74 T^{3} + 1369 T^{4} ) \))(\( ( 1 - 2 T + 37 T^{2} )^{2}( 1 + 12 T + 107 T^{2} + 444 T^{3} + 1369 T^{4} ) \))
$41$ (\( 1 - 60 T^{2} + 3494 T^{4} - 100860 T^{6} + 2825761 T^{8} \))(\( 1 - 60 T^{2} + 3494 T^{4} - 100860 T^{6} + 2825761 T^{8} \))
$43$ (\( 1 - 12 T + 72 T^{2} - 444 T^{3} + 2702 T^{4} - 19092 T^{5} + 133128 T^{6} - 954084 T^{7} + 3418801 T^{8} \))(\( 1 - 12 T + 72 T^{2} - 444 T^{3} + 2702 T^{4} - 19092 T^{5} + 133128 T^{6} - 954084 T^{7} + 3418801 T^{8} \))
$47$ (\( 1 + 12 T + 17 T^{2} + 396 T^{3} + 7152 T^{4} + 18612 T^{5} + 37553 T^{6} + 1245876 T^{7} + 4879681 T^{8} \))(\( 1 + 12 T + 17 T^{2} + 396 T^{3} + 7152 T^{4} + 18612 T^{5} + 37553 T^{6} + 1245876 T^{7} + 4879681 T^{8} \))
$53$ (\( 1 - 2 T + 101 T^{2} + 226 T^{3} + 4912 T^{4} + 11978 T^{5} + 283709 T^{6} - 297754 T^{7} + 7890481 T^{8} \))(\( 1 - 20 T + 101 T^{2} + 1072 T^{3} - 16076 T^{4} + 56816 T^{5} + 283709 T^{6} - 2977540 T^{7} + 7890481 T^{8} \))
$59$ (\( 1 - 28 T + 365 T^{2} - 3028 T^{3} + 22252 T^{4} - 178652 T^{5} + 1270565 T^{6} - 5750612 T^{7} + 12117361 T^{8} \))(\( 1 + 26 T + 365 T^{2} + 3506 T^{3} + 28624 T^{4} + 206854 T^{5} + 1270565 T^{6} + 5339854 T^{7} + 12117361 T^{8} \))
$61$ (\( 1 + 14 T + 53 T^{2} - 606 T^{3} - 9088 T^{4} - 36966 T^{5} + 197213 T^{6} + 3177734 T^{7} + 13845841 T^{8} \))(\( 1 - 4 T + 53 T^{2} - 624 T^{3} + 1892 T^{4} - 38064 T^{5} + 197213 T^{6} - 907924 T^{7} + 13845841 T^{8} \))
$67$ (\( 1 + 12 T + 45 T^{2} - 972 T^{3} - 11812 T^{4} - 65124 T^{5} + 202005 T^{6} + 3609156 T^{7} + 20151121 T^{8} \))(\( 1 + 6 T + 45 T^{2} - 738 T^{3} - 4576 T^{4} - 49446 T^{5} + 202005 T^{6} + 1804578 T^{7} + 20151121 T^{8} \))
$71$ (\( 1 - 228 T^{2} + 22310 T^{4} - 1149348 T^{6} + 25411681 T^{8} \))(\( 1 - 228 T^{2} + 22310 T^{4} - 1149348 T^{6} + 25411681 T^{8} \))
$73$ (\( 1 + 18 T + 277 T^{2} + 3042 T^{3} + 31116 T^{4} + 222066 T^{5} + 1476133 T^{6} + 7002306 T^{7} + 28398241 T^{8} \))(\( 1 - 18 T + 277 T^{2} - 3042 T^{3} + 31116 T^{4} - 222066 T^{5} + 1476133 T^{6} - 7002306 T^{7} + 28398241 T^{8} \))
$79$ (\( 1 - 16 T + 109 T^{2} + 176 T^{3} - 4856 T^{4} + 13904 T^{5} + 680269 T^{6} - 7888624 T^{7} + 38950081 T^{8} \))(\( 1 - 16 T + 109 T^{2} + 176 T^{3} - 4856 T^{4} + 13904 T^{5} + 680269 T^{6} - 7888624 T^{7} + 38950081 T^{8} \))
$83$ (\( 1 - 20 T + 200 T^{2} - 2180 T^{3} + 23086 T^{4} - 180940 T^{5} + 1377800 T^{6} - 11435740 T^{7} + 47458321 T^{8} \))(\( 1 - 20 T + 200 T^{2} - 2180 T^{3} + 23086 T^{4} - 180940 T^{5} + 1377800 T^{6} - 11435740 T^{7} + 47458321 T^{8} \))
$89$ (\( ( 1 - 9 T + 116 T^{2} - 801 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 + 9 T + 116 T^{2} + 801 T^{3} + 7921 T^{4} )^{2} \))
$97$ (\( ( 1 - 8 T + 162 T^{2} - 776 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 - 8 T + 162 T^{2} - 776 T^{3} + 9409 T^{4} )^{2} \))
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