Properties

Label 112.2.p
Level 112
Weight 2
Character orbit p
Rep. character \(\chi_{112}(31,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 8
Newform subspaces 3
Sturm bound 32
Trace bound 3

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Defining parameters

Level: \( N \) = \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 112.p (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 28 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 3 \)
Sturm bound: \(32\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

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

Total New Old
Modular forms 44 8 36
Cusp forms 20 8 12
Eisenstein series 24 0 24

Trace form

\( 8q - 4q^{9} + O(q^{10}) \) \( 8q - 4q^{9} + 4q^{21} - 8q^{25} - 24q^{29} - 36q^{33} - 4q^{37} + 36q^{45} + 32q^{49} + 12q^{53} + 56q^{57} + 36q^{61} - 12q^{65} - 12q^{73} - 48q^{77} + 8q^{81} - 72q^{85} - 36q^{89} + 4q^{93} + O(q^{100}) \)

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.p.a \(2\) \(0.894\) \(\Q(\sqrt{-3}) \) None \(0\) \(-1\) \(3\) \(4\) \(q+(-1+\zeta_{6})q^{3}+(2-\zeta_{6})q^{5}+(1+2\zeta_{6})q^{7}+\cdots\)
112.2.p.b \(2\) \(0.894\) \(\Q(\sqrt{-3}) \) None \(0\) \(1\) \(3\) \(-4\) \(q+(1-\zeta_{6})q^{3}+(2-\zeta_{6})q^{5}+(-1-2\zeta_{6})q^{7}+\cdots\)
112.2.p.c \(4\) \(0.894\) \(\Q(\sqrt{-3}, \sqrt{7})\) None \(0\) \(0\) \(-6\) \(0\) \(q+\beta _{1}q^{3}+(-2-\beta _{2})q^{5}-\beta _{3}q^{7}+4\beta _{2}q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(112, [\chi])\) into lower level spaces

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( \))(\( \))(\( \))
$3$ (\( 1 + T - 2 T^{2} + 3 T^{3} + 9 T^{4} \))(\( 1 - T - 2 T^{2} - 3 T^{3} + 9 T^{4} \))(\( 1 + T^{2} - 8 T^{4} + 9 T^{6} + 81 T^{8} \))
$5$ (\( 1 - 3 T + 8 T^{2} - 15 T^{3} + 25 T^{4} \))(\( 1 - 3 T + 8 T^{2} - 15 T^{3} + 25 T^{4} \))(\( ( 1 + 3 T + 8 T^{2} + 15 T^{3} + 25 T^{4} )^{2} \))
$7$ (\( 1 - 4 T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( ( 1 - 7 T^{2} )^{2} \))
$11$ (\( 1 + 3 T + 14 T^{2} + 33 T^{3} + 121 T^{4} \))(\( 1 - 3 T + 14 T^{2} - 33 T^{3} + 121 T^{4} \))(\( 1 + T^{2} - 120 T^{4} + 121 T^{6} + 14641 T^{8} \))
$13$ (\( ( 1 - 13 T^{2} )^{2} \))(\( ( 1 - 13 T^{2} )^{2} \))(\( ( 1 - 14 T^{2} + 169 T^{4} )^{2} \))
$17$ (\( 1 + 9 T + 44 T^{2} + 153 T^{3} + 289 T^{4} \))(\( 1 + 9 T + 44 T^{2} + 153 T^{3} + 289 T^{4} \))(\( ( 1 - 9 T + 44 T^{2} - 153 T^{3} + 289 T^{4} )^{2} \))
$19$ (\( ( 1 - T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))(\( ( 1 - 8 T + 19 T^{2} )( 1 + T + 19 T^{2} ) \))(\( 1 - 31 T^{2} + 600 T^{4} - 11191 T^{6} + 130321 T^{8} \))
$23$ (\( 1 - 15 T + 98 T^{2} - 345 T^{3} + 529 T^{4} \))(\( 1 + 15 T + 98 T^{2} + 345 T^{3} + 529 T^{4} \))(\( 1 + 25 T^{2} + 96 T^{4} + 13225 T^{6} + 279841 T^{8} \))
$29$ (\( ( 1 + 6 T + 29 T^{2} )^{2} \))(\( ( 1 + 6 T + 29 T^{2} )^{2} \))(\( ( 1 + 29 T^{2} )^{4} \))
$31$ (\( 1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4} \))(\( 1 - 5 T - 6 T^{2} - 155 T^{3} + 961 T^{4} \))(\( 1 - 55 T^{2} + 2064 T^{4} - 52855 T^{6} + 923521 T^{8} \))
$37$ (\( 1 - 5 T - 12 T^{2} - 185 T^{3} + 1369 T^{4} \))(\( 1 - 5 T - 12 T^{2} - 185 T^{3} + 1369 T^{4} \))(\( ( 1 + 7 T + 12 T^{2} + 259 T^{3} + 1369 T^{4} )^{2} \))
$41$ (\( 1 - 34 T^{2} + 1681 T^{4} \))(\( 1 - 34 T^{2} + 1681 T^{4} \))(\( ( 1 - 70 T^{2} + 1681 T^{4} )^{2} \))
$43$ (\( 1 - 74 T^{2} + 1849 T^{4} \))(\( 1 - 74 T^{2} + 1849 T^{4} \))(\( ( 1 - 2 T^{2} + 1849 T^{4} )^{2} \))
$47$ (\( 1 + 3 T - 38 T^{2} + 141 T^{3} + 2209 T^{4} \))(\( 1 - 3 T - 38 T^{2} - 141 T^{3} + 2209 T^{4} \))(\( 1 - 31 T^{2} - 1248 T^{4} - 68479 T^{6} + 4879681 T^{8} \))
$53$ (\( 1 - 9 T + 28 T^{2} - 477 T^{3} + 2809 T^{4} \))(\( 1 - 9 T + 28 T^{2} - 477 T^{3} + 2809 T^{4} \))(\( ( 1 + 3 T - 44 T^{2} + 159 T^{3} + 2809 T^{4} )^{2} \))
$59$ (\( 1 + 9 T + 22 T^{2} + 531 T^{3} + 3481 T^{4} \))(\( 1 - 9 T + 22 T^{2} - 531 T^{3} + 3481 T^{4} \))(\( 1 - 55 T^{2} - 456 T^{4} - 191455 T^{6} + 12117361 T^{8} \))
$61$ (\( ( 1 - 14 T + 61 T^{2} )( 1 - T + 61 T^{2} ) \))(\( ( 1 - 14 T + 61 T^{2} )( 1 - T + 61 T^{2} ) \))(\( ( 1 - 3 T + 64 T^{2} - 183 T^{3} + 3721 T^{4} )^{2} \))
$67$ (\( 1 - 9 T + 94 T^{2} - 603 T^{3} + 4489 T^{4} \))(\( 1 + 9 T + 94 T^{2} + 603 T^{3} + 4489 T^{4} \))(\( 1 + 113 T^{2} + 8280 T^{4} + 507257 T^{6} + 20151121 T^{8} \))
$71$ (\( 1 - 130 T^{2} + 5041 T^{4} \))(\( 1 - 130 T^{2} + 5041 T^{4} \))(\( ( 1 - 58 T^{2} + 5041 T^{4} )^{2} \))
$73$ (\( ( 1 - 10 T + 73 T^{2} )( 1 + 7 T + 73 T^{2} ) \))(\( ( 1 - 10 T + 73 T^{2} )( 1 + 7 T + 73 T^{2} ) \))(\( ( 1 + 9 T + 100 T^{2} + 657 T^{3} + 5329 T^{4} )^{2} \))
$79$ (\( ( 1 - 4 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} ) \))(\( ( 1 - 13 T + 79 T^{2} )( 1 + 4 T + 79 T^{2} ) \))(\( 1 + 137 T^{2} + 12528 T^{4} + 855017 T^{6} + 38950081 T^{8} \))
$83$ (\( ( 1 - 12 T + 83 T^{2} )^{2} \))(\( ( 1 + 12 T + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{4} \))
$89$ (\( 1 + 21 T + 236 T^{2} + 1869 T^{3} + 7921 T^{4} \))(\( 1 + 21 T + 236 T^{2} + 1869 T^{3} + 7921 T^{4} \))(\( ( 1 - 3 T + 92 T^{2} - 267 T^{3} + 7921 T^{4} )^{2} \))
$97$ (\( 1 - 146 T^{2} + 9409 T^{4} \))(\( 1 - 146 T^{2} + 9409 T^{4} \))(\( ( 1 - 182 T^{2} + 9409 T^{4} )^{2} \))
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