Properties

Label 252.2.b
Level 252
Weight 2
Character orbit b
Rep. character \(\chi_{252}(55,\cdot)\)
Character field \(\Q\)
Dimension 18
Newform subspaces 5
Sturm bound 96
Trace bound 7

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Defining parameters

Level: \( N \) = \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 252.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 28 \)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(96\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(5\), \(11\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(252, [\chi])\).

Total New Old
Modular forms 56 22 34
Cusp forms 40 18 22
Eisenstein series 16 4 12

Trace form

\( 18q + 3q^{2} - 3q^{4} + 9q^{8} + O(q^{10}) \) \( 18q + 3q^{2} - 3q^{4} + 9q^{8} - 5q^{14} + 5q^{16} + 14q^{22} - 22q^{25} - 3q^{28} + 20q^{29} - 7q^{32} - 4q^{37} - 42q^{44} - 22q^{46} + 2q^{49} - 33q^{50} - 12q^{53} + 29q^{56} - 50q^{58} - 39q^{64} - 16q^{65} + 48q^{70} + 34q^{74} - 12q^{77} - 16q^{85} + 74q^{86} + 34q^{88} + 42q^{92} - 37q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(252, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
252.2.b.a \(2\) \(2.012\) \(\Q(\sqrt{-7}) \) \(\Q(\sqrt{-7}) \) \(1\) \(0\) \(0\) \(0\) \(q+\beta q^{2}+(-2+\beta )q^{4}+(-1+2\beta )q^{7}+\cdots\)
252.2.b.b \(4\) \(2.012\) \(\Q(\sqrt{-2}, \sqrt{7})\) \(\Q(\sqrt{-21}) \) \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{2}-2q^{4}-\beta _{2}q^{5}+\beta _{3}q^{7}-2\beta _{1}q^{8}+\cdots\)
252.2.b.c \(4\) \(2.012\) \(\Q(i, \sqrt{7})\) \(\Q(\sqrt{-7}) \) \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-1+2\beta _{2}+\cdots)q^{7}+\cdots\)
252.2.b.d \(4\) \(2.012\) 4.0.2312.1 None \(1\) \(0\) \(0\) \(-2\) \(q+\beta _{1}q^{2}+\beta _{2}q^{4}+(-\beta _{1}-\beta _{3})q^{5}+\cdots\)
252.2.b.e \(4\) \(2.012\) 4.0.2312.1 None \(1\) \(0\) \(0\) \(2\) \(q+\beta _{1}q^{2}+\beta _{2}q^{4}+(\beta _{1}+\beta _{3})q^{5}+(1+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(252, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(252, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 - T + 2 T^{2} \))(\( ( 1 + 2 T^{2} )^{2} \))(\( 1 - 3 T^{2} + 4 T^{4} \))(\( 1 - T - 2 T^{3} + 4 T^{4} \))(\( 1 - T - 2 T^{3} + 4 T^{4} \))
$3$ 1
$5$ (\( ( 1 - 5 T^{2} )^{2} \))(\( ( 1 + 4 T^{2} + 25 T^{4} )^{2} \))(\( ( 1 - 5 T^{2} )^{4} \))(\( 1 - 6 T^{2} + 42 T^{4} - 150 T^{6} + 625 T^{8} \))(\( 1 - 6 T^{2} + 42 T^{4} - 150 T^{6} + 625 T^{8} \))
$7$ (\( 1 + 7 T^{2} \))(\( ( 1 - 7 T^{2} )^{2} \))(\( ( 1 + 7 T^{2} )^{2} \))(\( 1 + 2 T - 2 T^{2} + 14 T^{3} + 49 T^{4} \))(\( 1 - 2 T - 2 T^{2} - 14 T^{3} + 49 T^{4} \))
$11$ (\( ( 1 - 4 T + 11 T^{2} )( 1 + 4 T + 11 T^{2} ) \))(\( ( 1 - 20 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 - 6 T^{2} + 121 T^{4} )^{2} \))(\( 1 - 34 T^{2} + 514 T^{4} - 4114 T^{6} + 14641 T^{8} \))(\( 1 - 34 T^{2} + 514 T^{4} - 4114 T^{6} + 14641 T^{8} \))
$13$ (\( ( 1 - 13 T^{2} )^{2} \))(\( ( 1 - 13 T^{2} )^{4} \))(\( ( 1 - 13 T^{2} )^{4} \))(\( 1 - 12 T^{2} + 102 T^{4} - 2028 T^{6} + 28561 T^{8} \))(\( 1 - 12 T^{2} + 102 T^{4} - 2028 T^{6} + 28561 T^{8} \))
$17$ (\( ( 1 - 17 T^{2} )^{2} \))(\( ( 1 - 20 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 - 17 T^{2} )^{4} \))(\( 1 - 22 T^{2} + 682 T^{4} - 6358 T^{6} + 83521 T^{8} \))(\( 1 - 22 T^{2} + 682 T^{4} - 6358 T^{6} + 83521 T^{8} \))
$19$ (\( ( 1 + 19 T^{2} )^{2} \))(\( ( 1 + 10 T^{2} + 361 T^{4} )^{2} \))(\( ( 1 + 19 T^{2} )^{4} \))(\( ( 1 + 6 T + 30 T^{2} + 114 T^{3} + 361 T^{4} )^{2} \))(\( ( 1 - 6 T + 30 T^{2} - 114 T^{3} + 361 T^{4} )^{2} \))
$23$ (\( ( 1 - 8 T + 23 T^{2} )( 1 + 8 T + 23 T^{2} ) \))(\( ( 1 + 4 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 + 18 T^{2} + 529 T^{4} )^{2} \))(\( 1 - 82 T^{2} + 2722 T^{4} - 43378 T^{6} + 279841 T^{8} \))(\( 1 - 82 T^{2} + 2722 T^{4} - 43378 T^{6} + 279841 T^{8} \))
$29$ (\( ( 1 - 2 T + 29 T^{2} )^{2} \))(\( ( 1 + 29 T^{2} )^{4} \))(\( ( 1 - 54 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 2 T + 29 T^{2} )^{4} \))(\( ( 1 - 2 T + 29 T^{2} )^{4} \))
$31$ (\( ( 1 + 31 T^{2} )^{2} \))(\( ( 1 - 50 T^{2} + 961 T^{4} )^{2} \))(\( ( 1 + 31 T^{2} )^{4} \))(\( ( 1 + 31 T^{2} )^{4} \))(\( ( 1 + 31 T^{2} )^{4} \))
$37$ (\( ( 1 - 6 T + 37 T^{2} )^{2} \))(\( ( 1 - 8 T + 37 T^{2} )^{4} \))(\( ( 1 + 6 T + 37 T^{2} )^{4} \))(\( ( 1 + 6 T + 66 T^{2} + 222 T^{3} + 1369 T^{4} )^{2} \))(\( ( 1 + 6 T + 66 T^{2} + 222 T^{3} + 1369 T^{4} )^{2} \))
$41$ (\( ( 1 - 41 T^{2} )^{2} \))(\( ( 1 - 68 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 - 41 T^{2} )^{4} \))(\( 1 - 102 T^{2} + 5130 T^{4} - 171462 T^{6} + 2825761 T^{8} \))(\( 1 - 102 T^{2} + 5130 T^{4} - 171462 T^{6} + 2825761 T^{8} \))
$43$ (\( ( 1 - 12 T + 43 T^{2} )( 1 + 12 T + 43 T^{2} ) \))(\( ( 1 - 43 T^{2} )^{4} \))(\( ( 1 - 12 T + 43 T^{2} )^{2}( 1 + 12 T + 43 T^{2} )^{2} \))(\( 1 - 24 T^{2} + 3774 T^{4} - 44376 T^{6} + 3418801 T^{8} \))(\( 1 - 24 T^{2} + 3774 T^{4} - 44376 T^{6} + 3418801 T^{8} \))
$47$ (\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 + 47 T^{2} )^{4} \))(\( ( 1 + 47 T^{2} )^{4} \))(\( ( 1 + 4 T + 30 T^{2} + 188 T^{3} + 2209 T^{4} )^{2} \))(\( ( 1 - 4 T + 30 T^{2} - 188 T^{3} + 2209 T^{4} )^{2} \))
$53$ (\( ( 1 - 10 T + 53 T^{2} )^{2} \))(\( ( 1 + 53 T^{2} )^{4} \))(\( ( 1 - 6 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 + 8 T + 54 T^{2} + 424 T^{3} + 2809 T^{4} )^{2} \))(\( ( 1 + 8 T + 54 T^{2} + 424 T^{3} + 2809 T^{4} )^{2} \))
$59$ (\( ( 1 + 59 T^{2} )^{2} \))(\( ( 1 + 59 T^{2} )^{4} \))(\( ( 1 + 59 T^{2} )^{4} \))(\( ( 1 - 4 T + 59 T^{2} )^{4} \))(\( ( 1 + 4 T + 59 T^{2} )^{4} \))
$61$ (\( ( 1 - 61 T^{2} )^{2} \))(\( ( 1 - 61 T^{2} )^{4} \))(\( ( 1 - 61 T^{2} )^{4} \))(\( 1 - 132 T^{2} + 10710 T^{4} - 491172 T^{6} + 13845841 T^{8} \))(\( 1 - 132 T^{2} + 10710 T^{4} - 491172 T^{6} + 13845841 T^{8} \))
$67$ (\( ( 1 - 4 T + 67 T^{2} )( 1 + 4 T + 67 T^{2} ) \))(\( ( 1 - 67 T^{2} )^{4} \))(\( ( 1 - 4 T + 67 T^{2} )^{2}( 1 + 4 T + 67 T^{2} )^{2} \))(\( 1 - 144 T^{2} + 10830 T^{4} - 646416 T^{6} + 20151121 T^{8} \))(\( 1 - 144 T^{2} + 10830 T^{4} - 646416 T^{6} + 20151121 T^{8} \))
$71$ (\( ( 1 - 16 T + 71 T^{2} )( 1 + 16 T + 71 T^{2} ) \))(\( ( 1 + 100 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 + 114 T^{2} + 5041 T^{4} )^{2} \))(\( 1 - 114 T^{2} + 8418 T^{4} - 574674 T^{6} + 25411681 T^{8} \))(\( 1 - 114 T^{2} + 8418 T^{4} - 574674 T^{6} + 25411681 T^{8} \))
$73$ (\( ( 1 - 73 T^{2} )^{2} \))(\( ( 1 - 73 T^{2} )^{4} \))(\( ( 1 - 73 T^{2} )^{4} \))(\( 1 - 236 T^{2} + 24310 T^{4} - 1257644 T^{6} + 28398241 T^{8} \))(\( 1 - 236 T^{2} + 24310 T^{4} - 1257644 T^{6} + 28398241 T^{8} \))
$79$ (\( ( 1 - 8 T + 79 T^{2} )( 1 + 8 T + 79 T^{2} ) \))(\( ( 1 - 79 T^{2} )^{4} \))(\( ( 1 - 8 T + 79 T^{2} )^{2}( 1 + 8 T + 79 T^{2} )^{2} \))(\( 1 - 288 T^{2} + 33150 T^{4} - 1797408 T^{6} + 38950081 T^{8} \))(\( 1 - 288 T^{2} + 33150 T^{4} - 1797408 T^{6} + 38950081 T^{8} \))
$83$ (\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{4} \))(\( ( 1 + 83 T^{2} )^{4} \))(\( ( 1 + 4 T + 102 T^{2} + 332 T^{3} + 6889 T^{4} )^{2} \))(\( ( 1 - 4 T + 102 T^{2} - 332 T^{3} + 6889 T^{4} )^{2} \))
$89$ (\( ( 1 - 89 T^{2} )^{2} \))(\( ( 1 + 172 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 - 89 T^{2} )^{4} \))(\( 1 - 294 T^{2} + 36618 T^{4} - 2328774 T^{6} + 62742241 T^{8} \))(\( 1 - 294 T^{2} + 36618 T^{4} - 2328774 T^{6} + 62742241 T^{8} \))
$97$ (\( ( 1 - 97 T^{2} )^{2} \))(\( ( 1 - 97 T^{2} )^{4} \))(\( ( 1 - 97 T^{2} )^{4} \))(\( 1 - 204 T^{2} + 28950 T^{4} - 1919436 T^{6} + 88529281 T^{8} \))(\( 1 - 204 T^{2} + 28950 T^{4} - 1919436 T^{6} + 88529281 T^{8} \))
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