Properties

Label 225.2.f
Level 225
Weight 2
Character orbit f
Rep. character \(\chi_{225}(107,\cdot)\)
Character field \(\Q(\zeta_{4})\)
Dimension 12
Newform subspaces 2
Sturm bound 60
Trace bound 1

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Defining parameters

Level: \( N \) = \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 225.f (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 15 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 2 \)
Sturm bound: \(60\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(225, [\chi])\).

Total New Old
Modular forms 84 12 72
Cusp forms 36 12 24
Eisenstein series 48 0 48

Trace form

\( 12q + 8q^{7} + O(q^{10}) \) \( 12q + 8q^{7} - 4q^{13} - 28q^{16} - 8q^{22} - 8q^{28} - 8q^{31} - 4q^{37} + 32q^{43} + 32q^{46} - 4q^{52} - 12q^{58} - 24q^{61} - 16q^{67} - 4q^{73} + 160q^{76} + 4q^{82} + 24q^{88} - 88q^{91} + 44q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(225, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
225.2.f.a \(4\) \(1.797\) \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(8\) \(q+\zeta_{8}q^{2}-\zeta_{8}^{2}q^{4}+(2-2\zeta_{8}^{2})q^{7}+\cdots\)
225.2.f.b \(8\) \(1.797\) \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{24}^{6}q^{2}-4\zeta_{24}^{2}q^{4}+\zeta_{24}q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(225, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(225, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T^{4} + 16 T^{8} \))(\( ( 1 - 2 T + 2 T^{2} - 4 T^{3} + 4 T^{4} )^{2}( 1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4} )^{2} \))
$3$ 1
$5$ 1
$7$ (\( ( 1 - 4 T + 8 T^{2} - 28 T^{3} + 49 T^{4} )^{2} \))(\( ( 1 + 23 T^{4} + 2401 T^{8} )^{2} \))
$11$ (\( ( 1 - 6 T + 11 T^{2} )^{2}( 1 + 6 T + 11 T^{2} )^{2} \))(\( ( 1 - 4 T^{2} + 121 T^{4} )^{4} \))
$13$ (\( ( 1 - 4 T + 13 T^{2} )^{2}( 1 + 6 T + 13 T^{2} )^{2} \))(\( ( 1 - 337 T^{4} + 28561 T^{8} )^{2} \))
$17$ (\( 1 - 254 T^{4} + 83521 T^{8} \))(\( ( 1 + 206 T^{4} + 83521 T^{8} )^{2} \))
$19$ (\( ( 1 - 19 T^{2} )^{4} \))(\( ( 1 - 13 T^{2} + 361 T^{4} )^{4} \))
$23$ (\( 1 - 158 T^{4} + 279841 T^{8} \))(\( ( 1 + 542 T^{4} + 279841 T^{8} )^{2} \))
$29$ (\( ( 1 + 40 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 + 40 T^{2} + 841 T^{4} )^{4} \))
$31$ (\( ( 1 + 4 T + 31 T^{2} )^{4} \))(\( ( 1 - T + 31 T^{2} )^{8} \))
$37$ (\( ( 1 + 2 T + 2 T^{2} + 74 T^{3} + 1369 T^{4} )^{2} \))(\( ( 1 + 1106 T^{4} + 1874161 T^{8} )^{2} \))
$41$ (\( ( 1 - 80 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 - 10 T^{2} + 1681 T^{4} )^{4} \))
$43$ (\( ( 1 - 16 T + 128 T^{2} - 688 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 + 3191 T^{4} + 3418801 T^{8} )^{2} \))
$47$ (\( 1 - 3518 T^{4} + 4879681 T^{8} \))(\( ( 1 - 2818 T^{4} + 4879681 T^{8} )^{2} \))
$53$ (\( ( 1 - 56 T^{2} + 2809 T^{4} )( 1 + 56 T^{2} + 2809 T^{4} ) \))(\( ( 1 - 5518 T^{4} + 7890481 T^{8} )^{2} \))
$59$ (\( ( 1 + 46 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 - 44 T^{2} + 3481 T^{4} )^{4} \))
$61$ (\( ( 1 - 8 T + 61 T^{2} )^{4} \))(\( ( 1 + 7 T + 61 T^{2} )^{8} \))
$67$ (\( ( 1 + 8 T + 32 T^{2} + 536 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 + 2471 T^{4} + 20151121 T^{8} )^{2} \))
$71$ (\( ( 1 - 110 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 70 T^{2} + 5041 T^{4} )^{4} \))
$73$ (\( ( 1 + 2 T + 2 T^{2} + 146 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 + 7298 T^{4} + 28398241 T^{8} )^{2} \))
$79$ (\( ( 1 - 14 T^{2} + 6241 T^{4} )^{2} \))(\( ( 1 - 154 T^{2} + 6241 T^{4} )^{4} \))
$83$ (\( 1 + 8722 T^{4} + 47458321 T^{8} \))(\( ( 1 + 11822 T^{4} + 47458321 T^{8} )^{2} \))
$89$ (\( ( 1 + 16 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 + 106 T^{2} + 7921 T^{4} )^{4} \))
$97$ (\( ( 1 - 22 T + 242 T^{2} - 2134 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 - 16609 T^{4} + 88529281 T^{8} )^{2} \))
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