Properties

Label 450.2.f
Level 450
Weight 2
Character orbit f
Rep. character \(\chi_{450}(107,\cdot)\)
Character field \(\Q(\zeta_{4})\)
Dimension 12
Newform subspaces 3
Sturm bound 180
Trace bound 7

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Defining parameters

Level: \( N \) = \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 450.f (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 15 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 3 \)
Sturm bound: \(180\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(450, [\chi])\).

Total New Old
Modular forms 228 12 216
Cusp forms 132 12 120
Eisenstein series 96 0 96

Trace form

\( 12q - 8q^{7} + O(q^{10}) \) \( 12q - 8q^{7} + 12q^{13} - 12q^{16} - 8q^{22} - 8q^{28} + 48q^{31} + 12q^{37} - 32q^{46} - 12q^{52} - 4q^{58} - 80q^{61} + 16q^{67} - 4q^{73} - 16q^{76} + 28q^{82} - 8q^{88} + 96q^{91} + 12q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(450, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
450.2.f.a \(4\) \(3.593\) \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(-12\) \(q+\zeta_{8}q^{2}+\zeta_{8}^{2}q^{4}+(-3+3\zeta_{8}^{2})q^{7}+\cdots\)
450.2.f.b \(4\) \(3.593\) \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(-8\) \(q+\zeta_{8}q^{2}+\zeta_{8}^{2}q^{4}+(-2+2\zeta_{8}^{2})q^{7}+\cdots\)
450.2.f.c \(4\) \(3.593\) \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(12\) \(q+\zeta_{8}q^{2}+\zeta_{8}^{2}q^{4}+(3-3\zeta_{8}^{2})q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(450, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(450, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(150, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(225, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T^{4} \))(\( 1 + T^{4} \))(\( 1 + T^{4} \))
$3$ (\( \))(\( \))(\( \))
$5$ (\( \))(\( \))(\( \))
$7$ (\( ( 1 + 6 T + 18 T^{2} + 42 T^{3} + 49 T^{4} )^{2} \))(\( ( 1 + 4 T + 8 T^{2} + 28 T^{3} + 49 T^{4} )^{2} \))(\( ( 1 - 6 T + 18 T^{2} - 42 T^{3} + 49 T^{4} )^{2} \))
$11$ (\( ( 1 - 4 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 - 6 T + 11 T^{2} )^{2}( 1 + 6 T + 11 T^{2} )^{2} \))(\( ( 1 - 4 T^{2} + 121 T^{4} )^{2} \))
$13$ (\( ( 1 + 6 T + 18 T^{2} + 78 T^{3} + 169 T^{4} )^{2} \))(\( ( 1 - 6 T + 18 T^{2} - 78 T^{3} + 169 T^{4} )^{2} \))(\( ( 1 - 6 T + 18 T^{2} - 78 T^{3} + 169 T^{4} )^{2} \))
$17$ (\( ( 1 - 8 T + 32 T^{2} - 136 T^{3} + 289 T^{4} )( 1 + 8 T + 32 T^{2} + 136 T^{3} + 289 T^{4} ) \))(\( 1 - 254 T^{4} + 83521 T^{8} \))(\( ( 1 - 8 T + 32 T^{2} - 136 T^{3} + 289 T^{4} )( 1 + 8 T + 32 T^{2} + 136 T^{3} + 289 T^{4} ) \))
$19$ (\( ( 1 - 34 T^{2} + 361 T^{4} )^{2} \))(\( ( 1 + 26 T^{2} + 361 T^{4} )^{2} \))(\( ( 1 - 34 T^{2} + 361 T^{4} )^{2} \))
$23$ (\( 1 - 958 T^{4} + 279841 T^{8} \))(\( 1 - 158 T^{4} + 279841 T^{8} \))(\( 1 - 958 T^{4} + 279841 T^{8} \))
$29$ (\( ( 1 - 14 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 + 56 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 14 T^{2} + 841 T^{4} )^{2} \))
$31$ (\( ( 1 - 4 T + 31 T^{2} )^{4} \))(\( ( 1 - 4 T + 31 T^{2} )^{4} \))(\( ( 1 - 4 T + 31 T^{2} )^{4} \))
$37$ (\( ( 1 + 6 T + 18 T^{2} + 222 T^{3} + 1369 T^{4} )^{2} \))(\( ( 1 - 6 T + 18 T^{2} - 222 T^{3} + 1369 T^{4} )^{2} \))(\( ( 1 - 6 T + 18 T^{2} - 222 T^{3} + 1369 T^{4} )^{2} \))
$41$ (\( ( 1 - 64 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 + 16 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 - 64 T^{2} + 1681 T^{4} )^{2} \))
$43$ (\( ( 1 + 1849 T^{4} )^{2} \))(\( ( 1 + 1849 T^{4} )^{2} \))(\( ( 1 + 1849 T^{4} )^{2} \))
$47$ (\( ( 1 + 2209 T^{4} )^{2} \))(\( ( 1 + 2209 T^{4} )^{2} \))(\( ( 1 + 2209 T^{4} )^{2} \))
$53$ (\( 1 - 718 T^{4} + 7890481 T^{8} \))(\( ( 1 - 56 T^{2} + 2809 T^{4} )( 1 + 56 T^{2} + 2809 T^{4} ) \))(\( 1 - 718 T^{4} + 7890481 T^{8} \))
$59$ (\( ( 1 + 100 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 + 110 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 + 100 T^{2} + 3481 T^{4} )^{2} \))
$61$ (\( ( 1 + 10 T + 61 T^{2} )^{4} \))(\( ( 1 + 61 T^{2} )^{4} \))(\( ( 1 + 10 T + 61 T^{2} )^{4} \))
$67$ (\( ( 1 - 12 T + 72 T^{2} - 804 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 - 8 T + 32 T^{2} - 536 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 + 12 T + 72 T^{2} + 804 T^{3} + 4489 T^{4} )^{2} \))
$71$ (\( ( 1 - 70 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 110 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 70 T^{2} + 5041 T^{4} )^{2} \))
$73$ (\( ( 1 - 12 T + 72 T^{2} - 876 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 + 2 T + 2 T^{2} + 146 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 + 12 T + 72 T^{2} + 876 T^{3} + 5329 T^{4} )^{2} \))
$79$ (\( ( 1 - 94 T^{2} + 6241 T^{4} )^{2} \))(\( ( 1 - 14 T^{2} + 6241 T^{4} )^{2} \))(\( ( 1 - 94 T^{2} + 6241 T^{4} )^{2} \))
$83$ (\( 1 - 13294 T^{4} + 47458321 T^{8} \))(\( 1 - 13294 T^{4} + 47458321 T^{8} \))(\( 1 - 13294 T^{4} + 47458321 T^{8} \))
$89$ (\( ( 1 + 160 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 + 80 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 + 160 T^{2} + 7921 T^{4} )^{2} \))
$97$ (\( ( 1 - 24 T + 288 T^{2} - 2328 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 - 6 T + 18 T^{2} - 582 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 + 24 T + 288 T^{2} + 2328 T^{3} + 9409 T^{4} )^{2} \))
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