Properties

Label 300.2.h
Level $300$
Weight $2$
Character orbit 300.h
Rep. character $\chi_{300}(299,\cdot)$
Character field $\Q$
Dimension $32$
Newform subspaces $3$
Sturm bound $120$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 300.h (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 60 \)
Character field: \(\Q\)
Newform subspaces: \( 3 \)
Sturm bound: \(120\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(7\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(300, [\chi])\).

Total New Old
Modular forms 72 40 32
Cusp forms 48 32 16
Eisenstein series 24 8 16

Trace form

\( 32q + 8q^{4} - 6q^{6} + 4q^{9} + O(q^{10}) \) \( 32q + 8q^{4} - 6q^{6} + 4q^{9} - 8q^{16} + 16q^{21} - 22q^{24} - 52q^{34} - 6q^{36} - 16q^{46} + 8q^{49} + 76q^{54} - 56q^{61} + 80q^{64} - 22q^{66} - 36q^{76} - 60q^{81} - 112q^{84} - 48q^{94} + 6q^{96} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(300, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
300.2.h.a \(8\) \(2.396\) 8.0.342102016.5 None \(-2\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{2}-\beta _{4}q^{3}+(-\beta _{1}+\beta _{2}-\beta _{4}+\cdots)q^{4}+\cdots\)
300.2.h.b \(8\) \(2.396\) 8.0.342102016.5 None \(2\) \(0\) \(0\) \(0\) \(q-\beta _{1}q^{2}+\beta _{4}q^{3}+(-\beta _{1}+\beta _{2}-\beta _{4}+\cdots)q^{4}+\cdots\)
300.2.h.c \(16\) \(2.396\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{2}-\beta _{10}q^{3}-\beta _{4}q^{4}-\beta _{12}q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(300, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(300, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 + T + 2 T^{3} + 4 T^{4} )^{2} \))(\( ( 1 - T - 2 T^{3} + 4 T^{4} )^{2} \))(\( ( 1 - T^{2} - 2 T^{4} - 4 T^{6} + 16 T^{8} )^{2} \))
$3$ (\( 1 - 2 T^{2} + 2 T^{4} - 18 T^{6} + 81 T^{8} \))(\( 1 - 2 T^{2} + 2 T^{4} - 18 T^{6} + 81 T^{8} \))(\( ( 1 + T^{2} + 8 T^{4} + 9 T^{6} + 81 T^{8} )^{2} \))
$5$ 1
$7$ (\( ( 1 + 18 T^{2} + 162 T^{4} + 882 T^{6} + 2401 T^{8} )^{2} \))(\( ( 1 + 18 T^{2} + 162 T^{4} + 882 T^{6} + 2401 T^{8} )^{2} \))(\( ( 1 + 9 T^{2} + 108 T^{4} + 441 T^{6} + 2401 T^{8} )^{4} \))
$11$ (\( ( 1 + 24 T^{2} + 318 T^{4} + 2904 T^{6} + 14641 T^{8} )^{2} \))(\( ( 1 + 24 T^{2} + 318 T^{4} + 2904 T^{6} + 14641 T^{8} )^{2} \))(\( ( 1 + 15 T^{2} + 288 T^{4} + 1815 T^{6} + 14641 T^{8} )^{4} \))
$13$ (\( ( 1 - 16 T^{2} + 334 T^{4} - 2704 T^{6} + 28561 T^{8} )^{2} \))(\( ( 1 - 16 T^{2} + 334 T^{4} - 2704 T^{6} + 28561 T^{8} )^{2} \))(\( ( 1 - 31 T^{2} + 568 T^{4} - 5239 T^{6} + 28561 T^{8} )^{4} \))
$17$ (\( ( 1 - 2 T + 17 T^{2} )^{8} \))(\( ( 1 + 2 T + 17 T^{2} )^{8} \))(\( ( 1 + 9 T^{2} - 232 T^{4} + 2601 T^{6} + 83521 T^{8} )^{4} \))
$19$ (\( ( 1 - 56 T^{2} + 1438 T^{4} - 20216 T^{6} + 130321 T^{8} )^{2} \))(\( ( 1 - 56 T^{2} + 1438 T^{4} - 20216 T^{6} + 130321 T^{8} )^{2} \))(\( ( 1 - 50 T^{2} + 1183 T^{4} - 18050 T^{6} + 130321 T^{8} )^{4} \))
$23$ (\( ( 1 - 34 T^{2} + 514 T^{4} - 17986 T^{6} + 279841 T^{8} )^{2} \))(\( ( 1 - 34 T^{2} + 514 T^{4} - 17986 T^{6} + 279841 T^{8} )^{2} \))(\( ( 1 - 64 T^{2} + 1918 T^{4} - 33856 T^{6} + 279841 T^{8} )^{4} \))
$29$ (\( ( 1 - 80 T^{2} + 3214 T^{4} - 67280 T^{6} + 707281 T^{8} )^{2} \))(\( ( 1 - 80 T^{2} + 3214 T^{4} - 67280 T^{6} + 707281 T^{8} )^{2} \))(\( ( 1 - 80 T^{2} + 3118 T^{4} - 67280 T^{6} + 707281 T^{8} )^{4} \))
$31$ (\( ( 1 - 96 T^{2} + 4158 T^{4} - 92256 T^{6} + 923521 T^{8} )^{2} \))(\( ( 1 - 96 T^{2} + 4158 T^{4} - 92256 T^{6} + 923521 T^{8} )^{2} \))(\( ( 1 - 69 T^{2} + 2856 T^{4} - 66309 T^{6} + 923521 T^{8} )^{4} \))
$37$ (\( ( 1 - 112 T^{2} + 5806 T^{4} - 153328 T^{6} + 1874161 T^{8} )^{2} \))(\( ( 1 - 112 T^{2} + 5806 T^{4} - 153328 T^{6} + 1874161 T^{8} )^{2} \))(\( ( 1 - 64 T^{2} + 3598 T^{4} - 87616 T^{6} + 1874161 T^{8} )^{4} \))
$41$ (\( ( 1 - 112 T^{2} + 5886 T^{4} - 188272 T^{6} + 2825761 T^{8} )^{2} \))(\( ( 1 - 112 T^{2} + 5886 T^{4} - 188272 T^{6} + 2825761 T^{8} )^{2} \))(\( ( 1 - 49 T^{2} + 1656 T^{4} - 82369 T^{6} + 2825761 T^{8} )^{4} \))
$43$ (\( ( 1 + 110 T^{2} + 5890 T^{4} + 203390 T^{6} + 3418801 T^{8} )^{2} \))(\( ( 1 + 110 T^{2} + 5890 T^{4} + 203390 T^{6} + 3418801 T^{8} )^{2} \))(\( ( 1 + 101 T^{2} + 5008 T^{4} + 186749 T^{6} + 3418801 T^{8} )^{4} \))
$47$ (\( ( 1 - 178 T^{2} + 12322 T^{4} - 393202 T^{6} + 4879681 T^{8} )^{2} \))(\( ( 1 - 178 T^{2} + 12322 T^{4} - 393202 T^{6} + 4879681 T^{8} )^{2} \))(\( ( 1 - 136 T^{2} + 8878 T^{4} - 300424 T^{6} + 4879681 T^{8} )^{4} \))
$53$ (\( ( 1 - 8 T + 54 T^{2} - 424 T^{3} + 2809 T^{4} )^{4} \))(\( ( 1 + 8 T + 54 T^{2} + 424 T^{3} + 2809 T^{4} )^{4} \))(\( ( 1 + 176 T^{2} + 13198 T^{4} + 494384 T^{6} + 7890481 T^{8} )^{4} \))
$59$ (\( ( 1 - 16 T^{2} + 1518 T^{4} - 55696 T^{6} + 12117361 T^{8} )^{2} \))(\( ( 1 - 16 T^{2} + 1518 T^{4} - 55696 T^{6} + 12117361 T^{8} )^{2} \))(\( ( 1 + 59 T^{2} )^{16} \))
$61$ (\( ( 1 + 2 T + 106 T^{2} + 122 T^{3} + 3721 T^{4} )^{4} \))(\( ( 1 + 2 T + 106 T^{2} + 122 T^{3} + 3721 T^{4} )^{4} \))(\( ( 1 + 5 T + 118 T^{2} + 305 T^{3} + 3721 T^{4} )^{8} \))
$67$ (\( ( 1 + 222 T^{2} + 21282 T^{4} + 996558 T^{6} + 20151121 T^{8} )^{2} \))(\( ( 1 + 222 T^{2} + 21282 T^{4} + 996558 T^{6} + 20151121 T^{8} )^{2} \))(\( ( 1 + 234 T^{2} + 22503 T^{4} + 1050426 T^{6} + 20151121 T^{8} )^{4} \))
$71$ (\( ( 1 + 228 T^{2} + 22806 T^{4} + 1149348 T^{6} + 25411681 T^{8} )^{2} \))(\( ( 1 + 228 T^{2} + 22806 T^{4} + 1149348 T^{6} + 25411681 T^{8} )^{2} \))(\( ( 1 + 9918 T^{4} + 25411681 T^{8} )^{4} \))
$73$ (\( ( 1 - 78 T^{2} + 5329 T^{4} )^{4} \))(\( ( 1 - 78 T^{2} + 5329 T^{4} )^{4} \))(\( ( 1 - 231 T^{2} + 23168 T^{4} - 1230999 T^{6} + 28398241 T^{8} )^{4} \))
$79$ (\( ( 1 - 168 T^{2} + 19470 T^{4} - 1048488 T^{6} + 38950081 T^{8} )^{2} \))(\( ( 1 - 168 T^{2} + 19470 T^{4} - 1048488 T^{6} + 38950081 T^{8} )^{2} \))(\( ( 1 - 240 T^{2} + 26718 T^{4} - 1497840 T^{6} + 38950081 T^{8} )^{4} \))
$83$ (\( ( 1 - 82 T^{2} + 4834 T^{4} - 564898 T^{6} + 47458321 T^{8} )^{2} \))(\( ( 1 - 82 T^{2} + 4834 T^{4} - 564898 T^{6} + 47458321 T^{8} )^{2} \))(\( ( 1 - 319 T^{2} + 39208 T^{4} - 2197591 T^{6} + 47458321 T^{8} )^{4} \))
$89$ (\( ( 1 - 212 T^{2} + 25990 T^{4} - 1679252 T^{6} + 62742241 T^{8} )^{2} \))(\( ( 1 - 212 T^{2} + 25990 T^{4} - 1679252 T^{6} + 62742241 T^{8} )^{2} \))(\( ( 1 - 305 T^{2} + 39088 T^{4} - 2415905 T^{6} + 62742241 T^{8} )^{4} \))
$97$ (\( ( 1 - 158 T^{2} + 9409 T^{4} )^{4} \))(\( ( 1 - 158 T^{2} + 9409 T^{4} )^{4} \))(\( ( 1 - 199 T^{2} + 27888 T^{4} - 1872391 T^{6} + 88529281 T^{8} )^{4} \))
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